In the realm of green energy storage, the lithium-ion battery stands out as one of the most promising devices due to its high energy density, long lifespan, and environmental friendliness. However, during the usage of lithium-ion batteries, aging occurs under various stressors, which can significantly reduce the maximum available capacity and pose safety hazards. Therefore, accurately assessing the State of Health (SOH) of lithium-ion batteries is crucial for ensuring reliability and longevity. In practical applications, such as customs inspections at ports, rapid and accurate SOH detection methods are essential for identifying aged or potentially hazardous lithium-ion batteries, ensuring compliance with regulations, and facilitating global green energy cooperation. Traditional methods, including model-based approaches and full-range charge-discharge tests, often face limitations in accuracy, efficiency, or environmental constraints. In contrast, data-driven methods leverage machine learning to directly map battery features to SOH without delving into complex internal mechanisms, offering a viable solution for online rapid detection.
This article presents an integrated approach combining an Improved Whale Optimization Algorithm with Support Vector Regression (IWOA-SVR) for online rapid detection of lithium-ion battery SOH. The methodology involves collecting charge-discharge data under different strategies, extracting key aging features, validating their correlation with SOH, and optimizing the SVR model parameters via IWOA to enhance accuracy and robustness. Two distinct datasets under constant-current constant-voltage (CC-CV) charging and constant-current charging conditions are used for validation, demonstrating the superiority of the proposed method in terms of stability and precision.
Feature Parameter Extraction and Correlation Analysis
For data-driven SOH estimation, selecting features that strongly correlate with battery degradation is paramount. From the charge-discharge voltage curves of lithium-ion batteries, four key features are extracted to capture aging trends. In the charging phase, as the lithium-ion battery ages, the time required for the voltage to rise between specific levels decreases. Thus, Feature F1 is defined as the time taken for the voltage to increase from 3.8 V to 4.1 V during charging. Additionally, the slope of the charging voltage curve, denoted as F2, reflects changes in battery behavior and is computed as:
$$ \text{F2} = \frac{4.2 – V_{\text{start}}}{\Delta T} $$
where \( V_{\text{start}} \) is the starting voltage during charging, and \( \Delta T \) is the time interval until the voltage reaches 4.2 V. In the discharge phase, the slope of the discharge voltage curve, F3, is calculated as:
$$ \text{F3} = \left| \frac{V_{\text{min}} – V_{\text{start}}}{\Delta T} \right| $$
where \( V_{\text{min}} \) and \( V_{\text{start}} \) are the minimum and starting voltages during discharge, respectively, and \( \Delta T \) is the time to reach the minimum voltage. Feature F4 represents the time for the voltage to drop from 4 V to 3.5 V during discharge, indicating capacity fade.
To quantify the relationship between these features and SOH, Pearson correlation analysis is employed. For two datasets X (feature values) and Y (SOH values), the Pearson coefficient is given by:
$$ P_{X,Y} = \frac{\text{cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{\sum_{i=1}^{n} (X_i – \bar{X})(Y_i – \bar{Y})}{\sqrt{\sum_{i=1}^{n} (X_i – \bar{X})^2 \sum_{i=1}^{n} (Y_i – \bar{Y})^2}} $$
where \( \text{cov}(X,Y) \) is the covariance, \( \sigma_X \) and \( \sigma_Y \) are standard deviations, \( \bar{X} \) and \( \bar{Y} \) are means, and n is the sample size. The results for multiple lithium-ion batteries across two datasets show absolute correlation coefficients exceeding 0.95, confirming strong linear relationships, as summarized in Table 1.
| Battery | F1 | F2 | F3 | F4 |
|---|---|---|---|---|
| B5 | 0.997 | -0.983 | -0.996 | 0.998 |
| B6 | 0.980 | -0.975 | -0.987 | 0.968 |
| B7 | 0.988 | -0.986 | -0.997 | 0.998 |
| B18 | 0.988 | -0.962 | -0.996 | 0.998 |
| C1 | 0.999 | -0.996 | -0.998 | 0.983 |
| C2 | 0.995 | -0.995 | -0.995 | 0.985 |
| C3 | 0.999 | -0.997 | -0.998 | 0.985 |
| C4 | 0.997 | -0.997 | -0.998 | 0.988 |
| C5 | 0.998 | -0.978 | -0.979 | 0.950 |
| C6 | 0.996 | -0.998 | -0.996 | 0.979 |
| C7 | 0.998 | -0.997 | -0.998 | 0.982 |
| C8 | 0.998 | -0.997 | -0.998 | 0.986 |
These features provide a comprehensive basis for SOH estimation in lithium-ion batteries, enabling effective modeling of degradation patterns.
