# Real-Time Lithium Battery Aging Prediction Based on Capacity Estimation and Deep Learning Methods

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{max}/Q

^{rated}ratio, where Q

^{max}is the maximum charge the battery can hold at the present time and Q

^{rated}is the nominal or rated charge.

## 2. Methodology

_{n}is a key indicator of battery degradation. It is defined as the reduction in available battery capacity (cycle capacity) over time and is expressed as the difference between the initial or reference capacity of a fresh battery, i.e., C

^{ref}(reference capacity), and the remaining capacity at cycle n, i.e., ${C}^{n}$, as

^{remaining}(t), expressed in Coulombs [C], and the reference charge of a fresh battery (usually given by the cell manufacturer), i.e., Q

^{ref}[25], as follows:

_{0}is determined by using the Coulomb counting method. It integrates the instantaneous value of the cell current i(t) over time during the period being analyzed. SOC(t) can be determined if the initial state of charge SOC

_{0}at the initial time t

_{0}is known such that

_{0}. There are many methods to estimate this value, but in this study, we used the ModelGauge m5 EZ algorithm from Maxim Integrated [26].

**Phase 1.1.****Reference charge cycle characterization.**Several (SOC(t), v(t)) data points of the voltage–SOC curve of the battery are acquired, together with the input charge in the constant voltage (CV) stage of the reference charge cycle, i.e., when the battery is brand new. The initial or reference cycle determines the reference capacity of the battery.**Phase 1.2.****Nernst equation fitting and forecasting.**During the constant current (CC) stage of a generic charge cycle over the lifetime of the battery, a few data points (SOC(t), v(t)) corresponding to a portion of the generic charge cycle are used to fit the Nernst equation and forecast the behavior of the voltage–SOC curve during the remaining charge period.**Phase 1.3.****Capacity estimation.**The total charge that the battery can hold in a given cycle (cycle capacity) is estimated based on the forecasted charge cycle and the reference cycle data.

**Phase 2.1.****Phase 2.1 Capacity denoising and preprocessing.**To improve the accuracy of the forecasting techniques in future steps, the time-series data of the calculated cycle capacity are denoised using a discrete wavelet filter. In this step, various algorithms are applied to preprocess the input data of the forecast models.**Phase 2.2.****Phase 2.2 Aging behavior forecast.**Using the processed data from the previous steps as input, linear regression (LR), a long short-term memory neural network (LSTM-RNN), and a gated recurrent unit neural network (GRU-RNN) are used to predict the aging behavior and degradation of the battery.

#### 2.1. Phase 1.1. Reference Charge Cycle Characterization

^{ref}and the charge accumulated by the battery during the CV stage of the charge cycle ${Q}_{charge\_cycle,CV}^{ref}$. Note that the value of ${Q}_{charge\_cycle,CV}^{ref}$ remains almost constant throughout all charge cycles, regardless of the state of the battery, as shown in Section 4 (see Figure 5 and Table 2). A full discharge and a full charge are suggested to properly measure and characterize the initial or reference voltage–SOC profile of the battery.

^{ref}of the fresh battery is calculated using the data from the initial or reference cycle as follows:

#### 2.2. Phase 1.2. Nernst Equation Fitting and Forecast

^{n}(t) and i

^{n}(t) are the instantaneous values of the cell voltage and current at cycle n, E

_{0}is the standard cell potential, R

^{n}is the internal resistance of the cell calculated at each cycle, and μ

_{1}and μ

_{2}are constant parameters. Figure 3 shows the profile of the Nernst equation.

^{n}(t). The generalized least squares (GLS) method [29] is used to determine the values of the unknowns a, b, and c from experimental data using Equation (7).

#### 2.3. Phase 1.3. Capacity Estimation

_{ini,known},t

_{end,known}) of the voltage–SOC curve of the charge cycle, as shown in Figure 4. In this interval, the GLS method uses the Nernst equation to estimate the three unknowns a, b, and c. Once the fitted curve is obtained, it is used to forecast the remaining data points to the ending point of the CC stage of the charge cycle, as shown in Figure 4.

^{n}is the estimated cycle capacity at cycle n (it is an indicator of the SOH of the battery), $SO{C}_{known,0}^{n}$ is the SOC value at the first point of measure in cycle n, t

_{ini,known}and t

_{end,known}establish the time interval where the measured values of the voltage–SOC curve are taken during the CC stage, t

_{ini,forecast}and t

_{end,forecast}define the time forecast interval of the voltage–SOC curve, and ${Q}_{charge\_cycle,CV}^{ref}$ is the total input charge during the CV stage of the reference cycle.

