Journal of Industrial Engineering International (2019) 15:513–527

https://doi.org/10.1007/s40092-018-0272-8 (0123456789().,-volV)(0123456789().,-volV)

ORIGINAL RESEARCH

Evaluating different scenarios for Tradable Green Certificates by game

theory approaches
Meysam Ghaffari1 • Ashkan Hafezalkotob1

Received: 29 August 2017 / Accepted: 3 May 2018 / Published online: 1 June 2018
Ó The Author(s) 2018

Abstract
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.
1
Industrial Engineering College, Islamic Azad University, 2008).
South Tehran Branch, Entezari alley, Oskoui alley, Choobi
Bridge, Tehran 1151863411, Iran

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514 Journal of Industrial Engineering International (2019) 15:513–527

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 (Tamás 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; Tamás 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-
profit function of the power plants in electricity and TGC ches (Tamás et al. 2010). By using economic analysis,
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
Literature review deployment was analyzed by Heinzel and Winkler (2011).
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
TGCs system is usually operates as a market and is criteria of efficiency, equity and institutional feasibility.
based on demand and supply. The demand of TGCs is Some of the researchers analyzed the TGC system by using
determined by energy policies and the annual share of the system dynamic method. In recent researches, this
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

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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. Gürkan 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
practice, the TGC price is assumed to be constant and will be simplicity: renewable and thermal power plants. Electricity
determined by the government. They demonstrated that the producers compete in the electricity and TGC market under
relation between the electricity price and the TGC price is producer obligation. Thus, thermal power plant is obliged
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 pR the profit function of renewable producer;
intend to cooperate with each other for higher economic and pT the profit function of thermal producer;
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 pR þ pT );

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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

CT the cost function of thermal producer; TGC markets separately. The renewable producer cost
CR the cost function of renewable producer; CR ðqR Þ is a function of green electricity generated. The
U the consumer utility; cost of the renewable power plant is only dependent on the
D the function of environmental damages. green electricity generated qR . Therefore, the renewable
producer profit maximization problem will be as follows:

Decision variables Max pR ¼ Pe qR þ ð1  aÞPc qR  CR ðqR Þ

S:t: ð1Þ
Pe the end-user price of electricity ($/MWh), Pe [ 0; qR  0
Pc the price of TGC ($/MWh), Pc [ 0;
qT the electricity generated from thermal energy (MW), where Pe is the end user1 of each MW of generated elec-
qT  0; tricity, a is the minimum requirement of green electricity
qR the generated from renewable energy (MW), qR  0; and Pc is the price of certificate. This means that a
Q the total amount of electricity (MW), renewable producer can receive ð1  aÞPc for each unit in
Q  0ðQ ¼ qT þ qR Þ. addition to the electricity price. Under the TGC system, a
renewable producer would obtain per unit ‘‘subsidy’’
As shown in Table 1, we have modeled six scenarios to ð1  aÞPc .
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 CT ðqT Þ is a function of the
proposed models: thermal electricity qT .
1. There is no limitation for power plants in consumption Therefore, the thermal producer profit maximization
of the resources. problem will be as follows:
2. There is no limitation on the demand and supply Max pT ¼ Pe qT  Pc aqT  CT ðqT Þ
electricity and TGCs. S:t: ð2Þ
3. There is no excess demand and supply in the electricity
qT  0
and TGCs markets.
4. Parameters are deterministic and they are known in Thermal power plant can receive Pe 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 requirement ðqR  aQÞ. per unit ‘‘tax’’ aPc 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 1

plants by Tanaka and Chen (2013). Renewable power plant Amundsen and Nese (2009), discussed the relation of: Pe = wholesale
electricity price ? a Pc 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.

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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).
Skytte (2003) and for renewable and thermal power plants All players have no motivation to exit the equilibrium,
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
CR ðqR Þ ¼ aR q2R þ bR qR þ cR ð3Þ
follows:
CT ðqT Þ ¼ aT q2T þ bT qT þ cT ð4Þ The strategy profile in a (n) players game of P ¼
  
In Eqs. (1) and (2), it is assumed that aR , bR ; aT ; bT [ 0. P1 ; . . .; Pn is a Nash equilibrium (NE) if for all i 2
f1; . . .; ng there is:
   
Profit maximization problem for power plants Ui ¼ P1 ; . . .; Pn  P1 ; . . .; Pi1 ; Pi ; Piþ1 ; . . .Pn ð9Þ

