Stochastic Market Equilibrium Model for Generation Planning

Transcription

1 8 th Internatonal Conference on Probablstc Methods Appled to Power Systems, Iowa State Unversty, Ames, Iowa, September 1-16, 4 Stochastc Market Equlbrum Model for Generaton Plannng J. Barquín, E. Centeno, J. Reneses Abstract It s wdely accepted that medum-term generaton plannng can be advantageously modeled through market equlbrum representaton. There exst several methods to defne and solve ths knd of equlbrum n a determnstc way. Medum-term plannng s strongly affected by uncertanty n market and system condtons, thus extensons of these models are commonly requred. The man varables that should be consdered as subect to uncertanty nclude hydro condtons, demand, generatng unts falures and fuel prces. Ths paper presents a model to represent medum-term operaton of electrcal market that ntroduces ths uncertanty n the formulaton n a natural and practcal way. Utltes are explctly consdered to be ntendng to maxmze ther expected profts and bddngs are represented by means of a conectural varaton. Market equlbrum condtons are ntroduced by means of the optmalty condtons of a problem, whch has a structure that strongly resembles classcal optmzaton of hydro-thermal coordnaton. A tree-based representaton to nclude stochastc varables and a model based on t are ntroduced. Ths approach for market representaton provdes three man advantages: robust decsons are obtaned; techncal constrants are ncluded n the problem n a natural way, addtonally obtanng dual nformaton; and bg sze problems representng real systems n detal can be addressed.index Terms Electrcty Markets, Game Theory, Stochastc Programmng, Hydroelectrc- Thermal Power Generaton, Conectural Varaton. T I. INTRODUCTION he new framework of compettve generaton markets arsen n a large number of countres, s requrng the development of new operaton and plannng tools to ad n decson task tradtonally made by tools desgned for a centralzed envronment. One of these strategc tasks s the medum- and long-term hydrothermal coordnaton that s beng commonly addressed n deregulated envronments by fndng a Nash equlbrum. The equlbrum pont s defned as a set of prces, generator outputs, consumpton and other relevant numercal quanttes, whch no market agent could modfy unlaterally, by changng ts behavor, wthout a decrease n ts proft. Dfferent methods have been suggested to compute ths equlbrum. It s possble to fnd t by maxmzng a proft functon for one agent f some restrctve assumptons are accepted [1], []. Other alternatves nclude game theory [3], All the authors are wth Unversdad Pontfca Comllas (Insttuto de Investgacón Tecnológca), C/ Alberto Agulera, Madrd. SPAIN (e-mal: barqun@t.upco.es). teratve methods based on a representaton of the bddng process [4], or genetc algorthms [5]. As the market equlbrum s mathematcally characterzed by a set of equatons and nequaltes that nclude the so-called complementary condtons, t has been proposed to deal wth ths ssue usng technques as heurstcs [6], the lnear complementary problem [7], [8], and the ncluson of equlbrum constrants [9]. These approaches are characterzed by ts complexty. The focusng used n ths paper to solve equlbrum market s dfferent from the prevous. The market s represented usng a conectural varaton approach. The adustment of producton that generator companes wll produce as response to a prce varaton s assumed as lnear and known [1]. It s possble to show that, under some reasonable assumptons, the soluton of the market equlbrum condtons s equvalent to the soluton of a mnmzaton problem wth a structure that strongly resembles classcal optmzaton of hydro-thermal coordnaton [11]. Ths paper extends ths market equlbrum representaton technque to allow the consderaton of uncertanty n some of the market condtons. The structure of the optmzaton problem s sutable to ft n stochastc representatons for the operaton of hydro-thermal generaton systems [1], [13], [14]. Fve man sources of uncertanty can be dentfed when t comes to medum-term generaton operaton: hydro nflows, fuel prces, system demand, generatng unts falures and competence behavor. Any of those factors can be consdered as stochastc wth the proposed focusng based on a scenaro tree representaton. The proposed model s specally nterestng from a practcal pont of vew, because t s able to compute robust decsons for bg sze systems ncludng a detaled descrpton of techncal characterstc of the system that determne the shape of the agents cost functons. In secton, the determnstc case for market equlbrum computaton by optmzaton s summarzed and generalzed for the stochastc case; secton 3 presents a study case that shows the model capabltes and fnally the man conclusons are stated n secton 4. II. STOCHASTIC EQUILIBRIUM COMPUTATION A. Determnstc market equlbrum In [11] medum-term market equlbrum was defned from a determnstc pont of vew. It was proved that under some Copyrght Iowa State Unversty, 4