_{ini,known}, while t

_{end,known}corresponds to the time at which the SOC increased by 0.2 from t

_{ini,known}, and thus,

_{end,known}) = SOC(t

_{ini,known}) + 0.2

^{n}at cycle n. It is also noted that t

_{ini,forecast}= t

_{end,known}and t

_{end,forecast}corresponds to the time when the battery voltage reaches 4.2 V, i.e., the starting point of the CV stage.

#### 2.4. Phase 2.1. Capacity Denoising and Preprocessing

^{n}at each cycle n contains high-frequency noise components due to the test conditions, the experimental measurements, the nonlinear behavior of the battery, and even the GLS fitting procedure; therefore, a noise-reduction stage is required. Although other filtering approaches are possible [31], a four-stage discrete wavelet transform (DWT) using a discrete Meyer mother wavelet was applied in this study because it is well suited for signal denoising in forecasting applications [32]. As shown in Appendix A, this method decomposes the signals into different frequency components, allowing for the noise they contain to be selectively eliminated, thereby improving the accuracy and reliability of predictive models.

#### 2.5. Phase 2.2. Aging Behavior Forecast

## 3. Data Description

## 4. Experimental Results

_{cycle_CC,i}is the cycle capacity at cycle i obtained using Coulomb counting, C

_{estimated,i}is the value obtained by denoising the GLS–Nernst estimate at the same cycle, and m is the number of samples. The root-mean-square error is defined as

