In the context of global energy challenges, electric vehicles have emerged as a pivotal solution, with lithium-ion batteries serving as their primary power source due to high energy density and extended cycle life. However, without rigorous management, the performance of lithium-ion batteries can degrade significantly, posing safety risks. An efficient Battery Management System (BMS) is crucial for reliable operation, where accurate estimation of the State of Charge (SOC) and State of Health (SOH) is particularly vital. SOC represents the ratio of remaining capacity to total available capacity, while SOH indicates the ratio of maximum usable capacity to nominal capacity. Accurate estimation of these states ensures safe and optimal utilization of lithium-ion batteries in applications like electric vehicles and energy storage systems.
Traditional methods for SOC estimation, such as Coulomb counting and open-circuit voltage techniques, often suffer from limitations like dependency on initial conditions and lack of robustness under dynamic operating conditions. Model-based approaches, including Extended Kalman Filter (EKF) and Particle Filter (PF), have gained traction for their ability to handle nonlinearities. Yet, EKF may struggle with complex Jacobian calculations, and PF can face issues like particle degeneracy and high computational cost. To address these, adaptive mechanisms have been introduced to enhance filtering algorithms. In this paper, we propose a collaborative estimation method for SOC and SOH of lithium-ion batteries based on a Dual Adaptive Particle Filter (DAPF). This approach leverages an equivalent circuit model and incorporates adaptive noise adjustment to improve accuracy and robustness. We present a comprehensive analysis, including model derivation, algorithm design, and simulation validation, demonstrating the superiority of DAPF over conventional methods like EKF, Adaptive EKF (AEKF), and standard PF.
The foundation of our estimation methodology lies in establishing a precise model for the lithium-ion battery. We adopt a second-order RC equivalent circuit model, which balances complexity and accuracy effectively. This model comprises a voltage source \(U_{oc}\) representing the open-circuit voltage, an ohmic internal resistance \(R_0\), and two RC networks (\(R_1C_1\) and \(R_2C_2\)) that capture polarization effects. The terminal voltage \(U_b\) and current \(I_b\) are key measurable variables. The dynamic behavior of this lithium-ion battery model is described by the following equations:
\[
I_b(t) = \frac{U_1}{R_1} + C_1 \frac{dU_1}{dt}
\]
\[
I_b(t) = \frac{U_2}{R_2} + C_2 \frac{dU_2}{dt}
\]
\[
U_{oc} = U_1 + U_2 + R_0 I_b + U_b
\]
For digital implementation in BMS, we discretize these equations. Let \(\Delta t\) be the sampling interval, \(k\) denote the time step, and \(\tau_1 = R_1C_1\), \(\tau_2 = R_2C_2\) be the time constants. The discrete form is:
\[
U_1(k+1) = U_1(k) \cdot e^{-\Delta t / \tau_1} + R_1 I(k) (1 – e^{-\Delta t / \tau_1})
\]
\[
U_2(k+1) = U_2(k) \cdot e^{-\Delta t / \tau_2} + R_2 I(k) (1 – e^{-\Delta t / \tau_2})
\]
\[
U_b(k+1) = U_{oc}(SOC(k)) – U_1(k) – U_2(k) – R_0 I(k)
\]
The SOC of the lithium-ion battery is estimated using the Coulomb counting method, which integrates current over time:
\[
SOC(k+1) = SOC(k) – \frac{\eta I(k) \Delta t}{Q}
\]
where \(\eta\) is the charge/discharge efficiency, and \(Q\) is the battery capacity. Combining the state variables, we define the state vector \(x_k = [SOC(k), U_1(k), U_2(k)]^T\). The state-space representation for the lithium-ion battery model is:
\[
x_{k+1} =
\begin{bmatrix}
1 & 0 & 0 \\
0 & e^{-\Delta t / \tau_1} & 0 \\
0 & 0 & e^{-\Delta t / \tau_2}
\end{bmatrix}
x_k +
\begin{bmatrix}
-\eta \Delta t / Q \\
R_1 (1 – e^{-\Delta t / \tau_1}) \\
R_2 (1 – e^{-\Delta t / \tau_2})
\end{bmatrix}
I(k) + \omega_k
\]
\[
y_k = U_b(k) = \frac{\partial U_{oc}(SOC(k))}{\partial SOC(k)} \cdot SOC(k) – U_1(k) – U_2(k) – R_0 I(k) + v_k
\]
Here, \(\omega_k\) and \(v_k\) represent process and measurement noise, respectively. The open-circuit voltage \(U_{oc}\) is a nonlinear function of SOC, often obtained experimentally. For the lithium-ion battery used in our study, a polynomial fitting yields:
\[
U_{oc}(SOC) = -252.7629 \times SOC^8 + 1137.2 \times SOC^7 – 2117.6 \times SOC^6 + 2105.8 \times SOC^5 – 1214.1 \times SOC^4 + 411.3904 \times SOC^3 – 79.8901 \times SOC^2 + 8.4313 \times SOC + 2.8689
\]
This relationship is critical for accurate SOC estimation in lithium-ion batteries, as it captures the nonlinear voltage plateau characteristics.
