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

TL;DR

This paper develops a mean field game model for decentralized battery charging, incorporating state augmentation and price coupling, with proven existence and uniqueness of equilibrium solutions.

Contribution

It introduces a novel state-augmented MFG framework for battery charging with a monotonic price coupling, providing explicit solutions in the affine price case.

Findings

Existence and uniqueness of MFG equilibrium for any monotonic price function.

Simplified Riccati-based solutions for affine price functions.

The model captures decentralized charging dynamics with proven theoretical properties.

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.