Grey-box modelling is a possible approach to model the behaviour of dynamic systems. It combines physical and data-driven models to benefit from their respective advantages. Our focus is on using neural networks, especially neural ordinary differential equations (NODEs), for grey-box modelling of lithium-ion batteries.

Recurrent neural networks are widely used for time-series predictions. In these neural networks the learnable functions usually give the states at the considered time steps. Recently, residual networks with shared weights are used for time-series prediction of dynamic systems more often. In contrast to recurrent neural networks, the learnable function of residual networks gives the absolute change of the states during the considered time step. NODEs are understood as the continuous analogue of residual neural networks. Therefore, NODEs offer new possibilities for grey-box modelling. Differential equations governed from physical principles and NODEs can be modelled together in one framework. This simplifies the modelling approach and allows the consideration of irregularly-sampled data during training and evaluation of the model.

For the grey-box modelling of lithium-ion batteries, we use NODEs in combination with a simple equivalent circuit model. The chosen equivalent circuit model is a serial connection of an ideal voltage source, a serial resistor and an RC circuit. Additionally, we include a hysteresis voltage. We apply machine learning to parameterize the equivalent circuit model by using learnable functions and parameters to approximate the unknown dependencies. The parameterization follows two steps. In a first step, we use experimental data of constant current, constant voltage discharge and charge to train a simplified version of the grey-box model (without capacitance). The serial resistance is assumed state-of-charge dependent. In a second step, we include the capacitance in the pretrained model and use data from pulse tests to approximate the dynamics of the battery. The resulting grey-box model is able to reproduce experimental voltage during both constant and dynamic current operation.