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, Ph.D. Cornell University 1995
The objective of this work was to develop a formulation and an effi-cient 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 dis-tribution 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 effi-cient 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 meth-ods, backward/forward sweep methmeth-ods, and fast decoupled methods. Within each of the three classes, new algorithms were developed and exist-ing methods were extended to include the comprehensive modelexist-ing of the general formulation. Proofs of convergence for the backward/forward sweep and fast decoupled methods are also provided.
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 com-pensation 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 sys-tems range in size from 63 buses to over 1000 buses. The most efficient algorithm in each class was shown to require significantly less computa-tion 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.
iii
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 Univer-sity in Philadelphia, PA, he participated in a cooperative education pro-gram 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 Sci-ence degree in May, 1992, in the area of network reconfiguration in electric power distribution systems.
v
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.
vi
ABSTRACT
BIOGRAPHICAL SKETCH...iii
ACKNOWLEDGMENTS...v
TABLEOF CONTENTS... vi
LISTOF TABLES...x
LISTOF 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
vii
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
viii
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
ix
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
x
Table 2.1 Mathematical Notation...8
Table 2.2 Bus Indexing Implementation...14
Table 2.3 General Update Formulas ...17
Table 2.4 Implementation of (2.2) ...18
Table 2.5 Implementation of (2.3) ...18
Table 2.6 General Branch Update Formulas...20
Table 3.1 Load Parameters from Nominal Loads...25
Table 3.2 Load Admittance...27
Table 3.3 Current & Power Injected by Load...28
Table 3.4 Shunt Capacitor Admittance, Current & Power Injection ...29
Table 3.5 Cogenerator Admittance, Current & Power Injection ...30
Table 3.6 Update Formulas for Distribution Lines...32
Table 3.7 Update Formulas for Switches...34
Table 3.8 Admittance Matrices for Common Transformer Connections ...36
Table 3.9 Update Formulas for Class A Transformers...37
Table 3.10 Update Formulas for Class B Transformers ...39
Table 3.11 Update Formulas for Class C Transformers ...41
Table 4.1 Network Reduction Method...43
Table 4.2 Network Elements...44
Table 4.3 Notation for (4.5) and (4.6)...47
Table 4.4 Formulas for Combining with Incoming Branch...48
Table 4.5 Voltage Update Formulas ...50
Table 5.1 Backward/Forward Sweep Method...58
Table 5.2 Detail on Backward/Forward Sweeps...59
Table 5.3 Implementation of (5.1) ...60
Table 5.4 Implementation of (5.5) ...63
Table 5.5 Notation for V-I-PARS Convergence Proof...78
Table 6.1 Newton’s Method...86
Table 6.2 Fast Decoupled Power Flow Algorithm for Radial Systems...87
Table 6.3 VI-DePARS Jacobian Formation...97
Table 6.4 Jacobian Approximations for Transformers for VI-DePARS...106
Table 6.5 Various Formulations for DePARS...110
Table 6.6 I-DePARS Jacobian Formation ...118
Table 6.7 Jacobian Approximations for Transformers for I-DePARS ...121
xi
xii
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 Single Feeder Example ...88
Figure 6.2 Voltage Mismatch Calculation ...90
Figure 6.3 Structure of System Jacobian for VI-DePARS ...99
Figure 6.4 Approximation to the System Jacobian for VI-DePARS...108
Figure 6.5 Current Mismatch Calculation...114
Figure 6.6 Structure of System Jacobian for I-DePARS ...119
Figure 7.1 Loop Breakpoint ...129
Figure 7.2 Effect of Breakpoint Creation Method on Convergence ...131
Figure 7.3 Secondary Source Breakpoint...136
Figure 7.4 Effect of Power vs. Current Injection on Convergence...144
Figure 8.1 Iterations Required by Each Algorithm...156
Figure 8.2 Linear vs. Quadratic Convergence...157
Figure 8.3 Total Flops for Each Algorithm...158
Figure 8.4 Normalized Flops vs. Algorithm...159
Figure 8.5 Effect of Load Model on Number of Iterations ...161
Figure 8.6 Effect of Load Model on Number of Flops ...161
Figure 8.7 Effect of Load Factor on Number of Iterations ...162
Figure 8.8 Effect of Load Factor on Number of Flops...163
Figure 8.9 Effect of System Size on Number of Iterations ...164
Figure 8.10 Effect of System Size on Number of Flops ...165
Figure 8.11 Total PARS Iterations for Adaptive vs. Single Iterations ...167
Figure 8.12 Total Number of Flops for Adaptive vs. Single Iterations...167
Figure 8.13 Number of Iterations vs. Number of Loops...169
Figure 8.14 Number of Flops vs. Number of Loops ...169
Figure 8.15 Overall Comparison of Iteration Counts...170
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 infrastruc-ture 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 compo-nents: the generation system, the transmission system, and the distribu-tion 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 calcula-tion is the steady state power flow problem, also called load flow. It
essen-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 genera-tion points to the load points. The Newton-Raphson method itself is com-putationally 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 approxi-mations 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 prob-lem 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-decou-pled Newton method unsuitable for most distribution power flow problems. Since power flow is such a fundamental calculation for a power sys-tem, it is used in many applications in planning and operation. Some of the
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 formu-lation 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 effi-cient 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.
