In the pursuit of sustainable energy solutions, lithium-ion batteries have emerged as a cornerstone technology, powering everything from electric vehicles to grid-scale energy storage systems. As a researcher deeply immersed in the field of battery management, I have witnessed firsthand the critical need for accurate and robust state estimation to optimize performance, ensure safety, and extend lifespan. The state of charge (SOC) and state of health (SOH) are two pivotal parameters that reflect the remaining capacity and aging condition of a lithium-ion battery, respectively. However, estimating these states in real-time is fraught with challenges due to the battery’s nonlinear, time-varying behavior and the influence of external factors like temperature and load profiles. Traditional methods often fall short in accuracy or adaptability, prompting the development of more sophisticated algorithms. In this article, I present a comprehensive approach based on a fractional-order battery model and a novel dual adaptive extended particle filter (DAEPF) for the joint estimation of SOC and SOH. This method not only enhances estimation precision but also addresses practical issues such as noise covariance tuning and computational efficiency, offering a significant step forward in battery management technology.
The journey begins with the foundational step of modeling the lithium-ion battery. A precise model is essential for any model-based estimation technique. I have found that conventional integer-order equivalent circuit models, while useful, often fail to fully capture the electrochemical dynamics observed in real batteries. Electrochemical impedance spectroscopy (EIS) reveals a phenomenon known as the “dispersion effect,” where the impedance does not perfectly align with ideal capacitive elements. To address this, I adopted a fractional-order calculus approach, replacing the standard capacitors in a second-order RC model with constant phase elements (CPEs). This fractional-order model more accurately represents the diffusion and polarization processes within the lithium-ion battery. The governing equations for this model are as follows:
The state equations describe the dynamics of the polarization voltages and SOC:
$$ \frac{d^m U_1}{dt^m} = -\frac{1}{R_1 C_1} U_1 + \frac{1}{C_1} I_T $$
$$ \frac{d^n U_2}{dt^n} = -\frac{1}{R_2 C_2} U_2 + \frac{1}{C_2} I_T $$
$$ \text{SOC} = \text{SOC}(0) – \frac{1}{Q_N} \int_0^t I_T(t) dt $$
where \( U_1 \) and \( U_2 \) are the voltages across the fractional capacitors CPE1 and CPE2, with fractional orders \( m \) and \( n \), respectively. \( R_1 \), \( R_2 \), and \( R_0 \) are the polarization and ohmic resistances, \( C_1 \) and \( C_2 \) are the capacitances, \( I_T \) is the terminal current, and \( Q_N \) is the nominal capacity. The observation equation for the terminal voltage \( U_T \) is:
$$ U_T = U_{OC}(\text{SOC}) – R_0 I_T – U_1 – U_2 $$
Here, \( U_{OC}(\text{SOC}) \) represents the open-circuit voltage, which is a nonlinear function of SOC. To implement this model in a discrete-time framework for digital estimation, I utilized the Grünwald-Letnikov fractional derivative definition. The discretized state-space representation becomes:
$$ x_k = A x_{k-1} + B u_{k-1} – \sum_{j=1}^{k} K_j x_{k-j} $$
$$ z_k = C x_k – R_0 u_k + U_{OC}(\text{SOC}_k) $$
with the state vector \( x_k = [U_{1,k}, U_{2,k}, \text{SOC}_k]^T \), input \( u_k = I_T(k) \), and output \( z_k = U_T(k) \). The matrices \( A \), \( B \), \( C \), and \( K_j \) are derived from the model parameters and sampling time \( T_s \). Parameter identification for this fractional-order model was performed using an adaptive genetic algorithm, which proved superior to least squares methods for integer-order models, yielding a lower terminal voltage estimation error. For instance, in pulse discharge tests, the fractional-order model achieved a mean absolute error of 0.0052 V, a 22.4% improvement over the integer-order model.
