# Predicting the Remaining Useful Life of Lithium-Ion Batteries Using 10 Random Data Points and a Flexible Parallel Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. Datasets

## 3. Methodology

#### 3.1. Data Preprocessing

#### 3.2. Data Sparsification

#### 3.3. Hyperparameter Optimization

## 4. Results and Discussion

#### 4.1. Predictive Performance under Different Conditions

#### 4.2. Predictive Performance under 10 Data Points

#### 4.3. Ablation Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

LIBs | lithium-ion batteries |

ML | machine learning |

SVM | support vector machine |

KNN | k-nearest neighbors |

RUL | remaining useful life |

EIS | electrochemical impedance spectroscopy |

GPR | gaussian process regression |

SOTA | state of the art |

MAPE | mean absolute percentage error |

RNN | recurrent neural network |

CNN | convolutional neural network |

FPNN | flexible parallel neural network |

BMS | battery management system |

NOI | number of inceptionblock |

MAE | mean absolute error |

RMSE | root-mean-squared-error |

CC | constant current |

CV | constant voltage |

## Appendix A

**Figure A1.**Data point indices at the completion of each charging cycle for all batteries in the “2017-05-12_batchdata_updated_struct_errorcorrect.mat” file.

**Figure A2.**Data point indices at the completion of each charging cycle for all batteries in the “2017-06-30_batchdata_updated_struct_errorcorrect.mat” file.

Sampling Mode | Points | Detach | MAPE (%) | MAE (Cycles) | RMSE (Cycles) |
---|---|---|---|---|---|