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 pro-posed 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 configu-rations
• 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.
Chapters 4, 5, and 6, respectively, discuss in detail the network reduc-tion, backward/forward sweep, and fast decoupled algorithms. Each chap-ter presents first the basic concepts behind the corresponding class of methods, then a detailed description of a specific algorithm in the respec-tive 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 pre-sented 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 com-ments.
Each of the radial power flow algorithms, including all of the varia-tions presented, along with the extensions to weakly meshed networks, was implemented in a MATLAB® program for testing. In addition, the
New-ton-Raphson and Implicit Zbus Gauss methods were implemented for com-parison. 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
the network are examined. The chapter ends with a summary of the simu-lation 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.
7
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 con-ditions of load and cogeneration. This involves, first of all, finding all of the bus voltages. From these voltages, it is possible to directly compute cur-rents, 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 conven-tions. The conventions shown in Table 2.1 for complex vectors x and y, for
complex matrices A, X, and Y, and for functions f and g, will be used exten-sively throughout this dissertation.
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 equa-tions and variables can be reduced so that each set of equaequa-tions and
†The notation 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 pro-duces 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.
Table 2.1 Mathematical Notation The Expression: Is Used To Denote:
element-wise multiplication element-wise division
element-wise complex conjugate element-wise magnitude
element-wise squared magnitude
†solution to the equation †solution to the equation
computed value, as opposed to given value constant parameter, as opposed to a variable function composition, x.* y x. / y x∗ x x 2 x = A\ y Ax = y A–1 X = A\Y AX = Y x˜ x f • g( )x f g x( ( ))
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 lat-erals 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 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 tri-ple where n is the bus index. The ordered triple refers to the nth bus on the mth level l lateral. The source is given an index of . The number of levels in a network will be denoted by L, the num-ber of laterals on level l by , and the number of buses on lateral
by .
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
l m, ( ) l m n, , ( ) (l m n, , ) 1 1 0, , ( ) Ml (l m, ) Nl m,
Figure 2.1 Example of Bus & Lateral Indexing (1,1,0) (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6) (1,1,7) (2,3,1) (2,3,2) (3,6,1) (2,3,4) (2,3,3) (3,6,2) (3,6,3) (3,5,3) (3,5,2) (3,5,1) (4,9,1) (4,9,2) (2,1,1) (2,1,2) (2,1,3) (2,1,4) (2,1,5) (3,1,1) (3,1,2) Source (4,1,2) (4,1,1) (2,2,4) (2,2,3) (2,2,1) (2,2,2) (2,2,5) (2,2,6) (4,4,2) (4,4,1) (3,3,3) (3,3,2) (3,3,1) (3,2,1) (3,2,2) (3,2,3) (3,2,4) (3,2,5) (4,3,1) (4,3,2) (4,2,2) (4,2,1) (4,7,1) (4,7,2) (4,6,1) (4,6,2) (4,5,2) (4,5,1) (3,4,1) (3,4,2) (3,4,3) (3,4,4) (4,8,2) (4,8,1) (4,10,1) (4,10,2) 1 3 2 4 5 6 8 7 9 10 11 13 12 15 16 18 17 19 20 14
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 refers to the parent of lateral i, and bus refers to bus k’s parent bus. Unless specified otherwise, bus 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 , which is done fre-quently throughout the following chapters. In this case, bus 0 of lateral i refers to the bus on lateral from which lateral i emanates. For exam-ple, in the network in Figure 2.1, bus 0 of lateral (2,3) is also bus 4 of lat-eral (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 origi-nal network data.
