As the global demand for clean energy and electric mobility surges, li-ion batteries have become indispensable due to their high energy density, long cycle life, and low self-discharge rates. However, safety concerns, particularly internal short-circuit faults leading to thermal runaway, pose significant risks to energy storage systems and electric vehicles. In this article, I present a novel fault diagnosis method that integrates Whale Optimization Algorithm-optimized Variational Mode Decomposition (WOA-VMD) and Particle Swarm Optimization-optimized Support Vector Machine (PSO-SVM) to effectively detect internal short-circuit faults in li-ion batteries. This approach addresses the limitations of traditional model-based methods by leveraging data-driven signal processing and machine learning techniques, ensuring robust and accurate fault identification for enhanced safety in li-ion battery applications.
The reliability of li-ion batteries is critical in modern energy systems, but internal short-circuit faults can trigger catastrophic failures like thermal runaway, often with minimal warning signs. These faults are challenging to diagnose due to their latent and abrupt nature, necessitating advanced methods beyond conventional equivalent circuit models. My methodology focuses on extracting subtle features from voltage signals of li-ion batteries using optimized decomposition and intelligent classification. The core idea involves decomposing fault and normal signals into intrinsic mode functions (IMFs) via VMD, with parameters fine-tuned by WOA, then computing sample entropy values as feature vectors, and finally employing PSO-SVM for high-accuracy fault diagnosis. This data-driven framework not only improves diagnostic precision but also reduces reliance on complex physical models, making it suitable for real-time monitoring of li-ion battery health.
Variational Mode Decomposition (VMD) is a non-recursive signal processing technique that decomposes an input signal into a predefined number of IMFs, overcoming issues like mode mixing seen in empirical mode decomposition. For a signal \( f(t) \), VMD aims to minimize the sum of bandwidths of each mode \( u_k(t) \), subject to the constraint that the sum of modes equals the original signal. The variational problem is formulated as:
$$ \min_{\{u_k\},\{\omega_k\}} \left\{ \sum_k \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 \right\} $$
subject to:
$$ \sum_k u_k(t) = f(t) $$
where \( \delta(t) \) is the Dirac delta function, \( \partial_t \) denotes the time derivative, \( * \) represents convolution, \( j \) is the imaginary unit, and \( \omega_k \) are the center frequencies. To solve this, a Lagrangian multiplier \( \lambda(t) \) and penalty factor \( \alpha \) are introduced, leading to the augmented Lagrangian:
$$ \mathcal{L}(\{u_k\}, \{\omega_k\}, \lambda) = \alpha \sum_k \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 + \left\| f(t) – \sum_k u_k(t) \right\|_2^2 + \left\langle \lambda(t), f(t) – \sum_k u_k(t) \right\rangle $$
The solution is obtained iteratively in the frequency domain using alternate direction method of multipliers (ADMM), with updates for \( \hat{u}_k^{n+1}(\omega) \), \( \omega_k^{n+1} \), and \( \hat{\lambda}^{n+1}(\omega) \), where \( \hat{\cdot} \) denotes the Fourier transform. The iteration stops when the convergence criterion is met:
$$ \frac{\sum_k \| \hat{u}_k^{n+1}(\omega) – \hat{u}_k^n(\omega) \|_2^2}{ \| \hat{u}_k^n(\omega) \|_2^2 } < \epsilon $$
for a small tolerance \( \epsilon > 0 \). The performance of VMD heavily depends on two parameters: the number of decomposition layers \( K \) and the penalty factor \( \alpha \). Incorrect choices can lead to over-decomposition or loss of frequency information, which is why optimization is crucial for li-ion battery signal analysis.
To optimize \( K \) and \( \alpha \), I employ the Whale Optimization Algorithm (WOA), a metaheuristic inspired by the bubble-net hunting behavior of humpback whales. WOA explores the search space through three phases: encircling prey, spiral bubble-net attacking, and random search for prey. Each whale’s position represents a candidate solution \( [K, \alpha] \), and the fitness function is the minimum envelope entropy of the decomposed signal, which measures the sparsity and complexity—key indicators for effective decomposition in li-ion battery fault diagnosis. The WOA process involves initializing a population of whales, evaluating fitness, and updating positions based on random parameters \( A \) and \( C \), as shown in the pseudo-code below. After iterations, the optimal \( [K, \alpha] \) combination is derived, which for li-ion battery voltage signals typically yields \( K = 10 \) and \( \alpha = 1997 \), ensuring precise extraction of fault-related modes.
