- Email: [email protected]
An Effective Algorithm of Power Systems Reliability Calculation
AN EFFECTIVE ALGORITHM OF PORER S1STEii'S R.f!;LIABILITY CAICUIATION \ E. Zadrzynski Electroenergetical Institute of Warsaw Politechnical University, Warsaw, Poland Ab~tract. The classical algorithms of power system's reliabit ty calculation is presented and its cost is discussed. This is stated to be prohibitive to frequent LOLP calculation. Therefore some modification of the LOLP algorithm was realized and a new calculation program - called KONP - was carried into effect. This program employs the cumulative probabilities' . table, prepared during its previous run or produced by a NISP program, which works on a base of the classical LOLP algorithm. In consideration of computing time the KONP is more efficient than old algorithms. Then application area of the new &1go-, rithm is discussed. Such problems as: annual LOLP calculation., power system extension programming, generating units' maintenance yearly planning and calculations of a system stand-by capacity, are stUdied and surveyed. The exemplary programs' outputs are annexed to the paper.
Keywom. Power system reliability; LOLP calculations; extens on planning; Maintenance programming; stand-by capacity. CLASSICAL AWORITHIIS INTRODUCTION Since 194?, when the loss of 'load probability was proposed as a measure of power system reliability, the LOLP calculation became an indispensable instrllment for a power system quality evaJ.uation. Such problems &SI th~ extension of power systems, their daily conjrol and operation, equipment maintenance planning etc! t cOllld be pro'Derly resolved only U' the quality of the system could be discerned and various solutions compared. This key-function of LOLP computation is the basic reason for wide its employment in such plaI;miDg problems where the costs of comDuter time alOe relatively insignificant. However in an everyday planning .and control, 'the price issue may be sensitive enough to prevent the utilisation of the LOLP progr8JllS. In the large power systems of high capacity which include a large number of generating units of different kinda this computing cost is partiCUlar!, prohibitive. In o~er to minimize this obstacle a modificated algorithm was worked up by the author of this paper.
There are two means of LOLP calculation: The first, based on the probab1lity .dens1ty function of a system's generating capability PG: LOLP.=
~
P(PG=Pgj) ,P(L) Pgj) ,
j=1,2, •• ,n,n+1. (1) where P(PG:Pg~} means the probability that a generating capability will randomly attain the level of Pgj , P{~Pgj) meaDS the probability of the load greater than p~ and n is a number of oapacity in~irvals. The sec OM! which uses the distribution fUDCt on and whiwh is oalled otherwise "the frecpency and duration method": LOLP = P(PG
4
(2j
In both ways of computation is taken for granted that the transmission system is adequate in all times and that the system load ana oapacity are JlUtually independent. Assuming further that system capacity is divided into n equal. pans, the sec46
tional IDLPs can be computed as the products or the vrobabilities of two OccuraDOesl 1. that the system capability is equal to or lower than the bottOlll limit of the sect1on, . 2. that the system load is looated i.D8ic1e the section as it .i.s 1IIlple aented in Eq. (2). These products summed up for all load intervals produce in both methods the fiDa l valLLe tor the ~stea's LOLP. . To assure the condition of capacity and load independence, the calculation has to be performed only for specific periods of time in which the structure of the generating system is fixed. The problem of LOLP assessment involves two basic tasksl 1. construction of a system capacity probability table, 2. loss of load probability computation for all levels of the generating system's capability occurring In the range of loads or below them. Probability of load availability may be estimated from the average monotonio cu.rve, which has· been const1'Qcted for the examined work period T of the power system uDder cOD8ideration. The curve is presented on Fig. 1. When '1'=1, the ~rObabil1ty of a load greater than gj is represented by the segment (0,2). The segment (1,2) shows the probabil1.ty that the system load falls in-the capacity section of j aDdj-1. Qualification of different capacity level probabilities is more complicated. To achieve this end, in the first method a division of the generating system tn.to groups of like units is provided. Then in each group the probability of k units on simultaneous forced outage is calou lated with the help of Newtonian foraula. Finally the system capacity level is defined by the sum of kilowatts in an up state in every group and its availability by th~ product of group probabilities. The cost of such a computation 18 relatively high. This cost rises as the number of generating units B 1Dcreases aDd as the distribution of units between the groups becomes case more stea~. In the extreme when there are equal unit lWJIlbers in each of the twelve groups, used in the author's PBOO program, the cost of pdf computation iSI K1
= (~211)13.5D +
+
(b +1)12(14D+13A)
where D is the computer cost of one floating point mnlt1plication, A is the cost of one fl. addition and B aeaDs the number of generatiDg units in the system. In the case ot the opposite extre.. , when all system units belong to one .group , the cost of IDLP calculation is rather low: ~
= (Jl+1) -5»
+ (B+1)··f24D + 13A), (4)
This oost is qUite acceptable even for a large power system, but homogeneous generatiDg systems are wry scarce in power praotioe. A b.tter solution is offered by the next method which deals with the cumulative states of capaoity on forced outage instead of capacity at disposal. It comprises the computation of probabilities oumulated for all states of oapacity on forced outage e
P~um j j
= PCUJll
j (1-Q) - pcum (j-11Q,
= 1,2, •• ,n,n+1
(5)