2.2.2.1 Connectivity Data Structures
The network data which specifies the topology of the system is typi-cally given as a list of branches with information on which two buses the
i 1– k 1–
k 1+
k = n
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 index-ing 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
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 continua-tion of the same lateral and which children are considered to be on sub-lat-erals 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 main-tained during the search process. First, l, m, and n are used to denote the current level, lateral, and bus indices, respectively. The counters L, , and 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 , respectively. The algorithm used to generate the ordered tri-ple 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).
Ml Nl m,
l m,
The index for the source is set directly to . Each of the remain-ing 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
Table 2.2 Bus Indexing Implementation Algorithm for Setting Indices (l, m, n) for Bus k if bus k is the source
set
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 ) initialize
end if
increment by 1 and set initialize
end if
increment by 1 and set end if
set the index for bus k to l m n, , ( ) = (L M, 1,N1 1, ) = (1 1 0, , ) L = l Ml = 0 Ml m = Ml Nl m, = 0 Nl m, n = Nl m, l m n, , ( ) 1 1 0, , ( )
index m. Finally, in both cases, the bus counter for that lateral is incre-mented 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 correspond-ing lateral counter is initialized to zero. Likewise, each time a new level l lateral is first encountered, is incremented and assigned to m and the corresponding bus counter is initialized to zero.
When the entire network has been traversed by the search, all connec-tivity 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 correspond-ing lateral. These elements are sorted in descendcorrespond-ing order accordcorrespond-ing 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 con-nected 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 quanti-ties 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”.
Ml
Ml Nl m,
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 volt-ages, currents, and power flows along the paths from the source to the end buses. These are the backward/forward sweep and fast decoupled algo-rithms discussed in Chapter 5 and Chapter 6, respectively.
Figure 2.2 Basic Building Block
Bus k-1 Bus k distribution line, sub-laterals Vk–1 Ik Vk I k+1 SLk cogenerator shunt capacitor load SCk SGk (supplying) (receiving) Ik′ switch, or transformer Sk Sk′ Sk+1 ILk ICk IGk
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
and , (2.1)
these update formulas can be expressed mathematically by the equations in Table 2.3 for each of four different cases.
The functions and are inverses of each other and the functions and are inverses of each other. Each of the four functions is deter-mined by the respective load, shunt capacitor, cogenerator, and sub-later-als 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 con-sists 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
Table 2.3 General Update Formulas Based on Direction of
Calculation Update Formula current forward (2.2) backward (2.3) power flow forward (2.4) backward (2.5) wk Vk Ik 1+ = uk Vk Sk 1+ = wk = fk(wk 1– ) wk 1– = gk(wk) uk = ek(uk 1– ) uk 1– = hk( )uk fk gk ek hk
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 incom-ing branch4 can be represented by an admittance matrix, expressed in a general form as
. (2.6)
4 The incoming branch of bus k will also be referred to as branch k.
Table 2.4 Implementation of (2.2)
Procedure Details
1 Compute from and . see Tables 3.6 - 3.11 2 Compute from , and . see Tables 3.6 - 3.11 3 Compute , , and from . see Tables 3.3 - 3.5 4 Compute via KCL at bus k. see (2.20)
Table 2.5 Implementation of (2.3)
Procedure Details
1 Compute , , and from . see Tables 3.3 - 3.5 2 Compute via KCL at bus k. see (2.20)
3 Compute from and . see Tables 3.6 - 3.11 4 Compute from , and . see Tables 3.6 - 3.11
Vk Vk 1– Ik Ik′ Vk Vk 1– Ik IGk ICk ILk Vk Ik 1+ IGk ICk ILk Vk Ik′ Vk 1– Vk Ik′ Ik Vk 1– Vk Ik′ YkBR Yk 11 Y12k Yk21 Y22k =
The voltages and currents at either end of branch k are related by as follows:
(2.7)
From (2.7), can be calculated in the forward direction as a function of and or can be calculated in the backward direction as a function of and . If power flow is being used instead of current, the following are substituted for and , respectively:
(2.8) (2.9) Once the voltage, or depending on the direction of calcula-tion, 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:
(2.10)
(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
YkBR Ik Ik′ Yk11 Yk12 Yk21 Yk22 Vk 1– Vk = Vk Vk 1– Ik Vk 1– Vk Ik′ Ik Ik′ Ik = (Sk. /Vk 1– ) ∗ Ik′ = (Sk′. /Vk) ∗ Vk Vk 1– Sk = Vk 1– .* Ik∗ Sk′ = Vk.*Ik′∗
capacitors, and loads, represented by , , and , respectively. Each of these quantities is a function of and is hence designated with a tilde. It also requires the currents injected into sub-laterals branching off from bus k. Here and is the set of buses adjacent to bus k on sub-laterals.
