1 Cournot Equilibrium in Prie-apped Two-Settlement Eletriity Markets Jian Yao 1, Bert Willems 2, Shmuel S. Oren 1, Ilan Adler 1 1 Department of Industrial Engineering and Operations Researh 2 Unversity of California Energy Institute 4135 Etheverry Hall 2547 Channing Way University of California at Berkeley, Berkeley, CA 94720 Berkeley, CA 94720 {jyao, oren, adler}@ieor.berkeley.edu willems@berkeley.edu Abstrat We ompare two alternative mehanisms for apping pries in two-settlement eletriity markets. With suffiient lead time and ompetitive entry opportunities, forward market pries are impliitly apped by ompetitive pressure of potential entry that will our when forward pries rise above a ertain level. Another more diret approah is to ap spot pries through regulatory intervention. In this paper we explore the impliations of the two alternative mehanisms in a two settlement Cournot equilibrium framework. We formulate the market equilibrium as a stohasti equilibrium problem with equilibrium onstraints (EPEC) apturing ongestion effets, probabilisti ontingenies and market power. As an illustrative test ase we use the 53- bus Belgian eletriity network with representative generator ost but hypothetial demand and ownership assumptions. When ompared to two-settlement systems without prie aps we find that either of the prie apping alternatives results in redued forward ontrating. Furthermore the redution in spot pries due to forward ontrating is smaller. I. INTRODUCTION It is generally believed that forward ontrating mitigates generators horizontal market power in the spot markets and protet marketers against spot prie volatility resulting from system ontingenies and demand unertainty. Competitive entry in the forward market and regulatory aps on spot pries are further means of mitigating prie spikes and market power abuse. Previous works by [1], [2], [13], [20], fous on the impat of forward markets on spot pries and soial welfare under alternative assumptions regarding the relationship between forward and spot pries. It was shown that generators have inentives to ontrat in the forward markets whereas forward ontrating redues spot pries and inreases onsumption levels and soial welfare. These models, however, assume a fixed generation stok whih is an appropriate assumption for two-settlement system over short time intervals (e.g., day ahead and real time markets). For long term forward ontrats, potential ompetitive entries impose an impliit prie ap on forward ontrat pries sine new investment in generation apaity will our when forward pries rise above a ertain level. Alternatively, regulators in many restrutured eletriity markets have imposed prie or offer aps in the spot markets 1 Researh supported by the National Siene Foundation Grant ECS- 0224779 with the University of California. in an attempt to retify market imperfetions suh as demand inelastiity, barriers to entry, imperfet apital markets and loational market power. In this paper we extend our earlier model in [20] by onsidering, separately, the effets of these two ap types on the spot and forward pries. In partiular, we address the following questions: To what extent do the generators ommit forward ontrats under prie aps? How do aps on forward pries affet the spot market and how do aps on spot pries affet the forward market? We study these questions via a two-settlement Cournot equilibrium model where generation firms have horizontal market power. The system is subjet to ontingenies due to transmission and generation outages as well as to demand unertainties. Our model also aounts for network ongestion whih is represented through a apaity onstrained DC load flow approximation of the eletriity grid. We onvert our formulation into a stohasti equilibrium problem with equilibrium onstraints (EPEC) where eah generation firm faes a stohasti mathematial program with equilibrium onstraints (MPEC, see [14]). These MPEC problems have quadrati objetive funtions and share idential lower-level onstraints in the form of a parametri linear omplementarity problem (LCP, see [7]). The remainder of this paper is organized as follows. Related eletriity market models and MPEC algorithms are reviewed in the next setion, setion III presents the formulation for the two-settlement markets. In setion IV, we run a variety of experiments on our MPEC and EPEC algorithms. Finally, we investigate the impliations of hypothetial prie aps in a network desribing a reallized model of the Belgian eletriity market. II. RELATED RESEARCH In this setion, we review models for eletriity markets with forward ontrats and Cournot ompetition, as well as MPEC algorithms. Wei and Smeers [19] onsider a Cournot model with regulated transmission pries. They solve the variational inequalities to determine unique long-run equilibria in their models. In subsequent work, Smeers and Wei [18] onsider a separated energy and transmission market, where the system operator
