# Battery Capacity Estimation Based on Incremental Capacity Analysis Considering Charging Current Rate

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## Abstract

**:**

## 1. Introduction

## 2. Experiment Design and IC Curve Characteristic Analysis

#### 2.1. Battery Experiments

#### 2.2. Battery IC Curve Characteristics with Different Current Rates

## 3. Development of Battery Capacity Estimation Considering Charging Current

#### 3.1. Capacity Estimation Based on the Fitting Method

^{2}of the coefficients of peak II and peak III are greater than 0.95, which indicates the high correlation between the coefficients and capacity. In addition, it can be found that for peak II, the value of ${\alpha}_{1}$ gradually drops as the charging current increases, while that of ${\beta}_{1}$ shows an increasing trend. The coefficients of peak III show a complete opposite change rule. From Figure 4 below, we can have a more intuitive understanding of the regular.

#### 3.2. Capacity Estimation Based on the Data-Driven Method

_{i}and y

_{i}represent the input and output data.

_{n}to output value y

_{n}, as written in Equation (6):

_{n}follows the Gaussian process ξ

_{n}~N(0, σ

_{n}

^{2}).

_{0}, l

_{1},… l

_{d}] denotes hyperparameter in the kernel function. Substituting Equation (7) into Equation (6), we can obtain Equation (8).

## 4. Verification Results and Discussion

#### 4.1. Comparison of the Estimation Accuracy between the Fitting and Data-Driven Methods

#### 4.2. Comparison of the Accuracy between the GPR Method and Traditional Method

#### 4.3. The Influence of Charging Current Offset on Capacity Estimation

## 5. Conclusions

- Charging current has a significant impact on IC curves. With the decrease in the charging current, the IC curves have a moving trend to low voltage, and the height of all peaks increases, and the height of valleys decreases. In addition, the more serious the aging status of the battery, the more the IC curves of the battery will be affected by the charging current.
- The height of peak II, peak III has a strong linear relationship with the capacity degradation under all tested charging currents, which can be used in the fitting method or GPR model to build the capacity estimation model. Additionally, the data-driven method can solve the error generated from the fitting process well and is proved to have higher precision than the fitting method. The data-driven method has an accuracy improvement of up to 15.175% compared with the traditional method.
- Both methods have good robustness to the charging current interference under most working conditions. The data-driven method performs better compared with the fitting method, with an interference error under 0.1%, while that of the fitting method is up to 3%.

- In this paper, we did not consider the effect of charging temperature and initial charging SOC, which also have a severe influence on ICA. Future research work that takes these factors into consideration will be carried out to improve the accuracy on the basis of this study.
- Here, we simply use the average of the estimated results of two picked features as the final results in the fitting method, which may cause an increase in the capacity estimation error. Pearson correlation coefficient can be used to determine the weights of different features in order to optimize the estimation.
- Only the charging current smaller than 0.5 C is studied in this paper. However, fast charging is the research focus of the next-generation BMS. Hence, more work under the high charging current rate should be carried out to improve the applicability of this method.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

BMS | battery management system |

ETI | equal time interval |

EVI | equal voltage interval |

GPR | Gaussian process regression |

IC | incremental capacity |

ICA | incremental capacity analysis |

KF | Kalman filter |

LAM | loss of active material |

LIBs | lithium-ion batteries |

LLI | loss of lithium inventory |

RMSE | root mean squared error |

SOC | state of charge |

SOH | state of health |

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**Figure 2.**IC curves with different current rates under different aging cycles: (

**a**) cycle 0; (

**b**) cycle 200; (

**c**) cycle 400; (

**d**) cycle 600.

**Figure 3.**The correlation between the features of IC curves and battery capacity: (

**a**) peak II; (

**b**) peak III.

Step | Experiment Description |
---|---|

1 | Set the chamber temperature to 25 °C, discharge the battery to 2.5 V with 0.5 C current |

2 | Rest the battery for 1 h, and charge the battery to 4.2 V with 0.1 C current |

3 | Rest the battery for 1 h, and discharge the battery to 2.5 V with 0.5 C current |

4 | Rest the battery for 1 h |

5 | Charge the battery to 4.2 V with 0.2 C, 0.3 C, 0.4 C, 0.5 C current, repeat step 3 to step 4 |

Feature/Current | Peak II | Peak III | ||||
---|---|---|---|---|---|---|

${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | R^{2} | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | R^{2} | |

0.1 C | 0.215 | 1.478 | 0.993 | 0.906 | −0.359 | 0.945 |

0.2 C | 0.180 | 1.710 | 0.991 | 0.918 | −0.321 | 0.964 |

0.3 C | 0.169 | 1.806 | 0.995 | 1.109 | −0.849 | 0.984 |

0.4 C | 0.160 | 1.883 | 0.989 | 1.408 | −1.724 | 0.992 |

0.5 C | 0.169 | 1.870 | 0.990 | 1.640 | −2.3189 | 0.990 |

Error | Method | 0.1 C | 0.2 C | 0.3 C | 0.4 C | 0.5 C |
---|---|---|---|---|---|---|

Average error (%) | fitting method | 3.914 | 1.571 | 1.134 | 1.415 | 1.054 |

GPR method | 2.809 | 1.146 | 0.566 | 0.379 | 0.289 | |

RMSE (%) | fitting method | 0.104 | 0.044 | 0.034 | 0.040 | 0.030 |

GPR method | 0.077 | 0.032 | 0.017 | 0.011 | 0.010 |

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**MDPI and ACS Style**

Lin, Y.; Jiang, B.; Dai, H.
Battery Capacity Estimation Based on Incremental Capacity Analysis Considering Charging Current Rate. *World Electr. Veh. J.* **2021**, *12*, 224.
https://doi.org/10.3390/wevj12040224

**AMA Style**

Lin Y, Jiang B, Dai H.
Battery Capacity Estimation Based on Incremental Capacity Analysis Considering Charging Current Rate. *World Electric Vehicle Journal*. 2021; 12(4):224.
https://doi.org/10.3390/wevj12040224

**Chicago/Turabian Style**

Lin, Yiran, Bo Jiang, and Haifeng Dai.
2021. "Battery Capacity Estimation Based on Incremental Capacity Analysis Considering Charging Current Rate" *World Electric Vehicle Journal* 12, no. 4: 224.
https://doi.org/10.3390/wevj12040224