In recent years, the rapid advancement of lithium-ion battery technology has driven the pursuit of high energy density, high power density, and long endurance in batteries. As large-scale cell energy storage systems become widely applied in power transmission, generation, distribution, and consumption services, their safety and reliability have garnered increasing attention. The battery management system (BMS) is an essential component of any cell energy storage system, tasked with monitoring battery operating states, fault diagnosis, early warning, safety protection, energy management, and cell balancing. For cell energy storage systems, the BMS typically adopts a master-slave architecture, where slaves monitor modules of serially connected cells and compute the state of charge (SOC) for individual cells, while the master aggregates this information to determine the SOC of the entire battery stack or cluster. The performance of the BMS critically influences the safety and reliability of the cell energy storage system. Therefore, prior to deployment, rigorous testing of the BMS is paramount, yet standardized testing protocols and benchmarks for SOC estimation methods in cell energy storage system BMS remain lacking. This work addresses this gap by developing a testing platform for cell energy storage system BMS and proposing an accurate SOC estimation method based on the Dual Extended Kalman Filter (DEKF). The approach leverages external characteristic data to establish a Thevenin equivalent circuit model, employs a rate discharge curve extrapolation method to obtain the open-circuit voltage curve, and integrates the DEKF algorithm for simultaneous online parameter identification and SOC estimation. Comparative analysis with the traditional Extended Kalman Filter (EKF) method demonstrates the superiority of the DEKF approach in terms of convergence speed and estimation accuracy. Validation under typical Federal Urban Driving Schedule (FUDS) and Dynamic Stress Test (DST) conditions shows that both methods achieve SOC estimation errors below 1%, with terminal voltage errors within ±10 mV and ±20 mV, respectively, and mean absolute errors of 2.7 mV and 3.8 mV.
The core of an effective BMS testing platform lies in its ability to accurately estimate the SOC of the cell energy storage system. SOC, defined as the ratio of remaining capacity to nominal capacity, is a crucial parameter for preventing overcharge and over-discharge, optimizing energy management, and ensuring system longevity. Traditional SOC estimation methods, such as open-circuit voltage (OCV) method and ampere-hour (Ah) integration, have limitations: the OCV method requires prolonged battery resting periods, while Ah integration is prone to cumulative errors and dependency on accurate initial values. Model-based filtering techniques, like Kalman Filter (KF) and its nonlinear variant, the Extended Kalman Filter (EKF), offer robust alternatives by incorporating system dynamics and measurement noise. However, these methods often rely on offline parameter identification, which may not adapt to changing operating conditions or cell aging. To overcome this, we propose a DEKF-based approach that online estimates both the system state (SOC and polarization voltage) and model parameters (ohmic resistance, polarization resistance, and polarization capacitance). This dual estimation framework enhances adaptability and accuracy, making it particularly suitable for the dynamic and scalable nature of cell energy storage systems.
The foundation of our SOC estimation method is the mathematical modeling of the cell energy storage system. We adopt the Thevenin equivalent circuit model due to its balance between simplicity and fidelity in representing battery external characteristics. The model comprises an open-circuit voltage source \(U_o\) (a function of SOC), an ohmic internal resistor \(R_o\), and a parallel \(R_p\)-\(C_p\) network representing polarization effects. The governing equations are:
$$
C_p \frac{dV_p}{dt} + \frac{V_p}{R_p} = I
$$
$$
V_t = U_o + V_p + I R_o
$$
where \(V_p\) is the polarization voltage across the \(R_p\)-\(C_p\) network, \(I\) is the working current (positive for charge), and \(V_t\) is the terminal voltage. Discretizing these equations with a sampling time \(t_s\) yields:
$$
V_p(k+1) = V_p(k) e^{-\frac{t_s}{R_p C_p}} + I(k) R_p \left(1 – e^{-\frac{t_s}{R_p C_p}}\right)
$$
$$
V_t(k+1) = U_o(k) + V_p(k) + I(k) R_o
$$
For a cell energy storage system configured as an \(n \times m\) array ( \(n\) parallel strings, \(m\) series cells per string), we assume cell uniformity. Under this assumption, the parallel combination scales the capacity and current by a factor of \(n\), while the series combination scales the voltage by a factor of \(m\). Thus, the entire cell energy storage system can be treated as a single cell with scaled parameters for SOC estimation purposes, as the SOC remains identical across uniform cells. This simplification allows us to focus on single-cell modeling while maintaining applicability to larger systems.
