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arxiv: 2604.05269 · v1 · submitted 2026-04-07 · 📡 eess.SY · cs.SY· math.OC

Price-Coordinated Mean Field Games with State Augmentation for Decentralized Battery Charging

Nour Al Dandachly , Shuang Gao , Roland Malham\'e This is my paper

Pith reviewed 2026-05-10 20:11 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords mean field gamesdecentralized battery chargingstate augmentationprice coordinationRiccati equationsforward-backward differential equations
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The pith

Mean field games prove a unique equilibrium exists for decentralized battery charging under any continuous monotonic price signal, with no time-horizon restrictions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models a large population of battery agents whose charging is coordinated only through a price that rises with the gap between average charging power and a grid target. Each agent augments its state so that charging power itself becomes a state variable while its rate of change (ramp rate) is the control input. This produces a mean-field game whose equilibrium is characterized by a pair of nonlinearly coupled forward-backward differential equations. The authors prove existence and uniqueness of this equilibrium for any continuous and monotonically increasing price function, without any further conditions on the length of the time interval. In the special case of an affine price, the same equilibrium reduces to two independent Riccati equations that always possess unique positive semi-definite solutions.

Core claim

For agents obeying affine dynamics and coupled only through a continuous monotonically increasing price that depends on the deviation of mean charging power from a desired value, the associated mean-field game admits a unique equilibrium strategy for every finite time horizon. The equilibrium is characterized by two nonlinearly coupled forward-backward differential equations; when the price is affine these equations decouple into two Riccati equations, each of which possesses a unique positive semi-definite solution without additional assumptions.

What carries the argument

State-augmented mean-field game in which charging power is promoted to a state variable and its derivative (ramp rate) serves as the control, with population coupling realized solely through a monotonic price function of the average charging power.

If this is right

  • Decentralized charging policies for electric vehicles or residential batteries can be computed by solving the forward-backward system once the price function is specified.
  • When the price is linear in average power, each agent needs only to solve two independent Riccati equations to obtain its optimal ramp-rate strategy.
  • The framework extends immediately to any application in which agents have affine dynamics and are penalized through a monotonic function of their collective average action.
  • Coordination is achieved without a central dispatcher; each agent reacts only to the prevailing price signal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same state-augmentation technique could be applied to other ramp-constrained resources such as thermostatically controlled loads or distributed generators.
  • Because uniqueness holds for arbitrary horizons, the equilibrium can be used as a building block for receding-horizon implementations that react to real-time renewable fluctuations.
  • If the monotonicity assumption on price is relaxed, existence may fail; this supplies a clear boundary condition for market-design experiments.

Load-bearing premise

The agents obey affine dynamics and the population is large enough that the mean-field approximation accurately captures the effect of the average charging power on price.

What would settle it

A numerical example with a non-monotonic price function in which either no equilibrium exists or multiple distinct equilibria appear, or a finite-population simulation whose individual strategies deviate substantially from the mean-field prediction.

Figures

Figures reproduced from arXiv: 2604.05269 by Nour Al Dandachly, Roland Malham\'e, Shuang Gao.

Figure 1
Figure 1. Figure 1: Price functions as a function of the deviation [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: MFG equilibrium results for Q ̸= 0. (a)–(b) Optimal charging power under sigmoid and affine price coordination (solid), with individual trajectories (light blue) and uncoor￾dinated baseline (red dashed). (c)–(d) Corresponding SOC trajectories. (e) Equilibrium price signals. (f) Mean field consistency verification. C. Results: Price and Running Cost Coordination (Q ̸= 0) We next include a running cost (Q = … view at source ↗
Figure 2
Figure 2. Figure 2: MFG equilibrium results for Q = 0. (a)–(b) Optimal charging power under sigmoid and affine price coordination (solid), and uncoordinated profile (red dashed). (c)–(d) Cor￾responding SOC trajectories. (e) Equilibrium price signals. (f) Mean field consistency verification. produces a bell-shaped power profile that peaks well above rg, illustrating the benefit of price coordination for grid compliance. The co… view at source ↗
read the original abstract

