Jian Yao, Ilan Adler, Shmuel S. Oren,

Transcription

1 This artile was downloaded by: [ ] On: 13 July 216, At: 13:56 Publisher: Institute for Operations Researh and the Management Sienes (INFORMS) INFORMS is loated in Maryland, USA Operations Researh Publiation details, inluding instrutions for authors and subsription information: Modeling and Computing Two-Settlement Oligopolisti Equilibrium in a Congested Eletriity Network Jian Yao, Ilan Adler, Shmuel S. Oren, To ite this artile: Jian Yao, Ilan Adler, Shmuel S. Oren, (28) Modeling and Computing Two-Settlement Oligopolisti Equilibrium in a Congested Eletriity Network. Operations Researh 56(1): Full terms and onditions of use: This artile may be used only for the purposes of researh, teahing, and/or private study. Commerial use or systemati downloading (by robots or other automati proesses) is prohibited without expliit Publisher approval, unless otherwise noted. For more information, ontat permissions@informs.org. The Publisher does not warrant or guarantee the artile s auray, ompleteness, merhantability, fitness for a partiular purpose, or non-infringement. Desriptions of, or referenes to, produts or publiations, or inlusion of an advertisement in this artile, neither onstitutes nor implies a guarantee, endorsement, or support of laims made of that produt, publiation, or servie. Copyright 28, INFORMS Please sroll down for artile it is on subsequent pages INFORMS is the largest professional soiety in the world for professionals in the fields of operations researh, management siene, and analytis. For more information on INFORMS, its publiations, membership, or meetings visit

2 OPERATIONS RESEARCH Vol. 56, No. 1, January February 28, pp issn 3-364X eissn informs doi /opre INFORMS Modeling and Computing Two-Settlement Oligopolisti Equilibrium in a Congested Eletriity Network Jian Yao, Ilan Adler, Shmuel S. Oren Department of Industrial Engineering and Operations Researh, University of California, Berkeley, California 9472 {jianyao@al.berkeley.edu, adler@ieor.berkeley.edu, oren@ieor.berkeley.edu} A model of two-settlement eletriity markets is introdued, whih aounts for flow ongestion, demand unertainty, system ontingenies, and market power. We formulate the subgame perfet Nash equilibrium for this model as an equilibrium problem with equilibrium onstraints (EPEC), in whih eah firm solves a mathematial program with equilibrium onstraints (MPEC). The model assumes linear demand funtions, quadrati generation ost funtions, and a lossless DC network, resulting in equilibrium onstraints as a parametri linear omplementarity problem (LCP). We introdue an iterative proedure for solving this EPEC through repeated appliation of an MPEC algorithm. This MPEC algorithm is based on solving quadrati programming subproblems and on parametri LCP pivoting. Numerial examples demonstrate the effetiveness of the MPEC and EPEC algorithms and the tratability of the model for realisti-size power systems. Subjet lassifiations: nonooperative games: Cournot equilibrium; eletriity market, two settlements; programming: mathematial program with equilibrium onstraints, equilibrium problem with equilibrium onstraints, linear omplementarity problem. Area of review: Environment, Energy, and Natural Resoures. History: Reeived Deember 25; revisions reeived May 26, Otober 26; aepted Deember 26. Published online in Artiles in Advane November 19, Introdution Eletriity restruturing aims at reating new ompetitive environments that provide long-term onsumer benefits. A major obstale to this goal is market power, both vertial and horizontal. Vertial market power in eletriity markets has been substantially mitigated through the unbundling of the generation, transmission and distribution setors, and through open aess to transmission grids. However, horizontal and loational market power remains an important issue to poliymakers due to the nonstorability of eletriity, the lak of demand elastiity, high market onentration, and limited transmission apaities. Among the many proposed and implemented eonomi means of mitigating horizontal market power is a twosettlement approah, where forward ontrats and spot transations are settled at different pries. Both theoretial analysis and empirial evidenes in Allaz (1992), Allaz and Vila (1993), von der Fehr and Harbord (1992), Green (1999), Newbery (1998), and Powell (1993) have suggested that forward ontrating dereases sellers inentives for manipulating spot market pries beause, under two settlements, the volume of trading that an be affeted by spot pries is redued. Allaz (1992) assumes a two-period market and demonstrates that if all produers have aess to a forward market, it leads to a prisoners dilemma type of game among them. Allaz and Vila (1993) show that, as the number of forward trading periods inreases, produers lose their ability to raise energy pries above their marginal ost. Kamat and Oren (24) analyze two-settlement markets over two- and three-node networks and extend the results in Allaz (1992) and Allaz and Vila (1993) to a system with unertain transmission apaities in the spot market. Reent work in Yao et al. (24, 25) further extends the above results to more realisti multinode and multizone networks. Yao et al. (24) onsider flow onstraints, system ontingenies, and demand unertainties in the spot market. Their numerial tests show that, like in the simple ases, generation firms have inentives to engage in forward ontrating, whih inreases soial surplus and redues spot pries. Yao et al. (25) onsider two alternative mehanisms for apping pries. They observe that a forward ap, whih an be indued by free entry of new generation apaity, inreases firms inentives for forward ontrating, whereas a regulatory ap in the spot market redues suh inentives. This paper ontinues the study of two-settlement eletriity systems. Our objetive is twofold. First, we introdue a new model of Cournot equilibrium in two-settlement markets that overomes some shortomings of the formulations in Yao et al. (24, 25). As before, the model is formulated as an equilibrium problem with equilibrium onstraints (EPEC), where eah generation firm solves a mathematial problem with equilibrium onstraints 34

