# Operational Resilience Metrics for Complex Inter-Dependent Electrical Networks
## 1 Introduction
The problem is thus shifted to the identification of a _state function_ whose measure can be associated to the resilience property. This function should be able to reproduce the behavior of the system during a perturbation and it should consider all factors (or system properties) influencing the resilience level of the system. Similarly to [12], in this work the considered metric is represented in Figure 2 where the _quality of service_ (\(QoS\)) is defined as a quantitative metric for the resilience of EDNs on a temporal horizon. The temporal behavior of the state function showed in Figure 2 can be ideally divided into four phases:  
* Normal behavior and aging (\(t<t_{0}\)). During this phase, in the absence of perturbations, the QoS of the system is optimal. The service is delivered thus producing the \"wealth\" of all users. Maintenance actions are required to maintain this optimal (or near-optimal) behavior. During this phase, events prediction and the set up of preparedness actions should be considered. * Shock (\(t_{0}\leq t<t_{1}\)). This phase follows a perturbation. From this point on, the QoS typically decays rapidly following an exponential behavior. The decay rate depends on the infrastructure and might vary from a few seconds to hours. In an EDN, for instance, protection devices and/or procedures require the disconnection or isolation of system components to avoid the spreading of the impact. The decay behavior of this phase could be affected by several factors: involvement of a large number of components, and possible negative feedbacks from interdependent networks [17]. Shock phase ends up at time \(t_{1}\) corresponding to the time when CI operators starts acting on the network control and \(QoS(t_{1})=m\) where: \(0\leq m<1\) reaches the minimum. * Recovery (\(t_{1}\leq t<t_{2}\) where: \(m<QoS(t_{1})=r<QoS(t)\)). This phase starts when the activity of contingency management is implemented to recovery from the fault through automatic and manual interventions in order to increase the QoS. Success in the recovery phase, however, could be hindered, again, by negative feedback arising from the losses in other services; they could reduce the efficiency of recovery strategies (i.e., the absence of tele-control functionality, reduced by a severe electric outage, would force the use of longer manual restoration procedures). In general, this phase has the primary goal to start supplying services to critical users through the rapidly deployable contingency actions. * Restoration (\(t_{2}\leq t\leq t_{3}\) where: \(1\leq QoS(t)\leq QoS(t_{3})=M\)). This phase, which might extend on time scales larger than those of the previous phases, allows us to recover the full functionality of the network (i.e., the QoS before the perturbation, if possible). In principle, a wise and efficient restoration phase could also originate new properties of the network, which might produce a super-elastic behavior, i.e., allowing the system to gain a larger QoS with respect to that displayed before the crisis (i.e., a  
Figure 2: The system QoS during a contingency. \(t_{0}\): start time of shock and cascading, \(t_{1}\): degradation time, \(t_{2}\): recovery time, \(t_{3}\): restoration time. new network configuration could guarantee the service supply with lesser operational costs in term of electrical losses). In this work we do not consider other actions (i.e., the long term activities enabling to rebuild disrupted items and/or improve existing infrastructures) but the actions needed to return, at least, at the function level of the system before the occurrence of the perturbation. The proposed state function or system performance curve is similar to the resilience model proposed in [14; 18]. In our case, we clearly differentiate the recovery and restoration phase as this model is more suitable to describe the performance of an electrical network when affected by disturbances [5; 6].