Two-Settlement Electricity Markets with Price Caps and Cournot Generation Firms

Transcription

1 Two-Settlement Eletrty Markets wth Pre Caps and Cournot Generaton Frms Jan Yao, Shmuel S. Oren, Ilan Adler Department of Industral Engneerng and Operatons Researh 4141 Etheverry Hall, Unversty of Calforna at Berkeley Berkeley, CA 94720, US {jyao, oren, June 2005 We ompare two alternatve mehansms for appng pres n two-settlement eletrty markets. Wth suffent lead tme, forward market pres are mpltly apped by ompettve pressure of potental entry that wll our when forward pres rse above some bakstop pre. Another more dret approah s to ap spot pres through a performane-based regulatory nterventon. In ths paper we explore the mplatons of these two alternatve mehansms n a two-settlement Cournot equlbrum framework. We formulate the market equlbrum as a stohast equlbrum problem wth equlbrum onstrants (EPEC) apturng ongeston effets, probablst ontngenes and horzontal market power. As an llustratve test ase, we use the 53-bus Belgan eletrty network wth representatve generator osts but hypothetal demand and ownershp struture. Compared to a pre-unapped two-settlement system, a forward ap nreases frms nentves for forward ontratng, whereas a spot ap redues suh nentves. Moreover, n both ases, more forward ontrats are ommtted as the generaton resoure ownershp struture beomes more dversfed. keyword: OR n energy, Eletrty Markets, Cournot Equlbrum, Mathematal Program wth Equlbrum Constrants, Equlbrum Problem wth Equlbrum Constrants. 1 Introduton It s generally agreed that forward ontratng mtgates generators horzontal market power n the spot market and protets market partpants aganst spot pre volatlty resultng from system ontngenes and demand unertanty. Prevous ontrbutons, suh as [1, 2, 12, 19], fous on the mpat of forward markets on spot pres and soal welfare under alternatve assumptons regardng the relatonshp between forward and spot pres. It s shown that generators have nentves to trade n the forward markets whereas forward ontratng redues spot pres and nreases onsumpton levels and soal welfare. These models assume a fxed generaton stok whh s approprate for two-settlement system over short tme ntervals (e.g., day ahead and real tme markets). Compettve entry n the forward market and regulatory aps on spot pres are further means of mtgatng pre spkes and market power abuse. For long-term forward ontrats, 1 Researh supported by the Natonal Sene Foundaton Grant ECS and by the Power System Engneerng Researh Center (PSERC). The authors wsh to thank Dr. Bert Wllems who provded the network data for the Belgan eletrty system used n ths paper. 2 A prelmnary abrdged verson of ths paper appeared n the Proeedng of the 38th Hawa Internatonal Conferene on Systems Senes (HICCS38), Jan 4-7,

2 potental ompettve entry mposes an mplt pre ap on forward ontrat pres sne new nvestment n generaton apaty wll our when forward pres rse above the amortzed long-run generaton ost. Alternatvely, regulators n many restrutured eletrty markets have mposed pre or bd aps n the spot markets n an attempt to retfy market mperfetons suh as demand nelastty, barrers to entry, mperfet aptal markets and loatonal market power. In ths paper we extend our earler model n [19] by onsderng, separately, the effets of these two ap types on the spot and forward pres. In partular, we address the followng questons. To what extent do the generators ommt forward ontrats under pre aps? How do aps affet the spot and forward pres? What s the relatve mpat of spot aps versus ompettve entry n mtgatng market power? How senstve s the total forward ommtment wth respet to the resoure ownershp struture and the ap levels? We study these questons va a two-settlement Cournot equlbrum model where generaton frms have horzontal market power. The system s subjet to ontngenes n the spot market due to transmsson and generaton outages as well as to demand unertantes. Our model also aounts for network ongeston whh s represented through a apaty onstraned eletrty grd. We present our formulaton as a stohast equlbrum problem wth equlbrum onstrants (EPEC) where eah generaton frm solves a stohast mathematal program wth equlbrum onstrants (MPEC, see [13]). Related eonom results of the pre aps wll be analyzed based on a stylzed Belgan two-settlement system. The man purpose of ths paper s to present the formulaton and n partular, to explore the mpat of pre aps (on both spot and forward pres) n two-settlement markets. In addton, we dsuss the detals of how to norporate suh pre aps n the mathematal formulaton whle mantanng dfferentablty of the objetve funton. The algorthm detals of our new omputatonal approah employed n ths paper are developed n a sequel paper. There we ntrodue an effent algorthm to fnd an equlbrum pont whh s based on the observaton that the problem an be redued to a seres of parametr lnear omplementarty problems. The remander of ths paper s organzed as follows. Related eletrty market models are revewed n the next seton, seton 3 presents the formulaton for the two-settlement market. In seton 4, we nvestgate the mplatons of hypothetal pre aps n a stylzed verson of the Belgan eletrty system. Some remarks are drawn to onlude ths paper. 2 Related Researh In ths seton, we wll revew models wth forward ontrats and models of spot markets wth Cournot ompetton or transmsson onstrants. Most spot market models wth transmsson onstrants assume the Cournot onjetural varaton. Moreover, f the man purpose s to examne generators behavors n the energy market, assumng the agents at as pre takers n the transmsson market allows the models to formulated as omplementarty problems or varatonal nequaltes. We and Smeers [18] onsder a Cournot model wth regulated transmsson pres. They solve varatonal nequaltes to determne unque long-run equlbra n ther model. In subsequent work, Smeers and We [17] onsder a separated energy and transmsson market, n whh the system operator onduts a transmsson apaty auton wth power market partpants purhasng transmsson ontrats to support blateral transatons. They onlude that suh a market onverges to the optmal dspath wth a large number of market partpants. Hobbs [11] alulates a Cournot equlbrum under the assumpton of lnear demand and ost funtons, whh leads to a mxed lnear omplementarty problem. In a market wthout arb- 2

