# Fiducial Lower Confidence Limit of Reliability for a Power Distribution System
## 1 Introduction  
A power distribution system (PDS) is a power network system consisting of a variety of power distribution equipment (or components), and it carries electricity from the transmission system to individual consumers. The basic function of a PDS is to supply customers with electrical energy as reliably as possible [1]. The reliability and maintainability of a PDS have attracted some scholars' interests. For example, the authors of [2] discussed the protection devices for the aircraft power distribution system. The authors in [3] evaluated the reliability of a PDS subjected to hurricanes. A framework for the optimal maintenance of a PDS subjected to non-stationary hurricane hazard and decay is presented in [4]. The authors of [5] reviewed the technological perspective of cyber secure smart inverters used in PDS. The authors of [6] analyzed the reliability of a 20-KV electric PDS, while those in [7] and [8] calculated the reliability of two Nigeria PDSs, one of which was aging. The authors in [9] evaluated an Indian PDS and gave some suggestions to minimize the outages which can improve reliability. The authors of [10] presented an inverse reliability evaluation where some components' reliability parameters were unknown in a specific PDS. In general, there is a key subsystem named the solid state power controller (SSPC) in a PDS. The SSPC is a system protection device which provides protections for the electric installations from short circuits and overloads [11]. Based on the importance of SSPC, several researchers studied the structure and applications of the SSPC. The authors of [12] summarized the advantages of an SSPC, such as high reliability, smallsize and accessible remote control. The authors in [13] presented an optimizing scheme for behavioral modeling of an SSPC, while those in [14] proposed some models for some special SSPCs. The authors in [15] analyzed the reliability of a kind of SSPC from an engineering standpoint, and those in [16] proposed two fault detection methods with the analysis of only the half cycle data. Due to the characteristics of a long lifetime and high reliability, it is particularly important to analyze the reliability of the PDS and SSPC. Technically, the reliability is often defined as the probability that a system will perform its intended function under operating conditions for a specified period of time [17]. In system reliability assessment, the estimator of reliability is a major concern. The authors of [18] estimated the reliability of the multicomponent stress-strength based on a two-parameter exponentiated Weibull distribution. The authors in [19] compared different least squares methods for reliability of the Weibull distribution based on the right censored data, and those in [20] and [21] analyzed the reliability of Weibull distribution with zero-failure data and very little failure data, respectively. However, a point estimator makes little sense if the variance of the estimator is too high. Consequently, it is necessary to give the lower confidence limit (LCL) of reliability [22]. How to get the LCL of system reliability through the lifetimes of components is an important issue. Many scholars have conducted a lot of research on this. An effective reliability confidence bound for a multi-state system with binary-capacitated components is suggested in [23]. This is only applied for a series or parallel system. In addition, the authors of [24] considered the LCL of reliability under a nonparametric assumption, but the parametric method is often more accurate than the nonparametric method when the distribution type is known. The authors in [25] provided the maximum likelihood estimate (MLE) and a more accurate lower confidence limit for the SoS reliability, but the MLE method is more appropriate for a large sample size. Based on the small sample size, the authors of [26] applied the WCF approach to analyze the reliability of a special complicated system based on accelerated degradation data, but the WCF method is complicated for mathematical formula derivation.