2 onduts a transmission apaity aution with power marketers purhasing transmission ontrats to support bilateral transations. They onlude that suh a market onverges to the optimal dispath for a large number of marketers. Hobbs [12] alulates a Cournot equilibrium under the assumptions of linear demand and ost funtions, whih leads to a mixed linear omplementarity problem (mixed LCP). In a market without arbitrageurs, non-ost based prie differenes an arise beause the bilateral nature of the transations gives firms more degrees of freedom to disriminate between eletriity demand at various nodes. This is equivalent to a separated market as in Smeers and Wei [18]. In the market with arbitrageurs, any non-ost differenes are arbitraged by traders who buy and sell eletriity at nodal pries. This equilibrium is shown to be equivalent to a Nash-Cournot equilibrium in a POOLCO-type market. Hobbs, Metzler and Pang [11] present an oligopolisti market where eah firm submits a linear supply funtion to the Independent System Operator (ISO). They assume that firms an only manipulate the interepts of the supply funtions, but not the slopes, while power flows and priing strategies are onstrained by the ISO s linearized optimal power flow (OPF). Eah firm in this model faes a MPEC problem with spatial prie equilibrium as the inner problem. Work in forward markets has foused on the welfare enhaning properties of forward markets and the ommitment value of forward ontrats. The basi model in [1] assumes that produers meet in a two period market where there is some demand unertainty in the seond period. Allaz shows that generators have a strategi inentive to ontrat forward if other produers do not. This result an be understood using the onepts of strategi substitutes and omplements of Bulow, Geneakoplos and Klemperer [4]. In these terms, the availability of the forward market makes a partiular produer more aggressive in the spot market. Due to the strategi substitutes effet, this produes a negative effet on its ompetitors prodution. The produer with aess to the forward market an therefore use its forward ommitment to improve its profitability to the detriment of its ompetitors. Allaz shows, however, that if all produers have aess to the forward market, it lead to a prisoners dilemma type of effet, reduing profits for all produers. Allaz and Vila [2] extend this result to the ase where there is more than one time period where forward trading takes plae. For a ase without unertainty, they establish that as the number of periods when forward trading takes plae tends to infinity, produers lose their ability to raise market pries above marginal ost due to the ompetitive solution. von der Fehr and Harbord [8] and Powell [17] study ontrats and their impat on an imperfetly ompetitive eletriity spot market: the UK pool. von der Fehr and Harbord [8] fous on prie ompetition in the spot market with apaity onstraints and multiple demand senarios. They find that ontrats tend to put downward pressure on spot pries. Although, this provides disinentive to generators to offer suh ontrats, there is a ountervailing fore in that selling a large number of ontrats ommits a firm to be more aggressive in the spot market, and ensures that it is dispathed into its full apaity in more demand senarios. Powell [17] models expliitly re-ontrating by Regional Eletriity Companies (RECs) after the maturation of the initial portfolio of ontrats set up after deregulation. He adds risk aversion on the part of RECs to earlier models. Generators at as prie setters in the ontrat market. He shows that the degree of oordination has an impat on the hedge over demanded by the RECs, and points to a free rider problem whih leads to a lower hedge over hosen by the RECs. Newbery [15] analyzes the role of ontrats as a barrier to entry in the England and Wales eletriity market. He extends earlier work by modelling equilibria of supply funtions in the spot market. He further shows that if entrants an sign base load ontrats and inumbents have enough apaity, the inumbent an sell enough ontats to drive down the spot prie below the entry deterring level, resulting in more volatile spot pries if produers oordinate on the highest profit supply funtion equilibrium (SFE). Capaity limit however may imply that inumbents annot play a low enough SFE in the spot market and hene annot deter entry. Green [10] extends Newbery s model showing that when generators ompete in SFEs in the spot market, together with the assumption of Cournot onjetural variations in the forward market, imply that no ontrating will take plae unless buyers are risk averse and willing to provide a hedge premium in the forward market. He shows that forward sales an deter exess entry, and inrease eonomi effiieny and long-run profits of a large inumbent firm faed with potential entrants. Yao, Oren and Adler [20] study the Nash Equilibrium in the two-settlement ompetitive eletriity markets with unertainty of transmission, generation and demand in the spot market. The Cournot generators and the soial-welfare-maximizing system operator s behaviors are modelled in both forward and spot markets. The equilibrium is modelled as an EPEC in whih eah generation firm solves a MPEC problem. The model is applied to a six-bus illustrative example, and it is