# Assessing Voltage Stability in Distribution Networks:
## 1 Introduction  
With the large-scale integration of distributed generators, line losses in the power grid have significantly decreased, and voltage configuration has been optimized [1]. However, the modern power system [2] must not only incorporate a high proportion of renewable energy, but also achieve high resilience, intrinsic safety, and reliability. Nevertheless, because renewable energy is unstable and load demand fluctuates stochastically [3], the power system faces severe challenges in maintaining secure and reliable performance [4]. The distribution network, positioned at the terminal of the power grid, is essential for conveying electricity from the transmission network to users of various voltage levels, making it the most critical link in guaranteeing the secure functioning of the power grid [5]. Therefore, to address the blind spot issues caused by the integration of distributed generators in traditional distribution network state assessment methods, it is critically important to study the stability of distribution networks with distributed generators [6]. SVS is essential for ensuring the overall stability of the distribution network [7]. The LMI is commonly used as an assessment indicator for voltage stability due to its clear physical meaning and ability to account for the nonlinearity of the system. Methods for calculating the load margin include the CPF method, internal point method, etc. [8]. However, traditional SVS studies are based on deterministic models, which do not consider the uncertainties of new energy sources and loads, making it impossible to analyze system stability accurately and effectively [9]. Therefore, it is necessary to research probabilisticmethods for assessing the SVS of a distribution network with uncertain new energy outputs and load fluctuations. Currently, widely used probabilistic assessment methods include the Monte Carlo simulation (MCS) method, analytical method, and approximate method [10]. MCS primarily involves extensive stochastic sampling and repeated deterministic calculations for probabilistic analysis, providing clear physical significance, strong robustness, and high accuracy [11]. In [12], MSC was combined with dimensionally adaptive sparse grid interpolation to enhance computing efficiency. The MCS method is improved based on the ideas of dispersed sampling, which can significantly shorten calculation time by decreasing the sample size [13]. Although the above methods have improved computational efficiency while ensuring accuracy, the long processing time remains the biggest issue with the MCS method, and it generally serves as a control group to validate other methods. Widely used in analytical approaches are the semi-invariant method and the convolution method, the core of which is to linearize the relationships between stochastic variables for getting probabilistic characteristics of the variable values, resulting in faster solution speed [14]. [PERSON] [PERSON] [15] established a probabilistic power flow calculation method, taking into account load, and unit frequency characteristics based on the semi-invariant method. Reference [16] proposed an improved interval semi-invariant method to analyze the SVS margin for hybrid power grids with a large amount of new energy. Reference [17] used the maximum entropy principle and the semi-invariant method to research the comprehensive energy system of electric-gas interconnection with high efficiency, but it overlooked the influence of correlation factors among energy systems. Although the semi-invariant method has high computational efficiency, it involves complex mathematical calculations, and the convolution method is also difficult to implement, leading to suboptimal practical application results. The approximate method is to approximately analyze their numerical feature based on the distribution of the variables, with the point estimate method being a typical method [18].