3 Operations Researh 56(1), pp , 28 INFORMS 35 (MPEC) (see Luo et al. 1996) parameterized by the other firms forward ommitments. The model assumes linear demand funtions, quadrati generation ost funtions, and a lossless DC network, resulting in the preeding equilibrium onstraints in the form of a parametri linear omplementarity problem (LCP) (see Cottle et al. 1992). This EPEC model presents a omputational hallenge when applied to realisti-size systems. Therefore, our seond goal is to study the omputational aspet of this EPEC model and, by exploiting the problem struture, present in detail the solution approah for the EPEC and MPECs arising in our formulation. Solving an EPEC problem amounts to solving simultaneously a set of MPEC problems, eah parameterized by the other MPECs deision variables (see Pang and Fukushima 25 for more disussions on related topis). One solution approah is to derive the optimality onditions for the regularization sheme of the MPECs (see Flether and Leyffer 24, Sheel and Sholtes 2, Sholtes 21), and then either solve the nonlinear omplementarity onditions of the EPEC as a whole (Hu 22, Su 25) or iteratively solve the nonlinear omplementarity onditions of individual MPECs (Hu 22, Su 25). The seond approah we will follow in this researh is to iteratively solve MPECs using MPEC-based algorithms. There has been a growing literature on MPEC algorithms. The monograph by Luo et al. (1996) presents a omprehensive study of MPEC problems and provides first- and seond-order optimality onditions; it also desribes some iterative algorithms, suh as the penalty interior point algorithm (PIPA) and the pieewise sequential quadrati programming (PSQP) algorithm (see also Jiang and Ralph 1999). More reent advanes in MPEC algorithms an be found, for example, in Chen and Fukushima (24), Fahinei et al. (1999), Fukushima et al. (1998), Fukushima and Tseng (22), Fukushima and Lin (24), Hu and Ralph (24), and Ralph and Wright (24). Fukushima et al. (1998) present a sequential quadrati programming approah through a reformulation of the omplementarity ondition as a system of semismooth equations by means of Fisher-Burmister funtionals. This algorithm shares several ommon features with the PIPA in terms of omputational steps and onvergene properties; however, it differs from the PIPA in the way of updating the penalty parameters and determining the step sizes. Chen and Fukushima (24) onsider MPECs whose lower onstraints are a parametri P-matrix LCP. They smooth out the omplementarity onstraints through the use of Fisher- Burmister funtionals, from whih the state variables are viewed as impliit funtions of the deision variables. The MPECs an thus be solved by a sequene of well-behaved, though nononvex, nonlinear programs. Fukushima and Tseng (22) propose an -ative set algorithm for solving MPECs with linear omplementarity onstraints and establish onvergene to B-stationary points under the uniform linear independene onstraint qualifiation on the feasible set. This algorithm generates a sequene of variable value sets suh that the objetive value is almost dereasing, while maintaining the -feasibility of the omplementarity onstraints. The remainder of this paper is organized as follows. The next setion presents the EPEC model of two-settlement markets. In 3, we give a ompat representation of the omputational problem underlying our model. Setion 4 summarizes the proposed MPEC and EPEC algorithms, and 5 reports the omputational tests. Finally, we explore some eonomi impliations of our test ases and draw onlusions. More details of the algorithms are given in the online appendix. An eletroni ompanion to this paper is available as part of the online version that an be found at or.journal.informs.org/. 2. The Model We view two-settlement markets as a two-period Nash- Cournot game: the forward market (period ) and the spot market (period 1), and we haraterize the equilibrium of this game as a subgame perfet Nash equilibrium (SPNE) (see Fudenberg and Tirole 1991). In period, rational firms enter into forward ontrats, forming rational expetations regarding the forward ommitments of the rivals and the period 1 equilibrium outomes. Period 1 is a subgame with two stages. In stage 1, nature piks a state defined by a realization of the unertain demand and system ontingenies. In stage 2, the firms whose information sets inlude the state of nature and all forward ommitments ompete in a Nash-Cournot manner, while the independent system operator (ISO) transmits eletriity and sets ongestion pries to maximize soial surplus of the entire system. The dynamis of this model are illustrated in Figure 1, where the solid lines represent time progress and the dashed lines denote rational expetations. From a mathematial perspetive, the model is formulated as an EPEC, whih omprises a set of MPECs that haraterize the deisions of individual firms. In eah MPEC, the upper level is the firm s utility-maximization problem in the forward market, and the lower level, Figure 1. Period (forward market) Upper-level problems Firms write forward ontrats Forward pries The model dynamis. Period 1 (spot market) Lower-level problems Nature piks a state Expeted spot zonal settlement prie Time Firms produe ISO re-dispathes Spot nodal pries Spot zonal settlement pries