found that the generators have inentives for ommitting forward ontrats under spot market unertainties and ongestions, whereas two settlements inrease soial welfare, derease spot prie magnitudes and volatilities. III. THE MODEL In this setion, we introdue a generi model of the twosettlement eletriity system with both spot and forward prie aps. The system with either spot or forward market is a speial ase of this generi model by setting the forward or the spot prie aps, or both, to infinity. In this generi model, we formulate the two-settlement eletriity markets as a omplete-information two-period game with the forward market being settled in the first period, and the spot market being settled in the seond period. The equilibrium in either market is a sub-game perfet Nash equilibrium (SPNE, see [9]). In the forward market, the generation firms determine their forward ommitments while antiipating other firms forward quantities and the spot market outomes. The spot market is a subgame with three stages: in stage one, Nature piks the
3 state of the world so as to reveal the atual apaities of the generation failities and the transmission lines as well as the shape of the demand funtions at eah node; firms determine generation quantities to ompete in a Nash-Cournot manner in stage two; and the System Operator (SO) determines how to dispath eletriity within the network to maximize total soial welfare. Note that the generation firms take into onsideration the SO s ations, it is rational to assume that generation firms and the SO move simultaneous in the spot market. A. Notation: We onsider the set of nodes, transmission lines, zones, generation firms and their generation failities, and the states of the spot market. N: The set of all nodes or buses. Z: The set of all zones. Moreover, z(i) represents the zone where node i resides. L: The set of transmission lines whose probabilities of ongestion in the spot market are stritly positive. These lines are alled flowgates. C: The finite set of all states of the spot market. G: The set of all generation firms. N g denotes the set of nodes whih generation firm g has failities attahed to. In the two-settlement markets, generators determine how muh to ommit in the forward ontrats and how muh to generate in the spot market. The system operator determines how to adjust the onsumption level at eah node in the spot market. The variables related to the forward markets are: x g,z : The forward quantity ommitted from firm g to zone z. The variables related to the spot markets are: qi : The spot quantity generated at node i in state. ri : Adjustment quantity at node i by the system operator in state. The following exogenous parameters are onsidered in our formulation: q, i q i: lower and upper apaity bounds of generation faility at node i in state. ū: the spot prie ap. h: the forward prie ap. p i ( ): Inverse demand funtion at node i in state. We denote p as the ommon prie interept aross all nodes in eah state, and b i being the slopes: p i(q) = p b i q i N, C We assume that the inverse demand at eah node shifts inwards or outwards in different states, but the slope does not hange. C i ( ): Generation ost funtion at node i. We assume that the ost funtion is onvex quadrati where C i (q) = d i q + 1 2 s iq 2 with given positive d i and non-negative s i. Kl : apaity limit of line l in state. Dl,i : Power transfer distribution fator in state on line l with respet to node i. P r(): Probability of state in the spot market. δ i : (δ i 0, i:z(i)=z δ i = 1) the weights used to settle the zonal pries. B. The Spot Market For the sake of generality we allow different levels of granularity in the finanial settlements (with equal granularity being a speial ase). This is motivated by the fat that in real markets we observe different granularity levels in spot and long term forward markets (for example in PJM, the western hub representing the weighted average prie over nearly 100 nodes is the most liquid forward market). Speifially, the network underlying the nodal spot market is divided into a set of zones, eah of whih is a luster of onneted nodes. This suggests three priing shemes: spot nodal pries, spot zonal pries (used to settle zonal forward ontrats) and forward (zonal) pries. Spot nodal pries are the pries at whih generation and loads are settled at their respetive nodes. In state of the spot market, the total onsumption at node i is r i + q i, whih is sum of the quantity generated by the generator and the (export or import) adjustment made by the SO. Beause loads an never be negative, we restrit q i + r i 0 i N Consumers evaluate their onsumptions of qi + r i at prie p i (q i + r i ) aording to the inverse demand funtions. The atual spot nodal pries are min{ū, p i (q i + r i )} due to the spot prie aps. The spot zonal prie u z at a zone z in state is defined as the weighted average of nodal pries in the zone with predetermined weights δ i. Typially we expet theses weights to reflet historial load ratios and be updated periodially, however they are treated as onstants in our model so that ontrating, prodution and onsumption deisions do not affet these weights. In mathematial terms the zonal spot prie is given as: u z = i:z(i)=z δ i min{ū, p i(q i + r i )} z Z The forward zonal pries h z are the pries at whih forward ommitments are agreed upon in the respetive zones. We assume that in equilibrium no profitable arbitrage is possible between forward and spot zonal pries. This implies that the forward zonal prie is equal to the expeted spot zonal pries. That is: h z = E [u z] = C P r()u z z Z (2) In the existene of the forward prie ap, a threshold is put on the forward pries: h z h z Z. (3) In eah state of the spot market, generation firms deide variables qi : the output from eah of its plants. These outputs an not be below the minimal output, neither an they exeed
