PROBABILISTIC PRODUCTION COSTING OF TRANSMISSION CONSTRAINED POWER SYSTEMS UNDER GENERATION COST UNCERTAINTY

Transcription

1 PROBABILISTIC PRODUCTION COSTING OF TRANSMISSION CONSTRAINED POWER SYSTEMS UNDER GENERATION COST UNCERTAINTY P D C Wijaytunga Dept of Eectrica Eng University of Moratuwa Sri Lanka B J Cory E D Farmer C D Tembo Eectrica Energy Systems Section Imperia Coege, London ABSTRACT: A methodoogy for determining the expected production cost and reiabiity of a transmission constrained power system considering the uncertainties in the generator avaiabiities and generation costs are proposed. The method uses the group constrained muti-area representation of the transmission network with numerica convoution and is an extension to the approach in [3] to take into account generator cost uncertainties. A case study is presented with a sampe system and the resuts are compared with those obtained from Monte Caro simuation based cacuations. Key words: Group constained muti-area network mode, Monte Caro method, numerica convoution INTRODUCTION Accurate evauation of the production cost invoves the incusion of the uncertainties in the power system such as those associated with the demand eve, generation avaiabiity, the topoogy of the network and even the generation costs which are uncertain in the ong term. A these uncertainties can be readiy incuded in a Monte Caro based production cost or margina cost mode since it is a repeated performance of a deterministic cacuation under a set of simuated operating conditions [ 6]. But the computationa inefficiency in terms of time which accompanies the Monte Caro method is a major disadvantage in comparison to the anaytica probabiistic techniques for production costing (1,2]. These techniques have evoved consideraby since they werefirst introduced and arc mainy used for production costing in singe area power systems. References (3,4] discuss the use of anaytica techniques for muti-area systems and have given emphasis to the incusion of the uncertainties in generation avaiabiities. This paper presents a method of production costing in a muti-area system which is an extension to the mode presented in [3] by incuding generation cost uncertainties. approximation for a continuous distribution athough it takes a onger computation time. Therefore, depending on the computationa time requirements, the forecast generation cost distribution of each unit in the system can be discretised into severa generation cost eves with given probabiities. For the purpose of production costing, the costing period is divided into s number of time intervas corresponding to differing parts of the demand cyce or oad duration curve. The variations in individua generator costs are represented in terms of the expected mean costs for each generator and time interva, together with random components representing the cost uncertainties. In this mode the expected costs that vary between time intervas, representing variations due to fue price changes and the dependence of bidding strategy on season and oad eve. The random components, that are represented by discretised probabiity distributions, are taken to be statisticay independent. This simpification is justified because the correated cost components do not affect the merit order of the generators, or the energies produced by the generating units. Figure (1) shows a case where a given generation cost distribution is discretised into three cost eves. The vaues of C 1, C 2, C., p 1 and p 2 (p 3 = 1-(p, +p,}]) can be cacuated by equating the cumuants, k, of the origina probabiity distribution to those of the discretised distribution. This is a better way of cacuating these vaues since the cumuants represent certain aspects such as mean, variance, skew and kurtosis of the probabiity distribution. The arger the number of discrete eves the arger the number of cumuants that have to be considered to determine the unknown quantities. As can be seen in equation (1) this gives a set of non-inear simutaneous equations. Therefore it is more convenient to choose an arbitrary set of cost eves and determine the corresponding probabiities by equating the cumuants of the probabiity distributions as: GENERATION COST UNCERTAINTY In Engand and Waes the operation of the power system is now undertaken by the Nationa Grid Company(NGC), whereas other companies are invoved with generation and distribution in the system. The NGC does not have the direct knowedge of generating unit costs but ony the offer prices of the generators bidding into the common poo. In consequence, the offer prices avaiabe to the NGC contain uncertainties arising not ony from the pant avaiabities and costs but aso from the bidding strategy adopted by the generating companies. k1 = m1 k 3 m 3 -.znt k =m -3m 2 m 1 +2m~ whcrem,ith moment k 1 ith cumuant (1) When production costs are evauated for ong term generation panning in a power system, generation cost has to be considered as a quantity associated with uncertainties