Accurate state of charge (SOC) estimation is crucial for battery management systems (BMS) in electric vehicles, ensuring safety, efficiency, and longevity of lithium-ion batteries. Traditional methods often struggle with dynamic operating conditions and algorithmic robustness. In this article, I present a joint online SOC estimation approach combining time-variable forgetting factor recursive least squares (TVFRLS) and singular value decomposition unscented Kalman filter (SVD-UKF) to enhance accuracy and resilience. The methodology addresses limitations in parameter identification and covariance matrix issues, validated under urban dynamometer driving schedule (UDDS) conditions. Through detailed mathematical formulations, simulations, and analyses, I demonstrate the superiority of this approach for lithium-ion battery applications.
The growing adoption of electric vehicles hinges on reliable energy storage, with lithium-ion batteries being the preferred choice due to their high energy density and cycle life. A BMS monitors key parameters like SOC, which indicates remaining capacity but cannot be measured directly. SOC estimation faces challenges from nonlinear battery dynamics, varying operating conditions, and computational constraints. Existing methods include ampere-hour integration, open-circuit voltage mapping, model-based filters like Kalman variants, and machine learning techniques. However, offline parameter identification often lacks adaptability, while standard unscented Kalman filter (UKF) algorithms may fail due to non-positive-definite covariance matrices. My approach integrates TVFRLS for adaptive model parameter identification and SVD-UKF for robust SOC estimation, tailored for lithium-ion batteries in real-world scenarios.
I structure this article as follows: First, I describe the second-order RC equivalent circuit model for lithium-ion batteries. Then, I detail the TVFRLS algorithm for online parameter identification. Next, I explain the SVD-UKF algorithm for SOC estimation. Following that, I present simulation experiments using UDDS data, with results analyzed via tables and formulas. Finally, I conclude with insights and future directions. Throughout, I emphasize the importance of lithium-ion battery performance and repeatedly reference this technology to underscore its relevance.
Equivalent Circuit Model for Lithium-Ion Batteries
To represent the electrochemical behavior of a lithium-ion battery, I use a second-order RC equivalent circuit model, balancing complexity and accuracy. This model includes an open-circuit voltage source, ohmic resistance, and two RC networks capturing polarization effects. The circuit dynamics are governed by:
$$U_L = U_{OC}(SOC) – R_0 I_k – U_1 – U_2$$
where \(U_L\) is terminal voltage, \(U_{OC}(SOC)\) is open-circuit voltage as a function of SOC, \(R_0\) is ohmic resistance, \(I_k\) is current, and \(U_1\) and \(U_2\) are voltages across RC networks with resistances \(R_1, R_2\) and capacitances \(C_1, C_2\). The state-space representation is discretized for digital implementation. Defining state vector \(x_k = [SOC_k, U_{1,k}, U_{2,k}]^T\), input \(u_k = I_k\), and output \(y_k = U_{L,k}\), the discrete-time equations are:
$$x_{k} = A_k x_{k-1} + B_k u_k + w_k$$
$$y_k = U_{OC}(SOC_k) – R_0 u_k – U_{1,k} – U_{2,k} + v_k$$
with system matrix \(A_k = \text{diag}(1, e^{-\Delta t / (R_1 C_1)}, e^{-\Delta t / (R_2 C_2)})\), input matrix \(B_k = [-\Delta t / Q_n, R_1(1 – e^{-\Delta t / (R_1 C_1)}), R_2(1 – e^{-\Delta t / (R_2 C_2)})]^T\), where \(Q_n\) is battery capacity, \(\Delta t\) is sampling time, and \(w_k, v_k\) are process and measurement noises. The \(U_{OC}(SOC)\) relationship is obtained experimentally for lithium-ion batteries, often fitted with polynomial or lookup tables. This model forms the basis for parameter identification and SOC estimation.
