Parameterization of Linear Supply Functions in Nonlinear AC ...

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 2, MAY 2013

Parameterization of Linear Supply Functions in Nonlinear AC Electricity Market Equilibrium Models—Part I: Literature Review and Equilibrium Algorithm Andreas G. Petoussis, Xiao-Ping Zhang, Senior Member, IEEE, Savvas G. Petoussis, and Keith R. Godfrey

Abstract—The linear supply function equilibrium (SFE) model can be used to investigate bid-based electricity pool markets. Several studies assume that the market players construct optimal strategic bids by parameterizing the linear marginal cost function of their generating units. Four different types of parameterization are possible, for which the slope and/or the intercept of the marginal cost functions are varied. This paper reviews the existing literature and proposes a primal-dual nonlinear interior point algorithm to solve a bi-level market problem with AC network representation, using any of the parameterization methods. Part II of this paper uses the proposed algorithm to examine the effects of the different parameterizations on the resulting SFE solutions, the role of network complexity and the nature of the equilibrium points. Index Terms—AC network modeling, electricity market equilibrium, interior point algorithm, nodal prices, power system economics, SFE parameterization, supply function equilibrium.

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I. INTRODUCTION

LECTRICITY market reform was initiated during the 1990s in order to accommodate a more liberalized environment and introduce competition in centralized power markets. However, the electricity market has been proven to have a tendency to oligopolistic behavior due to a number of factors, such as the entry barriers and the potential market power of the participating generating firms [1]. Since then, several market models have been proposed for the prediction of the electricity market outcome under various network operating conditions. The concept of Nash equilibrium and non-cooperative games [2], in which the market players choose optimum strategies in order to maximize their individual profits, has been

Manuscript received July 06, 2011; revised November 05, 2011, February 21, 2012, May 03, 2012, and July 30, 2012; accepted August 06, 2012. Date of publication October 02, 2012; date of current version April 18, 2013. This work was supported in part by the UK Engineering and Physical Sciences Research Council (EPSRC). Paper no. TPWRS-00629-2011. A. G. Petoussis was with the School of Engineering, University of Warwick, Coventry, CV4 7AL, U.K., and now is with the Cyprus Transmission System Operator (TSO-Cyprus), Nicosia, CY-2057, Cyprus (e-mail: [email protected]; [email protected]). X.-P. Zhang is with the School of Electronic, Electrical and Computer Engineering, University of Birmingham, Birmingham, B15 2TT, U.K. (e-mail: [email protected]). S. G. Petoussis is with the Electricity Authority of Cyprus (EAC), Nicosia, CY-1399, Cyprus (e-mail: [email protected]). K. R. Godfrey is with the School of Engineering, University of Warwick, Coventry, CV4 7AL, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2012.2214243

extensively employed in different forms to study the behavior of centralized bid-based pool trading of electric power. The market outcome is a Nash equilibrium point if no player has an incentive to change strategy. The most popular equilibrium models for the electricity market have been the Cournot (quantity) competition [3] and the supply function equilibrium (SFE) model [4]. Since equilibrium analysis has become a frequent practice for investigating electricity market structures, the Cournot model was considered to be among the most appropriate tools for calculating electricity market equilibria. The reason for this was that quantity competition was an important step away from concentration measures techniques and monopoly solutions, being able to assess market power up to a certain degree, and the mathematical Cournot model was relatively simple to implement. However, the power generation market is a type of supply-curve competition and not a quantity-based one, since the generation costs are increasing with production. Therefore, market power behavior cannot be fully comprehended using Cournot theory, while other shortcomings arise due to the fact that the electricity demand is unknown and has very low elasticity. In contrast, the SFE competition meets the supply-curve modeling requirements and offers the possibility of developing insights into the bidding behavior of the market participants, which are able to adapt better in the uncertain demand environment by submitting bid functions that relate different power quantities to different prices. Several formats of supply functions have been employed for the bid offers of the market players. The original general SFE model in [4], which was first applied to the electricity market in [5], allowed any form of supply function to be used. Among the general supply function forms examined in [4], the case of linear SFE was observed. Later, the study in [6] proposes the application of an SFE model for which the market players parameterize their linear marginal cost function by varying its slope and/or intercept in order to construct the optimum supply function bids, to the electricity market. The investigation in [6], which solves a Klemperer-Meyer differential equation, shows that the linear solution is a reasonable approximation for any equilibrium function at low levels of demand, while for higher demand levels it is still a possible equilibrium solution. It has been reported that the linear SFE model has the distinct advantage of handling multiple asymmetric strategic firms [7]