This paper addresses the decentralized coordinated charging problem for a large population of battery storage agents (e.g. residential batteries, electrical vehicles, charging station batteries) using Mean Field Game (MFG). Agents are assumed to have affine dynamics and are coupled through a price that is continuous and monotonically increasing with respect to the difference between the average charging power and the grid's desired average charging power. An important modeling feature of the proposed framework is the state augmentation, that is, the charging power is treated as a state variable and its rate of change (i.e. the ramp rate) as the control input. The resulting MFG equilibrium is characterized by two nonlinearly coupled forward-backward differential equations. The existence and uniqueness of the MFG equilibrium is established for any continuous and monotonically increasing nonlinear price function without additional restrictions on the time horizon. Moreover, in the special case where the price is affine in the average charging power, we further simplify the characterization of the MFG equilibrium strategy via two separate Riccati equations, both of which admit unique positive semi-definite solutions without additional assumptions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a mean-field game (MFG) framework for decentralized coordination of battery charging among a large population of agents with affine dynamics. By augmenting the state to include charging power (with ramp rate as control), agents are coupled through a price that is continuous and monotonically increasing in the deviation of average charging power from a grid-desired value. The MFG equilibrium is characterized by two nonlinearly coupled forward-backward differential equations. Existence and uniqueness are claimed for any such price function and any finite time horizon T. In the affine-price special case, the equilibrium reduces to two Riccati equations that admit unique positive semi-definite solutions without additional assumptions.

Significance. If the existence/uniqueness results hold, the work provides a rigorous, scalable decentralized approach to large-scale battery/EV charging coordination that accommodates nonlinear price signals without time-horizon restrictions. The state-augmentation modeling choice naturally incorporates ramp-rate effects, and the Riccati reduction (with claimed parameter-free solutions) is a practical strength for implementation. This advances MFG applications in power systems and could inform real-time grid management under high renewable penetration.

major comments (2)
  1. [Abstract] Abstract (existence and uniqueness claim): the central result asserts existence and uniqueness of the MFG equilibrium for any continuous and monotonically increasing nonlinear price function without additional restrictions on T. Standard MFG theory for coupled FBDEs in augmented state space typically requires at least local Lipschitz continuity plus linear/sub-quadratic growth on the coupling to ensure compactness of the fixed-point map or well-posedness of the HJB; the manuscript should explicitly state and verify whether continuity alone suffices or if implicit regularity is used in the proof.
  2. [Affine price case] Affine price case (Riccati simplification): the claim that the two Riccati equations admit unique positive semi-definite solutions without additional assumptions is load-bearing for the practical simplification. The augmented dynamics are affine, so standard finite-horizon Riccati theory applies, but the manuscript must exhibit the explicit matrix forms and confirm that no controllability/detectability conditions on the augmented system are needed for uniqueness of the PSD solution.
minor comments (2)
  1. The abstract states the equilibrium is characterized by 'two nonlinearly coupled forward-backward differential equations' but does not cross-reference their explicit forms; add equation numbers in the main text for the coupled FBDEs to improve traceability.
  2. Notation for the mean-field term (average charging power) should be introduced with a clear distinction from individual agent power in the model section to avoid ambiguity when the price function is defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment point by point below, providing clarifications on the technical details and indicating revisions to improve the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract (existence and uniqueness claim): the central result asserts existence and uniqueness of the MFG equilibrium for any continuous and monotonically increasing nonlinear price function without additional restrictions on T. Standard MFG theory for coupled FBDEs in augmented state space typically requires at least local Lipschitz continuity plus linear/sub-quadratic growth on the coupling to ensure compactness of the fixed-point map or well-posedness of the HJB; the manuscript should explicitly state and verify whether continuity alone suffices or if implicit regularity is used in the proof.

