## 2 Day-ahead Scheduling Model for EGHS
### Natural Gas Network Model  
The transmission speed of natural gas in pipelines is related to various factors such as pressure and temperature. The natural gas steady-state model ignores the transmission delay of natural gas in the pipeline, and simply equates the pipeline injection gas flow with the outflow gas flow, resulting in an increase in calculation error. This paper uses gas pipeline partial differential equations to describe the dynamic characteristics of natural gas networks:  
\[\frac{\left(b_{ij,d+1}^{t}\right)^{2}-\left(b_{ij,d}^{t}\right)^{2}}{\Delta x _{ij}}+\text{M}_{1}\left(f_{ij,d}^{t}\right)^{2}=0 \tag{2}\]  
\[\frac{\partial\left(b_{ij,x}^{t}\right)^{2}}{\partial x}+\text{M}_{2}\left(f_ {i,j,x}^{t}\right)^{2}=0 \tag{3}\]  
In the above, \(b_{ij,x}^{t}\), \(f_{ij,x}^{t}\) are the gas pressure and gas flow at pipe \(x\) at time \(t\), respectively. \(\text{M}_{1}\) and \(\text{M}_{2}\) are pipe transmission constants and the specific expression of M1, M2 are provided in Appendix B; Equations (2)-(3) are partial differential equations, which are difficult to solve directly. Therefore, they are transformed into algebraic equations by [PERSON]'s finite difference method. The pipe _ij_ is divided into N segments, each segment length is \(\Delta x_{ij}=l_{ij}/N\), and the equation is expressed as follows:  
\[\frac{b_{ij,d}^{t}-b_{ij,d}^{t-1}}{\Delta t}+\text{M}_{1}\frac{f_{ij,d+1}^{t} -f_{ij,d}^{t}}{\Delta x_{ij}}=0 \tag{4}\]  
\[\frac{\left(b_{ij,d+1}^{t}\right)^{2}-\left(b_{ij,d}^{t}\right)^{2}}{\Delta x _{ij}}+\text{M}_{2}\left(f_{ij,d}^{t}\right)^{2}=0 \tag{5}\]  
In the above, \(\Delta t\) is the time step and \(\Delta x_{ij}\) is the space step. There is a delay effect in the transmission of natural gas in the pipeline, and some natural gas is retained in the pipeline, which is called \"pipe storage\". After completing a scheduling period, the storage should meet the following constraint:  
\[\sum_{t\in T}\sum_{(i,j)\in\Omega_{\text{GP}}}\left(f^{t}_{ij,0}-f^{t}_{ij,N} \right)\Delta t=0 \tag{6}\]  
In the above, \(\Omega_{\text{GP}}\) is a natural gas pipeline collection. The pressurization station must meet the following constraints:  
\[\left\{ \begin{array}{l}k^{\min}_{c}\leqslant\frac{b^{t}_{ij,0}}{b^{t}_{i}} \leqslant k^{\max}_{c}\\ 0\leq f^{t}_{i}\leq f^{\max}_{in}\end{array} \right. \tag{7}\]  
In the above, \(b^{t}_{i}\), \(b^{t}_{ij,0}\) is the pressure before and after boosting; \(k^{min}_{c},k^{max}_{c}\) are the lower and upper limits of the career ratio; \(f^{t}_{i}\) is the pipe airflow and \(f^{max}_{in}\) is the pipe airflow limit. Node air pressure and air source output constraints are as follows:  
\[\left\{ \begin{array}{l}b^{\min}_{i,t}\leqslant b^{t}_{i}\leqslant b^{\max} _{i,t}\qquad i\in\Omega_{\text{GB}}\\ f^{\min}_{g,i,t}\leqslant f^{\max}_{g,i,t}\quad j\in\Omega_{g}\end{array} \right. \tag{8}\]  
In the above, \(\Omega_{\text{g}}\) and \(\Omega_{\text{GB}}\) are gas source sets and gas node sets, respectively. Gas tank constraints are as follows:  
\[\left\{ \begin{array}{l}S_{n,t}=S_{n,t-1}+\frac{Q^{\min}_{s,t}\mu^{\min}_{n} \Delta t}{S^{\text{cap}}_{n}}-\frac{Q^{\text{out}}_{s,t}}{S^{\text{cap}}_{n} \mu^{\min}_{n}}\quad n\in\Omega_{s}\\ S^{\min}_{n}\leqslant S_{n,t}\leqslant S^{\max}_{n}\end{array} \right. \tag{9}\]  
In the above, \(\Omega_{\text{s}}\) is the collection of gas storage tanks; \(Q^{\text{in}}_{s,t}\), and \(Q^{\text{out}}_{S,n,t}\) are the natural gas input and output of the gas storage tank \(n\) at time \(t\), respectively; \(\mu^{\min}_{n}\), \(\mu^{\text{out}}_{n}\) are the inflation and deflation efficiency of the gas storage tank \(n\) at time \(t\); \(S^{\text{cap}}_{n}\) is the rated gas storage capacity of the gas storage tank \(n\). The node traffic balancing constraint is as follows:  
\[