Cournot equilibria in twosettlement electricity markets with system contingencies. Jian Yao, Shmuel S. Oren* and Ilan Adler


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1 142 Int. J. Crtal Infrastrutures, Vol. 3, Nos. 1/2, 2007 Cournot equlbra n twosettlement eletrty markets wth system ontngenes Jan Yao, Shmuel S. Oren* and Ilan Adler Department of Industral Engneerng and Operatons Researh Unversty of Calforna Berkeley, CA 94720, USA Emal: Emal: Emal: *Correspondng author Abstrat: We study Nash equlbrum n twosettlement ompettve eletrty markets wth horzontal market power, flow ongeston, demand unertantes and probablst system ontngenes. The equlbrum s formulated as a stohast Equlbrum Problem wth Equlbrum Constrants (EPEC) n whh eah frm solves a stohast Mathematal Programme wth Equlbrum Constrants (MPEC). We assume a noarbtrage relatonshp between the forward pres and the spot pres. We fnd that, wth two settlements, the generaton frms have nentves to ommt forward ontrats, whh nrease soal surplus and derease spot energy pres. Furthermore, these effets are amplfed when the markets beome less onentrated. Keywords: two settlements; Cournot; equlbrum problem wth equlbrum onstrants. Referene to ths paper should be made as follows: Yao, J., Oren, S.S. and Adler, I. (2007) Cournot equlbra n twosettlement eletrty markets wth system ontngenes, Int. J. Crtal Infrastrutures, Vol. 3, Nos. 1/2, pp Bographal notes: Jan Yao s a PhD anddate n the IEOR department at the Unversty of Calforna at Berkeley. He was Researh Assstant on the projet Extended Enterprse oalton for ntegrated Collaboratve Manufaturng Systems (EECOMS), funded by NIST Advane Tehnology Program (ATP), and Software Engneer for Advaned Plannng and Shedulng produts at Orale Corporaton. He reeved an MS n Computer Sene from the Unversty of North Carolna, and an MS and a BS n Mehanal Engneerng from Shangha Jao Tong Unversty. Yao s a member of the INFORMS. Dr. Shmuel S. Oren s Professor of IEOR at the Unversty of Calforna at Berkeley. He s the Berkeley Ste Dretor of PSERC (the Power System Engneerng Researh Center). He publshed numerous artles on aspets of eletrty market desgn and has been a onsultant to varous prvate and government organsatons, nludng the Brazlan regulatory ommsson, The Alberta Energy Utlty Board, the Publ Utlty Commsson, the Polsh system operator and the Publ Utlty Commsson of Texas, where he s urrently a Senor Advsor to the Market Oversght Dvson. He holds a BS and MS n Mehanal Engneerng and Materals Engneerng, respetvely, from the Tehnon n Israel, and he reeved an MS and PhD n Engneerng Eonom Systems n 1972 from Stanford. Dr. Oren s a Fellow of INFORMS and the IEEE. Copyrght 2007 Indersene Enterprses Ltd.
2 Cournot equlbra n twosettlement eletrty markets 143 Dr. Ilan Adler s Professor and Char of the IEOR department at the Unversty of Calforna at Berkeley. Hs man researh atvtes are n the area of optmsaton n general and nteror pont methods n partular. He has also been a onsultant to varous ompanes on large, omplex optmsaton problems. He holds a BA n Eonoms and Statsts from the Hebrew Unversty n Israel, an MS n Operatons Researh from the Tehnon n Israel and a PhD n Operatons Researh from Stanford. Dr. Adler s a member of INFORMS and the Mathematal Assoaton of Amera. 1 Introduton The last deade has wtnessed a fundamental transformaton of the eletr power ndustry around the world, from one domnated by regulated vertally ntegrated monopoles to an ndustry where eletrty s produed and traded as a ommodty through ompettve markets. In the USA, ths transformaton was poneered n the late 1990s by Calforna and the northeastern power pools, nludng the PennsylvanaNew JerseyMaryland (PJM) Interhange, New York and New England. A reent arrval s the Eletral Relablty Counl of Texas (ERCOT) market. Whle there are sgnfant dfferenes among the many mplemented and proposed market desgns that vary n terms of ownershp struture, level of entralsaton and authorty of the system operator, the prmary ratonale for eletrty restruturng n most markets has been to reap welfare gans by supplantng regulaton wth ompetton. Both theory and experene from other formerly regulated ndustres suggest that these gans wll nlude nreased effeny of shortrun produton, of resoure alloaton and of dynam nvestment. A potentally sgnfant obstale to these welfare gans s market power. Market power exersed by supplers typally