Optimization-Based Approach for Price Multiplicity in Network ...

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Optimization-Based Approach for Price Multiplicity in Network-Constrained Electricity Markets Natalia Alguacil, Senior Member, IEEE, José M. Arroyo, Senior Member, IEEE, and Raquel García-Bertrand, Senior Member, IEEE

Abstract—This paper addresses the problem of price multiplicity in network-constrained pool-based electricity markets under marginal pricing. This problem consists in the existence of multiple vectors of locational marginal prices for the same optimal marketclearing dispatch solution. As a consequence, delicate conflicts of interest may arise among market participants. Price multiplicity may take place even in auction designs based on convex formulations for the market-clearing procedure. An effective duality-based solution approach is proposed in this paper to handle the issue of price multiplicity. Once the optimal market-clearing dispatch is known, we propose the subsequent solution of a simple pricing problem in order to determine the vector of locational marginal prices. Unlike previously reported methodologies, the proposed approach presents two salient features: 1) it is based on sound mathematical programming, and 2) it is computationally inexpensive since it relies on the solution of a linear program. Numerical results are provided to illustrate the performance of the proposed tool.

Sets: Set of indices of generation offer blocks. Set of indices of demand bid blocks. Set of indices of generators. Set of indices of loads. Set of indices of buses. Set of indices of buses connected to bus . Set of indices of loads located at bus . Set of indices of generators located at bus . Constants:

Index Terms—Duality theory, electricity auctions, locational marginal prices, mathematical programming, price multiplicity.

NOTATION

Upper limit of energy block

bid by load .

Upper limit of energy block generator .

offered by

Capacity of the line connecting buses

The notation used throughout this paper is presented below. Functions:

and

Weighting factor associated with the consumer surplus. Weighting factor associated with the merchandising surplus.

Objective function of the pricing problem. Indices:

Weighting factor associated with the producer surplus.

Generation offer block index.

Reactance of the line connecting buses

Demand bid block index.

Price of energy block

Generator index.

Price of energy block offered by generator .

Load index.

and

.

bid by load .

Variables:

Bus index.

Consumer surplus.

Bus index. Index of the bus where generator is located. Index of the bus where load

.

is located.

Manuscript received October 31, 2012; revised February 11, 2013 and April 29, 2013; accepted June 05, 2013. This work was supported in part by the Ministry of Science of Spain, under CICYT Project ENE2012-30679; and by the European Commission, under Grant Agreement Number 309048. Paper no. TPWRS-01219-2012. The authors are with the Departamento de Ingeniería Eléctrica, Electrónica, Automática y Comunicaciones, E.T.S.I. Industriales, Universidad de CastillaLa Mancha, Ciudad Real E-13071, Spain (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRS.2013.2267213 0885-8950/$31.00 © 2013 IEEE

Merchandising surplus. Power consumed by load . Power consumed in block

by load .

Power output of generator . Power produced in block by generator . Producer surplus. Social welfare. Dual variable associated with the upper bound for the power produced in block by generator .

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Dual variable associated with the upper bound for the power consumed in block by load . Voltage angle at bus . Dual variable associated with the power balance equation at bus , representing the marginal market-clearing price at that bus. Dual variable associated with the power flow capacity of the line connecting buses and . I. INTRODUCTION

