Reliability Evaluation of Bulk Power Systems using Analytical and Equivalent Approaches

A thesis Submitted to the College of Graduate Studies and Research

in Partial Fulfhent of the Requirements for the Degree of Doctor of Philosophy

in the Department of Electrical Engineering University of Saskatchewan

Saskatoon

Wei Zhang

Fa11 1998

O Copyright Wei Zhang, 1998. AU rights reserved.

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UNLVERSITY OF SASlKATCIB[EWAN College of Graduate Studies and Research

SUMMARY OF DISSERTATION Submitted in partial fulfibent

of the requirements for the

DEGREE OF DOCTOR OF PHILQSOPBY

Wei Zhang

Department of Electrical Engineering University of Saskatchewan

S u m e r 1999

Examining Cornmittee:

Dr. P. Browne

Dr. T.S, Sidhu

Dr. R. Billinton Dr. N. Chowdhwy Dr. P. Pramanick Dr. M. Hosain

W ~ d I W & ~ e a n ' s Designate, Chair CoUege of Graduate Studies and Research Chair of Advisory Conmittee, Department of Electrical Engineering SupeMsor, Department of Electrical Engineering Department of Electrical Engineering Departrnent of Electrical Engineering Department of Civil Engineering

Extemal Examiner:

Dr, R.W. Menzies Head of the Department of Electrical & Computer Engineering University of Manitoba 15 Gillson Street Winnipeg, Manitoba, Canada R3T 5V6

EtELIABILWY EVALUATION OF BULK POWER SYSTEMS USING ANALYTICAL AND EQWALENT APPROACHlES

Electric power systems throughout the world are undergokg considerable changes due to the converging forces of deregdation, technologicd revolution and evolving customer expectations. Quantitative reliability evaluation plays an important role in the develo pment, design and operation o f composite generation and transmission or buk po wer systems. This thesis presents research conducted on the develo pment and examination of concepts, techniques and pertinent factors in the reliability evaluation of composite power systems using direct analytical and equivalent analyticat approaches.

A major dificulty in composite system reliability evduation using analytical methods is the long computation times required to investigate the extremely large number of possible system outage events in an actual composite system. O d y the credible system states are usually investigated. The credible system states are those that rnake significant contributions to the reliability indices and are d l y detennined by considering outages up to a certain level. The computation t h e increases rapidly with increase in system size and the defined outage level. The obtained adequacy indices can ofien be inaccunite due to limited depth of analysis.

Computation times can be significantly reduced when the unchanging portion of a systern can be replaced by a reduced equivalent reliability model. Equivalents can prove very usefl in the evaluation of large systems where sensitivity studies are to be performed on a portion of the systern or when the systern is to be intercomected to a M e r system, which is to be snidied in detail. Reliability equivalent concepts also have many other applications in composite system evaluation.

This thesis illustrates the mathematical foundations, evaluation procedures, pertinent factors, reliability indices and cornputer pro gram design concepts associated with composite system evaluation using the analytical approacb The thesis presents three advanced algorithms which effectively improve the accuracy of the reliability indices without considerably increasing the requixed computation time. The three advanced algorithms are illustrateci using numerical examples and are applied to the evaluation of two reliability test systems. The thesis provides a complete description of adequacy

equivalent concepts and their applications in composite system evaluation. System studies on two reliability test systems and inte'K0~eCted forms of the two reliability test systems using the equivalent techniques are provided. The concepts of reliability costhenefit analysis and the utilkation in this area of the equivalent techniques and the new algorithm for annual adequacy indices are presented in tl& thesis.

Born in Weihai, China B-Eng., Electncal Engineering, Chongqing University, China M.Eng., Electncal Engineering, Chongqing University, China M. Sc., Elecrical Engineering, University of Saskatchewan, Canada

R. Billinton and W. Zhang, "Algorithm for filure eequency and duration assessment of composite power systerns", IEE Proc. -Gener. Tronsm. Distrib., vol. 145, no. 2, March 1998, pp. 1 17-122.

W. Zhang and R. Billinton, "AppIication of an adequacy equivalent method in bulk power system reliability evaluation", IEEE Tram. Power Systems, VOL 13, no. 2, May 1998, pp. 66 1-666.

R. Billinton and W. Zhang, "Enhanced adequacy equivalent for composite power system reliability evaluation", IEE Proc. -Gener. Transm. Distrh, vol. 1 43, no. 5, September 1996, pp. 420-426.

R. Billinton and W. Zhang, 'State extension for adequacy evaiuation of composite power systems", accepted for publication in EZechic Power Systems Research.

R. Billinton and W. Zhang, "Equivalents in adequacy evaluation of power systems", PMAPS Conference, Vancouver, Canada, 1997.

R. Biliinton and W. Zhang, "An adequacy quivalent approach for composite power system reliability evaluation", IEEE FK?XCANEX Proceedings, Canada, 1 995.

R. Billinton, P. Wang and W. Zhang, "Reliability assessment of power systems by a network equivalent approach", IEEE W E S C m Proceedings, Canada, 1997.

W. Zhang, An Adequacy Equiwlent Approach for Reliability Evaluation of Composite Power Systems, MSc. Thesis, University of Saskatchewan, Canada, 1995.

Permission to Use

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Requests for permission to copy or to make other use of material in this thesis in whole or in part should be addressed to:

The Head D epartment of Electrical Engineering University of Saskatchewan Saskatoon, Saskatchewan Canada S7N 5A9

Abstract

Electric power systems throughout the world are undergoing considerable changes due to the converging forces of deregulation, technological revolution and evolving customer expectations. Quantitative reliability evaluation plays an important role in the develo pment, design and operation of composite generation and transmission or bulk power systems. This thesis presents research conducted on the development and examination of concepts, techniques and pertinent factors in the reliability evaluation of composite power systems using direct analytical and equivalent analytical approaches.

A major difficulty in composite system reliability evaiuation using analytical methods is the long computation times required to investigate the extrernely large number of possible system outage events in an actual composite system. Only the credible system states are usually investigated. The credible systero states are those that make signiscant contriibutions to the reliabsty indices and are usually determined by considering outages up to a certain level. The computation time increases rapidly with increase i systm size and the defmed outage level- The obtained adequacy indices can &en be inaccurate due to limited depth of analysis.

Computation times c m be significantly reduced when the unchanging portion of a system can be replaced by a reduced equivalent reliabiiity model. Equivalents can prove very useful in the evaluation of large systems where sensitivity studies are to be performed on a portion of the system or when the system is to be intercomected to a M e r system, which is to be studied in detail. Reliability equivalent concepts &O have many O ther applications in composite system evaluation.

This thesis illustrates the mathematical foundations, evduation procedures, pertinent fctors, reiiability indices and compter program design concepts associated with composite system evaluation using the analyticd approach. The thesis presents three advanced dgo~thms which effectively hprove the accuracy of the reliability indices without considerably increasing the required computation tirne. The three advanced algorithms are illustrated using numerical examples and are applied to the evaluation of two reliability test systems. The thesis provides a complete description of adequacy equivalent concepts and their applications in composite systern evaluation. System studies on two reliability test systems and intercomected forms of the two reliability test systems using the equivalent techniques are provided. The concepts of reliability cotlbenefit analysis and the utilization in this area of the equivdent techniques and the new algorithm for annual adequacy indices are presented in this thesis.

Acknowledgments

The author would like to express sincere thanks and appreciation to his supervisor Prof Roy Bi1l.int on for his guidance, discussion, criticism and encouragement throughout the course of this work. His assistance in the preparation of this thesis is phankflly achowledged. It has been a wondef i opportunity and expenence to work under hi supervision.

Financial assistance provided by Prof Roy Billinton in the form o f research support eom the Natural Sciences and Engineering Research Council (NSERC) of Canada is gratefully acknowledged.

The author takes the opportunity to thank his d e , Yaling, bis daughters, Haijing and Haili, his relatives and his fiends for the^ moral support, encouragement and love.

Table of Contents

Page

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permission to use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Acbowledgments

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures

. . . . . . . . . . . . . . . . . . . List of Symbols and Abbreviations

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !.1 Introduction

. . . . . . . . . . . . . . . . . . . . . : -2 Basics of Reliability S tudies . . . . . . . . . . . . . . . . . . . .. 3 Basics of Adequacy Evaluation

. . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Deregulation Impacts

. . . . . . . . . . . . . . . . . . . . . . . . . .. 5 Research Objectives

. . . . . . . . . . . . . . . . . . . . . . . . . .. 6 Outline of the Thesis

2 Basic Concepts and Evaluation Techniques . . . . . . . Introduction . . . . . . . . .

. . . . StateSpace Technique Basic Procedure and Concepts Network Solution Techniques

. . . . . . Rernedial Actions Load Curtailment Policies . .

. . . . . . . . . Load Modeis . . . . Basic Adequacy Indices

Cornputer Programmhg . . . S- . . S . . . . . . . .

