| Issue |
ESAIM: COCV
Volume 28, 2022
|
|
|---|---|---|
| Article Number | 40 | |
| Number of page(s) | 37 | |
| DOI | https://doi.org/10.1051/cocv/2022034 | |
| Published online | 29 June 2022 | |
Extended Mckean-Vlasov optimal stochastic control applied to smart grid management*,, **
1
Centre de Mathématiques Appliquées (CMAP), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
2
Electricité de France, Palaiseau, France
*** Corresponding author: emmanuel.gobet@polytechnique.edu
Received:
23
January
2021
Accepted:
22
April
2022
We study the mathematical modeling of the energy management system of a smart grid, related to a aggregated consumer equipped with renewable energy production (PV panels e.g.), storage facilities (batteries), and connected to the electrical public grid. He controls the use of the storage facilities in order to diminish the random fluctuations of his residual load on the public grid, so that intermittent renewable energy is better used leading globally to a much greener carbon footprint. The optimization problem is described in terms of an extended McKean-Vlasov stochastic control problem. Using the Pontryagin principle, we characterize the optimal storage control as solution of a certain McKean-Vlasov Forward Backward Stochastic Differential Equation (possibly with jumps), for which we prove existence and uniqueness. Quasi-explicit solutions are derived when the cost functions may not be linear-quadratic, using a perturbation approach. Numerical experiments support the study.
Mathematics Subject Classification: 93E20 / 49N10 / 49N80
Key words: Mean-field control / energy management / numerical methods for stochastic control
This work has benefited from several supports: Siebel Energy Institute (Calls for Proposals #2, 2016), ANR project CAESARS (ANR-15-CE05-0024), Association Nationale de la Recherche Technique (ANRT), Electricité De France (EDF), Finance for Energy Market (FiME) Lab (Institut Europlace de Finance), Chair “Stress Test, RISK Management and Financial Steering” of the Foundation Ecole Polytechnique.
This work has been presented at the FOREWER conference Paris-June 2017, at the MCM2017 conference Montreal-July 2017, at the conference “Stochastic control, BSDEs and new developments” Roscoff-September 2017, at the conference “Advances in Stochastic Analysis for Risk Modeling” CIRM-November 2017, at the Workshop “Mean-Field Games, Energy and Environment” London-February 2018, at the conference “Advances in Modelling and Control for Power Systems of the Future” Paris-September 2018, at the conference “APS Informs” Brisbane-July 2019, at the MCM2019 conference Sydney-July 2019. The authors wish to thank the participants for their feedbacks.
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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