## 2 Methods for Determining the Number of Power Line and Transformer Outages
### Factors Influencing the Transformer Outage Rate
In [57], the authors proposed using historical data to calculate the failure rate of a transmission line:  
\[\lambda_{k}(x)=\frac{\sum_{\mu\
eq kyx}}{YT_{x}L_{k}}\times 100\ x=1,2,\ldots,12 \tag{5}\]  
where \(\lambda_{k}(x)\) is the line failure rate in a specific historical month, \(n_{kyx}\) is the line failure rate in a given historical month, \(T_{x}\) is the month, \(Y\) is the total number of years taken into account in the calculations, and \(L_{k}\) is the line length expressed in kilometers. A similar relationship can also be used to calculate the failure rate for transmission lines with the same voltage level, as well as with similar meteorological conditions in historically corresponding months. This relationship will then appear as follows:  
\[\lambda(x)=\frac{\sum_{k}(\lambda_{kx}\times L_{k})}{\sum_{k}L_{k}} \tag{6}\]  
where \(\lambda(x)\) is the transmission line failure rate with the same voltage level in its historically corresponding month. The failure rate of a transmission line can also be calculated using the failure distribution function. When a month is assumed as the time interval scale, the failure rate can be determined by the relationship:  
Figure 2:0  
where \(\lambda(x)\) is the line failure rate in a specific historical month, \(\lambda{ave}\) is the average failure rate over several statistical years, while \(\lambda^{\prime}{ave}\) is the average monthly failure rate, \(f(x)\) is the normalized time distribution function of the failure rate in the corresponding historical month \(x\). Transmission line reliability can also be determined using the Markov distribution [57; 104]:  
Figure 2:1  
where \(\lambda(x)\) is the failure rate in month \(x\), and \(\mu\) is the repair rate. The authors of [56] determine the reliability coefficient of transmission lines using the following relationship:  
Figure 2:2  
where \(L_{i}\) is the length of the line, and \(f_{i}\) is the number of failures of the tested line during the year. An interesting approach was proposed by the authors of [43,45,100], where they attempted to determine the empirical form of the variability of the frequency of failures of 110 kV lines using a fourth-order polynomial. For this purpose, the authors used the following relationship:  
Figure 2:3  
where \(i\) is the consecutive number of the month, while \(a\), \(b\), \(c\), \(d\), \(e\) are the coefficients of the approximating function. In the article, their values were determined based on historical data. The values of these coefficients are \(a=0.0210\), \(b=-0.6306\), \(c=6.5081\), \(d=-25.9092\), and \(e=37.9207\). In addition, the author also proposed a logarithmic-normal distribution for the time of emergency power line outages. This distribution has the following form:  
Figure 2:4  
where \(m\) is the expected value of the random variable \(\log t_{\text{wa}}\), and \(\sigma\) denotes the standard deviation of the random variable \(\log t_{\text{wa}}\). The determined values of these parameters are \(m=2.1244\) and \(\sigma=1.0947\). Based on the conducted analyses, the authors also determined the hypothetical distribution of undelivered electricity as a result of power line outages. This distribution is exponential. To verify the hypothesis concerning the distribution types, the [PERSON] \(\lambda\) test and [PERSON]'s chi-squared test were employed \(\chi^{2}\). In [105] the authors use the following relationship to determine the failure rate of the transmission line in Nigeria:  
Figure 2:5  
where \(F_{\text{oi}}\) is the number of recorded failures, \(T_{\text{DT}}\) is the total interruption, and \(T_{\text{OH}}\) is the time of correct operation. In [11] the authors use the Weibull distribution to estimate the failure rate of 110 kV lines depending on the prevailing weather conditions:  
\[\lambda(t)= \begin{cases}\lambda_{I},&0\leq V(t)<T_{U}\\ \lambda_{C},&T_{U}\leq V(t)<T_{w}\\ \infty,&V(t)\geq T_{w}\end{cases}