where . P~WIl j' Pcum j are the cwaulative probabilities of foroed outage state j, before and after the unit addition,Pcum j-r means the previous cumulative probability of the forced outage state lowered by the capacity of the unit beiDg added, Q is the forced outage rate of the unit and r is a quotient of the unit capac1 ty and capacity interval between the states. The Eq. (5) oan be rendered in words as follows: . "The cumulative probab1.l~ty of the state with defined capacity on forced outage, after a unit is added to the generating system, eQuals to cumulative probability ~. the S8ll8 state before the addition increased by the unit availabi l ity. To this product IDl1St be added the cumulative probability of thestate.with forced outage lowered by the capaoity of the un! t, DJ. tiplyed by the unit FOR." Using this formula, the final table of cpf values can be arranged by ad~ng each of the units in suooessianto the zero capacity syste. and then computins the interim 47
tables. The cost o~ the computation, as it is realised in the author's'BISP program, is equal tOl
~
= n2B (5D+2A)
(6)
Then the coat is direotly dependent on the DUabel' o~ unit8 aDd the nuaber o~ oapaoity 8tates de~1Ded 1n the 81stem. IJ1 large power 818te. this oost OaA be prohibitive to ~re quent coaputation which is 1DdispeDr.able ~or 4&i11 nliabiUty as8es8JIl8nt. :NEW SUGGliSTIOB
The dom 1 DeDt part o~ LOLP computation cost belongs to the probability list o~ systea oapaoit;y states. On the other haDd, the usual LOLP or "~requeno1 and dure.tion" programs 1Dclude computation of ~ull list at ever,y program run. This expense is not a problem i f the :vrogr8J118 are deisgned ~or a s1stea 8 reliabi11t;y assessaent, usal11 aocomplished by one program ran. But in the oase of ~requent LOLP oalculat10D8, repeated runs of the progrea OaA be a serious expense, particular11 in large power s1steme. IJ1 these' oases the notioD. of ut1l1zing the probab1l11·list calculated at the former reliab1l1t;y computation is qU1te well-founded. The author of th1s paper - baviDS th1s purpose 1n view - worked up the new J..IJU? progr8J118, 1e. limp end KOHP, provided for fr.quent reliability computation in large power systeas. To .'Dimi •• the cost of oalculat10n the program KONP us.s the probabilit;y table produo.d at its previous 1'Wl aDd duaped on _gDet1c tape. I f the coaputation is the ~irst 1n the 'series, the Jiagnetio tape i8 prepared b;y the HlBP p~graa using the usual algoritbIL o~ LOLP oalculation, provided ~or larse power s18teJIB. GeD.eral~ there 18 no chang. in the generatiDs system structure duriDg the short time periods between two c9DSeoutive progrea rune .. aDd table CaA be emplO1.d direct11 to the computation 1n accordanoe with DO~l IDLP routine. However i t the st:ruoture is'ohanged the data conc.rn1Dg all the generat!n8unit. traDSm1ttea to or reoe1ved ~rOll maintenance t..... are to be put into the c caputer. Then, i t a given unit is to be added to the s1stea, the' KOlIP algoritbll treats it ·1n accordaDce with Eq. (5). Otherwise, the value8 of the probabilit1 tab~e are corrected following the ~01'laula.
P~UIl 3
:11
-
1~ POWl ~ ~ P~wa(3-r)"
(6) where P~UIl is the alrea~ corrected cWlUlative probabilit;y o~ the state ot ~oroed outage le8sened b1 the unit capacity 1n relation to the 8tate j . The other 81J1bols were de~1D.ed in Eq. (51. The probabllit1 P~ua(3-r IlWIt have the value alrea~ corrected! wheD. it is used 1n Eq. (61. ~o have ts proper value the computation o~ state probabilitie8 has to be pertoraed consecutively atart1D8 from the state o~ sero capaoity on ~oroed outage or ayre thaA zero. Then while the PCUll 3 is calculated the new, corrected value o~ p~WIl(;j-r) is alread;y
mown.
Block diagraa of the new algoritba 1s pre8ented on Fig. 2. Here the C08t of the probabilit;y table computation i8 .uch lOWer than 1n clas8ical LOLP prograJll8. For R UDit8 aoved to or from plaiilled repairs it i8 about. 1:4 • R-n(5D+2A)
(.1)
Then the cost reduction, as it relates to the old programs, is about
(8) In large power 81steme this reduction OaA be qU1te .UbstaAtial. FEATlJRE8 OF HEW PROGRAII3
TheKOBP .aDd' BlBP programs provide the follow1nf5 1~0ru.t10n about pow.r systea s reliabilit;ya 1. Loss o~ load probability in the fora o~ .aD. abstact lI.WIber, 2. The 88J1le, but .easured in day. o~ capacib de~ic lene1 by year! ,~ Expectea value o~ energy wh ch i. not supplied dur1.Dg one ye'ar in the S1ste. owiD8 to forced outages, 4. Average demand curtailment in the pOWer system, 5. Coef~ic1ent o~ the 8ystea's unreliabi11b de~1ned also aB "a syate. unrellabillty rate"; th1s ls a quotient o~ ene:rgy curtailment and eners;:v deaD4. The calculatlon o~ the three last parameters was iapl. .ented on the base o~ algoritbll described in the author'. paper, re~ereneed iD. the eighth position. . The prog1"8Jl8 ~eature the meaDS of checking the input data adequac1. 48
They test the. following data relationships: is the sum of the loan probabilities equal to one? - is the cap$city interval equal to its value in kilowatts as written on the mBgnetic tape? are the units which were withdr~wn from the system really eXisting in it? . In the case of a negative answer to any of the qu~stions a data error i~ signaled, and proper message is printed by the computer. Besides this, a program user setti!~ the definite b1.l;s of J!t'ogrammer wor-d in an CD position, cgn induce ~ll output froviding the .followi.ng uJ.;'or mation: - thecumuJ.ative probaoilit;y <;able created by the program - if bit 1 is set, ' a list of interval LOLPs - if bit 2 is set, - a schedule of ,the load probabili·· ties read into the computer - if bit 3 is set, ' - all information introduced to the program by magnetic tape - when bit 9 is set. If all the mentioned bits are set together the complete additional information is printed. The programs NIBP and KONP were employed by the author of this paper1 to resolve some practical problems • One of the application is presented in a s~stem stUdy. Besides this the programs were employed to test the precision of the KONP algorithm. During the test nearly a hundred generating units of different c~paci ties and forced outage rat9s were moved to and from an experimental syste~ /defined in the system study/ and proper corrections of the cumulative probability table were made. Finally the system returned to the situation at the start of the experiment and the probability tables were compared. They were identical. Outputs of the programs are presented in an appendix. A SUbstantial portion of the output text is provided in translation on English. S YS1'EM S'IUDy
An example of power system extension programming is here presented. This problem is not new and was already treated in several publications. In this stUdy it is used only to illu1 Unfortunately, there is not, as yet the experience of the program empl~yment for a system daily control and operation•
.
