An Oligopolistic Model of an Integrated Market for Energy and Spinning Reserve

Transcription

1 132 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006 An Oligopolistic Model of an Integrated Market for Energy and Spinning Reserve Guillermo Bautista, Student Member, IEEE, Víctor H. Quintana, Fellow, IEEE, and José A. Aguado, Member, IEEE Abstract In this paper, a model for oligopolistic competition in electricity markets is presented. Most previous proposed models have been static and focused only on the energy market incentives for strategic behavior. In contrast, in this paper, a multiperiod market for energy and spinning reserve (SR) is considered. By including such factors, the competition among participants is modeled with more realism. Competition in the energy market is modeled by means of conjectured supply functions, while conjectured reserve-price response functions are used to consider the generators ability to alter the SR prices. The resulting equilibrium problem is modeled in terms of complementarity conditions. Based upon a complementarity model, the opportunity cost between the energy and SR markets is derived for oligopolistic markets. The proposed model is illustrated by a six-node network using a dc approximation. Index Terms Complementarity, conjectured function, energy market, oligopoly, spinning reserve (SR). NOTATION The following notation is introduced as preliminary to the description of the model presented in this paper. Acronyms FTR Financial transmission right. GAMS General algebraic modeling system. GenCo Generation company. ISO Independent system operator. LMP Locational marginal pricing/price. MLCP Mixed linear complementarity problem. KKT Karush Kuhn Tucker. SR Spinning reserve. Indices and Sets Indices for generation units. Indices for nodes in the system. Index for transmission lines in the system. Index for time period. Index for GenCos. Set of generation units. Set of nodes. Set of transmission lines. Set of trading periods. Set of GenCos. Constants Reserve-price response conjectured by [($/MWh)/MW]. Rivals-supply response conjectured by at [MW/($/MWh)]. Quantity intercept for demand curve at (MW). Total energy supplied by unit (MWh). Distribution factor for transmission line with respect to node (p.u.). Parameter of the linear SR cost function of unit ($/MWh). Parameter of the quadratic energy cost function of unit ($/MWh). Parameter of the quadratic energy cost function of unit ($/MW h). Price intercept for demand curve at node ($/MWh). Variables Power arbitraged at (MW). Power sold by at (MW). Power produced by unit, owned by and placed at (MW). SR from unit (MW). Power injection at (MW). Real power flow through (MW). Profit for given by ($/h). Energy price at in hour ($/MWh). Energy price seen by at ($/MWh). Energy price at the slack node ($/MWh). SR price in ($/MWh). SR price seen by in ($/MWh). Congestion price at ($/MWh). Up-ramp rate for unit (MW/h). Down-ramp rate for unit (MW/h). Manuscript received August 4, 2004; revised May 25, This work was supported in part by CONACYT, Mexico, and in part by NSERC, Canada. Paper no. TPWRS G. Bautista and V. H. Quintana are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada ( bautista@kingcong.uwaterloo.ca; quintana@kingcong.uwaterloo.ca). J. A. Aguado is with the Department of Electrical Engineering, University of Málaga, Málaga E-29013, Spain ( jaguado@uma.es). Digital Object Identifier /TPWRS Symbols Maximum and minimum value for. Value for in equilibrium. Cardinality for the finite set. Complementarity condition between and. equals. equals. equals /$ IEEE

2 BAUTISTA et al.: OLIGOPOLISTIC MODEL OF AN INTEGRATED MARKET 133 I. INTRODUCTION THE implementation of electricity markets requires not only a market for energy but also markets for other goods and services, such as transmission and ancillary services. In an integrated market, different products can be simultaneously priced and procured. The advantages of using an integrated market, based on locational marginal prices (LMPs), can be found in [1] [3]. Locational marginal prices are based upon the theory of spot pricing [4]; they reflect the locational value of energy, which depends not only on the generation cost but also on the transmission system characteristics and demand willingness to pay. Within a market environment, one of the main concerns is about the ability of some market participants to behave strategically. An alternative for modeling imperfect competition is the one based upon market equilibrium conditions [5] [12]. An equilibrium model can be defined by a set of optimization problems, one per market participant suppliers, consumers and market operator which relates prices, generations, demands, and power flows to satisfy every market participant s first-order optimality conditions, plus a coordination condition to clear the market, i.e., to match supply and demand of goods and services. If a market solution exists and satisfies such requirements, then no market participant will unilaterally alter its current position, a Nash equilibrium [9]. However, nonconvexities within the mathematical models can result in the lack of uniqueness or even existence of equilibrium. For instance, nonconvexities arise if it is considered that market participants recognize their impact