Right now employment of polices and tools to decrease the carbon emission through electricity generation from renewable resources is one of the most important problem in energy policy. Tradable Green Certiﬁcate (TGC) is an economics mechanism to support green power generation. Any country has the challenge to choose an appropriate policy and mechanism for design and implementation of TGC. The purpose of this study is to help policy makers to design and choose the best scenario of TGC by evaluating six scenarios, based on game theory approach. This study will be useful for increasing the effectiveness of TGC system in interaction with electricity market. Particularly, the competition between thermal and renewable power plants is modeled by mathematical modeling tools such as cooperative games like Nash and Stackelberg. Each game is modeled by taking into account of the two following policies. The results of the six scenarios and the sensitivity analysis of some key parameters have been evaluated by numerical studies. Finally, in order to evaluate the scenarios we calculated the level of social welfare in the all scenarios. The results of all models demonstrate that when the green electricity share (minimum requirement) increases the TGC price decreases. Moreover, in all scenarios when the minimum requirement is 100% then the maximum level of social welfare is not met. Also when the minimum requirement is less than 50%, the scenarios with the market TGC price policy have more social welfare in comparison with the scenarios with ﬁxed TGC price policy. Keywords Green electricity Tradable Green Certiﬁcate Game theory Mathematical modeling Introduction develop renewable energy many countries have set road map, goals and mandatory targets to reduce greenhouse The policies of energy sector are one of the most effective gases emissions. The share of the renewable energy (RE) policies in development of countries. Climate change and should be increased from the current 17–30 or 75% or even energy security are the most important factors in the energy to 90% in some countries by 2050. Also, European Union policies, setting regulations and energy models of invest- (EU) has set a minimum target of 20% by 2020 in total ment (REN21 2012; Bazilian et al. 2011). It is necessary to energy consumption (GEA 2012; Zhou 2012). reduce the greenhouse gases emissions in order to control The signiﬁcant outcome of using the RE will be the climate change (Buchner and Carraro 2005). Hence, to strengthening the economic growth by creating employ- ment, developing clean environment by reducing carbon emissions, enhancing technological innovation systems and curbing the volatility of fuel prices. On the other hand, RE can boost economic growth and it can mitigate pollutant & Ashkan Hafezalkotob emissions. Moreover, it can increase the supply adequacy a_hafez@azad.ac.ir; Hafezalkotob@iust.ac.ir and it might facilitate the access to electricity in order to Meysam Ghaffari promote the rural development and social welfare (Tiba st_m_ghaffari@azad.ac.ir; mghaffari7@yahoo.com et al. 2016; Azuela and Barroso 2011; Fargione et al. 2008). Industrial Engineering College, Islamic Azad University, South Tehran Branch, Entezari alley, Oskoui alley, Choobi Bridge, Tehran 1151863411, Iran 123 Journal of Industrial Engineering International One of the most important factors in reducing the carbon as supplier, transmitter, distributer, retailer and consumer emissions is electricity generation from renewable sources. of electricity (except the green electricity producers). This Currently, tendency of different countries to generate is obligated to purchase a certain share of the TGCs from electricity from renewable sources is increasing by using electricity producers based on the energy policies of every TGC systems and feed-in tariff (Tamas et al. 2010). country (Mitchell and Anderson 2000). Many researches have addressed feed-in tariffs. As a Certiﬁcates are usually issued by the government and in case in point, Oderinwale and van der Weijde (2016) used exchange for 1 MW/h or higher units or higher produced by an input–output table to analyze a next-generation energy the renewable power plant. Renewable power plant can be system to evaluate economic impacts of Japan’s renewable proﬁtable by selling certiﬁcates and physical electricity. TGC energy sector and the feed-in tariff system. market as ﬁnancial market is created by an interaction The previous researches indicate that the TGC system between the supplier of TGC (renewable power plant) and has better results in comparison with feed-in tariffs (Ciar- demandant of TGC (thermal power plant in this study). As a reta et al. 2014; Tama´s et al. 2010). case in point, Denmark has set obligation on customers The TGC system as an economic mechanism is intro- (Nielsen and Jeppesen 2003). In this policy, TGCs market duced to supply electricity from RE with the least cost for creates an interaction between the green electricity producers government. In this system, any entity of electricity supply and electricity consumers where the consumers are obliged to chain can require a certain share in the production or buy certiﬁcates or consume a certain proportion of the consumption of electricity from RE (Aune et al. 2010). renewable electricity based on minimum requirement. In this study, we will model the interaction between The countries may employ different mechanisms to thermal and renewable producers in the electricity and TGC organize the demand certiﬁcates by markets where the thermal producer is an obligation to supply 1. Setting a ﬁxed price at certiﬁcates, a certain share of green electricity by buying TGC from 2. Creating an obligation at every entity of the electricity renewable producer. The models will be analyzed based on supply chain to purchase certiﬁcates within a certain imperfect competitive/cooperative situations like Nash and period, Stackelberg equilibriums. The impact of minimum require- 3. Establishing a mechanism to tender purchasing ment and the TGCs price on total electricity and electricity certiﬁcates, price will be investigated by a numerical study. 