To estimate SOC and SOH simultaneously, we propose the Dual Adaptive Particle Filter (DAPF) algorithm. First, we develop an Adaptive Particle Filter (APF) that enhances the standard PF by incorporating an adaptive noise adjustment mechanism. The APF addresses the limitations of fixed noise covariances in PF, which can lead to slow convergence or large errors. Consider a general nonlinear state-space model for the lithium-ion battery:
\[
x_{k+1} = f(x_k, u_k) + \omega_k, \quad y_k = h(x_k, u_k) + v_k
\]
where \(f\) and \(h\) are nonlinear functions, \(u_k\) is the input (e.g., current), and \(y_k\) is the output (e.g., terminal voltage). The APF algorithm proceeds as follows with \(N\) particles:
1. Initialization: For \(i = 1, 2, \ldots, N\), sample initial particles \(x_0^i\) from the prior distribution \(p(x_0)\), and set initial weights \(q_0^i = 1/N\). Define noise variance bounds \(\hat{\delta}_{\text{max}}\) and \(\hat{\delta}_{\text{min}}\).
2. Prediction: At each time step \(k\), propagate particles using the state equation:
\[
x_{k+1}^i = f(x_k^i, u_k) + \omega_k^i
\]
where \(\omega_k^i\) is drawn from a distribution with adaptive variance.
3. Update: Compute the measurement residual for each particle:
\[
e_{k+1}^i = y_{k+1} – h(x_{k+1}^i)
\]
Update the weights based on the likelihood:
\[
q_{k+1}^i = \frac{1}{\sqrt{2\pi \upsilon_k}} \exp\left(-\frac{(e_{k+1}^i)^2}{2\upsilon_k}\right)
\]
Normalize weights: \(q_{k+1}^i = q_{k+1}^i / \sum_{j=1}^N q_{k+1}^j\).
4. Resampling: Perform resampling to avoid degeneracy, generating a new set of particles \(\hat{x}_{k+1}^i\).
5. Noise Variance Adaptation: Estimate the noise variance using a sliding window approach. For each state component \(m\), compute the adjusted error:
\[
e_{m,k+1} = \lambda^{-1} \sum_{i=k+1}^{k+1} | e_m^i |
\]
where \(\lambda\) is a forgetting factor. Then, predict the next noise variance:
\[
\hat{\delta}_{m,k+1} =
\begin{cases}
\min(e_{m,k+1}, \hat{\delta}_{m,\text{max}}), & \text{if } e_{m,k+1} > \hat{\delta}_{m,k} \\
\min(\beta \hat{\delta}_{m,k}, \hat{\delta}_{m,\text{min}}), & \text{if } e_{m,k+1} \leq \hat{\delta}_{m,k}
\end{cases}
\]
where \(\beta\) is a decay factor. This adaptation allows the filter to dynamically adjust to changing conditions in the lithium-ion battery.