4 36 Operations Researh 56(1), pp , 28 INFORMS shared by all MPECs, onsists of the period 1 equilibrium onditions. The following summarizes the main features of our model that will be elaborated in the rest of this setion. We onsider a lossless DC approximation of an eletriity network, where flows on transmission lines are onstrained by thermal apaities and random line outages. This simplifies the ISO s deision problem in the spot market. Suh approximations are reasonable for an eonomi-oriented model onsidering that spot and dayahead market settlements in major loational-marginalprie (LMP)-based systems like PJM, NYISO, and ISO-NE apply linear programming algorithms to the linearized models of the transmission system. The demand side is prie taking with elasti demand funtions subjet to unertainty (in the form of quantity shifts) at eah node. Suh an essential assumption in a Cournot type model might be problemati if the spot market represents a real-time balaning market. On the other hand, if the spot market represents a day-ahead market, there is suffiient time for demand response to justify this assumption. Alternative models that do not require the assumption of elasti demand are onjetural variation models that lak theoretial support, or supply funtion equilibrium (SFE) models that have not yet been suffiiently developed for appliation in the ontext of a uneated network. The supply side onsists of Cournot produers with multiple generators at various nodes that are subjet to random outages, who sell energy to a pool at uniform LMPs set by the ISO. Generator outages, transmission line outages, and demand unertainty are represented in terms of system ontingent states that have known probabilities in the forward market and are realized before the spot market ommenes. 1 The forward market is organized at zonal hubs as finanial ontrats traded at uniform market-learing pries and settled at spot hub settlement pries based on the nodal LMPs. (Suh different granularity in the forward and spot markets is disussed in ) In the spot market, produers engage in a Nash- Cournot ompetition (i.e., setting quantities) while the ISO, who maintains the feasibility of the transmission onstraints, behaves a la Bertrand by setting nodal prie premiums or, equivalently, ongestion harges between nodes. The market is effiient; i.e., risk-neutral speulators will arbitrage away any differene between the forward hub pries and expeted spot hub settlement pries (see 2.2). To failitate the omputation, we will also assume that all the agents, i.e., the firms and the ISO, are risk neutral, as will be disussed later Period 1: The Spot Market Eletriity restruturing in different markets has been following different blueprints. In the United States, one prevailing design is the so-alled entrally dispathed market. This type of market usually onsists of a pool run by an ISO that serves as a broker, or autioneer, for wholesale spot eletriity market transations. The ISO leases the transmission system from transmission owners and ontrols flows so as to maintain the feasibility of the network. It also sets nodal prie premiums and implied ongestion harges for bilateral energy transations. We onsider a entrally dispathed wholesale spot market with demand unertainty, flow onstraints, and system ontingenies. The network underlying the spot market onsists of a set N of nodes and a set L of transmission lines. There is a set G of ompetitive firms, eah operating the units at a subset of loations N g N. We assume that at most one generation firm operates at a node, and if neessary, we an introdue artifiial nodes to meet this assumption. We also assume, for onveniene, that there is elasti demand at eah node so that pure generation nodes are represented by a demand funtion interseting the quantity axis at a very small value The ISO s Deision Making. In eah state C, the ISO ontrols the import/export ri at eah node i N (using the onvention that positive quantities represent imports) and sets the orresponding loational marginal pries. These quantities must satisfy the network feasibility onstraints, that is, the resulting power flows should not exeed the thermal limits Kl of the transmission lines in both diretions. We model the transmission network via a lossless DC (i.e., linear) approximation of Kirhhoff s laws (see Chao and Pek 1996). Speifially, flows on lines an be alulated using power transfer distribution fator (PTDF) Dl i, whih speifies the proportion of flow on a line l L resulting from an injetion of one-unit eletriity at a node i N and a orresponding one-unit withdrawal at some fixed referene node (also known as the slak bus). Moreover, beause eletriity is not eonomially storable, the load and generation must be balaned at all times so all import and export quantities must add up to zero. The ISO s objetive is to maximize soial welfare of the entire system. That is the aggregated area under the nodal inverse demand funtions (IDFs) P i, whih represent the total onsumer willingness-to-pay, less the sum of all generation osts C i. Mathematially, the ISO solves the following problem parametri on the firms prodution deisions qi : max r i ( r i +qi P i i d i C i q i ) subjet to r i = (1) D l i r i K l l L (2) D l i r i K l l L (3) In the above formulation, we have exluded the nonnegativity onstraints r i + q i, i N, by impliitly assuming an interior solution with respet to these onstraints.

