Power system reliability evaluation

Power system reliability evaluation A D Patton, A K A y o u b and C Singh Flectric Power Institute, Texas A & M University, Colle~le Station, Texas 77843, U S A

..i review is provMed (d" indices attd methods fi,r evaluating pcm'er system reliability per/bmTance Itsiltg probability flmory. Particular emphasis is placed on deveh,pment shu'e about 1960, when a stochastic process view o f p o w e r systems was u s e d / o r reliability evaluatbm. The present status o f such evaluation is given, and desirable areas o f turther research are outlbwd.

I. I n t r o d u c t i o n Reliability evaluation methods based on probability theory are useful tools in the planning and design of power systems. Such methods permit computation of quantitative indices of system reliability performance from tire reliability perlormance data of the system components. The present paper reviews these indices and methods, which are useful in planning and design studies for generation, transmission and distribution. Also reviewed are methods of determining an acceptable, adequate, or economic level o f system reliability. The paper concludes with an outline of promising a r e a s o f ]ill Life research. Twu areas which arc important but not covered in tiffs i)a]:oer ale:

•

component data for use in system reliability calculations,

• security assessment of system operation using probabilistic methods.

II. R e l i a b i l i t y indices Regardless of the purpose of a reliability study, it is essential to establish one or more reliability measures or indices upon which .iudgem'ents and decisions can be based. The reliability sludy itself is then concerned with the appropriate models and data needed for computation of those indices for the particular problem at hand. In general, a meaningful and useful index of system reliability should: •

be measurable lTom the historical records of an operating sySlClll

Vol 1 No3October 1979

•

be calculable for a proposed system using data available on the reliability performance of system components

•

be respm]sive in a predictable and consistent manner to differences in study alternatives.

A number of indices meet these general criteria and have been applied with success. Some o f the practical reliability indices fall into three basic classes which are now discussed.

I I. 1. Probability of system failure Generally speaking, the probability of system failure is the long-term average proportion of total lime for which lhe system is in a stale of failure, i.e. not performing satisfactorily. This index of reliability has been used for both generation and transmission/distribution, but has found its main application m generation capacity studies. In generation studies, system failure is frequently defined as when available installed capacity is insufficient to supply the peak load of a day. When failure is so defined, tile probability of system failure is the long-term average proportion o f days on which the daily peak load exceeds lhe available installed capacity. A closely related index is tile expected or long-term average number of days on which daily peak load exceeds tile available installed capacity. This latter index is commonly called loss-of-load probability (LOLP), even though il is an expectation and not a probability. LOLP is probably the most commonly used reliability index in generation capacity studies. It should be noted here that LOLP gives no information on tile magnitude of load lost m a capacity shortage event. Further, this index, as usually computed, ignores the effects of operating policies and constraints and therefore does not give the actual risk of capacity shortage which would be experienced in operation.

111 studies of Iransmission/distribution systenrs, failure is usually defined as loss of continuity of supply, but may also be defined as events o f low wqtage or component ovelload dmingwhich c{ ' i n u i t y i s n m losl. Time in transmis-

0142-0615/79/030139-12g02.00

d) 1979 IPC Business Press

139

sion/distributitm studies is usually measured continuously rather than on a daily basis as in generation studies. Thus, the probability of system faihue in transmission/distribution systems is usually the hmg-term average proportion of total time tha( lhe system is in a slale (/1 failure.

II I. 1. Combinatorial methods Ihc Iil>,t melh(yds for reli;ibilit,, cv:lhl:lti()l/ th:lt '.,,,.' ' devel(>ped were based (m c(mflfin'atori:)] c{mc'epts ()i l?,,i,-

11.2. Frequency and duration of system failure The long-term average frequency (>]"system failures and the

lion e l days per year (el which available gc/e at - c~ll)acil~ is less Ihan daily peak load ,)r as the expected numhL) ,)l

average duration of such events provide a very useful reliability index pair. It is generally believed that, from the consumer's point of view, this index pair is more meaningful than the single index of system failure probability. Further, the frequency and duration index pair allows more detailed study of tile reliability consequences of alternatives than does the probability index.

&LVS pe~ ',car tm which capacity is insufficient the expected magnitude of unserved energ~ demaml per year: and lhc probalfility (>) prt)por(i(m ~1 time a distributitm s3, >,ten/ will Iail to supply service to a particular coi/SUlller. The lll(>J

ability t h c o r v . S(>lUe lit l h e s e h i e ] h o d s :llst) use the c(qlccp', (~l mathematical expectation. These lundamentat clement~ t+l" probability theory allow comput:ltion of such faluilizll reliability imlices as: L O t P , expressed either :p, the pi,q>,,I-

COlI/lUOllIV used c o m b i n a t o r i a l e x p r e s s ] o i l s a l C: P( A I U /| 2 ) = to( A I ) + P( A 2 )

i l , l I and .I 2 a l e

Frequency and duration are by far tile most popular and commonly used indices in transmission/distribution studies. It is usual to compute frequencies both for short-duration momentary system failures and for sustained failures. System performance is normally measured :it a particular bus or point of consumption bill, if desired, may be integrated over all consumption points of a syslem IO give a general measure of network reliability. As mentioned previously, failure in transmission/distribution systems is most otte~ defined as loss of continuity, but iilay also be defined as low voltage or overload of components. Only in recenl years have frequency and duration indices become practical for generation capacity studies, and their use is probably not widespread as yet. It is expected that. as their benefits are appreciated, these indices will feature

increasingly in generati(m studies. 11.3. Consequences of system failure Two important indices fall into the category t)f consequences of system failure. The first is the expected or brag-term average amount of denranded energy not supplied due to failure. This expected energy loss index has obvious physical importance and appeal and has seen application in both generation and transmission/distribution studies. The index may be expressed in energy units or in pc) unit of tile total energy demanded. Sometimes an energy index of reliability is used which is unity minus the pet unit expected energy not supplied.

The second index to be mentioned here is the expected lossof-load magnitude (XLOL), given that a load loss event (capacity deficiency) has occurred. This index is considered a supplentent (o the LOLP index. Both expected energy loss and XLOL are nornmlly calculated assuming steady-state system conditions. That is, excess load shedding which might occur in practice after loss of a bh)ck of capacity is not considered. Ahernat ive, dymunic, wdues of these indices have been suggested ,,vhicll recognize the influence of system physical dynanrics on rite X[ e L , given a load loss event, and on tile expected energy not st, pplied.

III.

Mathematical

methods

This section describes rite basic concepts of probability theory which have been applied in power system reliability evaluation.

