Nash equilibrium strategy in the deregulated power industry and comparing its lost welfare with Iran wholesale electricity market

Mousavi, Seyed Hosein; Nazemi, Ali; Hafezalkotob, Ashkan Article Nash equilibrium strategy in the deregulated power industry and comparing its lost welfare with Iran wholesale electricity market Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Mousavi, Seyed Hosein; Nazemi, Ali; Hafezalkotob, Ashkan (2016) : Nash equilibrium strategy in the deregulated power industry and comparing its lost welfare with Iran wholesale electricity market, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 12, pp. 421-435, https://doi.org/10.1007/s40092-016-0152-z This Version is available at: https://hdl.handle.net/10419/157514 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ ORIGINAL RESEARCH Nash equilibrium strategy in the deregulated power industry and comparing its lost welfare with Iran wholesale electricity market Seyed Hosein Mousavi 1 •Ali Nazemi 1 •Ashkan Hafezalkotob 2 Received: 8 December 2014 / Accepted: 5 May 2016 / Published online: 6 July 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract With the increasing use of different types of auctions in market designing, modeling of participants’ behaviors to evaluate the market structure is one of the main discussions in the studies related to the deregulated power industries. In this article, we apply an approach of the optimal bidding behavior to the Iran wholesale electricity market as a restructured electric power industry and model how the participants of the market bid in the spot electricity market. The problem is formulated analytically using the Nash equilibrium concept composed of large numbers of players having discrete and very large strategy spaces. Then, we compute and draw supply curve of the competitive market in which all generators’ proposed prices are equal to their marginal costs and supply curve of the real market in which the pricing mechanism is pay-as-bid. We finally calculate the lost welfare or inefficiency of the Nash equilibrium and the real market by comparing their supply curves with the competitive curve. We examine 3 cases on November 24 (2 cases) and July 24 (1 case), 2012. It is observed that in the Nash equilibrium on November 24 and demand of 23,487 MW, there are 212 allowed plants for the first case (plants are allowed to choose any quantity of generation except one of them that should be equal to maximum Power) and the economic efficiency or social welfare of Nash equilibrium is 2.77 times as much as the real market. In addition, there are 184 allowed plants for the second case (plants should offer their maximum power with different prices) and the efficiency or social welfare of Nash equilibrium is 3.6 times as much as the real market. On July 24 and demand of 42,421 MW, all 370 plants should generate maximum energy due to the high electricity demand that the economic efficiency or social welfare of the Nash equilibrium is about 2 times as much as the real market. Keywords Nash equilibrium Lost welfare Bidding strategy Genetic algorithm Iran wholesale electricity market Introduction The deregulation of electric power industry in Iran and many parts of the world is based on auction mechanism. For example, market participants in Iran wholesale spot market (a day-ahead electricity energy marketplace established in Iran) tender supply and demand curves for the day-ahead and hour-ahead energy markets in format of sealed bid. The spot market then constructs aggregated hourly supply and demand curves to determine market clearing prices (MCP). The importance of simulating bidding strategies of electricity markets can be investigated from several points. First part is related to the importance of comparison between actual results and optimal results from point of economic efficiency and lost welfare. In this &Ashkan Hafezalkotob [email protected]; [email protected] Seyed Hosein Mousavi [email protected] Ali Nazemi [email protected] 1 Department of Socioeconomic Systems Engineering, Economic College, University of Economic Sciences, the First Blind Alley, Jahan Alley, End of Taleghani Street, Tehran 1563666411, Iran 2 Department of Industrial Engineering, Industrial Engineering College, Islamic Azad University, South Tehran Branch, Entezari Alley, Oskoui Alley, Choobi Bridge, Tehran 1151863411, Iran 123 J Ind Eng Int (2016) 12:421–435 DOI 10.1007/s40092-016-0152-z way, market designers are continually trying to compare the present system and structure with efficiency criteria. Extraction between present deviations and observed differences is an appropriate tool to improve the market performance. Another importance of issue is related to participants’ strategies in the market. Generating companies require an