on the transmission constraints. If such an impact is disregarded, or modeled by means of conjectured functions, the equilibrium problem can be defined as a mixed linear complementarity problem (MLCP) [10], [11]. For this kind of formulation, a global (and even a strict) equilibrium can be guaranteed [10], [13]. However, most models on imperfect competition have been based upon static models, i.e., they ignore temporal constraints such as up- and down-ramp rate limits. Hence, this kind of model may give unrealistic market outcomes, thus misleading conclusions of market power [14], [15]. In [16], a model for strategic behavior is proposed, where first-order optimality conditions of firms are introduced as constraints into a cost production problem; this model does not take into account transmission constraints. In [17], a multiperiod model for the oligopolistic energy-only market is presented. Traditionally, the concern of market power has been focused on the strategic behavior of generators within an energy-only market. However, within electricity markets, generation companies (GenCos) procure not only energy; indeed, GenCos have incentives from other market activities. For instance, a GenCo can have profits from participating in both the energy and spinning reserve (SR) markets. Due to the fact that the levels of energy and SR that a generation unit can provide are limited by the maximum capacity of the unit, a GenCo has to define the optimal levels of both of them simultaneously. On the side of the market, if the energy and SR markets are simultaneously cleared, then there can be an interaction between energy and SR; prices in one market will affect prices in the other. SR (which can be considered the most expensive and critical reserve [1]) may motivate generators to behave differently within the energy market due to the opportunity cost between producing and spinning. Sometimes a GenCo can shift power from one market to the other as it is more profitable. As this shift of power can affect the generation schedule, it may impact the energy market efficiency. For instance, if a cheap generation unit decreases its generation due to an SR incentive, more expensive generation will have to be used. Although the interaction energy-sr is well known in competitive markets, few works have addressed such an interaction within an oligopolistic market. In [18], oligopolistic GenCos are considered in separate energy and SR markets; in these markets, the transmission system is not included. In [19], a one-period Cournot model is used, with a transportation-network-like transmission system, to show the interaction of energy and ancillary services. In [20], oligopolistic competition is modeled in the energy market, while competitive competition is considered in the SR market; the transmission system is modeled by means of a dc approximation. Another interaction may occur between energy and pollution-permit markets. The usage of such permits to exacerbate GenCos market power has been studied for the California [21] and for the Pennsylvania New Jersey Maryland (PJM) [22] markets. Furthermore, the introduction of hedging instruments, such as financial transmission rights (FTRs), may add more room for market power. The FTR impact on market participants behavior has been analyzed in [5] and [23] [28]. The contribution of this paper to the analysis of imperfect competition is the derivation of an equilibrium model for an integrated market of energy and SR, where GenCos can influence the energy and reserve prices. The opportunity cost between the energy and spinning reserve markets is derived for an oligopolistic market. The model also includes temporal constraints; however, commitment decisions, such as those arising from modeling startup cost, are not considered into the model since it would require the use of binary variables [29]. This paper is organized as follows. In Section II, the proposed model for oligopolistic competition is described. Numerical examples are shown in Sections III and IV. Conclusion in Section V closes the paper. II. PROPOSED MODEL In this section, a spatial and dynamic oligopoly model for an electricity market is formulated. The proposed model takes into account the influence that GenCos can exert upon the energy and SR prices, within an integrated market [20]. We follow and extend the model proposed by Hobbs et al. [10] to define the oligopoly model as an equilibrium problem. A. GenCo Model For a GenCo participating in an integrated market for energy and SR, its objective is to maximize profits from both activities. Moreover, within an oligopolistic market, a GenCo also has to consider its rivals decisions within its optimization problem. Its decision variables are the sales, generation, and SR levels. Thus,

