# Operational Resilience Metrics for Complex Inter-Dependent Electrical Networks
## 2 Operational Resilience Metric for Electrical Distribution Networks
### Resilience Assessment of EDN in the Case of Metropolitan Contingency Scenarios  
In this analysis, RecSIM simulates network restoration procedures adopted in case of extraordinary network failures due to, for example, extreme weather conditions. These scenarios consider the failure of secondary substations and the possible restoration procedures that may be applied considering the available resources. Different failures cases have been considered:  
* \(N-1\). This case represents the usual \(N-1\) power grid analysis where one substation is considered in failure for each simulation run. * \(N-2\) worst case. In this case, for each MV line in the EDN, all the combinations of double failures on a single MV line have been considered. In general, each MV line can be fed by two or more primary substations. Then, if it is not possible to restore some substations on the normally configured MV line, the electrical operator can operate the electrical network switches to feed the isolated substations using the available _next_ MV line(s). If this happens, the electrical network switches from the _normal_ to a _temporary_ configuration. The \(N-2\) worst-case analysis considers double failures on a single MV line to prevent the electrical operator from switching from a normal to a temporary configuration. * Heuristic case. In this case, the substations configured in failure state have been chosen through an educated guess considering their effective rate of faults (as declared by the electrical operator). Indeed, statistics have been collected along several years and the number of observed faults normalized over the number of days of observation has been defined. This value is \(\rho_{i}\) and indicates the rate of faults per day that can be assimilated to the daily probability that the specific substation goes in a damaged state. The heuristic perturbation scheme has thus been applied to the network by simulating \(M\) working days: in each day of operations, the damaged state of each substation has been sampled (as in a Monte Carlo scheme) by extracting a random number \(r_{i}\) (\(r_{i}\) = [0,1]) and by comparing it with the \(\rho_{i}\) value: if \(r_{i}<\rho_{i}\) the \(i\)-th substation is put in the damaged state, whereas it remains unperturbed elsewhere. The substation set in the damage state has been put simultaneously in the damaged state, in order to simulate the worst-case scenario. This procedure is repeated \(N\) times to scan each substation and then repeated \(M\) times to simulate different working days. For each case, the resilience indicator is represented by the mean value of \(\Gamma\) with respect to the total number of simulation runs representing the total number of combinations. In particular, considering the \(N-2\) worst-case analysis, we define the set \(Next=(l_{i},l_{j})\) containing all pairs of MV line identifiers that can be used to switch to temporary network configurations for service restoration during a crisis (i.e., \(Next\) represents the set of suitable backup feeders). \(C\) is the total number of combinations for the \(N-2\) case; the resilience indicator is computed considering all possible double failures:  
\[\mathbf{R}_{N-2}^{-1}\propto\frac{\sum_{l\in L}\sum_{(v_{i},p_{j})\in I}\Gamma ^{l}(v_{i},v_{j})+\sum_{(l_{i},l_{j})\in Next}\sum_{v_{n}\in l_{x}\wedge v_{m} \in l_{y}}\Gamma^{l_{x}l_{y}}(v_{n},v_{m})}{C} \tag{7}\]  
For the \(N-1\) analysis, the total number of combinations coincides with the total number of substations in the network, whereas in the heuristic case, the total number of combinations coincides with the total number of damaged configurations obtained through the Monte Carlo method, such as the simulation described above. In particular, this procedure generates very few damaged states, as the rate of faults of the substations is usually small. However, it generates cases where one (or even more than one) substation will result in a damaged state.