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ESTRUCTURED power systems worldwide have typically adopted a market design integrating 1) a pool-based market cleared on the basis of bids and offers submitted by market participants, and 2) a location-dependent marginal pricing scheme by which generators and consumers are respectively paid and charged at the corresponding locational market-clearing prices [1], [2]. Examples of this market design are commonly found in the US electricity markets [3], [4]. The pool-based market is run by the system operator (ISO), which implements a market-clearing procedure similar to the network-constrained unit commitment problem solved in centralized systems [5]. As a result, the set of accepted bids and offers maximizing the declared social welfare is determined. In addition, nodal market-clearing prices result from the optimization process based on marginal pricing and are denoted as locational marginal prices [6]. Under the assumption of perfect competition, the optimal market-clearing solution is equal to the equilibrium solution where the marginal value to consumers is equal to the marginal cost to producers, thus constituting the marginal market-clearing price. This equilibrium maximizes the social welfare, which is the sum of the consumer surplus, the producer surplus, and the merchandising surplus. As a consequence, suppliers and consumers have no incentive to respectively offer and bid different from their production costs and consumption utilities. Thus, the declared social welfare based on submitted offers and bids reflects the true social welfare, and hence, maximizing the declared social welfare is commonly accepted as the right goal in a market setting [1], [2]. In addition, the marginal pricing framework provides appropriate long-term economic signals to market participants [6]. Although widely adopted, current market design is characterized by several implementation difficulties. Most of these issues have been extensively addressed in the technical literature, such as those related to 1) the lack of perfect competition, which questions the suitability of declared social welfare maximization as the right goal [7], [8]; 2) the existence of multiple market-clearing schedules with the same optimal level of declared social welfare [9]–[11]; and 3) the presence of nonconvexities and their impact on pricing [12]–[16]. However, little attention has been paid to the issue of price multiplicity, also known as primal degeneracy. Price multiplicity consists in the existence of multiple vectors of market-clearing prices for the same optimal market-clearing dispatch solution [17], [18]. This issue may arise even in the

absence of nonconvexities and is due to the discontinuities in the aggregate supply offer and demand bid curves, which are typically stepwise. Primal degeneracy may raise delicate conflicts of interest due to the different surplus allocations to market agents as a result of different clearing prices. In [17], price multiplicity was analyzed for a single-bus system. Madrigal et al. showed that under degenerate conditions the uniform clearing price can be set at any value within a specific range defined by the intersection of the aggregate supply offer and demand bid curves. In order to resolve this ambiguity, the authors proposed choosing the maximum value of that range as the clearing price. However, this approach is not directly applicable to the network-constrained case. In [18], the degeneracy issue has been recently addressed considering the effect of the transmission network. Feng et al.’s approach was based on the proportional allocation to producers and consumers of the social surplus under the price multiplicity condition. In addition, the method was inspired by Karush-Kuhn-Tucker optimality conditions through the consideration of the dual feasibility equalities. As a result, an iterative procedure based on the resolution of a set of nonlinear equations was implemented. The method proposed by Feng et al. is incentive-based and features relevant properties such as boundary requirement, consistency requirement, neutrality requirement, sensitivity requirement, and limit appropriateness, which were defined in [18]. Aside from its interest from a theoretical point of view, price multiplicity is a relevant practical issue. In [18], Feng et al. highlighted the potential impact of price multiplicity in real-world electricity markets with few dominant participants. Moreover, the significance of price multiplicity is backed by the explicit consideration received in practice. For example, the operation rules of the electricity markets of New England [4] and the Iberian Peninsula [19] explicitly address the price multiplicity issue by ad hoc mechanisms similar to that suggested in [17]. Other electricity markets avoid the controversy associated with price multiplicity by defining locational marginal prices through a sensitivity analysis based on pre-specified perturbations [3], which may not be consistent with optimality. Additional market rules related to the numerical precision of submitted offers and bids as well as the imposition of effective constraint limits may help mitigate the impact of price multiplicity [20]. This paper proposes an alternative approach to address primal degeneracy in network-constrained electricity markets, thus broadening the scope of the single-bus solution in [17]. Moreover, the salient feature of the proposed approach over those of [3], [4], and [17]–[20] is the use of a sound, transparent, and computationally efficient mathematical programming framework. The proposed methodology consists in the formulation and solution of a new and computationally inexpensive problem based on duality theory. This problem, hereinafter referred to as pricing problem, is solved once the optimal market-clearing dispatch solution is known. While keeping the optimal market-clearing dispatch attained by a conventional network-constrained procedure driven by social welfare maximization, the pricing problem yields the optimal vector of locational marginal prices through the optimization of a