3 Advanced Algorithm . . . . . . . . . . . . . . . . . . . . . . . 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 55

3.2 Algorithm for FF & FD Evduation . . . . . . . . . . . . . . . . . . 56 3.3 Numenc Example and Non-coherence Effect . . . . . . . . . . . . 62 3.4 State Extension Technique . . . . . . . . . . . . . . . . . . . . . . 67

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Numeric Example 76 3.6 Algorithm for Annual Adequacy Indices . . . . . . . . . . . . . . . 79 3.7 NumericalExample . . . . . . . . . . . . . . . . . . . . . . . . . . 85

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary 88

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 90 4.2 Two Test Systems: the RBTS and the IEEE-RTS . . . . . . . . . . 90 4.3 Improved FF & FD Evaluation . . . . . . . . . . . . . . . . . . . . 94

. . . . . . . . . . . . . . . . . . . . . . 4.4 State Extension Technique 100 4.5 Annual Adequacy Indices . . . . . . . . . . . . . . . . . . . . . . . 114 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5 Concepts and Methods of Adequacy Equivalents . . . 127 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

. . . . . . . . . . . . . . . . . . . . . . . . . . . . BasicConcepts 128 . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Models 136 . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Roundhg 143

. . . . . . . . . . . . . . . . . . Network Solutions for Equivalents 146 . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example 148

. . . . . . . . . . . . . . . . . . . . . . . Utilization of Equivalents 153 . . . . . . . . . . . . . . . . . . Common-cause Failure Equivalents 156

Station Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

. . . . . . . . . . 6 Applications of Equivalent Techniques 166 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.2 Effect o f Network Solution Techniques . . . . . . . . . . . . . . . 167 6.3 Effect of Equivdent Rounding Increments . . . . . . . . . . . . . . 179

. . . . . . . . . . . . . . . . . . . . . 6.4 Effect of High Level Outages 190 6.5 Calculation of Annual Adequacy Indices . . . . . . . . . . . . . . . 197

6.6 Effect of the Station Originated Events . . . . . . . . . . . . . . . . 203 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.2 Basic Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7.3 Customer Damage Function . . . . . . . . . . . . . . . . . . . . . 221 7.4 Expected Customer Damage Costs (ECOST) . . . . . . . . . . . . 231 7.5 Annualized and Annual ECOST for the RBTS and the EEE-RTS . 233 7.6 ECOST for the Two-area RBTS and the IEEE Two-area RTS

Using the Equivalent Approach . . . . . . . . . . . . . . . . . . . . 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Summary 248

. . . . . . . . . . . . . . . . . . . 8 Summary and Conclusions 250

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 259

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices 266 Appendix A Data of the RBTS . . . . . . . . . . . . . . . . . . . . . . 266 Appendix B Data of the IEEE-RTS . . . . . . . . . . . . . . . . . . . . 268 Appendix C Collection of Reliability Data . . . . . . . . . . . . . . . . . 273

List of Tables

Table 2- 1 . Table 3 . l . 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 3.1 0 . Table 3.11- Table 3.1 2- Table 3.13. Table 4.1. Table 4.2. Table 5.1. Table 5.2 . Table 5.3. Table 5.4. Table 5.5. Table 5.6. Table 5.7. Table 5.8. Table 5.9. Table 5.1 0 . Table 5.1 1 . Table 5- 12 . Table 5.13. Table 5.14- Table 5.15. Table 5.16, Table 5- 17 . Table 5.18.

Seven step load data for the IEEE-RTS . . . . . . . . . . . . . . 47 . . . . . . . . . Basicstateindicesforthetwo-componentsystem 58 . . . . . . . . . Failure fiquencies using the dif5erent equations 63

Essentialparametersforthethree-componentsystem . . . . . . . 66 . . . . . . . . . Basic parameters for the three-component system 70

Basic parameters for the three-cornponent system, when enumerating . . . . . . . . . . . . . . . . . . . . . . system states up to level 1 72

Boundary level subtree parameters for the five-component system . 73 S2+ and Sg+ parameters. assuming S2 and S3 are the investigated

. . . . . . . . system Mure states for the three-component systern 75 FEbndFFwithoutapplyingthestateextensiontechnique . . . . . 77 FP and FF when applying the state extension algorithm . . . . . 78 Effect of the state extension technique . . . . . . . . . . . . . . 79

. . . . . . . . . . . . . . . . . . . Annualized adequacy indices 86 . . . . . . . . . . . . . . . . . . . . . h n u a l adequacy indices 87

Cornparison of the adequacy indices . . . . . . . . . . . . . . . 87 General uiformation of the four given studied cases . . . . . . . . 107

. . . . . . . . . . . . . . . . . . . . . . A seven-step load mode1 116 Essential state parameters for the two-component system . . . . . 135

. . . . . . . . . . . . . . . . . . . . . . A three-state equivalmt 135 . . . . . . . . . . . . . . . . . . . Essentid parameters for the IA 139

. . . . . . . . . . . . . . . . . . . . . . . . . Mode1 1 equivalent 139 . . . . . . . . . . . . . . . . . Including outage level information 140

. . . . . . . . . . . . . . . . . . . . . . . . . Mode1 II equivalent 141

. . . . . . . . . . . . . . . . . . . . . . . . . Mode1 III equivdent 142 . . . . . . . . . . . . . . . . . . . . . . . . Mode1 N quivalent 143

. . . . . . . . . . . . . . . . Equivalent mode1 1 before rounding 144 . . . . . . . . . . . . . . . . . . . . . . . . . . Rounding process 144

. . . . . . . . . . . . . . . . . . Equivalent mode1 1 afier roundmg 145 . . . . . . . . . . . . The RBTS quivalent using d.c. load flow 149

. . . . . . . . The reduced RBTS equivdent using d.c. load flow 150 . . . . . . . . . The reduced RBTS quivalent using nekvork flow 151

The rounded RBTS equivalent using d.c . load flo w . . . . . . . 152 . . . . . . . . The rounded RBTS equivalent ushg network low 152

. . . . . . . . . . . . . Adequacy d y s i s of the system in Fig 5.13 156 . . . . . . . . . Essential parameters for the mode1 in Fig . 5.16(a) 158

Table 5.19. Table 5.20. Table 5.2 1 . Table 6.1.

Table 6.2. Table 6.3, Table 6.4. Table 6.5. Table 6.6- Table 6.7. Table 6.8. Table 6.9. Table 6.10. Table 7.1. Table 7.2. Table 7.3. Table 7.4. Table 7.5. Table 7.6. Table 7.7. Table 7.8. Table 7.9. Table 7.1 0 . Table 7.1 1 . Table 7.12. Table A- l . Table A-2 . Table A-3 . Table A 4 . Table B- 1 . Table B-2 . Table 8-3 . Table B-4 . Table B-5 . Table B-6 .

EquivalentforthemodelinFig.5.16(a) . . . . . . . . . . . . . . 158 . . . . . . . . . . Essential parameters for the mode1 in Fig 5.16(b) 159

Equivalent for the mode1 in Fig . 5.16(b) . . . . . . . . . . . . . . 159 Data on the three tie lines in the IEEE two-area RTS X and B/2 are in p.u . with a base power equal to 100 MVA) . . . . . . . . . 173 Cornparison of the required cpu times . . . . . . . . . . . . . . . 194 Comparison of the required CPU times for the two-area RBTS . . 20 1 Comparison of the required CPU times for the IEEE two-area RTS 202 Equivalent generator reliability data for the RBTS . . . . . . . . . 205

. . . . . EquivalenttransmissionIinereliabiltydatafortheEU3TS 205 . . . . . . . . Equivalent load feeder reliability data for the RBTS 205 . . . . . . . Equivalent generator reliability data for the IEEE.RTS 210

. . . Equivalent transmission line reliability data for the IEEE-RTS 2 1 1 . . . . . . Equivalent load feeder reliability data for the EEE-RTS 211

Sector CDF expressed in ($/kW) . . . . . . . . . . . . . . . . . . 222 Assumed load compositions . . . . . . . . . . . . . . . . . . . . . 223

. . . . . . . . . . . . . . . . CCDF in (%/kW) for the test systerns 223 Load factors of the seven sectors . . . . . . . . . . . . . . . . . . 225

. . . . . . . . . . . . . Sector peak load docation for the RBTS 226 . . . . . . . . . . . . Sector peak load percentages for the RBTS 226

. . . . . . . Sector energy consumption percktages for the RBTS 226 . . . . . . . . . . . . CCDF for the RBTS load buses in ($/kW) 227 . . . . . . . . . . . Sector peak load allocation for the IEEE-RTS 228

Sector peak load percentages for the IEEE-RTS . . . . . . . . . . 228 . . . . Sector energy consumption percentages for the IEEE-RTS 229

. . . . . . . . . . . CCDF for the IEEE-RTS load buses in ($/kW) 229 Bus data ( in p.u.) for the RBTS . . . . : . . . . . . . . . . . . . 266 Lnedata(inp.u.) fortheRBTS . . . . . . . . . . . . . . . . . . 266 Generator data for the RBTS . . . . . . . . . . . . . . . . . . . . 267

. . . . . . . . . . . Tenninal station equipment data for the RBTS 267 . . . . . . . . . . . . . . . Bus data ( in p.u.) for the IEEE-RTS 268

. . . . . . . . . . . . . . . . Line data ( in pu.) for the IEEE-RTS 269 Generator data for the EEE-RTS . . . . . . . . . . . . . . . . . 270

. . . . . . . . . . . . Weekly peak load in percent of annual peak 271 . . . . . . . . . . . . . D d y peak load in percent of weekly peak 271

. . . . . . . . . . . . . . Hourly peak load in percent of daily peak 272

List of Figures

Fig . 1.1. Fig . 1.2. Fig . 1.3. Fig . 1.4. Fig . 1.5. Fig . 1.6. Fig . 1.7. Fig . 2.1. Fig . 2.2, Fig . 2.3. Fig . 2.4. Fig . 2.5. Fig- 2.6. Fig . 2.7. Fig . 2.8. Fig . 2.9. Fig . 2.10. Fig . 2.1 1 .

