حداقل پرداخت مصرف کننده تحت قیمت گذاری یکنواخت: یک روش برنامه ریزی خطی عدد صحیح مختلط

This paper presents a multi-period auction for a day-ahead pool-based electricity market in which consumer payment for energy is minimized under uniform pricing. This optimization problem has been recently characterized as a non-separable, non-linear, mixed-integer, and combinatorial problem for which exact solution techniques are unavailable. We present a novel approach suitable for existing mixed-integer linear solvers. A major contribution of this paper is the explicit characterization of uniform market-clearing prices as primal decision variables. The proposed methodology allows considering both quadratic and piecewise linear supply offers. In addition, the market-clearing procedure also takes into account inter-temporal operational constraints such as start-ups, ramp rates, and minimum up and down times, which may be part of generation offers. This approach provides the system operator and market agents with a valuable tool to assess consumer payment minimization versus currently used declared social welfare maximization. This conclusion is backed by simulation results obtained with off-the-shelf software.

The electric power industry is currently immersed in a competitive context wherein market-clearing procedures play a crucial role. In a pool-based electricity market for energy, the independent system operator (ISO) collects the offers and bids respectively submitted by producers and consumers. Generation offers may comprise not only economic terms but also technical features such as production limits, ramp rates, and minimum up and down times. Based on the application of a market-clearing procedure, the ISO determines the market-clearing prices, the power productions, and the consumption levels for all time periods [1] and [2]. Market-clearing procedures should be transparent, fair, and incentive-compatible, while allowing investment cost recovery [1]. Additionally, they should preserve the privacy of the corporate information belonging to market participants. The market-clearing procedures implemented in current electricity markets [2], [3], [4] and [5] are essentially identical to the unit commitment problem that is solved in centralized non-competitive power systems [6], [7], [8], [9], [10], [11] and [12]. The main conceptual difference between both problems lies in the maximization of the so-called declared social welfare by the ISO, which replaces the conventional minimization of the operating cost. Declared social welfare is a measure of social welfare that depends on the bids and offers submitted by market participants. When the demand is inelastic, the objective function only includes information on generation offers and the resulting problem becomes an offer cost minimization. Under ideal conditions including perfect competition and convexity, the optimal market-clearing solution represents an equilibrium solution at which suppliers have no incentive to offer different from their production costs. Moreover, this equilibrium maximizes the social welfare. Thus, the declared social welfare based on generation offers reflects the true social welfare, and hence, maximizing the declared social welfare is commonly accepted as the right goal in a market setting [1]. However, conventional market clearing presents several shortcomings [13] and [14]. In general, the assumption of perfect competition does not hold. As a result, generation offers may not reflect actual production costs and hence market-clearing solutions may be far from those maximizing the social welfare often to the detriment of consumers. This distortion may be stressed by the presence of non-convexities characterizing the operation of generating units [2], [6] and [15] such as start-up offers, ramping rates, minimum generation limits, and minimum up and down times. Therefore, the fairness and economic efficiency of currently used market-clearing procedures may be questioned. International experience in mitigating the effects of dishonest behavior and market power abuse includes attempts to improve the efficiency of offering in market-clearing procedures. Relevant examples can be found in several US electricity markets [16] and [17]. This practice however is not sufficient to completely disable the impact of market power. Therefore, structural changes in electricity markets are required, as suggested in [16]. Such need for renovation has motivated extensive research on alternative market-clearing procedures based on consumer payment minimization [14], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] and [30]. It is worth mentioning that this body of research relies on the pioneering work by Jacobs [13], who first proposed consumer payment minimization as an alternative goal in the auction design. These works have given rise to the recently coined notion of price-based market clearing [29], which embeds auction designs modeling the payment to market participants for the commodities provided such as energy [14], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] and [30], reserves [18], [23], [24] and [27], and reactive power [31]. Consumer payment minimization is a particular instance of this new class of auctions. Bearing in mind that actual production costs are private corporate information, consumer payment minimization has been proposed as a solution to the lack of incentives for suppliers to offer their actual costs in current auctions driven by the maximization of the declared social welfare. Thus, this alternative auction scheme may be considered as an instrument to protect consumers against the exercise of market power by suppliers via offering above their actual costs. However, consumer payment minimization faces obstacles for its practical implementation in current electricity markets. First, consumer payment minimization raises concerns related to (i) the discrimination of generators in favor of consumers, (ii) the potential for gaming, and (iii) the investment cost recovery. Those issues regarding the short- and long-term economic efficiency of this auction design have resulted in an open debate that is beyond the scope of this paper. The interested reader is referred to [13], [14], [20] and [28] for further details on this discussion. Notwithstanding, it should be noted that those works rely on simplified models and approximate solution approaches. Therefore, no conclusive answer to the above concerns is available and hence new tools are required to either prove or disprove the economic efficiency of consumer payment minimization. Moreover, alternative auction models based on consumer payment minimization [14], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] and [30] result in complex optimization problems as a consequence of their two main differences with respect to standard market-clearing procedures: 1. The objective function is modified to model the minimization of the consumer payment. Hence, the new objective function is explicitly dependent on the vector of market-clearing prices. 2. The constraint set is extended to include the mathematical expressions explicitly defining market-clearing prices. Market-clearing prices are decision variables that may themselves result from an optimization process. As a consequence, the presence of market-clearing prices in the problem formulation complicates the solution of the optimization problem. Moreover, the implementation of price-based market clearing depends on the selected pricing scheme. Unlike previous works based on marginal pricing [19], [24], [26] and [29], this paper considers a uniform pricing scheme for energy based on the price of the highest accepted generation energy offer. Thus, market-clearing prices do not rely on the economic interpretation of dual variables in mathematical programming as the sensitivity of the optimal value of the objective function with respect to a small perturbation in the corresponding constraint [32]. Rather, market-clearing prices become primal decision variables of the optimization process that are explicitly constrained in the problem formulation. This latter pricing scheme has received a great deal of attention in the technical literature related to consumer payment minimization [14], [18], [20], [21], [22], [23], [25], [27], [28] and [30] due to the following appealing characteristics as compared to marginal pricing: (i) it is a simple and transparent pricing framework suitable for markets with non-convexities [33], (ii) it guarantees price adequacy even under non-convexity, i.e., the price for energy paid to each supplier in each period is greater than or equal to the respective offer price for the level of energy produced in that period, and (iii) it avoids the price multiplicity issue [34] and hence the price profile is uniquely defined. Moreover, this pricing scheme yields market-clearing prices that are greater than or equal to those provided by marginal pricing thereby behaving similarly in terms of investment cost recovery. It is worth emphasizing that, as a result of the above relevant features, this pricing scheme is the standard in several European electricity markets [35], [36] and [37], where single-bus models are used in the auction designs. Notwithstanding, we acknowledge that Europe is immersed in the transition to location-dependent settings (see [38] and references therein) that might require changes in energy pricing and auction models. From a mathematical viewpoint, the consumer payment minimization problem under uniform pricing is a non-separable, non-linear, mixed-integer, and combinatorial problem. Unfortunately, exact solution techniques are currently unavailable. Relevant approaches include the approximate method described in [20], mixed-integer non-linear programming [21] and [27], dynamic programming [22], augmented Lagrangian relaxation and surrogate optimization [14] and [25], game theory [28], and genetic algorithms [30]. The combination of augmented Lagrangian relaxation with surrogate optimization [14] and [25] allows addressing large-scale problems and is thus regarded as the state of the art. However, this method requires heuristic procedures to find feasible solutions, which may be suboptimal, as recognized in [25]. Moreover, this heuristic manipulation may be viewed as a source of discrimination among market participants. Therefore, its practical implementation within a competitive context may be difficult. Besides, this technique requires the tuning of case-dependent control parameters, which may be a complex task itself. Consequently, new solution procedures for consumer payment minimization under uniform pricing are yet to be explored. Despite consumer payment minimization is not currently implemented in practice, the unavailability of accurate models and exact solution methodologies as well as the lack of precise knowledge of the economic efficiency of this auction design have both called for substantial research effort [14], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] and [30]. The significance and timeliness of consumer payment minimization are both evidenced by recent work [30], where wind power generation was considered in the alternative auction under a uniform pricing scheme. It is within this framework of growing interest in consumer payment minimization that this paper contributes to the state of the art by presenting a new market-clearing procedure for day-ahead energy trading in an electricity pool driven by consumer payment minimization under a uniform pricing scheme. The salient feature of the proposed method with respect to previously reported works [14], [18], [20], [21], [22], [23], [25], [27], [28] and [30] is the use of mixed-integer linear programming (MILP). MILP guarantees convergence to the optimal solution in a finite number of steps [39] while providing a flexible and accurate modeling framework. Moreover, the development of efficient solution techniques such as the branch-and-cut algorithm [40] has allowed the availability of optimized commercial solvers with large-scale capabilities [41]. As a consequence, mixed-integer linear programming has been applied to manifold problems including electric power generation scheduling [3], [8], [9], [12], [15], [29] and [42], energy systems management [43], waste management [44] and [45], and flood diversion planning [46], just to name a few. In [29], MILP was recently applied to the consumer payment minimization problem under marginal pricing. Such different pricing scheme results in substantial differences between the proposed MILP approach and that described in [29]. Fernández-Blanco et al. [29] presented a bi-level programming framework to incorporate marginal pricing in the alternative auction design. In order to convert the original bi-level program into a single-level equivalent, a primal–dual transformation was applied. Such transformation yielded non-linear terms involving dual variables. The subsequent linearization of those terms required setting bounds for the corresponding dual variables, which were selected on a trial-and-error basis. In contrast, the consumer payment minimization problem addressed here is formulated as a single-level optimization thereby precluding the use of the MILP-based approach described in [29]. Moreover, since a single-level transformation is no longer required, the proposed MILP framework is characterized by two salient features over Fernández-Blanco et al.’s previous work: (i) it solely relies on the primal space, and (ii) it does not need the use of big-M-based linear schemes where the values of sufficiently large parameters have to be adjusted. Furthermore, the proposed MILP approach allows considering both quadratic and piecewise linear supply offers for energy, unlike [14], [18], [19], [20], [21], [22], [23], [24], [25], [26], [28], [29] and [30]. Another distinctive feature of this paper with respect to [14], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] and [30] is the inclusion of inter-temporal constraints modeling ramping limits as well as minimum up and down times. Such technical aspects are critical in generation scheduling [2], [3], [6] and [47]. Hence, we argue that inter-temporal constraints should also be accounted for in the optimization process when the consumer payment is minimized. As a result, the new auction design also attains the desired goals of efficiency and feasibility in energy production [2]. The main contributions of this paper are threefold: 1. From a methodological perspective, MILP is proposed to schedule energy in a day-ahead pool-based electricity market driven by consumer payment minimization under uniform pricing. To the best of the authors’ knowledge, there is no current literature referring to the application of mixed-integer linear programming to such problem. 2. From a modeling perspective, market-clearing prices for energy are characterized as primal decision variables by a new mixed-integer linear formulation. Under this modeling framework, generation energy offers are not restricted to single-block linear curves but quadratic and piecewise linear functions are also handled. An additional novelty of the consumer payment minimization model is the consideration of inter-temporal constraints such as ramping limits and minimum up and down times. 3. Numerical experience is reported to show the impact of inter-temporal constraints and to illustrate the performance of the proposed approach. Rather than suggesting that declared social welfare maximization should be replaced by consumer payment minimization in currently used electricity auctions, the ultimate purpose of this paper is to provide the ISO and market agents with a tool that allows a comprehensive assessment of both market-clearing procedures. Moreover, the proposed approach may be useful for the ISO and the regulator in two respects, namely (i) to monitor the behavior of market agents participating in a day-ahead pool-based electricity market driven by consumer payment minimization under uniform pricing, and (ii) to examine the long-term implications of this trading scheme. Such analyses, which are beyond the scope of this paper, may require dynamic game theoretical models or equilibrium-based models wherein the proposed tool would play a key role. Relevant related works are those by Vázquez et al. [20] and Zhao et al. [28] where simplified approaches were applied for consumer payment minimization under the same uniform pricing scheme considered here. The remaining sections are outlined as follows. In Section 2, a general formulation for the consumer payment minimization problem under uniform pricing is presented. In Section 3, the proposed MILP model is described. In addition, computational considerations are discussed. In Section 4, numerical results are presented and analyzed. Finally, relevant conclusions are drawn in Section 5.

This paper has presented the application of mixed-integer linear programming to day-ahead energy clearing in a pool-based electricity market based on consumer payment minimization under uniform pricing. Exact solution techniques are unavailable for this non-separable, non-linear, mixed-integer, and combinatorial optimization problem. Mixed-integer linear programming is proposed here as an appropriate framework to accurately model the complexities associated with the presence of market-clearing prices for energy as primal decision variables in the problem formulation. As another salient feature, our market-clearing procedure allows accounting for quadratic and piecewise linear supply offers for energy, as well as inter-temporal operational constraints. The proposed models have been successfully tested on two case studies. Simulation results show that either optimal solutions or high-quality near-optimal solutions are attained in moderate computing times. Moreover, for the case studies analyzed, the consideration of inter-temporal constraints increases payment reductions over the optimal solutions from conventional offer cost minimization without significantly affecting offer costs. The proposed tool may be useful for the system operator and market participants to comprehensively assess consumer payment minimization versus currently used auctions based on declared social welfare maximization and to analyze the economic efficiency of the alternative auction design. The proposed approach features two main limitations, namely (i) the branch-and-cut algorithm may find difficulties in proving optimality within a practical time frame for large-scale systems, and (ii) the methodology is tailored to power systems wherein the transmission network does not restrict electric energy trading. One potential extension of this research is to develop a tighter formulation for the purposes of improved computational performance. Moreover, in view of the future transition in Europe toward accounting for the effect of the transmission network in electricity auctions, it should be noted that the main steps used in the proposed solution approach for the single-bus formulation are readily applicable to the network-constrained instance. Using a well-known dc load flow model based on sensitivity factors would render the network-constrained auction essentially identical to a single-bus model with one power balance equation in addition to a set of constraints imposing line flow limits. Furthermore, the uniform pricing scheme could be applied by defining the market-clearing price for energy at a specific location as the price of the highest accepted generation energy offer associated with the corresponding bus or zone. Hence, the resulting network-constrained market-clearing procedure would also be formulated as a mixed-integer linear program. Notwithstanding, we recognize that such model needs further numerical studies. We are presently investigating the consumer payment minimization problem in joint energy and reserve day-ahead electricity markets. Further research will be devoted to examining piecewise quadratic functions for generation energy offers. Another interesting avenue of research is the analysis of the impact of demand elasticity and the effect of the transmission network. Research will also be conducted to analyze the strategic behavior of market agents under the alternative market design and to examine the long-term effects of this trading scheme.