Finding Multiple Nash Equilibria in Pool-Based Markets: A Stochastic EPEC Approach David Pozo, Student Member, IEEE, and Javier Contreras, Senior Member, IEEE

Abstract—We present a compact formulation to find all pure Nash equilibria in a pool-based electricity market with stochastic demands. The equilibrium model is formulated as a stochastic equilibrium problem subject to equilibrium constraints (EPEC). The problem is based on a Stackelberg game where the generating companies (GENCOs) optimize their strategic bids anticipating the solution of the independent system operator (ISO) market clearing. A finite strategy approach both in prices and quantities is applied to transform the nonlinear and nonconvex set of Nash inequalities into a mixed integer linear problem (MILP). A procedure to find all Nash equilibria is developed by generating “holes” that are added as linear constraints to the feasibility region. The result of the problem is the set of all pure Nash equilibria and the market clearing prices and assigned energies by the ISO. A case study illustrates the methodology and proper conclusions are reached.

Minimum offer price of the th block of the th generating unit in period . Maximum offer price of the th block of the th generating unit in period . Minimum offer quantity of the th block of the th generating unit in period . Maximum offer quantity of the th block of the th generating unit in period . Discretization gap of the offer price for the th block of the th generating unit in period . Discretization gap of the offer quantity for the th block of the th generating unit in period .

Index Terms—Bilevel programming, equilibrium problems with equilibrium constraints (EPEC), pool-based electricity market, pure Nash equilibrium.

Number of offer price intervals for the th block of the th generating unit in period .


Number of offer quantity intervals for the th block of the th generating unit in period .

The mathematical symbols used throughout this paper are classified below as follows. A. Indexes

Marginal cost of the th block of the th generating unit. Offer blocks.

Maximum power generation of the th generating unit.

Generating unit. Time index.

Fixed offer price of the th block by the th generating unit in period .

Generating company.

Fixed offer quantity of the th block by the th generating unit in period .

Discrete strategy for generating company . Scenario.

Probability of scenario .

Index of Nash equilibria found. B. Constants ,

C. Positive Variables Sufficiently large constants. Demand in period and scenario .

Manuscript received July 28, 2010; revised November 02, 2010; accepted December 02, 2010. Date of publication January 13, 2011; date of current version July 22, 2011. This work was supported in part by the Junta de Comunidades de Castilla-La Mancha Formación del Personal Investigador (FPI) grant 402/09. Paper no. TPWRS-00611-2010. The authors are with the Escuela Técnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain (e-mail: David. [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRS.2010.2098425 0885-8950/$26.00 © 2011 IEEE

Offer price of the th block by the th generating unit in period . Offer quantity of the th block by the th generating unit in period . Vector of price-quantity strategies available to the th GENCO. Power assigned by the ISO to the th block of the th generating unit in period and scenario .


Shadow price of the constraint of the power assigned to the th block of the th generating unit in period and scenario . product. product. D. Free Variables Market clearing price (MCP) in period and scenario . E. Binary Variables th binary variable of the discretization of the continuous variable . th binary variable of the discretization of . the continuous variable F. Superscripts Fixed offer by the competitors of the th GENCO. Variable in the equilibrium. Variable related to the th generating company th offer strategy. choosing the Variable related with the pure Nash equilibria found. G. Sets Set of all offer blocks. Set of all periods. Set of all generation units. Set of all generating companies. Set of all discrete strategies of the th generating company. Set of all scenarios. H. Functions