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PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

and it can be implemented into large power system models with relative simplicity. 1 Furthermore, under certain circumstances, the only stable SFE is the linear solution [8]. Since the linear SFE model conforms to the features of the electricity market and in addition exhibits the aforementioned advantages, it has received considerable attention. Despite the many advantages for modeling the electricity market behavior, the linear SFE model exhibits a certain disadvantage. This is related to the fact that supply offers in practical electricity markets exhibit a hockey-stick characteristic, i.e., the offers are fairly competitive at low demand levels while their prices increase steeply near the market capacity [38]. Such behavior can be captured by the general SFE framework (or by a piece-wise linear counterpart) but not by the linear SFE model. The linear SFE models and the proposed numerical algorithms that contributed to the literature have employed different network representations. The early developments [6], [9]–[11] consider generation capacity constraints but they ignore the electricity network. Later studies have introduced transmission constraints [12], the DC network [13]–[15] and the AC network representation [16]–[20], [39] to show that the market equilibrium depends on the constraints considered.2 As the conclusions drawn from each investigation were based only on a single parameterization method, it has been argued in [21] that the choice of parameterization method also has an impact on the market outcome, based on numerical results for a system with two firms separated by a transmission limit. Other studies that compare the different parameterization methods are very limited and the employed models consider no constraints or the linearized DC representation, as discussed in Section II-B. The study in this paper begins by reviewing the existing literature on the linear SFE models in order to discriminate between the features of the different parameterization methods. Since there has been no investigation of the impact of the parameterization chosen on the equilibrium, in applications with AC meshed network or large systems, an advanced AC-network market equilibrium algorithm is proposed for this purpose. Part II of this paper examines several case studies that involve a variety of test systems under different network operational conditions to further contribute to the subject. The investigation in this two-paper series follows the analysis of [21] and illustrates how the introduction of AC network constraints can affect the market results for the four parameterization methods and the relationships between them. The structure of Part I of this paper is as follows. The existing literature is presented in Section II, where the equilibrium solutions for the different supply function parameterization methods are discussed. The market assumptions and the mathematical derivation of the proposed AC-network market equilibrium algorithm are outlined in Section III. Section IV discusses the nature of the different equilibria that may result, while Section V illustrates the convergence characteristics of the proposed algorithm. Section VI provides some concluding remarks. 1The general SFE

model can also handle asymmetric firms, e.g., see [40], but the implementation of the mathematical modeling becomes complicated. 2For

further details on transmission-constrained equilibria refer to [41].

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Part II builds on the analysis of Part I by performing several case studies on small and larger meshed-network systems. The case studies investigate the effect of the different parameterization methods on the equilibrium solutions in regard of the system conditions and network complexity. Then, the nature of the resulting SFE points is analyzed. II. PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS A. Description of the Four Parameterization Methods Four parameterization methods are available for the construction of the optimal linear supply function bids, each one considering different strategic variables. For the general description that follows, a linear marginal cost function and a supply function bid are defined, where and are the parameterized bid terms that correspond to the true generation cost coefficients and , with , and is the active power generation. The different parameterization methods can then be described as follows: 1) Intercept-parameterization: the strategic players adjust the intercept of their marginal cost functions to construct the profit-maximizing bids to be submitted to the pool, while keeping the slope constant. Most models keep the bid slope fixed at the value of the marginal cost function slope, but a model may assume that the players are allowed to fix the slope at a pre-assigned non-negative value that might be other than the true slope. In this paper this approach is referred to as -parameterization. 2) Slope-parameterization: the strategic players’ behavior is modeled by varying the slope of the marginal cost functions while keeping constant the intercept . Most models set the bid intercept equal to the true value of the marginal cost function intercept, but pre-assigned intercepts at any other value may be allowed. The slope-parameterization method is referred to as -parameterization. 3) (Slope intercept)-parameterization: in this method the strategic players adjust both the slope and intercept in the supply function, but in a fixed linear relationship as the one between the true and parameters of the marginal cost function. This can be interpreted as multiplying the marginal cost function by an arbitrary non-negative constant, say , in order to construct the supply function bid, such that . The strategy of the players is defined by adjusting the factor. In this paper the term is termed as the bidding parameter and this parameterization method is referred to as -parameterization. 4) (Slope-and-intercept)-parameterization: this approach allows more degrees of freedom for the choice of the strategic supply function by arbitrarily parameterizing both slope and intercept independently. This represents the true flexibility in strategies available to bidders in the context of the linear SFE model (for positive bid slopes). This method is referred to as -parameterization. B. Comparison of the Different Parameterization Methods The -parameterization is considered as the most realistic of the four since it allows any non-decreasing linear supply

