ELECTRIC POWER SYSTEMS Volume I ELECTRIC ...

Electric Power Systems Volume I Electric Networks

Sisteme Electroenergetice Volumul I Reţele Electrice

This book was financially supported by: S.C. ROMELECTRO S.A., Romania Richard Bergner Elektroarmaturen & Co. KG – RIBE, Germany Washington Group International, U.S.A. Power & Lighting Tehnorob S.A., Romania C.N. Transelectrica S.A., Romania S.C. “ELECON PLUS” S.R.L., Romania

Mircea Eremia (Editor) Yong Hua Song • Nikos Hatziargyriou Adrian Buta • Gheorghe Cârţină • Mircea Nemeş Virgil Alexandrescu • Ion Stratan • Bucur Luştrea Hermina Albert • George Florea • Georgel Gheorghiţă Cătălin Dumitriu • Maria Tudose • Constantin Bulac • Sorin Pătrăşcoiu Ion Triştiu • Lucian Toma • Laurenţiu Nicolae

ELECTRIC POWER SYSTEMS Volume I ELECTRIC NETWORKS

EDITURA ACADEMIEI ROMÂNE Bucureşti, 2005

LIST OF CONTRIBUTORS Mircea Eremia University “Politehnica” of Bucharest 313, Spl. Independenţei 060032 Bucharest, Romania

Yong-Hua Song Brunel University Uxbridge, Middlesex, UB8 3PH London, United Kingdom

Nikos Hatziargyriou National Technical University of Athens 9, Heroon Polytechniou 15773 Zografou, Athens, Greece

Adrian Buta University “Politehnica” of Timişoara 2, Vasile Pârvan Blv. 300223 Timişoara, Romania

Gheorghe Cârţină Technical University “Gh. Asachi” of Iaşi 22, Copou Str. 700497 Iaşi, Romania

Mircea Nemeş University “Politehnica” of Timişoara 2, Vasile Pârvan Blv. 300223 Timişoara, Romania

Virgil Alexandrescu Technical University “Gh. Asachi” of Iaşi 22, Copou Str. 700497 Iaşi, Romania

Ion Stratan Technical University of Moldova 168, Ştefan cel Mare Blv. MD2004 Chişinău, Republic of Moldova

Bucur Luştrea University “Politehnica” of Timişoara 2, Vasile Pârvan Blv. 300223 Timişoara, Romania

Hermina Albert Institute for Energy Studies and Design 1-3, Lacul Tei Blv. 020371 Bucharest, Romania

George Florea Power & Lighting Tehnorob S.A. 355-357, Griviţei Av. 010717, Bucharest, Romania

Georgel Gheorghiţă, Laurenţiu Nicolae Fichtner Romelectro Engineering 1-3, Lacul Tei Blv. 020371, Bucharest, Romania

Cătălin Dumitriu University “Politehnica” of Bucharest 313, Spl. Independenţei 060032 Bucharest, Romania

Maria Tudose University “Politehnica” of Bucharest 313, Spl. Independenţei 060032 Bucharest, Romania

Constantin Bulac University “Politehnica” of Bucharest 313, Spl. Independenţei 060032 Bucharest, Romania

Sorin Pătrăşcoiu TRAPEC S.A. 53, Plevnei Av. 010234 Bucharest, Romania

Ion Triştiu University “Politehnica” of Bucharest 313, Spl. Independenţei 060032 Bucharest, Romania

Lucian Toma University “Politehnica” of Bucharest 313, Spl. Independenţei 060032 Bucharest, Romania

CONTENTS Volume I: ELECTRIC NETWORKS Foreword ............................................................................................................................ XV Preface............................................................................................................................. XVII Acknowledgements ............................................................................................................ XX Part one: BASIC COMPUTATION 1.

ELECTRIC POWER SYSTEMS CONFIGURATION AND PARAMETERS (Adrian Buta, Maria Tudose, Lucian Toma)..................................................................3 1.1. Classification and architecture of electric networks ..............................................3 1.1.1. Types of electric networks..........................................................................3 1.1.2. Architecture of electric networks ...............................................................5 1.2. Electric power systems components modelling under steady-state conditions ...23 1.2.1. Loads (consumers) modelling ..................................................................24 1.2.2. Electric lines modelling ...........................................................................28 1.2.3. Transformers modelling ...........................................................................62 1.2.4. Electric generators modelling ..................................................................76 Chapter references .......................................................................................................80

2.

RADIAL AND MESHED NETWORKS (Mircea Eremia, Ion Triştiu) .....................83 2.1. General considerations ........................................................................................83 2.2. Radial and simple meshed electric networks.......................................................85 2.2.1. Current flows and voltage drops calculation under symmetric regime...........................................................................85 2.2.2. Radial electric line with unbalanced loads on phases ..............................91 2.2.3. Simple meshed electric networks.............................................................94 2.2.4. Load flow calculation of radial electric networks ....................................97 2.3. Complex meshed electric networks...................................................................110 2.3.1. Transfiguration methods ........................................................................110 2.3.2. Load flow calculation of meshed networks............................................121 2.4. Reconfiguration of the distribution electric networks .......................................139 2.4.1. Operating issues .....................................................................................139 2.4.2. Mathematical model of the reconfiguration process ..............................141 2.4.3. Reconfiguration heuristic methods ........................................................146 Appendix 2.1. Existence and uniqueness of the forward/backward sweep solution..156 Appendix 2.2. The active power losses variation as a result of a load variation in a radial network ........................................................................................160 Chapter references .....................................................................................................162

3.

AC TRANSMISSION LINES (Mircea Eremia, Ion Stratan, Cătălin Dumitriu) .....165 3.1. Operating equations under steady state .............................................................165 3.2. Propagation of voltage and current waves on a transmission line .....................169 3.2.1. Physical interpretation............................................................................169

VIII 3.2.2. Apparent characteristic power. Natural power (SIL – surge impedance loading) ...........................................................173 3.3. Coefficients of transmission lines equations .....................................................176 3.3.1. Numerical determination of propagation coefficient .............................176 3.3.2. Numerical determination of characteristic impedance ...........................179 3.3.3. Numerical calculation of A, B, C and D coefficients.............................181 3.3.4. Kennelly’s correction coefficients .........................................................182 3.4. Transmitted power on the lossless line..............................................................185 3.5. Transmission lines operating regimes ...............................................................187 3.5.1. Transmission lines equations expressed in per unit ...............................187 3.5.2. Loading only with active power ( pB ≠ 0 , qB = 0 ) ...............................188 3.5.3. Loading with active and reactive power ( pB ≠ 0 , qB ≠ 0 )...................195 3.5.4. Operating regime with equal voltages at both ends ...............................199 3.6. Series and shunt compensation of transmission lines........................................201 3.6.1. Influence of power system lumped reactance ........................................202 3.6.2. Series compensation with capacitors......................................................205 3.6.3. Natural power control by capacitors ......................................................209 3.6.4. Shunt compensation with reactors .........................................................213 3.6.5. Mixed compensation of transmission lines ............................................219 3.7. Transmitted power on the line with losses ........................................................221 3.7.1. Power formulae ......................................................................................221 3.7.2. Performance chart (Circle diagram).......................................................224 3.7.3. Power losses...........................................................................................225 3.8. Application on AC long line ...............................................................................226 Chapter references .....................................................................................................238 4.

HVDC TRANSMISSION (Mircea Eremia, Constantin Bulac) ...............................239 4.1. Introduction .......................................................................................................239 4.2. Structure and configurations .............................................................................242 4.2.1. Structure of HVDC links........................................................................242 4.2.2. HVDC configurations ............................................................................247 4.3. Analysis of the three-phase bridge converter ....................................................256 4.3.1. Rectifier equations .................................................................................256 4.3.2. Inverter equations...................................................................................266 4.4. Control of direct current link ..............................................................................270 4.4.1. Equivalent circuit and control characteristics ........................................270 4.4.2. Control strategies of HVDC systems .....................................................275 4.4.3. Control implementation .........................................................................277 4.5. Reactive power and harmonics..........................................................................280 4.5.1. Reactive power requirements and sources .............................................280 4.5.2. Sources of reactive power ......................................................................283 4.5.3. Harmonics and filters .............................................................................285 4.6. Load flow in mixed AC-DC systems ................................................................293 4.7. Interaction between AC and DC systems ..........................................................297 4.7.1. AC systems stabilization........................................................................297 4.7.2. Influence of AC system short-circuit ratio.............................................299 4.7.3. Effective inertia constant .......................................................................301 4.7.4. Reactive power and the strength of the AC system................................301

IX 4.8. Comparison between DC and AC transmission ................................................302 4.9. Application on HVDC link................................................................................309 Appendix 4.1. HVDC systems in the world .............................................................319 Chapter references .....................................................................................................323 5.

NEUTRAL GROUNDING OF ELECTRIC NETWORKS (Adrian Buta) ..............325 5.1. General considerations ......................................................................................325 5.2. Basic electric phenomena in grounded neutral networks ..................................327 5.2.1. Network neutral potential relative to ground .........................................327 5.2.2. Single-phase-to-ground fault current .....................................................329 5.3. Isolated neutral networks...................................................................................334 5.4. Grounded neutral networks ...............................................................................338 5.4.1. Solidly grounded neutral networks ........................................................338 5.4.2. Resistor grounded neutral networks.......................................................338 5.4.3. Arc-suppression coil grounded networks (resonant grounding).............346 5.5. Neutral point situation in electric networks.......................................................358 5.5.1. Neutral grounding abroad ......................................................................359 5.5.2. Neutral grounding in Romania...............................................................361 Chapter references .....................................................................................................365

6.

ELECTRICAL POWER QUALITY (Adrian Buta, Lucian Toma) ...........................367 6.1. Introduction .......................................................................................................367 6.2. Short-duration voltage variations. Voltage dips and interruptions ....................374 6.2.1. Origins of dips and interruptions............................................................374 6.2.2. Voltage dips characterization and classification ....................................375 6.2.3. Voltage dips calculation.........................................................................379 6.2.4. Mitigation solutions ...............................................................................380 6.3. Transients and overvoltages ..............................................................................392 6.3.1. Sources...................................................................................................383 6.3.2. Mitigation methods ................................................................................385 6.4. Long-duration voltage variations .....................................................................386 6.4.1. Origin and effects...................................................................................386 6.4.2. Voltage level assessment .......................................................................388 6.4.3. Mitigation solutions for the voltage regulation ......................................390 6.5. Harmonics in power systems.............................................................................395 6.5.1. Sources...................................................................................................395 6.5.2. Fundamental concepts............................................................................396 6.5.3. Effects of harmonic distortion................................................................411 6.5.4. Modelling and analysis ..........................................................................419 6.5.5. Mitigation solutions to controlling harmonics .......................................437 6.6. Voltage unbalances ...........................................................................................446 6.6.1. Unbalance indices ..................................................................................447 6.6.2. Origin and effects...................................................................................449 6.6.3. Voltage unbalance and power flow under non-symmetrical conditions 450 6.6.4. Practical definitions of powers in system with non-sinusoidal waveforms and unbalanced loads...........................................................452 6.6.5. Mitigation solutions to the unbalanced operation ..................................456 Chapter references .....................................................................................................467

X 7.

POWER AND ENERGY LOSSES IN ELECTRIC NETWORKS (Hermina Albert) ......................................................................................................471 7.1. Introduction .......................................................................................................471 7.1.1. Background ............................................................................................471 7.1.2. Evolution and structure of the losses in the Romanian electric networks ....................................................................................474 7.1.3. Comparison between losses in the Romanian electric networks and other countries .................................................................................476 7.2. Own technologic power consumption ...............................................................477 7.3. Own electric energy technologic consumption .................................................481 7.3.1. Basic notions and data............................................................................481 7.3.2. Diagram integration method ..................................................................483 7.3.3. Root-mean-square current method .........................................................486 7.3.4. Losses time method................................................................................488 7.3.5. Technologic consumption in transmission installations.........................494 7.4. Economic efficiency of the electric network losses reducing............................496 7.5. Measures to reduce the own technologic consumption and the active energy and power losses................................................................................................502 7.5.1. Measures to cut the technical losses requiring no investments ..............502 7.5.2. Measures to cut the own technologic consumption requiring investments ............................................................................................504 Chapter references .....................................................................................................505

Part two: LOAD FLOW AND POWER SYSTEM SECURITY 8.

PERFORMANCE METHODS FOR POWER FLOW STUDIES (Virgil Alexandrescu, Sorin Pătrăşcoiu) ...................................................................509 8.1. Introduction .......................................................................................................509 8.2. Mathematical models ........................................................................................510 8.2.1. The balance of the nodal currents ..........................................................511 8.2.2. The balance of the nodal powers............................................................511 8.2.3. Power flow per unit computation...........................................................512 8.3. Newton-Raphson (N-R) method .......................................................................514 8.3.1. Theoretical aspects.................................................................................514 8.3.2. Computational algorithm for power flow study by N-R method ...........516 8.4. Decoupled Newton method ...............................................................................531 8.5. Fast decoupled method......................................................................................532 8.6. Direct current (DC) method ..............................................................................541 8.7. Improvements of power flow analysis methods ................................................543 8.8. Static equivalents of the power systems ............................................................545 8.8.1. Introduction............................................................................................545 8.8.2. Ward equivalent .....................................................................................546 8.8.3. REI – Dimo equivalent ..........................................................................548 8.8.4. Equivalent with ideal transformers (EIT)...............................................554 8.8.5. Updating possibilities of the static equivalents ......................................556 Appendix 8.1. Specific aspects of the power flow computation of large electric networks ...........................................................................................................558 Appendix 8.2. Structure and steady state data of the network test.............................563 Chapter references .....................................................................................................564

XI 9.

STATE ESTIMATION OF ELECTRIC POWER SYSTEMS (Mircea Nemeş) ......567 9.1. Some general aspects ........................................................................................567 9.2. Simple application.............................................................................................569 9.3. The estimator.....................................................................................................572 9.4. Two-node system ..............................................................................................574 9.5. Detection and identification of bad data. The procedure of performance index.577 9.6. The procedure of standard deviation multiple “ bˆ ”...........................................580 9.7. The correction of large errors............................................................................580 9.8. The procedure of test identification...................................................................582 9.9. Application with HTI method ...........................................................................585 9.10. Power system observability .............................................................................586 9.10.1. Test of observability P ~ θ..................................................................587 9.10.2. The structure of the gain matrix Q, E ~ U ..........................................590 9.10.3. Test of observability Q, E ~ U ............................................................591 Chapter references .....................................................................................................595

10. STEADY STATE OPTIMIZATION (Gheorghe Cârţină, Yong Hua Song).............597 10.1. Horizon of the power system optimization problems ......................................597 10.1.1. Minimization of the total generation cost (MTGC) ...........................599 10.1.2. Minimization of the active power losses (MAPL) .............................600 10.1.3. Optimization of the voltage-reactive power control (VQ) ................601 10.1.4. Optimal unit commitment (OUC) .....................................................602 10.1.5. Optimization of the strategies in deregulated market (OSDM)..........603 10.2. Optimization techniques in power systems......................................................604 10.2.1. Nonlinear programming (NLP)..........................................................604 10.2.2. Lagrange relaxation techniques (LRT) ..............................................607 10.2.3. Multiobjective optimization techniques .............................................612 10.2.4. Modern optimization techniques in operating planning.....................619 10.3. Optimal power flow (OPF) .............................................................................623 10.3.1. Optimization model............................................................................623 10.3.2. Minimization of the active power losses (MAPL) ............................626 10.3.3. Newton – Lagrange method (NL) .....................................................632 10.3.4. Interior–point methods (IPMs)...........................................................635 10.4. Optimal unit commitment (OUC) ...................................................................643 10.4.1. Introduction........................................................................................643 10.4.2. Lagrangian relaxation – genetic algorithms method (LRGA) ...........644 10.5. Optimal unit commitment in deregulated market ............................................653 10.5.1. Dynamic optimal power flow by interior-point methods ...................653 10.5.2. Power market oriented optimal power flow .......................................658 10.6. Optimization strategies in deregulated market..................................................660 10.6.1. Bidding problem formulation.............................................................660 10.6.2. Ordinal optimization method .............................................................662 10.6.3. Numerical results and discussions......................................................663 Chapter references .....................................................................................................666 11. LOAD FORECAST (Bucur Luştrea) .......................................................................671 11.1. Background......................................................................................................671 11.2. Factors that influence the energy consumption ...............................................672

XII 11.3. Stages of a forecast study ................................................................................673 11.3.1. Initial database selection, correlation and processing ........................673 11.3.2. Mathematical model of the load.........................................................673 11.3.3. Analysis of results and determining the final forecast .......................679 11.4. Error sources and difficulties met at load forecast...........................................680 11.5. Classical methods for load forecast .................................................................681 11.5.1. General aspects ..................................................................................681 11.5.2. Cyclical and seasonal components analysis .......................................682 11.5.3. Trend forecast ....................................................................................685 11.5.4. Load random component analysis......................................................697 11.6. Time series methods for load forecast .............................................................699 11.6.1. General aspects ..................................................................................699 11.6.2. Principles of methodology of the time series modelling ....................700 11.6.3. Time series adopted pattern. Components separation ........................701 11.6.4. Establishing of the time series model using the Box – Jenkins method..................................................................704 11.6.5. Time series model validation .............................................................707 11.6.6. Time series forecast ...........................................................................710 11.7. Short term load forecast using artificial neural networks ................................712 11.7.1. General aspects ..................................................................................712 11.7.2. ANN architecture ...............................................................................714 11.7.3. Case study ..........................................................................................716 Chapter references .....................................................................................................719 Part three: TECHNICAL AND ENVIRONMENTAL COMPUTATION 12. ELECTRIC NETWORKS IMPACT ON THE ENVIRONMENT (George Florea) ........................................................................................................723 12.1. Introduction .....................................................................................................723 12.2. Constructive impact.........................................................................................723 12.2.1. Visual impact ...................................................................................723 12.2.2. Impact on land use ...........................................................................724 12.2.3. Impact during erection and maintenance works...............................725 12.2.4. Direct impact on ecological systems................................................726 12.2.5. Final considerations .........................................................................727 12.3. Electric field impact.........................................................................................727 12.3.1. General considerations.....................................................................727 12.3.2. Induced currents in conductive objects............................................729 12.3.3. Voltages induced in not connected to ground objects......................730 12.3.4. Direct perception in humans ............................................................730 12.3.5. Direct biological effects on humans and animals ............................730 12.3.6. Effects on vegetation .......................................................................732 12.3.7. Audible noise ...................................................................................733 12.3.8. Interference on AM reception..........................................................735 12.3.9. Interference on FM reception...........................................................736 12.3.10. Ions and ozone generating ...............................................................737 12.3.11. Final considerations and recommendations .....................................739 12.3.12. Mitigation techniques ......................................................................742 12.4. Magnetic field impact ......................................................................................742

XIII 12.4.1. 12.4.2.

General considerations.....................................................................742 Induced voltages on long metallic structures parallel to inducting currents ........................................................................745 12.4.3. Direct biological effects on humans and animals ..............................745 12.4.4. Indirect biological effects ..................................................................748 12.4.5. Direct perception on humans .............................................................748 12.4.6. Effects on vegetation .........................................................................748 12.4.7. Final considerations and recommendations.......................................749 12.4.8. Mitigation techniques ........................................................................752 12.5. Conclusions......................................................................................................763 Chapter references .....................................................................................................764 13. OVERHEAD TRANSMISSION LINES TECHNICAL DESIGN (Georgel Gheorghiţă, Laurenţiu Niculae) .................................................................767 13.1. Introduction .....................................................................................................767 13.1.1. Changes of the operational environment ...........................................767 13.1.2. Environmental changes......................................................................768 13.1.3. Changing business environment ........................................................768 13.1.4. New technological possibilities .........................................................769 13.2. Opportunities and threats.................................................................................769 13.3. Objectives and strategy....................................................................................770 13.3.1. Ambitions and objectives ..................................................................770 13.3.2. Strategic technical directions.............................................................770 13.4. The future of overhead transmission lines (OTL)............................................771 13.4.1. Overhead transmission lines today ....................................................771 13.4.2. Overhead transmission lines – medium and long terms forecasting. New overhead transmission lines ......................................................799 Chapter references .....................................................................................................802 14. DISTRIBUTED GENERATION (Nikos Hatziargyriou) .........................................805 14.1. General issues ..................................................................................................805 14.2. Technical issues of the integration of DG in distribution networks ................808 14.2.1. Introduction........................................................................................808 14.2.2. Network voltage changes ...................................................................809 14.2.3. Increase in network fault levels..........................................................812 14.2.4. Effects on power quality ....................................................................813 14.2.5. Protection issues.................................................................................815 14.2.6. Effects on stability .............................................................................816 14.2.7. Effects of DG connection to isolated systems....................................817 14.3. Commercial issues in distribution systems containing DG .............................818 14.3.1. Introduction........................................................................................818 14.3.2. Present network pricing arrangements ...............................................819 14.4. Conclusions .....................................................................................................827 Chapter references .....................................................................................................827

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Volume II: POWER SYSTEMS STABILITY AND ELECTRICITY MARKETS Part one: POWER SYSTEM STABILITY AND CONTROL Part two: ELECTRICITY MARKETS. DEREGULATION Volume III: ADVANCED TECHNIQUES AND TECHNOLOGIES Part one: FACTS TECHNOLOGIES Part two: ARTIFICIAL INTELLIGENCE TECHNIQUES Part three: INFORMATION AND COMPUTING TECHNOLOGIES

FOREWORD Electric power systems are an integral part of the way of life in modern society. The electricity supplied by these systems has proved to be a very convenient, safe, and relatively clean form of energy. It runs our factories, warms and lights our homes, cooks our food and powers our computers. It is indeed one of the important factors contributing to the relatively high standard of living enjoyed by modern society. Electricity is an energy carrier; energy is neither naturally available in the electrical form nor is it consumed directly in that form. The advantage of the electrical form of energy is that it can be transported and controlled with relative ease and high degree of efficiency and reliability. Modern electric power systems are large complex systems with many processes whose operations need to be optimized and with millions of devices requiring harmonious interplay. Efficient and secure operation of such systems presents many challenges in today’s competitive, disaggregated business environment. This is increasingly evident from the many major power grid blackouts experienced in recent years, including the 14 August 2003 blackout of power network in the north-east of the American continent and the 28 September 2003 blackout of the Italian power network. The technical problems that the power engineers have to address today appear to be very complex and demanding for the students of the subject. They will need both the experience of the past generations and a new enlightened approach to the theory and practice of power generation, transmission, distribution and utilization taking into account the techniques that have evolved in other fields. The present book includes a comprehensive account of both theoretical and practical aspects of the performance of the individual elements as well as the integrated power system. The contributing authors are all recognized experts in power system engineering, either working for the electric power industry or for universities in Romania and abroad. Together they have had a total of many decades of experience in the technologies related to electric power systems. Upon invitation from Professor Mircea Eremia, I had the pleasure of visiting the Electrical Power Engineering Department at University “Politehnica” of Bucharest in May 2003. I found there a powerful school of Electric Power Systems from which about 50 students graduate yearly. During my visit, I also had the opportunity to review and discuss the proposal for preparing this book. I am very impressed with the outcome. I am truly honoured to write the foreword of this book, which I believe will be an invaluable source of reference for students of power engineering as well as practicing engineers. Prabha Kundur, Ph.D., FIEEE, FCAE President & CEO, Powertech Labs Inc. Surrey, British Columbia, Canada August 2005

PREFACE Modern power systems are the result of continuous development and improvement which, over the years, have led to highly sophisticated and complex technologies. Their reliable operation is a tribute to the work of dedicated scientists, innovative engineers and experienced business leaders. The relatively fast development of the electrical systems and networks has given rise to ceaseless discussions regarding safe operation and provision of power quality at the customer. Moreover, the energy policy concerning the promotion of renewable energy sources as well as the electricity market creation to stimulate the competition among generation companies have caused new problems in the transmission and distribution networks. It is clear that the initial destination of electrical networks to ensure the unidirectional transmission of power from the power plants towards consumers has changed, since by the installation of dispersed generation sources into the distribution network the power flow became bidirectional, with the possibility of injecting power into the transmission network. In the present work the authors tried to cover in the best way possible the basic knowledge that the experienced engineer as well as the young graduate student in electrical power systems should be able to handle. The work “Electric Networks” is the first volume of the treatise “Electric Power Systems”. It consists of 14 chapters grouped in 3 parts. The first part entitled “Basic Computation” introduces the basic topics related to electrical energy transmission to the reader. In chapter 1, the architecture of electrical networks and the steady state mathematical modelling of the network elements are described in detail. The student should be aware that the network modelling represents the starting point for any application. The electrical networks are designed for transmission, repartition and distribution of electrical energy, so that they present various structures. The transmission and repartition networks operate in complex meshed structure while distribution networks operate in simple meshed but mostly in radial configuration. In chapter 2, issues related to radial networks are presented, such as voltage drop or currents flow calculation. Also this chapter deals with issues related to meshed networks, such as nodal admittance matrix construction and steady state formulation and calculation by using the Seidel-Gauss method as well as the forward/backward sweep adapted for distributed generation. To use the energetic potential of the Earth, especially the might of water, as efficiently as possible, but also because of the continuous increase of the inhabited or industrialized areas, we are forced to transmit the electrical energy for longer and longer distances. However, the alternating current transmission for long distances presents special concerns related to voltage. With the exception of the ideal case when the reactive power consumed in the series inductive component of the lines is compensated by the reactive power generated by the shunt component, formed between the line conductors and earth, while the natural power is transmitted on the line, the voltage can vary in a wide range with respect to the

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nominal value. Chapter 3 presents the theory of alternating current long transmission lines, together with the problems related to their operation and, correspondingly, their solution by means of series or shunt compensation with capacitor banks or reactors. The advanced technology in power electronics proves to be the necessary support for power transmission at long distances, but at direct current. Although the direct current discovered by Edison, which constituted the revolution of the electricity industry, has lost the ”race” to electrify the world over the alternating current discovered by Tesla, however, the direct current transmission and power electronics based devices, respectively, is the only solution to make undersea links or to interconnect two systems operating at different frequencies. Chapter 4 reveals the technical and economical secrets of the direct current transmission. The modern power systems are probably the most complex systems man has ever built. Secure operation of the power systems is a very important issue since unwanted interruptions of power delivery have a large economical impact on customers and utilities. Chapters 5 and 6 deal with issues related to electrical network security and also to the quality of the power supplied to the consumers. The electrical network undergoes permanently disturbances, having different causes and consequences. Chapter 5 presents the efficient measures, which must be taken so that the network could cope with the faults, that is, the neutral grounding of the electric networks. It is disagreeable to find that the intensity of the light provided by the incandescent lamp flickers, or that our refrigerator is out of order due to an overvoltage, or even worse, to get stuck in the elevator due to an interruption in electricity. Chapter 6 deals with problems related to power quality and electromagnetic compatibility issues giving at the same time mitigation solutions. The power systems engineer must be a very good technician but also an economist. The calculation of power losses, presented in Chapter 7, as well as their reducing methodologies, supplements the knowledge of the engineer in designing and optimising of the system operation. To ensure a proper operation of power systems and the continuity in supplying the consumer, the engineer is challenged and at the same time stimulated to develop efficient concepts and technologies. This is the reason why magnificent ideas of some great engineers, as is the case of the famous Paul Dimo’s suggestion to interconnect the power systems at a large scale, where the partners are based on Trust, Solidarity and Common Interest, are nowadays put into practice. But, the power networks have become more meshed, and the specialists’ activities in the analysis of power system operation more complicated. Therefore, there is a need for analytical and simulation tools in the power systems operation and planning. In the second section of the book entitled ”Load flow and power system security”, performance methods for off-line assessment of a power system operating state, by using Newton-Raphson type methods and network static equivalents, are presented – Chapter 8. Due to the fast speed at which the electric phenomena are evolving, the system operator needs powerful tools that can

XIX

respond in real time; state estimation, provided in Chapter 9, is an efficient approach useful in this respect. The electric utility industry is undergoing unprecedented change in its structure worldwide. With the open market environment and competition in the electrical industry, after the restructuring of the system into separate generation, transmission and distribution entities, new issues in power system operation and planning are inevitable. One of the questions the engineer ponders over is: “How to minimize the production and transmission costs to obtain the lowest price possible for the final consumer?”. The answer is tentatively given in Chapter 10 that offers proposals for optimisation of power system operation in the conditions imposed by the competitive environment present in the electricity market. The maintenance of system frequency at a certain value is performed by a permanent balance between generation and consumption. System operator performs the power balancing by appropriate auxiliary services. That is the system operator must know a priori the load consumption from the system so that to appropriately contract with the qualified producers. Distribution companies are faced with the same situation since they function as intermediates between generators and captive consumers. Chapter 11 comes up in help with load forecasting methods. The power increase has led to the expansion of networks, including the high voltage and medium voltage powered transmission and distribution systems. However, for reasons concerning the conservation of the environment, the protection against electromagnetic fields or for aesthetic reasons, the installation of new electrical lines or supplementary capacities in mammoth power plants is, as much as possible, avoided. The third part of the book, entitled “Technical and environmental computation” comes up with ideas of how to solve the possible problems concerning the environment, presented in Chapter 12, by efficient and ecological solutions for overhead lines designing, given in Chapter 13. The negative impact on the environment along with the limited resources of the Earth, such as coal, oil, natural gas and nuclear fuel, have forced the man to think of tireless resources, such as wind and the sunlight. The evolution of technology facilitated the manufacturing of small size high efficiency sources of power, which can be installed into the distribution network, and in the majority of cases are even found at the consumer’s disposal. Chapter 14 presents issues concerning the impact of distributed generation on the electrical network. The work is addressed to the undergraduate and graduate students in the electrical engineering fields but also to specialists from design, research and services companies. Hoping that exploring this book will be an exciting endeavour, the authors apologize to the reader and native English speaker for the language and printing inaccuracies that inevitably exist. Mircea Eremia, Professor University “Politehnica” of Bucharest

ACKNOWLEDGEMENTS The authors wish to take this opportunity to acknowledge all persons that contributed directly or indirectly to carrying out this book, either by technical or editorial support. For some chapters the authors benefited from the kindness of some institutions or companies, which permitted the reprinting or adapting of some figures, equations or text excerpts. In this regard, special thanks are addressed to IEEE, CIGRE – particularly to the General Secretary Jean Kowal, EPRI – particularly to Dr. Aty Edris who facilitated the cooperation with EPRI, as well as to Copper Development Association. Reprinting permission was allowed by Dr. Roger Dugan, from EPRI, also with the kind acceptance from McGraw-Hill Companies, for some excerpts in Chapter 6, and Ing. Daniel Griffel, from EdF, for some excerpts in Chapter 5, to whom the authors address their deepest gratitude. The authors are deeply grateful to Acad. Gleb Drăgan for the constant support and encouragements during writing the book and cooperation with the Publishing House of the Romanian Academy. Special thanks are addressed to some persons who contributed to the content of the final manuscript. The authors wish to express warm thanks to Dr. Mohamed Rashwan (President of TransGrid Solutions Inc.), who provided valuable contributions in Chapter 4, to Prof. Petru Postolache (from University “Politehnica” of Bucharest), for constructive comments in Chapter 1 and Chapter 11, and Dr. Fănică Vatră (from Institute for Energy Studies and Design – ISPE S.A.) for valuable suggestions to Chapter 5. Special thanks are addressed to Prof. Nicolae Golovanov (from University “Politehnica” of Bucharest) who reviewed the Chapter 6 and provided many valuable contributions. The printing of this book was made possible by the financial support of some companies. The authors wish to express their gratitude to Ing. Viorel Gafiţa (Manager of Romelectro S.A.) and Dr. Dan Gheorghiu (General Manager of ISPE S.A.). Special thanks are also addressed to Washington Group International Inc. for the financial support granted for printing the book, and to RIBE Group (Germany) as well as its subsidiary in Romania, which provided valuable technical materials and financial support for printing the book. For contributions concerning the translation into English or the electronic editing of some chapters, the authors wish to extend their gratitude to Dr. Andrei Făgărăşanu, Dr. Monica Făgărăşanu, Dr. Cristina Popescu, Ing. Silviu Vergoti, Ing. Ioan Giosan, Ing. Mircea Bivol and Ing. Laurenţiu Lipan. The authors gratefully acknowledge the good cooperation with the Publishing House of the Romanian Academy, and address many thanks to Dr. Ioan Ganea, Ing. Cristina Chiriac, Mihaela Marian and Monica Stanciu, for their patience and professionalism in carrying out the printed book. The authors

Chapter

1

ELECTRIC POWER SYSTEMS CONFIGURATION AND PARAMETERS 1.1. Classification and architecture of electric networks 1.1.1. Types of electric networks The power flow from power plants to consumption areas and customers is ensured by transmission and distribution networks. The criteria that determine the types of electric networks are: – destination: transmission, interconnection, distribution and utility networks; – nominal voltage *): extra high voltage (400 kV, 750 kV), high voltage (35 to 220 kV), medium voltage (1 to 35 kV) and low voltage (< 1 kV); – covered area: national, regional, urban and rural networks; – configuration: radial, looped and complex looped (meshed) networks; – the situation of the neutral point: networks with earth insulated neutral, networks with solidly earthed neutral and networks with treated neutral; – the presence of the neutral wire: networks with available/not available neutral; – current frequency: alternative current networks and direct current networks. a. The destination criterion networks takes into account the functional role of the electric networks. Thus, transmission networks provide systematic power transfer from the production areas to the consumption areas. The power transferred in time corresponds to the determinist component of the forecasting values. The *)

Several Standards [EN50160, IEEE P1564, HD 637 S1] give definitions of the voltage either as reference quantity or for equipment designing. The term nominal voltage of the system Un, as a whole, is defined as a suitable value of voltage used to designate or identify a system and to which certain operating characteristics are referred. The term rated value represents a quantity value assigned, generally by the manufacturer, for a specified operating condition of a component, device or equipment. Because the nominal voltage is a reference value it is further used in definitions and formulae from theories to identify the voltage level of the network to which equipments or installations are connected.

4

Basic computation

interconnection networks provide the compensation transfer (Fig. 1.1). This transfer is realized in different direction and sense as compared to the systematic transfer and it corresponds to the random component of the forecasting powers, as well as to some fault situations of the generator groups or of the elements of systematic transmission networks. The Repartition networks supply the distribution networks or big customers (which have their own distribution networks). The distribution networks provide power to the loads and their components – power customers. The distribution networks belonging to the customers are called utility networks (industrial, domestic). b. The networks configuration (topology). The radial networks consist of elements (lines, substations, transformer points) beginning in a power injection node and ending in a consumption node. As a result, the loads cannot be supplied on more than a single path (Fig. 1.2). The looped networks are composed from many loops. The consumption nodes of these networks are supplied from two sides. Thus, the network in Figure 1.3 becomes a looped network when the circuit breaker CB is closed and both lines L1 and L2 are supplied. In this case, the supply continuity of the loads is provided even during the disconnection of a source or the failure of some network sections. If the network is supplied by two sources placed at the network’s ends then the network is called supplied from two ends (Fig. 1.4) and it may be regarded as a particular case of the looped network.

A Systematic transport

A, B – PP – IS –

B Systematic transport

Compensation

~

~

~

PP

IS

PP

consumption areas power plants interference station

Systematic transport

HV / MV transformer substation MV / LV transformer substation

Fig. 1.1. Definition of systematic and compensation power transfer.

Fig. 1.2. Radial networks.

The main feature of the complex looped networks is that the loads are supplied from more than two sides (Fig. 1.5), therefore on several paths and from

Electric power systems configuration and parameters

5

several sources. These networks present great supply reliability and economical operation, but they require more equipment and they are more expensive. The topology of a network can be modified by the position of the circuit breakers placed on the lines (Fig. 1.3) or on the substations busbars (Fig. 1.6). Therefore, the opening of the circuit breaker CB from Figure 1.3 transforms the looped network into a radial one; the same situation is also possible for the network from Figure 1.6.

Source 2

Source 1

~

~

CB

Fig. 1.3. Looped network.

Fig. 1.4. Network supplied from two ends.

~ ~ Fig. 1.5. Complex looped network.

~ ~

a. b. Fig. 1.6. Open ring network topology altering in transformer substations: a. without open ring; b. with open ring.

1.1.2. Architecture of electric networks The architecture of electric networks includes the electric networks configuration and structure, the voltage levels, as well as the calculated loads, the specific consumptions and the safety degree, all in tight relation to the functional role of the network elements, the design features, but also with the other elements of the power system. It is useful to emphasize some architecture features of whole power system before presenting the electric networks configuration [1.1], [1.2]. a. Architecture of power system features. The main element that can be taken into consideration at power system configuration analysis is the voltage level. Within the limits of a system, several voltage levels emerge from this point of view (Fig. 1.7), each of these having a well-determined role.

6

Basic computation

Voltage level [kV] 750 400

220

Transformer substation

110

Block transformer

20

~ System ~ power plant

Industrial networks

~

Local power plant

Rural area

Urban area

Urban networks

(6) 10

Transformer points

0.4

Low voltage distribution networks

Fig. 1.7. The architecture of the national power system.


7

The analysis of the planes sequence emphasizes the following features of the power system configuration: – the system elements (generators, transformers, lines, loads etc.) are placed in different parallel planes, according to their nature and functional role; the distance between planes is determined by the difference between the neighbouring voltage levels; – the connection between planes is achieved through the magnetic couplings of the transformers (in the case of different voltage levels, or in the case of not successive planes at the same voltage level) or autotransformers (in the case of successive planes belonging to the same voltage level); – a plane includes the longitudinal elements of the networks; the transversal elements are connected between these planes and the neutral point; – the networks from the upper planes serve for power transmission, while those from the lower planes serve for power distribution; – the generators of the central power plants inject the power in the system through block-transformers and transmission networks at medium voltage (Fig. 1.8); if the generators power is higher, the injection is carried out at higher voltage level; – the lower level buses and networks connected to these buses constitute a load for the higher level networks (excepting the generator buses); – the power consumption takes place at high, medium and low voltage level through the network coupling transformers; – moving to higher levels, the networks cover larger areas and the powers transfer rises, while the density of the networks decreases; – the networks at the lower levels are denser, they transfer less power on shorter distances; – the power system customers are in transversal connection between the buses of the distribution networks (the medium voltages planes) and the plane 0 kV; the power absorbed by the customer and the bus voltage are smaller. 750 kV, 400 kV 220 kV

Generator 110 kV

~

Power transmission Power repartition

20 kV 10 kV

Local consumption

Power distribution

0.4 kV 0 kV

Fig. 1.8. Main power transmission and distribution chart of the Romanian power system.

8

Basic computation

b. Transmission networks structure. The reason why transmission networks are a necessary part of the power system is the big (physical and electrical) distance between consumption centres and power plants. The transmission network is an inherent part of the power system because it insures the significant power flow from power plants to consumption centres and it constitutes the support for the other electric networks. The transmission network’s structure, components and arrangement varies in time, evolving from one development stage of the system to another, influenced mainly by the development of power consumption. Thus, the 110 kV, 220 kV and 400 kV Romanian networks first operating as radial transmission networks, were later (since 1985) transformed into repartition networks (110 kV) and looped networks (220 and 400 kV). The looped network is the optimum solution for transmission networks in countries with well-determined power flow, considering both the consumption and the power sources placement. The loop structure provides several paths of power transfer from one bus to another and it allows better coordination of power plants. This way, the generators operating at any given moment are the economical ones, and further more, during failure of some generators, the required power is still supplied by the big number of operating generators. The design also insures network operation even with a cut-off connection between two buses, because there’s always a backup path of supply for those two buses. c. Distribution networks architecture. The distribution network must ensure the same requirements as any other electric network (reliability, supply continuity, power transfer quality, adaptation capability during operation, possibility of future development, cost-effective operation, minimal impact on the environment), but the neighbouring customers networks raise special problems concerning the supply continuity and the power quality. The electric networks present certain design features depending on the customer or the consumption area properties: the load and population density, the urban or rural area, the network’s impact on the environment, and so on. The main features of a distribution network are the nominal voltage, the transfer capacity and its length. The nominal voltage is adopted according to: the power quantity requested by the loads, the consumers position relative to the existing electric networks, the utility type. The International Electrotechnics Commission recommends the following levels for the public distribution networks: – low voltage: 400 V (230 V); – medium voltage: 10…13 kV; 20…25 kV; 33…35 kV (the voltage of 6 and 10 kV is mainly used to supply big engines and industrial networks); – high voltage: 110 kV. The amount of power transferred through distribution networks depends on the nominal voltage of the network and on the loads it supplies. Thus:


9

– low voltage: under 50 kW, it supplies domestic, residential and tertiary customers; – medium voltage: up to 2000 kW, it supplies customers from tertiary and commercial domains; – high voltage: customers with absorbed powers exceeding 5 MW. As for the length of the distribution networks, it depends on the number and the arrangement of the distribution stations as well as on the number and the location of the transformer points. If the number of transformer substations and points is increased, then the maximum length of the distribution lines is reduced. On the other hand the length of the distribution networks is in tight relation with the voltage level. The solution for energy loss reduction is to adopt shorter lengths for the low and medium voltage networks, bringing the consumer as close as possible to the high voltage level (deep connection), but this solution isn’t always cost-effective. Two important elements considered for the design of distribution networks are the dynamics of the power consumption and the increased concern for the environment. Regarding the consumption dynamics, the network structure must be adaptable, especially for the medium voltage networks, providing extension possibilities through the addition of new lines and the connection of new injection points, while maintaining the initial and unitary character of the network. Usually the solution is not costly, necessary changes are an integrating part of the configuration development. The distribution networks must comply with the environment protection. This raises special problems concerning the aesthetic protection of the landscape and the elimination of accident hazards determined by the electric current influence (presence). The distribution networks architecture can be analyzed from two points of view: the voltage level and the network’s location. The architecture of the distribution networks is presented in which follows considering the voltage levels and emphasizing the features of urban and networks. c1. High voltage distribution networks. They usually include networks with 110 kV voltage, but in some countries higher voltage levels may also appear (for example 225 kV in France). These networks serve for: the supply of some urban and rural areas presenting several customer types, the supply of some concentrated customers with a power demand that requests a 110 kV/MV substation, the power evacuation from local power plants generating medium powers (from 50 up to 200 MW). The transmission networks supply the high voltage networks from the 400 kV/110 kV or 220 kV/110 kV substations called injection substations/ points. The 110 kV distribution system consists of lines and transformer substations of 110 kV/20 kV and 110 kV/10 kV that supply the urban and rural consumption areas. Its features are determined by the customer’s power, the surface of the supplied area, and by the configuration of the adjacent networks (transmission and distribution).

10

Basic computation

The high voltage distribution networks have a basic looped design, but they operate in open (radial) arrangement. The network’s technological consumption determines the separation point, but other restrictions are also considered. The loop is supplied from two different injection points or from the sectionalized busbars of the same injection point. Figures 1.9 to 1.11 illustrate the above-mentioned aspects. Figure 1.9 presents the architecture of a supply network requiring at the same time substations with one transformer unit and substations with two transformer units, connected in derivation to a mains 110 kV line. The network from Figure 1.10 uses 110 kV overhead lines (OEL) to supply rural areas through transformer substations with two transformer units. The A and C substations are source substations and ensure the supply of the medium voltage network. A

110 kV

Fig. 1.9. Architecture of supply networks in areas where the substations are equipped with one or two transformer units supplied from the same 110 kV line.

OEL 110 kV

A

110 kV 20 kV

Fig. 1.10. Network architecture in rural areas supplied from 110 kV lines connected to substations equipped with two transformer units.

C

Figure 1.11 presents the architecture of a repartition network of 110 kV that supplies an urban settlement. Some concentrated loads are supplied by means of deep joint transformer substations and lines. The lines have double circuit structure; each of these circuits is connected to another bus-bar section of the substation (Fig. 1.12).


11

Rural 110 kV/MV 110 kV/MV

400 kV/110 kV substation

110 kV/MV 110 kV/MV

6 kV/110 kV substation

~

~

Fig. 1.11. Architecture of 110 kV repartition networks supplying urban settlements.

110 kV ring Transformer substation Deep joint substation

400 kV/110 kV substation 110 kV System

Fig. 1.12. Architecture of a 110 kV urban network supplying a settlement with more than 150,000 inhabitants.

Regarding the HV/MV substations structure and the installed power, these depend on the destination of that substation namely: public distribution, supply of concentrated customers or both (public distribution and concentrated customers). The public distribution substations supply domestic and residential customers. They have a simple structure, and their dimensions depend on the requested power, the safety level they must provide and the configuration of the 110 kV networks. Big cities are supplied from one or two rings of 110 kV. The structure of the

12

Basic computation

110 kV/MV transformer substations is presented in Figure 1.13. The transformer substation in Figure 1.13,a is equipped with two transformer units of 10…25 MVA rated power, backing up each other. The 110 kV busbars are simple, sectionalized by bus breakers or by isolators only, depending on the network’s operating diagram and the protection system. Figure 1.13,b presents the situation of a repartition network with two 110 kV lines. In this case each busbar section is connected to a different ring. Ring 1, 110 kV Ring 2, 110 kV

110 kV

110 kV / MV transformer substation

a.

b.

Fig. 1.13. Architecture of transformer substations supplying big cities: a. equipped with two transformers; b. supplied from two 110 kV rings.

If the locality has great surface and load density, the supplying of some concentrated customers can be carried out through a deep joint substation (Fig. 1.12). c2. Medium voltage distribution networks. High voltage substations supply these networks in direct or indirect connection. In the first case medium voltage lines connect the transformer points directly to the MV busbars of the supply substations. In the second case, MV lines departing from a HV/MV substation supply the busbars of a MV connection substation, which in turn supplies the MV junctions through other MV lines. The medium voltage levels, varying in the 3 to 60 kV range, are chosen in tight relation to the network’s load density and to the economical and technical criteria. Romania adopted the 20 kV voltage as optimal. There are several configurations of distribution systems in use today, considering the phase number and the situation of the neutral point: – the north-American system (Fig. 1.14,a) employs a distributed and solidly earthed neutral; the main line is three-phase: three phases and neutral; the derivations are single-phase or three-phase according to the transmitted power; the distribution is single-phased, between phase and neutral;


13

– the English system (Fig. 1.14,b), has no distributed neutral, the main line is three-phase; the derivations have three or two phases; – the Australian system (Fig. 1.14,c), is an economical system; the main line has only three phases, and no distributed neutral; the derivations have one, two or three phases; the return path is insured by the ground; – the European system (Fig. 1.14,d) has no distributed neutral; the main line and the derivations are three-phase.

N LV

HV/MV

MV/LV

HV/MV CB

N 3 2 1

CB

N

DS

DS

LV

3 2 1

MV/LV

LV 1P

MV/LV

N 3 2

LV 3P

N

N 3 2

N LV

LV 1P MV/LV

N

1 2 3 N 3 1 2 3 N

a.

b.

Fig. 1.14. Medium voltage distribution types: a. with distributed neutral; b. without distributed neutral, mixed with two or three phases.

14

Basic computation

HV/MV

CB

DS

HV/MV

3 2 1 LV 1P

MV/LV

CB LV 3P N

DS 3 2 1

N 3 1

MV/LV

LV 1P

LV 3P N

1 2 3 N

IT

LV 1P

1 2 3

N

c.

d.

CB − circuit breaker; DS – disconnector switch; N – neutral; IT – insulating transformer Fig. 1.14. Medium voltage distribution types: c. without distributed neutral, mixed with one, two, three phases; d. without distributed neutral, three phase.

The medium voltage networks are composed of MV/0.4 kV lines and transformer points. Urban areas have mainly underground lines, while the suburbs and rural areas have overhead lines. The transformer points in urban environment are encased, while in rural environment they are placed on poles or on the ground. The medium voltage networks with underground lines and direct distribution can have backed-up from two substations (Fig. 1.15) or from the same substation


15

(Fig. 1.16). Under normal operating conditions the networks operate in radial connection. The sectioning point is imposed by the network’s technological consumption or by the network’s automation. The network presents separation capability in the middle or at its ends, according to the requirements. MV

MV

S1

S2

Fig. 1.15. MV direct distribution through cables, with backup from two transformer substations. MV

110 kV / MV substation

Fig. 1.16. MV direct distribution through cables, with backup from the same transformer substation.

Urban areas with big load densities of 5 to 10 MVA/km2 use cable networks in grid type direct distribution arrangement (Fig. 1.17) or in double derivation (Fig. 1.18). For both arrangements the backup supply can be made from the same substation or from different substations. This configuration of electrical drawings can be developed when the load growing on the consumers sides.

16

Basic computation MV

S2 Future 110 kV / MV substation

S1 Existing 110 kV / MV substation

Fig. 1.17. MV direct distribution through grid type cables.

MV Working cable Backup cable

Fig. 1.18. MV direct distribution through underground lines in double derivation.


17

The French MV underground distribution network uses interrupted artery arrangements. A transformer substation cable feeder supplies several MV/0.4kV transformer points and then it connects to another substation, to the same substation or to a backup cable. The interrupted artery is used for the loop, spindle and ear arrangements (Fig. 1,19). The cable from each MV/0.4 kV transformer point is passed through two circuit breakers (CB1 and CB2) in series connection. An artery has all its circuit breakers in normal close position, except the one corresponding to the loop’s normal opening (interruption) bus, thus avoiding parallel operation of two supply sources (transformer substations). In the case of the loops (daisy petals, Fig. 1.19,a), each artery (cable) returns to the same HV/MV source substation, and there aren’t any transversal connections between the loops. In the case of the spindle structure (Fig. 1.19,b), all cables are supplied from the same transformer substation source (S) and their ends converge to a common bus, called reflecting bus (R). A specialized complex device allows the connection of a backup cable that “brings” the source substation to the reflecting node. In the case of the ear structure (Fig. 1.19,c), the source substation supplies one end of the working cable while the other end lies on a backup cable. The network’s using degree is good. The development around the same transformer substation is economic and it’s performed in time. The length of the cables, their number and the number of transformer points can be adopted according to the load’s evolution. Direct distribution arrangements with overhead lines are used in rural areas. They supply transformer points in derivation (Fig. 1.20,a) or radial − tree arrangement (Fig. 1.20,b). In the first case different transformer substations back up the supply and there is the possibility of separation at the middle of network. In the second case there is only one supply source; during network faults the number of customers not supplied is limited with the aid of circuit breakers and isolators mounted along the main line or on some derivations. The North American medium voltage network presents special features. They result from the existence of single-phased MV/LV transformer points and from the presence of the neutral wire and its ground connection. The medium voltage network tree begins with a three-phase structure (3P+N), continues through threephase or double-phase ramifications (antennas), which in turn develop into singlephase lines (1P+N). The MV and LV network, the public lighting and the telephony network use the same poles. The MV/0.4 kV transformer points can be of network type, which supply domestic consumers or the low voltage public network, and of customer type, which supply a single consumer: industrial, commercial, public utility or mixed. Figure 1.21 presents the diagrams of some customers or mixed type transformer points.

18

Basic computation Power supply

TP

Head of petal transformer point

CB2

CB1

Supply feeder

CB F MV/0.4 kV

CB1 , CB2 , CB - circuit breaker F - fuse TP - MV/0.4 kV transformer point

a. Working feeder Working feeder TP S

R S

R

R

R

R

TP

Backup feeder S - power supply (source) R - reflection point TP - transformer points

Backup feeder

b.

c.

Fig. 1.19. MV underground networks in interrupted artery arrangement (EdF).

MV

MV

S1

S2



a. Fig. 1.20. Direct distribution through medium voltage overhead lines: a. with backup from two substation and derivation supply of the transformer points.


19

HV/MV source (circuit breaker bay) Main line Secondary line ACB DS

ACB DS

DS

ACB - Automatic circuit breaker DS - Disconnector switch

b. Fig. 1.20. Direct distribution through medium voltage overhead lines: b. radial-tree diagram. MV

MV MV cable MV cable

1x 400 kVA or 1x 1600 kVA

2 x 400 kVA or 2 x1600 kVA ABT

ABT

0.4 kV

0.4 kV MV

a.

b. ABT – Automatic Bus Transfer

MV cable

0.4 kV

MV cable

ABT

c.

Fig. 1.21. Diagram of customer and mixed type transformer point: a. one transformer; b. two transformers and common low voltage busbars; c. two transformers and sectionalised low voltage busbars.

20

Basic computation

c3. Low voltage distribution networks. The low voltage networks structure is imposed by the load’s density, by the medium voltage network configuration, by the number of MV/LV transformer points and by the consumer’s requirements (allowed outages number and duration). The rural and urban low voltage networks from the suburbs operate in normal conditions in radial arrangement (Fig. 1.22,a) with overhead and underground lines. A more recent solution uses aerial mounted insulated twisted wires. The solution is very economical in rural areas, suburbs or small cities. Central areas of the cities and some significant loads are supplied from looped networks (Fig. 1.22,b). The diagram from Figure 1.22,b1 has the main disadvantage that a line fault causes the outage of the entire LV line. Figure 1.22,b2 avoids this inconvenient with the help of fuse (F). If a fault occurs, the fuse separates the network in two parts, and by melting at the faulted end, insures the fault’s isolation. The diagram in Figure 1.22,b3 insures better safety. The network operates in radial arrangement and if a fault occurs in the transformer point, the corresponding fuse is removed and the circuit breaker is closed. The way the transformer’s medium voltage side is supplied divides the looped networks into three types: longitudinal looped (Fig. 1.23,a), transversal looped (Fig. 1.23,b) and mixed (Fig. 1.23,c). Big cities, with big load densities (1015 MVA/km2) have mesh type looped networks. In mesh networks the low voltage lines are connected to all possible buses and the network is supplied through medium voltage distribution cables departing from the busbars of the same substation. The special technical and economical advantages of the complex looped networks are: – high degree of supply continuity; – high quality of delivered power, since the consumers voltages are “levelled” on the entire area; – proper balancing of the network, due to the load’s even repartition; – adaptability to load development, because the looped networks are designed from beginning to meet future rises in power consumption. TP1 TP1

TP2

TP1

TP2

TP1

TP2

TP2

a.

b1.

b2.

Fig. 1.22. Low voltage networks: radial (a); looped (b1, b2, b3).

b3.


21

MV

0.4 kV

a. MV

0.4 kV

0.4 kV

MV

b. Fig. 1.23. Looped low voltage networks: a. longitudinal; b. transversal.

22

Basic computation

c. Fig. 1.23. Looped low voltage networks: c. mixed.

V

ca ble s

Substation busbars

M

CB

Transformer points

Distribution box

Branch

Fig. 1.24. Complex looped network of mesh type.


23

1.2. Electric power systems components modelling under steady-state conditions Generally, the components of an electric circuit are resistors, reactors, capacitors and conductors to which correspond the following parameters: resistance R, inductance L, capacitance C and conductance G. The same elements can be found in various combinations in electric receivers, electric lines, transformers and generator units. The property of a circuit element to absorb electromagnetic energy and transform it into thermal energy is called resistance. When an electric current is passed through a conductor, this is getting heated and in the same time along it a voltage drop occurs. In the case of an iron core coil (i.e., transformers, autotransformers, generators, electric motors, etc.), this is getting heated also due to the iron losses through hysteresis phenomenon and eddy current. Heat losses also occur in the dielectric of capacitors pertaining of a circuit element when an alternating current is passed through it. Any circuit section is linked by a magnetic flux when an electric current passes the circuit. In alternating-current circuits, the magnetic flux varies in time and therefore in every circuit section appears a self-induced or mutual e.m.f. and consequently, this circuit section is characterised by self and/or mutual inductance. In any dielectric insulation surrounding one of the electric circuit elements operating at alternating voltage there is always an alternating potential difference, which generates an electric field of density D, which varies in time, and therefore a displacement current δ = ∂D ∂t appears. Likewise, in the case of an electric line powered at alternating voltage, the alternating current value along the conductors is not constant because the current is split in every line section as displacement currents. Similarly with what happened into electric capacitors we can say that, in the case of an electric line, there are electric phase-to-phase and phase-to-ground capacitances. The same situation can be observed at the winding wires of transformers and autotransformers. Between windings there is an alternating potential difference and therefore there is an electric field variable in time and thus a displacement currents also appear. This situation is similar to the loads characterised by pure resistance where, due to the voltage drop across their conductors generates an electric field around the conductors and thus displacement currents. Because the resistance of the dielectric insulation is not infinite and also due to corona discharge, other active power losses occur, different from the losses in resistance R, and thus another parameter is defined. This is called conductance, noted by G, having inverse dimension of the electric resistance. This parameter is due to the leakage currents in the insulators of the overhead electric lines, between phases or between phases and ground, and in the insulation of underground cables. The intensity of these leakage currents is highly dependent on the state of

24

Basic computation

conductors, atmosphere condition or operating conditions. When these currents are negligible, the conductance G is also negligible. The parameters of an electric circuit are dependent on the characteristics of materials from which the circuit is made: resistivity ρ, magnetic permeability μ and electric permittivity ε. These material quantities are not constant and depend on other values such as temperature, the current passing through the circuit or the terminal voltage (especially, in the case of iron core saturation of electric equipment). In some cases this dependence is small and can be negligible, so it can be assumed that the electric circuit parameters are irrespective of the current or voltage. Under these considerations the circuits are called linear circuits; otherwise the circuits are nonlinear. In this chapter we will consider only the case of linear circuits.

1.2.1. Loads (consumers) modelling An electric consumer consists of a set of electric receivers. In an electric receiver the electromagnetic energy is transformed into other forms of energy. The load modelling is a difficult issue due to some objective factors such as: – huge number of electric receivers from complex consumer structure; – lack of accurate information related to the consumer components; – fluctuation of consumer structure in terms of day time, climatic conditions, evolution of technology, etc.; – uncertain characteristics of the consumer components, respectively of electric receivers in terms of voltage and frequency variation. In terms of the goal of the proposed analysis, the load modelling has distinct forms and specific approaches. In power systems practice three classes of issues are identified, every one necessitating an adequate modelling of the load: steady states calculation, emergency conditions calculation and the power systems planning and design, respectively. In this chapter the load modelling under steady state conditions is considered, being symmetrical and balanced regime. Nowadays, in most of the cases the load modelling for the steady state calculation is based on deterministic approach, neglecting the random character of the power consumption. Instead, there are also studies based on probabilistic approaches. The power receivers of a complex consumer have variations of demanded power, more or less time pronounced, by the hours of the day (hourly variations) or by the days of the year (seasonal variations). These variations in time of the power demand by a consumer are illustrated in daily load curves, respectively in yearly load curves. Based on these curves and using specific analysis, it can be determined the calculation load of a certain consumer. It should be mentioned that it is about a constant calculation load value determined for a balanced consumer, at a certain instant of time and under nominal voltage and frequency conditions as well.


25

Basically, the deterministic model of the load assumes providing information related to the power demand by the consumer at a certain instant of time. Although the determination procedure uses static analysis and admits the randomness of the nodal consumer, the calculation load obtained in this way is specific only for a certain instant of time, decreasing the accuracy of the model. Because in most cases the voltage at the consumer terminals is different from the nominal voltage, and also the frequency is different from the nominal one, the load modelling through static load characteristics is performed. In terms of the dependence of the power on the terminal voltage and frequency, two load models are defined: – Static load model, which expresses active and reactive powers at any instant of time, in terms of voltage and frequency at that instant of time, either as polynomial form or exponential form. The static model is currently used for normal steady state calculation; – Dynamic load model, which expresses active and reactive powers at any instant of time, in terms of voltage and frequency at that instant of time as well as at the foregoing times, by using differential equations. The dynamic models are used for emergency operating conditions. The static characteristics have the following general form:

P = P ( f , U ) , Q = Q( f , U )

(1.1)

The most used expressions of the active and reactive powers have one of the following forms, called polynomial model (1.2):

( Q( f ,U ) = Q (d U

)

P( f ,U ) = P0 a U 2 + b U + c (1 + g Δf ) 0

2

)

+ e U + q (1 + h Δf )

(1.2)

respectively exponential model (1.3) and (1.4): ⎛ U ⎞ ⎟⎟ P( f , U ) = A⎜⎜ ⎝ U nom ⎠

αU

⎛ U ⎞ ⎟⎟ Q( f , U ) = B⎜⎜ ⎝ U nom ⎠

βU

⎛ f ⎞ ⎜⎜ ⎟⎟ ⎝ f nom ⎠ ⎛ f ⎞ ⎜⎜ ⎟⎟ ⎝ f nom ⎠

αf

βf

(1.3)

or ⎛ U ⎞ ⎟⎟ P = P0 ⎜⎜ ⎝ U nom ⎠

αu

⎛ U ⎞ ⎟⎟ Q = Q0 ⎜⎜ ⎝ U nom ⎠

βu

(1 + g Δf ) (1.4)

(1 + h Δf )

26

Basic computation

where: a, d

are constants deriving from the load modelling through constant impedance; b, e – constants deriving from the load modelling through constant current; c, q – constants deriving from the load modelling through constant power; g, h – constants indicating the variation of P and Q with frequency; A, B – quantities calculated with the expressions (1.3), in terms of the steady state results when U = Un, f = fn; αu, βu – coefficients that take into account the variation of the active and reactive powers with the voltage; αf, βf – coefficients that take into account the variation of the active and reactive powers with the frequency.

In literature, complex studies for the determination of a, b, …, h and αf, αu, βf, βu coefficients are given. The values of these coefficients depend upon: – consumer type: complex, residential, commercial, industrial, agricultural, fluorescent lamps, arc bulbs, air-conditioned installations, domestic consumption, asynchronous motors, synchronous motors, inductive loads, electric heating, electrochemistry factories, arc furnace, static converters, and so on; – period of the year: summer or winter; – geographical area: north, south, east, west; – load power factor. If information about the electric consumer is not available, average value of the coefficients might be used (Table 1.1). Table 1.1 coefficient consumer Complex Residential Commercial Industrial

αf

αu

βf

βu

0.7 ... 1.2 1 ... 1.5 1.2 0.7 ... 1.5

0.6 ... 1.5 0.5 – 0.185

1 ... 2 1 ... 1.4 1.17 1 ... 2

– 0.6 ... 0 – 0.7 – 0.488

Generally, active power consumption modelling is a compromise between the consumption by resistive loads type and by electric motors type (closer to constant power modelling). In terms of the complex consumer structure (preponderance of resistance consumption or of motors) impedance or constant power modelling might be used. From the general model (1.2) the following particular static models results: • constant impedance model, where the power vary direct-proportional with the voltage magnitude square; P ~U2; Q ~U2

(1.5)


27

• the model where the power varies direct-proportional with the voltage magnitude;

P ~U ; Q ~U

(1.6)

• constant power model, where power demanded by the consumer is independent by the voltage;

P ~ const. ; Q ~ const.

(1.7)

Usually, the loads are modelled through its active and reactive powers, as shown in Figure 1.25,a. It is possible to consider the same load through series or parallel combinations of resistance and reactance of constant values (Fig. 1.25,b,c). U

U

U

RS

IS

Xp

IXp IRp

Rp

XS

P+jQ a.

b.

c.

Fig. 1.25. Load modelling: a. constant powers; b. series impedance; c. parallel impedance.

In the case of complex apparent power S s = P + jQ modelling through series impedance Z s = R s + jX s , the values of resistance Rs and of reactance Xs can be inferred from the expressions of the current: Is =

U R s + jX s

and of the complex power: *

S s = P + jQ = U I s =

U2 U2 (Rs + jX s ) = 2 R s − jX s R s + X s2

Equating the real and imaginary parts it results: P=

U 2 Rs R s2 + X s2

then: P2 + Q2 =

(

; Q=

U 4 Rs2 + X s2

(

Rs2

+

)

2 X s2

U 2Xs R s2 + X s2

) =U

2

P Q =U 2 Rs Xs

28

Basic computation

In consequence:

where: Rs Xs Zs U P Q

is – – – – –

Rs =

U 2P P2 + Q2

(1.8)

Xs =

U 2Q P2 + Q2

(1.9)

resistance of the load series connected, [Ω] ; reactance of the load series connected, [Ω] ; impedance of the load series connected, [Ω] ; phase-to-phase (line-to-line) voltage, [V]; single-phase active power of the load, [W]; single-phase reactive power of the load, [VAr].

If the load is modelled through a resistance in parallel with an inductive reactance:

where: Rp is Xp –

I Rp =

U U2 U2 ; P = UI Rp = ; Rp = Rp P Rp

(1.10)

IXp =

U U2 U2 ; Q = UI Xp = ; Xp = Xp Q Xp

(1.11)

load resistance parallel connected, [Ω] ; load reactance parallel connected, [Ω] .

1.2.2. Electric lines modelling An electric line is characterised by four parameters, having different physical causes: resistance R, caused by electric resistivity of current’s paths, inductance L, which is the effect of the magnetic field, capacitance C, which is the effect of the electric field, conductance G, caused by defective insulation and corona discharge losses. The resistance and inductance are included in the series impedance z = R + jωL , and the conductance and capacitance are included in the shunt admittance y = G + jωC . The presence of the impedance consisting of resistance and inductive reactance leads to voltage variations along the line so the impedances are so called series parameters, while the presence the admittance consisting of conductance and capacitive susceptance ωC, modifies the currents flowing through the line’s conductors, through leakage currents appearance, and therefore they are also called shunt parameters.


29

In Figure 1.26 are illustrated the series and shunt elements of a single-circuit three-phase overhead line, operating under normal steady state conditions, for which the following expression can be written: ΔV a = z aa I a + z ab I b + z ac I c = (Raa + jωLaa )I a + jωLab I b + jωLac I c ΔV b = z ba I a + z bb I b + z bc I c = jωLba I a + (Rbb + jωLbb )I b + jωLbc I c

(1.12)

ΔV c = z ca I a + z cb I b + z cc I c = jωLca I a + jωLcb I b + (Rcc + jωLcc )I c Δ Ia

Δ Va

Ia

yab

zaa Lab Ib

Δ Vc

Lbc Lca

Δ Ib

Δ Vb yca

zbb

ybc

Δ Ic zcc

Ic

Va yag

ycg

Vb ybg

Vc

Fig. 1.26. Series and shunt parameters of a single-circuit overhead line without shield wire.

Δ I a = y aa V a + y ab (V a − V b ) + y ac (V a − V c ) = Y aa V a + Y ab V b + Y ac V c Δ I b = y ba (V b − V a ) + y bb V b + y bc (V b − V c ) = Y ba V a + Y bb V b + Y bc V c (1.13) Δ I c = y ca (V c − V a ) + y cb (V c − V b ) + y cc V c = Y ca V a + Y cb V b + Y cc V c Due to unbalance of some loads the mutual inductance has different values, and then asymmetrical voltages appear. By manufacturing and operation (phase transposition) means, the equality of self-impedances and self-admittances respectively, and the equality of phase-to-phase mutual impedances and phase-tophase mutual admittances, are achieved: z aa = z bb = z cc = z = R + jωLself z ab = z bc = z ca = jωLm

and

(1.14,a)

30

Basic computation

Y ab = Y bc = Y ca = Y m = − y ab = − y bc = − y ca Y aa = y aa + y ab + y ac

(1.14,b)

Y bb = y ba + y bb + y bc Y cc = y ca + y cb + y cc

Note that, also, under normal steady state conditions, electric generators generate an e.m.f. of a, b, c sequence also called positive or direct sequence. The voltages, currents, impedances and admittances of this operating regime are considerate of positive/direct sequence. The normal steady state is considered as perfect symmetrical and balanced; therefore, the following equalities can be defined: V a +V b +V c = 0

(1.15)

Ia + Ib + Ic = 0

Taking into account (1.14) and (1.15), expressions (1.12) and (1.13) become:

( ( (

) ) )

+ ΔV a = ( z − z m ) I a = z I a = ⎡⎣ R + jω ( Lself − Lm ) ⎤⎦ I a = R + jωL+ I a + ΔV b = ( z − z m ) I b = z I b = ⎡⎣ R + jω ( Lself − Lm ) ⎤⎦ I b = R + jωL+ I b (1.16) + ΔV c = ( z − z m ) I c = z I c = ⎡⎣ R + jω ( Lself − Lm ) ⎤⎦ I c = R + jωL+ I c

Δ I a = (Y − Y m ) V a = Y V a +

Δ I b = (Y − Y m ) V b = Y V b +

(1.17,a)

Δ I c = (Y − Y m ) V c = Y V c +

If the conductance is neglected, the admittance y becomes of jωC form, and thus the expressions (1.17,a) become: Δ I a = jω ( Cself + 3Cm ) V a = jωC + V a Δ I b = jω ( Cself + 3Cm ) V b = jωC + V b

(1.17,b)

Δ I c = jω ( Cself + 3Cm ) V c = jωC + V c

Equations (1.16) and (1.17) shows that under perfect symmetrical conditions, by design, the scheme from Figure 1.26 can be replaced by a three-phase network, where the phases are electrically and magnetically decoupled. Under these considerations, the series and shunt elements from Figure 1.26 are replaced as shown in Figure 1.27, where positive/direct sequence impedances and admittances are considered.


31

Δ Ia Ia

Δ Ib

Δ Ic

z z

Ib

z

Ic

a.

y

y

y

b.

Fig. 1.27. Three-phase line of direct/positive sequence: a. series elements; b. shunt elements.

Therefore, under steady state conditions, the three-phase line can be replaced by three independent single-phase lines, with no electric or magnetic coupling, and with currents and voltages shifted only by 120 and 240 degrees. In consequence, steady state analysis can be performed only for a single-phase line, whose parameters are called service line parameters. The service inductance of a three-phase electric line represents the ratio of the total magnetic field flux, generated by the currents on the three phases linking the conductor of one phase, to the current flowing through it. The service inductance is the positive/direct sequence inductance and is denoted by L+ or more simple by L. The service capacitance of a three-phase electric line represents the ratio of electric charge of one phase conductor (to which contributes the electric charges from all the others conductors) to the potential of respective conductor, measured with respect to a reference potential (earth or metallic shell of a cable). In this way, the three-phase system (or in the general case, a multi-phase system) can be replaced with a single-phase system, which has only one capacitance with respect to the reference potential, determined in terms of all capacitances of the real three-phase (multi-phase) system. The service capacitance represents the direct/positive sequence capacitance and is noted by C + or more simple by C. The service parameters or direct/positive sequence parameters are, in general, given in per unit length, usually 1 km, called per unit length parameters and are noted by r0 , l0 , c0 , g 0 .

1.2.2.1. Electric resistance The electric resistance of electric line’s conductors is the most important cause of the active power losses ( ΔP ) in electric lines. The effective resistance of a conductor is given by formula: ΔP (1.18,a) R = 2 [Ω ] I

32

Basic computation

where ΔP is expressed in watts, and I represents the actual value of the current, in ampere. The effective resistance is equal to direct-current resistance of the conductor only if the distribution of current throughout the conductor is uniform. In direct current, the expression of resistance Rdc is as follows: l l Rdc = ρ = s γs

where: ρ γ l s

is – – –

[Ω]

(1.18,b)

electric resistivity of conductor, [Ω mm2/m]; conductivity of material, [m/Ω mm2]; conductor’s length, [m]; cross-sectional area, [mm2].

On the hypothesis of specific resistance varying linear with temperature, its value at a certain temperature θ °C is determined in terms of electric resistivity at 20 °C:

ρ θ = ρ 20 [1 + α 20 (θ − 20 )]

(1.19)

where α20 is temperature coefficient at 20 °C. For usual calculations the following values are considered: α Cu = 0.00393 ; α Al = 0.00403 ; α Fe = 0.0062 . The electric resistance R2 of a conductor at the temperature θ 2 can also be determined with formula: R 2 T0 + θ 2 = R1 T0 + θ1

where: R1 is R2 – θ1, θ2 – T0 –

(1.19')

resistance of conductor at temperature θ1, in Ω ; resistance of conductor at temperature θ2, in Ω; conductor temperatures, in °C; constant depending on the material type having the values: 234.5 °C for annealed copper, 241 °C for hard-drawn copper, 228 °C for hard-drawn aluminium.

At temperature of 20 °C, the electric resistivity of annealed copper is 1/58 Ωmm2/m, and of hard-drawn aluminium is 1/33.44 Ωmm2/m. The electric resistivity of hard-drawn conductors is different form the above-mentioned values due to the treatment applied. In general, the conductors are manufactured in a stranded form (aluminium conductor, steel reinforced − ACSR). A stranded conductor is made from wires disposed in layers, tight and spiralled in opposite directions. The resistance in direct current of such conductor is calculated taking into account that the average length of wire is 2 ÷ 4% greater than the real length of conductor, for overhead lines, and 2 ÷ 5% greater for underground cables, and the cross-sectional area used


33

in equation (1.18,b) is determined by multiplying the cross-sectional area of one wire with the number of wires. The alternating current resistance is the different from direct current resistance due to the following issues [1.16], [1.17]: • Skin effect consisting in non-uniformity distribution of current in the cross-section of an electric conductor mostly when this is made from one or more concentric circular elements. When a current is passed through a conductor this generates a magnetic field that is stronger inside the conductor and its intensity decrease towards the surface. In consequence, the self-induced e.m.f. in conductor is higher towards centre of cross-section. Accordingly to Lenz’s law the magnetic field push the current to flow near the outer surface of conductor, thus total density of current is smaller in interior and higher towards the surface. As the frequency of alternating current increases so does the effect becomes more pronounced. Thus, skin effect increases the effective resistance of conductor by reducing the effective cross-section of conductor from which flow the current. Skin effect depends upon the cross-sectional area, material type of conductor, frequency and magnetic permeability. In literature, the influence of skin effect on the electric resistance is given by Rac Rdc ratio, in curves or tables, in terms of the above-mentioned parameters. For instance, for steel-reinforced conductor (ACSR) of 400 mm2 area, Rac Rdc is equal to 1.0055. • Proximity effect leads also to a non-uniformity distribution of current, in cross-sectional area of an electric conductor. For instance, in a single-phase electric line consisting of two conductors (go and return conductors), the nearest parts of the conductors are sweep by a stronger magnetic field than in the farthest parts. Therefore, the parts of the conductor nearest each other have a lower inductance value compared with the farthest ones. The result is an increased current density in the parts of the conductor nearest each other, and a lower one in farthest ones. This non-uniformity density of current increases the effective resistance. For usual distances of the overhead lines operating at frequency of 50 Hz, this effect is negligible. • The resistance of the conductor made from magnetic materials varies with current magnitude because in the steel-core of the Al-Ol rope conductor the power losses increase, especially when the number of layers is odd. Magnetic conductor tables, such as ACSR conductors, contain values of electric resistance for two values of the carried current, emphasizing this effect. As regards the underground cables with high cross-sectional area, the abovementioned causes have a more pronounced character. Therefore, in alternating current, power losses in conductors can be calculated with expression:

ΔPac = ΔPdc + ΔPs + ΔPp + ΔPsc + ΔPsh + ΔPt

(1.20)

Expression (1.20) can be rewritten as:

(

ΔPac = 3Rac I 2 = 3I 2 Rdc + ΔRs + ΔR p + ΔRsc + ΔRsh + ΔRt

)

(1.21)

34

Basic computation

where: Rac is effective alternating current resistance; Rdc – direct current resistance; I – effective value of current flowing through conductor; ΔRs , ΔR p , ΔRsc , ΔRsh , ΔRt are incremental resistances due to skin effect, proximity effect, screen wire losses, outer sheath losses and protection tube losses. In practice, for the calculation of ΔRs and ΔRp quantities, having a prevailing effect, semiempiric formulae are used. When the cables are disposed in steel tubes, due to the increase in magnetic flux, skin effect and proximity effect become more pronounced.

1.2.2.2. Inductive reactance For determination of inductive reactance, the following simplifying hypotheses are considered: (i) the distances between phase conductors are bigger compared with their radius; (ii) the current is uniformly distributed within the cross-sectional area of each conductor; (iii) there is no ferromagnetic material inside or outside the conductor; (iv) the current in every conductor is constant along its length; (v) the sum of instantaneous currents flowing through the n conductors is zero, that is: i a + ib + ic + .... + i n = 0 Under these assumptions, we may apply superposition for the magnetic circuit of a solenoid with N turns. The inductive reactance of a single phase from a multiphase electric line is given by formula: (1.22) X = ωL = 2πfL where: f is L –

frequency, [Hz]; inductance of a single phase, [H].

We next consider a multiphase electric line as a set of turns passed by reciprocal influenced currents; first, the basic formula necessary for calculation of the inductance of a solenoid with N turns (Fig. 1.28) [1.18] is determined, then the general case of a multiphase system is treated. Ψ2 Ψ1 Ψ3 Fig. 1.28. Flux linkage into a solenoid.


35

Starting from the magnetic circuit law we express the magneto-motive force:

∫

mmf = H dl = i

(1.23)

where: H is magnetic field intensity, [ampere-turns/m]; dl – length element, [m]; i – instantaneous current linking the solenoid. According to Biot-Savart’s law, the magnetic field intensity in any spatial point is a linear function of all the currents that generates the respective field and therefore it can be calculated by using superposition. The relationship law between the magnetic flux density and magnetic field intensity (material law), in magnetic medium with linear characteristic, is: B = μH

(1.24)

where: B is magnetic flux density, [Wb/m2]; μr – relative magnetic permeability, [H/m]; μ0 – magnetic permeability of vacuum, [H/m]; μ = μ r μ 0 – absolute magnetic permeability of medium, [H/m]. The total flux through a surface of area A is the surface integral of the magnetic field B, that is:

∫

(

∫

)

φ = B d a = B cos B, d a da A

(1.25)

A

where d a is a vector perpendicular to the area element da. When the vector B is perpendicular to the surface of area A, the relationship (1.25) becomes: (1.26) φ = BA Due to the solenoid shape several magnetic fluxes appear, some of theme linking only some parts of the solenoid, called leakage fluxes, as it can be seen in Figure 1.28. The total leakage flux is determined by considering the contribution of each turn: φt =

N

∑φ

k

(1.27)

k =1

where φk is the sum of all fluxes linking the turn k of the solenoid. As illustrated in Figure 1.28 this could have the expression φ k = Ψ1 + Ψ2 + Ψ3 . The inductance of the overhead lines can be calculated starting from the fundamental relationship representing the ratio of the total magnetic flux, linking the surface bounded by the contour of a circuit, to the current passing through the respective circuit, having a linear dependency:

36

Basic computation

L=

φt i

(1.28)

When there exist more circuits, in the same medium, that influence each other, under the condition of linearity mentioned earlier, self- and mutual inductances are defined. Specific inductance calculation for an infinite straight conductor In order to obtain an accurate value of the inductance of an electric transmission line, the flux linkage inside as well as outside of each conductor must be taken into consideration.

a. Flux linkage inside the conductor In Figure 1.29 the cross-section of a cylindrical conductor in shown. ds x

dx

r

Fig. 1.29. Cross-section of the conductor.

Let the magnetic field lines at a distance x meters from the centre of the conductor and concentric distributed relative to the axis of the conductor. The magnetic field intensity Hx is constant along these field lines and tangent to it. In consequence, the equation (1.23) becomes:

∫ H ds = H x

x ( 2πx )

= ix

(1.29)

where ix is the current flowing through inside the area πx 2 . On the basis of assumption of uniformly distribution of current, the current ix is obtained by multiplying the total current i from the conductor with the ratio of the cross-sectional area, passed by the current ix, to the total cross-sectional area of the conductor:

ix =

πx 2 i πr 2

(1.30)

from where it results the expression of the magnetic field intensity:

Hx = i

x 2πr 2

(1.31)


37

The value of internal flux is a percent of the linkage flux due to the total current from conductor and is calculated with formula: r

φint = μ

∫ 0

x πx 2 μμi i 2 dx = 0 r 2 8π 2πr πr

(1.32)

Results thus the internal inductance of a conductor: Lint =

φint μ 0μ r = i 8π

(1.33)

b. Flux linkage outside the conductor Consider a cylinder of radius R∞ , made from magnetic flux lines surrounding an infinite length conductor of radius r. The flux linkage outside the conductor is determined as volume integral inside the cylinder considered from radius r to distance R∞ where the magnetic field intensity has no longer influence.

1m

B

i r x

R

dx

Fig. 1.30. Flux crossing surface element da = dx · 1.

If we denote by Hx the magnetic field intensity of the tubular element which is x meters from the centre of the conductor, the magneto-motive force becomes: 2πx H x = i

(1.34)

where i is the current passed through the conductor. Knowing that the relative magnetic permeability of air is μ r ≈ 1 and taking into consideration equations (1.24), (1.25) and (1.34), the flux linkage outside the conductor is:

∫

φext = B d a = μ 0 A

R∞

i

μ 0i

∫ 2πx dx = 2π ln r

R∞ ⎡ Wb ⎤ r ⎢⎣ m ⎥⎦

(1.35)

where: R∞ is the distance meters from the axis of the conductor to the point where H = 0 ; r – radius of the conductor.

38

Basic computation

The inductance corresponding to the flux outside the conductor is calculated with expression: φ μ R ⎡H⎤ (1.36) Lext = ext = 0 ln ∞ ⎢ ⎥ i 2π r ⎣m⎦ Therefore, the generalised self-inductance of a massive conductor is:

L = Lext + Lint =

1⎞ μ 0 ⎛ R∞ + μr ⎟ ⎜ ln 2π ⎝ r 4⎠

(1.37)

Consider a system of two conductors of radii equal to r situated at the distance D from each other, where one conductor is the return path of current for the other. The two-conductors system forms a contour that bounds a surface linked by the magnetic flux.

h

i

i

r

r D ground

Fig. 1.31. Two-conductors system.

Assuming that D is much greater than r, the inductance can be calculated with formula: 1⎞ μ ⎛ D (1.38) L = 0 ⎜ ln + μ r ⎟ 2π ⎝ r 4⎠ Relation (1.38) can also be written as:

L=

μ0 ⎛ D μ D μ 4⎞ ⎜ ln + ln e r ⎟ = 0 ln −μ r 2π ⎝ r ⎠ 2π r e

4

(1.39)

The expression of inductivity can be put in a more simple form noting by re = r e − μ r 4 the equivalent radius of conductor. Therefore:

L=

μ0 D ln 2π re

(1.40)

For μ r = 1 (non-magnetic material) and massive conductor, it results re = 0.7788 r . For stranded conductor, made from non-magnetic material, the following ratios are obtained:


39

– conductors with 7 wires re r = 0.725 ; – conductors with 19 wires re r = 0.757 ; and for aluminium-steel conductors: – conductors with 7 wires re r = 0.770 ; – conductors with 30 wires re r = 0.826 . Using the international system of units SI, where μ 0 = 4π10 −7 H/m, μ r = 1 for dry air, and l = 1000 m, the inductance per unit length is obtained:

L0 = 0.2 ln

D D = 0.46 lg [mH/km phase] re re

At frequency of 50 Hz the reactance per unit length is:

x0 = ωL0 = 314 ⋅ 0.46 lg

D D = 0.1445 lg [Ω/km phase] re re

(1.41)

Thus, we can draw the conclusion that the reactance of an electric line does not depend upon the current passing through the line. Total flux linkage of a conductor, from an n-conductors system Consider the general case of n parallel conductors (Fig. 1.32), representing a multi-phase system under normal steady state conditions. Conductor 1

H=0 d1

D12

Conductor 2

dk D1k

Conductor k

Fig. 1.32. Contribution of current ik to the total linkage flux of conductor 1.

In order to define a contour, consider first an imaginary conductor that constitutes the return path for the sum of currents from all phases (under normal operating conditions the intensity of current flowing through the imaginary conductor is zero). For symmetry reasons, this imaginary conductor is considered parallel with the other conductors. The distance between the real conductors is small enough so that they influence each other. For simplicity, consider only the calculation of magnetic flux linking the conductor 1, due to existence of the current from conductor k, the

40

Basic computation

currents from the other conductors are considered zero. Finally, we apply superposition, which takes into account the influence of magnetic flux due to the currents from the others conductors. As it can be seen in Figure 1.32 the magnetic field lines due to currents ik are concentric circles. Having the previous assumptions made, we may alike the general case of a multi-phase system with the case studied earlier of a system with two conductors. We denote by dk the distance from the centre of conductor k to the imaginary conductor, where the magnetic flux intensity is zero, and by D1k the distance from the centre of conductor 1 to the centre of conductor k. By applying equation (1.35), the self-flux linking the circuit composed by conductor 1 and imaginary conductor, due to current i1 in conductor 1, is:

φ11 =

μ 0i1 d1 ln [Wb/m] re 2π

(1.42,a)

and the flux linking the conductor 1, due to the current flowing through conductor k, has the following expression:

φ1k =

μ 0ik d ln k D1k 2π

(1.42,b)

Therefore, the total flux linking the conductor 1 due to the contributions of all currents flowing through the n conductors is:

φ t1 =

μ0 ⎡ d d ⎤ d1 d2 + ... + ik ln k + ... + in ln n ⎥ ⎢i1 ln + i2 ln re D12 D1k D1n ⎦ 2π ⎣

(1.43)

Expression (1.43) can be written as:

φt1 =

1 1 1 1 ⎤ μ0 ⎡ + ... + ik ln + ... + in ln ⎥ ⎢i1 ln + i2 ln 2π ⎣ re D12 D1k D1n ⎦

μ + 0 [i1 ln d1 + i2 ln d 2 + ... + ik ln d k + ... + in ln d n ] 2π

(1.44)

Knowing that the sum of instantaneous currents through the n conductors is zero: i1 + i2 + ... + ik + .... + in−1 + in = 0 we may express the current in in terms of the other n-1 currents [1.10]: in = −i1 − i2 − ... − ik − .... − in−1

(1.45)

Substituting the current in from (1.45) into (1.44) the following expression is obtained:


φ t1 =

41

μ0 ⎡ 1 1 1 1 ⎤ + ... + ik ln + ... + in ln ⎥ ⎢i1 ln + i2 ln re D12 D1k D1n ⎦ 2π ⎣

+

μ0 ⎡ d d ⎤ d1 d + i2 ln 2 + ... + ik ln k + ... + in−1 ln n−1 ⎥ ⎢i1 ln dn dn dn dn ⎦ 2π ⎣

Considering the imaginary conductor is located at infinite distance with respect to the real conductors, the logarithms of ratios of distances from real conductors to the imaginary conductor tend toward zero so that the expression of the total flux linking the conductor 1 get the form: φt1 =

μ0 ⎡ 1 1 1 1 ⎤ + ... + ik ln + ... + in ln ⎢i1 ln + i2 ln ⎥ re D12 D1k D1n ⎦ 2π ⎣

(1.46)

Using self and mutual inductances, we can see that the expression (1.46) represents the Maxwell’s law referring to inductances: φt1 = L11i1 + L12i2 + ... + L1k ik + ... + L1nin

(1.47)

Therefore, in the general case of a system with n conductors, where the sum of all currents is zero, the self-inductance Ljj and the mutual inductance Ljk are given by expressions: L jj =

μ0 ⎛ 1 ⎞ ln⎜ ⎟ 2π ⎜⎝ re ⎟⎠

L jk =

μ 0 ⎛⎜ 1 ln 2π ⎜⎝ D jk

[H/m] ⎞ ⎟ ⎟ ⎠

[H/m]

(1.48,a)

(1.48,b)

The above equations constitute the basis of practical evaluation of reactance of the electric lines. Inductance of a single-circuit three-phase overhead electric line Let us consider a single-circuit three-phase overhead electric line (Fig. 1.33) with unequal distances between phases. Due to unequal spacing between phases it cannot be defined the inductivity La assigned to the phase a that could be constant in time and cannot be say that the flux φa is proportional to the current ia (both in instantaneous quantities). For this reason, we next consider the sinusoidal steady state so that we must express the inductivity of a phase into the complex space:

La =

Φa Ia

(1.49)

By expressing the flux linking a phase it can be defined an operator, constant in time, and an inductivity that allow coherent operations, into a complex space (of

42

Basic computation

pulsation ω ), that establish relationships between flux and current, electromotive voltage and the derivative of current, etc. D12 1

D23

2 D13

3

Fig. 1.33. Tower phase spacing of a single-circuit three-phase overhead line.

Applying (1.46) for this case it results the expression of flux linkage of phase a per unit length: Φa =

μ0 ⎡ 1 1 1 ⎤ + I c ln ⎢ I a ln + I b ln ⎥ 2π ⎣ re D12 D13 ⎦

(1.50)

In a three-phase electric system, symmetrical and balanced, the dependency between the currents on the three phases is given by relationships: I a = I a ; I b = a2 I a ; I c = aI a

(1.51)

where

1 3 1 3 a = e j 2π / 3 = − + j ; a 2 = e− j 2π / 3 = − − j 2 2 2 2 Substituting expressions (1.51) into (1.50) obtain: Φa = =

⎤ ⎞ ⎛ 1 ⎛ 1 μ0 ⎡ 1 3 ⎞⎟ 1 ⎜ − + j 3 ⎟ ln 1 ⎥ = ln I + ⎢ I a ln + I a ⎜⎜ − − j a ⎜ 2 2π ⎣⎢ re 2 ⎟⎠ D12 2 ⎟⎠ D13 ⎦⎥ ⎝ ⎝ 2 D12 D13 μ 0 ⎛⎜ 3 D13 ⎞⎟ −j ln I a ln 2π ⎜⎝ 2 D12 ⎟⎠ re

Therefore, the inductance per kilometre of phase a get the form: La =

D12 D13 Φ a μ 0 ⎛⎜ 3 D13 ⎞⎟ ' = −j ln ln ≡ La − jL"a I a 2π ⎜⎝ re 2 D12 ⎟⎠

(1.52)

Observation: For the case when D12 = D13 = D the imaginary part becomes zero, so that an expression similar with (1.40), corresponding to the case of the two-conductor system, is obtained.


43

From (1.52) we can draw the conclusion that in the expression of inductance, an imaginary part appears which leads to a supplementary resistance on phase:

(

)

Z a = R + jωL a = R + jω L'a − jL"a = R + ωL"a + jωL'a

(1.53)

By increasing the real part of the impedance, supplementary active power and energy losses on the line appear. In order to avoid this, for symmetry of the circuit and equilateral spacing between phases, the phase transposition method is used for an electric line of length l (Fig. 1.34). l/3 a

l/3 c

l/3 b

Ib

b

a

c

Ic

c

b

a

Section 2

Section 3

Ia

Fig. 1.34. Transposition of the phases, without return to the initial state, for a single-circuit threephase overhead line.

Section 1

Transposition cycle

Therefore, if apply equation (1.46), the average linkage flux of phase a is:

φa , av

⎡ 1 ⎢μ ⎛ 1 1 1 ⎞ ⎟+ = ⎢ 0 ⎜⎜ ia ln + ib ln + ic ln 3 ⎢ 2π ⎝ re D12 D13 ⎟⎠

⎢ (1) ⎣ +

1 1 1 ⎞ μ0 ⎛ ⎜ ia ln + ib ln ⎟+ + ic ln ⎜ 2π ⎝ re D23 D21 ⎟⎠

(2 )

⎤ ⎛ ⎞ μ 1 1 1 ⎥ ⎟⎥ = + ic ln + 0 ⎜⎜ i a ln + ib ln 2π ⎝ re D31 D32 ⎟⎠⎥

⎥ (3 ) ⎦ =

μ0 ⎛ 1 1 1 ⎞ ⎜⎜ ia ln + ib ln ⎟= + ic ln re GMD GMD ⎟⎠ 2π ⎝ =

μ0 GMD ia ln 2π re

(1.54)

where GMD = 3 D12 D23 D31 is the geometric mean distance between positions 1, 2 and 3.

44

Basic computation

Based on equation (1.54), the expression of average inductance per phase becomes: La , av =

μ 0 GMD ln re 2π

(1.55,a)

Knowing μ 0 = 4π10 −7 H/m and considering the unit length of l = 1000 m, the inductance per phase and kilometre is obtained: L0 = 0.2 ln

GMD GMD = 0.46 lg re re

[mH/km phase]

(1.55,b)

At frequency of 50 Hz the reactance per unit length is: x0 = ωL0 = 0.1445 lg

GMD re

[Ω/km phase]

Inductance of a double-circuit three-phase overhead electric line Consider a double-circuit electric line with phase transposition, as shown in Figure 1.35. Considering that the two circuits are identical from manufacturing and loading point of view then i a = i a ' , ib = ib ' , ic = ic ' . The total linkages flux of phase conductor a from the circuit abc and of phase conductor a' from the circuit a'b'c' on all the three transposition sections, is:

φ (a1) =

μ0 ⎡ 1 1 1 1 1 1 ⎤ + i c ln + ia' + ib ' ln + ic ' ⎢i a ln + ib ⎥ 2π ⎣ re D12 D13 D11' D12' D13' ⎦

φ (a1') =

μ0 ⎡ 1 1 1 1 1 1 ⎤ + ic ' ln + ia + ib ln + ic ⎢i a ' ln + ib ' ⎥ 2π ⎣ re D1'2' D1'3' D1'1 D1'2 D1'3 ⎦

φ (a2 ) =

μ0 ⎡ 1 1 1 1 1 1 ⎤ + i c ln + ia' + ib ' ln + ic ' ⎢i a ln + ib ⎥ 2π ⎣ re D23 D21 D22' D23' D21' ⎦

φ (a2' ) =

μ0 ⎡ 1 1 1 1 1 1 ⎤ + ic ' ln + ia + ib ln + ic ⎢i a ' ln + ib ' ⎥ re D 2 '3 ' D2'1' D2 ' 2 D 2 '3 D2'1 ⎦ 2π ⎣

φ (a3) =

μ0 ⎡ 1 1 1 1 1 1 ⎤ + ic ln + ia' + ib ' ln + ic ' ⎢i a ln + ib ⎥ 2π ⎣ re D31 D32 D33' D31' D32' ⎦

φ (a3') =

μ0 ⎡ 1 1 1 1 1 1 ⎤ + i c ' ln + ia + ib ln + ic ⎢i a ' ln + ib ' ⎥ 2π ⎣ re D3'1' D3 ' 2 ' D3 ' 3 D3'1 D3'2 ⎦ (1.56)


l/3 Ia Ib Ic Ic Ib Ia

a

b

3

l/3

a

c

l/3

b

a

b c

c

b

a

c

b

b

a

a c

a

c

1 2

45

1

3 1 a

2

2

b

a

Section1

b 3

2

b

a

c

3

1

a

b

3 b

3

1

1

2 a

b

Section 2 Transposition cycle

2

Section 3

Fig. 1.35. Double-circuit line with phase transposition. The position of transposition sections 1, 2 and 3.