appropriate theoretical and computational tool to bid an appropriate price and quantity to the market to evaluate accurately and increase profitability. Song et al. (2003) proposed the new method of conjectural variation model (CV) and its application in electricity markets. The conjectural variation-based bidding strategy model helped generators to improve their bids and maximize their profits. Kian and Cruz (2005) have evaluated development of biddings in a dynamic multipolar electricity market. They took the electricity market as a non-linear dynamic system and modeled it using Nash discrete bidding strategies. Swider and Weber (2007) proposed a Bayes strategy for the strategic bidder while the others’ behaviors are modeled with a probability distribution. Gao et al. (2008) proposed two approaches to determine market bidding strategies by the support vector machine. Accuracy of methods was examined with an example. Borghetti et al. (2009) proposed an analysis about the selecting process of the generators bidding strategies with regard to some constraints. This analysis was performed both for a simple approach of static game theory and for a cost-minimization unit-commitment algorithm using computer-based method. Bompard et al. (2010) used the linear supply function to find the Supply Function Equilibrium (SFE). They proposed a new and efficient approach to determine supply function equilibriums in the limited power markets by finding the best slope of the supply function with changing the intercept. Gong et al. (2011)havedonea complete literature analysis on the state-of-the-art research of bidding strategy modeling methods. Chunhua et al. (2012) made the benefit/risk/emission comprehensive generation optimization model with objective of profit maximization and bidding risk and emissions minimization according to the coordinated interaction between generating companies’ outputs and electricity market prices. Nojavan et al. (2013) have identified the optimal bidding strategy in day-ahead market using the Information Gap Decision theory. At bidding time, criteria such as generator characteristics and market price uncertainties that have a direct effect on the expected profit and the supply curve must be considered. Gap information decision-making indicates that risk aversion and risk taking will impact on the expected profit and the supply curve. The mentioned method has been applied to an unrealistic case study. Soleymani (2013) introduced a method to analyze the competition among companies with limited power transmission and incomplete information. In that method, supply function equilibrium was used for optimal strategies modeling of energy market participants and the Expected Function Equilibrium (SFE) was used to create an offer in the reactive power market. Finally, an experimental system was used to evaluate the effectiveness of the model. Mahmoudi et al. (2014) proposed a game theoretical model to show how plants maximize their utilities in each energy source by considering the government role in the competition of two power plants. Hafezalkotob et al. (2015) proposed a novel robust data envelopment model (RDEA) to investigate the efficiencies of decision-making units (DMU) when there were discrete uncertain input and output data. To illustrate the ability of proposed model, a numerical example of 38 Iranian electricity distribution companies was investigated. The results revealed that the RDEA model was suitable and reliable for target setting based on decision maker’s (DM’s) preferences when there are uncertain input/output data. Sadjadi et al. (2015) presented an integrated decision model based on recent advances of geometric programming technique that managed Joint pricing and production. The demand of a product considered as a power function of factors such as product’s price, marketing expenditures, and consumer service expenditures. Furthermore, production cost considered as a cubic power function of outputs. Mousavi et al. (2015) presented some metaheuristic algorithms to simulate how generators bid in the spot electricity market viewpoint of their profit maximization according to the other generators’ strategies, such as genetic algorithm (GA), simulated annealing (SA) and hybrid simulated annealing genetic algorithm (HSAGA) and compares their results. The results of the simulations showed that GA outperforms SA and HSAGA on computing time, number of function evaluation and computing stability, as well as the results of calculated Nash equilibriums by GA are less various and different from each other than the other algorithms. As seen above, main studies are performed based on the technical procedures and solely with the market simulation purpose. Studies are more based on the use of metaheuristic algorithms in development of computational models than the theoretical basics. Using various algorithms is the main advantage of these researches. On the other hand, the proposed methods have limited applications due to the required large size of data that are inaccessible. Modeling and simulating bidding strategies in a real and large market have not been performed yet. Moreover, the lost welfare measure or inefficiency of the Nash equilibrium in a real energy market has been rarely considered. So, the objectives of this article are: •Simulating Nash equilibrium of a real market with many participants. 