# Linear Regression-Based Procedures for Extraction of Li-Ion Battery Equivalent Circuit Model Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation and Challenges

#### 1.2. Literature Review

#### 1.3. Main Contribution

#### 1.4. Article Organization

## 2. Materials and Methods

#### 2.1. Model Structure

#### 2.2. Parameter Extraction Procedures

#### 2.2.1. Parameter Extraction Using the ARX Model Format and Least Squares Linear Regression (LS-ARX)

#### 2.2.2. Parameter Extraction Based on Linearization and Least Squares Linear Regression (LS-ECM)

- Step 1

- Step 2

#### 2.2.3. Extraction Based on Differential Evolution (DE-ECM)

#### 2.3. Experimental

## 3. Results

^{−4}%, for resistance values the average was 0.532%, while the highest average, at a value of 2.533%, was presented for time constants.

## 4. Discussion

#### 4.1. Analysis of LS-ARX Procedure

#### 4.2. Analysis of LS-ECM Procedure

## 5. Conclusions

^{−7}V

^{2}and 7.069 × 10

^{−6}V

^{2}was obtained over short and long pulses of the current part of cell-level data obtained from a Sony VTC6 cell. The match between the results of the process and the results of the global optimization also suggests that the error is caused by the capability of the model to replicate the system (linear response assumption) rather than the parameter identification method employed.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

- Kirchhoff’s First Law applied to an RC element (Equation (A1))

- Kirchhoff’s Second Law applied to an RC element (Equation (A2))

- Capacitive current definition (Equation (A3))

## Appendix B

## Appendix C

## References

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ARX Parameter | Equation |
---|---|

${a}_{1}$ | $\frac{\u2206t}{{\tau}_{2}}}+{\displaystyle \frac{\u2206t}{{\tau}_{1}}}-2$ |

${a}_{2}$ | $\frac{{\u2206t}^{2}}{{\tau}_{1}{\tau}_{2}}}-{\displaystyle \frac{\u2206t}{{\tau}_{1}}}-{\displaystyle \frac{\u2206t}{{\tau}_{2}}}+1$ |

${b}_{0}$ | ${R}_{0}$ |

${b}_{1}$ | $\frac{{R}_{0}\u2206t}{{\tau}_{1}}}+{\displaystyle \frac{{R}_{0}\u2206t}{{\tau}_{2}}}+{\displaystyle \frac{{R}_{1}\u2206t}{{\tau}_{1}}}+{\displaystyle \frac{{R}_{2}\u2206t}{{\tau}_{2}}}-2{R}_{0$ |

${b}_{2}$ | $\frac{{R}_{0}{\u2206t}^{2}}{{\tau}_{1}{\tau}_{2}}}+{\displaystyle \frac{{R}_{1}\u2206{t}^{2}}{{\tau}_{1}{\tau}_{2}}}+{\displaystyle \frac{{R}_{2}{\u2206t}^{2}}{{\tau}_{1}{\tau}_{2}}}-{\displaystyle \frac{{R}_{0}\u2206t}{{\tau}_{1}}}-{\displaystyle \frac{{R}_{0}\u2206t}{{\tau}_{2}}}-{\displaystyle \frac{{R}_{1}\u2206t}{{\tau}_{1}}}-{\displaystyle \frac{{R}_{2}\u2206t}{{\tau}_{2}}}+{R}_{0$ |

${c}_{min/max}$ | $\left(1+a1+a2\right)\times {V}_{OC,\text{}min/max}$ |

Sample No. | SOC Range | Discharge Time (s) | LS-ARX | LS-ECM | DE-ECM (Average) |
---|---|---|---|---|---|

Sample 1 | 24.6–24.3% | 10 s | 5.349 × 10^{−7} | 5.900 × 10^{−8} | 5.890 × 10^{−8} |

Sample 2 | 64.8–64.5% | 10 s | 4.164 × 10^{−7} | 4.661 × 10^{−8} | 4.660 × 10^{−8} |

Sample 3 | 95.0–94.7% | 10 s | 5.971 × 10^{−7} | 3.798 × 10^{−7} | 3.794 × 10^{−7} |

