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Electrical Vehicles in the Smart Grid: A Mean Field Game Analysis
Romain Couillet, Samir Medina Perlaza, Hamidou Tembine, Merouane Debbah
TL;DR
The paper studies how large populations of EV and PHEV owners compete in smart-grid electricity markets while balancing private operating costs and grid reliability. It formulates this interaction as a mean field game, derives equilibrium equations, and uses simulations to examine prices and demand. The simulations suggest that appropriate electricity pricing can significantly reduce daily peak demand.
Problem
Large numbers of interacting EVs and PHEVs make finite-player game analysis difficult, while charging and discharging must account for both owner revenue and electricity-supply reliability.
Method
The paper models vehicle owners as selfish players in a Cournot market and uses a mean field game with feedback Nash equilibrium equations for large populations.
Results
Simulations quantify electricity prices and instantaneous demand and suggest that appropriate pricing can significantly reduce daily electricity peak demand.
Takeaways & Limitations
The framework provides a way to analyze EV and PHEV market behavior, electricity-price evolution, and grid demand at mean field equilibrium.
Abstract
from arXiv · showhide
In this article, we investigate the competitive interaction between electrical vehicles or hybrid oil-electricity vehicles in a Cournot market consisting of electricity transactions to or from an underlying electricity distribution network. We provide a mean field game formulation for this competition, and introduce the set of fundamental differential equations ruling the behavior of the vehicles at the feedback Nash equilibrium, referred here to as the mean field equilibrium. This framework allows for a consistent analysis of the evolution of the price of electricity as well as of the instantaneous electricity demand in the power grid. Simulations precisely quantify those parameters and suggest that significant reduction of the daily electricity peak demand can be achieved by appropriate electricity pricing.
I. INTRODUCTION
The paper models EV and PHEV participation in smart-grid electricity markets as a large-population competition. It uses a mean field game to analyze equilibrium behavior, electricity prices, and grid demand.
- Motivation: EV and PHEV can consume electricity or act as mobile energy sources, creating reliability challenges for electricity distribution.They can store, transport, buy, or sell energy through the grid.
- Motivation: Optimal charging and discharging policies must balance vehicle owners’ revenues with reliability for fixed consumers.The paper frames optimality in both economic and energy-supply terms.
- Market interaction: The proposed framework allows EVs and PHEVs to buy and sell energy in a Cournot competition with the smart grid.Electricity prices depend on grid demand and the aggregate demand and supply of connected vehicles.
- Research problem: Finite-player game analysis may provide insufficient insight when the number of vehicles is very large.The paper therefore studies the limit in which vehicles are numerous and alike.
- Mean field formulation: The paper formulates this large-population interaction as a mean field game in which players respond to aggregate behavior rather than individual actions.The equilibrium is called a mean field equilibrium and is characterized by coupled HJB and FPK equations.
- Contribution: Vehicle owners are the selfish players and electricity price is the mean field variable in a finite-horizon framework for EVs and PHEVs.Numerical simulations are used to draw conclusions about future smart-grid EV penetration.
II. ELECTRICAL VEHICLES
This section formulates EV energy trading over a finite time horizon using battery state, provisioning controls, dynamic prices, and vehicle-specific costs. Each vehicle chooses admissible consumption rates to minimize total operating cost.
- Model setup: The model considers a finite set of EVs trading energy with an underlying electricity distribution grid.Each vehicle has a battery level and a provisioning rate over a predefined period [0, T].
- Cost components: Vehicle costs include electricity transactions, consumption rates, preferences over trading times, battery reserves, and terminal battery load.The terminal cost discourages owners from selling all stored energy by the end of the trading period.
- State and control: The provisioning rate represents the rate at which an EV buys or sells electricity.Admissible controls keep the battery trajectory within the allowed range.
- Pricing: The electricity price is a time-dependent function of the provisioning-rate profile and represents dynamic pricing in the distribution network.The profile depends on vehicle energy reserves and initial reserves through the game dynamics.
- Optimization: Each EV chooses admissible consumption rates over [0, T] to minimize its total cost given the rates selected by other vehicles.The resulting problem is a continuous-time differential game.
B. Classical Game Formulation
The classical formulation represents smart-grid energy trading as a finite-player continuous-time differential game. Each vehicle uses an own-state feedback strategy, while players interact through the electricity price.
- Game formulation: The game has K players over a fixed duration T > 0, with each player’s battery trajectory determined by the initial state and controls.Player k minimizes a cost function over its control trajectory.
- Strategies: Players use non-anticipative own-state feedback strategies based on the information available about their individual states.The strategy maps the individual information set to admissible controls.
