The relentless pursuit of sustainable transportation has cemented the position of high-energy-density lithium-ion batteries as the cornerstone of modern electric propulsion systems, from personal vehicles to mass transit. Ensuring their operational safety and reliability is paramount, a responsibility that falls squarely on the Battery Management System (BMS). While safety risk prediction is a core BMS function, conventional algorithms predominantly hinge on the direct monitoring of macroscopic signals such as cell voltage, pack current, and temperature. These methods, however, face significant limitations. During the incipient stages of a fault within a lithium-ion battery, the deviations in these observable parameters are often exceedingly subtle or indistinguishable from normal operational noise. This inherent subtlety forces a trade-off: setting sensitive thresholds leads to an unacceptably high rate of false alarms, while conservative thresholds result in critically short warning times, often mere minutes before a catastrophic thermal runaway event. This fundamental gap between the need for early intervention and the capabilities of existing diagnostics presents a critical challenge for the safe deployment of lithium-ion battery technology across all transportation sectors.
To transcend these limitations, I propose a paradigm shift from direct thresholding to a probabilistic, information-theoretic assessment of system safety. This research introduces and elaborates on the concept of “Safety Entropy” as a foundational metric for quantifying the risk state of a lithium-ion battery system. By translating the likelihood of safe operation (or its converse, failure probability) into an entropic measure, this approach provides a continuous, scalar value that reflects the system’s “degree of disorder” or uncertainty regarding its safety. A rising safety entropy indicates increasing risk. Building upon this core concept, I develop an integrated early-warning algorithm that dynamically fuses multiple fault indicators. Crucially, this algorithm incorporates an adaptive weighting mechanism that adjusts the influence of each diagnostic feature based on the operational context and the aging state of the lithium-ion battery. This dynamic fusion stands in contrast to static models, allowing the diagnostic system to evolve with the battery throughout its lifecycle. The proposed methodology addresses the interpretability issues of “black-box” machine learning models while offering significantly enhanced early-warning capability compared to traditional direct parameter checks. Validation using extensive field data from electric vehicles demonstrates the algorithm’s efficacy in identifying latent faults days in advance of potential failure, showcasing its potential to become a robust safety sentinel for lithium-ion battery packs in diverse applications.
Theoretical Foundation: Quantifying Risk with Safety Entropy
The core innovation of this diagnostic framework is the formalization of Safety Entropy. In information theory, entropy measures the uncertainty or randomness in a system. By analogy, Safety Entropy quantifies the uncertainty regarding the safe operation of a system—in this case, a lithium-ion battery pack. A perfectly safe, deterministic state corresponds to minimum entropy, while increasing probabilities of various failure modes increase the entropy, signaling heightened risk.
The fundamental definition starts with a safety event \( x \). Let \( P(x) \) represent the probability density of this event remaining in a safe state. The basic Safety Entropy \( S(x) \) is then defined as:
$$ S(x) = k \log \frac{1}{P(x)} = -k \log P(x) $$
where \( k \) is a normalization or scaling coefficient. The sign of \( k \) determines the interpretative scale. If \( k \) is negative, a higher safety probability \( P(x) \) leads to a larger \( S(x) \), with maximum entropy \( S(x)=0 \) at \( P(x)=1 \), indicating maximum “disorder” or uncertainty from a safety perspective—paradoxically meaning the system’s safety is not guaranteed by a single dominant safe state. For intuitive risk indication, we can treat \( k \) as positive, making \( S(x) \) increase as \( P(x) \) decreases, directly correlating higher entropy with higher risk.
In practical application for a complex system like a lithium-ion battery, safety is not a single event but a confluence of multiple, potentially independent factors (e.g., voltage uniformity, thermal stability, charge efficiency). For \( n \) independent safety-related events \( x_1, x_2, …, x_n \), the joint safety probability is the product of individual probabilities:
$$ P(x | x_1, x_2, …, x_n) = P(x_1)P(x_2)···P(x_n) $$
Therefore, the total Safety Entropy for the system becomes the sum of individual entropies:
$$ S_{total}(x) = -k \log \left( \prod_{i=1}^{n} P(x_i) \right) = \sum_{i=1}^{n} \left( -k \log P(x_i) \right) = \sum_{i=1}^{n} S(x_i) $$
This additive property is crucial for building a composite risk score from multiple diagnostics.
Not all indicators contribute equally to the overall risk assessment of a lithium-ion battery. An over-temperature event might be more critical than a slight deviation in average voltage. To account for this, I introduce weighted Safety Entropy. Each event \( x_i \) is assigned a weight \( \alpha_i \), reflecting its relative importance to the system’s safety assessment:
$$ S_{weighted} = \sum_{i=1}^{n} \alpha_i S(x_i) = -\sum_{i=1}^{n} \alpha_i k \log P(x_i) $$
The determination of these weights \( \alpha_i \) is not static and forms a critical part of the adaptive mechanism, detailed later.
Two practical issues must be resolved for numerical stability. First, probability densities can be zero, leading to singularities in the logarithm. Second, not all safety features are best expressed as probabilities (e.g., maximum voltage delta is a physical measurement, not a probability). To address these, the final, generalized Safety Entropy model for a lithium-ion battery system incorporates a probability offset constant \( a \) and a set of constants \( b_j \) for non-probabilistic features:
$$ S = -\sum_{i=1}^{m} \alpha_i \log \left( a + P(x_i) \right) – \sum_{j=1}^{p} b_j $$
Here, \( a > 0 \) ensures the logarithm’s argument is always positive, and \( b_j \) represents the direct entropic contribution from the j-th non-probabilistic safety feature. This formulation provides a robust and calculable metric for real-time risk evaluation.