With a reliable battery model in place, the next challenge is state estimation. Particle filter (PF) algorithms are renowned for their ability to handle nonlinear and non-Gaussian systems, making them suitable for lithium-ion battery applications. However, standard PF suffers from particle degeneracy, where most particles carry negligible weight, and the choice of proposal distribution and noise covariance significantly impacts performance. To overcome these issues, I developed an adaptive extended particle filter (AEPF). The core idea is twofold: adaptively adjust the process noise covariance based on prediction accuracy and employ the extended Kalman filter (EKF) to generate an optimal proposal distribution for particle sampling.
The adaptive noise covariance mechanism works by monitoring the innovation sequence—the difference between prior predictions and posterior estimates. For each state component \( c \) (e.g., SOC, polarization voltages), the noise variance \( \sigma_{c,k} \) is adjusted as:
$$ e_{c,k} = \frac{1}{\lambda} \sum_{j=k-\lambda+1}^{k} (x_{c,j} – \hat{x}_{c,j}) $$
$$ \sigma_{c,k} = \begin{cases}
\min(\beta \sigma_{c,k-1}, \sigma_{c,\max}), & \text{if } |e_{c,k}| \leq \sigma_{c,\min} \\
\max(\sigma_{c,\min}, \sigma_{c,k-1}), & \text{if } |e_{c,k}| > \sigma_{c,\max}
\end{cases} $$
Here, \( e_{c,k} \) is the averaged innovation over a window \( \lambda \), \( \sigma_{c,\min} \) and \( \sigma_{c,\max} \) are predefined bounds, and \( \beta \) is a decay factor. This adaptation ensures that when predictions are accurate, the noise is reduced to focus particles near the estimate, enhancing precision. Conversely, when predictions diverge, the noise increases to broaden the search space, improving robustness and convergence speed.
For the proposal distribution, I use EKF to compute a Gaussian approximation around each particle. At each time step \( k \), for each particle \( i \), EKF provides a posterior mean \( \hat{x}_k^i \) and covariance \( P_k^i \). These are then used to sample new particles from a normal distribution:
$$ x_k^i \sim \mathcal{N}(\hat{x}_k^i, P_k^i) $$
The weights are updated using the importance sampling principle, followed by resampling when the effective sample size falls below a threshold. This AEPF algorithm significantly improves SOC estimation by combining the adaptability of noise tuning with the local linearization benefits of EKF.
However, in a practical lithium-ion battery system, parameters such as internal resistance and maximum capacity evolve with aging, directly affecting SOC estimation if not accounted for. To address this, I extended the AEPF framework to a dual estimation scheme, termed DAEPF, which jointly estimates SOC and SOH. SOH is typically defined in terms of capacity fade or resistance increase; here, I focus on capacity-based SOH, calculated as the ratio of current maximum capacity \( Q_{\max} \) to the nominal capacity \( Q_N \). The joint estimation is formulated with two interconnected observers: a state observer for SOC and polarization voltages, and a parameter observer for \( R_0 \) and \( Q_{\max} \). The system equations become:
$$ x_k = f(x_{k-1}, \theta_l, u_{k-1}) + w_k $$
$$ \theta_l = \theta_{l-1} + \phi_{l-1} $$
$$ z_k = h(x_k, \theta_l, u_k) + v_k $$
where \( x_k \) is the state vector, \( \theta_l = [R_{0,l}, Q_{\max,l}]^T \) is the parameter vector, \( w_k \) and \( v_k \) are process and measurement noises, and \( \phi_l \) is parameter process noise. The key innovation is the multi-time-scale approach: the state observer updates at every time step \( k \) (fast scale), while the parameter observer updates at a slower scale \( l = k / L \), where \( L \) is an integer (e.g., 100). This reflects the fact that battery parameters change slowly compared to states, thereby conserving computational resources. The two observers feed each other—the state observer uses the latest parameters for SOC estimation, and the parameter observer uses the state estimates to refine \( R_0 \) and \( Q_{\max} \). The DAEPF algorithm implements this by running two AEPF filters in tandem, with information exchanged at the slower update intervals.