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Term | Symbol | Unit |

Battery capacity | C | Ah |

Reference battery capacity (fresh battery) | C^{ref} | Ah |

Remaining battery capacity at cycle n | ${C}^{n}$ | Ah |

Constant current phase | CC | - |

Constant voltage phase | CV | - |

Battery charge | Q | C |

Max. battery charge | Q^{max} | C |

Rated or nominal battery charge | Q^{rated} | C |

Remaining battery charge | Q^{remaining} | C |

Reference battery charge (fresh battery) | Q^{ref} | C |

Current | i | A |

Generalized least squares | GLS | - |

Gated recurrent unit | GRU | - |

Linear regression | LR | - |

Long short-term memory | LSTM | - |

State of charge | SOC | - |

Initial state of charge | SOC_{0} | - |

State of health | SOH | - |

Remaining useful life | RUL | h |

Internal cell resistance | R | Ω |

Radial neural network | RNN | - |

Cell voltage | v | V |

## Appendix A

## References

- Huang, Y. The discovery of cathode materials for lithium-ion batteries from the view of interdisciplinarity. Interdiscip. Mater.
**2022**, 1, 323–329. [Google Scholar] [CrossRef] - Spitthoff, L.; Øyre, E.S.; Muri, H.I.; Wahl, M.; Gunnarshaug, A.F.; Pollet, B.G.; Lamb, J.J.; Burheim, O.S. Thermal management of lithium-ion batteries. In Micro-Optics and Energy: Sensors for Energy Devices; Springer: Berlin/Heidelberg, Germany, 2020; pp. 183–194. [Google Scholar]
- Meng, J.; Ricco, M.; Luo, G.; Swierczynski, M.; Stroe, D.I.; Stroe, A.I.; Teodorescu, R. An Overview and Comparison of Online Implementable SOC Estimation Methods for Lithium-Ion Battery. IEEE Trans. Ind. Appl.
**2018**, 54, 1583–1591. [Google Scholar] [CrossRef] - Meng, J.; Luo, G.; Gao, F. Lithium polymer battery state-of-charge estimation based on adaptive unscented kalman filter and support vector machine. IEEE Trans. Power Electron.
**2016**, 31, 2226–2238. [Google Scholar] [CrossRef] - Tao, R.; Gu, Y.; Sharma, J.; Hong, K.; Li, J. A conformal heat-drying direct ink writing 3D printing for high-performance lithium-ion batteries. Mater. Today Chem.
**2023**, 32, 101672. [Google Scholar] [CrossRef] - Teodorescu, R.; Sui, X.; Vilsen, S.B.; Bharadwaj, P.; Kulkarni, A.; Stroe, D.I. Smart Battery Technology for Lifetime Improvement. Batteries
**2022**, 8, 169. [Google Scholar] [CrossRef] - Zhu, J.; Xu, W.; Knapp, M.; Dewi Darma, M.S.; Mereacre, L.; Su, P.; Hua, W.; Liu-Théato, X.; Dai, H.; Wei, X.; et al. A method to prolong lithium-ion battery life during the full life cycle. Cell Reports Phys. Sci.
**2023**, 4, 101464. [Google Scholar] [CrossRef] - Ruan, H.; Barreras, J.V.; Engstrom, T.; Merla, Y.; Millar, R.; Wu, B. Lithium-ion battery lifetime extension: A review of derating methods. J. Power Sources
**2023**, 563, 232805. [Google Scholar] [CrossRef] - Jin, S.; Huang, X.; Sui, X.; Wang, S.; Teodorescu, R.; Stroe, D.I. Overview of Methods for Battery Lifetime Extension. In Proceedings of the 2021 23rd European Conference on Power Electronics and Applications (EPE’21 ECCE Europe), Ghent, Belgium, 6–10 September 2021. [Google Scholar]
- Harper, G.; Sommerville, R.; Kendrick, E.; Driscoll, L.; Slater, P.; Stolkin, R.; Walton, A.; Christensen, P.; Heidrich, O.; Lambert, S.; et al. Recycling lithium-ion batteries from electric vehicles. Nature
**2019**, 575, 75–86. [Google Scholar] [CrossRef] - Woody, M.; Arbabzadeh, M.; Lewis, G.M.; Keoleian, G.A.; Stefanopoulou, A. Strategies to limit degradation and maximize Li-ion battery service lifetime—Critical review and guidance for stakeholders. J. Energy Storage
**2020**, 28, 101231. [Google Scholar] [CrossRef] - Chung, C.H.; Jangra, S.; Lai, Q.; Lin, X. Optimization of Electric Vehicle Charging for Battery Maintenance and Degradation Management. IEEE Trans. Transp. Electrif.
**2020**, 6, 958–969. [Google Scholar] [CrossRef] - Shen, L.; Li, J.; Meng, L.; Zhu, L.; Shen, H.T. Transfer Learning-based State of Charge and State of Health Estimation for Li-ion Batteries: A Review. IEEE Trans. Transp.
**2023**. [Google Scholar] [CrossRef] - Koleti, U.R. Fast Charging Strategies in Lithium-Ion Batteries: Detection and Control of Lithium Plating; University of Warwick: Coventry, UK, 2020. [Google Scholar]
- de la Vega, J.; Riba, J.R.; Ortega-Redondo, J.A. Mathematical Modeling of Battery Degradation Based on Direct Measurements and Signal Processing Methods. Appl. Sci.
**2023**, 13, 4938. [Google Scholar] [CrossRef] - Wenzl, H. BATTERIES|Capacity. In Encyclopedia of Electrochemical Power Sources; Newnes: Sebastopol, CA, USA, 2009; pp. 395–400. [Google Scholar]
- Peng, J.; Meng, J.; Chen, D.; Liu, H.; Hao, S.; Sui, X.; Du, X. A Review of Lithium-Ion Battery Capacity Estimation Methods for Onboard Battery Management Systems: Recent Progress and Perspectives. Batteries
**2022**, 8, 229. [Google Scholar] [CrossRef] - Jiang, J.; Zhang, C. Fundamentals and Application of Lithium-Ion Batteries in Electric Drive Vehicles; Sons, J.W., Ed.; Wiley: Singapore, 2015. [Google Scholar]
- IEC IEC 62660-2:2018; Secondary Lithium-Ion Cells for the Propulsion of Electric Road Vehicles—Part 2: Reliability and Abuse Testing. International Electrotechnical Commission: Geneva, Switzerland, 2018.
