Citation: Brucker, J.; Behmann, R.;
Bessler, W.G.; Gasper R. Neural
Ordinary Differential Equations for
Grey-Box Modelling of Lithium-Ion
Batteries on the Basis of an
Equivalent Circuit Model. Energies
2022, 15, 2661. https://doi.org/
10.3390/en15072661
Academic Editors: Luis
Hernández-Callejo, Sergio
Nesmachnow and Sara Gallardo
Saavedra
Received: 24 February 2022
Accepted: 25 March 2022
Published: 5 April 2022
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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 data-
driven 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; lithium-
ion batteries
1. Introduction
Lithium-ion batteries are a key technology for electric vehicles, portable devices and
stationary applications such as home-storage systems. With the increasing usage of lithium-
ion batteries in complex fields of application, the demand for battery models is growing as
well. Battery models are necessary to predict the dynamic voltage and current behaviour
and to monitor internal states, particularly the state of charge (SOC) and the state of health
(SOH). There are many different types of battery models [
1
,
2
]. Depending on the required
purpose, they can be selected as a compromise between accuracy and simplicity. We
introduce here a grey-box (GB) modelling approach that uses a simple equivalent circuit
model (ECM) as a basis.
Digitisation has been progressing rapidly in the past decades, and with it the amount
of available data increases. This has boosted the development of artificial intelligence and
especially neural networks. Neural networks are an important representative of black-
box (BB) models. They learn relations between inputs and outputs of systems based on
data
[3–6]
. However, BB models require a huge amount of training data. Therefore, it is
reasonable to consider other modelling techniques. White-box (WB) modelling uses prior
physical, chemical or engineering knowledge in the form of mathematical equations to
describe the behaviour of the corresponding system. WB models are therefore limited to
the understanding of the underlying processes. GB models combine WB and BB modelling
techniques to benefit from their respective advantages [3–6].
Energies 2022, 15, 2661. https://doi.org/10.3390/en15072661 https://www.mdpi.com/journal/energies
