## 2 Background
### Analytical Assessment of the Probability of Adequacy (PoA)  
The load and generation models were combined in [5] to assess the PoA of an island \(j\). Assuming that there are \(L_{j}\) loads, \(R_{j}\) RDGs, and \(C_{j}\) CDGs in the considered island, the number \(M_{j}\) of combinations depends on the number of levels in each load model, as well as in the generation model of each DG. Therefore, \(M_{j}\) is equal to the number of working points at which the island can operate, that is:  
\[M_{j}=\prod_{i=1}^{L_{j}}N_{i}^{D}\cdot\prod_{r=1}^{R_{j}}N_{r}^{G}\cdot 2^{C_{ j}}. \tag{1}\]  
For a given combination \(m\), the total available local generation \(G_{j,m}^{E}\) is compared with the whole power demand \(D_{j,m}^{E}\) of the island's LPs to assess the load fraction that can be supplied by the island's DGs when combination \(m\) occurs. Weighting such a fraction by the occurrence probability, the related PoA (\(\rho_{A,j}\)) is computed as follows:  
\[\rho_{A,j}=\sum_{m=1}^{M_{j}}R_{j,m}\cdot\pi_{m}^{(R_{j})}, \tag{2}\]  
where:  
\[R_{j,m}=\min\left(1;\frac{G_{j,m}^{E}}{D_{j,m}^{E}}\right) \tag{3}\]  
and \(\pi_{m}^{(R_{j})}\) is the occurrence probability of the \(m\)-th combination. This approach neglects the second-order statistics of the time variations of load and generation that usually occur when the island mode persists for some hours. In other words, the previous approach considers that \(R_{j,m}\) does not change during the interval in which the portion of the network works in autonomous mode. To this purpose, in Section 4 a Markov chain will be introduced to model the time evolution of \(R_{j,m}\), and its dependence on the time-variant behavior of both load and generation processes. Analytical Evaluation of Distribution System Reliability Accounting for Islanding and Telecontrolled Switches  
In the following, the term _node_ will indicate either a point (bus) in the medium-voltage network where customers and/or generators are connected, or a switching substation. A node is an LP if at least one customer is connected to it. The electrical equipment connecting two nodes will be \begin{table}
\begin{tabular}{c c c} \hline \hline
**Level (I)** & **Power Output (\% of Rated Power)** & **Probability \(\left(\pi_{r,l}^{(RG)}\right)\)** \\ \hline
1 & 20 & 0.36 \\
2 & 40 & 0.28 \\
3 & 60 & 0.20 \\
4 & 80 & 0.11 \\
5 & 100 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Example of annual generation model for a renewable distributed generator \(r\) with five levels (\(N_{r}^{G}=~{}5\)). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Level (I)** & **Power Output (\% of Rated Power)** & **Probability \(\left(\pi_{c,l}^{(CG)}\right)\)** \\ \hline
1 & 0 & FOR \\
2 & 100 & 1-FOR \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Example of annual generation model for a conventional distributed generator \(c\), considering its forced outage rate (FOR). indicated as a _branch_. The branches are numbered sequentially and each node takes the number of the upstream branch, with reference to the primary substation (PS) position, as shown in Figure 1. Moreover, it is assumed that a sectionalizer is always installed where a circuit breaker is placed (in the following a CB and sectionalizer taken as a whole are referred to as a CBS). The switches placed in the network are CBSs and sectionalizers only, and a potential island is related to each switch. The CBS operations are delayed in order to telecontrol them, so as to enable a logic-selectivity-based protection scheme. Also, some sectionalizers are telecontrolled to enable faster restoration of the network portions outside of the faulted zone. The distribution management system (DMS) receives information from the switches detecting the fault and, firstly, sends an opening authorization to some CBSs while it forces the others to remain closed; then the DMS performs the restoration procedures, operating the telecontrolled sectionalizers.