Equivalent Relaxations of Optimal Power Flow

Journal Article · Sun Mar 01 00:00:00 EST 2015 · IEEE Transactions on Automatic Control

DOI:https://doi.org/10.1109/TAC.2014.2357112· OSTI ID:1211406

Bose, S; Low, SH; Teeraratkul, T; Hassibi, B

Several convex relaxations of the optimal power flow (OPF) problem have recently been developed using both bus injection models and branch flow models. In this paper, we prove relations among three convex relaxations: a semidefinite relaxation that computes a full matrix, a chordal relaxation based on a chordal extension of the network graph, and a second-order cone relaxation that computes the smallest partial matrix. We prove a bijection between the feasible sets of the OPF in the bus injection model and the branch flow model, establishing the equivalence of these two models and their second-order cone relaxations. Our results imply that, for radial networks, all these relaxations are equivalent and one should always solve the second-order cone relaxation. For mesh networks, the semidefinite relaxation and the chordal relaxation are equally tight and both are strictly tighter than the second-order cone relaxation. Therefore, for mesh networks, one should either solve the chordal relaxation or the SOCP relaxation, trading off tightness and the required computational effort. Simulations are used to illustrate these results.

• Sponsoring Organization: USDOE Advanced Research Projects Agency - Energy (ARPA-E)
• DOE Contract Number: DE-AR0000226
• OSTI ID: 1211406
• Journal Information: IEEE Transactions on Automatic Control, Vol. 60, Issue 3; ISSN 0018-9286
• Country of Publication: United States
• Language: English