Fig . 2.12. Fig . 3.1- Fig . 3.2. Fig . 3.3. Fig . 3.4. Fig . 3.5. Fig . 3.6. Fig . 3.7. Fig . 3.8. Fig . 4.1. Fig . 4.2. Fig . 4.3. Fig . 4.4. Fig . 4.5. Fig . 4.6. Fig . 4.7. Fig . 4.8.

Subdivision of power system reliability . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . Basic fiinction zones and hierarchical levels 5

Two fimdamentdy diffrent approaches . . . . . . . . . . . . . . . 5 Generation evaluation mode1 . . . . . . . . . . . . . . . . . . . . . 7 Basic generation modeling approach . . . . . . . . . . . . . . . . . 7 Investment. damage and to ta1 cost as a function of system reliability 1 2 Unbundled structure . . . . . . . . . . . . . . . . . . . . . . . . . . 14 A single component two-state state space diagram . . . . . . . . . 21

. . . . . . . . . A single component three-date state space diagram 21 A two-component system state space diagram . . . . . . . . . . . 22 State space diagrams including common cause filures . . . . . . . 23 Basic amlytical procedure . . . . . . . . . . . . . . . . . . . . . . . 27 State enurneration of the three-component system . . . . . . . . . . 29 State space diagram of a po wer system with n independent components 3 0 Equivalent conversion . . . . . . . . . . . . . . . . . . . . . . . . 35 The EEE-RTS hourly load duration curve . . . . . . . . . . . . . 46 A seven step load model based on the IEEE-RTS load duration curve 47 A tweive-intemal Ioad model based on the IEEE-RTS chronological hourlyloadvariation . . . . . . . . . . . . . . . . . . . . . . . . . 48 Computa programming strategy . . . . . . . . . . . . . . . . . . . 52 State space diagram for the two-component systern . . . . . . . . . 58 Simple netwurk configuration . . . . . . . . . . . . . . . . . . . . 62 S tate enumeration for the three-component system (non-CO herence) . 65 State space of the three-component system . . . . . . . . . . . . . 69 System States of the five-component system . . . . . . . . . . . . . 73 A radial configuration . . . . . . . . . . . . . . . . . . . . . . . . . 84 Simple network co&guration . . . . . . . . . . . . . . . . . . . . 85 Assumed hourly load duration curve . . . . . . . . . . . . . . . . . 86 Single h e diagram of the RBTS . . . . . . . . . . . . . . . . . . . 92 Single h e diagram of the IEEE-RTS . . . . . . . . . . . . . . . . 93 Failure fiequencies at the RBTS load buses at the 185 MW load level 95 Failure fiequencies at the RBTS load buses at the 148 MW load level 96 Failure fiequencies at bus 2 for different load levels in the RBTS . . 97 Failure fkequencies at bus 6 for ciiffirent load levels in the RBTS . . 97 Failure fkequencies at the IEEE-RTS load buses . . . . . . . . . . . 98 Failure fkquencies at bus 2 for different load levels in the IEEE-RTS 99

Fig. 4.9. Fig. 4.10. Fig. 4.1 1. Fig 4.12.

Fig. 4- 13.

Fig. 4- 14. FIg. 4.15. Fig. 4.16.

Fig. 4.17.

Fig. 4.18. Fig. 4.1 9. Fig. 4.20.

Fig. 4.2 1.

Fig. 4.22. Fig. 4.23. Fig. 4.24.

Fig. 4.25.

Fig. 4.26. Fig. 4.27. Fig. 4.28. Fig. 4.29. Fig. 4.30.

Fig. 4.3 1.

Fig. 4.32. Fig. 4.33. Fig. 4.34. Fig. 4.35.

Fig. 4.36.

Fig. 4.37.

Fig. 4.38.

Failure fiequencies at bus 15 for different load levels in the IEEE-RTS Failme probabilities at the RBTS load buses (Case 1) . . . . . . . . Failure fiequencies at the RBTS load buses (Case 1) . . . . . . . . . Failure probabilities for the overall RBTS at various load levels (Casel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fdure fiequencies for the overall RBTS at various load Ievels (Casel) . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . .

Failure probabiIities at the RBTS load buses (Case 2) . . . . . . . . Fdure fiequencies at the RBTS load buses (Case 2) . . . . . . . . . Failure probabilities for the overall RBTS at various load Levels (Case2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Failure fkquencies for the overall RBTS at various load Levels (Case 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failwe probabilities at the IEEE-RTS load buses (Casel) . . . . . . Fdure eequencies at the IEEE-RTS toad buses (Casel). . . . . . . Faure probabilities for the overall IEEE-RTS at various load levels (Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Failure fiequencies for the overall IEEE-RTS at various load levels (Casel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Failure probabilities at the IEEE-RTS load buses (Case 2) . . . . . . Fadure f?equencies at the IEEE-RTS load buses (Casez) . . . . . . Failme probabilities for the overall IEEE-RTS at various load Ievels (Case 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure fkequencies for the overall IEEE-RTS at various load levels (Case 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failme probabilities for the overd IEEE-RTS for the four cases . . Failure nequemies for the overall IEEE-RTS for the four cases . - . Failure probabilities at the RBTS load buses using the three methods Failure fiequencies at the RBTS load buses uskg the three methods . Expected load curtailed at the RBTS load buses using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expected energy not supplied at the RBTS load buses using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure probabilities for the overall RBTS ushg the three methods. . Failure fiequemies for the overd RBTS using the three methods . . Expected load curtailed for the overall RBTS using the three methods Expected energy not supplied for the overall RBTS using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure probabdities at the IEEE-RTS load buses using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure fiequencies at the IEEE-RTS load buses using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expected load curtded at the IEEE-RTS load buses using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig . 4.40. Fig . 4.4 1 . Fig . 4.42.

Fig . 4.43.

Fig . 5.1. Fig . 5.2. Fig . 5.3. Fig . 5.4. Fig . 5.5. Fig . 5.6. Fig . 5.7. Fig . 5.8. Fig . 5.9. Fig . 5.10. Fig . 5.1 1 . Fig . 5.12. Fig . 5.13. Fig . 5.14 . Fig . 5.15. Fig . 5.16. Fig . 5.17. Fig . 5.18 . Fig . 5.19. Fig . 5.20. Fig . 6.1. Fig . 6.2.

Fig . 6.3.

Fig . 6.4.

Fig . 6.5.

Fig . 6.6.

Fig . 6.7.

Fig . 6.8.

Fig . 6.9.

Expected energy not supplied at IEEE-RTS the load buses using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure probabilities for the overall IEEE-RTS using the three methods Failure fiequencies for the overd IEEE-RTS using the three methods Expected load curtded for the overall IEEE-RTS using the three methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expected energy not supplied for the overall IEEE-RTS using the three rnethods . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . Simple series system . . . . . . . . . . . . . . . . . . . . . . . . Simple pardel system

. . . . . . . . . . . . . . . . . . . Single component %th no spare . . . . . . . . . . . . . . . . . . . . . . Equivalent two-state model

. . . . . . . . . . . . . . . . . . . Single component with one spare . . . . . . . . . State space diagram for the two-component system

. . . . . . . . . . . . . . . A power network divided into two areas . . . . . . . . . A network represented by two interconnected areas

. . . . . . . . . . . A network represented by two-IA and one AI . . . . . . . . . . . . . . . . . . . . . A simple equivalent exarnple

The RBTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RBTS equivalent state probabilities . . . . . . . . . . . . . . . . . . RBTS equivalent state fiequencies

. . . . . . . . . . . . . . . . Topo 10 gical equivalent represent ation Topological equivalent representation for the example system . . . .