3 trageurs, non-ost based pre dfferenes an arse beause the blateral nature of the transatons gves frms more degrees of freedom to dsrmnate between eletrty demands at varous nodes. Suh result s smlar to that n a separated market n [17]. In the market wth arbtrageurs, any non-ost dfferenes are elmnated by speulators who buy and sell eletrty at nodal pres. Ths equlbrum s shown to be equvalent to a Nash-Cournot equlbrum n a POOLCO-type market. Hobbs, Metzler and Pang [10] present an olgopolst market where eah frm submts a lnear supply funton to the Independent System Operator (ISO). They assume that the frms an only manpulate the nterepts of the supply funtons, but not the slopes, whle the power flows and the prng strateges are onstraned by the ISO s lnearzed optmal power flow. Eah frm n ths model faes an MPEC problem wth the spatal pre equlbrum as the nner problem. Work on forward markets has foused on the welfare enhanng propertes of forward ontrats and the ommtment value. The bas model n [1] assumes that produers meet n a two-perod market where there exsts some demand unertanty n the seond perod. Allaz shows that the generators have a strateg nentve to ontrat forward f other produers do not. Ths result an be understood usng the onepts of strateg substtutes and omplements of Bulow, Geneakoplos and Klemperer [3]. The avalablty of the forward market makes a partular produer more aggressve n the spot market. Due to the strateg substtute effet, ths produes a negatve effet on ts ompettors produton. The produer havng aess to the forward market an therefore use ts forward ommtment to mprove ts proftablty to the detrment of ts ompettors. Allaz shows, however, that wth all produers havng aess to the forward market, t leads to a prsoners dlemma type of effet, redung the profts for all produers. Allaz and Vla [2] extend ths result to the ase wth more than one tme perod for forward tradng to take plae. For a ase wthout unertanty, they establsh that as the number of perods, when forward tradng takes plae, nreases to nfnty, the produers lose ther ablty to rase energy pres above ther margnal ost. von der Fehr and Harbord [7] and Powell [16] study ontrats and ther mpat on an mperfetly ompettve eletrty spot market: the UK pool. von der Fehr and Harbord [7] fous on pre ompetton n the spot market wth apaty onstrants and multple demand senaros. They fnd that the ontrats tend to put downward pressure on spot pres. Although, ths provdes dsnentve to the generators to offer suh ontrats, there s a ountervalng fore n whh sellng a large number of ontrats drves a frm to be more aggressve n the spot market, and ensures full dspath n more demand senaros. Powell [16] models expltly re-ontratng from Regonal Eletrty Companes (RECs) after the maturaton of the ntal portfolo of ontrats whh s set up after deregulaton. He adds rsk averson on the part of RECs to the earler models. Generators at as pre setters n the ontrat market. He shows that the degree of oordnaton has an mpat on the hedge over demanded by the RECs, and ponts to a free rder problem whh leads to a lower hedge over hosen by the RECs. Newbery [14] analyzes the role of ontrats as a barrer to entry n the England and Wales eletrty market. He extends the earler work by modellng supply funton equlbra (SFE) n the spot market. He further shows that f entrants an sgn base load ontrats and numbents have enough apaty, the numbent an sell enough ontrats to drve the spot pre below the entrydeterrng level, resultng n more volatle spot pres f the produers oordnate on the hghest proft supply funton equlbrum. Capaty lmt however may mply that numbents annot play a low enough SFE n the spot market and hene annot deter entry. Green [9] extends Newbery s model showng that when generators ompete n SFE n the spot market, together wth the assumpton of Cournot onjetural varatons n the forward market, no ontratng wll take plae unless the buyers are rsk averse and wllng to provde a hedge premum n the forward market. He shows that forward sales an deter exess entry, and nrease eonom effeny and long-run profts of a large numbent frm faed wth potental entrants. 3