4. Using a voluntary demand mechanism for certiﬁcates The reminder of the paper is organized as follows: (Schaeffer et al. 2000). Literature review is presented in Sect. 2. Section 3 describes the prerequisites and assumptions. In Sect. 4, In TGC system content, there are a few formal resear- ches (Tama´s et al. 2010). By using economic analysis, proﬁt function of the power plants in electricity and TGC markets is set up. Section 5 presents six scenarios based on Jensen and Skytte (2003) modeled the interaction of the the game theory models and TGC pricing policies. Sec- electricity market (with the assumption monopolistic tion 6 introduces the pricing system of electricity and TGC competition) and the TGC market (with the assumption of in six scenarios. Section 7 discusses the evaluation of a perfect competition). They showed that relationship policies by a numerical study and sensitivity analysis. between the TGC price and electricity price is linear. With Finally, Conclusion is provided in ‘‘Appendix’’ section. the same method, the polish scheme with regard to its economic functioning and its justiﬁcation with reference to solve common obstacles for renewable technology deployment was analyzed by Heinzel and Winkler (2011). Literature review The results demonstrate that the scheme is not mandatory TGCs have been introduced as ﬁnancial assets and they are to solve obstacles on the legal or institutional level. After allocated to the renewable power plants in exchange for the their liberalization, social acceptance might rather decrease amount of green electricity generated from renewable when power price for consumers goes up. sources. The outcome of this would be that the renewable By using the quality methods, Verhaegen et al. (2009) producers will beneﬁt from sale of physical electricity in described and analyzed the details of the TGCs system in electricity market and sale green certiﬁcates in TGCs Belgium. With the same method, Verbruggen and Lauber market (Farinosi et al. 2012). (2012) evaluated the feed-in tariff and TGC system in three criteria of efﬁciency, equity and institutional feasibility. TGCs system is usually operates as a market and is based on demand and supply. The demand of TGCs is Some of the researchers analyzed the TGC system by using the system dynamic method. In recent researches, this determined by energy policies and the annual share of electricity production from renewable sources. Obligation method has been used for conceptualizing, analyzing, can be set on any point of the electricity supply chain such designing and evaluating issues in energy sectors such as 123 Journal of Industrial Engineering International energy policy, power pricing, strategies of electricity There is a little comprehensive research about modeling market, and interaction between electricity and TGC mar- of the TGC system. Most previous studies analyzed the kets (Ahmad and bin Mat Tahar 2014). Ford et al. (2007) electricity and TGC markets by economic, and a few predicted the price of certiﬁcates to aid green electricity dynamic system methods investigated the implementation from the wind resources. The results showed that after a of this policy in a speciﬁc country. To the best of our few years the wind power exceeds the requirements knowledge, almost TGC system has not been analyzed by a because in the early years when a market opens the price of game theoretical approach under pricing policies. How- TGC will be increased rapidly. Recently, Hasani-Marzooni ever, in this study six different scenarios are analyzed and Hosseini (2012) modeled the TGC system by based on two common pricing policies in the TGC system employing the system dynamics to identify the potential to enhance the knowledge of designers and policy makers investment in the wind energy. They showed that the sys- in designing and deploying the TGC system. tem dynamics can be used as an appropriate tool to The contributions of this paper are as follows: investigate TGC market and help the regulatory authorities 1. We analyzed game theory models to achieve appro- to choose the appropriate policies in the energy sector. priate mechanisms to design market structure for TGC To analyze the TGC system, a number of mathematical market. We showed some outcomes and impacts. models are used by some researchers. Marchenko (2008) 2. We modeled the market structure for electricity and through a simple mathematical model simulated the bal- TGC markets in case of imperfect competition Cournot ance of supply and demand in electricity and the TGC oligopoly and monopoly under ﬁxed and variable TGC markets. He showed that the TGC system is not an price policy. appropriate policy to minimize the negative effects of 3. We used social welfare function for evaluating the energy production in the environment. Gu¨rkan and developed scenarios so policy makers and government Langestraat (2014) analyzed the renewable energy obliga- will be enable for choice the ﬁnest of energy policies. tions and technology banding in the UK by a nonlinear mathematical model. They studied three policies to apply the TGC and showed that the obligation target by UK banding policy cannot be achieved necessarily. Prerequisites and assumptions Recently Ghaffari et al. (2016) investigated a game theo- retical approach research to analyze the TGC system. In this We concentrate on the interaction of two producers for simplicity: renewable and thermal power plants. Electricity practice, the TGC price is assumed to be constant and will be determined by the government. They demonstrated that the producers compete in the electricity and TGC market under producer obligation. Thus, thermal power plant is obliged relation between the