140

( } )

mtua Iv exclusive

)D( A I ' : /1 2 } = ff(A I }if( "] 2 )

( ?)

if A I and .,1, are independent P(A ) = I

P(,t )

4.~ )

In the above expressions,P(A i) is ttle p r o b a b i l i t y o l evc))l

,.1 I,/~ is the complement of event A, tile symb.,>l (J denute,, unioll :111,.1lllealls q,>l", alld the symbol ~ denotes inters e c t i o n aud lneallS "and'. More general expressions lo/ equations (1) and (2) are required if the appended qualilications do not bold. but tim given expressions together with their qualifications are those used in practice for reliabilit\ evaluation. File independence qualification o1 equatiou ( '1 should be commented tin, Ilowever, for it translates in reliability evaluaiion to the assumption that system con> ponents fail and are repaired independently (11 each other. This assumption is widely' believed to be reasonable m most cases, but should always be viewed with caution, because lack of actual independence may result in drastic comput;i tiollal elt{llS.

The expected value of a random variable, such as tile amount of energy not served pet year, is simply the hmf_,term average value of the variable. Thus, presuming a discrete random variable X, Ihe expected value of X, t:'(X), is giveH by

t:'(X)= ~-

x PI.v)

(4)

all .v

where_v is a particular value t)l the randonl variahle .\ and P(x) is the probability of thai value. 111.2. Stochastic processes The hchaviour t)f stochastic processes is c(mven ien[I} described iu terms o f frequency h:dance. The simplest ',rod IUOSl powerful coucepl o1 frequency is all expectation ~)I interstale transition rate. i This definition o f frequency

simplifies tile computation of average and time-specili~ frequency in both Markovian and non-Markovian systems, Let Y+ and Y - be distain] subsets o f the state space Y. These subsets can be used to define events of interest such ;is system failure and system success. Let J+(/) be the timespecific frequency of encountering tile subset Y~. This is the expected rate at which the subset Y~ is encountered ;~1 time 1. For ),t ,()+,f÷(t)At represents the expected)lumber o1 times the subset Y~ is ei~countered in the interval (r. t + All. Let I/El(t) be the time-specific frequency el encountering s t a ( e / f r o m state L This is the expected rate at which s t a t e / i s encountered from state L Let

Electrical Power & Energy Systems

1',(I) bc the tnobability of the system being in state i at time z, fol the given initial condition, leer P+(I) be the probability ot tile system being in subset Y*, at time / for tire given in~itial condition, equal to ",.2 +Pi(l)" and let Z ( I ) be tim state o f tile system at t i m e I.

The encotmter rate of s t a t e / f r o m stale i can be represented by a discrete random variable ~ii(l) such that

I '~i/ I : i / ( l ) = ~ii

if Z(/) # i p~-

t .,.~

A very useful concept in reliability studies is that of the equivalent transition rate. The equivalent transition rate from a subset X a to subset Xp~ is given by )t,~, = F r { a ~ b" l } / P . ( t )

= E

= i} + O . P { Z ( t )

IZ(:)

P~ = P+ (t)

¢il

= )~ijPi(t)

Y~ P~(t)x~j

i~ X a /~ X h

(5)

and

The states being mutually exclusive

e.(t) = Y~ il }

+

l:

(1 3)

where t')'{a ~- b: l} is the frequency of encountering X/, from X a at time t

if Z ( : ) = i

fl~J(/l = t 0

where

i' Y

/=

P~(t)

i{: X a

.F

The equivalent transition rate concept has been used for deriving conditions of mergeability, i

and

./I:)= Y.

~

Pi(:)Z,.i

(7)

i ~ Y + /~ }

h can be further proved ' that d

111.3. Applications to p o w e r systems

(g)

:-(:)

-- &(:)::~(:)

dl Equation (8) describes the frequency balance between Y+ and Y . It states that the frequency of encountering a state (,~r subset of states) minus the frequency of exiting from the s/ate (or subset of states) equals the rate of change of probability of being in the state (or subset of states). Equation (8) can be used to write the state differential equations which m the familiar matrix form are d

- - P(:) = APu)

(9)

Jl

where A is the transition rate matrix and P(t) is the vector of stale probabilities. As t ~" oo. equation (8) yields

::+:

X

j,

)-

.

.

.

.

z

,oi

/.(y

Equation (10) describes the frequency equilibrium of }I+ v, Jth the rest of the state space: it states that under steady ~tate. the frequency of encountering Y+ equals the frequency ,~t encountering the rest of state space from Y+. Equations {(~) and (10) are the basic relationships tk)r calculating the time specific and steady state frequency of Y+. it should be m~ted that, although under steady state J+ and ) are numerically equal, conceptually they are different. The relati~mships for the mean cycle time and mean duration can m~w he derived. As [~oo

j~(/, ! + T)-

T f'+

where ] , ( t , t + T) is the expected number of times Y+ is encountered m the interval (I, I + 7") T, - T / / : J : , t

+ T) = I/t~

(ll)

where 7"+ is the expected time between successive encounters of Y+, i.e. tile mean cycle time. Also d+ is the mean duration of Y+, i.e. expected time of stay in Y+ in one cycle, and d~ = T~P~ - P~/];

Vol 1 No 3 O c t o b e r 1 9 7 9

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The first application of stochastic processes and of tire Markov process was suggested in 10592 when the application was to the modelling of generator 'up' and 'down' times for use in a Monte Carlo simulation approach to reliability evaluation. A 1963 paper 3 gave tire formation of the Markov process in terms of a set of linear I st-order differential equations and suggested the Markov process as a model lklr transmission and distribution systems. Papers in 1 9 ( 3 g 4 and 1060 s further clarified and illustrated applications of the Markov process to power system reliability modelling and showed how to compute steady state probabilities, mean time to failure, and mean time between failures (or other states) through mathematical operations utilizing the transition rate matrix of the Markov process. A very importaut series of papers beginning in 19686 10 developed the application of the Markov process to the modelling of generation capacity states and of load for computing frequency and duration indices in generation capacity studies. These papers, extending and generalizing the work of earlier autllors, I1 t5 showed that the frequency of a system state can be given in terms of the steady-state probability and the departure rates from the state. They also developed computationally efficient algorithms for computing the frequencies of simple or compound (cumulative) system states by adding one generator at a time to the system. A 1970 paperl(' observed that the results of Markov models hold in the mean sense even if the normal Markov restriction requiring state residence times to be exponentially distributed does not hold, providing state residence time distributions are not time varying. This observation greatly extends the field of application of the Markov process because l[]any c o m p o n e l ] l stale residence t illles :ire, in fact, not exponentially distributed. A 1972 paper iv discussed methods fin modelling systems with time-varying state residence time distributions. A particularly appealing method discussed in this and a later paper is the °method of stages' through which the non-Markov process is converted to air equivalent Markov process. The concept of frequency for compound states was formalized in a 1972 paper, iS A 1973 papern9 presented the concept of frequency of a minimal cut state which is very useful in tim computation of frequency and duration indices in transmission and distribution systems.

