SOME ASPECTS OF POWER SYSTEM NETWORK RELIABILITY STUDIES by VELAMURY SANKAR Department of Electrical Engineering THESIS SUBMITTED IN FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY to the INDIAN INSTITUTE OF TECHNOLOGY, DELHI INDIA MAY, 1992 Brbirtitta to fitg parrnts (Katt) Sri tirlanturu littralltt-tangarn anb iCaks4tni Narasanutta CERTIFICATE This is to certify that the thesis entitled 'SOME ASPECTS OF POWER SYSTEM NETWORK RELIABILITY STUDIES" which is being submitted by Mr. VELAMURY SANKAR to the Indian Institute of Technology, Delhi, for the award of the degree of Doctor of Philosophy in Electrical Engineering, is a record of bona fide research work carried out by him. He has worked under our guidance and supervision and has fulfilled the requirements for the submission of this thesis, which has attained the standard required for a Ph.D. degree of this institute. The results in this thesis have not been submitted elsewhere for the award of any degree or diploma. C,(V. C. PRASAD) Professor (K. S. PRAKASA RAO) Professor Department of Electrical Engineering Indian Institute of Technology, Delhi New Delhi- 110 016, INDIA. ACKNOWLEDGEMENTS I have great pleasure in expressing my deep sense of gratitude to my supervisors Prof. K. S. Prakasa Rao and Prof. V. C. Prasad for their guidance, supervision and encouragement during the course of this work. I in grateful to the authorities of Jawaharlal Nehru Technological University, Hyderabad, for sponsoring me under QIP to pursue my research work. I am very much thankful to my colleagues at J. N. T. U. College of Engineering, Anantapur who have been very co-operative during this period. In particular, I am very much grateful to Prof. T. B. Krishna Swamy, Prof. T. B. Parthasarathy, Prof. K. V. Desikachar, Prof. K. Raja Reddy, Prof. S. Kamakshaiah, Dr. N. Sreenivasulu, Dr. P. Vivekananda and Sri K. Prabhakara Rao for their advice and encouragement. My special thanks are due to Sri K. S. R. Anjaneyulu for his help and co-operation. I am very much grateful to Prof. C. S. Indulkar, Head of the Department of Electrical Engineering at I. I. T. Delhi for his encouragement and providing all necessary facilities. Special thanks are due to Sri N. D. R. Sarma for his keen interest in the work and for several helpful suggestions. I wish to thank Dr. V. Bapi Raju, Dr. S. Raghu Kumar and Sri J. Satish for their co-operation at various stages of the work. I am very much thankful to my friends Dr. D. Nageswara Rao and Sri K. V. Subba Rao for their suggestions and encouragement. iii I am very much grateful to Sri K. V. R. Murthy, Dr. V. V. L. Kantha Rao and Sri K. Bhattacharya for their help during the final stages of this work. My co-researchers Dr.K.L.Puttabuddhi, Dr. D.Das, Dr.(Ms.) M.S. Thomas, Sri L. Hari, Sri G.G.Bhise, Sri K. Ramalingam and others provided a congenial research atmosphere. I am very much thankful to Mr. Fateh Singh for his cooperation in the Computing laboratory. I wish to acknowledge Mr. Saraswat and Mr. Arora for preparing the diagrams of this thesis. I am thankful to Mr. Sanjay for printing the thesis neatly. I am very much grateful to Sri J. G. Sastry and his family for their support and encouragement. I wish to appreciate my wife Savitri, for her patience, understanding and encouragement throughout the period of this work. Finally, I appreciate my sons Sravan and Sravanth for their lively company which provided me the necessary relaxations. VOLAe—jok-49 .% V. SANKAR iv ABSTRACT Analysis of systems from reliability point of view is of great interest currently. Several systems are representable by graphs. Of late, various graph theoretic techniques are increasingly being used to evaluate the reliability of a system. Most of these methods require generation of paths/cutsets. In this thesis, some new techniques are developed for generating pathsets and several types of cutsets for single-input singleoutput networks and multi-input multi-output networks with specific reference to power system networks. In Chapter 1, a review of various methods of reliability analysis is presented. In Chapter 2, all basic paths are generated without generating all paths. This is done for single-input single-output networks. Boolean algebra is then used on these basic paths to generate all minimal vertex cutsets. A new type of cutset called "system minimal vertex cutset" is also introduced in this chapter. This is useful for a power system network with probabilities assigned to