# Evolutionary Game Analysis of Cooperation between Microgrid and Conventional Grid

Authors: Chengrong Pan; Yong Long
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper employs the evolutionary game theory to analyze the cooperation between microgrid and conventional grid, with the discussion focusing on the factors that influence the game players’ selection of strategies, and additionally makes a numerical simulation analysis. The results of analysis based on the evolutionary game model suggest that the two parties’ undertaking of cooperative strategies and uncooperative strategies during the game is the stable point of their evolution and that the probability of their unanimous choice of cooperative strategies positively correlates with the direct benefits and indirect benefits of their cooperation as well as the government subsidies but negatively correlates with the cooperation costs and spending on cooperation risks. Based on the findings, the paper puts forward a series of policy suggestions aiming to deepen the cooperation between microgrid and conventional grid.


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## Body

## 1. Introduction

Since the mid-20th century, in the wake of the great demand of electricity by large-scale industrial production and the stimulation of cheap fossil energy, centralized and unitary power supply systems which feature “high-capacity generator, large power system, and high voltage” have been constructed in major economies and even internationally [1, 2]. As power networks expand day by day, the drawbacks of the conventional grid, such as high cost and difficult operation, become more prominent and can no longer meet users’ increasing need for safety, reliability, and diversity [3–6]. At the same time, energy shortage and environmental deterioration resulting from the rapid development of the power industry turn out to be issues that urgently need to be addressed [7, 8]. Therefore, developing low-carbon economy characterized by low energy consumption, low emission, and low pollution, reducing carbon emission in the electrical industry, and improving the structure of power source have attracted particular attention all around the world [9–11]. Against this backdrop, researchers in 21st century proposed the concept of aMicrogrid [12], which is defined to be a group of interconnected loads and distributed energy resources within clearly defined electrical boundaries which acts as a single controllable system providing heat and power for a local region. Most importantly, a microgrid can realize the reliable supply of energy in various forms, especially distributed energy and clean energy [13]. Microgrid, if collaborating with conventional grid in power supply, can reasonably allocate electric power within a broader scope and increase the utilization rate of distributed energy and clean energy with the help of the sound power transmission and distribution (PTD) system of large power networks [14]. On the other hand, if conventional grid connects to microgrid, it can give more impetus to the construction of Smart Grid and the technological innovation in this field, meanwhile helping to solve the power supply problems in remote areas and autonomous regions [13]. In addition, the cooperation between microgrid and conventional grid can enlarge the proportion of distributed energy and clean energy in energy formation, which is conductive to the optimization of energy structure, energy conservation, and emission reduction in the power sector, as well as the enhancement of social welfare [15].At present, the research and construction of microgrid are put in an accentuated position in more and more countries. Taking the practical issues of the domestic power systems into consideration, these countries are actively involved in the construction of microgrid pilot projects along with the establishment of microgrid technical standards and management norms. In the United States, since the first microgrid project,Madriver, undertaken by North American Corporation was accomplished, a sequence of microgrid demonstration projects have been built up at University of Wisconsin-Madison, Sandia National Laboratories, Lawrence Berkeley National Laboratory, and other places. The development of microgrids in the United States is committed to increasing the reliability of power supply to critical loads, providing customized levels of high quality of electric energy, cutting down costs, and making power generation and transmission truly smart [16]. Microgrid in Europe is represented by Bornholm Microgrid in Denmark, Microgrid at National Technical University of Athens, and Bronsbergen Microgrid in Netherlands. The main purposes of these pilot projects are to test the ability of microgrid to switch between island mode and grid-connected mode, communication protocol verification, and demand-side management [17]. Typical microgrids in Japan include demonstration projects in Kyoto and Sendai as well as the one undertaken by Tokyo Gas. They are aimed at testing power quality control, load following, optimal operation, and load prediction of microgrid [18]. More microgrid pilot projects are currently underway in Canada, Australia, and some other countries.In China, with the introduction of the standards and incentive measures concerned with microgrid construction and grid connection, there emerges an improving environment for microgrid construction, which redounds to the flourishing of microgrid demonstration projects nationwide. According to the12th Five-Year Plan for Renewable Energy Development released by National Energy Administration, 30 new-energy microgrid demonstration projects should be completed by 2015. So far, some projects have been finished and grid connections have been put into practice, for example, the first phase of Zhangjiakou National Wind-Solar Hybrid Power Storage and Transmission Demonstration Project, Old Barag Banner Microgrid Demonstration Project, Shiquan River Water-Solar Hybrid Power Microgrid Demonstration Project, and Luxi Island Microgrid Demonstration Project. On September 10, 2013, Ministry of Finance issued theNotice on Adjusting the Standards for the Collection of Surcharges on Renewable Energy Power Prices, and on March 12, 2014, National Energy Administration released theNotice on Making Sufficient Efforts in Wind Power Grid Connection and Consumption. Later in December,Test Specification for Microgrids’ Access to Distribution Network andSystem DebuggingandAcceptance Specification for Microgrids’ Access to Distribution Network were both approved. These moves have greatly improved the status quo of microgrid in China and facilitated the construction, operation, and management of new-energy microgrid all around the country. It is safe to say that the institutional and field constructions of microgrid are becoming increasingly mature.Restricted by natural monopolies of conventional grids both home and abroad, research on grids cooperation has long been left out by the academic circle. In recent years, however, with