## 2 Methods for Determining the Number of Power Line and Transformer Outages
### Factors Influencing the Time of Failure Removal  
The factors influencing the time of failure removal include primarily the type and extent of the failure, as well as the technical capabilities of the operating teams of the operators in whose area the damage occurred. Factors influencing the time of failure removal will also be the availability of spare parts and weather conditions in which maintenance work will have to be carried out. Methods Used in the Literature for Determining the Time of Failure Removal in Transmission Lines and Transformers  
An important factor that should be taken into account in the case of determining the time of failure removal is the determination of the diagnostic time [117]. Because, as a result of diagnostics, the damage to the object is identified, and only after a thorough diagnosis is it possible to start repairing the components. The total time of failure removal \(T_{a}\) can therefore be divided into the time spent on diagnosis \(T_{d}\) and the time spent on repairing damaged elements \(T_{n}\):  
\[q=\frac{T_{a}}{T}=\frac{T_{d}+T_{n}}{T} \tag{31}\]  
Figure 5: Failure rate of transformers about age [112]. In [71; 118] the following relationship is used to determine the average annual repair time of transformers:  
\[AART=\frac{TRT}{N\times T} \tag{32}\]  
where \(AART\) is the average annual repair time of the transformer, \(TRT\) is the total repair time in the period under study, \(N\) is the average number of transformers accepted for testing, and \(T\) is the period from which the data was accepted for testing. In [119; 120] fuzzy logic is used to estimate the time of failure removal (\(r_{\text{E}}\)), as well as to determine the outage rate (\(\lambda_{\text{E}}\)) within a given timespan. The authors use a general relationship for this purpose, which allows them to determine these quantities:  
\[ \begin{array}{l}\lambda_{\text{E}}=\lambda_{\text{N}}\cdot R+\lambda_{ \text{A}}\cdot(1-R)\\ r_{\text{E}}=\frac{\lambda_{\text{N}}\cdot R\cdot\gamma_{\text{N}}+\lambda_{ \text{A}}\cdot(1-R)\cdot r_{\text{A}}}{\lambda_{\text{N}}\cdot R+\lambda_{ \text{A}}\cdot(1-R)}\end{array}  \tag{33}\]  
where \(\lambda_{\text{N}}\), \(r_{\text{N}}\), \(\lambda_{\text{A}}\) and \(r_{\text{A}}\) denote, respectively, failure rates and failure removal times of the line in normal states (index \(N\)) and unfavorable weather conditions (index \(A\)). Individual failure rates \(\lambda\) and failure removal times \(r\) can then be calculated from the relationship:  
\[ \begin{array}{l}\lambda=\left[\lambda_{1},\lambda_{2}\right]=\left[X_{1- \alpha/2}^{2}\cdot(2\cdot m)/(2\cdot T),\right.\\ X_{\alpha/2}^{2}\cdot(2\cdot m+2)/(2\cdot T)\right]\\ r=\left[r_{1},r_{2}\right]=\left[r_{\text{avg}}-z_{\alpha/2}\cdot s/\sqrt{n}, \right.\\ r_{\text{avg}}-z_{\alpha/2}\cdot s/\sqrt{n}]\end{array}  \tag{34}\]  
where \(m\) is the number of line outages in time \(T\), \(r_{\text{avg}}\) denotes the average time of failure removal (based on \(n\) historical data), \(\alpha\) is a given significant level, \(z_{\alpha/2}\) is such a value that the integral of the standard normal density function from \(z_{\alpha/2}\) to \(\infty\) equals \(\alpha/2\), and \(s\) is the standard deviation. The total time to remove failures in transmission lines is always much shorter than a period of one year. Therefore, the interruption rates and the numerical frequency of the interruption are very similar to each other and are often used interchangeably in the reliability assessment.