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Economics of Electric Vehicle Charging: A Game Theoretic Approach

Wayes Tushar, Walid Saad, H. Vincent Poor, David. B. Smith

arXiv:1208.0631v1cs.GTcs.IT

TL;DR

The paper studies how a smart grid and plug-in electric vehicle groups should exchange limited energy while balancing grid revenue and charging benefits. It formulates their interaction as a noncooperative Stackelberg game, analyzes equilibrium using variational inequalities, and proposes distributed and time-varying extensions. The analysis establishes a socially optimal equilibrium, with simulations showing the proposed scheme outperforming two comparison schemes in average utility.

  • Problem

    Limited grid capacity and PEV energy demands create conflicting pricing and charging decisions that require a distributed grid-to-vehicle model.

  • Method

    The paper models the SG as a Stackelberg leader and PEVGs as followers, analyzes their generalized Nash and variational equilibria, and proposes a distributed algorithm with a time-varying extension.

  • Results

    The proposed game admits a socially optimal Stackelberg equilibrium, and its scheme achieves average utility 1.6 times that of PSO and up to 3.5 times ED at N = 25 PEVGs.

  • Takeaways & Limitations

    The equilibrium framework provides a socially optimal price-and-charging solution for constrained SG–PEVG energy exchange and can handle slowly varying environments.

Abstract

from arXiv · show

In this paper, the problem of grid-to-vehicle energy exchange between a smart grid and plug-in electric vehicle groups (PEVGs) is studied using a noncooperative Stackelberg game. In this game, on the one hand, the smart grid that acts as a leader, needs to decide on its price so as to optimize its revenue while ensuring the PEVGs' participation. On the other hand, the PEVGs, which act as followers, need to decide on their charging strategies so as to optimize a tradeoff between the benefit from battery charging and the associated cost. Using variational inequalities, it is shown that the proposed game possesses a socially optimal Stackelberg equilibrium in which the grid optimizes its price while the PEVGs choose their equilibrium strategies. A distributed algorithm that enables the PEVGs and the smart grid to reach this equilibrium is proposed and assessed by extensive simulations. Further, the model is extended to a time-varying case that can incorporate and handle slowly varying environments.

I. INTRODUCTION

The paper frames PEV charging as a coordinated energy-allocation problem with conflicting grid and vehicle objectives under limited capacity. It motivates a distributed game-theoretic model for pricing, charging decisions, and energy exchange.

  • Motivation: Widespread PEV adoption creates charging-strategy, communication, and grid-management challenges.Simultaneous charging can overload the network and potentially double average load.
  • Motivation: The central modeling need is to capture the grid’s revenue objective and PEVs’ charging objectives when limited energy must be allocated by demand.The grid seeks revenue, while PEVGs balance charging benefits against costs and physical constraints.
  • Contribution: The proposed framework models an SG and multiple PEV groups as a generalized Stackelberg game with the SG as leader and PEVGs as followers.PEVGs choose charging profiles, while the SG chooses price in response to their strategies.
  • System model: Charging is divided into time slots, with grid capacity C and price p determining how surplus energy is sold to N PEVGs.Time slots last between 5 minutes and half an hour depending on changing PEV traffic conditions.
  • System model: PEVG demand depends on battery capacity, initial battery energy, price, and usage needs, while the grid selects a price that balances revenue and participation.A very high price may cause a PEVG to withdraw demand, whereas a very low price may reduce revenue.
  • Scope: The model is intended to support distributed decision-making and can extend beyond the considered period or to multiple energy sources.The paper notes that the proposed scheme can apply to any time duration and can be extended beyond the main grid.

III. NONCOOPERATIVE GENERALIZED STACKELBERG GAME

The game formulation assigns price-setting to the smart grid and energy-demand decisions to PEV groups, whose utilities and actions are coupled by shared capacity.

  • Game formulation: The SG is the leader, and PEVGs are followers that respond to the price set by the grid.This is formulated as a noncooperative Stackelberg game for multilevel decision-making.
  • Follower strategies: Each PEVG chooses demanded energy x_n from its feasible strategy set while satisfying the shared constraint Σ_n x_n ≤ C.The shared capacity couples the followers’ decisions.
  • Utilities: Each PEVG utility captures the benefit of consuming its demanded energy, while the SG utility captures total profit from selling surplus energy at price p.The leader’s price is the grid’s strategic variable.
  • Utilities: A PEVG’s utility depends on its energy consumption, other PEVGs’ strategies, satisfaction parameter, battery capacity, and grid price.The satisfaction parameter reflects factors such as battery state, available grid energy, and travel plans.

1) Utility Function of a PEVG:

The PEVG utility models charging satisfaction against price and vehicle-specific conditions, while imposing diminishing benefits and a maximum satisfaction level.

