# Online Parameters Identification and State of Charge Estimation for Lithium-Ion Battery Using Adaptive Cubature Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Battery Model and Parameter Identification

#### 2.1. Battery Model

_{p1}, R

_{p2}) and the polarization capacitance (C

_{p1}, C

_{p2}) are used to simulate electrochemical polarization and concentration polarization, respectively.

_{S}is the sample time.

#### 2.2. Model Parameter Identification

#### 2.2.1. Identification of Open Circuit Voltage

- Charge the battery fully. At this time SOC = 1 and the measured battery voltage is recorded as OCV.
- Discharge at 1 C until SOC = 0.95 and rest for 30 min. Measure the battery voltage and record it as the open circuit voltage of SOC = 0.95.
- Step (2) is repeated until SOC = 0.05 and the battery voltage corresponding to each SOC is recorded.

#### 2.2.2. Parameter Identification with Vector Forgetting Factor Recursive Least Squares Algorithm

_{s}), is:

_{S}= 4; ${\Delta \mathrm{U}}_{\mathrm{tv}}={\mathrm{U}}_{\mathrm{tv}}\left(\mathrm{k}\right)-{\mathrm{U}}_{\mathrm{tv}}\left(\mathrm{k}-\mathrm{Ts}\right)$; the other expressions of ${\Delta \mathrm{U}}_{0\mathrm{cv}}\left(\mathrm{k}\right)$, $\Delta \mathrm{I}\left(\mathrm{k}\right)$ in Equation (11) are obtained in a similar manner.

## 3. Adaptive Cubature Kalman Filter Algorithm

_{K}and R

_{K}

_{K}and R

_{K}are shown in Equations (34) and (35).

_{k}is the approximate estimate of the voltage residual covariance, which is expressed as:

## 4. Results and Discussion

#### 4.1. Experimental Setup

#### 4.2. Comparison of Two Battery Models

#### 4.3. Comparison of Two Parameter Identification Methods

#### 4.4. Comparison of SOC Estimation with Different Algorithms

## 5. Conclusions

- (1)
- By comparing the estimated terminal voltage of the Thevenin model and DP model, we can conclude that on the one hand, the Thevenin model and DP model can be adopted to the dynamic and complex condition. On the other hand, the DP model has higher accuracy and better dynamic performance compared with the Thevenin model.
- (2)
- Online parameter identification based on VRLS has an improvement for voltage estimation over the offline parameter identification of the HPPC test, and it is shown that the proposed VRLS has a more accurate parameter identification ability.
- (3)
- Experiments based on the two typical dynamic operating cycles are used to evaluate the superiority of the proposed algorithm compared with EKF and CKF in terms of accuracy and stability. The maximum errors of ACKF are 1.85% in UDDS and 1.64% in DST, the MAEs are 0.59% in UDDS and 0.39% in DST, and the RMSEs are 0.72% in UDDS and 0.49% in DST. The estimation accuracy is relatively high and the maximum error, MAE, RMSE of ACKF are all smaller than those of EKF and CKF. The results show that the ACKF has a satisfactory performance in SOC estimation, and has better accuracy and stability than EKF and CKF. To conclude, the VRLS-ACKF is capable of obtaining accurate SOC estimation and terminal voltage prediction with satisfying stability, which is suitable to implement in the real application. For future work, the proposed algorithm will be applied in BMS to verify its practicality.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACKF | Adaptive cubature Kalman filter |

AUKF | Adaptive unscented Kalman filter |

BMS | Battery Management System |

CKF | Cubature Kalman filter |

DP | Dual polarization |

DST | Dynamic Stress Test |

ECMs | Equivalent circuit models |

EKF | Extended Kalman filter |

EVS | Electric vehicles |

HPPC | Hybrid pulse power characterization |

KF | Kalman filter |

MAE | Mean Absolute Error |

OCV | Open Circuit Voltage |

PNGV | Partnership for a New Generation of Vehicle |

RLS | Recursive least square |

RMSE | Root Mean Square Error |

SOC | State of charge |

UKF | Unscented Kalman filter |

VRLS | Vector forgetting factor recursive least squares |

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**Figure 4.**The estimated terminal voltages and corresponding errors based on different battery models under UDDS condition: (

**a**) voltage comparison; (

**b**) error of voltages.

**Figure 5.**The estimated terminal voltages and corresponding errors based on different battery models under DST condition: (

**a**) voltage comparison; (

**b**) error of voltages.

**Figure 6.**The estimated terminal voltage and corresponding errors based on different battery parameter identification under UDDS condition: (

**a**) voltage comparison; (

**b**) error of voltage.

**Figure 7.**The estimated terminal voltage and corresponding errors based on different battery parameter identification under DST condition: (