    Authors: We appreciate the referee pointing out the need for explicit regularity details. In the proof of existence and uniqueness (Section 3), the state augmentation renders the individual dynamics linear, allowing us to apply a fixed-point argument on the space of probability measures. The monotonicity of the continuous price function ensures the coupling operator is monotone, which combined with the linear growth from the affine dynamics yields compactness and uniqueness without requiring additional Lipschitz continuity on the price itself. Continuity and monotonicity are sufficient in this augmented setup for any finite T, as the forward-backward structure avoids the general growth restrictions. We will revise the abstract, introduction, and proof to explicitly state and verify these conditions. revision: yes

  2. Referee: [Affine price case] Affine price case (Riccati simplification): the claim that the two Riccati equations admit unique positive semi-definite solutions without additional assumptions is load-bearing for the practical simplification. The augmented dynamics are affine, so standard finite-horizon Riccati theory applies, but the manuscript must exhibit the explicit matrix forms and confirm that no controllability/detectability conditions on the augmented system are needed for uniqueness of the PSD solution.

    Authors: We agree that the explicit matrix forms and justification are needed for rigor. In the revised manuscript, we will derive and display the explicit matrix Riccati differential equations obtained after substituting the affine price into the augmented-state Hamiltonian. For the finite-horizon case, the backward Riccati equation starting from the zero terminal condition admits a unique solution by standard existence theory for linear-quadratic problems; positive semi-definiteness follows directly from the quadratic cost structure without invoking controllability or detectability, which are typically required only for infinite-horizon or steady-state analysis. We will add this explicit derivation and confirmation to Section 4. revision: yes

Circularity Check

0 steps flagged

No circularity: standard MFG fixed-point argument applied to new model

full rationale

The derivation begins from affine agent dynamics and a continuous monotone price coupling, augments the state with charging power, obtains the standard MFG forward-backward system, and invokes monotonicity to conclude existence and uniqueness for arbitrary finite T. This is a direct application of classical MFG theory (monotone coupling yields uniqueness; existence follows from standard compactness or fixed-point arguments under the stated continuity). No parameter is fitted to data and then renamed a prediction, no self-citation supplies the uniqueness theorem, and the Riccati reduction in the linear-price case is the ordinary LQ-MFG result. The central claim therefore does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on standard mean-field game assumptions plus the explicit modeling choice of affine dynamics and monotonic price coupling; no free parameters are introduced and no new entities are postulated.

axioms (2)
  • domain assumption Battery agents have affine dynamics
    Stated directly in the abstract as the basis for the state and control formulation.
  • domain assumption Price is continuous and monotonically increasing in the deviation of average charging power from the grid target
    Core coupling mechanism used to establish the equilibrium characterization.

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Global EV outlook 2024,

    International Energy Agency, “Global EV outlook 2024,” IEA, Paris, Tech. Rep., 2024

    work page 2024

  2. [2]

    The role of energy storage in deep decarbonization of electricity production,

    M. Arbabzadeh, R. Sioshansi, J. Johnson, and G. Keoleian, “The role of energy storage in deep decarbonization of electricity production,” Nature communications, vol. 10, no. 1, p. 3593, 2019

    work page 2019

  3. [3]

    The role of transmission and energy storage in European decarbonization towards 2050,

    R. Golombek, A. Lind, H. Ringkjøb, and P. Seljom, “The role of transmission and energy storage in European decarbonization towards 2050,”Energy, vol. 239, p. 121811, 2022

    work page 2050

  4. [4]

    Decentralized charging control of large populations of plug-in electric vehicles,

    Z. Ma, D. S. Callaway, and I. A. Hiskens, “Decentralized charging control of large populations of plug-in electric vehicles,”IEEE Transactions on Control Systems Technology, vol. 21, no. 1, pp. 67–78, 2013

    work page 2013

  5. [5]

    Optimal decentralized protocol for electric vehicle charging,

    L. Gan, U. Topcu, and S. H. Low, “Optimal decentralized protocol for electric vehicle charging,”IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 940–951, 2013

    work page 2013

  6. [6]

    Mean field games,

    J.-M. Lasry and P.-L. Lions, “Mean field games,”Japanese Journal of Mathematics, vol. 2, no. 1, pp. 229–260, 2007

    work page 2007

  7. [7]

    Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle,

    M. Huang, R. P. Malham ´e, and P. E. Caines, “Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle,”Communications in Information & Systems, vol. 6, no. 3, pp. 221–252, 2006

    work page 2006

  8. [8]

