## 4 Case Analysis
### Case Analysis of Closing Angle Regulation Scheme Based on Probability Density Prediction
The fuzzy multi-attribute decision-making method mentioned above is used to select the optimal solution from the Pareto set. The calculation results of the decision index values under each scheme are shown in Figure 11.  
Figure 11: Multi-attribute decision indicators of each scheme. Figure 10: Pareto front. According to Figure 11, the decision index value of scheme 57 is the largest, so it is the optimal solution, and the corresponding objective function values are \(f_{1}=2.06\) and \(0.1189\). The corresponding unit output adjustment is shown in Table 5, and the restored load outgoing line and load are shown in Table 6.  
Table 5 only shows the units which exhibit a change in output. It can be seen from the table that three units are participating in the output adjustment in the regulation, namely units 31, 37, and 38--the adjustment amounts for these are 0.55, 1.02, and 0.49, respectively, and the accumulated output increase is 206 MW. According to the operating parameters of the generator, the generator units involved in the adjustment still have a large adjustable capacity, which indicates that the regulatory scheme in this paper has a strong ability to regulate the closing angle and can also achieve effective regulation of the closing angle in the case of insufficient standby capacity of the system. According to Table 6, the total restored load of this regulation scheme is 193.6 MW, of which the total number of feeders restored at nodes 8, 16, 20, 21, 23, and 24 is 9, 6, 8, 3, 7, and 4, respectively. According to the weight of the restored feeder load, the restoration scheme formulated by the model established in this paper can give priority to the restoration of more important loads while the total restored load is the largest, which is helpful to improve the load recovery efficiency and reduce the outage loss. The power flow results of lines 15-16 before and after regulation are shown in Table 7 (at this time, the wind-power output is 100 MW). According to the data in Table 7, the transmission active power of the system interconnection line 15-16 is as high as 800.76 MW before the regulation, which is close to the power limit of the interconnection line. After the regulation of the system generator output and feeder load, the transmission active power of lines 15-16 is reduced to 650.63 MW, which \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Node** & **Feeder** & **Weight** & **Load** & **Node** & **Feeder** & **Weight** & **Load** \\ \hline \multirow{8}{*}{8} & 1 & 0.418 & 0.042 & & 16 & 0.448 & 0.017 \\  & 2 & 0.62 & 0.067 & & 17 & 0.719 & 0.025 \\  & 3 & 0.837 & 0.039 & & 19 & 0.679 & 0.016 \\  & 7 & 0.608 & 0.056 & & 20 & 0.746 & 0.067 \\ \cline{2-8}  & 13 & 0.646 & 0.056 & & & 1 & 0.951 & 0.05 \\  & 15 & 0.752 & 0.016 & & 21 & 2 & 0.526 & 0.035 \\  & 16 & 0.75 & 0.026 & & 9 & 0.963 & 0.064 \\ \cline{2-8}  & 17 & 0.47 & 0.052 & & & 1 & 0.922 & 0.085 \\  & 18 & 0.545 & 0.065 & & & 3 & 0.994 & 0.013 \\ \hline \multirow{8}{*}{16} & 2 & 0.65 & 0.05 & & 5 & 5 & 0.523 & 0.096 \\  & 3 & 0.523 & 0.073 & & & 6 & 0.832 & 0.052 \\ \cline{1-1}  & 4 & 0.661 & 0.049 & & & 7 & 0.998 & 0.036 \\ \cline{1-1}  & 5 & 0.832 & 0.097 & & & 8 & 0.715 & 0.075 \\ \cline{1-1}  & 6 & 0.661 & 0.047 & & & 10 & 0.666 & 0.031 \\ \cline{1-1}  & 9 & 0.921 & 0.033 & & & 2 & 0.748 & 0.078 \\ \hline \multirow{8}{*}{20} & 1 & 0.699 & 0.081 & & 5 & 0.663 & 0.062 \\ \cline{1-1}  & 11 & 0.697 & 0.065 & & 6 & 0.871 & 0.039 \\ \cline{1-1}  & 12 & 0.823 & 0.038 & & 7 & 0.795 & 0.086 \\ \cline{1-1}  & 15 & 0.509 & 0.059 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Feeder load recovery. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Node** & \(\Delta P_{G1}\) \\ \hline \multirow{8}{*}{8} & 31 & 0.55 \\  & 37 & 1.02 \\  & 38 & 0.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Adjustment of generator active output. effectively alleviates the heavy load problem of lines 15-16 and is conducive to improving the stability of the system parallel process.