Probabilistic load flow plays a crucial role in the uncertainty analysis of distribution networks with a high proportion of distributed renewable energy sources and highly stochastic loads. However

due to the non-linearity of flow equations and probability distributions

as well as the correlation between random variables

analytical computation of probabilistic load flow is challenging. Existing methods often derive analytical expressions for probabilistic load flow from linear flow equations. However

the high resistance- reactance ratio of distribution networks makes it difficult for linear flow equations to accurately describe the relationships between flow variables

leading to distorted probability distributions. To address this issue

this paper presents an AC probabilistic load flow model for distribution network and an analytical computation method that can consider both the non-linearity of flows and the correlation of node power injection by expressing the AC flow equations as a quadratic form of node voltages and utilizing the relationship between the normal distribution and the generalized non-central chi-squared distribution. Firstly

it is proven that when node voltages follow a multi-variate normal distribution

the power flow quantities such as node power injection

line power

and network losses

follow a generalized non-central chi-squared distribution. Secondly

accurate expressions for the mean

variance

covariance

moment generating function

and characteristic function of power flow quantities are provided. Lastly

considering the complexity of the probability distribution function of power flow quantities

which makes it difficult to solve directly

a numerical method for calculating their probability density function and cumulative distribution function is proposed. Case studies based on the IEEE 33 bus distribution networks demonstrate that the proposed AC probabilistic load flow analytical algorithm can accurately describe the probabilistic relationships between node voltages and power quantities

and is suitable for calculating the probability distribution of distribution network flows in highly stochastic environments.