entals the wthholdng of output and an upward dstorton n the market pre. Market power s generally assoated wth varous forms of eonom neffeny. Among the many proposed and mplemented eonom tools for mtgatng market power s a multplesettlement approah wheren forward transatons, dayahead transatons and realtme balanng transatons are settled at dfferent pres. The rss n Calforna n 2001 has drawn more attenton to the role of forward markets n mtgatng market power and n managng pre rsk n the eletrty supply han. Theoretal analyss (Allaz, 1992; Allaz and Vla, 1993; Kamat and Oren, 2004) and empral evdenes (von der Fehr and Harbord, 1993; reen, 1999; Newbery, 1998; Powell, 1993) suggest that forward ontratng and multsettlement systems redue the nentves of sellers to manpulate spot market pres, sne under a multsettlement approah, the volume of tradng that an be affeted by an nrease n spot pres s redued substantally. Thus, forward tradng s vewed as an effetve way of mtgatng market power at real tme. Allaz (1992) assumes a twoperod market and demonstrates that, f all produers have aess to a forward market, t leads to a prsoners dlemma type of game among them. Allaz and Vla (1993) show that, as the number of forward tradng perod s nreases, produers lose ther ablty to rase energy pres above ther margnal ost. Kamat and Oren (2004) analyse twosettlement markets over two or threenode networks, and extend Allaz s results to a system wth unertan transmsson apates n the spot market. It s also argued that settng pres at ommtment tme
3 144 J. Yao, S.S. Oren and I. Adler provdes nentves for aurate foreastng and provdes exante pre dsovery that faltates tradng. Aurate foreastng and advaned shedulng of generaton and load also mprove system operaton and relablty whle redung the ost of reserves to handle unexpeted devatons from shedule. Whle ntutvely the above arguments n favour of forward tradng and multsettlement systems are ompellng, these studes typally gnore network effets, flow ongeston, generator outages and other system ontngenes. When flow ongeston, system ontngenes and demand unertantes are all present n the spot market, t s not lear to what extent produers are wllng to engage n forward transatons, or how ther nentves wll be affeted. Furthermore, t s not well understood whether forward tradng may n fat help produers exerse market power n the spot market to lok n or even nrease ther olgopoly rents. If ndeed forward tradng an be used to mtgate the exerse of market power but generators have lttle nentve to engage n suh tradng, a natural publ poly queston would be whether forward ontratng should be mposed as a regulatory requrement and the market be desgned to mnmse spot transatons. As a matter of fat, the urrent market rules n Calforna and n Texas are desgned to lmt the sope of the realtme balanng markets through penaltes or added harges. In ths paper, we formulate the twosettlement ompettve eletrty markets as a twoperod game, and ts equlbrum as a subgameperfet Nash equlbrum (see Fudenberg and Trole, 1991) expressed as an Equlbrum Problem wth Equlbrum Constrants (EPEC), n whh eah frm faes a Mathematal Programme wth Equlbrum Constrants (MPEC, see Luo et al., 1996; Karush, 1939), parametr on other frms forward ommtments. We apply the model to an IEEE 24bus test network. Wth the spef data and smplfyng assumptons of the example, t s shown that n equlbrum, frms ommt ertan quanttes n forward transatons and adjust ther postons n the spot market n respondng to ontngenes and demand realsaton. Whle ths paper overs muh of the same ground and employs the same modellng framework as Yao et al. (2004), t extends the prelmnary work reported n Yao et al. (2004) n both ontent and detal. Furthermore, the prelmnary omputatonal tests reported n Yao et al. (2004) for a stylsed sxbus network have been extended to a more realst IEEE 24bus network, whh demonstrates the omputatonal feasblty of the proposed model. The rest of ths paper s organsed as follows. The next seton presents the model assumptons and the mathematal formulaton. An example and numeral results are dsussed n Seton 3. 2 The model 2.1 Modellng approah We shall desrbe now our model for alulatng the equlbrum quanttes and pres of eletrty over a gven network wth two settlements. We vew the two settlements n the eletrty market as a NashCournot game wth two perods: a forward market (Perod 1), and a spot market (Perod 2).
