State of Charge (SOC) estimation is a critical parameter for optimizing the performance, safety, and longevity of LiFePO4 battery, especially under high-rate operating conditions. This paper proposes a robust SOC estimation framework combining a second-order equivalent circuit model (ECM) with the Extended Kalman Filter (EKF) algorithm. The methodology addresses challenges such as internal temperature rise, chemical reaction rate variations, and nonlinear dynamics inherent to high-rate LiFePO4 battery.
1. Challenges in SOC Estimation for LiFePO4 Battery
LiFePO4 battery exhibit unique characteristics under high current loads, including rapid polarization effects and voltage hysteresis. Traditional SOC estimation methods, such as open-circuit voltage (OCV) and ampere-hour integration (AHI), face limitations:
- OCV Method: Requires prolonged rest periods to stabilize voltage, making it impractical for dynamic applications.
- AHI Method: Accumulates errors due to sensor inaccuracies and uncertain initial SOC values.
- Neural Networks: Demand extensive datasets and computational resources, limiting real-time applicability.
The proposed approach leverages the simplicity and physical interpretability of ECMs while incorporating the EKF algorithm to handle nonlinearities and noise.
2. Second-Order Equivalent Circuit Model for LiFePO4 Battery
The second-order ECM (Figure 1) captures both short-term and long-term dynamic behaviors of LiFePO4 battery. The model parameters include:
- Ohmic resistance (R0R0)
- Electrochemical polarization resistance (R1R1) and capacitance (C1C1)
- Concentration polarization resistance (R2R2) and capacitance (C2C2)
The state-space equations are derived as:{U1,k+1=U1,ke−TsC1R1+R1(1−e−TsC1R1)IkU2,k+1=U2,ke−TsC2R2+R2(1−e−TsC2R2)IkSOCk+1=SOCk−ηTsIkg(T,I)CmaxUL,k=UOC(SOCk)−U1,k−U2,k−IkR0⎩⎨⎧U1,k+1=U1,ke−C1R1Ts+R1(1−e−C1R1Ts)IkU2,k+1=U2,ke−C2R2Ts+R2(1−e−C2R2Ts)IkSOCk+1=SOCk−g(T,I)CmaxηTsIkUL,k=UOC(SOCk)−U1,k−U2,k−IkR0
where TsTs is the sampling time, IkIk is the current, and g(T,I)g(T,I) is a temperature- and current-dependent capacity coefficient.
3. Parameter Identification and SOC-OCV Relationship
Key parameters (R0,R1,R2,C1,C2R0,R1,R2,C1,C2) were identified using Hybrid Pulse Power Characterization (HPPC) tests. The SOC-OCV relationship was modeled through segmented polynomial fitting:UOC(SOC)={a1SOC7+a2SOC6+⋯+a80.00≤SOC<0.10b1SOC5+b2SOC4+⋯+b60.10≤SOC<0.17⋮⋮g1SOC3+g2SOC2+⋯+g40.99≤SOC≤1.00UOC(SOC)=⎩⎨⎧a1SOC7+a2SOC6+⋯+a8b1SOC5+b2SOC4+⋯+b6⋮g1SOC3+g2SOC2+⋯+g40.00≤SOC<0.100.10≤SOC<0.17⋮0.99≤SOC≤1.00
Coefficients (ai,bi,…,giai,bi,…,gi) were determined via least-squares optimization, achieving R2>0.999R2>0.999.
4. Extended Kalman Filter for SOC Estimation
The EKF algorithm linearizes the nonlinear ECM using Taylor series expansion. The state vector xk=[U1,k,U2,k,SOCk]Txk=[U1,k,U2,k,SOCk]T evolves as:xk=Ak−1xk−1+Bk−1Ik−1+wk−1xk=Ak−1xk−1+Bk−1Ik−1+wk−1yk=Ck−1xk−1+Dk−1Ik−1+vk−1yk=Ck−1xk−1+Dk−1Ik−1+vk−1
where AA, BB, CC, and DD are Jacobian matrices, and ww, vv represent process and measurement noise. The EKF recursively updates SOC estimates by minimizing prediction errors.
5. Experimental Validation
The EKF algorithm was tested under varying temperatures (0°C, 25°C, 50°C) and discharge rates (1C, 2C, 3C). Key metrics included:
- MAE (Mean Absolute Error)
- MAX (Maximum Error)
- RMSE (Root Mean Square Error)
- IVSC (Initial Value Self-Calibration Time)
Table 1: Performance of EKF Under Different Initial SOC Values
| Initial SOC | Discharge Rate | MAE (%) | MAX (%) | RMSE (%) | IVSC (s) |
|---|---|---|---|---|---|
| 0.9 | 1C | 1.03 | 1.15 | 1.14 | 37 |
| 0.5 | 1C | 1.43 | 0.99 | 4.05 | 250 |
| 0.9 | 3C | 0.81 | 0.83 | 0.96 | 29 |
| 0.5 | 3C | 1.46 | 0.87 | 4.29 | 194 |
Results demonstrate that the EKF algorithm maintains SOC estimation errors below 5%, even with initial SOC deviations. Higher discharge rates slightly degrade accuracy but remain within acceptable limits.
6. Discussion and Future Directions
The EKF-based approach effectively addresses the nonlinear dynamics of LiFePO4 battery, offering rapid convergence and robustness against measurement noise. Future work will focus on:
- Aging Effects: Incorporating capacity fade and resistance growth due to cycle aging.
- Low-Temperature Performance: Enhancing SOC estimation under sub-zero conditions.
- Multi-Model Fusion: Integrating thermal and electrochemical models for improved accuracy.
7. Conclusion
This study establishes a reliable framework for SOC estimation in high-rate LiFePO4 battery. By combining a second-order ECM with the EKF algorithm, the method achieves MAE < 2%, MAX < 5%, and RMSE < 5% across diverse operating conditions. The results underscore the practicality of EKF for real-time battery management systems, ensuring safe and efficient utilization of LiFePO4 battery in electric vehicles and renewable energy storage applications.