Following Newbery (1998) and Tamás et al. (2010), we where Ui is the utility function of the ith player.
assume that the demand function for electricity is a linear In this section, we consider a Cournot-NE game under a
function, TGC system.
It can be seen that by solving NE , from Eqs. (7) and (8)
Pe ¼ c  bQ ¼ c  bðqR þ qT Þ; ð5Þ
qT and qR will be obtained. Now, with substitution of qT
Meanwhile, Q ¼ ðqR þ qT Þ is the total electricity. On and qR into pR and pT , the maximum profit of the pro-
the other hand, we assume that the price of electricity is a ducers (pR and pT ) will be reached. Propositions 1 and 2
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
Pc ¼ h  uqR ð6Þ TGC price policy and the market TGC price policy,
respectively.
With substitution of Eq. (5) into Eqs. (1) and (2), the
profit maximization problem can be formulated as follows. Proposition 1 Under the fixed TGC price policy, the
Renewable producer is given as below: optimum amounts of production for the renewable and
thermal producers in the Nash model can be given as
Max pR ¼ ðc  bðqR þ qT ÞÞqR þ Pc qR  aR q2R  bR qR  cR below:
s:t:
Pc ð2aaT þ ab  2aT  2bÞ þ A1
qR  0 qR½NF ¼  ð10Þ
2D þ 3b2
ð7Þ
Pc ð2aaR þ 2ba þ bÞ þ A2
Thermal producer is given as below: qT½NF ¼  ð11Þ
2D þ 3b2
Max pT ¼ ðc  bðqR þ qT ÞÞqT  Pc qT  aT q2T  bT qT  cT
where A1 ¼ 2bR aT  2aT c þ 2bbR  bbT  bc; A2 ¼ 2bT
S:t:
aR 2aR c þ 2bbT  bbR bc; D ¼ 2aR aT þ 2aR b þ 2aT b:
qT  0
ð8Þ All propositions have been proven in ‘‘Appendix’’. With
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
Game theory models producers in the Nash solution can be given as below:
2aaT h þ abh þ F1
qR½NM ¼ ð12Þ
Noncooperative Nash game 4aaT u þ 3abu þ F2
a2 hu  2aaR h  abR u þ 2abT u  acu  ahu þ E1
If no player has anything to gain by changing his strategy, qT½NM ¼ 
4aaT u þ 3abu þ F2
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

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where F1 ¼ 2bR aT  2aT c þ 2bT h þ 2bbR  bbT  bc ð2aaT h þ abh þ F1 Þðau  bÞ
qT ½SM ¼
2bh; F2 ¼ 4aR aT  4aR b  4aT b  4aT u  3b2  4b 2ðb þ aT Þð4aaT u þ 2abu  b2 þ F2 Þ
u; E1 ¼ 2aR bT þ 2aR c þ bbR  2bbT þ bc  bh  2bT ah  bT þ c
 : ð17Þ
u þ 2cu. 2ðb þ aT Þ
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.
Since the objective of the TGC system is supporting the Max p ¼ ðc  bðqR þ qT ÞÞqR þ Pc qR  aR q2R  bR qR
increasing share of the electricity generated by RE pro-  cR þ ðc  bðqR þ qT ÞÞqT
ducer, in this research we only examine renewable pro-  Pc qT  aT q2T  bT qT  cT
ducer—Stackelberg model where the renewable producer S:t:
is leader and the thermal power plant is the follower. In this
qR ; qT  0
model, the renewable producer first sells its generated
electricity in electricity market. Then the follower as ð18Þ
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  2aR 2b
Propositions 3 and 4 present the optimum production of and the utility function p is a
2b 2b  2aT
electricity from renewable and thermal producers in
concave function on (qR ; qT ) if and only if the Hessian
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.
Proposition 3 Under fixed TGC price policy, the optimal Subscripts [CF] and [CM] denote the optimum values in
amount of electricity generated from renewable and fossil the cooperative game model under fixed TGC price and
sources in renewable producer—Stackelberg model—is: market TGC price polices, respectively.
Pc ð2aaR þ ab  2aR  2bÞ þ E2 Proposition 5 Since ð2b  2aR Þð2b  2aT Þ
qR½SF ¼  ð14Þ
K ð2bÞð2bÞ [ 0, the optimal amount of electricity gen-
 
