## 4 Case Analysis
### Case Analysis of Closing Angle Regulation Scheme Based on Probability Density Prediction  
Taking the New England 10-machine 39-bus system as an example, this paper analyzes the closing angle regulation example to verify the effectiveness of the proposed model. The model is established in matlabr2019a, and the model is solved by the NSGA-II algorithm. The simulation computer condition is inter-core i5 and the memory is 8 GB. As shown in Figure 9, the 39 nodes of New England's 10 machines are assumed to be disconnected from the lower-right subnetwork after a power outage caused by a failure of the lower-right subnetwork. After a power outage, the main network transmits power to the subnetwork through lines 15-16 to realize the black start of the subnetwork. To realize the stable operation of the subnetwork, reduce the risk of subnetwork load recovery, and reduce the transmission power of lines 15-16. The next step is to restore lines 16-17 and perform the parallel operation of the ring network. To share the power flow of heavy load lines 15-16, strengthen the connection between the main network and the subnetwork, and improve the operation stability of the whole system. To ensure the smooth operation of parallel operation, the phase angle difference at both ends of lines 16-17 to be parallel shall be limited within the maximum allowable value. When the wind-power output is the predicted value of 100 MW, the phase angle difference at both ends of the line is 23.32\({}^{\circ}\). \(\theta_{set}=\)20\({}^{\circ}\) is set to ensure the safety of the closing process. The operating parameters of the generator and the recovered load data under the initial operating state are shown in Tables 3 and 4 (both are standard unit values). Node 26 is connected to the wind farm, with an installed capacity of 150 MW, and the output probability is shown in Table 2. At nodes 8, 16, 20, 21, 23, and 24, there are 20, 10, 20, 10, 10, and 10 feeders, respectively, whose loads are to be restored. The load and weight on each feeder are generated randomly and the power factor is consistent. The upper- and lower-unit values of the node voltage are 1.1 and 0.95, respectively. The NSGA-II parameters are set as follows: the population size is 120, the crossover probability is 0.9, the mutation probability is 0.1, and the maximum number of iterations is 150. Figure 8: Wind-power output probability interval division. The optimal regulation model of closing angle is solved based on NSGA-II, and the cumulative time is 187 s. The obtained multiobjective Pareto solution is shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node** & \(P_{\textit{dresi}}\) & \(Q_{\textit{dresi}}\) & **Node** & \(P_{\textit{dresi}}\) & \(Q_{\textit{dresi}}\) \\ \hline
1 & 2.52 & 1.19 & 20 & 2.80 & 1.03 \\
3 & 3.22 & 0.03 & 21 & 2.24 & 1.15 \\
4 & 5.00 & 1.84 & 23 & 1.98 & 0.85 \\
7 & 2.34 & 0.84 & 24 & 2.09 & \(-0.92\) \\
8 & 2.22 & 1.76 & 25 & 2.24 & 0.47 \\
9 & 1.04 & 1.25 & 26 & 2.39 & 0.17 \\
12 & 0.09 & 0.90 & 27 & 2.81 & 0.76 \\
15 & 3.20 & 1.53 & 28 & 3.06 & 0.28 \\
16 & 2.79 & 0.33 & 29 & 2.04 & 0.27 \\
18 & 3.08 & 0.30 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial operating load data. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node** & \(\textbf{U}_{\textit{G0}}\) & \(P_{\textit{G0}}\) & \(P_{\textit{G1}}\)max & \(Q_{\textit{G1}}\)min & \(Q_{\textit{G1}}\)max \\ \hline
30 & 1.048 & 3.4 & 3.5 & \(-2.8\) & 3.15 \\
31 & 0.982 & 3.45 & 11.46 & \(-9.17\) & 10.3 \\
32 & 0.983 & 3.5 & 7.5 & \(-6\) & 6.75 \\
33 & 0.997 & 4.5 & 7.32 & \(-5.86\) & 6.59 \\
34 & 1.012 & 6 & 6.08 & \(-4.86\) & 5.47 \\
35 & 1.049 & 4 & 7.5 & \(-6\) & 6.75 \\
36 & 1.064 & 5.6 & 6.6 & \(-5.28\) & 5.94 \\
37 & 1.028 & 4.6 & 6.4 & \(-5.12\) & 5.76 \\
38 & 1.027 & 3.8 & 7.6 & \(-7.44\) & 8.37 \\
39 & 1.03 & 8 & 11 & \(-8.8\) & 9.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial operating parameters of the generator. Figure 9: Schematic diagram of IEEE 39-node system.