Random sampling | 10 | None | 2.36 | 3.15 | 4.13 |

10 | Initial layers | 3.23 | 3.87 | 5.06 | |

10 | Residual | 2.20 | 3.12 | 4.04 | |

10 | 3D conv | 3.88 | 5.61 | 7.75 | |

10 | 1 block | 2.54 | 3.72 | 4.83 | |

10 | 2 blocks | 4.00 | 4.38 | 5.62 | |

Random sampling | 10 | 3 blocks | 2.68 | 3.72 | 5.02 |

10 | A branch | 99.86 | 484.65 | 619.75 | |

100 | None | 2.31 | 3.01 | 3.92 | |

100 | Initial layers | 6.07 | 7.21 | 8.87 | |

100 | Residual | 2.39 | 3.16 | 4.08 | |

100 | 3D conv | 4.46 | 5.61 | 7.32 | |

100 | 1 block | 3.37 | 4.73 | 6.01 | |

100 | 2 blocks | 4.37 | 5.37 | 6.51 | |

100 | 3 blocks | 11.17 | 13.62 | 14.35 | |

100 | A branch | 99.84 | 484.56 | 619.63 | |

200 | None | 2.62 | 3.21 | 4.36 | |

200 | Initial layers | NaN | NaN | NaN | |

200 | Residual | 1.87 | 2.69 | 3.45 | |

200 | 3D conv | 4.36 | 5.92 | 7.44 | |

200 | 1 block | 2.7 | 3.99 | 4.85 | |

200 | 2 blocks | 2.56 | 3.46 | 4.45 | |

200 | 3 blocks | 7.07 | 7.43 | 8.75 | |

200 | A branch | 99.85 | 484.58 | 619.65 | |

300 | None | 2.86 | 3.43 | 4.34 | |

300 | Initial layers | NaN | NaN | NaN | |

300 | Residual | 2.24 | 2.87 | 3.84 | |

300 | 3D conv | 5.07 | 7.58 | 9.07 | |

300 | 1 block | 2.76 | 3.32 | 4.32 | |

300 | 2 blocks | 2.59 | 3.28 | 4.29 | |

300 | 3 blocks | 3.99 | 5.43 | 6.8 | |

300 | A branch | 99.84 | 484.57 | 619.63 | |

400 | None | 2.2 | 2.8 | 3.7 | |

400 | Initial layers | NaN | NaN | NaN | |

400 | Residual | 2.07 | 3.11 | 3.96 | |

400 | 3D conv | 3.75 | 5.48 | 7 | |

400 | 1 block | 2.88 | 3.5 | 4.61 | |

400 | 2 blocks | 5.42 | 8.24 | 10.29 | |

400 | 3 blocks | 6.47 | 7.12 | 8.66 | |

400 | A branch | 99.85 | 484.63 | 619.72 | |

Uniform sampling | 10 | None | 2.28 | 3.09 | 4.04 |

10 | Initial layers | 2.80 | 3.40 | 4.50 | |

10 | Residual | 2.52 | 3.22 | 4.50 | |

10 | 3D conv | 4.40 | 6.06 | 8.24 | |

10 | 1 block | 2.48 | 2.97 | 3.98 | |

10 | 2 blocks | 2.63 | 3.31 | 4.39 | |

10 | 3 blocks | 3.44 | 3.91 | 5.09 | |

10 | A branch | 99.86 | 484.67 | 619.77 | |

100 | None | 2.49 | 3.51 | 4.42 | |

100 | Initial layers | 4.53 | 4.93 | 6.17 | |

100 | Residual | 2.13 | 2.96 | 3.8 | |

100 | 3D conv | 3.95 | 4.96 | 6.56 | |

100 | 1 block | 2.6 | 4.05 | 5.2 | |

100 | 2 blocks | 2.48 | 3.23 | 4.21 | |

100 | 3 blocks | 4.71 | 7 | 8.66 | |

100 | A branch | 99.83 | 484.53 | 619.6 | |

200 | None | 2.65 | 3.12 | 4.18 | |

200 | Initial layers | NaN | NaN | NaN | |

200 | Residual | 1.92 | 2.48 | 3.27 | |

200 | 3D conv | 3.77 | 5.2 | 6.77 | |

200 | 1 block | 2.42 | 2.97 | 3.95 | |

200 | 2 blocks | 2.53 | 3.15 | 4.16 | |

200 | 3 blocks | 5.84 | 6.39 | 7.82 | |

200 | A branch | 99.84 | 484.58 | 619.64 | |

300 | None | 3.31 | 3.44 | 4.39 | |

300 | Initial layers | NaN | NaN | NaN | |

300 | Residual | 2.07 | 2.83 | 3.77 | |

300 | 3D conv | 3.54 | 5.05 | 6.64 | |

300 | 1 block | 3.65 | 4.04 | 5.12 | |

300 | 2 blocks | 2.98 | 3.51 | 4.55 | |

300 | 3 blocks | 3.42 | 5.08 | 6.53 | |

300 | A branch | 99.84 | 484.56 | 619.61 | |

400 | None | 2.24 | 2.92 | 3.77 | |

400 | Initial layers | NaN | NaN | NaN | |

Uniform sampling | 400 | Residual | 2.29 | 2.72 | 3.56 |

400 | 3D conv | 3.35 | 4.51 | 6.01 | |

400 | 1 block | 2.72 | 3.38 | 4.48 | |

400 | 2 blocks | 3.78 | 5.62 | 7.15 | |

400 | 3 blocks | 6.75 | 8.75 | 10.17 | |

400 | A branch | 99.85 | 484.66 | 619.74 |

Sampling Mode | Points | Detach | MAPE (%) | MAE (Cycles) | RMSE (Cycles) |
---|---|---|---|---|---|

