## 2 Methods for Determining the Number of Power Line and Transformer Outages
### Factors Influencing the Transformer Outage Rate  
Transformers are a group of devices whose repair and renewal process most often takes place at the transformer's place of operation. Only when significant damage or wear  
Figure 1: Percentage of factors influencing damage to 110 kV lines according to [96]. Figure 2: Percentage of the most frequent damages of 110 kV lines according to [16]. occurs does a complete replacement become necessary. According to [43; 98], the most common causes of failure include damage to the tap changer or damage to the Buchholz relay. A graphical summary of factors influencing the occurrence of transformer failures is presented in Figure 3, while the percentage summary of the failure frequency in transformers is presented in Figure 4. Regarding the factors causing failures, as much as 51% were aging processes, and 18% were caused by lightning. [PERSON] published the first major study on failures and reliability factors for power transformers in 1983 [68]. The research conducted by him and his team shows that the average failure rate for transformers is 2%, regardless of the voltage category. The authors of [99] disagree with this statement. According to the results obtained by them, the failure rate due to aging in the case of smaller power transformers (up to 100 MVA) is less than 6% for transformers up to 25 years old, while it is less than 10% for those above 25 years. In the case of larger power transformers (above 100 MVA), the failure rate is insignificant (below 1%) in the first 10 years, while with age, the failure rate increases to almost 18%. It should be noted, however, that the studies conducted by [PERSON], [PERSON], and [PERSON] were conducted on a much smaller number of transformers. Methods Used in the Literature to Determine the Number of Outages of Transmission Lines and Transformers  
The number of outages (failure rate) can be defined as the number of random events (failures) that prevent the execution of a task, divided by the operating time of the device. To determine the reliability properties of individual network elements, it is crucial to determine their parameters and functions. Determining the reliability function \(R(t)\), unreliability function \(F(t)\), failure probability function\(f(t)\), as well as failure intensity functions \(\overline{\lambda}\) and the expected uptime function \(r(t)\) is essential for comprehensive reliability analysis [43; 59]. One defines the reliability function through the following relationship:  
\[R(t)=\frac{n(t)}{N} \tag{2}\]  
Figure 4: Percentage of the most common transformer failures by [98; 101; 102; 103]. Figure 3: Percentage of factors influencing transformer damage according to [100]. where \(n(t)\) is the number of elements that have not failed in the time interval \((0,t>)\) and \(N\) is the number of elements that are included in the system. Researchers define the unreliability function through the following relationship:  
\[F(t)=\frac{m(t)}{N} \tag{3}\]  
where \(m(t)\) is the number of elements that have failed in the time interval \((0,t>)\), and \(N\) is the number of elements that are included in the system. Researchers define the failure intensity function through the following relationship:  
\[\overline{\lambda}=\frac{2\cdot m}{(n_{p}+n_{k})\cdot\Delta t} \tag{4}\]  
where \(m\) is the observed number of failures in the time interval \(\Delta t\), \(n_{p}\) is the number of elements at the beginning of the observation period, \(n_{k}\) is the number of elements at the end of the observation period, and \(\Delta t\) is the total time of the observation.