## 6 Case Study

Table 4: PoA computed considering (\(PoAR_{j};PoAD_{j}\)) and not considering (\(\rho_{A,j}\)); load and generation fluctuations when correlation between load and generation is neglected and not neglected. Figure 5: Overestimation of the Probability of Adequacy computed not considering (\(\rho_{A,j}\)) load and generation fluctuations during islanding with respect to the value obtained accounting for them. Table 5 reports the values of the distribution system reliability indices usually adopted for the reward/penalty mechanisms, that is, the System Average Interruption Frequency Index (SAIFI):  
\[SAIFI=\frac{\sum\limits_{i=1}^{n_{LP}}N_{C,i}\cdot\lambda_{i}}{\sum\limits_{i=1}^ {n_{LP}}N_{C,i}} \tag{17}\]  
and the System Average Interruption Duration Index (SAIDI):  
\[SAIDI=\frac{\sum\limits_{i=1}^{n_{LP}}N_{C,i}\cdot U_{i}}{\sum\limits_{i=1}^{n _{LP}}N_{C,i}}, \tag{18}\]  
where \(n_{LP}\) is the number of LPs in the network, \(N_{C,i}\) is the number of customers connected to LP \(i\), and \(\lambda_{i}\) and \(U_{i}\) are the outage rate and duration of LP \(i\) that can be computed by using the formulation reported in Appendix A, respectively. SAIFI and SAIDI have been computed for the network in Figure 1, where manual and telecontrolled sectionalizers are installed. Three different cases have been considered:  
1. islanding is not permitted by regulation;
2. islanding is permitted by regulation, but load and generation fluctuations are neglected;
3. islanding is permitted by regulation and both load and generation fluctuations are considered by using of the proposed analytical model. The results presented in Table 5 show that both network reliability indices are overestimated when the aforementioned fluctuations occurring during islanding are neglected. For example, the actual improvement in SAIFI is about 20% when islanding is permitted by regulations (1.014 _vs._ 1.270), but the improvement is overestimated when load and generation fluctuations are neglected. More specifically, an improvement of about 25% appears when considering the index value evaluated by neglecting fluctuations (0.947 _vs._ 1.270). Therefore, the overestimated improvement is 26.17% greater than the actual one. As is expected from the results in Table 4, the overestimation of SAIFI improvement is greater than for SAIDI. Similar considerations hold when only manual sectionalizers are installed, as highlighted by the results in Table 6. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Index** & **I** & **II** & **III** & \begin{tabular}{c} **Overestimated** \\ **Improvement** \\ \end{tabular} & \
\begin{tabular}{c} **Actual** \\ **Improvement** \\ \end{tabular} &
\begin{tabular}{c} **Overestimation on** \\ **the Improvement** \\ \end{tabular} \\ \hline SAIFI & 1.270 & 0.929 & 0.992 & 26.85\% & 21.89\% & 22.66\% \\ SAIDI & 6.020 & 3.146 & 3.389 & 40.96\% & 36.40\% & 12.53\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Distribution system reliability indices values when only manual sectionalizers are installed in the network. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Index** & **I** & **II** & **III** & \begin{tabular}{c} **Overestimated** \\ **Improvement** \\ \end{tabular} & \
\begin{tabular}{c} **Actual** \\ **Improvement** \\ \end{tabular} &
\begin{tabular}{c} **Overestimation on** \\ **the Improvement** \\ \end{tabular} \\ \hline SAIFI & 1.270 & 0.947 & 1.014 & 25.43\% & 20.16\% & 26.17\% \\ SAIDI & 5.329 & 2.312 & 2.568 & 56.61\% & 51.81\% & 9.27\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Distribution system reliability indices values when both manual and telecontrolled sectionalizers are installed in the network. The results have confirmed that in a network with both manual and telecontrolled sectionalizers, islanding improves SAIFI less (20.16% _vs._ 21.89%) than when only manual sectionalizers are installed. On the other hand, when both kinds of sectionalizers are installed, the SAIDI improvement is greater than that achievable with only manual sectionalizers (51.81% _vs._ 36.40%).