# Quantification of the Benefits for Power System of Resilience Boosting Measures
## 2 A Framework to Model Power System Resilience
### RELIEF: The Tool Implementing the Bow Tie Model  
The bow tie conceptual model was translated into a methodology and implemented into a tool for power system resilience assessment named RELIEF [20; 22; 23; 24]. Threats and component vulnerability models are combined thus leading to the failure probabilities of components, thus also the probability of occurrence of outages which can potentially provoke load disconnections [20]. RELIEF allows to assess power system resilience over different time frames. In the operational context, it combines the probabilistic models of the threats (e.g., lightning, fire, and wet snow) with the probabilistic models of component vulnerability in order to calculate the component failure probabilities, with a time horizon ranging from few hours to few minutes ahead. The step following the computation of the component failure probabilities consists in creating a quite exhaustive set of multiple, common mode and dependent contingencies affecting the components more prone to fail. After a fast screening of the generated contingencies, the retained ones are simulated via a quasi-static cascading outage simulator to assess possible cascading outages provoked by the initiating events. This allows to evaluate impact indicators such as the amount of MW not supplied or the energy not served. The risk based resilience indicator in case of generic contingency \(j\) in [20] is given by Equation (2).  
\[\textit{Resilience}_{ctg,j}=\frac{1}{\textit{Risk}_{ctg,j}}=\frac{1}{\Big{(} \textit{p}_{ctg,j}\times\textit{Imp}_{ctg,j}\Big{)}} \tag{2}\]  
where \(\textit{p}_{ctg,j}\) is the probability of occurrence of the contingency, the impact indicator \(\textit{Imp}_{ctg,j}\) is given by the loss of load (MW). The total risk of loss of load can be defined as the sum of the risk indicators \(\textit{Risk}_{ctg,j}\) for all the most dangerous contingencies. This means that the adopted risk metric \(\Theta\) in the general formulation is given by the product of contingency probability and impact.  
The system resilience can be defined as the minimum of the resilience indicators among the set \(\Omega\) of most dangerous contingencies to which the system is exposed, see Equation (3).  
\[Resilience_{syst}=\min_{j\in\Omega}Resilience_{cg,j} \tag{3}\]  
More comprehensive indicators include information about the restoration process, but the proposed indicator is currently adopted by the Italian National Regulator in cost-benefit analyses as reference indicator to be assessed by operators to justify the effectiveness of their investments. Thus, in the sequel this indicator will be calculated for all the measures under study.  
RELIEF also includes a submodule which computes the time for infrastructure and electric service recovery depending on the weather forecasts: some applications of the submodule to a real emergency scenario due to wet snow in North of Italy are reported in [22].  
The tool allows to characterize the threat geospatial models in the following ways:  
1. In engineering mode, where a simple analytical model with few parameters is used to characterize the intensity and the extent of the threat as well as their uncertainties [20].
2. In operational planning mode, by integrating the \(k\) hour-ahead weather forecasts in order to build the probabilistic model of weather based threats.
3. In long term planning mode, by considering the weather-related geospatial distributions as the distributions of the annual extreme values of the stress variables (for each location). In this mode, the combination of extreme value distributions with the vulnerability models of grid components leads to the calculation of the return periods of component outages [24].