# Operational Resilience Metrics for Complex Inter-Dependent Electrical Networks
## 1 Introduction
An analytical expression to measure resilience \(R\) as for the above-mentioned definition, is a piecewise function that captures the reduction of the quality of service as a function of time \(QoS(t)\); \(QoS(t)\) ranges from 0 (total loss of functionality) to 1 (no reduction in functionality). The equation for resilience proposed hereafter is modified after [PERSON] and [PERSON] [12] and represented mathematically as:  
Figure 1: Total number of reported natural disasters between 1950 and 2018. \[R^{-1}=\int_{t_{0}}^{t_{0}+T_{RE}}[1-Qos(t)]dt \tag{1}\]  
where \(t_{0}\) is the time of occurrence of the event and \(T_{RE}\) is the recovery time (i.e., the time at which the full functionality of the system is regained and the \(QoS\) takes the unit value. According to (1) resilience R is essentially a measurement of the total functionality lost in a system and recovered over time, when subjected to a crises event. Resilience can be estimated in terms of \(QoS(t)\) at the individual component level, or at the system level. There are four key properties outlined by [12] that can influence \(QoS(t)\) and therefore the extent to which a system will be more or less resilient, namely: (i) _robustness_, a system's or component's ability to withstand stress; (ii) _redundancy_, the substitutability of different elements within the system; (iii) _resourcefulness_, the ability of a system to adapt in order to prevent or reduce disruption of the system and rapidity and/or the availability of necessary resources; (iv) _rapidity_, the ability to respond to and mitigate disruption in a timely manner. In this work, we propose an _operational_ metric to evaluate the resilience \(R\) of electrical distribution networks that encompass all the above-mentioned resilience-building properties. The term \"operational\" indicates that the framework aims to assess the resilience of a network defining operational models that consider the different factors affecting the resilience of the network itself. Such factors span from the technological ones (e.g., the network topology, the functioning status of the SCADA system) to network management procedures adopted by the operators. In particular, the proposed framework can be considered an electric utility-centric approach for the assessment of the resilience of _large_ power distribution grids _also_ under normal, daily network operations (in contrast with the approach proposed in [3] that considers the resilience of a power distribution grid limited to natural extreme events). According to [13], where a system is said to perform in a resilient manner when \"it can sustain required operations under both expected and unexpected conditions by adjusting its functioning prior to, during or following events (changes, disturbances and opportunities)\", the proposed resilience assessment approach also considers opportunities in the assessment process. Indeed, the proposed framework can be used to estimate the impact of: (i) technological improvements (e.g., a larger density and a wiser positioning of automatic devices along the network topology) and (ii) changes in operational configurations, as for example increasing the number of emergency technical crews available to manage and solve an EDN crisis. Moreover, as improving the resilience of a system considering the interdependent infrastructures systems and services [14; 15] is a fundamental aspect, the proposed framework allows us to quantify the resilience of an EDN considering the availability of other services and infrastructures that may impact the performance of the power grid as, for instance, the telecommunication network providing the communication services to the SCADA system and the urban mobility infrastructure allowing the operations of the emergency technical crews. In general terms, as it has been stressed in [16] \"the assessment of resilience should therefore identify the critical functionality of a system and evaluate the temporal profile of system recovery in response to adverse events\".