# Reliability of Supply and the Impact of Weather Exposure and Protection System Failures +
## 3 Method
### Time Series Method  
Time series of failure probability can be used to incorporate varying failure probabilities due to exposure to external threats, e.g., wind, lightning, icing etc., leading to failure bunching effects. For time series, unavailability and associated reliability indices for cut-sets are calculated using the time series of different fault types, dependent on adjacency, as explained in this subsection. The algorithmic method outlined in [10] is used to calculate resulting unavailability and further reliability indices.  
The addition of a time dimension to failure rates makes it necessary to make alterations to the equations in Section 3.1. The probability matrices established in Section 3 can still be used, assuming that the probability of a failure propagating is time independent. Time-series of hourly failure probabilities for a set of lines, as presented in (22), is now considered.  
When referring to the time-series of failure probability for a single line, the column vector is denoted \(\lambda_{;,\downarrow}\). The fault types for a single target line \(i\) are given in (23)-(27).  
\[\lambda^{8}=\begin{bmatrix}\lambda_{i,\downarrow}^{s}\end{bmatrix}=\begin{bmatrix} \lambda_{1,\downarrow}&\ldots&\lambda_{1,\downarrow}\\ \vdots&\ddots&\vdots\\ \lambda_{t,\downarrow}&\ldots&\lambda_{t,\downarrow}\end{bmatrix} \tag{22}\]  
\[\text{FT1}_{;i}=\lambda_{;\downarrow} \tag{23}\]  
\[\text{FT2}_{;i}=[ft2_{t,i}]\text{, where }ft2_{t,i}=\frac{\sum_{s}\lambda_{ l}^{s}}{8760} \tag{24}\]  
\[\text{FT3}_{;i}=\sum_{j}[\lambda_{;\downarrow}\cdot pt3_{i,j}] \tag{25}\]  
\[\text{FT4}_{;i}=\sum_{j}[\lambda_{;\downarrow}\cdot pt4_{i,j}] \tag{26}\]  
\[\lambda_{;i}^{\prime}=\text{FT1}_{;i}+\text{FT2}_{;i}+\text{FT3}_{;i}+\text{FT4 }_{;i} \tag{27}\]  
When calculating the unavailability and failure probability of second order cut-sets it is necessary to adjust FT3 and FT4 to account for adjacency as was done in the approximate equations, for both lines \(x\) and \(y\), as seen in (28) and (29). A time series of dependency mode failure probability for the two lines in conjunction is calculated in (30). Again, the adjustments and dependency mode failure rate are equal to zero if the lines are not adjacent.  
\[\text{FT3}^{\prime}_{;x}=\text{FT3}_{;x}-\text{FT1}_{;y}\cdot pt3_{x,y} \tag{28}\]  
\[\text{FT4}^{\prime}_{;x}=\text{FT4}_{;x}-\text{FT1}_{;y}\cdot pt4_{x,y} \tag{29}\]  
\[\lambda_{;\downarrow}^{\text{D}}_{;\downarrow}(\text{x,y})=\text{FT1}_{;y} \cdot pt3_{x,y}+\text{FT1}_{;y}\cdot pt4_{x,y}+\text{FT1}_{;x}\cdot pt3_{y,x} +\text{FT1}_{;x}\cdot pt4_{y,x} \tag{30}\]  
For the cut-set itself, time-series of unavailability of individual lines with updated fault type values for FT3 and FT4 is calculated. The unavailability due to dependent faults is calculated as a separate time series, using restoration times according to the relevant line and the fault type. The calculation of unavailability is done using the method outlined in [10]. The result is a time series of the probability of the component being unavailable at a given point in time, which together with the unavailability due to the dependency mode failure probability is combined to the unavailability of the cut-set in (31). Once the time-series of unavailability is established, this is used to calculate the failure probability of the cut-set in (32).  
\[\text{U}^{\prime}_{;\downarrow}(\text{x,y})=\text{U}^{\prime}_{;x}\circ\text {U}^{\prime}_{;y}+\text{U}^{\text{D}}_{;\downarrow}(\text{x,y}) \tag{31}\]  
\[\lambda_{;\downarrow}^{\prime}=\text{U}^{\prime}_{;x}\circ\lambda_{;y}+\text {U}_{;y}\circ\lambda_{;x}+\lambda_{;\uparrow}^{\text{D}}(\text{x,y}) \tag{32}\]