    A mean-field game approach to price formation,

    D. A. Gomes and J. Sa ´ude, “A mean-field game approach to price formation,”Dynamic Games and Applications, vol. 11, no. 1, pp. 29–53, 2021

    work page 2021

  9. [9]

    Mean field games for renewable energy development,

    L. Campi and Z. Wu, “Mean field games for renewable energy development,”arXiv preprint arXiv:2603.23156, 2026

    work page arXiv 2026

  10. [10]

    An extended mean field game for storage in smart grids,

    C. Alasseur, I. Ben Taher, and A. Matoussi, “An extended mean field game for storage in smart grids,”Journal of Optimization Theory and Applications, vol. 184, no. 2, pp. 644–670, 2020

    work page 2020

  11. [11]

    Decen- tralized convergence to Nash equilibria in constrained deterministic mean field control,

    S. Grammatico, F. Parise, M. Colombino, and J. Lygeros, “Decen- tralized convergence to Nash equilibria in constrained deterministic mean field control,”IEEE Transactions on Automatic Control, vol. 61, no. 11, pp. 3315–3329, 2016

    work page 2016

  12. [13]

    Mean-field games and dynamic demand management in power grids,

    F. Bagagiolo and D. Bauso, “Mean-field games and dynamic demand management in power grids,”Dynamic Games and Applications, vol. 4, no. 2, pp. 155–176, 2014

    work page 2014

  13. [14]

    Data-driven mean-field game approx- imation for a population of electric vehicles,

    D. Bauso and T. Namerikawa, “Data-driven mean-field game approx- imation for a population of electric vehicles,” in2019 IEEE Data Science Workshop (DSW), 2019, pp. 285–289

    work page 2019

  14. [15]

    Price coordination for electric vehicle fleet using mean field game,

    J. Hedel, S. Grammaticoet al., “Price coordination for electric vehicle fleet using mean field game,”IEEE Transactions on Control Systems Technology, 2024

    work page 2024

  15. [16]

    A mean-field game method for decentralized charging coordination of a large population of plug-in electric vehicles,

    M. A. Tajeddini and H. Kebriaei, “A mean-field game method for decentralized charging coordination of a large population of plug-in electric vehicles,”IEEE Systems Journal, vol. 13, no. 1, pp. 854–863, 2019

    work page 2019

  16. [17]

    Electrical vehicles in the smart grid: A mean field game analysis,

    R. Couillet, S. M. Perlaza, H. Tembine, and M. Debbah, “Electrical vehicles in the smart grid: A mean field game analysis,”IEEE Journal of Selected Topics in Signal Processing, vol. 6, no. 4, pp. 388–399, 2012

    work page 2012

  17. [18]

    On aggregative and mean field games with applications to smart grids,

    D. Paccagnan, M. Kamgarpour, S. Zampieri, S. Bolognani, and J. Lygeros, “On aggregative and mean field games with applications to smart grids,” in2016 European Control Conference (ECC), 2016, pp. 387–392

    work page 2016

  18. [19]

    A mean-field game approach to charging of plug-in electric vehicles,

    S. Gong and T. Ba s ¸ar, “A mean-field game approach to charging of plug-in electric vehicles,” in2016 American Control Conference (ACC), 2016, pp. 1978–1983

    work page 2016

  19. [20]

    Optimal scheduling management of the parking lot and electric vehicle charging based on mean field game,

    R. Lin, L. Li, Y . Li, and X. Zhang, “Optimal scheduling management of the parking lot and electric vehicle charging based on mean field game,”Energy, vol. 254, p. 124359, 2022

    work page 2022

  20. [21]

    On a class of linear quadratic gaussian quantilized mean field games,

    R. F. Tchuendom, R. Malham ´e, and P. E. Caines, “On a class of linear quadratic gaussian quantilized mean field games,”Automatica, vol. 170, p. 111878, 2024

    work page 2024

  21. [22]

    Large- scale EV charging coordination: a detailed exploration of mean-field and reinforcement learning approaches,

