## 5 Case Studies
### Influence of Gas Properties on Convergence  
In addition to the initial values, the operating conditions of the MES also influence the convergence of the energy flow model. To simulate the convergence behavior of the proposed models under different gas qualities, the injection proportion of hydrogen--representing the most significant variation in gas quality compared to natural gas--is adjusted in the DGN. This creates MES operation scenarios with varying gas quality conditions while minimizing changes to the hydraulic conditions of the DGN. Let the original hydrogen injection amount \(\hat{f}_{\rm{ini}}^{\rm{H_{2}}}\) of the test system be the baseline; the hydrogen blending level is varied according to \(k_{3}\hat{f}_{1}^{\rm{H_{2}}}\), where \(k_{3}\in[0,3000]\) is the hydrogen injection coefficient. In addition, since the gas quality conditions of the DGN have changed, the effects of gas quality initial values \(\boldsymbol{\rho}_{0}\) and \(\boldsymbol{q}_{0}\) on model convergence must also be considered. Therefore, the gas quality initial values are set to \(k_{2}\rho_{\rm{ini}}\) and \(k_{2}\eta_{\rm{ini}}\), respectively. Similarly, to generate 2500 MES operation scenarios at equal intervals, \(k_{2}\) and \(k_{3}\) are iterated within their value ranges with step sizes of 0.02 and 60, respectively, while \(k_{1}\) is fixed at 0.2. The resulting convergence domains of Models II-IV under different gas quality conditions are shown in Figure 13.  
As observed in Figure 13, due to its high sensitivity to initial values, the traditional model almost fails to converge under any gas quality conditions. However, after incorporating the deviation flow, the convergence range of Model IV has been significantly expanded compared to Model II. Within a reasonable range of gas quality initial values, Model IV can reliably converge under all gas quality conditions (with convergence rates above 80% defined as reliable convergence). Notably, when \(k_{3}=60\), the model reliably converges  
Figure 12: The convergence domains of Models II–IV with variable \(\mathbf{Z}\) under different initial values. (**a**) Model II with variable \(\mathbf{Z}\); (**b**) Model IV with variable \(\mathbf{Z}\); (**c**) Model III with variable \(\mathbf{Z}\). within the range \(k_{2}\in[0.5,1]\), whereas when \(k_{3}=3\times 10^{3}\), the reliable convergence range of \(k_{2}\) shifts to [0.4,0.75]. This indicates that as the hydrogen concentration increases, the reliable convergence range of gas quality initial values in Model IV shifts left (i.e., the values decrease), and the range narrows. This is because, with higher hydrogen concentrations, the actual gas quality parameters in the DGN decrease, making smaller gas quality initial values closer to the actual values and thus facilitating reliable convergence. Furthermore, under high hydrogen permeability, the probability of ill conditioning in Model IV with inappropriate gas quality initial values also increases, which leads to a narrower reliable convergence range. In contrast, Model III, which accounts for the uncertainty in gas flow direction, shows an expanding reliable convergence range as the hydrogen injection amount increases. The reliable convergence range for \(k_{2}\) changes from [0.5,1] at \(k_{3}=60\) to [0.4,1] at \(k_{3}=3\times 10^{3}\), further demonstrating the superiority of the proposed modified model. Figure 14 shows the convergence curves of Models II-IV with the same initial values when \(k_{2}=0.9\) and \(k_{3}=3\times 10^{3}\). At this point, the average values of \(\rho\) and \(q\) in the DGN are 0.46 and 32.28 MJ/m\({}^{3}\), respectively. It can be observed that Models II and IV terminate their iterations after 33 and 32 steps, respectively. At the termination points, in Model II, the flow in gas pipeline 8 reverses, causing the gas quality tracking equation at node 9 to become invalid. In Model IV, reverse flow occurs in gas pipelines 1 and 2 and in the slack gas source, with flow values of \(-8.58\) MSCMD, \(-7.56\) MSCMD, and \(-20.54\) MSCMD, respectively.