141

111.4. Monte Car/o simu/ation Methods for computi,ilg reliability il,idiccs Iall into two categories: analylical m0thiods and Monte ('arlo sinn,ila(ittl,i inethods. Ai,ialytical i,i~0tllods lise tile rules of probcll)iliiy iheoly dh-0ctly to clm'ipt,ite syslein r01iability indic0s: Monic Carlo sJnlt,ilatJo,il I11et],iods create histories of system op0ralion by simulation on :t digital c o m p u l e r , a,i,id th0n conlpuic rcliability indiccs l'ro,il] thcsc sintulatcd hislories clotoperaticm. Thle advantag0 of Menlo Carlo meihlods is thiat thlc t)crlorill;into of syst0111s that are illflt,ioncod by prol~abilislic ev0i,ils cal,i be studied ,,vithl the rain)nit,ill1 of asslnllptioi,is and without explicit iroat,ill0nl of lhe probabilistic laws which ,iinderlie the slochastic i,iatui0 o~ thle systoiil. The prinlary drawback of Morlt0 Carlo sin//,ilation is )he length of COml)UlOr tirrie usually required to simukit0 sufficient hi)stories io produce statistically significant values l\)r reliability indices. Anolh0r drawback l/lay bc all inahiiity io ai,ialyse (hc infhl0nce of changes hi pi-obl0n/pararn0iers withe,ill a full-scale repetition of the simulalion. Two general approaches have seen some application. The fhst al)pro:lch , whlich may be called sequential siinulation, produces a chu-onological or sequential history of alternating "tip" ai,icl "down' times for each compo,i~enl of the system being studied. This is done by drawing r a n d o m 'tip" and ' d o w n ' ti,incs Jrolll the specified distribtltioil el" thiese times. Comp
142

have becn ~,uggesled for :i ilUl/lber o1 applicative>,. ~-' ~ -' : AI presel,il. Ihe use o1 M o n t e ('ailo ii~eth//)ds >,ceil>, lii,>'-,t prevalenl in |{,ill-O[-le.

IV. Generation systems In the literature tm the applicatio,it of probability mcth~d~ to power system reliability evaluation, generative systems arc by Im the most frequently discussed topic. The p~>blcm ,i,isually considered is tile adequacy of installed gencrat ill<'_' capacity, with the i,inplicit assumption that the ability ~)1 t hi<.: system to st,ipply :ill loads is never influenced by a shol/u~c of basic energy resources. Methctds are also available, however, loi sludying systems inwtlving limitcd-cneig}, sources such as hydro plants where generation i,nadcq,ituc 5 l,il;iy ticc,iir throudl,, basic el,)orgy shortages• us well as mst:illed capac'ity deficiencies. In tMs section inelllocls :ire discussed lhiat apply to reliability ovah,ialioi] ii,i iniercomlOCted ,,; stcm~, as well as sii,igl0-ar0a syste,ills, b,i,it ii,i gei;oial the ctt,iesli~m ,)1 transmission l,iCtwork adequacy is not co,i,isideied, l'hu-,, iii elf0cl, '<,isinglc-:.ire:i gcil0iatioll systcill is modelled alid >,Iudicd as a single bus to \vhlich are ci.),ili]ected all ti,ie gcilelatslems arc available: analytical methods and Monic (k,,l(, smmlation n,iethods. Both approaches have been applied in tile lISA, h,ul analytical methods have been and remain lh<: l,ilorC pOplllar of lhio twit. The pfesell{ lise e l Mr))lie (';lll,~ niclhotls seeillS It) I)O c<)nfincd largel}) t<) l{utopc. In general, the analytical c o m p u t a l i o u ¢~1 rcliabilit} indict,, i o r a generation system requires the de\,elopnleni ,~i :t generating capacity model, the developme,iit of a load model, and finally the nlcrging ~ d t h e t\~..o models. Fhe l ~ n l , ' generating capacity models and load models lequiicd i:., vci ', much/a function of lhe rcliabiliw ) e l i t e s lil~tl :lie t~, b , . coi,i,ip,iilcd. IV. 1. Generatheg capacity models (;el|crating capacity models describe lhc vulicm~, ~lv'1 generating units. The simplest capaciLv tll*)doJs 2G gi\c tile probability of existencc ~>i"the va,iious available c:q~lcii~. (or c:q~acity on forced or,)rage)stales of the system. I he-,c m:ty be exact models, which give the probabitii} that Ih,,: capacity el/ [oiced o,iilage is cqtiaJ it) a stated al,i){ till[ ,~l cumulative models, whictl give lhe probability [hal /hc capacity on l~rced otilL,i,~e [S ~reatcr than or cq,ilal It) :l Mated ;.ill|to)ill). St,icl,i models arc usei.i,il i11 the calcu];lti~li t,i :ill rcii:lbilit,<' indices excel)/ ihose ,)i J'reqtlOilcv 'dlld d,ur',lti,m t{ITicieni algolithmls for the development of e×a<,'t <~ cumlt I',ttivc capacity outage probability m~dels (,u l',d~les);lic ~li! ill/p()rlal,il developi/leill. 27,2'<';Thcsc 31g()lithlllS tl:-,t.'{l\ 13:l>,ic dutu the Iolced emtage existence probzibilitics (C~mlm~,ul3 called Iorccd o,utagc rates)el the gcneltiting traits td iii,' st. :-,tctn and ptt)cced tI\vo c4pucil}< -.l:tic,, either ltllb< :,vztilablc o , rL)tally t]mivailahle duo !,> i~,i .:, ,,.

Electrical Power & Energy Systems

oulage, is as follows:

P(X)- P'(X)(I

p) + P'IX

C)I,

(14)

where P(X) is the probability of exact or cumulative outage of magnitude X after a new generator is added, P'(JO is the probahility of outage state before a new generator is added, ("is the capacity of the generator added, and p is the pr~.bability that the generator added is unavailable because of forced outage (forced outage rate). Special attention is called to the fact that equation (!_4) can be used for building exact or cumulative capacity outage probability tables. Note also that when X C < O, P'(X C) = 0 for exact tables and P'(X C) = 1 for cumulative tables. Equation (14) is easily generalized for generators with more than two availability states as required in the treatment of forced deratings. ('apacily models for use in computing frequency and duration reliability indices must give the l~requencies and effective departure rates front the capacity states as well as the probabilities of such states. Depending on the method used for the conaputing of frequency and duration indices, exact or cumulative state models are required. Efficient recursive algorithms, like the algorithm of equation (14), form the capacity model by adding generating units one at a time. Assuming a two-state generator, the expressions for forming an exact capacity model which are needed in addition to equation (1 4) are given as fl)lh)ws: < 9 P'IX)(1 ,o+(X)

p)p+(X) +P'(X - C)pIPI(X

-

C) +/Jr

P(X) (15)

o

(x)-

P'(X)(I

P ) I p ' ( X ) + Xl +P'(X

C)pp'(X

C)

IV.2. Load models D)ad models used in generation system studies vary in form and detail depending on tire reliability indices desired and the calculation methods used. Methods for computing LOLP, expected energy not supplied, and XLOL use the sinlplest load models: load duration curves, daily peak load variation curves (load duration curves of daily peak loads), and chronological sequence of daily peak loads. 30 The load model consisting of the chronological sequence of daily peak loads is commonly used in connection with exact treatment of maintenance outages as will be described later. D)ad models for use in connection with methods for computing frequency and duration indices are more complex. Two types of load models for this application have been suggested. The first type 7,3i views the system load cycle as a sequence of discrete h)ad levels each possessing a probability of existence and transition rates to higher and lower load levels. The continuous nature of load can be modelled by the techniques discussed in reference 31. The second type, 32,33 which may be called a cumul.ative state model, treats system load as a continuous variable which can be described by two continuous functions: i

the load duration curve which gives the probability that load is greater than or equal to airy specified value

•

a load frequency curve which gives the frequency with which load equals or exceeds any specified wilue.

