# State of Charge Estimation of Lithium-Ion Batteries Based on an Improved Sage-Husa Extended Kalman Filter Algorithm

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## Abstract

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## 1. Introduction

## 2. Equivalent Circuit Model and Parameter Identification

#### 2.1. Equivalent Circuit Model

#### 2.2. Parameter Identification

- Identification of ${R}_{0}$: ${R}_{0}$ is mainly composed of electrolyte resistance. Once the discharge current is performed or stopped, the terminal voltage of the battery model presented in Figure 1 decreases or increases. According to the discharge-static experiment, in the first moment of battery resting and the last moment of applying discharge current, the instantaneous drop of terminal voltage (${U}_{a}$ drops rapidly to ${U}_{b}$, ${U}_{c}$ rises sharply to ${U}_{d}$) reflects the ohmic characteristics of the battery. Hence, the ohmic resistance ${R}_{0}$ is calculated by the Equation (4).$${R}_{0}=\frac{({U}_{a}-{U}_{b})+({U}_{d}-{U}_{c})}{2I}$$
- Parameter identification of RC parallel circuits: To begin, it is necessary to establish the values for the time constants ${\tau}_{1}$, ${\tau}_{2}$, and ${\tau}_{3}$. And then based on the determined time constants, the parameters of ${R}_{1}$, ${R}_{2}$, ${R}_{3}$, ${C}_{1}$, ${C}_{2}$ and ${C}_{3}$ are determined in detail. Furthermore, crucial for identification is the response of a first-order RC circuit classically consisting of a resistor R, a capacitor C, and a constant current I, which is given by Equation (5) below:$$U\left(\mathrm{t}\right)=U\left({t}_{0}\right)+IR(1-{e}^{\frac{{t}_{0}-t}{\tau}})$$
- The previous equation states that $\tau =RC$, and ${t}_{0}$ is the time that represents the beginning of the process. The following is the specific process to identify the parameters ${R}_{1}$, ${R}_{2}$, ${R}_{3}$, ${C}_{1}$, ${C}_{2}$, and ${C}_{3}$. Firstly, identify the time constants during the process ${U}_{b}$-${U}_{c}$-${U}_{d}$-${U}_{e}$. Take note that there is no current flowing at all during the relaxation stage of the procedure. Then, the voltages ${U}_{1}$, ${U}_{2}$, and ${U}_{3}$ can then be computed using the following calculation based on Equation (5):$$\left\{\begin{array}{c}{U}_{1}\left(\mathrm{t}\right)={U}_{1}\left({t}_{c}\right){e}^{\frac{{t}_{c}-t}{{\tau}_{1}}}\hfill \\ {U}_{2}\left(\mathrm{t}\right)={U}_{2}\left({t}_{c}\right){e}^{\frac{{t}_{c}-t}{{\tau}_{2}}}\hfill \\ {U}_{3}\left(\mathrm{t}\right)={U}_{3}\left({t}_{c}\right){e}^{\frac{{t}_{c}-t}{{\tau}_{3}}}\hfill \end{array}\right.$$We can get Equation (7) from the output Equation (2).$$U\left(t\right)={U}_{oc}\left(soc\right)-{U}_{1}\left({t}_{c}\right){e}^{\frac{{t}_{c}-t}{{\tau}_{1}}}-{U}_{2}\left({t}_{c}\right){e}^{\frac{{t}_{c}-t}{{\tau}_{2}}}-{U}_{3}\left({t}_{c}\right){e}^{\frac{{t}_{c}-t}{{\tau}_{3}}}$$Additionally, rewrite Equation (7) as Equation (8).$$U\left(t\right)={\alpha}_{1}-{\alpha}_{2}{e}^{\frac{{t}_{c}-t}{{\beta}_{1}}}-{\alpha}_{3}{e}^{\frac{{t}_{c}-t}{{\beta}_{2}}}-{\alpha}_{4}{e}^{\frac{{t}_{c}-t}{{\beta}_{3}}}$$$$\left\{\begin{array}{c}{U}_{1}\left(t\right)=I{R}_{1}(1-{e}^{\frac{{t}_{c}-t}{{\tau}_{1}}})\hfill \\ {U}_{2}\left(t\right)=I{R}_{2}(1-{e}^{\frac{{t}_{c}-t}{{\tau}_{2}}})\hfill \\ {U}_{3}\left(t\right)=I{R}_{3}(1-{e}^{\frac{{t}_{c}-t}{{\tau}_{3}}})\end{array}\right.$$
- Parameter identification of OCV-SOC curve: The steady voltage measured at both ends of the positive and negative electrodes when the battery is left in one location for an extended period of time is the open-circuit voltage (OCV). However, the functional connection between OCV and SOC is not linear. The analytical complexity of SOC estimation is increased by this nonlinearity. In most publications, polynomial fitting is used to obtain OCV-SOC curves. The precision of the SOC estimation is directly effected by the fitting order chosen because it determines the quality of the fitting effect. To enhance the precision of SOC estimation, consider the functional relationship between OCV and SOC as a piecewise linear relationship [27], and refine the fitting curve. The test points are divided into seven segments, each of which is a function of SOC in relation to OCV. The following linear Equation (10) can be used to represent each component. Table 1 shows the values of $k1$ and ${k}_{2}$ that correspond to each curve. The 8th-order polynomial fitting curve was compared to the 7-segment linear fitting curve. As shown in Figure 3, the 7-segment linear fitting outperforms the 8th-order polynomial fitting at the start and finish of the discharge.$${U}_{oc}=f\left(SOC\right)={k}_{1}+{k}_{2}SOC$$

