Power System ReliabilityI-Measures of Reliability and Methodsof CalculationD. P. Gaver F. E. Montmeat, Senior Member IEEE A. D. Patton, Member IEEE

An increasing amount of attention is being focused on the been shown to be influenced by frequency and duration ofreliability or continuity of service afforded by transmission outages as well as other more elusive factors.' Thus, at leastand distribution systems. This increased attention primarily two basic measures of reliability are significant. Any onestems from (1) the need to supply improved service as cus- customer is, of course, primarily interested in his own servicetomers become more dependent on their electric service, and goodness. That is, he is interested in system reliability as(2) the desire to use new system voltages and designs whose seen from his point of service. The utility may well be in-reliability is not well-known to supply the heavier loads of the terested in other measures of reliability such as the averagefuture. In the past, the reliability of proposed systems has service goodness provided all customers on the system or thebeen estimated by extrapolating the experience obtained from poorest service provided any customer. Such measures ofexisting systems and using rule-of-thumb methods. In the reliability will have meaning when comparing alternative sys-future, however, more precise methods of predicting and eval- tems which serve many customers.uating reliability will be required. Listed here are some measures of reliability which are of

Recognizing the need for precise methods of predicting the interest and significance and which can be calculated by thereliability of transmission and distribution systems, Public methods of the paper:Service Electric and Gas Company and Westinghouse Elec-tric Corporation have developed analytical techniques em-

F per yearploying probability theory which permit calculation of a R = average customer restoration timenumber of important measures of reliability or service "good- H= average total interruption time per customer served per yearness" in general power system networks. These techniques Fmax = maximum expected number of interruptions experiencedhave built on and extend the work of other authors.'-3 by any one customer per yearRmax=maximum expected restoration time experienced by anyThe techniques which are presented in this paper were de- one customerveloped as part of a general study of distribution system P=probability that any customer will be out of service at anyplanning undertaken jointly by Public Service and Westing- one time longer than a specified timehouse. This general study, designed to assure economical The first three quantities in this list express measures of aver-future expansion of distribution systems, is introduced in a age service reliability for a system serving large numbers ofcompanion paper.4 customers. Note that these quantities could also be expressedThe methods presented in the paper will permit the cal- in terms of load units rather than customer units. The next

culation of various measures of reliability in power systems two quantities indicate the poorest service reliability affordedfrom basic system component parameters and a characteriza- any customer on the system. The last measure of reliability,tion of environmental severity variations. This makes possi- P, is also a measure of the poorest service afforded any cus-ble the comparison of alternative system designs to discover tomer. The measure P may be of interest if a goal of systemthe lowest cost system yielding the desired reliability proper- design is to assure, with some probability, that no customerties. The method will also permit a rational evaluation of will be out of service longer than a certain specified time.alternative operating procedures, such as amount of tree The quantities F, R, H, Fmax, and Rmax, can be estimatedtrimming done and number of line crews available to perform for existing systems if records are kept of the duration ofrepairs. each outage and the number of customers affected. The

following are expressions for estimates of P, R, and H. Thevalues which should be assigned to Fmax and Rmax are obvious

The determination of an adequate measure of reliability or from their definitions.service goodness for a transmission or distribution system isitself a difficult problem. Indeed, it seems certain that no F =

single measure of system reliability is completely descriptive No. of customers servedof the system's ability to supply satisfactory service. Satis- No. of customer interruption hours during yearfactory service as defined by the electric power customer has R =No. of customer interruptions during year (2)Paper 64-90, recommended by the IEEE Powver System Engineering H

o fcsoe neruto or uigyaCommittee and approved by the IEEE Technical Operations Corn- No. of customers servredmittee for presentation at the IEEE Winter Power Meeting, NewYork, N. Y., February 2-7, 1964. Manuscript submitted November Note that the "hat" symbol (A/) denotes an estimate of the value4, 1963; made available for printing December 2, 1963. of a parameter.D. P. GAyER is with the Westinghouse Electric Corporation, Pitts-burgh, Pa.; F. E. MONTMEAT is with Public Service Electric and The average degree of customer satisfaction is a function ofGas Company, Newark, N. J.; and A. D. PATTON is with Westing- the quantities F~, R, and ft. Ultimately, energy sales andhouse Electric Corporation, East Pittsburgh, Pa.The authors wish to express their appreciation to R. M. Sigley of prft ar loafnto fteeqatte.UfruaeyLehigh University for his contributions toward the development of however, customer satisfaction and energy sales and profitsthe methods descriked in the paper. cannot be evaluated for most utility systems given the various

JULY 1964 Gaver, M71ontmeat, Patton-Power System Reliability-I 727

Fig. 1. A 2-state up a system simultaneously, several component failures mayS WEATHERE ' m fluctuating en- occur during a short period of time. This bunching of failures

I: |l vironment due to a common event can have important effects on systemii| ^ reliability. To illustrate, consider a "system" made up of

E5 two parallel components equally capable of carrying full sys-II I'|: tem load. This system will fail only if both components are

o down at the same time. Ignoring maintenance outages, both° NORMAL EJ components will be down at the same time only if (a) both fail8WEATHER simultaneously, or (b) one component fails and is not repaired

TIME - before the other component fails. It should be obvious thatthe chance of overlapping component outages and consequentsystem failure is greater when component failures are inducedby severe environmental conditions to bunch than when com-

measures of reliability. The relationships involved are not ponent failures occur randomly and independently.well enough understood at the present time to permit the Previous methods of estimating power system reliabilityrequired analytical expressions to be written. In some in- have presumed that constant average component failure ratesdustrial systems, howi-ever, relationships between reliability apply at all times. That is, previous methods presume thatand product output have been formulated in such a way that a components fail randomly and independently. In a laterdefinite dollar value can be assigned to various degrees of section, a numerical example will be presented to show thatreliability.3 Since, in general, no definite economic value can the independence assumption can yield an appreciable under-be assigned to degree of reliability in utility systems, utilities estimate of the outage rate of parallel systems. The inde-are for the most part forced to rely on history and experience pendence assumption in calculating the outage rate of seriesto establish acceptable levels of system reliability. There- systems results in a slight overestimation of system outagefore, in studying uitility systems, it is usually necessary to rate. The error incurred due to an assumption of inde-have estimates of the various measures of reliability for pendence appears negligible, however, for reasonable numbersexisting systems. These estimates of the various measures of of components in series and for representative failure rates,reliability define levels of service goodness against which since single component failures stillgreatly outnumber over-the calculated reliability of proposed systems can be compared lapping failures. Furthermore, a series system is at leastmeasure for measure. partially de-energized as soon as a single component fails, thus