A market equilibrium requires optimal bidding by the generation companies (GENCOs) seeking for the maximum profit based on their costs and the expectations about the other competitors. Bid optimization can be solved using bilevel models [3], [4] that propose a leader-follower scheme (Stackelberg games). An application of a bilevel model to electricity markets is presented in [5], where a binary expansion approach for the strategic offer problem is shown. In addition, [6] and [7] present bilevel models that assume step-wise optimal strategic bidding considering demand uncertainties or generation offer uncertainties. In equilibrium problems with equilibrium constraints (EPEC) [8], the GENCOs optimize their profits simultaneously subject to coupling equilibrium constraints [9]. EPECs arise when single GENCOs face utility maximization problems in the form of mathematical programs with equilibrium constraints (MPEC). In this regard, transmission constraints and market power are analyzed in [10] under an MPEC setting. A similar model considering network constraints uses MILP with disjunctive constraints and linearization [11]. A bilevel noncooperative model with locational marginal prices and line constraints is proposed in [12] as part of an EPEC, where conditions for the existence of Nash equilibria are examined. Game theory models can capture the strategic behavior of market agents in order to find Nash-Cournot equilibria. In [13], a Nash-Cournot equilibrium of a three-bus system is solved efficiently with a linear complementarity problem (LCP) formulation. Other schemes use an iterative process [14], considering that the offers of all agents are known per iteration and solving the profit maximization for each agent. In [15], the authors show how a Nikaido-Isoda relaxation algorithm can converge to the Nash-Cournot in a bilateral market. Another type of equilibrium model formulation is related to the concept of a supply function equilibrium. In [16], the equilibrium is found by simulation in a simple two-bus system, where the pure strategy equilibrium is eliminated by the inclusion of a network constraint. In another paper by the same authors [17], they simulate the same two-bus system and find that there are many Nash equilibria under line constraints. Similarly, a three-bus system with network constraints is analyzed in [18] to find the Nash equilibrium using a supply function equilibrium model (SFE). B. Aims and Contributions

Profit of the th generating company per scenario. Euclidean distance between



I. INTRODUCTION A. Literature Review LECTRICITY market equilibrium models are commonly used to replicate the behavior of the agents and to mimic the rules of the market [1], [2]. Regulators use these models to monitor the market; buyers and sellers use them to refine their bids and offers, respectively.


In this paper, we propose a stochastic EPEC model based on a bilevel programming approach where the GENCOs bid strategically acting as price-makers. In the bilevel model, the lower-level problem represents the market clearing mechanisms and the upper level the optimal bids by the GENCOs. Uncertainty is incorporated to the load demand in the lower-level problem. Note that stochastic EPEC models are applied when all the GENCO’s MPEC problems are solved simultaneously under uncertain demand. In general, the EPEC problems are nonlinear and nonconvex systems of inequalities. In our approach, the lower-level problem is transformed into a set of primal constraints, dual constraints, and the strong



duality condition, which are sufficient conditions for the linear lower-level problem to have a global optimum. A Fortuny-Amat reformulation [19] and a binary expansion approach as in [5] are applied to express the MPEC as an MILP. Note that a fine-grained binary expansion is able to convexify the previously nonlinear and nonconvex formulation without losing much accuracy. We model the EPEC as a mixed integer linear system of inequalities based upon [20]. However, not only quantities but also prices are the strategic variables of our model, as done in [7]. The outcomes derived from the model are all the pure Nash equilibria. This is achieved including additional linear constraints in the feasible region centered around the iteratively found pure Nash equilibria. The contributions of this paper are threefold: 1) formulation of a bilevel MILP model focusing on the strategic price and quantity bidding variables of a GENCO in a multi-period and multi-block (bid) setting; 2) formulation of a stochastic EPEC using an MILP model with uncertainty associated with the demand; 3) addition of new linear constraints to find all the pure Nash equilibria of the stochastic EPEC.

subject to

C. Paper Organization

The model in (1)–(6) consists of: 1) the upper-level problem (1) and (2) of the GENCO and 2) the lower-level problem (3)–(6) of the ISO for each scenario . The upper-level problem (1) maximizes the expected profit of the GENCO from selling in the pool market for all periods. The profit comes , solved in the from the difference between the spot price, lower level and the marginal cost, , which depends on each production block of the GENCO. Note that the GENCO can be the owner of several generating units. Equation (2) represents the maximum production of each unit of the th GENCO. The lower-level objective function (3) that is minimized for each scenario is the cost of the energy dispatched. This cost minimization is equivalent to the social welfare maximization when the demand is inelastic, as assumed here. Without loss of generality, demand elasticity can be included using stepwise functions and the lower-level problem remains linear. The constraints of the ISO problem come from the energy balance and the limits on the quantity offers (5) and (6) of all GENCOs. We disregard line constraints in our model as in [12]. Note that the lower-level problem can be decoupled per scenario and per period. The price and quantity offers of the lower-level problem are split into two parts: 1) the strategic offers of the upper-level problem solved by the th GENCO and 2) the fixed offers of its competitors.