  • Utility properties: The marginal benefit of charging is non-increasing, so satisfaction gradually saturates as more energy is consumed.The model assumes a PEVG does not consume energy beyond its maximum satisfaction level.
  • Utility properties: For fixed consumption, larger battery capacity increases utility, while a larger satisfaction parameter decreases utility under the paper’s formulation.These comparative properties are stated for Un(x_n, x_-n, s_n, b_n, p).
  • Utility properties: Higher electricity prices decrease PEVG utility.Price is therefore part of the tradeoff governing each group’s requested energy.
  • Utility function: PEVG utility is modeled as a function of requested energy, other groups’ demands, satisfaction parameter, battery capacity, and price.The satisfaction parameter may reflect battery state, available grid energy, or travel plans.
  • Charging cost: The PEVG pays p x_n for consuming x_n MWh at price p per MWh.This payment is the cost imposed by the SG on the PEVG.

2) Utility Function of the Power grid:

The grid’s utility is its revenue from selling surplus energy, optimized through price selection against PEVG demands and the shared capacity constraint.

  • Grid utility: The grid chooses price p to maximize revenue from selling surplus energy after serving primary consumers.Its revenue depends on the energy required by all PEVGs and the price per unit of energy.
  • Follower response: For a fixed price, PEVGs choose demands that maximize their utilities subject to the common energy constraint.Their coupled actions form a resource-sharing generalized Nash equilibrium problem.
  • Equilibrium: At the GSE, the grid’s price is optimized given the followers’ equilibrium demands.This places leader optimization after the followers’ equilibrium response.
  • Equilibrium: The shared constraint makes PEVG demands interdependent, so the followers seek a generalized Nash equilibrium rather than a classical Nash equilibrium.Each demand depends on both its own strategy and the demands of other PEVGs.
  • Equilibrium: A generalized Stackelberg equilibrium pairs the grid’s optimal price with followers’ GNE demands, leaving no player able to improve unilaterally.The equilibrium strategies satisfy the paper’s defining inequalities.

B. Existence and efficiency of GSE

For a fixed grid price, the followers’ game has a socially stable variational equilibrium, and this equilibrium supports a socially optimal Stackelberg equilibrium when the grid optimizes its price.

  • A variational equilibrium is targeted because it is socially more stable than other possible generalized Nash equilibria.
  • The resulting Stackelberg equilibrium is socially optimal when the smart grid sets its optimal price in response to the followers’ variational-equilibrium demands.
  • The followers’ generalized Nash equilibrium problem reduces to a variational inequality over their jointly constrained strategy set.
  • The operator’s positive-definite Jacobian makes the variational inequality strictly monotone, yielding a unique global variational equilibrium.
  • Because the game is jointly convex, the variational equilibrium is the unique global maximizer of the corresponding social objective.

IV. PROPOSED SOLUTION AND ALGORITHM

The paper formulates the followers’ game as a strongly monotone variational inequality and proposes solving it to obtain their unique equilibrium and, subsequently, the game’s socially optimal equilibrium.

  • The followers’ generalized Nash equilibrium problem is formulated as a variational inequality whose solution is the socially optimal variational equilibrium.
  • For every fixed price, the associated variational inequality is strongly monotone, so the followers’ variational equilibrium is unique.
  • The variational inequality is defined by finding z∗ in X such that ⟨F(z∗), z − z∗⟩ ≥ 0 for every z in X.
  • The proposed procedure first computes the followers’ equilibrium for a price, then uses that equilibrium to optimize the grid’s price and reach the game equilibrium.

A. GNE for a fixed p

For a fixed price, PEVG demands are coupled by the shared grid-energy constraint; at the variational equilibrium, total demand exhausts available energy under the stated capacity condition.

  • A. GNE for a fixed p: At the variational equilibrium, the sum of all connected PEVGs’ demands equals the grid’s available energy C.
  • A. GNE for a fixed p: The shared constraint couples PEVG demands, creating a jointly convex generalized Nash equilibrium problem in which groups compete for scarce grid energy.
  • A. GNE for a fixed p: The total capacities of the connected PEVGs must exceed their total variational-equilibrium demand by pN to achieve maximum utilities.
  • A. GNE for a fixed p: With heterogeneous capacities and a common satisfaction parameter, the model imposes the condition b_n > pN + sC.
  • A. GNE for a fixed p: The algorithm has each PEVG submit an initial demand, compute projections, and iteratively update its demand toward the equilibrium.

B. Price optimization

After determining the followers’ variational-equilibrium demands, the smart grid selects the price that maximizes its revenue subject to the equilibrium conditions.

  • B. Price optimization: The smart grid determines its optimal price after analyzing the followers’ game and obtaining their variational-equilibrium demands.
  • B. Price optimization: The grid’s revenue-maximizing price is the upper limit imposed by the equilibrium condition.

