## 5 Numerical Simulations
### Results and Discussions  
Table 2 summarizes the globally optimal solutions of the traditional and proposed frameworks. Figure 8 illustrates the line currents in each restoration configuration.  
As summarized in Table 2, different configurations were selected, although each problem framework judged that its optimal solution could restore all outage loads excluding those in the load sections directly connected to the faulty feeding point. The number of changes of the switch states in the traditional framework was the smallest; however, as shown in Figure 8a, the overload occurred by the extra power supply necessary to compensate the disconnected PVs. In Figure 8c, which shows the result of pure robust  
\begin{table}
\begin{tabular}{l c} \hline \hline Total number of distribution substations & 3 \\ Total number of feeding points & 53 \\ Total number of switches & 266 \\ Total number of all configuration candidates & 1.19 \(\times\) 10\({}^{80}\) \\ Total amount of section loads & 30.97 MW \\ Total amount of measured outage loads & 3.14 MW \\ Total amount of PV capacities in target area & 1.32 MW \\ Total amount of actual PV outputs in target area & 0.72 MW \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specification of Figure 5.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Comparison Index** & \begin{tabular}{c} **Traditional** \\ **Framework** \\ \end{tabular} & \
\begin{tabular}{c} **Two-Stage Stochastic Programming** \\ \(\xi\) = 0.7 \\ \end{tabular} &
\begin{tabular}{c} **\% = 1.0** \\ \end{tabular} \\ \hline Estimated total outage loads & 206.57 kW & 248.45 kW & 260.98 kW \\ Expected value of extra outage loads & - & 1.11 kW & 0 \\ Number of changes of switch states & 10 (5 pairs) & 12 (6 pairs) & 14 (7 pairs) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of numerical simulation results.  
Figure 6: Line currents in normal condition.  
Figure 7: PV output and its probability distribution: (**a**) sample of actual and predicted PV outputs; (**b**) probability distribution for assumed PV output (55% of PV capacity).  
optimization, there is no operational constraint violation in exchange for an increase in the number of changes of the switch states. On the other hand, Figure 8b shows characteristics intermediate between the above two frameworks. It indicates that the relaxation of conservativeness, namely, setting \(\xi=0.7\) rather than \(\xi=1.0\), brought a decrease in the number of changes of the switch states. In Figure 8b, the PV-originated uncertainty (\(\geq\xi\)) was reflected in the expected value of extra outage loads.  
The authors carried out several numerical simulations under different conditions, and as a result, found that the solutions derived in the traditional framework can cause operational constraint violations. For example, when we increased the level of PV penetration, the PV-originated uncertainty was intensified, associated with (10-12), and it brought more violations to the optimal solution of the traditional framework. In this section, the typical results were shown to discuss the possibility of constraint violations by the traditional framework in a realistic situation. Under these circumstances, we can conclude that the proposed problem framework and its solution method functioned appropriately.