PhD Dissertation - Power Systems Engineering Research Center

COMPREHENSIVE DISTRIBUTION POWER FLOW: MODELING, FORMULATION, SOLUTION ALGORITHMS AND ANALYSIS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by Ray Daniel Zimmerman January 1995

© Ray Daniel Zimmerman 1995 ALL RIGHTS RESERVED

COMPREHENSIVE DISTRIBUTION POWER FLOW: MODELING, FORMULATION, SOLUTION ALGORITHMS AND ANALYSIS Ray Daniel Zimmerman, Ph.D. Cornell University 1995 The objective of this work was to develop a formulation and an efficient solution algorithm for the distribution power flow problem which takes into account the detailed and extensive modeling necessary for use in the distribution automation environment of a real world electric power distribution system. The formulations for the three classes of existing algorithms for radial systems were generalized and were extended to handle the comprehensive modeling already presented in the context of more traditional but less efficient methods, such as Newton-Raphson and Implicit Zbus Gauss. The modeling includes unbalanced three-phase, two-phase, and single-phase branches, constant power, constant current, and constant impedance loads connected in wye or delta formations, cogenerators, shunt capacitors, line charging capacitance, switches, and three-phase transformers of various connection types. The three classes of algorithms explored are: network reduction methods, backward/forward sweep methods, and fast decoupled methods. Within each of the three classes, new algorithms were developed and existing methods were extended to include the comprehensive modeling of the general formulation. Proofs of convergence for the backward/forward sweep and fast decoupled methods are also provided.

In addition to the radial algorithms, the compensation method used to handle weakly meshed systems was generalized to encompass three-phase networks with loops, multiple sources, and three-phase PV buses. This compensation method can be applied in conjunction with any of the radial power flow solvers. Termination of the radial solver, at each iteration, is based on an adaptive criterion. A generalized correction step for the compensation method was also developed. All of the proposed methods were evaluated and compared on various test systems based on data from real distribution systems. The test systems range in size from 63 buses to over 1000 buses. The most efficient algorithm in each class was shown to require significantly less computation than both the Newton-Raphson and the Implicit Zbus Gauss methods, with the backward/forward sweep and fast decoupled methods typically showing an improvement of more than a factor of three.

BIOGRAPHICAL SKETCH Ray Daniel Zimmerman was born in Ephrata, PA on December 17, 1965. Four years later he moved with his family to a chicken farm in rural Lancaster County, PA, where he lived until he began studying Electrical Engineering in September of 1984. As an undergraduate at Drexel University in Philadelphia, PA, he participated in a cooperative education program which involved working for six month periods at each of the following companies: IBM Corporation, Research Triangle Park, NC, Evaluation Associates, Bala Cynwyd, PA, and UNISYS Corporation, Tredyffrin, PA. He received a Bachelor of Science degree in Electrical Engineering from Drexel University in June, 1989. In August of the same year he began graduate studies in Electrical Engineering at Cornell University in Ithaca, NY, where he received a Master of Science degree in May, 1992, in the area of network reconfiguration in electric power distribution systems.

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to my wife, Esther and my daughter, Ana

iv

ACKNOWLEDGMENTS Pero habiendo obtenido auxilio de Dios, persevero hasta el día de hoy. — Hechos 26:22

I would like to express my appreciation to my advisor, Dr. Hsiao-Dong Chiang, for his support and direction for this work. I would also like to thank Dr. James S. Thorp and Dr. Lloyd N. Trefethen for serving on my committee. My appreciation also goes to Gary Darling of New York State Gas & Electric and Matt Downey of Rochester Gas & Electric for providing the data used for testing the methods developed in this work. Several friends have been helpful throughout the various stages of this work, whether through discussions of technical issues or simply with helpful perspective on the process of getting a doctorate. In particular, I would like to acknowledge Guerney Hunt, Jen-Lun Yuan, Yi-Jen Chiu, Jianzhong Tong, and Karen Nan Miu. A special thanks to Karen for taking the time to read this dissertation and make helpful comments to improve its readability. I would also like to express my appreciation to Ernie for his help in proofreading. Most of all, I appreciate the constant support of my family, especially during the final months of writing. Quisiera agradecer primero a Esther por su amor y apoyo constante. Y gracias, Anita, por el ánimo que me das solo verte crecer cada día. Gracias también por ser la compañerita de mamá durante este tiempo difícil.

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TABLE OF CONTENTS ABSTRACT BIOGRAPHICAL SKETCH ...............................................................................................iii ACKNOWLEDGMENTS .....................................................................................................v TABLE OF CONTENTS ................................................................................................... vi LIST OF TABLES..............................................................................................................x LIST OF FIGURES ......................................................................................................... xii 1

Introduction 1 1.1 Background.....................................................................................................1 1.2 Objectives and Contributions..........................................................................3

2

Basic Problem Framework 7 2.1 Mathematical Notation....................................................................................7 2.2 Bus and Lateral Indexing................................................................................8 2.2.1 Indexing Scheme.................................................................................9 2.2.2 Indexing Implementation..................................................................11 2.2.2.1 Connectivity Data Structures................................................11 2.2.2.2 Breadth-First Search .............................................................12 2.3 Basic System Model .....................................................................................15 2.3.1 Voltage and Current/Power Flow Update for Branch k ...................18 2.3.2 Application of KCL ..........................................................................19

3

Detailed Component Models 21 3.1 Load Model...................................................................................................23 3.1.1 Admittance Matrix for the Load .......................................................26 3.1.2 Current and Power Injected by the Load ..........................................27 3.2 Shunt Capacitor Model .................................................................................28 3.3 Cogenerator Model .......................................................................................29 3.4 Distribution Line Model ...............................................................................31 3.5 Switch Model................................................................................................33 3.6 Transformer Model.......................................................................................34 3.6.1 Class A: Primary and Secondary both Grounded or both Ungrounded ..........................................35 3.6.2 Class B: Grounded Primary—Ungrounded Secondary ....................35 3.6.3 Class C: Ungrounded Primary—Grounded Secondary ....................39

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4

Network Reduction Power Flow Algorithms for Radial Systems (NR-PARS) 42 4.1 Detailed Solution Algorithm.........................................................................43 4.1.1 Linearization .....................................................................................44 4.1.2 Build Driving Point Equivalents.......................................................45 4.1.3 Calculate Voltages and Currents.......................................................49 4.1.4 Termination Criterion .......................................................................51 4.2 Implementation .............................................................................................51 4.2.1 Linearity Check.................................................................................51 4.2.2 Improved Line Update......................................................................52 4.2.3 Storage of Intermediate Variables ....................................................52 4.3 Variations......................................................................................................53 4.4 Convergence Analysis ..................................................................................55 4.5 Comments .....................................................................................................55

5

Backward/Forward Sweep Power Flow Algorithms for Radial Systems (BFS-PARS) 57 5.1 Detailed Solution Algorithm.........................................................................58 5.1.1 Backward Sweep...............................................................................60 5.1.2 Forward Sweep .................................................................................62 5.1.3 Termination Criterion .......................................................................65 5.2 Implementation .............................................................................................66 5.2.1 Class B Transformers........................................................................66 5.2.1.1 Backward Sweep...................................................................66 5.2.1.2 Forward Sweep .....................................................................67 5.2.2 Class C Transformers........................................................................69 5.2.2.1 Forward Sweep .....................................................................69 5.2.2.2 Backward Sweep...................................................................70 5.3 Variations......................................................................................................72 5.3.1 VI-VI-PARS .......................................................................................72 5.3.2 VS-VS-PARS......................................................................................73 5.3.3 V-VI-PARS ........................................................................................75 5.3.4 V-VS-PARS........................................................................................75 5.3.5 VI-I-PARS .........................................................................................76 5.3.6 VS-S-PARS ........................................................................................76 5.3.7 V-I-PARS...........................................................................................76 5.3.8 V-S-PARS ..........................................................................................77 5.4 Convergence Analysis ..................................................................................77 5.5 Comments .....................................................................................................83

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6

Fast Decoupled Power Flow Algorithms for Radial Systems (DePARS) 84 6.1 Detailed Solution Algorithm.........................................................................85 6.1.1 Reduced Power Flow Equations.......................................................87 6.1.1.1 Single Feeder ........................................................................88 6.1.1.2 General Radial Structure.......................................................89 6.1.1.3 Class B and Class C Transformers........................................93 6.1.2 Update of Independent Variables......................................................94 6.1.2.1 Structure of the System Jacobian..........................................95 6.1.2.2 Numerical Properties of the System Jacobian ....................100 6.1.2.3 Transformers.......................................................................103 6.1.2.4 Solving for the Update........................................................107 6.1.3 Termination Criterion .....................................................................107 6.2 Implementation ...........................................................................................109 6.3 Variations....................................................................................................110 6.3.1 VI-DePARS .....................................................................................111 6.3.2 VS-DePARS.....................................................................................111 6.3.3 I-DePARS........................................................................................112 6.3.3.1 Reduced Power Flow Equations.........................................113 6.3.3.2 Update of the Independent Variables..................................116 6.3.3.3 Implementation ...................................................................120 6.3.4 S-DePARS .......................................................................................122 6.4 Convergence Analysis ................................................................................123 6.5 Comments ...................................................................................................126

7

Power Flow Algorithms for Weakly Meshed Systems (PAWMS) 127 7.1 Detailed Solution Algorithm.......................................................................128 7.1.1 Loop Breakpoint Creation...............................................................130 7.1.2 Breakpoint Voltage Mismatch........................................................131 7.1.3 Breakpoint Impedance Matrix ........................................................132 7.1.4 Breakpoint Injections......................................................................134 7.1.5 Multiple Sources.............................................................................135 7.1.6 PV Buses.........................................................................................136 7.1.7 Summary.........................................................................................140 7.1.8 Termination Criterion .....................................................................140 7.2 Implementation ...........................................................................................141 7.2.1 Modeling Limitations and Simplifying Assumptions.....................141 7.2.2 Termination of Radial Power Flow.................................................141 7.3 Variations....................................................................................................142 7.3.1 Power Injection for Loop Breakpoints............................................143 7.3.2 Correction Step ...............................................................................144 7.4 Comments ...................................................................................................146

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8

Simulation Results 148 8.1 Summary of Algorithms Tested..................................................................150 8.1.1 Newton-Raphson Method...............................................................150 8.1.2 Implicit Zbus Gauss Method............................................................152 8.2 Description of Test Systems .......................................................................154 8.3 Power Flow Algorithms for Radial Systems (PARS) .................................155 8.3.1 Effect of Load Model and Load Factor on Convergence ...............160 8.3.2 Effect of System Size on Convergence...........................................163 8.4 Power Flow Algorithms for Weakly Meshed Systems (PAWMS)..............165 8.4.1 Effect of PARS Termination Criterion on Convergence.................166 8.4.2 Effect of Number of Loops on Convergence..................................168 8.4.3 Effect of Load Model on Convergence...........................................171 8.5 Summary.....................................................................................................172

9

Conclusions 175 9.1 Contributions...............................................................................................175 9.2 Future Work................................................................................................178

BIBLIOGRAPHY ...........................................................................................................180

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LIST OF TABLES Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5 Table 2.6

Mathematical Notation..............................................................................8 Bus Indexing Implementation.................................................................14 General Update Formulas .......................................................................17 Implementation of (2.2) ..........................................................................18 Implementation of (2.3) ..........................................................................18 General Branch Update Formulas...........................................................20

Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 3.6 Table 3.7 Table 3.8 Table 3.9 Table 3.10 Table 3.11

Load Parameters from Nominal Loads...................................................25 Load Admittance.....................................................................................27 Current & Power Injected by Load.........................................................28 Shunt Capacitor Admittance, Current & Power Injection ......................29 Cogenerator Admittance, Current & Power Injection ............................30 Update Formulas for Distribution Lines.................................................32 Update Formulas for Switches................................................................34 Admittance Matrices for Common Transformer Connections ...............36 Update Formulas for Class A Transformers...........................................37 Update Formulas for Class B Transformers ...........................................39 Update Formulas for Class C Transformers ...........................................41

Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5

Network Reduction Method....................................................................43 Network Elements...................................................................................44 Notation for (4.5) and (4.6).....................................................................47 Formulas for Combining with Incoming Branch....................................48 Voltage Update Formulas .......................................................................50

Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5

Backward/Forward Sweep Method.........................................................58 Detail on Backward/Forward Sweeps.....................................................59 Implementation of (5.1) ..........................................................................60 Implementation of (5.5) ..........................................................................63 Notation for V-I-PARS Convergence Proof............................................78

Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7

Newton’s Method....................................................................................86 Fast Decoupled Power Flow Algorithm for Radial Systems..................87 VI-DePARS Jacobian Formation.............................................................97 Jacobian Approximations for Transformers for VI-DePARS................106 Various Formulations for DePARS.......................................................110 I-DePARS Jacobian Formation .............................................................118 Jacobian Approximations for Transformers for I-DePARS ..................121

Table 7.1

Power Flow Algorithms for Weakly Meshed Systems........................129

x

Table 8.1 Table 8.2

Summary of Distribution Power Flow Algorithms...............................151 Summary of Test Systems ....................................................................154

Table 9.1

Summary of Radial Power Flow Algorithms........................................176

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LIST OF FIGURES Figure 2.1 Example of Bus & Lateral Indexing.......................................................10 Figure 2.2 Basic Building Block..............................................................................16 Figure 3.1 Grounded Wye Connected Load.............................................................24 Figure 3.2 Ungrounded Delta Connected Load........................................................26 Figure 3.3 Three-Phase Distribution Line Model ....................................................31 Figure 4.1 Norton Equivalent at Bus k.....................................................................46 Figure 4.2 Combine with Incoming Branch.............................................................47 Figure 4.3 Admittance Equivalent at Bus k .............................................................54 Figure 5.1 Single Feeder Example ...........................................................................59 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6

Single Feeder Example ...........................................................................88 Voltage Mismatch Calculation ...............................................................90 Structure of System Jacobian for VI-DePARS ........................................99 Approximation to the System Jacobian for VI-DePARS.......................108 Current Mismatch Calculation..............................................................114 Structure of System Jacobian for I-DePARS ........................................119

Figure 7.1 Figure 7.2 Figure 7.3 Figure 7.4

Loop Breakpoint ...................................................................................129 Effect of Breakpoint Creation Method on Convergence ......................131 Secondary Source Breakpoint...............................................................136 Effect of Power vs. Current Injection on Convergence........................144

Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4 Figure 8.5 Figure 8.6 Figure 8.7 Figure 8.8 Figure 8.9 Figure 8.10 Figure 8.11 Figure 8.12 Figure 8.13 Figure 8.14 Figure 8.15 Figure 8.16

Iterations Required by Each Algorithm................................................156 Linear vs. Quadratic Convergence........................................................157 Total Flops for Each Algorithm............................................................158 Normalized Flops vs. Algorithm...........................................................159 Effect of Load Model on Number of Iterations ....................................161 Effect of Load Model on Number of Flops ..........................................161 Effect of Load Factor on Number of Iterations ....................................162 Effect of Load Factor on Number of Flops...........................................163 Effect of System Size on Number of Iterations ....................................164 Effect of System Size on Number of Flops ..........................................165 Total PARS Iterations for Adaptive vs. Single Iterations .....................167 Total Number of Flops for Adaptive vs. Single Iterations....................167 Number of Iterations vs. Number of Loops..........................................169 Number of Flops vs. Number of Loops ................................................169 Overall Comparison of Iteration Counts...............................................170 Overall Comparison of Flop Counts.....................................................171 xii

Figure 8.17 Convergence of V-I-PAWMS for Various Load Models.......................172 Figure 8.18 Effect of Load Model on Number of PARS Iterations..........................173 Figure 8.19 Effect of Load Model on Number of Flops ..........................................173

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Chapter 1

Introduction The supply of electric power to homes, offices, schools, factories, stores, and nearly every other place in the modern world is now taken for granted. Electric power has become a fundamental part of the infrastructure of contemporary society, with most of today’s daily activity based on the assumption that the desired electric power is readily available. The power systems which provide this electricity are some of the largest and most complex systems in the world. They consist of three primary components: the generation system, the transmission system, and the distribution system. Each component is essential to the process of delivering power from the site where it is produced to the customer who uses it.

1.1 Background One of the most fundamental calculations related to any system is the determination of the steady state behavior. In power systems, this calculation is the steady state power flow problem, also called load flow. It essen-

1

2 tially involves finding the steady state voltages at each node, given a certain set of generation and loading conditions. The majority of power flow algorithms in wide use in industry today, most notably, the Newton-Raphson method and its variants [25; 28], have been developed specifically for transmission systems which have a meshed structure, with parallel lines and many redundant paths from the generation points to the load points. The Newton-Raphson method itself is computationally expensive for large systems, due primarily to the size of the Jacobian and the resulting system of linear equations which must be solved to find the Newton step. For transmission systems, some approximations can typically be made which allow for the decoupling of real and reactive power from and voltage magnitude and angle, respectively. The Jacobian can also be approximated by a constant matrix, resulting in the fast-decoupled Newton method [26] which has proven to be a great improvement over the standard Newton-Raphson power flow for many cases. The focus of this dissertation is on the solution of the power flow problem for the distribution system. Typically, a distribution system originates at a substation where the electric power is converted from the high voltage transmission system to a lower voltage for delivery to the customers. Unlike a transmission system, a distribution system typically has a radial topological structure. Unfortunately, this radial structure, along with the higher resistance/reactance (R/X) ratio of the lines, makes the fast-decoupled Newton method unsuitable for most distribution power flow problems. Since power flow is such a fundamental calculation for a power system, it is used in many applications in planning and operation. Some of the

3 optimization problems related to distribution automation, such as network reconfiguration, service restoration, and capacitor placement, require the solution of hundreds or even thousands of power flow problems. These applications place two primary requirements on a distribution power flow program. First, the modeling must reflect the actual behavior of the system components. Second, the solution algorithm must be robust and efficient. Various efficient distribution power flow algorithms which exploit the radial structure have been proposed in the literature. These algorithms can be classified into three groups: • network reduction methods [4] • backward/forward sweep methods [3; 11; 18; 19; 20; 23] • fast decoupled methods [12; 17; 32] All of the proposed methods, as presented, have some limitations. Many are only applied to single-phase representations of the network and cannot handle unbalanced distribution systems or networks with a mixed number of phases. Most of the methods are also proposed in the context of limited network modeling. In particular, none of the algorithms in the literature include modeling for transformers which are grounded on one side and ungrounded on the other. Unlike the extension from a single-phase to a three-phase representation, the addition of such modeling into the formulation is not straightforward. Line charging capacitance, cogeneration, and general load models are also typically not considered.

1.2 Objectives and Contributions The objective of this work was to develop a formulation and an efficient solution algorithm for the distribution power flow problem which

4 takes into account the detailed and extensive modeling necessary for use in the distribution automation environment of a real world power system. A general framework was developed for each of the three classes of existing algorithms, and a common set of network component models was chosen. The general framework for each class helps in relating the proposed algorithms to one another and also reveals variations of each class that have not previously been explored. Within each class, new algorithms were developed and, where necessary, the existing algorithms were extended to remove limitations and generalized to handle the following: • general radial structure1 • unbalanced three-phase operation, including single-phase and two-phase branches • general load models, including constant power, constant current, and constant impedance loads, connected in wye or delta configurations • cogenerators • shunt capacitors • line charging effects • switches • three-phase transformers of various connection types The basic problem framework and some common notation used throughout the dissertation are introduced in Chapter 2. In Chapter 3, detailed models for loads, shunt capacitors, cogenerators, distribution lines, switches, and transformers are presented, along with some of the specific equations needed to implement these models in the algorithms which follow.

1 Some

existing methods only handle a main feeder with laterals.

5 Chapters 4, 5, and 6, respectively, discuss in detail the network reduction, backward/forward sweep, and fast decoupled algorithms. Each chapter presents first the basic concepts behind the corresponding class of methods, then a detailed description of a specific algorithm in the respective class. Following this detailed description of the algorithm are some comments on the implementation of the method. Each of the three classes includes several variations which are discussed relative to the version presented in detail. Each of these chapters concludes with a discussion of the convergence characteristics followed by some general comments. Chapter 5 and Chapter 6 include proofs of convergence for the respective algorithms. Chapter 7 explores an extension of the radial power flow algorithms discussed in the previous three chapters to handle weakly meshed systems with certain modeling restrictions. The extension described is based on a radial power flow solver imbedded within a compensation method. It extends the formulation to address systems with loops, secondary sources, and PV buses. The structure of the chapter is similar to the pattern of the previous three, discussing the basic concepts, followed by a detailed description of the algorithm, implementation notes, variations, and comments. Each of the radial power flow algorithms, including all of the variations presented, along with the extensions to weakly meshed networks, was implemented in a MATLAB® program for testing. In addition, the Newton-Raphson and Implicit Zbus Gauss methods were implemented for comparison. Chapter 8 analyzes the relative performance of the various methods on test systems ranging in size from 63 buses to over 1000 buses. The effects of system size, load models, load factor, and number of loops in

6 the network are examined. The chapter ends with a summary of the simulation results and some general conclusions about the relative merits of the different approaches. The final chapter discusses the conclusions drawn from this work, outlines a summary of the contributions made, and mentions some ideas for possible areas of future research to extend the work in this dissertation.

Chapter 2

Basic Problem Framework The distribution power flow problem is the problem of finding the operating point of a distribution network at steady state under given conditions of load and cogeneration. This involves, first of all, finding all of the bus voltages. From these voltages, it is possible to directly compute currents, power flows, system losses and other steady state quantities. This chapter presents some of the fundamental concepts which are general in nature and apply to all or at least several of the approaches discussed in later chapters.

2.1 Mathematical Notation Since this dissertation deals with three-phase unbalanced power flow, vectors are typically used to represent voltages, currents, power flows, and admittances. Many of the formulas presented in this work can be expressed more clearly and compactly by using certain notational conventions. The conventions shown in Table 2.1 for complex vectors x and y, for

7

8 complex matrices A, X, and Y, and for functions f and g, will be used extensively throughout this dissertation. Table 2.1 Mathematical Notation The Expression:

Is Used To Denote:

x .* y

element-wise multiplication

x./ y

element-wise division

x∗

element-wise complex conjugate

x

element-wise magnitude

x2

element-wise squared magnitude

x = A\ y

†

X = A\Y

†solution

solution to the equation Ax = y to the equation AX = Y

x˜

computed value, as opposed to given value

x

constant parameter, as opposed to a variable

f • g( x)

function composition, f ( g( x))

†The

notation A –1 is not used since it is restricted to cases where A is square and non-singular. The \ notation is used only when the corresponding equation produces a unique solution. In other words, if A is not a square non-singular matrix, then A must have more rows than columns and the corresponding equation must have the appropriate number of redundant rows.

2.2 Bus and Lateral Indexing In most typical power flow formulations, a set of equations and unknowns is associated with each bus in the network, and these equations and unknowns are organized by a particular bus ordering. Because of the radial structure of the systems under consideration, the number of equations and variables can be reduced so that each set of equations and

9 unknowns corresponds to an entire lateral instead of an individual bus. Such a formulation therefore calls for an appropriate lateral indexing to order these equations and variables.

2.2.1

Indexing Scheme

A radial system can be thought of as a main feeder with laterals. These laterals may also have sub-laterals, which themselves may have sub-laterals, etc. First, the level of lateral i is defined as the number of laterals which need to be traversed to go from the end of lateral i to the source. For example, the main feeder would be level 1, its sub-laterals would be level 2, their sub-laterals level 3, etc. Second, the laterals within level l are indexed according to the order seen during a breadth-first traversal of the network. Each lateral can be uniquely identified by an ordered pair ( l, m ) where l is the lateral level and m is the lateral index within level l. Third, buses are indexed within each lateral, starting with the first bus on the lateral, so that each bus is uniquely identified by an ordered triple ( l, m, n ) where n is the bus index. The ordered triple ( l, m, n ) refers to the nth bus on the mth level l lateral. The source is given an index of ( 1, 1, 0 ) . The number of levels in a network will be denoted by L, the number of laterals on level l by M l , and the number of buses on lateral ( l, m ) by N l, m . Figure 2.1 shows an example of this indexing scheme on a sample 63-bus system. The boxed numbers show the reverse breadth-first (RBF) ordering of the laterals found by sorting the lateral indices in descending order, first by level, then by lateral index. The RBF ordering is typically

10

19

10

1

(2,1,5)

(4,1,2)

(2,1,4)

(4,1,1)

(4,10,2) 17

(4,10,1)

(2,3,4) 16

(2,1,3)

(3,1,1)

(3,1,2)

(2,1,2)

(2,3,3)

(3,6,1)

(3,6,2)

11 (3,6,3)

(3,5,1)

(3,5,2)

(3,5,3) 12

(2,3,2)

(2,1,1)

(2,3,1)

Source (1,1,0)

(4,8,1)

(4,9,1)

(4,8,2)

(4,9,2)

3 (1,1,1)

(1,1,2)

2

(1,1,3)

(1,1,4)

(1,1,5)

(1,1,6)

(3,2,1)

(3,2,2)

(3,2,3)

(3,2,4)

20 (1,1,7)

(2,2,1)

(2,2,2)

(3,3,3) 14

(3,3,2)

(3,3,1)

(4,2,1)

(4,3,1)

(4,2,2)

(4,3,2)

(2,2,3)

9 (4,4,1)

8

(2,2,4)

13 (3,4,1)

(3,4,2)

(3,4,3)

(3,4,4)

(4,4,2) 7

(2,2,5)

(4,5,1)

(4,6,1)

(4,7,1)

(2,2,6)

(4,5,2)

(4,6,2)

(4,7,2)

18

6

5

4

Figure 2.1 Example of Bus & Lateral Indexing

(3,2,5) 15

11 used for backward sweep type operations. If the laterals are sorted in ascending order, the result is a breadth-first (BF) ordering, typically used for forward sweep type operations. The following shorthand notation will also be used when i is an ordered pair referring to a lateral and k is an ordered triple referring to a bus. Lateral i – 1 refers to the parent of lateral i, and bus k – 1 refers to bus k’s parent bus. Unless specified otherwise, bus k + 1 is used to refer to the bus following bus k on the same lateral. This is consistent with using k as a simple bus index, in other words letting k = n , which is done frequently throughout the following chapters. In this case, bus 0 of lateral i refers to the bus on lateral i – 1 from which lateral i emanates. For example, in the network in Figure 2.1, bus 0 of lateral (2,3) is also bus 4 of lateral (1,1). This notation is used in indexing voltages, currents, power flows, impedances, etc.

2.2.2

Indexing Implementation

All of the algorithms discussed in Chapters 4, 5, 6, and 7 use the ordering of the buses and laterals presented in the previous section. They also require the ability to traverse any given lateral from its source to its end bus or vice versa. Certain data structures are therefore needed in the program to store information about connectivity and ordering. Also required is a process by which this information is generated from the original network data. 2.2.2.1

Connectivity Data Structures

The network data which specifies the topology of the system is typically given as a list of branches with information on which two buses the

12 branch connects. In order to efficiently traverse a feeder, it is important to store information with each bus k indicating the incoming branch, the bus which follows bus k on the same lateral, the number of sub-laterals branching off at bus k, and the first bus on each of these sub-laterals. The first three of these quantities will be denoted inbranch(k), next(k), and nsubs(k), respectively. The first bus on the sub-laterals branching from bus k will be called subbus1(k,1), subbus1(k,2), …, subbus1(k,nsubs(k)), respectively. Assuming the source bus is known, these data structures can be built up, along with the ordered triples of the previous section, during the process of a breadth-first search. 2.2.2.2

Breadth-First Search

The initial traversal of the network is done via a breadth-first search algorithm. This traversal can be useful for many things such as detecting isolated sub-networks, checking the consistency of the phase data,1 and marking sections of the network as grounded or ungrounded. However, the primary purpose is to build up the connectivity data structures and assign the bus and lateral indices. It should be noted that a depth-first search works equally well for building the connectivity data structures and indexing the buses. The breadth-first approach was chosen for convenience in dealing with weakly meshed networks as discussed in Chapter 7. The breadth-first search, as described in [22], requires the ability to find all of the “children” associated with a given node of the tree. Since the connectivity structures are not yet available, the children of bus k must be found via brute force by searching all branches for those connected to

1 How

many and which phases are present at each bus and branch and do they match.

