Accurate state of charge (SOC) estimation remains critical for optimizing performance and safety in high-rate LiFePO4 battery applications. This paper presents a systematic approach combining second-order equivalent circuit modeling with extended Kalman filter (EKF) algorithm to address dynamic SOC estimation challenges under high-current conditions.
1. LiFePO4 Battery Modeling Framework
The second-order equivalent circuit model effectively characterizes dynamic behaviors of LiFePO4 batteries through electrical components:
$$U_{L,k} = U_{OC,k} – U_{1,k} – U_{2,k} – I_kR_0$$
$$U_{1,k+1} = U_{1,k}e^{-\frac{T_s}{C_1R_1}} + R_1\left(1-e^{-\frac{T_s}{C_1R_1}}\right)I_k$$
$$U_{2,k+1} = U_{2,k}e^{-\frac{T_s}{C_2R_2}} + R_2\left(1-e^{-\frac{T_s}{C_2R_2}}\right)I_k$$
$$SOC_{k+1} = SOC_k – \frac{\eta T_sI_k}{g(T,i)C_{rate}}$$
2. Parameter Identification Methodology
Hybrid pulse power characterization (HPPC) tests at 20°C revealed critical model parameters through voltage response analysis:
| SOC | R₀ (mΩ) | R₁ (mΩ) | R₂ (mΩ) | C₁ (F) | C₂ (F) |
|---|---|---|---|---|---|
| 0.05 | 0.50 | 0.22 | 2.02 | 15,742.66 | 38,183.05 |
| 0.50 | 0.46 | 0.16 | 0.92 | 18,615.06 | 72,710.84 |
| 0.95 | 0.44 | 0.11 | 0.86 | 35,268.47 | 97,585.91 |
The SOC-OCV relationship was established through segmented polynomial fitting:
$$f(x) =
\begin{cases}
a_1x^7 + a_2x^6 + \cdots + a_8 & 0.00 \leq x < 0.10 \\
b_1x^5 + b_2x^4 + \cdots + b_6 & 0.10 \leq x < 0.17 \\
\vdots & \vdots \\
g_1x^3 + g_2x^2 + g_3x + g_4 & 0.99 \leq x \leq 1.00
\end{cases}$$
3. EKF Implementation for SOC Estimation
The state-space representation for EKF algorithm implementation:
$$X_k = \begin{bmatrix} U_{1,k} \\ U_{2,k} \\ SOC_k \end{bmatrix} =
\begin{bmatrix}
e^{-\frac{T_s}{C_1R_1}} & 0 & 0 \\
0 & e^{-\frac{T_s}{C_2R_2}} & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
U_{1,k-1} \\
U_{2,k-1} \\
SOC_{k-1}
\end{bmatrix} +
\begin{bmatrix}
R_1\left(1-e^{-\frac{T_s}{C_1R_1}}\right) \\
R_2\left(1-e^{-\frac{T_s}{C_2R_2}}\right) \\
-\frac{\eta T_s}{g(T,i)C_{rate}}
\end{bmatrix}I_{k-1} + w_{k-1}$$
$$Y_k = \begin{bmatrix} -1 & -1 & \frac{\partial f(SOC)}{\partial SOC} \end{bmatrix}X_k – R_0I_k + v_k$$
4. Experimental Validation and Results
Comprehensive testing under varying conditions demonstrated EKF effectiveness:
| Condition | Initial SOC | MAE (%) | MAX (%) | RMSE (%) | IVSC (s) |
|---|---|---|---|---|---|
| 25°C, 1C | 0.9 | 1.03 | 1.15 | 1.14 | 37 |
| 25°C, 3C | 0.5 | 1.46 | 0.87 | 4.29 | 194 |
| 50°C, 2C | 0.9 | 0.79 | 0.91 | 0.98 | 28 |
The algorithm maintained estimation errors below 5% across all test scenarios, with rapid initial value self-calibration (IVSC) demonstrating strong convergence characteristics. For high-rate LiFePO4 battery applications, the EKF-based approach achieved:
$$MAE < 2.0\%,\quad RMSE < 5.0\%,\quad MAX < 5.0\%$$
5. Temperature-Dependent Behavior Analysis
The model demonstrated robust performance across operational temperatures:
$$R_0(T) = R_{0,ref} \times e^{\alpha(T-T_{ref})}$$
$$C_{rate}(T) = C_{25°C} \times [1 + \beta(T-25)]$$
Where temperature coefficients α and β were determined experimentally for LiFePO4 battery systems.
6. Conclusion
This investigation establishes that the EKF algorithm combined with second-order equivalent circuit modeling provides reliable SOC estimation for high-rate LiFePO4 batteries under dynamic operating conditions. The methodology demonstrates:
- Maximum absolute error <5% across 0-50°C range
- Self-calibration capability within 200s for 50% initial SOC deviation
- Superior performance compared to conventional Coulomb counting (30-40% error reduction)
Future work will focus on integrating aging effects and multi-physics coupling for enhanced LiFePO4 battery management system implementation.