electricity price and the TGC price is reverse, whereas the relation between the electricity price and to buy a certain amount of TGCs based on minimum the minimum requirement is direct. Also in renewable power requirement. plant Stackelberg model, the production of total electricity The government sets the minimum requirement. We and the renewable electricity is at the maximum, while the consider two policies for the price of certiﬁcates. In the ﬁrst price of electricity is at the minimum. policy, the price of certiﬁcates is ﬁxed and is set by the Game theory is the one of the most important tools in government. In the second policy, the price of certiﬁcates is decision-making. Game theory focuses on the interaction determined by market conditions and supply and demand among the players in a game by assuming the conditions mechanisms. that each player chooses to rationalize their preferences (Myerson 1991; Jørgensen and Zaccour 2002). Notations According to game theory, all the players can use from pure or mixed strategies for their own interests. The reaction In this study, parameters and decision variables are as of an actor in a critical situation in a game can deﬁne a pure follows: strategy. Each combination of different player strategies will have a speciﬁc payoff for these players. The numbers of the Parameters desirability of possible outcomes show the payoffs in the game. These payoffs are dependent on the applied strategies a the minimum requirement of renewable electricity, of the players. There are two types of games such as coop- 0 a 1; erative and noncooperative games. In the ﬁrst one, the players p the proﬁt function of renewable producer; p the proﬁt function of thermal producer; intend to cooperate with each other for higher economic and environmental beneﬁts. In the second one, the system might p the total payoff of centralized producer, reach an equilibrium state (Lou et al 2004). (p ¼ p þ p ); R T 123 Journal of Industrial Engineering International Table 1 Scenarios of TGC Game theory models Market price of certiﬁcate Fixed price of certiﬁcate implementation Nash NM scenario NF scenario Stackelberg SM scenario SF scenario Cooperative CM scenario CF scenario TGC markets separately. The renewable producer cost C the cost function of thermal producer; C ðÞ q is a function of green electricity generated. The R R C the cost function of renewable producer; cost of the renewable power plant is only dependent on the U the consumer utility; green electricity generated q . Therefore, the renewable D the function of environmental damages. producer proﬁt maximization problem will be as follows: Max p ¼ P q þð1 aÞP q C ðÞ q R e R c R R R Decision variables S:t: ð1Þ P the end-user price of electricity ($/MWh), P [ 0; q 0 e e P the price of TGC ($/MWh), P [ 0; c c where P is the end user of each MW of generated elec- q the electricity generated from thermal energy (MW), tricity, a is the minimum requirement of green electricity q 0; and P is the price of certiﬁcate. This means that a q the generated from renewable energy (MW), q 0; R R renewable producer can receiveðÞ 1 a P for each unit in Q the total amount of electricity (MW), addition to the electricity price. Under the TGC system, a Q 0ðÞ Q ¼ q þ q . T R renewable producer would obtain per unit ‘‘subsidy’’ As shown in Table 1, we have modeled six scenarios to ðÞ 1 a P . implement the TGC system based on the game theory approach and government policies regarding the control Thermal producer price of certiﬁcates. Thermal producers can fulﬁll their obligation by either Assumptions production of the renewable electricity or buying the TGCs from renewable producer. The following assumptions have been considered in the Thermal producer cost C ðÞ q is a function of the T T proposed models: thermal electricity q . Therefore, the thermal producer proﬁt maximization 1. There is no limitation for power plants in consumption problem will be as follows: of the resources. 2. There is no limitation on the demand and supply Max p ¼ P q P aq C ðq Þ T e T c T T T electricity and TGCs. S:t: ð2Þ 3. There is no excess demand and supply in the electricity q 0 and TGCs markets. 4. Parameters are deterministic and they are known in Thermal power plant can receive P for each unit of advance. electricity. The thermal producer is obliged to pay for 5. The demand function of TGC is similar to demand buying the TGC from the renewable producer in order to function of electricity. compensate the unfulﬁlled requirements. Therefore, the 6. In all models, the supply of certiﬁcate meets the thermal producer under the TGC system virtually pays a minimum requirementðÞ q aQ . R per unit ‘‘tax’’ aP as in Eq. (2). In the developed model, there is just one thermal power plant that is obliged to hold a number of the TGCs equal to a times its production. Model formulation Renewable producer In this practice, we adopted proﬁt functions for the power Amundsen and Nese (2009), discussed the relation of: P = wholesale plants by Tanaka and Chen (2013). Renewable power plant electricity price ? a P must be established in the competitive can sell the electricity and the TGCs on the electricity and equilibrium market with a large number of retailers. 