Support Vector Regression Model for Lithium-Ion Battery SOH Estimation
Support Vector Regression (SVR) is a powerful machine learning technique for regression tasks, particularly suited for nonlinear relationships and high-dimensional data. In the context of lithium-ion battery SOH detection, SVR maps input features to SOH values by transforming data into a higher-dimensional space via kernel functions. Given a sample set \( S = \{ (\mathbf{x}_i, y_i) \}_{i=1}^{N} \) where \( \mathbf{x}_i \in \mathbb{R}^n \) is the feature vector and \( y_i \in \mathbb{R} \) is the SOH value, the SVR function is defined as:
$$ f(\mathbf{x}) = \boldsymbol{\omega} \cdot \phi(\mathbf{x}) + b $$
where \( \boldsymbol{\omega} \) is the weight vector, \( \phi(\cdot) \) is a nonlinear mapping, and \( b \) is the bias term. The optimization problem minimizes the risk function:
$$ \min R(f) = \frac{1}{2} \|\boldsymbol{\omega}\|^2 + C \sum_{i=1}^{n} L(f(\mathbf{x}_i), y_i) $$
with the loss function \( L(f(\mathbf{x}_i), y_i) = \max\{0, |f(\mathbf{x}_i) – y_i| – \varepsilon\} \), where \( C \) is the penalty factor, \( \varepsilon \) is the maximum error tolerance, and \( n \) is the number of samples. Introducing slack variables \( \xi_i \) and \( \xi_i^* \), the problem becomes:
$$ \min R(\boldsymbol{\omega}, b, \boldsymbol{\xi}) = \frac{1}{2} \|\boldsymbol{\omega}\|^2 + C \sum_{i=1}^{n} (\xi_i + \xi_i^*) $$
subject to:
$$ \begin{cases} y_i – f(\mathbf{x}_i) \leq \varepsilon + \xi_i \\ f(\mathbf{x}_i) – y_i \leq \varepsilon + \xi_i^* \\ \xi_i, \xi_i^* \geq 0 \end{cases} $$
Using the dual principle and Lagrange multipliers \( \alpha_i \) and \( \alpha_i^* \), the solution is expressed as:
$$ f(\mathbf{x}) = \sum_{i=1}^{n} (\alpha_i – \alpha_i^*) K(\mathbf{x}_i, \mathbf{x}) + b $$
where \( K(\mathbf{x}_i, \mathbf{x}) \) is the kernel function. For nonlinear SOH estimation in lithium-ion batteries, the Radial Basis Function (RBF) kernel is commonly used:
$$ K_{\text{RBF}}(\mathbf{x}_i, \mathbf{x}) = \exp\left(-\frac{1}{2\sigma^2} \|\mathbf{x}_i – \mathbf{x}\|^2\right) $$
Here, \( \sigma \) is the width parameter. The performance of SVR heavily depends on the selection of hyperparameters \( C \) and \( \sigma \). Inappropriate choices can lead to overfitting or underfitting, compromising SOH detection accuracy for lithium-ion batteries.
Improved Whale Optimization Algorithm for Hyperparameter Tuning
The Whale Optimization Algorithm (WOA) is a metaheuristic inspired by the bubble-net hunting behavior of humpback whales. It balances exploration and exploitation to find optimal solutions. However, traditional WOA may suffer from premature convergence and local optima stagnation. To enhance its efficacy in tuning SVR parameters for lithium-ion battery SOH detection, two improvements are incorporated: adaptive weight adjustment and Levy flight strategy.
In standard WOA, the position of a whale \( \mathbf{X} \) is updated based on the best solution \( \mathbf{X}^* \). The distance \( \mathbf{D} = |\mathbf{X}^*(k) – \mathbf{X}(k)| \) is computed, and with probability \( p \), the update follows either a spiral or shrinking encircling mechanism:
$$ \mathbf{X}(k+1) = \begin{cases} \mathbf{D}^* \cdot e^{bl} \cdot \cos(2\pi l) + \mathbf{X}^*(k), & p \geq 0.5 \\ \mathbf{X}^*(k) – \mathbf{A} \cdot \mathbf{D}, & p < 0.5 \end{cases} $$
where \( b \) is a constant, \( l \) is a random number in \([-1,1]\), and \( \mathbf{A} = 2\mathbf{a} \cdot \mathbf{r} – \mathbf{a} \) with \( \mathbf{a \) decreasing linearly from 2 to 0 over iterations. If \( |\mathbf{A}| \geq 1 \), a random search is performed: \( \mathbf{X}(k+1) = \mathbf{X}_{\text{rand}} – \mathbf{A} \cdot \mathbf{D} \).
