# 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
* The impact of a contingency (e.g., cascading outages depend on post-contingency power flow). Figure 3 reports the conceptual relationship among the variables. Starting from the probability distributions associated to the weather variables, the methodology can derive the probability distributions of threat severity and combine them with the component vulnerability models in order to evaluate the probabilities associated to the faults of components (\(Fa\)) and to the occurrence of contingency \(Co\). Given a suitable metrics \(Imp\) to describe the extent, the duration and the severity of system degradation (e.g., the ENS or LOL) which depends on the specific contingency \(Co\) and on power system state \(S\) and on the measures deployed to counteract it, one can define the resilience metrics in two forms:  
* Directly, as a function of a risk index given by a combination \(\Theta\) of contingency probability \(\mathrm{P}_{\mathrm{CO}}\) and of its impact \(Imp\) (\(Co\), \(S\), \(Ba_{d}\)), with \(Co=Co\) (\(Fa\) (\(Th_{\Delta}\), \(S\), \(Bp_{d}\)), \(S\), \(Ba_{pr}\)) and \(Th_{\Delta}=Th_{\Delta}\) (\(W\), \(Bp_{t}\), \(S\)) by considering only the uncertainties related to the fault probability of components. \(Th_{\Delta}\) is the probability distributions of the stress variables over the time interval of analysis \(\Delta\). Figure 3: Conceptual relationship among the variables derived from the bow tie. * In probabilistic terms, as the probability that the risk metric above is lower than a specified threshold \(\gamma\) (established by TSO guidelines) over an interval of analysis \(\Delta\) (ranging from few minutes for quasi real time control to years for long term planning), as follows: \[\textit{Resilience}_{\Delta}=\textit{Prob}\Bigg{(}\Theta\Bigg{(} \begin{array}{c}P_{ \text{CO}}\Big{(}\textit{Co}\big{(}\textit{Fa}(\textit{Th}_{\Delta},\,S_{ \Delta},\textit{Bp}_{f}\big{)},\,S_{\Delta},\,\textit{Bap}_{pr}\big{)},\\ \textit{Imp}\Big{(}\big{(}\textit{Co}\big{(}\textit{Fa}(\textit{Th}_{\Delta},\,S_{ \Delta},\,\textit{Bp}_{f}\big{)},\,S_{\Delta},\,\textit{Bap}_{pr}\big{)},S_{ \Delta},\,\textit{Bad}\Big{)}\end{array} \Bigg{)}<\gamma\Bigg{)}\] (1) where \(\textit{Th}_{\Delta}\) and \(S_{\Delta}\) are the probability distributions of system state variables and of the stress variables over the time interval of analysis \(\Delta\). This form accounts also for the uncertainties on power system state. These indexes can be calculated by accounting either for an individual contingency or for a set of contingencies. Thanks to this new conceptualization of risk as a quadruple (threat, fault, contingency, impact), the property of vulnerability is much more general than in the previous definition of risk as the quadruple (threat, vulnerability, contingency, impact): In fact, in the latter one vulnerability only refers to the ability of components to resist to the threat, while in the new conceptualization of risk, the vulnerability is interpreted as the property which links any couple of quantities in the process of Figure 2, i.e., threat, fault, contingencies and impacts, following the cause-effect sequence (Th \(\rightarrow\) Fa \(\rightarrow\) Co \(\rightarrow\) Imp). Next sections will classify these measures and propose their mathematical models with some application examples to realistic grid models. The sequel of the paper will consider the former formulation of the resilience index. A brief description of the resilience index adopted in the simulations is given in the next subsection. The analytical development of Equation (1) is out of the scope of the present paper.