# Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology for RUL Prediction

#### 2.1. Long-Range Dependence of Fractional Lévy Stable Motion

#### 2.2. Analysis of Performance Degradation

#### 2.3. Adaptive Evaluation of Nonlinear Drift Coefficient

#### 2.4. Semi-Analytic Solution of the RUL Distribution for AfLSM Prediction Models

## 3. Estimation of Degradation Model Parameters

## 4. Case Study

#### 4.1. Data Sets and Predictive Evaluation Indicators

#### 4.2. RUL Prediction Based on the AfLSM Model

#### 4.3. Comparison and Discussion with Other Methods

- (1)
- Method 3 (M3): This is the EMD-LSTM model [35,36], and we also quantify the uncertainty of the MMA-LSTM model by using the Dropout method. The optimal value of Dropout was set to 0.3, the initial learning rate was set to 0.01, the maximum step size was 210, the learning rate reduction factor was 0.4, and the learning rate period was 40.
- (2)
- (3)
- Method 5 (M5): This is the fBM model without adaptive drift coefficient λ, and the drift function is $\mu \left(s;\Theta \right)=A{Bt}^{B-1}$. The degradation model satisfies the following formula,

- (4)
- Method 6 (M6): This is the Wiener model with adaptive drift coefficient λ, and the drift function is $\mu \left(s;\Theta \right)=A{Bt}^{B-1}$. The degradation model satisfies the following formula:

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RUL | Remaining useful life |