13 bus k. If the bus at the other end of such a branch has not yet been visited during the search,2 it is a child of bus k. When the search is at bus k and all of bus k’s children have been found, the inbranch information can be set for each child. The value of nsubs is typically set to one less than the number of children since one of the children is generally selected to be the next bus on the same lateral and the rest are assigned to the elements of subbus1. The decision of which child, if any, is considered to be on the continuation of the same lateral and which children are considered to be on sub-laterals has a significant effect on the resulting bus and lateral indexing. Since some of the power flow algorithms require that each lateral have at most one transformer, a child whose inbranch is a transformer is never assigned to next. Transformers are always assigned to a sub-lateral, even if it leads to the bus’s only child. Similarly, if the branch leading to a child of bus k has fewer phases than bus k, then the child is put on a sub-lateral. This simplifies some of the implementation since it means that the phases present are consistent throughout an entire lateral. After setting the connectivity data structures at bus k, the bus’s ordered triple is generated. This requires several counters to be maintained during the search process. First, l, m, and n are used to denote the current level, lateral, and bus indices, respectively. The counters L, M l , and N l, m keep track of the number of levels encountered during the search, the number of laterals on level l, and the number of buses on lateral ( l, m ) , respectively. The algorithm used to generate the ordered triple for bus k is shown in Table 2.2. 2 i.e.

the branch is not the incoming branch of bus k (assuming a radial network).

14 Table 2.2 Bus Indexing Implementation Algorithm for Setting Indices (l, m, n) for Bus k if bus k is the source set ( l, m, n ) = ( L, M 1, N 1, 1 ) = ( 1, 1, 0 ) otherwise (for all other buses) set l to the level of k’s parent bus if bus k lies on the same lateral as its parent bus set m to lateral index of k’s parent bus otherwise (it is the first bus on a new lateral) increment l by 1 if this is the first level l lateral encountered increment L by one (i.e. set L = l ) initialize M l = 0 end if increment M l by 1 and set m = M l initialize N l, m = 0 end if increment N l, m by 1 and set n = N l, m end if set the index for bus k to ( l, m, n ) The index for the source is set directly to ( 1, 1, 0 ) . Each of the remaining buses falls into one of two classes based on whether it is the first bus on a lateral or a continuation of a lateral previously encountered during the search. For a bus k in the second class, the level and lateral indices, l and m, are copied directly from the bus’s parent. For the first class, the level index l is set to one more than that of the bus’s parent, and the lateral counter for that level is incremented by one and assigned to the lateral

15 index m. Finally, in both cases, the bus counter for that lateral is incremented by one and assigned as the bus index n. Each time a new level l is first encountered, L is incremented and assigned to l and the corresponding lateral counter M l is initialized to zero. Likewise, each time a new level l lateral is first encountered, M l is incremented and assigned to m and the corresponding bus counter N l, m is initialized to zero. When the entire network has been traversed by the search, all connectivity structures have been built and all bus indices have been assigned. These bus indices are then used to form a list of laterals in RBF3 order. Each element of the list contains the first and last buses on the corresponding lateral. These elements are sorted in descending order according to the indices associated with the corresponding buses, first by the lateral level, then by the lateral index.

2.3 Basic System Model For the purposes of power flow studies, a radial distribution system can be modeled as a network of buses connected by distribution lines, switches, or transformers to a voltage specified source bus. Each bus may also have a corresponding load, shunt capacitor, and/or cogenerator connected to it. The model can be represented by a radial interconnection of copies of the basic building block shown in Figure 2.2. Since a given branch may be single-phase, two-phase, or three-phase, each of the labeled quantities is, respectively, a complex scalar, a 2 x 1, or a 3 x 1 complex vector. Figure 2.2 establishes a consistent notation, which will be used extensively throughout this dissertation, for the voltages, currents, and power flows 3 See

page 9 under Section 2.2.1, “Indexing Scheme”.

16

Bus k-1 (supplying)

Bus k (receiving) sub-laterals

Vk –1

Ik

I k′

distribution line, switch, or transformer

S k′

Sk

I Gk cogenerator

S Gk

I Ck

Vk

S Ck

shunt capacitor

Ik + 1 Sk + 1

I Lk

S Lk

load

Figure 2.2 Basic Building Block related to a given bus k. V is used to denote voltage, I to denote current, and S to denote power flow. The dotted lines from the cogenerator, shunt capacitor, and load to ground are to indicate that these elements may be connected in an ungrounded delta configuration. The radial structure implies that there are no loops in the network and each bus is connected to the source via exactly one path. It is this structure that makes possible the three classes of power flow algorithms discussed in this dissertation. The first is based on the network reduction methods presented in Chapter 4. The remaining two involve updating voltages, currents, and power flows along the paths from the source to the end buses. These are the backward/forward sweep and fast decoupled algorithms discussed in Chapter 5 and Chapter 6, respectively.

17 Each of the algorithms in these three classes depends on the ability to compute the voltage and current (or power flow) at a given bus from the corresponding quantities at an adjacent bus. Letting wk =

Vk Ik + 1

Vk

and u k =

Sk + 1

,

(2.1)

these update formulas can be expressed mathematically by the equations in Table 2.3 for each of four different cases. Table 2.3 General Update Formulas Based on

Direction of Calculation

Update Formula

forward

w k = f k(w k – 1)

(2.2)

backward

w k – 1 = g k(w k)

(2.3)

forward

u k = e k(u k – 1)

(2.4)

backward

u k – 1 = h k(u k)

(2.5)

current

power flow

The functions f k and g k are inverses of each other and the functions e k and h k are inverses of each other. Each of the four functions is determined by the respective load, shunt capacitor, cogenerator, and sub-laterals attached at bus k as well as the incoming distribution line, switch, or transformer. The implementation of each of the update formulas in Table 2.3 consists of a four step procedure. The steps for the implementation of (2.2) are shown in Table 2.4. The backward calculation of (2.3) is accomplished by the steps shown in Table 2.5. These steps are nearly the same as those in Table 2.4, but in a different order. In the case of (2.4) and (2.5), where

18 Table 2.4 Implementation of (2.2) Procedure

Details

1

Compute V k from V k – 1 and I k .

see Tables 3.6 - 3.11

2

Compute I k ′ from V k , V k – 1 and I k .

see Tables 3.6 - 3.11

3

Compute I Gk , I Ck , and I Lk from V k .

see Tables 3.3 - 3.5

4

Compute I k + 1 via KCL at bus k.

see (2.20)

Table 2.5 Implementation of (2.3) Procedure

Details

1

Compute I Gk , I Ck , and I Lk from V k .

see Tables 3.3 - 3.5

2

Compute I k ′ via KCL at bus k.

see (2.20)

3

Compute V k – 1 from V k and I k ′ .

see Tables 3.6 - 3.11

4

Compute I k from V k – 1 , V k and I k ′ .

see Tables 3.6 - 3.11

power flow is used instead of current, all of the I’s in Tables 2.4 and 2.5 are replaced by S’s.

2.3.1

Voltage and Current/Power Flow Update for Branch k

The voltage updates, as detailed in Tables 3.6 - 3.11, depend only on the parameters of the branch itself and the voltage and injected current (or power) at one end of the branch. Typically, the parameters of bus k’s incoming branch4 can be represented by an admittance matrix, expressed in a general form as BR Yk

4 The

=

11

12

21 Yk

22 Yk

Yk Yk

.

incoming branch of bus k will also be referred to as branch k.

(2.6)

19 BR

The voltages and currents at either end of branch k are related by Y k

as

follows: Ik Ik′

=

11

12

21 Yk

22 Yk

Yk Yk

Vk – 1 Vk

(2.7)

From (2.7), V k can be calculated in the forward direction as a function of V k – 1 and I k or V k – 1 can be calculated in the backward direction as a function of V k and I k ′ . If power flow is being used instead of current, the following are substituted for I k and I k ′ , respectively: I k = ( Sk . / V k – 1) ∗

(2.8)

I k ′ = ( Sk ′ . / V k) ∗

(2.9)

Once the voltage, V k or V k – 1 depending on the direction of calculation, has been updated, the updated current can also be computed directly from (2.7). If the power flow is needed it can be computed from the updated current and voltage by one of the following: S k = V k – 1 .* I k∗

(2.10)

S k ′ = V k .* I k ′ ∗

(2.11)

The resulting general branch update formulas are summarized in Table 2.6.

2.3.2

Application of KCL

The current or power flow component of the update formulas (2.2)(2.5) is based on the current or power lost in bus k’s incoming branch and the application of Kirchhoff’s Current Law (KCL) at bus k. The application of KCL at bus k requires the currents injected by cogenerators, shunt

20 Table 2.6 General Branch Update Formulas Based on

Direction of Function Calculation of forward

V k – 1, I k

current backward

forward

V k, I k ′

V k – 1, S k

power flow backward

V k, S k ′

Update Formula 12 – 1 11 V˜ k = ( Y k ) ( I k – Y k V k – 1 )

(2.12)

21 22 I˜ k ′ = Y k V k – 1 + Y k V˜ k

(2.13)

21 – 1 22 V˜ k – 1 = ( Y k ) ( I k ′ – Y k V k )

(2.14)

11 12 I˜ k = Y k V˜ k – 1 + Y k V k

(2.15)

12 – 1 11 V˜ k = ( Y k ) ( ( S k . / V k – 1 ) ∗ – Y k V k – 1 )

(2.16)

21 22 S˜ k ′ = V˜ k .* ( Y k V k – 1 + Y k V˜ k ) ∗

(2.17)

21 – 1 22 V˜ k – 1 = ( Y k ) ( ( S k ′ . / V k ) ∗ – Y k V k )

(2.18)

11 12 S˜ k = V˜ k – 1 .* ( Y k V˜ k – 1 + Y k V k ) ∗

(2.19)

capacitors, and loads, represented by I˜ Gk , I˜ Ck , and I˜ Lk , respectively. Each of these quantities is a function of V k and is hence designated with a tilde. It also requires the currents I j injected into sub-laterals branching off from bus k. Here j ∈ A k and A k is the set of buses adjacent to bus k on sub-laterals. KCL at bus k can then be written as I˜ Gk + I˜ Ck + I˜ Lk –  

∑

j ∈ Ak

I j – I k′ – I k + 1 = 0 . 

(2.20)

In the case where power flow is used there is an analogous equation expressing the conservation of complex power at bus k. S˜ Gk + S˜ Ck + S˜ Lk –  

∑

j ∈ Ak

S j – Sk ′ – Sk + 1 = 0 

(2.21)

Chapter 3

Detailed Component Models In any problem where mathematics and numerical algorithms are used to analyze a physical system, the results are only as accurate as the mathematical models used. In power systems analysis, the solutions found by any power flow algorithm are only useful to the user if they provide results which are meaningful with respect to some real system. It is therefore important to model each component of the system as accurately as possible. On the other hand, care must be taken to avoid using models which are overly detailed and therefore either computationally impractical or unusable due to unavailability of parameter data. The algorithms presented in this dissertation are based on models which attempt to meet these two requirements. Most are based on standard three-phase models as presented in [2; 8; 10]. This chapter describes in detail the models used for loads, shunt capacitors, cogenerators, distribution lines, switches, and transformers. These models provide relationships between the relevant voltages, currents, and power flows. By convention, injected currents and power flows

21

22 are always used for loads, shunt capacitors, and cogenerators, as shown in Figure 2.2. a

b

c

Bus voltages are typically the phase voltages V , V , and V referenced to ground. However, it is possible to have floating sections of the network in which there is no reference to ground. For example, there might be a feeder connected to the secondary side of a grounded wye to delta transformer which has only ungrounded, i.e. delta connected, loads. The terms grounded and ungrounded, respectively, will be used to distinguish between parts of the network which have a reference to ground and those floating sections which do not. Which buses are in grounded sections and which are in ungrounded sections is determined according to the grounding of the transformer connections during the initial traversal of the network described in Section 2.2.2.2, “Breadth-First Search”. It is assumed that any part of a network supplied through an ungrounded transformer connection will be entirely ungrounded.1 In the ungrounded sections, to avoid arbitrarily picking a particular phase as the voltage reference, the line-to-line voltages V used. In this case, the third line voltage V equal to – ( V

ab

ca

ab

and V

bc

are

is redundant since it is always

bc

+ V ) , and the dimension of the equations is reduced by

one. Similarly, the current in one of the phases is redundant since c

a

b

I = – ( I + I ) , so typically only phase a and phase b currents are used for calculation.

1A

sub-network, supplied through an ungrounded transformer connection, which does have some grounded elements could be handled by the network reduction methods of Chapter 4, though the details of such a case are not discussed. In their current forms, however, the methods of Chapter 5 and Chapter 6 are unable to handle this case.

23 When computing power flows as opposed to currents, ground is used a a a∗ as a reference in grounded sections. For example, S = V I . However, in ungrounded sections, the power used is that defined by the line-to-line voltages V

ab

and V

bc

a

b

, and the currents I and I . S

ab

S

bc

= V

ab a∗

(3.1)

I

bc b∗

(3.2)

= V I

It is important to note that, although it is possible to calculate total power a

b

c

flows in grounded sections of the network by the sum S + S + S , the total power flow in an ungrounded section is not equal to S

ab

bc

+ S . Total

power flows must be calculated using a common voltage base. For example, using phase c as a voltage reference, the total power can be computed as S

total

= S

ac

bc

+S .

(3.3)

3.1 Load Model The model used for loads is a flexible one. It includes constant complex power, constant current, and constant impedance types.2 Three-phase loads can be balanced or unbalanced and can be connected in a grounded wye configuration or an ungrounded delta configuration. It is also possible to have single-phase or two-phase grounded loads. Typically, the load values are given as nominal power delivered to the load and must be converted into the appropriate constant model parameters. Depending on the type of power flow algorithm being utilized, it may be necessary to compute

2

Each load could actually be a linear combination of these three types. In fact, it is straightforward to generalize the model presented here to a current injection expressed as an arbitrary function of voltage.

24 the following quantities from the bus voltage V k and the constant model parameters: • admittance matrix Y Lk • injected current I Lk = – Y Lk V k • injected power S = V .* I ∗ Lk

k

Lk

With a grounded wye connected load as shown in Figure 3.1, for each p

phase p, the parameter S Lk , nom is given. This is the nominal complex power absorbed by the element connected between phase p and ground. In other words, for a three-phase load, the nominal load is a

S Lk , nom S Lk , nom = S b . Lk , nom

(3.4)

c

S Lk , nom

a

Vk

V bk a

c

Vk b

I Lk

c

I Lk S bLk S aLk

I Lk

y bLk c

S Lk y aLk

y cLk

Figure 3.1 Grounded Wye Connected Load

These values are converted to the appropriate constant model parameters I Lk , S Lk , or y Lk , according to the type of load and the nominal voltage V k, nom , using the equations in Table 3.1. I Lk , S Lk , and y Lk are n x 1 com-

25 plex vectors of current, power, and admittance, respectively, where n is the number of phases present. Note that I Lk and S Lk are injected quantities, hence the negative sign in (3.5) and (3.6). Table 3.1 Load Parameters from Nominal Loads Connection

grounded wye

V k, nom

a V k, nom b V k, nom c

V k, nom

ungrounded delta V ab k, nom bc

V k, nom ca

V k, nom

†where

U =

Load Type

Parameter Calculation

constant S

S Lk = – S Lk , nom

(3.5)

constant I

I Lk = – ( S Lk , nom . / V k, nom ) ∗

(3.6)

constant Z

y Lk = S Lk , nom . / V k, nom 2

(3.7)

constant S

S Lk = – S Lk , nom

(3.8)

constant I†

I Lk = ( S Lk , nom . / ( UV k, nom ) ) ∗

(3.9)

constant Z†

y Lk = S Lk , nom . / UV k, nom 2

(3.10)

constant S

S Lk = – S Lk , nom

(3.11)

constant I

I Lk = ( S Lk , nom . / V k, nom ) ∗

(3.12)

constant Z

y Lk = S Lk , nom . / V k, nom 2

*

*

*

(3.13)

1 –1 0 0 1 –1 . –1 0 1

Figure 3.2 shows an ungrounded delta connected load for which the nominal power given is the power absorbed by the elements between each phase. In this case, the nominal load is ab

S Lk , nom S Lk , nom = S bc . Lk , nom ca

S Lk , nom

(3.14)

26 a

Vk

b

Vk a

c

Vk b

I Lk

c

I Lk

I Lk

bc

I Lk

S ab Lk ab y Lk

I ab Lk

bc

y bc Lk S ca Lk ca y Lk

S Lk

I ca Lk

Figure 3.2 Ungrounded Delta Connected Load In the conversion from the nominal load to the appropriate constant model parameters shown in (3.8)-(3.13) in Table 3.1, the voltages used are phaseto-ground or line-to-line voltages, respectively, depending on whether the load is in a grounded or ungrounded section of the network.

3.1.1

Admittance Matrix for the Load

Some power flow algorithms require an admittance equivalent for each load. For constant impedance loads, the admittance matrix can be built directly from the constant element admittances y Lk given. For constant current or constant PQ loads, equivalent admittances y Lk for each element are computed first. These admittances are equivalent in the sense that they yield the appropriate current or power flow, respectively, at the given bus voltage. Table 3.2 gives the element admittance equivalents and the admittance matrix for each load and connection type.

27

Table 3.2 Load Admittance Connection

grounded wye

Vk

Load Type

b

Vk c

Vk

ungrounded delta

constant S

– ( S Lk ∗ . / V k 2 )

constant I

– I Lk . / V k

constant Z

y Lk

constant S†

– ( S Lk ∗ . / UV k 2 )

y Lk + y Lk

constant I†

I Lk . / ( UV k )

– y Lk

constant Z

y Lk

– y Lk

constant S

– ( S Lk ∗ . / V k 2 )

ab

Vk

bc

Vk

3.1.2

U =

Admittance Matrix Y˜ Lk = y aLk 0

constant I constant Z

I Lk . / V k y Lk

0

0 y bLk 0

a

Vk

†where

Element Admittance y Lk =

c

0 ca

ab

0 y Lk ab

ca

– y Lk

ab

ab

– y Lk bc

bc

y Lk + y Lk

ca

bc

bc

– y Lk

ca

– y Lk ca

y Lk + y Lk

ab

ca

y Lk + y Lk y Lk ab

– y Lk

bc

y Lk

1 –1 0 0 1 –1 . –1 0 1

Current and Power Injected by the Load

Some power flow algorithms require the computation of the current or power injected by the load at bus k based on the bus voltage V k . Table 3.3 shows how the injected current and power can be computed from the voltage and the load parameters for each of the different cases. Some of the calculations are based on the admittance matrix from Table 3.2.

28 Table 3.3 Current & Power Injected by Load Connection

grounded wye

Vk

a Vk b Vk c Vk

Injected Current I˜ Lk =

Injected Power ˜ Lk = S

constant S

( S Lk . / V k ) ∗

S Lk

constant I

I Lk

constant Z

– y Lk .* V k

constant S†

( U [ S Lk . / ( UV k ) ] ) ∗

constant I†

– U I Lk

constant Z

– Y˜ Lk V k

Load Type

T

constant S

∗ V k .* I˜ Lk

T

– ungrounded delta

ca S Lk -----------------------ab bc Vk + Vk

–

ab

bc

ab S Lk --------ab Vk

S Lk S Lk --------– --------ab bc Vk Vk

ab

Vk

bc Vk

constant I

∗ V k .* I˜ Lk

ca

ab

ab

bc

∗

∗ V k .* I˜ Lk

I Lk – I Lk I Lk – I Lk

constant Z

†where

U =

– Y˜ Lk V k

1 –1 0 0 1 –1 . –1 0 1

3.2 Shunt Capacitor Model Shunt capacitors, often used for reactive power compensation in a distribution network, are modeled as constant capacitance devices. As with loads, they can be connected in a grounded wye configuration or an ungrounded delta configuration as shown in Figure 3.1 and Figure 3.2, respectively. In fact, they are treated in exactly the same way as a purely reactive constant impedance load. It is assumed that shunt capacitors in

29 grounded sections of the network are wye connected and those in ungrounded sections are three-phase and delta connected. The constant model parameter, in this case, is the admittance y Ck which is computed from the given nominal reactive power injection Q Ck , nom . The nominal voltage V k, nom is the phase-to-ground voltage in grounded sections and the full 3-dimensional line-to-line voltage for ungrounded sections.3 The admittance y Ck is then given by y Ck = jQ Ck , nom . / V k, nom 2 .

(3.15)

From y Ck it is possible to compute the necessary admittance matrix and injected current and power as shown in Table 3.4. Table 3.4 Shunt Capacitor Admittance, Current & Power Injection Connection

Vk a

grounded wye

ungrounded delta

Vk

Admittance Matrix Injected Current Injected Power ˜ Y˜ Ck = I˜ Ck = S Ck = a

y Ck 0

b Vk

0

c Vk

0

V ab k V bc k

b y Ck

0 0 c

0 y Ck

ca

ab

– Y˜ Ck V k

∗ V k .* I˜ Ck

ca

y Ck + y Ck y Ck ab

– y Ck

bc

y Ck

3.3 Cogenerator Model Depending on its particular control parameters, a cogenerator in a distribution system may be set to output power at either a constant power factor or a constant terminal voltage. In other words, some cogenerators are modeled as constant complex power elements, treated as constant PQ 3 i.e.

b c bc ca V k, nom is V a or V ab , respectively. k, nom V k, nom V k, nom k, nom V k, nom V k, nom

30 loads with positive, as opposed to negative, real power injection. Others are modeled as PV buses. This second type of cogenerator cannot be handled directly by the radial power flow programs presented here. However, Chapter 7 presents some extensions to the radial power flow methods which do handle PV buses. Even in this case, each PV bus is treated as a constant complex power element during any given iteration of the power flow algorithm. It is therefore sufficient to present only the relevant formulas for constant PQ cogenerators. Cogenerators in grounded sections of the system are assumed to be wye connected, and in ungrounded sections they are assumed to be threephase and delta connected. Typically S Gk , the complex power supplied by each element, is given and is used to compute the necessary admittance matrix and injected current and power as shown in Table 3.5. Table 3.5 Cogenerator Admittance, Current & Power Injection Connection

Vk

Element Admittance y Gk =

a

grounded wye

a

y Gk 0

Vk

b Vk c

Admittance Matrix ˜ Y Gk =

Vk

ab

Vk

0

0

0

0

y Gk

bc Vk

ca

ab

( S Gk . / V k ) ∗

S Gk

c

ca

y Gk + y Gk y Gk ab – y Gk

Injected Power ˜ S Gk =

0

b y Gk

– ( S Gk ∗ . / V k 2 ) ungrounded delta

Injected Current I˜ Gk =

bc y Gk

ca – S Gk -----------------------ab bc Vk + Vk ab S Gk ---------– V ab k

–

ab S Gk --------ab Vk

bc S Gk --------V bc k

∗ ∗ V k .* I˜ Gk

31

3.4 Distribution Line Model The model used to represent a distribution line connecting two buses is the standard π-model shown in Figure 3.3. The impedance of distribu-

Bus k-1 (supplying)

Bus k (receiving)

a

a

Vk –1 b

Vk b Vk c Vk

Series Impedance

Vk –1 c

Vk –1

Zk Shunt Capacitance

1 --- Y 2 k

1 --- Y 2 k

Shunt Capacitance

Figure 3.3 Three-Phase Distribution Line Model

tion line k4 is represented as a series impedance Z k , and the line charging effects are divided between the two shunt arms, each with an admittance of Y k ⁄ 2 . The impedance Z k and the admittance Y k are both n x n complex matrices, where n is the number of phases in the line. The branch admittance matrix for this model is

BR Yk

4 The

1 Z k–1 + --- Y k 2 = – Z k–1

distribution line entering bus k.

– Z k–1 Z k–1

1 + --- Y k 2

.

(3.16)

32 BR

Substituting this value for Y k

in (2.7) and solving for the appropriate

variables yields the equations for voltage, current and power flow updates given in Table 3.6. Table 3.6 Update Formulas for Distribution Lines

Function of

V k – 1, I k

V k, I k ′

V k – 1, S k

V k, S k ′

Update Formula

1 V˜ k = V k – 1 + Z k  --- Y k V k – 1 – I k  2

(3.17)

1 I˜ k ′ = --- Y k ( V˜ k + V k – 1 ) – I k 2

(3.18)

1 V˜ k – 1 = V k + Z k  --- Y k V k – I k ′  2

(3.19)

1 I˜ k = --- Y k ( V k + V˜ k – 1 ) – I k ′ 2

(3.20)

1 V˜ k = V k – 1 + Z k  --- Y k V k – 1 – ( S k . / V k – 1 ) ∗  2

(3.21)

∗ 1 S˜ k ′ = V˜ k .*  --- Y k ( V˜ k + V k – 1 ) – ( S k . / V k – 1 ) ∗  2

(3.22)

1 V˜ k – 1 = V k + Z k  --- Y k V k – ( S k ′ . / V k ) ∗  2

(3.23)

∗ 1 S˜ k = V˜ k – 1 .*  --- Y k ( V k + V˜ k – 1 ) – ( S k ′ . / V k ) ∗  2

(3.24)

For a three-phase line in an ungrounded section of the network, the line charging effects are assumed to be negligible5 so Y k is set to zero. The series impedance Z k is reduced to 2 x 2 for use with the line-to-line voltages. The series impedance Z k which is given is a 3 x 3 matrix expressed in per unit based on the nominal phase-to-ground voltages. This matrix,

5

Otherwise, the section would not be ungrounded since the shunt capacitance of the π-model would give a reference to ground.

33 phase

denoted by Z k

, relates the phase-to-ground voltages to the phase cur-

rents. a

Ik

b

Ik c

Ik

 a a Vk – 1  Vk phase   Vb – Vb = Zk k–1  k  c c Vk – 1  Vk line

The desired series impedance, denoted by Z k

      

(3.25)

, is a 2 x 2 matrix relating

the line-to-line voltages to the phase currents as follows: a

Ik

b Ik

 ab ab Vk – 1 line  V k = Zk  –  bc bc Vk – 1  Vk

    

(3.26)

This impedance is expressed in per unit based on the nominal line-to-line phase

voltage and can be computed from Z k line

Zk

as follows:

1 phase = 1 – 1 0 ⋅ --- Z k ⋅ 0 1 –1 3

1 0 0 1 , –1 –1

(3.27)

1 where the factor of --- is due to the change of per unit voltage base. 3

3.5 Switch Model Sectionalizing switches are modeled as branches with zero impedance. For a switch between bus k – 1 and bus k, the voltage and current (or power flow) at bus k can be computed directly from the voltage and current (or power flow) at bus k – 1 and vice versa from the formulas given in Table 3.7, with no need to express the model using an admittance matrix.

