In the context of the global transition to clean energy, battery energy storage systems (BESS) have emerged as a critical component for enhancing grid resilience and stability. The International Energy Agency (IEA) projects that by 2030, the global installed capacity of battery energy storage will surge from 90 GW in 2023 to over 850 GW, with average discharge durations extending from 2 hours to 3 hours. This rapid expansion underscores the importance of advanced safety management and monitoring techniques for lithium-ion batteries, which dominate the BESS landscape. As a key parameter, the cell voltage of a lithium-ion battery exhibits complex nonlinear and non-stationary characteristics, making it challenging to detect anomalies—such as those caused by internal short circuits—that may be masked by normal aging processes. Therefore, in-depth time-frequency domain analysis is essential for accurate performance evaluation and fault diagnosis.
Traditionally, non-stationary signal decomposition methods like Empirical Mode Decomposition (EMD), wavelet transform, and Local Characteristic-scale Decomposition (LCD) have been employed for analyzing battery voltage signals. However, these methods often lack real-time capabilities and robustness against noise, limiting their effectiveness in online monitoring scenarios. In contrast, Variational Mode Decomposition (VMD) has proven to be a powerful adaptive technique for extracting intrinsic mode functions (IMFs) from dynamic signals, offering superior frequency resolution and noise resistance. To enable real-time analysis in edge computing environments, we propose an online VMD method based on a sliding window approach. This method captures the time-varying features of lithium-ion battery cell voltage signals by processing data segments incrementally, thereby facilitating timely fault detection and health assessment.
The voltage signal from a lithium-ion battery cell is inherently non-stationary due to factors such as charge-discharge cycles, temperature variations, and aging. Mathematically, this signal can be represented as a time-series $v(t)$, where $t$ denotes time. The non-stationarity implies that statistical properties like mean and variance change over time, complicating direct analysis. To address this, signal decomposition methods aim to break down $v(t)$ into simpler components. For instance, EMD decomposes a signal into IMFs based on local oscillations, but it suffers from mode mixing and lacks a rigorous mathematical foundation. Wavelet transform uses basis functions scaled and translated in time, yet it requires prior selection of mother wavelets and may not adapt well to signal-specific features. LCD improves upon EMD by considering local characteristic scales, but it still shares similar limitations in noise handling. In comparison, VMD formulates decomposition as a variational optimization problem, ensuring robustness and adaptability.
Let us delve into the VMD method. Given a signal $v(t)$, VMD aims to decompose it into $K$ discrete modes $u_k(t)$, each with limited bandwidth around a center frequency $\omega_k$. The optimization problem is constructed as follows:
$$ \min_{\{u_k\},\{\omega_k\}} \left\{ \sum_{k=1}^K \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 \right\} \quad \text{subject to} \quad \sum_{k=1}^K u_k(t) = v(t) $$
Here, $*$ denotes convolution, $\delta(t)$ is the Dirac delta function, and $\partial_t$ is the partial derivative with respect to time. The objective minimizes the bandwidth of each mode, while the constraint ensures that the sum of modes reconstructs the original signal. To solve this constrained problem, we introduce a quadratic penalty term $\alpha$ and a Lagrangian multiplier $\lambda(t)$, transforming it into an unconstrained form:
$$ \mathcal{L}(\{u_k\},\{\omega_k\},\lambda) = \alpha \sum_{k=1}^K \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 + \left\| v(t) – \sum_{k=1}^K u_k(t) \right\|_2^2 + \left\langle \lambda(t), v(t) – \sum_{k=1}^K u_k(t) \right\rangle $$
The solution is obtained via the Alternating Direction Method of Multipliers (ADMM), which iteratively updates $u_k$, $\omega_k$, and $\lambda$. In the frequency domain, the update equations are:
$$ \hat{u}_k^{n+1}(\omega) = \frac{\hat{v}(\omega) – \sum_{i \neq k} \hat{u}_i(\omega) + \frac{\hat{\lambda}(\omega)}{2}}{1 + 2\alpha (\omega – \omega_k)^2} $$
$$ \omega_k^{n+1} = \frac{\int_0^\infty \omega |\hat{u}_k(\omega)|^2 d\omega}{\int_0^\infty |\hat{u}_k(\omega)|^2 d\omega} $$
where $\hat{\cdot}$ denotes the Fourier transform, and $n$ is the iteration index. The penalty factor $\alpha$ balances reconstruction accuracy against mode bandwidth; typically, $\alpha = 2000$ is used initially. The Lagrangian multiplier $\lambda$ enforces constraint satisfaction. This formulation allows VMD to effectively separate signal components even in noisy environments, making it suitable for analyzing lithium-ion battery voltage signals.