The average linkage flux of any of the two phase conductors a or a' is:

φ a , av = φ a ', av

(1) (1) (2 ) (2 ) φ (a3) + φ (a3') ⎞ 1 (1) 1 ⎛⎜ φ a + φ a ' φ a + φ a ' ⎟ = φ a , av + φ (a2, )av + φ (a3,)av = ⎜ + + ⎟ 3 3⎝ 2 2 2 ⎠

(

)

(1.57) φ (a1,)av = φ (a1'), av = + ib ln

φ a + φ a ' μ 0 1 ⎛⎜ 1 = i a ln 2 2 + ⎜ 2 2π 2 ⎝ re D11'

φ (a2 ) + φ (a2' ) μ 0 1 ⎛⎜ 1 = i a ln 2 2 + 2 2π 2 ⎜⎝ re D22'

⎞ 1 1 ⎟ + ic ln D23 D2'3' D23' D2 '3 D21D2'1' D21' D2 '1 ⎟⎠

φ (a3,)av = φ (a3',)av = + ib ln

(1)

⎞ 1 1 ⎟ + ic ln D12 D1'2' D12' D1'2 D13 D1'3' D13' D1'3 ⎟⎠

φ (a2, )av = φ (a2',)av = + ib ln

(1)

φ (a3) + φ (a3') μ 0 1 ⎛⎜ 1 = i a ln 2 2 + ⎜ 2 2π 2 ⎝ re D33'

⎞ 1 1 ⎟ + ic ln D31D3'1' D31' D3'1 D32 D3'2' D32' D3'2 ⎟⎠

The following notations are adopted:

46

Basic computation

Dab , eq = 4 D12 D1'2' D12' D1'2 Dbc , eq = 4 D23 D3'2 ' D23' D2'3 Dac , eq = 4 D31 D3'1' D31' D3'1

GMRa = re D11' GMRb = re D22' GMRc = re D33' Taking into account that i a + ib + ic = 0 , we obtain: φ a , av = φ a ', av = 2 ⋅10 − 7 ia ln

GMD [H/m phase] GMR

(1.58)

where geometric mean distance GMD is defined by: GMD = 3 Dab , eq Dbc , eq Dac , eq

(1.59)

and geometric mean radius GMR is:

GMR = 3 GMRaGMRbGMRc

(1.60)

According to equation (1.58), the expression of average inductance per phase becomes: GMD GMD [mH/km phase] (1.61) La , av = 0.2 ln = 0.46 lg GMR GMR Inductance of a single-circuit three-phase overhead electric line, with bundle conductors In the case of extra high voltage powered electric lines, power losses due to corona discharge and their influence on telecommunication lines become excessively high if a single conductor per phase is used. The voltage gradient is considerable decreased if instead of a single conductor per phase more conductors are used, and the distance between the conductors of each phase is small as compared with the distance between phases. Such conductors are called bundled conductors. Assume that instead of a single conductor, there are f conductors on each phase, also called sub-conductors. In the following we consider the case of a phase consisting of f = 5 sub-conductors, as shown in Figure 1.36,a [1.14]. The following assumptions are considered: – all the sub-conductors from the bundle have the same radius r, and the current in each phase splits equally among the f parallel sub-conductors; – the distance D between bundle centres is much greater than the distance df between the sub-conductors of the same phase.

Electric power systems configuration and parameters 7 r

6

2

3

df

1

D

11 15

D D1

2

D

3

4

2 13

9

D1

5

3

8

10

D

47

12

2π/n 13

A

1

14

a.

b.

Fig. 1.36. Electric line with bundled conductors: a. Phase spacing for the case of line with 5 sub-conductors per bundle; b. sub-conductors spacing for the case of line with f sub-conductors per bundle.

For the calculation of the total linkage flux of sub-conductor 1, from phase a, by applying expression (1.46) it results: φt 1 =

=

μ 0 ⎡ ia ⎛ 1 1 1 1 1 ⎞ ⎟+ + ln + ln + ln ⎢ ⎜⎜ ln + ln D12 D13 D14 D15 ⎟⎠ 2π ⎢⎣ 5 ⎝ re +

ib ⎛⎜ 1 1 1 1 1 ⎞⎟ + ln + ln + ln + ln + ln ⎜ 5 ⎝ D16 D17 D18 D19 D1,10 ⎟⎠

+

1 1 1 1 1 ⎞⎟⎤ ic ⎛⎜ + ln + ln + ln + ln ln ⎥= 5 ⎜⎝ D1,11 D1,12 D1,13 D1,14 D1,15 ⎟⎠⎥ ⎦

(1.62)

1 1 μ 0 ⎡ ia i + b ln + ⎢ ln 2π ⎣⎢ 5 (re D12 D13 D14 D15 ) 5 D16 D17 D18 D19 D1,10

(

+

1 ic ln 5 D1,11D1,12 D1,13 D1,14 D1,15

(

)

⎤ ⎥ ⎥⎦

)

where D1j is the distance from the sub-conductor 1 to sub-conductor j, j = 2, 3, ..., 15. φt1 =

μ 0 ⎛⎜ 1 1 1 ⎞⎟ + ib ln + ic ln ia ln 2π ⎜⎝ GMR f GMD1b GMD1c ⎟⎠

(1.63)

where the following definitions have been introduced: GMR f = 5 re D12 D13 D14 D15 is geometric mean radius of the bundle; GMD1b = 5 D16 D17 D18 D19 D1 10

– geometric mean distance from conductor 1 to phase b;

GMD1c = 5 D1,11D1,12 D1,13 D1,14 D1,15 – geometric mean distance from conductor 1 to phase c.

48

Basic computation

Considering the phases are located in an equilateral triangle corners that is GMD1b = GMD1c = D , and ia + ib + ic = 0 , the expression (1.63) becomes:

φt 1 =

D μ0 ia ln 2π GMR f

(1.64)

In order to calculate the inductance per kilometre of conductor 1, we take into account the fact that the intensity of current passed through the conductor 1 is 1/5 from the total current per phase. Thus: L1 =

φ t1 5μ 0 D = ln ia RMG f 2π 5

(1.65)

Next, calculating the total linkage flux φ t 2 of conductor 2, we find the same geometric mean radius GMRf. Due to the large distances between phases, we can consider that the inductances of the five parallel conductors of a phase are approximately equal L1 ≅ L2 ≅ L3 ≅ L4 ≅ L5 , so that:

La =

L1 μ 0 D D = ln = 2 ⋅10− 4 ln [H/km phase] 5 2π GMR f GMR f

(1.66)

In the particular case of f conductors per bundle, symmetrical spaced along a circle of radius A, the distances between the sub-conductors of a phase, from equation (1.62), can be calculated with:

⎛π⎞ D12 = 2 A sin ⎜⎜ ⎟⎟ ⎝f ⎠ ⎛ 2π ⎞ D13 = 2 A sin ⎜⎜ ⎟⎟ ⎝ f ⎠ # ⎛ ( f − 1)π ⎞ ⎟⎟ D1 f = 2 A sin ⎜⎜ f ⎝ ⎠ respectively ⎡ ⎛ ( f − 1)π ⎞⎤ ⎛ 2π ⎞ ⎤ ⎡ ⎛ π ⎞⎤ ⎡ ⎟⎟⎥ = A f −1 f (1.67) D12 D13 " D1 f = A f −1 ⎢2 sin ⎜⎜ ⎟⎟⎥ ⎢2 sin ⎜⎜ ⎟⎟⎥ ⎢2 sin ⎜⎜ f f f ⎝ ⎠⎦ ⎝ ⎠⎦ ⎣ ⎝ ⎠⎦ ⎣ ⎣ where the following trigonometric identity has been used: ⎡ ⎛ π ⎞⎤ ⎡ ⎛ 2π ⎞ ⎤ ⎡ ⎛ ( f − 1)π ⎞⎤ ⎟⎟⎥ = f ⎢2 sin ⎜⎜ ⎟⎟⎥ ⎢2 sin ⎜⎜ ⎟⎟ ⎥ " ⎢2 sin ⎜⎜ f ⎝ f ⎠⎦ ⎣ ⎝ f ⎠⎦ ⎣ ⎝ ⎠⎦ ⎣

(1.68)


49

Therefore, in this case, for the geometric mean radius and geometric mean distances, the following expressions are obtained: GMR f = f re A f −1 f GMD1b = f D1, f +1D1, f + 2 " D1, 2 f

(1.69)

GMD1c = f D1, 2 f +1D1, 2 f + 2 " D1,3 f

Considering that practical, the distances D1,f+1 . . . D1,2f . . . D1,3f are equal to the distance between bundles centres, for the inductance La the same expression as in (1.66) is obtained, where GMRf and GMDs are given by (1.69). Application Calculate the reactance of 750 kV single-circuit three-phase overhead electric line. Each phase consists of Al-Ol conductors of 5×300/69 mm2, and the conductors of bundles are of radius r = 2.515 / 2 cm . The distance between the conductors of bundles, situated in the corners of a pentagon, is df = 40 cm (Fig. 1.36,a). The mean distance between the bundle centres of two different phases is D =17500 mm. Thus: D12 = D15 = 40 cm , D13 = D14 = 2 ⋅ 40 ⋅ cos 36° = 64.72136 cm 1/ 5

⎛ 2.515 −1 / 4 ⎞ ⋅ 40 2 ⋅ 64.72136 2 ⎟ GMR f = ⎜ e 2 ⎝ ⎠

= 23.09 cm

Therefore, the inductance per kilometre, noted by La 0 , is: La 0 = 2 ⋅10− 4 ln

17500 = 0.86559 mH km 230.9

xa 0 = ωLa 0 = 314 ⋅ 0.86559 ⋅ 10−3 = 0.272 Ω km Considering the distances D between bundles centres are not equal and perform phase transposition, obtain an average reactance of 0.2861 Ω /km.

Underground electric lines have the same parameters as overhead electric line: series impedance consisting of a resistance and an inductive reactance and shunt admittance consisting of a conductance and a susceptance. As compared with overhead lines there are some important differences such as: – cables are much closer to each other; – in the most cases the cross-section of underground cables is not of circle form, being of circle sectors form more or less regular; – conductors are surrounded by other metallic objects (usually grounded), such as screens, protection sheaths or steel tubes; – insulation material between conductors is solid (in the most cases) or gas. This insulation material is mostly mixed than uniform because, in fact, it consists of each phase insulation, as well as the filling material between phases.

50

Basic computation

All factors enumerated earlier tend to complicate the calculation of line’s constants. The closeness between phase conductors and the irregular forms of crosssectional areas tend to make non-uniform the distribution of field lines into crosssectional area as well as the displacement currents around the dielectric surface. The non-uniform distribution of currents, eddy currents, secondary currents induced in screens, sheaths, tubes, etc., modify the inductive, respectively capacitive reactance, and adds supplementary active power losses. When the cables have a mixed insulation, in addition to power losses by corona discharge that occur in insulating gas, power losses in dielectric also occur. The complexity of constants’ line calculation for underground cables is compensate by the fact that all dimensions are kept at standard values and thus, the constants once determined are available in tables and charts. Therefore, if for overhead lines, in terms of the voltage level, the average inductive reactance ranges in the interval: x0 = 0.306 ÷ 0.45 Ω /km for underground lines this value is: x0 = 0.074 ÷ 0.154 Ω /km

1.2.2.3. Capacitive susceptance Two conductors of an overhead electric line, have a capacitance, which once connected to an alternating voltage leads to the appearance of a current even for no-load conditions. This current is bigger at the sending-end of the line and decrease to zero towards the receiving-end of the line. Consequently, the receivingend of an electric line operating under no-load conditions has a capacitive power factor. An electric line has an intricate structure, such as the double-circuit threephase overhead line, with two bundled conductors, forming an assembly of 12 conductors, one or two shield conductors as well as the ground, which has capacitances between pairs of conductors and between conductors and ground. No matter how intricate is the geometry of a line an effective capacitance to ground can be obtained. The determination manner of this effective capacitance is presented in the following. The capacitive susceptance per phase of a multi-phase line, is given by formula: (1.70) B = ωC = 2πfC where: f is frequency, [Hz]; C – effective capacitance per phase, [Farad]. Consider first the case of an infinite straight conductor of radius r (Fig. 1.37), belonging to an electric line, and then define a Gaussian cylindrical surface coaxial with conductor axis, charged with instantaneous electric charge q.


51

h

E

r

D ds

R

Fig. 1.37. Cylindrical surface coaxial with conductor centre.

In homogeneous medium, the law of dependency between the electric field intensity E [Volt/metre] and the electric field density D [Coulomb/metre2], is given by relationship: D = εE

(1.71)

where: ε = ε 0 ε r is electric permittivity, [F/m]; 1 F/m; 4 π 9 ⋅109 – relative permittivity of medium, ε r = 1 F/m for dry air.

ε0

– permittivity of vacuum, ε 0 =

εr

The value of electric flux density D, called also electric field induction, is determined by applying the electric flux law along the cylindrical surface of radius R, surrounding the conductor, as shown in Figure 1.37:

∫ D d s = D ⋅ 2πRh = q

(1.72)

A

with q = ql h where: D ds A q ql h

is

– – – – –

electric flux density vector, [C/m2]; element area vector (perpendicular to the surface A), [m2]; closed surface area, [m2]; algebraic sum of all the line charges enclosed within surface A, [C]; charge density per unit length, [C/m]; length of conductor, [m].

For R ≥ r , from expression (1.72) it results the magnitude of per length electric flux density: q (1.73) D= l 2πR Let us imagine two points A and B located at the distance d A and d B respectively from the centre of conductor in question. The point A is farther away

52

Basic computation

than B relative to the conductor. The potential difference between the two points A and B is the line integral of the electric field E along any curve path joining the two points (the potential difference is independent on the path followed in irrotational field): dB

∫

VBA = VB − VA = − E d s

(1.74)

dA

Taking account of equations (1.71) and (1.73) equation (1.74) becomes: dB

VBA = VB − VA = −

ql

ql

dA

∫ 2πεR dR = 2πε ln d

dA

(1.75)

B

Consider now two parallel conductors a and b of equal radii ra = rb = r located at the distance D from each other (Fig. 1.38). electric field lines

equipotential lines qa

r

V =0

r

qb

D Fig. 1.38. Cross section of a two-conductor system.

The voltage between the two conductors is determined considering the contribution of the electric charge from each conductor: Vab =

qa D q r ln + b ln 2πε r 2πε D

(1.76)

Since the two conductors are charged with electric charges of equal values but opposite in sign, that is qa = − qb , obtain: Vab =

qa ⎛ D r⎞ q D ⎜ ln − ln ⎟ = a ln D ⎠ πε r 2πε ⎝ r

(1.77,a)

If consider that conductor b is the image of conductor a then the potential of conductor a relative to the point of potential zero (or to ground), that is at the half distance between the two conductors (Fig. 1.38), is: Va =

Vab q D = a ln 2 2πε r

The capacitance between the two conductors is given by:

(1.77,b)


53

πε qa = Vab ln D r We can also express the capacitance to ground: Cab =

(1.78,a)

qa 2πε (1.78,b) = Fm Va ln D r The real situations consists of multiphase systems so that let us consider the case of n parallel conductors that carry the line charges q1 , q2 , …, qn , located above a perfectly conducting earth plane as shown in Figure 1.39. CaN = CbN =

conductor 2

M DM1 q1

q2 DM2

DMk

conductor k

DMn qn

qk conductor 1

conductor n

V=0 Fig. 1.39. n conductors system.

In this case the following assumptions are considered: – the distance between conductors is much greater than their radius; in consequence, the distribution of a charge on a conductor is not influenced by the presence of the charges from the other conductors; – charges are uniformly distributed on conductors; – the dielectric is assumed to be linear, so that superposition for fields and potentials can be applied; – the sum of instantaneous values of the n electric charges from the n conductors is zero: q1 + q 2 + ... + q k + ... + q n = 0 (1.79) because the regime studied is assumed to be normal steady state. The real overhead electric lines have the parallel conductors so that the cross sections are located in the same plane (two-dimensional space), therefore, we may define the potential of a point M relative to the n conductors system by formula:

54

Basic computation

VM =

1 n 1 qk ln DMk 2πε k =1

∑

(1.80)

The individual terms in expression (1.80) are referred to as logarithmic potentials and can be viewed as the individual contributions of each conductor charge to the total potential of point M [1.8, 1.18]. Notice that the theory of the logarithmic potential is valid in a plane-parallel space. The main application of (1.80) is to calculate the transmission line capacitance in terms of potentials and electric charges of conductors. For example, the voltage of a point on the surface of conductor 1 with respect to the whole system of charges is easily determined if consider that the point M is located on conductor 1:

V1 =

1 ⎛ 1 1 1 1 ⎞ ⎜⎜ q1 ln + q2 ln ⎟ + ... + qk ln + ... + qn ln r D12 D1k D1n ⎟⎠ 2πε ⎝

(1.81)

where: r is the conductors’ radius; D12 – distance from the considered point on conductor 1, to the centre of conductor 2, and so on. A similar expression as (1.81) can be written for any from the n conductors (Fig. 1.39). Therefore, for the general case, we obtain n expressions as in (1.81), which can be written as matrix form:

[V ] = [α]⋅ [q]

(1.82)

where: [V] is column vector of potentials with components V1, V2, ..., Vn; [q] – column vector of electric charges with components q1 , q2 , …, qn ; [α] – matrix of Maxwell’s potential coefficients having the terms:

α jj =

1 1 1 1 ln ln ; α jk = 2πε Dij 2πε r

(1.83)

The capacitances can be calculated by calculating the inverse of matrix [α]:

[q ] = [α ]−1 [V ] = [C ] ⋅ [V ] Notice that the matrix [C ] includes off-diagonal terms, that is, there are also mutual capacitances. Under symmetrical and balanced steady state conditions, the capacitance of one conductor in the presence of the others can be expressed as an equivalent capacitance. Capacitance of a single-circuit three-phase overhead electric line For better understanding of calculation of electric lines capacitance consider now the simple case of a single-circuit three-phase overhead electric line with transposed conductors (Fig. 1.34).


55

Consider the expression of the average potential on the entire length of a transposition cycle: 1 (1.84) Va , av = Va(1) + Va(2 ) + Va(3 ) 3

(

)

where Va(1) is the potential in a point situated on conductor a from the first transposition section. The conductors radii of the three phases are equal each other and equal to r. Va , av = + qa ln

1 ⎛ 1 1 1 ⎜⎜ qa ln + qb ln + qc ln + 3 ⋅ 2πε ⎝ r D12 D13

1 1 1 1 1 1 ⎞ ⎟ + qb ln + qc ln + qa ln + qb ln + qc ln r D23 D21 r D31 D32 ⎟⎠

or Va , av =

1 ⎛⎜ 1 1 1 qa ln + qb ln + qc ln 3 D D D 3 D D D 2πε ⎜⎝ r 12 23 31 13 21 32

⎞ ⎟ ⎟ ⎠

(1.85)

Under normal steady state conditions the per unit length electric charges qa , qb and qc of phases a, b and c satisfy the equality: qa + qb + qc = 0 , that is qa = −(qb + qc )

(1.86)

so we obtain: Va , av =

1 GMD qa ln 2πε r

(1.87)

where GMD = 3 D12 D23 D31 . Therefore, the per length unit average capacitance to ground is given by: Ca , av =

qa 2πε = Va , av ln GMD r

(1.88)

1 F/m, ε r = 1 , length l = 1000 m and using lg 4 π 9 ⋅ 10 9 instead of ln, expression (1.88) becomes:

Knowing that ε 0 =

Ca , av =

0.02415 ⋅ 10 −6 GMD lg r

[F/km]

(1.89)

In a similar manner the average capacitance for a single-circuit or doublecircuit electric line with bundle conductors can be determined. In all these cases the same mean distances and radii are used as for inductance calculation. The only one

56

Basic computation

difference is that the radius r of the conductor is replaced with an equivalent radius Rf . Application For the same example of the 750 kV single-circuit three-phase overhead electric line with bundled conductors let us calculate now the capacitive susceptance. The average capacitance of phase a is given by:

Ca , av = Cb, av = Cc , av =

0.02415 ⋅ 10 − 6 GMD lg Rf

(1.90)

where the equivalent radius Rf is: 1/ 5

⎛ 2.515 ⎞ R f = (rD12 D13 D14 D15 )1 5 = ⎜ ⋅ 40 2 ⋅ 64.72136 2 ⎟ ⎝ 2 ⎠

= 242.7398 mm

Hereby obtain: ba , av = bb, av = bc , av = ω ⋅ Ca , av =

314 ⋅ 0.02415 − 6 10 = 4.0765 ⋅ 10 − 6 S/km 17500 lg 242.7398

that is a value very closed to the recommended one for an 750 kV overhead electric line, of 4.12 ⋅10 −6 S/km, evaluated taking into account the inequality of the distances between bundle centres and using the geometric mean distances.

Effect of earth on the capacitance Consider also the case of single-circuit overhead electric line (Fig. 1.40). The effect of earth can be taken into account by using the method of electric charges images. These have a charge equal but opposite in sign and are symmetrically located below the surface of earth. By applying the phase transposition in sections 2 and 3 of the line, only the conductors change their positions, the distances remaining the same. By applying expression (1.81), the potential of the conductor phase a, throughout the section 1, is:

Va(1) = =

1 ⎛ 1 1 1 1 1 1 ⎞ ⎟= ⎜⎜ qa ln − qa ln + qb ln − qb ln + qc ln − qc ln 2πε ⎝ r H1 D12 H12' D13 H13' ⎟⎠ 1 2πε0

⎛ H ⎞ H H ⎜⎜ qa ln 1 + qb ln 12' + qc ln 13' ⎟⎟ r D12 D13 ⎠ ⎝

(1.91,a)

Likewise, the potentials of conductor phase a, throughout the transposition sections 2 and 3, can be obtained:

Va(2 ) =

1 2πε 0

⎛ H H H ⎜⎜ q a ln 2 + q b ln 23' + q c ln 21' r D23 D21 ⎝

⎞ ⎟⎟ ⎠

(1.91,b)


Va(3) =

1 2πε 0

⎛ H H H ⎜⎜ q a ln 3 + q b ln 31' + q c ln 32' D32 r D31 ⎝

qa

1

a

Cab

Cac

qb b Cb0

Ccb Ca0

c

qc

qb

3

qa

2

qc

3

3

b

qa

3 a

2

a

a

1 Section 1

qc

2

2

(1.91,c)

qb 3q

qb

a

H2 H1 H3

H2 H1 H 3 c

⎞ ⎟⎟ ⎠ 1

qc

qc

H12 H13

ground

qb b

2

1

qa

H2 H1 H3

Cc0

c

57

b

3 c

c

1 Section 2

a

2

b 1 Section 3

Fig. 1.40. Single-circuit three-phase line (with phase transposition) considering the influence of earth.

The average potential of phase a, taking into consideration the expressions (1.84) and (1.86), is given by: Va , av =

3 D D D 3 H H H qa 12 23 31 1 2 3 ln 3 2πε 0 r H12 ' H13' H 23'

(1.92)

From expression (1.92), the resultant capacitance of phase a can be expressed as: Ca =

2πε0 GMD 3 H1H 2 H 3 ln r 3 H12 ' H13' H 23'

If we define the geometric mean distances: H mean, s = 3 H1H 2 H 3 H mean, m = 3 H12' H13' H 23'

(1.93)

it results: Ca = or

2πε0 H mean, s GMD ln + ln r H mean, m

(1.94)

58

Basic computation

Ca =

2πε0 H mean, m GMD ln − ln r H mean, s

(1.95)

Notice that, by considering the influence of earth in expression (1.95), a bigger value for capacitance compared with (1.88) is obtained, since H mean, m > H mean, s . Likewise, we can determine the influence of earth on service capacitance, of a double-circuit line, of a line with bundled conductor, etc. For the determination of service capacitance of a three-phase underground line, the equipotential surface of conductors (plumbum or aluminium cover surrounding the three phases) is replaced with a system of charges qa ' , qb ' , qc ' (the images of charges qa , qb , qc with respect to a surface S), so that, in the resulted electric field of the real charges and of their images, the surface S to remain equipotential. The calculation goes on using the method presented earlier. The service capacitance of a three-phase overhead line is much smaller than the service capacitance of a three-phase underground line. If the average capacitance per unit length for a single-circuit overhead line is: C = (8 ÷ 9.5) μF/km then for a three-phase underground electric line this has a value of: C ≅ 23 μF/km

1.2.2.4. Conductance The conductance is the shunt parameter from the equivalent circuit of an electric line and it corresponds to shunt active power losses, due to imperfect insulation and corona discharge [1.1]. If note these losses by ΔPins , respectively by ΔPcor , and line nominal voltage by U n , the conductance G L of the line is determined with formula: GL =

ΔPins + ΔPcor U n2

[S]

(1.96)

a) Active power losses due to imperfect insulation In the fixing points of the conductor on the electric tower, current leakages through insulation towards ground occur, being more intensive as the atmospheric conditions are worst. Consider an insulators chain from an overhead electric line of nominal phaseto-phase voltage Un = 220 kV that can be replaced with an insulating resistance, under normal atmospheric conditions, of about 2.4·109 Ω/phase. Taking into consideration that such line is equipped along one kilometre with 3 support chains, it results that the insulating resistance is 0.8·109 Ω/phase, and the corresponding


59

conductance is GL=1.25 nS/km. In consequence, this conductance produces losses −9 ⎛

2

220 ⋅103 ⎞ ⎟ ≅ 20 W/km. = 1.25 ⋅10 ⎜⎜ per phase of 3 ⎟⎠ ⎝ During unfavourable atmospheric conditions (rain, moist), the values of losses increase 5-6 times, but remain negligible for calculations of operating regimes. In polluted areas, due to intensive dirt deposition on the line’s elements, the conductance increases very much, up to 20 ÷ 40 nS/km, but, taking into consideration that by designing insulators chain that do not favour dirt deposition are chosen, with self-cleaned glazed surface, or are periodically washed, in practice the value of ΔPins is negligible. b) Power losses due to corona discharge These losses must be taken into account from the designing stage of the line. Corona phenomenon is an incomplete and autonomous electric discharge and occurs at the conductor’s surface, as a luminous corona accompanied by a characteristic noise. This electric discharge appears when the electric field intensity among conductor’s surface exceeds the critical value Ecr=21.1 kVrms/cm. It must be mentioned that local increments of electric field may occur due to non-uniform surface of conductors caused by mechanical damages, dust deposition, spots of rain, wires spiralling or even by roughness of conductor’s surface. Corresponding to critical electric field intensity, the critical phase-to-phase voltage at which corona phenomenon occurs can be calculated with the following formula: GMD (1.97) U cr = 84 m1 m2 δ r n lg [ kV ] re where: m1 is coefficient that take into account the conductor’s surface state, being equal to 1 for smooth surface, 0.88 ÷ 0.99 for roughness surface and 0.72 ÷ 0.89 for stranded conductor; m2 – coefficient that takes into account the atmospheric conditions; is equal to 1 for nice weather and 0.8 for moisture and rainy weather; δ – air relative density; under standard conditions of temperature and pressure ( t = 25 ºC, p = 760 mmHg), δ =1; r – conductor radius; n – number of conductors from bundle; GMD – geometric mean distance between phases, [cm]; re – equivalent radius of the conductor, [cm]. ΔPins = GLU 2f

Expression (1.97) is valid when phases are equilateral spaced in the corners of a triangle. Whether the conductors are placed in the same plane, the critical voltage for the conductor from middle is with 4% less than, and for the conductors from exterior is with 6% greater than the value calculated with expression (1.97). In designing of overhead electric lines, corona discharge is verified for operating regimes at voltages above 60 kV. The standards indicate that on dry

60

Basic computation

weather, the condition for which there are no power losses by corona discharge is formulated as U n < U cr . The calculation of power losses due to corona discharge, by using experimental empiric formula is performed. For voltages above 110 kV and large diameters, Peek’s formula is the most used in this evaluation:

ΔPc = where: f is U, Ucr –

241 ( f + 25) re (U − U cr )2 ⋅10−5 [kW/km] δ GMD

(1.98)

the frequency of electric network, [Hz]; the phase-to-phase network voltage and critical voltage, [kV], respectively.

Peek’s formula provides good results only for overhead lines operating at voltages up to 110 kV and with not too large diameters of conductor. For voltages above 110 kV and large diameters, the Peterson’s formula is used: ΔPc = 14,7 ⋅10− 6 f F

U2 [kW/km] GMD ln re

(1.99)

where F is Peterson’s function and is dependent on the value of U/Ucr ratio. For 400 kV lines, the power losses due to corona discharge reach 5 ÷ 7 % from Joule’s losses, and for 750 kV lines, these are 4 times bigger compared with 400 kV lines. Corona phenomenon leads to: – increase in power and energy losses; – decreasing of life time of conductors, fittings, clamps, caused by the corrosion process, high frequency disturbances and slight hissing noises. The avoidance of corona discharge appearance needs increasing of critical voltage Ucr by: – increasing in conductors radii, leading to assembling and operating difficulties of the line; – using bundle conductors, obtaining on this approach the increasing of apparent surface of the sub-conductors group and the decreasing of the critical field intensity at the conductor’s surface; this is the most used method being the most widespread. For cables, the conductance appears due to the power losses by ionization phenomenon in the dielectric of cables, current leakages due to imperfect insulation or to power losses due to magnetic hysteresis loop. For power losses assessment in dielectric material the tangent of the angle of dielectric losses tan δ is used. For 110 kV and 220 kV cables the power losses in insulation increase up to 5 ÷ 10 kW/km. From the above presented issues, results that the conductance GL is a value that can be determined only through experimental approaches; it varies generally


61

along the line, caused by line state, meteorological conditions and the voltage variations as well. In practice, the value of the conductance is considered within the interval:

(

GL = 0.97 ⋅ 10 −8 ÷ 27 ⋅ 10 −8

)

S/km

1.2.2.5. Equivalent circuit of the electric lines So far, the calculation manner of electrical lines’ parameters on length unit has been shown, instead, the total impedance and admittance of the line is calculated in terms of length l and the number of circuits n operating in parallel: z=

1 (r0 + jx0 )l = R + jX n

(1.100)

y 0 = n(g 0 + jb0 )l = G0 + jB0

The electric lines are classified by voltage level, its length as well as the environment. In the modelling of electric line the most used is П four-terminal network, where the shunt admittance, which represents corona losses, leakage current and shunt capacitive currents, is split equal to both input and output ends of the equivalent circuit. X

R

i B 2

k B 2

G 2

G 2

ground

Fig. 1.41. Equivalent model of electric lines.

In literature, overhead lines of length less than 80 km are classified as short lines, for which the conductance and capacitive susceptance can be neglected without influencing the accuracy of operating regimes of power systems calculation. Thereby, the equivalent circuit of electric line is: i Vi

zik=Rik+jXik

k Vk

Fig. 1.42. Series equivalent line model.

For medium and long lines, due to high value of shunt capacitive currents or for the cases when corona losses become significant, the shunt admittance is no

62

Basic computation

longer neglected. Obtain thus the equivalent П circuit, which represents, with good accuracy, the electric line. zik

i Vi

k

yik0

yki0

2

2

Vk

Fig. 1.43. Equivalent π circuit.

A more detailed theory of long lines parameters determination is presented in Chapter 3.

1.2.3. Transformers modelling The existence of electric transformers and autotransformers in the electric networks makes possible obtaining different voltage levels. In actual electric power systems, the transmitted power can suffer 4-5 voltage and current transformations that make the rated power of all transformers from the system be 4-5 times bigger than the rated power of all generators. An important part of the transformers and autotransformers are manufactured with two or more three-phase windings disposed either on common magnetic cores constituting three-phase units or on magnetic cores individual to each phase, constituting single-phase units. Many transformers from the system are used for voltage and reactive power control and because of that one winding is tapped. In Figure 1.44, the simplified equivalent circuits of transformer and autotransformer are presented.

V1

N1

N2 a.

V2

V1

N1

N2

V2

b.

Fig. 1.44. Simplified equivalent circuits of transformers (a) and autotransformers (b).

Autotransformers are used when the transformer turns ratio is small. These have also a third winding of small rated power, delta-connected, constituting closing path for currents of the 3rd harmonic and multiple of 3, reducing on this way the flowing of these harmonics in the network. Often, the third winding of


63

autotransformers is used for connecting synchronous compensators for reactive power compensation. Nowadays electric power systems, high voltage and ultra high voltage loops often occur. The power flow control in these loops and preventing of electric lines overloading are performed by means of special transformers with complex turns ratio that modifies not only the voltage magnitude but also voltage phase angle. The transformers from electric power systems can be ordered on three categories [1.11]: – step-up transformers, by means of which the generators are connected to transmission network and transformers supplying auxiliary services; – coupling transformers which links different parts of the transmission network, usually with different voltage levels, or which links the transmission and distribution networks; – distribution step-down transformers which decrease the voltage level according to the desired consumer’s voltage level.

1.2.3.1. Mathematical model and equivalent circuits To study the two-winding three-phase transformers the model of a singlephase transformer can be considered. This approach is based on the fact that the magnetic core and the electric circuits are symmetrically manufactured, so that the study of three-phase transformers, under symmetrical regime of phases a, b and c can be performed by using the direct-sequence equivalent circuit of a single-phase transformer. Let us consider the magnetic circuit of a transformer with two windings disposed on a common magnetic circuit called magnetic core (Fig. 1.45). One part from the magnetic field lines are focused in the magnetic core, made from ferromagnetic material with magnetic permeability μ > μ0, constituting the utile magnetic flux, and one part of them are closing through air constituting leakage magnetic flux. By convention the winding that receive the energy from the network is called primary winding and the one that send is toward the network is called secondary winding. Φ

Ni Ii

Nk k

i Vi

Ik Vk

ΓEi

Γ Ek

Fig. 1.45. Two-winding transformer model.

64

Basic computation

By applying the law of electromagnetic induction along the magnetic circuits ΓEi and ΓEk respectively, crossing the paths of primary and secondary windings, obtain: dϕ E ds=− (1.101) dt Γ

∫

E

Applying (1.101) for the two paths corresponding to the two windings and taking as reference the direction of current I i , for instantaneous quantities the following system of equations can be written: d ϕi ⎧ ⎪ − vi + Ri ii = − d t ⎪ ⎨ ⎪ − v + R i = − d ϕk k k ⎪⎩ k dt

(1.102)

The magnetic fluxes ϕi , ϕk are the sum of the utile and leakage fluxes: ⎧⎪ϕ i = N i Φ + Li ,σ ii ⎨ ⎪⎩ϕ k = N k Φ + Lk ,σ i k

where: Ni, Nk are – Ri, Rk Li , σ , Lk , σ – Φ –

(1.103)

number of primary and secondary turns; resistances of primary and secondary windings; leakage inductances of primary and secondary windings; fascicular flux common to the two windings.

Considering the sinusoidal steady state and expressing (1.102) as phasor form, obtain:

( (

)

⎧⎪− V i + Ri I i = − jω N i φ + Li ,σ I i ⎨ ⎪⎩− V k + Rk I = − jω N k φ + Lk ,σ I k

)

(1.102')

The mathematical model of the electric transformer, under sinusoidal steady state conditions, is described by the phasor equations of the two electric circuits: ⎧− V i + z i I i = N i E ⎨ ⎩− V k + z k I k = N k E

(1.102'')

where E = − jωφ is e.m.f. (electromagnetic force) per turn of winding, and the impedances of the windings are: ⎧⎪ z i = Ri + jω Li ,σ ⎨ ⎪⎩ z k = Rk + jω Lk ,σ

(1.104)

Based on the system of equations (1.102"), we can draw a simplified model of the two-winding transformer (Fig. 1.46).


zk

zi

Ii Vi

65

Ni E

Ik Vk

Nk E

Fig. 1.46. Simplified transformer model.

On the hypothesis of z i = z k = 0 , from (1.102") the equations of the ideal transformer are obtained: ⎧− V i 0 = N i E ⎨ ⎩− V k 0 = N k E

(1.105)

Based on the equations (1.105) it can be defined the transformer turns ratio Nik, which is equal to the ratio between the number of turns of the two windings or to the ratio of the no-load voltages at the two terminals: N i V i0 = Nk V k0

N ik =

(1.106)

The transformer turns ratio defined by (1.106) is, in this case, real; normally, this is complex because there is a phase shift between secondary and primary voltages. In practice, the two-winding transformer is represented either as equivalent circuit with magnetic coupling (Fig. 1.47,a) or as equivalent circuit with transformer operator (Fig. 1.47,b). Ii Vi

zk

zi V i0

Vk 0

Ik Vk

Ii Vi

a.

Nik

zi

zk Vk 0

V i0

Ik Vk

b.

Fig. 1.47. Two-winding transformer equivalent circuit: a. circuit with magnetic coupling; b. circuit with transformer operator.

Further, if the Kirchhoff’s theorem for magnetic circuits is applied along the contour linking the magnetic circuit (Fig. 1.45):

∫ H ds = Θ

ΓM

66

Basic computation

we obtain the expression of intensity of the total current (ampere-turns): Θik = N i I i + N k I k

(1.107)

If assume that Θ ik remain constant, because the permeability of magnetic core is assumed infinite, for the load regime ( I k ≠ 0 ) as well as for no-load regime ( I k = 0 ), we can write Θ ik ≅ N i I i 0 and Θ ik ≅ N k I k 0 respectively. In the previous relationships I i 0 is the no-load current if the transformer is supplied at the winding i, and I k 0 is no-load current if the transformer is supplied at winding k. Because the no-load current can be negligible compared with the load current, we achieve: Ni I i ≅ −N k I k and the turns ratio get the expression

N ik =

I Ni ≅− k Nk Ii

(1.108)

N ki =

I Nk ≅− i Ni Ik

(1.108')

or

If the first equation from (1.102'') is divided to the second one, and taking into consideration (1.108) and (1.108'), the mathematical equations of the twowinding transformer becomes: ⎧ z ik I i − V i = − N ik V k ⎨ ⎩ z ki I k − V k = − N ki V i

(1.109)

⎧⎪ z ik = z i + N ik2 z k ⎨ ⎪⎩ z ki = z k + N ki2 z i

(1.110)

where:

Under these circumstances, the two-winding transformer can be modelled through an impedance series with an ideal transformer, for which two cases are defined: a. Equivalent circuit with transformer operator Nik and impedance z ik referred to winding i (impedance z ik is galvanically connected to node i) (Fig. 1.48,a); b. Equivalent circuit with transformer operator Nki and impedance z ki referred to winding k (impedance z ki is galvanically connected to node k) (Fig. 1.48,b);


i

Nik

z ik

Vi

V i0

k

67

i Vk

Nki

Vi

z ki Vk0

a.

k Vk

b.

Fig. 1.48. Two-winding transformer equivalent circuit, with transformer operator: a. step-up transformer; b. step-down transformer.

In consequence, the two-winding transformer is drawn as a branch characterised by two parameters: either impedance z ik and turns ratio Nik or impedance z ki and turns ratio Nki satisfying the system of equations (1.110). In literature, the turns ratio is also defined as a:1 where a = N ik , respectively 1:a 1 where = N ki . Conventionally, the transformer operator replaces the ideal a transformer (without power losses). If express z k from the second equation of (1.110) and substitute it in the first one, obtain:

(

)

z i + N ik2 z ki − N ki2 z i = z ik Knowing that N ik ⋅ N ki = 1 we can obtain the relationship between impedances and admittances referred to the two windings: ⎧⎪ z ik = N ik2 z ki ⎨ ⎪⎩ z ki = N ki2 z ik

(1.111,a)

⎧ y = N ki2 y ⎪ ik ki ⎨ 2 ⎪⎩ y ki = N ik y ik

(1.111,b)

and

where z ik = 1 y ik and z ki = 1 y ki . If the admittance y i 0 = I i 0 V i 0 , noted with a supplementary subscript 0, which represents the no-load power losses, is connected on the primary winding side of the ideal transformer, then the turns ratio becomes (Fig. 1.49): N ik =

Ik Ni ≅− Nk I i − I i0

(1.112)

68

Basic computation

zi

i

i

zk

i0

k

i0

Vi

Vk

y i0

Fig. 1.49. The modelling of no-load power losses in the two-winding transformer.

Further, if the admittance y i 0 is moved from the ideal transformer terminals at the real transformer terminals, the product z i I i 0 can be negligible as compared with the product z i I i . Keeping the no-load admittance y i 0 connected at i − 0 terminals, then we obtain the equivalent Γ circuit of two-winding single-phase transformer (Fig. 1.50).

Vi

Nik

zik

i yi0

Nki

i

k Vi

Vk

z ki

y i0

a.

k Vk

b.

Fig. 1.50. Equivalent Γ circuit of two-winding transformer, with transformer operator: a. step-up transformer; b. step-down transformer.

Into large electric power systems, into transformers from connecting substation, during different operating regimes, the power flow can change its direction. In this case, for more accurate assessment of power losses, the equivalent Π circuit of transformer is used, where the admittance modelling the power losses is located at the input and output terminals (Fig. 1.51). The two shunt admittances have different values because the admittance y ik 0 will be referred to winding i while the admittance y ki 0 will be referred to winding k. z ik

i Vi

yik 0 2

N ik yki 0 2

k Vk

Fig. 1.51. Equivalent Π circuit of two-winding transformer with transformer operator.

Autotransformers are usually installed into electric network loops where the direction of power flow can be changed. Three-phase transformers and


69

autotransformers can be manufactured so that to provide both voltage magnitude and phase angle regulation. Phase-shift transformers provide a phase angle shift of secondary vectors V k , I k related to the primary ones V i , I i (Fig. 1.52). Under these conditions, a transformer has complex turns ratio N ik and provides phasor shift of angles determined by the connection class: N ik = N ik e jΩ ik = N ik (cos Ω ik + j sin Ω ik )

(1.113)

where Nik is absolute value of turns ratio: N ik =

Vi ,n

(1.114)

Vk ,n

Ωik – transformer turns ratio angle, [radians]. wmax

wc

Ωik wn

Vregulated

ΔV

wmin

V

Fig. 1.52. Regulating voltage diagram of phase-shift transformer.

Phase shift transformers (known also as phase shifters), having a particular connection of windings, are installed into electric network loops in order to change the active and reactive powers flows. Two cases are defined [1.12]: a) Step-up transformer, with the secondary winding k tapped, having the possibility to modify the number of turns (regulated winding), the regulated voltage and the turns ratio values being calculated with formula: V k , regulated = Vk + (wa − wn )Vk

N ik =

ΔV (cos Ω ik + j sin Ω ik ) 100

Vi ,n V k , regulated

(1.115,a) (1.116,a)

b) Step-down transformer, with the primary winding i tapped, having the possibility to modify the number of turns (regulated winding), the regulated voltage and the turns ratio values being calculated with formula:

70

Basic computation

V i , regulated = Vi + (wa − wn )Vi

N ik = where: Vi, Vk

wa wn ΔV

ΔV (cos Ω ik + j sin Ω ik ) 100

V i , regulated

(1.115,b) (1.116,b)

Vk ,n

are rated voltages of transformer windings connected at nodes i and k; – actual tap label; – median tap label; – percentage voltage step size.

• If Ω ik = 0 , transformer provides only voltage control; • If Ωik = π 2 , transformer provides only shift in phase angle control and thus active power redispatching; • If 0 < Ωik < π 2 , transformer provides both active power and voltage control. To be mentioned that, usually, the winding with voltage control possibilities is that of higher voltage because this is more accessible, and the current is lower. For a better understanding of drawing manner of the two-winding transformer equivalent circuit, let us consider the equivalent Π circuit case with impedance z ik , referred to i node, and complex turns ratio N ik , being the case of step-up transformer, then we can design the generalised equivalent circuit (Fig. 1.53). Ii Vi

m z ik

i

N ik

(1-m) Nik2 z ik

k

yik 0

y ki 0

2

2

Ik Vk

Fig. 1.53. Generalised equivalent circuit with transformer operator.

The generalised model of the transformer with transformer operator consists of an ideal transformer with complex turns ratio N ik , in series with an impedance or admittance. The series impedance consists of two terms: m z ik , referred to

winding i, and (1 − m ) N ik2 z ik which is proportional with the impedance z ik , referred to winding i. It can be seen that by using the generalised equivalent circuit we can achieve the transformer equivalent circuits corresponding to the two cases: m = 1 for stepup transformer and m = 0 for step-down transformer. For easier implementation of two-winding transformer mathematical model into professional software for load flow calculation, galvanic equivalent circuit can


71

be used. In this respect, we consider the equivalent circuits of two-winding transformer, with real turns ratio Nik (Fig. 1.48,a,b) and equations (1.109). If the current I i is expressed from the first equation of (1.109) to which we add and subtract the term y ik N ik V i , after rearranging the terms we obtain:

I i = y ik (1 − N ik )V i + y ik N ik (V i − V k ) ≡ I i 0 + I ik

(1.117)

In a similar manner, if from the second equation of (1.109) we express the current I k to which we add and subtract the term y ik N ik V k , then we obtain:

I k = y ik N ik (N ik − 1)V k + y ik N ik (V k − V i ) ≡ I k 0 + I ki

(1.118)

where the equality y ik N ik = y ki N ki has been used. Following the equations (1.117) and (1.118), the galvanic equivalent circuit of two-winding transformer is achieved (Fig. 1.54). yik Nik

Ii i Vi

yik (1-Nik )

yik Nik (Nik -1)

k Ik Vk

Fig. 1.54. Galvanic equivalent circuit of two-winding transformer.

1.2.3.2. Transformer parameters Two-winding transformer Let us consider the equivalent circuit with transformer operator of the transformer with parameters referred to winding i, for which we consider the two operating regimes: no-load and short-circuit (Fig. 1.50,a). Transformer parameters, series impedance z ik = Rik + jX ik and shunt

admittance

y i 0 = Gi 0 − jBi 0

respectively, are calculated in terms of its

manufacturing parameters. In general, in catalogues the following specific parameters of the transformer are given: Sn is rated power of transformer, respectively autotransformer; Ui,n – rated phase-to-phase voltage of winding i; Uk,n – rated phase-to-phase voltage of winding k; nom ΔPsc – active power losses under short-circuit test; usc [%] – percentage voltage under short-circuit test; ΔP0 – active power losses under no-load test;

72

Basic computation

i0 [%] Δu p

– –

percentage current under no-load test; percentage voltage variation on tap;

wn

–

median tap label.

In order to calculate the equivalent resistance Rik of the transformer consider the short-circuit test, that is k − 0 winding is short-circuited and the transformer is supplied at i − 0 terminals, so that the current from winding i is equal to the rated current I i ,n . Based on the above-mentioned hypothesis, we obtain the expression of active power losses:

ΔPscnom = 3 Rik I i2, n Taking into consideration the relationship:

Ii, n =

Sn Sn = 3Vi , n 3 U i, n

it results that the transformer resistance is calculated with formula: Rik = ΔPscnom

U i2, n S n2

[Ω]

(1.119)

In order to calculate the equivalent reactance X ik of the transformer, we depart from the short-circuit voltage value: Vsc =

usc [%] Vi , n 100

and taking into consideration that: Vsc = zik I i , n

it results: 2

zik =

u [%] U i , n usc [%] U i , n 1 = sc [Ω] 100 100 S n 3 Ii, n

(1.120)

Knowing the two terms Rik and zik we can obtain the equivalent reactance of transformer with expression:

X ik = zik2 − Rik2 ≅ zik

[ Ω]

(1.121)

Observation: For transformers of large rated power zik >> Rik , so the reactance X ik is identified by impedance zik . In the case of autotransformers with tapped windings, in the calculation of the short-circuit percentage voltage, the tap position is considered:


73

u sc = A(wn − wa ) + B(wn − wa ) + C 2

where A, B, C are constants given into autotransformer’s catalogues. Concerning the equivalent conductance Gi of the transformer, consider the no-load test characterised by the fact that k – 0 winding operates under no-load conditions and at i – 0 terminals the voltage Vi , n = U i , n 3 is applied. For this

regime, the three-phase active power losses are given by: 2

ΔP0 =

3 Gi 0 Vi ,2n

⎛ U i, n ⎞ ⎟⎟ = Gi 0 U i2, n = 3 Gi 0 ⎜⎜ 3 ⎝ ⎠

from where it results the conductance of the two-winding transformer:

Gi 0 =

ΔP0 U i2, n

[S]

(1.122)

In practice, the equivalent inductive susceptance Bi0 is calculated departing from the transformer magnetising losses, therefore:

yi 0 =

I0 i [%] i [%] 3 U i , n I i , n i [%] S n 1 [S] = 0 Ii, n = 0 = U i, n Vi , n 100 100 100 U i2, n U i2, n 3

(1.123)

and the susceptance is determined with formula: Bi 0 = yi20 − Gi20 ≅ yi 0 [S]

(1.124)

Three-winding transformer In the catalogues of these transformers the following characteristics are given: rated powers of the three windings SnI, SnII, SnIII, power losses under no-load nom conditions ΔP0, rated power losses under short-circuit test ΔPscnom I − II , ΔPsc II − III ,

ΔPscnom I − III , percentage short-circuit voltages u sc I − II [%] , u sc II − III [%] , u sc I − III [%] . In calculating the equivalent resistances of transformer windings, the rated powers of the three windings must be taken into account. Hereby, three types of transformers are defined [1.1]: Type a. Case S n I = S n II = S n III At this type of transformer, the winding resistances are referred to the same voltage level and are calculated departing from the expression of power losses under short-circuit test ΔPscnom , that are maximum at the rated loading of windings I and II, winding III being no-loaded:

74

Basic computation

ΔPscnom = 3 R I I I2 + 3 R II′ I II′ 2

where: R I , R II′ are resistances of windings I and II referred to the same voltage level; I I , I II′ – rated currents of windings I and II referred to the same voltage level. Since the rated powers of the two windings (I and II) are equal, the secondary current I II′ referred to the primary winding is equal to the primary current. Therefore, the expression of the rated power losses under short-circuit conditions becomes: ΔPscnom = 2 ⋅ 3 RT I n2

We can obtain, thus, the expression of the resistance RT =

ΔPscnom = ΔPscnom 6I2 2

(

U n2 3Un In

= ΔPscnom

)

2

U n2 2 S n2

(1.125)

2 Sn I 3 At this type of transformer, the rated power losses under short-circuit test nom ΔPsc are maximum when the transformer is full loaded on the windings I and II, while the winding III operates under no-load conditions. The winding resistances get the expression:

Type b. Case S n II = S n I , S n III =

R I = R II = RT = ΔPscnom

U n2 2 S n2

(1.126)

and R III =

2 RI 3

Type c. This case is defined by two situations: S n II =

2 2 S n I , S n III = S n I 3 3

S n II =

2 1 S n I , S n III = S n I 3 3

or

Similar to the previous cases, ΔPscnom is calculated as it follows:

(1.127)


75

′ 2 R III ′ = ΔPscnom = 3 I I2 R I + 3 I II′ 2 R II′ + 3 I III 2

2

= 3 I I2 =

3 I I2 R I

⎛I ⎞ ⎛3 ⎞ ⎛2 ⎞ ⎛3 ⎞ RI + 3 ⎜ I I ⎟ ⎜ RI ⎟ + 3 ⎜ I ⎟ ⎜ RI ⎟ = ⎝3 ⎠ ⎝2 ⎠ ⎝ 3 ⎠ ⎝2 ⎠

I I2 3 4 23 1 ⎞ 11 ⎛ R I = I I2 R I ⎜ 3 + 2 + ⎟ = I I2 R I + 3 I I RI + 3 9 2 9 2 2⎠ 2 ⎝

The resistance of the primary winding RI is given by: 2 ΔPscnom 2 RI = = ΔPscnom 11 I I2 11

(

(

3Un

)

2

3Un In

)

2

=

6 U2 ΔPscnom n2 11 Sn

(1.128)

and ′ = R II′ = R III

2 RI 3

(1.129)

It should be mentioned that in the case of three-winding transformers, RI, RII, RIII define the winding resistances (primary, secondary and tertiary windings), referred to the same voltage level, different by the case of two-winding transformers were R define the total resistance of the two windings per one phase referred to the same voltage level. In calculation of inductive reactance, in catalogues are given the percentage short-circuit voltages between two terminals being determined as follows: shortcircuit voltage between the terminals I and III ( u sc I − III ) is obtained by supplying the primary winding, the tertiary one being short-circuited, and the secondary one operating under no-load conditions. Similarly, short-circuit voltages u sc I − II and u sc II − III are determined. By analogy with the two-winding transformers, the

inductive reactances of the transformer windings can be expressed as: X I − II = X I − III = X II − III =

u sc I − II [%] U n2 100

Sn

usc I − III [%] U n2 100

Sn

(1.130)

u sc II − III [%] U n2 100

Sn

where: Un is voltage level at which the transformer parameters are referred; Sn – rated apparent power of the winding with the greatest value. Knowing that:

X I − II = X I − X II ; X I − III = X I − X III and X II − III = X II − X III

76

Basic computation

it results:

X I − II + X I − III − X II − III 2 X + X I − II − X I − III X II = II − III (1.131) 2 X + X II − III − X I − II X III = I − III 2 The conductance and the susceptance of these types of transformers are calculated in the same manner as for two-winding transformers. XI =

1.2.4. Electric generators modelling The electric generators are synchronous machines that represent the main source of energy from the electric power plants. These can be divided into two categories by the design model: hydro-generators, with isotropic rotor on the directions of d and q axes, and turbo-generators, with anisotropic rotor on the directions of the two axes. To simplify, we next consider a turbo-generator where the synchronous reactances along the direct and quadrature axes are equal, X S = X d = X q . The synchronous reactance of the generator is calculated with formula: XS =

where: x[%] is Un – Sng –

x[%] U n2 100 S ng

[Ω ]

(1.132)

percentage synchronous reactance; phase-to-phase rated stator voltage, [kV]; rated apparent power of generator, [MVA].

The synchronous generator is represented, in the direct-sequence circuit, through an impedance, where the armature resistance is neglected, in series with an electromotive force (Fig. 1.55,a). By optimum operating reasons in the system, the generator is represented, under steady state conditions, through constant active power, P = ct. , and constant terminal voltage, U = ct. (Fig. 1.55,b). jXS

jXS

P

I E

U

a.

U

E

b.

Fig. 1.55. Direct-sequence circuit representation of electric generator: a. XS = ct., E = ct.; b. XS = ct., P = ct. and E = ct. or U = ct.


77

The operating equation under normal steady state is: E = U + jX S I

(1.133)

Based on this equation, the phasor diagram is drawn, where the terminal voltage of the generator is taken as reference (Fig. 1.56) [1.12]. P E

N C (I=ct) I jXS I

O’

ϕ

δ

δ O’

I

Pn

UnIn

CE (E=ct)

U

N

A

UnEn XS

Un2 XS

a.

O

O” Qn

b.

Fig. 1.56. Phasor diagram of the synchronous generator under normal steady state.

To study the generator, the inductive operating regime is analysed, when voltage lags the current by ϕ degrees. Under the hypothesis of constant terminal voltage, a semicircle CE of centre O' and radius O'N is drawn, which represents the geometric locus of the operating points with constant electromotive force. Likewise, the circle CI of centre O and radius ON, which represents the geometric locus of the operating points with constant stator current I = I or with constant apparent power, is drawn. Considering the rated operating regime, the powers diagram is obtained by multiplying each of phasors, U n , X S I n and En by U n X S , where the phasor ON becomes equal to the rated apparent power Sn. The intersection point of the two circles represents the rated operating point, for which both the internal voltage E and the armature current I are maximum. In order to draw the powers diagram, the Cartesian coordinates system of axes P and Q of centre O, where the imaginary axis is overlapped on the phasor U = U , has been chosen. The projections of Sn on the horizontal and vertical axes represent the rated reactive power Qn , and the rated active power Pn , having the expressions (Fig. 1.56,b): EnU n sin δ n XS

(1.134)

EnU n U2 cos δ n − n XS XS

(1.135)

Pn = U n I n cos ϕn =

Qn = U n I n sin ϕ n =

The main electric quantities that characterise a synchronous generator are rated (active or apparent) powers, rated voltage and rated power factor. When the

78

Basic computation

generator operates under a given regime different from the rated one, the previous enumerated quantities are large scaled, and comprised in a domain constrained by the loading limits (Fig. 1.57) referred to as loading capability curve of the synchronous generator. This is important to the power plant operators who are responsible for proper loading operation of the generator [1.11, 1.13]. Mechanical limit of turbine (L3) P Field current (L2) Pmax

Underexcitation limit (L4)

N

Minimum active power limit

Armature current (L1)

ϕn

(L5)

Pmin O

Q O

Qmin Leading

Qmax Lagging

Fig. 1.57. Loading capability curve of a synchronous generator.

Taking into consideration the complexity of the processes from inside the synchronous machine, in order to draw the loading capability curve of synchronous generator, the following hypothesis are considered [1.19]: • Armature resistance R=0 is neglected; • The magnetising characteristic is assumed linear E0 = f (I ex ) ; • Power losses by Joule’s effect in the armature windings as well as the power losses in the armature core are neglected; • Synchronous reactance is constant X S = ct. Starting from these hypotheses, the following operating limits of the synchronous generator are defined. a) Armature current limit (L1), Is,max, imposed by the heating limit of stator windings. This limit is a circle of centre O and of radius UnIn that represents the geometric locus of the operating points given by the expression: S n2 = Pn2 + Qn2 =

(

3UnIn

)

2

(1.136)

Taking into account that the apparent power S must not exceed the rated value, that is S ≤ S n , the operating point must be situated inside or on the limit circle L1.


79

For a value of armature current greater than the limit value, the generator can operate under secure conditions for a short period of time depending on the measure of how much the limit value is exceeded. b) Field current limit (L2), Ir,max. Providers of electric equipments specify the maximum value of the excitation current Iex, imposed by the heating limit of rotor windings. Likewise, there is also a limit value of the excitation voltage equal to the rated one. Also, by secure operating reasons at motor torque shocks a minimum value of the excitation current is imposed. The limit curve of field current is a circle of centre O' and of radius proportional to the rated internal voltage En. As it can be seen in Figure 1.57, for an active power less than the rated power Pn, field current limit is more restrictive than the armature current limit. The rated operating point of the generator is the intersecting point of the two limits L1 and L2 when the generator is used at maximum from the generated apparent power point of view. c) Mechanical limit of turbine (L3), Pmax, imposed by the maximum shaft torque of turbine. Taking into consideration that, in general, the mechanical power of turbine is greater than the electric power of generator, this limit is a horizontal line drawn at an active power value greater than the rated power output Pn of the generator. Under leading regime, the operating domain of the synchronous generator is constrained by another three limits: – core end heating limit which is a curve determined through experimental tests; – static stability reserve chosen so that a certain value of the internal angle δ is maintained; – minimum excitation current limit that ensure a motor torque reserve to the generator. The generator operating point near to the three limits previous defined can be avoided by using the underexcitation limiter so that the 4th limit can be defined: d) Underexcitation limit (L4). By means of automatic control systems of the generator, the operating at leading power factor is constrained by the characteristic shown in Figure 1.57. e) Minimum active power limit (L5), Pmin. In thermal power plants a minimal power, Pmin, is required by combustion reasons. If the operating point is different from the rated one, under lagging regime, due to the limits L2 and L3, the maximum reactive power is determined with formula:

Q = Qmax

⎡⎛ E U = ⎢⎜⎜ n n ⎢⎝ X s ⎣

2 ⎤ ⎞ ⎟ − P2 ⎥ ⎟ ⎥ ⎠ ⎦

1/ 2

−

U n2 when P ≤ Pn Xs

(1.137)

80

Basic computation

[

2

Q = Qmax = S n − Pn

]

2 12

when P ≥ Pn

(1.138)

If information about En and XS are not available, an approximate limit of the maximum reactive power is calculated: Qmax = 0.9 Qn where Qn = S n sin ϕ n .

Appendix Table A1 Average values of the per kilometre parameters of the overhead electric lines fn [Hz] 50 (Romania) 60 (USA)

Un [kV] 220 400 750 230 345 500 765 1100

r0 [Ω/km] 0.070 0.034 0.017 0.050 0.037 0.028 0.012 0.005

x0 [Ω/km] 0.421 0.328 0.275 0.488 0.367 0.325 0.329 0.292

b0 [μS/km] 2.920 3.611 4.082 3.371 4.518 5.200 4.978 5.544

ZC [Ω] 380 300 260 380 285 250 257 230

PN [MW] 127 535 2160 140 420 1000 2280 5260

Note: the quantities ZC and PN are explained in Chapter 3.

Chapter references [1.1] [1.2] [1.3] [1.4] [1.5] [1.6] [1.7] [1.8] [1.9] [1.10] [1.11]

Poeată, A., Arie, A., Crişan, O., Eremia, M., Alexandrescu, A., Buta, A. − Transportul şi distribuţia energiei electrice (Electric energy transmission and distribution), Editura Didactică şi Pedagogică, Bucureşti, 1981. Crişan, O. − Sisteme electroenergetice (Electric power systems), Editura Didactică şi Pedagogică, Bucureşti, 1979. Ionescu, T.G., Pop, O. – Ingineria sistemelor de distribuţie a energiei (Energy distribution systems engineering), Editura Tehnică, Bucureşti, 1998. Meslier, F., Persoz, H. − Réseaux de transport et d’interconnexion, D070, Techniques de l’Ingénieur, Traité de Génie électrique, EdF, Paris, 1992. Carrive, P. − Réseaux de distribution. Structure et planification, D4210, Techniques de l’Ingénieur, Traité de Génie électrique, EdF, Paris, 1992. Gros, M., Righezza, P. − Réseaux de distribution. Exploitation, D4230, Techniques de l’Ingénieur, Traité de Génie électrique, EdF, Paris, 1992. Bergen, A.R. – Power Systems Analysis, Prentice Hall, Inc. Englewood Cliffs, New Jersey, 1986. Elgerd, O.I. – Electric energy systems theory: An introduction, McGraw-Hill, 1971. Bercovici, M., Arie, A.A., Poeată, A. – Reţele electrice. Calculul electric (Electric networks. Electric Calculation), Editura Tehnică, Bucureşti, 1974. Grainger, J.T., Stevenson, W.D. – Power Systems Analysis, McGraw-Hill, 1994. Mackowski, J., Bialek, J.W., Bumby, J.R. – Power Systems Dynamics and Stability, John Wiley and Sons, Chichester, New York, 1997.

Electric power systems configuration and parameters [1.12] [1.13] [1.14] [1.15] [1.16] [1.17] [1.18] [1.19]

81

Potolea, E. – Regimurile de funcţionare a sistemelor electrice (Operating regimes of electric power systems), Editura Tehnică, Bucureşti, 1977. Adibi, M.M., Milanicz, D.P. – Reactive Capability limitation of synchronous machine, IEEE Trans. on Power Systems, Vol. 9, No.1, February 1994. El-Hawary, M. – Electrical power systems. Design and analysis (Revised printing), IEEE Press, New York, 1995. Persoz, H., Santucci, G., Lemoine, J.C., Sapet, P. – La planification des réseaux électriques, Edition Eyrolles, 1984. Morgan, V.T., Findlay, F.D. – The effect of frequency on the resistance and internal inductance of bare ACSR conductors, IEEE Trans. on Power Delivery, Vol. 9, No. 3, pp. l391–l396, July 1991. Morgan, V.T., Zhang, B., Findlay, R.D. – Effect of magnetic induction in a steelcored conductor on current distribution, resistance and power loss, IEEE Trans. on Power Delivery, Vol. 12, pp. 1299–1308, July 1997. Mocanu, C.I. – Teoria cîmpului electromagnetic (The theory of electromagnetic field), Editura Didactică şi Pedagogică, Bucureşti, 1984. Ghiţă, C. – Maşini şi acţionări electrice (Electric machines and operation), Volume I, Institutul Politehnic din Bucureşti, Bucureşti, 1992.

Chapter

2

RADIAL AND MESHED NETWORKS

2.1. General considerations In order to ensure a proper operation of the load, a certain level of power quality is required, respectively the continuity in supplying, keeping the frequency and voltage near the nominal values and a waveform of the voltage as sinusoidal as possible as well. One of the restrictive conditions for the networks operation is the magnitude voltage deviation with respect to the reference voltage, called nominal voltage. Of importance is how much the voltage in a point of the network is deviated from the nominal value, and also the voltage drop between two nodes, galvanically connected to an electric network. In this respect, there are two notions used: phasor voltage drop and algebraic voltage drop. The phasor voltage drop refers to the phasor difference of two voltages, from two different nodes of the network. The algebraic voltage drop refers to the algebraic difference between the rms voltages into two nodes of the network, of the same nominal voltage. For simplicity, in this paperwork, the algebraic voltage drop is further called voltage drop and will not be underlined. In terms of the type and the importance of the load, the admissible deviations of the voltage in a node of the network are given in standards. These deviations must not be exceeded during exploitation, because they would lead to an unsuitable operation of the load. The electric networks operation is strongly influenced by the loads behaviour to different changes. The load modelling through static characteristics presents importance for network analysis. In this respect, some simplifying hypotheses are used [2.1]: a. Constant impedance (the values of the impedance will be constant in time and independent of the currents passed through them or the terminal voltage). The active and reactive powers absorbed by these loads are proportional with the square of the terminal voltage; b. Constant active and reactive powers (these are independent of the terminal voltage and current passing through the load); c. Constant active and reactive currents.

84

Basic computation

It should be mentioned that these hypotheses of load modelling through static characteristics are ideal conditions. In practical cases, the network loads are complex, including electric engines, arc furnaces, rectifiers, illumination, etc., which leads to a non-linear load characteristic. Generally, the network calculation under constant impedance hypothesis leads to more optimistic results than the real ones. Instead, the hypothesis of constant powers leads to more pessimistic results than the real ones. For short electric lines, the third hypothesis of the load modelling through constant current leads to results closer to reality. The modelling of the power sources – generators, in the calculation of the normal operation regime, by using one of the following simplifying hypotheses can be performed: − Constant electromotive voltage characteristic; − Constant current characteristic. In this case, the network must have a specific node – equilibrium node – where the currents generated or absorbed into different nodes of the network are closing; − Furthermore, in one of the network’s nodes, must be fixed a voltage related to the neutral conductor. This node, called reference node, can coincide or not with the equilibrium node; − Constant active power and constant voltage magnitude characteristic. In this case, in one of the network nodes must be applied a voltage source, constant as magnitude and phase angle, which is considered reference voltage. If in this node the active and reactive powers generated by the source are left to vary freely, this node coincides with the equilibrium node; − Constant reactive power characteristic. If the constant voltage or current characteristics for the power sources as well as for the loads are used, then systems of linear equations for the steady state calculation are obtained. This type of modelling does not express the real situation, where the generators from the power system operates according to a characteristic

Pg = ct. and V g = ct . For the other hypotheses, closer to reality for large power systems, systems of non-linear equations (of second degree) result. For short electric networks the linear hypothesis that leads to results closer to reality are used. The electric lines can be classified, in terms of their length, into short lines respectively long lines (generally longer than 250 km). The long electric lines operate at 220 kV, 400 kV or 750 kV and serve for the transmission of the electric energy. The short electric lines usually operate at voltages below 110 kV being used for the repartition and distribution of electric energy. Consider a three-phase electric line that satisfies the conditions of homogeneity and symmetry as well as symmetric voltages and balanced currents on all the three phases. Under these conditions it is sufficient to study the operation of a single phase, with a double-wire circuit, where the going conductor represents

Radial and meshed networks

85

the conductor of the phase, with the service parameters, and the return conductor is a fictitious neutral conductor, which ensures the closing of the current. For the short overhead lines, powered at low nominal voltages, the intensities of the shunt currents − the capacitive currents as well as the leakage ones − have low values as compared with those of the conduction current that passes through the phase conductor. Therefore, in the case of short lines, the shunt currents can be neglected and the corresponding equivalent circuit is a dipole with lumped parameters (Fig. 2.1,a), where the shunt admittances have been neglected. For more accurate results, the equivalent π (or T) circuit, with lumped parameters, is used (Fig. 2.1,b). Phase conductor

Phase conductor

Z Y 2

Neutral conductor

a.

Z Y 2

Neutral conductor

b.

Fig. 2.1. Equivalent circuits for short lines.

For the underground electric lines, powered at high nominal voltages, even for small length cases, the leading leakage currents should be taken into account so a proper circuit, either of lumped or uniformly distributed parameters, is chosen.

2.2. Radial and simple meshed electric networks 2.2.1. Current flows and voltage drops calculation under symmetric regime Assume a radial electric network operating at alternating voltage, supplying only one load, represented through a dipole with the impedance Z = R + jX (Fig.2.2,a). Being given the current at the receiving end i B , of i B = ct. and source voltage V A = ct. characteristic, it is required to determine the current I A at the sending end and the voltage at the receiving end V B , which can be kept within admissible limits only if the voltage drop does not exceed the recommended values. In Figure 2.2,b the fundamental phasor diagram of the voltage drop is plotted.

86

Basic computation +j

IA A

I

θ

B

Z=R+jX VA

IB=iB

0

VB

ϕ

Ia

B ∆VA

VB

A R I

-jIr iB=IB=I

C

δVAB

jX I

VA

ϕ B

D

E

∆VAB DVAB

a.

b.

Fig. 2.2. The radial electric network supplying one load: a. equivalent circuit; b. fundamental phasor diagram of the voltage drops.

The voltage V B is chosen as phase reference, and the current i B = I B = I (lagging load) lags behind the voltage with an angle ϕ B = ϕ . Due to the current passing through the line, an active phase-to-neutral voltage drop R I in phase with I occurs, and an inductive voltage drop jX I , which leads the current by 90° as well. The sum of these two phase-to-neutral voltage drops is the phasor voltage drop represented in the diagram by the segment AC , which represents the phasor difference between the voltage at the sending and at the receiving end of the line, that is: ∆V AB = V A − V B = Z I (2.1)

Its projections on the two axes correspond to the segments AD = ∆VAB and CD = δVAB , and represent the longitudinal and the transversal components of the voltage drop, having the following expressions: ∆VAB = RI cos ϕ + XI sin ϕ = RI a + XI r

(2.2)

δV = XI cos ϕ − RI sin ϕ = XI a − RI r

(2.3)

where: I a = I cos ϕ is active component of the current passing through the line; I r = I cos ϕ − reactive component of the current passing through the line; R − ohm resistance of the line; X − inductive reactance of the line. Consider the circle sector of radius equal to the supply voltage VA , which intersects the horizontal axis in the point E. The algebraic difference between the voltages magnitudes (or the effective values)


87

DVAB = VA − VB

(2.4)

is called voltage drop (phase-to-neutral). For lower values of the θ phase angle between the two voltages, the transversal component of the phasor voltage drop can be neglected, and the longitudinal component is identified with the voltage drop: DVAB ≅ ∆V AB

If the θ phase angle has great values, the voltage drop can be determined directly, with the expression: DVAB = VA − VB =

(VB + ∆VAB )2 + (δVAB )2 − VB

Since δVAB 1 and S (S ) − S (S ) ≤ ε then go

to the next step, else update p = p + 1 and go to step 3. 7. Calculation of power losses through the network branches. In literature, there are also others calculation algorithms for the unknown state quantities by means of the backward/forward sweep. The principle of one of these algorithms consists in the use of a recursive set of equations to calculate all the unknown state quantities (nodal voltages and power flows) after processing the forward and backward sweeps [2.13]. In order to test this algorithm, consider the radial network with n loads in Figure 2.9, with the line sections represented by series impedances z k = rk + jxk , and the loads by complex constant powers s k = pk + jqk . The powers Pn and Qn represents the components of the complex power flowing through a fictive branch outgoing from the node n. The voltage V A of the source node of the network is constant. A VA=

PA+jQA

1

...

k-1

z1

UA 3

Pk-1+jQk-1

k

Pk+jQk

zk s1

k+1

Pk+1+jQk+1

n

Pn+jQn

zk+1

sk-1

sk

sk+1

sn

a.

Pgk+jQgk A VA=

UA 3

PA+jQA

1

...

k-1

z1

Pk-1+jQk-1 zk

s1

Pk+jQk

k

sk-1

k+1

Pk+1+jQk+1

n

Pn+jQn

zk+1 sk

sk+1

sn

b. Fig. 2.9. Distribution electric network: a. simple radial network b. radial network with one distributed generator.

In the forward sweep, the state quantities Pk , Qk and Vk of the node k are used to calculate the state quantities at the node k + 1 using the set of equations: ⎧ Pk2 + Qk2 Pk2 + Qk2 p Q Q x − = − − qk +1 ; ⎪ Pk +1 = Pk − rk +1 k +1 k +1 k k +1 3Vk2 3Vk2 ⎪ ⎨ 2 2 ⎪3V 2 = 3V 2 − 2 r P + x Q + r 2 + x 2 Pk + Qk ( ) + + + + + 1 1 1 1 1 k k k k k k k k ⎪ 3Vk2 ⎩

(

)

(2.41)

102

Basic computation

Considering that the state quantities PA , QA and VA at the node A are known or estimated, the state quantities at the other nodes can be calculated by successive applications of equations (2.41) starting from the first node and going toward to the node n. In the backward sweep, the state quantities Pk , Qk and Vk at the node k are used to calculate the state variables at the node k − 1 , using the set of equations: ⎧ Pk'2 + Qk'2 Pk'2 + Qk'2 + pk +1 ; Qk −1 = Qk + xk + qk +1 ⎪ Pk −1 = Pk + rk 3Vk2 3Vk2 ⎪ ⎨ '2 '2 ⎪3V 2 = 3V 2 + 2 r P ' + x Q ' + r 2 + x 2 Pk + Qk k k k k k k k ⎪⎩ k −1 3Vk2

(

) (

)

(2.42)

where Pk' = Pk + pk and Qk' = Qk + qk . Similarly to the forward sweep, in the backward sweep, considering the state quantities Pn , Qn and Vn at the node n as known, the state quantities at the others nodes can be calculated by successive applications of equations (2.42) starting from the node n − 1 and going toward the node A. By successive applications of the backward and forward sweeps the load flow solutions are achieved. The following boundary constraints are considered during the calculation process [2.10]: – to voltage magnitude VA at the sources node A is known, and considered constant; – the components of the apparent power flowing through a hypothetical branch outgoing from the node n are: Pn = 0 and Qn = 0 .

2.2.4.3. Backward/forward sweep adaptation for the case of distributed generation Usually, the distributed generators are used to produce locally, in consumption areas, relatively reduced amounts of power and are connected in medium and low voltage distribution networks. The main differences with respect to the classical power plants (thermal, nuclear, and hydro) are related to the location and the installed capacity. These sources can generate active power and sometimes can generate or consume reactive power, having the possibility to maintain the nodal voltage at a set value by means of an automatic voltage regulator. The distributed generators capable to vary their output active power can contribute to the frequency control into the power system. Taking into account these considerations, the nodes to which these generators are connected can be classified in: – PQ nodes, to which the specified quantities are the generated active Pgsp and reactive Qgsp (capacitive or inductive) powers, and the unknown quantities are the components of the complex voltage U ;


103

– PU nodes, to which the specified quantities are the generated active power Pgsp and voltage magnitude U sp , and the unknown quantities are the generated reactive power Qg and the voltage phase θ; – Uθ nodes, to which the specified quantities are the components of the sp complex voltage U (magnitude U sp and phase θsp ), and the unknown quantities are the generated active Pg and reactive Qg powers. In order to consider these types of nodes in the backward/forward sweep for the radial electric networks which include distributed generators, some specifications and adaptations are necessary. Therefore, the distributed generators should be modelled by PQ nodes, where the specified quantities Pgsp and Qgsp are considered as being components of a constant complex power with negative sign S = − ( Pgsp + jQgsp ) , because the backward/forward sweep, presented above, cannot be applied for the PU and Uθ nodes. This inconvenience is due to the fact that for these types of nodes, one or both components of the voltage are specified, which is not appropriate to the backward/forward sweep algorithm where the voltage components are specified only at the source node. Starting from the voltage of this node, chosen as reference, the voltages of the others nodes are calculated in terms of the voltage drops on the line sections. However, in order to apply the backward/forward sweep algorithm for the PU and Uθ nodes, some adaptations are required, which are based on the decoupling of the four state quantities, i.e. the interdependences P – θ and Q – U, respectively. These adaptations are presented below. As previously explained, the PU nodes are characterized by the specification of the generated active power Pgsp and the voltage magnitude U sp . In the load flow calculation process, by backward/forward sweep, these nodes are assimilated with PQ type nodes. The active and reactive powers are equal to the specified values Pgsp and Qg , considered with negative sign. To maintain the nodal voltage at the specified value U sp , the interdependence relationship between the voltage and the reactive power is used, i.e. appropriate change of the reactive power Qg , between the limits Qgmin and Qgmax , is adopted. Depending on the model used for the nodes, two situations could be encountered: – in the case of modelling by constant currents, the reactive component of the complex current I gr is determined based on the condition that the nodal voltage should be equal to the specified value U sp [2.24]; for the backward/forward sweep algorithm, the nodal current is considered as

(

)

sp sp I = − I ga + jI gr , where I ga represents the specified value of the

generated active current;

104

Basic computation

– in the case of modelling by constant powers, the generated reactive power Qg is determined based on the condition that the nodal voltage should be equal to the specified value U sp ; in the backward/forward sweep

(

)

algorithm, the nodal power is considered as S = − Pgsp + jQg . The use of the second model for the load modelling requires a non-linear mathematical model for load flow calculation. Like the global load flow calculation methods, for the backward/forward sweep algorithm, the calculation of the generated reactive power and its comparison with the capability limits at every step is performed. For better understanding of the modified backward/forward method applied when PU nodes are present within the network, a radial electric network with only one generator located at the node k is considered (2.9,b). The calculation steps are presented in the following: 1. Initialise the iterative step p = 0 and establish the initial value of the

(

)

reactive power Qg( ,k) = 0 , so that S k = Pc ,k + jQc ,k − Pgsp, k + jQg( , k) , 0

(0)

0

where Pc ,k and Qc ,k are the components of a complex constant power consumed at the node k, and Pgsp,k is the specified active power generated at the node k; 2. Update the iterative step p = p + 1 ; 3. Perform load flow calculation by backward/forward sweep; p 4. If U k( ) − U ksp < εU the iterative process stops;

5. Calculate the generated reactive power Qgcalc , k necessary to achieve the specified voltage U ksp at the node k. Establish the new value of the generated reactive power Qg( ,k) in terms of its value with respect to the p

min max capability limits Qg,k and Qg,k : min calc max ⎧Qg( p, k) = Qgcalc , k if Qg , k ≤ Qg , k ≤ Qg , k ⎪ ⎪ ( p) min min calc ⎨Qg , k = Qg ,k if Qg ,k < Qg , k ⎪ max calc ⎪Qg( p, k) = Qgmax , k if Qg , k > Qg , k ⎩

(2.43)

6. Calculate the new value of the complex power at the node k by formula

(

)

S k = Pc ,k + jQc ,k − Pgsp,k + jQg( ,k) and go to step 2. ( p)

p

There are several possibilities to calculate the value of the generated reactive sp power Qgcalc , k necessary to achieve the specified voltage U k at the node k, i.e.:


105

(i) using the voltage sensitivity to the reactive power variation ∂U k ∂Q k , obtained from the sensitivity matrix: U ( ) −U ) ( + p

Qgcalc ,k

sp k

k

( p −1)

= Qg , k

⎛ ∂U k ⎞ ⎜ ⎟ ⎝ ∂Qk ⎠

( p)

(ii) using the secant method [2.25]: ( ) ( ) p ( p −1) Qg , k − Qg , k Qgcalc = Q + U k( ) − U ksp ,k g ,k ( p −1) ( p −2) Uk −Uk p −1

p−2

(

)

(iii) using a calculation formula based on the generated reactive current I gr , k , calculated by considering the constant currents model for the load: ( p −1) Qgnec I gr ,k , k = 3U k

2.2.4.4. Backward/forward sweep adaptation for meshed distribution electric network The backward/forward algorithm presented earlier can be used only for load flow calculation of arborescent networks. By convenient changes that imply additional computation, the method can be expanded for load flow calculation of simple or complex meshed electric networks. Consider a simple meshed electric network, supplied at two ends (Fig.2.10,a). A VA

A VA

A

k-1

z2 s1

s2

1

2

z1

a.

s1

s2

1

2

b.

s2

k-1 sk-1

zn sk

Ik-1,k

k' k'' sk' sk''

VB

n zn

sk

zk

B zn+1

sn

k

zk

sk-1

z2 s1

k-1

n

k zk

sk-1

z2

z1 VA

2

1 z1

B zn+1

sn

VB

n zn

B zn+1

sn

c. Fig. 2.10. Calculation steps for a simple meshed electric network.

VB

106

Basic computation

To apply the backward/forward sweep for the load flow calculation of this network, one of the loop’s nodes is split (for example node k) obtaining two radial sub-networks [2.14]. After splitting the node k, we require that the total consumed power at the two resulted nodes k′ and k″ to be constant and equal to the power at the node k before splitting:

s k ' + s k" = s k

(2.44)

The calculation of the powers consumed at each node is performed in terms of the balancing current that appears at the loop closing, given by the relation:

I k −1,k =

V k −1 − V k Z AB

(2.45)

where Z AB is the cumulated impedance between nodes A and B, and the voltages V k −1 and V k correspond to the operating state when the branch between the nodes k − 1 and k is in “out of service” state (Fig. 2.10,b). These voltages are calculated by successively applying of the backward/forward sweep for the sub-networks supplied from the source nodes A and B, respectively. The load flow calculation is iteratively performed, by setting at each iteration the power consumed at each node resulted after splitting and by calculating the load flow for each radial sub-network. The iterative process goes on until the difference between the voltages of the two nodes k′ and k′′ is less than an specified value ε. The steps followed for load flow calculation of a simple meshed network are: 1. One of the loop’s branches is switched to “out of service” state (for instance, the branch between the nodes k − 1 and k, Fig. 2.10,b), and for the new configuration the load flow is calculated by means of the backward/forward sweep; 2. The node k is split into nodes k′ and k′′ and a new branch is introduced between nodes k − 1 and k′, having the same parameters as the branch between the nodes k − 1 and k (Fig. 2.10,c). The voltages and the powers consumed at the split nodes are set to:

V k ' = V k −1 ; V k '' = V k s k ' = 0; s k '' = s k

(2.46)

3. The balancing current through the loop is calculated as:

I k ', k '' =

V k ' - V k '' Z AB

n +1

where Z AB =

∑Z

i

is the total impedance of the loop;

i=1

4. Updating the powers at the nodes k′ and k″ with the relations:

(2.47)


107

s k ' = s k ' + 3V k ' I *k ',k '' s k '' = s k - s k '

(2.48)

5. The load flow is calculated for the network configuration in Figure 2.10,c; 6. If V k ' − V k '' > ε then go to step 3, else return to the initial configuration (Fig. 2.10,a) considering V k = V k '' and I k −1,k = I k −1,k ' . When the network has a complex meshed configuration, the load flow calculation by means of the backward/forward sweep is performed by introducing a number of supplementary nodes (resulted after splitting) equal to the number of loops. At each step, the equality of voltages at the split nodes for one single loop has to be achieved. The calculation is repeated until the differences of the voltage magnitudes at the split nodes in all loops is less than the specified value ε.

2.2.4.5. Advantages of the backward/forward sweep The backward/forward sweep load flow algorithm has some advantages as compared to the global methods based on the nodal voltages: – for the nonlinear model of the network, the number of the iterations is smaller than the one required by Seidel-Gauss and Newton-Raphson methods, and the calculation effort during each iteration is smaller, too; – the calculation of the nodal admittance matrix is not necessary; – the introducing of switches and shunts (elements of reduced impedance) do not cause convergence problems. Application Consider the radial electric network from Figure 2.11,a, given that: – the voltage at the source node U 1 = 20 kV ;

– the complex powers at the loads: s 2 = 0 kVA , s 3 = ( 250 + j150 ) kVA and s 4 = ( 75 + j 50 ) kVA .

The one-line diagram of the electric network and the branch parameters are shown in Figure 2.11,b. The voltages at the load nodes, and current flows for this network have to be determined. For simplicity, the phase-to-phase voltages are used. The initial values of voltages at the load nodes are: (0) U (0) 2 = U 3 = U 1 = 20 kV

U (0) 4 = N 42 U 1 = 0.02 ⋅ 20 = 0.4 kV

Only the calculation of the first iteration is detailed below, the results of the whole iterative process being presented in Table 2.1.

108

Basic computation

S1 1

2

3

4

s3

s4 a.

I12 (1.4+j0.1) Ω

S1 1 I120

j30 µS

I23 (2.1+j0.15) Ω

2 I210

j30 µS

I230

j45 µS

j45 µS

i3 3 I320 s3=(250+j150) kVA

I240 (0.8-j8.72) µS

I24 N42=0.02 I'24

(0.028+j0.058) Ω 4

s4 =(75+j50) kVA b.

Fig. 2.11. Radial electric network: a. one-line diagram; b. equivalent circuit. Backward sweep Calculation of the current in the branch 2 − 3 i 3(1) =

s*3 3U 3(0)*

(1) I 320 =y

=

U 3(0) 320

3

( 250 − j150 ) ⋅103 3 ⋅ 20 ⋅103 = j 45 ⋅10−6 ⋅

= ( 7.217 − j 4.330 ) A

20 ⋅103 3

= j 0.520 A

(1) (1) I (1) 23 = i 3 + I 320 = ( 7.217 − j 3.810 ) A

Calculation of the current in the branch 2 − 4 i (1) 4 =

s*4 3U (0)* 4

=

( 75 − j50 ) ⋅103 3 ⋅ 0.4 ⋅103

= (108.253 − j 72.169 ) A

(1) I '(1) 24 = i 4 = (108.253 − j 72.169 ) A '(1) I (1) 24 = N 42 I 24 = 0.02 (108.253 − j 72.169 ) = ( 2.165 − j1.443 ) A


109

Calculation of the current in the branch 1 − 2 s*2

i (1) 2 =

3U (0)* 2

I (1) 210 = y I (1) 230 = y I (1) 240 = y

=0A

U (0) 2 210

3 U (0) 2

230

3 U (0) 2

240

3

= j 30 ⋅10−6 ⋅ = j 45 ⋅10−6 ⋅

20 ⋅103 3 20 ⋅103 3

= j 0.346 A = j 0.520 A

= ( 0.8 − j8.72 ) ⋅10−6 ⋅

20 ⋅103 3

= ( 0.009 − j 0.101) A

(1) (1) (1) (1) (1) (1) = i (1) I 12 2 + I 23 + I 230 + I 24 + I 240 + I 210 = ( 9.391 − j 4.488 ) A

Forward sweep Calculation of the voltage at the node 2 (1) (1) ∆U 12 = 3 z12 I 12 = 3 (1.4 + j 0.1)( 9.391 − j 4.488 ) ⋅10−3 = ( 0.024 − j 0.009 ) kV (1) U (1) 2 = U 1 − ∆U 12 = 20 − ( 0.024 − j 0.009 ) = (19.976 + j 0.009 ) kV

Calculation of the voltage at the node 3 (1) −3 ∆U (1) = ( 0.027 − j 0.012 ) kV 23 = 3 z 23 I 23 = 3 ( 2.1 + j 0.15 )( 7.217 − j 3.810 ) ⋅ 10 (1) U 3(1) = U (1) 2 − ∆U 23 = (19.976 + j 0.009 ) − ( 0.027 − j 0.012 ) = (19.949 − j 0.021) kV

Calculation of the voltage at the node 4 '(1) −3 ∆U (1) = ( 0.013 + j 0.007 ) kV 24 = 3 z 24 I 24 = 3 ( 0.028 + j 0.058 )(108.253 − j 72.169 ) ⋅ 10 (1) (1) U (1) 4 = N 42 U 4 − ∆U 24 = 0.02 (19.949 − j 0.021) − ( 0.013 + j 0.007 ) = ( 0.386 − j 0.007 ) kV

Calculation of the power injected at the source node (1) I 120 =y

U1

120

3

= j 30 ⋅10−6 ⋅

(

(1) (1) S 1(1) = 3U 1 I 12 + I 120

)

*

20 ⋅103 3

= j 0.346 A

= 3 ⋅ 20 ⋅ ( 9.391 − j 4.142 ) = ( 325.314 + 143.483) kVA

S1(1) = 355.551 kVA

Table 2.1 Results of the iterative process Quantity 0 i3

1 A

Iteration 1 2 7.217 − j 4.330

Iteration 2 3 7.240 − j 4.334

Iteration 3 4 7.240 − j 4.334

I 320

A

j 0.520

0.001 + j 0.518

0.001 + j 0.518

I 23

A

7.217 − j 3.810

7.241 − j 3.816

7.241 − j 3.816

110

Basic computation

A

2 108.253 − j 72.169

3 113.498 − j 72.728

Table 2.1 (continued) 4 113.681 − j 72.430

I 24

A

2.165 − j1.443

2.270 − j1.455

2.274 − j1.455

i2

0

1

I '24 = i 4

A

0

0

0

I 210

A

j 0.346

j 0.346

j 0.346

I 230

A

j 0.520

j 0.519

j 0.519

I 240

A

0.009 − j 0.101

0.009 − j 0.101

0.009 − j 0.101

I 12

A

9.391 − j 4.488

9.520 − j 4.507

9.524 − j 4.507

∆U 12

kV

0.024 − j 0.009

0.024 − j 0.009

0.024 − j 0.009

U2

kV

19.976 + j 0.009

19.976 + j 0.009

19.976 + j 0.009

∆U 23

kV

0.027 − j 0.012

0.027 − j 0.012

0.027 − j 0.012

U3

kV

19.949 − j 0.021

19.949 − j 0.021

19.949 − j 0.021

∆U 24

kV

0.013 + j 0.007

0.013 + j 0.008

0.013 + j 0.008

U4

kV

0.386 − j 0.007

0.386 − j 0.008

0.386 − j 0.008

S1

kVA

325.314 + 143.483

329.782 + 144.141

329.921 + 144.141

S1

kVA

355.551

359.907

360.021

Notes: The load flow results were achieved after 3 iterations by applying the backward/forward sweep. The difference between voltages at the last two iterations is less than 0.001 kV, and the difference between apparent powers at the source node is 0.1 kVA. The same results were achieved using Seidel-Gauss (12 iterations) and Newton-Raphson (3 iterations) methods.

2.3. Complex meshed electric networks 2.3.1. Transfiguration methods In the following, several commonly used transfiguration methods will be presented. a) The reduction of a conductor of a certain length and cross-sectional area, to an equivalent conductor of a different length and cross-sectional area. In calculation of a network, sometimes it is advantageous that portions of line with different cross-sectional areas be transformed into sections of line with the same cross-sectional area. Thus, the conductor of cross-sectional area s1 and length l1 can be substituted with another conductor of cross-sectional area s2 and length l2, provided that the distribution of the loads and the voltage drop along the conductors remains the same. In other words, the resistances of the two conductors must remain unchanged; thereby the equivalencing condition emerges:


111

s1 s2

l1 = l2

As equivalencing cross-sectional area, the most frequent cross-sectional area from the respective network will be chosen. b) Loads throwing at the nodes. Composing branches in parallel needs the loads to be situated only at their ends, in nodes. If the loads are connected everywhere along the branches, first their throwing (moving) at the ends is performed, with the condition of keeping the voltage drop constant, in the initial circuit as well as in the transformed circuit. In Figure 2.12 an electric line to which the loads i1 and i 2 are connected is represented. Z Z2

Z2 Z1

Z1

A

1

VA

i1

Z B

2

VA

VB

i2

B

A

iA

iB

a.

VB

b.

Fig. 2.12. Electric network diagram for the throwing of the loads at the nodes: a. initial circuit; b. transfigured circuit.

For instance, to throw at the ends the two currents i1 and i 2 from Figure 2.12, a, two loads i A and i B applied at the line’s ends in the transformed network (Fig. 2.12,a) will be determined, such that the same voltage drop as in the initial network is obtained: ∆V AB = Z 1 i1 + Z 2 i 2 = Z i B '

'

∆V BA = Z 1 i1 + Z 2 i 2 = Z i A from where it results: n

iA =

' Z 1 i1

+ Z

' Z 2 i2

=

∑Z

' k ik

k =1

Z

(2.49)

n

Z i + Z 2 i2 iB = 1 1 = Z '

∑Z

k ik

k =1

Z '

'

where Z 1 , Z 2 , ..., Z k and respectively Z 1 , Z 2 , ..., Z k represents the impedances from the two ends to the connection points of the k loads.

112

Basic computation

In the particular case of moving only one load, the consumed current component, moved at one of the ends, is proportional to the impedance of the line from the point of consumption to the other end, and inversely proportional to the line impedance. From the expression (2.49) results that load throwing at the nodes is performed according to the rule determined for distribution of the currents (powers) in the case of the networks supplied from two ends, that is considering the electric moments of the loads referred to the supplying points. For the case of the homogenous network, in the relationships of transformation, the impedances are substituted with the corresponding lengths. c) Composing of several branches of different supplying voltages which debit into a node, in a single equivalent branch. Consider the branches A, B, C of an electric network that has different phase-to-neutral voltages V A , V B , V C at the ends and debits into a node O (Fig. 2.13). VA A

VE

IE

E

Fig. 2.13. Ramified electric network with different voltages at the ends.

IA YE

YA O

IE

VB

IB YB Y VC I C C B

VO

C

These branches can be substituted with a single equivalent branch of admittance Y E and voltage V E at the end E. In order to determine the quantities of the equivalent branch E-O, the relations of equivalencing between the real circuit with three branches and the equivalent circuit with a single branch are written:

IE = I A + IB + IC

(2.50)

Kirchhoff’s first theorem will be written as:

(V E − V O )Y E = (V A − V O )Y A + (V B − V O )Y B + (V C − V O )Y C or

V E Y E − V O Y E = V A Y A + V B Y B + V C Y C − V O (Y A + Y B + Y C ) Equating the left and right terms, obtain:

Y E =Y A +Y B +YC =

n

∑Y k =1

respectively

k


113 n

V Y +V BY B +V CY C = VE = A A Y A +YB +YC

∑V

kYk

k =1 n

∑Y

(2.51) k

k =1

It should be mentioned that the replacing of the parallel branches with an equivalent branch is possible only if along them there are no derivations with supplementary loads. Instead, in the case of inverse transformations, the current passed through the equivalent branch is known and the currents passed through the branches of the initial network, not transfigured, are required. In this case, voltage drops expressions are written: I I I I V A −V O = A ; V B −V O = B ; V C −V O = C ; V E −V O = E YC YA YB YE From the last relationship, the voltage of the node O can be determined: I VO =VE − E YE which, after substituting in the other three equations, enable us to determine the currents passed through the component branches: Y I A = I E A + (V A − V E )Y A (2.52,a) YE

IB = IE

YB + (V B − V E )Y B YE

(2.52,b)

IC = I E

YC + (V C − V E )Y C YE

(2.52,c)

It is obvious that if the voltages of the branches are equal, the relations are still correct, with the observation that the voltage of the equivalent branch is equal to that of the component branches. In this case, in expressions (2.52) the second term of the right side will disappear. d) Star − delta transformation. Another structure, which comes as a subassembly into a meshed network, is the star structure, in the simplest case powered from three nodes (Fig. 2.14). The condition of equivalencing of the two circuits (Fig. 2.14,a,b): the impedances measured at the pairs of terminals 1 – 2, 2 – 3 and 3 – 1 of the starshape network must be equal to the impedances measured at the same pairs of terminals of the delta-shape network:

Z1 + Z 2 =

Z 12 (Z 23 + Z 31 ) Z (Z + Z 12 ) Z (Z + Z 23 ) ; Z 3 + Z 1 = 31 12 ; Z 2 + Z 3 = 23 31 Z 12 + Z 23 + Z 31 Z 12 + Z 23 + Z 31 Z 12 + Z 23 + Z 31

114

Basic computation

I1

I1

1

1 Z1 Z31

I3

I31

Z2

Z3 3

2

I12

Z12

I23 I2

I3

2

3 Z23

a.

I2

b. Fig. 2.14. Star and delta circuits.

Next, solve this system of three equations with unknown quantities Z 1 , Z 2 ,

Z3 : Z1 =

Z 12 Z 13 Z 12 + Z 23 + Z 31

;

Z2 =

Z 23 Z 12 Z 12 + Z 23 + Z 31

; Z3 =

Z 31 Z 23 Z 12 + Z 23 + Z 31

(2.53)

If the system of equations is solved in terms of the unknown quantities Z 12 ,

Z 23 , Z 31 obtain: Z 12 = Z 1 + Z 2 +

Z1 Z 2 ; Z3

Z 23 = Z 2 + Z 3 +

Z2 Z3 Z1

; Z 31 = Z 1 + Z 3 +

Z1 Z 3 Z2

(2.54)

In terms of admittances, from the equations (2.53) and (2.54) we obtain the transformation relationships of a delta-shape network into a star-shape network with three branches: Y 1 = Y 12 + Y 13 +

Y 12 Y 13 ; Y 23

Y 2 = Y 12 + Y 23 +

Y 12 Y 23 Y Y ; Y 3 = Y 31 + Y 23 + 31 23 (2.53') Y 31 Y 12

respectively, the expressions of the admittances resulted from the star-delta transformation: Y 12 =

Y 1Y 2 ; Y1 + Y 2 + Y 3

Y 23 =

Y 2Y 3 Y 3Y1 ; Y 31 = Y1 + Y 2 + Y 3 Y1 + Y 2 + Y 3

(2.54')

Observation: In the case of general transformation, a star network having 1, 2, …, n supplying terminals, can be transformed into a polygon with n(n − 1) 2 branches, connecting its terminals two by two. Since the two networks (Fig. 2.15,a,b), initial and transformed, are equivalent, it results that the terminal voltages V 1 , V 2 , ..., V n and the currents I 1 , I 2 , ..., I n , which enter into the terminals, must be identical in the two cases.


