(Battery Student Slam 8 Award Winner) A Comparison of Different Convergence Criteria to Improve the Performance of a P2D Modelling Algorithm of a Lithium-Ion Battery
Authors: Wickramanayake Toshan , Mehran Kamyar 
Published: August 2024

Lithium-ion batteries (LiBs) are used as the storage for electric vehicles, grid energy and consumer electronics mainly due to the LiBs’ high energy density, long cycle life, and reliable manufacturing. Effective performance and diagnostic management of LiBs is important and is implemented via Battery Management Systems (BMS) [1]. A BMS should also maximise the safety and lifetime of battery packs and therefore, needs to estimate the internal battery states including the state of charge (SoC), state of health (SoH), state of power (SoP), state of function (SoF), and state of safety (SoS) [2]. Accurate estimation of the battery SoX can be realized with the help of a battery model; an algorithm representing the electrochemical, mechanical, electrical, and thermal behaviour of the LiB cells.
Cell-level battery models typically fall into three categories: (i) Empirical Models, (ii) Equivalent Circuit Models (ECM) and (iii) Physics-based Electrochemical Models (EM). Electrochemical Models represent the internal cell processes within a cell based on its governing fundamental physical and electrochemical equations [3].
The Partial two-dimensional (P2D) model is considered a high-fidelity Electrochemical Model (EM), which in its most basic form, captures three major physio-chemical processes in a cell: mass transfer via reduction/oxidisation (redox) reactions, diffusion of neutral lithium in the electrodes and diffusion of lithium-ions in the electrolyte. The P2D model captures the different physio-chemical phenomena using a set of coupled Partial Differential Algebraic Equations (PDAEs) [4].
Despite accuracy, P2D models are complex mathematical models with typically no analytical solution [5]. Therefore, to implement the full-order version of the P2D model, numerical-iterative solver algorithms are required. Multiple works have looked at different methods of optimizing and solving the P2D model. These methods include using different discretisation schemes, such as the finite difference method and the finite volume method, using different iterative solvers and different convergence criteria [5]. However, the different works on iterative solvers we have reviewed to date [5], haven’t investigated the effects of the different criteria on improving the performance of P2D iterative solvers.
Hence, in this work, we aim to build on our previous work [6], where we proposed an algorithm to implement the P2D model using the finite volume spatial discretisation method (FVM) and Euler time integration method. We then analyse how the role of (i) the choice of convergence criteria, (ii) the choice of initial guess and (iii) the choice of root-finding method used to converge to the convergence criteria, can be varied to improve the performance of the iterative solver.
Therefore, the contributions of this work can be summarised as:

An algorithm that implements the isothermal P2D model of a LiB using the Finite Volume Method and Euler integration methods.

An analysis of the changes in algorithm performance when the following performance factors are varied; (i) The convergence criteria, (ii) the choice of initial guess and (iii) the choice of root-finding method.

An optimal configuration of performance criteria that solves the isothermal-P2D model for a 1C discharge simulation, with an average speed of 3 seconds and a root mean square error of less than 1% when compared to the commercial solver COMSOL.

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[2] L. Lu, X. Han, J. Li, J. Hua, and M. Ouyang, “A review on the key issues for lithium-ion battery management in electric vehicles,” Journal of Power Sources, vol. 226. pp. 272–288, Mar. 15, 2013. doi: 10.1016/j.jpowsour.2012.10.060.
[3] S. Abada, G. Marlair, A. Lecocq, M. Petit, V. Sauvant-Moynot, and F. Huet, “Safety focused modeling of lithium-ion batteries: A review,” J Power Sources, vol. 306, pp. 178–192, Feb. 2016, doi: 10.1016/J.JPOWSOUR.2015.11.100.
[4] M. Doyle, T. F. Fuller, and J. Newman, “Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell,” J Electrochem Soc, vol. 140, no. 6, pp. 1526–1533, Jun. 1993, doi: 10.1149/1.2221597/XML.
[5] A. M. Ramos, “On the well-posedness of a mathematical model for Lithium-ion batteries,” Appl Math Model, vol. 40, no. 1, pp. 115–125, May 2015, doi: 10.48550/arxiv.1506.00605.
[6] T. Wickramanayake, M. Javadipour, and K. Mehran, “A Novel Root-Finding Algorithm to Solve the Pseudo-2D Model of a Lithium-ion Battery,” 2023 IEEE International Conference on Electrical Systems for Aircraft, Railway, Ship Propulsion and Road Vehicles and International Transportation Electrification Conference, ESARS-ITEC 2023, 2023, doi: 10.1109/ESARS-ITEC57127.2023.10114840.