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EV Charging in Smart Grids: Mean Field Equilibrium and Approximate Non-Cooperative and Cooperative Strategies

Lijun Bo, Fengcheng Liu, Shihua Wang

TL;DR

This work models large-scale EV charging in smart grids as a finite-horizon mean-field game with sticky price dynamics, deriving both decentralized (epsilon-Nash) and centralized (epsilon-optimal social) strategies. It proves existence and uniqueness of the mean-field consistency solution and constructs approximately optimal strategies for finite populations, with convergence rates of O(1/√N). The analysis employs HJB equations, Riccati-type ODEs, and coercivity results via Positive Real Lemma to ensure well-posedness and stability. Numerical experiments corroborate the theoretical findings, showing distinct dynamics under cooperative vs. non-cooperative setups and validating the mean-field approximations for large N.

Abstract

We study the optimal charging strategies for large-scale electric vehicles in smart grids within a finite-horizon mean field game framework. We first establish the existence and uniqueness of the solution to the consistency condition equation, which characterizes the optimal charging behavior in the mean field limit. Building on this result, we construct approximate optimal charging strategies for a finite population of vehicles in both non-cooperative and cooperative settings. Finally, we provide numerical analyses that illustrate and compare the approximate strategies in the non-cooperative and cooperative games.

EV Charging in Smart Grids: Mean Field Equilibrium and Approximate Non-Cooperative and Cooperative Strategies

TL;DR

This work models large-scale EV charging in smart grids as a finite-horizon mean-field game with sticky price dynamics, deriving both decentralized (epsilon-Nash) and centralized (epsilon-optimal social) strategies. It proves existence and uniqueness of the mean-field consistency solution and constructs approximately optimal strategies for finite populations, with convergence rates of O(1/√N). The analysis employs HJB equations, Riccati-type ODEs, and coercivity results via Positive Real Lemma to ensure well-posedness and stability. Numerical experiments corroborate the theoretical findings, showing distinct dynamics under cooperative vs. non-cooperative setups and validating the mean-field approximations for large N.

Abstract

We study the optimal charging strategies for large-scale electric vehicles in smart grids within a finite-horizon mean field game framework. We first establish the existence and uniqueness of the solution to the consistency condition equation, which characterizes the optimal charging behavior in the mean field limit. Building on this result, we construct approximate optimal charging strategies for a finite population of vehicles in both non-cooperative and cooperative settings. Finally, we provide numerical analyses that illustrate and compare the approximate strategies in the non-cooperative and cooperative games.

Paper Structure

This paper contains 8 sections, 16 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Model and the Optimal Strategies
  3. Non-cooperate Game
  4. The Cooperate Games
  5. Numerical experiments
  6. Conclusion
  7. Proofs of Auxiliary Lemmas
  8. Proofs of Auxiliary Propositions

Key Result

Proposition 1

The solution to (Nash HJB eq1), (Nash HJB eq2) and (Nash HJB eq3) exists and is unique.

Figures (3)

  • Figure 1: Non-cooperate case: evolution of $\bar{P}, P$ and $\bar{Q}, Q$
  • Figure 2: Cooperate case: evolution of $\bar{p}, P$ and $\bar{q}, Q$
  • Figure 3: Comparison of the evolution process of $v^i$ under both cases

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Proposition 1
  • Remark 2
  • Theorem 3.1
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • ...and 27 more