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Mathematical programming formulations for the alternating current optimal power flow problem

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Abstract

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

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Notes

  1. We remark that most of the power grid literature uses i to indicate current, and therefore resorts to j to indicate \(\sqrt{-1}\). We chose to keep notation in line with mathematics and the rest of the physical sciences, namely we use \(i=\sqrt{-1}\), and employ I to denote current.

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Acknowledgements

We are grateful to Cedric Josz and to Kundan Guha for interesting technical discussions.

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Authors and Affiliations

  1. IEOR, Columbia University, New York, USA

    Dan Bienstock

  2. LIX CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France

    Mauro Escobar & Leo Liberti

  3. IASI, CNR, Rome, Italy

    Claudio Gentile

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  1. Dan Bienstock
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  2. Mauro Escobar
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  3. Claudio Gentile
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  4. Leo Liberti
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Correspondence to Leo Liberti.

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CG was partly supported by the Italian Ministry of Education under the PRIN 2015B5F27W project “Nonlinear and conditional aspects of complex networks”. DB and LL benefitted from an exchange between Ecole Polytechnique and Columbia University financed by Columbia Alliance. CG and LL have received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 764759 “MINOA”. LL was partially supported by CNR STM Program Prot. AMMCNT–CNR No. 16442 dated 05/03/2018 and by INDAM Visiting Professors program 2018 prot. U-UFMBAZ-2017-001577 dated 22/12/2017.

Appendix A: Computational consistency check

Appendix A: Computational consistency check

Many of the formulations discussed in this survey have been tested computationally. These tests were not designed to establish whether a formulation can be solved faster, or whether a relaxation is tighter, than another. The sheer complication of these formulations can be a formidable hurdle to their successful deployment, and previous experience from all of the authors confirmed that it is extremely difficult to remove all of the bugs. We therefore employed computational tests as a validity and consistency check.

1.1 Modelling platforms

We used three modelling platforms for implementing our formulations.

  1. 1.

    AMPL (Fourer and Gay 2002) is a commercial, command-line interpreter which offers an incomparably elegant language, yielding code which is very similar to the formulations as they are presented mathematically on the written page. Its expression terms, objectives, and constraints may be quantified by indices varying on a set. AMPL has two serious limitations: (a) the amount of post-processing which can be expressed by its imperative sublanguage is limited (e.g. there is no function for computing eigenvalues/eigenvectors or the inverse of a matrix); (b) there is no interface with most SDP solvers.

  2. 2.

    Python (van Rossum et al. 2019) is a de facto standard in “scripting programming”. It is an interpreted language, with relatively low interpretation overhead CPU costs, and with a considerably large set of external modules (both interpreted and compiled), which allow the user to rapidly code almost anything. An equally large corpus of online documentation makes it possible to solve and issues using a simple internet query. We used the cvxpy (Diamond and Boyd 2016) MP modelling interface, which allows the coding and solution of SDPs and SOCPs using a range of solvers.

  3. 3.

    Matlab (2017) is a well-known commercial “general-purpose” applied mathematical software package. It offers very good MP capabilities through a range of interfaces. We used YALMIP (Löfberg 2004), which, notably, also connects to SDP solvers.

Specifically, we implemented and tested:

  • the real cartesian (S, I, V)-formulation (Sect. 5.1), the real cartesian voltage-only QCQP formulation (Sect. 5.2), the real polar NLP formulation (also in Sect. 5.2), and Jabr’s (real) relaxation (Sect. 5.4) using AMPL;

  • the complex SDP relaxation Eq. (33) (Sect. 4.4) using cvxpy on Python3;

  • the real matrix formulation (Sect. 5.6) using YALMIP on Matlab.

1.2 Solvers

We solved our formulations with a variety of solvers, some global and some local (used within a multi-start heuristic): Baron (Sahinidis and Tawarmalani 2005), Couenne (Belotti et al. 2009), ECOS (Domahidi et al. 2013), Mosek (Mosek 2016), IPOpt (COIN-OR 2006), Snopt (Gill 1999). We remark that Baron cannot deal with trigonometric functions. We found that cvxpy was able to pass complex number SDPs to ECOS correctly, but not (always) to Mosek. The global solvers for NLP (Baron, Couenne), could never certify global optima, even for the smallest instances, testifying to the practical hardness of the ACOPF.

On the other hand, Baron’s “upper bounding heuristic”, consisting in a multi-start on various local NLP solvers, yielded best solutions. Our benchmark was formed by small instances in Matlab’s MatPower’s (Zimmermann and Murillo-Sánchez 2018) data folder, and our comparison stone by the optima found by MatPower’s own local NLP solver—a Matlab implementation of a standard interior point method algorithm—from the local optima stored in the instances.

1.3 Notes

We also implemented a symbolic computation code [provided by sympy (Meurer et al. 2017)] in order to derive the the real expressions of formulations in real numbers from the corresponding complex expressions occurring in their complex counterparts.

Most of the code we used is available in github.com/leoliberti/acopf. Note this is research rather than production code. It is imperfect and may contain bugs.

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Bienstock, D., Escobar, M., Gentile, C. et al. Mathematical programming formulations for the alternating current optimal power flow problem. 4OR-Q J Oper Res 18, 249–292 (2020). https://doi.org/10.1007/s10288-020-00455-w

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