- ISO ISO 6469-1:2019; Electrically Propelled Road Vehicles—Safety Specifications—Part 1: Rechargeable Energy Storage System (RESS). ISO: Geneva, Switzerland, 2019.
- IEEE IEEE Std 450-2020; Recommended Practice for Maintenance, Testing, and Replacement of Vented Lead-Acid Batteries for Stationary Applications. IEEE: Piscataway Township, NJ, USA, 2020.
- Meng, J.; Cai, L.; Luo, G.; Stroe, D.I.; Teodorescu, R. Lithium-ion battery state of health estimation with short-term current pulse test and support vector machine. Microelectron. Reliab.
**2018**, 88–90, 1216–1220. [Google Scholar] [CrossRef] - Song, Y.; Li, L.; Peng, Y.; Liu, D. Lithium-Ion Battery Remaining Useful Life Prediction Based on GRU-RNN. In Proceedings of the 2018 12th International Conference on Reliability, Maintainability, and Safety (ICRMS), Shanghai, China, 17–19 October 2018; pp. 317–322. [Google Scholar]
- Park, K.; Choi, Y.; Choi, W.J.; Ryu, H.Y.; Kim, H. LSTM-Based Battery Remaining Useful Life Prediction with Multi-Channel Charging Profiles. IEEE Access
**2020**, 8, 20786–20798. [Google Scholar] [CrossRef] - Zhao, J.; Zhu, Y.; Zhang, B.; Liu, M.; Wang, J.; Liu, C.; Hao, X.; Kowal, J.; Zhao, J.; Zhu, Y.; et al. Review of State Estimation and Remaining Useful Life Prediction Methods for Lithium–Ion Batteries. Sustainability
**2023**, 15, 5014. [Google Scholar] - MaximIntegrated. MAX1726x ModelGauge m5 EZ User Guide; MaximIntegrated: San Jose, CA, USA, 2018. [Google Scholar]
- Plett, G.L. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification. J. Power Sources
**2004**, 134, 262–276. [Google Scholar] [CrossRef] - Hussein, A.A.H.; Batarseh, I. An overview of generic battery models. In Proceedings of the 2011 IEEE Power and Energy Society General Meeting, Detroit, MI, USA, 24–28 July 2011. [Google Scholar]
- Wang, Y.; Meng, D.; Chang, Y.; Zhou, Y.; Li, R.; Zhang, X. Research on online parameter identification and SOC estimation methods of lithium-ion battery model based on a robustness analysis. Int. J. Energy Res.
**2021**, 45, 21234–21253. [Google Scholar] [CrossRef] - Pavković, D.; Kasać, J.; Krznar, M.; Cipek, M. Adaptive Constant-Current/Constant-Voltage Charging of a Battery Cell Based on Cell Open-Circuit Voltage Estimation. World Electr. Veh. J.
**2023**, 14, 155. [Google Scholar] [CrossRef] - Meng, J.; Azib, T.; Yue, M. Early-Stage end-of-Life prediction of lithium-Ion battery using empirical mode decomposition and particle filter. Proc. Inst. Mech. Eng. Part A J. Power Energy
**2023**. [Google Scholar] [CrossRef] - Wang, X.; Xu, J.; Zhao, Y. Wavelet Based Denoising for the Estimation of the State of Charge for Lithium-Ion Batteries. Energies
**2018**, 11, 1144. [Google Scholar] [CrossRef] - Dogan, A.; Cidem Dogan, D. A Review on Machine Learning Models in Forecasting of Virtual Power Plant Uncertainties. Arch. Comput. Methods Eng.
**2023**, 30, 2081–2103. [Google Scholar] [CrossRef] - Kwon, S.J.; Han, D.; Choi, J.H.; Lim, J.H.; Lee, S.E.; Kim, J. Remaining-useful-life prediction via multiple linear regression and recurrent neural network reflecting degradation information of 20Ah LiNixMnyCo1−x−yO
_{2}pouch cell. J. Electroanal. Chem.**2020**, 858, 113729. [Google Scholar] [CrossRef] - Hochreiter, S.; Schmidhuber, J. Long Short-Term Memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] [PubMed] - Cho, K.; van Merriënboer, B.; Bahdanau, D.; Bengio, Y. On the Properties of Neural Machine Translation: Encoder-Decoder Approaches. arXiv
**2014**, arXiv:1409.1259. [Google Scholar] - Preger, Y.; Barkholtz, H.M.; Fresquez, A.; Campbell, D.L.; Juba, B.W.; Romàn-Kustas, J.; Ferreira, S.R.; Chalamala, B. Degradation of Commercial Lithium-Ion Cells as a Function of Chemistry and Cycling Conditions. J. Electrochem. Soc.
**2020**, 167, 120532. [Google Scholar] [CrossRef] - Available online: https://www.batteryarchive.org/ (accessed on 27 November 2023).
- Lu, J.; Xiong, R.; Tian, J.; Wang, C.; Hsu, C.W.; Tsou, N.T.; Sun, F.; Li, J. Battery degradation prediction against uncertain future conditions with recurrent neural network enabled deep learning. Energy Storage Mater.
**2022**, 50, 139–151. [Google Scholar] [CrossRef] - Costa, N.; Sánchez, L.; Anseán, D.; Dubarry, M. Li-ion battery degradation modes diagnosis via Convolutional Neural Networks. J. Energy Storage
**2022**, 55, 105558. [Google Scholar] [CrossRef] - Wang, Y.; Zhu, J.; Cao, L.; Gopaluni, B.; Cao, Y. Long Short-Term Memory Network with Transfer Learning for Lithium-ion Battery Capacity Fade and Cycle Life Prediction. Appl. Energy
**2023**, 350, 121660. [Google Scholar] [CrossRef] - Lee, G.R.; Gommers, R.; Waselewski, F.; Wohlfahrt, K.; O’Leary, A. PyWavelets: A Python package for wavelet analysis. J. Open Source Softw.
**2019**, 4, 1237. [Google Scholar] [CrossRef] - Matlab Introduction to Wavelet Families. Available online: https://www.mathworks.com/help/wavelet/gs/introduction-to-the-wavelet-families.html (accessed on 17 December 2023).