- Interaction: The players’ interaction occurs through the electricity price, which depends on all players’ provisioning decisions.A player’s cost therefore depends on the global price generated by the full control profile.
- Equilibrium: The equilibrium concept is an own-state feedback Nash equilibrium in which no EV has reason to change its control policy.The equilibrium is defined over admissible controls satisfying the state evolution.
- Challenge: Finite-player Nash analysis becomes difficult because a change in one vehicle’s battery level affects other players and requires them to react.The paper introduces additional conditions to reduce this complexity.
C. Mean Field Game Formulation
The mean field formulation replaces detailed finite-player interactions with a population distribution over vehicle states. Under large-population and indistinguishability assumptions, the problem becomes a stochastic control problem coupled to aggregate demand and price.
- Mean field assumptions: The formulation assumes a sufficiently large population and indistinguishable players with similar batteries and objectives.These assumptions permit removal of individual player labels and use of a limiting state distribution.
- Population state: The state of vehicles is represented by a random battery level x_t with distribution m(t, x).The distribution is the limiting distribution of player states as the number of vehicles grows.
- State dynamics: A stochastic consumption process and reflecting boundary term keep the battery state within [0, 1].The stochastic formulation avoids assuming that all vehicles consume energy at the same rate at every instant.
- Reduced problem: Under player indistinguishability, the game reduces to a single-player stochastic control problem coupled to the population distribution.The individual control is an own-state feedback function of the battery state.
- Mean field price: The mean field electricity price is modeled as a function of the state distribution and can be related to aggregate instantaneous EV demand.The pricing formulation reflects regulators’ access to the state distribution rather than instantaneous demand.
D. Mean Field Equilibrium
The paper defines mean field equilibrium through own-state feedback strategies and derives coupled HJB and FPK equations for its analysis. The framework is extended from purely electrical vehicles to plug-in hybrid vehicles with two energy sources.
- Mean field equilibrium is defined through an equilibrium control whose induced state distribution is m⋆ for a given initial distribution.
- The equilibrium analysis uses a value function and characterizes the equilibrium through backward HJB and forward Fokker-Planck-Kolmogorov equations.
- The control cost is assumed quadratic, with H_t representing the car owner’s unwillingness to buy or sell energy at time t.
- The quadratic control cost implies that larger energy transactions are more costly, so only part of the population buys or sells energy on average.
- The resulting HJB and FPK equations are the two fundamental differential equations solved explicitly or numerically for the mean field equilibrium.
- The framework is extended to plug-in hybrid vehicles by allowing players to select between two alternative energy sources.
III. PLUG-IN HYBRID VEHICLES
The plug-in hybrid vehicle model represents each vehicle with battery and oil reserves and allows interaction with electricity and oil markets. A source-mixing function determines the relative battery contribution, but its optimization by owners is excluded for simplicity.
- Each plug-in hybrid vehicle can operate with an electrical energy source and an alternative source such as oil.
- Electricity has an elastic price determined by aggregate controls, whereas oil is traded at a fixed price.
- The state z^(k) tracks battery energy z_1,t and oil-tank level z_2,t, both represented within [0,1].
- The model assigns provisioning controls μ_1,t^(k) and μ_2,t^(k) to electricity and oil, respectively.
- The function β^(k)(t,z) determines the relative proportion of energy drawn from the battery, with β^(k)(t,z)=z_1/(z_1+z_2) representing indistinguishable source consumption.
- The model allows β to depend on travel patterns such as weekdays versus weekends, but does not let owners optimize this additional control.
- The finite-player stochastic differential game specifies costs for market transactions, vehicle states, and terminal energy reserves.
B. Classical Game Formulation
The classical formulation models vehicles as players in a fixed-duration stochastic differential game using non-anticipative own-state feedback strategies. Each player minimizes its cost given the other players’ controls, while the mean field formulation tracks the limiting state distribution.
- B. Classical Game Formulation: The interaction among PHEVs is modeled as a K-player continuous-time stochastic differential game over a fixed duration T>0.
- B. Classical Game Formulation: Each player chooses admissible controls to minimize its cost given the initial conditions and the controls adopted by all other players.
- B. Classical Game Formulation: At time t, a player’s instantaneous control is based on its available information set η_t^(k).
- B. Classical Game Formulation: The strategy class consists of non-anticipative own-state feedback strategies, with the joint admissible strategy set formed from the individual sets.
- C. Mean Field Game Formulation: Asymptotic player indistinguishability yields a weak limiting distribution as K→∞, and individual states are modeled as noisy versions of deterministic trajectories.
- C. Mean Field Game Formulation: The mean field analysis reduces the game to a single player with an identical cost function, and the next section determines the mean field equilibrium.
