A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

Ponce, C.; Bindel, D. S.; Vassilevski, P. S.

doi:10.1137/16m1109965

Title: A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

Abstract

Multigrid is a highly scalable class of methods most often used for solving large linear systems. In this paper we develop a nonlinear algebraic multigrid framework for the power flow equations, a complex quadratic system of the form $${diag}({v})\overline{Y{v}}={s}$$, where $$Y$$ is approximately a complex scalar rotation of a real graph Laplacian. This is a standard problem that needs to be solved repeatedly during power grid simulations. A key difference between our multigrid framework and typical multigrid approaches is the use of a novel multiplicative coarse-grid correction to enable a dynamic multigrid hierarchy. We also develop a new type of smoother that allows one to coarsen together the different types of nodes that appear in power grid simulations. In developing a specific multigrid method, one must make a number of choices that can significantly affect the method's performance, such as how to construct the restriction and interpolation operators, what smoother to use, and how aggressively to coarsen. In this paper, we make simple but reasonable choices that result in a scalable and robust power flow solver. Experiments demonstrate this scalability and show that it is significantly more robust to poor initial guesses than current state-of-the-art solvers.

Authors:

Ponce, C. ^[1]; Bindel, D. S. ^[2]; Vassilevski, P. S. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Cornell Univ., Ithaca, NY (United States). Dept. of Computer Science

Publication Date:: Thu May 24 00:00:00 EDT 2018

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE Advanced Research Projects Agency - Energy (ARPA-E); US Air Force Office of Scientific Research (AFOSR)

OSTI Identifier:: 1671184

Report Number(s):: LLNL-JRNL-717563
Journal ID: ISSN 1064-8275; 861365

Grant/Contract Number:: AC52-07NA27344; AR0000230; 32-CFR-168a

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Scientific Computing

Additional Journal Information:: Journal Volume: 40; Journal Issue: 3; Journal ID: ISSN 1064-8275

Publisher:: SIAM

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; power flow; algebraic multigrid; nonlinear multigrid; multiplicative correction

Citation Formats


                    Ponce, C., Bindel, D. S., and Vassilevski, P. S. A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations.  United States: N. p., 2018. 
Web.  doi:10.1137/16m1109965.

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                    Ponce, C., Bindel, D. S., & Vassilevski, P. S. A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations.  United States.  https://doi.org/10.1137/16m1109965

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                    Ponce, C., Bindel, D. S., and Vassilevski, P. S. Thu .  
"A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations".  United States.  https://doi.org/10.1137/16m1109965.  https://www.osti.gov/servlets/purl/1671184.

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@article{osti_1671184,

  title        = {A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations},

  author       = {Ponce, C. and Bindel, D. S. and Vassilevski, P. S.},

  abstractNote = {Multigrid is a highly scalable class of methods most often used for solving large linear systems. In this paper we develop a nonlinear algebraic multigrid framework for the power flow equations, a complex quadratic system of the form ${diag}({v})\overline{Y{v}}={s}$, where $Y$ is approximately a complex scalar rotation of a real graph Laplacian. This is a standard problem that needs to be solved repeatedly during power grid simulations. A key difference between our multigrid framework and typical multigrid approaches is the use of a novel multiplicative coarse-grid correction to enable a dynamic multigrid hierarchy. We also develop a new type of smoother that allows one to coarsen together the different types of nodes that appear in power grid simulations. In developing a specific multigrid method, one must make a number of choices that can significantly affect the method's performance, such as how to construct the restriction and interpolation operators, what smoother to use, and how aggressively to coarsen. In this paper, we make simple but reasonable choices that result in a scalable and robust power flow solver. Experiments demonstrate this scalability and show that it is significantly more robust to poor initial guesses than current state-of-the-art solvers.},

  doi          = {10.1137/16m1109965},

  journal      = {SIAM Journal on Scientific Computing},

  number       = 3,

  volume       = 40,

  place        = {United States},

  year         = {Thu May 24 00:00:00 EDT 2018},

  month        = {Thu May 24 00:00:00 EDT 2018}

}

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Figures / Tables:

Table 1: Jacobi weighting schedules for various real-world networks. Also listed is the maximum number of hierarchy levels for each of the networks.

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