# A Fast Lithium-Ion Battery Impedance and SOC Estimation Method Based on Two-Stage PI Observer

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## Abstract

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## 1. Introduction

_{0}is difficult to determine. (3) The actual available capacity of the battery will change dynamically due to factors such as aging, discharge rate, temperature and so on. Many other methods were proposed to solve the drawbacks of the Coulomb counting method. One of the effective methods to improve the accuracy of SOC estimation is to use the unique correlation between the measurable parameters of battery and SOC, such as open-circuit voltage method [11] or electrochemical impedance spectroscopy [12]. However, these methods require long periods of resting for batteries, which is not suitable for practical control and these measurable parameters are also affected by internal or external factors, such as battery aging [13]. The model-based method is another popular SOC estimation method. The purpose of the model-based method [14] is to gradually converge the output parameters (such as voltage) of the model to the target value during the recursive process and then obtain an estimated value of the battery state parameters. The model-based method requires an accurate battery model to correctly reflect the battery characteristics and they estimate battery SOC using typical regression algorithms of modern control theory, including least squares [15], filters [16,17], neural networks [18], fuzzy control algorithm [19], sliding mode observer [20] and other methods. Meanwhile, charge imbalance is a very common problem in multi-battery SOC estimation. The module-based battery charge balancing system [21] proposes a mathematical model to accelerate the battery charge balancing and improve the system performance.

## 2. SOC Estimation Based on PI Observers

#### 2.1. Introduction of the Electrochemical Battery Model

#### 2.2. First-Level PI Observer for Impedance Estimation

_{ζ,k}is the battery dynamic impedance. ${U}_{k}$, ${U}_{k-1}$, ${I}_{k}$,${I}_{k-1}$ are the battery measured voltages and current at k and k − 1, respectively. Figure 2 is the result of battery impedance obtained from the relevant experiments at different sampling intervals, taking SOC = 0.5. As shown in Figure 2, when the time interval is between 0 and 0.3 s, the impedance changes gently, but after 0.3 s, the impedance changes sharply. The result declares that the short sampling interval can reduce the influence of current on the calculation result of dynamic impedance. The sampling interval adopted in this paper is 0.1 s.

_{k}is the feedback gain. ${K}_{p}$ is the scaling factor that adjusts the feedback gain. ${T}_{i}$ represents the time of integration. err

_{k}means the error between the model output and the measured parameters at time k, as shown in Equation (4).

#### 2.3. Second-Level PI Observer for SOC Estimation

^{th}battery discharge cycle, to update the observer compensation factor under the (n+1)

^{th}battery discharge cycle.

## 3. Verification and Discussion

#### 3.1. Experiment Design

#### 3.2. Verification and the Results

#### 3.3. Analysis for Fault Tolerance of the Factor ξ

## 4. Conclusions

- The experimental results show that the two-stage PI observer method can obtain reliable data results in the presence of unknown initial SOC, current drift, measurement noise, or inaccurate battery capacity.
- The compensation factor can adjust the model parameters online according to the battery usage, compensate part of the capacity loss and keep the system robust.
- The proposed SOC estimation method is capable of obtaining satisfactory accuracy in different use states for test batteries. The SOC error can be kept within 2%.
- The proposed SOC and battery impedance estimation have a simple structure and are easy to implement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Dynamic impedance (${R}_{\zeta})$ and static impedance (${R}_{real,k}$ ) in the case of different current changes.

**Figure 6.**Dynamic stress test standard and modified dynamic stress test for a pure electric vehicle.

**Figure 11.**Battery SOC estimation results and errors (including with $\mathsf{\xi}$ and without $\mathsf{\xi}$ ).

Group | Discharge Condition | |
---|---|---|

1 | $\mathsf{\Delta}{I}_{1}$ | ${I}_{4}-{I}_{3}$ |

$\mathsf{\Delta}{I}_{2}$ | ${I}_{4}-{I}_{2}$ | |

$\mathsf{\Delta}{I}_{3}$ | ${I}_{4}-{I}_{1}$ | |

2 | $\mathsf{\Delta}{I}_{4}$ | ${I}_{3}-{I}_{1}$ |

$\mathsf{\Delta}{I}_{5}$ | ${I}_{2}-{I}_{1}$ | |

3 | $\mathsf{\Delta}{I}_{6}$ | ${I}_{4}-0$ |

$\mathsf{\Delta}{I}_{7}$ | ${I}_{1}-0$ |

Battery type | ISR18650PC |

Battery capacity | 2.6 Ah |

Working voltage | 4.2–2.75 V |

Maximum continuous discharge current | 15 A |

Initial Impedance | ≤30.0 mΩ |

Parameter | Value |
---|---|

E_{0} | 3.459 |

k_{0} | −0.039 |

k_{1} | 0.001 |

k_{2} | 0.066 |

k_{3} | −0.070 |

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**MDPI and ACS Style**

Chen, T.; Huo, M.; Yang, X.; Wen, R.
A Fast Lithium-Ion Battery Impedance and SOC Estimation Method Based on Two-Stage PI Observer. *World Electr. Veh. J.* **2021**, *12*, 108.
https://doi.org/10.3390/wevj12030108

**AMA Style**

Chen T, Huo M, Yang X, Wen R.
A Fast Lithium-Ion Battery Impedance and SOC Estimation Method Based on Two-Stage PI Observer. *World Electric Vehicle Journal*. 2021; 12(3):108.
https://doi.org/10.3390/wevj12030108

**Chicago/Turabian Style**

Chen, Tao, Mengmeng Huo, Xiaolong Yang, and Rui Wen.
2021. "A Fast Lithium-Ion Battery Impedance and SOC Estimation Method Based on Two-Stage PI Observer" *World Electric Vehicle Journal* 12, no. 3: 108.
https://doi.org/10.3390/wevj12030108