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function to be implemented as an equilibrium strategy. It illustrates the true potential of the firms to exercise market power, as far as the linear SFE solution is concerned, and is expected to yield superior profits compared with the other parameterization methods, as demonstrated by numerical results in [22] for a 5-firm unconstrained system, and in [21] and [23] for linearized 2-bus and 6-bus DC systems. However, such high profits may be limited and subject to the network and market constraints present, as shown by numerical results in Part II of this study. The -parameterization has been employed in only a limited number of studies for two reasons. Firstly, a unique -parameterized SFE exists only in rare cases under very restrictive conditions. Therefore, it will be difficult to perform mathematical analysis on test systems for the comparison of different market situations, as multiple equilibria may be encountered. Such comparisons would be fraught, even though the market outcome predictions are more realistic, because it will not be apparent which equilibrium to select. The second reason is related to the fact that systems that consider the electrical network and multiple players cannot be solved analytically and numerical algorithms are required. As the -parameterization may have multiple solutions, the algorithm employed will have convergence difficulties if the SFE points are not (at least) locally unique, due to oscillations in the convergence process caused by a continuum of equilibria [24], [25]. Yet, other reasons related to AC network modeling may also be responsible for these complications and further discussions take place in Part II of this paper. Other studies that use the -parameterization include [9] that uses the differential equation approach and [13], [25], and [26] that apply numerical methods on DC or relatively small systems. The other three parameterization methods have been criticized to be less predictive than the -parameterization in describing the market behavior, since they may provide equilibrium results that can be artifacts of their restrictive assumptions. In [21], it has been shown with a simple example that the SFE solution of the -parameterization differs from those of the other three methods and in certain cases even the existence, non-existence or multiplicity of equilibria can be an artifact of the numerical framework. Nevertheless, the - and -parameterization methods tend to have locally unique equilibrium points when modeling markets with transmission network representation, and the numerical algorithms employed will converge to comparable equilibrium solutions for different network operating conditions. Furthermore, if only intercepts or only slopes are manipulated by fixing the other coefficient to its true value, the resulting local equilibrium point will also be an equilibrium for the -parameterization game, as long as the strategic variables do not reach any limits specified in the market model [24], [25]. Therefore, in either case, the coefficient held constant can be chosen accordingly to result in a (very probably unique) Nash equilibrium that corresponds to the most desirable equilibrium under -parameterization. Further elaboration on this can be found in [21]. Clarification between the nature of different equilibrium points is provided in Section IV-A. The investigation in [13] presents numerical results on a 2-firm market structure with no constraints to report that the choice between - , - , or -parameterizations does

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 2, MAY 2013

not show appreciable difference in the unconstrained SFE market outcome, while the investigation in [22] for a 5-firm unconstrained market presents comparable numerical results for the - , -, and -parameterizations. Nonetheless, it seems that this is not the case if more complicated meshed network constrained systems are considered (as it is shown in Part II of this paper). However, each of the aforementioned parameterization methods may be used for qualitative comparisons of different market situations (at least for simplified systems) and the employment of each particular method can be justified by the purpose and requirements of the apparent oligopolistic analysis as discussed below. The slope-parameterization is closely related to the unconstrained linear SFE model resulting from the Klemperer-Meyer formulation, in which revealing the true intercept in the strategic bid is incentive compatible for the generating firms in a multi-period SFE game, as observed in [7]. Therefore, since this type of parameterization is naturally related to the original SFE model by Klemperer and Meyer [4], it provides the ability for an extension to model situations that incorporate network or other constraints if multiple pricing periods are considered. Notable applications for this parameterization method, which is the most popular one, include [6], [10], [11], [13], [21], [22], [24], [25], and [27]–[29]. The intercept-parameterization has been regarded in some studies as more appropriate than the slope-parameterization, because slopes of marginal cost functions are usually very shallow and the steep slopes that would result from the strategic behavior would not be credible.3 The intercept manipulation approach can be utilized instead, since the steepness of an aggregate bid curve for an entire multi-unit firm can be manipulated by diffor different generating units [14]. A ferent markups major advantage of the -parameterization is that it can be employed for market power assessment, since the manipulation of the intercept provides a clear picture of how the strategic firms shift their supply functions up or down, specifying the price at which a generator is willing to start producing [25]. This concept was primarily considered in [30] to assess the extent of market power in the Nordic power market. Among others, the intercept-parameterization method has also been employed in [13], [14], [21], [22], [24], and [25]. -parameterization can be regarded as exercise of The market power by offering optimized supply functions that are scaled versions of the true marginal cost functions. Hence, this model depicts successfully the strategic behavior in the parameter is a relaoligopolistic environment, since the tive measure for market power that can be easily interpreted without referring to the actual generation costs. This method is eligible for comparisons of different market situations for qualitative purposes, such as the investigation of the impact of network constraints on the equilibrium market outcome and the interactions between the strategic firms. Examples 3However, since practical firms usually enter the pool market after they have already sold a large share of their output in the forward market, the slope-parameterization approach can be appropriately adjusted by implementing this consideration into the model. This will mitigate the strategic players’ market power and the resulting supply function bid offers would be expected to be more realistic.

PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

of such studies that employ the -parameterization include [12] and [17]–[20]. Furthermore, it appears to be a convenient way to simplify the numerical computations in SFE models in order to deal with complicated systems. An aspect that can be regarded as a drawback (which may also be shared with the - and -parameterization methods if the non-manipulated term is fixed to its true value) is that it cannot predict the focal equilibrium if multiple SFE solutions exist [21]. However, the investigation in [27] shows that existence of multiple equilibria should not be expected in large numbers for realistic cases, but only for small trivial systems. Other investigations that employ the -parameterization include [15], [16], [22], and [39]. C. Relation of Linear Supply Functions and Realistic Bids Apart from implementing theoretical equilibrium models with linear bid functions, a linear supply function can be useful in calculating the prices related to the quantities of discontinuous supply function block offers as those required in some real electricity markets. For example, if the market mechanism requires the strategic firms to submit step-wise supply functions that relate each MW block to a price, each bid segment can be defined according to a linear supply function of the form , where is the price for the MW block segment under consideration and is the quantity of the particular block, by choosing appropriate profit-maximizing equilibrium values for the strategic bidding parameter . Then bid pairs are submitted to the pool, depending on how many blocks are required by the market protocols or chosen by the strategic firms if is not restricted. Further information on such a scheme can be found in [31]. Note that in contrast to the theoretical linear supply function bidding model that involves only one or two strategic decision variables for each firm, a step-wise supply function (as in practical applications) may involve several decision variables, such as the number of blocks, the power quantity for each block and the associated bidding parameters that define the block price. As the -parameterization has more degrees of freedom than the other three methods of the linear supply function bidding model, it can be regarded as the one closer to practical applications, since the -bid can be freely adjusted in the function space to resemble as closely as possible a step-wise block offer strategy if the price-quantity steps are not distributed in an inconsistent manner. For details on gaming with multi-block supply function offers refer to [32]–[34]. III. MARKET EQUILIBRIUM ALGORITHM The proposed electricity market equilibrium algorithm can be used to calculate the Nash equilibrium point for bid-based pool markets for the different parameterization methods. The implementation of the algorithm is based on the primal-dual nonlinear interior point method. The electrical network is represented using AC power flow analysis. The power network constraints and controls considered include active and reactive transmission power flows, transmission line and transformer branches capacity limits and losses, transformer tap-ratio control, bus voltage limits, active and reactive power generation