123 Journal of Industrial Engineering International Cost functions Nash equilibrium is a vector of participation decisions so that no player has an incentive to deviate from his chosen strategy We adopt cost functions for the power plants by Jensen and after considering an opponent’s choice (Urpelainen 2014). All players have no motivation to exit the equilibrium, Skytte (2003) and for renewable and thermal power plants it can be described as follows: because the outcome of this will be reduction in proﬁt of players. Krause et al. (2006) deﬁned the Nash equilibrium as C ðq Þ¼ a q þ b q þ c ð3Þ R R R R R R follows: C ðÞ q ¼ a q þ b q þ c ð4Þ T T T T T T The strategy proﬁle in a (n) players game of P ¼ P ; ...; P is a Nash equilibrium (NE) if for all i 2 1 n In Eqs. (1) and (2), it is assumed that a , b ; a ; b [ 0. R R T T fg 1; ...; n there is: Profit maximization problem for power plants U ¼ P ; ...; P P ; ...; P ; P ; P ; ...P ð9Þ i i 1 n 1 i1 iþ1 n where U is the utility function of the ith player. Following Newbery (1998) and Tamas et al. (2010), we In this section, we consider a Cournot-NE game under a assume that the demand function for electricity is a linear TGC system. function, It can be seen that by solving NE , from Eqs. (7) and (8) P ¼ c bQ ¼ c bðÞ q þ q ; ð5Þ e R T q and q will be obtained. Now, with substitution of q T R T and q into p and p , the maximum proﬁt of the pro- Meanwhile, Q ¼ðq þ q Þ is the total electricity. On R T R T the other hand, we assume that the price of electricity is a ducers (p and p ) will be reached. Propositions 1 and 2 R T decreasing function of amount of renewable electricity. present the optimum electricity production quantities in Moreover, based on sixth assumption the demand function Nash equilibrium under ﬁxed TGC price and market TGC of TGC is similar to electricity assumed. The inverse price polices, respectively. Subscripts [NF] and [NM] demand function of TGC is as follows: denote the equilibrium points in the Nash game under ﬁxed TGC price policy and the market TGC price policy, P ¼ h uq ð6Þ c R respectively. With substitution of Eq. (5) into Eqs. (1) and (2), the Proposition 1 Under the ﬁxed TGC price policy, the proﬁt maximization problem can be formulated as follows. optimum amounts of production for the renewable and Renewable producer is given as below: thermal producers in the Nash model can be given as Max p ¼ðc bðq þ q ÞÞq þ P q a q b q c R R T R c R R R R R below: s:t: P ð2aa þ ab 2a 2bÞþ A c T T 1 q 0 q ¼ ð10Þ RN½ F 2D þ 3b ð7Þ PðÞ 2aa þ 2ba þ b þ A c R 2 q ¼ ð11Þ Thermal producer is given as below: TN½ F 2 2D þ 3b Max p ¼ðc bðq þ q ÞÞq P q a q b q c T R T T c T T T T T where A ¼ 2b a 2a c þ 2bb bb bc; A ¼ 2b 1 R T T R T 2 T S:t: a 2a c þ 2bb bb bc; D ¼ 2a a þ 2a b þ 2a b: R R T R R T R T q 0 All propositions have been proven in ‘‘Appendix’’. With ð8Þ substituting the optimal quantities and Cournot TGC price Note that with substitution of Eq. (6) into Eqs. (7) and into Eqs. (7) and (8), optimal proﬁt of the power plants can (8), the problems of producers under market TGC price be calculated. policy will be obtained. Proposition 2 Under market TGC price policy, the opti- mal amounts of production for the renewable and thermal producers in the Nash solution can be given as below: Game theory models 2aa h þ abh þ F T 1 q ¼ ð12Þ RN½ M Noncooperative Nash game 4aa u þ 3abu þ F T 2 a hu 2aa h ab u þ 2ab u acu ahu þ E R R T 1 q ¼ If no player has anything to gain by changing his strategy, TN½ M 4aa u þ 3abu þ F T 2 when the other players do not change their strategies, then the ð13Þ set of strategies for all the players and the corresponding payoffs constitute a Nash equilibrium (Lou et al 2004). The 123 Journal of Industrial Engineering International ð2aa h þ abh þ F Þðau bÞ T 1 where F ¼ 2b a 2a c þ 2b h þ 2bb bb bc 1 R T T T R T q ¼ TS½ M 2 2ðb þ a Þð4aa u þ 2abu b þ F Þ T T 2 2bh; F ¼ 4a a 4a b 4a b 4a u 3b 4b 2 R T R T T ah b þ c u; E ¼2a b þ 2a c þ bb 2bb þ bc bh 2b 1 R T R R T T : ð17Þ 2ðb þ a Þ u þ 2cu. Cooperative game Noncooperative Stackelberg Games In this section, a cooperative relationship between thermal We investigated a noncooperative structure for interaction and renewable producers is investigated. In this model, between the thermal and renewable producers where the power plants collaborate together in electricity and TGCs initiative is the possession of one of the power plants, i.e., markets. We investigate this situation to increase our the leader. This can enforce its strategy on its rival, i.e., the knowledge about how to divide thermal producer capacity follower. The ﬁrst move is made by leader to maximize its to generate in competition with the renewable producer. proﬁt and then in return the follower reacts by choosing the Summation of Eqs. (7) and (8) gives cooperative model: best strategies. Max p ¼ðc bðq þ q ÞÞq þ P q a q b q R T R c R R R R Since the objective of the TGC system is supporting the c þðc bðq þ q ÞÞq R R T T increasing share of the electricity generated by RE pro- ducer, in this research we only examine renewable pro- P q a q b q c c T T T T T ducer—Stackelberg model where the renewable producer S:t: is leader and the thermal power plant is the follower. In this q ; q 0 R T model, the renewable producer ﬁrst sells its generated ð18Þ electricity in electricity market. Then the follower as thermal producer sells its generated electricity in electricity A Hessian matrix of p in Eq. (18) is: H ¼ market and buys certiﬁcates from renewable producer. 2b 2a 2b and the utility function p is a Propositions 3 and 4 present the optimum production of 2b 2b 2a electricity from renewable and thermal producers in concave function on (q ; q ) if and only if the Hessian R T Stackelberg equilibrium under ﬁxed TGC price and market matrix H is negative deﬁnite. Propositions 5 and 6 present TGC price polices, respectively. Subscripts [SF] and [SM] the optimum production quantities of green and thermal refer to optimal values of Stackelberg models under the electricity of producers in cooperative game under ﬁxed ﬁxed TGC price and market TGC price, respectively TGC price and market TGC price polices, respectively. Subscripts [CF] and [CM] denote the optimum values in Proposition 3 Under ﬁxed TGC price policy, the optimal the cooperative game model under ﬁxed TGC price and amount of electricity generated from renewable and fossil market TGC price polices, respectively. sources in renewable producer—Stackelberg model—is: P ð2aa þ ab 2a 2bÞþ E c R R 2 Proposition 5 Since ð2b 2a Þð2b 2a Þ R T q ¼ ð14Þ R½SF K ð2bÞð2bÞ [ 0, the optimal amount of electricity gen- 2 2 erated from