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pre-specified objective function. As a result, potential conflicts of interest among market participants are ruled out. It is worth mentioning that the resulting pricing problem is formulated as a linear program that can be straightforwardly solved with little computational effort. The mathematical programming framework proposed in this paper relies on a primal-dual formulation of the market-clearing procedure. Recent examples of successful application of primal-dual formulations in the context of electricity markets can be found in [15], [21], and [22], where the issue of price multiplicity was disregarded. In [15], nonconvexities associated with a unit-commitment-based auction were priced by minimizing the duality gap between the corresponding primal and dual problems. In [21], a primal-dual formulation was suggested for the purposes of congestion pricing in a multiperiod security-constrained economic dispatch. Finally, Fernández-Blanco et al. [22] applied a primal-dual-based methodology to convert the bilevel programming formulation of a price-based market-clearing procedure into a single-level equivalent. The proposed approach provides a flexible framework to address the price multiplicity issue through the pre-specification of the objective function to be optimized. The discussion on the choice of the objective function to be used is an important policy issue that is beyond the scope of this paper. Notwithstanding, this paper analyzes the impact of different objective functions in terms of the producer surplus, the consumer surplus, and the merchandising surplus. This analysis provides valuable information for the system operator, market agents, and the regulator in order to make informed decisions. The main contributions of this paper are threefold: 1) A new optimization-based pricing model is proposed to address the existence of multiple market-clearing prices in network-constrained electricity markets. 2) The model allows optimizing different objective functions, whose impact on pricing is analyzed. 3) The effective performance of the proposed tool is illustrated with the solution of several case studies. The remaining sections are outlined as follows. Section II describes a simple network-constrained market-clearing model. Based on such model, Section III illustrates the problem of price multiplicity. Section IV presents the formulation of the proposed pricing problem. Section V provides and discusses results from two case studies. Finally, some relevant conclusions are drawn in Section VI. II. NETWORK-CONSTRAINED AUCTION MODEL This section presents the formulation of the network-constrained market-clearing procedure used to analyze price multiplicity. Both primal and dual formulations are described. A. Primal Formulation For expository purposes, we use a single-period auction based on a dc optimal power flow (OPF) with no discrete variables associated with generation scheduling. The auction accounts for demand elasticity bearing in mind that such general model can be straightforwardly tailored to the particular instance of inelastic nodal demands. As is common in electricity markets

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[23], supply offer functions and demand bid functions are both stepwise, i.e., they are each characterized by a set of energy blocks and their corresponding prices. For the sake of simplicity, we consider that offers and bids submitted by market agents respectively represent the actual production costs and consumption benefits. Thus, the declared social welfare is equal to the actual social welfare. The resulting convex auction model is simple to describe and analyze, yet bringing out the issue of market-clearing price multiplicity. The dc-OPF-based auction is formulated as the following linear program: (1) subject to

(2) (3) (4) (5) where dual variables are shown in parentheses. The objective function to be maximized (1) represents the social welfare. This objective function comprises the sum of the bid benefits for consuming power minus the offered costs for generating power. Using a dc load flow model, constraints (2) represent the nodal power balance equations. Constraints (3) enforce the line flow limits. Finally, constraints (4) and (5) respectively set the limits of the energy blocks for generation offers and consumption bids. The total power output of generator can be computed as . Likewise, the total power consumed by load can be obtained by . Under marginal pricing [6], nodal market-clearing prices for energy are defined as the dual variables or Lagrange multipliers associated with the nodal power balance equations (2). Thus, once problem (1)–(5) is solved, the optimal values of the producer surplus, the consumer surplus, and the merchandising surplus are computed as follows: (6) (7) (8) where superscript denotes the value at the optimum. This work is focused on the degeneracy of problem (1)–(5), in particular, on the multiplicity of the optimal values of the market-clearing prices for the same optimal dispatch solution, and . Such multiplicity may yield different values of , , and for the same optimal value of the social welfare, .

