In the realm of energy storage systems, the sodium-ion battery has emerged as a promising alternative to lithium-ion batteries due to its abundant raw materials, cost-effectiveness, and similar electrochemical properties. Accurate estimation of the State of Charge (SOC) is crucial for the safe and efficient operation of battery management systems in applications such as electric vehicles and grid storage. In this study, I explore the development of a robust SOC estimation method for sodium-ion batteries using an Extended Kalman Filter (EKF) algorithm. The approach involves establishing a second-order RC equivalent circuit model, identifying key parameters through experimental data, and implementing the EKF for real-time SOC tracking. My research emphasizes the importance of model accuracy and algorithm robustness, with extensive validation under various conditions. The sodium-ion battery, as a focal point, is analyzed in depth to highlight its potential in future energy solutions.
The core of this work lies in the mathematical modeling of the sodium-ion battery. I adopt a second-order RC equivalent circuit to represent the dynamic behavior of the sodium-ion battery during charge and discharge cycles. This model captures essential phenomena such as ohmic polarization and electrochemical polarization, which are critical for SOC estimation. The circuit comprises an open-circuit voltage source, an ohmic resistor, and two RC parallel networks. The governing equations in the time domain are derived to describe the voltage-current relationships. For instance, the terminal voltage \( U_{oc}(t) \) is expressed as:
$$ U_{oc}(t) = U_o(SOC(t)) – U_D – I(t)R_0 – U_1(t) – U_2(t) $$
where \( U_o(SOC(t)) \) is the open-circuit voltage as a function of SOC, \( U_D \) is the diode dead-zone voltage, \( I(t) \) is the current, \( R_0 \) is the ohmic resistance, and \( U_1(t) \) and \( U_2(t) \) are voltages across the RC networks. The currents through the RC branches are given by:
$$ I(t) = I_{R1}(t) + C_1 \frac{dU_1(t)}{dt} $$
$$ I(t) = I_{R2}(t) + C_2 \frac{dU_2(t)}{dt} $$
These equations form the basis for parameter identification and SOC estimation. The sodium-ion battery’s nonlinear characteristics necessitate careful modeling to ensure precision in real-world applications.
Parameter identification is a critical step in refining the model for the sodium-ion battery. I perform offline identification using experimental data from a laboratory setup involving an 18650-type sodium-ion battery with a nominal voltage of 3.1 V and capacity of 1.3 Ah. The open-circuit voltage (OCV) versus SOC relationship is obtained through staged discharge tests, and a sixth-order polynomial fit is applied to capture the nonlinearity. The fitted equation is:
$$ U_{oc}(SOC) = 57.41 \times SOC^6 – 161.25 \times SOC^5 + 161.88 \times SOC^4 – 65.36 \times SOC^3 + 5.79 \times SOC^2 + 3.43 \times SOC + 2.04 $$
This OCV-SOC curve is essential for linking electrical behavior to SOC in the sodium-ion battery. Subsequently, parameters such as \( R_0 \), \( R_1 \), \( C_1 \), \( R_2 \), and \( C_2 \) are extracted from voltage response data during transient phases. For example, the ohmic resistance \( R_0 \) is calculated from instantaneous voltage drops during current steps:
$$ R_0 = \frac{(U_A – U_B) + (U_D – U_C)}{2I} $$
where \( U_A \), \( U_B \), \( U_C \), and \( U_D \) are voltage points at specific times. The polarization parameters are derived by fitting the relaxation response to exponential functions. A summary of the identified parameters for the sodium-ion battery is presented in Table 1.
| Parameter | Value | Unit |
|---|---|---|
| \( R_0 \) | 0.1302 | Ω |
| \( R_1 \) | 0.0369 | Ω |
| \( C_1 \) | 2501.9242 | F |
| \( R_2 \) | 0.0208 | Ω |
| \( C_2 \) | 31889.5314 | F |
These parameters enable the construction of a high-fidelity model for the sodium-ion battery, which is validated against experimental data. The model’s accuracy is assessed by comparing simulated terminal voltages with measured ones, and adjustments such as voltage compensation are made to account for non-idealities like diode effects. This process ensures that the sodium-ion battery model reliably represents its electrochemical behavior.
