Reliability Evaluationof Power SystemsSecond Edition

Roy BillintonUniversity of SaskatchewanCollege of EngineeringSaskatoon, Saskatchewan, Canada

andRonald N. AllanUniversity of ManchesterInstitute of Science and TechnologyManchester, England

PLENUM PRESS NEW YORK AND LONDON

Library of Congress Catilogtng-tn-PublIcatfon Data

Btlltnton, Roy.Reliability evaluation of paver systens / Roy Btlllnton and Ronald

N. AlIan. 2nd ed.p. c.

Includes bibliographical references and Index.ISBN 0-306-45259-61. Electric power systensReliability. I. Allan, Ronald N.

(Ronald Klornan) II. Title.TK1010.B55 1996621.31dc20 96-27011

CIP

ISBN 0-306-45259-6

0.1984 Roy Billimon and Ronald N. AllanFirst published in England by Pitman Books Limited

1996 Plenum Press, New YorkA Division of Plenum Publishing Corporation233 Spring Street, New York, N. Y. 10013

10 9 8 7 6 5 4 3 2 1

All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise,without written permission from the Publisher

Printed in the United States of America

Preface to the first edition

This book is a seque! to Reliability Evaluation of Engineering Systems: Conceptsand Techniques, written by the same authors and published by Pitman Books inJanuary 1983.* As a sequel, this book is intended to be considered and read as thesecond of two volumes rather than as a text that stands on its own. For this reason,readers who are not familiar with basic reliability modelling and evaluation shouldeither first read the companion volume or, at least, read the two volumes side byside. Those who are already familiar with the basic concepts and only require anextension of their knowledge into the power system problem area should be ableto understand the present text with little or no reference to the earlier work. In orderto assist readers, the present book refers frequently to the first volume at relevantpoints, citing it simply as Engineering Systems.

Reliability Evaluation of Power Systems has evolved from our deep interest ineducation and our long-standing involvement in quantitative reliability evaluationand application of probability techniques to power system problems. It could nothave been written, however, without the active involvement of many students inour respective research programs. There have been too many to mention individu-ally but most are recorded within the references at the ends of chapters.

The preparation of this volume has also been greatly assisted by our involve-ment with the IEEE Subcommittee on the Application of Probability Methods, IEECommittees, the Canadian Electrical Association and other organizations, as wellas the many colleagues and other individuals with whom we have been involved.

Finally, we would like to record our gratitude to all the typists who helped inthe preparation of the manuscript and, above all, to our respective wives, Joyce andDiane, for all their help and encouragement.

Roy BilliaJooRon Allan

'Second edition published by Plenum Press in 1994.

Preface to the second edition

We are both very pleased with the way the first edition has been received inacademic and, particularly, industrial circles. We have received many commenda-tions for not only the content but also our style and manner of presentation. Thissecond edition has evolved after considerable usage of the first edition by ourselvesand in response to suggestions made by other users of the book. We believe theextensions will ensure that the book retains its position of being the premierteaching text on power system reliability.

We have had regular discussions with our present publishers and it is a pleasureto know that they have sufficient confidence in us and in the concept of the bookto have encouraged us to produce this second edition. As a background to this newedition, it is worth commenting a little on its recent chequered history. The firstedition was initially published by Pitman, a United Kingdom company; the mar-keting rights for North America and Japan were vested in Plenum Publishing ofNew York. Pitman subsequently merged with Longman, following which, completepublishing and marketing rights were eventually transferred to Plenum, our currentpublishers. Since then we have deeply appreciated the constant interest and com-mitment shown by Plenum, and in particular Mr. L. S. Marchand. His encourage-ment has ensured that the present project has been transformed from conceptualideas into the final product.

We have both used the first edition as the text in our own teaching programsand in a number of extramural courses which we have given in various places. Overthe last decade since its publication, many changes have occurred in the develop-ment of techniques and their application to real problems, as well as the structure,planning, and operation of real power systems due to changing ownership, regula-tion, and access. These developments, together with our own teaching experienceand the feedback from other sources, highlighted several areas which neededreviewing, updating, and extending. We have attempted to accommodate these newideas without disturbing the general concept, structure, and style of the originaltext.

We have addressed the following specific points: Acomplete rewrite of the general introduction (Chapter 1) to reflect the changing

scenes in power systems that have occurred since we wrote the first edition.vii

vffi Preface to the second edition

Inclusion of a chapter on Monte Carlo simulation; the previous edition concen-trated only on analytical techniques, but the simulation approach has becomemuch more useful in recent times, mainly as a result of the great improvementin computers.

Inclusion of a chapter on reliability economics that addresses the developing andvery important area of reliability cost and reliability worth. This is proving tobe of growing interest in planning, operation, and asset management.

We hope that these changes will be received as a positive step forward and thatthe confidence placed in us by our publishers is well founded.

Roy BillintonRon Allan

Contents

1 Introduction 11.1 Background 11.2 Changing scenario 21.3 Probabilistic reliability criteria 31.4 Statistical and probabilistic measures 41.5 Absolute and relative measures 51.6 Methods of assessment 61.7 Concepts of adequacy and security 81.8 System analysis 101.9 Reliability cost and reliability worth 121.10 Concepts of data 141.11 Concluding comments 151.12 References 16

2 Generating capacitybasic probability methods 182.1 Introduction 182.2 The generation system model 21

2.2.1 Generating unit unavailability 212.2.2 Capacity outage probability tables 242.2.3 Comparison of deterministic and probabilistic criteria 272.2.4 A recursive algorithm for capacity model building 302.2.5 Recursive algorithm for unit removal 312.2.6 Alternative model-building techniques 33

2.3 Loss of load indices 372.3.1 Concepts and evaluation techniques 37

ix

x Contents

2.3.2 Numerical examples 402.4 Equivalent forced outage rate 462.5 Capacity expansion analysis 48

2.5.1 Evaluation techniques 482.5.2 Perturbation effects 50

2.6 Scheduled outages 52if

2.7 Evaluation methods on period bases 552.8 Load forecast uncertainty 562.9 Forced outage rate uncertainty 61

2.9.1 Exact method 622.9.2 Approximate method 632.9.3 Application 632.9.4 LOLE computation 642.9.5 Additional considerations 67

2.10 Loss of energy indices 682.10.1 Evaluation of energy indices 682.10.2 Expected energy not supplied 702.10.3 Energy-limited systems 73

2.11 Practical system studies 752.12 Conclusions 762.13 Problems 772.14 References 79

3 Generating capacityfrequency and duration method 833.1 Introduction 833.2 The generation model 84

3.2.1 Fundamental development 843.2.2 Recursive algorithm for capacity model building 89

3.3 System risk indices 953.3.1 Individual state load model 953.3.2 Cumulative state load model 103

3.4 Practical svstem studies 105

Contents xi

3.4.1 Base case study 1053.4.2 System expansion studies 1083.4.3 Load forecast uncertainty 114

3.5 Conclusions 1143.6 Problems 1143.7 References 115

4 Interconnected systems 1174.1 Int roduct ion 1174.2 Probabi l i ty array method in two interconnected systems

4.2.1 Concepts 1184.2 ,2 Evaluat ion techniques 119

4.3 Equivalent assisting unit approach to two interconnectedsystems 120

4.4 Factors affecting the emergency assistance available through theinterconnections 1244.4.1 Introduction 1244.4.2 Effect of tie capacity 1244.4.3 Effect of t ie l ine re l iab i l i ty 1254.4.4 Effect of number of tie l ines 1264.4.5 Effect of tie-capacity uncertainty 1294.4.6 Effect of interconnection agreements 1304.4.7 Effect of load forecast uncertainty 132

4.5 Variable reserxe versus maximurn peak load reserve 1324.6 Rel iabi l i ty evaluation in three interconnected systems 134

4.6.1 Direct assistance from two systems 1344.6.2 Indirect assistance from two systems 135

4.7 Mult i -connected systems 1394.8 Frequency and durat ion approach 1 4 1

4.8.1 Concepts 1414.8.2 Applications 1424.8.3 Period analysis 145

idl Contents

4.9 Conclusions 1474.10 Problems 1474.11 References 148

5 Operating reserve 1505.1 General concepts 1505.2 PJM method 151

5.2.1 Concepts 1515.2.2 Outage replacement rate (ORR) 1515.2.3 Generation model 1525.2.4 Unit commitment risk 153

5.3 Extensions to PJM method 1545.3.1 Load forecast uncertainty 1545.3.2 Derated (partial output) states 155

5.4 Modified PJM method 1565.4.1 Concepts 1565.4.2 Area risk curves 1565.4.3 Modelling rapid start units 1585.4.4 Modelling hot reserve units 1615.4.5 Unit commitment risk 1625.4.6 Numerical examples 163

5.5 Postponable outages 1685.5.1 Concepts 1685.5.2 Modelling postponable outages 1685.5.3 Unit commitment risk 170

5.6 Security function approach 1705.6.1 Concepts 1705.6.2 Security function model 171

5.7 Response risk 1725.7.1 Concepts 1725.7.2 Evaluation techniques 1735.7.3 Effect of distributing spinning reserve 1745.7.4 Effect of hydro-electric units 175