IV.3. Computation of reliability indices Reliability indices for single-area systems are computed by merging the appropriate capacity and load models liar the system. General nlethods for C(mlputing common reliability indices are outlined as follows. 3°,34 LOLP =

P(X) (16)

where p+(X) is the effective departure rates to states with more (less) available capacity after a new unit is added, P i (X) is the effective departure rates before a new unit is added, ~, is the transition rate from 'up' state to 'down' stale of the added unit, and/2 is the transition rate from 'down' state to 'up" state of the added unit.

(I 7)

The frequency of cumulative capacity outage states can be l,mnd by recursive use of the above exact state parameters, 6 but if ,rely the cumulative state capacity model is needed, the formation of the exact state model can be bypassed and the cumulative state model can be formed directly, i°,29 Model parameters needed in addition to the state probabilities are cumulative state frequencies. These frequencies, ){.¥). c a n be efficiently computed as follows:

fiX) = (i

P) lJ'(X)

+ p [J'"(X

1:'=

P (awiilabte capacity = Ci)I: )

~

(20)

e x a c t capaci t', states,j

where k' is the expected energy not supplied and t:) is the energy not supplied if the available capacity is Cj.

~.

)_], (L i

days, i Q

Ci)P(availahle capacity

Ci)

I. i

XLOL =

1.0 LP

where XLOL is the expected magnitude of load loss at time of daily peak given that a load loss event occurs. The equations ik)r computing the frequency and duration of a ctunulative margin M, as utilized in reference 26, can be stated in the following basic f o r m P(M') =

~

p(L )P~........ti,,, ( C

L

M)

(22)

load

Ievel s, 1,

and

XP'(X)I

C)+IaP'(X

(7)1

(18)

where all quantities are as defined previously and where J"(X C) = 0 for X C < 0. The expressions of equations (15) (18) can be readily generalized to generators with more than two capacity states.

Vol 1 No 3 October 1979

(19)

where L i is the peak load on day i. Note that LOLP can also be calculated using a daily peak load variation curve load model.

In the above expressions, p~ •X 6") = 0 when X - C < O. If required, the frequency of capacity outage state X is readily obtained by

/ i X ) - P(X)IP+(X) + p (X)I

P (available capacity ~ L i )

~ d a y s o f ve lr, i

t i M ) = Y ' ~,1 (L ffg" . . . . . . . . . t i o , , ( C L

I 'P generation(C N i

+ -- EI

Pg........tion( C

Li,,,i,,

1,

M)

Limax M) M)}

(23)

143

where P(M)./'(My is the probability and frequency of nmrgiu M or less;p(L) is the probability of load level L; Pgenenitio,,{ {" L My, )'{C L M I is th0 probability and frequoncy of capacity outage equal to or greater than (C L My: C is the system installed capacity less any capacity on scheduled outage: L/max, Limin is the ith pair of consecutive maxima and mininla on the load cycle: and N is the total nurnber of(L/max , Limin ) pairs. Another method 33 combines an exact-state capacity n~odel with a cumulative-state load model to yieht:

P(M) = y(M)-

~ exact capaci ty ~u rage states, .v

~.

exact capacity outage states, x

x ]Jhl(("

X

"¥

P~e. . . . . tion(X)PId( ("

Pgenerati,,n(X){lP+(X) M) +fia(C

X

il't)

(24)

p (X)l

My}

(25)

where/;'generation(X) iS the probability of capacity outage of magnitude equal to X, p~(X) is the departure rates from capacity outage state X to states with more (less) available capacity; Pl
P(MIIJiM). Methods are also available for computing the various reliability indices in interconnected systems, ttere, reliability indices may be computed separately for each area of the interconnection and for the interconnected system as a whole. Such studies are ttseful in assessing the benefits of possible interconnection, planning generating capacity additions, and choosing the capacities of tic lines. The capacity models for each area of interconnected systems are the same as for a single-area study. The choice of load models lor interconnected system studies depends on tile reliability indices desired and the degree of correlation betweeu area loads which is to be rect/gnized. Methods are available to treat area loads as: uncorrelated or independent ; and correlated to the degree actually observed. Treating loads as correlated is tile most accurate and seemingly the mosl used approach. Methods for computing LOLP or frequency and duration indices 1°,3s in an interconnected system composed of two areas are well established and relatively straightlkn-ward. Ti-eMment of three areas has also been described, -~s,-~' bul is complicated and requires approximations in the representation of tie lines if tie lines from a loop. A promising new method -~v for handling an N-area system with a general tie line network is discussed in tile section on combined systems. I V.4. Limited-energy generation Evaluation of systems containing limited-energy generators, such as hydro units, must recognize tile possibility of f-tilure to supply loads because of basic energy shortages as well as capacity outages. Two approaches to this problem have been suggested. The first, and simplest, approach :v assumes that the energy producthm capabilities of a limited-energy generator are independent of the operation of the other generators of the system. [Jnder this assumption, an available capacity im~del can be constructed for the generator

144

which gives the probcibility of eXiSICUCC<~1vali(m.~ c~ipr each w001, o! rhc Sltldy period to yield the ieliability index ot 0,:peeled eileig) lll)l supplied. Methods lot cotnputii/g lrequency and duration indices ill systems with limited-energy generation clo nol seem t,, he available. IV.5. Sche&ded outages Scheduled or maintenauce outages of generating units have an important effect o n system reliability and must be considered when computing reliability indices. Scheduled outages can be treated approximately -~° by adding the capacity of units on scheduled outage to system load dm in<,,: the period of such outages to create an effective load reliability index calculations are then made using the capacity model which represents all system units including those on scheduled outage. This metlmd has tile advantage ~d computational speed, since the capacity model is not altered for scheduled outages, but always results in some pessimistic error in computed reliability indices. f h e e m u has been shown to be large in some cases. t-]xacl treatment e l scheduled outages inw>lves retnowil ~,J generators on scheduled outage from the capacity model during the period of scheduled outage. This is easily and efficiently performed through reversing the capacity model building process which was described earlier, l tence, exacl treatment e l scheduled outages through capacity model modification is considered the superior approach For m~s! studies today. Note also that exact treatment of scheduledmita<,es> ~equi~es the use oJ'load models that permit the lnaintenance ~1 the proper time correlation between loads and scheduled ,mtages. The load model used in LOLP studies is usuall3 the