5 Operations Researh 56(1), pp , 28 INFORMS 37 The numerial results in 5 validate this simplifiation, but that might not be true in general. Let p, l, and l+ be the Lagrange multipliers orresponding to (1) (3). The first-order neessary onditions (the Karush-Kuhn-Tuker (KKT) onditions) for the ISO s problem are P i q i + r i p i = i N i = l L l+ D l i l D l i i N r i = l D l i r i + K l l L l+ K l D l i r i l L The first KKT ondition herein implies that q i + r i = P i 1 p + i i N and onsequently, due to (1), q i = P i 1 p + i (4) This equation represents the aggregate demand funtion in the network relating the total onsumption quantity to the referene node prie p and the nodal prie premiums i, whih determine the relative nodal pries. The orresponding ongestion harges for transmission from node i N to node j N that will prevent arbitrage between nodal energy transations and bilateral transations among nodes must be j i The Firms Deision Making. In the spot market, eah firm g G determines the outputs from its units at N g. A variety of modeling approahes have been proposed to simulate generation firms deision making (see, for example, Hobbs 21, Neuhoff et al. 25, Smeers and Wei 1997, and Wei and Smeers 1999). One modeling onsideration regarding the suppliers strategi behaviors in these models is whether or not they game the ongestion pries set by the ISO. Following Hobbs (21) and Neuhoff et al. (25), we lassify spot market models into two basi approahes. The first approah assumes that generation firms antiipate the impat of their prodution on the ongestion pries set by the ISO and take that effet into aount in their prodution deisions. The resulting formulation of the spot market is a multileader, one-follower Stakelberg game (Neuhoff et al. 25). Eah produer g solves the following MPEC, in whih the optimality onditions for the ISO s program are the onstraints shared by all the firms: max P qi i r i + q i q i C i q i g g g subjet to q i q i i N g P i q i + r i p + l L l D l i l+ D l i = r i = l D l i r i + K l l L l+ K l D l i r i l L i N The equilibrium problem among the above MPECs represents a generalized Nash game (see Harker 1991), and it ould have zero or multiple equilibria (see Borenstein et al. 2). On the other hand, even if some pure-strategy equilibrium is found, it an be degenerate; that is, firms will find it optimal to barely ongest some transmission lines to avoid ongestion harges (see Oren 1997). Moreover, this formulation would lead to a two-settlement model with three deision levels, whih makes an equilibrium solution for the two-settlement market omputationally intratable. The seond approah assumes that the firms do not fully antiipate the impat of their prodution deisions on ongestion harges (see, for example, Metzler et al. 23), whih an be interpreted as a bounded rationality assumption. In this approah, the ISO is a Nash player that moves simultaneously with the generation firms. The firms determine their supply quantities to maximize their profits, but they at as prie takers with respet to transmission osts. The market equilibrium is then determined by aggregating the optimality onditions for the firms and the ISO s problems, whih result in a mixed omplementarity problem or a variational inequality problem. There are still two modeling options within this simultaneous-move framework. The first option assumes that the ISO, like the generation firms, is a Cournot player whose strategi variables are the import/export quantities at the nodes (see Neuhoff et al. 25; and Yao et al. 24, 25). Hene, eah firm g G solves the following profitmaximization problem: max q i g g P subjet to q i q i i r i + q i q i i N g g C i q i Note that beause this program is parameterized by r i, it an be deomposed into N g subproblems, eah orresponding to the prodution deision at one node. Therefore, this model will yield a spot market equilibrium that is invariant to the generation resoure ownership struture (i.e, it does not matter whether a firm owns one or multiple generators). Moreover, under this formulation, when the network onstraints (2) (3) are nonbinding, the equilibrium solution predits uniform nodal pries that are systematially higher than the Cournot equilibrium prie orresponding to a single market with the aggregated system demand

6 38 Operations Researh 56(1), pp , 28 INFORMS funtion (Neuhoff et al. 25). These aspets make the hoie of import/export quantities as the ISO s strategi variables (whih we have used in our previous work; see Yao et al. 24, 25) unsatisfatory. The seond option we employ in this paper is to use the loational prie premiums as the ISO s strategi variables. This option an be viewed as a mixed Cournot-Bertrand model, where the ISO behaves a la Bertrand while the generation firms are Cournot players with respet to eah other (i.e., set quantities), but treat the ISO as a prie setter. Thus, eah firm hooses its prodution quantities to maximize profits with respet to the residual demand defined impliitly by (4). In this formulation, the referene bus prie p is determined impliitly by the aggregate prodution deisions of all the generation firms, just as in a regular Cournot game. However, these prodution deisions and the implied referene node prie also depend on the nodal premiums i set by the ISO. The resulting problem solved by eah generation firm is max p + qi g p i q i C i q i g g subjet to q i q i i N g q i = P i 1 p + i This modeling option takes aount of the resoure ownership struture and, when the network onstraints are relaxed, the loational prie premiums go to zero so that the model produes the same equilibrium solution as the Cournot equilibrium applied to the aggregate system demand. Unfortunately, this approah has another shortoming whih manifests itself if we redue the apaity of a radial transmission line to zero or, more realistially, if it is ommon knowledge that a radial line is onstantly ongested. In suh situations, subnetworks onneted by saturated radial lines are effetively deoupled from a ompetitive interation point of view. The demand funtions on both sides of the saturated line will be shifted by the import/export quantities, but their slope stays the same so generators will behave as loal monopolists. For example, in the ase of a symmetri two-node one-line network, reduing the line apaity to zero reates two symmetri loal monopolies. However, in this situation, our model will produe a symmetri duopoly equilibrium with pries that