Abstract
We consider a stylized model for a power network with distributed local power generation and storage. This system is modeled as a network connection of a large number of nodes, where each node is characterized by a local electricity consumption, has a local electricity production (photovoltaic panels for example) and manages a local storage device. Depending on its instantaneous consumption and production rate as well as its storage management decision, each node may either buy or sell electricity, impacting the electricity spot price. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. In a noncooperative game setting, we are led to the analysis of a nonzero sum stochastic game with N players where the interaction takes place through the spot price mechanism. For an infinite number of agents, our model corresponds to an extended mean field game. We are able to compare this solution to the optimal strategy of a central planner and in a linear quadratic setting, we obtain and explicit solution to the extended mean field game and we show that it provides an approximate Nash equilibrium for N-player game.
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References
Delft, C.E.: The potential of energy citizens in the European Union. CE Delft Publications (2016)
Couillet, R., Medina Perlaza, S., Tembine, H., Debbah, M.: Electrical vehicles in the smart grid: a mean field game analysis. IEEE J. Sel. Areas Commun. 30, 1086–1096 (2012)
de Paola, A., Angeli, D., Strbac, G.: Distributed control of micro-storage devices with mean field games. IEEE Trans. Smart Grid 7(2), 1119–1127 (2016)
Huang, M., MalhamŽ, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop Mckean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)
Huang, M., Caines, P.E., Malhame, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)
Lasry, J.M., Lions, P.L.: Jeux à champ moyen. i–le cas stationnaire. Comptes Rendus Mathématique 343(9), 619–625 (2006)
Lasry, J.M., Lions, P.L.: Jeux à champ moyen. ii–horizon fini et contrôle optimal. Comptes Rendus Mathématique 343(10), 679–684 (2006)
Cardaliaguet, P.: Notes on mean field games from p.-l. lions? lectures at coll‘ege de france. In: Preprint (2013)
Lions, P.-L.: Théorie des jeux de champ moyen et applications (MFG). https://www.college-de-france.fr/site/pierre-louis-lions/course-2007-11-16-09h00_1.htm
Carmona, R., Delarue, F., Lachapelle, A.: Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games. Probability Theory and Stochastic Modelling. Springer, New York (2017). https://doi.org/10.1007/978-1-4614-8508-7
Carmona, R., Delarue, F., Lacker, D.: Mean field games with common noise. Ann. Probab. 44(6), 3740–3803 (2016). https://doi.org/10.1214/15-AOP1060
Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springer Briefs in Mathematics. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-8508-7
Bensoussan, A., Frehse, J., Yam, S.C.P.: On the interpretation of the Master Equation. Stoch. Process. Appl. 127(7), 2093–2137 (2017). https://doi.org/10.1016/j.spa.2016.10.004
Carmona, R., Delarue, F., Lachapelle, A.: Control of McKean–Vlasov dynamics versus mean field games. Math. Financ. Econ. 7(2), 131–166 (2013). https://doi.org/10.1007/s11579-012-0089-y
Carmona, R., Delarue, F.: Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43(5), 2647–2700 (2015). https://doi.org/10.1214/14-AOP946
Carmona, R., Delarue, F., Lacker, D.: Mean field games of timing and models for bank runs. Appl. Math. Optim. 76(1), 217–260 (2017). https://doi.org/10.1007/s00245-017-9435-z
Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Graber, P.J.: Linear quadratic mean field type control and mean field games with common noise, with application to production of an exhaustible resource. arXiv:1607.02130 [math.OC] 7 Jul 2016
Cardaliaguet, P., Lehalle, C.A.: Mean field game of controls and an application to trade crowding. arXiv:1610.09904 (2017)
Carmona, R., Delarue, F.: Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51(4), 2705–2734 (2013)
Yong, J.: Linear forward? Backward stochastic differential equations. Appl. Math. Optim. 39, 93–119 (1999)
Paraschiv, F., Erni, D., Pietsch, R.: The impact of renewable energies on EEX day-ahead electricity prices. Energy Policy 73, 196–210 (2014)
Hu, Y., Huang, J., Nie, T.: Linear-quadratic-gaussian mixed mean-field games with heterogeneous input constraints. arXiv:1710.02916v1 (2017)
Acknowledgements
The authors wish to thank the anonymous referees for all the pertinent remarks she/he made. The author’s research is part of the ANR Project CAESARS (ANR-15-CE05-0024) and PACMAN (ANR-16-CE05-0027) and of PANORISK project. The third author was partially supported by chaire Risques Financiers de la fondation du risque, CMAP-Ecole Polytechniques and Chair Risques Emergents ou Atypiques en Assurance supported by Mutuelle du Mans Assurance, Le Mans University, Risk Foundation and Ecole Polytechnique.
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Communicated by Nizar Touzi.
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Alasseur, C., Ben Taher, I. & Matoussi, A. An Extended Mean Field Game for Storage in Smart Grids. J Optim Theory Appl 184, 644–670 (2020). https://doi.org/10.1007/s10957-019-01619-3
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DOI: https://doi.org/10.1007/s10957-019-01619-3
Keywords
- Smart grid
- Distributed generation
- Stochastic renewable generation
- Optimal storage
- Stochastic control
- Mean field games
- Nash equilibrium
- Extended mean field game