Accurate estimation of the State of Charge (SOC) in energy storage lithium battery systems is critical for ensuring operational efficiency, safety, and longevity. Among various lithium-ion batteries, lithium iron phosphate (LiFePO4) batteries are widely adopted in energy storage applications due to their superior thermal stability, long cycle life, and cost-effectiveness. However, the flat voltage plateau characteristic of LiFePO4 batteries during operation poses significant challenges for precise SOC estimation using conventional voltage-based methods. This limitation often leads to inaccuracies in battery management systems, potentially affecting the performance and reliability of energy storage lithium battery setups. To address this issue, this paper proposes a novel SOC estimation method that leverages expansion force signals, which exhibit a more distinguishable relationship with SOC compared to voltage signals. The integration of expansion force data with advanced computational techniques offers a promising solution to enhance SOC estimation accuracy for energy storage lithium battery systems.
The proposed method combines a Quantum Particle Swarm Optimization (QPSO) algorithm with a Least Squares Support Vector Machine (LSSVM) to model the nonlinear relationship between SOC, current, and expansion force in energy storage lithium battery. This QPSO-LSSVM model is then incorporated into an Adaptive Extended Kalman Filter (AEKF) framework, forming a closed-loop estimation structure that dynamically corrects SOC predictions based on real-time expansion force measurements. By fusing machine learning and adaptive filtering strategies, this approach mitigates the effects of model uncertainties and noise, resulting in robust and accurate SOC estimates for energy storage lithium battery applications. The following sections detail the theoretical foundations, experimental validation, and performance analysis of the QPSO-LSSVM-AEKF method, demonstrating its superiority over traditional techniques in handling the complexities of energy storage lithium battery systems.
Methodology: QPSO-LSSVM-AEKF Framework
The QPSO-LSSVM-AEKF framework is designed to address the nonlinearities and uncertainties inherent in energy storage lithium battery dynamics. The core idea involves using expansion force as an observable variable to infer SOC, as it provides a more sensitive indicator compared to voltage during the flat plateau region. The framework consists of three main components: QPSO for parameter optimization, LSSVM for nonlinear mapping, and AEKF for state estimation. Each component is elaborated below with mathematical formulations.
Quantum Particle Swarm Optimization (QPSO)
Traditional Particle Swarm Optimization (PSO) is prone to premature convergence and local optima traps when applied to high-dimensional problems. QPSO introduces quantum behavior to enhance global search capabilities, making it suitable for optimizing the hyperparameters of LSSVM in energy storage lithium battery modeling. In QPSO, particles are described by quantum mechanical principles, where their positions are updated based on a probability density function derived from a delta potential well model. The probability density function for a particle’s position is given by:
$$ |\psi(x, t)|^2 = \frac{1}{L} e^{-\frac{2|x – p|}{L}} $$
where $L$ is the characteristic length and $p$ is the potential well center, computed as a weighted average of the particle’s historical best position ($pBest_i$) and the global best position ($gBest$):
$$ p(t) = \phi \cdot pBest_i + (1 – \phi) \cdot gBest $$
Here, $\phi$ is a random weight uniformly distributed between 0 and 1. The new position of a particle is generated through Monte Carlo sampling:
$$ x_i(t+1) = p(t) \pm \frac{L_i(t)}{2} \ln\left(\frac{1}{u}\right) $$
where $u$ is a uniform random number in [0, 1], and $L_i(t)$ is the dynamic characteristic length defined as:
$$ L_i(t) = 2\beta \cdot |mBest(t) – x_i(t)| $$
The term $mBest(t)$ represents the mean best position of all particles:
$$ mBest(t) = \frac{1}{N} \sum_{i=1}^{N} pBest_i $$
and $\beta$ is the contraction-expansion coefficient that decreases linearly over iterations to balance exploration and exploitation:
$$ \beta(t) = \beta_0 – (\beta_0 – \beta_f) \cdot \frac{t}{T_{\text{max}}} $$
where $\beta_0$ and $\beta_f$ are initial and final values, typically set to 1.2 and 0.5, respectively, and $T_{\text{max}}$ is the maximum number of iterations. This quantum-inspired mechanism enables QPSO to efficiently search the parameter space for LSSVM, ensuring optimal model performance for energy storage lithium battery SOC estimation.
