# Valuing Resilience Benefits of Microgrids for an Interconnected Island Distribution System
## 2 Valuation Approach
### Duration of Outage Events  
For reliability events, outage durations are established based on historic data as indicated in Table 2. Figure 1 shows that reliability events commonly cause outages lasting less than four hours. For major and exceptional hurricane outage events, outage durations must be assumed using inference from the available historic data. First, the most recent exceptional hurricane outage event is examined. A resilience curve is established that quantifies the baseline performance in the two smaller islands to past hurricane outage events. The rate of recovery to a loss suffered by the system during _black sky days_ (i.e., following a major outage) is a precursor metric to evaluate historic performance. This can be obtained from the difference between the outage rate \(O(t)\) and the recovery rate \(R(t)\) at time \(t\). These instantaneous quantities can be defined as  
\[O(t)=OOS_{component}(t), \tag{1}\]  
\[R(t)=R_{component}(t), \tag{2}\]  
where OOS stands for a component that is Out Of Service. These can be assigned to the number of outages, the number of unserved customers, or another pertinent parameter. As an example, cumulatively, these can be defined as:  
\[O(t_{i})=\sum_{i=0}^{t1}OOS_{customers}(t_{i}), \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \begin{tabular}{c} **Maximum Strength** \\ **within 50 nm** \\ \end{tabular} & \begin{tabular}{c} **Number of** \\ **Occurrences over** \\ **170-Year Interval** \\ \end{tabular} & \begin{tabular}{c} **Annual Frequency** \\ **of Occurrence** \\ \end{tabular} & \
\begin{tabular}{c} **Recurrence Interval** \\ **(Years)** \\ \end{tabular} \\ \hline
\begin{tabular}{c} Tropical \\ Depression/Storm \\ \end{tabular} & 27 & 0.159 & 6 \\ \hline Hurricane-I & 9 & 0.053 & 19 \\ \hline Hurricane-II & 5 & 0.029 & 34 \\ \hline Hurricane-III & 9 & 0.053 & 19 \\ \hline Hurricane-IV & 3 & 0.018 & 57 \\ \hline Hurricane-V & 3 & 0.018 & 57 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hurricane Count at Various Strengths. \[R(t_{i})=\sum\
olimits_{ti=t0}^{t1}R_{ customers}(t_{i}), \tag{4}\]  
where these equations represent the customer outages following an event and those recovered, both at time \(t_{i}\). The difference of these two equations is the remaining customers not served. The metric used to quantify the resilience of the system is the resilience curve, defined as \(C(t_{i})\), which can be calculated as the difference between customer outages and those recovered: [17]  
\[C(t_{i})=R(t_{i})-O(t_{i}), \tag{5}\]  
The outage and restoration rates, as well the resilience curves were populated for Vieques and Culebra. The calculation was applied to the summer of 2017 when the system suffered during two category 5 hurricanes. This is the only recent representation of an exceptional hurricane outage event in electric utility data records. The first hurricane (Hurricane Irma) grazed the northeast corner of Puerto Rico, but its wind activity had a significant impact on the smaller islands of Vieques and Culebra. The second hurricane (Hurricane Maria), on the other hand, was devastating, as its path was directly through the center of the main island and over the two small islands, entering the southeast corner and leaving through the northwest. Both of the smaller islands have emergency backup diesel plants to serve the community following these events. In Vieques, the plant was inoperable, and the U.S. Federal Emergency Management Agency (FEMA) hauled in mobile diesel units, as reflected in the outage duration shown in Figure 2, which depicts the outage, restoration, and resilience curves for the timeframe of the two hurricanes and for the two islands combined.