An asymptotic derivation of a single particle model with electrolyte

The standard continuum model of a lithium-ion battery, the Doyle-Fuller-Newman (DFN) model, is computationally expensive to solve. Typically simpler models, such as the single particle model (SPM), are used to provide insight for control purposes. Recently, there has been a move to extend the SPM to include electrolyte effects, which increase the accuracy and range of applicability. However, these extended models are derived in an ad-hoc manner, which leaves open the possibility that important terms may have been neglected, resulting in the model not being as accurate as possible. In this paper, we provide a systematic asymptotic derivation of both the SPM and a correction term that accounts for the behaviour in the electrolyte. Firstly, this allows us to quantify the error in the reduced model in terms of ratios of key parameters in the model, from which the range of applicable operating conditions can be determined. Secondly, in comparing our model with the ad-hoc models from the literature, we show that previous models have neglected a key set of terms. In particular, we make the crucial distinction between writing the terminal voltage in pointwise and electrode-averaged form, which allows us to gain additional accuracy whilst maintaining the same computational complexity as the existing models.