3 134 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006 can be mathemat- the profit-maximization problem for GenCo ically stated as follows: to the strategy of, e.g., how expects that changes in the nodal price alter the level of power supplied by its rivals s.t. The maximization of profit is subject to the supply and market constraints specifications and. Each component of problem (1) is next described. 1) Profit Components: The profit from participating in the energy market is given by (1) For the standard Cournot outcome. The higher the parameter, the stronger the rivals response when attempts to jack up the price and, consequently, the more competitive the market. For a competitive outcome,. On the other hand, the price seen by at is given by the inverse demand function (5) (6) By introducing (5) into (6), the price function becomes The first term stands for the revenue from selling power throughout the system s nodes at the corresponding locational price. The second term is the cost for supplying power from all the generation units. The generation cost is denoted by ; hence, the marginal cost is. Both revenues and costs include the congestion cost, by means of, to move power among the system nodes. A second money stream may come from participating in the SR market; such profit is represented by the difference between the revenue from selling reserve at the market price and the cost of providing such reserve, i.e., In this model, as the SR price is seen as an endogenous variable, will adjust its SR levels based upon its belief of how it can alter the SR price see (8) below. 2) Constraints: The firm decisions are subject to different operational constraints, such as the following. Power balance. This is an auxiliary constraint used within the model to have a GenCo produce, by its own, all the energy it has to supply, i.e., this constraint avoids arbitrage from Conjectured supply function [10]. Let the demand at node be satisfied by both GenCos and independent system operator (ISO) (by means of the arbitraged power ), i.e.,. As the power supplied by GenCos is composed by the own power of,, and its rivals power, the demand at node is. Under the pure Cournot strategy, GenCo considers that rivals response is fixed. In a step further, a conjecture function can be used to model the rivals response (2) (3) (4) Conjectured reserve-price function. For a given period, let be the net spinning reserve to be provided by GenCo ; thus, a conjectured reserve-price function can be defined as The conjectured function (8) allows one to model the ability of to influence the SR prices, i.e., how expects the SR price varies if modifies its provision of reserves. The parameter can be varied to see the potential impact of different degree of price manipulation by ; negative values of can be used to see a potential increase in the SR price. For the standard competitive market outcome,. The conjectured reserve-price function is an analogy to what has been used for energy markets (conjectured supply function [10]), for transmission prices manipulation (conjectured transmission price response [11]), and for pollution permits markets (conjectured price function [22]). Both conjectured supply and reserve-price response functions are used to substitute and into the objective function of GenCo ; hence, (5) and (8) are not explicitly used in the model as constraints, and no dual variables are associated to them. Generation, sales, and SR limits. These constraints follow the classical limits of generation capacit, and also a maximum limit to provide SR (7) (8) (9) (10) (11) (12) Up- and down-ramp rates. These constraints represent the physical limitations of the thermal units to increase/

4 BAUTISTA et al.: OLIGOPOLISTIC MODEL OF AN INTEGRATED MARKET 135 decrease their output levels. Such constraints couple together every period with the previous and following ones: market price of energy ; this constraint matches supply and demand (13) (14) Maximum energy supply in the trading horizon. This is to consider that some generators may be limited due to other constraints such as fuel or total emission limits (15) In this paper, hydroelectric generators are not considered. B. ISO In the proposed equilibrium model, an independent entity, say, an ISO, operates the transmission system to efficiently allocate the transmission. The congestion prices are considered exogenous variables within the ISO problem in order to have the ISO behave competitively. This entity also carries out spatial arbitrage of power in order to eliminate any price difference that is beyond transmission costs. Consequently, the arbitrage leads the transmission price between two nodes be defined as the price difference between such nodes [8], i.e., to obtain LMPs. The ISOs objective is given by (16) Furthermore, the ISO s decisions have to be feasible for the transmission system constraints, i.e., (17) Expression (17) stands for the dc power-flow equations 1 in terms of the distribution factors. A market requirement is also that the ISO is only an arbitrager, i.e., (18) Within the ISO problem, the variables and are both unconstrained. In order to derive the Karush Kuhn Tucker (KKT) conditions for the ISO problem, the dual variables, and are associated with constraints (17) and (18), respectively. C. Market-Clearing Conditions In addition to the GenCo and ISO problems, there need to be a set of conditions for clearing the markets. These conditions will balance demand and supply of energy, transmission, and SR; they are described next. 1) Energy Services: This condition establishes that the price assumed by every market participant is the actual 1 A transmission line is modeled by means of two power flows constraints, one constraint per power-flow direction, denoted as z ;z. (19) 2) Transmission