Power System Robust State Estimation As a Layer: An Optimization-embedded End-to-end Learning Approach

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 potentially yields solutions that are statistically accurate yet physically inconsistent. To bridge this gap, this work proposes a novel E2E learning based RSE framework, where the convex-relaxed RSE problem is innovatively constructed as an explicit differentiable layer into an NN as the first trial. This optimization-embedded layer (termed as `Opt-Layer` in our work) serves as a solver of the RSE problem. Then, the relaxed solutions are recovered through post-processing layers. Through seamlessly embedding the underlying KKT conditions into the gradients during backward propagation, the physical consistency in the estimated states could be significantly enhanced, realizing lower measurement residuals. Also, the measurement weights are treated as learnable parameters of NN to enhance estimation robustness, enabling the Opt-Layer to actively denoise. A hybrid loss function is formulated to pursue accurate and physically consistent solutions. Extensive simulations have been carried out to demonstrate that the proposed framework can significantly improve the SE performance especially in terms of physical consistency on eight test systems, in comparison to classical E2E learning models, physics-informed NN (PINN) models, graph-based learning models, and conventional optimization-based approaches. The estimation performances under partial observability, severe noise contamination are systematically evaluated. Computational complexity and runtime analysis are also comprehensively demonstrated.