4 Cournot equlbra n twosettlement eletrty markets 145 In Perod 1, frms enter nto forward ontrats by ompetng n a Cournot fashon, antpatng one another s forward ommtments and the ommon knowledge of the expeted spot market outome n Perod 2. In the spot market, the unertan ontngenes are realsed and the generaton frms at as Cournot ompettors, hoosng ther spot produton quanttes for the generaton unts. In dong so, they take as gven the revealed forward ommtments of all other generaton frms, the onjetured spot produton desons of all other generators, and the redspath desons of the Independent System Operator (ISO) spefyng the mport/export quantty at eah node. Smultaneously wth the generators produton desons, the ISO makes ts redspath deson, determnng mports and exports at eah node so as to maxmse total soal welfare based on ts onjetured spot produton at eah node, the transmsson onstrants and the energy balane onstrant. Our model permts dfferent levels of loatonal granularty n the forward and spot markets. Spefally, we wll assume that n the forward market nodes are lustered nto zones and frms enter nto forward ontrats, whh spefy forward zonal quantty ommtments at agreedupon zonal pres. Another key assumpton underlyng our formulaton s that the forward market s suffently lqud, so that the forward pre n eah zone s unform aross all frms operatng n the zone whle the forward ommtments are publ knowledge n the spot market. All forward ontrats are settled fnanally n the spot market based on the dfferene between the forward zonal pre and the spot zonal pre, whh s a weghted average of all spot nodal pres n the zone. The weghts used n determnng the spot zonal pres are onstants that reflet hstoral load shares but are not endogenously determned based on atual load shares n the spot market. We also assume that rskneutral speulators take opposte postons to the generaton frms and explot any arbtrage opportuntes so that the forward pre n a zone equals the orrespondng expeted spot zonal pres over all possble ontngenes (no arbtrage assumpton). The avalable apates of generaton unts and transmsson lnes n the spot market are unknown n Perod 1 and are subjet to stohast varatons n Perod 2. We model the transmsson network onstrants n the spot market n terms of a lossless DC approxmaton of Krhhoff s laws. Spefally, flows on lnes an be alulated usng Power Transfer Dstrbuton Fators (PTDFs), whh spefy the proporton of flow on any partular lne resultng from an njeton of one unt at a partular node and a orrespondng wthdrawal at an arbtrary, but fxed, slak bus (Chao and Pek, 1996). Unertanty regardng the realsed network topology n the spot market s haratersed by dfferent PTDF matres wth orrespondng probabltes. In order to avod omplatons due to dsontnuous payoffs n the spot market, we follow the ommon assumpton (see Hobbs, 2001; Smeers and We, 1997; We and Smeers, 1999) that agents do not game the transmsson pres or onsder the mpat of ther produton desons on ongeston pres. Assumng that produers an game transmsson pres may yeld multple spot market equlbra or may result n no pure strategy equlbrum (see Cardell et al., 1997; Hobbs et al., 2000), whh makes the equlbrum omputaton ntratable. Moreover, when gamng the transmssonmarket, generaton frms wll typally fnd t optmal to barely ongest some lnes so as to avod ongeston rent; ths wll lead to degenerate spot market equlbrum (see Oren, 1997), f any. Empral evdene suggests that gnorng potental gamng of transmsson pres by generators s qute realst, sne t s pratally mpossble for
5 146 J. Yao, S.S. Oren and I. Adler multplegeneraton frms to oordnate ther produton so as to avod ongeston harges by barely deongestng transmsson lnes. Indeed, the ntal desgn of the Calforna market, whh attempted to ontrol ongeston by relyng on suh oordnaton n response to advsory ongeston harges, proved to be unworkable. For smplty, we further assume that there s at most one generaton falty at a node (ths assumpton an be easly relaxed by aggregatng unts owned by a sngle frm and splttng nodes wth multple frms). 2.2 Model notatons Sets: N : The set of nodes (or buses) Z : The set of zones. Moreover, z() represents the zone where node resdes L : The set of transmsson lnes whose ongeston n the spot market s under onsderaton. These lnes are alled flowgates. C : The fnte set of states n the spot market : The set of generaton frms. N g denotes the set of nodes where generaton faltes of frm g are loated Parameters: q, q : The lower and upper apaty bounds of generaton falty at node N n state C. p (): The lnear Inverse Demand Funton (IDF) at node n state : p ( q) = p bq N, C We assume that n eah state the pre nterepts of the nverse demand urves are unform aross all nodes, and that, for eah node, the nodal demand shfts nward and outward n dfferent states, but the slope remans unhanged. C (): The ost funton at node. In ths model, the ost funtons are assumed lnear. C ( q) = d q N K : The flow apaty of lne l L n state C. l D l, : The power transfer dstrbuton fator n state C on lne l L wth respet to node N. Pr(): The probablty of state C of the spot market : δ The weghts used to settle the spot zonal pres ( δ : z( ) = zδ = ) 0, 1.