are systematially lower than the loational monopoly pries. Unfortunately, there is no satisfying solution to this problem beause a Nash equilibrium in a ongestion-prone network depends on the onjetured ommon knowledge with regard to the extent of possible ompetition aross transmission lines. Suh onjetures affet the pereived elastiity of the residual demand by the ompeting firms and hene their strategi behaviors. The disontinuities in reation funtions and the resulting equilibrium pries when a single transmission line separating two ompetitors in a two-node system swithes from a ongested to an unongested regime, have been eloquently demonstrated in Borenstein et al. (2). Suh disontinuities beome intratable in a meshed system with multiple nodes. We partially address the above issue in a sequel paper (Yao et al. 26) through a hybrid approah that requires a prior identifiation of systematially ongested links (e.g., path 15 in California or the link between Frane and the United Kingdom aross the English Channel), whih effetively deouples the network into strategi subnetworks. In this paper, however, we will assume that the network is fully onneted physially and strategially so that ompeting firms behave as if the demand at all nodes is ontestable. Our two-settlement model permits different granularity in the forward and spot markets. This is ahieved by dividing the network into a set Z of zones (or trading hubs), eah onsisting of a luster of nodes. By allowing different granularity in the forward and spot markets, we are able to apture the ase where the two settlements represent long-term forward ontrats typially traded at hubs and nodal day-ahead spot markets, as well as the ase where the forward market is in the day ahead and the spot market is at real time, both of whih are typially nodal. In partiular, our model assumes that the spot market supply and demand at eah node are settled at the nodal pries, whereas the forward ontrats are traded at zonal hub forward pries and settled at the orresponding spot hub, or zonal settlement, pries u z z Z, whih are defined as weighted averages of the nodal pries in the respetive zones. Thus, the nodal spot pries resulting from the strategi interation in the spot market will affet the settlement of the forward ontrats, whih is debited from the firms spot market profits, through these hub pries. The nodal weights i are assumed to be exogenous parameters based on historial load shares at the nodes. This assumption is onsistent with ommon pratie, for example, at the Pennsylvania-Jersey-Maryland (PJM) western hub. In mathematial terms, eah firm g G solves in the spot market the following profit-maximization problem parametri on the loational prie premiums i and on its own forward ontrats x gz z Z : max p + qi g p i q i u z x gz C i q i g z Z g subjet to u z = p + i i i z i =z q i i N g (5) q i q i i N g (6) q i = P i 1 p + i (7) Let i, i+, and g be the Lagrange multipliers orresponding to (5) (7). Then, the KKT onditions for firm g s

7 Operations Researh 56(1), pp , 28 INFORMS 39 program are (after substituting the first onstraint into the objetive funtion) p + i g dc i qi + dqi i i+ = i N g d Pi 1 p + i g + q dp i i x gz i = g q i = P i 1 p + i i q i i N g i+ q i q i i N g Here, the first two onditions are the derivatives of the Lagrangian funtion with respet to qi and p, respetively Period 1 Equilibrium Conditions. Aggregating the KKT onditions for the firms and the ISO s programs yields the spot market equilibrium onditions, whih, in general, form a mixed nonlinear omplementarity problem. It beomes a mixed LCP when both the nodal demand funtions and the marginal ost funtions are linear, as is assumed in the remainder of this paper. Let the inverse demand funtions and the ost funtions be, respetively, P i q = a b i q i N C i q = d i q s iq 2 i N Then, the market equilibrium onditions beome p + i g d i s i q i + i i+ = g g G (8) p = a i /b i qi (9) 1/bi 1/bi 1 g + q bi i i x gz i = g G (1) g i q i i N (11) i+ q i q i i N (12) r i = (13) a q i + r i b i p i = i N (14) i = l L l+ D l i l D l i i N (15) l D l i r i + K l l L (16) l+ K l D l i r i l L (17) Here, (8) (12) are the aggregated KKT onditions for the firms problems, and (13) (17) are the KKT onditions for the ISO s problem. Under the assumption of linear demand funtions and quadrati onvex ost funtions, the firms and the ISO s programs are stritly onave-maximization problems, so (8) (17) are also suffiient. Note that (9) an be exluded from the preeding market equilibrium onditions beause it is implied by (13) and (14) Period : The Forward Market The forward market is assumed to be standardized and liquid suh that all forward ontrats in a zone are settled at equal pries. It is also assumed that there are enough risk-neutral arbitrageurs in the markets, and they will eliminate any arbitrage opportunity arising between the forward pries and the expeted spot zonal settlement pries. Consequently, the forward prie (h z ) in eah zone z Z is equal to the expeted values of the orresponding spot hub pries over all ontingent states ( C) with respetive probabilities (Pr ) (we assume for simpliity that the state probabilities and the market risk-neutral probabilities are idential). This is referred to as a no-arbitrage or perfet-arbitrage ondition. The risk-neutral firms simultaneously determine their forward ontrat quantities x gz g G z Z so as to maximize the total profit from both the forward ontrats and the spot produtions, while antiipating the forward ommitments of the rivals as well as the equilibrium outome in period 1. In mathematial terms, eah firm g G solves the following MPEC program, where {(8) (17)} C form the inner problem: max h z x g z + Pr g g z Z C subjet to g = x g z z Z r i q i i+ i C l l+ l L C g = p + i q i u z x g z C i q i g z Z g C h z = Pr u z z Z C u z = i z i =z 8 17 p + i i z Z C and C The equilibrium problem among the preeding MPECs is an EPEC. A solution to this EPEC is a set of the variables, inluding the firms forward and spot deisions, the ISO s redispath deisions, and the aforementioned Lagrange multipliers, at whih all firms MPEC problems are simultaneously solved, and no market partiipant is willing to unilaterally hange its deisions in either market. It is worth noting that, from a philosophial point of view, the above formulation might appear internally inonsistent beause firms seem to base their deisions in the forward market on information that is not available to them in the spot market. To resolve this inonsisteny, we might assume that forward ommitments are based on orret foreast of the expeted spot market outomes rather than on the detailed information we use to repliate that foreast. Furthermore, it is also reasonable to assume that forward ontrating deisions and spot market prodution deisions are made by funtionally independent entities within a firm

