The widespread adoption of lithium-ion batteries in electric vehicles (EVs) and grid-scale energy storage systems underscores their pivotal role in the modern energy landscape. However, their operational safety remains a paramount concern, primarily due to critical failure modes such as thermal runaway and severe cell inconsistency. These faults, if undetected, can lead to catastrophic safety incidents, significant financial loss, and erosion of consumer confidence. Consequently, developing reliable, early-stage fault diagnosis methods is an urgent and critical research challenge. This article presents a novel, data-driven diagnostic framework designed to preemptively identify and localize such faults by leveraging real-world vehicle operational data.
Existing diagnostic approaches often fall short in practical, complex driving scenarios. Model-based methods rely on precise electrochemical or equivalent circuit models, which struggle to adapt to the dynamic aging and varied operating conditions of a lithium-ion battery. Knowledge-based systems, such as expert systems, are limited by their predefined rule bases and may fail to recognize novel or complex fault patterns. While data-driven methods show promise, many exhibit limitations in feature extraction stability, sensitivity to early fault signatures, or the ability to provide a unified diagnostic mechanism for different fault types. For instance, methods relying on simple statistical thresholds or correlation coefficients can suffer from high false alarm rates under normal voltage fluctuations common in EV operation.
To address these gaps, we propose an integrated fault diagnosis methodology centered on adaptive signal processing and longitudinal anomaly analysis. The core innovation lies in the synergistic use of a Trigonometric Topology Aggregation Optimization (TTAO) algorithm to fine-tune the Variational Mode Decomposition (VMD) parameters, followed by a Longitudinal Outlier Average (LOA) analysis applied within a sliding window framework. This combination enables the stable extraction of subtle fault features from noisy operational voltage data and the precise quantification of a cell’s deviation from the pack’s normal behavior. The proposed method demonstrates high robustness, low false-positive rates, and provides effective early warning for thermal runaway while accurately pinpointing cells with severe inconsistency.
Theoretical Foundations of the Proposed Methodology
1. Variational Mode Decomposition (VMD) and Parameter Optimization Challenge
VMD is a non-recursive, fully intrinsic signal decomposition technique that adaptively decomposes a complex signal into a discrete number of band-limited Intrinsic Mode Functions (IMFs). For a given signal \( f(t) \), VMD seeks to find a set of modes \( \{u_k(t)\} \) and their corresponding center frequencies \( \{\omega_k\} \) by solving the following constrained variational problem:
$$
\min_{\{u_k\},\{\omega_k\}} \left\{ \sum_{k=1}^{K} \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 \right\}
$$
$$
\text{subject to} \quad \sum_{k=1}^{K} u_k(t) = f(t)
$$
where \( K \) is the predefined number of modes, \( \delta(t) \) is the Dirac delta function, and \( * \) denotes convolution. To solve this, the constrained problem is transformed into an unconstrained one using a quadratic penalty term \( \alpha \) and a Lagrangian multiplier \( \lambda(t) \), leading to the augmented Lagrangian \( \mathcal{L} \):
$$
\begin{aligned}
\mathcal{L}(\{u_k\},\{\omega_k\},\lambda) = &\alpha \sum_k \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 \\
&+ \left\| f(t) – \sum_k u_k(t) \right\|_2^2 + \left\langle \lambda(t), f(t) – \sum_k u_k(t) \right\rangle
\end{aligned}
$$
The solution is found iteratively using the Alternating Direction Method of Multipliers (ADMM) in the Fourier domain. The performance of VMD is highly sensitive to the selection of two key parameters: the number of modes \( K \) and the penalty factor \( \alpha \). An inappropriate choice can lead to mode mixing (insufficient decomposition) or overly smooth, meaningless components. Manually setting these parameters is subjective and often suboptimal for the diverse voltage profiles of a lithium-ion battery under different conditions.
2. Trigonometric Topology Aggregation Optimization (TTAO) Algorithm
The TTAO algorithm is a novel metaheuristic inspired by geometric principles of triangular topology. It efficiently explores the search space through phases of formation, global aggregation, and local aggregation. For optimizing VMD parameters \( (K, \alpha) \), each potential solution is represented as a point in a 2D search space. The algorithm’s effectiveness stems from its balanced exploration-exploitation strategy, modeled mathematically through triangle vertices and weighted aggregations.
1. Population Initialization & Triangle Formation: From an initial random population, solutions are grouped into triangular units. The first vertex \( \mathbf{X}_{i,1} \) is randomly generated. The second and third vertices are created based on directional vectors derived from spherical coordinate transformations:
$$
\mathbf{X}_{i,2} = \mathbf{X}_{i,1} + l \cdot \mathbf{f}(\vec{\theta}), \quad \mathbf{X}_{i,3} = \mathbf{X}_{i,1} + l \cdot \mathbf{f}(\vec{\theta} + \frac{\pi}{3})
$$
where \( l \) is a decreasing size factor over iterations, and \( \mathbf{f} \) is the directional vector function using cosine terms.