4 Kamat and Oren [12] analyze the welfare and dstrbutonal propertes of a two-settlement market, whh onssts of a nodal spot market over 2-node and 3-node networks wth a sngle energy forward market. The system s subjet to ongeston wth unertan transmsson apates n the spot market, and to generators market power. Yao, Oren and Adler [19] extend Kamat and Oren s model to more realst mult-node and mult-zone systems and onsders the unertanty n transmsson apaty, generaton avalablty and demand n the spot market. The Cournot equlbrum s modeled as an EPEC n whh eah generaton frm solves an MPEC problem. The model s appled to a sx-bus llustratve example, and t s shown that, lke the smple ases, the frms have nentves for forward ontratng, and that the present of the forward market nreases soal welfare, and dereases spot pre magntudes and volatltes. 3 The Model We ntrodue a gener model of the two-settlement eletrty system wth both spot and forward pre aps. The system wth ether spot or forward pre aps an be treated as a speal ase of ths gener model by relaxng the forward or spot pre aps. In ths gener model, we model the two-settlement eletrty markets as a omplete-nformaton two-perod game wth the forward market beng settled n the frst perod, and the spot market beng settled n the seond perod. The equlbrum s a sub-game perfet Nash equlbrum (SPNE, see [8]). In the forward market, the generaton frms determne ther forward ommtments n order to maxmze ther expeted utlty n the spot market whle antpatng other frms forward quanttes and the spot market outomes. The spot market s a subgame wth two stages: n stage one, Nature pks the state of the world realzng the atual apates of the generaton faltes and flow apates of the transmsson lnes as well as the shape of the demand funtons at all nodes. In stage two, frms determne generaton quanttes to ompete n a Nash-Cournot manner whle the System Operator (SO) determnes how to re-dspath eletrty wthn the network so as to maxmze total soal surplus subjet to the transmsson onstrants. Note that the generaton frms and the SO take nto onsderaton eah other s atons, ther desons are treated as smultaneous moves n our model. From a mathematal perspetve, the model s formulated as an EPEC defnng an equlbrum soluton among MPECs that haraterze the problems of the dfferent ompetng frms. Eah MPEC has two levels where the upper level s a proft maxmzaton problem of the frm n the forward market takng nto onsderaton the spot market equlbrum defned by the lower level problem. Ths spot market equlbrum s haraterzed by the set of frst order optmalty ondtons of the frms and the S), where eah frm selets ts output level so as to maxmze spot net revenue whle the SO redspathes eletrty so as to maxmze soal welfare subjet to the transmsson onstrants. 3.1 Notatons We onsder the sets of nodes, transmsson lnes, zones, generaton frms and ther generaton faltes, as well as the states of world n the spot market. N: The set of nodes Z: The set of zones. We denote by z() as the zone where node N resdes. L: The set of avalable transmsson lnes. 4

5 Fgure 1: Deson herarhy C: The fnte set of possble states of nature n the spot market. G: The set of generaton frms. generaton faltes. The deson varables related to the forward markets are: x g,z : The forward ommtment from frm g G to zone z Z. The deson varables related to the spot markets are: q : Spot quantty generated at node N n state C. N g denotes the set of nodes at whh frm g G owns r : Import/export quantty at node N by the SO n state C. The followng exogenous parameters are onsdered n our formulaton: q, q : The lower and upper apaty bounds of generaton falty at node N n state C. ū: The spot pre ap. h: The forward pre ap. p ( ): The nverse demand funton (IDF) at node N n state C. We denote by p the ommon pre nterept aross all nodes n eah state, and b as the slopes: p (q) = p b q N, C We assume that the demand urve at eah node shfts nwards or outwards n dfferent states, but ts slope does not hange. C ( ): The generaton ost at eah node N. We assume that the ost funton s quadrat onvex: C (q) = d q s q 2. 5

6 K l : The flow apaty lmt of lne l L n state C. Dl, : The power transfer dstrbuton fator (PTDF) n state C on lne l L wth respet to node N. P r(): The probablty of state C n the spot market. δ : The non-negatve weght for node N used to defne the spot zonal pres. For eah zone z Z, t holds that :z()=z δ = Pres and Pre Caps For the sake of generalty, we allow dfferent levels of granularty n the fnanal settlements, wth equal granularty beng a speal ase. Ths s motvated by the fat that, n real markets, we observe dfferent granularty levels n the spot and long-term forward markets for example n PJM, the western hub representng the weghted average pre over nearly 100 nodes s the most lqud forward market. Spefally, the network underlyng the nodal spot market s parttoned nto a set of zones, eah of whh ontans a luster of nodes. Ths suggests three loatonal pres: spot nodal pres, spot zonal pres (used to settle zonal forward ontrats) and forward (zonal) pres. Spot nodal pres ˆp are the pres at whh the loads are settled at ther respetve nodes. In eah state C of the spot market, the onsumpton at node N s the sum of the quantty generated by the generator loated at that node (q ), and the (export or mport) adjustment (r ) made by the SO. Beause the eletrty s not eonomally storable, loads are restrted to be negatve,.e. q + r 0, N. Based on the IDFs, the nodal onsumpton q + r whenever aps are gven are expressed as ˆp = mn{ū, p (q + r )}, N. s pred at p (q + r ). Atual nodal pres The spot zonal pre u z of a zone z Z n a state C s used to fnanally settle zonal forward ontrats. It s defned as the weghted average of the nodal pres n the zone wth predetermned weghts δ for the nodes n the zone. These weghts are assumed to be gven onstants n our model (typally these weghts reflet the hstoral load ratos). In mathematal terms, the zonal spot pre s gven by: u z = δ ˆp, z Z :z()=z The forward zonal pres h z are the pres at whh forward ommtments are agreed upon n the respetve zones z Z. We assume that n equlbrum no proftable arbtrage s possble between the forward and spot zonal pres. Ths mples that the forward zonal pre s equal to the expeted spot zonal pre; that s, h z = C P r()u z, z Z. Wth a forward pre ap, an upper bound s mposed on these forward pres: h z h, z Z. In a long-run equlbrum, ths upper bound on forward pres typally reflets a bakstop pre at whh new entry beomes ex ante proftable. 6