Random sampling | 10 | None | 0.75 | 5.99 | 7.69 |

10 | Initial layers | 0.70 | 5.57 | 7.62 | |

10 | Residual | 0.76 | 5.41 | 6.24 | |

10 | 3D conv | 1.17 | 9.86 | 13.38 | |

10 | 1 block | 0.52 | 3.50 | 4.33 | |

10 | 2 blocks | 0.91 | 6.21 | 6.96 | |

10 | 3 blocks | 0.90 | 7.80 | 10.89 | |

10 | A branch | 99.60 | 820.09 | 931.36 | |

100 | None | 0.65 | 5.93 | 9.85 | |

100 | Initial layers | 1.22 | 9.97 | 12.93 | |

100 | Residual | 0.74 | 5.61 | 7.76 | |

100 | 3D conv | 0.86 | 6.84 | 10.31 | |

100 | 1 block | 0.83 | 7.04 | 9.96 | |

100 | 2 blocks | 1.08 | 8.49 | 10.45 | |

100 | 3 blocks | 1.7 | 11.44 | 12.41 | |

100 | A branch | 99.58 | 819.93 | 932.21 | |

200 | None | 0.75 | 5.67 | 6.79 | |

200 | Initial layers | NaN | NaN | NaN | |

200 | Residual | 0.57 | 4.49 | 6.58 | |

200 | 3D conv | 1.2 | 9.8 | 14.38 | |

200 | 1 block | 0.62 | 4.5 | 5.49 | |

200 | 2 blocks | 0.97 | 7.33 | 9.27 | |

200 | 3 blocks | 1.55 | 11.37 | 13.39 | |

200 | A branch | 99.58 | 819.95 | 931.22 | |

300 | None | 0.48 | 4.41 | 6.53 | |

300 | Initial layers | NaN | NaN | NaN | |

300 | Residual | 0.64 | 4.29 | 5.67 | |

300 | 3D conv | 1.6 | 12.54 | 15.29 | |

300 | 1 block | 0.78 | 6.43 | 8.97 | |

300 | 2 blocks | 0.7 | 5.99 | 8.02 | |

300 | 3 blocks | 0.86 | 7.5 | 11.64 | |

300 | A branch | 99.58 | 819.93 | 931.19 | |

400 | None | 0.68 | 6.02 | 8.17 | |

400 | Initial layers | NaN | NaN | NaN | |

400 | Residual | 0.56 | 4.48 | 6.48 | |

400 | 3D conv | 1.23 | 9.4 | 12.16 | |

400 | 1 block | 0.74 | 6.51 | 9.82 | |

400 | 2 blocks | 1.37 | 12.49 | 17.37 | |

400 | 3 blocks | 1.08 | 8.45 | 11.23 | |

400 | A branch | 99.6 | 820.04 | 931.34 | |

Uniform sampling | 10 | None | 0.71 | 4.94 | 5.82 |

10 | Initial layers | 0.62 | 4.69 | 6.22 | |

10 | Residual | 0.51 | 3.63 | 4.25 | |

10 | 3D conv | 1.30 | 10.96 | 16.38 | |

10 | 1 block | 0.69 | 4.91 | 5.82 | |

10 | 2 blocks | 0.52 | 3.50 | 4.54 | |

10 | 3 blocks | 0.76 | 6.77 | 10.16 | |

10 | A branch | 99.60 | 820.10 | 931.40 | |

100 | None | 0.78 | 5.77 | 7.07 | |

100 | Initial layers | 0.75 | 6.79 | 9.49 | |

100 | Residual | 0.66 | 5.3 | 7.64 | |

Uniform sampling | 100 | 3D conv | 1.34 | 10.46 | 15.11 |

100 | 1 block | 1 | 8.49 | 11.73 | |

100 | 2 blocks | 0.76 | 6.92 | 10.09 | |

100 | 3 blocks | 0.77 | 5.77 | 7.76 | |

100 | A branch | 99.58 | 819.9 | 931.19 | |

200 | None | 0.7 | 5.58 | 7.52 | |

200 | Initial layers | NaN | NaN | NaN | |

200 | Residual | 0.71 | 5.28 | 7.36 | |

200 | 3D conv | 0.94 | 7.73 | 11.04 | |

200 | 1 block | 0.89 | 6.74 | 8.93 | |

200 | 2 blocks | 0.87 | 7.77 | 10.85 | |

200 | 3 blocks | 1.01 | 8.93 | 12.87 | |

200 | A branch | 99.58 | 819.94 | 931.22 | |

300 | None | 0.63 | 5.29 | 8.02 | |

300 | Initial layers | NaN | NaN | NaN | |

300 | Residual | 0.66 | 4.8 | 6.08 | |

300 | 3D conv | 1.42 | 11.78 | 16.94 | |

300 | 1 block | 0.66 | 5.54 | 8.09 | |

300 | 2 blocks | 0.6 | 5.3 | 8.41 | |

300 | 3 blocks | 1.2 | 10.16 | 12.86 | |

300 | A branch | 99.58 | 819.93 | 931.19 | |

400 | None | 0.61 | 4.5 | 6.15 | |

400 | Initial layers | NaN | NaN | NaN | |

400 | Residual | 0.63 | 4.76 | 6.36 | |

400 | 3D conv | 1.08 | 9.03 | 14.65 | |

400 | 1 block | 0.74 | 6.28 | 9.73 | |

400 | 2 blocks | 0.97 | 9.77 | 15.15 | |

400 | 3 blocks | 1.21 | 8.06 | 9.55 | |

400 | A branch | 99.72 | 820.65 | 931.67 |

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**Figure 1.**Data point indices at the completion of each charging cycle for all batteries in the “2018-04-12_batchdata_updated_struct_errorcorrect.mat” file.

