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Branch Flow Model: Relaxations and Convexification (Parts I, II)

Farivar, Masoud and Low, Steven H. (2012) Branch Flow Model: Relaxations and Convexification (Parts I, II). . (Unpublished)

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We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) problems that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact and we characterize when the angle relaxation may fail. We propose a simple method to convexify a mesh network using phase shifters so that both relaxation steps are always exact and OPF for the convexified network can always be solved efficiently for a globally optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network graph and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Since power networks are sparse, the number of required phase shifters may be relatively small.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Low, Steven H.0000-0001-6476-3048
Additional Information:We are grateful to S. Bose, K. M. Chandy and L. Gan of Caltech, C. Clarke, M. Montoya, and R. Sherick of the Southern California Edison (SCE), and B. Lesieutre of Wisconsin for helpful discussions. We acknowledge the support of NSF through NetSE grant CNS 0911041, DoE’s ARPA-E through grant de-ar0000226, the National Science Council of Taiwan (R. O. C.) through grant NSC 101-3113-P-008-001, SCE, the Resnick Institute of Caltech, Cisco, and the Okawa Foundation.
Group:Resnick Sustainability Institute
Funding AgencyGrant Number
National Science Council (Taipei)101-3113-P-008-001
Southern California EdisonUNSPECIFIED
Resnick Sustainability InstituteUNSPECIFIED
Okawa FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20190628-073720381
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:96804
Deposited By: Tony Diaz
Deposited On:28 Jun 2019 14:43
Last Modified:02 Jun 2023 00:14

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