## 5 Case Studies and Discussions
### Comparison between DO, ADMM and CO  
Figure 7 shows the variation of the original residuals and dual residuals with the iteration progress under the synchronous ADMM algorithm. In the 129 iterations, the residuals have a significant downward trend, showing a fluctuating decline. At the beginning of iterations, the magnitude of residual error is about 1e4, which indicates that there is a large gap in the coupling part of gas turbine. In the middle and later stage of the iteration, the residuals do not increase or decrease monotonically, but decrease gradually under small oscillation, and finally reach the convergence requirement. The main idea of the proposed step-by-step cooperative operation method for a power gas interconnected system is the constructed credit degree and credit value parameters, and the credit value is directly determined by the credit degree. Therefore, this part of the example mainly studies the influence of credit degree on the effectiveness of the proposed method. First, it should be noted that, because the cost, the coefficient of G2 is higher than that of other units. The minimum output is maintained, so the credit degree is set to a fixed value of 1. It mainly analyzes the change and influence of credit degree of G1 and G3. Credit degree indicates the quantitative degree of the relationship between the actual gas consumption of gas units and the constraints generated by gas units. It is suitable for gas network managers to evaluate large gas users. Figure 8 shows the output of gas turbine G1 under different credit combinations under step-by-step coordinated operation, and the credit degree of G1 is 1-1. 9, the interval is 0. 1, G3 credit rating is 1-1. 9, the interval is 0. 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pipeline Number** & **Start Node** & **End Node** &
\begin{tabular}{c} **Pipeline Constant** \\ **(kcf/Psig)** \\ \end{tabular} \\ \hline
1 & 2 & 1 & 50.6 \\
2 & 4 & 2 & 50.1 \\
3 & 5 & 2 & 37.5 \\
4 & 5 & 3 & 43.5 \\
5 & 6 & 5 & 45.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural gas pipeline characteristics. Figure 6: The topology of electricity and natural gas networks. A one-hour operation between electricity and the natural gas network was analyzed. An economic dispatch model was adopted in electricity system and an hourly steady-state gas flow model was adopted in the natural gas system [19]. Concurrently, the gas flow upper limit is similar for the comparison. This study illustrates the overall advantage obtained from the proposed framework CO, as compared to the other usual methods for coordinated operation of natural gas and electricity networks. Table 3 shows that performance comparison between DO, alternating direction method of multipliers (ADMM) and CO. It is observed that ADMM requires a longer time to obtain a solution. The convergence time for DO and CO are nearly the same. However, DO  
Figure 8: Residual change of synchronous ADMM. Figure 7: Residual change of synchronous ADMM. cannot produce better solutions under congestions and demand response situations, so CO gives the best overall performance. Due to the information asymmetry of the coupled gas units in the discrete operation, the natural gas required by the gas units in the power grid dispatching has no solution in the gas grid dispatching, that is, the dispatching is unbalanced. The dispatching unbalance of the discrete operation in the table comes from G1, and the specific reasons have been explained in the previous paper. In the operation cost of the two networks, due to the reduction in power generation and gas supply, the operation cost of discrete operation of the two networks is lower than that of other operation modes, but the existence of dispatching imbalance requires the power grid to pay a higher price. In terms of solution time, the optimization time of step-by-step collaborative operation is 1.04s, which is larger than that of discrete operation and much smaller than that of ADMM operation.