In modern power systems, the integration of renewable energy sources like wind and solar has highlighted the need for efficient energy storage solutions. Among these, lithium-ion batteries are widely adopted due to their high safety, excellent electrochemical performance, long cycle life, and environmental friendliness. Accurate estimation of the State of Charge (SOC) for energy storage cells is crucial to ensure the safe and stable operation of battery management systems. SOC represents the remaining capacity of a battery and is a key parameter for monitoring its health and performance. However, the highly nonlinear nature of energy storage cells, coupled with uncertainties in model parameters and environmental noise, poses significant challenges to precise SOC estimation. Traditional methods, such as the open-circuit voltage (OCV) method and Ampere-hour (Ah) integration, often suffer from limitations like long resting times, dependency on initial conditions, and cumulative errors. To address these issues, we propose a novel approach combining the Forgetting Factor Recursive Least Squares (FFRLS) method for online parameter identification and the Unscented Kalman Filter (UKF) for robust SOC estimation. This method leverages the Thevenin equivalent circuit model to capture the dynamic behavior of energy storage cells, enabling real-time adaptation to changing conditions. In this article, we detail the model development, parameter identification process, SOC estimation algorithm, and experimental validation, demonstrating the superiority of our approach in terms of accuracy and convergence under various initial SOC conditions.
The foundation of our SOC estimation method lies in the accurate modeling of the energy storage cell. We adopt the Thevenin equivalent circuit model, which effectively represents the electrical characteristics of lithium-ion batteries. This model includes components such as an internal resistance, a polarization resistance, and a capacitance to simulate the transient responses during charging and discharging. The governing equations for the Thevenin model are derived from circuit theory, providing a mathematical framework for SOC estimation. The terminal voltage of the energy storage cell can be expressed as a function of the OCV, which is intrinsically related to SOC, and the voltage drops across the internal components. The dynamics of the polarization voltage are described by a first-order differential equation, capturing the relaxation effects observed in real energy storage cells. By discretizing these equations, we obtain a state-space representation suitable for digital implementation. The model parameters, including the ohmic resistance, polarization resistance, and polarization capacitance, are time-varying due to factors like temperature, aging, and operating conditions. Therefore, online identification of these parameters is essential to maintain model accuracy. Our approach uses the FFRLS algorithm to continuously update the parameters based on measured current and voltage data, ensuring that the model adapts to the evolving characteristics of the energy storage cell.
The parameter identification process begins with the derivation of the discrete-time transfer function from the continuous-time model. Using bilinear transformation, we convert the s-domain representation to the z-domain, resulting in a difference equation that relates the input current to the output voltage. The FFRLS algorithm is then applied to estimate the coefficients of this difference equation, which are subsequently used to compute the physical parameters of the energy storage cell. The recursive nature of FFRLS allows for efficient computation and memory usage, making it suitable for real-time applications. Moreover, the inclusion of a forgetting factor enables the algorithm to prioritize recent data, reducing the impact of outdated information and improving tracking of parameter variations. The performance of the parameter identification is validated by comparing the model-predicted voltage with experimental measurements, showing minimal error and high fidelity. This accurate parameter identification forms the basis for the subsequent SOC estimation using the UKF algorithm.
For SOC estimation, we employ the UKF, which is well-suited for nonlinear systems like energy storage cells. Unlike the Extended Kalman Filter (EKF), which linearizes the system model, the UKF uses a deterministic sampling technique known as the unscented transform to approximate the state distribution. This avoids the truncation errors associated with linearization and provides better accuracy for highly nonlinear systems. The state vector in our UKF formulation includes SOC and the polarization voltage, while the input is the load current and the output is the terminal voltage. The state-space equations incorporate the OCV-SOC relationship, which is characterized by a polynomial function derived from experimental data. The UKF algorithm involves steps such as sigma point generation, state prediction, measurement update, and covariance correction. By iterating these steps, the UKF provides an optimal estimate of SOC, even in the presence of noise and model uncertainties. We compare our FFRLS-UKF method with traditional approaches like EKF and Ah integration, using metrics such as mean error (ME), mean square error (MSE), and accuracy to evaluate performance. The results demonstrate that our method achieves higher accuracy and faster convergence, making it a reliable solution for online SOC estimation of energy storage cells.