The DAPF algorithm extends APF by employing two interconnected APF observers: one for state estimation (APF1) and another for parameter estimation (APF2). APF1 estimates the state vector \(x_k = [SOC, U_1, U_2]^T\) and the terminal voltage, while APF2 estimates battery parameters such as capacity \(Q\) and internal resistance \(R_0\), which relate to SOH. SOH is defined as:
\[
SOH = \frac{Q_{\text{current}}}{Q_{\text{nominal}}}
\]
where \(Q_{\text{current}}\) is the estimated maximum capacity, and \(Q_{\text{nominal}}\) is the nominal capacity. The two observers exchange information: APF1 outputs estimated SOC to APF2, which uses it to update parameters, and APF2 feeds back updated parameters to APF1 for improved state prediction. This collaborative scheme enhances accuracy, as SOC and SOH are interdependent in lithium-ion batteries. Moreover, to optimize computational efficiency, we employ multi-time-scale updates. APF1 operates at a faster time scale (every time step \(k\)) to track rapid SOC changes, while APF2 updates at a slower scale (every \(K = nk\) steps, where \(n\) is an integer) since parameters like capacity degrade gradually over cycles. The DAPF flowchart illustrates this interaction, ensuring robust joint estimation for lithium-ion battery management.
To validate the DAPF algorithm, we conducted simulations using MATLAB, with a lithium-ion battery test system. The battery under test is a lithium iron phosphate (LiFePO4) cell with a nominal capacity of 36 Ah, rated voltage of 3.2 V, charge cutoff voltage of 3.7 V, and discharge cutoff voltage of 2.5 V. We applied Dynamic Stress Test (DST) profiles to emulate real-world operating conditions, including dynamic charge-discharge cycles and rest periods. The DST involves recording initial states, performing dynamic current profiles until SOC reaches 0, and allowing rest intervals for voltage stabilization. This comprehensive testing assesses the lithium-ion battery’s performance under varied stresses.
We compared DAPF against established algorithms: EKF, AEKF, and standard PF. Key metrics include Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) for terminal voltage prediction, SOC estimation, and SOH estimation. The open-circuit voltage versus SOC curve was fitted using experimental data, as shown in the polynomial equation above, which is essential for model accuracy. Simulations were run for two SOH levels: 1.0 (fresh battery) and 0.5 (aged battery), to evaluate algorithm robustness across different aging states of the lithium-ion battery.
The results for terminal voltage prediction are summarized in Table 1. Under SOH = 1.0, DAPF achieved the lowest errors, with RMSE of 0.03% and MAE of 0.009%, outperforming EKF (RMSE 0.12%, MAE 0.043%), AEKF (RMSE 0.11%, MAE 0.041%), and PF (RMSE 0.08%, MAE 0.014%). For SOH = 0.5, DAPF maintained superiority with RMSE of 0.10% and MAE of 0.04%, while other methods showed higher errors, such as EKF at RMSE 0.49% and MAE 0.31%. This demonstrates DAPF’s enhanced accuracy, especially as the lithium-ion battery ages.
| Algorithm | SOH = 1.0 RMSE (%) | SOH = 1.0 MAE (%) | SOH = 0.5 RMSE (%) | SOH = 0.5 MAE (%) |
|---|---|---|---|---|
| EKF | 0.12 | 0.043 | 0.49 | 0.31 |
| AEKF | 0.11 | 0.041 | 0.21 | 0.17 |
| PF | 0.08 | 0.014 | 0.18 | 0.11 |
| DAPF | 0.03 | 0.009 | 0.10 | 0.04 |
SOC estimation results further highlight DAPF’s efficacy. For SOH = 1.0, DAPF’s maximum SOC error was below 0.1%, compared to 1.0% for EKF, 0.5% for AEKF, and 0.9% for PF. Under SOH = 0.5, DAPF’s error was 0.7%, whereas EKF, AEKF, and PF had errors of 2.5%, 1.5%, and 0.9%, respectively. The adaptive mechanism in DAPF effectively compensates for model uncertainties and noise variations in lithium-ion batteries, leading to more stable and accurate SOC tracking across diverse conditions.
SOH estimation is critical for assessing the long-term health of lithium-ion batteries. Using the parameter estimation observer (APF2) in DAPF, we monitored SOH over DST cycles. As shown in Table 2, DAPF achieved rapid convergence to the true SOH values. For SOH = 1.0, the maximum estimation error was within 2%, and for SOH = 0.5, the error was within 1%. In contrast, other methods exhibited larger deviations, especially under aged conditions. The multi-time-scale update in DAPF allows parameter adjustments to reflect gradual degradation, ensuring reliable SOH monitoring without excessive computational load. This capability is vital for predicting the remaining useful life of lithium-ion batteries and planning maintenance.