6 Cournot equlbra n twosettlement eletrty markets 147 Deson varables: x gz, : Forward quantty ommtted by frm g to zone z Z. q : eneraton level at node N n state C of the spot market r : Import/export quantty at node N by the ISO n state C of the spot market 2.3 The formulaton The zonal forward market, whh s onduted at zonal tradng hubs (e.g., PJMs Western hub), gnores ntrazonal transmsson ongeston (although suh ongeston s aounted for mpltly through the ratonal expetaton of the spot zonal pre, whh s based on a weghted nodal pre average). The spot market, on the other hand, s organsed at a nodal level wth all transmsson onstrants reognsed n the ISO redspathes. In eah state C, the spot nodal pre at eah node N s gven by the nodal nverse demand funton p ( q + r ) appled to the net loal onsumpton that results from the loal produton deson by the generatng frms and the redspath deson by the ISO. The spot zonal (settlement) pre u z at a zone z n eah state s defned as the weghted average of the nodal pres n that zone wth predetermned weghts δ. Mathematally, the zonal spot settlement pres are gven by: u = δ p ( r + q ), z Z. z z : ( ) = z The forward zonal pres h z are the pres at whh forward ommtments are traded n the respetve zones. The noarbtrage assumpton mples that the forward zonal pres are equal to the expeted spot zonal settlement pres: hz = Pr() uz, z Z. (1) C In eah state of the spot market, the frms hoose the produton levels. Eah frm g s revenue n eah state s the sum of ts forward ommtment settlement at the spot zonal settlement pres, and the payment for ts produton quanttes at the spot nodal pres. So ts proft s: π = p ( r + q ) q u x C ( q ) g z g, z Ng z Z Ng Eah frm g s objetve n the spot market s to maxmse ts proft π g. It solves the followng proft maxmsaton problem parametr on ts forward ommtments x gz, and the ISO s redspath quanttes r : :max π g q : Ng g subjet to: q q, N (2) g q q, N. (3) g
7 148 J. Yao, S.S. Oren and I. Adler In ths programme, onstrants (2) and (3) ensure that the produton levels q fall between the apaty bounds of the generaton faltes n eah state. The ISO determnes mport/export quanttes r at the nodes. Its objetve s to maxmse the soal surplus defned by the onsumers wllngness to pay mnus the total generaton ost. It solves a soalwelfaremaxmsaton problem: r q S :max + p ( ) ( ) 0 τ dτ q r N subjet to: r = 0 (4) N Dl, r Kl, l L (5) N Dl, r Kl, l L (6) N Here, onstrant (4) represents energy balane (assumng no losses), whereas onstrants (5) and (6) enfore the network feasblty,.e., the power flows resultng from the ISO redspath must not exeed the thermal lmts. Sne the nodal nverse demand funtons as well as the ost funtons are assumed lnear, problems g and S are both strtly onavemaxmsaton programmes, whh mples that ther frstorder neessary ondtons (ommonly referred to as the KarushKuhnTuker, or KKT ondtons; see Karush, 1939; Kuhn and Tuker, 1951) are also suffent. The spot market outomes an thus be haratersed by the KKT ondtons of the frms and the ISO s problems. Let α, λl and λ l + be the Lagrange multplers orrespondng to (4) (6), then the KKT ondtons derved from problem S are: rj = 0 (7) j N p ( q + r ) b α + ( λ D λ D ) = 0 N (8) t t, t+ t, t L 0 D r + K 0 l L (9) λl l, j l j N 0 K D r 0 l L (10) λ l + l l, j j j N Here and heneforth, we use the onventonal notaton x y to represent the T omplementarty ondton x y = 0. Smlarly, let ρ and ρ + be the Lagrange multplers orrespondng to (2) and (3), then the KKT ondtons for problem are: p 2bq br d + δ bx + ρ ρ = 0 N (11) g, z( ) + g ρ g 0 q q 0 N (12) 0 q q 0 N (13) ρ + g g