4 the respetive apaities of the plants in that state. Hene the generators fae the onstraints: q i q i q i i N g Eah generator g s revenue in state of the spot market is her generation quantities paid at spot nodal pries and the finanial settlement of her forward ommitments settled at the differene between the forward zonal pries and spot zonal pries. Her profit π g is given by: πg = qi min{ū, p i(ri + qi )} + (h z u z)x g,z g z Z C i (qi ) g To avoid disontinuity in generators profit funtion (see for example [16]), We have assumed here that the generators do not onsider the impat of their deisions on the settlement of transmission rights. Eah generator g solves the following program in state of the spot market: max π qi g subjet to: q i q i q i, i N g (4) q i + r i 0, i N g (5) The SO deides in eah state of the spot market how to dispath the energy within the network (i.e., import and export quantities at eah node) given the prodution deisions by generators. She makes the adjustment ri at eah node i. Her dispath must satisfy the network thermal onstraints on power flows. We model eletriity flows on transmission lines through Power Transfer Distribution Fators (PTDFs) using a Diret Current (DC) approximation of Kirhoff s law [6]. The PTDF is the proportion of flow on a partiular line resulting from an injetion of one unit at a partiular node and a orresponding one-unit withdrawal at the referene slak bus. The network feasibility onstraints are K l D l,ir i K l, l L Meanwhile, the SO should also maintain real time balane of loads and outputs, that is or equivalently (q i + r i ) = ri = 0 The SO s objetive is to maximize the soial welfare defined by the area under the onsumers inverse demand funtion q i minus total generation ost. She solves the following mathematial program in eah state of the spot market: max r i [ subjet to: r i +q i 0 p i(τ i )dτ i C i (q i )] ri + qi 0, i N (5) ri = 0 (6) K l D l,ir i K l, l L (7) Sine the generators deision variables qi are treated as onstant parameters in the SO s deision, the term C i (qi ) an be dropped from this problem without affeting its optimal solution. The onstraint (3) is exluded in spot market deision problems beause it has been onsidered by the generators in the forward market. C. Spot market smooth formulation The generators and the system operator s deision problems in the spot market do not have straightforward optimality onditions due to the non-smooth funtion haraterizing the spot pries. In this sub-setion, we reformulate these problems by removing the minimization terms of spot nodal pries. It is aomplished by onsidering separately two ases. 1) High spot aps: The first ase is of high spot aps, i.e. ū p. Due to the onstraint (5), it must hold that Thus the spot nodal pries are p p i(q i + r i ) min{p i(q i + r i ), ū} = p i(q i + r i ), and the spot zonal pries are u z = δ i min{ū, p i(qi + ri )} i:z(i)=z = i:z(i)=z δ i p i(q i + r i ) z Z. (8) High spot prie aps are hene not binding, and we drop them from the generators deision problems. So the generators spot deision problems beome G g : max π qi g subjet to: q i q i q i, i N g (4) q i + r i 0, i N g (5) where πg = p i(ri + qi )qi + (h z u z)x g,z C i (qi ) g z Z g
5 Prie p u d i 0 Inverse demand funtion Spot prie ap Marginal ost v i Quantity The system operator s soial welfare maximization problem beomes: Ŝ : max r i,v i ( v i 0 v p i +r i +q i v i i(τ i )dτ i + p i(τ i )dτ i ) v i subjet to: ri = 0 (6) K l D l,ir i K l, l L (7) 0 v i v i, i N (9) r i + q i v i 0, i N (10) Fig. 1. Low spot prie aps For the system operator, she still faes the deision problem: S : max r i subjet to: r i +q i 0 p i(τ i )dτ i ri + qi 0, i N (5) ri = 0 (6) K l D l,ir i K l, l L (7) 2) Low spot prie aps: When the spot prie aps are low, i.e. ū < p, they interset the inverse demand funtions, and are possibly binding (see figure 1). Though spot nodal prie min{p i (q i + r i ), ū} is a minimization funtion and the generators objetive is to be maximized, we an nevertheless apply a min-max formulation to it. This is beause of the unertainty of the relative magnitude between x g,z and qi in πg, whih we should not restrit. To overome this issue, we introdue artifiial variables vi : inframarginal quantity from q i + r i at node i whih the generators evaluate at the spot prie ap. Following from its definition, vi must satisfies 