# Lithium-Ion Battery Modeling and State of Charge Prediction Based on Fractional-Order Calculus

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## Abstract

**:**

## 1. Introduction

## 2. Fractional-Order Modeling of Lithium-Ion Batteries

#### 2.1. Thevenin Equivalent Circuit Model

#### 2.2. Fractional Calculus Modeling

#### 2.2.1. Fractional-Order Calculus Definition

#### 2.2.2. Fractional-Order Impedance Model

_{1}, $W$, and ${R}_{0}$ are the parameters to be identified.

#### 2.3. Parameter Identification

- (1)
- Identification of parameters $\alpha $, $\beta $, ${R}_{1}$, ${C}_{1}$, $W$

- (2)
- Identification of parameter ${R}_{0}$

## 3. SOC Prediction Based on the Fractional-Order Extended Kalman Filter

- (1)
- Initialization

- (2)
- Time update

- (3)
- Measurement update

_{k}is the gain matrix of Kalman filter; ${R}_{k}$ is the covariance matrix of process noise ${v}_{k}$; $E$ is the identity matrix of $3\times 3$.

## 4. Experiment Results

#### 4.1. Data Acquisition

#### 4.2. Battery Model Verification

#### 4.3. SOC Prediction and Analysis of Lithium-Ion Batteries

#### 4.3.1. DST Working Condition Simulation Verification

#### 4.3.2. FUDS Working Condition Simulation Verification

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Parameter | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{R}}_{1}$ | ${\mathit{C}}_{1}$ | $\mathit{W}$ | ${\mathit{R}}_{0}$ |
---|---|---|---|---|---|---|

FOIM | 0.998 | 0.982 | 0.2452 Ω | 15,879 F | 526,190 Ω | 0.2129 Ω |

IM | 0.995 | - | 0.2834 Ω | 18,527 F | - | 0.2129 Ω |

Thevenin model | - | - | 0.3521 Ω | 25,413 F | - | 0.2129 Ω |

Model | FOIM | IM | Thevenin |
---|---|---|---|

Root mean square error | 0.0139 V | 0.0152 V | 0.0195 V |

Conditions | EKF | FEKF |
---|---|---|

DST | 0.0352 | 0.0121 |

FUDS | 0.0348 | 0.0233 |

Conditions | EKF | FEKF |
---|---|---|

DST | 0.042 | 0.020 |

FUDS | 0.060 | 0.045 |

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**MDPI and ACS Style**

Zhang, X.; Li, X.; Yang, K.; Wang, Z.
Lithium-Ion Battery Modeling and State of Charge Prediction Based on Fractional-Order Calculus. *Mathematics* **2023**, *11*, 3401.
https://doi.org/10.3390/math11153401

**AMA Style**

Zhang X, Li X, Yang K, Wang Z.
Lithium-Ion Battery Modeling and State of Charge Prediction Based on Fractional-Order Calculus. *Mathematics*. 2023; 11(15):3401.
https://doi.org/10.3390/math11153401

**Chicago/Turabian Style**

Zhang, Xinfeng, Xiangjun Li, Kaikai Yang, and Zhongyi Wang.
2023. "Lithium-Ion Battery Modeling and State of Charge Prediction Based on Fractional-Order Calculus" *Mathematics* 11, no. 15: 3401.
https://doi.org/10.3390/math11153401