Online Parameter Identification with TVFRLS
Accurate model parameters are vital for SOC estimation. Traditional recursive least squares (RLS) with fixed forgetting factors may not adapt to lithium-ion battery dynamics under varying loads. I propose TVFRLS, which adjusts the forgetting factor based on estimation error statistics. The standard RLS update equations are:
$$e(k) = y(k) – \phi^T(k) \hat{\theta}(k-1)$$
$$\hat{\theta}(k) = \hat{\theta}(k-1) + K(k) e(k)$$
$$K(k) = \frac{P(k-1) \phi(k)}{\lambda + \phi^T(k) P(k-1) \phi(k)}$$
$$P(k) = \frac{1}{\lambda} \left[ P(k-1) – K(k) \phi^T(k) P(k-1) \right]$$
where \(e(k)\) is prediction error, \(\hat{\theta}(k)\) parameter vector, \(K(k)\) gain, \(P(k)\) covariance matrix, \(\phi(k)\) regressor vector, and \(\lambda\) forgetting factor. In TVFRLS, \(\lambda\) varies dynamically using a data window approach:
$$H(k) = -\lambda(k-1) \frac{\sum_{i=k-M+1}^{k} e_i e_i^T}{M}$$
$$\lambda(k) = \lambda_{\text{min}} + (\lambda_{\text{max}} – \lambda_{\text{min}}) \cdot 2^{H(k)}$$
with window size \(M=2\), error coefficient \(H(k)\), and bounds \(\lambda_{\text{min}}=0.95, \lambda_{\text{max}}=1.00\). This enhances tracking of lithium-ion battery parameter variations, such as \(R_0, R_1, R_2, C_1, C_2\), identified from UDDS data. The benefits include reduced bias and improved convergence, critical for lithium-ion battery management in EVs.
SVD-UKF Algorithm for SOC Estimation
For SOC estimation, I employ SVD-UKF, which combines unscented transformation with singular value decomposition to ensure numerical stability. Standard UKF uses Cholesky decomposition for covariance matrix square roots, but this can fail if the matrix becomes non-positive-definite due to noise or rounding errors in lithium-ion battery applications. SVD provides a robust alternative. Consider a nonlinear system:
$$x_k = f(x_{k-1}, u_k) + w_k$$
$$y_k = g(x_k, u_k) + v_k$$
with state \(x_k\), input \(u_k\), output \(y_k\), and noises \(w_k \sim \mathcal{N}(0, Q), v_k \sim \mathcal{N}(0, R)\). The SVD-UKF steps are:
- Initialization: Set initial state estimate \(\hat{x}_0\) and covariance \(P_0\).
- Weight Calculation: For \(n\) state dimensions, compute sigma point weights:
$$\lambda_f = \alpha^2 (n + \kappa) – n$$
$$\omega_m^{(0)} = \frac{\lambda_f}{n + \lambda_f}, \quad \omega_c^{(0)} = \frac{\lambda_f}{n + \lambda_f} + 1 – \alpha^2 + \beta$$
$$\omega_m^{(i)} = \omega_c^{(i)} = \frac{1}{2(n + \lambda_f)}, \quad i=1,\dots,2n$$
with scaling parameters \(\alpha=10^{-3}\), \(\kappa=0\), \(\beta=2\). - Sigma Point Generation: At time \(k-1\), perform SVD on \(P_{k-1} = U_{k-1} S_{k-1} V_{k-1}^T\). Generate sigma points:
$$\mathcal{X}_{0,k-1} = \hat{x}_{k-1}$$
$$\mathcal{X}_{i,k-1} = \hat{x}_{k-1} + \rho U_{k-1} \sqrt{S_{k-1}}, \quad i=1,\dots,n$$
$$\mathcal{X}_{i+n,k-1} = \hat{x}_{k-1} – \rho U_{k-1} \sqrt{S_{k-1}}, \quad i=1,\dots,n$$
where \(\rho = \sqrt{n + \lambda_f}\). - Time Update: Propagate sigma points:
$$\mathcal{X}_{i,k|k-1} = f(\mathcal{X}_{i,k-1}, u_k)$$
$$\hat{x}_{k|k-1} = \sum_{i=0}^{2n} \omega_m^{(i)} \mathcal{X}_{i,k|k-1}$$
$$P_{k|k-1} = \sum_{i=0}^{2n} \omega_c^{(i)} (\mathcal{X}_{i,k|k-1} – \hat{x}_{k|k-1})(\mathcal{X}_{i,k|k-1} – \hat{x}_{k|k-1})^T + Q$$ - Measurement Update: Update sigma points and compute output:
$$\mathcal{Y}_{i,k|k-1} = g(\mathcal{X}_{i,k|k-1}, u_k)$$
$$\hat{y}_{k|k-1} = \sum_{i=0}^{2n} \omega_m^{(i)} \mathcal{Y}_{i,k|k-1}$$
$$P_{yy,k} = \sum_{i=0}^{2n} \omega_c^{(i)} (\mathcal{Y}_{i,k|k-1} – \hat{y}_{k|k-1})(\mathcal{Y}_{i,k|k-1} – \hat{y}_{k|k-1})^T + R$$
$$P_{xy,k} = \sum_{i=0}^{2n} \omega_c^{(i)} (\mathcal{X}_{i,k|k-1} – \hat{x}_{k|k-1})(\mathcal{Y}_{i,k|k-1} – \hat{y}_{k|k-1})^T$$ - Estimation: Compute Kalman gain and update:
$$K_k = P_{xy,k} P_{yy,k}^{-1}$$
$$\hat{x}_k = \hat{x}_{k|k-1} + K_k (y_k – \hat{y}_{k|k-1})$$
$$P_k = P_{k|k-1} – K_k P_{yy,k} K_k^T$$
This algorithm ensures covariance matrices remain positive-definite via SVD, crucial for lithium-ion battery SOC estimation under harsh conditions. The joint TVFRLS-SVD-UKF approach adapts model parameters online and robustly estimates SOC.
