Linear Models of Non-linear Power System ...

TRITA-ETS-2001-04 ISSN 1650-674X

Linear Models of Non-linear Power System Components Jonas Persson

Linear model

Non-linear component

Stockholm 2002 Licentiate Thesis Royal Institute of Technology Department of Electrical Engineering Electric Power Systems

© Jonas Persson, February 2002 Department of Electrical Engineering Electric Power Systems, Stockholm, 2002

To My Parents

Abstract Simulation of electric power systems becomes more and more important in order to, among other things see through scenarios before they become real. Since power systems are more integrated and contain more complex components nowadays, the simulation is today a bigger task than decades ago. In parallel with the expansion of electric power systems, computers have made it possible to create models of distribution networks and of national power systems. This thesis is focused on the development of a detailed model of one of the newest products on the power system market: the Thyristor-Controlled Series Capacitor. From the detailed model later on a linear model is created. The linear model represents the fundamental behavior of the ThyristorControlled Series Capacitor. When using the linear model, simulations can be carried out faster and the whole power system can also be linearized in a more proper way since the developed, simpler model of the component is linear in contrast to the original one. The developed linear model is compared to the original model both in time domain simulations and by linearizations of a power system. TRITA-ETS-2001-04 · ISSN 1650-674X

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Sammanfattning Simulering av elkraftsystem blir viktigare och viktigare för att bl a genomskåda scenarior innan de äger rum i verkligheten. I och med att elkraftsystem numer är mer integrerade och innehåller mer komplexa komponenter än tidigare, är simulering av elkraftsystem idag en svårare uppgift än för bara några årtionden sedan. Parallellt med expansionen av elkraftsystem har datorer gjort det möjligt att skapa modeller av såväl distributionssystem som länders stamnät. I denna avhandling behandlas en av de senaste produkterna på elkraftmarknaden: den tyristorstyrda seriekondensatorn. En detaljerad modell av den tyristorstyrda seriekondensatorn tas fram och av den skapas senare en linjär modell. Den linjära modellen representerar den tyristorstyrda seriekondensatorns fundamentala, låg-frekventa uppförande. Vid användandet av den linjära modellen kan simuleringar göras snabbare och elkraftsystemet kan dessutom linjäriseras på ett mer korrekt sätt, eftersom den förenklade komponentmodellen är linjär till skillnad från den ursprungliga modellen. Den framtagna linjära modellen jämförs med den ursprungliga, dels genom tidssimuleringar och dels genom linjäriseringar av ett elkraftsystem. TRITA-ETS-2001-04 · ISSN 1650-674X

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Acknowledgements This report completes the work carried out since December 1998 at Electric Power Systems, department of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden, and constitutes my licentiate thesis. First of all, I want to thank my supervisors, Professor Lennart Söder at Royal Institute of Technology and Dr. Kjell Aneros at ABB Utilities, for their advice, support and encouragement during the project. I also want to thank Professor Göran Andersson who was my supervisor the first one and a half year of the project and who invited me to start this project. I would like to thank the members of the reference group for always discussing and supporting the project in our meetings: Dr. Bertil Berggren at Corporate Research within ABB, Magnus Danielsson at Svenska Kraftnät, Dr. Mehrdad Ghandhari at Royal Institute of Technology, Dr. Sture Torseng and Lennart Ängquist at ABB Utilities. I want to thank my inspiring colleagues at ABB Utilities that are always around and to whom I can ask anything I think about: Ture Adielson, Eva Fredricson-Sjögren, Anders Frost, Jean-Philippe Hasler, Bertil Klerfors, Dr. David Larsson, Lars Lindkvist, Bengt Lundin, Tech. Lic. Ann Palesjö, Jytte Pedersen, Tore Petersson, Bo Poulsen, Sune Sarri, Sten Stenemar, and Dr. Ricardo Tenório. Also Tech. Lic. Inger Segerqvist at ABB Utilities I want to thank. My one-day-a-week in Västerås is always a highlight for me. One extra thanks goes to Lars Lindkvist for all anytime-Simpow-support, to Lennart Ängquist for instructing me in the TCSC-model that was used in the thesis and to Tore Petersson for sharing his knowledge. I also would like to thank the staff of Electric Power Systems for providing the stimulating atmosphere. Special thanks to my roommates: the mathematician Tech. Lic. Valery Knyazkin and Magnus Öhrström, the computer expert Dr. Erik Thunberg for all stimulating thoughts and talks about "fundamental electric relations everyone should easily know", the former secretary Lillemor Hyllengren, Margaretha Surjadi, Tech. Lic. Mikael Amelin for late night sessions, Ingemar Jonasson, Thomas Ackermann, Dr. Peter Bennich, Tech. Lic. Lina Bertling, and Dr. Niclas Schönborg for all inspiration. The master thesis students I have had the opportunity to meet beside my project I also want to thank: Erik Ek at Svenska Kraftnät, Anna Eriksson at Bombardier Transportation, Eva Centeno López at Endesa, Íris Baldursdóttir at ABB, and Emil Johansson at ABB.

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I want to thank ABB, Elforsk, Energimyndigheten, Svenska Kraftnät, and Teknikvetenskapliga Forskningsrådet for providing funding for this licentiate project. Finally, I would like to thank those closest to me: My lovely Eva, for all her love, support, spell-checking of the thesis, and always understanding. My parents Hasse and Ingrid, for their love and believing in me and for always giving me permission to do whatever I have wanted, and my sister Ewa with her family Glenn, Alexander, and Rebecca for all love and support whenever I have to be inspired. Without you this would never have been done! Many thanks also to my singing quartet "Good People" for getting me relaxed and thinking in harmonies instead of power systems. Stockholm February 2002

Jonas Persson.

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Contents

1 Introduction .................................................................................................. 1 1.1 Background............................................................................................. 1 1.2 Problem formulation............................................................................... 3 1.3 Main contributions of the thesis.............................................................. 4 1.4 Outline of the thesis ................................................................................ 5 1.5 List of publications ................................................................................. 5 2 Concepts of dynamic systems ...................................................................... 7 2.1 Basic formulation of dynamic systems ................................................... 7 2.2 Definition of a linear system................................................................... 9 2.3 Definition of a time-invariant system ..................................................... 9 2.4 Linear and time-invariant system ......................................................... 10 2.5 Linearization ......................................................................................... 11 2.6 Formulations in the thesis ..................................................................... 15 2.6.1 The dq0-representation ...................................................................... 17 2.6.2 The input and output signals to the linear model............................... 20 3 Methods of linearizing a component ......................................................... 23 3.1 Introduction........................................................................................... 23 3.2 Estimate an ARX-model....................................................................... 24 3.3 Prony analysis....................................................................................... 25 3.4 Newton-Raphson algorithm using Discrete Fourier Transform ........... 26 3.5 Describing functions............................................................................. 27 3.6 Linearized discrete-time model of a thyristor-controlled series compensator......................................................................................... 28 3.7 Dynamic modeling of TCSCs for subsynchronous resonance studies .................................................................................................. 31 3.8 Dynamic phasor model of a TCSC ....................................................... 33 4 Implementation of the Thyristor-Controlled Series Capacitor .............. 35 4.1 Introduction to the TCSC...................................................................... 35 4.2 The structure of the TCSC .................................................................... 37 5 Linearization of the Thyristor-Controlled Series Capacitor .................. 39 5.1 Introduction........................................................................................... 39

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Contents 5.2 The studied power system..................................................................... 40 5.3 Steady-state of the system..................................................................... 43 5.4 Perturbing the TCSC in a stiff power system ....................................... 45 5.4.1 Perturbing the d-component of the current ........................................ 47 5.4.2 Perturbing the q-component of the current ........................................ 49 5.5 Building a linear model of the TCSC ................................................... 50 5.5.1 Preparation of the four signals ........................................................... 51 5.5.2 Identification of the four signals........................................................ 52 5.5.3 How a transfer function is formulated on ABCD-form ..................... 55 5.5.4 The final linear TCSC-model ............................................................ 55 5.6 A reference system interfacing the linear TCSC-model ....................... 56

6 Comparisons in time domain simulations ................................................ 59 6.1 Introduction........................................................................................... 59 6.2 Description of the three dynamic simulation cases............................... 59 6.3 Case a.................................................................................................... 61 6.4 Case b.................................................................................................... 64 6.5 Case c.................................................................................................... 67 6.6 Conclusions of time domain simulations.............................................. 70 7 Comparisons in linear analysis.................................................................. 71 7.1 Introduction........................................................................................... 71 7.2 Linearizations of the studied power system.......................................... 72 7.2.1 Linear analysis of the system in phasor mode with a fixed series capacitor............................................................................................. 72 7.2.2 Linear analysis of the system in instantaneous value mode with a fixed series capacitor....................................................................... 73 7.2.3 Linear analysis of the system in instantaneous value mode with the original TCSC-model ................................................................... 75 7.2.4 Linear analysis of the system in instantaneous value mode and the linear TCSC-model ...................................................................... 77 7.3 Conclusions of linear analysis of the power system ............................. 78 8 Conclusions and future work..................................................................... 79 8.1 Conclusions........................................................................................... 79 8.2 Future work........................................................................................... 80 8.3 Personal comments on simulation of power systems ........................... 81 Appendix A Newton-Raphson algorithm using Discrete Fourier Transform................................................................................................... 83 A.1 Introduction.......................................................................................... 83 A.2 Setup of the algorithm.......................................................................... 83 A.3 The structure of the algorithm.............................................................. 85 A.4 Calculation order of the Newton-Raphson algorithm using Discrete Fourier Transform ................................................................. 90 A.5 Comments about using the algorithm .................................................. 91

Contents

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A.6 An example when running the algorithm............................................. 92 Appendix B Eigenvalues, eigenvectors, and participation factors in a linearized system........................................................................................ 95 B.1 Eigenvalues .......................................................................................... 95 B.2 Eigenvectors......................................................................................... 96 B.3 An example of a linearization .............................................................. 96 B.4 Participation factors ............................................................................. 99 Appendix C Implementation of the TCSC-control................................... 101 C.1 Structure of the TCSC-control ........................................................... 101 C.1.1 Phase Locked Loop......................................................................... 103 C.1.2 Booster ............................................................................................ 106 C.1.3 Thyristor Pulse Generator ............................................................... 107 C.1.4 Thyristors ........................................................................................ 109 C.2 The rhythm of the thyristors............................................................... 110 References...................................................................................................... 111

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List of Figures Figure 1.1. Thyristor-Controlled Series Capacitor ........................................................3 Figure 2.1. A dynamic system with a vector of input signals u, a vector of output signals y, and an internal state vector x ............................................8 Figure 2.2. Input and output signals of a time-invariant system..................................10 Figure 2.3. The closer interval around x0, the better the linear approximation holds ..........................................................................................................12 Figure 2.4. A dynamic system with a vector of input signals ∆u, a vector of output signals ∆y, and an internal state vector ∆x .....................................15 Figure 2.5. A non-linear component............................................................................15 Figure 2.6. A Thyristor-Controlled Series Capacitor ..................................................16 Figure 2.7a. Momentary phase voltages ua, ub, and uc of a three-phase voltage .........18 Figure 2.7b. dq0-components ud, uq, and u0 of a three-phase voltage .........................18 Figure 2.8. A slow power oscillation in both active and reactive power for a transmission line........................................................................................18 Figure 2.9a. ud, uq, and u0 for one of the line nodes ....................................................19 Figure 2.9b. id, iq, and i0 for a transmission line ..........................................................19 Figure 2.10. A phase current containing a slow power oscillation when simulating in instantaneous value mode ....................................................19 Figure 2.11. Frequency spectrum of a phase current containing a slow power oscillation when simulating in instantaneous value mode .........................20 Figure 2.12. The dq0-components of the current into the TCSC in steady-state.........21 Figure 2.13. The dq0-components of the capacitor voltage of the TCSC in steady-state ................................................................................................22 Figure 3.1. Voltage and current waveforms of a TCSC ..............................................28 Figure 3.2. Basic scheme of one phase of the Thyristor-Controlled Series Capacitor ...................................................................................................29 Figure 3.3. The same A-matrix is used in time intervals α and γ (A1) ........................30 Figure 3.4. One-line diagram of one phase of a Thyristor-Controlled Series Capacitor ...................................................................................................31 Figure 3.5. One phase line current and an associated thyristor current .......................32 Figure 3.6. One phase of the TCSC-representation.....................................................33 Figure 4.1. Basic scheme of the Thyristor-Controlled Series Capacitor .....................35 Figure 4.2. The TCSC from a fundamental frequency point of view ..........................36 Figure 4.3. The series capacitor voltage and the current through the series reactor in steady-state. The figure describes the situation for one phase of the TCSC.....................................................................................36 Figure 4.4. Control system of the TCSC .....................................................................38 Figure 5.1. A simplified model of the TCSC-compensated south-north link in Brazil .........................................................................................................40 Figure 5.2. Block diagram of excitation system with DC commutator exciter ...........42

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List of Figures

Block diagram of the governors for S1 and S2 .........................................42 Block diagram of the steam turbines for S1 and S2 ..................................43 id, iq, and i0 into the TCSC when the system is reaching steady-state .......43 uC d, uC q, and uC 0 over the TCSC when the system is reaching steady-state, unfiltered values....................................................................44 Figure 5.7. The current into the TCSC in steady-state ................................................45 Figure 5.8. System used for observing the dynamic behavior of the TCSC................45 Figure 5.9. The current into the TCSC is disturbed with 10% of id 0 in its d- and q-components ............................................................................................46 Figure 5.10. Filter characteristic of the low-pass filter, magnitude and phase ............47 Figure 5.11. uC d when id is increased with 10%..........................................................47 Figure 5.12. uC q when id is increased 10%..................................................................47 Figure 5.13. u and i scheduled in the same diagram. The marked square is focused below............................................................................................48 Figure 5.14. u zoomed.................................................................................................48 Figure 5.15. uC d when iq is increased 10% of id 0 ........................................................49 Figure 5.16. uC q when iq is increased with 10% of id 0 ................................................49 Figure 5.17. u and i scheduled in the same diagram....................................................49 Figure 5.18. u zoomed.................................................................................................50 Figure 5.19. Periodic sampling of the signals .............................................................51 Figure 5.20. The filtered and unfiltered value of uC q when id is increased 10% .........52 Figure 5.21. uC d identified when id is perturbed..........................................................53 Figure 5.22. uC q identified when id is perturbed..........................................................53 Figure 5.23. uC d identified when iq is perturbed..........................................................53 Figure 5.24. uC q identified when iq is perturbed..........................................................53 Figure 5.25. The linear TCSC-model ..........................................................................56 Figure 5.26. The surrounding reference system for the linear model, H(s).................57 Figure 6.1. The power system wherein the TCSC is modeled with the original TCSC-model..............................................................................................60 Figure 6.2. The power system wherein the TCSC is modeled with the linear model including an interfacing reference system ......................................61 Figure 6.3. id resp. iq for both the original TCSC-model and the linear model............62 Figure 6.4. iq(id) for the two models............................................................................62 Figure 6.5. uC d resp. uC q for both the original TCSC-model and the linear model in case a .....................................................................................................63 Figure 6.6. uC q(uC d) for the two models......................................................................63 Figure 6.7. id resp. iq for both the original TCSC-model and the linear model............64 Figure 6.8. The d- and q-components of the current through the TCSC viewed in one diagram, iq(id) .................................................................................65 Figure 6.9. uC d resp. uC q for both the original TCSC-model and the linear model in case b .....................................................................................................65 Figure 6.10. uC q(uC d) for the two models....................................................................66 Figure 6.11. Impedance of the original TCSC-model and the linear model during the time interval 5.5 s < t < 7 s.......................................................67 Figure 6.12. id resp. iq for both the original TCSC-model and the linear model..........68 Figure 6.13. The d- and q-components of the current through the TCSC viewed in one diagram, iq(id) for case c .................................................................68 Figure 6.14. uC d resp. uC q for both the original TCSC-model and the linear model in case c ..........................................................................................69 Figure 6.15. uC q(uC d) for the two models....................................................................69

List of Figures Figure A.1. Figure A.2. Figure A.3. Figure A.4. Figure A.5. Figure B.1. Figure B.2.

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A test signal containing nine modes.........................................................92 The algorithm has found 6 eigenvalues....................................................93 The algorithm has found 7 eigenvalues....................................................93 The algorithm has found 8 eigenvalues....................................................94 The algorithm has found all 9 eigenvalues...............................................94 A dynamic system ....................................................................................95 A regulator in unlimited operation, i.e. Min1 < x2 < Max1 and Min2 < x4 < Max2......................................................................................97 Figure C.1. Control system of the TCSC ..................................................................101 Figure C.2. Basic scheme of the Thyristor-Controlled Series Capacitor (TCSC) .....103 Figure C.3. Block diagram of the PLL......................................................................104 Figure C.4. The signal TETAPLL and the integer signals COUNTFOR and COUNTBACK are calculated inside the PLL.........................................105 Figure C.5. Basic scheme of the Thyristor-Controlled Series Capacitor (TCSC) .....106 Figure C.6. Block diagram of the BOO ....................................................................107 Figure C.7. Block diagram of the TPG .....................................................................108

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Chapter 1

Introduction 1.1 Background During the last decades power systems have been equipped with complex components such as Static Var Compensators (SVCs) and High Voltage Direct Current (HVDC) links. These components have introduced new possibilities to control power systems, SVCs can almost continuously change the amount of reactive power from capacitor banks and HVDC has made it possible to transfer power over long distances, and connect unsynchronized AC systems. The behavior of these new components is different from 'old' technologies since they contain power electronics and detailed representations of them are non-linear. Due to their complexity they are difficult also to simulate. When studying power systems including such components, the analyzer must decide whether it is necessary to include a detailed model of them or not, i.e. to represent each event that takes place, or ignore them. Power system studies must include high accuracy and at the same time, it cannot take too long time to carry out the simulations. These two wishes are impossible to fulfill since the faster a simulation is, the more simplified the representation of the power system is. A faster computer would speed up a detailed simulation but still the simplified simulation would be faster to perform. The title of this thesis "Linear Models of Non-linear Power System Components" seems like a contradiction. Non-linear components are and behave nonlinear and that is of course true. But, what to do if the non-linearities of such a component are not necessary to represent in a simulation and it is preferable to have a low-order, linear model that is much faster to simulate with reasonable accuracy? The aim of the thesis is to study the possibilities and limits of using linear models of non-linear power system components. In the thesis we will not try to model each event (as e.g. a thyristor switch) during a time simulation, like high-frequent, small changes of signals. Instead, the aim is to find linear models that describe fundamental behaviors of non-linear components. These linear models will simplify time domain simulations of a power system when they once have been developed. Even to linearize such power system will be done more adequate, compared to linearize the power system including the non-linear components.

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Chapter 1. Introduction

We have focused on slow dynamics, which means that the developed linear model is valid in the frequency range of a few Hertz or up to approximately 0.3 times the power frequency of the network. With slow dynamics are meant power oscillations that are of big interest when simulating power systems. A non-linear component is any equipment that often changes between different sets of differential equations, i.e. the relation between its output signals and input signals changes with time. It can be a Thyristor-Controlled Series Capacitor (TCSC), a Static Var Compensator (SVC), or any regulator or component that, in a short time-interval, passes several sets of differential equations. To include such components in the study of a power system, the researcher must decide whether it is necessary or not to include all the transitions between different sets of differential equations. All transitions will keep the time-step low in the simulation and to simulate ten seconds of such a power system can take hours. The time it takes to simulate is more related to how many power system components that switch between different sets of differential equations than to the size of the power system. In the used power system simulation software, the time-step is automatically adjusted1. In case of a fixed time-step, this must be set to a value small enough so that the simulation correctly includes all transitions in the numerical process2. Also with a fixed time-step, the simulation would certainly take hours. The standard quasi-static approximation models the TCSC as a variable fundamental-frequency reactance: the line and the TCSC dynamics are omitted. That approach is widely used because of its simplicity, but relies on the assumption that the transmission system is operating in sinusoidal steady-state, with only the generators' dynamics represented. This thesis presents a method to build a linear model of a non-linear component simulated in instantaneous value mode and to include the linear model in a power system study with reasonable accuracy. Instantaneous value mode means that the system is not operating in sinusoidal steady-state. When using the linear model in a simulation, it is possible to get a better linearization of the whole power system. This is the case if the researcher or software user wants to analyze the eigenvalues of the power system or do other activities that are based upon that the power system is linearized. When not using a linear model of a non-linear component, the orientation of the eigenvalues will have different setups depending on in which instant the power system is linearized in steady-state. These instants can be very close, but still the results from the linearizations differ.

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Examples of such power system simulation software are DIgSILENT, Eurostag, and Simpow. 2 Examples of such power system simulation software are EMTDC and EMTP.

