On-Line Remaining Useful Life Prediction of Lithium-Ion Batteries Based on the Optimized Gray Model GM(1,1)
Abstract
:1. Introduction
2. Related Algorithms
2.1. Gray Model GM(1,1)
- Step 1:
- The non-negative historical sequence X is expressed as
- Step 2:
- When this sequence is subjected to the accumulating generation operation (AGO), the following sequence is acquired. The AGO is expressed as:
- Step 3:
- The whitening gray dynamic model can be acquired by a first order differential equation with coefficient b, where b represents the influence of the external impact on the development of an event.
- Step 4:
- To obtain the forecasting values of , the inverse accumulated generating operation (IAGO) is used to establish the following gray model:
2.2. Optimized Gray Model GM(1,1)
- Step 5:
- Using the GM(1,1) model from Equations (7) and (8) to forecast the values of the original data, the error is then written as follows:
- Step 6:
- Establish the GM(1,1) model for the above mentioned error by the same procedures as described in Equations (1)–(8), and the get the following gray model:
- Step 7:
- Combining Equations (8) and (11), obtain Equation (12).
2.3. Making Predictions
- Step 1:
- We assume that the raw data are , then the data are predicted by an optimized GM(1,1) model. This is can be represented as:
- Step 2:
- When new data are obtained, the previous data should be removed, and a new GM(1,1) model established by the series . Later, by using the new model, the series is predicted.
- Step 3:
- Steps 1 and 2 should be implemented iteratively until the last data point has been utilized.
2.4. Box–Cox Transformation
Parameter Estimation for Box–Cox transformation
3. HI Extraction and Optimization
3.1. HI Extraction
- Step 1:
- Extract the monitoring voltage and cycle index in each discharging cycle.
- Step 2:
- Define the discharging voltage interval ( and ), and extract the health indicating time series. Thus, the time interval corresponding to the discharging voltage between and can be obtained by Equation (30).
- Step 3:
- Combine the in every discharging cycle to form the HI series .
3.2. HI Optimization
3.2.1. Qualitative Analysis
3.2.2. Correlation Analysis and Evaluation
3.3. HI Performance Evaluation
3.3.1. Evaluation Criterion
3.3.2. Evaluation Result
4. Battery RUL Prediction
5. Conclusions
Author Contributions
Conflicts of Interest
References
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Time | Pearson Correlation | Spearman Rank Correlation |
---|---|---|
Before Box–Cox transformation | 0.9984 | 0.9937 |
After Box–Cox transformation | 0.9988 | 0.9937 |
λ | RSEM | R2 |
---|---|---|
1.5 | 0.0297 | 0.9753 |
Index | λ | RSEM | R2 |
---|---|---|---|
#6 | 2.0 | 0.0200 | 0.9937 |
#7 | 1.0 | 0.0379 | 0.9384 |
#18 | 0.5 | 0.0273 | 0.9686 |
Starting Point | Prediction Number | RUL | Predicted RUL | AE | RE |
---|---|---|---|---|---|
10 | 10 | 130 | 123 | 7 | 5.38% |
15 | 20 | 125 | 115 | 10 | 8.00% |
20 | 5 | 120 | 122 | 2 | 1.67% |
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Zhou, D.; Xue, L.; Song, Y.; Chen, J. On-Line Remaining Useful Life Prediction of Lithium-Ion Batteries Based on the Optimized Gray Model GM(1,1). Batteries 2017, 3, 21. https://doi.org/10.3390/batteries3030021
Zhou D, Xue L, Song Y, Chen J. On-Line Remaining Useful Life Prediction of Lithium-Ion Batteries Based on the Optimized Gray Model GM(1,1). Batteries. 2017; 3(3):21. https://doi.org/10.3390/batteries3030021
Chicago/Turabian StyleZhou, Dong, Long Xue, Yijia Song, and Jiayu Chen. 2017. "On-Line Remaining Useful Life Prediction of Lithium-Ion Batteries Based on the Optimized Gray Model GM(1,1)" Batteries 3, no. 3: 21. https://doi.org/10.3390/batteries3030021