34 Table 3.7 Update Formulas for Switches

Function of

V k – 1, I k

V k, I k ′

V k – 1, S k

V k, S k ′

Update Formula

V˜ k = V k – 1

(3.28)

I˜ k ′ = – I k

(3.29)

V˜ k – 1 = V k

(3.30)

I˜ k = – I k ′

(3.31)

V˜ k = V k – 1

(3.32)

S˜ k ′ = – S k

(3.33)

V˜ k – 1 = V k

(3.34)

S˜ k = – S k ′

(3.35)

3.6 Transformer Model Three-phase transformers are modeled by an admittance matrix which depends upon the connection type, the primary and secondary side taps,

and

the

leakage

admittance.

This

admittance

matrix

for

transformer k6 is BR Yk

=

pp

ps

sp Yk

ss Yk

Yk Yk

.

(3.36)

For a grounded wye to grounded wye transformer, this is a 6 x 6 complex matrix relating primary and secondary side currents and primary and secondary side phase-to-ground voltages. In the case where one side of the

6 The

transformer entering bus k.

35 transformer is ungrounded, such as a delta or ungrounded wye connection, line-to-line voltages are used and the dimension of the admittance matrix is reduced to 5 x 5. If both sides are ungrounded, line-to-line voltages are BR

used on both sides and the dimension of Y k

is 4 x 4.

In the following sections, the primary side taps for transformer k are denoted by α k , the secondary side taps by β k , and the per unit leakage admittance per phase by y k . The admittance matrices for common transformer connections are given in Table 3.8. To simplify the presentation of the relevant update formulas, the various transformer types are divided into three classes based on the grounding of their connections.

3.6.1

Class A: Primary and Secondary both Grounded or both Ungrounded

The simplest class of transformer connections will be presented first. This is the class of transformers which are either grounded on both sides or ungrounded on both sides. This includes connection types 1, 5, 6, 8, and 9. BR

In this case, each submatrix of Y k

is square and non-singular so (3.36)

can be substituted into (2.7) to solve directly for the appropriate variables. The resulting update formulas are given in Table 3.9.

3.6.2

Class B: Grounded Primary—Ungrounded Secondary

The second class of transformer connections to be presented is the class of transformers with grounded primary side and ungrounded secondary side. This includes connection types 2 and 3. For these transformers, the voltage, current, and power flow on the primary side are three-dimensional quantities, but on the secondary side they are two-dimensional quantities. There is a constraint, however, on the primary side currents

36 Table 3.8 Admittance Matrices for Common Transformer Connections Transformer Connection Type Primary

1

2

3

A

Secondary

Grounded Wye

B

Grounded Wye

B

pp

ps

Yk

Yk

sp

ss

Yk

Yk

y 100 -----k2- 0 1 0 αk 001

– yk 1 0 0 ----------αk βk 0 1 0 001

– yk 1 0 0 ----------αk βk 0 1 0 001

yk 1 0 0 ----β k2 0 1 0 001

yk 2 –1 –1 --------3α k2 – 1 2 – 1 –1 –1 2

2 1 – yk ------------------- –1 1 3α k β k –1 –2

– yk ------------------- 2 –1 –1 3α k β k – 1 2 – 1

yk 2 1 ----β k2 – 1 1

Grounded Wye

Ungrounded Wye

Grounded Wye

y 100 -----k2- 0 1 0 αk 001

Delta

– yk 1 0 –1 ----------αk βk –1 1 0 4

5

6

C

A

A

Ungrounded Wye

Grounded Wye

– yk ----------αk βk

yk 2 1 ----β k2 – 1 1

opposite† of type 2 y -----k2- 2 1 αk –1 1

– yk 2 1 ----------αk βk –1 1

– yk 2 1 ----------αk βk –1 1

yk 2 1 ----β k2 – 1 1

y -----k2- 2 1 αk –1 1

– 3 yk 1 0 ----------------αk βk 0 1

– 3 yk 1 1 ----------------αk βk –1 0

yk 2 1 ----β k2 – 1 1

Ungrounded Wye Ungrounded Wye

Ungrounded Wye

Delta

7

C

Delta

Grounded Wye

opposite† of type 3

8

A

Delta

Ungrounded Wye

opposite† of type 6

9

A

Delta

same as type 5

†i.e.

1 0 0 1 –1 –1

swap

Delta pp Yk

and

ps Yk

with

ss Yk

and

sp Yk

, respectively, then swap α k with β k .

37 Table 3.9 Update Formulas for Class A Transformers

Function of

V k – 1, I k

V k, I k ′

V k – 1, S k

V k, S k ′

Update Formula

ps – 1 pp V˜ k = ( Y k ) ( I k – Y k V k – 1 )

(3.37)

sp ss I˜ k ′ = Y k V k – 1 + Y k V˜ k

(3.38)

sp – 1 ss V˜ k – 1 = ( Y k ) ( I k ′ – Y k V k )

(3.39)

I˜ k = Y kpp V˜ k – 1 + Y kps V k

(3.40)

ps – 1 pp V˜ k = ( Y k ) ( ( S k . / V k – 1 ) ∗ – Y k V k – 1 )

(3.41)

ss ˜ ∗ S˜ k ′ = V˜ k .* ( Y sp k V k – 1 + Y k V k)

(3.42)

sp – 1 ss V˜ k – 1 = ( Y k ) ( ( S k ′ . / V k ) ∗ – Y k V k )

(3.43)

pp ps ∗ S˜ k = V˜ k – 1 .* ( Y k V˜ k – 1 + Y k V k )

(3.44)

which effectively restricts it to two degrees of freedom. This constraint can be expressed in terms of the sum of the currents on the primary side, Σ

a

b

c

denoted by I k = I k + I k + I k . For the type 2 grounded wye to ungrounded wye case, the primary currents must sum to zero. Σ

Ik = 0

(3.45)

For type 3 grounded wye to delta connections, the sum of the primary currents is related to the sum of the primary voltages as follows: y Σ Σ I k = -----k2- V k – 1 , αk Σ

a

b

(3.46)

c

where V k – 1 = V k – 1 + V k – 1 + V k – 1 . In attempting to solve for the secondary voltage V k , (2.7) yields ps

pp

Y k V k = Ik – Y k V k – 1 ,

(3.47)

38 Σ

ps

where Y k is 3 x 2. This equation has a unique solution for V k only if I k

satisfies the appropriate constraint above, in which case one of the three rows of (3.47) becomes redundant. Using the notation from Table 2.1, the solution to this equation can be written as in (3.50) in Table 3.10. Consider trying to solve for the primary side voltage from the lower half of (2.7). sp

ss

Y k V k – 1 = Ik′ – Y k V k

(3.48)

sp

In this case, Y k is 2 x 3 and, consequently, the secondary voltage and current do not uniquely specify the primary voltage. It is only possible to solve for V k – 1 if it is assumed that some information is already given, such as Σ

the sum of the primary side voltages V k – 1 . If this information is given, the solution to (3.48) can be written as in (3.52). In this case, w k used in (2.3) Σ

must include V k – 1 . Vk wk = V Σ k–1

(3.49)

Ik + 1 So, for forward calculation, there is a constraint on the sum of the primary side currents. For backward calculation, information about the sum of the primary side voltages must be available. Assuming these conditions are met, the expressions for the updates can be written as shown in Table 3.10.

39

Table 3.10 Update Formulas for Class B Transformers

Function of

V k – 1, I k

V k, I k ′, V Σk – 1

V k – 1, S k

Update Formula

ps pp V˜ k = Y k \ ( I k – Y k V k – 1 )

(3.50)

sp ss I˜ k ′ = Y k V k – 1 + Y k V˜ k

(3.51)

V˜ k – 1 =

sp

ss

Ik′ – Y k V k

111

Vk – 1 (3.53)

ps pp V˜ k = Y k \ ( ( S k . / V k – 1 ) ∗ – Y k V k – 1 )

(3.54)

sp ss S˜ k ′ = V˜ k .* ( Y k V k – 1 + Y k V˜ k ) ∗

(3.55)

V˜ k – 1 =

–1

Yk

( Sk ′ . / V k) ∗ – Y k V k

111

Vk – 1

ss

Σ

pp ps S˜ k = V˜ k – 1 .* ( Y k V˜ k – 1 + Y k V k ) ∗

3.6.3

(3.52)

Σ

pp ps I˜ k = Y k V˜ k – 1 + Y k V k

sp

V k, S k ′, V Σk – 1

–1

Yk

(3.56)

(3.57)

Class C: Ungrounded Primary—Grounded Secondary

The third class consists of transformers with ungrounded primary side and grounded secondary side. This includes connection types 4 and 7. Transformers in this class have two-dimensional quantities for the voltage, current, and power flow on the primary side, but three-dimensional quantities on the secondary side.

40 In this case, the constraint on the sum of the currents is on the secΣ

a

b

c

ondary side, i.e. on I k ′ = I k ′ + I k ′ + I k ′ , and thus affects the backward calculation. For the type 4 ungrounded wye to grounded wye case, Σ

Ik′ = 0 .

(3.58)

For type 7 delta to grounded wye connections, the constraint on the current Σ

a

b

c

is related to the sum of the secondary voltages V k = V k + V k + V k . yk Σ Σ I k ′ = -----V β k2 k

(3.59)

sp

Since Y k is 3 x 2, the appropriate constraint above must be satisfied in order to solve (3.48) for the primary voltage V k – 1 . The solution can be expressed as in (3.63) in Table 3.11. Forward calculation with transformers of class C is similar to backps

ward calculation with class B transformers. In this case, Y k is 2 x 3 and more information is needed to solve for a unique secondary voltage V k from the primary voltage and current. If the sum of the secondary voltage Σ

V k is known, that is sufficient to specify the secondary voltage completely. However, in the power flow algorithms proposed in this dissertation, it is actually current information, rather than voltage information, which is available at the secondary side. For type 7 delta to grounded wye transΣ

formers, (3.59) can be used to solve for V k from the sum of the secondary Σ

current I k ′ , allowing V k to be expressed as in (3.61). Here w k – 1 of (2.2) Σ

must include I k ′ . Vk – 1 wk – 1 =

Ik Σ

Ik′

(3.60)

41 Σ

For type 4 ungrounded wye to grounded wye connections, I k ′ gives no information about V k , so algorithms based on forward calculation using this formulation cannot handle this type of transformer connection. To summarize, the update formulas for class C transformers apply only to type 7 connections. In this case, there is a constraint on the sum of the secondary side currents for backward calculation. For forward calculation, it is necessary to have available the sum of the secondary side currents. Under these conditions, the voltage, current, and power flow updates can be expressed as shown in Table 3.11. Table 3.11 Update Formulas for Class C Transformers

Function of

V k – 1, I k, I Σk ′

V k, I k ′

Update Formula

V˜ k =

ps

V k, S k ′

pp

Ik – Y k V k – 1

(3.61)

β k2 Σ ------I k ′ yk

111

sp ss I˜ k ′ = Y k V k – 1 + Y k V˜ k

(3.62)

sp ss V˜ k – 1 = Y k \ ( I k ′ – Y k V k )

(3.63)

pp ps I˜ k = Y k V˜ k – 1 + Y k V k

(3.64)

ps

Σ V k – 1 , S k, I k ′

–1

Yk

V˜ k =

Yk

111

–1

( Sk . / V k – 1) ∗ – Y k V k – 1 pp

β k2 Σ ------I k ′ yk

(3.65)

S˜ k ′ = V˜ k .* ( Y sp V k – 1 + Y ss V˜ k ) ∗ k k

(3.66)

V˜ k – 1 = Y sp \ ( ( S k ′ . / V k ) ∗ – Y ss V k) k k

(3.67)

pp ps S˜ k = V˜ k – 1 .* ( Y k V˜ k – 1 + Y k V k ) ∗

(3.68)

Chapter 4

Network Reduction Power Flow Algorithms for Radial Systems (NR-PARS) The solution to the power flow problem is typically viewed as the solution to a set of power balance equations. It is possible, however, to look at the problem from a circuit theory point of view. The distribution system, in this context, is a circuit with one compound (three-phase) voltage source, many constant impedance elements, and possibly some constant current elements and some elements with a non-linear relationship between voltage and current (the constant power elements). If each non-linear element is replaced by a linear element which is equivalent in some sense, the new system can be solved using the standard methods of nodal or loop analysis for linear circuits. Recomputing the linear equivalents at the new solution and repeating yields a basic iterative method for solving the distribution power flow problem. In fact, this is exactly the idea behind some of the Z-matrix methods in [9; 10; 25; 27].

42

43 However, the Z-matrix methods do not directly exploit the radial topology of the typical distribution network. The network reduction methods presented in this chapter take advantage of the radial structure in the solution of the “equivalent” linear circuits. Instead of using general nodal or loop methods, network reduction techniques are applied recursively to find a driving point linear equivalent at each bus. These equivalents are then used to solve for the network voltages and currents. The acronym NR-PARS will be used to refer to the network reduction algorithms of this class. A method proposed by Berg, Hawkins and Pleines [4] falls into the NR-PARS class. This chapter proposes a more general approach which also includes the comprehensive modeling presented in Chapter 3, “Detailed Component Models”.

4.1 Detailed Solution Algorithm The general algorithm (NR-PARS), shown in Table 4.1, consists of three basic steps which are repeated until convergence is achieved. The Table 4.1 Network Reduction Method NR-PARS - The Algorithm Initialize all bus voltages. 1

Linearize system based on current bus voltages.

2

Build driving point equivalent circuit at each bus.

3

Compute all voltages & currents.

Repeat steps 1 to 3 until convergence is achieved.

44 specific network reduction technique presented in this section is based on Norton equivalent circuits and will be referred to simply as N-PARS.

4.1.1

Linearization

The first step of this algorithm is to replace all non-linear circuit element models by some linear equivalent. According to the models presented in Chapter 3, “Detailed Component Models”, a distribution system can be thought of as a radial interconnection of the elements shown in Table 4.2. Table 4.2 Network Elements Network Element

Element Type

source (substation)

constant voltage sources

Class

lines switches constant impedance elements

transformers

linear

shunt capacitors constant Z loads

constant I

constant current sources

constant S

constant complex power elements

cogenerators

non-linear

All of the elements have a linear relationship between current and voltage, except the constant complex power devices: cogenerators and constant PQ loads. In this case, the current as a function of voltage is I(V ) = ( S . / V ) ∗ , where S is the constant complex power.

(4.1)

45 The desired linear approximation to (4.1) would consist of a constant current, a constant impedance, or a combination of the two. In other words, the goal is to approximate (4.1) by a linear expression of the form I˜ (V ) = I + YV .

(4.2)

Unfortunately, (4.1) is not complex differentiable so it is not possible to use a truncated Taylor series to find Y and I in (4.2). Instead, Y is set to 0 and the current voltage is used to compute the equivalent injected current. I = (S . / V) ∗

(4.3)

This simple constant current equivalent is then used to solve the resulting linear system.

4.1.2

Build Driving Point Equivalents

The second step of the algorithm is to build up the driving point reduced equivalent circuit at each bus. Specifically, the driving point Norton equivalent circuits are constructed for the sending end of the incoming branch of each bus. In other words, for each bus, the Norton equivalent is constructed for the bus’s incoming branch combined with the sub-network supplied through that branch. Since the networks being dealt with are multi-phase1 networks, these are compound Norton equivalents where an admittance matrix is used to represent the constant impedance part and a vector to represent the current injections. The laterals are processed starting at the end buses and working backward toward the source in the RBF order described on page 9 under Section 2.2.1, “Indexing Scheme”. For each bus this process consists of two parts:

1 i.e.

single-phase, two-phase, and three-phase.

46 1. Build the Norton equivalent at the bus. 2. Combine with the incoming branch. First, the appropriate currents and admittances are summed to build the Norton equivalent at bus k. Each bus could have one or more of the following connected in parallel: • load • shunt capacitor • cogenerator • Norton equivalent for outgoing branches

Ik′ Vk

YEQk ′

IEQk ′

Figure 4.1 Norton Equivalent at Bus k Using the notation from Figure 4.1, the current I k ′ is written as a function of the bus voltage V k . I k ′ = I EQk ′ – Y EQk ′ Vk

(4.4)

The two parameters of the Norton equivalent are given by Y EQk ′ = Y Lk + Y Ck + ∑ Y EQj

(4.5)

I EQk ′ = I Lk + I Gk + ∑ I EQj

(4.6)

j

j

where the meaning of each variable is given in Table 4.3.

47 Table 4.3 Notation for (4.5) and (4.6) Variable

Interpretation

From

Y Lk

admittance of constant impedance load

Table 3.2

Y Ck

admittance of shunt capacitor

Table 3.4

I Lk

current injection of constant current or constant PQ load

Table 3.3

I Gk

current injection of cogenerator

Table 3.5

Y EQj , I EQj

Norton equivalent for incoming branch of bus j

Table 4.4

j

index of bus at receiving end of an outgoing branch of bus k

network connectivity

Secondly, this Norton equivalent at bus k is combined with the bus’s incoming branch as illustrated in Figure 4.2. The parameters Y EQk and I EQk are computed from their primed values and the parameters of the

Ik′

Ik Vk –1

incoming branch

Vk

YEQk ′

IEQk ′

Ik Vk –1

YEQk

IEQk

Figure 4.2 Combine with Incoming Branch

48 branch. To accomplish this, (2.7) and (4.4) are combined so as to eliminate the variables V k and I k ′ , yielding the following expression of I k as a function of V k – 1 . 11

12

22

–1

12 [ –Y k

22 (Yk

–1

I k = [ Y k – Y k ( Y k + Y EQk ′ ) –

+ Y EQk ′ )

21

Yk ] Vk – 1

(4.7)

I EQk ′ ]

This is in the same form as the relationship between current and voltage for the combined Norton equivalent I k = Y EQk V k – 1 – I EQk ,

(4.8)

making it possible to extract directly from (4.7) the expressions for Y EQk and I EQk . Table 4.4 gives the resulting formulas for each of the three types of branches. Since a switch is modeled as a zero impedance connection, it is not necessary to use (4.7). The parameters computed at bus k are not changed by combining them with an incoming switch. Table 4.4 Formulas for Combining with Incoming Branch Branch Type

line

switch

Combination Formula –1 1 1 Y EQk = Z k–1 + --- Y k – Z k–1  Z k–1 + --- Y k + Y EQk ′  Z k–1 2 2 –1 1 I EQk = Z k–1  Z k–1 + --- Y k + Y′ EQk  I EQk ′ 2

(4.10)

Y EQk = Y EQk ′

(4.11)

I EQk = I EQk ′

(4.12)

pp

transformer

(4.9)

ps

ss

Y EQk = Y k – Y k ( Y k + Y EQk ′ )

–1

I EQk = – Y kps ( Y ss + Y EQk ′ ) –1 I EQk ′ k

sp

Yk

(4.13) (4.14)

To summarize, the driving point Norton equivalent is computed for the incoming branch of each bus starting with the most deeply nested sub-

49 laterals and working back toward the main feeder. At each bus the procedure consists of two parts. First, all of the current sources and admittances connected in parallel at the bus are summed together to give the Norton equivalent at the bus for the part of the network supplied through that bus. Then, this Norton equivalent is combined with the incoming branch to determine the driving point Norton equivalent for that branch. At the end of this second step of the algorithm, the driving point Norton equivalent for the incoming branch of each bus is known. In other words, any branch and the sub-network it supplies may be replaced by its Norton equivalent for the purposes of computation.

4.1.3

Calculate Voltages and Currents

The result of the second step of the algorithm given in Table 4.1 is that, given the voltage at a bus, the current to each of its outgoing branches can be easily computed from the corresponding driving point equivalent. The third step of the algorithm is based on this result. It involves starting at the source, where the voltage is known, and computing voltages and currents toward the end buses. Once again, the entire network is traversed, this time in the BF order described on page 11 under Section 2.2.1, “Indexing Scheme”. As in the previous step, at each bus there is a two part calculation to be performed. First, the current is computed for the bus’s incoming branch, then the bus voltage is updated. For bus k this means calculating I k then V k . The new value of I k is computed directly from the voltage at bus k – 1 and the Norton equivalent associated with the incoming branch of bus k. The formula is given in (4.8) which is repeated here.

50 (4.15)

I k = Y EQk V k – 1 – I EQk

The second part, updating the bus voltage V k , is dependent on V k – 1 , I k , and the parameters of the branch. The upper portion of (2.7) gives 12

11

Y k V k = Ik – Y k V k – 1 .

(4.16)

12

If Y k is square and non-singular this is sufficient, as in the case of a dis12

tribution line. However, for some transformer types Y k is 2 x 3, and more information is needed: specifically, information about the part of the network connected to the secondary side of the transformer. This information is available in the form of the Norton equivalent at bus k, that is, I EQk ′ and Y EQk ′ . These parameters were computed in the previous step and can be stored for each bus whose incoming branch is a transformer. Combining the lower half of (2.7) with (4.4) by eliminating I k ′ yields an equation which is typically solvable for V k . For switches, since they are zero impedance, the voltage does not drop so V k can be taken directly from the value at the parent bus. The resulting voltage update formulas for lines, switches, and transformers are summarized in Table 4.5. Table 4.5 Voltage Update Formulas Branch Type line switch transformer

Voltage Update Formula 1 V k = V k – 1 – Z k  I k – --- Y k V k – 1  2

(4.17)

Vk = Vk – 1

(4.18)

ss

V k = ( Y k + Y EQk ′ )

–1

sp

( I EQk ′ – Y k V k – 1)

(4.19)

51

4.1.4

Termination Criterion

The three steps, linearization, building driving point equivalents, and computing voltages and currents, are repeated until convergence is achieved. The algorithm is terminated when the norm of the change in voltage from one iteration to the next is smaller than some predefined tolerance.2

4.2 Implementation Several specific techniques can be used in the implementation of the Norton reduction algorithm (N-PARS) to improve the efficiency and robustness.

4.2.1

Linearity Check

It is conceivable that some networks do not contain any cogenerators or constant PQ loads. In this case, the circuit is already a linear circuit. Not only is the first step of the algorithm, linearization, unnecessary, but there is no need to iterate. The circuit is solved directly in one iteration. In the implementation, a check for non-linear components can be performed at the beginning. If none are found, the program proceeds directly to Norton reduction and terminates after one iteration. Without this check, the program would do two iterations and find that there is no change between iterations. The consequence of implementing this check is a 50% savings in computation for linear networks.

2

Alternatively, the termination criterion could be based on the norm of the mismatch between the given power injection and the power injection computed from the voltage and current injection at the current iteration.

52

4.2.2

Improved Line Update

The formulas given in (4.9) and (4.10) for combining the Norton equivalent at bus k with its incoming line involve two matrix inversions. Inversion of a matrix is an expensive operation in general, and may be illconditioned as well, depending on the matrix. The formulas below offer a more efficient and robust implementation. They are equivalent to (4.9) and (4.10) but require less computation (only one inverse instead of two) and are better conditioned for some line parameters. 1 1 Y EQk =  --- Y k + I +  --- Y k + Y EQk ′  Z k 2 2 I EQk =

1 I +  --- Y k + Y EQk ′  Z k 2

–1  

1 - Y + Y EQk ′    -2 k

–1

I EQk ′

(4.20) (4.21)

Here I is the appropriately sized identity matrix.

4.2.3

Storage of Intermediate Variables

In the case where the circuit contains non-linear elements, the only parameters which change from iteration to iteration are the currents injected by these non-linear elements. In particular, as a new iteration begins, only I Lk (for constant PQ loads) and I Gk in (4.6) take on new values based on the updated bus voltages. These changes affect only I EQk ′ and ultimately I EQk , but not Y EQk ′ and Y EQk . These admittances remain constant throughout all iterations and therefore need not be recomputed. This means that (4.5), (4.9), 3 (4.11), and (4.13) need only be computed once during the first iteration.

3 Or

(4.20) in actual implementation.

53 A further savings in computation is achieved by storing the constant coefficient of I EQk ′ in (4.10)4 and (4.14). This reduces the second step of the algorithm to a simple summation of the injected currents at a bus and matrix-vector multiplication to combine with the bus’s incoming branch. Saving these intermediate quantities and avoiding the recomputation of the admittances is somewhat analogous to factoring Ybus only once and saving it in a factored form in an Implicit Zbus Gauss type of power flow algorithm such as those used in [9; 10; 27]. A more intelligent implementation could even detect areas of the network which do not contain constant PQ elements and only perform the updates on the areas with quantities which actually change.

4.3 Variations There are other variations of the general network reduction power flow method (NR-PARS) outlined in Table 4.1, “Network Reduction Method”, on page 43. A method based on admittance equivalent only (Y-PARS) was also implemented. Berg, Hawkins and Pleines initially presented a method based on this idea in [4], which has been extended to include the more general modeling of Chapter 3. Instead of Norton equivalent circuits, admittance equivalents are used. The first step of the algorithm, linearization, now also includes converting constant current elements to their constant impedance equivalents. Constant current loads, constant PQ loads, and cogenerators are all converted to the equivalent admittance via the formulas in Table 3.2 and

4 Or

(4.21) in actual implementation.

54 Table 3.5. These admittances are summed together to form the admittance equivalent at bus k.

Ik′ YEQk ′

Vk

Figure 4.3 Admittance Equivalent at Bus k

Using the notation from Figure 4.3, the current I k ′ is written as a function of the bus voltage V k and, since the model only includes admittances, (4.4) becomes I k ′ = – Y EQk ′ Vk

(4.22)

and (4.5) becomes Y EQk ′ = Y Lk + Y Ck + Y Gk + ∑ Y EQj .

(4.23)

j

The formulas for combining the equivalent circuit at bus k with its incoming branch remain the same except that only the admittance formulas are needed. For the third step of the algorithm, calculating voltages and currents, similar changes are made. For computing the new value of I k , (4.15) becomes simply I k = Y EQk V k – 1 .

(4.24)

The formulas in Table 4.5 for updating the bus voltages remain the same except for the case of transformers. For transformers, (4.19) is replaced by ss

V k = – ( Y k + Y EQk ′ )

–1

sp

Yk Vk – 1 .

(4.25)

55 Though the algorithm works fine, there seems to be little or no advantage of Y-PARS over N-PARS. It is slightly more efficient for the case where iteration is not necessary. However, the set of systems for which iteration is not required has now been reduced to cases with no cogenerators and only constant impedance loads. In the general iterative case, each iteration requires all parameters to be recomputed. This is much more computationally expensive than N-PARS implemented using the techniques described in Section 4.2.3, “Storage of Intermediate Variables”.

4.4 Convergence Analysis The convergence characteristics of N-PARS are exactly the same as those of the familiar Z-matrix methods, including the Implicit Zbus Gauss method [9; 10; 27]. Both N-PARS and the Z-matrix methods repeatedly solve the linear circuit obtained by replacing all constant PQ devices with their equivalent current injections at each iteration. In a circuit with no constant power components, the solution is therefore obtained in a single iteration and convergence is not an issue. For the more general case, including constant power devices, convergence behavior is discussed in [25]. The general conclusion is that convergence is best when load powers are small, a condition which is met by the typical distribution system.

4.5 Comments The power flow algorithms described in this chapter, particularly the Norton reduction method (N-PARS), have several characteristics which make them uniquely suited to certain situations. First of all, N-PARS is a general method for solving the radial power flow problem which can handle

56 all of the transformer connection types listed in Table 3.8, “Admittance Matrices for Common Transformer Connections”, on page 36. It is particularly efficient for networks which contain no constant PQ elements. The equivalent circuit parameters themselves, computed during the solution process, may be useful for certain applications. They could be used, for example, to replace a section of the network with a simple equivalent linear model, while focusing on the effects of changes in another part of the network. Since the NR-PARS approaches exploit the radial structure of the network, the work required for each iteration is proportional to the number of buses. If the number of iterations remains constant, the work grows linearly with the size of the system, making NR-PARS suitable for very large scale systems.

