Convex Relaxations of the Network Flow Problem Under Cycle Constraints

Zholbaryssov, Madi; Dominguez-Garcia, Alejandro D.

doi:10.1109/tcns.2019.2915390

Title: Convex Relaxations of the Network Flow Problem Under Cycle Constraints

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Abstract

In this paper, we consider the problem of optimizing the flows in a lossless flow network with additional nonconvex cycle constraints on nodal variables; such constraints appear in several applications, including electric power, water distribution, and natural gas networks. This problem is a nonconvex version of the minimum-cost network flow problem (NFP), and to solve it, we propose three different approaches. One approach is based on solving a convex approximation of the problem, obtained by augmenting the cost function with an entropy-like term to relax the nonconvex constraints. We show that the approximation error, for which we give an upper bound, can be made small enough for practical use. An alternative approach is to solve the classical NFP, that is, without the cycle constraints, and solve a separate optimization problem afterwards in order to recover the actual flows satisfying the cycle constraints; the solution of this separate problem maximizes the individual entropies of the cycles. The third approach is based on replacing the nonconvex constraint set with a convex inner approximation, which yields a suboptimal solution for the cyclic networks with each edge belonging to at most two cycles. We validate the practical usefulness of the theoretical results through numericalmore »« less

Authors:

^[1];

^[1]

Univ. of Illinois at Urbana-Champaign, IL (United States)

Publication Date:: Wed May 08 00:00:00 EDT 2019

Research Org.:: Univ. of California, San Diego, CA (United States)

Sponsoring Org.:: USDOE Advanced Research Projects Agency - Energy (ARPA-E)

OSTI Identifier:: 1799058

Grant/Contract Number:: AR0000695

Resource Type:: Accepted Manuscript

Journal Name:: IEEE Transactions on Control of Network Systems

Additional Journal Information:: Journal Volume: 7; Journal Issue: 1; Journal ID: ISSN 2325-5870

Publisher:: IEEE

Country of Publication:: United States

Language:: English

Subject:: 42 ENGINEERING; Automation & Control Systems; Computer Science; Power systems; Cost function; Control systems; Natural gas; Entropy; Network topology

Citation Formats


                    Zholbaryssov, Madi, and Dominguez-Garcia, Alejandro D. Convex Relaxations of the Network Flow Problem Under Cycle Constraints.  United States: N. p., 2019. 
Web.  doi:10.1109/tcns.2019.2915390.

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                    Zholbaryssov, Madi, & Dominguez-Garcia, Alejandro D. Convex Relaxations of the Network Flow Problem Under Cycle Constraints.  United States.  https://doi.org/10.1109/tcns.2019.2915390

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                    Zholbaryssov, Madi, and Dominguez-Garcia, Alejandro D. Wed .  
"Convex Relaxations of the Network Flow Problem Under Cycle Constraints".  United States.  https://doi.org/10.1109/tcns.2019.2915390.  https://www.osti.gov/servlets/purl/1799058.

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@article{osti_1799058,

  title        = {Convex Relaxations of the Network Flow Problem Under Cycle Constraints},

  author       = {Zholbaryssov, Madi and Dominguez-Garcia, Alejandro D.},

  abstractNote = {In this paper, we consider the problem of optimizing the flows in a lossless flow network with additional nonconvex cycle constraints on nodal variables; such constraints appear in several applications, including electric power, water distribution, and natural gas networks. This problem is a nonconvex version of the minimum-cost network flow problem (NFP), and to solve it, we propose three different approaches. One approach is based on solving a convex approximation of the problem, obtained by augmenting the cost function with an entropy-like term to relax the nonconvex constraints. We show that the approximation error, for which we give an upper bound, can be made small enough for practical use. An alternative approach is to solve the classical NFP, that is, without the cycle constraints, and solve a separate optimization problem afterwards in order to recover the actual flows satisfying the cycle constraints; the solution of this separate problem maximizes the individual entropies of the cycles. The third approach is based on replacing the nonconvex constraint set with a convex inner approximation, which yields a suboptimal solution for the cyclic networks with each edge belonging to at most two cycles. We validate the practical usefulness of the theoretical results through numerical examples, in which we study the standard test systems for water and electric power distribution networks.},

  doi          = {10.1109/tcns.2019.2915390},

  journal      = {IEEE Transactions on Control of Network Systems},

  number       = 1,

  volume       = 7,

  place        = {United States},

  year         = {Wed May 08 00:00:00 EDT 2019},

  month        = {Wed May 08 00:00:00 EDT 2019}

}

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