Strengthened SOCP Relaxations for ACOPF with McCormick Envelopes and Bounds Tightening

Abstract

Abstract The solution of the Optimal Power Flow (OPF) and Unit Commitment (UC) problems (i.e., determining generator schedules and set points that satisfy demands) is critical for efficient and reliable operation of the electricity grid. For computational efficiency, the alternating current OPF (ACOPF) problem is usually formulated with a linearized transmission model, often referred to as the DCOPF problem. However, these linear approximations do not guarantee global optimality or even feasibility for the true nonlinear alternating current (AC) system. Nonlinear AC power flow models can and should be used to improve model fidelity, but successful global solution of problems with these models requires the availability of strong relaxations of the AC optimal power flow constraints. In this paper, we use McCormick envelopes to strengthen the well-known second-order cone (SOC) relaxation of the ACOPF problem. With this improved relaxation, we can further include tight bounds on the voltages at the reference bus, and this paper demonstrates the effectiveness of this for improved bounds tightening. We present results on the optimality gap of both the base SOC relaxation and our Strengthened SOC (SSOC) relaxation for the National Information and Communications Technology Australia (NICTA) Energy System Test Case Archive (NESTA). For the cases where the SOC relaxation yields an optimality gap more than 0.1 %, the SSOC relaxation with bounds tightening further reduces the optimality gap by an average of 67 % and ultimately reduces the optimality gap to less than 0.1 % for 58 % of all the NESTA cases considered. Stronger relaxations enable more efficient global solution of the ACOPF problem and can improve computational efficiency of MINLP problems with AC power flow constraints, e.g., unit commitment.