Chapter 5

Backward/Forward Sweep Power Flow Algorithms for Radial Systems (BFS-PARS) One of the distinguishing features of the radial distribution network is that there is a unique path from any given bus back to the source. This is the key feature exploited by the backward/forward sweep class of algorithms (BFS-PARS) presented in this chapter. These methods are based on updating voltages and currents (or power flows) along these unique paths. There are many variations to BFS-PARS. This chapter will first present in detail a fairly general one, VI-VI-PARS, then discuss several possible modifications which lead to the other variations. Several methods based on the backward/forward sweep concept have been proposed by others [3; 11; 18; 19; 20; 23]. In general, however, the approaches proposed in the literature do not include the comprehensive modeling covered by the general framework presented in this chapter. In fact, none of the methods proposed in [3; 11; 18; 19; 20; 23] handle transformers of type 2, 3, 4, or 7. In addition, [3; 19; 20; 23] only propose single-phase implementations,

57

58 although most of the extension to three-phase is straightforward. In this chapter, existing variations of the backward/forward sweep approach are extended to handle the comprehensive modeling of Chapter 3, and some new variations are introduced.

5.1 Detailed Solution Algorithm The general algorithm (BFS-PARS), shown in Table 5.1, consists of two basic steps, backward sweep and forward sweep, which are repeated until convergence is achieved. The backward sweep is primarily a current or power flow summation with possible voltage updates. The forward sweep is primarily a voltage drop calculation with possible current or power flow updates. Table 5.1 Backward/Forward Sweep Method BFS-PARS - The Algorithm Initialize all bus voltages. 1

Backward Sweep:

Sum currents or power flows (and possibly update voltages).

2

Forward Sweep:

Calculate voltage drops (and possibly update currents/power flows).

Repeat steps 1 and 2 until convergence is achieved.

Using the boundary condition of zero current and power flow out of the end of each lateral, the backward sweep computes the currents or power flows injected into the beginning of each lateral as a function of the end voltages. The forward sweep is a function of these currents or power

59 flows injected into each lateral, and computes the end voltages using the specified source voltage as a boundary condition. Table 5.2 Detail on Backward/Forward Sweeps Backward Sweep

Forward Sweep

currents or power flows injected into each lateral

end voltages

end voltages

currents or power flows injected into each lateral

By:

current or power flow summation†

voltage drop calculation‡

Based on boundary condition:

zero current or power flow out of end of lateral

specified source voltage

Update: As a function of:

†With ‡With

optional voltage updating. optional current/power flow updating.

The specific method presented in this section (VI-VI-PARS) is based on current (as opposed to power flow), and updates both current and voltage in each of the backward and forward sweeps. Figure 5.1 shows a single feeder example system with notation that will be useful for describing the backward and forward sweeps. This notation will also be used for a general lateral i, where bus zero is not the source in a strict sense but rather the

V0 = V0

VN

source

end bus I1

IN + 1 = 0 Figure 5.1 Single Feeder Example

60 bus where lateral i branches off of lateral i – 1 . The subscripts only refer to the bus index and V 0 refers to the voltage as calculated from lateral i – 1 .

5.1.1

Backward Sweep

Once the bus voltages are initialized, the algorithm begins with a backward sweep, processing the laterals in the RBF 1 order. Each lateral is traversed from the end bus toward the source and the currents and voltages are updated at each bus. At each bus k, the update formula (2.3) from Table 2.3 is applied to compute the voltage and current at the previous bus. w k – 1 = g k(w k)

(5.1)

This update formula, repeated here in (5.1), is implemented according to the four steps detailed in Table 2.5, which is also repeated here as Table 5.3 for convenience. Table 5.3 Implementation of (5.1) Procedure

Details

1

Compute I Gk , I Ck , and I Lk from V k .

see Tables 3.3 - 3.5

2

Compute I k ′ via KCL at bus k.

see (2.20) and (5.2)

3

Compute V k – 1 from V k and I k ′ .

see Tables 3.6 - 3.11

4

Compute I k from V k – 1 , V k and I k ′ .

see Tables 3.6 - 3.11

First, the currents I˜ Lk , I˜ Ck , and I˜ Gk , injected by a load, shunt capacitor, and cogenerator, respectively, are computed from the value of the bus voltage V k at the current iteration. These injected currents are found based on the appropriate equations from Tables 3.3, 3.4, and 3.5.

1 See

page 9 under Section 2.2.1, “Indexing Scheme”.

61 The next step is to apply KCL at bus k to find the current I k ′ injected by bus k into its incoming branch. The application of KCL at bus k is described in Section 2.3.2, “Application of KCL”, on page 19. In this case, (2.20) is solved for I k ′ to yield I k ′ = I˜ Gk + I˜ Ck + I˜ Lk –  

∑

j ∈ Ak

I j – I k + 1 

(5.2)

where the currents I j are the currents injected into sub-laterals branching off from bus k, and I k + 1 is the current injected into the outgoing branch leading to the next bus on the same lateral. The third and fourth steps compute the voltage V k – 1 and current I k at the previous bus using the update formulas in Tables 3.6 through 3.11. For each lateral, the backward sweep updates the current injected into the lateral as a function of the end voltage. Using the notation from the single feeder example in Figure 5.1, I 1 is updated as a function of V N . Given the boundary condition I N + 1 = 0 and a value for V N , the update formula (5.1) is applied recursively, starting at the end bus and ending at the lateral’s supplying bus with w 0 , as shown in (5.3).  V  N  w N – 1 = g N (w N ) = g N   0 

 V  N  w N – 2 = g N – 1(w N – 1) = g N – 1 • g N   0 

w0 =

V0 I1

(5.3)

 V  N  = g 1(w 1) = g 1 • … • g N – 1 • g N   0 

The lower half of the composite function g 1 • … • g N – 1 • g N , which yields w 0 , is the desired current as a function of the end voltage and will be denoted

62 I 1 = I˜ 1(V N ) .

(5.4)

This value of the current is stored for use during the KCL calculation at this lateral’s supplying bus during the current backward sweep and the succeeding forward sweep. The upper half, which is the voltage part, is V˜ 0(V N ) , and is used only for mismatch calculation. At the solution, the mismatch V˜ 0(V N ) – V 0 between the computed voltage V˜ 0(V N ) and the specified voltage V 0 must be zero. To evaluate (5.4) for a given lateral, the currents injected into all of its sub-laterals must be known, since they are needed for the application of KCL in (5.2). This means the laterals must be processed in a specific order, such as the RBF1 order used here. Since the level L laterals have no sublaterals, they are processed first. Next, the currents injected into the sublaterals of all level L – 1 laterals are known, so the level L – 1 laterals can be processed. Each lateral is processed in this manner, starting with level L, then moving to level L – 1 , then L – 2 , etc. Finally, the main feeder is processed, completing the backward sweep. In the process, all voltages and currents in the network are updated. In particular, the currents injected into each lateral are updated as a function of the end voltages.

5.1.2

Forward Sweep

The second half of each iteration of the BFS-PARS algorithm is the forward sweep which starts at the source and moves toward the end buses. During the forward sweep, the laterals are processed in the BF2 order.

2 See

page 11 under Section 2.2.1, “Indexing Scheme”.

63 Each lateral is traversed from the supplying connection toward the end bus. Once again, a voltage and current update are performed at each bus. The corresponding update formula for forward calculation is w k = f k(w k – 1) ,

(5.5)

which is (2.2) from Table 2.3. The formula (5.5) is also implemented in four steps. These steps are detailed in Table 2.4 which is repeated here as Table 5.4 for convenience. Table 5.4 Implementation of (5.5) Procedure

Details

1

Compute V k from V k – 1 and I k .

see Tables 3.6 - 3.11

2

Compute I k ′ from V k , V k – 1 and I k .

see Tables 3.6 - 3.11

3

Compute I Gk , I Ck , and I Lk from V k .

see Tables 3.3 - 3.5

4

Compute I k + 1 via KCL at bus k.

see (2.20) and (5.6)

The first two steps are to compute the voltage V k and current I k ′ at the current bus from the voltage and current at the previous bus, V k – 1 and I k , respectively. These updates are done according to the formulas given in Tables 3.6 through 3.11. The third step is to update the currents I˜ Lk , I˜ Ck , and I˜ Gk injected by a load, shunt capacitor, and cogenerator, respectively. These injected currents are updated from the bus voltage V k according to the appropriate equations from Tables 3.3, 3.4, and 3.5. The last step is to apply KCL to update the current I k + 1 on the branch going out to the next bus on the lateral. In this case, (2.20) of Section 2.3.2, “Application of KCL”, is solved for I k + 1 , yielding

64 I k + 1 = I˜ Gk + I˜ Ck + I˜ Lk –  

∑

j ∈ Ak

I j – I k′ . 

(5.6)

The values for the currents I j injected into sub-laterals are those computed during the preceding backward sweep. For each lateral, the forward sweep updates the end voltages as a function of the currents injected into the beginning of each lateral. In the notation of the single feeder example of Figure 5.1, V N is updated as a function of I 1 . Given the boundary condition V 0 = V 0 and a value for I 1 , the update formula (5.5) is applied recursively starting at the lateral’s supplying bus and terminating at the end bus with w N as shown in (5.7).   V   0 w 1 = f 1(w 0) = f 1    I1    V   0 w 2 = f 2(w 1) = f 2 • f 1    I1  wN =

VN IN + 1

(5.7)

  V   0 = f N (w N – 1) = f N • … • f 2 • f 1    I1 

The upper half of the composite function f N • … • f 2 • f 1 , which yields w N , is the desired end voltage as a function of the current injected into the lateral and will be denoted as V N = V˜ N ( I 1) .

(5.8)

This end voltage is stored for use during the next backward sweep. The lower half of the function, I˜ N + 1( I 1) , is used only as a mismatch. At the solution, this current must be zero. To evaluate (5.8) for a given lateral, the voltage at its supplying bus must be known since it is needed for the first step in Table 5.4. This means

65 the laterals must be processed in a specific order, such as the BF2 order used here. Since the source voltage is given, the main feeder is processed first. Then, since the voltages at the supplying buses for all level 2 laterals are known, the level 2 laterals can be processed, followed by level 3, etc. Each lateral is processed in this manner until all the level L laterals have been updated, completing the forward sweep. In the process, all voltages and currents in the network are updated. In particular, all of the end voltages are updated as a function of the currents injected into each lateral.

5.1.3

Termination Criterion

The backward and forward sweeps are repeated until convergence is achieved. There are several ways of detecting convergence. The voltage mismatches V˜ 0(V N ) – V 0 at the beginning of each lateral, calculated during the backward sweep, must be zero at the solution. Likewise, the current mismatches I˜ N + 1( I 1) at the ends of the laterals, computed during the forward sweep, must also be zero. Requiring that the norm of either of these mismatches be smaller than some tolerance is a suitable termination criterion. It is also possible to use the termination criterion used for the network reduction methods of Chapter 4. This approach requires that the norm of the difference in bus voltages between iterations be smaller than a given tolerance. In fact, this is the only one of the three approaches which is suitable for some of the variations of BFS-PARS.

66

5.2 Implementation For the most part, the implementation of BFS-PARS is very straightforward. Special attention, however, is needed for laterals which contain the class B and class C transformers. In particular, in step 3 of Table 5.3 and step 1 of Table 5.4 it is necessary to make some modifications to the standard backward and forward sweeps described above. Consider lateral i whose first bus k has a transformer as its incoming branch.3

5.2.1

Class B Transformers

Transformers of class B have three dimensional voltages and currents on the primary side, while these quantities at the secondary side are two dimensional. These transformers are described in detail in Section 3.6.2, “Class B: Grounded Primary—Ungrounded Secondary”, where the relevant update formulas are given in Table 3.10. 5.2.1.1

Backward Sweep

In step 3 of Table 5.3, the primary voltage is being computed from the secondary voltage and current. For class B transformers, the corresponding update formula is V˜ k – 1 =

sp

Yk

111

–1

ss

Ik′ – Y k V k Σ Vk – 1

(5.9)

from (3.52) in Table 3.10. As explained in Section 3.6.2, “Class B: Grounded Primary—Ungrounded Secondary”, it is only possible to solve for V k – 1 if it

3 Due

to the implementation explained in Section 2.2.2.2, “Breadth-First Search”, transformers always appear as the first branch on a lateral.

67 Σ

is assumed that some information is already given, such as the sum V k – 1 of the primary side voltages. Σ

Since (5.9) assumes that V k – 1 is available, BFS-PARS must provide a reasonable estimate which becomes more accurate at each iteration as the algorithm moves toward convergence. The sum of the primary side voltage computed during the previous forward sweep provides such an estiΣ

mate and is stored as V k – 1 for use during the backward sweep. For the Σ

first iteration, the initial primary voltage is used to compute V k – 1 . For the typical flat start, with all voltages assumed to be balanced 1 per unit, this Σ

yields V k – 1 = 0 . For a lateral with a class B transformer, the typical backward sweep Σ

function of (5.4) is a function of V k – 1 as well as V N .  V N ˜ I 1 = I 1  Σ  Vk – 1

   

(5.10)

Since, in this case, V N is 2 x 1, this is still a 3 x 1 function of three variables. 5.2.1.2

Forward Sweep

During the forward sweep, step 1 of Table 5.4 computes the secondary voltage from the primary voltage and current. The corresponding update formula for class B transformers is ps pp V˜ k = Y k \ ( I k – Y k V k – 1 )

(5.11)

from (3.50) in Table 3.10. As explained in Section 3.6.2, “Class B: Grounded Primary—Ungrounded Secondary”, this formula yields a unique solution

68 for the secondary voltage V k only if the sum of the primary side currents Σ

I k meets the appropriate constraint in (3.45) or (3.46). Since the primary side current computed during the forward sweep does not, in general, satisfy the appropriate constraint, I k must be modified to meet the constraint before using it in (5.11). The value used in (5.11) should be as close to the computed I k as possible while meeting the necessary constraint. A reasonable approximation to this closest point can be found by adding or subtracting equal values from each element of I k to satisfy the constraint. This is done by computing the constraint mismatch, which is Σ

Σ

∆I k = I k

(5.12)

for type 2 grounded wye to ungrounded wye transformers and y Σ Σ Σ ∆ I k = I k – -----k2- V k – 1 αk

(5.13)

for type 3 grounded wye to delta transformers. A third of this mismatch is then subtracted from each element of I k to give a new I k which satisfies the relevant constraint and can be used in (5.11) to compute V k . Σ ∆I k 1 I k ← I k – --------- 1 3 1

(5.14)

The modified update formula, including the adjustment to the primary side current I k , can be written in a closed form as  ∆I Σ  1 ps pp k ˜ V k = Y k \ I k –  -------- 1 – Yk Vk – 1 .   3 1

(5.15)

69 This small modification to I k allows the forward sweep to continue. The typical forward sweep function of (5.8) is modified to reflect the imporΣ

tance of the update to V k – 1 which must be stored for the backward sweep. VN Σ

=

Vk – 1

V˜ N ( I 1) Σ V˜ k – 1( I 1)

(5.16)

Again, this is still a three dimensional function of three variables. A three dimensional mismatch can also be formed by taking the constraint misΣ match ∆ I k along with the current I˜ N + 1( I 1) out of the end of the lateral,

which in this case is 2 x 1. The relatively small adjustment to the primary side current, shown in (5.14), produces secondary side voltages and currents which are consistent with the primary side current constraint. During the succeeding backward sweep, these updated voltages produce a primary side current which more closely meets the constraint.

5.2.2

Class C Transformers

Transformers of class C have three dimensional voltages and currents on the secondary side, while these quantities at the primary side are two dimensional. These transformers are described in detail in Section 3.6.3, “Class C: Ungrounded Primary—Grounded Secondary”, where the relevant update formulas are given in Table 3.11. 5.2.2.1

Forward Sweep

For class C transformers, step 1 of Table 5.4 during the forward sweep is similar to the backward sweep for class B transformers. The relevant update formula is

70

ps Yk

V˜ k =

–1

pp

Ik – Y k V k – 1

(5.17)

β k2 Σ ------ I k ′ yk

111

from (3.61) in Table 3.11. Once again, additional information is needed for the evaluation of V k in (5.17). This additional information at the secondary side of the transformer must come from the previous backward sweep. Since the backward sweep is based on a boundary condition of zero end current, it is current information that is available at the transformer secondary. As explained in Section 3.6.3, “Class C: Ungrounded Primary— Grounded Secondary”, this eliminates type 4 ungrounded wye to grounded Σ

Σ

wye transformers which have no direct relationship between I k ′ and V k . Consequently, (5.17) is only valid for type 7 delta to grounded wye transformers and the rest of this discussion of the handling of class C transformers will be restricted to this type. Since it was computed during Σ

the previous backward sweep, the sum of the secondary side current I k ′ is available for the evaluation of (5.17). The typical forward sweep function of (5.8), for a lateral with a class C transformer, becomes

VN

 I 1 ˜ = V N  Σ  Ik′

 ,  

(5.18)

Σ

where V N is 3 x 1, I 1 is 2 x 1, and I k ′ is a scalar. 5.2.2.2

Backward Sweep

The backward sweep is similar to the forward sweep for class B transformers. The relevant update formula for step 3 of Table 5.3 is sp ss V˜ k – 1 = Y k \ ( I k ′ – Y k V k )

(5.19)

71 from (3.63) in Table 3.11, which requires that the constraint in (3.59) be satisfied. Since the secondary voltage and current sums computed in the current backward sweep typically do not meet this constraint, some modification is required to proceed with the backward sweep. The forward sweep Σ

requires an accurate value for I k ′ so it is stored as is and the voltage V k is chosen for adjustment. First, a constraint mismatch is computed. 2 Σ Σ βk Σ ∆V k = V k – -----I ′ yk k

(5.20)

A third of this mismatch is then subtracted from each element of V k to give a new value which satisfies (3.59) and can be used to solve for V k – 1 in (5.19). Σ ∆V k 1 V k ← V k – ----------- 1 3 1

(5.21)

The modified update formula, including the adjustment to the secondary side voltage V k , can be written in a closed form as

V˜ k – 1

  Σ β k2 Σ     V k – ------ I k ′  1  yk    sp ss  = Y k \ I k ′ – Y k  V k –  --------------------------- 1  .     3    1 

(5.22)

This small change in V k allows the backward sweep to continue. The typical backward sweep function of (5.4) is modified to reflect the imporΣ

tance of the update to I k ′ which must be stored for the forward sweep. I1 Σ Ik′

=

I˜ 1(V N ) Σ I˜ k ′(V N )

(5.23)

72 Both the function and the independent variable V N are still 3 x 1. A three dimensional mismatch is also formed by combining the constraint misΣ

match ∆V k of (5.20) with the two dimensional voltage mismatch at the beginning of the lateral, V˜ 0(V N ) – V 0 . The relatively small adjustment to the secondary side voltage, shown in (5.21), produces primary side voltages and currents which are consistent with the secondary side constraint. During the next forward sweep these updated currents produce a secondary voltage which more closely meets the constraint.

5.3 Variations There are many variations to the general BFS-PARS of Table 5.1 and Table 5.2. Some are based on current and others on power flow. Some update voltages during the backward sweep and some do not. Some update currents or power flows during the forward sweep and some do not. The naming convention used for the various methods indicates first the variables updated during the forward sweep, then those updated during the backward sweep. V is used for voltage, I for current, and S for power flow.

5.3.1

VI-VI-PARS

VI-VI-PARS is the particular variation described in detail in the preceding sections. It is based on current and it updates both voltages and currents in both the backward and forward sweeps.

73

5.3.2

VS-VS-PARS

VS-VS-PARS modifies the VI-VI-PARS approach, presented in detail in foregoing sections of this chapter, by using power flow rather than current. The approach presented in [3] is based on the same idea. The backward sweep computes the power flow injected into each lateral as a function of the end voltages by starting with the boundary condition of zero power flow out of the end of the laterals. The forward sweep starts with the specified source voltage as a boundary condition and computes the end voltages as a function of the power injected into each lateral. At each step of the algorithm, where a current is used in VI-VI-PARS, the corresponding power flow is used in VS-VS-PARS. The update function used at each bus during the backward sweep is (2.5) instead of (2.3), and for the forward sweep it is (2.4) instead of (2.2). The class B and class C transformers are the only exception to the change from current to power flow. They are still handled exactly as described in Section 5.2.1, “Class B Transformers”, and Section 5.2.2, “Class C Transformers”. Due to the fact that the constraints are constraints on current and voltage, power flows are converted to the equivalent currents and these currents are used as described. For instance, for class B transformers during the forward sweep these currents are adjusted as in (5.14). Likewise, the power at the secondary of class C transformers during the backward sweep is converted to the equivalent current and stored for use during the forward sweep. Hence, (5.18) becomes

VN

 S  1 ˜ = V N  Σ   Ik′ 

(5.24)

74 and (5.23) becomes S1 Σ

Ik′

=

S˜ 1(V N ) Σ I˜ k ′(V N )

(5.25)

There are two essential differences between VS-VS-PARS and VI-VI-PARS. One difference has to do with computational efficiency. VS-VS-PARS typically requires more computation than VI-VI-PARS. Only certain parts of the calculation, such as the computation of the power and current injections, respectively, for constant PQ loads are more efficient in VS-VS-PARS. Even in networks with all constant PQ loads these gains are more than offset by the increase in computation for other operations such as voltage drop calculation. The other essential difference in VS-VS-PARS is that the current injected into each lateral ( S 1 . / V 0 ) ∗ changes from the time S 1 is computed during the backward sweep to the time S 1 is used during the forward sweep. In VI-VI-PARS the current I 1 is held constant between the time it is computed in the backward sweep and the time it is used in the forward sweep. In VS-VS-PARS, it is the power S 1 that is held constant and the current I 1 changes depending on the change in the voltage. Although this does not seem to have a significant effect on convergence, the sequence of iterates does differ for the two algorithms. The voltage mismatch at the beginning of each lateral calculated during the backward sweep is the same as the voltage mismatch computed by VI-VI-PARS. The forward sweep yields power mismatches at the end of each lateral instead of current mismatches.

75

5.3.3

V-VI-PARS

This variation is identical to VI-VI-PARS in the backward sweep. The forward sweep updates only voltages. A method based on a slightly modified version of this idea was proposed in [18]. Table 5.4 shows the procedure performed at each bus during the forward sweep when both voltages and currents are updated. In V-VI-PARS, steps 2, 3, and 4 are not performed. The value used for I k in step 1 is the value computed during the previous backward sweep. For this method, it is necessary to store this current for each bus during the backward sweep. For VI-VI-PARS, it is only necessary to store the currents injected into each lateral, not the currents at each bus. For class B transformers it is still necessary to modify I k during the forward sweep according to (5.14) in order to satisfy the appropriate constraint in (3.45) or (3.46). Since the currents are not updated in the forward sweep, the current mismatch at the end of each lateral is not available as a termination criterion for this method.

5.3.4

V-VS-PARS

V-VS-PARS is the power flow based counterpart to V-VI-PARS. The backward sweep is identical to that of VS-VS-PARS and the forward sweep updates only voltages based on the power flow values S k computed during the backward sweep. Since no power flows are updated during the forward sweep, there are no power mismatches available at the ends of the laterals for use as a termination criterion.

76

5.3.5

VI-I-PARS

The forward sweep in this variation is identical to that of VI-VI-PARS. The backward sweep, however, does not update voltages, only currents. In Table 5.3, which details the procedure at each bus during the backward sweep, step 3 is skipped. For step 4, I k is computed using values of voltage computed in the previous forward sweep. For class C transformers, it is still necessary to modify the secondary voltage V k during the backward sweep according to (5.21) in order to satisfy the constraint in (3.59). Since no voltages are updated during the backward sweep, the voltage mismatch at the beginning of each lateral is not available as a termination criterion for this method.

5.3.6

VS-S-PARS

This is the power flow based version of VI-I-PARS. The forward sweep is identical to that of VS-VS-PARS, but the backward sweep, like VI-I-PARS, does not update voltages. Voltage values from the previous forward sweep are used where needed. Once again, no voltage mismatch is available as a termination criterion for this method.

5.3.7

V-I-PARS

V-I-PARS uses the backward sweep of VI-I-PARS, which updates only currents, and the forward sweep of V-VI-PARS, which updates only voltages. The methods proposed in [11] and [23] are based on this concept of simple current summation in the backward sweep and voltage drop calculation in the forward sweep.

77 This method computes neither current nor voltage mismatches, and the only convergence criterion available, of the three presented in Section 5.1.3, “Termination Criterion”, is the one based on the difference in voltage between iterations.

5.3.8

V-S-PARS

V-S-PARS is the power flow based counterpart to V-I-PARS, using the backward sweep of VS-S-PARS and the forward sweep of V-VS-PARS. This method uses simple power flow summation for the backward sweep and voltage drop calculation for the forward sweep, which is also the basis for the methods proposed in [19] and [20].

5.4 Convergence Analysis In this section, the convergence of V-I-PARS is considered for a distribution system with no ungrounded sections. Some necessary notation for the proof is given below in Table 5.5. With this notation, the backward sweep can be summarized by the following two equations: ( k)

In

( k)

= I n + YV n ( k)

Ib

( k)

+ f (V n )

T ( k)

(5.26) (5.27)

= A In

The forward sweep can be summarized by the following two equations: ( k + 1)

( k)

Vb ( k + 1)

Vn

(5.28)

= ZI b ( 0)

= Vn

( k + 1)

– AV b

(5.29)

where the initial node voltages at each bus are set to the corresponding source voltage. Combining these four equations gives an expression for the

78 Table 5.5 Notation for V-I-PARS Convergence Proof Symbol P(i) P(i, j) Q(i)

Interpretation set of branches between bus i and the source P(i) ∩ P( j) set of buses supplied through branch i

di

number of elements in P(i)

bi

number of elements in Q(i)

( k)

vector of node voltages at iteration k

( k)

vector of branch voltages drops at iteration k

( k)

vector of branch currents at iteration k

( k)

vector of node current injections at iteration k

In

vector of node current injections from constant current loads

f (V )

vector of node current injections from constant PQ elements, as a function of voltage V

Vn Vb Ib

In

†Or,

V*

vector of node voltages at the solution

zi

impedance matrix for branch i

yi

total admittance to ground at bus i, including line charging, shunt capacitors, and constant impedance loads

Z

block diagonal matrix of branch impedances z i

Y

block diagonal matrix of bus admittances to ground y i

A

matrix with an identity block in each block row i and block column j for all ( i, j ) in { ( i, j ) i ∈ Q( j) } †

si

complex power injection from constant PQ elements at node i

equivalently, for all ( i, j ) in { ( i, j )

j ∈ P(i) } .