the development of distributed energy and microgrid construction, studies on grids cooperation from technical perspective have yielded fruitful results. For the purpose of achieving a seamless transfer from grid-connected to island mode, paper [19] proposes a cooperative frequency control method which consists of microgrid central control and microgrid local control and can increase the frequency stability of islanded microgrid during both primary and secondary frequency control. Paper [20] comes up with a formation mechanism for the optimal microgrid coalition in the smart power distribution system, that is, HR coalition which boasts remarkable computational efficiency and is applicable to a wide range of microgrids. This mechanism is propitious for real-time processing and can bring down the power loss of a 500-microgrid-connected network by 26%–80%. In the meantime, in order to relieve time variation and intermittency imposed on conventional grid by renewable energy power generation, paper [21] finds out that grids can achieve the most benefits from cooperation if there are limited storage devices, but when there are adequate high-capacity storage devices, the benefits gained from cooperation are not desirable. Besides, on the basis of renewable energy power generation, paper [22] studies the real-time energy management under the cooperation between two microgrids, discusses the relationship between grid distance and power generation cost, and puts forward the allocation algorithm for optimal off-grid energy management. On top of that, it attempts to simulate this algorithm with the data from Tucson Electric Power. What is more, since there are varying coordination control strategies for the operation of microgrids under grid-connected and island mode, paper [23] presents the strategy for the coordination and control of different parts of microgrid and evaluates the management performance of this strategy through microgrid experimental projects. The results reveal that this strategy can guarantee the stability of microgrid operated under island mode by keeping the voltage of the public connection point within a reasonable range.The evolutionary game theory, based on limited rationality, dynamically analyzes the behavior strategy of human beings by using the method of studying the evolution and stability of the biological population. In an evolutionary game, different individual will get the appropriate degree of “fitness” through different behavior strategy. When the fitness of a strategy is higher than the average fitness of the population, this strategy will be developed in the population, until the fitness of the strategy is not longer than the average fitness of the population. The 1970s is the critical period for the formation and development of evolutionary game theory. Smith and Price [24] introduced the concept of evolutionary theory into the game theory and put forward the concept of evolutionary game theory and evolutionary stability strategy (ESS) in their paper “The Logic of Animal Conflicts” in 1973, which marks the birth of evolutionary game theory. In 1978, Taylor and Jonker [25] initiated the basic dynamic concept replicator dynamics so that evolutionary game theory has a clear research goal. Thus, the study on the theory and application of evolutionary game theory is developing rapidly.Research on the application of evolutionary game theory, in retrospect, mainly involves such areas as supply chain management, corporate governance, and financial investment. Paper [26] analyzes the game relationship between government and core enterprises in the green supply chain and discovers that the costs and benefits that core enterprises get from the implementation of green supply chain as well as the government subsidies and punishment will exert a direct impact on the result of the game. In the long run, government should enact and enforce more strict environmental regulations and increase relevant allowances and penalties so as to achieve a win-win result, while core enterprises should actively be devoted to environmental management practice to gain experience and set examples for enterprises in the upper and lower reaches. In developing green supplier-buyer relationship in the manufacturing industry, we find that, through establishing dynamic game models and observing the trend of cooperation between the stakeholders of green purchasing, trading behaviors can serve as policy and payoff functions of suppliers and producers. According to the analog experiment, manufacturing industry can obtain sustainable development, and the recycle ability of suppliers directly determines how green a supply chain is [27]. Some researchers have applied evolution mechanism to centipede game analysis, finding that evolution mechanism is conductive to delaying the risk of decision-making during the centipede game and even facilitates the full cooperation between the game players. This is particularly true to games that last for a fixed period of time, since human cooperative behaviors are characterized by “irrationality” [28]. As to the research on enterprise credit behaviors, when two companies form an alliance and carry out reasonable admission and exit mechanisms under the condition of sustainable development, credit risks can be effectively managed and controlled [29]. Paper [30] discusses how to take advantage of innovative financial tools to help enterprises raise fund with environmental protection technologies. The study concludes that as long as the expected revenue is greater than the average revenue, a financial market can be built up between economic agents and public administrations. The establishment of such a financial market will not only add to public benefits but also stimulate all enterprises to adopt environment-friendly technologies.From the literature review above, it can be seen that the cooperation between microgrid and conventional grid has rarely been studied before and there is little research on the evolutionary game analysis of strategy selection regarding cooperation between microgrid and conventional grid. However, as grids connection demonstration projects are in full swing, the issue on the cooperation between microgrid and conventional grid is arousing more academic concern. Based on the evolutionary game theory, this paper studies the choice behavior concerning the cooperation between microgrid and conventional grid, analyzes the factors that influence their cooperative game, and provides data for microgrid research and industrial decision-making for reference.The rest of the paper is arranged as follows: Section2 introduces the fundamental assumptions of modeling; Section 3 establishes the evolutionary game model and analyzes the evolutionary stable strategies of the two parties; Section 4 analyzes the factors influencing the cooperative game between microgrid and conventional grid; Section 5 makes a numerical simulation analysis of the evolutionary game strategies proposed in this paper; Section 6 is research conclusions.