**Figure 1.**Flowchart summarizing the steps involved in phase 1 and phase 2 of the proposed approach for estimating the cycle capacity (SOH) and forecasting the capacity fade.

**Figure 5.**Amount of charge accumulated in the battery during the CV stage of the charge cycle ${Q}_{charge\_cycle,CV}^{n}$. These data correspond to an NCA-type battery cycled at 15 °C and 0.5C/2C.

**Figure 6.**Evolution of the cycle capacity over the lifetime of the battery was obtained using Coulomb counting, the estimation was obtained using the GLS–Nernst approach, and its denoised version was obtained by applying the DWT filtering method. These data correspond to an NCA-type battery cycled at 35 °C and 0.5C during the charge cycle and 1C during the discharge cycle.

**Figure 7.**Forecasted cycle capacities using the LR, LSTM-RNN, and GRU-RNN methods at cycles corresponding to 6%, 15%, and 24% of capacity fade with respect to the reference capacity. These data correspond to an NCA-type battery cycled at 35 °C and 0.5C during the charge cycle and 1C during the discharge cycle.

**Table 1.**Technical specifications of the Sandia National Laboratories database batteries investigated in this study and their test conditions.

Cathode Chemistry | NCA | NMC |
---|---|---|

Manufacturer | Panasonic | LG Chem |

Manufacturer PN | NCR18650B | 18650HG2 |

Battery type | 18650 | 18650 |

Nominal capacity [Ah] | 3.2 | 3 |

Nominal voltage [V] | 3.6 | 3.6 |

Voltage range [V] | 2.5 to 4.2 | 2 to 4.2 |

Max discharge current [A] | 6 | 20 |

Temperature range [°C] | 0 to 45 | −5 to 50 |

Charge C-rate | 0.5C | 0.5C |

Discharge C-rate | 0.5C/1C/2C | 0.5C/1C/2C |

Test temperature [°C] | 15 °C/25 °C/35 °C | 15 °C/25 °C/35 °C |

Depth of discharge | 0% to 100% | 0% to 100% |

**Table 2.**Accumulated charge during the CV stage of the charge cycle over the life of the batteries. Average maximum differences for the 39 analyzed batteries of NCA and NMC cathode chemistries between the reference cycle and all cycles over the life of the batteries.

T_{ambient} | 15 °C | 25 °C | 35 °C | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cathode | NCA | NMC | NCA | NMC | NCA | NMC | ||||||||

C-Rate | 1 | 2 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 1 | 2 | 1 | 2 |

Average max. difference [Ah] | 0.08 | 0.10 | 0.04 | 0.03 | 0.05 | 0.06 | 0.08 | 0.21 | 0.14 | 0.09 | 0.09 | 0.09 | 0.07 | 0.07 |

Average max. difference [%] * | 2.73 | 3.41 | 1.46 | 1.08 | 0.44 | 2.03 | 2.62 | 7.14 | 4.63 | 3.06 | 2.50 | 2.80 | 2.38 | 2.38 |

**Table 3.**Average RMSE and MAPE errors between the Coulomb counting cycle capacity and the denoised estimated cycle capacity calculated for the 39 batteries analyzed over their full lifetime for CV stage starting at 3.7, 3.8, and 3.9 V.