because of factors such as changes in fue prices and transport costs. Aso generation cost uncertainty affects the charges for the use of the transmission network and therefore it is vita to the utiites that this uncertainty is incuded in the production cost and transmission cost cacuations. When numerica convoution is used, as in this paper, for production cost computations it is important that the probabiity distributions are discretised. The higher the number of discrete eves the better is the It is assumed in this paper that the predicted generation cost distributions arc avaiabe for the anaysis. MULTI-AREA MODEL FOR THE TRANSMISSION NETWORK When the intact transmission network is used in cost cacuations taking into account the security imits of tho transmission ines, the arge number of possibe contingency constraints which any given ine can have under different operating conditions make the cacuations amost an impossibe task. Aso the inabiity of the anaytica probabiistic techniques to address the probems with the intact 379

2 transmission network eads to the use of a Monte Caro based approach which is highy inefficient in terms of computing time. Moreover, even a Monte Caro based approach cannot easiy accommodate the arge number of possibe security imits for each ine. These reasons justify the use of a group constrained muti-area network mode for the transmission network in cost cacuations [3]. In this mode it is assumed that the transmission network can be divided into severa regions depending on the transmission security imits (figure 2). Each of these regions can then be assigned with group constraints (import and export imits) between the region and the rest of the system. With the import and export imits, such a system can be transformed into a mode as shown in figure 3 where a dummy node is used as a power poo. This dummy region may or may not exist in the rea system. If in the actua power system there exists a region which is connected to a the other areas and if that region's import and export imits are not binding, such a region can be seected as the dummy region. effective export imit, E 1 1 of an area i, is give by E~ = Ei +Min (Di, Ii) (4) The transformed network is shown in figure 3. Area 1 Area 2 Proba!Uty "'"'. An:!. 3 Fc'e 2 : uti-aroa mode for the transmimion network P2 PI P3 An:!. I D C2 Figure 1: Diocretioed probabiity distribution or aenntion cost Cost of Generation Nod< E 4' E 3' Determination of the boundaries of the regions and their import and export imits are outside the scope of this paper. PRODUCTION COST MODEL FOR A GROUP CONSTRAINED POWER SYSTEM In the methodoogy [3) which is used for production costing in a muti-area network mode, the first step is to transform the network into an equivaent mode by eiminating the import constraints. This is not an essentia process but is recommended because it greaty reduces the compexity of the cacuations. The transformation is carried out by importing power into each of the areas from the dummy node up to its import imit or sufficient eve to satisfy its tota area demand depending upon which of these is the smaer. The sum of a the imports to the areas gives the effective additiona oad introduced into the dummy region. After the transformation, the effective oad, IY; in an area 'i', is given by Area Area 3 Figure 3 : Transformed muti-aroa mode If it is assumed that each generator has ony one cost eve, It 1s possibe to maintain a unique merit order ist considering a the generators in the system. To determine the production cost, the generators are oaded onto the system in merit order. The probabiity distribution of power in each area can be determined by convoving the probabiity distributions of the generator avaiabiities, g(x), recursivey to the existing distribution starting with the demand distribution, -D/(x). The probabiity distribution of power, P.u in area 'i' after oading n generators onto the system is given by equation (5) when generator 'n' (g.(x)) is in area 'i' Pni (y) = J P<n-1) i (y-x) gn (x) dx (5) where D; - origina oad and I; - import imit of area 'i' and the oad of the dummy region is given by (2) (3) Negative vaues of this distribution represent the unserved demand and positive vaues give the excess power which is avaiabe for export to the dummy region subject to the export imit.if the export power probabiity density is p,,;p. for region 'i' after oading n generators, then P! (x) = Pni (x) for O<x<Ei 0 P! (x) [ J Pni (y) dy 6 (x) for x=o (6) If the dummy region actuay exists in the power system D 0 is its origina demand or ese D 0 is zero. The eimination of import constraints essentiay aters the export imits of the areas. The p~(x) CJ Pni (y) dy] 6 (x-e;) for x=ei 1!1 380

