## 2 Steady-State Model of Energy Flow in the Electric-Gas Interconnection System
### Energy Hub Model of Energy Flow in the Electric-Gas Interconnection System  
The establishment of steady-state model of energy flow of multi-type heterogeneous energy in the electric-gas interconnection system is the basic premise for decision-makers to make decision optimization. The energy hub model is widely used in the steady-state modeling of energy flow [14; 16]. In this model, for an energy conversion device with only single input and single output, the relationship between input and output is as follows:  
\[L_{\alpha}=\eta_{\alpha,\,\beta}P_{\beta} \tag{1}\]  
where, \(L_{\alpha}\) and \(P_{\beta}\) are the input and output of the steady-state system; \(\eta_{\alpha,\beta}\) is the coupling coefficient between the input and output. The integrated energy system consists of multiple energy conversion devices and various energy forms. The coupling relationship between input and output in the system can be described by the energy hub model. The formula is as follows:  
\[\left[ \begin{array}{c}L_{\alpha}\\ L_{\beta}\\ \vdots\\ L_{\omega}\\ L_{\omega}\\ \end{array} \right]=\left[ \begin{array}{cccc}\eta_{\alpha\alpha}&\eta_{ \beta\alpha}&\cdots&\eta_{\omega\alpha}\\ \eta_{\alpha\beta}&\eta_{\beta\beta}&\cdots&\eta_{\omega\beta}\\ \vdots&\vdots&\vdots&\vdots\\ \eta_{\alpha\omega}&\eta_{\beta\omega}&\cdots&\eta_{\omega\omega}\\ \eta_{\alpha\omega}&\eta_{\beta\omega}&\cdots&\eta_{\omega\omega}\\ \eta_{\alpha}&\eta_{\beta}&\end{array} \right]\left[ \begin{array}{c}P_{ \alpha}\\ P_{\beta}\\ \vdots\\ P_{\alpha\omega}\\ \end{array} \right] \tag{2}\]  
where, vectors \(P\) and \(L\) are the input and output variables of the system respectively. Each coupling coefficient in the coupling matrix \(\eta\) corresponds to the coupling relationship between a specific input and a specific output. By combining the energy concentrator conversion matrix and the flow direction of different energy sources in the electric-gas interconnection system shown in Figure 1, the corresponding energy input and output relations of natural gas, heat, cooling and the power systems in the electric-gas interconnection system can be obtained. Due to the bidding period \(t\)\(=1h\), the power and energy (electric quantity and natural gas quantity) of each bidding period are the same in numerical value. Without specification, the bidding power (Unit: MW) is adopted below to describe the model. The energy conversion model of the natural gas system is as follows:  
\[L_{\text{G,}t}=-P_{\text{G,}t}^{\text{bid}}+\eta^{\text{P2G}}P_{\text{E,}t}^{ \text{P2G}}-P_{\text{G,}t}^{\text{CHP}}-P_{\text{G,}t}^{\text{charge}}+P_{ \text{G,}t}^{\text{discharge}} \tag{3}\]  
where, \(L_{\text{G,}t}\)(Unit: MW) represents the output power of the natural gas system in the \(t\) period, and subscript \(t\) represents the time scale variable, the same below. \(P_{\text{G,}t}^{\text{bid}}\)(Unit: MW) represents the natural gas power purchased or sold, which is positive when sold and negative when purchased. \(\eta^{\text{P2G}}\)(Unit: %) represents the conversion efficiency of electricity to gas. \(P_{\text{E,}t}^{\text{P2G}}\)(Unit: MW) represents the power allocated by the power system to the P2G system. \(P_{\text{G,}t}^{\text{CHP}}\)(Unit: MW) represents the power allocated by the natural gas system to the CHP system. \(P_{\text{G,}t}^{\text{charge}}\)(Unit: MW) represents the air charge power of the air tank. \(P_{\text{G,}t}^{\text{discharge}}\)(Unit: MW) represents the air discharge power of the air tank. The energy conversion model of the heating system in the electric-gas interconnection system is as follows:  
\[L_{\text{H},t}=\eta_{\text{H}}^{\text{CHP}}P_{\text{G},t}^{\text{CHP}}-P_{\text{ H},t}^{\text{AR}}+\eta^{\text{HP}}P_{\text{E},t}^{\text{HP}} \tag{4}\]  
where, \(L_{\text{H},t}\)(Unit: MW) represents the output power of the heating system. \(\eta_{\text{H}}^{\text{CHP}}\)(Unit: %) represents the heating efficiency of CHP. \(P_{\text{H},t}^{\text{AR}}\)(Unit: MW) represents the power allocated by the heating system to the AR system.