**Figure 2.**(

**a**) Schematic diagram of the technical route for RUL prediction based on the FPNN; (

**b**) Detailed architecture and components of the FPNN: ① a 3D convolutional layer using 3 × 3 convolutional kernels and 64 channels; ② an InceptionBlocks module; ③ a 2D convolutional layer with a kernel size of 7 × 7 and 64 channels; ④ a max-pooling layer with a pooling kernel size of 3 × 3; ⑤ an InceptionBlock flexible unit; ⑥ a 2D convolutional layer with a kernel size of 1 × 1 and 16 or 24 channels (used as the target channel number for residual connections in other cases); ⑦ an average pooling layer with a pooling kernel size of 3 × 3; and a ⑧ 2D convolutional layer with a kernel size of 3 × 3 and 16 or 24 channels. The figure also shows I FPNN video-like data after preprocessing; II the overall architecture of the FPNN; III the detailed structure of the InceptionBlocks flexible module; and the IV specific details of the InceptionBlock flexible unit. Reprinted with permission from Ref. [45].

**Figure 3.**Voltage variations during each charging cycle for the “b1c23” battery. The black circles in the figure mark the areas of voltage rise and fall, highlighting the fluctuation characteristics of the voltage during the charging process. (

**a**) Uniform sampling of 10 points; (

**c**) Random sampling of 10 points, depicting the temperature change trend of the “b1c23” battery during the charging process, where temperature variations reflect the thermal management status at different charging stages. (

**b**) Uniform sampling of 10 points; (

**d**) Random sampling of 10 points.

**Figure 4.**RUL prediction under different sampling modes. “Comp” represents non-early predictions, “Early” stands for early predictions, “Rand” denotes random sampling, and “Unif” signifies uniform sampling. The figure includes heatmaps and box plots to visually present the prediction accuracy. The heatmap section includes the (

**a**) MAPE; (

**b**) MAE; and (

**c**) RMSE. The box plot section shows the (

**d**) MAPE and (

**f**) MAE. Additionally, the cycle life distribution of the samples in the test set is also presented, including (

**e**) the complete test set for non-early RUL predictions and (

**g**) the test set for early RUL predictions.

**Figure 5.**The specifics of RUL prediction when sampling 10 data points. “Comp” represents non-early predictions, “Early” stands for early predictions, “Rand” denotes random sampling, and “Unif” signifies uniform sampling. The figure includes (

**a**) random sampling for non-early RUL predictions; (

**b**) uniform sampling for non-early RUL predictions; (

**c**) random sampling for early RUL predictions; (

**d**) uniform sampling for early RUL predictions; (

**e**) “b1c1” battery: random sampling for non-early RUL predictions; and (

**f**) “b2c44” battery: random sampling for non-early RUL predictions. Figure 4 also includes a comprehensive display of early and non-early predictions, as well as the RUL predictions for random and uniform sampling, specifically including the (

**g**) MAPE; (

**h**) MAE and RMSE; and (

**i**) early prediction scenarios for “b1c1” and “b2c44” batteries with random sampling.

**Figure 6.**Results of ablation experiments for RUL prediction, including early and non-early predictions, as well as random and uniform sampling of different numbers of points. The figure includes heatmaps and bar charts to visually demonstrate prediction accuracy. “Comp” represents non-early predictions, “Early” stands for early predictions, “Rand” denotes random sampling, and “Unif” signifies uniform sampling. The heatmap section includes the (

**a**) MAPE; (

**b**) MAE; and (

**c**) RMSE. (

**d**) The bar chart section shows comparisons of the MAPE, MAE, and RMSE when sampling 10 data points. Notes: (1) “NaN” indicates missing data, which occurred in some cases where, after removing the initialization layer, the model training consumed excessive GPU memory, preventing experimentation. (2) “A branch” refers to a branch removed from the dual-stream network, specifically the differential feature branch.