115

I1star 1 Y1 V1

Instar

Yn n

Vn

I1p 1

2 I2star

Y2

I1n

O

Y3 VO

3

V2 I3star

I12 I1

Y12

2 I2p

3

Y1n

V3

3 I3p

Inp

a. b. Fig. 2.15. The general transformation of a network from star into a polygon: a. star-shape network; b. polygon-shape network.

For instance, for the current injected at the node 1 into the polygon-shape network: I 1 p = I 12 + I 13 + K + I 1n = Y 12 (V 1 − V 2 ) + Y 13 (V 1 − V 3 ) + K + Y 1n (V 1 − V n ) (2.55) respectively, into the star-shape network: Y Y Y Y Y Y I 1 star = Y 1 (V 1 − V O ) = n1 2 (V 1 − V 2 ) + n1 3 (V 1 − V 3 ) + K + n1 n (V 1 − V n ) Yk Yk Yk

∑

∑

k =1

∑

k =1

k =1

(2.56) where n

VO =

∑Y

kV k

k =1 n

∑Y

k

k =1

Equating the two currents from (2.55) and (2.56), obtain: Y 1Y j Y1 j = n Yk

∑ k =1

or, in the general case

Y ij =

Y iY j n

∑Y

(2.57)

k

k =1

Notice that the polygon has not all its branches independent. Thus, considering the transformed admittance between nodes i and j, given by (2.57), and dividing Y ij to Y iλ , where:

Y iλ =

Y iY λ n

∑Y k =1

whatever i might be, obtain:

k

116

Basic computation

Y ij Y iλ Therefore, it results that: Y1j

Y 1λ

=

Y2j Y 2λ

=

Yj Yλ

=K=

Y nj Y nλ

=

Yj Yλ

Now, we can draw the conclusion that any complete polygon, having all branches independent from each other, cannot be transformed into a star. The triangle (delta) is the only polygon that allows this transformation, having all branches dependent from each other. e) Electric networks equivalencing by using Kron elimination. In some cases, of interest is to hold only certain nodes in calculation (for instance: 1, 2 which are source nodes), whereas the other non-essential nodes (passive nodes, loads passivized through Z=ct.), since not of interest, are eliminated/reduced through star-delta transformation, taking benefit of the zero value of the current injected into the non-essential nodes (for instance: nodes 3 and 4) (Fig. 2.16,b). 1

Y42

Y14 4

2

1

4

Y43 3

1

a.

Y42

Y14 Y43

Y30 2

1

2

b.

3 0 2

4 0

0

c.

d.

Fig. 2.16. Exemplification of the non-essential nodes elimination: a. initial network; b. the network with the load from node 3 replaced with an impedance Z=ct.; c. the circuit after the elimination of node 3; d. the circuit after the elimination of nodes 3 and 4.

For the electric network from Figure 2.16,b, where node 4 was non-essential, and node 3 became passive, by replacing the consumption with an impedance Z=ct., the equation from the nodal voltages method∗) becomes:

Y 14 ⎤ ⎡U 1 ⎤ ⎡ I 1 ⎤ ⎡Y 11 ⎢ I ⎥ ⎢ Y 22 Y 24 ⎥⎥ ⎢⎢U 2 ⎥⎥ ⎢ 2 ⎥=⎢ ⎢ I 3 = 0⎥ ⎢ Y 33 Y 34 ⎥ ⎢U 3 ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎣ I 4 = 0⎦ ⎣Y 41 Y 42 Y 43 Y 44 ⎦ ⎣U 4 ⎦ ∗)

(2.58)

As it can be seen in §2.4.2., in the framework of nodal voltages method, phase-to-phase voltage is used.


117

In the first stage the node 3 is eliminated. In this regard, from the equation corresponding to the current from node 3, where I 3 = 0 , U 3 is obtained and then substituted in the equation of the current from node 4, resulting:

I 4 = 0 = Y 41U 1 + Y 42 U 2 + Y 43U 3 + Y 44 U 4 = ⎛ Y ⎞ = Y 41U 1 + Y 42 U 2 + Y 43 ⎜⎜ − 34 U 4 ⎟⎟ + Y 44 U 4 ⎝ Y 33 ⎠ or ⎛ Y Y ⎞ Y 41U 1 + Y 42U 2 + ⎜⎜ Y 44 − 34 43 ⎟⎟U 4 = 0 Y 33 ⎠ ⎝ It can be noticed that by eliminating node 3, the term 44 has been modified: '

Y 44 = Y 44 −

Y 34 Y 43 Y 33

resulting the reduction of the number of equations with one unit (Fig. 2.16,c): Y 14 ⎤ ⎡U 1 ⎤ ⎡ I 1 ⎤ ⎡Y 11 ⎢ I ⎥=⎢ Y 22 Y 24 ⎥⎥ ⎢⎢U 2 ⎥⎥ ⎢ 2 ⎥ ⎢ ⎢⎣ I 4 = 0⎥⎦ ⎢⎣Y 41 Y 42 Y '44 ⎥⎦ ⎢⎣U 4 ⎥⎦

(2.59)

Next, in order to eliminate the node 4, U 4 is obtained from the new equation corresponding to it and substituted in the first two equations in (2.59): ⎛Y Y ⎞ ⎛ Y Y ⎞ I 1 = ⎜⎜ Y 11 − 14 ' 41 ⎟⎟U 1 − ⎜⎜ 14 ' 42 ⎟⎟U 2 Y 44 ⎠ ⎝ Y 44 ⎠ ⎝ ⎛ Y Y ⎞ ⎛ Y Y ⎞ I 2 = ⎜⎜ − 24 ' 41 ⎟⎟U 1 + ⎜⎜ Y 22 − 24 ' 42 ⎟⎟U 2 Y 44 ⎠ Y 44 ⎠ ⎝ ⎝ or ' ' ⎡ I 1 ⎤ ⎡Y 11 Y 12 ⎤ ⎡U 1 ⎤ ⎥ ⎢I ⎥ = ⎢ ' ' ⎥⎢ ⎣ 2 ⎦ ⎢⎣Y 21 Y 22 ⎥⎦ ⎣U 2 ⎦

(2.60)

where: '

Y 11 = Y 11 − '

Y 21 = −

Y 14 Y 41 ' Y 44

Y 24 Y 41 ' Y 44

'

; Y 12 = − '

Y 14 Y 42

; Y 22 = Y 22 −

'

Y 44 Y 24 Y 42

respectively the new electric circuit from Figure 2.16,d.

'

Y 44

118

Basic computation

Observation: to minimize the calculation effort, the order of elimination of the nodes must not be random. Thus, first are eliminated the non-essential nodes with the smallest number of connections, that is with the smallest number of terms in the nodal admittance matrix; the nodes with many connections will be considered at the end of the elimination process. In the example shown, the elimination first of node 3 has been performed by connecting two admittances (Y 34 + Y 30 ) , followed by a transformation star-delta. If would have been eliminated first node 4, two transformations star-delta would have been necessary, which means an increased calculation effort. In the general case, by using the method of partitioning into blocks, the equation (2.58) can be written as: ⎡[I r ]⎤ ⎡[A] [B ]⎤ ⎡[U r ]⎤ ⎢[I ]⎥ = ⎢[C ] [D ]⎥ = ⎢[U ]⎥ ⎦ ⎣ e⎦ ⎣ e⎦ ⎣ I1 ⎤ ⎥; ⎣I 2 ⎦

[I r ] = ⎡⎢

where:

[U r ] = ⎡⎢

U1 ⎤ ⎥ for the preserved nodes, and ⎣U 2 ⎦

(2.61) I3⎤ ⎥, ⎣I 4 ⎦

[I e ] = ⎡⎢

[U e ] = ⎡⎢

U3⎤ ⎥ for the nodes that are being eliminated. ⎣U 4 ⎦

From (2.61) results:

[I r ] = [A][U r ] + [B][U e ] [I e = 0] = [C ][U r ] + [D][U e ] If [U e ] is expressed from latter equation and substituted into the previous equation, it results:

[I r ] = [A][U r ] − [B][D]−1[C ][U r ] = [Y rr ][U r ] where the admittance matrix reduced to the preserved nodes can be calculated with the expression:

[Y rr ] = [A] − [B][D]−1[C ]

(2.62)

f) A general reduction method [2.4]. The general method takes into consideration the expanding of the partitioning method in the case of expressing the nodal currents, from the nodal voltages method, in terms of nodal powers and reduction or equivalencing only a certain zone from the whole network, the reduced zone can contain load (consumer) and/or generator nodes. The electric network is divided into two sub-networks (Fig. 2.17): − Internal sub-network (having I nodes), where all quantities are known in real time (voltage Ui and phase angle θi , current flows on electric lines, generated powers − Pg , Qg , consumed powers − Pc , Qc , network structure);


119

− External sub-network (having E nodes) that will be reduced/equivalated, where there is a lack of on-line information (only the power flow on interconnection tie-lines, electric lines state and the most important generators are known); − A number of frontier nodes (F), connected to nodes I and E, from both sub-networks; there are no connections between nodes I and E. Internal System I

External System E

F

Fig. 2.17. The division of the electric network.

Because the state of the external sub-network is not fully known, its reducing or equivalencing is performed. The matrix equation from the nodal voltages method applied in the divided network is:

⎡[Y EE ] ⎢Y ⎢[ FE ] ⎢⎣ [ 0]

[Y EF ] [0] ⎤ [Y FF ] [Y FI ]⎥⎥ [Y IF ] [Y II ] ⎥⎦

⎡⎡⎣U E ⎤⎦⎤ ⎡[ I E ]⎤ ⎢ ⎥ ⎢ ⎥ ⎢⎡⎣U F ⎤⎦⎥ = ⎢[ I F ]⎥ ⎢⎣ ⎡⎣U I ⎤⎦ ⎥⎦ ⎢⎣ [ I I ] ⎥⎦

(2.63)

Next, equation (2.63) is expressed in terms of the vector [S ] of injected complex powers. In this regard, the vector [S ] is written as:

[ S ] = ⎣⎡U d ⎦⎤ ⎣⎡ I * ⎦⎤ = ⎣⎡U d ⎦⎤ ([Y nn ]⎡⎣U n ⎤⎦)

*

(2.64)

where [U d ] is a diagonal matrix whose elements correspond to the elements of the

vector [U n ] , but grouped into I, F and E:

⎡(U E ) ⎤ d ⎢ ⎥ ⎡⎣U d ⎤⎦ = ⎢ (U F )d ⎥ (U I )d ⎥⎦ ⎢⎣ By using (2.63) and (2.65), equation (2.64) becomes: ⎡⎣⎡S E ⎦⎤⎤ ⎡(U E )d ⎢ ⎥ ⎢ ⎢⎡⎣S F ⎤⎦⎥ = ⎢ ⎢ ⎣⎢ ⎣⎡S I ⎦⎤ ⎦⎥ ⎣

(U F )d

⎤ ⎥ ⎥ (U I )d ⎦⎥

⎡ ⎡Y * ⎤ ⎢ ⎣ EE ⎦ ⎢ ⎡Y *FE ⎤ ⎦ ⎢⎣ ⎢ [ 0] ⎣

⎡Y *EF ⎤ [0] ⎤⎥ ⎣ ⎦ ⎡Y *FF ⎤ ⎡Y *FI ⎤ ⎥ ⎣ ⎦ ⎣ ⎦⎥ * * ⎡Y IF ⎤ ⎡Y II ⎤ ⎥ ⎣ ⎦ ⎣ ⎦⎦

(2.65)

⎡ ⎡U * ⎤ ⎤ ⎢⎣ E ⎦ ⎥ ⎢ ⎡U * F ⎤ ⎥ (2.66) ⎦⎥ ⎢⎣ * ⎢ ⎡U I ⎤ ⎥ ⎦⎦ ⎣⎣

Note: If [A] is a square matrix with only diagonal terms, [X] being a vector, ⎡ x1 ⎢0 [A] = ⎢⎢ L ⎢ ⎣0

0 L 0⎤ x2 L 0 ⎥⎥ ; L L L⎥ ⎥ 0 0 xn ⎦

⎡ x1 ⎤ ⎢x ⎥ [X ] = ⎢⎢ 2 ⎥⎥ and noting through [Xd] the diagonal matrix M ⎢ ⎥ ⎣xn ⎦

120

Basic computation whose elements are those of the matrix [A], it can be shown that the vector ⎡ x1 y1 ⎤ ⎢x y ⎥ [W ] = ⎢⎢ 2 2 ⎥⎥ can be rewritten as ⋅⋅⋅ ⎥ ⎢ ⎣ xn y n ⎦

[W ] = [X d ][y ] = [Yd ][X ] The matrix equation (2.66) can be rewritten as a system of three vector equations: ⎡⎣S E ⎤⎦ = (U E )d ⎡⎣[Y EE ] ⎡⎣U E ⎤⎦ + [Y EF ] ⎡⎣U F ⎤⎦⎤⎦

*

⎡⎣S F ⎤⎦ = (U F )d ⎡⎣[Y FE ] ⎡⎣U E ⎤⎦ + [Y FF ] ⎡⎣U F ⎤⎦ + [Y FI ] ⎡⎣U I ⎤⎦⎤⎦ ⎣⎡S I ⎦⎤ = (U I ) d ⎡⎣[Y IF ] ⎣⎡U F ⎦⎤ + [Y II ] ⎣⎡U I ⎦⎤⎤⎦

*

(2.67,a) *

(2.67,b) (2.67,c)

The reduction procedure of the network consists in manipulating the equation (2.67,a):

(U E )d

−1

⎡⎣S E ⎤⎦ = ⎡⎣Y

* EE

⎤ ⎡U *E ⎤ + ⎡Y *EF ⎤ ⎡U *F ⎤ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

(2.68)

or −1

⎡U *E ⎤ = ⎡Y *EE ⎤ ⎡(U E )−1 ⎡⎣S E ⎤⎦ − ⎡Y *EF ⎤ ⎡U *F ⎤ ⎤ d ⎣ ⎦ ⎣ ⎦ ⎣ ⎣ ⎦ ⎣ ⎦⎦

(2.69)

Component i of the term (U E )d [S E ] is given by −1

S Ei U Ei

=

S Ei U E2i

*

⋅ U Ei

If we define the vector:

⎡ S E1 ⎤ ⎢U 2 ⎥ ⎢ E1 ⎥ [wE ] = ⎢ M ⎥ ⎢ S En ⎥ ⎢U 2 ⎥ ⎢⎣ E n ⎥⎦ then the expression from (2.68) takes another form:

(U E )d

−1

( ) *

⎡⎣S E ⎤⎦ = U E

d

* ⎡⎣w E ⎤⎦ = (W E )d ⎡⎣U E ⎤⎦

By substituting (2.70) in (2.69) it results:

( )

⎡U *E ⎤ = Y *EE ⎣ ⎦

−1

⎡(W ) ⎡U * ⎤ − ⎡Y * ⎤ ⎡U * ⎤ ⎤ ⎣ E d ⎣ E ⎦ ⎣ EF ⎦ ⎣ F ⎦ ⎦

(2.70)


121

or by conjugating:

( )

−1 ⎡ * ⎣⎡U E ⎦⎤ = (Y EE ) ⎣⎢ W E

d

⎤ ⎣⎡U E ⎦⎤ − [Y EF ] ⎣⎡U F ⎦⎤⎦⎥

If in the latter relation we multiply to the left with [Y EE ] we obtain:

[Y EE ] ⎣⎡U E ⎦⎤ = (W *E )d ⎣⎡U E ⎦⎤ − [Y EF ] ⎣⎡U F ⎤⎦ or

( ))

(

⎡⎣U E ⎤⎦ = − [Y EE ] − W E *

−1

d

[Y EF ] ⎡⎣U F ⎤⎦

(2.71)

By substituting (2.71) in (2.67,b) obtain:

[S F ] = (U F )d ⎪⎨[Y FF ][U F ] + [Y FE ] ⎡⎢− ([Y EE ] − (W *E )d ) [Y EF ][U F ]⎤⎥ + [Y FI ][U I ]⎫⎬ ⎧

*

−1

⎣

⎪⎩

⎦

⎭

or

( ))

(

*

−1 ⎡ ⎤ * * * * ⎣⎡S F ⎦⎤ = (U F )d ⎢[Y FF ] − [Y FE ] [Y EE ] − W E d [Y EF ]⎥ ⎡⎣U F ⎤⎦ + ⎡⎣Y FI ⎤⎦ ⎡⎣U I ⎤⎦ ⎣ ⎦ If in the latter equation we let [Y FF ] stand for the term that modifies the initial matrix, then:

(

( ))

⎡⎣Y eq ⎤⎦ = − [Y FE ] [Y EE ] − W *E

d

−1

[Y EF ]

(2.72)

where ⎡⎣Y eq ⎤⎦ corresponds to an equivalent network connected to the frontier nodes. If assume that the voltage magnitudes at the external nodes and the injected powers remain constant, then (W E )d = S E / U E2 is constant. As a consequence, the injections at the external nodes can be represented by equivalent admittances given by: *

yi = −

Si U i2

i ∈ (external network)

(2.73)

Observations: for the validity of the resulted equivalent it is necessary that all the terms Wi remain constant after considering a contingency inside the network. But this contradicts the classical approach where the load injections into the external nodes were represented through constant impedances.

2.3.2. Load flow calculation of meshed networks 2.3.2.1. Formulation of the load flow problem The load flow defines the state of a power system at a certain instant of time and corresponds to a given generation and consumption pattern. Taking into

122

Basic computation

consideration that the load varies from moment to moment, the values of the electric variables, characteristic of this state, vary also from scenario to scenario. The results obtained after the load flow calculation, being the starting point for any analysis of transmission and distribution networks, represent [2.4]: • Necessity in the planning strategies of electric networks development for the determination of the optimal configuration as well as in the exploitation activity for establishing the operating regime (overloading possibilities, voltage level, “weak” network areas identification, etc.); • Input data in the following activities: (i) contingency analyses, for testing the unavailability of an electric line, transformer or synchronous generator, known as security criteria with (N – 1) or (N – 2) availabilities; (ii) transmission capacity analysis, for testing the limits of transfer powers (thermal limit Imax adm); (iii) VAr – voltage analysis, for assessment of necessity of VAr – voltage equipment and its regulation manner; (iv) on-line control, of power system operation, using state estimators and process computers. • Starting point in the study and the selection of the protection relays and automations, also for static, transient and voltage stability analysis, the optimisation of operating regimes, etc. The mathematical model for steady state analysis is based on nodal voltages method using either nodal admittance matrix ⎛ ⎡

*

⎤⎞

[Y nn][U n] = [I n] ⎜⎜= ⎢ S n * ⎥ ⎟⎟

(2.74)

[Z nn][I n]=[U n]

(2.75)

⎝ ⎣⎢U n ⎥⎦ ⎠

or nodal impedance matrix:

For steady state analysis, assume that the electric network is symmetrical, load balanced, and there are no magnetic couplings between its elements. In consequence, the electric network can be modelled through a single-line diagram. In the framework of load flow calculations phase-to-phase voltage is used, which also represents the rated voltage of the network’s elements given in catalogues.

2.3.2.2. Nodal admittance matrix Generally, an electric network consists of branches – electric lines, transformers – and nodes to which the generators and/or loads are connected. The branches are represented through impedances/admittances, the generators through injected currents/powers at nodes and the loads through impedances or currents/powers coming out from the nodes.


123

In electric power systems and networks analysis in order to define the parameters and the structure of their elements the so-called system matrix is used – nodal admittance matrix ([Ynn]) or nodal impedance matrix ([Znn]). (i) Nodal admittance matrix in the case of the network without transformers Consider the single-line diagram and the equivalent circuit of an electric network, respectively (Fig. 2.18). y13

1

I1 1

3 I 3

V1 V2

I2

2

V3

I1

y130

y120

y12

3

y310 y23

y210

I3 y320

y230 2

I2 a.

b.

Fig. 2.18. Three-node network: a. Single-line diagram; b. Equivalent circuit.

In order to obtain the nodal admittance matrix [Ynn] and the equations from nodal voltages method, respectively, we apply the Kirchhoff’s first theorem at the independent nodes, conventionally adopting the sign “+” for injected nodal currents and the sign “–” for consumed nodal currents:

( ( (

)

⎧ y (V 1 – V 2 ) + y (V 1 – V 3 ) + y + y V 1 = I 1 13 120 130 ⎪ 12 ⎪ ⎨ y 21 (V 2 – V 1 ) + y 23 (V 2 – V 3 ) + y 210 + y 230 V 2 = – I 2 ⎪ ⎪ y (V 3 – V 1 ) + y (V 3 – V 2 ) + y + y V 3 = I 3 32 310 320 ⎩ 31

) )

(2.76)

If we group the latter system of equations in terms of the nodal voltages then:

(

)

⎧ y + y + y + y V 1 – y V 2 – y V 3 = I1 13 120 130 12 13 ⎪⎪ 12 ⎨ – y 21V 1 + y 21 + y 23 + y 210 + y 230 V 2 – y 23V 3 = – I 2 ⎪ ⎪⎩ – y 31V 1 – y 32 V 2 + y 31 + y 32 + y 310 + y 320 V 3 = I 3

(

(

)

)

(2.76')

or, as matrix form:

⎡Y 11 Y 12 Y 13 ⎤ ⎡V 1 ⎤ ⎡ I 1 ⎤ ⎢Y ⎥⎢ ⎥ ⎢ ⎥ ⎢ 21 Y 22 Y 23 ⎥ ⎢V 2 ⎥ = ⎢ – I 2 ⎥ ⎢⎣Y 31 Y 32 Y 33 ⎥⎦ ⎢⎣V 3 ⎥⎦ ⎢⎣ I 3 ⎥⎦ or

[Y nn ][V n ] = [I n ]

(2.77)

124

Basic computation

where:

Y 11 = y12 + y13 + y120 + y130 ;

Y 12 = − y12 ;

Y 21 = − y ;

Y 22 = y + y + y

Y 31 = − y ;

Y 32 = − y ;

21

21

31

23

32

+y

210

230

Y 13 = − y13 ; Y 23 = − y ;

;

23

Y 33 = y + y + y 31

32

310

+y

320

(ii) Nodal admittance matrix in the case of the networks with transformers [2.9, 2.22] Consider the case of the series ik branch of a transformer (shunt losses are neglected) with complex transformer turns ratio N ik (Fig. 2.19,a) where V N ik = i ' is the complex transformer turns ratio. Vk Ii i Iik Si

Vi

z ik

i Vi

N ik

k Ik Sk Vk

N ki

Ii i

z ki

k

Ik

Vk

Vi

a.

b.

Fig. 2.19. Equivalent circuit with transformer operator.

If apply Kirchhoff’s second theorem for the loop in Figure 2.19,a, obtain: – V i + z ik I ik + V i ' = 0 Knowing that V ik = z ik I ik ‚ the relationship between voltages results: V ik = V i – N ik V k

(2.78)

In order to establish the relationship between currents we start from the equality between the complex apparent powers from the input and those at the output terminals of the ideal transformer: *

*

S i = 3V i ' I ik = – S k = –3V k I k

(2.79)

Then, considering V i ' ≅ V i , obtain: ⎛I Vi = N ik = – ⎜⎜ k Vk ⎝ I ik

*

⎞ ⎟ ⎟ ⎠

(2.80)

respectively ⎧⎪ I k = – N *ik I ik ⎨ ⎪⎩ I i = I ik

(2.80')


125

Using the matrix form, the relationship of voltages becomes: ⎡V ⎤ ⎡V ⎤ V ik = [1 − N ik ] ⎢ i ⎥ = [ Aik ] ⎢ i ⎥ ⎣V k ⎦ ⎣V k ⎦

(2.78')

where [ Aik ] is known as quasi-incidence matrix of the ik branch to i and k nodes. In this case, when the branch is directional from i to k, that is the ideal transformer (of transformer turns ratio N ik ) is connected to the k node, the quasiincidence matrix is written as: i

k

[ Aik ] = ik [1 – N ik ] Expressing the relationship between currents as matrix form:

⎡Ii ⎤ ⎡ 1 ⎤ * ⎢ I ⎥ = ⎢ – N * ⎥ I ik = [ Aik ] t I ik ik ⎦ ⎣ k⎦ ⎣ and taking into account that I ik = y ik V ik and considering the expression (2.78') of

V ik , obtain:

[Aik ] *t I ik = [Aik ] *t y ik V ik = [Aik ] *t y ik [Aik ]⎡⎢

Vi ⎤ ⎡V ⎤ = [Y ik ] ⎢ i ⎥ ⎥ ⎣V k ⎦ ⎣V k ⎦

where:

[Y ik ] = [Aik ] *t y ik [Aik ] = ⎡⎢

1 ⎤ * ⎥ y ik [1 – N ik ] ⎣ – N ik ⎦

The nodal admittance matrix of the branch representing a transformer can be written as:

⎡

'

[Y ik ] = ⎢Y 'ii

⎢⎣Y ki

' Y ik ⎤ ⎡ y ik =⎢ * ' ⎥ Y kk ⎥⎦ ⎢⎣− y ik N ik

− y ik N ik ⎤ ⎥ y ik N ik2 ⎥⎦

(2.81)

with the mention that the admittance y ik is referred to winding i. If the branch is directional from k to i, that is the ideal transformer (of transformer turns ratio N ki ) is connected to i node (Fig. 2.19,b), [ Aki ] = ik [ N ki −1] , then nodal admittance matrix becomes:

⎡ y N2 [Y ik ] = ⎢ ki ki ⎣⎢ – y ki N ki

* – y ki N ki ⎤ ⎥ y ki ⎦⎥

(2.81')

with the mention that the admittance y ki is referred to winding k. It is clear that, in the case of a transformer with complex transformer turns ' ' ratio ( N ik or N ki ), the matrix [ Y ik ] is asymmetrical since Y ik ≠ Y ki , and for a

126

Basic computation

transformer with real transformer turns ratio ( N ik ), the matrix [ Y ik ] is symmetrical '

'

since Y ik = Y ki . (iii) General rules for writing the nodal admittance matrix [Ynn] • Any diagonal term Y ii is equal to the sum of the series and shunt admittances of the branches (lines, transformers, others) galvanically connected to i node. If at the respective node a transformer branch is connected, we have two cases: − if the ideal transformer is connected to i node, the series admittance of the transformer is multiplied by the square of transformer turns ratio: Y ii =

∑ (y

ik

) ∑y

+ y ik 0 +

ki

N ki2

(2.82)

where the first sum stands for lines elements, and the second sum stands for transformer elements; − if the series admittance of the transformer is galvanically connected to i node, the diagonal term is: Y ii = where

∑y

ik

∑y +∑y ik

(2.82')

ik 0

corresponds to series admittances of lines and transformers as well.

If we do not neglect the shunt components of the transformer, these are added to the term Yii since by hypothesis these are connected on the primary winding side. • The non-diagonal terms in the case of transformer branch are expressed as follows: − if the transformer operator N ik is connected to k node: '

Y ik = − y ik N ik

;

'

*

Y ki = – y ik N ik

(2.82")

− if the transformer operator N ki is connected to i node: '

*

Y ik = − y ki N ki

;

'

Y ki = – y ki N ki

(2.82"')

• The non-diagonal terms, in the case of line branch, are equal to the minus sign value of admittance of the incident line to the two i and k nodes. The non-diagonal term can be zero ( Y ik = 0 ) if there is no connection between i and k nodes. (iv) What is essential or specific for the [Ynn] matrix? First, for a network without transformers, this is a square and symmetrical matrix of size equal to the number of independent (n) nodes. Second, the module of self-admittance of the nodes – diagonal term – is bigger or at least equal to the sum of the modules of the non-diagonal terms:


127

Y ii ≥

∑Y

ik

i≠k

Sometimes, due to phase-shift transformers and capacitive susceptances of the lines it is possible that the previous inequality is not true. If we neglect the shunt components then Y ii = Y ik .

∑ i≠k

The number of null terms on rows and columns is equal to the number of branches incident to node plus 1 (corresponding to the self-admittance of the node). Third, in the real networks, the number of non-zero elements from matrix [Ynn] is low, being about 2%. The matrix [Ynn] is said to have a high degree of sparsity or to be a sparse matrix. Considering‚ for instance‚ a power system of 1000 nodes‚ the number of incident branches to a node does not exceed 10…15 (electric lines and transformers). In consequence‚ from 1000 terms of any row or column‚ only 11...16 terms are non-zero‚ the others 989...984 being equal to zero. For general case, of electric network, the relationship between nodal voltages and currents are expressed as:

[Y nn ][V n ] = [I n ]

(2.77)

In practice, in power systems operation analysis, three-phase powers and phase-to-phase voltages are used. In this respect, the expression (2.77) is multiplied by 3 resulting:

[Y nn ] 3 [V n ] =

3 [I n ]

Taking into consideration the relationship between phase-to-neutral voltages and phase-to-phase voltages U = 3 V , and noting I = 3 I , it results the known form of matrix equation (2.74) from nodal voltages method. Under these conditions, the expression of three-phase apparent power becomes: *

*

S = 3V I = 3 V ⋅ 3 I = U I mentioning that the currents I are

*

(2.83)

3 times bigger than the real ones I.

The nodal voltages method in the case of the three-phase models of the electric lines To calculate the asymmetrical load regimes of the phases it is useful to consider the equivalent Π circuit, thus emphasizing the self and mutual parameters (Fig. 2.20,a,b). Note that in the equivalent circuit from Figure 2.20,c, the hypothesis from equation (2.83) with phase-to-phase voltage and currents multiplied by 3 has been applied.

128

Basic computation

i

a

Ii

b

z acik

b

Ii

c

c

Ii

k

z aa ik

a

y aa ik

yab yac

ybc

y bb ik

y ccik

z bb ik

z ab ik

z ccik

z bc ik yab yac y aa ik

y bb ik

ybc

a

a

Ik

b

Ik

c

Ik

b

c

y cc ik

a. I

[I i ] = I I

a i b i c i

y aa y ab y ac = y ba y bb y bc 2 y ca y cb y cc

a

Vi

[Y ik]

b

[V i] = V i V

c i

a

z aa z ab z ac z ba z bb z bc z ca z cb z cc

i

[Z ik]

k

y aa y ab y ac y ba y bb y bc = [Y ik] 2 y ca y cb y cc

Ik b [I k] = I k c Ik V

[V k] = V V

a k b k c k

b.

[I i]

[U i]= 3 [V i]

i

[Z ik]

k [I k]

[Y ik] 2

[Y ik] 2

[U k]= 3 [V k]

c. Fig. 2.20. Equivalent circuit of a three-phase electric line.

Using direct writing rules of the nodal admittance matrix, the relationship between the nodal voltages and currents is (Fig. 2.20,c): [Y ik ] –[Z ]–1 ⎤ ⎡ –1 ik ⎥ ⎡ [U i ]⎤ ⎡ [I i ]⎤ ⎢[Z ik ] + 2 (2.84) ⎢[I ]⎥ = ⎢ [ Y ik ]⎥ ⎢⎣[U k ]⎥⎦ –1 ⎣ k ⎦ ⎢ –[Z ik ]–1 ⎥ [Z ik ] + 2 ⎦ ⎣


129

2.3.2.3. Active and reactive power flow Transmission line [2.3] In this section the expressions for the active and reactive power flows in transmission lines are derived. In this respect, consider the equivalent Π circuit from Figure 2.21. Sik

yik

i Iik

Vi

yik0

Ski

k yki0

Vk

Fig. 2.21. Equivalent Π circuit of the transmission line.

For the power flow calculation on a branch, we first consider the phase-toneutral voltages V i and V k and the current passed through the branch I ik , respectively. The apparent power at the sending-end has the expression: *

*

*

S ik = 3V i I ik = 3V i 3 I ik = U i I ik

(2.85)

where I ik is the value of current at the sending-end, determined by

I ik = V i y ik 0 + (V i − V k ) y ik =

[

1 U i y ik 0 + (U i − U k ) y ik 3

]

(2.86)

resulting: not

3 I ik = U i y ik 0 + (U i − U k ) y ik = I ik

(2.87)

where U i and U k are the phase-to-phase voltages. Next, we express the voltages as polar coordinates: U i = Uie

jθi

U k = Uke

= U i (cos θi + j sin θi )

jθk

(2.88,a)

= U k (cos θ k + j sin θ k )

and the series and shunt admittances as Cartesian or polar coordinates: y ik = g ik + jbik = yik e jγ ik = yik (cos γ ik + j sin γ ik )

(2.88,b)

y ik 0 = y ki 0 = g ik 0 + jbik 0

Thus, the expression of the power flow on a branch is calculated as follows: *

(

* S ik = U i I ik = U i ⎡U i y ik 0 + (U i − U k ) y ik ⎤ = U i2 y ik 0 + y ik ⎣ ⎦ 2 = U i ⎡⎣ yik ( cos γ ik − j sin γ ik ) + gik 0 − jbik 0 ⎤⎦ −

−U iU k yik ⎡⎣cos ( θi − θk − γ ik ) + j sin ( θi − θk − γ ik ) ⎤⎦

)

*

*

*

− U i U k y ik =

130

Basic computation

or S ik ≡ Pik + jQik

(2.85′)

Equating the real and imaginary parts, the active and reactive powers flowing on the transmission line from node i to node k are obtained: Pik = U i2 ( g ik 0 + yik cos γ ik ) − U iU k yik cos(θi − θ k − γ ik )

(2.89,a)

Qik = −U i2 (bik 0 + yik sin γ ik ) − U iU k yik sin (θi − θ k − γ ik )

The expressions of active and reactive powers flowing in opposite direction are: Pki = U k2 (g ki 0 + yki cos γ ki ) − U kU i yki cos(θ k − θi − γ ki )

(2.89,b)

Qki = −U k2 (bki 0 + yki sin γ ki ) − U kU i yki sin (θ k − θi − γ ki )

Transformer Consider the case of equivalent circuit with transformer operator and real turns ratio (Fig. 2.19,a,b) for which power losses, represented only through shunt admittance y i 0 located on the primary winding side, are taken into consideration.

The expressions of complex powers flowing through transformer depend on the side the taps are located. Consider first the case of the step-up transformer (Fig. 2.22,a) where the taps are located on the secondary winding side, and the series parameters are referred to the lower voltage side. Adopting the same convention as for transmission line, it can be written:

[

]

S ik = U i I i = U i y i 0 U i + y ik (U i − N ik U k ) *= *

= U i2 ( g i 0 − jbi 0 + yik ) − U i U k y ik N ik *

*

*

where: y i 0 = gi 0 − jbi 0 . Ii Sik

i Vi

Iik yi0

y ik

Nik k I k Ski Vk

i Vi

Ii Sik

i Vi

a.

Nki

k

y ki

yi0 V k

b.

Fig. 2.22. Equivalent circuits of transformer with shunt admittance. a. Step-up transformer; b. Step-down transformer.

Expressing in polar coordinates, obtain:

[

]

Pik + jQik = U i2 (g i 0 + jbi 0 ) + yik e − jγ ik − U iU k yik N ik e j (θ i − θ k − γ ik )

k Vk

Ik Ski


131

Separating the real and imaginary parts of the latter expression yields the active Pik and reactive Qik power flow expressions: Pik = U i2 (g i 0 + yik cos γ ik ) − U iU k yik N ik cos(θi − θ k − γ ik ) Qik = −U i2 (− bi 0 + yik sin γ ik ) − U iU k yik N ik sin (θi − θ k − γ ik )

(2.90,a)

The expressions of active Pki and reactive Qki powers flowing in opposite direction are:

[

]

S ki = U i ' I k = N ik U k y ik (N ik U k − U i ) *= U k2 y ik N ik2 − U k U i y ik N ik *

*

*

*

Expressing in polar coordinates, obtain: Pki + jQki = U k2 yik N ik2 e − jγ ik − U kU i yik N ik e j (θ k − θ i − γ ik ) then it results: Pki = U k2 yik N ik2 cos γ ik − U kU i yik N ik cos(θ k − θi − γ ik ) Qki = −U k2 yik N ik2 sin γ ik − U kU i yik N ik sin (θ k − θi − γ ik )

(2.90,b)

Likewise, if consider the case of the step-down transformer (Fig. 2.22,b), where the taps are located on the primary winding side, the expressions of active and reactive powers for both directions are:

(

)

Pik = U i2 g i 0 + yki N ki2 cos γ ki − U iU k yki N ki cos(θi − θ k − γ ki )

(

)

Qik = −U i2 − bi 0 + yki N ki2 sin γ ki − U iU k yki N ki sin (θi − θ k − γ ki ) Pki = U k2 yki cos γ ki − U kU i yki N ki cos(θ k − θi − γ ki ) Qki = −U k2 yki sin γ ki − U kU i yki N ki sin (θ k − θi − γ ki )

(2.91,a)

(2.91,b)

Analysing the expressions of powers flow through both step-up and stepdown transformers observe that these are identical, taking also into consideration the relationships (1.111,b). Furthermore, if consider the transformer operates on the median tap, and introducing the quantities in per units, that is N ik ≅ N ki ≅ 1 , obtain the expressions of powers flow similar to the ones for transmission line.

2.3.2.4. Nodal equations By definition, the nodal power is the difference between the generated and consumed powers into a node. Taking into consideration that one set of generator units (noted by g) and one set of loads (noted by c) are connected at i node, the expression of the nodal complex power is (Fig. 2.23): S i = Pi + jQi = S gi − S ci

(2.92)

132

Basic computation

node i Sgi=Pgi+jQgi

k α(i)

Sci=Pci+jQci Fig. 2.23. Nodal powers balance.

According to conservation of powers at node i it results:

∑S

Si − or Pi =

k ∈α ( i )

∑P

;

ik

ik

=0

Qi =

k ∈α ( i )

∑Q

ik

k ∈α ( i )

Taking into account the expressions of powers flow on a branch (2.89), the exchanged powers between the i node and remaining part of the network through the nodes directly connected with it, are: Pi = U i2

n

n

∑ (gik 0 + yik cos γ ik ) − U i ∑U k yik cos(θi − θk − γ ik ) k =1

Qi =

−U i2

k =1

n

∑ (b

ik 0

+ yik sin γ ik ) − U i

k =1

(2.93)

n

∑U

k

yik sin (θi − θ k − γ ik )

k =1

Next, we express equations (2.93) in terms of nodal admittance matrix elements. According to Figure 2.23, the terms of nodal admittance matrix are: Y ii =

∑ (y

k ∈α ( i )

ik

) ∑ [( y

+ y ik 0 =

n

cos γ ik + g ik 0 ) + j ( yik sin γ ik + bik 0 )]

ik

k =1

(2.94,a)

= Yii e jγ ii ≡ Gii + jBii Y ik = − y ik = − yik e jγ ik = Yik e jγ ik ≡ Gik + jBik

(2.94,b)

Therefore, the two equations of balance powers become: Pi = U i2 Re{Y ii } + U i

n

∑U Y

k ik

cos(θi − θ k − γ ik )

k =1

Qi =

−U i2

Im{Y ii } + U i

(2.95)

n

∑U Y

k ik

k =1

sin (θi − θ k − γ ik )


133

If equations (2.88,a) and (2.94) are used, the expression of nodal power becomes: *

Si =U i Ii =U i

n

∑Y

* * ik U k

≡ Pi + jQi

k =1

or Si =

n

∑U U e i

k

j ( θi −θk )

(Gik − jBik ) =

k =1

=

n

∑U U {G [cos(θ i

k =1

k

ik

i

− θ k ) + j sin (θi − θ k )] −

(2.96)

− jBik [cos(θi − θ k ) + j sin (θi − θ k )]}

The expressions for active and reactive power injections are obtained by identifying the real and imaginary parts of equation (2.96), yielding: Pi (U m , θm ) =

n

∑U U [G i

k

ik

cos(θi − θk ) + Bik sin (θi − θk )] =

k =1

=

n

∑U U GG i

k

ik

=

GiiU i2

+

k =1

n

∑U U GG i

k

(2.97,a)

ik

k =1, k ≠ i

respectively Qi (U m , θm ) = − BiiU i2 −

n

∑U U BB i

k

ik

(2.97,b)

k =1, k ≠ i

where:

GGik = Gik cos(θi − θk ) + Bik sin (θi − θk ) BBik = Bik cos(θi − θk ) − Gik sin (θi − θk )

(2.98)

and Um and θm are vectors of state variables. From (2.92) and (2.97) results the mathematical model of steady state: Pgi − Pci = Pi (U m , θ m )

Q gi − Qci = Qi (U m , θ m )

(2.99)

2.3.2.5. Power losses For a branch i-k, the total power losses are simply calculated with formula: ∆ S ik = S ik + S ki

(2.100)

or, by separating the real and imaginary parts obtain the active and reactive power losses:

134

Basic computation

∆Pik = Pik + Pki

(2.101,a)

∆Qik = Qik + Qki

(2.101,b)

For the transmission line, the reactive power losses can have negative sign, due to the capacitive shunt currents, that means the line generates reactive power.

2.3.2.6. Basic load flow problem Problem variables The load flow problem can be formulated as a set of non-linear algebraic equality/inequality constraints. These constraints represent both Kirchhoff’s theorems and network operation limits. In the basic formulation of the load flow problem, four variables are associated to each node i: Ui – voltage magnitude (node i); θi – voltage angle; Pi and Qi − net active and reactive powers (algebraic sum of generation and load). Basic node types Depending on which of the above four variables are known (given) and which ones are unknown (to be calculated), three basic types of nodes can be defined (Table 2.2). The major problem in the definition of node types is to guarantee that the resulting set of power flow equations contains the same number of equations and unknown quantities, as are normally necessary for solvability. For each node we have four unknown state quantities P, Q, U, θ, and only two equations (for active and reactive power balance). This requires that two of the state quantities to be specified, the other two resulting after the steady state calculation. Table 2.2 Node types Node type

Generator node

Passive node Slack (swing) node

Specified P, U

G pure

G hybrid

or

P,Q

Load node

Quantities

Symbol

C

(Q

min

, Q max )

Unknown θ, Q

P(Q) C

or

C

YC

S

BC(GC)

P, Q

θ, U

P = 0, Q = 0

θ, U

Us, θs = 0

P, Q


135

These nodes are explained as follows: • At generator node (PU node), active power P, voltage magnitude U as well as reactive power limits (Qmin and Qmax) are specified. Fixing a certain voltage level U sp at this type of node is possible due to the control possibilities through reactive power support from generators. After calculation, the generated reactive power Qg and the angle voltage θ are determined. At “hybrid” generator node, the injected power is equal to the algebraic sum of the power produced by the generator unit and the power absorbed by local load. • Load node (PQ node) must have either both active and reactive powers specified or only one of the powers plus a parameter such as conductance (Gc) or susceptance (Bc). The passive nodes of zero injected powers are also included in this category. In these nodes there are no connected loads or, if any, they are represented through constant admittance (Yc) or impedance (Zc). • Slack (swing) node (Uθ node), where the voltage magnitude Us and the phase angle θs = 0 are specified, has a double function in the basic formulation of the power flow problem: it serve as the voltage angle reference and since power losses ∆ S are unknown in advance, the active and reactive powers generation of the Uθ node are used to balance generation, load and losses, and are determined at the end of steady state calculation. The apparent power at the slack node should be: Ss =

∑S

cj

+ ∆S −

j∈c

∑S

gi

i∈ g

Therefore, the slack node must be chosen so that it can undertake the inaccuracies introduced by the power losses in the network. Usually, this role is performed by the most important power plant of the system.

2.3.2.7. Seidel – Gauss method Let us consider the expression of the injected current in i ∈ g ∪ c node [2.9]: *

Ii=

Si

= *

Ui

n

∑Y

ik U k

(2.102)

k =1

and by separating the current corresponding to one of the independent nodes, namely i node, it becomes: I i = Y ii U i +

n

∑Y

k =1, k ≠ i

ik U k

, i = 2,..., n ; i ≠ s (= 1)

Thus, the voltage at i node can be written as: Ui =

n ⎞ 1 ⎛⎜ Ii − Y ik U k ⎟, i = 2,..., n ⎟ Y ii ⎜⎝ k =1, k ≠ i ⎠

∑

which represents the fundamental equation of Gauss iterative method.

(2.102')

136

Basic computation

For the beginning, a set of initial voltages U i( 0) is considered. Usually these voltages are equal to the nominal ones, except for the specified voltages, as magnitude, at the slack node and generator nodes, which are constant during the calculation process. For the (p+1) step following any given p step in the iterative process, the linear relationship (2.102') becomes: n ⎛ ( p) ( p) ⎞ ⎜Ii − Y ik U k ⎟, i ≠ s ⎜ ⎟ k =1; k ≠ i ⎝ ⎠

( p +1 )

=

1 Y ii

( p)

=

Si

Ui

∑

(2.102")

where Ii

( p )* ( p )*

Ui

( p)

Si

;

= Pi + jQi( p )

In the framework of Seidel – Gauss method, the finding of solution is accelerated, by using in the (p+1) step the values of all the nodal voltages U k , with k Qimax , then Qi( p ) = Qimax . That means the reactive power support is not sufficient to maintain the voltage at the specified value, and the voltage at iteration (p) is the calculated value ( p) ( p) U i = U i , calc . Note that after the convergence test is satisfied, generator nodes must be again treated accordingly, that is depending on the reactive power calculated, the final voltages are established as explained earlier. Also, in the end of the calculation process, after the (p+1) iteration when the convergence test is satisfied, the apparent complex power at the slack node s is calculated with the expression: final

Ss

( p +1)

=Ss

*

= U s2 Y ss + U s

n

∑

*

( p +1)*

Y sk U k

(2.107)

k =1,k ≠ s

Observations [2.4]: a) If the calculated real power generation violates generator limits, the excess (or deficiency) of slack node generation is distributed among the remaining units, and more load flow iteration are carried out. This adjustment is repeated until slack node generation is within acceptable limits; b) Also, if slack node reactive power generation violates generator limits, then a number of possibilities may be considered. One possibility is to change the slack node to a different generator. Another is to change slack node voltage appropriately without violating its voltage limits. A third possibility is to introduce reactive generation and/or load by means of the switching of appropriate capacitor and/or inductor banks. In practice, for the control of the convergence of iterative process, there are also other criteria to use, the most common consisting in testing the module of the difference between the apparent powers at the slack node, calculated at two successive iterations: ( p +1)

Ss

( p)

− Ss ≤ ε


139

Once the state vector is calculated we could determine the injected powers and the load flow on the network branches. In chapter 8 “Performance methods for power flow studies”, methods of Newton-Raphson type for steady state calculation are presented.

2.4. Reconfiguration of the distribution electric networks Usually, the urban distribution electric networks consist of underground cables. These cables rise some problems concerning the repairing of insulation damages or braking of conductors. The time necessary to detect and repair the damages can be important, time in which many loads can remain not supplied. In order to cope with this inconvenient, a back-up supply is recommended, which implies the existence of at least two supply paths for each consumption point, from the same source or from different sources. In order to limit the number of loads affected by a short circuit emerging into the electric networks with such configurations, the networks are operated in radial configuration. The rural distribution electric networks mainly consist of overhead electric lines. This type of lines do not present special problems in detecting and repairing the damages. In addition, the density of the loads supplied by these networks is much smaller than the one in urban networks. A reserve in the power supply of these loads is not economically justified, the structure of the rural distribution electric network being usually arborescent or radial. Although these networks have a meshed structure they are operated in radial configuration. In practice, for short periods of time, the distribution electric networks can de operated in a meshed configuration, especially when reconfiguration manoeuvres are performed within the network.

2.4.1. Operating issues Generally, by reconfiguration of a physical system is understood the modification of the operational connections that exist among its components, in order to improve the system operation as a whole or just a part of it, without modifying the characteristic parameters of the system components. In the particular case of distribution electric networks, the reconfiguration aims at improving and optimising the operating state by changing only the topological state “in operation” / “out of service” of some electric lines. The network reconfiguration is possible only for meshed networks, for which the arborescent operation is recommended. For such configuration, the set of electric lines “in operation” and “out of service” have a well determined number of elements, the number of the electric lines “in operation” being equal to the number of load nodes. The elements of these two sets can be exchanged subject to the arborescent operation of the network. The advantage is the possibility of achieving the most suitable configuration in order to improve or optimise the operating state, in terms of the strategy of network configuration and of the electricity demand. For

140

Basic computation

example, consider a simple meshed electric network, which supplies n loads (Fig. 2.25). The arborescent configuration allows us to achieve n + 1 possible arborescent configurations in operation. Source A

1

2

...

k

...

n

Source B

Base network

Network sectionalization

...

Configuration 1

...

Configuration k

Configuration n+1

Fig. 2.25. Possible arborescent configurations for a simple meshed electric network.

The reconfiguration process can be applied for all the possible operating conditions of a distribution electric network: – normal conditions, characterized by the availability of all the network elements, the state quantities being within the admissible operating limits; – critical conditions, characterized by the availability of all the network elements, with some of the state quantities being at the limit of normal operation (the thermal limit, the voltage stability limit, etc.); – emergency operation, characterized by the unavailability of one or more elements of the network, due to operation under critical conditions on expanded period of time or to some accidental damages emerged from outside the network. For the normal and critical conditions, finding the optimal configuration of an electric network actually implies network reconfiguration, but for the emergency operation the process becomes one of reconstruction. Usually, under normal conditions, the purpose is to reach an optimum in operation in order to minimize the active power losses and energy losses and to improve the security in supplying the loads. For the critical conditions the goal of the reconfiguration process is to restore the network normal operating state, by load reducing and balancing the lines load as well as by reducing the voltage drops and also by obtaining uniformity of the voltage level at the loads. For the emergency operation the goal is to supply as many as possible loads after the detection and isolation of the fault. In this case, the optimisation is of lower interest, more important being the restoration of the power supply of all loads in a time as short as possible and the reducing of the financial penalties for the electricity not supplied.


141

At least two arguments are supporting of the reconfiguration process: – the operating state of the network can be improved by a reduced coordination effort, achieving considerable results. The advantages mainly consist in decreasing the active power losses and, in most cases, in decreasing the reactive power losses as well as the decreasing of the line load, the decreasing of the voltage drops and the improvement of the voltage level at loads. The effort done for network reconfiguration is related to the cost of the manoeuvres necessary to change the present configuration and, eventually, the cost of the electricity not supplied during these manoeuvres; – a second aspect refers to the dynamics of the power energy demanded by the loads. The load curve can be significantly changed either for long or for shorter periods of time, causing the change of the load gravity centre and thus of the operating state of the network. Therefore, specific (normal) operating configurations can be defined for each period of time in terms of the season and the characteristics of the consumer activity during the week-days.

2.4.2. Mathematical model of the reconfiguration process The reconfiguration process of a distribution electric network can be seen as an optimisation problem. To define the mathematic model, we start from the observation that to any electrical network, consisting of n nodes and l branches, a graph G ( X, A ) can be assigned, where X is the set of nodes and A is the set of branches. To these sets, state or operational quantities can be also assigned, which characterize the operating state of the network. Therefore, to the set A of the branches it can be assigned: – the set I of state quantities, representing the branches currents; – the set C of decision quantities, representing the topological states of the branches; for any branch l from the set A , the topological state can be: – cl = 1 , if the branch l is “in operation”; – cl = 0 , if the branch l is “out of service”. To the set X of the nodes, it can be assigned: – the set U of state quantities representing the nodal voltages; – the set F of quantities representing the reliability indices of the nodes. Based on these notations, the mathematical model of the reconfiguration optimisation problem has the general form [2.15], [2.16]: OPTIM ⎡⎣ f ( U, I, C, F ) ⎤⎦ subject to equality and inequality constraints:

(2.108)

142

Basic computation

g ( U, I, C, F ) = 0 h ( U, I, C, F ) > 0

(2.109)

In equation (2.108), f ( U, I, C, F ) represents the objective function, which, in the general case, it can be written as: f ( U, I, C, F ) = α1 f1 ( U, I, C, F ) + α 2 f 2 ( U, I, C, F ) + K + α n f n ( U, I, C, F ) (2.110) where: f1 , f 2 ,K, f n represent the weighting of criteria taken into account; α1 , α 2 ,K, α n – weight coefficients of every criterion. The criteria that can be used in the objective function for the electric distribution network reconfiguration problem are: – real power losses decrease; – decrease and balancing the branch load; – voltage drops decrease; – improve the safety in power supply of the loads; – decrease the manoeuvres cost. Analysing the criteria shown above, it can be seen that, in most of the cases, for the mathematic model solution, the main goal is the minimization of the objective function. There can be also situations when the goal is to find the maximum of the objective function. In terms of the number of criteria taken into account, the objective function is of single-criterion type, when only one criterion is considered, or multi-criterion type, when two or more criteria are considered. The constraints can be related to the network exploitation or operation: – the network connectivity or the supply of all loads, constraint checked by applying the Kirchhoff’s current law in all load nodes; – the arborescent configuration of the network; – the security in operation, which refers to branch load, voltage drops as well as nodal voltage level; – the reliability level in the power supply of the loads; – the possibility of the network branches to be subjected to manoeuvres; – the maximum number admitted for manoeuvres to change the network operating configuration. A synthesis on the issues that can be taken into consideration in the reconfiguration process is presented in Table 2.3. Of the many issues presented above, that can be taken into account in the mathematical model of the reconfiguration problem, only the following aspects are of interest in operation: – active power losses ( ∆P ); – branch load ( I I adm ); – voltage drops ( ∆U );


143

– maximum yearly number of interruptions ( N int ); – duration of the supply restoration ( Tdint ). Table 2.3 Issues used currently in the reconfiguration of the distribution electric networks

Power losses Manoeuvres cost

Normal Criterion Criterion

Security in power supply

Criterion

Branch load

Constraint

Voltage drops

Constraint

Arborescent configuration Configuration connectivity Executing manoeuvres on certain electric lines and transformers Admitted number of manoeuvres

Restriction Restriction

Operating state Critical Criterion/ Constraint Criterion/ Constraint Criterion/ Constraint Restriction Restriction

Restriction

Restriction

Restriction

Restriction

Restriction

-

Issue

Emergency Criterion/ Constraint Criterion/ Constraint Criterion/ Constraint Restriction Restriction

The objective functions assigned to these issues are:

∆P =

∑R I

2 l l cl ;

l∈A

{

}

I I adm = max I l I ladm ; l∈A

⎧⎪ Zl Il ∆U = max ⎨ k∈X ⎪⎩ l∈Dk

∑

⎫⎪ ⎬; ⎭⎪

(2.111)

N int = max { N ( λ ek , µ ek )} ; Tdint = max {Td ( λ ek , µ ek )} k∈X

k∈X

where: Z l = Rl + jX l is the impedance of the branch l;

I ladm – the admissible current (thermal limit) of the branch l; Dk – the path between the node k and the source node; λ e k – the equivalent failure rate of the node k with respect to the source µe k

node; – the inverse of the mean time to repair of the node k with respect to the source node.

The reconfiguration process constraints can be written under the form:

144

Basic computation

ik =

∑ I c , k ∈ X; l l

lR = l − n + nC ;

l∈A k

I l ≤ I ladm , l ∈ A;

∑Z

l

I l ≤ ∆U adm , k ∈ X;

l∈Dk

(2.112)

adm N man ≤ N man ; adm N int ≤ N int ;

where: i k Ak lR nC

is – – –

Tdint ≤ Tdadm max

the complex load current at the node k; the set of branches adjacent to the node k; the number of “out of service” branches; the number of load nodes;

∆U adm – the admissible voltage drop in the network; N man – the number of manoeuvres necessary to obtain the final

adm N int

configuration; – the maximum admissible number of manoeuvres necessary to obtain the final configuration; – maximum yearly number of interruptions in the load supply;

Tdadm max

– maximum admissible duration of the supply restoration.

adm N man

In order to identify the theoretical and practical possibilities for reconfiguration problem solution we start from some remarks regarding the mathematical model. In the case when the problem solution does not involve the voltage change at the source node, the size of the sets U , I and F implicitly depend on the quantities of the set C , so that the objective function can be written as:

OPTIM ⎡⎣ f ( U ( C ) , I ( C ) , F ( C ) ) ⎤⎦

(2.113)

Therefore, in solving the optimisation reconfiguration problem, the final goal is to explicitly determine the decision variables cl , l ∈ A . These are discrete binary variables, which can be equal to 1 or 0. Under these conditions, the mathematical model has the form of a general mathematical programming problem with discrete variables. Furthermore, for the previously discussed issues, the functions assigned to each criterion, as well as the ones that describe the constraints, have a convex character. Hereby, the mathematical model takes the form of a convex programming problem with discrete variables. Out of the previous remarks, the conclusion that comes out is that a theoretical possibility to solve the mathematical model consists in the use of tools specific to the mathematical programming: linear programming, convex programming, dynamic programming, etc.


145

A modern and relatively new possibility in solving the reconfiguration optimisation problem is based on artificial intelligence techniques, such as decision trees, genetic algorithms, fuzzy logic, expert systems, Petri nets, etc. A practical possibility to obtain the solution to the reconfiguration process consists in searching within the solutions’ space, which is the set of all arborescent configurations that can be generated for an electric network with a meshed structure. The number of elements of the solutions space is directly influenced by the complexity and the geographical spread of the electric network. Only a small part of the possible arborescent configurations of an electric network, that form the solutions’ space, fulfil the inequality constraints, and they form what is called the set of allowed operating configurations. Because the optimum is related to various issues, the final configurations obtained after the reconfiguration process can be different. The goal of reconfiguration is to identify those optimal configurations which fulfil all the technical and operational constraints. The methods based on searching within the solutions’ space, used for reconfiguration problem solution, can be systematic or heuristic. In the frame of the systematic methods (uninformed searching methods) all the possible arborescent configurations of a distribution electric network are individually generated and analysed. The configuration corresponding to the optimal operation subjected to the main objective is further considered. As far as this principle is concerned, the systematic search methods are optimal methods, which ensure finding the global optimal solution. This is the main advantage of the systematic searching methods. Although there are only two possible values for each variable, it is rather difficult to apply this kind of methods for most of the distribution networks because of the very large number of arborescent configurations which have to be generated and analysed. This is the main disadvantage on the systematic methods. Solving the mathematical model Mathematical Programming

Artificial Intelligence Techniques

Searching inside the Solution Space

Linear Programming

Decision trees

Systematic Search (uninformed)

Convex Programming

Genetic Algorithms

Heuristic Search (informed)

Dynamic Programming

Fuzzy Logic

Expert Systems

Petri Networks

Fig. 2.26. Possibilities of solving the reconfiguration problem.

The heuristic methods (informed searching methods) are used in order to decrease the number of configurations that should be analysed to achieve the

146

Basic computation

reconfiguration solution. These methods use a number of observations that allow filtering for analysing only the intermediary configurations that lead to a final solution close to or even identical to the optimal global solution. The advantage consists in a considerable reduced computation time and effort to the detriment of the fact that they are not optimal. The theoretical and practical possibilities of solving the reconfiguration optimisation problem are synthetically presented in Figure 2.26.

2.4.3. Reconfiguration heuristic methods A heuristic method is a searching procedure that allows for an easy solving of a combinative problem. The existence of the heuristic methods is based on the use of a set of observations, rules and knowledge, gained from previous experience or theoretically developed, that allow filtering for analysis only the solutions that lead to a final solution close to or even identical to the optimal global one. For the reconfiguration of the distribution electric networks, the main issue that allows the use of heuristic methods is related to the variation of the currents’ curve within network branches for different arborescent configurations. For a simple meshed network consisting of an electric line supplying n loads (Fig. 2.27), consider that the network splitting is performed by opening the line between the nodes k − 1 and k, the currents flowing through the other line sections increase toward the source nodes A and B. The increase of these currents has a convex form. When the line section chosen for network splitting is changed, the currents curve is moved up or down, whilst preserving the shape. Ik

Ik

| In+1|

| I1|

| In|

| I2|

| In-1| | Ik-1| | Ik+1| A I1 1 I2 2 VA

i1

Ik-1 k-1 ik-1

k Ik+1 k+1 ik

ik+1

In-1 n-1 In n In+1 B in-1

in

Fig. 2.27. Variation of currents flowing through the branches of a simple meshed network that operates radially.

VB


147

The heuristic methods starts from an initial configuration, chosen based on specific requirements, and scan for improved configurations. If there is at least one improved configuration, the selection criterion replaces the actual configuration with the improved one. The procedure ends when no improved configuration ca be found by applying the searching mechanism for the actual configuration. An improved configuration of the actual one is defined as being that configuration which leads to the evolution of the objective function value in the desired direction.

2.4.3.1. Reconfiguration strategies The heuristic reconfiguration methods of the distribution electric networks are based on three strategies [2.15], [2.17]: – “constructive” strategy, in which all the branches of the initial configuration are in “out of service” state. By successively transitions to the “in operation” state of some branches, the desired arborescent configuration is achieved (Fig. 2.28). Because each load node can be supplied by just one branch, the number of intermediary steps necessary to achieve the final configuration is equal to the number of the load nodes.

(1)

(3)

(2)

Fig. 2.28. Principle of the “constructive” strategy.

– “destructive” strategy, in which all branches of the initial configuration are “in operation” state. By successive transition to the “out of service” state of some branches the desired arborescent configuration is achieved (Fig. 2.29). The number of intermediary steps necessary to achieve the final configuration is equal to difference between the total number of branches and the number of the load nodes.

(1)

(2)

Fig. 2.29. Principle of the “destructive” strategy.

– the “branch exchange” strategy starts from an initial arborescent configuration and preserves the arborescent character during the process. For the transition from one configuration to another, a branch is switched “in operation” and than another one, from the loop resulted from this manoeuvre, is switched “out

148

Basic computation

of service” (Fig. 2.30). While for the previous strategies the number of intermediary steps necessary for achieving the final configuration is well defined, for this strategy the number of steps depends on many factors, out of which the most important are the searching manner of the substituting configuration and its selection criterion.

(1)

(2)

(3)

Fig. 2.30. Principle of the “branch exchange” strategy.

Since for the “branch exchange” reconfiguration strategy, the path between the initial and the final configuration is not unique, for identifying improved configurations, several strategies can be applied (Fig. 2.31).

Local

Descending

Ordered

Dynamic Strategy for search of improved configurations

Random

Maximal Irrevocable (irreversible)

Tentative (reversible)

General reversible

Backtracking

Fig. 2.31. Strategies for searching improved configurations [2.12].

Taking into account the convex variation of the curve of the branch currents for a simple meshed network, for the criteria based on these currents, the searching process is simplified. Therefore, starting from an “out of service” branch, branch exchange is subsequently performed with the two adjacent branches. If an improved configuration is found for one of these exchanges the search continues in this direction with the next branch, until no improved configurations can be found. Depending on the fact that the searches continue or not, the search is called descending or local.


149

In a complex meshed network the number of “out of service” branches can be greater or equal to 2. Examination of “out of service” branches can be random or ordered; in the second case various ordering criteria can be used, such as: the voltage drop at the terminals of these branches, the resistance of the loop to which the branch is assigned, etc. If for a complex meshed network, starting from the actual configuration, different improved configurations can be found, the substitution configuration can be decided upon by choosing the best of them (maximal strategy) or the first configuration encountered (dynamic strategy). The main disadvantage of the heuristic methods consists in the fact that the global optimal solution is not guaranteed. The search around local optimums is avoided by returning to a previous improved configuration and restarting the search on a different path. For this procedure the search strategies can be irrevocable (irreversible) or tentative (reversible) [2.18]. In an irreversible search once a certain improved configuration is found the search mechanism does not return to previous improved configurations. A tentative search strategy returns to a previous configuration by following either exactly the path between the initial and the final configuration (backtracking search) or any other path (general reversible search).

2.4.3.2. Heuristic methods for active power losses reducing Reducing the active power losses is the main objective of the reconfiguration process of the distribution electric networks operated under normal conditions. The improved configurations are strictly subjected to the inequality constraints, especially to those referring to line load, nodal voltage level and voltage drops. “Power losses” include three components, namely (see Chapter 7): – own technologic consumption; – technical losses; – commercial losses. The reduction that can be achieved by reconfiguration is aimed at Joule losses, that belong to own technologic consumption, the objective function of this criterion having the form: ⎡ ⎤ MIN [ ∆P ] = MIN ⎢ Rl I l2 cl ⎥ ⎣ l∈A ⎦

∑

(2.114)

Minimization of the active power losses by reconfiguration requires, irrespective of the strategy employed, to start from an initial configuration and adopting some intermediary configurations to reach the final configuration. Theoretically, the analysis of the objective function while passing from one configuration to another is performed by load flow calculation and evaluating the total active power losses of the network. For the linear model of the network, the evaluation of the active power losses variation can be performed without load flow calculation. In this regard, the results obtained in Appendix 2.2 are used further on to describe some heuristic methods for active power losses reduction.

150

Basic computation

A. Branch exchange strategy Consider a simple meshed electric network supplying n loads (Fig. 2.32,a). The initial radial configuration from which the “branch exchange” strategy starts is the one in which the splitting is done between the nodes k and k + 1 (Fig. 2.32,b). IA A

1

2

z1 VA

k-1

z2 i1

k zk

zk-1

zk+1

ik-1

i2

k+1

n

B IB zn+1

zn

ik

ik+1

in

k

k+1

In n

ik

ik+1

VB

a.

A

I1 z1

VA

1

2

I2

Ik-1 k-1 zk-1 ik-1

z2 i1

i2

Ik zk

zn

In+1 B zn+1

in

VB

b. Fig. 2.32. Simple distribution electric network: a. meshed network; b. meshed electric network with radial operation.

• The local load transfer of one load or of a group of loads between two neighbouring feeders is performed by doing an elementary exchange for an ”out of service” branch. The selection of this branch exchange is based on using the equation. (A2.2.8) to estimate the active power losses variation δP generated by the load transfer from one feeder to another. The condition for the load transfer is δP < 0 . For the network shown in Figure 2.32,b, consider the transfer of the load from the node k located on the feeder supplied from node A to the feeder supplied from node B (Fig. 2.33). ∆ ik=-ik A VA

I1-ik

1

I2-ik

2

z2

z1 i1

∆ ik=+ik Ik-1-ik

k-1

k

Ik=ik k+1 zk+1

zk-1 i2

ik-1

ik

ik+1

In+ik zn

n in

In+1+ik B zn+1

VB

Fig. 2.33. Simple meshed electric network after the transfer of the node k from the feeder A to the feeder B.

After the load transfer the currents through the line sections between the nodes A and k will decrease with the value of i k , and the currents through the line sections between the nodes B and k will increase with the same value. By applying equation (A2.2.8) for the mentioned line sections, for the current flows


151

corresponding to the situation previous to the load transfer, and considering that ∆i k = ±i k , one obtains: k k ⎡ k ⎤ δPA,k = 3 ⎢ik2 ∑ ri − 2ika ∑ ri I ia − 2ikr ∑ ri I ir ⎥ i =1 i =1 ⎣ i =1 ⎦

δPB ,k

n +1 n +1 ⎡ n ⎤ = 3 ⎢ik2 ∑ ri + 2ika ∑ ri I ia + 2ikr ∑ ri I ir ⎥ i = k +1 i = k +1 ⎣ i = k +1 ⎦

(2.115)

By summing up the above relations, the active power losses variation, as a result of the transfer of the node k from the feeder A to the feeder B, is [2.19]: k k ⎡ ⎛ n +1 ⎞ ⎛ n +1 ⎞⎤ δP = 3 ⎢ik2 RAB + 2ika ⎜ ∑ ri I ia − ∑ ri I ia ⎟ + 2ikr ⎜ ∑ ri I ir − ∑ ri I ir ⎟ ⎥ (2.116) i =1 i =1 ⎝ i = k +1 ⎠ ⎝ i = k +1 ⎠⎦ ⎣

where RAB is the resistance between nodes A and B. • The optimal currents pattern represents the current flows through the branches of a simple meshed electric network for which the active power losses are minimised, in comparison with any other operating state. For a simple meshed electric network, the optimal currents’ pattern corresponds to the natural repartition of currents through the line sections, considering only their resistances, given that the voltages at the two ends are equal [2.1, 2.14, 2.20]. Consider that the meshed electric network in Figure 2.32,a has equal voltages at both ends. The current flows through the network branches is determined starting from the current I A or I B and subsequently applying Kirkhoff’s current law in nodes 1, 2,K, n , or in the nodes n,K , 2,1 , respectively. The currents injected by the two supplying nodes are calculated with the relations (2.29,a) and (2.29,b) adapted for the situation in which only the branches resistances are considered: n

IA =

∑R i k =1

’ k k

RAB

; Rk’ =

n +1

∑r

i = k +1

n

IB =

∑R i k =1

k k

RAB

i

(2.117)

k

; Rk = ∑ ri i =1

To obtain the radial configuration from Figure 2.33, the line section between nodes k − 1 and k from the meshed network is switched “out of service” state (Fig. 2.33). The currents through the line sections between nodes A and k decrease with the value of I k , and the currents through the line sections between nodes B and k increase with the same value. By applying the relation (A2.2.8) for the mentioned line sections, corresponding to the current flows in meshed operation and considering that ∆i k = ± I k , the active power losses variation, after transforming the simple meshed networks into two radial sub-networks, is:

152

Basic computation k k ⎡ ⎛ n +1 ⎞ ⎛ n +1 ⎞⎤ δP = 3 ⎢ I k2 RAB + 2 I ka ⎜ ∑ ri I ia − ∑ ri I ia ⎟ + 2 I kr ⎜ ∑ ri I ir − ∑ ri I ir ⎟ ⎥ (2.118) i =1 i =1 ⎝ i = k +1 ⎠ ⎝ i = k +1 ⎠⎦ ⎣

Taking into account that the current flows for the line sections of the simple meshed network has been calculated considering only the branch resistances, the voltage drops of the nodes A and B with respect to the node k can be written as: k

k

k

i =1

i =1

∆V Ak = ∑ ri ( I ia + jI ir ) = ∑ ri I ia + j ∑ ri I ir ∆V Bk =

n +1

∑ r (I

i = k +1

i

ia

+ jI ir ) =

i =1

n +1

∑ rI

i = k +1

i ia

n+1

+ j ∑ ri I ir

(2.119)

i = k +1

Assuming equal voltages at both ends V A = V B , the voltage drops of the nodes A and B with respect to the node k are identical ∆V Ak = ∆V Bk , and subtracting the first row of equation (2.119) from the second one, one obtains: k ⎛ n +1 ⎞ − r I ⎜ ∑ i ia ∑ ri I ia ⎟ + i =1 ⎝ i = k +1 ⎠

k ⎛ n +1 ⎞ j ⎜ ∑ ri I ir − ∑ ri I ir ⎟ = 0 i =1 ⎝ i = k +1 ⎠

(2.120)

which leads to: n +1

∑ rI

i = k +1

i ia

n +1

k

− ∑ ri I ia = 0 i =1 k

∑ r I − ∑r I

i = k +1

i ir

i =1

i ir

(2.121)

=0

Based on the previous results, relation (2.118) becomes: δP = 3I k2 RAB

(2.122)

The previous equation shows that by transforming a simple homogeneous meshed network, which operates with equal voltages at both ends, into two radial sub-networks, the active power losses increase. The smallest increase is recorded on the line section with the smallest current. Consequently, applying the optimal current pattern method, in order to reduce the active power losses, three steps are necessary: a) Closing a loop by switching a line section from the “out of service” state to the “in operation” state; b) Calculating the optimal currents pattern in the closed loop achieved previously; c) Identifying the line section from the loop whose current magnitude is minimum and switching it in the “out of service” state.

B. Destructive strategy Employment of the destructive type strategy for active power losses reduction is based on the notion of optimal currents pattern in a loop. As presented


153

in section 2.4.3.1, the destructive strategy consists in subsequent openings of the loops within a complex meshed network, until the final radial configuration is achieved. The selection criterion at each computation step of the branch of a loop that should be switched “out of service” is based on the conclusions provided by equation (2.122). The following steps are required: a) Load flow calculation, considering only the branch resistances; b) All closed loops are individually analysed, by identifying the “origin” nodes and calculating the cumulative resistance of the line sections between them; c) Identifying for each loop, the branch with the minimal current and calculating the power losses variation after switching the branch “out of service”; d) Selecting the branch for which the variation of the power losses δP has the lowest value and switching it “out of service”.

C. Constructive strategy Employment of this strategy in the reconfiguration process for active power losses reducing is based on the results obtained in Appendix A2.2. This strategy consists in subsequently switching “in operation” some network branches, until the final arborescent configuration is achieved. The selection of the branch that should be switched “in operation” among the candidate branches at each computation step, is based on the active power losses increase minimization criterion [2.15], [2.17]. Consider the electric network in Figure 2.32, for which, at a certain computation step, a choice between introducing either the line section between the nodes k − 1 and k or the line section between the nodes q and q + 1 is necessary (Fig. 2.34). A

I1

1

z1 VA

i1

Ik-1 k-1 zk-1 ik-1

k

q

ik

iq

zk

q+1 zq+1

Iq+2

n

zq+2 iq+1

In+1

B zn+1

in

VB

Fig. 2.34. Selecting the branch that has to be added to the network.

By applying the expression (A2.2.8) to the current flows corresponding to Figure 2.34, the active power losses variation for both cases becomes: k −1 k −1 ⎡ k ⎤ δPA,k = 3 ⎢ik2 ∑ ri + 2ika ∑ ri I ia + 2ikr ∑ ri I ir ⎥ i =1 i =1 ⎣ i =1 ⎦

δPB ,q

n +1 n +1 ⎡ n +1 ⎤ = 3 ⎢iq2 ∑ ri + 2iqa ∑ ri I ia + 2iqr ∑ ri I ir ⎥ i =k + 2 i=k +2 ⎣ i = q +1 ⎦

The choice is done based on the criterion min {δPA,k , δPB ,q } .

(2.123)

154

Basic computation

2.4.3.3. Reducing and balancing the branch load This criterion is employed within the reconfiguration process for critical operating conditions, characterized by the fact that the power flows through branches closely to the technical capacity, which is the maximum load of the network branches (I I adm )max , has exceeded a specified value, usually equal to unity. The objective function for the branch load reducing criteria can be expressed as follows: MIN (I I adm )max (2.124)

[

]

If the objective is the balancing of branch loads, the objective function is expressed as follows:

⎡1 l ⎛ I ⎞ ⎤ MIN ⎢ ∑ ⎜ ⎟ ⎥ ⎣⎢ l k =1 ⎝ I adm ⎠ k ⎦⎥

(2.125)

While reducing the branch loads is a local criterion, when applied for a single branch of the network the load balancing on branches is a global criterion, which is used for the whole network. In the last case the goal is equivalent to obtaining more or less the same load for all branches. This can be represented mathematically as [2.21]:

⎛ I ⎞ ⎛ I ⎞ ⎛ I ⎞ 1 l ⎛ I ⎞ L = = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ I adm ⎠1 ⎝ I adm ⎠ 2 ⎝ I adm ⎠l l k =1 ⎝ I adm ⎠ k

∑

(2.126)

Theoretically, for branch load reduction, elementary branch exchanges for all branches in “out of service” state can be performed. To perform only the branch exchanges that leads to the proposed goal we consider the fact that at an elementary branch exchange in a certain loop only the current flow through the line sections of the loop in question will change (Fig. 2.35). Im

m

k n a.

-Im

m

k n

+Im b. Fig. 2.35. Variation of currents in a meshed network: a. before the branch exchange; b. after the branch exchange.


155

Starting from the previous observation, we conclude that for reduction a branch load, the necessary condition is to perform branch exchanges only in the loops that include that branch. Since for each “out of service” branch only two elementary exchanges are possible, the second necessary condition is that the chosen branch exchange should lead to the decrease of current flows through the branch in question. The sufficient conditions that ensure this objective are that, by a branch exchange, a non-zero load should be transferred from a feeder to a neighbouring one, and on the feeder to which the transfer was performed, no branch load exceeding the admissible value or the existing load of the branch in question should emerge. For the network in Figure 2.35, the second sufficient condition is given by [2.12]: I l + I m ≤ I adml ; l = k + 1,K , n (2.127) I m ≤ I admm−n For a better understanding, the possible branch exchanges in Figure 2.36 necessary to reduce the load of the branch in question, are presented. I > Iadm

Fig. 2.36. Possible branch exchanges performed to reduce the branch load.

2.4.3.4. Reducing the voltage drops This criteria is employed in the reconfiguration process for critical operating conditions, characterized by the fact that the voltage drop occurring along some branches is bigger than the admissible value, or that the minimum voltage node has a voltage level too low with respect to the voltage of the source node which supplies it. In this case, by performing branch exchanges, the goal is to achieve the reduction of the voltage drops along the branches on the path between the source node and the minimum voltage node. Likewise to the previous method, theoretically, in order to fulfil this objective, elementary exchanges for all “out of service” branches can be performed. In order to perform only those branch exchanges that can lead to this objective, it should be taken into account that, when performing an elementary exchange, the current values will change only for the branches of the loop to which the exchanged branches belong, which leads to voltage change for all nodes in the arborescent sub-networks that include the branches of the respective loop. Based on this observation, we conclude that the first necessary condition to reduce the voltages drop in a network is to perform a branch exchange that will include at least one of the branches on the path between the minimum voltage node and the source node that supplies it. From the same reason as the one when reducing the branch loads, the second necessary condition

156

Basic computation

is that the chosen branch exchange should lead to a voltage drop decrease on the path considered. The sufficient conditions that ensure this objective are: for a branch exchange, a non-zero load should be transferred from a feeder to a neighbouring one, and in the arborescent sub-network for which the transfer was performed no voltage drops exceeding the admissible value or the existing voltage drops occurring on the path in question should emerge [2.12]. For a better understanding, the possible branch exchanges performed in the Figure 2.37 to reduce the voltage drop between the source node and the node in question, are presented.

Umin Fig. 2.37. Possible branch exchanges performed to reduce the maximum voltage drop.

Appendix 2.1 EXISTENCE AND UNIQUENESS OF THE FORWARD/BACKWARD SWEEP SOLUTION Consider a simple electric network consisting of a source node and a load node, linked by an electric line (Fig. A2.1.1).

SA

Z=R+jX

A

1 S1

VA

V1

Fig. A2.1.1. Example of electric network with two nodes. For this network, the voltage V A at the source node and the complex power S 1 at the load node are known, and the goal is to establish the operating conditions in which, by applying the backward/forward sweep, to achieve the load flow results as well as the proof of their uniqueness. For the load flow calculation of this network, the voltage at the load node is first (0)

initialised, V 1 = V A , and then the following calculations are iteratively performed: I ( p) =

S 1*

3V 1( p −1)*

V 1( p ) = V A − Z I ( p ) S (Ap )

= 3V A I

( p )*

where I represents the line current, and p stands for the iteration index.

(A2.1.1)


157

In order to determine the convergence conditions of the iterative process, it is necessary to know the complex power S 1 . Because for the general equation (2.34) finding these conditions is extremely difficult, two particular load modelling cases will be considered in the following: by constant current or by constant power (these are the most frequently used models in electric network studies). Therefore: (1) In the case of load modelling by constant complex current, the line current I , obtained from the first relation of (A2.1.1), is constant and independent on the voltage level at the node 1. Under these conditions, the mathematical model is linear, and the load flow results are achieved after just a single iteration. Mathematically, V 1 and S A exist and are well determined for any value taken by the current I . Technically, low or negative values of the active component of the voltage V 1 , due to a too large voltage drop on the line, are not accepted. (2) For the case of load modelling by complex constant power, consider that the power has the expression S 1 = P1 + jQ1 , with P1 and Q1 constant. Therefore, the mathematical model is no longer linear, and in order to achieve the load flow results an iterative computation should be performed. In order to establish the conditions for which PA , Q A and V1 exist, we start from the relationship between the powers at the two nodes: PA = P1 + R

PA2 + QA2

QA = Q1 + X

3VA2

(A2.1.2)

PA2 + QA2 3VA2

The voltage V 1 at the load node can be calculated in terms of the voltage V A at the source node, by means of the relationship: V 1 = V A − ( R + jX )

P1 − jQ1

(A2.1.3)

3V 1*

The following relationship exists between the magnitudes of voltages V A and V 1 at the two ends of the line:

(

3VA2 = 3V12 + 2 ( RP1 + XQ1 ) + R 2 + X 2

)

P12 + Q12 3V12

(A2.1.4)

Next, the following notations are adopted: r=

R 3VA2

;

x=

X 3VA2

;

⎛V ⎞ v=⎜ 1 ⎟ ⎝ VA ⎠

2

(A2.1.5)

obtaining thus the system of equations that describes the network operation, in the form:

(

)

⎧P = P + r P2 + Q2 1 A A ⎪ A ⎪ 2 2 ⎨QA = Q1 + x PA + QA ⎪ ⎪v 2 − ⎡⎣1 − 2 ( rP1 + xQ1 ) ⎤⎦ v + r 2 + x 2 ⎩

(

)

(

(A2.1.6)

)( P

2 1

)

+ Q12 = 0

158

Basic computation

The previous system is non-linear, having the unknown variables PA , QA and v. The first two equations of the set (A2.1.6) define two curves in the system of PA − QA co-

ordinates. The crossing points ( PA1 , QA1 ) and ( PA2 , QA2 ) of these curves are in fact the solutions for the unknown variables PA and QA (Fig. A2.1.2).

Fig. A2.1.2. Defining the solutions of the single-load network [2.10].

To determine the solutions analytically, from the first two equations in (A2.1.6) we express the power losses on the line as:

(

∆P = PA − P1 = r PA2 + QA2 ∆Q = QA − Q1 = x

(

PA2

)

+ QA2

(A2.1.7)

)

and, by dividing the two equations, it results: ∆P r = ∆Q x

(A2.1.8)

From equations (A2.1.7), the power losses in terms of the load power components and the power losses components, are: 2 2 ∆P = r ⎡⎢( P1 + ∆P ) + ( Q1 + ∆Q ) ⎤⎥ ⎣ ⎦ 2 2 ∆Q = x ⎡⎢( P1 + ∆P ) + ( Q1 + ∆Q ) ⎤⎥ ⎣ ⎦

(A2.1.9)

Using the equation (A2.1.8) and substituting the unknown variable ∆Q from the second equation in (A2.1.9) into the first one, we obtain a second order equation where the variable is ∆P : ∆P 2 −

r 2

r +x

2

(1 − 2rP1 − 2 xQ1 ) ∆P +

r2 2

r + x2

(P

2 1

)

+ Q12 = 0

(A2.1.10)

The solutions are: ∆P11,2

⎡ r ⎢(1 − 2rP1 − 2 xQ1 ) ± = ⎣

(1 − 2rP1 − 2 xQ1 )2 − 4 ( r 2 + x 2 )( P12 + Q12 ) ⎤⎥

(

2 r 2 + x2

)

⎦

(A2.1.11)


159

Likewise, we proceed to the calculation of the unknown variable ∆Q :

∆Q11,2

(1 − 2rP1 − 2 xQ1 )2 − 4 ( r 2 + x 2 )( P12 + Q12 ) ⎤⎥

⎡ x ⎢(1 − 2rP1 − 2 xQ1 ) ± = ⎣

(

2

2 r +x

2

⎦

)

(A2.1.12)

Solving the third equation in (A2.1.6), the solutions for the voltage v are obtained: v1,2 =

(1 − 2rP1 − 2 xQ1 )2 − 4 ( r 2 + x 2 )( P12 + Q12 )

1 1 − rP1 − xQ1 ± 2 2

(A2.1.13)

The existence of these solutions is conditioned by the sign of the quantities under to be square-rooted, i.e:

(1 − 2rP1 − 2 xQ1 )2 − 4 ( r 2 + x 2 )( P12 + Q12 ) ≥ 0

(A2.1.14)

On the boundary, the above relation describes a parabola. Considering also the restriction P1 ≥ 0 (node 1 being of load type), the existence domain of the solutions of the system of equations (A2.1.1) is given by the hatched area in Figure A2.1.3.

Q1

1 2

1 4

2

2

P'1 r + x

θ P1

1 2 2

2

Q'1 r + x

Fig. A2.1.3. The existence domain of the solutions [2.12]. In

figure

P1′ = ( rP1 + xQ1 )

A2.1.3, 2

r +x

2

the

following

and Q1′ = ( xP1 − rQ1 )

notations 2

have

been

made:

2

r + x , representing components of a

rotation of angle θ = atan ( x r ) , of the system of axes P1 − Q1 around the origin. For technical and psychical reasons [2.9, 2.22, 2.23], of the two values of the voltage v, only the value with the sign “+“ is accepted. The quantities PA and QA depend upon the existence and the uniqueness of the quantity v, as expressed in equations (A2.2.1). The proof of existence and uniqueness of the load flow solution for the general case of a radial (arborescent) network consisting of n nodes can be obtained by generalizing the results previously achieved. Applying iteratively the process described earlier, it can be

160

Basic computation

demonstrated that for each branch i of the electric network there is a unique relationship between the active Pi and reactive Qi power entering (injected) into the branch and the voltage magnitude U i +1 at the other end of the branch.

Appendix 2.2 THE ACTIVE POWER LOSSES VARIATION AS A RESULT OF A LOAD VARIATION IN A RADIAL NETWORK Consider a radial electric network that supplies n loads, whose one-line diagram is illustrated in Figure A2.2.1. The loads are modelled by constant currents and the lines sections by series impedances. I1

A z1 VA

I2

1

2

Ik

k-1

k

zk

z2 i1

i2

In

n-1

n

zn

ik-1

ik

in-1

in

Fig. A2.2.1. Radial electric network supplying n loads. For this network, the active power losses variation to a change of the load current at any node k is of interest. In this respect, the complex power losses are firstly expressed with the relation: *

n

∑

∆ S = 3 [ I ]t [ Z ][ I ] = 3

n

∑

z k I k I *k = 3

k =1

n

∑

z k I k2 = 3

k =1

rk I k2 + j

k =1

n

∑x I

2 k k

(A2.2.1)

k =1

where z k is the impedance of the branch k, between the nodes k − 1 and k, and I k is the complex current flowing through the branch k. In the previous relation the active power losses can be expressed as: n

∑

∆P = 3

n

∑r (I

rk I k2 =3

k =1

k

k =1

2 ka

+ I kr2

)

(A2.2.2)

where I ka and I kr are the active and reactive components of the complex current I k . The current I k ca be expressed in terms of the load currents as: n

Ik =

∑i

(A2.2.3).

i

i=k

Expanding the equation (A2.2.2) and considering the equation (A2.2.3), it results:

{

2 2 ∆P = 3 r1 ⎡⎢( i1a + K + ika + K + ina ) + ( i1r + K + ikr + K + inr ) ⎤⎥ + K + ⎣ ⎦

+ rk ⎡⎢( ika + K + ina ) + ( ikr + K + inr ) ⎣ ⎦ 2

2⎤

(

2 ⎥ + K + rn ina

2 + inr

)}

(A2.2.4)

In the previous equation the line sections resistances are considered to be constant. Also, consider that the load currents are constant, except the current at the load connected


161

to the node k. In order to calculate the power losses variation in the whole network for a variation of the current i k , the expansion of expression of ∆P from equation (A2.2.4) using Taylor series is performed in the vicinity of the operating point, in terms of the quantities ika and ikr [2.12]: Not

∆ ( ∆P ) = δP =

∂ ( ∆P ) ∂ ika

∆ika +

∂ ( ∆P ) ∂ ikr

∆ikr +

2 1 ⎡ ∂ ( ∆P ) 2 ∆ika + ⎢ 2 2 ⎢⎣ ∂ ika

∂ ( ∆P ) ∂ ( ∆P ) 2 ⎤ +2 ∆ika ∆ikr + ∆ikr ⎥ + K 2 ∂ ika ∂ ikr ∂ ikr ⎥⎦ 2

(A2.2.5)

2

The partial derivatives that emerges in this expansion have the expressions: ∂ ( ∆P ) ∂ ika ∂ ( ∆P ) ∂ ikr

= 3 ⎡⎣ 2r1 ( i1a + K + ika + K + ina ) + K + 2rk ( ika + K + ina ) + K + 2rn ina ⎤⎦ = 3 ⎡⎣ 2r1 ( i1r + K + ikr + K + inr ) + K + 2rk ( ikr + K + inr ) + K + 2rn inr ⎤⎦

∂ 2 ( ∆P ) 2 ∂ ika

∂ 2 ( ∆P ) ∂ ika ∂ikr ∂ 2 ( ∆P ) 2 ∂ ika

= 3⎡⎣2r1 + K + 2rk + K + 2rn ⎤⎦

(A2.2.6) =0 = 3⎣⎡2r1 + K + 2rk + K + 2rn ⎦⎤

∂ p ( ∆P ) q ∂ ika ∂ikrp − q

= 0 (∀) p ≥ 3, q ≤ p

Replacing the equations from (A2.2.6) in (A2.2.5) and taking into account (A2.2.3), it results: δP = 3 ⎡⎣ 2 ( r1 I1a + r2 I 2 a + K + rk I ka ) ∆ika + 2 ( r1 I1r + r2 I 2 r + K + rk I kr ) ∆ikr +

(

2 2 + ( r1 + r2 + K + rk ) ∆ika + ∆ikr

)

(A2.2.7)

The latter equation express the active power losses variation in the network due to the change of current in node k, of the form ∆i k = ∆ika + j ∆ikr : ⎡ 2 2 δP = 3 ⎢ ∆ika + ∆ikr ⎢⎣

(

k

k

) ∑ r + 2∆i ∑ r I i

i =1

ka

i ia

i =1

k

+ 2∆ikr

∑r I

⎤

i ir ⎥

i =1

⎥⎦

(A2.2.8)

Likewise, the expression of the reactive power losses variation due to the change of current in node k is:

162

Basic computation ⎡ 2 2 δQ = 3 ⎢ ∆ika + ∆ikr ⎣⎢

(

k

)∑

k

xi + 2∆ika

i =1

∑

k

xi I ia + 2∆ikr

i =1

∑x I

⎤

i ir ⎥

i =1

⎦⎥

(A2.2.9)

Neglecting the reactance of the line sections, the voltage drop ∆V Ak between the nodes A and k becomes: k

∆V Ak = ∆VAk + jδVAk =

∑

k

ri I ia + j

i =1

∑r I

i ir

(A2.2.10)

i =1

and the expression of the active power losses variation becomes [2.19]: ⎡ δP = 3 ⎢ ∆ik2 ⎢⎣

k

⎤

∑ r + 2 Re {∆i ∆V }⎥⎥ i

i =1

k

* Ak

(A.2.2.11)

⎦

where Re stands for the real part, and * indicates the complex conjugate.

Chapter references [2.1] [2.2] [2.3] [2.4] [2.5] [2.6] [2.7] [2.8] [2.9] [2.10] [2.11] [2.12]

Bercovici, M., Arie, A.A., Poeată, A. – Reţele electrice. Calculul electric (in Romanian) (Electric networks. Electric calculation), Editura Tehnică, Bucureşti, 1974. Grainger, J.T., Stevenson, W.D. – Power systems analysis, Mc Graw-Hill, 1994. Eremia, M., Trecat, J., Germond, A. – Réseaux électriques. Aspects actuels, Editura Tehnică, Bucureşti, 2000. Debs, A. – Modern power systems control and operation: A study of real – time operation of power utility control centers, Kluwer Academic Publishers, 1992. Guill, A.E., Paterson, W. – Electrical power systems. Volume one. 2nd Edition, Pergamon Press, Oxford, New York, 1979. Weedy, B.M. – Electrical power systems. 3rd Edition, John Wiley & Sons, Chichester, New York, 1979. Poeată, A., Arie, A.A., Crişan, O., Eremia, M., Alexandrescu, V., Buta, A. – Transportul şi distribuţia energiei electrice (Transmission and distribution of electric energy), Editura Didactică şi Pedagogică, Bucureşti, 1981. El-Hawary, M. – Electrical power systems. Design and analysis (Revised printing), IEEE Press, New York, 1995. Eremia, M., Crişciu, H., Ungureanu, B., Bulac, C. – Analiza asistată de calculator a regimurilor sistemelor electroenergetice (Computer aided analysis of the electric power systems regimes), Editura Tehnică, Bucureşti, 1985. Chiang, H.D., Baran, M. – On the Existence and Uniqueness of Load Flow Solution for Radial Distribution Power Network, IEEE Transactions on Circuits and Systems, Vol. 37, No. 3, March 1990. Bart, A. – Reconfiguration des réseaux de distribution en régime critique et défaillant, Thèse 1176, Ecole Polytechnique Fédérale de Lausanne, 1993. Triştiu, I. – Reconfigurarea reţelelor electrice de distribuţie de medie tensiune (Reconfiguration of distribution electric networks of medium voltage), Ph.D. Thesis, Universitatea “Politehnica” din Bucureşti, 1998.

Radial and meshed networks [2.13] [2.14] [2.15] [2.16] [2.17] [2.18] [2.19] [2.20] [2.21] [2.22] [2.23] [2.24]

[2.25]

163

Baran, M., Wu, F. – Network Reconfiguration in Distribution Systems for Loss Reduction and Load Balancing, IEEE Transactions on Power Delivery, Vol.4, No.2, April 1989. Goswami, S.K., Bassu, S.K. – A new Algorithm for the Reconfiguration of Distribution Feeders for Loss Minimisation, IEEE Transactions on Power Delivery, Vol.7, No.3, July 1992. Cherkaoui, R. – Méthodes heuristiques pour la recherche de configurations optimales d'un réseau électrique de distribution. Thèse 1052, Ecole Polytechnique Fédérale de Lausanne, 1992. Triştiu, I., Eremia, M., Ulmeanu, P., Bulac, C., Bulac, A.I., Mazilu, G. – Un nouveau mode d’aborder la reconfiguration des réseaux de distribution urbaine, CIGRE, Black Sea El – Net Regional Meeting, Suceava, 10-14 June 2001. Cherkaoui, R., Germond, A. – Structure optimale de schéma d’exploitation d’un réseau électrique de distribution. Energetica Revue, Nr.5 B, 1994. Florea, A.M. – Elemente de Inteligenţă Artificială, Vol. I, Principii şi Modele. (Elements of artificial intelligence. Vol. I, Principles and models) Litografia Universităţii “Politehnica” din Bucureşti, Bucureşti, 1993. Cinvalar, S., Grainger, J.J., Yin, H., Lee, S.S.H. – Distribution feeder reconfiguration for loss reduction, IEEE Transactions on Power Delivery, Vol.3, No.3, April 1988. Shirmohammadi, D., Hong, H.W. – Reconfiguration on electric distribution networks for loss reduction and load balancing, IEEE Transactions on Power Delivery, Vol.4, No.2, April 1989. Kashem, M.A., Ganapathy, V., Jasmon, G.B. – Network reconfiguration for load balancing in distribution networks, Generation, Transmission and Distribution, IEE Proceedings, Volume: 146 Issue: 6, Nov. 1999. Potolea, E. – Calculul regimurilor de funcţionare a sistemelor electroenergetice (Calculation of the operating regimes of the power systems), Editura Tehnică, Bucureşti 1977. Barbier, C., Barret, J.P. – Analyse des phénomènes d’écroulement de tension sur un réseau de transport, Revue Générale d’Electricité, Tome 89, No.10, October 1980. Augugliaro, A., Dusonchet, L., Favuzza, S., Ippolito, M.G., Riva Sanseverino, E. – A new model of PV nodes in distribution networks backward/forward analysis, 39th International Universities Power Engineering Conference UPEC 2004, 6-8 September 2004, Bristol, England. Shirmohammadi, D., Hong, H.W., Semlyen, A., Luo, G.X. – A compensationbased power flow method for weakly meshed distribution and transmission networks, IEEE Transactions on Power Systems, Vol.3, No.2, May 1988.