    J. Caball ´e, P. Segovia, C. Ocampo-Martinez, and N. Quijano, “Large- scale EV charging coordination: a detailed exploration of mean-field and reinforcement learning approaches,”Control Engineering Practice, vol. 168, p. 106669, 2026

    work page 2026

  22. [23]

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

    M. Fabini, A. Pascucci, and A. Rondelli, “Trading in residential energy systems with storage: a kinetic mean-field approach,”arXiv preprint arXiv:2603.00713, 2026

    work page arXiv 2026

  23. [24]

    Linear- quadratic mean field games,

    A. Bensoussan, K. C. J. Sung, S. C. P. Yam, and S. P. Yung, “Linear- quadratic mean field games,”Journal of Optimization Theory and Applications, vol. 169, pp. 496–529, 2016

    work page 2016

  24. [25]

    Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ϵ-Nash equilibria,

    M. Huang, P. E. Caines, and R. P. Malham ´e, “Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ϵ-Nash equilibria,”IEEE Transactions on Automatic Control, vol. 52, no. 9, pp. 1560–1571, 2007

    work page 2007

  25. [26]

    Ramp-rate control approach based on dynamic smoothing,

    S. Sukumar, M. Mohanpurkar, and R. Jackson, “Ramp-rate control approach based on dynamic smoothing,”Applied Energy, vol. 220, pp. 726–737, 2018

    work page 2018

  26. [27]

    Optimal charging of electric vehicles in low-voltage distribution systems,

    P. Richardson, D. Flynn, and A. Keane, “Optimal charging of electric vehicles in low-voltage distribution systems,”IEEE Transactions on Power Systems, vol. 27, no. 1, pp. 268–279, 2012

    work page 2012

  27. [28]

    Yong and X

    J. Yong and X. Y . Zhou,Stochastic Controls: Hamiltonian Systems and HJB Equations, ser. Applications of Mathematics. New York: Springer, 1999, vol. 43

    work page 1999

  28. [29]

    Brezis,Functional Analysis, Sobolev Spaces and Partial Differential Equations, ser

    H. Brezis,Functional Analysis, Sobolev Spaces and Partial Differential Equations, ser. Universitext. New York: Springer, 2011

    work page 2011

  29. [30]

    W. H. Fleming and R. W. Rishel,Deterministic and Stochastic Optimal Control, ser. Applications of Mathematics. New York: Springer-Verlag, 1975

    work page 1975

  30. [31]

    Convex analysis for LQG systems with applications to major-minor LQG mean-field game systems,

    D. Firoozi, S. Jaimungal, and P. E. Caines, “Convex analysis for LQG systems with applications to major-minor LQG mean-field game systems,”Systems & Control Letters, vol. 142, p. 104734, 2020

    work page 2020

  31. [32]

    A mean-field game approach to equilibrium pricing in solar renewable energy certificate markets,

    A. V . Shrivats, D. Firoozi, and S. Jaimungal, “A mean-field game approach to equilibrium pricing in solar renewable energy certificate markets,”Mathematical Finance, vol. 32, no. 3, pp. 779–824, 2022

    work page 2022

  32. [33]

    Data-driven network LQG mean field games with heterogeneous populations via integral reinforcement learning,

    J. Zhu and S. Gao, “Data-driven network LQG mean field games with heterogeneous populations via integral reinforcement learning,” inthe 25th International Symposium on Mathematical Theory of Networks and Systems (MTNS), 2026, submitted

    work page 2026

  33. [34]

    Linear quadratic mean field games with quantile-dependent cost coefficients,

    S. Gao and R. P. Malham ´e, “Linear quadratic mean field games with quantile-dependent cost coefficients,”Journal of Systems Science and Complexity, vol. 38, no. 1, pp. 495–510, 2025

    work page 2025

  34. [35]

    Teschl,Ordinary Differential Equations and Dynamical Systems, ser

    G. Teschl,Ordinary Differential Equations and Dynamical Systems, ser. Graduate Studies in Mathematics. Providence, Rhode Island: American Mathematical Society, 2012, vol. 140

    work page 2012