C. Proposed algorithm

The proposed distributed algorithm iteratively aligns PEVG demands to a variational equilibrium and then has the grid optimize its price, reaching the game's generalized Stackelberg equilibrium.

  • Distributed implementation: The algorithm requires limited communication among PEVGs, the grid, and the SEM to reach the efficient generalized Stackelberg equilibrium.The SEM communicates the grid price and aggregates PEVG demands during the distributed process.
  • Demand optimization: The PEVGs iteratively update their demands using hyperplane projections onto X ∩ H_nk and submit each update to the SEM.The iteration continues until all λ_n converge to a common nonnegative value.
  • Price optimization: The grid receives the VE demand from the SEM and sets the optimal price p* using (28), completing the generalized Stackelberg equilibrium.The equilibrium consists of each PEVG's equilibrium demand together with the optimal price.
  • Convergence: Strong monotonicity guarantees convergence of the projection method to a unique game solution under the PEVGs' demand constraints and the grid capacity C.The convergence result implies that the proposed generalized Stackelberg game reaches its equilibrium.
  • Demand optimization: The SEM determines the variational-equilibrium demand when all PEVG parameters λ_n converge to the same λ ≥ 0.This common-value condition identifies the VE demand vector.

V. ADAPTATION TO TIME-VARYING CONDITIONS

The paper extends the Stackelberg game to slowly varying environments through a discrete-time feedback model in which grid supply, prices, and PEVG demands evolve across time slots. Each time step uses updated state information and a follower equilibrium, yielding a team-optimal solution under the stated conditions.

  • Model assumptions: The time-varying model represents changing vehicle populations and grid energy availability over moderate time intervals.Vehicle counts may change over roughly 5–30 minutes, while available grid energy may vary about hourly.
  • Time-indexed variables: The model includes time-varying battery capacities, satisfaction parameters, demands, prices, and strategy vectors for the PEVGs.These quantities are indexed by time, with satisfaction parameters allowed to change randomly between consecutive slots.
  • State dynamics: At time t, the state C_t records available energy supply and transitions according to C_{t+1} = f_t(C_t, p_t, x_t).The state depends on current PEVG demand and the energy available in the previous time slot.
  • Feedback structure: The grid estimates the next energy state from current information and feeds C_t back to the PEVGs through the SEM.The feedback uses parking-lot volume, prior demand, and the prior per-unit price.
  • Equilibrium property: The discrete-time Stackelberg solution is team optimal when each follower subgame reaches a variational equilibrium under the grid-capacity constraint.The paper treats the single-time game as the subgame at each time interval.

VI. NUMERICAL ANALYSIS

The simulations evaluate convergence to the generalized Stackelberg equilibrium (GSE), how network size and grid capacity affect prices and iterations, and how the proposed scheme compares with PSO and ED. Results also examine demand and utility in both fixed and time-varying settings.

  • Convergence: The proposed algorithm converges to the GSE after 9 iterations for a network of 5 PEVGs.The convergence analysis tracks λn approaching a common λ ≥0 as the PEVGs reach their variational equilibrium.
  • Convergence: The grid price converges to an approximate optimal value within 5 iterations for networks of 5, 10, and 15 PEVGs.The grid updates price in response to PEVG demand strategies.
  • Price effects: The average optimal price increases with the number of PEVGs but decreases as grid capacity increases.More PEVGs increase demand on limited resources, while greater capacity gives the grid more energy to sell.
  • Scalability: 52 to 79 average iterations are needed as the number of PEVGs increases from 15 to 25 at fixed grid capacity.The maximum iteration count shows similar behavior as network size increases.
  • Scheme comparison: The proposed scheme achieves average utility 1.3 times PSO utility and twice ED utility at the GSE.It improves utility for most PEVGs versus PSO, except PEVGs 9 and 10.
  • Demand and time variation: Average demand per PEVG decreases as the number of PEVGs increases, with the proposed scheme below PSO and ED for all N.The paper interprets this as better energy utilization than either comparison scheme.
  • Scheme comparison: Across network sizes, proposed-scheme average utility is 1.6 times PSO and reaches 3.5 times ED utility at N = 25 PEVGs.Average utility per PEVG decreases with N for all three schemes because more PEVGs share fixed grid energy.
  • Demand and time variation: In time-varying conditions, proposed-scheme utility averages 1.6 times PSO utility and 3.8 times ED utility across time slots.The improvement varies with available energy and the number of PEVs in the network.

VII. CONCLUSIONS

The paper formulates energy trading between a smart grid and PEV groups as a noncooperative Stackelberg game, with the smart grid setting price and PEVGs responding through charging demands.

  • VII. CONCLUSIONS: The smart grid chooses the energy price while PEVGs participate as followers in the noncooperative Stackelberg game.The game addresses energy trading between the grid and multiple PEV groups.
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