1.2 Problem formulation

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1.2 Problem formulation To simulate a power system containing a component with a detailed representation that includes several events during a period of the system's fundamental frequency, an automatically adjusted time-step will be decreased after each event. Such component can be a Thyristor-Controlled Series Capacitor (TCSC), see figure 1.1. The TCSC consists of a series capacitor with a reactor in parallel that is switched in by thyristors and out at the next zero-crossing of the current. By varying the start-conducting time for each thyristor, the fundamental reactance between Node A and Node B can be changed. Node A

Node B

Node D Figure 1.1. Thyristor-Controlled Series Capacitor

When simulating a TCSC the time-step will be decreased to its minimum after every instant when either one of the thyristors change status or if status of the control algorithm of the TCSC changes. These events slow down the simulation. When each thyristor is represented as above, the simulation is done in instantaneous value mode, i.e. when no harmonics exist in the power system and there are no un-symmetries in the network, each phase voltage is sinusoidal and contains only the fundamental frequency, see figure 2.7a. The TCSC is built under the assumption that the reactor for each phase is recurrently switched on two times per period and blocked at the next zerocrossing of the reactor current. This implies that the TCSC with its control algorithm passes several different setups of differential equations. Two problems arise when simulating power systems containing non-linear components such as a TCSC: •

Time domain simulations will take very long time to perform since the time-step is decreased to its minimum value if it is automatically adjusted

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Chapter 1. Introduction after each event in the TCSC-models. To simulate ten system-seconds can take hours.

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Linearizing such a power system is not trivial since the differential equation setup and the linearization of it change several times within each millisecond. The small-signal stability based on these linearized models will be untrustworthy. Depending on in which operating points the TCSCs are, they will be linearized differently. Depending on in which ms (millisecond) the linearization takes place, eigenvalues will have different locations in the complex plane.

To avoid these problems, relevant linear models of the TCSCs are needed. With these, the time domain simulation will be performed much faster and a linearization of the whole power system will, with these relevant linear models of the non-linear components included, be more adequate. To build a linear model of a non-linear component it is, in the suggested method in this thesis, disturbed in a small isolated system to focus on its lowfrequent dynamic behavior. Input signals and output signals are identified and for the TCSC the input signals are selected to the currents sent into the TCSC and the output signals are the voltage drop over the TCSC. By selecting the d- and q-components of both the current into the TCSC and the voltage drop over the TCSC, four relations are analyzed, i.e. each output signal as a function of each input signal. By analyzing the influence that the dcomponent of the current has on the d- and q-components of the voltage drop and the influence that the q-component of the current has on the d- and qcomponents of the voltage drop, a linear model for low-frequent dynamic behavior is built. All power system simulations are made with ABB's power system simulation software Simpow and when building the linear model, functions of the System Identification Toolbox provided by Matlab are used. In the linear model the high-frequency components of the voltage drop are omitted by low-pass filtering of the voltage drop. The linear model represents only the low-frequency behavior of the TCSC.

1.3 Main contributions of the thesis The main contributions of the thesis are: •

Formulation of a linearization procedure of non-linear power system components. This model is valid for low-frequent dynamics.

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Development of a linear model of a non-linear power system component, i.e. the Thyristor-Controlled Series Capacitor (TCSC).

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Application and analysis of the created linear model of a TCSC in time domain simulations.

1.4 Outline of the thesis •

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Application and analysis of the created linear model of a TCSC in linear analysis.

1.4 Outline of the thesis Chapter 2 in the thesis contains a background in concepts of dynamic systems. In chapter 3 different ways to linearize a component and model a ThyristorControlled Series Capacitor (TCSC) are discussed and viewed. Chapter 4 consists of a description of the TCSC-model that has been used in the thesis as an example of a non-linear power system component. Chapter 5 contains the linearization of the TCSC, and shows how a linear model is generated. In chapter 6, the original TCSC-model is compared in time domain simulations with the created linear model and in chapter 7 linear analysis with the original TCSC-model is compared with the linear model. Finally in chapter 8, conclusions and ideas for future work are given.

1.5 List of publications •

Jonas Persson: "Are there any Limits in Building Small-Signal Models of Dynamic Systems?", report, KTH, Stockholm, Sweden, 1999.

•

Jonas Persson: "On Linearization of non-linear components", A-EES-0014, report, KTH, Stockholm, Sweden, 2000.

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Jonas Persson: "A description of the Masta-model of the ThyristorControlled Series Capacitor in Simpow", Technical Report, TR H 00163A, ABB Power Systems, Västerås, Sweden, 2000-10-12.

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Jonas Persson, Lennart Söder: "Linear Analysis of a Two-Area System Including a Linear Model of a Thyristor-Controlled Series Capacitor", Presented at the IEEE Porto Power Tech Conference 2001, paper 289, Porto, Portugal, September 10th – 13th, 2001.

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Jonas Persson, Lennart Söder: "Validity of a Linear Model of a ThyristorControlled Series Capacitor for Dynamic Simulations", Submitted paper to Power Systems Computation Conference 2002, Sevilla, Spain, June 24th – 28th, 2002.

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J.G. Slootweg, J. Persson, A.M. van Voorden, G.C. Paap, W.L. Kling: "A Study of the Eigenvalue Analysis Capabilities of Power System Dynamics Simulation Software", Submitted paper to Power Systems Computation Conference 2002, Sevilla, Spain, June 24th – 28th, 2002.

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Emil Johansson, Jonas Persson, Lars Lindkvist, Lennart Söder: "Location of Eigenvalues Influenced by Different Models of Synchronous Machines",

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Chapter 1. Introduction Submitted paper to Sixth IASTED International Multi-Conference on Power and Energy Systems, Marina del Rey, USA, May 13th – 15th, 2002.

Chapter 2

Concepts of dynamic systems In this chapter, basic concepts of dynamic systems are described. Also the dq0representation is shown. The chapter ends with describing how we formulate and connect the linear model of the non-linear power system component to the rest of the power system.

2.1 Basic formulation of dynamic systems In section 2.1 and 2.5 we are following in most extent reference [1]. A dynamic system may be described by a set of n first order non-linear ordinary differential equations of the following form:

x& i = f i (x1 , x 2 , K , x n ; u1 , u 2 , K , u r ; t )

for

i = 1,2, K , n

(2.1)

where n is the order of the system and the number of the system's state variables xi, and r is the number of input signals. x& i is a state variable's time derivative. By using vector notation this can be written:

x& = f (x,u, t )

(2.2)

where

 x1  x  x =  2 M    x n 

 u1  u  u =  2 M   u r 

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 f1  f  f =  2 M    f n 

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Chapter 2. Concepts of dynamic systems

The column vector x is referred to as the state vector, and its entries xi as the system's internal state variables. The column vector u is the vector of the r input signals to the system and f is a vector of n non-linear functions relating the input signals and the internal state variables xi to the internal state variables time derivative x& i . Time is denoted by t in equation (2.2), and the derivative of a state variable x with respect to time is denoted by x& . If the time derivatives of the state variables are not explicit functions of time, i.e. x& is only a function of the state variables and the input signals, equation (2.2) can be simplified to

x& = f (x, u )

(2.3)

In that case the system is called time-invariant, see section 2.3 below3. Equation (2.3) shows how the internal state variables' time derivative develop with time as a function of both the input signals and the internal state variables. The output signals from a dynamic system may be expressed in terms of the input signals and the internal state variables on the following form:

y = g(x, u )

(2.4)

where

 y1  y  y= 2  M     y m 

 g1  g  g= 2  M     g m 

The column vector y is the vector of the m output signals and g is a vector of m non-linear functions relating the input signals and the internal state variables to the output variables. An internal state vector

u

x

y

Figure 2.1. A dynamic system with a vector of input signals u, a vector of output signals y, and an internal state vector x

3

Still

x& , x, and u are varying with time.

2.2 Definition of a linear system

9

2.2 Definition of a linear system A system is linear if, S u1 (t ) → y1 (t )  S  ⇒ c1u1 (t ) + c 2 u 2 (t ) → c1 y1 (t ) + c 2 y 2 (t ) S u 2 (t ) → y 2 (t )

(2.5)

for any values of the real constants c1 and c2 [2]. s → indicates that the expression on the left-hand side of the The notation  arrow passes a system s. The output from the system s is written on the righthand side of the arrow. Equation (2.5) is called superposition [7,35].

A system that holds for equation (2.5) can still vary with time, for instance y(t) = sin(t)u(t). Most real circuit elements are non-linear to some extent but they can often be accurately represented by a linear approximation [7]. Non-linear can be explained as in [8] for power systems in particular: "A component is called nonlinear if it draws non-sinusoidal current when energized with a sinusoidal voltage."

2.3 Definition of a time-invariant system If a system has the same time-response of an input signal, no matter when it happens, the system is called time-invariant. If, S S u (t )  → y (t ) ⇒ u (t − T )  → y (t − T )

(2.6)

for any real value of T [2]. Expression (2.6) is shown graphically in figure 2.2. Equation (2.6) means: if two input signals to a system are equal, except for a time shift T, the two output signals are equal, except for the same time shift T, see figure 2.2.

10

Chapter 2. Concepts of dynamic systems

u (t )

y (t )

t

t

u (t − T )

y (t − T )

t T

t T

Figure 2.2. Input signals u and output signals y of a time-invariant system

With time-invariant is meant that a system's dynamic behavior is not depending on the absolute time [35]. Most dynamic systems in power systems are timeinvariant. In some literature, time-invariant is defined similar as autonomous, see Aggarwal [3].4

2.4 Linear and time-invariant system For a system that is both linear and time-invariant (abbreviated as an LTIsystem), the vector-function f in equation (2.3), for each element, contains a sum of the state variables and a sum of the input signals, each term multiplied with factors that are constant with time. For such a system, equation (2.1) is formulated as: n

r

k =1

l =1

x& i = ∑ aik x k + ∑ bil u l

for i = 1,2, K , n

(2.7)

For an LTI-system, even the vector-function g contains for each element a sum of the state variables and a sum of the input signals, each term multiplied with factors that are constant with time. The elements of y in equation (2.4) is for an LTI-system formulated as:

4

In other literature than [3], autonomous is defined as a system without input signals, an unforced system, i.e. a homogenous system, see Schmidtbauer [4] and Lefschetz [5]. A system without input signals is for instance, y = x& + x . To avoid uncertainties, the name time-invariant will be used in this thesis.

2.5 Linearization

11

n

r

k =1

l =1

y j = ∑ c jk x k + ∑ d jl u l

for j = 1,2, K , m

(2.8)

The linear model developed and used in this thesis is both linear and timeinvariant, i.e. an LTI-model. Equation (2.7) and (2.8) can also be written on matrix-form as in the following equations:

x& = Ax + Bu

(2.9)

y = Cx + Du

(2.10)

where

 a11 L a1n    A= M L M  a n1 L a nn   

b11 L b1r    B= M L M  bn1 L bnr   

(2.11)

 c11 L c1n    C= M L M  c m1 L c mn   

 d 11 L d1r    D= M L M  d m1 L d mr   

(2.12)

2.5 Linearization If equation (2.3) is equal to the zero-vector:

x& = f (x, u ) = 0

(2.13)

the system is said to be in rest or in an equilibrium point since all variables are constant and not varying with time. Let x0 be the state vector and u0 be the input vector corresponding to the system in rest, so that

x& 0 = f (x 0 , u 0 ) = 0

(2.14)

Let us perturb the system from the above point of rest, by letting

x = x 0 + ∆x

(2.15)

u = u 0 + ∆u

(2.16)

in equation (2.3). The prefix ∆ denotes a small deviation in equation (2.15) and (2.16). The new state (and every state) must satisfy equation (2.3). Hence,

12

Chapter 2. Concepts of dynamic systems

x& = f (x 0 + ∆x, u 0 + ∆u )

(2.17)

By time-differentiating both sides of equation (2.15) we get

x& = x& 0 + ∆x&

(2.18)

and by exchanging the left-hand side in equation (2.17) with (2.18) we get

x& 0 + ∆x& = f (x 0 + ∆x, u 0 + ∆u )

(2.19)

For small deviations, the non-linear function f(x,u) in equation (2.19) can be expressed in terms of a Taylor expansion. A Taylor expansion for a general scalar function f(x) as a function of one variable x in a close interval around x0 is [6]:

f ( x ) = f (x0 ) + K+

f

(n )

f (x0 ) '

''

(x ) (x − x )

n

0

0

2

0

0

1!

n!

(x − x ) + f (x ) (x − x ) 2!

0

+K

(2.20)

+ Rn +1 ( x )

where

Rn +1 ( x ) =

x

∫

x0

( x − t )n n!

f

( n +1)

(t ) dt

(2.21)

If we omit second and higher orders of expression (2.20) we get,

f ( x ) = f (x 0 ) + f (x0 )(x − x0 ) '

(2.22)

The closer interval we choose around x0, the better the linear approximation (2.22) holds.

f (x ) f (x 0 ) '

x0

x

Figure 2.3. The closer interval around x0, the better the linear approximation holds

2.5 Linearization

13

For one single state variable xi, equation (2.19) is, when applying equation (2.22),

x& i 0 + ∆x& i = f i (x 0 + ∆x, u 0 + ∆u ) =

= f i (x 0 , u 0 ) +

(

∂f i ∂x1

∆x1 + K +

∂f i ∂x n

∆x n +

∂f i ∂u1

∂f i

∆u1 + K +

∂u r

∆u r

(2.23)

)

Since x& i 0 = f i x 0 , u 0 = 0 , see equation (2.9) above, these terms can be omitted in equation (2.23) as,

∆x& i =

∂f i ∂x1

∆x1 + K +

∂f i ∂x n

∆x n +

∂f i ∂u1

∆u1 + K +

∂f i

∆u r

∂u r

(2.24)

with i = 1, 2, …, n. The smaller perturbations the better approximation (2.24) holds. Equation (2.24) expresses the time derivative for one of the n state variables and its linear relations to the state variables' and the input signals' deviations from an equilibrium point. The linearization is possible since equation (2.3) is disturbed with so small deviations that it acts linear and can be formulated as with equation (2.24). Let us do the same with equation (2.4) as we did above with equation (2.3). Perturb equation (2.4) with equation (2.15), (2.16), and (2.25) below.

y = y 0 + ∆y

(2.25)

Then we can rewrite equation (2.4) as

y 0 + ∆y = g (x 0 + ∆x, u 0 + ∆u )

(2.26)

For one single output signal yj, equation (2.26) will be, when omitting second and higher orders in a Taylor expansion, y j 0 + ∆y j = g j (x 0 + ∆x, u 0 + ∆u ) = = g j (x 0 , u 0 ) +

∂g j ∂x1

∆x1 + K +

∂g j ∂x n

∆x n +

∂g j ∂u1

∆u1 + K +

∂g j ∂u r

∆u r

(2.27)

Since yj0 = gj(x0,u0), see equation (2.4) above, these terms can be omitted in equation (2.27) and then we get:

∆y j =

∂g j ∂x1

∆x1 + K +

∂g j ∂x n

∆x n +

∂g j ∂u1

∆ u1 + K +

∂g j ∂u r

∆u r

(2.28)

with j = 1, 2, …, m. The linearized forms of equations (2.3) and (2.4) around an equilibrium point, with using equation (2.24) and (2.28), are written on matrix-form as,

14

Chapter 2. Concepts of dynamic systems

∆x& = A∆x + B∆u

(2.29)

∆y = C∆x + D∆u

(2.30)

where

 ∂f 1   ∂x1 A= M  ∂f n  ∂x  1  ∂g1   ∂x1 C= M  ∂g m  ∂x  1

∂f 1   ∂x n  L M  ∂f n  L ∂x n  

 ∂f 1   ∂u1 B= M  ∂f n  ∂u  1

∂g1   ∂x n  L M  ∂g m  L ∂x n  

 ∂g 1   ∂u1 D= M  ∂g m  ∂u  1

L

L

∂f 1   ∂u r  L M  ∂f n  L ∂u r   L

∂g 1   ∂u r  L M  ∂g m  L ∂u r  

(2.31)

L

(2.32)

The matrices (2.31) and (2.32) are evaluated at the equilibrium point5 around which the small perturbation is being analyzed. In equations (2.29) – (2.32), ∆x is the state vector of dimension n ∆u is the input vector of dimension r ∆y is the output vector of dimension m A is the state matrix of size nxn B is the input matrix of size nxr C is the output matrix of size mxn D is the feedforward matrix of size mxr The aim of the thesis is to analyze how many state variables that are necessary to obtain a linear model, which represents the deviation from the original state of the non-linear power system component with an acceptable accuracy within a certain frequency span. For the linear model, the four matrices above have to be constructed.

5

The system is in rest.

2.6 Formulations in the thesis

15

The number of state variables n sets: the length of the state vector ∆x, the number of rows and columns of matrix A, the number of rows of matrix B, and the number of columns of matrix C. The number of selected input variables r sets: the length of the input vector ∆u and the number of columns of matrices B and D. The number of selected output variables m sets: the length of the output vector ∆y and the number of rows of matrices C and D. An internal state vector

∆u

∆x

∆y

Figure 2.4. A dynamic system with a vector of input signals ∆u, a vector of output signals ∆y, and an internal state vector ∆x

The linear model can be used for small-signal stability problems, i.e. linear analysis of the power system and small perturbations around an equilibrium point in time domain simulations. Appendix B describes how eigenvalues, eigenvectors and participation factors can be obtained from the state matrix A in equation (2.31) above.

2.6 Formulations in the thesis In the thesis we will create a low-order linear model of a non-linear power system component. The low-order linear model will be able to describe lowfrequent dynamic behavior within an acceptable accuracy of the power system component.

Non-linear component Figure 2.5. A non-linear component

In the thesis, a Thyristor-Controlled Series Capacitor (TCSC) represents the non-linear power system component. A TCSC is a series capacitor with a controllable capacitance. Varying the turnon time for the thyristors that are controlling the phase current through the parallel reactor does the change of the capacitance, see figure 2.6.

16

Chapter 2. Concepts of dynamic systems

The aim is to create a linear model of the TCSC, i.e. a model that can be used for small signal disturbances. The linear model shall be implemented between node A and node B in figure 2.6 in order to represent the whole component, i.e. the capacitor together with the controlled reactor. A

+ UC -

B

I

Figure 2.6. A Thyristor-Controlled Series Capacitor

The output signals from the linear model of the TCSC are the dq-components of the series capacitor voltage drop UC and the dq-components of the current into the TCSC, I in figure 2.6 are the input signals. Section 2.6.1 is explaining the dq0-transformation and section 2.6.2 is describing why it is possible to omit the 0-component of both the series capacitor voltage drop and the current in the linear model of the TCSC. When the original instantaneous value model of the TCSC6 is running in steady-state it creates 93 events in each period. A period is 16.67 ms since the fundamental frequency is 60 Hz in the studied network. For each phase, the TCSC and its control algorithm creates 31 events including the events when the thyristors start conducting and when the following zero-crossing of the current occurs. Other events are, commands to when different calculation routines should be activated or not and when internal variables are within certain intervals. Each event decreases the actual time-step used in the simulation to its minimum and therefore a time simulation takes considerable long time to complete.

6

Described in chapter 4.

2.6 Formulations in the thesis

17

2.6.1 The dq0-representation The electrical state of the TCSC is represented in the instantaneous value mode within the used software. When using this representation currents and voltages in the power system do not necessary have to be sinusoidal. To fasten the used computer time the software uses the dq0-transformation of phase quantities when solving the electric quantities during the simulation, see equation (2.33) below.  2π  2π     cosθ −  cosθ +    cosθ 3  3   u    ud   a 2π  2π      2   θ θ θ = − − − − + u u sin sin sin      q 3 3  3   b    u0    uc    1 1 1   2 2  2 

(2.33)

where

θ

= the co-ordinated system angle, referred to a reference machine anywhere in the power system

ua = the momentary value of phase voltage a, real ub = the momentary value of phase voltage b, real uc = the momentary value of phase voltage c, real ud = the d-component7 of the voltage, real uq = the q-component8 of the voltage, real u0 = the 0-component9 of the voltage, real By this transformation, a symmetrical three-phase voltage, see figure 2.7a, has the dq0-components showed in figure 2.7b below.

7

Component in d-axis. Component in q-axis. 9 Component in 0-axis. 8

Chapter 2. Concepts of dynamic systems

600

600

400

400

200

200

ud , uq , u0 (kV)

ua , ub , uc (kV)

18

0

-200

0 ud and u0

-200

-400 -600 0

uq

-400

0.0167 0.0333

-600 0

0.05 0.0667 0.0833 Time (s)

Figure 2.7a. Momentary phase voltages ua, ub, and uc of a threephase voltage

0.0167 0.0333

0.05 0.0667 0.0833 Time (s)

Figure 2.7b. dq0-components ud, uq, and u0 of a three-phase voltage

The coordinate system in the dq 0 -representation rotates with the rotor of a reference machine. The linear model is desired to describe low-frequent fundamental behavior of the TCSC. Low-frequent behavior is for instance power oscillations. In figure 2.8 such a slow oscillation of 1.5 Hz is shown in transmitted power, both active and reactive, for a three-phase transmission line. At 1.00 < t < 1.03 seconds, a threephase fault is applied to a bus in the system. 3

Active and reactive power (p.u.)