Complete/Early | Points | MAPE (%) | MAE (Cycles) | RMSE (Cycles) |
---|---|---|---|---|

Complete | 10 | 2.36 | 3.15 | 4.13 |

100 | 2.31 | 3.01 | 3.92 | |

200 | 2.62 | 3.21 | 4.36 | |

300 | 2.86 | 3.43 | 4.34 | |

400 | 2.20 | 2.80 | 3.70 | |

Early | 10 | 0.75 | 5.99 | 7.69 |

100 | 0.65 | 5.93 | 9.85 | |

200 | 0.75 | 5.67 | 6.79 | |

300 | 0.48 | 4.41 | 6.53 | |

400 | 0.68 | 6.02 | 8.17 |

Methods | MAPE (%) | MAE (Cycles) | RMSE (Cycles) | Requirements for Input Data |
---|---|---|---|---|

Linear model [7] | 9.1 | — | — | The dense data of the 100 cycles |

HPR CNN [44] | 5.16 | 46.69 | 64.52 | 20% sparse charging data from the first 10 cycles |

HPR CNN [44] | 4.15 | 16.09 | 27.47 | 20% sparse charging data from 10 cycles |

HCNN [43] | 3.55 | 9 | 11 | Dense charging data of the 60 cycles |

TOP-Net [42] | 3.37 | 8 | 11 | The dense data of the 50 cycles |

Proposed method | 2.36 | 3.15 | 4.13 | 10 random charging points from each of 10 cycles |

Proposed method | 0.75 | 5.99 | 7.69 | 10 random charging points from each of the first 10 cycles |

Complete/Early | Detach | MAPE (%) | MAE (Cycles) | RMSE (Cycles) |
---|---|---|---|---|

Complete | None | 2.36 | 3.15 | 4.13 |

Initial layers | 3.23 | 3.87 | 5.06 | |

Residual | 2.20 | 3.12 | 4.04 | |

3D conv | 3.88 | 5.61 | 7.75 | |

1 block | 2.54 | 3.72 | 4.83 | |

2 blocks | 4.00 | 4.38 | 5.62 | |

3 blocks | 2.68 | 3.72 | 5.02 | |

A branch | 99.86 | 484.65 | 619.75 | |

Early | None | 0.75 | 5.99 | 7.69 |

Initial layers | 0.70 | 5.57 | 7.62 | |

Residual | 0.76 | 5.41 | 6.24 | |

3D conv | 1.17 | 9.86 | 13.38 | |

1 block | 0.52 | 3.50 | 4.33 | |

2 blocks | 0.91 | 6.21 | 6.96 | |

3 blocks | 0.90 | 7.80 | 10.89 | |

A branch | 99.60 | 820.09 | 931.36 |

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## Share and Cite

**MDPI and ACS Style**

Jiang, L.; Huang, Q.; He, G.
Predicting the Remaining Useful Life of Lithium-Ion Batteries Using 10 Random Data Points and a Flexible Parallel Neural Network. *Energies* **2024**, *17*, 1695.
https://doi.org/10.3390/en17071695

**AMA Style**

Jiang L, Huang Q, He G.
Predicting the Remaining Useful Life of Lithium-Ion Batteries Using 10 Random Data Points and a Flexible Parallel Neural Network. *Energies*. 2024; 17(7):1695.
https://doi.org/10.3390/en17071695

**Chicago/Turabian Style**

Jiang, Lidang, Qingsong Huang, and Ge He.
2024. "Predicting the Remaining Useful Life of Lithium-Ion Batteries Using 10 Random Data Points and a Flexible Parallel Neural Network" *Energies* 17, no. 7: 1695.
https://doi.org/10.3390/en17071695