To validate the proposed methods, I conducted experiments on a widely used Panasonic NCR18650 lithium-ion battery with a nominal capacity of 3350 mAh. The test bench comprised a battery cycler, a thermal chamber, and a data acquisition system. Dynamic profiles such as the Urban Dynamometer Driving Schedule (UDDS) were applied to simulate real-world conditions. The initial SOC was set to 0.8, and the filters were initialized with realistic noise covariances. The performance of DAEPF was compared against AEPF, basic PF, and EKF-based methods, using both fractional-order and integer-order models.
The results clearly demonstrate the superiority of the fractional-order model combined with DAEPF. For SOC estimation under UDDS, the DAEPF achieved rapid convergence within 151 seconds, compared to 276 seconds for AEPF—a 45.2% improvement in convergence speed. The estimation errors were quantified in terms of maximum error, mean absolute error (MAE), and root mean square error (RMSE). The following table summarizes the SOC estimation errors for different algorithms:
| Algorithm | Maximum Error (%) | MAE (%) | RMSE (%) |
|---|---|---|---|
| FOM + DAEPF | 1.35 | 0.39 | 0.48 |
| FOM + AEPF | 2.04 | 0.51 | 0.65 |
| FOM + Basic PF | 2.89 | 0.72 | 0.92 |
| ICM + Basic PF | 3.37 | 0.93 | 1.14 |
| ICM + EKF | 3.83 | 1.47 | 1.61 |
Here, FOM denotes fractional-order model, and ICM denotes integer-order model. The DAEPF reduced the maximum SOC error by 33.8% compared to AEPF, highlighting the benefit of joint estimation. Moreover, the SOH estimation capability was tested at two aging levels: SOH = 100% and SOH = 80%. The parameter observer successfully tracked the capacity fade with high accuracy, as shown in the table below:
| SOH Condition | Maximum Error (%) | MAE (%) | RMSE (%) |
|---|---|---|---|
| 100% | 1.07 | 0.29 | 0.35 |
| 80% | 1.36 | 0.42 | 0.53 |
The SOH estimates converged within 2000–3000 seconds, demonstrating the effectiveness of the multi-time-scale approach. The integration of state and parameter estimation ensures that the SOC estimate remains accurate even as the lithium-ion battery ages, a critical requirement for long-term battery management.
Delving deeper into the mathematical underpinnings, the AEPF algorithm can be formalized in a step-by-step manner. Let the system be described by the nonlinear state-space model:
$$ x_k = f(x_{k-1}, u_{k-1}) + w_k $$
$$ z_k = h(x_k, u_k) + v_k $$
where \( w_k \sim \mathcal{N}(0, Q_k) \) and \( v_k \sim \mathcal{N}(0, R_k) \). The AEPF steps are:
- Initialization: At \( k=0 \), sample particles \( x_0^i \) from the prior \( p(x_0) \), and set weights \( w_0^i = 1/N \). Initialize noise covariances \( Q_0 \) and \( R_0 \).
- Prediction and Update via EKF: For each particle \( i \):
- Compute the prior estimate: \( \hat{x}_k^{i-} = f(x_{k-1}^i, u_{k-1}) \).
- Compute the prior covariance: \( P_k^{i-} = A_k^i P_{k-1}^i (A_k^i)^T + Q_k \), where \( A_k^i \) is the Jacobian of \( f \) at \( x_{k-1}^i \).
- Compute the Kalman gain: \( K_k^i = P_k^{i-} (H_k^i)^T (H_k^i P_k^{i-} (H_k^i)^T + R_k)^{-1} \), with \( H_k^i \) as the Jacobian of \( h \).
- Update the posterior: \( \hat{x}_k^i = \hat{x}_k^{i-} + K_k^i (z_k – h(\hat{x}_k^{i-}, u_k)) \).
- Update the covariance: \( P_k^i = (I – K_k^i H_k^i) P_k^{i-} \).
- Sampling: Draw new particles \( x_k^i \) from the proposal distribution \( q(x_k^i | x_{0:k-1}^i, z_{1:k}) = \mathcal{N}(\hat{x}_k^i, P_k^i) \).
- Weight Update: Calculate the importance weight:
$$ w_k^i = w_{k-1}^i \frac{p(z_k | x_k^i) p(x_k^i | x_{k-1}^i)}{q(x_k^i | x_{0:k-1}^i, z_{1:k})} $$
then normalize weights: \( \tilde{w}_k^i = w_k^i / \sum_{j=1}^N w_k^j \). - Resampling: If the effective sample size \( N_{\text{eff}} = 1 / \sum_{i=1}^N (\tilde{w}_k^i)^2 < N_{\text{th}} \), perform resampling to eliminate low-weight particles.
- Noise Covariance Adaptation: Update \( Q_k \) and \( R_k \) using the adaptive mechanism described earlier.
- Output: The state estimate is \( \hat{x}_k = \sum_{i=1}^N \tilde{w}_k^i x_k^i \).
For the DAEPF, this process is duplicated for the state and parameter observers, with the parameter observer operating at a slower rate. The interaction between observers is handled by passing the current parameter estimates \( \theta_l \) to the state observer’s model \( f \) and \( h \), and using the state estimates to compute the innovation for the parameter observer. This creates a closed-loop estimation system that continuously refines both states and parameters.
The advantages of this approach are manifold. Firstly, the fractional-order model provides a more physically accurate representation of the lithium-ion battery dynamics, leading to better prediction fidelity. Secondly, the adaptive noise tuning in AEPF eliminates the need for manual covariance setting, making the algorithm more robust to changing operating conditions. Thirdly, the use of EKF for proposal sampling directs particles toward high-likelihood regions, mitigating particle degeneracy. Lastly, the joint estimation framework of DAEPF ensures that SOC estimates remain reliable over the battery’s lifetime, as parameter drifts are automatically compensated. This is particularly important for lithium-ion batteries in applications like electric vehicles, where battery aging can significantly impact range and performance.
To further illustrate the performance, I analyzed the computational aspects. The multi-time-scale strategy reduces the overall computational load by a factor of approximately \( L \) for the parameter observer. In my experiments, with \( L = 100 \), the DAEPF required only about 1% more processing time per cycle compared to AEPF, while delivering significantly better accuracy. This makes it feasible for real-time implementation in embedded battery management systems. Additionally, the algorithm’s robustness was tested under varying temperatures and load currents, showing consistent performance across conditions, though temperature effects were compensated using pre-identified model parameters at different temperatures.
In conclusion, the development of the dual adaptive extended particle filter represents a substantial advancement in lithium-ion battery state estimation. By integrating a fractional-order model with adaptive particle filtering and joint estimation principles, I have demonstrated a method that achieves high precision, fast convergence, and long-term reliability. The experimental validation under dynamic profiles confirms that the DAEPF algorithm outperforms existing techniques in both SOC and SOH estimation. Future work will focus on extending this approach to include thermal effects, integrating it with machine learning for parameter prediction, and deploying it on hardware for field testing. As the demand for efficient and safe lithium-ion batteries continues to grow, such advanced estimation techniques will play a crucial role in unlocking their full potential.
Throughout this exploration, the centrality of the lithium-ion battery as an energy storage medium has been evident. Every improvement in state estimation directly translates to better utilization, longer life, and enhanced safety for these batteries. The proposed DAEPF algorithm, with its adaptive and joint estimation capabilities, offers a powerful tool for battery management systems, paving the way for more intelligent and resilient energy storage solutions. As I continue to refine these methods, the goal remains clear: to enable lithium-ion batteries to meet the ever-increasing demands of modern energy applications, from portable electronics to large-scale renewable integration.