8 Cournot equlbra n twosettlement eletrty markets 149 If we restrt { xgz, } = 0,.e., no frm ommts to forward ontrats, the solutons to z Zg, KKT ondtons (7) (13) haraterse the outomes of the snglesettlement market,.e., there s no forward market, and all frms at only n the spot market. If there s no flow ongeston n some state of the spot market, the shadow pres orrespondng to the transmsson apates wll all be zero,.e.: λl = λl+ = 0, l L Ths leads ondton (8) to: Thus: p ( q + r ) b α = 0, N, α = p N N q. 1/ b That s, all the spot nodal pres are dental and equal to α. The forward market s organsed as a transparent fnanal market at zonal tradng hubs and s settled n real tme at zonal settlement pres omputed as weghted averages of nodal pres. The expeted ongeston osts due to transmsson onstrants are hene propagated to the forward market through ratonal expetatons of the zonal settlement pres. In the forward market, eah frm g onjetures the other frms forward quanttes and determnes ts own forward quanttes. In general, the frms objetves are to maxmse ther respetve expeted utlty funton over total proft from spot produtons and forward settlements. For smplty, the frms are assumed here to be rsk neutral, so ther forward objetves are to maxmse the expeted jont profts n both the forward and the spot markets, subjet to the noarbtrage ondton and the preedng KKT ondtons. Eah frm g solves the followng stohast MPEC programme n the forward market: max hx + Prπ ( ) xgz, z Z z gz, g C subjet to: π = p ( r + q ) q u x C ( q ) g z g, z Ng z Z Ng h = Pr() u, z Z z j N C r = 0, C j z p ( q + r ) b α + ( λ D λ D ) = 0, N, C t t, t+ t, t L λl l, j l j N 0 D r + K 0, l L, C
9 150 J. Yao, S.S. Oren and I. Adler λ l + l l, j j j N 0 K D r 0, l L, C p 2bq br d + δ bx + ρ ρ = 0, m, N, C m, z( ) + m ρ 0 q q 0, N, C ρ + 0 q q 0, N, C Note that the forward settlement terms n the objetve funton are anelled owng to onstrant (1), so that the MPEC programme for eah frm g s redued to: subjet to: max Pr( ) p ( r + q ) q C( q ) xgz, C Ng Ng j N r = 0, C j p ( q + r ) b α + ( λ D λ D ) = 0, N, C t t, t+ t, t L λl l, j l j N 0 D r + K 0, l L, C λ l + l l, j j j N 0 K D r 0, l L, C p 2bq br d + δ bx + ρ ρ = 0, m, N, C m, z( ) + m ρ 0 q q 0, N, C ρ + 0 q q 0, N, C The general struture of eah frm s MPEC problem (after rearrangng and relabellng the varables) s of the form: mx f ( x, x, y, w) xg, y, w g g g subjet to: g g w = a+ A x g + A xg + My 0 w y 0 In ths programme, x g represents deson varables that are the frm s forward varables and x g are the orrespondng deson varables ontrolled by all other frms, whereas w and y are the shared state varables. Lkewse, a, A g, A g, and M represent sutable vetors and matres mpled by the system s parameters. Owng to the lnearty of the demand funtons and ost funtons, the objetve funtons n these MPEC problems are quadrat and the KKT onstrants (7) (13) are redued to a Lnear Complementarty Problem (LCP, see Cottle et al., 1992). Combnng all frms MPEC programmes, the
10 Cournot equlbra n twosettlement eletrty markets 151 equlbrum problem n the forward market s an EPEC, whh nvolves smultaneous solutons of the ndvdual frms MPECs. In the followng numeral example, we have employed a spealpurpose algorthm for suh problems that explots ther speal struture. Ths MPEC algorthm treats y and w as peewse lnear funtons of x g. Thus, eah frm g s MPEC problem s redued to a programme wth respet only to x g. The algorthm parttons the spae of x g nto a set of polyhedra aordng to the feasble omplementary bases of the LCP onstrant. Suh parttonng allows the MPEC algorthm to searh for a statonary pont of the MPEC problem va parametr LCP pvotng and fntely many quadrat programmes. A detaled desrpton of the algorthm s out of the sope of ths paper and wll be reported elsewhere. 