5.7.5 Effect of rapid start units 1765.8 Interconnected systems 1785.9 Conclusions 1785.10 Problems 1795.11 References 180

6 Composite generation and transmission systems 1826.1 Introduction 1826.2 Radial configurations 1836.3 Conditional probability approach 1846.4 Network configurations 1906.5 State selection 194

6.5.1 Concepts 1946.5.2 Application 194

6.6 System and load point indices 1966.6.1 Concepts 1966.6.2 Numerical evaluation 199

6.7 Application to practical systems 2046.8 Data requirements for composite system reliability

evaluation 2106.8.1 Concepts 2106.8.2 Deterministic data 2106.8.3 Stochastic data 2116.8.4 Independent outages 2116.8.5 Dependent outages 2126.8.6 Common mode outages 2126.8.7 Station originated outages 213

6.9 Conclusions 2156.10 Problems 2166.11 References 218

7 Distribution systems-basic techniques and radial networks 2207.1 Introduction 220

xiv Cofttonts

7.2 Evaluation techniques 2217.3 Additional interruption indices 223

7.3.1 Concepts 2237.3.2 Customer-orientated indices 2237.3.3 Load-and energy-orientated indices 2257.3.4 System performance 2267.3.5 System prediction 228

7.4 Application to radial systems 2297.5 Effect of lateral distributor protection 2327.6 Effect of disconnects 2347.7 Effect of protection failures 2347.8 Effect of transferring loads 238

7.8.1 No restrictions on transfer 2387.8.2 Transfer restrictions 240

7.9 Probability distributions of reliability indices 2447.9.1 Concepts 2447.9.2 Failure rate 2447.9.3 Restoration times 245

7.10 Conclusions 2467.11 Problems 2467.12 References 247

8 Distribution systemsparallel and meshed networks 2498.1 Introduction 2498.2 Basic evaluation techniques 250

8.2.1 State space diagrams 2508.2.2 Approximate methods 2518.2.3 Network reduction method 2528.2.4 Failure modes and effects analysis 253

8.3 Inclusion of busbar failures 2558.4 Inclusion of scheduled maintenance 257

8.4.1 General concepts 257

Contents xv

8.4.2 Evaluation techniques 2588.4.3 Coordinated and uncoordinated maintenance 2598.4.4 Numerical example 260

8.5 Temporary and transient failures 2628.5.1 Concepts 2628.5.2 Evaluation techniques 2628.5.3 Numerical example 265

8.6 Inclusion of weather effects 2668.6.1 Concepts 2668.6.2 Weather state modelling 2678.6.3 Failure rates in a two-state weather model 2688.6.4 Evaluation methods 2708.6.5 Overlapping forced outages 2708.6.7 Forced outage overlapping maintenance 2778.6.8 Numerical examples 2818.6.9 Application to complex systems 283

8.7 Common mode failures 2858.7.1 Evaluation techniques 2858.7.2 Application and numerical examples 287

8.8 Common mode failures and weather effects 2898.8.1 Evaluation techniques 2898.8.2 Sensitivity analysis 291

8.9 Inclusion of breaker failures 2928.9.1 Simplest breaker model 2928.9.2 Failure modes of a breaker 2938.9.3 Modelling assumptions 2948.9.4 Simplified breaker models 2958.9.5 Numerical example 296

8.10 Conclusions 2978.11 Problems ' 2988.12 References 301

xvi Contents

9 Distribution systems extended techniques 3029.1 Introduction 3029.2 Total loss of continuity (TLOC) 3039.3 Partial loss of continuity (PLOC) 305

9.3.1 Selecting outage combinations 3059.3.2 PLOC criteria 3059.3.3 Alleviation of network violations 3069.3.4 Evaluation of PLOC indices 3069.3.5 Extended load-duration curve 3099.3.6 Numerical example 310

9.4 Effect of transferable loads 3119.4.1 General concepts 3119.4.2 Transferable load modelling 3149.4.3 Evaluation techniques 3169.4.4 Numerical example 317

9.5 Economic considerations 3199.5.1 General concepts 3199.5.2 Outage costs 322

9.6 Conclusions 3259.7 Problems 3259.8 References 326

10 Substations and switching stations 32710.1 Introduction 32710.2 Effect of short circuits and breaker operation 327

10.2.1 Concepts 32710.2.2 Logistics 32910.2.3 Numerical examples 329

10.3 Operating and failure states of system components 33210.4 Open and short circuit failures 332

10.4.1 Open circuits and inadvertent opening of breakers 33210.4.2 Short circuits 333

Contents xvii

10.4.3 Numerical example 33410.5 Active and passive failures 334

10.5.1 General concepts 33410.5.2 Effect of failure mode 33610.5.3 Simulation of failure modes 33810.5.4 Evaluation of reliability indices 339

10.6 Malfunction of normally closed breakers 34110.6.1 General concepts 34110.6.2 Numerical example 34110.6.3 Deduction and evaluation 342

10.7 Numerical analysis of typical substation 34310.8 Malfunction of alternative supplies 348

10.8.1 Malfunction of normally open breakers 34810.8.2 Failures in alternative supplies 349

10.9 Conclusions 35210.10 Problems 35210.11 References 354

11 Plant and station availability 35511.1 Generating plant availability 355

11.1 .1 Concepts 35511.1.2 Generating units 35511.1.3 Including effect of station transformers 358

11.2 Derated states and auxiliary systems 36111.2.1 Concepts 36111.2.2 Modelling derated states 362

11.3 Allocation and effect of spares 36511.3.1 Concepts 36511.3.2 Review of modelling techniques 36511.3.3 Numerical examples 367

11.4 Protection systems 37411.4.1 Concepts 374

xvHi Contents

11.4.2 Evaluation techniques and system modelling 37411.4.3 Evaluation of failure to operate 31511.4.4 Evaluation of inadvertent operation 381

11.5 HVDC systems 38211.5.1 Concepts 38211.5.2 Typical HVDC schemes 38411.5.3 Rectifier/inverter bridges 38411.5.4 Bridge equivalents 38611.5.5 Converter stations 38911.5.6 Transmission links and filters 39111.5.7 Composite HVDC link 39211.5.8 Numerical examples 395

11.6 Conclusions 39611.7 Problems 39611.8 References 398

12 Applications of Monte Carlo simulation 40012.1 Introduction 40012.2 Types of simulation 40112.3 Concepts of simulation 40112.4 Random numbers 40312.5 Simulation output 40312.6 Application to generation capacity reliability evaluation 405

12.6.1 Introduction 40512.6.2 Modelling concepts 40512.6.3 LOLE assessment with nonchronological load 40912.6.4 LOLE assessment with chronological load 41212.6.5 Reliability assessment with nonchronological load 41612.6.6 Reliability assessment with chronological load 417

12.7 Application to composite generation and transmissionsystems 42212.7.1 Introduction 422

Contents xix

12.7.2 Modelling concepts 42312.7.3 Numerical applications 42312.7.4 Extensions to basic approach 425

12.8 Application to distribution systems 42612.8.1 Introduction 42612.8.2 Modelling concepts 42712.8.3 Numerical examples for radial networks 43012.8.4 Numerical examples for meshed (parallel)

networks 43312.8.5 Extensions to the basic approach 439

12.9 Conclusions 43912.10 Problems 44012.11 References 440

13 Evaluation of reliability worth 44313.1 Introduction 44313.2 Implicit'explicit evaluation of reliability worth 44313.3 Customer interruption cost evaluation 44413.4 Basic evaluation approaches 44513.5 Cost of interruption surveys 447

13.5.1 Considerations 44713.5.2 Cost valuation methods 447

13.6 Customer damage functions 45013.6.1 Concepts 45013.6.2 Reliability worth assessment at HLI 45113.6.3 Reliability worth assessment at HLII 45913.6.4 Reliability worth assessment in the distribution

functional zone 46213.6.5 Station reliability worth assessment 469

13.7 Conclusions 47213.8 References 473

xx Contnt

14 Epilogue 476

Appendix 1 Definitions 478

Appendix 2 Analysis of the IEEE Reliability Test System 481A2.1 Introduction 481A2.2 ffiEE-RTS 481A2.3 IEEE-RTS results 484

A2.3.1 Single system 484A2.3.2 Interconnected systems 486A2.3.3 Frequency and duration approach 486

A2.4 Conclusion 490A2.5 References 490

Appendix 3 Third-order equations for overlapping events 491A3.1 Introduction 491A3.2 Symbols 491A3.3 Temporary/transient failure overlapping two

permanent failures 492A3.4 Temporary/transient failure overlapping a permanent and a

maintenance outage 493A3.5 Common mode failures 495

A3.5.1 All three components may suffer a commonmode failure 495

A3.5.2 Only two components may suffer a common modefailure 495

A3.6 Adverse weather effects 496A3.7 Common mode failures and adverse weather effects 499

A3.7.1 Repair is possible in adverse weather 499A3.7.2 Repair is not done during adverse weather 499

Solutions to problems 500

Index 509

1 Introduction

1.1 Background

Electric power systems are extremely complex. This is due to many factors, someof which are sheer physical size, widely dispersed geography, national and inter-national interconnections, flows that do not readily follow the transportation routeswished by operators but naturally follow physical laws, the fact that electricalenergy cannot be stored efficiently or effectively in large quantities, unpredictedsystem behavior at one point of the system can have a major impact at largedistances from the source of trouble, and many other reasons. These factors are wellknown to power system engineers and managers and therefore they are notdiscussed in depth in this book. The historical development of and current scenarioswithin power companies is, however, relevant to an appreciation of why and howto evaluate the reliability of complex electric power systems.