| Algorithm | SOH = 1.0 Max Error (%) | SOH = 0.5 Max Error (%) | Convergence Speed |
|---|---|---|---|
| EKF | >5 | >10 | Slow |
| AEKF | 3-4 | 5-6 | Moderate |
| PF | 2-3 | 3-4 | Moderate |
| DAPF | <2 | <1 | Fast |
To delve deeper into the algorithm’s mechanics, we analyze the mathematical formulations. The adaptive noise variance in APF is derived from the prediction errors. Let the state vector for the lithium-ion battery be \(x = [SOC, U_1, U_2, Q, R_0]^T\) in the joint estimation framework. The DAPF algorithm minimizes the cost function \(J\) defined as the expectation of squared errors:
\[
J = \mathbb{E}[(x_k – \hat{x}_k)^T (x_k – \hat{x}_k)]
\]
where \(\hat{x}_k\) is the estimated state. The adaptive mechanism adjusts the process noise covariance matrix \(Q_k\) and measurement noise covariance matrix \(R_k\) online using the following relations:
\[
Q_k = \alpha Q_{k-1} + (1-\alpha) \Delta x_k \Delta x_k^T
\]
\[
R_k = \beta R_{k-1} + (1-\beta) (y_k – h(\hat{x}_k)) (y_k – h(\hat{x}_k))^T
\]
Here, \(\alpha\) and \(\beta\) are forgetting factors between 0 and 1, and \(\Delta x_k\) represents the state innovation. This adaptation enhances the filter’s responsiveness to changes in the lithium-ion battery dynamics, such as those caused by aging or temperature variations.
Furthermore, the particle weighting in DAPF incorporates the likelihood function based on the measurement model. For a lithium-ion battery, the measurement equation involves the terminal voltage, which is sensitive to SOC and internal parameters. The weight for particle \(i\) is computed as:
\[
w_k^i \propto \exp\left(-\frac{1}{2} (y_k – h(x_k^i))^T R_k^{-1} (y_k – h(x_k^i))\right)
\]
This ensures that particles aligning better with observed voltages receive higher weights, improving estimation accuracy. The resampling step then redistributes particles to focus on high-probability regions, mitigating sample impoverishment common in PF applications for lithium-ion batteries.
In our simulations, we also examined the impact of initial SOC uncertainty on DAPF performance. We set initial SOC errors up to 20% and observed that DAPF converged to within 2% error within a few time steps, thanks to the adaptive noise tuning and dual-observer structure. This robustness is crucial for practical BMS implementations, where initial conditions may be unknown. Additionally, we tested the algorithm under different current profiles, including constant current and federal urban driving schedules, to emulate varied usage of lithium-ion batteries. DAPF consistently maintained low errors, with SOC RMSE below 1% and SOH errors within 2% across scenarios, demonstrating its versatility.
The computational complexity of DAPF is higher than that of EKF due to particle-based sampling, but the multi-time-scale approach mitigates this. For a lithium-ion battery BMS with typical update rates (e.g., 1 Hz), DAPF runs efficiently on embedded platforms when particle counts are optimized (we used \(N=100\) particles in our tests). The adaptive mechanism reduces the need for manual tuning of noise covariances, which is a significant advantage in real-world deployments where lithium-ion battery characteristics evolve over time.
In conclusion, the Dual Adaptive Particle Filter (DAPF) algorithm proposed in this paper offers a robust solution for joint estimation of SOC and SOH in lithium-ion batteries. By integrating an equivalent circuit model with adaptive particle filtering and a dual-observer framework, DAPF achieves high accuracy and resilience to noise and aging effects. Simulation results under DST profiles show that DAPF outperforms conventional methods like EKF, AEKF, and PF in terms of terminal voltage prediction, SOC estimation, and SOH monitoring. For fresh lithium-ion batteries (SOH=1.0), DAPF keeps SOC errors below 0.1% and SOH errors within 2%; for aged batteries (SOH=0.5), it maintains SOC errors around 0.7% and SOH errors within 1%. These findings underscore DAPF’s potential for enhancing battery management systems, ensuring safer and more efficient operation of lithium-ion batteries in electric vehicles and energy storage applications. Future work will focus on experimental validation with real-time BMS hardware and extending the algorithm to estimate additional states like temperature and power capability for comprehensive lithium-ion battery management.