## 5 Case Studies
### Influence of Initial Value on Convergence  
In this subsection, the convergence behavior of the proposed matrix DGN model and the rationale behind its modified version are discussed from the perspective of initial values. The examined models include (1) Model II and (2) Model IV, where only the deviation flow is incorporated into Model II without considering the direction factor, and (3) Model III. Among the four types of initial values of the DGN, gas pressure \(\mathbf{\Pi}_{0}\) and equipment flow \(\boldsymbol{f}_{\text{E,0}}\) primarily influence the convergence of the hydraulic part, while specific gravity \(\rho_{0}\) and GCV \(\boldsymbol{q}_{0}\) mainly affect the convergence of the gas composition tracking part. To comprehensively explore the convergence domains of different models, the hydraulic initial values \(\mathbf{\Pi}_{0}\) and \(\boldsymbol{f}_{\text{E,0}}\) are set as random values in the intervals of \([(1-k_{1}/2)\Pi_{\text{ini}},\)\((1+k_{1}/2)\Pi_{\text{ini}}]\) and \([(1-k_{1}/2)\Psi_{\text{E,ini}},\)\((1+k_{1}/2)\Psi_{\text{E,ini}}]\), respectively, and the gas quality initial values \(\rho_{0}\) and \(\boldsymbol{q}_{0}\) are set to \(k_{2}\rho_{\text{ini}}\) and \(k_{2}\eta_{\text{ini}}\), respectively. \(k_{1}/k_{2}\in[0,1]\) is the coefficient of hydraulic/gas quality initial values. \(\Pi_{\text{ini}}\) is the pressure of the \(\pi\) node. \(f_{\text{E,ini}}/\rho_{\text{ini}}/\dot{\eta}_{\text{ini}}\) is the reference initial value of equipment flow/specific gravity/GCV. The values of \(k_{1}\) and \(k_{2}\) are traversed with a step size of 0.02 within \([0,1]\), forming 2500 groups of different initial value samples. For each group, the convergence rates of the models are calculated. Considering the randomness of the hydraulic initial value generation, the EFC is performed 200 times for each group of initial value samples, and the average convergence rate is counted. In this manner, the convergence domains of Models II-IV are illustrated in Figure 10. As shown in Figure 10a, Model II only converges with a very low probability in the region  
Figure 9: The EFC results of the DGN in the test system. (**a**) Nodal pressure; (**b**) pipeline gas flow and compressibility factor; (**c**) equipment gas flow and compressibility factor; (**d**) nodal specific gravity; (**e**) nodal GCV. where \(k_{1}\) takes small values and \(k_{2}\) takes large values, indicating that the traditional DGN model is highly sensitive to the selection of initial values. In contrast, the proposed modified DGN model effectively overcomes this issue and ensures reliable convergence within a reasonable range of initial values. Specifically, the convergence region of Model III is similar to that of Model IV, with the convergence rate decreasing as \(k_{1}\) increases or \(k_{2}\) decreases (because larger \(k_{1}\) and smaller \(k_{2}\) values cause the initial values to deviate further from the actual state of the DGN). However, because Model III simultaneously considers the imbalance of nodal inflow and outflow and the uncertainty of gas flow direction, it exhibits a higher convergence rate compared to Model IV. Taking \(k_{1}\) = 0.04 and \(k_{2}\) = 0.8 as an example, the convergence curves of Models II-IV based on the same initial values are presented in Figure 11. It can be observed that, except for Model III, the remaining models fail to converge. By analyzing the state variables at the convergence termination, it is found that after one iteration, the gas flow values in pipeline 8 and compressor 3 in Model II are \(-\)0.53 MSCMD and 2.48 MSCMD, respectively, leading to a nodal flow imbalance at node 9, as shown in Figure 4. This further causes the Jacobian matrix of the DGN to become singular, ultimately terminating the EFC iteration of Model II.