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capacity limits, and interrelated price-responsive active and reactive load demands. The employment of such complex network constraints aims to investigate the different parameterization methods under realistic assumptions. In addition, such restrictions that reflect the nature of practical power systems are expected to eliminate most (or even all but one) equilibria. The study in [27] shows that under very restrictive conditions in the presence of elastic load the possible equilibria may be narrowed down to a unique equilibrium point. Note that under the -parameterization all pure strategy equilibria may be eliminated due to the nature of the looped constrained network. This is further discussed in the numerical analysis of the companion paper Part II. A. Electricity Market Assumptions The wholesale electricity market model proposed considers strategic generating firms that are profit-maximizers. The market structure is based on the single-period bidding model, for which the firms submit optimal linear supply function bids to an independent system operator (ISO) for each bidding time interval. Each firm chooses strategy by anticipating the profit-maximizing actions of its rivals and this holds for all the individual firms. After the bidding interval is closed, the ISO clears the market and calculates the market clearing price for active power in terms of nodal prices, which include both the energy price and the short-term transmission costs. To do so, the ISO determines the power distribution and network parameters (including transformer tap-ratio settings, reactive power dispatch and bus voltages) by balancing supply and demand, while abiding by the network constraints, based on a social welfare maximization scheme. During this procedure the ISO takes into account the strategic actions of the firms, i.e., the submitted supply function bids. Each firm is paid from the ISO the nodal price at its bus for the active power sold to the pool, and each consumer (load) pays the nodal price at its bus to the ISO for the active power received. Nodal prices for reactive power are not considered in this study. It is assumed that the generators produce an amount of active power equal to that of the ISO schedule, which is entirely sold to the power pool, in order to cover the load demand and system losses. In addition, it is mandatory that the generators provide the required reactive power generation or absorption to support the voltage levels determined by the ISO, although the reactive power provision does not account for any profit or extra cost since the market is only for active power. The generating cost function of the firms is of the form (1) with a marginal cost function (2) where is the relevant bus. Depending on the chosen parameterization, the strategic bid can be constructed as (3)

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where is the bidding parameter of the generating unit at bus as defined in Section II-A. For the - or -parameterizations, the other cost coefficient in the bid function is held constant to the true value. The consumers’ utility function is of quadratic inverse form (4) which corresponds to an inverse load demand function of linear form with negative gradient, such that (5) where and are the linear and quadratic load demand coefficients, respectively, and is the active power load demand. B. Formulation of the Equilibrium Algorithm The algorithm is formulated based on the optimization problems of the ISO for social welfare maximization and of the strategic firms for the maximization of their individual profits. Since the ISO receives only the strategic bids from the firms and not the actual generating costs, its objective function can be represented by the quasi social welfare that incorporates the strategic variables. This function is equal to the consumers’ benefit minus the generating costs as reflected by the strategic bids. Therefore, the ISO problem, which includes the network constraints, can be represented as

subject to:

(6) and are the active and reactive power miswhere match equations; and are the active and reactive power generations; and , , are the active and reactive power load demands; and are the active and reactive power injections given by the power flow equations in rectangular coordinates; and are the power flows through a transmission or transformer branch of a maximum MVA capacity limit ; is the transformer tap-ratio; and are the real and imaginary components of bus voltage

in rectangular coordinates; and are the minimum and maximum voltage limits; is the bus load angle; and are the total number of transmission lines and transformer branches, respectively; and are the total number of load and generation buses, respectively; is the total number of system buses. Depending on the parameterization method chosen, one of the two bid variables in the objective function may be equal to the true value from the marginal cost function, or the term may be introduced. The ISO optimization problem (6) can be represented by its 1st-order Karush-Kuhn-Tucker (KKT) conditions as suggested by the primal-dual interior point theory. More details are provided in the Appendix. The optimization problem of the strategic firms is based on the maximization of their individual profits , given by the revenue minus the true generating costs, such that