0 v i q i + r i Moreover, quantity vi an not be to the right of the intersetion point of the prie ap and the inverse demand funtions, i.e. v i v i = p ū b i Due to the SO s objetive to maximize soial welfare, the optimal spot outomes will set vi as muh as possible up to min{ v i, q i + r i }, before setting the remaining quantity q i + ri v i to the dereasing line segment of the inverse demand funtions. In another word, it an not be true that v i < min{ v i, q i + r i } and q i + r i v i > 0. Otherwise, the soial surplus is not maximized. The spot nodal pries are thus funtions of q i, r i and v i : ū b i (q i +r i v i ). The generation firms spot profit maximization problems are: Ĝ g : max π qi g subjet to: where the generators spot profit π g = q i q i q i, i N g (4) q i + r i v i 0, i N g (10) g (ū b i (q i + r i v i ))q i + (h z u z)x g,z C i (qi ) z Z g The onstraint (5) is no longer inluded in problems Ŝ and Ĝ g beause it it implied by onstraints (10) and the left hand of onstraint (9). D. Spot market outomes The spot market is desribed by the pair of Gg and S if spot prie aps are no less than the prie interepts of the inverse demand funtions, or otherwise the pair of Ĝg and Ŝ. These four problems are all strit onave-maximization programming problems, whih implies that their first order neessary onditions (the KKT onditions) are also suffiient. Thus, the spot market outomes an be replaed by their KKT onditions. Let ρ i and ρ i+ to both diretions of onstraint (4), µ i be the Lagrangian multipliers to onstraint (5) and (10), α to onstraint (6), λl and λ l+ to both diretions of onstraint (7), and βi and β i+ to both diretions of onstraint (9), we have KKT onditions as follows:
6 1. For the pair of G g and S (KKT 1) : rj = 0 j N p b iq i b ir i α + µ i + t L(λ t λ t+)d t,i = 0 0 λ l j N D l,ir j + K l 0 0 λ l+ K l j N d l,jr j 0 0 µ i q i + r i 0 p 2b iq i b ir i d i s i q i + µ i + (1 P r())δ i b ix g,z(i) + ρ i ρ i+ = 0 0 ρ i q i q i 0 0 ρ i+ q i q i 0 2. for the pair of Ĝ g, and Ŝ (KKT 2) : 0 µ i qi + ri vi 0 rj = 0 j N p b iv i ū + b i (r i + q i v i ) α + µ i + t L(λ t λ t+)d t,i = 0 ū b i (r i + q i v i ) + β i β i+ µ i = 0 0 λ l j N D l,ir j + K l 0 0 λ l+ K l j N D l,jr j 0 0 β i v i 0 E. the Forward Market 0 β i+ v i v i 0 ū 2b iq i b i (r i v i ) d i s i q i + µ i + (1 P r())δ i b ix g,z(i) + ρ i ρ i+ = 0 0 ρ i q i q i 0 0 ρ i+ q i q i 0 (11) In the forward market, network feasibility is ignored and the forward ontrats are settled. Eah firm g takes all her rivals forward quantities as given, and determines her own best forward quantities to maximize her expeted spot utilities. Assuming the firms are risk neutral, their forward objetives are to maximize their expeted spot profits subjet to the KKT onditions whih represent the antiipated ations in the spot market. The generators should also onsider onstraint (3) in the forward market so that their forward ontrating deisions will not result in expeted spot zonal pries, or the forward pries, above the forward prie ap. Thus eah firm g s optimization problem in the forward market is a stohasti MPEC problem: max x g,z E [π g] = C P r()π g subjet to: z Z x g,z X g z Z h z = E [u z] h (3) and onstraints (KKT1) and (KKT2) where X g defines the set of allowable non-negative forward positions for firm g. Combining the generators MPEC problems, the equilibrium problem in the forward market is an EPEC. Furthermore, this EPEC problem is a stohasti EPEC due to the unertainty of exogenous data. Note that variables ri and α an be eliminated from the KKT onditions, we further define x g : The vetor of firm g s forward variables. x g = [x g,z, z Z] y: y = [y, C] where y is the vetor of lagrangian multipliers for all inequality onstraints in the generators and the SO s deision problems in the spot market. w: w = [w, C] where w is the slakness of the onstraints orresponding to y. Then KKT onstraint set of both (KKT1), and (KKT2) beomes the following parametri LCP with respet to w and y with x g being the parameters where w = a + g a = M = a 1 a 2... a C A g x g + My, 0 w y 0 (12), Ag = A g,1 A g,2... A g, C M 1 0 M 2... 0 M C, in whih a, A g,, and M, C, being suitable vetor and matries derived from onstraints (KKT1) and (KKT2). Alternatively, the LCP an be expressed in another way due to its separability: w = a + g Ag, x g + M y 0 w y 0 C (13) Now, the generators forward objetives E [π g] an be expressed as funtions f g (x g, x g, y, w) with respet to x g, y, and w, where g denotes G\{g} and x g denotes all other firms design variables exept for generator g s. The firms