The SOC is intrinsically related to the current integral. Using the Ah integration method:
$$
\text{SOC}(k) = \text{SOC}(k-1) + \frac{\eta I(k-1) t_s}{C_N}
$$
where \(\eta\) is the coulombic efficiency (assumed 1 for discharge in our study), and \(C_N\) is the nominal capacity. For the scaled cell energy storage system, \(C_N\) becomes \(n\) times the single-cell capacity, but the SOC update equation remains structurally similar due to the current scaling.
To implement the DEKF, we define the state vector \(\mathbf{x}_k = [V_p(k), \text{SOC}(k)]^T\) and the parameter vector \(\boldsymbol{\theta}_k = [R_o, R_p, C_p]^T\). The process equation for the state vector is derived from the discretized model and Ah integration:
$$
\mathbf{x}_k = \begin{bmatrix} e^{-\frac{t_s}{R_p C_p}} & 0 \\ 0 & 1 \end{bmatrix} \mathbf{x}_{k-1} + \begin{bmatrix} R_p (1 – e^{-\frac{t_s}{R_p C_p}}) \\ -\frac{\eta t_s}{C_N} \end{bmatrix} I(k-1) + \boldsymbol{\omega}_{k-1}
$$
The observation equation is the terminal voltage:
$$
z_k = V_t(k) = U_o(\text{SOC}(k)) + V_p(k) + I(k) R_o + v_k
$$
where \(\boldsymbol{\omega}_k\) and \(v_k\) are process and measurement noises, assumed zero-mean Gaussian with covariance matrices \(\mathbf{Q}_x\) and \(R_x\), respectively. The OCV-SOC relationship \(U_o(\text{SOC})\) is obtained offline and stored as a lookup table with linear interpolation.
The DEKF algorithm operates two intertwined EKFs: one for state estimation and another for parameter estimation. The steps are as follows:
1. Parameter Prediction: \(\boldsymbol{\theta}^-_k = \boldsymbol{\theta}_{k-1}\)
2. Parameter Covariance Prediction: \(\mathbf{P}^-_{\theta,k} = \mathbf{P}_{\theta,k-1} + \mathbf{Q}_{\theta}\)
3. State Prediction: \(\mathbf{x}^-_k = f(\mathbf{x}_{k-1}, \boldsymbol{\theta}^-_k, I(k-1))\)
4. State Covariance Prediction: \(\mathbf{P}^-_{x,k} = \mathbf{A}_x \mathbf{P}_{x,k-1} \mathbf{A}_x^T + \mathbf{Q}_x\)
5. State Kalman Gain: \(\mathbf{K}_{x,k} = \mathbf{P}^-_{x,k} \mathbf{H}_x^T (\mathbf{H}_x \mathbf{P}^-_{x,k} \mathbf{H}_x^T + R_x)^{-1}\)
6. State Update: \(\mathbf{x}_k = \mathbf{x}^-_k + \mathbf{K}_{x,k} (z_k – h(\mathbf{x}^-_k, \boldsymbol{\theta}^-_k, I(k)))\)
7. State Covariance Update: \(\mathbf{P}_{x,k} = (\mathbf{I} – \mathbf{K}_{x,k} \mathbf{H}_x) \mathbf{P}^-_{x,k}\)
8. Parameter Kalman Gain: \(\mathbf{K}_{\theta,k} = \mathbf{P}^-_{\theta,k} \mathbf{H}_{\theta}^T (\mathbf{H}_{\theta} \mathbf{P}^-_{\theta,k} \mathbf{H}_{\theta}^T + R_{\theta})^{-1}\)
9. Parameter Update: \(\boldsymbol{\theta}_k = \boldsymbol{\theta}^-_k + \mathbf{K}_{\theta,k} (z_k – h(\mathbf{x}^-_k, \boldsymbol{\theta}^-_k, I(k)))\)
10. Parameter Covariance Update: \(\mathbf{P}_{\theta,k} = (\mathbf{I} – \mathbf{K}_{\theta,k} \mathbf{H}_{\theta}) \mathbf{P}^-_{\theta,k}\)
where \(\mathbf{A}_x = \frac{\partial f}{\partial \mathbf{x}}\), \(\mathbf{H}_x = \frac{\partial h}{\partial \mathbf{x}}\), and \(\mathbf{H}_{\theta} = \frac{\partial h}{\partial \boldsymbol{\theta}} + \frac{\partial h}{\partial \mathbf{x}^-_k} \frac{d \mathbf{x}^-_k}{d \boldsymbol{\theta}}\). The recursive computation of \(\frac{d \mathbf{x}^-_k}{d \boldsymbol{\theta}}\) is detailed in the literature. The matrices \(\mathbf{Q}_{\theta}\) and \(R_{\theta}\) are the process and measurement noise covariances for the parameter filter. This dual structure enables real-time adaptation of model parameters, enhancing the robustness of SOC estimation for cell energy storage systems under varying operational conditions.
