Reduction of an Electrochemistry-Based Li-Ion Battery Model via Quasi-Linearization and Padé Approximation

Abstract

This paper examines an electrochemistry-based lithium-ion battery model developed by Doyle, Fuller, and Newman. The paper makes this model more tractable and conducive to control design by making two main contributions to the literature. First, we adaptively solve the model's algebraic equations using quasi-linearization. This improves the model's execution speed compared to solving the algebraic equations via optimization. Second, we reduce the model's order by deriving a family of analytic Padé approximations to the model's spherical diffusion equations. The paper carefully compares these Padé approximations to other published methods for reducing spherical diffusion equations. Finally, the paper concludes with battery simulations showing the significant impact of the proposed model reduction approach on the battery model's overall accuracy and simulation speed. This paper examines the problem of developing reduced, electrochemistry-based models of the dynamics of charging and discharging of lithium-ion batteries. The overarching goal of the paper is to develop lithium-ion battery models that satisfy two important but potentially conflicting objectives. First, the models must have the ability to accurately predict the performance of lithium-ion batteries in applications involving potentially complex and rapid charge/discharge cycles, e.g., hybrid vehicle applications. Second, the models must run with sufficient speed to enable battery system design, optimization, and control.Several models have been used to monitor battery state of charge ͑SOC͒ and state of health. 1-4 While these models are very desirable for control and estimation, they do not capture all of the high rate dynamics associated with hybrid vehicle drive cycles. For this one can use an electrochemical battery model. One such model is provided by Doyle, Fuller, and Newman, with the addition of a potential degradation mechanism provided by Ramadass et al. [5][6][7][8] There are two major numerical difficulties with this electrochemical model. The first is the large number of state variables: a finite difference discretization of the model with M points along the width of the cell and N points in the pseudospherical direction has approximately ͑2/3͒ * M * N state variables. The second challenge is the model's nonzero index, represented by approximately ͑2/3͒ * M * N algebraic equations, most of them involving a hyperbolic sine nonlinearity. This results in a large set of differential algebraic equations ͑DAEs͒. Ideally, one would like a model that: ͑i͒ runs quickly with a low number of state variables to enable optimal design and control studies, while ͑ii͒ still retaining the ability to accurately model complex, high rate charge/discharge cycles.Applying model reduction techniques to the above electrochemical battery model can bring it closer to the ideal speed and fidelity goals. Several reductions of this model are already presented in the literature. Some of these reductions pay special attention to the spherical diffusion submodel because it ap...

Enhanced Performance of Li-Ion Batteries via Modified Reference Governors and Electrochemical Models

“…4 of [19] and references therein). In our work the PDEs governing diffusion in the solid phase, (1), are discretized in the r-dimension via Padé approximates [20]. All the remaining PDEs are discretized in the x dimension via the central difference method, such that the moles of lithium are conserved.…”

A survey of mathematics-based equivalent-circuit and electrochemical battery models for hybrid and electric vehicle simulation

“…Other methods, including the Chebyshev polynomial method [77,78], the residue grouping method [79,80], proper orthogonal decomposition method [81], and Padé approximation [82] have also been used to derive reduced-order models for Li-ion batteries. In the methods using Chebyshev polynomials, the state variables are approximated by linear combinations of several Chebyshev polynomials, and then an approximate model is projected onto a subspace formed by these orthogonal Chebyshev polynomials to form a reduced-order model, which can then be solved for the unknown coefficients in the truncated expressions.…”

Modeling of Li-Ion Cells for Fast Simulation of High C-Rate and Low Temperature Operations

“…e,0 [18] In this work, κ and κ ef f D are assumed concentration-independent, and only temperature-dependent.…”

“…For this reason, several approximation methods based on Model Order Reduction (MOR) techniques have been recently applied to electrochemical models using various techniques, including Padé approximation, 18 residue grouping, 19 polynomial approximation 20 and Proper Orthogonal Decomposition 21 (POD). In recent times, the Galerkin Projection method has also been applied to analytically approximate the diffusion PDEs describing the Lithium transport in the solid and electrolyte phases.…”