Modeling battery cells under discharge using kinetic and stochastic battery models

TL;DR

This paper reviews and develops simple kinetic and stochastic models for battery discharge, emphasizing charge recovery and response behavior, and introduces new PDE and probabilistic representations to compare linear and nonlinear dynamics.

Contribution

It introduces a new PDE boundary value problem and probabilistic representation for kinetic battery models, and compares linear and nonlinear dynamics using Markov chains and scaling limits.

Findings

New PDE with Robin boundary conditions derived from kinetic models

Explicit solutions for nonlinear ODEs modeling battery capacity

Comparison of discharge profiles between linear and nonlinear models

Abstract

In this paper we review several approaches to mathematical modeling of simple battery cells and develop these ideas further with emphasis on charge recovery and the response behavior of batteries to given external load. We focus on models which use few parameters and basic battery data, rather than detailed reaction and material characteristics of a specific battery cell chemistry, starting with the coupled ODE linear dynamics of the kinetic battery model. We show that a related system of PDE with Robin type boundary conditions arises in the limiting regime of a spatial kinetic battery model, and provide a new probabilistic representation of the solution in terms of Brownian motion with drift reflected at the boundaries on both sides of a finite interval. To compare linear and nonlinear dynamics in kinetic and stochastic battery models we study Markov chains with states representing available and remaining capacities of the battery. A natural scaling limit leads to a class of nonlinear ODE, which can be solved explicitly and compared with the capacities obtained for the linear models. To indicate the potential use of the modeling we discuss briefly comparison of discharge profiles and effects on battery performance.