# Fiducial Lower Confidence Limit of Reliability for a Power Distribution System
## Materials and Methods
### LCL Analysis of Reliability for a Typical PDS  
In Section 2.2 and Section 2.3, the pivotal quantities were derived for the lifetime of an exponential-type component and a Weibull-type component. In this subsection, we analyze the LCL of reliability for a typical satellite intelligent PDS. The structure of this PDS is shown in Figure 1. The SSPC is a key subsystem in this PDS. Thus, we analyzed the LCL of reliability for the SSPC first. The lifetime of the MOSFET was assumed to be Weibull-distributed with unknown parameters \(\eta\) and \(m\), and the lifetime of the short circuit protector was assumed to be exponentially distributed with an unknown failure rate \(\lambda_{S2}\). The lifetimes of the voltage detector, current detector, sense resistor and drive circuit in the SSPC were assumed to be exponentially distributed with known failure rates \(\lambda_{S3}\), \(\lambda_{S4}\), \(\lambda_{S5}\) and \(\lambda_{S6}\), respectively. Let \(R_{S}(t)\) be the reliability of the SSPC at time \(t\). \(R_{S1}(t)\), \(R_{S2}(t)\), \(R_{S3}(t)\), \(R_{S4}(t)\), \(R_{S5}(t)\) and \(R_{S6}(t)\) represent the reliability of the MOSFET, short circuit protector, voltage detector, current detector, sense resistor and drive circuit, respectively. Thus, \(R_{S}(t)\) can be expressed as follows:  
\[ \begin{split} R_{S}(t)&=R_{S1}(t)R_{S2}(t)R_{S3}(t )R_{S4}(t)R_{S5}(t)R_{S6}(t)\\ &=\exp\left\{-(t/\eta)^{m}-\lambda_{S2}t-\lambda_{S3}t-\lambda_{ S4}t-\lambda_{S5}t-\lambda_{S6}t\right\}\\ &=\exp\left\{-\left(t\exp\left\{\frac{E_{1}}{E_{2}}S-\overline{X }\right\}\right)^{\frac{E_{2}}{S}}-\frac{t}{2T}E-\sum\limits_{i=3}^{6}\lambda _{Si}t\right\},...\end{split}  \tag{13}\]  
where \(\overline{X}\) and \(S\) are the logarithm mean and the logarithm standard deviation of the sample lifetime for the MOSFET, respectively. \(E\), \(E_{1}\) and \(E_{2}\) are independent random variables. \(E\) follows an \(\chi^{2}\) distribution with \(2n_{2}\) (\(n_{2}\) is the sample size of the short circuit protector) degrees of freedom. \(E_{1}\) and \(E_{2}\) have the same distribution as the sample mean and standard deviation of \(W_{1}\), \(\cdots\), \(W_{n1}\) (\(W_{1}\), \(\cdots\), \(W_{n1}\) follow the standard extreme value distribution, and \(n_{1}\) is the sample size of the MOSFET). By simulating \(\chi^{2}\) random numbers and standard extreme value random numbers, the simulations of the pivotal quantities \(E\), \(E_{1}\) and \(E_{2}\) were derived. Then, the simulated \(R_{S}(t)\) was computed by Equation (13). The estimator and the LCL of \(R_{S}(t)\) for a single SSPC were taken as the sample mean (or sample median) and the sample quantile of the simulations. In order to improve the reliability of an SSPC, a standby system is usually adopted, especially a hot standby system, which is in parallel with the main subsystem. The main subsystem and the standby subsystem were independent and identically distributed. The structure of the SSPC is shown in Figure 5.  
Similar to Equation (13), let \(R_{SS}(t)\) represent the reliability of the SSPC subsystem, which is shown in Figure 5. \(R_{SS}(t)\) is a function of \(E\), \(E_{1}\) and \(E_{2}\):  
\[ \begin{split} R_{SS}(t)&=1-\left[1-R_{S}(t)\right] ^{2}\\ &=1-\left[1-\exp\left\{-\left(t\exp\left\{\frac{E_{1}}{E_{2}}S- \overline{X}\right\}\right)\right.^{\frac{E_{2}}{S}}-\frac{t}{2T}E-\sum\limits _{i=3}^{6}\lambda_{Si}t\right\}\right]^{2}.\end{split}  \tag{14}\]Then, we analyzed the LCL of reliability for a satellite intelligent PDS, which is shown in Figure 1. The cold standby subsystem was adopted for the TM/TC, whose lifetime was assumed to be exponentially distributed with a failure rate \(\lambda_{T}=338.58\times 10^{-9}\). This constant could be used directly when the reliability was calculated. Then the reliability of the TM/TC subsystem with one cold standby subsystem at time \(t\) can be expressed as  
\[R_{TT}(t)=e^{-\lambda_{T}t_{0}}(1+\lambda_{T}t). \tag{15}\]  
For the DC/DC, a hot standby system was adopted.