Accurate state-of-charge (SOC) estimation is critical for ensuring the safety and reliability of energy storage battery systems. This work proposes a hybrid model-driven framework combining a forgetting factor recursive least squares (FFR-LS) algorithm with unscented Kalman filtering (UKF) to achieve high-precision SOC estimation under dynamic operating conditions.
1. Thevenin Equivalent Circuit Model
The Thevenin model captures the dynamic behavior of energy storage batteries using the following equations:
$$
\begin{cases}
U_{1} = I R_1 \left(1 – e^{-t/(R_1 C_1)}\right) \\
U_t = V_{OCV}(SOC) – I R_0 – U_1
\end{cases}
$$
where \(R_0\), \(R_1\), and \(C_1\) represent ohmic resistance, polarization resistance, and capacitance, respectively. The OCV-SOC relationship is modeled as:
$$
V_{OCV} = \sum_{i=0}^6 A_i SOC^i
$$
2. Online Parameter Identification via FFR-LS
The recursive identification process for energy storage battery parameters follows:
$$
\begin{cases}
y_k = V_{OCV,k} – U_{t,k} \\
\phi_k = [-y_{k-1} \quad I_k \quad I_{k-1}]^T \\
\theta_k = [c_1 \quad c_2 \quad c_3]^T
\end{cases}
$$
Parameter update equations with forgetting factor \(\lambda\):
$$
\begin{cases}
K_k = \frac{P_{k-1}\phi_k}{\lambda + \phi_k^T P_{k-1}\phi_k} \\
\theta_k = \theta_{k-1} + K_k(y_k – \phi_k^T\theta_{k-1}) \\
P_k = \frac{1}{\lambda}(I – K_k\phi_k^T)P_{k-1}
\end{cases}
$$
| Parameter | Identification Range | Convergence Time (s) |
|---|---|---|
| \(R_0\) | 0.8–1.2 mΩ | 42 |
| \(R_1\) | 1.5–3.5 mΩ | 58 |
| \(C_1\) | 4.2–8.7 kF | 63 |
3. UKF-Based SOC Estimation
The state-space model for energy storage battery SOC estimation:
$$
\begin{cases}
x_{k+1} = \begin{bmatrix} 1 & 0 \\ 0 & e^{-T/(R_1 C_1)} \end{bmatrix}x_k + \begin{bmatrix} -T/Q_n \\ R_1(1 – e^{-T/(R_1 C_1)}) \end{bmatrix}I_k + w_k \\
y_k = V_{OCV}(SOC_k) – R_0 I_k – U_{1,k} + v_k
\end{cases}
$$
Sigma point generation and update:
$$
\begin{cases}
\chi_{k|k}^{(i)} = \hat{x}_{k|k} \pm \left(\sqrt{(n+\kappa)P_{k|k}}\right)_i \\
W^{(m)} = \begin{cases} \frac{\kappa}{n+\kappa}, & i=0 \\ \frac{1}{2(n+\kappa)}, & \text{otherwise} \end{cases}
\end{cases}
$$
4. Experimental Validation
Comparative results under 1C discharge conditions:
| Method | MAE (%) | RMSE (%) | Convergence Steps |
|---|---|---|---|
| Ah Integration | 2.15 | 3.08 | N/A |
| EKF | 1.02 | 1.47 | 650 |
| FFR-LS-UKF | 0.41 | 0.68 | 120 |
Robustness analysis with initial SOC errors:
$$
\begin{cases}
\Delta SOC_0 = 25\% \Rightarrow |\epsilon| < 1\% \text{ in } 150 \text{ steps} \\
\Delta SOC_0 = 50\% \Rightarrow |\epsilon| < 1\% \text{ in } 300 \text{ steps}
\end{cases}
$$
5. Thermal-Electrochemical Coupling Effects
The temperature-dependent characteristics of energy storage batteries are modeled as:
$$
\begin{cases}
R_0(T) = R_{0,25^{\circ}\text{C}}e^{\alpha(T-25)} \\
V_{OCV}(SOC,T) = V_{OCV,25^{\circ}\text{C}}(SOC) + \beta(T-25)
\end{cases}
$$
where \(\alpha = 0.012\,^\circ \text{C}^{-1}\) and \(\beta = 0.3\,\text{mV}/^\circ \text{C}\) for typical lithium-ion energy storage batteries.
6. Aging Compensation Mechanism
Capacity fade model for energy storage batteries:
$$
Q_{loss} = A\cdot e^{-E_a/(RT)}t^{0.5}(SOC_{avg} – 0.5)^2
$$
Where \(A = 3.2\times10^4\) and activation energy \(E_a = 35\,\text{kJ/mol}\).
| Cycle Count | Capacity Retention (%) | \(R_0\) Increase (%) |
|---|---|---|
| 500 | 95.2 | 8.7 |
| 1000 | 88.6 | 18.3 |
| 1500 | 79.1 | 31.5 |
7. Multi-Time Scale Estimation Framework
A hierarchical estimation architecture for energy storage battery systems:
$$
\begin{cases}
\text{Micro-scale (ms):} & \text{Voltage/current sampling} \\
\text{Meso-scale (s):} & \text{Parameter identification} \\
\text{Macro-scale (min):} & \text{Capacity recalibration}
\end{cases}
$$
The proposed FFR-LS-UKF method demonstrates superior performance in energy storage battery management applications, achieving 99.2% estimation accuracy with proper initialization. Future work will focus on extending this framework to battery pack-level SOC estimation and remaining useful life prediction.