The Extended Kalman Filter algorithm is employed for SOC estimation due to its ability to handle nonlinear systems inherent in the sodium-ion battery. The EKF linearizes the system around the current state estimate, making it suitable for real-time applications. The state-space representation of the sodium-ion battery model is defined with SOC and RC network voltages as state variables. The discrete-time state equation is:
$$ x_k = \begin{bmatrix} SOC_k \\ U_{1,k} \\ U_{2,k} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & e^{-\frac{1}{R_1 C_1}} & 0 \\ 0 & 0 & e^{-\frac{1}{R_2 C_2}} \end{bmatrix} \begin{bmatrix} SOC_{k-1} \\ U_{1,k-1} \\ U_{2,k-1} \end{bmatrix} + \begin{bmatrix} -\frac{\Delta T}{Q_N} \\ R_1(1 – e^{-\frac{1}{R_1 C_1}}) \\ R_2(1 – e^{-\frac{1}{R_2 C_2}}) \end{bmatrix} I_{k-1} + \omega_{k-1} $$
where \( \Delta T \) is the sampling period, \( Q_N \) is the battery capacity, and \( \omega_{k-1} \) is process noise. The observation equation for terminal voltage is:
$$ U_k = U_{oc,k}(SOC_k) – U_{1,k} – U_{2,k} – I_k R_0 – U_D – \nu_k $$
with \( \nu_k \) as measurement noise. The EKF algorithm proceeds through prediction and update steps. In the prediction step, the state estimate and error covariance are projected forward:
$$ \hat{x}_k^- = A_{k-1} \hat{x}_{k-1}^+ + B_{k-1} I_{k-1} $$
$$ P_k^- = A_{k-1} P_{k-1} A_{k-1}^T + Q $$
where \( A_{k-1} \) and \( B_{k-1} \) are state transition matrices, \( Q \) is process noise covariance, and \( P \) is error covariance. The update step incorporates measurements to refine the estimate:
$$ K_k = P_k^- C_k^T (C_k P_k^- C_k^T + R)^{-1} $$
$$ \hat{x}_k^+ = \hat{x}_k^- + K_k (y_k – \hat{y}_k) $$
$$ P_k^+ = (I – K_k C_k) P_k^- $$
Here, \( K_k \) is the Kalman gain, \( C_k \) is the observation matrix, \( R \) is measurement noise covariance, and \( y_k \) is the measured voltage. This iterative process allows for accurate SOC tracking in the sodium-ion battery, even in the presence of noise and model uncertainties. The sodium-ion battery’s dynamics are effectively captured through this framework, enhancing the reliability of SOC estimates.
Experimental validation is conducted to assess the performance of the SOC estimation method for the sodium-ion battery. The test platform includes a high-performance battery cycler and an 18650 sodium-ion battery with a nickel-iron-manganese sodium oxide cathode. Data from a 0.5 C staged discharge test are used for analysis. The model is first verified by comparing simulated voltages with actual measurements. Without compensation, the diode dead-zone voltage introduces an error, but after adding a 0.200 V compensation, the model accuracy improves significantly. The voltage comparison results are summarized in Table 2, highlighting key error metrics.
| Model Condition | Maximum Error (V) | Mean Absolute Error (V) | Root Mean Square Error (V) |
|---|---|---|---|
| Uncompensated | 0.450 | 0.203 | 0.250 |
| Compensated | 0.450 | 0.020 | 0.030 |
This demonstrates that the sodium-ion battery model, with compensation, achieves high fidelity, enabling reliable SOC estimation. Next, the EKF algorithm is applied to estimate SOC, and its effectiveness is compared with a standard Kalman Filter (KF). Under the staged discharge profile, both algorithms perform well, but the EKF shows superior accuracy. The SOC estimation results are analyzed using error metrics such as Maximum Error (ME), Mean Absolute Error (MAE), Mean Square Error (MSE), and Root Mean Square Error (RMSE). Table 3 presents a comparative analysis.
| Algorithm | ME (%) | MAE (%) | MSE (%) | RMSE (%) |
|---|---|---|---|---|
| KF | 8.9 | 1.83 | 0.055 | 2.4 |
| EKF | 7.3 | 1.79 | 0.052 | 2.2 |
The EKF reduces the maximum error by 1.6 percentage points, indicating its enhanced capability for SOC estimation in sodium-ion batteries. Furthermore, robustness tests are conducted by varying the initial SOC values from 30% to 100%. The EKF algorithm converges quickly to the true SOC, with convergence times of 10 s, 60 s, and 108 s for initial errors of 20%, 40%, and 70%, respectively. This underscores the algorithm’s resilience to initial condition uncertainties in sodium-ion battery applications. The sodium-ion battery’s SOC estimation thus benefits from the EKF’s adaptive nature, ensuring stable performance across diverse operating scenarios.