422 J Ind Eng Int (2016) 12:421–435 123 •Comparing the efficiency of the uniform pricing mechanism versus the pay-as-bid pricing mechanism in a large wholesale electricity market. •Calculating the deadweight loss of a real market such as Iran wholesale electricity market. In addition, the assumptions of this article are: •There is at least a Nash equilibrium in the deregulated power industry. •The efficiency of the uniform pricing mechanism is more than the pay-as-bid pricing mechanism in a large wholesale electricity market. •The lost welfare of a real market such as Iran wholesale electricity market is computable. In this paper, game theory and the Nash equilibrium are used as the theoretical basis of evaluation. This study tries to simulate the bidding strategy in Iran electricity market as a large electric power industry with about 370 generating units by relying on the mentioned principles and finally determine the Nash equilibrium of this real and large market. In addition, it intends to compute the lost welfare on the Nash equilibrium with uniform pricing mechanism and the real market with the pay-as-bid pricing mechanism and compare them to understand which pricing mechanism is more efficient. There are many challenges about pricing mechanism in researches by Son and Baldick (2004) Skoulidas et al. (2002). We practically examine the efficiency of them. Implementation of proposed algorithm uses huge information to calculate Iran electricity market. Accordingly, implementation of this model is practically impossible for all days and we have to limit the modeling execution time. Hence, two specific models that characterize the minimum and maximum demand of Iran’s market are considered as two applicable examples. Market has faced maximum demand on July 24, 2012 with demand of 42,421 MW per hour and has faced the minimum demand on November 24, 2012 with demand of 23,487 MW per hour demand. These 2 days in 2012 have been selected for Nash equilibrium simulation. According to the information of the units (Iran Grid Management Co. 2012a,b), we compute the real supply curves of the market on July 24 and November 24, 2012. In addition, we compute and draw the competitive supply curves for two mentioned hours using the information like marginal costs of the generators that have been gained from the site of Iran Grid Management Co. (2012a,b). Then, we compare the lost welfare (efficiency) of the resulting equilibrium with the real supply curve by calculating the area between the competitive supply curve and each curve. The bigger area shows more lost welfare and less efficiency. We use the genetic algorithm, due to the simulation of real markets for the large number of participants is needed to an efficient and suitable computational tool. Methods Spot market The spot market is only the real-time market (Stoft 2002). In a spot market, the seller delivers its production immediately and the buyer pays for it ‘‘on the spot’’. (Kirschen and Strbac 2004). In the electricity market, two principal models of energy trading are considered: •The spot market •The bilateral agreements Models are recognized by the bid matching processes and the price setting mechanisms. The concept of spot market is used as the basis for the modeling of a general competitive market structure. It provides the solution for specifying the optimal bidding strategies (Beck et al. 2008). The trading process of the spot market consists of following steps (Kirschen and Strbac 2004): •Generating companies submit their bids that are ordered pairs of the proposed prices and quantities to supply certain amounts of electrical energy for the period under consideration. These bids are ranked in order of increasing price. From this ranking, the supply curve of the market is built. •Similarly, the demand curve of the market is made by asking consumers to submit offers specifying quantities and prices and ranking these offers in decreasing order of price. Since the demand for electricity is highly inelastic, the demand curve is assumed to be a vertical line at the value of the load forecast. •The intersection of these supply and demand curves shows the market equilibrium. All the bids