## 4 Case Studies
### Analysis of the Algorithm  
In order to verify the accuracy of the decentralized framework and algorithm for the OMEF problem proposed in this paper, the algorithm solving process and optimization results are analyzed in detail. The key parameters of the algorithm in this paper are shown in Table 1.  
In order to verify the accuracy and effectiveness of the decentralized algorithm proposed in this paper, the decentralized optimization results are compared with the centralized algorithm.  
As shown in Table 2, the NLP algorithm is to solve the OMEF problem of IEGS without processing non-convex equations. Meanwhile, the PCCP algorithm is to use the PCCP method proposed in this paper to model for the power system and the natural gas system and solve it centrally. Comparing the two results, although NLP can directly use the IPOPT solver to solve the OMEF problem, the results obtained are not optimal solutions, and the non-convex nature makes the decentralized algorithm unable to converge or obtain ideal results. The PCCP algorithm used to solve the OMEF problem can converge to an optimal solution. PCCP-ADMM algorithm is the decentralized algorithm proposed in this paper. Compared with the PCCP centralized algorithm, the results obtained by the decentralized algorithm are similar to the centralized algorithm, which verifies the feasibility of the decentralized algorithm. The difference between the two results is 0.4%, which is acceptable. For optimization time, the decentralized algorithm takes more time than the centralized algorithm. However, the decentralized algorithm only needs to exchange a small amount of boundary information in each area to coordinate the global optimal scheduling, which guarantees the privacy of the information in each area.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Algorithm**} & \multicolumn{4}{c}{**Parameters**} \\ \hline \multirow{4}{*}{PCCP} & power system & \(\gamma_{\text{e}}^{0}\) & \(\gamma_{\text{e,max}}\) & \(\mu_{\text{e}}\) & \(\omega_{\text{e}}\) \\  & & 10 & 1000 & 2 & 0.1 \\ \cline{1-1}  & natural gas system & \(\gamma_{\text{g}}^{0}\) & \(\gamma_{\text{g,max}}\) & \(\mu_{\text{g}}\) & \(\omega_{\text{g}}\) \\ \cline{1-1}  & & 0.1 & 100 & 2 & 0.01 \\ \cline{1-1}  & ADMM & & \(\rho\) & & \(\tau\) \\ \cline{1-1}  & & 1000 & & 0.01 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the algorithm.  
Figure 4: The curve of electric load and gas load.  
The iterative process of PCCP-ADMM algorithm is shown in Figure 5. It can be seen from (a) that the optimization process performed 7 PCCP iterations and 22 ADMM iterations. As shown in (b), at the boundary of each iteration of PCCP, the ADMM residual will basically increase, because the algorithm's solution mechanism is that the ADMM algorithm is performed at each iteration of PCCP, but the overall trend is declining. In addition, in the second iteration of PCCP and later, the number of ADMM iterations is small. The reason is that the energy flow results obtained by the PCCP-ADMM in these steps are close to the optimal solution, and the ADMM algorithm in \(k\)+1 th iteration of PCCP uses the Lagrange multiplier result in \(k\)th iteration as the initial iteration value. Therefore, the proposed algorithm can not only guarantee convergence to the optimal solution, but also speed up the convergence speed.