LIBs | Lithium-ion batteries |

LRD | Long-range dependence |

Probability density function | |

AfLSM | Adaptive fractional Lévy stable motion |

LSTM | Long short-term memory networks |

fBM | Fractional Brownian motion |

DCNN | Deep convolutional neural networks |

fGC | Fractional generalized Cauchy |

EMD | Empirical Mode Decomposition |

HD | Health degree |

COS | Cosine similarity |

RMSE | Root mean square error |

MAE | Mean absolute error |

SRD | Short-range dependence |

fLSM | Fractional Lévy stable motion |

## References

- Arshad, F.; Lin, J.; Manurkar, N.; Fan, E.; Ahmad, A.; Tariq, M.-N.; Wu, F.; Chen, R.; Li, L. Life Cycle Assessment of Lithium-Ion Batteries: A Critical Review. Resour. Conserv. Recycl.
**2022**, 180, 106164. [Google Scholar] [CrossRef] - Hu, X.; Zhang, K.; Liu, K.; Lin, X.; Dey, S.; Onori, S. Advanced Fault Diagnosis for Lithium-Ion Battery Systems: A Review of Fault Mechanisms, Fault Features, and Diagnosis Procedures. IEEE Ind. Electron. Mag.
**2020**, 14, 65–91. [Google Scholar] [CrossRef] - Zhu, X.; Wang, W.; Zou, G.; Zhou, C.; Zou, H. State of Health Estimation of Lithium-Ion Battery by Removing Model Redundancy through Aging Mechanism. J. Energy Storage
**2022**, 52, 105018. [Google Scholar] [CrossRef] - Wang, F.; Zemenu, E.A.; Chou, J.; Tseng, C. Online Remaining Useful Life Prediction of Lithium-Ion Batteries Using Bidirectional Long Short-Term Memory with Attention Mechanism. Energy
**2022**, 254, 124344. [Google Scholar] [CrossRef] - Wei, J.; Dong, G.; Chen, Z. Remaining Useful Life Prediction and State of Health Diagnosis for Lithium-Ion Batteries Using Particle Filter and Support Vector Regression. IEEE Trans. Ind. Electron.
**2018**, 65, 5634–5643. [Google Scholar] [CrossRef] - Liu, B.; Jia, Y.; Yuan, C.; Wang, L.; Gao, X.; Yin, S.; Xu, J. Safety Issues and Mechanisms of Lithium-Ion Battery Cell upon Mechanical Abusive Loading: A Review. Energy Storage Mater.
**2020**, 24, 85–112. [Google Scholar] [CrossRef] - Zhao, S.; Zhang, C.; Wang, Y. Lithium-Ion Battery Capacity and Remaining Useful Life Prediction Using Board Learning System and Long Short-Term Memory Neural Network. J. Energy Storage
**2022**, 52, 104901. [Google Scholar] [CrossRef] - Li, X.; Yuan, C.; Wang, Z. State of Health Estimation for Li-Ion Battery via Partial Incremental Capacity Analysis Based on Support Vector Regression. Energy
**2020**, 203, 117852. [Google Scholar] [CrossRef] - Li, Y.; Li, K.; Liu, X.; Wang, Y.; Zhang, L. Lithium-Ion Battery Capacity Estimation—A Pruned Convolutional Neural Network Approach Assisted with Transfer Learning. Appl. Energy
**2021**, 285, 116410. [Google Scholar] [CrossRef] - Lyu, Z.; Gao, R.; Li, X. A Partial Charging Curve-Based Data-Fusion-Model Method for Capacity Estimation of Li-Ion Battery. J. Power Sources
**2021**, 483, 229131. [Google Scholar] [CrossRef] - Gou, B.; Xu, Y.; Feng, X. State-of-Health Estimation and Remaining-Useful-Life Prediction for Lithium-Ion Battery Using a Hybrid Data-Driven Method. IEEE Trans. Veh. Technol.
**2020**, 69, 10854–10867. [Google Scholar] [CrossRef] - Ma, G.; Wang, Z.; Liu, W.; Fang, J.; Zhang, Y.; Ding, H.; Yuan, Y. A Two-Stage Integrated Method for Early Prediction of Remaining Useful Life of Lithium-Ion Batteries. Knowl. Based Syst.
**2023**, 259, 110012. [Google Scholar] [CrossRef] - Liu, K.; Shang, Y.; Ouyang, Q.; Widanage, W.D. A Data-Driven Approach with Uncertainty Quantification for Predicting Future Capacities and Remaining Useful Life of Lithium-Ion Battery. IEEE Trans. Ind. Electron.
**2020**, 68, 3170–3180. [Google Scholar] [CrossRef] - Wang, Z.; Liu, N.; Chen, C.; Guo, Y. Adaptive Self-Attention LSTM for RUL Prediction of Lithium-Ion Batteries. Inf. Sci.
**2023**, 635, 398–413. [Google Scholar] [CrossRef] - Peng, W.; Chen, Y.Q.; Xu, A.; Ye, Z. Collaborative Online RUL Prediction of Multiple Assets with Analytically Recursive Bayesian Inference. IEEE Trans. Reliab.
**2023**, 1–15. [Google Scholar] [CrossRef] - Li, X.; Yu, D.; Byg Vilsen, S.; Ioan Store, D. The Development of Machine Learning-Based Remaining Useful Life Prediction for Lithium-Ion Batteries. J. Energy Chem.
