## 2 Electricity-Gas-Heat Integrated Energy System Model
### P2G Model  
P2G can be used as the electric load in the power network and the gas source in the natural gas network. In the process of methane production by P2G, a large amount of waste heat is generated, which can be used as the heat source in the heat network by its heat release characteristics. At the same time, P2G can directly use the CO\({}_{2}\) captured by the carbon capture device for production in the integrated energy system considering carbon capture. The reaction process of P2G to produce methane is as follows:  
\[\left\{\begin{array}{l}2\mathrm{H}_{2}\mathrm{O}\to 2\mathrm{H}_{2} +\mathrm{O}_{2}\\ \mathrm{CO}_{2}+4\mathrm{H}_{2}\to\mathrm{CH}_{4}+2\mathrm{H}_{2}\mathrm{O} \end{array}\right. \tag{1}\]  
Figure 1: Model diagram of integrated energy system combined electricity, gas and heat.  
During the thermal reaction of methanation, 165.01 kJ of heat is released for every 1 mol of methane produced. The heat released by P2G can be collected through the heat exchanger and injected into the heat network.  
The conversion relationship between CH\({}_{4}\) production rate of P2G and reaction heat release in unit time is as follows [13]:  
\[V^{\text{CH}_{4}}\ =\ \frac{\rho_{\text{H}_{2}}\text{-}P_{\text{P2G}}/v_{\text{ H}_{2}}}{4M_{\text{H}_{2}}}\frac{M_{\text{CH}_{4}}}{\rho_{\text{CH}_{4}}}\cdot 10^{3} \tag{2}\]  
\[H_{\text{P2G},\text{k}}\ =\ \frac{\rho_{\text{H}_{2}}\text{-}P_{\text{P2G}}/v_{ \text{H}_{2}}}{4M_{\text{H}_{2}}}\frac{\Delta H\eta_{\text{P2G}}^{\text{heat}}} {3600} \tag{3}\]  
where \(V^{CH_{4}}\) is the methane generation rate, \(v_{H_{2}}\) is the hydrogen generation rate, \(v_{H_{2}}\) = 1 Nm\({}^{3}\)/ (3.47 kW\(\cdot\)h), \(M_{\text{CH}_{4}}\) and \(M_{\text{H}_{2}}\) are the molar mass of methane and hydrogen respectively, \(M_{\text{CH}_{4}}\) = 16 g/mol, \(M_{\text{H}_{2}}\) = 2 g/mol, \(\rho_{\text{CH}_{4}}\) and \(\rho_{\text{H}_{2}}\) are the density of methane and hydrogen, respectively, \(\rho_{\text{CH}_{4}}\) = 717.4 g/Nm\({}^{3}\), \(\rho_{\text{H}_{2}}\) = 89.9 g/Nm\({}^{3}\), \(P_{\text{P2G}}\) is the active power output of P2G power station, \(H_{\text{P2G}}\) is methanation heat reaction exothermic, \(\Delta H\) is the value of heat released by the reaction when 1 mol of methane is generated, \(\Delta H\) = 165.01 kJ, \(\eta_{\text{P2G}}^{\text{heat}}\) is the proportion of methanation heat reaction injected into the heat network which is set to \(\eta_{\text{P2G}}^{\text{heat}}\) = 0.8 in [PERSON] et al. [32].  
It can be calculated from the above equation that when the operating output of P2G is 1 MW, its reaction thermal output is:  
\[H_{\text{P2G}}\ =\ \frac{89.9\times 1/3.47}{4\times 2}\frac{165.01\times 0.8 }{3600}\ =\ 0.1188\text{ MW} \tag{4}\]  
The thermal efficiency of methanation reaction can reach more than 10% and in the case it is worth implementing the heat recovery. Because the volume of CO\({}_{2}\) input and methane output before and after the reaction is the same. Therefore, the consumption of P2G is expressed as follows:  
\[E_{\text{out},t}\ =\ \rho_{\text{CO}_{2}}V_{t}^{\text{CO}_{2}}\ =\ \rho_{\text{CO}_{2}}V_{t}^{\text{CH}_{4}} \tag{5}\]  
where \(E_{\text{out},t}\) is the amount of CO\({}_{2}\) consumed by P2G power station, \(\rho_{\text{CO}_{2}}\) is the density of CO\({}_{2}\), \(V_{t}^{\text{CO}_{2}}\) is the volume of CO\({}_{2}\) input to P2G power station at time \(t\), \(V_{t}^{\text{CH}_{4}}\) is the volume of methane output from P2G power station at time \(t\).