Least Squares Support Vector Machine (LSSVM)
LSSVM is a variant of Support Vector Machine (SVM) that simplifies the optimization problem by replacing inequality constraints with equality constraints, making it suitable for regression tasks in energy storage lithium battery modeling. The goal is to establish a mapping from input variables (current and SOC) to the output (expansion force) using a nonlinear function. The LSSVM model is expressed as:
$$ y_k = w^T \phi(x_k) + b $$
where $y_k$ is the output (expansion force), $x_k$ is the input vector [SOC, current], $w$ is the weight vector, $b$ is the bias term, and $\phi(\cdot)$ is a feature map that projects the input into a high-dimensional space. The optimization problem minimizes both the model complexity and prediction errors:
$$ \min_{w,b,e} \frac{1}{2} \|w\|^2 + \frac{\gamma}{2} \sum_{k=1}^{n} e_k^2 $$
subject to the constraints:
$$ y_k = w^T \phi(x_k) + b + e_k, \quad k = 1, \ldots, n $$
Here, $\gamma$ is the regularization parameter that controls the trade-off between model smoothness and fitting accuracy, and $e_k$ is the error variable. Applying Lagrange multipliers, the solution leads to a linear system where the kernel trick is employed to handle nonlinearities. The radial basis function (RBF) kernel is used due to its effectiveness in capturing complex patterns:
$$ K(x_i, x_j) = \exp\left(-\frac{\|x_i – x_j\|^2}{2\sigma^2}\right) $$
where $\sigma$ is the kernel width parameter. The resulting LSSVM model for expansion force prediction becomes:
$$ F_k = \sum_{k=1}^{\infty} \alpha_k K(\text{SOC}_k, i_k) + b $$
where $F_k$ is the predicted expansion force, $\alpha_k$ are Lagrange multipliers, and $K(\cdot)$ is the RBF kernel. The hyperparameters $\gamma$ and $\sigma$ are optimized using QPSO to minimize the prediction error on training data, ensuring the model accurately represents the behavior of energy storage lithium battery systems.
Adaptive Extended Kalman Filter (AEKF)
The AEKF algorithm integrates the QPSO-LSSVM model into a state estimation framework for SOC prediction in energy storage lithium battery. Unlike standard EKF, AEKF adapts the process and measurement noise covariances online using the Sage-Husa estimator, enhancing robustness against model inaccuracies and disturbances. The state equation, based on ampere-hour integration, is:
$$ x_k = x_{k-1} + \frac{\eta I_{k-1} \Delta t}{Q} + w_{k-1} $$
where $x_k$ is the SOC at time $k$, $\eta$ is the Coulombic efficiency (assumed 1 for simplicity), $I_{k-1}$ is the current, $\Delta t$ is the sampling interval, $Q$ is the battery capacity, and $w_{k-1}$ is the process noise. The observation equation, derived from the QPSO-LSSVM model, is:
$$ z_k = \sum_{k=1}^{\infty} \alpha_k K(x_k, I_k) + b + v_k $$
where $z_k$ is the measured expansion force and $v_k$ is the observation noise. The AEKF steps are as follows:
- Initialization: Set initial SOC estimate $\hat{x}_0$ and error covariance $P_0$.
- Prediction:
$$ \hat{x}_{k|k-1} = \hat{x}_{k-1} + \frac{\eta I_{k-1} \Delta t}{Q} $$
$$ P_{k|k-1} = F_{k-1} P_{k-1} F_{k-1}^T + Q_{k-1} $$
where $F_{k-1}$ is the Jacobian of the state function. - Observation Prediction:
$$ \hat{z}_{k|k-1} = \sum_{k=1}^{\infty} \alpha_k K(\hat{x}_{k|k-1}, I_k) + b $$ - Kalman Gain Calculation:
$$ K_k = P_{k|k-1} H_k^T (H_k P_{k|k-1} H_k^T + R_k)^{-1} $$
where $H_k$ is the Jacobian of the observation function. - Update:
$$ \hat{x}_k = \hat{x}_{k|k-1} + K_k (z_k – \hat{z}_{k|k-1}) $$
$$ P_k = (I – K_k H_k) P_{k|k-1} $$
The adaptive mechanism updates the noise covariances $Q_k$ and $R_k$ recursively based on residuals:
$$ Q_k = \frac{d_{k-1}}{d_k} Q_{k-1} + \frac{1}{d_k} (g_k g_k^T) $$
$$ R_k = \frac{d_{k-1}}{d_k} R_{k-1} + \frac{1}{d_k} (e_k e_k^T) $$
where $g_k$ is the state residual, $e_k$ is the observation residual, and $d_k$ is a forgetting factor (e.g., 0.95). This adaptability makes AEKF particularly suitable for energy storage lithium battery applications, where operating conditions and battery parameters may vary over time.
