# Assessing Voltage Stability in Distribution Networks:
## 1 Introduction
Reference [19] utilized the point estimate method to obtain samples of stochastic variables and combined it with maximum entropy to determine their probability distribution, analyzing the impact of stochastic variables on system harmonics during the calculation of harmonic power flow. In [20], a surrogate-assisted point estimate approach was proposed to analyze the influence of mixed uncertainties on electric systems probabilistically. The point estimate method is extensively employed in the uncertainty evaluation of electric systems due to its short solving time, high accuracy, and the fact that it does not require knowledge of the specific functional relationships between variables. In the operation of distribution networks, it is crucial to consider influences such as seasonal changes and weather conditions [21]. The solar radiation intensity at different locations in the same region and the load demands are not independent of each other but have certain correlations. These correlation factors significantly impact the operation of the distribution network [22]. Therefore, in conducting SVS analysis, it is imperative to not only account for the stochasticity of distributed generators but also for their correlations. Methods for addressing the correlation of stochastic variables include the Rosenblatt transformation [23], Orthogonal transformation [24], and the Nataf transformation [25]. Although the Rosenblatt transformation is applicable to any probability distribution, it requires detailed joint distribution data, which is challenging to acquire from practical engineering scenarios. Orthogonal transformation calculations are simple but only suitable for stochastic variables that obey normal distributions. The Nataf transformation has high computational accuracy, requiring only the marginal distributions and correlation coefficients of the stochastic variables, hence it has found widespread applications. Reference [26] combined the Nataf transformation with kernel density estimation to take the generator output power as input, and solve the probabilistic load flow using the point estimation method. However, the Pearson coefficient was used in the correlation treatment in the above literature, which cannot accurately describe the linear relationship between variables. The Copula function can accurately describe both linear and nonlinear relationships between variables. There is already research that combines MCS for probabilistic stability analysis [27], but it is time-inefficient. Therefore, researching a probabilistic stability assessment method with high efficiency and an accurate description of the correlation between stochastic variables is essential. Based on the above analysis, for the purpose of accurately characterizing the implications of uncertainty and correlation factors on SVS in distribution systems with distributed generators, a probability assessment method for SVS of distribution networks is proposed, which considers the uncertainties and correlations of the stochastic variables. The method presented in this article has the following main contributions:  
* Unlike the traditional probabilistic voltage stability method, which disregards correlations, the method presented uses the Copula function to obtain the correlation coefficients of stochastic variables. The Frank Copula function can describe both the non-negative and negative correlations of variables;
* The 3 PEM requires input variables to be mutually independent, thus it cannot handle correlated stochastic variables that follow arbitrary distributions in actual distribution networks. The proposed method uses the Nataf transformation to convert the correlated variable space into an independent standard normal space, which meets the applicability conditions of the 3 PEM;
* Traditional power flow calculation can only calculate the node voltage and branch power. The load margin and critical voltage are important indexes to analyze voltage stability.