Calculatio ofMeasuesofReiabireducing the chance of subsequent overlapping failures.CluaonerofstsMasrescompofsRelibiitofgroupsofelemTherefore, independence can usually be safely assumed in cal-Power systemas are composed of groups of elements or culating the reliability of series systems.

components that act in series or parallel or both with each The degree of sophistication used in representing the effectother to carry oweprfrsom generation sources to load buses. of fluctuating environment in parallel system reliability cal-In a system link comllposed of several components in series, culations is limited largely by the availability of required data.the failure of any comaponent will result in an outage of the Field trouble reports usually state whether or not weathersystem link. The series link will then be out of service until was a contributing or causative agent in the component failure.the failed component can be replaced or repaired. In a sys- Hence failures can readily be separated into "normal-weather"tem link composed of two or more components acting in or "random" failures and "storm-associated" failures.parallel, however, a system link outage is experienced only Trouble reports may also give a broad indication as to thewhen all parallel components are out of service or when load type of failure-causing weather such as lightning, wind, or ice,exceeds the capaclty of components remaining i service making possible classification of failures according to type ofThe parallel link will... 11be out until a component capable of storm. Given these data, together with statistics on the num-carrying the required load is restored to service. In the sec- ermo m nents in sieather savastme tea year* . . ~~~~~~~~~~~~berof components in servic- and the average time each yeartions to follow, mathematical models for the calculation of during which the various kinds of weather prevail, averagevarious measures of reliability in such systems are developed, component failure rates which prevail during normal weatherFirst, however, a unique feature of the power system reli- and the different kinds of stormy weather may be estimated.ability problem must be discussed. It appears, however, that the additional accuracy in reliabilityFLUCTUATING ENVIRONMENT calculations gained by identifying different types of storms

rather than lumping all storms together does not warrant theA power system at least that part of it that is exposed to considerable additional difficulty in data collection and cal-the weather, is subjected to a fluctuating environment.During stormy periods, for example, environmental conditionsmay be so severe as to result in equipment and line failurerates much higher than those prevailing during nonstormperiods. Fig. 1 illustrates an environment which fluctuates Table 1. Storm Definitionsbetween normal and sevrere states as a random process. The Minimum Minimum Minimum

r *7'11 1 * * * * 1 * t-O 1 ~~~Weather Bureau Wind Velocity, Temperature, Duration,failure rate process illiustrated in Fig. 1 is certainly simplified. Basic Indicator KnotsF HoursOne would expect the various occasions of "severe" conditions Thunderstormto result in failure rates with different values rather than the Heavy thunderstormconstant value assumed here. Not all storms have the same Tornadowind velocities, amount and kind of precipitation, amount of Moderate rain 20 .. 5lightning, etc. Hence the value of failure rate prevailing dur- Moderate freezing rain .. io 1ing a severe period might more properly he a random draw Heavy freezing rain .. .. 4from a population of values for "severe" failure rates. Hoeravy snow 20 28

If severe weather strikes a number of components making Wind only 25...

728 Gay)er, Montmeat, Patton-Power System Reliability-I JULY 1964

Fig. 2. Storm duration tions of thunderstorm and "other" storm durations. Thus adistribution for Newark, possible refinement would be to consider separately different

N. J. 8 -yt tI seasons or periods of the year; then combine the results of thett + - 1__ 1separate calculations to put measuresof reliabilitvon an annual

o6 t --I- - basis-their usual form. The methods of reliability calcula-OBSERVED DISTRIBUTION_\ EXP,gONENTIALDISTRIBUTION ITH_ tion described in the paper are applicable to any period of timeEXPONENTIAL DISTRIBUTION WITHI EXPECTED VALUE= 1.25 HR, THE-- but require the approximate fulfillment of certain mathe-

6 AVERAGE OF OBSERVED2 _ t--DURATIONS matical assumptions made in their derivation.

c SYSTEM COMPONENT DATA FOR RELIABILITY CALCULATIONSa_O-2 4 Two basic types of component data are required in making

DURATION, HOURS power system reliability calculations. These data are (1)com)onent failure and maintenance outage rates and (2) dis-tributions of component repair times. The detail with whichcomponent failure and repair time distributions are required

culation m-iiethiods requtiredl Therefore, the mi-iethod of this depends somewhat on the portion of the system which is tobe studied and the objective of the study, buit irn general the

paper presumes the siml)le 2-state model of Fig. 1 in which following data are required:the environmiient alterinates between norlmal and severeeconditions. The durations of normial and severe conditions 1. A normal weather (random) "lperianent" forced outageare randoim variables drawn from distriblutions of durations (failure) rate should be estimated for all types of apparatusobtained from historical weather data. lNote that disaster and lines which exhibit distinetive or characteristic failurestorms such as major hurricanes and tornadoes should not be rates. This failure rate, NX (where i labels the componentlumped with other less violent storms. Suheh stormi-s ale type), is expressed in units of failures per year of normalrelatively, rare, but give rise to com-iponent failure rates much weather per unit of apparatus or per mile of line. It may behigher than those exl)erienced during "average" storms. estimated as follows:Therefore, lumping disaster storms with average storms cwould greatly dilute the failure bunching effect of disaster Xi= - (4)

Ystorlms. Th-iel effect of disaster stormns on system reliabilitycan be estimated by separate consideration of such high stress wherel)eriods, again using the methods of the paper. C = number of nonstorm-associated comnponent failulres duriingHistorical weather records in the form of hourly and s-pecial observation periodobservations, fromni which required weather statisti(s can be Y=summation of n-ormal weather exposure times for each mileobtained, are available fromn the United States Weather 13u- of line or piece of apparatus duiring the observation periodreau for all maj or weathecrstations. Weather statistis 2. Astormy weather "permanent" forced outage rate shouldshoulld be obtained from data recorded at a stationl in the be obtained for components whose failure rate is affected byvicinity of the system being2, studied. For P'ublic Service, veather and which are used in a portion of the system, suchdiata from the New-ark and Tienton stations were analyzed.Weather data are scanned with a definition of wveather condi- parallel system. The stormy weather failure rate, .t', hastions constituting a stormi to deterlmine (1) the total amount the units of failures per year of stormy weatherrae, unit ofof timze each y-ear during which normal and storm con(litionls apparatus or per mile of line. It may be estimated b- theprevail, anid (2) the distributions of normal and severe weather equation:period durations.Definitions of weather cotnditions constitutinig stormis imiust ,=C(5)