Chapter

3

AC TRANSMISSION LINES

For long-distance energy transmission − hundred of kilometres, either AC or DC, long lines powered at high (HV) and extra high voltages (EHV) are used. These lines present a set of operating peculiarities and therefore their modelling is different from distribution short lines modelling (with respect to wavelength). Accurate analysis of phenomena that occur on EHV transmission long lines does not have to consider the line parameters as lumped, as performed in the case of medium or low voltage powered lines, instead of uniformly distributed along the line. The following assumptions can be made in this regard [3.1]: a) The leakage current and conduction current through dielectric are approximately equal to the value of current flowing through the series impedance and therefore they cannot be neglected anymore; b) For no-load conditions of the line, the conduction current at the source is non-zero. The no-load current is capacitive and varies from a cross-section to another: the current increases from the receiving-end (load) toward the sending-end (source), and the voltage increases from the source toward the receiving-end. This increase in voltage is known as Ferranti phenomenon and is more pronounced as the length of line increases (i.e. for L = λ 4 = 1500 km , the voltage for no-load conditions could theoretically reach infinite values). Therefore, if a transmission line with uniformly distributed parameters is supplied with a sinusoidal voltage, in every point of the line the voltage and current have a sinusoidal variation in time, but their magnitude depends on the position of the considered point along the line.

3.1. Operating equations under steady state Consider a very short section Δx, from a line of length L, at a distance x measured from the receiving-end (Fig. 3.1). By applying Kirchhoff’s theorems obtain: − Voltage drop in section Δx is: V (x + Δx ) − V (x ) = z 0 Δx I (x )

(3.1)

166

Basic computation

− Shunt current passing through y 0 Δx is:

I (x + Δx ) − I (x ) = y 0 ΔxV ( x ) I(x+Δx)

I(L) V(L)

I(x)

z0Δx y0Δx

V(x+Δx)

I(0)

V(x) Δx

I(L)

(3.2)

V(0) I(0)

L

Source

x

x=0 Load

Fig. 3.1. Equivalent circuit of a transmission line – telegraph equations.

Equation (3.1) can be written as: V (x + Δx ) − V (x ) = z 0 I (x ) Δx

In the limit, when Δx → 0 ,

V (x + Δx ) − V (x ) = z 0 I (x ) Δx → 0 Δx lim

or

dV (x ) = z 0 I (x ) dx

(3.3)

for current respectively obtain:

d I (x ) = y 0 V (x ) dx Differentiating with respect to x it results:

(3.4)

2 d I (x ) d V (x ) = z0 2 dx dx 2 d I ( x ) y dV ( x ) = 0 dx dx 2

or taking into account equations (3.3) and (3.4) obtain: 2 d V (x ) = z 0 y 0 V (x ) dx 2

(3.5)

2 d I (x ) = z 0 y 0 I (x ) dx 2

(3.6)

AC transmission lines

167

where: z 0 = r0 + jx0 = r0 + jωl0 y 0 = g 0 + jb0 = g 0 + jωc0

is per length complex impedance of the line; − per length complex admittance of the line.

Equations (3.5) and (3.6) are known as telegraph equations, which define the electromagnetic energy transfer along the “long” lines. The voltage V (x ) and current I (x ) are unique solutions of a second-order differential equation with constant coefficients. Knowing the form of a solution for V (x ) and I (x ) , we can deduce the other solution. The general solution for V (x ) from (3.5) can be written in exponential form as: V (x ) = A1 e γx + A2 e − γx

(3.7)

where A1 , A2 and γ are integration constants. Calculating the second order derivative: 2 2 d V (x ) γ 2 = (A1 e γx + A2 e − γx ) = γ V (x ) dx 2

(3.8)

and equating with (3.5) we obtain: 2

γ = z0 y0

Thus, the expression of the complex propagation coefficient will be:

(r0 + jx0 )(g0 + jb0 )

2

γ = z 0 y 0 or γ = ± z 0 y 0 = ±

(3.9)

The propagation coefficient can also be expressed as:

γ = α + jβ

(3.9')

where: α is the attenuation coefficient, [nepers/m], depending on the voltage and current magnitude variation on the line; β − the phase coefficient, [rad/m], expressing the voltage or current phase variation in two points on the line. By substituting (3.7) in (3.3) obtain: dV ( x ) d = (A1 e γx + A2 e − γx ) = γ (A1 e γx − A2 e − γx ) = z 0 I (x ) dx dx respectively: I (x ) = If consider

γ z0

(A1 eγx − A2 e− γx ) =

1 (A1 eγx − A2 e− γx ) ZC

(3.10)

168

Basic computation

z0 z0 z = =± 0 =± y0 γ ± z0 y 0

r0 + jx0 g0 + jb0

so the ratio

ZC = +

z0 y0

(3.11)

is called characteristic impedance (or surge impedance) of the electric line. Observation: For a lossless electric line, r0 ≈ 0 and g 0 ≈ 0 , the characteristic impedance has the dimension of a resistance. The minus sign has no meaning because there is no negative resistance. The propagation coefficient γ and the characteristic impedance Z C of the line reflect the geometrical and material properties (of the conductor and dielectric environment) and characterize the electromagnetic energy propagation. They do not depend on the line length. Parameters γ , α , β and Z C are called secondary

parameters of the electric line and they can be inferred from the primary parameters r0, l0, c0, g0. In order to determine the constants A1 and A2, the conditions in the limit imposed at the input and output terminals of the circuit are used. Therefore: • At the receiving-end, for x = 0, obtain: V (0 ) = V B = A1 + A2 I (0 ) = I B = resulting in:

1 ( A1 − A2 ) ZC

A1 = 1 2 (V B + Z C I B )

(3.12')

A2 = 1 2 (V B − Z C I B )

(3.12'')

Substituting (3.12') and (3.12'') in (3.7) obtain:

V (x ) = or

γx − γx γx − γx 1 (V B + Z C I B )eγx + 1 (V B − Z C I B )e− γx = V B e + e + Z C I B e − e 2 2 2 2

(

)

(

)

(3.13)

)

(3.14)

V ( x ) = cosh γx V B + Z C sinh γ x I B Likewise, the equation of current is obtained:

(

)

(

I (x ) = Y C sinh γx V B + cosh γx I B

The matrix equation that gives the voltage and current in terms of the output quantities, in a point placed at the distance x, is:


169

Z C sinh γ x ⎤ ⎡V B ⎤ ⎡V (x )⎤ ⎡ cosh γx ⎢ I (x ) ⎥ = ⎢Y sinh γx cosh γx ⎥⎦ ⎢⎣ I B ⎥⎦ ⎣ ⎦ ⎣ C • At the sending-end, for x = L, obtain: Z C sinh γL ⎤ ⎡V B ⎤ ⎡V A ⎤ ⎡ cosh γL ⎢ I ⎥ = ⎢Y sinh γL cosh γL ⎥⎦ ⎢⎣ I B ⎥⎦ ⎣ A⎦ ⎣ C

(3.15)

The coefficients of the long lines equations are:

A = D = cosh γL ; B = Z C sinh γL ; C = Y C sinh γL Since the coefficients fulfill the necessary condition of a passive fourterminal network, that is:

AD − BC = cosh 2 γL − sinh 2 γL = 1 it results that any electric long line can be represented through an equivalent fourterminal network (Fig.3.2). IA VA

A=coshγL

B= ZCsinhγL

C =Y CsinhγL

D=coshγL

IB VB

Fig. 3.2. Equivalent four-terminal network of an electric line.

In the case when the input quantities V A , I A are given and output quantities V B , I B are required we obtain: −1

− Z C sinh γL ⎤ ⎡V A ⎤ Z C sinh γL ⎤ ⎡V A ⎤ ⎡ cosh γL ⎡V B ⎤ ⎡ cosh γL =⎢ ⎥ ⎢ I ⎥ = ⎢Y sinh γL ⎢ ⎥ cosh γL ⎦ ⎣ I A ⎦ ⎣− Y C sinh γL cosh γL ⎥⎦ ⎢⎣ I A ⎥⎦ ⎣ B⎦ ⎣ C (3.16)

3.2. Propagation of voltage and current waves on a transmission line 3.2.1. Physical interpretation In order to emphasize the physical aspect of propagation of the voltage and current waves on a line, the following equations are written again:

170

Basic computation

V (x ) = A1 e γx + A2 e − γx I (x ) =

1 (A1 eγx − A2 e− γx ) ZC

(3.7) (3.10)

where the constant A1 will be determined in terms of the input quantities, that is for x=L: V A = A1 e γL + A2 e − γL Z C I A = A1 e γL − A2 e− γL Adding these equations the following expression results: A1 =

1 (V + Z C I A)e− γL 2 A

(3.17)

For the constant A2 the value from (3.12''), determined in terms of the output quantities, will be kept. Substituting (3.17) and (3.12'') in (3.7) obtain: V (x ) =

1 (V A + Z C I A)e− γLeγx + 1 (V B − Z C I B )e− γx 2 2

or taking into consideration Figure 3.3,a: V ( x ) = V A e − γx ' + V B e − γx '

'

(3.18)

where '

VA = '

VB =

1 (V + Z C I A) = V A' e j ψa 2 A 1 (V B − Z C I B ) = VB' e j ψb 2

Taking into account that γ = α + jβ it results that: V (x ) = V A e j (ψa −βx′ ) e− αx′ + V B e j (ψb −βx ) e− αx '

'

Expressing in instantaneous values, the voltage is a function of t and x: V (x,t ) = 2VA' sin (ωt − βx′ + ψ a )e − αx′ + 2VB' sin (ωt − β x + ψ b )e − αx

(3.19)

or V (x, t ) = Vd ( x' ,t ) + Vr ( x,t ) Thus, in any point and any instant of time, the voltage is a sum of two waves of decreasing phase angle:


171

• direct travelling wave, which propagates from the source toward the ' consumer, of preponderant magnitude, V A = (V A + Z C I A ) 2 and which is exponentially damped with the coefficient e − αx ' (Fig. 3.3,b); L x x=L

x=0 x'=L-x a.

Vd

t t+Δt

2 VA e-αx

2 VB e

v

Vr

-αx

v x'

x

Δx'=vΔt b. c. Fig. 3.3. Travelling waves propagation along a transmission line: a. defining of line section, b. direct travelling wave propagation, c. reflected travelling wave propagation.

• reflected travelling wave, which propagates from the consumer to the source (in the opposite direction of the energy transfer), of lower magnitude ' V B = (V B − Z C I B ) 2 with respect to the direct waves, and which is exponentially damped with the coefficient e − αx (Fig. 3.3,c). In any point x, there is a superposition of travelling waves resulting in a stationary wave. Velocity and direction of propagation In order to determine the velocity and direction of propagation of the waves, two successive points along the line, having the same phase angle, are considered. If the voltage phase angle at the instant t and the distance x' is equal, by definition, to the voltage phase angle at the instant ( t + Δt ) and the distance ( x'+ Δx' ), then it can be written: ωt − βx' + ψ a = ω(t + Δt ) − β(x'+Δx')+ ψ a

from where it results: ωΔt − β Δx' = 0 or in the limit: Δx' ω = =ν Δt β where v is the velocity of propagation of the wave.

172

Basic computation

It results that the direct travelling wave Vd (x' , t ) is moving in the positive direction along the x‫׳‬-axis, with the same velocity ν = ω / β ; for this reason Vd (x' , t ) is called direct wave. Likewise, for the second travelling wave, obtain: Δx ω =− =−v Δt β that is Vr (x, t ) is moving in the positive direction along the x-axis, thus in the opposite direction from the direct travelling wave, with the same velocity ν = ω / β . Therefore, Vr (x, t ) is a reflected or inverse wave. For lossless electric lines, that is for r0 ≈ 0 and g 0 ≈ 0 , from equation of propagation coefficient (3.9), it results: γ ≅ j x0b0 = jω l0c0 = jβ

(3.20)

where β = ω l0c0 . If substitute β in the expression of velocity of propagation of the waves obtain: ω 1 (3.21) v= = β l 0 c0 Returning to the equation (3.9) and taking into account (3.21) obtain: γ≅

jω 2πf 2π 2π = j = j = j ν ν νT λ

(3.20')

It should be mentioned that in the case of lossless overhead lines, the velocity of propagation of the waves is independent on the frequency and it is equal to the velocity of the light in vacuum, which is 300.000 km/s. In all the other cases, the velocity of propagation of the waves is lower than the velocity of the light. As a consequence, the wavelength λ of the AC powered electric lines (with frequency of 50 Hz) is equal to:

λ=

v 300.000 km/s = = 6000 km f 50 Hz

only when the power losses on the line can be neglected. Therefore, the propagation phenomenon is periodical in space after every 6000 km. Generally, of interest are the lines with lengths of l/4=1500 km and l/2=3000 km Proceeding in a similar manner for current travelling waves, obtain:


173

I (x ) = I A e − γx′ − I B e − γx = I A' e j ϕa e − (α + jβ )x′ − I B' e j ϕb e − (α + jβ )x '

'

and in instantaneous values respectively: I ( x,t ) = 2 I A' sin (ωt − βx'+ ϕa )e- αx ' − 2 I B' sin (ωt − βx + ϕb )e − αx

(3.22)

Observations: • The reflected waves of current have opposite sign with respect to the direct current waves as compared to the voltage waves that bear the same sign; • The damping factor attached to both the direct waves ( e − αx ' ) and the reflected waves ( e − αx ) shows that the propagation phenomenon on real lines, with resistance and shunt admittance, operates with electric energy losses.

3.2.2. Apparent characteristic power. Natural power (SIL – surge impedance loading) The operation of an electric line without reflected wave is more favourable, from the economic point of view, because in this case energy losses decrease and in consequence the transmission efficiency improves. Under these circumstances, ' the term V B from equation (3.18) becomes zero, that is: '

VB =

1 (V B − Z C I B ) = 0 2

so that IB =

VB V = B Z C Z load

By the notion of characteristic impedance Z C understand a value of the impedance of the consumer from the point B for which there can be no reflected waves, that is with minimum losses on the line. Thus, when the receiving-end B of the line is closed on an impedance of value equal to the characteristic impedance, the propagation phenomenon occurs as if the considered line is of infinite length ∗). In this case, the impedance measured at the sending-end of the line (source) is also equal to the characteristic impedance Z C , respectively to the impedance measured at the receiving-end of the line (load). The apparent power demanded by the consumer, under such circumstances – regime without reflected waves – is called characteristic apparent power ( S C ). The expression of the single-phase characteristic apparent power at the consumer is:

∗)

The ratio of voltage to current at any point along an infinite line is a constant equal to the characteristic impedance of the line [3.2].

174

Basic computation

*

S B 0,C = V B I B =

VB2 *

ZC

or, if ξ is the angle of characteristic impedance Z C : S B 0,C =

VB2

[ZC (cos ξ + j sin ξ)]*

=

VB2 V2 = B (cos ξ + j sin ξ ) Z C (cos ξ − j sin ξ ) Z C

In the regime without reflected wave, where V B = Z C I B , the equations (3.13) and (3.14) become:

(

)

(

)

V (x ) = V B cosh γx + sinh γx = V B e γx I (x ) = I B sinh γx + cosh γx = I B e γx The characteristic single-phase apparent power in a point x on the line can be determined from expression: S 0,C (x ) = V ( x )I ( x ) = V B I B e(γ + γ )x = S B 0,C e 2αx *

*

*

Since ξ is very small for overhead electric lines, the characteristic impedance having a high resistive component, the dominant term of the characteristic apparent power S B 0, C will be the active characteristic power:

PB 0,C =

VB2 cos ξ ZC

respectively: P0,C ( x ) = PB 0,C e 2 αx Since the attenuation coefficient α has a small value, the active power P0,C ( x ) does not vary much along the line, being almost of the same value as

characteristic active power absorbed by the consumer P0,C (x ) ≅ PB 0,C . Furthermore, in the case of lossless electric lines the attenuation coefficient is zero, that is α = 0 and thus γ ≅ jβ , and the characteristic impedance becomes a resistance; under these circumstances PB 0,C is conserved along the line, being a characteristic constant called natural power or surge impedance loading: PB 0,C = P0, N =

VB2 ZC


175

For a transfer of active power, the voltage is the same along the entire length of the line and assuming this is equal to the nominal voltage, then P0, N = Vn2 Z C . The three-phase natural power is: PN = 3 P0, N = U n2 Z C where: Un is phase-to-phase nominal voltage; ZC – characteristic impedance of the lossless line. The three-phase natural power is an index in designing the transmission capacity of the lines. In Table 3.1, several natural power values corresponding to different operating nominal voltages are given. Table 3.1 Un [kV] Overhead PN lines [MW] Underground lines

20

110

220

400

750

1

30

120

400÷500

1800

10

300

1200÷1400

2000÷2500

4000÷5000

Such regime of natural power has the following characteristics: – the equivalent impedance of the line and of the consumer, determined in every point of the line, is the same and equal to the characteristic impedance; – the phase angle between current and voltage in every point of the line has the same value. If consider a lossless line, then the voltage and current are in phase at the receiving-end of the line as well as in any point of the line; – the voltage and current values do not change much along the line, and if the line is without losses, they remain constant; instead only phase angle will shift in proportion to the line length; – the power transmitted on the line under this regime has a strong active characteristic. A phenomenon specific to long lines will appears: although on the line there are inductive and capacitive reactive power losses, the line absorbs from the source only active power. The explanation is as follows: the inductive reactive power losses occurring on the line reactance are compensated by the capacitive reactive power generated by the line. In this respect, for the line without losses on resistance and conductance ( r0 ≈ 0 , g 0 ≈ 0 ) consider the inductive and capacitive losses per length unit: ΔQind = I 2 x0 ; ΔQcap = V 2b0 From their ratio it results: 2

2 ΔQind I 2 x0 ⎛ I ⎞ ⎛⎜ x0 ⎞⎟ 1 = 2 =⎜ ⎟ = 2 Z C2 = 1 ⎜ ⎟ ΔQcap V b0 ⎝ V ⎠ ⎝ b0 ⎠ ZC

176

Basic computation

Therefore, on every segment of the line, inductive and capacitive reactive powers are reciprocally compensated. Thus, under natural power regime the line does not absorb reactive power at its ends; it is said that the line is selfcompensated.

3.3. Coefficients of transmission lines equations For the numerical solving of long lines equations (3.15) the determination of coefficients cosh γL , sinh γL , Z C sinh γL , Y C sinh γL is needed. Since the quantity γ is a complex number, the coefficients of long lines, equations are hyperbolic functions of complex quantities.

3.3.1. Numerical determination of propagation coefficient In order to determine the numerical values of the complex propagation coefficient for underground cables and overhead lines, either algebraic method or trigonometric method can be used. Algebraic method enables us to determine the constants α and β by considering the real parts of the square equation (3.9) and the magnitude square of γ coefficient. The following equations are obtained: r0 g 0 − x0 g 0 = α 2 − β2

(r

)(

2 0

)

+ x02 g 02 + b02 = α 2 + β2

then 1 2 1 = 2

α=

1 2 1 = 2

β=

(r

2 0

)(

)

+ x02 g 02 + b02 + (r0 g 0 − x0b0 ) =

z 0 y 0 + (r0 g 0 − x0b0 )

(r

2 0

)(

)

+ x02 g 02 + b02 − (r0 g 0 − x0b0 ) =

z 0 y 0 − (r0 g 0 − x0b0 )

Trigonometric method. In this respect, the expression of complex propagation coefficient is considered: γ =

z0 y0 =

(r0 + j x0 )(g0 + j b0 )

= α + jβ


177

By expressing: z 0 = z0∠ψ ; y 0 = y0∠ψ' where tan ψ = x0 r0 ; tan ψ ' = b0 g0 it results γ=

⎛ ψ ψ' ⎞ z 0 y 0 ∠⎜ + ⎟ = γ∠η ⎝2 2⎠

• The magnitude of propagation coefficient can be expressed as follows:

⎡ ⎛ r ⎞2 ⎤ ⎡ ⎛ g ⎞2 ⎤ 2 2 2 ⎢1 + ⎜ 0 ⎟ ⎥ ⎢1 + ⎜ 0 ⎟ ⎥ 4 x g + b = x b γ= + 0 0 0 0 0 ⎢ ⎜⎝ x0 ⎟⎠ ⎥ ⎢ ⎜⎝ b0 ⎟⎠ ⎥ ⎦ ⎣ ⎦⎣ 1 = x0b0 4 (1 + cot 2 ψ )(1 + cot 2 ψ′) = x0b0 sin ψ ⋅ sin ψ′ 4

(r

2 0

)(

)

(3.23)

Taking into account (3.20'), the expression of magnitude becomes: γ = 2π / λ

(3.23')

λ = λ 0 sin ψ ⋅ sin ψ′

(3.24)

where

is the equivalent wavelength of the line with losses, and λ 0 the wavelength corresponding to the electromagnetic waves of T period, propagating on a lossless line. In order to simply the calculation, the following assumption can be taken into consideration: − If neglect the shunt power losses ( g 0 ≈ 0 that is tan ψ' → ∞ , ψ' = 90° , sin ψ ' = 1 ) the expression (3.23') becomes: γ≅

2π λ 0 sin ψ

=

2π λ'

(3.23'')

where λ ' = λ 0 sin ψ

(3.24')

The value of λ′ can be expressed as:

λ' = λ 0 sin ψ =

λ0 4

1+ cot 2 ψ

λ0

= 4

⎛r ⎞ 1 + ⎜⎜ 0 ⎟⎟ ⎝ x0 ⎠

2

178

Basic computation

Furthermore, if consider r0 π 2 as inverter. The limit case α tr + λ 2 = π 2 corresponds to the situation when the converter takes from the network only reactive power. It results that the delay angle corresponding to start of inversion is α tr = 90° − λ 2 , always smaller than 90°. In Figure 4.18 there are presented the voltage waveforms (b), for the inversion mode. +

va

Id T4

Udi

vb

T6

vc T2

T6

T4

T2 v a

ωt

vb

By-pass

vc T1

-

T3

T5

T3

T1

T5 α

T3 λγ β

a.

b.

Fig. 4.18. Bridge connexion – inverter operation (a); Bridge connexion – inverter voltage and current waveforms (b).

In order to obtain the equations describing inverter’s operation we uses (4.12) and (4.13), where α is the delay angle of the thyristors while operating in rectifier mode and δ = α + λ is the extinction angle. Both angles are measured by the delay from the instant at which the commutating voltage is zero and increasing (ωt = 0º).

268

Basic computation

Although angles α and δ could have been used in inverter’s theory, in order to make a difference, other symbols will be employed. These angles are defined by their advance with respect to the instant (ωt = 180º for ignition of valve 3 and extinction of valve 1) when the commutating voltage is zero and decreasing, as shown in Figure 4.19. From the figure, we see that: β = 180° − α for the ignition advance angle, and γ = β − λ = 180° − δ for the extinction advance angle. Back to rectifier’s equation (4.12) we use opposite polarity for direct voltage and replaces α = 180° − β and λ = β − γ : U d 0, i U [cos α + cos(α + λ )] = − d 0,i [cos(180° − β) + cos(180° − γ )] 2 2 If we consider: cos α = cos(180° − β ) = − cos β U di = −

cos δ = cos(180° − γ ) = − cos γ we obtain: U d 0, i (cos β + cos λ ) 2 Similarly, the equation (4.13) becomes: U di =

(4.17)

I d = I sc 2 (cos γ − cos β )

(4.18)

uba Isc2

3 Vs

ωt

Isc2(cosα-cosωt) Rectifier i1 α δ

Inverter i1

i3 λ

i3 λ

α δ

γ β

π

Fig. 4.19. Relationships between the angles used in converter theory and why the curvature of the front of a current pulse of inverter differs from that of a rectifier. Reprinted with permission from IEEE 519–1992, Guide for harmonic control and reactive compensation of static power converters © 1992 IEEE.

HVDC transmission

269

Similarly, if one replaces cos α = − cos β in the equation (4.17), we obtain a first equation for converter’s operation in inversion mode: U di = U d 0,i cos β + Rci I d

(4.19)

Because inverters are commonly controlled so as to operate at constant extinction advanced angle γ , it is useful to have the relations between U d and I d for this condition. If we replaces U d 0, i cos β from the equation (4.17) in the equation (4.19), results another form of the equation: U di = U d 0,i cos γ − Rci I d

(4.19')

However it is to be noted that while α is directly controllable, γ is not. Accordingly to the operation equations (4.19) and (4.19'), two possible equivalent circuits could be build for the inverter operation (Fig. 4.20,a,b). -Rci

Rci

Id

Id Udi

Ud0,icosβ

Ud0,i

Udi

Ud0,icosγ

a.

Ud0,i

b.

Fig. 4.20. Equivalent circuits of the inverter bridge.

In Table 4.1 there are grouped together the equations necessary to calculate the parameters of an HVDC link. Table 4.1 HVDC link equations 0 Ideal no-load direct voltage Direct voltage with commutation overlap and ignition delay

Rectifier 1 U d 0, r =

Inverter 2

3 2 Nik ,rU kr π

U dr = U d 0,r cos α −

U d 0,i =

3 2 Nik ,iU ki π

3 3 X kr I d U di = U d 0,i cos γ − X ki I d π π

Current in the transformer secondary

6 Id π I pr = N ik , r I sr

6 Id π I pi = N ik , i I si

Active power on DC line

Pdr = U dr I d

Pdi = U di I d

Apparent power at AC system terminal bus

S kr = 3U kr I pr

S ki = 3U ki I pi

I sr ≅

I si ≅

270

Basic computation

0 Active power at AC system terminal bus Reactive power at AC system terminal bus

Table 4.1 (continued) 2

1 Pkr ≅ Pdr

Pki ≅ Pdi

Qkr = S kr2 − Pkr2

Qki = S ki2 − Pki2

U dr = U di + RL I d

DC line equation

Observation: The alternating voltages are phase-to-phase voltages, N ik , r and N ik , i are the transformation ratio of the transformers supplying rectifier, respectively inverter bridges. The parameters of DC lines, at rated voltages ranging from 200 to 500 kV and rated powers up to 2000 MW, take on the following typical values [4.10]: • for overhead sections: r0 = 0.015 ... 0.020 Ω / km ; l0 = 3 ... 4 mH / km ; c0 = 0.04 ... 0.05 µF / km ; • for cable (chiefly submarine) sections: r0 = 0.04...0.05 Ω / km ; l0 = 7...10 mH / km ; c0 = 0.8...1µF / km .

4.4. Control of direct current link 4.4.1. Equivalent circuit and control characteristics A DC transmission link with two ends, composed by two transformers, rectifier, inverter and the electric line might be represented through an equivalent circuit, where the baseline for all elements is the DC part (Fig. 4.21). Subscript r and i refer to rectifier and inverter respectively. Pr+jQr

Rcr

RL

Pi-jQi

-Rci

Id UAC,r

Ud0,r

Ud0,rcosα

Rectifier AC

Udi

Udr

DC line

Ud0,icosγ

Ud0,i

UAC,i

Inverter

DC Fig. 4.21. Equivalent circuit of the HVDC link.

AC

HVDC transmission

271

When the scheme is build, several issues should be considered: − the transformers have variable transforming ratios. The effect of leakage reactance on DC voltage was included through the commutation resistances Rcr on the rectifier side, respectively Rci on inverter’s side; – the DC overhead line is represented only through resistance RL; its capacity and reactance have been neglected; – harmonics filters and the elements for reactive power generation (capacitor banks, static or synchronous compensators) have not been included. A converter can be used to either convert AC power to DC power or viceversa. Only the relative DC voltage and current polarities determine the direction of power flow. A terminal, which supplies power to DC link, is termed the rectifier terminal; the terminal, which takes power from the DC line, is termed the inverter terminal. The direction of power flow and therefore the terminology for the terminals can change in less than a second if the converter voltage levels are changed by firing angle control. Power flow on a DC link is always from the terminal with the greater positive direct voltage to the lesser positive voltage or from the more negative terminal to the less negative terminal. However, a power direction reversal does not require a current direction reversal. The direct current I d , flowing through the line from the rectifier to inverter is: Id =

U d 0, r cos α − U d 0, i cos γ Rcr + RL − Rci

(4.20)

where: Rcr =

3 X kr ; π

Rci =

3 X ki π

(4.21)

DC power is simply the product of the current and the voltage at the particular location. Power transfer on the DC link can be increased by either increasing the rectifier voltage or decreasing the inverter voltage. Either of these increases the current and power in the DC link. The power at the rectifier terminals is: Pdr = U dr I d

(4.22')

and power at the inverter terminal is: Pdi = U di I d = Pdr − (RL + Rcr − Rci )I d2

(4.22'')

An essential characteristic of the transmission at direct voltage is the possibility to rigorously control the transmitted active power, in terms of magnitude and direction. The normal mode of power control on a DC link is to hold the inverter voltage constant and to control the current by changing the rectifier voltage level.

272

Basic computation

The rectifier direct voltage will usually be between 0% (no load) and 10% higher than the inverter DC voltage, depending upon line losses and loading level. The value of the direct current, I d , could be controlled by the change of U d 0, r and U d 0, i values or by the change of α or γ angles: – the values of U d 0, r and U d 0, i could be regulated by changing the converter transformer turn ratio N ik with a slow acting control; – the firing angle could be rapidly controlled by gate-control. Usually, both the rectifier and the inverter are operating with α = 12° ÷ 15° , respectively γ min = 18° . As a consequence, through firing angles control, a converter could operate as rectifier or as inverter; the power flow direction could be changed. This change could be obtained by reversal of polarity of the direct voltages at the both ends. The responsibilities for voltage regulation and current regulation are kept distinct and are assigned to separate terminals. Generally, in a DC transmission link, the inverter substations controls the direct voltage U d , keeping it at a constant value and rigorously dependent on the voltage on AC side. At the other terminal, the rectifier substation regulates the direct voltage so that I d current corresponds to the necessary active power Pd . The ideal voltage-current characteristics are presented in Figure 4.22. The voltage U d and the current I d forming the coordinates may be measured at the same common point on the DC line. The rectifier and inverter characteristics are both measured at the rectifier. The inverter characteristic thus includes the voltage drop across the line. With the rectifier maintaining constant current, its U d − I d characteristic is a vertical line. Ud

Operating point

Inverter (CEA)

Rectifier (CC) Id Fig. 4.22. Ideal steady-state U d − I d characteristic seen from the rectifier terminal.

Based on the inverter equation seen from the rectifier U d = U d 0, i cos γ + (RL − Rci ) I d

(4.23)

if the current is constant, also the voltage will be constant. This gives the inverter characteristic, with γ maintained at a fixed value. If the commutating resistance Rci is slightly larger than the line resistance RL , the

HVDC transmission

273

characteristic of the inverter has a small negative slope. Since an operating condition has to satisfy both rectifier and inverter characteristics, it is defined by the intersection of the two characteristics. The rectifier characteristic can be shifted horizontally by adjusting the “current command” or “current order”. If measured current is less than the command, the regulator advances the firing by decreasing α. The inverter characteristic can be raised or lowered by means of its transformer tap changer. When the tap changer is moved, the constant extinction angle (CEA) regulator quickly restores the desired γ. As a result, the direct current changes, which is then quickly restored by the current regulator of the rectifier. The rectifier tap changer acts to bring α into the desired range (10°…20°) to ensure a high power factor and adequate room for control [4.9]. The rectifier maintains constant current (CC) by changing α. However, α cannot be less than its minimum value (αmin). Once αmin is reached, no further voltage increase is possible, and the rectifier will operate at constant ignition angle (CIA). Ud A H A

Rectifier (CIA) Normal vo lt E Reduced volt

Inverter B (CEA) D G B

G Inverter (CC) O

F

C

I0i

I0r

∆Im

Rectifier (CC)

Id

Fig. 4.23. Actual steady-state characteristics.

The actual steady-state characteristics, based on the above description, are showed in Figure 4.23: (i) the rectifier characteristic consists of the two segments (AB and BC). The segment AB corresponds to minimum ignition angle and represents the CIA control mode; the segment BC represents the normal constant current (CC) control mode. The complete rectifier characteristic at normal voltage is defined by ABC. At a reduced voltage it shifts, as indicated by A'B'C; (ii) the inverter characteristic consists of two segments (DE and EF). The CEA characteristic of the inverter intersects the rectifier characteristic at G for normal voltage. However, the inverter CEA characteristic (HD) does not intersect the rectifier characteristic at a reduced voltage represented by A'B'C. Therefore, a big reduction in rectifier voltage would cause the current and power to be reduced to zero after a short time depending on the presence of the smoothing inductance. To avoid this situation, the inverter is equipped with a current regulator, whose reference I 0i is smaller than the reference I 0 r of the rectifier.

274

Basic computation

The variables I 0 r and I 0i are called rectifier current order and inverter current order, respectively, while their difference ∆I m = I 0 r − I 0i is called current margin and its usual value is ∆I m = 0.1 ... 0.15 p.u. from the rated current. Under normal operating conditions, the operating point is G, the rectifier controls the direct current and the inverter the direct voltage; this is CCR (Current control performed by the rectifier) operating mode. With a reduced rectifier voltage, the operating condition is represented by the intersection point G'. The inverter takes current control and the rectifier established the voltage; this is CCI (Current control performed by the inverter) operating mode. The change from a mode to another is referred to as a mode shift. In most HVDC systems, each converter is required to function as a rectifier as well as an inverter. Consequently, each converter is provided with a combined characteristic as shown in Figure 4.24. Ud

Converter A (CIA) Converter B (CEA)

G1

Id

0 (CC)

(CC)

G2

Converter A (CEA)

Converter B (CIA)

Fig. 4.24. Combined characteristics.

The characteristic of each converter consists of three segments: constant ignition angle (CIA) corresponding to α min , constant current (CC) and constant extinction angle (CEA). The power transfer is from converter A to converter B, when the characteristics are as shown in Figure 4.24 by solid lines. The operating condition in this mode of operation is represented by point G1 . The power flow is reversed when the characteristics are as shown by the dotted lines. This is achieved by reversing the “margin setting”, i.e. by making the current order setting of converter B exceed that of converter A. The operating condition is now represented by G 2 in

HVDC transmission

275

the Figure 4.24; the current I d is the same as before, but the voltage polarity has changed [4.9].

4.4.2. Control strategies of HVDC systems A DC link constitutes an electric system whose operating state is determined by means of the values of the electrical quantities associated to the converter station, called state variables, which are grouped into a vector [ X ] . For a simple approach of our problem, the one-line diagram (Fig. 4.25) is used: Uk φ

Nik

~ k

Ik

0

Ui

Xk i'

Ii 0

ϕ

Id α(γ)

+ Ud

i

Fig. 4.25. The one-line diagram of the converter station.

U k ∠φ is the alternating voltage at the converter station bus: U kr ∠φ r (rectifier) and U ki ∠φi (inverter); U i ∠ϕ – the fundamental component of the alternating voltage at the secondary winding of the converter transformer; I p , I s – the fundamental component of the alternating current at the

primary and secondary windings of the transformer; α , γ – the firing angle and extinction angle respectively; N ik = U i U k – the transformation ratio; – the direct voltage at every converter; Ud – the direct current. Id In order to simplify the expressions of the mathematical model and to improve the performances of the computation algorithms, the alternating current at the secondary winding of the transformer is chosen as reference. These 10 variables defined earlier, of which nine are associated to the converter and the voltage U k form a possible choice of the vector [ X ] of the quantities associated to the DC system. For the solution of the load flow problem with two terminal HVDC systems, two of ten variables are enough to be chosen as independent variables (the others, determined in terms of these two independent variables, form the assembly of dependent or output variables). This choice leads to more complicate expressions of the other state variables, making difficult their implementation into traditional load flow programs. Thus, we can use either a vector of 9 independent variables

[X ] = [U dr , φ r , N ik , r , cosα,U di , φi , N ik ,i , cosγ, I d ] t

(4.24)

276

Basic computation

or a vector of 7 independent variables

[X ] = [U dr , N ik , r , cosα,U di , N ik ,i , cosγ, I d ] t

(4.24')

If the vector of 7 components is chosen, besides the system of independent equations corresponding to the rectifier, the inverter and the DC line, four more equations are added, modelling the control strategy of the link by specifying the values of 4 independent variables and also the limit values (minimum and maximum) in terms of which the commutation into a operation mode is performed. A three position code will be used to identify the control mode. The first position indicates whether constant power, denoted by P, or constant current, denoted by I, is used. The second position indicates if the current control is in the rectifier (R) or in the inverter (I). The number in the third position of the operation mode code shows in which converter station the transformation ratio N ik is fixed: PR1 (no station); PR2 (rectifier); PR3 (inverter); PR4 (both stations). Taking into account the four main strategies PR, PI, IR, II, which are functions of the four variables chosen for the model, it results a total number of 16 possible operation modes (Table 4.2). Table 4.2 Possible control modes of a HVDC link Control mode PR 1

U di ; Pdi

Control mode IR 1

PR 2

N ik , r ; U di ; Pdi

IR 2

N ik , r ; U di ; I d

PR 3

Nik ,i ; Pdi

IR 3

Nik ,i ; I d

PR 4

N ik , r ; Nik ,i ; Pdi

IR 4

N ik , r ; Nik ,i ; I d

PI 1

U dr ; Pdi

II 1

U dr ; I d

PI 2

N ik , r ; Pdi

II 2

N ik , r ; I d

PI 3

Nik ,i ; U dr ; Pdi

II 3

Nik ,i ; U dr ; I d

PI 4

Nik ,i ; N ik , r ; Pdi

II 4

Nik ,i ; N ik , r ; I d

Specified variables

Specified variables

U di ; I d

The PR strategy having four operation modes is used, as it can be seen in Figure 4.26. The direct voltage U di is controlled by the tap changer at the inverter, and the firing angle α is controlled by the tap changer at the rectifier. The variables in boxes are those that are specified for respective control mode. The operation mode PR1 corresponds to operation with constant angles α and γ, ensured by tap changer actions in order to maintain constant the alternating voltage value applied to the converter station terminals. In this way, the values of U d 0, r and U d 0,i , and also the current I d on the DC line and the power Pdi at inverter are maintained constant.

HVDC transmission

277 sp

Udi>Udi

PR1 α, γ, Udi, Pdi ααsp

Udi 4 ; − intermediate power system, 2 < K sc , ef < 4 ; voltage support may have to be provided at the AC terminals of the converter station, by for example, static VAr compensators; − weak power system, K sc , ef < 2 ; synchronous condensers or static VAr capacitors may have to be use to strengthen of the AC system. These compensation devices, for the inversion station case, alongside generation/absorption of reactive power, are used on the AC side to reduce the temporary overvoltages after load shedding. Figure 4.38 illustrates the characteristics Pd − I d of an inverter feeding an AC system by infinite impedance (commutating reactance at 20% for γ = 18° fixed). Pd 1.2 3 1.6 1

1.0 0.8 0.6 0.4

Ksc Fig. 4.38. Pd − I d characteristics of an inverter in terms of shortcircuit ratio.

0.2 0

0.2 0.4 0.6 0.8 1.0 1.2

Id

Such classification of the AC system strength gives a preliminary evaluation of potential interactions between AC and DC systems. Also, the AC-DC interaction is influenced by the phase angle of the Thevenin equivalent (of the impedance

HVDC transmission

301

Z th ); this is also called damping angle and its value has significant influence on the DC system control stability.

4.7.3. Effective inertia constant The capability of the AC system to maintain the required voltage and frequency depends on its rotational inertia. For satisfactory performance, the AC system should have a minimum inertia relative to the size of the DC link [4.9]: H dc =

total rotational inertia of AC system, MW ⋅ s MW rating of DC link

For a normal operation, an effective inertia constant H dc of at least 2.0 to 3.0 seconds is required. Synchronous condensers have to be use in order to increase the AC system inertia.

4.7.4. Reactive power and the strength of the AC system It should be noted that because the AC systems are largely inductive, the reactive power exchange is mainly responsible for the effect of converter behaviour on the AC network voltage side. Many schemes in the past were designed with transformer reactance of the order of 20% or even more to limit the thyristors fault currents. On the other hand, the reduction of the transformer reactance has some advantages, such as: − the reduction of the consumed reactive power at the converter Qc ; − the AC system filters and any additional shunt capacitors are normally designed to supply at least all converter reactive power. By reducing the total reactive power Q f + Qbc the cost of the equipments decreases while K sc , ef increases; − temporary overvoltages will be reduced, due to smaller shunt capacitors; The bigger the reactive power consumed by any converter is, the transferred power increase. As it has been shown, for angles α and γ in the range of 15° to 18° and a commutating reactance X k = 15% , a converter consumes 50% to 60% reactive power. The reactive power necessary for converters operation is mainly provided by the capacitors from the filters constitution and by the capacitor banks. Since the consumed reactive power varies with the transferred DC power Pd , capacitors must be provided in appropriate sizes of switchable banks, so that the voltage is maintained in acceptable range at all load levels. The voltage level is influenced, also, by the short-circuit ratio of the AC system. Generators, if located near the DC terminals, can provide some of the required reactive power to maintain the voltage in acceptable range.

302

Basic computation

For weak AC systems, it may be necessary to provide reactive power by means of static VAr compensators or synchronous condensers. Furthermore, at the operation of HVDC system when connected to a weak AC system, other problems appear [4.9]: − Dynamic overvoltage, when sudden interruption of the transferred power Pd through the DC link occurs. With sudden decrease of consumed reactive power of converter station the voltage increase suddenly due to shunt capacitors and harmonic filters. − Reduction of voltage stability reserve. For a HVDC link connected to a weak AC system, the alternating as well as direct voltages, especially on the inverter side are very sensitive to changes in DC line loading. Therefore, an increase in direct current is accompanied by a decrease in alternating voltage. In such cases, the voltage control and recovery after faults presents problems; the DC system behaviour can contribute to reduction in stability reserve or even to AC system collapse. − Voltage flicker due to the temporary switching off the capacitors and reactors. Also, the harmonic resonance at low frequency appearance are due to parallel resonance between capacitors and harmonic filters, on one hand, and the inductive components from the AC system; consequently, dangerous overvoltages can appear.

4.8. Comparison between DC and AC transmission Normally, with bulk power transmission interconnections there is a choice between AC and DC, and the determination may in some cases be a matter of economics. In the case of DC a large investment is required in terminal equipments for conversion, and this is mostly independent of the length of transmission. However, the DC overhead lines are cheaper than AC lines for the same power transfer and DC lines losses are less than those for AC lines. Recent trends indicate that the costs of overhead lines are increasing at a higher rate than the costs of terminal equipment. Generally, the construction cost of AC lines, for the same transmitted power and the same insulation is higher than direct current technology. This is due, on one hand, to the fact that for the same rated voltage of the line, the insulation level is greater at AC than DC and, on other hand, the transmitted power, for the same per unit power losses, at AC is half from the power transmitted at DC [4.6]. Case I: Consider a single-circuit three-phase line and a bipolar DC line, in different assumptions (Fig. 4.39). a) For the same transmitted power and the same pick voltage to neutral, the ratio of power losses at AC to the ones at DC is 1.33. The active power on the three-phase line is: PAC = 3Vs I AC

HVDC transmission

where Vs = Vmax, s

303

2 ( Vmax, s = Vˆs being the peak voltage to neutral of the AC

line); the power factor cos ϕ = 1 has been considered.

AC

AC

AC

a. AC

AC + +

DC

b. Fig. 4.39. One line diagrams of a single-circuit three-phase line (a) and of a bipolar DC line (b).

The active power on the bipolar DC line is: Pd = U d I d where U d 2 = U max, d for the DC line. Assuming that PAC = Pd obtain: 3Vs I AC = U d I d or in other form

(3 2 ) V

max, s I AC

(4.47)

= 2U max, d I d

Assume also the insulators withstand the same peak voltage to neutral in both cases, Vmax, s = U max, d , it results:

(

)

I d = 3 2 2 I AC Power losses for the two cases are given by: 2 ∆PAC = 3 I AC R ; ∆Pd = 2 I d2 R

where R = Rd = RAC is the ohm resistance of one phase.

(4.48)

304

Basic computation

Taking into account (4.48) it results the ratio of power losses as: 2

∆PAC 3 ⎛ 1 ⎞ = ⎜ ⎟ = 1.33 ∆Pd 2 ⎝ 1.06 ⎠ b) For the same transmitted power and considering the same power losses and the same conductor cross-sectional area, the insulation level at direct current is only 87% with respect to the one at alternating current. The power losses for the two cases are: 2 ∆PAC = 3 I AC R ; ∆Pd = 2 I d2 R

Equating the expressions of power losses obtain: Id =

(

)

2 I AC = 1.225 I AC

3

(4.49)

From (4.47) and (4.49) it results: U d = 3 ⋅ 2 Vs

(4.50)

Assuming the discharge voltage of DC insulators is equal to the peak value of the alternating voltage that generates the discharging, it results: insulation level in AC line is k1 ⋅ 2 Vs and k 2 ⋅ (U d 2 ) in DC line respectively, where k1 and k 2 are multiplication factors. To simplify, assume k1 = k2 ; the ratio of insulation levels in DC and AC will be: DC insulation level U d 2 = AC insulation level 2 Vs

(4.51)

For the chosen case, taking into account (4.41), it results: DC insulation level 3 = = 0.87 AC insulation level 2 We can conclude the DC line is more economic, besides it having only two conductors compared with AC line that has three conductors and presents an insulation level of 87% from the AC one. c) If y stand for the ratio of DC and AC power losses: y = ∆Pd ∆PAC for Rd = RAC , obtain: Id 3y = I AC 2 For cos φ = 1 and assuming the transmitted power is the same for DC and AC as well, we can write:

HVDC transmission

305

3Vs I AC = U d I d By compounding the last two equations it results: Ud 3⋅ 2 = Vs y The ratio of the two voltages given by relationship (4.51) becomes: DC insulation level U d 2 3 0.87 = = = AC insulation level y 2 Vs 2 y

Fig. 4.40. Dependency between the ratio of insulation levels and the ratio of power losses for the same transmitted power [4.6].

AC insulation level

DC insulation level

This latter dependency is provided graphically in Figure 4.40.

y

∆Pd ∆PAC

Case II: Double-circuit three-phase line transformed into three DC circuits, having the same insulation level. The transmitted power through the double-circuit AC line ( cos ϕ = 1 ) is: PAC = 2 ⋅ 3 Vs I AC The double-circuit AC line is transformed into three DC circuits, each one having two conductors on the polarities (+) and (-) respectively, and the potential U d 2 referred to ground. The power transmitted through the three DC circuits is: Pd = 3U d I d a) For the case

I AC = I d

and considering the same voltage level

U d 2 = 2Vs , the ratio of transmitted powers should be: Pd 3U d I d 2 2 Vs = = = 2 PAC 2 ⋅ 3 Vs I AC 2Vs

306

Basic computation

The ratio of percentage powers is: DC power losses % ∆Pd PAC = Pd ∆PAC AC power losses % or taking into account the above mentioned ∆Pd [%] 3 ⋅ 2 R I d2 1 = = 0.71 2 ∆PAC [%] 2 ⋅ 3 R I AC 2 that is ∆Pd [%] = 0.71 ∆PAC [%] b) If consider the same percentage power losses and the same insulation level ∆Pd ∆PAC = ; Pd PAC

Ud = 2 Vs 2

it results 2 3 ⋅ 2 R I d2 2 ⋅ 3 R I AC = 3U d I d 2 ⋅ 3Vs I AC

or

2I d U d = I AC Vs

that is I d = 2 I AC . In this case the ratio of the transmitted powers will be: Pd 3 Ud Id = =2 PAC 2 ⋅ 3 Vs I AC that is: Pd = 2 PAC

40.1 m

40.5 m

From the above it results that for the same rated voltage the DC lines leads to lower investments, respectively softer constructions than AC ones: with less conductors and insulators and softer electric towers (Fig. 4.41) [4.25].

Fig. 4.41. Comparison between towers sizes of 800 kV AC and ±500 kV DC, having the same transmission capacity (2000 MW).

HVDC transmission

307

For efficient utilization of the area occupied, a line operating at alternating current of rated 220 kV with transmission capacity of 480 MVA can be transformed into a line operating at direct current of rated ±380 kV, obtaining on this way a triple transmission capacity (Fig. 4.42). The AC three-phase line with double conductor on phase is converted into a DC bipolar line with triple conductor on pole. 480 MVA

1440 MVA

Unchanged tower body and foundations 220 kV AC

380 kV DC

Fig. 4.42. Transformation of a tower operating at 220 kV AC voltage into a one operating at ±380 kV DC.

The DC transmission lines have a bigger transmission capacity for the same right of way (Fig. 4.43).

Fig. 4.43. Transmission power vs. right of way for HVDC and HVAC.

Transmitted power [MW]

10000

HVAC

HVDC 1000

30

30

40

50

60

70

80

Right of way [m]

The costs of terminal conversion substations being higher than of AC ones, the total costs become comparable around the equilibrium distances (Fig. 4.44). The costs of overhead lines for powers bigger than 100 MW and transmission distances of 500 ÷ 800 km have the same size for DC and AC as well. The transmission distance increases the DC links will be more efficient. The cost advantage of HVDC increases with the length but decreases with the capacity of a link.

308

Basic computation

A

st co l cost ta to DC total C

Fig. 4.44. Total investments for an OEL and a DC link in terms of transmission distance.

DC line DC terminal AC line cost AC terminal cost Break even distance

Distance

OEL 800 km UEL 50 km

A comparison between the transmission capacities of some AC and DC lines, based of technical-economic considerations, is given in Table 4.4 [4.16]. Table 4.4 Transmission capacities of some AC and DC lines Economic loading of a AC line [kV] [MW] 230 240 345 580 500 1280 765 2700

Equivalent DC line The same insulation level The same right of way [kV] [MW] [kV] [MW] ± 200 400 ± 300 900 ± 300 900 ± 500 2500 ± 400 1600 ± 700 4500 ± 600 3600 ± 1000 8000

These costs can be used to explore development options but confirmatory figures obviously need to be obtained from manufacturers. Each power system is different with respect to voltage, system strength, harmonic and reactive power limits. Each owner has different requirements concerning overloads, availability, reliability, etc. Each HVDC scheme is therefore unique and caution is needed when utilizing the DC turnkey costs and additional facility cost variations discussed above for competing options. It is extremely important to consider all options on the same relative cost basis and also on approximately the same system scope basis (same capacity, dynamic performance, reliability, loss analysis, etc.). The 177 kilometres long Murraylink underground high voltage interconnection, uses HVDC Light technology, is the world’s longest [4.23]. The project of ABB connects the electricity grids in the states of Victoria and South Australia, allowing power to be traded directly between the two states. Underground cables were used because a large proportion of terrain between the two states is made up of national parks with sensitive wildlife, as well as large privately owned agricultural areas. In addition to the visual and environmental impact, underground cables offer protection against Australia’s traditional causes of power outages, such as lightning and damage caused by wildlife or bush fires. In Table 4.5, the cost values, given in 1998 US$/kW/bipole (both ends), for one converter per pole, are presented.

HVDC transmission

309 Table 4.5 Historical HVDC turnkey cost division [4.8]

Back-Back 200-500 MW

Valve groups Converter transformers DC switchyard & filtering AC switchyard & filtering Control / Prot / Comm. Civil / Mech. works Auxiliary power Project eng. & admin. Total per kW

± 250 kV 500 MW

± 350 kV 1000 MW

± 500 kV 2000 MW

± 600 kV 3000 MW

$/kW

%

$/kW

%

$/kW

%

$/kW

%

$/kW

%

38

19

50

21

42

21

35

22

33

22

45

22.5

50

21

44

22

35

22

33

22

6

3

14

6

12

6

10

6

9

6

22

11

25

10

19

9.5

14

9

13

9

17

8.5

19

8

16

8

13

8

12

8

26

13

33

14

28

14

21

13.5

20

13.5

4

2

6

2.5

5

2.5

5

2.5

4

2.5

42

21

43

17.5

34

17

27

17

26

17

$200

100

$240

100

$200

100

$160

100

$150

100

3.8. Application on HVDC link Let us consider the DC bipolar line from Figure 4.45, having the following characteristics: • The rated power and voltage values are 1200 MW and ±300 kV , respectively; • The conversion stations consists of four bridges 6-pulse configurations, each of them having the commutation reactance X kB ,r = X kB ,i = 6 Ω ; • The total resistance of the line is RL = 15 Ω ; • The converter transformers are equipped with on-load tap change mechanisms to provide an appropriate level of the three-phase voltage to the valve bridge. This aims to restore, after disturbances, the values of the angles α and γ in intervals appropriate to the normal operating state ( α ∈ [15°, 21°] and γ ∈ [18°, 21°] , respectively). The rated transformation ratio has the same value at both the rated rectifier and inverter stations ( Nikrated , r = N ik ,i = 0.320 ) and may vary in the

310

Basic computation min max max interval from Nikmin , r = N ik ,i = 0.256 to N ik , r = N ik ,i = 0.384 , corresponding to the interval (0.8...1.2), in per unit, with increasing step of 0.01 p.u., in order to maintain the ignition angle α in appropriate range in normal operating state and the inverter voltage in the range ±2.5% of the rated voltage ( U di ∈ [585, 615] kV ).

• The minimum value of the ignition delay angle is α min = 6° ; • The current margin is set to ∆I m = 15% ; • Under normal operating conditions the link operates in control mode 1, so that: − the rectifier controls the current (mode CC), operates with α = 18° and provides the power at terminals Pdr = 1260 MW ; − the inverter controls the voltage level, operates with constant extinction angle (mode CEA) γ = 19° and has the terminal voltage U di = 600 kV ; − the transformation ratio values are Nik ,r = Nik ,i = 0.32 ;

RL/2 400 kV

400 kV

+300 kV

-300 kV RL/2

Rectifier (r)

Inverter (i)

Fig. 4.45. Bipolar line.

Ukr φr Nik,r Ikr 0 kr

Xk,r ir’

Uir

Id

ϕr α

Iir 0

ir

Ud,r

Uii Ud,i

γ

ii

ϕi

Xki Iii 0

Nik,i ii’

Uki φi

Iki 0 ki

Fig. 4.46. Equivalent circuit of the bipolar line. I. Determine for normal operation: (i) rectifier voltage U dr and direct current I d ; (ii) overlap angles λ r and λ i ; (iii) phase-to-phase voltages values U kr and U ki , and currents I kr and I ki , respectively, at high voltage terminals of the conversion stations;

HVDC transmission

311

(iv) active, reactive and apparent powers as well as the power factor at high voltage terminals of the conversion stations ( S kr , Pkr , Qkr and cos φr , respectively Ski , Pki , Qki and cos φi ). II. Considering that the AC voltage at rectifier decrease by ∆U = 10% with respect to the voltage in normal operating state and the AC voltage at inverter remains unchanged, and the voltage control by means of tap changing is inactive, determine: (i) control strategy and the new value of the direct current I d ; (ii) rectifier and inverter voltages U dr and U di ; (iii) extinction angle γ and the overlap angles λ r and λ i after the grid control system acts; (iv) active, reactive and apparent powers as well as the power factor at high voltage terminals of the conversion stations ( S kr , Pkr , Qkr and cos φr , respectively Ski , Pki , Qki and cos φi ) before the control system of the tapchange position at transformer is activated; (v) transformation ratio value Nik ,r necessary to restore the normal operating conditions. III. Supposing that, with respect to the normal operating state, the AC voltage at inverter, increase by 2.5%, and the AC voltage at rectifier remain unchanged, under the hypothesis that the voltage control by means of tap changing is activated, determine: (i) rectifier and inverter voltages U dr and U di and the direct current I d before the grid control system acts; (ii) ignition delay α after the grid control system acts; (iii) transformation ratio values Nik ,r and Nik ,i after the action of the control system of tap-changer position; (iv) active, reactive and apparent powers as well as the power factor at high voltage terminals of the conversion stations ( S kr , Pkr , Qkr and cos φr , respectively Ski , Pki , Qki and cos φi ) under the new operating conditions. Generally, for the operating state analysis of a bipolar DC link the equivalent monopolar circuit is used (Fig. 4.46). Let us denote by nBr the number of rectifier bridges, and by nBi the number of inverter bridges and knowing that these are series connected on the DC side and parallel connected on the AC side, then the calculation relationships of the monopolar link, given in Table 4.1, get the form given in Table 4.6. Table 4.6. Operating equations of a bipolar line equivalated through a monopolar line Rectifier 0 3 2 nBr Nik ,rU kr π 3 = U d 0,r cos α − nBr X kB , r I d π

Inverter 1

U d 0, r = U dr

3 2 nBi N ik ,iU ki π 3 = U d 0,i cos α − nBi X kB ,i I d π

U d 0,i = U di

2 (A1) (A2)

312

Basic computation

0 U dr =

U d 0, r 2 I kr ≅

Table 4.6. (continued) 2

1

⎡⎣ cos α + cos ( α + λ r ) ⎤⎦

U di =

6 nBr N ik , r I d π

U d 0,i 2

⎡⎣ cos γ + cos ( γ + λ i ) ⎤⎦

I ki ≅

6 nBi N ik ,i I d π

(A3) (A4)

Skr = 3U kr I kr

Ski = 3U ki I ki

(A5)

Pkr ≅ Pdr = U dr I d

Pki ≅ Pdi = U di I d

(A6)

Qkr = Skr2 − Pkr2

Qki = Ski2 − Pki2

(A7)

cos φr =

Pkr Skr

cos φi =

Pki Ski

U dr = U di + RL I d

(A8) (A9)

Solution I. Normal operating state (i) Calculation of rectifier voltage U dr and direct current I d . From relation (A6) we express the direct current as: Id =

Pdr U dr

which is substituted in relation (A9) resulting the second order equation: 2 U dr − U diU dr − RL Pdr = 0

(A10)

having the solutions: U dr =

U di ± U di2 + 4 RL Pdr 2

Of the two solutions of equation (A10), the one with positive “+” sign is kept because for the no-load conditions ( I d = 0 and Pdr = 0 , respectively) this provides the value U dr = U di which is physically correct. Thus U dr =

600 + 6002 + 4 ⋅15 ⋅1260 = 630 kV 2

and Id =

Pdr 1260 = = 2 kA 630 U dr

(ii) Calculation of the overlap angles. From relation (A2) we determine:

HVDC transmission

313

U d 0, r =

3 3 nBr X kB ,r I d 630 + ⋅ 4 ⋅ 6 ⋅ 2 π π = = 710.62 kV cos α cos18°

U dr +

and U d 0,i =

U di +

3 3 nBi X kB ,i I d 600 + ⋅ 4 ⋅ 6 ⋅ 2 π π = = 683.05 kV cos γ cos19°

respectively. Taking into account relation (A3) it results: cos(α + λ r ) =

2U dr 2 ⋅ 630 − cos α = − cos18° = 0.822 710.62 U d 0,r

cos( γ + λ i ) =

2U di 2 ⋅ 600 − cos γ = − cos19° = 0.811 683.05 U d 0,i

and λ r = arccos 0.822 − 18° = 16.71° λ i = arccos 0.811 − 19° = 16.81°

(iii) The AC phase-to-phase voltages U k ,r and U k ,i at the terminal buses are determined using the relation (A1), from which it results: U kr =

U ki =

πU d 0, r 3 2 nBr N ik , r πU d 0,i 3 2 nBi Nik ,i

=

=

π ⋅ 710.62 3 2 ⋅ 4 ⋅ 0.32 π ⋅ 683.05 3 2 ⋅ 4 ⋅ 0.32

= 411.09 kV

= 395.14 kV

(iv) The active, reactive and apparent powers as well as the power factor on the AC sides are determined in the following. From relation (A4) it results: I kr =

6 6 nBr Nik , r I d = ⋅ 4 ⋅ 0.32 ⋅ 2 = 1.996 kA π π

I ki =

6 6 nBi Nik ,i I d = ⋅ 4 ⋅ 0.32 ⋅ 2 = 1.996 kA π π

Next, using the relations (A6), (A7), (A8) and (A9) we proceed to the calculation of: • Apparent powers: Skr = 3 U kr I kr = 3 ⋅ 411.09 ⋅1.996 = 1421.21 MVA Ski = 3 U ki I ki = 3 ⋅ 395.14 ⋅1.996 = 1366.07 MVA

314

Basic computation • Active powers: Pkr ≅ Pdr = U dr I d = 630 ⋅ 2 = 1260 MW Pki ≅ Pdi = U di I d = 600 ⋅ 2 = 1200 MW

• Power losses on the DC line: ∆PDC = Pdr − Pdi = RL I d2 = 15 ⋅ 22 = 60 MW

• Reactive powers absorbed at the terminal buses: Qkr = Skr2 − Pkr2 = 657.45 MVAr Qki = Ski2 − Pki2 = 652.80 MVAr

• Power factors: cos φr =

Pkr = 0.887 Skr

cos φi =

Pki = 0.878 Ski

II. Analysis of the disturbed state caused by the decrease in voltage by ∆U = 10% on the AC side of the rectifier terminal. (i) The control strategy and direct current I d . Due to the decrease in voltage at the terminal bus, the rectifier voltage decreases. In order to maintain the current at the specified value, the control system of the link will trigger, in a first stage, the decrease of the ignition angle α in order to increase the voltage U dr . Subsequent, the angle α will be brought in the normal operating range by changing the transformation ratio (shifting the tap-changer position). In this section, the operation of the HVDC link will be analysed for the time interval before to activation of the control system of the tap-changer position. Assume that the HVDC link will keep on operation in normal conditions with: rectifier on CC control with I d = 2 kA and inverter on CEA control with U di = 600 kV . Therefore, the ignition angle α should be changed so that:

U dr = U di + RL I d = 600 + 15 ⋅ 2 = 630 kV From relation (A2) written under the form U dr =

3 2 3 ⎛ ∆U ⎞ nBr Nik , r ⎜ 1 + U kr cos α − nBr X kB , r I d ⎟ π π ⎝ 100 ⎠

where ∆U stands for the percent voltage variation at the terminal bus with respect to the value corresponding to the normal operating state, we determine: 3 nBr X kB ,r I d π cos α = 3 2 ⎛ ∆U ⎞ nBr N ik , r ⎜ 1 + ⎟ U kr π ⎝ 100 ⎠ U dr +

HVDC transmission

315

For ∆U = −10% it results: cos α =

630 +

3 ⋅4⋅6⋅2 π

3 2 ⋅ 4 ⋅ 0.32 ⋅ (1 − 0.1) ⋅ 411.09 π

= 1.057 > 1

Therefore, the normal operating conditions cannot be fulfilled, and the shift in the control mode is chosen: rectifier on CIA control mode with α = α min = 6o and inverter on CC control mode with: I d = I dsp − ∆I m = (1 − 0.15) I dsp = 0.85 ⋅ 2 = 1.7 kA

where I dsp = 2 kA is the direct current value under normal operating conditions. (ii) The rectifier and inverter direct voltages now are: U dr = =

3 2 3 ⎛ ∆U ⎞ nBr N ik , r ⋅ ⎜ 1 + ⎟ U k ,r cos α min − π nBr X kB ,r I d = π ⎝ 100 ⎠ 3 2 3 ⋅ 4 ⋅ 0.32 ⋅ 0.9 ⋅ 411.09 ⋅ cos 6° − ⋅ 4 ⋅ 6 ⋅1.7 = 597.09 kV π π

U di = U dr − RL I d = 597.09 − 15 ⋅1.7 = 571.59 kV

(iii)

The angles γ , λ r and

λi

Taking into account that the new AC rectifier voltage value is U kr = 0.9 ⋅ 411.09 kV (reduced by 10%) and that the AC inverter voltage remained unchanged ( U ki = 395.14 kV ), from relation (A1) it results: U d 0, r =

3 2 3 2 nBr Nik ,rU kr = ⋅ 4 ⋅ 0.32 ⋅ 0.9 ⋅ 411.09 = 639.55 kV π π

which is 0.9 of the normal operating state value, while the inverter voltage remained unchanged, U d 0,i = 683.05 kV . Next, from relations (A2) and (A3) we determine: cos γ =

U di +

3 3 nBi X kB ,i I d 571.59 + ⋅ 4 ⋅ 6 ⋅1.7 π π = = 0.894 683.05 U d 0,i

that is γ = arccos 0.894 = 26.64° cos(α min + λ r ) = cos( γ + λ i ) =

2 U dr 2 ⋅ 597.09 − cos α min = − cos 6° = 0.873 639.55 U d 0,r

2 U di 2 ⋅ 571.59 − cos γ = − cos 26.64° = 0.780 683.05 U d 0,i

316

Basic computation

and λ r = arccos 0.873 − 6° = 23.191° λi = arccos 0.780 − 26.64° = 12.10°

respectively. (iv) The active, reactive and apparent powers as well as the power factors at the terminal buses are determined in a similar manner like at the point (iv) from the case I. It results: • Currents at the terminal buses: I kr =

6 6 nBr Nik ,r I d = ⋅ 4 ⋅ 0.32 ⋅1.7 = 1.697 kA π π

I ki =

6 nBi Nik ,i I d = 1.697 kA π

• Apparent powers: Skr = 3 U kr I kr = 3 ⋅ 0.9 ⋅ 411.09 ⋅1.697 = 1087.48 MVA Ski = 3 U ki I ki = 3 ⋅ 395.14 ⋅ 1.697 = 1161.43 MVA

• Active powers: Pkr ≅ Pdr = U dr I d = 597.09 ⋅1.7 = 1015.05 MW Pki ≅ Pdi = U di I d = 571.59 ⋅1.7 = 971.70 MW

• Reactive powers: Qkr = Skr2 − Pkr2 = 390.24 MVAr Qki = Ski2 − Pki2 = 636.18 MVAr

• Power factors: cos φr =

Pkr = 0.933 Skr

cos φi =

Pki = 0.837 Ski

Observations: a) The transmitted power (the power at the inverter terminal) decreases by: ∆P =

1200 − 971.70 ⋅100 ≅ 19% 1200

with respect to the power transmitted in normal operating state. b) the decrease in absorbed reactive power and the increase in power factor at rectifier terminal are due to the decrease of the ignition delay angle up to the value α min = 6° .

HVDC transmission

317

c) the reduction of the power factor at the inverter terminal bus is due to the increase of the extinction angle γ . (v) In order to restore the normal operating conditions, the change of the transformation ratio Nik ,r is required, so that for α = 18° the voltage U dr to (n) restore to the value U dr = 630 kV .

From relations (A1) and (A2) it results that the normal operating state is restored if: U d( n0,) r =

3 2 (n) nBr Nik ,r ⋅ 0.9 ⋅ U kr π

where the superscript (n) designate the values corresponding to the normal operating state. The new value of the transformation ratio is: Nik ,r =

π U d( n0,) r 3 2

(n) nBr ⋅ 0.9 ⋅U kr

=

π ⋅ 710.62 3 ⋅ 2 ⋅ 4 ⋅ 0.9 ⋅ 411.09

= 0.356

respectively Nik ,r =

0.356 = 1.1125 p.u. 0.320

III. Analysis of the disturbed state caused by the increase in voltage by ∆U = 2.5% on the AC side of the inverter terminal. (i) Because the inverter operates in mode CEA ( γ = ct. ) and the control system of the tap-changer is not activated (this will act after a certain period of time which is of the seconds order), the increase in AC voltage at inverter terminal bus will determine an increase in inverter voltage U di . Therefore: U d 0,i =

3 2 ⎛ ∆U ⎞ ( n ) 3 2 nBi Nik ,i ⎜ 1 + ⎟ U ki = π 4 ⋅ 0.32 ⋅1.025 ⋅ 395.14 kV = 700.12 kV π ⎝ 100 ⎠

and thus: U di = U d 0,i cos γ −

3 3 nBi X kB ,i I d = 700.12 ⋅ cos19° − ⋅ 4 ⋅ 6 ⋅ 2 = 616.14 kV π π

This change in voltage U di will results in change of the direct current I d because (n) the operating conditions at rectifier remained unchanged ( U dr = U dr = 630 kV ).

The new value of the direct current is: Id =

(n) U dr − U di 630 − 616.14 = = 0.924 kA RL 15

(ii) Detecting this decrease in current, the grid control system will trigger the decrease of the ignition angle α so that the current will restore to the normal operating state value I d( n ) = 2 kA . The new value of the angle α is determined as it follows:

318

Basic computation • The voltage necessary at rectifier is calculated as: U dr = U di + RL I d( n ) = 616.14 + 15 ⋅ 2 = 646.14 kV

• From relation (A2) it results: cos α =

U dr +

3 3 nBr ⋅ X kB , r I d 646.14 + ⋅ 4 ⋅ 6 ⋅ 2 π π = = 0.974 710.62 U d( n0,) r

and α = 13.09° , respectively. (iii) After a period of time (seconds) from the disturbance occurrence, the control system of the transformer tap-changer will be activated. In this way, the rectifier transformer tap-changer acts to hold α between 15° and 21° while the inverter transformer tap-changer acts to hold U di between 585 kV and 615 kV. Maintaining the current on the DC line at the value I d = I d( n ) = 2 kA , according to relations (A1), (A2) and (A9), the voltages U dr and U di fulfil the following relations: U dr =

3 2 3 (n) nBr Nik , rU kr cos α − nBr X kB , r I d = π π

(A11)

3 2 3 = ⋅ 4 ⋅ Nik , r ⋅ 411.09 ⋅ cos α − ⋅ 4 ⋅ 6 ⋅ 2 = 2220.67 Nik , r cos α − 45.84 π π

U di =

3 2 3 nBi Nik ,i ⋅1.025 ⋅ U ki( n ) cos γ − nBi X kB ,i I d = π π

3 2 3 = ⋅ 4 ⋅1.025 ⋅ 395.14 ⋅ Nik ,i cos19° − ⋅ 4 ⋅ 6 ⋅ 2 = 2068.67 N ik ,i − 45.84 π π

U dr − U di = RL I d = 30 kV

(A12)

(A13)

Analysing the expressions (A11), (A12) and (A13) we see that in order to restore the normal operating conditions, the increasing of the ratio Nik ,r and angle α , and the decrease of ratio Nik ,i , respectively, are necessary. In Table 4.7 the ignition angle α values and the voltages U dr and U di for the analysed case, are presented, where the transformation ratio Nik ,r increase from Nikrated , r to rated Nikmax to Nikmin , r , and the transformation ratio N ik ,i decrease from N ik ,i ,i .

Table 4.7 Voltages U dr and U di angle α variations in terms of the and transformation ratio Nik ,r

Nik ,i

U dr

U di

0 0.320 0.323

1 0.320 0.317

kV 2 646.13 639.51

kV 3 616.13 609.51

α [º] 4 13.15 17.27

HVDC transmission

319

0 0.326 0.330 … 0.374 0.378 0.381

1 0.314 0.310 … 0.266 0.262 0.259

2 632.89 626.28 … 533.60 526.98 520.36

Table 4.7 (continued) 3 4 602.89 20.54 596.28 23.32 … … 503.60 45.82 496.98 46.91 490.36 47.97

We see that the shift in tap-changers with one position in increasing direction of Nik ,r and decreasing direction of Nik ,i , respectively, will restore the operation of the link with values of α and U di in admissible range, that is α = 17.17° ∈ [15°, 21°] and U di = 609,51 kV ∈ [585, 615] kV.

Appendix 4.1 HVDC SYSTEMS IN THE WORLD Line

Year

Power Voltage

Length [km] Main reason for choosing Over- Under[MW] [kV] HVDC system head ground a. Mercury-arc valve systems Kashira - Moscow 1950 30 ±200 115 1954Long sea crossing; frequency Gotland - Sweden I 20-30 100-150 96 1970 control 7+ Sea crossing; asynchronous link France - Great 1961 160 ±100 50+ 50/60Hz, out of service since Britain I 1984 8 Volgograd 1964 750 ±400 475 Long distance Donbass Asynchronous link, rapid Sakuma (Japan) 1965 300 2×125 control, low losses (ASEA) Konti - Skan Sea crossing; asynchronous link 1965 250 250 86 87 (ASEA) (Denmark-Sweden) New-Zeeland 1965 600 ±250 575 42 Long line including sea crossing Sardinia - Corsica The first multi-terminal link 1967 200 200 290 116 Italy (SACOI-1) (English Electric, ASEA) The first line operating in Vancouver, 1968/ 312 +260 41 28 parallel with an AC circuit; Pole I (Canada) 1969 undersea cable (ASEA) Pacific - Intertie I, 1970 1600+ Long line in parallel with two AC lines; fast control (General (Columbia River- 1985 400+ ±500 1360 Electric, ASEA) 1989 1100 Los Angeles) Long distance; stability Nelson - River 19731620 ±450 890 (GEC ALSTOM) Bipole I (Canada) 1977

320

Basic computation Appendix (continued) b. Thyristor valve systems

Eel River (Canada) Skagerrak I...III (Norway Denmark) Square - Butte (USA) Cabora Bassa Apollo (Mozambique – South Africa) Skin - Shinano (Japan) Vancouver Pole II (Canada) Nelson - River Bipole II Bipole III(Canada) Hokkaido-Honshu (Japan) Acaray (Paraguay-Brazil) Vyborg (Russia Finland) Inga Shaba (Rep. Congo) Durnrohr (Austria) Gotland - Sweden II/III Itaipu, Bipole I and II (Brazil)

1973

320

2×80

1974/ 1977/ 1993

500 ... 1500

±250 85+28

1977

500

±250

749

19771979

1930

±533

1420

1977

300

2×125

-

-

Frequency converter 50/60 Hz

1978

792

±260

41

32

-

1978 - 900 1992/ 2000 1997 1500

+250 ±500 ±500

940 -

Long distance (ABB/Siemens/AEG)

1979

300

±250

1981

50

26

-

-

1982

1070

3×±85

-

-

1982

560

±500

1700

-

1983

550

±145

-

-

19831987

130

150

7

96

930 27 +97

1986

Intermountain 1986 (USA) Ekibastuz - Centre 1987 (Russia)

2000 2×±270

-

Back-to-back link

127

Sea crossing (ASEA, ABB) DC active filter

-

Long distance; stability

-

44

Long distance

Sea crossing; finally 600MW (Hitachi) Asynchronous link 50/60 Hz Back-to-back link

-

-

-

-

Long distance, finally, 1120 MW Back-to-back link, out of service since 1996 Undersea cable, asynchronous link The highest transmitted AC power at long distance (18432 thyristors), (ASEA) Back-to-back link (ABB / Siemens) Back-to-back link (ABB) Back-to-back link (ABB)

-

18 +46 +6

Undersea cable; asynchronous link 60/50Hz; peak load (CGEE ALSTHOM / GEC)

785+ 1985- 3150+ 2×±600 805 1987 3150

Chateauguay 1984 2×500 2×140.6 (Canada) Blackwater(USA) 1985 200 56.8 Highgate (USA) 1985 200 57 Great Britain France II

-

-

1920

±500

785

-

Long distance (ASEA)

6000

±750

2400

-

The biggest overhead DC line

HVDC transmission

321 Appendix (continued)

McNeil (Canada)

1989

150

42

-

-

Fenno - Skan 1989 (Finland - Sweden)

500

400

33

200

2×70

-

-

1200

±500

1000

-

1990

1500

±500

814

1991/ 1992

560

-350

575

42

1480

-

Vindhyachal (India)

1989/ 2×250 1990

Gezhouba 1991 Shanghai (China) Rihand - Delhi (India) New Zeeland DC Hybrid link Quèbec New England (Canada) South Vienna (Austria) Etzenricht (Germany -Tchéquie) Baltic Cable (Sweden Germany) Kontek (DenmarkGermany) Chandrapur-Padge (India) Haenam - Cheju Island (Korea) Chandrapur – Ramgundam (India)

19901992

2×2000 + ±450 2×690

Back-to-back link between East and West weak sub-systems (ALSTOM) Monopolar line, sea crossing (ASEA) Back-to-back link between North and West sub-systems (ASEA) Long distance; stability benefits (ABB/Siemens) Long distance; stability (ABB) Long distance including sea crossing (ABB) Long distance, asynchronous, with 5 terminals (ABB)

1992

550

145

-

-

Back-to-back link, out of service since 1996.

1993

600

-

-

-

Back-to-back link, out of service since 1996.

1994

600

450

12

250

The biggest undersea link of high capacity for a single cable (ABB)

1995

600

400

-

170

Sea crossing; asynchronous system (ABB)

1998

1500

±500

753

-

1998

300

±180

-

101

1998

1000

205

-

-

Vijayawada 1999 Gajuwaka (India)

500

205

-

-

DC link in parallel with an AC line; stability (ABB) Undersea cable; asynchronous link (GEC ALSTOM) Back-to-back link between West and South systems (ALSTOM) Back-to-back link between the East and North systems. ALSTOM) Long distance, undersea cable (ABB)

Leyte - Luzon (Philippine)

1999

880

±350

22

440

Shin - Shinano (Japan)

1999

53/ 53/ 53

10.6

-

-

Three terminal BTB 60/50/50 Hz

Minami 1999 Fukumitsu (Japan)

300

±125

-

-

Asynchronous link (BTB)

322

Basic computation Appendix (continued)

Ecosse-North Ireland Malaysia Thailand Rihant - Sasaram (India) Kii Channell (Japan) Tian - Guang (China) Higashi-Shimizu (Japan) Greece-Italy Three Gorges Guangdong (China) NorNedkabel Link (Norway-Holland) Bangalore-Talcher (India) Argentina-Brazil Sarawak Malaysia SWEPOL (Sweden - Poland) Eurokabel Vikingcable (NorwayGermany) Three Gorges Changzhou (China)

2000

250

250

-

-

2000

300/ 600

±300

110

-

2000

500

-

-

2000

2800 2×±500

51

51

2000

1800

±500

986

-

2000

300/ 600

±125

-

-

2001

500

400

105

163

2001

3000

±500

890

-

Long distance

2002

600/ 800

-

-

570

Undersea cable; asynchronous line

2003

2000

±500

1450

-

The biggest DC line from Asia

2003

1000

±70 2×±500 1500 660 4×+400

-

Back-to-back link (ABB) Undersea DC link (ABB) (Project cancelled) Long distance and sea crossing (ABB) Undersea cable; asynchronous line

2003 2003

600

±450

-

2002 2×600

600

-

2003 2×800

600

-

2003

±500

-

670 230

540

3000

-

Undersea cable Thyristor 8 kV, 1 kA (Siemens) Back-to-back link between East and North systems Thyristor 8 kV, 3500 A DC link in parallel with an AC line (Siemens) Asynchronous link between 50Hz and 60Hz systems Undersea cable; asynchronous line (ABB)

Undersea cable; asynchronous line Long distance

c. Systems HVDC in construction or in designing Three Gorges 2007 3000 ±500 Long distance Shanghai (China) IbValley-Jaipur 3000 ±600 1500 Long distance (India) Karamsad-Korba 3000 ±600 1450 Long distance (India) Dehang-Bareilly 5000 ±600 1500 Long distance (India) Balipara5000 ±600 1500 Long distance Ballabhgarh (India)

HVDC transmission

323 Appendix (continued)

Pancheswar (Nepal)Vadodra (India) Karnali - Vadodra (India) Vadodra - Pune (India) Pune - Madras (India) East - West High 2010 Power Link (Russia-Germany)

5000

±600

1100

5000

±600

1000

3000

±600

900

3000

±600

1050

4000

±500

2000

-

Multi-terminal system

-

Long distance, asynchronous, with 5 terminals

Chapter references [4.1] [4.2] [4.3] [4.4] [4.5] [4.6] [4.7] [4.8] [4.9] [4.10] [4.11] [4.12] [4.13] [4.14] [4.15]

Kimbark, E.W. – Direct current transmission, Vol. I, John Wiley and Sons Inc., New York, 1971. Uhlmann, E. – Power transmission by direct current, Springer-Verlag, Berlin, 1975. Weedy, B.M. – Electric power systems, Third Edition, John Wiley & Sons, Chichester, New York, 1979. Arrillaga, J. – High Voltage direct current transmission, Peter Peregrinus, London, 1983. EPRI – Methodology for integration of HVDC links in large AC systems, Phase I: Reference manual. Final Report El-3004, RP 1964-1, Prepared by Ebasco Services Incorporated, New York, March 1983. El-Hawary, M.E. – Electrical power systems. Design and analysis, Reston Publishing Company, 1983. Padiyar, K.R. – HVDC Power transmission systems. Technology and systems interactions, John Wiley & Sons, New York, 1991. CIGRE Working Group 14.20 – Economic assessment of HVDC links, Electra revue, No. 196, June 2001. Kundur, P. – Power systems stability and control, McGraw-Hill, Inc., New York, 1994. Marconato, R. – Electric power systems. Vol.1. Background and basic components, CEI Italian Electrotechnical Committee, Milano, 2002. Eremia, M. – Tehnici noi în transportul energiei electrice. Aplicaţii ale electronicii de putere (New techniques in electric power transmission. Power electronic applications), Editura Tehnică, Bucureşti, 1997. Eremia, M., Trecat, J., Germond, A. – Réseaux électriques. Aspects actuels, Editura Tehnică, Bucureşti, 2000. IEEE Std. P1030.1-2000 – IEEE Guide for specification of HVDC systems. Linder, S. − Power semiconductors. At the centre of a silent revolution, ABB Review, No. 4, 2003. England, L., Lagerkvist, M., Dass, R. − HVDC superhighways for China, ABB Review, Special report, 2003.

324 [4.16] [4.17] [4.18] [4.19] [4.20] [4.21] [4.22] [4.23] [4.24] [4.25] [4.26] [4.27]

Basic computation *** − Comparison between AC and DC transmission systems, Report CIGRE no. 37-94, WG 12-06, June 1994. *** − AC/DC Transmission-Interactions and comparisons, CIGRE Symposium, Boston, September 1987. Zhang, W., Asplund, G. − Active DC filter for HVDC systems, IEEE Computer Applications in Power, January 1994. Adamson, C., Hingorani, N.G. − High voltage direct current power transmission, Garraway Limited, London, 1960. Cory, B.J. − High voltage direct current converters and system, Mac Donald, London, 1965. Miller, T.J.E. − Reactive power control in electric systems, John Wiley and Sons, 1982. Gőnen, T. − Electric power systems engineering. Analysis and design, A Wiley − Interscience Publication, 1986. Asplund, G. et al. – HVDC Light – DC transmission based on voltage sourced converters. ABB revue, No. 1, 1998. *** – DC and AC Configurations (Chapter 3), CIGRE WG 14.20, Johanesburg, 1997. *** – High voltage direct current transmission, Technical Reports, AEGTelefunken. IEEE Std. 519-1992 – IEEE Guide for harmonic control and reactive compensation of static power converters. Smed, T. – Interaction between high voltage AC and DC systems, PhD thesis, Royal Institute of Technology, Sweden, 1991.

5

Chapter

NEUTRAL GROUNDING OF ELECTRIC NETWORKS 5.1. General considerations The neutral grounding of electric networks is one of the earliest concerns of power engineers; this is due to the effects of an accidental touch between a phase and the ground. Neutral grounding is intended to quickly and safely eliminate the electric arc that occurs during phase-to-ground faults, in order to avoid the network disconnection and to prevent the incident from turning into a breakdown of insulation (short-circuit between two or three phases and ground). The neutral grounding of electric networks is consequently one of the factors that condition the supplied power quality. The electric neutral point of a balanced positive- or negative-sequence threephase system corresponds to the gravity centre of the equilateral triangle formed by the phase-to-phase voltages of the network (Uab, Ubc, Uca) (Fig. 5.1). The connection point between the windings of a generator with star connection or secondary windings of a transformer with star or zigzag connection represents the physical neutral point. It is possible to provide the neutral to the terminals or to distribute it or neither of these. When distributed, a neutral conductor is wired to it. a

Fig. 5.1. Defining the neutral point N for a three-phase network.

U ca Vc c

Va

U ab N

U bc

Vb b

In terms of the situation of the neutral point with respect to the ground, the Standards give the classification of the following types of networks (Fig. 5.2): − isolated neutral networks; − solidly grounded neutral networks and impedance (resistor or reactance) grounded neutral networks; − resonant grounding neutral networks (arc suppression or Petersen coil).

326

Basic computation

N

N

N

XN

ZN

ZN=

Ground

Ground a.

b.

Ground c.

Fig. 5.2. Neutral grounding of electrical networks: a. isolated neutral network; b. solidly and impedance grounded neutral network; c. resonant grounded network.

Actually, the term “Neutral Grounding” designates a much general situation and it takes into consideration the neutral situation even in the case when there is no physical impedance between the ground and the neutral point. If the star-point is not available at transformer, an “artificial neutral point” can be created by means of a Bauch transformer, a special coil for neutral grounding or the primary winding of an auxiliary services transformer. The neutral grounding raised many discussions during the evolution and development of the electric networks because of the large number of factors to be considered when applying a solution, and the implications of that particular solution. As far as the network is concerned, the following factors should be considered [5.1, 5.2]: − fault current magnitude; − overvoltage magnitude; − technical-economic characteristics of the grounding device; − voltages induced in the neighbourhood of the faulty line; − constructive design of the line; − selective fault location; − automatic fault clearance; while in terms of the consumers supplied by that network the considerations are: − uninterrupted supply; − compatibility with the supply for the industrial processes of other consumers. The experience gained in operation and the tradition should also be added to the above-mentioned factors. In order to specify the grounding degree of the network in a certain point, the grounding coefficient of a three-phase network is used. For a certain network configuration, this coefficient, denoted by KG [5.16], represents the ratio between the network’s highest rms voltage measured between a healthy phase and the ground, in the considered area, during one (or more) phase-to-ground fault, and the rms phase-to-phase voltage, that could be measured in the same section without fault.

Neutral grounding of electric networks

327

Generally speaking, this coefficient shows the grounding conditions of a network seen from the considered location, regardless of the network’s operating voltage in that location. This coefficient can be determined in terms of the network and generator characteristics.

5.2. Basic electric phenomena in grounded neutral networks The first phenomenon associated to a phase-to-ground fault occurring in electric networks is the insulation breakdown (or flashover), a phenomenon not depending on the neutral grounding method. The phenomena following the insulation failure depend on the neutral grounding and influence the magnitude potential of the network neutral relative to ground, the value of the ground fault current, the voltages of the faulty phase and of the sound phases, the duration of the electric arc forming the ground fault, the restoring conditions to steady state, etc.

5.2.1. Network neutral potential relative to ground The phase potential of the network can be measured relative to the neutral of the network or to the ground. In the first case, the phase-to-neutral voltages (Va, Vb, Vc) are supplied by the power source: a synchronous generator or the secondary winding of a supply transformer (Fig. 5.3,a). These voltages are practically equal in magnitude and assure a symmetrical positive-sequence system (Fig. 5.3,b). The neutral potential relative to ground depends on the admittances-to-ground of the line: Y 0 = G0 + jωC0 . a a b

N

U ca

c ZN

C 0c

a.

G0c C0b

G0b C0a

Vc

G0a

U ab Vb

ZN c

Ground

Va N

U bc

b

b.

Fig. 5.3. Symmetric network with grounded neutral point operating in steady state: a. the emphasizing of line admittances-to-ground; b. the phasor diagram of the phase voltages relative to the neutral.

If the network is symmetrical and operates under steady state conditions, the three admittances-to-ground are equal, i.e. Y 0 a = Y 0b = Y 0c or G0 a = G0b = G0 c and

328

Basic computation

C0 a = C0b = C0c . Notice that, in the neutral grounding of electric networks theory and ground fault current calculation, the phase-to-phase capacitances are neglected. Consequently, the neutral potential relative to ground is: VN =

V a Y 0 a + V b Y 0b + V c Y 0c Y 0 (V a + V b + V c ) =0 = 1 1 3Y 0 + Y 0 a + Y 0b + Y 0c + ZN ZN

(5.1)

Therefore, the neutral potential of the networks operating under normal symmetrical conditions is zero (identical with the ground potential) regardless of the neutral grounding method (i.e. the value of the impedance ZN). The symmetry, as the one shown above, is seldom met in practice because, usually, there are some differences between the admittances of the line phases. The term “non-symmetry degree” has been introduced to describe this situation. It is defined as follows [5.9]:

u=

C0 a + a 2C0b + aC0c C0 a + C0b + C0c

(5.2)

where a = exp(j2π/3) is the complex rotation operator. For single-circuit MV overhead electric lines, the u value is about 1 ÷ 5%, while for the double-circuit lines this value is up to 20%. On the other hand, for underground cables, the non-symmetry degree value is less than 0.5% [5.9]. Considering that a ground fault occurs on a certain phase (Fig. 5.4), the admittance corresponding to that phase becomes zero and the neutral potential relative to ground depends on the grounding neutral impedance ZN rating. a b

N

ZN

α β

c C0c

G0c C0b

G0b C0a

G0a

Ground Fig. 5.4. Three-phase network with single-phase-to-ground fault

If this impedance has high rating or does not exist ( Z N = ∞ ), then the neutral potential takes the value of the phase voltage supplied by the power source, and if the impedance ZN is low or zero, the neutral potential value is also low or zero. In the first case, the potential of the sound phases increases to the phase-to-phase voltage value overstressing the network insulation; in the second case, a high shortcircuit current occurs overloading the network current paths.


329

In terms of the magnitude and character (inductive or capacitive) of the impedance ZN, the neutral point N (representing the neutral potential) can move to any position inside the triangle abc, and its potential of the neural point can take any value between 0 and the phase-to-neutral voltage V. The same situation could happen if we consider Z N = ∞ and the admittances-to-ground Y 0a , Y 0b , Y 0c could take different values. Therefore, it can be said that under steady state conditions as well as during a phase-to-ground fault caused by a partial or complete failure of the insulation at one or more points, the neutral potential is not identical to that of the ground.

5.2.2. Single-phase-to-ground fault current Let us consider that the network from Figure 5.4,a, with ground fault occurring on phase c, can be schematically drawn as in Figure 5.5, where α and β designate the terminals where the electric arc appears. Under normal operating conditions the circuit is considered opened (not-loaded) while during the ground fault, the arc impedance ZF (fault impedance) is superposed between terminals α and β. IF V =0 Z αβ

ZF

IF

IF 0 α

α V =0 Z αβ

ZF Vαβ

β

β

V =0 Z αβ

α ZF V αβ β

Fig. 5.5. Determination of the single-phase-to-ground fault current IF: the decomposition of the real network into an active network and a passive one.

If the superposition and ideal voltage source (Thévenin) theorems are applied, the real network can be decomposed into two fictive networks: an active one with voltages forced by the supply source, with a zero current through the fault impedance between terminals α−β, and a passive one with zero forced electromotive voltages but with a ground fault current flowing through the fault impedance. The following relation can be written for the last case: IF =

V αβ Z αβ + Z F

where: Vαβ is the open circuit voltage at terminals α and β, when the network operates under normal consitions; the current through the fault impedance is zero for the active network because an ideal voltage source of electromotive voltage -Vαβ has been introduced on the path of arc from phase to ground;

330

Basic computation

Zαβ − the impedance of the passivized network seen between terminals α and β. For its calculation, assume that the impedances of the transformer windings are much smaller than those corresponding to the capacitances to ground.

Unsymmetrical operation due to various faults on the transmission system, such as short-circuits, phase-to-ground faults, phase-to-phase faults, open conductors are studied by means of method of symmetrical components, introduced by Fortescue in 1918. Under normal operating conditions, the generator is designed to supply balanced three-phase voltages and therefore only the positive-sequence exists. The positive-sequence network consists of a positive-sequence impedance in series with an e.m.f. designated by V. The single-phase-to-ground fault is an unsymmetrical phase to ground operation that can be analyzed using the method of symmetrical components. In this respect, the real network can be decomposed into three sequence networks: positive, negative and zero (Fig. 5.6). +

+

Z a b c

Ia=0

V

ZF

I

V

~

Ib=0

Z

ZF

I

Ic

+

V Z0

V c = 0 ; Ia = 0 ; Ib = 0

I0

V0

3 ZN

a.

ZF

b.

Ic

Ib+ Ia+

Ic+

Ia

+

Ib

Ic

0

+

Ia Ib0 Ic0

c. Fig. 5.6. The main circuit used for the calculation of the ground fault current by means of the method of symmetrical components: a. fault representation; b. sequence networks connection; c. decomposition of capacitive currents system into sequence components.

Upon a ground fault occurrence, zero-sequence currents flow only if a return path exists, that is, if there is a connection between the neutral point and ground. If the neutral point of the network is grounded via resistor or reactor, an impedance


331

3Z N must be also inserted in the zero-sequence network, in series with the zerosequence impedance. The zero-sequence voltage, which has as reference the ground potential at the point of interest in the system, is applied at the zerosequence network terminals. In order to balance the sequence networks, the negative-sequence network has to be defined, which is, in fact, the passivized positive-sequence network. The equations between phase and sequence voltages are given below: V a = aV + a 2 V + V

+

−

0

+

−

0

V b = a 2 V + aV + V +

−

V c =V +V +V

therefore: +

( = ( aV

0

V = a 2 V a + aV b + V c V

−

a

+ a2V b + V c

) )

3 3

V = (V a + V b + V c ) 3 0

where the equations are referred to the reference phase c. A similar set of equations can be written for phase and sequence currents. + − 0 Knowing that the three sequence currents are equal, I = I = I , the fault current (on the phase c) can be inferred from: 3V + − 0 + (5.3) IF = I + I + I = 3I = + − 0 Z + Z + Z + 3Z N +

−

or, if the fault impedance Z F is taken into account and knowing that Z = Z , the expression (5.3) becomes: IF = where: V

Z

+

Z

−

Z

0

3V +

0

2 Z + Z + 3Z N + 3 Z F

(5.4)

is the equivalent electromotive force corresponding to the fault section, without fault. In other words, it is the phase-to-neutral voltage of the phase c; – equivalent impedance of the positive-sequence network, seen from the fault section; – equivalent impedance of the negative-sequence network, seen from the fault section; – equivalent impedance of the zero-sequence network, seen from the fault section.

332

Basic computation

For the sequence voltages, the following relations can be written: 0

0 0

+

+

V = −Z I

V =V − Z I −

−

V = −Z I

+

(5.5)

−

from where it results the phase-to-neutral voltages in the faulty section: 0 + ⎡ Z −Z + − 0 V a = aV + a 2 V + V = V ⎢ a − + 0 2 Z + Z + 3Z N + 3Z F ⎢⎣

⎤ ⎥ ⎥⎦

0 + ⎡ ⎤ Z −Z + − 0 V b = a 2 V + aV + V = V ⎢ a 2 − ⎥ + 0 2 Z + Z + 3Z N + 3Z F ⎥⎦ ⎣⎢ 3V Z F Vc = ZF IF = + 0 2 Z + Z + 3Z N + 3Z F

(5.6)

Obviously, one of the most important problems in calculating the fault current I F is the forming of the sequence networks but mostly, the zero-sequence network. Healthy feeder

Transformer HV

MV YH0

α

ZN

ZF β

0 ZTL

Faulty feeder

IF

YF0

ZTL0 3ZN

I0

α β

1/Y

0

3ZF b.

a. Fig. 5.7. The calculation circuit of the single-phase-to-ground fault current, in a MV network: a. three-phase circuit; b. zero-sequence equivalent single-phase circuit.

Therefore, for the network in Figure 5.7,a, the three-phase and single-phase equivalent circuits can be drawn. The zero-sequence impedance can be calculated, from Figure 5.7,b, using the expression:


0

Z =

(3 Z

N

+

0

0

0

0 ⎛ Z ⎞ 1 3 ⎜ Z N + TL ⎟ 0 ⎜ 3 ⎟⎠ 3 Y ⎝ = = 0 1 1 0 + 0 3Y 0 ⎛⎜ Z + Z TL ⎞⎟ + 1 3Y + 0 ⎜ N Y Z 3 ⎟⎠ ⎝ Z N + TL 3

0 Z TL

3 Z N + Z TL

333

)

0

where: Z TL = Z T + Z L is the zero-sequence impedance of the network; 0

Z T – zero-sequence impedance of the transformer; 0

Z L – zero-sequence impedance of the line; Y

0

– zero-sequence admittance of the line. It consists of two terms: 0

0

0

0

0

Y = Y H + Y F ; where Y H corresponds to the healthy line and Y F corresponds to the faulty line. Denoting by: 1 0 1 + + Z NT = Z N + Z TL , Y NT = , Z = 2 Z = 2 Z TL 3 Z NT the expression (5.4), of the fault current, becomes:

IF =

Z+

Y NT

3V V = 3 Z 1 + 3Z F + + ZF 0 0 3 + 3Y Y NT + 3Y

To the above expression, the equivalent circuit from Figure 5.8 can be attached. This circuit shows us that the fault current has two components: one due to the 0 admittance to ground Y of the network, and another one due to the neutral admittance Y NT . α Fig. 5.8. Equivalent circuit for the calculation of the singlephase-to-ground fault current.

