Implementation and Model Order Reduction of a Lithium-Ion Battery Single Particle Model with Electrolyte Dynamics
Authors: Li Liuying , Brosa Planella Ferran , Widanage Widanalage Dhammika , Marco James 
Published: February 2020

Electrochemical models of batteries are typically developed from first principles and aim to accurately capture the internal battery dynamics such as diffusion, intercalation kinetics, and electric potentials. Underpinned by such models, battery state or parameter behaviour can be better understood and manipulated by sophisticated estimation and control approaches. However, the use of electrochemical models makes it computationally intractable for online implementation as the model is subject to a complicated mathematical structure including partial-differential equations (PDE), ordinary-differential equations (ODE) and algebraic equations. In order to devise a control-oriented model, to support real-time monitoring and control, model order reduction techniques can be used to reduce the model order while maintaining the desired level of accuracy.
This paper is based on the single particle model with electrolyte dynamics (SPMe) derived by Scott J. Moura in [1]. The PDEs in the governing equation of the SPMe are solved by finite difference method (FDM) and finite element method (FEM). The process and differences of applying these two methods are analysed. After solving the PDEs, the full order model would be high (e.g. 350th order). Model order reduction techniques are used to develop a low order Li-ion battery model derivative that is suitable for real-time implementation, for example within a battery control system. Residue grouping method is an order reduction technique for high order systems. It can be applied either analytically (from a transcendental transfer function) or numerically (from finite element method). In this paper, the residue grouping method is applied to the SPMe model to reduce the model order from 350 to 13. The solid-state diffusion equations of the two electrodes employ residue grouping analytically; conversely, the liquid-state diffusion applied residue grouping numerically.
Both the full order SPMe model (FOM) and the reduced order model (ROM) have been implemented within the commercially available software Matlab and Simulink. From the simulation results presented in the figure below, it is observed that the ROM correlates well with the SPMe model in the time domain, with a peak relative difference in the order of 10-3 and the root mean square (RMS) of the relative error of 3.09x10-4 in the terminal voltage. In addition to time domain analysis, the ROM is analysed in the frequency domain, by comparing the Bode graphs for each model type. Finally, the application of the ROM within a battery state estimation strategy for the purpose of estimating the lithium ion concentrations and the optimal order number of the model will be discussed.

References:

[1] S. J. Moura, F. B. Argomedo, R. Klein, A. Mirtabatabaei, and M. Krstic, "Battery State Estimation for a Single Particle Model With Electrolyte Dynamics," IEEE Transactions on Control Systems Technology, vol. 25, no. 2, pp. 453-468, 2017.

Figure 1