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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.