# Assessing Voltage Stability in Distribution Networks:
## 2 Modeling of Stochastic Variables Considering Correlation
### Stochastic Variable Correlation Modeling  
In the distribution network with integrated, distributed PV, the stochasticity of the variables mentioned is affected by multiple factors, including geographical location, climate conditions, and seasonal changes. These factors not only independently affect each variable but also lead to certain correlations among them. In the same distribution network, the locations of distributed PV are close to load points, and the external conditions are similar, so there is also a correlation within the variables. PV generators located in similar geographical areas are affected by similar climate conditions, showing certain correlations in their power output. Similarly, loads in neighboring areas may also show correlations due to similar climate conditions. To accurately simulate the correlations between these uncertain variables, including their linear and nonlinear dependencies, Copula functions are introduced for modeling correlation coefficients [27], thus flexibly describing and modeling various complex dependency relationships. Copula theory is based on [PERSON]'s theorem, which provides an accurate definition of Copula functions by [PERSON]. Assuming there are \(n\) stochastic variables, \(X_{1}\), \(X_{2}\), \(\cdots\), \(X_{n}\), with their respective marginal distribution functions \(F_{1}(x_{1})\), \(F_{2}(x_{2})\), \(\cdots\), \(F_{n}(x_{n})\), and their joint distribution function \(H(X_{1}\), \(X_{2}\), \(\cdots\), \(X_{n})\), the Copula function, which serves as a connector between marginal and joint distributions, is defined as follows [31]:  
\[H(X_{1},X_{2}\cdots X_{n})=C(F_{1}(x_{1}),F_{2}(x_{2}),\cdots,F_{n}(x_{n})) \tag{8}\]  
where \(C()\) is the Copula function. If there are stochastic variables with continuous marginal distribution functions, then a unique Copula function exists:  
\[C(u_{1},u_{2},\cdots,u_{n})=F\Big{(}F_{1}^{-1}(u_{1}),F_{2}^{-1}(u_{2}), \cdots,F_{n}^{-1}(u_{n})\Big{)} \tag{9}\]  
Copula theory provides a method that allows for accurately describing dependencies between variables, even when they have different marginal distributions. This approach offers greater precision compared to other methods. Common Copula functions take many forms, as shown in Figure 1. The Archimedean Copula function is simple in form and can be uniquely determined by a convex function \(\varphi\)So, the problem can be simplified as follows: when used in high-dimensional situations, the rate and speed of convergence are higher [32]. Among them, the Gumbel and Clayton Copula functions cannot describe positive correlations, whereas the Frank Copula function can describe both positive and negative correlations [33]. Therefore, the Frank Copula function in Figure 1 is chosen to describe the correlation of stochastic variables in distribution networks with distributed generators [34]. Its cumulative distribution function (CDF) and PDF are, respectively, shown in Equations (10) and (11):  
\[C_{F}(u,v,\theta)=-\frac{1}{\theta}\ln\left[1+\frac{\left(e^{-\theta u}-1 \right)\left(e^{-\theta v}-1\right)}{\left(e^{-\theta}-1\right)}\right] \tag{10}\]  
\[c_{F}(u,v,\theta)=\frac{-\theta\left(e^{-\theta}-1\right)e^{-\theta(u+v)}}{ \left[\left(e^{-\theta}-1\right)+\left(e^{-\theta u}-1\right)\left(e^{-\theta v }-1\right)\right]^{2}} \tag{11}\]  
where \(\theta\) is the correlation parameter of stochastic variables \(u\) and \(v\). The Kendall rank correlation coefficient \(\tau\) can be derived instantly from these parameters of Copula function, making it especially suitable for application in Copula correlation models.