Electrical Power & Energy Systems

chronological sequence of daily peak loads. Frequency and duration methods break the study period into series of periods within each of which scheduled outages can be assunred constant. Each such period is studied separately and tile results of each period are then combined to give total study period indices. -7

where reliability is affected by operating policy and limitations, such as in systems containing significant proportions of limited-energy generators and generators which operate in a peaking or cycling mode. The general method of Motrte Carl() simulation which is appropriate in such studies is that of sequential simulation. 2,2° 22,24,25

IV.6. Load uncertainty System plannitlg studies are always made using load forecasts which are subject to uncertainty. This uncertainty in loads can be readily factored into reliability evaluations to produce the expected wdues of reliability indices, given load uncertainty. Tile procedure can be expressed as follows: 3°

IV. 10. Operating considerations in reliability evaluation Methods currently used in the USA for cotnputing measures of systetn reliability tk)r generation planning purposes generally assume that generators run continuously unless shut down due to forced or scheduled outages, l lence, these tnethods do not reflect the impact on reliability of such operating considerations as: generator econouly shutdowns, spinning reserve policy, generator start-up tithes, generator start-up failures, and generator outage postponability. The results of simulation study examining tile reliability influence of these considerations have recently been reported 23 and suggest that these considerations can have significant effects on reliability indices.

l:([)

-

~_

p(LIlIL)

(2(0

load levels. I.

where t:(1) is the expected value of index I, p(l) is the probability of load level L, and I(L) is the computed value <~1index 1, given tile load level L. IV.7. Peaking and cycling service units (;enerator capacity models in conventi(mal use assunre, in effect, that generators are either running or in a state of forced outage. Such models are appropriate for base-loaded units or units which run for a large proportion of tile time, and the forced outage rates needed for use in these models arc readily estimated from available field data records. I h)wever, the conventional models are not particularly realistic for generators in peaking or cycling service because of the length ill'time such units spend shut down and also because of the large number of start-ups with associated st resses whicll such units experience. Nevertheless, it is desirable from a computational viewpoint to retain tile c~mventional generator capacity model for use in forming It/(; system capacity model, tlence, it has been suggested 39 that setting the forced outage rates of peaking and cycling units equal to tile probability of unit forced outage during a f,eak load period may give reasonable results. It is noted that this treatment of peaking and cycling units gives no recognition (ll start-ups in response to forced outages of other units, tlence the modelling of such units is still incomplete. IV.8. Confidence Ihnits on LOLP The ftqced outage rates of generating units considered in planning studies ale subject to uncertainly because there arc often no, or only lilnited, appropriate opcrating data on the traits, l lcnce, tile [ D L P index computed using a capacity model derived from uncertain parameters is also uncertain. A method has recently been described whicit f, ctmits the uncertainty in LOLP for a single-area system to be displayed through placing confidence limits on

LOLP. 40 Tile confidence limits on LOLP give tile range ~)I vatiation in LOLP with specified probability and should help to place the computed index in proper persp,eclive for making pkmnmg udgements. IV.9. Monte Code simulation Monte Carlo simulation methods offer an alternative to analytical methods for the calculation of reliability indices in any type (flgenerati
Vol 1 No 3 October 1979

V . Transmission and d i s t r i b u t i o n systems Reliability calculation methods for transmission and distribution systems can be broadly categorized as analytical methods or Monte Carlo simulation methods. Analytical metht)ds can be further categorized as tnethods based (m a direct solution of a Markov model and methods using a step-by-step analysis usually based on the minimal cut-sets of the system. V.1. Direct solution o f Markov model Explicit formation of the Markov model of a system and the straightforward solution of this model for reliability indices provides a general method l'oJ reliability ewduation. In practice, the method is applied only where system con> ponents cannot be treated as independent and where the number of system stales is not "large'. If components can be treated as independent, tile minimal cut-set approach, which will be described in the next section, provides a much easier method ()l calculating reliability indices. Ttle Markov model is defined by a transition rate matrix of order equal to tile nUlllber o f system states, and wltere C()lllpt)nelllS ale not independent a tnt~,dified form of this matrix having tile same order urust be inverted to find the steady-state probability of system states. This consideration limits direct use of tile Markov model to systems with relatively small lltlmbers o f COllrponents even t h o u g h p r o c e d u r e s are available for reducing the ntnnber of system stales by merging ()r combining some of them. Algorithms for the automatic formation TMof the transition rate matrix have also been described which reduce the alnounl o f labt)ur associated with the nrethod. The state-space truncation and sequential truncation I can be used to alleviate tile problenl t)l dimensionality of tile state space. V.2. Step-by-step methods Methods using a step-by-step analysis are tile most con> monly used in practice. Such methods utilize concepts o f

Markov processes but do not explicitly form tile Marker model of the system ttlereby avoiding the computational difficulties of the direct solution method. Step-by-step methods are theoretically limited to applications where components can be assumed independent, but in practice they provide good approximations where there are certain

145

linlited l~ornls t)l conlponellt dependencc t)ccasJoiled by 111aintenancooutages, stofnls, and c()i111nolr-ll]t)deJaJlLlres. Step-by-step methods are, in principle, applicable to any size of system and to tile ct~mputalion of reliability indices considering continuity only or considering tolerable voltage levels. In p~actice, reliability ewtluatitm becomes increasingly dift]cult as the number of system ctmlptments and the degree (fl networking of the system increases. Treatment of reliability on other than a c o n t i n u i t y basis also acids appreciahly t,> tile time and cost of analysis since hmdJlow calculations are usually required to determine ltiose system states and load levels cons( itul ing failure.

sets of tile system defines a hiilure event, the iaihn c frequency of the system is jUSt tile sum t)f the lrcqtlCllcJc>, computed I~r the niinimal cut-sets. The average failure ,,~ {mtage duration for the sysleni IS sin/ply the weighled average
l . > t ..... /".}.st..... -

';-,.

