A Hybrid Electrochemical Multi-Particle Model for Lithium-Ion Batteries
Authors: Ali Haider Adel Ali , Raijmakers Luc H.J. , Danilov Dmitri L. , Notten Peter H.L. , Eichel Rüdiger-A. 
Published: February 2024

Over the years, physics-based models have proven to be effective tools in understanding the behavior of lithium-ion batteries, which is favorable for enhancing their design and performance. Among the various physics-based models, the Doyle-Fuller-Newman (DFN) model [1] has emerged as the most widely used model due to its ability to simulate battery behavior accurately. However, it has been observed that the DFN model performs poorly in the relaxation region after discharge, which might result in inaccurate battery simulations. The Multiple-Particle DFN (MP-DFN) model [2] has been proposed to address this issue. The MP-DFN model uses multiple electrode particle sizes to allow internal concentration heterogeneities, accurately capturing the slow diffusion processes. However, the MP-DFN model comes at a relatively high computational cost. The diffusion process within the solid electrode particles, typically modeled by Fick’s 2nd law, is the most computationally expensive part of the DFN model. Multiple particle sizes will increase the computational burden, leading to a longer calculation time. Padé approximations have been suggested to replace the commonly used finite difference method (FDM), reducing the computational burden [3]. However, this simplification may not work for larger particle sizes and/or low diffusion coefficients. To address these challenges, a Hybrid-Multiple-Particle DFN (HMP-DFN) model has been developed in this study. The model uses a combination of the FDM and Padé approximation for the various particle sizes in the electrodes. The scaled diffusion length [4] determines whether the FDM or Padé method for a specific particle in the electrode should be applied.
A simulative comparison has been performed between MP-DFN, HMP-DFN, and Padé approximation using the fourth order for all particles (PMP-DFN) and the DFN with one particle size using the average value from the particle distribution. In this comparison, the MP-DFN model has been used as a reference as it is considered the most accurate. The main results are shown in Fig.1, which reveal that the DFN model gives the largest simulation errors, especially at the cut-off and relaxation parts. The PMP-DFN shows a relatively good agreement with the MP-DFN and is 60% faster in simulation time. However, relatively large errors can be observed at the beginning of discharge and in the relaxation region, resulting in a root mean square error (RMSE) of 0.4 mV and a maximum error of 8 mV. The HMP-DFN model shows remarkable accuracy in comparison to the MP-DFN model. It neglects the errors from the PMP-DFN model at the initial stage and decreases the error in the relaxation significantly, giving an RMSE below 0.06 mV, a maximum error below 1 mV, and a more than 40% faster simulation time in comparison to the MP-DFN model. Therefore, due to the decreased computation time with respect to the MP-DFN and the low simulation error, the HMP-DFN model is a suitable and practical tool for accurately simulating battery behavior and optimizing battery design.
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Fig. 1. Discharge voltage curves (1.5C) and corresponding relaxation parts simulated with various model types (a); the error curves of the comparison between the HMP-DFN, PMP-DFN, and DFN with the MP-DFN (b).

Figure 1