. . . . . . . . . . . . . . . . . . . Two common-cause failure models . . . . . . . . . . . . . . . . . . . . . niree-state component mode1

EqUvalent of a single bus configuration statio a . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent of a ring configuration station

Equivalent of a 1 112 breaker configuration station . . . . . . . . . . . . . . . . . . . . . . . . . Symbolic interconnected two-area RBTS

Failure probabilities for the AI load buses in the two-area RBTS . . . . . . . . . . . . . . . . . . . . . using the two IA equivalents

Failure fiequencies for the AI load buses in the two-area RBTS . . . . . . . . . . . . . . . . . . . . . using the two IA equivalents

Expected load curtailed for the AI load buses in the two-area RBTS . . . . . . . . . . . . . . . . . . . . . using the two IA equivalents

Expected energy not supplied for the AI load buses in the two-area . . . . . . . . . . . . . . . . . RBTS using the two IA equivalents

Failure probabilities for the overall AI in the two-area RBTS using . . . . . . . . . . . . . . . . . . . . . . . . the two IA equivalents

Failure fiequencies for the overd AI in the two-area RBTS using . . . . . . . . . . . . . . . . . . . . . . . . the two LA equivalents

Expected load curtaiied for the overall AI in the two-area RBTS . . . . . . . . . . . . . . . . . . . . . using the two IA equivalents

Expected energy not supplied for the overall AI in the two-area . . . . . . . . . . . . . . . . . RBTS using the two IA equivalents

xii

Fig. 6.10. Symbolic IEEE two-area RTS . . . . . . . . . . . . . . . . . . . . 173 Fig. 6.1 1. Failure probabilities for the AI load buses in the IEEE two-area

RTS using the two IA equivalents . . . . . . . . . . . . . . . . . 174 Fig. 6.12. Faure fiequemies for the AI load buses in the for the IEEE

two-area RTS using the two IA equivalents . . . . . . . . . . . . . 175 Fig. 6.13. Expected load curtailed for the AI load buses in the IEEE

two-area RTS using the two IA equivalents . . . . . . . . . . . . . 176 Fig. 6.14. Expected energy not supplied for the AI load buses in the IEEE

two-qea RTS using the two IA equivalents . . . . . . . . . . . . . 1 76 Fig. 6.15. Failwe probabilities for the overall AI in the IEEE two-area

RTS using the two IA equivalents . . . . . . . . . . . . . . . . . . 1 77 Fig. 6.16. Failure fkequencies for the overall AI in the IEEE two-area

RTSusingthetwoIAequivalents . . . . . . . . . . . . . . . . . 177 Fig. 6.1 7. Expected load curtailed for the overall AI in the IEEE two-area

RTS using the two IA equivdents . . . . . . . . . . . . . . . . . . 178 Fig. 6.18. Expected energy not supplied for the overall AI in the IEEE

two-area RTS using the two LA equivalents . . . . . . . . . . . . . 1 78 Fig. 6.19. Failure probabilities for the AI load buses in the two-area RBTS

using difEerent rounding increments for the IA quivalent . . . . . . 180 Fig. 6.20. Failure fiequencies for the AI load buses in the two-area RBTS

using dBerent rounding increments for the lA equivalent . . . . . . 1 8 1 Fig. 6.2 1. Expected load curtailed for the AI load buses in the two-area RBTS

using dBerent rounding increments for the IA equivalent . . . . . . 182 Fig. 6.22. Expected energy not supplied for the AI load buses in the two-area

RBTS using dEerent rounding increments for the IA equivalent . . . 182 Fig. 6.23. Failure probabilities for the overall AI in the two-area RBTS

using diCerent rounding increments for the IA equivalent . . . . . . 183 Fig. 6.24. Failme fiequencies for the overd AI in the two-area RBTS

using dXerent rounding increments for the IA equivalent . . . . . . 183 Fig. 6.25. Expected load curtailed for the overd AI in the two-area RBTS

using dinerent rounding increments for the equivalent LA . . . . . . 184 Fig. 6.26. Expected energy not supplied for the overall AI in the two-area

B T S usjng different rounding increments for the IA equivalent. . . 184 Fig. 6.27. Failure probabilities for the AI load buses in the IEEE two-area

RTS using diffrent rounding increments for the IA equivalent . . . 185 Fig. 6.28. Failure fiequencies for the AI load buses in the IEEE two-area

RTS using diffrent rounding increments for the IA equivalent. . . . 186 Fig. 6.29. Expected load curtailed for the AI load buses in the IEEE two-area

RTS using dserent rounding increments for the IA equivalent. . . . 187 Fig. 6.30. Expected energy not supplied for the AI load buses in the IEEE two

-area RTS using different rounding mcrenients for the IA equivalent 187 Fig. 6.3 1. Fdure probabilities for the overall AI in the IEEE two-area RTS

using different rounding iacrements for the LA quivalent . . . . . . 188 Fig. 6.32. Failure fkquencies for the overall AI in the IEEE two-area RTS

using different rounding increments for the IA quivalent . . . . . . 188

Fig. 6.33.

Fig. 6.34.

Fig. 6.35.

Fig. 6.36.

Fig. 6.37.

Fig. 6.38.

Fig. 6.39.

Fig. 6.40.

Fig. 6.41.

Fig. 6.42.

Fig. 6.43.

Fig. 6.44.

Fig. 6.45.

Fig. 6.46.

Fig. 6.47.

Fig. 6.48.

Fig. 6.49.

Fig. 6.50.

Fig. 6.51.

Fig. 6.52.

Fig. 6.53.

Fig. 6.54.

Expected load curtailed for the overall AI in the IEEE two-area RTS using different rounding increments for the IA quivalent. . . . Expected energy not supplied for the overall AI in the IEEE two-area RTS using different rounding incrments for the IA quivalent. . . . Failure probabilities for the overd AI in the IEEE two-area RTS

. . . . . . . . . . . . . . . . . . . . . . . . . at diffrent load levek Failure fkquencies for the overall AI in the IEEE two-area RTS

. . . . . . . . . . . . . . . . . . . . . . . . . at diffrent load levek Expected load curtailed for the AI load buses in the IEEE

. . . . . . . . . . . . . two-area RTS at the system peak load level Expected load curtailed for the AI load buses in the IEEE

. . . . . . . . . . . . two-area RTS at the system peak load level Effects of high level generator outages on the expected load

. . . . . . . . . . . . . . . . . . . . . . . . . . . . curtailed index Effects of hi& level generator outages on the expected energy not supplied index . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of hi& level transmission h e outages on the failure probability index . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of high level transmission line outages on the failure fiequency index . . . . . . . . . . . . . . . . . . . . . . . . . . . Annual f'ailure probabilities for the AI load buses in the two-area

. . . . . . . . . . . . . . . . . . . . . RBTS using the two methods Annual failure fiequencies for the AI load buses in the two-area RBTS using the two methods . . . . . . . . . . . . . . . . . . . . . Annual expected load curtailed for the AI load buses in the two-area RBTS using the two rnethods . . . . . . . . . . . . . . . . Annual expected energy not supplied for the AI load buses in the two-area RBTS using the two methods . . . . . . . . . . . . Annual expected load curtailed for the AI load buses in the IEEE two-area RTS using the two methods . . . . . . . . . . . . . Annual expected energy not supplied for the AI load buses in the IEEE two-area RTS using the two methods . . . . . . . . . . . Failure probabilities for the RBTS load buses at the peak load level in the three cases. . . . . . . . . . . . . . . . . . . . . . . . . Failure fiequencies for the RBTS load buses at the peak load level in the three cases. . . . . . . . . . . . . . . . . . . . . . . . . Expected load curtailed for the RBTS load buses at the peak

. . . . . . . . . . . . . . . . . . . . . . load level in the three cases Expected energy not upplied for the RBTS load buses at the

. . . . . . . . . . . . . . . . . . . peak Ioad level in the three cases Failure probabities for the overd RBTS at difEerent load levels in the three cases . . . . . . . . . . . . . . . . . . . . . . . . . . . Faure fiequemies for the overall RBTS at diffrent load levels in the three cases . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

Fig . 6.55.

Fig . 6.56.

Fig . 6.57.

Fig . 6.58.

Fig . 6.59.

Fig . 6.60.

Fig . 6.62.

Fig . 6.62.

Fig . 6.64.

Fig . 6.65.

Fig. 6.66.

Fig . 7.1. Fig . 7.2. Fig . 7.3. Fig . 7.4. Fig . 7.5.

Fig . 7.6.

Fig . 7.7.

Fig . 7.8.

Fig . 7.9.

Fig . 7.10.

Fig . 7.1 1 .

Fig . 7.12.