renewable and fossil sources in the coopera- P 2aa b þ ab aE 2a b 2b þ bE þ Kðc b Þ c R 2 R 2 T q ¼ T½SF tive game model under ﬁxed TGC price policy will be: 2E ðb þ a Þ 2 R ð15Þ PðÞ aa a bþ B c T T 1 q ¼ ð19Þ R½CF where E ¼ 2a b 2a c þ 2bb bb bc; K ¼ 2 R R R R T 2 PðÞ aa þ bþ B c R 2 4a þ 8a b þ 2b . q ¼ ð20Þ T½CF Proposition 4 Under market TGC price policy, the opti- where B ¼ a b a c þ bb bb ; B ¼ a b a c 1 T R T R T 2 R T R mal amount of electricity generated from renewable and bb þ bb : R T fossil sources in renewable producer—Stackelberg model—is: Substituting Eq. (4) into Eq. (18), the problem of proﬁt centralized power plant under market TGC price policy 2aa h þ abh þ F T 1 q ¼ ð16Þ RS½ M 2 yields: 4aa u þ 2abu b þ F T 2 123 Journal of Industrial Engineering International Table 2 Price of TGC in six Game models TGC price policy scenarios Market price Fixed price ð2aa hþabhþF Þu T 1 Nash P ¼ cte P ¼ h cN½ F cN½ M 4aa uþ3abuþF T 2 ð2aa hþabhþF Þu T 1 Stackelberg P ¼ cte P ¼ h cS½ F cS½ M 4aa uþ2abub þF T 2 ða uhþ2aa hþab hacuþG Þu Cooperative T T 1 P ¼ cte P ¼ h cC½ F 2 2 cC½ M a u þ4aa uþF þ3b T 2 ‘‘cte’’ represents a ﬁxed value transaction level focuses on managing the implementation Max p ¼ðc bðq þ q ÞÞq þðh uq Þq R T R R R of discounts away from the reference or the price list which a q b q c þðc bðq þ q ÞÞq R R R R R T T occur both on and off the invoice or receipt. ðh uq Þq a q b q c R T T T T T T In this section, the pricing at the electricity market level S:t: is considered in oligopoly and monopoly market structures. Oligopoly is a common form of market where a number of q ; q 0 R T ﬁrms are in competition with each other. Based on the ð21Þ game theory approach, the Cournot–Nash and Cournot– A Hessian matrix of the proﬁt function in the TGC Stackelberg models are the oligopoly models. The oligo- market price policy is polies are in fact price setters rather than price takers (Perloff 2008). By substituting the optimal amounts of 2au 2a 2b 2uau 2b H ¼ and the utility green and black electricity production quantities in the 2b 2a 2b 2a T T payoff functions of the renewable and black power plants, function in the cooperative model is a concave function on the optimum prices of the electricity and TGC are achieved (q ; q ) if and only if the Hessian matrix H is negative R T in six scenarios. Tables 2 and 3 depict the electricity price deﬁnite. and TGC price in each scenario. Proposition 6 Since detðHÞ¼ð2au 2a 2b 2uÞ ð2b 2a Þðau 2bÞð2b 2a Þ [ 0, under market T T TGC price policy the optimal amount of electricity gener- Evaluation policies and sensitivity analysis ated from renewable and fossil sources in the cooperative game model are: Comparison price and production a hu þ 2aa h þ ab u acu þ G T T 1 q ¼ ð22Þ R½CM 2 2 In this section, sensitivity analysis is performed by a u þ 4aa u þ 3b þ F T 2 numerical examples to illustrate performance differences a hu 2aa h ab u þ 2ab u acu ahu þ G R R T 2 q ¼ between different models. T½CM 2 2 2 a u þ 4aa u þ 3b þ F T 2 We present numerical studies by assuming that the ð23Þ marginal costs and other parameters of the cost function in renewable power plant are higher than nonrenewable where G ¼þ2a b 2a c 2a u þ 2bb 2bb 1 T R T T R T power plant. 2bh; G ¼2a b þ 2a c þ 2bb þ 2bb 2bb 2 R T R R R T Cost function of the renewable and nonrenewable pro- 2bh 2b u þ 2cu. ducers is assumed as below: Pricing is the most effective proﬁt lever (Dolan and ðÞ Cq ¼ 0:06q þ 11q þ 100 and R R Simon 1996). This is a process for determining what a CqðÞ¼ 0:04q þ 8q þ 20: T T company will receive in exchange for its products or ser- T vices. Pricing can be considered in industry, market, and The price elasticity of the electricity supply and TGC transaction levels. At the industry level, the main focus is supply is assumed as below: b ¼ 0:4 and u ¼ 0:3. It is on the overall economics of the industry, including price assumed that c ¼ 150 and h ¼ 50: In ﬁxed TGC price changes of the supply and demand changes of the cus- policies, the TGC price is set equal to average of the TGC tomer. On the other hand, in the market level the com- market prices per different amounts of the minimum quota. petitive situation of the price in comparison with the value Figure 1 illustrates the changes of total electricity sup- differential of the product to that of the comparative ply, green electricity supply and black electricity supply competing products will be considered. Pricing at the versus the minimum requirement of green electricity. 123 Journal of Industrial Engineering International Figure 2 shows the changes of electricity and TGC price versus the minimum requirement of green electricity. It can be seen from Fig. 1 that in every six scenarios of Table 1 when a increases Q decreases. However, supply of the green electricity increases in the market price policy and Nash model in the ﬁxed price policy. Moreover, when a increases the black electricity decreases in every six scenarios. In the CM scenario, when a increases, supply of the total electricity in the ﬁrst step decreases but then it starts to increase. But in the CF scenario when a increases, supply of total electricity consistently decreases. This means that contrary to the other ﬁve scenarios, in the CM scenario when minimum requirement of green electricity (aÞ is almost 60% the electricity generated is at minimum amount. The maximum amounts of the green electricity are generated in the CF scenario. The maximum amounts of the black electricity are generated in the NF scenario and the minimum amounts of the black electricity are supplied in CF scenario. Generally speaking, with changes of the minimum mandatory quota, supply of the total electricity in the SF scenario has the least changes in comparison with the other scenarios. Figure 1 demonstrates that supply of the total electricity in the SM scenario is greater than the SF scenario con- sistently. Moreover, the trend of electricity