Services: This condition is related to the efficient rationing of transmission services, such that the nodal power balance is preserved at every system node, i.e., it matches supply and demand of transmission (20) 3) Reserve Services: This condition establishes that the SR requirement of the system must be satisfied (21) The SR,, can be defined based upon the ISO demand forecast and other operating conditions [3]. The SR can be set equal to some percent of the demand, as in the Spanish system [30], or equal to the largest loss of power due to a single (generator or line) contingency, as in the Ontario system [31]. SR can also be provided by demands that can decrease their consumption [32], [33]. In this paper, the requirement of SR is defined as a percent of the demand; it can be modeled as a given value or can be implicity computed within the equilibrium model. For the latter, one has (22) D. Equilibrium Model As each GenCo and ISO problem can be defined via KKT optimality conditions, an equilibrium will be a point that simultaneously satisfies their first-order optimality conditions and clears the markets. Furthermore, as the KKT conditions of each GenCo and ISO problem are given by affine functions (see the Appendix), it follows that each set of KKT conditions compose an MLCP [13]. This set of MLCPs together with the marketclearing conditions can be written as a single MLCP. : : free : (23) (24) (25)

5 136 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006 : free (26) : : (27) (28) (29) has been formulated in GAMS and solved using the PATH solver [34]. E. Energy and SR Interaction When a generation unit is constrained by its maximum capacity limit, there can be a tradeoff between energy and SR. This opportunity cost can be derived from the equilibrium conditions in Section II-D. Taking into account only the terms related to energy and SR for a static case, by complementarity conditions, (23), (24), and (25), respectively, become (41) (42) : (30) (31) (32) (43) Without any loss of generality, assume that generation unit is not constrained by its minimum generation limit, such that. By substituting (41) and (42) into (43), the market price for SR is given by free (33) (34) (35) free (36) free (37) free (38) (44) Notice that in equilibrium,. The first term in brackets is the marginal cost of SR affected by the ability of to manipulate the SR price, while the second term in brackets represents the opportunity cost for supplying SR instead of energy. This term is composed by the price difference between the nodal price where unit is placed and the marginal cost of this unit and the impact that has on the price by means of its sales at. Only when the generation unit has reached its SR limit, the term may count. For the Cournot outcome, the spinning reserve price is (45) free (39) On the other side, for competitive energy and SR markets, one gets (40) Simultaneously solving (23) (40) for the primal and dual variables, with prices, and, an equilibrium point of the multiperiod market is obtained. In this paper, the MLCP (46) In this case, the opportunity cost is given by the differential between the corresponding LMP and the energy marginal cost. A

6 BAUTISTA et al.: OLIGOPOLISTIC MODEL OF AN INTEGRATED MARKET 137 TABLE II ENERGY AND SR INTERACTION. COMPETITIVE CASE Fig. 1. Six-node system. TABLE I SIX-NODE SYSTEM DATA similar relationship has been derived in [3] for a competitive energy market. The terms related to ramp-rates and energy constraints can also be included to analyze their impact on the opportunity cost. III. A SIX-NODE SYSTEM Let us consider the six-node system shown in Fig. 1. The related data are given in Table I. The network is modeled as lossless and with equal transmission line reactances. Transmission limits for lines 1 6 and 2 5 are 200 MW; these lines are labeled as 1 and 2, respectively. The limits of the others transmission lines are large enough to be neglected. All generation units have unlimited capacity (otherwise specified); generation units and demands are accordingly labeled as shown in Fig. 1. A. Competitive Energy Market Let us consider a static case for an integrated energy and SR market. The energy market is considered to be competitive through all the cases; meanwhile, the SR market is considered as competitive for all cases but Case D. The reserve requirement is set to be 10% of the total demand. The market outcomes are summarized in Table II. 1) Case A: No Generation and SR Limits: Under competitive conditions in the energy market, all generation units are providing energy. Since the power flow limit of transmission line 1 becomes active, locational price discrimination arises, and the ISO collects congestion rents of $4800/h. In this case, all generation units are marginal, i.e., the nodal prices where they are placed equal their corresponding marginal cost. Due to the fact that no generator is constrained by its maximum capacity limit, the introduction of the SR market does not modify the energy-only market outcome. The SR requirement of MW is to be provided by the cheapest generator at a price of $4/MWh. 2) Case B: No SR Limits and MW: As is the cheapest supplier of both energy and SR, its full capacity is used. If there were no opportunity cost between energy and SR, would supply 672 MW for the energy market (as in Case A), and its remaining capacity (77.92 MW) would be to provide reserves. Nonetheless, as such an opportunity costs exists, provides more SRs ( MW) by reducing the power to be supplied in the energy market up to MW. This shift of power is actually due to an incentive of the SR market, not an exercise of market power. Moreover, this decrease in generation causes a different market outcome (see prices and