8 4 Operations Researh 56(1), pp , 28 INFORMS operating on different time horizons and employing different foreasting tools. So while the deisions made in the spot market are informed of the forward ontrating positions of the firm, they do not neessarily aount for all the global information that led to these ontrating deisions. 3. A Compat Representation of the Model In this setion, we ompat the notation to streamline the subsequent algorithmi presentation by grouping and relabelling the variables, inluding the dual variables, as follows: x g ( R Z : vetor of the forward ommitments by firm g G. r ( R N : vetor of the ISO s import/export quantities in state C. q ( R N : vetor of the firms generation quantities in state C. + ( R N : vetors of the Lagrange multipliers assoiated with the generation apaity onstraints in state C. + ( R L : vetors of the Lagrange multipliers assoiated with the flow apaity onstraints in state C. In addition, the parameters are relabelled as R N Z : a matrix where the i z th element is 1 ifz i = z, and otherwise. q R N : vetor of the generator apaity bounds in state C. B R N N : a diagonal matrix for state C, where the i i th element is bi. d R N : vetor of the marginal generation osts. D R L N : A PTDF matrix for state C, where the l i th element is Dl i. k R L : vetor of the flow apaities of the transmission lines in state C. X g R Z : feasible region of x g for eah firm g G Compat Representation of the Inner Problem 8 17 C Let e R N be a vetor with all 1s. Then, (13) and (14) beome [ ] [ ] [ ][ ] a e B B e r q e T p [ ] [ ] [ ] D T D T + + = Solving r and p yields B [ ] r Q = p 1 1 ([ ] [ ] [ ] [ ] ) a e B D T q D T + + where Q = B 1 Hene, B 1 e 1 r = Q B q + Q D T DT + et et B 1 p = a q + DT DT + Now, onsolidating onditions (8) (1), we have = a e + d + H q + B Q D T DT x g g G where H is a matrix suh that 2 + s i if i = j 2 h ij = if i j and the units at nodes i and j belong to the same firm, 1 otherwise. Next, let w and y be two variable vetors, and t, A, and M be onstants suh that q q + w = k + D r q y = k D r + q t a e + d = A k = k I I H B Q D T B Q D T M = D Q B D Q D T D Q D T D Q B D Q D T D Q D T The preeding applied to (8) (17) leads to w = t + A x g + M y g G w y y T w = (18) Finally, aggregating (18) for all states C, we present the inner problem {(8) (17)} C as w = t + A g G x g + My w y y T w =

9 Operations Researh 56(1), pp , 28 INFORMS 41 where y and w are variables, and t, A, and M are onstants as follows: y = y C w = w C t = t C M 1 M 2 A = A C M = M C 3.2. Compat Representation of the MPEC Problems In period, eah firm g G solves the following MPEC problem: F g x g min x g y w f g x g y w x g subjet to x g X g w = t + A x g + Ax g + My w y y T w = (19) In this program, x g is the deision variable, y w are the state variables, and x g = k G\ g x k is a parameter that is the sum of other firms forward ontrat quantities. We denote the EPEC problem in period as F g g G. An equilibrium of this EPEC problem in period zero is a set x g g G y w that solves F g x g for all g G, i.e., x g y w SOL F g x g, where SOL F g x g denotes the solution set of F g x g. 4. Solution Approah To solve the EPEC as stated above, we propose an iterative sheme that solves in turn the MPEC problems by holding fixed the deision variables of the other MPEC problems The MPEC Algorithm The MPEC algorithm is motivated by the following properties of F g x g. (1) f g x g y w x g is quadrati with respet to x g y w. (2) M is positive semidefinite. To show this, we first note that H is symmetri positive-definite. Seond, v T Q v = v T B 1 v vt B 1 v = B 1/2 v 2 B 1/2 e 2 v T B 1/2 B 1/2 e 2 B 1/2 e 2 v R N Hene, Q is symmetri positive semidefinite. Now, beause T M + M T H = 2 + D Q D D D we onlude that M is positive semidefinite. (3) Given x g, the onstraint set (19) is an LCP parameterized by x g. Moreover, for any x g, q ( a e + d + x ) g + x + g w = k k ( a e + d + x g + x g y = ) C satisfy the linear onstraints of this LCP. By Theorem in Cottle et al. (1992), the LCP problem (19) is always solvable. Reall that suh solvability is ahieved by assuming that the demand funtions are linear and unonstrained and that the ost funtions are quadrati. In addition, we assume that for eah state in period 1, the ative onstraints at the optimal solutions to the period 1 problems are linearly independent. By Theorem in Cottle et al. (1992), (19) is always uniquely solved for all x g X g, i.e., its solution y w is an impliit funtion of x g, and F g x g an be redued to an optimization problem with respet only to x g. We developed an algorithm for solving F g x g via a divide-and-onquer approah (see Figure 2). The proposed MPEC algorithm is a variant of the PSQP algorithm in Jiang and Ralph (1999) and Luo et al. (1996), but it speializes the PSQP algorithm by taking advantage of the preeding properties of F g x g. Speifially, the partition of X g is determined by the feasible omplementary bases (see Cottle et al. 1992, Definition 1.3.2) of the LCP problem (19). In eah polyhedron, we derive the expliit affine funtions for the state variables in terms of x g, and solve a quadrati program involving only x g for determining a stationary point within this polyhedron. An x g is a stationary point of F g x g if and only if it is a stationary point Figure 2. A typial partition of X g. X g A P g (α 1 ) P g (α 2 )