strate the mauy P')::;s~~;il ~ tl'-'':.'. [·n' 13 pplica tion of the [i)NT a J.~•.J ... i thm. The usual purpose of 30 8xt;(,ns:Lon program is t(, pr0v:1dA fixed cl'3tes for 8ddition to tnr; system of new units and to make I t ssrIy enough to allow for the ttUle nt;,flded for construction work. rt'l~~ ft ,J.~'; 25swne [;ha-t the lOA.M of a ,,»V;fC;' €,:,stem during Ctll p',('ogrR.mmed r.i.:n'" n'8 eVf'lnly distributed between ::he :n9.x·~1j),u.m an.d m:1.'l:i.mUlll eqlnl to 'tl"t, half <)f the former. It mei.'11J3 cb,st "~lle shRpe of load lllonoton~.(' is assu-·· med to remain tillchanged an:; to preserve cheform as presented on Fig. 3. Let lie furth~r nssume that maximol load grows every year by 1000 MV{ and that the generating system at the start, in 19~ includes the units listed in ~£E 1. System capacity is increased if nece ssarJ' by increments of 1500 MW units commissions. For simplification, maintenance of the generating units is not taken into account. At the stflrt, the IDLP is calculated with a help of NISP program for conditions of the year 1978 i.e. for capacity of 22000 MN and maximal loed 13 000 ~tV. The resulting LOLP is 0.000'2414,1e. much below the criterion level of 0.001. During the program run, the cumUlative probabilities table of the Bystem capacity on forced outage is d4mped on a magnetic tape and other parameters /besides of LOLP/ of system~s reliability sre calculated. This ~arame ters are listed in the last three columns of T.o\.BIB 2. For the next year, 1979, the system load grows up to 14000 MW. LOLP and system's parameters are then computed in a short run of KONP~rogram using the table from the magnetic tape previously. produced. Its value is 0.OO10137lJ, a little above the criterion line.' Then the new 1500 M~ unit should be added and a new LOLP calculated. Now the KONP corrects the tape from magnetic tape and writes it on another MT. tiy using KONP program 1n this way, all interesting points of the extension program can be calculated. The results of the computation are specified in'the middle column of TABLE 2 and presented on Fig. 4.
The' line segments on the figure: 1-2,3-4-5.6-?1_etc. are the pieces of curves:IDLk' = f(L). traced in semilogarithmical soale. where the system oapacity is a parameter. The curves crossings with the horiontal line: IDLP 0.001, 'appoint the maxi mal loads which can be carried on by generating system conserving still' the reliability level. The distance beween neighbouring points constitutes the load covered by the unit addition and is oa;J.led its "effect~ ve load carrying capability". EWC. The difference between maximal load and system capacity is the system margin 11. Capacities of the generating systems created by the consecutive additions of the units IC/. the maximal loads carried on on the reliability level ILl, system margins, the reserve percentage andEWC of the units being added are jwctaposed-in Table 3. The system extension program shows that to assure the given reliability level a commission of the units has to be fixed for every year of the programmed period exclUding the years 1980 and 198? The task of this programming process included 1 run of NlBP and 23. sucess ful runs of the KONP program. There were also additional runs not speoified 1n the table because their results were too distant from the LOLP criterion. The NlBP took 15 minutes of computer times • The average KONP run took 1.5 min. only. A quite important saving I.
=
OTHER APPLICATIONS The KONP program has many more poss1 bilitiesfor employment than daily planning and system extension programming only. It oan be also helpful in the solution of .the following problems. Annual 'LOLP Caloulatiops The WLP algorithm, as it is known., assumes the independence of capacity and load. This condition reqUires that the generating system remain uncbaDged during the time interval for which the LOLP is calculated. Such reqUirement is rather hard in practice. In real systems there are frequent cbaDges of the generating capacity. Consequently. to compute tht annual • LOLP the. course of a year Has to be divided into intervals with a constant structure of generating oapacity. Then the LOLP is computed .for each interval. .. It is evident that the intervals'
11Jl1ts depend on the moments when some units are transmitted for planned repairs or back from maintenance to the system.1 they oan be also designed by the dates of the new units commissions. The exemplary division of the year is presented' on Fig. 5, where it is divided on 16 intervals. In practioe the number ot intervals is much higher than that. YearlY llaintenanoe Planning If the system is to stand by reliability criterion•. the WLP must be computed for several variants of the maintenance plan. The condition of the calculation is the stability of the generating system structures, , create.d in consequence of the uxut s transmission to or from planned repairs. Then the time intervals of a system work for which the seotional WLP is computed are limited by the dates of oonsecutive generating units withdrawals and/or additions. They are usually nombreous and ~ program runs are necessary. The resulting IDLP is time dependent average of partitionsl values. It is calculated for every maintenance variant. Then the optimal solution for planned repairs oan be choosen. Caloulation Of The System's Reserves Calculation of the capacity margin whiohis needed to keep up the power system's reliabilitj level - this is the real problem here discussed. The stand-by capacity computation includes the following tasks: After the structure of a generating system and cri~~rion level are defined, the system 8 LOLP is callputed . for maximal load optionally choosen. I f resulting LOLP is greater than the oriterion the loads are made lower, if smaller than criterion - higher. Then a new computation is performed for cbaDged load. Approaching step by step·the reqUired level and eventually using the ~terpolation technios in final phase of the task, the proper load is defined. Then the margin is found as the difference between the system's oapacity and determined load. As can be easily seen, this job too inc ludes the recurrent LOLP calc ulations, the activity where the KONP program is a great assistance. CONCIDSIcmB 1. The KONP algodthm by providing the correction of the cumulative probability table instead of costly ab ovo calculation gives evident computer time saving in the daily or repeated calculations of 50
power system's reliability. . 2. The KONP algorithm can also be effeotively 'used in such tasks of power system planning as: - power system extension programming, - annual WLl? assessment, - yearly maintenance planning, calculation of the stand-by oapacity necesljary to sustain the power system s reliability level. ~. The KONP algorithm is partic ular. ly effective when applied to the problems of reliability computa. tion for la~e power systems. 4. Theacouraoy of the KONP and NlJ:)P a~orithm's results are comparable with yhose of the WLl? programs even in the case af a big number of table oorrections.