## 2. Fundamental Assumptions

Modeling and analysis will be conducted on the premise of the following assumptions.(1)
There are two subjects, namely, microgrid and conventional grid, for the game, denoted asM and C, respectively. The sets for the strategies of a microgrid and a conventional grid are both cooperation, noncooperation, and one side can choose its optimum strategy in response to the strategy of the other side. Suppose the probability of a conventional grid’s choice of cooperation with a microgrid is x and the probability of its choice of noncooperation is 1-x, while the probability of a microgrid’s choice of cooperation with a conventional grid is y and the probability of its choice of noncooperation is 1-y (0≤x≤1, 0≤y≤1).(2)
Suppose that a microgrid and a conventional grid choose to cooperate on the basis of resources sharing and complementation; both parties can benefit more from cooperation strategy. When a microgrid and a conventional grid both resort to noncooperation strategy, their respective income from operation isA1, A2; when they both adopt cooperation strategy, their respective income from the cooperative operation is U1, U2.(3)
When both microgrid and conventional grid adopt cooperation strategy, they collaborate with each other to ensure power supply. Under the umbrella of conventional grid’s sound power transmission and distribution system, the phenomena of abandoned wind power and abandoned solar power in microgrid can be relieved. More importantly, more electric energy can be generated with the use of renewable energy, which enables microgrid to yield more direct income. Meanwhile, the flexibility and decentralization of microgrid make it possible for conventional grid to supply economical electric power in islands and remote areas, saving the high costs for grid construction and operation. Suppose the direct income of a microgrid and a traditional grid gained from their cooperation isD1 and D2.(4)
Apart from direct income, indirect income can also be generated from the cooperation between microgrid and conventional grid. Their cooperation cannot only increase the proportion of renewable energy and clean energy in power generation but also improve the social energy structure and ecological environment. Besides, the cooperation between microgrid and conventional grid is not confined to power supply but can be extended to technological innovation for grid connection and exploration of different cooperation modes, providing technical support and management experience for grid-connection projects. Therefore, we presume the indirect income of a microgrid and a conventional grid gained from their cooperation isI1 and I2.(5)
Given the high cost of electricity production with renewable energy and clean energy, Chinese government will give price allowances to electricity generated by renewable energy and clean energy to encourage the sale and purchase of this kind of electric power. At the same time, following the central government’s opinions on the reformation of electric power system, Chinese power grid enterprises will change their previous profit model of price difference between purchase and sale into a cost-plus-benefit model in charging wheeling cost. And in order to motivate conventional grids to renovate their facilities for grid access and control and to transmit and distribute more electricity generated by renewable energy and clean energy, a certain amount of compensation of value should be offered to conventional grid. Suppose the subsidies for a microgrid and a conventional grid areG1 and G2, respectively.(6)
When a microgrid and a conventional grid choose to cooperate, their cooperation costs are closely related to their respective input of production factors. Before the cooperation is reached, cost on partner selection and negotiation will arise. After the cooperation is nailed down, if one side pulls back from it, the other side who is cooperative is likely to trap in a difficult situation where all the possible opportunities and benefits are gone, because the assets it has invested can hardly be used for other purposes due to the high level of asset specificity (Williamson, 1975) [31] of the resources input into the power industry. Consequently, compared with the uncooperative side, the cooperative side needs to jack up its spending but cannot gain the cooperation benefit it deserves. Suppose the cost of a microgrid during the cooperation is C1 and that of a conventional grid is C2.(7)
Microgrid and conventional grid will meet risks particular to their cooperation, such as uncertain incidents, opportunism, and free rider problems, and thus they have to pay the costs arising from supervision, coordination, and control to cope with such kind of risks. Suppose the risk costs of a microgrid and a conventional grid areF1 and F2.