2 reasonable assumptons, t can be formulated as an equvalent mnmzaton problem. Ths result s summarzed heren n a smplfed verson, for a sngle untary perod and wthout contracts sgned by generaton companes. Let us assume a generaton market n whch offers are a lnear functon of prce: P = P + α λ = 1,..., n (1) P s the offered power by the generaton company, λ the demanded prce, and P and α two constants whch characterze the lnear offer. Let us also assume that demand can be descrbed by the followng equaton: D= D α λ () D s the ntercept of the demand, whch s assumed to be known. α represents the demand slope. Gven P, α, D and α, the market clearng prce λ, generated powers P and demand D are computed by solvng the system formed by (1), () and the generaton-demand balance equaton: D = P (3) In order to compute ts offer (.e., the P and α coeffcents) each generaton company must make some assumptons on the offers that the other utltes are gong to send (.e., t must have some dea on the coeffcents P and α, ). For each partcular assumpton on these coeffcents, there s a resdual demand functon (4). Note that the sum of α where ncludes α. P = D P = D P λ α (4) Gven ths functon, companes offer must maxmze ts proft, whch s assumed to be: B = λ ( P) P C( P) (5) The frst term represents the market ncome, and the second one the producton cost. By dfferentatng and zerong: C 1 λ P = P α (6) Equatons (), (3) and (6) consttute a Nash market equlbrum that can be obtaned from the soluton of the mnmzaton problem: mn C P U D P, D ( ( ) ) ( ) = st.. P D : λ The effectve cost functons C ( ) P are defned as: (7) P C( P) = C( P) + α (8) The utlty functon U(D) s defned as: D 1 D U( D) = λ ( D) dd= D D α (9) The clearng prce λ s the Lagrange multpler of the constrant (3). B. Scenaro tree defnton If some of the parameters ncluded n the prevous representaton of market equlbrum are consdered to be subect to uncertanty, the approach has to be extended. A basc alternatve s takng t nto account by analyzng a set (normally a reduced one, due to the sze of the problem) of representatve scenaros. Market equlbrum s computed for each of them separately. Results obtaned from these scenaros are analyzed together. The man drawback of ths method s ts lack of robustness for short-term operaton: dfferent decsons are obtaned for the frst tme perod n each scenaro. Besdes, the reduced number of scenaros makes the analyss not accurate enough, f treated as a Montecarlo smulaton. A more advanced possblty to nclude uncertanty, specally when the sze of the problem prevents from an extensve Montecarlo analyss s usng a scenaro tree. It has the man advantage of allowng both to nclude stochastc varables and to compute a sngle robust decson for the frst perod of the study. A sample tree s shown n Fg. 1. Perods are numbered consecutvely and branches are also numbered consecutvely startng by b1 wthn each perod. b1 b1 b b1 b3 b4 b b5 p=1 p= p=3 p=4 Fg. 1 Sample scenaro tree Tree structure s establshed usng a correspondence and a set. The correspondence a(p,b) relatons the branch b of perod p wth the one that s mmedately before t. The set B(p) ncludes all the branches that are defned n perod p. In the tree of the fgure, for example, a(p 3,b 5 )=b and B(p 3 )={b 1, b, b 3, b 4, b 5 } A probablty w s defned for each branch. To guarantee tree coherence, the total probablty of the branches n a perod must add one: w = 1 p (1) b B( p) Besdes, the probablty of all the branches followng a sngle one must add the probablty of the prevous branch:

3 b*/ a( p, b*) = b w = w p> 1 b B( p) * p 1, b (11) The use of a scenaro tree mples the defnton of an obectve functon to be maxmzed along the whole tree for every agent n the market. In ths paper the mean proft s used defned as: B = lp w λ P C ( P ) (1) Ths expresson ncludes the duraton of each perod, l p. Alternatve obectve functons could be defned, ncludng those non-neutral to rsk modfyng the value of probabltes. Ths new defnton of proft mples that prces, companes productons, or costs may be dfferent for each branch tree. The use of dfferent cost functons for each branch tree allows a probablstc representaton of any parameter that affects ths costs, such as fuel costs, hydro nflows or unts avalablty. C. Stochastc market equlbrum defnton The prevous market representaton requres an extenson of market equlbrum concept. Frst, offers have dfferent slope n each branch of the tree, representng dfferent behavors of the companes under dfferent crcumstances. P = P + α λ = 1,..., n (13) Addtonally, demand s also dfferent for each branch that can be used to ntroduce uncertanty n demand value. D = D α λ (14) D = P (15) Smlarly to the determnstc case (6), the maxmzaton of the proft functon for each company leads to: C 1 λ P = P α (16) Equatons (14), (15) and (16) defne the stochastc market equlbrum. D. Stochastc equlbrum computaton by optmzaton The newly defned equlbrum can be obtaned from the soluton of the followng mnmzaton problem that s a natural extenson of the determnstc one. mn lp w ( C ( P) ) U( D) P, D (17) st.. P = D : η The effectve cost functons ( ) C P are defned as: P C ( P ) = C ( P ) + α (18) The utlty functon U(D ) s defned as formerly (9): D ( ) 1 D U D = λ ( D) dd= D D α (19) The clearng prce for each perod and branch λ s now obtaned from the Lagrange multpler of the constrant as follows. η λ = () lp w Optmalty condtons of the presented optmsaton problem are the same as equlbrum market equatons. Ths can be easly shown for frst order condtons. The Lagrange functon for the prevous optmsaton problem s: L( P, D, η ) = lp w C ( P) U( D) + (1) + η psb D P The soluton to the optmzaton problem s obtaned by dfferentatng and zerong ths functon. L( P, D, η ) = = D P () η L( P, D, η ) = = P (3) C( P) P = w lp η P α L( P, D, η ) lp w = = ( D D ) η (4) D α These three expressons are equvalent to those that defne market equlbrum. Equaton () corresponds to (15), the balance of generaton and demand. Equaton (3) s equal to (16), the market equlbrum condton, f prce s computed usng (). Fnally, equaton (4), wth the same prce expresson leads to (14): the lnear relatonshp between prce and demand. Thus, stochastc market equlbrum can be computed solvng an optmsaton problem. A further analyss shows that second order condtons are also the same for both formulatons. E. Inelastc demand representaton In some cases, demand can be treated as a known value for each branch and ndependent of prce. If demand s nelastc, t can be easly shown that market equlbrum can be obtaned as the soluton to a smplfed optmsaton problem: mn P lp w ( C ( P) ) st.. P = D : η (5) F. Cost functons representaton The structure of the problem allows addng varables and constrants to represent the cost functons and the techncal constrants related to t wthn the same optmzaton problem.

4 To llustrate ths pont, the power system wll be consdered as a group of thermal and hydro unts owned by dfferent companes. The followng new varables have to be ncluded: t Power generaton of thermal unt n branch b of perod p. h m Power generaton of hydro unt m n branch b of perod p. b m Power consumpton of pumped-hydro unt m n branch b of perod p. r m Energy reservor level of hydro unt m n branch b at the end of perod p. The value for the last perod s consdered to be known. s m Energy spllage of hydro unt m n branch b of perod p. Generated power, as wll be shown later, may be computed as a lnear combnaton of decson varables. Some addtonal parameters must also be consdered. Hydro nflows have been assumed as stochastc, and so they have a dfferent value n each branch. t Maxmum power generaton of thermal unt. δ Varable cost of thermal unt. o Owner utlty of thermal unt. b m Maxmum pumpng power consumpton of hydro unt m. r Maxmum energy reservor storage of hydro unt m n mp branch at the end of perod p. r Mnmum energy reservor storage of hydro unt m at mp the end of perod p. f Run-off-the-rver hydro energy for utlty n branch b of perod p. I m Hydro nflows (except run-off-the-rver) of hydro unt m n branch b of perod p. r m Intal energy reservor level of hydro unt m. ρ m Performance