# Fiducial Lower Confidence Limit of Reliability for a Power Distribution System
## 3 Results  
This section uses the datasets of the MOSFET and short circuit detector, which are the key components in a typical satellite intelligent PDS made by the Beijing Satellites Casting Factory, to illustrate the proposed method. The lifetimes of the MOSFET came from the accelerated lifetime test, and the lifetimes of the short circuit detector came from the simulated data. The datasets of the MOSFET and short circuit detector are shown in Table 5. The failure rates of the voltage detector, current detector, sense resistor and drive circuit were 0.2 \(\times\) 10\({}^{-9}\), 0.5 \(\times\) 10\({}^{-9}\), 0.4 \(\times\) 10\({}^{-9}\) and 0.6 \(\times\) 10\({}^{-9}\), respectively, which were known from experience. From Equation (13), the simulated reliability with 2000 repetitions was obtained. The estimates were averaged by 1000 simulation trials. The estimator of reliability at 131,400 h (15 years) was taken as the sample mean or the sample median of simulations. Then, we ranked the reliability values in order from small to large. The LCL of reliability at 15 years was the reliability of the ordered simulations. The results are shown in Table 6.  
From Table 6, we found that the mean of simulated reliability was 0.9960092, which was close to the LCL (0.9952635). Therefore, it was more appropriate to take the median (0.9989452) as the estimator of reliability. It is illustrated that the LCL of reliability at 15 years with a confidence level of 0.8 was 0.9952635. Figure 6 gives a plot of the estimator and the LCL of reliability for the SSPC. The real line represents the estimator of reliability for the SSPC. We took the median of the simulated sample as the estimator instead of the mean. The dashed line represents the LCL of reliability for the SSPC. \begin{table}
\begin{tabular}{c c c c} \hline
**Reliability** & **Estimator (Mean)** & **Estimator (Median)** & **Lower Confidence Limit** \\ \hline \(R(t_{0})\) & 0.9960092 & 0.9989452 & 0.9952635 \\ \hline \end{tabular}
\end{table} 
Table 6: Estimators and LCL of reliability for the SSPC (\(t_{0}\) = 131,400 h). Figure 6: Curves of the estimator and the LCL of reliability for the SSPC. From Figure 6, we can see the rate of decline was slow when t ranged from 131,400 h (15 years) to 350,400 h (40 years). The LCLs of reliability for the SSPC at 15 years and 40 years with a confidence level of 0.8 were 0.995 and 0.930, respectively. Both the estimator and the LCL of reliability for the SSPC were very high, which could meet the factory's requirement for reliability. Next, the reliability for the SSPC with hot standby equipment was calculated. The structure is shown in Figure 5. The estimators and the LCL of reliability for the SSPC with hot standby equipment could also be obtained by Monte Carlo simulation from Equation (14). The results are shown in Table 7.  
Table 7 illustrates that the estimator and the LCL of reliability at 15 years were 0.9999989 and 0.9999775, respectively. When comparing Table 6 with Table 7, it is shown that the reliability and the LCL were improved when the hot standby systems were adopted. In fact, if the cold or warm standby systems were adopted, the reliability was also improved similarly. Next, the reliability of the whole PDS with at least one SSPC was calculated. The estimators and the LCL of the whole PDS reliability with one SSPC subsystem were obtained by Equation (17) and are shown in Table 8.  
From Table 8, it is shown that the estimators and the LCL of reliability were 0.9988801 and 0.9988588, respectively. In the same way, we could calculate the LCL of reliability for a PDS with more than one SSPC. For example, Table 9 calculates the estimators and the LCL of PDS reliability at 15 years with 20 SSPCs. Figure 7 gives two curves of the estimator and the LCL of reliability for the PDS. The real line and dashed line represent the estimator and the LCL of reliability for the PDS, respectively. As shown in Figure 7, the rate of decline was slow when \(t\) ranged from 131,400 h (15 years) to 350,400 h (40 years).