The OCV-SOC relationship is a critical component in SOC estimation, as it links the electrical behavior of the energy storage cell to its charge level. We obtain this relationship through experimental tests involving constant current discharge pulses followed by relaxation periods. The OCV values at different SOC points are recorded and fitted to a polynomial function. This function is then used in the state-space model to relate SOC to the OCV. The polynomial coefficients are determined using least squares fitting, ensuring a smooth and accurate representation. The OCV-SOC curve exhibits nonlinear characteristics, particularly at low and high SOC levels, which must be captured precisely to avoid estimation errors. In our work, we use a sixth-order polynomial to model this relationship, as it provides a good balance between complexity and accuracy. The equation for the OCV-SOC curve is given by:
$$ V_{OCV} = A + A_0 S_{OC}^0 + A_1 S_{OC}^1 + A_2 S_{OC}^2 + A_3 S_{OC}^3 + A_4 S_{OC}^4 + A_5 S_{OC}^5 + A_6 S_{OC}^6 $$
where \( V_{OCV} \) is the open-circuit voltage, \( S_{OC} \) is the state of charge, and \( A, A_0, A_1, A_2, A_3, A_4, A_5, A_6 \) are the polynomial coefficients obtained from experimental data. This equation is integrated into the UKF algorithm to facilitate accurate SOC estimation.
The Thevenin equivalent circuit model for the energy storage cell consists of a voltage source representing the OCV, a series resistance accounting for ohmic losses, and a parallel RC network modeling the polarization effects. The differential equations governing the model are:
$$ \dot{U_1} = \frac{I}{C_1} – \frac{U_1}{R_1 C_1} $$
$$ U = V_{OCV} – I R_0 – U_1 $$
where \( U_1 \) is the voltage across the polarization capacitance, \( I \) is the current, \( U \) is the terminal voltage, \( R_0 \) is the ohmic resistance, \( R_1 \) is the polarization resistance, and \( C_1 \) is the polarization capacitance. Discretizing these equations with a sampling time \( T \) yields the discrete-time state-space model:
$$ x_{k+1} = A x_k + B u_k + w_k $$
$$ y_k = C x_k + D u_k + v_k $$
with the state vector \( x_k = [SOC_k, U_{1,k}]^T \), input \( u_k = I_k \), and output \( y_k = U_k \). The matrices are defined as:
$$ A = \begin{bmatrix} 1 & 0 \\ 0 & \exp\left(-\frac{T}{R_1 C_1}\right) \end{bmatrix} $$
$$ B = \begin{bmatrix} -\frac{T}{Q_0} \\ R_1 \left(1 – \exp\left(-\frac{T}{R_1 C_1}\right)\right) \end{bmatrix} $$
$$ C = \begin{bmatrix} \frac{\partial F(SOC_k)}{\partial SOC_k} & -1 \end{bmatrix} $$
$$ D = -R_0 $$
where \( Q_0 \) is the nominal capacity of the energy storage cell, and \( F(SOC_k) \) is the OCV-SOC function. The process noise \( w_k \) and measurement noise \( v_k \) are assumed to be Gaussian with zero mean and known covariances.
Parameter identification using FFRLS involves expressing the model in a form suitable for recursive estimation. The discrete transfer function is derived as:
$$ G(z^{-1}) = \frac{U_{oc}(z^{-1}) – U_t(z^{-1})}{I(z^{-1})} = \frac{c_2 + c_3 z^{-1}}{1 + c_1 z^{-1}} $$
where \( c_1, c_2, c_3 \) are coefficients related to the physical parameters. The time-domain equation is:
$$ y_k = -c_1 y_{k-1} + c_2 I_k + c_3 I_{k-1} $$
with \( y_k = U_{oc,k} – U_{t,k} \). The parameter vector \( \theta_k = [c_1, c_2, c_3]^T \) and regressor vector \( \phi_k = [-y_{k-1}, I_k, I_{k-1}]^T \) are defined, leading to the linear form \( y_k = \theta_k^T \phi_k \). The FFRLS update equations are:
$$ \theta_k = \theta_{k-1} + K_k (y_k – \theta_{k-1}^T \phi_k) $$
$$ K_k = \frac{P_{k-1} \phi_k}{\lambda + \phi_k^T P_{k-1} \phi_k} $$
$$ P_k = \frac{1}{\lambda} \left( P_{k-1} – \frac{P_{k-1} \phi_k \phi_k^T P_{k-1}}{\lambda + \phi_k^T P_{k-1} \phi_k} \right) $$
where \( \lambda \) is the forgetting factor (typically between 0.95 and 1), and \( P_k \) is the covariance matrix. The physical parameters are then computed as:
$$ R_0 = \frac{c_2 – c_3}{1 – c_1} $$
$$ R_1 = \frac{c_2 + c_3}{1 + c_1} – R_0 $$
$$ C_1 = \frac{T (1 – c_1)}{2 R_1 (1 + c_1)} $$
This online identification ensures that the model parameters reflect the current state of the energy storage cell, enhancing SOC estimation accuracy.
The UKF algorithm for SOC estimation involves the following steps:
- Initialization: Set initial state estimate \( \hat{x}_0 \) and error covariance \( P_0 \).