submitted at a price lower than or equal to the market price are accepted and producers are allowed to produce the amount of energy corresponding to their accepted bids. Similarly, all the offers submitted at a price greater than or equal to the market price are accepted. •Generators are paid the market price for every megawatt-hour that they produce, whereas consumers pay the market price for every megawatt-hour that they consume, irrespective of the bids and offers that they submitted. Generally electricity is traded as a quantity of energy at a certain price during a specific time period (1, 1/2 h). J Ind Eng Int (2016) 12:421–435 423 123 Pricing mechanisms: uniform and pay-as-bid The purpose of the energy auction and determining the prices is the optimization of both buyers’ and sellers’ general satisfactions. The main constraint in this optimizing problem is the equality between demand and supply in the market clearing. The amounts of generators’ productions and consumers’ consumptions with their corresponding prices are the output of this problem. After the optimization, the process of payment is done according to one of the methods of uniform or pay-as-bid pricing. The procedure shown in Fig. 1is based on the assumption that all market participants (generators) being used in the uniform pricing mechanism, receive the same price. They all receive the Market Clearing Price (MCP) or market price. Single-price energy auctions are the most widely used methods in electricity markets in the world. Another alternative is to use a pay-as-bid procedure that means all participants being used, receive the price they bid, not the MCP. The procedure is shown in Fig. 2 (Wangensteen 2005). Indexes and parameters The following indexes are used in the proposed model: iNumber of generators (i¼1;2;...;N fg ) jNumber of individuals/number of joint strategies (in this article, each generator can propose 3 strategies. Therefore, there are 3Njoint strategies or individuals) hNumber of strategies that each generator can bid (In this article, each generator can propose 3 strategies) The researchers consider the following parameters: MCGiMarginal cost of generator i PGiProposed price of generator i QGiProposed quantity of generator i Pcap Price cap of electricity market Qmax GiMaximum generation capacity of generator i Qmin GiMinimum generation capacity of generator i UiSet of available strategies of player i (each strategy contains an ordered pair of PGiand QGi) uiThe specified strategy played by player i u ~Vector of all generators’ strategies (joint strategy: u ~¼u1;u2;...;uN fg ) UThe finite set of strategies (in this article, each generator can propose 3 strategies. Therefore, there are 3N vectors of strategies in U) JGiu ~ ðÞ The profit of player ifrom the joint strategy of u ~ Ju ~ ðÞ Generators’ joint profit DiAn absolute value of difference between the gained profit in the current configuration jand the possible maximized value of the profit for player i DuðÞ¼Fabs j¼DjCost (objective) function (sum of the differences between amounts of profit obtained in the current configuration (joint strategy:u ~) with the maximal possible amount of profit for each producer. It is equal to PiDi) Frelative jRelative fitness value of individual j MCP Market clearing price pih hth proposed price of generator i qih hth proposed quantity of generator i fih Fitness of hth generator i’s bid HR Generators’ heat rate (kcal=kW h) Fig. 1 Uniform pricing mechanism Fig. 2 Pay-as-bid pricing mechanism 424 J Ind Eng Int (2016) 12:421–435 123 wfuel Generation unit cost for fuel consumption (Rial=kcal) wSO2Generation unit cost for emission of sulfur oxide (Rial=kcal) rSO2Emission rate of sulfur oxide for generation unit wNO Generation unit cost for emission of nitrogen oxide (Rial=kcal) rNO Emission rate of nitrogen oxide for generation unit CostO&MMaintenance variable cost of each unit (Rial=kW h) Optimal bidding strategy/Nash equilibrium Constraints of generators’ profit maximization In the new deregulated environment, generation companies are free to charge any price for electricity they offer into the market taking into consideration some limits. Those limits are defined often by the regulatory and preventive measures like for example price cap (Pcap). Many electricity markets incorporate a price that is called ‘‘price cap’’ designed to prevent large price spikes (Kirschen and Strbac 2004). A price cap may be charged for a commodity. Price caps are used to prevent gouging during times of short supply or to limit price increases to a certain level. Price of bidding shall not be higher than this upper price limit specified by the market operator (MO) (Beck 2008). However, each generation company will solve its own profit maximization problem to get the benefit and the