KCL at bus k can then be written as
. (2.20)
In the case where power flow is used there is an analogous equation expressing the conservation of complex power at bus k.
(2.21) Table 2.6 General Branch Update Formulas
Based on Direction of Calculation Function of Update Formula current forward (2.12) (2.13) backward (2.14) (2.15) power flow forward (2.16) (2.17) backward (2.18) (2.19) Vk 1– ,Ik V˜k Yk 12 ( )–1 Ik–Yk11Vk 1– ( ) = I˜k′ Yk 21 Vk 1– Yk 22 V˜k + = Vk,Ik′ V˜k 1– Yk 21 ( )–1 Ik′–Yk22Vk ( ) = I˜k Yk 11 V˜k 1– Yk 12 Vk + = Vk 1– ,Sk V˜k Yk 12 ( )–1 Sk. /Vk 1– ( ) ∗–Yk11Vk 1– ( ) = S˜k′ V˜k Yk 21 Vk 1– +Yk22V˜k ( ) ∗ .* = Vk,Sk′ V˜k 1– Yk 21 ( )–1 Sk′. /Vk ( ) ∗–Yk22Vk ( ) = S˜k V˜k 1– Yk 11 V˜k 1– Yk 12 Vk + ( ) ∗ .* =
I˜Gk I˜Ck I˜Lk Vk
Ij j∈ Ak Ak
I˜Gk I˜Ck I˜Lk Ij
j∈Ak
∑
– – Ik′–Ik 1+ + + = 0 S˜Gk S˜Ck S˜Lk Sj j∈Ak∑
– –Sk′–Sk 1+ + + = 021
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 there-fore 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 pre-sented 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, cur-rents, and power flows. By convention, injected currents and power flows
are always used for loads, shunt capacitors, and cogenerators, as shown in Figure 2.2.
Bus voltages are typically the phase voltages , , and refer-enced to ground. However, it is possible to have floating sections of the net-work 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 trans-former 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 ground-ing of the transformer connections durground-ing the initial traversal of the net-work 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 and are used. In this case, the third line voltage is redundant since it is always equal to , and the dimension of the equations is reduced by one. Similarly, the current in one of the phases is redundant since , so typically only phase a and phase b currents are used for calculation.
1 A 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.
Va Vb Vc Vab Vbc Vca Vab+Vbc ( ) – Ic = –(Ia+Ib)
When computing power flows as opposed to currents, ground is used as a reference in grounded sections. For example, . However, in ungrounded sections, the power used is that defined by the line-to-line voltages and , and the currents and .
(3.1)
(3.2) It is important to note that, although it is possible to calculate total power flows in grounded sections of the network by the sum , the total power flow in an ungrounded section is not equal to . 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
. (3.3)
3.1 Load Model
The model used for loads is a flexible one. It includes constant com-plex 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 val-ues are given as nominal power delivered to the load and must be con-verted 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.
Sa = VaIa∗ Vab Vbc Ia Ib Sab = VabIa∗ Sbc = VbcIb∗ Sa+Sb+Sc Sab+Sbc Stotal = Sac+Sbc
the following quantities from the bus voltage and the constant model parameters:
• admittance matrix • injected current • injected power
With a grounded wye connected load as shown in Figure 3.1, for each phase p, the parameter 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
. (3.4)
These values are converted to the appropriate constant model parameters , , or , according to the type of load and the nominal voltage , using the equations in Table 3.1. , , and are n x 1
com-Figure 3.1 Grounded Wye Connected Load Vk YLk ILk = –YLkVk SLk = Vk.*ILk∗ SLk nomp , SLk nom, SLk noma , SLk nomb , SLk nomc , = yLkb yLka yLkc Vkc ILka SLkb SLkc SLka ILkb ILkc Vka Vkb ILk SLk yLk Vk nom, ILk SLk yLk
plex vectors of current, power, and admittance, respectively, where n is the number of phases present. Note that and are injected quantities, hence the negative sign in (3.5) and (3.6).
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
. (3.14)
†where .
