A grey-box model with neural ordinary differential equations for the slow voltage dynamics of lithium-ion batteries: Application to single-cell experiments

Abstract

Lithium-ion batteries exhibit a complex, nonlinear and dynamic voltage behaviour. Modelling their slow dynamics is a challenge due to the multiple potential causes involved. We present here a neural equivalent circuit model for lithium-ion batteries including slow voltage dynamics. The model uses an equivalent circuit with voltage source, series resistor, and diffusion element. The series resistance is parameterized using neural networks. The diffusion element is based on a discretized form of Fickian diffusion, parameterized using a neural network and learnable parameters. It is flexible to represent not only Warburg behaviour, but also resistor-capacitor-type dynamics. Mathematically, the resulting model is given by a differential–algebraic equation system combining ordinary and neural differential equations. Therefore, the model combines concepts of both physical theory (white-box model) and artificial intelligence (black-box model) to a combined framework (grey-box model). We apply this approach to a lithium iron phosphate based lithium-ion battery cell. The experimental voltage behaviour during constant-current cycles as well as the dynamics during pulse tests are well reproduced by the model. Only at very high and very low states of charge the simulation significantly deviates from experiments, which might result from insufficient training data in these regions.