## 3 P-M Model
### C-P Sub-Model  
#### 3.1.1 Construction of C-P Sub-Model  
The cost price of cooling, heating, and electricity equals the total cost divided by the total energy at time \(t\), as shown in Equation (1), where \(fval^{c}_{t}\), \(fval^{h}_{t}\), and \(fval^{c}_{t}\) represent the total costs of cooling, heating, and electricity, respectively. \(Q^{c}_{t}\) can be obtained by summing the output of all the cooling devices at time \(t\), as well as \(Q^{h}_{t}\) and \(Q^{c}_{t}\) for the heating and electrical devices, respectively, as shown in Equation (2). \[P^{c}_{t}=\frac{fval^{c}_{t}}{Q^{c}_{t}},P^{h}_{t}=\frac{fval^{h}_{t}}{Q^{h}_{t} },P^{e}_{t}=\frac{fval^{e}_{t}}{Q^{e}_{t}} \tag{1}\]  
\[Q^{c}_{t}=\sum_{i}Q^{c}_{t,i},\;Q^{h}_{t}=\sum_{j}Q^{h}_{t,j},\;Q^{c}_{t}=\sum _{k}Q^{e}_{t,k} \tag{2}\]  
Parameter \(fval^{e}_{t}\) comprises the operating costs, initial investment, and maintenance costs of all power generation equipment, as shown in Equation (3). \[fval^{e}_{t}=\sum_{k}(P_{t,k}*\frac{Q^{e}_{t,k}}{\eta^{e}_{k}}+a*IV_{k}*\frac{Q ^{e}_{t,k}}{\sum_{\tau=1}^{8760}Q^{e}_{\tau,k}}+M_{t,k} \tag{3}\]  
The large power grid, which supplies electricity to the PIES in the time-of-use mode, is treated as power generation equipment in this study. For the operating costs, \(P_{t,k}\) represents the price of fossil energy consumed by each power generation device at time \(t\), and \(P_{t,k}=0\) represents renewable energy power generation. Meanwhile, \(\eta^{e}_{k}\) represents the efficiency of power generation equipment \(k\), which varies with the part load ratio. The initial investment of every device is apportioned at each moment throughout the life cycle according to the time-by-time output, where \(a\) is the coefficient of the uniform annual value; \(IV_{k}\) is the initial investment of power generation device \(k\), which can be calculated by the maximum capacity of device \(k\) for the entire year; and \(\sum_{\tau=0}^{8760}Q^{e}_{\tau,k}\) represents the total output of all power generation equipment over the entire year. \(M_{t,k}\) represents the maintenance costs, which can be obtained by ref. [23]. Different from the micro-grid, the process of cooling and heating production consumes the internal electricity of park \(P^{e}_{t}\). \(P^{e}_{t}\), calculated by Equation (1), is different from the electricity price of the large power grid. In addition, \(P_{t,i}\) and \(P_{t,j}\) represent the price of fossil energy consumed by each cooling and heating device at time \(t\), respectively. Therefore, \(fval^{c}_{t}\) and \(fval^{h}_{t}\) are expressed as Equations (4) and (5), where \(CO^{c}_{i}\) and \(CO^{p}_{j}\) represent the efficiencies of cooling device \(i\) and heating device \(j\), respectively, which vary with the part load ratio. \[fval^{c}_{t}=\sum_{i}\left(P_{t,i}*\frac{Q^{c}_{t,i}}{\eta^{e}_{i}}+P^{e}_{t}* \frac{Q^{c}_{t,i}}{CO^{c}_{i}}+a*IV_{i}*\frac{Q^{c}_{t,i}}{\sum_{\tau=1}^{8760 }Q^{c}_{\tau,i}}+M_{t,i}\right) \tag{4}\]  
\[fval^{h}_{t}=\sum_{j}\left(P_{t,j}*\frac{Q^{h}_{t,j}}{\eta^{h}_{j}}+P^{e}_{t}* \frac{Q^{h}_{t,j}}{CO^{h}_{j}}+a*IV_{j}*\frac{Q^{h}_{t,j}}{\sum_{\tau=1}^{8760 }Q^{h}_{\tau,j}}+M_{t,j}\right) \tag{5}\]  
The objective function is the minimum total cost at time \(t\), as shown in Equation (6). This is a multivariable constrained nonlinear programming problem that can be solved by FMINCON in Matlab2018b. The constraints include those on supply and demand balance, energy conversion efficiency, equipment, renewable energy, and other nonlinear constraints [23; 24], as shown in Equation (7). \[min(fval_{t}^{e}+fval_{t}^{c}+fval_{t}^{h}\big{)} \tag{6}\] \[const.\{\left\{ \begin{array}{l}\vspace{0.2 cm}A\vspace{0.2 cm}A \vspace{0.2 cm}Q\leq\vspace{0.2 cm}b\\ \vspace{0.2 cm}A_{eq}\vspace{0.2 cm}Q=\vspace{0.2 cm}b_{eq}\\ \vspace{0.2 cm}lb\leq Q\leq\vspace{0.2 cm}ub\\ c(\vspace{0.2 cm}Q)\leq 0\\ c_{eq}(\vspace{0.2 cm}Q)=0\\ Q_{0}\end{array} \right.