of pumpng for hydro unt m. o m Owner utlty of hydro unt m. u m, v m Constant and lnear term of the relatonshp between reservor storage and maxmum power for hydro unt m (run-off-the-rver hydro energy s not ncluded). Generated power for utlty n branch b of perod p s computed as a lnear combnaton of decson varables. P = t + ( hm bm ) + f (6) / o= m/ om= The followng constrants are added to the problem. Decson varables bounds: t t (7) m m m, p 1, b m h u + r v (8) b m b (9) m rmp rm rmp (3) Power balance for each branch and perod. It s the result of ncludng (6) n (15). t + h + f = D + b (31) m m m m Energy balance for each perod, branch and hydro unt. rm r, 1, (, ) = lb mp a ( hm ρm bm ) + Im s m (3) And fnally, cost functon can now ncluded explctly n the obectve functon as a lnear combnaton of decson varables, keepng the structure of the optmzaton problem. C = δ t (33) III. STUDY CASE A. System descrpton Ths study case represents operaton n a market dvded nto nne perods representng January to September. It has been consdered that hydro energy storage at the end of September s the lowest n the year and easy to forecast. Table I shows demand, whch has been consdered nelastc. TABLE I DEMAND (GWH) Dem Seven generaton companes are consdered to be competng n the market. Table II shows ts generaton structure and Table III the ranges for varable costs. Ths case s an example of a real sze case, and loosely represents Spansh daly market, where a large number of groups s owned by a reduced group of companes. The model could be also used f the same set of groups would be property of a large number of companes. TABLE II INSTALLED POWER (MW) AND NUMBER (IN PARENTHESES) OF GROUPS OWNED BY GENCOS Company Nuclear Coal Gas Hydro Pumpng (3) 5517 (15) 359 (1) 443 (7) 1431 (5) 3169 (5) 1167 (5) 4879 (11) 541 (5) 8 () 3 75 (1) 1888 (7) 113 (3) 1518 (3) 16 (1) (5) 381 (1) 314 (1) 115 (1) (5) 731 () 651 (1) 36 (1) (3) (1) 114 (3) - - TABLE III VARIABLE COST RANGES ( /MWH) Company Nuclear Coal Gas Mn Max Mn Max Mn Max Conectural varatons have been assgned accordng to companes sze, rangng from.5 to 5. for the largest ones (companes 1, and 3), between 1 and for the medum-szed utltes (4, 5 and 6) and wth a sngle value of.5 for the smallest one (company 7). Unts for ths parameter are

5 (( /MWh)/GW). Three dfferent scenaros have been consdered for hydro nflows: wet, medum and dry wth probabltes.1,.4 and.5 respectvely. The hghest probablty for the worst case from the company pont of vew (dry scenaro leads to hghest operaton cost) represents a non-neutral to rsk pont of vew. Table IV shows total hydro nflows consdered for each one. TABLE IV SCENARIOS TOTAL HYDRO INFLOWS (GWH) Scenaro Wet Medum Dry Hydro Run-off-the-rver Total Det. Wet Det. Medum Det. Dry Sto. Wet Sto. Medum Sto. Dry The structure of the tree s shown n Fg.. The value of nflows s the same for the frst three months; the rest of the year makes the dfference. These three scenaros have been solved separately (consderng a determnstc market equlbrum) and also together, solvng a stochastc market equlbrum. The obectve of ths tree structure s to decde operaton for the frst three months. Dfferences between determnstc and stochastc approaches wll be studed. Stochastc hydro condtons Known hydro condtons Jan - Mar Apr - Sep Fg. Scenaro tree for study case Wet Medum Dry The model has been coded n GAMS 1. language and solved by usng CPLEX 8.1 solver. Executon tme for the presented case s about 15 mnutes n a 1.7 GHz PC wth Pentum IV processor. B. Results Prces are presented n Fg. 3. Substantal dfferences appear between stochastc and determnstc analyss. Prces n dry and medum determnstc scenaro are smlar to those n dry and medum branch of the stochastc case. Shortage of hydro resources prevents companes from dfferent hydro resources management. However, prces n the wet branch of the stochastc case are lower n comparson to determnstc case because more hydro energy s stored n ths case. Medum determnstc case and medum stochastc branch are very smlar although these smlar prces produce dfferent hydro resources operaton for the frst tme perods. Fg. 3 Prces for determnstc and stochastc cases ( /MWh) Energy reservor levels are shown n Fg. 4. For the shake of clarty, determnstc and stochastc reservor levels have been represented agan separately n Fg. 5 and Fg. 6. The