## 3. Evolutionary Game Analysis of Cooperation between Microgrid and Conventional Grid

### 3.1. Modeling

Power grid enterprises, as commercial organizations, have natural economic aspirations. Their willingness to choose cooperation strategy depends on whether this strategy can maximize their own benefits. Meanwhile, as providers of electricity, a necessity of social life, they also have noneconomic appeals including social benefits, environmental benefits, and political benefits, driven by the public voice and corporate social responsibilities. It follows that their choice of cooperation is a result of the evolutionary game. From the evolutionary game theory’s perspective, the payoff matrix in the gaming can be expressed as shown in Table1.Table 1
Payoff matrix in the gaming between microgrid and conventional grid.
Conventional gridC
Cooperationx
Noncooperation1-x
MicrogridM
Cooperationy
U
1
=
A
1
+
D
1
+
I
1
+
G
1
-
C
1
-
F
1, U2=A2+D2+I2+G2-C2-F2
A
1
-
C
1, A2
Noncooperation1-y
A
1, A2-C2
A
1, A2The necessary condition for a microgrid and a conventional grid to simultaneously choose cooperation strategy is that the sum of their direct benefits, indirect benefits, and government subsidies is greater than the sum of their cooperation costs and risk expenses; that is, the following conditions should be satisfied:(1)A1+D1+I1+G1>C1+F1,A2+D2+I2+G2>C2+F2.The benefits for a microgrid in choosing cooperation strategy are(2)Mc=A1+xD1+I1+G1-F1-C1.The benefits for a microgrid in choosing noncooperation strategy are(3)Mn=xA1+1-xA1.The average benefits for a microgrid in choosing mixed strategies can be derived as(4)M∗=yMc+1-yMn=A1+yxD1+I1+G1-F1-C1.Similarly, the average benefits for a conventional grid in choosing mixed strategies can be derived as(5)C∗=xCc+1-xCn=A2+xyD2+I2+G2-F2-C2.Based on these, if a microgrid and a conventional grid both choose cooperation strategy, the replicator dynamics differential equation (variant with time) is(6)dydt=yMc-M∗=y1-yxD1+I1+G1-F1-C1,dxdt=xCc-C∗=x1-xyD2+I2+G2-F2-C2.From (6), the evolution strategy matrix concerning the cooperation between a microgrid and a conventional grid can be conducted. The stability of the equilibrium point of the evolution system can be obtained by analyzing the local stability of the Jacobian matrix of this system. Suppose the Jacobian matrix of (6) is J, which can be expressed as(7)J=1-2yxD1+I1+G1-F1-C1y1-yD1+I1+G1-F1x1-xD2+I2+G2-F21-2xyD2+I2+G2-F2-C2.Thereof det⁡(J) is(8)det⁡J=1-2yxD1+I1+G1-F1-C11-2xyD2+I2+G2-F2-C2-y1-yD1+I1+G1-F1x1-xD2+I2+G2-F2and trJ is(9)trJ=1-2yxD1+I1+G1-F1-C1+1-2xyD2+I2+G2-F2-C2.From formula (8) and formula (9), the values and denotations of determinant and trace of J can be obtained.