7 3.3 The Spot Market and the Lower Level Problem We shall now be ready to derve our model. To faltate the representaton, we start from the spot market formulaton The Gener Spot Model In eah state C of the spot market, a generaton frm g G determnes the output of ts generaton unts q ( N g ). These outputs are restrted wthn the range determned by the mnmal and maxmal possble outputs of the plants n that state: q q q N g. To avod omputatonal ntratablty due to dsontnuty n the frms best-response funtons, we follow the ommon prate n the lterature (see [11]) by assumng that the frms gnore the mpat of ther produton desons on the ongeston harges and the settlement of transmsson rghts. Eah frm g G reeves the revenue of the quanttes pad at the spot nodal pres and the fnanal settlement of ts forward ontrats settled at the spot zonal pres. In addton to ts revenue from the froward ontrat settlements, frm g s spot market proft πg s gven by: πg = q ˆp u zx g,z C (q ). N g z Z N g Eah frm g thus solves the followng program n eah state of the spot market: max π q g subjet to: πg = q ˆp u zx g,z C (q ). N g z Z N g ˆp = mn{ū, p (q + r )}, u z = δ ˆp, z Z :z()=z h z = C P r()u z, z Z h z h z Z q q q, q + r 0, N g N g N g Gven the produton desons by the generators, the SO determnes for eah state n the spot market the adjustment r at eah node N (.e. the mport and export quanttes at eah node). These adjustments are subjet to the network thermal onstrants on the power flows. We model the eletrty flows on the transmsson lnes n term of the Power Transfer Dstrbuton Fators (PTDF) based on a Dret Current (DC) approxmaton of the Krhhoff s law (see [5]). The PTDF D l, s the proporton of a flow on a partular lne l resultng from an njeton of one unt of energy at a partular node and a orrespondng one-unt wthdrawal at a referene slak bus. The network feasblty onstrants are K l N D l, r K l, l L. 7

8 The SO also mantans the real tme balane of loads and generaton, that s r = 0. N The SO s objetve s to maxmze the soal welfare whh s defned as the total area under the IDFs mnus the total generaton ost. Hene, the SO problem n eah state C of the spot market s: max r [ N subjet to: r + q 0, r = 0 N r +q 0 N p (τ )dτ C (q )] K l N D l, r K l, l L Beause the frms deson varables q are treated as gven onstants n the SO s deson program, the term C (q ) an be dropped from the SO s objetve funton wthout affetng the optmal soluton Spot Market Smooth Formulaton The frms and the SO deson problems n the spot market do not have straghtforward explt optmalty ondtons due to the non-smooth funtons haraterzng the spot pres. In ths seton, we reformulate these problems by removng the mn operator n the haraterzaton of the apped spot nodal pres. It s aomplshed by onsderng separately two ases. In the frst ase, the spot pre ap s hgher than the pre nterepts of the IDFs. The spot pre ap s thus not bndng. Therefore, the spot market formulaton s redued to a varant of that n [19]. The frms spot deson problems are: G g : π g max q subjet to: πg = p (q + r )q u zx g,z C (q ). N g z Z N g u z = δ p (q + r ), z Z :z()=z h z = C P r()u z, z Z h z h z Z q q q, N g q + r 0, N g 8

9 Pre p Inverse demand funton u Spot pre ap d Margnal ost 0 v Fgure 2: spot pre aps Quantty The SO deson problem s: S : max r N subjet to: r + q 0, r = 0 N r +q 0 N p (τ )dτ K l N D l, r K l, l L When the spot pre ap s below the pre nterepts of IDFs, t ould be bndng (see fgure 2). Note that the standard tehnque of relaxng the equalty ˆp = mn{ū, p (q + r )} wth two nequalty onstrants, ˆp ū and ˆp p (q + r ), may not work here sne the sgn of the terms ontanng ˆp n the frms objetve funtons may be postve or negatve, dependng on the relatve magntude of x g,z and q. To overome ths omplaton, we ntrodue auxlary varables v to represent mn{q + r, v }, so that 0 v q + r and v v = p ū b, and rewrte the spot nodal pres (onsderng of the lnearty of the demand funton) as ˆp = ū (q + r v )b. However, wrtng the spot nodal pres n suh a way s orret only f v = mn{q + r, v }. To 9

