## 4 Model of Operation Strategy for EGCS
### Constraints  
(1) Power flow constraints:  
\[\left\{ \begin{array}{l}P_{i}=V_{i}\sum\limits_{j\in i}V_{j}\left(G_{ij}\cos \theta_{ij}+B_{ij}\sin\theta_{ij}\right)\\ Q_{i}=V_{i}\sum\limits_{j\in i}V_{j}\left(G_{ij}\sin\theta_{ij}-B_{ij}\cos \theta_{ij}\right)\end{array} \right. \tag{19}\]  
where \(P_{i}\) and \(Q_{i}\) are, respectively, the active and reactive power of node \(i\). \(V_{i}\) is the voltage of node \(i\). \(V_{j}\) is the voltage of node \(j\). \(G_{ij}\) and \(B_{ij}\) are the line parameters between nodes \(i\) and \(j\). \(\theta_{ij}\) is the power factor of the line between nodes \(i\) and \(j\). (2) Power balance constraint:  
\[\sum\limits_{i=1}^{N_{\rm P}}P_{\rm G}^{i,t}+\sum\limits_{w=1}^{N_{\rm EG}}P_{ \rm DG}^{w,t}+\sum\limits_{v=1}^{N_{\rm G2P}}P_{\rm G2P}^{v,t}-P_{L}^{t}+E_{\rm F }^{t}=0 \tag{20}\]  
where \(N_{\rm DG}\) is the total amount of DG units. \(P_{\rm DG}^{w,t}\) is the output power of the \(w\)th DG unit when the unit runs in \(t\)th period. \(N_{\rm G2P}\) is the total amount of G2P units. \(P_{\rm G2P}^{v,t}\) is the output power of the \(v\)th G2P unit when the unit runs in \(t\)th period. \(P_{\rm L}^{t}\) is the electric load. \(E_{\rm F}^{t}\) is the expected value of the system when the power is insufficient in \(t\)th period. (3) Upper and lower limit constraints of conventional unit output:  
\[\left\{ \begin{array}{l}P_{\rm G}^{i,min}\leq P_{\rm G}^{i}\leq P_{\rm G}^{i, max}\\ Q_{\rm G}^{i,min}\leq Q_{\rm G}^{i,max}\end{array} \right. \tag{21}\]  
where \(P_{\rm G}^{i}\) and \(Q_{\rm G}^{i}\) are, respectively, the active and reactive power output of the \(i\)th conventional unit. \(P_{\rm G}^{i,max}\) and \(Q_{\rm G}^{i,max}\) are respectively the upper limits of the active and reactive power output of the \(i\)th conventional unit. \(P_{\rm G}^{i,min}\) and \(Q_{\rm G}^{i,min}\) are, respectively, the lower limits of the active and reactive power output of the \(i\)th conventional unit. 4 Constraint of DG output:  
\[P_{\text{DG}}^{\mu,\text{min}}\leq P_{\text{DG}}^{\mu,t}+P_{\text{DG}}^{\mu,t} \times\varepsilon_{\text{DG}}^{\mu,t}\leq P_{\text{DG}}^{\mu,\text{max}} \tag{22}\]  
where \(P_{\text{DG}}^{\mu,\text{min}}\) and \(P_{\text{DG}}^{\mu,\text{max}}\) are, respectively, the upper and lower limits of the active power output of the \(w\)th DG unit. \(\varepsilon_{\text{DG}}^{\mu,t}\) is the prediction error of active power output of the \(w\)th DG unit in \(t\)th period. 5 Ramping constraints:  
\[\left\{ \begin{array}{l}P_{\text{G}}^{i,t+1}-P_{\text{G}}^{i,t}\leq u_{i,t} \cdot r_{\text{G},j}^{\mu}\cdot\Delta T+P_{\text{G}}^{i,\text{max}}(1-u_{i,t}) \\ P_{\text{G}}^{i,t}-P_{\text{G}}^{i,t+1}\leq u_{i,t+1}\cdot r_{\text{G},i}^{d} \cdot\Delta T+P_{\text{G}}^{i,\text{max}}(1-u_{i,t+1})\end{array} \right. \tag{23}\]  
where \(u_{i,t}\) is the switching state of the \(i\)th conventional unit in \(t\)th period, which is 0-1 vector, turn on is 1, turn off is 0. \(r_{\text{G},j}^{\mu}\) is the uphill ramping rate of the \(i\)th conventional unit. \(u_{i,t+1}\) is the switching state of the \(i\)th conventional unit in (\(t\)+1)th period. \(r_{\text{G},i}^{d}\) is the downhill ramping rate of the \(i\)th conventional unit. \(\Delta T\) is the dispatching time. 6 Constraint of node voltage:  
\[U_{i}^{\text{min}}\leq U_{i,t}\leq U_{i}^{\text{max}} \tag{24}\]  
where \(U_{i,t}\) is the voltage of node \(i\) in \(t\)th period. \(U_{i}^{\text{max}}\) and \(U_{i}^{\text{min}}\) are, respectively, the upper and lower limits of allowable voltage for node \(i\). 7 Constraint of line transmission power:  
\[P_{l}^{\text{min}}\leq U_{i,t}U_{j,t}\big{(}G_{ij}\cos\theta_{ij}+B_{ij}\sin \theta_{ij}\big{)}-U_{i,t}^{2}G_{ij}\leq P_{l}^{\text{max}} \tag{25}\]  
where \(P_{l}^{\text{max}}\) and \(P_{l}^{\text{min}}\) are, respectively, the upper and lower limits of active power allowable for transmission on line \(l\).