# Operational Resilience Metrics for Complex Inter-Dependent Electrical Networks
## 5 Results
### Resilience Assessment of EDN in Case of Metropolitan Contingency Scenarios  
The present work extends the results presented in [29] describing the \(N-1\) case analysis of the Rome EDN. In the present work, RecSIM was used for the assessment of the resilience of the EDN by: (1) considering the \(N-2\) worst case analysis of the EDN and (2) considering more realistic crisis scenarios obtained through Monte Carlo simulations by appropriately weighting the probability of occurrence of a given fault with the effective substations rate of faults resulting from historical data provided by the EDN operator.  
First of all, the proposed resilience assessment approach provides a quantitative approach to _measure_ the global operational resilience of a network and it provides a mean for the electrical operator to evaluate technological and/or operational improvements to _increase_ the resilience of the network. At the same time, the proposed framework allows for the identification of the most critical components (e.g., MV lines) of the network in terms of impacts in case of contingencies. Then, the electrical operator can use the framework to optimize and to simulate different mitigation strategies.  
Figure 10 reports the distribution function \(D^{(1)}(\Gamma)\) for all resulting \(\Gamma_{i}\) considering the simulation parameters showed in Table 2 with two crews available. The network configuration (topology, position of switches along the MV lines) is referred to as a _normal configuration_.  
Different distributions will be obtained applying different perturbations to the network. For example, we used the proposed approach for the assessment of the \(N-2\) worst case and heuristic contingency analysis. Figure 11 shows the comparison among the different distribution functions.  
Figure 10: \(D^{(1)}(\Gamma)\) distribution for the \(N-1\) analysis.  
As pointed out in [30], the overall system resilience could be estimated as a series of terms, each one representing the contribution towards resilience for different (and progressively large) perturbations.  
\[R=a^{(1)}R^{(1)}+a^{(2)}R^{(2)}+a^{(3)}R^{(3)}+... \tag{8}\]  
where the terms \(R^{(i)}\) will be achieved by applying equations similar to and to the different \(D^{(i)}(\Gamma)\). The terms \(a^{(i)}\) can be related to the probability of the event; this would produce a series of progressively smaller terms, which would reduce the impact of the high order contributions to the total value of the global resilience score.  
For the three cases considered in our analysis, the corresponding \(R^{(1)}\), \(R^{(2)}\) and \(R^{(h)}\) are \(R^{(1)}=2.1710^{-2}\), \(R^{(2)}=7.6010^{-3}\) and \(R^{(h)}=1.7810^{-2}\). It is interesting, in turn, to notice that crises produced in the heuristic case (i.e., involving substations that have shown a large propensity to fault), although in some cases involving more than a substation, produce impacts which, even in the largest cases, are of the same dimension of those produced by worst cases in the \(\left(N-1\right)\) simulation. This is probably due to the fact that more vulnerable substations are located along lines, which do not produce relevant outages in case of faults (either for the presence of a few non-remotely controlled substations and/or for the presence of a small number of connected customers).