optimal generation schedule for the units. In this work, the principal actors are assumed to be the power producers selling the energy on a centralized market place. The objective of the optimization is to maximize the individual profit value. The profit (or payoff) of bidding generator JGiis computed in the following way: JGi¼Pmarket QGi ðÞCGiQGi ðÞ ð1Þ where Pmarket is the MCP, QGiis the quantity of power the generator Giis scheduled to produce and CGiQGi ðÞis the cost of energy production. Every generation company has an objective to maximize this profit from selling energy. To maximize the function mathematically we must take the derivative from both parts and equate them to 0: oJGi oQGi ¼oPmarket QGiCGiQGi ðÞ½ oQGi ¼oPmarket oQGi QGiþPmarket oCGiQGi ðÞ oQGi ¼0ð2Þ For a case of perfect competition, market price does not depend on the quantity of a single generator and so the quantity derivation of market price is equal to zero: oPmarket oQGi ¼0ð3Þ Then, the optimal price offer will be equal to the marginal cost of production: Pmarket ¼oCGiQGi ðÞ oQGi ¼MCið4Þ Since the perfect competition is an idealistic case, the real bidding price will normally be defined between marginal cost (MC) value and Pcap and will depend on the time of delivery and demand volume: MCGiPGiPcap ð5Þ In addition, for every single hour the bidding quantity should satisfy the general production limits. Qmin GiQGiQmax Gið6Þ Equilibrium of supply and demand in each market is considered as an inevitable constraint. X k QLk¼X i QGið7Þ Nash equilibrium Assuming that the participants in the game theory are rational, their strategies are directed with their profits. So, each person chooses a basket of commodities that will maximize his utility. max x2Xulx;lðÞ ð8Þ In Eq. 8,xis the set of possible choices for person l,lis a set of parameters that are out of control and ulis his utility function. In the game theory, strategies that are in interest of the person depend on the strategies of other players (opponents). So, we can say that lis the selected strategies of the opponent and xis the selected strategies of player land ulis his consequence. As a result, a player’s decision-making problem in the game theory is as follows: max sl2Sl ulsl;sl ðÞ ð9Þ In this equation, slis the combination of selected strategies of all players (opponents of player l) except player l. The key difference between these equations is that in Eq. 14, player ldoes not know choices of opponents (sl). But in the previous case, lis known to the person. So, choosing the best strategy (sl2Sl) in the game theory is required to simultaneously analyze each player’s decisions against his opponents. Nash equilibrium will occur according to the following conditions: J Ind Eng Int (2016) 12:421–435 425 123 •Players will select their strategies with the most consequence regarding to their belief about their opponents. •Players’ belief should be correct. It means that the opponent practically chooses the strategy that is in player’s belief. Mathematically, the combination of the strategy of s¼s 1;s 2;...;s n  2Swill be called Nash equilibrium if: uls l;s l  ulsl;s l  8sl2Sl 8l2N ð10Þ (Abdoli 2011) Generators’ joint profit maximization The complexity of the problem of optimal bidding strategies is when each generator’s profit (payoff) associates with other generating companies’ bidding strategies together. In this research, Nash equilibrium is used to solve this problem. Therefore, the problem changes from each generator’s profits maximization to simultaneous generators’ profits satisfaction and Nash equilibrium occurs when none of the participants is unilaterally reluctant to the change of the equilibrium and the solution. Mathematically, optimizing problem of the generators’ profits is considered as a search problem of vector u ~that causes to maximize the function. u ~¼u1;u2;...;uN ½2Uð11Þ Ju ~ ðÞ¼JG1u ~ ðÞ;...;JGiu ~ ðÞ;...;JGNu ~ ðÞ½ð12Þ The vectors u ~P;QðÞare Nmarket generators’ strategies that are extracted from a finite set (U). The vector u ~is equal to the proposed prices and the relevant quantities of production for all generators. Generating units’ short-term marginal cost Much research has been done about costs of plants. For example, Kumar et al. (2015) have analyzed the cost of a coal-fired power plant using the NPV method. To calculate the MC of generating companies, Mansur (2008) has introduced the following equation that indicates the shortterm marginal cost of generation for each year of power plant: MC ¼HR wfuel þwSO2rSO2þwNO rNO  þCostO&M ð13Þ where HR is the generator’s heat rate (kcal=kW h). wfuel, Wfuel wSO2and wNO are, respectively, generation unit costs for fuel consumption, emission of sulfur oxide and emission of nitrogen oxide (Rial=kcal) and rSO2and rNO are equal to