[Paper Review] Economics of Electric Vehicle Charging: A Game Theoretic Approach

Wayes Tushar, Walid Saad+2 more|arXiv (Cornell University)|Aug 2, 2012

Electric Vehicles and Infrastructure9 citations

TL;DR

This paper proposes a noncooperative Stackelberg game model to optimize electricity pricing and charging strategies between a smart grid (leader) and plug-in electric vehicle groups (followers). Using variational inequalities and a distributed algorithm, it achieves a socially optimal equilibrium, demonstrating 1.6× higher average utility per PEVG than particle swarm optimization and 3.5× improvement over equal distribution in static settings, with similar gains in dynamic environments.

ABSTRACT

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.

Motivation & Objective

• Address the conflict between smart grid revenue maximization and plug-in electric vehicle group (PEVG) cost-benefit optimization in grid-to-vehicle energy exchange.
• Model the interaction between the smart grid and multiple PEVGs as a noncooperative Stackelberg game with strategic pricing and charging decisions.
• Develop a distributed algorithm enabling PEVGs and the grid to reach a socially optimal Stackelberg equilibrium without centralized coordination.
• Extend the model to handle slowly varying, time-varying environments with stochastic fluctuations in grid energy availability and PEVG battery capacity.
• Evaluate performance against benchmark schemes (PSO and equal distribution) in terms of utility, energy demand, and system efficiency.

Proposed method

• Formulate a generalized Stackelberg game where the smart grid acts as the leader setting electricity prices to maximize revenue, and PEVGs act as followers choosing charging amounts to optimize a utility function balancing charging benefits and costs.
• Use variational inequality theory to prove the existence and uniqueness of a socially optimal Stackelberg equilibrium in the proposed game model.
• Design a distributed algorithm that enables PEVGs and the grid to iteratively converge to the equilibrium solution through local information exchange and price updates.
• Extend the model to a time-varying environment by modeling grid energy availability and PEVG battery capacities as independent, uniformly distributed random variables over time.
• Simulate the system under dynamic conditions with time slots representing 30-minute intervals during peak hours, incorporating stochastic variations in energy supply and demand.
• Compare the proposed scheme against two baselines: particle swarm optimization (PSO) and equal distribution (ED) in terms of average utility, energy demand, and system performance.

Experimental results

Research questions

• RQ1 How can a smart grid and multiple plug-in electric vehicle groups (PEVGs) strategically interact in a grid-to-vehicle energy exchange under conflicting objectives?
• RQ2 Does the proposed Stackelberg game model admit a socially optimal equilibrium that balances grid revenue and PEVG utility?
• RQ3 Can a distributed algorithm enable PEVGs and the grid to reach this equilibrium without centralized control?
• RQ4 How does the system perform under time-varying conditions with stochastic fluctuations in energy availability and PEVG capacity?
• RQ5 What performance gains does the proposed scheme achieve compared to PSO and equal distribution in terms of average utility per PEVG?

Key findings

• The proposed Stackelberg game model admits a socially optimal generalized Stackelberg equilibrium, proven via variational inequality analysis.
• The distributed algorithm enables PEVGs and the grid to converge to this equilibrium using only local information and price feedback.
• In static environments, the proposed scheme achieves an average utility per PEVG that is 1.6 times higher than the PSO scheme and 3.5 times higher than the equal distribution (ED) scheme at N=25 PEVGs.
• In time-varying environments, the proposed scheme maintains superior performance, achieving an average utility per PEVG that is 1.6 times higher than PSO and 3.8 times higher than ED across all time slots.
• The average demand per PEVG remains well above minimum battery requirements even under low energy availability, ensuring operational feasibility.
• The model successfully captures dynamic fluctuations in grid energy and PEVG capacity, with energy demand per PEVG varying across time slots due to stochastic changes in satisfaction parameters and available energy.