Garver, D.P. (1968). Effective load oarr,ting oa~ability of generating units. IEEE TraDd. on PAS, VIII. .Garver, D.P., F,E. Montmeat, and A. D. Fatton (1965). Power system reliability: I - Measure o~ reliability methods ·of caloulation IEEE Trans. on PAS., Vol. 85. Montmeat, F.E" A.D. Patwon, J, Zekomski, and D.J, Cummings (1968) Power system reliabili~y: 11 Application on computer program• 1EEE Trans. on PAS., Nr 9. ' zadrzynski E. (1977). Power system reliab 1lity assessment by meaDs of loss of load probability, /in polish/. Prooeedings of International Symposium on Power Sytem Reliability, Politechnical Univarsity of Wroclaw,
Vol. ILlS.
REFERENcm Ayoub, A.;· J. Gguy, and A. Patton (1970. Evaluation and oomparison of some methods for caloulating generating system reliability. IEEE Trans. on Power. Ap:raratus and S~stems, Nr 4. Bill nton R., R•• Ringlee, and A. Wood t197~). Power system reliabili:' calculations, /book!. The ~T Press. .
.
Zadrzynski, E, (1977). Reourrent algorithm of power system reliability oaloulation, /in polish/. Proceedings of International Scientifio Conference, Politech- ' nioal uniVetSit~ of Gliwice, Zadrzynski, 1.197 ). More 1Dformation from IDLl? oalculation. Expe cted value of unsatisfied energy demand. Proceedings of IEEE PES Winter .eeting, New York.
TABLE 1 Structure of Generating System
~er Qf Unit"s Unit's ts in oapac. F.O.R', MW It;he grouJi
-
12 5 5 5
1500 500 200 100
0.14 0.12 0.09 0.05
Capao'ity of the grou~ MW
.....---,~--~1
_--.---1- 2
18 000 2500 1 000 500
load monoton1c
i
~1'r-.....;) .. ~
Capaoity'cd!
~.
P(PG~Pgj)
'"""
n
1.0
T Fig.~.
Load monotonio for time interval T.
•
0
J
or
Cumulativp time probability Fig, 1. Capacity and load models for system's LOLP oalculation•
51.
(
ST~
READ TABLE FEOM M
READ
U~I!'S
F.C.R
AND SA.PACITIES
=1
i
It
YES
?
.. -".I
..../1-=: \~J
. . . ., ... _.... L
..... .
"'0
./
i
A (i)
-- ..1U.ll. 1-Q (j) - -9..W..
,
~ - i\.~.~:.,."
--_
.•.•......... _.. ,.,
1-QCj J
Fig. 2. Block diagram of the new algorithm. NOTATION: lSTEP.
= Length of capacity interval in MW,
IC(j) /Q(j) = Capacity/forced outage rate of unit j
moved to or from generating system, B (i) /A (i) = Probability of the state i before and after a unit addition.
52
TABLE 2
Data of power syste. extension program,
pon. Year Load limits Nr
Syst~;:l
cc;pec. MW
MW
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1978 13000- 6500 1979 14000-7000 11
.
1980 15000- 7500 1981 16000- 8000
.
"
1982 17000- 8500
"
"
22000 It 23500
."
25000
"
26500
" . 28000 1984 19000" . 9500 29500 " . 1985 20000-10000 . 31000 " . 1983 18000- 9000
"
1986 21000-10500
"
"
32500
't
"
34000 n
"
35500
1987 22000-11000 1988 23000-11500 1989 24000..12000
"
1990 25000-12500
"
"
1991 26000-13000
WLP
"
"
"
37000
"
0,00032414 0.00101379 0,00026760 0,00089268 0,00254473 0,000?4984 0,00206054 0,00061388 0,00178430 0.00054061 0.00146916 0.00045297 0.00120190 0.00037338 0.00104137 0.00032975 0.00087225 0.00211085 0.00071727 0.00182437 0.00062620 0.00151505 0.00052452 0.00125298
E curt. L curt,' MWh MW year
W.
699,010
0,23·10-4 .
1766,34 6176.15
753,496 789.802
0,19'10-4 0.62'10-4
5261,17
800,952
0.50'10-4
4567,24
849'.312
0,40'10-4'
3958,71
835.924 0.33'10-4
3359~49
846.632
0.27'10-4
2913.04
890.608
0.22'10-4
2523.89, 873.742 7156.02 936.545
0,18'10:: 0.40'10
6123.72
974.80
0.40'10-4
5229.51
953,326
0,33'10-4
4439,29
966,139
0.27'10-4
M/L
Ewe
1648.97
'
TABIB C
L
11
MW
MW
MW
22000 23500 25000 26500 28000 29500 31000 32500 34000 35500
13990 15130 16280 17440 18610 19790 20980 22180 23385 24595
8010 8370 8720 9060 9390 9710 10020 10320 10615 10950
% 36.4 35.6 34.9 34,2 33.6 33.0 32.3 31.8 ;;1.2 30.8
MW
1140 1150 1160 1170 1180 1190 1200 1205 1210 1210
c r-~_-:.L
1.1
Fig, 5.
1
2
.XII.
Annuald1agram of system's capacity C and daily peak load L. 53
LOliP
I
•
1/
I
0.002
I
time . ·23 0.0005 0.0003 MW
19
Fig. 4.
loads/
/years/
Power system extension diagram.
ANlfflP' EZ PROGRAM NIBP
/mBX.