To improve local search capability, an adaptive weight \( \omega \) is introduced:
$$ \omega = 1 – \sin\left(\pi \cdot \left(1 – \frac{k}{2T}\right)\right) $$
where \( k \) is the current iteration and \( T \) is the maximum iterations. The position update becomes:
$$ \mathbf{X}(k+1) = \begin{cases} \mathbf{D}^* \cdot e^{bl} \cdot \cos(2\pi l) + \omega \mathbf{X}^*(k), & p \geq 0.5 \\ \omega \mathbf{X}^*(k) – \mathbf{A} \cdot \mathbf{D}, & p < 0.5 \end{cases} $$
This weight starts large for global exploration and decreases for local refinement, adapting to the search process. Additionally, Levy flight is integrated to enhance global exploration and avoid local optima. After updating positions, a Levy flight step is applied:
$$ \mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \alpha \oplus \text{Levy}(s) $$
where \( \alpha \) is a constant, and the step length \( s \) follows a Levy distribution: \( s = \frac{u}{|v|^{1/\beta}} \), with \( u \) and \( v \) from normal distributions and \( \beta = 1.5 \). This stochastic movement allows escaping local regions and exploring broader spaces, crucial for optimizing SVR parameters in complex lithium-ion battery SOH estimation tasks.
The IWOA algorithm optimizes the SVR hyperparameters \( C \) and \( \gamma \) (where \( \gamma = \frac{1}{2\sigma^2} \)) within bounds [0.01, 1000]. The fitness function is the prediction error on validation data. The overall flow involves initializing a whale population, evaluating fitness, updating positions with adaptive weight and Levy flight, and iterating until convergence. This integrated approach ensures robust parameter selection for accurate lithium-ion battery SOH detection.
Experimental Validation and Results
Two lithium-ion battery datasets are used to validate the IWOA-SVR method. Dataset 1 comprises 18650-type lithium-ion batteries with a nominal capacity of 2 Ah, subjected to CC-CV charging and constant-current discharging cycles. Dataset 2 includes Kokam lithium-ion batteries with a nominal capacity of 740 mAh, undergoing constant-current charging and dynamic discharging. The SOH is defined based on capacity fade:
$$ \text{SOH} = \frac{C_k}{C_0} \times 100\% $$
where \( C_k \) is the available capacity at cycle \( k \), and \( C_0 \) is the initial capacity. For each lithium-ion battery, the extracted features (F1–F4) serve as inputs to the SVR model, with SOH as the output.
In the experimental setup, the IWOA-SVR model is configured with a population size of 30, dimension 2 (for \( C \) and \( \gamma \)), and maximum iterations of 100. The search ranges for \( C \) and \( \gamma \) are [0.01, 1000]. For comparison, baseline methods include Gaussian Process Regression (GPR), standard SVR, and WOA-SVR. Each lithium-ion battery is evaluated using a train-test split; for instance, when testing battery B5, batteries B6, B7, and B18 form the training set, and similarly for dataset 2. This ensures generalizability across different lithium-ion batteries.
The optimized hyperparameters from IWOA for both datasets are listed in Tables 2 and 3, demonstrating the algorithm’s ability to find suitable values for each lithium-ion battery.
| Battery | C | γ |
|---|---|---|
| B5 | 3.798 | 0.997 |
| B6 | 989.784 | 0.028 |
| B7 | 5.401 | 0.357 |
| B18 | 96.384 | 0.065 |
| Battery | C | γ |
|---|---|---|
| C1 | 500.650 | 0.058 |
| C2 | 3.405 | 0.010 |
| C3 | 165.245 | 0.010 |
| C4 | 720.104 | 9.279 |
| C5 | 40.538 | 0.020 |
| C6 | 12.471 | 0.011 |
| C7 | 1.735 | 0.782 |
| C8 | 432.558 | 0.284 |
The SOH estimation results are evaluated using error metrics: Absolute Error (AE), Mean Absolute Percentage Error (MAPE), and Root Mean Square Error (RMSE). MAPE and RMSE are defined as:
$$ \text{MAPE} = \frac{100\%}{n} \sum_{i=1}^{n} \left| \frac{\hat{y}_i – y_i}{y_i} \right| $$
$$ \text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (\hat{y}_i – y_i)^2} $$
where \( \hat{y}_i \) is the predicted SOH and \( y_i \) is the actual SOH. For lithium-ion batteries in dataset 1, the maximum AE values are below 1.3%, while for dataset 2, they are under 1.4%, showcasing the high accuracy of IWOA-SVR. The error distributions, represented via boxplots, indicate that IWOA-SVR has median errors near zero and tighter interquartile ranges compared to GPR, SVR, and WOA-SVR, highlighting its stability.