Experimental Setup and Data Analysis
To validate the QPSO-LSSVM-AEKF method, experiments were conducted on a 5Ah LiFePO4 energy storage lithium battery. The battery was subjected to Dynamic Stress Test (DST) and Federal Urban Driving Schedule (FUDS) cycles to simulate real-world operating conditions. A pressure sensor was installed at the battery bottom to measure expansion force, while current and voltage data were recorded simultaneously. The battery specifications are summarized in Table 1.
| Parameter | Value |
|---|---|
| Rated Capacity | 5 Ah |
| Nominal Voltage | 3.2 V |
| Charge/Discharge Cut-off Voltage | 3.65 V / 2.0 V |
| Operating Temperature | -20°C to 60°C |
The DST cycle data, including current, voltage, and expansion force, are shown in the figures below. The expansion force exhibits a non-monotonic relationship with SOC, initially decreasing due to anode delithiation, then increasing as cathode lithiation dominates, and finally decreasing again as saturation occurs. This behavior underscores the complexity of using expansion force for SOC estimation in energy storage lithium battery systems, necessitating advanced modeling techniques.
The DST data were used to train the QPSO-LSSVM model, with current and SOC as inputs and expansion force as output. The QPSO algorithm optimized the LSSVM hyperparameters $\gamma$ and $\sigma$ by minimizing the mean absolute error (MAE) on training data. The optimized values were $\gamma = 125.6$ and $\sigma = 0.85$. The model was then tested on FUDS data to evaluate its generalization capability. The prediction results, illustrated in Figure 1, show close agreement between measured and predicted expansion force, with an MAE of 0.2385% and root mean square error (RMSE) of 0.1766%, confirming the model’s accuracy for energy storage lithium battery applications.
SOC Estimation Results and Comparison
The trained QPSO-LSSVM model was embedded into the AEKF algorithm to estimate SOC under FUDS conditions. The performance was compared against traditional methods, including PSO-LSSVM-AEKF, QPSO-LSSVM-EKF, and voltage-based approaches. The SOC estimation results are summarized in Table 2, with errors calculated as MAE and RMSE.
| Method | MAE (%) | RMSE (%) |
|---|---|---|
| QPSO-LSSVM(F)-EKF | 9.4487 | 20.9211 |
| PSO-LSSVM(F)-AEKF | 0.7508 | 0.7762 |
| QPSO-LSSVM(V)-AEKF | 1.7020 | 1.7511 |
| QPSO-LSSVM(F)-AEKF | 0.5212 | 0.5542 |
| QPSO-LSSVM Data-Driven | 3.4327 | 2.2032 |
The QPSO-LSSVM-AEKF method achieved the lowest errors, with MAE = 0.5212% and RMSE = 0.5542%, demonstrating its superiority in handling the nonlinearities of energy storage lithium battery systems. In contrast, the voltage-based approach (QPSO-LSSVM(V)-AEKF) yielded higher errors due to the flat voltage plateau, while the non-adaptive EKF version suffered from noise amplification. The pure data-driven method, which directly maps current and expansion force to SOC without filtering, exhibited significant errors, highlighting the importance of the closed-loop AEKF structure for robust SOC estimation in energy storage lithium battery applications.
Further analysis of the SOC trajectories reveals that the QPSO-LSSVM-AEKF method maintains stability throughout the discharge cycle, with minimal deviation from the reference SOC. The adaptive noise tuning in AEKF effectively compensates for model inaccuracies and sensor noise, ensuring reliable performance even under dynamic operating conditions. These results validate the proposed method as a viable solution for accurate SOC estimation in energy storage lithium battery systems, particularly where voltage-based methods fall short.
Conclusion
This paper presented a novel SOC estimation method for energy storage lithium battery systems using the QPSO-LSSVM-AEKF framework and expansion force signals. The integration of QPSO-optimized LSSVM modeling with adaptive Kalman filtering addresses the challenges posed by the flat voltage plateau and nonlinear expansion force characteristics of LiFePO4 batteries. Experimental results on a 5Ah energy storage lithium battery under FUDS conditions confirmed the method’s accuracy, with SOC estimation errors below 0.56% RMSE. The comparison with conventional techniques demonstrated significant improvements in robustness and precision, making it suitable for real-world energy storage lithium battery management systems. Future work will focus on extending the method to account for battery aging effects and validating it under broader operational scenarios to further enhance its applicability in energy storage lithium battery applications.