However, standard VMD processes the entire signal batch, which is impractical for real-time monitoring of lithium-ion batteries. To enable online analysis, we integrate a sliding window technique. Sliding windows are a fundamental concept in stream processing, where a fixed-size window moves over the data sequence, processing each segment independently. This approach reduces computational complexity and memory usage, as each data point is processed only twice—once when entering and once when leaving the window. The key parameters are window size $N$ and overlap size $O$, with step size $S = N – O$. For a signal of length $L$, the number of windows is approximately $\lceil (L – O) / S \rceil$.
We propose an online VMD framework based on sliding windows for lithium-ion battery cell voltage monitoring. The algorithm proceeds as follows: for each window indexed by $i$, we extract the voltage signal segment $v_i(t)$ of length $N$. Then, VMD is applied to $v_i(t)$ to obtain IMFs $u_{i,k}(t)$ for $k=1,\dots,K$. These IMFs capture local time-frequency features, such as low-frequency trends related to state-of-charge variations and high-frequency fluctuations indicative of faults. By analyzing IMFs across successive windows, we can track dynamic changes in the lithium-ion battery’s behavior.
The choice of window size $N$ is critical. A small $N$ enhances real-time responsiveness but may miss long-term patterns, while a large $N$ improves frequency resolution at the cost of increased latency. To guide selection, we evaluate different $N$ values using voltage data from a BESS with 224 lithium-ion battery cells, sampled at 0.1 Hz (10-second intervals). We focus on 15 representative cells and set VMD parameters as $K=7$ and $\alpha=2000$. The table below summarizes the effects of varying $N$ on decomposition quality and computational load.
| Window Size ($N$) | Decomposition Quality | Real-Time Performance | Remarks |
|---|---|---|---|
| 10 | High: Smooth IMFs, clear cell inconsistency | Excellent: Low latency | Optimal for early detection |
| 20 | Moderate: Some smoothing, reduced detail | Good: Moderate latency | Balanced trade-off |
| 30 | Moderate: Further smoothing | Acceptable: Increased latency | Less suitable for fast dynamics |
| 40 | Low: Over-smoothed, loss of features | Poor: High latency | Not recommended for online use |
| 50 | Low: Very smooth, minimal variability | Poor: Very high latency | Unsuitable for real-time monitoring |
Based on this analysis, we select $N=10$ for our online system, as it preserves signal features while minimizing delay. The step size is set to $S=5$ (50% overlap) to ensure continuity between windows. For each window, the VMD output includes seven IMFs: IMF1 (lowest frequency) to IMF7 (highest frequency). IMF1 typically reflects long-term trends like state-of-health degradation, while higher-order IMFs capture short-term anomalies such as voltage dips or spikes caused by internal shorts in the lithium-ion battery.
To illustrate, consider the voltage signal $v(t)$ from a single lithium-ion battery cell over a window. After applying VMD, we obtain IMFs that satisfy:
$$ v(t) = \sum_{k=1}^7 u_k(t) + r(t) $$
where $r(t)$ is a residual component. The energy distribution across IMFs can be quantified using:
$$ E_k = \int |u_k(t)|^2 dt $$
This energy metric helps identify dominant modes. For instance, in a healthy lithium-ion battery, IMF1 may hold most energy, whereas fault conditions might shift energy to higher modes. We can also compute instantaneous frequencies $\phi_k'(t)$ for each IMF via Hilbert transform, enabling time-frequency visualization.
