Two-Stage Robust Quadratic Optimization with Equalities and its Application to Optimal Power Flow

TL;DR

This paper introduces an iterative method for solving two-stage robust quadratic optimization problems with equalities under ellipsoidal uncertainty, with applications to optimal power flow, demonstrating improved solutions on power system instances.

Contribution

It presents a convergent iterative algorithm that handles non-linear second-stage constraints with affine approximations, specifically tailored for quadratic problems in power systems.

Findings

Algorithm converges to approximately robust feasible solutions

Demonstrates effectiveness on Matpower power system instances

Applicable to broader classes of non-linear optimization problems

Abstract

In this work, we consider two-stage quadratic optimization problems under ellipsoidal uncertainty. In the first stage, one needs to decide upon the values of a subset of optimization variables (control variables). In the second stage, the uncertainty is revealed and the rest of the optimization variables (state variables) are set up as a solution to a known system of possibly non-linear equations. This type of problem occurs, for instance, in optimization for dynamical systems, such as electric power systems as well as gas and water networks. We propose a convergent iterative algorithm to build a sequence of approximately robustly feasible solutions with an improving objective value. At each iteration, the algorithm optimizes over a subset of the feasible set and uses affine approximations of the second-stage equations while preserving the non-linearity of other constraints. We implement our approach and demonstrate its performance on Matpower instances. This paper focuses on quadratic problems, but the approach is suitable for more general setups.