Simulation Experiments and Analysis
I validate the proposed method using data from a lithium-ion battery (INR18650-30Q) tested under UDDS, mimicking real-world driving. The setup includes a battery cycler, thermal chamber, and data acquisition system. Current and voltage profiles are sampled at 1 Hz. TVFRLS identifies model parameters, while SVD-UKF estimates SOC. For comparison, I also implement extended Kalman filter (EKF), standard UKF, and SVD-UKF alone. Key metrics are absolute estimation error (AEE), mean absolute error (MAE), and root mean square error (RMSE).
The parameter identification results show TVFRLS achieves lower voltage errors than offline methods. For instance, the mean voltage error is 0.011 V, indicating better adaptation for lithium-ion battery dynamics. SOC estimation results under UDDS are summarized in Table 1, with formulas for error metrics:
$$AEE = \max |SOC_{\text{true}} – SOC_{\text{est}}|$$
$$MAE = \frac{1}{N} \sum_{k=1}^{N} |SOC_{\text{true},k} – SOC_{\text{est},k}|$$
$$RMSE = \sqrt{\frac{1}{N} \sum_{k=1}^{N} (SOC_{\text{true},k} – SOC_{\text{est},k})^2}$$
| Algorithm | AEE (%) | MAE (%) | RMSE (%) |
|---|---|---|---|
| TVFRLS-SVD-UKF (Proposed) | 1.31 | 0.56 | 0.75 |
| SVD-UKF Only | 2.16 | 1.02 | 1.28 |
| Standard UKF | 3.24 | 1.40 | 1.56 |
| EKF | 6.20 | 1.95 | 2.27 |
The proposed TVFRLS-SVD-UKF reduces MAE by 60.0% and RMSE by 51.9% compared to standard UKF, highlighting its accuracy for lithium-ion battery SOC. Robustness tests with erroneous initial SOC values (0.8, 0.6, 0.4, 0.2) show convergence within 100 seconds, whereas UKF fails with non-positive-definite covariance. This underscores the reliability of SVD in lithium-ion battery management systems.
Further analysis involves sensitivity studies on noise statistics and temperature effects. For lithium-ion batteries, temperature variations impact parameters; future work could integrate thermal models. The joint algorithm’s computational complexity is moderate, suitable for embedded BMS implementation.
Discussion and Implications
The integration of TVFRLS and SVD-UKF addresses key challenges in lithium-ion battery SOC estimation. TVFRLS adapts to changing battery parameters, such as internal resistance and capacitance, which degrade over time in lithium-ion batteries. SVD-UKF prevents algorithmic divergence, ensuring continuous operation. This is vital for electric vehicles where safety and range prediction depend on accurate SOC. The methodology can be extended to other battery chemistries, but lithium-ion batteries remain the focus due to their dominance in EVs.
Practical considerations include calibration of forgetting factor bounds and SVD tolerance. For lithium-ion batteries, experimental data from diverse cycles (e.g., highway, city driving) can refine the algorithm. Additionally, combining with health estimation could enhance BMS functionality. The proposed approach demonstrates that adaptive filtering and robust linear algebra can significantly improve lithium-ion battery performance monitoring.
Conclusion
In this article, I presented a joint online SOC estimation method for lithium-ion batteries using TVFRLS and SVD-UKF. The TVFRLS algorithm dynamically adjusts the forgetting factor for precise parameter identification, while SVD-UKF ensures numerical stability in SOC estimation. Simulations under UDDS conditions show superior accuracy and robustness compared to traditional EKF and UKF. The proposed method achieves an MAE of 0.56% and RMSE of 0.75%, with strong convergence under initial errors. Future work will explore temperature adaptation and real-time implementation for lithium-ion battery packs. This research contributes to advancing BMS technology for electric vehicles, emphasizing the critical role of lithium-ion batteries in sustainable transportation.
Overall, the TVFRLS-SVD-UKF framework offers a reliable solution for SOC estimation, addressing nonlinearities and uncertainties inherent in lithium-ion batteries. By leveraging adaptive control and matrix decomposition techniques, it paves the way for more intelligent and resilient energy management systems.