subject to (7) where is the set of strategic bidding variables, i.e., the set of , or , or and , or , for all generators, depending on the chosen parameterization method; and are the minimum and maximum limits for the strategic variables, respectively; and is the nodal price for active power at bus . In order to obtain a combined problem that represents both optimization problems, the following arguments are brought into attention. The active power generation and the nodal prices used in the objective function of the firms’ optimization problem (7) that can be expressed as implicit functions of all generating firms’ strategies, are produced by the ISO nonlinear program given in (6). Therefore, the values for and should satisfy the ISO KKT conditions, which represent the original ISO optimization problem. Hence, the ISO KKT conditions can be incorporated into the strategic firms’ optimization problem as equality constraints, to form a combined problem that can be solved to give the market equilibrium solution. To do so, the 1st-order KKT conditions for the combined problem must be obtained. The mathematical formulation is outlined in the Appendix. The KKT conditions of the combined market problem can be solved simultaneously to give the electricity market SFE solution. For efficiency enhancement, sparse matrix techniques are applied to construct a symmetric Newton matrix equation, which can be solved iteratively, as shown in the Appendix. The procedure for the initialization and update of the variables during the iterative process is similar to that in [19]. Note that the optimal solution produced by the KKT system may not exist or may not be a global optimization solution as the formulation of the problem is nonconvex. If a solution exists, its uniqueness cannot be guaranteed. However, experience has verified the effectiveness of the algorithm, which, at least for the -, -, and -parameterizations, can converge to an equilibrium point, as long as a pure Nash equilibrium exists. If

PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

more than one equilibrium exists, a global solution may be obtained if the initial values of the bidding variables are appropriately adjusted. The algorithm shows some difficulties in convergence when the -parameterization is employed for large systems, but this may be due to the nonexistence of pure strategy equilibria (see discussion in Part II). Furthermore, the results in [25] have shown that such formulations yield at least local Nash equilibria, which even though they are less credible than their global counterparts, still have meaning as they can be sufficient for the satisfaction of the players due to difficulties in identifying a global SFE among themselves or due to limitations on their rationality and knowledge. Part II of this study examines whether the solutions obtained from the proposed interior point algorithm are unique or not. IV. CLASSIFICATION OF EQUILIBRIUM POINTS A market equilibrium problem may have a unique equilibrium, multiple equilibria, or no pure strategy equilibria (in which case the players can play mixed strategies). As already mentioned in Section III-B, the market solution obtained from the primal-dual nonlinear interior point algorithm may not be a global equilibrium and other equilibrium points may exist. A. Nash Equilibria of Different Nature A solution obtained from the formulation of a KKT system for the bi-level market problem that results from its Lagrange function (A4) may fall into one or more of three categories. By solving a KKT system (see [19] for the analytical presentation), a Nash stationary equilibrium point that satisfies the necessary conditions of the market problem (the 1st-order KKT conditions) will result. In order to determine if this solution point is a local Nash equilibrium, the 2nd-order sufficient conditions for isolated local optimization at that point must be checked. If it can be suggested that more than one local Nash equilibrium points may exist, an appropriate method must be employed to identify all the local Nash equilibria. Then it can be checked if the surpluses in one of those points appear to be superior or if a point is mutually beneficial for all players (focal point), which can be identified as the global Nash equilibrium. Equilibria that are dominated by others may be ignored because none of the strategic players will gain anything by playing them. These are termed as local Nash equilibrium traps in [35]. In general, such equilibria are not considered to be preferred by market players but in certain situations, when lacking knowledge or rationality (due to, say, difficulties in interpreting alterations in the market rules or network configuration), the players may resort to these as the identification of a global Nash equilibrium may not be easy. However, it is expected that in the long-term their strategies should converge to a more globally preferred equilibrium. If multiple equilibria in the form of a continuum or as isolated local points are found to yield the same prices and surpluses for different strategies, they can be regarded as equivalent Nash equilibria. In this case the market equilibrium can be predicted without particular difficulties, since unique market prices and profits are obtained [31]. Such results are presented in [22] for a 5-firm unconstrained market and in [26] for a 3-bus looped AC network constrained system.

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B. Searching for Nash Equilibria A systematic procedure for finding multiple pure (if any) or mixed strategies equilibria in non-cooperative games has been proposed in [27], where the decision space of bidding strategies is divided into regions. The boundaries of these regions are composed of points where the network constraints change status, i.e., the market operating conditions are changing. When forcing the system to operate in a particular region by enforcing the relevant constraints the existing pure Nash equilibria can be identified while mixed strategies Nash equilibria are defined from the cyclical behavior of the implemented algorithm across a given set of network constraints. Filtering processes are employed in order to identify the relevant feasible regions and enhance the efficiency of the search procedure for solving large systems with an unmanageable number of search regions. The analysis and case studies in [27] have shown that the expected equilibria (both in pure and mixed strategies) for realistic constrained systems are very few or unique. Instead of searching for Nash equilibria by eliminating the feasible regions of the problem, a parallel-and-global approach may be employed. The study in [35] applies a genetic algorithm-based hybrid coevolutionary programming approach in order to distinguish between the global Nash equilibrium solution and the possible local Nash equilibrium traps. Case studies performed on 2-bus and 3-bus systems by employing the Cournot model have provided a unique Nash equilibrium or resulted in no pure strategy Nash equilibria. V. CONVERGENCE CHARACTERISTICS In order to check if the convergence characteristics of the interior point algorithm are different for each parameterization and to illustrate the efficiency of the proposed computational method, the average required CPU time and average number of iterations for each test system from the case studies examined in Part II of this paper are presented in Table I. The break point for the algorithm is based on the minimization of the active and reactive power mismatches and from problem (6) and of the complementary gap of the interior point method, given by ; see the Appendix for the definition of the variables. The convergence tolerances are set to p.u. for the maximum absolute bus active and reactive mismatches, and to for the complementary gap. The algorithm was implemented and run on a Pentium 4 CPU 3.20 GHz, 0.99 GB of RAM. From Table I it can be seen that the CPU times and number of iterations for the -, -, and -methods are similar (except for the case of the -method in the 57-bus system where the computational requirements are higher due to the extra congested transmission lines), while the -parameterization requires a CPU time and number of iterations about twice those for the other three methods, when converging. However, all the CPU times are relatively small, ranging from 16 ms to 4.6 s. To further illustrate the convergence characteristics of the algorithm, Fig. 1 compares graphically the minimization of the absolute power mismatches for the four parameterization methods for a case where simple constraints, as well as functional inequality constraints, are binding (this is Case 5 from Part II of the paper). From the graph, it can be seen that the -