10 42 Operations Researh 56(1), pp , 28 INFORMS with respet to all polyhedra ontaining itself. Through parametri LCP pivoting, the proposed MPEC algorithm searhes in the spae of feasible x g for a B-stationary point of F g x g along adjaent polyhedra. Details of the algorithm are desribed in online Appendix The EPEC Sheme To find a B-stationary equilibrium of F g g G, we start with an arbitrary set x g X g g G. At eah outer iteration k, we ompute x g k from F g x g k for eah g G while taking x g k as given. The algorithm terminates when the improvement of the design variables in two onseutive iterations is redued to a predetermined limit, or when the number of iterations reahes a predetermined upper bound. Online Appendix 2 disusses this sheme. Unlike the approah desribed in Hobbs et al. (2), the urrent sheme arries y w, whih always solves (19), among the MPEC problems. This offers the flexibility of terminating the MPEC algorithm before it reahes a B-stationary point. 5. Computational Results We implemented in MATLAB the MPEC and EPEC algorithms that utilize the optimization toolbox for solving quadrati programs. In the implementation, we treat any number below 1 16 as zero to aount for round-off errors. Tests of the algorithms are performed on both randomly generated problems and representative test ases speifi to the ontext of eletriity markets Tests of the MPEC Algorithm The main omputational effort involved in the EPEC sheme is to solve the MPECs. While our MPEC algorithm is guaranteed to terminate in finite steps, its atual performane is not known. Indeed, linear and also quadrati programs with linear omplementarity onstraints are shown to be NP-hard in Luo et al. (1996). In this setion, we test the algorithm on a randomly generated set of generi MPEC problems with quadrati objetive funtions. Speifially, these MPEC problems are of the form [ ] [ ] 1 x x min x y P + T x y 2 y y subjet to Ax + a w = Nx + My + q w y w T y = where P, A, M (a positive semidefinite matrix), N,, a, and q are onstant matries and vetors with suitable dimensions. We use the QPECgen pakage by Jiang and Ralph (1999) to generate these MPEC programs. In the tests, we launh the MPEC algorithm from random starting points. Table 1 summarizes the test results. The first three olumns list the dimensions of the deision and Table 1. Test results of the MPEC algorithm. Iterations Total Dim(x) Dim(y) Dim(w) dimension Min Max Average state variables, and olumns 5 to 7 report the minimum, maximum, and average numbers of iterations, respetively. We observe that: The average number of iterations inreases moderately as the dimension of the MPEC problems grows (exept for the ase of n = 15 and m = 1), but there does not exist suh a trend for the minimum and maximum numbers of iterations. The algorithm is able to effetively solve MPEC problems with relatively large dimensions. Note that all instanes in Table 1 have greater dimensions than those reported in Jiang and Ralph (1999) Tests of the EPEC Sheme We now test the MPEC/EPEC algorithms on an EPEC problem derived from the stylized Belgian eletriity system whih was also used in our previous work (Yao et al. 25). This system is originally omposed of kv and 22 kv transmission lines inluding some lines in neighboring ountries for apturing the effet of loop flow. Parallel lines between the same pairs of nodes have been ollapsed into single lines with equivalent eletri harateristis. In total, the stylized network omprises 71 transmission lines and 53 nodes (see Figure 3). Generation units in this system are loated, respetively, at the nodes , The ownership struture, zonal aggregation in the forward market, and ontingeny states are fititious and so are the nodal demand funtions, although they are alibrated to atual demand information. Table 2 lists the nodal information for this test problem, inluding the IDF slopes, the marginal generation osts (marginal osts are onstant in this example), and the apaity bounds of the generation units. Table 3 summarizes the impedane of the transmission lines and the orresponding thermal limits. Only lines 22-49, 29-45, 3-43, and are assumed prone to ongestion in this example. The method for alulating the state-dependent PTDF matries from the network data an be found in standard eletrial engineering textbooks (e.g. Hambley 24) and will be omitted here due to spae limitation. We assume six independent ontingeny states in the spot market. 2 The first three states orrespond to demand

11 Operations Researh 56(1), pp , 28 INFORMS 43 Figure Belgian high-voltage network unertainty, while all generation units and all transmission lines are rated at their full apaities. States 4, 5, and 6 have the same demand levels as state 2, but they represent the system ontingenies resulting from transmission or generator outages. State 4 denotes a transmission breakdown on lines States 5 and 6 apture the unavailability of two generation units at nodes 1 and 41, respetively. The prie interepts of the hypothetial IDFs and the probabilities of the six states are given in Table 4. The stylized system has the dimension 2 C N + L = 684 of y (and w), and the total number of possible partitions is In this implementation, we terminate the EPEC algorithm at an outer iteration k if the relative improvement of the MPECs deision variables (forward ommit- Table 2. Nodal data ments) is no greater than 1 8, i.e., x g k xk 1 g g G 1 8 x g k 1 g G We ran the tests with different numbers of zones and firms, and for eah test we start with randomly generated deision variables of the MPECs. In the implementation, the MPEC algorithm was limited to exeute a single inner iteration. We also tried some other rules for terminating the MPEC algorithm; however, they do not provide omparable results. 3 The test results are summarized in Table 5. Columns 4 to 9 list the minimum, maximum, and average numbers of outer iterations and quadrati programs, respetively. In addition, Tables 6 and 7 report the outer iterations of the firms total forward ommitments for the ases of two zones and two or three firms. We find that: For all test problems, the EPEC sheme onverges rapidly. There exists no lear relationship between the problem dimension and the number of iterations. However, the total number of quadrati programs grows as the number of firms inreases. In the tests, the EPEC sheme quikly reahes the proximity of the B-stationary equilibrium, after whih it only improves the signifiant deimal digits (see, for example, Tables 6 and 7). 6. Eonomi Interpretation of the Results The EPEC algorithm is not guaranteed to loate a (global) Nash equilibrium; however, as we will demonstrate in this setion, it produed results that are onsistent with eonomi intuition. In partiular, we onsidered two hypothetial generator ownership strutures with two zones in the stylized Belgian network: nodes 1 through 32 belong to zone 1, and the remaining nodes to zone 2. The first struture has two firms, IDF slope Marg. ost Capa. IDF slope Marg. ost Capa. IDF slope Marg. ost Capa. Node $/MW 2 $/MW MW Node $/MW 2 $/MW MW Node $/MW 2 $/MW MW , These numbers are zeros in states 5 and 6, respetively.

