## 4 Case Studies
### Algorithm Performance Comparision
#### 4.2.2 Convergence Comparison between ADMM-FE and Directly Generalized ADMM  
Taking the iterative process of obtaining the initial point as an example, the convergence of the ADMM-FE algorithm and the directly generalized ADMM algorithm is compared and analyzed. According to the algorithm parameter settings, under the constraints of \(a=0.001\) and \(b=0.0001\), the maximum imbalance amount \(\varepsilon^{k}\) in the optimization process is shown in Figure 2. The ratio of the target difference between the two iterations is shown in Figure 3.  
As can be seen from Figures 2 and 3, in Example System 1, both ADMM and ADMM-FE can obtain convergence solutions, but in Example System 2, when the 66 th iteration of ADMM-FE is completed, the boundary variable imbalance has reached the convergence condition, but the boundary variables cannot converge in ADMM. It can be seen that in solving the 3-Block problem, the directly popularized ADMM does not necessarily converge, but the ADMM-FE algorithm proposed in this paper has good convergence characteristics and convergence speed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Example** & **Algorithm Type** & **Algorithms** & **Value/10\$** & **Time/s** \\ \hline \multirow{2}{*}{Example 1} & centralized optimization & IPOPT & 654.46124 & 27.8724 \\  & centralized optimization & CPLEXD & 654.46119 & 13.3160 \\  & distributed optimization & PCCP-ADMM-FE & 654.21669 & 454.3120 \\ \hline \multirow{2}{*}{Example 2} & centralized optimization & IPOPT & 692.31667 & 29.3883 \\  & distributed optimization & PCCP-ADMM-FE & 691.88816 & 767.8922 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results comparison of centralized optimization and alternating direction method of multipliers (ADMM)-FE (forced equality) algorithms. Figure 3: Unbalanced amount of objective function. Figure 2: The absolute maximum value of the unbalanced amount. It can be seen from Figure 4 that in the power network, in the early morning hours, the power load is low, and the wind turbine output is high. At this time, the power of the P2G device can convert the wind power into natural gas storage to absorb the wind power. After 7 o'clock, the daytime work starts, the electric load increases, the grid can absorb all the wind power, so the power of the P2G device drops to 0. In the evening, the power load reaches a peak, and the output of the coal-fired unit reaches the upper limit. At this time, the gas turbine output with higher output cost is called to maintain the power balance. At other times, the coal-fired unit's own output can meet the load demand, so the gas turbine output has been maintained at the lowest level. Therefore, the gas turbine output will maintain the lowest value during the period except for fluctuations from 17 to 19 h. The electrical output of the CHP device is determined by the thermal output. Therefore, in addition to the CHP heating for the demand of heating networks during 6-12 h and 16-22 h, the CHP output grid power is zero. It can be seen from Figure 5 that in the gas network, in the early morning hours, the gas load and the gas consumption of the gas turbine are low, and the grid inputs high natural gas for the wind power converted by P2G, so the gas tank input is high with 0 output, and the surplus natural gas is stored in the gas storage tank used as a slow storage energy. During the daytime working hours, as the air load increases and the heating-network-demand CHP increases, the gas tank input decreases and the output increases. At 9 h, the gas load peaks and the gas tank output peaks. It can be seen from Figure 6 that in the heating network, in the early morning hours, the external heat source energy supply cost is lower than the CHP and is at a low price. Therefore, the heat load is satisfied by the external heat source, and the energy supply is greater than the heat load, so the excess heat is used as the heat storage energy.