# Probabilistic Resilience Analysis of the Icelandic Power System under Extreme Weather
## 3 Methodology
### Weather-Dependent Failure Probabilities  
We begin by defining weather dependent failure rates for each time step in the analysis horizon and component of the transmission system. To do so, we rely on the methodology introduced in detail in [3; 20]. Briefly, the methodology involves categorizing the observed failures of each individual component according to the identified primary cause and mapping this data to the time series describing the intensity of the identified primary cause. The end results are (i) a set of time-invariant (average) failure rates per component and identified primary case and (ii) a set of correction factors to account for the momentary intensity of each identified primary cause.  
In the present application, we focus on the impact of an extreme wind-storm on the Icelandic power system. To do so, we model two distinctive primary causes of failure, namely wind (\(w\)and other \((o)\). Further, we use the location-specific forecasted wind speed as the descriptor of the anticipated intensity of the wind primary cause. Accordingly, we express the time-dependent failure rate per transmission component \((i)\) and time moment \((t)\), as:  
\[\lambda_{i,t}=\hat{\lambda}_{i}^{o}+c_{i}(v_{i,t})\cdot\hat{\lambda}_{i}^{w}, \tag{2}\]  
where,  
\(\lambda_{i},t\) time-dependent failure rate of component \(i\) at moment \(t\);  
\(\hat{\lambda}_{i}^{w}\) time-invariant failure rate of component \(i\) for the wind primary cause;  
\(\hat{\lambda}_{i}^{o}\) time-invariant failure rate of component \(i\) for all other causes;  
\(v_{i,t}\) forecasted wind speed at the location of component \(i\) and moment \(t\); and,  
\(c_{i}(v_{i,t})\) correction factor for the intensity of the wind facing component \(i\) at moment \(t\).  
Typical values for \(c_{i}(v_{i,t})\) range from 0.1 at low wind speeds up to \(10^{4}\) for high wind speeds. For wind speeds that are greater than those experienced historically by a line, the value of \(c_{i}(v_{i,t})\) becomes significantly higher. Note that the time-invariant failure rate of a component for all other causes, \(\hat{\lambda}_{i}^{o}\), represents all other possible causes of failure on the Icelandic power system, as summarized by [PERSON] [16]. It should be noted that some of these causes represent time-varying phenomena that may be modeled separately, but this however requires additional data that was not available at the time of the study.  
Given the time-dependent failure rates, we use the classical models of [PERSON] [8] to enumerate weather-dependent survival and failure probabilities per component and time instant. Precisely, we define the former as \"probability that component \(i\) has not experienced any failure during the first \(t\) hours\" and the latter as \"probability that component \(i\) may sustain a failure within \([t,t+dt]\) hours\". Denoting these probabilities \(\pi_{i,t}^{s}\) and \(\pi_{i,t}^{f}\), respectively, we compute them as:  
\[\pi_{i,t}^{s}=\int_{t}^{\infty}\lambda_{i,\tau}\cdot e^{-\lambda_{i,\tau}\tau }d\tau, \tag{3}\]  
\[\pi_{i,t}^{f}=\int_{t}^{t+dt}\lambda_{i,\tau}\cdot e^{-\lambda_{i,\tau}\tau}d\tau. \tag{4}\]