Power systems have evolved over decades. Their primary emphasis has beenon providing a reliable and economic supply of electrical energy to their customers[1]. Spare or redundant capacities in generation and network facilities have beeninbuilt in order to ensure adequate and acceptable continuity of supply in the eventof failures and forced outages of plant, and the removal of facilities for regularscheduled maintenance. The degree of redundancy has had to be commensuratewith the requirement that the supply should be as economic as possible. The mainquestion has therefore been, "how much redundancy and at what cost?"

The probability of consumers being disconnected for any reason can bereduced by increased investment during the planning phase, operating phase, orboth. Overinvestment can lead to excessive operating costs which must be reflectedin the tariff structure. Consequently, the economic constraint can be violatedalthough the system may be very reliable. On the other hand, underinvestment leadsto the opposite situation. It is evident therefore that the economic and reliabilityconstraints can be competitive, and this can lead to difficult managerial decisionsat both the planning and operating phases.

These problems have always been widely recognized and understood, anddesign, planning, and operating criteria and techniques have been developed overmany decades in an attempt to resolve and satisfy the dilemma between theeconomic and reliability constraints. The criteria and techniques first used in

1

2 Chapter 1

practical applications, however, were all deterministically based. Typical criteriaare:

(a) Planning generating capacityinstalled capacity equals the expected maxi-mum demand pius a fixed percentage of the expected maximum demand;

(b) Operating capacityspinning capacity equals expected load demand plus areserve equal to one or more largest units;

(c) Planning network capacityconstruct a minimum number of circuits to a loadgroup (generally known as an (n - 1) or (n - 2) criterion depending on theamount of redundancy), the minimum number being dependent on the maxi-mum demand of the group.Although these and other similar criteria have been developed in order to

account for randomly occurring failures, they are inherently deterministic. Theiressential weakness is that they do not and cannot account for the probabilistic orstochastic nature of system behavior, of customer demands or of componentfailures.

Typical probabilistic aspects are:(a) Forced outage rates of generating units are known to be a function of unit size

and type and therefore a fixed percentage reserve cannot ensure a consistentrisk.

(b) The failure rate of an overhead line is a function of length, design, location,and environment and therefore a consistent risk of supply interruption cannotbe ensured by constructing a minimum number of circuits.

(c) All planning and operating decisions are based on load forecasting techniques.These techniques cannot predict loads precisely and uncertainties exist in theforecasts.

1.2 Changing scenario

Until the late 1980s and early 1990s, virtually all power systems either ha\'e beenstate controlled and hence regulated by governments directly or indirectly throughagencies, or have been in the control of private companies which were highlyregulated and therefore again controlled by government policies and regulations.This has created systems that have been centrally planned and operated, with energytransported from large-scale sources of generation through transmission and distri-bution systems to individual consumers.

Deregulation of private companies and privatization of state-controlled indus-tries has now been actively implemented. The intention is to increase competition,to unbundle or disaggregate the various sectors, and to allow access to the systemby an increased number of parties, not only consumers and generators but alsotraders of energy. The trend has therefore been toward the "market forces" concept,with trading taking place at various interfacing levels throughout the system. Thishas led to the concept of "customers" rather than "consumers" since some custom-

Introduction 3

ers need not consume but resell the energy as a commodity. A consequence of thesedevelopments is that there is an increasing amount of energy' generated at localdistribution levels by independent nonutility generators and an increasing numberof new types of energy sources, particularly renewables, and CHP (combined heatand power) schemes being developed.

Although this changing scenario has a very large impact on the way the systemmay be developed and operated and on the future reliability levels and standards,it does not obviate the need to assess the effect of system developments oncustomers and the fundamental bases of reliability evaluation. The needto^ssessthe present performance and predict the future behavior of systems remains and isprobably even more important given the increasing number of players in the electricenergy market.

1.3 Probabilistic reliability criteria

System behavior is stochastic in nature, and therefore it is logical to consider thatthe assessment of such systems should be based on techniques that respond to thisbehavior (i.e., probabilistic techniques). This has been acknowledged since the1930s [25], and there has been a wealth of publications dealing with the develop-ment of models, techniques, and applications of reliability assessment of powersystems [6-11 ]. It remains a fact, however, that most of the present planning, design,and operational criteria are based on deterministic techniques. These have beenused by utilities for decades, and it can be, and is, argued that they have served theindustry extremely well in the past. However, the justification for using a prob-abilistic approach is that it instills more objective assessments into the decision-making process. In order to reflect on this concept it is useful to step back intohistory and recollect two quotes:

A fundamental problem in system planning is the correct determination of reserve capacity.Too low a value means excessive interruption, while too high a value results in excessivecosts. The greater the uncertainty regarding the actual reliability of any installation thegreater the investment wasted.

The complexity of the problem, in general makes it difficult to find an answer to it byrules of thumb. The same complexity, on one side, and good engineering and soundeconomics, on the other, justify "the use of methods of analysis permitting the systematicevaluations of all important factors involved. There are no exact methods available whichpermit the solution of reserve problems with the same exactness with which, say. circuitproblems are solved by applying Ohm's law. However, a systematic attack of them can bemade by "judicious" application of the probability theory.

(GIUSEPPE CALABRESE (1947) [12]).The capacity benefits that result from the interconnection of two or more electric generatingsystems can best and most logically be evaluated by means of probability methods, and suchbenefits are most equitably allocated among the systems participating in the interconnectionby means of "the mutual benefits method of allocation," since it is based on the benefitsmutually contributed by the several systems. (CARL WATCHORN (1950) [ 13])

4 CbaptoM

These eminent gentlemen identified some 50 years ago the need for "prob-abilistic evaluation," "relating economics to reliability," and the "assessment ofbenefits or worth," yet deterministic techniques and criteria still dominate theplanning and operational phases.

The main reasons cited for this situation are lack of data, limitation ofcomputational resources, lack of realistic reliability techniques, aversion to the useof probabilistic techniques, and a misunderstanding of the significance and meaningof probabilistic criteria and risk indices. These reasons are not valid today sincemost utilities have valid and applicable data, reliability evaluation techniques arevery developed, and most engineers have a working understanding of probabilistictechniques. It is our intention in this book to illustrate the development of reliabilityevaluation techniques suitable for power system applications and to explain thesignificance of the various reliability indices that can be evaluated. This bookclearly illustrates that there is no need to constrain artificially the inherent prob-abilistic or stochastic nature of a power system into a deterministic domain despitethe fact that such a domain may feel more comfortable and secure.

1.4 Statistical and probabilistic measures

It is important to conjecture at this point on what can be done regarding reliabilityassessment and why it is necessary. Failures of components, plant, and systemsoccur randomly; the frequency, duration, and impact of failures vary from one yearto the next. There is nothing novel or unexpected about this. Generally all utilitiesrecord details of the events as they occur and produce a set of performancemeasures. These can be limited or extensive in number and concept and includesuch items as: system availability; estimated unsupplied energy; number of incidents; number of hours of interruption; excursions beyond set voltage limits; excursions beyond set frequency limits.These performance measures are valuable because they:(a) identify weak areas needing reinforcement or modifications;(b) establish chronological trends in reliability performance;(c) establish existing indices which serve as a guide for acceptable values in future

reliability assessments;(d) enable previous predictions to be compared with actual operating experience;(e) monitor the response to system design changes.

The important point to note is that these measures are statistical indices. Theyare not deterministic values but at best are average or expected values of aprobability distribution.

Introduction 5

The same basic principles apply if the future behavior of the system is beingassessed. The assumption can be made that failures which occur randomly in thepast will also occur randomly in the future and therefore the system behavesprobabilistically, or more precisely, stochastically. Predicted measures that can becompared with past performance measures or indices can also be extremelybeneficial in comparing the past history with the predicted future. These measurescan only be predicted using probabilistic techniques and attempts to do so usingdeterministic approaches are delusory'.

In order to apply deterministic techniques and criteria, the system must beartificially constrained into a fixed set of values which have no uncertainty orvariability. Recognition of this restriction results in an extensive study of specifiedscenarios or "credible" events. The essential weakness is that likelihood is neglectedand true risk cannot be assessed.

At this point, it is worth reviewing the difference between a hazard and riskand the way that, these are assessed using deterministic and probabilistic ap-proaches. A discussion of these concepts is given in Engineering Systems but isworth repeating here.

The two concepts, hazard and risk, are often confused; the perception of a riskis often weighed by emotion which can leave industry in an invidious position. Ahazard is an event which, if it occurs, leads to a dangerous state or a system failure.In other words, it is an undesirable event, the severity of which can be rankedrelative to other hazards. Deterministic analyses can only consider the outcome andranking of hazards. However, a hazard, even if extremely undesirable, is of noconsequence if it cannot occur or is so unlikely that it can be ignored. Risk, on theother hand, takes into account not only the hazardous events and their severity, butalso their likelihood. The combination of severity and likelihood creates plant andsystem parameters that truly represent risk. This can only be done using prob-abilistic techniques.