lililliiliail cut st?t:,,

.ti

!3~7 t

Z .t}F,/Jsyst .... nfiNim;tl cut sets

O n c e Jsystcm a n d rsystem have b e e n f o u n d , t h e s t e a d y - s l a l c

probability of systeni failure, Ps_~ste,n, can be c~mHmted Step-by-step methods will first be discussed assuming that system faihlre is defined as loss tfl" continuity. The first step in computation of system reliability indices is the performance el' a failure modes arid effects analysis ( F M E A ) 4t on the system. The FMIiA basically enumerates those con> ponenl faihn-e events which result in system faihlre. More specifically, tile FMEA amounts to determining the minimal cut-sets tH tile system. Minimal cut-sets are defined us sets of the minimum number tH ctmq~onents whose outage will result in loss o l c t m t m u i t y for the system. Also, a minimal cut-set is nol a sub-set of any other minimal cut-set. The FMEA l~r a transmission and distributitm system Must recognize the dilTerenl types of component failures that are pt~ssible, 1o~ example transient and persistent forced outages nf transnlJssion lines and faults, false trips, and taihnes t¢~ trip for ciicuil breakers. 1(>,42,43 Additionally, tile FMI(A llltlSt r e c o g n i z e t h e effect of a u l t ) n l a l i c a n d 111al/tlai switchJng t)peratituls after Ct/l/li~lt)l]en[ faJhnes, i t,, i %42 Ill prac[ ice, tile FMFA ilia).' be terminated after some specified contingency level, usually the second or third, is studied, since t h e COlliributions of highit/rder contingencies 1o system tailure prt~bability or frequency are invariably negligible coinpared with lower t)rder contingencies. The results of tile FMI-~Acan be displayed in the form of netwurks called reliability blt~ck diagrams, 42 but a more flexible apprtmch is simply Io list lhe nlJnimal cut-sets found.it~.l° The FMEA of a simple system is easily conducted by hand, but manual determination of minimal cut-sets for a large, complicated system is a formidable task. I lence, it was recognized as early as ] 0(:,744 th:lt the FMEA should be automated, and work in recent years has resulted in the developnlent {fl some excellent programs for this purpose. 10,43,45 These progralllS, t)l course, oh) not stop with tile FMEA but continue to compute system reliability indices, as wilt be descl ibed below. The second stage in tile step-by-step method is the con> putatitm of system reliability indices using the list oF mininlal cut-sets provided by tile FMEA. The list can be used to Ctmlpute system /allure probability, frequeucy, and average dtnaiion exactly, but, in practice, approximations ale used to reduce conq)utational burden. These apl~roximalions lest on the fact that transmission and distribution component 'success" probabilities are ahnost always very close to unity. That is, the time u T & D comptment spends on outage is very small compared to tile time it is operating satisfactt)rily. With this assumptitln, the frequency and duration contributions of each minimal cutset can be computed by considering the coniponenls in tile cut-set to he functitmally in parallel and ignoring the other con/ponents of lhe system. Since each of lhe minimal cul-

146

['l(llli

Psystenl - J'systcn,/'s}.'steni

(2~) )

where r a n d r are in the same time units. Ahernativel), Psystem can be estimated directly as tile sunl {*f the pl~babilities {~1 the mininml cut-sets. Much work has been done on tile develt~pment e l explessions, usually approximate, that can be used for conlpul Jng the frequency aud average duration of faihne events in parallel systen/s and that are suitable for use with tile stopby-step melhod. Expressions are available for: •

overlapping t(/rced outages t ) t C O l l / p o n e n t s , 1(',4(~

II overlapping forced and n/aintenance outages ol COlllpollent s, I 6,4(~

•

overlapping outages considering failure-bunching hecausc {)l" Stt)Inls, 4~'4 7

•

modelling the effects of switching after faults ioi service restorat ion. i 6,42,48

•

modelling overload outages of colllponents. 32,4°

('alculation of system reliability indices, considering w4tagc level as well as ctmtinuity in the definition of systeni failure, proceeds in much tile satne manner as outlined above except that physical calculations such as Ioadl]mv bcconle an integral part ol'the FMEA. The genclal pi<~ cedure is as follows: (a)

asstlllle a given component or glOtlp e l c o n l p { ) n e n l s oUt o f service due to forced outage and/or maintenance

(hi determine tile load level given the outage event {~I (at which just causes unacceptable wHlages tt> appeal. Repetititm of steps (a) and (hi for different assumed con/ponent nutages defines the nlinimal cut-sets of lhc systetn ill terms tfl" conq~oneitt outages arid ()l" load Icvcl~. A usual assumption is that all system loads scale up and down together. Thus, a single per-unit load model applJc,, For tile entire systetn. A general CUnlulative 2-state l<~ad model allows tile computation of system failure prtH~ubitit), frequency, and average duration by sumnfing lesults {wcl tile minimal cut-sets determined from the FMEA. -~~ V.3. Monte Carlo simulation methods Both sequential simulation and independent increment~ simulatitm nlethods 2s have been suggested and used t<~ stm/c extent, altar)st exchlsively in Eurtq)e, For CtmlptJting rcliahilil_'+

Electrical Power 84 Energy Systems

indices in transmission and distribution systems, particularly in large and complex systems where the determination of minimal cut-sets may be difficult. Evidently, independent increments simulation is the most widely used method of Monte Carlo simulation for transmission and distribution systems because of its speed adwmtage over the alternative method, although, tmfortunately, it yields only tile index of system failure probability rather than the more popular system failure frequency and average duration indices.

(a) Determine the minimal cut-sets of the system reflecting both component outages and load level. Such cut-sets define those minimal sets o f components whose outage at or above a particular system load level will result in an inability to fully supply tile requirements of one or more consunrption points in tile system. Determination of these minimal cut-sets requires, in general, that power t]ow calculations be made that consider tile postulated component outages and load levels.

(h) Once the minimal cut-sets of tile system have been Vl.

Total system

Methods have been described in preceding sections for computing reliability indices either for geueration or for transmission and distribution. Frequently, however, there is a desire to compute reliability indices reflecting both power production capability and tile ability to transfer power to the various points of consumption. Such total system studies are particularly relevant when tradeoffs between investments in power production facilities and tr:msmission facilities are to be investigated. As in generation and ttansmissiot~/distribution, two basic nrethods of reliability index computation are available: analytical methods and Monte Carlo simulation methods.

V I. 1. Analytical methods

Analytical methods may be subdivided into methods that evaluate generation and transmission failure events independently, and met/~ods that evaluate failure events through simultaneous consideration o f generation and transmission. Tile method 49 that assumes independent treatment of generathm and transmission makes the basic assumption that transmission power flow patterns are not a function of the status of the generating units of tile system. Where this assumption is reasonable, total system reliability indices can be found by computing separately the reliability index contributions of each portion of the system using the techniques already described and then combining these contributions. Thus, the steady-state probability, frequency, and avera,,e dural ion of system failure events are given by: Psystem - Pgenerati,m + P t r a n s m i s s i , m

The major practical difficulty in computing reliability indices in a combined system, whether or not generation and transmission can be treated independently, is the determination of minimal cut-sets which must be identified througll use of loadl]ow calculations. Direct enumeration of possible cut-sets and their evaluation by all AC loadflow, or even a DC loadl]ow approximation, is not feasible for a large system. Methods that represent tim system as a linear flow graph and that aw)id direct enumeration of system states have, however, been used with success. 3v, Sl Such methods appear to offer the only practical approach to determination of minimal cut-sets in large highly networked systems. V l.2. M o n t e Carlo shnulation methods Monte Carlo simulation offers a means of computing reliability indices in conrbined systems that awfids sonre of the computational difficulties associated with analytical metlrods at tile expense of considerable computer time. Prior to the development o f efficient schemes for determining the minimal cut-sets ol a system, Monte Carlo simulation was probably tile only practical method for studying a large, complex systenr. Now, however, the trend seems to be towards analytical methods. Tire independent increments approactr to Monte Carlo simulation has been used with success in studying combined systems. 2s Here the approach is as follows:

" PgencrationPtransmissi,m

I~e ....... ti .... + Pt ......... issi ....