IF

Z 3

~

YNT

3Y 0

ZF β

It should be mentioned that the currents flowing through the network elements are obtained by adding up the corresponding currents from the active and passive network. The currents in the active circuit are exactly the currents from the steady state because the current through the fault impedance is zero. Each solution for the neutral grounding will be analyzed in the following sections, considering the following issues:

334

Basic computation

− − − − − −

the current flow through the faulty network; the magnitude of the ground fault current; the displacement of the neutral point; the voltages of the faulty phase and of the healthy phases; the design of the system grounding; the connection to the network of the grounding device when the star-point does not exist; − the line influence on the neighbouring circuits during the fault; − the faulty line detection and fault clearance; − the modern trends in promoting the solution;

5.3. Isolated neutral networks Let us consider the isolated neutral network from Figure 5.9. Assume that the capacitances to ground are equal on each phase, C0 a = C0b = C0 c = C0 , while Il,a, Il,b and Il,c designate the load currents, that form a symmetrical system, situation similar for the charging capacitive currents flowing through the natural capacitances to ground, Ic,a, Ic,b, Ic,c. For easier understanding of the phenomena, the line conductance G0 is further not represented in the one-line diagrams. Therefore, the neutral potential VN of the network will be identical to that of the ground (Fig. 5.10, the diagram drawn with solid line). If a ground fault occurs on one of the phases, for instance the phase c (Fig. 5.9), the voltage and current values will change. It can be seen that the capacitance to ground of the faulty phase is shortcircuited by the fault and the resulting unbalance causes capacitive currents to flow into the path of fault via sound phases, then through phase capacitances to ground. a

Il, a

b N

Il, b

c

Il, c

α Vαβ=-Vc ZF β Ground

Ic,c=0

C0

Ic,b

IF Fig. 5.9. Isolated neutral network.

C0

Ic,a

C0


335

a

Vαβ a Va

Va

N

Vαβ Vb

N G

c

b

c

Vc Vαβ

Vαβ

Vb b

Fig. 5.10. Phasor diagram of the network’s voltages from Figure 5.9.

In order to determine the new values, the Thévenin’s theorem is applied. Thus, considering the passive network with an electromotive force Vαβ = –Vc included on the phase-to-ground path α-β, the potential of the phase c becomes equal to that of the ground, and the network neutral point changes its position to N' (Fig. 5.10), having practically the voltage Vαβ relative to the ground. In practice, it slightly differs from that value by the voltage drop due to the capacitive currents flowing through the transformer windings and the impedance to ground of phase c. Likewise, the voltages of the sound phases (a, b) change their positions with the same value with respect to ground, getting the values Va′ and Vb′. The magnitudes of these voltages are equal to 3 V , where V is the phase-to-neutral voltage under steady state conditions. Therefore, the value of the grounding coefficient is: KG =

Va ' V 3V ⋅ 100 = b ' ⋅ 100 = ⋅ 100 = 100% Un Un Un

The currents flowing through the passive network, due to the voltage Vαβ = –Vc, are closing through the capacitances to ground of the line phases, through the transformer windings and through the ground fault path (the path of these currents was drawn with dashed line in Fig. 5.9). These are sine currents and charge the transformer with a zero-sequence charge therefore resulting in non-symmetry between voltages and currents. Overlapping on these zero-sequence currents the capacitive and load currents from the normal operating conditions determines the currents flowing through the ground-fault network. If the load currents are neglected, the phasor diagram of the network currents can be drawn (Fig. 5.11,a).

336

Basic computation

a

IC

Va

Vc c

N Ic, c

-Ic, c

G Vb

Ic, b

30

Ic, a

N -Ic, c

Va

Ic, b

-Ic, c

Ic, b

Ic, a

Ic, b Ic, a

IC

Ic, a N

Ic, c

Vb

b

a.

b.

c.

Fig. 5.11. Capacitive currents position before and after ground fault on the phase c; phasor diagram: a. normal operating conditions; b. ground fault conditions; c. capacitive currents composition during ground fault conditions.

Figure 5.11,b shows the composition of the zero-sequence (ground fault) capacitive currents − I c ,c due to the voltage Vαβ with those from the normal operating conditions ( I c ,a and I c ,b ), obtaining thus the capacitive currents from '

'

the state of ground fault on phase c, I c ,a and I c ,b . The ground fault current I F is obtained by composing the last two currents. The magnitude of this current is determined as follows:

I F = I C = 2 I c' cos30° = 2 ⋅ ( 2 I c cos30° )

3 = 3I c 2

where Ic is the capacitive current of one phase under normal operating conditions. Its value is rendered below: I c = V ωC 0 where C 0 = C0 is the zero-sequence capacitance (relative to ground) of all lines connected to the transformer’s secondary of the substation. The same result can be obtained in two ways: − by composing the capacitive currents corresponding to the sound phases (Fig. 5.9,c); − by sticking to Thévenin’s theorem and neglecting the impedances of the line and transformer windings for the passive network where voltage Vαβ exists. Having the three capacitances C0 in parallel connection, the equivalent capacitance is 3C0 and the additional capacitive ground fault current is I F = Vαβ 3ωC 0 . The last method is the most handy because it allows selection of three components within the ground fault current, as follows: one component due to the


337

voltage existing at the fault location (Vαβ), another one due to capacitance to ground of each phase and the last one due to the load currents. Approximate relations can be used for informative purposes [5.9]:

I FOEL =

UnL U L [A], I FUEL = n [A] 300K 500 2K10

where Un is the nominal voltage of the line, in kV, and L is the length of the network lines, in km. The per kilometre ground fault current values for various categories of lines are given in tables and charts. For underground electric lines the higher values of the ground fault current correspond to the polychloride vinyl insulation and the smaller ones correspond to the polyethylene insulation. The analysis of the above expressions reveals that the ground fault current ' values are relatively low. The currents I c are in the same situation, remaining much smaller than the load currents of the network. The change in phase-to-neutral voltage will therefore be insignificant and the network will continue to operate, supplying the consumers in satisfactory conditions, even with one faulty phase. Although the phase-to-neutral voltages remain appreciatively the same as in the case of the network without fault, the potential of the sound phases with respect to the ground changes. Therefore, the phase-to-ground voltage of the faulty phase is zero, while the phase-to-ground voltage of the healthy phases increase by 3 becoming equal to the phase-to-phase voltage. Upon ground fault occurrence, an electric phase-to-ground arc forms, through which the resulting capacitive current flows. The electric arc extinguishes at each zero crossing of the current. Depending on the intensity of the current, the arc can re-strike after the current has passed through zero and when the current reappears, the arc may extinguish and re-strike successively, thus creating an intermittent arc (arcing fault). This can result in dangerous overvoltages on the sound phases that can be 3 ÷ 4 times higher than the phase-to-neutral voltage. If the intermittent electric arc and the associated overvoltages persist for a high number of periods, the insulation can breakdown causing the phase-to-ground fault to turn into a double-phase- or three-phase-to-ground fault. For this reason, protective measures must be taken to avoid the intermittent arc. The researches performed in Romania showed that the intermittent arc emerges when the current intensity from the fault location is higher than 5 ÷ 10 A. For values smaller than 5 A, the arc is extinguished at the first zero crossing and the network resumes to normal operation [5.19]. Based on these remarks it can be said that the overhead lines with nominal voltages up to 35 kV are able to operate with isolated neutral. The 6 ÷ 10 kV underground cables can operate with isolated neutral if the current does not exceed 10 A at the fault location. To limit the ground fault current in MV networks with isolated neutral, the lines are galvanically separated by sectionalising the busbars from the supplying

338

Basic computation

substation, a certain number of lines, respectively a certain line length, being assigned to each busbar. This assures an admissible level for the capacitive current. From the above-mentioned, it results that the isolated neutral networks have the great advantage that the immediate disconnection of the line is not necessary in case of a single-phase-to-ground fault, assuring thus the continuity in supplying of the symmetrical consumers. Moreover, during failure (phase-to-ground fault) these lines have low influence on the telecommunication lines located in the neighbourhood. The disadvantage of these networks is that the increase by 3 of the voltages of the sound phases during a ground fault can cause the breakdown of the insulation on the sound phases. Also, if the capacitive current exceeds the admissible limit, the electric arc space is no longer deionised during the zero crossing of the current and can persist for a longer time, thus causing dangerous overvoltages. Application Determine the maximum length of the 10 kV overhead lines that can be supplied from a distribution substation, so that to have the network neutral isolated from the ground, given that c0 = 5 ⋅10−9 F/km. The ground fault current is: I F < 5 A ; I F = 3 I c = 3Vn b0 l = 3 U n ωc0l ; It results: l
> X T 1 and Isc tends toward V / X T 1 . Otherwise, if the transformer T2 has a relatively high rated power, then X T 2 < X T 1 and I sc = 3V X T 1 = 3 I sc3 P , I sc3 P being the value of the three-phase-to-ground shortcircuit current. This situation is not appropriate and therefore, in order to reduce the value of the single-phase-to-ground short-circuit current the star-points of some transformers are not grounded. The single-phase-to-ground short-circuit current charge the network asymmetrically causing a significant voltage decrease. If the thermal and dynamic stresses of the current paths are added to these, a relay protection is obviously necessary to disconnect the faulty outgoing line. An advantage of the solidly grounded neutral is that, during the fault period, the parts of the transformer’s windings located in the vicinity of the neutral will have a potential of value almost equal to that of the ground. For this reason, the insulation level of these windings is possibly lower than that of the windings from the vicinity of the higher voltage terminals of transformer. The main disadvantage of the solidly grounded neutral is the generation of disturbances on the telecommunication lines and radio transmissions during phaseto-ground faults. Furthermore, due to erroneous operation, protection will trigger the tripping of the line so that the consumer will be disconnected.


341

From the phase-to-neutral voltages point of view, during the occurrence of a short-circuit the symmetry is assured only if the ground electrode resistance is zero. In practice, the symmetry cannot be assured because of the resistance of the mesh ground electrode, although is very low, it is passed by a high intensity current that causes a displacement of the neutral point, thus “altering” the phase-to-neutral voltages symmetry; the voltages of the sound phases increase slightly.

5.4.2. Resistor grounded neutral networks In order to limit the phase-to-ground short-circuit current, the network neutral can be connected to ground via a limiting resistor. This method of grounding has the advantages of both the ungrounded and effectively grounded system, while eliminating most of their disadvantages. For example, the potentially dangerous system overvoltages caused by arcing-type ground faults are suppressed by dissipating the energy in the resistor. Safety to personnel and system stability are significantly improved. The mitigation of the ground fault’s damaging effects and resulting hazards to personnel are even more pronounced when compared to solid grounding. A useful rule of thumb: the energy released and the damage done by the fault are approximately proportional to the square of the fault current multiplied by the fault duration. If the fault current is reduced form 10000 A with solid grounding to 100 A with resistive grounding, the magnitude of the fault is reduced by a factor of 10000 [5.17]. The resistance value of the limiting resistor is determined by imposing a certain value for the short-circuit current in order to have acceptable thermal and electrodynamic stresses. + Expressing the sequence impedances in complex form, Z = R + + jX + and 0

Z = R 0 + jX 0 , the single-phase-to-ground fault current from (5.4) can be written under the form: I sc =

3V 2

(

⎡3 ( RN + RF ) + 2 R + + R 0 ⎤ + 2 X + + X 0 ⎣ ⎦

)

2

and 2 ⎡ 1 ⎢ ⎛ 3V ⎞ + 0 RN = ⎜ ⎟ − 2X + X 3 ⎢ ⎝ I sc ⎠ ⎣⎢

(

) − ( 2R 2

+

⎤ + R 0 ⎥ − RF ⎥ ⎦⎥

)

In order to emphasize some qualitative issues of the calculation of singlephase-to-ground fault currents, the capacitance to ground and the resistance of the line, as well as the resistance of the supply transformer can be neglected with respect to the limiting resistor. Taking into account that 3RN 3RF + 2 R + + R 0

342

Basic computation

and X + ≅ X 0 = X , the expression of the short-circuit current written above becomes: V I sc ≅ (5.9) 2 RN + X 2 where X is the inductive reactance of the whole circuit passed by the short-circuit current and RN is the resistor’s resistance. Under these conditions, the neutral potential relative to ground is: VN = I sc RN

(5.10')

or 2

VN = I sc

⎛V ⎞ 2 ⎜ ⎟ −X I ⎝ sc ⎠

(5.10")

If RN = 0 , the single-phase-to-ground short-circuit current corresponding to the solidly grounded neutral I sc is maximum, that is: I sc , max =

V X

(5.11)

and VN from the formulae (5.10') and (5.10") is divided to the phase-to-neutral voltage, the following relation can be obtained: ⎛ I ⎞ VN = 1 − ⎜ sc ⎟ ⎜ I sc , max ⎟ V ⎝ ⎠

2

or 2

2 ⎛ VN ⎞ ⎛⎜ I sc ⎞⎟ =1 ⎜ ⎟ +⎜ ⎝ V ⎠ ⎝ I sc , max ⎟⎠

(5.12)

The expression (5.12), represented in the quarter circle diagram from Figure 5.15, allows us to determine the neutral potential versus the decrease of shortcircuit current. When reduced to half of its maximum value, the neutral potential becomes: VN = 1 − ( 0.5 ) ⋅ V = 0.867 ⋅V 2

Besides the reduction of the neutral potential, the introduction of a resistance in the circuit makes it easier to interrupt short-circuit current and to return to normal operating conditions, because of the significant attenuation of transient and resonance phenomena.


343

VN / V

1 0.75

Fig. 5.15. The neutral potential variation in terms of the single-phase-to-ground short-circuit current in the case of the neutral grounding via resistor.

0.5 0.25 0

0

0.25

0.5

0.75

1

Isc / Isc, max

Three issues should be observed related to the limiting resistor: the determination of the resistance value, the constructive design of resistor and its connection to the network when the network neutral point is not available. a) The resistor sizing is set up in terms of the value of the single-phase-toground short-circuit current desirable to appear at a fault occurrence in the network, Isc. If the voltage VN from (5.10') is replaced in (5.12) the expression of the resistance can be obtained: RN = V

1 1 − 2 2 I sc I sc , max

(5.13)

More restrictions can be added to this requirement, as follows: − minimization of the overvoltage factor; − assurance of protection sensibility for zero-sequence overcurrent for the lines and transformers; − assurance of sensitivity for the differential protection; − assurance of protection sensitivity for overcurrents when emerges in the limiting resistor; − limitation of the step and touch voltage. In Romania, the resistance value of the limiting resistor is established according to the data from Table 5.1 [5.19] (the resistances are given in Ω). Table 5.1 Limiting resistors for MV networks Nominal voltage of the network [kV] Maximum singlephase-to-ground fault current [A] 1000 600 300

6

10

15

20

3.4 5.8 11.6

5.8 9.7 19.3

8.7 14.4 28.8

11.6 19.3 38.5

344

Basic computation

Application Calculate the resistance value of the resistor used to ground the star-point of a transformer that supply a 20 kV underground cable so that the single-phase-to-ground shortcircuit current to be reduced to 400 A. Using the expression (5.13) knowing the Isc, max = 600 A and V = 20000 obtain: RN =

20000 3

1 400

2

−

1 600 2

3 V, we

= 21.517 Ω

b) Constructively, the resistor may be built in the shape of rolled angle or chromium – nickel tapes and wires. The grids are series connected, forming the socalled resistor banks. The banks are mounted on angle iron metallic frames, and the support insulators provide the necessary insulation. These installations contain measuring and protection systems, and their metallic parts are grounded. A much cheaper solution is the manufacturing of the grounding resistor in a special shape ground electrode separated from the mesh ground electrode of the substation in which it is embedded. It is manufactured in a metallic board shape buried at a suitable chosen depth. The board is embedded in concrete, which is essential to assure the thermal stability of the mesh ground electrode, to decrease the leakage resistance of the metallic board and to assure a resistance of value as constant as possible. c) For resistor connecting, when there is no star-point available for connection to ground, the artificial neutral coil (ANC) or the primary of an auxiliary service transformer with zigzag connection (TZC) are used (Fig. 5.16). These elements are connected to the medium voltage busbars; the ANC coils can be also connected directly to the supply transformer terminals, operating together. The most useful ANCs have a zigzag connection. In this case, the value of the zerosequence impedance is very low, so that when a single-phase-to-ground fault occurs, the entire phase-to-neutral voltage is distributed on the resistance connected to the neutral. The core coil is lighter; the windings and the core coil are designed to withstand only the phase-to-neutral voltages.

ANC

TZC

RN

RN

a.

b.


345

Fig. 5.16. Connecting schemes of the artificial neutral: a. artificial neutral coil; b. transformer with zigzag connection.

The limiting resistor used for neutral grounding via resistor is one of the solutions implemented in the last period (20 ÷ 25 years) in Romania, and it seems to be useful enough especially in underground MV networks, since it turns the ground fault occurring in isolated or grounded neutral networks via a relatively high impedance into a controlled short-circuit, limiting the transient phenomenon and the accompanying overvoltages, preventing such that the evolution of the fault into double or extended faults. The neutral grounding via resistor has been improved by employment of a special measuring and protection transformer SMPT [5.10] or a shunt circuit breaker SCB. The first solution (Fig 5.17) is used for shorter underground cables where there is no star-point available. Usually, in Europe, the SMPT has two secondary windings, one of 100 3 volts and other one of 500 3 volts or greater voltage, which are series connected forming an open triangle where the resistor R is connected. Consequently, during a single-phase-to-ground fault occurring on an outgoing feeder, the circuit forces an additional active current, determined by the value of R, which triggers the opening of the circuit breaker CB, and therefore the disconnection of the faulty feeder. After the fault clearance, the circuit breaker CB will be re-closed, and the network resumes the normal operation. The second solution, the one with the shunt circuit breaker, repeats the phaseto-ground fault that occurred on an outgoing feeder, by connecting a single pole of the circuit breaker SCB after a short delay, which connects the respective phase to the ground electrode of the supply substation (Fig. 5.18) [5.14].

346

Basic computation Fig. 5.17. Usage of a special measuring and protection transformer for neutral grounding via resistor.

SMPT

Measurement

R

CB

breaker coil

This method has been used in France since 1975, by EdF, with very good results. The current at the fault location is practically cancelled, the fault disappearing without disconnecting the faulty feeder, and the network returns to normal operation after the opening of the shunt circuit breaker. The shunt circuit breaker should meet certain requirements: − upon the occurrence of a phase-to-ground fault it connects only the corresponding pole of the circuit breaker; − it provides phase selection logic and shunt locking system for multiple faults; − it does not allow repeated operation (shunting) if the fault did not extinguish after the first opening of the circuit breaker.

RN

SCB

Fig. 5.18. Usage of a shunt circuit breaker in networks grounded via resistor.


347

The advantages of the shunt circuit breaker are: − transient faults do not trigger the disconnection of the outgoing feeders; − the phase-to-phase voltages remain unchanged during the circuit breaker operation; − the electric arc is not eliminated by disconnection but by shunting; − shunting duration can be increased up to the thermal stability limit (about 5 seconds); − it is a simple and economical solution. The operation of shunt circuit breaker was also investigated for islanded MV networks within the context of dispersed generation [5.25].

5.4.3. Arc-suppression coil grounded networks (resonant grounding) 5.4.3.1. General considerations The neutral point grounding via coil aims to compensate the capacitive fault currents. The particular case of this method is the grounding of the neutral point via arc-suppression (Petersen) coil of an appropriate chosen reactance tuned at resonance. This solution is applied for MV overhead lines or short underground cables. Thévenin’s theorem can be applied for the determination of the additional currents that emerge during phase-to-ground fault (e.g. phase c, Fig. 5.19), considering only the passive network and the voltage Vαβ applied at the ground fault location. a b

N

c α ZN Ground

IL

Vαβ β

Ic,c=0 C

0

Ic,b

Ic,a

0

C0

C

IC

Fig. 5.19. Grounded neutral network via arc-suppression coil.

Two currents emerge as consequence of the voltage V αβ :

− a capacitive charging current I C , corresponding to the capacitances to ground of the line. This current closes through these capacitances to ground and through the transformer windings; − an inductive current IL, which represents the sum of the currents of the parallel arc-suppression coils, forced through the faulty phase.

348

Basic computation

The two currents, one of them leading the voltage and the other lagging behind the voltage Vαβ by almost 90°, overlap each other at the ground fault location, resulting thus the fault of residual current Ir (Fig. 5.20): I F = Ir = IC + I L

(5.14)

or neglecting the active components of the currents, we obtain: I r ≅ jV αβ 3ωC 0 +

V αβ

(5.15)

j ωL

The expression (5.15) is written taking into consideration only the reactances of the two circuits. Vαβ Fig. 5.20. The phasor diagram of the currents from the circuits of the passive network from Figure 5.19.

Ir

IL

IC

The residual current has the following features: − its value is much smaller than that of the capacitive fault current, therefore all its effects will be very much diminished; − its phase shift with respect to the voltage is small, favouring the instantaneous extinguishing of the arc at the ground fault location; due to the improvement of the extinguishing conditions, the residual current can have much higher values (20 ÷ 30 A), avoiding the danger of arc restriking. To gain a maximum advantage, that is a low residual current, the coil will be chosen so that: V αβ jωL

+ jV αβ 3ωC 0 = 0

resulting

ωL = X N =

1 = Xc 3ωC 0

(5.16)

The expression (5.16) indicates the resonance condition for the equivalent parallel circuit formed by the two branches: the reactance of the arc-suppression coil and the capacitances to ground, at the frequency f = ω 2π . It should be observed that due to the various elements (resistances and reactances) of the circuit, a total compensation of the capacitive current cannot be achieved and therefore I r ≠ 0 . This can also be seen from the analysis of Table 5.2


349

[5.9] where the ratio between the active (real) and capacitive (imaginary) components of the ground fault current is shown. Table 5.2 The ratio between the active and capacitive components of the ground fault current Insulation state

Overhead electric lines: wet or polluted insulation Underground cables: aged insulation 10% 10% 10%

Normal

Line type 15 ÷ 35 kV 60 ÷ 110 kV Underground cables: aged insulation

5% 3% 2 ÷ 4%

Overhead electric lines

The equivalent circuit of the network from Figure 5.19 is represented in Figure 5.21,a, where LN and RN denote the inductance and the resistance of the coil, while G0 and C0 are the zero-sequence conductance and capacitance to ground of the line. Figure 5.21,b shows the phasor diagram of the voltages and currents of the line. IL

a

IC Ir

LN

RN

Va Va

0

3C

V

0

Vb

N

Vc Ic,b

3G

Vb I L=V YN

c

0

IC=j3ωC V

b

Ic,c

a.

V IL= jωL

b.

Fig. 5.21. Network neutral grounded via arc-suppression coil: a. the equivalent one-line diagram; b. the phasor diagram for voltages and currents.

In reality, because the arc suppression coil is not 100% inductive, the current flowing through it I 'L , is shifted with respect to the voltage V c by an angle smaller than 90º. In the case of resonance tuning of the coil, the residual current consists only of the active (real) components of the two currents: I r = Re{I C } + Re{I L } , where Re{I C } = 3G 0V and Re{I L } = V R

(5.17)

The expressions (5.16) and (5.17), corresponding to the resonance tuning, are valid only for the fundamental harmonic. In practice, the ground fault current can have higher harmonic components, mainly odd series (the order of 3, 5, 7, etc.) that inhibit the extinguishing of the electric arc even when the resonance ideal tuning is achieved. In this case, the residual current had the expression: I r2 = (Re{I r }) + I 32 + I 52 + I 72 + ... 2

(5.18)

350

Basic computation

where, Re{I r } denotes the active component of the residual current on the fundamental harmonic. By the same token, it is worth mentioning that in many practical situations, significant values of the residual current were due not only to the active component but also to the higher harmonics.

5.4.3.2. Arc-suppression coil issues The issues concerning the arc-suppression coil are the following: a) the establishing of the coil reactance value; b) the construction of the arcsuppression coil; c) the coil connection to the network when the star-point does not exist; d) the detection of the faulty outgoing feeder. The establishing of the coil reactance value The value of the coil reactance XN is determined by taking into consideration the failure conditions of the network (phase-to-ground fault) and the normal operating ones (steady state). From the failure operating state point of view it is desirable to have a value as small as possible for the residual current, theoretically zero. The necessary condition for this to happen is I L = I C , that is, the parallel resonance condition [5.11]. Nevertheless, under normal operating conditions, because of the inequality of the capacitances (especially for overhead electrical lines), the uneven dirty deposition on insulators, the asymmetric installing of the measuring transformers, etc., a slight displacement of the neutral potential relative to ground takes place, generating a voltage source in the series circuit: the coil, the secondary winding of the transformer and the capacitances to ground (zerosequence) of the line. This phenomenon causes the emergence of very high currents in the network when the tuned arc-suppression coil is connected. This is due to the series resonance phenomenon (knowing that the parallel resonance condition coincides with that of the series resonance). For this reason, the value of XN is set so that I L ≠ I C ( I C = 3I c ). In fact, the network can operate with either under-compensation, i.e. I L < 3I c , or over-compensation I L > 3I c . An overcompensation by 10÷15% is preferred in Romania because adopting undercompensation can result in reaching the resonance condition after the disconnection of a line. It is not the case for the underground cables where the non-symmetry degree is too low. In this situation the network can operate very close to resonance. The coil tuning to the resonance or to a certain value is established taking into account the voltage emerging at the coil terminals under normal operating conditions due to non-symmetry of its capacitances. Therefore, if we consider for one phase (e.g. phase a) that Ca0 = C 0 + ΔC , and for the other phases

Cb0 = Cc0 = C 0 , and taking also into account the line conductance G 0 , the expression of the voltage applied at the coil terminals is [5.20]:


V N =V

351

(

jω C 0 + ΔC + a 2C 0 + aC 0

(

0

)

)

(5.19)

0

jω 3C + ΔC + 3G + 1/ jωL

or expressed in absolute value and referred to the phase-to-neutral voltage: VN n = V (3d )2 + (3v + n )2

ΔC C0 d = G0

where: n =

∑

(5.20)

is the network non-symmetry degree;

∑ ωC

v =1− k =1−

IL IC

0

– the damping factor of the compensated network; it takes values in the interval 1÷2 %; – the off tuning degree of the coil.

The relation (5.20) can be graphically plotted as in Figure 5.22. VN /V n1 > n2 > n3 n1

Fig. 5.22. Variation of the neutral point displacement voltage in terms of parameters n and v.

n2 n3 -5 -4 -3 -2 -1

0

1

2

3

4

5 V [%]

One can notice that the voltage on the coil is practically at its peak during resonance. Accordingly, the value of the current flowing through the coil (and thus the total capacitive current) can be established in practice for the resonance situation. Some errors appear in reality due to both the measuring system and the maximum’s shifting relative to the tuning situation, and also due to the very low value of the voltage at the coil terminals, especially for underground cables. One of the solutions is to artificially increase the non-symmetry between phase capacitance by introducing an additional coil during the measurement. Recently, automatic adjustments assured the increase of the arc-suppression coil efficiency. The following automation systems can be mentioned: – a system using the resonance curve of the zero-sequence circuit: coil – zero-sequence capacitances; – a system that monitors the magnitude and the phase angle of the zerosequence voltage; – the GENEPI system.

352

Basic computation

Application of the GENEPI system. The GENEPI system aims to identify the impedance value of the zero-sequence circuit, without handling the coil [5.1, 5.2, 5.4, 5.21]. In this respect, a current is injected into the neutral grounding circuit for few seconds (Fig. 5.23). By measuring the injected current and the zero-sequence voltage, before and during the injection, the system is able to determine the parameters of the zero-sequence circuit and thus the total capacitive current. Accordingly, the coil is switched on the tuned position. GENEPI has the advantage of a fast tuning, its action being also compatible with a fault occurrence during the measurements.

N

0 VN (V ) L

0

C

0

C

0

C

Ii

Fig. 5.23. The tuning system of the coil – GENEPI.

Construction of the arc-suppression coil The arc-suppression coil has a similar design to a single-phase transformer having the primary winding placed on an iron core introduced in a metallic tank cooled by transformer oil. In order to linearize the voltage-current characteristic of the coil, the core is provided with air gap sections and the coil is sized for the magnetization curve bend portion. A current density much greater than the economic one is adopted for the winding because the coil operates a reduced number of hours and intermittently. The core position is variable in order to modify the reactance value of the coil, and sometimes the winding is tapped. An additional winding used for relay protection is provided on the coil’s core. It will indicate the presence of a zero-sequence component of the current. Connection to the network of the arc-suppression coil Special arrangements with ANC and TZC should be used when a star-point does not exist at transformer and also when a resistor is employed. The neutral grounding via arc-suppression coil is not efficient in large networks, especially for underground cables with high active power losses (6 or 10 kV networks with polyvinyl chloride insulation) or during ground faults with high fault impedance. In the recent years, the improvement of the performances of neutral grounded MV networks via arc-suppression coil has been experienced; some of them being designed for accurate selection of the faulty outgoing feeder while others for efficiently compensation of the ground fault current. The coil – resistor mixed grounding is another often used solution.


353

The detection of the faulty outgoing feeders As regards the detection of the faulty outgoing feeders, the systems promoted by Electricité de France, namely WHAT and DESIR, should be mentioned [5.13]. The WHAT and DESIR approach may also be used, in a restricted way and with less accuracy, in networks grounded via limiting impedance [5.2, 5.11]. The WHAT system consists in zero-sequence protective relays, while the DESIR system is used for resistive fault detection. Application of the WHAT system. One of the major advantages of neutral compensation is to achieve the self-extinguish of the arc. Unfortunately this makes certain faults more difficult to detect when re-striking faults occur. The current in the faulty feeder is not sinusoidal and the fundamental frequency component is sometimes rather small. Therefore, the voltage waveform presents amplitude variations (Fig. 5.24). The WHAT system is based on analysis of common mode energy transfer between the fault and the remainder of the network (Fig. 5.25) [5.1, 5.13]. Extinguished fault voltage Neutral point displacement voltage Present fault voltage (RF IF)

20000 15000 Voltages [V]

10000 5000 0 -5000

0

0.05

0.1

0.15

0.2

0.25 Time [s]

-10000 -15000 -20000

Fig. 5.24. Voltages during a re-striking ground fault. Reprinted with permission from Griffel, D., Leitoff, V., Harmand, Y., Bergeal, J. – A new deal for the safety and the quality on MV networks, IEEE Transactions on Power Delivery, Vol. 12, October 1997 © IEEE 1997. Energy in the last 20 ms [J]

20000 Healthy feeder Faulty feeder

10000 0 0

0.05

0.1

0.15

0.2

0.25 time [s]

-10000 -20000

Fig. 5.25. Energy transfer during a restriking ground fault. Reprinted with permission from Griffel, D., Leitoff, V., Harmand, Y., Bergeal, J. – A new deal for the safety and the quality on MV networks, IEEE Transactions on Power Delivery, Vol. 12, October 1997 © IEEE 1997.

354

Basic computation

In the fault absence the zero-sequence circuit formed by the capacitances to ground of the electric line, the coil inductance and resistance is unloaded. When the fault occurs, the energy is transferred from the fault to the zero-sequence circuit, which is charged with electric and magnetic energy through the fault resistance. The energy transfer direction on the healthy feeders is positive, and on the faulty feeder it is negative (Fig. 5.26,a). At the fault extinction, this energy charged in the line capacitances and coil inductance is released over the resistance of the grounding coil. Therefore, the energy transfer on the healthy feeders changes the direction (Fig. 5.26,b). The reactive power exchanged between the neutral point coil and the zero-sequence circuit capacitance adds some oscillations. E>0 HV/MV E>0 E 10 if resonance in the high voltage supply is possible in the studied frequency range; - X T X S > 4 if resonance in the high voltage supply is unlikely in the studied frequency range; • The total capacitance connected to the secondary system is low so that the resonance frequency is at least 2.5 times the highest studied harmonic frequency. The resonance frequency is calculated as f r = 1 / 2π LC , where L is the inductance per phase corresponding to Z1 if the capacitance is omitted, and C is the total capacitance per phase, with both power factor capacitors and cable capacitances taken into consideration. b) Zh with single resonance. If the total capacitance is higher than what is stated above, but all capacitive components can be regarded as connected to the same electrical point, the resulting value of Zh can be calculated as L in parallel with C, L and C being defined above. Close to resonance point, however, this method gives a far too high value of the resulting impedance. In order to calculate the correct value, the resistive component of the network impedances must be taken into consideration. (iii) Computer programs. The calculation of the harmonic current distribution and the harmonic voltages in the network requires the definition of the impedances that describes the behaviour at the harmonic frequencies of each network component. Several computer programs have been developed for the purpose of those calculation possibilities, such as [6.1]: – the harmonic impedance seen from a given node of the network; – a prediction of current and voltage harmonics propagation origin from given non-linear loads; – a prediction of network changes influence on the existing harmonics; – a prediction of ripple control signals propagation. In practice, the assessment method to be chosen depends on several factors, mainly the kind of power system, the kind of disturbing load and the available method.

6.5.4.3. Harmonic sources representation Harmonic sources can be divided schematically into two categories depending on their origins: they can be intrinsic to systems or due to the nature of the connected loads.

Electrical power quality

427

Although, system lines do not introduce harmonics and often act as filters reducing distortions, other components contribute intrinsically to the deformation of the voltage wave. The generators, which, despite the optimised construction, do not provide perfectly sinusoidal voltage, and mainly transformers, which act as harmonic sources during their operation in saturated conditions, can be given as example. Load with non-linear current-voltage characteristics connected to the system and fed by a practically pure voltage absorb non-sinusoidal currents. These currents cross the Thevenin impedance of the system and generate voltage. These voltages are more deformed and more intense and higher than the currents. Among all these loads, a distinction can be drawn between two broad categories [6.34]: – loads such as arc furnaces; – loads supplied with power from devices including semiconductors such as power converters and electrical domestic appliances. Arc furnaces Arc furnaces are the most difficult to study as their power can reach extremely high values. It is nevertheless possible to determine experimentally empirical models of harmonic injections produced by these loads. The arc furnace is modelled according to the equivalent circuit diagram in Figure 6.36.

R Fig. 6.36. Equivalent circuit of an arc furnace.

Ih X

The source of harmonic current Ih is defined by [6.34]: – even order h:

Ih =

Sn 3U n

( 0.15 + 3.5exp ( −0.4 ( h − 2) )) 100

– odd order h:

Ih =

S n (0.15 + 7.5 exp(− 0.45(h − 3))) 100 3U n

where Sn is the apparent power of the furnace and Un is the rated phase-to-phase voltage.

428

Basic computation

Load supplied by semiconductor-based devices The main reason for the success of semiconductors lies in the non-linearity of their current-voltage characteristics. This special feature allows them to perform basic functions such as rectifying or even dynamic power and velocity control. These harmonic current producing devices are classified into two categories as a function of the power of the loads they feed: power converters and electrodomestic loads [6.34]. • Power converters This category comprises all equipment which perform the industrial systemload interface. Their power levels are high and their applications manifold: electric traction, electrolysis, induction rolling mill, etc. All these applications require electronic switches (diodes, thyristors) as shown in the configuration in Figure 6.37. Id

i(t)

Load

Fig. 6.37. Supply of a DC load via a Gräetz three-phase rectifier bridge.

Figure 6.38 presents the shape (a) and spectrum (b) of the current absorbed by a conversion bridge. i Id

Ih 100 I1 80

[%] 60

0 α

π

2π

a.

t

law 1/h (for α = 30 )

40 20 0

1

5

7

11

13

h

b.

Fig. 6.38. Shape over time (a.) and spectrum of the current absorbed (b.) of a six-pulse rectifier.

For example, let us take a static converter of apparent power Sn, with rated phase-to-phase voltage Un and pulse order p (6 or 12). Such a converter can be modelled by a source of harmonic currents Ih, such that: – for instant switching of thyristors:


Ih =

429

Sn , with h = pk ± 1 , k = 1, 2, ..., n 3U n h

– for non-instant switching of thyristors:

Ih =

Sn 1.2

5⎞ ⎛ 3U n ⎜ h − ⎟ h⎠ ⎝

, with h = pk ± 1 , k = 1, 2, ..., n

• Electro-domestic loads All electrical household devices (television sets, video recorders, etc.), connected in millions of units to the low voltage distribution system, contribute to the greatest extent to harmonic pollution of the system. The first supply step of these appliances is formed of a diode bridge followed by capacitive filtering; other appliances are motorized via an electronic controller (in general a triac switch) or use discrete power regulation from a single diode placed in series. They all generate substantial harmonic currents and contribute to a large extent to the voltage wave distortion. For example, Figure 6.39 presents the spectrum of the current absorbed by a computer monitor.

Fig. 6.39. Spectrum of the current absorbed by a computer monitor.

These loads produce high harmonic currents. In addition, despite the advantage of these appliances of being low power units, the drawback is their substantial number distributed throughout the system. Due to their extremely variable characteristics, they appear and disappear from the system at random. The great variation of these loads and lack of information on their numbers lead to their representation not with current sources of given amplitude, but with harmonic current sources which are statistically defined by density laws.

6.5.4.4. Techniques for harmonic analysis The problem of harmonic analysis can be mathematically evaluated as solution of a network equations set at fundamental and harmonic frequencies. The

430

Basic computation

network equations can be formulated in an admittance matrix form or in a power flow equation form. The model used can be simple or more complex as the data are more or less available. An important consideration in harmonic analysis is to use a method to commensurate the input data accuracy [6.21]. • “Frequency scan” is the simplest and the most commonly used technique for harmonic analysis. The input data requirements are minimized. It calculates the frequency response of a network seen at a particular bus or node. Typically, a one per unit sinusoidal current (or voltage) is injected into the bus of interest and the voltage (or current) response is calculated. This calculation is repeated using discrete frequency steps throughout the range of interest. Mathematically, the following network equation, at frequency h f 1, have to be solved:

[Y h ] [V h ] = [I h ]

(6.64)

where [Ih] is the known current vector (for current injection scan) and [Vh] is the nodal voltage vector to be solved. The current has a magnitude determined from typical harmonic spectrum and rated load current of the harmonic-producing equipment under study:

Ih = Irated Ih−spectrum I1−spectrum

(6.65)

where h is the harmonic order and the subscript spectrum indicates the typical harmonic spectrum of the element. Equation (6.65) is then solved only for the harmonic frequency. The load flow computer programs, which model the harmonic-producing devices as constant power loads, calculate the fundamental frequency current injected from the load toward the system. Assuming that the current has a phase angle of θi1, the phase angle of the harmonic current θih corresponding to the nonlinear element can be determined by:

(

θih = θih − spectrum + h θi1 − θi1h − spectrum

)

(6.66)

where θih-spectrum is the typical phase angle of the harmonic source current spectrum. This approach is very efficient for analysing the power system with power electronic devices. The fundamental frequency load flow solution is also beneficial for providing more accurate information such as base voltages that can be used for distortion index calculations. • “Harmonic iteration method” is another used method [6.35]. In this method, a harmonic-producing device is modelled as a supply voltage-dependent current source: (6.67) I h = F (V1 ,V2 ,...,VH , c) ; h = 1, ..., H where (V1, V2, ...., VH) are the phase harmonics of the supply voltage and c is a set of control variables such as converter firing angle or output power. This equation is first solved using an estimated supply voltage. The results are used like the current sources in equation (6.64), from which nodal harmonic voltages are then obtained.


431

The voltages are used to calculate more accurate harmonic current sources from equation (6.67). This iterative process is repeated until convergence is achieved. • The Newton based method takes into account the voltage-dependent nature of non-linear devices to solve simultaneously the systems of equations (6.64) and (6.67). This method generally requires that the device models to be available in a closed form where its derivatives can be efficiently computed. An important step of this method is the formulation of the system equation. In [6.36], the equation (6.64) is formulated as a power flow equation and the control variables (firing angles) are solved based on the converter specifications. Furthermore, the phase-shifting effects of transformers on harmonics can be easily represented using three-phase modelling. • Besides the frequency-domain methods described above, others techniques have also been developed for harmonic analysis in the time-domain [6.37]. The simplest approach is to run a time simulation until a steady-state is reached. Electromagnetic transient programs such as EMTP have been used as such a tool. Complex techniques, such as the shooting method, have been proposed to accelerate the convergence to steady-state. One of the main disadvantages of the time-domain based methods is the lack of load flow constraints at the fundamental frequency. • The “state variables analysis” is another method used for study of harmonic disturbances [6.38, 6.39], which is based on the network admittance matrix inversion technique. The variable analysis as a concept, which was initially introduced in automatics, has been soon extended to many other domains. It supposes that the development of a linear system depends on its past and on the input signals applied at a given moment. The introduction of a number of variables judiciously selected allows the future of the system to be forecasted using these data only. Its application to power networks allows to effectively analyse their frequency behaviour by trying to find the series (zeroes of the system) and parallel (poles of the system) resonance frequencies. In the framework of the state variable method, the system behaviour is described by the classical equations: x& = A ⋅ x + B ⋅ j (6.68') u =C ⋅x

where: A B C x u j

is – – – – –

(6.68")

the state matrix of the system, with dimension n×n; the control matrix, with dimension n×q; the output matrix, with dimension m×n; the state vector, with dimension n; the output vector, with dimension m; the control vector, with dimension q;

Besides the equations (6.68), the initial values of the quantities xi (i = 1, n) and ji (i = 1, q) are known.

432

Basic computation

In the case of an electric distribution network, harmonically polluted, the components of the above-defined matrices are determined as follows: – the state variables: the independent harmonic currents through the series inductance ( iij ) or shunt inductance ( ii , i = 1, nl ) from the equivalent circuits of the consumers, lines and/or transformers, as well as the harmonic voltages at the capacitors’ terminals ( vi , i = 1, nc ). Assumes that capacitor banks, used for power factor correction and/or filtering, are installed into the system’s nodes. The choice of the state variables is performed by taking into account that the resonance harmonic frequencies are calculated in terms of the values of the capacitances and inductances of the equivalent circuit; obviously, the state variables are the currents through inductances and the voltages at the capacitor’s terminals; – the control variables: the harmonic currents injected in each node of the network ji (i = 1, q) ; – the output quantities: the harmonic voltages resulted in each node of the network ui (i = 1, m) . The matrices A and B can be written based on the Kirchhoff’s theorems for the considered network. Thus, for the network section between nodes i and j (Fig. 6.40), we may write: dui ui ⎧ ⎪ii + ji = Ci dt + R + iij i ⎪ dii ⎪ ⎨ −ui = Li dt ⎪ ⎪u − u = L diij + R i j ij ij ij ⎪ i dt ⎩

(6.69)

⎧ dii 1 ⎪ = − ui d t L i ⎪ Rij ⎪ diij 1 1 = − iij + ui − u j ⎨ Lij Lij Lij ⎪ dt ⎪ dui u 1 1 1 = ii − iij − i + ji ⎪ Ci Ci Ri Ci ⎩ dt Ci

(6.70)

or

iij ui

i ii Ci

Li

Lij

Ri

Rij j

ji

Fig. 6.40. Example of network section.

uj


433

Therefore, the matrices A, B and C have the following structure:

nl 64748 R ⎡ ⎢ terms ± L A=⎢ ⎢ terms ± 1 ⎢⎣ C

m 8 64 4744 1 ⎤ terms ± ⎥ L ⎥ 1 ⎥ terms ± RC ⎥⎦

⎫ ⎬ nl ⎭ ⎫ ⎬m ⎭

(6.71)

nl m 8 6474 8 64447444 1 ⎤ ⎡ 0 L 0 ⎥ ⎫ ⎢0 L 0 C 1 ⎥ ⎪ ⎢ ⎥ ⎪ 1 ⎢M L M L 0 0 ⎥ ⎪⎪ C2 B=⎢ ⎥ ⎬q ⎢ M L M M 0 L M ⎥ ⎪ ⎢ ⎥ ⎪ ⎢ 1 ⎥ ⎪ ⎢0 L 0 0 0 L ⎪ ⎢⎣ Cm ⎥⎦ ⎭ nl m 8 6474 8 644744 ⎡0 L 0 1 0 L 0 ⎤ ⎫ ⎪ ⎢ C = ⎢ M L M 0 1 L 0 ⎥⎥ ⎬ m M L M M 0 L M ⎢⎣0 L 0 0 0 L 1 ⎥⎦ ⎪⎭

Applying the Laplace transform to the equation (6.68'), it results: s X ( s) = A X (s) + B J ( s)

(6.72)

or, if we denote by IN the unity matrix of order n, the equation (6.72) becomes:

X ( s ) = ( sI N − A) −1 B J ( s )

(6.73)

Using the Ohm’s low, the relation between the harmonic nodal voltages ui and the injected currents ji, is:

U ( s) = Z (s) J ( s)

(6.74)

where Z ( s ) is the Laplace plane impedance matrix of the network. Replacing the expression (6.74) in (6.68"), we obtain:

Z = C ( sI N − A) −1 B or Z = C

adj( sI N − A) B det ( sI N − A)

(6.75)

The Laplace plane equivalent impedance seen from a node k is represented by the diagonal term from the position k+nl of the matrix Z, nl being the number of currents flowing through inductances considered as state variable. Therefore, if Ak

434

Basic computation

is the matrix obtained from A by eliminating the line and the column k+nl, it results: 1 det ( sI N −1 − Ak ) ⋅ (6.76) Z k ( s) = C k det ( sI N − A) Analysing the expression (6.76) we see that the poles of the system correspond to the eigenvalues of the matrix A, and the zeroes of the system seen from the node k correspond to the eigenvalues of the matrix Ak. The equivalent impedance seen from any node of the network will therefore have n poles and n-1 zeroes. In most cases the injection of the harmonic disturbance is modelled as a current source, the presence of impedance peaks (poles) at frequencies multiple of f1 generate high harmonic voltages at these frequencies. However, too low impedance at the frequency that corresponds to the tariff setting remote control signals may give raise to certain problems. The impedance, in fact, seen from the nodes of the network must be high enough to ensure the proper propagation of the signals. From these remarks, we can infer three rules for placing the impedance poles and zeroes: a pole must be set far away from the harmonic frequencies if it is “blocked” by a zero; the zeroes must be set close to each harmonic frequency or to poles so as to “compensate” them; the case of the central remote control frequency must be taken into consideration. Application As a practical application of the above-presented method, one of the most frequently study network is considered. Passive filters for harmonics cancellation are installed on the medium voltage bus from a transformer station, supplying a harmonic polluted distribution network (Fig. 6.41). These filters have a double role: to absorb the corresponding harmonic currents and to compensate the reactive power on the fundamental frequency up to the desired level. S u1

1

110 kV

20 kV

LL

Lf 3

Cf5

1/j kωC 1

i12

Lf 4

Cf7

Fig. 6.41. Simple distribution network (LL – linear load; NL – non-linear load).

R12

2

u2

i2 j kωL 2

NL

1 j1

j kωL12

T

2

j kωL1 i1

R2 j kωL 23

3 j3

i23

u3

i24

1/j kωC 3

1/j kωC 2

j2

j kωL24

u4 j4

4 1/j kωC 4

Fig. 6.42. Harmonic equivalent circuit of the network.


435

The filtering-compensation device contains a resonant circuit for the 5th and 7th harmonics, present in the current absorbed by the non-linear component of the load. The sizing criteria used for the filter’s components leads to the same inductance for the two filters. Table 6.6 presents the rated parameters of the network’s elements and the corresponding equivalent parameters (referred to 20 kV), while Figure 6.42 shows the equivalent electrical circuit, used for writing the state equations and the matrix A. Table 6.6 Rated parameters and equivalent parameters of the network’s elements Network elements System

Transformer

Linear load

Filtering/ Compensation devices

Rated parameters SSC = 1000 MVA US = 110kV Sn = 25 MVA ΔPsc = 130 kW Un MV = 22 kV usc = 11 % Pc Qc [MVAr] [MW] a 4 b 6 4 c 8 d 10 Qk1 [MVAr] a 2.833 b 2.25 c 1.667 d 1.083

Equivalent parameters XS = 0.4 Ω

LS =1.273⋅10-3 H

L1 =LS

RT = 0.101 Ω R12 = RT XT = 2.13 Ω LT = 6.779⋅10-3 H L12 = LT Rc [Ω]

R 2 = Rc

100 66.667 50 40 Lf [H] L23 = L24 = Lf 0.028 0.035 0.048 0.073

Xc = 100 Ω Lc = 0.318 H

L2 = L c

Cf5 [μF] 14.43 11.46 8.488 5.517

Cf7 [μF] 7.362 5.847 4.331 2.815

The active load, corresponding to the linear load, supplied from the medium voltage bus of the substation is considered by its four values (a÷d cases). In order to obtain the desired value of the power factor (0.96), the necessary values for capacitances and inductances of the passive filters are determined. The connection points between the coils and capacitors of the two filters are considered buses of the equivalent network (numbered by 3 and 4 in Fig. 6.42); therefore, the filter’s coil can be considered series connected and the capacitors can be considered shunt connected. In each of the network’s buses a current source (even fictive because the state matrix does not change) and a capacitor (even of fictive value C1 = C2 = 10 −10 F ) are connected. For this application, this consideration is not necessarily, so that by applying the Kirchhoff’s laws on the equivalent circuit, we obtain the following state matrix of the network:

436

Basic computation ⎡ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ A=⎢ 0 ⎢ ⎢1 ⎢C ⎢ 1 ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢⎣⎢

−

1 Ls

0

0

0

0

0

0

0

0

0

R12 L12

0

0

1 L12

0

0

0

0

0

0

0

0

0

0

0

0

0

1 C2

0

0 1 C4

0

−

1 C2

1 C1 1 C2

0

0

0

0

0

−

1 C2 1 C3

−

0

−

0 1 Lc 1 − L12 1 L23 1 L24

0

−

0 0 −

1 L23 0

0

0

1 C2 R2

0

0

0

0

0

0

0

−

⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 1 ⎥ ⎥ L24 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥⎦⎥

The frequencies corresponding to the network’s poles are obtained by dividing by 2π the imaginary positive parts of the complex conjugate eigenvalues pairs of the matrix A. The frequencies corresponding to the network’s zeroes seen from the four buses are obtained applying the same procedure, but to the four matrices Ak obtained from matrix A by removing by turn the pairs line-column corresponding to the indices from 6 to 9. The results corresponding to the four cases are presented in Table 6.7. The values of the frequencies of the network’s poles and zeroes are determined based on the frequency dependence between the diagonal elements of the inverse of the network admitance matrix – the classical method. Table 6.7 The frequencies of the network’s poles and zeroes.

Case

a b c d

Frequencies poles [Hz] classical method state variables bus number method 1 2 3 4 216.4 214 216 216 215 324.3 323 324 325 324 223.2 220 223 223 221 327.5 325 327 329 328 230.3 226 229 230 227 331.6 328 331 333 332 237.5 233 236 238 234 337.0 332 336 338 337

Frequencies zeroes [Hz] state variables method classical method bus number bus number 1 2 3 4 1 2 3 4 221.2 250 – 226.6 222 250 – 227 326.4 350 317.4 – 327 350 316 – 227.0 250 – 230.2 228 250 – 231 329.5 350 322.7 – 331 350 319 – 233.1 250 – 234.5 235 250 – 235 333.5 350 328.9 – 335 350 324 – 239.2 250 – 239.4 240 250 – 240 338.4 350 335.9 – 340 350 330 –

Each the four cases, the frequency dependence of the impedances seen from the network’s buses, determined with the classical method, is presented in Figure 6.43.


437 Zk2 [Ω]

Zk1[Ω] 4

100

3.33

80

a b

2.67 2

a

c

60

d

b c

40

1.33

d

20

0.67

0

0

150 175 200 225 250 275 300 325 350 375 400

150 175 200 225 250 275 300 325 350 375 400

f [Hz]

f [Hz] b. node 2

a. node 1

Zk3 4000

Zk4 [Ω]

[Ω]

7000

d

3200

5600

a 4200

2400

c 1600

b a

2800

b c

1400

800

d

150 175 200 225 250 275 300 325 350 375

0 150 175 200 225 250 275 300 325 350 375 400

f [Hz]

f [Hz]

0

c. node 3

d. node 4

Fig. 6.43. Frequency dependence of the impedance seen from the network’s buses. By analysing the obtained values, the following conclusions can be drawn: – the values of the frequencies of the parallel and series harmonic resonances that can appear into the considered network, obtained by means of the state variables method, are almost identically with those obtained by means of the classical method, for all the four cases of the load and filtering-compensation devices connected; – the zeroes seen from the bus 2, where the filtering-compensation devices are installed, are obtained on the frequencies of 250 Hz and 350 Hz, meaning that the passive filters are correctly sized in order to absorb the 5th and 7th current harmonics injected by the pollution source; – in the presence of the filtering-compensation devices, there is no risk of causing any parallel resonance because the network’s poles are positioned at nondangerous frequencies.

438

Basic computation

6.5.5. Mitigation solutions to controlling harmonics Basically, the harmonics can be significantly present under the form of strongly distorted voltage or current waveform becoming a problem if [6.1]: – the harmonics generated by some devices are greater than a tolerable threshold; – there is no controlling device to mitigate the harmonics so that a spread network area is affected by the distorted voltage; – the system can magnify one or more harmonics generated by the loads to a non tolerable level. The possible mitigation solutions of the harmonics are: – add filters to correct the current waveform to a sinusoidal shape; – use of delta or zigzag connected transformers to block the triplen harmonics; – modify the frequency response of the system by filters, inductors and capacitors.

6.5.5.1. Reducing the harmonic currents at the consumers Usually, the power supply utility imposes the level of harmonic current distortions produced by the load. This level should not be greater than the thresholds recommended by standards for the point of common coupling. In order to reduce the harmonic distortion level special methods are used in terms of nonlinear load type [6.1]: – for rectifiers, supplementary reactors at the AC entrance or bridges with increased number of pulses should be used; – for PWM inverters, reactors at the entrance circuit are recommended; – for arc furnaces, high reactance transformers or series reactors to increase the short-circuit impedance can be employed, only as long as this method will not affect the furnace operation; – for residential consumers, supply transformers with delta connection on the MV side can be used. Delta connected transformers can block the flow of zero-sequence harmonics (3rd order harmonics) from the line. Zigzag and grounding transformers can shunt the triplen harmonics. The international standard IEC 61000-3-2 [6.40] recommend limits for the harmonic emissions produced by non-linear consumer equipments. Tables 6.8 and 6.9 gives admissible limits for equipments operating with a current lower than 16 A. Accordingly, the devices are classified in four classes: – A class: three-phase balanced devices and all other equipments not included in B, C, and D classes; – B class: portable devices; – C class: lightning devices, including the setting-up lamp current devices; – D class: devices absorbing rectangular shape currents having a power lower than 600 W.


439

Table 6.8 Table 6.9 Maximum admissible limits for harmonic Maximum admissible limits for harmonic currents of the devices supplied by the public currents of the devices supplied by the LV networks: A, B, D classes public LV networks: C class Harmonic Harmonic Device Device Harmonic order current Harmonic order h current I [A] Ih Type Type h h [%] I 3 2.3 2 2 30 ⋅ PF 4 0.43 3 5 1.14 5 10 C class 6 0.30 7 7 7 0.77 9 5 8 ≤ h ≤ 40 11 ≤ h ≤ 39 1.84/h 3 h – even number h – odd number 9 0.4 11 0.33 13 0.21 15 ≤ h ≤ 39 2.25/h h – odd number The above value B class multiplied by 1.5 A class 2 1.08 Like A class, but only D class for odd number harmonics

6.5.5.2. Filtering harmonic distortion When the measures taken, concerning the equipment structure and its operating states as well as the connection configurations, are not sufficient to bring the harmonics at an allowable level, special filters for harmonics cancelling should be installed. In principle, the filters are installed for each harmonic to be limited. They are connected to the busbar common with the disturbing load. There are three general classes of filters: passive filters, active filters and hybrid filters. (i) The passive filters are a combination of inductors, capacitors, and resistors designed to block the flow of harmonic currents toward the distribution system. However, their performance is limited to a few harmonics, and they can introduce resonance in the power system. The passive filters are usually custom designed for the application. The shunt filter is the most common filtering application in use due to economical reasons and its advantage of smoothening the supplied voltage. The series filter can be also used to limit the harmonic currents but they have the disadvantage of distorting the supply voltage, and also being difficult to insulate. Figure 6.44 shows several types of common filtering circuits.

440

Basic computation

C1

R

Fig. 6.44. Common passive shunt filter configuration: a. single-tuned; b. first order high pass; c. second order high pass; d. third order high pass.

R

L

R L

R C

C a.

C2

C c.

b.

L

d.

The single-tuned filter is the most common shunt filter in use. Its main characteristics on the 5th harmonic, when connected to low voltage inductive load, are shown in Figure 6.45,a, while Figure 6.45,b presents a typical frequency response of the filter connected to the network. Zh [p.u.]

Zh [p.u.]

0.6

capacitive

0.5

inductive

0.6 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 1

2

3

4

5

6

7 h

1

a.

2

3

4

5

6

7 h

b.

Fig. 6.45. Typical frequency response of notch filter: a. filter alone; b. filter and sysyem.

The frequency response of the filter is just the harmonic impedance seen at its terminals. Examination of the filter response reveals the following characteristics [6.41]: – its harmonic impedance has a very low value at the frequency for which it is tuned; – when the source impedance is inductive, there is a resonance peak, which always occurs at a frequency lower than the frequency for which the filter is tuned; – there is a sharp increase in impedance below the tuned frequency due to the proximity of the resonant frequency; – the impedance increase with frequency for frequencies above that at which the filter is tuned. The filter in question can be characterized by impedance and quality factor. The filter’s impedance is given by the following relation: Z = R + j ( ωL − 1/ ωC ) where R, L and C are the filter’s parameters, i.e. resistance, inductance and capacitance.


441

At resonance, the imaginary part is equal to zero, and the impedance Z becomes a resistance given by the value of R. The quality of the filter, q, is a measure of the sharpness of tuning. Mathematically, the factor q is defined as: q=

L / C ωr L 1/ ωr C = = R R R

were ωr L and 1/ ωr C are the reactances at the resonance frequency. The following problems concerning the factor q for single-tuned filters are of interest: a) the value of q is seldom taken into account at filtering action. This is due to the fact that the values of R usually result in a significant increase in losses within the filter; b) the higher the value of q, the more pronounced is the valley at the tuned frequency; c) typically the value of R is given only by the resistance of the coil. In this case, q is equal to R times the X/R ratio of the tuning reactance. When the shunt filter is connected to a node of the system, due to the system’s impedance Ls, the resonance frequency of the equivalent filter (filtersystem) has a small displacement, having the expression: fr =

1 1 2π ( Ls + L)C I1 Load

Xs VC,1

V1

Fig. 6.46. Shunt filter.

The value of the capacitance C of the filter is established taking into consideration the following two cases: a) the filter is used only for harmonic filtering; b) the filter provides both filtering and reactive power supply. In the first case, the goal is to minimize the size of the capacitance C but chosen from a range given by the manufacturer. In this respect, we start from the expression of the reactive power: Q = Q1 + Qh = ω1CVC2,1 +

I h2 ω1hC

where: Q1 is the reactive power provided at the fundamental frequency; Qh – the reactive power provided on the hth harmonic;

442

Basic computation

VC ,1 – the voltage at the bank’s terminals at the fundamental frequency; I h – the current absorbed by the filter at the hth harmonic; ω1 = 2πf1 and f1 are the angular frequency and fundamental frequency, respectively. From Figure 6.46, we can write the relationship between the voltage VC ,1 and the voltage V1 , applied at the filter’s terminals: V V1 = C ,1 X C ,1 − X L ,1 X C ,1 At resonance, for ωr = 1 VC ,1 =

or

V1 1 − ω1 L ω1C

=

VC ,1 1 ω1C

LC , the last equation becomes:

V1 1 − ω12 ωr2

or

VC ,1 = V1

h2 h2 − 1

Differentiating Q with respect to C and equating to zero, and taking into account the latter expression, it results the minimum value of the capacitance: Cmin =

(h 2 − 1) I h ⋅ V1ω1 h2 h

1

⋅

(6.77)

In the second case (b), we start from the condition that, at the fundamental frequency, the reactive power Q1 provided by the filter should be equal to the rated value QCn . Therefore: Q1 = QCn =

V12 V 2ω C h2 = 1 21 = 2 V12 ω1C X C ,1 − X L ,1 1 − ω1 LC h − 1

(6.78)

From the latter relation it results the value of C, and from the resonance condition at the hth harmonic, we can obtain the value of L. For multiple parallel single-tuned filters ( h = 5, 7, 11, ... ), establishing the capacity Cn of the capacitor bank is performed based on the following two conditions: – the total reactive power of the capacitor bank should be minimum: ⎡ 2 I h2 ⎤ ⎢VC ,1ω1Ch + ⎥ = min hω1Ch ⎦ h = 5,7,... ⎣ the reactive power at fundamental frequency should have the average value QCn , therefore: Q1 =

–

∑

QCn =

h2 V12 ω1C 2 1 − h h =5,7,...

∑


443

The Lagrange method allows us to solve the above equations. One important side effect of adding a filter is that it creates a sharp parallel resonance point at a frequency below the notch frequency. This resonant frequency must be safely away from any significant harmonic. Filters are commonly tuned slightly lower than the harmonic to be filtered to provide a margin of safety in case there is some change in system parameters. For this reason, filters are added to the system starting with the lowest problem harmonic [6.1]. (ii) Active filters are relatively new types of devices used for the elimination of harmonics [6.1]. They are based on power electronics and are much more expensive than passive filters. They have the distinct advantage that they do not resonate with the system. They can also address more than one harmonic simultaneously and combat other power quality problems such as flicker. They are particularly useful for large, distorting loads fed from relatively weak points of the power system. The basic idea is to replace the portion of the sine wave that is missing in the current in a non-linear load. Figure 6.47 presents this concept. An electronic control monitors the line voltage and/or current, switching the power electronics very precisely to track the load current or voltage and force it to be sinusoidal. As shown, there are two fundamental approaches: one that uses an inductor to store up current to be injected into the system at the appropriate instant and one that uses a capacitor. Therefore, while the load current is distorted to the extent demanded by the non-linear load, the current seen by the system has a much more sinusoidal shape.

is

~

+

il

Non-linear load

Ls Iaf

Control

Fig. 6.47. Application of a filter at a non-linear load.

Active filters can be classified in a number of ways [6.42]: – application: AC or DC systems, AC or DC side of converters, transmission or distribution systems; – connection to the system: shunt (Fig. 6.47 and Fig. 6.49), series (Fig. 6.48) or series/shunt (Fig. 6.50,a) and shunt/series (Fig. 6.50,b); – type: active only or hybrid, i.e. combination of active and passive (Fig. 6.50); – function: harmonic mitigation, reactive power compensation, voltage regulation, flicker compensation; – topology: voltage source or current source converter. Active filters are fundamentally static power consumers configured to synthesize a current source (Fig. 6.47) or a voltage source (Fig. 6.48).

444

Basic computation

It should be noted that the active filter must satisfy the basic laws of association of current and voltage sources; for example, in Figure 6.48, the active filter current is defined by the network, and can therefore be controlled only in voltage. Vaf Non-linear load (Electronic)

~ Control

Active filter

Fig. 6.48. The active filter as a voltage source.

is

~

il iaf

~ ~

Non -linear load (Rectifier) Reactive element

Active filter

Fig. 6.49. Implementation of shunt connected active filter

V af

~ ~

Reactive element

iaf

Load bus

~

a.

V af

~ iaf

~

Load bus Reactive element

~

b. Fig. 6.50. Connection type of active filter: a. series/shunt; b. shunt/series.

The basic active filter module consists of: – a power converter, or matrix of switches, the more common being a threephase bridge converter, in either a current or, more often, a voltage source converter;


445

– a DC bus having a reactive element, an inductor or a capacitor as a storage component. In this configuration, the energy is only stored for a short period and the DC bus reactive element essentially absorbs the ripple produced by the converter operation. However, energy storage elements, capable of storing significant amounts of energy have been suggested. A shunt active filter is connected in parallel with the non-linear load, to detect its harmonic current and to inject into the system a compensating current. A series active filter is connected in series with a supply source and the load. It presents high impedance for the harmonic current, blocking their flow from load to source and from source to load. Regarding the active filter classification, a comparison of shunt active filters and series active filters is presented in Table 6.10. Table 6.10 Comparison of shunt active filters and series active filters. System configuration

Shunt active filter

Series active filter

Basic operating principle

Current source Voltage source Inductive, current-source Capacitive, voltage-source Non-linear load type loads or harmonic current loads or harmonic voltage sources sources Independent on the source Independent on Zs, and/or ZL impedance Zs, or currentCompensation for voltage loads, but source but dependent on Zs dependent on ZL when the characteristics when the load impedance ZL load is current source type is low A low impedance shunt branch Injected current flows into the (e.g. a shunt passive filter or a load side and may cause Application considerations capacitor bank) is needed when overcurrent when applied to a applied to an inductive or capacitive or voltage source current source load Current harmonics filtering, Load on the current unbalance, reactive Current harmonics, Solutions to AC supply current compensation, voltage reactive current power flicker quality Voltage unbalance, distortion, problems AC supply on flicker, notching, interruptions the load reactive current, dips, swells

In addition, the active filters can be combined with passive filter to enforce filter effectiveness and reduce active filter rating. Taking into account the advantages and the disadvantages of active and passive filters a mixed solution can be used. In Figure 6.51, the block diagram using an active series filter and a shunt passive filter with three branches is presented: two branches tuned on the 5th and 7th harmonic orders and a high-pass

446

Basic computation

filter. This filter mitigates parallel harmonic resonance between the passive filter and the supply system, blocking the access into the filter or load of the existing harmonics; this implies low cost operation. iL

iaf if

Non-linear load

Active filter h=5 h=7 high-pass filter

Fig. 6.51. Hybrid filter: active series and passive shunt.

6.5.5.3. Modification of the system frequency response The system frequency response is very important in the problem of reactive power compensation in non-sinusoidal conditions, with capacitor banks. Thus, all circuits containing capacitances have one or more self-resonance frequencies. When one of those frequencies lines up with a frequency that is being produced into the power system, the resonance can develop the voltages and current at that frequency persistent at very high values. The system frequency response can be modified by different methods [6.1]: (i) Adding a shunt filter. Introducing a shunt filter could change the system response; (ii) Adding a reactor to detune the system. Harmful resonances appear generally between the system inductance and shunt capacitors used for power factor correction. The reactor must be added between the capacitor and the system. One method is to simply put a reactor in series with the capacitor, to remove the system resonance without actually tuning the capacitor to create a filter; (iii) Changing the size of the capacitor. This is one of the less expensive options for both utilities and industrial customers; (v) Removing the capacitor and simply accepting the increase of losses, the decrease of voltage, and the power factor penalty. If technically feasible, this is occasionally the best economic choice.

6.6. Voltage unbalances According to the IEC, the expression unbalanced voltage is defined as a phenomenon caused by the differences of voltage deviation between the phases in a point of a multiphase system. In some papers, inbalance is used instead of unbalance.


447

A balanced three-phase power system presents equal voltage magnitude on each phase, the voltages being separated by the same phase-shift value. The voltage at the terminals of the generator is balanced and sinusoidal in shape. If the impedance of the various system components is linear and equal for each phase and all loads are balanced on the three phases, the voltage at the terminals components of the system remain balanced. Single-phase load currents and unbalanced three-phase loads currents determine unequal voltage drops on the three phases of the supply system. Consequently, the phase-to-neutral voltages within the supply system will be unbalanced because the system voltage at any point is the difference between the generated voltage and voltage drops due to the load current. The unbalanced voltages can be represented by the sum of three sets of symmetrical voltage components, namely: – the positive-sequence voltage component, consisting of three phases all equal in magnitude and symmetrically spaced, at 2π / 3 intervals in timephase, their phase order being equal to the phase order of the system generated voltages; – the zero-sequence voltage component, consisting of three phases, all equal in magnitude and phase; – the negative-sequence voltage component, consisting of three phases, all equal in magnitude and symmetrically spaced, at 2π / 3 intervals in timephase, their phase order being the reverse of the positive sequence phaseorder.

6.6.1. Unbalance indices Voltage unbalance is defined in the U.S. Standards as the maximum deviation from the average of the three phase voltages, divided by the average of three phase-to-neutral voltages expressed as a percentage [6.1]:

(

)

max V − V ΔVmax ⋅ 100 [%] → ⋅ 100 [%] Vaverage V where:

(

)

(

)(

)(

)

max V − V = max ⎡ Va − V , Vb − V , Vc − V ⎤ ⎣ ⎦ In sinusoidal operation or for harmonics analysis, the voltage unbalance − factor uV is defined by the ratio of the negative sequence voltage V to the positive +

sequence voltage V : uV = V

−

V = uV exp ( ψV ) +

and is usually expressed as a percentage, given by:

448

Basic computation

uV = V − V + ⋅ 100 [%]

(6.79)

where ψV is the phase shift between the two voltages. Alternatively, simultaneous measurement of the three rms phase-to-phase voltages can be also used to calculate the unbalance factor, for isolated neutral networks: 1 − 3 − 6β

uV =

1 + 3 + 6β

⋅ 100 [%]

(6.80)

where: β=

4 4 U ab + U bc4 + U ca

(U

2 ab

2 + U bc2 + U ca

)

2

U ab being the voltage between phases a and b at the fundamental frequency. Another unbalance index is the voltage nonsymmetry factor u0, defined by + the ratio between the zero-sequence V 0 and the positive-sequence V component: V0 (6.81) ⋅ 100 [%] V+ The voltage unbalance due to any load connected between two of the three phases of the line or between one phase and the neutral, at the point of connection of the load, can be evaluated by the following factor [6.43]: u0 =

uV' =

S S sc

(6.82)

where S is the load power, in MVA, and Ssc is the three-phase short circuit power level at the point of connection of the load, in MVA. The voltage unbalance occurring at the point of common coupling due to a combination of unbalanced three-phase loads or phase-to-phase loads also calculated in terms of the negative-sequence current I − is: uV'' =

3 ⋅ I − ⋅U S sc

(6.83)

Likewise, the unbalance current factor is defined as the ratio of the fundamental negative-sequence current to the fundamental positive-sequence current component: iV =

I1− ⋅ 100 [%] I1+

(6.84)

Note that, in practice, determination of the factors uV, u0 and iV is possible only after the determination of the harmonics spectrum.


449

6.6.2. Origin and effects An electrical power system is expected to operate in a balanced three-phase condition, but some causes that produce voltage unbalance exist. These causes can be split in two components. The first component results from the own unbalance structure of the network (lines, transformers, capacitor bank, etc.) and has a constant value; the second component has a fluctuating value due to temporary unbalance fluctuating loads. Some of the most common causes of the unbalanced voltage are: – unbalanced incoming utility supply; – unequal transformer tap setting; – large single-phase distribution transformers in the system; – faults or grounds in the power transformer; – a blown fuse in a three-phase capacitor bank for power factor improvement; – unequal impedance in conductors of the power supply wiring; – unbalanced distribution of single-phase loads such as lightning; – heavy reactive single-phase loads such as welders. The most common symptoms of unbalanced voltages are the damaging effects on electric motors, power supply wiring, transformers and generators. Unbalanced voltages at motor terminals cause phase current unbalance ranging from 6 to 10 times the percent voltage unbalance for a fully loaded motor. As an example, if the voltage unbalance is 1%, then the current unbalance could be anywhere from 6% to 10%. This causes motor overcurrent resulting in excessive heat that shortens the motor life, and hence, eventual motor burnout. The Figure 6.52 shows the typical percentage increases in motor losses and heating for various levels of voltage unbalance. Other effects on motors are that locked rotor stator winding current will be unbalanced proportional to the voltage unbalance, full load speed will be slightly reduced, and the torque will be reduced. When a motor continues to operate with unbalanced voltages its efficiency is reduced as well. Both increased current and resistance, due to heating, cause the reduction of efficiency. 150 effect of unbalance [%] 100

motor heating motor losses

50

0

1

2

3

4

5

6

7 9 10 8 voltage unbalance [%]

Fig. 6.52. Increase in motor heating and losses in terms of voltage unbalance.

450

Basic computation

The increase in resistance and current stack-up contribute to the exponential increase in motor heating. Essentially, this means that as the resulting losses increase, the heating intensifies rapidly. This may lead to a condition of uncontrollable heat size, called “terminal runaway”, which results in a rapid deterioration of the winding insulation concluding with failure of the winding. Under normal operating conditions, during each period of one week, 95% of the 10 minute mean rms value of the negative phase sequence component of the supply voltage shall be within the range 0 to 20% of the positive phase sequence component [6.14].

6.6.3. Voltage unbalance and power flow under non-symmetrical conditions In order to establish the power flows in three-phase networks operating under sinusoidal conditions, but with unbalanced load, let us consider a simple power system, where an ideal three-phase supplier, having the emf of positive sequences, supplies, through a balanced network LN, a balanced load BL connected in parallel with an unbalanced load, UBL (Fig. 6.53) [6.25]. The presence of the unbalanced load in the system is, evidently, the cause of the non-symmetric operation of the system.

G

Linear Network LN

BL

UBL

Fig. 6.53. A power system in which an ideal generator supplies through a linear network a balanced load and an unbalanced load.

The active and reactive powers balance is satisfied for each sequence, separately: + + + Pg+ = PLN + PBL + PUBL − − − 0 = PLN + PBL + PUBL 0 0 0 0 = PLN + PBL + PUBL

+ + + Qg+ = QLN + QBL + QUBL

(6.85)

− − − 0 = QLN + QBL + QUBL 0 0 0 0 = QLN + QBL + QUBL

The network and the balanced load being passive loads, absorb positive active power on each sequence, and thus: + − 0 + PUBL = PUBL + PUBL + PUBL ≤ PUBL

(6.86)


451

+ Consequently, this load absorbs from the supplier an active power PUBL , which is greater than it would “need”, uses PUBL and reinjects the difference into the balanced load causing supplementary losses:

(

+ − 0 + − 0 + + − 0 PLN + PBL = PLN + PLN + PLN + PBL + PBL + PBL = PLN + PBL − PUBL + PUBL

)

(6.87)

These relations suggest the power flow diagram given in Figure 6.54. BL PBL , QBL Pg , Qg

G

PLN , QLN

LN

PUBL , QUBL UBL

0 PBL + PBL

QBL + QBL0 0 PLN + PLN

QLN + QLN0 0 PUBL + PUBL 0 QUBL + QUBL

Fig. 6.54. The power flow diagram under non-symmetrical conditions of electrical power systems.

The fact that the unbalanced loads are the sources of negative- and zerosequence powers, although intuitively known, has not been explicitly stated before. The complex, active and reactive powers are conservative for each sequence (+, -, 0) separately. Generally, the negative- and zero-sequence active powers reinjected in the network by the unbalanced load represents supplementary losses. In all this reasoning the analogy with the non-sinusoidal conditions is evident. The flow of reactive power can be similarly interpreted; this produces a supplementary reactive load on the generator and the balanced loads. All these considerations clearly show that the positive-, negative- and zero-sequence power flows must be separated and distinctly accounted. Under these conditions the power factor of a balanced or unbalanced load can be defined as the ratio between the total active power absorbed by the load and the apparent power of symmetry: kp =

P P+ + P− + P0 P = = cos ϕ+ + U+ = k p+ + k pn + + S S S

(6.88)

is the apparent power of symmetry (positive sequence); where: S + − 0 PU = P + P – the real power of the unbalance (non symmetry); k p+

– the power factor of symmetry;

k pn

– the power factor of the unbalance (non symmetry).

452

Basic computation

6.6.4. Practical definitions of powers in system with non-sinusoidal waveforms and unbalanced loads *) In [6.20] and [6.27], definitions for power terms that are practical and effective when voltage and/or currents are distorted and/or unbalanced are proposed. Unbalanced systems may be analysed using the approach of equivalent apparent power, that is: S e = 3Ve I e For a four-wire system, the equivalent voltage is: Ve =

Va2 + Vb2 + Vc2 3

(6.89,a)

I a2 + I b2 + I c2 3

(6.89,b)

and the equivalent current is: Ie =

where Va , Vb and Vc are the phase-to-neutral rms voltages. For a three-wire system, the equivalent voltage Ve may be calculated using the relation: Ve =

2 2 U ab + U bc2 + U ca 9

where the rms voltages U ab , U bc and U ca are measured from phase to phase. The equivalent current I e is calculated in terms of the rms currents I a , I b , I c . For a four-wire system the above relation becomes: Ve =

1 ⎡ 2 + U bc2 + U ca2 ⎤ 3 Va2 + Vb2 + Vb2 + U ab ⎦ 18 ⎣

(

)

(6.90,a)

I a2 + I b2 + I c2 + I 02 3

(6.90,b)

and Ie = *)

Reprinted with permission from IEEE Standard 1459-2000 – Definition for the measurement of electric power quantities under sinusoidal, non-sinusoidal, balanced or unbalanced conditions © IEEE 2000, and IEEE Working Group on Non-sinusoidal Situations: Effects on meter performance and definitions of power – Practical definitions for powers in systems with non-sinusoidal waveforms and unbalanced loads: A discussion, IEEE Trans. on Power Delivery, Vol. 11, No. 1, pp. 79 – 87, January 1996 © IEEE 1996.


453

where I0 is the neutral rms current. Similarly to the single-phase case, the equivalent voltage and current may be separated into two components: 2 Ve2 = Ve21 + VeH

and

2 I e2 = I e21 + I eH

where the index 1 marks the fundamental rms components: Ve21 =

I2 + I2 + I2 Va21 + Vb21 + Vc21 ; I e21 = a1 b1 c1 3 3

and the index H marks the totalised non-fundamental rms components: 2 VeH =

⎡Vah2 + Vbh2 + Vch2 ⎤ 2 ⎥ ; I eH = ⎢ 3 h ≠1 ⎣ ⎦

∑

2 2 2 ⎤ ⎡ I ah + I bh + I ch ⎥ ⎢ 3 h ≠1 ⎣ ⎦

∑

The equivalent apparent power is separated into two components: the fundamental apparent power S e1 and the non-fundamental apparent power: 2 2 2 S e2 = S e21 + S eN = S e21 + DeI2 + DeV + S eH

where: DeI = 3Ve1 I eH is the current distortion power; DeV = 3VeH I e1 – the voltage distortion power; 2 2 SeH = 3VeH I eH = PeH + QeH – the apparent harmonic power.

The ratio between the squares of SeN and Se1 can be written: 2

⎛ SeN ⎞ 2 2 2 ⎜ ⎟ = ( ITHDe ) + (VTHDe ) + ( ITHDe VTHDe ) ⎝ Se1 ⎠ where: VTHDe =

VeH Ve1

and

I THDe =

I eH I e1

In the case of unbalanced systems, the definition of another power component becomes inevitable. The unbalanced loads convert part of the fundamental positive-sequence active power into fundamental negative- and zerosequence active power. This is true for reactive power also. The unbalance degree in the fundamental apparent power S e1 can be divided into two terms: 2

S e21 = S1+ + Su21 where: S1+ = 3V1+ I1+ is the positive-sequence fundamental apparent power; V1+ , I1+ S u1

– the rms values of the positive-sequence fundamental voltage and current; – the unbalanced fundamental apparent power.

454

Basic computation

This approach of decomposing the apparent power, Se, has the following useful features: – conveniently separates the fundamental apparent power and its active and reactive components from the non-fundamental apparent power; – provides a useful measure of the degree of harmonic pollution in the normalized ratio S N Se1 ; – provides a useful measure of the degree of unbalance pollution in the normalized ratio Su1 Se1 . The IEEE standard 1459-2000 [6.20] defines an arithmetic apparent power, vector apparent power and Budeanu’s apparent power. The per phase apparent powers are given by: S a = Va I a ; Sb = Vb I b ; Sc = Vc I c and S a2 = Pa2 + Qa2 ; Sb2 = Pb2 + Qb2 ; Sc2 = Pc2 + Qc2 The arithmetic apparent power is: S A = Sa + Sb + Sc and the vector apparent power is: SV = P 2 + Q 2 = (Pa + Pb + Pc ) + j (Qa + Qb + Qc ) = P + jQ

(6.91)

A geometrical interpretation of SV is presented in Figure 6.55. SV Sc

Sb Sa Qa 0

Pa

Qb

Qc

Pc

Pb SV SA

Fig. 6.55. Arithmetic and vector apparent powers under sinusoidal conditions. Reprinted with permission from IEEE Standard 1459-2000 – Definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced or unbalanced conditions © IEEE 2000.

The definition of the arithmetic apparent power is an extension of Budeanu’s apparent power resolution for single-phase systems. Thus, for each phase the apparent power is given by:


455

S a = Pa2 + Qa2 + Da2 ; S b = Pb2 + Qb2 + Db2 ; S c = Pc2 + Qc2 + Dc2 and for the three-phase system, the total apparent power is: S A = S a + S b + S c or SV = P 2 + Q 2 + D 2

(6.92)

with P = Pa + Pb + Pc ; Q = Qa + Qb + Qc ; D = Da + Db + Dc where: Pa, Pb, Pc are the per phase active powers; Qa, Qb, Qc – per phase reactive powers, according to Budeanu; Da, Db, Dc – per phase distortion powers, according to Budeanu. Additional to the already defined power factor, due to the unbalance in voltage, other definitions should be given: the effective power factor PFe = P Se ; the arithmetic power factor PFa = P S A ; the geometric power factor PFV = P SV ; and the positive-sequence power factor PF+1 = P1+ S1+ , where S1+ is the fundamental positive-sequence apparent power and P1+ is the fundamental positive-sequence active power. SA

SV

SV

0

Pa

P Da b

Dc

Pc Qc Db

Sb Sa

Sc

Qb

Qa

Fig. 6.56. Arithmetic SA and vector SV apparent powers: unbalanced nonsinusoidal conditions. Reprinted with permission from IEEE Standard 1459-2000 – Definition for the measurement of electric power quantities under sinusoidal, non-sinusoidal, balanced or unbalanced conditions © IEEE 2000.

456

Basic computation

The major drawback of the power factor definition stems from the difference between the quantities [Qa + Qb + Qc ] + [ Da + Db + Dc ] and S A2 − P 2 (Fig. 6.56). 2

2

6.6.5. Mitigation solutions to the unbalanced operation Because the voltage and current unbalances are mainly due to the nonsymmetrical loads, appropriate measures are taken to balance the current on the three phases. One of the measures for preventing the unbalances is the natural symmetrizing. Two methods can be mentioned [6.45]: – equaly repartition of the single-phase loads on the three phases of the supply system. This is the case of urban consumers, and mostly the low voltage supplied loads; – connecting the unbalanced loads to an superior voltage level, which results in the increase of the short-circuit power. This is the case of industrial consumers, of high rated power (few MVA or tens of MVA – induction furnace, welders, electric traction, etc.), which are fed via own transformers. In these conditions, the nonsymmetry degree will decrease proportional with the short-circuit power (Fig. 6.57). MV

MV

PCC

LV

LV single-phase loads

single-phase loads

Fig. 6.57. Mitigation of unbalance generated by single-phase loads.

Considering that the system provides pure sinusoidal voltages, some measures for limiting the unbalances are mentioned: – symmetrization configurations with single-phase transformers (Scott connection, V connection); – Steinmetz symmetrization circuit; – reactive power compensation systems. The Scott connection consists in connecting two single-phase transformers ' ' (Fig. 4.58,a), which provides two voltages U 1 and U 2 in the secondary of transformers, of equal magnitudes but phase shifted by π/2 (Fig. 6.58,b). The two transformers have identical secondary, but their primary has different number of turns. If the load is connected only to the secondary terminals of the transformer T1, the three-phase electric network is loaded only on two phases, i.e. b and c in the


457

example from Figure 6.58,c, and if the load is connected only to the secondary of the transformer T2, all the three phases of the network will be unequally loaded (Fig. 6.58,d). a b

c

c

VcN

c M T1

c

N/2

N/2

b

N 3/2

a T2

b M a.

c

a

a

a

b

b c

Ia

Ib

M

a

b

VbN b.

Ic

Ia

T2

T1

I T1

U2

b

c

a

U1

U2

U1

VaN

N

M

M

M I/ 3 I/ 3

Ic

Ib 2I / 3

T2

I1=I

I2=I c.

a

a

b c

b c Ia

Ib

M

T1

d.

Ic

Ia

T2

T1

I1=I2=I

I C

R

L e.

L1

Ic

Ib

M

I1

I2

C1

C2 L2

R1

T2

R2

f.

Fig. 6.58. The Scott connection for single-phase load supply.

In practical situations, the Scott connection is used to supply a single load (Fig. 6.58,e) or for two loads (Fig. 6.58,f). The equivalent circuits of the loads are represented through the series impedance Z = R + jωL in parallel with the capacitor bank C, used for power factor correction.

458

Basic computation

For the case of a single-phase linear load, the currents absorbed from the network, determined by superposing the currents from Figures 6.48,c and 6.48,d, results: ⎛ ⎛ 1 ⎞ 1 ⎞ I , I b = ⎜1 + ⎟ I and I c = ⎜ 1 − ⎟I 3 3⎠ 3⎠ ⎝ ⎝

2

Ia =

When two loads are considered (Fig. 6.48,f), for which reactive power is ' provided by the capacitor bank for full power factor correction (the phasors I 1 and '

'

'

I 2 are orthogonal), and denoting I 2 I 1 = β , it results: 2

Ia = −

3

I 2 , I b = − I 1 − I 2 and I c = I 1 −

1 3

I2

or in absolute values: Ia =

2 3

β I1 , I b = I1 1 +

β2 3

and I c = I1 1 +

β2 3

The V connecting of two identical single-phase transformers (Fig. 6.59) and the supply of single-phase loads with voltage obtained by cross-connecting the secondary of transformers allows loading all the three phases of the supply network. The currents are thus given by: I a = I ; I b = −2 I and I c = I a b c 2I

Ic

Ib

Ia

I

I

I

I

T1

T2 C

L

R

Fig. 6.59. The V connection for single-phase load supply.

The load supply by means of single-phase transformer configurations, although it does not provide equal loading of the three-phase network, allows in many cases to limit the unbalances at an acceptable level. The loads balancing by reactive power compensation is based on the Steinmetz connection (Fig. 6.60). Consider that the transformer T, connected between phases a and c, provides the power P, at lagging power factor, to the


459

industrial load. A capacitor bank C1, sized so that to provide a unity power factor (Ica is in phase with Uca), is connected in parallel with the secondary of the transformer T that supplies the two-phase load (e.g., arc furnace or welder). The symmetrizing circuit (Fig. 6.60) consists of the capacitor bank C2 and the coil L2. These devices are sized so that the three currents I a , I b and I c form a symmetrical system (of equal magnitudes but shifted in time by 2π/3) and are in phase with the phase-to-neutral voltages (resistive behaviour). a b c

Ia Ib Ic

Ica

a

T

Iab

C2

b Ibc

L2 c

Ic Ica

two-phase load

C1

Vc

Ia

Ubc

Uca Iab

Ib Ibc

Va N

Vb Uab

Ica a.

b.

Fig. 6.60. The Steinmetz circuit for loads balancing.

If the system of voltages is considered symmetrical, with equal voltages on the three phases U ab = U bc = U ca = 3 V (where V is the phase-to-neutral voltage), and the current is sinusoidal, the phasor diagram (Fig. 6.60,b) for the circuit in Figure 6.60,a can be drawn. In order that the three currents absorbed from the network Ia, Ib, Ic to form a symmetrical system, the negative-sequence component has to be zero, so that: I a = I b = I c = I ab = I bc = I ca

3

Knowing that: I ab =

3V 3V P and I ca = , I bc = X L2 X C2 3V

it results: 3V 3V P = = 3V X C2 X L2 where X C2 is the capacitive reactance of the capacitor bank C2, and X L2 is the inductive reactance of the coil L2. We then obtain: 1 P P = and ωC2 = 2 ωL2 3 3 V 3 3V 2

460

Basic computation

or: BL = G

3 and BC = −G

3

where G, BL and BC are the admittances corresponding to R, X L2 and X C2 (given that B > 0 for lagging load, and B < 0 for leading load, respectively). The capacitance value necessary for power factor correction to unity is: C1 =

P tan ϕ or BC1 = − BLoad 3ωV 2

where tan φ corresponds to the natural power factor of the two-phase load. Practical cases show that the active power P is variable in time. In order to ensure the adaptive symmetrization of the load, the capacitor banks C1 and C2 and the coil L2 should provide self-regulation of their parameters according to the load. The Steinmetz connection provides very good symmetrization of the load on the three phases. Instead, it has the following disadvantages: – at the fundamental frequency, the assembly load – compensator can be equivalent with a perfectly balanced impedance but at other frequencies; these corresponds to the superior order harmonics produced by the same load or the neighbouring loads, leading to a strong unbalance. Furthermore, the compensator must be sized so that to avoid the resonance with the network, at the harmonics present under normal operating conditions. – manually or automatic control of C1, C2 and L2 leads to frequent voltage oscillations into the network, due to commutations. These inconveniences can be corrected with modern applications of power electronics. It is known for long time that the static reactive power compensation system (SVS) can be used to balance the load currents and to improve the power factor of unbalanced power systems [6.45, 6.46]. Each phase of the SVSs can be independently controlled and it can provide a different amount of reactive power compensation. In [6.46] the derivation of load balancing theorem is based on the symmetrical component method. However, the same result can also be obtained by minimizing the quadratic sum of three-phase currents. Most of the studies about SVSs concentrate on application techniques, such as using high speed programmable controllers or microprocessors and solving sub optimal solutions when discrete-tap compensators are used. In the three-phase four-wire distribution networks, unbalanced loads may produce negative- and zero-sequence currents. In order to improve the load bus power factor these currents have to be reduced accordingly. The technique uses a wye (Y) connected SVS and a delta (Δ) connected SVS to give different amount of reactive power compensation to each phase. While the Δ-SVS is used to eliminate the negative-sequence currents, the Y-SVS is used to eliminate the zero-sequence currents and the imaginary part of the positive sequence currents. Figure 6.61


461

[6.47] illustrates the simplified connection circuit of the SVS to the three-phase four-wire system, in which the electronic part was not represented. a b c N

l

Ia

Ia

Ib

l b

Ic

Ic

IN

IN

Unbalanced load

l

Ia

Ib

IYc

Ca

Cb

Cc

Y

Y

I

I Δb

I aΔ

Y N

I

I Δc

Δ ab

I Δbc

Cab Cbc I Δca Cac Fig. 6.61. An unbalanced three-phase four-wire distribution system with SVSs connected at the load terminals.

In order to achieve the load balancing and full reactive power compensation, we start from the following equalities:

( ) = 0 ; Re ( I ) = 0 ; Im ( I ) = 0 ; Re ( I ) = 0 ; Im ( I ) = 0

Im I

+

−

−

0

0

(6.93)

Equations (6.93) have infinite number of solutions because there are six unknowns (susceptances of the Y-SVS and the Δ-SVS) and five equations. An additional constraint together with the five ones in (6.93) will offer a unique solution. The additional equation needs an additional constraint. The first suggested constraint is that the imaginary part of the positive-sequence component of the load current should be eliminated by the Y-SVS because the Δ-SVS does not generate imaginary part of positive sequence currents. The new constraint is given by: I abΔ + I bcΔ + I caΔ = 0

(6.94)

Substituting equation (6.94) into equations (6.93), we get the on-line compensation formulae for the currents of the SVSs [6.48]: 1 a 1 1 a I aY = I b − I ca + I ar ; I bY = − I aa + I ca + I cr ; I cY = I c − I ca + I cr 3 3 3 2 2 2 I abΔ = I aa − I ba ; I bcΔ = I ba − I ca ; I caΔ = I ca − I aa 3 3 3 (6.95) a r where I p and I p , with p = a, b, c , are the active and reactive (imaginary) parts of

(

)

(

the load currents, i.e.:

(

)

)

(

)

(

(

)

)

462

Basic computation l

l

l

I a = I aa − jI ar ; I b = I ba − jI br ; I c = I ca − jI cr

(6.96)

Other two constraints may be used. The first is the condition of minimum squared sum of the SVS currents ( ( I c ) 2 = min ) and the second is the minimum

∑

squared sum of the reactive powers provided by the SVS (

∑ (I V ) c

2

= min ).

The obtained load bus voltages and currents are used to calculate the on-line compensation susceptance values of the SVSs. The SVSs can provide a different amount of reactive power to each phase such that, although the load is unbalanced, the system will be balanced, seen from the load bus. The SVSs can also maintain the load bus voltage at a certain value in addition to the balancing effect. The problem is difficult because the network operates under a non-sinusoidal steady-state. Under these conditions, the problem can be solved by means of simultaneous operation of the reactive power compensation devices. One of the most efficient methods consists in using passive shunt filters. These are, in fact, single-tuned filters on the harmonic current to be cancelled. For their frequency the equivalent impedance is practically zero, but it has capacitive character for low frequencies, and so does for the fundamental frequency. For each harmonic current which is to be filtered, a three-phase symmetrical wye-connected single-tuned filter is needed. This will have a double effect: firstly, it will cancel the corresponding harmonic current, and secondly, it will compensate the reactive power at the fundamental frequency. Furthermore, the value of the reactive power to compensate is one of the designing criteria for the filter elements. Therefore, the problem of the reactive power compensation in distribution networks that supply unbalanced and non-linear loads has to be solved taking into account the correlation with the effects of the present or future electrical devices used for load balancing and for the non-sinusoidal steady-states amelioration, respectively. In this respect, the case of unbalanced and non-linear equivalent load, connected to a three-phase network is considered. Ia a b c Ia

Unbalanced and nonlinear load

IaΔ

IaY BbcΔ

CkY LkY

BabΔ BcaΔ

Y compensator (filters)

Δ compensator

Fig. 6.62. Simplified electric circuit of equivalent load and compensation-balancing-filtering device.


463

The aim is to achieve a full compensation of the load reactive power and complete cancellation of its distributing effects. These corrections can be made by means of a simultaneously balancing-filtering device. This device consists in a simplified configuration of a branch with three-phase filters (single-tuned wyeconnected filters) and a delta-connected compensator that contains only inductive susceptances (Fig. 6.62). The analytical form of the compensator functions can be inferred using the component parts of the current supplied by the network to the load-compensator system. Thus, it is possible to write the relations: +

−

−

Im(I 1 ) = 0; Re( I 1 ) = 0; Im(I 1 ) = 0; I h (h > 1) = 0

(6.97)

In [6.47], two designing criteria for the compensator elements are proposed (that are in fact criteria for reactive power compensation). The single-tuned filter (here called wye-compensator) and the delta-compensator can fulfil both their special functions of filtering and balancing and also the reactive power compensation: • criterion a – the designing of the filtering elements under the condition of full compensation of reactive power at fundamental frequency, the load balancing function being accomplished by the delta-compensator; • criterion b – the designing of the filtering elements under the condition of full compensation of reactive power injection minimization (at fundamental frequency), this time the delta-compensator having to accomplish both the load balancing and compensation functions, up to unity power factor at fundamental frequency; Star (Y) compensator design a) Assuming that a single harmonic current is filtered, the capacitance and inductance is given by the following relations:

Ch =

h 2 − 1 I1c 1 and Lh = 2 2 ⋅ 2 V1 ω1 h ω1 Ch h

(6.98)

where I1c is the current needed to compensate the imaginary part of the positivesequence current of the load; V1 – rated phase-to-neutral voltage of the network; ω1 – fundamental angular frequency. b) The capacitance resulted from the minimization of the installed reactive power is (equation (6.77)): Ch =

I 1 ⋅ h h Vc ω1

where Vc is the voltage at the capacitor terminals:

(6.99)

464

Basic computation

Vc =

V1 h 2 ; Vc > V1 h2 − 1

(6.100)

The capacitive current on each phase is then given by: I1c = Vc ω1 Ch

(6.101)

Delta (Δ) compensator design The susceptances values of the Δ compensator branches will be computed using the appropriate compensatory currents, BΔ = I Δ V . Depending on the sign of BΔ , the appropriate capacitance and inductance are determined: LΔ = 1 ω 1 BΔ if BΔ > 0 and CΔ = − BΔ ω1 if BΔ < 0 , respectively. The compensation currents are determined by establishing for the Δ compensator phases the following issues: a) a negative-sequence currents system, equal and with the opposite sense of rotation with respect to the negative-sequence currents system of the load (at fundamental frequency); b) full compensation of reactive power at fundamental frequency.

