Robust Fail-Safe Iteration Free Solvers for Battery Models

Abstract

Mathematical models of batteries1, 3, 4 are dynamic systems. which are typically solved by using a finite difference or other discretization technique in the spatial direction and solving the resulting system of nonlinear differential algebraic equations (DAEs) in time2. The resulting DAEs are stiff, often times ill-conditioned and many standard solvers fail to solve them. The complications in solving the system of DAEs also include the inability of the solvers to find consistent initial conditions for the algebraic variables5,6. Most stiff solvers include a Newton-Raphson type iterative methods to solve these stiff equations. (DASKR, RADAU, IDA7). In this work, we propose a single-step iteration free approach to system initialization and simulation allowing for systems of DAEs to be solved using explicit ODE solvers without exact knowledge of the initial conditions for the algebraic variables. We will present this new solver which is guaranteed to not fail for any meaningful physical situations. Acknowledgements The work presented herein was funded in part by the Advanced Research Projects Agency – Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0000275. References Methekar, R. N., Ramadesigan, V., Pirkle, J. C. and Subramanian, V. R. A perturbation approach for consistent initialization of index-1 explicit differential-algebraic equations arising from battery model simulations. Computers & Chemical Engineering 2011; 35(11): 2227-2234.Michelsen, M. L. Application of Semi-Implicit Runge-Kutta Methods for Integration of Ordinary and Partial-Differential Equations. Chem Eng J Bioch Eng 1977; 14(2): 107-112.M. Doyle, T. F. Fuller and J. Newman, J. Electrochem. Soc., 140, 1526 (1993).T. F. Fuller, M. Doyle and J. Newman, J. Electrochem. Soc., 141, 1 (1994).Cash, J. R. Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations. P Roy Soc a-Math Phy 2003; 459(2032): 797-815.E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential–algebraic Problems, second ed., Springer-Verlag, 1996.IDA,[http://computation.llnl.gov/casc/sundials/main.html]