# Reliability of Supply and the Impact of Weather Exposure and Protection System Failures +
## 3 Method
### Approximate Method  
This approach starts by calculating the equivalent failure rate of each line by considering each line a target line. The equivalent failure rate is calculated based on the failure rate of the line itself (FT1) and its protection systems (FT2), and the fault types FT3 and FT4 propagating from adjacent lines, following the method in [17]. FT1 and FT2 are only dependent on information of the line itself and is repeated for reference here (7) and (8). FT3 and FT4 are calculated by using the associated probability matrices (9) and (10). The multiplication of the matrices with the line failure rate vectors gives vectors of FT3 and FT4 failure rate contributions from all source lines at the target line. \[f\mathbf{t1}_{i}=\mathbf{\lambda}_{I} \tag{7}\]  
\[f\mathbf{t2}_{i}=\mathbf{\sum_{s}}\mathbf{\lambda}_{I}^{s} \tag{8}\]  
\[f\mathbf{t3}_{i}=\text{PT3}\cdot\mathbf{\lambda}_{I} \tag{9}\]  
\[f\mathbf{t4}_{i}=\text{PT4}\cdot\mathbf{\lambda}_{I} \tag{10}\]  
Equivalent failure rate, incorporating protection system failures is then calculated as \(\lambda^{\prime}\) in (11). \[\mathbf{\lambda^{\prime}}=f\mathbf{t1}+f\mathbf{t2}+f\mathbf{t3}+f\mathbf{t4} \tag{11}\]  
From this, equivalent unavailability and outage durations for all target lines is found in (12) and (13), where the vectors of fault type contributions are paired with vectors of the associated repair or switching times using element-wise Hadamard operations (\(\circ\) signifies the Hadamard product, where \(c_{ij}=a_{ij}\cdot b_{ij}\), and \(\oslash\) is used for Hadamard division, see e.g., [35] for an introduction). A slight adjustment is made when incorporating unavailability due to FT3. As opposed to the other fault types, FT3 is dependent on the switching time at the source line, \(j\), rather than that at the target line \(i\). To include this, the contribution of the unavailability from the source line to target line is carried through the probability matrix for FT3. \[\mathbf{U^{\prime}}=f\mathbf{t1}\circ\mathbf{r^{FT1}}+f\mathbf{t2}\circ\mathbf{r^{ FT2}}+\text{PT3}\cdot[\mathbf{\lambda}\circ\mathbf{r^{ FT3}}]+f\mathbf{t4}\circ\mathbf{r^{FT4}} \tag{12}\]  
\[\mathbf{r^{\prime}}=\mathbf{U^{\prime}}\oslash\mathbf{\lambda^{\prime}} \tag{13}\]Vectors containing reliability indices for single lines are now established. Second order cut-sets involving two lines \(x\) and \(y\) can now be calculated in a general form, avoiding a distinction in equations between adjacent and non-adjacent lines by utilizing the matrix of adjacency adjusted probabilities. If two lines in a cut-set are adjacent and they experience a FT3 or a FT4 where the source line is the other line in the cut-set, they will both surely be unavailable. This means that these dependent failures should be treated separately from the independent failures. Adjustments for dependent failures between individual lines in the cut-set are created in (14) and (15) before calculating the new expected restoration time of the cut-set due to independent failures in (16). \[\lambda_{x}^{a}=\left(pt3_{x,y}+pt4_{x,y}\right)\cdot ft1_{y} \tag{14}\]  
\[U_{x}^{a}=\left(pt3_{x,y}\cdot r_{y}^{FT3}+pt4_{x,y}\cdot r_{x}^{FT4}\right) \cdot ft1_{y} \tag{15}\]  
\[r_{x}^{n}=(U_{x}^{\prime}-U_{x}^{a})/(\lambda_{x}^{\prime}-\lambda_{x}^{a}) \tag{16}\]  
However, to account for both lines in the cut-set failing simultaneously due to the occurrence of a FT3 or FT4 of an adjacent line also in the cut-set, an added dependency mode failure rate, \(\lambda^{D}\), is created in (17). The dependency mode failure rate and the expected unavailability due to these dependent events is used to calculate the associated expected restoration time in (18). It is important to note that if the lines are not adjacent, the elements \(\{x,y\}\) and \(\{y,x\}\) in the probability matrices will be zero, and hence all adjustments and the dependency mode failure rate will be zero. The failure rate of the cut-set will be calculated on the basis of independent component failures if the components are not adjacent.