1 /TRANSLATED FRCIl POLISH/
DATE 04/12/78 1
GENERATING SYSTEM'S STRUC1URE NUMBER UNI1B UNIT'S GROOP UNIT'S OF THE IN A CAPAC. CAPAC. F.O.R. GROUP GROOP MN MW . '1 ·12 1500 18000 .140 2 5500 2500 .120 3 5 200 1000 .090 4 ·5 100 500 .050 SYSTEM'S CAPACITY !WITHOOT PLANNED REPAIRS/ 22000 MW MAXIMAL LOAD IN THE COMRJTED ~IOD z 13000 MW .MmIMA L " 11 tf " " 6500 MW LOOS OF LOAD PROBABILITY =0.3241,3758-03 DEFICIENCY DAYS!DAY OR =0.1183102E 00 DEFICIENCY DAYS/YEAR . EIBCTRICAL ENERGY UNSUPPLIED= .19849?E 04 MWh/lEAR AVERAGE LOAD CURTAIII4ENT = 699.070 i.iW SYSTEM's UNRELIABILITY lUTE .229989E-Q4
=
=
=
smTiM";s-STRUC1URE
S
EZ PROGRAM KOiP- DATE-04/12/5' 1- - - - - - _RATmG GROUP' UNI!lB UNIT'S NUMBER. IN CAPAC. ~ GROOP MW 1 13 1500
2
5
500
GROUP CAPAC.
UNIT"S F.O.R.
MW 18000
2500 1000
3 5 200 '. 4 ,5 100 500 CAPACITY OF THE PREVIOUS S ~TE14 = 22000 MW UNIT"S TRANSMISSION IN THE LAST PERIOD OF TIME TO P~NED REPAlBS FRCIl PLANNED REPAIBS UNIT'S CAPACITY F.O.R. UNIT'S CAPAC:ITY F.O.R. . 1~ 0.140
.140 .120
.090
.050
1 No bits set on in this program.
54
Keywom. Power system reliability; LOLP calculations; extens on planning; Maintenance programming; stand-by capacity. CLASSICAL AWORITHIIS INTRODUCTION Since 194?, when the loss of 'load probability was proposed as a measure of power system reliability, the LOLP calculation became an indispensable instrllment for a power system quality evaJ.uation. Such problems &SI th~ extension of power systems, their daily conjrol and operation, equipment maintenance planning etc! t cOllld be pro'Derly resolved only U' the quality of the system could be discerned and various solutions compared. This key-function of LOLP computation is the basic reason for wide its employment in such plaI;miDg problems where the costs of comDuter time alOe relatively insignificant. However in an everyday planning .and control, 'the price issue may be sensitive enough to prevent the utilisation of the LOLP progr8JllS. In the large power systems of high capacity which include a large number of generating units of different kinda this computing cost is partiCUlar!, prohibitive. In o~er to minimize this obstacle a modificated algorithm was worked up by the author of this paper.
There are two means of LOLP calculation: The first, based on the probab1lity .dens1ty function of a system's generating capability PG: LOLP.=
~
P(PG=Pgj) ,P(L) Pgj) ,
j=1,2, •• ,n,n+1. (1) where P(PG:Pg~} means the probability that a generating capability will randomly attain the level of Pgj , P{~Pgj) meaDS the probability of the load greater than p~ and n is a number of oapacity in~irvals. The sec OM! which uses the distribution fUDCt on and whiwh is oalled otherwise "the frecpency and duration method": LOLP = P(PG
4
(2j
In both ways of computation is taken for granted that the transmission system is adequate in all times and that the system load ana oapacity are JlUtually independent. Assuming further that system capacity is divided into n equal. pans, the sec46
tional IDLPs can be computed as the products or the vrobabilities of two OccuraDOesl 1. that the system capability is equal to or lower than the bottOlll limit of the sect1on, . 2. that the system load is looated i.D8ic1e the section as it .i.s 1IIlple aented in Eq. (2). These products summed up for all load intervals produce in both methods the fiDa l valLLe tor the ~stea's LOLP. . To assure the condition of capacity and load independence, the calculation has to be performed only for specific periods of time in which the structure of the generating system is fixed. The problem of LOLP assessment involves two basic tasksl 1. construction of a system capacity probability table, 2. loss of load probability computation for all levels of the generating system's capability occurring In the range of loads or below them. Probability of load availability may be estimated from the average monotonio cu.rve, which has· been const1'Qcted for the examined work period T of the power system uDder cOD8ideration. The curve is presented on Fig. 1. When '1'=1, the ~rObabil1ty of a load greater than gj is represented by the segment (0,2). The segment (1,2) shows the probabil1.ty that the system load falls in-the capacity section of j aDdj-1. Qualification of different capacity level probabilities is more complicated. To achieve this end, in the first method a division of the generating system tn.to groups of like units is provided. Then in each group the probability of k units on simultaneous forced outage is calou lated with the help of Newtonian foraula. Finally the system capacity level is defined by the sum of kilowatts in an up state in every group and its availability by th~ product of group probabilities. The cost of such a computation 18 relatively high. This cost rises as the number of generating units B 1Dcreases aDd as the distribution of units between the groups becomes case more stea~. In the extreme when there are equal unit lWJIlbers in each of the twelve groups, used in the author's PBOO program, the cost of pdf computation iSI K1
= (~211)13.5D +
+
(b +1)12(14D+13A)
where D is the computer cost of one floating point mnlt1plication, A is the cost of one fl. addition and B aeaDs the number of generatiDg units in the system. In the case ot the opposite extre.. , when all system units belong to one .group , the cost of IDLP calculation is rather low: ~
= (Jl+1) -5»
+ (B+1)··f24D + 13A), (4)
This oost is qUite acceptable even for a large power system, but homogeneous generatiDg systems are wry scarce in power praotioe. A b.tter solution is offered by the next method which deals with the cumulative states of capaoity on forced outage instead of capacity at disposal. It comprises the computation of probabilities oumulated for all states of oapacity on forced outage e
P~um j j
= PCUJll
j (1-Q) - pcum (j-11Q,
= 1,2, •• ,n,n+1
(5)
where . P~WIl j' Pcum j are the cwaulative probabilities of foroed outage state j, before and after the unit addition,Pcum j-r means the previous cumulative probability of the forced outage state lowered by the capacity of the unit beiDg added, Q is the forced outage rate of the unit and r is a quotient of the unit capac1 ty and capacity interval between the states. The Eq. (5) oan be rendered in words as follows: . "The cumulative probab1.l~ty of the state with defined capacity on forced outage, after a unit is added to the generating system, eQuals to cumulative probability ~. the S8ll8 state before the addition increased by the unit availabi l ity. To this product IDl1St be added the cumulative probability of thestate.with forced outage lowered by the capaoity of the un! t, DJ. tiplyed by the unit FOR." Using this formula, the final table of cpf values can be arranged by ad~ng each of the units in suooessianto the zero capacity syste. and then computins the interim 47