- Sigma Point Calculation: Generate sigma points \( \chi_{i,k} \) around the current state estimate using:
- State Prediction: Propagate sigma points through the state transition function:
- Measurement Update: Transform sigma points to measurement space:
- Kalman Gain and Update: Compute the Kalman gain and update the state estimate and covariance:
$$ \chi_{0,k} = \hat{x}_k $$
$$ \chi_{i,k} = \hat{x}_k + \sqrt{(n + \kappa)} \cdot \text{col}_i(\sqrt{P_k}), \quad i = 1, \dots, n $$
$$ \chi_{i+n,k} = \hat{x}_k – \sqrt{(n + \kappa)} \cdot \text{col}_i(\sqrt{P_k}), \quad i = 1, \dots, n $$
where \( n \) is the state dimension, and \( \kappa \) is a scaling parameter.
$$ \chi_{i,k+1|k} = f(\chi_{i,k}, u_k) $$
and compute the predicted state and covariance:
$$ \hat{x}_{k+1|k} = \sum_{i=0}^{2n} w_i^m \chi_{i,k+1|k} $$
$$ P_{k+1|k} = \sum_{i=0}^{2n} w_i^c (\chi_{i,k+1|k} – \hat{x}_{k+1|k}) (\chi_{i,k+1|k} – \hat{x}_{k+1|k})^T + Q $$
where \( w_i^m \) and \( w_i^c \) are weights, and \( Q \) is the process noise covariance.
$$ \gamma_{i,k+1} = h(\chi_{i,k+1|k}, u_k) $$
and compute the predicted measurement and innovation covariance:
$$ \hat{y}_{k+1} = \sum_{i=0}^{2n} w_i^m \gamma_{i,k+1} $$
$$ P_{yy} = \sum_{i=0}^{2n} w_i^c (\gamma_{i,k+1} – \hat{y}_{k+1}) (\gamma_{i,k+1} – \hat{y}_{k+1})^T + R $$
$$ P_{xy} = \sum_{i=0}^{2n} w_i^c (\chi_{i,k+1|k} – \hat{x}_{k+1|k}) (\gamma_{i,k+1} – \hat{y}_{k+1})^T $$
where \( R \) is the measurement noise covariance.
$$ K_{k+1} = P_{xy} P_{yy}^{-1} $$
$$ \hat{x}_{k+1} = \hat{x}_{k+1|k} + K_{k+1} (y_{k+1} – \hat{y}_{k+1}) $$
$$ P_{k+1} = P_{k+1|k} – K_{k+1} P_{yy} K_{k+1}^T $$
This iterative process enables robust SOC estimation for energy storage cells under varying operating conditions.
Experimental validation is conducted using a lithium-ion energy storage cell subjected to constant current discharge tests. The voltage and current data are collected, and the FFRLS-UKF method is applied to estimate SOC. For comparison, the EKF and Ah integration methods are also implemented. The performance metrics, including ME, MSE, and accuracy, are calculated to evaluate the estimation quality. The results show that the FFRLS-UKF method outperforms the others, with higher accuracy and faster convergence, especially when initial SOC errors are present. The following table summarizes the performance comparison under different initial SOC conditions:
| Initial SOC | Method | ME (%) | MSE (%) | Accuracy (%) |
|---|---|---|---|---|
| 1.00 | Ah Integration | 0.9337 | 0.1082 | 97.93 |
| EKF | 0.7051 | 0.0602 | 98.61 | |
| FFRLS-UKF | 0.3997 | 0.0201 | 99.20 | |
| 0.75 | EKF | 0.8500 | 0.0800 | 98.20 |
| FFRLS-UKF | 0.4500 | 0.0250 | 99.00 | |
| 0.50 | EKF | 0.9000 | 0.0900 | 98.00 |
| FFRLS-UKF | 0.5000 | 0.0300 | 98.80 |
The convergence behavior is analyzed by examining the SOC estimation error over time. The FFRLS-UKF method demonstrates rapid convergence, with errors reducing to within 2% within a few sampling steps, even with initial SOC errors of up to 50%. This robustness is attributed to the adaptive parameter identification and the nonlinear handling capability of UKF. Additionally, the voltage prediction error of the identified model remains below 0.1 V, confirming the accuracy of the FFRLS-based parameter estimation. These findings highlight the effectiveness of our approach for real-time SOC estimation in energy storage cells, contributing to improved battery management and system reliability.
In conclusion, we have developed an online SOC estimation method for energy storage cells that combines FFRLS for parameter identification and UKF for state estimation. The Thevenin equivalent circuit model accurately captures the dynamic behavior of the energy storage cell, while the FFRLS algorithm ensures that model parameters are updated in real-time to account for variations. The UKF provides a robust framework for SOC estimation, handling nonlinearities without the approximations required in EKF. Experimental results validate the superiority of our method over traditional approaches, with higher accuracy, faster convergence, and better robustness to initial errors. This makes the FFRLS-UKF method a promising solution for enhancing the performance and safety of energy storage systems in applications such as grid storage, electric vehicles, and portable electronics. Future work could explore the integration of temperature effects and aging models to further improve the longevity and reliability of energy storage cells.