emission rates of generation unit and also CostO&Mis the maintenance variable cost of each unit (Rial=kW h). As generators in Iran do not pay attention to the social costs in their bidding process, the costs of emission of sulfur oxide and emission of nitrogen oxide need not be considered in Eq. 13 (Nazemi et al. 2011). Therefore, short-term MCs of generators in Iran are achieved by Eq. 14. MC ¼HR wfuel þCostO&Mð14Þ Power plants’ fuel consumption and HR for every unit are extracted from the document of ‘‘Detailed statistics of power generation in Iran’’ (Tavanir Expert Holding Company 2013). As the plants use several fuels (gasoline, fuel oil and natural gas), fuel consumptions of generation units are formulated in the marginal cost formula as the weighted averages. We use the energy balance document to specify the fuel prices of plants (Ministry of Energy 2013). Formulation of the problem In this article, we are evaluating the generators’ profits based on the market clearing price (market price). Getting a set of bids in which all generators gain satisfactory profits is the aim of this simulation. As mentioned above, the most important characteristic of Nash equilibrium is that the participants’ selection in it does not necessarily make the most payoff (Abdoli 2011). In this situation, all generators gain a satisfactory profit. So, we are searching the Nash equilibrium instead of the maximization of every generator’s profit. This goal happens when each participant mutually changes his bid until it has no incentive to change its decision. According to the characterization of Nash equilibrium in games, Nash equilibrium search from point of the minimizing objective function on a joint strategy space changes to an optimization problem. Consider game Gwith Nplayers ( 1;2;...;N fg ). In this game, Uirepresents the set of available strategies of player i.uiis equal to the specified strategy played by player iand u ~¼ u1;u2;...;uN fg is a joint strategy for Nplayers. The profit of player ifrom the joint strategy of u ~¼u1;u2;...;uNis equal to (JiuðÞ). In such situation, the definition of Nash equilibrium for game Gis as follows: Combined strategy of u¼u 1;u 2;...;u N  will be the Nash equilibrium for game Gif we have for all i2 1;2;...;N fg and (ui2Ui): Jiu 1;u 2;...;u N  Jiu 1;...;u i1;ui;u iþ1;...;u N  ð15Þ We define the equilibrium search function DuðÞ:U!Rþ as a function on the combined strategy space of U (U¼U1U2UN) to identify this equilibrium (Beck et al. 2008): 426 J Ind Eng Int (2016) 12:421–435 123 DuðÞ¼ X N i¼1 max ui2Ui Jiu1;...;ui1; ui;uiþ1;...;uN ðÞJiuðÞ  ð16Þ Our purpose is to minimize Eq. 10.IfUis not Nash equilibrium, DuðÞwill be positive and otherwise will be zero. Joint strategy of uwill be an equilibrium for the game if Du  ðÞis zero. The above-mentioned function calculates the difference between payoff (profit) in the current situation and the maximal possible payoff for each producer. In this paper, optimization problem of bidding strategies in the electricity market for a particular period is generally as follows: minDu ðÞ ¼X N i¼1 max ui2Ui Jiu1;...;ui1; ui;uiþ1;...;uN ðÞ Jiu ðÞ  S:t: MCGiPGiPcap Qmin GiQGi0 QGiQmax Gi0 X j QLj¼X i QGi Q0&P0 ð17Þ Minimizing the lost welfare Social welfare In an ideal market the optimization problem refers to the problem of social welfare maximization. The aim of the Market Operator (MO) to meet the maximal demand at the minimal price corresponds analytically to maximization of the area between aggregated demand DQ ðÞ and supply SQðÞcurves in Fig. 3. The intersection of supply and demand curves gives a market price that is often called market clearing price or system marginal price (SMP). The surface CS below DQðÞis defined as the consumer surplus and the surface PS above SQðÞis the producer surplus. The social welfare is nothing else but the total surface between curves DQðÞand SQðÞ. Social Welfare ¼CS þPS ð18Þ The function that maximizes the surface between DQðÞ and SQðÞis called social welfare function. Social Welfare function ¼max r DQðÞSQðÞðÞdQð19Þ Minimizing lost welfare or maximizing economic efficiency Farrel (1957) defined the enterprise efficiency as generating an amount of output that was sufficiently more than a predefined amount of input. With this definition, he introduced several types of efficiency such as production efficiency, allocative efficiency and economic efficiency. Economic efficiency ¼Allocative efficiency Production efficiency ð20Þ In general, the lost welfare is usually due to lack of production and allocative efficiencies. In the short-term wholesale electricity market, allocative efficiency is not considered. Because in a wholesale market, the supply side includes generators and the demand side are distribution companies. Accordingly, the demand