The remainder of the paper is organized as follows. In Section II, we formulate the individual MPEC optimization model of a GENCO, establish the EPEC Nash equilibrium model as nonlinear and nonconvex problem, transform it into an MILP problem, and provide an algorithm to find all pure equilibria. Section III presents an illustrative example of the proposed methodology. The main conclusions are summarized in Section IV. The lower level linearization is shown in Appendix A and the linear constraints methodology to find multiple pure equilibria is explained in Appendix B. II. MODELS A. Bilevel Problem Formulation We assume that the th GENCO optimizes its strategy offers as profit maximizer in a pool market. The offer strategies are both in quantities and prices. Assuming that the strategies of the competitors are estimated, a GENCO can anticipate the results of the market. To consider all the above, we use a bilevel model where a GENCO maximizes its profits in the upper level with information from the lower level, which has the information of the market clearing by the ISO. The resulting problem is equivalent to a Stackelberg game. In the upper level, the GENCO acts as price-taker, while the joint solution of both levels is equivalent to a price-maker model. Hence, each th agent can get its expected optimal value, , with the following stochastic model:


(2) where

(3) subject to

(4) (5) (6)

B. MPEC Mixed Integer Linear Reformulation We take the bilevel formulation in section A and replace the lower level part by a set of constraints composed of the primal constraints, the dual constraints, and the strong duality condition, which yields an equivalent MPEC formulation to the bilevel problem. Note that the lower-level problem is linear and the KKT conditions are equivalent to this set of constraints. Then, we apply the Fortuny-Amat linearization with yields an MILP formulation of the MPEC.




D. Stochastic EPEC MILP Reformulation For each GENCO, the strategy vector consists of a discrete set of bids where is the available number of combinations of the discrete strategies set. The utility function is evaluated in the inequality system for each discrete strategy (see [20] for further details):

(7) (10) subject to (8) The decision variables of the problem (7) and (8) are: the and binary variables from the upper-level problem and the free variable positive variables and from the lower-level problem. Variables and result from the linearization of the bilinear term of the upper- and lower-level variables. Only two of the decision variables of the MPEC-MILP model are strategic variables . Both variables come from the binary expansion . approach of All variables are controlled by the leader. The leader’s target is to anticipate the reaction of the other agents (implicit in the fixed bids of the other agents). If the competitors behave as rational agents, they must choose their optimal bids. Consequently, they choose the strategies that are the best ones against the ones of their competitors (assumed fixed); this represents the set of (pure) Nash equilibria. Thus, we use the MPEC model within an equilibrium model where we set the competitor’s strategies to a fixed value.

The problem set in (10) can be solved by simple enumeration of the strategies available to each GENCO. In this set, the left-hand side (LHS) represents the equilibrium point and the right-hand side (RHS) each available strategy of the GENCO. , which is a The number of inequalities is given by better alternative than solving the combinatorial game creating combinations in the payoff matrix. The LHS expected profit of the GENCOs can be transformed as shown in (11), where the expected value is given with the linear objective function of the MPEC-MILP in the equilibrium:

(11) C. Stochastic Nash Equilibrium The vector of strategies available for the th GENCO is de. The stochastic fined as Nash equilibrium [21] is defined from the set of inequalities (9), . The for any feasible strategy vector feasible region is defined with the set of constraints of the MPEC-MILP problem:

where the feasibility region of the LHS is the set of constraints (12)–(22). This constraint set is the same as the one of the MPEC-MILP, but for the Nash equilibrium in this case:





The resulting problem (9) is a nonlinear and nonconvex set of inequalities that represent an EPEC problem. In this setting, all GENCOs solve their MPEC-MILP problems simultaneously and the fixed strategies offers in prices and quantities result from the solution of the MPEC-MILP problem of the other GENCOs.






same constraint for the rest of the GENCOs who are fixed in the equilibrium. Constraints (27) and (28) show the dual constraint set for company and the competitors, respectively. The strong duality constraint (33) consists of the terms related to company and to its competitors. Equations (29)–(32) set represents the Fortuny-Amat linearization:

(24) (25) (18)









Constraints (12) and (13) provide the limits of the binary expansion on quantity and price offers. Equation (14) shows the maximum production of each GENCO. Equations (15)–(18) are the lower-level constraints in the equilibrium and constraints (19)–(22) are necessary in the Fortuny-Amat approach [19]. In the RHS, the expected profit of the GENCOs is defined in


(31) (32)

(23) The RHS feasibility constraints are given by (24)–(32). These constrains are based on the set of equations given by the MPECMILP constraints, similarly to the LHS constraints. The RHS available for each constraints are defined for each strategy company who chooses this strategy when the competitor companies are fixed in the equilibrium . Constraint (24) models represents the power asthe load balance. The variable signed by the ISO to the th generating unit when company chooses strategy . Note that each th generating unit can belong to the GENCO or to the competitors. Equations (25) and (26) model the quantity offer limits: (25) refers to company who chooses the (known) strategy indexed by , and (26) shows the


The stochastic EPEC-MILP model is defined by: 1) the set of inequalities in (10) that uses the LHS and RHS expected profits in (11) and (23), respectively, and 2) the LHS and RHS feasible regions, (12)–(22) and (24)–(32), respectively. Note that the EPEC model is represented as a system of equations without any objective function.






is simplified to one bid offer block (in quantities and prices) per generating unit and one period of study. The strategic price bids range from marginal cost to the limit shown in the second column of Table II. The third column provides the strategies in quantity, ranging from the minimum to the maximum production values. Price and quantity offers bids are equally divided into four and two levels, respectively (see Table II). Thus, there are eight different step-wise offer curves per unit and 64 combinations per GENCO. Consequently, the payoff matrix for this combinations. game has Three different scenarios are provided in Table III to describe possible demands.


B. Results

E. Finding All Pure Nash Equilibria The stochastic EPEC-MILP model solution is a pure Nash equilibrium but, usually, there is more than one pure Nash equilibrium. We propose a new methodology to find these equilibria by creating “holes” in the feasible region of the stochastic EPEC-MILP model. The holes are centered around each Nash equilibrium. For each newly found equilibrium, we add a new linear constraint (hole) in the feasible region, as shown in

(34) Note that and are constant values for the stochastic EPEC-MILP model. The radius must be small enough so as not to lose any solutions inside the hypersphere hole and the solution must not belong to the boundary of the hypersphere hole. Note that the discrete space. Therestrategic variables belong to the fore, the limits of limits: . See Appendix B for details. III. CASE STUDY

The game is solved by two methods: 1) constructing the expected payoff matrix and searching for the equilibria with conventional methods and 2) with a stochastic EPEC-MILP formulation. The proposed model can be solved generating the corresponding payoff matrix for each scenario. For each combination of strategic bids for all GENCOs, the ISO problem is solved and the solutions (profits of the GENCOs) fill out the payoff matrix cells. The three payoff matrixes generate a new expected payoff matrix taking into account the probability of each scenario. This new matrix is used to find all Nash equilibria. Table IV provides all the pure Nash equilibria found with this method. The equilibria can be grouped according to profits and expected spot prices since many offer bids do generate the same prices and energy commitments. The stochastic EPEC-MILP model is also solvable with conventional MILP solvers. For each Nash equilibrium found, a new hole constraint (34) is added in order to find the next equilibrium. The algorithm stops when there are no more feasible solutions. We solve the stochastic EPEC-MILP model as an optimization problem and we use the maximization of the expected spot price as the objective function. We obtain the results shown in Table IV. To reduce CPU running time, we add a new set of constraints (35) for sets of strategies that produce the same price and energy, obtaining a unique pure Nash equilibrium for each set. We can use an ex-post heuristic methodology to find all plausible pure Nash equilibria given by the quantity and price offers that are valid for the expected price found in each group of equilibria:

A. Data Model data are presented in Tables I–III. Table I shows three GENCOs with two generating units per company. The market





primal problem constraints (A1), the dual problem constraints (A2) and (A3), and the strong duality condition (A4). The equivalent set of constraints is linear except for the strong duality condition, where there are two bilinear terms. Lower-Level Primal Constraints: (A1)