7 forward problems an be represented as: F g : min x g,y,w subjet to : f g (x g, x g, y, w) x g X g (14) F (x g, x g, y, w) 0 (15) w = a + A g x g + A g x g + My (16) 0 w y 0 (17) Here, x g are design variables, y, w are state variables, x g are parameters. The onstraint (15) is onstraint (3) represented in terms of x g, x g, y, and w. The onstraints (16) and (17) are a rewriting of the LCP onstraint (12) by separating x g and x g. Moreover, the onstraints (15), (16) and (17) are shared among all generators. We observe from the problems S, Gg, Ŝ and Ĝ g that M is positive semi-definite. Note that both problems Gg and S have optimal solutions for any set of x g, the KKT onditions (KKT1) and (KKT2) have thus feasible solutions, so do the LCP onstraints (12) as a transformation of the KKT onditions. Furthermore, due to theorem 3.1.7 in [7], the LCP problem (12) satisfies the w-uniqueness ondition. For simpliity, we assume that the exogenous parameter in our formulation is perturbed in suh way that the spot deision problems have non-degenerate optimal solutions. Following again from theorem 3.1.7 in [7], the LCP onstraints (12) have a unique y solutions as well. This property is stated as follows. A1: The LCP onstraints (12) have unique solutions of w and y orresponding to all feasible {x g }. Another observation from problems F g is that A2: the objetive funtion are quadrati. Properties A1 and A2 guarantee that the solution approahes in [21] are also appliable here. IV. THE BELGIAN ELECTRICITY MARKET We use the Belgian eletriity network to illustrate the eonomis results. For the ompleteness of the network, we inorporate some lines in the Netherlands and Frane. The network has 92 380kV and 220kv transmission lines, some of whih are parallel lines between the same pair of nodes. For omputational purpose, parallel lines are ombined to single lines with adjusted thermal apaities and resistanes, thus the network is redued to 71 transmission lines linking 53 nodes (see figure 2). Insignifiant lower voltage lines and small generation plants have been exluded from this example. We assume that there are six states in the spot market. The first state is a state in whih the demands are at the shoulder, all generation plants operate at their full apaity, all transmission lines rated at full thermal limits. The seond state is the same as the first state exept that it has on-peak demand. Off-peak state 3 differs from state 1 by the very low demand levels. State 4 denotes the ontingeny of the transmission line [31,32] being out of servie. State 5 and 6 apture the unavailability of two plants at node 10 and 41 respetively. The assumed probabilities of these states are given in table I. Fig. 2. 53 19 38 39 40 33 37 47 34 35 41 36 46 52 43 42 44 45 18 30 31 29 28 Belgian high voltage network 27 17 51 48 22 9 10 21 8 7 25 24 11 6 32 26 23 20 50 12 4 1 2 3 We assume that there are two generation firms ompeting in the forward market and the spot market, and two zones in the network with nodes 1 through 32 being in zone 1 and the remaining nodes being in zone 2. Firm 1 owns plants at nodes 7, 9, 11, 31, 32, 33, 35, 37, 41, 47, and 53, and firm 2 owns plants at nodes 10, 14, 22, 24, 40, 42, 44, and 48. There are no generation plants on other nodes. The orresponding information of generation plants is listed in table II. More details are given in table II desribing the nodal information on inverse demand funtions, first-order marginal generation osts, generation apaities and historial load ratios (in this example, marginal osts are assumed onstant). As to thermal limits, we ignore the intra-zonal flows and fous only on the flowgates of lines [22,49], [29,45], [30,43], and [31,52]. We onsider five test ases: Case 1: prie-unapped single-settlement system, i.e. single settlement without (spot) prie aps. Case 2: prie-apped single-settlement system, i.e. single settlement with (spot) prie aps. Case 3: prie-unapped two-settlement system, i.e. two settlements without spot and forward prie aps. Case 4: forward-apped two-settlement system, i.e. two settlements with only forward prie aps. We set the forward prie ap to 465. Case 5: spot-apped two-settlement system, i.e. two settlements with only spot prie aps. The spot prie ap is set to 600. Case 1 and 2 are essentially equivalent to a two-settlement system with the allowable forward ontrats limited to only zeros. We restrit the generators forward positions to their total generation apaity in the respetive zones. Table III reports the spot zonal pries of these five senarios. We find that Both spot and forward prie aps redue the magnitude of spot pries as ompared to the orresponding singlesettlement market. 5 13 15 16 49 14
8 TABLE I STATES OF THE BELGIAN NETWORK State Probability Type and desription 1 0.50 Shoulder state: Demands are at shoulder. 2 0.20 On-peak state: All demands are on the peak. 3 0.20 Off-peak state: All demands are off-peak. 4 0.03 Shoulder demands with line breakdown: Line [31,32] goes down. 5 0.03 Shoulder demands with generation outage: Plant at node 10 goes down. 6 0.04 Shoulder demands with generation outage: Plant at node 41 goes down. TABLE II NODAL INFORMATION. Node demand marginal apa- Node demand marginal apa- Id slope ost ity Id slope ost ity 1 1 N/A 0 28 1 N/A 0 2 0.82 N/A 0 29 0.93 N/A 0 3 1.13 N/A 0 30 0.85 N/A 0 4 1 N/A 0 31 1 180 712 5 0.93 N/A 0 32 1 580 95 6 0.85 N/A 0 33 0.88 20 496 7 1 450 70 34 0.5 N/A 0 8 1 N/A 0 35 1 250 1053 9 0.88 180 460 36 0.73 N/A 0 10 0.9 180 121 37 1 100 1399 11 1 200 124 38 0.85 N/A 0 12 0.73 N/A 0 39 1 N/A 0 13 1 N/A 0 40 1.15 100 1378 14 0.85 130 1164 41 1 210 522 15 1 N/A 0 42 0.79 