supply in both scenarios is descending with increase in the minimum quota. This result is supported by Jensen and Skytte (2003) and Tamas et al. (2010). Supply of the total electricity in the Nash model of both policies has a descending trend with increase in the minimum quota. Nevertheless, the total electricity generated in the NF scenario is greater than that of the NM scenario. About the cooperative model, it can be stated that the total electricity generated in the CF scenario with increase in the minimum quota has absolutely descending trend, whereas the CM scenario shows a convex shape. When the minimum requirement of green electricity is less than 60%, the total electricity generated in the CF scenario is greater than that of the CM scenario. Figure 2 shows that there is a reverse relation between the minimum requirement of green electricity and TGC price. However, the relation between the minimum requirement of green electricity and electricity price is direct. In other words, when a increases P increases and P decreases in all scenarios. This matter represents there is a reverse relation between price of TGC and electricity price. This result is supported by Jensen and Skytte (2003), Fristrup (2003), Tama´s et al. (2010) and Marchenko (2008). In the CM scenario, when a increases, there is a rapid reduction in the price of TGC in comparison with the other Table 3 Price of electricity in six scenarios Game TGC price policy Models Market price Fixed price ahðau2a 2a 2buÞþauðb þ2b cÞþE F P ð2aa þ2aa þ2ab2a bÞþA þA Nash R T R T 1 1 c R T T 1 2 P ¼ c b P ¼ c b eN½ M 4aa uþ3abuþF eN½ F T 2 2Dþ3b 0 1 2 2 2 2 ð2aa hþabhþF Þðauþbþ2a Þ Stackelberg T 2 T cahb T P að4a þ 4a b þ b þ KÞ 2P ð2a þ 3a b þ b Þ P ¼ c b þ c R c R R R eS½ M 2ðÞ bþa 2ð4aa uþ2abub þF ÞðÞ bþa T T 2 T B C þ E ð2a þ bÞþ Kðb þ cÞ B 2 R T C P ¼ c bB C eS½ F 2KðÞ bþa @ A 2bðP aa þP bþB Þ að2a hþ2a hþb ub uþhuÞþG G c R c 2 Cooperative R T R T 1 2 P ¼ c þ P ¼ c b 2 2 eC½ F D eC½ M a u þ4aa uþF þ3b T 2 Journal of Industrial Engineering International Market TGC price policy Fixed TGC price policy 240 240 220 220 200 200 180 180 160 160 140 140 120 120 100 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( percentage) ( percentage) 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( percentage) ( percentage) 0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( percentage) ( percentage) Nash Stackelberg Cooperave Fig. 1 Changes of total, green and black electricity versus minimum quota scenarios. Among the game theory models, the Stackelberg Changes of payoffs of thermal, renewable and central- model in the ﬁxed TGC price results in the minimum ized power plants are depicted in Fig. 3. The results of electricity price. However, the cooperative model has the numerical study show that by increasing a total payoff of maximum electricity price in both ﬁxed TGC price and the centralized power plant decreases in all scenarios. Cen- market policy price. The price of electricity in the ﬁxed tralized power plant payoff in cooperative model is higher TGC price policy is less than that of the same game theory than the other scenarios. By increasing a, the payoff of model in the market TGC price policy. green electricity producer decreases in all scenarios except CM scenario. Note that in Nash and Stackelberg models by Black electricity supply Green electricity supply Total electricity supply Total supply of electricity (Mwh) Supply of black electricity (Mwh) Supply of green electricity (Mwhr) Supply of black electricity (Mwh) Supply of green electricity (Mwh) Total supply of electricity Mwh) Journal of Industrial Engineering International Market TGC price policy Fixed TGC price policy 100 100 90 90 0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1 0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1 ( percentage) ( percentage) 0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1 0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1 ( percentage) ( percentage) Nash Stackelberg Cooperave Fig. 2 Changes of TGC and electricity price versus minimum quota increasing a the payoff of green electricity producer relation to inverse demand function in Eq. (4), and it is decreases under market TGC price policy. Remarkably, by assumed to be equal to 100. Moreover, it is assumed that increasing a, the payoff of black electricity producer the cost function of the green and black power plants is 2 2 decreases in all scenarios, but in CM scenario, it decreases 5q þ 30q þ 100 and 3q þ 10q þ 20, respectively, R T R T faster than the other scenarios. It can be concluded that the where b ¼ 1:2 and u ¼ 1:2. It is assumed that: c ¼ use of CM scenario will lead to elimination of thermal 150; h ¼ 100 and k ¼ 0:4. Figure 4 depicts the results of power plants more quickly. this example in six scenarios. The evaluation of these polices reveals that in each six Social welfare scenarios by increasing the minimum quota, social welfare increases at ﬁrst and decreases later. In other words, in all Social welfare is an appropriate criterion to evaluate any scenarios the maximum of social welfare does not happen policy or program (Tama´s et al. 2010). To evaluate the six when all the electricity supply is generated from the green proposed scenarios in this paper, we use the equation of sources (a ¼ 100%Þ. This result is in accordance with social welfare proposed by Currier (2013). In this case, the Currier (2013) and Currier and Sun (2014). In the ﬁxed social welfare is equal to the total utility minus the all costs TGC price polices, in the ﬁrst, by increasing of the mini- including the environmental damages and production costs. mum quota, the social welfare will increase with a fas- Here, U represents the consumer utility and D denotes the ter slope compared with the market TGC price polices. function of environmental damages. Generally, when the minimum requirement of renew- able energy sources in the electricity supply