# Prediction of Lithium-Ion Battery Capacity by Functional Principal Component Analysis of Monitoring Data

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Functional Principal Component Analysis (fPCA)

#### 3.2. Prediction Model Building and Testing

- (a).
- The monitoring data of the Li-ion battery comprise 168 cycles of voltage, current, and temperature. To begin, we considered the initial $N=100$ cycles as training data to build a prediction model.
- (b).
- The original discretized measurements are transformed to smooth curves by applying the B-spline basis expansion. The number of knots is chosen, for each of voltage, current, and temperature, such that the fitted model has the lowest BIC value.
- (c).
- The fPCA technique is performed on each monitoring variable, and the mean function $\mu \left(t\right)$, fPC scores ${\xi}_{ik},k=1,\dots ,K$, and corresponding eigenfunctions are obtained for each predictor. These fPC scores characterize the status of the battery at the corresponding cycle.
- (d).
- The LASSO regression model is trained based on fPC scores. The LASSO complexity parameter is chosen by the $\mathrm{K}$-fold cross-validation with 5 folds.
- (e).
- To test the model, the monitoring variables’ measurements for the forthcoming $P=20$ cycles are predicted. Simple linear regression was used for this task for each time point. For the voltage measurements for a time point of $j=1$, for example, we fit the following model to the initial 100 cycles of data:$${v}_{i1}={\beta}_{0}+{\beta}_{1}i+{\epsilon}_{i1},\hspace{1em}i=1,\dots ,100$$

- (f).
- Using the eigenfunctions obtained from the training phase, the fPC scores are extracted from each predicted curve.
- (g).
- The prediction of ${\widehat{y}}_{i},i=101,\dots ,120$ are obtained by LASSO model obtained in (d) and compared with the true battery capacity values to evaluate the model performance.

## 4. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Graphical demonstration of the fPCA for monitoring data. The mean functions and eigenfunctions of (

**a**) voltage, (

**b**) current, and (

**c**) temperature are displayed.

**Table 1.**Literature of data-driven models’ classification. The check mark indicates corresponding models used in each reference item.

Authors | Data-Driven Models | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Bayesian Regression | Gaussian Process | Kalman Filter | Particle Filter | Particle Swarm Optimization | Autoregressive Based | Neural Network | Support Vector Machine | Relevance Vector Machine | Functional PCA | LASSO Regression | |

Hu et al., 2016 [20] | ✓ | ||||||||||

Patil et al., 2015 [21] | ✓ | ✓ | |||||||||

Zheng et al., 2018 [22] | ✓ | ✓ | |||||||||

Mavroforakis et al., 2006 [23] | ✓ | ✓ | |||||||||

Long et al., 2013 [24] | ✓ | ✓ | |||||||||

Kirk, 2014 [25] | ✓ | ✓ | |||||||||

Nuhic et al., 2013 [26] | ✓ | ✓ | |||||||||

Qin et al., 2015 [27] | ✓ | ✓ | |||||||||

Zhao et al., 2018 [28] | ✓ | ✓ | |||||||||

Richardson et al., 2017 [29] | ✓ | ✓ | |||||||||

Xian et al., 2014 [30] | ✓ | ✓ | |||||||||

Cheng et al., 2015 [6] | ✓ | ✓ | |||||||||

Lin et al., 2017 [31] | ✓ |

Criteria | Case 1 | Case 2 | Case 3 |
---|---|---|---|

Training cycle | Training cycle | Training cycle | |

1–100 | 1–120 | 1–140 | |

Testing cycle | Testing cycle | Testing cycle | |

101–120 | 121–140 | 141–160 | |

RMSE | 0.009 | 0.02 | 0.04 |

MAPE (%) | 0.44 | 1.74 | 3.18 |

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**MDPI and ACS Style**

Shoriat Ullah, M.; Seo, K.
Prediction of Lithium-Ion Battery Capacity by Functional Principal Component Analysis of Monitoring Data. *Appl. Sci.* **2022**, *12*, 4296.
https://doi.org/10.3390/app12094296

**AMA Style**

Shoriat Ullah M, Seo K.
Prediction of Lithium-Ion Battery Capacity by Functional Principal Component Analysis of Monitoring Data. *Applied Sciences*. 2022; 12(9):4296.
https://doi.org/10.3390/app12094296

**Chicago/Turabian Style**

Shoriat Ullah, MD, and Kangwon Seo.
2022. "Prediction of Lithium-Ion Battery Capacity by Functional Principal Component Analysis of Monitoring Data" *Applied Sciences* 12, no. 9: 4296.
https://doi.org/10.3390/app12094296