180 385 16 1.3 N/A 0 43 0.68 N/A 0 17 1 N/A 0 44 1.03 200 538 18 0.79 N/A 0 45 1 N/A 0 19 0.68 N/A 0 46 1 N/A 0 20 1.05 N/A 0 47 1 100 32 21 1 N/A 0 48 0.73 220 258 22 1.1 190 602 49 1.2 N/A 0 23 1 N/A 0 50 1.5 N/A 0 24 0.75 100 2985 51 1 N/A 0 25 1 N/A 0 52 1 N/A 0 26 0.8 N/A 0 53 1 200 879 27 1.13 N/A 0, : these numbers are zeros in state 5 and 6 respetively. N/A: the marginal osts are not appliable to zero apaities. TABLE III SPOT ZONAL PRICES COMPARISONS 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 1 428.29 429.04 428.29 429.04 423.95 424.79 417.69 418.12 425.27 425.60 state 2 849.75 844.72 600.00 600.00 845.18 841.02 838.67 834.48 600.00 600.00 state 3 232.30 232.30 232.30 232.30 224.24 224.24 213.98 213.98 229.46 229.46 state 4 428.20 429.14 428.20 429.14 419.75 420.97 409.02 406.23 424.38 424.56 state 5 429.96 430.78 429.96 430.78 421.55 422.53 412.08 405.65 426.11 426.12 state 6 431.35 432.31 431.35 432.31 423.34 424.49 413.18 413.65 428.77 428.14 Expeted 473.55 473.01 423.56 424.07 468.03 467.73 460.53 459.65 421.19 421.34 Spot prie ap redues generators inentives to ommit forward ontrats ompared to prie-unapped twosettlement system. The spot prie ap auses the generators to ommit about 70% ompared to the prieunapped two-settlement markets. This is beause the spot prie aps themselves already redue spot pries leaving less inentives for generators to ommit forward ontrats. Forward prie aps inrease generators inentives for forward ontrating. The generators ommitment in the forward-apped two-settlement markets is about 180% ompared to the same markets without aps. Under
9 the forward prie aps, the generators are in fat net eletriity buyer in the spot market. This is beause that, knowing that entries would our if the forward pries are too high, the inumbents have to at more ompetitively to deter entries. They will only play an equilibrium in whih the forward pries are below the forward prie ap. Compared to the prie-unapped two-settlement system, the spot ap results in more prodution in on-peak states, therefore the on-peak pries are lower; but, the redued forward ontrating due to spot aps ause different results in off-peak states, i.e. lower prodution and higher nodal pries. On the ontrary, the inreased forward ontrating due to forward prie aps results in more prodution and lower spot pries ompared to the prieunapped two-settlement markets. Note that the forward pries are dereasing funtions in forward ommitments, the generators will sueed in deterring entries by playing an equilibrium with forward pries lower than the forward ap whenever then have suffiient apaity. On the other hand, if the generators don t have enough apaity, the forward ontrats will be signed at pries higher than the forward prie ap, and entries to the markets are inevitable. This result is onsistent to Newbery [15]. V. CONCLUDING REMARKS In this paper, we extend our model in [20] to the ase in whih either the forward pries or the spot pries are apped. The forward aps represent an approximation for ompetitive entry by new generators. We formulate the Cournot equilibrium in the prie-apped two-settlement markets as a stohasti equilibrium problem with equilibrium onstraints. We also onsider the spot market with unertainty of demands, generation apaities, and thermal limits of transmission lines. The main goal in this paper is the development of an effetive formulation to analyze apping alternatives under a variety of senarios in the framework of two-settlement markets. We run test ases based on the Belgian eletriity market. The resulted equilibrium reveals less inentives from the generators to ommit forward ontrats due to the spot prie aps, and more inentives due to forward prie aps. However, the spot zonal pries under both ap types still derease as ompared to the respetive single-settlement ases. Moreover, these two ap types result in different behaviors of spot prodution and spot energy pries. Additional interesting analysis onerning soial welfare, generator s profits as well as other aspets of the markets affeted by the presene of prie aps, will be the subjet of a future report. REFERENCES [1] B. Allaz, Oligopoly, Unertainty and Strategi Forward Transations, Internal Journal of Industrial Organization, vol. 10, pp. 297-308,1992 [2] B. Allaz, and J.-L. Vila, Cournot Competition, Forward Markets and Effiieny, Journal of Eonomi Theory, vol. 59, pp. 1-16, 1993 [3] S. Borenstein and J. Bushnell, An Empirial Analysis of Potential for Market Power in California s Eletriity Industry, Journal of Industrial Eonomis, vol. 47, no. 3, pp. 285-323, 1999 [4] J. Bulow, J. Geneakolos and P. Klemperer, Multimarket Oligopoly: Strategi Substitutes and Complements, Journal of Politial eonomy, vol. 93, pp. 488-511, 1985 [5] J. Cardell, C. Hitt and W. W. Hogan, Market Power and Strategi Interation in Eletriity Networks, Resoure and Energy Eonomis, vol. 19, pp. 109-137, 1997 [6] H.-P. Chao and S. C. Pek, A market mehanism for eletri power transmission, Journal of Regulatory Eonomis, vol 10, no. 1, pp. 25-60, 1996 [7] R. W. Cottle, J.S. Pang and R. E. Stone, the Linear Complementarity Problem, Aademi Press, Boston, MA, 1992 [8] N.