Network flow and copper plate relaxations for AC transmission systems

Convex relaxations of the power flow equations and, in particular, the Semi-Definite Programming (SDP), Convex Quadratic (QC), and Second-Order Cone (SOC) relaxations, have attracted significant interest in recent years. Thus far, little attention has been given to simpler linear relaxations of the power flow equations, which may bring significant performance gains at the cost of bounding strength. To fill the gap, this paper develops two intuitive linear relaxations of the power flow equations, one based on classic network flow models (NF) and another inspired by copper plate approximations (CP). Theoretical results show that the proposed NF model is a relaxation of the established nonlinear SOC model and the CP model is a relaxation of the NF model. An experimental study on 94 real-world test cases demonstrates that the linear relaxations are very fast and only reduce the bounding strength by a few percent. Consequently, considering the linear NF and CP relaxations alongside established relaxations (SDP, QC, SOC) provides a rich variety of tradeoffs between relaxation strength and performance.