Tables 4 and 5 present the MAPE and RMSE values for all lithium-ion batteries across the two datasets. IWOA-SVR consistently achieves the lowest errors, affirming its superiority in lithium-ion battery SOH detection.
| Method | B5 | B6 | B7 | B18 |
|---|---|---|---|---|
| GPR | 2.7492 / 2.2893 | 6.7844 / 6.8747 | 2.6160 / 2.2030 | 2.3905 / 2.3494 |
| SVR | 3.7973 / 3.2329 | 8.3384 / 8.3900 | 2.0578 / 2.0408 | 1.6606 / 1.5209 |
| WOA-SVR | 0.6278 / 0.5948 | 2.1833 / 1.8691 | 1.0916 / 0.9928 | 1.6606 / 1.5209 |
| IWOA-SVR | 0.2949 / 0.3089 | 0.2316 / 0.2795 | 0.4908 / 0.4449 | 0.2024 / 0.1794 |
| Method | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
|---|---|---|---|---|---|---|---|---|
| GPR | 2.7628 / 2.5274 | 3.8424 / 3.5550 | 2.5295 / 2.3390 | 1.8788 / 1.8994 | 1.4402 / 2.1231 | 1.7122 / 1.8560 | 1.9951 / 1.9567 | 2.6589 / 2.4432 |
| SVR | 3.0297 / 2.5746 | 3.3788 / 2.8200 | 2.7694 / 2.3796 | 2.5810 / 2.2949 | 3.1435 / 3.0944 | 2.2931 / 2.1060 | 2.6817 / 2.3609 | 2.4762 / 2.1321 |
| WOA-SVR | 0.5813 / 0.7073 | 1.0324 / 0.8492 | 0.4979 / 0.6530 | 0.7737 / 0.8624 | 1.2440 / 1.0889 | 0.6805 / 0.8648 | 0.8353 / 0.8852 | 0.6632 / 0.7453 |
| IWOA-SVR | 0.1622 / 0.1578 | 0.8291 / 0.6964 | 0.1913 / 0.1856 | 0.1846 / 0.1929 | 0.1780 / 0.1962 | 0.4269 / 0.3910 | 0.2388 / 0.2435 | 0.3426 / 0.3123 |
The results indicate that IWOA-SVR reduces MAPE and RMSE significantly, with improvements up to 90% compared to baseline methods. This underscores the effectiveness of the adaptive weight and Levy flight strategies in enhancing optimization for lithium-ion battery SOH estimation.
Discussion on the Robustness and Applicability
The proposed IWOA-SVR method demonstrates high robustness across diverse lithium-ion battery datasets with varying charging protocols. The extracted features—F1 (voltage rise time), F2 (charging slope), F3 (discharging slope), and F4 (voltage drop time)—prove to be universally indicative of aging in lithium-ion batteries, as evidenced by strong Pearson correlations. This feature selection strategy eliminates the need for invasive measurements, aligning with practical online detection requirements.
The integration of IWOA addresses key limitations in traditional optimization algorithms. The adaptive weight mechanism ensures a balanced trade-off between exploration and exploitation during the search for SVR parameters, while Levy flight introduces randomness to escape local optima, crucial for handling the nonlinearities in lithium-ion battery degradation data. Consequently, the optimized SVR model achieves precise mapping between features and SOH, with errors consistently below 1.4% across all tested lithium-ion batteries.
In real-world scenarios, such as customs inspections, this method offers a rapid, non-destructive solution for assessing lithium-ion battery health. By requiring only partial charge-discharge data—instead of full cycles—it reduces testing time and resource usage. Moreover, the model’s adaptability to different lithium-ion battery types and conditions enhances its applicability in global trade environments, where battery specifications may vary.
Future work could explore expanding the feature set to include temperature effects or internal resistance metrics, further improving accuracy for lithium-ion batteries under extreme operating conditions. Additionally, deploying the IWOA-SVR model in embedded systems for real-time monitoring could advance predictive maintenance strategies for lithium-ion battery packs in electric vehicles and grid storage.
Conclusion
This article presents a comprehensive framework for online rapid detection of lithium-ion battery State of Health using an Improved Whale Optimization Algorithm with Support Vector Regression. The method leverages key aging features extracted from charge-discharge curves, validated via Pearson correlation analysis, and optimizes SVR hyperparameters through enhanced global and local search capabilities. Experimental results on two distinct lithium-ion battery datasets show that IWOA-SVR achieves superior accuracy and stability, with maximum absolute errors under 1.4% and significant reductions in MAPE and RMSE compared to conventional approaches. The proposed technique provides a reliable, efficient solution for lithium-ion battery SOH assessment, supporting applications in customs inspection, safety compliance, and energy management. As the demand for lithium-ion batteries grows in green energy systems, such advanced detection methods will play a pivotal role in ensuring performance and safety worldwide.