### 3.2. Analysis of the Evolutionary Stable Strategies of the Game Players

According to the stability condition of Jacobian matrix, whendy/dt=0, dx/dt=0, there are five local equilibrium points that can be found on the plane S=[(y,x)∣0≤y,x≤1]. They are O(0,0), A(0,1), C(1,0), B(1,1), and E(y∗,x∗)(C2/(D2+I2+G2-F2), C1/(D1+I1+G1-F1)). Put them, respectively, into formula (8) and formula (9), and the list of local stability is obtained (see Table 2).Table 2
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
+
+
Unstable point
y
=
1, x=0
+
+
Unstable point
y
=
1, x=1
+
−
ESS
y
=
C
2
D
2
+
I
2
+
G
2
-
F
2 x=C1D1+I1+G1-F1
−
0
Saddle point
Note: this table is valid whenC2<D2+I2+G2-F2, C1<D1+I1+G1-F1.In Table2, there are five equilibrium points of which O and B are in evolutionary steady state, A and C are unstable points, and E is a saddle point. The process of the evolutionary game is shown as Figure 1.Figure 1
Schematic diagram of dynamic evolution (C2<D2+I2+G2-F2, C1<D1+I1+G1-F1).As seen in Figure1, when the benefits that a microgrid and a conventional grid gain from cooperation are greater than the benefits gained from noncooperation, in their evolutionary game, they either simultaneously adopt cooperation strategy or adopt noncooperation strategy. When the initial state falls in the bottom-left area of the system formed byAOCE, the system converges to O(0,0). That means both microgrid and conventional grid adopt noncooperation strategy. When the initial state falls in the upper right area of the system formed byAECB, the system converges to B(1,1). That means both the microgrid and the conventional grid adopt cooperation strategy. When one party adopts cooperation strategy and the other party adopts noncooperation strategy, the system is in an unstable point. At this point, one party gets the maximum benefits and the other suffers from the maximum loss, so the cooperation fails to reach Pareto Optimality. In the long run, if one party is found to commit “free-riding,” “fleecing,” and some other optimistic behaviors, the other party will also choose noncooperation strategy in order to minimize its loss. As a result, both parties will abandon cooperation strategy and O(0,0) indicates a stable confrontation state. If both parties choose to cooperate, a microgrid can fully exploit its flexibility and ability to utilize clean energy and distributed energy, and a conventional grid can take advantage of its sound power transmission and distribution network, ultrahigh voltage grid, and financial strength to provide fund and experience for the construction and technological innovation of microgrid. Only in this way can their cooperation reach Pareto Optimality and their benefits be maximized, as the point of convergence B(1,1) shows.However, when there are changes in the direct benefits, indirect benefits, government subsidies, cooperation cost, and risk expense, the total income of the bilateral cooperation might be less than the sum of cooperation cost and risk cost. Detailed analysis is as follows:(1)
WhenC2>D2+I2+G2-F2, C1<D1+I1+G1-F1 (i.e., the income of a conventional grid is less than its cooperation cost and risk expense when it adopts cooperation strategy, while the income of a microgrid is greater than the cooperation cost and risk expense when it adopts cooperation strategy), there are four equilibrium points in the system which converges to O(0,0), as seen from Table 3 and Figure 2. It suggests that noncooperation strategy that both parties adopt is actually an evolutionary stable strategy.(2)
WhenC2<D2+I2+G2-F2, C1>D1+I1+G1-F1 (i.e., the income of a large grid is greater than its cooperation cost and risk expense when it adopts cooperation strategy, while the income of a microgrid is less than the cooperation cost and risk expense when it adopts cooperation strategy), there are four equilibrium points in the system (see Table 4), and it can be inferred from Figure 3 that noncooperation strategy that both parties adopt is actually an evolutionary stable strategy, as the system converges to O(0,0).(3)
WhenC2>D2+I2+G2-F2, C1>D1+I1+G1-F1 (i.e., the income of a large grid is less than its cooperation cost and risk expense when it adopts cooperation strategy, and this is the same for a microgrid), there are four equilibrium points in the system (see Table 5), and it can be inferred from Figure 4 that the way that both parties are adopting noncooperation strategy is an evolutionary stable strategy, as the system converges to O(0,0).Table 3
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
+
+
Unstable point
y
=
1, x=0
−
Uncertain
Saddle point
y
=
1, x=1
−
Uncertain
Saddle point
Note: this table is valid whenC2>D2+I2+G2-F2, C1<D1+I1+G1-F1.Table 4
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
−
Uncertain
Saddle point
y
=
1, x=0
+
+
Unstable point
y
=
1, x=1
−
Uncertain
Saddle point
Note: this table is valid whenC2<D2+I2+G2-F2, C1>D1+I1+G1-F1.Table 5
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
−
Uncertain
Saddle point
y
=
1, x=0
−
Uncertain
Saddle point
y
=
1, x=1
+
+
Unstable point
Note: this table is valid whenC2>D2+I2+G2-F2, C1>D1+I1+G1-F1.Figure 2
Schematic diagram of dynamic evolution (C2<D2+I2+G2-F2, C1>D1+I1+G1-F1).Figure 3
Schematic diagram of dynamic evolution (C2<D2+I2+G2-F2, C1>D1+I1+G1-F1).Figure 4
Schematic diagram of dynamic evolution (C2>D2+I2+G2-F2, C1>D1+I1+G1-F1).


## 3.1. Modeling

Power grid enterprises, as commercial organizations, have natural economic aspirations. Their willingness to choose cooperation strategy depends on whether this strategy can maximize their own benefits. Meanwhile, as providers of electricity, a necessity of social life, they also have noneconomic appeals including social benefits, environmental benefits, and political benefits, driven by the public voice and corporate social responsibilities. It follows that their choice of cooperation is a result of the evolutionary game. From the evolutionary game theory’s perspective, the payoff matrix in the gaming can be expressed as shown in Table1.Table 1
Payoff matrix in the gaming between microgrid and conventional grid.
Conventional gridC
Cooperationx
Noncooperation1-x
MicrogridM
Cooperationy
U
1
=
A
1
+
D
1
+
I
1
+
G
1
-
C
1
-
F
1, U2=A2+D2+I2+G2-C2-F2
A
1
-
C
1, A2
Noncooperation1-y
A
1, A2-C2
A
1, A2The necessary condition for a microgrid and a conventional grid to simultaneously choose cooperation strategy is that the sum of their direct benefits, indirect benefits, and government subsidies is greater than the sum of their cooperation costs and risk expenses; that is, the following conditions should be satisfied:(1)A1+D1+I1+G1>C1+F1,A2+D2+I2+G2>C2+F2.The benefits for a microgrid in choosing cooperation strategy are(2)Mc=A1+xD1+I1+G1-F1-C1.The benefits for a microgrid in choosing noncooperation strategy are(3)Mn=xA1+1-xA1.The average benefits for a microgrid in choosing mixed strategies can be derived as(4)M∗=yMc+1-yMn=A1+yxD1+I1+G1-F1-C1.Similarly, the average benefits for a conventional grid in choosing mixed strategies can be derived as(5)C∗=xCc+1-xCn=A2+xyD2+I2+G2-F2-C2.Based on these, if a microgrid and a conventional grid both choose cooperation strategy, the replicator dynamics differential equation (variant with time) is(6)dydt=yMc-M∗=y1-yxD1+I1+G1-F1-C1,dxdt=xCc-C∗=x1-xyD2+I2+G2-F2-C2.From (6), the evolution strategy matrix concerning the cooperation between a microgrid and a conventional grid can be conducted. The stability of the equilibrium point of the evolution system can be obtained by analyzing the local stability of the Jacobian matrix of this system. Suppose the Jacobian matrix of (6) is J, which can be expressed as(7)J=1-2yxD1+I1+G1-F1-C1y1-yD1+I1+G1-F1x1-xD2+I2+G2-F21-2xyD2+I2+G2-F2-C2.Thereof det⁡(J) is(8)det⁡J=1-2yxD1+I1+G1-F1-C11-2xyD2+I2+G2-F2-C2-y1-yD1+I1+G1-F1x1-xD2+I2+G2-F2and trJ is(9)trJ=1-2yxD1+I1+G1-F1-C1+1-2xyD2+I2+G2-F2-C2.From formula (8) and formula (9), the values and denotations of determinant and trace of J can be obtained.