## 3. Sage-Husa Extended Kalman Filter Algorithm

#### 3.1. SHEKF Algorithm

#### 3.2. Improvement of the SHEKF Algorithm

#### 3.2.1. Simplifying Noise Covariance Matrix

#### 3.2.2. Setting Two Improved Forgetting Factors

## 4. Experiments and Analysis

#### 4.1. Model Verification

#### 4.2. Evaluation of Correct Initial SOC Value

#### 4.3. Evaluation of Incorrect Initial SOC Value

#### 4.4. Impact of the Forgetting Factor on SOC Estimation

#### 4.5. The Algorithm Performance under DST Working Condition

#### 4.6. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BMS | Battery management system |

ECM | Equivalent circuit model |

SOC | State of charge |

OCV | Open circuit voltage |

KF | Kalman filter |

EKF | Extended Kalman filter |

SHEKF | Sage-Husa Extended Kalman filter |

DST | Dynamic Stress Test |

MAE | Mean absolute error |

RMSE | Root mean square error |

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**Figure 2.**Discharge data and voltage amplification curve: (

**a**) Discharge pulse voltage and current, (

**b**) voltage amplification.

**Figure 5.**Voltage comparison and its error curve under the mode of pulse discharge: (

**a**) Voltage, (

**b**) Error.

**Figure 6.**Terminal voltage comparison and its error curve under DST operating conditions: (

**a**) Voltage, (

**b**) Error.

**Figure 8.**Algorithms comparison and its error curve with correct SOC initial value: (

**a**) SOC, (

**b**) Error.

**Figure 9.**Algorithms comparison and its error curve with incorrect SOC initial value: (

**a**) SOC with the initial value of SOC is 0.5, (

**b**) Error with the initial value of SOC is 0.5.

**Figure 10.**Algorithms comparison and its error curve with incorrect SOC initial value: (

**a**) SOC with the initial value of SOC is 0.2, (

**b**) Error with the initial value of SOC is 0.2.

**Figure 11.**MAE and RMSE of different algorithms with different SOC initial value: (

**a**) MAE, (

**b**) RMSE.

**Figure 13.**Complete DST curve and current amplification curve in DST: (

**a**) current, (

**b**) current amplification.

**Figure 14.**Comparison curves of algorithms and their error when running at 25 °C DST: (

**a**) SOC, (

**b**) Error.

**Figure 15.**Comparison curves of algorithms and their error when running at 0 °C DST: (

**a**) SOC, (

**b**) Error.

**Figure 16.**Comparison curves of algorithms and their error when running at 45 °C DST: (

**a**) SOC, (

**b**) Error.

SOC | 0.1–0.2 | 0.2–0.3 | 0.3–0.4 | 0.4–0.5 | 0.5–0.7 | 0.7–0.8 | 0.8–1 |