Table 3.1 Load Parameters from Nominal Loads
Connection Load Type Parameter Calculation
grounded wye constant S (3.5) constant I (3.6) constant Z (3.7) ungrounded delta constant S (3.8) constant I† (3.9) constant Z† (3.10) constant S (3.11) constant I (3.12) constant Z (3.13) ILk SLk Vk nom, Vk noma, Vk nomb, Vk nomc, SLk = –SLk nom, ILk = –(SLk nom, . /Vk nom, ) ∗ yLk = SLk nom* , . / Vk nom, 2 SLk = –SLk nom, U 1 1– 0 0 1 1– 1 – 0 1 = ILk = (SLk nom, . / (UVk nom, )) ∗ yLk = SLk nom* , . / UVk nom, 2 Vk nomab, Vk nombc, Vk nomca, SLk = –SLk nom, ILk = (SLk nom, . /Vk nom, ) ∗ yLk = SLk nom* , . / Vk nom, 2 SLk nom, SLk nomab, SLk nombc, SLk nomca, =
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 phase-to-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 given. For con-stant current or concon-stant PQ loads, equivalent admittances 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.
Figure 3.2 Ungrounded Delta Connected Load
Vkc ILka ILkb ILkc Vka Vkb yLkbc yLkab yLkca SLkbc SLkca SLkab ILkab ILkbc ILkca yLk yLk
3.1.2
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 . Table 3.3 shows how the injected current and power can be computed from the volt-age and the load parameters for each of the different cases. Some of the calculations are based on the admittance matrix from Table 3.2.
†where .
Table 3.2 Load Admittance
Connection Load Type
Element
Admittance Admittance Matrix
grounded wye constant S constant I constant Z ungrounded delta constant S† constant I† constant Z constant S constant I constant Z Vk yLk = Y˜Lk = Vka Vkb Vkc SLk∗. / Vk2 ( ) – y Lk a 0 0 0 yLkb 0 0 0 ycLk ILk. /Vk – yLk U 1 1– 0 0 1 1– 1 – 0 1 = SLk∗. / UVk2 ( ) – y Lk ca yabLk + –yLkab –yLkca y – Lkab yabLk+ yLkbc –yLkbc y – Lkca –yLkbc yLkbc +ycaLk ILk. /(UVk) yLk Vkab Vkbc SLk∗. / Vk2 ( ) – yLkca + yLkab ycaLk yabLk – ybcLk ILk. /Vk yLk Vk
3.2 Shunt Capacitor Model
Shunt capacitors, often used for reactive power compensation in a dis-tribution 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
†where .
Table 3.3 Current & Power Injected by Load
Connection Load Type Injected Current Injected Power
grounded wye constant S constant I constant Z ungrounded delta constant S† constant I† constant Z constant S constant I constant Z Vk I˜Lk = S˜Lk = Vak Vkb Vkc SLk. /Vk ( ) ∗ SLk ILk Vk.*I˜Lk∗ yLk.*Vk – UT[SLk. / (UVk)] ( ) ∗ Vk.*I˜Lk∗ UTILk – Y˜LkVk – Vkab Vkbc SLk ca Vkab+Vkbc ---– SLk ab Vabk ---– SLk ab Vkab --- SLk bc Vkbc ---– ∗ Vk.*I˜Lk∗ ILk ca ILk ab – ILk ab ILk bc – Y˜LkVk – U 1 1– 0 0 1 1– 1 – 0 1 =
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 which is computed from the given nominal reactive power injection
. The nominal voltage is the phase-to-ground voltage in grounded sections and the full 3-dimensional line-to-line voltage for ungrounded sections.3 The admittance is then given by
. (3.15)
From it is possible to compute the necessary admittance matrix and injected current and power as shown in Table 3.4.
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. is or , respectively.
Table 3.4 Shunt Capacitor Admittance, Current & Power Injection
Connection Admittance Matrix Injected Current Injected Power
grounded wye ungrounded delta yCk QCk nom, Vk nom, yCk Vk nom, V k nom, a
Vbk nom, Vk nomc, Vabk nom, Vk nombc, Vk nomca,
yCk = jQCk nom, . / Vk nom, 2 yCk Vk Y˜Ck = I˜Ck = S˜Ck = Vak Vkb Vkc yCka 0 0 0 yCkb 0 0 0 yCkc Y˜CkV k – Vk.*I˜Ck∗ Vkab Vkbc yCkca + yCkab yCkca yCkab – yCkbc
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 formu-las 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 three-phase and delta connected. Typically , 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 Element Admittance Admittance Matrix Injected Current Injected Power grounded wye ungrounded delta SGk Vk yGk = Y˜Gk = I˜Gk = S˜Gk = Vka Vkb Vkc SGk∗. / Vk2 ( ) – yGka 0 0 0 yGkb 0 0 0 yGkc SGk. /Vk ( ) ∗ SGk Vkab Vkbc yGkca +yGkab yGkca yGkab – yGkbc S – Gk ca Vkab+Vkbc --- SGk ab Vkab ---– SGk ab Vkab --- SGk bc Vkbc ---– ∗ Vk.*I˜Gk∗
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-tion line k4 is represented as a series impedance , and the line charging effects are divided between the two shunt arms, each with an admittance of . The impedance and the admittance are both n x n complex matrices, where n is the number of phases in the line. The branch admit-tance matrix for this model is
. (3.16)
