# A Lithium-ion Battery RUL Prediction Method Considering the Capacity Regeneration Phenomenon

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Algorithms

#### 2.1. Wavelet Decomposition

#### 2.2. NAR Neural Network

## 3. WDT–NARNN Prediction Method

#### 3.1. Experiment Data Analysis

#### 3.2. WDT- NARNN Modeling Process

Algorithm 1. The integrated method with WDT and NARNN |

(1) Initialization: |

Select the wavelet function $\psi (t)$ and the decomposition levels $l$; |

(2) Decomposition: |

Decompose the capacity series ${\left\{Cap(i),i\right\}}_{i=1}^{T-1}$ for the $l$ levels to obtain the low and high-frequency signals at different scales by Equations (4) and (5); |

(3) Initialize the NAR neural network: |

Initialize the parameters of NAR neural network, the numbers of input layer, hidden layer, and output layer are set to ${N}_{i}$, ${N}_{h}$, and ${N}_{o}$ respectively, the delay of the network is set to d, and the training function is set to ‘trainbr’; |

(4) Output the prediction results: |

Input the decomposed signals ${\left\{WDT\_Cap(i),i\right\}}_{i=1}^{T-1}$ into the NAR models to predict the following changes after time T, then prediction results ${\left\{WDT\_Cap(i),i\right\}}_{i=T}^{N}$ are obtained; |

(5) Wavelet reconstruction: |

The signals ${\left\{WDT\_Cap(i),i\right\}}_{i=T}^{N}$ are reconstructed from 1 to $l$ levels by Equation (6) to obtain the fusing predicted series corresponding to capacity series, and then RUL value can be calculated by Equation (8); |

(6) Evaluate the prediction results: |

The evaluation is given with original testing data and prediction results through some criteria to evaluate the performance of the integrated method WDT–NARNN. |

#### 3.3. Performance Analysis

- (1)
- Root Mean Square Error (RMSE) to evaluate the prediction accuracy. The smaller the RMSE is, the better the prediction performance:$$RMSE=\sqrt{\frac{{\displaystyle \sum _{i=1}^{n}{({y}_{i}-{\widehat{y}}_{i})}^{2}}}{n}}$$
- (2)
- R
^{2}to evaluate the prediction performance. If the fitting degree between the prediction curve and real curve is high, R^{2}will be close to 1:$${R}^{2}(y,\widehat{y})=1-\frac{{\displaystyle \sum _{i=0}^{n-1}{({y}_{i}-{\widehat{y}}_{i})}^{2}}}{{\displaystyle \sum _{i=0}^{n-1}{({y}_{i}-{\overline{y}}_{i})}^{2}}}$$ - (3)
- Absolute Error (AE) to evaluate the RUL accuracy of the prediction model:$$AE=|R-\widehat{R}|$$
- (4)
- Prediction Accuracy Improvement Ratio (${\eta}_{AE}$) to evaluate the RUL prediction accuracy improvement ratio of two different methods. If ${\eta}_{AE}>0$, the first method is more accurate, on the contrary, the second method has higher prediction accuracy:$${\eta}_{AE}=\frac{A{E}_{2}-A{E}_{1}}{R}$$

## 4. Results and Discussion

#### 4.1. RUL Prediction of Lithium-Ion Battery

^{2}are all larger than 0.9, indicating that model M1 has good prediction performance and high fitting degree to the original capacity curve. Meanwhile, as can be seen from columns 3–6 in Table 2, for RMSE, model M1 of each battery is significantly smaller than model M2 and M3. For R

^{2}, M1 is significantly larger than model M2 and M3. The above analysis illustrates that model M1 can effectively capture the regeneration of capacity and has good prediction performance, and the prediction performance of model M1 is significantly better than that of model M2 and M3.

#### 4.2. Different Starting Point Predictions and Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The structure of the Nonlinear Auto Regressive (NAR) neural network, where the w

^{1}is the connection matrix between input layer and hidden layer, w

^{2}is the connection matrix between hidden layer and output layer.

**Figure 4.**Procedures of lithium-ion battery Remaining Useful Life (RUL) prediction based on the proposed method.

**Figure 6.**Capacity series prediction of model M1 at different prediction starting points: (

**a**) battery #5; (

**b**) battery #6; (

**c**) battery #7; (

**d**) battery #18.

**Figure 7.**Uncertainty of predicted End of Life (EOL) at different prediction starting points. (

**a**) battery #5; (

**b**) battery #6; (

**c**) battery #18.

Model | Model Description |
---|---|

M1 | WDT combine with NARNN |

M2 | NARNN without using WDT |

M3 | WDT combine with BPNN |

Evaluate Criteria | Model | #5 | #6 | #7 | #18 |
---|---|---|---|---|---|

RMSE | M1 | 0.0270 | 0.0087 | 0.0175 | 0.0064 |

M2 | 0.0949 | 0.0436 | 0.0678 | 0.0260 | |

M3 | 0.0500 | 0.0616 | 0.0234 | 0.0253 | |

R^{2} | M1 | 0.9226 | 0.9933 | 0.9460 | 0.9751 |

M2 | 0.4151 | 0.8457 | 0.4611 | 0.6494 | |

M3 | 0.7745 | 0.7298 | 0.9035 | 0.6091 |

Battery | Prediction starting point | Predicted RUL | RUL AE |
---|---|---|---|

#5 | 60 | 96 | 28 |

70 | 70 | 12 | |

80 | 48 | 0 | |

90 | 37 | 1 | |

#6 | 60 | 57 | 5 |

70 | 42 | 0 | |

80 | 33 | 1 | |

90 | 22 | 0 | |

#18 | 60 | 42 | 2 |

70 | 30 | 0 | |

80 | 20 | 0 | |

90 | 10 | 0 |

**Table 4.**Comparisons of the average RUL prediction accuracy at four prediction starting points under model M1 and other three methods (M2, M3, and M-LG).

Battery | Method | Average RUL AE | Average ${\mathit{\eta}}_{\mathit{A}\mathit{E}}$ |
---|---|---|---|

#5 | M1 | 10.3 | - |

M2 | 14 | 34.2% | |

M3 | 12 | 21.1% | |

M-LG | 16.3 | 14.3% | |

#6 | M1 | 1.5 | - |

M2 | 17.8 | 37.8% | |

M3 | 12 | 34.6% | |

M-LG | 22.3 | 54.9% | |

#18 | M1 | 0.5 | - |

M2 | 6.5 | 35.6% | |

M3 | 14.8 | 50% | |

M-LG | 6 | 25.8% |

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**MDPI and ACS Style**

Pang, X.; Huang, R.; Wen, J.; Shi, Y.; Jia, J.; Zeng, J.
A Lithium-ion Battery RUL Prediction Method Considering the Capacity Regeneration Phenomenon. *Energies* **2019**, *12*, 2247.
https://doi.org/10.3390/en12122247

**AMA Style**

Pang X, Huang R, Wen J, Shi Y, Jia J, Zeng J.
A Lithium-ion Battery RUL Prediction Method Considering the Capacity Regeneration Phenomenon. *Energies*. 2019; 12(12):2247.
https://doi.org/10.3390/en12122247

**Chicago/Turabian Style**

Pang, Xiaoqiong, Rui Huang, Jie Wen, Yuanhao Shi, Jianfang Jia, and Jianchao Zeng.
2019. "A Lithium-ion Battery RUL Prediction Method Considering the Capacity Regeneration Phenomenon" *Energies* 12, no. 12: 2247.
https://doi.org/10.3390/en12122247