./'s,,t~'n, - / ' ~ ......r~ti .... + Jt~a.,smissi.n rge .... ati,,nJ'F.enerati,,n

determined, the probability, frequency, and average duratkm of the system failure state is readily found using previously described techniques.

(30)

(31)

+ rtrans,nissionftra,,sm ission

l'% '.~ hie 111

(a) draw a load level, (h) draw the status o f generalors and transmission network COlllpollell is.

(c) evaluate tile ability of the system configuration as

.tsystem

(32) The above equations are best understood by regarding failure of generation and failure of transmission as tile minimal cut-sets of the total system. If transmission flow patterns are significantly related to

defined in step (h) to supply the load as found in step (a) using exact or approximate power flow calculations,

(d) repeat steps (a) (c) until sufficient data nave been generated to permit reliability indices to be calculaled with acceptable accuracy.

~ellerator status, get]craters and trallslnission components

must be considered simuhaneously in the computation of reliability indices. 4~ This markedly increases tire difficulty of computing system reliability indices because of the increased number of system states that are possible and because of the need to recotnpute transmission network power flows for each change in generation status, s° The general approach to reliability index computation is as foJ[Icws.

Vol 1 No 3 October 1979

V I I . Assessment of system r e l i a b i l i t y It was recognized as early as 1938 s2 that tim computation of the expected reliability perfi)rmance of a system is only one step in reliability evaluation. The next step is assessment of the computed value of reliability: is system reliability more or less than that desired or .justified? Reliability assessment is easy if only the relative reliability of alterna-

147

rives is needed, but is much more difficuh if an absohite assessment is needed, as is always the case. The following threc basic methods of absohile reliability assessment have seen some application. The siinplest meth,.)d (}f reliabiliiy assessmem c{/inpares expected system reliability with a fixed level of reliability that is cl/nsidered acceptable fr{)m public reaction t{i past reliability perf{irmance and/or upon professional j u d g e m e n t . The widely quoted generation reliability standard o f ' o n e day {)l load loss in ten years" falls into this category of reliability assessment. Note thai a historical or judgenl0ntal criteri{ul does not dircclly recognize tile c{}sl (if provMing :,igivcn level o f rcliability or o f costs incurrcd by pow0r C{)llSlii/lers because {)f power iilterruplions. MosI reliability assessments in the USA ar0 madc usiilg this general method. A second hie]hod {/1 reliability assessment is based {)it the C{)st efl'ecIlVelless ill" System investnlenls in improving rcliabilily pcrl{unlancc. 1'ypically, incr0nleilt0d invesl menl s in s3,;len/ hnpiovem0nts arc vet). cosl-iliot]0ctive thereafter. That i->.ihe cilrv0 i}f syst0ill cos[ :.ig,:.iillst reliability tis/lally has ct rath0i sharp "kiloC'. Thlls, Ihere is good i0aS{}li h} choose :l ~elhibilily goal ill the reg,il!tl {if the "kiloo'. Thi,,, appioach direclly rcc{igniz0s ih0 cos[ o1 provicihig reliability al/d 'Jsy;lil~ ,. ,_,o{/{I pcrl{}riilailce Ior moli0y speiil, bill d{}os u{}l clhccli~ rcc{}gnizo the r01iabilily noecl~; {>1 Ihc c{~n,-;tllilcr. A thild n;01hod llli[izes tlala {)it l i l t COSl {~1 lu>WC'l i n l ¢ i i t l p ti{ms I{~ c.'lillSlllllers lo deleriuh/e a level <,r SySIClII rcliabilir,' ]hal iesuhs in lhe h}wesi ovcrall cosl Io the coilSillllor col/ sid0rii/g bolh sysleill iilv0stru0nts as i0ilecl0d it/ rates and lhe cos] o f hiierrur, tions. ~3.~4 This method, in theo:y ihc ]]lost equitable me]hod o f reliabiliiy ass0ssmcnl, has becii difficuh to apply because o f a hick i d data {)it Ih0 c{isl lo Ct/llSlllllPiS tti iiHcrrupli~uls, hi rece11[ y0al,S, hovv'cv01, SOllle data have tR'coine available, and the m c l h o d is ll{l~A v~ich_'h tls0d in t!ulopc. D5 5"7 The nleihod has also been Lised hi ihc planning and design o f induslrial plan] syst0nts in the IJSA. but has seemingly s0011 little or it{} applicalioi~ 1o eleciric utilil,,, systems in g01]cral in ]hal c o u i / l i y . VIII. Areas of future research and development [ h 0 a/libel'-; hay0 idcnlified a lit]]i/her {/1 areas in which additional iesearcIi allot develotmlenl w{uk should he very bcneliciai, and th0se are outlincd in this seciiou. The plesen[ high c{/sls of capital e(tuipmeni and fuel suggcst lhat now, 111010than 0vor, p{iw0r s),'stcms sh{~uh.1 b0 planned with direci r0cogniti{}n o f the service r e l i a b i l i l y needs o f C{/lISUlll0rs. This r0quircs a kllowlet]g0 {iJ"th0 ecoilOlllic coilseqtlCllCOS |{/ l[I,i." COI1SUI11eI{11"service i n l e r r u p l i o l l s . Thus, st]idles should be madc of thc impaci of service reliabiliiy level on various s0gmel]ts of th0 ccotloilly alld upotl Ct)lisulners in gcncral to identify the cost of inteltupti{ms and to estabti.sh overall o p t i n m l n levcts el +reliability. Methods presenlly in general ]is0 foi c o m p u t i n g wirious l{}llg-t0rIll averaoe measures of reliability in gcileratioll, transnfission, and distribution systems presumc that thc parameters of the problem, such as c q u i p m e n l failure and repair rales, arc conslant and klloWll. Thus, present m0tt]ods cotnpute poinl estimates {)f system reliability measures which do not recognize variability due to uncertainty or variation in the parameters of the problem. Two methods

148

exist For assessing tile variability {>1 sbstem iL'liahilit\ measures dtic to paramctm u n c e r t a i n l y : SCl/Silivily :llKi]),-,i5 and confidence limits. Sensitivity :lmtlysis has been I;ttlw] widely used in a crude Iorm, namely repeat]u# the rcli:lbili{~ coll/pkilati{m USillg different vaHiest}[ pal~_lllleler~ dlHI observing [lie ellccts on system reliability mea~u~c~. Ihia direct appr{}ach to sensitivi/y analysis is usually not le:L~ible il" the iltlliiber of par:,illieters subic¢l to tllIcerl'diilty la I;ligC Some linfited work h:,is been done {in more .<,<}phisticatcd illdh-cct tile]hods, IMit 11111ch I+Cill:,dilSl{e bc el{me iu Itli,, :t~c:~. Tile second illclh{}d {}f lle:,llii/~ tLlraiIlelcr tlilCCi I~Jiu[\ ~,. lhr{}ugh the use of c.'twc~ ~nc!lu+{t-, {)!c{unpulilkL, c(mfidence limits lol leliabilil', i l i d h c , f,~ ~cneral.