10 p Pre Inverse demand funton u Pre ap v q + r v v + q + r v Load Fgure 3: the SO s objetve when q + r v guarantee ths, we reformulate the SO s problem as: Ŝ : max r,v ( N v 0 v p +r +q v (τ )dτ + p (τ )dτ ) v subjet to: r = 0 (1) N Dl, r, Kl, l L (2) N Dl, r Kl, l L (3) N v 0, N (4) v v, N (5) r + q v 0, N (6) The two omponents of the SO s objetve funton are llustrated n fgures 3 and 4. Note that, due to the dfferene of the heghts of the two ntervals, the maxmzaton of the SO objetve funton wll assgn postve load (q +r v ) n the seond nterval (beyond v ) only f the load on the frst nterval reahes ts lmt of v. Thus, at the optmal, t holds v = v f q + r v, and v = q + r f q + r < v. In another word, the optmal soluton wll set v to mn{ v, q + r }. 10

11 p Pre Inverse demand funton u Pre ap v v q + r v + q + r v Load Fgure 4: the SO s objetve when q + r > v The generaton frms spot proft maxmzaton programs an now be wrtten as: Ĝ g : π g max q subjet to: πg = (ū (q + r v )b )q u zx g,z C (q ) N g z Z N g u z = (ū (q + r v )b )δ, z Z :z()=z h z = C P r()u z, z Z q + r v 0, N g (6) h z h z Z (7) q q, N g (8) q q, N g (9) It s worth a menton that, f v are set to zero, programs Ĝ g and Ŝ also apture the generaton frms and the SO s programs when the spot pre ap s hgher than the pre nterepts of IDFs. Therefore, problems Ĝ g and Ŝ along wth sutable v for all nodes N and states C fully represent the agents spot market deson problems Spot Market Equlbrum Outomes To obtan the smultaneous solutons to the frms and SO s spot market problems (Ĝ g and Ŝ ), we ombne the Karush-Kuhn-Tuker (KKT) ondtons, nto one problem. Sne all the problems nvolved are strtly onave-maxmzaton problems, we are assured that the soluton to the KKT ondtons provdes the global solutons to all the problems at hand. In the spot market equlbrum, the shadow pres (the Lagrangan multplers) orrespondng to the onstrants (7) for eah zone (η z ), must be equal aross all states for eah frm. If not, the frms ould nrease ther proft by dereasng ther outputs n states wth lower shadow pres whle 11

12 nreasng the outputs n states wth hgher shadow pres. Moreover, all frms should fae equal shadow pres on these onstrants n eah zone, beause these shadow pres reflet the margnal proft of new entry va the forward pre aps. Mathematally, sngle Lagrangan multplers should be assgned to (7) for ndvdual zones beause ths onstrant s a oupled onstrant shared among all frms spot market problems n all states. Let α, λ l, λ l+, β, β +, µ, ρ and ρ + be the Lagrangan multplers orrespondng to onstrants (1)-(6) and (8)-(9) respetvely, we obtan the followng KKT ondtons: rj = 0 (10) j N p b v ū + (r + q v )b α + µ + t L(λ t λ t+)d t, = 0, N (11) ū (r + q v )b + β β + µ = 0, N (12) ū 2b q (r v )b d s q + µ + δ b x g,z() + ρ ρ + + P r()δ b η z = 0, N (13) λ l 0, Dl, r j + Kl )λ l = 0, l L (14) λ l+ 0, Dl, r j Kl, ( j N j N j N D l, r j K l, (K l j N Dl, r j)λ l+ = 0, l L (15) β 0, v 0, v β = 0, N (16) β + 0, v β +, v β + = 0, N (17) µ 0, q + r v 0, (q + r v )µ = 0, N (18) η z 0, h z h, ( h h z )η z = 0, z Z (19) ρ 0, q q, (q + q )ρ = 0, N (20) ρ + 0, q q, (q q )ρ + = 0, N (21) 3.4 The Forward Market and the EPEC Problem As was dsussed earler, we assume that the forward ontrats are settled fnanally. Eah frm g G takes all ts ompettors forward quanttes as gven, and determnes ts own forward quanttes to maxmze ts expeted spot utlty. Assumng the frms are rsk neutral, ther forward objetves are to maxmze ther revenues from the forward ontrats plus ther expeted profts from the spot market, subjet to the spot market KKT ondtons whh represent the antpated equlbrum n the spot market. Thus for frm g, ts optmzaton problem n the forward market s 12