**2023**. [Google Scholar] [CrossRef] - Liu, Y.; Zhao, G.; Peng, X. Deep Learning Prognostics for Lithium-Ion Battery Based on Ensembled Long Short-Term Memory Networks. IEEE Access
**2019**, 7, 155130–155142. [Google Scholar] [CrossRef] - Pang, X.; Zhao, Z.; Wen, J.; Jia, J.; Shi, Y.; Zeng, J.; Dong, Y. An Interval Prediction Approach Based on Fuzzy Information Granulation and Linguistic Description for Remaining Useful Life of Lithium-Ion Batteries. J. Power Sources
**2022**, 542, 231750. [Google Scholar] [CrossRef] - Xi, X.; Chen, M.; Zhang, H.; Zhou, D. An Improved Non-Markovian Degradation Model with Long-Term Dependency and Item-To-Item Uncertainty. Mech. Syst. Signal Process.
**2018**, 105, 467–480. [Google Scholar] [CrossRef] - Xu, X.; Yu, C.; Tang, S.; Sun, X.; Si, X.; Wu, L. Remaining Useful Life Prediction of Lithium-Ion Batteries Based on Wiener Processes with Considering the Relaxation Effect. Energies
**2019**, 12, 1685. [Google Scholar] [CrossRef] - Song, W.; Liu, H.; Zio, E. Long-Range Dependence and Heavy Tail Characteristics for Remaining Useful Life Prediction in Rolling Bearing Degradation. Appl. Math. Model.
**2022**, 102, 268–284. [Google Scholar] [CrossRef] - Wang, H.; Song, W.; Zio, E.; Kudreyko, A.; Zhang, Y. Remaining Useful Life Prediction for Lithium-Ion Batteries Using Fractional Brownian Motion and Fruit-Fly Optimization Algorithm. Measurement
**2020**, 161, 107904. [Google Scholar] [CrossRef] - Hong, G.; Song, W.; Gao, Y.; Zio, E.; Kudreyko, A. An Iterative Model of the Generalized Cauchy Process for Predicting the Remaining Useful Life of Lithium-Ion Batteries. Measurement
**2022**, 187, 110269. [Google Scholar] [CrossRef] - Weron, A.; Burnecki, K.; Mercik, S.; Weron, K. Complete Description of All Self-Similar Models Driven by Lévy Stable Noise. Phys. Rev. E
**2005**, 71, 016113. [Google Scholar] [CrossRef] [PubMed] - Liu, H.; Song, W.; Zio, E. Fractional Lévy Stable Motion with LRD for RUL and Reliability Analysis of Li-Ion Battery. ISA Trans.
**2021**, 125, 360–370. [Google Scholar] [CrossRef] - Duan, S.; Song, W.; Zio, E.; Cattani, C.; Li, M. Product Technical Life Prediction Based on Multi-Modes and Fractional Lévy Stable Motion. Mech. Syst. Signal Process.
**2021**, 161, 107974. [Google Scholar] [CrossRef] - Williard, N.; He, W.; Osterman, M.; Pecht, M. Comparative Analysis of Features for Determining State of Health in Lithium-Ion Batteries. Int. J. Progn. Health Manag.
**2020**, 4. [Google Scholar] [CrossRef] - He, W.; Williard, N.; Osterman, M.; Pecht, M. Prognostics of Lithium-Ion Batteries Based on Dempster–Shafer Theory and the Bayesian Monte Carlo Method. J. Power Sources
**2011**, 196, 10314–10321. [Google Scholar] [CrossRef] - Xing, Y.; Ma, E.W.M.; Tsui, K.-L.; Pecht, M. An Ensemble Model for Predicting the Remaining Useful Performance of Lithium-Ion Batteries. Microelectron. Reliab.
**2013**, 53, 811–820. [Google Scholar] [CrossRef] - Laskin, N.; Lambadaris, I.; Harmantzis, F.C.; Devetsikiotis, M.H.; Devetsikiotis, M. Fractional Lévy Motion and Its Application to Network Traffic Modeling. Teletraffic Sci. Eng.
**2002**, 40, 363–375. [Google Scholar] [CrossRef] - Jumarie, G. On the Representation of Fractional Brownian Motion as an Integral with Respect to (DT)A. Appl. Math. Lett.
**2005**, 18, 739–748. [Google Scholar] [CrossRef] - Gawronski, W. On the Unimodality of Geometric Stable Laws. Stat. Risk Model.
**2001**, 19, 417–419. [Google Scholar] [CrossRef] - Blachowicz, T.; Ehrmann, A.; Domino, K. Statistical Analysis of Digital Images of Periodic Fibrous Structures Using Generalized Hurst Exponent Distributions. Phys. A Stat. Mech. Its Appl.
**2016**, 452, 167–177. [Google Scholar] [CrossRef] - Duan, S.; Song, W.; Cattani, C.; Yasen, Y.; Li, H. Fractional Levy Stable and Maximum Lyapunov Exponent for Wind Speed Prediction. Symmetry
**2020**, 12, 605. [Google Scholar] [CrossRef] - Guo, R.; Wang, Y.; Zhang, H.; Zhang, G. Remaining Useful Life Prediction for Rolling Bearings Using EMD-RISI-LSTM. IEEE Trans. Instrum. Meas.
**2021**, 70, 1–2. [Google Scholar] [CrossRef] - Wang, F.-K.; Mamo, T. Gradient Boosted Regression Model for the Degradation Analysis of Prismatic Cells. Comput. Ind. Eng.
**2020**, 144, 106494. [Google Scholar] [CrossRef] - Li, X.; Ding, Q.; Sun, J.-Q. Remaining Useful Life Estimation in Prognostics Using Deep Convolution Neural Networks. Reliab. Eng. Syst. Saf.
**2018**, 172, 1–11. [Google Scholar] [CrossRef] - Jiao, J.; Zhao, M.; Lin, J.; Liang, K. A Comprehensive Review on Convolutional Neural Network in Machine Fault Diagnosis. Neurocomputing
**2020**, 417, 36–63. [Google Scholar] [CrossRef]

