Nonlinear State-Variable Method for Solving Physics-Based Li-Ion Cell Model with High-Frequency Inputs

Abstract

A nonlinear state-variable method is presented and used to solve the pseudo-2D (P2D) Li-ion cell model under high-frequency input current and temperature signals. The physics-based governing equations are formulated into a nonlinear state variable method (NSVM), in which the mass transfer variables are evaluated using a 1 st order exponential integrator approach at each discrete time point and the electrochemical kinetics (Butler-Volmer) equations are solved by either an iterative or an explicit method. This procedure provides an accurate, computationally efficient method to develop physics-based simulations of the performance of a dual-foil Li-ion cell during practical drive cycles. Physics-based Li-ion cell models are useful tools for analyzing the battery cell performances, characterizing material properties, supporting cell design, and developing battery management systems (BMS). 5 in their work, a dual-foil Li-ion cell was represented by a pseudo-two-dimensional domain; and therefore, this model is usually referred as the "pseudo-2D model (P2D)".3 This model contains several coupled transport phenomena described by nonlinear partial differential equations (PDEs). Due to the numerical complexity of these models, various reduced-order models (ROMs) have been proposed as substitutes for either the single particle (SP) or the P2D full-order model (FOM) in practical online simulation applications.6-16 Also, ROMs have been published to simulate capacity fading 17,18 and for use in parameter estimations. presented an approximate solution to the spherical diffusion problem with a fixed flux which provides high accuracy with only a few terms in a series when compared to the analytic solution with at least 100 terms. This approximate solution provides significant savings in computation time.In order to predict the performance of lithium ion cells operating at higher rates, more complicated models were developed. In 1982, West et al. 4 presented a Nernst-Planck model for porous insertion electrodes for a Li/TiS 2 cell. They pointed out the importance of coupling the transport of lithium ions between the insertion electrode and the electrolyte. In 1993, Mao and White 35 presented a Nernst-Planck model of the Li/TiS 2 cell which included transport of lithium ions through the separator and demonstrated quantitatively how the utilization of the TiS 2 electrode could be improved by decreasing the thickness of the separator. Also in 1993, Doyle et al. 36 published a concentrated solution theory model for a lithium/polymer/TiS 2 cell. Shortly after that in 1994 Fuller et al.5 published a concentrated solution theory model for a dual lithium ion insertion cell ("dual foil"), which is known as "pseudo-2D model (P2D)". In 1996, Doyle et al. 37 published a comparison of their model predictions to experimental data from a Bellcore plastic lithium ion cell. Unfortunately, the computation burden associated with using the program dual foil is too great to use their program for extensive simulations needed for parameter estimation an...

Nonlinear State-Variable Method (NSVM) for Li-Ion Batteries: Finite-Element Method and Control Mode

“…In the previous work, 15 we have examined the accuracy of our NSVM algorithm for current control mode through model-to-model comparisons. In this work, we focus on model vs experimental validation and simulation of different control modes.…”

“…The solution approach for the solid particle diffusion equation (r dimension) was discussed previously. 15 Consequently, the following sections will focus on the equations in the x dimension.…”

“…Accordingly, the overpotential of the electrodes can be expressed in terms of the modified potentials: [14] Note that the SEI resistance R film is neglected for the cathode and the definition of θ * is given in [15] The boundary current density for the solid phase is defined as follows:…”

“…The solution approach for the solid particle diffusion equation (r dimension) was discussed previously. 15 Consequently, the following sections will focus on the equations in the x dimension.Formulation of the governing equations.-Three partial differential equations (PDEs) in the x dimension in the electrode domains (diffusion and charge conservation in the electrolyte and charge conservation in the solid phase) are solved using the finite element method, and all the PDEs can be written in the following standard format:where t is time, w(x, t) is a field variable, ρ(x) is the mass function, μ(x, t) is the transport coefficient, γ(x) is the forcing function, and j s (x, t) is the current density for surface electrochemical reactions. The governing equation for the electrolyte charge conservation is given by:…”

“…In our previous work, 15 we presented a nonlinear state variable modeling (NSVM) algorithm to reduce the computational cost for solving the P2D model over a high-frequency time domain. According to that NSVM work, the state variables in the diffusion equations are solved using the following approach: [1] where x k and x k−1 are vectors for the concentration states at current and previous time instances, d k−1 is the source term vector at the previous time instance, and , , and B are matrix operators.…”