## 2 Basics of AEPS Reliability Analysis
### AEPS Reliability Design  
System reliability design refers to taking reliability into consideration in the system design stage, which requires designers not only to consider that the designed system can provide sufficient electrical power to complete the power supply function, but also to evaluate the power supply reliability of the designed system [42; 43]. When the evaluation results are unsatisfactory, the design needs to be revised: re-build the structure of the network or re-select better performance components until the reliability requirements are met (see Figure 2). Figure 1: Typical architecture of an AEPS. As shown in Figure 2, the core of system reliability design lies in two points: (1) how to compute the reliability of the designed system; and (2) how to modify the system that does not meet the reliability requirements. For point (1), unlike the general system having one probability to quantify its reliability, since the function of AEPS is to distribute power from source nodes to sink nodes, the reliability computation of AEPS is to calculate the probability of each sink node that obtains electrical power successfully from source nodes under stated conditions for a given system time \(T\). That is to say, the system-level reliability of AEPS means the power supply reliability of each sink node. For example, the AEPS in Figure 1 contains sink nodes (e.g., left AC power supply busbar (LG BUS), right AC power supply busbar (RG BUS), essential three-phase alternating current busbar (AC ESS BUS), essential single-phase alternating current bus (ESS BUS 1 PHASE), etc. ), so the system reliability of the AEPS refers to the power supply reliability of these sink nodes under system time \(T\). Moreover, on the basis of Figure 2, the method for computing the power supply reliability of the designed AEPS is illustrated as follows. Assume that an AEPS is composed of \(n\) components with failure rates \(\lambda=[\lambda_{1},\lambda_{2},\cdots\lambda_{n}]\), of which \(Num\) sink nodes are included. Failure rate \(\lambda(t)\) refers to the probability of the component that has not failed at time point \(t\) to fail in the next per unit time. The failure rate \(\lambda(t)\) of a component is often described by the Bath-Tub curve. It has three periods: early failure period, accidental failure period, and wear out period. In AEPS, components usually operate during the accidental failure period (so this period is also called using life), where \(\lambda(t)\) can be roughly regarded as a constant value \(\lambda\) if the operation environment is steady and unchanged. Therefore, the failure probability of the \(i\)-th component can be expressed as if we do not consider the operation environment changes:  
\[P_{i}=1-e^{-\lambda_{i}T} \tag{1}\]  
Furthermore, in theory, in combination with the failure/function logical relationship between the failure of components and power loss of sink nodes, formula \(R_{sys,s}=F(\lambda,T)\) can be obtained and computed, \(1\leq s\leq Num\). \(R_{sys,s}\) represents the power supply reliability of the sink nodes \(s\), and \(F(\lambda,T)\) represents the power supply reliability function with the component failures rate as variables. When each component failure rate is constant, power supply reliability is a function of time \(T\), as shown in Figure 3. In addition, since system reliability model can effectively express the logics between the failures of the component-level and system-level, the formula \(R_{s}=F(\lambda,T)\) can be obtained with the help of the system reliability model. The commonly used system reliability model of AEPS includes minimal path sets, minimal cut sets, fault tree, and reliability block diagram. Equation (2) is the formula for calculating the power supply reliability of the \(s\)-th power supply sink node by using the minimal path sets [32].