1.5 Absolute and relative measures

It is possible to calculate reliability indices for a particular set of system data andconditions. These indices can be viewed as either absolute or as relative measuresof system reliability.

Absolute indices are the values that a system is expected to exhibit. They canbe monitored in terms of past performance because full knowledge of them isknown. However, they are extremely difficult, if not impossible, to predict for thefuture with a very high degree of confidence. The reason for this is that futureperformance contains considerable uncertainties particularly associated with nu-merical data and predicted system requirements. The models used are also notentirely accurate representations of the plant or system behavior but are approxi-mations. This poses considerable problems in some areas of application in which

6 Chapter 1

absolute values are very desirable. Care is therefore vital in these applications,particularly in situations in which system dependencies exist, such as commoncause (mode) failures which tend to enhance system failures.

Relative reliability indices, on the other hand, are easier to interpret andconsiderable confidence can generally be placed in them. In these cases, systembehavior is evaluated before and after the consideration of a design or operatingchange. The benefit of the change is obtained by evaluating the relative improve-ment. Indices are therefore compared with each other and not against specifiedtargets. This tends to ensure that uncertainties in data and system requirements areembedded in all the indices and therefore reasonable confidence can be placed inthe relative differences. In practice, a significant number of design or operatingstrategies or scenarios are compared, and a ranking of the benefits due to each ismade. This helps in deciding the relative merits of each alternative, one of whichis always to make no changes.

The following chapters of this book describe methods for evaluating theseindices and measures. The stress throughout is on their use as relative measures.

The most important aspect to remember when evaluating these measures isthat it is necessary to have a complete understanding of the engineering implicationsof the system. No amount of probability theory can circumvent this importantengineering function. It is evident therefore that probability theory is only a toolthat enables an engineer to transform knowledge of the system into a prediction ofits likely future behavior. Only after this understanding has been achieved can amodel be derived and the most appropriate evaluation technique chosen. Both themodel and the technique must reflect and respond to the way the system operatesand fails. Therefore the basic steps involved are: understand the ways in which components and system operate; identify the ways in which failures can occur; deduce the consequences of the failures; derive models to represent these characteristics; only then select the evaluation technique.

1.6 Methods of assessment

Power system reliability indices can be calculated using a variety of methods. Thebasic approaches are described in Engineering Systems and detailed applicationsare described in the following chapters.

The two main approaches are analytical and simulation. The vast majority oftechniques have been analytically based and simulation techniques have taken aminor role in specialized applications. The main reason for this is because simula-tion generally requires large amounts of computing time, and analytical models andtechniques have been sufficient to provide planners and designers with the resultsneeded to make objective decisions. This is now changing, and increasing interest

Introduction 7

is being shown in modeling the system behavior more comprehensively and inevaluating a more informative set of system reliability indices. This implies theneed to consider Monte Carlo simulation. {See Engineering Systems, Ref. 14, andmany relevant papers in Refs. 6-10.)

Analytical techniques represent the system by a mathematical model andevaluate the reliability indices from this model using direct numerical solutions.They generally provide expectation indices in a relatively short computing time.Unfortunately, assumptions are frequently required in order to simplify the problemand produce an analytical model of the system. This is particularly the case whencomplex systems and complex operating procedures have to be modeled. Theresulting analysis can therefore lose some or much of its significance. The use ofsimulation techniques is very important in the reliability evaluation of such situ-ations.

Simulation methods estimate the reliability indices by simulating the actualprocess and random behavior of the system. The method therefore treats theproblem as a series of real experiments. The techniques can theoretically take intoaccount virtually all aspects and contingencies inherent in the planning, design, andoperation of a power system. These include random events such as outages andrepairs of elements represented by general probability distributions, dependentevents and component behavior, queuing of failed components, load variations,variation of energy input such as that occurring in hydrogeneration, as well as alldifferent types of operating policies.

If the operating life of the system is simulated over a long period of time, it ispossible to study the behavior of the system and obtain a clear picture of the typeof deficiencies that the system may suffer. This recorded information permits theexpected values of reliability indices together with their frequency distributions tobe evaluated. This comprehensive information gives a very detailed description,and hence understanding, of the system reliability.

The simulation process can follow one of two approaches:(a) Randomthis examines basic intervals of time in the simulated period after

choosing these intervals in a random manner.(b) Sequentialthis examines each basic interval of time of the simulated period

in chronological order.The basic interval of time is selected according to the type of system under

study, as well as the length of the period to be simulated in order to ensure a certainlevel of confidence in the estimated indices.

The choice of a particular simulation approach depends on whether the historyof the system plays a role in its behavior. The random approach can be used if the

. history has no effect, but the sequential approach is required if the past historyaffects the present conditions. This is the case in a power system containinghydroplant in which the past use of energy resources (e.g., water) affects the abilityto generate energy in subsequent time intervals.

8 Chapter 1

It should be noted that irrespective of which approach is used, the predictedindices are only as good as the model derived for the system, the appropriatenessof the technique, and the quality of the data used in the models and techniques.

1.7 Concepts of adequacy and security

Whenever a discussion of power system reliability occurs, it invariably involves aconsideration of system states and whether they are adequate, secure, and can beascribed an alert, emergency, or some other designated status [15], This is particu-larly the case for transmission systems. It is therefore useful to discuss the signifi-cance and meaning of such states.

The concept of adequacy is generally considered [ 1 ] to be the existence ofsufficient facilities within the system to satisfy the consumer demand. Thesefacilities include those necessary to generate sufficient energy and the associatedtransmission and distribution networks required to transport the energy to the actualconsumer load points. Adequacy is therefore considered to be associated with staticconditions which do not include system disturbances.

Security, on the other hand, is considered [1] to relate to the ability of thesystem to respond to disturbances arising within that system. Security is thereforeassociated with the response of the system to whatever disturbances they aresubjected. These are considered to include conditions causing local and widespreadeffects and the loss of major generation and transmission facilities.

The implication of this division is that the two aspects are different in bothconcept and evaluation. This can lead to a misunderstanding of the reasoning behindthe division. In reality, it is not intended to indicate that there are two distinctprocesses involved in power system reliability, but is intended to ensure thatreliability can be calculated in a simply structured and logical fashion. From apragmatic point of view, adequacy, as defined, is far easier to calculate and providesvaluable input to the decision-making process. Considerable work therefore hasbeen done in this regard [610]. While some work has been done on the problemof "security," it is an exciting area for further development and research.

It is evident from the above definition that adequacy is used to describe asystem state in which the actual entry to and departure from that state is ignoredand is thus defined as a steady-state condition. The state is then analyzed anddeemed adequate if all system requirements including the load, voltages, VARrequirements, etc., are all fully satisfied. The state is deemed inadequate if any ofthe power system constraints is violated. An additional consideration that maysometimes be included is that an otherwise adequate state is deemed to be adequateif and only if, on departure, it leads to another adequate state; it is deemedinadequate if it leads to a state which itself is inadequate in the sense that a networkviolation occurs. This consideration creates a buffer zone between the fully ade-quate states and the other obviously inadequate states. Such buffer zones are better

Introduction 9

known [14] as alert states, the adequate states outside of the buffer zone as normalstates, and inadequate states as emergency states,

This concept of adequacy considers a state in complete isolation and neglectsthe actual entry transitions and the departure transitions as causes of problems. Inreality, these transitions, particularly entry ones, are fundamental in determiningwhether a state can be static or whether the state is simply transitory and verytemporary. This leads automatically to the consideration of security, and conse-quently it is evident that security and adequacy are interdependent auApari ofIhesame problem; the division is one of convenience rather than of practical experi-ence.

Power system engineers tend to relate security to the dynamic process thatoccurs when the system transits between one state and another state. Both of thesestates may themselves be acceptable if viewed only from adequacy; i.e., they areboth able to satisfy all system demands and all system constraints. However, thisignores the dynamic and transient behavior of the system in which it may not bepossible for the system to reside in one of these states in a steady-state condition.If this is the case, then a subsequent transition takes the system from one of theso-called adequate states to another state, which itself may be adequate or inade-quate. In the latter case, the state from which the transition occurred could bedeemed adequate but insecure. Further complications can arise because the statefrom which the above transition can occur may be inadequate but secure in the sensethat the system is in steady state; i.e., there is no transient or dynamic transitionfrom the state. Finally the state may be inadequate and insecure.

If a state is inadequate, it implies that one or more system constraints, eitherin the network or the system demand, are not being satisfied. Remedial action istherefore required, such as redispatch, load shedding, or various alternative waysof controlling system parameters. All of these remedies require time to accomplish.If the dynamic process of the power system causes departure from this state beforethe remedial action can be accomplished, then the system state is clearly not onlyinadequate but also insecure. If, on the other hand, the remedial action can beaccomplished in a shorter time than that taken by the dynamic process, the state issecure though inadequate. This leads to the conclusion that "time to perform" aremedial action is a fundamental parameter in determining whether a state isadequate and secure, adequate and insecure, inadequate and secure, or inadequateand insecure. Any state which can be defined as either inadequate or insecure isclearly a system failure state and contributes to system unreliability. Presentreliability evaluation techniques generally relate to the assessment of adequacy.This is not of great significance in the case of generation systems or of distributionsystems; however, it can be important when considering combined generation andtransmission systems. The techniques described in this book are generally con-cerned with adequacy assessment.