2.5

2

P 1.5

1

0.5

Q 0

-0.5 0.8

1

1.2

1.4

1.6 1.8 Time (s)

2

2.2

2.4

Figure 2.8. A slow power oscillation in both active and reactive power for a transmission line

2.6 Formulations in the thesis

19

In the left figure below, a node voltage is viewed for one of the line nodes of the transmission line. In the d- and q-component of the node voltage it is possible to see that the same oscillation frequency as in figure 2.8 also occur in the d- and q-component of the voltage. In figure 2.9b the line current is viewed for the transmission line and even the d- and q-component of the line current contains the same oscillation frequency. 1.2

4

d-, q-, and 0-components of current (p.u.)

d-, q-, and 0-components of voltage (p.u.)

1 0.8

u

q

0.6 0.4

u

0

0.2 0 -0.2 -0.4 0.8

u 1

1.2

1.4

3

i

q

2

1

i

-1

i d

1.6 1.8 Time (s)

2

2.2

-2 0.8

2.4

Figure 2.9a. ud, uq, and u0 for one of the line nodes

0

0

1

1.2

1.4

d

1.6 1.8 Time (s)

2

2.2

2.4

Figure 2.9b. id, iq, and i0 for a transmission line

In figure 2.10 a phase current is shown for the transmission line. Within that figure it is possible to see how a power oscillation occurs in the phase current when the power system is simulated within instantaneous value mode. The power oscillation frequency 1.5 Hz occurs as an interference frequency between 58.5, 60, and 61.5 Hz and creates a pulsing round mean square value of the current with the frequency 1.5 Hz. See also the frequency spectrum in figure 2.11. The power frequency of the system is 60 Hz. 2

One phase current (p.u.)

1

0

-1

-2

-3

-4

-5 0.8

1

1.2

1.4

1.6 1.8 Time (s)

2

2.2

2.4

Figure 2.10. A phase current containing a slow power oscillation when simulating in instantaneous value mode

20

Chapter 2. Concepts of dynamic systems

1.4

|X(f)| of one phase current (p.u.)

1.2

1

0.8

0.6

0.4

58.5 Hz

61.5 Hz

0.2

0

0

10

20

30

40 50 60 Frequency (Hz)

70

80

90

100

Figure 2.11. Frequency spectrum of a phase current containing a slow power oscillation when simulating in instantaneous value mode

2.6.2 The input and output signals to the linear model As explained earlier in this section we will use the current into the TCSC as the input signal to the linear model and the voltage drop over the TCSC as the output signal. In figure 2.12 the current into the TCSC is shown. At t = 0.2 s the TCSC is activated. Since there are no frequencies lower than the third harmonic (180 Hz) in the 0-component of the current into the TCSC, we will omit the 0component when creating a model which aim is to represent the fundamental behavior of the TCSC. The magnitude of the third harmonic is 0.025 Ampere, and this shall be compared to the d- and q-components, which have the average level 0.094 p.u. (200 Ampere) and 0.169 p.u. (358 Ampere) respectively, see figure 2.12.10 During disturbances in the power system, the magnitude of the third harmonic of the 0-component varies. However, the magnitude is still small enough to omit it when representing the input signals as a vector in expression (2.34).

10

See section 5.3 for values of the base current.

2.6 Formulations in the thesis

21

0.2 0.18 0.16

i

q

id, iq, i0 (p.u.)

0.14 0.12 0.1

0.08

i

d

0.06 0.04

i

0

0.02 0 0

1

2

3 Time (s)

4

5

6

Figure 2.12. The dq0-components of the current into the TCSC in steady-state

The d- and q-component of the current are the input signals to the linear model:

∆id  ∆u =    ∆iq 

(2.34)

where ∆id = ∆iq =

deviation from initial operating point in d-axis of the current into the TCSC deviation from initial operating point in q-axis of the current into the TCSC

Locally in the TCSC, a rotating third harmonic (180 Hz) in the 0-component current exists and causes a third harmonic 0-component voltage drop over the TCSC. Since such a high frequency is not interesting when describing the fundamental, low-frequency behavior of the TCSC also the 0-component of the voltage drop over the series capacitor is omitted in the linear model of the TCSC, see figure 2.13 and equation (2.35) below.

22

Chapter 2. Concepts of dynamic systems

0.06 uC d

uC d, uC q, uC 0 (p.u.)

0.04 0.02 0

-0.02 uC 0

-0.04 uC q -0.06 0

1

2

3 Time (s)

4

5

6

Figure 2.13. The dq0-components of the capacitor voltage of the TCSC in steady-state

The output signal of the linear model of the TCSC is modeled as:

∆u C d  ∆y =    ∆uC q 

(2.35)

where ∆uC d = deviation from initial operating point in d-axis of the voltage drop over the series capacitor ∆uC q = deviation from initial operating point in q-axis of the voltage drop over the series capacitor In chapter 4 the TCSC is described in detail.

Chapter 3

Methods of linearizing a component This chapter briefly describes different methods to linearize components and in particular non-linear components such as a TCSC.

3.1 Introduction In the literature, methods can be found to identify linear models. In this chapter some of them are briefly described. Also a new method that has been developed in the thesis project is presented. The following seven methods will be described: 1. Estimate an ARX-model11 2. Prony analysis 3. Newton-Raphson algorithm using Discrete Fourier Transform 4. Describing functions Methods suggested for TCSCs in particular: 5. Linearized discrete-time model of a thyristor-controlled series compensator 6. Analytical modeling of TCSCs for subsynchronous resonance studies 7. Dynamic phasor model of a TCSC Methods 1-3 can more or less be applied to any system component since they ignore the electrical topology of the studied component, i.e. the component is treated as a "black box". With them a linear model is provided, see subsections below for comments for each of them12. They provide a Linear and TimeInvariant model, an LTI-model and it has to be proved by the researcher if the studied power system component is an LTI-model, in other words, if the linear model is a good substitute for the studied power system component. If so, then, once a linear model has been obtained by the methods, the same linear model can be used in all operating points and for all time intervals. 11 12

ARX is an abbreviation of AutoRegressive-model with eXtra inputs. Method 3 is not complete in providing a linear model.

23

24

Chapter 3. Methods of linearizing a component

Therefore, the studied system component has to be compared with the developed linear model to prove if the linear model behaves like the studied component, i.e. a linear model has to be tested versus the studied system component to prove the validity of the linear model. Method 4 is commented upon below. Methods 5-7 analyze the electrical topology of the TCSC and provide a linear model, which is dependent on among other parameters, actual operating point of the TCSC.

3.2 Estimate an ARX-model This method works in the time-discrete domain. The input to the method is uniformly spaced sampled values of input and output signals of an unknown system. The following difference equation is then identified: y (t ) + a1 y (t − 1) + K + a n y (t − n a ) = b1 u (t − 1) + K + bn u (t − n b ) + e(t ) (3.1) a

b

where u(t-1), …, u(t-nb) are the delayed input signals to the model, y(t), y(t-1), …, y(t-na) are the present and delayed output signals, and e(t) is a white-noise term.13 Equation (3.1) can be rewritten as: y (t ) = − a1 y (t − 1) − K − a n y (t − n a ) + b1 u (t − 1) + K + bn u (t − n b ) + e(t ) a

(3.2)

b

where the adjustable parameters in the method can be given in a vector θ as:

[

θ = a1 a 2 K a n

a

b1 K bn

b

]

T

(3.3)

and the known values of the input and output signal given in a vector as:

ϕ (t ) = [− y (t − 1) K − y (t − na ) u (t − 1) K u (t − nb )]T

(3.4)

(3.1) is called an ARX-model, where AR refers to the auto-regressive part A(q)y(t) and X to the extra inputs B(q)u(t) [21]. The method solves values of the vector θ so that expression (3.1) fit as good as possible with the real values of y(t). This is done by squaring the errors of mismatching and find values of θ that minimizes the sum of all squared errors as:

13

Another method that is describing the error e(t) with more terms is the ARMAXmodel.

3.3 Prony analysis

V=

1 N

25

∑ (y(t ) − ϕ (t )θ ) N

T

2

(3.5)

t =1

N in equation (3.5) are the number of known values of u(t) and y(t). Values of θ are found by differentiating the sum of squared errors according to θ and set it equal to 0 as:

2 ∂ V= 0= ∂θ N

∑ ϕ (t )(y(t ) − ϕ (t )θ ) N

T

(3.6)

t =1

The method is implemented in Matlab's System Identification Toolbox [21,24]. When using this method, the required order of the parameters in (3.3) must be given, i.e. na and nb. Also the delay of the first input signal should be given when using the method, i.e. how many samples the input signal is delayed in expression (3.1). The generated discrete time model (3.1) is converted to a continuous time model and then the continuous time model is converted to a state-space model [24]. The ARX-method will be used in chapter 5 to identify the transfer functions of the TCSC.

3.3 Prony analysis In Prony analysis, N samples of an equi-distant sampled time signal is used to create a dynamic model. The existing signal {y[0], y[1], …, y[N-1]} is approximated to the equivalent p

yˆ [k ] = ∑ Ai e

σ i ∆tk

i =1

cos(2πf i ∆tk + φi )

(3.7)

where σi and fi in equation (3.7) describes the eigenvalues as λi below,

λ i = σ i + j 2πf i

(3.8)

are the poles, also called eigenvalues. When using this method the order of the required model p must be given and it must be less or equal to

p≤

N 2

N [23], i.e. 2 (3.9)

For a given p the 4p unknown constants Ai, σi, fi and φi will be determined.

26

Chapter 3. Methods of linearizing a component

In the case of large values of N , the results are least mean squared, so that the calculated Ai, σi, fi and φi fit the whole time interval as good as possible. Prony analysis demands that an oscillation should exist in the whole studied time interval. If not, the calculated result contains an equivalent of a higher order [23]. In most cases, the studied system is nonlinear, but it is assumed that it behaves in a relatively linear manner about its current operating point [25]. To this algorithm a method also must be added that calculates values of the elements of the matrices B, C, and D, see (2.31) and (2.32).

3.4 Newton-Raphson algorithm using Discrete Fourier Transform Within this project attempts have been made to develop a method that could calculate eigenvalues in an equi-distant sampled time signal by using Discrete Fourier Transform. The studied signal f(t) is assumed to consist of the following N terms,

[

N

]

N

σ t σ t j (ϕ + 2π f k t ) fˆ (t ) = ∑ Ak e k Re e k = ∑ Ak e k cos(ϕ k + 2π f k t ) k =1

(3.10)

k =1

fˆ (t ) is an estimate of the correct f(t). σk in (3.10) is the real part, i.e. the damping of an eigenvalue, fk is the imagi-

nary part, i.e. the oscillation frequency of an eigenvalue, Ak is the initial magnitude, and ϕk is the initial phase of a predicted eigenvalue. The studied signal is an output signal from a non-linear component.

For each expected eigenvalue in the signal, four Fourier coefficients are calculated as:

2 aˆ n = T

a +T

∫ a

  N σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) cos nΩtdt 1 1k =4 444244444 3

(3.11)

= fˆ (t )

2 bˆn = T aˆ n +1 =

a +T

2 T

∫ a

 N  σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) sin nΩtdt  k =1 

a +T

∫ a

 N  σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) cos(n + 1)Ωtdt  k =1 

(3.12) (3.13)

3.5 Describing functions 2 bˆn +1 = T

a +T

∫ a

27

  N σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) sin (n + 1)Ωtdt   k =1

(3.14)

 f peak  2π , and T is the length of the studied time in, Ω = T  ∆f 

where n = int  terval.

All σk, fk, Ak, and ϕk in the right-hand sides of equations (3.11) – (3.14) are predicted values and will be improved as the algorithm goes on. Instead of solving the integrals above, Discrete Fourier Transform is used. When the four Fourier coefficients (3.11) – (3.14) are equal the correct Fourier coefficients, i.e.

aˆ n = a n

bˆn = bn

aˆ n+1 = a n +1 bˆn +1 = bn +1

(3.15)

for all N peak frequencies, the correct number of modes with parameters σk, fk, Ak, and ϕk have been found. To improve the estimates of the parameters, the Newton-Raphson method is used. Then (3.11) – (3.14) are differentiated relative σk, fk, Ak, and ϕk. The found eigenvalues define the A-matrix of the linear model. To this algorithm a method also must be added that calculates values of the elements of the matrices B, C, and D, see (2.31) and (2.32). The algorithm has been tested on signals that contain eigenvalues as described in equation (3.10). In real-life cases, the method has not been successful. The fail of the method can be a result of that in real-life there are noise added to the studied signal. The method has to be further improved to work properly. A detailed description of this algorithm can be found in appendix A.

3.5 Describing functions The method "Describing functions" produces a transfer function that is varying according to the magnitude of the input signal(s) [30]. Describing function calculates the output signal's magnitude and phase relatively the sinusoidal input signal. A describing function is time-invariant. If we implement the ideas of describing functions to the TCSC-case we would end up with a model that will change according to the magnitude of the input signal in a time simulation. This may speed up a time domain simulation compare to use the original TCSC-model.

28

Chapter 3. Methods of linearizing a component

It could be interesting to apply the ideas of describing functions and to develop one describing function per phase. The input signal could be, for instance the voltage drop over the TCSC for one phase and the output signal could be the current through the TCSC. This setup could be used in steady-state and maybe also expanded to be valid within transient phenomena.

3.6 Linearized discrete-time model of a thyristorcontrolled series compensator In [14] a method called "Linearised discrete-time model of a thyristorcontrolled series-compensated transmission line" is presented. The authors build up two linear models by analyzing the two different circuits that a phase circuit of the TCSC-compensated line can model: either that none of the thyristors are conducting or that one of them does. Four points of time (t0, t1, t2, and t3) are defined as in the figure below.

1.5 The voltage drop over the capacitor. The current through the reactor.

1

0.5

0 t0

t1

t2

t3

-0.5

-1

-1.5 1.21

1.215

1.22 Time (s)

1.225

1.23

Figure 3.1. Voltage and current waveforms of a TCSC

3.6 Linearized discrete-time model of a thyristor-controlled series compensator

+ E

R IL

Gene-

-

VD L

29

+ VC −

V Infinite bus

C

rator

LP

I P RP

Figure 3.2. Basic scheme of one phase of the Thyristor-Controlled Series Capacitor

For each circuit that can exist, a state equation is formulated. With the definitions as in figure 3.2, the circuit is written with the following state space form when none of the thyristors are conducting,

 I&L  − R L − 1 L 0   I L   1      L &   1 V = 0 0   VC  +  0 V D  C C  & I  0 − κ   I P   0   P   0   

(3.16)

or shorter as

x& = A 1 x + BV D

(3.17)

where κ is a large positive number14, and VD is the forcing voltage of the whole TCSC-compensated line, see figure 3.2. When one of the thyristors is conducting the following state space form represents the circuit.

  0  I   1   I&L  − R L − 1 L   L   L &   1 1 0 − V + 0 V VC  =  C C  C    D   I&   I   0  R 1  P  0 − P  P    LP LP  

(3.18)

or shorter as

The use of κ is equivalent to modelling the open switch as a large series resistance; this model is required to ensure numerical accuracy of modelling the zero current state, and is common in power electronic simulation. 14

30

Chapter 3. Methods of linearizing a component

x& = A 2 x + BVD

t0

(3.19)

A1

A2

A1

α

β

γ

t1

t2

t3

Figure 3.3. The same A-matrix is used in time intervals α and γ (A1)

In time intervals α and γ, none of the thyristors are conducting and in time interval β one of the thyristors conduct. In [14] a method is described that predict the values of the coming instants t0, t1, t2, and t3. In steady-state the time interval in figure 3.3 is equal to one half-period, i.e. t3 – t0 = π/ω, but during transients it is not. Each time interval α, β, and γ is dependent on the time-interval just before itself. The model takes care of what happens at the junctions of the time-intervals, i.e. when the TCSC transits between equation (3.16) and (3.18). The aim of the method is to predict what the change of the line current in the middle of time interval β above will be.15 It is not obvious to understand how the firing angles to the thyristors are regulated in [14]. Change in the line current or change in the real power through the TCSC can be modeled as the output signal from the model. The model is a fundamental frequency model and it predicts either a round mean square value of the line current IL or the real power through the TCSC one half-period later, a discrete-time model. [14] provides a linear model of the system in figure 3.2 when it is in steadystate. The linear model varies as a function of operating point of the system.

15

The idea of predicting the voltage at t3 is used in the method described in section 3.7.

3.7 Dynamic modeling of TCSCs for subsynchronous resonance studies

31

3.7 Dynamic modeling of TCSCs for subsynchronous resonance studies Subsynchronous resonance (SSR) is a phenomenon that can occur when adding a capacitor Ccomp in series with the inductance of a transmission line Lline. The oscillation frequency f osc = 1 2π

Lline C comp can interact with natural fre-

quencies of shaft mechanical systems of nearby steam turbine generating units [1,11]. Such frequencies are below fundamental frequency. In [28] a proposed method is described. This method remains of the method described in section 3.6 but unlike in the sense that the authors use the dq0representation.

+V − I line

C L

IT

Figure 3.4. One-line diagram of one phase of a Thyristor-Controlled Series Capacitor

The two possible circuits that the TCSC can represent in one phase are described as follows. When one of the thyristors is conducting, the following state space form describes the diagram in figure 3.4:

 V&   0 − 1 C   V  1 cos ωt &  =   + 0   I T  C  0  I T  1 L

I  − sin ωt 1  d  I 0 0  q  I0   

(3.20)

where,

I dq 0

I d    = Iq  I0   

(3.21)

are the dq0-components of the line current, Iline, and V is the voltage drop of the series capacitor for one phase and IT is the phase current through the reactor, L. When none of the thyristors conduct, the diagram representing one phase in figure 3.4 is described by:

32

Chapter 3. Methods of linearizing a component

cos ωt C V& = [1 0]  0

[]

I  − sin ωt C 1 C   d  I 0 0   q   I0   

(3.22)

The inputs to the TCSC-model are the capacitor voltage V at the present instant t0, the dq0-components of the line current Iline, and the upcoming thyristor triggering instant φ½, see figure 3.5. The output of the TCSC-model is the capacitor voltage V after one half cycle at time t½, see figure 3.5. The capacitor voltage at t½ is obtained by integrating equation (3.22) during the time interval [t0 φ½], then integrating equation (3.20) during the time interval [φ½ τ½], and finally integrating equation (3.22) again during the time interval [τ½ t½]. Then an expression V(t½) is obtained. The expression V(t½) is non-linear in the dependence of the thyristor triggering instant φ½. Expression V(t½) is linearized around an operating point and the obtained linearized model is dependent on the thyristor-triggering instant φ½ and the current operating point. 0.6

Iline

0.4

Ithyr

Iline, Ithyr (p.u.)

0.2

0

-0.2

t0

-0.4

φ1/2 τ1/2

-0.6

-0.8 2.2

t1/2

2.21

2.22

2.23

2.24

2.25

Time (s)

Figure 3.5. One phase line current Iline and an associated thyristor current Ithyr

The analytical linearized model of the single-phase TCSC seeks to predict the changes in the capacitor voltage V at time t½ from its equilibrium value. The value of V(t½) is a function of the change in the capacitor voltage V at time t0, the change in the thyristor triggering instant φ½ and the change in the line current Iline during the half cycle between t0 and t½.

3.7 Dynamic modeling of TCSCs for subsynchronous resonance studies

33

Later, the single-phase model of the TCSC is expanded straightforward to a three-phase model assuming that the line current Iline is symmetric and that the thyristor triggering instants are equally spaced for the three phases with π/3, i.e. 1/6 of the fundamental period. The instant t½ in figure 3.5 is the same instant as t3 in figure 3.1. The instant t0 in figure 3.5 is the same as in figure 3.1. The linear model is validated in [28].

3.8 Dynamic phasor model of a TCSC This method develops a continuous-time model for TCSC. The model are based on the representation of voltages and currents as time-varying Fourier coefficients. The Fourier coefficients (Fourier series) are truncated to keep only the fundamental component of it.

+v− il

C L

i

Figure 3.6. One phase of the TCSC-representation

Reference [15] derives a TCSC-model based on time-varying Fourier coefficients that capture the phasor dynamics of the TCSC. The approach can be viewed as intermediate between instantaneous value mode and the quasi-static sinusoidal steady-state approximation. The TCSC's phasor dynamics represent the dynamics of the fundamental currents and voltages. The developed model of the TCSC is a phasor model of fourth-order.16 The model can be linearized and in different operating points it provides different linear models. See [15] for further details.

16

Fundamental frequency mode.

34

Chapter 4

Implementation of the ThyristorControlled Series Capacitor In this chapter the implementation of the Thyristor-Controlled Series Capacitor (TCSC) in the power system simulation software Simpow is described. The TCSC is modeled in instantaneous value mode. In chapter 5 it is described how a linear model is built of the original nonlinear TCSC-model from this chapter. Appendix C contains a detailed description of the implementation of the TCSC.

4.1 Introduction to the TCSC The basic scheme of the Thyristor-Controlled Series Capacitor (TCSC) is shown in figure 4.1. It consists of a series compensating capacitor shunted by a Thyristor-Controlled Reactor (TCR).