3 This export distribution, p.. E is then convoved with the existing power distribution of the dummy node to determine its new distribution. The expected unscrved energy UE.. in each area after oading n generators is given by 0 UEni T J Pni (x) xdx (7) PROGRAM DESCRIPTION Read data corresponding the generators and each area. Repace each of the generators with generating units having the decared generating capacity for each cost eve and cacuate the probabiity distribution of their avaiabiities. where T is the time period concerned. The energy contributed by each generator is cacuated as in the case of other estabished production costing methods by taking the difference in unscrvcd energies before and after oading that generator. INCLUSION OF GENERATION COST UNCERTAINTY IN THE ALGORITHM As discussed before, the cost distribution for each generator can be approximated with a discrete distribution having a few cost eves at given probabiities. This means that a given generator can reside at the corresponding cost eves with certain probabiities during the period of investigation. Additionay, this same generator has a forced outage rate which is independent of the cost eves. In the proposed method the merit order ist is prepared for the production cost cacuations assuming that there exists a generating unit with the decared capacity for each cost eve of the generator. This means that if there are three discrete cost eves assigned to a given generator, three separate generating units are assumed to be cx1stmg. If a two state avaiabiity mode is assumed for the generators, the avaiabiity of each of these generating units, Nu can be determined as foows :- where CumuaU.. Pn>babiiy 1.0 f (8) avaiabiity of the generator i cumuative probabiity of the cost eve j of generator i (figure 4) s Arrange the generators in merit order aocating a position for each of the cost eves of the generators. To oad the generators in merit order onto the system, proceed as foows: i. deconvove any previous distribution corresponding to the same generator oaded at a different cost, from the oca area ('i ') power distribution. ii. convove the distribution of the new generating unit with the oca power distribution. Determine the export power distribution of the area. To determine the new power distribution at the dummy node, proceed as foows:- i. deconvove the previous export distribution of the area 'i' from the existing dummy node power distribution. ii. convove the new export distribution of the area 'i' with the dummy node distribution. Cacuate the unserved energies in a the areas incuding the dummy node. Cacuate the expected energy production corresponding to the generator at the present cost eve by taking the difference in unscrved energies before and after oading the generating unit. Cacuate the generation cost corresponding to each generating unit. Pi j f ,-----; Pi I r---r I ~ 4 : CWnuai.. probo.biiy distribution o...,...ion oos Cj Initiay the merit order ist is formed on the basis of the cost eves of the generators. The generators are then oaded onto the system in merit order to cacuate the production cost, as previousy described. When a generator which has been aready oaded under a different cost is reached in the merit order ist, the previous probabiity distribution is deconvoved before convoving the new distribution of the generator. This ensures that the present distribution contains the probabiistic properties of the previous &. present states and represents the same generator. 10. Repeat from step 4 to step 9 unti a the schedued generators arc oaded. 11. Determine the tota production cost corresponding to each area. CASE STUDY An arbitrary system with 21 generating units and three areas has been seected for the case study. Each generator can have up to three different cost eves at given probabiities. The data corresponding to the generators are given in tabe A and those of the areas are given in tabe A2. The production cost cacuation is aso carried out for the same system after reaxing the group constraints. The resuts obtained from this system using the proposed methodoogy are compared with those obtained using Monte Caro Simuation on the same muti-area mode in Tabe I SOOO simuations are carried out for this purpose. Aso a comparison is made with the resuts obtained assuming that each generator has ony one cost eve at the average generation cost. To acheivc comparabe resuts, it was found that a conventiona 381

4 Monte-Caro method required 5000 simuations., taking over one hour of CPU time, as compared with the convoution method that required about 2 minutes. With the appication of variance reduction techniques, incuding stratified and antithetic samping, the computationa requirement of the Monte-Caro method was reduced by a factor of two. However, the reative computationa efficiency of the convoution method, as compared with the Monte-Caro technique, increases with system size and, in particuar is expected to increase in proportion to the number of generating units and the number of transmission-constrianedzoncs of the network. A further advantage of the anaytica approach stems from its abiity to take accurate account of rare events. such as overa generation deficiency. This resuts in reiabe estimates of