7 138 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006 demand levels). Because less cheap power is produced, more power flows through line 2, resulting in less congestion in line 1. GenCos surplus increases mainly by the higher prices at which GenCos 1 and 2 now sell; in contrast, the loads surplus is reduced. Although the net surplus increases, the social welfare decreases since less congestion rents (welfare transfer from demands and GenCos to the ISO) are collected. In addition, sets a higher SR price of $5.84/MWh. This price is composed by s SR marginal cost ($4/MWh) plus an opportunity cost of $1.84/MWh. Such an opportunity cost is simply the difference between the corresponding price at node 1 ($15.46/MWh) and the energy marginal cost ($13.61/MWh) of see (46). Since the SR price is above the SR marginal cost of, this GenCo has a profit of $270.2/h by providing reserves. 3) Case C: MW and MW: Let us now consider that, besides MW, has also a maximum SR limit. If, it will alter the energy market outcome. 2 Taking into account this fact, consider an SR limit of, say, 100 MW. In this case, now provides 650 MW for energy and 100 MW for SR. To satisfy the SR requirement, now provides MW and sets the SR market price at $6/MWh. The introduction of the SR limit curbs the decrease (motivated from the interaction of the energy and reserve markets) in generation of, which produces 650 MW rather than MW. As more cheap generation is used (in comparison to Case B), more congestion occurs in line 1; however, the market is better off as the welfare increases. Consequently, the incentive from the SR market will impact less severely on the energy market. 4) Case D: Market Power in the SR Market: In this case, GenCos can manipulate the SR market price; this is included by means of the conjectured reserve-price function described in Section II-A2. Assume that all GenCos behave strategically. The market outcome is computed by including both energy and SR limits for such that Case C is the benchmark. The best strategy for is to shift power ( MW) from the SR market to the energy market. Such an increase of generation makes decrease its generation (leading to lower energy prices at nodes 1 3) and increase its generation. As more cheap power from is produced, transmission line 1 becomes more congested, increasing the congestion rents by $169.3/h. The s reduction of SR is mainly compensated by, which leads to a higher SR price ($8.1/MWh). On the other hand, this manipulation in the SR market has been not profitable for. This GenCo is now obtaining a higher profit from the SR market; nonetheless, such a profit is not enough to compensate the lost profit from the energy market. 3 The other GenCos have seen a net profit increase. 2 If an SR limit r 2 [0; 77:92] MW is chosen, the energy market outcome will be as that of Case A. This is because r would be lower than the SR level that g has available ( MW). Then the SR that g provides would be such a chosen SR limit. On the other hand, if an SR limit r 2 [146:68; 1) is chosen, the energy market outcome will be as that of Case B. This is because r would be higher than the optimal SR level of g. Then the SR that g provides would be MW. 3 This fact is due to g s position in the SR market. This generator is the cheapest unit to provide SR and cannot supply all of it; hence, this generator is not the marginal unit for SR, and its strategies are limited by the other GenCos strategies. TABLE III ENERGY AND SR INTERACTION. OLIGOPOLISTIC CASE B. Oligopolistic Energy Market In this section, all generators are considered to compete à la Cournot in the energy market, while the SR market remains competitive for all cases but Case D. The interaction becomes more complex for the oligopolistic market, although the logic is similar to the competitive case. The market outcomes comparison is presented in Table III. 1) Case A: No Generation Limits: In comparison to the competitive case, there is a reduction of generation from all GenCos that jacks up the market prices. Due to higher energy prices, less demand is served; this causes not only less congestion in the system 4 but also a lower requirement of spinning reserves (from to MW). Although GenCos produce less power, they earn higher profits since they are charging much higher energy prices. The strategic behavior increases the GenCos surplus by 365%, while the demands surplus is reduced by 34%; this represents a large welfare transfer from consumers to GenCos. In comparison to the competitive case, the net welfare decreases from $ /h to $ /h. On the other hand, as no generator is constrained by a maximum capacity, the SR market does not affect the energy market outcome; then, by the complementarity principle,, 4 The dual variable for the transmission line 1 decreases from $24/MWh to $9.8/MWh, shrinking the congestion rents by 58.8%.

