# Operational Resilience Metrics for Complex Inter-Dependent Electrical Networks
## 2 Operational Resilience Metric for Electrical Distribution Networks
The electrical utility operator using the SCADA system (if available) is able to restore the service in a number of substations of the line \(l\). The time-to-restore by SCADA value is, in general, short. According to the electrical utility operator of the metropolitan area of Rome (Areti SpA) this value is approximately 3 min. The value \(r\) will depend on the number of substations that can be restored by remote actions. More of these substations are closer to the \(r\) value than they will be to \(M\). Anyway, in general, there exist substations that cannot be restored using the SCADA system. In such case the electrical utility operator has to coordinate manual operations (e.g., to isolate a failure) using the available technical emergency teams. As is shown in the next section, the time-to-restore by manual operations depends on the availability of the technical crews and the state of congestion of the urban viability infrastructure. According to the operator, a value for this quantity is approximately 45 min. Let us assume that the only way to repair a **damaged** node is to replace it with a power generator (PG) to ensure electrical continuity to the node's customers. Thus, in the proposed model, a damaged node will be not repaired during the simulation. In particular, the functioning status of a damage node is restored through the settlement of a PG (operation that requires time \(T\) in the previous QoS formula). The disconnected nodes, in turn, are reconnected either through a tele-control operation (if available) or by dispatching technical crews to provide manual reconnection. Such interventions may require specific times, which are considered when defining a restoration sequence of interventions. The \(QoS_{l}(\mathcal{S}(t))\) function of a line \(l\), over the time horizon \([0,t_{f}]\), can be represented by the following function:  
\[QoS_{l}(t)=N_{l}-\sum_{j=1}^{m}d_{j}\delta_{j}(t) \tag{2}\]  
where \(N_{l}\) is the total number of customers connected to the line \(l\), \(m\) is the number of substations on the line \(l\), \(d_{j}\) represents the number of customers of the \(v_{j}\) and \(\delta_{j}(t)\) is a function that equals 1 if \(s_{j}(t)=\{\textit{disc},\textit{dam}\}\) and 0 if \(s_{j}(t)=\textit{on}\). It is worth recalling that \(\sum_{j=1}^{m}d_{j}=N_{l}\). The \(QoS_{l}(t)\) metric can be appropriately normalized in order to be bounded to the \([0,1]\) interval. Figure 3: The QoS behavior of a medium-voltage (MV) line. In other terms, in order to measure the impact of a perturbation on an MV line and in general on an EDN, our framework uses a parameter similar to the CMI index introduced in Section 1, that measures the total loss area due to a perturbation as showed in Figure 4. This index is denoted by \(\Gamma(kmin)\) in the following. Indeed, the quantity \(\Gamma\) expressed in \(kmin\) for a line \(l\) is computed as follows:  
\[\Gamma^{l}(\tau_{i})=(N_{l})*\tau_{llc}+(d*\tau_{crew}) \tag{3}\]  
where \(\tau_{llc}=t_{1}-t_{0}\), \(\tau_{crews}=T-t_{1}\) and \(d=N_{l}-r\) (\(d\) represents the customers that has to be reconnected through manual restoration procedures). In case of a double failure on two distinct secondary substations on the same medium voltage line the \(\Gamma\) value can be computed as follow:  
\[\Gamma^{l}(v_{i})=(N_{l})*\tau_{llc}+(d_{1}*\tau_{crews})+(d_{2}*\tau_{pg}) \tag{4}\]  
where \(\tau_{llc}=t_{1}-t_{0}\), \(\tau_{crews}=t_{2}-t_{1}\), \(\tau_{pg}=T-t_{2}\), \(d_{1}=N_{l}-r_{1}\) and \(d_{2}=N_{l}-r_{2}\)  
A double failure can be considered the worst case as, in general, this will result in a number of isolated substations. In this case, the only possibility to restore the power delivery to the final customers is to deploy mobile power generators. As indicated by the electrical operator, mobile power generator deployment is a time-consuming operation (the mean time to deploy a mobile power generator has been indicated as \(\tau_{pg}\)).