3 The 24bus system In ths seton, we apply our model to the IEEE 24bus test network wth dfferent, fttous generatorownershp strutures, and observe the eonom results of two settlements. The 24bus network s omposed of 24 nodes and 38 lnes (see Fgure 1). There are ten generators n ths system, eah loated at one node. The resoure ownershps, zonal strutures, demand funtons and ontngeny states are hypothetal. (When applyng ths model to a real system, one an generate the demand funtons and ontngeny states by samplng hstoral data.) Table 1 lsts the nodal nformaton, nludng nverse demand funton slopes, margnal generaton osts and full apates of generaton plants. We assume two zones n the system wth node 1 through 13 n zone 1 and the rest of the nodes n zone 2. As to the thermal lmts, we gnore the ntrazonal flows and fous only on flowgates 3 24, 11 14, and We assume seven states n the spot market (see Table 2). In the frst state, the demands are at peak, all generaton plants operate at ther full apates and all transmsson lnes are rated at ther full thermal lmts. The seond state s the same as the frst state exept that t has shoulder demands. States 3 through 6 also have shoulder demands, but represent the ontngenes of unavalablty of the four flowgates, respetvely. Offpeak state 7 dffers from states 1 and 2 n havng very low demand levels. Table 2 also llustrates the pre nterepts of IDFs as well as the probabltes of the states. We run tests on the systems wth sngle settlement and two settlements, and observe the lkelhood of ongeston, generator output hanges, soal welfare hanges and the behavours of the spot nodal and zonal pres due to forward ontratng. For the ase of two settlements, we test dfferent generator ownershp strutures wth two, three, four or fve frms. The detals of the ownershps are lsted n Table 3. We observe that wth two settlements, the frms have strateg nentves for ommttng to forward ontrats. Furthermore, the nentves for forward ontratng are strengthened by dereased market onentraton. Fgure 2 ompares the total forward ontratng quanttes wth dfferent numbers of frms. It shows that the total forward ontrat quantty nreases from 60 MW wth two frms to 640 MW wth fve frms.
11 152 J. Yao, S.S. Oren and I. Adler Fgure 1 The 24bus network Table 1 Nodal nformaton Margnal ost ($/MWh) Capaty Margnal ost ($/MWh) Node IDF slope (MW) Node IDF slope Capaty (MW)
12 Cournot equlbra n twosettlement eletrty markets 153 Table 2 States of the spot market State Prob. IDF nterept ($/MWh) Desrpton Onpeak state: The demands are on the peak Shoulder state: The demands are at shoulder Shoulder demands wth lne breakdown: Lne 3 24 goes down Shoulder demands wth lne breakdown: Lne goes down Shoulder demands wth lne breakdown: Lne goes down Shoulder demands wth lne breakdown: Lne goes down Offpeak state: The demands are off peak. Table 3 enerator ownershp struture Number of frms Node Two Three Four Fve 1 #1 #1 #1 #1 4 #1 #2 #2 #2 7 #1 #2 #3 #3 11 #2 #3 #4 #4 13 #2 #3 #4 #5 15 #2 #1 #1 #1 17 #2 #2 #2 #2 21 #2 #2 #3 #3 22 #2 #3 #3 #4 23 #1 #3 #4 #5 In all states of the spot market, the aggregated spot outputs nrease under two settlements; moreover, the more frms n the markets, the greater s ths effet (see Fgure 3). Despte ths phenomenon, some generators stll derease the outputs n some states. Ths s beause, when fang ntensve ompetton, some frms have to redue produton of the generators loated n the nodes wth lower spot pres, so as to sustan ther profts by nreasng ther outputs from other plants. For example, when there are fve frms n the markets, generators at nodes 15 and 21 nrease ther produton levels only n the peak state, but redue them n the other sx states (see Table 4). Consequently, the expeted spot outputs from these generators mght be lower under two settlements than those under a sngle settlement. The expeted generaton quanttes are shown n Fgure 4, wth the dark bars denotng the outputs for a sngle settlement and the grey and whte bars for two settlements wth two frms and fve frms respetvely. It s shown that, under two settlements wth fve frms, the generators n both nodes 15 and 21 operate at expeted levels that are lower than those under a sngle settlement.