A

+U C −

B

C D

L → IL

Figure 4.1. Basic scheme of the Thyristor-Controlled Series Capacitor

35

36

Chapter 4. Implementation of the Thyristor-Controlled Series Capacitor

By regulating the conducting time for the thyristors, the total fundamental capacitive reactance between node A and B can be varied. So, from a fundamental frequency point of view the inserted reactance of the TCSC is controllable like,

C

A

B

Figure 4.2. The TCSC from a fundamental frequency point of view

When successfully varying the inserted reactance in figure 4.2, power oscillation damping can be carried out in a transmission system. Power oscillations have oscillation frequencies around 0.2 < f < 2 [Hz]. To damp oscillations, the TCSC-model in the thesis must be completed with a reactance unit that provides a reactance reference value, see parameter Ref in table C.1 in appendix C. 1.5 The series capacitor voltage 1

0.5

0

-0.5

-1 The current through the reactor -1.5 1.2

A

1.21 B

1.22

1.23

1.24

1.25

Figure 4.3. The series capacitor voltage and the current through the series reactor in steady-state17. The figure describes the situation for one phase of the TCSC

17

The current through the reactor is the same as the current through the corresponding forward- or reverse-thyristor.

4.2 The structure of the TCSC

37

Figure 4.3 gives a characteristic picture on how the TCSC works. As can be seen, the series capacitor voltage is “deformed” as a result of that the thyristors are conducting. Of course the series capacitor voltage will be sinusoidal if the thyristors would be blocked, not conducting. In figure 4.3 it can also be seen that if the series capacitor voltage U C is positive when the thyristor starts to conduct, the current through the reactor I L will be positive as defined in figure 4.1. At point of time A in figure 4.3, the forward-thyristor of the phase is conducting. The forward-thyristor is the lower thyristor depicted in figure 4.1. At point of time B in figure 4.3, the reverse-thyristor of the phase is conducting. The reverse-thyristor is the upper thyristor depicted in figure 4.1. The implementation of the TCSC is following the Synchronous Voltage Reversal-control algorithm18 used by ABB [9]. The instantaneous value mode in Simpow enables the user to deal with a TCSC-model that is not only represented as a continuously variable capacitor. Instead, the model does include the switches of the thyristors, as can be seen in figure 4.4, two thyristors for each phase. The instantaneous value mode in Simpow is similar to the simulation mode that the power system software EMTDC uses. The three-phase TCSC with it's control algorithm is split up in one part for each phase. Each part is consisting of four underlying parallel functions: •

Phase Locked Loop (PLL)

•

Booster (BOO)

•

Thyristor Pulse Generator (TPG)

•

Thyristors (THY)

see figure 4.4. These functions are documented in appendix C.

4.2 The structure of the TCSC The series capacitor C and the series reactor L, see figure 4.1, are modeled using the corresponding standard components in the power system simulation software [12,20]. The rest of the three-phase TCSC (the thyristors and the control algorithm) are split up in three parts, one for each phase. Such a part is implemented as a userdefined system that has four underlying functions: Phase Locked Loop (PLL), Booster (BOO), Thyristor Pulse Generator (TPG), and Thyristors (THY). 18

Synchronous Voltage Reversal, SVR.

38

Chapter 4. Implementation of the Thyristor-Controlled Series Capacitor

PLL

BOO

TPG

+ uC A − iA

Figure 4.4. Control system of the TCSC

The control algorithm is formulated by using the in-built simulation language Dynamic Simulation Language (DSL) in Simpow. The aim of the control algorithm is to calculate in what point of time the thyristors should start conducting. They are automatically blocked at the following zero-crossing of the phase current through the series reactor. The thyristors are included in the programmed control algorithm. The nature of the DSL-language is of that kind that all functions are solved in parallel. In each time-step the functions are calculated until a solution has been reached. Therefore the four functions, as well as the rest of the power system, are calculated at the same time, in parallel. See appendix C for a detailed description of the implementation of the TCSC.

Chapter 5

Linearization of the ThyristorControlled Series Capacitor This chapter describes the procedure of how the Thyristor-Controlled Series Capacitor is linearized.

5.1 Introduction The linearization of the Thyristor-Controlled Series Capacitor (TCSC)19 will take place in a realistic network in a realistic working point. Since we are not sure of if the TCSC behaves with the same linear response in all kinds of working points, we will linearize it in a realistic working point. The proposed method is to: I

run a simulation of a power system with the original TCSC-model, described in chapter 4,

II

when steady-state exists in the power system20, isolate the TCSC with the steady-state conditions in a stiff power system, see section 5.4 and perturb the d- and q-component of the current through the TCSC, one by one and observe the response of the d- and q-component of the voltage drop over the TCSC. In total, four voltage signals are recorded,

III

low-pass filter the d- and q-component of the voltage drop with the filter time T = 0.01 s,

IV

sample the d- and q-component of the voltage drop equi-distant since a variable time-step is used in point 2 above. In point V the signal must be equi-distant sampled. TS = 0.0001 [s],

V

again the signals are low-pass filtered to decrease the amount of harmonics, filter time T = 0.005 s,

19

From now on the detailed model of the TCSC that was described in chapter 4 will be called 'original TCSC-model'. 20 In fact, steady-state never exist in the network. In the reached equilibrium point still &x ≠ 0 . This is a consequence of the detailed representation of the thyristor switchings.

39

40

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

VI

the signals are re-sampled with the sample interval, TS = 0.002 [s],

VII

identify ARX-models of the four transfer functions, see equation (5.5), to create a low-order linear and time-invariant model,

VIII prove the validity of the linear model by simulating both of them in similar cases in time domain simulations, IX

perform linear analysis of the power system for the two cases: with the linear model and with the original TCSC-model.

In this chapter the first seven steps will be described. In chapter 6, step VIII is presented and in chapter 7, step IX is presented.

5.2 The studied power system The system below has been selected to study the original TCSC-model and the linear TCSC-model. The system is a simplified model of the TCSCcompensated south-north link in Brazil. This section 5.2 contains point I in the list given in section 5.1.

Bus A Bus B, S1 Imperatriz

North

Bus C

Bus D Bus E, S2 Serra da Mesa Figure 5.1. A simplified model of the TCSC-compensated south-north link in Brazil

The two synchronous machines, S1 and S2, are modeled with realistic eighthorder machine models including saturation in both d- and q-axis. The settings of them are viewed in table 5.1; see also [38] for a description of the machine models.

5.2 The studied power system

41

Table 5.1: Data for synchronous machines S1 and S2. Per unit (p.u.) is on machine basis

SN [MVA] H [MWs/MVA] UN [kV] X D [p.u.] X Q [p.u.] X D' [p.u.] X Q' [p.u.] X l [p.u.] X D'' = X Q'' [p.u.] R A [p.u.] T D0' [s] T Q0' [s] TD0'' [s] TQ0'' [s] KD [p.u. torque/p.u. speed]

S1 900 6.5 500 1.8 1.7 0.3 0.55 0.2 0.25 0.0025 8.0 0.4 0.03 0.05 0.0

S2 900 6.5 530 1.8 1.7 0.3 0.55 0.2 0.25 0.0025 8.0 0.4 0.03 0.05 0.0

The machines have different rated voltages and are assumed to be connected directly to the 500 kV level: UNS1 = 500 kV and UNS2 = 530 kV. Also exciters, governors, and turbines are modeled; see figures 5.2, 5.3, and 5.4. No transformers are included in the power system. The capacitance of the series capacitor is C = 0.2 mF, that produces a capacitive series reactance in steady-state of XC = 13.27 Ω (=1/(2πf0C)). The series reactor has an inductance of L = 5.63 mH, that produces an inductive series reactance in steady-state of XL = 2.12 Ω (=2πf0L). The control function of the TCSC has the same settings as in appendix C and it will produce a capacitive reactance that is two times the fundamental capacitive reactance of the series capacitor, i.e. XTCSC = 2*XC = 2*13.27 = 26.54 Ω. The impedance of the line between Bus D and E is R = 17.7 Ω and X = 266 Ω. It represents a 1000 km long transmission line. The load at Bus A is P = 500 MW and Q = 0 Mvar and it is modeled with constant impedance character. The impedance of the line between Bus A and Bus B is X = 0.266 Ω. Both synchronous machines are using the exciter shown in figure 5.2.

42

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

VRMAX REF +

VC

1 1+sTR

-

+

1+sTC 1+sTB

+ -

K + 1+sT

-

1 KE+sTE -

VRMIN

VS

EFD

Table saturation S E (UF) SE(UF)= TAB1

sK F 1+sTF

Figure 5.2. Block diagram of excitation system with DC commutator exciter (IEEE Type = DC1)

VC in figure 5.2 is the terminal voltage, VS is the output from a power system stabilizer (not used in the simulation), and EFD is the field voltage that is an input signal to the synchronous machine. The exciters have the following data: Table 5.2: Data for exciters KA TA TE KF KES1

20 0.055 0.36 0.125 0.034

TF TR VRMAX VRMIN KES2

1.8 0.05 4.0 -4.0 0.025

Both synchronous machines are using the same governor setup, see figure 5.3.

W PEG

+

P0

K 1+sT

+

YMAX

+

Y YMIN

K = 100 T = 0.1 YMAX = 1 YMIN = -1 Figure 5.3. Block diagram of the governors for S1 and S2

P0 in figure 5.3 is the initial mechanical power at t = 0 s, PEG is the active power in the machine, W is the speed of the machine and Y is the gate opening which is the input signal to the steam turbine shown in figure 5.4. Both synchronous machines are using the same steam turbine model except for different setups, see figure 5.4.

5.3 Steady-state of the system

Y

43

1 1+sTC

+

K

TM +

1-K

1 1+sTR

TC = 0.1 TR = 7.0 KS1 = 0.6 KS2 = 0.3 Figure 5.4. Block diagram of the steam turbines for S1 and S2

TM is the mechanical torque of the turbine and is an input signal to the synchronous machine. As viewed in figure 5.4, the steam turbines have different settings of K as indicated with KS1 and KS2 in the figure.

5.3 Steady-state of the system When steady-state exists in the power system, the d- and q-component of the current into the TCSC and the voltage drop over the TCSC have the following curve forms. The thyristors of the TCSC are activated at t = 0.2 sec. 0.2 0.18 0.16

iq

id, iq, i0 (p.u.)

0.14 0.12 0.1

0.08

i

d

0.06 0.04

i0

0.02 0 0

1

2

3 Time (s)

4

5

6

Figure 5.5. id, iq, and i0 into the TCSC when the system is reaching steady-state

44

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

0.06 u

Cd

uC d, uC q, uC 0 (p.u.)

0.04 0.02 0

-0.02 u

C0

-0.04 u

Cq

-0.06 0

1

2

3 Time (s)

4

5

6

Figure 5.6. uC d, uC q, and uC 0 over the TCSC when the system is reaching steady-state, unfiltered values

It can be seen in the two figures above that after approximately 1.5 seconds the TCSC has went through an initialization process. In the figures in the thesis, the currents are plotted in per unit of the following base current:

I base =

S base * 2 U base− phase −to − phase * 3

=

1300 * 2 500 * 3

= 2.12 kA

(5.1)

and the voltages are plotted in per unit of the following base voltage:

U base =

U base− phase−to − phase * 2

3

=

500 * 2 3

= 408.25 kV

(5.2)

Since we are simulating in instantaneous value mode, the definitions of the per unit bases above are different from the ones used when simulating in the fundamental frequency mode (phasor domain). Especially in figure 5.6 it can be seen that harmonics are included in all the voltage components. Steady-state is assumed to exist in the power system at t = 4 s, see figures 5.5 and 5.6.

5.4 Perturbing the TCSC in a stiff power system

45

5.4 Perturbing the TCSC in a stiff power system Before the system is disturbed it runs until it has reached steady-state. Since the simulation is made using an instantaneous value representation and since the power system contains power electronics, signals are still varying when steady-state has been reached, as can be seen in figures 5.5 and 5.6 above, i.e. they contain harmonics. This section 5.4 contains point II in the list given in section 5.1. After t > 4.0 s, i.e. in steady-state, the current into the TCSC, in figure 5.5 is:

iq = 0.169 p.u.

id = 0.094 p.u.

(5.3)

In a dq-frame the current can be scheduled as,

iq =

i = id + jiq

i

= id Figure 5.7. The current into the TCSC in steady-state

Figure 5.5 shows that the 0-component of the current into the TCSC is equal to zero. The absolute value of the steady-state current i in the figure above is, 2

2

2

2

i = i d + iq = 0.094 + 0.169 = 0.193 p.u.

(5.4)

To perturb the TCSC and to analyze nothing else than the TCSC's dynamic behavior it is isolated in a small, "stiff" power system, see figure 5.8.

Node 1

Node 2

+ uC −

Node 3

∞

i

Figure 5.8. System used for observing the dynamic behavior of the TCSC

46

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

The stiff power system depicted in figure 5.8, consists of, from left to right: •

an infinite bus,

•

a low-impedance line,

•

the TCSC, and

•

a load of constant current character.

The steady-state current in equation (5.4) is in the following assumed to be of pure d-component character, given as id 0 in the figure below. The load in Node 3 in the stiff power system in figure 5.8 will model this load. By turning the current in the dq-frame and let it have the same magnitude but be of pure dcomponent character makes the analysis of the disturbed TCSC easier. Physically it is the same situation as the one in section 5.3 since the same magnitude of the current is used. In the stiff power system, the TCSC is perturbed both in the current's d- and qcomponent, one by one as in figure 5.9:

∆iq id 0

∆id

Figure 5.9. The current into the TCSC is disturbed with 10% of id 0 in both its d- and q-components

By disturbing the current into the TCSC in its d- and q-component separately it is possible to identify the four elements of the [2x2]-matrix in expression (5.5).

∆u C d  G11 (s ) G12 (s ) ∆i d   ∆u  =     C q  G21 (s ) G22 (s )  ∆iq 

(5.5)

In figures 5.5 and 5.6 unfiltered values of id, iq, i0, uC d, uC q, and uC 0 are shown. As can be seen, the signals contain harmonics, mostly third and sixth harmonic. In order to omit harmonics in the signals that will be studied, the following low-pass filter is applied to the studied d- and q-components of the voltages in sections 5.4.1 and 5.4.2. The time constant of the first-order filter is T = 0.01 [s].

5.4 Perturbing the TCSC in a stiff power system

1

0

0.9

-10

0.8

-20

0.7

-30

arg(K(f))

0.6

|K(f)|

47

0.5 0.4

-40 -50 -60

0.3 -70

0.2

-80

0.1 0

-90

0

10

20

30

40

50

0

60

10

20

30

40

50

60

Frequency (Hz)

Frequency (Hz)

Figure 5.10. Filter characteristic of the low-pass filter, magnitude and phase

The following sections 5.4.1 and 5.4.2 contain point III in the list given in section 5.1.

5.4.1 Perturbing the d-component of the current

0.01

-0.015

0.005

-0.02

uC q (p.u.)

uC d (p.u.)

In the following four figures, the two responses of the voltage drop of the TCSC, uC d and uC q, are shown when the current id has been increased (perturbed) with a step of 10% of the steady-state current. As described in section 5.4, the signals uC d and uC q are filtered through a first-order filter with the time constant T = 0.01 [s] to decrease the amount of harmonics. Below the filtered signals are shown.

0

-0.025

-0.005

-0.01 3.5

-0.03

4

4.5

5

5.5

6

Time (s)

Figure 5.11. uC d when id is increased with 10%

-0.035 3.5

4

4.5

5

5.5

6

Time (s)

Figure 5.12. uC q when id is increased 10%

Figure 5.11 and 5.12 can also be viewed with a complex representation of the voltage and current, i.e. u = ud + juq and i = id + jiq.

48

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

0.15

0.1

q-component

0.05

i 0

u -0.05

-0.1

-0.05

0

0.05

0.1 d-component

0.15

0.2

0.25

Figure 5.13. u and i scheduled in the same diagram. The marked square is focused below

When i is increased with a 10%-step in its d-component in the figure above, u responds as in the lower left corner in figure 5.13. As shown in figure 5.13, the voltage over the TCSC is 90 degrees behind the current through the TCSC, correlated to the origin. This is characteristic for a capacitor. Figure 5.13 can be zoomed to get a better view of how u follows a change in id, see figure 5.14. -0.02

q-component

-0.022

-0.024

t = 4.0 s -0.026

-0.028 t = 5.0 s -0.03 -5

-3

-1

1 d-component

Figure 5.14. u zoomed

3

5 -3

x 10

5.4 Perturbing the TCSC in a stiff power system

49

In figure 5.14 we see the dynamic response of the TCSC's voltage drop when the d-component of the current through it is increased with a 10%-step.

5.4.2 Perturbing the q-component of the current

0.01

-0.015

0.005

-0.02

uC q (p.u.)

UC d (p.u.)

In figure 5.15 - 5.18 the response of the TCSC's uC d and uC q are shown when the current iq has been increased (perturbed) with a size that is 10% of id 0. The following voltage signals are filtered through a first-order filter with the time constant T = 0.01 [s].

0

-0.025

-0.005

-0.03

-0.01 3.5

4

4.5

5

5.5

-0.035 3.5

6

4

4.5

Time (s)

5

5.5

6

Time (s)

Figure 5.15. uC d when iq is increased 10% of id 0

Figure 5.16. uC q when iq is increased with 10% of id 0

As in section 5.4.1, figures 5.15 and 5.16 can also be viewed with a complex representation of the voltage and current, i.e. u = ud + juq and i = id + jiq. 0.15

0.1

q-component

0.05

i 0

u -0.05

-0.1

-0.05

0

0.05

0.1 d-component

0.15

0.2

0.25

Figure 5.17. u and i scheduled in the same diagram. The marked square is focused below

50

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

When i is increased with a 10%-step of id 0 in its q-component, u responds as in the lower left corner in figure 5.17. As shown in figure 5.17, the voltage over the TCSC is 90 degrees behind the current through the TCSC, correlated to the origin. This is characteristic for a capacitor. Figure 5.17 can be zoomed to get a better view of how u follows a change in iq, see figure 5.18. -0.02

q-component

-0.022

-0.024

t = 5.0 s -0.026

t = 4.0 s -0.028

-0.03 -5

-3

-1

1

3

d-component

5 -3

x 10

Figure 5.18. u zoomed

In figure 5.18 we see the dynamic response of the TCSC's voltage drop when the q-component of the current through it is increased with 10% of id 0. Figure 5.14 and 5.18 are describing the dynamics of the TCSC and how the voltage drops are reaching a new steady-state after the TCSC has been disturbed, i.e. the path the voltage u selects from t = 4.0 s to reach a new steadystate at approximately t = 5.0 s. In the following, the purpose is to build a linear model that will contain the same dynamic responses as the curves in figure 5.14 and 5.18.

5.5 Building a linear model of the TCSC The dynamic responses showed in figures 5.11, 5.12, 5.15, and 5.16 of the TCSC are used in this section to build a linear model. This section 5.5 contains points IV, V, VI, and VII in the list given in section 5.1. By subtracting the initial values of the voltages shown in figures 5.11, 5.12, 5.15, and 5.16, the dynamic response of the changed voltage drop over the TCSC according to steps in both the currents d- and q-component can be generated, i.e. the relations,

5.5 Building a linear model of the TCSC

51

∆uC d ∆id , ∆uC q ∆id , ∆uC d ∆iq , ∆uC q ∆iq

(5.6)

can be scheduled. The relations in (5.6) have been pointed out earlier in (5.5). In (5.5) they were written as:

G11 (s ), G21 (s ), G12 (s ), G22 (s )

(5.7)

5.5.1 Preparation of the four signals The signals in figures 5.5 – 5.6, and 5.11 – 5.18 are non-uniformly spaced, i.e. they are not given with a fixed time-step from the used power system software [12,20]. Since the linear model will be developed in the discrete-time domain the signal must be equi-distant sampled. So, first the four voltage signals have to be sampled with uniformly spaced sampling. By studying the signals it can be observed that they, after have been filtered, still contain harmonics, but no frequencies higher than the 6th harmonic (360 Hz) exist. Therefore, the selected sampling frequency is set to 10 kHz to be sure of that no frequencies can be folded into the frequency spectrum of interest. With sampling frequency FS selected to 10 kHz, the Nyquist frequency will be 5 kHz according to the sampling theorem, i.e. frequencies above 5 kHz could be folded – aliased21 into the desired frequency spectrum [22]. Non-uniformly spaced Signal

Uniformly spaced Signal

x( n) = x a ( nTS )

xa ( t ) FS = 10 kHz

TS = 1 FS

Figure 5.19. Periodic sampling of the signals

Before continuing, the signals are again filtered with a first-order filter, this time with the time constant T = 0.005 [s], to decrease the magnitude of harmonics, see figure 5.20. After selecting every 20th sample of the filtered values of the signals, the twotimes filtered uC q relative the one-time filtered uC q is viewed in the figure be-

21

Alias means: with false name. That synonym gives a good picture of the importance of avoiding alias.

52

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

low. The two-times filtered signal is after re-sampling given with the time-step 0.002 s.22 -0.025

-0.026

uC q (p.u.)

Two-times filtered -0.027

-0.028

-0.029

One-time filtered -0.03 3.8

3.9

4

4.1

4.2

4.3

Time (s)

Figure 5.20. The filtered and unfiltered value of uC q when id is increased 10%

The last re-sampling is made since the number of samples is unnecessary large to describe the slow dynamic of the input signals.

