## 4 The GSA in Probabilistic Evaluation of ALSC
### Theory of GSA
#### 4.1.3 Computational Strategy  
According to the idea of single-loop Monte Carlo (SMC) [36], [PERSON] et al. provided estimation formula for (24) [35].  
\[K\text{-}ST_{i}=\frac{1}{2N_{\text{sam}}\hat{V}(Y_{s})}\sum_{j=1}^{N_{\text{ sam}}}\Big{[}g_{s}(\mathbf{x}^{(j)})-g_{s}(\overline{x}_{i}^{(j)},\mathbf{x}_{\sim i}^{(j)} )\Big{]}^{2} \tag{25}\]  
where \(\mathbf{M}_{\text{A}}=\{\mathbf{x}^{(1)},\mathbf{x}^{(2)}\ldots,\mathbf{x}^{(N_{\text{sam}})}\}\) and \(\mathbf{M}_{\text{B}}=\{\mathbf{x}^{\prime}{}^{(1)},\mathbf{x}^{\prime}{}^{(2)}\ldots,\mathbf{ x}^{\prime}{}^{(N_{\text{sam}})}\}\) are two different input sample sets generated from the joint probability density function of the input random variables \(\mathbf{X}\); \(\hat{\mathbf{x}}_{\sim i}^{(j)}\) is generated from the conditional probability density function when \(X_{i}=x_{i}^{(j)}\); \(\hat{V}(Y_{s})\) is the estimated value of the output variance.  
Moreover, when the input variables are correlated, it is necessary to discuss the conditional distribution of the input vector \(\tilde{\mathbf{X}}_{w}\) when \(\mathbf{X}_{v}\) takes a fixed value. For simplicity, we first analyze correlated standard normal distribution variables. Assume that the variable \(\mathbf{Z}=[Z_{1}\text{,--},\,Z_{m}]=[Z_{i},\,\mathbf{Z}_{\sim i}]\) follows a standard normal distribution with mean vector \(\mathbf{\mu}\) and covariance matrix \(\mathbf{\Sigma}\). The components \(Z_{i}\) and \(\mathbf{Z}_{\sim i}\) also follow standard normal distributions with mean vectors \(\mu_{i}\) and \(\mathbf{\mu}_{\sim i}\), and covariance matrices \(\Sigma_{i}\) and \(\mathbf{\Sigma}_{\sim i}\), respectively. Consequently, the mean vector \(\mathbf{\mu}\) and covariance matrix \(\mathbf{\Sigma}\) can be expressed as follows:  
\[\mathbf{\mu}=\begin{bmatrix}\mu_{i}\\ \mathbf{\mu}_{\sim i}\end{bmatrix},\,\,\mathbf{\Sigma}=\begin{bmatrix}\Sigma_{i}&\mathbf{ \Sigma}_{i\sim i}\\ \mathbf{\Sigma}_{\sim i}&\mathbf{\Sigma}_{\sim i}\end{bmatrix} \tag{26}\]  
Let the variable \(\tilde{Z_{i}}\) represent the conditional variable when \(\mathbf{Z}_{\sim i}\) takes a fixed value \(\mathbf{z}_{\sim i}\). Then, the variable \(\tilde{Z_{i}}\) will follow a normal distribution with a mean \(\tilde{\mu}_{i}\) and covariance \(\tilde{\Sigma}_{i}\), where \(\tilde{\mu}_{i}\) and \(\tilde{\Sigma}_{i}\) are given by:  
\[\overline{\mu}_{i}=\mu_{i}+\mathbf{\Sigma}_{\sim i}\mathbf{\Sigma}_{\sim i}^{-1}(\mathbf{ z}_{\sim i}-\mathbf{\mu}_{\sim i}) \tag{27}\]  
\[\overline{\mathbf{\Sigma}}_{i}=\Sigma_{i}-\mathbf{\Sigma}_{i\sim i}\mathbf{\Sigma}_{\sim i }^{-1}\mathbf{\Sigma}_{\sim i,i} \tag{28}\]  
According to (25), to obtain the first-order sensitivity indices and total sensitivity indices corresponding to all input random variables, a total of \(N_{\text{sim}}=N_{\text{MCS}}+m\times N_{\text{MCS}}=(m+1)N_{\text{MCS}}\) model evaluations are required, where \(N_{\text{MCS}}\) is the Monte Carlo simulation number. Since \(N_{\text{MCS}}\) typically needs to be a large value to ensure the convergence of the probabilistic calculation results, the computational burden of the SMC method becomes particularly significant in power system analysis applications with high-dimensional input variables.