---|---|---|---|---|---|---|---|

${k}_{1}$ | 3.387 | 3.453 | 3.525 | 3.477 | 3.216 | 3.22791 | 3.00967 |

${k}_{2}$ | 0.84 | 0.51 | 0.27 | 0.39 | 0.916 | 0.8847 | 1.17 |

Pulse Discharge | DST | |||
---|---|---|---|---|

Experiment Number | Third Order/s | Second Order/s | Third Order/s | Second Order/s |

1 | 7.016 | 6.824 | 2.008 | 2.139 |

2 | 6.901 | 7.098 | 2.131 | 2.366 |

3 | 6.865 | 6.819 | 2.115 | 2.478 |

4 | 7.098 | 6.890 | 2.149 | 2.222 |

5 | 6.791 | 6.9474 | 2.308 | 2.278 |

Average value | 7.149 | 7.101 | 2.142 | 2.296 |

Algorithm | SHEKF1 | SHEKF2 | SHEKF3 | SHEKF4 | EKF |
---|---|---|---|---|---|

$MAE$(%) | 0.20 | 0.92 | 0.38 | 0.89 | 0.43 |

$RMSE$(%) | 0.16 | 0.73 | 0.35 | 0.71 | 0.40 |

Algorithm | Difference = 0.047 | Difference = 0.045 | Difference = 0.04 | Difference = 0.03 |
---|---|---|---|---|

$MAE$(%) | 0.20 | 0.20 | 0.52 | 0.70 |

$RMSE$(%) | 0.18 | 0.16 | 0.31 | 0.51 |

Algorithm | SHEKF1 | SHEKF2 | SHEKF3 | SHEKF4 | EKF |
---|---|---|---|---|---|

$MAE$(%) | 0.27 | 0.97 | 0.90 | 1.02 | 2.42 |

$RMSE$(%) | 0.27 | 0.86 | 0.78 | 0.91 | 0.88 |

Algorithm | SHEKF1 | SHEKF2 | SHEKF3 | SHEKF4 | EKF |
---|---|---|---|---|---|

$MAE$(%) | 1.81 | 3.25 | 2.37 | 2.99 | 2.36 |

$RMSE$(%) | 0.90 | 3.16 | 2.09 | 2.63 | 2.07 |

Algorithm | SHEKF1 | SHEKF2 | SHEKF3 | SHEKF4 | EKF |
---|---|---|---|---|---|

$MAE$(%) | 0.67 | 1.34 | 1.16 | 3.12 | 2.01 |

$RMSE$(%) | 0.67 | 0.90 | 1.08 | 2.23 | 1.88 |

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## Share and Cite

**MDPI and ACS Style**

Xiang, L.; Cai, L.; Dai, N.; Gao, L.; Lei, G.; Li, J.; Deng, M.
State of Charge Estimation of Lithium-Ion Batteries Based on an Improved Sage-Husa Extended Kalman Filter Algorithm. *World Electr. Veh. J.* **2022**, *13*, 220.
https://doi.org/10.3390/wevj13110220

**AMA Style**

Xiang L, Cai L, Dai N, Gao L, Lei G, Li J, Deng M.
State of Charge Estimation of Lithium-Ion Batteries Based on an Improved Sage-Husa Extended Kalman Filter Algorithm. *World Electric Vehicle Journal*. 2022; 13(11):220.
https://doi.org/10.3390/wevj13110220

**Chicago/Turabian Style**

Xiang, Lihong, Li Cai, Nina Dai, Le Gao, Guoping Lei, Junting Li, and Ming Deng.
2022. "State of Charge Estimation of Lithium-Ion Batteries Based on an Improved Sage-Husa Extended Kalman Filter Algorithm" *World Electric Vehicle Journal* 13, no. 11: 220.
https://doi.org/10.3390/wevj13110220