## 3 Reliability Models of Electric Distribution System Components
### Distributed Generation Sources
#### 3.2.1 Conventional Energy Sources  
The conventional electric energy sources are:  
* Engine-driven generators,
* Turbine-driven generators,
* Microturbines,
* Fuel cells. Depending on the type of service, the aforementioned energy sources can be modeled as the following Markov chain:  
* Two-state model--in case of continuous service; determined by the failure rate \(\lambda\) and the repair rate \(\upmu\),
* Four-state model--in case of peak service; shown in Figure 1  
Both the standby anticipation rate \(\rho\) and operation rate \(\
u\) should be determined individually depending on the analyzed electric distribution system. The reliability parameters of engine-driven generators (EDG) and turbine-driven generators (TDG) can be found in Table 2. The values of parameters \(\
u\) and \(\rho\) have been arbitrarily selected. One can observe a much lower failure rate \(\lambda\) for generation units (both EDG and TDG) operating in peak service. This fact obviously results from less wear of individual components of a generation unit. Therefore, the possibility of failure is decreased. The TDG exhibit the lowest failure rates of units in peak service. Simultaneously, the lower repair rate is observed in comparison to other cases. It is attributable to a relatively small number of long-duration events. In recent years, a new type of gas turbine, microturbine (MT), has become a fully developed technology. As MT's have only relatively recently been used as the commercial generation sources, there is not wide access to reliability data obtained from a long-time operation of this DG type. The same problem concerns the fuel cells (FC) as a relatively new technology in an industrial and commercial usage. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Distribution System Component**} & \multicolumn{2}{c}{**Probability Distribution of Time to Failure**} & \multicolumn{2}{c}{**Probability Distribution of Time to Repair**} \\ \cline{2-5}  & **Type** & **Parameters** & **Type** & **Parameters** \\ \hline Overhead lines (100 km) & EXP & \(\lambda=25.0\,1/\text{a}\) & EXP & \(\lambda=0.071\,1/\text{h}\) \\ Underground cable with paper-insulation & WEI & \(\lambda=12.0\,1/\text{a}\) & \(\beta=2.24\) & EXP & \(\lambda=0.69\,1/\text{h}\) \\ used in urban networks (100 km) & & & & & \\ Underground cables with polythene-insulation & WEI & \(\lambda=12.0\,1/\text{a}\) & \(\beta=1.85\) & EXP & \(\lambda=0.72\,1/\text{h}\) \\ used in urban networks (100 km) & & & & & \\ Underground cables with paper-insulation & WEI & \(\lambda=13.0\,1/\text{a}\) & \(\beta=2.33\) & EXP & \(\lambda=0.30\,1/\text{h}\) \\ used in rural networks (100 km) & & & & & \\ Underground cables with polythene-insulation & & & & & \\ used in rural networks (100 km) & & & & & \\
110 kV/MV transformers & EXP & \(\lambda=0.06\,1/\text{a}\) & EXP & \(\lambda=0.08\,1/\text{h}\) \\ Circuit-breakers & EXP & \(\lambda=0.132\,1/\text{a}\) & NOR & \(\mu=5.5\,\text{h}\) & \(\sigma=1.0\,\text{h}\) \\ Disconnectors & EXP & \(\lambda=0.0055\,1/\text{a}\) & EXP & \(\lambda=0.115\,1/\text{h}\) \\ Busbars in MV switchgear (bay) & EXP & \(\lambda=0.003\,1/\text{a}\) & EXP & \(\lambda=0.10\,1/\text{h}\) \\ Current transformers & EXP & \(\lambda=0.009\,1/\text{a}\) & EXP & \(\lambda=0.047\,1/\text{h}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reliability parameters of selected types of MV distribution system elements; elaborated on the basis of [59,60]. As in case of EDG and TDG, MT and FC can be modeled using a two-state or four-state Markov chain with the failure rate \(\lambda\) and repair rate \(\mu\) as well as a probability of unsuccessful DG unit starting \(P_{S}\). For the reliability calculation purpose \(\lambda\), \(\mu\) and \(P_{S}\) have been obtained from the manufacturers data available only for peak service. The same values of parameters \(\
u\) and \(\rho\) as for EDG and TDG have been assumed.