## 2 Methods for Determining the Number of Power Line and Transformer Outages
### Factors Influencing the Time of Failure Removal
Therefore, according to [119], the total failure removal time can also be calculated from the relationship:  
\[U_{E}=\frac{\lambda_{E}\cdot r_{E}}{8760} \tag{35}\]  
In [121] the authors propose a relationship that allows determining the line renewal time depending on weather conditions:  
\[r(t)=w_{w}(t)\times w_{d}(d)\times w_{h}(t)\times r \tag{36}\]  
where \(w_{w}\) is the weather weighting factor dependent on the current conditions, \(w_{h}\) is the hourly weighting factor established based on experience in repairs and switching, \(\tau\) is the renewal time in normal weather conditions, and \(w_{d}\) is the weekly weighting factor established based on experience in repairs and switching. In [122] the authors adopt a similar approach to determining the failure removal time in overhead lines, dependent on weather conditions:  
\[r(t)=f_{w}(w(t),N_{g}(t))f_{d}(t)f_{h}(t)r_{norm} \tag{37}\]  
where \(f_{w}(w(t),N_{g}(t))\) are constants dependent on weather conditions, \(f_{h}(t)\) is a constant dependent on hourly changes, \(f_{d}(t)\) is a constant dependent on daily changes, and \(r_{norm}\) is the reference renewal time in normal weather conditions. In [50] the following relations are used to determine the failure recovery time coefficient:  
\[r_{SL}=\frac{\lambda_{fe}r_{fe}+\lambda_{e}^{\prime\prime}r_{e}^{\prime\prime}}{ \lambda_{SL}} \tag{38}\]  
where \(\lambda_{fe}\) is the general forced interruption index in the system calculated from the relation  
\[\lambda_{fe}=\sum\limits_{i=1}^{n}\lambda_{fi} \tag{39}\]  
\(r_{fe}\) is the expected interruption value for the system calculated from the relation  
\[r_{fe}=\frac{\sum\limits_{i=1}^{n}\lambda_{fi}r_{i}}{\lambda_{fe}} \tag{40}\]  
\(\lambda_{SL}\) is the annual interruption index calculated from the relation  
\[\lambda_{SL}=\lambda_{fe}+\lambda_{e}^{\prime\prime} \tag{41}\]  
\(\lambda_{e}^{\prime\prime}\) is the maintenance interruption index calculated from the relation  
\[\lambda_{e}^{\prime\prime}=\sum\limits_{i=1}^{n}\lambda_{i}^{\prime\prime} \tag{42}\]  
and \(r_{e}^{\prime\prime}\) is the expected value of maintenance interruption calculated from the relation  
\[r_{e}^{\prime\prime}=\frac{\sum\limits_{i=1}^{n}\lambda_{i}^{\prime\prime}r_ {i}^{\prime\prime}}{\lambda_{e}^{\prime\prime}} \tag{43}\]  
Table 1 summarizes the methods used to determine the failure rate and failure removal time. \begin{table}
\begin{tabular}{c c c} \hline
**Method Used** & **Continent** & **References** \\ \hline \multirow{5}{*}{Empirical formula} & Europe & [2; 3; 4; 5; 6; 7; 89; 90; 110; 111; 112; 113; 114; 115; 116; 117; 118; 119; 120; 121; 122; 123; 124; 125; 126; 127] \\ \cline{2-3}  & Asia & [5; 8; 30; 31; 44; 56; 108; 110; 118; 122] \\ \cline{2-3}  & North America & [44; 50; 103; 113; 128] \\ \cline{2-3}  & Africa & [49; 71; 80; 105; 118; 129] \\ \cline{2-3}  & Australia and Oceania & [70; 76] \\ \hline \multirow{5}{*}{Weibull distribution} & Europe & [11; 81] \\ \cline{2-3}  & Asia & [50; 54] \\ \cline{2-3}  & North America & [82] \\ \hline \multirow{5}{*}{Markov processes} & Asia & [51; 52; 53; 55; 65] \\ \cline{2-3}  & North America & [58; 62; 82] \\ \cline{2-3}  & Australia and Oceania & [64] \\ \hline \multirow{2}{*}{Poisson distribution} & Asia & [97; 99] \\ \cline{2-3}  & North America & [98] \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of methods used to determine the failure rate and failure removal time. Methods of Determining the Reliability Factor and the Time of Failure Removal of HV Lines and Transformers  
The authors analyzed the relationships available in the literature and discussed about them in the previous sections of this article. In this section, the data obtained by the authors, concerning transmission lines, were used, while the data on the number of transformers operating in individual years were obtained based on the report of the Polish Society for Transmission and Distribution of Electricity, as well as data provided by the Polish Power Grids [130,131].