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TABLE I CONVERGENCE CHARACTERISTICS OF THE ALGORITHM

literature and the analysis in Part II contributes to the subject by providing discussions and conclusions regarding the equilibrium market solutions of AC power systems. APPENDIX Reformulation of the ISO problem: The Fiacco-McCormick logarithmic barrier method [36] is used to transform the ISO maximization problem (6) into an equivalent minimization and , where problem by introducing the slack variables , . By grouping the inequalities from (6) into a set , the equivalent problem is

subject to:

(A1) where is the barrier parameter. The Lagrange function of this problem is

Fig. 1. Convergence characteristics of the four parameterizations for Case 5 of Part II of this paper.

and -methods show exactly the same quadratic converging behavior, while the - and -parameterization algorithms demonstrate quadratic convergence characteristics near the final solution. VI. CONCLUSION This paper has discussed the choice of parameterization method for the linear SFE model and the possible impact on the electricity market equilibrium solution, based on observations from the existing literature. The four available parameterization methods have been compared with regard to the properties of the numerical market algorithms implemented and the resulting equilibria. A market equilibrium algorithm able to solve bi-level market problems for AC meshed power systems has then been proposed. The algorithm can use any of the four available parameterization methods for linear supply functions. The ability of the algorithm to provide SFE solutions for realistic systems is demonstrated by several case studies in Part II of this paper. These case studies follow the observations from the existing

(A2) , , , and are the Lagrange multipliers (dual where variables) for the equality constraints of (A1). By differentiating with respect to all of its primal and dual variables, the ISO 1st-order KKT conditions can be obtained. For the analytical procedure, refer to [19]. Formulation of the combined market problem: By incorporating the ISO KKT conditions into the firms’ problem as equality constraints and then applying the Fiacco-McCormick barrier method, the following equivalent combined problem for the market solution results:

subject to:

(A3)

PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

By introducing Lagrange multipliers for each equality constraint in (A3), the following Lagrange function yields:

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As the Newton matrix equation depends on the type of parameterization chosen, two different algorithms were coded based on the above mathematical derivations. The first algorithm corresponds to the -parameterization and the second one to the -parameterization. Using the -algorithm, the - and -parameterization methods can be implemented by adjusting the limits of the or parameters, respectively, to force them to equal the true cost coefficient values from the marginal cost functions. ACKNOWLEDGMENT The authors would like to thank the six anonymous referees for their valuable comments and discussions. REFERENCES

(A4) where the terms are the Lagrange multipliers, represents , , the network variables, , , and . By differentiating the new Lagrange function with respect to all of its variables, the KKT conditions for the combined optimization problem, which represent the overall market equilibrium problem, can be obtained. Depending on the parameterization method chosen, different KKT conditions will result, since the Lagrange function (A4) contains the bidding variables and the term that is dependent on the submitted bids. Hence, the resulting market solutions for the four methods are expected to differ from each other. The structure of the Newton matrix: The KKT conditions of the combined problem are linearized using Taylor series expansion as in [19]. The resulting equations are rearranged into a matrix equation for which the configuration of the majority of its elements attains the same sparse structure as the admittance matrix used in the conventional Newton power flow as proposed in [37]. The basic structure of the Newton matrix is shown in the following:

(A5) terms correspond to the contribution of the slack where the variables and the Lagrange multipliers and ; the terms to the contribution of the system variables and the Lagrange terms to the contribution of the strategic multipliers ; the variables ; and are the sets of primal and dual variables, respectively. The detailed element structure for the , , and terms (depending on the type of parameterization) can be determined using the guidelines from [19]. Relevant mathematical analysis can be found in [42].