Lower-Level Dual Constraints: (A2)


Lower-Level Strong Duality Condition: Equation (35) is transformed into a set of linear constraints using an additional binary variable and the Fortuny-Amat linearization. Table V shows the running time required for solving the problem. The first column shows the traditional payoff matrix result, the second one the stochastic EPEC-MILP model solved, and the last one the same EPEC-MILP model with the new constraint set (35). We use MATLAB [22] for solving the payoff matrix and CPLEX 11 under GAMS [23] for the EPEC-MILP models. We have used a Dell PowerEdge R910x64 computer with four processors at 1.87 GHz and 32 GB of RAM. The complexity of the EPEC-MILP formulation is shown in Table VI for each iteration of the case study. Based on the case study, we show the complexity for 100 scenarios and 24 h in the third and fourth columns, respectively.

(A4) is a positive variable and where the dual variable is a free variable. The bilinear terms, and , are approximated by an equivalent expression of the binary expansion approach [5] applied to . The new pair of the controllable variables represent each discrete strategy binary variables :

IV. CONCLUSIONS This paper presents a compact formulation for the strategic bidding problem in pool-based electricity markets considering jointly price and quantity strategic bids in a multi-agent, multiperiod, and multi-block game. In addition, we consider stochasticity of the demand in several scenarios. The stochastic EPEC is formulated as an MILP. To do that we consider: 1) the strong duality constraint, instead of the KKT conditions; 2) the Fortuny-Amat representation; and 3) a binary expansion to eliminate the bilinear terms of the problem. Since multiple Nash equilibria can be expected in this problem, we propose an iterative procedure is applied to find all (pure) Nash equilibria by including successive linear constraints to the stochastic EPEC-MILP model. The MILP models are suitable for application in large-scale systems solvable with commercial solvers. An illustrative example shows the results of the methodology proposed.


(A6) Substituting (A5) and (A6) in (A4), the latter becomes

APPENDIX A LOWER-LEVEL MILP TRANSFORMATION The lower-level optimization problem (3)–(6) is transformed into an equivalent representation (A1)–(A4), made up of the



where we have replaced the nonlinear products and . Adding up the equivalent Fortuny-Amat linear constraints, this yields


The expression in (B1) can be rewritten as shown in (B2):


(B2) where the quadratic term can be converted into a linear term discrete (B3) by taking into account the properties of the variables:




(A11) The equivalent lower-level set of the MILP is defined by (A12)–(A18). Lower-Level Primal Constraints:

Using (B3) to replace the nonlinear terms in (B2), we obtain a linear constraint formulation:

(B4) (A12) Lower-Level Dual Constraints:


Lower-Level Strong Duality Condition: (A15) Binary Expansion Limits: (A16)

(A17) Fortuny-Amat Linearization: (A18)

APPENDIX B HYPERSPHERE LINEAR CONSTRAINTS IN THE STOCHASTIC EPEC-MILP MODEL Let a Nash equilibrium vector solution of the stochastic . If there exists EPEC-MILP point be another feasible Nash equilibrium , with radius it will be outside the hypershpere centered at point . Thus, the distance between and must be greater than the radius . We define this distance as the Euclidean distance between two points (B1): (B1)

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David Pozo (S’09) received the B.S. degree in electrical engineering from the University of Castilla-La Macha, Ciudad Real, Spain, in 2006, where he is currently pursuing the Ph.D. degree. His research interests include power systems economics and electricity markets.

Javier Contreras (SM’05) received the B.S. degree in electrical engineering from the University of Zaragoza, Zaragoza, Spain, in 1989, the M.Sc. degree from the University of Southern California, Los Angeles, in 1992, and the Ph.D. degree from the University of California, Berkeley, in 1997. His research interests include power systems planning, operations and economics and electricity markets. He is currently Full Professor at the University of Castilla-La Mancha, Ciudad Real, Spain.

Finding Multiple Nash Equilibria in Pool-Based Markets

companies (GENCOs) optimize their strategic bids anticipating the solution of the .... straints of the ISO problem come from the energy balance and the limits on the ... better alternative than solving the combinatorial game creating combinations ...

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