2. Global Aggregation: A new solution \( \mathbf{X}_{i,new1} \) is generated by combining the best individual from the current triangle \( (\mathbf{X}_{i,best}^t) \) with the best from a randomly selected other triangle \( (\mathbf{X}_{rand,best}^t) \):
$$
\mathbf{X}_{i,new1}^{t+1} = r_4 \cdot \mathbf{X}_{i,best}^t + (1 – r_4) \cdot \mathbf{X}_{rand,best}^t
$$
This promotes information exchange and global exploration.
3. Local Aggregation: To refine the search, a temporary triangle is formed using the current best and second-best solutions within a unit. A new candidate is generated by perturbing the best solution along the vector towards the second-best:
$$
\mathbf{X}_{i,new2}^{t+1} = \mathbf{X}_{i,best}^{t+1} + \eta(t) \cdot (\mathbf{X}_{i,best}^{t+1} – \mathbf{X}_{i,sbest}^{t+1})
$$
where \( \eta(t) \) is a time-varying scaling factor that decreases, focusing the search locally around promising areas.
3. TTAO-VMD Optimization Framework
We employ TTAO to automatically find the optimal pair \( (K^*, \alpha^*) \) for decomposing the voltage signal of a lithium-ion battery. The fitness function for TTAO is chosen as the Minimum Average Sample Entropy of the reconstructed signal from the IMFs. Sample entropy quantifies signal complexity and regularity; minimizing its average across modes encourages a decomposition that effectively separates noise and meaningful components, which is ideal for subsequent fault feature extraction. The optimization process is outlined below:
| Step | Action | Objective |
|---|---|---|
| 1 | Initialize TTAO. Set bounds: \( K \in [3, 10] \), \( \alpha \in [100, 2500] \). | Define the search space for VMD parameters. |
| 2 | For each candidate \( (K, \alpha) \), perform VMD on the voltage signal. | Decompose the signal into K IMFs. |
| 3 | Calculate the average Sample Entropy of the first few (e.g., 3) IMFs. | Evaluate decomposition quality (fitness). |
| 4 | Run TTAO iterations (global/local aggregation) to minimize fitness. | Find parameters yielding the most informative, noise-separated decomposition. |
| 5 | Output optimal \( (K^*, \alpha^*) \) and the corresponding set of IMFs. | Obtain adaptively optimized signal components for analysis. |
4. Feature Extraction via Sliding Window and LOA Analysis
Post decomposition, the first two IMFs (IMF1 and IMF2) are empirically selected as they typically capture the most significant trend and primary oscillatory components of the lithium-ion battery voltage signal, which are most sensitive to incipient faults.
To enable online, real-time capable diagnosis, a sliding window (MW) approach is applied. For a time series \( V_j = \{v_j(1), v_j(2), …, v_j(N)\} \) of cell \( j \), a window of size \( W \) (e.g., 30 samples) moves with a step of 1. This creates a sequence of data segments, allowing local and recent behavior to be analyzed independently of distant historical data, enhancing sensitivity to developing faults in the lithium-ion battery.
Within each window for a given IMF component, the Longitudinal Outlier Average (LOA) is computed for every cell. The LOA quantifies the consistent deviation of a single cell’s voltage from the pack’s normal reference, defined as the median voltage across all cells at each timestep. For cell \( j \) over a window of \( n \) samples, LOA is calculated as:
$$
\text{LOA}_j = \frac{ \sum_{i=1}^{n} \left( V_{ij} – V_{i}^{\text{mid}} \right) }{n \times AF}
$$
where \( V_{ij} \) is the voltage (or IMF value) of cell \( j \) at sample \( i \) within the window, \( V_{i}^{\text{mid}} \) is the median voltage of all cells at that sample, and \( AF \) is an amplification factor (e.g., 100) for clearer visualization. A consistently high positive or negative LOA value indicates that the cell is behaving anomalously compared to the pack majority. A threshold \( \Theta \) (empirically set to 1) is used to trigger a fault alarm. The diagnostic decision logic is:
$$
\text{Fault Alert for Cell } j \iff \text{LOA}_j^{(t-W+1:t)} > \Theta \quad \text{for a sustained period}
$$
Experimental Validation and Diagnostic Results
The proposed methodology was validated using real-world operational data from an EV fleet management platform. Data from three distinct vehicles were analyzed: one experiencing thermal runaway (C1), one with a cell showing severe inconsistency (C2), and one normal vehicle (C3). The specifications are summarized below:
| Vehicle ID | Number of Cells | Fault Type | Faulty Cell ID | Total Samples |
|---|---|---|---|---|
| C1 | 97 | Thermal Runaway | 23 | 488 |
| C2 | 97 | Severe Inconsistency | 65 | 3244 |
| C3 | 94 | None (Normal) | N/A | 962 |
1. Diagnosis of Thermal Runaway in Vehicle C1
The raw voltage of the lithium-ion battery pack in vehicle C1 shows a sharp drop for cell #23 near sample 474. Applying TTAO-VMD yielded optimal parameters \( K^*=10, \alpha^*=1103 \). Analysis revealed that IMF2 was the most diagnostic component for this fault type. While IMF1 showed the overall downward trend, IMF2 exhibited pronounced, abnormal oscillations for the faulty cell well before the final voltage collapse.