13 154 J. Yao, S.S. Oren and I. Adler Fgure 2 Total forward ontratng Fgure 3 Total spot generaton
14 Cournot equlbra n twosettlement eletrty markets 155 Table 4 Output level hanges (MW) Node State Fgure 4 Expeted spot nodal generaton Spot nodal and zonal pres under two settlements derease n all states. Ths follows dretly from the fat that the aggregate output s nreased n the spot market under two settlements. Fgure 5 ompares the expeted spot nodal pres under a sngle settlement to those orrespondng to two settlements for ether two and for fve frms. The pres under a sngle settlement are drawn dark, whle the pres under two settlements wth two frms are grey, and wth fve frms, whte. In Table 5, we report the spot zonal pres under a sngle settlement n olumns 2 and 3, and the spot zonal pres under two settlements n olumns 4 through 7. The last row of ths table lsts the forward zonal pres. It s also shown n Fgure 5 and Table 5 that as more frms ompete n the twosettlement system, the lower are the spot nodal and zonal pres.
15 156 J. Yao, S.S. Oren and I. Adler Fgure 5 Expeted spot nodal pres Table 5 Zonal pres Two settlements Sngle settlement Two frms Fve frms Market Zone 1 Zone 2 Zone 1 Zone 2 Zone 1 Zone 2 Spot: state Spot: state Spot: state Spot: state Spot: state Spot: state Spot: state Forward Soal surplus nreases under two settlements. Moreover, the soal welfare of the twosettlement system nreases as the number of frms nreases. The expeted soal welfare of a sngle settlement s $7796/h, whh, under two settlements, s nreased to $8133/h wth two frms and $9383/h wth fve frms. Fgure 6 shows that the same trend apples to the onsumer surplus. These results are qualtatvely onsstent wth those of Allaz and Vla (1993). Lnes not ongested n the snglesettlement system mght be ongested n the spot market of the twosettlement system, or ve versa. Ths follows from the fat that the frms adjust ther outputs, whh alters the flows on the transmsson lnes. For example, n state 3, the snglesettlement market has only lne ongested; however, the only ongested lne under two settlements wth fve frms s lne (see Table 6).
16 Cournot equlbra n twosettlement eletrty markets 157 Fgure 6 Soal surplus Table 6 Lne Flow ongeston State Sngle settlement Two settlements Lne State Sngle settlement Two settlements 1 Congested Congested Congested Congested 2 Unongested Unongested 2 Unongested Unongested 3 Breakdown Breakdown 3 Unongested Congested 4 Unongested Unongested 4 Unongested Unongested 5 Unongested Unongested 5 Breakdown Breakdown 6 Unongested Unongested 6 Congested Congested 7 Unongested Unongested 7 Unongested Unongested 1 Congested Congested Unongested Unongested 2 Unongested Unongested 2 Unongested Unongested 3 Congested Unongested 3 Unongested Unongested 4 Breakdown Breakdown 4 Congested Unongested 5 Unongested Unongested 5 Congested Congested 6 Unongested Unongested 6 Breakdown Breakdown 7 Unongested Congested 7 Unongested Unongested Fnally, Fgure 7 llustrates generator outputs for the seven states under two settlements wth two frms. We note that the generators produe at levels between 40 MW and 70 MW n the peak state, and that only three generators operate n the offpeak state. Compared to ther forward ontrats, the frms are n fat net buyers n the offpeak state, and net supplers n the peak state.
17 158 J. Yao, S.S. Oren and I. Adler Fgure 7 Statewse spot generatons 4 Conludng remarks In ths paper, we model the twosettlement eletrty system as a twoperod game wth multple states of the world n the seond perod. The Cournot equlbrum s a subgameperfet Nash equlbrum represented n the format of an EPEC. We assume lnear demand funtons and onstant margnal generaton osts, so the spot market equlbrum an be omputed as a lnear omplementarty problem. In the forward market, frms solve MPECs subjet to the noarbtrage relatonshp between the forward pres and the expeted spot zonal pres, and the spot market equlbrum ondtons. We apply our model to the 24bus network, and observe from t the strateg nentves of the frms for forward ontratng, the lkelhood of ongeston, nreased generaton quanttes, nreased soal surplus and dereased spot pres wth the ntroduton of a forward market. We also fnd that these effets are amplfed when there are more frms n the network. Fnally, t should be ponted out that our numeral tests are lmted, and that our goal s not to reah onlusve eonom results, whh requre far more extensve smulatons, but to valdate our modellng methodology. Moreover, one should expet quanttatvely dssmlar results f the model s appled to a dfferent settng of resoure ownershp strutures, demand funton dstrbutons or ontngeny states, or to dfferent networks. We plan to relax the noarbtrage assumpton between the forward and spot pres wth a marketlearng ondton that sets the forward pres based on the expeted demands n the spot market. Suh an analyss wll attempt to apture how lak of lqudty (or hghrsk averson) on the buyers sde mght be refleted n a hghrsk premum embedded n the forward pres. We expet that suh a ondton enhanes frms market power and enables them to rase forward pres above the expeted spot pres whle nreasng ther profts.