79 nodal voltages at iteration k + 1 as a function of the same voltages at iteration k. ( k + 1)

( 0)

Vn

= Vn

( k)

T

– AZA [ I n + YV n

( k)

+ f (V n ) ]

(5.30)

If the node voltage vector at the solution is V* , the node voltage at iteration k can be expressed as a sum of the solution plus the error. ( k)

Vn

( k)

= V* + ∆V n

(5.31)

Expressing the node voltage vector in this way for both iteration k and iteration k + 1 , (5.30) becomes ( k + 1)

V* + ∆V n

( 0)

= Vn

( k)

T

( k)

– AZA [ I n + Y ( V* + ∆V n ) + f (V* + ∆V n ) ] . (5.32)

Since the solution is a fixed point of (5.30), that is, ( 0)

V* = V n

T

– AZA [ I n + YV* + f (V* ) ] ,

(5.33)

(5.33) can be subtracted from (5.32) to give an expression for the error at iteration k + 1 as a function of the error at iteration k. ( k + 1)

∆V n

T

( k)

= – AZA [ Y ∆V n

( k)

+ f (V* + ∆V n ) – f (V* ) ]

(5.34)

To prove linear convergence of V-I-PARS, it is sufficient to show that the ratio of the error magnitudes is smaller than one. Specifically, if it can be shown that T

( k)

( k)

AZA [ Y ∆V n + f (V* + ∆V n ) – f (V* ) ] µ = ------------------------------------------------------------------------------------------------------------------ ≤ κ < 1, ( k) ∆V n

(5.35)

then V-I-PARS converges linearly. First, consider the case with no constant power devices in the network. In this case,

80 ( k)

T

AZA Y ∆V n T µ = ------------------------------------------≤ AZA Y . ( k) ∆V n

(5.36)

A simple bound on the ratio µ can be found by taking the product of the T

norms of the individual matrices A, Z, A , and Y. For each of the p-norms, where p is 1, 2, or infinity, this product is equal to µ≤ A Z A

T

Y = ( max b j ) ( max d j ) ( max z j ) ( max y j ) . j

j

j

j

(5.37)

This bound, however, is too loose and is not always satisfied in a typical distribution system. A better bound can be found by considering the norms of the matrices T

AZ and A Y , since T

T

µ ≤ AZA Y ≤ AZ A Y ≤ A Z A

T

Y .

(5.38)

The matrix AZ is simply the matrix A with each identity block in block T

column j replaced by z j . Likewise, the matrix A Y is just the matrix A

T

with each identity block in block row j replaced by y j . It was found that using the infinity-norm yields a tighter bound than the 1-norm. µ ≤ AZ

∞

T

A Y

∞

=

max  ∑ i 

zk

k ∈ P ( i)

=

max  ∑ i 

zk

k ∈ P ( i)



∞



∞

max  ∑ i 

yk

k ∈ Q ( i)

∑ i

yi



∞

(5.39)

∞

It is easy to see that, in a typical system, this is much smaller than the bound given in (5.37), since not all load magnitudes are equal to the maximum value and not all of the branch impedances along the highest impedance path are equal to the maximum branch impedance. In fact, in all of the available cases based on data from real systems, this quantity

81 was significantly smaller than one, guaranteeing the linear convergence of V-I-PARS on these systems. If constant PQ loads and cogenerators are considered in the network, then the ratio µ , as given in (5.35), can be bounded as follows: T

( k)

AZA [ f (V* + ∆V n ) – f (V* ) ] µ ≤ AZA Y + --------------------------------------------------------------------------------------, ( k) ∆V n T

(5.40)

where the first term has just been dealt with. In the second term, f (V* ) is the nodal current injected by constant power devices at the solution, and ( k)

f (V* + ∆V n ) is the injection at iteration k. At a particular node i, the magnitude of the difference between the two injections is si   ∗  si  ∗ ( k) fi (V* + ∆V n ) – fi (V* ) =  ------------------------------------–  --------------- ( k)   [ V* ] i   [ V* + ∆V n ] i  ( k)

s i [ ∆V n ] i = ----------------------------------------------------( k) [ V* ] i [ V* + ∆V n ] i

(5.41)

si ( k) ≤ --------------------------------------- [ ∆V n ] i . ( k) [ V* ] i [ V n ] i For real systems, it is reasonable to assume that voltage magnitudes at the solution and at each iteration4 are larger than 0.7 per unit, implying that ( k) 1 [ V* ] i [ V n ] i > --2

(5.42)

and therefore ( k)

( k)

fi (V* + ∆V n ) – fi (V* ) ≤ 2 s i [ ∆V n ] i . 4 Assuming

(5.43)

a flat start, the assumption holds for the voltages at the first iteration. The arguments following show a decrease in the magnitude of the error, indicating that voltages during subsequent iterations lie closer to the solution than the initial voltages did.

82 Let S be a diagonal matrix whose ith diagonal element is equal to 2 s i .5 Since the elements of A, and therefore of its transpose, are all positive, the following can be said about the second term of (5.40): ( k)

T

T ( k) A f (V* + ∆V n ) – f (V* ) AZA [ f (V* + ∆V n ) – f (V* ) ] AZ -------------------------------------------------------------------------------------------------------------------------------------------------------------≤ ( k) ( k) ∆V ∆V n n T

( k)

A S ∆V n ≤ AZ -----------------------------------( k) ∆V n

(5.44)

T

≤ AZ A S The bound given in (5.39) then becomes µ ≤ AZ =

∞

T

( A Y

max  i 

∑

∞

zk

k ∈ P ( i)

T

+ A S  ∞

∞)

 ∑ yi  i

 +  ∑ 2 si  ∞  

(5.45)

i

In words, this says that V-I-PARS will converge linearly if the product of the following two terms is smaller than one. Roughly speaking, the first term is the total impedance of the highest impedance path and the second term is the total admittance to ground plus twice the total power injection from all constant power devices.6 In a typical distribution system, all z i , y i , and s i are much smaller than one, yielding a value of µ which is also smaller than one. When this condition holds, linear convergence from a flat start is guaranteed.

5

The matrix S actually has units of admittance since the factor of 2 comes from an inverse squared voltage quantity. 6 The seeming inconsistency in units is due to the hidden voltage units in the factor of 2.

83

5.5 Comments The backward/forward sweep methods presented in this chapter are applicable to most radial distribution networks. The one modeling limitation is that BFS-PARS cannot handle type 4 ungrounded wye to grounded wye connected transformers. Some general observations with regard to the many variations indicate two things. First, the methods based on current generally require less computation per iteration than their power flow based counterparts. Second, the methods which do not update extra variables such as voltage during backward sweep and current/power flow during forward sweep, require less computation per iteration. Based on these two observations, V-I-PARS appears to be the most attractive of the BFS-PARS class of algorithms presented, assuming that the number of iterations required for convergence is comparable among the various methods. The comparisons of the algorithms are investigated in more detail in Chapter 8, “Simulation Results”. It should also be noted that the amount of work per iteration is proportional to the number of buses. Therefore, if the number of iterations remains constant, the computational complexity increases linearly with the size of the network, making BFS-PARS suitable for very large radial distribution systems.

Chapter 6

Fast Decoupled Power Flow Algorithms for Radial Systems (DePARS) One of the most widely used power flow algorithms throughout the power industry is the fast decoupled Newton method proposed in 1974 in [26]. This method exploits some of the numerical properties of the standard power flow formulation to make simplifying assumptions which allow significant savings in computation over the standard Newton method. Unfortunately, this approach is not typically suitable for radial distribution networks. There are often ill-conditioning problems due to the formulation and, in addition, the assumptions necessary for the simplifications used in the standard fast decoupled Newton method are often not valid for these types of systems. Some work, however, has been done to address these problems [16; 31; 21]. This chapter explores a class of algorithms which exploits the radial topological structure to reduce the number of equations and unknowns in the formulation. These algorithms also take advantage of the special

84

85 numerical structure of the new formulation to further reduce the computation required for each iteration, in the spirit of the standard fast decoupled method for meshed transmission systems. This class of algorithms will be referred to as fast Decoupled Power flow Algorithms for Radial Systems or DePARS. This chapter presents four main variants of DePARS. The first of the four methods, VI-DePARS, is a generalization of the method proposed in [32] and will be presented in detail. The other variations, one of which is an extension of the methods proposed in [12], will then be discussed with respect to VI-DePARS.

6.1 Detailed Solution Algorithm The standard fast decoupled methods used in transmission systems are based on the well-known Newton’s method [28] for solving a non-linear set of equations. In this case, the non-linear equations being solved are the power balance equations which specify that, at each bus, the complex power generated, minus the power absorbed by load, must equal the power injected into the rest of the network. In a distribution system with one source bus and many load buses, the traditional power flow formulation would have six equations for each load bus, balancing the real and reactive part of the power at each of the three phases. The traditional fast decoupled method for transmission systems improves on the standard Newton method, shown in Table 6.1, by making simplifying approximations which reduce the computational burden for step 5 and step 6.

86 Table 6.1 Newton’s Method Solution of F( x) = 0 by Newton’s Method 1

Choose an initial guess for the solution, x

2

Set i = 0 .

3

Evaluate F

4

Stop if F

5

Evaluate the Jacobian, J

( i)

( i)

( i)

( i)

= F(x

( i)

≤ some tolerance .

( i)

( i)

=

Solve J

7

Let x

8

Let i = i + 1 and go to step 3.

( i + 1)

.

).

6

s

( 0)

= –F

= x

( i)

+s

∂F ∂x

. x

( i)

.

( i)

.

The DePARS approach is also based on the Newton method. Like the traditional fast decoupled method, it exploits the numerical structure of the Jacobian to greatly reduce the computation required by step 5 and step 6 in Table 6.1. However, DePARS also uses a different formulation of the power flow equations which exploit the radial topological structure of the network resulting in a reduced number of equations and unknowns. Table 6.2 gives a high level view of DePARS and its basic steps. Steps 5, 6, and 7 from the original Newton method have been simplified and grouped together into step 2 of DePARS. At the right side of Table 6.2 the details have been kept general enough to cover all four variations of the method. The first option in each step, however, is the one used by VI-DePARS, which will be the focus of the remainder of this section.

87 Table 6.2 Fast Decoupled Power Flow Algorithm for Radial Systems DePARS - The Algorithm

Initialize independent variables:

Initialize end voltages. or … Initialize currents or power flows injected into each lateral.

1

Evaluate power flow equations:

Compute voltage mismatch at each lateral’s source. or … Compute current or power flow mismatch at end of each lateral.

Compute updated solution:

Update end voltages. or … Update currents or power flows injected into each lateral.

2

Repeat steps 1 to 2 until convergence is achieved. For VI-DePARS, the independent variables are the voltages at the end of each lateral. The power flow equations state that the voltage mismatch at the beginning of each lateral, calculated as a function of the end voltages, must be zero at the solution. If this mismatch is not zero it can be used to update the end voltages for the next iteration.

6.1.1

Reduced Power Flow Equations

The traditional power flow problem can be expressed mathematically as a non-linear set of equations of the form F ( x) = 0 .

(6.1)

This section presents a reduced set of non-linear equations which can be expressed in the same form.

88 6.1.1.1

Single Feeder

Consider first a network consisting of a single three-phase lateral with N load buses, as shown in Figure 6.1. The traditional formulation of the power flow equations would have 6N real equations and 6N real unknowns. However, it is not necessary to solve such a large system of nonlinear equations just to find all of the steady-state bus voltages. Specifying V N also determines the remaining voltages and currents in the system.

V0 = V0

VN

source

end bus I1

IN + 1 = 0 Figure 6.1 Single Feeder Example

Using the same procedure used by BFS-PARS in Section 5.1.1, “Backward Sweep”, on page 60, all of the voltages and currents are updated by applying (2.3) from Table 2.3 at each bus, starting at the end bus and moving toward the source. This procedure is written in a closed form in (5.3) which is repeated here for convenience.  V  N  w N – 1 = g N (w N ) = g N   0 

 V  N  w N – 2 = g N – 1(w N – 1) = g N – 1 • g N   0 

w0 =

V0 I1

(6.2)

 V  N  = g 1(w 1) = g 1 • … • g N – 1 • g N   0 

Here both the source voltage V 0 and the source current I 1 are expressed as functions of the end voltage V N . For BFS-PARS, the current

89 I˜ 1(V N ) is of primary interest, since it is used during the forward sweep to update the end voltage V N . For VI-DePARS, the voltage V˜ 0(V N ) is of primary importance, since it is used for mismatch calculation. At the solution, the difference between the calculated source voltage V˜ 0(V N ) and the specified source voltage V 0 must be zero. The power flow equations can therefore be written F (V N ) = V˜ 0(V N ) – V 0 = 0 .

(6.3)

This is a reduced formulation of the power flow problem for a single feeder which has only three complex equations and three complex unknowns, or six real equations and six real unknowns. The dimension of this set of nonlinear equations is independent of the number of buses on the lateral and is a factor of N smaller than the traditional formulation. 6.1.1.2

General Radial Structure

To generalize this formulation to handle an arbitrary radial structure, first note that the voltage mismatch at the beginning of any lateral can be computed as a function of the end voltage, as in (6.3), if the currents injected into each sub-lateral are known. These currents are necessary for the application of KCL at the branching buses. In a system with L levels of laterals, the level L laterals have no sub-laterals and can therefore be updated first. For each lateral V 0 and I 1 are computed as functions of the end voltage V N as follows: V0 I1

 V  N  = g1 • … • g N – 1 • g N   0 

(6.4)

After all level L laterals have been updated, the currents injected into the sub-laterals of each level L – 1 lateral are known (the I 1 recently cal-

90 culated via (6.4) for level L laterals), hence the level L – 1 laterals can be updated in the same way. Next the level L – 2 laterals are updated, and so on, until the main feeder has been updated. This is the same RBF1 order of processing the laterals used by the backward sweep of BFS-PARS. The current I 1 injected into lateral i is needed for doing the update of lateral i – 1 . The voltage V 0 is needed for mismatch calculation. This voltage mismatch calculation for each lateral can be completed once all the laterals have been updated by taking the difference between the two voltages computed for each branching bus. Suppose lateral i branches off of lateral i – 1 at bus q as shown in Figure 6.2. The value of the voltage at lateral i bus N xi = V N

V˜ q lateral i – 1

from source bus q

Vq

Figure 6.2 Voltage Mismatch Calculation

bus q computed from the end voltage of lateral i will be denoted V˜ q( x) and corresponds to V˜ 0(V N ) from the single feeder example. Here x is a vector

1 See

page 9 under Section 2.2.1, “Indexing Scheme”.

91 containing the end voltages of all laterals and x i denotes the end voltage V N for lateral i. These x i are put into x in RBF1 order. x ( L, M x =

V ( L, M

L)

x ( l, m )

=

x ( 1, 1 )

L,

N L, M )

V ( l, m, N

l, m )

V ( 1, 1, N

1, 1 )

(6.5)

To compute V˜ q , it is necessary to use x instead of just V N since V˜ q is affected by the currents injected into the sub-laterals of lateral i and these currents in turn are functions of the end voltages of the corresponding sublaterals. In fact, V˜ q is a function of the voltages of all end buses supplied through lateral i, not just of its own end voltage. The other value of V q corresponds to the specified source voltage V 0 from the single feeder example and will be denoted V q( x) . This is the voltage at bus q, as computed from the end voltage of lateral i – 1 . Once again, the variable x is used since V q is a function of the voltages of all end buses supplied through the section of lateral i – 1 beyond bus q, not just of the end bus of lateral i – 1 itself. Subtracting the two voltages V˜ q( x) and V q( x) yields an equation similar to (6.3) for lateral i starting at bus q, but V N is replaced by the vector x of the end voltages of all laterals. F i( x) = V˜ q( x) – V q( x) = 0

(6.6)

Note that V q( x) is not actually constant except when i is the main feeder and q is the source bus, i.e. the last set of equations when in RBF1 order. The function F i does not depend on all elements of x, but only on the voltages of end buses supplied through bus q. This relationship will be seen

92 more clearly in Section 6.1.2.1, “Structure of the System Jacobian”, especially in Figure 6.3 which illustrates the sparsity structure of the system Jacobian. Taking (6.6) for each lateral and combining these equations in RBF1 order, F ( L, M F =

L)

F ( l, m )

,

(6.7)

F ( 1, 1 ) yields the new reduced power flow equations which can be expressed compactly as F ( x) = 0 .

(6.8)

Assuming real and imaginary parts are separated, for a system with M laterals, (6.8) is a set of 6M non-linear equations in 6M real unknowns. This is also assuming that all buses are three-phase; single and two-phase laterals would reduce these numbers accordingly. The reduced power flow equations for VI-DePARS are expressed mathematically by the non-linear set of equations in (6.8). The evaluation of these equations is performed by traversing each lateral in RBF1 order, updating voltages and currents via (6.4), then computing the voltage mismatches via (6.6).

93 6.1.1.3

Class B and Class C Transformers

If lateral i, emanating from bus q on lateral i – 1 , has a class B or class C transformer as the incoming branch of bus k,2 the corresponding equations in (6.4), and consequently in (6.6), may take on a slightly different form. For a class B transformer entering bus k, the primary voltage Σ

and current are functions of the w k of (3.49) which includes V k – 1 , the sum of the voltages at the primary side of the transformer, not just the two dimensional secondary voltage V k . This is described in Section 5.2.1.1, “Backward Sweep”, on page 66. In this case, the independent variable x i associated with lateral i is

xi =

VN Σ

.

(6.9)

Vk – 1

The variable x which contains the end voltages of all laterals will also conΣ

tain V k – 1 for laterals with class B transformers. This allows (6.6) to still be used in its present form. For a lateral with a class C transformer, (6.4) remains unchanged, although the voltage update formula used is the one in (5.22) which includes the modified secondary voltage. The mismatch function in (6.6) is a set of two equations in three unknowns. The additional equation required to solve the problem is obtained by setting to zero the secondary side constraint mismatch presented in Section 5.2.2.2, “Backward Sweep”, as (5.20).

2 Due

to the implementation explained in Section 2.2.2.2, “Breadth-First Search”, transformers always appear as the first branch on a lateral. So, although the more general notation is used here, restricted only to one transformer per lateral, in practice k = 1 and k–1 = q.

94 2 Σ Σ βk Σ ∆V˜ k( x) = V k – -----I ′ = 0 yk k

(6.10)

This mismatch is a function of x, since the secondary voltage and current sums are functions of x. In this case, the function F i( x) of (6.6), used to compute the mismatch for lateral i, becomes

F i( x) =

V˜ q( x) – V q( x) Σ ∆V˜ k( x)

= 0.

(6.11)

With these small changes in the independent variable and the mismatch computed for laterals with class B and class C transformers, respectively, the compact form of the reduced power flow equations in (6.8) can still be used.

6.1.2

Update of Independent Variables

The second step of DePARS, after the evaluation of the power flow equations, is to update the independent variables, that is, the elements of x. In the case of VI-DePARS, these are the voltages V N at the ends of the laterals.3 In the standard Newton method, this involves the following three steps: 1. Evaluate the Jacobian at the current solution. 2. Solve a linear system of equations to find the update step. 3. Add the update step to the current solution. The system Jacobian J ( x) for the reduced set of power flow equations in (6.8) is

3 For

Σ

laterals with class B transformers this also includes V k – 1 , the sum of the voltages at the primary side of the transformer.

95

J ( x) =

∂F ∂x

(6.12)

and has a special numerical structure which can be exploited to greatly reduce the computation involved in finding a good update step. 6.1.2.1

Structure of the System Jacobian

Since the function F consists of the mismatch functions F i for each lateral i, and the variable x is made up of the individual x i associated with each lateral i, the system Jacobian can be thought of in terms of block elements associated with a pair of laterals. The block element ∂ F i ⁄ ∂ x j , in block row i and block column j, is the sensitivity of the mismatch at lateral i to changes in the independent variables corresponding to lateral j. First consider a single feeder system where the mismatch function is F (V N ) as given in (6.3). The Jacobian for this simple system is just a single block element and can be expressed using the chain rule and the branch Jacobians. The branch Jacobian G k for bus k’s incoming branch is the Jacobian of the branch update equation in (2.3) from Table 2.3. This branch update function is w k – 1 = g k(w k)

(6.13)

and therefore the branch Jacobian is  ∂V˜ k – 1    ∂g k  ∂V k  Gk = = ∂ wk  ∂I˜ k     ∂ V k

 ∂V˜ k – 1     ∂ Ik + 1  .  ∂I˜ k     ∂ I k + 1

(6.14)

The system Jacobian for this simple single feeder case gives the sensitivity of the calculated source voltage mismatch to variations in the end

96 voltage. It can be written, using the chain rule, as the product of the branch Jacobians from the source bus to the end bus ∂g ∂V˜ 0 ∂V˜ 0 ∂F J (V N ) = = = ⋅ G2 ⋅ … ⋅ G N – 1 ⋅ N , ∂V N ∂V N ∂ w1 ∂V N

(6.15)

where the first term in the product is just the upper half of G 1 and the last term is the left half of G N . Extending this to the general radial structure, each block element of the system Jacobian can be expressed, using the chain rule, as a product of branch Jacobians along the path between the supplying bus of lateral i and the end bus of lateral j. If lateral i branches off from its parent lateral at bus q and the end bus of lateral j is bus n, then changes in x j at bus n can only affect the mismatch F i at bus q if bus n is supplied through bus q. In other words, if bus q does not lie on the path between bus n and the source, then ∂ F i ⁄ ∂ x j = 0 . This determines the sparsity structure of the system Jacobian J ( x) . If bus q does lie on the path between bus n and the source, then the corresponding block element ∂ F i ⁄ ∂ x j is non-zero and fits into one of the four categories listed in Table 6.3. In order to simplify the presentation, the equations given in Table 6.3 are for the case with no class B or class C transformers present. The first type of non-zero blocks are those on the block diagonal of the system Jacobian. These diagonal blocks are formed via (6.16) in a manner analogous to the single feeder case in (6.15). The second type is the set of non-zero blocks above the diagonal and is similar to the first except that q + 1 refers to the bus following bus q on lateral j, hence it is V q that is affected instead of V˜ q . For non-zero blocks of type 3 and type 4, let bus p

97 Table 6.3 VI-DePARS Jacobian Formation Non-Zero Blocks of the System Jacobian for VI-DePARS Type 1

Type 2

Type 3

Type 4

i and j are the same lateral

i is a direct sub-lateral of j

j is supplied through i

j is supplied through i – 1

source

source

source

source

q q +1 q +1 q i&j

∂ wq + 1

i j

n

non-zero blocks above diagonal

∂F i = ∂x j ∂F i

q

q +1 i j

diagonal blocks (all non-zero)

q +1 p –1

q

n

(6.16) ⋅ Gq + 2 ⋅ … ⋅ Gn – 1 ⋅

∂g n

∂V n

where q + 1 refers to the first bus on lateral i and ∂F i ∂V˜ q = ∂ wq + 1 ∂ wq + 1

where q + 1 refers to the bus following q on lateral j and ∂F i – ∂V q = ∂ wq + 1 ∂ wq + 1

approximately identity

approximately minus identity

p –1

p n

n

p

i j

non-zero blocks below diagonal ∂F i = ∂x j ∂F i

(6.17) ⋅ Gq + 2 ⋅ … ⋅ G p – 2 ⋅

∂g p – 1

∂ wq + 1 ∂I p ∂g ∂I˜ ⋅ p ⋅ G p + 1 ⋅ … ⋅ Gn – 1 ⋅ n ∂w p ∂V n where q + 1 refers to the first bus on lateral i and ∂F i ∂V˜ q = ∂ wq + 1 ∂ wq + 1

where q + 1 refers to the bus following q on lateral j and ∂F i – ∂V q = ∂ wq + 1 ∂ wq + 1

approximately zero

98 be the first bus on lateral j. These types are comprised of the non-zero blocks below the diagonal and require products of branch Jacobians from more than one lateral. In the examples shown, the path from bus q to bus n includes only two laterals, but in general there could be more. If, for example, this path included four laterals and the first bus on each new lateral encountered along this path were labelled consecutively as p 1 , p 2 , and p 3 , then (6.17) would look like the following: ∂g p – 1 ∂F i ∂F i 1 = ⋅ Gq + 2 ⋅ … ⋅ G p – 2 ⋅ 1 ∂x j ∂ wq + 1 ∂I p 1 ˜ ∂g ∂I p1 p2 – 1 ⋅ ⋅ Gp + 1 ⋅ … ⋅ Gp – 2 ⋅ 1 2 ∂w p ∂I p 1 2 ˜ ∂g ∂I p2 p3 – 1 ⋅ ⋅ G p2 + 1 ⋅ … ⋅ G p3 – 2 ⋅ ∂w p ∂I p 2 3 ˜ ∂I p3 ∂g ⋅ ⋅ G p + 1 ⋅ … ⋅ Gn – 1 ⋅ n 3 ∂ w p3 ∂V n

(6.18)

For a general radial distribution network, the structure of the system Jacobian is determined by the RBF1 order of the equations and variables and the application of the rules in Table 6.3 for the non-zero blocks. This Jacobian structure is illustrated in Figure 6.3 for the sample radial network shown in Figure 2.1 on page 10. As was mentioned earlier, a given F i does not depend on all elements of x. The non-zero blocks in block row i correspond to the x j on which F i depends. For lateral i branching off of bus q, block row i will have non-zero elements in the block columns corresponding to the laterals whose end buses are supplied through bus q.

99

zero block element non-zero block element of Type 1 non-zero block element of Type 2 non-zero block element of Type 3 non-zero block element of Type 4

xj 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9

Fi

10 11 12 13 14 15 16 17 18 19 20

Figure 6.3 Structure of System Jacobian for VI-DePARS

100 6.1.2.2

Numerical Properties of the System Jacobian

The process of building the system Jacobian requires the application of (6.16) or (6.17) for each non-zero block as determined by the topology of the network. Each non-zero block is a product of branch Jacobians which must be evaluated first. The branch Jacobian G k of (6.14) relates the sensitivity of voltages and currents at bus k – 1 to small variations in the voltages and currents at bus k. For a three-phase branch, it is typically a 12 x 12 real matrix.4 Even for a very simple distribution line represented with the standard π-model, each element of the matrix is a very complex expression depending not only on the parameters of the line but also on any load, shunt capacitor, or cogenerator at bus k. For a transformer the expression can be even more complicated. To include in a computer program an analytical expression for each of the 144 terms of each variation of the branch Jacobian is not practical due to the number of different expressions and their extreme complexity. The building of the true system Jacobian via products of true branch Jacobians is therefore not a viable option either. Some numerical approximation is needed. In order to determine the numerical structure of the system Jacobian it is necessary to first examine carefully the numerical properties of the branch Jacobians for each type of branch. Deriving an analytical expression for each element and analyzing it to determine which terms can be neglected and what approximations can be made would also be prohibi-

4

If one or both sides of the branch are in an ungrounded section of the network, this dimension is reduced to 10 x 10 or 8 x 8, respectively. Note that it is necessary to separate each complex variable into two real variables in order to differentiate to find the true branch Jacobian.

101 tively tedious. However, some reasonable approximations can be made by examining the relative magnitudes of certain quantities when expressed in per unit. For the incoming branch of bus k, the branch Jacobian G k contains the sensitivities of V˜ k – 1 and I˜ k to small changes in V k and I k + 1 . Suppose this branch is a switch, modeled as a zero impedance connection. In this case, small changes in V k cause identical changes in V˜ k – 1 and small changes in I k + 1 cause identical changes in I˜ k . In other words, the diagonal blocks of G k are identity blocks. The off-diagonal block based on the sensitivity of V˜ k – 1 to I k + 1 is a zero block since changes in I k + 1 have no effect on the voltage V˜ k – 1 . The other off-diagonal block is nearly zero as well, under the assumption that the admittances and power injections of loads, shunt capacitors, and cogenerators are small relative to voltage magnitudes when expressed in per unit. From the KCL equation in (5.2) it can be seen that V k does have a small effect on the currents injected by loads, shunt capacitors, and cogenerators. However, assuming the admittance or power parameters are small with respect to voltage, the sensitivity of I k ′ , and therefore of I˜ k also, to V k can be neglected. This is equivalent to approximating loads, shunt capacitors, and cogenerators by their corresponding current injections. For a switch, then, the branch Jacobian G k can be approximated by the appropriately sized identity matrix.5 Using I to denote an identity block Gk ≈ I 0 . 0 I 5

(6.19)

When using this approximation, it is not necessary to split up complex variables into two real variables. Hence, for a three-phase branch in a grounded section of the network G k is a 6 x 6 identity matrix.

