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Trading in residential energy systems with storage: a kinetic mean-field approach

Margherita Fabini, Andrea Pascucci, Alessio Rondelli

Abstract

We study a stochastic optimal control problem motivated by the operation of a large ensemble of residential storage devices coordinated by an energy aggregator. The aggregator remunerates prosumers in exchange for direct control of their batteries and seeks to jointly (i) reduce local supply-demand imbalances and (ii) exploit intraday price fluctuations through energy arbitrage. The core modeling feature is a kinetic mean-field formulation: the state of charge is treated as a position, the charging/discharging power as a velocity, and the control as an acceleration, thus encoding ramp-rate limitations and producing smooth power trajectories. This leads to a controlled McKean-Vlasov Langevin-type system in which both the drift and the objective functional depend on the time-marginal law of the state, allowing one to capture endogenous interaction effects and population-level stabilization incentives. The performance criterion combines the cost of grid exchange with convex penalties representing degradation and control effort, and includes mean-field terms that promote alignment with the population average; terminal contributions account for residual energy value and end-of-horizon coordination. The resulting control problem is Markovian and hypoelliptic, and naturally connects mean-field control with ultraparabolic operators of kinetic type. This viewpoint provides a coherent bridge between physically constrained storage actuation and law-dependent incentives in large-scale energy management. Numerical experiments based on deep learning solvers are presented to validate the model. From a computational standpoint, the problem is particularly challenging, as it yields a fully coupled forward-backward stochastic system associated with a five-dimensional Hamilton-Jacobi-Bellman equation.

Trading in residential energy systems with storage: a kinetic mean-field approach

Abstract

We study a stochastic optimal control problem motivated by the operation of a large ensemble of residential storage devices coordinated by an energy aggregator. The aggregator remunerates prosumers in exchange for direct control of their batteries and seeks to jointly (i) reduce local supply-demand imbalances and (ii) exploit intraday price fluctuations through energy arbitrage. The core modeling feature is a kinetic mean-field formulation: the state of charge is treated as a position, the charging/discharging power as a velocity, and the control as an acceleration, thus encoding ramp-rate limitations and producing smooth power trajectories. This leads to a controlled McKean-Vlasov Langevin-type system in which both the drift and the objective functional depend on the time-marginal law of the state, allowing one to capture endogenous interaction effects and population-level stabilization incentives. The performance criterion combines the cost of grid exchange with convex penalties representing degradation and control effort, and includes mean-field terms that promote alignment with the population average; terminal contributions account for residual energy value and end-of-horizon coordination. The resulting control problem is Markovian and hypoelliptic, and naturally connects mean-field control with ultraparabolic operators of kinetic type. This viewpoint provides a coherent bridge between physically constrained storage actuation and law-dependent incentives in large-scale energy management. Numerical experiments based on deep learning solvers are presented to validate the model. From a computational standpoint, the problem is particularly challenging, as it yields a fully coupled forward-backward stochastic system associated with a five-dimensional Hamilton-Jacobi-Bellman equation.
Paper Structure (16 sections, 5 theorems, 67 equations, 7 figures, 1 table)

This paper contains 16 sections, 5 theorems, 67 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. A kinetic mean-field approach
  3. Related literature and numerical aspects
  4. General strategy and main results
  5. Semilinear kinetic equations
  6. McKean-Vlasov optimal control problem
  7. Numerical tests
  8. Electricity price
  9. Net home load
  10. Battery power and cost functional
  11. FBSNN and the optimal control
  12. Numerical experiments
  13. Proof of Theorem \ref{['th3']}
  14. Kinetic FBSDEs
  15. Frozen optimal control problem
  16. ...and 1 more sections

Key Result

Theorem 2.3

Under Assumption ass7, the Cauchy problem Cauchy admits a unique classical solution $u\in \mathbf{C}^{2+{\alpha}}_{\mathcal{L},t}$ for any ${\alpha}\in(0,\bar{{\alpha}})$ and $t\in[0,T)$, such that for any $z,z'\in{\mathbb {R}}^{2}$ and $t,t'\in(0,T)$.

Figures (7)

  • Figure 1: Second week of June 2025 and first week of December 2025: observed PUN price (red) and 80% confidence bands of the calibrated model (light blue).
  • Figure 2: Representative weekly input profiles in 2025 (second week of June and first week of December): household consumption (gray), photovoltaic production (green), and net home load (red).
  • Figure 3: Second week of June 2025 and first week of December 2025: observed net load (red) and $80\%$ confidence bands of the calibrated model (light blue).
  • Figure 4: Second week of June 2025 and first week of December 2025: comparison of the accumulated cost. Monte Carlo mean for the uncontrolled benchmark (blue) and for the optimal control strategy (red), with a $90\%$ percentile band (shaded) and three representative sample trajectories under optimal control.
  • Figure : (a) June 2025
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 2.3
  • Definition 2.4: Anisotropic Hölder spaces
  • Definition 2.5: Intrinsic Hölder spaces
  • Remark 2.6
  • Theorem 2.8: Well-posedness for the McKean-Vlasov optimal control problem \ref{['MKVCP']}
  • Remark 2.9
  • Definition 4.1: Decoupling field
  • Remark 4.2
  • Theorem 4.5
  • proof
  • ...and 2 more