**Figure 7.**RUL prediction and PDF analysis for M1, M3, M4, M5, and M6 at different observation times. (

**a**) RUL prediction and PDF analysis with a prediction starting point of 321 cycles; (

**b**) RUL prediction and PDF analysis with a prediction starting point of 361 cycles; (

**c**) RUL prediction and PDF analysis with a prediction starting point of 401 cycles; (

**d**) RUL prediction and PDF analysis with a prediction starting point of 441 cycles; (

**e**) RUL prediction and PDF analysis with a prediction starting point of 481 cycles.

Start Cycle | Method | Actual RUL | Predicted RUL | 95% Confidence Interval | Error |
---|---|---|---|---|---|

321 | M1 | 216 | 221 | [207,236] | 6 |

321 | M2 | 216 | 229 | [214,245] | 13 |

321 | M3 | 216 | 289 | [238,372] | 73 |

321 | M4 | 216 | 278 | [238,402] | 62 |

321 | M5 | 216 | 264 | [207,236] | 48 |

321 | M6 | 216 | 262 | [214,245] | 46 |

361 | M1 | 176 | 180 | [167,192] | 4 |

361 | M2 | 176 | 189 | [175,203] | 13 |

361 | M3 | 176 | 231 | [192,310] | 55 |

361 | M4 | 176 | 239 | [196,348] | 63 |

361 | M5 | 176 | 168 | [175,203] | −6 |

361 | M6 | 176 | 241 | [186,298] | 65 |

401 | M1 | 136 | 141 | [129,151] | 5 |

401 | M2 | 136 | 149 | [138,157] | 13 |

401 | M3 | 136 | 174 | [136,230] | 38 |

401 | M4 | 136 | 172 | [128,238] | 36 |

401 | M5 | 136 | 162 | [133,187] | 26 |

401 | M6 | 136 | 168 | [130,240] | 32 |

441 | M1 | 96 | 101 | [92,110] | 5 |

441 | M2 | 96 | 88 | [80,97] | −8 |

441 | M3 | 96 | 117 | [91,157] | 21 |

441 | M4 | 96 | 110 | [80,172] | 14 |

441 | M5 | 96 | 93 | [79,106] | −3 |

441 | M6 | 96 | 121 | [100,156] | 25 |

481 | M1 | 56 | 61 | [54,68] | 5 |

481 | M2 | 56 | 52 | [46,59] | −4 |

481 | M3 | 56 | 70 | [60,92] | 14 |

481 | M4 | 56 | 75 | [70,98] | 19 |

481 | M5 | 56 | 46 | [34,61] | −10 |

481 | M6 | 56 | 76 | [54,68] | 20 |

Start Cycle | A | B | H | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\eta}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|---|

321 | 0.006699 | 0.508152 | 0.9090 | 1.999982 | 0 | 7.5267 × 10^{−06} | 0.0003946 |

361 | 0.005570 | 0.545937 | 0.9300 | 1.999835 | 0 | 7.6433 × 10^{−06} | 0.0004058 |

401 | 0.003967 | 0.613273 | 0.9500 | 1.999986 | 0 | 8.2174 × 10^{−06} | 0.0004322 |

441 | 0.002917 | 0.672244 | 0.9624 | 1.999985 | 0 | 8.6561 × 10^{−06} | 0.0004889 |

481 | 0.001761 | 0.766351 | 0.9693 | 1.999987 | 0 | 8.4561 × 10^{−06} | 0.0004891 |

HD | COS | RMSE | MAE | |
---|---|---|---|---|

M1 | 0.9885 | 0.9999 | 6.0663 | 6.0000 |

M2 | 0.9617 | 0.9990 | 11.0725 | 10.2000 |

M3 | 0.4148 | 0.9981 | 43.2759 | 36.4000 |

M4 | 0.2868 | 0.9996 | 47.7724 | 43.0000 |

M5 | 0.7973 | 0.9933 | 25.4716 | 18.4000 |

M6 | 0.4247 | 0.9983 | 42.9045 | 40.0000 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Song, W.; Chen, J.; Wang, Z.; Kudreyko, A.; Qi, D.; Zio, E.
Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon. *Fractal Fract.* **2023**, *7*, 827.
https://doi.org/10.3390/fractalfract7110827

**AMA Style**

Song W, Chen J, Wang Z, Kudreyko A, Qi D, Zio E.
Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon. *Fractal and Fractional*. 2023; 7(11):827.
https://doi.org/10.3390/fractalfract7110827

**Chicago/Turabian Style**

Song, Wanqing, Jianxue Chen, Zhen Wang, Aleksey Kudreyko, Deyu Qi, and Enrico Zio.
2023. "Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon" *Fractal and Fractional* 7, no. 11: 827.
https://doi.org/10.3390/fractalfract7110827