AA4 – Energy and Markets



Recovery of Battery Ageing Dynamics with Multiple Timescales

Project Heads

Martin Eigel, Martin Heida, Manuel Landstorfer

Project Members

Alireza Selahi

Project Duration

01.04.2021 − 31.03.2024

Located at



Electrochemical electricity storage is a central pillar for a large variety of industrial goods, ranging from power sources for medical devices to electric vehicles and large scale battery plants. In 2019 this was honored by the Nobel price in chemistry, awarded “for the development of lithium-ion batteries (LIBs)”. The central innovation is the concept of intercalation, the physico-chemical process by which a lithium ion is stored within some solid host material. This process is essential for the safety, durability and energy density of modern LIBs.


However, all current and future LIBs face a common issue: they degrade in their lifetime upon usage. This degradation is in general a superposition of various ageing effects and depends on external (time dependent) parameters, e.g. the rate at which a battery is charged and discharged. Quantitative and qualitative knowledge of the degradation is of ultimate importance to estimate the lifespan of a battery, set up control engineering and ensure safety.


The project aims at developing a data-driven methodology to recover the dynamics of battery ageing on the basis of a parametrized mathematical model and experimental data. We want to determine the evolution of certain parameters of the model as function of the cycling number N. This is to be achieved by setting up a two time-scale PDE model, where the small time scale covers one charge/discharge cycle and the large time scale the number of such cycles.


To solve this statistical inverse problem for the degrading parameters, we will use recent ideas from invertible neural networks . Moreover, low-rank surrogate models for parametric PDE solutions will be employed in order to efficiently cope with the high model complexity [1,3].


In the long term, our approach can lead to real-time tracking and estimation of the battery health, which is of paramount importance for electric mobility. It can help to determine the residual value of an aged battery as well as its future lifetime, which is crucial for stationary energy storage devices and thus the “Energiewende”. The solution of statistical inverse problems with machine learning techniques is still in its infancy. We hope that we will contribute to the understanding of how to use Deep Neural Networks for these common problems, which will then be applicable in many fields.

Related Publications

  • [1] M. Eigel, R. Schneider, P. Trunschke, and S. Wolf. Variational Monte Carlo – bridging concepts of machine learning and high dimensional pdes. Adv Comput Math, 2019.
  • [2] M. Landstorfer. A discussion of the reaction rate and the cell voltage of an intercalation electrode during discharge. J. Electrochem. Soc., 167(1):A2573–A2589, 2019.
  • [3] M. Eigel, M. Marschall, and R. Schneider. Sampling-free bayesian inversion with adaptive hierarchical tensor representations. Inverse Problems, 34(3):035010, 2018.
  • [4] M. Heida and A. Mielke. Averaging of time-periodic dissipation potentials in rate- independent processes. Discrete & Continuous Dynamical Systems-Series S, 10(6), 2017.
  • [5] M. Landstorfer. Boundary conditions for electrochemical interfaces. J. Electrochem. Soc., 164(11):E3671–E3685, 2017.
  • [6] M. Heida. An extension of the stochastic two-scale convergence method and application. Asymptotic Analysis, 72(1-2):1–30, 2011.
  • [7] M. Landstorfer, B. Prifling, and V. Schmidt. Mesh generation for periodic 3d microstructure models and computation of effective properties. Journal of Computational Physics, 431:110071, 2021.

Related Pictures

BMBF Timescales

Time and length scales of a typical battery system. The scale transition is carried out with asymptotic and homogenization methods.


The microstructure of a typical battery electrode consists of intercalation particles (blue), which are surrounded by a liquid electrolyte through which lithium ions travel an electric current (red arrows). For the purpose of computer simulations, digital representations of such microstructures are required, which can be obtained with proper meshing techniques [7].

Ageing Particle

Upon repetitive charging and discharging, called cycling, the capacity of a battery decreases. This originates from various superimposed ageing phenomena, for instance due to microscopic crack formations within a particle.


The different ageing mechanisms can be expressed mathematically in terms of cycle number N dependent parameters. Microscopic cracks within particles yield a cycle number dependent diffusion coefficient D=D(N), while a degradation of the solid electrolyte interphase (SEI) yields a degradation of the intercalation reaction rate L=L(N). These yield, by numerical simulations, qualitatively different spectra of the cell voltage, which can be exploited to determined as an inverse problem the specific ageing effect in a real battery system.