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

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.