Offline parameter identification is necessary to initialize the DEKF. We employ the Hybrid Pulse Power Characterization (HPPC) test, which applies current pulses at various SOC levels to extract model parameters. For a discharge pulse of magnitude \(I_B\), the ohmic resistance \(R_o\) is computed from the instantaneous voltage drop:
$$
R_o = \frac{|U_A – U_B|}{I_B}
$$
where \(U_A\) and \(U_B\) are voltages before and at the start of the pulse. The polarization resistance \(R_p\) is derived from the voltage difference between the end of the pulse and after a relaxation period:
$$
R_p = \frac{|U_C – U_D|}{I_B}
$$
The polarization capacitance \(C_p\) is calculated from the time constant \(\tau = R_p C_p\), where \(\tau\) is the time for the polarization voltage to reach 95% of its steady-state value during relaxation:
$$
C_p = \frac{\tau}{R_p}
$$
We perform HPPC tests at SOC intervals of 10% from 100% to 0% to capture parameter variations. The results are summarized in Table 1, showing the dependence of \(R_o\), \(R_p\), and \(C_p\) on SOC for a typical lithium-ion cell used in cell energy storage systems.
| SOC (%) | \(R_o\) (mΩ) | \(R_p\) (mΩ) | \(C_p\) (F) |
|---|---|---|---|
| 100 | 25.1 | 15.3 | 1,520 |
| 90 | 24.8 | 14.9 | 1,540 |
| 80 | 24.5 | 14.7 | 1,560 |
| 70 | 24.3 | 14.5 | 1,580 |
| 60 | 24.2 | 14.4 | 1,600 |
| 50 | 24.1 | 14.3 | 1,620 |
| 40 | 24.2 | 14.5 | 1,610 |
| 30 | 24.5 | 14.8 | 1,590 |
| 20 | 25.0 | 15.2 | 1,560 |
| 10 | 25.8 | 15.8 | 1,520 |
| 0 | 27.0 | 16.5 | 1,480 |
The OCV-SOC relationship is critical for accurate SOC estimation. Instead of time-consuming static methods, we use a rate discharge curve extrapolation technique. Discharge curves at multiple rates (e.g., 1C, 0.75C, 0.5C, 0.25C, 0.1C) are obtained, and for each SOC point, voltage values at different rates are linearly fitted against the discharge rate. The intercept at zero current gives the OCV. This method efficiently generates a high-resolution OCV-SOC table with 0.5% SOC intervals. The relationship is nonlinear and can be approximated by a polynomial, but we use lookup table with linear interpolation for precision. The OCV-SOC curve for our cell energy storage system cell is shown in functional form as:
$$
U_o(\text{SOC}) = 3.0 + 1.2 \cdot \text{SOC} – 0.5 \cdot \text{SOC}^2 + 0.1 \cdot \text{SOC}^3 \quad (\text{example approximation})
$$
In practice, we store 201 data points from SOC 0 to 1 for accurate interpolation.
To validate the DEKF-based SOC estimation method, we conduct experiments using a high-precision battery test system. The cell is a ternary lithium-ion battery with nominal voltage 3.6 V, rated capacity 2.2 Ah, and voltage range 2.75–4.2 V. The test system has voltage and current accuracies of 0.02%, serving as ground truth. Experiments are performed at 25°C. The cell is fully charged, then discharged at 0.5C to 90% SOC, followed by application of FUDS and DST current profiles to discharge to 0% SOC, with a final 30-minute rest. Sampling time is 1 s. The FUDS and DST profiles simulate dynamic load conditions typical in cell energy storage system applications, featuring variable current magnitudes and signs.
The DEKF algorithm is implemented in MATLAB, with initial SOC set to 0 (100% error) to test convergence. Initial parameters are from Table 1 at 90% SOC. Noise covariances are tuned empirically: \(\mathbf{Q}_x = \text{diag}(10^{-6}, 10^{-6})\), \(R_x = 10^{-4}\), \(\mathbf{Q}_{\theta} = \text{diag}(10^{-8}, 10^{-8}, 10^{-8})\), and \(R_{\theta} = 10^{-4}\). For comparison, a standard EKF with fixed parameters (from Table 1) is also run. The computational load is minimal, with average execution time per step around \(5.6 \times 10^{-5}\) seconds, suitable for real-time BMS in cell energy storage systems.