The online decomposition process is implemented in an edge computing module. As new voltage samples arrive from the lithium-ion battery management system (BMS), the sliding window shifts, and VMD is applied to the latest segment. The extracted IMFs are then analyzed for fault indicators. For example, a sudden increase in the amplitude of IMF3 or IMF4 could signal an emerging short circuit. By comparing IMFs across multiple cells, we can detect inconsistencies that precede failures. This proactive approach enhances the safety and reliability of lithium-ion battery packs in BESS applications.
In practice, the computational efficiency of online VMD is crucial. The ADMM algorithm converges within a few iterations per window, and the sliding window strategy limits data processing to $O(N)$ operations per step. To further optimize, we can adaptively adjust $K$ based on signal complexity, but for consistency, we fix $K=7$. The table below compares computational aspects of different signal decomposition methods for lithium-ion battery voltage analysis.
| Method | Real-Time Capability | Noise Robustness | Mathematical Rigor | Suitability for Lithium-Ion Batteries |
|---|---|---|---|---|
| EMD | Low: Iterative, sensitive to sampling | Poor: Mode mixing | Heuristic | Limited due to noise sensitivity |
| Wavelet Transform | Moderate: Fixed basis, fast transforms | Moderate: Depends on wavelet choice | High: Well-established theory | Good for stationary segments |
| LCD | Low: Similar to EMD | Poor: Limited noise handling | Heuristic | Not ideal for dynamic signals |
| VMD (Batch) | Low: Processes full signal | High: Explicit noise tolerance | High: Variational formulation | Excellent for offline analysis |
| Online VMD (Sliding Window) | High: Incremental processing | High: Inherits VMD robustness | High: Based on VMD theory | Optimal for real-time monitoring |
Our experiments with lithium-ion battery data confirm that online VMD effectively decomposes voltage signals into interpretable modes. For instance, in a window where a cell exhibits abnormal behavior, IMF2 through IMF5 show pronounced oscillations, while IMF1 remains stable. This separation allows for targeted fault diagnosis. Moreover, the method is scalable to large BESS installations, as each lithium-ion battery cell can be monitored independently via distributed edge nodes.
The integration of sliding windows with VMD also facilitates adaptive parameter tuning. For example, if signal noise increases due to environmental interference, we can dynamically adjust $\alpha$ to maintain decomposition quality. A higher $\alpha$ strengthens bandwidth constraints, suppressing noise in the IMFs. This adaptability is vital for lithium-ion batteries operating in diverse conditions, such as renewable energy farms or grid-scale storage facilities.
From a theoretical perspective, the online VMD method aligns with time-frequency analysis principles. The Wigner-Ville distribution of an IMF $u_k(t)$ provides insights into energy concentration:
$$ W_k(t,\omega) = \int u_k\left(t + \frac{\tau}{2}\right) u_k^*\left(t – \frac{\tau}{2}\right) e^{-j\omega\tau} d\tau $$
However, for computational simplicity, we rely on the IMFs directly. The sliding window ensures that the analysis localizes in time, capturing transient events in the lithium-ion battery voltage signal. This is particularly important for detecting early-stage faults, such as micro-shorts, which manifest as brief voltage deviations.
In conclusion, the proposed online VMD method based on sliding windows offers a robust solution for real-time monitoring of lithium-ion battery cell voltages. By decomposing signals into meaningful modes, it enables early fault detection and health assessment, contributing to the safety and efficiency of battery energy storage systems. Future work could explore deep learning integration for automated anomaly classification or extend the method to other battery parameters like temperature and current. As the demand for lithium-ion batteries grows in clean energy applications, such advanced monitoring techniques will play a pivotal role in ensuring grid resilience and sustainability.
To summarize key equations, the VMD optimization and sliding window process are encapsulated below. For a windowed signal $v_i(t)$:
$$ \text{VMD: } \min_{u_k,\omega_k} \mathcal{L}(u_k,\omega_k,\lambda) \quad \text{with updates as per ADMM} $$
$$ \text{Sliding Window: } v_i(t) = v(t) \cdot \mathbf{1}_{[iS, iS+N]}(t) $$
where $\mathbf{1}$ is the indicator function. The overall framework ensures that lithium-ion battery voltage signals are analyzed continuously, with minimal latency, supporting the transition to a cleaner energy future.