## 3 Importance Measure Index and Calculation Method
### Calculation Method  
Based on the Monte Carlo simulation theory [20; 32], this paper proposed a method to estimating \(\eta_{s,i}^{2}\) (\(i=1,2,\,\ldots,\,n,s=1,\,2,\,\ldots,\,Num\)), which includes six steps. The pseudocode of the algorithm is shown in Algorithm 1, and the steps are explained below. Step 1: Establish a failure rate sample vector \(\mathbf{\lambda}_{i}=\left[\lambda_{i}^{1},\,\,\lambda_{i}^{2},\,\ldots,\,\, \lambda_{i}^{M}\right]\) for component \(i\). According to the probability distribution function \(F(\lambda_{i})\) that variable \(\lambda_{i}\) obeys, the inverse transformation method is adopted to generate \(M\) samples to form the sample vector \(\mathbf{\lambda}_{i}=\left[\lambda_{i}^{1},\,\,\lambda_{i}^{2},\,\ldots,\,\, \lambda_{i}^{M}\right]\). Specifically, generate \(M\) random variables \(U_{1}\), \(U_{2}\), \(\ldots,\,U_{M}\) from the uniform distribution in the interval [0, 1] first, which is \(U\)-_Uniform_(0, 1). Then, use the inverse distribution function \(F^{-1}(U)\) to calculate the random variable \(\lambda_{i}^{j}\): \(\lambda_{i}^{j}=F^{-1}(U_{j})\), \(j=1,\,2,\,\ldots,\,M\). Step 2: Establish a component failure rate sample matrix \(\mathbf{\lambda}_{\
eg i}|\lambda_{i}\). Similarly, for \(\lambda_{i}^{j}(j=1,\,2,\,\ldots,\,M)\) in vector \(\mathbf{\lambda}_{i}\), use the inverse transformation method to generate sample \(\lambda_{\
eg i}^{j,k}\) (\(k=1,\,2,\,\ldots,\,\,C\)) from the conditional probability distribution \(f_{\lambda_{\
eg i}}(\mathbf{\lambda}_{\
eg i}|\lambda_{i}^{j})\) to form the component failure rate sample matrix \(\mathbf{\lambda}_{\
eg i}|\lambda_{i}\) (\(j=1,\,2,\,\ldots,\,M\)), see Equation (4). \[\mathbf{\lambda}_{\
eg i}|\lambda_{i}= \begin{array}{c}\lambda_{i}^{1}\\ \lambda_{i}^{2}\\ \vdots\\ \lambda_{i}^{M}\end{array}\begin{bmatrix}\mathbf{\lambda}_{\
eg i}^{1,1}\\ \mathbf{\lambda}_{\
eg i}^{2,1}\end{bmatrix}\begin{array}{c}\mathbf{\lambda}_{ \
eg i}^{1,2}\\ \mathbf{\lambda}_{\
eg i}^{2,1}\end{array}\begin{bmatrix}\mathbf{\lambda}_{\
eg i}^{1,C }\\ \mathbf{\lambda}_{\
eg i}^{2,1}\end{bmatrix}\begin{array}{c}\mathbf{\lambda}_{\
eg i}^{1,2}\\ \mathbf{\lambda}_{\
eg i}^{2,1}\end{array}\begin{bmatrix}\mathbf{\lambda}_{\
eg i}^{1,C }\\ \mathbf{\lambda}_{\
eg i}^{2,1}\end{bmatrix}\begin{array}{c}\mathbf{\lambda}_{\
eg i}^{ 2,1}\\ \mathbf{\lambda}_{\
eg i}^{2,1}\end{array}\begin{bmatrix}\mathbf{\lambda}_{\
eg i}^{2, 1}\\ \mathbf{\lambda}_{\
eg i}^{2,1}\end{bmatrix}  \tag{4}\]  
In the reliability analysis of AEPS, the failures between different components are generally considered to be independent of each other, so formula \(f_{\lambda_{\
eg i}}(\lambda_{\
eg i}|\lambda_{i}^{j})=f_{\lambda_{\
eg i}} (\lambda_{\
eg i})\) is established. In the component failure rate sample matrix \(\mathbf{\lambda}_{\
eg i}|\lambda_{i}\), the element \(\mathbf{\lambda}_{\
eg i}^{jk}\) is the \(k\)-th vector composed of the failure rates of the remaining components of the system under the failure rate \(\lambda_{i}^{j}\). For example, for the power supply bus \(s\), if there are four components that affect its power supply reliability in the system, then for the 2 nd component (\(i=2\)), the vector in the proposed matrix is donated as \(\mathbf{\lambda}_{\
eg i}^{jk}=\left[\lambda_{1}^{jk},\lambda_{3}^{jk},\lambda_{ 4}^{jk}\right]\). Step 3: Calculate the power supply reliability matrix \(\mathbf{R}_{s}\) of the sink nodes \(s\). Substitute the sample point \([\lambda_{i^{\prime}}^{j}\mathbf{\lambda}_{\
eg i}^{jk}]\) of the obtained component failure rate sample matrix into Equation (2) to calculate the power supply reliability \(R_{s}^{j,k}=F_{s}(\lambda_{i^{\prime}}^{j}\mathbf{\lambda}_{\
eg i}^{jk})\) of the sink nodes \(s\), of which \(F_{s}\) represents the function with the state of each component in the system as a variable and the power supply state of the sink nodes \(s\) is a dependent variable.