10 Chapter 1

1.8 System analysis

As discussed in Section 1.1, a modern power system is complex, highly integrated,and very large. Even large computer installations are not powerful enough toanalyze in a completely realistic and exhaustive manner all of a power system as asingle entity. This is not a problem, however, because the system can be dividedinto appropriate subsystems which can be analyzed separately. In fact it is unlikelythat it will ever be necessary or even desirable to attempt to analyze a system as awhole; not only will the amount of computation be excessive, but the results arelikely to be so vast that meaningful interpretation will be difficult, if not impossible.

The most convenient approach for dividing the system is to use its mainfunctional zones. These are: generation systems, composite generation and trans-mission (or bulk power) systems, and distribution systems. These are therefore usedas the basis for dividing the material, models, and techniques described in this book.Each of these primary functional zones can be subdivided in order to study a subsetof the problem. Particular subzones include individual generating stations, substa-tions, and protection systems, and these are also considered in the followingchapters. The concept of hierarchical levels (HL) has been developed [1] in orderto establish a consistent means of identifying and grouping these functional zones.These are illustrated in Fig. 1.1, in which the first level (HLI) refers to generationfacilities and their ability on a pooled basis to satisfy the pooled system demand,the second level (HLII) refers to the composite generation and transmission (bulkpower) system and its ability to deliver energy to the bulk supply points, and thethird level (HLIII) refers to the complete system including distribution and itsability to satisfy the capacity and energy demands of individual consumers. Al-

hierarchical level IHLI

hierarchical level IIHLII

hierarchical level IIIHLIII

Fig. I.I Hierarchical Levels

Introduction 11

though HLI and HLI! studies are regularly performed, complete HLIII studies areusually impractical because of the scale of the problem. Instead the assessment, asdescribed in this book, is generally done for the distribution functional zone only.

Based on the above concepts and system structure, the following main subsys-tems are described in this book:(a) Generating stationseach station or each unit in the station is analyzed

separately. This analysis creates an equivalent component, the indices of whichcan be used in the reliability evaluation of the overall generating capacity ofthe system and the reliability evaluation of composite systems. The componentstherefore form input to both HLI and HLII assessments. The concepts of thisevaluation are described in Chapters 2 and 11.

(b) Generating capacitythe reliability of the generating capacity is evaluated bypooling all sources of generation and all loads (i.e., HLI assessment studies).This is the subject of Chapters 2 and 3 for planning studies and Chapter 5 foroperational studies.

(c) Interconnected systemsin this case the generation of each system and the tielines between systems (interconnections) are modeled, but the network in eachsystem (intraconnections) is not considered. These assessments are still HLIstudies and are the subject of Chapter 4.

(d) Composite generation/transmissionthe network is limited to the bulk trans-mission, a'nd the integrated effect of generation and transmission is assessed(i.e., HLII studies). This is the subject of Chapter 6.

(e) Distribution networksthe reliability of the distribution is evaluated by con-sidering the ability of the network fed from bulk supply points or other localinfeeds in supplying the load demands. This is the subject of Chapters 7-9. Thisconsiders the distribution functional zone only. The load point indices evaluatedin the HLII assessments can be used as input values to the distribution zone ifthe overall HLIII indices are required.

(f) Substations and switching stationsthese systems are often quite complicatedin their own right and are frequently analyzed separately rather than includingthem as complete systems in network reliability evaluation. This createsequivalent components, the indices of which can be used either as measures ofthe substation performance itself or as input in evaluating the reliability oftransmission (HLII) or distribution (HLIII) systems. This is the subject ofChapter 10.

(g) Protection systemsthe reliability of protection systems is analyzed sepa-rately. The indices can be used to represent these systems as equivalentcomponents in network (transmission and distribution) reliability evaluation oras an assessment of the substation itself. The concepts are discussed in Chapter11.The techniques described in Chapters 2-11 focus on the analytical approach,

although the concepts and many of the models are equally applicable to thesimulation approach. As simulation techniques are now of increasing importance

12 Chapter 1

and increasingly used, this approach and its application to all functional zones of apower system are described and discussed in Chapter 12.

1.9 Reliability cost and reliability worth

Due to the complex and integrated nature of a power system, failures in any part ofthe system can cause interruptions which range from inconveniencing a smallnumber of local residents to a major and widespread catastrophic disruption ofsupply. The economic impact of these outages is not necessarily restricted to lossof revenue by the utility or loss of energy utilization by the customer but, in orderto estimate true costs, should also include indirect costs imposed on customers,society, and the environment due to the outage. For instance, in the case of the 1977New Year blackout, the total costs of the blackouts were attributed [ 16] as: Consolidated Edison direct costs 3.5% other direct costs 12.5% indirect costs 84.0%As discussed in Section 1.1, in order to reduce the frequency and duration of theseevents and to ameliorate their effect, it is necessary to invest either in the designphase, the operating phase, or both. A whole series of questions emanating fromthis concept have been raised by the authors [17], including: How much should be spent? Is it worth spending any money? Should the reliability be increased, maintained at existing levels, or allowed to

degrade? Who should decidethe utility, a regulator, the customer? On what basis should the decision be made?

The underlying trend in all these questions is the need to determine the worthof reliability in a power system, who should contribute to this worth, and whoshould decide the levels of reliability and investment required to achieve them.

The major discussion point regarding reliability is therefore, "Is it worth it?"[17]. As stated a number of times, costs and economics play a major role in theapplication of reliability concepts and its physical attainment. In this context, thequestion posed is: "Where or on what should the next pound, dollar, or franc beinvested in the system to achieve the maximum reliability benefit?" This can be anextremely difficult question to answer, but it is a vital one and can only be attemptedif consistent quantitative reliability indices are evaluated for each of the alterna-tives.

It is therefore evident that reliability and economics play a major integratedrole in the decision-making process. The principles of this process are discussed inEngineering Systems. The first step in this process is illustrated in Fig. 1.2, whichshows how the reliability of a product or system is related to investment cost; i.e.,increased investment is required in order to improve reliability. This clearly shows

Introduction 13

Investment cost CFie. !? Incremental cost of reliability

the general trend that the incremental cost AC to achieve a given increase inreliability AR increases as the reliability level increases, or, alternatively, a givenincrease in investment produces a decreasing increment in reliability as the reliabil-ity is increased. In either case, high reliability is expensive to achieve.

The incremental cost of reliability, AC/M, shown in Fig. 1.2 is one wayof deciding whether an investment in the system is worth it. However, it doesnot adequately reflect the benefits seen by the utility, the customer, or society.The two aspects of reliability and economics can be appraised more consistentlyby comparing reliability cost (the investment cost needed to achieve a certainlevel of reliability) with reliability worth {the benefit derived by the customerand society).

This extension of quantitative reliability analysis to the evaluation of serviceworth is a deceptively simple process fraught with potential misapplication. Thebasic concept of reliability-cost, reliability-worth evaluation is relatively simple andcan be presented by the cost/reliability curves of Fig. 1.3. These curves show thatthe investment cost generally increases with higher reliability. On the other hand,the customer costs associated with failures decrease as the reliability increases. Thetotal costs therefore are the sum of these two individual costs. This total cost exhibitsa minimum, and so an "optimum" or target level of reliability is achieved. Thisconcept is quite valid. Two difficulties arise in its assessment. First, the calculatedindices are usually derived only from approximate models. Second, there aresignificant problems in assessing customer perceptions of system failure costs. Anumber of studies and surveys have been done including those conducted inCanada, United Kingdom, and Scandinavia. A review of these, together with adetailed discussion of the models and assessment techniques associated withreliability cost and worth evaluation, is the subject of Chapter 13.

14 Chapter 1

system reliability

fig. 1.3 Total reliability costs

1.10 Concepts of dataMeaningful reliability evaluation requires reasonable and acceptable data. Thesedata are not always easy to obtain, and there is often a marked degree of uncertaintyassociated with the required input. This is one of the main reasons why relativeassessments are more realistic than absolute ones. The concepts of data and thetypes of data needed for the analysis, modeling, and predictive assessments arediscussed in Ref. 18. The following discussion is an overview of these concepts.

Although an unlimited amount of data can be collected, it is inefficient andundesirable to collect, analyze, and store more data than is required for the purposeintended. It is therefore essential to identify how and for what purposes it will beused. In deciding which data is needed, a utility must make its own decisions sinceno rigid rules can be predefined that are relevant to all utilities. The factors thatmust be identified are those that have an impact on the utility's own planning,design, and asset management policies.

The processing of this data occurs in two distinct stages. Field data is firstobtained by documenting the details of failures as they occur and the various outagedurations associated with these failures. This field data is then analyzed to createstatistical indices. These indices are updated by the entry of subsequent new data.The quality of this data depends on two important factors: confidence and rele-vance. The quality of the data, and thus the confidence that can be placed in it, isclearly dependent on the accuracy and completeness of the compiled information.

introduction 15

It is therefore essential that the future use to which the data will be put and theimportance it will play in later developments are stressed. The quality of thestatistical indices is also dependent on how the data is processed, on how muchpooling is done, and on the age of the data currently stored. These factors affect therelevance of the indices in their future use.

