## 6 Case Study  
The analytical formulation proposed in this paper is applied to the example distribution network reported in Figure 1 with the aim of providing an accurate measure of the reliability improvement that can be achieved thanks to islanding in distribution networks where telecontrolled CBs, as well as telecontrolled and manual sectionalizers, are installed. For each branch, the failure rate (_i.e._, the number of faults per year) and the repair time have been assumed as 0.05 and 8 h, respectively. Both the PS and the switches are considered fully reliable. Each node is considered as a LP with 100 customers. Switching times of 0.1 h and 2 h are assumed for telecontrolled and manual sectionalizers, respectively. The local DGs of each island need 0.08 h to become available (DGs' time to be available). A set of branches (SOB) that affect a given LP in the same way can be considered as an equivalent branch. In detail, the branches located between switch \(j\) and the switches placed downstream from \(j\) belong to SOB \(j\). An equivalent branch failure rate and repair time for SOB \(j\) are obtained by summing, respectively, the branch failure rates and the normalized repair times of its branches. The normalized repair time of a branch belonging to SOB \(j\) is computed by multiplying the failure rate by the repair time of the branch and, subsequently, by dividing the result by the equivalent branch failure rate of \(j\). Similarly, set of nodes (SON) \(j\) is defined as the set of nodes located between switch \(j\) and the switches placed downstream from it. An SOB affects all the LPs belonging to an SON in the same way, so all the LPs within the same reliability zone have the same annual outage rate and duration [31]. Appendix B reports information about the branches and the nodes belonging to each SOB and SON, respectively; the scenario related to each 2-tuple SON/SOB; the average and max power demand for each LP; the position, average, and max power level for each renewable distributed generator; and the position and rated power of each conventional generator. As a term of comparison, first the PoA is calculated by means of \(\underline{\pi}^{(R_{j})}\), neglecting load and generation fluctuations, thus obtaining \(\rho_{A,j}\). On the other hand, the proposed indices for assessing PoA more accurately, \(PoAR_{j}\left(T,N\right)\) and \(PoAD_{j}\left(T,N\right)\), are obtained by means of \(P^{(R_{j})}\). Table 4 shows the PoA values for each island of the distribution network reported in Figure 1. If \(T=8\) h and \(N=8\) steps, then the duration of each step is one hour. In this case study, the results show that neglecting the correlation between load and generation does not involve inaccurate values. Moreover, the effect of these values on the system indices is negligible because they are slightly overestimated in some cases and slightly underestimated in others. On the other hand, neglecting load and generation fluctuations leads to an overestimation of the ability of DGs to meet the island load, as highlighted by Figure 5. The error in the PoA affecting the LP outage rate is mainly in the interval 15%-20%, and is greater than the error on the PoA affecting outage duration (8%-10%). In a few cases, the error is negligible, such as for islands 15, 25, and 34, because the load is almost always met by local generation. In other words, the Markov process \(r_{j}(p)\) often assumes a value of 1 and consequently generation and load fluctuations do not significantly affect the PoA value. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Island j**} & \multicolumn{3}{c}{**Correlation Neglected**} & \multicolumn{3}{c}{**Correlation Not Neglected**} \\ \cline{2-7}  & \(\rho_{A,j}\) & \(PoAR_{j}\left(8,8\right)\) & \(PoAD_{j}\left(8,8\right)\) & \(\rho_{A,j}\) & \(PoAR_{j}\left(8,8\right)\) & \(PoAD_{j}\left(8,8\right)\) \\ \hline  
\hline \hline \end{tabular}
\end{table}