Derivation and Tuning of a Compact Differential-Algebraic Equations Model for LiFePO4-Graphite Li-Ion Batteries
Authors: Lee Cheol W. , Hong Yao , Hayrapetyan Mushegh , Xi Zhimin 
Published: February 2020

This paper presents a procedure for deriving and tuning a compact and solvable differential-algebraic equations (DAE) model for the LiFePO4-graphite battery cell. Electrochemical models of the battery cell are typically represented by complex partial differential equations of state variables and parameters whose relationships can be highly nonlinear. Several software packages are available for numerically solving these equations for simulations. However, their long computing time due to the complexity of the model is a major bottleneck for control and monitoring applications. A reduced order model (ROM) can drastically decrease the simulation time with a minimal loss of the prediction accuracy. A ROM is constructed by simplifying or approximating parameters of the full-order model of the battery cell. For on-line diagnostics and real-time control purposes, a lower-order compact representation of complicated transport and diffusion phenomena of a lithium-ion battery is highly desirable.
In this paper, the psedo-2D (P2D) electrochemical model of the battery cell is simplified based on polynomial representations of pore-wall flux and lithium concentrations in a similar manner to previous studies. However, instead of adopting the standard Galerkin method which tends to increase the number of equations, this paper adopts Subramanian et al.’s method which demonstrated a potential for producing a compact description of a lithium-ion battery cell. While a traditional Galerkin method requires selecting particular polynomial representations which satisfy the boundary conditions, Subramanian et al.’s method selects a simple polynomial which can be analytically integrated foregoing setting up separate polynomials for solid and electrolyte potentials. This technique, therefore, has a potential to drastically reduce the size of simplified system. However, since the boundary conditions should be satisfied by additional analytical equations, one requires judicious choice of Galerkin formulations for electrolyte concentration in order to build a solvable DAE system. This paper introduces a systematic way of assessing a Galerkin formulation for the electrolyte concentration based on the theorem of Weierstrass. It is shown that each different Galerkin formulation converts the governing equation and its boundary conditions for the electrolyte concentration into a different set of linear DAE’s. However, not all DAE systems are solvable and some may induce numerical instabilities. It is shown that a full rank of the matrix pencil can ensure that a certain Galerkin formulation produces a solvable DAE system. Usefulness of the proposed method is illustrated by comparing several different Galerkin formulations for the LiFePO4-graphite system.
In addition, this paper improves the accuracy of the model by accounting for the hysteresis of open-circuit voltage in LiFePO4 using a differential equation model. When the order of the polynomials are chosen at fours, the P2D model of the cell is simplified into a system of 24 DAE’s. This size is significantly smaller than those of comparable ROMs in the literature. The developed DAE system is solved by using SUNDIAL’s IDA solver where a typical charge/discharge cycle can be simulated under 10 seconds on a regular personal computer.
The parameters of the developed DAE model are tuned based on charge/discharge experimental data from a commercial LiFePO4-graphite battery. In this paper, a systematic tuning of the model parameters is investigated by exploiting the fast simulation capability of the developed model. Based on a sensitivity study, six model parameters are chosen for tuning. These tuning parameters, thus selected, include the solid phase diffusivity of the negative electrode, initial hysteresis parameters, reaction rate constants, and the contact resistance. The genetic algorithm is applied to simultaneously find the six model parameters that minimize the sum of squared errors in charge/discharge voltages. The tuned model shows a good agreement with the experimental data at rates upto 4C.
Contributions of this paper can be summarized as follows. This paper proposes a new method to ensure a compact DAE system is solvable when it is driven from the original higher-order model of the Lithium-ion battery cell. Moreover, a systematic tuning of the model parameters using the global optimization method is demonstrated by exploiting the computational efficiency of the simplified model. When coupled with the model for describing the hysteresis of the battery cell, the tuned model shows a good agreement with the experimental data from a LiFePO4-graphite battery cell.