4 The distribution line entering bus k.
Figure 3.3 Three-Phase Distribution Line Model
Vkb Vkc Series Impedance Shunt Capacitance Vka–1 Vkb–1 Vkc–1 Vka Shunt Capacitance Bus k-1 Bus k (supplying) (receiving) 1 2 ---Y k Zk 1 2 ---Y k Zk Yk⁄2 Zk Yk YkBR Zk 1 – 1 2 ---Yk + –Zk–1 Zk–1 – Zk–1 1 2 ---Yk + =
Substituting this value for in (2.7) and solving for the appropriate variables yields the equations for voltage, current and power flow updates given in Table 3.6.
For a three-phase line in an ungrounded section of the network, the line charging effects are assumed to be negligible5 so is set to zero. The series impedance is reduced to 2 x 2 for use with the line-to-line volt-ages. The series impedance 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.
Table 3.6 Update Formulas for Distribution Lines
Function of Update Formula (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) YkBR Vk 1– ,Ik V˜k Vk 1– Zk 1 2 ---YkVk 1– –Ik + = I˜k′ = 2---Y1 k(V˜k+Vk 1– ) –Ik Vk,Ik′ V˜k 1– Vk Zk 1 2 ---YkVk–Ik′ + = I˜k = 2---Y1 k(Vk+V˜k 1– ) –Ik′ Vk 1– ,Sk V˜k Vk 1– Zk 1 2 ---YkVk 1– – (Sk. /Vk 1– ) ∗ + = S˜k′ = V˜k.*2---Y1 k(V˜k+Vk 1– ) –(Sk. /Vk 1– ) ∗∗ Vk,Sk′ V˜k 1– Vk Zk 1 2 ---YkVk– (Sk′. /Vk) ∗ + = S˜k V˜k 1– 1 2 ---Yk(Vk+V˜k 1– ) – (Sk′. /Vk) ∗ ∗ .* = Yk Zk Zk
denoted by , relates the phase-to-ground voltages to the phase cur-rents.
(3.25)
The desired series impedance, denoted by , is a 2 x 2 matrix relating the line-to-line voltages to the phase currents as follows:
(3.26)
This impedance is expressed in per unit based on the nominal line-to-line voltage and can be computed from as follows:
, (3.27)
where the factor of is due to the change of per unit voltage base.
3.5 Switch Model
Sectionalizing switches are modeled as branches with zero imped-ance. For a switch between bus 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 and vice versa from the formulas given in Table 3.7, with no need to express the model using an admittance matrix.
Zkphase Ika Ikb Ikc Zkphase Vka Vkb Vkc Vk 1a– Vk 1b– Vk 1c– – = Zkline Iak Ikb Zkline Vk ab Vkbc Vk 1ab– Vk 1bc– – = Zkphase Zkline 1 1– 0 0 1 1– 1 3 ---Zkphase 1 0 0 1 1 – –1 ⋅ ⋅ = 1 3 ---k 1– k 1–
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
. (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 sec-ondary side phase-to-ground voltages. In the case where one side of the
6 The transformer entering bus k.
Table 3.7 Update Formulas for Switches
Function of Update Formula (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) Vk 1– ,Ik V˜k = Vk 1– I˜k′ = –Ik Vk,Ik′ V˜k 1– = Vk I˜k = –Ik′ Vk 1– ,Sk V˜k = Vk 1– S˜k′ = –Sk Vk,Sk′ V˜k 1– = Vk S˜k = –Sk′ YkBR Yk pp Ykps Yksp Ykss =
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 used on both sides and the dimension of is 4 x 4.
In the following sections, the primary side taps for transformer k are denoted by , the secondary side taps by , and the per unit leakage admittance per phase by . The admittance matrices for common trans-former 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. In this case, each submatrix of 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 second-ary side. This includes connection types 2 and 3. For these transformers, the voltage, current, and power flow on the primary side are three-dimen-sional quantities, but on the secondary side they are two-dimenthree-dimen-sional quantities. There is a constraint, however, on the primary side currents
YkBR
αk βk
yk