13 the followng MPEC problem: max h z x g,z + P r()π x g g,z z Z C subjet to: z Z x g,z X g, z Z πg = (ū (q + r v )b )q u zx g,z C (q ) N g z Z N g u z = (ū (q + r v )b )δ, z Z :z()=z h z = C P r()u z, z Z and the KKT ondtons (10) (21) for C where the predefned set X g onssts of the allowable forward postons for frm g. Combnng the frms MPEC problems, the equlbrum problem n the two-settlement markets s an EPEC, where the KKT ondtons (10)-(21) denote the lower level problem. An equlbrum of suh EPEC problem s a tuple of the varables, nludng all frms forward and spot deson varables, the SO s redspath desons, as well as the aforementoned Lagrangan multples, at whh eah frm s MPEC problem s solved, and no frm wants to unlaterally hange ts desons n both the forward and spot markets. Next, we aggregate some of the varables so that we an smplfy the notatons for the frms MPEC programs by presentng the lower level problem n ths EPEC as parametr lnear omplementarty problem (LCP, see [6]). Defne x g : The vetor of eah frm g s forward quanttes. x g = [x g,z, z Z]. y: y = [y, C] where y s the vetor of the Lagrangan multplers for the nequalty onstrants n the frms and the SO s deson problems wth respet to state of the spot market. w: w = [w, C] where w s the vetor of the slak varables for the onstrants orrespondng to y. Thus the spot market KKT ondtons beome the followng parametr LCP wth respet to w and y wth x g beng the parameters w = a + g A g x g + My, y 0, w 0, y T w = 0, where a, A g, and M are sutable vetor and matres derved from the KKT ondtons (We omt here the detaled representatons of a, A g, and M). Now, the frms objetves n ther MPECs an be expressed as funtons f g (x g, x g, y, w), where x g denotes forward ommtments of all frms wth the exepton of frm g. 13

14 Fgure 5: Belgan hgh voltage network Usng ths notaton, frm g forward problems s presented as: mn x g,y,w subjet to : f g (x g, x g, y, w) x g X g w = a + A g x g + A g x g + My (22) y, w 0, y T w = 0 (23) Here, x g are the deson varables, y, w are the state varables, x g are the parameters. The EPEC problem n the forward market s hene to fnd a tuple of ({x g } g G, y, w) at whh all frms MPEC problems are smultaneously solved. We observe from the programs Ŝ and Ĝ g that M s postve sem-defnte and symmetr. Note that sne both programs Ŝ and Ĝ g have optmal solutons for all feasble x g, the KKT ondtons (10)-(21) are solvable as well, and as a result so do the LCP onstrants (22)-(23). To smplfy the omputaton, we assume that the exogenous parameters n our model are perturbed n suh a way that the spot deson problems have non-degenerate optmal solutons. Thus, by Theorem n [6], we are assured that for any x g X g, g G, the LCP onstrants (22)-(23) have unque (y, w) soluton. The preedng property together wth the quadrat objetve funton f g (x g, x g, y, w) allows us to apply an MPEC soluton sheme whh makes use of the soluton unqueness of the parametr LCP onstrants (22)-(23). The detals of the soluton approah s reported n a sequel paper [21]. 4 The Belgan Eletrty Market We use a stylzed verson of the Belgan eletrty network to test our model and llustrate the eonom results. For ompleteness of the network, we norporate some lnes n the Netherlands and Frane. The network has kV and 220kv transmsson lnes, some of whh are parallel 14

15 Table 1: Nodal nformaton. Node demand margnal apa- Node demand margnal apa- Id slope ost ty Id slope ost ty 1 1 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0, : these numbers are zeros n states 5 and 6 respetvely. N/A: the margnal osts are not applable to zero apates. between the same par of nodes. For omputatonal purpose, these parallel lnes are ombned nto sngle lnes wth adjusted thermal apates and resstanes, the network s hene redued to 71 transmsson lnes onnetng 53 nodes (see fgure 5). 19 nodes n ths system have attahed generaton plants. Insgnfant lower voltage lnes and small generaton plants have been exluded from ths example. The orrespondng nformaton of the generaton plants s lsted n table 1. More detals are also gven n ths table desrbng the nodal nformaton on the slopes of the nverse demand funtons, frst-order margnal generaton osts, the generaton apates and the hstoral load ratos (n ths example, all margnal generaton osts are assumed onstant). As to the thermal lmts, we gnore the ntra-zonal flows and fous only on the flowgates of lnes [22,49], [29,45], [30,43], and [31,52]. We assume that there are sx states n the spot market. The frst state s a state n whh the demands are at the shoulder, all generaton plants operate at ther full apates, all transmsson lnes are rated at ther full thermal lmts. The seond state s the same as the frst state exept that t has on-peak demand. Off-peak state 3 dffers from state 1 by the very low demand levels. State 4 denotes the ontngeny of the transmsson lne [31,52] beng out of serve. State 5 and 6 apture the unavalablty of two plants at node 10 and 41 respetvely. The assumed probabltes 15