102 If bus k’s incoming branch is a distribution line modeled as described in Section 3.4, “Distribution Line Model”, then the numerical properties of (3.19) and (3.20) from Table 3.6 must be examined in order to derive an approximation for G k . For practical distribution lines the elements of the line impedance matrix Z k and the line charging admittance matrix Y k are typically small with respect to voltage magnitudes when expressed in per unit. When differentiating, under this assumption, the last term of (3.19) and the first term of (3.20) can be neglected. This is equivalent to approximating the distribution line as a zero impedance branch with no charging. As with the switch, this implies that the branch Jacobian G k can be approximated by the appropriately sized identity matrix.5 In a system with only lines and switches but no transformers, each branch Jacobian can be approximated by an identity matrix. Voltages and currents are essentially decoupled from one another. It is important to note that these approximations are only for simplifying the Jacobian used to compute the update step in Newton’s method. They have no effect on the final solution, which is still based on the full models described in Chapter 3, “Detailed Component Models”. The diagonal blocks of the system Jacobian are under type 1 in Table 6.3 and are formed via (6.16). Using the identity approximations for the branch Jacobians yields an identity block in the system Jacobian as well. ∂F i ≈ I 0 ⋅ I 0 ⋅…⋅ I 0 ⋅ I ∂x j 0 I 0 I 0 = I

(6.20)

103 Type 2 non-zero blocks, those above the diagonal, are formed by the same equation except that the first term has a negative sign, resulting in a negative identity block in the system Jacobian. ∂F i ≈– I 0 ⋅ I 0 ⋅…⋅ I 0 ⋅ I ∂x j 0 I 0 I 0

(6.21)

= –I Using the identity approximation to the branch Jacobian in (6.17) for type 3 non-zero blocks yields ∂F i ≈ I 0 ⋅ I 0 ⋅…⋅ I 0 ⋅ 0 ⋅ 0 I ⋅ I 0 ⋅…⋅ I 0 ⋅ I ∂x j 0 I 0 I I 0 I 0 I 0 = I 0 ⋅ 0 ⋅ 0 I ⋅ I I 0

(6.22)

= 0 Here the result is a zero block in the system Jacobian. This is due to the fact that the only coupling between the voltages involved is through the current I p , but the approximation decouples voltages and currents. The only difference for type 4 non-zero blocks is a leading negative sign. These blocks can therefore also be approximated by zero. The last row of Table 6.3 summarizes these approximations. 6.1.2.3

Transformers

Building non-zero blocks of the system Jacobian which involve transformers is somewhat more complicated. First, note that there can be no transformers on the relevant part of lateral j for a type 2 non-zero block, so the approximations for these blocks are never affected by transformers.

104 Second, note that if a transformer is approximated by its ideal equivalent with no leakage admittance, then currents are completely decoupled from voltages and vice versa. Changes in secondary voltage have no effect on primary current and changes in secondary current have no effect on primary voltage. This means that the off-diagonal blocks of G k can be approximated by a zero block for a transformer. Given that a transformer always appears as the first branch on the lateral,6 the lower right block of G k is not even used in the formation of the system Jacobian for type 1 blocks since the first term of (6.16) includes only the left half of G k . The same is true about the first term of (6.17) for type 3 and type 4 blocks. In fact, for a network with a transformer entering bus p, the identity approximations for the lines and switches disappear and (6.17) becomes ∂F i  ˜  ≈ I 0 ⋅ 0 ⋅ 0  ∂I p  ⋅ I ∂x j I 0  ∂ I p + 1

(6.23)

= 0. Even without knowing the value of ∂ I˜ p ⁄ ∂ I p + 1 these sub-diagonal blocks can still be approximated by zero. This means that it is not necessary to evaluate the lower right block of G k for a transformer since it is never used in the formation of the approximated system Jacobian. The only non-zero blocks of the approximated system Jacobian affected by transformers are the type 1 diagonal blocks. Suppose bus k7 is the first bus on lateral i and its incoming branch is a transformer. The 6 See

Section 2.2.2.2, “Breadth-First Search”. k is also equal to q + 1 in the figure for type 1 blocks in Table 6.3.

7 Here

105 diagonal element in block row i can be approximated by the upper left block element of G k . ∂F i  ˜  ≈  ∂V k – 1 0 ⋅ I 0 ⋅ … ⋅ I 0 ⋅ I ∂x j 0 I 0 I 0  ∂V k 

(6.24)

∂V˜ k – 1 = ∂V k

This relationship between secondary and primary voltages, however, depends on the connection type. For class A transformers, this partial derivative is computed from (3.39) of Table 3.9 and yields ∂F i ∂V˜ k – 1 sp – 1 ss ≈ = – (Yk ) Yk . ∂ xi ∂V k

(6.25)

Σ

For class B transformers, g k is a function of V k – 1 as well as V k . Differentiating (3.52) of Table 3.10 gives the relevant expression:  ∂V˜ k – 1   ∂V˜ k – 1  ∂F i ≈     = ∂ xi  ∂V k   ∂V Σ  k–1

sp Yk

111

–1

0

ss

–Y k

0

0

1

0

(6.26)

For class C transformers, the corresponding block diagonal element of the approximated Jacobian consists of the partial derivatives of (5.22) and (6.10).

∂F i ≈ ∂ xi

 ∂V˜ k – 1     ∂V k   ∂V˜ Σ  k   ∂ V k

=

sp –Y k

  ss 1 2 – 1 – 1 \  Y k ⋅ --- – 1 2 – 1 3  –1 –1 2  1

1

    

(6.27)

1

Table 6.4 shows, for each transformer type, the constant matrix depending only on α k and β k which results from the evaluation of (6.25),

106 Table 6.4 Jacobian Approximations for Transformers for VI-DePARS Transformer Connection Type Primary

1

A

∂F i ∂ xi

Secondary

αk 1 0 0 ------ 0 1 0 βk 001

Grounded Wye Grounded Wye

αk 1 0 -----βk 0 1

5

Ungrounded Wye

Ungrounded Wye

6

Ungrounded Wye

Delta

8

Delta

Ungrounded Wye

αk ------------ 21 3β k – 1 1

9

Delta

Delta

αk 1 0 -----βk 0 1

sp – 1

– (Yk )

αk 1 –1 -----------3β k 1 2

ss

Yk

2α k α k 1 ------------ ------------- --3β k 3β k 3 2

Grounded Wye

–αk αk 1 ------------ ------------- --3β k 3β k 3

Ungrounded Wye sp

–1

Yk

B

– Y ss k

111

3

Grounded Wye

0

0

0 0 1

– α k – 2α k 1 ------------ ------------- --3β k 3β k 3 αk -----βk

1 0 --3

0

αk 1 ------ --βk 3

Delta

–αk –αk 1 --------- --------- --βk βk 3 4

Ungrounded Wye

Grounded Wye

C

not applicable

sp

7

Delta

Grounded Wye

–Y k

   ss 1 2 – 1 – 1  \  Y k ⋅ --- – 1 2 – 1  3   –1 –1 2  

1

1

1

2α k – α k – α k ---------- --------- --------3β k 3β k 3β k – α k 2α k – α k --------- ---------- --------3β k 3β k 3β k 1

1

1

107 (6.26), and (6.27). The appropriate constant matrix from the table is used to replace the identity block on the diagonal of the approximated system Jacobian for a lateral with a transformer. 6.1.2.4

Solving for the Update

The approximation to the system Jacobian is a block upper-triangular matrix as shown in Figure 6.4 for the sample radial network of Figure 2.1 on page 10. Neglecting the near zero type 3 and type 4 blocks below the diagonal is essentially saying that the mismatch for lateral i depends only on the end voltages of laterals i and i – 1 . Typically, each iteration i of the Newton method requires the evaluation of the system Jacobian J ( x the update step s

( i)

( i)

) and its factorization for the solution of

from J(x

( i)

)s

( i)

= – F(x

( i)

).

(6.28)

However, when using the approximation to J described above, the matrix is constant and need not be reevaluated at each iteration. Furthermore, due to its block upper-triangular structure, it is not necessary to factor the matrix. Instead, (6.28) can be solved very efficiently for s

( i)

via a simple

block backward substitution.

6.1.3

Termination Criterion

The power flow equations F ( x) are evaluated to determine the mismatch for each iteration of DePARS. This mismatch is used to update the value of x for the next iteration. These two steps are repeated until convergence is achieved. Once the norm of the mismatch F ( x) becomes smaller than some tolerance it can be said that the algorithm has converged.

108

zero block element identity block (or from Table 6.4) negative identity blocks

xj 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9

Fi

10 11 12 13 14 15 16 17 18 19 20

Figure 6.4 Approximation to the System Jacobian for VI-DePARS

109 It is also possible to use the termination criterion used for the network reduction methods of Chapter 4. This approach requires that the norm of the difference in bus voltages between iterations be smaller than a given tolerance.

6.2 Implementation In many respects DePARS is very similar to the backward/forward sweep methods of Chapter 5. The function evaluation for VI-DePARS in the implementation is, in fact, the same routine as the one used for backward sweep in VI-VI-PARS and V-VI-PARS. In the fast decoupled method, the end voltages are updated directly via an approximate Newton step as opposed to the complete forward sweep used in the backward/forward sweep methods. Because of the simple structure of the approximated Jacobian, it is not even necessary to explicitly form the matrix. Once the mismatches F i have been calculated, the update step for each lateral can be computed directly as si = si – 1 – F i

(6.29)

for laterals without transformers, and  ∂F i  –1  ( si – 1 – F i) si =   ∂ xi 

(6.30)

for laterals with transformers. Here s i – 1 refers to the update step for lateral i – 1 , the parent of lateral i. The calculation of the update steps must therefore proceed in BF8 order, starting with the main feeder. The

8 See

page 11 under Section 2.2.1, “Indexing Scheme”.

110 term ( ∂ F i ⁄ ∂ x i ) –1 can be computed beforehand for each lateral with a transformer. Since the package used for the implementation had built-in sparse matrix routines which were “smart” enough to do the block backward substitution automatically, the approximate Jacobian was formed explicitly. This allowed for the flexibility of plugging it directly into an existing standard Newton solver routine without sacrificing efficiency.

6.3 Variations There are four main variations to the generic DePARS given in Table 6.2. The first two methods, VI-DePARS and VS-DePARS, use a backward sweep for function evaluation and an approximate Newton update to the independent variables in place of the forward sweep of BFS-PARS. The last two methods, I-DePARS and S-DePARS, use a forward sweep for function evaluation and the approximate Newton step is in place of a backward sweep. Table 6.5 summarizes the four variations. Table 6.5 Various Formulations for DePARS Algorithm

Based on

VI-DePARS

current

VS-DePARS

power flow

I-DePARS

current

S-DePARS

power flow

Independent Variable

Function Evaluated

voltage at end of each lateral

voltage mismatch at each lateral’s source

current at each lateral’s source

current mismatch at end of each lateral

power flow at each lateral’s source

power mismatch at end of each lateral

111

6.3.1

VI-DePARS

VI-DePARS is the variation presented in detail in Section 6.1, “Detailed Solution Algorithm”. It computes voltage mismatches at the beginning of each lateral as a function of the end voltages and is based on current as opposed to power flow. This is the algorithm presented in [32] for networks which have only type 1 grounded wye to grounded wye transformers. The approach proposed in [17] is based on a similar idea. For a system consisting of a single feeder, it is equivalent to VI-DePARS, using (6.29) to update the end voltages directly from the voltage mismatch at the source. However, in the extension to a general radial structure, instead of updating the end voltage of all laterals at each iteration, the equations associated with each lateral are solved one by one to completion. Once convergence is achieved for all level l laterals, the level l – 1 are solved. When all the laterals have been solved the process is repeated. This nested looping, of course, is inefficient and, fortunately, not necessary, as has been shown.

6.3.2

VS-DePARS

VS-DePARS is the power flow based counterpart to VI-DePARS. It uses the same independent variables and the mismatches computed are still the voltage mismatches at the beginning of each lateral. The function evaluation, however, is based on the backward sweep used by VS-VS-PARS described in Section 5.3.2.9 Many of the differences between VI-VI-PARS

9 This

backward sweep is also used by V-VS-PARS of Section 5.3.4.

112 and VS-VS-PARS described in this section are also relevant to the differences between VI-DePARS and VS-DePARS. Since this method computes the same mismatch as a function of the same variables, the same approximate Jacobian can be used. An extensive analysis of the branch Jacobians, now based on h k from (2.5), lead to the same approximations yielding the same approximate system Jacobian.

6.3.3

I-DePARS

In the single feeder example of Figure 6.1, there are two boundary conditions: the source voltage is a specified constant and the end current is zero. The first two variations of DePARS are based on using the boundary condition at the end of the lateral to compute backward toward the source, obtaining a voltage mismatch at the source as a function of the end voltage. I-DePARS takes the opposite approach. The source voltage boundary condition is used to calculate in the forward direction to obtain a current mismatch at the end of the lateral as a function of the current injected at the source. In this case, the independent variables are the currents injected into the beginning of each lateral. The function evaluation is equivalent to a forward sweep and the approximate Newton update of the independent variables is in place of a backward sweep. In fact, the routine used to do the function evaluation is the same as the one used for the forward sweep of VI-VI-PARS and VI-I-PARS, which is described in detail in Section 5.1.2, “Forward Sweep”.

113 6.3.3.1

Reduced Power Flow Equations

For a single lateral, the branch update function f k of (2.2) is applied repeatedly starting from the source and moving toward the end of the lateral as shown in (6.31).   V   0 w 1 = f 1(w 0) = f 1    I1   w 2 = f 2(w 1) = f 2 • f 1  V 0  I1 wN =

VN IN + 1

   

(6.31)

   V  0 = f N (w N – 1) = f N • … • f 2 • f 1    I1 

The lower half of the composite function f N • … • f 2 • f 1 is used as the mismatch function for the lateral. F ( I 1) = I˜ N + 1( I 1) = 0

(6.32)

To extend this to a general radial structure, note that the voltage V 0 must be known in order to apply (6.31). This implies that lateral i – 1 must be evaluated before lateral i. The BF 8 order used by the standard forward sweep, described in Section 5.1.2, “Forward Sweep”, meets this requirement. Suppose bus k is the first bus on lateral i as shown in Figure 6.5. The current I˜ N + 1 at the end of lateral i is computed as a function of I k and V q , and will be written I˜ N + 1( x) , where x is the vector containing the currents injected into all laterals and x i denotes the current I k injected into lateral i. These x i are put into x in RBF1 order.

114

lateral i bus N

I˜ N + 1

bus k xi = Ik lateral i – 1

from source

Vq

bus q = k – 1

Figure 6.5 Current Mismatch Calculation

x ( L, M L ) x =

x ( l, m ) x ( 1, 1 )

I ( L, M L, 1 ) =

I ( l, m , 1 )

(6.33)

I ( 1, 1, 1 )

To compute I˜ N + 1 , it is necessary to use x instead of just I k since I˜ N + 1 also depends on the currents injected into the sub-laterals of i. Furthermore V q is affected by the currents injected into the sub-laterals of lateral i – 1 which are closer to the source. Because of this, I˜ N + 1 is a function of the currents injected into all sub-laterals which branch off of the path between the source and the end of lateral i. This yields an equation similar to (6.32) for lateral i whose end bus is bus N, where I 1 is replaced by the vector x of the currents injected into all laterals. F i( x) = I˜ N + 1( x) = 0

(6.34)

115 Taking (6.34) for each lateral and combining these equations in RBF1 order as in (6.7) yields the new reduced power flow equations which can be still be expressed compactly as F ( x) = 0 .

(6.35)

Note that the implementation of (6.35) in this case requires that the laterals be evaluated in BF8 order, not in the order they appear in F and x.10 As with VI-DePARS, a class B or class C transformer entering bus k causes the variables or mismatch functions for lateral i to change slightly from the standard I k and I˜ N + 1 . For a lateral with a class B transformer, the voltage update formula used is the one in (5.15) which includes the modified primary current. The mismatch function in (6.34) is a set of two equations in three unknowns. The additional equation required to solve the problem is obtained by setting to zero the primary side constraint mismatch presented in Section 5.2.1.2, “Forward Sweep”, as (5.12) or (5.13). For type 2 grounded wye to ungrounded wye transformers, (5.12) gives Σ

Σ

∆ I˜ k( x) = I k = 0

(6.36)

and for type 3 grounded wye to delta transformers, (5.13) gives y Σ Σ Σ ∆ I˜ k( x) = I k – -----k2- V k – 1 = 0 . αk

(6.37)

This mismatch is a function of x since the primary voltage and current sums are functions of x. In this case, the mismatch function F i( x) becomes F i( x) =

10 The

I˜ N + 1( x) Σ ∆ I˜ k( x)

= 0.

(6.38)

ordering of F and x need only satisfy the constraint that they be sorted by lateral level. RBF order was chosen primarily for consistency with VI-DePARS.

116 For a class C transformer entering bus k, the secondary voltage and Σ

current are functions of the w k – 1 of (3.60) which includes I k ′ , the sum of the currents at the secondary side of the transformer, not just the two dimensional primary current I k . This is described in Section 5.2.2.1, “Forward Sweep”, on page 69. In this case, the independent variable x i associated with lateral i is

xi =

Ik Σ

Ik′

.

(6.39)

The variable x, which contains the currents injected into all laterals, will Σ

also contain I k ′ for laterals with class C transformers. This allows (6.34) to be used in its present form. With these small changes in the mismatch and the independent variable for laterals with class B and class C transformers, respectively, the compact form of the reduced power flow equations in (6.35) can still be used. 6.3.3.2

Update of the Independent Variables

As with the formulation for VI-DePARS, the elements of the system Jacobian can be formed via the chain rule as the product of branch Jacobians. The branch Jacobian G k for I-DePARS is the Jacobian of the branch update function in (2.2) from Table 2.3. This branch update function is w k = f k(w k – 1) and the branch Jacobian is therefore

(6.40)

117

Gk =

∂f k ∂ wk – 1

 ∂V˜ k   ∂V˜ k       ∂ V k – 1  ∂ I k  = .  ∂I˜ k + 1   ∂I˜ k + 1       ∂ V k – 1  ∂ I k 

(6.41)

Suppose lateral j branches off of its parent at bus q and lateral i’s end bus is bus n.11 If bus q does not lie on the path from the source to bus n, then the corresponding block element ∂ F i ⁄ ∂ x j of the system Jacobian is a zero block. If bus q does lie on the path from the source to bus n, then the corresponding block is non-zero. Table 6.6 summarizes the formation of the non-zero blocks of the system Jacobian. In order to simplify the presentation, the equations given in Table 6.6 are for the case with no class B or class C transformers present. This leads to the structure shown in Figure 6.6 for the sample radial network shown in Figure 2.1 on page 10. This structure is the transpose of the Jacobian structure for VI-DePARS shown in Figure 6.3. For lines and switches, the branch Jacobian in (6.41) can once again be approximated by an identity matrix, under the same assumptions about the relative magnitudes of per unit circuit parameters and per unit voltages. Following a development similar to that of Section 6.1.2.2, “Numerical

Properties

of

the

System

Jacobian”,

and

Section 6.1.2.3,

“Transformers”, leads to similar approximations. The type 3 and type 4 blocks, now above the diagonal, are approximated by zero, the type 2 blocks below the diagonal are approximated by negative identity, and the diagonal type 1 blocks, except when dealing with transformers, are approximated by identity blocks. 11 Here

i and j are exchanged from the scenario used in Table 6.3.

118 Table 6.6 I-DePARS Jacobian Formation Non-Zero Blocks of the System Jacobian for I-DePARS Type 1

Type 2

Type 3

Type 4

i and j are the same lateral

j is a direct sub-lateral of i

i is supplied through j

i is supplied through j – 1

source

source

source

source

q q +1 q +1 q i&j

∂ wn – 1

i j

n

non-zero blocks below diagonal

∂F i = ∂x j ∂I n + 1

q

q +1 i j

diagonal blocks (all non-zero)

q +1 p –1

q

n

(6.42) ⋅ Gn – 1 ⋅ … ⋅ Gq + 2 ⋅

∂f q + 1 ∂ Iq

where q + 1 refers to the first bus on lateral j

where q + 1 refers to the bus following q on lateral i

approximately identity

approximately minus identity

p –1

p n

n

p

i j

non-zero blocks above diagonal ∂F i = ∂x j ∂I n + 1

∂ wn – 1

(6.43) ⋅ Gn – 1 ⋅ … ⋅ G p + 1 ⋅

∂f p

∂V p – 1 ∂f ∂V p – 1 ⋅ ⋅ G p – 2 ⋅ … ⋅ Gq + 2 ⋅ q + 1 ∂w p – 2 ∂ Iq where q + 1 refers to the first bus on lateral j

where q + 1 refers to the bus following q on lateral i

approximately zero

For a lateral i, with a transformer entering bus k,2 the corresponding diagonal block is approximated by one of the following three expressions for class A, class B, and class C transformers, respectively.

119

zero block element non-zero block element of Type 1 non-zero block element of Type 2 non-zero block element of Type 3 non-zero block element of Type 4

xj 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9

Fi

10 11 12 13 14 15 16 17 18 19 20

Figure 6.6 Structure of System Jacobian for I-DePARS

120 ∂F i – ∂ I˜ k ′ ss ps – 1 ≈ ------------- = – Y k ( Y k ) ∂ xi ∂I k

∂F i ∂ xi

≈

 – ∂ I˜ k ′   -------------  ∂I k   ∂ I˜ Σ   -------k-   ∂I k 

∂F i  – ∂ I˜ k ′  ≈  ------------ ∂ xi  ∂I k 

(6.44)

 2 –1 –1 ss  ps 1 – Y k  Y k \ --- – 1 2 – 1 3  =  –1 –1 2 1

   – ∂ I˜ k ′  ss  -  = –Y k   -----------  ∂ I Σk ′   

1

    

(6.45)

1

ps

Yk

111

–1

10 0 01 0 β k2 0 0 -----yk

      

(6.46)

When (6.44), (6.45), and (6.46) are evaluated for each transformer type, the result is a constant matrix depending only on α k and β k as shown in Table 6.7. The appropriate constant matrix from the table is used to replace the identity block on the diagonal of the approximated system Jacobian for a lateral with a transformer. These approximations lead to a lower block triangular matrix which means that, for I-DePARS, the update step in (6.28) can be solved efficiently by block forward substitution. 6.3.3.3

Implementation

As with the other variations of DePARS, the Jacobian need not be formed explicitly due to the simple structure of the approximated Jacobian. Once the mismatches F i have been calculated, the update step for each lateral can be computed directly as si =

∑ s j – Fi j

(6.47)

121 Table 6.7 Jacobian Approximations for Transformers for I-DePARS Transformer Connection Type

A

∂F i ∂ xi

Primary

Secondary

1

Grounded Wye

Grounded Wye

αk 1 0 0 ------ 0 1 0 βk 001

5

Ungrounded Wye

Ungrounded Wye

αk 1 0 -----βk 0 1

6

Ungrounded Wye

Delta

8

Delta

Ungrounded Wye

αk 1 –1 -----------3β k 1 2

9

Delta

Delta

αk 1 0 -----βk 0 1

2

Grounded Wye

Ungrounded Wye



 

B

ps Yk

 2 –1 –1  1 \ --- – 1 2 – 1  3  –1 –1 2 

1 Grounded Wye

αk ------------ 21 3β k – 1 1

2α k – α k – α k ------------ ------------- ------------3β k 3β k 3β k

ss  –Y k 

3

ps – 1

ss

–Y k ( Y k )

1

– α k 2α k – α k ------------ ------------- ------------3β k 3β k 3β k

1

Delta

1

1

αk –α ------ 0 --------kβk βk –αk αk --------- ------ 0 βk βk 1

4

Ungrounded Wye

C 7

Delta

Grounded Wye

Grounded Wye

1

1

1

not applicable

  ss  –Y k    

ps

Yk

111

–1

10 0 01 0 β k2 0 0 -----yk

      

αk –αk –1 -------- --------- -----3β k 3β k 3 α k 2α k – 1 -------- ---------- -----3β k 3β k 3 –2 αk –αk –1 ------------- --------- -----3β k 3β k 3

122 for laterals without transformers, and  ∂F i  –1   s –F si =  j i  ∂ xi   ∑ j

(6.48)

for laterals with transformers. Here s j refers to the update step for lateral j, where j is a sub-lateral of lateral i. The calculation of the update steps must therefore proceed in RBF1 order. One disadvantage to I-DePARS as compared to VI-DePARS is the availability of a good starting value for the independent variable x. For VI-DePARS the end voltages are set directly to balanced 1 per unit. However, some computation is required to find an initial value for the currents injected into each lateral. This is typically done by current summation during some type of backward sweep such as the one used in V-I-PARS.

6.3.4

S-DePARS

S-DePARS is the power flow based counterpart to I-DePARS and is a generalization of the reduced power flow equations and fast decoupled algorithm for radial systems first proposed in [12]. This approach is restricted to transformers of types 1, 5, and 9 which have the same connection on both primary and secondary. For an ideal transformer, there is no change in power from the primary to the secondary so an identity block is used for the corresponding diagonal block of the system Jacobian. Unfortunately, for the other transformer connection types there is no simple way to approximate the Jacobian, so they have not been included in this formulation.

123 The approximate Jacobian for S-DePARS is identical to that of I-DePARS. The update step in the Newton method can therefore be solved by a block forward substitution.

6.4 Convergence Analysis In this section, it will be shown that the fast decoupled algorithms described here fall into the class of inexact Newton methods discussed in [14]. In the classical Newton’s method shown in Table 6.1, the update step s

( i)

at iteration i is the solution to J

( i)

s

( i)

= –F

( i)

.

(6.49)

In the inexact Newton methods, the corresponding equation is J

( i)

s

( i)

= –F

where the size of the residual r

( i)

( i)

+r

( i)

,

(6.50)

is restricted so that the relative residual

is bound by some forcing sequence. Specifically, ( i)

r ( i) --------------≤η , ( i) F where { η

η

( i)

(6.51)

} is a forcing sequence which is uniformly less than one. Here n

.

denotes an arbitrary norm in IR . Notice that the special case where

( i)

≡ 0 gives the exact Newton’s method. A sequence { x

( i)

} of iterates produced by the inexact Newton

method is locally and linearly convergent. This result is stated and proved as Theorem 2.3 in [14]. Using the present notation, this theorem asserts the following:

124 ( i)

max

Assume that η ≤ η < t < 1 . There exists ε > 0 such that, if ( 0) x – x∗ ≤ ε , then the sequence of inexact Newton iterates ( i) { x } converges to x∗ . Moreover, the convergence is linear in the sense that x where y

J∗

( i + 1)

– x∗

J∗

≤t x

( i)

– x∗

J∗

(6.52)

≡ J ( x∗) y .

For the fast decoupled methods presented here, the update step can be expressed as the solution to ( i) ( i) J˜ s = –F ,

(6.53)

where J˜ is the constant approximation, shown in Figure 6.4,12 to the Jacobian J

( i)

. With the definition ∆J

( i)

≡J

( i)

– J˜ ,

(6.54)

(6.53) can be rewritten in the form of the inexact Newton equation of (6.50), J

( i)

where the residual is r

s

( i)

( i)

= –F

= ∆J

( i)

( i)

s

( i)

+ ∆J

( i)

s

( i)

(6.55)

.

To prove that DePARS is, in fact, an inexact Newton method, and therefore locally and linearly convergent, it remains to be shown that there is some sequence { η

( i)

} , uniformly less than one, which bounds the rela-

tive residual. This relative residual can now be expressed as ( i)

( i)

( i)

r ∆J s -------------- = ----------------------------. ( i) ( i) ˜ F Js

12 Or,

(6.56)

for I-DePARS and S-DePARS, the transpose of the matrix shown in Figure 6.4.

