## 2 Methods for Determining the Number of Power Line and Transformer Outages
### Factors Influencing the Time of Failure Removal
Due to the large number of transformers operating in the transmission network (several hundred units), as well as the low damage rate, statistically around 0.11 disturbance per 100 units, for illustrative purposes, the occurrence of 1 damage per year was assumed for the example calculations. The transformer reliability function can then be defined using the relationship (2) presented, among others, in [59]:  
\[R(t)=\frac{n(t)}{N}=\frac{212}{213}=0.99 \tag{44}\]  
The unreliability function can be defined using the following relationship:  
\[F(t)=\frac{m(t)}{N}=\frac{1}{213}=0.0047 \tag{45}\]  
The authors determined the average failure intensity for a specific year, knowing the number of transformer failures that occurred in a given year, as well as the population size at the beginning and end of the period under study, using the relationship (4):  
\[\overline{\lambda}=\frac{2\cdot m}{(n_{p}+n_{k})\cdot\Delta t}=\frac{2\cdot 1 }{(213+213)\cdot 1}=0.0047 \tag{46}\]  
In order to calculate the transmission line failure rate, the authors used, among others, the formula proposed by [PERSON] et al. For this purpose, the authors assumed a month duration of 30 days and an average length of a 110 kV line of 11.3 km. The average annual number of failures, according to the obtained data, was 5 failures per year. \[\lambda_{k}(x)=\frac{\sum_{y^{a}kyx}}{YT_{x}L_{k}}\times 100=\frac{5}{9\cdot 30 \cdot 11.3}\times 100=1.64 \tag{47}\]  
The average monthly failure rate of a 110 kV line based on the data obtained by the authors was 1.64 failures per 100 km. If the lines are at the same voltage levels, the following relationship can be used:  
\[\lambda(x)=\frac{\sum_{k}(\lambda_{kx}\times L_{k})}{\sum_{k}L_{k}} \tag{48}\]  
The data needed to calculate this indicator are the average data for 110 kV lines obtained by the authors. Then, taking into account 5 lines with a voltage of 110 kV with lengths of 6 km, 10 km, 15 km, 20 km, and 25 km, the average annual reliability indicator of a section consisting of 5 lines is 1.13 failures per 100 km of line. Based on Formula (1) and data presented in [56], the authors assumed an average failure rate of 0.92 in the case of variant 1 shown by [PERSON] et al., and 0.97 in the case of variant 2. The failure rate ranged from 0.5881 to 1.6431 in the case of the first variant and from 0.4271 to 1.7500 in the case of the second variant. In the case of 220 kV lines in the German transmission network, the reliability factor is between 0.5 and 3.2 failures per 100 km. In the case of 400 kV lines, the factor was 2.1 failures per 100 km, while in the case of 110 kV lines it is 1.6 failures per 100 km [44]. In the case of 110 kV lines in the Austrian power system, the failure factor is 1.29 failures per 100 km of line. In the case of 220 kV lines, the failure factor is 0.53 failures per 100 km of line. In the case of the Japanese transmission network, the failure factor is between 0.52 and 1.64 failures per 100 km of line. In the case of the UK power system, the authors of [123] present data from 3 different operators. The data they obtained demonstrate that lightning discharges and gusty winds exerted the greatest influence on the reliability of transmission lines. The reliability index of transmission lines ranged from 0.55 to 1.34 outages per 100 km of line, depending on the Operator. The authors also compared the coefficients on other continents. The average index for transmission lines without dividing into voltage levels in the United States should be assumed to be at the level of 0.6 outages per 100 km of line [128]. In the case of South America, the country where the information was obtained was Chile, and the reliability index for transmission lines without dividing into voltage levels is 2.66 outages per 100 km of line [129]. In the case of Africa, the Republic of South Africa was used as a reference point, where this coefficient is 3.13 outages [124].