## 3.2. Analysis of the Evolutionary Stable Strategies of the Game Players

According to the stability condition of Jacobian matrix, whendy/dt=0, dx/dt=0, there are five local equilibrium points that can be found on the plane S=[(y,x)∣0≤y,x≤1]. They are O(0,0), A(0,1), C(1,0), B(1,1), and E(y∗,x∗)(C2/(D2+I2+G2-F2), C1/(D1+I1+G1-F1)). Put them, respectively, into formula (8) and formula (9), and the list of local stability is obtained (see Table 2).Table 2
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
+
+
Unstable point
y
=
1, x=0
+
+
Unstable point
y
=
1, x=1
+
−
ESS
y
=
C
2
D
2
+
I
2
+
G
2
-
F
2 x=C1D1+I1+G1-F1
−
0
Saddle point
Note: this table is valid whenC2<D2+I2+G2-F2, C1<D1+I1+G1-F1.In Table2, there are five equilibrium points of which O and B are in evolutionary steady state, A and C are unstable points, and E is a saddle point. The process of the evolutionary game is shown as Figure 1.Figure 1
Schematic diagram of dynamic evolution (C2<D2+I2+G2-F2, C1<D1+I1+G1-F1).As seen in Figure1, when the benefits that a microgrid and a conventional grid gain from cooperation are greater than the benefits gained from noncooperation, in their evolutionary game, they either simultaneously adopt cooperation strategy or adopt noncooperation strategy. When the initial state falls in the bottom-left area of the system formed byAOCE, the system converges to O(0,0). That means both microgrid and conventional grid adopt noncooperation strategy. When the initial state falls in the upper right area of the system formed byAECB, the system converges to B(1,1). That means both the microgrid and the conventional grid adopt cooperation strategy. When one party adopts cooperation strategy and the other party adopts noncooperation strategy, the system is in an unstable point. At this point, one party gets the maximum benefits and the other suffers from the maximum loss, so the cooperation fails to reach Pareto Optimality. In the long run, if one party is found to commit “free-riding,” “fleecing,” and some other optimistic behaviors, the other party will also choose noncooperation strategy in order to minimize its loss. As a result, both parties will abandon cooperation strategy and O(0,0) indicates a stable confrontation state. If both parties choose to cooperate, a microgrid can fully exploit its flexibility and ability to utilize clean energy and distributed energy, and a conventional grid can take advantage of its sound power transmission and distribution network, ultrahigh voltage grid, and financial strength to provide fund and experience for the construction and technological innovation of microgrid. Only in this way can their cooperation reach Pareto Optimality and their benefits be maximized, as the point of convergence B(1,1) shows.However, when there are changes in the direct benefits, indirect benefits, government subsidies, cooperation cost, and risk expense, the total income of the bilateral cooperation might be less than the sum of cooperation cost and risk cost. Detailed analysis is as follows:(1)
WhenC2>D2+I2+G2-F2, C1<D1+I1+G1-F1 (i.e., the income of a conventional grid is less than its cooperation cost and risk expense when it adopts cooperation strategy, while the income of a microgrid is greater than the cooperation cost and risk expense when it adopts cooperation strategy), there are four equilibrium points in the system which converges to O(0,0), as seen from Table 3 and Figure 2. It suggests that noncooperation strategy that both parties adopt is actually an evolutionary stable strategy.(2)
WhenC2<D2+I2+G2-F2, C1>D1+I1+G1-F1 (i.e., the income of a large grid is greater than its cooperation cost and risk expense when it adopts cooperation strategy, while the income of a microgrid is less than the cooperation cost and risk expense when it adopts cooperation strategy), there are four equilibrium points in the system (see Table 4), and it can be inferred from Figure 3 that noncooperation strategy that both parties adopt is actually an evolutionary stable strategy, as the system converges to O(0,0).(3)
WhenC2>D2+I2+G2-F2, C1>D1+I1+G1-F1 (i.e., the income of a large grid is less than its cooperation cost and risk expense when it adopts cooperation strategy, and this is the same for a microgrid), there are four equilibrium points in the system (see Table 5), and it can be inferred from Figure 4 that the way that both parties are adopting noncooperation strategy is an evolutionary stable strategy, as the system converges to O(0,0).Table 3
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
+
+
Unstable point
y
=
1, x=0
−
Uncertain
Saddle point
y
=
1, x=1
−
Uncertain
Saddle point
Note: this table is valid whenC2>D2+I2+G2-F2, C1<D1+I1+G1-F1.Table 4
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
−
Uncertain
Saddle point
y
=
1, x=0
+
+
Unstable point
y
=
1, x=1
−
Uncertain
Saddle point
Note: this table is valid whenC2<D2+I2+G2-F2, C1>D1+I1+G1-F1.Table 5
Stability of local equilibrium point.
Equilibrium point
det(J)
trJ
Local stability
y
=
0, x=0
+
−
ESS
y
=
0, x=1
−
Uncertain
Saddle point
y
=
1, x=0
−
Uncertain
Saddle point
y
=
1, x=1
+
+
Unstable point
Note: this table is valid whenC2>D2+I2+G2-F2, C1>D1+I1+G1-F1.Figure 2
Schematic diagram of dynamic evolution (C2<D2+I2+G2-F2, C1>D1+I1+G1-F1).Figure 3
Schematic diagram of dynamic evolution (C2<D2+I2+G2-F2, C1>D1+I1+G1-F1).Figure 4
Schematic diagram of dynamic evolution (C2>D2+I2+G2-F2, C1>D1+I1+G1-F1).


