Coordinate Transformation, Orthogonal Collocation, Model Reformulation and Simulation of Electrochemical-Thermal Behavior of Lithium-Ion Battery Stacks

Abstract

In this paper, a simple transformation of coordinates is proposed that facilitates the efficient simulation of the non-isothermal lithium-ion pseudo 2-D battery model. The transformed model is then conveniently discretized using orthogonal collocation with the collocation points in the spatial direction. The resulting system of differential algebraic equations (DAEs) is solved using efficient adaptive solvers in time. A series of mathematical operations are performed to reformulate the model to enhance computational efficiency and programming convenience while maintaining accuracy even when non-linear or temperature dependent parameters are used. The transformed coordinate allows for efficient simulation and extension from cell sandwich to stack models. Furthermore, the transformation and reformulation techniques are used to simulate operation of an 8-cell battery stack subject to varying heat transfer coefficients as well as specified temperature boundary conditions. Mathematical modeling and simulation of the operation of lithiumion batteries is not trivial, as concentration and potential fields must be evaluated simultaneously in both solid and liquid phases. This is complicated by the fact that the transport and kinetic parameters which determine battery behavior are highly nonlinear, leading to very complex governing equations. Doyle et al. 1 developed a general model based on concentrated solution theory to describe the internal behavior of a lithium-ion sandwich consisting of positive and negative porous electrodes, a separator, and current collectors. 2 This model proved generic enough to incorporate further advancements in battery systems understanding, leading to the development of a number of similar models. 3-13 Reviews of models for lithium-ion batteries can be found elsewhere in the literature. [10][11][12] Table I depicts a pseudo-twodimensional isothermal model for a lithium-ion battery. 14-16 Table II presents the various expressions used in the model, while Table III shows the physical parameters used in this paper. For analysis and control of lithium-ion batteries in hybrid environments (e.g. with a fuel cell, capacitor, or other electrical components), there is a need to simulate state of charge, state of health, and other parameters of lithium-ion batteries in milliseconds. Full-order physics-based models may simulate discharge curves in several seconds to minutes, depending on the solvers, routines, computers, etc. In contrast, empirical models (based on correlations of past data) can simulate specific operating scenarios in milliseconds. However, use of these models under a different operating condition than for which they were developed may cause abuse or underutilization of electrochemical power sources. This paper presents a coordinate transformation and mathematical analysis for reformulation of physics-based models to solve them quickly, conveniently, and accurately in a way that is valid for a wide range of operating conditions and parameters. The porous electrode model as given in Ta...

LIONSIMBA: A Matlab Framework Based on a Finite Volume Model Suitable for Li-Ion Battery Design, Simulation, and Control

“…For example, orthogonal collocation can be used with an efficient coordinate transformation to solve the set of resulting DAEs. 20 In this paper, in order to exploit the properties of variable-step solvers, MOL is used to reformulate the original set of PDAEs. In particular, the finite volume method (FVM) is employed.…”

“…All the parameter values have been taken from the real battery data in Ref. 20, and are summarized in Table IV. LIONSIMBA validation.-The P2D model has been experimentally validated numerous times since, 14 this section validates the numerical implementation of LIONSIMBA by comparing the results coming from the proposed framework with COMSOL MultiPhysics and DUALFOIL. While COMSOL has been supplied with the same model used in our framework, where a heat diffusion PDE is used to describe the thermal dynamics, DUALFOIL neglects the spatial distribution of the temperature and averages the heat generation rates over the cell.…”

“…20 with a cutoff voltage of 2.5 V and environmental temperature of 298.15 K. For the proposed chemistry, the 1C value is ≈30 A/m 2 . The effectiveness and ease of use of the proposed framework are shown in a series of simulations.…”

“…The package comes with the experimental parameters of the battery reported in Ref. 20. An initialization file allows changes in battery and simulator parameters.…”

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

“…Cai and White [38][39][40] used the proper orthogonal decomposition method and the orthogonal collocation method to solve the model equations for lithium ion cells to reduce the computation needed for simulation. Subramanian and his coworkers 7,8,10,41,42 applied coordinate transformation, orthogonal collocation and model reformulation techniques to facilitate the efficient simulation of the P2D model equations. In later work, Subramanian et al 43 extended the polynomial approximation approach to the diffusion equation in electrolyte phase.…”