Risk Levelized Maintenance Scheduling in Electric Power Systems

Transcription

1 Risk Levelized Maiteace Schedulig i Electric Power Systems LÁSZLÓ VARGA Directorate of Eergy Ecoomics E.ON Hugária Ltd. H-1051 Budapest, Roosevelt tér 7-8. GYÖRGY DÓSA Pao Uiversity H-8200 Veszprém, Egyetem utca 10. LÓRÁNT ORMAI Pázmáy Péter Catholic Uiversity H-2087 Piliscsaba, Egyetem utca 1. ZSOLT SOÓS Szet Istvá Uiversity H-1146 Budapest, Thököly út 74. HUNGARY laszlo..varga@eo-hugaria.com Abstract: - Schedulig of geeratig uits for prevetive maiteace plays a importat role i the capacity plaig of power systems. Optimal schedulig ca be obtaied i expoetial time, cosequetly, efforts have bee directed toward devisig approximatio algorithms which fid ear-optimal schedule. Based o the observatio that the maiteace schedulig task ad the makespa miimizatio o parallel machies are equivalet problems, i this paper effective approximatio algorithms are preseted for the levelized reserve as well as the levelized risk schedulig. These heuristic algorithms are compared with local search methods (simulated aealig ad geetic algorithms). A case study for the Hugaria Power System shows that the developed algorithms ca achieve a substatial levelizatio i the reliability idices over the plaig horizo. Keywords: maiteace schedulig, reliability of power systems, parallel machie schedulig, local search 1 Itroductio A reliable power system should have eough reserve capacity to overcome service iterruptios caused by radom equipmet failures. The availability of geeratig uits ca be improved by careful prevetive maiteace. At the same time this maiteace icreases the outage capacity i the power system i those periods whe the maiteace is carried out. Therefore the maiteace schedulig of geeratig uits is implicitly related to power system reliability [1]. With the appearace of decetralized power systems there are ew characteristic i the maiteace schedulig task [2]. A deregulated power system ca be divided ito three mai segmets: geeratio, trasmissio ad distributio. The mai task i these three segmets remaied the same as before, however, there are rules to comply with ubudlig requiremets which are established to guaratee the fair competitio amog the market participats. At the same time each segmet has certai resposibilities for the power system reliability. Therefore, they are resposible for performig the ecessary maiteace o their equipmet. O the liberalized market uit maiteace schedulig is decided by geeratio compaies where the goal is to maximize their profit. Co- ISSN: ISBN:

2 sequetly, these compaies try to carry out equipmet maiteaces i those time periods whe the market clearig price is lowest. The aim of the Idepedet System Operator (ISO) is to provide a required reliability level maximizig the reserve capacity i the power system. I this paper the maiteace schedulig problem is cosidered from the ISO poit of view. I this cotext it is assumed that geeratio compaies specify their optimal maiteace schedulig pla ad hads it over to ISO. The ISO performs its ow optimal maiteace schedulig cosiderig the adequate reliability of the power system. If the two maiteace strategies are close eough i terms of reliability, it will be approved; otherwise the uits, which violate the costraits, will be rescheduled. 2 Basic Models of Maiteace Schedulig Assume that, the plaig horizo is oe year, which ca be divided ito (e.g. weekly) time itervals. Basically, there are two differet obective fuctios i maiteace schedulig [1]. Let c, c, 1 2, c deote the capacity of the geeratig uits. The istalled capacity of the power system is c. If the forced outage rate (FOR) of uit i th is based o the twostate Markovia availability model we ca defie X i as the uavailability capacity for uiti. X i s (i 1,, ) form a sequece of idepedet radom variables where X i c i with probability f ad X with probability 1. i i 0 i c i i Levelized Reserve Method Let p be the peak load i the th time icremet. The gross reserve i this particular period is s c p, 1,2,, m, where m is the umber of time itervals i a year. Let uits be scheduled o maiteace i the time iterval, cosequetly are f i f i available. The et reserve i this particular iterval is r s c where is the idex of k 1 k the uit i the k th positio i sequece scheduled i the th time icremet. The reliability level of the power system is proportioal to the system reserve, so we have to maximize the miimum et reserve over the time itervals of the schedulig horizo. The obective fuctio of the levelized reserve method is as follows: r opt max mi r. (1) 1 m 2.2 Levelized Risk Method Let P deote the system load i the th time period. I the power system the electric loads vary i a give time iterval so it ca be regarded as radom variable. Let uits be scheduled for maiteace i the time iterval, cosequetly uits are available for geeratio. The the LOLP (Loss of Load Probability) idex for this particular iterval is: LOLP Pr 1, 2,,m, X k 1 k k k 1 P c where k is the idex of the uit i the k th positio i sequece scheduled i the th time iterval. A appropriate schedulig esures a sufficiet reliability level over the whole year. Cosequetly, i the case of risk levelized schedulig, we have to miimize the largest LOLP idex over the time itervals of the schedulig horizo: LOLP opt mi max LOLP. (2) 1 m Aother commoly used reliability idex is the Loss of Load Expectatio (LOLE): k LOLE T * LOLP, 1,2,,m, ISSN: ISBN:

3 where T is the legth of time period i hours (e.g. 168 hours i weekly periods). The LOLE is the expected time durig which the power deficit greater tha the available capacity of the power system. Reliability idices ca be determied from the probability distributios of the outage capacities ad the system loads i each time iterval. I order to calculate the outage capacity, its oit distributio has to be computed from the idividual uit uavailability distributios determied by FOR of the geeratig uits [1]. 3 Modelig Approaches I the literature several criteria ad techiques have bee employed to develop maiteace plas. Levelized reserve methods seek to equalize the reserve for each time iterval of a year (1). I [3] a implicit eumeratio algorithm is applied, a commoly used method for solvig mixed-iteger programs. This is oe of the most frequetly cited papers i the literature of maiteace schedulig ad the first attempt to formulate the problem as a exact discrete optimizatio task. Recetly, may kids of computatioal itelligece methods have bee applied to solve the uit maiteace schedulig problem. I [4] the simulated aealig method was used with the result that the algorithm succeeded i fidig a solutio to large problems. Followig this success, i [5] the authors developed a techique to iclude a geetic algorithm ad the icorporated tabu search elemets ito their method. I order to solve the maiteace schedulig task [6] suggests a appropriate chromosome represetatio i the geetic algorithm to accommodate costraits of the schedule ad a adequate evaluatio fuctio is offered to satisfy optimum criteria. Due to the large search space it is difficult to reach a optimal solutio usig traditioal geetic algorithm. This solutio is improved i [7] where the task is divided ito several layers. The power system plaig i restructured eviromet ca be formulated for selfiterested etities [2]. I the coordiatio of differet goals a optimal trade-off should be achieved betwee the profit maximizatio for geeratio compaies ad power system reliability. I [8] the maiteace schedulig of geeratig uits i a deregulated power system is formulated i differet aspects. It uses geetic algorithm to fid the optimum schedule for the prevetive maiteace cocerig the forecasted market clearig price. [9] suggests schedules usig the criterio of maximum profit for the geeratio compay. The obective fuctio icludes a pealty compoet for violatio of reliable supply. The risk-levelig techiques seek to spread the risk evely as measured by some kid of reliability idex (2). This task is much more complex from the computatioal poit of view sice i each step of the schedulig algorithm we should calculate the reliability idex, which also eeds cosiderable ruig time, furthermore the obective fuctio will ot be liear. I [10] the author took the first step by developig a approximatio algorithm based o the otio of the load carryig capacity of geeratig uits. Utilizig the aalogy betwee the maiteace schedulig ad the makespa miimizatio o parallel machies a effective heuristic algorithm was described i [11], which ca be applied for both levelig strategies. A applicatio of the method is preseted for the Hugaria Power System i [12]. 4 Parallel machie problem Based o the observatio that the maiteace schedulig of geeratig uits i power systems ad the makespa miimizatio o parallel machies i productio schedulig are equivalet problems, i this paper effective approximatio algorithms are preseted where heuristic as well as local search algorithms will be applied. A overview o mathematical schedulig is preseted i [13]. A state-of-the art review of the parallel machie problem ca be foud i [14]. The task is to schedule idepedet obs with p 1, p2,, p processig times o m idetical machies. Each ob ca be processed by exactly oe machie; o machie ca work o more tha oe ob simultaeously. There are o ISSN: ISBN:

4 precedece costraits ad preemptio is ot allowed. A schedule is a assigmet of the ob to a machie. A ob sequece determies the completio times of the obs c 1, c2,, c. The makespa, defied as c max c 1, c2,, c is equal to the completio time of the last ob. For ay schedule, the load of machies is the total processig requiremet of the obs assiged to this machie. The load of the th machie is equal to (3) p k 1 k where is the umber of obs executed o the th machie, k is the idex of ob i the k th positio o the th machie. Sice, the makespa is the load o the most heavily loaded machie, the obective fuctio of the makespa miimizatio is for all schedules. mi max 1 m 4.1 Heuristic algorithms A effective heuristic for the makespa optimizatio o parallel machies is the MultiFit algorithm. The makespa miimizatio is i a sese the dual problem to the well-kow bi-packig task. The tasks are i bi-packig thought of as items with weights, which are placed i bis of a give capacity. Our goal is to miimize the umber of bis used i the packig. The First- Fit-Decreasig (FFD) algorithm is a attempt to fid a ear-optimal packig very effectively. I the first step of the FFD heuristic weights are put i decreasig order. The a packig is built up by treatig each item i successio, ad addig it to the lowest idexed bi ito which it will fit without violatig the capacity costrait. The makespa miimizatio i the parallel machie schedulig we iterest i a differet questio amely, give the umber of bis, how large does the bi capacity have to be. To determie the bi capacity a biary search is executed betwee the estimated upper ad lower bouds of the bi capacity. Aother approach to the makespa miimizatio came from the two dimesioal orthogoal packig. Let a set of rectagular figures be ito a give "ope-eded" rectagular area. Each rectagular piece is defied by a ordered pair correspodig to the horizotal ad vertical dimesios of rectagle. We are cocered with the packig of the pieces i the list ito the "big" rectagle so as to miimize the height of the packig; i.e. the maximum height, measured from the bottom edge of the "big" rectagle, of the space occupied by ay piece i the packig. Obviously, there ca be wasted spaces ("holes") i the big rectagle. I the makespa miimizatio task we eed a relaxatio of this problem where those parts of the last placed rectagle, which do ot cotact the upper edge of packed rectagles, are allowed to fall dow. That meas we push them dow while they touch the top of replaced rectagle or the bottom edge of the "big" rectagle. It is clear that if the horizotal dimesios i the relaxed task are the same, the rectagular packig is equivalet to the makespa miimizatio for the parallel machies. I the first step we costruct a priority list i decreasig order correspodig to height or width of the rectagles. For the solutio of packig, a good heuristic is the so-called Bottom-up Left-ustified (BL) algorithm. The BL algorithm packs the pieces oe at a time as they are draw i sequece from the list. Whe piece is packed ito "big" rectagle it is first placed ito the lowest possible locatio, ad it is left ustified at this vertical positio. A geeralizatio of the BL algorithm: i each steps we determie which rectagles cause the most serious problem that is which piece will icrease most seriously the height i the actual packig. This rectagle is placed accordig to the BL strategy. Observe that this algorithm does ot eed a fixed priority list. I this paper this schedulig strategy is amed as MiMax algorithm. ISSN: ISBN:

5 4.2 Local Search Methods Variatios of local search algorithms have bee proposed based o aalogues with processes i ature such as statistical physics (simulated aealig), biological evolutio (geetic algorithms), ad europhysiology (artificial eural etworks) [15]. A commoly used techique i optimizatio is the threshold algorithm, which starts from a feasible solutio, the selects a solutio from the eighbourhood of the curret value. Uder the procedure it compares the differece i obective fuctio. If the differece is below a give threshold, the eighbour replaces the curret solutio. Otherwise, the search cotiues with the last solutio. I the course of the algorithm's executio, the threshold values are determiistic ad gradually lowered. Simulated aealig uses radomized thresholds where solutios correspodig to large icreases i obective fuctio have a small probability of beig accepted, whereas solutios correspodig to small icreases have a larger probability of beig accepted. Geetic algorithm starts with a large commue of chromosomes (e.g. biary strigs) kow as the iitial populatio. The iitial values of chromosomes correspod to discrete values of a matrix (populatio matrix). The iitial populatio is passed to the cost fuctio (fitess) evaluatio. From this iitial populatio a large portio of the high cost chromosomes are discarded through atural selectio (survival of the fittest). First, the costs associated chromosomes are raked from lowest cost to highest cost. The, oly the best members of the populatio are kept for each iteratio while the others are discarded. Two chromosomes are selected from the mate i pool to produce two ew offsprig. Pairig takes place i the matig populatio util offsprig are bor to replace the discarded chromosomes. Radom mutatios are performed with some small probability to make variety ito the populatio ad to avoid fast covergece to a local optimum. The ext step raks the ew populatio ad selects a ew matig pool. The process cotiues util some stoppig criteria are satisfied. 5 Maiteace Schedulig Agorithms I the previous sectio several approximatio algorithm were described, which are developed essetially for the makespa miimizatio of parallel idetical machies ad other resource schedulig tasks. 5.1 Equivalecy with Parallel Machie Problem If the duratio of maiteace is oe time period for ay geeratig uits; ay geeratig uits ca be scheduled i ay time periods ad we assume that the peak loads are costat durig the year the maiteace schedulig problem is equivalet to the makespa miimizatio o idetical parallel machies [11]. Note that the time itervals i the maiteace schedulig task correspod to the machies i the parallel machie schedulig. The capacities of the geeratig uits correspod to the processig times of the obs o the parallel machies so the plaed outages i the several time periods correspod to the load o the machies. Cosequetly, the obective fuctio of the levelized reserve schedulig (1) is the same as the obective fuctio (3) for the parallel machie problem where the makespa is miimized. I the levelized risk method usig LOLP reliability idex we should miimize the probability that the outage capacity of the power system greater tha a fixed value (the available capacity of the power system). I the parallel machie problem this obective fuctio is equivalet to miimizig the probability whe the makespa is greater tha a fixed value. 5.2 Approximatio Algorithms Basically, the task is to place two-dimesioal rectagles with give width (maiteace requiremets i weeks) ad give legth (capacity of the uits i MW) cosiderig the weekly peak loads ad satisfyig the costraits of the maiteace (e.g. fixed maiteace periods, ot allowed time itervals for uit maiteace, ISSN: ISBN:

6 umber of uits that allowed to be simultaeously o outage). The first step is to make a priority list i decreasig order correspodig to the legth of maiteace requiremets ad the capacities of geeratig uits. The ext step is to select a packig strategy. I this paper we applied Bottom-up Left-ustified (deoted by BL), MiMax (MM) ad MultiFit (MF) heuristics. I some cases we modified the obtaied result i such a way that we alter the schedulig of oe or few uits (these versios are distiguished deoted by "r"). This reschedulig of the uits ca be carried out i radom or determiistic way. I the geetic algorithm (deoted by GA) the pool is geerated calculatig a certai permutatio of feasible solutios. The geetic operators operate these idex sequeces from which we costruct schedules usig ay heuristic algorithms. I the simulated aealig approach (deoted by SA) we started a radom solutio, which is a feasible schedulig of the geeratig uits. This solutio is altered accordig to the scheme of local search strategy. 6 Results A demostrative example from the Hugaria Power System is preseted to show the applicatio of the approximatio algorithms described i Sectio 5. The system cosists of 41 thermal ad 4 uclear uits located i 17 differet power plats. The total capacity of the cosidered power system is 7070 MW. The yearly peak load is 5800 MW. The year was divided ito 53 weekly time icremets. I the first test calculatio schedulig costraits are ot cosidered i.e. ay geeratig uits ca be scheduled i ay time periods. Two local search (GA ad SA) ad three heuristic methods (BL, MM, MF) were applied i reserve levelig optio where i case of BL ad MF heuristics we calculated the obective fuctio without ad with reschedulig. I Table 1. the results are summarized where the peak load, the gross reserve ad the et reserve are show characterized by their miimum, maximum, average values ad their stadard deviatio over the schedulig horizo. The best et reserve ca be obtaied usig MultiFit heuristic with replacig (MFr) where the optimal et reserve is 1270 MW with stadard deviatio of 45 MW. The MM ad the BL algorithms ca achieve much more uevely et reserve (stadard deviatios are 105 MW ad 104 MW, respectively) ad lower optimal system reserve (1220 MW). I the secod versio of the test calculatio from the 45 geeratig uits 24 uits are ot allowed to maitai i witer seasos (Jauary, February, November, ad December) ad there is o maiteace i the New Year eve ad Christmas period. Obviously, uder these costraits more disadvatageous et reserve patter ca be obtaied as we performed i the previous calculatio. The best algorithm as i the first versio is the MFr (the optimum value of the et reserve is 1250 MW with the stadard deviatio of 160 MW). The MiMax heuristic ca achieve 1200 MW system reserve; the stadard deviatio is 164 MW. Moreover i this versio the LOLE idices were calculated. Usig the MFr heuristic the highest LOLE idex is hour while the MM ca achieve hour (see Table 2). I Fig. 1. the et reserves ad LOLE idices are show over the schedulig horizo applyig the MFr ad MM algorithms. The MM heuristic i levelized risk optio exhibits more uevely distributed et reserves i compariso with the MFr strategy but it ca achieve more uiformly distributed risk values. Usig the MFr i the levelig reserve optio the highest (worst) LOLE idex is hour (stadard deviatio is hour) while the MM algorithm i levelized risk optio performed hour (stadard deviatio is hour). 7 Coclusio Based o the observatio that the maiteace schedulig of geeratig uits i power systems ad the makespa miimizatio o parallel machies i schedulig theory are equivalet problems, i this paper effective approximatio algorithms are preseted. Usig a demostrative example from the Hugaria Power System, these heuristics are compared with each other ad ISSN: ISBN:

7 with the local search methods i the levelized reserve schedulig strategy. The heuristic (costructive) methods are much more effective from the computatioal poit of view eve they achieved better system reserve i compariso with local search techiques. Usig the MM heuristic i the levelized risk optio evertheless the et reserve showed more irregular fluctuatio, the algorithm ca achieve more uiformly distributed reliability idices over the year which results i a more reliable power system. The applicatio of the method yielded higher reliability level at critical time itervals of the year (see Fig. 1). Refereces: [1] Wag, X, McDoald, J.R. (eds), Moder Power System Plaig, McGraw-Hill Book Compay, Lodo, [2] Shahidehpour, M, ad Marwali, M. (eds), Maiteace Schedulig i Restructured Power Systems, Kluwer, [3] Dopazo, J.F., Merrill, H.H., Optimal Geerator Maiteace Schedulig Usig Iteger Programmig, IEEE Tras. o Power Apparatus ad Systems, vol. PAS-94, No. 5, Sept/Oct 1975, pp [4] Satoh, T. ad Nara, K., Maiteace schedulig by usig the simulated aealig method, IEEE Tras. o Power Systems, vol. 6, No. 2, 1991, pp [5] Kim, H., Hayasi, Y. ad Nara, K., A algorithm for thermal uit maiteace schedulig, through combied use of GA, SA, ad TS, IEEE Tras. o Power Systems, vol. 12, No. 1, 1997, pp [6] Negevitsky, M. ad Kelareva, G., Applicatio of geetic algorithms for maiteace schedulig i power systems, IEEE Tras. o Power Systems, vol. 9, No. 3, August 2001, pp Coferece ad Expositio, April 2008, pp [8] Eshraghia, R. Modir Shaechi, M.H., Raabi Mashadi, H., A New Approach for Maiteace Schedulig of Geeratig Uits i power Market, 9 th Iteratioal Coferece o Probabilistic Methods Applied to Power Systems, KTH, Stockholm, Swede, [9] Fedotova, G.A, Voropai, N.I., A techique of geeratio equipmet maiteace schedulig i deregulated electric power systems, Third Iteratioal Coferece o Electric Utility Deregulatio ad Restructurig ad Power Techologies, 6-9 April 2008, Nauig, Chia, pp [10] Garver, L.L, Adustig Maiteace Schedules to Levelize Risk, IEEE Tras. o Power Apparatus ad Systems, vol. PAS-91, No. 5, Sept/Oct 1972, pp [11] Varga, L., Determiistic ad Stochastic Schedulig for Prevetive Maiteace i Electric Power Systems, i Arkeryd, L., Bergh, J., Breer, P. ad Pettersso, P. (eds): Progress i Idustrial Mathematics at ECMI 98, B.G. Teuber Stuttgart, Leipzig, 1999, pp [12] Varga, L., Soós, Zs., Zsigmod, J., Levelized Risk Schedulig for Prevetive Maiteace, IEEE PowerTech99 Coferece, Budapest, 1999, BP [13] Dempster, M.A.H., Lestra, J.K., Riooy Ka (eds), Determiistic ad Stochastic Schedulig, Reidel, Dodrecht, [14] Cheg, T.C.E., Si, C.C.S., A state-of-the-art review of parallel-machie schedulig research, EJOR, vol. 47, 1990, pp [15] Aarts, E., Lestra, J.K. (eds), Local Search i Combiatorial Optimizatio, Joh Wiley & Sos, [7] Negevitsky, M., Developmet of a multi-layer geetic algorithm for maiteace schedulig i power systems, Trasmissio ad Distributio ISSN: ISBN:

8 Table 1 Results of the Levelized Reserve Schedulig PeakL GrossRes Net Reserve (MW) (MW) MM GA SA BL BLr MF MFr Mi Max Average StDev Table 2 Results of the Levelized Reserve ad Levelized Risk Schedulig PeakL GrossRes (MW) ReserveL Net Reserve RiskL (MW) (MW) MM GA SA BL BLr MF MFr MM Mi Max Avg StDev Loss of Load Expectatio ReserveL RiskL (hours) (hours) MM GA SA BL BLr MF MFr MM Mi Max Avg StDev LOLE (hours) LOLE_LRisk LOLE_LRes RES_LRes RES_LRisk Weeks Net Reserve (MW) Fig. 1: Net Reserves ad LOLE idices over the weeks of the year ISSN: ISBN: