# Prediction for the Remaining Useful Life of Lithium–Ion Battery Based on RVM-GM with Dynamic Size of Moving Window

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Data Description

#### 2.2. Relevance Vector Machine

#### 2.2.1. Relevance Vector Regression

#### 2.2.2. Bayesian Inference

#### 2.2.3. Updating Hyper-Parameters and Outputting Prediction Result

#### 2.3. Grey Predictive Model

#### 2.4. RVM-GM Framework Based on Dynamic Window

- (1)
- The estimated values or the actually measured capacities of lithium–ion batteries are taken as the capacity history data. The measured capacities obtained from the NASA lithium–ion battery aging experiment datasets are used to train the RVM algorithm.
- (2)
- Initialize the data window size and pre-process the data in the window. The window should be first initialized to select history capacity data. Then, the data in the window need to be preprocessed. If there is a large increment, the capacity sequence after this cycle shall be selected as the capacity historical data.
- (3)
- Train the RVM algorithm with data in the window and save the relevant vector. After preprocessing, the history capacity sequence can be input into the RVM algorithm as a training sample. The relevance vectors representing the history capacity sequence in the window can then be found.
- (4)
- Obtain the prediction trend of the capacity by grey prediction based on the relevance vector saved in (3). Because the capacity displays a decrement trend, the relevance vector reduction is adopted to generate grey data. The obtained predicted data point can be recognized as the relevance vector of the prediction trend curve.
- (5)
- Interpolate all the grey predicted points by spline curve to figure out the full prediction trend curve of the capacity. The relevance vectors in the window and the relevance vectors obtained by grey prediction form a series of characteristic discrete points representing the degradation curve. By means of spline curve interpolation, the existing discrete points are interpolated to obtain a continuous curve.
- (6)
- Refit the prediction trend curve to obtain the predicted function by the RVM algorithm. Combining the advantage that RVM can fit the equation and give the probability output, the curve is refitted to reach a capacity degradation trend curve equation.
- (7)
- Substitute the capacity failure threshold into the predicted function to obtain the predicted cycle and its probability distribution. For a certain lithium–ion battery, 80% of its nominal capacity can be regarded as the failure threshold. The threshold can be substituted into the capacity degradation trend curve equation. The predicted RUL can be expressed as ${N}_{RUL}={N}_{EOL}-{N}_{ECL}$, where ${N}_{RUL}$ is the current RUL value, ${N}_{EOL}$ is the predicted cycle corresponding to the capacity degradation threshold and ${N}_{ECL}$ is the current predicted starting point.
- (8)
- Dynamically reduce the window size and move the window forward for circular prediction. Before the next cycle, the predicted starting point needs to be examined. If the point is smaller than the predicted failure cycle, then dynamically reduce the window size, move the window forward with a certain step length, and go back to (2) for the next prediction. Otherwise, it is considered that the point has exceeded the predicted failure cycle and the operation ends.

## 3. Results

#### 3.1. Experiment on No. 36 Battery with Window Size 40

#### 3.2. Experiment on No. 05/No. 32/No. 36/No. 47 with Dynamic Window Size

#### 3.3. Algorithm Comparison

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Battery | PredictionStarting Cycle | Win_Size_40 | Win_Size_30 | Win_Size_20 | Dynamic Win_Size | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | ||

#5 | 45 | 66 | 18 | [52, 80] | 127 | 43 | [110, 154] | 125 | 41 | [95, 155] | 66 | 18 | [52, 80] |

50 | 63 | 16 | [47, 79] | 116 | 37 | [95, 137] | 118 | 39 | [99, 147] | 65 | 14 | [50, 80] | |

55 | 52 | 22 | [39, 65] | 86 | 12 | [71, 101] | 91 | 17 | [66, 116] | 53 | 21 | [37, 69] | |

60 | 38 | 31 | [23, 53] | 39 | 30 | [20, 58] | 39 | 30 | [14, 54] | 40 | 29 | [25, 55] | |

65 | 40 | 24 | [28, 52] | 50 | 14 | [34, 66] | 47 | 17 | [27, 67] | 43 | 21 | [26, 60] | |

70 | 37 | 22 | [23, 51] | 50 | 9 | [36, 64] | 34 | 25 | [14, 54] | 49 | 10 | [37, 61] | |

75 | 35 | 19 | [24, 46] | 46 | 8 | [34, 58] | 43 | 11 | [28, 58] | 45 | 9 | [32, 58] | |

80 | 28 | 21 | [16, 40] | 42 | 7 | [32, 52] | 40 | 9 | [28, 52] | 43 | 6 | [33, 53] | |

85 | 20 | 24 | [10, 30] | 31 | 13 | [19, 43] | 30 | 14 | [16, 44] | 32 | 12 | [20, 44] | |

90 | 20 | 19 | [2, 38] | 22 | 17 | [12, 32] | 26 | 13 | [14, 38] | 28 | 11 | [18, 38] | |

95 | 18 | 16 | [5, 31] | 19 | 15 | [9, 29] | 19 | 15 | [6, 32] | 20 | 14 | [10, 30] | |

100 | 12 | 17 | [2, 32] | 12 | 17 | [2, 22] | 12 | 17 | [2, 22] | 12 | 17 | [0, 24] | |

105 | 24 | 0 | [16, 32] | 24 | 0 | [18, 30] | 24 | 0 | [19, 29] | 24 | 0 | [16, 32] | |

110 | 10 | 9 | [4, 16] | 10 | 9 | [3, 17] | 10 | 9 | [3, 17] | 10 | 9 | [2, 18] | |

115 | 1 | 13 | [0, 10] | 1 | 13 | [0, 12] | 11 | 3 | [6, 16] | 16 | 2 | [11, 21] | |

MAE | 18.1 | 16.3 | 17.3 | 12.9 | |||||||||

RMSE | 19.3 | 19.8 | 20.8 | 14.8 | |||||||||

STD | 7.2 | 11.7 | 11.9 | 7.6 | |||||||||

MAPE | 17% | 13% | 14% | 11% |

Battery | Prediction Starting Cycle | Win_Size_8 | Win_Size_6 | Dynamic Win_Size | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | ||

#32 | 14 | 13 | 7 | [3, 23] | 11 | 9 | [2, 20] | 13 | 7 | [3, 23] |

16 | 12 | 6 | [2, 22] | 10 | 8 | [2, 18] | 12 | 6 | [2, 22] | |

18 | 7 | 9 | [1, 13] | 8 | 8 | [2, 14] | 7 | 9 | [1, 13] | |

20 | 6 | 8 | [0, 12] | 7 | 7 | [2, 12] | 7 | 7 | [2, 12] | |

22 | 5 | 7 | [1, 9] | 14 | 2 | [9, 19] | 14 | 2 | [9, 19] | |

24 | 15 | 5 | [10, 25] | 11 | 1 | [7, 15] | 11 | 1 | [7, 15] | |

26 | 11 | 3 | [8, 14] | 10 | 2 | [7, 13] | 10 | 2 | [7, 13] | |

28 | 3 | 3 | [0, 6] | 8 | 2 | [6, 10] | 8 | 2 | [6, 10] | |

MAE | 6.0 | 4.9 | 4.5 | |||||||

RMSE | 6.3 | 5.8 | 5.3 | |||||||

STD | 2.2 | 3.4 | 3.2 | |||||||

MAPE | 21% | 18% | 16% |

Battery | Prediction Starting Cycle | Win_Size_40 | Win_Size_30 | Win_Size_20 | Dynamic Win_Size | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | ||

#36 | 60 | 118 | 4 | [96, 140] | 177 | 55 | [155, 199] | 68 | 54 | [49, 87] | 115 | 7 | [99, 131] |

70 | 148 | 36 | [128, 168] | 145 | 33 | [125, 165] | 66 | 46 | [48, 84] | 128 | 16 | [113, 143] | |

80 | 118 | 16 | [100, 136] | 126 | 24 | [109, 143] | 62 | 40 | [46, 78] | 122 | 20 | [110, 134] | |

90 | 71 | 21 | [58, 84] | 62 | 30 | [49, 75] | 61 | 31 | [46, 76] | 72 | 20 | [62, 82] | |

100 | 85 | 3 | [73, 97] | 59 | 23 | [48, 70] | 101 | 19 | [90, 112] | 72 | 10 | [64, 80] | |

120 | 61 | 1 | [51, 71] | 62 | 0 | [52, 72] | 41 | 21 | [33, 49] | 50 | 12 | [45, 55] | |

140 | 50 | 8 | [44, 56] | 38 | 4 | [30, 46] | 34 | 8 | [29, 39] | 42 | 0 | [37, 47] | |

160 | 2 | 20 | [0, 5] | 2 | 20 | [0, 3] | 2 | 20 | [0, 6] | 6 | 16 | [4, 7] | |

MAE | 13.6 | 23.6 | 29.9 | 12.6 | |||||||||

RMSE | 17.6 | 28.6 | 33.3 | 14.2 | |||||||||

STD | 11.2 | 16.1 | 14.7 | 6.4 | |||||||||

MAPE | 7.3% | 12% | 21% | 7.1% |

Battery | Prediction Starting Cycle | Win_Size_20 | Win_Size_15 | Win_Size_10 | Dynamic Win_Size | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | Prediction | Error | 95% Confidence Bound | ||

#47 | 21 | 24 | 1 | [20, 28] | 17 | 8 | [9, 25] | 23 | 2 | [19, 27] | 24 | 1 | [20, 28] |

23 | 19 | 4 | [13, 25] | 15 | 8 | [8, 22] | 16 | 7 | [8, 24] | 19 | 4 | [13, 24] | |

25 | 9 | 12 | [4, 14] | 4 | 17 | [0, 12] | 6 | 15 | [0, 18] | 11 | 10 | [4, 18] | |

27 | 9 | 10 | [3, 16] | 10 | 9 | [3, 17] | 9 | 10 | [5, 13] | 12 | 7 | [6, 18] | |

29 | 12 | 5 | [7, 17] | 6 | 11 | [0, 15] | 13 | 4 | [9, 17] | 14 | 3 | [8, 20] | |

31 | 8 | 7 | [3, 13] | 9 | 6 | [4, 14] | 8 | 7 | [3, 13] | 9 | 6 | [4, 14] | |

33 | 3 | 10 | [0, 11] | 3 | 10 | [0, 11] | 5 | 8 | [2, 8] | 8 | 5 | [3, 13] | |

35 | 9 | 2 | [4, 14] | 9 | 2 | [4, 14] | 9 | 2 | [4, 14] | 9 | 2 | [4, 14] | |

37 | 6 | 3 | [2, 10] | 6 | 3 | [2, 10] | 6 | 3 | [2, 10] | 6 | 3 | [2, 10] | |

39 | 5 | 2 | [1, 9] | 5 | 2 | [1, 9] | 5 | 2 | [1, 9] | 5 | 2 | [1, 9] | |

41 | 3 | 2 | [0, 6] | 3 | 2 | [0, 6] | 3 | 2 | [0, 6] | 3 | 2 | [0, 6] | |

MAE | 5.3 | 7.1 | 5.6 | 4.1 | |||||||||

RMSE | 6.4 | 8.4 | 6.9 | 4.8 | |||||||||

STD | 3.9 | 4.8 | 4.2 | 2.7 | |||||||||

MAPE | 14% | 20% | 15% | 10% |

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**Figure 2.**RUL prediction result of No. 36 with a fixed window size 40: (