Pc 2aaR b þ ab2  aE2  2aR b  2b2 þ bE2 þ Kðc  bT Þ erated from renewable and fossil sources in the coopera-
qT½SF ¼ 
2E2 ðb þ aR Þ tive game model under fixed TGC price policy will be:
ð15Þ Pc ðaaT  aT  bÞ þ B1
qR½CF ¼  ð19Þ
where E2 ¼ 2aR bR  2aR c þ 2bbR  bbT  bc; K¼ D
4aR þ 8aR b þ 2b2 .
2 Pc ðaaR þ bÞ þ B2
qT½CF ¼  ð20Þ
D
Proposition 4 Under market TGC price policy, the opti-
where B1 ¼ aT bR  aT c þ bbR  bbT ; B2 ¼ aR bT  aR c
mal amount of electricity generated from renewable and
bbR þ bbT :
fossil sources in renewable producer—Stackelberg
model—is: Substituting Eq. (4) into Eq. (18), the problem of profit
2aaT h þ abh þ F1 centralized power plant under market TGC price policy
qR½SM ¼ ð16Þ yields:
4aaT u þ 2abu  b2 þ F2

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Table 2 Price of TGC in six

Game models TGC price policy
scenarios
Market price Fixed price

Nash Pc½NM ¼ h  ð2aaT hþabhþF1 Þu Pc½NF ¼ cte

4aaT uþ3abuþF2

Stackelberg ð2aaT hþabhþF1 Þu

Pc½SM ¼ h  4aa Pc½SF ¼ cte
uþ2abub2 þF
T 2

Cooperative Pc½CM ¼ h  ða
2
uhþ2aaT hþabT hacuþG1 Þu Pc½CF ¼ cte
a2 u2 þ4aaT uþF2 þ3b2

‘‘cte’’ represents a fixed value

Max p ¼ ðc  bðqR þ qT ÞÞqR þ ðh  uqR ÞqR transaction level focuses on managing the implementation
of discounts away from the reference or the price list which
 aR q2R  bR qR  cR þ ðc  bðqR þ qT ÞÞqT
occur both on and off the invoice or receipt.
 ðh  uqR ÞqT  aT q2T  bT qT  cT In this section, the pricing at the electricity market level
S:t: is considered in oligopoly and monopoly market structures.
qR ; qT  0 Oligopoly is a common form of market where a number of
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
 
2au  2aR  2b  2u au  2b (Perloff 2008). By substituting the optimal amounts of
H¼ and the utility
2b  2aT 2b  2aT green and black electricity production quantities in the
function in the cooperative model is a concave function on payoff functions of the renewable and black power plants,
(qR ; qT ) if and only if the Hessian matrix H is negative the optimum prices of the electricity and TGC are achieved
definite. in six scenarios. Tables 2 and 3 depict the electricity price
and TGC price in each scenario.
Proposition 6 Since detðHÞ ¼ ð2au  2aR  2b  2uÞ
ð2b  2aT Þ  ðau  2bÞð2b  2aT Þ [ 0, under market
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
a2 hu þ 2aaT h þ abT u  acu þ G1
qR½CM ¼ ð22Þ
a2 u2 þ 4aaT u þ 3b2 þ F2 In this section, sensitivity analysis is performed by
numerical examples to illustrate performance differences
a2 hu  2aaR h  abR u þ 2abT u  acu  ahu þ G2
qT½CM ¼  between different models.
a2 u2 þ 4aaT u þ 3b2 þ F2
We present numerical studies by assuming that the
ð23Þ marginal costs and other parameters of the cost function in
where G1 ¼ þ2aT bR  2aT c  2aT u þ 2bbR  2bbT  renewable power plant are higher than nonrenewable
2bh; G2 ¼ 2aR bT þ 2aR c þ 2bbR þ 2bbR  2bbT  power plant.
2bh  2bT u þ 2cu. Cost function of the renewable and nonrenewable pro-
ducers is assumed as below:
Pricing is the most effective profit lever (Dolan and
Simon 1996). This is a process for determining what a C ðqR Þ ¼ 0:06q2R þ 11qR þ 100 and
company will receive in exchange for its products or ser- CðqT Þ ¼ 0:04q2T þ 8qT þ 20:
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.

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1
C
C
C
A
Figure 2 shows the changes of electricity and TGC price

Pc að4a2R þ 4aR b þ b2 þ KÞ  2Pc ð2a2R þ 3aR b þ b2 Þ

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

þ E2 ð2aR þ bÞ þ KðbT þ cÞ
scenarios.

bþa R Þ

T bÞþA1 þA2

In the CM scenario, when a increases, supply of the

2K ð
total electricity in the first step decreases but then it starts
to increase. But in the CF scenario when a increases,
Pe½NF ¼ c  b  Pc ð2aaR þ2aaT þ2ab2a
2Dþ3b2

supply of total electricity consistently decreases. This

means that contrary to the other five scenarios, in the CM
Pe½CF ¼ c þ 2bðPc aaRDþPc bþB2 Þ 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
Pe½SF ¼ c  bB

generated in the CF scenario. The maximum amounts of

0
B
B
@


the black electricity are generated in the NF scenario and

Fixed price

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)

R þ2bT cÞþE1 F1

and Tamás et al. (2010). Supply of the total electricity in

the Nash model of both policies has a descending trend

T
2ðbþaT Þ
þ cahb

with increase in the minimum quota.


4aaT uþ3abuþF2

T hþbR ubT uþhuÞþG1 G2

Pe½NM ¼ c  b  ahðau2aR 2aT 2buÞþauðb

Nevertheless, the total electricity generated in the NF

scenario is greater than that of the NM scenario. About the
uþ2abub2 þF Þðbþa Þ
ð2aaT hþabhþF2 Þðauþbþ2aT Þ

a2 u2 þ4aa uþF þ3b2

T

cooperative model, it can be stated that the total electricity

2
2

generated in the CF scenario with increase in the minimum

T

quota has absolutely descending trend, whereas the CM

scenario shows a convex shape. When the minimum
Pe½CM ¼ c  b að2aR hþ2a

requirement of green electricity is less than 60%, the total

T
Pe½SM ¼ c  b 2ð4aa

electricity generated in the CF scenario is greater than that

TGC price policy




of the CM scenario.
Market price

Figure 2 shows that there is a reverse relation between

Table 3 Price of electricity in six scenarios

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 Pe increases and
Pc 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), Tamás et al. (2010) and Marchenko
(2008).
Cooperative
Stackelberg

In the CM scenario, when a increases, there is a rapid

Models
Game

reduction in the price of TGC in comparison with the other

Nash

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Journal of Industrial Engineering International (2019) 15:513–527 521

Market TGC price policy Fixed TGC price policy

Total supply of electricity (Mwh)

Total supply of electricity Mwh)

240 240
220 220
Total electricity supply

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)
Supply of green electricity (Mwhr)

Supply of green electricity (Mwh)

160 160
140 140
Green electricity supply

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)

140 120
Supply of black electricity (Mwh)

Supply of black electricity (Mwh)

120
Black electricity supply

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)

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

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522 Journal of Industrial Engineering International (2019) 15:513–527

Market TGC price policy Fixed TGC price policy

100 100
90 90

Electricity price ($/Mwh)

Electricity price ($/Mwh) 80 80
Electricity price

70 70
60 60
50 50
40 40
30 30
20 20
10 10
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)

40 40
TGC price ($/Mwh)

35 35

TGC price ($/Mwh)

30 30
25 25
TGC price

20 20
15 15
10 10
5 5

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)

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
decreases in all scenarios, but in CM scenario, it decreases 5q2R þ 30qR þ 100 and 3q2T þ 10qT þ 20, respectively,
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 (Tamás 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-
SW ¼ UðQÞ  CðqT Þ  CðqR Þ  Dðk; qR Þ ð24Þ able energy sources in the electricity supply is less than
almost 50%, the market TGC price polices lead to a higher
Currier and Sun (2014) assumed that D ¼ q2R =2 and level of welfare. The welfare in Stackelberg model with the
ðQÞ ¼ cQ  Q2 =2. Here c represents the parameter in

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Journal of Industrial Engineering International (2019) 15:513–527 523

Market TGC price policy Fixed TGC price policy

Total payoff of centralized power plant

14000 14000

Total porofit of power plants($)

Total porofit of power plants($)
12000 12000

10000 10000

8000 8000

6000 6000

4000 4000

2000 2000

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)

12000 12000
Green electricity producer payoff

Payoff of renewable power plant ($)

