# Fiducial Lower Confidence Limit of Reliability for a Power Distribution System
## Materials and Methods
### Fiducial LCL of Reliability for an Exponential-Type Component  
In this subsection, we discuss the LCL of reliability based on a typical component whose lifetime follows the exponential distribution commonly used in a PDS. Except for the MOSFET, the lifetimes of the other components in an SSPC were assumed to be exponentially distributed. Thus, we derived the LCL of reliability for an exponential-type component first. Let \(T\) be a random variable which follows an exponential distribution with an unknown parameter \(\lambda\). The cumulative distribution function of \(T\) is  
\[F(t)=1-e^{-\lambda t},\ t>0. \tag{1}\]  
Suppose \(T_{1}\), \(T_{2}\), \(\cdots\), \(T_{n}\) is a random sample from the population \(T\), where \(n\) is the sample size. Then, the total test time \(T_{sum}=T_{1}+T_{2}+\cdots+T_{n}\) follows a Gamma distribution with parameters \(n\) and \(\lambda\). Therefore, \(2\lambda T_{sum}\) follows an \(\chi^{2}\) distribution with \(2n\) degrees of freedom. Then, parameter \(\lambda\) can be represented as  
\[\lambda=\frac{E}{2T_{sum}},\ E\sim\chi^{2}(2n), \tag{2}\]  
where \(E\) is the pivotal quantity which follows a completely known distribution. We can derive the estimator and the confidence interval (CI) of parameter \(\lambda\) by solving the expectation and the quantile of \(\chi^{2}\) distribution, respectively, or by generating a random number. The CI for \(\lambda\) with the confidence level is given by  
\[\left[\chi^{2}_{\alpha/2}(2n)/2T_{sum},\chi^{2}_{1-\alpha/2}(2n)/2T_{sum} \right], \tag{3}\]  
where \(\chi^{2}_{\alpha/2}(2n)\) is the \(\alpha/2\) quantile of a Chi-square distribution with \(2n\) degrees of freedom. Equation (3) is the same as the result of the traditional method. However, the quantity of interest is the reliability, which is a function of parameter \(\lambda\). The reliability of the component whose lifetime follows an exponential distribution at time \(t\) can be expressed as  
\[R(t)=\exp\biggl{\{}-\frac{t}{2T_{sum}}E\biggr{\}},\ E\sim\chi^{2}(2n). \tag{4}\]  
Then, the LCL of reliability with the confidence level \(1-\alpha\) is  
\[R_{L}(t)=\exp\biggl{\{}-\frac{t}{2T_{sum}}\chi^{2}_{1-\alpha}(2n)\biggr{\}}. \tag{5}\]  
The fiducial algorithm for the CI of parameter \(\lambda\) and the LCL of reliability based on an exponential distribution is shown in Algorithm 1. ```
Input: \(x\): the lifetime of the exponential-type component; \(1-\alpha\): the confidence level; \(t\): time;  M: the sample size of simulated pivotal quantity;  rep: the number of simulation trials. Output: The CI of parameter \(\lambda\) and the LCL of reliability. for i\(\leftarrow\) 1 to rep do for j \(\leftarrow\) 1 to M do \(\lambda_{\text{ij}}\)\(\leftarrow\) a random number follows \(\chi^{2}(2n)\), in which n is the sample size of \(x\). \(\text{Rij}\) (\(t\)) \(\leftarrow\)\(\exp\{-\lambda_{\text{ij}}t\}\). end Sort (\(\lambda_{\text{11}}\), \(\lambda_{\text{22}}\),,, \(\lambda_{\text{M}}\)) from small to large: (\(\text{N}_{\text{1}\,(\text{1})}\), \(\lambda_{\text{12}}\)), \(\ldots\), \(\lambda_{\text{M}}\)). Sort (\(\text{Ri}\) (\(t\)), \(\text{R}_{\text{2}}\) (\(t\)), \(\ldots\), \(\text{R}_{\text{M}}\) (\(t\))) from small to large: (\(\text{Ri}_{\text{1}\,(\text{1})}\) (\(t\)), \(\text{Ri}_{\text{2}}\) (\(t\)), \(\ldots\), \(\text{Ri}_{\text{M}}\)(\(t\))). The ith LCL of reliability is \(\text{R}_{\text{Li}}\) (\(t\)) \(\leftarrow\)\(\text{R}_{\text{i}}\)(\(\text{M}\)(\(-\alpha\))) (\(t\)). The ith CI of parameter \(\lambda\) is \(\lambda_{\text{CI}}\)\(\leftarrow\) (\(\lambda_{\text{i}}\)(\(\text{M}\)(\(\alpha\)/2)), \(\lambda_{\text{i}}\)(\(\text{M}\)(\(-\alpha\)/2)). end The LCL of reliability is \(\text{R}_{\text{L}}\) (\(t\)) \(\leftarrow\) mean of (\(\text{R}_{\text{LL}}\)(\(t\)), \(\ldots\), \(\text{R}_{\text{L}\text{P}}\)(\(t\))). The CI of parameter \(\lambda\) is (\(\lambda_{\text{L}}\), \(\lambda_{\text{U}}\)) \(\leftarrow\) mean of (\(\lambda_{\text{CI1}}\), \(\lambda_{\text{C2}}\), \(\ldots\), \(\lambda_{\text{CL}\text{P}}\)).