12 44 Operations Researh 56(1), pp , 28 INFORMS Table 3. Belgian transmission network data. Imped. Capa. Imped. Capa. Imped. Capa. Line (Ohm) (MW) Line (Ohm) (MW) Line (Ohm) (MW) This line breaks down in state 4. where the units at the node set belong to the first firm and the remaining units to the seond firm. The seond struture is omposed of three firms, operating the units at , , and , respetively. We observe that, under both resoure strutures, firms have strategi inentives for forward ontrating as reported in Tables 6 and 7. However, unlike in the single-node ase where all firms have strategi inentives for short forward positions (i.e., to sell forward) as shown in Allaz (1992), here some firms might take long forward positions (i.e., buy forward). In the duopoly ase, for example, firm 1 Table 4. States of the Belgian spot market. IDF interept State ($/MW 2 ) Probability Type and desription Demand unertainty: Demands are on the peak Demand unertainty: Demands are at shoulder Demand unertainty: Demands are off-peak Contingeny of line breakdown: Line goes down Contingeny of generation outage: Plant at node 1 goes down Contingeny of generation outage: Plant at node 41 goes down. buys 552 MWh in the forward market, but firm 2 is short 1,747 MWh, so the supply side as a whole sells forward, whih is qualitatively onsistent with the result in Allaz (1992). A generation firm ould take long forward positions when an arbitrage opportunity is raised from a disrepany between its expeted generator-loation-weighted average nodal prie in a zone and the zonal settlement prie. In suh a ase, a firm s speulative inentive for long positions may overwhelm the strategi Allaz-Vila -type inentive for selling forward. Suh an ation, however, will typially indue rival firms to inrease forward sales due to the inreased treat of low spot pries, as observed in our duopoly example. In Figure 4, we plot the expeted spot nodal pries under two settlements and ontrast them with the orresponding Table 5. Test results of the EPEC algorithm. Outer iterations Quadrati programs Z G Dim(y) Min Max Average Min Max Average

13 Operations Researh 56(1), pp , 28 INFORMS 45 Table 6. Iterations of the firms total forward ommitments (two firms). Outer iteration Firm 1 Firm nodal pries in the equilibrium of a single-settlement market whih is obtained by onstraining all firms forward positions to zero. We first note that, whether or not there exists a forward market, the three-firm struture yields lower spot equilibrium pries than the Duopoly struture, as one would expet. Moreover, under both the two- and three-firm strutures, a two-settlement equilibrium results in lower spot equilibrium pries at most nodes than a single settlement. However, nodes 29 and 31 do not follow this trend. Consequently, two settlements inrease soial welfare and onsumer surplus. These results suggest that the welfare-enhaning effet desribed in Allaz (1992) and Allaz and Vila (1993) generalizes to the ase with flow ongestion, system ontingeny, and demand unertainty, although that effet is quantitatively different due to generator apaities and transmission limits. 7. Conluding Remarks We study the Nash-Cournot equilibrium in two-settlement eletriity markets. We develop an EPEC model of this equilibrium, in whih eah firm solves an MPEC problem parameterized by the design variables of the other MPECs. We propose an MPEC algorithm by taking advantage of the speial properties of the problems at hand. This algorithm partitions the feasible region of the deision variables into a set of polyhedra, and projets the state variables into the spae of the deision variables. The algorithm solves a quadrati program for a stationary point in eah polyhedron and pivots through adjaent polyhedra while maintaining Table 7. Iterations of the firms total forward ommitments (three firms). Outer iteration Firm 1 Firm 2 Firm Figure 4. Prie Prie Expeted spot nodal pries Node Single settlement firms 3 firms Node Two settlements feasibility of the linear omplementarity onstraints. We establish the finite global onvergene of this MPEC algorithm. An EPEC sheme is onstruted by deploying the MPEC algorithm iteratively. Numerial tests on randomly generated quadrati MPECs and on the EPEC derived from a stylized Belgian eletriity network demonstrate the effetiveness of the algorithms. One limitation of our model is the assumption of risk neutrality on the part of the generating firms. Unfortunately, introduing risk aversion will make the objetive funtions of the MPECs nonquadrati, whih signifiantly inreases the omputational omplexity of the model. On the other hand, we like to point out that although the MPEC and EPEC algorithms are presented here in the ontext of two-settlement eletriity markets, they an be applied to other quadrati EPEC problems, provided that the linear omplementarity onstraints yield unique values of the state variables. 5 5