buses. Normally, methods for generation of minimal edge cutsets, using paths, multiply all paths from input to output using Boolean algebra- However, in Chapter 3 it is shown that a subset of all paths called "independent pathset" is enough from this point of view. When these paths are multiplied in the Boolean sense, all minimal edge cutsets are shown to be contained in the product expansion. A similar dual relationship between independent minimal edge cutsets and all paths is also established. Further, new bounds on the cardinalities of minimal edge cutsets and minimal vertex cutsets are explored in this chapter. Some new bounds on the number of paths and number of minimal edge cutsets are derived. These are sometimes better than the well known bounds of appropriate powers of 2. Some examples are given to illustrate that these new bounds are reached in practice for some graphs. In the existing literature, not much attention was given to the problem of enumeration of system minimal edge cutsets. A similar problem corresponding to the vertices is not pursued at all. In Chapter 4, two new algorithms are presented, one for generation of system minimal edge cutsets and the other for the generation of system minimal vertex cutsets for power system networks. All paths, from all inputs, to all load points are generated simultaneously using what is called "all path tree". By multiplying all paths from the all path tree using Boolean algebra, all system minimal edge cutsets are generated. Some new generalized Boolean simplification rules are developed to simplify the product terms. Similarly, all basic paths to all load points from every input are determined using "all basic path tree". System and load point minimal vertex cutsets are determined by multiplying the basic paths of the basic path tree, using Boolean algebra and the generalized Boolean simplification rules mentioned earlier. Further, bounds on the cardinalities of vi system minimal edge cutsets and system minimal vertex cutsets are also derived in this chapter. The path and basic path concepts of single-input singleoutput networks are generalized to multi-input multi-output networks, with application to power system networks in Chapter 5. These are called "minimal edge system connected subgraph" and "minimal vertex system connected subgraph" respectively. Algorithms are presented for generating these system connected subgraphs. The all path tree, the all basic path tree and Boolean algebra are used in a different way for generating these subgraphs. In Chapter 6, an algorithm is developed for generating all system minimal edge cutsets directly without using paths for power system networks. This algorithm makes use of layering the graph where, each layer contains vertices adjacent to its preceding layer. This helps to generate sets of vertices containing the input without duplication. Capacities of elements are considered in Chapter 7 to evaluate the reliability of a system which has to meet the required flow from the input to the output. In this context, paths and edge cutsets are called weighted pathsets and weighted cutsets respectively. A comprehensive algorithm is developed for generating all minimal weighted pathsets that give the required flow from the input to the output for directed networks. This makes use of only a subset of all paths called "forward paths". Boolean algebra is shown to be useful to determine minimal vi i weighted cutsets from minimal weighted pathsets. These concepts are applied to HVDC converter equipment failure analysis. The a.c. to d.c. converter equipment circuit is treated as a directed network by incorporating capacity constraints. Weighted cutsets of this graph are then obtained using the above methods. Further, the weighted cutset approach is also adopted for power system networks. The undirected edges in the graph of a power system network are replaced by a pair of anti-parallel edges. The minimal weighted cutsets are then generated. A summary of the conclusions of the thesis is presented in Chapter 8. Scope for further research work is also mentioned. viii CONTENTS Page No. ABSTRACT LIST OF FIGURES xv LIST OF TABLES xx CHAPTER 1: INTRODUCTION 1 1.1 INTRODUCTION 1 1.2 LITERATURE REVIEW 2 1.2.1 State enumeration methods 4 1.2.2 Network modification Methods4 a)Series-parallel systems4 b) Star-delta transformations5 c)Hayes' theorem 6 1.2.3 Topological formulae 6 1.2.4 Path generation methods 7 1.2.5 Cutset generation methods8 