T_{ambient} | 15 °C | 25 °C | 35 °C | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cathode | NCA | NMC | NCA | NMC | NCA | NMC | |||||||||

V | C-Rate | 1 | 2 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 1 | 2 | 1 | 2 |

3.7 | RMSE | 0.11 | 0.12 | 0.11 | 0.07 | 0.09 | 0.09 | 0.11 | 0.35 | 0.34 | 0.17 | 0.18 | 0.17 | 0.17 | 0.16 |

MAPE | 4.46 | 5.16 | 5.27 | 3.05 | 2.54 | 3.16 | 3.78 | 13.80 | 13.78 | 7.02 | 5.71 | 6.40 | 6.61 | 6.15 | |

3.8 | RMSE | 0.06 | 0.11 | 0.07 | 0.04 | 0.04 | 0.06 | 0.1 | 0.18 | 0.21 | 0.22 | 0.09 | 0.12 | 0.17 | 0.21 |

MAPE | 2.23 | 4.08 | 1.81 | 1.4 | 1.34 | 1.75 | 2.77 | 6.26 | 7.23 | 9.05 | 2.18 | 3.86 | 6.69 | 8.23 | |

3.9 | RMSE | 0.04 | 0.05 | 0.13 | 0.12 | 0.05 | 0.07 | 0.08 | 0.11 | 0.07 | 0.11 | 0.12 | 0.14 | 0.13 | 0.15 |

MAPE | 1.14 | 1.89 | 6.30 | 5.84 | 1.87 | 2.32 | 3.02 | 2.67 | 2.82 | 4.54 | 3.99 | 5.55 | 5.22 | 5.62 |

**Table 4.**Average RMSE and MAPE errors between the Coulomb counting cycle capacity and the forecasted cycle capacity calculated at cycles corresponding to 6%, 15%, and 24% of capacity fade with respect to the reference capacity. Errors averaged for the 39 batteries analyzed over their full lifetime.

Forecast Errors Calculated at the Cycle Corresponding to 6% Capacity Fade | ||||||||||||||||

T_{Amb} | 15 °C | 25 °C | 35 °C | |||||||||||||

Cathode | NCA | NMC | NCA | NMC | NCA | NMC | ||||||||||

C-Rate | 1 | 2 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 1 | 2 | 1 | 2 | Average | |

LR | RMSE | 0.09 | 0.13 | 0.19 | 0.19 | 0.17 | 0.13 | 0.14 | 0.16 | 0.26 | 0.24 | 0.23 | 0.20 | 0.22 | 0.28 | 0.19 |

MAPE | 3.54 | 5.14 | 9.68 | 9.41 | 6.10 | 4.77 | 5.51 | 6.57 | 10.50 | 10.55 | 7.75 | 7.71 | 9.29 | 11.49 | 7.72 | |

GRU | RMSE | 0.48 | 0.80 | 0.29 | 0.27 | 0.05 | 0.28 | 0.38 | 0.75 | 0.40 | 0.30 | 0.15 | 0.31 | 0.34 | 0.15 | 0.35 |

MAPE | 19.20 | 40.47 | 13.61 | 10.48 | 1.59 | 9.70 | 15.66 | 28.82 | 15.60 | 10.63 | 4.65 | 11.15 | 12.30 | 5.60 | 14.25 | |

LSTM | RMSE | 0.08 | 0.15 | 0.17 | 0.11 | 0.10 | 0.08 | 0.11 | 0.14 | 0.24 | 0.24 | 0.16 | 0.16 | 0.20 | 0.27 | 0.16 |

MAPE | 3.09 | 5.66 | 8.63 | 5.28 | 3.40 | 2.83 | 3.27 | 5.87 | 9.68 | 10.57 | 4.92 | 5.59 | 8.18 | 11.11 | 6.29 | |

Forecast Errors Calculated at the Cycle Corresponding to 15% Capacity Fade | ||||||||||||||||

T_{Amb} | 15 °C | 25 °C | 35 °C | |||||||||||||

Cathode | NCA | NMC | NCA | NMC | NCA | NMC | ||||||||||

C-Rate | 1 | 2 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 1 | 2 | 1 | 2 | Average | |