5.5.2 Identification of the four signals Here follows the graphs and the expressions of the identified transfer functions. In the figures, both the identified and the original signals are viewed.

22

Re-sampling have here been made with the Nyquist frequency 250 Hz, the small amounts of the sixth harmonic 360 Hz that still exists in the signals may have been folded into the frequency spectrum and appears as 140 Hz.

5.5 Building a linear model of the TCSC

53

-4

5

-3

x 10

0.5

x 10

4 -0.5

Identified

uC q (p.u.)

uC d (p.u.)

3

-1.5

2

Identified

1

-2.5

0 -3.5

4

4.5

5

5.5

4

6

4.5

5

6

Figure 5.22. uC q identified when id is perturbed

Figure 5.21. uC d identified when id is perturbed

-3

3.5

5.5

Time (s)

Time (s)

-3

x 10

x 10

6 2.5

uC q (p.u.)

uC d (p.u.)

Identified Identified

1.5

1

0.5

-0.5 4

4.5

5

5.5

6

-4

4

4.5

5

Time (s)

5.5

6

Time (s)

Figure 5.23. uC d identified when iq is perturbed

Figure 5.24. uC q identified when iq is perturbed

By using Matlab’s System Identification Toolbox [21] to build ARX-models for the four relations, transfer functions can be found as follows, see also section 3.2. The method works in the time-discrete domain. The input to the method is uniformly spaced sampled values of input and output signals of an unknown system. The following difference equation is then identified: y (t ) + a1 y (t − 1) + K + a n y (t − n a ) = b1 u (t − 1) + K + bn u (t − n b ) + e(t ) a

(5.8)

b

When using the method, the required order of the polynomials in (5.8) must be given, i.e. na and nb. By varying the settings of na and nb, re-sample the input signal, and also change the length of the identified time-interval of the signals, values of parameters a1, …, a n and b1, …, b n can be found for each of the a

four relations G11(s), G21(s), G12(s), and G22(s).

b

The four generated time-discrete models (5.8) are converted to continuous time models and after that the continuous time models are converted to state-space models [21].

54

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

The signals above have been identified with the following four transfer functions. G11(s):

G11 (s ) =

∆u C d (s ) ∆id (s )

= d1

(s + a ) (s − (b + jc ))(s − (b 1

1

1

(5.9)

− jc1 ))

1

where a1 =-0.6561

b1 =-23.7283

c1 =29.8200

d1 =1.1963

G21(s):

G21 (s ) =

∆u C q (s ) ∆id (s )

= d2

(s + a ) (s − (b + jc ))(s − (b 2

2

2

2

(5.10)

− jc 2 ))

where a2 =90.4735

b2 =-14.9999

G12(s):

G12 (s ) =

∆u C d (s ) ∆i q ( s )

= d3

c2 =27.0205

(s + a ) (s − (b + jc ))(s − (b 3

3

3

3

d2 =-1.4829

(5.11)

− jc3 ))

where a3 =15.5650

b3 =-12.8007

c3 =13.8145

d3 =3.1194

G22(s): G22 (s ) =

∆uC q (s ) ∆iq (s )

= d4

(s − (b

(s − (b

7

+ jc7 ))(s − (b7 − jc7 ))(s + a6 )

+ jc5 ))(s − (b5 − jc5 ))(s − (b6 + jc6 ))(s − (b6 − jc6 )) 5

(5.12)

where a6 =-0.3737

b5=-36.7106

c5 =96.1461

b6 = -9.8140

c6 =20.9656

b7 =-203.15

c7 =113.23

d4 =2.2057

Since there are in total ten poles in the transfer functions above, it is found that a tenth-order model is enough to describe the behavior of the four relations, i.e. n =10 in equations (2.31) – (2.32). All constants c1, …, c7 are given in rad/s.

5.5 Building a linear model of the TCSC

55

Since there are two input signals and two output signals to the linear model, the dimension of the other two matrices becomes: r = 2 and m = 2 in equations (2.31) – (2.32).

5.5.3 How a transfer function is formulated on ABCD-form The transfer function G11(s) in equation (5.9) is formulated on the ABCD-form as:

 x&1  b1  x&  = c  2  1

a + b − c1   x1   1 1 + c b1   x 2   1  1

 0 ∆id    ∆i q  0  

(5.13)

∆u C d  0 d1   x1  0 0 ∆id   ∆u  =      +   C q  0 0   x 2  0 0  ∆i q 

(5.14)

The roots of the transfer function (5.9) define the A-matrix and the gain of the transfer function, d1, is placed in the C-matrix.

5.5.4 The final linear TCSC-model When including all the other transfer functions, (5.10) – (5.12) into equations (5.13) and (5.14), the final linear form of the TCSC is described with the following ABCD-model: 23  x&1  b1  x&  c  2  1  x&3   0 &    x4   0  x&   0  5=  x& 6   0  x&   0  7   x&8   0  x&   0  9   x&10   0

23

− c1

0

0

0

0

0

0

0

b1

0

0

0

0

0

b2

0 − c2

0

0

0

0

0

0

0

0

c2

b2

0

0

0

0

0

0

0

b3

0 − c3

0

0

0

0 0

0 0

0 0

c3 0

b3 0

0 b5

0 − c5

0 0

0

0

0

0

0

c5

b5

0

0

0

0

0

0

0

A98

b6

0

0

0

0

0

0

A108

c6

 a1 + b1  0   x1   c1 0   x2   1  a2 + b2 0   x3   c2   0   x4   1  0   x5   0   + 0   x6   0   x7   0    0   x8   0 − c6   x9   0    b6   x10   0   0

   0   0   0  a3 + b3  ∆i  d c3   ∆i   q  1  a6 + b5  c5  1   0   0  0

(5.15)

It should be mentioned that in equations (5.13) - (5.16), ∆ has been omitted in the state vector x. ∆ was earlier included in the linear formulation in chapter 2.

56

Chapter 5. Linearization of the Thyristor-Controlled Series Capacitor

∆uC d  0 d1 0 0  ∆u  =   C q  0 0 0 d 2

0 d3

0

0

0 d4

0

0

 x1  x   2  x3     x4  0 0   x5  0 0 ∆id   +   0 d 4   x6  0 0  ∆iq  x   7  x8  x   9  x10 

(5.15)

where

A98 =

(b

6

− b7 ) + c 7 − c6 2

2

2

A108 = 2(b6 − b7 )

c6

Now, the small-signal behavior of the detailed system in figure 4.4 can be modeled as a tenth-order system according to the figure below. An internal state vector

∆u

x

∆y

Figure 5.25. The linear TCSC-model

5.6 A reference system interfacing the linear TCSC-model To model the deviations in the current from steady-state as an input signal to the TCSC, a surrounding reference system has to be built. In figure 5.26 the deviation from steady-state current, models the input signal to the linear model, H(s). The signal i0 is the d- and q-components of the current through the TCSC in steady-state. ∆i, immediately to the left of the block diagram H(s) in figure 5.26, is the input signal to the linear model. u0 is the d- and q-components of the voltage drop over the TCSC in steadystate. ∆u, immediately to the right of the block diagram H(s) in figure 5.26, is the output signal from the linear model H(s) which was developed earlier in this chapter.

5.6 A reference system interfacing the linear TCSC-model

57

-1.

i

i = id + j*iq

i=

+

-j*TETA

e

-

id + j

H(s)

iq u=

ud + j

uq

TETA -j*TETA

e i0

i 0 = id0 + j*iq0

u = ud + j*uq

+

u

+

-j*TETA

e

u0

u0 = ud0 + j*uq0

Figure 5.26. The surrounding reference system for the linear model, H(s). All block diagrams in the figure are named as M(s) in chapters 6 and 7

In chapters 6 and 7 the original TCSC-model will be exchanged with the block diagram in figure 5.26 when the system has reached steady-state. The block diagrams in figure 5.26 will be named as M(s) in chapters 6 and 7.

58

Chapter 6

Comparisons in time domain simulations This chapter compares time domain simulations with the original TCSC-model and the created linear model. Comparisons are made with three cases. The event in the first case is in the same range as was used when the linear model was developed in chapter 5. The second case contains a larger disturbance and the third case simulates a disturbance when the power system is in another operating point.

6.1 Introduction To control the validity of the linear model of the TCSC, comparisons are made with the linear and the original TCSC-model. Since the linear model has been developed for being valid for small-signal perturbations, the perturbations cannot be too large. The assumption on which a linearization is built is that in a narrow interval of the input signals, the component behaves linear. Therefore, the created linear model is valid for such small disturbances and the larger disturbances that are applied to the input signals, the more unsure it is that the linear model represents the behavior of the original model.

6.2 Description of the three dynamic simulation cases Three cases have been performed to compare the linear model with the original TCSC-model. In the first case, case a, the event in the power system reminds of the event that took place when the linear model was identified, i.e. the load is increased with 10% at t = 6 s. In case b, a larger disturbance takes place in the power system than in case a. In the last case, case c, the event is the same as in case a, but the power system is running in another operating point. Case c has been constructed to check the validity of the linear model in another operating point of the power system. For all cases two simulations are done, one with the original TCSC-model, see figure 6.1, and one with the linear model, see figure 6.2.

59

60

Chapter 6. Comparisons in time domain simulations

One difference between the system used during the linearization procedure and the system in cases a, b, and c is that the system in the three test cases contains dynamic models in the surrounding power system, for instance the two synchronous machines. In the linearization procedure in section 5.4 the power system was modeled as stiff, i.e. no dynamics outside the TCSC. Therefore, the input signal to the linear model will not be a step-function, it will instead vary during the disturbance, see figure 6.3 below.

Bus A Bus B, S1 Imperatriz

North

Bus C

Bus D Bus E, S2 Serra da Mesa Figure 6.1. The power system wherein the TCSC is modeled with the original TCSCmodel

The original TCSC-model produces not only a pure capacitive reactance in steady-state but also a small resistance. The TCSC has the setup to produce a capacitive reactance XTCSC = 26.54 Ω, however the produced impedance also contains a small resistance of R = 0.070 Ω, (which is 0.3% of the capacitive reactance) see figure 6.2. In order to have the same initial settings when comparing the original TCSC-model with the linear model, the impedance between BUS B and Bus E is expanded to include that resistance in the simulations with the linear model, see figure 6.2 below. In case a and case b, the steady-state conditions are the same as was used when the linear model was developed in sections 5.2 - 5.5.

6.3 Case a

61

Bus A Bus B, S1 Imperatriz

XC R

North

M (s )

At t = 4 s

Bus D Bus E, S2 Serra da Mesa Figure 6.2. The power system wherein the TCSC is modeled with the linear model including an interfacing reference system, see section 5.6

In steady-state, at t = 4 s, the linear TCSC-model replaces the series impedance as the switches are indicating in the figure above. After t = 4 s the block diagram marked with M(s), including the linear model and the interfacing reference system described in section 5.6, is switched on. It replaces the series impedance R + jXC. In all following diagrams, results are compared from simulations with the original TCSC-model and the linear model, i.e. figure 6.1 and 6.2.

6.3 Case a In case a, the pure active load in Bus A is increased at t = 6 s with 10%, so that ∆P = 50 MW. The load is modeled with constant impedance character and therefore the load increment will be less than 50 MW since the bus voltage in the load node will decrease when the load is changed at t = 6 s. Before t = 6 s, the load is initially P0 = 500 MW and Q0 = 0 Mvar. Two simulations are done, one with the original TCSC-model and one with the linear model. Since the load is increased with 10% and the steady-state of the power system before the load is increased is the same as when the linear model of the TCSC

62

Chapter 6. Comparisons in time domain simulations

was identified, this case is in the same range as the situation when the linear model of the TCSC was identified in section 5.4. In figures 6.3 – 6.6, the current into the TCSC and the voltage drop over the TCSC are shown. The signals in these figures have been low-pass-filtered with a first-order filter with the time constant T = 0.01 s. In each figure the results from both the original TCSC-model and the linear model is viewed. 0.104

0.188

Linear model

0.186

0.102

0.184

Original TCSC-model

0.182

iq (p.u.)

id (p.u.)

0.1

Original TCSC-model

0.098

0.18

0.178 0.176

0.096 0.174 0.172

0.094

Linear model

0.17 0.092 5.5

6

6.5

7

7.5

8

8.5

0.168 5.5

9

6

6.5

7

Time (s)

7.5

8

8.5

9

Time (s)

Figure 6.3. id resp. iq for both the original TCSC-model and the linear model in case a

In the left figure above, id is plotted for both the simulations with the original TCSC-model and with the linear model. In the right figure, iq is plotted for the two simulations. In the two graphs in figure 6.3 it is hard to find any discrepancies between the two simulations. In figure 6.4, iq is plotted relative id. In steady-state (t < 6 s), the current is in the center of the circle and during the disturbance the change in current never gets larger than 10% of the steady-state value. The change in current is therefore in the same range as when the TCSC was linearized. 0.2

i q( i d ) 0.15

iq (p.u.)

radius = 0.10 * | i | 0.1

| i | = 0.193 p.u. (steady-state) 0.05

0

0

0.05

0.1

0.15

0.2

i (p.u.) d

Figure 6.4. iq(id) for the two models

6.3 Case a

63

-0.0126

0.027

-0.0128

0.0265

-0.013

Linear model

0.026

Linear model

-0.0132

Original TCSC-model

uC q (p.u.)

uC d (p.u.)

0.0255

-0.0134

0.025

-0.0136

0.0245

-0.0138 0.024

-0.014

Original TCSC-model

0.0235 0.023 5.5

-0.0142

6

6.5

7

7.5

8

8.5

-0.0144 5.5

9

6

6.5

7

7.5

8

8.5

9

Time (s)

Time (s)

Figure 6.5. uC d resp. uC q for both the original TCSC-model and the linear model in case a

In the left figure above, uC d is plotted for both the simulations with the original TCSC-model and the linear model. In the right figure, uC q is plotted for the two simulations. The thin line is the result from the simulation with the linear model. The signals in these figures have been low-pass-filtered with a firstorder filter with the time constant T = 0.01 s. The linear model follows very much the original TCSC-model in the left diagram. That is not the case in the right diagram, but the scale is much more detailed in the right diagram and therefore it shows more discrepancies between the two simulations. In figure 6.6, uC q is plotted relative uC d. In the upper left corner t is equal to 6 s and in the middle, t is equal to 9 s. -0.012

Original TCSC-model

uC q (p.u.)

-0.013

-0.014

Linear model

-0.015

-0.016 0.023

0.024

0.025

u

Cd

0.026

(p.u.)

Figure 6.6. uC q(uC d) for the two models

0.027

64

Chapter 6. Comparisons in time domain simulations

6.4 Case b In case b, both the active and reactive load in Bus A are increased at t = 6 s with ∆P = 100 MW and ∆Q = 100 Mvar. The load is modeled with constant impedance character and therefore the load increment will be less than 100 MW and 100 Mvar respectively since the bus voltage in the load node will decrease when the load is changed at t = 6 s. For t < 6 s, the load is P0 = 500 MW and Q0 = 0 Mvar, i.e. the same steadystate as in case a. The load increment is in this case larger than for the event that took place when the identification of the linear model was done in chapter 5. Two simulations are done, one with the original TCSC-model and one with the linear model. In figure 6.7 – 6.10, the current into the TCSC and the voltage drop over the TCSC are shown. The signals in these figures have been low-pass-filtered with a first-order filter with the time constant T = 0.01 s. 0.12

0.2

Linear model

Linear model 0.195

0.115

0.19

iq (p.u.)

0.11

id (p.u.)

0.185

0.105

Original TCSC-model

Original TCSC-model 0.18

0.1 0.175 0.095

0.09 5.5

0.17

6

6.5

7

7.5

Time (s)

8

8.5

9

0.165 5.5

6

6.5

7

7.5

8

8.5

9

Time (s)

Figure 6.7. id resp. iq for both the original TCSC-model and the linear model in case b

In the left figure above, id is plotted for both the simulation with the original TCSC-model and with the linear model. In the right figure, iq is plotted for the two simulations. In figure 6.8, iq is plotted relative id. In the middle of the circle, t < 6 s and in the upper right corner, t is equal to 9 s. In figure 6.8 it is viewed that the changes of the current is larger than in the linearization procedure since the graph of iq(id) passes the border of the circle. In figure 6.8, iq(id) are shown for both the simulations with the original TCSC-model and the linear model. Also in case b, the steady-state current has the magnitude |i| = 0.193 p.u.

6.4 Case b

65

0.2

iq ( id ) 0.19

iq (p.u.)

0.18

0.17

0.16

0.15

radius = 0.10 * | i | 0.08

0.09

0.1

0.11

0.12

0.13

id (p.u.) Figure 6.8. The d- and q-components of the current through the TCSC viewed in one diagram, iq(id). In the figure, |i| is 0.193 p.u.

0.029

-0.012

Original TCSC-model

-0.013

0.028

Original TCSC-model -0.014

uC q (p.u.)

uC d (p.u.)

0.027 -0.015

0.026

-0.016

Linear model

0.025

Linear model

-0.017 0.024

0.023 5.5

-0.018

6

6.5

7

7.5

Time (s)

8

8.5

9

-0.019 5.5

6

6.5

7

7.5

8

8.5

9

Time (s)

Figure 6.9. uC d resp. uC q for both the original TCSC-model and the linear model in case b

In the left figure above, uC d is plotted for both the simulation with the original TCSC-model and with the linear model. In the right figure, uC q is plotted for the two simulations. The signals in these figures have been low-pass-filtered with a first-order filter with the time constant T = 0.01 s. The linear model follows the original TCSC-model as well as it did in case a.

66

Chapter 6. Comparisons in time domain simulations

In figure 6.10, uC q is plotted relative uC d. In the upper left corner t is equal to 6 s and in the upper right corner, t is equal to 9 s.

-0.013

Linear model

uC q (p.u.)

-0.014

-0.015

-0.016

-0.017

-0.018

Original TCSC-model

0.023

0.024

0.025

0.026

0.027

0.028

0.029

uC d (p.u.) Figure 6.10. uC q(uC d) for the two models

The voltage drop of the linear model follows very well the voltage drop of the original TCSC-model in figure 6.9 – 6.10. That proves that the linear model is valid even for disturbances larger than were used in the linearization in section 5.4. Figure 6.10 is also viewed on the front cover of the thesis. In figure 6.11 the impedance of the TCSC is pictured. The impedance of the TCSC is calculated with the expression,

Z TCSC =

u d + ju q id + ji q

(6.1)

where, ud + juq = the voltage drop over the TCSC, uBUSD - uBUSB in figure 6.1. id + jiq = the current from Bus D to Bus B in figure 6.1. The control algorithm of the TCSC control ZTCSC so that ZTCSC = -j26.54 Ω. In figure 6.11 the impedances of both the original TCSC-model and the linear model are viewed.

6.5 Case c

67

-24

Linear model -25

X (Ohm)

-26

-27

-28

-29

Original TCSC-model -30

-31 -2

-1

0

1

2

3

4

5

R (Ohm)

Figure 6.11. Impedances of both the original TCSC-model and the linear model during the time interval 5.5 s < t < 7 s

6.5 Case c It is important to check if the linear model is valid in another initial steady-state (operating point) than was used during the linearization. For t < 6 s, the load is initially P0 = 700 MW and Q0 = 0 Mvar. In case c, the pure active load in Bus A is increased at t = 6 s with ∆P = 50 MW (7.1%). The load is modeled with constant impedance character and therefore the load increment will be less than 50 MW since the bus voltage in the load node will decrease when the load is changed at t = 6 s. No change in reactive power is made at t = 6 s. Two simulations are done, one with the original TCSC-model and one with the linear model. In figure 6.12 – 6.15, the current into the TCSC and the voltage drop over the TCSC are shown. The signals in these figures have been low-pass-filtered with a first-order filter with the time constant T = 0.01 s.

68

Chapter 6. Comparisons in time domain simulations

0.094

0.37

Linear model 0.092

0.365

0.09 0.36

Original TCSC-model

iq (p.u.)

id (p.u.)

Original TCSC-model 0.088

0.355

0.086

0.35 0.084 0.345

0.082

Linear model 0.08 5.5

6

6.5

7

7.5

8

8.5

0.34 5.5

9

6

6.5

Time (s)

7

7.5

8

8.5

9

Time (s)

Figure 6.12. id resp. iq for both the original TCSC-model and the linear model in case c

In the left figure above, id is plotted for both the simulation with the original TCSC-model and with the linear model. In the right figure, iq is plotted for the two simulations. In figure 6.13, iq is plotted relative id. In the middle of the circle, t < 6 s and in the upper right corner, t is equal to 9 s. In figure 6.13, iq(id) are shown for both the simulations with the original TCSC-model and the linear model. 0.37

radius = 0.10 * | i | 0.36

iq (p.u.)

0.35

0.34

iq( id )

0.33

0.32 0.06

0.07

0.08

0.09

0.1

0.11

id (p.u.) Figure 6.13. The d- and q-components of the current through the TCSC viewed in one diagram, iq(id) for case c. In the figure, the radius is 0.0193 p.u.

6.5 Case c

69

0.0515

-0.01

0.051 -0.0105

0.0505 0.05

-0.011

uC q (p.u.)

uC d (p.u.)