.<'wslcm ~eJmI',ility nic:.lsmcs a~ plebeutly c,.}mputcd h,i j)Jaltliil!g a n d desigIl purp{}sc>; are tile lollS-tell// ~i~,,-'iLi~2c?h,~ expected values {}l peif{}i nlal/cc indicai{}rs such :,~ lh, l/llmbcl {}1 davs pel ycal on Bhich 2clleraliuN ,.;it~:i, i ?.. ' ii],;iltlicieiil 1{> cat+i+} {]ail,< peak It>el {l e l P. the iltltllt,t'] <+1 i imcs por }ca] t hal a ctlsh}mOi suflor:, a sot vice ii]l ci t upt h ~i] etc.). IhtL'>. tho pieseutly c{)nq+utod +xpccled vclili<_':, Ol inlerrtiption~, whicli iili~hl t+CClll ill 'Cltl t~!;l<:ti<_< ]night load I{} incoi10ct ¢{mchisioi>;. t}iedicli{m limits pl{wide the nlissin 7 i l f l o l n ] a l i { m by giving Ihc inlel;'tll v, iihiil ;~hicl] perliunl:_liiC0 ill a given year will fall with specified pi{}i]~ :ibilit?,. li i>, ,>ugge>,tei]. thcl0h}rc, thai resoaich hc, c~uidliclcd inh} wa}s {>1 c{}inl}uiing tu0diction limit.', on leliabilit', r~cilorl/iaiiee iilea',,tires tiscd in r}lal/llillg :.lilt] desigu ~1 7ei/el211ioli. llailSi/liSSioi/, add d i s t r i l m l i o n 5} 5tClll', /~,nai} lica] tncth{}d~, ]iii c{ul]ptltillg relialJilit}, iudicc:> iii gCllU'lalillg capacity sic]dies c{}nirJtiic rcliat}iliI? essenlicllb, {)i! ;.ill mstalled capclcity basis, lhat b. ,aiih lniilt:i cxcepli{}ns, tile prose]t[ ailalvlicaJ lilelh{R]>, ch~ iii;t lelJecl +)DCl:iIilig t}{;licies <>~ such gcilel{lhu opctaiillg f~ic:/,}i,, :l,. :llti{>lilll {~1 capacil} hold in vaiious it?S{?lVe slaie~ :illct Ihc , ,~i-,c{iticHt elTecl tql lhe expostir0 o f tlliils t~} lailurc; sl;irl-tll~ iiHic> ~Jl gCllc_'ralt)is ill Hen-spinning reserve :.;l:.l{cs ;lilt] !{;<~,.ii;] : ;.,1< . elTo]It]atollS. Thus, pleSCllI 111etl/t}tis tciicl it, ~i\c {)pllllli',llL vahles <}1 i c l i a b i l i i y indicc~ alld, ill :iddiiiou. d<~ il{,l pcI inl! udequaie evaluation o f ahernatives which ilia} dilie~ h; fort/is ~.)l opcralillg characteristics. Tilerelore, a need e×i:-;l>, l{u iicw analytical nloth{.)ds aild in{}dol:-; which ICC%L'ni/c Ol}oraliil g policies, chaiactorislics, alld ¢ol/sllaiul-, il! lhc c o m p u t a t i o n o f reliability indices f o i use in planniu 7 and design. (It i~ iiotcd that Monte ('al Io methods have bccii described which 0n¢ompass tile features n{)10d, bul Ih0sc mclhocls have proved ctilllbersolilc and t_'xpeilsive iu :lpplicati{m. ) Tile n0w[y intr,.iduc0d techniques based {)l] lineal llCtV~{}rk~, foi the c o n i p u i a t i o n o f rcliability in,ticcs in ¢outbined systems should b0 further deveh/ped and additional ar}p]icait{ills {/1 thes0 and related techniqucs expIor0d.

Electrical Power & Energy Systems

IX.

Acknowledgement

The paper from which this material is derived was originally prepared under a contract to the US Department of Energy and appeared in ,~vstems en#ineerin% for power: status attd prospects, ltennikcr, USA (August 17 22, 1975). X.

3

4

Baldwin, C J, Gaver, D P and Hoffman, C H 'Mathematical models for use in the simulation of power generation outages part 1 : fundamental considerations' AIEE Trans. Power Appar. & Syst. Vol 78 part III (1969) pp 1251-1258 DeSieno, C F and Stine, L L 'A probability method for determining the reliability of electric power systems' IEEE Trans. Power Appar. & Syst. (February 1964) pp 174-179 Billington, R and Bollinger, K E 'Transmission system reliability evaluation using Markov processes" IEEE Trans. Power Appar. & Syst. (February 1968) pp 5 3 8 547

5 Stanton, K N 'Reliability analysis for power system applications' IEEE Trans. Power Appar. & Syst. (April 1969) pp 4 3 1 - 4 3 7 6 Hall, J D, Ringlee, R J and Wood, A J 'Frequency and duration methods for power system reliability calculations part I: generation system model' IEEE Trans. Vol PAS-87 No g (September 1968) pp 1787 1796 7

Ringlee, R J and Wood, A J 'Frequency and duration methods for power system reliability calculations part I1: demand model and capacity reserve model' IEEE Trans. Vol PAS-88 No 4 (April 1969) pp 3 5 7 388

8 Galloway, C D, Garver, L L, Ringlee, R J and Wood, A J 'Frequency and duration methods for power system reliability calculations part II1: generation system planning' IEEE Trans. Vol PAS-88 No 8 (August 1969) pp 1216-1223 9 Cook, V M, Ringlee, R J and Wood, A J 'Frequency and duration methods for power system reliability calculations part IV: models for multiple boilerturbines and for partial outage states' IEEE Trans. Vol PAS-88 No 8 (August 1969) pp 1224 1232 10

Ringlee, R J and Wood, A J 'Frequency and duration methods for power system reliability calculations part V: models for delays in unit installations and two interconnected systems' IEEE Trans. Power Appar. & Syst. (January/February 1971) pp 79--88

Halperin, H and Adler, H A 'Determination of reservegenerating capability' AIEE Trans. Power Appar. & Syst. Vol 77 (August 1958) pp 5 3 0 - 5 4 4

15 Falk, A K 'The effects of availability upon installed

reserve requirements' IEEE Trans. Power Appar. & Syst. Vol PAS-85 No 11 (November 1966) pp 1 1 3 5 1144

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38

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'Report on reliability survey of industrial plants part I1: cost of power outages, plant restart time, critical service loss duration time, and type of loads los{ versus time of power outages' Committee Report, IEEE Trans. Ind. Appl. (March/April 1974) pp 236-241

Electrical Power 84 Energy Systems