**a**) voltage comparison; (

**b**) error of voltage.

i | λ_{0i} | λ_{1i} | τ_{i} |
---|---|---|---|

1 | 0.998 | 0.98 | 500 |

2 | 0.995 | 0.99 | 500 |

3 | 0.99 | 0.97 | 500 |

4 | 0.998 | 0.98 | 500 |

5 | 0.995 | 0.99 | 500 |

Step 1. Initialize the parameters:${\omega}_{0}=E\left[{\omega}_{0}\right],{P}_{0}=E\left[\left({\omega}_{0}-{\hat{\omega}}_{0}\right){\left({\omega}_{0}-{\hat{\omega}}_{0}\right)}^{T}\right]$ |

Step 2. Calculate the state variables and the predicted value of covariance matrix:${\hat{\omega}}^{-}\left(k\right)={\hat{\omega}}^{+}\left(k-1\right){P}^{-}\left(k\right)={P}^{+}\left(k-1\right)$ |

Step 3. Update the gain: ${L}_{\omega}\left(k\right)={P}^{-}\left(k\right)\phi \left(k\right){\left[1+{\phi}^{T}\left(k\right){P}^{-}\left(k\right)\phi \left(k\right)\right]}^{-1}$ |

Step 4. Update the state variables and covariance matrix: $\begin{array}{l}{\hat{\omega}}^{+}\left(k\right)={\hat{\omega}}^{-}\left(k\right)+{L}_{\omega}\left(k\right)\left[y\left(k\right)-{\hat{\omega}}^{-T}\left(k\right)\phi \left(k\right)\right]\\ {P}^{+}\left(k\right)={\Lambda}^{-1}\left[E-{L}_{\omega}\left(k\right){\phi}^{T}\left(k\right)\right]{P}^{-}\left(k\right){\Lambda}^{-1}\end{array}$ where $\Lambda =diag\left(\left[\sqrt{{\lambda}_{w1}}\sqrt{{\lambda}_{w2}}\sqrt{{\lambda}_{w3}}\sqrt{{\lambda}_{w4}}\sqrt{{\lambda}_{w5}}\right]\right)$, and E is a unit vector of the fifth order. |

Items | Parameters |
---|---|

Cathode materials | LiNi_{1-x-y}Co_{x}Mn_{y}O_{2} |

Nominal capacity (Ah) | 35 |

Rated voltage (V) | 3.7 |

Maximal continuous discharge current (C) | 3 |

Maximal pulse discharge current (C) | 5 (30 s) |

Upper/lower cut-off voltage (V) | 4.2/2.5 |

Condition | Algorithm | Maximum Error/V | MAE/V | RMSE/V |
---|---|---|---|---|

UDDS | DP-VRLS | 0.0754 | 0.0087 | 0.0126 |

Thevenin-VRLS | 0.0525 | 0.0106 | 0.0149 | |

DP-offline | 0.4014 | 0.0486 | 0.0907 | |

DST | DP-VRLS | 0.0389 | 0.0123 | 0.0147 |

Thevenin-VRLS | 0.0606 | 0.0185 | 0.0227 | |

DP-offline | 0.0612 | 0.0152 | 0.0192 |

Condition | Algorithm | Maximum Error | MAE | RMSE |
---|---|---|---|---|

UDDS | ACKF | 0.0185 | 0.0059 | 0.0072 |

CKF | 0.0268 | 0.0067 | 0.0084 | |

EKF | 0.0399 | 0.0156 | 0.0188 | |

DST | ACKF | 0.0164 | 0.0039 | 0.0049 |

CKF | 0.0283 | 0.0056 | 0.0077 | |

EKF | 0.0181 | 0.0085 | 0.0096 |

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## Share and Cite

**MDPI and ACS Style**

Li, W.; Luo, M.; Tan, Y.; Cui, X. Online Parameters Identification and State of Charge Estimation for Lithium-Ion Battery Using Adaptive Cubature Kalman Filter. *World Electr. Veh. J.* **2021**, *12*, 123.
https://doi.org/10.3390/wevj12030123

**AMA Style**

Li W, Luo M, Tan Y, Cui X. Online Parameters Identification and State of Charge Estimation for Lithium-Ion Battery Using Adaptive Cubature Kalman Filter. *World Electric Vehicle Journal*. 2021; 12(3):123.
https://doi.org/10.3390/wevj12030123

**Chicago/Turabian Style**

Li, Wei, Maji Luo, Yaqian Tan, and Xiangyu Cui. 2021. "Online Parameters Identification and State of Charge Estimation for Lithium-Ion Battery Using Adaptive Cubature Kalman Filter" *World Electric Vehicle Journal* 12, no. 3: 123.
https://doi.org/10.3390/wevj12030123