unserved demand and ost oad probabiities. DISCUSSION As can be seen from Tabe I the production costs cacuated from the proposed method cosey foow the resuts from a Monte Caro Simuation. Whereas the cost eves of a given generator arc repaced with a singe average cost equivaent some of the generators (6, 7, 11, 12, 21) have different production costs compared to when these cost uncertainties arc considered expicity. This is mainy due to the different oading points they can assume in the merit order ist under these two scenarios. Aso it can be seen from Tabe I that some of the generators have the same production cost under both these scenarios. This is mainy because the cost uncertainty does not affect the production cost of generators higher in the merit order since they tend to generate at their maximum possibe capacity a the time. When it comes to the margina generators the oading order significanty affects their energy contribution and hence the production cost. Therefore the incusion of the cost uncertainty of the margina generators is very important in these cacuations. Th;s observation can be effectivey utiised to improve the computationa efficiency of the proposed method especiay to reduce the number of deconvoutions invoved. But it is aso important to note that the margina generators arc determined party by the import and expon constraints in the system. Therefore the identification of these margina generators in advance is not an easy task. As shown in the Tabe II, when the import and export constraints are reaxed the difference in the energy production of the margina generators under the two scenarios is considerabe compared to that of the constrained case. This is mainy because the change in the oading point of a given margina generator due to the cost uncertainty can cause a greater change in the energy production. This is as a resut of its generay arger energy contribution towards satisfying the oad outside its oca area unike in a heaviy constrained system. CONCLUSION A method to incude generation cost (or offer price) uncertainty in anaytica production cost cacuations bas been proposed in this paper. It bas been argued that the incusion of production cost uncertainty is important not ony in production cost cacuations but aso in determini:ig the margina costs especiay in a power system where these uncertainties are reativey high. 7he proposed method is computationay efficient compared to the use of the Monte Care technique since it uses convoution me.thods and for a arger system this efficiency gain can be more apparent. Moreover the Monte Caro based methods accompany an embedded uncertainty in the cacuated expected costs. Hence the accuracy of the resuts are greaty affected especiay when the variances of the interested quantities are arger. Therefore the anaytica technique proposed in the paper can be very usefu in accuratey determining the expected production costs as we as margina costs when the transmission network can be approximated with an equivaent mutiarca mode. ACKNOWLEDGEMENT The financia assistance of the Trustees of Bie Feowships. Imperia Coege, London, Committee of Vice Chanceors and Principas. UK and the Nationa Grid Company, UK is gratefuy acknowedged. Tabe I : Production cost with active import & export constraints G A B c * LUO Generation with cost different between co A and Co B G Generator number A - Production cost from proposed mode (x 10' /MWh) B - Production cost with average generation cost(x 10' /M\\'1:) C Production cost with Monte Caro method (x 10' /MV/h) Tabe II : Production cost with reaxed import & export constraints G A B C * * *

5 Generation affected by import/export constraints REFERENCES 1. H Baeriaux et a, "Simuation de 'expoitatioo d'uo pare de machines thermiques de production d 'eectricite coupes a des stations de pompage", Revue Eectricite, Vo) 5, No 7, R R Booth, "Power system simuation mode based on probabiity anaysis", IEEE transactions on Power Apparatus and Systems, PAS 91, pp 62-69, Tabe A2 Data corresponding to areas Area 1 Area 2 Arca 3 Load (MW) Import Constraint (MW) Export Constraint (MW) E D Farmer, "Probabiistic production costing of transmission-constrained power systems", Oth Power Systems Computationa Conference, pp 1-6, Austria, August Fred N Lee, " A new muti-area production costing method", IEEE Transactions on Power Systems, Vo 3, No 3, August Priyaotha D C Wijayatuoga, Brian J Cory, E Dion Farmer, "Probabiistic costing of transmission services", IEE Iotematiooa Conference on Probabiistic Methods Appied to Power Systems, pp 1-6, London Juy Priyaotha D C Wijayatuoga, Brian J Cory, "Sampe size reduction in Moote Caro based use-of-system costing", IEE otematiooa Conference on Advances in Power System Contro, Operation and Management, pp 1-6, Hong Kong, November APPENDIX Tabe A : Generator data G A B, C2, C3 P, P2, P3 R G A B Generator number capacity of the generators avaiabiity possibe cost eves ( /MWb) corresponding probabiities Area number C2 C3 P P2 P3 R