Results under FUDS and DST conditions are summarized in Table 2. The SOC estimation errors are computed relative to the reference SOC from coulomb counting with initial SOC corrected. Terminal voltage errors are compared against measured voltages.
| Condition | Method | Max SOC Error (%) | RMSE SOC (%) | Max Voltage Error (mV) | MAE Voltage (mV) | Convergence Time (s) |
|---|---|---|---|---|---|---|
| FUDS | DEKF | 0.95 | 0.32 | 9.8 | 2.7 | 120 |
| EKF | 1.98 | 0.78 | 15.2 | 4.5 | 300 | |
| DST | DEKF | 0.89 | 0.28 | 19.5 | 3.8 | 100 |
| EKF | 3.87 | 1.23 | 28.7 | 6.2 | 350 |
The SOC estimation curves show that both methods converge, but DEKF achieves stable estimation faster. For FUDS, DEKF reduces SOC error to below 1% within 120 seconds, while EKF takes about 300 seconds. Under DST, DEKF converges in 100 seconds, compared to 350 seconds for EKF. The terminal voltage estimation errors are within ±10 mV for FUDS and ±20 mV for DST using DEKF, with mean absolute errors (MAE) of 2.7 mV and 3.8 mV, respectively. EKF exhibits larger errors due to fixed parameters. Voltage error spikes occur at the beginning and end of discharge, attributed to initial parameter mismatches and rapid voltage changes at low SOC. The online parameter adaptation in DEKF mitigates these issues.
To further illustrate the parameter adaptation, Figure 1 plots the online estimates of \(R_o\), \(R_p\), and \(C_p\) during FUDS operation. The parameters evolve with SOC and current, reflecting dynamic changes in cell behavior. For instance, \(R_o\) increases slightly as SOC decreases, consistent with offline HPPC data. This adaptability is crucial for long-term accuracy in cell energy storage systems, where parameters may drift due to aging or temperature variations.
The scalability of the method to larger cell energy storage systems is validated through simulation. For an \(n \times m\) system, we assume identical cells and use scaled current and voltage. The SOC estimation error remains consistent, as the SOC is uniform across cells. In practice, cell imbalances may occur, but the method can be extended to estimate individual cell SOCs by applying DEKF to each cell or module, provided current and voltage measurements are available. This modular approach aligns with the distributed architecture of BMS in cell energy storage systems.
Several factors influence the performance of the DEKF-based SOC estimation. The accuracy of the OCV-SOC curve is paramount; any error propagates to SOC estimates. Our extrapolation method balances speed and precision. The assumption of cell uniformity holds for new systems, but aging may introduce divergences, necessitating periodic recalibration or adaptive algorithms. Temperature effects are not addressed here but can be incorporated by including temperature-dependent parameters or using a thermal- coupled model. Future work will focus on integrating temperature compensation and aging detection to enhance robustness for real-world cell energy storage system deployments.
In conclusion, we have developed a DEKF-based SOC estimation method tailored for cell energy storage system BMS testing platforms. The approach combines a Thevenin equivalent circuit model with dual online estimation of states and parameters, offering improved convergence speed and accuracy over traditional EKF. Experimental validation under FUDS and DST conditions confirms SOC errors below 1% and terminal voltage errors within practical limits. The method supports scalable deployment for cell energy storage systems by leveraging cell uniformity assumptions and modular design. This work contributes to the standardization of BMS testing for cell energy storage systems, providing a reliable benchmark for SOC estimation performance. Future directions include extending the algorithm to handle temperature variations, cell aging, and imbalances, ensuring long-term reliability for grid-scale cell energy storage systems.
The implementation of such advanced estimation techniques is essential for optimizing the operation and lifespan of cell energy storage systems. By accurately monitoring SOC, BMS can prevent harmful operating conditions, balance energy flows, and provide critical data for grid management. As cell energy storage systems become more prevalent in renewable integration and grid stabilization, robust SOC estimation methods will play a pivotal role in ensuring their economic and operational viability. The DEKF framework presented here offers a flexible and accurate solution that can be adapted to various battery chemistries and system configurations, making it a valuable tool for researchers and engineers working on next-generation cell energy storage systems.
From a practical standpoint, the testing platform incorporating this method can automate BMS validation, reducing time and costs. The platform can simulate diverse operational scenarios, from steady-state to extreme dynamics, assessing BMS performance under realistic conditions. This is crucial for certifying BMS for grid-connected cell energy storage systems, where reliability standards are stringent. Moreover, the online parameter estimation capability reduces the need for frequent manual calibration, lowering maintenance overhead. As the demand for cell energy storage systems grows, such automated testing and calibration tools will become indispensable for ensuring system safety and efficiency.
In summary, the integration of DEKF into SOC estimation for cell energy storage systems represents a significant advancement in BMS technology. It addresses key challenges in accuracy and adaptability, providing a solid foundation for future innovations. We anticipate that continued research in this area will further enhance the performance and reliability of cell energy storage systems, contributing to a more sustainable and resilient energy infrastructure.