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Power System Robust State Estimation As a Layer: A Novel End-to-end Learning Approach

Yibo Ding, Wenzhuo Shi, Mengzhao Duan, Yuhong Zhao, Jiaqi Ruan, Jian Zhao, Zhao Xu

TL;DR

This work tackles robust power system state estimation (RSE) in the presence of outliers by embedding a differentiable optimization layer that enforces physical constraints via KKT conditions. It relaxes the non-convex AC power-flow constraints to a convex SDP/SOCP and recovers capable voltage estimates, while learning measurement weights to enhance robustness. A hybrid loss balances estimation accuracy with physical consistency, and the framework demonstrates significant improvements in physical validity and competitive accuracy across six test systems compared to classical E2E and PINN approaches. Sparsification and a differentiable differentiation scheme enable scalable, real-time application with rigorous constraint satisfaction.

Abstract

Serving as an essential prerequisite for modern power system operation, robust state estimation (RSE) could effectively resist noises and outliers in measurements. The emerging neural network (NN) based end-to-end (E2E) learning framework enables real-time application of RSE but cannot strictly enforce the physical constraints involved, potentially yielding solutions that are statistically accurate yet physically inconsistent. To bridge this gap, this work proposes a novel E2E learning based RSE framework, where the RSE problem is innovatively constructed as an explicit differentiable layer of NN for the first time, ensuring physics alignments with rigors. Also, the measurement weights are treated as learnable parameters of NN to enhance estimation robustness. A hybrid loss function is formulated to pursue accurate and physically consistent solutions. To realize the proposed NN structure, the original non-convex RSE problem is specially relaxed. Extensive numerical simulations have been carried out to demonstrate that the proposed framework can significantly improve the SE performance while fulfilling physical consistency on six testing systems, in comparisons to the classical E2E learning based approach and the physics-informed neural network (PINN) approach.

Power System Robust State Estimation As a Layer: A Novel End-to-end Learning Approach

TL;DR

This work tackles robust power system state estimation (RSE) in the presence of outliers by embedding a differentiable optimization layer that enforces physical constraints via KKT conditions. It relaxes the non-convex AC power-flow constraints to a convex SDP/SOCP and recovers capable voltage estimates, while learning measurement weights to enhance robustness. A hybrid loss balances estimation accuracy with physical consistency, and the framework demonstrates significant improvements in physical validity and competitive accuracy across six test systems compared to classical E2E and PINN approaches. Sparsification and a differentiable differentiation scheme enable scalable, real-time application with rigorous constraint satisfaction.

Abstract

Serving as an essential prerequisite for modern power system operation, robust state estimation (RSE) could effectively resist noises and outliers in measurements. The emerging neural network (NN) based end-to-end (E2E) learning framework enables real-time application of RSE but cannot strictly enforce the physical constraints involved, potentially yielding solutions that are statistically accurate yet physically inconsistent. To bridge this gap, this work proposes a novel E2E learning based RSE framework, where the RSE problem is innovatively constructed as an explicit differentiable layer of NN for the first time, ensuring physics alignments with rigors. Also, the measurement weights are treated as learnable parameters of NN to enhance estimation robustness. A hybrid loss function is formulated to pursue accurate and physically consistent solutions. To realize the proposed NN structure, the original non-convex RSE problem is specially relaxed. Extensive numerical simulations have been carried out to demonstrate that the proposed framework can significantly improve the SE performance while fulfilling physical consistency on six testing systems, in comparisons to the classical E2E learning based approach and the physics-informed neural network (PINN) approach.

Paper Structure

This paper contains 25 sections, 3 theorems, 42 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Framework Overview
  3. Problem Formulation
  4. Classical Robust State Estimation Problem
  5. Convex Relaxation
  6. Second-order Conic Relaxation
  7. Solution Recovery
  8. End-to-end Learning for Power System Robust State Estimation
  9. Loss Function Design
  10. Differentiating the RSE Problem
  11. Problem Reformulation
  12. Differentiation Decomposition
  13. Pre-processing Problem Parameters
  14. Homogeneous Self-Dual Embedding
  15. Solution Mapping
  16. ...and 10 more sections

Key Result

Theorem 1

The equality $\bm{X}=\bm{V}\bm{V}^{*}$ holds if and only if the following two conditions are met: (1) $\bm{X}\succeq \bm{0}$; and (2) $\mathrm{rank}(\bm{X})=1$vandenberghe1996semidefinite.

Figures (3)

  • Figure 1: The Proposed End-to-end Learning Framework for Power System Robust State Estimation.
  • Figure 2: Estimation Robustness Comparison on IEEE-9 Bus System.
  • Figure 3: Radar Chart of Estimation Performance Under Different Weights.

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 1
  • Theorem 2