[1] A. K. David and F. Wen, “Market power in electricity supply,” IEEE Trans. Energy Convers., vol. 16, no. 4, pp. 352–360, Dec. 2001. [2] J. Nash, “Non-cooperative games,” Ann. Math., vol. 54, no. 2, pp. 286–295, Sep. 1951. [3] A. Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richesses. Paris, France: L. Hachette, 1838. [4] P. D. Klemperer and M. A. Meyer, “Supply function equilibria in oligopoly under uncertainty,” Econometrica, vol. 57, no. 6, pp. 1243–1277, Nov. 1989. [5] R. Green and D. M. Newbery, “Competition in the British electricity spot market,” J. Polit. Econ., vol. 100, no. 5, pp. 929–953, Oct. 1992. [6] R. Green, “Increasing competition in the British electricity spot market,” J. Ind. Econ., vol. 44, no. 2, pp. 205–216, Jun. 1996. [7] R. Baldick, R. Grant, and E. Kahn, “Theory and application of linear supply function equilibrium in electricity markets,” J. Regulat. Econ., vol. 25, no. 2, pp. 143–167, 2004. [8] R. Baldick and W. Hogan, “Capacity constrained supply function equilibrium models of electricity markets: Stability, non-decreasing constraints, and function space iterations,” in Working Paper PWP-089. Berkeley, CA: Univ. California Energy Inst., 2002. [9] R. Green, “The electricity contract market in England and Wales,” J. Ind. Econ., vol. 47, no. 1, pp. 107–124, Mar. 1999. [10] R. W. Ferrero, J. F. Rivera, and S. M. Shahidehpour, “Application of games with incomplete information for pricing electricity in deregulated power pools,” IEEE Trans. Power Syst., vol. 13, no. 1, pp. 184–189, Feb. 1998. [11] F. Wen and A. K. David, “Optimal bidding strategies and modeling of imperfect information among competitive generators,” IEEE Trans. Power Syst., vol. 16, no. 1, pp. 15–21, Feb. 2001. [12] Z. Younes and M. Ilic, “Generation strategies for gaming transmission constraints: will the deregulated electric power market be an oligopoly?,” Decision Support Syst., vol. 24, pp. 207–222, 1999. [13] C. A. Berry, B. F. Hobbs, W. A. Meroney, R. P. O’Neil, and W. R. Stewart Jr, “Understanding how market power can arise in network competition: a game theoretic approach,” Util. Policy, vol. 8, pp. 139–158, 1999. [14] B. F. Hobbs, C. B. Metzler, and J.-S. Pang, “Strategic gaming analysis for electric power systems: An MPEC approach,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 638–645, May 2000. [15] X. Wang, Y. Li, and S. Zhang, “Oligopolistic equilibrium analysis for electricity markets: A nonlinear complementarity approach,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1348–1355, Aug. 2004. [16] J. D. Weber and T. J. Overbye, “A two-level optimization problem for analysis of market bidding strategies,” in Proc. IEEE PES Summer Meeting 1999, Jul. 18–22, 1999, vol. 2, pp. 682–687. [17] J. D. Weber and T. J. Overbye, “An individual welfare maximization algorithm for electricity markets,” IEEE Trans. Power Syst., vol. 17, no. 3, pp. 590–596, Aug. 2002. [18] S. G. Petoussis, X.-P. Zhang, A. G. Petoussis, and K. R. Godfrey, “The impact of reactive power on the electricity market equilibrium,” in Proc. IEEE PES Power System Conf. Expo. (PSCE), Atlanta, GA, Nov. 2006, pp. 96–102. [19] S. G. Petoussis, X. P. Zhang, and K. R. Godfrey, “Electricity market equilibrium analysis based on nonlinear interior point algorithm with complementarity constraints,” IET Gener. Transm. Distrib., vol. 1, no. 4, pp. 603–612, Jul. 2007.