8 BAUTISTA et al.: OLIGOPOLISTIC MODEL OF AN INTEGRATED MARKET 139 and there is no opportunity cost between producing and spinning. The SR price is set again by at $4/MWh. 2) Case B: No SR Limits and MW: For this case, assume has a maximum generation limit of, say, 550 MW; because of this constraint, will not be able to provide all the SR, as in Case A. Due to the opportunity cost, reduces generation (from to MW) in order to provide a larger amount of SR (77.6 MW instead of MW). Now, provides the remaining amount of SR (37.6 MW) and becomes the marginal unit for it, setting the SR market price at $6/MWh. The SR price $6/MWh is composed by the SR marginal cost of ( $4/MWh) and the opportunity cost between energy and SR ($2/MWh). This opportunity cost is not only defined by the difference between the locational price at node 2 and s marginal cost (like in the competitive case) but also by the price that sets upon its power sales at node 1 ( $15.74/MWh) see (45). Generator is producing less power than it would be in the case of an energy-only market; such a reduction causes an increase of the energy prices at nodes 1 3. This further increase in prices is due to the incentives from the SR market; nonetheless, such an incentive also depends upon the s ability to influence the energy market. 3) Case C: MW and MW: Consider now that has also a maximum SR limit of 70 MW. Due to this limit, is constrained to provide 70 MW for SR, while its remaining capacity (480 MW) is used for the energy market. The SR limit makes increase its generation from MW (Case B) to 480 MW; such an increase leads lower energy prices at nodes 1 3 but higher prices at nodes 4 6. As more (cheap) power from is used to supply the demands, transmission line 1 becomes more congested, increasing the congestion rents by 9.7%. On the other hand, the SR price, from the point of view of,is now composed by its SR marginal cost ( 4/MWh), the opportunity cost ($1.32/MWh), and the dual variable of the SR limit $0.67/MWh). 4) Case D: Market Power in the SR Market: Consider Case C of this section and assume now that all GenCos behave strategically in the SR market. The optimal strategy of and is to reduce the provision of SR. Such reductions of SR make provide 16.5 MW of SR at price of $7.82/MWh. For, the SR price is composed by its SR marginal cost ( $4/MWh), the SR price manipulation ($3.14/MWh), and the opportunity cost between the provision of energy and SR ($0.68/MWh). Such an opportunity cost is reduced (from $1.32/MWh to $0.68/MWh) because is shifting power from the SR market to the energy market. As its strategic behavior in the SR market increases, the opportunity cost reduces to zero. Hence, at some point, will get the energy market outcome from Case A (this is equivalent to having no limits in generation and SR) as its manipulation of the SR market will no longer have an effect on the energy market. That is, depending on its degree of strategic behavior in the SR market, the generation level of can fall between 480 MW (competitive in the SR market) and MW (highly strategic in the SR market). This is a tradeoff Fig. 2. Demand profile. TABLE IV RAMP-RATE LIMITS EFFECT ON THE MARKET OUTCOME. STATIC VERSUS DYNAMIC CASE between strategic behavior in the SR market and the incentive of the opportunity cost between the energy and SR markets. C. Multiperiod Market In another simulation, consider an extension for 24 trading periods of the test system described above. The demand profile is as shown in Fig. 2; these demands are defined with a reference price of $20/MWh and a demand function slope of 0.2. Upper generation limits are MW, MW, and MW, while all generating units are assumed to have up- and down-ramp limits of 50 MW/h. Through all cases, assume GenCos behave strategically in both the energy and SR markets, with. The main results are presented in Table IV. In the comparisons, only the mean of the nodal prices are shown. 1) Case A: No Ramp-Rate Limits: For the sake of comparison, a market outcome is obtained by relaxing the ramp-rate

9 140 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006 Fig. 3. Fig. 4. GenCos profits. No ramp-rate limits. GenCos profits considering ramp-rate limits. limits; this is equivalent to having 24 independent and static market outcomes. The optimal solution for the whole trading horizon is the set of all the static (hourly) optima. When ramprate limits are neglected, generation units can be scheduled to any level from one period to another, and hence, any unit can avoid low prices before incurring generation losses. 