Ibe carefully correlated vith conditions observed to cause an Y'miereased rate of component failure. If storm definitions are wheretoo stringent, too few storms will be recog,nized. On the otherland, if storm definitions allow too imany mild, relatively un- C =number of stormn-associated componenit failuires durtinlgdestructive storims to be recognized, the imi1portant failure- observation period

bui-hingeffectof mor sever e storis will be diluted. Using P=summation of stormny weather exposure times for each milebummehing effect of mlore severe stormns Will be dilutedl. Using of line or piece of apparatus during observation periodthe storm definitions given in Table I, Newark weather datawere analyzed for the period 1955-1961. Storms were foundto occupy an average of 0.65%o of the time a year. Figs. 2and 3 show, resl)ectively, the distributions of durationis ofstormy and normal weather periods found for the 1955-1961period. Note that these distributions do not differ greatlv O DISTRIBUTIONfrom the expgonential. Thus it apopears reasonable to utilizeOS DISTRIBUTIONthe exp)onential distribution to describe severe and normlal 3j x 85k EXPONEN12ALTD VAW,E 191 HR,wveather l)eriod durations in making reliabilityr calculations. hgeO\|!SDLRATIONREEO OSREFOr a discussion of the exponential distribution, see Feller .6 -;-r \ li

WVhile stormy and normal wveather period diurations applear < 4 t\sto be appIroximlately exponentially distributed ov-er anl enltire <> I t

the same for different seasons of the year. Fig^. 4 showrs the Fig. 3. Normal weather taverage frequenlcy of occurrence of (1) thunderstorms and period duration distribu- CS 00 20 30 40 00(2) all other stormns for Newark. Fig. 5 shows the distribu- tion for Newark, N. J. DURATION HOJRS

JULY 1(364 Gayer, MWtmct, Patton-Power System Reliability-I 729

3. A temporary forced outage rate for various types of coIn- Fig. 5. Distribu- x -

ponenits is necessary if system temporary outages are to be tions of thunder-II 0calculated. In general, a temporary outage does not require storm and "other"i 8

repair or replacement of facilities but can be remedied by a sorm durato Jreclosing operation or by replacing a fuse. Component f N N J T-temporary forced outage rate is estimated by dividing the C THUNDERSTORMOS, AVERAGE OF

- BSERVED DURATIONS .95 HR.'number of component temporary outages during the observa- ER STORMS AVER Otion l)eriod by the number of unit-calendat years of compo- -- - _-ZOBSERVED DURATIONS=1-93HR.nent exposure durina the observation period. It is not con- .2- -4c& -----! -sidered necessary to separate temporary outages into normal -Xand stormy weather components because temporary outages , 2 3 A S Fare usually quite short, making the chance of overlapping DURATION, HOURSoutages negligibly small.4. AMaintenance outage rates should be obtained for typesof components wAhich are used in portions of the systemwhich operate as a parallel system. MAlaintenance outages of If it is desired to investigate the ossible economies ofcoln)onents in radial systems, if such occur, can be lumped g2 . ~~~~~~~parallel systems with less than 100%7 redundancy, that is,with normal weather forced outages. MIaintenancee outag,e . ...............witnrmawateroredoutge. ainennc o e parallel components not individuallv capable of carrying sys-rate Xi" has the units of outages per calendar year per unit t

of aparaus opermileof ine.It i estmate bytem peak load, then certain additional data are required.These data consist of component capabilities, loadings under

Cv"=f (6) contingency conditions, and curves giving the probability ofyi( carrying load. A more complete discussion of overload out-

ages in parallel systems is deferred to a later section of thepaper.

C" = number of component mainteniance outages durinig observa-tion period AssuMPTIONS IN RELIABILITY CALCULATIONS

Y= summation of observation periods for each imile of linle or Some of the assumptions which were made in derivin thepiece of apparatus (exposure to maintenance is assumed to b mnbe essentially the same each year) reliability calculation methods; of the paper have been men-

tioned in preceding sections. These assumptions together5. Distributions of component repair times prevailing during with other pertinent assumptions, are listed as follows fornormal and stormoy weather and for maintenance are obtained ease of reference:from historical records. Repair time is here taken to meanthe duration of a period during which a component is out of 1. Timeis to failure (periods beteeen failures) and repairservice being repaired or replaced following a forced outage, or times are exponentially distributed durino 1)th normal andIthe time a component is out of service for maintenance o01 other stormy eather. That iS,work. Repair timie distributions seem to be exlponential to probability [time to failure (during niorial weather)>t] e-xta reasonable approximation. That is, probability (repair probability [time to failure (during st-ormy weather)>tl = e -'ttime>t)= e-'/". For examp)le see Fig. 6. Trhe expectedTt

2. The durations of periods of normal and stormy weather arevalues of repair times during normal and stormy weather n d>, ~~~exponentially distributed.and for maintenance, denoted respectivelv by ri, ri and ri",are estimated by the means (averages) of observed repair 3. Repair times are typically very short coimpared withtimes. Because it appears that repair times during normal times to failure and times betwveen storms.and stormy weather are very difficult to separate and are not 4. Storms are very short in duration compared with timesgreatly different, the calculation methods to follow presume to failure for components. Storms are also short cornparedthat the same repair time distribution prevails duriing bothl with typical repair times.normal and stormy weather. This combined repair time dis- 5. Maintenance outages occur at random during normaltribution is obtained from records of repair times following all weather periods except that components are not taken out forforcedz outages.

maintenance if (a) such action would cause the remainingcomponents in a parallel system to become overloaded, or (b)maintenance could not be completed before a storm struck.6. Maintenance down times are exponentially distributed.

I_______________ 7. In computing parallel svstem down times resulting froniEo loC --- ; | overload outages it is assumed that once a line is over-9 1 -~ --. nloaded, it remains overloaded and down uintil a failed l)arallel