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B. Dual Formulation Problem (1)–(5) is a linear program, hence, its dual counterpart can be formulated as [24]

(9) subject to (10) (11)

Fig. 1. Example of primal degeneracy.

(12) (13) (14) (15) , Moreover, the strong duality theorem [24] states that if ( , ) is a feasible solution to the primal problem (1)–(5), and ( , , , ) is a feasible solution to the dual problem (9)–(15), then both solutions are respectively primal and dual optimal if and only if

(16) These well-known results will be used later in the definition of the proposed pricing problem. III. PRICE MULTIPLICITY Price multiplicity is a result of primal degeneracy, which consists in the existence of multiple optimal dual solutions ( , , , ) for the same optimal primal solution ( , , ). Thus, although the optimal value of the social welfare is unique, different values of the producer surplus, the consumer surplus, and the merchandising surplus can be obtained. In general, primal degeneracy arises when the primal objective function is non-differentiable at the optimal solution. In other words, at the optimum, the slope of the objective function in one direction is different from the slope in the other direction. Such different slopes yield multiple solutions for the dual variables. As a consequence, there is ambiguity in the definition of marketclearing prices based on marginal pricing. There are many circumstances that could give rise to primal degeneracy in market clearing with the associated ambiguity and discontinuity in market-clearing prices. As an example, let us consider the simple auction model (1)–(5) where the effect of the network is neglected by considering sufficiently large line capacities. Therefore, a single uniform price is defined for all buses. The clearing of such simple auction can be graphically

represented by the intersection of the aggregate supply offer and demand bid curves. Fig. 1 illustrates a case of price multiplicity for the considered market-clearing procedure. As shown in Fig. 1, the intersection of the aggregate supply offer and demand bid curves takes place in vertical segments of both curves. In this case, the market-clearing price could be set anywhere in , ] of the overlapped vertical segments. In the range [ other words, any price in the range would support the auction solution and there would be no incentive to produce or consume different from the cleared quantities. It is worth mentioning that primal degeneracy can be straightforwardly generalized to the network-constrained case. Moreover, a graphical representation of primal degeneracy similar to that of Fig. 1 can be obtained if the dc network equations (2) and (3) are replaced by their equivalents based on sensitivity factors [1]. Under the equivalent model, the market-clearing price represented in Fig. 1 corresponds to the reference bus used to derive the sensitivity factors. Examples of such graphical illustration can be found in [4] and [18]. Primal degeneracy is a relevant issue since different energy prices may exist for the same auction solution thereby leading to different surplus allocations and undesired discrimination among market participants. Although this problem is illustrated in this paper with the convex auction model (1)–(5), the likelihood of occurrence is increased when nonconvexities associated with generation scheduling are considered, as done in actual electricity markets [1]–[4], [19], [20]. In the presence of primal degeneracy, the question is how to choose the set of locational marginal prices eventually used in the settlement process. If the effect of the transmission network is neglected, the implication of the discontinuity of the single market-clearing price can be easily addressed by previously agreeing upon the direction of the derivative of the objective function, as suggested in [17]. When the network is congested, this simple approach however may yield market-clearing prices that are not consistent with optimality conditions. Here we propose an optimization-based approach as an alternative to a recently reported method [18]. IV. SOLUTION METHODOLOGY This section presents a methodology to address price multiplicity in a network-constrained electricity market. Based on du-

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ality theory, the method consists in solving the pricing problem associated with the market-clearing problem (1)–(5). Locational marginal prices are thus obtained by solving an optimization problem in which 1) the objective function accommodates a pre-specified criterion, and 2) the set of constraints is built on duality-based expressions (10)–(16). In order to guarantee that the resulting prices correspond to the optimal primal solution, and are considered as inputs of the the optimal values pricing problem. It should be noted that in the absence of price multiplicity the proposed method yields the optimal values of the market-clearing prices according to marginal pricing. The proposed optimization-based pricing model is formulated as

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tees that such optimal vector of decision variables maximizes the social welfare thereby complying with the boundary requirement discussed in [18]. The interested reader is referred to [26] for further details on this primal-dual formulation. The objective function (17) reflects the interests of market participants and the regulator thereby requiring a beforehand agreement. Although the choice of a specific criterion is beyond the scope of this paper, the following expressions might be conin (17): sidered by market participants to replace (25) (26)

(17) (27) subject to (28) (18) (19) (20) (21) (22) (23)

(24) The pricing problem (17)–(24) is a linear program parameterized in terms of the optimal values of the primal problem, and , which are known once problem (1)–(5) is solved. Constraints (18)–(23) respectively correspond to the dual feasibility constraints (10)–(15), whereas the equality (24) is the strong duality condition (16) for the optimal solution to the primal problem (1)–(5). Based on duality theory and linear programming [24], dual feasibility constraints (18)–(23) and the strong duality condition (24) are equivalent to Karush-Kuhn-Tucker optimality conditions of the primal problem. This equivalence guarantees the consistency of the resulting vector of decision variables of problem (17)–(24) as dual variables of the primal problem. Thus, the proposed pricing model complies with the consistency requirement described in [18]. For the particular case of inelastic nodal demands, the resulting prices would be solely set by generation offers. Hence, the limit appropriateness analyzed in [18] is also met. In addition, the sensitivities of the decision variables of problem (17)–(24) are identical to the sensitivities of the dual variables of the primal problem, which are readily available [25]. Therefore, such sensitivities are bounded by the same values associated with the original market-clearing procedure. Moreover, the imposition of primal optimality in (24) guaran-

Expression (25) represents the negative producer revenues. Since the social welfare is set at its optimal value, minimizing such objective function leads to maximizing the producer surplus. This criterion, denoted as MXPS, is inspired by the approach proposed in [17] for single-bus systems, and favors producers to the detriment of consumers. Similarly, minimizing the consumer payments (26) is equivalent to maximizing the consumer surplus, which is referred to as MXCS. Analogous to MXPS, MXCS lacks neutrality since it favors consumers. Notwithstanding, it is worth mentioning that MXCS has received a great deal of attention from the power system community as an alternative goal in the design of day-ahead electricity auctions. Further details on MXCS or consumer payment minimization can be found in [7], [8], [22], and references therein. Expression (27) represents the merchandising surplus whereas (28) is the negative merchandising surplus. As per (6)–(8), minimizing the merchandising surplus (27) is equivalent to maximizing the overall surplus of producers and consumers. Hence, this criterion is likely to appeal both market participants since it favors them as a whole. In contrast, minimizing the negative merchandising surplus (28), i.e., maximizing the merchandising surplus, may be of interest for the ISO since this criterion allows partially recovering the fixed costs associated with the transmission network. Acronyms MNMS and MXMS are hereinafter used to denote the criteria related to minimizing and maximizing the merchandising surplus, respectively. It should be noted that MNMS and MXMS are only applicable when the optimal solution to the conventional auction (1)–(5) yields network congestion. In the uncongested cases, the merchandising surplus is equal to zero for all feasible solutions. Therefore, in the event of price multiplicity, optimizing the merchandising surplus would not allow making a distinction among all dual solutions leading to the maximum social welfare. For the sake of neutrality and acceptability by market agents, alternative criteria jointly accounting for the interests of all parties involved may also be considered. One possible implementa-

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tion of such joint criterion might be in the form of the following weighted average objective function :

Fig. 2. Four-bus system.

(29)

TABLE I ILLUSTRATIVE EXAMPLE—DATA FOR OFFERS AND BIDS

which can be simplified by dropping the constant terms as

TABLE II ILLUSTRATIVE EXAMPLE—LINE DATA

(30) In (29) and (30), , , and represent weighting factors respectively associated with the consumer surplus, the merchandising surplus, and the producer surplus. Each parameter models the relative significance of the corresponding individual surplus in the objective function. Therefore, the values of such parameters depend on the beforehand agreement reached and by market agents. For example, the choice yields MXPS, while setting and is equivalent to MXCS. Similarly, MNMS would be obtained by specifying and , whereas the choice and results in MXMS. A detailed discussion on how to obtain appropriate values for the weighting factors is beyond the scope of this paper. Note that similar models parameterized in terms of factors that are dependent on the preferences of decision makers are common in the power system literature [27], [28]. Moreover, specific constraints on the surplus allocations to market participants can be straightforwardly imposed in the formulation of the proposed pricing problem. Admittedly, the choice of the criterion driving the optimization is a source of arbitrariness inherent to the ambiguity in price definition from a strict mathematical viewpoint. Notwithstanding, we argue that the soundness, transparency, and simplicity of the proposed optimization-based approach are appealing features in order for market agents to agree upon a specific criterion. V. NUMERICAL RESULTS Results from two case studies are presented in this section. For didactical purposes, the pricing model (17)–(24) has been first applied to an illustrative example comprising four buses.

The second case study is based on the IEEE One Area Reliability Test System (RTS) [29]. For both systems, the effect of the transmission network has been investigated by considering an uncongested case and a congested case. The criteria examined in the uncongested cases are MXPS and MXCS. For the congested cases, the criteria analyzed are MXPS, MXCS, MNMS, and MXMS. The proposed methodology has been implemented on a PC with an Intel Core Duo processor at 2.93 GHz and 3.46 GB of RAM using MINOS 5.51 under GAMS [30]. The computing times required to achieve optimality were less than 1 s for all simulations. A. Illustrative Example This example consists of two generating units, two loads, and a transmission network comprising four buses and four lines. The topology of this system is depicted in Fig. 2. Generating units and consumers respectively submit two-block energy offers and bids, as presented in Table I. The data of the transmission network are provided in Table II, where line reactances are expressed on a 100-MVA base. 1) Uncongested Network: We first analyze the four-bus system with line capacities set at 1000 MW so that power flow limits (3) are not binding at the optimal solution. The optimal social welfare attained by problem (1)–(5) is $2668. Fig. 3 represents the market clearing. It can be observed that price multiplicity arises for the optimal power dispatch (567 MW). For this case, the market-clearing price ranges between $/MWh and $/MWh. It should be noted

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TABLE V ILLUSTRATIVE EXAMPLE—NODAL RESULTS FOR THE CONGESTED NETWORK

TABLE VI ILLUSTRATIVE EXAMPLE—SURPLUS ALLOCATIONS ($) FOR THE CONGESTED NETWORK

Fig. 3. Illustrative example. Market clearing for the uncongested case.

TABLE III ILLUSTRATIVE EXAMPLE—NODAL RESULTS FOR THE UNCONGESTED NETWORK

TABLE IV ILLUSTRATIVE EXAMPLE—SURPLUS ALLOCATIONS ($) FOR THE UNCONGESTED NETWORK

that producer revenues and consumer payments in this example, although identical for each possible price in the range, arbitrarily lie in the interval between $11 340 and $11 907. Table III summarizes the nodal results attained by the proposed methodology under MXPS and MXCS for this uncongested case. As listed in Table III, the lack of congestion yields a uniform price for all buses. However, it should be noted that, for the same levels of power generation and consumption, two different marginal prices are determined: $21/MWh under MXPS and $20/MWh under MXCS. Both prices respectively correspond to the aforementioned limits and . Note also that, in the absence of congestion, maximizing the producer surplus is equivalent to selecting as the market-clearing price, as suggested in [17]. For comparison purposes, the method presented in [18] has also been implemented. The resulting market-clearing price is equal to $20.67/MWh, which is an intermediate value within the range [ , ]. Table IV shows the optimal values for individual surplus allocations (rows 1 to 4) and total surpluses (rows 5 to 6), as well as the optimal social welfare (row 7) attained by the proposed approach for both objective functions being optimized. As expected, the surplus allocations vary with the objective function

selected for the same optimal social welfare ($2668). From the generator perspective, MXPS is the most advantageous, as this method leads to the largest producer surplus. In contrast, MXCS provides the best results for consumers. 2) Congested Network: In this case, the line capacities presented in Table II are imposed. The optimal social welfare is $2217.93, which is lower than that of the uncongested case. Table V lists nodal results in terms of power output, power consumption, and locational marginal prices. As shown in this table, marginal prices vary across location for each criterion. In addition, three different price vectors are identified. Table VI shows the individual and the total surplus allocations. As expected, the optimal social welfare is the same ($2217.93) irrespective of the selected objective function. From the producer perspective, MXPS is the preferred option. In contrast, consumers would be equally inclined toward MXCS and MNMS. Note that MXPS results in no consumer surplus. Finally, MXMS is an undesirable alternative for both producers and consumers since it yields individual surplus allocations equal to $0.00. For the sake of comparison, the sensitivity-based pricing scheme adopted in real-world electricity markets [3] has also been applied. The resulting locational market-clearing prices are $/MWh, $/MWh, $/MWh, and $/MWh, which yield a different surplus allocation to market participants: , , and . As can be seen, the producer surplus is greater than the maximum attainable level satisfying optimality conditions yielded by MXPS, which is equal to $400.00. This boost in producer surplus is obtained by reducing the merchandising surplus with respect to that obtained by MXPS. This optimality-inconsistent result stems from the infeasibility of the dual problem (9)–(15) with dual variables set equal to the aforementioned sensitivity-based vector of market-clearing prices. It is worth mentioning that, in the absence of price multiplicity, the proposed approach provides marginal prices identical to those resulting from the sensitivity-based method. In

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TABLE VII RTS-BASED SYSTEM—GENERATION OFFERS

TABLE VIII RTS-BASED SYSTEM—GENERATION RESULTS FOR THE UNCONGESTED NETWORK

TABLE IX RTS-BASED SYSTEM—CONSUMER RESULTS FOR THE UNCONGESTED NETWORK

contrast, when price multiplicity arises, the proposed pricing problem does not feature the optimality inconsistency characterizing the sensitivity-based pricing scheme. B. RTS-Based System The case based on the 24-bus RTS [29] comprises 32 generating units and 17 loads. Generation offers are structured in four blocks, as presented in Table VII. The load profile corresponds to Tuesday of week 51 at 18:00 [29]. For the sake of simplicity, the same bidding price of $45/MWh is considered for all loads. The data for the transmission network can be found in [29]. 1) Uncongested Network: The RTS-based system has been first solved assuming that line capacities are equal to 1000 MW, so that no congestion is experienced and the same uniform price characterizes all buses. In this case, the optimal value of the social welfare is $107 118.14. The proposed approach yields a price of $22.13/MWh for producer surplus maximization, whereas a price of $21.67/MWh results from maximizing the consumer surplus. Table VIII shows the results for those generating units with a power output greater than 0.00 MW. It should be noted that the total producer surplus under MXPS is 4.2% larger than under MXCS. Table IX shows the results for consumers. In this case, the total consumer surplus is 2.0% larger under MXCS. 2) Congested Network: The RTS-based system is analyzed considering the line capacities provided in [29], except for lines 3-24 and 14-16, for which their capacities have been respectively reduced from 400 MW and 500 MW to 193.333 MW and 272.280 MW. The optimal solution to the market-clearing problem (1)–(5) yields a social welfare of $105 555.69.

Table X lists locational marginal prices provided by the proposed approach under all criteria (25)–(28). For each criterion, different nodal prices are obtained as lines 3-24 and 14-16 are congested. Although nodal prices depend on the selected criterion, the same trend can be observed in all price profiles. As can be seen in Table X, for all criteria, prices at buses 1–14 are greater than $21/MWh, whereas prices at buses 15–24 are less than $21/MWh. This result is explained by the fact that the system can be split into two areas: an upper area comprising buses 15–24, with excess of generation; and a lower area including buses 1–14, with a deficit of generation capacity. Finally, Table XI lists the producer, consumer, and merchandising surplus allocations, as well as the social welfare for all criteria. In this particular case, MXPS improves the producer surplus achieved by MXCS by 9.6%. In contrast, the consumer surplus determined by MXCS is a 5.6% larger than that identified by MXPS. Note also that the maximum level of merchandising surplus achieved by MXMS is a 12.6% larger than the minimum level determined by MNMS. As imposed by the proposed approach, the social welfare remains identical to the optimal value of $105 555.69 no matter what objective function is selected.

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TABLE X RTS-BASED SYSTEM—LOCATIONAL MARGINAL PRICES FOR THE CONGESTED NETWORK

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For the purposes of wide consensus, a candidate criterion is recommended to simultaneously consider the interests of all market agents. Thus, rather than maximizing the producer surplus or the consumer surplus, which favors a single group of market participants and thereby lacks neutrality, an alternative criterion weighting the individual surplus allocations in the objective function might be an acceptable choice. Further research will be devoted to identifying the most appropriate criterion in terms of acceptability by market participants. Particular attention will be paid to the selection of suitable values for the weighting factors in the proposed joint criterion. REFERENCES

TABLE XI RTS-BASED SYSTEM—OVERALL SURPLUS ALLOCATIONS ($) FOR THE CONGESTED NETWORK

VI. CONCLUSIONS This paper has presented a mathematical formulation to effectively address price multiplicity in network-constrained pool-based electricity markets under marginal pricing. The ultimate goals of this paper are to raise awareness of the issue of primal degeneracy in pool-based electricity markets and to provide the ISO with a computationally inexpensive optimization-based tool to resolve the potential conflicts of interest that may arise among market participants. The proposed pricing problem is a linear program based on duality theory that is solved once the optimal market-clearing solution is known. The formulation presented in this paper models rigorously physical constraints while meeting maximum social welfare. Moreover, it is transparent for all participants and it is simple to implement and to interpret. The resulting linear programming problem is suitable to be solved by efficient off-the-shelf software with little computational effort. Different case studies were analyzed to appraise the performance of the proposed procedure, which presents an accurate and efficient behavior. Numerical results also shed light on the impact of different objective functions on the surplus allocation. This piece of information is relevant for the ISO and market agents in order to make informed decisions.

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Natalia Alguacil (S’97–M’01–SM’07) received the Ingeniero en Informática degree from the Universidad de Málaga, Málaga, Spain, in 1995, and the Ph.D. degree in power systems operations and planning from the Universidad de Castilla-La Mancha, Ciudad Real, Spain, in 2001. She is currently an Associate Professor of electrical engineering at the Universidad de Castilla-La Mancha. Her research interests include operations, planning, and economics of power systems, as well as optimization.

José M. Arroyo (S’96–M’01–SM’06) received the Ingeniero Industrial degree from the Universidad de Málaga, Málaga, Spain, in 1995, and the Ph.D. degree in power systems operations planning from the Universidad de Castilla-La Mancha, Ciudad Real, Spain, in 2000. From June 2003 through July 2004 he held a Richard H. Tomlinson Postdoctoral Fellowship at the Department of Electrical and Computer Engineering of McGill University, Montreal, QC, Canada. He is currently a Full Professor of electrical engineering at the Universidad de Castilla-La Mancha. His research interests include operations, planning, and economics of power systems, as well as optimization and parallel computation.

Raquel García-Bertrand (S’02–M’06–SM’12) received the Ingeniera Industrial degree and the Ph.D. degree from the Universidad de Castilla-La Mancha, Ciudad Real, Spain, in 2001 and 2005, respectively. She is currently an Associate Professor of electrical engineering at the Universidad de Castilla-La Mancha. Her research interests include operations, planning, and economics of electric energy systems, as well as optimization and decomposition techniques.