2) Case B: Inclusion of Ramp-Rate Limits: As every period is linked with the previous and subsequent ones, the optimal solution for every period may not be the same as the static optimal from Case A (see Figs. 3 and 4). This fact becomes evident in the periods around the peak. For instance, the generation of becomes limited up and down by the ramp-rate limits. On one side, generation levels will be lower (in comparison to those of Case A) for the up and peak periods (16 20), leading to higher energy prices and higher profits. On the other side, generation levels are higher for the down periods (21 24), leading to lower prices and profits. However, the money collected from the up periods is much higher than the money lost from the down periods; consequently, sees a net increase in profit. The decrease in generation (in comparison to those from Case A) in up and peak hours arises from intertemporal constraints. As expected, misleading conclusions on market power could be derived from an analysis built upon a static model [15]. In the case of, the ramp-rate limits make it follow a different strategy. For the up periods, has no changes in generation, and it sells the same amount of power at higher prices, earning higher profits. For the down periods, reduces generation in order to compensate the s effect on prices, selling less power to avoid lower prices. Therefore, cheap power is substituted by more expensive generation. This logic is more vivid for ; since this generator is the most expensive, changes in its generation have a stronger impact on prices. In peak periods, as finds it very profitable to produce, it is at high generation levels. However, in subsequent periods, the demand becomes low and so do prices; then this generator cannot freely ramp down to be scheduled at static optima generation levels, and it has to produce at prices below its marginal costs, incurring losses. Nonetheless, the high profits earned from peak periods offset the losses from down periods, resulting in a net increase of profits for. Furthermore, the change in generation also impacts on the SR level to be provided by each GenCo. In periods 15 20, less demand, and, hence, less SR, is required; the reduction in SR is from and. In periods 21 24, there is an increase in demand and, thus, in SR; such an increase is provided by. There is also an extra shift of SR; due to ramp-rate constraints, reduces generation and now has more capacity for reserves (this generator has cheaper SR); thus, will substitute SR from, leading to a lower SR price in these periods. In the remaining periods, there is no change in the profile of the SR provision. This results in a net increase of SR profit for and a decrease for the other GenCos. 3) Case C: MWh: Let us consider Case B, but is now limited by the maximum energy that it can provide in the trading horizon. From Case B, the net energy supplied by is MWh; to see how the energy limit affects its strategies, assume an energy limit of, say, MWh. Due to this constraint, has to reduce throughout the horizon a net amount of 499 MWh. For periods 17 20, with the highest prices and also the most profitable ones, there is no change in generation. For most remaining periods, the trend is to have a larger reduction where both generation levels and prices are higher; a generation decrease where prices are already high will lead to higher prices and, hence, higher profits. The extra available capacity of is now used for providing SR; therefore, cheaper prices (except for periods 17 20) of SR are obtained. On the other hand, more expensive power from has to be used to compensate for the reductions of, leading higher prices at nodes 1 3 (except at periods 17 20). This is followed by a decrease in generation by in order to avoid selling power at a lower price. The inclusion of more constraints may lead to complicated interactions within the market. Although more constraints are considered into the system, the GenCos profits becomes higher;

10 BAUTISTA et al.: OLIGOPOLISTIC MODEL OF AN INTEGRATED MARKET 141 this counterintuitive result comes from the fact that the cheapest generation is more limited; hence, power from more expensive units is used, resulting in higher prices and profits. The decrease of cheap power also causes less congestion in the system. On the other hand, the inclusion of more constraints results in a decrease of the social welfare. The simulations presented in this paper have been performed using GAMS on a Pentium IV 1.6-GHz PC. The maximum time required to find an equilibrium of the static market cases was 0.01 s, while for the dynamic market cases, the maximum time required was 2.35 s. The proposed model has been also implemented with the standard IEEE-based test power system of 57 nodes and 80 transmission lines, for 24 trading periods. For this system, the time required to find an equilibrium was 25 min. A slightly different version of this case study with the inclusion of FTRs can be found in [20]. IV. CONCLUSION A model to analyze imperfect competition in an integrated market for energy and SR has been presented. Based