-H. M. von der Fehr and D. Harbord, Long-term Contrats and Imperfetly Competitive Spot Markets: A Study of UK Eletriity Industry, Memorandum no. 14, Department of Eonomis, Univeristy of Oslo, Oslo, Sweden, 1992 [9] D. Fudenberg and J. Tirole, Game Theory, the MIT Press, Cambridge, MA, 1991 [10] R. J. Green, the Eletriity Contrat Market in England and Wales, Jounal of Industrial Eonomis, vol. 47, no. 1, pp.107-124, 1999 [11] B.F. Hobbs, C. B. Metzler, and J.S. Pang, Strategi Gaming Analysis for Eletri Power Systems: An MPEC Approah, IEEE Transations on Power Systems, vol 15, no. 2, 638-645, 2000 [12] B.F. Hobbs, Linear Complementarity Models of Nash-Cournot Competition in Bilateral and POOLCO Power Markets, IEEE Transations on Power Systems, vol. 16, no. 2, pp. 194-202, 2001 [13] R. Kamat, and S. S. Oren, Multi-Settlement Systems for Eletriity Markets: Zonal Aggregation under Network Unertainty and Market Power, Proeeding of the 35th Hawaii International Conferene on Systems Sienes (HICCS 35). Big Island, Hawaii, January 7-11, 2002 [14] Z.Q. Luo, J.S. Pang and D. Ralph, Mathematial Programs with Equilibrium Constraints, Cambridge University press, Cambridge, 1996 [15] D. M. Newbery, Competition, Contrats, and Entry in the Eletriity Spot Market, Rand Journal of Eonomis, vol. 29, no. 4, pp. 726-749, 1998 [16] S. S. Oren, Eonomi Ineffiieny of Passive Transmission Rights in Congested Eletriity Systems with Competitive Generation, the Energy Journal, vol. 18, pp. 63-83, 1997 [17] A. Powell, Trading Forward in an Imperfet Market: the Case of Eletriity in Britain, the Eonomi Journal, vol. 103, pp. 444-453, 1993 [18] Y. Smeers and J.-Y. Wei, Spatial Oligopolisti Eletriity Models with Cournot Firms and Opportunity Cost Transmission Pries, Center for Operations Researh and Eonometris, Universite Catholique de Louvain, Louvain-la-newve, Belgium, 1997 [19] J.-Y. Wei and Y. Smeers, Spatial Oligopolisti Eletriity Models with Cournot Firms and Regulated Transmission Pries, Operations Researh; vol. 47, no. 1, pp. 102-112, 1999 [20] J. Yao, S. S. Oren, and I. Adler, Computing Cournot Equlibria in Twosettlement Eletriity Markets with Transmission Contraints, Proeeding of the 37th Hawaii International Conferene on Systems Sienes (HICCS 37). Big Island, Hawaii, January 5-8, 2004 [21] J. Yao, S. S. Oren, and I. Adler, Two-settlement Eletriity Markets with Prie Caps: Formulation and Computation, Manusript. Department of Industrial Engineering and Operations Researh, University of California at Berkeley, Berkeley, California, February, 2004 Jian Yao is a Ph.D. andidate in the department of IEOR at the University of California at Berkeley. He was researh assistant on projet EECOMS (Extended Enterpriser oalition for integrated COllaborative Manufaturing Systems) funded by NIST ATP (Advane Tehnology Program), and software engineer for Advaned Planning & Sheduling produts at Orale Corporation. He has reeived his M.S. in Computer Siene from the University of North Carolina at Charlotte, and M.S. and B.S in Mehanial Engineering from Shanghai Jiao Tong University. Yao is a member of the INFORMS. Dr. Bert Willems reeived his Master degrees in Mehanial Engineering and in Eonomis in 1998 and 2000 from the K.U.Leuven, Belgium. In 2004, he obtained a Ph.D. in Eonomis from the same institution with a thesis on eletriity networks and generation market power. Dr. Willems worked a year at the K.U.Leuven Energy Institute, and four years at the Energy, Transportation and Environmental Eonomis researh
10 group. He is urrently a visiting researh assoiate of the University of California Energy Institute and his researh interests inlude energy eonomis, regulation, industrial eonomis, and ontrat theory. Dr. Shmuel Oren is Professor of IEOR at the Univ. of California, Berkeley. He is the Berkeley site diretor of PSERC (the Power System Engineering Researh Center). He published numerous artiles on aspets of eletriity market design and has been a onsultant to various private and government organizations inluding the Brazilian regulatory ommission, The Alberta Energy Utility Board the Publi Utility Commission, the Polish system operator and to the Publi Utility Commission of Texas were he is urrently a Senior Advisor to the Market Oversight Division. He holds a B.S. and M.S. in Mehanial Engineering and in Materials Engineering from the Tehnion in Israel and he reeived an MS. and Ph.D in Engineering Eonomi Systems in 1972 from Stanford. Dr. Oren is a Fellow of the INFORMS and the IEEE. Dr. Ilan Adler is a Professor and Chair of the IEOR department at the University of California at Berkeley. His main researh ativities are in the the area of optimization in general and Interior Point methods in partiular. He has also been a onsultant to various ompanies on large, omplex optimization problems. He holds a B.A. in Eonomis and Statistis from the Hebrew University in Israel, M.S. in Operations Researh from the Tehnion in Israel and a Ph.D in Operations Researh from Stanford. Dr. Adler is a member of INFORMS and the Mathematial Assoiation of Ameria.