LR | RMSE | 0.05 | 0.12 | 0.05 | 0.03 | 0.06 | 0.06 | 0.12 | 0.15 | 0.19 | 0.2 | 0.14 | 0.15 | 0.17 | 0.21 | 0.12 |

MAPE | 1.7 | 3.84 | 2.16 | 1.54 | 2.49 | 2.03 | 3.79 | 6.21 | 7.34 | 8.93 | 4.33 | 5.74 | 7.18 | 8.35 | 4.69 | |

GRU | RMSE | 0.24 | 0.3 | 0.28 | 0.32 | 0.1 | 0.21 | 0.12 | 0.18 | 0.12 | 0.33 | 0.15 | 0.41 | 0.14 | 0.1 | 0.21 |

MAPE | 10.38 | 13.95 | 14.85 | 15.25 | 3.82 | 8.03 | 4.05 | 7.43 | 4.8 | 14.16 | 4.89 | 17.67 | 5.26 | 3.09 | 9.12 | |

LSTM | RMSE | 0.04 | 0.15 | 0.06 | 0.02 | 0.03 | 0.07 | 0.12 | 0.14 | 0.16 | 0.21 | 0.12 | 0.15 | 0.15 | 0.19 | 0.12 |

MAPE | 1.51 | 5.53 | 2.69 | 0.78 | 0.86 | 2.21 | 3.57 | 5.85 | 6.28 | 9.12 | 3.3 | 5.84 | 6.12 | 7.75 | 4.39 | |

Forecast Errors Calculated at the Cycle Corresponding to 24% Capacity Fade | ||||||||||||||||

T_{Amb} | 15 °C | 25 °C | 35 °C | |||||||||||||

Cathode | NCA | NMC | NCA | NMC | NCA | NMC | ||||||||||

C-Rate | 1 | 2 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 1 | 2 | 1 | 2 | Average | |

LR | RMSE | 0.04 | 0.15 | 0.06 | 0.02 | 0.01 | 0.07 | 0.15 | 0.12 | 0.12 | 0.21 | 0.09 | 0.15 | 0.11 | 0.19 | 0.11 |

MAPE | 1.83 | 5.1 | 3.23 | 0.88 | 0.43 | 2.67 | 5.71 | 4.86 | 5.16 | 9.28 | 2.8 | 5.85 | 4.69 | 7.41 | 4.28 | |

GRU | RMSE | 0.09 | 0.24 | 0.06 | 0.17 | 0.05 | 0.13 | 0.13 | 0.13 | 0.08 | 0.2 | 0.1 | 0.21 | 0.09 | 0.17 | 0.13 |

MAPE | 3.88 | 12.12 | 3.07 | 8.93 | 1.61 | 5.18 | 2.49 | 5.71 | 3.12 | 9.25 | 3.62 | 8.23 | 3.72 | 6.66 | 5.54 | |

LSTM | RMSE | 0.05 | 0.17 | 0.08 | 0.02 | 0.02 | 0.06 | 0.15 | 0.12 | 0.11 | 0.22 | 0.08 | 0.17 | 0.12 | 0.2 | 0.11 |

MAPE | 2.04 | 6.48 | 4.64 | 0.78 | 0.63 | 2.34 | 4.62 | 5.24 | 4.79 | 9.9 | 2.41 | 7.47 | 4.87 | 7.76 | 4.57 |

Method | Average Computation Time per Cycle [s] |
---|---|

Capacity estimation | 0.00158 |

DWT denoising | 0.00053 |

LR | 0.00188 |

GRU | 19.28116 |

LSTM | 21.13642 |

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**MDPI and ACS Style**

de la Vega, J.; Riba, J.-R.; Ortega-Redondo, J.A.
Real-Time Lithium Battery Aging Prediction Based on Capacity Estimation and Deep Learning Methods. *Batteries* **2024**, *10*, 10.
https://doi.org/10.3390/batteries10010010

**AMA Style**

de la Vega J, Riba J-R, Ortega-Redondo JA.
Real-Time Lithium Battery Aging Prediction Based on Capacity Estimation and Deep Learning Methods. *Batteries*. 2024; 10(1):10.
https://doi.org/10.3390/batteries10010010

**Chicago/Turabian Style**

de la Vega, Joaquín, Jordi-Roger Riba, and Juan Antonio Ortega-Redondo.
2024. "Real-Time Lithium Battery Aging Prediction Based on Capacity Estimation and Deep Learning Methods" *Batteries* 10, no. 1: 10.
https://doi.org/10.3390/batteries10010010