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Cited by 1,541Optimal Power Flow DistributionElectric Power System OptimizationPower System Optimization and Stability

Branch Flow Model: Relaxations and Convexification—Part I

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Farivar, Masoud, Low, Steven H.·2013·IEEE Transactions on Power Systems
DOI 10.1109/tpwrs.2013.2255317
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Abstract

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) 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 but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

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  • The exactness of the second-order conic relaxation has been verified for the radial distribution network when the power injection at each bus is not too large, and the bus voltage maintains around the nominal value.

    “The exactness of relaxation has been verified for the radial distribution network [30] when the power injection at each bus is not too large, and the bus voltage maintains around the nominal value.”

    From Distributed Coordination of Charging Stations Considering Aggregate EV Power Flexibility (2023) · cites this paper as [30]

    The Summary states that the conic relaxation is exact for radial networks, but gives a different condition: no upper bounds on real and reactive loads. The cited paper does not establish exactness specifically when each bus’s power injection is not too large and voltage stays near nominal; its Extensions section instead discusses other sufficient conditions involving load and voltage bounds. (LLM, conf=0.88)

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Authors
Farivar, M., & Low, S. H.
Affiliations
California Institute of Technology (US)
ORCID
Venue
IEEE Transactions on Power Systems
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Published
2013-08
Volume
28
Issue
3
Pages
2554-2564
ISSN
  • 0885-8950 (ISSN-L)
  • 1558-0679
DOI
10.1109/tpwrs.2013.2255317
OpenAlex
W2100197017
Cited by
1,541
References
1
Citations by year
2026: 154 · 2025: 153 · 2024: 172 · 2023: 172 · 2022: 169 · 2021: 183 · 2020: 129 · 2019: 117 · 2018: 96 · 2017: 83 · 2016: 56 · 2015: 29 · 2014: 25 · 2013: 2
FWCI
22.40
Type
journal article
Language
en
Open access
Licence
ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html (vor)
Locations
  • IEEE Transactions on Power Systems (publishedVersion)
  • CaltechAUTHORS (California Institute of Technology) (submittedVersion)
Funders
  • National Science Foundation (0911041)
  • U.S. Department of Energy (DEAR0000226)
  • Cisco Systems
  • California Institute of Technology
  • Resnick Sustainability Institute for Science, Energy and Sustainability, California Institute of Technology
  • National Science Council (NSC 101-3113-P-008-001, 101-3113-P-008-001-)
  • Okawa Foundation for Information and Telecommunications
  • Advanced Research Projects Agency - Energy (DE-AR0000226)
  • Advanced Research Projects Agency
Topics
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Optimal Power Flow Distribution
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Electric Power System Optimization
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Power System Optimization and Stability
Keywords
branch flow model, optimal power flow, conic relaxation, mesh networks, radial networks, phase shifters, convexification, global optimality, power system optimization

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