Payoff of renewable power plant($)

10000 10000

8000 8000

6000 6000

4000 4000

2000 2000

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)

9000 9000
Payoff of thermal power plant($)

Payoff of thermal power plant($)

Black electricity producer payoff

8000 8000
7000 7000
6000 6000
5000 5000
4000 4000
3000 3000
2000 2000
1000 1000
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)

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

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Market TGC price policy Fixed TGC price policy

450 450
400 400
350 350
Social Welfare

Social Welfare
300 300
250 250
200 200
150 150
100 100
50 50
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)

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
sources less than 50 percent because not only a higher modeling of the TGC system seems to be useful.
social welfare in this sector is created but also by using this
policy the profit of thermal power plants will be decreased Open Access This article is distributed under the terms of the Creative
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
power plants with abrupt withdrawal, then using the CM link to the Creative Commons license, and indicate if changes were
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
have the lowest levels of social welfare for a values above
Appendix
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

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Journal of Industrial Engineering International (2019) 15:513–527 525

opR Therefore, the profit function of the green producer is

¼ ðPc þ cÞ  ðbqT þ 2bqR þ 2aR qR þ bR Þ ¼ 0: concave. Similarly, the first-order derivative for Eq. (8) is
oqR
ð25Þ as follows:
opT
The second-order derivative for Eq. (7) is as follows: ¼ qT ð2b  2aT Þ þ qR ðb þ uaÞ þ c  ah  bT ¼ 0:
oqT
o2 p R ð31Þ
¼ ðþ2bþ2aR Þ: ð26Þ
o2 qR
The second-order derivative is as follows:
Since the amounts of b and aR are positive, the second- 2
2  o pT
order derivative is negative oo2 pqR \0 : ¼ 2aT  2b ¼ 0: ð32Þ
R o2 qT
Therefore, the profit function of the green producer is
Since it is assumed that b; aT [ 0, the second-order
concave. Similarly, the first-order derivative for Eq. (8) is 2 
as follows: derivative is negative oo2 pqR \0 .
R

opT Therefore, the profit function of the green producer is

¼ c  ð2bqT þ bqR þ Pca þ 2aT qT þ bT Þ ¼ 0: ð27Þ concave.
oqT
Solving Eqs. (29) and (31), it follows that the optimal
The second-order derivative for Eq. (8) yields: production of power plants is:
2
o pT 2aaT h þ abh þ F1
¼ ð2bþ2aT Þ: ð28Þ qR½NM ¼ ;
o2 qT 4aaT u þ 3abu þ F2
a2 hu  2aaR h  abR u þ 2abT u  acu  ahu þ E1
qT½NM ¼ :
Since the amounts of b and aT are positive, the second- 4aaT u þ 3abu þ F2
2 
order derivative is negative oo2 pqR \0 : h
R

Hence, the profit function of the thermal producer will

be concave. Solving Eqs. (25) and (27), it follows that the Proof of Proposition 3 To solve the model, qT is first
optimal production of power plants is: obtained as a function of qR and then the first-order
derivative is first examined for a profit function of the
Pc ð2aaT þ ab  2aT  2bÞ þ A1
qR½NF ¼  ; thermal power plant of Eq. (8). The best response strategy
2D þ 3b2 for a thermal power plant is computed as follows:
Pc ð2aaR þ 2ba þ bÞ þ A2
qT½NF ¼ : aPc þ qR b þ bT  c
2D þ 3b2 qT ¼  : ð33Þ
2ðb þ aT Þ
Substituting Eq. (33) into Eq. (7) gives:
h   
aPc þ bqR þ bT  c
pR ¼ Pc qR þ c  b   þ qR qR
Proof of Proposition 2 If the second-order derivative of 2ðb þ aT Þ
Eq. (7) under market TGC price policy is negative, the  aR q2R  bR qR  cR :
profit function of the green producer will be concave. The ð34Þ
first-order derivative for Eq. (7) is:
The first-order derivative for Eq. (34) yields:
opR 
¼ qR ð2u  2b þ au þ ua  2aR Þ opR b
oqR ¼ Pc  b  þ 1 qR þ c
oqR 2ðb þ aT Þ
þ h þ c  bqT  ah  bR ¼ 0: ð29Þ 
aPc þ bqR þ bT  c
b  þ qR  2aR qR  bR ¼ 0:
The second-order derivative for Eq. (7) under market 2ðb þ aT Þ
TGC price policy is as follows: ð35Þ
2
o pR The profit function of the renewable power plant is
¼ 2au  2aR  2b  2u ¼ 0: ð30Þ
o2 qR 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; aR [ 0, and 0  a  1. We
power plant gives:
know (au\aR þ b þ uÞ, then the second-order derivative
2 
is negative oo2pqR \0 : o2 p R 2aR aT þ 2aR b þ b2
R 2
¼ : ð36Þ
o qR b þ aT

