## 5 Case Studies
### Efficiency Analysis  
Based on the analysis of the convergence process above, it is evident that different initial values and operating conditions affect the convergence rate of the model, which, in turn, affects the computational time. To comprehensively assess the convergence efficiency of the proposed models, the computational time of Models I-III under various initial values and operating conditions is calculated. The iteration steps and computational time for Models I-III across multiple operating scenarios are shown in Table 6. Similarly, all results presented are the average values of 200 simulation runs.  
In Scenario 1, where \(k_{1}\) = 0.2, \(k_{2}\) = 1, \(k_{3}\) = 1, and \(k_{4}\) = 1, the DGN operates under the original condition, and the initial values are close to the actual system operating state. It can be found from Table 6 that, when the variation in \(Z\) is not considered, Model I takes approximately 40 steps to converge, with a computation time of 0.076s. In contrast, owing to the Jacobian matrix with the gradient descent direction, Models II-III converge in less than 20 steps. Furthermore, due to the matrix calculation structure rather than conventional recursive loops, the computational time for Models II-III is only one-third of that of Model I. When variable \(\mathbf{Z}\) is considered, Models II-III still exhibit good convergence rates, but due to the increased number of model equations, the computation time is slightly longer compared to the models without \(\mathbf{Z}\) variation. In Scenario 2, where \(k_{1}\) = 0.2, \(k_{2}\) = 0.2, \(k_{3}\) = 3 \(\times\) 10\({}^{3}\), and \(k_{4}\) = 1, the DGN operates under high hydrogen penetration conditions, with unfavorable gas quality initial values. In Scenario 3, where \(k_{1}\) = 1, \(k_{2}\) = 1, \(k_{3}\) = 1, and \(k_{4}\) = 0.8, the DGN operates under light load conditions with inappropriate  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**Without \(\mathbf{Z}\) Variation**} & \multicolumn{3}{c}{**With \(\mathbf{Z}\) Variation**} \\ \cline{3-6}  & & **Steps** & **Time (s)** & **Steps** & **Time (s)** \\ \hline \multirow{3}{*}{Scenario 1} & Model I & 39.47 & 0.07573 & - & - \\  & Model II & 18.94 & 0.02515 & 17.48 & 0.03477 \\  & Model III & 18.44 & 0.02372 & 16.74 & 0.02387 \\ \hline \multirow{3}{*}{Scenario 2} & Model I & - & - & - & - \\  & Model II & - & - & - & - \\  & Model III & 31.37 & 0.03607 & 27.59 & 0.03652 \\ \hline \multirow{3}{*}{Scenario 3} & Model I & - & - & - & - \\  & Model II & - & - & - & - \\ \cline{1-1}  & Model III & 42.63 & 0.04773 & 37.65 & 0.04762 \\ \hline \hline \end{tabular}  
* Abbreviation—\(\cdot\)-’ represents non-convergence or non-existence.  
\end{table}
Table 6: Iterations and computational time for Models I-III in Scenarios 1–3.  
hydraulic initial values. As shown in Table 6, only Model III successfully converges in these two scenarios, with both the number of iterations and computational time larger than those in Scenario 1. Additionally, when considering \(\mathbf{Z}\) variation, the convergence steps of these two scenarios become lower while the computation time remains longer due to the increased model complexity. Last but not least, although the iterations of Model III in Scenarios 2 and 3 are similar to those of Model I in Scenario 1, the computation time is still much shorter for Model III. This is still attributed to the proposed matrix formulation, which is crucial for improving the efficiency of the NR-based EFC framework that requires repeated calculation of the same equations.  
In addition, by comparing the iteration steps and computation time of Model III in different scenarios, it can be seen that the computational efficiency of the proposed model will be attenuated under inappropriate initial value conditions. Therefore, how to select appropriate initial values for the NR iteration to ensure the EFC computational efficiency is also one of the potential research directions in the future.