tables. The cost o~ the computation, as it is realised in the author's'BISP program, is equal tOl
~
= n2B (5D+2A)
(6)
Then the coat is direotly dependent on the DUabel' o~ unit8 aDd the nuaber o~ oapaoity 8tates de~1Ded 1n the 81stem. IJ1 large power 818te. this oost OaA be prohibitive to ~re quent coaputation which is 1DdispeDr.able ~or 4&i11 nliabiUty as8es8JIl8nt. :NEW SUGGliSTIOB
The dom 1 DeDt part o~ LOLP computation cost belongs to the probability list o~ systea oapaoit;y states. On the other haDd, the usual LOLP or "~requeno1 and dure.tion" programs 1Dclude computation of ~ull list at ever,y program run. This expense is not a problem i f the :vrogr8J118 are deisgned ~or a s1stea 8 reliabi11t;y assessaent, usal11 aocomplished by one program ran. But in the oase of ~requent LOLP oalculat10D8, repeated runs of the progrea OaA be a serious expense, particular11 in large power s1steme. IJ1 these' oases the notioD. of ut1l1zing the probab1l11·list calculated at the former reliab1l1t;y computation is qU1te well-founded. The author of th1s paper - baviDS th1s purpose 1n view - worked up the new J..IJU? progr8J118, 1e. limp end KOHP, provided for fr.quent reliability computation in large power systeas. To .'Dimi •• the cost of oalculat10n the program KONP us.s the probabilit;y table produo.d at its previous 1'Wl aDd duaped on _gDet1c tape. I f the coaputation is the ~irst 1n the 'series, the Jiagnetio tape i8 prepared b;y the HlBP p~graa using the usual algoritbIL o~ LOLP oalculation, provided ~or larse power s18teJIB. GeD.eral~ there 18 no chang. in the generatiDs system structure duriDg the short time periods between two c9DSeoutive progrea rune .. aDd table CaA be emplO1.d direct11 to the computation 1n accordanoe with DO~l IDLP routine. However i t the st:ruoture is'ohanged the data conc.rn1Dg all the generat!n8unit. traDSm1ttea to or reoe1ved ~rOll maintenance t..... are to be put into the c caputer. Then, i t a given unit is to be added to the s1stea, the' KOlIP algoritbll treats it ·1n accordaDce with Eq. (5). Otherwise, the value8 of the probabilit1 tab~e are corrected following the ~01'laula.
P~UIl 3
:11
-
1~ POWl ~ ~ P~wa(3-r)"
(6) where P~UIl is the alrea~ corrected cWlUlative probabilit;y o~ the state ot ~oroed outage le8sened b1 the unit capacity 1n relation to the 8tate j . The other 81J1bols were de~1D.ed in Eq. (51. The probabllit1 P~ua(3-r IlWIt have the value alrea~ corrected! wheD. it is used 1n Eq. (61. ~o have ts proper value the computation o~ state probabilitie8 has to be pertoraed consecutively atart1D8 from the state o~ sero capaoity on ~oroed outage or ayre thaA zero. Then while the PCUll 3 is calculated the new, corrected value o~ p~WIl(;j-r) is alread;y
mown.
Block diagraa of the new algoritba 1s pre8ented on Fig. 2. Here the C08t of the probabilit;y table computation i8 .uch lOWer than 1n clas8ical LOLP prograJll8. For R UDit8 aoved to or from plaiilled repairs it i8 about. 1:4 • R-n(5D+2A)
(.1)
Then the cost reduction, as it relates to the old programs, is about
(8) In large power 81steme this reduction OaA be qU1te .UbstaAtial. FEATlJRE8 OF HEW PROGRAII3
TheKOBP .aDd' BlBP programs provide the follow1nf5 1~0ru.t10n about pow.r systea s reliabilit;ya 1. Loss o~ load probability in the fora o~ .aD. abstact lI.WIber, 2. The 88J1le, but .easured in day. o~ capacib de~ic lene1 by year! ,~ Expectea value o~ energy wh ch i. not supplied dur1.Dg one ye'ar in the S1ste. owiD8 to forced outages, 4. Average demand curtailment in the pOWer system, 5. Coef~ic1ent o~ the 8ystea's unreliabi11b de~1ned also aB "a syate. unrellabillty rate"; th1s ls a quotient o~ ene:rgy curtailment and eners;:v deaD4. The calculatlon o~ the three last parameters was iapl. .ented on the base o~ algoritbll described in the author'. paper, re~ereneed iD. the eighth position. . The prog1"8Jl8 ~eature the meaDS of checking the input data adequac1. 48
They test the. following data relationships: is the sum of the loan probabilities equal to one? - is the cap$city interval equal to its value in kilowatts as written on the mBgnetic tape? are the units which were withdr~wn from the system really eXisting in it? . In the case of a negative answer to any of the qu~stions a data error i~ signaled, and proper message is printed by the computer. Besides this, a program user setti!~ the definite b1.l;s of J!t'ogrammer wor-d in an CD position, cgn induce ~ll output froviding the .followi.ng uJ.;'or mation: - thecumuJ.ative probaoilit;y <;able created by the program - if bit 1 is set, ' a list of interval LOLPs - if bit 2 is set, - a schedule of ,the load probabili·· ties read into the computer - if bit 3 is set, ' - all information introduced to the program by magnetic tape - when bit 9 is set. If all the mentioned bits are set together the complete additional information is printed. The programs NIBP and KONP were employed by the author of this paper1 to resolve some practical problems • One of the application is presented in a s~stem stUdy. Besides this the programs were employed to test the precision of the KONP algorithm. During the test nearly a hundred generating units of different c~paci ties and forced outage rat9s were moved to and from an experimental syste~ /defined in the system study/ and proper corrections of the cumulative probability table were made. Finally the system returned to the situation at the start of the experiment and the probability tables were compared. They were identical. Outputs of the programs are presented in an appendix. A SUbstantial portion of the output text is provided in translation on English. S YS1'EM S'IUDy
An example of power system extension programming is here presented. This problem is not new and was already treated in several publications. In this stUdy it is used only to illu1 Unfortunately, there is not, as yet the experience of the program empl~yment for a system daily control and operation•
.