Expected load curtailed for the overall RBTS at different load . . . . . . . . . . . . . . . . . . . . . . . . levels in the three cases 209

Expected energy not supplied for the overall RBTS at different . . . . . . . . . . . . . . . . . . . . . load levels in the tbree cases 209

Failure probabilities for the IEEE-RTS load buses at the peak load . . . . . . . . . . . . . . . . . . . . . . . . . level m the three cases 212

Failure fiequencies for the IEEE-RTS load buses at the peak load . . . . . . . . . . . . . . . . . . . . . . . . . level in the three cases 212

Expected load curtailed for the IEEE-RTS load buses at the peak . . . . . . . . . . . . . . . . . . . . . . load level in the three cases 213

Expected energy not supplied for the IEEE-RTS load buses at . . . . . . . . . . . . . . . . . the peak load level in the three cases 213

Failure probabilities for the IEEE-RTS toad buses at the load . . . . . . . . . . . . . . . . . level of 2565 MW in the three cases 214

Expected energy not supplied for the IEEE-RTS load buses at the . . . . . . . . . . . . . . . load Ievel of 2565 MW in the three cases 214

Failure probabilities for the overall IEEE-RTS at different load . . . . . . . . . . . . . . . . . . . . . . . . levels in the three cases 215

Failure frequencies for the overall IEEE-RTS at different load . . . . . . . . . . . . . . . . . . . . . . . . levels in the three cases 215

Expected load curtailed for the overd IEEE-RTS at different . . . . . . . . . . . . . . . . . . . . . load levels in the three cases 216

Expected energy not supplied for the overall IEEE-RTS at . . . . . . . . . . . . . . . . . different load levels in the three cases 216

. . . . . . . . . . . . Sector customer damage functions in (%/kW) 222 . . . . . . . . . The composite custorner damage function in ($/kW) 224

. . . . . . . . . . . . . . CCDF for the RBTS load buses in ($/kW) 227 . . . . . . . . . . . CCDF for the EEE-RTS load buses in ($/kW) 230

Expected customer damage costs for the RBTS load buses at . . . . . . . . . . . . . . . . . . . . . . . . . . variable load levels 234

Expected customer darnage costs for the overall RBTS at various . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . load levels 235

Intempted energy assessment rate for the RBTS load buses at . . . . . . . . . . . . . . . . . . . . . . . . . . variable load levels 235

Intempted energy assessment rate for the overall RBTS at various . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . load levels 236

Distribution of the annual expected custorner damage cost in the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RBTS 236

Annual intempted energy assessment rate for the overall RBTS . . . . . . . . . . . . . . . . . . . . . . . . . andattheloadbuses 237

Expected customer damage costs for the IEEE-RTS load buses . . . . . . . . . . . . . . . . . . . . . . . . . at variable load levels 238

Expected customer damage costs for the overall EEE-RTS . . . . . . . . . . . . . . . . . . . . . . . . . at various load Ievels 238

Fig. 7.13. lntemrpted energy assessment rate for the IEEE-RTS load buses . . . . . . . . . . . . . . . . . . . . . . . . . at variable load levels

Fig. 7.1 4. Interrupted energy assessment rate for the overall IEEE-RTS at . . . . . . . . . . . . . . . . . . . . . . . . . . . various load levels

Fig. 7.1 5. Disfibution of the annual expected customer damage cost in the IEEE-RTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 7.1 6. Annual i n t m p t ed energy assessrnent rate for the overd IEEE-RTS . . . . . . . . . . . . . . . . . . . . . . . . . and at the load buses

Fig. 7.17. Disfibution of the m u a l expected customer damage cost in the . . . . . . . . . . . . . . . . . . . . IEEE-RTS considering 5Gt3L

Fig. 7.18. h u a i intemipted energy assessment rate for the overd IEEE-RTS . . . . . . . . . . . . . . and at the load buses considering 5Gf-31.

Fig. 7.19. Distnbution of the annualized expected customer damage costs in the . . . . . . . . . . . . . . . . . . . . . . AI for the two-area RBTS

Fig. 7.20. Distribution of the annual expected customer damage costs in the AI . . . . . . . . . . . . . . . . . . . . . . . . for the two-area RBTS

Fig. 7.2 1. Aonualized intempted energy assessment rates for the overall AI . . . . . . . . . . . and at the AI load buses for the two-area RBTS

Fig. 7.22. Annualized interrupted energy assessment rates for the overall AI . . . . . . . . . . . and at the AI load buses for the two-area RBTS

Fig. 7-23. Distnbution of the a~ualized expected customer damage costs in . . . . . . . . . . . . . . . . . . the AI for the IEEE two-area RTS

Fig. 7.24. Distribution of the annual expected customer damage costs in the AI . . . . . . . . . . . . . . . . . . . . . . for the IEEE two-area RTS

Fig. 7.25. Annualized intempted energy assessment rates for the overall AI and at the AI load buses for the IEEE two-area RTS . . . . . . . . .

Fig. 7.26. Annual interrupted energy assessment rates for the overd AI and at the AI load buses for the IEEE two-area RTS . . . . . . . . . . .

xvi

Psi

%i

dsi

a. c.

AI CCDF CDF

List of S p b o l s and Abbreviations

Markov state transition rate

Average M u r e rate

Average repair rate

Average installation rate

Average repair tirne

Set of in-service components Set of out-of-service components Set of system normal states Set of systern failure states Link fiom bus i to bus j Flow on link (i, j) Flow f?on source to si&

Availability of component i

Unavailability of component i

Probability of system state si

Frequency of system state si

Duration of system state si

Altemating Current Area of Interest

Composite Customer Damage Function Customer Damage Function

xvii

CEA COMREL d-c.

ECOST EENS FF&FD ELC FP

HLVC HL1 HL 11

HLrn IA

IEAR IEEE

ISO LDC occ./y RBTS RTS

Cana&m Electrkity Association Campo site System Reliability Evaluation Program. Direct Current Expected Customer Damage Cost Expected Energy Nat Suppfied F d ~ e Frequency and Failure DUrati~n

Expected Load Cuaailed Failure Probabm Hourly Load Variation *e

Hierarchical LevelI Hierarchical Level 11

Hierarhical Level Intercomectd Ares Inten-up t ed Energy Assessrnent Rate Institute of ~lectricd and Electronic Engineers, Inc. Independent S y t a Operator Load Duration Qrve Occurrences per year Roy Billinton Test System Reliabilit y Test S yst em

1

Introduction

1 . Introduction

The primary objective of an electnc power system is to supply its custorners with high quality reliable electric energy at the lowest possible cost. Reliability is basically a quality characteristic. It is however a specific quality characteristic, which is difEerent fiom the ordinary quality attriiutes such as the voltage and fkequency of electricity which can be instantly measured using particular devices in detenninistic terms. Reliability is probabilistic in nature and can only be measured in a probabilistic format. This suggests that reliability measurernent c m only be done over a long time and reliability assurance is more complex than that of vokage or fiequency. Reliability engineering is concerned with the measurement and the prediction of reliability and has become a recognized profession.

The reliability of electnc supply is directly associated with the quality of life in modem society. The importance of supplying highly reiiable electric energy cannot be overstated and is constantly growing. Reliability is, however, not fkee. Reliability and economic constraints will always codlict with each other and lead to dScult managerid decisions. Reliability evaluation provides the O pportunity to quantitatively evaluate the reliability of the system and c m be used to provide valuable input to the decision making process [ l-31.

The conventional approach to creating a satisfactory degree of reliability for many decades was achieved through empirical methods and policies. The aiteria and techniques that form the basis for these empincal methods and policies were all deteTmiI3istically based, and many of them are stiU in use today. The basic weakness of

. . these d e t e r d c criteria is that they do not respond to or reflect the probabilistic or stochastic nature of system behavior, customer demands and component fdures.

DeterminiStic techniques also cannot respond systematically to many of the parameters which actudy Innuence system reliability, especidy in modern electric power systems which are large and extremely complicated.

The need for ngorous and quantitative reliability analysis has become increasingly

more evident and urgent, as electric utility deregulation forces the electric power supply industry into market cornpetition. In recent years, the formal concepts and methods of reliability theory have been applied to almost every aspect of power system planning and operation. Reliability evaluation can provide effective infiormation in regard to the identification of system weaknesses, cornparison of alternative system designs, and the justification of new facilties [4-81.

The basics of power system reliability evaluation are desmbed in the follo wing two sections. This is then followed by a bnef description of the impacts of ebctnc utility deregulation in t erms of power system reliability evaluation activities. The research objectives and the outline of the thesis are provided in the last two sections of this chapter.

1.2 Basics of Reliability Studies

Reliability is, in a general sense, a measure of the ability of an item or a systern to perform its intended f'unction. In a mission onentated sense, reliabity is dehed as the ability of the item to perfonn a required function under certain conditions for a stated period of tirne. In the case of an electnc power systern, reliability is usually defined in a quite different way. A power systern is designed to perform its function not for a lirnited short p&od but for a relatively long t h e . A power system cm have faimes at dinerent points in the system at certain thes, but these faiures cm always be repaired, and the system can be constantly developed in order to satisfL the changing demands and to improve the performance of its fiuiction. It is therefore important for a power system to

recognize and control the various possible systern fdures and to minmiize the fdure effects that the customer experiences. One dennition of systern reliability is simply the control of fdures [l]. The rneasurement of power system reliability includes the unavailability, the expected faiure fiequency and duration, and the expected magnitude of adverse effects on consumer service.

