Certified and accurate SDP bounds for the ACOPF problem

We propose a new method for improving the bound tightness of the popular semidefinite programming (SDP) relaxation for the ACOPF introduced in Lavaei and Low (2012), Molzahn and Hiskens (2019). First, we reformulate the ACOPF Lagrangian dual as an unconstrained concave maximization problem with a clique decomposition induced sparse structure. We prove that this new formulation has the same optimal value as the SDP relaxation. We then use the solution of the SDP relaxation as a starting point for a tailored structure-aware bundle method. This post-processing technique significantly improves the tightness of the SDP bounds computed by the state-of-the-art solver MOSEK, as shown by our computational experiments on large-scale instances from PGLib-OPF v21.07. For ten of the tested instances, our post-processing decreases by more than 50% the optimality gap obtained with MOSEK.