AC Power Flow Feasibility Restoration via a State Estimation-Based Post-Processing Algorithm

TL;DR

The paper tackles restoring AC power flow feasibility from outputs of simplified OPF formulations (relaxations, approximations, and ML surrogates). It introduces a state estimation–inspired post-processing framework that treats simplified outputs as noisy measurements of the true AC quantities, with weight $\mathbf{\Sigma}$ and bias $\mathbf{b}$ learned offline via Adaptive Stochastic Gradient Descent (ASGD) and applied online. The method supports merging multiple simplified OPF solutions and achieves AC-feasible voltages and injections that closely approximate the true AC OPF solution, often reducing the squared two-norm loss by orders of magnitude. This approach offers a scalable, flexible restoration mechanism and provides insight into the relative reliability of different OPF simplifications for practical power-system operation.

Abstract

This paper presents an algorithm for restoring AC power flow feasibility from solutions to simplified optimal power flow (OPF) problems, including convex relaxations, power flow approximations, and machine learning (ML) models. The proposed algorithm employs a state estimation-based post-processing technique in which voltage phasors, power injections, and line flows from solutions to relaxed, approximated, or ML-based OPF problems are treated similarly to noisy measurements in a state estimation algorithm. The algorithm leverages information from various quantities to obtain feasible voltage phasors and power injections that satisfy the AC power flow equations. Weight and bias parameters are computed offline using an adaptive stochastic gradient descent method. By automatically learning the trustworthiness of various outputs from simplified OPF problems, these parameters inform the online computations of the state estimation-based algorithm to both recover feasible solutions and characterize the performance of power flow approximations, relaxations, and ML models. Furthermore, the proposed algorithm can simultaneously utilize combined solutions from different relaxations, approximations, and ML models to enhance performance. Case studies demonstrate the effectiveness and scalability of the proposed algorithm, with solutions that are both AC power flow feasible and much closer to the true AC OPF solutions than alternative methods, often by several orders of magnitude in the squared two-norm loss function.

Figures (5)

• Figure 1: Flowchart of the proposed algorithm.
• Figure 2: Diagonal elements of the weight parameters $\Sigma$ for the 5-bus system as obtained using Algorithm \ref{['alg:proposed-method']}. Higher values (in blue) indicate more trustworthy quantities for reconstructing AC feasible points. (a) QC and LPAC (left). (b) SOCP and SDP (right).
• Figure 3: Contour plot of weight parameters for voltage magnitudes in the Illinois 200-bus system for SOCP, QC, SDP, and LPAC.
• Figure 4: Differences in the density distributions for the voltage magnitudes and angles for the restored points relative to the true OPF solutions. The vertical axis represents the difference in density from the true OPF solution, with positive values indicating higher density and negative values indicating lower density compared to true OPF solution for the 5-bus system over the test set of 2000 scenarios. (a) Voltage magnitudes. (b) Voltage angles (in degrees).
• Figure 5: Cumulative proportion of the absolute error in the voltage magnitudes and angles for various restoration methods for the 5-bus system over $2,000$ test scenarios. The vertical axis displays the cumulative proportion of the absolute error and the horizontal axis shows differing levels of the absolute error. Both the horizontal and vertical axes are on logarithmic scales. Each curve shows the cumulative proportion of errors up to a certain level, with higher curves thus indicating a larger proportion of smaller errors (i.e., better performance). (a) Voltage magnitudes. (b) Voltage angles (in degrees).