2 THUNDERSyORMS-> componiplent iSrep)aired.° 6 X H--|-tx ~~~~~~~~SERIESSYSTEMISLOLTHESRMS- In this and the followinlg sectionl, thte necessary mlathle-

~~~~~~~~~~~reliability in simple series systems and in simple parallel2L}-n-- i .4Average systems are presented. Because it is possible to regard mnost

.t L ~L I-- number of storms pzowxer system networks ass a group of series and parallel sub-J00 A M JJA S O N9 o per month In syJstems, these expressions can be ulsed to calcuilate measures

MONTIH Newark, N. J. of reliability in general systems.

730 Gay!er, Mio?tnInwl Patton--Pno7er Sjystemn Ieliabdmlty-I Jumxz 1964

Consider a system coImposed of n dissimiiilar components Conisidlerationi of outages caused by overloading in. a seriesconnected in series. A component may be a single piece of svstem is not meaningful unless the series system operates inapparatus, several miles of line, or even an equivalent com- parallel with other components. If such is the case and ifponent composed of several other compoonents connected in overloa(l outages aie to be evaluated, then the capacity andseries or parallel or both. The followinig reliability paraml- contingency loading of the equivalent element e, representingeters are required: the seriecs system, are requiired. The capacity of the equiv-

alent elemeint is obviously the minimum of the capacitiesX,, X,, , X,= normal weathler comp)oneat failulre rawte, of the comp)onenlts making up)the series systemn. The con-failures/year of normal weatherJ, Xa *--, X,,'=stormy weather comllponenlt failur7e ralte, tingency loading( of the equiivalent element depenids oni the

failures/year of stormy weather configuration of the parallel system being, sttudied.XII", X2, .,. X,nt= component mainteniance outage rate, muilll- If a load was attached to the last element of the seriestenance outages/calendar year snsten- ancd a source to the first elemet tlie- the significantri, r2, ., rn = expecte(l valtue of repair time for all forced out- s a a s,ages, years mieasures of reliability for the load wirould be calculated asr1 r2 , .,.r,,"= explected value of dlown time for maintenancile followss:outages, years

N =expected value of normal weatlher pe-riod duration, years 1. Annual outage rate:S =expected value of stormny weather period duration, years

vt XlSt-,L=Xfe+ e (oultages/cailend(iatr XPt)(14)The required valuies of component failuire or outage rates just 2. Expected value of outage duration (reStiOn time):listed are obtained fromn norinalized failure rate data by miiulti-plying the normalized values by, the appropriate nuimber of 7, Xferfe+Xe"re (yeas) (15)units, such as miiles of line. XSLThe over-all forced outage rate (normal and stormy which may )e converted to houlrs by multiplying by 8760.

weather) for the ith component is

Ag S X, (forced3. Average total outage tiimie per year:Xft-N+S ou+j\T+S tt (force(lonitagfes/ealenlldart year) (7) r.,

us+ +ff-- XSLTS r (yekars/ye'ar) (16)

This is an appropriate approximation wlheni XiN and Xj'S are rs +very small conmipared with unity, as is usually the case in XSLutility practice. The over-all forced outage rate of the series which may be converted to lhouirs per year by multiplyingsystem is by 8,760.

4. Probability that a single ouitage wvill last longer than IXfe= L Xfi (forced otifages/calenwar yert) (8) hoursi-1m

Again, this approximation is useful wlivei N2,Xj tand S2Xi' aren

Xf,e t/ (876 i)+X,ifCti(8760 Tr")small compared with uniity; see referetnce 7. Similarly, the P (Otltalge > t lhours) =s-- (17)maintenance outage rate of the series system is

n PARALL-EL SYSTENS)e EXs (mzillte -ice ou1tages/calentdar year) (9

Xe

Xi" (maintenance outages/calendar year) (9) The equations to be given for parallel systemes are limllitedto two components, or equivalent components, in parallel.If the sesries system acts in parallel wvith other components, it If three or more components are in l)arallel, it appears that, for

is necessary to calculate normal and stormy weather failure typical characteristics, thev mav be treated two at a timerates for an element c which is the equivalenit of the series i.e., two of the three parallel components are combined; thensystem. This equivalent elemnent is used in furlther calcula- the equivalent combined with the third. This simplificationtions. For the equivalent element implies a certain additional degree of independence between

n the three or more components not considered when oinly twoXe = xi (failures/year of norn.tl weatler) (10) componenits are in parallel. In other words, not quite as

much failure bunching due to storms is recognized in the cal-and cullation when three or more components are in parallel.

n

Xe' = Xi (failures/year of stormy weather) (11)

I.0Expected values of down- timeit for series sy,stem as a result x0of forced outages and maintenance outages are, respectively, . E DISTRIB I ___

OBSFT VFI nT Int

,'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.6.1r,o EXPONENTIAL DISTRIBUTION WITH________12_F R iti cC|| ( EXPECTED VALUE-775 HR,Tf je (yas 1) distribution for 26- H L11 REPAIR TIMES

and kv open-wire sub- H |Itransmission cir- m'.W

E~~~~~~~~~~~~~~~~~evc El""ectric LL_retw --- (years) (13) and Gas Company 024sBI 24I Bs 22e ~~~~~ ~~ ~~~~~~~~~~system HOURS

JULJIY 196i4 Goere, AIo?utrn? cat, Pattonl- -Pom ere System IReliczhility--I 731

This is consisten-t with the fact that a systemimiiade uI) of X NF=§(x x. +x,x'i)+2 X2three or inore components in parallel is likely to be more dis- SN Npersed geographically than a 2-component svstem with the 11-1~~~~~(failures /year of stor'IlVy weather ) (20 )result that all components Are somnewhat less likely to beaffected by the same storm. Xe" = x1\2Tr1" +X2'X?.2' (mainteniance outages/calendar year)Component or equivalent componient data req(uired in (21)

miiaking reliability calculations in parallel systems, excludingoutages due to overloads, are the same as that listed previously The failure rates A., k'I , an-d X, for the equivalent componentfor series systems. representing the parallel systemn are derived frnom the variousThe calculation of reliability in parallel systems may be terms of the expressioni for XSL. The nature of the weather

approached in several ways. The most accurate inethod, wAThenthe second component of the )arallel system fails result-at least theoretically, involves treatment of systemn experience inig in system failure determines whether a term contributesas a Markov process. The imiathematical background for this to equivalent coMponent normal or stormy weather forcedapproach is discussed by Felleri6 and the approach used by De outage rate. The termxs in the exp)ression for XSL accountingSieno and Stine2 in their paper on power system reliability. for the effect of imaintenance outages yield the maintenanceThe AIarkov process approach generates a series of linear equa- outage rate of the equivalent component directly.tions which must be solved simultaneously to find system Given that the timnes to failure for components are X-properties such as long-run average availability and failure ponentially distibutedaine that coml)onent repair times arerate. Explicit consideration of a 2-state fluctuating eniviron- independently and exponentially distributed, the expectedn-cnt for a parallel system composed of two nonidentical value of parallel system downi time as a result of system out-components requires the solution of eight simultaneous equa- ages caused by overlapping component forced outages onily istions. Such a solution is, of course, easy on a high-speeddigital computer, but reconsideration of the l)roblem lpovides 1 - 1 r2 (years) (22)an intuitively appealing and simple approximation which camn 11 rl+r2be quickly computed by hand. The approximate method cani r r2be expected to give results that are withiii a few per cemit of Tthose obtained by using the Markov method. It should be hedeptae oveal s a componentDeremarked that the Markov method itself is an approximiiatiolonand does depend upon certain specific distributional assunp- le"=2-1'1-_'2'_it_+tions. \1,'°+>tl7/-2+rtThei simplifid approximate method yields the followxing 2