125 For a network with only grounded wye to grounded wye transformers some conclusions can be drawn about the size of this residual under the following assumptions: • All voltage magnitudes are close to 1 per unit. • All transformer tap ratios are close to one. • All per unit network parameters13 are small compared to voltage magnitudes (i.e. they are « 1 ). Given these assumptions, J˜ is of the form shown in Figure 6.4,14 where all diagonal terms are equal to or nearly equal to one and the offdiagonal non-zero terms are equal to minus one. The matrix ∆J

( i)

in the

residual term is the difference between the true Jacobian, whose structure is shown in Figure 6.3, and J˜ . This elements of this matrix are all small compared to one. This implies that ( i)

( i)

∆J s ---------------------------- « 1, ( i) J˜ s hence η

( i)

(6.57)

can be chosen such that ( i)

( i)

∆J s - < η ( i ) < 1 , ---------------------------( i) J˜ s

(6.58)

thereby completing the proof that DePARS is a locally and linearly convergent inexact Newton method.

13

This includes all line and transformer impedances, line charging admittances, shunt admittances, constant Z load admittances, constant PQ load power injections, and cogenerator power injections. 14 Figure 6.6 for I-DePARS and S-DePARS.

126

6.5 Comments As with the backward/forward sweep methods of Chapter 5, the fast decoupled methods are applicable to a wide range of radial distribution networks. The one modeling limitation of the general DePARS formulation is that it does not include type 4 ungrounded wye to grounded wye transformers. One variation, S-DePARS, is further restricted to transformers of types 1, 5, and 9, all of which have identical connection and grounding on both primary and secondary sides. As with BFS-PARS, the power flow based variations typically require more computation per iteration than their current based counterparts. Considering this difference and the fact that S-DePARS is limited to only three types of transformer connections, VI-DePARS and I-DePARS appear to be the most attractive of the four fast decoupled methods. VI-DePARS has the added advantage of readily available initial values for the independent variable x. As with NR-PARS and BFS-PARS, the amount of computation required per iteration is proportional to the number of buses. For a constant number of iterations for convergence, the computational complexity increases linearly with the size of the system, making DePARS effective for very large radial distribution networks.

Chapter 7

Power Flow Algorithms for Weakly Meshed Systems (PAWMS) The previous chapters have dealt with electric distribution systems with a radial topological structure. The algorithms developed and presented in these chapters are specific to networks which do not contain any loops. This chapter investigates an approach for extending these radial algorithms to handle systems with a limited number of loops. The resulting class of algorithms will be referred to as Power flow Algorithms for Weakly Meshed Systems, or PAWMS. The approach taken by PAWMS requires that the meshed system be converted to a radial structure by breaking each of the loops. The current or power injections at each breakpoint are adjusted in order to balance the voltages at either side using a compensation method [30]. Variations of this method have been presented in [23], [19], and [20] for single-phase networks, and more recently in [11] for three-phase systems. The approach presented here generalizes these to work with the various three-phase radial power flow algorithms presented in this dissertation.

127

128 Apart from handling cases with loops, the compensation method also makes it possible to solve systems with more than one voltage controlled bus. These can be secondary sources with both voltage magnitude and angle specified, or PV buses which specify voltage magnitude and real power injection. First, the method of dealing with loops is described in detail, followed by the modifications necessary for secondary sources and PV buses.

7.1 Detailed Solution Algorithm Since PAWMS is based on a radial power flow solver, the weakly meshed system must first be converted to a radial structure. This is done by choosing a bus for each loop to serve as the breakpoint. Figure 7.1 shows a system with a loop containing bus k. This loop can be broken by splitting bus k to create a new artificial bus k′ . The solution of the original meshed network is equivalent to the solution of the resulting radial system under the constraints that V k = V k′ and the current injection at bus k′ is the negative of the injection at bus k. The actual creation of the artificial buses to break the loops is performed once during the initialization process. The algorithm, after initialization, consists of two steps which are repeated until convergence is achieved, as shown in Table 7.1. First, the breakpoint voltages are updated via a radial power flow method. Then, the current injections, I BPj at bus k and – I BPj at bus k′ , are adjusted according to the breakpoint voltages in order to eliminate any mismatch. The adjustment is based on the sensitivity of the breakpoint voltage mismatch to changes in the breakpoint current injections. This sensitivity is approximated by the breakpoint

129 Table 7.1

Power Flow Algorithms for Weakly Meshed Systems PAWMS - The Algorithm

Break loops. Form and factor breakpoint impedance matrix. Initialize breakpoint injections, initialize PARS. 1

Update PARS, compute breakpoint voltage mismatch.

2

Update breakpoint injections.

Repeat steps 1 and 2 until convergence is achieved.

– +

primary source

– +

primary source

rest of network

rest of network

bus q

bus q

loop j

loop j

bus k

bus k ′

bus k

I BPj

I BPj negative path

I BPj positive path

Figure 7.1 Loop Breakpoint impedance matrix Z BP , which is a constant linear approximation to the sensitivity matrix. Since it is constant, it is necessary to form and factor Z BP only once during the initialization of the algorithm.

130

7.1.1

Loop Breakpoint Creation

The primary issue in converting the meshed system to a radial network is the choice of the loop breakpoint locations. As shown in Figure 7.1, each loop can be divided into two pieces, both of which are supplied through some common bus q. The positive path refers to the path between bus q and the breakpoint bus k, and the negative path is the path between bus q and the artificial bus k′ . Some of the power supplied to the loop through bus q will enter through the positive path and some through the negative path depending on the branch impedances and the distribution of the loads on the loop. For good convergence, the breakpoint location for a loop should be chosen so as to minimize the breakpoint current I BPj . Although it is not possible to choose a priori the breakpoint location which will result in the smallest breakpoint current I BPj , it is easy to see that a depth-first approach to finding and breaking loops would be a worst case. In this case, the loop is detected after traversing the entire loop in one direction and arriving again at bus q, which becomes the site of the breakpoint. All of the load on the entire loop would be put on the negative path and none on the positive path. Not only would this approach make it difficult to estimate reasonable initial values for the breakpoint injections, it would also cause unnecessary convergence problems for the radial power flow solver due to the potentially tremendous load on one lateral. Figure 7.2 shows that a much more reasonable and successful method is to use a breadth-first approach to detecting and breaking loops. In this case, both the positive and negative paths are traversed simultaneously and the breakpoint is chosen as the bus farthest from the source. If bus k is visited twice, coming from two different directions, a loop has been

131

1e+0 1e-1 1e-2 Breadth-First 1e-3

Depth-First

1e-4 1e-5 1e-6 1e-7 1e-8 1e-9 0

5

10

15

20 25 Iterations

30

35

40

45

Figure 7.2 Effect of Breakpoint Creation Method on Convergence detected and an artificial bus k′ is created. One of the branches entering bus k is removed and connected as the incoming branch of the new bus k′ . The loop breakpoint creation process is easily included in the initial traversal of the network during bus and lateral indexing as described in Section 2.2.2.2, “Breadth-First Search”. Using this approach, it is reasonable to set the initial values of the breakpoint injections to zero.

7.1.2

Breakpoint Voltage Mismatch

At each iteration of PAWMS, the breakpoint voltages V k and V k′ are updated, via one of the radial power flow algorithms of the previous chapters, based on the current value of the breakpoint injections. This could take the form of a complete power flow solution for the radial system or alternatively, one or several iterations of a radial power flow solver. The

132 resulting breakpoint voltages are then used to compute the mismatch for each loop j with breakpoint k. ∆V BPj = V k′ – V k

(7.1)

Each complex three-phase voltage V = E + jF can be represented by a 6 x 1 vector containing its real and imaginary parts. V = E F

(7.2)

The breakpoint voltage mismatch vector ∆V BP is formed by combining the mismatches ∆V BPj for each loop j as follows: ∆E BP 1 ∆F BP 1 ∆V BP =

∆E BP 2 ∆F BP 2

(7.3)

∆E BPnlp ∆F BPnlp This is a (6 ⋅ nlp) x 1 real vector, where nlp is the number of loops in the network.

7.1.3

Breakpoint Impedance Matrix

In order to update the breakpoint injections in a way which eliminates the breakpoint voltage mismatches, it is necessary to know the corresponding sensitivity information. A linear approximation to these sensitivities is given by the breakpoint impedance matrix Z BP . For a three-phase system, Z BP is a real, non-sparse, square matrix of dimension (6 ⋅ nlp).

133 Let each complex three-phase current I = C + jD also be represented by a 6 x 1 vector of its real and imaginary parts, however, with the imaginary part negated. I =

C –D

(7.4)

The reason for this separation into real and imaginary parts will be seen clearly in Section 7.1.6, “PV Buses”. The breakpoint currents I BPj for each loop j are then combined to form the (6 ⋅ nlp) x 1 real breakpoint injection vector C BP 1 – D BP 1 C BP 2 I BP =

– D BP 2

(7.5)

C BPnlp – D BPnlp The breakpoint impedance matrix Z BP is an approximation of the sensitivity of ∆V BP to changes in I BP . This relationship can be written ∆V BP = Z BP ∆I BP

(7.6)

The matrix Z BP consists of 6 x 6 block elements Z BPij corresponding to loops i and j. These blocks can be constructed directly according to the following observations:

134 • For diagonal blocks where i = j , Z BPij is the sum of the branch impedances in loop i. • For off-diagonal blocks where i ≠ j , the magnitude of Z BPij is the sum of the impedances of branches common to loops i and j. The sign is negative if the loops have opposite direction and positive if they have the same direction.1 To be consistent with the representations of voltage and current in (7.2) and (7.4), the branch impedances Z = R + jX are represented as Z =

R X X –R

(7.7)

Since Z BP is constant, it is only formed once at the beginning of the algorithm. It can also be factored once via LU decomposition as Z BP → L BP ⋅ U BP

(7.8)

and its triangular factors L BP and U BP stored for later use in solving for ∆I BP from (7.6).

7.1.4

Breakpoint Injections

The second part of each iteration of PAWMS is to update the breakpoint current injections. This involves solving for ∆I BP from the set of linear equations in (7.6) via backward and forward substitution using the triangular factors from (7.8). ∆I BP = U BP \ ( L BP \ ∆V BP )

(7.9)

The appropriate element ∆I BPj is then added as a correction to the current value of the injection for loop j. I BPj ← I BPj + ∆I BPj 1

(7.10)

The direction of a loop is arbitrarily defined to be toward the source on the positive path and away from the source on the negative path. This is also the convention used for the direction of the breakpoint current as shown in Figure 7.1.

135 These new current injections are used with positive and negative signs, respectively, at the corresponding breakpoint buses k′ and k during the next radial power flow update.

7.1.5

Multiple Sources

Throughout the preceding chapters of this dissertation, it has always been assumed that the network under consideration has a single voltage specified bus denoted as the source. The compensation method used to deal with loops in a system can also be applied to the problem of multiple sources. In a network with multiple sources, one of them is arbitrarily chosen as the primary source and is used as the source for the purposes of the radial load flow algorithms. Each of the other secondary sources is treated as a normal load bus with an extra current injection. For a secondary source at bus k, the voltage constraint creates a loop j through ground, which can be broken as shown in Figure 7.3. In this case, the artificial bus k′ need not be created explicitly. The voltage mismatch ∆V BPj for a secondary source at bus k is simply specified

∆V BPj = V k

– Vk .

(7.11)

The corresponding breakpoint current injection I BPj is the current injected into the system at bus k by the secondary source. Although the negative path of a loop formed by a secondary source has no impedance, the formation of the breakpoint impedance matrix Z BP does not change from the method described above for simple loops.

136

– secondary + source I BPj

– +

primary source

bus k

– secondary + source I BPj

– +

primary source

bus k ′ bus k

loop j

loop j

rest of network

rest of network

positive path

Figure 7.3 Secondary Source Breakpoint

7.1.6

PV Buses

Some cogenerators are modeled as devices which deliver a specified real power while maintaining a given voltage magnitude.2 This describes the typical PV bus used for generator buses in transmission systems. As with a secondary source, the voltage constraint at the PV bus can be viewed as creating an artificial loop through the primary source. The sensitivity of the voltage at the PV bus to the current injection is given by the same breakpoint impedance matrix. However, only the voltage magnitude is specified and the current injection must be constrained so as to keep the real power injection constant. The breakpoint impedance matrix must therefore be modified to express directly the sensitivity of the voltage magnitude to the reactive power injection for PV buses. This requires some additional approxima-

2 See

Section 3.3, “Cogenerator Model”, on page 29.

137 tions and a change of basis for voltage and current vectors V and I, and impedance matrices Z, as given in (7.2), (7.4), and (7.7). Define a rotation matrix U to rotate phases b and c by 120° and -120°, respectively, 1 0 0 U = 0α 0 , 0 0 α2 where α =

2π j -----e 3

(7.12)

1 3 = – --- + j ------- , and note that 2 2 U –1

1 0 0 ∗ = U = 0 α2 0 . 0 0 α

(7.13)

For 3 x 1 complex vectors V and I, and 3 x 3 complex matrix Z, let V′ ≡ UV

(7.14)

I′ ≡ UI

(7.15)

Z′ ≡ UZU –1

(7.16)

This set of definitions corresponds to a change of basis or rotation of the phase b and phase c voltages and currents. Assuming a three-phase network with a balanced 1 per unit primary source and no phase shifts, the magnitude of each element of V′ is equal to the magnitude of the corresponding element of V and is therefore close to 1 per unit. The angles θ′ of the elements of V′ , however, are all nearly zero, leading to the following approximations to the real and imaginary parts of V′ = E′ + jF ′ .

138

1 E′ ≈ V′ = V ≈ 1 1

(7.17)

0 F′ ≈ θ′ ≈ 0 0

(7.18)

If V is the voltage of a PV bus and I is the corresponding current injection, the change in complex power injection ∆S = ∆P + j∆Q can be expressed in terms of V′ and the change in I′ , and separated into is real and reactive parts as follows: ∆S = V .* ∆I∗ = V .* ( UU ∗ ∆I∗ ) = UV .* ( U∆I ) ∗ = V′ .* ∆I′∗

(7.19)

= ( E′ + jF ′ ) .* ( ∆C′ – j∆D′ ) =

( E′ .* ∆C′ + F′ .* ∆D′ ) + j ( F′ .* ∆C′ – E′ .* ∆D′ ) ∆P

∆Q

Applying the approximations to E′ and F′ from (7.17) and (7.18) yields ∆P ≈ ∆C′

(7.20)

∆Q ≈ – ∆ D′ .

(7.21)

If Z BPij is a sensitivity matrix relating the breakpoint voltage mismatch ∆V BPi for loop i to the breakpoint current injection I BPj for loop j, then Z′BPij is the corresponding sensitivity matrix for ∆V′BPi and I′BPj . This is shown by the following equation where the subscripts have been dropped to simplify the presentation.

139 ∆V′ = = = =

U∆V UZ ∆I UZU –1 U∆I Z′∆ I′

(7.22)

This complex equation can be represented by the following real equation: ∆E′ = R′ X′ ∆C′ ∆F′ X′ – R′ – ∆D′

(7.23)

If loop i is due to a PV bus, the approximations from (7.17) and (7.18) are applied and only the first block row of (7.23) is considered since there is no constraint on the voltage angles. ∆ V ≈ ∆E′ = R′ X′

∆C′ – ∆D′

(7.24)

If loop j is due to a PV bus, the approximations from (7.20) and (7.21) are applied and, since the real power injection is fixed, ∆P is zero so only the second block column is needed. ∆E′ ≈ X′ ( – ∆D′ ) ≈ X′ ∆Q ∆F′ – R′ – R′

(7.25)

If both loops are due to PV buses, both sets of approximations apply and the relevant equation is ∆ V ≈ ∆E′ ≈ X′ ( – ∆D′ ) ≈ X′∆ Q .

(7.26)

The value actually used for the voltage magnitude mismatch at a PV bus k is ∆E′k = ( V k

specified

– V k ) .* E′k . / V k .

(7.27)

This gives the change in the real part of V′k due to the difference between the specified and calculated values of the voltage magnitudes, assuming the angle is left as calculated.

140

7.1.7

Summary

To yield a unified approach for dealing with loops, secondary sources, and PV buses, a change of basis is made according to (7.14)-(7.16) for all breakpoint voltages, all breakpoint current injections, and all branch impedances used in the formation of the breakpoint impedance matrix, now denoted Z′BP . With the inclusion of PV buses, the matrix Z′BP is formed as before, with the exception of block rows and columns corresponding to PV buses. These block rows contain only the rows corresponding to ∆E′ (the first three), and the block columns contain only the columns corresponding to ∆Q (the last three). Each additional loop or secondary source adds six rows and columns to Z′BP , while a PV bus only adds three. A simple example of a system with one loop, one secondary source, and one PV bus illustrates the structure of the breakpoint mismatch and injection vectors and the breakpoint impedance matrix. ∆E′1

R′11 X′11 R′12 X′12

∆F′1

X′11 – R′11 X′12 – R′12 – R′13 – ∆D′1

∆E′2 = R′12 X′12 R′22 X′22

7.1.8

X′13 X′23

∆C′1 ∆C′2

∆F′2

X′12 – R′12 X′22 – R′22 – R′23 – ∆D′2

∆E′3

R′13 X′13 R′23 X′23

X′33

(7.28)

∆Q 3

Termination Criterion

At each step of PAWMS, the breakpoint current injection is updated according to the breakpoint voltage mismatch. Convergence is achieved and the algorithm terminated when the norm of this mismatch is reduced below a pre-determined constant tolerance.

141

7.2 Implementation Since the compensation method used to eliminate breakpoint voltage mismatches is independent of the radial power flow algorithm used to compute these mismatches, this independence was preserved in the implementation. The program was structured in a way which allows any of the variations of NR-PARS, BFS-PARS, or DePARS to be used as the radial solver.

7.2.1

Modeling Limitations and Simplifying Assumptions

The algorithm used to form the breakpoint impedance matrix is based closely on the one from [20]. Since this method builds the matrix directly from sums of branch impedances, it is not suited to loops which contain some grounded and some ungrounded sections. For this reason, the implementation was restricted to networks with only type 1 grounded wye to grounded wye transformers. To simplify the code, the following additional assumptions were made. All loops, including those formed by secondary sources and PV buses, are three-phase only. This eliminates the need to deal with different dimensions for breakpoint voltages and injections. It is also assumed that no two loops result in a breakpoint at the same bus, preventing the need for a variable number of breakpoint injection vectors for any given bus.

7.2.2

Termination of Radial Power Flow

Each iteration of PAWMS requires the computation of the breakpoint voltage mismatch via one of the radial power flow algorithms presented in previous chapters. However, depending on the tolerance used for the radial

142 power flow convergence criterion, a completely converged result may not be necessary, especially in the first several iterations of PAWMS. In fact, in some cases, a single iteration of the radial power flow may be sufficient to achieve satisfactory convergence for PAWMS. As implemented, the program allows for three possible modes of operation. In addition to converged sets and single iterations of the radial power flow, an adaptive method is also included. This adaptive strategy is based on a heuristic which attempts to set the tolerance for the radial power flow for a given iteration such that the breakpoint voltage mismatches computed are only as accurate as necessary. Using the maximum change in bus voltage as the mismatch for the radial solver, a tolerance of 10 –n typically results in bus voltages accurate to about 10 – ( n + 1 ) . This is generally sufficient for the computation of breakpoint mismatches on the order of 10 –n . An estimate of the breakpoint voltage mismatch at the next iteration can therefore be used as a reasonable tolerance for the current radial power flow. This estimate is readily computed from the current breakpoint voltage mismatch and the current rate of convergence, as calculated from the current and previous mismatches. This is the approach used by the adaptive method implemented in the program.

7.3 Variations Several modifications to the basic PAWMS presented here have been considered. They are based primarily on the various approaches presented in [23], [19], and [20] for single-phase power flow in weakly meshed systems.

143

7.3.1

Power Injection for Loop Breakpoints

It is possible to use power injection instead of current injection at the loop breakpoints. This, obviously, does not affect the treatment of PV buses and, in fact, has no effect for secondary sources as well, since their voltages are fixed. For a simple loop j with breakpoint k, however, the voltage mismatch is used to compute a correction to the injected current which is converted to a correction to the injected power according to the following equation:  V k + V k′  ∆S BPj =  ---------------------  .* ∆I BPj ∗ 2

(7.29)

 V′k + V′k′  =  ---------------------- .* ∆I′BPj ∗ 2 In practice, this produces much better results than the more obvious update ∆S = ∆C′ – j∆D′ ,

(7.30)

as shown in Figure 7.4. For a system with a mixed load model, there is little difference in performance between the method based on breakpoint power injection and that based on current injection. In general, using current injection yields equivalent or slightly better convergence results. The largest difference is seen in a system with purely constant current loads, and no shunt components or cogenerators. In this case, the current-based PAWMS converges in a single iteration since the network is linear and the breakpoint impedance matrix is the exact sensitivity matrix.

144

1e+0 ∆I = ∆I U (=∆C′ + j∆D′ ) U(∆C'+j∆D') V′k′   V′=k + ∗ (V1+V2)/2 ∆S = ∆S  ------------------- .* ∆I′* 2 conj(∆I)

1e-1 1e-2 1e-3

∆S = ∆S ∆C′ = –∆C'-j∆D' j∆D′

1e-4 1e-5 1e-6 1e-7 1e-8 1e-9 0

5

10

15

20

25

Iterations

Figure 7.4 Effect of Power vs. Current Injection on Convergence

7.3.2

Correction Step

In each iteration of PAWMS, the breakpoint injections are updated. In the basic PAWMS presented above, the radial power flow iteration immediately following this breakpoint update uses, as its initial condition, the result of the iteration preceding the update. This method can be improved by updating this initial condition to reflect the changes in the breakpoint injections. For NR-PARS and BFS-PARS, the initial condition is specified by the bus voltages. The voltages at the end of each lateral are sufficient to specify the initial condition for VI-DePARS and VS-DePARS. For the other two fast decoupled methods, I-DePARS and S-DePARS, the initial condition is comprised of the currents or powers, respectively, injected into the beginning of each lateral.

145 First consider the correction for NR-PARS and BFS-PARS. This voltage correction step adjusts the bus voltages to reflect the change in the breakpoint injections according to the method presented in [20]. A backward/forward sweep of the radial network is performed with: • all loads, shunts, and cogenerators disconnected • breakpoint injections set to their incremental values (the change in injection just computed) • source voltage set to zero The bus voltages computed from this process are incremental values reflecting the change in bus voltages due to the change in breakpoint injection. These incremental voltages are added to the bus voltages from the previous radial power flow iteration, resulting in a better starting point for the new radial power flow. For VI-DePARS and VS-DePARS, each iteration consists of a mismatch evaluation followed by an update to the end voltages. The voltage correction step described above can be used to calculate the incremental voltages. In this case, it is only necessary to add the incremental values to the end voltages since they are used to update all other bus voltages in the succeeding mismatch evaluation. When using I-DePARS and S-DePARS, the breakpoint injections are updated immediately following a function evaluation. Each iteration of the radial power flow then begins with the update to the independent variables, based on the mismatch from the previous iteration, and ends with the evaluation of a new mismatch. It is convenient to switch this order from that used by the other two DePARS methods above to allow for a mismatch correction to adjust for the change in breakpoint injections.

146 The effect of a change in the injection at a particular breakpoint then is simply to increase the corresponding lateral’s mismatch by the amount of the change. The only other detail is that, for I-DePARS, incremental power injections are converted to their approximate current equivalents3 before adding them to the appropriate mismatch. Likewise, for S-DePARS, incremental current injections are converted to their approximate power equivalents.3 The effect of adding a correction step to PAWMS is to improve the overall convergence characteristics of the algorithm. Using the correction step typically reduces the total number of radial power flow iterations required to solve the weakly meshed power flow.

7.4 Comments For a distribution system with a small number of loops, secondary sources, and PV buses, the size of the breakpoint impedance matrix is relatively small. The computation involved in forming and factoring the matrix is still relatively small compared to the work required for the solution of the radial system. However, as the number of loops grows, the computational burden associated with the breakpoint impedance matrix grows. Since Z BP is, in general, not necessarily sparse, at some point, as the number of loops increases, the work associated with Z BP becomes so large that it is more efficient to use a general nodal approach, such as the traditional Newton-Raphson or Implicit Zbus Gauss methods. For this reason, PAWMS is well suited to weakly meshed systems, but less well suited to the highly connected structure of a typical transmission network. 3 Based

on balanced 1 per unit voltages.

147 Another issue raised by extending a radial power flow technique to handle weakly meshed systems is that of the existence and uniqueness of solutions. According to [13], a typical radial distribution network always has a unique feasible power flow solution. On the other hand, it is a wellknown fact that a meshed transmission network may have many feasible steady-state equilibrium points or none at all. The solution found by PAWMS for a weakly meshed distribution network may therefore not be a unique feasible solution. Presumably, choosing a different initial value for the breakpoint injections could result in a different solution to the power flow problem. Furthermore, divergence of the algorithm in certain cases could be due to the lack of a feasible solution.

Chapter 8

Simulation Results All of the algorithms under consideration were implemented in a program written for MATLAB®.1 The program reads the network data once from a text file and then stores it in binary format for later use. After reading the network data, either from the original text file or from the binary file, the network is traversed as described in Section 2.2.2.2, “BreadthFirst Search”. During this traversal the nodes and laterals are indexed, data is verified for consistency, and sections are marked as grounded or ungrounded. Since it is common to all of the methods, this preprocessing step is omitted from the comparison of computational effort associated with each algorithm. MATLAB® is a high-level interpreted language designed with matrix manipulation in mind. The version used for this implementation, version 4, includes sparse matrix storage and manipulation as built-in functions. These capabilities made it an attractive choice for quick implementation and testing of ideas during the stage of algorithm development. 1 MATLAB®

is a trademark of The MathWorks, Inc.

148

149 For the analysis of the algorithms, the interpreted nature of MATLAB® makes it ideal for observing the behavior of individual parameters. On the other hand, an interpreted language is usually quite a bit slower in execution than the compiled languages typically used for power flow, such as Fortran or C. For this reason, MATLAB® is probably not the language of choice for a program intended for use in industry. On a sufficiently fast workstation, however, computation time for this implementation was not a problem even with networks of more than 1000 buses. The goal of this analysis of the results of the MATLAB® simulation is to draw some meaningful conclusions about the behavior of the algorithms in a compiled language, such as Fortran or C. This entails comparisons of the effectiveness of each of the algorithms presented in Chapter 4 through Chapter 7 relative to one another and relative to other relevant power flow algorithms. For this purpose, computation time in MATLAB® is nearly meaningless. In MATLAB®, solving a set of linear equations is a built-in function and therefore executes at approximately the same speed as Fortran or C. Simple loops, however, are many times slower since each line is interpreted each time through the loop. Consequently, the run time of an algorithm in MATLAB® may be completely unrelated to the run time of the same algorithm in Fortran or C. The number of floating point operations (flops) required, though not a perfect measure, is chosen as a much better indicator of relative run time in a compiled language.