## 4. Analysis of the Impact of Model Parameters on Cooperative Game

It can be concluded through the above analysis that the stable strategies for evolutionary game between a microgrid and a conventional grid include (cooperation, cooperation) and (noncooperation, noncooperation). Pareto Optimality can be achieved when the two parties choose the cooperation strategy but a stable state can also be attained if the two parties do not choose the cooperation strategy, so factors influencing the changes in the area ofAOCE andAECB need to be analyzed in order to determine towards which direction the evolutionary results will develop, thus obtaining the factors influencing cooperation between the two parties. It can be observed through model analysis that factors influencing system convergence encompass cooperation cost, spending on cooperation risk, direct earnings on cooperation, and indirect earnings on cooperation as well as government subsidies, and the impact of each factor on the state of cooperative game will be discussed in the following.(1)
Cooperation costCi: the greater the cooperation cost is, the larger the area ofAOCE in the system is, the greater the system’s probability of evolving towards O(0,0) is, and the smaller the probability of adopting cooperation strategy between microgrid and conventional grid is. Cooperation costs of microgrid and conventional grid are associated with their respective inputs, and therefore, for the sake of cost reduction, a platform and mechanism for information communication firstly should be established to eliminate information asymmetry and lower spending on communication and selection and secondly close collaboration and management mode should be adopted to increase penalty cost (restraining noncompliance behaviors institutionally and legally) and decrease risk on the investment of specific asset.(2)
Spending on cooperation riskFi: the increase of cooperation risk will lead to considerable costs of coordination and control. The larger the area ofAOCE is, the greater the system’s probability of evolving towards O(0,0) is and the smaller the probability of adopting cooperation strategy between microgrid and conventional grid is. Cooperation risk primarily arises from the potential opportunistic behaviors of cooperative members and conflicts of interest between the two parties. Opportunistic behaviors are primarily manifested by partners’ breaking promise, evading duties, and misappropriation of public resources as well as information distortion, while conflicts of interest are mainly demonstrated as mergers or acquisitions by partners, thus giving rise to unfair distribution of interest. Therefore, it is necessary to enhance the transparency of bilateral cooperation by means of establishing a mechanism for distribution of interests, a risk-sharing mechanism, and a penalty and oversight mechanism as well, thus reducing behaviors of information asymmetry occurring in bilateral cooperation and lowering the risk of cooperation.(3)
Direct earnings on cooperationDi: the greater the direct earnings on bilateral cooperation are, the larger the area ofAECB in the system is, the greater the system’s probability of evolving towards B(1,1) is, and the greater the probability of adopting cooperation strategy between microgrid and conventional grid is. Microgrid and conventional grid can give full play to their comparative advantages in cooperation and boost the benefits of bilateral cooperation, thereby building trust through cooperation and helping to form a sound partnership. Based on this analysis and on comprehensive and repeated demonstration, great efforts will be made to construct microgrid and renovate conventional grid in appropriate areas; in the meantime, power supply capacity of microgrid and conventional grid will be given an overall consideration. What is more, scientific and reasonable scheduling will be achieved through coordination and arrangement, so more accommodation of wind power, photovoltaic power, and other renewable clean energy and power can be provided.(4)
Indirect earnings on cooperationIi: the greater the indirect earnings on bilateral cooperation are, the larger the area ofAECB in the system is, the greater the system’s probability of evolving towards B(1,1) is and the greater the probability of adopting cooperation strategy between microgrid and conventional grid is. Further cooperation between microgrid and conventional grid can not only enhance technological innovation such as the connection, control, protection, and scheduling of microgrid and conventional grid but also provide appropriate experiences and references for exploring the cooperation model of grids and establishing industrial standards, thus promoting the optimization of energy allocation of the power industry and improvement in ecological environment and eventually increasing the well-being of the whole society. As a result, microgrid and conventional grid need to raise awareness of cooperative effect and the government will provide better policy guidance for the cooperation and further encourage microgrid and conventional grid to cooperate in more areas.(5)
Government subsidiesGi: microgrid and conventional grid offer renewable energy, clean energy and electricity, and so forth to the society. The greater the government subsidies the two parties gain from this practice are, the larger the area ofAECB in the system is, the greater the system’s probability of evolving towards B(1,1) is, and the greater the probability of adopting cooperation strategy between microgrid and conventional grid is. In view of the characteristics of microgrid construction and conventional grid renovation including high investment and cost but low earnings in the early stage, higher earnings in mid-late stage, and unclear price formation mechanism, the government will, through offering subsidies to electricity price of renewable energy and clean energy, encourage microgrid and conventional grid to explore and utilize through cooperation distributed energy, clean energy, energy of transmission and distribution, clean energy and power, and other projects, thus further improving electricity services of the whole society.


