## 4 Model of Operation Strategy for EGCS
### Constraints
8 Grid reliability level constraint:  
\[E_{\text{F}}^{t}\leq E_{\text{F}}^{\text{max}} \tag{26}\]  
where \(E_{F}^{\text{max}}\) is the maximum allowable load shedding value of the system, indicating the reliability level of the system operation. 9 Natural gas flow balance constraint:  
\[\sum_{r=1}^{N_{\text{G}}}C_{\text{gas}}^{r,t}+\sum_{m=1}^{N_{\text{P2G}}}C_{ \text{P2G}}^{m,t}-\sum_{n=1}^{N_{\text{L}}}C_{\text{line}}^{n,t}-C_{L}^{t}=0 \tag{27}\]  
where \(C_{\text{gas}}^{r,t}\) is the gas flow output of the \(r\)th natural gas source in \(t\)th period. \(N_{\text{P2G}}\) is the total amount of P2G units. \(C_{\text{P3G}}^{m,t}\) is the gas flow output of the \(m\)th P2G unit when the unit runs in \(t\)th period. \(N_{\text{L}}\) is the total amount of natural gas pipelines. \(C_{\text{line}}^{n,t}\) is the gas flow loss of the \(n\)th pipeline in \(t\)th period. \(C_{\text{L}}^{t}\) is the total natural gas load in \(t\)th period. 10 Upper and lower limit constraint of nature gas unit output:  
\[C_{\text{gas}}^{r,\text{min}}\leq C_{\text{gas}}^{r}\leq C_{\text{gas}}^{r, max} \tag{28}\]  
where \(C_{\text{gas}}^{r}\) is the output of the \(r\)th natural gas unit. \(C_{\text{gas}}^{r,\text{min}}\) and \(C_{\text{gas}}^{r,\text{max}}\) are, respectively, the upper and lower limits of the output of the \(r\)th natural gas unit. (11) Pipeline flow constraints:  
\[\left\{ \begin{array}{l}C_{\text{line}}^{n}+C_{\text{L,line}}^{n}\leq C_{\text{ line}}^{n,\text{max}}\\ C_{\text{line}}^{n}=v_{n}s_{n}\sqrt{s_{n}\big{(}\Pi_{n_{1}}^{2}-\Pi_{n_{2}}^{2} \big{)}}\\ s_{n}=\left\{\begin{array}{l}+1,\Pi_{n_{1}}-\Pi_{n_{2}}\geq 0\\ -1,\Pi_{n_{1}}-\Pi_{n_{2}}<0\end{array}\right.\end{array} \right. \tag{29}\]  
where \(C_{\text{line}}^{n}\) is the flow of natural gas in pipe \(n\). \(C_{\text{L,line}}^{n}\) is the total natural gas load supplied by pipeline \(n\). \(C_{\text{line}}^{n,\text{max}}\) is the maximum flow of natural gas in pipe \(n\). \(v_{n}\) is constant. \(s_{n}\) reflects the direction of the pipeline flow. \(n_{1}\) and \(n_{2}\) are node numbers. \(\Pi_{n1}\) is the pressure of node \(n_{1}\). \(\Pi_{n2}\) is the pressure of node \(n_{2}\). (12) Electric-gas coupling constraints:  
\[\left\{ \begin{array}{l}H_{\text{g}}=\beta_{\text{g}}^{1}+\beta_{\text{g}}^{2 }P_{\text{g}}+\beta_{\text{g}}^{3}P_{\text{g}}^{2}\\ C_{\text{gas}}=\frac{H_{\text{g}}}{GHV}\end{array} \right. \tag{30}\]  
where \(H_{\text{g}}\) is the input heat value of the gas turbine. \(P_{\text{g}}\) is the output power of the gas turbine. \(C_{\text{gas}}\) is the equivalent natural gas load of the node connected to the gas turbine in NGS14. \(GHV\) is a fixed high heat value. \(\beta_{\text{g}}^{1}\), \(\beta_{\text{g}}^{2}\), \(\beta_{\text{g}}^{3}\) are constants determined by the heat consumption rate curve of the gas turbine. (13) Constraints of energy conversion efficiency  
\[\left\{ \begin{array}{l}P_{\text{G2P}}^{v,t}=\eta_{\text{G2P}}^{v}C_{\text{G2P }}^{v,t}\\ C_{\text{P2G}}^{m,t}=\eta_{\text{P2G}}^{m}P_{\text{P2G}}^{m,t}\\ H_{\text{Heat}}^{s,t}=\eta_{\text{Heat}}^{s}P_{\text{Heat}}^{s,t}\\ H_{\text{Heat}}^{u,t}=\eta_{\text{Heat}}^{u}C_{\text{Heat}}^{u,t}\end{array} \right. \tag{31}\]  
where \(P_{\text{G2P}}^{v,t}\) is the output power of the \(v\)th G2P unit when the unit runs in \(t\)th period. \(C_{\text{G2P}}^{v,t}\) is the gas flow input of the \(v\)th G2P unit when the unit runs in \(t\)th period. \(C_{\text{P2G}}^{m,t}\) is the gas flow output of the \(m\)th P2G unit when the unit runs in \(t\)th period. \(P_{\text{P2G}}^{m,t}\) is the input power of the \(m\)th P2G unit when the unit runs in \(t\)th period. \(H_{\text{Heat}}^{s,t}\) is the heat output of the \(s\)th power to heat unit when the unit runs in \(t\)th period. \(H_{\text{Heat}}^{k,t}\) is the heat output of the \(k\)th gas to heat unit when the unit runs in \(t\)th period. \(C_{\text{Heat}}^{k,t}\) is the gas flow input of the \(k\)th gas to heat unit when the unit runs in \(t\)th period.