# Evolutionary Game Analysis of Power Generation Groups Considering Energy Price Fluctuation
## 3 Analysis of Power Generation Cost under Primary Energy Price Fluctuation
### Gas-Fired Power Generation Cost Analysis
At this time, the quotation of coal-fired and gas-fired power generation groups depends on the relationship between \(\varepsilon\) and \(\mu_{gas}/\
u_{coal}\) the two situations will be discussed as follows. If \(\varepsilon<\mu_{gas}/\
u_{coal}\), then \(P_{coal}^{H}<P_{gas}^{L}\), at this time, the price quoted by the strategic bidding of the coal-fired power generation group is still lower than the price quoted by the cost bidding of the gas-fired power generation group, the payment parameters of the two types of generation groups can be expressed as Equation (21). \[\left\{ \begin{array}{l}u_{3}=P_{coal}^{H}-C_{coal}=(\
u_{coal}-1)C_{coal} \\ v_{3}=\frac{Q_{DM}-\sum\limits_{i=1}^{m}Q_{coal,i}}{\sum\limits_{i=1}^{n}Q_{ gas,i}}(P_{gas}^{L}-C_{gas})=\frac{Q_{DM}-\sum\limits_{i=1}^{m}Q_{coal,i}}{ \sum\limits_{i=1}^{n}Q_{gas,i}}(\mu_{gas}-1)C_{gas}\end{array} \right. \tag{21}\]  
If \(\varepsilon>\mu_{gas}/\
u_{coal}\), then \(P_{coal}^{H}>P_{gas}^{L}\), at this time, the price quoted by the strategic bidding of the coal-fired power generation group is higher than the price quoted by the cost bidding of the gas-fired power generation group, the payment parameters of the two types of generation groups can be expressed as Equation (22):  
\[\left\{ \begin{array}{l}u_{3}=\frac{Q_{DM}-\sum\limits_{i=1}^{n}Q_{gas,i}}{ \sum\limits_{i=1}^{n}Q_{gas,i}}(P_{coal}^{H}-C_{coal})=\frac{Q_{DM}-\sum \limits_{i=1}^{n}Q_{gas,i}}{\sum\limits_{i=1}^{n}Q_{coal,i}}(\
u_{coal}-1)C_{coal} \\ v_{3}=P_{gas}^{L}-C_{gas}=(\mu_{gas}-1)C_{gas}\end{array} \right. \tag{22}\]When both coal-fired and gas-fired power generation groups choose strategic bidding \(u_{4}\) and \(v_{4}\) are calculated by Equation (23):  
\[\left\{ \begin{array}{l}u_{4}=P_{coal}^{H}-C_{coal}=(\
u_{coal}-1)C_{coal}\\ \
u_{4}=\frac{Q_{DM}-\sum\limits_{i=1}^{m}Q_{coal,i}}{\sum\limits_{i=1}^{n}Q _{gas,i}}(P_{gas}^{H}-C_{gas})=\frac{Q_{DM}-\sum\limits_{i=1}^{m}Q_{coal,i}}{ \sum\limits_{i=1}^{n}Q_{gas,i}}(\
u_{gas}-1)C_{gas}\end{array} \right. \tag{23}\]  
Based on this, when the gas-fired power generation group chooses cost bidding and the coal-fired power generation group adopts strategic bidding, there will be two situations in the payment matrix. The first situation will be discussed below:  
Scenario 1.1-a: the price quoted by the strategic bidding of the coal-fired power generation group is lower than the price quoted by the cost bidding of the gas-fired power generation group (\(P_{coal}^{H}<P_{gas}^{L}\))\(\epsilon<\mu_{gas}/\
u_{coal}\). At this time, the replicator dynamics equation is Equation (24):  
\[\left\{ \begin{array}{l}\dot{p}=\frac{dp}{dt}=p(1-p)(\mu_{coal}-\
u_{coal})C _{coal}\\ \dot{q}=\frac{dq}{dt}=q(1-q)\frac{Q_{DM}-\sum\limits_{i=1}^{m}Q_{coal,i}}{ \sum\limits_{i=1}^{n}Q_{gas,i}}(\mu_{gas}-\
u_{gas})C_{gas}\end{array} \right. \tag{24}\]  
Each element in the corresponding Jacobi matrix is calculated by Equation (25):  
\[\left\{ \begin{array}{l}\frac{\partial\dot{p}}{\partial p}=(1-2p)(\mu_{coal}- \
u_{coal})C_{coal}\\ \frac{\partial\dot{p}}{\partial q}=0\\ \frac{\partial\dot{p}}{\partial q}=0\\ \frac{\partial\dot{q}}{\partial q}=(1-2q)\frac{Q_{DM}-\sum\limits_{i=1}^{m}Q_{ coal,i}}{\sum\limits_{i=1}^{n}Q_{gas,i}}(\mu_{gas}-\
u_{gas})C_{gas}\end{array} \right. \tag{25}\]  
At this time, the equilibrium point E5 is not located in the space range \([0,1]\times[0,1]\). Thus, there are four equilibrium points in this evolutionary game. According to the Jacobi matrix Formula (25) in this Scenario, the stability of each equilibrium point is shown in Table 2.  
Condition 1 is as stated in inequality (26):  
\[(\mu_{coal}-\
u_{coal})C_{coal}<\frac{Q_{DM}-\sum\limits_{i=1}^{m}Q_{coal,i}} {\sum\limits_{i=1}^{n}Q_{gas,i}}(\mu_{gas}-\
u_{gas})C_{gas} \tag{26}\]