Linear model

Linear model

0.0495 0.049

Original TCSC-model

-0.0115

0.0485

-0.012

0.048 0.0475

-0.0125

0.047

Original TCSC-model 0.0465 5.5

6

6.5

7

7.5

8

8.5

9

-0.013 5.5

6

Time (s)

6.5

7

7.5

8

8.5

9

Time (s)

Figure 6.14. uC d resp. uC q for both the original TCSC-model and the linear model in case c

In the left figure above, uC d is plotted for both the simulation with the original TCSC-model and with the linear model. In the right figure, uC q is plotted for both the simulation with the original TCSC-model and with the linear model. The linear model does not follow the original TCSC-model as good as in case a. uC d shows more discrepancies between the models than in case a and b. In figure 6.15, uC q is plotted relative uC d. In the upper left corner, t is equal to 6 s and in the middle part; t is equal to 9 s. -0.01

Original TCSC-model

uC q (p.u.)

-0.011

-0.012

-0.013

Linear model -0.014

-0.015 0.046

0.047

0.048

0.049

0.05

uC d (p.u.) Figure 6.15. uC q(uC d) for the two models

0.051

70

Chapter 6. Comparisons in time domain simulations

6.6 Conclusions of time domain simulations In case a, the steady-state and the disturbance in the power system had the same size as the disturbance when the linear identification of the TCSC was done in section 5.4. The voltage drop over the TCSC shows good agreement between the original TCSC-model and the linear model. In figure 6.4 we can see that the current through the TCSC is within the circle that indicates the size of the applied disturbance during the linearization. In case b, the event is a larger disturbance than in case a. Figure 6.8 shows that the change in the current through the TCSC is almost two times larger than the current change in case a. However, the voltage drop over the TCSC shows good agreement between the two models and that proves that the linear model is valid even for disturbances larger than were used in the linearization in section 5.4. In case c, it is checked whether the linear model is valid or not in another initial steady-state (operating point) than was used during the linearization. In case c the linear model follows quite well the original TCSC-model but it is hard to state if the linear model is valid or not since it shows more discrepancies in case c than in case a and b. The conclusions are: •

The linear model shows corresponding results with the original TCSCmodel.

•

Agreements in simulation results have been found for simulation cases with a disturbance in the same range as was used when the linear model was developed.

•

The linear model is also valid for larger disturbances than the one used when the linear model was developed.

•

The linear model of the TCSC shows discrepancies for steady-states other than the one that was used when linearization was done. Case c shows discrepancies larger than the other two cases.

•

The simulation with the linear model takes seconds to complete. This should be compared with that the simulation of the original TCSC-model takes hours to complete.

Chapter 7

Comparisons in linear analysis This chapter contains comparisons of linear analysis of the two models, the original TCSC-model and the linear model. The used power system is the same as in chapter 6.

7.1 Introduction Eigenvalues of a power system indicates whether it is stable or not. It also gives valuable information about the behavior of the power system such as, existing modes (eigenvalues) that are excited when the power system is disturbed. Appendix B describes eigenvalues, eigenvectors, and participation factors for a simple exciter. When performing linear analysis of a power system, it is linearized in the current operating point. To do so, the state matrix (the A-matrix, see section 2.5) has to be constructed. For a power system simulation software, a numerical process has to be implemented to obtain such an A-matrix. In reference [37] it is outlined how the A-matrix is constructed numerically in two power system simulation software. The eigenvalues of the A-matrix are then calculated and in this section different representations of the studied power system will be outlined. In section 7.2, linearizations of the studied power system are done. The following setups (representations) of the power system are studied: •

Simulation in phasor mode with a fixed series capacitor.

•

Simulation in instantaneous value mode with a fixed series capacitor.

•

Simulation in instantaneous value mode with the original TCSC-model.

•

Simulation in instantaneous value mode with the linear model.

In the following, complex pair of eigenvalues with real part greater than –40 are listed. Real-valued eigenvalues and complex eigenvalue-pairs with real part less than –40 will be omitted in the following tables.

71

72

Chapter 7. Comparisons in linear analysis

7.2 Linearizations of the studied power system Here follows linear analysis performed to the power system that was used in chapter 5 and 6. The system is linearized when it has reached steady-state, just before the load at Bus A is increased with 10% at t = 6 s, see figures 6.1 and 6.2.

7.2.1 Linear analysis of the system in phasor mode with a fixed series capacitor To get a fast overview of the system we start with a linearization of the power system in phasor mode24. This simulation mode is used when linearizing power systems to study their transient stability [32]. Phasor mode is used for studies of power flows in networks and changes of relative rotor angles between different machines. Voltages and currents are approximated to be sinusoidal and for instance, reactances are used in the calculations instead of inductances and capacitances for a line. Since we do not have any developed model of the TCSC in phasor mode we model it as a constant series capacitor, as in figure 6.2 when t < 4 s. The following complex pair of eigenvalues with real part greater than -40 are derived. Real-valued eigenvalues will not be listed in the tables of this chapter. Table 7.1: Complex eigenvalues with real part greater than -40 in phasor mode with the TCSC represented as a fixed series capacitor

λ

σ ± jω [1/s, Hz]25

Comment

λ1, 2

-1.32 +/- j1.14

Inter-area oscillation

λ3, 4

-17.05 +/- j0.28

λ5, 6

-35.33 +/- j0.03

λ7, 8

-3.95 +/- j0.18

λ9, 10

-1.42 +/- j0.24

λ11, 12

-0.54 +/- j0.06

λ13, 14

-2.54 +/- j0.59

The eigenvalue-pair λ1, 2 in table 7.1 is an inter-area oscillation. After a disturbance26, power will oscillate between the north and the south part of the power 24

Fundamental frequency mode Note that the imaginary part of the eigenvalues is given in Hz. 26 It should be noted that the disturbance mentioned here cannot be too large. 25

7.2 Linearizations of the studied power system

73

system with the frequency 1.14 Hz during the recovery of the power system. That oscillation can also be observed in time domain simulations, see all figures in chapter 6. Eigenvalues λ3 - λ12 are introduced when the synchronous machines are equipped with exciters and eigenvalue-pair λ13, 14 is introduced when the synchronous machines are equipped with turbines and governors. The less damped pair of the eigenvalues in table 7.1 is λ11, 12. However, the imaginary part of that eigenvalue-pair is very slow (0.06 Hz) and therefore such slow oscillation is impossible to detect in a time domain simulation.

7.2.2 Linear analysis of the system in instantaneous value mode with a fixed series capacitor In this section, linearization is done when simulating the power system in the instantaneous value mode, i.e. voltages and currents are represented with instantaneous values and for instance a line is represented by its inductance and capacitance. As in section 7.2.1 the TCSC is modeled as a constant series capacitor, as in figure 6.2 when t < 4 s. When linearizing the power system in instantaneous value mode the following complex pair of eigenvalues is derived. Table 7.2: Complex eigenvalues with real part greater than -40 in instantaneous value mode with the TCSC represented as a fixed series capacitor

λ

σ ± jω [1/s, Hz]

Comment

λ1, 2

-1.31 +/- j1.14

Inter-area oscillation

λ3, 4

-17.04 +/- j0.27

Already found in phasor mode

λ5, 6

-35.28 +/- j0.05

Already found in phasor mode

λ7, 8

-3.96 +/- j0.18

Already found in phasor mode

λ9, 10

-1.41 +/- j0.25

Already found in phasor mode

λ11, 12

-0.54 +/- j0.06

Already found in phasor mode

λ13, 14

-2.54 +/- j0.59

Already found in phasor mode

λ15, 16

-7.85 +/- j44.85

Subsynchronous resonance eigenvalues

λ17, 18

-9.33 +/- j75.12

Subsynchronous resonance eigenvalues

Eigenvalues λ1 - λ14 are recognized from table 7.1 in section 7.2.1.

74

Chapter 7. Comparisons in linear analysis

Eigenvalue-pairs λ15, 16 and λ17, 18 are eigenvalues not recognized from section 7.2.1. These are eigenvalues in the system that is a result of subsynchronous resonance. A fixed capacitor in series with the total circuit inductance of the transmission line (including generator and transformer leakage inductances) forms a series resonant circuit with the natural frequency fe:

fe =

1 2π

LC = f 0

XC

(7.1)

X

where XC is the series reactance of the series capacitor and X is the total reactance of the line at power frequency f0 [11]. Subsynchronous resonance cannot be detected in phasor mode and that is why it did not occur in section 7.2.1. The natural frequency fe creates the following imaginary parts for two eigenvalue-pairs:

fs = f0 ± fe

(7.2)

In the studied power system we get the following natural frequency by using equation (7.1):

XC

fe = f0

X

= f0

XC "

"

X line + X S1 + X S 2 "

(7.3)

"

The subtransient reactances, X S 1 and X S 2 , for synchronous machines S1 and S2 in equation (7.3) are given in Ω and are calculated as: "

"

X Ω = X p.u.

2

UN SN

(7.4)

where "

X p.u. = Subtransient reactance in d- or q-axis of the synchronous machine in per unit of machine ratings (XD" or XQ"). UN

= Rated voltage of the synchronous machine, kV.

SN

= Rated power of the synchronous machine, MVA.

In our synchronous machines the subtransient reactances are the same in d- and q-axis, i.e. XD" = XQ". With (7.4) in (7.3) we get:

7.2 Linearizations of the studied power system

75

XC

fe = f0

2

"

X line + X S1 p.u.

UN S 1 SN S1

(7.5)

2

"

+ X S 2 p.u .

UN S 2 SN S 2

and with system values from section 5.2, we obtain the following natural frequency with equation (7.5):

f e = 60

26.54 2

2

500 530 266 + 0.25 + 0.25 900 900

= 15.20 Hz

(7.6)

The natural frequency fe in equation (7.6) creates the following two eigenvaluepairs' imaginary parts by using equation (7.2):

f s = f 0 ± f e = 60 ± 15.20 = 44.80 and 75.20 Hz

(7.7)

These two frequencies can be found in the imaginary parts of the eigenvaluepairs λ15, 16 and λ17, 18, see table 7.2 above. After a disturbance in a time simulation using instantaneous value mode, the eigenvalue-pair λ15, 16 is possible to detect as a frequency oscillation on transmitted power from the north to the south with a frequency of 44.80 Hz.

7.2.3 Linear analysis of the system in instantaneous value mode with the original TCSC-model In this section, linearizations have been done in four different instants with the original TCSC-model. With one millisecond difference in time, four different setups of eigenvalues are derived as in table 7.3. Table 7.3: Complex eigenvalues with real part greater than -40 in instantaneous value mode with the original TCSC-model

λ

σ ± jω [1/s, Hz]

σ ± jω [1/s, Hz]

σ ± jω [1/s, Hz]

σ ± jω [1/s, Hz]

Comment

λ1, 2

-1.30 +/- j1.13

-1.30 +/- j1.13

-1.30 +/- j1.13

-1.30 +/- j1.13

IAO27

λ3, 4

-17.04 +/- j0.27

-17.04 +/- j0.27

-17.04 +/- j0.27

-17.04 +/- j0.27

AFPM28

λ5, 6

-35.28 +/- j0.04

-35.28 +/- j0.04

-35.28 +/- j0.04

-35.28 +/- j0.04

AFPM

λ7, 8

-3.93 +/- j0.18

-3.93 +/- j0.18

-3.93 +/- j0.18

-3.93 +/- j0.18

AFPM

λ9, 10

-1.41 +/- j0.25

-1.41 +/- j0.25

-1.41 +/- j0.25

-1.41 +/- j0.24

AFPM

27 28

t = 5.900 s

t = 5.901 s

t = 5.902 s

t = 5.903 s

IAO = Inter-Area Oscillation, see section 7.2.1. AFPM = Already Found in Phasor Mode, see section 7.2.1.

76

Chapter 7. Comparisons in linear analysis

λ11, 12

-0.56 +/- j0.06

-0.57 +/- j0.06

-0.56 +/- j0.06

-0.56 +/- j0.06

AFPM

λ13, 14

-2.54 +/- j0.59

-2.54 +/- j0.59

-2.54 +/- j0.59

-2.54 +/- j0.59

AFPM

λ15, 16

-0.22 +/- j158.65 -0.21 +/- j158.65 -0.22 +/- j158.65 -0.21 +/- j158.65

λ17, 18

-15.74 +/- j61.77 -15.75 +/- j61.77 -15.74 +/- j61.77 -15.79 +/- j61.77

λ19, 20

-24.09 +/- j60.00 -24.09 +/- j60.00 -24.09 +/- j60.00 -24.09 +/- j60.00

λ21, 22

-1.03 +/- j30.78

-1.01 +/- j30.78

-1.03 +/- j30.78

-0.98 +/- j30.78

λ23, 24

-1.39 +/- j0.39

-

-

-

OS29

λ25, 26

-2.89 +/- j0.48

-

-

-

OS

λ27, 28

-

+0.89 +/- j0.40

-

-

OS

λ29, 30

-

-5.73 +/- j0.38

-

-

OS

λ31, 32

-

-5.25 +/- j0.46

-

-

OS

λ33, 34

-

-

-3.29 +/- j0.50

-3.32 +/- j0.51

OS

λ35, 36

-

-

-1.60 +/- j0.34

-

OS

λ37, 38

-

-

-

-0.47 +/- j0.36

OS

The original TCSC-model gives different eigenvalues since the system topology changes with the control algorithm of the non-linear TCSC. The eigenvalues that differ between the linearizations are marked with bold in table 7.3. These belong to the non-linear power system component in the system. Eigenvalues λ1 - λ14 are recognized from earlier sections. Eigenvalue-pair λ19, 20 occur in every linearization and comes from the system's power frequency (60 Hz). Even eigenvalue-pair λ17, 18 comes from the system's power frequency. Eigenvalue-pairs λ15, 16 and λ21, 22 are obtained from the TCSC. Eigenvalues λ23 - λ38 are not permanent in the linearizations. Eigenvalue-pair λ33, 34 exist in two of the linearizations and the others exist in one of the linearizations. Eigenvalue-pair λ27, 28 is unstable since it has a positive real part. Table 7.3 shows how difficult it is to analyze the result from linear analysis of a power system containing a non-linear power system component. A number of linearizations have to be done to understand which of the eigenvalue-pairs that are generated from the non-linear power system component, in this case the TCSC. The inter-area oscillation can be seen as eigenvalue-pair λ1, 2.

29

OS = Occurs only in Some linearizations.

7.2 Linearizations of the studied power system

77

The subsynchronous resonance eigenvalues cannot be detected when the TCSC is modeled as a non-linear component.

7.2.4 Linear analysis of the system in instantaneous value mode and the linear TCSC-model When doing linear analysis of the power system containing the linear model, different linear analysis generates the same eigenvalues in steady-state, no matter in which millisecond the linear analysis is executed. As can be seen, five of the complex eigenvalue-pairs are generated from the linear model of the TCSC, see section 5.5, and the comments in table 7.4. Table 7.4: Complex eigenvalues in instantaneous value mode with the linear model of the TCSC

λ

σ ± jω [1/s, Hz]

Comment

λ1, 2

-1.25 +/- j1.14

Inter-area oscillation

λ3, 4

-17.04 +/- j0.27

Already found in phasor mode

λ5, 6

-35.29 +/- j0.05

Already found in phasor mode

λ7, 8

-3.96 +/- j0.18

Already found in phasor mode

λ9, 10

-1.41 +/- j0.25

Already found in phasor mode

λ11, 12

-0.54 +/- j0.06

Already found in phasor mode

λ13, 14

-2.54 +/- j0.59

Already found in phasor mode

λ15, 16

-9.79 +/- j3.25

∆uq/∆iq-relation from the linear model

λ17, 18

-12.04 +/- j2.19

∆ud/∆iq-relation from the linear model

λ19, 20

-14.51 +/- j4.17

∆uq/∆id-relation from the linear model

λ21, 22

-23.76 +/- j4.76

∆ud/∆id-relation from the linear model

λ23, 24

-37.20 +/- j15.39

∆uq/∆iq-relation from the linear model

λ25, 26

-24.09 +/- j60.00

λ27, 28

-17.45 +/- j60.09

Eigenvalues λ1 - λ14 are recognized from earlier sections. Eigenvalues λ15 - λ24 are eigenvalues introduced by the linear model, see section 5.5. These modes describe the fundamental behavior of the TCSC that was found in section 5.5 and this linearization is the only linearization that includes these modes.

78

Chapter 7. Comparisons in linear analysis

Eigenvalue-pairs λ25, 26 and λ27, 28 come from the system's power frequency (60 Hz). The system with the linear model contains the same eigenvalues, no matter when linear analysis is done in steady-state.

7.3 Conclusions of linear analysis of the power system In chapter 7, four linear analysis of the studied power system have been made in both phasor mode and instantaneous value mode. The conclusions are: •

In phasor mode the dynamics of the generators are found.

In instantaneous value mode, three representations of the TCSC were analyzed: •

When representing the TCSC as a fixed series capacitor, the modes for subsynchronous resonance were detected.

•

When representing the TCSC with the original TCSC-model, it was shown how eigenvalues existed in only a few of the linear analysis sessions of the power system. That shows how difficult it is to draw conclusions of one single linear analysis of a power system containing a non-linear power system component.

•

When representing the TCSC with the linear model, all eigenvalues exist, no matter when linear analysis was done in steady-state. With the linear model the fundamental behavior of the TCSC is included in the linearization.

Chapter 8

Conclusions and future work This chapter contains conclusions, future work, and personal comments regarding simulation.

8.1 Conclusions In simulation of power systems there is a need of linearizing non-linear components to speed up simulations and to make it possible to perform linear analysis on the whole power system. In the thesis the work has mainly been focused on building a linear model for the non-linear Thyristor-Controlled Series Capacitor (TCSC). The control algorithm of the TCSC is set to create a constant reactance and the linear model is valid for low frequencies. The thesis contains: •

A method to create a linear model of a non-linear power system component: the Thyristor-Controlled Series Capacitor.

•

Comparisons between the developed linear model and the original TCSCmodel, both for small-signal and relatively large-signal disturbances.

•

Comparisons of linearizations of a power system containing the linear model and a power system containing the original TCSC-model.

The conclusions are: •

Time domain comparisons show resemblance. Also in another operating point than the one used when the linear model was developed, the linear model shows resemblance with the original TCSC-model.

•

The linearizations show discrepancies, which is expected since the original TCSC-model, which is non-linear, generates different eigenvalues depending on the current operating point, i.e. the current setup of differential equations. Even in steady-state, the non-linear TCSC generates different eigenvalues from time to time. The linear model generates the same eigenvalues during the whole simulated time-interval.

79

80

Chapter 8. Conclusions and future work

Beside what have been presented in the thesis, two additional papers have been written, [37] and [39]. In [37] the linear analysis tool is evaluated for two power system simulation software. The two software use different methods to linearize the power system and therefore discrepancies occur. Even models of the synchronous machines differ between the software, which influence the result of linearizing the power system as well. In [39] it is documented how different models of synchronous machines influence the location of a power system's eigenvalues. The machine model is expanded from a classical machine model of 2nd order, in step by step to a 6th order model. It is shown how the damping constant D can be used for low-order models to obtain the same damping as for high-order models.

8.2 Future work In a work like this, several methods for providing linear models can be tested on several non-linear power system components valid for different frequency intervals. However, the following list describes more in detail, tasks that can be studied in a continuation of the project. •

Translate the linear TCSC-model to a fundamental frequency model. To convert the model from dq0-representation to +-0-representation, i.e. from instantaneous value mode to phasor mode.

•

Include the reference signal to the input vector of the linear model of the TCSC and build an expanded linear model. Later on, add a regulator that changes the wished value of the produced reactance of the TCSC. Until now that reference signal has been constant in the created linear TCSC-model. By doing this expansion of the model it will be possible to see how the linear TCSC-model can damp power oscillations.

•

Analyze differences in building linear models with the method described in this thesis and other methods in the literature.

•

Apply the method for different systems and other system components than the TCSC.

•

Continue the validation of the linear model. How valid is the created linear model for large perturbations?

•

Explore the limits for linear analysis with the linear model.

•

Add a Phase Locked Loop inside the reference system of the linear model, see section 5.6. Then it is possible to apply repeating, periodic disturbances when validating the linear model. The Phase Locked Loop will turn the current through the TCSC so that it is in steady-state of dcomponent character.

8.3 Personal comments on simulation of power systems

81

•

Improve the Newton-Raphson algorithm using Discrete Fourier Transform.

•

Prove whether the linear model is valid in other power systems than the power system that was used during the linearization.

8.3 Personal comments on simulation of power systems At the end of this thesis I would like to add some lines regarding simulation of power systems. My personal experience is that one should not give too much attention to one single simulation, linear analysis, or short-circuit calculation of a power system. Instead it is important to understand which assumptions are made in the used power system simulation software, which mathematical routines are used, how the used models are formulated, and to compare the results with (hopefully) real experiences within the studied system. Since approximations (or even guesses) often have been made when system parameters have been specified, results from simulations should not be analyzed with too many decimals. Not many simulations of power systems contain settings of the system components that have been measured or that are known from where they are produced. In the world of 'simulation of power systems' engineers believe in one software and take for sure that the provided results are correct. Other software must bring up the same results to prove that they are correct, no matter whether the used models or mathematical techniques are the same or not in the different software. The only way to understand behavior of power systems is to understand provided results from simulation tools or real experiences. During this project and the time before, I feel that I understand better and better what's behind graphs. That is the best any work can give, that you develop your understanding. Where no questions are raised, no knowledge is exchanged.