18 Cournot equlbra n twosettlement eletrty markets 159 Aknowledgements Ths researh s supported by the Natonal Sene Foundaton rant ECS , by the Power System Engneerng Researh Center (PSERC), by the Unversty of Calforna Energy Insttute (UCEI) and by the Department of Energy through the Consortum for Eletr Relablty Tehnology Solutons (CERTS). Referenes Allaz, B. (1992) Olgopoly, unertanty and strateg forward transatons, Internal Journal of Industral Organzaton, Vol. 10, pp Allaz, B. and Vla, JL. (1993) Cournot ompetton, forward markets and effeny, Journal of Eonom Theory, Vol. 59, pp Cardell, J., Htt, C. and Hogan, W.W. (1997) Market power and strateg nteraton n eletrty networks, Resoure and Energy Eonoms, Vol. 19, pp Chao, HP. and Pek, S.C. (1996) A market mehansm for eletr power transmsson, Journal of Regulatory Eonoms, Vol. 10, No. 1, pp Cottle, R.W., Pang, J.S. and Stone, R.E. (1992) The Lnear Complementarty Problem, Boston, MA: Aadem Press. Fudenberg, D. and Trole, J. (1991) ame Theory, Cambrdge, MA: The MIT Press. reen, R.J. (1999) The eletrty ontrat market n England and Wales, Journal of Industral Eonoms, Vol. 47, No. 1, pp Hobbs, B.F., Metzler, C.B. and Pang, J.S. (2000) Strateg gamng analyss for eletr power systems: an MPEC approah, IEEE Transatons on Power Systems, Vol. 15, No. 2, pp Hobbs, B.F. (2001) Lnear omplementarty models of NashCournot ompetton n blateral and POOLCO power markets, IEEE Transatons on Power Systems, Vol. 16, No. 2, pp Kamat, R. and Oren, S.S. (2004) Multsettlement systems for eletrty markets: zonal aggregaton under network unertanty and market power, Journal of Regulatory Eonoms, Vol. 25, No. 1, pp Karush, W. (1939) Mnma of funtons of several varables wth nequaltes as sde onstrants, MS Thess, Department of Mathemats, Unversty of Chago, Chago, IL. Kuhn, H.W. and Tuker, A.W. (1951) Nonlnear programmng, J. Neyman (Ed.) Proeedngs of the Seond Berkeley Symposum, Berkeley, CA: Unversty of Calforna Press. Luo, Z.Q., Pang, J.S. and Ralph, D. (1996) Mathematal Programmes wth Equlbrum Constrants, Cambrdge: Cambrdge Unversty Press. Newbery, D.M. (1998) Competton, ontrats, and entry n the eletrty spot market, Rand Journal of Eonoms, Vol. 29, No. 4, pp Oren, S.S. (1997) Eonom neffeny of passve transmsson rghts n ongested eletrty systems wth ompettve generaton, The Energy Journal, Vol. 18, pp Powell, A. (1993) Tradng forward n an mperfet market: the ase of eletrty n Brtan, The Eonom Journal, Vol. 103, pp Smeers, Y. and We, JY. (1997) Spatal Olgopolst Eletrty Models wth Cournot Frms and Opportunty Cost Transmsson Pres, Center for Operatons Researh and Eonometrs, Unverste Catholque de Louvan, Louvanlanewve, Belgum. von der Fehr, NH.M. and Harbord, D. (1993) Longterm ontrats and mperfetly ompettve spot markets: a study of UK eletrty ndustry, Memorandum No. 14, Department of Eonoms, Unversty of Oslo, Oslo, Sweden.
19 160 J. Yao, S.S. Oren and I. Adler We, JY. and Smeers, Y. (1999) Spatal olgopolst eletrty models wth Cournot frms and regulated transmsson pres, Operatons Researh, Vol. 47, No. 1, pp Yao, J., Oren, S.S. and Adler, I. (2004) Computng Cournot equlbra n twosettlement eletrty markets wth transmsson onstrants, Proeedng of the 37th Hawa Internatonal Conferene on Systems Senes (HICCS 37), Bg Island, Hawa, 5 8 January.
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