## 5. Numerical Analysis

Numerical simulation is carried out through Matlab programming to make an in-depth analysis of the evolution of cooperation between microgrid and conventional grid. The parameters of the payoff matrix for the game between microgrid and conventional grid are as follows:C1=200, C2=400, D1=300, D2=500, I1=300, I2=200, G1=300, G2=200, F1=400, and F2=200. The evolution of the system towards point B(1,1) is described when the proportion (y∗,x∗) of initial strategy of microgrid and conventional grid varies in the plane of [0,1]∗[0,1]. When (0.2, 0.6), (0.5, 0.5), and (0.6, 0.2) are taken as the initial values and the time period is [0,100], it can be seen from Figure 5 that the greater the possibility of choosing initial cooperation strategy by microgrid and conventional grid is, the faster the evolution towards the point B(1,1) is and that cooperation willingness of conventional grid plays a vital role in facilitating the cooperation between microgrid and conventional grid. In order to verify the impact of changes in a certain parameter on the model convergence when other parameters remain unchanged, let C1=100, C2=200, D1=400, D2=600, G1=500, and G2=400 when (0.5, 0.5) is taken as the initial value of (y∗,x∗) and [0,100] as the time period. As shown in Figure 6, compared with the initial values of Ci, Di, and Gi, lowering cooperation cost and increasing direct earnings and government subsidies will accelerate the evolution of the system towards point B(1,1) and increase the probability of cooperation between microgrid and conventional grid, which lends credence to the early hypothesis.Figure 5
Schematic diagram of dynamic evolution of cooperation under different initial values of(y∗,x∗).Figure 6
Schematic diagram of dynamic evolution of cooperation under differentCi, Di, and Gi values.


## 6. Conclusions and Suggestions

This paper employs the evolution game theory to model the selection of cooperation strategies between microgrid and conventional grid. It is concluded that the probability of their unanimous choice of cooperative strategies positively correlates with the direct benefits and indirect benefits of their cooperation as well as the government subsidies but negatively correlates with the cooperation costs and spending on cooperation risks. Based on the research results, the paper proposes the following suggestions:(1)
Government should not only formulate management norms for the planning, design, construction, operation, and maintenance of microgrid but also improve the technical standards and regulations for grids connection, in an effort to create a better policy environment for the cooperation between microgrid and large grid.(2)
More backups should be given to the construction of microgrid and the reformation of large grid in order to ignite their willingness for cooperation and especially encourage the participation of large grid. A series of measures such as electricity price subsidies, preferential tax, and allowances for research and development can be taken to motivate large grid to transport renewable energy and clean energy and deepen the cooperation on technological innovation and the exploration of different cooperation modes.(3)
Communication, coordination, and risk prevention mechanisms for the cooperation between microgrid and large grid should be established to eliminate information asymmetry, increase mutual trust, lower the risk of opportunism, and reduce cooperation costs.



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*Source: 103206-2015-08-31.xml*