upon complementarity conditions, a general expression of the opportunity cost between both markets has been derived. Such a derivation allows one to identify the components of the opportunity cost between generating and spinning within a range of strategies, going from the Cournot setting to the competitive one, and also to identify the impact of manipulation from the SR market. In addition, the effect of intertemporal constraints and energy constraints over both markets has been studied. It has been shown that even a competitive SR market may have an effect on the energy market efficiency. For instance, when a capacity-limited generation unit procures both energy and SR, such a unit shifts power from the energy market to the SR market. Due to this fact, more expensive energy has to be used, with a decrease of the social welfare. However, if this unit attempts to increase the SR price, it has to reduce its SR capacity. This fact can be an opposite incentive to that of the opportunity cost, as now such a unit would have more capacity available for the energy market. On the other hand, the inclusion of intertemporal and energy constraints may result in different market outcomes from what could be computed with static and simpler oligopolistic models. The proposed temporal and integrated market for energy and SRs can be useful to gain insights into more realistic conditions that GenCos face in an electricity market. APPENDIX COMPLEMENTARITY CONDITIONS The Lagrangian for the problem (2) (15) can be written as $ (47) (48) (49) free (50) where are the corresponding dual variables associated with the GenCos constraints described supra in Section II-A. For illustrative purposes, let us consider only the KKT conditions with respect to the sales variables, i.e., where $ (51) (52) (53) (54) Because of the non-negativity requirement on, its corresponding KKT equations are defined by complementarity conditions. In order to compactly denote such conditions, the symbol is used throughout this paper. Thus, the first-order optimality conditions (51) (53) can be casted as follows. For (55) which gives (23). The same notation is used for the KKT conditions of the other nonnegative primal and dual variables. REFERENCES [1] S. Stoft, Power System Economics: Designing Markets for Electricity. New York: Wiley-Interscience, [2] R. P. O Neill, U. Helman, B. Hobbs, W. R. Stewart, and M. Rothkopf, A joint energy and transmission rights auction: Proposal and properties, IEEE Trans. Power Syst., vol. 17, no. 4, pp , Nov [3] T. Wu, M. Rothleder, Z. Alaywan, and A. D. Papalexopoulos, Pricing energy and ancillary services in integrated market systems by an optimal power flow, IEEE Trans. Power Syst., vol. 19, no. 1, pp , Feb [4] F. C. Schweppe, M. C. Caramanis, and R. D. Tabors, Spot Pricing of Electricity. Norwell, MA: Kluwer, [5] J. B. Cardell, C. C. Hitt, and W. W. Hogan, Market power model and strategic interaction in electricity networks, Res. Energy Econ., vol. 19, pp , [6] W. J. Yuan and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., vol. 47, no. 1, pp , Jan. Feb

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(hons.) degree in 2001 in electrical engineering, majoring in power systems, from the Instituto Tecnológico de Oaxaca, Oaxaca, Mexico, and Instituto Politécnico Nacional, Mexico City, Mexico, respectively. Currently, he is working toward the Ph.D. degree at University of Waterloo, Waterloo, ON, Canada. His main research interests are optimization applications for the operation and control of power systems and markets. Victor H. Quintana (F 01) received the Dipl. Ing. degree from the State Technical University of Chile, Santiago, Chile, in 1959 and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Wisconsin, Madison, in 1965 and the University of Toronto, Toronto, ON, Canada, in He has lectured and given seminars and/or short courses in several countries around the world, among them, Australia, Brazil, Belgium, Canada, Chile, China, Colombia, Egypt, Mexico, Trinidad and Tobago, Spain, and the U.S. Since 1973, he has been with the University of Waterloo, Waterloo, ON, Canada, Department of Electrical and Computer Engineering; currently, he is an Emeritus Professor. His main research interests are in the areas of numerical optimization techniques, state estimation, control theory, and deregulated power systems. José A. Aguado (M 01) received the Ingeniero Eléctrico and Ph.D. degrees from the University of Málaga, Málaga, Spain, in 1997 and 2001, respectively. Currently, he is an Associate Professor in the Department of Electrical Engineering, University of Málaga. He was a Visiting Researcher at the University of Waterloo, Waterloo, ON, Canada. His research interests include operation, planning, and deregulation of electric energy systems and numerical optimization techniques.