Abstract
We consider an energy system with n consumers who are linked by a Demand Side Management (DSM) contract, i.e. they agreed to diminish, at random times, their aggregated power consumption by a predefined volume during a predefined duration. Their failure to deliver the service is penalised via the difference between the sum of the n power consumptions and the contracted target. We are led to analyse a non-zero sum stochastic game with n players, where the interaction takes place through a cost which involves a delay induced by the duration included in the DSM contract. When \(n \rightarrow \infty \), we obtain a Mean-Field Game (MFG) with random jump time penalty and interaction on the control. We prove a stochastic maximum principle in this context, which allows to compare the MFG solution to the optimal strategy of a central planner. In a linear quadratic setting we obtain a semi-explicit solution through a system of decoupled forward-backward stochastic differential equations with jumps, involving a Riccati Backward SDE with jumps. We show that it provides an approximate Nash equilibrium for the original n-player game for n large. Finally, we propose a numerical algorithm to compute the MFG equilibrium and present several numerical experiments.
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Notes
see Energy prices and costs in Europe, European Commission, 2019.
See, for instance, Theorem 2.12 in Carmona and Delarue book (Carmona et al., 2017) Vol. II, which can be easily extended to our case with jumps exploiting in particular the fact that the intensities are constant.
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The authors thank the two anonymous referees for their insightful comments which helped to make the model more realistic.
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Alasseur, C., Campi, L., Dumitrescu, R. et al. MFG model with a long-lived penalty at random jump times: application to demand side management for electricity contracts. Ann Oper Res 336, 541–569 (2024). https://doi.org/10.1007/s10479-023-05270-0
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DOI: https://doi.org/10.1007/s10479-023-05270-0