# Assessing Voltage Stability in Distribution Networks:
## 2 Modeling of Stochastic Variables Considering Correlation
### 3 PEM Considering Variable Correlation  
Considering the superiority of the point estimation method in dealing with uncertain factors, this paper adopts 3 PEM for probabilistic voltage stability analysis. The basic idea of the 3 PEM is: Given the probability distributions of the input variables on the condition that they are mutually independent, three sampling points are taken from each input stochastic variable to reflect probability and statistical characteristics of the variables. The output samples are calculated according to the functional relationship between the input and output variables. Subsequently, origin moments at each order of the output variables are determined, and probability distributions of the output variables are acquired through fitting [19]. Assuming \(G\) is the m-dimensional output variable and \(x_{i}\) (\(i=1\), \(2\), \(\cdots\), \(n\)) are the input stochastic variable, then:  
\[G=P(X)=P(x_{1},x_{2},\cdots,x_{n}) \tag{18}\]  
In Equation (18), \(P\) is the correlation function of \(G\) and \(X\), where \(G=[g_{1},g_{2},\cdots,g_{n}]^{\mathrm{T}}\). According to the principle of 3 PEM, three sampling values of each stochastic variable \(x_{i}\) in independent standard normal space are obtained, \(x_{i,k}\) (\(k=1\), \(2\), \(3\)), and the sampling calculation equation is as follows [20]:  
\[x_{i,k}=\mu_{xi}+\xi_{xi,k}\sigma_{xi}\quad k=1,2,3 \tag{19}\]  
where \(\mu_{xi}\), \(\xi_{xi,k}\), and \(\sigma_{xi}\) represent the expected value, location coefficient, and standard deviation of stochastic variable \(x_{i}\), respectively, where location coefficient \(\xi_{xi,k}\) can be expressed as:  
\[\left\{ \begin{aligned} \xi_{xi,k}=\frac{\lambda_{i,3}}{2}+(-1)^{3-k} \sqrt{\lambda_{i,A}-\frac{3}{4}\lambda_{i,3}^{2}}& k=1,2\\ \xi_{xi,i}=0&\end{aligned} \right. \tag{20}\]  
where \(\lambda_{i,3}\) represents the third-order moment of variable \(x_{i}\), known as the skewness coefficient. \(\lambda_{i,A}\) represents the fourth-order moment, known as the kurtosis coefficient, and is expressed as:  
\[\left\{ \begin{aligned} \lambda_{i,3}&=\frac{\mathbb{E} \left[\left(x_{i,k}-\mu_{xi}\right)^{3}\right]}{\sigma_{xi}{}^{3}}\\ \lambda_{i,A}&=\frac{\mathbb{E}\left[\left(x_{i,k}- \mu_{xi}\right)^{4}\right]}{\sigma_{xi}{}^{4}}\end{aligned} \right. \tag{21}\]  
The weight coefficient corresponding to each sampling point is:  
\[ \begin{aligned} w_{xi,k}&=\frac{(-1)^{3-k}}{\xi_{xi,i,k}(\xi_{xi,i-1}-\xi_{xi,i}z)}\quad k=1,2\\ w_{xi,3}&=\frac{1}{n}-\frac{1}{\lambda_{i,A}- \lambda_{i,3}^{2}}\end{aligned}  \tag{22}\]  
Through the above process, \(3n\) sampling points can be obtained, so \(3n\) deterministic evaluations need to be performed. However, when \(k=3\) and \(x_{i,k}=\mu_{xi}\), only one calculation is needed, hence there are only \((2n+1)\) samples, and only \((2n+1)\) evaluations need to be performed. Using the Nataf inverse transformation, which is employed in Section 3.1, to transform the sample correlations before input, and integrating the above results [37], the \(l\)-order origin moment of the output variable \(g_{j}\) is \(E(g_{ij})\):  
\[E\left(g_{j}^{l}\right)\approx\sum_{i=1}^{n}\sum_{k=1}^{2}w_{xi,k}\left(g_{j} (i,k)\right)^{l}+w_{2n+1}g_{2n+1}^{l} \tag{23}\]  
From Equations (24) and (25), the mean \(\mu_{gi}\) and standard deviation \(\sigma_{gi}\) can be calculated as:  
\[\mu_{gj}=E\left(g_{j}\right)\quad j=1,2,\cdots,n \tag{24}\]  
\[\sigma_{gj}=\sqrt{E\left(g_{j}^{2}\right)-\mu_{gj}{}^{2}}\quad j=1,2,\cdots,n \tag{25}\]The steps for the 3 PEM based on the Nataf inverse transformation are as follows:  
Step 1: Determine the input stochastic variable \(X=[x_{1},x_{2},\cdots,x_{n}]\) and the correlation coefficient matrix \(p_{x}\), solve the correlation coefficient matrix \(p_{y}\) of standard normal stochastic variable \(Y\), and obtain matrix \(L\) through Cholesky decomposition. Step 2: Determine the location coefficients and weight coefficients for the sampling points in the independent standard space based on the principle of 3 PEM.