strate the mauy P')::;s~~;il ~ tl'-'':.'. [·n' 13 pplica tion of the [i)NT a J.~•.J ... i thm. The usual purpose of 30 8xt;(,ns:Lon program is t(, pr0v:1dA fixed cl'3tes for 8ddition to tnr; system of new units and to make I t ssrIy enough to allow for the ttUle nt;,flded for construction work. rt'l~~ f
The' line segments on the figure: 1-2,3-4-5.6-?1_etc. are the pieces of curves:IDLk' = f(L). traced in semilogarithmical soale. where the system oapacity is a parameter. The curves crossings with the horiontal line: IDLP 0.001, 'appoint the maxi mal loads which can be carried on by generating system conserving still' the reliability level. The distance beween neighbouring points constitutes the load covered by the unit addition and is oa;J.led its "effect~ ve load carrying capability". EWC. The difference between maximal load and system capacity is the system margin 11. Capacities of the generating systems created by the consecutive additions of the units IC/. the maximal loads carried on on the reliability level ILl, system margins, the reserve percentage andEWC of the units being added are jwctaposed-in Table 3. The system extension program shows that to assure the given reliability level a commission of the units has to be fixed for every year of the programmed period exclUding the years 1980 and 198? The task of this programming process included 1 run of NlBP and 23. sucess ful runs of the KONP program. There were also additional runs not speoified 1n the table because their results were too distant from the LOLP criterion. The NlBP took 15 minutes of computer times • The average KONP run took 1.5 min. only. A quite important saving I.
=
OTHER APPLICATIONS The KONP program has many more poss1 bilitiesfor employment than daily planning and system extension programming only. It oan be also helpful in the solution of .the following problems. Annual 'LOLP Caloulatiops The WLP algorithm, as it is known., assumes the independence of capacity and load. This condition reqUires that the generating system remain uncbaDged during the time interval for which the LOLP is calculated. Such reqUirement is rather hard in practice. In real systems there are frequent cbaDges of the generating capacity. Consequently. to compute tht annual • LOLP the. course of a year Has to be divided into intervals with a constant structure of generating oapacity. Then the LOLP is computed .for each interval. .. It is evident that the intervals'
11Jl1ts depend on the moments when some units are transmitted for planned repairs or back from maintenance to the system.1 they oan be also designed by the dates of the new units commissions. The exemplary division of the year is presented' on Fig. 5, where it is divided on 16 intervals. In practioe the number ot intervals is much higher than that. YearlY llaintenanoe Planning If the system is to stand by reliability criterion•. the WLP must be computed for several variants of the maintenance plan. The condition of the calculation is the stability of the generating system structures, , create.d in consequence of the uxut s transmission to or from planned repairs. Then the time intervals of a system work for which the seotional WLP is computed are limited by the dates of oonsecutive generating units withdrawals and/or additions. They are usually nombreous and ~ program runs are necessary. The resulting IDLP is time dependent average of partitionsl values. It is calculated for every maintenance variant. Then the optimal solution for planned repairs oan be choosen. Caloulation Of The System's Reserves Calculation of the capacity margin whiohis needed to keep up the power system's reliabilitj level - this is the real problem here discussed. The stand-by capacity computation includes the following tasks: After the structure of a generating system and cri~~rion level are defined, the system 8 LOLP is callputed . for maximal load optionally choosen. I f resulting LOLP is greater than the oriterion the loads are made lower, if smaller than criterion - higher. Then a new computation is performed for cbaDged load. Approaching step by step·the reqUired level and eventually using the ~terpolation technios in final phase of the task, the proper load is defined. Then the margin is found as the difference between the system's oapacity and determined load. As can be easily seen, this job too inc ludes the recurrent LOLP calc ulations, the activity where the KONP program is a great assistance. CONCIDSIcmB 1. The KONP algodthm by providing the correction of the cumulative probability table instead of costly ab ovo calculation gives evident computer time saving in the daily or repeated calculations of 50
power system's reliability. . 2. The KONP algorithm can also be effeotively 'used in such tasks of power system planning as: - power system extension programming, - annual WLl? assessment, - yearly maintenance planning, calculation of the stand-by oapacity necesljary to sustain the power system s reliability level. ~. The KONP algorithm is partic ular. ly effective when applied to the problems of reliability computa. tion for la~e power systems. 4. Theacouraoy of the KONP and NlJ:)P a~orithm's results are comparable with yhose of the WLl? programs even in the case af a big number of table oorrections.
Garver, D.P. (1968). Effective load oarr,ting oa~ability of generating units. IEEE TraDd. on PAS, VIII. .Garver, D.P., F,E. Montmeat, and A. D. Fatton (1965). Power system reliability: I - Measure o~ reliability methods ·of caloulation IEEE Trans. on PAS., Vol. 85. Montmeat, F.E" A.D. Patwon, J, Zekomski, and D.J, Cummings (1968) Power system reliabili~y: 11 Application on computer program• 1EEE Trans. on PAS., Nr 9. ' zadrzynski E. (1977). Power system reliab 1lity assessment by meaDs of loss of load probability, /in polish/. Prooeedings of International Symposium on Power Sytem Reliability, Politechnical Univarsity of Wroclaw,
Vol. ILlS.
REFERENcm Ayoub, A.;· J. Gguy, and A. Patton (1970. Evaluation and oomparison of some methods for caloulating generating system reliability. IEEE Trans. on Power. Ap:raratus and S~stems, Nr 4. Bill nton R., R•• Ringlee, and A. Wood t197~). Power system reliabili:' calculations, /book!. The ~T Press. .
.
