Time Stepping Methods and Solvers for Battery Models

Abstract

There are a wide range of battery models at different scales, from empirical models to molecular dynamics models, that can describe the battery behavior. The Pseudo 2-Dimensional (P2D) model considers the porous electrode theory, concentrated solution theory, Ohm’s law, kinetic relationships, as well as charge and material balances.1 These physic-based behaviors are described by a set of stiff nonlinear partial differential algebraic equations (DAEs) which can be only solved numerically. In this talk, we discuss and review different methods to simulate the battery models, specifically about integration in the time domain. As the P2D model discretized in spatial coordinates by using any suitable method such as finite difference2-3, finite volume4 and spectral methods5-8, it results in a system of nonlinear DAEs. Typically, nonlinear DAEs can be solved based on Runge-Kutta (RK) methods (explicit or implicit) or multistep methods. The different orders of accuracy will affect the accuracy of the numerical solutions for each time step and it will further affect the computational efficiency. However, most of the time the stability region becomes smaller when the order of accuracy of the method increases. For the multistep method, Backward Differentiation Formula (BDF) method is commonly used, because we can get more than second order of accuracy without increasing the number of variables.9 Both RK and BDF methods will be reviewed for simulating battery models. Subtle differences among different methods and efficiency improvement and robustness of all these methods will be analyzed and discussed. In particular, the compromise between, stability, accuracy, and ease of programming will be discussed. Implementation in different programming languages and platforms will also be compared. Acknowledgements The authors are thankful for the financial support of this work by the Clean Energy Institute (CEI) at the University of Washington. References 1. W. Tiedemann and J. Newman, J. Electrochem. Soc., 122, 1482–1485 (1975) 2. M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc., 140, 1526–1533 (1993). 3. G. G. Botte, V. R. Subramanian, and R. E. White, Electrochim. Acta, 45, 2595–2609 (2000). 4. M. Torchio, L. Magni, R. B. Gopaluni, R. D. Braatz, and D. M. Raimondo, J. Electrochem. Soc., 163, A1192–A1205 (2016). 5. V. R. Subramanian, V. Boovaragavan, V. Ramadesigan, and M. Arabandi, J. Electrochem. Soc., 156, A260–A271 (2009). 6. P. W. C. Northrop, V. Ramadesigan, S. De, and V. R. Subramanian, J. Electrochem. Soc., 158, A1461–A1477 (2011). 7. V. Ramadesigan, V. Boovaragavan, J. C. Pirkle, and V. R. Subramanian, J. Electrochem. Soc., 157, A854–A860 (2010). 8. V. R. Subramanian, V. D. Diwakar, and D. Tapriyal, J. Electrochem. Soc., 152, A2002–A2008 (2005). 9. K. Brenan, S. Campbell, and L. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (1995).