82

Appendix A

Newton-Raphson algorithm using Discrete Fourier Transform This appendix contains a description of a method that can be used to find eigenvalues in a time-simulated signal.

A.1 Introduction Within this project lots of time has been spent on trying to find an algorithm that can find eigenvalues in a time-simulated signal generated from a system's recovery after a disturbance. The aim is to apply the algorithm to a non-linear component as for instance a Thyristor-Controlled Series Capacitor (TCSC). This has been done very briefly and was not successful. In a continuation of the project, more time can be spent upon how to deal with real situations when a signal is constructed including noise. The idea is to identify eigenvalues and further on, a transfer function for a nonlinear power system component. By disturbing the component with a known input signal and analyzing the output signal, the eigenvalues of the component can be identified and a transfer function can later be built. The identification algorithm is using the Fast Fourier Transform-algorithm30 (FFT) and the Discrete Fourier Transform-algorithm (DFT) [22,26]. By subtracting the studied signal with suggestions of eigenvalues it contain, the eigenvalues can be found when correct damping, frequency, magnitude, and phase of each mode (oscillation) have been found.

A.2 Setup of the algorithm The algorithm starts with that an initial FFT is performed for the studied signal. In the frequency spectrum the number of peaks gives a first guess of how many eigenvalues the signal contains, N in equation (A.1) below. 30

Published 1965 by J. W. Cooley and J. W. Tukey.

83

84

Appendix A. Newton-Raphson algorithm using Discrete Fourier Transform

The studied signal f(t) is assumed to consist of the following terms, N

[

]

N

σ t σ t j (ϕ + 2π f k t ) fˆ (t ) = ∑ Ak e k Re e k = ∑ Ak e k cos(ϕ k + 2π f k t ) k =1

(A.1)

k =1

fˆ (t ) is an estimate of the correct f(t). σk in equation (A.1) is the damping of an eigenvalue, fk is the oscillation frequency of an eigenvalue, Ak is the initial magnitude of an eigenvalue, and ϕk is

the initial phase of eigenvalue.

The eigenvalues do not have to be very good isolated, they can be orientated relatively close in the complex plane, i.e. they can be close in both damping, σ, and frequency, f. The length of the studied time interval T sets the frequency resolution as,

∆f =

1 T

(A.2)

For an existing frequency in the studied signal, the discrete Fourier coefficients below and above that frequency are further analyzed. For instance, if ∆f is 1 [Hz] and 10.3 Hz exists in the signal, then the discrete Fourier coefficients for 10 and 11 Hz will be used, i.e. a10, b10, a11, and b11 in equations (A.3) and (A.4). FFT demands less computer operations than DFT, but in the case that we are not interested in all discrete Fourier coefficients it is more economical to use the DFT only for the coefficients that are needed. The expression of the Fourier coefficients of a function f(t) are given in equations (A.3) and (A.4)

2 an = T

a +T

2 bn = T

a +T

∫ f (t )cos nΩtdt

(n = 0,1,2,K)

(A.3)

(n = 1,2,3,K)

(A.4)

a

∫ f (t )sin nΩtdt a

where an and bn is the real and imaginary part of Fourier coefficient n, Ω=2π/T, and T is the length of the studied time interval [2].

A.3 The structure of the algorithm

85

The Fourier coefficients an and bn have the following relation to Discrete Fourier Transform (DFT)31:

 T M −1 nk  a n = Re[ X (n )] = Re 2 S ∑ f (k )W  T k =0 14 442444 3

(n = 0,1,2,K)

(A.5)

 T M −1 nk  bn = − Im[ X (n )] = − Im 2 S ∑ f (k )W  T k =0 14 442444 3

(n = 1,2,3,K)

(A.6)

= X (n )

= X (n )

where X(n) is the Discrete Fourier coefficient obtained by Discrete Fourier Transform and f(k) is f(t) sampled with the sample interval TS as

f (k ) = f (kTS )

W =e

− j 2π

(A.7)

M

(A.8)

and M is the number of samples in the studied time interval as

M =

T TS

(A.9)

In the following, every expression containing an integral is exchanged by using the corresponding DFT as above.

A.3 The structure of the algorithm The algorithm finds an eigenvalue for each of the N peaks from the initial frequency spectrum that is calculated with FFT. The estimation of the eigenvalues are improved by minimizing the difference between the 4N Fourier coefficients

aˆ n , bˆn , aˆ n+1 , and bˆn +1 , see below, and the real Fourier coefficients an, bn, an+1, and bn+1 calculated with equations (A.3) and (A.4) above. The initial values of fk and Ak are set according to the magnitudes of the initially calculated Fourier coefficients with FFT above. The damping is assumed to initially be σk = 1, and the phase is assumed to initially be ϕk = 0. 31

Here the DFT is defined as

X (n ) = 2

TS T

M −1

∑ f (k )W

nk

. The factor 2 makes a si-

k =0

nusoidal function with the magnitude 1 having the magnitude of the Discrete Fourier coefficient equal to 1 for exact that frequency. The factor TS/T can also be written as 1/M that can be found in some literature [26].

86

Appendix A. Newton-Raphson algorithm using Discrete Fourier Transform

aˆ n =

2 T

a +T

∫ a

  N σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) cos nΩtdt 1 1k =4 444244444 3

(A.10)

= fˆ (t )

2 bˆn = T

a +T

 N  σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) sin nΩtdt  k =1 

∫ a

2 T

a +T

2 bˆn +1 = T

a +T

aˆ n +1 =

∫ a

∫ a

(A.11)

 N  σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) cos(n + 1)Ωtdt  k =1 

(A.12)

  N σ t  ∑ Ak e k cos(ϕ k + 2π f k t ) sin (n + 1)Ωtdt   k =1

(A.13)

 f peak  .  ∆f 

for n = int 

All σk, fk, Ak, and ϕk in the right-hand sides of equations (A.10) – (A.13) are predicted values of the eigenvalues and will be improved as the algorithm continues. When the four Fourier coefficients,

bˆn = bn

aˆ n = a n

aˆ n+1 = a n +1

bˆn +1 = bn +1

(A.14)

for all N peak frequencies, the correct number of modes with parameters σk, fk, Ak, and ϕk have been found. To improve the estimates of the parameters σk, fk, Ak, and ϕk the NewtonRaphson method is used. Equations (A.10) – (A.13) are differentiated relative the predicted eigenvalue m, see σm, fm, Am, and ϕm below. Equations (A.10) – (A.13) differentiated relative σm become: a +T ∂aˆn σ t 2 = ∫ tAme m cos(ϕ m + 2π f mt )cos nΩtdt ∂σ m T a

(A.15)

a +T ∂bˆn 2 σ t = ∫ tAm e m cos(ϕ m + 2π f mt )sin nΩtdt ∂σ m T a

(A.16)

∂aˆ n+1 ∂σ m

=

2 T

a +T

σ mt

∫ tA e m

a

cos(ϕ m + 2π f mt )cos(n + 1)Ωtdt

(A.17)

A.3 The structure of the algorithm

∂bˆn+1 2 a+T σ t = ∫ tAm e m cos(ϕ m + 2π f mt )sin (n + 1)Ωtdt ∂σ m T a

87

(A.18)

Equations (A.10) – (A.13) differentiated relative fm become: a +T

∂aˆn

sin (ϕ m + 2π f mt )cos nΩtdt

(A.19)

a +T ∂bˆn 2 σ t = − ∫ 2πtAm e m sin (ϕ m + 2π f mt )sin nΩtdt ∂f m T a

(A.20)

2 =− ∂f m T

∂aˆn+1 ∂f m

=−

∫ 2πtA e

σ mt

m

a

a +T

2 T

σ mt

∫ 2πtA e m

sin (ϕ m + 2π f mt )cos(n + 1)Ωtdt

(A.21)

a

a +T ∂bˆn+1 σ t 2 = − ∫ 2πtAm e m sin (ϕ m + 2π f mt )sin(n + 1)Ωtdt ∂f m T a

(A.22)

Equations (A.10) – (A.13) differentiated relative Am become:

∂aˆ n

a +T

cos(ϕ m + 2π f mt )cos nΩtdt

(A.23)

∂bˆn 2 a+T σ mt = e cos(ϕ m + 2π f mt )sin nΩtdt ∂Am T ∫a

(A.24)

∂Am

=

2 T

∂aˆ n+1

∫e

σ mt

a

a +T

cos(ϕ m + 2π f mt )cos(n + 1)Ωtdt

(A.25)

∂bˆn+1 2 a+T σ mt = e cos(ϕ m + 2π f mt )sin (n + 1)Ωtdt ∂Am T ∫a

(A.26)

2 = ∂Am T

σ mt

∫e a

Equations (A.10) – (A.13) differentiated relative ϕm become:

∂aˆn

a +T

sin (ϕ m + 2π f mt )cos nΩtdt

(A.27)

a +T ∂bˆn 2 σ t = − ∫ Am e m sin (ϕ m + 2π f mt )sin nΩtdt ∂ϕ m T a

(A.28)

2 =− ∂ϕ m T

∂aˆn+1 ∂ϕ m

=−

2 T

σ mt

∫A e m

a

a +T

∫A e m

a

σ mt

sin (ϕ m + 2π f mt )cos(n + 1)Ωtdt

(A.29)

88

Appendix A. Newton-Raphson algorithm using Discrete Fourier Transform a+T ∂bˆn+1 σ t 2 = − ∫ Am e m sin (ϕ m + 2π f mt )sin (n + 1)Ωtdt ∂ϕ m T a

(A.30)

By Taylor developing up to the first order for each calculation of Fourier coefficients, we can come closer to the correct values of an, bn, an+1, and bn+1. For iteration υ we can almost get to correct values, υ

υ

υ

N  N  N  N  ∂a  ∂a  ∂a  ∂a υ υ υ υ an ≈ an + ∑  n  ∆σ k + ∑  n  ∆Ak + ∑  n  ∆f k + ∑  n  k =1  ∂σ k  k =1  ∂Ak  k =1  ∂f k  k =1  ∂ϕ k N  ∂b υ bn ≈ bn + ∑  n k =1  ∂σ k

an +1

υ

N   ∂b  ∆σ υ + ∑  n k   ∂A k = 1   k

υ

υ

υ

  ∆ϕ υ k  

N  N   ∂b  ∂b  ∆Aυ + ∑  n  ∆f υ + ∑  n k k     ∂ϕ ∂ f k k = 1 = 1   k  k

υ

  ∆ϕ υ k  

υ

υ

υ

υ

υ

υ

υ

υ

N  N  N   ∂a  ∂a  ∂a  ∂a  υ υ υ υ ≈ an +1 + ∑  n +1  ∆σ k + ∑  n +1  ∆Ak + ∑  n+1  ∆f k + ∑  n +1  ∆ϕ k ∂ σ ∂ ∂ ∂ ϕ A f k =1  k =1  k =1  k =1  k  k  k  k 

υ

N

N  N  N  N  ∂b  ∂b  ∂b  ∂b  υ υ υ υ υ bn+1 ≈ bn +1 + ∑  n +1  ∆σ k + ∑  n +1  ∆Ak + ∑  n +1  ∆f k + ∑  n+1  ∆ϕ k σ ∂ ϕ ∂ ∂ ∂ A f k =1  k =1  k =1  k =1  k  k  k  k 

(A.31) (A.32) (A.33) . (A.34)

Above and in the following the sign ^ has been replaced by the index υ . When formulating the errors between the correct Fourier coefficients and the Fourier coefficients calculated from the estimated parameters σm, fm, Am, and ϕm, we get υ

υ

υ

υ

∆an = an − an

(A.35)

∆bn = bn − bn

(A.36)

υ

υ

υ

υ

∆an +1 = an +1 − an +1

(A.37)

∆bn +1 = bn +1 − bn +1

(A.38)

Now we can put up the Newton-Raphson algorithm by using equations (A.35) – (A.38) in equations (A.31) – (A.34): υ

N  ∂a υ ∆an ≈ ∑  n  k =1  ∂σ k

N   ∂a  ∆σ υ + ∑  n k   k =1  ∂Ak 

N  ∂b υ ∆bn ≈ ∑  n  ∂ k =1  σ k

N   ∂b  ∆σ υ + ∑  n k   ∂ k =1  Ak 

υ

υ

υ

υ

  ∆ϕ υ k  

υ

υ

  ∆ϕ υ k  

N  N   ∂a  ∂a  ∆Aυ + ∑  n  ∆f υ + ∑  n k k     k =1  ∂f k  k =1  ∂ϕ k  N  N   ∂b  ∂b  ∆Aυ + ∑  n  ∆f υ + ∑  n k k     ∂ ∂ f k =1  k =1  ϕ k k  

υ

υ

υ

(A.39)

υ

(A.40) υ

N  N  N  N  ∂a  ∂a  ∂a  ∂a  υ υ υ υ υ ∆a n +1 ≈ ∑  n +1  ∆σ k + ∑  n +1  ∆Ak + ∑  n+1  ∆f k + ∑  n +1  ∆ϕ k         k =1  ∂σ k  k =1  ∂Ak  k =1  ∂f k  k =1  ∂ϕ k 

(A.41)

A.3 The structure of the algorithm υ

89

υ

υ

υ

N  N  N  N  ∂b  ∂b  ∂b  ∂b  υ υ υ υ υ ∆bn +1 ≈ ∑  n +1  ∆σ k + ∑  n+1  ∆Ak + ∑  n +1  ∆f k + ∑  n +1  ∆ϕ k         k =1  ∂σ k  k =1  ∂Ak  k =1  ∂f k  k =1  ∂ϕ k 

(A.42)

In matrix-form we get:

 ∂a υ  ∆aν   ∂σ  ν= υ  ∆b   ∂b  ∂σ 

υ

∂a ∂A υ ∂b ∂A

υ

∂a ∂f υ ∂b ∂f

ν υ ∂a  ∆σ   ν  ∂ϕ   ∆A  υ ∂b   ∆f ν    ∂ϕ   ∆ϕ ν 

(A.43)

The matrix containing the partial derivatives is a Jacobian-matrix. ∆aυ and ∆bυ are vectors of dimension [2Nx1]:

 ∆aυ   υn   ∆an +1  υ ∆a =  M   υ   ∆am  ∆aυ   m +1 

 ∆bυ   υn   ∆bn +1  υ ∆b =  M   υ   ∆bm  ∆bυ   m +1 

(A.44)

∆συ, ∆Aυ, ∆fυ, and ∆ϕυ are vectors of dimension [Nx1]:

 ∆σ υ   1 υ ∆σ =  M  ∆σ υ   N

 ∆Aυ   1 υ ∆A =  M  ∆Aυ   N

(A.45)

∆f υ   1  υ ∆f =  M  ∆f υ   N

 ∆ϕ υ   1 υ ∆ϕ =  M  ∆ϕ υ   N

(A.46)

One of the sub-matrices of the Jacobian-matrix in (A.43) looks like:

90

Appendix A. Newton-Raphson algorithm using Discrete Fourier Transform   ∂a υ  n   ∂σ 1   υ   ∂an +1    ∂σ  υ 1  ∂a  = M υ ∂σ   ∂a  m       ∂σ 1  υ   ∂am +1   ∂σ 1   

L L L L L

υ  ∂an       ∂σ    N υ  ∂an +1      ∂σ   N    M  υ  ∂am      ∂σ    N  υ  ∂am +1      ∂σ   N   

(A.47)

Equation (A.43) inverted gives the errors in the estimation of the parameters:

 ∆σ ν   ∂a υ  ν   ∆A  =  ∂σ  ∆f ν   ∂b υ  ν   ∆ϕ   ∂σ

υ

∂a ∂A υ ∂b ∂A

υ

∂a ∂f υ ∂b ∂f

−1

υ ∂a  ν  ∂ϕ  ∆a    υ ∂b   ∆bν  ∂ϕ 

(A.48)

As each iteration improves the parameters for each eigenvalue, the errors in the left-hand side in equation (A.48) will decrease. The iteration stops when the errors are smaller than a specified tolerance. When the errors are smaller than a specified tolerance a new FFT is made on the difference between the original signal and the estimated one with the calculated parameters, to check if the original signal contains any more eigenvalues, see example in section A.6 below.

A.4 Calculation order of the Newton-Raphson algorithm using Discrete Fourier Transform 1.

Calculate an FFT of the original f(t) and predict starting values of σm, fm, Am, and ϕm for the N peaks in the FFT-calculation.

2.

Calculate (new) values of all four aυ and bυ, for the N assumed eigenvalues.

3.

Calculate the errors ∆aυ = a - aυ respectively ∆bυ = b - bυ.

4.

Calculate the elements of the Jacobian-matrix:

 ∂a υ   ∂σ υ  ∂b  ∂σ 

υ

∂a ∂A υ ∂b ∂A

υ

∂a ∂f υ ∂b ∂f

υ ∂a   ∂ϕ  υ . ∂b  ∂ϕ 

A.5 Comments about using the algorithm

91

5.

∆σ ν   ∂a υ  ν  ∆A ∂σ Calculate  ν  =   ∆f   ∂b υ  ν   ∆ϕ   ∂σ

6.

Update new values of the parameters: υ +1

σ

k

υ +1

A

f

ϕ 7.

k

υ +1 k

υ +1 k

υ

= σ + ∆σ k

k

υ

k

k

υ

υ

= f + ∆f k

k

υ

υ

k

k

= ϕ + ∆ϕ

υ

∂a ∂f υ ∂b ∂f

−1

υ ∂a  ν  ∂ϕ  ∆a    υ ∂b  ∆bν  ∂ϕ 

υ

υ

= A + ∆A

υ

∂a ∂A υ ∂b ∂A

ν

Return to point 2 above and interrupt the routine when ∆σ k < ε σ , ν

ν

ν

∆Ak < ε A , ∆f k < ε f , and ∆ϕ k < ε ϕ ∨k = 1, K , N , then go to point 8 below. 8.

Calculate an FFT of the difference between the predicted fˆ (t ) and the original f(t). Is the magnitude of any of the Fourier coefficients larger than a specified ε? In that case, increase N with the number of found peaks among the Fourier coefficients and go to point 2 above. Otherwise, exit the algorithm.

A.5 Comments about using the algorithm If a second eigenvalue are close in frequency to an already predicted eigenvalue, the Fourier coefficients, close to the first frequency, are used to find the second eigenvalue. Only the upper half plane is used when searching for the eigenvalues, i.e. only positive frequencies are considered. When an eigenvalue is expected to be real-valued, two rows and two columns in the Jacobian-matrix are omitted since no phase and no frequency is estimated for a real-valued eigenvalue.

92

Appendix A. Newton-Raphson algorithm using Discrete Fourier Transform

A.6 An example when running the algorithm In the following a signal contains nine signal-components (eigenvalues). Table A.1: Signal-components 1

2

3

4

5

6

7

8

9

σ [1/s]

-3.6 -2

-2.05

-2

-3

-3

-8

-8

-6

A [p.u.]

1

0.5

0.5

0.42

0.5

1

0.11

0.4

0.2

ϕ [degrees]

-179 90

90

-90

20

10

85

12

24

f [Hz]

10.4 30.3

50.9

70.9

20.5 22.9

70.9

30.2

50

9

f (t ) = ∑ Ak e

σ kt

k =1

[

j (ϕ k + 2π f k t )

0.2

0.3

Re e

]= ∑ A e 9

k =1

σ kt

k

cos(ϕ k + 2π f k t )

(A.49)

4 3 2

f(t)

1 0 -1 -2 -3 -4 0

0.1

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s) Figure A.1. A test signal containing nine modes

By using the algorithm described in appendix A we get the following result in point 8, see section A.4, of eigenvalues that the signal contains. The algorithm reaches in this example point 8 four times.

A.6 An example when running the algorithm

93

100 80 60 40

w (Hz)

20 0 -20 -40 -60 -80 -100 -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Sigma (1/sec) Figure A.2. The algorithm has found 6 eigenvalues; the others are still “hidden” 100 80 60

w (Hz)

40 20 0 -20 -40 -60 -80 -100 -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Sigma (1/sec) Figure A.3. The algorithm has found 7 eigenvalues; the others are still “hidden”

94

Appendix A. Newton-Raphson algorithm using Discrete Fourier Transform

100 80 60

w (Hz)

40 20 0 -20 -40 -60 -80 -100 -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Sigma (1/sec) Figure A.4. The algorithm has found 8 eigenvalues; the last eigenvalue is still “hidden”

100 80 60

w (Hz)

40 20 0 -20 -40 -60 -80 -100 -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

Sigma (1/sec) Figure A.5. The algorithm has found all 9 eigenvalues

0

Appendix B

Eigenvalues, eigenvectors, and participation factors in a linearized system This appendix contains a description of eigenvalues, eigenvectors, and participation factors.