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Regarding the assumptions and parameter values, 2aaT h þ abh þ F1

qR½SM ¼ :
Eq. (36) is negative. Therefore, the profit function of the 4aaT u þ 2abu  b2 þ F2
renewable power plant is found to be concave. From Substituting qR½SM  into Eq. (37), the optimal black
Eq. (35), it follows that the optimal green electricity electricity production is:
production is:
qT½SM
Pc ð2aaR þ ab  2aR  2bÞ þ E2  
qR½SF ¼ : ¼
Pc 2aaR b þ ab2  aE2  2aR b  2b2 þ bE1  bT E2 þ E2 c
:
K 2E2 ðb þ aR Þ
Substituting qR½SF into Eq. (33), the optimal black
electricity production is:
ð2aaT h þ abh þ F1 Þðau  bÞ h
qT½SF ¼
2ðb þ aT Þð4aaT u þ 2abu  b2 þ F2 Þ
ah  bT þ c Proof of Proposition 5 The first-order derivative for the
 : profit function of the power plants in Eq. (25) yields (in the
2ðb þ aT Þ
fixed TGC price policy):
h
opR
¼ Pc þ c  2bðqT þ qR Þ  2aR qR  bR ¼ 0; ð41Þ
Proof of Proposition 4 To solve the model, qT is first oqR
obtained as a function of qR and then the first-order opT
derivative is examined for a profit function of the thermal ¼ c  2bðqT þ qR Þ  aPc  2aR qT  bT ¼ 0: ð42Þ
oqT
power plant of Eq. (8) under market TGC price policy; the
best response strategy of the thermal power plant is com- Solving Eqs. (41) and (42), they give:
puted as follows: Pc ðaaT  aT  bÞ þ B1
qR½CF ¼  :
qR au  qR b  ah  bT þ c D
qT ¼ : ð37Þ
2ðb þ aT Þ Pc ðaaR þ bÞ þ B2
qT½CF ¼  :
D
Substituting Eq. (37) into Eq. (7) gives:
h
pR ¼ ðq
 R u þ hÞq
R 
qR au  bqR  ah  bT þ c Proof of Proposition 6 The first-order derivative for the
þ cb þ qR qR
2ðb þ aT Þ profit function of the power plants in Eq. (26) yields (in the
 ðqR u þ hÞaqR  aR q2R  bR qR  cR : market TGC price policy):
ð38Þ op
¼ qR ð2au  2aR  2b  2uÞ þ qT ðau  2bÞ
The first-order derivative for Eq. (38) yields: oqR
  ah  bR þ c þ h ¼ 0; ð43Þ
opR au  b
¼hb  þ 1 qR
oqR 2ðb þ aT Þ op
 ¼ qR ð2au  2bÞ þ qT ð2aT  2bÞ
qR au  bqR  ah  bT þ c : oqT
þcb  þ qR
2ðb þ aT Þ  ah  bT þ c ¼ 0; ð45Þ
þ qR ð2u þ au þ ua  2aR Þ  bR ¼ 0 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
concave if the second-order derivative for Eq. (38) is a2 hu þ 2aaT h þ abT u  acu þ G1
qR½CM ¼ ;
negative. The second-order derivative for the renewable a2 u2 þ 4aaT u þ 3b2 þ F2
power plant gives: a2 hu  2aaR h  abR u þ 2abT u  acu  ahu þ G2
qT½CM ¼ :
 a2 u2 þ 4aaT u þ 3b2 þ F2
o2 p R au  b
¼ 2u  2b þ 1 þ 2au  2aR : ð40Þ
o2 qR 2ðb þ aT Þ h
Regarding the assumptions and parameter values,
Eq. (40) will be negative. From Eq. (39), it follows that
the optimal green electricity production is:

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Journal of Industrial Engineering International (2019) 15:513–527 527

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