System reliabilxty evduation is based on reliability data for the items which rnake up the system. The collection of systern cornponent reliability data is therefore a

fundamental task in system reliability evaluation. Many electric power utilities have traditionally recorded the performance of power system components. These recordings are the primary sources of the component reliability data required for the purpose of conducting power system reliability evaluation. The data collection systems established

and managed by the Canadian Electricity Association (CEA) are extrernely valuable repositories of power system equipment and system data, and sources of data definitions and methodologies, which are briefly desmied in Appendix C of the thesis. Although the reliability data provided by the CEA may not be directly applicable in other countries, the data collection systerns of the CEA provide a very usefl example of how to collect reliability data in electric power systerns [9- 121.

Reliability as applied to a power system is divided into the two general categories of system adequacy and system security. This division is shown in Fig. 1.1. System adequacy relates to the existence of sufficient generation, transmission and distribution

System Reliability

Fig. 1.1 . Subdivision of power system relisbility

facilities within the system to satisfy the customer load demnd. Adequacy evaluation is therefore associated with system steady state conditions. System security, on the other

hand, relates to the ability of the system to cope with disturbances and is consequently associated with transient system conditions. System adequacy is usually associated with

system planning for both long and short terms, but is also very important in system operation. S ystem security is ccncemed with both system planning and operation [2,6].

System adequacy precedes system security. Satisfactory system security cannot be

obtained without acceptable systern adequacy. Comprehensive probabilistic techniques

for reliability evaluation in the adequacy domain have been developed and are coiitinually

developing [2,13- 1 61. The need for the application of adequacy evaluation techniques in efectric power industries is growing. The ability to assess security is relatively iimited at

the presmt tirne compared with adequacy evaluation, due to the complexities associated

with the required modeling. Effective work on security assessrnent has been done and

efforts are contiming. It is worth noting that many adequacy evaluation techniques are

now being used in the security domain to evaluate security based reliability or to establish risk based security limits instead of using traditional deterministic security Limits

in system operation. In these cases, risk is defined as the product of probability and consequence [17-251. This thesis is concemed with system adequacy evaluation.

Electnc power systems are generally categorized into the three segments or

functiond zones of generation, transmission and distniution. This division is an

appropriate one as most utilities are either divided into these zones for the purpose of

organization, planning, operation and analysis or are solely responsible for one of these

functions. The three fwictional zones can be combined to give three hierarchical levels,

as shown in Fig. 1.2. Hierarchical Lxvel 1 (HL 1) is concemed only with the generation facilities. Hierarchical Level II (Ea II) includes both generation and transmission facilities and HL III uicludes aU three functional zones. System adequacy analysis is

usually conducted in each of the three functional zones or in the three hierarchical levels

~6~71.

Fig. 1.2. B s i c fiinction zones and hierarchical levels

HL 1 analysis using probability rnethods is the oldest and most extensively developed area. Considerable effort has been applied over the last two decades [14-161 to develop acceptable techniques and criteria for HL II analysis, which is also designated as composite power system or bulk power system evaluation HL III studies are not usually done directly due to the enormity of the problem in a practical system and instead the dstri'bution hctional zone is analyzed as an independent entity.

There are two basicdy and conceptually different methodo logies, the analytical approach and the Monte Car10 simulation approach, used in power system reliability evaluation [2,26]. This is shown in Fig. 1.3. The analytical approach represents the system by a mathematical model and evaluates the reliability indices fkom this model

I Basic Approaches

Fig. 1 -3. Two fiindamentally different approaches

5

using analytical solutions. The Monte Car10 simulation approach, however, esthates the reliability indices by simulating the achial process and random behavior of the system and treats the problem as a series of real experiments [26-291.

There are merts and demerits in both approaches. Monte Car10 simulation usually requires much longer computation times than analytical methods and it provides approximate estimates of the reliability indices. The analytical approach, however, requires relatively less computation t h e and c m be used to provide accurate reliability indices. The analytical approach is n o d y selected given suitable methods are available for specsc problems. In theory, the simulation approach can include any system effect or system process which may have to be approximated in an andyticd method. The simulation approach cm also provide probability distributions that analytical methods canno t. Both analyticd and simulation approaches are undergohg continuous M e r development and being used in specific applications. This thesis is devoted to the development and extension of the analytical approach to power systern adequacy evaluation.

1.3 Basics of Adequacy Evaluation

The basics of adequacy evaluation at the three hierarchical levels are provided in this section. This includes the general concepts and the basic adequacy indices at the three levefs. The concepts of reliabilty cost/worth assessment are ais0 uirroduced in this section-

HL 1 Evaluation

HL 1 assessment, more commonly known as generating capacity adequacy evaluation, is concemed with assessing the generating capacity that must be Istalled in order to meet the system load with an acceptable degree of risk. The HL I system evaluation mode1 is shown in Fig. 1.4. The major concm in HL 1 evaluation is the total generation required to satisfy the total system demand and to have sufncient capacity to perform corrective and preventive maintenance on the generation facilities. The

@+dLoad Generation

Fig. 1.4. Generation evaluation model

transmission systern and its ability to move the generated energy to the consumer load points are not considered in an HL 1 evaluation.

The basic modeling approach is shown in Fig. 1.5. The generation model and the load model are convolved to form probabilistic risk or adequacy indices. The basic generation model is a capacity outage probability table which contains the capacity outage States of the generating system together with the probability, fkquency and duration of each state. The load model can either be the daily peak load variation c w e (DPLVC), which includes the peak loads of each day, or the load duration c w e (LDC), which represents the hourly variation of the load.

HL 1 adequacy evaluation is widely applied in electnc power utilities, especidy in North Amerka. The evaluation of the generating capacity risk level provides effective input in the planning decision makmg process and in the expansion of installed generating capacity to mset fiiture load growth. Research is still active in this area with regard to technique extensions, modifications, new algonthms and their applications in a changing system environment. The basic adequacy indices [2] include Loss Of Load Expectation (LOLE), Loss Of Energy Expectation (LOEE), and Expected Failure Frequency and Duration (FF&FD). They are dehed as follows:

Convoive

Mode1 Mode1 Indices

Fig. 1 .S. Basic generation modeling approach

LOLE -- Expected number of days (or hours) per year in which available capacity is less than the daily peak load (or hourly load).

LOEE -- Expected energy not supptied by the generation system due to the load demand exceeding the available generating capacity.

FF&FD -- Expected fiequency of capacity shortage events in a year and expected duration of a shortage event.

The LOLE is the most Widely used probabilistic critenon in generation planning because of its simplicity. It indicates the expected number of days or hours in which a load loss or a deficiency will occur. It does not indicate, however, the severity of the deficiency, the eequency or the duration of loss of load.

The LOEE is an appealing index as it meanires the seventy of deficiencies rather than just the number of time units and therefore the impact of energy shortfalls as well as their likelhood is evaluated. As it is an energy based index, it also reflects the basic fact that a power systern is an energy supply syst&n. This index therefore is well suited for use in situations in which alternative energy sources are being considered. The expected energy not supplied is sometimes divided by the total energy demand. This gives a normalized index which can be used to compare the adequacy of systerns that dSer considerably in size.

The FF&FD approach identifies the expected frequency of encountering a deficiency and the expected duration of a deficiency. It contains additional physical characteristics which make it sensitive to the generating system parameters and therefore provides more information to a power system plmer. FF&FD indices have been used extensively in network studies. They have not, however, been usai very widely in generating system adequacy analysis.

HL II Evaluation

Adequacy analysis at HL II is uusually termed as composite power systern or bulk power system evduation. The term composite refers to the consideration of both

generation and oansmission system contingencies, including the modehg of the operating policies necessary in order to dispatch generating units, assesment of power flows on the transmission system, alleviation of network violations, and load shedding if required. The basic modeling approach for HL II is the same as that for HL 1 shown in Fig. 1.5, except that the generation model is replaced by a composite generation and transmission system model. Adequacy andysis at HL II is much more complicated than that at HL 1, as the composite system is much more complex than a single generation systern and the system loads and generators a-re C O M ~ C ~ ~ at buses distriiuted

throughout the systern. Considerable effort has been expended during the last two decades in developing techniques and criteria for HL II shidies [2,16-251. The many complications in HL II analysis include large required computation times, network solution techniques, generation redispatch, overload alleviation, load curtailment policies and the consideration of independent, dependent, cornmon-cause and station-asso ciated outages. Many of these aspects have not yet been Mly resolved and there is no universally accepted method of analysis.

One major dfficulty in adequacy evaluation at HL II is the long computation times required to solve the large number of system outage events or system states. For a system with n components, the total number of system states is 2", when a two-date model of each component is used. This is a v q large number for a normal power system with several hundred components. The large number of system states in a composite system &plies that a very large size sample of system states is required to obtain an acceptable standard deviation for the adequacy index estimates when using Monte Car10 simulation. In the andytical approach, only credible events are norrndy investigated, where a credible outage is dehed as an outage event that has s i w c a n t contribution to the adequacy indices. Credibility is u s d y determined by considering outages up to a certain contingency level defined in terms of the nurnber of simultaneous outages.