| Parameter | Description | Value/Range |
|---|---|---|
| Population Size | Number of whales in the search | 30 |
| Iterations | Maximum number of generations | 30 |
| \( K \) Range | Decomposition layers | [2, 15] (integers) |
| \( \alpha \) Range | Penalty factor | [100, 10000] (integers) |
| Fitness Function | Minimum envelope entropy | Minimized |
Once the li-ion battery signal is decomposed using WOA-VMD, the resulting IMFs are analyzed for feature extraction. I use sample entropy (SampleEn) as a robust measure of signal complexity, which is sensitive to changes caused by internal short-circuit faults in li-ion batteries. For a time series \( \{x(i), i = 1, 2, \dots, N\} \), sample entropy is defined as:
$$ \text{SampleEn}(m, r, N) = -\ln \left[ \frac{B^{m+1}(r)}{B^m(r)} \right] $$
where \( m \) is the embedding dimension (usually set to 2), \( r \) is the tolerance (typically 0.2 times the standard deviation of the data), \( B^m(r) \) is the probability that two sequences match for \( m \) points, and \( B^{m+1}(r) \) is for \( m+1 \) points. A lower sample entropy indicates more regularity, which often correlates with normal operation of li-ion batteries, while higher values may signify fault-induced irregularities. By computing sample entropy for each IMF, I construct a feature vector that encapsulates the diagnostic information from the li-ion battery voltage signals. For instance, with \( K = 10 \), each signal yields a 10-dimensional feature vector, as summarized in the table below for both faulty and normal li-ion battery states.
| Battery State | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 | IMF7 | IMF8 | IMF9 | IMF10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Internal Short-Circuit Fault | 0.0050 | 0.0559 | 0.0231 | 0.1050 | 0.0123 | 0.0151 | 0.0033 | 0.0087 | 0.0006 | 0.0012 |
| Normal Operation | 0.0037 | 0.1308 | 0.0185 | 0.0632 | 0.0098 | 0.0018 | 0.0042 | 0.0065 | 0.0011 | 0.0009 |
The feature vectors derived from li-ion battery signals are then fed into a classification model for fault diagnosis. I use Support Vector Machine (SVM), a supervised learning algorithm that finds an optimal hyperplane to separate classes with maximum margin. For a dataset with features \( \mathbf{x}_i \) and labels \( y_i \in \{-1, 1\} \) (where -1 denotes normal and 1 denotes fault for li-ion batteries), the primal optimization problem is:
$$ \min_{\mathbf{w}, b, \xi} \frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^n \xi_i $$
subject to:
$$ y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \geq 1 – \xi_i, \quad \xi_i \geq 0, \quad i = 1, \dots, n $$
Here, \( \mathbf{w} \) is the weight vector, \( b \) is the bias, \( \xi_i \) are slack variables allowing misclassifications, and \( C \) is the penalty parameter controlling trade-off between margin and error. Using the Lagrangian dual, the problem transforms to:
$$ \max_{\alpha} \sum_{i=1}^n \alpha_i – \frac{1}{2} \sum_{i,j=1}^n y_i y_j \alpha_i \alpha_j K(\mathbf{x}_i, \mathbf{x}_j) $$
subject to:
$$ \sum_{i=1}^n y_i \alpha_i = 0, \quad 0 \leq \alpha_i \leq C $$
where \( \alpha_i \) are Lagrange multipliers and \( K(\mathbf{x}_i, \mathbf{x}_j) \) is the kernel function. I employ the radial basis function (RBF) kernel, which is effective for nonlinear classification in li-ion battery fault data:
$$ K(\mathbf{x}_i, \mathbf{x}_j) = \exp\left(-\gamma \|\mathbf{x}_i – \mathbf{x}_j\|^2\right) $$
with \( \gamma \) as a kernel parameter. The decision function for new li-ion battery samples becomes:
$$ f(\mathbf{x}) = \text{sgn}\left( \sum_{i=1}^n y_i \alpha_i K(\mathbf{x}_i, \mathbf{x}) + b \right) $$
However, direct SVM may suffer from suboptimal performance due to irrelevant features in the li-ion battery dataset. To enhance this, I integrate Particle Swarm Optimization (PSO) for feature selection, optimizing which IMF-based features to include. PSO is a population-based algorithm where particles (representing feature subsets) move through the solution space, adjusting their positions based on personal and global bests. The velocity and position updates for particle \( i \) are:
$$ \mathbf{v}_i^{t+1} = w \mathbf{v}_i^t + c_1 r_1 (\mathbf{p}_{\text{best},i} – \mathbf{x}_i^t) + c_2 r_2 (\mathbf{g}_{\text{best}} – \mathbf{x}_i^t) $$
$$ \mathbf{x}_i^{t+1} = \mathbf{x}_i^t + \mathbf{v}_i^{t+1} $$
where \( w \) is inertia weight, \( c_1 \) and \( c_2 \) are acceleration coefficients, \( r_1 \) and \( r_2 \) are random numbers, \( \mathbf{p}_{\text{best},i} \) is the best position of particle \( i \), and \( \mathbf{g}_{\text{best}} \) is the global best. The fitness function for PSO is the classification accuracy of SVM on a validation set from li-ion battery data. After optimization, PSO selects a subset of features (e.g., IMFs 1, 2, 4, 6, 7, and 9), reducing dimensionality and improving diagnostic precision for li-ion battery internal short-circuit faults.
To validate my approach, I conduct simulations using voltage signals from ternary li-ion batteries, with data sourced from experimental studies. The dataset includes 30 internal short-circuit fault samples and 30 normal samples from li-ion batteries, each of length 1000 points. I split the data into training and testing sets: 15 fault and 15 normal samples for training, and the remaining 30 for testing. The signals are decomposed via WOA-VMD with optimal parameters, sample entropy features are extracted, and classification is performed using SVM and PSO-SVM. The results demonstrate the superiority of the optimized method for li-ion battery fault diagnosis, as shown in the table below.
| Diagnosis Method | Features Used | Accuracy on Test Set (%) | Remarks |
|---|---|---|---|
| SVM (Direct) | All 10 IMFs | 66.667 | Baseline classification without optimization |
| PSO-SVM | Selected IMFs (e.g., 1,2,4,6,7,9) | 96.667 | Enhanced accuracy via feature selection |
The high accuracy of PSO-SVM underscores the effectiveness of combining optimized signal decomposition with machine learning for li-ion battery internal short-circuit fault diagnosis. Compared to traditional methods like EMD or EEMD, WOA-VMD avoids mode mixing and虚假 components, providing cleaner IMFs for feature extraction in li-ion battery signals. Moreover, the PSO-based feature selection mitigates the curse of dimensionality, ensuring that only the most discriminative sample entropy values from li-ion battery IMFs are used, which is critical for real-world applications where computational efficiency matters.
In terms of computational complexity, my method involves iterative processes in WOA and PSO, but these are offline optimizations that can be performed during system calibration. For online monitoring of li-ion batteries, the trained PSO-SVM model offers fast inference, making it suitable for embedded systems in electric vehicles or energy storage stations. The robustness of this approach is further validated by its ability to handle noisy signals common in li-ion battery operations, as VMD’s bandwidth separation properties enhance noise resilience.
Looking ahead, future work could extend this methodology to other fault types in li-ion batteries, such as overcharge or degradation, by incorporating additional sensors like temperature or current. Moreover, deep learning models could be integrated with WOA-VMD for end-to-end fault diagnosis in li-ion batteries, potentially improving accuracy further. The scalability of this approach also allows adaptation to large battery packs, where early detection of internal short-circuit faults can prevent cascading failures, ensuring safety and longevity of li-ion battery systems.
In conclusion, my proposed fault diagnosis method for li-ion batteries, based on WOA-VMD and PSO-SVM, offers a reliable and accurate solution for detecting internal short-circuit faults. By optimizing signal decomposition parameters and feature selection, it achieves a diagnostic accuracy of 96.667%, significantly outperforming conventional SVM. This data-driven framework not only enhances the safety of li-ion battery applications but also contributes to the advancement of predictive maintenance in clean energy systems. As the adoption of li-ion batteries continues to grow, such intelligent diagnostic tools will play a pivotal role in mitigating risks and promoting sustainable energy utilization.