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Equivalent relaxations of optimal power flow

Bose, Subhonmesh and Low, Steven H. and Teeraratkul, Thanchanok and Hassibi, Babak (2015) Equivalent relaxations of optimal power flow. IEEE Transactions on Automatic Control, 60 (3). pp. 729-742. ISSN 0018-9286. doi:10.1109/TAC.2014.2357112.

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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.

Item Type:Article
Related URLs:
URLURL TypeDescription DOIArticle Paper
Bose, Subhonmesh0000-0002-3445-4479
Low, Steven H.0000-0001-6476-3048
Additional Information:© 2013 IEEE. Received: September 7, 2014. We are thankful to Prof. K. Mani Chandy and Lingwen Gan at Caltech for helpful discussions. We also acknowledge the support of NSF through NetSE grant CNS 0911041, CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, ARPA-E through grant DE-AR0000226, the National Science Council of Taiwan (R. O. C.) through grant NSC 103-3113-P-008-001, Southern California Edison, the Los Alamos National Lab (DoE), Resnick Institute at Caltech, Office of Naval Research under the MURI grant N00014-08–0747, Qualcomm Inc., and King Abdulaziz University.
Group:Resnick Sustainability Institute
Funding AgencyGrant Number
NSFCNS 0911041
National Science Council of TaiwanNSC 103-3113-P-008-001
Southern California EdisonUNSPECIFIED
Los Alamos National Laboratory (LANL)UNSPECIFIED
Caltech Resnick InstituteUNSPECIFIED
Office of Naval Research (ONR)N00014-08-0747
King Abdulaziz UniversityUNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20150112-101936328
Persistent URL:
Official Citation:Bose, S.; Low, S.H.; Teeraratkul, T.; Hassibi, B., "Equivalent Relaxations of Optimal Power Flow," Automatic Control, IEEE Transactions on , vol.60, no.3, pp.729,742, March 2015 doi: 10.1109/TAC.2014.2357112 URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:53561
Deposited By: Tony Diaz
Deposited On:13 Jan 2015 17:20
Last Modified:10 Nov 2021 20:04

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