150

8.1 Summary of Algorithms Tested The simulations performed involve 16 different algorithms which can be classified into the following four categories: • traditional algorithms for the standard Y bus formulation (NewtonRaphson, Implicit Zbus Gauss) • network reduction methods (NR-PARS) • backward/forward sweep methods (BFS-PARS) • fast decoupled methods (DePARS) These algorithms are summarized in Table 8.1. The first class, included for the sake of comparison, consists of standard power flow methods applied to the traditional power flow formulation for general meshed systems. The remaining three classes are for radial networks only and are collectively referred to as PARS (Power flow Algorithms for Radial Systems). Each variation of PARS can be extended, as described in Chapter 7, to solve the power flow for weakly meshed networks. When a particular version of PARS is used in conjunction with these extensions, the “PARS” in the name of the algorithm is simply changed to “PAWMS” (Power flow Algorithms for Weakly Meshed Systems). Since the two traditional distribution power flow algorithms tested, Newton-Raphson and Implicit Zbus Gauss, have not been discussed in previous chapters, each of them will be described briefly.

8.1.1

Newton-Raphson Method

The traditional formulation of the distribution power flow problem is a set of power balance equations at each load bus as a function of the bus voltages. Let S bus be the vector of net power injected into the system at

151 Table 8.1 Summary of Distribution Power Flow Algorithms ID

Class

1

traditional algorithms, standard Ybus formulation

2 3

linear network reduction (NR-PARS)

Algorithm NewtonRaphson Implicit Zbus Gauss

Fwd Swp

Bwd Swp

not applicable

V, I

Y, I

Y-PARS

V, I

Y

5

VI-VI-PARS

V, I

V, I

6

V-VI-PARS

V

V, I

V, I

I

V

I

V, S

V, S

V

V, S

V, S

S

V

S

VI-DePARS

direct

V, I

VS-DePARS

direct

V, S

I-DePARS

V, I

direct

S-DePARS

V, S

direct

7 8 9 10

VI-I-PARS back/forward V-I-PARS sweep (BFS-PARS) VS-VS-PARS V-VS-PARS

11

VS-S-PARS

12

V-S-PARS

13 14 15 16

fast decoupled (DePARS)

[25], [28]

general meshed all types

N-PARS

4

Network Transformer Related Structure Connections Refs

[27], [9], [10]

[4]

[18]

radial structure only (or weaklymeshed using PAWMS)

[23], [11] [3]

all except type 4†

[19], [20] [17], [32]

types 1, 5, 9‡

[12]

†All

except ungrounded wye to grounded wye connections. those with identical connection and grounding on both primary and secondary sides. ‡Only

each bus by constant power elements, let I bus be the vector of net current injection by constant current elements, and let Y bus be the bus admittance matrix containing all constant impedance elements. The power balance equation can then be written in terms of the bus voltage vector V. S bus = V .* ( Y bus V – I bus ) ∗

(8.1)

The power flow problem is to find a bus voltage vector V which satisfies this power balance. Roughly speaking, the Newton-Raphson method [25; 28] solves this problem by setting the power mismatch function

152 F (V ) = V .* ( Y bus V – I bus ) ∗ – S bus = 0

(8.2)

to zero and solving for the roots via Newton’s method.2 To be more accurate, the set of equations given to Newton’s method consists of only the rows of (8.2) associated with the load buses. Similarly, the unknowns are the bus voltages at the load buses, since the source voltage is assumed to be given.

8.1.2

Implicit Zbus Gauss Method

The Implicit Zbus Gauss method [27; 9; 10] also uses a standard Y bus formulation. At all times, the state of a power system must satisfy Kirchhoff’s current law at every bus. This constraint can be expressed as I = Y bus V

(8.3)

where I is the vector of current injected by all constant current and constant power elements. The voltage and current vectors are separated into two parts, the first corresponding to the source and the second corresponding to the remaining buses. I1 I2

=

Y 11 Y 12 V 1 Y 21 Y 22 V 2

(8.4)

In this case, the power flow problem is to solve for V 2 , given the source voltage V 1 . If the circuit contains no constant power elements, I 2 is a known constant injection and V 2 can be found directly from the lower part of (8.4). –1 V 2 = Y 22 ( I 2 – Y 21 V 1 )

2 See

Table 6.1, “Newton’s Method”, on page 86.

(8.5)

153 This is a simple linear circuit solution via a nodal method. If the network contains constant power devices, they can be linearized by replacing them with equivalent current injections based on an estimate of the bus voltages. In this case, the current injection I 2 becomes a function of the bus voltage vector V 2 . –1 V 2 = Y 22 ( I 2(V 2) – Y 21 V 1 )

(8.6)

The Gauss method can be applied to solve this equation by repeatedly updating V 2 , evaluating the right hand side using the most recent value of V 2 . When the change in V 2 between iterations is smaller than some tolerance the algorithm is terminated. This is called a “Zbus” method since it is –1 equivalent to repeatedly multiplying by the impedance matrix Y 22 . In

practice, since this matrix is not sparse, it is not necessary or desirable to explicitly form it, hence the “implicit” in the name of the algorithm. Instead, Y 22 is factored once via an LU decomposition and its factors are stored. The evaluation of the right hand side of (8.6) then consists of computing I 2(V 2) and doing a backward and forward substitution using the factored matrix. The primary differences between the implementation used for the tests presented in this chapter and the version described in [9] and [10] are the following. In this implementation, all constant impedance elements, including transformers, shunt capacitors, and constant impedance loads, are included in Y bus . It is possible to include transformers without encountering the ill-conditioning problems mentioned in [10] since this implementation is based on only solving for line-to-line voltages in ungrounded sections of the network.

154

8.2 Description of Test Systems A variety of test systems were used to evaluate the performance of the power flow algorithms under consideration. Table 8.2 gives a summary of the test systems used. Networks G and J originate from two different modTable 8.2 Summary of Test Systems ID Buses Nodes Laterals

Transformers

Power Input† P (kW)

Real Minimum Power Voltage Q † † Loss Magnitude (kVAR)

A

63

135

32

—

710.4

-276.8

1.04%

0.98 p.u.

B

125

249

26

—

1673.9

-650.7

1.74%

0.97 p.u.

C

204

348

101

1

1154.2

-115.0

2.49%

0.93 p.u.

D

242

423

66

—

4007.7

1676.8

2.13%

0.94 p.u.

E

380

754

97

—

4524.6

1187.1

0.74%

0.98 p.u.

F

446

811

123

—

6807.7

2042.5

2.58%

0.94 p.u.

G

293

829

85

6

29,277.2 11,214.7

1.34%

0.91 p.u.

H

552

1031

139

—

4696.7

830.6

2.34%

0.89 p.u.

I

599

1063

149

—

3962.8

-1603.9

1.88%

0.94 p.u.

J

396

1133

108

6

29,116.3

9757.4

2.15%

0.99 p.u.

K

1064

1976

284

—

18,405.0

7799.9

2.95%

0.87 p.u.

†With

mixed load model.

els (J more detailed) of NYSEG’s3 distribution system in Elmira, NY. Test system J uses the peak loading data from the summer of 1993, and is the primary set of data used for the analysis. Unless specifically indicated, the results presented are from test system J. The other systems are used to study the behavior of the algorithms in relation to the size of the system being solved. The data from these systems

3 New

York State Gas & Electric.

155 are derived from models provided by Rochester Gas & Electric in NY. Except for the tests which explicitly examine the effect of the load model, all of the tests are run with a random mixture of approximately equal numbers of constant PQ, constant current, and constant impedance loads.

8.3 Power Flow Algorithms for Radial Systems (PARS) This section examines the performance of each of the power flow algorithms described in this dissertation as applied to strictly radial systems. As certain variations of the algorithms are found to be inferior, they are then excluded from further tests and analysis. First, consider a comparison of the number of iterations required by the various algorithms to solve test systems B and J for a mixed load model, as shown in Figure 8.1. With a few exceptions, the algorithms require between about five and ten iterations to converge. The NewtonRaphson method on average requires fewer iterations due to its quadratic convergence property. The remaining algorithms exhibit what appears to be linear convergence. Figure 8.2 illustrates this difference for a representative subset of the methods. It should be noted that the number of iterations required by Implicit Zbus Gauss and N-PARS are always identical since both methods solve the same linear approximation to the circuit at each iteration. Despite its superior performance with regard to number of iterations, Figure 8.3 shows that the total number of flops required by the NewtonRaphson method is approximately an order of magnitude greater, on average, than required by the remaining methods. In the network reduction

156

Newton-Raphson

Implicit Zbus Gauss Test System J N-PARS

Test System B

Y-PARS

VI-VI-PARS

V-VI-PARS

VI-I-PARS

V-I-PARS

VS-VS-PARS

V-VS-PARS VS-S-PARS

V-S-PARS

VI-DePARS

VS-DePARS

I-DePARS

S-DePARS

0

2

4

6 Iterations

8

10

Figure 8.1 Iterations Required by Each Algorithm

12

157

I-DePARS 1e+0 1e-1

N-PARS

1e-2

V-I-PARS

1e-3

VI-DePARS

1e-4

Newton-Raphson

1e-5 1e-6 1e-7 1e-8 1e-9 1e-10 1e-11 1e-12 1

2

3

4 Iterations

5

6

7

8

9

Figure 8.2 Linear vs. Quadratic Convergence class of algorithms, Y-PARS performs significantly worse than N-PARS, as expected, and is therefore not considered in further comparisons. Figure 8.4 excludes Newton-Raphson and Y-PARS and normalizes the number of flops with respect to V-I-PARS, showing more clearly the comparison for the remaining methods. N-PARS requires fewer flops per iteration and therefore fewer total flops than the Implicit Zbus Gauss method. For the backward/forward sweep methods, updating voltages during backward sweep and currents or power flows during forward sweep on seems to hurt the overall performance. Doing both actually increases the number of iterations required. So the winners in current and power based backward/ forward sweep methods are V-I-PARS and V-S-PARS, respectively. In the remaining tests, BFS-PARS will be restricted to these two variations. In

158

Newton-Raphson

Implicit Zbus Gauss

N-PARS

Y-PARS

VI-VI-PARS

V-VI-PARS Test System J VI-I-PARS

Test System B

V-I-PARS

VS-VS-PARS

V-VS-PARS VS-S-PARS

V-S-PARS

VI-DePARS

VS-DePARS

I-DePARS

S-DePARS

0.01

0.1 1 Floating Point Operations (Mflops)

Figure 8.3 Total Flops for Each Algorithm

10

159

Implicit Zbus Gauss

N-PARS

VI-VI-PARS

V-VI-PARS

VI-I-PARS

V-I-PARS

VS-VS-PARS

V-VS-PARS

VS-S-PARS

V-S-PARS

Test System J Test System B

VI-DePARS

VS-DePARS

I-DePARS

S-DePARS

0

0.5

1 1.5 2 2.5 3 Floating Point Operations (normalized)

Figure 8.4 Normalized Flops vs. Algorithm

3.5

4

160 the fast decoupled class, I-DePARS appears most attractive. In fact, due to the limitations on the types of transformers handled by S-DePARS, it is unable to solve test system J and is therefore also removed from further consideration.

8.3.1

Effect of Load Model and Load Factor on Convergence

From Figure 8.4, the current based methods appear to perform better than the corresponding power based methods, and I-DePARS appears to be more efficient than VI-DePARS, despite the fact that the starting point for VI-DePARS is available without any computation. However, before eliminating the power based methods or VI-DePARS from further consideration, the effect of load models and load factor are examined. First consider the effect of different load models as illustrated in Figure 8.5 and Figure 8.6. For a network with no constant power loads (and no cogenerators), both Implicit Zbus Gauss and N-PARS require a single iteration since the system is linear, making N-PARS the method of choice. For BFS-PARS and DePARS, considering all of the load models, the results show a maximum of one iteration difference between the current based methods and their power flow based counterparts. Even when requiring an extra iteration, Figure 8.6 shows that the current based methods require fewer flops than the corresponding power based methods. The superiority of the current based methods does not seem to be affected by the load model. Consider the effect of increasing the load, for a mixed load model, by multiplying all loads in the base case by a scalar load factor. Figure 8.7 shows the Newton-Raphson method to be least affected by high load fac-

161

12 10 8 6

Constant PQ Constant I Constant Z

4

Mixed

2 0

Figure 8.5 Effect of Load Model on Number of Iterations

10

Constant PQ 1

Constant I Constant Z Mixed

0.1

Figure 8.6 Effect of Load Model on Number of Flops

162

Newton-Raphson 100 V-S-PARS VS-DePARS

90 80

N-PARS

70

V-I-PARS

60

VI-DePARS

50

I-DePARS

40 30 20 10 0

Load Factor

Figure 8.7 Effect of Load Factor on Number of Iterations tors in terms of required iterations. It also appears that the power based BFS-PARS and DePARS methods require fewer iterations at high load factors than their current based versions. It should be noted, however, that all seven of the algorithms shown in Figure 8.7, including Newton-Raphson, diverged when the load factor was increased to 3.75. Looking at the number of flops required by each algorithm reveals that, in spite of the larger number of iterations required by the current based methods at high load factors, the number of flops required is still smaller until the load factor reaches approximately three. It is only beyond this point, indicated by the arrows in Figure 8.8, that the power based methods show any real advantage.

163

7

6

V-I-PARS VI-DePARS

5

V-S-PARS VS-DePARS

4

N-PARS 3

2

1

0

0

0.5

1

1.5 2 Load Factor

2.5

3

3.5

Figure 8.8 Effect of Load Factor on Number of Flops It could be argued that, since the base case load is already a peak loading condition, the number of typical power flow cases where power based methods are better than current based methods is insignificant. Though it is not shown in the plot, the curve for I-DePARS falls between V-I-PARS and VI-DePARS at each load factor, making it the best of the DePARS methods below a load factor of about three.

8.3.2

Effect of System Size on Convergence

An important requirement for any power flow algorithm is that it scale well to very large systems. In order to examine the performance of algorithms proposed in this dissertation, all of the test systems listed in Table 8.2 were used. Along with Newton-Raphson and Implicit Zbus Gauss, only the best version of each class of PARS was evaluation in this test.

164 It is well-known that the number of iterations required by the Newton-Raphson method is typically not affected by the size of the power system. Figure 8.9 shows that, although there is variation in the number of

Newton-Raphson 20

0.5

N-PARS

18

V-I-PARS

0.55

16

I-DePARS

0.6

14

Minimum |V|

0.65

12

Size of Systems Tested

0.7

10

0.75

8

0.8

6

0.85

4

0.9

2

0.95

0

1 0

500

1000 1500 Size of System (# of nodes)

2000

Figure 8.9 Effect of System Size on Number of Iterations

iterations required for the different size systems, there seems to be no correlation between the system size and the number of iterations for any of the algorithms tested. Since the analysis has shown that the number of iterations is affected by load factor, it seems reasonable to conclude that differences in loading could account for the variation in number of iterations from one test system to another. Assuming the minimum voltage magnitude in the system gives some indication of the loading of the sys-

165 tem, the shape of the minimum voltage curve in Figure 8.9 seems to confirm this conjecture. Aside from the variation in number of iterations caused by loading differences, the number of flops required by each algorithm grows approximately linearly with the number of nodes in the system, as illustrated in Figure 8.10.

Implicit Zbus Gauss

2.5

N-PARS V-I-PARS

2

I-DePARS Size of Systems Tested

1.5

1

0.5

0 0

500

1000 Size of System (# of nodes)

1500

2000

Figure 8.10 Effect of System Size on Number of Flops

8.4 Power Flow Algorithms for Weakly Meshed Systems (PAWMS) The system used to test the extension of the radial power flow algorithms to handle weakly meshed systems is derived from the same data as test system J. For these tests, some transformer types have been changed

166 and the open/closed status of some of the switches is changed to create loops in the system. The implementation of PAWMS used for the results presented here includes the extra correction described in Section 7.3.2, “Correction Step”. It was found that this correction step nearly always reduced the total number of flops required for convergence. Sometimes adding this correction turned a divergent case into a convergent one. In particular, some of the DePARS methods did not converge without this step.

8.4.1

Effect of PARS Termination Criterion on Convergence

As mentioned in Section 7.2.2, “Termination of Radial Power Flow”, the program, as implemented, allows for three modes of operation with regard to the termination criterion for PARS during each iteration of PAWMS.4 The adaptive mode typically results in the same number of iterations of PAWMS as the case where converged sets of PARS solutions are used at each PAWMS iteration. The adaptive method, however, requires fewer overall iterations of the radial power flow. In comparing the adaptive method with the mode which performs a single iteration of PARS for each breakpoint injection update, consider Figure 8.11 which shows the total number of PARS iterations for each case. For V-I-PAWMS, the adaptive method typically requires a few more total PARS iterations than the single iteration approach. However, Figure 8.12 shows that the total number of flops is nearly the same for both modes. This is because the single iteration mode requires more PAWMS iterations

4 One

PAWMS iteration refers to one iteration of the outer loop, i.e. one breakpoint injection update.

167

V-I-PAWMS (Adaptive)

45

35

V-I-PAWMS (Single Iterations)

30

I-DePAWMS (Adaptive)

40

I-DePAWMS (Single Iterations)

25 20 15 10 5 0 0

5

10 Number of Loops

15

20

Figure 8.11 Total PARS Iterations for Adaptive vs. Single Iterations

2 1.8 Adaptive

1.6

Single Iterations

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10 Number of Loops

15

20

Figure 8.12 Total Number of Flops for Adaptive vs. Single Iterations

168 and therefore performs more correction steps than the adaptive mode. The results for N-PAWMS and VI-DePAWMS are not shown, but are quite similar with a slightly higher number of flops required for the adaptive mode. For I-DePAWMS, however, Figure 8.11 shows an enormous increase in the number of PARS iterations required when the number of loops is increased beyond about eight. This is due to an increase in the number of PAWMS iterations caused by inaccurate breakpoint voltage mismatches being used for the calculation of the breakpoint injection update. The number of required flops also increases dramatically. The adaptive mode does not suffer from this problem since the breakpoint voltages are always solved to the necessary precision. The adaptive mode then is a “safer” approach, particularly for I-DePAWMS, where the number of flops required is often significantly less than for single iteration mode. For the other methods tested, N-PAWMS, V-I-PAWMS, and VI-DePAWMS, the number of flops for the two modes are typically comparable. For this reason, the adaptive mode is used throughout the remaining tests.

8.4.2

Effect of Number of Loops on Convergence

As the number of loops in the network is increased, the number of total radial power flow iterations typically settles to some constant number, as illustrated in Figure 8.13. The number of iterations required by the Implicit Zbus Gauss method also settles on some constant. Figure 8.14 shows the relationship between the number of loops in the system and the number of flops required for each algorithm. Although the number of iterations stops growing with the number of loops, the number of flops does not.

169

Implicit Zbus Gauss

35

N-PAWMS 30 V-I-PAWMS 25

VI-DePAWMS I-DePAWMS

20 15 10 5 0 0

5

10 Number of Loops

15

20

Figure 8.13 Number of Iterations vs. Number of Loops

Implicit Zbus Gauss

3

N-PAWMS 2.5

V-I-PAWMS VI-DePAWMS

2 I-DePAWMS 1.5

1

0.5

0 0

5

10 Number of Loops

15

Figure 8.14 Number of Flops vs. Number of Loops

20

170 This is because the size of the breakpoint impedance matrix, and therefore the work involved in factoring it, continues to grow. In the case of the Implicit Zbus Gauss method, the increase is more slight and is probably due to the extra non-zeros in Y bus and the extra fill-in during factoring. Because of this slower increase in flops as the loops are increased, Implicit Zbus Gauss becomes more attractive, compared to the methods based on PAWMS, as the number of loops becomes large. In this test, however, even with 19 loops, V-I-PAWMS is slightly more efficient than Implicit Zbus Gauss. Considering the overall performance of the various methods on cases with 3 to 19 loops, illustrated in Figure 8.15, shows that the total number of PARS iterations for I-DePAWMS is the worst of the algorithms tested. However, due to its cheaper radial power flow iteration and cheaper correc-

Newton-Raphson 19 loops 15 loops

Implicit Zbus Gauss

11 loops 7 loops

N-PAWMS

3 loops V-I-PAWMS

VI-DePAWMS

I-DePAWMS

0

10

20

30

40 50 Iterations

60

70

Figure 8.15 Overall Comparison of Iteration Counts

80

90

171 tion step, Figure 8.16 shows that it is only marginally worse than V-I-PAWMS in terms of overall flop count. Here the flop counts have been normalized for each case so that the total for V-I-PAWMS is equal to one. Not surprisingly, N-PAWMS and VI-DePAWMS are less efficient, falling in the same range as Implicit Zbus Gauss in terms of total number of flops.

Implicit Zbus Gauss

N-PAWMS

V-I-PAWMS

19 loops 15 loops 11 loops

VI-DePAWMS

7 loops 3 loops

I-DePAWMS

0

0.2

0.4 0.6 0.8 1 1.2 1.4 Floating Point Operations (normalized)

1.6

1.8

Figure 8.16 Overall Comparison of Flop Counts

8.4.3

Effect of Load Model on Convergence

A case with 19 loops was used to study the effects of various load models on the convergence of V-I-PAWMS and I-DePAWMS. As shown in Figure 8.17, the convergence of PAWMS is approximately linear for each of the load models tested. Of the three models tested, the constant current load model offers the best convergence. This is reasonable since the true

172

1e+0 1e-1 Constant PQ 1e-2 Constant I 1e-3

Constant Z

1e-4

Mixed

1e-5 1e-6 1e-7 1e-8 1e-9 1

2

3

4

5 6 Iterations

7

8

9

10

Figure 8.17 Convergence of V-I-PAWMS for Various Load Models breakpoint sensitivity matrix is not affected by the presence of constant current loads. The corresponding plot for I-DePAWMS is quite similar, though each case requires a few more PARS iterations as illustrated by Figure 8.18. In spite of the extra iterations, Figure 8.19 indicates that the number of flops required is only slightly higher than for V-I-PAWMS.

8.5 Summary All of the results given in this chapter and the conclusions drawn from them are based on the MATLAB® implementation and the test systems described. Much of the code was reused from one algorithm to the next and it is possible that any given algorithm may not have been implemented in

173

Constant PQ

I-DePAWMS V-I-PAWMS

Constant I

Constant Z

Mixed

0

5

10

15

20

25

Iterations

Figure 8.18 Effect of Load Model on Number of PARS Iterations

Constant PQ

I-DePAWMS Constant I

V-I-PAWMS

Constant Z

Mixed

0

0.5

1 1.5 2 Floating Point Operations (Mflops)

2.5

Figure 8.19 Effect of Load Model on Number of Flops

3

174 the most efficient manner. A different implementation could yield slightly different results. In particular, an analysis of the run time of a Fortran or C implementation could produce results which differ from the conclusions drawn from the flop counts given by the MATLAB® implementation. Most of the qualitative results, however, should be similar. For radial power flow, N-PARS is clearly the better of the two network reduction methods. For the backward/forward sweep and fast decoupled methods, the variations based on current typically require less computation than those based on power flow. At very high load factors, however, this is not always true. In general however, for the typical power flow cases tested, V-I-PARS and I-DePARS were superior to the other BFS-PARS and DePARS methods, respectively. They also proved to be quite comparable to one another in performance, showing significant improvements over the traditional methods based on a Y bus formulation. For weakly meshed systems, the adaptive mode was chosen as the best choice for the termination criterion for the radial power flow solver. Out of the PAWMS methods, V-I-PAWMS and I-DePAWMS offered the best performance, showing a significant improvement over Implicit Zbus Gauss for networks with a small number of loops. As the number of loops increases, the more general Y bus formulation shows little increase in computation and the PAWMS approach loses its advantage due to the increasing size of the breakpoint impedance matrix.

Chapter 9

Conclusions The objective of this work was to develop a comprehensive formulation and an efficient solution algorithm for the distribution power flow problem which takes into account the detailed and extensive modeling necessary for use in the distribution automation environment of a real world power system. This objective was achieved through extensions and generalizations of existing power flow algorithms as well as through the development of new methods.

9.1 Contributions A general framework was developed which encompasses existing radial power flow algorithms. This framework consists of the three main classes of algorithms summarized in Table 9.1. Within each class, the existing methods were generalized and extended to include more comprehensive modeling, and new algorithms for each class were introduced.

175

176 Table 9.1 Summary of Radial Power Flow Algorithms Number of Variations

Number of New Variations

Network Reduction (NR-PARS)

2

1†

Backward/Forward Sweep (BFS-PARS)

8

4

Fast Decoupled (DePARS)

4

3†

Class of Algorithm

†The

best variation in this class is new.

In particular, the general formulation includes: • general radial structure • unbalanced three-phase operation, including single-phase and two-phase branches • general load models, including constant power, constant current, and constant impedance loads, connected in wye or delta configurations • cogenerators • shunt capacitors • line charging effects • switches • three-phase transformers of various connection types Some of the extensions required by the above list are straightforward. The handling of general transformer connections, however, required significant modifications to the existing methods. In the first class of algorithms, the network reduction methods or NR-PARS, the method based on Norton equivalent reductions (N-PARS) proved to be the best. This method is capable of handling all nine of the transformer connection types listed in Table 3.8. In the second and third classes, V-I-PARS and I-DePARS offer the best performance in their

177 respective classes and have very similar computational requirements which are significantly less than those of N-PARS. Both V-I-PARS and I-DePARS, however, are restricted to systems which do not have any type 4 ungrounded wye to grounded wye transformer connections. All three methods, N-PARS, V-I-PARS, and I-DePARS, require significantly less computation than the traditional Newton-Raphson or Implicit Zbus Gauss methods. Proofs of convergence have been given for the backward/forward sweep and fast decoupled algorithms, indicating that they are locally and linearly convergent. Furthermore, the simulation results indicate that the number of iterations required for convergence is not a function of the system size. Therefore, since the amount of work for each iteration is proportional to the size of the system, the computational burden of each algorithm grows only linearly with the size of the system, making them suitable for very large distribution systems. In order to solve weakly meshed systems, various extensions were also made to the compensation method, previously applied only in conjunction with various backward/forward sweep methods. The general structure proposed, with certain modeling restrictions, includes the following contributions: • general three-phase radial power flow • general correction step • secondary sources • three-phase PV buses • adaptive mode of radial power flow termination

178 Out of the PAWMS methods, V-I-PAWMS and I-DePAWMS offer the best performance, showing a significant improvement over Implicit Zbus Gauss for networks with a small number of loops.

9.2 Future Work As with any work of research, there is always more that can be done. Aside from further testing of the code and the algorithms as they stand, there are several extensions and modifications which can be explored. These include: • expand network modeling • remove limitations on formulation • explore possibilities for improved contingency analysis • implement for industry use Though the modeling presented in this dissertation is quite general, there are certainly improvements that can be made. As monitoring devices become cheaper, more detailed information will become available on the behavior of the components of a distribution system. In particular, with more data it may be possible to use more accurate load models which are not combinations of constant power, constant current, and constant impedance. The inclusion of core loss in the transformer models is also important for some applications, but was not included in this formulation due to a lack of verifiable data. The automatic tap changes of voltage regulators might also be considered. Further extensions to the formulation of PAWMS could be explored as well. In particular, it may be possible to extend the current formulation to

179 remove the limitations discussed in Section 7.2.1, making it possible to handle systems with a mixture of grounded and ungrounded sections. One common use of a power flow algorithm is to study various possible contingencies to determine the most profitable configuration for the operation of the network. In such an application, the contingencies are typically specified with respect to some base case. If the power flow solution for the base case is known, it may not be necessary to run a complete power flow for each contingency. It is possible that the concepts behind the proposed algorithms could be applied to compute partial or approximate power flow solutions for the contingencies, given the base case solution. The exploration of this possibility might yield more efficient approaches to contingency analysis in distribution systems. One of the most obvious ways of building upon the work presented in this dissertation is to convert the MATLAB® program, used for the study of the algorithms, to a compiled C or C++ program suitable for everyday use in by a power engineer in industry. Such a program could be a very useful tool for many applications in distribution planning and operation.

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