Zadrzynski, E, (1977). Reourrent algorithm of power system reliability oaloulation, /in polish/. Proceedings of International Scientifio Conference, Politech- ' nioal uniVetSit~ of Gliwice, Zadrzynski, 1.197 ). More 1Dformation from IDLl? oalculation. Expe cted value of unsatisfied energy demand. Proceedings of IEEE PES Winter .eeting, New York.
TABLE 1 Structure of Generating System
~er Qf Unit"s Unit's ts in oapac. F.O.R', MW It;he grouJi
-
12 5 5 5
1500 500 200 100
0.14 0.12 0.09 0.05
Capao'ity of the grou~ MW
.....---,~--~1
_--.---1- 2
18 000 2500 1 000 500
load monoton1c
i
~1'r-.....;) .. ~
Capaoity'cd!
~.
P(PG~Pgj)
'"""
n
1.0
T Fig.~.
Load monotonio for time interval T.
•
0
J
or
Cumulativp time probability Fig, 1. Capacity and load models for system's LOLP oalculation•
51.
(
ST~
READ TABLE FEOM M
READ
U~I!'S
F.C.R
AND SA.PACITIES
=1
i
It
YES
?
.. -".I
..../1-=: \~J
. . . ., ... _.... L
..... .
"'0
./
i
A (i)
-- ..1U.ll. 1-Q (j) - -9..W..
,
~ - i\.~.~:.,."
--_
.•.•......... _.. ,.,
1-QCj J
Fig. 2. Block diagram of the new algorithm. NOTATION: lSTEP.
= Length of capacity interval in MW,
IC(j) /Q(j) = Capacity/forced outage rate of unit j
moved to or from generating system, B (i) /A (i) = Probability of the state i before and after a unit addition.
52
TABLE 2
Data of power syste. extension program,
pon. Year Load limits Nr
Syst~;:l
cc;pec. MW
MW
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1978 13000- 6500 1979 14000-7000 11
.
1980 15000- 7500 1981 16000- 8000
.
"
1982 17000- 8500
"
"
22000 It 23500
."
25000
"
26500
" . 28000 1984 19000" . 9500 29500 " . 1985 20000-10000 . 31000 " . 1983 18000- 9000
"
1986 21000-10500
"
"
32500
't
"
34000 n
"
35500
1987 22000-11000 1988 23000-11500 1989 24000..12000
"
1990 25000-12500
"
"
1991 26000-13000
WLP
"
"
"
37000
"
0,00032414 0.00101379 0,00026760 0,00089268 0,00254473 0,000?4984 0,00206054 0,00061388 0,00178430 0.00054061 0.00146916 0.00045297 0.00120190 0.00037338 0.00104137 0.00032975 0.00087225 0.00211085 0.00071727 0.00182437 0.00062620 0.00151505 0.00052452 0.00125298
E curt. L curt,' MWh MW year
W.
699,010
0,23·10-4 .
1766,34 6176.15
753,496 789.802
0,19'10-4 0.62'10-4
5261,17
800,952
0.50'10-4
4567,24
849'.312
0,40'10-4'
3958,71
835.924 0.33'10-4
3359~49
846.632
0.27'10-4
2913.04
890.608
0.22'10-4
2523.89, 873.742 7156.02 936.545
0,18'10:: 0.40'10
6123.72
974.80
0.40'10-4
5229.51
953,326
0,33'10-4
4439,29
966,139
0.27'10-4
M/L
Ewe
1648.97
'
TABIB C
L
11
MW
MW
MW
22000 23500 25000 26500 28000 29500 31000 32500 34000 35500
13990 15130 16280 17440 18610 19790 20980 22180 23385 24595
8010 8370 8720 9060 9390 9710 10020 10320 10615 10950
% 36.4 35.6 34.9 34,2 33.6 33.0 32.3 31.8 ;;1.2 30.8
MW
1140 1150 1160 1170 1180 1190 1200 1205 1210 1210
c r-~_-:.L
1.1
Fig, 5.
1
2
.XII.
Annuald1agram of system's capacity C and daily peak load L. 53
LOliP
I
•
1/
I
0.002
I
time . ·23 0.0005 0.0003 MW
19
Fig. 4.
loads/
/years/
Power system extension diagram.
ANlfflP' EZ PROGRAM NIBP
/mBX.
1 /TRANSLATED FRCIl POLISH/
DATE 04/12/78 1
GENERATING SYSTEM'S STRUC1URE NUMBER UNI1B UNIT'S GROOP UNIT'S OF THE IN A CAPAC. CAPAC. F.O.R. GROUP GROOP MN MW . '1 ·12 1500 18000 .140 2 5500 2500 .120 3 5 200 1000 .090 4 ·5 100 500 .050 SYSTEM'S CAPACITY !WITHOOT PLANNED REPAIRS/ 22000 MW MAXIMAL LOAD IN THE COMRJTED ~IOD z 13000 MW .MmIMA L " 11 tf " " 6500 MW LOOS OF LOAD PROBABILITY =0.3241,3758-03 DEFICIENCY DAYS!DAY OR =0.1183102E 00 DEFICIENCY DAYS/YEAR . EIBCTRICAL ENERGY UNSUPPLIED= .19849?E 04 MWh/lEAR AVERAGE LOAD CURTAIII4ENT = 699.070 i.iW SYSTEM's UNRELIABILITY lUTE .229989E-Q4
=
=
=
smTiM";s-STRUC1URE
S
EZ PROGRAM KOiP- DATE-04/12/5' 1- - - - - - _RATmG GROUP' UNI!lB UNIT'S NUMBER. IN CAPAC. ~ GROOP MW 1 13 1500
2
5
500
GROUP CAPAC.
UNIT"S F.O.R.
MW 18000
2500 1000
3 5 200 '. 4 ,5 100 500 CAPACITY OF THE PREVIOUS S ~TE14 = 22000 MW UNIT"S TRANSMISSION IN THE LAST PERIOD OF TIME TO P~NED REPAlBS FRCIl PLANNED REPAIBS UNIT'S CAPACITY F.O.R. UNIT'S CAPAC:ITY F.O.R. . 1~ 0.140
.140 .120
.090
.050
1 No bits set on in this program.
54