# Quantification of the Benefits for Power System of Resilience Boosting Measures
## 2 A Framework to Model Power System Resilience
### A Mathematical Formulation Derived from CIGRE Definition  
Considering the CIGRE definition of power system resilience helps build a general mathematical formulation to fully quantify the system resilience by suitable metrics and the benefits to resilience obtained by deploying measures. The formulation in this paper proposes a generalization of the risk concept reported in [20] by interpreting the risk as a quadruple (threat, fault, contingency, impact). Such a general mathematical framework includes the following variables:  
* Threat, _Th_, characterized in probabilistic terms using a suitable probability distribution for the related stress variables as a function of time. * The fault of the component, _Fa_, with a fault probability \(\mathrm{P_{Fa}}(t)\) which is a function of time. * The contingency, _Co_, meant as any combination of component faults with the response of the relative protection and automation schemes, with a probability of occurrence \(\mathrm{P_{CO}}(t)\) which is again a function of time. * The impact of contingency _Co_ on the system, _Imp_, which can be expressed in terms of LOL (Loss of Load), ENS (Energy Not Served) and static and dynamic instability indicators, and described with a suitable probability distribution of the abovementioned continuous quantities, depending on the characteristics (load and generation patterns) of system state defined through appropriate stochastic variables \(S\). * Barriers \(B\) which can reduce the vulnerability of the grid components to the threat or the severity of a contingency. The barriers in Figure 2 are implemented by the key actionable measures in the CIGRE C4.47 resilience definition presented in Section 2.1. Many factors of influence affect the intensity of threat _Th_, the probability of component faults _Fa_, the probability of occurrence and the impact _Imp_ of contingency _Co_, as well as the power system state \(S\). The most important factors consist in environmental factors _W_: e.g., the time evolution of weather variables and slowly varying or fixed characteristics of terrain (orography, tree coverage) determine the severity, the extent and duration of the threat stress on the grid infrastructure. Other significant factors may be organizational factors \(Z\), such as the number of maintenance teams and the way to schedule their jobs, which affect, e.g., the duration of the recovery phase. This means in mathematical terms that _Th_, _Fa_, _Imp_, and _Co_ are functions of \(W\) and \(Z\), as well as of the deployed measures \(B\). These measures can be classified into two categories, as described in detail in Section 3:  
* Passive measures, defined with binary variables _Bp_. * Active measures, defined with binary variables _Ba_. In particular, passive measures mainly affect:* The intensity of threat stress (indicated with \(Bp_{t}\)). * The probability of having component faults (indicated with \(Bp\)), given a specific threat stress intensity. Active measures typically affect:
* The extent, the probability of occurrence of a contingency (e.g., redundancy in protection systems, indicated with \(Ba_{pr}\)). * The impact of a contingency, given a specific set of component faults (e.g., the defense systems, indicated with \(Ba_{d}\)). * The state of the system \(S\) (e.g., the redispatching of conventional generators can bring the line current above anti-icing current which avoids wet snow sleeve formation). These measures are indicated by binary variables \(Ba_{s}\). System state \(S\) influences in turn all the other variables because it may affect:
* The threat intensity (e.g., if the line current is above a minimum anti-icing current, then the wet snow sleeve does not accrete on the conductor). * The fault probability (e.g., a high current on a line causes an augmented sag and increases the probability of tree contact). * The extent and the probability of occurrence of a contingency given a set of component faults (e.g., the probability of cut-out of wind farms depends on the system conditions).