# Online Broadband Impedance Identification for Lithium-Ion Batteries Based on a Nonlinear Equivalent Circuit Model

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

## 3. Modeling

## 4. Methodology

#### 4.1. Discretization of Impedance Model

#### 4.2. Online Identification Algorithm for Impedance Models

## 5. Results and Discussion

#### 5.1. Verification of Impedance Model

^{®}(version 3.1) was used to fit the broadband impedance measured based on M-sequence excitation when the battery in Figure 1 was charged and discharged at different rates and at different temperatures and SOC. The frequency range was 1000 Hz~1 Hz. The impedance fitting results are shown in Figure 8.

#### 5.2. Verification of Parameter Identification Methods

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Broadband impedance measurement results based on M-sequence excitation under different battery states: (

**a**) charge at 25 °C and 30% SOC; (

**b**) discharge at 25 °C and 30% SOC; (

**c**) charge at 25 °C and 50% SOC; (

**d**) discharge at 25 °C and 70% SOC; (

**e**) charge at 5 °C and 50% SOC; (

**f**) discharge at 35 °C and 50% SOC.

**Figure 5.**Simulation model for studying the current variation of electric double-layer capacitor branch during the charging process.

**Figure 7.**The identification process of the current dependence coefficient of charge transfer resistance.

**Figure 8.**Comparison of impedance measurement results and impedance model fitting results under different battery states: (

**a**) Charge at 25 °C and 50% SOC; (

**b**) discharge at 25 °C and 50% SOC; (

**c**) charge at 25 °C and 70% SOC; (

**d**) discharge at 5 °C and 50% SOC.

**Figure 9.**In this figure, schemes follow the same formatting. Current and voltage changes within one cycle of the M-sequence during charging at a rate of 0.5C at 25 °C and 50% SOC: (

**a**) current; (

**b**) terminal voltage.

**Figure 10.**Comparison of predicted terminal voltage and measured terminal voltage obtained from the model based on the parameter identification algorithm: (

**a**) without ${k}_{\mathrm{I}}$ correction; (

**b**) with ${k}_{\mathrm{I}}$ correction; (

**c**) comparison of terminal voltage errors.

**Figure 12.**Comparison between impedance model reconstruction, impedance spectrum, and impedance spectrum based on M-sequence calculation.

**Figure 13.**Time and frequency domain identification results of discharge at 1C rate at 25 °C and 30% SOC: (

**a**) model prediction terminal voltage without ${k}_{\mathrm{I}}$ correction; (

**b**) model prediction terminal voltage with ${k}_{\mathrm{I}}$ correction; (

**c**) comparison of terminal voltage errors; (

**d**) reconstructed impedance spectrum.

**Figure 14.**Time and frequency domain identification results when charging at a 1C rate at 5 °C and 50% SOC: (

**a**) model prediction terminal voltage without ${k}_{\mathrm{I}}$ correction; (

**b**) model prediction terminal voltage with ${k}_{\mathrm{I}}$ correction; (

**c**) comparison of terminal voltage errors; (

**d**) reconstructed impedance spectrum.

Temperature | SOC | Charge/Discharge Rate | RMSE/mΩ | Temperature | SOC | Charge/Discharge Rate | RMSE/mΩ |
---|---|---|---|---|---|---|---|

25 °C | 50% | 0.3C | 0.425 | 25 °C | 70% | 0.3C | 0.458 |

0.6C | 0.478 | 0.6C | 0.427 | ||||

−0.3C | 0.450 | 5 °C | 50% | 0.3C | 0.406 | ||

−0.6C | 0.493 | 0.6C | 0.379 | ||||

30% | 0.3C | 0.462 | 35 °C | 0.3C | 0.495 | ||

0.6C | 0.397 | 0.6C | 0.484 |

Temperature | SOC | Charge/ Discharge Rate | Terminal Voltage RMSE/mV | Impedance RMSE/mΩ |
---|---|---|---|---|

25 °C | 50% | 0.5C | 0.228 | 0.358 |

1C | 0.154 | 0.324 | ||

−0.5C | 0.286 | 0.426 | ||

−1C | 0.215 | 0.379 | ||

30% | 0.5C | 0.239 | 0.361 | |

1C | 0.173 | 0.412 | ||

70% | 0.5C | 0.112 | 0.344 | |

1C | 0.237 | 0.466 | ||

5 °C | 50% | 0.5C | 0.258 | 0.438 |

1C | 0.279 | 0.435 | ||

35 °C | 50% | 0.5C | 0.281 | 0.472 |

1C | 0.292 | 0.493 |

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**MDPI and ACS Style**

Pan, H.; Wang, X.; Zhang, L.; Wang, R.; Dai, H.; Wei, X.
Online Broadband Impedance Identification for Lithium-Ion Batteries Based on a Nonlinear Equivalent Circuit Model. *World Electr. Veh. J.* **2023**, *14*, 168.
https://doi.org/10.3390/wevj14070168

**AMA Style**

Pan H, Wang X, Zhang L, Wang R, Dai H, Wei X.
Online Broadband Impedance Identification for Lithium-Ion Batteries Based on a Nonlinear Equivalent Circuit Model. *World Electric Vehicle Journal*. 2023; 14(7):168.
https://doi.org/10.3390/wevj14070168

**Chicago/Turabian Style**

Pan, Hongyu, Xueyuan Wang, Luning Zhang, Rong Wang, Haifeng Dai, and Xuezhe Wei.
2023. "Online Broadband Impedance Identification for Lithium-Ion Batteries Based on a Nonlinear Equivalent Circuit Model" *World Electric Vehicle Journal* 14, no. 7: 168.
https://doi.org/10.3390/wevj14070168