-
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)
+