There are two sets of adequacy indices which can be used in composite system evaluation. They are indices at the system load points and indices for the overall system [2] . These indices are complementary, not alternatives. The overd system indices give

an assessment of ovaall system adequacy, while the load point indices indicate the performance at the individual load buses and provide input values to the next hierarchical level. The basic adequacy indices for both the overall system and the individual load points in HL II assessment include the fouowing:

Failure probability Expected failure fiequency Expected failure duration Expected load curtailed Expected energy not supplied

The following adequacy indices are used in an overd system assessment of a composite power system.

Bulk power supply disturbances Bulk power interruption index Buk power supply average MW curtailment Bulk power energy curtailment index (or severity index) Modifed bulk power energy curtailment index

Application of HL II adequacy evaluation is not as extensive as that of HL I analysis, as rnany conceptual, modeling and computational difficulties are still unsolved. The interest or need for applying HL II adequacy evaluation techniques is increasing and urgent. This is due to electric utility deregdation and the resulting market cornpetition forces. Adequacy evaluation of a composite generation and transmission srjtem can provide realistic and comprehensive information with regard to the identification of system weaknesses, cornparison of alternative system designs and the justification of new expansion plans [3W].

HL 111 Evaluation

HL III includes al l three functional zones, staaing at the generating points and tenniDating at the individual consumer load points. It is not usual or practical to conduct

adequacy evaluation directly on actual HL III systems due to their large size and complexity. Instead, the distrr'bution functional zone is n o d y analyzed as a separate entity. An adequacy analysis of the distribution functional zone provides valuable evaluation on the strength of the distribution system. HL III indices c m be evaluated by usEg the HL II load-point indices as input values to the distribution fnctional zone being studied. The objective of an HL III study is to obtain suitable adequacy indices at the actud consumer load points.

Distribution systems are comparatively simple networks, and therefore their reliability evaluation is not always as complex a task as that of generation and transmission systems. The analflical methods for distn'bution systems are highly developed 121. The usual techniques are bas& on the minmial-cut-set method or on Mure-modes analysis in conjunction with sets of analytical equations which account for all the realistic faiure and restoration processes. The actud application of these techniques, however, is not extensive. This is rnainfy because of the lack of appropriate data and the fact that distribution outages have very localized eEects.

Analysis of customer fdure statistics shows that a major part of the service interruptions experienced by an individual customer is due to faiures in the distribution system, which clearly points out the need to irnprove the reliability of this area. Distriution reliability applications are steadily inmeashg with the collection of better data on component m u r e rates and custorner interruption costs. These applications allow planners to incorporate reliability cost/worth concepts in system design and in the cost justification of operation and maintenance policies. Advanced work is also being conducted oi; the collection of cost of customer interruption data and the evaluation of the effects of transmission systems on overall customer reliability.

The primary indices in HL III analysis are the expected frequency of failwe, the average duration of filure and the annual unavailability (or outage time) of the load points. In order to give a complete representation of the system behaviour and response, additional indices are usudy required. The most commonly used additional indices at HL III are as follows [2].

11

System average interruption fiequency index, SAFI Systern average intmption duration index, SMDI Customer average interruption fiequency index, CAIFI

4 Custorcer average intemption duration index, CAIDI Average savice availability index, ASAI Energy not supplied index, ENS

4 Average energy not supplied, AENS 4 Average customer curtaiIment index, ACCI

Co srnenefit Assessrnent

Economics plays a major role in the application of reliabiIity concepts and the attainment of an acceptable level of reliability- Reiiability is not fiee, but poor reliabity of electric power supply usudy costs much more than good reliability. It is therefore important to detemiine the optimal reliability level at which the reliability investment achieves the best results in reducing the customer damage costs due to power supply intemiptions. This optimal reliability level concept cm be lIustrated as shown in Fig. 1.6.

The reliability investment cost shown in Fig. 1.6 generdy increases as consumers are provided with higher reliability. On the O ther hand, the consumer costs associated

Investment cost

System Relia bility

Fig. 1.6. Investment, darnage and total cost as a function of system reliability

12

with power supply interruptions wiU decrease as the reliability increases. The total cost to society will therefore be the sum of these two individual costs. This to ta1 cost exhibits a mUiimu111, at which an optimum or target level of reliabilty is achieved.

One aspect of reliability costhenefit malysis is to develop techniques which can appropriately evaluate the co sts of the various system planning alternatives that are associated with the dif3erent reliability levels. These planning alternatives include diffkrent reinforcanent schemes and improvements in maintenance and operating policies.

An important requirement in reliability cost/benefit andysis is the ability to quantitatively evaluate the customer damage costs due to supply interruptions. It is important to note that the customer damage costs due to supply intemiptions include both direct and indirect components. The assessmect of the indirect costs is usually a difncult task as many non-technical factors and the uncertainties are involved. There have been many studies of interruption or outage costs [45-491- The commonly used method is to derive a composite customer damage function (CCDF) fiom surveys to individual customers and to calculate the expected total cost (ECOST) of power supply interruptions to the customers. It is also important to note that the customer damage costs vary with societal development and increasing reliance on electncity.

The market cornpetition due to utility deregulation at the present t h e has forced many electric utilities to reconsider their strategies. Most have reexamined capital investments, on the theory that there is no point investing fnds that may not be recovered fiom customers within some reasonable time in the fture. Reliabity costhenefit concepts and techniques are of great importance in this situation [50-551.

1.4 Deregulation Impacts

The electric power system industry has been traditiody organized and operated as a regulated monopoly. Traditional electric power utilities own and operate all three functional zones of the power system and therefore control all aspects of the system

planning, design and operation. The power system industry is now undergoing considerable change throughout the world due to the forces of deregulation, technological revo lution and evolving customer expectations [56-601. The intent of the change or restnicturing is to push the electric power systern mdustry into competitive markets in order to mprove cost effectiveness and to improve customer service and satisfaction,

There are myriads of ways to restructure the power system supply industry. In the new deregulated and competitive environment, the three functional mnes of a power system will not be controlled by single verticdy integrated utilities, but will be controlled independently by different entities. This is shown in Fig. 1.7. Despite the separation in control for the three functional zones, an electric power system is a total energy system in which generation and consurnption must rernain continuously and instantaneously in balance. The GENCOs, TRANSCOs and DISCOs, while pursuing their own interests, must work CO-operatively to provide cost effective and a reliable high quaty electric power supply. Great varieties exist concemuig faccilities ownership, systern operation and system development in the three functional zones. While the details of restmcturing are diffrent in different jurisdictions, there are common features in these restructuring activities. A prominent common feature is to provide undiscriminated open access to

DISCOs r- l Fig. 1 -7. Unbundled structure

essential transmission Gicilities. The comrnon features of deregulation in North Amena are as follows:

0 Cornpetitive market for electric generation, 0 Independent S ystem Operat or (ISO) controls transmission, 9 No change in distriiution fianchises, and 9 Customer choices of electricity providers.

The deregulation of the power system industry will also change the environment associated with system reliability evaluation activities. New reliability criteria and analytical tools will be required, while some of the traditional criteria may no longer be appropnate in the new environment. The traditional reliability criteria based on deterministic considerations will become increasingly dScult to apply as the traditional utility functions are unbundled. Quantitative probabilistic reliability evaluation techniques should be extensively utlized in the new competitive environment and will become important tools in many aspects of power system operation and development in order to irnprove CO st effectiveness and competitiveness in the market. Conventional generation expansion planning based on both determinitic criteria and probabilistic methods may disappear or entirely change as the generating stations will be built when and where investors decide and wiu be controlled by many diffrent participants. nie various probabilistic reliability techniques in generating capacity evaluation dl, however, still be ver- important, but will be used in a different way in the new restructurecl environment. In the distribution functional zone, the existing aiteria and techniques for probabilistic reliability evaluation will remain and be more actively utilized in the new environment.

Reliability evaluation applied strictly to the transmission fnctional zone may decrease, while composite generation and transmission s yst ern reliability evaluation will increase in order to recognize the integration of the generation and transmission systems. The ISO is a key element in the effective operation of an electric power system in the new competitive environment. The primary f ict ion of the ISO is to ensure that the composite or bulk electric power system is operating in both a safe and cost effective

mariner. Probabilistic reliability evaluation of composite power systems will play an important role in the restnictured competitive environment of electnc power supply.

1.5 Research Objectives This thesis is concemed with the adequacy evaluation of composite power systems

using analytical rnethods. A major dinicuky in the adequacy evaluation of composite power systems using analytical methods is the long computation times required to investigate the extrernely large number of possible system outage events in a practical composite system. Only the credible system states are usually investigated, where the credi'ble system states are those that have sigmficant contniution to the adequacy indices and are u d y detennined by considering outages up to a certain level defined in ternis of the number of sirnultaneous outages. The computation t h e inmeases rapidly with increase in the system size and the dehed system outage levels. The adequacy indices obtained can ofien be inaccurate or uncertain due to iimiting the investigated system states.

