## 3 Reliability Modeling of PV Systems
### Reliability Modeling of a Central PV System  
The multi-state model shown in Table 1 is combined with a two-state model of central PV system components described in this section. The main components of a typical central PV system are illustrated in Figure 6. This central PV system consists of a solar array, bulk DC-link capacitance, inverter, line filter, AC switch, AC circuit breaker and transformer. The reliability analysis can be performed at the component level or system level. The functional block diagram of this PV system is shown in Figure 7.  
The reliability research of power electronic components has focused on failure rate models of conductors, capacitors and magnetic devices [17; 18]. However, field experience has demonstrated that electrolytic capacitors and switch devices are the most vulnerable components [19]. There are several reliability models available for power electronic and high power voltage. MIL-HDBK-271F second edition is the military handbook for reliability component prediction for power electronic components [20]. This handbook provides an extensive reliability database for power electronic components. This database is used in this work to evaluate the failure rate of power electronic components. Then, the results of this step are used in the reliability analysis at the system level. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**States** & **Capacity (pu)** & **Probability** & **States** & **Capacity (pu)** & **Probability** \\ \hline
1 & 0 & 0.4774 & 12 & 0.525 & 0.023 \\
2 & 0.00417 & 0.0763 & 13 & 0.575 & 0.0223 \\
3 & 0.0375 & 0.0313 & 14 & 0.625 & 0.0255 \\
4 & 0.1042 & 0.0217 & 15 & 0.675 & 0.0300 \\
5 & 0.175 & 0.0195 & 16 & 0.725 & 0.0274 \\
6 & 0.225 & 0.0252 & 17 & 0.775 & 0.0239 \\
7 & 0.275 & 0.0212 & 18 & 0.825 & 0.0223 \\
8 & 0.325 & 0.0201 & 19 & 0.875 & 0.0244 \\
9 & 0.375 & 0.0172 & 20 & 0.925 & 0.0193 \\
10 & 0.425 & 0.0219 & 21 & 0.975 & 0.0182 \\
11 & 0.475 & 0.0235 & 22 & 1 & 0.0081 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Multi-state Model of PV capacity without considering PV system topology. Figure 5: The power output of solar cell. Solar panel  
Solar panels consist of solar cells. These solar cells have a very low failure rate. Most manufactures offer a warranty from 20 to 25 years on their solar module [21] and have demonstrated very high reliability in the field with mean time between failure (MTBF) of 522 and 6666 years for residential and utility systems, respectively [22]. The effect of PV architecture on the overall system reliability is therefore not considered on this work. The failure rate of capacitors is considered one of the major factors leading to the failure of PV systems. Capacitances can be made of different materials such as an electrolytic capacitor, paper, plastic film, tantalum and ceramic [23]. Each material has a different failure rate [23]. The electrolytic capacitor is used in this work. The inductance is not considered since it has a low failure rate [24]. Equations (14)-(16) are used to evaluate the failure rate of capacitance (\(\lambda_{cap}\)) [25; 26; 27], where \(n\) is the total number of components in the system; \(\lambda_{base}\) is the base failure rate of capacitance and equal to 0.0314 occur/year; \(\tau_{E}\) is the effect of environmental stress and is equal to 1; \(\tau_{Q}\) is the quality factor and is equal to 1; \(C\) is the capacitance value in microfarad (\(\mu\)F); and \(T_{j}\) is the junction temperature, which is 50 \({}^{\circ}\)C. The total failure rate of DC-link capacitor is 0.4449 occur/year. \[\lambda_{cap}=n*\lambda_{base}*\tau_{Cv}*\tau_{Q}*\tau_{E}*\tau_{T} \tag{14}\]  
\[\tau_{cv}=0.34*C^{0.18} \tag{15}\]  
Figure 6: Schematic of a typical central PV system. Figure 7: Functional block diagram of a central PV system. \[\pi_{T}=\exp^{-4061.74*(\frac{1}{T_{j}+2\gamma}-\frac{1}{198})} \tag{16}\]  
Inverter  
The MTBF of PV-inverter is between 1 to 16 years [22]. Inverter is considered as another major factor leading the failure of PV systems.