## 3 Hybrid Game Theory-Based Optimization for the Energy Trading Strategy of VPP
### Master-Slave Game Model Between VPP and PVP Alliance  
The members within the PVP alliance, as the game follower, should be optimized to obtain the amount of heat and electricity transactions between the PVP alliance and the VPP, as well as the amount of electricity transactions among the members within the PVP alliance. The model of the game leader VPP has been given in Section 3.1.2. The game follower, PVP alliance model can be constructed as following contents. 1. Objective function Each member within the PVP alliance aims to minimize its own costs, which specifically include the cost of purchasing electricity and heat, the benefits of interaction between members, and the cost of demand response. \[\text{min}F_{i}^{PVP}=C_{i}^{PVP}+C_{i}^{DR}-I_{i}^{TRA},\] (16) where \(F_{i}^{PVP}\) is the comprehensive benefit of the \(i\)th member in the PVP alliance, \(C_{i}^{PVP}\), \(I_{i}^{TRA}\) and \(C_{i}^{DR}\) are the cost of purchased electricity, and heat of the \(i\)th member in the PVP alliance, the benefit of the \(i\)th member caused by internal P2P electricity trading among PVP members, and the cost of load participation in demand response within the \(i\)th member, respectively. The cost of purchased electricity and heat of the \(i\)th member in the PVP alliance is given as follows  
\[C_{i}^{PVP}=\sum_{t=1}^{24}\left(c_{t}^{ss}P_{i,t}^{buy}+c_{t}^{hs}H_{i,t}^{buy }\right), \tag{17}\]  
where \(c_{t}^{cs}\) and \(c_{t}^{hs}\) represent the prices of electricity and heat purchased from VPP by the PVP alliance at time \(t\), respectively; \(P_{i,t}^{buy}\) and \(H_{i,t}^{buy}\) represent the purchased electricity and heat load from VPP by the \(ith\) member in PVP alliance at time \(t\), respectively. The benefit of the \(i\)th member caused by internal P2P electricity trading among PVP members is  
\[I_{i}^{TRA}=\sum_{j=1,j\
eq i}^{N}\sum_{t=1}^{24}c_{t}^{i-j}\cdot P_{t}^{i-j, P2P}, \tag{18}\]  
\[c_{\text{min}}^{i-j}\leq c_{t}^{i-j}\leq c_{\text{max}}^{i-j}, \tag{19}\]  
where \(c_{t}^{i-j}\) is the price of electricity traded between member \(i\) and member j of the PVP alliance at time t, \(c_{\text{min}}^{i-j}\) and \(c_{\text{max}}^{i-j}\) are the lower limit and upper limit of the permissible trading price stipulated in the electricity market policy, respectively; \(P_{t}^{i-j,P2P}\) is the amount of electricity traded between member \(i\) and member \(j\) at time \(t\). While demand response reduces the overall cost of purchased power and heat for the PVP consortium, there are corresponding implementation operating costs for the equation as follows:  
\[C_{i}^{DR}=c_{e}^{p,cut}\cdot\sum_{l=1}^{24}P_{i,t}^{p,cut}+c_{h}^{p,cut} \cdot\sum_{l=1}^{24}H_{i,t}^{p,cut}, \tag{20}\]where \(P_{i,t}^{p,cut}\) and \(H_{i,t}^{p,cut}\) represent the electric heat load reduction of PVP alliance member \(i\) at time \(t\); \(c_{e}^{p,cut}\) and \(c_{h}^{p,cut}\) are the electric heat load reduction cost coefficients of the \(ith\) member in the PVP alliance at time \(t\), respectively. (2) Constraint conditions  
(1) Demand response constraint  
The \(i\)th member in the PVP alliance can reduce the electric heating load at times \(t\) to meet the following constraints:  
\[0\leq P_{i,t}^{cut}\leq P_{i}^{cut,\max}, \tag{21}\]  
\[0\leq H_{i,t}^{cut}\leq H_{i}^{cut,\max}, \tag{22}\]  
where \(P_{i}^{cut,\max}\) and \(H_{i}^{cut,\max}\) are the maximum value of the reduced electricity and heating load for the \(i\)th member in the PVP alliance, respectively. (2) P2P electrical energy trading constraints  
P2P power transactions among members of the PVP alliance should be limited to the capacity of the power grid, that is  
\[-P_{t}^{P2P,\max}\leq P_{t}^{i-j,P2P}\leq P_{t}^{P2P,\max}, \tag{23}\]  
\[P_{t}^{j-j,P2P}=P_{t}^{j-i,P2P}, \tag{24}\]  
where \(P_{t}^{P2P,\max}\) is the maximum allowed interaction power among members within the PVP alliance at time \(t\); \(P_{t}^{j-j,P2P}\) and \(P_{t}^{j-i,P2P}\) represent the electricity transaction from member \(i\) to \(j\), from member \(j\) to \(i\), respectively.