Analytical Probabilistic Power Flow and Global Sensitivity Analysis of Distribution Systems Based on Gaussian Mixture Model of Input-Output Variables

This paper proposes a computationally efficient and accurate method to solve probabilistic power flow (PPF) and its global sensitivity analysis (GSA) problems of power distribution systems. A distinguishing feature of this study is that a closed-form expression of the joint probability distribution function (PDF) of random input and output variables of distribution system power flow is derived based on the Gaussian mixture model (GMM), and then both PPF and GSA problems are calculated analytically. First, we combine kernel density estimation with the density-preserving hierarchical expectation maximization algorithm to obtain the GMM of random input variables, which describes their non-Gaussian distributions and correlations. Second, to characterize the nonlinear mapping relationship from inputs to outputs in power flow calculation, we establish the joint PDF of random input and output variables using the GMM of input variables and a piecewise linear power flow model, and then derive the closed-form PPF and semi-closed-form GSA solutions. Third, we develop a GMM-based decomposition method of global sensitivity indices to identify correlation effects and individual effects of correlated input variables on power flow variations. The proposed method is tested on the IEEE 33-bus and 123-bus distribution systems integrated with photovoltaic power units, wind turbines, and electric vehicle charging stations. Numerical results show that the proposed method is both fast and accurate, and superior over state-of-the-art surrogate model-based PPF and GSA methods.