16 Table 2: States of the Belgan spot market State Probablty Type and desrpton Shoulder state: Demand at shoulder On-peak state: Demand on the peak Off-peak state: Demand off-peak Shoulder demand wth lne breakdown: Lne [31,52] out of serve Shoulder demand wth generaton outage: Plant at node 10 down Shoulder demand wth generaton outage: Plant at node 41 down. of these states are gven n table 2. For these states, the pre nterepts for the nodal nverse demand funtons are set to 500, 1000, 250, 500, 500, and 500 respetvely. We frst onsder fve test ases: Case 1: The pre-unapped sngle-settlement system,.e. sngle settlement wthout (spot) pre aps. Case 2: The pre-apped sngle-settlement system,.e. sngle settlement wth (spot) pre aps. Case 3: The pre-unapped two-settlement system,.e. two settlements wthout spot and forward pre aps. Case 4: The forward-apped two-settlement system,.e. two settlements wth only forward pre aps. Case 5: The spot-apped two-settlement system,.e. two settlements wth only spot pre aps. Case 1 and 2 are essentally equvalent to a two-settlement system wth the allowable forward ommtments fored to zeros. In these fve ases, we restrt the frms forward postons to not exeed ther total generaton apates n the respetve zones. We frst run the test ases wth a fxed number of frms and some fxed pre aps. We assume that there are two frms ompetng n the system, and two zones n the network wth nodes 1 through 32 beng n the frst zone and the remanng nodes belong to the seond zone. The frst frm owns the plants at nodes 7, 9, 11, 31, 32, 33, 35, 37, 41, 47, and 53, and the seond frm owns the plants at nodes 10, 14, 22, 24, 40, 42, 44, and 48. We also assume the spot pre ap n ases 2 and 5 s 600, and the forward pre ap n ase 4 s 425. We fnd that, when both the spot and the forward pres are not apped, the generaton frms have nentves for forward ontratng. Moreover, two settlements nrease spot generaton, derease the spot pres, and thus nrease soal welfare as shown n the table 3. Ths result s onsstent wth that n [1, 2, 12] and [19]. Spot and forward pre aps reate opposte effets on the frms nentves for forward ommtments. Dfferent nentves further mply dfferent results of spot produton and pres as ompared to the unapped two-settlement system. Tables 3 and 4 report the spot generaton quanttes and spot zonal pres under dfferent ases. The spot pre ap redues the frms nentves to ommt forward ontrats as ompared to the unapped two-settlement system. Spefally, n our smulaton, the spot pre ap auses frms to ommt totally about 90% of the orrespondng forward ontrat n the unapped two-settlement 16

17 Table 3: Spot zonal pres Case 1 Case 2 Case 3 Case 4 Case 5 zone 1 zone 2 zone 1 zone 2 zone 1 zone 2 zone 1 zone 2 zone 1 zone 2 state state state state state state Expeted Table 4: Aggregated spot generaton Case 1 Case 2 Case 3 Case 4 Case 5 state state state state state state Expeted markets. Ths observaton an be explaned n terms of the postve orrelaton between expeted spot output and forward ommtments. Spot pre aps ndue frms to nrease produton n the on-peak states when the ap s bndng and to offset ther proft losses by redung output and nreasng pres n the states where the aps are not bndng. The net effet s a reduton n expeted output aross all states and onsequently n redued forward ommtments. Forward pre aps, on the other hand, nrease the frms nentves for forward ontratng. In our smulaton frms ommtments n the forward-apped two-settlement markets are about 80% hgher than those n the same markets wthout aps. Ths observaton s appealng sne t s onsstent wth the ntuton that ompettve entry wll ndue more forward ontratng by numbents n order to deter suh entry. Thus, t valdates the use of forward aps as a proxy for ompettve entry. The more dret explanaton s agan based on the orrelaton between expeted output and forward ommtment. Unlke spot aps that lmt pres n on-peak states, forward aps lmt expeted spot pres aross all states va the no arbtrage relatonshp. Suh a lmt ndues pre reduton through an nrease n output aross all the states resultng n a total nrease n expeted output and a resultng nrease n forward ommtment. We then run more tests to fnd the senstvty of forward ontratng on the number of frms. We retan the pre aps n ases 4 and 5, but vary the number of frms n the system from two to sx. We fnd that the total forward ontratng s nreasng wth the number of frms. Ths s true for all types of two-settlement systems, no matter whether or not the forward or the spot pres are apped. Fgure 6 shows the relatonshp between the ontratng quantty relatve to total generaton apaty and the number of frms n the market. Agan, t s shown that the spot-apped system results n less forward ontratng than the unapped system, whh auses less forward ontratng than the forward-apped system. The nrease n forward ontratng wth 17

18 unapped spot apped forward apped Fgure 6: omparson of total forward ontratng the number of frms an be explaned by the fat that the prsoners dlemma effet drvng forward ontratng nentves s amplfed as the number of players nreases. Fnally, we nvestgate the forward ontratng at dfferent levels of the spot and forward pre aps. Fgures 7 and 8 plot the total forward ontratng wth respet to the pre aps and the number of frms. Wth all other parameters fxed, we fnd that the lower the spot ap s, the less quantty frms are wllng to ommt n the forward market. On the other hand, the forward ommtments nrease as the forward pre ap dereases. For any gven spot or forward ap, the total forward ontratng quantty s nreasng n the number of frms. Note the forward pres are dereasng funtons n the forward ommtments and the spot generaton output quantty, n the ase wth potental entry the frms wll sueed n deterrng entry by playng an equlbrum wth the forward pres just below the forward ap when they have suffent apates and the forward pre ap s hgh enough. However, when the forward pre ap s too low or the frms don t have enough apaty, the forward pres wll attempt to rse above the forward pre ap, and entry to the markets s nevtable as noted by Newbery n [14]. 5 Conludng Remarks In ths paper, we extend our model presented n [19] to the ase n whh ether the forward pres or the spot pres are apped. We formulate the Cournot equlbrum n the pre-apped two-settlement markets as a stohast equlbrum problem wth equlbrum onstrants. We also onsder the spot market wth demand unertanty and probablst ontngenes of generaton apaty and thermal lmts on the transmsson lnes. The man goal of ths paper s to develop an effetve model for analyzng the appng alternatves under a varety of senaros n the framework of two-settlement markets. We run test ases based on the stylzed Belgan eletrty market. The resultng equlbrum reveals fewer nentves for the frms to ommt forward ontrats due to spot pre aps, but more nentves for forward ontratng under forward pre aps ndued through ompettve entry. However, spot 18

