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Analytical Probabilistic Power Flow Approximation Using Invertible Neural Networks

Weijie Xia, James Ciyu Qin, Edgar Mauricio Salazar Duque, Hongjin Du, Peter Palensky, Giovanni Sansavini, Pedro P. Vergara

Abstract

Probabilistic power flow (PPF) is essential for quantifying operational uncertainty in modern distribution systems with high penetration of renewable generation and flexible loads. Conventional PPF methods primarily rely on Monte Carlo (MC) based power flow (PF) simulations or simplified analytical approximations. While MC approaches are computationally intensive and demand substantial data storage, analytical approximations often compromise accuracy. In this paper, we propose a novel analytical PPF framework that eliminates the dependence on MC-based PF simulations and, in principle, enables an approximation of the analytical form of arbitrary voltage distributions. The core idea is to learn an explicit and invertible mapping between stochastic power injections and system voltages using invertible neural networks (INNs). By leveraging the Change of Variable Theorem, the proposed framework facilitates direct approximation of the analytical form of voltage probability distributions without repeated PF computations. Extensive numerical studies demonstrate that the proposed framework achieves state-of-the-art performance both as a high-accuracy PF solver and as an efficient analytical PPF estimator.

Analytical Probabilistic Power Flow Approximation Using Invertible Neural Networks

Abstract

Probabilistic power flow (PPF) is essential for quantifying operational uncertainty in modern distribution systems with high penetration of renewable generation and flexible loads. Conventional PPF methods primarily rely on Monte Carlo (MC) based power flow (PF) simulations or simplified analytical approximations. While MC approaches are computationally intensive and demand substantial data storage, analytical approximations often compromise accuracy. In this paper, we propose a novel analytical PPF framework that eliminates the dependence on MC-based PF simulations and, in principle, enables an approximation of the analytical form of arbitrary voltage distributions. The core idea is to learn an explicit and invertible mapping between stochastic power injections and system voltages using invertible neural networks (INNs). By leveraging the Change of Variable Theorem, the proposed framework facilitates direct approximation of the analytical form of voltage probability distributions without repeated PF computations. Extensive numerical studies demonstrate that the proposed framework achieves state-of-the-art performance both as a high-accuracy PF solver and as an efficient analytical PPF estimator.

Paper Structure

This paper contains 26 sections, 34 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Analytical PPF
  4. NN-based PF Solver
  5. Invertible NNs
  6. Problem Formulation
  7. Methodology
  8. Power Flow (PF) Formulation
  9. Change of Variable Theorem
  10. Adjustment of the Change of Variable Theorem for PPF
  11. Proposed Framework Overview
  12. Invertible Mixed Neural Flow (IMNF)
  13. Simplified FCPFlow (SFCP)
  14. Spline Flow (SF)
  15. Combining SFCP and SF
  16. ...and 11 more sections

Figures (3)

  • Figure 1: Overview of the proposed probabilistic power flow framework. (A) Mathematical foundation of the density estimation. (B) Construction of the IMNF model. (C) GAT for topology-aware feature aggregation. (D) Transformer-style block structure used within $f_{sp}$. (E) LSS for efficient numerical integration.
  • Figure 2: Example of voltage density estimation for the 39-bus system at bus 1 using LSS. Increasing the number of scenarios yields progressively more accurate density estimates.
  • Figure 3: Exemplar density evaluation results from the proposed framework and benchmarks.