# Flexible Dispatch for Integrated Power and Gas Systems Considering Power-to-Gas and Demand Response
## Nomenclature  
A. Indexes and Sets  
\(i,p,k,m/q\)  
\(l,b,t\)  
\(\Omega_{G}\), \(  
\begin{tabular}{l l l} \(\sigma\) & Conversion factor for electric power and heat. \\ \(\eta\) & Conversion efficiency of the P2G facility. \\ C. Variables & \\ \(u_{i,t}\), \(u_{\text{P2g,p,t}}\), \(v_{m,t}\) & Commitment Statuses of unit \(i\), P2G facility \(p\), demand response user \(m\) \\  & at time \(t\). \\ \(P_{i,t}\), \(P_{\text{P2g,p,t}}\), \(P_{\text{DRE,m,t}}\) & Dispatch of unit \(i\), P2G facility \(p\), demand response user \(m\) at time \(t\). \\ \(P_{GT,i}\) & Dispatch of gas-fired unit \(i\) at time \(t\). \\ \(f^{\text{u},t}_{\text{p,t}}\), \(f^{\text{d}}_{\text{g,t}}\) & Upward and downward adjustable capacity of conventional units. \\ \(f^{\text{u}}_{\text{P2g,p,t}}\), \(f^{\text{d}}_{\text{P2g,p,t}}\) & Upward and downward adjustable capacity of P2G facility. \\ \(f^{\text{u}}_{\text{d,m,t}}\), \(f^{\text{d}}_{\text{d,t,t,t,t}}\) & Upward and downward adjustable capacity of demand response. \\ \(SU_{i,t}\), \(SD_{i,t}\) & Start-stop status variable of generator units. \\ \(X^{on}_{i,t-1}\), \(X^{off}_{i,t-1}\) & On/off time counter of unit \(i\) at time \(t\). \\ \(f_{\text{g,t}}\), \(f_{\text{P2G,t}}\), \(f_{\text{d,t}}\) & Adjustable capacity provided by the units, P2G facilities, demand response at time \(t\). \\ \(f^{\text{ur,s}}_{\text{P2g,p,t}}\), \(f^{\text{dr,s}}_{\text{P2g,p,t}}\) & Called upward and downward adjustable capacity of P2G facility. \\ \(\text{C}^{\text{ur}}_{\text{P2g,p,t}}\), \(\text{C}^{\text{ur}}_{\text{P2g,p,t}}\) & P2G called cost. \\ \(u_{\text{P2g,p,t}}\) & Start-stop status variable of P2G facility. \\ \(R^{\text{u}}_{\text{P2g,p}}\), \(R^{\text{d}}_{\text{P2g,p}}\) & Upward and downward climbing rate of P2G facility. \\ \(P_{\text{DRE,j,m,t}}\), \(\text{P}_{j,m}\) & j-level response capacity, price of demand response user \(m\). \\ \(P_{w,k}\) & Dispatch of wind farm \(k\) at time \(t\). \\ \(\Delta P_{w,k}\) & Wind power curtailment of wind farm \(k\) at time \(t\). \\ \(LX_{m}\) & Cumulative unregulated time of the demand response user \(m\). \\ \(f_{w,t}\), \(f_{\text{P2G,p,t}}\), \(f^{out}_{s,t}\) & Natural gas flow from gas well \(w\), P2G facility \(p\), storage facility \(s\). \\ \(f_{\text{N,i,t}}\) & Non-served gas loads caused by the gas-fired unit \(i\) at time \(t\). \\ \(p_{i}\) & Pressure of gas node \(i\) at time \(t\). \\ \(\widetilde{f}_{i,t}\) & Average flow of pipeline \(ij\) at time \(t\). \\ \(f^{int}_{i,t}\), \(f^{out}_{j,t}\) & Inflow/outflow of gas pipeline \(ij\) at time \(t\). \\ \(f^{int}_{s,t}\), \(f^{out}_{s,t}\) & Inflow/outflow of storage facility \(s\) at time \(t\). \\ \(L_{ij,t}\) & Linepack of pipeline \(ij\) at time \(t\). \\ \(p_{c,m,t}\), \(P_{c,n,t}\) & Pressure of the incoming node \(m\), the out-coming node \(n\) of the compressor \(c\). \\ \(E_{s,t}\) & Gas storage capacity of the storage facility \(s\) at time \(t\). \\ \(f_{\text{L,b,t}}\) & Gas load of node \(b\) at time \(t\). \\ \(f_{\text{GT,i,t}}\) & Gas flow consumed by the gas-fired unit \(i\) at time \(t\). \\ \(f_{\text{P2G,p,t,max}}\) & Max injection flow of P2G facility \(p\) at time \(t\). \\ \(\lambda^{u(s)}_{i,t}\), \(\mu^{u(s)}_{i,t}\), \(\epsilon^{u(s)}_{b,t}\), & 6 dual variables of Benders cut. \\ \(\lambda^{d,s(p)}_{i,t}\), \(\theta^{d,s(p)}_{q,t}\), \(\xi^{d,s(p)}_{k,t}\) & \\ D. Functions & \\ \(f(P_{i,t})\) & Production cost of the coal-fired unit \(i\) at time \(t\). \\ \(W(P_{\text{GT,i,t}})\) & Gas purchase cost of the gas-fired unit \(i\) at time \(t\). \\ \end{tabular}