is less than SW ¼ UðQÞ Cðq Þ Cðq Þ Dðk; qÞð24Þ T R R almost 50%, the market TGC price polices lead to a higher Currier and Sun (2014) assumed that D ¼ q =2 and R level of welfare. The welfare in Stackelberg model with the ðÞ Q ¼ cQ Q =2. Here c represents the parameter in Electricity price TGC price TGC price ($/Mwh) Electricity price ($/Mwh) Electricity price ($/Mwh) TGC price ($/Mwh) Journal of Industrial Engineering International Market TGC price policy Fixed TGC price policy 14000 14000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( percentage) ( percentage) 12000 12000 10000 10000 8000 8000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( percentage) ( percentage) 9000 9000 8000 8000 7000 7000 6000 6000 4000 4000 3000 3000 2000 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( percentage) ( percentage) Nash Stackelberg Cooperave Fig. 3 Changes of power plants Payoff versus minimum quota market TGC price policy (SM scenario) is consistently TGC price policy (NM and SM scenarios) and 70–80% of greater than the ﬁxed TGC price policy. But comparing the the electricity supply is generated from the RE sources. In two control price of certiﬁcates policies among other game contrast, the minimum welfare is obtained when that theory models (Nash and cooperative) shows that there is market structure follows the Nash or Stackelberg model not a constant trend in terms of welfare created. The with the ﬁxed TGC price policy (i.e., NF and SF scenarios) maximum welfare is obtained when that market structure when minimum quota is zero (a = 0). When a =0, the follows the Nash or Stackelberg model with the market maximum welfare is obtained by CM scenario. Among six Black electricity producer payoff Green electricity producer payoff Total payoff of centralized power plant Payoff of renewable power plant ($) Total porofit of power plants($) Payoff of thermal power plant($) Payoff of thermal power plant($) Payoff of renewable power plant($) Total porofit of power plants($) Journal of Industrial Engineering International Market TGC price policy Fixed TGC price policy 350 350 300 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( percentage) ( percentage) Nash Stackelberg Cooperave Fig. 4 Comparison of social welfare acquired in each scenario scenarios, SF scenario creates minimum welfare for all social welfare. In this policy, the use of NF scenario will be amounts of a. more beneﬁcial in terms of social welfare and high power It seems that the results of this practice are useful for supply in comparison with three other scenarios. private and public investors, energy policy makers, gov- There are several directions for the future research. ernment and other active players in the electricity supply Firstly, this study considers the national trade in the elec- chain. It is an undeniable fact that pricing the TGC is a tricity market and the TGC system. Game theory formu- challenging problem for the government. Therefore, ana- lation of international TGC trade in the internal and lyzing these models with various scenarios can improve the external markets is interesting. Secondly, other approaches effectiveness of designing and implementing TGS system. of game theory to analyze the implementation of the TGC system can be considered. For example, modeling the TGC system in the incomplete information mode by Bayesian Conclusion models is both interesting and challenging. Thirdly, we only consider the producer’s obligation option in the TGC This study demonstrates that using market TGC price system, but other obligations in the TGC system can also policy is more beneﬁcial when a country intends to deploy be considered. Finally, no time constraint was considered a system of credentials with a share of renewable energy to validate the certiﬁcates. Using the time variables in modeling of the TGC system seems to be useful. sources less than 50 percent because not only a higher social welfare in this sector is created but also by using this Open Access This article is distributed under the terms of the Creative policy the proﬁt of thermal power plants will be decreased Commons Attribution 4.0 International License (http://creative with a modest slope and it will not lead to an abrupt commons.org/licenses/by/4.0/), which permits unrestricted use, dis- withdrawal from the market and lack of power supply. tribution, and reproduction in any medium, provided you give Moreover, if the goal is accelerating the removal of these appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were power plants with abrupt withdrawal, then using the CM made. scenario is beneﬁcial where the proﬁt of fossil fuels is reduced more steeply. This scenario also will have the lowest power supply among the six scenarios, and it will Appendix have the lowest levels of social welfare for a values above 50%. Proof of Proposition 1 If the second-order derivative for If a country is so much developed that can provide more Eq. (7) is negative, the proﬁt function of the green pro- than 50% of its electricity from renewable sources, then ducer will be concave. The ﬁrst-order derivative for Eq. (7) using ﬁxed TGC price policy can be beneﬁcial too because is: at this point it acts like market TGC price policy in creating Social Welfare Social Welfare Journal of Industrial Engineering International Therefore, the proﬁt function of the green producer is op ¼ðP þ cÞðbq þ 2bq þ 2a q þ b Þ¼ 0: c T R R R R concave. Similarly, the ﬁrst-order derivative for Eq. (8)is oq as follows: ð25Þ op ¼ q ð2b 2a Þþ q ðb þ uaÞþ c ah b ¼ 0: The second-order derivative for Eq. (7) is as follows: T T R T oq o p ð31Þ ¼ ðþ2bþ2a Þ: ð26Þ o q The second-order derivative is as follows: Since the amounts of b and a are positive, the second- o p o p R ¼2a 2b ¼ 0: ð32Þ order derivative is negative \0 : 2 o q R o q Therefore, the proﬁt function of the green producer is Since it is assumed that b; a [ 0, the second-order concave. Similarly, the