expression for parallel system over-all outage rate due to ( X."X1r2 9_ (years) (23)normal and stormy weather forced outages and maintenanceX X2? ll'+X2'X1r2 )'I+r2

'

outages: The reasoninig used to derive re" may be explained by comlsider-N s ing the first term of the expression for re'. The first factor of

XSL=~ XIX2(TI+r2)+-.(Xl'X2r,+ X2 1?12)+,AT= L ± this term is the fraction of system outages involving coma-S 1,1,,,,,, 5ponent mainteinance outages in which component 2

,(X,X2t1r1+X2X1r2)+2 XIIX21x '+xI'X2_r1+X2'Xd_.12 failed while component 1 was out for maintenance. The sece-AjI+Al ond factor of the first term gives the expected system down(outages/calendar year) (18) timiie given that component 2 fails while component 1 is out

for miaintenance. The second term of the expression forA derivation of expression 18 is given in Appendix I. If the " accounts in like manner for the situation where a forcedp)arallel system operates in lparallel with other components, outage ofcoioimponent 1 overlaps a maintenance outage of com-then normal weather, stormy weather, and maintenance out- Ponent 2.age rates are required for the equivalent component represent- The significant measures of reliability for a load fed by aing the parallel system in further calculationis. These r-ates parallel system are: annual ouitage rate, X; expected value ofaI e expressed as follows:

restoration time, rSL; average total outage time per year,s Us; and the probability that a single outage will last longer)e = XX2(r,+r2)+- (Xi'Xoro+Xo'Xo2) than t hours. The annual outage rate, XSL, is given by equa-

tion 18 of this section. The expected value of restoration(failures/year of nlormlal weather) (19) -time, rSL, and the average total outage time per year, Us,5:are obtained using the same equations (15 and 16) as for seriessystems. The probability that a single outage will last longerthan t hours is given by the following expression:

Table ll. Component Failure Rates P (ouitage>t hours)=-[(XsL-X01)e lX (O?6orf,)+XSLL

Failures Per Calendar Year

Daiuring During Normal jlXrte (7+l)X , _/86t 2)

Storms Storms Weather 'X

0 0 0.441 0 0.444 Tlo illustrate the difference in systeml failure rates calculated20 0084821 0.39528 i3.59 0.355 using the method of the paper anld a method p)resuming in-40 0.1764 0.2646 27.16 0.266 dependence, consider a "system" composed of two identical80 0235248 0°08824 54.324 0.08878 10-mile sections of overhead circuit acting in parallel. Sup-100 0.441 0 67.90 0 pose that the average numnber of forced outages in both

732 Gayer, MVontm.eat, Patton-Power System Reliabii;ty-I JULY 1964

25 Fig. 7. A comparison of of reducing system cost is to reduce the capacities of parallelt23 r-: X S 4 parallel system failure system components relative to the highest loads the com-21 S=___ / rates calculated (1) as- ponents may be called on to carry under contingency condi-LL. uiS2.50HR~N=382 HR suming independence, tions. Obviously, this procedure will introduce the possi-FZa17a - _/ and (2) using the method bility of system failure due to component overload and will,w _ . of the paper therefore, reduce system reliability to some extent. A

13 component with less than 100% capacity may or may not bew u) / /<< Il f /: i,/ S able to carry contingency load depending on time of day,

S=125 HRi day of week, season of year, duration of contingency, and=JD 7 / / N=l91 HR. omponent capacity relative to the peak value of load. The

method which is to be described permits an estimate of theV)>_

3 ! ;reduction in system reliability caused by overload outages to0 20 40 60 80 1oc be calculated so that systems with less than 100% redundancy

PER CENT OF COMPONENT FAILURESca evltd ''"rduanOCCURRING DURING STORMS can be evaluated relative to 100% redundant systems.

The first step in the procedure for estimating the number ofoverload outages a parallel system is likely to experience is thedetermination of the probability that a component will be

stormy and normal weather observed per year per mile for the able to carry a given contingency load. Assuming thatstormy *

cuXq

contingency is equally likely to occur at any time and thattype of circuit in question iS 0.04451. ThIen ..component capacity is constant, the probability of carryingf,= Xf2= 10(0.0441) = 0.441 failure/calendar year contingency load for a given time is found by randomly

Let r1 = r2 =7.75 hours = 8.84X l0-4 years. Then, assuming sampling the load cycle which would be appropriate if the

independence and ignoring maintenance outages the system component were called on to carry contingency load con-

failure rate is tinuously. A given sample try is termed a success if loaddoes not exceed component capacity during the assumed con-

XfiXf2(rl+r2) = 3.44 X 10-4outages/calendar year tingency period. The probability of carrying contingency

Let the expected values of stormy and normal weather periods, load successfully is estimated by the ratio of the number ofS and N, be, respectively, 1.25 hours and 191 hours. With successes to the total number of samples. The totalthese values of S and N storms occupy an average of 0.65% load cycle period sampled should, in general, be at least

of the time each year. The values of component failure rates a year to insure that all types of load cycle variationappropriate during stormy and normal weather periods when are included: daily, weekly, and seasonal. If com-

storms occupy 0.65% of time are given in Table II as a fune- ponent capacity for design purposes is not constant through-tion*of the percentage oftotal c f t out the year, it is necessary to sample the annual load cycletiong torms.taralle sytem failuresmaybccu- by seasons. A family of curves giving the probability ofdluring storms. Parallel system failure rates may be calcu-

caryn cotn.n loa asafnto fcnignylalated from the component failure rates given in Table II with o ylequation 18. Maintenance outages are ignored. Expressing duration and component capacity can be generated by sam-

the resultant values of system failure rate in per unit of the plng the load cycle with different values of contingency loadvalue of system failure rate obtained assuming independence duration and component capacity. Fig. 8 shows a typicaland plotting the per-unit values as a function of the percentage family of probability curves. The Public Service system load

r 1 * * 11 W- ww ~~~~~~cycleshape for the year 1961 was used in obtaining theseof total component outages occurring in storms yields Fig. 7. yFig. 7 indicates that the error in parallel system failure rate curves. Component capacities were assumed constant overcalculated assuming independence can be appreciable and is the year. Total system load cycle shape is thought to bemost serious when a sizable percentage of component failures representative of load cycle shapes in transmission and sub-occur during storms. Fig. 7 also shows the effect on parallel transmission systems serving a combination of industrial,system failure rate of doubling storm durations while keeping commercial, and residential loads. It is worthwhile to note