a) Edge cutset generation methods9 b) Vertex cutset generation methods12 c)Mixed cutsets 13 1.2.6 Reliability expression 13 1.2.7 Multi-output networks 16 1.2.8 Power system networks 18 1.2.9 Networks with flows 20 1.3 NOTATION AND DEFINITIONS 22 1.4 SCOPE OF THE THESIS 25 CHAPTER 2: MINIMAL VERTEX CUTSETS 29 2.1 INTRODUCTION 29 ix 2.2 BASIC PATHS 31 2.2.1 Algorithm 2.1: Generation of basic paths 31 2.2.2 Properties of basic paths35 2.2.3 Justification of Algorithm 2.137 2.2.4 Discussion 2.3 MINIMAL VERTEX CUTSETS 38 45 2.3.1 Algorithm 2.2: Generation of minimal vertex cutsets 47 2.3.2 Justification of Algorithm 2.249 2.3.3 Discussion 2.4 RELIABILITY EXPRESSIONS 53 59 2.4.1 Reliability expression using basic paths 59 2.4.2 Unreliability expression using minimal vertex cutsets 62 2.5 SYSTEM MINIMAL VERTEX CUTSETS 64 2.6 CONCLUSIONS 68 CHAPTER 3: INDEPENDENT PATHS AND EDGE CUTSETS70 3.1 INTRODUCTION 70 3.2 INDEPENDENT PATHS 73 3.2.1 Properties of independent paths73 3.2.2 Method for obtaining independent paths74 3.3 RELATIONSHIP BETWEEN INDEPENDENT PATHS AND MINIMAL EDGE CUTSETS 77 3.3.1 Discussion 83 3.4 RELATIONSHIP BETWEEN INDEPENDENT MINIMAL EDGE CUTSETS AND PATHS 86 3.5 CARDINALITY OF MINIMAL CUTSETS AND PATHS90 3.6 NUMBER OF EDGE CUTSETS AND PATHS96 3.6.1 Number of edge cutsets 96 3.6.2 Number of paths 98 3.7 CONCLUSIONS 103 CHAPTER 4: SYSTEM MINIMAL CUTSETS FROM PATHS104 4.1 INTRODUCTION 104 4.2 ALL PATH TREE 106 4.2.1 Algorithm 4.1: Generation of all path tree 108 4.2.2 Properties of all path tree111 4.2.3 Justification of Algorithm 4.1113 4.3 SYSTEM MINIMAL EDGE CUTSETS 114 4.3.1 Generalized Boolean simplification rules 117 4.3.2 Algorithm 4.2: Generation of system minimal edge cutsets using all path tree 124 4.3.3 Justification Algorithm 4.2128 4.3.4 Generation of load point minimal edge cutsets 4.4 ALL BASIC PATH TREE 131 133 4.4.1 Algorithm 4.3: Generation of all basic path tree xi 136 4.4.2 Properties of all basic path tree139 4.4.3 Justification of Algorithm 4.3141 4.5 GENERATION OF SYSTEM AND LOAD POINT MINIMAL VERTEX CUTSETS 143 4.5.1 Algorithm 4.4: Generation of multioutput minimal vertex cutsets145 4.5.2 Properties of multi-output minimal vertex cutsets 147 4.5.3 Justification of Algorithm 4.4150 4.5.4 Algorithm 4.5: System minimal vertex cutsets 153 4.5.5 Justification of Algorithm 4.5156 4.5.6 Algorithm 4.6: Load point minimal vertex cutsets 158 4.5.7 Justification of Algorithm 4.6160 4.6 CARDINALITY OF SYSTEM MINIMAL CUTSETS161 4.6.1 Cardinality of a system minimal edge cutset 163 4.6.2 Cardinality of a system minimal vertex cutset 4.7 CONCLUSIONS CHAPTER 5: SYSTEM CONNECTED SUBGRAPHS 5.1 INTRODUCTION 166 172 175 175 . 5.2 MINIMAL EDGE SYSTEM CONNECTED SUBGRAPHS177 5.2.1 Algorithm 5.1: Generation of ,minimal edge system connected subgraphs181 xii 5.2.2 Properties of minimal edge system connected subgraphs 185 5.2.3 Justification of Algorithm 5.1188 5.3 MINIMAL VERTEX SYSTEM CONNECTED SUBGRAPHS190 5.3.1 Boolean Simplification Rules194 5.3.2 Algorithm 5.2: Generation of minimal vertex system connected subgraphs196 5.3.3 Properties of minimal vertex system connected subgraphs 200 5.3.4 Justification of Algorithm 5.2206 5.4 CONCLUSIONS CHAPTER 6: EDGE CUTSETS BY LAYERING 6.1 INTRODUCTION 209 211 211 6.2 MINIMAL EDGE CUTSETS BY LAYERING212 6.2.1 Description of the Algorithm of Arun Kumar and Lee 212 6.2.2 Drawbacks of the algorithm of Arun Kumar and Lee 215 6.2.3 Notation,definitionsand modifications 217 6.2.4 Algorithm 6.1: Generation of minimal edge cutsets by layering220 6.2.5 Justification of Algorithm 6.1227 6.3 SYSTEM MINIMAL EDGE CUTSETS BY LAYERING233 6.3.1 Algorithm 6.2: Generation of system minimal edge cutsets by layering237 6.3.2 Justification of Algorithm 6.2241 6.3.3 Discussion 6.4 CONCLUSIONS 249 258 CHAPTER 7: PATHSETS OF NETWORKS WITH FLOWS259 7.1 INTRODUCTION 259 7.2 MINIMAL WEIGHTED PATHSETS 264 7.2.1 Algorithm 7.1: Generation of all paths using Breadth-First-Search technique266 7.2.2 Algorithm 7.2: Generation of minimal weighted pathsets 267 7.2.3 Justification of Algorithm 7.2278 7.2.4 Relationship between minimal weighted cutsets and pathsets 282 7.3 PROPERTIES OF MINIMAL WEIGHTED PATHSETS AND CUTSETS 286 7.4 DISCUSSION 296 7.5 APPLICATIONS 301 7.5.1 HVDC networks 303 7.5.2 Power system networks 304 7.6 CONCLUSIONS CHAPTER 8: CONCLUSIONS 306 309 8.1 INTRODUCTION 309 8.2 CONCLUSIONS 309 8.3 FURTHER SCOPE 314 REFERENCES 317 APPENDIX 336 CURRICULAM VITAE 343 xiv

© Copyright 2022