ﬁrst-order derivative for Eq. (8)is o p derivative is negative \0 . as follows: 2 o q Therefore, the proﬁt function of the green producer is op ¼ c ð2bq þ bq þ Pca þ 2a q þ b Þ¼ 0: ð27Þ T R T T T concave. oq Solving Eqs. (29) and (31), it follows that the optimal The second-order derivative for Eq. (8) yields: production of power plants is: o p T 2aa h þ abh þ F T 1 ¼ð2bþ2a Þ: ð28Þ q ¼ ; T RN½ M 4aa u þ 3abu þ F T 2 o q a hu 2aa h ab u þ 2ab u acu ahu þ E R R T 1 q ¼ : TN½ M Since the amounts of b and a are positive, the second- T 4aa u þ 3abu þ F T 2 o p order derivative is negative \0 : 2 h o q Hence, the proﬁt function of the thermal producer will Proof of Proposition 3 To solve the model, q is ﬁrst be concave. Solving Eqs. (25) and (27), it follows that the obtained as a function of q and then the ﬁrst-order optimal production of power plants is: derivative is ﬁrst examined for a proﬁt function of the P ð2aa þ ab 2a 2bÞþ A c T T 1 thermal power plant of Eq. (8). The best response strategy q ¼ ; RN½ F 2D þ 3b for a thermal power plant is computed as follows: PðÞ 2aa þ 2ba þ bþ A c R 2 q ¼ : aP þ q b þ b c TN½ F 2 c R T 2D þ 3b q ¼ : ð33Þ 2ðb þ a Þ Substituting Eq. (33) into Eq. (7) gives: aP þ bq þ b c c R T p ¼ P q þ c b þ q q R c R R R 2ðb þ a Þ Proof of Proposition 2 If the second-order derivative of a q b q c : R R R R Eq. (7) under market TGC price policy is negative, the ð34Þ proﬁt function of the green producer will be concave. The ﬁrst-order derivative for Eq. (7) is: The ﬁrst-order derivative for Eq. (34) yields: op op b ¼ q ð2u 2b þ au þ ua 2a Þ R R R ¼ P b þ 1 q þ c oq c R oq 2ðb þ a Þ R T þ h þ c bq ah b ¼ 0: ð29Þ T R aP þ bq þ b c c R T b þ q 2a q b ¼ 0: R R R R 2ðb þ a Þ The second-order derivative for Eq. (7) under market TGC price policy is as follows: ð35Þ o p The proﬁt function of the renewable power plant is ¼ 2au 2a 2b 2u ¼ 0: ð30Þ o q R concave if the second-order derivative for Eq. (34)is negative. The second-order derivative for the renewable Since it is assumed that u; b; a [ 0, and 0 a 1. We power plant gives: know (au\a þ b þ uÞ, then the second-order derivative 2 2 o p o p 2a a þ 2a b þ b R R T R is negative \0 : ¼ : ð36Þ o q b þ a o q T 123 Journal of Industrial Engineering International 2aa h þ abh þ F T 1 Regarding the assumptions and parameter values, q ¼ : RS½ M 4aa u þ 2abu b þ F Eq. (36) is negative. Therefore, the proﬁt function of the T 2 renewable power plant is found to be concave. From Substituting q into Eq. (37), the optimal black RS½ M Eq. (35), it follows that the optimal green electricity electricity production is: production is: T½SM P ð2aa þ ab 2a 2bÞþ E c R R 2 2 2 P 2aa b þ ab aE 2a b 2b þ bE b E þ E c c R 2 R 1 T 2 2 q ¼ : R½SF ¼ : 2E ðb þ a Þ 2 R Substituting q into Eq. (33), the optimal black RS½ F electricity production is: ð2aa h þ abh þ F Þðau bÞ T 1 q ¼ TS½ F 2ðb þ a Þð4aa u þ 2abu b þ F Þ T T 2 Proof of Proposition 5 The ﬁrst-order derivative for the ah b þ c proﬁt function of the power plants in Eq. (25) yields (in the 2ðb þ a Þ ﬁxed TGC price policy): op ¼ P þ c 2bðq þ q Þ 2a q b ¼ 0; ð41Þ c T R R R R oq Proof of Proposition 4 To solve the model, q is ﬁrst obtained as a function of q and then the ﬁrst-order R op ¼ c 2bðq þ q Þ aP 2a q b ¼ 0: ð42Þ T R c R T T derivative is examined for a proﬁt function of the thermal oq power plant of Eq. (8) under market TGC price policy; the Solving Eqs. (41) and (42), they give: best response strategy of the thermal power plant is com- puted as follows: PðÞ aa a bþ B c T T 1 q ¼ : R½CF q au q b ah b þ c R R T q ¼ : ð37Þ PðÞ aa þ bþ B 2ðb þ a Þ c R 2 q ¼ : T½CF Substituting Eq. (37) into Eq. (7) gives: p ¼ðq u þ hÞq R R R q au bq ah b þ c R R T Proof of Proposition 6 The ﬁrst-order derivative for the þ c b þ q q R R 2ðb þ a Þ proﬁt function of the power plants in Eq. (26) yields (in the ðq u þ hÞaq a q b q c : R R R R R R market TGC price policy): ð38Þ op ¼ q ð2au 2a 2b 2uÞþ q ðau 2bÞ R R T oq The ﬁrst-order derivative for Eq. (38) yields: ah b þ c þ h ¼ 0; ð43Þ op au b ¼ h b þ 1 q op oq 2ðb þ a Þ R T ¼ q ð2au 2bÞþ q ð2a 2bÞ R T T oq q au bq ah b þ c T R R T : þ c b þ q 2ðb þ a Þ ah b þ c ¼ 0; ð45Þ T T þ q ð2u þ au þ ua 2a Þ b ¼ 0 R R R Since Hessian matrix for this function is negative ð39Þ deﬁnite, the proﬁt function is concave. Thus, Solving Eqs. (43) and (44) yields: The proﬁt function of the renewable power plant is a hu þ 2aa h þ ab u acu þ G T T 1 concave if the second-order derivative for Eq. (38)is q ¼ ; R½CM 2 2 2 a u þ 4aa u þ 3b þ F negative. The second-order derivative for the renewable T 2 power plant gives: a hu 2aa h ab u þ 2ab u acu ahu þ G R R T 2 q ¼ : T½CM 2 2 2 a u þ 4aa u þ 3b þ F T 2 o p au b ¼2u 2b þ 1 þ 2au 2a : ð40Þ 2ðb þ a Þ o q T Regarding the assumptions and parameter values, Eq. (40) will be negative. From Eq. (39), it follows that the optimal green electricity production is: 123 Journal of Industrial Engineering International Heinzel C, Winkler T (2011) Economic functioning and politically References pragmatic justiﬁcation of tradable green certiﬁcates in Poland. Environ Econ Policy Stud 13:157–175 Ahmad S, bin Mat Tahar R, (2014) Using system dynamics to Jensen S, Skytte K (2003) Interactions between the power and green evaluate renewable electricity development in Malaysia. Kyber- certiﬁcate markets. Energy Policy 30:425–435 netes 43(1):24–39 Jørgensen S, Zaccour G (2002) Time consistency in cooperative Amundsen E, Nese G (2009) Integration of tradable green certiﬁcate differential games. In: Zaccour G (ed) Decision & Control in markets: what can be expected? J Policy Model 31:903–922 Management Science Advances in Computational Management Aune F, Dalen H, Hagem C (2010) Implementing the EU renewable Science, vol 4. Springer. 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Journal of Industrial Engineering International – Springer Journals
Published: Jun 1, 2018
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