# Operational Resilience Metrics for Complex Inter-Dependent Electrical Networks
## 1 Introduction  
Electrical transmission/distribution networks provide the essential power delivery service to the principal core infrastructures and services of a country. Disturbances affecting electrical networks in a particular area will affect its services provided to citizens and, in general, through the perturbation produced to other infrastructures and services (e.g., telecommunication networks, urban mobility, banking systems) to a large part of the community in an area that can also be much larger than the initial ones. Depending on the magnitude of disturbances, the impacts on the dependent services and on citizen activities and well-being can be significant. In general, the disturbances of an electrical network can be the result of different events that may have more or less pronounced effects on the system functionality, from a light perturbation up to the complete disruption of one or more of its components. Perturbations may arise from different sources (natural or anthropic) and associated with different agents: physical (e.g., lightning, floods [1], earthquakes, fires), or cyber (e.g., denial of services attacks affecting the RTU of an electric substation). These aspects are further worsened by the fact that perturbations could also affect other systems, which may generate negative feedbacks, thus amplifying the effects and reducing system capability to return to an equilibrium state. The electrical network disturbances may span from minor or routine disturbances as in normal conditions, that, in general, are mitigated and restored through protection and automation devices and/or by the operators' interventions, to major disturbances, due to extreme meteorological, catastrophic events and cyber-attacks. In general, this kind of disturbance causes electrical network disruptions that last for hours. As reported in [2], in Italy in February 2015, over 360,000 customers were left withoutpower for more than 8 h, and more recently in the Abruzzo and Marche regions in January 2017, with disruptions that lasted over 72 h affecting 39,000 customers. Nowadays, frameworks for resilience assessment are relevant for electrical operators as the demand of a resilient electrical delivery service is increasing and the requirements imposed by the electrical authorities are more and more strict. Moreover, electrical operators have to improve the efficiency of their operational procedures in order to adapt to the increase of user demand, the adoption of smart operative modes and the increase of competition among operators. The requested resilience assessment approaches can be divided in short term or electric utility-centric approaches where the objective is the evaluation of actual or hypothetical power grid design and the optimisation of normal, daily network operations and configurations) and long-term, strategic resilience assessment to help decision makers, infrastructure operators, public agencies which drive investments to establish better policies for the resilience improvement of the power grid and to improve the resilience of a specific region against natural disasters [3; 4; 5; 6]. Moreover, resilience assessment frameworks will play a central role in the analysis of mitigation measures to contrast the increase of extreme and catastrophic events (Figure 1) and the increase of the number and the dangerousness of cyber-attacks on industrial control systems (in Ukraine, 230,000 customers were without power for 6 h in one of the first attacks on a nation's power grid [7]). Different definitions of resilience can be currently found [8; 9; 10], each reflecting a different domain where the concept of resilience can be applied. An exhaustive literature review is out of the scope of the paper, however, in the context of this work, resilience can be defined as \"the ability of a Critical Infrastructure (CI) system exposed to hazards to resist, absorb, accommodate to and recover from the effects of a hazard in a timely and efficient manner, for the preservation and restoration of essential societal services\" [11].