In this work, I present a comprehensive methodology for the online estimation of the state of charge (SOC) of lithium-ion batteries used in battery energy storage systems. Accurate SOC estimation is critical for ensuring the safe and reliable operation of the battery energy storage system, as it directly influences charge/discharge control, thermal management, and overall system stability. The proposed approach integrates the Thevenin equivalent circuit model with a forgetting factor recursive least squares (FFRLS) method for online parameter identification, and employs the unscented Kalman filter (UKF) for SOC estimation. I conducted experiments on 18650 lithium-ion cells under interval constant current discharge profiles to validate the method. The results demonstrate that the proposed FFRLS-UKF method achieves high estimation accuracy (up to 99.2%) and strong robustness against initial SOC errors, outperforming traditional ampere-hour integration and extended Kalman filter (EKF) methods.
The battery energy storage system is a key component in modern power grids, enabling the integration of intermittent renewable energy sources such as wind and solar. However, the complex electrochemical behavior of lithium-ion batteries, combined with uncertainties in operating conditions and measurement noise, poses significant challenges for accurate SOC estimation. Model-driven approaches, which combine an equivalent circuit model with filtering algorithms, offer a balance between computational efficiency and estimation accuracy. Among various filters, UKF is particularly suitable for nonlinear systems because it approximates the probability distribution using a set of sigma points, avoiding the linearization errors inherent in EKF.
To begin, I establish the open-circuit voltage (OCV) versus SOC characteristic curve through pulsed discharge experiments. The battery is fully charged and then discharged at 1C rate in steps of 5% rated capacity, with a 900-second rest after each step to allow voltage relaxation. The measured OCV values for SOC from 1.0 to 0 are fitted with a sixth-order polynomial:
$$V_{OCV}(SOC) = A + A_0 SOC + A_1 SOC^2 + A_2 SOC^3 + A_3 SOC^4 + A_4 SOC^5 + A_5 SOC^6$$
The polynomial coefficients obtained from the fitting are listed in the table below.
| Coefficient | Value |
|---|---|
| A | 240.4653 |
| A0 | 2.4823 |
| A1 | 21.5399 |
| A2 | -162.5463 |
| A3 | 622.9878 |
| A4 | 1308.5389 |
| A5 | 1538.6869 |
| A6 | -950.8749 |
Next, I adopt the Thevenin equivalent circuit model to represent the dynamic behavior of the battery. The model consists of an OCV source, an ohmic resistance R0, and a parallel RC network (R1, C1) representing polarization effects. The electrical equations are:
$$\dot{U}_1 = \frac{I}{C_1} – \frac{U_1}{R_1 C_1}$$
$$U = V_{OCV} – I R_0 – U_1$$
where U1 is the voltage across the polarization capacitor, I is the load current (positive for discharge), and U is the terminal voltage. The unknown parameters R0, R1, and C1 must be identified online to track the time-varying characteristics of the battery.
To achieve online parameter identification, I apply the FFRLS method. The transfer function of the Thevenin model in the s-domain is:
$$G(s) = \frac{V_{OC}(s) – U_t(s)}{I(s)} = R_0 + \frac{R_1}{s R_1 C_1 + 1}$$
Using the bilinear transform with sampling period T:
$$s = \frac{2(1 – z^{-1})}{T(1 + z^{-1})}$$
the discrete transfer function becomes:
$$G(z^{-1}) = \frac{c_2 + c_3 z^{-1}}{1 + c_1 z^{-1}}$$
where coefficients c1, c2, c3 are related to the physical parameters as follows:
$$c_1 = \frac{T – 2R_1 C_1}{T + 2R_1 C_1}$$
$$c_2 = \frac{R_0 T + R_1 T + 2R_0 R_1 C_1}{T + 2R_1 C_1}$$
$$c_3 = \frac{R_0 T + R_1 T – 2R_0 R_1 C_1}{T + 2R_1 C_1}$$
In the time domain, the regression equation is:
$$y_k = (V_{OC,k} – U_{t,k}) = -c_1 y_{k-1} + c_2 I_k + c_3 I_{k-1}$$
Define the parameter vector and regressor:
$$\theta_k = [c_1, c_2, c_3]^T, \quad \phi_k = [-y_{k-1}, I_k, I_{k-1}]^T$$
The FFRLS algorithm updates the parameter estimates recursively with a forgetting factor λ (set to 0.95 in this study). After obtaining c1, c2, c3, the physical parameters are recovered by:
$$R_0 = \frac{c_2 – c_3}{1 – c_1}$$
$$R_1 = \frac{c_2 + c_3}{1 + c_1} – R_0$$
$$C_1 = \frac{T(1 – c_1) / [2(1 + c_1)]}{R_1}$$
I perform interval constant current discharge experiments on a 18650 lithium-ion battery (rated capacity 2.0 Ah) at room temperature. The discharge current is 1C (2.0 A), with 5% capacity discharge steps followed by 900 s rest. The measured current and voltage profiles are used for parameter identification and subsequent SOC estimation. The identified parameters are shown in the following table over the discharge process.
| Time (s) | R0 (mΩ) | R1 (mΩ) | C1 (F) |
|---|---|---|---|
| 0 | 18.2 | 12.5 | 2500 |
| 500 | 17.9 | 13.1 | 2470 |
| 1000 | 17.6 | 13.8 | 2420 |
| 2000 | 17.2 | 14.5 | 2350 |
| 3000 | 16.9 | 15.2 | 2280 |
To validate the identified model, I compare the terminal voltage predicted by the model using the identified parameters with the measured voltage. The voltage error remains within ±0.1 V throughout the entire discharge, confirming the accuracy of the parameter identification.
For SOC estimation, I formulate the state-space model of the battery using SOC and U1 as state variables, current I as input, and terminal voltage U as output. The discrete state equations are:
$$\begin{bmatrix} SOC_{k+1} \\ U_{1,k+1} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & e^{-T/(R_1 C_1)} \end{bmatrix} \begin{bmatrix} SOC_k \\ U_{1,k} \end{bmatrix} + \begin{bmatrix} T/Q_n \\ R_0(1 – e^{-T/(R_1 C_1)}) \end{bmatrix} I_k + w_k$$
$$U_k = V_{OCV}(SOC_k) – U_{1,k} – I_k R_0 + v_k$$
where Qn is the nominal capacity, wk and vk are process and measurement noise with covariances Q and R, respectively. The UKF algorithm proceeds as follows:
- Initialize state estimate x̂0 and covariance P0.
- Generate sigma points χi from x̂k and Pk.
- Propagate sigma points through state transition function to obtain predicted sigma points and compute predicted state x̂k+1|k and covariance Pk+1|k.
- Propagate sigma points through measurement function to obtain predicted measurement ŷk+1|k and innovation covariance Pyy, as well as cross-covariance Pxy.
- Compute Kalman gain K = Pxy Pyy⁻¹.
- Update state estimate x̂k+1 = x̂k+1|k + K (yk+1 – ŷk+1|k) and covariance Pk+1 = Pk+1|k – K Pyy Kᵀ.
The UKF is particularly effective for the nonlinear battery energy storage system because it captures the mean and covariance of the state distribution through sigma points, providing higher accuracy than EKF for strongly nonlinear systems.
I compare the proposed FFRLS-UKF method with two conventional approaches: ampere-hour (AH) integration and EKF. The experiments are conducted under the same discharge profile with a total of 13,000 sampling points (sampling period T = 1 s). The initial SOC is set to 1.0 for all methods. The estimated SOC curves and absolute errors are shown in the analysis.
The evaluation metrics are defined as:
Mean Error (ME): $$ME = \frac{1}{N} \sum_{k=1}^N |SOC_{true,k} – SOC_{est,k}|$$
Mean Squared Error (MSE): $$MSE = \frac{1}{N} \sum_{k=1}^N (SOC_{true,k} – SOC_{est,k})^2$$
Accuracy: $$Accuracy = \left(1 – \frac{ME}{1.0}\right) \times 100\%$$
The quantitative results are listed in the table below.
| Method | ME (%) | MSE (%) | Accuracy (%) |
|---|---|---|---|
| AH Integration | 0.9337 | 0.1082 | 97.93 |
| EKF | 0.7051 | 0.0602 | 98.61 |
| FFRLS-UKF (Proposed) | 0.3997 | 0.0201 | 99.20 |
As shown, the proposed FFRLS-UKF method achieves the lowest ME and MSE, and the highest accuracy of 99.2%, significantly outperforming AH integration and EKF. This improvement is attributed to the online parameter identification that adapts to battery aging and operating conditions, and the UKF’s superior handling of nonlinearities in the battery energy storage system.
To evaluate robustness, I test the methods under different initial SOC errors. The battery is discharged from an initial SOC of 1.0, but the algorithms are initialized with deliberately incorrect values: 0.75 and 0.50. The convergence behavior is examined. The table below summarizes the convergence steps (number of samples required for the estimated SOC to settle within 2% of the true value).
| Initial SOC Error | EKF (steps) | FFRLS-UKF (steps) |
|---|---|---|
| 0.75 (error -0.25) | 690 | 104 |
| 0.50 (error -0.50) | 722 | 133 |
The results demonstrate that the proposed method converges much faster than EKF when the initial SOC is incorrect. This robustness is due to the UKF’s ability to adjust the Kalman gain based on the voltage error, quickly correcting the state estimate. The AH integration method is not applicable here because it requires an accurate initial value.
Furthermore, I analyze the impact of model parameter errors on SOC estimation. Even when the initial SOC is correct, if the identified parameters contain errors (e.g., due to battery aging), the UKF can still provide reliable estimates because it continuously updates the covariance and uses the measurement to correct the state. In contrast, EKF, which relies on linearization, may accumulate errors over time. The proposed FFRLS-UKF framework effectively mitigates this issue by combining online parameter identification with a robust filter.
The above figure illustrates a typical battery energy storage system installation, where the proposed SOC estimation algorithm can be deployed for real-time monitoring and control. The integration of online parameter identification and UKF ensures that the battery energy storage system operates safely and efficiently under varying conditions.
In addition to the main experiments, I perform a sensitivity analysis to understand the effect of forgetting factor λ in FFRLS. With λ = 0.95, the algorithm responds quickly to parameter changes without being too sensitive to noise. A smaller λ (e.g., 0.90) leads to faster adaptation but introduces higher variance; a larger λ (e.g., 0.98) provides smoother estimates but slower tracking. The choice of λ = 0.95 balances these requirements for the battery energy storage system application.
The computational complexity of the proposed method is also evaluated. The FFRLS update requires O(3²) operations per step, and the UKF requires O(2³) operations (since state dimension is 2). The total computational load is modest and suitable for real-time implementation on embedded controllers used in battery management systems for battery energy storage systems.
Finally, I summarize the key contributions of this work:
- An online parameter identification method using FFRLS for the Thevenin model of lithium-ion batteries in battery energy storage systems.
- A robust SOC estimation method based on UKF that accounts for nonlinearities and measurement uncertainties.
- Experimental validation showing that the proposed FFRLS-UKF method achieves 99.2% accuracy under accurate initial conditions and converges within ~100 steps under large initial SOC errors.
- The method outperforms traditional AH integration and EKF in terms of both accuracy and robustness.
The developed approach can be directly applied to real-world battery energy storage systems to improve safety, extend battery life, and optimize energy management. Future work will focus on extending the method to consider temperature effects and cell balancing in large-scale battery energy storage systems.