There is a wide range of data which can be collected and most utilities collectsome, not usually all, of this data in one form or another. There are many differentdata collection schemes around the world, and a detailed review of some of theseis presented in Ref. 18. It is worth indicating that, although considerable similaritiesexist between different schemes, particularly in terms of concepts, considerabledifferences also exist, particularly in the details of the individual schemes. It wasalso concluded that no one scheme could be said to be the "right" scheme, just thatthey are all different.

The review [18] also identified that there are two main bases for collectingdata: the component approach and the unit approach. The latter is considered usefulfor assessing the chronological changes in reliability of existing systems but is lessamenable to the predictive assessment of future system performance, the effect ofvarious alternative reinforcements schemes, and the reliability characteristics ofindividual pieces of equipment. The component approach is preferable in thesecases, and therefore data collected using this approach is more convenient for suchapplications.

1.11 Concluding comments

One point not considered in this book is how reliable the system and its varioussubsystems should be. This is a vitally important requirement and one whichindividual utilities must consider before deciding on any expansion or reinforce-ment scheme. It cannot be considered generally, however, because different sys-tems, different utilities, and different customers al! have differing requirements andexpectations. Some of the factors which should be included in this decision-makingconsideration, however, are:(a) There should be some conformity between the reliability of various pans of the

system. It is pointless to reinforce quite arbitrarily a strong part of the systemwhen weak areas still exist. Consequently a balance is required betweengeneration, transmission, and distribution. This does not mean that the reliabil-ity of each should be equal; in fact, with present systems this is far from thecase. Reasons for differing levels of reliability are justified, for example,because of the importance of a particular load, because generation and trans-mission failures can cause widespread outages while distribution failures arevery localized.

(b) There should be some benefit gained by an improvement in reliability. Thetechnique often utilized for assessing this benefit is to equate the incremental

16 Chapter 1

or marginal investment cost to the customer's incremental or marginal valu-ation of the improved reliability. The problem with such a method is theuncertainty in the customer's valuation. As discussed in Section 1.9, thisproblem is being actively studied. In the meantime it is important for individualutilities to arrive at some consistent criteria by which they can assess thebenefits of expansion and reinforcement schemes.It should be noted that the evaluation of system reliability cannot dictate the

answer to the above requirements or others similar to them. These are managerialdecisions. They cannot be answered at all, however, without the application ofquantitative reliability analysis as this forms one of the most important inputparameters to the decision-making process.

In conclusion, this book illustrates some methods by which the reliability ofvarious parts of a power system can be evaluated and the types of indices that canbe obtained. It does not purport to cover every known and available technique, asthis would require a text of almost infinite length. It will, however, place the readerin a position to appreciate most of the problems and provide a wider and deeperappreciation of the material that has been published [6-11] and of that which will,no doubt, be published in the future.

1.12 References

1. Billinton, R., Allan, R. N., 'Power system reliability in perspective', IEEJ.Electronics Power, 30 (1984), pp. 231-6.

2. Lyman, W. J., 'Fundamental consideration in preparing master system plan',Electrical World, 101(24) (1933), pp. 788-92.

3. Smith, S. A., Jr., Spare capacity fixed by probabilities of outage. ElectricalWorld, 103 (1934), pp. 222-5.

4. Benner, P. E., 'The use of theory of probability to determine spare capacity',General Electric Review, 37(7) (1934), pp. 345-8.

5. Dean, S. M., 'Considerations involved in making system investments forimproved service reliability', EEJ Bulletin, 6 (1938), pp. 491-96.

6. Billinton, R., 'Bibliography on the application of probability methods inpower system reliability evaluation', IEEE Transactions, PAS-91 (1972), pp.649-6CX

7. IEEE Subcommittee Report, 'Bibliography on the application of probabilitymethods in power system reliability evaluation, 1971-1977', IEEE Transac-tions, PAS-97 (1978), pp. 2235-42.

8. Allan, R. N., Biliinton, R., Lee, S. H., 'Bibliography on the application ofprobability methods in power system reliability evaluation, 1977-1982',IEEE Transactions, PAS-103 (1984), pp. 275-82.

introduction 17

9. Allan, R. N., Biliinton, R.. Shahidehpour, S, M., Singh, C, 'Bibliography onthe application of probability methods in power system reliability evaluation,1982-1987', IEEE Trans. Power Systems, 3 (I988), pp. 1555-64.

10. Allan, R. N., Biliinton, R., Briepohl, A. M, Grigg, C. H., 'Bibliography onthe application of probability methods in power system reliability evaluation,1987-199 \\IEEE Trans. Power Systems, PWRS-9(1)(1994).

11. Biliinton, R., Allan, R. N., Salvaderi. L. (eds.). Applied Reliability Assessmentin Electric Power Systems, IE EE Press, New York (1991).

12. Calabrese, G., 'Generating reserve capability determined by the probabilitymethod'. ALEE Trans. Power Apparatus Systems, 66 (1947), 143950.

13. Watchorn, C. W., 'The determination and allocation of the capacity benefitsresulting from interconnecting two or more generating systems', AIEE Trans.Power Apparatus Systems, 69 (1950), pp. 1180-6.

14. Biliinton, R., Li, W., Reliability Assessment of Electric Power Systems UsingMonte Carlo Methods, Plenum Press, New York (1994).

15. EPRI Report, 'Composite system reliability evaluation: Phase 1scopingstudy', Final Report, EPRI EL-5290, Dec. 1987.

16. Sugarman, R., 'New York City's blackout: a S350 million drain', IEEESpectrum Compendium, Power Failure Analysis and Prevention, 1979, pp.48-50.

17. Allan, R. N., Biliinton, R., 'Probabilistic methods applied to electric powersystemsare they worth it', IEE Power Engineering Journal, May (1992),121-9.

18. CIGRE Working Group 38.03, 'Power System Reliability AnalysisAppli-cation Guide, CIGRE Publications, Paris (1988).

2 Generating capacitybasicprobability methods

2.1 Introduction

The determination of the required amount of system generating capacity to ensurean adequate supply is an important aspect of power system planning and operation.The total problem can be divided into two conceptually different areas designatedas static and operating capacity requirements. The static capacity area relates to thelong-term evaluation of this overall system requirement. The operating capacityarea relates to the short-term evaluation of the actual capacity required to meet agiven load level. Both these areas must be examined at the planning level inevaluating alternative facilities; however, once the decision has been made, theshort-term requirement becomes an operating problem. The assessment of operat-ing capacity reserves is illustrated in Chapter 5.

The static requirement can be considered as the installed capacity that must beplanned and constructed in advance of the system requirements. The static reservemust be sufficient to provide for the overhaul of generating equipment, outages thatare not planned or scheduled and load growth requirements in excess of theestimates. Apractice that has developed over many years is to measure the adequacyof both the planned and installed capacity in terms of a percentage reserve. Animportant objection to the use of the percentage reserve requirement criterion is thetendency to compare the relative adequacy of capacity requirements provided fortotally different systems on the basis of peak loads experienced over the same timeperiod for each system. Large differences in capacity requirements to provide thesame assurance of service continuity may be required in two different systems withpeak loads of the same magnitude. This situation arises when the two systems beingcompared have different load characteristics and different types and sizes ofinstalled or planned generating capacity.

The percentage reserve criterion also attaches no penalty to a unit because ofsize unless this quantity exceeds the total capacity reserve. The requirement that areserve should be maintained equivalent to the capacity of the largest unit on thesystem plus a fixed percentage of the total system capacity is a more valid adequacycriterion and calls for larger reserve requirements with the addition of larger unitsto the system. This characteristic is usually found when probability techniques areused. The application of probability' methods to the static capacity problem provides18

Generating capacitybasic probability methods 19

an analytical basis for capacity planning which can be extended to cover partial orcomplete integration of systems, capacity of interconnections, effects of unit sizeand design, effects of maintenance schedules and other system parameters. Theeconomic aspects associated with different standards of reliability can be comparedonly by using probability techniques. Section 2.2.3 illustrates the inconsistencieswhich can arise when fixed criteria such as percentage reserves or loss of the largestunit are used in system capacity evaluation.

A large number of papers which apply probability techniques to generatingcapacity reliability evaluation have been published in the last 40 years. Thesepublications have been documented in three comprehensive bibliographies pub-lished in 1966,1971, and 1978 which include over 160 individual references [ 1-3].The historical development of the techniques used at the present time is extremelyinteresting and although it is rather difficult to determine just when the firstpublished material appeared, it was almost fifty years ago. Interest in the applicationof probability methods to the evaluation of capacity requirements became evidentabout 1933. The first large group of papers was published in 1947. These papersby Calabrese [4], Lyman [5]. Seelye [6] and Loane and Watchorn [7] proposed thebasic concepts upon which some of the methods in use at the present time are based.The 1947 group of papers proposed the methods which with some modificationsare now generally known as the 'loss of load method', and the 'frequency andduration approach'.