- C. Mean Field Game Formulation: The mean field model uses the distribution m_t=m(t,·) of the state variable as the population-level state.
D. Mean Field Analysis
The mean field analysis reduces equilibrium behavior to a single-player optimal control problem with a quadratic cost. Its value function and HJB formulation provide necessary conditions for an equilibrium generating regular value-distribution couples.
- The equilibrium formulation is an optimal control problem for a single player under the mean field game.
- The analysis introduces a value function with initial value z_u=y to represent the controlled state problem.
- The cost function is quadratic, with coefficients (Q_1,t,Q_2,t)∈R^2.
- The HJB equation provides a necessary condition for an MFE generating a regular couple (v,m).
- The equilibrium distribution and aggregate quantities are represented by m⋆ and μ⋆ in the mean field formulation.
- Assuming σ_t=0 yields more compact equations, including the HJB equation and its final expression v⋆.
IV. SIMULATIONS
Simulations of a three-day EV scenario show that price-responsive charging shifts consumption toward low-demand periods, reducing critical demand peaks while increasing off-peak consumption. The analysis also characterizes battery evolution, electricity purchases, and price formation under fixed non-EV demand.
- Price formation: The price is mostly driven by the deterministic demand function for non-EV services, which the analysis assumes does not change with EV electricity prices.This assumption treats the EV electricity market as independent of the outer electricity trading market.
- Model specification: The model constrains battery energy to [0, 1] and uses f(t, x) = (1 − x)^2 plus κ(x) = (1 − x)^2 to discourage low reserves and last-minute sales.The initial battery distribution is triangular, centered at 0.5, with support [0.3, 0.7].
- Battery dynamics: Battery levels increase during nighttime, indicating nighttime electricity purchases followed by daytime consumption.The simulated battery distribution exhibits daily increases and decreases in average battery levels.
- Electricity purchases: EV electricity purchases peak at night and are lowest during peak-demand periods because prices are higher during those periods.Despite this response, the amplitude difference between the lowest and highest purchases remains limited because owners avoid empty batteries.
- Demand response: Price incentives reduce demand during critical periods while simultaneously increasing consumption during low-consumption periods.The comparison is between evenly distributed EV purchases without incentives and consumption under the modeled price incentives.
B. PHEV analysis
The PHEV mean field analysis models vehicles with coupled oil and electricity states and examines their equilibrium transactions, distributions, consumption, and prices. Under the stated pricing and energy-cost assumptions, simulations show increased battery levels, near-equivalence of oil and electricity, and state-dependent electricity purchases.
- Model and setup: The PHEV model solves coupled HJB and FPK differential equations with a fixed-point algorithm under time-independent consumption and transaction parameters.The discretization uses 12 time samples and 16 samples for each spatial scale.
- Grid outcomes: The study compares electricity purchased by vehicles, total consumption with or without EV regulation, and the evolution of the electricity price over time.The electricity-purchase peaks and regulated-versus-unregulated consumption are presented as time-dependent simulation outputs.
- Model and setup: The initial vehicle distribution is a truncated, scaled Gaussian with mean (0.4, 0.6)^T and covariance 0.02I_2, indicating more oil than electricity initially.The distribution is evolved freely under the model constraints.
- Distribution evolution: The final distribution shifts toward higher oil and electricity levels, with a stronger increase in mean battery level.It also stretches along the z_1 = z_2 diagonal, indicating that oil and electricity are treated as almost equivalent under the loosely constraining energy-cost policy.
- Optimal transactions: At t = 0+, the electricity price is r_1,t = 0.706 ≃ r_2, so vehicles with low electricity reserves purchase electricity to reach price parity with oil.For vehicles with larger oil quantities, electricity is also attractive for increasing total energy; when battery and tank levels match, purchased electricity equals purchased oil.
- Scope: The simulations provide a rational-vehicle-owner framework, but applying its interpretations to real-life conditions requires extreme care.The authors note that many additional scenarios are needed to assess actual EV and PHEV impacts in realistic smart-grid settings.
V. CONCLUSION
The article models electrical and hybrid vehicle owners as selfish players in a large-population mean field game. Its equilibrium equations and numerical analysis provide insights for optimizing future electric-vehicle penetration in the smart grid.
- The framework models electrical and hybrid electricity-oil vehicle owners as selfish agents minimizing their operating costs.
- Because the number of similar selfish players is large, the competition is formulated as a mean field game.
- The analysis derives fundamental differential equations describing the game's mean field equilibrium.
- Numerical methods produce conclusions offering new insights into the modeled vehicle interaction.
- The framework is intended to support optimization of electrical vehicle penetration in future smart grids.