Nonlinear State-Variable Method (NSVM) for Li-Ion Batteries: Finite-Element Method and Control Mode

Abstract

The finite element method (FEM) was used in our nonlinear state-variable method (NSVM) presented recently (J. Electrochem. Soc., 164, E3001 (2017)). The details of the application of the FEM to solve the lithium ion pseudo-2D (P2D) model equations using the NSVM are presented here for several control modes (constant current, voltage, power, or load). Validation of the method was performed by comparison to rigorous full-order models and experimental data. The FEM based NSVM shows excellent performance, and the estimated cell parameters are determined with a high confidence level. The pseudo-2D (P2D) model for Li-ion cells 1 has been widely applied. This model includes two spatial dimensions and several partial differential equations (PDE) which are coupled by nonlinear electrochemical kinetics.2 The basic solution approach for the P2D model is to discretize the spatially-distributed PDE into a differential-algebraic equation (DAE) system. The finite difference method (FDM), finite volume method (FVM), and finite element method (FEM) have been used to discretize the P2D model.3-11 For physics-based simulation of Li-ion batteries, the FEM is an excellent solution approach especially for complex multi-scale cell domains.12-14 Although there are commercial finite element software packages (e.g., COMSOL Multiphysics) with the capability of solving the model equations, in-house code developed with scripting languages (i.e., MATLAB, FORTRAN, and Python) are still important due to their low cost and flexibility.In our previous work, 15 we presented a nonlinear state variable modeling (NSVM) algorithm to reduce the computational cost for solving the P2D model over a high-frequency time domain. According to that NSVM work, the state variables in the diffusion equations are solved using the following approach:where x k and x k−1 are vectors for the concentration states at current and previous time instances, d k−1 is the source term vector at the previous time instance, and , , and B are matrix operators. The source terms are solved by a nonlinear algebraic constraint:where u k and y k are respectively the inputs and outputs of the model. The above mentioned approach enables the P2D model to be solved under a high-frequency current signal with significant time efficiency (50+ times faster than a COMSOL baseline model). However, Reference 15 does not show the full details of the NSVM due to the limits on article length; in Equation 1, operators , , and B come from a specific FEM formulation which was not presented in that article; in Equation 2, the detailed solution procedure is also based on the FEM formulation. Also, the published NSVM algorithm can only simulate current control and the results have not been compared with real data. Furthermore, some derivation steps involved in the FEM solution procedure can be applied potentially to develop a reduced-order-modeling (ROM) approach. Consequently, the following is presented in this article: * Electrochemical Society Member. * * Electrochemical Society Fellow. z E-mail: wh...

An Adaptive Reduced-Order-Modeling Approach for Simulating Real-Time Performances of Li-Ion Battery Systems

“…Also, the gain and pole values used by the SVM or DRA method need to be pre-determined through implicit approaches and there are parameterization issues to implement these algorithms. Due to the above-mentioned limitations, we recently developed nonlinearstate-variable-modeling (NSVM) method, 8,9 which maintains all the nonlinear features while solving the P2D model on discrete highfrequency time space. The accuracy and time-efficiency of NSVM have been confirmed at cell-level by the work in References 8 and 9.…”