## 4 Markov Models for Adequacy Evaluation
### Markov Model of the Ability of the Generation to Meet the Island Load  
This section defines the Markov process \(r_{j}(p)\) to represent the ability of the local generation to meet the load in island \(j\). More specifically, by considering the generic combination \(m\) of load and generation, the value of the process \(r_{j}(p)\) is computed by means of Equation (3). a given amount of power; local DGs can supply a part of this power, but the PoA neglects the time fluctuations of \(R_{j,m}\). The first step to devising a new computational approach in order to account for such fluctuations is to choose how they have to be managed. The best strategy in terms of minimization of the average outage duration is to supply as many customers as possible, whenever the load and/or generation change. However, this choice is very disadvantageous from the average outage rate point of view, since a customer could be cyclically left unsupplied and resupplied many times during islanding, thus increasing the number of interruptions. Therefore, during islanding, the best choice in terms of average outage rate is not to resupply (when \(R_{j,m}\) increases) some customers previously left unsupplied. Obviously, these customers will be reconnected to the network after the fault is repaired. When the latter strategy is adopted, the PoA to be used for the outage rate assessment is related to the minimum value of \(R_{j,m}\) during islanding. It is indicated by an arrow in Figure 4 and it represents the improvement, _i.e._, the reduction of outage rate that can actually be obtained, thanks to islanding. The PoA to be considered for outage duration assessment is related to the customers not left unsupplied during islanding. It is represented by the dotted area in Figure 4 and it is the reduction, in terms of outage duration, that can actually be obtained through islanding. The bricked area represents the further reduction achievable when the former strategy is adopted, although it involves a worsening in the outage rate that can be even worse than the value obtained when islanding is not adopted. However, both strategies reduce the outage duration. In [4] it has been proved that telecontrolled sectionalizers strongly reduce (improve) the outage duration but slightly increase (worsen) the outage rate when islanding is permitted. Therefore, considering this, and also the inconvenience of using the first strategy, the second one is considered in the following in order to improve both outage rate and duration by means of islanding. An analytical model to compute the PoA index to be used for evaluating LP reliability indices when the second strategy is adopted is described in the following. To this purpose, a given time interval \(T\) during which the portion of the network downstream from switch \(j\) works in island mode of operation is considered. Assuming that there are \(N\) time slots in the interval \(T\), a path is defined as the evolution of process \(r_{j}(p)\), and it is given by the sequence of values assumed by \(r_{j}(p)\) in each time slot. The set of all the possible paths that process \(r_{j}(p)\) can assume in the \(N\) time slots is called \(\Psi_{j}\), and \(\wp_{j}=\big{<}r_{j}(1)\),..., \(r_{j}(N)\big{>}\) is the generic element of \(\Psi_{j}\). By applying the Chapman-Kolmogorov equation  
Figure 4: Improvement in term of outage rate (arrow) and duration (dotted area) achievable thanks to islanding for a path when the island management strategy aims at maximizing the reduction of the outage rate. The bricked area represents the further reduction achievable when the strategy aiming at maximizing the reduction of the outage duration is adopted, although it involves a worsening in the outage rate.