In accordance with (6.97), we can write the expression of compensatory currents at fundamental frequency, taking into account the active and reactive components of the phase currents of the load, given by: 1 a 1 ( I b ,1 − I aa,1 ) + (− I ar,1 − I br,1 + 2 I cr,1 ) 3 3 3 1 a 1 Δ ( I c ,1 − I ba,1 ) + (2 I ar,1 − I br,1 − I cr,1 ) I bc ,1 = 3 3 3 1 a 1 Δ ( I a ,1 − I ca,1 ) + (− I ar,1 + 2 I br,1 − I cr,1 ) I ca ,1 = 3 3 3 Δ I ab ,1 =

(6.102)

Application A numerical application is used to evaluate the quantitative effects of the compensation and to perform a comparison between the two compensation criteria. Consider an equivalent load, connected to a 20 kV network, for which the phase currents at fundamental frequency and at the fifth harmonic are taken into account. Both the currents system at the fundamental frequency and the currents system at the fifth harmonic are considered unbalanced, but without the zero-sequence (absent into the three-phase distribution networks). As known, the three-phase currents system at the fifth harmonic has only negative-sequence, for the positive-sequence they appear only in cases of unbalanced loads, but even in these cases its amplitude is usually lower than the amplitude at the corresponding negative-sequence. A computer program implemented under Borland Pascal has been used for this application. This program includes a procedure that generates the three-phase currents system both for the fundamental and fifth harmonic with adequate features, as mentioned above. Table 6.11 presents two patterns for load currents (both for fundamental and fifth harmonic) and the designed parameters of the compensator’s elements, respectively the


465

currents flow (on phase components and sequence components) inside the loadcompensator assembly, as a result of taking into consideration the two criteria. Table 6.11 Results of the numerical application

S

Y

Δ

E

first pattern Ia1s*= 90.000 - j 32.757 A Ib1s*= 83.599 - j 90.301 A Ic1s*= 36.965 - j 55.980 A

second pattern Ia1s*= 72.800 - j 42.031 A Ib1s*= 84.282 - j 70.043 A Ic1s*= 54.282 - j 65.980 A

I(+)1s= 70.184 - j 59.679 A I(-)1s= 19.815 + j 26.922 A

I(+)1s= 70.454 - j 59.351 A I(-)1s= 2.345 + j 17.320 A

I(+)1s= 92.128 A I(-)1s= 33.428 A

cos ϕ(+)1s= 0.7618 kn(-)1s= 0.3628 s a5 s b5 s c5

I(+)1s= 92.122 A I(-)1s= 17.478 A

cos ϕ(+)1s= 0.7648 kn(-)1s= 0.1897

I *= 9.685 + j 23.275 A I *= 20.000 - j 1.749 A I *= -6.830 + j 1.830 A

Ia5s*= 1.650 + j 22.141 A Ib5s*= 20.000 - j 5.359 A Ic5s*= -12.990 - j 7.500 A

I(+)5s= 7.618 + j 7.785 A I(+)5s= 10.892 A I(-)5s= 2.066 + j 15.490 A I(-)5s= 15.627 A

I(+)5s= 2.886 + j 3.094 A I(+)5s= 4.231 A I(-)5s= -1.236 + j 19.047 A I(-)5s= 19.087 A

γI(+)5s= 0.1182 γI(-)5s= 0.4675 criterion a criterion b CY5= 15.7936 μF CY5= 3.2375 μF LY5= 0.0257 H LY5= 0.1252 H IYa1*= 0.0 + j 59.679 A IYa1*= 0.0 + j 12.233 A IYb1*= 0.0 + j 59.679 A IYb1*= 0.0 + j 12.233 A IYc1*= 0.0 + j 59.679 A IYc1*= 0.0 + j 12.233 A Y I (+)1= 0.0 + j 59.679 A IY(+)1= 0.0 + j 12.233 A IY(-)1= 0.0 - j 0.0 A IY(-)1= 0.0 - j 0.0 A IY(+)1= 59.679 A IY(+)1= 12.233 A IY(-)1= 0.0 A IY(-)1= 0.0 A Iab1= -4.271 A Iab1= -31.664A Ibc1= -31.087 A Ibc1= -58.480 A Ica1= 35.359 A Ica1= 7.965 A Cab= 0.67982 μF Cab= 5.03958 μF Cbc= 4.94774 μF Cbc= 9.30750 μF Lca= 1.80045 H Lca= 7.99191 H IΔa1*=-19.815-j 26.922 A IΔa1*= -19.815+j 20.523 A IΔb1*=-13.408+j 30.621 A IΔb1*= -13.408+j 78.068 A IΔc1*= 33.223 -j 3.699 A IΔc1*= 33.223 + j 43.747 A IΔa1= 33.428 A IΔa1= 28.528 A IΔb1= 33.428 A IΔb1= 79.211 A IΔc1= 33.428 A IΔc1= 54.932 A IΔ(+)1= 0.0 - j 0.0 A IΔ(+)1= 0.0 + j 47.446 A IΔ(-)1= -19.815 - j 26.922 A IΔ(-)1= -19.815 - j 26.922 A IΔ(+)1= 0.0 A IΔ(+)1= 47.446 A IΔ(-)1= 33.428 A IΔ(-)1= 33.428 A Ia1*= 70.184 - j 0.0 A Ia1*= 70.184 - j 0.0 A Ib1*= 70.184 - j 0.0 A Ib1*= 70.184 - j 0.0 A Ic1*= 70.184 - j 0.0 A Ic1*= 70.184 - j 0.0 A I(+)1= 70.184 - j 0.0 A I(+)1= 70.184 - j 0.0 A I(-)1= 0.0 - j 0.0 A I(-)1= 0.0 - j 0.0 A I5= 0.0 A I5= 0.0 A

γI(+)5s= 0.0459 γI(-)5s= 1.0920 criterion a criterion b CY5= 15.7067 μF CY5= 2.8512 μF LY5= 0.0258 H LY5= 0.1421 H IYa1*= 0.0 + j 59.351 A IYa1*= 0.0 + j 10.773 A IYb1*= 0.0 + j 59.351 A IYb1*= 0.0 + j 10.773 A IYc1*= 0.0 + j 59.351 A IYc1*= 0.0 + j 10.773 A Y I (+)1= 0.0 + j 59.351 A IY(+)1= 0.0 + j 10.773 A IY(-)1= 0.0 - j 0.0 A IY(-)1= 0.0 - j 0.0 A IY(+)1= 59.351 A IY(+)1= 10.773 A IY(-)1= 0.0 A IY(-)1= 0.0 A Iab1= -20.391 A Iab1= 7.654 A Ibc1= -20.000 A Ibc1= -48.046 A Ica1= 12.345 A Ica1= -15.701 A Lab= 8.31673 H Cab= 3.24543 μF Cbc= 3.18310 μF Cbc= 7.64681 μF Lca= 5.15677 H Cca= 2.49890 μF IΔa1*= -2.345 - j 17.320 A IΔa1*= -2.345+j 31.257 A IΔb1*= -13.827+j 10.691 A IΔb1*=-13.827+j 59.269 A IΔc1*= 16.172 + j 6.629 A IΔc1*= 16.172+j 55.206 A IΔa1= 17.478 A IΔa1= 31.345 A IΔb1= 17.478 A IΔb1= 60.860 A IΔc1= 17.478 A IΔc1= 57.527 A IΔ(+)1= 0.0 - j 0.0 A IΔ(+)1= 0.0 + j 48.577 A IΔ(-)1= -2.345 - j 17.320 A IΔ(-)1= -2.345 - j 17.320 A IΔ(+)1= 0.0 A IΔ(+)1= 48.577 A IΔ(-)1= 17.478 A IΔ(-)1= 17.478 A Ia1*= 70.454 - j 0.0 A Ia1*= 70.454 - j 0.0 A Ib1*= 70.454 - j 0.0 A Ib1*= 70.454 - j 0.0 A Ic1*= 70.454 - j 0.0 A Ic1*= 70.454 - j 0.0 A I(+)1= 70.454 - j 0.0 A I(+)1= 70.454 - j 0.0 A I(-)1= 0.0 - j 0.0 A I(-)1= 0.0 - j 0.0 A I5= 0.0 A I5= 0.0 A

The abbreviation “S” was used for load, and the abbreviation “E” was used for the load-compensator assembly. Some other quantities are also given for the load: cos ϕ(+)1L is the power factor of the positive sequence at fundamental frequency, k n(−)1L is the

466

Basic computation

coefficient of non-symmetry of the load at fundamental frequency, γ I (+ )5 L and γ I (−)5 L is the currents level for positive- and negative-sequence at fifth harmonic. The calculation has been performed taking into account the following simplifications hypotheses: • the non-symmetrical and non-sinusoidal steady-state concerns the currents only, the voltage system at the load terminals being symmetrical and sinusoidal; • the filters cancel the fifth harmonic; • the filters and the compensators use ideal reactive elements, i.e. all the electrical resistances and corresponding the active power losses were neglected, the filters being considered linear (having no unbalancing effect); • the Δ compensator is designed so as to compensate at fundamental frequency only the positive- and negative-sequence currents, the influence on the current flows at superior harmonics being neglected. The filter (the Y compensator) cancel the fifth harmonic of the current through the load-compensator assembly; in the same time, at fundamental frequency, in order to obtain symmetry, it acts only for the positive-sequence, achieving full compensation of the reactive power (correcting to unity the power factor) for the criterion a, and only partial compensation of the reactive power for the criterion b; Although the compensator consists only of susceptances, the Δ compensator also affects the active power flow in the network. In both cases it symmetrizes the load at the fundamental frequency, its negative-sequence components of the phase currents at fundamental frequency being the same for both criteria, equal in amplitude and with the opposite sense of rotation with respect to the negative-sequence of the phase currents of the load. The difference between the two criteria appears in the positive sequence at fundamental frequency: in the first case the Δ compensator does not have any effects, the reactive power compensation being performed entirely by the filter; in the second case it provides compensation only for the difference between the required reactive power to correct to unity power factor and the reactive power provided by the filter.

Many techniques have been proposed to improve the supply side power factor, to balance the load and to cancel out the harmonics generated by power electronic loads [6.49]. These schemes usually employ single/three-phase voltage source inverters that are supplied from a DC storage capacitor and operate in current control mode to track a specified reference current waveform. The single most important issue in such a scheme is the generation of the reference current waveforms that, when injected into the power systems, cancel out the load harmonics and/or improve the supply power factor. Of the various methods that have been proposed for generating the reference current waveforms, the instantaneous p-q theory has gained considerable attention. This theory is extremely versatile and can be utilized to compensate either the fluctuating or constant part of the load reactive power as well as the fluctuating part of the real power. In [6.49], the theory of instantaneous symmetrical components for generating the instantaneous reference current waveforms to balance a given load is used. It has been observed that the instantaneous power in an unbalanced system contains an oscillating component that rides a DC value. The objective of the compensating system is to supply this zero-mean oscillating power such that the DC component


467

can be supplied by the source. The structure of the compensating system depends on the manner in which the load is connected (Fig. 6.63). vsa

N

~v ~v ~

sb

sc

isa

ila

isb

ilb

isc

ilc

vsa

n

~v ~v ~ sb

sc

ifa

ifb

isa isb

ifab ilab

isc

ifc

a.

ifca ilca ilbc

ifbc

b.

Fig. 6.63. Schematic diagram of the compensation scheme for: a. wye-connected load; b. delta-connected load [6.49].

One of the major advantages of the scheme is that the desired source power factor angle can be explicitly defined. Furthermore, it is easy to implement on-line since the desired compensator currents are directly computed. The drawing also is computationally simple since it does not require complicated transformations.

Chapter references [6.1] [6.2] [6.3] [6.4] [6.5] [6.6] [6.7] [6.8] [6.9] [6.10]

Dugan, R.C., McGranaghan, M.F., Santoso, S., Beaty, H.W. – Electrical power systems quality, 2nd edition, McGraw-Hill, New York, 2003. Grigsby, L.L. – The electric power engineering handbook, CRC Press, 2000. Bhattacharya, K., Bollen, M.H.J., Daalder, J.E. – Operation of restructured power systems, Kluwer Academic Publishers, London, 2001. West, K. – Power quality application guide: Harmonics, True RMS – the only true measurement. Copper Development Association, IEE Endorsed Provider, July 2004. Domijan, A., Heydt, G.T., Meliopolos, A.P.S., Venkata, S.S., Wert, S. – Directions of research on electric power quality, IEEE Trans. on Power Delivery, Vol. 8, No. 1, pp. 429 – 436, January 1993. IEC 61000-4-30 – Electromagnetic compatibility (EMC). Part 4-30: Testing and measurement techniques - Power Quality Measurement Methods, 2000. EN 61000-2-5 – Electromagnetic Compatibility: Environment – Classification of electromagnetic environments, 1995. WG14.31 – Custom Power - State of the art, July 2000. IEEE Standard 446-1987 – IEEE Recommended practice for emergency and standby power systems for industrial and commercial applications (IEEE orange book). IEC 61000-2-8 – Electromagnetic compatibility (EMC). Part 2: Environment. Section 8: Voltage dips and short interruptions on public electric power supply systems with statistical measurement results, 2000.

468 [6.11] [6.12] [6.13] [6.14] [6.15] [6.16] [6.17] [6.18]

[6.19] [6.20] [6.21]

[6.22] [6.23] [6.24]

[6.25]

[6.26] [6.27]

Basic computation *** – Network protection & automation, Guide, Alstom, 2002. Bollen, M.J. – Algorithms for characterizing measured three-phase unbalanced voltage dips, IEEE Transactions on Power Delivery, Vol. 18, No. 3, pp. 937–944, July 2003. Zhang, L., Bollen, M.J. – Characteristic of voltage dips (sags) in power systems, IEEE Trans. on Power Delivery, Vol. 15, No. 2, pp. 827–832, April 2000. EN 50160 – Voltage characteristics of electricity supplied by public distribution system, CENELEC, 1999. Poeată A., Arie A.A., Crişan O., Eremia M., Alexandrescu V., Buta A. – Transportul şi distribuţia energiei electrice (Transmission and distribution of electric energy), Editura Didactică şi Pedagogică, Bucureşti, 1981. Pelissier, R. – Les réseaux d’énergie électrique, Tome I, Dumod, Paris, 1971. IEC 61000-3-6, “Assessment of harmonic emission limits for the connection of distorting installations to MV, HV and EHV power systems (draft)”, May 2005. IEC 61000-4-7, Ed. 2 – Electromagnetic compatibility (EMC); Part 4-7: Testing and measurement techniques – General guide on harmonics and interharmonics measurements and instrumentation, for power supply systems and equipment connected thereto, 2002. Hanzelka, Z., Bień, A. – Power quality application guide: Harmonics, Interharmonics. Copper Development Association, IEE Endorsed Provider, July 2004. IEEE Standard 1459-2000 – Definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced or unbalanced conditions, 2000. IEEE Task Force on Harmonics modelling and simulation – Modelling and simulation of the propagation of harmonics in electric power networks, Part I: Concepts, models and simulation techniques, IEEE Trans. on Power Delivery, Vol. 11, No. 1, pp. 452 – 465, January 1996. Budeanu, C. – Puissance reactives et fictives, Editura IRE, Bucureşti, 1927. Golovanov, Carmen, Albu, Mihaela, et al. – Probleme moderne de măsurare în electroenergetică (Modern measurement problems in power systems), Editura Tehnică, Bucureşti, 2001. Arie, A., Neguş, C., Golovanov, Carmen, Golovanov, N. – Poluarea cu armonici a sistemelor electroenergetice funcţionând în regim permanent simetric (Harmonic pollution of power systems operating under symmetrical steady-state conditions), Editura Academiei Române, Bucureşti, 1994. Ţugulea, A. – Power-flows under non-sinusoidal and non-symmetric periodic and almost periodic steady-states of electrical power systems, 6th Into, Proceedings of IEEE International Conference on Harmonics in Power Systems, Bologna, pp. 388 – 395, 1994. Ţugulea, A. – Criteria for the definition of the electric power quality and its measurement systems, ETEP, Vol. 6, No. 5, pp. 357 – 363, September/October 1996. IEEE Working Group on Non-sinusoidal Situations: Effects on meter performance and definitions of power – Practical definitions for powers in systems with nonsinusoidal waveforms and unbalanced loads: A discussion, IEEE Trans. on Power Delivery, Vol. 11, No. 1, pp. 79 – 87, January 1996.

Electrical power quality [6.28] [6.29]

[6.30] [6.31]

[6.32] [6.33] [6.34] [6.35] [6.36] [6.37] [6.38] [6.39]

[6.40] [6.41] [6.42] [6.43] [6.44]

469

IEEE Task Force on The effects of harmonics on equipment – Effects of harmonics on equipment, IEEE Trans. on Power Delivery, Vol. 8, No. 2, pp. 672– 680, April 1993. Fuchs, E.F., Roesler, D.J., Alashhab, F.S. – Sensitivity of electrical appliances to harmonics and fractional harmonics of the power system’s voltage, Part II: Television sets, induction watt-hour meters and universal machines, IEEE Trans. on Power Delivery, Vol. 2, No. 2, pp. 445 – 453, April 1987. Robert, A., Deflandre, T. – Groupe de Travail CIGRE/CIRED CCO2: Guide pour l’évaluation de l’impédance harmonique du réseau, ELECTRA, No. 167, pp. 96 – 131, Août 1996. Albert, Hermina et al. – Probleme privind măsurarea şi facturarea energiei active şi reactive cu ajutorul actualelor sisteme de măsurare (Problems concerning the measurement and invoice of active and reactive energy by means of modern measurement systems), Proceedings of CEE, Târgovişte, România, pp. 98–105, 2003. Arrillaga, J., Arnold, C.P. – Computer analysis of power systems, John Wiley & Sons, 1990. Capasso, A. et al. – Rotating load modelling for steady-state harmonic analysis, Proceedings on the 7th ICHQP, Las Vegas, pp. 400 – 405, October 16 – 18, 1996. Barret, J.P., Bornard, P., Meyer, B. – Power system simulation, Chapman & Hall, London, 1997. Xu, W., Marti, J., Dommel, H. – A multiphase harmonic load flow solution technique, IEEE Trans. on Power Systems, Vol. 6, No.1, pp. 174 – 182, February 1991. Xia, D., Heydt, G.T. – Harmonic flow studies, Part I: Formulation and solution, Part II – Implementation on practical application, IEEE Trans. on Power Systems, Vol. 101, pp. 1275 – 1270, June 1982. Kitehin, R.H. – Convector harmonics in power systems using variable analysis, IEC Proc., Part C, Vol. 128, No. 4, pp. 567 – 572, July 1981. Martinon, J., Fauquemberque, P., Lachaume, J. – A state variable approach to disturbances in distribution networks, Proceedings on the 7th ICMQP, pp. 293 – 298, Las Vegas, 1996. Buta, A., Pană, A., Ticula, E. – Stabilirea frecvenţelor de rezonanţă armonică în reţelele de distribuţie folosind metoda variabilelor de stare. (Establishment of harmonic resonance frequencies in the distribution networks using the state variable analysis method), Energetica Revue, Vol. 51, No. 1, pp. 14 – 18, 2003. IEC 61000-3-2 – Electromagnetic compatibility (EMC): Limits for harmonic current emissions (equipment input current up to and including 16 A per phase), 2000. Gonzalez, A.D., McCall, J.C. – Design of filters to reduce harmonic distortion in industrial power systems, IEEE Trans. on Industry Applications, Vol. 11–23, pp. 504 – 511, No. 3, May/June 1987. Moran, L.T., Joos, G. – Principles of active power filters, IEEE-IAS’98, Tutorial course, St. Louis, Missouri, 1998. *** – Guide to quality of electrical supply for industrial installations. Part 4: Voltage unbalance, WG2 “Power Quality”, International Union for Electroheat, Paris, 1998. Buta, A., Pană, A., Milea, L. – Calitatea energiei electrice, Editura AGIR, Bucureşti, 2001.

470 [6.45] [6.46] [6.47]

[6.48] [6.49]

Basic computation Lee, S.Y., Wu, C.J. – On-line reactive power compensation schemes for unbalanced three phase four wire distribution feeders, IEEE Trans. on Power Delivery, Vol. 8, No. 4, pp. 1958 – 1965, October 1993. Gyugyi, L., Otto, R.A., Putman, T.H. – Principles and applications of static thyristor-controlled shunt compensators, IEEE Trans. on Power Apparatus and Systems, Vol. 97, pp. 1935 – 1945, 1978. Pană, A., Buta, A., Ticula, E. – Criteria for reactive power compensation in power distribution networks with unbalanced and nonlinear loads, Proceedings of the 3rd International Power Systems Conference, Timisoara, pp. 96 – 101, November 6-7, 1999. Buta, A., Pană, A. – Criterii de compensare a puterii reactive în reţele cu sarcini dezechilibrate (Reactive power compensation criteria in networks with unbalanced loads), Energetica Revue, Vol. 45, No. 5 – 6, pp. 273 – 289, 1997. Ghosh, A., Joshi, A. – A new approach to load balancing and power factor correction in power distribution system, IEEE Trans. on Power Delivery, Vol. 15, No. 1, pp. 417 – 422, January 2000.

Chapter 7 POWER AND ENERGY LOSSES IN ELECTRIC NETWORKS 7.1. Introduction 7.1.1. Background As with any physical process, electric power transmission and distribution requires energy consumption associated with irreversible thermodynamic conversions. This consumption, referred to as “losses in electric networks” is referenced as such in the technical literature and international statistics. In the supplying process of consumers with energy, losses occur during generation, transmission and distribution stages. Generally, it is considered that, an average of 8% from the energy produced by the all sources is lost during generation stage and 10% during transmission and distribution. Figure 7.1 shows the power balance of a developed power system. Energy generated in power plants

Internal consumption in power plants ~ 8% Losses in the electric networks ~ 10%

Total sources 100% Energy delivered to the consumers ~ 82 %

Energy imported

Energy exported

Domestic energy consumers Fig. 7.1. Electric energy balance in a complex power system.

472

Basic computation

In establishing the tariffs for the power transmission and distribution, the cost of the losses in the electric networks represents a significant component and, in a market economy, it is a weighty element in the competition between companies. Taking into consideration that these “network losses” range in various countries between 8 to 15% of the consumed electric energy, the energy saving is one of the main energy sources, and special concerns has been dedicated, in all the power systems, to their reducing as much as possible. Studies performed emphasized that most often, the losses reduction is more economicaly than the corresponding increase of the generating capacities [7.1]. A decisive element in changing the losses outlook was the evolution of the oil cost compared to the copper and aluminium cost, which are components of the lines and transformers. Between 1970 and 1994, the cost of oil increased more than 3 times, while the copper and aluminium costs practically kept unchanged or even had decreased. According to the actual forecast [7.1, 7.2], this situation is likely to be unchanged all through 2010. Statistically, the energy losses in the electric networks result from the difference between the energy injected into the networks by the power plants, including the energy imported from neighbouring systems, and the energy sold to consumers including the exported energy. These losses include three components: − Own technologic consumption (o.t.c.) associated with the power transmission and distribution processes in compliance with the installation design requirements; − Technical losses due to deviations from the designed operation state, either through incomplete development of the installations or through improper operation; − Commercial losses (positive or negative) resulting from errors introduced by the inaccuracy of the metering units and unorganised electric energy records, including unaccounted consumption of the metering transformers and meters, as well as the electric power theft. Therefore, the reduction studies of what we improperly call “losses in the electric networks”, require three distinct analyses: (i) Optimization of the transmission and distribution process in the designing stage and setting the theoretical technologic consumption for various operation states of the installations; (ii) Elimination of the technical losses in the networks by framing with the optimum operation state of the installations, by observing the investment program and through the optimum operation of the installations; (iii) Improve the electric energy record within the administrative organization of the enterprises so that the influence of some calculation or measurement errors on the values reported for the network losses to be minimal. Each of the three direction studies should aim the achievement of a maximum gain on the whole system, while satisfying the safety conditions of supply required

Power and energy losses in electric networks

473

by all consumers. This desideratum, for a given pattern of the sources and consumers, leads to the concerns of reducing the power delivery costs by reducing the network power losses. Physically, the technologic consumption of active power (energy) in system’s networks is the sum of the technologic consumptions located in: − line conductors and the windings of the transformers or autotransformers through Joule effect. The losses by thermal effect (Joule) are due to the current passing through the conductors and also to the active and reactive powers, and can be reduced, in a given case, by increasing the cross sectional area of the wire or by reducing the amperage (through voltage increase, reactive power compensation, etc.). In appropriate sized electric networks, loaded at their rated capacities, these losses represent the main rate; − the magnetic core of the transformers or autotransformers, due to the magnetic field presence, through eddy currents (flux) and hysteresis phenomenon. The power losses due to the transformers (autotransformers) magnetization do not depend upon the load. Their weight can be decreased by optimal sizing and utilization, as well as avoidance of useless operation. − lines of 220 kV and higher rated voltage, due to the presence of the electric field through corona phenomenon; − the dielectric of the high voltage insulation, mainly in polluted areas or during fog time. Of the total amount of power losses, in appropriate sized networks, the loadfree losses due to corona phenomenon, losses through dielectric and in improper insulations have a low weight. The meters consumption, in terms of their quality and number, can increase the losses into the networks by 0.5…3% (for instance in Germany, the consumption of the 25,000,000 installed meters represents 2% of all the losses in the networks [7.3]). Losses determination for an electric network, based on measurements, represents both technically and economically, a difficult issue. Given the low percentage of the losses in various elements of network and the accuracy of the metering units mounted in installations, the determination, by measuring per element of network losses is obviously not feasible. Therefore: ● In high and very high voltage networks, the losses determination is achievable by comparing the injected and the withdrawal energy from a contour, since the number of the input-output points is relatively small and a simultaneous reading of the meters is feasible mostly under the actual conditions of the market economy when the transmission network dispose of a metering systems. Also, given the reduced number of elements, it is possible to perform post calculation in terms of the elements loading and even in real time, thanks to an appropriate computerized system existence; ● In medium and low voltage networks this comparison can only be done in special situations, since in many countries it is not possible an accurate

474

Basic computation

determination of the energy sold for a given period, because, generally, the reading of all the meters takes more than a week. In the medium voltage networks it is possible to perform a post calculation of the technologic losses on those elements whose loading is monitored (recorded) operatively or even in real time if an appropriate computerized system exists; ● For low voltage networks, the post calculation method is not widely used because it is rational only for the study of smaller parts of the network, the manpower and costs involved for metering being less than for the entire low voltage network analysis. The obtained results can be used as guiding values for the networks with comparable configurations and consumption. The creation of the electricity market has imposed the development of a simultaneous centralized reading system of the meters of all consumers within a company, making possible a more accurate ascertainment of the losses and their costs, respectively, for the power distribution. Taking into consideration the above-mentioned aspects, it can be said that the amount of the power losses is an indicator that characterizes the operation of a power system. The values of this indicator are determined even in the early planning, designing and sizing of equipments stage, when the level of the justified technologic consumption is established. This leads to an optimum of the whole power system – subjected to the actual market economy framework for the transmission network and each electricity company – but not being an achievable minimum. During operation, the optimum level of the technologic losses in the electric networks, for a certain network topology and generation scenario, can be reached by optimal reconfiguration of the network according to the real conditions or correct voltage adjustment in terms of the load behaviour and weather conditions (for the reduction of the losses caused by corona phenomenon). As a result of the powerful hardware and software from the operative control centres, of the automatic voltage control, the supervising of the network loading, local reactive power generation etc., operating regimes near to optimum can be obtained. Also, with appropriate software, the technical losses achieved in the chosen configuration can be determined in real time.

7.1.2. Evolution and structure of the losses in the Romanian electric networks The total amount of losses in the electric networks is directly influenced by the distance between sources and consumption area, as well as the structure and characteristic parameters of the network that connects them. Table 7.1 shows the losses evolution, in percentage of the aggregate energy transmitted (sum of the energy injected into the networks by the power plants plus the imported electricity). This evolution should be correlated with the evolution of the maximum power consumption, which, during 1970–1989, increased from 5245 MW to 11270 MW, while between 1989–2000 decreased to 8161 MW [7.10].


475 Table 7.1

Evolution of power losses in the Romanian electric networks (percentage of the aggregate energy transmitted) Total NPS Transmission networks Distribution networks

1982 6.69

1985 6.70

1989 7.73

1990 9.00

1995 11.98

2000 13.93

2001 13.81

2.82

2.84

3.55

3.74

2.43*

2.08*

2.12*

3.87

3.86

4.18

5.26

9.55**

11.85**

11.69**

*without the 110 kV network, ** with the 110 kV network

The evolution after 1989 is characterized by two significant factors: − at once with the consumption decrease, the operation of a transmission installation, dimensioned for a transit much higher than the achieved one, has lead to a remarkable decrease of the efficiency (the no-load losses weight prevailing in the aggregate amount of losses); − alteration of the consumption pattern, which shifted from 110 kV and 220 kV (large industrial enterprises) to MV and LV. Table 7.2 Electric power balance. Year 1982 1989 1990 1994 1995 2000 2001

Energy generated in the network [%] 100 100 100 100 100 100 100

Energy delivered to the consumers [%] 93.31 92.27 91.01 89.96 88.02 86.07 86.18

Loss in the networks [%] 6.69 7.73 8.99 10.04 11.98 13.93 13.82

From the analysis of the electric power balance before 1989 and after 1990 the following conclusions can be drawn: − the change of the generation pattern by the increase of the energy weight delivered at 400 kV voltage (from ≈ 9.50% to 15.25); − the electricity import reduction from 2.3% in 1989 up to 1.5% in 1994; − the change of electricity consumption structure through export variation, the consumption decrease at 220 kV (mainly by reducing and rationalizing the consumption of the aluminium factories), the variation at 110 kV correlated with the activity of large industrial enterprises (the increase during 1976–1989 and tremendous decrease after 1989) and the increase of the importance of the MV consumption (mainly after 1989 through the development of the small and average enterprises) and LV.

476

Basic computation

The above mentioned factors have various effects on the losses in the electric networks: the increase of sources share at 400 kV leads to losses increase, while the increase of sources at 110 kV leads to losses decrease, the consumption reduction at 220 kV and its increase al MV and LV having the same effect, the losses increase.

7.1.3. Comparison between losses in the Romanian electric networks and other countries Generally, a comparison of the losses in the electric networks of various power systems is difficult, because: − the network losses depend on the power system structure (the distance of sources to the consumers, constructional type of the electric lines and equipment, static load characteristics, etc.); − the economic losses level justified by the sizing depends on the power policy of each system, on specific circumstance factors (such as, availability of primary resources, possibility to assimilate new equipments, etc.); − the possibility of real optimization of the operation state varies from one system to another according to the existing reserves in sources and networks, the level of technical endowment used in the operative management of power installations (power and voltage controls automation, centralized organization of the surveillance through process computers) and also the structure in terms of ownership form; − the organization of energy record and accuracy of input data as well as calculation method are different from one power system to another. Table 7.3 shows the variation of losses in the electric networks in some power systems, in terms of the percentage of total transmitted energy (net generated energy plus import), according the data presented in [7.6, 7.8, 7.9]. It can be noticed a low level of the losses in Romania until 1989. The power systems with lower amounts of losses than in the Romanian Power Grid are either systems developed on small geographic areas and implicitly with short networks (e.g. Belgium, The Netherlands), or systems with different generation patern, with a very large share of the autoproducers in the total energy production (e.g. Germany). The situation for Romania is quite different in 1993, the losses level being – due to the causes dealt earlier – among the highest. The data in Table 7.3 reveal that: − losses amount varies between countries within wide limits from 5% to 16%; − in some systems the losses level is practically around the same value, determined by the power characteristic of the respective system (such as Belgium, France, Germany, England); − during 1989–1993, in all power systems of the former socialist countries, the losses in the electric networks increased very much (Bulgaria, Czechoslovakia, Hungary, Poland, Romania);


477

− there exists no clear continuous tendency towards increase or decrease of losses amount and networks efficiency, the variations being accountable through the development of the networks which are performed differently on stages (with larger or smaller capacity reserves), and the development and policy of resources utilization. Table 7.3 Variation of the losses in electric networks of some power systems (% of the transmitted energy). Austria Belgium Bulgaria Czechoslovakia Denmark France Germany Greece Hungary Italy Romania Turkey England Swiss Sweden Poland

1980 7.14 5.71 10.04 7.85 8.19 6.93 4.70 7.29 9.21 9.05 6.53 12.16 8.08 7.99 9.04 10.78

1989 6.31 5.32 10.49 6.94 6.77 8.80 4.35 7.84 10.83 7.45 7.73 13.58 7.92 7.19 8.68 9.73

1990 6.13 5.52 10.75 6.83 5.13 7.51 4.58 8.72 10.85 6.85 9.00 12.48 7.81 7.09 6.60 8.40

1991 6.24 5.30 14.49 6.74 4.35 7.61 4.71 8.61 10.79 7.06 10.31 13.37 8.01 7.04 6.64 11.40

1992 5.96 5.21 14.45 7.40 6.59 7.07 4.11 7.65 9.19 6.72 10.88 14.28 8 .55 7.08 6.67 12.63

1993 9.08 6.77 13.27 9.66 6.23 8.40 5.37 8.41 13.34 8.72 10.76 13.90 7.90 9.35 7.99 16.19

It is worth mentioning that UNIPEDE [7.2] considers technically correct a network efficiency having 0.98 for U nom ≥ 100 kV (transmission) respectively 0.92 for U nom < 100 kV (distribution). Within the distribution networks the economical level is 3…5% (efficiency 0.95…0.97), out of which 10% in sub-stations HV/MV, 55% in the medium voltage networks and 35%, in the stations and low voltage network. The economical level of the losses in the transmission networks is 2…3% of total sources (efficiency 0.97…0.98).

7.2. Own technologic power consumption In order to evaluate the losses associated with the operation of various components of an electric network (lines, transformers) it is necessary to know their technical characteristics and to establish a representation model. In this regard, the following technical data are required:

478

Basic computation

− for electric lines: rated voltage, network type (overhead, cable, pole type, cross sectional area of the active and protection conductors, cable characteristics etc.) − for transformers: rated voltages, rated power, unit type (transformer with single or several windings, autotransformer), transformation ratio, magnetization current, no-load losses, load losses at rated power, short circuit voltage; − for reactors: rated voltage, rated current, rated power, losses of active power through coil; − for capacitors: rated voltage, rated power, losses through dielectric or dissipation factor (loss factor). The information obtained help us to establish the parameters associated to the model that, for various operation states, will provide the no-load and load losses. Further on, besides the theoretical approaches in chapters 2 and 3, aspects related to the calculation of losses on the “long” transmission lines are detailed. Thus, the power losses can be determined: i) as the difference between the powers at the line’s ends: *

*

*

*

ΔP = Re[U 1 I 1 − U 2 I 2 ] ΔQ = Im[U 1 I 1 − U 2 I 2 ]

(7.1)

ii) using the value of the mean-square current on the line: l

I2 =

1 2 Ix d x l0

∫

where I x can be expressed as per parameters from one of the line’s end: I x = I 2 cosh γx +

U2 sinh γ x Zc

I x = I 1 cosh γx −

U1 sinh γx Zc

or

where γ is the propagation constant and Z c is the characteristic impedance. Thus, the following expressions are obtained [7.15]: ⎡ P2 + Q2 ⎛ sinh2αl sin 2βl ⎞ U22 ⎛ sinh2αl sin 2βl ⎞ ⎟+ ⎜ ⎟+ ΔP(2) = r0 ⎢ 2 2 2 ⎜⎜ + − 2β ⎟⎠ 2 Zc2 ⎜⎝ 2α 2β ⎟⎠ ⎣ 2U2 ⎝ 2α

+

P2 cosh 2αl − 1 Q2 cosh 2βl − 1⎤ + ⎥ Zc Zc 2α 2β ⎦

(7.2,a)


479

⎡ P2 + Q2 ⎛ sinh 2αl sin 2βl ⎞ U 22 ⎛ sinh 2αl sin 2βl ⎞ ⎟+ ⎜ ⎟+ ΔQ(2) = x0 ⎢ 2 2 2 ⎜⎜ + − 2β ⎟⎠ 2 Zc2 ⎜⎝ 2α 2β ⎟⎠ ⎣ 2U 2 ⎝ 2α +

P2 cosh 2αl − 1 Q2 cosh 2β l − 1 ⎤ + ⎥ Zc Zc 2α 2β ⎦

(7.2,b)

⎡ P 2 + Q 2 ⎛ sinh 2αl sin 2βl ⎞ U12 ⎛ sinh 2αl sin 2βl ⎞ ⎟+ ⎜ ⎟− ΔP(1) = r0 ⎢ 1 2 1 ⎜⎜ + − 2β ⎟⎠ 2 Z c2 ⎜⎝ 2α 2β ⎟⎠ ⎣ 2 U 1 ⎝ 2α −

P1 cosh 2αl − 1 Q1 cosh 2βl − 1⎤ − ⎥ Zc Zc 2α 2β ⎦

(7.2,c)

⎡ P2 + Q2 ⎛ sinh 2αl sin 2βl ⎞ U12 ⎛ sinh 2αl sin 2βl ⎞ ⎟+ ⎜ ⎟− ΔQ(1) = x0 ⎢ 1 2 1 ⎜⎜ + − 2β ⎟⎠ 2 Zc2 ⎜⎝ 2α 2β ⎟⎠ ⎣ 2U1 ⎝ 2α −

P1 cosh 2αl − 1 Q1 cosh 2βl − 1⎤ − ⎥ Zc Zc 2α 2β ⎦

(7.2,d)

where ΔP(1) , ΔQ(1) respectively ΔP( 2 ) , ΔQ( 2 ) are the active and reactive power losses, respectively, calculated according to the power at the sending and receiving end , respectively. It is worth mentioning that ΔP( 2 ) ≠ ΔP(1) , ΔP(1) < ΔP , ΔP( 2 ) > ΔP(1) and ΔP( 2) + ΔP(1)

= ΔP . 2 Figure 7.2 shows the percentage error between power losses calculated with relations (7.2) and the one determined with relation (7.1), for various loads of a line of 750 kV, 400 km, 5×400 mm2, A1-Ol, where λ stands for the power factor. The determination of the power losses in the networks, dependent on both the network configuration and consumers’ state and the loading of various sources (power plants and reactive power sources), is carried out through a steady state calculation. In this respect, it should be mentioned that: − it is very important to model no-load losses for all types of transformers and mainly of the distribution ones (110 kV/MV and MV/LV); − it is required the modelling of 220 kV networks in the calculation of the operating states and more of the losses caused by corona phenomenon. For a certain type of overhead line (OHL), losses variation due to corona phenomenon in terms of the operation voltage can be represented by a polynomial, whose coefficients are determined by regression. In Romania,

480

Basic computation

for the existing lines, with cross sectional area of 2×450 mm2 and 3×300 mm2, respectively, the relation:

ΔPcorona = a + bU 2

(7.3)

proved to be suitable (correlation factor 0.998). In order to consider the corona discharge losses, modelled according to (7.3), in computer programs for steady state calculation, at the line ends, a lumped load (a/2×L) and a conductance (b/2×L) will be introduced. It is important to relive that the introduction of these parameters has a little influence in the voltage value (< 0.05%) determined by load flow calculation, but the voltage value may have a big influence on the value of the corona losses. [%] 4 ΔP2-ΔP ΔP 3

ΔP1-ΔP ΔP

2

λ=1 λ=0.9 λ=0.8 λ=0.9 λ=0.7

1

0

-1

1000

2000

leading state

3000 λ=0.7 4000 λ=0.9 λ=0.8 λ=0.9 λ=1

P2[MW]

leading state

-2

-3

Fig. 7.2. Percentage errors between power losses determined by means of relations (7.1) and (7.2) for a line of 750kV, 400km, 5×400mm2 Al-Ol.


481

The relation (7.3) allows us to obtaine guiding average values, taking into consideration the actual values for each line (configuration, route, average multiannual weather conditions). In steady-state calculation, for the optimization of the voltage level and reactive power compensation, the modelling of the static characteristic of the load is also required. Notice that the simple switching of the taps, disregarding the load characteristics, can lead to other effects than the ones intended. Thus, switching the taps towards the direction of the voltage increase on the consumption bus-bar can result in increased consumption of reactive power that may determine the voltage drop at bus-bars, supply and consumption.

7.3. Own electric energy technologic consumption 7.3.1. Basic notions and data The calculation of the own technologic consumption with high accuracy, carried out for the analysis of steady states of complex networks, allows the determination of o.t.c. (active and reactive) both on the entire network and its elements. Further on, equal loads of the three phases in every instant are assumed. Taking into consideration that the determination of the own electric energy technologic consumption by using an integral in time of the current square flowing through the element: T

∫

ΔW = 3R I t2 dt

(7.4)

0

constitutes a very difficult issue, as in practice the accurate knowledge of It for the entire period analyzed is impossible; also, in many cases even the accurate knowledge of the value R is difficult, R being line’s electric resistance. This issue necessitated the use of simplified ways for determination of the own technologic consumption. To this effect, two distinct calculation cases are considered. a. Own technologic consumption practically load independent corresponds to the product between the own technologic consumption of power and the duration the respective element is powered (obviously, disregarding the voltage variations influence, either by considering the rated voltage or a medium operation voltage); b. Own technologic consumption dependent of the load that appears at: − series elements that supply a radial consumer, when the operation state of the element is determined by the consumption characteristics and its annual variation; − basic network elements of the power system, whose state is influenced by a large number of factors (network structure, consumption level and

482

Basic computation

repartition on the system nodes, generation dispatching etc.) and therefore the state of an element cannot be characterized independently by all the other elements of the network in which it is integrated. The o.t.c. determination for the basic networks of the system, of the meshed network with variable states, is carried out on the basis of state calculations for characteristic operation levels (generally, four levels for one working day in winter and summer and two levels for holiday in winter and summer) and assignment of a number of hours for the carrying out of each one. For each element of an electric network, the calculation of the o.t.c. is performed considering its two components, the one dependent on the load and the one independent of load. • For a three-phase electric line, of length L, considered “long”, that is with distributed parameters, the energy losses in an interval T are: LT

ΔW = 3R0

∫∫ I

2 lt dldt

(7.5)

0 0

where I lt is the current at distance l and instant t, while R0 is the resistance per length unit of the line. The overall energy loss is made up of no-load losses (losses by corona phenomenon and insulation) and load losses. Neglecting the losses through insulation, the relation can be written as: LT

ΔW = 3R0

∫ ∫ (I

' 2 lt ) dldt

+ ΔWcorona

(7.6)

0 0

where ΔWcorona are the energy losses caused by corona phenomenon, established by considering weather condition, I lt' the current through the line, minus the shunt currents through line conductance. For the relatively short lines, in the equations of the long line the hyperbolic sine can be replaced with the circular one, and the cosine with 1, and the relation (7.6) becomes: T

∫

ΔW = 3R I t2dt + ΔWcorona

(7.7)

0

where I t is the constant current along the entire line at instant t. • For a power transformer, knowing the no-load losses ( ΔP0 ) and load power losses at rated load ( ΔPscc ), the o.t.c. will be (assuming that the voltages at the terminals are maintained constant): T

2

⎛ S ⎞ ΔW = ΔPscc ⎜⎜ t ⎟⎟ dt + ΔW0 S 0 ⎝ nom ⎠

∫

(7.8,a)


483

or: 2

T

⎛ I ⎞ ΔW = ΔPscc ⎜⎜ t ⎟⎟ dt + ΔW0 I 0 ⎝ nom ⎠

∫

(7.8,b)

where ΔW0 = ΔP0T is the electric power own technologic consumption (e.p.o.t.c.) independent of load (dependent only on the applied voltage); S t ( I t ) – transformer loading at instant t, that determines the power losses in transformer; • For a synchronous compensator having the rated power Qnom , a technologic consumption of power at rated loading ΔPcs,nom out of which K P ΔPcs,nom represents the technologic consumption independent of the load, resulting: T

ΔWc.s

⎛ Q = K P ΔPcs,nomT + (1 − K P )ΔPcs,nom ⎜⎜ Q 0 ⎝ cs,nom

∫

2

⎞ ⎟ dt ⎟ ⎠

(7.9)

• For the installations with series capacitors (for the series compensation) T

ΔWc.c

⎛ I = Δpc Qnom,c ⎜⎜ I 0 ⎝ nom,c

∫

2

⎞ ⎟ dt ⎟ ⎠

(7.10)

where Δpc is the o.t.c. of power at rated load. • For shunt reactors and capacitor banks whose technologic consumption is not dependent of the load, the e.p.o.t.c. is obtained as follows (the voltage at terminals is considered to be kept unchanged):

ΔWb = ΔpbQnom,bT ΔWc = ΔpcQnom,cT

(7.11)

where Δpb and Δpc are the o.t.c. of power of the coil, respectively capacitor. In order to determine the own technologic consumption of energy by means of relations (7.5) and (7.6) it is required to know the law of time variation of the current flowing through the element. In the general case, this law cannot be mathematically expressed. That is why, for the consideration of current variation in time, various assumptions are taken into consideration and various calculation methods have been elaborated for the calculation of the integral value.

7.3.2. Diagram integration method It is considered that the time variation of the root-mean-square value of the current passing through the element is known, and that it is represented in a diagram (Fig.7.3). The time period T is divided into n equal Δt periods.

484

Basic computation

I, S

0 2 4 6 8 10 12 14 16 18 20 22 24

t [h ]

Fig. 7.3. A daily load curve.

The areas between two neighbouring y-coordinates can either be considered as rectangles or trapezes. For the loads indicated in the points of intersection with the load curve the following can be written: − approximating with rectangles: T

∫I

2 t dt

=

n

∑I

2 t Δt

T n

=

t =1

0

n

∑I

(7.12)

2 t

t =1

− approximating with trapezes: T

∫ 0

I t2 dt =

n −1 ⎞ T ⎛ 2 ⎜ I 0 + I n2 + 2 I t2 ⎟ ⎜ ⎟ 2n ⎝ t =1 ⎠

∑

(7.13)

For I 0 = I n , the relation (7.13) is reduced to (7.12). Consequently, the energy losses will be: − for loads expressed in amperes:

ΔW = 3 R

T n

n

∑I

2 t

(7.14)

t =1

or

ΔW = 1.5 R

n −1 ⎤ T⎡ 2 2 I I 2 I t2 ⎥ + + ⎢ 0 n n⎣ t =1 ⎦

∑

(7.15)

− for loads expressed in powers T ΔW = R n

n

∑ t =1

St U t*

2

(7.16)

or 2 2 n −1 T ⎡ S0 Sn St ⎢ ΔW = R + + 2 * * * n ⎢ U0 Un t =1 U t ⎣

∑

2

⎤ ⎥ ⎥ ⎦

(7.17)


485

Even if the graphical integration method leads to an enhanced accuracy, it has the disadvantage of a large amount of work. Generally, in practice, to simplify the calculations, load curves characteristic for working days and holydays, winter, summer, spring and autumn are used instead of the load variation curve for the entire year. Sometimes, the calculation is limited only to the winter and summer working days. For these days, their number is established within the analyzed T period. Energy losses, determined by means of relations (7.14…7.17) for a characteristic day, allow the determination of the energy lost within one year, by means of the formula:

ΔW = ΔWwin , w nwin , w + ΔWsum , w nsum, w + ΔWspr , w nspr , w + + ΔWaut , w naut , w + ΔWwin , h nwin , h + ΔWsum, h nsum, h +

(7.18)

+ ΔWspr , h nspr , h + ΔWaut , h naut , h where win, sum, spr, aut represent winter, summer, spring, autumn, respectively, w and h representing the working day and holiday, respectively. The determination of the energy losses can be further simplified, within the characteristic day, not hour by hour, but on consumption levels (generally: night off peak, morning peak, day off peak, evening peak), establishing the duration, in hours, of each level. This deficiency of this method relies in its failure to render the required accuracy due to the regular and irregular deviations that occur in the analysed network, which makes the diagram, states and energy losses determined for one day not to be kept unchanged for the entire characteristic period. By regular deviations it understands the dynamics of the electricity consumption, determined by electricity consumption variation, weather conditions influence, load actual growth etc. Irregular deviations mean disconnection of some elements in the network for planned repairs, change of the producer’s operation state. In order to reduce the errors when using the diagram integration method, to an admissible level, the engineering practice, usually, introduces states correction factors: equivalence factor k Π from the energy losses and irregularity factor k i point of view. In the general case when the voltage is constant, load losses in the electric networks, within a time period t, are: t

∫

ΔW SΠ = S t2 dt

(7.19,a)

0

ΔW SΠ =

n

m

∑∑ S i =1 j =1

2 ij t ij

(7.19,b)

486

Basic computation

where i is the number of the day in the characteristic period, i ∈ (1, n ) ; j − number of the level in the load curve; i, j ∈ (1, m ) ; S ij − load on the level j in the day i; Π − calculation period in the studied period T, Π ∈ (1, N ) .

In relations (7.19) it is assumed that the voltage of the line is constant. If the analysis is performed in a period Π only one day, we have:

ΔWS' Π = nΠ

m

∑S

2 j tj

(7.20)

j =1

In order to obtain the equivalence between (7.19) and (7.20) it is required: m

kΠ nΠ

∑

S 2j t j =

j =1

n

m

∑∑ S

2 ij tij

i =1 j =1

From the last equality, it is possible to obtain the equivalence factor of the operating states during the period Π : n

kΠ =

m

∑∑ S i =1 j =1 m

nΠ

∑

2 ij tij

(7.21) S 2j t j

j =1

The energy losses in the networks for the period T will be: Δ W ST = k i

N

∑k 1

' Π Δ W SΠ

(7.22)

The graphical integration method can be applied relatively easily with the aid of computers. It can be considered a standard method for the elimination of errors introduced by other methods. It is a correct method for the complex meshed networks as well.

7.3.3. Root-mean-square current method It is supposed that through a network element flows a constant current of magnitude I, which, in a period T, can produce on the line the same energy losses as in the case of flowing in the given time period of the alternating current corresponding to the real load curve: T 2

∫

3 RI T = 3 R I t2dt 0

from which:


487 T

I=

∫I

2 t dt

0

(7.23)

T

From the expression (7.22) it results that, for the calculation of the rootmean-square current it is necessary to know the real load curve. The expression under the radical can be calculated by means of relations (7.12) and (7.13). If the value of I is known, e.p.o.t.c. in the element is determined by means of the formula: ΔW = 3 RI 2T

(7.24)

or when the loads are given through powers: ΔW = RT

Sm, p

2

U nom

where S m , p is the root-mean-square of the power that flows through the element; U nom − the rated voltage, considered constant; R – the element resistance; T – the calculation period. This calculation method is called the method of the root-mean-square current, and, when the relations (7.12) and (7.13) are used for his determination, this method can be considered as another way of applying the diagram integration method. In this form of application it provides no advantage with respect to the diagram integration method. In practice, the method of the root-mean-square current is appropriate for distribution networks, mainly MV. In this case, the root-mean-square current I is determined by means of empirical relations, in terms of the mean value of the current I mean and the quadratic dependence factor k p : I = I mean k p

(7.25)

or in terms of the maximum current I max and the number of hours of the maximum power utilization, e.g. through a dependence of the form:

(

I = I max 0.12 + Tmax ⋅10 −4

)

(7.26)

or the calculation is performed in terms of the maximum current I max and the losses time τ : I = I max

Tmax τ

(7.27)

The values of Tmax , k and τ have a probabilistic character and can be evaluated on the basis of the samples and measurements performed only for certain

488

Basic computation

types of consumers or for radial supplying lines of groups of consumers, electric lines operating at nominal voltage of 110 kV or less, where the capacitive current can be ignored. For this reason, the method of the root-mean-square current must be considered a statistical-probabilistic determination of the energy losses in the radial networks with rated voltage 110 kV or less. This method cannot be applied in the simple or complex meshed networks, since in this case there is no tight correlation between Tmax , k p , τ and the quantities that determinate them.

7.3.4. Losses time method One considers the annual classified curve of active powers flowing through the element, obtained from the daily curves. The area under the curve Pt represents, at a certain scale, the energy W transmitted through the element within a period of time T. The same quantity of energy could be transmitted at constant power Pmax , within a period of time Tmax < T : T

∫

W = Pt dt = PmaxTmax P

(7.28)

0

from which: T

∫ P dt t

Tmax P =

0

(7.29)

Pmax

In Figure 7.4,a the expressions given above are represented, the area of the rectangle determined by Pmax and Tmax P being equal to the area determined by axes and the curve Pt ; Tmax P is called the number of hours of the maximum active power utilization. Obviously, such times, representing a period conventionally adopted, when on the line operating at maximum loading an energy equal to the actual one is transmitted, can be also determined for the reactive power (Fig. 7.4,b): T

∫ Q dt t

Tmax Q =

0

(7.30)

Pmax

and for the current (and for the apparent power, respectively), (Fig. 7.4,c): T

Tmax =

∫

T

I t dt

0

I max

∫ S dt t

≈

0

S max

(7.31)


489

It is worth noticing that, because the curve S (t ) is fuller than that of P(t ) , Tmax P < Tmax . This observation should be taken into consideration at the application of various formulae for the losses time calculation. The determination of the energy losses per element requires, naturally, losses calculation for each curve point I t = f (t ) and then their summation. The same energy losses occurs in the element considered with a constant loading, equal to the maximum loading, during a period of time τ , smaller than the calendar period of operation, taking into account that, in this case, during the entire period, the maximum power losses occur. From this results the name for τ – time of maximum losses or, more often, losses time. Pt

Qt 2 t

P 2 max

P

Pmax

2 max

Qt2

Q

Pt τp TmaxP a.

Qt

Qmax τQ

t

TmaxQ

t

b.

St St 2

2 Smax

St

Smax

τ Tmax

t

c. Fig. 7.4. Determination of values Tmax and τ .

To determine the value τ it is required, in Figure 7.4, to draw the rectangles 2 with I max as coordinate, and abscisa determined by means of the relation: T

τ=

∫I

2 t dt

0

(7.32)

2 I max

Thus, losses time is a conventional time during which, within the element, operating at maximum loading, the same energy losses as during operation at actual loading, variable loading, within period of time T, occur. Consequently, the energy losses will have the expression: 2

ΔW

2 = 3 RImax τ=R

Smax τ U

(7.33)

490

Basic computation

where I max is the maximum load current, S max − apparent maximum power, U − voltage at maximum load, most frequently approximated by U n . In order to use the formula (7.33), firstly, it is necessary to evaluate in each concrete case the losses time value τ . Usually, the value τ is determined empirically or through regressions, in terms of the number of hours of the maximum active power utilization and the power factor. Basically, there is a tight correlation between the number of hours of the maximum power utilization, Tmax , and the maximum power only in the networks of voltage up to 110 kV (220 kV), where the influence of the capacitive load of the respective network can be neglected. In order to calculate the losses time, many formulas are established in terms of at least the maximum current value (or of the maximum apparent, active or reactive power), mean current value (active, reactive or apparent power) and losses calculation duration. To enhance the accuracy for the losses time determination, some authors introduced additional information, such as the minimum value of the current (power). In West European countries and American companies [7.17, 7.18], considering that a sufficiently accurate determination of the maximum value ( I max , Pmax , Qmax , S max ) is not feasible in practice unless there are appropriate recording installations (printmaxigraph, computerized systems, up-to-date meters), the analysis is based on the load factor with reference to the apparent power kU or active power kUP . The load factor is defined as the ratio between the average apparent power and the maximum apparent power: kU =

S med I med = S max I max

(7.34)

Pmed Pmax

(7.35)

and kUP =

In (7.34), the voltage is assumed to be the same in both operating states. For a consumption centre, the load curve keeps its shape, respectively the value of kU . This can be determined by means of recordings – during the characteristic period of the load curve. The rate will be kept until a new large consumer shows up, when the recordings will be performed again. At the same time, the notion loss factor τ* , is introduced, for which relations in terms of kU (or kUP ) are established: τ T where τ is the losses time and T the operating period. τ* =

(7.36)


491

For the calculation of the o.t.c. associated with the active power, the literature provides several relations for the determination of τ p* (Table 7.4). In [7.17], it is shown that for τ p* , the following general relation can be used: τ p * = pkU + (1 − p ) kU2 with p = (0.15K 0.3) according to the consumption type.

The work [7.19] deals with values of the loss factor τ ∗ in distribution – obtained as average of the values used in many power systems – corresponding to a curve type B (Fig. 7.5,b) with loading ranging between 4.2% and 10%. The values are given in Table 7.4 (formula 6’). Figure 7.6 presents the dependencies between loss factor and load factor for an A and B load type. Pmax[%]

Pmax[%] 100

100

75

75

Consumption 100 % or 0%

50

Constant load 23 hours (any value between 0 % and 100 %)

50

25

25

5

10

15

[h]

24

20

[h] 10

5

a.

15

20

24

b.

Fig. 7.5. Extreme types of load curves: a. load type A; b. load type B. Losses factor [%]

100

Distribution feeder

80

60

Type A load

40

Distribution transformer Type B load

20

Load factor [%]

0 0

20

40

60

80

100

Fig. 7.6. Dependency between the load factor and loss factor for a type B curve (distribution).

492

Basic computation Table 7.4 Expression for the loss factors in terms of the parameters α P and kUP . Author

Formula τ P* = k UP

Wolf [7.27] Fleck and Rahn [7.29] Langrehn [7.29]

τ P* =

( 1 = (k 2

(

1 2 kUP + kUP 2

2 UP

1

Superior limit curve corresponding to a unsmooth load curve (type A Fig. 7.5,a)

3

)

τ P * = 0.124 + Tmax 10 − 4 τ P*

Observations

2

1.6 τ P * = kUP

Iansen [7.29] Kezevici [7.28] Militaru [7.34]

kUP 2 − kUP

τ P* =

Curve no.

4

)

2

5

+ αkUP + kUP − α

)

Average curve

τ P * = 2Tmax* − 1 +

[7.26]

+

1 − Tmax* 1 + Tmax* − 2

Pmin Pmax

⎛ P ⎜⎜1 − min Pmax ⎝

Glazunov [7.31]

⎞ ⎟⎟ ⎠

2

Fig. 7.8

[7.30]

τ P * = kU (0.66 + 0.34kU )2

[7.30]

τ P* = kU kU

VDEW [7.32]

τ P* = 0.17 kU + 0.83 kU2

[7.33]

τ P* = 0.7 kU2 + 0.3 kU

[7.1]

τ P* = 0.85 kU2 + 0.15 kU

Wolf [7.27]

τ P* = kU2

[7.17]

τ P* = 0.8 kU2 + 0.2 kU

*) α P = Pmin Pmax ; kUP = Pmed Pmax

Indicate τ P in terms of

Tmax P and λ Valid for variable and discontinuous diagrams The same, if kU > 0.25

1’

Valid for apparent powers Used for distribution networks and transformers Inferior limit curve corresponding to a smooth load curve (type B Fig. 7.5,b). ( kU = 0.0417 K1 ) Used for distribution networks

Power and energy losses in electric networks τP* 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

493

8000 τs [h]

1

3

2

5

λ = 0.8 4000

4 5

λ = 0.6

6000

1’

λ=1

2000

2 kU, max

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 7.7. Dependencies between load factors of load curves and loss factors.

2000 4000 6000 8000 TmaxP [h]

Fig. 7.8. The curves τ P = f (Tmax P ) .

Figure 7.7 presents the dependencies of τ*P in terms of kU , and Figure 7.8 represents the dependence of τs in terms of Tmax P , for some expressions from Table 7.4. The dependencies described for losses time evaluation played an important role in the development of theory for the determination of the power losses in the electric networks. However, the average error within the limits ± (10…25)% is not suitable for the selection of the optimum variant of a network development and much less for performing analyses on operation, this error should not exceed ±5%. In this respect, for the determination of the losses time it is required to take into account the load curve, power factor dynamics, possibility that during the analyzed period the maximum of active power may not coincide with the reactive power. By separating the losses time for active and reactive power, the accuracy degree increases. The analysis of the load duration curves in radial networks allowed to obtain satisfactorily accurate dependencies between the active and reactive power. Such dependency is [7.21]:

Q*i = cP*ib

(7.37)

where P*i and Q*i are the actual ordinates of the active and reactive power expressed in per units, the base value being their maximum value; c and b are constants, coefficients of a regression. By considering this dependency, the relationship between the times T*max P and T*max Q corresponding to maximum active and reactive loads are determined by: T*max P = T*bmax Q

between T*max P , τ*P and τ*Q , respectively, existing the relationships: τ* P = (0.7 T*max P + 0.3)T*max P

(

)

τ*Q = 0.7 T*bmax P + 0.3 T*bmax P

(7.38) (7.39)

494

Basic computation

If the maximum values of the active and reactive power coincide in time, which is characteristic to the distribution networks, the total losses time is: τ* = τ*P cos 2 ϕ max + τ*Q sin 2 ϕ max

(7.40)

τ* = τ* P cos 2 ϕmax + K q τ*Q sin 2 ϕmax

(7.41)

and in the general case:

where K q is the factor of no overlapping of the active and reactive power maximums in the daily load curve, while ϕ the phase shift angle between the voltage and current curves, considered as sinusoidal quantities. For the municipal and rural networks with voltages of 20 kV and less, the exponent b is 0.75, while for the 110...220 kV networks, b = 0.5 . Losses time method is recommended to determine power losses in radial networks, while diagram integration method is recommended for the meshed networks. Under the presented form, due to the relations (7.38) and (7.39), the method allows the separate computation of the technologic losses in the networks due to the active and reactive powers flow. Power factor dynamics and possibility of no overlapping of the active and reactive power maximums are also taken into consideration. By means of this method, accuracy equal to the basic information ( ± 2.5K3% ) was obtained for the power losses into the distribution networks.

7.3.5. Technologic consumption in transmission installations The “long” transmission lines are used for the transmission of large amount of electricity from large power plants to the consumption areas, as well as for the parallel connection of large power plants and interconnection of power systems. The operating state of such electric lines is different, the load curves being determined in terms of the power and energy exchanges resulting from the approval of the generation bids to meet power demand and imports/exports of electricity, as well as the voltages state. Usually, the long lines are considerate to operate with fixed voltage drop. For this reason, the reactive load curve for a fixed state of the active power is determined by the possibilities of the adjustment and compensation devices. Load curves of the apparent power on these lines vary within very large limits during one year. For the losses calculation with a suitable level of credibility, the analysis of 25…30 characteristic days is required. The calculation effort can be reduced if characteristic states corresponding to characteristic levels of the daily load curve, instead of all the states for characteristic days, are used. Power technologic losses in the case of long distance transmission consist in no-load losses and load losses. The no-load losses are determined from the corona phenomenon losses on the lines, iron loss in the step-up and step-down


495

transformers, as well as losses in shunt compensation devices. Running load losses occur in lines, transformer windings and in circuits of series compensation devices connected along the transmission path. In order to determine the power losses it is required to sum up the power losses in the elements of the transmission equipments corresponding to the selected characteristic states, for their performance duration during the chosen calculation period (by means of the diagram integration method). The duration of state j during the chosen calculation period is determined on the basis of the analysis of n characteristic load curves:

tj =

n

∑ Δt

ji ni

(7.42)

t =1

where: t j is the duration of the j characteristic state; Δt ji − duration of the j state in the characteristic day of i type; ni − characteristic days of i type. For electricity transmission with power injections in the paths, the calculation is performed on each section. The calculation algorithm of the technologic consumption, in long distance transmission, by means of the diagram integration method, consists of the following steps: 1. Formation of the computerized system for the electricity transmission and the basic scheme with technical data related to equipments. 2. Parameters determination of the four-terminal network Ai , Bi , Ci , Di of each element i, by means of which the equivalent constants A, B, C, D are determined. 3. Analysis of the daily load curves and voltages states on parts of the transmission network. For the planning and forecasting of the technologic consumption, the operation states of the active power and corresponding voltage drops are used. As result of the analysis, the calculation states are established while, by means of formula (7.42), their duration within the calculation period limits is determined. 4. On the basis of the operation states concerning the active power and the limits given for the voltage drop, the state concerning reactive powers, for each part is determined. 5. Power losses on each network section and for each operating state are calculated: ΔPj = ( A'C' + A"C" )U 22 j + (B'D' + D"B" )3 I 22 j + + 2( A"D" + B′C ′) P2 j + 2(B"C ′ − A′D" )Q2 j

(7.43)

The first term of these expressions refers to the no-load losses, the second to the load losses, third and fourth terms to the losses determined by the model with distributed parameters, as well as the passage of the capacitive current of the line.

496

Basic computation

6. Power losses during period t are equal to: ΔWt =

m

∑ ΔP t

(7.44)

j j

t =1

where m is the number of the calculation states, and tj is the period corresponding to the state j; 7. The structure of the transmission power losses is expressed by means of relations (7.43). Technologic consumption associated with losses caused by corona phenomenon can be determined as follows: − by modelling also, within the characteristic states, the losses caused by corona phenomenon (7.3), these are included in the technologic consumption of each level; − considering their average annual value: ΔWcorona = ΔPc med T '

(7.45,a)

where T ' is the duration of keeping the respective line energized. − assuming differentiated values for the losses due to corona phenomenon according to the weather conditions, which require the knowledge of coefficients a and b (eq. 7.3) in compliance with the weather conditions: ΔWcorona =

n

∑ ΔP T

ci i

(7.45,b)

i =1

where i refers to weather condition considered, while Ti is its duration (

∑ T = T ' ). i

In the end the total technologic consumption ΔWT will be: ΔWT = ΔWS + ΔWcorona

(7.46)

where ΔWS represents the losses caused by the apparent power transmitted on the line.

7.4. Economic efficiency of the electric network losses reducing Generally, for the economic analysis of several variants (options), two criteria categories are used: • ranking criterion of the options, which considers that under any conditions the electricity must be supplied without pointing out the investment profitability, its economical effectiveness; • determination criterion of the economical efficiency of the electricity supply installation and chosing of the optimal solution according to this criterion.


497

For the problem studied, it is considered that the safety in operation of the electric networks of various types and maintenance conditions are the same and therefore, the damages at the consumers following failures in the network generally are not taken into consideration. In the first criteria category the well known criterion of present total discounted costs (PTDC), is included which, in the last time, as for the practice in the countries with developed power systems (e.g. EdF), ignores the equivalencing investment afferent to power losses differences. Actually, this investment is considered non-necessary since it leads to a false result of the analysis. Furthermore, for an enterprise that supplies electricity (respectively purchased from producers), even if required at power system level, it has no influence on the economical efficiency of the respective enterprise. PTDC value is established as referred to one updating year selected (for instance, the investment commencement year, start-up year, etc.). The calculation relation is [7.15]: PTDC =

Ds

∑

I t (1 + a) −t +

t =1

−

Ds

Ds

∑ (C

t

+ Cet + Dt )(1 + a) −t −

t = d +1

∑ (W

' ni

(7.47)

+ Wrt )(1 + a) −t − Wn (1 + a) − Ds

t = d +1

where: I t Ct Cet

Dt Ds

are − − −

investments carried out in year t; annual operation expenses in year t ( Ct = kI ); equivalent annual expenses (corresponding to power losses); presumable annual damages, in the year t (as per specification above, in our case Dt = 0 ); − study period;

Wn (Wni ') − remanent value after the period Ds and after a limited operating period, respectively; Wrt − residual value (in our case Wrt = 0 ); d − execution period. a − discount rate, with: a = ad + ar + a s ad − activity interest rate (10 − 25%); ar − risk rate (1 − 1.5%); as − safety rate (for social security) − (0.5%) Applications The following input data are considered: − study period, 10 years; − annual operation expenses for substations, 6% of investment value;

498

Basic computation

− the cost of kWh for the equivalencing of the power losses in the medium voltage networks in the expression of PTDC is e = 65 Є/MWh;

[

{

]

Wn = I (1 + α )− Ds m(1 + α )Dv + 1 − m(1 + α )Dv ⋅ ⋅

where:

[(1 + α )(1 + a )]D − [(1 + α )(1 + a )]D −D [(1 + α )(1 + a )]D − 1 v

v

s

v

(7.48)

⎫⎪ ⎬ ⎪⎭

α is the average annual factor of equipment wear = 1%; m − residual value upon lifetime expiry, % of investment value = 10%; Ds − study period = 10 years; Dv − lifetime = 20 years; D S =10

a

− discount rate = 12% and

∑ (1 + a )

−1

≈ 5.65 .

i =1

For these data from (7.48) it results Wn = 0.726 I . Considering the residual value on the lifetime expiry of only 5%, the following is proposed: ⎡ D − Ds ⎤ VRM = I ⎢0.05 + (1 − 0.05) v ⎥ = 0.525 I Dv − d ⎦ ⎣

(7.49)

where d is the execution period ≈ 0. Further on, the relation (7.48) will be used. Mention that, for example in the USA, for the PTDC calculation, the remanent value is not subtracted, on the ground that the recovery of these at the equipment decommissioning date is improbable. Knowing these information, the PTDC from expression (7.47) becomes: PTDC = I + I k

n

n

1

1

∑ (1 + a )−i + ΔP0Te ∑ (1 + a )−i +

n

+ τe

∑ ΔP (1 + a )

−i

i

(7.50,a)

− 0.726 I (1 + a )

− Ds

1

or ⎡ PTDC = I ⎢1 + k ⎣⎢

n

−i

1

n

+ ΔP0Te

⎤ − 0.726(1 + a )− Ds ⎥ + ⎦⎥

∑ (1 + a )

∑ (1 + a )

−i

n

+ τe

1

PTDC = I + (0.06 I ) ⋅ 5.65 +

∑ ΔP (1 + a ) i

1

∑ ΔW (1 + a )

−i

i

(7.50,b)

−i

⋅ 65 Є/ MWh − 0.726 ⋅ (1.12)−10 I

or post calculations: PTDC = 1.105 ⋅ I +

∑ ΔP (1 + a )

where ΔPi is the power losses in the year t.

i

−i

τ ⋅ 65 [Є]

(7.51)


499

If we denote by ΔPs the power losses under running load in the year 1 and by β the annual percentage of the load growth, it results the sum: n

∑

ΔPsi (1 + a )

−i

=

i =1

ΔPs ΔPs (1 + β )2 ΔPs (1 + β )4 + + +K = 1+ a (1 + a )2 (1 + a )3 n

⎡ (1 + β)2 ⎤ ⎢ ⎥ −1 ΔPs ⎢⎣ 1 + a ⎥⎦ = ⋅ = 1 + a (1 + β)2 −1 1+ a ΔPs (1 + β )2 n − (1 + a )n = ⋅ (1 + a )n (1 + β)2 − (1 + a ) For the calculation of τ the following relation can be used [7.17]:

[

(7.52)

]

τ = T τ* = T (1 − p )ku2 + pK u with p ∈ (0.15 ÷ 0.3)

S med ; T is the operation period. S max In the end, the option with the minimum PTDC will be chosen. The second category of the efficiency criterion consists of the followings: (i) Net present income (NPI), which evaluates the annual cumulated, proceeds:

where ku =

NPI =

Ds

Vt − (Ct + I t ) (1 + a)t t =1

∑

(7.53)

where: Vt is the total income in the year t; Ct − investment in the year t; a − discount rate; t − current year; Ds − study period. Obviously, the economically efficient options have NPI ≥ 0 and the options with maximum NPI are choosen. If the total investments are not equal, the net presents income rate (NPIR) will be used.

NPIR =

NPI Ds

∑ t =1

It (1 + a )t

(7.54)

and the option with maximum NPIR is selected. (ii) Internal rate of return (IRR) represents a relative profit criterion. Actually, this is that discount rate for which the net present income is zero:

500

Basic computation Ds

Vt − Ct

∑ (1 + IRR) t =1

=0

t

(7.55)

The solutioning is done by means of an iterative process. The investment is considered profitable if IRR is larger than a set minimum return rate (regularly, with the discount rate 12%). In order to compare two options (A and B) the relative profit criterion is applied: Ds

∑

(V

t =1

t

B

) (

)

− CtB − Vt A − CtA =0 (1 + a )t

(7.56)

(iii) Pay back period (PBP) represents the time period required for the recovery of the initial investment value (financed from own funds) from the annual returns associated with it. − Simple pay back period (SPBP) I=

T'

∑ (V

t

− Ct )

(7.57)

t =1

The value T′, obtained from (7.57), represents the recovery period. If the difference ( Vt − Ct ) is constant and the investment is carried out in a single year, then the recovery period is: T '=

I Vt − Ct

The investment is, regularly, considered efficient, T '≤

(7.58) 1 a

− Present pay back period (PPBP), T " : 1

∑

I t (1 + a ) t =

T"

∑ t =1

d

(Vt − Ct ) (1 + a) t

(7.59)

where d is the period of carrying out the investment (between the year d and the start-up year). The pay back period can also be calculated under the form of pay back period of investment differences through annual expenses differences. It is proposed the application under this form, as the revenue-sum cashed from the selling of a fixed electricity quantity, cannot be accurately determined when performing the analysis. Therefore: PBP =

I 2 − I1 C1 − C 2

C being composed of the energy losses cost and operation expenses.

(7.60)


501

For the cost of losses under running load, if consider a variation in time of the costs, to apply the simple pay back period criterion, the utilization of the “levelizing factor” β is required, to obtain a constant value of them during the entire analyzed period [7.13, 7.18]: ΔPlevelized = ΔPs

(1 + β)2 N − (1 + a )N ⋅ a 2 (1 + β) − (1 + a ) (1 + a )N − 1

(7.61)

Considering an average value for 10 years of study N = 10 years β = 2% and a = 12% , it results: ΔPlevelized ≈ 1.16 ΔPs (iv) Annual recovery of investment differences. If one considers that the supply of a consumer is compulsory and therefore it is not subjected to a cost-profit analysis, it is necessary to only check the conditions of recovering the investment differences between options, of existence of conditions for credits reimbursement. It is known that a loan of 1Є with i interest for N years leads to an annual constant rate of reimbursement as follows: Ra = 1 Є ⋅

i 1−

1 (1 + i ) N

Generally, the reimbursement is accomplished after commissioning of the equipment (execution period of d years). In this case, the annual recovery of the investment differences can be assured for the option K with respect to the option with minimum investments: K=

I K RaNK − I min RaN min 0) , the dual function (DF) is defined as: (DF)

Φ (λ , μ) = minimum L( X , λ, μ) X subject to: h1 ( X ) = 0

(10.22)

g1 ( X ) ≤ 0 The dual function is concave and, in general, non-differentiable, [10.43]. This is a fundamental fact in the algorithm of the LR methods. The dual problem (DP) is then defined as: (DP) Maximize Φ(λ, μ) λ, μ subject to: μ ≥ 0

(10.23)

The LR decomposition procedure is attractive if, for fixed values of λ and μ, the dual function is easily solved, namely, if the dual function is easily evaluated ~~ of multiplier vectors λ and μ respectively. The problem to be for given values λ, μ ~~ solved for given values λ, μ is the so-called relaxed primal problem (RPP),

Steady state optimization

(RPP)

611

~ ~ Minimize L( X , λ, μ ) X subject to: h1 ( X ) = 0 g1 ( X ) ≤ 0

(10.24)

The above problem typically decomposes into sub problems. This decomposition facilitates its solutions, that is n

(DPP)

Minimize

~

∑ L( X , λ, μ~) i

i =1

Xi subject to: h1i ( X i ) = 0 ; i = 1,..., n

(10.25)

g 1i ( X i ) ≤ 0; i = 1,..., n This problem is called the decomposed primal problem (DPP). The resulting sub problem can be solved in parallel. Under local convexity assumptions, the local duality theorem says that: F ( X * ) = Φ (λ* , μ* ) (10.26) where X* is the minimizer for the PP and (λ* , μ * ) , the maximizer for the DP. In the nonconvex case, given a feasible solution for the PP, X, and a feasible solution for the DP, (λ,μ), the weak duality theorem says that: (10.27) F ( X ) ≥ Φ (λ, μ) LR generates a separable problem by integrating some constraints into the objective function, through “penalty factors”, which are functions of the constraints violations, [10.20]. By assumption the Lagrangian problem is relatively easy to solve, there are three major questions in using the Lagrangian Relaxation method, [10.46]: • Which constraints should be relaxed? • How to compute good multipliers? • How to deduce a good feasible solution to the original problem, given a solution to the relaxed problem? The penalty factors, referred to the Lagrangian multipliers, are determined iteratively. The procedure is dependent on the initial estimates of the Lagrangian multipliers and the method used to update these. The most used technique for estimating the Lagrangian multipliers relies on sub gradient algorithms or heuristic, [10.20]. Application Consider the problem [10.20] Minimize ( x 2 + y 2 ) x, y subject to: ( x + y ) − 2 2 = 0; x, y ≥ 0

612

Load flow and power system security Whose solution is x*= y*=

2 , F( 2 , 2 ) = 4. The Lagrange multipliers associated to the equality constraint has the optimal value λ*= – 2

2 . The Lagrangian function is: L( x, y , λ) = x 2 + y 2 + λ( x + y − 2 2 )

This problem is solved by LR follows. Step 0. Initialization. λ = λ0. Step 1. Solution of the relaxed primal problem. The problem is decomposed into two sub problems: Minimize ( x 2 + λ x − 2 λ )

Minimize ( y 2 + λ y − 2 λ )

x whose solutions are x and y.

y

Step 2. Multiplier updating. Use a sub gradient procedure with proportionality constant equal to δ.

λ ← λ + δ( x + y − 2 2 ) Step 3. Convergence checking. If multiplier λ does not change sufficiently, stop; the optimal solution is x* = x, y* = y. Otherwise the procedure continues in Step 1. In this algorithm the following values can be considered: δ = 1 and an initial multiplier value λ = 1.

Augmented Lagrangian decomposition The augmented Lagrangian (AL) function of problem (10.20) has the form: 1 α h2 ( X ) 2

AL ( x, λ, μ, α, β) = F ( X ) + λT h2 ( X ) + μ T g~2 ( X , Z ) +

2

+

1 ~ 2 β g 2 ( X , e) 2 (10.28)

Penalty parameters α and β are large enough scalars to ensure local convexity, and the component i of function g~2 ( X , e) is defined as g~ ( X , e ) = g ( X ) + e 2 . 2i

i

i

2i

i

i

10.2.3. Multiobjective optimization techniques 10.2.3.1. Introduction to multiobjective optimization (MO) Practical optimization problems, especially the engineering design optimization problems, seem to have a multiobjective nature much more frequently than a single objective one. Practically, real-world decision making problems with only one objective are rare.


613

Despite of that, solving single objective optimization problems is far more common than solving multiobjective problems, since there appears to be no generally effective and efficient method available for solving multiobjective problems directly as they are. Typically a multiobjective problem is to be effectively converted to a single objective problem before applying an optimization algorithm. This conversion can be done easily by first deciding the relative importance for each objective a priori. Then, for example, the Decision-Maker (DM) may combine the individual objective functions into a scalar cost function (linear or nonlinear combination), which effectively converts a multiobjective problem into a single objective one. When considering multiobjective optimization (MO) problems, the most frequently applied evolutionary optimization algorithms are genetic algorithms. Anyway, single objective problems are only a subclass of multiobjective problems. Thus finding a method for solving the multiobjective problems as multiobjective problems, without any a priori preference decisions, and without first converting the problem into a single objective one, is one of the most important optimization research objectives at the moment, [10.40]. This focusing can be justified by considering two facts: • The Pareto-optimization approach does not require any a priori preference decisions between the conflicting objectives. • Evolutionary optimization approach has proven to be potential for solving Pareto-optimization problems. A wide variety of approaches have been applied for attacking MO problems. As well as in case of single objective optimization problems, these approaches can be classified into three classes: enumerative, deterministic and stochastic methods. Each approach has some advantages and some fundamental limitations. The engineer/economist Vilfredo Pareto made (1886) one of the most important findings in the field of (MO) by finding that “Multiple criteria solutions could be partially ordered without making any preference choices a priori”. Optimal solutions for a multiobjective problem, defined by applying Pareto’s idea, are currently called as Pareto-optimal solutions due to obvious reasons. The Pareto relatively simple idea of optimality in case of multiple objectives can be verbally described as follows, [10.40]: “A solution is Pareto-optimal if it is dominated by no other feasible solution, which means that there exists no other solution that is superior at least in case of one objective function value, and equal or superior with respect to the other objective functions values”. It is clear that in case of conflicting objectives, the Pareto optimal solutions are rather a class of solutions, forming a surface in objective function space, than a single solution. This surface is commonly called as a Pareto-front. This decision can be done applying one of the following approaches: 1. A priori preference articulation: The Decision-Maker (DM) selects the weighting before running the optimization algorithm. In practice it means that the DM combines the individual objective functions into a scalar cost

614

Load flow and power system security

function (linear or nonlinear combination). This effectively converts a multiobjective problem into a single objective one. 2. Progressive preference articulation: DM interacts with the optimization program during the optimization process. Typically the system provides an updated set of solution and let the DM consider whether or not change the weighting of individual objective functions. 3. A posteriori preference articulation: No weighting is specified by the user before or during the optimization process. The optimization algorithm provides a set of efficient candidate solutions from which the DM chooses the solution to be used. Currently, in connection with evolutionary algorithms, there exist clearly two mainstream approaches for appropriate definition of MO problem also in case of conflicting objectives: 1. Weighted sum of objective functions: Converting the multiobjective problem to a single objective one using weighted sum of objective functions as a representative objective function, and then solve the problem as a single objective one represents an a priori preference articulation. 2. Pareto-optimization: Solving the multiobjective problem by applying Pareto-optimization approach. DM selects the solution from the resulting Pareto-optimal set. This represents an a posteriori preference articulation. The problems with multiple objectives do not have a unique optimal solution, but a set of Pareto-optimal solutions. The set of Pareto-optimal solutions can be characterized by Pareto-front – a hypersurface in the objective function space in which the Pareto-optimal points are located.

10.2.3.2. Multiobjective decisions in power systems In the operating and expansion planning of the power systems the various structures are assessed and compared according to various type of criteria: • economic criterion; • supply quality criterion; • robustness criterion; • environment criterion and so on. The priority of the criteria has changed dramatically in recent years. In the economic criterion a comparison of the various structures adopted is made on the basis of costs calculation. Regarding the reliability of customers supply as a quality criterion, by means of a special network topology, in Figure 10.4 is indicated the general possibilities for applying reliability criterion. A key criterion to the optimization problem is that of robustness. A given plan is robust regarding a specific constraint, if the constraint holds true for every possible value of the uncertain variables and constants. In the last time the importance of the environmental criterion is greater and greater and a very important aspect that must be taking into account is the networks (lines) impact with the environment.


615 Methods for applying reliability criterion to power system planning

Treatment as a restriction

(n-1) criterion topology

Limits for the reliability indices

Treatment within the goal function

Multiobjective optimization

Interruption costs

Fig. 10.4. Methods for using reliability in the power systems planning.

In contrast to traditionally planning methods, in the last time, the (MO) methods show their advantages. As example, in OPF and OUC case, the economic, the security, environmental and reliability objectives must be considered too. Usually, the objective function to be considered is the sum of start-up, shut-down and running cost. The security refers to the maximization of the static and dynamic security of the system. In environment criterion, the objective is the minimization of the total emission from the fossil fuel plants. For reliable operation of the power system, a sufficient amount of operating reserve expressed as a percentage of the total generation must be maintained at each generating station. Optimization according to multiple objectives can be done using the following methods: • definition of a scalar function as the weighted sum of the results of each objective; • transforming the multiobjective problem into a single objective one using fuzzy technique; • optimization of one objective while treating the others as restrictions; • ε - constrained technique; • goal – programming. Fundamental to the MO problems is the Pareto-optimal concept (known as a noninferior solution). Assuming that the Decision-Maker (DM) has imprecise or fuzzy goal for each of objectives, a fuzzy goal expressed can be quantified by drawing out a corresponding membership function. To elicit a membership function for each objective, we first estimate the individual minimum and maximum value of each objective function under given constrains by the experiences of the DM. The problem of Economic Dispatching (ED) of the thermal power plants tries to find the optimal commitment of generating units, to meet the demand, transmission losses and reserve requirements. An equally important objective is to minimize the adverse effects on the environment. In this case, the objective is to find the commitment of generating units so as to comply with environmental constraints, such as limitation of emission of oxides of sulfur, oxides of nitrogen or the heat discharge into watercourse. The schedule that achieves this last objective may be called Controlled/ Minimum Emission Dispatching (C/MED). In the MO problems, multiple objectives are usually non-commensurable and cannot be combined into a single objective. Moreover, any improvement of one

616


objective can be reached only at the loss of another. Consequently, it is necessary to design a DM for the MO problems. The aim of the MO is to find a compromise between all the DM solutions. This means that the DM must select the compromise solution between global non-inferior solutions.

10.2.3.3. Application on the case of a distribution station This section presents a MO technique, based on fuzzy dynamic programming (FDP) for the voltage-reactive power (VQ) control on the case of distribution stations. The main two objectives are the improvement of the voltage profile and the reduction of the active power losses. Using the hourly forecast of the demanded load for the next day, the VQ control optimizes the control actions for the transformer tap changing (tc) and the capacitor switching (CS) of capacitor banks (CBs), so as to maintain the secondary bus voltage, as close as possible to the specified value and to keep the active power losses in the station transformers, as small as possible, at all hours of the day. Coordination of tc and CS is achieved through the maximization of a fuzzy objective function by fuzzy dynamic programming (FDP). The strategy for preventing unnecessary tap operations is based on the prediction of the near future voltage and reactive power. Most control actions are conditioned by the tendency of the load (to increase, decrease, or remain steady). The operators will not be willing to change the voltage set point, if they take for granted that the opposite action will be necessary in the near future, in twenty minutes, for example, [10.44]. The short-term reactive scheduling (24 hours) is executed one day in advance, using an adaptive artificial neural network. The optimal voltage profiles are those, which minimize the real losses for the most representative time intervals of the daily load diagram. The attainment of the planned optimal profile is possible only if the current network situation exhibits the characteristics of the day before forecast. On the contrary, the very short-term (15 – 30 minutes) scheduling will be executed, on the basis of the actual state, [10.23]. If the tc and the CS are operated independently, it may happen that the tc move too often and, consequently, it is necessary to coordinate the tc and CS actions. To solve this problem, an FDP approach is used to find the proper dispatching strategy for the tc movements and CS actions, so as to improve the bus voltage profile and to reduce the transmission losses in main transformer. Hard limits on the maximum allowable number of control actions of CS and tc are also imposed.

Optimization mathematical model Using the demanded load forecasting for the next day, the status of the tc and CS must be determined, so as to keep the active power losses in the main transformer as small as possible and to maintain the secondary bus voltage as close as possible to the specified value, at all hours in the day. In practice, the following values are imposed: cos ϕ ≥ 0.8 , ΔU 2 ≤ 5% and the daily switching number for the tc and CB is: N tc ≤ 30 , respectively, N CB ≤ 6 [10.11, 10.42]. If the control variables for CB (CB1, CB2) and tc are defined: X i ( X 1i , X 2i ) = 0 (CB1 (CB2) is off); i = h = 1, K, N = 24 ; X i ( X 1i , X 2i ) = 1 (CB1 (CB2) is on); i = h = 1, K, N = 24 ; tci = −9, K, 0, K, 9 ; i = h = 1, K, N = 24 ; ΔU 2 = ΔU 2 ; Δ Q = Q2 − QCB


617 μcosφ

μΔU2 1

1

0.5

0.5 cosφ

ΔU2 1

2

3

4

0.5

5

Fig. 10.5. Membership function for ΔU2.

0.6 0.7 0.8 0.9 1

Fig. 10.6. Membership function for cos ϕ .

The problem is to find a set of control variables X i and tci , so that the objective function: N

F=

∑ i =1

N

μΔ~

u 2i

+

∑μ i =1

~ ΔQ

= F1 + F2

(10.29)

is maximized subject to: N

N tc =

∑ tc − tc

≤ 30

i −1

i

(10.30)

i =1 N

N CB1 =

∑ X1 − X1 i

i −1

≤6

(10.31)

≤6

(10.32)

i =1 N

N CB 2 =

∑ X2 − X2 i

i −1

i =1

The membership functions regarding the voltage deviation and, respectively, reactive power transit, are shown in Figures 10.5 and 10.6. In the objective function (10.29), the term F1 quantifies the desirability of maintaining the secondary bus voltage, as close as possible to the specified value, and the term F2 quantifies the desirability of keeping the reactive power flow in the main transformer, as small as possible at all hours during the day. The optimization model (10.29) – (10.32) enables us to determine the optimal control values of X i and tci at each hour during the observed period of time. The dynamic programming principle will be applied to solve the optimization problem. The recurrence formula to reach the status (H,QCB) at hour H has the form:

{

F (H , QCB ) = max FH (H , X i ) + F (H − 1, QCB ) Xi

}

(10.33)

where: F (H , QCB ) is the total value of the objective function to reach the status (H, QCB ); FH (H , X i ) – the value of the objective function for the transition from the status (H–1, QCB ), to the status (H, QCB ).

618


To obtain the second bus voltage, the calculated transformer ratio (CTR) must be rounded at the real transformer ratio (RTR) at each step of the FDP, by adjusting the correspondent value of the tci . Each step of the FDP supposes two possibilities for CB (on/off).

Numerical results The proposed method has been applied to several cases. For example, we present in Figure 10.7 a distribution station case.

U1=110 kV Sn=25 MVA 110 kV / 22 kV + 9 x 1.78% U2=110 kV QCB

P2+jQ2

Fig. 10.7. Test distribution station. The optimal value of the objective function in the case of QCB = 2.4 MVAr is F = F1 + F2 = 37.25 , N tc = 6 , N CB = 2 . Using fuzzy logic, logical decisions can be made in problems exhibiting ambiguity. The coordination of voltage-reactive power control represents an example of such a problem. The fuzzy variables associated with the coordination are the transformer taps and the capacitor switching of the capacitor banks.

Table 10.2 Numerical Results for case QCB = 2.4 MVAr H 1 2 3 4 5 6 7 8 9 10 11 12

Q2 P2 [MW] [MVAr] 5.75 5.15 5.25 5.75 6.00 7.00 8.75 12.5 15.0 15.5 15.7 16.0

1.2 1.00 0.90 1.20 1.30 2.00 4.00 7.00 11.25 11.62 11.80 12.00

tc 2 2 2 2 2 2 1 0 -1 -1 -1 -1

QCB = 2.4 MVAr F2 X F1 0 0.90 0.96 0 1.78 1.94 0 2.69 2.94 0 3.75 3.90 0 4.65 4.85 0 5.04 5.75 1 6.24 6.67 1 6.98 7.37 1 7.80 7.63 1 8.62 7.67 1 9.46 7.89 1 10.32 8.09

H 13 14 15 16 17 18 19 20 21 22 23 24

Q2 P2 [MW] [MVAr] 15.50 15.00 14.50 12.50 10.50 7.75 6.75 6.25 6.00 5.75 5.75 5.75

11.60 11.25 10.80 7.00 6.00 3.50 3.00 1.75 1.30 1.20 1.20 1.20

tc -1 -1 0 0 1 1 2 2 2 2 2 2

QCB = 2.4 MVAr F2 X F1 1 11.14 8.35 1 11.94 8.63 1 12.84 8.95 1 13.56 9.53 1 14.42 10.29 1 15.00 11.25 1 15.88 12.25 0 16.76 13.15 0 17.66 14.10 0 18.56 14.97 0 19.46 15.93 0 20.36 16.89


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10.2.4. Modern optimization techniques in operating planning 10.2.4.1. Sequential quadratic programming (SQP) In the most general formulation, the Optimal Power Flow (OPF) is a nonlinear, nonconvex, large-scale, static optimization problem with both continuous and discrete control variables, [10.3, 10.6, 10.10, 10.11, 10.20, 10.51, 10.53, 10.54, 10.61 ÷ 10.63, 10.65 ÷ 10.67, 10.71 ÷ 10.74]. The literature OPF is vast and presents the major contributions in this area. Mathematical programming approaches, such as nonlinear programming (NLP), quadratic programming (QP), linear programming (LP), etc. have been used for the solution of the OPF problems. Some methods, instead of solving the original problem, solve the problem Karush-Kuhn-Tucker (KKT) optimality conditions. For equality constrained optimization problem, the KKT conditions are a set of nonlinear equations, which can be solved using a Newton type algorithm. The inequality constraints can be added as penalty terms to the problem objective, multiplied by appropriate penalty multipliers. Over the past three decades, there has been a great interest in heuristic search methods for complex optimization problems. In this section, an overview of these optimization techniques in power systems is presented. Dommel and Tinney proposed a reduced gradient technique. This method combines projection for control variables and penalty for dependent variables and functional constraints. The method contains many uncertainties over convergence. Other authors, more recently, proposed sequential quadratic programming (SQP) technique to solve OPF by Newton method. In the many other papers, the Newton method (for unconstrained optimization) is combined with a Lagrange multipliers method (for optimizations with equalities) and penalty functions (for handling inequalities) to solve large scale OPF, in nonlinear manner. Desirable properties of a quadratic function are, [10.61]: • its Hessian is constant; • the higher order term is easily evaluated. There are two reasons for the successful applications of this method (LR Lagrange relaxation), [10.54]: • many real problems are complicated by the addition of some constraints; • practical experience with LR has indicated that it performs well at reasonable computational cost. The important aspect of this approach is in identifying the set of binding constraints; these are the inequality constraints, which need to be considered as equality, [10.53]. The dual augmented Lagrangian approach handles all constraints using penalty and dual variables. The dual variables are updated using an intrinsic rule. With this method, the increase in penalty terms improves the dual convergence, but it makes the Hessian matrix ill conditioned. The optimization of this trade off is the critical aspect of this method.

620


[10.53] combines the sequential quadratic programming (Newton) and the augmented Lagrangian methods, so the difficulties, which arise from these methods are eliminated. The binding constraints need not be identified and the equalities constraints are not associated with the penalty functions. Thus one avoids the need to create more nonlinearity in the problem (augmented Lagrangian). The necessary first-order optimality conditions (FONC) of the OPF model are the stationary conditions of the augmented Lagrangian function. These conditions are reached by a primal-dual procedure in which, [10.53]: • Newton method is applied to FONC with respect to the primal and dual variables related to the equality constraints. • The dual variables related to the inequality constraints are updated by a dual rule and the penalty factor is slightly increased.

10.2.4.2. Interior-point methods (IPMS) In the past years, research on interior-point (IP) methods, theory and computational implementation, have evolved extremely fast. Interior-point method variants are being extended to solve all kind of programs: from linear to nonlinear and from convex to non-convex (the latter with no guarantee regarding their convergence). In the same way, they are also being applied to solve all sorts of practical problems. Optimization of power system operations is one of the areas, where IP methods are being applied extensively, due to the size and special features of these problems, [10.51, 10.61 ÷ 10.63, 10.65 ÷ 10.67, 10.73]. Interior-point methods (IPMs) are a central, striking feature of the constrained optimization landscape today. They have led a fundamental shift in thinking about continuous optimization. Today, in complete contrast to the era before 1984, researchers view linear and nonlinear programming from a unified perspective. Also, IPMs provide an alternative to active set methods for the treatment of inequality constraints, which permits the effective and efficient handling of large sets of equality and inequality constraints, [10.65]. IPMs convert the inequalities by introduction of nonnegative slack variables. A logarithmic barrier function of the slack variables is then added to the objective function, multiplied by a barrier parameter, which is gradually reduced to zero during the solution process. The computational effort in each iteration of an IP algorithm is dominated by the solution of large, sparse linear system. The greatest breakthrough in the IPMs research field took place in 1984, when Karmarkar [10.37] came up with a new IP method for LP, reporting solution time up to 50 times faster than the simplex method. Karmarkar algorithm is based on nonlinear projective transformations.

