## 5 Case Studies
### Correctness Analysis  
The correctness of the proposed matrix MES model and Jacobian matrix is first verified. Three types of models are considered in this part: (1) Model I (benchmark model): the traditional MES model with the DHN model of [11] and the DGN model of [27]; (2) Model II: the proposed matrix-based MES model with the universal DGN model; (3) Model III: the proposed matrix-based MES model with the modified DGN model. It should be noted that, in Models I-III, the DEN models are the same, and the gas is assumed to behave ideally. The absolute EFC errors between Model I and Model II/III on the test system are given in Table 4, and the typical iteration process of Models I-III is shown in Figure 8a. It can be found that the maximum absolute errors of Model II and Model III from the benchmark model are 4.01 \(\times\) 10\({}^{-9}\) MJ/m\({}^{3}\) and 3.95 \(\times\) 10\({}^{-9}\) MJ/m\({}^{3}\), respectively, demonstrating the accuracy of the proposed matrix model. In addition, Model II and Model III converge in 18 and 17 steps, respectively, while it takes Model I 40 steps to meet the accuracy criterion. Thus, the proposed Jacobian matrix offers a faster convergence rate owing to its gradient descent direction. Additionally, it can also be found that the convergence processes of Model II and Model III are similar, which means that the proposed modification strategy does not affect the convergence speed of the matrix-based models. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**DEN Results**} & \multicolumn{3}{c}{**DHN Results**} & \multicolumn{3}{c}{**DGN Results**} \\ \cline{3-10}  & **Error** & \(1\)**V1 (p.u.)** & \(\theta\) (\(\Upsilon\)) & \(\boldsymbol{m}\) (kg/s)** & \(\boldsymbol{T^{5}}\) (\(\Upsilon\)) & \(\boldsymbol{T^{6}}\) (\(\Upsilon\)) & \(\boldsymbol{\pi}\) (bar) & \(\boldsymbol{f_{\xi}}\) (\(\boldsymbol{\Omega}\)/CMD) & \(\boldsymbol{\rho}\) & \(\boldsymbol{\boldsymbol{q}}\) (MJ/m\({}^{3}\)) \\ \hline \multirow{3}{*}{Model II} & Max & \(3.99\times 10^{-5}\) & \(5.40\times 10^{-13}\) & \(1.09\times 10^{-11}\) & \(9.95\times 10^{-14}\) & 0 & \(2.78\times 10^{-11}\) & \(6.18\times 10^{-11}\) & \(7.76\times 10^{-11}\) & \(4.01\times 10^{-9}\) \\  & Average & \(4.66\times 10^{-16}\) & \(1.98\times 10^{-13}\) & \(1.63\times 10^{-12}\) & \(1.55\times 10^{-14}\) & 0 & \(7.60\times 10^{-12}\) & \(1.63\times 10^{-11}\) & \(1.86\times 10^{-11}\) & \(9.15\times 10^{-10}\) \\  & Min & 0 & 0 & 0 & 0 & 0 & 0 & 1.61 \(\times 10^{-12}\) & \(2.00\times 10^{-15}\) & \(2.10\times 10^{-12}\) \\ \hline \multirow{3}{*}{Model III} & Max & \(6.11\times 10^{-15}\) & \(1.67\times 10^{-12}\) & \(5.10\times 10^{-11}\) & \(1.14\times 10^{-13}\) & 0 & \(7.98\times 10^{-11}\) & \(6.31\times 10^{-11}\) & \(7.92\times 10^{-11}\) & \(3.95\times 10^{-9}\) \\  & Average & \(1.10\times 10^{-15}\) & \(8.80\times 10^{-13}\) & \(7.84\times 10^{-12}\) & \(4.40\times 10^{-14}\) & 0 & \(2.57\times 10^{-11}\) & \(2.55\times 10^{-11}\) & \(1.53\times 10^{-11}\) & \(8.02\times 10^{-10}\) \\ \cline{1-1}  & Min & 0 & 0 & 0 & 0 & 0 & 0 & 9.99 \(\times 10^{-14}\) & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The EFC errors of the proposed models on the test system. Figure 7: Structure schematic of the test system. With the accurate basic model, the validity of Model II/III is further discussed considering the volatility of the gas compressibility factor \(\mathbf{Z}\). The iteration process of Model II/III with variable \(\mathbf{Z}\) is depicted in Figure 8b. It can be seen that the proposed models can still converge at a faster speed when the updating of \(\mathbf{Z}\) is added during the EFC iteration process. Figure 9 displays the EFC results of the DGN characterized by Model I and Model II/III with variable \(\mathbf{Z}\). Obviously, the EFC results of Model II and Model III are consistent, while Model I has a more pronounced difference. This is because the energy flow distribution of the DGN has changed after accounting for the varying \(\mathbf{Z}\).