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[20] S. G. Petoussis, A. G. Petoussis, X. P. Zhang, and K. R. Godfrey, “Impact of the transformer tap-ratio control on the electricity market equilibrium,” IEEE Trans. Power Syst., vol. 23, no. 1, pp. 65–75, Feb. 2008. [21] R. Baldick, “Electricity market equilibrium models: The effect of parametrization,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 1170–1176, Nov. 2002. [22] H. Chen, K. P. Wong, C. Y. Chung, and D. H. M. Nguyen, “A coevolutionary approach to analyzing supply function equilibrium model,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1019–1028, Aug. 2006. [23] H. Haghighat, H. Seifi, and A. R. Kian, “Gaming analysis in joint energy and spinning reserve markets,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 2074–2085, Nov. 2007. [24] X. Hu, D. Ralph, E. K. Ralph, P. Bardsley, and M. C. Ferris, “Electricity generation with looped transmission networks: Bidding to an ISO,” in CMI Working Paper 65. Cambridge, U.K.: Univ. Cambridge, 2004. [25] X. Hu and D. Ralph, “Using EPECs to model bilevel games in restructured electricity markets with locational prices,” Oper. Res., vol. 55, no. 5, pp. 809–827, Sep.-Oct. 2007. [26] G. Bautista, M. F. Anjos, and A. Vannelli, “Numerical study of affine supply function equilibrium in ac network-constrained markets,” IEEE Trans. Power Syst., vol. 22, no. 3, pp. 1174–1184, Aug. 2007. [27] P. F. Correia, T. J. Overbye, and I. A. Hiskens, “Searching for noncooperative equilibria in centralized electricity markets,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1417–1424, Nov. 2003. [28] Y. Liu and F. F. Wu, “Impacts of network constraints on electricity market equilibrium,” IEEE Trans. Power Syst., vol. 22, no. 1, pp. 126–135, Feb. 2007. [29] L. Xu and Y. Yu, “Transmission constrained linear supply function equilibrium in power markets: method and example,” in Proc. IEEE PowerCon 2002, 2002, vol. 3, pp. 1349–1354. [30] A. Halseth, “Market power in the Nordic electricity market,” Util. Policy, vol. 7, pp. 259–268, 1998. [31] T. Li and M. Shahidehpour, “Strategic bidding of transmission-constrained GENCOs with incomplete information,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 437–447, Feb. 2005. [32] A. Rudkevich, M. Duckworth, and R. Rosen, “Modeling electricity pricing in a deregulated generation industry: The potential for oligopoly pricing in a poolco,” Energy J., vol. 19, no. 3, pp. 19–48, 1998. [33] E. Hasan, F. D. Galiana, and A. Conejo, “Electricity markets cleared by merit order—Part I: Finding the market outcomes supported by pure strategy Nash equilibria,” IEEE Trans. Power Syst., vol. 23, no. 2, pp. 361–371, May 2008. [34] E. Hasan and F. D. Galiana, “Electricity markets cleared by merit order—Part II: Strategic offers and market power,” IEEE Trans. Power Syst., vol. 23, no. 2, pp. 372–379, May 2008. [35] Y. S. Son and R. Baldick, “Hybrid coevolutionary programming for Nash equilibrium search in games with local optima,” IEEE Trans. Evol. Computat., vol. 8, no. 4, pp. 305–315, Aug. 2004. [36] A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques. New York: Wiley, 1968. [37] D. I. Sun, B. Ashley, B. Brewer, A. Hughes, and W. F. Tinney, “Optimal power flow by Newton approach,” IEEE Trans. Power App. Syst., vol. PAS-103, no. 10, pp. 2864–2880, Oct. 1984. [38] P. Holmberg and D. Newbery, “The supply function equilibrium and its policy implications for wholesale electricity auctions,” Util. Policy, vol. 18, pp. 209–226, 2010. [39] A. G. Petoussis, X. P. Zhang, S. G. Petoussis, and K. R. Godfrey, “Electricity market equilibrium of nonlinear power systems with reactive power control,” Elect. Power Syst. Res., vol. 80, no. 5, pp. 537–546, May 2010.

[40] E. J. Anderson and X. Hu, “Finding supply function equilibria with asymmetric firms,” Oper. Res., vol. 56, no. 3, pp. 697–711, May-Jun. 2008. [41] S. Borenstein, J. Bushnell, and S. Stoft, “The competitive effects of transmission capacity in a deregulated electricity industry,” RAND J. Econ., vol. 31, no. 2, pp. 294–325, 2000. [42] A. G. Petoussis, “Supply function equilibrium analysis for electricity markets,” Ph.D. dissertation, Sch. Eng., Univ. Warwick, Coventry, U.K., 2009.

Andreas G. Petoussis received the B.Eng. (1st Class Honors) and Ph.D. degrees in electrical engineering from the University of Warwick, Coventry, U.K., in 2005 and 2009, respectively. He is a Consultant Electrical Engineer with the Cyprus Transmission System Operator, Nicosia, Cyprus. His research interests include electricity market equilibrium analysis, dynamic modeling of isolated power systems, and grid-connected renewable energy systems.

Xiao-Ping Zhang (M’95–SM’06) received the B.Eng., M.Sc., and Ph.D. degrees in electrical engineering from Southeast University, China, in 1988, 1990, and 1993, respectively. He is currently a Professor in Electrical Power Systems at the University of Birmingham, Birmingham, U.K., and he is also Director of the University Institute for Energy Research and Policy. Before joining the University of Birmingham, he was an Associate Professor in the School of Engineering at the University of Warwick, Coventry, U.K. From 1998 to 1999, he was visiting UMIST. From 1999 to 2000, he was an Alexander-von-Humboldt Research Fellow with the University of Dortmund, Germany. He worked at China State Grid EPRI on EMS/DMS advanced application software research and development between 1993 and 1998. He is co-author of the monograph Flexible AC Transmission Systems: Modeling and Control (New York: Springer, 2006 and 2012). Prof. Zhang is an Editor of the IEEE TRANSACTIONS ON POWER SYSTEMS.

Savvas G. Petoussis received the B.Eng. and Ph.D. degrees in electrical engineering at the University of Warwick, Coventry, U.K., in 2002 and 2006, respectively. He is an Electrical Network Engineer with the Electricity Authority of Cyprus (EAC), Networks Business Unit, Nicosia, Cyprus. He received the IEE Prize Award for his excellent performance achievement in his B.Eng. degree studies.

Keith R. Godfrey received the B.Sc. and Ph.D. degrees from the University of London, London, U.K., and the D.Sc. degree from the University of Warwick, Coventry, U.K. He is an Emeritus Professor and member of the Systems, Measurement and Modelling Research Group in the School of Engineering at the University of Warwick, Coventry, U.K. He is author/co-author of more than 200 papers. Prof. Godfrey was awarded the IEE Snell Premium (2000) and the Honeywell International Medal (2000–2001) of the Institute of Measurement and Control, for distinguished contributions to control engineering. He is a member of the IFAC Technical Committees on Biomedical Engineering and Control, and on Modeling, Identification and Signal Processing.