Mathematical programming formulations for the alternating current optimal power flow problem
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Mathematical programming formulations for the alternating current optimal power flow problem Dan Bienstock1 · Mauro Escobar2
· Claudio Gentile3
· Leo Liberti2
Received: 1 July 2020 / Revised: 9 August 2020 / Published online: 15 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords ACOPF · Smart grid · Complex numbers Mathematics Subject Classification 90C90 · 90C26
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.
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Leo Liberti [email protected] Dan Bienstock [email protected] Mauro Escobar [email protected] Claudio Gentile [email protected]
1
IEOR, Columbia University, New York, USA
2
LIX CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
3
IASI, CNR, Rome, Italy
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dealing with the time dependency . . . . . . . . . . . . . . 2.1 Change of coordinates . . . . . . . . . . . . . . . . . . 3 Modelling the ACOPF . . . . . . . . . . . . . . . . . . . . 3.1 The power grid as a graph . . . . . . . . . . . . . . . . 3.2 The π -model of a line . . . . . . . . . . . . . . . . . . 3.3 Informal description of ACOPF formulations . . . . . 4 Complex formulations . . . . . . . . . . . . . . . . . . . . 4.1 The (S, I , V )-formulation . . . . . . . . . . . . . . . . 4.1.1 Sets, parameters and decision variables . . . . . 4.1.2 Objective and constraints . . . . . . . . . . . . . 4.2 A modelling issue with the power flow equations . . . 4.3 Voltage-only formulation . . . . . . . . . . . . . . . . 4.4 Semidefinite relaxation . . . . . . . . . . . . . . . . . 5 Real formulations . . . . . . . . . . . . . . . . . . . . . . 5.1 Cartesian (S, I , V )-formulation . . . . . . . . . . . . . 5.2 Cartesian voltage-only QCQP . . . . . . . . . . . . . . 5.
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