*Article* **Neural Ordinary Differential Equations for Grey-Box Modelling of Lithium-Ion Batteries on the Basis of an Equivalent Circuit Model**

**Jennifer Brucker \*, René Behmann, Wolfgang G. Bessler and Rainer Gasper**

Institute of Sustainable Energy Systems, Offenburg University of Applied Sciences, Badstraße 24, 77652 Offenburg, Germany; rene.behmann@hs-offenburg.de (R.B.); wolfgang.bessler@hs-offenburg.de (W.G.B.); rainer.gasper@hs-offenburg.de (R.G.)

**\*** Correspondence: jennifer.brucker@hs-offenburg.de

**Abstract:** Lithium-ion batteries exhibit a dynamic voltage behaviour depending nonlinearly on current and state of charge. The modelling of lithium-ion batteries is therefore complicated and model parametrisation is often time demanding. Grey-box models combine physical and datadriven modelling to benefit from their respective advantages. Neural ordinary differential equations (NODEs) offer new possibilities for grey-box modelling. Differential equations given by physical laws and NODEs can be combined in a single modelling framework. Here we demonstrate the use of NODEs for grey-box modelling of lithium-ion batteries. A simple equivalent circuit model serves as a basis and represents the physical part of the model. The voltage drop over the resistor–capacitor circuit, including its dependency on current and state of charge, is implemented as a NODE. After training, the grey-box model shows good agreement with experimental full-cycle data and pulse tests on a lithium iron phosphate cell. We test the model against two dynamic load profiles: one consisting of half cycles and one dynamic load profile representing a home-storage system. The dynamic response of the battery is well captured by the model.

**Keywords:** neural ordinary differential equations; grey-box model; equivalent circuit model; lithiumion batteries