**a**) inspected cycle of 40; (

**b**) inspected cycle of 72; (

**c**) inspected cycle of 122; (

**d**) inspected cycle of 162.

**Figure 3.**RUL prediction results of No. 05 with a dynamic window size: (

**a**) prediction starting cycle of 42; (

**b**) prediction starting cycle of 82; (

**c**) prediction starting cycle of 118; (

**d**) RUL prediction results.

**Figure 4.**RUL prediction results of No. 32 with a dynamic window size: (

**a**) prediction starting cycle of 13; (

**b**) prediction starting cycle of 25; (

**c**) RUL prediction results.

**Figure 5.**RUL prediction results of No. 36 with a dynamic window size: (

**a**) starting cycle of 60; (

**b**) starting cycle of 120; (

**c**) starting cycle of 150; (

**d**) starting cycle of 160; (

**e**) RUL prediction results.

**Figure 6.**RUL prediction results of No. 47 with a dynamic window size: (

**a**) prediction starting cycle of 21; (

**b**) prediction starting cycle of 30; (

**c**) prediction starting cycle of 40; (

**d**) RUL prediction results.

Battery Number | Discharge Current | End Voltage(V) | Charge Current | End-of-Life (Capacity Fade (%)) | Operating Temperature (°C) | Number of Cycles |
---|---|---|---|---|---|---|

No. 05 | 2A constant current | 2.7 | 1.5CC mode then change to 4.2V CV mode | 30% | 24 | 168 |

No. 06 | 2A constant current | 2.5 | 30% | 24 | 168 | |

No. 32 | 4A constant current | 2.7 | 20% | 43 | 40 | |

No. 36 | 2A constant current | 2.7 | 20% | 24 | 197 | |

No. 47 | Fixed loaded-1A | 2.5 | 30% | 4 | 72 |

Battery | Window Size | MAE | RMSE | STD | MAPE |
---|---|---|---|---|---|

No. 05 | Dynamic | 12.9 | 14.8 | 7.6 | 11.5% |

20 | 17.3 | 20.8 | 11.9 | 13.8% | |

30 | 16.3 | 19.8 | 11.7 | 12.8% | |

40 | 18.1 | 19.3 | 7.2 | 16.7% | |

No. 32 | Dynamic | 4.5 | 5.3 | 3.2 | 16.1% |

6 | 4.9 | 5.8 | 3.4 | 17.9% | |

8 | 6.0 | 6.3 | 2.2 | 21.3% | |

No. 36 | Dynamic | 12.6 | 14.2 | 6.4 | 7.1% |

20 | 29.9 | 33.3 | 14.7 | 20.5% | |

30 | 23.6 | 28.6 | 16.1 | 12.4% | |

40 | 13.6 | 17.6 | 11.2 | 7.3% | |

No. 47 | Dynamic | 4.1 | 4.8 | 2.7 | 10.2% |

10 | 5.6 | 6.9 | 4.2 | 15.3% | |

15 | 7.1 | 8.4 | 4.8 | 20.0% | |

20 | 5.3 | 6.4 | 3.9 | 13.9% |

Algorithm | Prediction Starting Cycle | RUL Prediction Error | 95% Confidence Bound |
---|---|---|---|

RVM-GM | 15 | 40 | [89, 111] |

40 | 17 | [116, 130] | |

70 | 19 | [115, 127] | |

100 | 15 | [121, 128] | |

PF | 15 | 49 | [78, 103] |

40 | 23 | [105, 128] | |

70 | 21 | [108, 129] | |

100 | 19 | [115, 127] | |

CNN | 15 | 71 | [68, 70] |

40 | 55 | [73, 96] | |

70 | 45 | [86, 104] | |

100 | 21 | [153, 168] |

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## Share and Cite

**MDPI and ACS Style**

Nan, J.; Deng, B.; Cao, W.; Tan, Z.
Prediction for the Remaining Useful Life of Lithium–Ion Battery Based on RVM-GM with Dynamic Size of Moving Window. *World Electr. Veh. J.* **2022**, *13*, 25.
https://doi.org/10.3390/wevj13020025

**AMA Style**

Nan J, Deng B, Cao W, Tan Z.
Prediction for the Remaining Useful Life of Lithium–Ion Battery Based on RVM-GM with Dynamic Size of Moving Window. *World Electric Vehicle Journal*. 2022; 13(2):25.
https://doi.org/10.3390/wevj13020025

**Chicago/Turabian Style**

Nan, Jinrui, Bo Deng, Wanke Cao, and Zihao Tan.
2022. "Prediction for the Remaining Useful Life of Lithium–Ion Battery Based on RVM-GM with Dynamic Size of Moving Window" *World Electric Vehicle Journal* 13, no. 2: 25.
https://doi.org/10.3390/wevj13020025