# A Predictive Reliability Model to Assess the Performance of Photovoltaic Systems
## 3 Results and Discussion
### Results of Reliability Analysis  
Based on the results obtained from the simulation experiments, the PV system efficiency depends mainly on the ambient temperature, and the produced energy depends mainly on the global effective radiation. However, the PV system performance ratio (\(PR\)) depends on multiple factors. Thus, \(PR\) is the PV system response selected for reliability analysis. The ReliaSoft Weibull++ software estimated a probability density function of \(PR\) with time \(f(t)\) to be an exponential function given by:  
\[f(t)\ =\ \lambda e^{-\lambda(t\ -\ \gamma)},\ f(t)\ \geq\ 0,\ \lambda\ >\ 0,t\ \geq\ \gamma \tag{8}\]  
While the scale parameter \(\lambda\) indicates the stretch of the curve along the time axis, the location parameter \(\gamma\) indicates the shift of the curve from the origin. For exponential pdf, \(\lambda\) is related to \(\gamma\) by:  
\[\lambda=\frac{1}{m-\gamma} \tag{9}\]  
where \(t\) is the operating time (or life) and \((m-\gamma)\) is the mean of the pdf [21]. From Equations (4) and (8), \(R(t)\) is given by:  
\[R(t)\ =\ \int\limits_{t}^{\infty}\lambda e^{-\lambda(k\ -\ \gamma)}dk\]  
\[R(t)=e^{-\lambda(t-\gamma)}\]  
Substitute for \(\lambda\) from Equation (9),  
\[R(t)\ =\ e^{-\frac{(t\ -\ \gamma)}{(m\ -\ \gamma)}} \tag{10}\]  
For experiment 1 (\(30^{\circ}\), \(0^{\circ}\)) and experiment 2 (\(30^{\circ}\), \(20^{\circ}\)) of the near Cairo region, the obtained value of the parameter \(\gamma\) is zero, and hence, the pdf is called the 1-paramter (1-p) distribution, and the corresponding reliability function is given by:  
\[R(t)\ =\ e^{-\frac{1}{m}}\]  
However, for the location parameter, \(\gamma=1\) for the remaining four experiments. Thus, the distribution is a 2-p distribution, and the corresponding reliability function \(R(t)\) is given by:  
\[R(t)=e^{\frac{-(t-1)}{(m-1)}}\]  
* The reliability functions \(R(t)\) of the PV system performance ratio for the six experiments are shown in Figure 6 with an assumed target reliability of 90%, indicated by the red horizontal line. * As an example, the reliability at time \(t=100\) months is given in the first row of Tables 6 and 7 for the two regions. This value is greater for near Cairo for the two experiments (azimuth angles \(0^{\circ}\), \(20^{\circ}\)), while it is greater for near Riyadh for the remaining experiments. * The values of the \(Mean\_Life\) are shown in the second row of Tables 6 and 7 for the two regions. The obtained \(Mean\_Life\) values of the near Cairo region are 768.4, 772.2 and 659.4 months for experiments 1, 2 and 3, while those of near Riyadh are 685.63, 687.59 and 688.36, respectively. Thus, the field orientation (\(30^{\circ}\), \(20^{\circ}\)) gives the longest time for the near Cairo region, and the field orientation (\(30^{\circ}\), \(30^{\circ}\)) gives the longest lifetime for the near Riyadh region. The lifetimes of the systems in near Cairo are longer for orientations (\(30^{\circ}\), \(0^{\circ}\)) and (\(30^{\circ}\), \(20^{\circ}\)), while that of near Riyadh is longer for orientation (\(30^{\circ}\), \(30^{\circ}\)). The _Failure_Rate_ equals the reciprocal of the \(Mean\_Life\). The failure rate is given by the parameter \(\lambda\), and the exact values are shown in the third row of Tables 6 and 7 for both regions. If we consider an acceptable limit of reliability to be 90%, then the \(Reliable\_Life\) for the six experiments are less than their \(Mean\_Life\) as shown in the fourth row of Tables 6 and 7. This result is logic as the reliability at the \(Mean\_Life\) is usually much less than 90%. Considering a reliability of 50%, the system may have working reliably for 532.63, 535.28 and 457.38 months for the three orientations in the near Cairo region as shown in the fifth row of Table 6. This also shows that the field orientation (30\({}^{\circ}\), 20\({}^{\circ}\)) is the best among the three orientations for near Cairo, as mentioned above, as it gives the longest time for a 50% reliability of the performance ratio.