stochastc analyss suggests a more conservatve resources management n order to face the dry scenaro n case t would happen. In January, February and March a larger amount of hydro energy s reserved n the stochastc case than n the wet and medum determnstc cases. The analyss of determnstc cases shows that management of dry scenaro s unrealstc. Hgh levels of reservors levels are kept n order to take advantage of the hgh prces that wll happen from June to September. Ths would not be a good decson f a wet scenaro happened from Aprl (for example) and prces fell down. In a smlar way, wet determnstc case s not realstc. Decsons taken n the stochastc case for wet and dry branches are more robust because n ths case, only three months of nflows are supposed to be known n advance. 6 4 Det. Wet Det. Medum Det. Dry Sto. Wet Sto. Medum Sto. Dry Fg. 4 Reservor levels for company for determnstc and stochastc cases (GWh)

6 Det. Wet Det. Medum Det. Dry Fg. 5 Reservor levels for company n determnstc cases (GWh) Sto. Wet Sto. Medum Sto. Dry Fg. 6 Reservor levels for company n stochastc case (GWh) IV. CONCLUSIONS A stochastc representaton of market equlbrum has been ntroduced, ncludng ts equvalence wth an optmzaton problem to solve t effcently. Market equlbrum has been represented by means of a conectural varaton, and has been computed along a scenaro tree that allows ncludng stochastc varables. A study case wth stochastc nflows has been solved and analyzed showng ths model advantages. Other varables could have been consdered as stochastc usng the same method. The prevous features make ths approach very nterestng from a practcal pont of vew: frst, robust decsons are obtaned; second, techncal constrants are ncluded n the market equlbrum representaton n a natural way obtanng dual nformaton; and thrd, bg sze problems representng real systems wth a large number of companes (or alternatvely wth a reduced number of companes ownng a large number of groups each), can be addressed. [3] Ferrero, W., Shahdehpour, S.N. and Ramesh, V.C.: Transacton Analyss n Deregulated Power Systems Usng Game Theory, IEEE Trans., 1997, PWRS Vol. 1(3), pp [4] Otero-Novas, I., Meseguer, C., Batlle, C. and Alba, J.J.: Smulaton Model for a Compettve Generaton Market,, IEEE Trans. PWRS Vol.15(1), pp [5] Nguyen, D.H.M., Wong, K.P.: Natural Dynamc Equlbrum and Multple Equlbra of Compettve Power Markets, IEE Proc. Gener., Transm., Dstrb.,, Vol. 149, (), pp [6] Bushnell, J.: Water and Power: Hydroelectrc Resources n the Era of Competton n the Western US. Power Conference on Electrcty Restructurng, Unversty of Calforna, Energy Insttute, [7] Hobbs, B.F.: Lnear Complementary Models of Nash-Cournot Competton n Blateral and POOLCO Power Markets, 1, IEEE Trans. PWRS Vol.16(), pp [8] Rver M., Ventosa, M. and Ramos, A.: A Generaton Operaton Plannng Model n Deregulated Electrcty Markets based on the Complementary Problem. Internatonal Conference on Complementary Problems ICCP, 1999, Wsconsn. [9] Ventosa, M., Ramos A. and Rver M.: Modelng Proft Maxmzaton n Deregulated Power Markets by Equlbrum Constrants Proceedng of 13th Power System Computng Conference PSCC, June 1999, Trodhem. [1] Day, C.J., Hobbs, B.F. and Pang J.S.: Olgopolstc Competton n Power Networks: A Conetured Supply Functon Approach, IEEE Trans.,, PWRS Vol. 17(3), pp [11] J. Barquín, E. Centeno, J. Reneses, Medum-term generaton programmng n compettve envronments: A new optmzaton approach for market equlbrum computng, IEE Proceedngs - Generaton, Transmsson and Dstrbuton. Vol. 151(1). pp , January 4. [1] D. P. Morton, Algorthmc Advances n Mult-stage Stochastc Programmng, PhD thess, Stanford Unversty, Standford, [13] B.G. Gorestn, N.M. Campodonco, J.P. Costa and M.V.F. Perera, Stochastc Optmzaton of a Hydrothermal System ncludng Network Constrants, IEEE Trans. Power Systems 7, 199. [14] M.V.F. Perera and L.M.V.G. Pnto, Mult-Stage Stochastc Optmzaton Appled to Energy Plannng, Mathematcal Programmng 5. pp , V. REFERENCES [1] Yu Z., Sparrow, F.T. and Bowen B. H.: A New Long-Term Hydro Producton Schedulng Methods for Maxmzng the Proft of Hydroelectrc Systems, IEEE Trans., 1998, PWRS Vol. 13(1), pp [] Barquín, J., Centeno, E., Malllos, E. and Román J.: Medum-Term Hydro Operaton n a Compettve Electrcty Market. IEEE Internatonal Conference on Electrc Power Engneerng PowerTech, September 1999, Budapest.