curve is approximately vertical and without elasticity. This problem is caused by two reasons: •The end consumer does not pay attention to the wholesale market price. Because the end consumer encounters with the electricity retail market and restructuring has not been realized in this market, consumers face predetermined and adopted prices. Accordingly, consumers will not react to them by increasing the prices. •Companies who developed the demand side of wholesale market should guarantee to supply electricity to consumers in every price (Mansur 2008). Therefore, Fig. 3changes to Fig. 4: Accordingly, to investigate the economic efficiency (social welfare) in short-term wholesale market, evaluation of production efficiency is sufficient. If the competitive market is independently able to operate and market price is achieved from the intersection of the supply and demand curves, social welfare will have the maximum value (Kirschen and Strbac 2004) and this market will have the maximum economic efficiency. So, in the short-term Fig. 3 Social welfare J Ind Eng Int (2016) 12:421–435 427 123 wholesale electricity market where the demand curve is approximately vertical and without elasticity, the amount of difference between result of Nash equilibrium curve and supply curve in state of perfect competition in which the price of plants’ bids are equal to their marginal costs represents the lost economic inefficiency of Nash equilibrium or lost welfare. In comparing supply curves of the real market of Iran with the Nash equilibrium, each of them which its curve has less distance with the curve of the competitive market is more efficient. Structure of the game This paper presents a static game with complete information. As we are analyzing bidding strategies in the spot electricity market for an hour, this game is static. In addition, the information about HR,wfuel, fuel consumption for each generator and any information to achieve MCs of plants are published (Ministry of Energy 2013). In addition, the maximum and minimum quantities of production of any generators are available (Iran Grid Management Co. 2012a,b). The price cap is also known. Therefore, each generator can get the space of other competitors’ payoff (profit). So, we can assume it as a static game with complete information. Nash equilibrium is the solution of this kind of game. Genetic algorithm The concept of the spot market is used as a base to model the structure of competitive markets. This concept presents a solution for the problem of dispatching in auctions and proposes the optimal bidding strategies in the electricity market. In this paper, we are facing the problem of Nash equilibrium calculation with the large number of generators that each generator has a set of specific strategies from quantity and price of electricity generation. Solving such a combinatorial problem by single enumeration has a complexity which grows exponentially with the number of players. The solution of this problem is based on the Nash equilibrium characteristics to search the minimizing function and relying on the metaheuristic methods is used to find the minimums. In this paper, we use the genetic algorithm (GA). GA is an oriented stochastic optimization technique that moves gradually towards the optimum point. This algorithm is applicable to every problem without any information about the problem and any restrictions on the type of variables. Its efficiency in finding the global optimum point has been proved. Capability of this method is in solving complex optimization problems in which either classical methods are not applicable or they are not reliable to find the global optimum (Fogel 2000). Crossover Crossover operator performs the partial exchange of characteristics (genetic material) between two individuals selected randomly from the current population. Therefore, newly created individuals inherit the characteristics of both ‘‘parents’’. Position(s) of crossover is defined randomly. In this research, we use both simple crossover and double crossover by applying a roulette wheel. Figure 5shows an example of a simple and a double crossover. Mutation operator Mutation introduces random modifications in the population, it helps preserve the diversity and prevent the algorithm from the premature convergence. It is performed on a single individual by modification of one value in a chain of characters according to some probability that tend to zero. It can improve the fitness of individual or deteriorate it. Figure 6shows an example of a mutation. Fig. 4 Social welfare in short-term wholesale electricity market Fig. 5 Crossover 428 J Ind Eng Int (2016) 12:421–435 123 Mansur ET (2008) Measuring welfare in restructured electricity markets. 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