10.2.4.3. Lagrange relaxation - evolutionary programming (LREP) Computational intelligence seeks, as its main goal, to create artificial systems, which mimic aspects of human behaviour, such as perception, evolution, learning, adaptability and reasoning. It involves a number of intelligent techniques,


621

which are inspired by a well established and successful system, the nature. Several different types of evolutionary search methods were developed independently: • Genetic programming (GP); • Evolutionary programming (EP), which focuses on optimizing continuous functions without recombination; • Evolutionary strategies (ES), which focuses on optimizing continuous functions with recombination; • Genetic algorithms (GA), which focuses on optimizing general combinatorial problems. The main applications developed worldwide in the last years: • Load forecasting; • Optimal power flow; • Optimal unit commitment; • Operating and expansion planning; • Optimal capacitor placement and control; • Tariff selection; • Alarm processing and fault diagnosis; • Control of power consumption. The Table 10.3 presents the intelligent techniques and their main properties. The Evolutionary programming (EP) is a stochastic optimization method in the area of evolutionary computation, which uses the mechanics of evolution to produce optimal solutions to a given problem, [10.68]. It works by evolving a population of candidate solutions toward the global minimum through the use of a mutation operator and selection scheme. The EP technique is particularly well suited to non-monotonic solution surfaces, where many local minima may exist. EP seeks the optimal solution by evolving a population of candidate solutions, over a number of generations or iterations. During each iteration, a second new population is formed, from an existing population, through the use of mutation operator. This operator produces a new solution, by perturbing each component of an existing solution, by a random amount. The degree of optimality of each of the candidate solution or individuals is measured by their fitness, which can be defined as an objective function of the problem. Through the use of a competition scheme, the individuals in each population compete with each other. The winning individuals form a resultant population, which is regarded as next generation. For optimization to occur, the competition scheme must be such that the more optimal solutions have great chance of survival that the poorer solutions. As example, enhancement of the optimization methods by using genetic algorithms, [10.20] proposes a method, where Lagrangian relaxation incorporates genetic algorithms to solving the optimal commitment problems. The Artificial Neural Networks (ANN) are used too to improvement of the optimization methods. Most gradient-based optimization methods involve a major difficulty – the derivation of the objective function. Usually, a great number of objective functions cannot be derivable and, frequently, are not continuous. A method for computing the network output sensitivities, with respect to the input

622


variations for multilayer perceptron (MLP) using differentiable activation functions is presented in [10.9]. The method is applied to obtain the values of the first and second order sensitivities. These sensitivities, with the conjugated gradient, can be used as a basis for the process optimization. As an illustration, the minimization of losses in a power system is presented. In this case three controls are used, namely the voltages at the generation buses, the positions of the transformers tap changing, and the reactive power of switchable capacitors/reactors. The fuzzy set theory, introduced by Zadeh and investigated further by many researchers, provides the tools for representing and manipulating inexact concepts, prelevant in human interpretation and in the reasoning process. Uncertainty in fuzzy logic is a measure of no specificity that is characterized by possibility distributions. This is somehow similar to the use of probability distributions, which characterize uncertainty in the theory of probability. A fuzzy set A is characterized by a membership function μ A (x) relating each element x to its compatibility degree with set X:

A = { x, μ A ( x ) x ∈ X }

The membership function, which represents the possibility distribution, assigns a real number, between 0 and 1, to every value in the fuzzy set. In the real-world problems, there exist uncertainties in both the objective functions and constraints. By using the fuzzy set theory, these uncertainties can be considered. So, using the fuzzy logic, a multipleobjective problem can be transformed into a single objective problem. This approach simplifies the solving of the complex optimization problems. Artificial Neural Networks, Fuzzy Logic and Evolutionary Computing have shown capability on many problems, but have not yet been able to solve the really complex problems that their biological counterparts can (e.g., vision). It is useful to fuse Artificial Neural Networks, Fuzzy Logic and Evolutionary Computing techniques for offsetting the demerits of one technique by the merits of another techniques, [10.14]. Table 10.3 Intelligent techniques and their properties Techniques

Primary Application

Genetic Optimization Algorithm Neural Pattern Network Recognition Fuzzy Logic

Control

Expert System

Decision Making

Strengths

Weaknesses

Parallelism. Easy Implementation Easy Implementation. Learning with data. Manipulates inaccurate concepts such as: bigger, smaller, high, old. Explains the results obtained. Explains completely the process of solution.

Hard to represent some problems. Does not explain the results obtained. Needs an expert to formulate the necessary rules. Needs an expert to precisely formulate the rules.


623

10.2.4.4. Strategies optimization techniques in energy markets The utility restructuring has enhanced the role and the importance of OPF tools. Some of the basic business functions cannot be performed without OPF. In general, the objective of an energy market is to maximize market benefits. This is equivalent to minimizing the payments to energy offers and the revenues from demand bid. There are two sources for negotiating models: Game Theory (GT) and Distributed Artificial Intelligence (DAI). GT is concerned with interactions between human agents, while DAI deals with similar issues concerning software agents. Unfortunately, DAI negotiation protocols developed to date are only suitable for tasks where agents have a common goal. For such “coordination” problems, protocol such, as “contract net” are applicable. In competitive markets, agents may form coalitions, but clearly they have different goals. GT, on the other hand, is limited by assumptions about requisite knowledge of the others agents, [10.45]. Game theory is a branch of economics focused on behaviour related to interactive decision problems. Research within this field of study includes analysis of pricing strategy and behaviour that applies to suppliers (market incumbents and new entrants) and customers. More than 50 years ago when Game theory first emerged, attempts were made to develop theories of bargaining that would predict outcomes. John Nash, the 1994 Nobel laureate, theorized in two papers (1950, 1953) that suppliers competing for the same business could cooperate and set high prices. Research reveals that as the number of bidders increase, the more likely it is that the winner will overestimate the actual value. Individual bidders will logically adjust bids to reflect their own market expectations, assuming over time that any errors will average out, with the high estimates cancelling the low estimates. However, the law of averages does not apply to competitive bidding. Suppliers who underestimate the value of the product lose the deal and the one who overestimates the value the most ends up a winner who overpaid, [10.45]. In deregulated market, ordinal optimization techniques often are used for optimization of the bidding strategies, [10.29].

10.3. Optimal power flow (OPF) 10.3.1. Optimization model 10.3.1.1. Introduction The optimal power flow (OPF) is a steady state operation of the power system that minimizes the costs of meeting the load demand for a power system,

624


while maintaining the security of the system, [10.71]. The costs associated with the power system may depend on the situation, but in general they can be attributed to the cost of generating power at each generator. From the viewpoint of an OPF, the maintenance of system security requires keeping each device in the power system within its desired operation range at steady state. This will include maximum and minimum outputs for generators, maximum MVA flows on transmission lines and transformers. The transmission and distribution of the electric energy at a desired voltage profile is a measure of not only the quality, but also the security of supply. The optimal voltage profile is defined as the profile corresponding to the minimum active power losses, [10.11, 10.28]. In this order, the OPF will perform all the steady-state control functions of the power system, including generator control and transmission system control. For generators, the OPF controls generator MW outputs, as well as generator voltages. For the transmission system, the OPF may control the tap ratio or phase-shifter angle of the transformers, switched shunt control, and all other flexible ac transmission system (FACTS) devices. Another goal of an OPF is the determination of system marginal cost data. This marginal cost data can aid in the pricing of MW transactions, as well as the pricing ancillary services such as voltage support through MVAr support. In solving the OPF using Newton method, the marginal cost data are determined as a by-product of the solution technique, [10.71].

10.3.1.2. Objective function The objective function for the OPF reflects the costs associated with generating power in the system. The cost is assumed to be approximated by a quadratic function of generator active power output as:

Ci = ai + bi PGi + ci PG2i where: PGi [MW]

(10.34)

is the active output of generator i, subject to lower/upper

bounds constraint; ai , bi , and ci – the coefficients of the consumption characteristics. Therefore, this objective function will minimize the total system costs:

F(X ) =

∑ (a + b P i

i

i Gi

+ ci PG2i

)

(10.35)

10.3.1.3. Equality and inequality constraints The equality constraints of the OPF, generally, reflect the power balance at the buses of the system. The power flow equations require that the injection of real and reactive power at each bus sum to be zero:


Pi = U i

625

N

∑ U [G k

ik

k =1

Qi = U i

cos( θ i − θ k ) + Bik sin( θ i − θ k ) ] − PG i + PL i = 0

N

∑ U [G k

ik

k =1

(10.36)

sin( θ i − θ k ) − Bik cos( θ i − θ k ) ] − QG i + Q L i = 0

where: PGi , QGi are active and reactive power of the generator i;

PLi , Q Li

– active and reactive power of the load i.

It is common for OPF problems to be formulated in polar form, since voltage magnitude limits are treated easily as simple variable limits. Despite this merit, OPF in rectangular form has advantages in that the second derivatives of power flow equations are constants and trigonometric functions are not including, [10.73]. Pi =

N

∑ (U (G U ' i

ik

' k

k =1

Qi =

N

∑

(U i" (GikU k'

− BikU k" ) + U i" (U k" Gik + U k' Bik )) − PGi + PLi = 0

(10.37) −

BikU k" ) − U i' (U k" Gik

k =1

+ U k' Bik )) − QGi

+ Q Li = 0

For each generator, a voltage set of points (Ω) can be enforced. In this case, an equality constraint for each generator is added, [10.71]:

U Gi − U Gi Ω = 0

(10.38)

A special attention must be paid to the inequality constraints of this problem. The inequality constraints of the OPF reflect the devices limits in the power system, as well as the limits created to ensure system security. Generators have maximum and minimum output active powers and reactive powers, which add inequality constraints:

PGi min ≤ PGi ≤ PGi max QGi min ≤ QGi ≤ QGi max

(10.39)

Transformer tap changing have a maximum and a minimum tap changing, which can be achieved:

tcik min ≤ tcik ≤ tcik max

(10.40)

For the maintenance of system security, power systems have transmission line, as wells transformer MVA ratings. These ratings may come from thermal ratings of conductors, or they may be set to a level due to the system stability concerns. To make the mathematics less complex, the constraint used in the OPF will limit the square of the MVA flow on a transformer or transmission line. 2

2

Sik − Sik max ≤ 0

(10.41)

626


To maintain the quality of electric service and system security, buses voltages, usually, have a desired voltage profile (maximum and minimum magnitudes): U i min ≤ U i ≤ U i max (10.42) All variables can be assumed to be continuous. The OPF algorithm also assumes this for the tap ratios of transformers, although this is not true for them. One possible solution for this problem is to round the optimal setting found assuming a continuous tap to the nearest discrete tap. This could be done for all transformers. However, three problems arise from this methodology. First, there is no guarantee that the rounded solution is the optimal solution. Second, the solution may become infeasible after rounding, i.e., some constraints may be violated. Finally, this methodology will not work well for discrete variables that have very large step sizes such as switched capacitor banks, [10.71].

10.3.1.4. Optimal power flow variables In order to handle the variables in the OPF problem efficiently, it is convenient to separate them into three categories: controls, states, and constraints: • The control variables correspond to quantities that can be arbitrarily manipulated, within their limits, in order to minimize the costs. These include generator MW outputs ( PGi ), transformer tap changing ( tcik );

• The states variables correspond to quantities that are set as a result of the controls, but must be monitored. They are also of interest at the solution. The states include all system voltages ( U i ) and angles ( θ i ); • The constraint variables are variables associated with the constraints. These include all the Lagrange multipliers.

10.3.2. Minimization of the active power losses (MAPL) 10.3.2.1. Introduction The minimization of the active power losses (MAPL) in power systems is a very important research issue. The generators, transformer taps, and switchable capacitors/inductors are, by far, the most useful devices in power losses minimization, [10.4, 10.10, 10.32, 10.44]. The difficulties encountered were basically the following: • A high number of variable devices. The dispatcher is faced with the problem of selecting a subset of effective controls to shift; • The desired corrective actions must be effected sometimes within a very short period; • The discrete character of certain control variables (the objective function can not be derived); • Unexpected convergence problems in certain cases.


627

It is very important to reduce the number of required adjustments, making the approach suitable for practical applications. Many problems can be quickly ascertained using first order sensitivity factors. These sensitivities can be used as a basis for inferences about input-output relationships. In this field several sophisticated and robust computational tools have been developed. This section investigates the minimization of losses by voltage-reactive power control, using the sequential quadratic programming (SQP) method. Consider the MAPL problem, [10.10, 10.32], Minimize F ([θ], [U ], [tc ]) = ΔP ([θ], [U ], [tc ]) +

+

1

∑ QGC (Q ) + T ∑ TSC i

i

i∈q

subject to: Pi ([θ], [U ], [tc ]) − PGi + PLi = 0

i ∈ n \ e , dimension N –1

Qi ([θ], [U ], [tc ]) − QGi + Q Li = 0 U i min ≤ U i ≤ U i max ,

(10.43)

i ∈ q , dimension Q,

ij ∈ r , dimension R,

0 ≤ I ij2 ([θ], [U ], [tc ]) ≤ I ij2 max , where: F ([θ], [U], [tc]) ΔP([θ],[U],[tc]) QGCi (Qi) TSCi j

i ∈ C , dimension C,

i ∈ n , dimension N,

Qi min ≤ Qi ([θ], [U ], [tc ]) ≤ Qi max ,

tci min ≤ tcij ≤ tci max ,

ij∈r ( Δtcij )

ij∈r

ij ∈ b , dimension B,

is – – –

objective function; active power losses; reactive generation cost at bus i; tapes shift cost at transformers (ij), determined by statistical data (tape shift wear and break-down risk); [U] – voltage magnitude vector, dimension N; [θ] – voltage angles vector, dimension (N – 1); [tc] – transformers tap changing matrix, dimension R; e – network slack bus; T – time period considered, [hours]; c, q, r – the sets of the consumers buses, buses with reactive control, respectively, buses with tapes swift. The MAPL variables are: [ Z ] = [[ X ][Y ]]t ; [ X ] = [[U q ][tc ]]t ; [Y ] = [[θ ] [U c ]]t

(10.44)

where: [X] is independent (free/control) variables; [Y] – dependent (basic/state) variables. The general minimization problem can be written in the following form:

628


Minimize F ( Z1 , Z 2 , ..., Z M ) , M = I + D subject to: hi ( Z1 , Z 2 , ..., Z M ) = 0 , i = 1, 2, ..., D g i ( Z1 , Z 2 , ..., Z M ) ≤ 0 , i = 1, 2, ..., M c

(10.45)

Most discrete-time iterative methods for this problem involve generating a sequence of search points Z k , via the iteration procedure: Z k = Z k + λk d k ; k = 0, 1, 2, K

(10.46)

The iterative procedure (10.46) in the case of the conjugated gradient can be written as, [10.10, 10.32]: d ( k ) = − g k + βk d k −1 , i =1, 2,K

(10.47)

d 0 =− g0

(10.48)

βk =

( gk , gk ) ( g k −1 , g k −1 )

(10.49)

The equality constraints arise from the operation equations at buses. The admissible limits of the powers, of the voltages at the buses and of the power flows give the inequality constraints. The number of the variables is equal to the size of the vector Z, M = I + D , where I and D are the number of the independent, respectively, dependent variables.

10.3.2.2. Sequential quadratic programming (SQP) Considering an initial point [Z0] and separating the constraints in two sets, active and inactive (passive) constraints, by development in Taylor series, the objective function and the constraints of the above nonlinear model, the following quadratic model (QM) is obtained: 1 Minimize FQ ([ Z]) = F 0 + [∇F]t [ΔZ] + [ΔZ]t [H ][ ΔZ] 2 (10.50) subject to: f a0 + [Jf a ][ΔZ ] = 0

[ ]

[ f ]+ [Jf ][ΔZ ] ≤ 0 0 1

1

where: [ΔZ] is the new variables vector (for the QM); [H] – Hessian of the objective function; [fa], [fl], [Jfa], [Jfl] – the vectors and matrices of the active (equality) and inactive (inequality) constraints. Using the above quadratic model (QM) a sequential quadratic programming (SQP) can be formulated. The solving of the initial problem (10.45), by sequential transforming in problems of the type (10.50), is performed with the algorithm given below.


629

Step 1. Initialize [Z], [Z] = [Z0]; Step 2. Calculate: [H], [∇F], [Ja], [Jl], [F0], [fa], [fl]; Step 3. Calculate [ΔZ], as solution of the QM; Step 4. [Z] = [Z] + [ΔZ]; Step 5. If ⎮[ΔZ]⎮> ε1 or ⎮[ΔF]⎮> ε2 Go to Step 2; Step 6. STOP.

The algorithm seems simple, but it includes a point key: the solution of the model (10.50). The model (10.50) belongs of the sequential quadratic programming, with linear equality and inequality constraints. There are many methods for solving these models, such conjugated reduced gradient, Newton, quasi-Newton and so on. In order to use the conjugated reduced gradient, the constraints of the above model will be handled by separating them in two sets: active and inactive:

[ ] 1 1 + [ΔX ] [H ][ΔX ] + [ΔY ] [H ][ΔY ] + [ΔX ] [H ][ΔY ] 2 2

Minimize FQ([ΔX ], [ΔY ]) = F 0 + [∇Fx ][ΔX ] + ∇Fy [ΔY ] + t

xx

t

yy

t

xy

[ f ]+ [Jf ][ΔX ] + [Jf ][ΔY ] = 0 [ f ]+ [Jf ][ΔX ] + [Jf ][ΔY ] ≤ 0

subject to: 0 1

0 a

ax

ay

1x

(10.51)

1y

The reduced gradient of FQ(ΔX,ΔY) model has the following form:

] [Jf ] [∇FQ ]

(10.52)

[∇FQx ] = [∇Fx ] + [H xx ][ΔX ] + [H xy ][ΔY ]

(10.53)

[∇FQx ] = [∇Fy ]+ [H yy ][ΔY ] + [H xy ][ΔX ]

(10.54)

[∇FQx ] = [∇FQx ] − [Jf a

where:

x

t

−1 ay t

y

After the elimination of the variable [ΔY], from (10.51) ÷ (10.54), the QM achieves the final form: 1 Minimize FQR([ΔX ]) = FQR 0 + [G ][ΔX ] + [ΔX ] t [H r ][ΔX ] (10.55) 2 [ΔY ] = Ty [ΔX ] + [T ] (10.56)

[ ]

where, the reduced gradient is: with:

[G ] = [∇FQR] = [G0 ] + [H x ][ΔX ]

(10.57)

[G0 ] = [ΔFx ] + [Ty ] t [ΔFy ] + [ [H xy ] + [Ty ] t [H yy ] ] [T ]

(10.58)

630


[H r ] = [H xx ] + [H xy ][Ty ]+ [Ty ] t [H xy ] t + [Ty ] t [H yy ][Ty ]

(10.59)

[T ] = −[Jf ] [Jf ] ; [T ] = −[Jf ] [ f ]

(10.60)

y

−1

ar

ax

−1

ar

0 a

10.3.2.3. Algorithm for solving SQP problem To solve this model, (10.55) ÷ (10.60), the following iterative process is used, [10.10, 10.32]:

[ΔX ] = [ΔX ] + λ [D ] ; [D ] = [[D ]] ; D k −1

k

k

0k

k

0k

k

k = 1,2,…

(10.61)

where: [Dk], λk – the direction of the motion from point [Xk] in the iteration (k), respectively, the length of the step in direction [Dk]. The length of the step is calculated with conjugated (reduced) gradient method:

[D ] = −[G ]+ [G[G ]] [[GG ] ][D ]; [D ] = −[G ] ; k > 1 [G ] [D ] λ =− [D ] [H ][D ] k

k

k

k −1

k

t

k −1

k −1

t

k>1

k

k

opt

0k

t

(10.62) (10.63)

0k

t

r

0k

(10.64)

With the constraint (10.51), the length of the step becomes: λ ad = min (λ j ) ; λ j = j

f l j (ΔX k −1 ) 0k

[∇f1x ]t [ D ]

; λk = min (λ opt , λ ad )

(10.65)

j

In order to reduce the number of control variables, in the beginning, the first order sensitivities of losses to all control variables are computed. Then, an “Efficient Coefficient” (EC) for power losses reduction can be defined as: max EC i = (∇ u P losses )i Δ u i

(10.66)

which represents the product of the ith component of the reduced gradient by the maximum amount that control i can be shifted, without causing any limit violation. Discarding those devices whose ECs are lower than a threshold, the remaining ones are considered in the optimization process, [10.9, 10.10, 10.44]. This is a quasioptimal algorithm, which has the following advantages, [10.10, 10.44]: • Control actions are sequential rather than simultaneous. This is more adapted to the operators needs and to the way they work (sequential, by nature).


631

• The number of control actions performed can be chosen by the operator, on the basis of incremental savings attainable, cost of operations and time availability. • Easy integration within the expert systems and related heuristics. • Moderate computational effort. Thus a simple gradient technique can be adopted in the power losses minimization, sacrificing optimality for the sake of usefulness, reliability and quick response. Step 1. Initialize: k = 0, [ΔX] =0, calculate [Yk] with (10.56); Step 2. k = k + 1; Step 3. Calculate: [G], [Dk], [λk] with (10.57), (10.62), (10.65); Step 4. Calculate [ΔXk], with (10.61); Step 5. Calculate [ΔY], with (10.56); Step 6. If all the direction are uselessly, Go to Step 7; Step 7. If one of the constraint becomes active, determine a new set of the active inequality and Go to Step 2; Step 8. STOP.

10.3.2.4. Tests and results The method outlined above for sensitivity analysis and optimization in this section has been applied to a test network of 220/110 KV (10 buses, 12 branches), Figure 10.8.

3

7 10

220 kV

110 kV

9+ j2

6+ j4

4 1

-20.2 – j11 (-20.2-j7.2)

2 49.8+ j78.5

-11+ j2.9

8

6 13+ j6 35+ j20 9 5

Fig. 10.8. Test network of 220 kV /110 kV.

15+ j25

632

Load flow and power system security Table 10.4 The objective function and control variables while iterative optimization process

Variable F[MW] F [%] tc 1-2 tc 5-6 tc 8-9 F[MW] F [%] tc 1-2 tc 5-6 tc 8-9

Number of QP Iteration

Initial amount

Conjugated reduced gradient iteration

Final amount

1

2

3

4

5

1

0.6184 100 12 9 5

0.5943 96.10 11.29 9.59 4.96

0.5879 95.07 11.76 10.17 2.03

0.5878 95.05 11.78 10.17 2.03

-

-

0.5945 96.13 12 10 2

2

0.6184 100 12 9 5

0.5943 96.10 11.29 9.59 4.96

0.5879 95.07 11.76 10.17 2.03

0.5878 95.05 11.78 10.17 2.03

0.5921 95.74 11.79 10.17 1.92

0.5920 95.73 12.04 10.47 1.92

0.5945 96.13 12 10 2

0.62

F [MW]

0.6 1

0.6 0

0.5 9

0.5 8

0

1

2

3 4 Iteration N um b er

5

6

Fig. 10.9. F [MW] as function of iterations number for a test network 220/110 kV.

10.3.3. Newton – Lagrange method (NL) 10.3.3.1. Introduction In this section the approach of the OPF problem is based on Newton method, which operates with Lagrangian function associated with the original problem. The Lagrangian function aggregates all the equality and inequality constraints, [10.53]. Newton method is well known in the area of power systems. It has been the standard solution algorithm for the power flow problem. Newton method is a very powerful solution algorithm, because of its rapid convergence near the solution. The solution of this problem by Newton – Lagrange method requires the creation of the Lagrangian as shown below, [10.71]. (10.67) L( Z ) = F ( X ) + μ t h( X ) + λ t g ( X ) where: Z = [X μ λ]t, λ and μ are vectors of the Lagrange multipliers; g(X) only includes the active (or binding) inequality constraints.


633

A gradient and Hessian of the Lagrangian may then be defined: ⎡ ∂L(Z ) ⎤ Gradient = ∇ Z L(Z ) = L z (Z ) = ⎢ ⎥ ⎣ ∂Z ⎦ L Xμ L Xλ ⎤ ⎡L ⎡ ∂ 2 L(z ) ⎤ ⎢ XX Hessian = [H ] = ⎢ 0 0 ⎥⎥ ⎥ = ⎢ LμX Z Z ∂ ∂ ⎣⎢ i j ⎦⎥ ⎢ L 0 0 ⎥⎦ ⎣ λX

(10.68)

(10.69)

The scarcity of the Hessian matrix will be exploited in the solution algorithm. From this, according to optimization theory, the Karush-Kuhn-Tucker necessary conditions of optimality are, [10.43, 10.71]: L X ( Z * ) = L X ([ X * , λ* , μ * ]) = 0; Lλ ( Z * ) = Lλ ([ X * , λ* , μ * ]) = 0; *

*

*

(10.70)

*

Lμ ( Z ) = Lμ ([ X , λ , μ ]) = 0;

λ*i ≥ 0 if g ( X * ) = 0 λ*i = 0 if g ( X * ) ≤ 0 μ *i = Real

where: Z* = [X*, λ*, μ*] is the optimal solution. Thus solving the equation Lz (Z*) = 0 will yield the optimal problem solution.

10.3.3.2. Inequality constraints handling A special attention must be paid to the inequality constraints of this problem. Lagrangian only includes those inequalities that are being enforced. For example, if a bus voltage is within the desired operating range, then there is no need to activate the inequality constraint associated with that bus voltage. For this Newton Lagrange method formulation, the inequality constraints will be handled by separating them into two sets: active and inactive, [10.43, 10.71]. Determination of those inequality constraints that are active is of utmost importance.

10.3.3.3. Soft constraints by using penalty functions When trying to solve a minimization problem, the nonexistence of a feasible solution can be encountered. Essentially, this means that too many constraints have been added to the problem. One way to avoid this issue is to implement soft inequality constraints in the form of penalty functions. The word “soft” signifies that the constraint is not absolutely enforced. The soft constraint only encourages the solution to meet the constraint by enforcing a penalty if the constraint is not met. In the OPF problem, soft equality constraints are not used, because of the nature of the equality constraints in the OPF problem. If the power flow equations cannot be violated, for the inequality constraints, the penalty functions offer a viable option.

634


Penalty functions are added to the objective function of the minimization problem. Ideally, a penalty function will be very small near a limit and increase rapidly as the limit is violated more. A well-suited penalty function for use in Newton method is the quadratic penalty function, [10.43, 10.71], which meets the requirements of a penalty function and is also easily differentiated for use by Newton - Lagrange method. 2 2 ⎧ 2 ⎛ ⎞ ⎪Wik = k ⎜⎝ Sik − Sik max ⎟⎠ ; ⎪ 2 ⎪ ⎧k U U i < U i min Penalty Functions ⎨ i min − U i ; ⎪⎪ ⎪W = ⎨0 ; U i min ≤ U i ≤ U i max ⎪ i ⎪ 2 ⎪ ⎩⎪k U i − U i max ; U i > U i max ⎩

(

)

(

)

(10.71)

While the inequality constraint is not violated, the penalty function has a value of zero. As the constraint begins to be violated, the penalty function quickly increases. Another advantage of the quadratic penalty function is the ability to control how hard or soft to make the constraint.

10.3.3.4. Model and algorithm for solving Newton – Lagrange problems In the above conditions, the optimal power flow problem can be written in the following form. Minimize

∑ (a + b P

i i ( generators )

i Gi

)

+ ci PG2i +

∑

αi i ( penalties )

gi ( X )

(10.72)

Subject to:

⎧ Pi = 0 ⎪ h( X ) = 0 ⎨Qi = 0 ⎪U − U = 0 i set ⎩ i

2 ⎧S 2 − S ≤0 max ik ik ⎪ ⎪ PG − PG max ≤ 0 i ⎪ i ⎪ PG i min − PGi ≤ 0 ⎪⎪ g ( X ) ≤ 0 ⎨U i − U i max ≤ 0 ⎪ ⎪U i min − U i ≤ 0 ⎪tc − tc ik max ≤ 0 ⎪ ik ⎪tcik min − tcik ≤ 0 ⎪⎩

The constraints on the reactive power at each generator are not included in the problem as stated above. These constraints will be taken care of by treating a generator bus at a Q limit as a load bus. This is commonly done in a power system, when modelling generator reactive power limits, [10.71].


635

The application of Newton-Lagrange method to the OPF algorithm can be as follows, [10.71]: Step 1. Initialize the OPF solution. a) Initial guess at which inequalities are violated. b) Initial guess Z vector (bus voltages and angles, generator output power, transformer tap ratios, all Lagrange multipliers). Step 2. Evaluate those inequalities that have to be added or removed using the information from Lagrange multipliers for hard constraints and direct evaluation for soft constraints. Step 3. Determine viability of the OPF solution. Presently this ensures that at least one generator is not at a limit. Step 4. Calculate the gradient and Hessian of the Lagrangian. Step 5. Solve the equation [H]ΔZ = ∇ZL(Z). Step 6. Update solution Zk+1 = Zk - ΔZ. Step 7. Check whether ||ΔZ || < ε. If not, go to Step 4, otherwise continue. Step 8. Check whether correct inequalities have been enforced. If not go to Step 2. If so, problem is solved.

10.3.3.5. Information gained from the OPF solution The OPF is capable of performing all of the control functions necessary for the power system. While the economic dispatch of a power system does control generator MW output, the OPF controls transformer tap changing. The OPF also is able to monitor system security issues including line overloads and low or high voltage problems. If any security problems occur, the OPF will modify its controls to fix them, i.e., remove a transmission line overload, [10.71]. Besides performing these enhanced engineering functions, the greatest advantage of the OPF is the great wealth of knowledge it yields concerning the economics of the power system. In studying the Lagrange multipliers associated with each constraint, one can show that they can be interpreted as the marginal costs associated with meeting the constraint. Therefore, the Lagrange multipliers, μ Pi and μQi , can be seen as the marginal cost of real and reactive power generation at bus i in [$/MWh] and [$/MVArh], respectively.

10.3.4. Interior-point methods (IPMs) 10.3.4.1. Introduction In the past years, research on interior-point (IP) methods, theory and computational implementation, have evolved extremely fast. Interior-point method

636


variants are being extended to solve all kind of programs: from linear, to nonlinear and from convex, to non-convex (the latter with no guarantee regarding their convergence). In the same way, they are also being applied to solve all sorts of practical problems. Optimization of power system operations is one of the areas, where IP methods are being applied extensively, due to the size and special features of these problems, [10.51, 10.61 ÷ 10.63, 10.65 ÷ 10.67, 10.73]. The computational effort for each iteration of an IP algorithm is dominated by the solution of large, sparse linear system. Therefore, the performance of any IP code is highly dependent on the linear algebra kernel. Interior-point methods are usually classified into three main categories, [10.15]: projective methods, affine-scaling methods and primal-dual methods. Projective methods include Karmarkar original algorithm, and are responsible for the great interest set to the area. Affine-scaling methods were obtained as simplifications of projective methods. They do not share all the good theoretical qualities of projective methods, but their reduced computational complexity and simplicity made them become very popular at the time and the most effective in practice. Primal-dual methods include path-following methods and potential reduction methods. The primal-dual algorithms that incorporate predictor and corrector steps are currently accepted as the computationally most effective variants. The first theoretical results for primal-dual path-following methods are due to Megiddo (1986), who proposed to apply a logarithmic barrier method to the primal and dual problems simultaneously. The path-following algorithms that incorporate Mehrotra predictor-corrector technique are, at present, accepted as the most computationally effective IP algorithms. Additional improvements to Mehrotra algorithm are usually achieved by applying multiple corrector steps. Although logarithmic barrier IP methods were devised to solve general NLP problems, research on IP methods for NLP has been lately motivated mainly by the superb performance of the IP variants for LP, an area that has received much attention and enjoyed incredible progress. The computational effort of each iteration of an IP method is dominated by the solution of large, sparse linear systems of the form

⎡ D −2 ⎢ ⎣A

At ⎤ ⎡ X ⎤ ⎡ r ⎤ ⎥⎢ ⎥=⎢ ⎥ 0 ⎦ ⎣ Y ⎦ ⎣ w⎦

(10.73)

Then, it is essential to consider efficient methods for their solution, which can be either direct or iterative methods. Typically, the linear systems (10.73) are solved using direct factorization. Direct methods usually consider the normalequations form,

[

Y = AD 2 At

] (AD r − w) −1

2

X = − D 2 At y + D 2 r

(10.74)

involving the symmetric positive definite matrix AD2At, or the augmented-system form, involving the symmetric indefinite matrix in (10.73).


637

In the next section we present the primal-dual and predictor-corrector method for the solution of the nonlinear programming. Forwards, an implementation issue in optimal unit commitment is addressed.

10.3.4.2. Primal-dual interior-point method (PDIP) A typical nonlinear programming problem that frequently arises in power engineering (for example, optimal power flow problem) has following general mathematical formulation: Minimize

F(X ) h( X ) = 0

subject to:

(10.75)

g min ≤ g ( X ) ≤ g max

where: X ∈ Rn

is a vector of decision variables, including the control and nonfunctional dependent variables; n F : R → R – a scalar function that represents the power system operation optimization goal; n m → R – a vector function with conventional power flow equations h:R and other equality constraints; g : Rn → Rp – a vector of functional variables, with lower bound gmin and upper bound gmax, corresponding to operating limits in the system.

Henceforth, we assume that F(X), hi(X) and gi(X) are twice continuously differentiable. The IP method described in this section transforms all inequality constraints in (10.75) into equalities, by adding non-negative slack vectors. The non-negativity conditions (s, z) ≥ 0 are handled by incorporating them into logarithmic barrier terms: p

Minimize F ( X ) − μ k

∑ (ln s + ln z ) ; i

i

s, z > 0

i =1

subject to: h( X ) = 0 − s − z + g max − g min = 0

(10.76)

− g ( X ) − z + g max = 0

where: μk > 0 is a barrier parameter that is forced to decrease to zero as iteration progress. To solve the equality-constrained problem (10.76), we use a LagrangeNewton method. Associated with the problem (10.76) is Lagrange function L( X , s, z , π, v, λ) that is given by:

638


L(Y ) = L( X , s, z , π, ν, λ ) = F ( X ) − μ k

p

∑ (ln s + ln z ) − λ h( X ) − i

i =1

i

t

(10.77)

− πt (− s − z + g max − g min ) − ν t (− g ( X ) − z + g max ) where: Y = ( X , s, z , π, v, λ ) . If X* is a local minimize of (10.75), then there exist vectors of Lagrange multipliers, say, (λ*, π*, v*), that satisfy the Karush-Kuhn-Tucher (KKT) optimality conditions: (10.78) Sπ = μ k e Zνˆ = μ k e

(10.79)

s + z − g max + g min = 0

(10.80)

g ( X ) + z − g max = 0

∇ X F ( X ) − J h ( X )t λ + J g ( X )t ν = 0

(10.82)

− h( X ) = 0

(10.83)

where: s ∈ R(p) and z ∈ R(p)

are

S = diag (s1, …, sp); Z = diag (z1, …, zp); ∇X F : Rn → Rn Jh : Rn → Rm x Jg : Rn → Rp x n λ ∈ Rm, π ∈ Rp and v ∈ Rp

) v = v + π; e = [1, 1, …, 1]t

(10.81)

slack vectors that transform the inequalities in (10.75) into equality (10.80), (10.81); – – – –

the gradient of F; the Jacobian of h; the Jacobian of g; vectors of Lagrange multipliers, called dual variables;

– a vector of ones.

The main steps of the primal-dual algorithm are: Step 0. Set k = 0, choose μ0 > 0 and a starting point Y0 = (s0, z0, π0, v0, X0, ) λ0), with (s0, z0, π0, v 0 ) > 0. Step 1. From the Newton system (10.78) – (10.80) at the current point Yk, solve for the Newton direction ΔY. Step 2. Compute the steps length α kP and α kD along ΔY and obtain a new

solution estimate as Yk+1=Yk+ α k ΔY . Step 3. If Yk+1 satisfied the convergence criteria, then stop. If not, then set k ← k + 1, compute the barrier parameter μk < μk-1, and return to Step 1. In Step 1, the Newton direction ΔY is obtained as the solution to the indefinite system of linear equations


639

⎤ ⎤ ⎡ Δs ⎤ ⎡ − Sπ + μ k e ⎢ ⎥ ) ⎥ ⎢ ⎥ k Z − Zv + μ e ⎥ ⎥ ⎢ Δz ⎥ ⎢ ⎥ ⎥ ⎢ Δπ ⎥ ⎢ 0 − s − z + g max − g min ⎥ ⎥⎢ ⎥ = ⎢ 0 Jg − g ( X ) − z + g max ⎥ ⎥ ⎢ Δv ⎥ ⎢ 0 J gt ∇ 2X L − J ht ⎥ ⎢ ΔX ⎥ ⎢ − ∇ X F ( X ) + J h ( X )t λ − J g ( X )t ν ⎥ ⎥ ⎥⎢ ⎥ ⎢ 0 0 − Jh 0 ⎥⎦ ⎢⎣ Δλ ⎥⎦ ⎢⎣ h( X ) ⎥⎦ (10.84) ) where: Π = diag (π1 , ..., π p ) , Υ = diag(vˆ1 , ..., vˆ p ) ⎡∏ 0 ⎢ 0 Υ) ⎢ ⎢I I ⎢ ⎢0 I ⎢0 0 ⎢ ⎢⎣ 0 0

S Z 0 0

0

0 0 0

0 0 0 0

The computation of HL involves a combination of the objective function and constraints Hessian: ∇ 2X L(Y ) = ∇2X F ( X ) −

m

∑

λ j ∇ 2X h j ( X ) +

j =1

p

∑ν ∇ g ( X ) j

2 X

j

(10.85)

j =1

We have dropped the iteration index k to simplify the presentation. The Newton direction can be obtained by solving (10.84) directly, or by solving the reduced system: ⎡ Jd ⎢− J ⎣ h

− J ht ⎤ ⎡ΔX ⎤ ⎡rX ⎤ = 0 ⎥⎦ ⎢⎣ Δλ ⎥⎦ ⎢⎣ rλ ⎥⎦

(10.86)

rX = −∇F ( X ) + J h ( X )t λ − J g ( X )t ν

(10.87)

rλ = h( X )

(10.88)

(

)

J d = ∇ 2X L(Y ) + μ k J g ( X )t S −2 + Z −2 J g ( X )

(10.89)

for ΔX and Δλ first, and then computing Δz = − J g ( X ) ΔX Δs = − Δz Δπ = −μ k S − 2 Δs

(10.90)

Δv = −μ k Z − 2 Δz − Δπ

In Step 2, a new solution estimate Yk +1 is obtained by X k +1 = X k + α kP ΔX

λk +1 = λk + α kD Δλ

s k +1 = s k + α kP Δs

π k +1 = π k + α kD π

z k +1 = z k + α kP Δz

v k +1 = v k + α kD Δv

(10.91)

640


where the scalars α kP ∈ (0, 1] and α kD ∈ (0, 1] are the step length that can be taken along Δy, given by: ⎧⎪ ⎧⎪ − s k ⎫⎪⎫⎪ − zk α kP = min ⎨1 , γ min ⎨ i Δsi < 0 , i Δzi < 0⎬⎬ i ⎪ Δs Δzi ⎪⎭⎪⎭ ⎪⎩ ⎩ i ) ⎧⎪ ⎫⎪⎫⎪ ⎧⎪ − π k − vk ) α kD = min ⎨1 , γ min ⎨ i Δπ i < 0 , )i Δvi < 0⎬⎬ i ⎪ Δπ Δvi ⎪⎭⎪⎭ ⎪⎩ ⎩ i

(10.92)

where γ ∈ (0,1) is a safety factor to ensure that Yk+1 will hold the strict positivity ) condition (s, z, π, v ) > 0. In Step 3, the kth iteration is considered converged if

{ {

}

{

}

ν1 = max max g − g ( X k ) , max g ( X k ) − g , h( X k ) ν2 =

ν3 =

∇ X F ( X k ) − J h ( X k )t λk + J g ( X k )t v k 1+ x

k

2

+ λ

2

+ v

∞

k

}≤ ε

1

(10.93)

≤ ε1

2

F ( X ) − F ( X k −1 ) k

ρk 1+ X k

k

∞

≤ ε 2 ; υ4 =

1 + F(X k )

2

≤ ε2

or μ k ≤ εμ ΔX

∞

g( X k )

≤ ε2 ∞

(10.93')

≤ ε1

ν4 ≤ ε2 where ε1, ε2 and εμ are predetermined tolerances, and ρk is the residual of the complementary conditions, obtained by: ) ρ k = ( s k )t πk + ( z k )t v k (10.94) k If not converged, then Y is reduced based on an expected decrease of the average complementary residual, as ρk μ k +1 = σ k , σ k = max 0.99σ k −1 , 0.1 (10.95) 2p where σ ∈ (0,1) is expected (not necessarily realized) decrease in ρ k, known as the centring parameter.

{

}

10.3.4.3. Predictor-Corrector Interior-Point Method (PCIP) Computing ΔY from (10.84) involves factorization of the coefficient matrix and two triangular systems solutions that follow the factorization.


641

What makes the Mehrotra predictor-corrector IP method very efficient is that a more successful search direction is obtained by solving two systems of linear equations, in each iteration. The two systems solutions, which define the predictor and corrector steps, involve the same coefficient matrix with two different righthand sides. Thus, only one matrix factorization is required and little additional work is needed to compute the corrector step using the matrix factorization from the predictor step, [10.63]. In the predictor-corrector method, rather that applying Newton method to (10.78) – (10.83) to generate correction terms to current estimate, the new point Y k +1 = Y k + α k +1ΔY is substituted into (10.78) ÷ (10.83) directly, to obtain the approximation: k − Sπ ⎤ ⎡μ e⎤ ⎡− ΔSΔπ⎤ ⎡ ) ⎥ ⎢ k ⎥ ⎢− ΔZΔvˆ ⎥ ⎢ − Zv ⎥ ⎥ ⎢μ e⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎢ − s − z + g max − g min 0 ⎥ 2 ∇ y L(Y ) ⋅ ΔY = ⎢ ⎥ (10.96) ⎥+⎢ ⎥+⎢ − g ( X ) − z + g max 0 ⎥ ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢− ∇F ( X ) + J h ( X ) t λ − J g ( X ) t v⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ h( X ) 0 ⎦⎥ ⎦⎥ ⎢⎣ 0 ⎥⎦ ⎣⎢ ⎣⎢

where: ∇Y2 L(Y ) is the coefficient matrix in (10.84); ΔS and ΔZ – diagonal matrices defined by the components of the vectors Δs and Δz. The major difference between the system (10.96) and (10.84) is the righthand side of the (10.96) cannot be determined beforehand, because of the nonlinear delta terms. The direction ΔY obtained from (10.96) consists of three direction components, [10.51]: ΔY = ΔYaf + ΔYce + ΔYco

(10.97)

where: ΔYaf is an affine direction, the pure Newton direction that is obtained when we set μk = 0 in (10.84). The affine direction is responsible for “optimization”, that is, reducing primal and dual infeasibility and complementary gap; ΔYce – a centring direction, whose size is governed by the adaptively chosen barrier parameter μk. The centring direction, given by the second vector from the right-hand side of (10.96), keeps the current point away from the boundary of the feasible region and ideally close to the barrier trajectory to improve the changes for a long step to be made in the next iteration; ΔYco – is a corrector direction, defined by the last vector in the right-hand side of (10.96) that attempts to compensate for some of the nonlinearity in the affine direction.

642


The Predictor Step To determine a step that approximately solves (10.96), first it drops the μ terms and the delta terms from the right-hand side of (10.96) and then solve for the affine direction: ⎡ Δsaf ⎤ ⎡ − Sπ ⎤ ) ⎢ Δz ⎥ ⎢ ⎥ − Zv ⎢ af ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Δ π s z g g − − + − af max min ∇Y2 L(Y ) ⋅ ⎢ ⎥=⎢ ⎥ − g ( X ) − z + g max ⎢ Δvaf ⎥ ⎢ ⎥ ⎢ΔX af ⎥ ⎢− ∇F ( X ) + J ( X ) λ − J ( X ) v ⎥ h t g t ⎢ ⎥ ⎢ ⎥ h( X ) ⎥⎦ ⎢⎣ Δλ af ⎥⎦ ⎢⎣ (10.98)

The affine direction ΔYaf is then used in two distinct ways: • to approximate the delta terms in the right-hand side of (10.96); • to dynamically estimate the barrier parameter μ. To estimate μ, first it considers the standard step length rule (10.92) to determine the step that would actually be taken if the affine direction ΔYaf were used. Second, an estimate of the complementary gap is computed from:

ρkaf = ( s k + α afP Δsaf )t (πk + α afD Δπaf ) + ( z k + α afP Δzaf )t (vˆ k + α afD Δvˆaf ) (10.99) Finally, an

estimate μ kaf

k+1

for μ

μ kaf

can be obtained from:

⎧⎛ ρ k ⎪ af = min ⎨⎜ k ⎜ ⎪⎩⎝ ρ

2 ⎫ ρk ⎞ ⎟ , 0.2⎪⎬ af ⎟ ⎪⎭ 2 p ⎠

(10.100)

This procedure chooses μ kaf to be small when the affine direction produces a large decrease in complementary, μ kaf 10 −3

Iteration 2 ⎡169.1651⎤ X = ⎢⎢155.8348⎥⎥ ; ⎢⎣ 300 ⎥⎦ 2

⎡0.1034 ⋅ 10 −3 ⎤ ⎢ ⎥ π 2 = ⎢0.0988 ⋅ 10 − 3 ⎥ ; ⎢ −3 ⎥ ⎢⎣0.0425 ⋅ 10 ⎥⎦

⎡87.9151⎤ s = ⎢⎢40.8348⎥⎥ ; ⎢⎣ 240 ⎥⎦ 2

⎡ 57.0848 ⎤ ⎢ ⎥ z = ⎢ 139.1651 ⎥ ; ⎢0.2875 ⋅ 10 −6 ⎥ ⎣ ⎦ 2

⎡ 0.0688 ⋅ 10 −3 ⎤ ⎢ ⎥ v 2 = ⎢− 0.0988 ⋅ 10 −3 ⎥ ; ⎢ −2 ⎥ ⎢⎣ 0.5924 ⋅ 10 ⎥⎦

λ 2 = 1.0138

α 2P = 0.22198 ; α 2D = 0.01308 ; F ( X 2 ) = 411.2703 ; μ 2 = 0.5527 ⋅ 10 −3 < 10 −3

ΔX

∞

= 26.4963 > 10 −3 ;

h( X 2 )

∞

= 0 < 10−3 ;

ν 24 = 0.004373 > 10 −3


653

Iteration 3 ⎡189.7707⎤ X = ⎢⎢135.2292⎥⎥ ; ⎢⎣ 300 ⎥⎦ 3

⎡ 36.4792 ⎤ ⎡108.5207⎤ ⎢ ⎥ ⎢ ⎥ 3 s = ⎢ 20.2292 ⎥ ; z = ⎢ 159.7707 ⎥ ; ⎢0.2874 ⋅ 10 −6 ⎥ ⎢⎣ 240 ⎥⎦ ⎣ ⎦ 3

⎡ 0.0688 ⋅ 10 −3 ⎤ ⎡0.1034 ⋅ 10 −3 ⎤ ⎢ ⎥ ⎢ ⎥ π3 = ⎢0.0988 ⋅ 10 − 3 ⎥ ; v 3 = ⎢− 0.0988 ⋅ 10 − 3 ⎥ ; λ 3 = 1.01382 ⎢ ⎢ ⎥ −2 ⎥ 0.0425 ⋅ 10 − 3 ⎥ ⎢⎣ 0.5924 ⋅ 10 ⎦⎥ ⎣⎢ ⎦ α 3P = 1 ; α 3D = 0.94464823 ⋅ 10 -6 ; F ( X 3 ) = 408.5067 ; μ 3 = 0.4951 ⋅ 10 −3 < 10 −3

ΔX

∞

= 20.6056 > 10 −3 ; h( X 3 )

∞

= 0 < 10 −3 ; υ34 = 0.006748 > 10 −3

Iteration 4 ⎡189.77⎤ X = ⎢⎢135.23⎥⎥ ; ⎢⎣ 300 ⎥⎦ 4

⎡ 36.4782 ⎤ ⎡ 108.52 ⎤ ⎢ ⎥ ⎢ ⎥ 4 s = ⎢ 20.23 ⎥ ; z = ⎢ 159.7717 ⎥ ; ⎢0.2874 ⋅ 10 −6 ⎥ ⎢⎣ 240⎥⎦ ⎣ ⎦ 4

⎡0.1034 ⋅ 10 −3 ⎤ ⎢ ⎥ π 4 = ⎢0.0988 ⋅ 10 − 3 ⎥ ; ⎢ ⎥ 0.0425 ⋅ 10 − 3 ⎥ ⎣⎢ ⎦ λ 4 = 1.01382 ;

α 4P = 1 ;

α 4D = 0.94464823 ⋅ 10 -5 ; F ( X 4 ) = 408.5067 ;

μ 4 = 0.4951 ⋅ 10 −3 < 10 −3 ;

h( X 4 )

∞

⎡ 0.0688 ⋅ 10 −3 ⎤ ⎢ ⎥ v 4 = ⎢− 0.0988 ⋅ 10 − 3 ⎥ ; ⎢ ⎥ 0.5924 ⋅ 10 − 2 ⎥ ⎣⎢ ⎦

= 0 < 10−3 ;

ΔX

∞

= 0.9527 ⋅ 10 −3 < 10 −3 ;

ν 44 = 0.6748 ⋅ 10−9 < 10−3

10.5. Optimal unit commitment in deregulated market 10.5.1. Dynamic optimal power flow by interior-point methods 10.5.1.1. Problem formulation With deregulation and open access in the utility industry occurring internationally, there are pressures to not only optimize the operation of generation resources, but also transmission systems. During the past decade, some researchers

654


have incorporated the simplified network model into the scheduling problem formulation by using Lagrangian relaxation (LR), augmented Lagrange relaxation (ALR) and so on. The OPF considers only power system at a particular instant and generally is unable to model time-related constrains (e.g. ramping rates) or energy related constraints (e.g. generation contract, fuel storage and reservoir capacity). With time related and energy related constraints considered, OPF becomes a dynamic OPF (DOPF) problem, [10.66]. However, the OPF problem for a large system in itself is a very large complex nonlinear programming problem. Adding time-related constraints and new decision variables creates a very large scale complex problem encompassing both time and the network features. The solution method is therefore very crucial for the success of this integration. Other papers propose an iterative multi-stage method, to integrate power system scheduling and OPF together. The optimization problem is then decoupled into a primary stage, of minimizing the cost of active power generation, and a secondary stage, of minimizing the power system losses. The result of one-stage is used to update the parameters of another stage. However, like other iterative methods, this method may have convergence problem. By introducing a set of duplicating variables and relaxing the duplicating equations with a set of Lagrangian multipliers, the problem can be decomposed into a standard power system scheduling problem and a series of OPF problems, one for each hour. In this case, the genetic algorithms can be used to transfer a population of dual feasible solutions into a good primal feasible solution. A direct nonlinear interior-point method (IPM) is adopted [10.66], to solve the DOPF problem as a single optimization problem, rather than many separated optimization sub-problems coordinated by LR or other techniques. The proposed method takes advantage of both the super sparsity technique of Newton OPF and the advantage of IPMs to handle inequality constraints. Consider a power system with NG units, the objective is to minimize the total generation cost over a scheduling horizon. This minimization is subject to the time separated constraints, such as load demand and reserve requirement, and the time related constraints which couple the generation of individual unit across hours. The time unit is one hour and the planning horizon T may vary from several hours to a week. A nonlinear DOPF model can therefore be formulated as follows, [10.66]. Objective function The DOPF extends the objective function of the OPF problem to minimize the generation cost across the whole time horizon:

Minimize F =

Nt NG

∑∑(F (P t=1 i=1

Pi

Git ) + FQi (QGit ))

where: NG is the number of generators; Nt – the number of time periods for the research horizon.

(10.114)


655

In general, it is more difficult to obtain an expression of FQi (QGit ) than FPi ( PGit ) . However, it is very important to compensate generators for their MVAr contribution in an electricity market. The quadratic curve expression that is similar to the active power generation cost is adopted in this section. An extra advantage brought by the introduction of FQi (QGit ) is that reactive power generations can be treated as basic variables like active power naturally, it therefore enhances the numerical stability of the DOPF. Time-separated constraints

Within every single time period, the following constraints should be satisfied (subscript t is omited for simplicity): • Power output limits of generator i: PGi min ≤ PGi ≤ PGi max QGi min ≤ QGi ≤ QGi max

(10.39)

• Spinning reserve for generator i:

⎧⎪ PGi max − PGi if PGi ≥ (1 − f i ) PGi max (Type 1) Ri = ⎨ if PGi < (1 − f i ) PGi max (Type 2) ⎪⎩ f i PGi max

(10.115)

The system minimum spinning reserve constraint:

∑R ≥ R i

min

(10.116)

i

• Bus voltages limits for node i: U i min ≤ U i ≤ U i max

(10.42)

• Power flow equations for node i: Pi = U i

N

∑ [U [G k

ik

k =1

Qi = U i

(10.36)

N

∑ [U [G k

k =1

cos(θ i − θ k ) + Bik sin(θ i − θ k )]] − PGi + PLi = 0

ik

sin(θ i − θ k ) − Bik cos(θ i − θ k )]] − QGi + QLi = 0

• Branch flow limits for branch i-k:

Pik min ≤ Pik ≤ Pik max

(10.117)

Other control variables like taps of LTC transformers and shunt capacitance can be included as well.

656


Time-related constraints The time-related constraints, like ramping rates, are dynamic operational constraints. As a result of these dynamic operational constraints, the operational decision at a given hour may affect the operational decisions at a later hour.

• Ramping rates constraints: The loading and unloading rates of a generator should be less than or equal to certain maxim. These ramping rate constraints can be stated as: PGit − RampGi ≤ PGi ,t +1 ≤ PGit + RampGi

(10.118)

In (10.118), upper limit and lower limit of ramping rates are in the same manner. For the notation of simplicity, only the upper limit is shown in the following derivation:

PGi ,t +1 − PGit ≤ PR max i

(10.119)

where: PR maxi = RampGi • Generation contract In a power market, generators may sign a contract with pool (or customers directly) to guarantee the amount of generation output for a time horizon. Pc min i ≤

∑P

Git

t

≤ Pc max i

(10.120)

In some cases, generators have fixed generation contract for the scheduling horizon:

∑P t

where: Pcmin i

= Pc i

G it

(10.121)

is lower limit of generation contract i;

Pcmax i

–

upper limit of generation contract i;

Pci

–

active power of generation contract i;

Pcit = PGit –

active power of generation i for time period t.

Here Pci means the maximum available energy of unit i for the contract period. In [10.66], constraint (10.121) is handled as inequality constraint, in order to utilize the advantage of interior-point method of dealing with inequality constraints: Pci − δ ≤

∑P t

where δ is a very small number. • Other energy related constraints

Gi t

≤ Pci + δ

(10.122)

657


Notably, other energy limited constraints, such as total hydro energy in a watershed, total fuel constraints and emission constraints can be treated in the similar manner to (10.120). These constraints couple the generation of individual units across hours. DOPF Model In summary, the DOPF problem can be formulated as, [10.66]:

Minimize

∑ Ft ( X t ) t

subject to:

ht ( X t ) = 0 ;

g min t ≤ g t ( X t ) ≤ g max t PGt +1 − PGt ≤ PR max Pc min i ≤

∑P t

Gi t

≤ Pc max i

t = 1,K, N t t = 1,K, N t t = 1,K, N t − 1

(10.123)

i = 1,K, N c

where: Nc is the number of generation contract. The first two constraints in equations (10.123) are time separated constraints and the last two are corresponding to ramping rates constraints and generation contracts constraints.

10.5.1.2. Algorithm of the DOPF method A nonlinear primal dual interior-point method (PDIP) is applied to the DOPF problem in (10.123). The key idea is to construct a border blocked matrix in which every diagonal sub matrix is an augmented Hessian matrix in Newton OPF of the corresponding time period and can be decoupled from each other. DOPF algorithm can be summarized as follows, [10.66]. Step 1. Change inequality constraints to equality by introducing slack variables in (10.123); Step 2. Construct the corresponding Lagrangian function; Step 3. Apply first order optimality conditions to Lagrangian function; Step 4. Find Newton search direction; Step 5. Reduce system scale by eliminating time separated slack variables and Lagrangian multipliers from the obtained system; Step 6. Reduce system scale by eliminating time-related slack variables; Step 7. Rearrange the remaining variables orders by period, construct the border blocked system; Step 8. Solve the obtained system by the primary-first-secondary phase solution process; Step 9. Update barrier factor, primal and dual variables; Step 10. Check convergence, if not converged, go back to Step 7.

658


The above multiperiod DOPF problem is solved as a single large sparse optimization problem, rather than many separated optimization sub-problems coordinated by Lagrangian relaxation (LR) or other techniques. Theoretically, the proposed method is superior to the LR method both in the sense of CPU time and iteration numbers. The approach described involves heavy computational burdens. Performing OPFs for all periods in a scheduling horizon, and representing this operation at each dual iteration, is the major task. The computational effort for the OPF sub problem can be expected to increase at least linearly with the number of periods in the scheduling horizon and more than linearly, with the number of buses and generators represented in the network model.

10.5.2. Power market oriented optimal power flow 10.5.2.1. Problem statement With deregulation and open access of electricity supply industry occurring internationally, there are increasing pressures to develop power market oriented OPF algorithms, which are expected to be able to handle new added functional type inequality constraints, e.g. network congestion, generation bundling and pricedependent loads. A nonlinear interior-point method based OPF is proposed in [10.67] for this purpose, which inherits both the super-sparsity technique of Newton OPF and the advantages of interior point methods in handling inequality constraints efficiently. Moreover, prices of various components are obtained directly from corresponding Lagrangian multipliers. Optimal power flow is predicted to be playing a crucial role in the future power market. However, the trend towards competitive power market also imposes great pressure to turn the conventional OPF to a Power Market Oriented OPF due to the following challenges, [10.67]: • While power systems are becoming increasingly large and complex, OPF problems are becoming more and more difficult. Advanced solution methods with reliable, robust and fast iterative process, which are capable to efficiently handle both equality and inequality constraints, are the key factors for a successful OPF application. • Power market imposes more and more coupling type constraints (functional inequality constraints): e.g. network congestion (coupling between nodes), generation bundling (coupling between units, such as spinning reserve) and price-dependent consumption (coupling between demand and price). • It is highly desirable to account for all these constraints by making minor changes to the standard OPF formula in order to most utilize the existing OPF code. • Security pricing helps the operator to maintain the system security as well as gives the market participators an indication, which can correctly reflect the cost of security. However, this is not an easy task as the security price

659


obtained from OPF is usually highly volatile. A power market oriented OPF solver should be able to output smoother security price signals. However, in spite of the extensive bibliography on this subject, only a few methods have been considered with potential enough to solve the complex problem of OPF in power market environment with the inherit difficulties of its dimension. Among the nonlinear programming algorithms proposed for the solution of OPF, interior-point methods have shown good properties in terms of fast convergence and numeric robustness, besides the flexibility in the treatment of inequality constraints. Moreover, IPMs based OPF can output smooth security price signals due to the application of logarithmic barrier function.

10.5.2.2. Interior-point algorithm for OPF Emphasis on the solver is placed in [10.67] – a power market oriented OPF algorithm employing a primary-dual interior point (PDIP) method. Three typical coupling type constraints, i.e. network congestion, spinning reserve and demandprice elasticity (DPE) are modelled. The Lagrangian function of the problem is given by, [10.67] L = F ( X ) − λ t h( X ) + πt g ( X )

(10.124)

where λ and π are the Lagrangian multipliers of the equality and inequality constraints respectively. Using interior-point technique for the this problem, the model for OPF is obtained, those algorithm is presented below, [10.67]: Step1. Initialization; Step 2. Choosing step size; Step 3. Barrier factor; Step 4. Convergence criteria.

In this model, λ P and λ Q in OPF can be interpreted as short-term marginal cost of active power and reactive power respectively. This algorithm, based on interior-point method, is developed in order to cope with the new features of OPF introduced by the emerging power markets. Although the models for three typical coupling type constraints, i.e. network congestion, spinning reserve and demand-price elasticity, are constructed and included in the OPF, and the calculation burden of the modification of Hessian matrix is very limited. This method inherits all the advantages of original Newton OPF the primary-dual interior-point method. The obtained security price signals are smoother and more predictable due to the applications of logarithmic penalty function. One attractive topic for future research may be an integration of the proposed model and multiperiod power system scheduling problems to set up real time prices.

660


10.6. Optimization strategies in deregulated market 10.6.1. Bidding problem formulation The electric power industry worldwide is experiencing unprecedented restructuring. The core of the restructuring is deregulation of the industry and introduction of competition among power suppliers and consumers, [10.29, 10.67]. In deregulated power systems, a free market structure is advocated for competition among participants as generators and consumers. To ensure nondiscriminatory access to the transmission networks by all participants, an Independent System Operator (ISO) is created for each power system. The ISOs are mandated to maintain the physical integrity of transmission system, and to be independent of financial concerns of the market participants. Several different ISOs exist for different regions and market structures. However, the common responsibility of all ISOs is to find an optimal operation schedule, taking into account physical constraints of the transmission networks, particularity of management of congestion. There are two distinct models for ISOs: the pool model and the bilateral/multilateral model. In the pool model, all suppliers and consumers transact with the pool, where the least expensive is dispatched. Therefore, the responsibility of a power exchange (PX) – determining generators outputs by economic dispatch functions − is managed by the ISO. This idea is based on the concept that a natural monopoly would ensure a least cost dispatch of all generators in the system. In the bilateral model all suppliers and consumers transact witch each other, independently. This model is based on the concept that the regulation of the commercial market should be minimized, and the market efficiency is achieved by consumers choosing their own suppliers, [10.47]. The pool is an e-commerce market place. The ISO uses a market-clearing tool, to clear the market, which is normally based on a single round auction, [10.49]. In the deregulated market based on the pool, participants submits bids energy, as PX piece-wise linear and monotonically increasing (suppliers)/decreasing (consumers) to ISO, who decides Marginal Clearing Price (MCP) and hourly generation levels of each participant over a 24-hours period. The relationship between ISO and participants is shown in Figure 10.11, [10.29, 10.69]. The supply bid curves are aggregated by the ISO, to create a single “supply bid curve”. The demand bid curves are also aggregated, to create a single “demand bid curve”, Figure 10.12. Although most electricity is bought and sold under long-term bilateral contracts, perhaps 5 to 10% will be traded day-ahead and in real-time. These shortterm trades could be a consequence of changed circumstances (e.g., a competitive retail provider signed up more customers than it anticipated or a generator completed its planned maintenance outage faster than it expected to), or they could be part of a company risk management strategy. In this case, ISO calls for suppliers and buyers to submit hourly bid by 10:00 a.m., on the day before the operating day.

661


The ISO then evaluates these bids using its security constrained unit commitment optimization computer model. This model schedules generation and price responsive demand hour-by-hour for the operating day so as to respect all generator and security (reliability) constraints and to minimize operating cost. E n ergy Price [$/M W h ]

A ggregated D em an d Bid C urve

ISO Generation Level & ECP

Generation Level & ECP Bidding Strategies Participant 1

Participant 2

A ggregated Supply Bid C urve

MCP

E n ergy [M W h ]

Participant 3 System D em an d

Fig. 10.11. Relationship between ISO and participants.

Fig. 10.12. Market Clearing Price.

Based on supply bid curve and demand bid curve, ISO determines Market Clearing Price (MCP), Figure 10.12, [10.29]. The suppliers in the market energy are paid at the MCP. For the bid curves, which are the true marginal ones, the following remarks can be performed. If a generator unit is on the margin (MCP) during all the hours of scheduling period, total payment it receives will equal its energy bid. If a generator unit is not on the margin, the total payment it receives will exceed its energy supply bid, [10.69]. Therefore, the MCP must be calculated for each hour of the scheduling period so that to be sufficient to completely recover all costs of committed generator units, [10.30]. MCP is public information made available by the ISOs, but aggregate offers and demands are not available in many electricity markets, [10.49]. Producers and consumers rely on price forecast information to prepare their corresponding bidding strategies. If a producer has a good forecast of next day MCP, it can develop a strategy to maximize its own benefit and establish a pool bidding technique, to achieve its maximum benefit. Similarly, once a good next-day price forecast is available, a consumer can derive a plan to maximize its own utility, using the electricity purchased from the pool. In the current literature, approaches based on neural networks, time series are used in the forecast of next-day electricity price, [10.49]. In the literature, the simulation has been used for its simplicity, and bidding strategies are discretized, such as “bidding high”, “bidding low”, or “bidding medium”. With discrete bidding strategies, payoff matrices are constructed by enumerating all possible combinations. It is very interesting to find a model and a method for optimization bidding strategies from the viewpoint of a utility, say Participant i, [10.69]. We assume that for all generators, the production cost is a quadratic function of generated power, Ci = ai + bi Pi + ci Pi 2 is the active output of generator i; where: Pi [MW] ai , bi , and ci – the coefficients of the cost function.

(10.34)

662


Since unit commitment is incomplete without the consideration of the effective constraints, really, for the optimum determination in a commitment problem, the following information are required, too: • minimum up-time and minimum down-time; • ramp-up limit and ramp-down limit; • start-up ramp rate and shut-down ramp rate and so on. The incremental cost (IC) at any point of the cost curve (CC) is the derivative evaluated at this point, ICi = 2 ci Pi + bi

(10.125)

10.6.2. Ordinal optimization method The ISO receives energy bids from suppliers and consumers and determines, for every hour, the MCP, the power production of every bidding generator, and the consumption level of every consumer. The power to be awarded to each bidder is then determined based on individual bid curves and the MCP. All the power awards will be paid at the MCP. After the auction closes, each bidder aggregates all its power awards as its system demand, and performs a unit commitment/scheduling, to meet its obligations at the minimum, cost over the bidding horizon, [10.29]. The aggregated supply bid curve and the powers awards are made in the ascending order of the bids, the goal of the ISO being the maximization of the total social welfare of generators and consumers, [10.47]. From ISO point of view, its problem is deterministic, [10.69]. When the supplier i solves the ISO problem, it only has distributions parameters for the others participants and its parameters, which must be optimized, so to maximize its own benefit. A survey of literature on unit commitment reveals that various numerical optimization techniques (integer programming, dynamic programming, Lagrangian relaxation, genetic algorithms) have been employed to approach this problem. Since bidding problems are generally associated with the uncertainty and complexity of the market and with the computational difficulties, it is more desirable to ask which solution is better as opposed to looking for an optimal solution. A systematic bid selection method based on ordinal optimization is developed in [10.29] to obtain “good enough bidding” strategies for generation suppliers. Ordinal optimization provides a way to obtain reasonable solution with much less effort. The ordinal optimization method has been developed to solve complicated optimization problems possibly with or without uncertainties. Ordinal optimization is based on the following two tenets: • It is much easier to determine “order” than “value”. To determine whether A is better or worse than B is a simpler task than to determine how much better is A than B (i.e. the value of A-B) especially when uncertainties exist.

663


• Instead of asking the “best for sure”, we seek the “good enough with high probability”. This goal softening makes the optimization problem much easier. To apply ordinal optimization framework to integrated generation scheduling and bidding, major efforts are needed to build power market simulation models and to devise a strategy to construct a small but good enough select set. The basic idea is to use a rough model that describes the influence of bidding strategies on market clearing price, MCP. In this section we assume that each generator bids more values (slices) in the market, one for each generator slice, corresponding to the each portion of cost curve. This is equivalent to assuming that each generator owns more slices (production units), with “bidding low”, “bidding medium”, “bidding high”, and so on.

10.6.3. Numerical results and discussions To illustrate the simulating action and MCP determination, a simple pool model with four generators is considered. Each generator can be equalized with one or more generator slices. Using the characteristics presented in Table 10.7, the bid prices are calculated, Table 10.8, Table 10.9, respectively, Table 10.10. Table 10.7

Economic Characteristics for the Generators, [10.71] Generat or Unit 1 2 3 4

a [$/h] 105 96 105 94

b [$/MW h] 12 9.6 13 9.4

c [$/MW2 h] 0.0120 0.0096 0.0130 0.0094

Min Generation [MW] 50 50 50 50

Max Generation [MW] 250 250 250 250

C1 = 105 + 12.0 P + 0.0120 P 2 C 2 = 96 + 9.60 P + 0.0096 P 2 C 3 = 105 + 13.0 P + 0.0130 P 2

(10.126)

C 4 = 94 + 9.40 P + 0.0094 P 2 Table 10.8

Generator Bid Prices (50 MW Slices) [$/MWh] – bidding low Slices Number 1 2 3 4 5

Size [MW] 50 50 50 50 50

Generated Power GP [MW] 50 50 - 100 100 - 150 150 - 200 200 - 250

Generator Bid Price [$/MWh] BP1 BP2 BP3 BP4 15.30 12.48 16.40 12.22 15.45 12.48 16.65 12.22 16.30 13.12 17.60 12.85 17.32 13.92 18.73 13.63 18.42 14.78 19.92 14.48

664

Load flow and power system security Table 10.9 Generator Bid Prices (50 and 100 MW Slices) [$/MWh] – bidding medium

Slices Number 1 2 3

Size [MW] 50 100 100

Generated Power GP [MW] 50 50 - 150 150 - 250

Generator Bid Price [$/MWh] BP1 BP2 BP3 BP4 15.30 12.48 16.40 12.22 16.30 13.12 17.60 12.85 18.42 14.78 19.92 14.48

Table 10.10 Generator Bid Prices (250 MW Slices) [$/MWh] – bidding high

Slices Number 1

Size [MW] 250

Generated Power GP [MW] 50 - 250

Generator Bid Price [$/MWh] BP1 BP2 BP3 BP4 18.42 14.78 19.92 14.48

Other combinations can be performed, apart from the considered cases, “bidding low”, “bidding medium” and “bidding high”, simultaneously, for all the suppliers. The unit commitment and aggregated supply bid curve (50 and 100 MW Slices) for “bidding medium” are given in Table 10.11 and Figure 10.13. Table 10.11 Unit Commitment for various Demands (with 50 and 100 MW slices) – bidding medium – Merit Order 1 2 3 4 5 6 7 8 9 10 11 12

Bid Price [$/MWh] 12.22 12.48 12.85 13.12 14.48 14.78 15.30 16.30 16.40 17.60 18.42 19.92

Generator Unit 4 2 4 2 4 2 1 1 3 3 1 3

Number Slices 1 1 2 2 3 3 1 2 1 2 3 3

Demand [MW] 50 100 200 300 400 500 550 650 700 800 900 1000

Awards Powers [MW] AP1 AP2 AP3 AP4 0 0 0 50 0 50 0 50 0 50 0 150 0 150 0 150 0 150 0 250 0 250 0 250 50 250 0 250 150 250 0 250 150 250 50 250 150 250 150 250 250 250 150 250 250 250 250 250

Table 10.12 Awards and Revenues for the Generators Units (50 MW Slices) – bidding low Demand [MW]

MCP [$/MWh]

200 400 600 800

12.48 13.92 15.45 17.32

Awards Powers [MW]

Generators Revenue [$/h]

AP1 AP2 AP3 AP4

GR1 GR2

0 100 0 100 0 200 0 200 100 250 0 250 200 250 100 250

0 0 1545 3465

GR3

1248.00 0 2784.00 0 3862.50 0 4331.25 1732.5

GR4 1248.00 2784.00 3862.50 4331.25

Total Revenues [$/h] 2496 5568 9270 13860

665

Steady state optimization 22

18

3 MCP_800

3

Bid Price [$/MWh]

17.60 MCP_600 16.30

1

MCP_200 4

12.85

4

12

3

4

MCP_400

14.48

1

1

2

2

2

8

4

0 16 0

200

400

600

800

1000

Generators Bid [MWh]

Fig. 10.13. Aggregated Supply Bid Curve with 50 and 100 MW Slices – bidding medium. Table 10.13 Awards and Revenues for the Generators Units (50 and 100 MW Slices) – bidding medium Demand [MW]

MCP [$/MWh]

200 400 600 800

12.85 14.48 16.30 17.60

Awards Powers [MW] AP1 AP2 AP3 AP4 0 50 0 150 0 150 0 250 100 250 0 250 150 250 150 250

Generators Revenue [$/h] GR1 GR2 GR3 GR4 0 642.33 0 1927 0 2171.40 0 3619 1630 4075.00 0 4075 2640 4400.00 2640 4400

Total Revenues [$/h] 2569.33 5790.40 9780.00 14080.00

Table 10.14 Awards and Revenues for the Generators Units (250 MW Slices) – bidding high Demand [MW]

MCP [$/MWh]

200 400 600 800

14.48 14.78 18.42 19.92

Awards Powers [MW] AP1 AP2 AP3 AP4 0 0 0 200 0 150 0 250 100 250 0 250 250 250 50 250

Generators Revenue [$/h] GR1 GR2 GR3 GR4 0 0 0 2896 0 2217 0 3695 1842 4065 0 4065 4980 4980 996 4980

Table 10.15 Benefits for the Generators Units (50 MW Slices) – bidding low Demand [MW] 200 400 600 800

Generators Benefit [$/h] GB1 GB2 GB3 GB4 0 96.00 0 120.00 0 384.00 0 434.00 120 766.50 0 831.00 480 1235.25 197.50 1299.75

Total Benefits [$/h] 216.00 818.00 1717.50 3212.50

Total Revenues [$/h] 2896 5912 9972 15936

666

Load flow and power system security Table 10.16 Benefits for the Generators Units (50 and 100 MW Slices) – bidding medium Demand [MW] 200 400 600 800

Generators Benefit [$/h] GB1 GB2 GB3 GB4 0 42.33 0 211.5 0 419.40 0 587.5 205 979.00 0 1043.5 465 1304.00 292.50 1368.5

Total Benefits [$/h] 253.83 1006.90 2227.50 3430.00

Table 10.17 Benefits for the Generators Units (250 MW Slices) – bidding high Demand [MW] 200 400 600 800

Generators Benefit [$/h] GB1 GB2 GB3 GB4 0 0 0 546.5 0 465 0 663.5 417 969 0 1033.5 1125 1884 208.5 1948.5

Total Benefits [$/h] 546.5 1128.5 2419.5 5166.0

The comparison of results for the three strategies shows the influence bidding low/medium/high on MCP, awards and revenues for the generator units. Ideally, each generator participant in the energy market will select the binding strategy that maximizes its profits. Using a discrete bidding strategy, a simulation bidding can be performed for multiple possible combinations. The number of combinations depends on the available information. Using an ordinal optimization technique, a good enough bidding strategy is obtained, with reasonable computation effort. In a typical short-term forward wholesale electricity market where products are auctioned sequentially, one often observes significant market inefficiency and price volatility – thus the recent growing impetus in developing integrated short-term forward markets where electric energy, reserves, and transmission capacity are auctioned simultaneously. Such markets need new computational methods and models for determining market clearing price and physical (delivery/consumption) schedules, [10.31].

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Chapter 11 LOAD FORECAST

11.1. Background Since the electric energy storage on a large scale is not possible, the main role of the power network is to transport the demanded energy to consumers. Therefore, it is very important to study and analyse the evolution of the load in order to operate and design the power network, because all the other decisions are based on the consumed energy [11.3, 11.16]. The importance of accurate load estimation for a long time period is also due to the following aspects [11.12]: • the designing and construction period of power plants, substations, electric lines, etc. is quite long (5 – 15 years); • the power equipment life is very long (35 – 150 years); • the investment costs are very high (million, even billion dollars); • the damages caused by the power supply interruption are very important; • the optimisation cost is a basic condition for the exploitation of a power system, it being possible to be achieved if accurate present and future amount of power consumption is known. The load forecast is the scientific activity that targets the estimation of the energy and power consumption based on the analysis of miscellaneous information, so that the estimated consumption can finally match the real one [11.5]. The load forecast has the following characteristics [11.13]: • it is a dynamical activity, which is strongly influenced by the time factor; • the correct appraisal of some uncertain factors’ evolution is essential for the realistic forecast; • the forecast results are strongly necessary in order to justify the decisions related to the power system development and operation; • the load forecast errors imply high extra costs: – if the load was underestimated, extra costs will be caused by the damages due to lack of energy, or by the overloading of the system elements; – if the load was overestimated, the network investment costs overtake the real needs, and the fuel stocks are overvalued, locking up, in an unjustified way, capital investment.

672


11.2. Factors that influence the energy consumption The experience in this field emphasised that the main causes generating the load modification are [11.5, 11.13]: • The weather conditions: the season, the daily temperatures (average, minimum, maximum), the wind speed, the rain-fall quantity, the cloudiness, etc. [11.11]; • The demographic factors: the population rate of growth, the number of the inhabitants in a certain area or in a certain country, the birth rate (the number of child bearings per 1000 inhabitants), the population growth (the difference between the birth rate and the death rate) etc.; • The economical factors: the gross national product, the labour productivity, the mean specific incoming, the economy development rate, the endowment level, and the life quality level. A very important element is represented by the energy price, this one being related to the supply/demand ratio on one hand, and to the energy resources and the economical policy on the other; • Other factors are: – the length of the day compared to the length of the night, influencing directly the load demanded for artificial lightening; – the day of the week to which the energy consumption is referred knowing that on holidays the load is reduced as compared to a working day, when the production activity leads to an increased load demand; the economical activity is more intense in the days in the middle of week (Tuesday, Wednesday, Thursday) than in the other days (Monday, Friday). The evolution in time of these parameters has a strong random character. At a certain moment, the more or less accidental realisations of these parameters (explanatory variables) influence in a direct way the load (explained variable) and their variation tendency change influences in a decisive way the load variation tendency. The load forecast implies the building of a mathematical model that uses as inputs the determinant factors that influence the load, and as output, the corresponding consumed energy. The load forecast implies the solving of the following aspects: • Identification and classification, according to their importance, of the causes that influence the load; • Determination from the qualitative and the quantitative point of view the relationship between the causes and the effect; • Utilization the relationship established above for the load forecast, based on the estimation of the cause parameters’ evolution; • The gradual checking of the forecast results, as time goes on, correlating the energy consumption to its causes in order to remake the forecast by using the corrected correlation rule that can offer improved prediction.

Load forecast

673

11.3. Stages of a forecast study The forecast survey methodology implies the following main stages: 1. The selection, the correlation and the processing of the initial database; 2. The building of the load mathematical model; 3. The solution analysis and the final decision making.

11.3.1. Initial database selection, correlation and processing In order to obtain an accurate forecast, a quite large and correct database must be used [11.4]. It must contain: • The amount of the global load and, if possible, detailed on components, for a quite long period of time (at least 5 years); • The evolution of load influencing factors. This leads to the processing of the huge database, which has to be easily stocked, accessed, visualised, and modified. The database setting up and updating represents a permanent work that needs a huge quantity of information and implies an important responsibility. If the database is correct, there is the probability that the forecast results are verified by the reality, but if the database is erroneous, the probability to obtain an accurate forecast is extremely reduced, even completely erroneous. The primary data introduction can be made easier by using the automatic data acquisition systems based on micro-controllers. The present technologies allow the data acquisition on a large area, by remote measurement and transmission, to a unique computer situated in a control centre. In this forecast step, a preliminary data selection and modification takes place. The data are visualised under the graphical form. Then they are followed by a statistical computation, which allows the wrong data elimination. The correlation setting between the load and one or more parameters considered as independent variables represents another problem. If there is a correlation, its nature interests us (linear, non-linear, etc.). The graphical visualisation of the dependence between the variables gives us quite quick and correct intuitive answers, which are given, more precisely, by a competent person.

11.3.2. Mathematical model of the load 11.3.2.1. Model components The load curve represents the energy variation in terms of the determinant parameter. If the parameter taken into consideration is the time (t), the curve can be divided in several components. The experience underlines the fact that there are four main components that induce the load profile (W) (Fig. 11.1) [11.10].

674


W T C t

0 S

t

0 ε

t

0

Fig. 11.1. The considered components of the load mathematical model.

1. The trend (T) is the main load component, establishing the main load variation form. 2. The cyclic component (C). It is due to some slow-varying causes such as the correlation supply – demand, which lasts more than a year. 3. The seasonal component (S) is caused by certain parameters, which represent seasonal fluctuations. This component’s variation period lasts only a few months and it is almost the same for all the years. 4. The random component ( ε ) is due to accidental causes that have not been mentioned above. So, the load is due to the summing up of the above mentioned components:

W (t ) = T (t ) + C (t ) + S (t ) + ε(t )

(11.1)

The practical experience proved us that, in general, the relation (11.1) underlines, correctly, the rule through which the components of the load curve combine. However, there are other rules according to which the components of the load are associated. For instance, one of the most popular rules is the rule given by the all components product, or of some of them:

W (t ) = T (t ) ⋅ C (t ) ⋅ S (t ) ⋅ ε(t )

(11.2)

The relation (11.1) is obtained by applying the logarithm on the relation (11.2). In general, the forecast methods are given by summing up the load components. Therefore, the model is converted to its standard form (11.1) in the first period of the forecast, using the adequately chosen functions and variables transformations. There are two main criteria that guide us how to choose the right transformations: • The load graphics visualisation gives to a competent person enough information, so that she/he can make the correct choice of the solution, based on intuition;

Load forecast

675

• The statistical indices, which can be obtained by some computations on the load profile, as it results from the time series theory, give us enough information to find the correct transformations that can lead to the components outline and to the way they are associated. If a load forecast is performed, the most important thing to do is to calculate, separately, the variation of each load component, the final result being obtained by the addition of the results of the components forecast. The separation of components represents one of the main problems of the Statistics Theory. The separation process is made easier by the cyclical component and by the seasonal one when using load curve standardisation methods and it becomes more difficult (sometimes even impossible) for the random component. For this reason, for situations when the random component form is not directly necessary, forecast probabilistic methods have been developed that calculate the limits of the so-called confidence interval, in which the tendency can vary from the value deterministically estimated, as a result of the accidental factor’s action, without overtaking the pre-established confidence level. If the cyclical, seasonal and random components are small as compared to the trend then their influence over the load variation can be neglected, and everything is limited to tendency forecast. This is the most usual situation that occurs in practice and it represents one of the main subjects treated in books.

11.3.2.2. Mathematical models used in load forecast studies The models that will be described next characterise, generally, the tendency of the load profile, except for the cases when introducing, explicitly, the random component through the doubt factor. In fact, it can be treated if necessary, in the same way the other models are treated, by the simply addition of the doubt factor. A large variety of mathematical models are used for the load forecast [11.13]: a) econometric models, which are characterised by mathematical rule issued as a result of technical and economical analysis, followed by a statistical check-up; b) analytical models that take into consideration the energy sources and types of loads; c) conditional models that take into consideration the energy purchase price and the equipment price; d) models that take into consideration the electricity market games; e) models that take into consideration the influence of the electric installations used; f) models that incorporate the production factors. Among the enumerated models, the most usual are the econometric models, such as: • autonomous tendential models, whose only one variable is the time, and whose aim is the load variation tendency extrapolation. These models have the doubt factor as random variable ε(t ) . The most usual ones are:

– linear:

W (t ) = a + bt + ε(t )

(11.3)

W (t ) = W0 (1 + α ) + ε(t )

(11.4)

– exponential: t

where: a, b, α and W0 are the models coefficients. These coefficients are determined from the regression analysis and some of them can be regarded from a physical point of view. A more detailed description of these models can be seen in Table 11.1. • adaptive conditional models which are based on the use of the explicative variables (the causal factors that determine the load) and on the respecting of some evolution scenarios (future events series, nodal events traces); according to the mathematical rule of the model there are: – linear models: W (t ) = a + b ⋅ GNP(t ) (11.5) where GNP is the Gross National Product. Other explicative variables can also be used instead of this variable. – non linear models: without elasticity implementation:

W (t ) = a(t )[1 + r (t )]GNP(t )

(11.6)

with elasticity implementation:

W (t ) = k ⋅ e ⋅ GNP(t )

(11.7)

The elasticity is defined by: e=

ΔW (t ) W (t ) ΔGDP(t ) GDP(t )

(11.8)

and k represents the model constant. • adaptive models: without elasticity implementation: ΔW =a+ W

Δx i +δ xi

(11.9)

ΔGNP (t , t + 1) W (t ) GNP

(11.10)

∑b

i

i

with elasticity implementation: ΔW (t , t + 1) = e ⋅

The following relation defines the elasticity in the above formula:

e=

∑ ΔW (t, t + 1) W (t ) ΔGNP(t , t + 1) N

(11.11)

672

Load flow and power system security Table 11.1 The main types of tendency extrapolation mathematical models [11.5]. Function type

With continuous growth

linear

Mathematical expression W (t ) = a + bt

Graphical representation

Substitution y = a 0 + a1 x y =W x = t a 0 = a a1 = b y = a0 + a1 x +

polynomial

W (t ) = a + bt + ct 2 + L

+ a2 x 2 + L y =W a2 = c

exponential

W (t ) = a ⋅ e

x=t

a0 = a a1 = b

L

y = a 0 + a1 x bt

y = ln W

x=t

With limited growth

a 0 = ln a a1 = b

logarithmic

W (t ) = ln (a + bt )

y = a 0 + a1 x y = eW

x=t

a0 = a

a1 = b

y = a 0 + a1 x

logistic

W (t ) =

a b + c ⋅ e − dt

y = ln[W (a − Wb )] x=t a 0 = ln 1 c a1 = d

The summing up is performed for all the N periods, of the time taken into consideration. The two terms in brackets define the limits of the interval on which the variation of the number, symbolised by Δ, is calculated. There are two important situations: • linear model relative to its parameters; • non-linear model relative to its parameters. when its parameters influence the structure of the model. Since it is very difficult, from the mathematical point of view, to identify a non-linear model, linear models are generally used. However, if the model that has to be used is non-linear with respect to its parameters, it should be converted to obtain a linear model, by changing the input and the output data. The logistical

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model (Table 11.1) is the most adequate example for this situation; as found from the “Substitution” column. Calculating the coefficients a and b, using the existing data, then, using the above transformations, a linear model is obtained. After finding the a0 , a1 coefficients, by using the existing transformation relations, it is very easy to obtain the d, c parameters. According to its variables, a mathematical model can be: • endogenous when the load is considered to be a time function W (t ) = f (t ) ; the load forecast is performed by function extrapolation from the past to the future, method called direct forecast; the time can be considered as a: – continuous variable, because the models are continuous; – discrete variable, whose values are known at equal time samples; in general, discrete endogenous models are called time series. • Exogenous, when the variables on which the load depends are the climatic, the economical, the demographic factors, etc., the time not being taken into account in this structure W (x1 ,K, xn ) = f (x1 ,K, xn ) ; since the load forecast uses an exogenous model it is called indirect forecast; every parameter is a time function: xi = gi (t ) ; • combined: W (t , x1 (t ),K, xn (t )) = f ( x1 (t ),K, xn (t )) . From the load point of view, the forecast models used can be: • global (synthetic) models, when the load is seen as a whole; • analytical models, when the load is obtained by summing up some of the components, the forecast being realised at the components level.

11.3.2.3. The choice of the mathematical model Many suggestions concerning the choice of the mathematical model can be done, and if they are taken into consideration, the errors are avoided in principle, and so, the time and effort for their correction are saved [11.13]. First of all, the following elements must be exactly defined: • the forecast aim implies the knowledge of the load location, of the forecast time horizon, of the available initial data, and of its application field; • the forecast quality has to be accomplished, in conformity to the aim, and to reflect the exactitude of the results and the establishment of the accepted hypotheses; • explicative variables that can be introduced in the model; • scenarios that involve a possible load development; • the forecast types that are going to be taken into consideration knowing that: – simple, but not simplified models are the favourite; – endogenous models, done by extrapolation, without taking into consideration the power and equipment price, the production factors, or the prices in general, are simple, and need only the knowledge of the

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past load, are adequate in the preliminary step of the long term forecast; however, the results have a low accuracy; – exogenous models are more complex, need the knowledge of some extra data and even of the explicative parameters evolution in future, so that the energy forecast to be possible; the results are useful for the medium term forecast; – the energy price is the most important variable which has to be taken into consideration into the standard models; – global models are less recommended for the normative studies; – the introduction of some limits is very important for medium and long term studies; – the models that use the elasticity coefficient of the energy price give acceptable results only for short and medium periods of time. • it is useful to include limits which take into consideration the environmental restrictions and energy preserving requirements; when the limits are set, the following aspects must be analysed: – have these limits any role at all? – is there any asymptotic conduct of the load evolution which justifies the setting of the limits? – can models with limited growth be introduced in the study, models that take into account the existence of the expected limits? • the models setting must take into account also the methodology used for the identification of the model: – if it is suitable for the actual analysed situation; – if the objective (technical) and subjective (related to the staff competence) conditions are accomplished, related to the personal qualification, for their application.

11.3.3. Analysis of results and determining the final forecast In order to increase the coincidence between the forecasted results and the actual ones, every study analyses more possibilities. The number of data taken into consideration for the past, the type of the used mathematical model and the adopted work hypothesis differentiate these possibilities. As there are more possibilities, there are also more results that may differ, more or less, one from each other [11.18]. The staffs that makes the forecast has the duty to establish which of the analysed possibilities has the biggest chances to be corrected, and on their basis, the final results of the load forecast to be provided. In order to compare these possibilities, different criteria are used, such as: • The use of tendency quality indices (the correlation coefficient, the correlation ratio, the variance, the sum of the absolute square deviations, etc.). The better they are, the biggest the chances for the forecast to reflect correctly the future real data;

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675

• The checking of the forecast possibilities in some known points. In order to have such checking points, the last 2 – 3 years (the latest ones) are being taken into account for the future, the forecast sphere moves towards the past, 2 – 3 years ago. The forecast results for the next 2 – 3 years can be checked, and if there is a resemblance between the actual results and the expected ones for this same period, then, it is very possible that the results to respect the load increase for the next years. This method has also disadvantages, that is, in order to establish the forecast, the tendencies which occurred in the latest years and which are crucial for the future evolutions are neglected; • The use of another information to evaluate the obtained values by comparison, such as: previous forecasts, forecasts performed by other teams, and already existing data; • The heuristic approaches based on experience and on the experts’ intuition can be very useful and can offer us amazing results. After the comparison of the forecast possibilities and setting of the final approach, the forecast results are transmitted under a very flexible form [11.14]. In this regard are developed: • Values intervals where the future load should be found, with the mention that this interval should be respected; it is the so-called the presentation of the results under a probabilistic form, described in the section 11.5.4; • Different scenarios with afferent results, where assume the load evolution takes place according to some general hypotheses, which take into account the concrete considered conditions.

11.4. Error sources and difficulties met at load forecast The load forecasted values will differ due to more or less from the future real load values especially due to multiple random factor that influence the load, such as: climatic and demographic, the international economic circumstances, and due to more or less controllable error sources. The random factors can be less controllable, so they are grouped into the socalled inherent error sources. It is necessary to reduce the other error sources – called specific sources – in order to increase the precision of the forecast results. The main sources of errors are [11.13]: • load data error for the past caused by their bad processing or storing; • arbitrary reductions introduced by accepting too optimistic or too pessimistic work hypotheses; • neglecting of important factors or consideration of insignificant factors; • computational error in determining the mathematical model coefficients; • imprecise estimation of some initial data due to their unknowing; • wrong interpretation of the obtained results.

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The most important inherent source of errors is the sudden and significant change of conditions that leads to the work hypothesis violation. This fact is difficult to predict, but its consequences are, in general, dramatic for the forecast result. In such cases, the best solution is to update the load forecast using the new initial conditions. Many difficulties can occur in the load forecast and they are caused by: • incomplete data or too short time series for the past; • the difficulty to combine more different systems (technical, economical, social systems) in one mathematical model; • the number of variables may be very important; • the existence of some qualitative parameters difficult to quantify; the fuzzy set theory is very useful in such cases; it is more and more amplified in our field.

11.5. Classical methods for load forecast 11.5.1. General aspects The methods that use continuous analytical mathematical models and are based on classical elements of Mathematical Analysis and Statistics belong to the load forecast classical methods [11.15, 11.19]. The load components separation, i.e. the trend, the periodical components (the cyclical, the seasonal components) and random components, represents the first decisive load forecast stage. The periodical components separation can be made using the load data Fourier analyse outlining the following parameters: • the component amplitude; • each component frequency (or period); • the component angle phase. Using the relation (11.1), the trend and the random component sum are: y (t ) = T (t ) + ε(t ) = W (t ) − [C (t ) + S (t )]

(11.12)

Now one of the two following strategies can be adopted: 1) Calculate the trend mathematical coefficients by the least squares method. Assuming that the load deviation towards the tendency corresponds to the random component, it results: ε(t ) = y (t ) − T (t )

(11.13)

and the random component estimation follows. 2) We accept a probabilistic model for y (t ) that allows its coefficients determination, future load estimation using the extrapolation and estimation of the domain where they will be included with a certain probability that the forecast is not revealed.

11.5.2. Cyclical and seasonal components analysis A Fourier analysis of W (t ) function is necessary for the determination of the load components that have a harmonic time variation Y (t ) – the cyclical C (t ) and the seasonal S (t ) components [11.21]. The mathematical model used for the harmonic components representation is:

Y (t ) = S (t ) + C (t ) =

n2

n2

i =1

i =1

∑ (ai sin ωi t + bi cos ωi t ) = ∑ Pi sin(ωi t + φ i )

(11.14)

where: a i , bi , Pi are the component amplitudes, ωi = 2π Ti is the angular frequency, Ti is the period, φi is the component phase angle: φi =

2π n i and Ti = i n

(11.15)

The estimated values of the a i and bi are established through Fourier analysis of the function Y (t ) using the formulae:

) 2 ai = n ) 2 bi = n

n

∑ W (k ) sin ω k i

k =1 n

∑ W (k ) cos ω k

, i = 1, 2, K , n 2

(11.16)

i

k =1

By plotting the periodogram, that is: Pi = ai2 + bi2 = f (i )

(11.17)

it following variables can be estimated (see Fig. 11.2): • the Pi amplitudes of the ith components, whose values are important and which are going to be kept into the model (for Fig. 11.2, i = 2 and 5);

Fig. 11.2. Load variation periodogram.