Several excellent papers appeared each year until in 1958 a second large groupof papers was published. This group of papers modified and extended the methodsproposed by the 1947 group and also introduced a more sophisticated approach tothe problem using 'game theory' or 'simulation' techniques [8-10]. Additionalmaterial in this area appeared in 1961 and 1962 but since that time interest in thisapproach appears to have declined.

A third group of significant papers was published in 1968/69 by Ringlee, Woodet al. [1115]. These publications extended the frequency and duration approachby developing a recursive technique for model building. The basic concepts offrequency and duration evaluation are described in Engineering Systems.

It should not be assumed that the three groups of papers noted above are theonly significant publications on this subject. This is not the case. They do, however,form the basis or starting point for many of the developments outlined in furtherwork. Many other excellent papers have also been published and are listed in thethree bibliographies [13] referred to earlier.

The fundamental difference between static and operating capacity evaluationis in the time period considered. There are therefore basic differences in the dataused in each area of application. Reference [16] contains some fundamentaldefinitions which are necessary for consistent and comprehensive generating unit-reliability, availability, and productivity. At the present time it appears that the lossof load probability or expectation method is the most widely used probabilistictechnique for evaluating the adequacy of a given generation configuration. There

20 Chapter 2

Fig. 2.1 Conceptual tasks in generating capacity reliability evaluation

are, however, many variations in the approach used and in the factors considered.The main elements are considered in this chapter. The loss of energy expectationcan also be decided using a similar approach, and it is therefore also included inthis chapter. Chapter 3 presents the basic concepts associated with the frequencyand duration technique, and both the loss of load and frequency and durationmethods are detailed in Chapter 4 which deals with interconnected system reliabil-ity evaluation.

The basic approach to evaluating the adequacy of a particular generationconfiguration is fundamentally the same for any technique. It consists of three partsas shown in Fig. 2.1.

The generation and load models shown in Fig. 2.1 are combined (convolved)to form the appropriate risk model. The calculated indices do not normally includetransmission constraints, although it has been shown [39] how these constraints canbe included, nor do they include transmission reliabilities; they are therefore overallsystem adequacy indices. The system representation in a conventional study isshown in Fig. 2.2.

The calculated indices in this case do not reflect generation deficiencies at anyparticular customer load point but measure the overall adequacy of the generationsystem. Specific load point evaluation is illustrated later in Chapter 6 under thedesignation of composite system reliability evaluation.

Total systemgeneration Total system load

Fig. 2.2 Conventional system model

G-r-.aritidfc jpacrtybasic probability methods 21

2,2 The generation SYstem model

2.2.1 Generating unit unavailability

The basic generating unit parameter used in static capacity evaluation is theprobability of finding the unit on forced outage at some distant time in the future.This probability was defined in Engineering Systems as the unit unavailability, andhistorically in power system applications it is known as the unit forced outage rate(FOR). It is not a ratein modern reliability terms as it is the ratio of two time values.As shown in Chapter 9 of Engineering Systems,

Unavailability (FOR) = C/= = -L = - = ^A, + ^ m+r T u

[down time] 2.1(a)Zfdown time] + S[up time]

Availability = A=-A.

[up time] 2.1(b)Ifdown time] + Z[up time]

where X = expected failure rateu = expected repair ratem = mean time to failure = MTTF = I/A.r = mean time to repair = MTTR = 1/u

m + r= mean time between failures = MTBF = l/f/= cycle frequency = l/TT= cycle time = l/f.

The concepts of availability and unavailability as illustrated in Equations2.1 (a) and (b) are associated with the simple two-state model shown in Fig. 2.3(a).This model is directly applicable to a base load generating unit which is eitheroperating or forced out of service. Scheduled outages must be considered separatelyas shown later in this chapter.

In the case of generating equipment with relatively long operating cycles, theunavailability (FOR) is an adequate estimator of the probability that the unit undersimilar conditions will not be available for service in the future. The formula doesnot, however, provide an adequate estimate when the demand cycle, as in the caseof a peaking or intermittent operating unit, is relatively short. In addition to this,the most critical period in the operation of a unit is the start-up period, and incomparison with a base load unit, a peaking unit will have fewer operating hoursand many more start-ups and shut-downs. These aspects must also be included inarriving at an estimate of unit unavailabilities at some time in the future. A working

22 Chapter 2

(a)

(b)

F/g. 2.3 (a) Two-state mode! for a base load unit(b) Four-state model for planning studies

7" Average reserve shut-down time between periods of needD Average in-service time per occasion of demandjs Probability of starting failure

group of the IEEE Subcommittee on the Application of Probability Methodsproposed the four-state model shown in Fig. 2.3(b) and developed an equationwhich permitted these factors to be considered while utilizing data collected underthe conventional definitions [17].

The difference between Figs 2.3(a) and 2.3(b) is in the inclusion of the 'reserveshutdown' and 'forced out but not needed' states in Fig. 2.3(b). In the four-statemodel, the 'two-state' model is represented by States 2 and 3 and the two additionalstates are included to model the effect of the relatively short duty cycle. The failureto start condition is represented by the transition rate from State 0 to State 3.

This system can be represented as a Markov process and equations developedfor the probabilities of residing in each state in terms of the state transition rates.These equations are as follows:

where

A = (m + ps)

Geoerating capacitybasit probability mets

p,=-

3 A

The conventional FOR = -

i.e. the 'reserve shutdown' state is eliminated.In the case of an intermittently operated unit, the conditional probability that

the unit will not be available given that a demand occurs is P, where

l / T ) + Ps/TThe conditional forced outage rate P can therefore be found from the generic

data shown in the model of Fig. 2.3(b). A convenient estimate of P can be madefrom the basic data for the unit.

Over a relatively long period of time,* service time ST

2 available time + forced outage time AT + FOT

v '

3/ AT + FOT

Defining

where r = 1 f\i.The conditional forced outage rate P can be expressed as

^3) /(FOT)P3) Sr+/(FOT)

The factor/serves to weight the forced outage time FOT to reflect the timethe unit was actually on forced outage when in demand by the system. The effectof this modification can be seen in the following example, taken from Reference[17].

Average unit data

24 Chapter 2

Service time . ST = 640.73 hoursAvailable time = 6403.54 hours

No. of starts = 38.07No. of outages = 3.87

Forced outage time FOX = 205.03 hoursAssume that the starting failure probability Ps = 0

ft A 6403.54 , ,_.^=^OT~ =168 hours

A 205.03 .r - ,

0- = 53 hoursJ.O/

A 640.73 ,,,,m = .

0_ = 166 hoursJ.O/

Using these values

,_P_,_J__1 /( ' l , * | mf~ 53+155.2 / i 16.8^ 53 + 151.2 I-03

The conventional forced outage rate = : x 100640.73 + 205.03

= 24.24%

The conditional probability P = ' : x 100F 640.73 + 0.3(205.03)

= 8.76%

The conditional probability P is clearly dependent on the demand placed uponthe unit. The demand placed upon it in the past may not be the same as the demandwhich may exist in the future, particularly under conditions of generation systeminadequacy. It has been suggested [18] that the demand should be determined fromthe load model as the capacity table is created sequentially, and the conditionalprobability then determined prior to adding the unit to the capacity model.

2.2.2 Capacity outage probability tables

The generation model required in the loss of load approach is sometimes known asa capacity outage probability table. As the name suggests, it is a simple array ofcapacity levels and the associated probabilities of existence. If all the units in thesystem are identical, the capacity outage probability table can be easily obtainedusing the binomial distribution as described in Sections 3.3.7 and 3.3.8 of Engi-neering Systems. It is extremely unlikely, however, that all the units in a practical

Ganerating sapaertybasic probability methods 25

Table 2.1

Capacity out of serviceOMW3MW6MW

Probability0,96040.03920.00041.0000

system will be identical, and therefore the binomial distribution has limited appli-cation. The units can be combined using basic probability concepts and thisapproach can be extended to a simple but powerful recursive technique in whichunits are added sequentially to produce the final model. These concepts can beillustrated by a simple numerical example.

A system consists of two 3 MW units and one 5 MW unit with forced outagerates of 0.02. The two identical units can be combined to give the capacity outageprobability table shown as Table 2.1.

The 5 MW generating unit can be added to this table by considering that it canexist in two states. It can be in service with probability 1 0.02 = 0.98 or it can beout of service with probability 0.02. The two resulting tables (Tables 2.2,2.3) aretherefore conditional upon the assumed states of the unit. This approach can beextended to any number of unit states.

The two tables can now be combined and re-ordered (Table 2.4). The prob-ability value in the table is the probability of exactly the indicated amount ofcapacity being out of service. An additional column can be added which gives thecumulative probability. This is the probability of finding a quantity of capacity onoutage equal to or greater than the indicated amount.

