## 5 Case Studies
### Influence of Load Level on Convergence  
Finally, the convergence performance of different models under varying load conditions is examined. Using the original gas load energy demand \(E_{\text{D,ini}}\) of the test system as the baseline, the load level of the DGN is varied according to \(k_{4}E_{\text{D,ini}}\), where \(k_{4}\in[0.5,1.5]\) is the load level coefficient. Considering the significant differences in hydraulic state variables of the DGN under different load conditions, the hydraulic initial values \(\mathbf{\Pi}_{0}\) and \(f_{\text{E,0}}\) are set as random values within the intervals \([(1-k_{1}/2)\Pi_{\text{ini}},(1+k_{1}/2)\Pi_{\text{ini}}]\) and \([(1-k_{1}/2)f_{\text{E,ini}},(1+k_{1}/2)f_{\text{E,ini}}]\), respectively. \(k_{2}\) is fixed at 1. Similarly, the values of \(k_{1}\) and \(k_{4}\) are iterated with a step size of 0.02 within their respective ranges, resulting in 2500 different MES operation scenarios. The calculated convergence domains of Models II-IV under different load levels are shown in Figure 15.  
As shown in Figure 15a, Model I has a low probability of convergence only in the top-left corner of the convergence domain. When \(k_{4}>0.8\) and the hydraulic initial values are appropriately chosen, Model IV can achieve reliable convergence. Specifically, as the load level increases, the reliable convergence range of the hydraulic initial values narrows from [0.04, 0.44] at \(k_{4}=0.9\) to [0, 0.24] at \(k_{4}=1.5\). This is because the pressure within the DGN decreases as the load becomes heavier, so only smaller \(k_{1}\) values can bring the gas pressure initial values closer to the actual pressure levels in the gas network. When \(k_{4}<0.8\), i.e., when the DGN operates in a light load mode, Model IV completely fails. Figure 15c presents the full convergence domain of the test system. It is evident that Model III converges under all load levels, with the convergence range expanding as the load decreases. Additionally, in the overlap of the convergence domains with Model IV, Model III exhibits a higher convergence rate.  
Herein, we focus on the operating conditions of the DGN when \(k_{4}<0.8\). The convergence performance of Models II-IV with the same initial values for \(k_{1}=0.2\) and \(k_{4}=0.8\) is shown in Figure 16. In the first iteration, Model II encounters an ill-conditioned state due to the flow imbalance at gas node 9. Model IV also fails after 12 iterations due to insufficient precision in solving the inverse of the Jacobian matrix. Model III successfully converges after 27 iterations. However, it should be noted that in the results calculated by Model III, the flow in compressor 3, the gas consumption of compressor 3, and the flows in gas pipelines 6 and 8 are all negative. This indicates that compressor 3 and pipelines 6 and 8 are operating in reverse flow, which is clearly not permissible in a real system. Therefore, the proposed model not only applies to normal operating scenarios but also reveals some infeasible extreme conditions, which helps outline the operational boundaries of the MES and mitigate potential operational risks.  
Figure 15: The convergence domains of Models II–IV under different load levels. (**a**) Model II; (**b**) Model IV; (**c**) Model III.