Efficient Conservative Reformulation Schemes for Lithium Intercalation

Abstract

Porous electrode theory coupled with transport and reaction mechanisms is a widely used technique to model Li-ion batteries employing an appropriate discretization or approximation for solid phase diffusion with electrode particles. One of the major difficulties in simulating Li-ion battery models is the need to account for solid phase diffusion in a second-radial-dimension r, which increases the computation time/cost to a great extent. Various methods that reduce the computational cost have been introduced to treat this phenomenon, but most of them do not guarantee mass conservation. The aim of this paper is to introduce an inherently mass conserving yet computationally efficient method for solid phase diffusion based on Lobatto III A quadrature. This paper also presents coupling of the new solid phase reformulation scheme with a macro-homogeneous porous electrode theory based pseudo 2D model for Li-ion battery. Lithium-ion chemistry has been identified as a good candidate for high-power/high-energy secondary batteries which are expected to play a vital role in the future of automobile, power storage, military, mobile, and space applications. Significant efforts have been made and reported in literature regarding the modeling and understanding of Lithium-ion batteries using physics based first-principles models. The most widely used first principles model for the lithium-ion battery is the porous electrode pseudo two dimensional (P2D) model, 1 which is based on the fundamentals of electrochemistry and transport phenomena. These models are represented by coupled nonlinear PDEs in 1-2 dimensions, are typically solved numerically and require few minutes to hours to simulate.For the P2D model, the diffusion of Lithium ion into the solid electrode particles is solved in a pseudo dimension r, which is coupled to the macro-homogenous model at the surface of the intercalation particles. This pseudo 2 dimensional approximation avoids the need for a solution of a full 2 dimensional model and hence the name. Accurate predictions of the concentration at the surface of the particle are therefore important, as it contributes to the exchange current density for the reaction at the particle-electrolyte interface. Typically, solid phase diffusion in the micro-scale is modeled using Fick's law of diffusion. More detailed schemes involving pressure induced diffusion along with solid phase diffusion have also been reported in literature. 2These models are important especially for high capacity materials where the stress developed affects the concentration profile inside the intercalation particle. For phase changing materials, the shrinking core model 3 has been used and approximate solutions have been proposed. 4 Cahn-Hilliard models 5 have also been employed to track the phase boundary within active material particles during Lithium intercalation.6,7 One of the major difficulties in the electrochemical engineering models is that even the inclusion of a simple Fickian model for solid phase diffusion in a second dimension r increa...

A Computationally Efficient Multi-Scale Model for Lithium-Ion Cells

“…Thus, a high number of nodes is necessary when a concentration dependent diffusion coefficient is employed. 48 To minimize the influence of the electrochemical model on the potential and temperature distribution, the number of p2D models is set to one for the comparison of the mesh size. In consequence, the transfer current density is identical for the reformulated and the FEM model with a value of 126.26 A m −2 for a 5C discharge.…”

“…At particle level we apply Lobatto IIIA method which has shown promising performance in p2D models. 48 In order to achieve high accuracy with minimal computational costs, we extend our previous work 26 by introducing a node point coupling method based on interpolation instead of volume average technique which can be used to reduce the number p2D models and enhance model preciseness. Hence, this work aims to foster the development of spatially resolved MSMD models that can be applied in fields such as online estimation or cell-design optimization where high accuracy and fast computation is required.…”

Efficient electrochemical model for lithium-ion cells

“…The computation can be compared with the reported time in [26] for the current rate 1 C. The reported MAPLE computation time in [26] is using a 3.33 GHz Intel processor with 24 GB RAM for the degrees of freedom 136 and 72 are respectively 28.361 s and 9.812 s which is comparable with 36 s obtained in this paper for a degree of freedom 88. The computational time is also less than the one introduced in [34]; the reported MAPLE computation time in [34] is 174.71 s. It should be noted that the nonlinearity of the equations in this paper is more than the nonlinearity in either A likely cause of the discrepancy between the simulation results and experimental data is errors in the modeling parameters. A more accurate model was obtained by including rate dependency in the diffusivity.…”

“…A computationally efficient method, a control volume method, is developed in [33] for solving the diffusion equation with the variable diffusion equation. The approximation of the solid phase diffusion model with the variable diffusion coefficient is also considered in [34] based on Lobatto IIIA quadrature to approximate the solid concentration. In this paper, eigenfunction based Galerkin collocation technique, which has shown an adequate result for constant diffusivity ( [31]) and keeps key dynamical behaviour of the system, is extended to approximate the solid diffusion equation in which the diffusion coefficient is not constant.…”

Efficient Reformulation of Linear and Nonlinear Solid-Phase Diffusion in Lithium-ion Battery Models using Symmetric Polynomials: Mass Conservation and Computational Efficiency

“…Leakage or gain in mass due to numerical errors has consequences for long cycle simulations, wherein potentially serious errors in the estimation of critical cell-level quantities like temperature, SOC and voltage may be introduced. 36,37 Analytical solutions are often preferred since they permit a closed form evaluation of the relevant variables and further reduce computational cost by eliminating the assembly, preprocessing and initialization steps that are often necessary for numerical methods. 3,38 However, despite these advantages, analytical solutions are generally valid only for linear solid diffusion problems, rendering them inapplicable for describing transport through electrode materials such as graphite and Lithium Iron Phosphate (LFP), which exhibit complex phase behavior and phase separation dynamics.…”