Improving Optimal Power Flow Relaxations Using 3-Cycle Second-Order Cone Constraints

This paper develops a novel second order cone relaxation of the semidefinite programming formulation of optimal power flow, that does not imply the `angle relaxation'. We build on a technique developed by Kim et al., extend it for complex matrices, and apply it to 3x3 positive semidefinite matrices to generate novel second-order cone constraints that augment upon the well-known 2x2 principal-minor based second-order cone constraints. Finally, we apply it to optimal power flow in meshed networks and provide numerical illustrations.