The LOA values calculated from the IMF2 component within a sliding window (W=30) provided a clear diagnostic signal. The LOA for cell #23 exceeded the threshold \( \Theta=1 \) starting at window 437 (corresponding to samples 437-466), providing an early warning approximately 7-8 sample points (70-80 seconds) before the major voltage plunge observed in the raw data. All other cells maintained LOA values below the threshold, demonstrating precise fault localization and low false alarms for the lithium-ion battery pack.
2. Diagnosis of Severe Inconsistency in Vehicle C2
For vehicle C2, cell #65 exhibited a persistently lower voltage profile compared to the pack. In this case, the optimized VMD decomposition (\( K^*=8, \alpha^*=1800 \)) and subsequent analysis indicated that the IMF1 component, representing the primary trend, was more effective for diagnosis. The LOA values derived from IMF1 clearly identified cell #65, with its LOA consistently above the threshold of 1 from window 655 onwards, while all other cells remained below it. This confirms the method’s capability to reliably identify gradual, consistency-based faults in a lithium-ion battery.
3. Analysis of Normal Vehicle C3
To test the false-positive rate, data from the normal vehicle C3 was processed. The LOA values for all cells, calculated from both IMF1 and IMF2 components, fluctuated but remained consistently below the fault threshold throughout the entire operational cycle. This result validates the specificity and reliability of the proposed diagnostic framework, showing that it does not generate spurious alarms under normal operating conditions for a lithium-ion battery system.
Comparative Analysis and Advantages
A common baseline method for fault detection involves calculating the Pearson correlation coefficient between a target cell’s voltage and the pack average or median. A significant drop in correlation is interpreted as a fault. We implemented this method using the same IMF components (IMF2 for C1, IMF1 for C2) and an identical sliding window for a direct comparison.
The results highlighted key shortcomings of the correlation-based approach. For the thermal runaway case, while the faulty cell’s correlation dropped notably, several normal cells also experienced correlation dips below a typical threshold (e.g., 0.5) during certain windows due to normal pack dynamics, leading to potential false positives. For the inconsistency fault, the correlation coefficient for the faulty cell was unstable and did not provide a clear, sustained diagnostic signal.
The advantages of the proposed TTAO-VMD-LOA framework are systematically summarized below:
| Aspect | Traditional Correlation Method | Proposed TTAO-VMD-LOA Method | Advantage |
|---|---|---|---|
| Parameter Sensitivity | Low; requires setting a correlation threshold. | Adaptive; TTAO auto-tunes VMD for optimal feature extraction. | Enhanced robustness and adaptability to different lithium-ion battery signals. |
| Feature Stability | Directly uses raw/lightly processed voltage, sensitive to noise and trends. | Uses decomposed IMFs (e.g., IMF1, IMF2), isolating fault-sensitive components from noise. | Produces cleaner, more stable features for diagnosis. |
| Fault Quantification | Provides a correlation score (bounded between -1 and 1). | Provides LOA value, a direct measure of longitudinal deviation magnitude. | LOA offers a more intuitive and scalable measure of anomaly severity. |
| False Alarm Rate | Potentially high during normal pack voltage fluctuations. | Significantly lower; LOA is based on deviation from the robust median, less affected by co-varying fluctuations. | Higher reliability in real-world, noisy operational environments. |
| Early Warning Capability | Limited; often detects fault only after significant deviation occurs. | Strong; sensitive to incipient oscillations (IMF2) or trend shifts (IMF1) before major failure. | Enables proactive safety interventions for the lithium-ion battery. |
Conclusion
This article has presented a comprehensive and robust fault diagnosis framework for lithium-ion battery packs in electric vehicles. By integrating the adaptive signal decomposition capability of TTAO-optimized VMD with the sensitive anomaly detection power of sliding-window LOA analysis, the method effectively addresses the challenges of early thermal runaway warning and accurate identification of severely inconsistent cells. The framework operates directly on real vehicle operational data, requiring no precise battery models or extensive historical fault libraries.
The experimental validation on real-world datasets confirms the method’s practical efficacy. It successfully provided early warning for an impending thermal runaway event and reliably pinpointed a cell with progressive inconsistency, all while maintaining a near-zero false alarm rate on normal vehicle data. The comparative analysis against a conventional correlation-based method unequivocally demonstrated superior robustness and diagnostic clarity.
The proposed approach signifies a meaningful step towards more reliable and intelligent battery management systems. Future work will focus on expanding the diagnostic scope to include other fault types such as internal short circuits and connection faults, as well as exploring the integration of this signal-based method with lightweight model-based observers for a hybrid diagnostic architecture, further solidifying the safety foundation for lithium-ion battery applications.