1. Introduction
The rapid growth of renewable energy systems, such as wind and solar power, has underscored the critical role of energy storage battery in stabilizing grid operations. Among these, lithium-ion batteries dominate due to their high energy density, long cycle life, and environmental friendliness. However, ensuring the safe and efficient operation of energy storage battery hinges on accurately estimating their State of Charge (SOC), a parameter that reflects the remaining capacity of the battery. Traditional SOC estimation methods, such as the Ampere-hour (Ah) integration and Extended Kalman Filter (EKF), suffer from cumulative errors, sensitivity to initial conditions, and poor adaptability to nonlinear systems. To address these challenges, this work proposes a hybrid approach combining Forgetting Factor Recursive Least Squares (FFRLS) for parameter identification and Unscented Kalman Filter (UKF) for SOC estimation. The proposed method achieves high accuracy (99.2%) and robustness under varying initial SOC conditions, making it ideal for real-world energy storage battery management.
2. Equivalent Circuit Model and Parameter Identification
2.1 OCV-SOC Relationship
The Open-Circuit Voltage (OCV) of an energy storage battery is intrinsically linked to its SOC. Through controlled pulse discharge experiments, the OCV-SOC relationship is modeled using a 6th-order polynomial:VOCV=A+∑i=06AiSOCiVOCV=A+i=0∑6AiSOCi
where SOCSOC is the SOC, and A,A0,…,A6A,A0,…,A6 are coefficients obtained via curve fitting. Experimental data (Table 1) validate this relationship.
| SOC | 1.00 | 0.95 | 0.90 | … | 0.00 |
|---|---|---|---|---|---|
| OCV (V) | 3.65 | 3.58 | 3.52 | … | 2.80 |
2.2 Thevenin Equivalent Circuit Model
A first-order Thevenin model (Figure 1) is adopted to balance accuracy and computational efficiency. The model equations are:U˙1=−U1R1C1+IC1U˙1=−R1C1U1+C1IU=VOCV−IR0−U1U=VOCV−IR0−U1
Here, R0R0, R1R1, and C1C1 represent ohmic resistance, polarization resistance, and polarization capacitance, respectively.
2.3 Online Parameter Identification via FFRLS
To dynamically track parameter variations in energy storage battery, FFRLS with a forgetting factor (λ=0.95λ=0.95) is employed. The discrete-time system is transformed into a linear regression form:yk=θkϕkyk=θkϕk
where yk=VOCV,k−Ukyk=VOCV,k−Uk, θk=[c1,c2,c3]θk=[c1,c2,c3], and ϕk=[−yk−1,Ik,Ik−1]Tϕk=[−yk−1,Ik,Ik−1]T. The parameters R0R0, R1R1, and C1C1 are derived as:R0=c2−c31−c1,R1=c2+c31+c1−R0,C1=T(1−c1)2R1(1+c1)R0=1−c1c2−c3,R1=1+c1c2+c3−R0,C1=2R1(1+c1)T(1−c1)
Experimental results confirm the accuracy of FFRLS, with voltage prediction errors below 0.1 V (Table 2).
| Parameter | Initial Value | Identified Value | Error (%) |
|---|---|---|---|
| R0R0 | 0.05 Ω | 0.048 Ω | 4.0 |
| R1R1 | 0.15 Ω | 0.152 Ω | 1.3 |
| C1C1 | 2000 F | 1980 F | 1.0 |
3. FFRLS-UKF Algorithm for SOC Estimation
3.1 State-Space Model
The system state vector xk=[SOC,k,U1,k]Txk=[SOC,k,U1,k]T and measurement UkUk are governed by:xk+1=[100e−T/(R1C1)]xk+[−T/Q0R1(1−e−T/(R1C1))]Ik+wkxk+1=[100e−T/(R1C1)]xk+[−T/Q0R1(1−e−T/(R1C1))]Ik+wkUk=VOCV(SOC,k)−U1,k−IkR0+vkUk=VOCV(SOC,k)−U1,k−IkR0+vk
where Q0Q0 is the battery capacity, and wkwk, vkvk are process and measurement noise.
3.2 UKF Implementation
The UKF employs sigma points to capture nonlinear dynamics without linearization. Key steps include:
- Sigma Point Generation:
Xk=[x^k,x^k±(n+κ)Pk]Xk=[x^k,x^k±(n+κ)Pk]
- Measurement Update:
Kk=PxyPyy−1,x^k=x^k−+Kk(Uk−U^k−)Kk=PxyPyy−1,x^k=x^k−+Kk(Uk−U^k−)
4. Experimental Validation
4.1 Accuracy Under Ideal Conditions
The FFRLS-UKF method is compared against EKF and Ah integration using an 18,650 energy storage battery. With zero initial SOC error, FFRLS-UKF achieves superior performance (Table 3).
| Algorithm | ME (%) | MSE (%) | Accuracy (%) |
|---|---|---|---|
| Ah | 0.9337 | 0.1082 | 97.93 |
| EKF | 0.7051 | 0.0602 | 98.61 |
| FFRLS-UKF | 0.3997 | 0.0201 | 99.20 |
4.2 Robustness to Initial SOC Errors
Tests with erroneous initial SOC values (0.75 and 0.50) demonstrate FFRLS-UKF’s rapid convergence (Figure 2). For SOC,initial=0.50SOC,initial=0.50, FFRLS-UKF converges in 133 steps, while EKF requires 722 steps.
| Initial SOC | Convergence Steps (FFRLS-UKF) | Convergence Steps (EKF) |
|---|---|---|
| 1.00 | 0 | 0 |
| 0.75 | 104 | 690 |
| 0.50 | 133 | 722 |
5. Conclusion
This work presents a novel FFRLS-UKF framework for real-time SOC estimation in energy storage battery. By integrating FFRLS for adaptive parameter identification and UKF for nonlinear state estimation, the method achieves 99.2% accuracy under ideal conditions and demonstrates robustness against initial SOC uncertainties. The algorithm’s computational efficiency and adaptability make it a promising solution for enhancing the reliability of energy storage battery systems in renewable energy applications. Future work will explore its integration with cloud-based battery management systems for large-scale deployments.