cs.LGMar 10, 2026

Probabilistic Hysteresis Factor Prediction for Electric Vehicle Batteries with Graphite Anodes Containing Silicon

Authors: Runyao Yu, Viviana Kleine, Philipp Gromotka, Thomas Rudolf, Adrian Eisenmann, Gautham Ram Chandra Mouli, Peter Palensky, Jochen L. Cremer

Organizations: Delft University of Technology, Delft, The Netherlands · AIT Austrian Institute of Technology, Vienna, Austria · Porsche Engineering Group, Weissach, Germany

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Abstract

Batteries with silicon-graphite-based anodes, which offer higher energy density and improved charging performance, introduce pronounced voltage hysteresis, making state-of-charge (SoC) estimation particularly challenging. Existing approaches to modeling hysteresis rely on exhaustive high-fidelity tests or focus on conventional graphite-based lithium-ion batteries, without considering uncertainty quantification or computational constraints. This work introduces a data-driven approach for probabilistic hysteresis factor prediction, with a particular emphasis on applications involving silicon-graphite anode-based batteries. A data harmonization framework is proposed to standardize heterogeneous driving cycles across varying operating conditions. Statistical learning and deep learning models are applied to assess performance in predicting the hysteresis factor with uncertainties while considering computational efficiency. Extensive experiments are conducted to evaluate the generalizability of the optimal model configuration in unseen vehicle models through retraining, zero-shot prediction, fine-tuning, and joint training. By addressing key challenges in SoC estimation, this research facilitates the adoption of advanced battery technologies. A summary page is available at: https://runyao-yu.github.io/Porsche_Hysteresis_Factor_Prediction/