Computation times can however be significantly reduced when the unchanging portion of a system c m be replaced by a reduced equivalent model. These reduced models can be designated as reliability or adequacy equivalents. They c m prove very usefiil in the evaluation of large systems where sensitivity studies are to be performed on a portion of the system or when the system is to be interconnected to a M e r system which is to be studied in detail. These situations arise quite Eequently Ki power system applications. Adequacy equivalent concepts can also have many other applications in adequacy evaluation of composite power systems.

The primary objective of the research was to investigate and develop techniques for the adequacy evaluation of large composite power systems in order to provide improved adequacy indices with feasible or acceptable reqired computation times. A major activity in the research was the investigation of the applications of adequacy equivalent concepts in composite system evaluation in order to achieve the primary research objective. Previous research work on adequacy equivaent techniques has been mainly

conducted by the Po wer S ystem Research Group at the University of Saskatchewan [6 1 - 701. The research in this thesis is based on and extends the previous research work. Specincally, the following fou. major research objectives have been studied and are detailed in the thesis.

Investigate the possible techniques which cm improve the accuracy of the adequacy indices, provide error estimation of the obtained indices, and reduce the required computation times.

0 Develop the techniques and investigate the effects of using different solution

techniques in composite system adequacy evaluation using the quivalent approach.

- Investigate various applications of the equivaient concepts and their effects in adequacy evaluation of large composite power systems.

* Investigate the techniques for reliability cost/benefit analysis and the pertinent factors in the costhenefit analysis when the adequacy equivalent approach is used.

1.6 Outline of this Thesis

This introductory chapter provides the philo sophy and basic concepts of reliability evaluation in general and particularly for eiectric power systems. The concepts and the indices in the adequacy evaluation of power systems at the three hierarchical levels together with the general framework of reliability costhenefit assessment are bnefly described. The impacts of power systern utility deregulation on reliability evaluation activities and the research objectives of the thesis are also descriied in this chapter.

In Chapter 2, the fundamental concepts, techniques and the basic procedure of adequacy evaluation in composite power systems are descriied in detail. Many of these concepts are applicable to both the analytical approach and the Monte Carlo simulation approach. Three advanced algorithms are developed and illustrated in Chapter 3. These advanced algorithms can generally provide improved adequacy indices without

considerabIy increasing the required computation times and they can be effectively utilized in the adequacy equivalent. techniques Uustrated in the thesis. System studies using two reliabity test systerns designatecl as the Roy Billinton Test System W T S ) [7 11 and the IEEE Reliability Test S ystem (IEEE-RTS) [72,73] are iuustrated in Chapter 4. These studies use the basic evaluation techniques and the advanced algonthms developed in Chapters 2 and 3. The studies provide furdier insight into the concepts, techniques, adequacy indices and the new advanced algorithms presented in Chapters 2 and 3.

Chapter 5 provides a complete description of the adequacy equivalent concepts, techniques and applications. System studies on an interconneted two-area RBTS and an interconnected two-area EEE-RTS using the adequacy quivalent techniques are provided in Chapter 6. The effects of various fctors in the composite systern adequacy evaluation using the equivalent techniques are investigated and iuustrated h Chapters 5 and 6 . The major use of the adequacy equivalent techniques is to establish an equivalent for the unchanged portion of a composite system and to use this equivalent in composite system analysis in which the required computation tirnes are effectively reduced. The equivalent techniques cm also be used to represent common cause outages and station originated events by equivalents and to effectively investigate the effects of these complicated events. This application of equivalent techniques is also illustrated in Chapters 5 and 6.

Costhenefit analysis of composite power systems is becorning an essential factor in the detemimation of the system reinforcement and expansion projects, particularly in the e l e c ~ c utility dereplation and market cornpetition environment. The concepts and techniques for CO sthenefit malysis of composite po wer systems, combined with the equivalent techniques are descnbed in Chapter 7. System studies on both one and two area RBTS and IEEE-RTS which consider the cost related indices in different situations and investigate the effects of various factors on the variation of the indices when using the equivalent techniques are also illustrated in Chapter 7. Chapter 8 provides the surmnary and conclusions of this research.

Basic Concepts and Evaluation Techniques

2.1 Introduction

A primary objective in adequacy evaluation of composite generation and transmission systems is to calculate a set of adequacy indices for the overd system and for the load points distnbuted throughout the system. These indices indicate the composite system adequacy fiom different points of view. The indices provide important numerical input in the identification of the system weaknesses, in the cornparison of alternative system designs or reinforcement schemes and in the justification of new expansion plans.

The requirement of large computation times is a cornon problem in composite power system adequacy evaluation when using either an analytical technique or a Monte Car10 simulation approach. This is d y because of the large size associated with a practical composite system and the very large nurnber of possible systern states due to component outages. An analytical approach usually requires less computation time than a simulation approach and is able to provide accurate adequacy indices, when mitable models and techniques are developed. The simulation approach however can be used to provide approximate estimates of the adequacy indices and their standard deviations.

This chapter provides a description of the theoretical foundation, the basic concepts and the practical evaluation techniques for composite generation and transmission system adequacy evaluation using the andytical approach Many of these concepts are &O effective in the simulation approach as the difference between the analytical and simulation approaches is mainly in the process of generating the sytern states, which is one aspect of composite system assessment.

2.2 State Space Technique

The state space technique cm be generally used for a wide range of engineering system reliability assessments. The technique is based on the mathematical theory of Markov processes. A process that is govemed by probabilistic laws is usually referred to as a stochastic process. A Markov process is a specinc stochastic process and is characterized by a lack of memory, that is, a system state is independent of all past states except the immediately preceding one. A Markov process is therefore a process in which the fture random behaviour of a wern only depends on where it is at the present, not on where it has been in the past or how t arrived at its present position. A stochastic process can also be designated as stationary or non-stationary. A stationary process is a process in which the system random behaviour in a k e d time interval is the same at all times in the past and fiiture regardless of the time being considered. The state space technique is applied to stationary Markov processes.

The two features of lack of memory and being stationary c m be satisfied in those systerns whose random behaviours can be desmbed by probability distn'butions that are characterked by constant transition rates, such as Poisson and exponential distributions. Power systems and many other engineering systems can often be descnbed as having discret e system states with constant transition rates between them; their random behaviours are therefore stationary Markov processes. The state space technique using state space visualization and basic Markov solution concepts is generally applicable to a wide range of electric power system reliability problems and other engineering systems whose random behaviours can be descned by stationary Markov processes [3].

A power system state space diagram provides the discrete or identifiable states in which the system and its components can reside, and the transition rates between the states. The establishment of a system state space mode1 is the &st step in using the state space technique. AU the relevant system states and the known transitions between the states should be included in a specifc system reliability malysis. The state space diagram of a single component system is shown in Fig. 2.1, Ui which the system can reside in

Fig. 2.1. A single component two-state state space diagram

either state 1 (the up or operating state) or state 2 (the down or faied state). The transition rates between the two states are the component fidure rate h and repair rate p, which are considerd to be constant values.

In some situations, a single component may be best represented by more than two states. A unit in a nin-of-the-river hydro plant cm be a simple example of this situation, in which the river flow rate detemines the unit output capacity. The unit is then represented as a d t i -s ta te unit in which the capacity states correspond to the river flow rates. Fig. 2.2 shows the state space diagram of a single component represented by three states.

For a two-cornponent system, there are four possible states in which the system can exist, if component independence is assumed and each component is represented by a two-state mode1 The state space diagram of the two-component system is s h o w in Fig.

Fig. 2.2. A single component three-state Gate space diagram

Fig. 2.3. A two-cornponent system state space diagram

2.3, in which hl, pl, h2 and p2 are the Mure and repair rates of components 1 and 2 respectively. It should be noted that there c m be many diffrent state space diagrams for a two-component systern. Fig. 2.3 is one of the sirnplest. The state space rnay contain more than four states when, for instance, each or both components are represented by more than two states.

If the two components are not independent of each other, as is the situation of two transmission h e s on a common tower, then the two components may fail at the same time due to filure of the tower. This lcind of fiiilure is usudy refmed to as a common cause or common mode Mure . In Zn situation, states 1 and 4 in Fig. 2.3 should have direct transitions to represent the possibility of common cause Mures and repairs. Fig. 2.4 shows two typical common cause fidure models. In Fig. 2.4 (a), the two components on outage c m be repaired and returned to service separately or simultaneously (if pI2 >O). In Fig. 2.4 @), the two components out of service due to a common cause filure are repaired and rehinied to seMce simultaneously.

The creation of a state space diagram translates a physical problem into a mathematical model which requires engineering judgment and a thorough understanding of the physical and logical operation of the system. The importance of engineering

Ca)

Fig. 2.4. State space diagams

2 1 5 1 3 1 I Down 1 Down 1 UP 2 UP 2 D o m 2 Down

(b)

including common cause filures

judgernent and a thorough understanding of the systern physicai and logical operation cannot be overemphasized in this phase of the problem solution. A state space diagram indudes all the possible system states, the way in which these states comUI1icate and the transition rate values. There are no basic restrictions on the number of states or the type and number of tran