the percentage of time that is stormy constant at 0.65%. The that the probability of carrying load for a given contingencyconclusion is that a smaller number of long storms is more load duration and component capacity increases as load cyclelikely to cause system outages than a larger number of shortstorms even though total storm time is the same in eithercase.

OUTAGEs DUE TO OVERLOADS IN PARALLEL SYSTEMSIf each component in a parallel system is capable of carry- 8

ing the highest load to which it may be subjected in any BO ~ : AACTY -contingency, the system will suffer an outage only if allparallel components are out of service at the same time. -Such a system will never suffer an outage as a result of over- 06zn5 -~ ___. !2I4 -- --loading of a sound component when a parallel component °or components fail. A parallel system which can never fail . tt-==because of component overload is termed fully or 100% I 5 + v jtt -

has a capacity which is greater than or equal to the highest CONTINGENCY LOAO OURATION, HOURSload that component can ever have to carry. A fully re-dundant system while quite reliable is also relatively expen- Fig. 8. Probability that contingency load can be carriedsive because of high component capacities. A possible means successfully

JULY 1964 Gaver, Montmeat, Palton-Power System Reliability-I 733

variability increases and load factor decreases. This follows to approximate the integral in equation 28 or 29 as a summa-because load is below component capacity more of the time tion overj. Thus, for example, P1 becomeswhen the load cycle is found to be highly variable and the loadfactor low. P1=1- Ql(Xj)M2(Xj)-M2(Xj-l1) (30)

Consider a system composed of elements 1 and 2 in parallel.Presuming that component maintenance outages never pre- A practical approach to the calculation of P1 and P2 wouldcipitate system overload outages, system outage rates during seem to be to split time into 1-hour intervals and to read valuesnormal and stormy weather due to overloads are approximated for Qi (Xj) off the probability curves that have been prepared.as follows: We can then put Xj=j where j =0, 1, 2, . .. Presuming re-

pair times are exponentially distributed, P1 is given as follows:Xoe = XiP2+X2Po (overload outages/year of normal weather)

(25) P1= 1- , Q'(j)[e-(i-1)/(8760r2) - e-J/(8760r2)a (31)j =1

andOr perhaps better,

X,e' = X1TP2+X2'P1 (overload outages/year of stormy weather)(26) Pi =I1-Ql(j -1/2) [e 1) 876012) e -j/(8760r2) ] (32)

where =1

P= probability that component i will not be able to carry Because there is reason to believe that Q1(X) should equal-contingency load Q, (24) for all X greater than 24 hours,

The system over-all outage rate due to overloads is Xofe where P1= 1 - Qj(24)e 24/(876072)-(1 -e-1(876072)x

Xofe S= Xf+ oe+ ?xoe' (overload outages/calendar year) E Q,(j-1/2)e (l)/(876or2) (33)N+S N+S=i

(27)and

This expression for system overload outage rates slightly over-estimates these rates because some component forced outages PI = 1-Q2(24)e -24/ (S7607j) - ( -e - l (876071) ) Xoverlap causing system forced outages and hence are not avail- 24able to cause overload outages. Since the number of com- E Q2(j 1/2)e -1)(8760ri) (34)ponent forced outages that overlap is small compared with j=1the total number of component forced outages, the error in Systems with three or more components in parallel are han-the approximation should be negligible. dled in much the same way as 2-component systems. ForThe probability Pi that a component will not be able to instance, in a 3-component system

carry contingency load for a given load cycle shape is relatedto: (a) component capacity, (b) the value of contingency load Xoe = X1P2,3+X2P1,3+X3P1,2+X1 ,2P3+X1,3P2+X2,3PIat annual peak, and (c) the distribution of repair times for the (overload outages/year of normal weather) (35)component(s) which failed and thus precipitated the contin- wheregency. In a 2-element parallel system, peak contingencyload is simply the sum of the normal peak loads on the two Xi=normal weather failure rate of ith componentelements. In more complex systems, however, peak con- X1.i=normal weather failure rate of equivalent component com-tingency load can only be found by a load flow analysis. posed of components i and j in parallel (see equation 19Component capacity expressed in per unit of contingency load for method of calculation)Component capacity expressed in per unit of contingency load Pi = probability that ith component will not be able to carry loadat annual peak, the distribution of failed component repair when other components are outtimes (identical to contingency durations), and a set of curves Pi,j= probability that components i and j will not be able tosuch as those shown in Fig. 8 can be used to estimate Pi. If carry load when other component is outcontingency load duration were a constant value rather than a It appears that the terms in equation 35 involving forceddistribution of values, the quantity Q = (1 -Pi) could be read outages of more than one component may be disregarded withdirectly by entering the probability curves with component an error of less than 5% in typical cases.capacity and contingency load duration. If, however, con- System failure rates due to all types of outages (overlappingtingency load duration is not constant but is exponentially forced outages, maintenance associated outages, and overloaddistributed as appears reasonable then Pi and P2 are givendistributed aspparrasnabe,thn nd2routages) can be obtained by adding the values found for eachas follows: type of outage. Thus, defining TXSL as parallel system

Pi= 1- f0 Q,(X)dM2(X) (28) failure rate as a result of all types of outages,

and TYASL =-XSL+ Xofe (total outages/calendar year) (36)

P=1 -fJ0 Q2(X)dM1(X) (29) The expected value of 2-component parallel systeml downtime as a result of all types of outages including overloads is

where T~~~~~~~~~~~~~rTSL,whereQi (N) =probability that component i can successfully carry Xfe Xe" Xf1PI XVIPO

contingency load for a timeX TrsL= rf6-1 r¢"+- r1+ r2 (years) (37)MkI(X) =probability that repair of component i is completed in TXSL TXSL TXSL TXSL

time X=1-e-X/r6 for exponentially distributed repair times The value of TrsL for a parallel system with three or more

components is not indicated here, but is found in a similarSince the function Q1(X) is difficult to work with, it is useful manner. The average total outage time per year for the

734 Garer, Montmeat, Patton-Power System Reliability-I JULY 1964

failure will follow if the second component fails during the repairtime of the first component. Failure of the second componentmay occur during (a) normal or (b) stormy weather giving riseto the following two cases.

V S r _4 1.1. SECOND FAILURE Is DURING NORMAL WEATHER

F_ Presuming that repair times are typically very short comparedwith times between storms such that, at most, one weatherchange is likely during a repair time, system failure rate per

B A calendar year as a result of component failures during normal(2 CUSTOMERS) (I CUSTOMER) weather only is:

Failure rate= (long-run fraction of time that weather is normal)Fig. 9. A sample system [(normal weather failure rate of component 1)

(probability that a storm does not occur duringrepair of component 1) (probability that com-ponent 2 fails during repair of component 1)+

system as a result of all types of outages is TUSL. The quan- (normal weather failure rate of component 2)tity TUSL can be obtained from equation 16 by substituting (probability that a storm does not occur duringTrSL for rSL and TXSL for XSL in that equation. repair of component 2) (probability that compo-

Conslidering all types of system outages, the probability nent 1 fails during repair of component 2)]that a single system outage in a 2-component parallel system N F ri\ r2 \will last longer than t hours is NN+S L N1Nr2r1)+x2 N1-/(Xlr2)J

1 t 876or Since r, and r2 are typically very small compared with N, it canP (outage>t hours)= [Xfee- /( fe)+ be shown thatTXSL

18780T2__TI"Ntl 8760~ "

X, "x,r, "e- '/ ( +"7J)+7X2"X1r2ei- +r2 Failure rate= NS X,X2(ri+r2)Xf,P2e t/(876orl) +Xf2Ple-t/(6760r2)1 (38) 1.2. SECOND FAILURE Is DURING STORMY WEATHER

The probability of a single outage longer than t for systems System failure rate per calendar year as a result of first com-with more than two components is found in like manner. ponent failure during normal weather and second component

failure during stormy weather isEXAMPLE SOLUTION Failure rate=(long-run fraction of time that weather is normal)The various measures of reliability for a simple series- [(normal weather failure rate of 1) (probability a

parallel system will now be calculated to illustrate the method storm occurs during repair of 1) (probability 2ofapplyingtfthe preceding sections to complex fails during that storm)+(term giving other

of applying the equations of th rcdgscln oemlxpossibility)]networks. The system to be studied is shown in Fig. 9. Thesource is considered to be 100% reliable. Two customers are Nr_ r2served at B and one at A. The computations for the example N+S N Nsystem are contained in Appendix II. T S\

NS rV} (X1X2'r±+X2X1'r2)Conclusions N+S N

The probability methods presented in the paper will permit 2. INITIAL FAILURE Is DURING STORMY WEATHERimportant measures of reliability in general power systemnetworks to be calculated from basic system component data. If the initial component failure is during stormy weather,

ml 1 1-1*1- > 1 . 1 . ~~~~~~system failure will occur if the other component fails during (a)Thus, the reliability of alternative proposed systems can be stormy or (b) normal weather before the failed component iscompared to discover the system which yields the highest restored to service. Thus, presuming that repair times arereliability or a desired level of reliability at lowest cost. Im- typically short compared with the time between storms, systemportant features of the method of the paper include the ability failure may occur as a result of failure of one component duringto consider failure bunching caused by storms and outages as a storm and failure of the other component during normalweather following that storm.a result of component overloading in parallel systems.Future papers will describe experience in application of the

techniques which have been developed, data for reliability cal- 2.1. SECOND FAILURE IS DURING STORMY WEATHERculations, and a digital computer program to facilitate calcula- Presuming that storm durations are typically short comparedtions. with repair times, system failure rate per calendar year as a

result of component failures during the same storm is

Failure rate = (expected number of storms per year) [(probabilityAppendix I 1 fails during a storm) (probability 2 fails during

When the conditions listed in the section of the paper titled remainder of storm)+(term giving other possi-"Assumptions in Reliability Calculations" are reasonable, it is bility)]possible to obtain an approximate expression for parallel system 1faiur rae hic wllyield results very close to those obtained = [(SX1')(S)X,')+(SX,')(S)X1')lusing the Mlarkov approach. In the derivation of the simplifiedN+approximate expression for system failure rate to follow, discus- N /Ssion is limited to a 2-component system. = {\\The derivation proceeds in tbe following stages:N+ /

1. INITIALFAILURE ISDURING NORAL WEATHERNote that the probability of a second component failure during a1. INITIALFAIURE Is DURING ORMAL WEATHERstormn iS the same as the probability of the first failure because ofSuppose that, at some initial instant, the system is entirely opera- the "memoryless" nature of the exponential distribution describ-

tive. Then if one component fails during normal weather, system ing storm durations. If storm durations were a constant value

.JUSLY 1964 Gayner, MIontmeat, Patton-Power System Reliability-I 735

rather than being exponentially distributed, the probability of The expected values of repair times following forced and main-aS n tenance outages for the equivalent component are given bya second component failure during a storm would be 2 Xi) and equations 12 and 13.

the system failure rate would be half that given above. rfel =10 years

2.2. SECOND FAILURE Is DURING NORMAL WEATHER rei = 10 3 yearsSystem failure rate per calendar year when the first component 2. An equivalent can now be found for the system link made

fails during stormy weather and the second during normal up of components el and 3 in parallel. Call this equivalentweather is component e2. From equations 19, 20, and 21, the failure rate

of the equivalent, not involving overloads, areFailure rate = (expected number of storms per year) [(probability '

t

1 fails during a storm) (probability 2 does not fail X,2= 8.50X 10-4 failures/year of normal weatherduring that storm) (probability 2 fails during repairof 1)+(term giving other possibility)] X22' = 0.261 failure/year of stormy weather

I [(SX,')( 1-SX2' )(X2rl)+(SX2')( 1-SX,')(Xir2)] Xe2 = 3.20 X 10 3maintenance outages/calendar yearN+S The over-all forced outage rate for the equivalent is given by

Usually SXi'<<1, hence equation 7:

N (S\ Xfe2=2.55 X 10-I forced outages/calendar yearFailure rate = -- -l- (VAX2ri+ 2 'X1 r2 ) N+SF r +The expected values of repair times followed forced and main-tenance (associated) outages are given by equations 22 and 23.

3. MIAINTENANCE OUTAGES rfe2=5X 10 4 yearsIt is presumed that a componient will never be taken out for rf12 5X 10-4 yearsmaintenance if this would precipitate a system outage. Like-

wise, a component is not taken out for maintenance if a storm Since the capacities of parallel components el and 3 are notis impending. A system failure can occur, however, if a com- 100% of the contingency loads they may have to carry, theponent fails during normal weather while the other component possibility of system link outages as a result of overloads exists.is out for maintenance. System failure rate as a result of this Because the system link does not operate in parallel with anysituation is as follows: other component, it is not necessary to separate overload outagesFailure rate=(maintenance outage rate of 1) (probability 2 fails occurring in stormy and normal weather. Therefore, equations,

while 1 is out)+(term giving other possibility) 25, 26, and 27 reduce to the following for this case:

= XI "(X2ri")+ X2"(Xir2') Xofe2= Xfe1P3+i+X3Pei (overload outage/calendar year)