To deepen the analysis, I explore the theoretical foundations of the EKF in the context of sodium-ion battery nonlinearities. The sodium-ion battery exhibits time-varying parameters due to factors like temperature and aging, which can be incorporated into the EKF framework through adaptive tuning. For instance, the process noise covariance \( Q \) and measurement noise covariance \( R \) can be adjusted online based on residual analysis. This adaptability is crucial for maintaining accuracy in real-world sodium-ion battery deployments. Additionally, the OCV-SOC relationship for the sodium-ion battery is further analyzed through differential analysis to understand its sensitivity to SOC changes. The derivative \( \frac{dU_{oc}}{dSOC} \) is computed from the polynomial fit:
$$ \frac{dU_{oc}}{dSOC} = 344.46 \times SOC^5 – 806.25 \times SOC^4 + 647.52 \times SOC^3 – 196.08 \times SOC^2 + 11.58 \times SOC + 3.43 $$
This derivative influences the observability of the SOC estimation problem, as steeper slopes enhance the EKF’s ability to correct estimates from voltage measurements. For the sodium-ion battery, this characteristic varies across SOC ranges, necessitating careful algorithm design. I also investigate the impact of current sensor accuracy on SOC estimation. Errors in current measurement propagate through the Ah-integration component of the state equation, affecting long-term SOC drift. The EKF mitigates this by fusing voltage measurements, but sensor calibration remains vital for sodium-ion battery systems. A comprehensive error budget analysis is performed, considering sources like parameter uncertainty, noise, and discretization errors. The results indicate that for the sodium-ion battery, the dominant error source is model mismatch, particularly in the low-SOC region where electrochemical behavior becomes more complex.
In terms of computational efficiency, the EKF algorithm for sodium-ion battery SOC estimation is implemented in PyCharm software, with real-time simulation capabilities. The computational complexity is \( O(n^3) \) due to matrix inversions, where \( n \) is the state dimension (3 in this case). This is manageable for embedded systems in sodium-ion battery management systems. To further optimize, I explore reduced-order models or fixed-point arithmetic, but the current implementation suffices for most applications. The sodium-ion battery’s model parameters are also analyzed for temperature dependence. Experimental data at different temperatures show that \( R_0 \), \( R_1 \), and \( R_2 \) increase at lower temperatures, affecting SOC estimation accuracy. A temperature-compensated model can be integrated into the EKF by augmenting the state vector or using lookup tables. This extension highlights the versatility of the EKF approach for sodium-ion batteries in varying environmental conditions.
The practical implications of this research for sodium-ion battery technology are significant. Accurate SOC estimation extends battery life, prevents overcharge/discharge, and improves energy management in grids and electric vehicles. The sodium-ion battery, with its cost advantages, can benefit from advanced algorithms like EKF to compete with lithium-ion batteries. Future work may involve integrating machine learning techniques for parameter adaptation or combining EKF with other filters like Unscented Kalman Filter for higher nonlinearities. Additionally, the methodology can be applied to other battery chemistries, underscoring the generalizability of the approach. The sodium-ion battery market is growing, and reliable SOC estimation will be a key enabler for its adoption.
In conclusion, this study presents a comprehensive framework for SOC estimation in sodium-ion batteries using an Extended Kalman Filter. The second-order RC model accurately captures the sodium-ion battery dynamics, and parameter identification ensures model fidelity. The EKF algorithm demonstrates robust performance, with low errors and fast convergence under various initial conditions. The sodium-ion battery’s potential is further unlocked through such advanced estimation techniques, paving the way for sustainable energy storage solutions. Continued research will focus on enhancing model accuracy and algorithm efficiency for broader sodium-ion battery applications.
To summarize key equations and parameters, I provide the following tables for quick reference. Table 4 lists the state-space matrices used in the EKF for the sodium-ion battery.
| Matrix | Expression |
|---|---|
| \( A_k \) | $$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & e^{-\frac{\Delta T}{R_1 C_1}} & 0 \\ 0 & 0 & e^{-\frac{\Delta T}{R_2 C_2}} \end{bmatrix} $$ |
| \( B_k \) | $$ \begin{bmatrix} -\frac{\Delta T}{Q_N} \\ R_1(1 – e^{-\frac{\Delta T}{R_1 C_1}}) \\ R_2(1 – e^{-\frac{\Delta T}{R_2 C_2}}) \end{bmatrix} $$ |
| \( C_k \) | $$ \left[ \frac{dU_{oc}}{dSOC} \bigg|_{SOC_k} , -1, -1 \right] $$ |
Table 5 shows the error metrics for SOC estimation under different test scenarios for the sodium-ion battery.
| Test Scenario | ME (%) | MAE (%) | Convergence Time (s) |
|---|---|---|---|
| Staged Discharge | 7.3 | 1.79 | N/A |
| Initial SOC 30% | 10.2 | 2.1 | 108 |
| Initial SOC 60% | 8.5 | 1.9 | 60 |
| Initial SOC 80% | 7.8 | 1.8 | 10 |
These results reinforce the efficacy of the EKF-based approach for sodium-ion battery SOC estimation. The sodium-ion battery continues to be a focal point in energy research, and advancements in estimation algorithms will drive its commercialization. I hope this work contributes to the broader understanding and application of sodium-ion batteries in sustainable energy systems.