B.1 Eigenvalues By disturbing the system in figure B.1 with small deviations in both the state vector ∆x and the input vector ∆u, the linearized form of the dynamic system can be written as,

∆u

∆x

∆y

Figure B.1. A dynamic system

∆x& = A∆x + B∆u

(B.1)

∆y = C∆x + D∆u

(B.2)

A in equation (B.1) is the state matrix of size [nxn] and the poles of the dynamic system are the roots of the equation,

det( sI − A ) = 0

(B.3)

The values of s which satisfy equation (B.3) are known as eigenvalues32, λ, of the matrix A, and equation (B.3) is referred to as the characteristic equation of matrix A, see [1], page 707.

32

The prefix eigen has been borrowed from German to English.

95

96

Appendix B. Eigenvalues, eigenvectors, and participation factors in a linearized system

The number of eigenvalues is always the same as the dimension of matrix A. Some of them can be multiple. The eigenvalues can be complex and therefore it is useful to depict them in the complex plane.

B.2 Eigenvectors For every eigenvalue it exists two eigenvectors, right and left eigenvector. For any eigenvalue λi, the column vector of dimension n, Φi, which satisfies equation (B.4) is called the right eigenvector of A , see [1], page 707.

AΦ i = λi Φ i

(B.4)

The right eigenvector determines the distribution of mode33 i among the components of the state vector (the mode shape), see [27], page 5. For any eigenvalue λi, the column vector of dimension n, Ψi, which satisfies equation (B.5) is called the left eigenvector of A .

Ψ Ti A = λi Ψ Ti

(B.5)

In the linearization example in section B.3 both eigenvectors are calculated. The eigenvectors can be normalized so that the product of them will be:

Ψ Ti Φ i = 1

(B.6)

Note that the product of a left and right eigenvector that belong to different eigenvalues is 0, i.e.: T

Ψ j Φi = 0

for

i≠ j

(B.7)

See section B.4 how participation factors can be constructed from the eigenvalues.

B.3 An example of a linearization Below follows one example on how a simple regulator is linearized [31]. The regulator has four possible operating points. One operating point wherein none of the limited block functions are limited, two operating points wherein either one of them are limited and one where both of them are limited. In the following linearization of the regulator in figure B.2, both limiters are unlimited, i.e. the linearization takes place in an equilibrium point when the limiters are not activated.

33

Modes are similar to the eigenvalues of the system.

B.3 An example of a linearization

97 Max1

u

x1 -

1 1 + sTR

+

Σ

+

Σ

-

KA 1 + sTA

x2

1 KE + sTE

x3

y

Min1 Max2

x4

VREF

sKF 1 + sTF

Min2 Figure B.2. A regulator in unlimited operation, i.e. Min1 < x2 < Max1 and Min2 < x4 < Max2

The system contains of four state variables, i.e. four variables that has to be differentiated within a time simulation. When applying expressions (B.1) and (B.2) on the regulator the following linear form appear,  1 −T  R  ∆x&1   K A ∆x&  −  2  =  TA  ∆x&3      0 ∆x&4    0 

0 1 TA 1 TE KF

0

−

TETF

0 − −

KE

TE KE KF TETF

 0   1 K A   ∆x1    − T   TA  ∆x2   R   +  0  ∆u  ∆x  0  3   0   ∆x     4   0  1 −  TF 

 ∆x1  ∆x  2 ∆y = [0 0 1 0] + [0]∆u  ∆x3    ∆x4 

(B.9)

By assuming realistic values of the constants in the regulator,

K A = 200.0

KE = 1.0

KF = 0.05

TA = 0.89

TE = 0.80

TF = 0.60

TR = 0.02 expression (B.8) becomes:

(B.8)

98

Appendix B. Eigenvalues, eigenvectors, and participation factors in a linearized system 0 0 0  ∆x&1   − 50   ∆x1  50  ∆x&     0 − 224.72 ∆x2   0   2  =  − 224.72 − 1.12 + ∆u  ∆x&3     ∆x3   0  0 1.25 − 1.25 0  &       0 0.10 − 0.10 − 1.67  ∆x4   0   ∆x4  

(B.10)

By using equation (B.3)

det( sI − A ) = 0

(B.11)

the eigenvalues of A can be calculated by identifying the values of s that satisfies equation (B.11). The eigenvalues in the system will be,

λ1 = −50.0

(= −1 TR )

(B.12)

λ 2 = −0.0823

(B.13)

λ 3, 4 = −19790 . ± j 0.7882

(B.14)

The real part of the eigenvalues are given in sec-1 and the imaginary part of eigenvalues λ 3, 4 are given in Hz. Together with each eigenvalue the right and left eigenvectors are calculated, see below. The eigenvectors are normalized so that the product of the left and right eigenvector is equal to 1, see equation (B.6). The right eigenvectors will be, see equation (B.4):

 1   4.66   Φ1 =  −0.12    −0.01

 0   − 9.33  Φ2 =  − 9.99    0.04 

(B.15)

 0.00∠0 o    21.45∠77 o   Φ3 = 5.35∠ − 21o   o   0.48∠ − 3 

 0.00∠0 o    21.45∠ − 77 o   Φ4 =  5.35∠21o    o  0.48∠3 

(B.16)

The left eigenvectors will be, see equation (B.5):

Ψ1T = [1 0.00 0.00 0.00]

(B.17)

ΨT2 = [0.03 −0.01 −0.09 1]

(B.18)

Ψ T3 = [0.10∠88° 0.02∠ − 86° 0.02∠82° 1∠0°]

(B.19)

Ψ T4 = [0.10∠ − 88° 0.02∠86° 0.02∠ − 82° 1∠0°]

(B.20)

B.4 Participation factors

99

See [31] for a more comprehensive development of the linear equations (B.8) and (B.9) of the regulator.

B.4 Participation factors By multiplying the right and left eigenvectors for each eigenvalue, element by element, we can calculate the participation factors of the linearized system. In the following this is done for eigenvectors corresponding to each eigenvalue.  p11   Φ 11 Ψ11   1*1  1  p  Φ Ψ   4.66 * 0  0 =  p 1 =  21  =  21 12  =   p 31  Φ 31 Ψ13  − 0.12 * 0 0          p 41  Φ 41 Ψ14   − 0.01* 0  0

(B.21)

0 * 0.03  p12   Φ 12 Ψ21     0   p  Φ Ψ   − 9.33 * (− 0.01) 0.07  =  p 2 =  22  =  22 22  =   p 32   Φ 32 Ψ23  − 9.99 * (− 0.09 ) 0.89         0.04 *1  0.04  p 42  Φ 42 Ψ24  

(B.22)

 p13   Φ 13 Ψ31   p  Φ Ψ  p 3 =  23  =  23 32  =  p 33  Φ 33 Ψ33       p 43  Φ 43 Ψ34 

o    0  0 * 0.10∠88  0.47∠ − 9 o  o o 21.45∠77 * 0.02∠ − 86  =    5.35∠ − 21o * 0.02∠82 o   0.11∠61o     o o 0.48∠ − 3 *1   0.48∠ − 3 

 p14   Φ 14 Ψ41   p  Φ Ψ  p 4 =  24  =  24 42  =  p 34  Φ 34 Ψ43       p 44  Φ 44 Ψ44 

o    0  0 * 0.10∠ − 88   0.47∠9 o  o o 21.45∠ − 77 * 0.02∠86  =    5.35∠21o * 0.02∠ − 82 o  0.11∠ − 61o     o  o 0.48∠3 * 1   0.48∠3  

(B.23)

(B.24)

The participating columns (B.21) – (B.24) are combined to one participating matrix. The resulting participation matrix contains all participation factors as:

P = [p 1

p2

p3

0 0 1 0 0.07 0.47∠ − 9 o p4 ]=  o 0 0.89 0.11∠61  o 0 0.04 0.48∠ − 3

 0.47∠9  (B.25) o 0.11∠ − 61  o  0.48∠3  0

o

The first column shows how mode 1 involves each state variable, i.e. only state variable x1 is influenced by (or participates in) λ1.

100

Appendix B. Eigenvalues, eigenvectors, and participation factors in a linearized system

The second column shows how mode 2 involves each state variable, i.e. mainly state variable x3 is influenced by (or participates in) λ2 ( p32 = 0.89 ). The sum of each row and column in a participation matrix is equal to 1. The following matrix explains how a participation matrix should be understood.

 x :  1 P =  x2 :   x3 : x :  4

λ1

λ2

λ3

p11 p 21 p 31 p 41

p12 p 22 p 32 p 42

p13 p 23 p 33 p 43

λ4 

p14  p 24   p 34  p 44 

(B.26)

For each mode λ1, λ2, λ3, and λ4, the participation matrix shows how each state variable is involved.

Appendix C

Implementation of the TCSC-control This appendix contains a detailed description of the implemented original TCSC-model.

C.1 Structure of the TCSC-control The series capacitor C and the series reactor L, sketched in figure 4.1, are modeled using the corresponding standard components, in this case the power system simulation software Simpow [12,20]. The rest of the three-phase TCSC (the thyristors and the control algorithms) are split up in three parts, one for each phase. Such a part is implemented as a userdefined system that has four underlying functions34: Phase Locked Loop (PLL), Booster (BOO), Thyristor Pulse Generator (TPG), and Thyristors (THY).

PLL

BOO

TPG

+ uC A − iA

Figure C.1. Control system of the TCSC

34

In the used programming language, functions are called processes.

101

102

Appendix C. Implementation of the TCSC

The implementation of the TCSC follows the Synchronous Voltage Reversalcontrol algorithm35 used by ABB [9]. The system and functions are formulated by using the in-built simulation language Dynamic Simulation Language (DSL) in Simpow. The name of the system is TCSC and such a system is modeling one phase. The aim of the total control algorithm is to calculate in what point of time the thyristors should start conducting and automatically block them at the following zero-crossing of the phase current through the series reactor. The thyristors are included in the here sketched algorithm. Currents and voltages in the following sections are phase currents and phase voltages. The input parameters to the TCSC are given in table C.1. Table C.1: The input parameters to the TCSC-system

PARAMETER

VALUE

FUNCTION

Lambda

2.5

Lambda explains how the TCSC is tuned. 2.5 means that the oscillation frequency of the TCSC is 2.5 times the fundamental frequency. In this case 2.5*60 = 150 [Hz].

KPLL

18.8496 [rad/s/rad]

A proportional gain in PLL.

TPLL

0.3 [s]

Time constant in PLL.

Fas

1

What phase the TCSC represents.

Capac

1.999E-4 [F]

The capacitance of the series capacitor, input parameter to the BOO and the TPG.

Ref

2 [p.u.]

The reference, the value of the fundamental reactance that the TCSC will create. When Ref=2, the fundamental reactance between node A and node B will be two times the fundamental reactance of the series capacitor (=1 p.u.). Ref is an input parameter to BOO.

35

Synchronous Voltage Reversal, SVR.

C.1 Structure of the TCSC

103

TAU1 36

6.366E-3 [s]

Filter time in PLL and BOO for phasors.

TAU2 37

10.61E-3 [s]

Filter time in PLL and BOO for average.

TBOO

0.15 [s]

Time constant in BOO.

KBOO

0.1 [rad/p.u.]

A proportional gain in BOO.

Freeze

0 [s]

Freeze the TCSC at a specific point of time. If Freeze is 0, the TCSC is never frozen.

TimeOn

1 [s]

Point of time when the TCSC should be activated.

C.1.1 Phase Locked Loop Into the Phase Locked Loop-function, the PLL-function, the phase current I is sent, see figure below. E

A

+U C −

I

B

D

→ IL Figure C.2. Basic scheme of the Thyristor-Controlled Series Capacitor (TCSC)

The results of the PLL-function are the following: Whether the Thyristor Pulse Generator (TPG) should start calculating the point of time when the forward thyristor should start conducting (COUNTFOR=1 in figure C.3), whether the TPG should start calculating the point of time when the reverse-thyristor should start conducting (COUNTBACK=1 in figure C.3), the complex current X5, and the angle TETAPLL, see figure C.3. Both the output signals COUNTFOR and COUNTBACK are integers.

36 37

3 dB cutoff frequency = 1/TAU1. 3 dB cutoff frequency = 1/TAU2.

104

Appendix C. Implementation of the TCSC

Also a first prediction of the time points when the next zero-crossing of the phase voltage drop over the capacitor will take place, point of time A respectively B, see figure 4.3, are calculated in the PLL-function. Expression (C.1) is the formula for the prediction of time A, TIMEFOR. Expression (C.2) is the formula for the prediction of time B, TIMEBACK. At these points of time the current through the reactor is at its maximum and its minimum respectively, see figure 4.3. These first predictions are later improved in the BOO-function; see section C.1.2. The output signals COUNTFOR, COUNTBACK, TIMEFOR, and TIMEBACK are calculated only if simulated time is greater than TimeOn, i.e. TIME > TimeOn.

2

CURRENT

CURRENT

+

X1

K

X2

+

-

1 1+sT

-j*TETAPLL X4

X3

e

-

TAU1

Parameter

X7

X5

-j*TETAPLL

X6

e

conj

Re[X ] X8

+

1 1+sT

X9

Parameter

Parameter

X

1 sT

ARG

Parameter

X11

+

TAU2

KPLL X12

K

X13

+

TPLL

+

OMEGAPLL

1 s

TETASEED

+ Parameter

2*PI*F

COUNTBACK

TETAPLL

Mod 2PI

X10

COUNTBACK

TETAPLL time

COUNTFOR

COUNTFOR

Figure C.3. Block diagram of the PLL

In figure C.3 the bold-marked signals X3, X4, X5, X6, and X7 are complexvalued.

C.1 Structure of the TCSC

105

The block diagram with the two output signals COUNTFOR and COUNTBACK is working as follows, If the signal TETAPLL is in the interval [π(λ-1)/λ < TETAPLL < π] then the integer output signal COUNTFOR is equal to 1. With λ = 2.5 the interval becomes [1.88 < TETAPLL < 3.14], see figure C.4. If the signal TETAPLL is in the interval [-π/4 < TETAPLL < 0] then the output signal COUNTBACK is equal 1, see figure C.4. 5

TETAPLL, COUNTFOR, COUNTBACK

TETAPLL 4

3

COUNTBACK

COUNTFOR

2

1

0

-1

-2 1.2

1.21

1.22

1.23

1.24

1.25

Time (s)

Figure C.4. The signal TETAPLL and the integer signals COUNTFOR and COUNTBACK are calculated inside the PLL

TETAPLL is varying in the interval [-π/2 ≤ TETAPLL < 3π/2]. If the integer output COUNTFOR = 1, then the prediction of next instant A, see figure 4.3, is calculated with the formula:

TIMEFOR = TIME +

π − TETAPLL 2πf 0

(C.1)

where TIME is actual simulated time and fo is the fundamental frequency of the power system. If the integer output COUNTBACK = 1, then the prediction of next time B, see figure 4.3, is calculated with the formula:

106

Appendix C. Implementation of the TCSC

TIMEBACK = TIME −

TETAPLL 2πf 0

(C.2)

Expression (C.1) and (C.2) are included in the PLL but are not viewed in figure C.3. The predictions of the time points TIMEFOR and TIMEBACK are improved in the BOO-function. The output signals COUNTFOR, COUNTBACK, TIMEFOR, and TIMEBACK are calculated only if simulated time is greater than TimeOn, i.e. TIME > TimeOn.

C.1.2 Booster Into the Booster-function, BOO, among other input signals, the series capacitor voltage UC is sent.

E

A

I

+U C −

B

D

→ IL Figure C.5. Basic scheme of the Thyristor-Controlled Series Capacitor (TCSC)

The results from BOO are the boosted points of time for the next two zerocrossings of the series capacitor voltage, TZFORNEW and TZBACKNEW, see figure C.6. TZFORNEW is the adjusted point of time A and TZBACKNEW is the adjusted point of time B in figure 4.3. The Booster is therefore boosting (improving) the predicted points of time TIMEFOR and TIMEBACK from the PLL, see section C.1.1. The BOO-function is in stand-by mode as long as TIME0). If FREEZE is specified, the BOO-function will at that moment stop calculating new values of adjustments from the PLL predictions of TIMEFOR and TIMEBACK. In other words, when TIME > FREEZE, the signal X27 in the block diagram, see figure C.6, will be frozen. If the input parameter FREEZE is not specified then it is set to the default value 0 and in that case the BOO-function is active through the whole simulation. TETAPLL

TETAPLL 2

+

VOLTAGE

VOLTAGE

X1

K

X2

+

X3

-

-j*TETAPLL

e

-

Parameter

X7

1 1+sT

X4

X5

TAU1

-j*TETAPLL

e

X6

conj

Re[X ] X8

+

1 1+sT

X9

Parameter

X10

TAU2

Ref

X5FROMPLL

2

Z_pu

*

*

OMEGA

Parameter 1 sT

Parameter

TBOO

X24

Im[X +]

X_Ohm

:

X22

+

-

X23

Boos t f actor

-1

TWOPI Frequency

+

UBASE 2 Z_Ohm SBASE

CAPAC_FARAD

CAPAC F

:

X5_FROM_PLL

:

PROD

X21

KBOO X25

K

X26

:

X27

+

TIMEBACK

+

TIMEBACK

TZBACKNEW

TZBACKNEW

TIMEFOR

TIMEFOR +

TZFORNEW

TZFORNEW

Figure C.6. Block diagram of the BOO

In the block diagram in figure C.6 the signals X3, X4, X5, X6, X7, X5_FROM_PLL, Z_pu, and Z_Ohm are complex-valued signals.

C.1.3 Thyristor Pulse Generator The aim of this function is to calculate the points of time when the thyristors should start conducting. Pulse signals are generated to fire each thyristor

108

Appendix C. Implementation of the TCSC

(FireFor and FireBack) in such a moment. Such a signal is set when the forward- respectively the reverse-thyristor should start to conduct. As long as TIME < TimeOn, FireFor and FireBack are both equal to 0. Input signals to the function TPG are: TZFORNEW, TZBACKNEW, the phase voltage drop over the series capacitor UC (VOLTAGE in figure C.7), the line phase current I (CURRENT in figure C.7), Lambda, and the capacitance of the series capacitor (CAPAC). BlowBack and BlowFor are also input signals but these are not viewed in figure C.7, see the text below figure C.7. When the calculated conducting time TFFOR or TFBACK, see the lower right part of figure C.7, are equal actual time, the output signals FireFor and FireBack are set equal to 1. These output signals are input signals to the fourth DSL-function THY and they are commands to the thyristors to start conducting. Once a thyristor has start conducting it will conduct until next zero-crossing of the current. TZFORNEW

TZFORNEW +

Y3

TIME

TIME

*

Y4

+

UZFOR

UZFOR2

UBASE

+ CURRENT

CURRENT

: 2

UBASE Y1 SBASE

CAPAC_FARAD

CAPAC

VOLTAGE

2

F

TZBACKNEW

UZFOR2 IL X0

*

TWOPI

*

Frequency

OMEGA

+

IL UZBACK2

*

Y5

Y8

+

UZBACK

U

UBASE

+

IL BF: U = IL*X0*(Lambda*BF-tan(Lambda*BF))

BF

X0

:

-

Y9

+

X0 IL BK: U = IL*X0*(Lambda*BK-tan(Lambda*BK)) U

Y7

:

X0

+

BK

:

Y10

-

UZBACK2

TZFORNEW TFFOR

TIME X0

*

Y6

SBASE IL UBASE

VOLTAGE

TZBACKNEW

1 Lambda

Lambda

Y2

a=b

FireFor

FireFor

a=b

FireBack

FireBack

TIME

TZBACKNEW TFBACK

Figure C.7. Block diagram of the TPG

FireFor and FireBack are equal to 1 as long as the corresponding thyristor is conducting.

C.1 Structure of the TCSC

109

FireFor and FireBack remains 1 until they are reset by the signals BlowFor and BlowBack generated in the function THY. When BlowFor or BlowBack is equal to 1 the corresponding thyristors are blocked. In figure C.7, the block diagram

U IL BF: U = IL*X0*(Lambda*BF-tan(Lambda*BF)) X0

solves a value of BF in the implicit equation,

U = IL * X 0 * (λ * BF − tan (λ * BF ))

(C.3)

The output signal from that block diagram is BF. This is done both for forward values, that is BF, and reverse values, that is BK, see figure C.7.

C.1.4 Thyristors Into the Thyristor-function, THY, the signals FireFor and FireBack are input signals. Those are the signals that command the thyristors to start to conduct. The thyristors will be conducting until next zero-crossing of the current through the thyristors. When the current through the thyristor is equal 0 the signals BlowBack resp. BlowFor is set equal 1. BlowFor and BlowBack are input signals to the TPG and when they are equal 1 it resets FireFor and FireBack in the TPG.

110

Appendix C. Implementation of the TCSC

C.2 The rhythm of the thyristors As mentioned in the beginning of section C.1, the TCSC is split in three parts, one for each phase. In total the thyristors will turn on within one period in the following order: Table C.2: The order of the thyristor switches

Phase

Thyristor

A

Forward

C

Reverse

B

Forward

A

Reverse

C

Forward

B

Reverse

Every thyristor is automatically turned off when the current through it has its zero-crossing, see figure 4.3.

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