Adding the system failure rates given under 1, 2, and 3 above The probabilities that components el and 3 cannot carry con-gives system failure rate per calendar year due to all causes. tingency load successfully are given by equations 33 and 34,

making use of the curves of Fig. 8.

Pei=P3= 0.019Appendix 11 then

All components of the example system of Fig. 9 are overhead Xofe2 = 0.030 overload outage/calendar yearlines. The characteristics of the components are as follows:

(a). XI = X2 = Xi = X4 =0.4 failure/year of normal weather The total number of outages per year of the parallel system link(b). XI'= X2'= X3'= X4'= 20 failures/year of stormy weather represented by equivalent component e2 is given by equation 36.(C). X1'=X2"=X3"=2 maintenance outages/calendar year TXSLe2 = 3.575 X 10-i outages/calendar year(d). r,= r2 = r3 = r4 = 8.76 hours= 10- years

(e). ri'=r2'=r3'=l0-3 years The expected value of repair for component e2 considering al(f). Component 4 is not taken out for maintenance; hence types of outages is given by equation 37.4'=0 and r4' has no significance.(g). The capacities of components 1, 2, and 3 are 90% of the TrSLe2= 9.35X 10-4 yearspeak contingency loads that they may be called on to carry.

=-8.20 hoursThe expected values of stormy and normal weather periods are

The customers served at point B in the system experience theS=1.25 hours= 1.43X 10-a years same outage (interruption) rate and expected value of down

N 191 hours=2.18X10-2 years time (restoration time) as equivalent component e2. Theaverage total interruption time per year experienced by thel. The first step in the reliability calculations for the example customers at B as given by equation 16 is 0.293 hour. Thesystem is to compute an equivalent for the system link made up probabilitv that the customers at B will experience a singleof components 1 and 2 in series. Call the equivalent component interruption longer than t, say 24 hours, is given by equation 38el. From equations 9, 10. and 11 the failure rates of the equiva- as follows:lent are

P(B out>24 hours) = 0.0553Xei" =4 maintenance outages/calendar year

3. The customer at point A is served by components e2 and 4Xei =0.8 failure/year of normal weather in series. The interruption rate at A is given by equations 8,, . ~~~~~~~~~~~~~9,and 14 and is 0.562 interruption per calendar year. The

Xei' =40failures/yerof stormy weatherexpected value of restoration time for the customer at A is givenThe over-all forced outage rates for single components is given by equations 12, 13, and 15 and is 8.71 hours. The averageby equation 7: total interruption time per year for the customer at A is given

by equation 16 and is 4.9 hours. The probability that theXf =Xfi =0.527 forced outage/calendar year customer at A will experience a single interruption longer than

then 24 hours as given by equation 17 is 0.0645.4. The measures of reliability for the sample system then are

X\fei= XJl+)Xfi= 1.054 forced outages/calendar year as follows:

736 Gayer, Moutmealt, Patton.-Power System Reliability-I JULY 1964

(a). Average number of interruptions per customer served Referencesper year

1. A PROBABILITY METHOD FOE TRANsMISSION AND DISTRIBUTION2(3.575 X 10 2)+I(0.562) OUTAGE CALCULATIONS, Z. G. Todd. IEEE Transactions on PowerF= = 0.211 interruption/year Apparatus and Systems, vol. 83, 1964.

2. A PROBABILITY METHOD FOR DETERMINING THE RELIABILITY(b). Average customer restoration time OF ELECTRIC POWER SYSTEMS, C. F. DeSieno, L. L. Stine. Ibid.,2(8.20)+1(8.71) vol. 83, Feb. 1964, pp. 174-81.

3 = =8.37 hours 3. ECONOMIC EVALUATION OF INDUSTRIAL POWER-SYSTEM RE-LIABILITY, W. H. Dickinson. AIEE Transactions, pt. II (Applica-

(c). Average total interruption time per custonmer served per tions and Industry), vol. 76, 1957, pp. 264-72.year 4. AUTOMATED DISTRIBUTION SYSTEM PLANNING, R. F. Lawrence,

2(0.293)-1(4.9) F. E. Montmeat, A. D. Patton, D. Wappler. IEEE Transactions on\/2(.23)+\]49)= 1.83 hours Power Apparatus and Systems, vol. 83, Apr. 1964, pp. 311-16.3

5. THE APPLICATION OF PLANNING CRITERIA TO THE DETERMINA-(d). Maximum expected number of interruptions experienced TION OF GENERATOR SERVICE DATES BY OPERATIONAL GAMING,by any one customer per year C. A. DeSalvo, C. H. Hoffman, R. G. Hooke. AIEE Transactions,

pt. III (Power Apparatus and Systems), vot. 78, 1959, pp. 1752-58.Fmax = 0.562 interruption/year 6. AN INTRODUCTION TO PROBABILITY THEORY AND ITS APP,ICA-

* . . ~~~~~~~TIONS,W. Feller. John Wiley & Sos'n. New Yr,N .(e). Maximum expected restoration time experienced by any 1957, vol. Sons, Inc., York, N.one customer

7. RANDOM HAZARD IN PRELIABILITY PROBLEMS, D. P. Gaver.I?max=8.71 hours Technometrics, vol. 5, no. 2, May 1963.

A Review of Some Basic Characteristics ofProbability Methods as Related to PowerSystem ProblemsC. W. Watchorn, Fellow IEEE

Summary: This paper presents some basic concepts of the THE PROBABILITY OF Loss OF LOADphysical significance of various aspects of probability methods asapplied to power system generating capacity problems. It also At the time of the daily peak loads, the probability of loaddiscusses the physical meaning and interrelation of various stand- loss may be:ards of service reliability, and the determination of the installedcapacity benefits of an interconnection by mneans of the applica- 1. As a pure number, ortion of probability methods. 2. As a rate in terms of years per day.

At any time at all, it may be:Although the literature is replete with papers on the applica- Asa pure number, ortion of probability methods to generating capacity problems,1 1. As a pure number, orit nevertheless appears appropriate that the principles in- 2. As a rate in terms ofyears per hour.volved could be reviewed to provide a more basic under- THE PROBABILITY OF Loss OF VARIOUS MAIAGNITUDES OF LOADstanding of these problems than is generally prevalent. Awider use of such methods might then be expected, resulting At the time of daily peak loads, this probability may be:in benefits not obtainable otherwise. Probability methods can 1. As a pure number, orserve as a tool for measuring the reliability performance of an 2. As a rate in terms of years per day.electric power system, thus providing sound basis for judg-ment as to when additional facilities are needed. A basis for At any time at all, it may be:estimating capacity benefits of an interconnection and for 1. As a pure number, ormeasuring relative merits of various other general design fea- 2. As a rate in terms of vears per hour.tures of electric power systems is also afforded.

THE ENERGY REQUIREMENTS NOT SUPPLIEDAvailable Measuring UnitsThese may be:Some of the more common measuring units available for

such purposes are outlined here. 1. In termas of the average amount of such energy per year, or2. As a percentage of the total annual energy requirements.

Paper 64-27, recommenlded by the IEEE Power System Engineering TEEOOI RTROCommittee and approved by the IEEE Technical Operations Com- TEEOOI RTROmittee for presentation at the IEEE Winter Power Meeting, New The single economic criterion isYork, N.Y., FEebruary 2-7, 1964. Manuscript submitted October 31,1963; made available for printing November 19, 1963. 1. As the ratio of the incremental cost of providing facilities toC. W. WATCHORN is with the Pennsylvania Powrer and Light Coin- reduce the loss of load to the incremental economic value of thepany, Allentown, Pa. average increased energy thus made available to the service area.

JUL,Y 1964 Watchorn-Characteristics of Probabilit?y Methods as Related to Powzer 737