Figures & tables

Fig. 1: Illustration of OCV-SoC curves. a The OCV-SoC relationship provided by the cell manufacturer. b The OCV-SoC relationship with voltage hysteresis, which is particularly significant for silicon–graphite anode-based batteries.
Fig. 2: Overview of harmonization framework and exemplary driving cycles of two vehicle models. a Visualization of the harmonization framework. Heterogeneous driving cycle data are harmonized to produce unified representations: X~s\tilde{X}_{s} and X~d\tilde{X}_{d} , which are fed into corresponding machine learning models. b Distribution of the hysteresis factor (range in [-1, 1]), showing a highly imbalanced distribution for vehicle mode A and vehicle model B. Less than 70% of the data available for vehicle model A is present in vehicle model B. c Scatter plot of maximum versus minimum current. The dashed lines indicate the zero-current baseline, separating charging (negative current) and discharging (positive current) regions, where the vehicle model A exhibit a more aggressive charging and discharging behavior. d Scatter plot of normalized OCV against cell temperature, capturing diverse thermal conditions and spanning a wide range from –18.3 °C to 48.9 °C. Most samples are concentrated between 20–35°C, suggesting that typical operations occur within standard ambient or thermally managed conditions. e Distribution of sequence lengths of driving cycles, exhibiting a heterogeneous temporal structure. The vehicle model A tends to be shorter on average, with a mean sequence length of 30.4 h, while the vehicle model B shows longer sequences, averaging 47.4 h. This suggests that different vehicle models are associated with different operating regimes.
ModelHysteresis FactorBattery CurrentCapacitySoCOCV Hysteresis CurvesCell Voltage
Plett [ 23 ]✓✓    
Roscher [ 25 ] ✓✓   
Xie [ 27 ] ✓  ✓ 
Ko [ 15 ] ✓✓✓  
Xie [ 28 ]✓✓✓✓  
Plett [ 22 ]   ✓✓ 
TABLE I: Relevant features according to different hysteresis models.
Fig. 3: Example illustration of segmentation. A valid segment starts at the 4th hour, where a relaxation phase is followed by a SoC correction. The segment between hours 16 and 20 is not retained, as it lacks a closing SoC correction.
Fig. 4: Illustration of two transformation approaches. a Transformation for statistical learning models. b Transformation for deep learning models.
NameMathematical definition
Minimum valuemin⁡ℓ=0Lxℓ\displaystyle\min^{L}_{\ell=0}x_{\ell}
Maximum valuemax⁡ℓ=0Lxℓ\displaystyle\max^{L}_{\ell=0}x_{\ell}
Absolute minimum valuemin⁡ℓ=0L∣xℓ∣\displaystyle\min^{L}_{\ell=0}|x_{\ell}|
Absolute maximum valuemax⁡ℓ=0L∣xℓ∣\displaystyle\max^{L}_{\ell=0}|x_{\ell}|
Sum of absolute changes∑ℓ=0L−1∣xℓ+1−xℓ∣\displaystyle\sum^{L-1}_{\ell=0}|x_{\ell+1}-x_{\ell}|
Mean of changes1L∑ℓ=0L−1(xℓ+1−xℓ)\displaystyle\frac{1}{L}\sum^{L-1}_{\ell=0}(x_{\ell+1}-x_{\ell})
TABLE II: Statistical features and their mathematical formulation.
ModelHyperparameterRange
LQRReg. L1[0, 1]
QXGBLearning Rate[0.005, 0.1]
 Max. Depth[3, 5]
 Subsample[0.6, 1]
 Reg. Lambda[0.1, 10]
 Reg. Alpha[0.1, 3]
TABLE III: Hyperparameter search range.
Fig. 5: Comparison of model performance and computational efficiency for different modeling strategies on sequence-level and subsequence-level prediction tasks. a PCA + LQR on sequence-level prediction. b PCA + QXGB on sequence-level prediction. c F-Reg + LQR on sequence-level prediction. d F-Reg + QXGB on sequence-level prediction. e QGRU on sequence-level prediction. Different resampling rates are explored for each sequence size. f Scatter plots of true label versus predicted (median) label (left) and residuals (right) for QGRU on subsequence-level prediction. Tight clustering along the diagonal and low residuals demonstrate the strong predictive performance of QGRU.
Sequence-Level Prediction
Dim. Red.ModelSeq. SizeResamp. RateAQL ↓\downarrowROM ↓\downarrowRAM ↓\downarrow
PCALQR600–7.67×10−27.67\times 10^{-2}2.67×10−52.67\times 10^{-5}2.29×10−52.29\times 10^{-5}
 QXGB3000–5.34×10−25.34\times 10^{-2}2.67×10−22.67\times 10^{-2}1.35×10−31.35\times 10^{-3}
F-RegLQR600–8.06×10−28.06\times 10^{-2}1.91×10−5\bm{1.91\times 10^{-5}}1.53×10−5\bm{1.53\times 10^{-5}}
 QXGBAll–2.67×10−22.67\times 10^{-2}1.79×10−21.79\times 10^{-2}6.80×10−46.80\times 10^{-4}
–QGRUAll2402.15×10−2\bm{2.15\times 10^{-2}}2.00×10−22.00\times 10^{-2}7.00×10−27.00\times 10^{-2}
TABLE IV: Model performance comparison on testing data.
StrategyScalerAQL ↓\downarrowChange ↓\downarrow
B/BOld7.86×10−37.86\times 10^{-3}+0.0%
 New8.38×10−38.38\times 10^{-3}+6.2%
A/BOld4.06×10−24.06\times 10^{-2}+415.9%
 New6.49×10−26.49\times 10^{-2}+726.0%
A →\rightarrow B/BOld8.62×10−38.62\times 10^{-3}+9.7%
 New8.23×10−38.23\times 10^{-3}+4.7%
TABLE V: Comparison of testing loss under different training strategies and scalers.
LibraryVersion
scikit-learn1.4.2
tsfresh0.20.2
xgboost2.0.3
torch2.3.1+cu121
ray2.34.0
TABLE VI: Key Python libraries and their respective version.
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