19 ontratng number of frms spot pre ap 800 Fgure 7: total forward ontrat wth spot aps ontratng number of frms forward pre ap Fgure 8: total forward ontrat wth forward aps 19

20 zonal pres, under both ap types, stll derease when forward ontratng s possble. Senstvty studes show that the results are robust and that forward ontratng nreases as the forward pre ap dereases and dereases as the spot ap dereases. We also fnd that the total level of forward ontratng and the senstvtes are amplfed as the number of ompetng frms nreases. Fnally, t should be ponted out, that our numeral smulatons are lmted and ther objetve was amed prmarly at valdatng our modellng methodology rather than reahng onlusve eonom results whh would requre far more extensve test studes. Referenes [1] Allaz, B., Olgopoly, Unertanty and Strateg Forward Transatons. Internal Journal of Industral Organzaton 10, [2] Allaz, B., Vla, J.-L., Cournot Competton, Forward Markets and Effeny. Journal of Eonom Theory 59, [3] Bulow, J., Geneakolos, J., Klemperer, P., Multmarket Olgopoly: Strateg Substtutes and Complements. Journal of Poltal eonomy 93, [4] Cardell, J., Htt, C., Hogan, W. W., Market Power and Strateg Interaton n Eletrty Networks. Resoure and Energy Eonoms 19, [5] Chao, H.-P., Pek, S. C., A market mehansm for eletr power transmsson. Journal of Regulatory Eonoms 10(1), [6] Cottle, R. W., Pang, J.S., Stone, R. E., the Lnear Complementarty Problem. Aadem Press, Boston, MA, US. [7] von der Fehr, N.-H. M., Harbord, D., Long-term Contrats and Imperfetly Compettve Spot Markets: A Study of UK Eletrty Industry. Memorandum no. 14, Department of Eonoms, Unverstyof Oslo, Oslo, Sweden. [8] Fudenberg, D., Trole, J., Game Theory. the MIT Press, Cambrdge, MA, U.S. [9] Green, R. J., the Eletrty Contrat Market n England and Wales. Jounal of Industral Eonoms 47(1), [10] Hobbs, B.F., Metzler, C. B., Pang, J.S., Strateg Gamng Analyss for Eletr Power Systems: An MPEC Approah. IEEE Transatons on Power Systems 15(2), [11] Hobbs, B.F., Lnear Complementarty Models of Nash-Cournot Competton n Blateral and POOLCO Power Markets. IEEE Transatons on Power Systems 16(2), [12] Kamat, R., Oren, S. S., Mult-Settlement Systems for Eletrty Markets: Zonal Aggregaton under Network Unertanty and Market Power. Journal of Regulatory Eonoms 25(1), [13] Luo, Z.Q., Pang, J.S., Ralph, D., Mathematal Programs wth Equlbrum Constrants. Cambrdge Unversty press, Cambrdge, MA, US. [14] Newbery, D. M., Competton, Contrats, and Entry n the Eletrty Spot Market. Rand Journal of Eonoms 29(4),

21 [15] Oren, S. S., Eonom Ineffeny of Passve Transmsson Rghts n Congested Eletrty Systems wth Compettve Generaton. the Energy Journal 18, [16] Powell, A., Tradng Forward n an Imperfet Market: the Case of Eletrty n Brtan. the Eonom Journal 103, [17] Smeers, Y., We, J.-Y., Spatal Olgopolst Eletrty Models wth Cournot Frms and Opportunty Cost Transmsson Pres. Center for Operatons Researh and Eonometrs, Unverste Catholque de Louvan, Louvan-la-newve, Belgum. [18] We, J.-Y., Smeers, Y., Spatal Olgopolst Eletrty Models wth Cournot Frms and Regulated Transmsson Pres. Operatons Researh 47 (1), [19] Yao, J., Oren, S. S., Adler, I., Computng Cournot Equlbra n Two-settlement Eletrty Markets wth Transmsson Contrants. Proeedng of the 37th Hawa Internatonal Conferene on Systems Senes (HICCS 37). Bg Island, Hawa. [20] Yao, J., Wllems, B., Oren, S. S., Adler I., 2005a. Cournot Equlbrum n Pre-apped Two- Settlement Eletrty Markets. Proeedng of the 38th Hawa Internatonal Conferene on Systems Senes (HICCS 38). Bg Island, Hawa, January 4-7 [21] Yao, J., Adler, I., Oren, S.S., 2005b. An EPEC algorthm for an Equlbrum Model of Twosettlement Eletrty Markets. Manusrpt. Department of Industral Engneerng and Operatons Researh, Unversty of Calforna at Berkeley, Berkeley, Calforna, U.S. 21