# A Joint Estimation Method Based on Kalman Filter of Battery State of Charge and State of Health

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## Abstract

**:**

## 1. Introduction

## 2. Battery Modeling and Parameter Identification

#### 2.1. Equivalent Circuit Model of Lithium Battery

_{OC}represents the ideal voltage source, R

_{0}represents the battery’s ohmic resistance, U

_{1}is the voltage between R

_{1}and C

_{1}, U

_{2}is the voltage between R

_{2}and C

_{2}, R

_{1}and R

_{2}are the battery’s polarization resistance, C

_{1}and C

_{2}are the battery polarization capacitance, and U

_{T}indicates battery terminal voltage. The system and observation equations can be obtained according to Kirchhoff’s law. The charging current is negative, and the discharge current direction is positive.

#### 2.2. Online Parameter Identification of Lithium Battery

_{m}(t). The terminal voltage U

_{m}(t) output of the model is different from the actual measured result U(t) due to the inaccurate parameter identification, resulting in the errors U

_{e}(t) and U

_{e}(t) = $\left|U\left(t\right)-{U}_{m}\left(t\right)\right|$. The error U

_{e}(t) was used to correct the model parameters through an identification algorithm (recursive least square algorithm). When the voltage U(t) and U

_{e}(t) of the two terminals were very close, the battery model’s online parameter identification results were finally obtained.

## 3. Joint State Estimation of Battery Power

#### 3.1. UKF Algorithm Principle

#### 3.2. UKPF Algorithm Principle

- (1)
- PF algorithm principle

- (2)
- UKPF algorithm principle

#### 3.3. Multi-Time Scale Joint Estimation of Battery power State

- (1)
- Input the battery voltage and current data into the battery model for online parameter identification;
- (2)
- Determine whether the time scale transformation is met, if so, step (3) is carried out, otherwise, step (4) is carried out;
- (3)
- UKF estimates SOH and uses estimated results to update system parameters;
- (4)
- UKPF cycle estimation SOC;
- (5)
- Output the SOC and SOH estimation results.

## 4. Test Results and Analysis

#### 4.1. Test Platform Building and Test Data Collection

#### 4.2. Parameter Identification Results

_{0}estimate result tends to be stable. Accompanying the discharge process, the battery’s SOC decreases and the value of estimated R

_{0}increases. It is highly consistent with the conclusion that the ohmic resistance decreases with an increasing battery temperature, but it increases with the decrease of SOC.

#### 4.3. Multi-Time Scale Joint Estimation Results

- (1)
- UKPF algorithm estimation of SOC

- (2)
- UKF algorithm estimation of SOH

- (3)
- Multi-time scale joint estimation results

## 5. Conclusions

- (1)
- The battery parameters can be identified online. The error of the parameter identification results is less than 5%, which verifies the validity and accuracy of the model. Therefore, this model can accurately represent the working process of a lithium battery and lays a foundation for the subsequent estimation of its battery state.
- (2)
- Compared with the UKF and the PF algorithm, the UKPF algorithm has higher robust accuracy and stability, and its estimation error of a lithium battery’s state of charge is less than 3.4%. The SOH error of the UKF algorithm is less than 2.5%, which can accurately and effectively estimate the SOH of the battery.
- (3)
- The multi-time scale joint estimation error is within 2.2%, which significantly improves the estimation accuracy of a battery’s SOC and ensures the long-term estimation performance of a battery.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**Result of SOC_discharge simulation based on different algorithms. (

**a**) SOC_discharge estimation based on the UKPF, PF and UKF algorithms; (

**b**) SOC_discharge estimation error based on the UKPF, PF and UKF algorithms.

**Figure 12.**Joint estimation simulation graph. (

**a**) SOC_discharge estimation based on the UKPF and multi-time scale joint estimation algorithms; (

**b**) SOC_discharge estimation error based on the UKPF and multi-time scale joint estimation algorithms.

Battery Parameters | Nominal Capacity (mAh) | Charge Cut-Off Voltage (V) | Discharge Cut-Off Voltage (V) | Nominal Voltage (V) |
---|---|---|---|---|

INR18650-30Q | 3000 | 4.2 | 2.5 | 3.6 |

SOC Estimated Method | SOC Estimated Mean Error | SOC Estimated MAXIMUM error |
---|---|---|

Joint estimation algorithm | 0.74% | 2.11% |

UKPF algorithm | 1.19% | 3.37% |

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**MDPI and ACS Style**

Yang, Q.; Ma, K.; Xu, L.; Song, L.; Li, X.; Li, Y.
A Joint Estimation Method Based on Kalman Filter of Battery State of Charge and State of Health. *Coatings* **2022**, *12*, 1047.
https://doi.org/10.3390/coatings12081047

**AMA Style**

Yang Q, Ma K, Xu L, Song L, Li X, Li Y.
A Joint Estimation Method Based on Kalman Filter of Battery State of Charge and State of Health. *Coatings*. 2022; 12(8):1047.
https://doi.org/10.3390/coatings12081047

**Chicago/Turabian Style**

Yang, Qingxia, Ke Ma, Liyou Xu, Lintao Song, Xiuqing Li, and Yefei Li.
2022. "A Joint Estimation Method Based on Kalman Filter of Battery State of Charge and State of Health" *Coatings* 12, no. 8: 1047.
https://doi.org/10.3390/coatings12081047