The cumulative probability values decrease as the capacity on outage in-creases: Although this is not completely true for the individual probability table,the same general trend is followed. For instance, in the above table the probabilityof losing 8 MW is higher than the probability of losing 6 MW. In each case onlytwo units are involved. The difference is due to the fact that in the 8 MW case, the3 MW loss contribution can occur in two ways. In a practical system the probabilityof having a large quantity of capacity forced out of service is usually quite small,

Table 2.2 5 MW unit in service

Capacity ota Probability

0 + Q = OMW (0.9604) (0.98) = 0.9411923+0=3MW " (0.0392) (0.98) = 0.0384166 + 0 = 6MW (0.0004) (0.98) = 0.000392

0.980000

26 Chapter 2

Table 2.3 5 MW unit out of service

Capacity out Probability

0 + 5 = 5MW (0.9604) (0.02) = 0.0192083+5 = 8MW (0.0392) (0.02) = 0.0007846 + 5 = 1 1 MW (0.0004) (0.02) = 0.000008

0.020000

as this condition requires the outage of several units. Theoretically the capacityoutage probability table incorporates all the system capacity. The table can betruncated by omitting all capacity outages for which the cumulative probability isless than a specified amount, e.g. KT8. This also results in a considerable saving incomputer time as the table is truncated progressively with each unit addition. Thecapacity outage probabilities can be summated as units are added, or calculateddirectly as cumulative values and therefore no error need result from the truncationprocess. This is illustrated in Section 2.2.4. In a practical system containing a largenumber of units of different capacities, the table will contain several hundredpossible discrete capacity outage levels. This number can be reduced by groupingthe units into identical capacity groups prior to combining or by rounding the tableto discrete levels after combining. Unit grouping prior to building the tableintroduces unnecessary approximations which can be avoided by the table roundingapproach. The capacity rounding increment used depends upon the accuracydesired. The final rounded table contains capacity outage magnitudes that aremultiples of the rounding increment. The number of capacity levels decreases asthe rounding increment increases, with a corresponding decrease in accuracy. Theprocedure for rounding a table is shown in the following example.

Two 3 MW units and one 5 MW unit with forced outage rates of 0.02 werecombined to form the generation model shown in Table 2.4. This tabie, when

Table 2.4 Capacity outage probability table forthe three-unit system

Capacity outof service

02

Generating capacitybasic probability methods 27

Table 2.5

Capacityon outagei MW) Individual probability0 0.941192 + |(0.0384!6) =0,95655845 0.019208 + |(0.038416)

+1(0.000392) +1

28 Chapter!

system 1,24 x 10 MW units each having a FOR of 0.01system 2, 12 x 20 MW units each having a FOR of 0.01system 3, 12 x 20 MW units each having a FOR of 0.03system 4,22 x ] 0 MW units each having a FOR of 0.01

All four systems are very similar but not identical. In each system, the unitsare identical and therefore the capacity outage probability table can be easilyconstructed using the binomial distribution. These arrays are shown in Table 2.6

Table 2.6 Capacity Outage Probability Tables for systems 1-4

System I Capacity (MW) ProbabilityOut

01020304050System

Out0

20406080System

Oil!

020406080

100120

Svstem

Out0

1020304050

In

240230220210200190

2 Capacity (MW)In

240220200180160

3 Capacity (MW)In

240220200180160140120

4 Capacity (MW)

In

220210200190180170

Individual0.7856780.1904670.0221250.0016390.0000870.000004

Individual0.8863840.1074410.0059690.0002010.000005

Individual

0.6938410.2575090.0438030.0045160.0003140.0000160.000001

Individual

0.8016310.1781400.0188940.0012720.0000610.000002

Cumulative1.0000000.2143220.0238550.0017300,0000910.000004

Probability

Cumulative1.0000000.1136160.0061750.0002060.000005

Probability

Cumulative1.0000000.3061590.0486500.0048470.0003310.0000170.000001

Probabi/in'

Cumulative1.0000000.1983690.0202290.0013350.0000630.000002

Generating cap^tyba*=c probability methods 29

and have been truncated to a cumulative probability of 10". It can be seen that aconsiderable number of capacity outage states have oeen deleted using this trunca-tion technique.

The load level or demand on the system is assumed to be constant. If the riskin the system is defined as the probability of not meeting the load, then the true riskin the system is given by the value of cumulative probability corresponding to theoutage state one increment below that which satisfies the load on the system. Thetwo deterministic risk criteria can now be compared with this probabilistic risk asin Sections (a) and (b) following.(a) Percentage reserve marginAssume that the expected toad demands in systems 1,2,3 and 4 are 200,200,200and 183 MW respectively. The installed capacity in all four cases is such that thereis a 20% reserve margin, i.e. a constant for all four systems. The probabilistic ortrue risks in each of the four systems can be found from Table 2.6 and are:

risk in system 1 = 0.000004risk in system 2 = 0.000206nsk in system 3 = 0.004847risk in system 4 = 0.000063These values of risk show that the true risk in system 3 is 1000 times greater

than that in system I. A detailed analysis of the four systems will show that thevariation in true risk depends upon the forced outage rate, number of units and loaddemand The percentage reserve method cannot account for these factors andtherefore, although using a 'constant' risk criterion, does not give a consistent riskassessment of the system.

(b) Largest unit reserveAssume now that the expected load demands in systems 1,2,3 and 4 are 230,220,220 and 210 MW respectively. The installed capacity in all four cases is such thatthe reserve is equal to the largest unit which again is a constant for all the systems.In this case the probabilistic risks are:

risk in system 1 = 0.023855risk in system 2 = 0.006175risk in system 3 = 0.048650risk in system 4 = 0.020229The variation in risk is much smaller in this case, which gives some credence

to the criterion. The ratio between the smallest and greatest risk levels is now 8:1and the risk merit order has changed from system 3-2-4-1 in the case of percentagereserve' to 31-4-2 in the case of the 'largest unit' criterion.

It is seen from these comparisons that the use of deterministic or 'ruie-of-thumb' criteria can lead to very divergent probabilistic risks even for systems thatare very similar. They are therefore inconsistent, unreliable and subjective rnethodsfor reserve margin planning.

30 Chapter 2

2.2.4 A recursive algorithm for capacity model building

The capacity model can be created using a simple algorithm which can also be usedto remove a unit from the model [19]. This approach can also be used for amulti-state unit, i.e. a unit which can exist in one or more derated or partial outputstates as well as in the fully up and fully down states. The technique is illustratedfor a two-state unit addition followed by the more general case of a multi-state unit.

Case 1 No derated states

The cumulative probability of a particular capacity outage state of ^ MW after aunit of capacity C MW and forced outage rate U is added is given by

P(X) = (1 - U)F(X) + (U)F(X- C) (2.2)where P"(X) and P(X) denote the cumulative probabilities of the capacity outagestate of JfMW before and after the unit is added. The above expression is initializedby setting P(X) = 1.0 for X< 0 andF(X) = 0 otherwise.

Equation (2.2) is illustrated using the simple system shown in Table 2.7. Eachunit in Table 2.7 has an availability and unavailability of 0.98 and 0.02 respectively(Equation 2.1).

The system capacity outage probability is created sequentially as follows:Step I Add the first unit

P(0) =(1-0.02)(1.0) +(0.02X1-0) =1.0P(2S) = (1-0.02)(0) + (0.02)(1.0) =0.02

Step 2 Add the second unitP(0) = (1 - 0.02X 1.0) + (0.02)(1.0) = 1,0/3(25) = (1-0.02)(0.02) + (0.02)(1.0) =0.0396P(SQ) = (1-0.02)(0) + (0.02)(0.02) =0.0004

Step 3 Add the third unitP(0) =(1-0.02)(1.0) + (0.02)(1.0) =1.0P(25) = (1 -0.02)(0.0396) + (0.02X1.0) = 0.058808P(50) = (1 -0.02)(0.0004) + (0.02)(1.0) = 0.020392P(75) - (1-0.02)(0) +(0.02)(0.0396) =0.000792P(100) =(1-0.02)(0) +(0.02)(0.0004) =0.000008

The reader should utilize this approach to obtain Table 2.4.

Table 2.7 System data

Unit no. Capacity (MW) Failure rate (f / day) Repair rate (r / day)123

252550

0.010.010.01

0.490.490.49

Generating capacitybasic probability methods 31

Table 2,8 50 MW unitthree-state representation

State123

Capacity ou!0

2050

State probability (pj0.9600.0330.007

Case 2 Derated states includedEquation (2,2) can be modified as follows to include multi-state unit repre-sentations.

" W= Z/A*-Qi=i

where n = number of unit statesCj = capacity outage of state j for the unit being addedPi = probability of existence of the unit state i.

when n = 2, Equation (2.3) reduces to Equation (2.2).Equation (2.3) is illustrated using the 50 MW unit representation shown in Table2.8.

If the two-state 50 MW unit in the previous example is replaced by thethree-state unit shown in Table 2.8, Step 3 becomes

P(0) =(0.96X1.0) +(0.033)(1.0) +(0.007X1.0) =1.0P(20) =(0.96)(0.0396) +(0.033X1-0) +(0.007)(1.0) =0.078016P(25) = (0.96)(0.0396) + (0.033)(0.0396)+ (0.007)(1.0) =0.0463228P(45) =(0.96)(0.0004) +(0.033)(0.0396)+(0.007)(LO) =0.0086908P(50) = (0.96)(0.0004) +(0.033X0.0004)+(0.007X1-0) =0.0073972P(70) =(0.96X0) + (0.033X0.0004)+(0.007X0.0396) =0.0002904P{75) =(0.96X0) +(0.033X0) +(0.007X0.0396) =0.0002772P( 100) =(0.96X0) + (0.033)(0) +