Polynomial SDP Cuts for Optimal Power Flow

Abstract: The use of convex relaxations has lately gained considerable interest in Power Systems. These relaxations play a major role in providing global optimality guarantees for non-convex optimization problems. For the Optimal Power Flow (OPF) problem, the Semi-Definite Programming (SDP) relaxation is known to produce tight lower bounds. Unfortunately, SDP solvers still suffer from a lack of scalability. In this work, we introduce a new set of polynomial SDP-based constraints, strengthening weaker quadratic convex relaxations. The SDP cuts, expressed as polynomial constraints, can be handled by standard Nonlinear Programming solvers, enjoying better stability and computational efficiency. The new cut-generation procedure benefits from recent results on tree-decomposition methods, reducing the dimension of the underlying SDP matrices. As a side result, we present the first formulation of Kirchhoff's Voltage Law in the SDP space and reveal the existing link between these cycle constraints and the original SDP relaxation for three dimensional matrices. Numerical results on state-of-the- art benchmarks show a significant gain both in computational efficiency and optimality bound quality.