• the periods of the symmetrical components kept in the model:

Ti = n i

(11.18)

• the phase angle shifting of the considered components:

tan φi = bi ai

(11.19)

Application 1 The load of a residential area is represented in Table 11.2. Determine the periodical components of the considered load and then separate the trend and the random component. Table 11.2 The cyclical and the random components data Month W [MWh] Month W [MWh] Month W [MWh] Month W [MWh] Month W [MWh] Month W [MWh]

1 2.572 7 1.960 13 2.978 19 1.942 25 2.730 31 1.126

2 2.900 8 1.763 14 3.305 20 1.403 26 3.482 32 0.9493

3 2.547 9 2.668 15 2.376 21 1.794 27 2.478 33 2.073

4 2.511 10 2.146 16 2.817 22 1.749 28 2.797 34 1.719

5 2.891 11 1.599 17 3.170 23 1.471 29 3.729 35 0.9443

6 3.117 12 1.855 18 3.313 24 1.523 30 3.371 36 1.626

Figure 11.3 shows the evolution of the residential area load during the 36 months that have been taken into consideration. [MWh]

4 3

yk

2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

k

[month]

Fig. 11.3. The load evolution. In order to obtain the symmetrical components (the cyclical one and the seasonal one), calculate the ai , bi coefficients using the relations (11.16) and the Pi amplitudes of the harmonics up to the 18th order, using the relation (11.17), obtaining the values from Table 11.3. The phase angle shifting and the periods corresponding to the cyclical and the seasonal components are calculated using the formulae (11.18) and (11.19), and have the following values: • for the cyclical component: φ3 = −0.415 rad; T3 = 12 months;

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Load flow and power system security • for the seasonal component: φ3 = −0.766 rad; T3 = 4 months. Table 11.3 The harmonic values i ai

1 0.09011

2 -0.0368

3 0.6878

4 0.04385

5 0.01302

6 -0.02071

bi

-0.1199

-0.1726

-0.3028

0.1383

0.03057

0.03636

Pi i ai

0.15000 7 -0.07506

0.1765 8 -0.001041

0.7515 9 0.4534

0.1451 10 0.09771

0.03323 11 -0.01665

0.04184 12 -0.0269

bi

-0.108

-0.03823

-0.4364

0.03729

0.1403

0.07851

Pi i ai

0.1316 13 0.02427

0.03824 14 0.0343

0.6293 15 -0.09003

0.1046 16 0.0243

0.1413 17 0.0206

0.08299 18 0.0000

bi

0.02264

0.01112

-0.0601

0.02607

-0.009395

0.07215

Pi

0.1439

0.1143

0.1015

0.1446

0.1286

0.1846

Using this information, the periodogram represented in Figure 11.4 can be plotted. It can be seen that the most important values of the amplitudes correspond to the 3rd and 9th harmonics: P3 = 0.7515 and P9 = 0.6293 . It is obvious, even from Table 11.3, that these values are threefold bigger than the amplitudes of the others harmonics. [MWh] 0.8 0.6

Pi

0.4 0.2 0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

i Fig. 11.4. The periodogram. Figure 11.5 represents the evolution in time of the cyclical and the seasonal components, calculated with relation (11.14). Using the relation (11.12), the trend variation and the load random component variation are obtained, represented in Figure 11.6.

Load forecast

673 [MWh]

2 1.2

yk

0.4 -0.4 -1.2 -2

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

[month] k Fig. 11.5. The cyclical and the seasonal components. [MWh]

3 2.7

Tk+εk

2.4 2.1 1.8 1.5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

k

[month]

Fig. 11.6. The trend and the random component. Note: The Mathcad application, product under licence from Mathsoft Corporation, from Canada, had been used as help for the calculation of the problem solution.

11.5.3. Trend forecast After separation of the symmetrical components from the load curve, and according to paragraph 11.3.2.2, in order to forecast the trend, a linear model that contains also random component is adopted. Therefore, a probabilistic model of the multiple linear regression is found [11.15, 11.19].

11.5.3.1. Probabilistic model of the multiple linear regression We start from the assumption that y, x1 , K, x j , K, x p is a sample of n independent observations of the p + 1 random variable where ψ is the load and ϕ 1 , ϕ 2 , K, ϕ p the factors that influence the value of the consumed energy. The y vector is called explainable variable or the criteria, and the vectors x1 , L, x j , L, x p are called explanatory variables or the predictors. In general, it is assumed that the explanatory variables form a system of independent linear vectors, but this does not mean that they are statistically independent, but, on the contrary, there can be some statistical correlation between the predictors. That is why, in order to avoid any confusion, it is suitable to use the symbol y for the dependent variable and the symbol x j for the independent one.

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(

)

The expectation E ψ, ϕ1 , ϕ2 , ..., ϕ p gives the best approximation for ψ in the case of the function ϕi . Now, if the hypothesis of the multiple linear regression is taken into account:

(

)

p

E ψ, ϕ1 , ϕ2 , ..., ϕ p = β0 +

∑β ϕ j

j

(11.20)

j =1

the following relation is obtained if we add a random variable ε , whose average is 0, and is not correlated to ϕi : ψ = β0 +

p

∑β ϕ j

j

+ε

(11.21)

j =1

where the symbol σ 2 is using for the variance of the random variable ε . In most cases the coefficients β0 , β1 , K, β p and the variance σ 2 are unknown, so they are expected to be evaluated as better as possible. Between the realisation yi , xi1 , K, xip , ei of ψ, ϕ1 , ϕ2 , ..., ϕ p , ε , the following relation inferred from the hypothesis of the multiple linear regression exists: p

yi = β 0 +

∑β x

j ij

j =1

+ ei with i = 1, 2,K, n

(11.22)

The relation (11.22) written under a matrix form represents the probabilistic model of the multiple linear regression: y = Xβ + e

(11.23)

where:

[

X = 1 x1

⎡1 X 11 ⎢1 X 21 L xp = ⎢ ⎢M M ⎢ ⎢⎣1 X n1

]

L X1 p ⎤ L X 2 p ⎥⎥ O M ⎥ ⎥ L X np ⎥⎦

(11.24)

y = [ y1 , y2 , K, yn ] t

[

β = β0 , β1 , K, β p

]

t

e = [e1 e2 L en ] t where [ ] t stands for the transposition of a vector (or a matrix), therefore y, β and e are column vectors.

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11.5.3.2. Assessment of the multiple linear regression coefficients The vector of coefficients β estimation is required so that the assessed values ) of the criterion’s variables y approximate as better as possible the y vector achievements, which correspond to the predictor variables included in the X matrix – the relation (11.24). For this purpose, the objective function is used [11.19]: S (β ) = e t e = (y − Xβ ) (y − Xβ ) = y t y − 2β t Xy + β t X t Xβ t

(11.25)

which has to be minimised to get the best values of the regression coefficients. So, ) the best values β of the regression coefficients in the sense of the least squares (LS) correspond to the situation when the sum of the errors squares is minimal. Therefore, the imposed condition is that the derivative of function S to be zero.

) ⎛ ∂S (β ) ⎞ ⎜⎜ ⎟⎟ = −2 X t y + 2 X t Xβ = 0 ⎝ ∂β ⎠ β =β) The following equation is derived from the function above: ) X t Xβ − X t y = 0

(11.26)

(11.27)

The solution of equation (11.27) can be obtained almost immediately: ) −1 β = Xt X Xt y = X + y (11.28)

(

where

)

(

X + = Xt X

)

−1

Xt

(11.29)

is the pseudo-inverse of the matrix X . Of course, X t X is invertible, therefore it is positively defined, meaning that X has independent linear columns. The forecasted values for the variable of the criterion are: ) −1 ) y = Xβ = X X t X X t y = Ay (11.30)

(

)

with

A = XVXt

(

A = XX t

)

−1

(11.31) (11.32)

11.5.3.3. Features of the multiple linear regression The following statements can be proved [11.19]: ) • For the multiple linear regression probabilistic model the β values calculated through the LS method are non-biased estimations of the parameters β : ) Eβ = β (11.33) ) • β is the best estimation for β because, from all the estimators of the regression coefficients, it has the smallest variance; • The regression coefficient variance is: ) V β = σ2V (11.34)

()

()

• The difference between the achievements and the estimations is: ) y − y = (I - A)y

(11.35)

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• The sum of squared errors (SSE), where the errors represent the differences between the achievements and the estimation, is [11.19]: n

SS E =

)

∑(y − y ) i

i

) = y t y − βt Xt y

2

(11.36)

i =1

formula used to calculate the SS E from the initial data of the problem; • The variance estimator is:

) σ 2 = MS E =

SS E n − p −1

(11.37)

it is not moved and has a minimal variation; • The multiple correlation coefficient R represents the highest value of the coefficient of the simple linear correlation between the y vector’s co) ) ordinates and the co-ordinates of any other vector y = Xβ . Its square is equal to the ration between the variance explained through regression and the variance of y : n

R2 =

∑

( yi − y ) 2 −

i=1

n

∑(y

i

) − y)2

i=1

n

∑(y

i

− y)

2

= 1−

SS E s yy

(11.38)

i =1

where y = E ( y ) is the average of the yi values and [11.15] s yy =

n

∑(y

i

− y)2

(11.38')

i=1

is the sum of the square of the differences toward the average value. If for ) every i we have got yi = yi , then the adjustment of the experimental data linear model is perfect and the result is: R 2 = 1 ; • Eliminating the SS E between the relation (11.37) and (11.38), a practical relation used to calculate the estimated variance is obtained: 1 − R2 ) s yy σ2 = n − p −1

(11.39)

) It can be seen that if the adjustment is perfect ( R = 1 ) then σ 2 = 0 ; • If ei ∈ N (0, σ 2 ) , so it has a normal distribution with zero average and the variance σ, then the confidence interval 100 (1 − α ) for the regression coefficient is given by:

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673

) βi − t α 2

where: – tα 2

, n − p −1

, n − p −1

( )

) ) ) S E βi ≤ βi ≤ βi + t α 2

, n − p −1

( )

) S E βi

(11.40)

represents α the bilateral quantile and n − p − 1 represents the

degree of freedom for the t distributed statistics; – S E is called standard error or typical deviation of the regression ) coefficient βi , being rendered by the formula: ) ) S E βi = σ 2Vii (11.41)

( )

– Vii represents the diagonal element of the coefficients covariation matrix V – the relations (11.32) and (11.34); • If the explained variable real value is y0 , which corresponds to the x 0 predictor, then the confidence intervals having the probability 100 (1 − α)% for the y0 value will be:

) y0 − t α 2

where:

, n − p −1

) ) σ 1 + x t0 Vx 0 ≤ y 0 ≤ y 0 + t α 2

[

x'0 = 1 x01 L x0 p

,n − p −1

) σ 1 + x t0 Vx 0

]

(11.42,a)

(11.42,b)

11.5.3.4. Testing of model coefficients The following two contrary hypotheses are desirable to be tested [11.19]: H0: β j = 0 H1: β j ≠ 0 In this regard must be calculated: t0 j =

) βj

( )

) SE β j

(11.43)

If: t 0 j > t α 2,n −2 then the hypothesis H0: β j = 0 is eliminated and the hypothesis H1: β j ≠ 0 is accepted. α represents the distribution threshold value. It is worth mentioning that if a significant R 2 can be found, so that no regression coefficient, taken separately, is much different from zero: this can happen only if there is a strong correlation between the predictors.

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11.5.3.5. Residua and observations analysis The residua method is a very important one and, sometimes, the only one used to check the base hypothesis of the model (its linearity, homoscedasticity, etc.) and to discover the wrong data that are influenced by great errors. In a good situation, the residua do not have to be correlated to the explanatory variables: in other words, the residua diagram, according to the predictors, does not have to allow the presence of any tendency [11.19]. We call the criterion variable residuum the difference between the real value ) ) and its estimation: y i − y i . The components of the residua y − y vector are the residua of all the explanatory variables. The formula for the residua variance matrix is: ) V ( y − y ) = σ 2 (I − A ) (11.44) It can be seen that, in general, the residua are correlated between them, because the matrix A has not a diagonal form. As a consequence, the estimation of the residuum variance can be approximated for the ith term by taking into account only the diagonal terms: ) ) V ( yi − yi ) = σ 2 (1 − Aii ) (11.45) Aii is the diagonal term of the A projector, achieving the following conditions: 1 • ≤ Aii ≤ 1 n •

n

∑A

ii

= p +1

i =1

The following formula is called studentised residuum: ) yi − yi rsti = ) σ 1 − Aii

(11.46)

When n is large, the studentised residua values have to be situated between -2 and +2, otherwise a big residuum can indicate a wrong value. Wrong values can be obtained without having a big residuum. Such case is illustrated in Figure 11.7. x x x x

x

x

x x x

x x

Aberrant value with null residuum

x

Fig. 11.7. The case of an aberrant value with null residuum.

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In such situations, the influence of the observations on the prediction must be analysed. Thus, from the completed data model is removed the ith component, whose influence must be estimated and the reduced model (n − 1) is estimated for ) the explanatory variables. With the vector y ( −i ) , the observations obtained for the explanatory variables in a reduced model, are noted. ) The forecasted residuum is the difference y j − y( − i ) j . If the influence of the ith observation is small, there should be no large difference between the residua and the forecasted residua. Carefulness should be paid to the observations whose term Aii is large (close to the value 1).

Application 2 Forecast, taking into account the influence of the random component, the industrial load for the period 2000 – 2005, using the direct extrapolation method for the exponential trend. The annual load for the period 1995 – 2001 is illustrated in the table below: Table 11.4 The load data Year W [GWh]

1995 1.06

1996 1.37

1997 1.61

1998 1.97

1999 2.24

2000 2.69

2001 3.01

The exponential model W (t ) = a ⋅ ebt is reduced to a linear model by finding the

logarithm of the number: y (t ) = ln W (t ) = ln a + b ⋅ t = β0 + β1 ⋅ t . The mathematical model that corresponds to the relation (11.23) has the following formula: ⎡ y1 ⎤ ⎡1 X 1,1 ⎤ ⎡ e1 ⎤ ⎢ y ⎥ ⎢1 X ⎥ ⎢e ⎥ 2,1 ⎥ ⎢ 2⎥ = ⎢ [ β0 β1 ] + ⎢ 2 ⎥ ⎢ M ⎥ ⎢M ⎢M⎥ M ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 1 X ⎥ 7 ,1 ⎦ ⎣ y7 ⎦ ⎣⎢ ⎣e7 ⎦

According to the relation (11.28) the following formula is obtained: ⎡ ) ⎢ n ) ⎡β 0 ⎤ ⎢ −1 t t β = X X X y = ⎢) ⎥ = ⎢ 7 ⎣⎢ β1 ⎦⎥ ⎢ X i ,1 ⎢⎣ i =1

( )

∑

7

∑ i =1 7

∑ i =1

⎤ X i ,1 ⎥ ⎥ ⎥ X i2,1 ⎥ ⎥⎦

−1

⎡ 7 ⎤ yi ⎥ ⎢ ⎢ i =1 ⎥ ⎢ 7 ⎥ ⎢ yi X i ,1 ⎥ ⎢⎣ i =1 ⎥⎦

∑

∑

In order to render the calculation easier the following co-ordinates transformation is performed: X k = X k ,1 − X 1 Yk = yk − y

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with X 1 =

1 n

n

∑

X k ,1, Y =

k =1

1 n

n

∑y

k

representing the averages of the explanatory and the

k =1

explained variables. As a consequence, all the sums in which the variables are at an odd power become zero and the relations for the parameter calculus are the following: ) ⎡n ) ⎡b0 ⎤ ⎢ B = ⎢) ⎥ = 0 ⎢ ⎣⎢ b1 ⎦⎥ ⎢ ⎣

0 ⎤ ⎡ ⎥ ⎢ 7 ) 0 ⎤ ⎡ 0 ⎤ X Y ⎥ ⎢ i i ⎡ ⎤ 7 7 B0 ⎥ ⎢ ⎥ ⎥ ⎢ X i2 ⎥ ⎢ X i Yi ⎥ or ⎢ B) ⎥ = ⎢ i =71 ⎥ 1⎦ ⎣ i =1 ⎦⎥ ⎣⎢ i =1 ⎦⎥ ⎢ X i2 ⎥ ⎥ ⎢ i =1 ⎦ ⎣ ) ) ) ) β1 = b1 and β0 = Y − b1 X 1 . −1

∑

∑

∑

∑

The number of the degrees of freedom is n L = n − p − 1 = 7 − 1 − 1 = 5 . Having p = 95% for the probability to achieve the forecast, the result is t α = 2.57 . The 2

, n − p −1

limits of the confidence interval are calculated according to the relation (11.42): 1 ) ) ) ) y0 ± 2,57 ⋅ σ ⋅ 1 + xt Vx 0 = y0 ± 2.57 ⋅ σ ⋅ 1 + + 7

X 02 7

∑X

2 i

i =1

The calculation is performed in a spreadsheet; the detailed results are indicated in Tables 11.5 and 11.6. Table 11.5 The solution data Year

x

W

1995 1996 1997 1998 1999 2000 2001 SUM

0 1 2 3 4 5 6 21

1.51 1.62 1.76 1.89 2.10 2.30 2.60

y = ln (W ) X

0.4121 0.4824 0.5653 0.6366 0.7419 0.8329 0.9555 4.6268

-3 -2 -1 0 1 2 3

X2 Y = y − y 9 4 1 0 1 4 9 28

-0.2489 -0.1785 -0.0957 -0.0244 0.0810 0.1719 0.2945

Intermediary and final results are: • the average values: X 1 = 3 ; y = 0.661 ;

X*Y 0.7466 0.3571 0.0957 0.0000 0.0810 0.3439 0.8836 2.5078

) y

) y− y

0.3923 0.4818 0.5714 0.6610 0.7505 0.8401 0.9297

0.0198 0.0006 -0.0061 -0.0244 -0.0086 -0.0072 0.0258

( y − y$ ) 2

3.93E-04 3.42E-07 3.71E-05 5.95E-04 7.39E-05 5.17E-05 6.68E-04 1.82E-03

) ) • the estimated coefficients of the linear regression: B1 = 0.0896 ; B0 = 0 ; ) ) β1 = 0.0896 ; β 0 = 0.3923 ;

Load forecast

677

) ) • the estimated parameters of the exponential model: a = 1.4803 ; b = 0.0896 ; ) • the estimated variance for the linear regression: σ 2 = 0.364 ⋅10 −3 ;

• the correlation coefficient for the linear regression: R 2 = 0.991 . The final results for the forecasted load are obtained by inverse logarithm of linear model results. Table 11.6 Final results Year 2000 2001 2002 2003

yˆ 1.0192 1.1088 1.1984 1.2879

x 7 8 9 10

εˆ 0.0287 0.0313 0.0342 0.0373

yˆ + εˆ 1.0479 1.1401 1.2325 1.3252

yˆ − εˆ 0.9905 1.0775 1.1642 1.2506

Wˆ

Wˆ min

2.7710 3.0307 3.3147 3.6252

2.6926 2.9373 3.2033 3.4925

Wˆ max

2.8517 3.1270 3.4299 3.7630

The load evolution for the past period and its forecast, accompanied by the minimal and the maximal variation limits caused by the random component are illustrated in Figure 11.8 [11.11]. 4

Real load

3.5

Load forecast

3

Maximum load forecast Minimum load forecast

2.5 2

20 00

19 98

19 96

19 94

19 92

19 90

1.5

Fig. 11.8. The load evolution. Note: Microsoft Excel was used as a support for the solution of the application.

Application 3 Table 11.7 illustrates the daily load for an industrialised area according to 4 main parameters: • the industrial production daily quantity; • the number of the inhabitants from the area on the period the energy was measured; • the atmospheric temperature of that day; • the power cost prices the moment it was measured. It is requested: a) to establish a linear mathematical model including the calculation of its coefficients; b) to estimate the variance of the measured values compared to the linear mathematical model; c) to find out the multiple correlation coefficient;

678

Load flow and power system security d) to forecast one-day load characterised by the following parameters: x10 = 1.55 ; x20 = 2.105 ; x30 = 25 ; x10 = 50 ;

e) f) g) h)

to find out the confidence interval for the forecasted energy value from point c); to find out the confidence interval for the mathematical model coefficients; to check if there can be null coefficients; to find out the studentised residua of the daily consumed energy. Table 11.7 The daily load data

W [GWh] Daily load

x1 Daily quantity production [107 $]

x2 Number of inhabitants [106 loc.]

36.998 55.055 69.780 44.912 43.012 61.736 33.696 43.798 54.275 54.403

1.15 1.05 1.30 1.92 1.47 1.23 1.76 1.85 1.62 1.58

2.135 2.203 2.071 2.178 2.123 2.035 2.117 2.192 2.094 2.015

x3 Daily atmospheric temperature [o C] 35 11 -13 32 20 10 37 27 15 5

x4 The energy cost price [$/MWh] 35 42 55 30 61 38 49 52 32 57

a) The mathematical model for the load in the area corresponds to the relation (11.1), particularised for p = 4 : W ( x1, x2 , x3 , x4 ) = β0 + β1x1 + β2 x2 + β3 x3 + β4 x4

where the meaning of the variables corresponds to the columns of Table 11.4. It is now possible to create the matrix X , according to its definition from relation (11.24): 1 1 1 1 1 1 1 1 1 ⎤ ⎡ 1 ⎢ 1.15 1.05 1.30 1.92 1.47 1.23 1.76 1.85 1.62 1.58 ⎥ ⎢ ⎥ Xt = ⎢ 2.135 2.203 2.071 2.178 2.123 2.035 2.117 2.192 2.094 2.015⎥ ⎢ ⎥ 11 32 20 10 37 27 15 5 ⎥ − 13 ⎢ 35 ⎢⎣ 35 42 55 30 61 38 49 52 32 37 ⎥⎦

Using some routine calculations for relation (11.32) the following matrix is obtained:

Load forecast

679

− 4.37 − 90.366 ⎡ 197 ⎢ − 4.37 1.635 1.217 ⎢ t -1 V = (X X) = ⎢− 90.366 0.217 42.421 ⎢ − 0.018 − 0.105 ⎢ 0.223 ⎢⎣ − 0.064 − 0.0075 0.012

0.223 − 0.018 − 0.105 0.00085 0.00024

− 0.064 ⎤ − 0.0075⎥⎥ 0.012 ⎥ ⎥ 0.00024 ⎥ 0.00102 ⎥⎦

According to relation (11.31) the projector matrix is obtained: ⎡ 0.685 −0.154 0.217 −0.221 −0.031 0.111 0.172 0.223 ⎢ 0.679 0.226 0.022 0.111 0.042 − 0.165 0.137 ⎢ 0.217 ⎢ − 0.221 0.226 − 0.157 0.513 − 0.072 0.119 0.21 0.094 ⎢ 0.108 0.274 ⎢ − 0.031 0.022 − 0.072 0.586 − 0.147 0.013 ⎢ − 0.084 0.237 0.111 0.111 0.119 − 0.147 0.4 0.218 A = X ⋅ V ⋅ X' = ⎢ ⎢ 0.172 − 0.218 0.042 0.21 0.013 − 0.084 0.436 − 0.044 ⎢ 0.237 − 0.044 0.401 0.167 ⎢ 0.223 − 0.165 − 0.157 0.108 ⎢− 0.154 0.137 − 0 . 094 0 . 274 0 . 218 0 . 218 0 . 167 0.471 ⎢ ⎢ 0.044 − 0.029 0.067 0.343 − 0.167 0.266 0.028 2.902 ⋅10 −3 ⎢ − 0.097 0.201 0.208 0.208 8.622 ⋅10 −3 ⎢⎣− 0.046 − 0.241 0.22

0.044 − 0.029 0.067 0.343 − 0.167 0.266 0.028 2.902 ⋅10 −3 0.363 0.08

−0.046 ⎤ ⎥ − 0.241 ⎥ 0.22 ⎥ ⎥ − 0.097 ⎥ ⎥ 0.201 ⎥ 0.208 ⎥ ⎥ 0.201 ⎥ −3 ⎥ 8.622 ⋅10 ⎥ 0.08 ⎥ ⎥ 0.465 ⎥⎦

The determination of the mathematical model coefficients is performed by using some elements from relation (11.30):

(

)

βt = VXt W t = [β0 L β 4 ] t = [13.131

5.014

25.467

− 0.813 − 0.245]

The determination of the mathematical model coefficients is performed by using some elements from relation (11.30): W (x1 , x2 , x3 , x4 )=13.131 + 5.041 ⋅ x1 + 25.467 ⋅ x2 − 0.813 ⋅ x3 − 0.245 ⋅ x4

b) In order to assess the variance of the real values for the load W vector towards v ) the established mathematical model, the energy forecasted values y = W elements are firstly calculated according to relation (11.30). ˆ t = ( AW)t = W = [36.246 55.268 69.481 44.866 43.367 61.813 33.791 43.545 54.549 54.339]

Now, the relation (11.36) is used to calculate the value: ) SS E = W t W − β t X t W = 0.544 Using the relation (11.37) the value of the estimated variance is obtained: ) σ 2 = MS E =

SS E 0.544 = = 0.109 n − p − 1 10 − 4 − 1

So, the estimation of the mean square deviation is: ) σ = MS E = 0.109 = 0.33

680

Load flow and power system security c) Beforehand, the relation (11.38') is used to calculate the value: n

s yy =

∑ ( y − y) i

2

= 1150

i =1

Then, calculate the multiple correlation coefficient using the relation (11.38): R2 = 1−

SS E 0.544 = 1− = 0.98855 1150 s yy

It can be seen that R 2 is very close to 1, which means that the modelled process – the daily load – can be correctly represented through a multiple linear regression, which depends on the 4 chosen explanatory variables. d) The one-day load forecast, characterised by the following parameters x1 0 = 1.55 ; x2 0 = 2.105 ; x3 = 25 ; x4 0 = 50 is calculated by introducing the values of parameters in 0

the established mathematical model: ) W0 =13.131 + 5.041 ⋅ x10 + 25.467 ⋅ x20 − 0.813 ⋅ x30 − 0.245 ⋅ x40 = ) W0

= 13.131 + 5.041 ⋅1.55 + 25.467 ⋅ 2.105 − 0.813 ⋅ 25 − 0.245 ⋅ 50 =41.941 GWh

e) The value of the confidence interval band for the above forecasted load results from relation (11.42,a): ΔW0 = t α 2

) σ 1 + x t0 Vx 0 = 2.01 ⋅ 0.33 ⋅ 1.206 = 0.728

, n − p −1

where α = 10% is considered to be the distribution significance threshold, and n − p − 1 = 5 represents the number of the degrees of freedom. Therefore, having a probability of 90%, the real energy (W) for that day has to meet the following inequalities: W0 − ΔW0 ≤ W ≤ W0 + ΔW0 41.212 ≤ W ≤ 42.668

f) The confidence intervals are established for the mathematical model coefficients.

)

Prior, the quantity S E (βi ) is obtained using the relation (11.41): S E (β)t = [21.451 0.178 4.619 9.287 ⋅10 −5 1.114 ⋅10−4 ]

Assuming that α = 10% represents the distribution significance threshold and n − p − 1 = 5 the number of the degrees of freedom, so, having t = 2.01 , the value of the confidence interval band is obtained using the relation (11.40):

Load forecast

681 ) Δβ i = t α 2

, n − p −1

) Δβ t = [9.309 0.848

) S E (βi ) 4.32

0.019 0.021]

The lower and the upper limits for the confidence interval of a coefficient of the mathematical model is obtained by subtracting, respectively by summing, the estimated value of the coefficient in question with the confidence band that has already been calculated. Therefore: ) β min t = [ 3.821 4.166 21.47 − 0.832 − 0.266] ) β max t = [ 22.44 5.862 29.787 − 0.793 − 0.234]

( ) ( )

g) Checking if one of the coefficients might be zero implies the calculation of the statistical variables using the relation (11.43): ) βj t0 j = ) SE β j

( )

t t0 = [3.33 7.974 4.822 47.641 13.463]

It can be seen that all the components of the t 0 vector are bigger than the t = 2.57 threshold ( α = 5% , n − p − 1 = 5 ). Therefore, for every coefficient of the multiple linear regression the hypothesis H1: β j ≠ 0 is true, in other words, using a probability of 95%, every coefficient differs from zero. h) The studentised residua of the consumed energy are calculated using the relation (11.46): ) W − Wi rsti = ) i σ 1 − Aii rstt = [ 0.16

0.54

− 0.55 − 1.99 − 0.95 − 0.61

0.03

1.19

1.51

0.51

]

Notice that all the studentised residua of the daily load are included in the interval (-2; +2), therefore none of the forecasted load values can be included in the category of aberrant errors, they all represent plausible values. Note: Calculations were performed in Mathcad software tool, product under licence from Mathsoft Corporation, from Canada, as a support to our application.

11.5.4. Load random component analysis 11.5.4.1. Random component separation The first step to be performed is to extract the random component from the load profile using the relation (11.1) under the form:

682


y (t ) = ε(t ) = W (t ) − T (t ) − C (t ) − S (t )

(11.47)

If the value of the random component is small as compared to the value of the other components of the load, its analysis cannot be justified from a practical point of view. It is recommended only to appreciate its highest variation interval, according to what it has already been said [11.21]. If the random component is important, then for their modelling it must be performed a preliminary analysis that should investigate: a) the stationarity level of that time period which indicates if the random component was correctly separated from the other components; b) the predictability level of the time period, which establishes if the random component can be forecasted, or if it has to be assimilated to the white noise.

11.5.4.2. Random component stationarity and predictability In order to estimate the stationarity and the predictability of the time period that represents the load random component, the covariation function of the series is calculated: n− j

rˆj = 1 n

∑(y

k

j = 0, 1,K, n 4

yk + j )

k =1

(11.48)

where k represents the discrete time: k =t τ

(11.49)

and τ represents the time period patterning. a) If the series covariation function is amortised, that is rˆj tends to 0 when j raises, the series is steady, the random component being correctly and completely separated from the other load components; b) If the series covariation rˆj is not amortised and consequently the cyclic, seasonal and trend components are not completely separated. Indeed, if there exists a symmetrical component based on the period of time p then yk ≈ yk + p , from (11.48) it results that rˆj + p ≈ rˆj so rˆj is not amortised. c) The covariation series damping speed gives us important information on its level of predictability: – if the damping is slow, there is a correlation between the past and the future values and so the random component can be forecasted; – if the damping is fast, the series cannot be forecasted in time and it is assimilated to the white noise. It was proved that if the time series is white noise, then: rˆj ≤ 2

rˆ0 n

(11.50)

Load forecast

683

for j ≥ 1 , with the level of probability equal to 95%. If it is possible to forecast the random component, assuming there is a correlation between the past and the future values of the load, then, in general, the autoregressive model (AR) is used in the following form: m

∑a y i

k −i

= zk

(11.51)

i =0

where: k represents the discrete time; m – the number of terms of the AR model; ai – the mathematical model coefficients, having a0 = 1 ; yk − i – the autoregressed value of the load random component; zk – the non-correlated white noise, component of the random variable yk. The best forecast for the white noise is zˆ j = 0 , from where it results the following calculation formula for the random component forecast with forward step: yˆ ( k |k − 1) = −

m

∑a y

(11.52)

i k −i

i =1

respectively with j forward steps: j

yˆ (k + j|k − 1) = −

m

∑ a yˆ (k − 1 + i|k − 1) − ∑ a y

i k −i

i

i =1

(11.53)

i = j +1

The ai coefficients are determined through the least squares method, which leads to the solving of the overdeterminated system of equations formed by the relations (11.52) where yˆ ( k|k − 1) is replaced by yk for k = m + 1,K, n (with n >> m ) and having ai as unknown quantity. The solution for the overdeterminated system of equations can accurately be obtained using the orthogonal transformation method. The determination of the model term number (m) is performed through unidirectional research. The appreciation criterion is given by minimising the square deviation sum of the estimated values towards the real values.

11.6. Time series methods for load forecast 11.6.1. General aspects A sequence of observations, which were carried out on a physical quantity that changes in time, and which succeed each other in the order of their apparition

684


is called a time series. This time series can be also considered a finite sequence of random components y (1), y (2), K , y (n ) the components of a multidimensional random variable [11.18, 11.21]. Technically speaking, the time series { y (t )} is a data sample also called achievement (realisation), extracted from a random process, illustrated by a finite sequence of random variables (we use the term of statistical population from the Mathematical Statistics): y (− ∞ ) , K , y (− n ), K , y (− 2 ), y (− 1), y (0 ), y (1), y (2 ), K , y (n ), K , y (+ ∞ ) An important property of this time series is the fact that all the random variables taking part at the series report on the same physical, biological, social fact. Thus, there are some interdependencies between the time series values and they have to be underlined when analysing the time series to obtain a regularity synthesis that allows the prediction of the future values of the random process y (n + 1), y (n + 2 )K on the basis of the existing reality y (1), y (2), K , y (n ) . For instance, if the analysed random process is the hourly electric energy – numerically equal to the electric power average for one hour – of the load belonging to an electric network bus, the process is achieved by measuring, for a day, the hourly average electric power: P(1), P(2 ), K , P(24 ) . The electric power of the bus is a random process because we cannot state for certain, only knowing the time series, which the required electric power values for the next hours will be P(25), P(26 ), etc . If the achievements for the past few hours are analysed, “an average anticipation” only if taking into account the load “medium evolution” can be obtained. The average tendency can be growing if the load is growing as a result of a positive conjuncture, or conversely, if the conjuncture is negative. Oscillations caused by the weather can superpose on the tendency, and they influence the normal running of the climate control systems connected to that bus. If more electric loads from the same area are analysed, they may be similar or different. In the first case, the properties of the random variables – the distribution law, the average value and the variance – can be alike, in the second case, they can partially or totally differ.

11.6.2. Principles of methodology of the time series modelling The main objective of a time series analyse is the finding of a “good” illustration, in general a mathematical one (function, differential equation, etc.) of the mechanism which governs the process that allowed the given realisation. This illustration is called a model [11.20]. The model quality implies the fulfilment of some conditions: a reduced number of parameters, the residua independence (the difference between the real values and the ones obtained through the model), a satisfying approximation of realisation’s data, the production of enough precise values for the forecast, etc.

Load forecast

685

Figure 11.9 illustrates the chart of the existing relations between the three main elements taken into consideration: the process, the realisation and the model. Its structure establishment represents the first step of the model identification. To do this, the following elements are used as tools [11.18]: • The AutoCorrelation Function (ACF); • The Partial AutoCorelation Function (PACF). The validation and the diagnosing represent the third step in the model realisation. It is achieved by checking the model using different methods proposed in literature, by simulating in the same conditions as those of the represented process, and by comparing the results from the calculation to the realisations of the process. If the model has not gone through the validation step, it can be improved by taking again the identification – estimation – validation cycle, until the expected results are obtained. The identification, the estimation and the validation steps are covered as many times as necessary until the quality of the model fulfils all the required conditions. PROCESS Stochastic mechanism for series values generation Validation & Diagnosis

Measurement

MODEL Process mathematical representation

REALISATION Disposable series values Identification & Estimation

Implementation

APPLICATIONS Simulation, Forecast, Decision making

Fig. 11.9. The relations existing between the process, the realisation and the model.

After having finally obtained the model, its implementation follows, which includes the forecast, the simulation, the decision-making, the management, etc. In order to forecast the future values of the time series, the available values must be introduced in the model at that moment. The future interest values are established based on the future values of the time series and using a quite simple recursive calculation.

11.6.3. Time series adopted pattern. Components separation

686


Next, a discrete and a quite general model are adopted for the random process. It is available for the majority of the applications, including those related to the estimation of the future values. A discrete model is preferable in order to avoid the difficulties caused by the introduction of the continuous white noise. The discrete models can be easily implemented in a numerical calculation program. Even more, the choice of this discrete model does not restrict the approach generality, the obtained results being available also for the continuous models.

ε(t)

RATIONAL STABLE FILTER

y(t)

Fig. 11.10. The chart of a time series model.

According to the spectral representation theorem a steady random process can be obtained from a white noise ε(t ) , whose null average is E (ε ) = 0 and whose

( )

variance is V (ε ) = E ε 2 = σ 2 filtrated through a stable rational filter:

( )

y (t ) = H q −1 ε(t − κ )

(11.54)

where: • t represents the discrete time: t = 1, 2, K , k , K , ∞ . The continuous time is calculated using the relation t c = t ⋅ T , where T represents the signal sample time. T has to fulfils the condition given by sampling theorem T < 0.5 ⋅ Tmin , where Tmin represents the least time constant afferent to the process; • The lag operator q −1 means: y (t − 1) = q −1 y (t )

(11.55)

• κ represents the discrete dead time of the analysed process. It respects the causality principle: κ ≥ 0 . It is obvious that ε(t − κ ) = q − κ ε(t ) ;

( )

• H q −1 represents the Discrete Transfer Function (DTF):

( ) B(q ) A(q )

H q −1 =

−1 −1

(11.56)

• A ( q −1 ) and B ( q −1 ) are prime polynomials between them, the degree of polynomial B being less then or at most equal to that of the polynomial A.

Load forecast

687

In order to ensure the stability condition, the module of the roots of the characteristic equation A(x ) = 0 must be greater than 1, (they have to be found outside the unity circle of the complex plane). The model given by the relations (11.54) and (11.56) is too general. It has to be adapted to the desired purpose. In order to forecast the trend of the load, it is generally used the AutoRegressive Integrated Moving Average (ARIMA) model. It has two components: − the integrator one which allows the presence in the A q −1 polynomial of a

(1 − q )

−1 d

( )

factor that leads to the relation:

( )

( )(

A q −1 = A1 q −1 1 − q −1

)

d

(11.57)

− the other component is

(

y1 (t ) = 1 − q −1

)

d

y (t ) = ∇ d [ y (t )]

(11.58)

It corresponds to the AutoRegressive Moving Average (ARMA) model:

( )

( )

A1 q −1 y1 (t ) = B q −1 ε(t − κ )

(11.59)

For d = 1 , the relation (11.58) becomes:

(

)

y1 (t ) = 1 − q −1 y (t ) = y (t ) − y (t − 1)

(11.58')

because the polynomials from the relation (11.59) have the form:

( )= 1+ a q

A1 q

−1

1

( )= b

Bq

−1

0

−1

+ b1 q

−1

+ L + a na q

− na

+ L + bnb q

= 1+

na

∑a q i

−i

i =1

− nb

= b0 +

(11.60)

nb

∑b q i

−i

i =1

With na ≥ nb the model (11.59) has the explanatory form: y1 (t ) = −a1 y1 (t − 1) − L − ana y1 (t − na ) + b0 ε(t − κ ) + b1ε(t − κ − 1) + (11.61) + L + bnb ε(t − κ − nb )

The real time series contain some periodical components that cannot be included in the ARIMA adopted model ( na , d , nb ). The elimination of the periodical component is a simple process. Notice that if this component has the T C period, it can reappear over T C intervals:

688


y (t ) = y (t − TC ) = q −TC y (t )

(11.62)

It is obvious now that if the time series contains a cyclical component whose period is T C , then a 1 − q −TC factor has to correspond to the A1 q −1 polynomial from relation (11.59). So the polynomial can be factorised in the following product:

(

)

( )

(

)

A1 (t ) = 1 − q −TC A2 (t )

(11.63)

and (11.59) relation can be rewritten as it follows:

( )( ( ) ) A (q ) y (t ) = B (q )ε(t − κ )

A2 q −1 1 − q −TC y1 (t ) = B q −1 ε(t − κ ) therefore :

−1

2

−1

2

(11.64)

y2 (t ) represents the time series from which the cyclical component of period T C has been eliminated:

(

)

y2 (t ) = 1 − q −TC y1 (t ) = y1 (t ) − y1 (t − TC )

(11.65)

11.6.4. Establishing of the time series model using the Box – Jenkins method The modelling and the forecast time series methodology based on the ARIMA model is also know as the Box-Jenkins methodology, according to the names of the persons who created and proposed it. Practically speaking, the using of the Box-Jenkins method implies the using of some calculus programs dedicated to this purpose. One of the most familiar program created for the mathematical calculation, Matlab, a product of MathWorks Company, contains some toolboxes for the time series analysis and identification: Financial Time Series, GARCH, System Identification that can be also used for the energy forecast based on the Box-Jenkins method. The main elements of this complicated technique of system classes’ identification are going to be briefly introduced. For further details, the reader is advised to make use of monographic works and books mentioned in the list of references [11.18, 11.21]. The problem of the identification of a time series model consists of selecting a subclass from the general model ARIMA ( na , d , nb ). The starting point is represented by the existence of a realisation y (1), y (2), K , y (n ) that contains an important observation number n, of a onedimensional variable, in general n ≥ 50 . The main steps in the identification of the time series model are: 1) The graphical representation of the data model allows us to identify some important proprieties of the random process. Several useful aspects can be

Load forecast

689

obtained through a visual, simple and efficient analysis, based on the researcher experience: • if there is a statistical correlation between the observations; • if the process that caused the time series is steady or if it is not; • if the process is not steady, then must be appreciated the way in which the average, the variance, or both of them are influenced; • if the model contains other cyclical components, except for the trend component. As a result, decisions can be taken concerning: • the separation of components; • providing the steady character, by finding the logarithm of a number, by derivation, etc. For the identification methodology elaborated by Box and Jenkins to be applied, the time series has to be brought to the typical form of the ARMA ( na , nb ) models, using the transformations (11.58) and (11.59) adequately chosen, performed on the initial realisation. 2) Determine the estimated auto-correlation function (ACF). In this regard, ) the estimated correlation coefficients are calculated ( rk ) from the observations, ) shifted by k periods of time, belonging to the same realisation. The rk (k ) coefficient represents an important statistical measure of ordered pairs [ y(t ), y(t + k )] , being an ACF estimation. It is a dimensionless number situated in ) the interval (-1, +1). If r)k = 0 then the variables are not correlated. The rk = ±1 value indicates a perfectly positive correlation, respectively a negative one between ) the variables. The estimated value rk of the auto-correlation coefficient supplies estimation, more or less accurate, on the theoretical auto-correlation coefficient rk . The standard formula for the calculation of the auto-correlation coefficients is: n−k

) rk (k ) =

n−k

∑ [y(t ) − y ][y(t − k ) − y ] ∑ [y(t ) − y ][y(t − k ) − y ] t =1

n

∑ [y(t ) − y ]

2

=

t =1

s yy

(11.66)

t =1

where y represents the average value of the model observations. In the case of a steady process, it can be an estimation of the average value of the statistical population, in the first approximation. In general, it is recommended that k ≤ n 4 . ) 3) Determine the partial auto-correlation function (PACF) estimated to ϕkk , ) which is, in the main, similar to the ACF estimated to rk . The partial autocorrelation refers to the correlation degree between y (t ) and y (t + k ) taking into account the effects of the series values y (t ) , which are between the two time ) periods t and t + k . The partial correlation coefficient estimated to ϕkk represents a

690


statistical size that supplies an estimation for the theoretical partial auto-correlation coefficient ϕkk . In the case of a steady time series, the calculation of the partial auto-correlation coefficients is performed using the method proposed by Durbin: ) ) ϕ1,1 = r1

) ϕk , k =

) k −1 ) ) rk − ϕk −1, j rk − j

∑

1−

j =1 k −1

∑

k = 2, 3, K

) ) ϕk −1, j r j

(11.67)

j =1

) ) ) ) ϕkj = ϕk −1, j − ϕk , k ϕk −1, k − j Á

k = 3, 4, K; j = 1, 2, K, k − 1

) Mention that, by definition, the estimated PACF is ϕkk (k ) . ) 4) After the calculation of the auto-correlation coefficients, estimated to rk , ) and the partial auto-correlation coefficients, estimated to ϕkk , the following hypothesis is checked: might they be equal to zero? In this regard, the statistics from Table 11.8 are assessed. Table 11.8 The necessary formulae for the estimation of the auto-correlation coefficients nullity, and of the estimated partial auto-correlation coefficients nullity ) ) rk ϕkk Coefficient ) ) ϕkk − ϕkk rk − rk ) ) t = t = The t statistics ) ) ϕ kk rk s (rk ) s (ϕkk ) The approximation of the standard error of the values distribution

) s(rk ) =

k −1 1 ⎛⎜ ) ⎞ 1 + 2 r j2 ⎟ ⎟ n⎜ j =1 ⎝ ⎠

∑

) s (ϕkk ) =

1 n

The estimated auto-correlation coefficients and the partial auto-correlation coefficients, for which the absolute value of the test t statistics is bigger than 2, are statistically different from 0, at a significance level value around 5%. Table 11.9 The detailed features of the common steady random processes The process

AR (1)

ACF Exponential damping: • Positive values if a1 < 0 ; • Values whose sign is alternative, the first one is negative if a1 > 0 .

PACF Peak for the lag value 1, and then it is cancelled: • The peak is positive if a1 < 0 ; • The peak is negative if a1 > 0 ;

Load forecast

AR (2)

MA (1)

691 Sum of exponential or symmetrical functions depending on the signs and the values of the a1 , a2 parameters. Peak for the lag value 1, then it is cancelled: • The peak is positive if b1 > 0 ; • The peak is negative if b1 < 0 .

MA (2)

Peaks for the lag values 1 and 2, then they are cancelled

ARMA (1,1)

Exponential damping starting from the lag value 1: • sign (r1 ) = sign (b1 − a1 ) • all the values have the same sign if a1 < 0 ; • the values have an alternative sign if a1 > 0 .

Peaks for the lag values 1, 2 then they are cancelled. Exponential damping: • Values whose sign is alternative, the first value is positive if b1 > 0 ; • Negative values if b1 < 0 ; Sum of exponential functions or symmetrical functions. Their exact form depends on the b1 , b2 parameters signs and values. Exponential damping starting from the lag value 1: • ϕ11 = r1 • all the values have the same sign if b1 < 0 ; • the values have an alternative sign if b1 > 0 .

The establishment of the auto-correlation coefficients number, and of the partial auto-correlation number, which differ for zero, is essential for the choice of the adopted model form ARMA. 5) Find the ARMA model structure for the problem. In this regard, the following principles and results are used: • Try to establish very simple models for the following “classical” types: AR(1), AR(2), MA(1), MA(2), ARMA(1,1), …, ARMA(2,2). This fact avoids the parameters redundancy, supplies their quality estimation and avoids the model instability with respect to its coefficients values; • Compare the ACF and the PACF estimated for the studied realisation to the theoretical ACF and PACF, corresponding to the above models, in order to find a better match of their model. For the analysed realisation, the theoretical model whose model matching is the best is maintained. • In the case of a random process AR( na ), the PACF will have the elements ϕkk ≠ 0 for k ≤ na and ϕkk = 0 for k > na ; • In the case of a random process MA( nb ), the ACF will have the elements rk ≠ 0 for k ≤ nb and rk = 0 for k > nb ; • In the case of a random process ARMA( n a , nb ), the elements ϕkk from PACF will tend to 0 after the first na − nb lag values, while the elements rk from ACF will tend to 0 after the first nb − na lag values;

692


• The detailed properties of the simple steady random processes, illustrated in Table 11.9, are a real help for the model type identification of the analysed process. 6) The model parameters, whose structure has already been identified, have to be estimated on the ground of some very well known techniques, but which do not represents the object of the Box – Jenkins identification methodology. If the model is linear in parameters, just like the presented case, a multivariable linear random model should be used. If the model is not linear in parameters, we can apply linearization techniques described in the substitutions from Table 11.1. On the contrary, we have to use a gradient method for the parameter estimation. The most common is the Marquardt method, well known in literature.

11.6.5. Time series model validation The validation of the time series model implies the approaching of the following problems [11.13]: 1) The checking of the model steadiness is ensured by the fulfilment of some requirements for the AR coefficients ( ai with i = 1,K, na ). The imposed steadiness required is very important because it is the only way we obtain quality estimation for the models parameters. An unsteady model provides forecast values of the time series containing variances that grow unlimited, result undesirable. The steadiness checking can be performed by: • The visual examination of the time series to see if the average value or the variance change in time; • The examination of the estimated ACF to see if its values tend quickly to 0 ( rj ≤ 1.6 for j ≥ 5,6 ); • The checking of the theoretical steadiness conditions where the coefficient ai with i = 1, K, na must be respected. These conditions depend on the na order of the model and become very complicated for na > 2 . 2) The checking of the model unsteadiness means the respecting of some requirements for the moving average coefficients (SA) bi with i = 1, K, nb . These requirements are similar, from the algebraic point of view, to the steadiness conditions set for the AR coefficients. The model inversion is required due to the fact that, in the case of an ARIMA non-invertible model, old observation importance does not decrease as they become older. In reality, as it is physically normal, the latest values have to influence the latest observations. 3) The checking of the statistical significance of the model parameters is performed using a test of t type. We calculate the value: t=

estimated coefficient value − hypothesis coefficient value coefficient estiameted standard error

Load forecast

693

The value of the coefficient from hypothesis is set to 0 to verify if the coefficient has or has not to be excluded from the model. Every coefficient for which t > 2 (the test confidence level is 5%) will be accepted by the model, else it is excluded from the model if its statistical significance is not important. 4) The model correlation matrix calculation gives important information on its quality. The very high correlation values between the coefficients suggest that the parameters estimations are bad. Practically speaking, the coefficient estimations are unsteady if the correlation coefficient absolute value is equal or bigger than 0.9. The adoption of a different model structure is the only one solution for the model steadiness. 5) The appreciation of model capacity to represent the series data is absolutely necessary because there is no guarantee that a well-built model will represent quite precisely the realised data it has been built on. We have to calculate the following indicators: • Tuned square average error (TSAE): ) TSAE = σ 2a =

1 n − na − nb

)

∑ [ y ( t ) − y ( t )]

2

(11.68)

As the random noise ε(t ) from the input of the random process-generating filter cannot be directly analysed, the hypothesis that TSAE is the noise variance estimation is introduced. If we have to choose from two models having the same realisations and proprieties, but differing by TSAE, we choose the model for which TSAE has a smaller value because it better estimates the data series used for its building. • Mean Absolute Percentage Error (MAPE): ) 100 y (t ) − y (t ) [%] (11.69) MAPE = ⋅ n y (t )

∑

is useful for the model precision and its forecast precision characterisation. In practice, we choose to express the forecast precision using the forecasted confidence interval values. 6) We analyse if the model matches for the studied series. If the answer is negative, we go back to the identification step to select other models that can approximate the series data more correctly. We build these new models using the recommendations and the conclusions obtained in the validity – diagnose stage. The main objective of the analysis is the obtained residua calculated as difference between the realisation values { y (t )} and the values provided by the identified model of the process in question. ) e(t ) = y (t ) − y (t ) (11.70)

694


a) The graphical representation of the residua is the first step. The visual analysis of the graphic, accompanied by a large experience, can quickly mark out some significant issues regarding the determined model: • big data error detection: the existence of some residua bigger than the double standard deviation, make very probable the existence of incorrect acquired data or the presence of a large perturbation due to the external identifiable causes; • the variance value modification suggests the necessity to apply data series logarithmic transformations. b) The residua independence checking represents the most important ARIMA model statistic relevance test. If the identified model residua are correlated we can find a better model, which identifies more precisely the process dynamics and which has better AR and MA coefficients combination. During the model validation – diagnose stage, the Residua Auto-correlation Function RACF (k ) represents the main analytical instrument. The residua autocorrelation coefficient is: n−k

) rk (e ) =

n−k

∑ [e(t ) − e ] [e(t − k ) − e ] ∑ [e(t ) − y ] [e(t − k ) − e ] t =1

n

∑ [e(t ) − e ]

2

=

t =1

see

(11.71)

t =1

) In the case of a well-built ARIMA model, all the RACF (k ) = rk (e ) components are theoretically zero. In reality, even in the case of a correctly identified model we can have RACF values different from zero, as a result of the sample error or of the fact that time-limited realisations are used. So, in practice, it is very important to see if, for RACF, its components are or not important. In this respect we can make use of: • the t test that uses the formula proposed by Bartlett for the calculation of the residua auto-correlation coefficients variance; • the Ljung – Box test. c) The model oversizing represents another validation technique. We introduce a new parameter in the model we have to validate. We check if the extended model is better for the process. As a rule, it should be avoid the simultaneous introduction of two AR and MA parameters, because it could cause serious estimation and steadiness problems due to the parameters redundancy. We can also reduce the number of parameters, one by one. Although it is a good thing from many points of view (the easiness, the steadiness, the calculation speed, etc.), we have to analyse, to justify and to appreciate the parameters we want to reduce. d) The time series splitting into data under-sets and the identification of a single model for each series that represents another validation technique. If the

Load forecast

695

established models do not significantly differ between them and are not very different from the complete series model, then the analysed process does not change its properties in time, that can be verified using an adequate test, and the new models are important for the analysed stochastic process. If there are big differences between the new models, we have to analyse the following two possibilities: • either the identification process failed and we have to resume and to modify it, in order to obtain the expected result; • or the models are correct, but the studied stochastic process varies in time. It is important to mention that the time series splitting is possible only if the database is large, so that the under-sets contain each, enough observations. The validation – diagnose stage has to end either with a clear conclusion or with model modification recommendations. They contain generally: • extended or reduced models; • the model structure changing by reanalysing the FAC and FACP; • replacing the residua model in the original model; • the form of the y1 (t ) realisation, etc.

11.6.6. Time series forecast The time series forecast using the general model ARMA having the form (11.61) implies the determination of the time series future values y (t + l ) with l ≥ 1 [11.18]. The moment t represents the forecast origin, l represents the forecast horizon, and (t − k ) represents the previous moment, situated at k time interval. Everything is given in natural numbers, multiples of the sample period T. Any observation, noted by y (t + l ) , generated by the ARMA process can be expressed under the form of the recursive equation with differences: y (t + l ) = −

na

∑ i =1

ai y (t + l − i ) +

nb

∑ b ε(t + l − κ − i ) i

(11.72)

i =0

In conclusion, we obtain the series forecasted values if we take into consideration: • The series y (t − i ) terms, corresponding to the past, known at the t moment, are replaced by the corresponding real values of the series; • The series y (t + i ) terms, corresponding to the future, which are NOT known at the t moment, are replaced by the corresponding values ) forecasted for that moment y (t + i ) ; • The noise series terms ε(t − i ) , for the past, are determined in the following way:

696


) ε(t − i ) = y (t − i ) − y (t − i − 1)

(11.73)

• The noise series terms ε(t + i ) that cannot be determined in the future are 0. In that case, the following estimator is found: ) y (t + l t ) = −

+

l −1

na

)

∑ a y(t + l − i ) − ∑ a y(t + l − i ) + i

i =1 nb

∑

i

i =l

) bi [ y (t + l − κ − i ) − y (t + l − κ − i − 1)]

(11.74)

i =1

The estimation error variance value, for a horizon of l sample periods is obtained using the relation: ⎛ l −1 ⎞ ) V [e (t + l )] = ⎜⎜ ψ i2 ⎟⎟ σ 2a ⎝ i =0 ⎠

∑

(11.75)

where its values are replaced by the estimated corresponding values. The weighting coefficients ψ i are determined by solving the equation system that results from the polynomial identity:

( )∑ ψ q

A q −1

∞

i

−i

( )

= B q −1

(11.76)

i =0

If the noise that generates the series has a normal probability distribution and the ARMA is fit for the analysed time series then the forecasted error will be distributed normally. The confidence intervals can be found for every forecasted value, if all these are taken into account. For α significance degree the confidence interval limits are: ) ) ) y min,max (t + l ) = y (t + l ) ± u α 2 ⋅ V [e (t + l )] (11.77) where uα

2

represents α – bilateral quantile.

If some transformations were made to obtain the time series model, the results refer to the transformed series data. Therefore, in order to obtain the original series forecast results, we have to make appropriate transformations, on the estimated values and on the confidence intervals.

11.7. Short term load forecast using artificial neural networks

Load forecast

697

11.7.1. General aspects Short term load forecast (STLF), with time horizons from a few hours to several days, plays a key role in the economic and secure operation of power systems. This type of forecasting is required by basic operating activities such as unit commitment, hydro-thermal coordination, interchange evaluation and security assessment. The load forecast is mainly performed starting from historic load data, recorded systematically and processed by specific methods. The lack of accurate load forecasts leads to erroneous schedule results, no matter how accurate the methods of power generation scheduling are [11.8]. The load is influenced more or less by factors related to the type of consumers. The consumers are sensitive to external factors, which are different for each forecast horizon. Factors, such as the economic growth, influence the long term electricity demand, instead, it can be ignored in short term load forecast. Weather conditions, having an immediate effect on the load demand, are included in the STLF model. There are special events such as strikes, special TV shows, which affect the load, being very hard to model in STLF mainly due to the lack of historical data regarding their random influence upon the load demand. The weather-related factors have an important influence on the power consumption. The parameter that has the strongest effect on the load is the temperature. For example, a temperature range ensures the human comfort, outside of which the heating or cooling devices (air conditioning) are used to restore the temperature in the desired range. The day of the week is another load influencing factor, since the human beings have a weekly working cycle. Due to the factors mentioned above, the load is a non-linear function of many variables. STLF is easily performed through statistical methods, but it needs a large volume of past information, a complex model and an important computational effort, in order to obtain an accurate forecast. In the last years, the research in this field, aiming at forecast accuracy improvement, leaded to the use of artificial intelligence, with new techniques and algorithms. The Artificial Neural Networks (ANN), highly parallel computational tools, implemented software or hardware, are a reliable alternative to the classical methods for load forecast. Their use requires less input data and it leads to the increase of the forecast accuracy and the decrease in the forecast time. The artificial neural network (ANN) is a complex interconnected set of information processing elements, called Artificial Neurons (AN). The ANN is a domain of the Artificial Intelligence, based on the of the living nervous system structure. Conceived as the technical model of the human nervous system, the prove to be useful due to its features [11.6, 11.7]: • possibility of parallel processing of information, ensuring the achieving of optimal architectures, the answers being offered in real time;

698


• the memorising ability using an adequate training process; a problem is described and then solved by self-learning and not by a program; • the process modelled by the ANN does not have to be described by clearlystated rules. This characteristic makes an easy modelling of some complex processes whose functioning rules are either too complicated of too ambiguous from analytical point of view; • good behaviour in the case of a partially incomplete input data set, a performance due to the ability to associate the available input data with the complete training data set(s) to which it “resonates” the best; • good behaviour in the case of a partially wrong input data set; • relatively correct operation also when some neurons are “damaged”, due to the distributed memorizing of the information into the network. The possible applications of ANN are very useful and interesting, they aim many fields and tend to replace the specific human activity, such as voice and pattern recognition, the diagnose and the forecast in many fields or, in our case, the load forecast. However, the ANN present some disadvantages such as: • the learning process is, in general, long and complicated; • the ANN requires a large database for training; • the way the answer is provided is not explainable from a deterministic point of view. Although the present achievements are promising, sometimes even amazing, it takes long time to reach the human brain performances. There are many ANN types and architectures, among which the most important one can be mentioned: the Kohonen map, the Hopfield network, the Hamming network, the Multi Layer Perceptron with one or more layers, etc [11.2, 11.7].

11.7.2. ANN architecture The load forecast is one of the many activities performed in power system operation, the most used forecast interval being 24 hours, belonging to the short term load forecast. This type of forecast can be performed by using a Multi Layer Perceptron (MLP) [11.2]. It has a simple structure (one input layer, one output layer and one hidden layer) as presented in Figure 11.13.

Load forecast

699 x0 1

y1,0 w[1] 1,0

w[2] 1,0

1 y1,1

x1

y2,1

x2

y2,2

x3

y2,3

xI0

w[1] I1,I0 s=0

y1,I1

y2,I2

w[2] I2,I1

s=1

s=2

Fig. 11.13. Multi Layer Perceptron.

In Figure 11.13, xi represents the input values; y represents the output values; Is is the number of neurons on the layer s; w[jis ] is the synaptic weight for the connexion between the neuron i from the layer s and the neuron j from the layer s − 1 ; s is the layer index. Before the employing of MLP be possible for the load forecast, it is necessary that its weights are initialised with values so that for a certain input vector to obtain through the MLP an output vector as close as possible to the output vector. This process of the weights computation represents the training step of the MLP. Once trained, the network can be tested for the architecture and weights validation or subsequent for employment in practice. The training process consists in finding the right weights to produce a desired ANN output from a given input. A well-known algorithm is the back-propagation, which can be simplified in the following form [11.2]: Step 1. Initialise the weights with small random values. Step 2. Propagate forward an example through the MLP. A model is chosen from the training data set (input – desired output). In some implementations, the models can be shuffled. Based on the weights of neurons, the output value is computed using the normalized data: ⎛ I L −1 [ s ] [ s −1] ⎞ o[js ] = Ψ[js ] ⎜⎜ w ji oi ⎟⎟ ⎝ i =0 ⎠ [s] where o j is the output of neuron i on the layer s and Ψ[js ] is the sigmoid activation function of the neuron i on the layer s. Step 3. Update the weights through back-propagation. An iterative algorithm is used, by propagating backward the error gradient from the output layer toward the input layer, using the relations:

∑

w[jis ] ( t + 1) = w[jis ] ( t ) + Δw[jis ]

700


Δw[jis ] = η δ[js ] xi[ s ] where η is the learning rate and δ[js ] is the error for the layer s.

Step 4. A new example is presented to the network by going to step 2. If there are no models left, go to step 5. Step 5. Convergence test. If the global error is below a chosen threshold or the number of epochs (complete iteration through the training set) is too high then STOP, else re-initialise the training set and go to step 2. A STLF software * has been developed to provide the forecast of the average hourly loads for the next 24 hours, having as inputs past load data as well as external variables with influence on the load (temperature, cloudiness, etc.) [11.22, 11.23]. Therewith, the software provides also the forecast of an 24/48 hours time horizon, for the near future (for instance, a forecast performed Friday for the next Monday), but with less accuracy. From statistical observations and also from the SCADA operators’ experience has been concluded that the load in a certain day is strongly correlated with the load from the previous day and with the load from the same day one week before. Although the software provides acceptable forecasts using as input data only the load from the past, in order to increase the forecast accuracy, external variables (meteorological) were also used from areas with strong influence on the load (large cities from different geo-climatic zones). In the case of load forecast at the country scale, data from meteo stations can be used (such as Bucharest, Cluj and Iaşi). As regards the training and validation period, two different directions were analysed. In the first direction, the training was performed on an extended period (one year), and the validation of the achieved model was performed on a comparable period (6 months to 12 months). The second direction consists in training and validation on short periods. The validation period ranges from two weeks to one month, and the training period consists of two months before the validation one and the same month one year before the validation (forecast) period. In order to differentiate better the load profile of the weekend days from the working days, 7 binary inputs was introduced, where “1” marks the input corresponding to the forecasted day, and “0” marks the remainder of the days. The output variables of ANN represents 24/48 average hourly loads. The number of neurons from the hidden layer is established by a trial-anderror process, and the optimal architecture is chosen by successive trials. All the input and output variable are normalized in order to use the linear domain (unsaturated) of the activation functions of the artificial neurons. The STLF software allows the construction of many forecast models and the choosing of different MLP architectures and training sets. *

The software STLF v1.3 has been developed within the Doctoral School in Electrical Power Engineering from University “Politehnica” of Bucharest, by Silviu Vergoti.

Load forecast

701

11.7.3. Case study The STLF 1.3 software was used for the load forecast at two different levels: at the Romanian power system level (National Dispatching Centre) and at the distribution system level of Bucharest (which has more than two millions inhabitants).

(i) Short term load forecast at the Romanian power system level During the configuration and validation period of the forecast model, using the training on a long time horizon (one year), the optimal architecture was chosen from more than 70 configurations. It has 57 input neurons, 12 hidden neurons and 24 output neurons. The inputs correspond to 48 hourly loads (the day before and the same day one week before), 2 temperatures (from Bucharest and Cluj) and 7-bit day encoding. All other ANN configurations tested resulted in slightly higher errors. A forecast example can be seen in Figure 11.14,a,b. The forecasts are usually less accurate in winter than in summer since the electric heating is more developed than the air conditioning. The day encoding allows forecasting weekend days with greater accuracy within the same model. During the use of the program, the operators have built separate models for weekends and Mondays, slightly improving the forecast quality. Finding the optimal configuration was a time-consuming job since each architecture had to be trained and tested. The goal was to find a structure that does not overfit the data and provides forecast with an acceptable error. The size of the training set was one year, the special days being excluded automatically by the software (fixed holidays, such as May 1st and Christmas, and moving holidays, such as Easter). The accuracy of the ANN based forecasts was tested on one year of data excluding the holidays. The daily load profile was forecasted using the trained network, and subsequent it was compared to the real profile. The average percentage absolute error was used to evaluate the accuracy of the forecast per each hour. The errors resulted for the case analysed are mostly less than 2…4%, as shown in Figure 11.15. (note that, the first term in Figure 11.15 represents the error, and the second term represents the percentage of examples for which the error appear).

702

Load flow and power system security 8500 8000

Real Real Forecasted Prog

7500

MW

7000 6500 6000 5500 5000 4500 4000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

h

a. 7000

Real Real Forecasted Prog

6500

MW

6000 5500 5000 4500 4000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

h

b. Fig. 11.14. Load forecast example for the Romanian power system: a. winter day; b. summer day. 4-6% 4-6% 10% 10%

>6% >6% 3% 3%

1 h-1 2 d-1 < r < 1 h-1 r < 2 d-1 Steady-state change, dc Maximum change, dmax

r > 100 h-1

≤3% ≤4%

≤ 5.5 %

10 h < r ≤ 100 h-1 r ≤ 1 h-1 -1

≤7%

≤2%

≤3%

≤4 %

The magnitude of the current transients can, to a large extent, be limited by careful design of the DG plant, although for single generators connected to weak systems the transient voltage variations caused may be the limitation on their use

814

Technical and environmental computation

rather than steady-state voltage rise. Synchronous generators can be connected to the network with negligible disturbance if synchronised correctly and anti-parallel soft-start units can be used to limit the magnetising inrush current of induction generators to values lower than rated current. However, disconnection of the generators when operating at full load may lead to significant voltage drops. Also, some forms of prime mover (e.g., fixed speed wind turbines) may cause cyclic variations in the generator output current, which can lead to flicker nuisance, if not adequately controlled. Conversely however, the addition of DG plant acts to raise the distribution network fault level. Once the generation is connected any disturbances caused by other customers’ loads, or even by remote faults, will result in smaller voltage variations and hence improved power quality. It is interesting to note that one conventional approach to improving the power quality of sensitive high value manufacturing plants is to install local generation. Similarly, incorrectly designed or specified DG plants, with power electronic interfaces to the network, may inject harmonic currents which can lead to unacceptable network voltage distortion. For LV systems specific compatibility levels are given in IEC 61000-2-2 [14.21] and IEC 61000-3-6 [14.22], which also serve as planning levels, and are included in Table 14.4. Planning levels for harmonic voltages in MV and HV systems are not presented here since similar data was given in Table 6.4, Chapter 6. However, the generators can also lower the harmonic impedance of the distribution network and so reduce the network harmonic voltage at the expense of increased harmonic currents in the generation plant and possible problems due to harmonic resonances. This is of particular importance if power factor correction capacitors are used to compensate induction generators. Table 14.4 Planning levels for harmonic voltages in LV networks (IEC 61000-3-6, [14.22]) Odd harmonics ≠3k Harmonic Order voltage h (%) 5 6 7 5 11 3.5 13 3 17 2 19 1.5 23 1.5 25 1.5 25 >25 0.2+ 1.3 ⋅ h

Odd harmonics = 3k Harmonic Order voltage h (%) 3 5 9 1.5 15 0.3 21 0.2 >21 0.2

Note: THD at LV = 8 %

Even harmonics Harmonic Order voltage h (%) 2 2 4 1 6 0.5 8 0.5 10 0.5 12 0.2 >12 0.2

Distributed generation

815

A rather similar effect is shown in the balancing of the voltages of rural MV systems by induction generators. The voltages of rural MV networks are frequently unbalanced due to the connection of single-phase loads. An induction generator has very low impedance to unbalanced voltages and will tend to draw large unbalanced currents and hence balance the network voltages at the expense of increased currents in the generator and consequent heating. Power quality is an increasingly important issue and generation is generally subject to the same regulations as loads. This tends to work well in practice and it is generally possible to meet the required standards by careful design. The effect of increasing the network fault level by adding generation often leads to improved power quality. A notable exception is that a single large dispersed generator, e.g., a wind turbine, on a weak network may lead to power quality problems particularly during starting and stopping.

14.2.5. Protection issues A number of aspects regarding protection can be identified: • Protection of the generation equipment from internal faults; • Protection of the faulted distribution network from fault currents supplied by the DG; • Anti-Islanding or loss-of-mains protection; • Impact of DG on existing distribution system protection. Protecting the DG from internal faults is usually fairly straightforward. Fault current flowing from the distribution network is used to detect the fault and techniques used to protect any large spinning load are generally adequate. In rural areas, a common problem is ensuring that there will be adequate fault current from the network to ensure rapid operation of the relays or fuses. Protection of the faulted distribution network from fault current from the DGs is often more difficult. Induction generators cannot supply sustained fault current to a three-phase close up fault and their sustained contribution to asymmetrical faults is limited. Small synchronous generators require sophisticated exciters and field forcing circuits if they are to provide sustained fault current significantly above their full load current. Thus, for some installations it is necessary to rely on the distribution protection to clear any distribution circuit fault and hence isolate the dispersed generation plant, which is then tripped on over/under voltage, over/undervoltage protection or loss-of-mains protection. Loss-of-mains protection is a particular issue in a number of countries particularly where auto-reclose is used on the distribution circuits. For a variety of reasons, both technical and administrative, the prolonged operation of a power island fed from the dispersed generator but not connected to the main distribution network is considered unacceptable. Thus a relay is required which will detect when the dispersed generator, and perhaps a surrounding part of the network, has become islanded and will then trip the generator. This relay must work within the

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dead-time of any auto-reclose scheme to avoid out-of-phase reconnection. Although a number of techniques are used, including rate-of-change-of-frequency (rocof) and voltage vector shift, these are prone to mal-operation, if set sensitively to detect islanding rapidly. There is considerable scope for improvement in this area, particularly with the emergence of digital integrated protection, control and monitoring systems. The neutral grounding of the generator is a related issue as in a number of countries it is considered unacceptable to operate an ungrounded system and so care is required as to where a neutral connection is obtained and grounded. Finally, DG may effect the operation of existing distribution network by providing flows of fault current which were not anticipated when the protection was originally designed. The fault contribution from the dispersed generator can support the network voltage and lead to relays under-reaching.

14.2.6. Effects on stability Traditionally, distribution network design did not need to consider issues of stability as the network was passive and remained stable under most circumstances, provided the transmission network was itself stable. Further, for the early DG schemes, whose objective was to generate kWh from new renewable energy sources, considerations of generator transient stability tended not to be of great significance. If a fault occurred somewhere on the distribution network to depress the network voltage and the dispersed generator tripped, then all that was lost was a short period of generation. The DG tended to overspeed and trip on its internal protection. The control scheme of the DG would re-start it automatically, once the network conditions were restored. Of course if the generation scheme is intended mainly as a provider of steam for a critical process, then more care is required to try to ensure that the generator does not trip for remote network faults. However, as the inertia of dispersed generation plant is often low and the tripping time of distribution protection long, it may not be possible to ensure stability for all faults on the distribution network. A particular problem in some countries is nuisance tripping of rate of change of frequency (rocof) relays. These are set sensitively to detect islanding but, in the event of a major system disturbance, e.g., loss of a large central generator, mal-operate and trip large amounts of dispersed generation. The effect of this is, of course, to depress the system frequency further. Synchronous generators will pole-slip during transient instability but when induction generators overspeed they draw very large reactive currents, which depress the network voltage further and lead to voltage instability. The steady-state stability limit of induction generators can also limit their application on very weak distribution networks as a very high source impedance, or low network short circuit level, can reduce their peak torque to such an extent that they cannot operate at rated output. At present, stability is hardly considered when assessing dispersed renewable generation schemes. However, this is likely to change as the penetration of these


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schemes increases and their contribution to network security becomes greater. The areas that need to be considered include transient (first swing stability), as well as long term dynamic stability and voltage collapse.

14.2.7. Effects of DG connection to isolated systems Isolated power systems, like the ones operating in islands, face increased problems related to their operation and control. In most of these systems, the cost of electricity production is much higher than in interconnected systems due to the high operating costs of their thermal generating units, mainly diesel and gas turbines, and the import and transportation costs of the fuel used. In these systems the production of electric energy from renewable energy sources, mainly wind, presents particular interest, especially when the wind energy potential allows significant displacement of conventional fuels. The large volatility of these sources makes accurate wind forecasting a very important EMS function and the high degree of resource uncertainty makes economic scheduling a challenging task. Moreover, mismatches in generation and load or unstable system frequency control might lead to system failures much easier than interconnected systems. Thus, next to the more common angle and voltage stability concerns, frequency stability [14.23, 14.24] must be ensured. This depends on the ability of the system to restore balance between generation and load following a severe system upset with minimum loss of load. The control of frequency and the management of system generation reserves are of primary importance [14.24]. The introduction of a high penetration from wind energy causes additional difficulties, since the majority of Wind Turbines cannot participate in frequency control. In Figure 14.5 the frequency variation and the thermal Unit response of a large island system during disconnection of a 20 MW Gas Turbine for operation without and with Wind Production and different types of spinning reserves is simulated. The need for spinning reserves optimisation is obvious. Moreover, fast wind power changes and very high wind speeds might result in disconnection of wind turbines and thus frequency excursions and dynamically unstable situations. In addition, frequency oscillations might easily trigger the frequency protection relays of the wind parks, thus causing further imbalance in the system generation/load. Advanced control systems can substantially help operators to manage efficiently these systems allowing increased penetration of renewable energy sources in a secure way, as shown in a series of EC projects [14.25 ÷ 14.27]. In particular, the control system CARE has been developed comprising advanced software modules for load and wind power forecasting, unit commitment and economic dispatch of the conventional and renewable units and on-line security assessment capabilities integrated in a friendly Man-Machine environment. CARE has been fully installed in Crete, the largest Greek island and Madeira and its wind forecasting modules in Ireland. Its on-line dynamic security assessment functions allow operators to retain acceptable security levels by assessing in a fast way

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expected system frequency excursions and df dt values, for selected critical disturbances and by helping them in defining robust operating strategies [14.28 ÷ 14.30]. In this way penetration of renewable energy sources in isolated systems can be increased in a secure and reliable way.

Fig. 14.5. Frequency variation and thermal production in a large island system during disconnection of a 20 MW gas turbine for operation without and with wind power and different types of spinning reserves.

14.3. Commercial issues in distribution systems containing DG 14.3.1. Introduction In order to facilitate the competition between various generators, central and dispersed, setting of appropriate connection and use of transmission and distribution tariffs, as well as an equitable loss allocation policy is essential. Due to its location, DG not only acts as another source of electricity, but it can potentially substitute for transmission and high voltage distribution facilities, as well as reduce losses in those networks. It should be noted, that electricity prices at wholesale electricity market in Europe, average at about 20 – 30 $/MWh while the retail price of electricity is currently about 60 – 100 $/MWh. Transmission and distribution networks, together with the supply business are responsible for the difference between retail and wholesale prices. This indicates that kWh produced by DG has a


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higher value than kWh generated at transmission level. As in a fully competitive environment DG competes directly with central generation, the importance of a consistent commercial framework for pricing of network services is essential for the establishment of fair competition among generators. In the following, the impact of DG on distribution network operating and capital costs is discussed. Particular emphasis is placed on the relationship between network pricing practices and the economic impact of DG on power networks.

14.3.2. Present network pricing arrangements 14.3.2.1. Connection costs and charges Legislation requires Distribution Companies to provide a supply of electricity required. In meeting such a request, the company may set connection charges at a level which enables it to recover the cost incurred in carrying out any works, the extension or reinforcement of the distribution system including a reasonable rate of return on the capital represented by the costs. From DG perspective, two questions related to the policy of connecting DG are of considerable importance: a. voltage level to which generation should be connected, as it has a major impact on the overall profitability of generation projects, and b. question of whether connection policy is based on “shallow” or on “deep” charges. Voltage level related connection cost The overall connection costs may considerably alter the cost base of a DG. Primarily the voltage level to which the generator is connected drives these costs: the higher the voltage, the larger the connection cost. Generally, in order to secure the viability of a generation project, developers and operators of DG would prefer to be connected at the lowest possible voltage level. On the other hand, the higher the connection voltage level, the lower the impact that DG has on the performance of the local network. Therefore, network operators may prefer such solutions. These two conflicting objectives need to be balanced appropriately, and may require not only an in depth technical and economic analysis of the connection design but also the presence of an appropriate network pricing policy. The determination of the voltage level to which a generator should be connected to is driven by its impact on the voltage profile of the local network. In the majority of European countries the accepted steady state voltage variations are much stricter than the European Norm, EN 50160. In this respect, particularly critical is the voltage rise effect which generator connected to a weak network could produce. It is important to remember that adequate reactive or/and active power control could be used as a means for controlling the voltage rise. However, the commercial framework for the voltage regulation policy through active or reactive power control is not yet very well developed. For example, VAr management as a means of reducing the voltage fluctuations in distribution networks is not supported by appropriate pricing mechanisms. Instead, the majority

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of distribution companies charge for the consumption of reactive power implicitly through peak kVA or explicitly in terms of penalising reactive power consumption above a threshold. Furthermore, VAr injections are not considered to be useful. In fact present reactive power pricing in some distribution networks is that generators which absorb reactive power are charged by the distribution company on the basis of the active power demand taken by the plant, not its generation. As active power input is much lower than the active power output, reactive power taken by the generator is seen as reactive excess, and consequently, the generator is expected to pay the highest possible excess reactive charges. There are two different approaches to charging for reactive power/energy. The majority of distribution companies charge with respect to kVArh (reactive energy) in excess of 40-50% of the total unit consumption in that month. Some other companies charge for maximum kVAr (reactive power) of demand in excess of the value obtained by multiplying the maximum kW of demand registered in any time during the month by 0.4. There are also companies who base their distribution use-of-system charges on kVA demand, which discourages consumption of reactive power. Absorbing reactive power can be very beneficial to controlling voltage rise effect in weak overhead networks with dispersed generation. Although this would normally lead to an increase in network losses, DG does not have the opportunity to balance the connection costs against cost of losses and make an appropriate choice. Clearly, the above tariff structure discourages generators from participating in voltage regulation. Conversely, synchronous generators are offered no incentive by the distribution company to provide reactive support and take part in voltage regulation. The philosophy behind the excess reactive charges has been derived for passive distribution networks and cannot easily be justified in the context of a distribution network with dispersed generation. This mechanism does not, however, encourage the development of a reactive power management as a component of the voltage regulation service. Consequently, the inability of the present reactive power pricing concept to support provision of voltage regulation may unnecessarily force generators to connect to a higher voltage level, imposing significant connection costs. It is important to remember that reactive power flows may have a significant impact on active power losses. This fact is sometimes used to argue that VAr absorption as a means of reducing the voltage rise effect may be not desirable, as it generally leads to increase in losses. However, the generators are not given the opportunity to balance the benefit from connecting to a low voltage level against the cost incurred from increase in losses. Furthermore, presently used loss allocation factors are related to active power and are not capable of capturing the impact of reactive power consumption or absorption. Similarly, constraining generation off could be used as a means of reducing the voltage rise effect. The generator may find it profitable to shed some of its output for a limited period if allowed to connect to a lower voltage level. However, this option has not yet been offered to generators. It follows from the above discussion that the inability of the present reactive power pricing concept to support provision of voltage regulation may unnecessarily


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force generators to connect to a higher voltage level, imposing significant connection costs. Further development of market mechanisms and pricing policies may lead to the development of a market for the provision of voltage regulation services in distribution networks and provide more choice for dispersed generation to control its connection costs. This is an area which has only recently received noticeable attention. It is expected that it will develop in the near future and open up further possibilities for dispersed generation to participate not only in the energy market but also in a market for the provision of ancillary services. Appropriate tools, which would enable the establishment of an ancillary services market in distribution networks, are not currently available to network operators. This area is increasingly receiving considerable attention by both industry and academia. Deep versus shallow connection charges Another issue that can significantly influence the profitability of a generation project is related to whether connection charges should reflect only costs exclusively associated with making the new connection or should also include the additional costs which are indirectly associated with the reinforcement of the system. In other words, should connection charges be based on so-called shallow or deep connection costs. One such situation is illustrated in Figure 14.6. Cost associated with connecting the DG to the nearest point on the local distribution network system is referred to as shallow connection. Clearly, the line between the new DG and the system is used only by the generator, the generator is therefore required to cover the cost of the connection through connection charges. These could be imposed over a period of time if the distribution company invests and owns the connection, or alternatively, as one off-payment in which case the generator is effectively the owner of the line. It has been argued that the advantages of shallow connection charges is in the simplicity of their definitions, as it is relatively straightforward to identify cost exclusively related to connecting the generator to the nearest point on the network. On the other hand, the cost associated with a new connection might not be fully reflected in the connection charge made, as such a connection may require reinforcement of the system further away from the connection itself. The majority of network operators charge new entries for the cost of connection itself and cost incurred for any upstream reinforcement. For the purpose of the illustration of the concept it is assumed that the new connection requires the circuit breaker to be replaced, as the presence of the generator increases the fault level, as indicated in Figure 14.6. If the DG is required to cover the cost of the reinforcement of the circuit breaker, this would be referred to as deep connection cost. It is, however, important to stress that this circuit breaker is installed because of all generators, both central and embedded. The individual contribution of each generator to the size of the circuit breaker can be readily computed using conventional short circuit analysis tools. These contributions to the short circuit current then may be used to allocate the cost of replacing the circuit breaker. This is indicated in Figure 14.7.

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DG

CB Shallow Connection Deep Connection

Fig. 14.6. Illustration of shallow and deep connection.

CB

Contribution from embedded generation

Contribution from central generation

Fig. 14.7. Contribution of central and dispersed generators to fault level.

It is important to emphasise that an argument such as ‘there would not be a need to replace the breaker if the new generator did not appear’ cannot be credibly used to require the new entry to recover all system reinforcement cost. In accordance with conventional economic theory it can be argued that in this case distribution network owner should replace the circuit breaker, and recover its cost through charging all generators with respect to their respective contribution. This can be achieved by adjusting the use of system charges accordingly to all generators in the following price review period. Clearly, the circuit breaker in question should be considered as a system related investment and its cost recovered through the use of system charges rather than through the connection charges. Under such scenario, large generators connected to the transmission system would be likely required to compensate the majority of these costs. It is, however, important to observe that, in accordance with the present practice for pricing of network services, central generators are not charged for the use of the distribution network, as if these networks were not required for the use of their output.

14.3.2.2. Distribution use of system charges The distribution network business is dominated by capital cost and it operates in a near monopolistic or highly restricted competitive environment. Tariffs for the


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use of distribution networks are set so as to recover the cost involved in serving network users and to facilitate competition in the supply and generation of electricity. This defines the following primary objectives of setting tariffs for the use of distribution networks: • Revenue generation – tariffs should yield adequate revenue to cover network operating and capital costs but also should encourage justifiable network investment and discourages over-investment. • Economic efficiency – tariffs should reflect cost streams and should send appropriate economic messages to users of the network avoiding any temporal and spatial cross-subsidies. Although these objectives have been well understood and recognised, the implementation of such a tariff structure was always far from being straightforward. Determining an appropriate tariff structure is a complex procedure mainly due to the mass of technical detail involved, constraints imposed by available metering technology and the necessity to take into consideration various conflicting standards of fairness and efficiency in the choice of a tariff structure. One of the major challenges in tariff setting is establishing the trade off between the various objectives of tariff making: the ability to reflect accurately cost streams, efficiency in responding economically to changing demand and supply conditions, effectiveness in delivering appropriate revenue requirements, stability and predictability of the revenue and tariffs themselves, simplicity in terms of their practical implementability. Clearly, the economic efficiency requirement may lead to tariff complexity requiring a considerable amount of data handling, since strictly speaking each node in the network would have its own unique set of tariffs for each hour of the day. Tariffs also should provide a transparent framework in which regulatory agencies can exercise their statutory responsibilities in terms of monitoring the revenue, expenditure and performance of the distribution businesses. The question whether or not to adopt a particular tariff structure is settled by assessing the cost and benefits of its implementation. From DG perspective, the fundamental issue is the economic efficiency of these tariffs and their ability to reflect cost streams imposed by the users. The impact of DG on the networks (in terms of costs and benefits) is very site specific. It varies in time and depends on the availability of the primary sources (important for some forms of renewable generation), size and operational practice of the plant, proximity of the load, layout and electric characteristics of the local network. It is not, therefore, surprising that the relatively simplistic tariff structures, with network charges being averaged across customer groups and various parts of the network do not appropriately reflect the economic impact of DG on distribution network costs. This is because these tariffs have no real ability to capture the temporal and spatial variations of cost streams. As indicated in the introduction, the impact of DG on network operating and capital costs is location specific. It is, therefore, essential to recognise that only location specific tariff regimes are candidates for adequately recognising the

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benefit/cost of DG to power networks. Furthermore, network operating and capital costs are known as being influenced by variation of demand and generation in time. Therefore, a further requirement of the ideal pricing policy is that it must possess a time-of-use dimension. Consequently, in the development of the ideal pricing mechanism for networks with DG the key requirements of spatial and time of use discrimination must rank very highly.

14.3.2.3. Network security, service quality and distributed generation As stated already, system security in distribution networks tends to be measured deterministically. The most widely applied security assessment method is the so-called (N-1) security criterion. This method is conceptually simple to understand and also relatively easy to programme. Unfortunately, evaluation of system security by such deterministic approaches is inappropriate for the deregulated environment. It is particularly unsuitable for systems with DG. DG by definition is located deep in the distribution network, and often very close to end customers. Therefore, apart from being another source of energy, DG can potentially replace transmission and distribution network facilities. From this perspective, DG can be regarded as a competitor to transmission and distribution in the provision of network services. However, DG, particularly renewable with stochastic output, is not and cannot be available at all times. Consequently the potential of this type of generation to replace transmission or distribution network facilities is mostly ignored in current transmission and distribution planning. The argument used to justify this attitude is that transmission facilities would in any case be required to ship power from central generation at times when DG is not available. Therefore the level of security (and therefore capacity) designed into transmission and distribution systems cannot be altered by the presence of DG. This argument is, however, only valid in the narrow context of deterministic security standards. For systems with significant penetration of DG with a wide diversity of primary sources, it would be inconceivable and indeed most improbable that all the generating units would be unavailable at exactly the same time. And yet this is the assumption made to discount the value of distributed generation as a potential provider of network service. In order to quantify and evaluate the true value of DG in terms of its contribution to improvement of quality of supply, it is necessary to abandon present deterministic security assessment methods in favour of probabilistic ones. In probabilistic security assessment each network element including all dispersed generating units, is assigned an availability index. The required system generation reserve and network capacity margin, which is optimal in probabilistic sense, is then determined as a trade off between additional capital investments as well as operating costs and the reduction in outage costs incorporating operating constraints for the contingent systems into the optimisation process. This problem can be solved within a stochastic optimisation framework for which methodologies are available.


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14.3.2.4. Allocation of losses in networks with DG The presence of DG alters the power flows and hence losses that are incurred in transporting electricity across transmission and distribution networks. Several methods have been proposed in the literature to allocate losses, including the proportional sharing loss formula, pro rata, incremental transmission loss, and incremental bilateral contract path [14.31 ÷ 14.36]. Amongst them, the ones based on marginal losses come closer to an ideal policy that fulfils the following requirements: • Economic efficiency; Losses must be allocated so as to reflect the true cost that each user imposes on the network with respect to the cost of losses i.e. it must avoid cross subsidies between users and between different times of use; • Equity, accuracy and consistency; The loss allocation method must be equitable, accurate and consistent; • Must utilise metered data; From a practical standpoint, it is desirable to base allocation of losses on actual metered data; • Must be simple and easy to implement; In order for any proposed loss allocation method to find favour, it must be easy to understand and implement. Because power system load as well as generation output vary in time and space, any proposed loss allocation scheme must have the ability to capture and accurately convey the spatial and temporal impacts that each user has on losses. Figure 14.8 summarises graphically the impact of DG on marginal losses for a particular study case network. It is evident that minimum losses in this network occur when the DG output equals approximately 250 kW. Beyond this level of output, DG ceases to have a beneficial effect on losses. 4

Total system Loss (kW)

3.5 3 2.5 2 1.5 1 0.5 0 0

100

200

300

400

500

DG output (kW)

Fig. 14.8. Variation of total system loss with DG output.

Figure 14.9 depicts the variation of marginal loss coefficients (MLCs) with DG output at a certain DG bus for a whole day [14.38]. Notice that the MLC at this node has a negative value for most of the day, indicating that this DG is

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contributing to system loss reduction and should therefore be rewarded. In the period 4.30-10.30 however, MLC becomes positive, and DG should be penalised. This is shown in the lower curve of Figure 14.10, illustrating the revenue variations at two DG buses.

Fig. 14.9. Daily variation of Marginal Loss Coefficients (MLC’s) related to active injections and payment factors at a DG bus.

Fig. 14.10. Daily revenue variations at two DG buses.

It is shown [14.37], that methods based on the evaluation of marginal contributions that each user makes to the total system losses provide a consistent policy for allocating series losses in distribution and transmission networks, that ensures economic efficiency. These methods can be proven to be applicable for a fully competitive electricity market.


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It is important to emphasise that the above methods may not be appropriate to allocate losses that do not depend on system loading, such as voltage driven core losses in transformers. These charges associated should be dealt within the same framework as charges for availability, and not as a part of half hourly settlement process.

14.4. Conclusions In this chapter the main technical and economic implications from the connection of DG to the distribution networks is discussed. In particular, the effects on steady state voltages, on fault level increase, on power quality, protection and stability are discussed. The technical criteria and limitations are critically reviewed and where possible alternative approaches proposed. Commercial issues and tariff structures are discussed next, focusing on connection costs and charges, network security and service quality and allocation of losses. It is argued that the adoption of non-discriminatory tariffs would enable efficient operation and long term development of power systems with significant presence of DG.

Chapter references [14.1] CIRED, Working Group WG04 – Dispersed generation, June 1999. [14.2] CIGRE, Working Group WG 37-23, Impact of increasing contribution of dispersed generation on the power system, 1997. [14.3] *** – Communication from the European Commission: Energy for the future, Renewable sources of energy. White Paper for a community strategy and action plan, COM (97) 559. [14.4] *** – Directive of the European Parliament and of the Council on the Promotion of Electricity from Renewable Energy Sources in the Internal Electricity Market, European Commission, 2000. [14.5] Hatziargyriou, N., Zervos, A. – Wind power development in Europe, Proceedings of the IEEE , Vol. 89, Issue 12, pp. 1765 – 1782., December 2001. [14.6] *** – Wind Directions, Vol. XXII, No. 1, November 2002. [14.7] Sanchez, M. – Cluster – Integration RES+DG as European research activities, presentation at Beta Session 4b: Integration of RES+DG, CIRED, Barcelona, May 12–15, 2003. [14.8] Strbac, G. (convener) et al. – Economic and technical implications from the connection of dispersed generation to the distribution network, CIGRE Task Force 38.06.03, August 2002. [14.9] *** – Electricity tariffs and embedded renewable generation, Contract JOR3CT98-0201, Final Report, July 2000. [14.10] Hatziargyriou, N.D., Papathanassiou, S.A. – Technical requirements for the connection of dispersed generation to the grid, IEEE Power Engineering Society Summer Meeting 2001, Vol. 2, pp. 749 – 754, Vancouver, Canada, July 15-19, 2001.

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[14.11] European Norm EN 50160 – Voltage characteristics of electricity supplied by public distribution systems, CENELEC, 1999. [14.12] Hatziargyriou, N.D., Karakatsanis, T.S., Papadopoulos, M. – Probabilistic load flow in distribution systems containing wind power generation, IEEE Trans. on Power Systems, Vol. 8, No. 1, pp. 159 – 165, February 1993. [14.13] Hatziargyriou, N., Karakatsanis, T., Strbac, G. – Connection criteria for renewable generation based on probabilistic analysis, 6th Intern. Conference on Probabilistic Methods Applied to Power Systems, PMAPS’2000, Funchal, Madeira, Portugal, September 25 – 28, 2000. [14.14] IEC 868-0, Part 0 – Evaluation of flicker severity, 1991. [14.15] IEC 868 (1986) – Flickermeter. Functional design and specifications. Amendment No. 1, 1990. [14.16] IEC 61000-4-15 – Part 4: Testing and measurement techniques – Section 15: Flickermeter-Functional and design specifications, 1997. [14.17] IEC 61000-3-3 – Part 3: Limits – Section 3: Limitation of voltage fluctuations and flicker in low-voltage supply systems for equipment with rated current ≤ 16Α, 1994. [14.18] IEC 61000-3-5 – Part 3: Limits – Section 5: Limitation of voltage fluctuations and flicker in low-voltage power supply systems for equipment with rated current greater than 16 Α, 1994. [14.19] IEC 61000-3-11 – Part 3: Limits – Section 11: Limitation of voltage changes, voltage fluctuations and flicker in low voltage supply systems for equipment with rated current < 75 Α and subject to conditional connection, 2000. [14.20] IEC 61000-3-7 – Part 3: Limits – Section 7: Assessment of emission limits for fluctuating loads in MV and HV power systems – Basic EMC publication, 1996. [14.21] IEC 61000-2-2 – Part 2: Environment – Section 2: Compatibility levels for lowfrequency conducted disturbances and signalling in public supply systems, 1990. [14.22] IEC 61000-3-6 – Part 3: Limits – Section 6: Assessment of emission limits for distorting loads in MV and HV power systems, 1996. [14.23] Hatziargyriou, N.D., Karapidakis, E.S., Hatzifotis, D. – Frequency stability of power systems in large islands with high wind power penetration, Proceedings of the 1988 Bulk Power Systems Dynamics and Control Symposium – IV Restructuring, Santorini, Greece, August 23 – 28, 1998. [14.24] Dialynas, E.N., Hatziargyriou, N.D., Koskolos, N., Karapidakis, E. – Effect of high wind power penetration on the reliability and security of isolated power systems, 37th Session, CIGRE, pp. 38-302, Paris, 30th August-5th September, 1998. [14.25] *** – Development and implementation of an advanced control system for the optimal operation and management of medium-sized power systems with a large penetration from renewable power sources, Final report of JOULE II project JOU2-CT92-0053. Edited by the Office for Official Publications of the European Communities, Luxembourg, 1996. [14.26] *** – CARE: Advanced control advice for power systems with large scale integration of renewable energy sources, JOR3-CT96-0119, Final Report, September 1999. [14.27] *** – MORE CARE: More advanced control advice for secure operation of isolated power systems with increased renewable energy penetration and storage, NNE5-1999-00726, Final Report, 2003.


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[14.28] Hatziargyriou, N. – Guest Editorial: Secure wind power penetration in isolated systems, Wind Engineering, Vol. 23, No. 2, 1999. [14.29] Hatziargyriou, N., Bakirtzis, A., Contaxis, G., Cotrim, J.M.S., Dokopoulos, P., Dutton, G., Fernandes, M.J., Figueira, A.P., Gigantidou, A., Halliday, J., Kariniotakis, G., Lopes, J.A.P., Matos, M., Mayer, D., McCoy, D., O'Donnell, P., Stefanakis, J. – Energy management and control of island power systems with increased penetration from renewable sources, IEEE Power Engineering Society Winter Meeting 2002, Vol. 1, pp. 335 – 339, 2002. [14.30] Hatziargyriou, N.D.; Karapidakis, E.S. – Online preventive dynamic security of isolated power systems using decision trees, IEEE Trans. on PWRS, Vol. 17, Issue 2, pp. 297 – 304, May 2002. [14.31] Bialek, J. – Tracing the flow of electricity, IEE Proceedings, Generation, Transmission and Distribution, Vol. 143, No. 4, pp. 313 – 320, July 1996. [14.32] Kirschen, D., Allan, R., Strbac, G., – Contributions of individual generators to loads and flows, IEEE Trans. on PWRS, Vol. 12, No. 2, pp. 52-60, February 1997. [14.33] Bialek, J.W., Ziemianek, S., Abi-Samra, N. – Tracking-based loss allocation and economic dispatch, Proceedings of 13th Power Systems Computation Conference, pp. 375-381, Trondheim, Norway, July 1999. [14.34] Schweppe, F., Caramanis, M., Tabors, R., Bohn, R. – Spot pricing of electricity, Kluwer Academic Publishers, Boston, 1988. [14.35] Zobian, A., Ilic, M. – Unbundling of transmission and ancillary services, Part I: Technical issues, IEEE Trans. on PWRS, Vol. 12, No. 2, pp. 539 – 548, May 1997. [14.36] Gross, G., Tao, S. – A loss allocation mechanism for power system transactions, presented at Bulk Power System Dynamics and Controls IV – Restructuring, Santorini, Greece, 1998. [14.37] Mutale, J., Strbac, G., Curcic, S., Jenkins, N. – Allocation of losses in distribution systems with embedded generation, IEE Proceedings, Generation, Transmission and Distribution, Vol. 147, No. 1, pp. 7 – 14, January 2000. [14.38] Hatziargyriou, N., Karakatsanis, T., Papadogiannis, K. – Probabilistic cost allocation of losses in networks with dispersed renewable generation, PMAPS2002, Naples, September 22 – 26, 2002.