Sample 4 | 24.3–14.6% | ~334 s | 1.082 × 10^{−4} | 4.825 × 10^{−6} | 4.826 × 10^{−6} |

Sample 5 | 64.5–54.8% | ~334 s | 2.120 × 10^{−5} | 8.061 × 10^{−7} | 8.058 × 10^{−7} |

Sample 6 | 94.7–84.9% | ~334 s | 2.391 × 10^{−5} | 7.069 × 10^{−6} | 7.073 × 10^{−6} |

Parameter | LS-ARX | LS-ECM | DE-ECM (Average) |
---|---|---|---|

${V}_{OC,max}$ (V) | 4.103 | 4.108 | 4.108 |

${V}_{OC,min}$ (V) | 4.075 | 4.076 | 4.076 |

${R}_{0}$ (Ω) | 2.513 × 10^{−2} | 2.666 × 10^{−2} | 2.670 × 10^{−2} |

${R}_{1}\text{}$(Ω) | 1.011 × 10^{−2} | 1.434 × 10^{−2} | 1.437 × 10^{−2} |

${\tau}_{1}$ (s) | 3.850 | 13.788 | 13.938 |

${R}_{2}$ (Ω) | 2.240 × 10^{−2} | 1.668 × 10^{−2} | 1.664 × 10^{−2} |

${\tau}_{2}$ (s) | 95.961 | 183.044 | 184.345 |

Sample No. | LS-ARX (% of DE-ECM) | LS-ECM (% of DE-ECM) | DE-ECM (Average) |
---|---|---|---|

Sample 1 | 19 (0.15%) | 311 (2.42%) | 12,836 (100%) |

Sample 2 | 25 (0.17%) | 161 (1.12%) | 14,398 (100%) |

Sample 3 | 20 (0.13%) | 223 (1.48%) | 15,107 (100%) |

Sample 4 | 9 (0.08%) | 121 (1.10%) | 10,983 (100%) |

Sample 5 | 7 (0.05%) | 163 (1.15%) | 14,129 (100%) |

Sample 6 | 18 (0.16%) | 77 (0.67%) | 11,367 (100%) |

Sample No. | Default MSE | Best Result MSE | Filter Window Size | DS Factor | Global Minimum MSE |
---|---|---|---|---|---|

Sample 1 | 1.611 × 10^{−6} | 5.349 × 10^{−7} | 2048 | 2 | 5.316 × 10^{−7} |

Sample 2 | 2.334 × 10^{−6} | 4.164 × 10^{−7} | 2048 | 16 | 4.043 × 10^{−7} |

Sample 3 | 2.994 × 10^{−6} | 5.971 × 10^{−7} | 1024 | 4 | 5.971 × 10^{−7} |

Sample 4 | 1.085 × 10^{−4} | 1.082 × 10^{−4} | 1 | 2 | 1.082 × 10^{−4} |

Sample 5 | 2.120 × 10^{−5} | 2.120 × 10^{−5} | 1 | 1 | 1.705 × 10^{−5} |

Sample 6 | 3.150 × 10^{−5} | 2.391 × 10^{−5} | 64 | 8 | 2.226 × 10^{−5} |

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**MDPI and ACS Style**

Savu, V.-I.; Brace, C.; Engel, G.; Didcock, N.; Wilson, P.; Kural, E.; Zhang, N.
Linear Regression-Based Procedures for Extraction of Li-Ion Battery Equivalent Circuit Model Parameters. *Batteries* **2024**, *10*, 343.
https://doi.org/10.3390/batteries10100343

**AMA Style**

Savu V-I, Brace C, Engel G, Didcock N, Wilson P, Kural E, Zhang N.
Linear Regression-Based Procedures for Extraction of Li-Ion Battery Equivalent Circuit Model Parameters. *Batteries*. 2024; 10(10):343.
https://doi.org/10.3390/batteries10100343

**Chicago/Turabian Style**

Savu, Vicentiu-Iulian, Chris Brace, Georg Engel, Nico Didcock, Peter Wilson, Emre Kural, and Nic Zhang.
2024. "Linear Regression-Based Procedures for Extraction of Li-Ion Battery Equivalent Circuit Model Parameters" *Batteries* 10, no. 10: 343.
https://doi.org/10.3390/batteries10100343