Feature Extraction and Dynamic Weight Allocation
The effectiveness of the Safety Entropy model hinges on the selection of informative features and their adaptive integration. Through failure mode analysis and historical data mining, I have identified and defined a comprehensive set of 21 safety features for the lithium-ion battery system. These can be categorized into two types: Indicative Features, which directly signal an abnormal state (e.g., excessive temperature), and Influential Features, which modulate the risk landscape over the battery’s lifespan (e.g., state of health). A subset of these key features is summarized in the table below.
| Feature ID | Feature Name | Symbol | Calculation Formula | Type |
|---|---|---|---|---|
| 1 | Maximum Cell Voltage | \(V_{max}\) | \( \max\{V_1, V_2, …, V_n\} \) | Indicative |
| 2 | Minimum Cell Voltage | \(V_{min}\) | \( \min\{V_1, V_2, …, V_n\} \) | Indicative |
| 3 | Cell Voltage Inconsistency | \(V_{inc}\) | \( \max\{V_i – V_j\} \) for all \(i, j\) | Indicative |
| 4 | Maximum Cell Temperature | \(T_{max}\) | \( \max\{T_1, T_2, …, T_n\} \) | Indicative |
| 5 | Temperature Inconsistency | \(T_{inc}\) | \( \max\{T_i – T_j\} \) for all \(i, j\) | Indicative |
| 6 | Maximum Temperature Rise Rate | \(\dot{T}_{max}\) | \( \Delta T / \Delta t \) for the fastest-rising cell | Indicative |
| 7 | System State of Health (SOH) | \(ASOH_{sys}\) | Estimated capacity / Initial capacity | Influential |
| 8 | Minimum Cell SOH | \(ASOH_{min}\) | \( \min\{ASOH_1, ASOH_2, …, ASOH_n\} \) | Influential |
| 9 | SOH Decline Rate | \(\dot{ASOH}\) | \( \Delta ASOH / \Delta t \) | Influential |
The static sum-of-entropies model is insufficient because the significance of a feature evolves. For instance, a small voltage inconsistency might be tolerable in a new lithium-ion battery but a strong precursor to failure in an aged one. Therefore, the weights \( \alpha_i \) in the Safety Entropy equation must be dynamic. I propose a dual-strategy weighting mechanism: knowledge-based fixed functions and data-driven adaptive updates.
The knowledge-based weights utilize specific mathematical functions whose shape is chosen based on the failure mode physics associated with each feature for the lithium-ion battery. Three primary function types are employed:
- Exponential Function: Used for critical indicative features where exceeding a threshold signifies rapidly escalating risk.
$$ w_i = a \cdot e^{-b(x_i – x_0)} $$
Here, \(x_i\) is the feature value, \(x_0\) is a reference threshold, and \(a, b\) are scaling parameters. When \(x_i > x_0\), the weight increases exponentially. - Sigmoid Function: Used for features with a gradual influence, such as average parameters or efficiency metrics.
$$ w_i = \frac{a}{1 + e^{-b(x_i – x_0)}} $$
This provides a smooth transition from a lower to an upper weight bound. - Softmax-type Normalization: Used for distribution-based features (e.g., count of cells at minimum voltage) to emphasize outliers.
$$ w_i = \frac{e^{x_i – \bar{x}}}{\sum e^{x_i – \bar{x}}} $$
This amplifies the weight of features that deviate significantly from the pack’s average state.
The following table outlines the weighting strategy for selected features of the lithium-ion battery diagnostic system.
| Feature (Symbol) | Weight Function Type | Weight Formula | Typical Bounds |
|---|---|---|---|
| Voltage Inconsistency (\(V_{inc}\)) | Exponential | \(w = a e^{-b(V_{inc} – V_0)}\) | 1 to 3 |
| Max. Temp. Rise Rate (\(\dot{T}_{max}\)) | Exponential | \(w = a e^{-b(\dot{T}_{max} – \dot{T}_0)}\) | 1 to 3 |
| Average Pack Voltage (\(\bar{V}\)) | Sigmoid | \(w = \frac{a}{1 + e^{-b(\bar{V} – V_0)}}\) | 0 to 1 |
| Cell SOH (\(ASOH_{min}\)) | Softmax | \(w = \frac{e^{ASOH_{min} – \overline{ASOH}}}{\sum e^{ASOH_i – \overline{ASOH}}}\) | 0 to 3 |
Furthermore, the overall influence of certain weight categories can be linked to the global aging of the lithium-ion battery. As the battery’s State of Health (SOH) degrades, features related to inconsistency and performance fade become more significant. This is modeled by making the scaling factor \(a\) in some weight functions a variable dependent on \(ASOH_{sys}\):
$$ a(ASOH) = A_{SOH_0} – \frac{A_{SOH_{Cur}}}{1 + \frac{(A_{SOH_0} – A_{SOH_n})}{f \cdot A_{SOH_0} A_{SOH_n}} \cdot A_{SOH_j}} $$
Where \(A_{SOH_0}\), \(A_{SOH_{Cur}}\), and \(A_{SOH_n}\) represent the initial, current, and end-of-life SOH values, respectively, and \(f\) is a tuning parameter. This equation ensures that as the lithium-ion battery ages (\(A_{SOH_{Cur}}\) decreases), the weights for critical indicative features can grow non-linearly, enhancing the sensitivity of the Safety Entropy to incipient faults in an aged pack.
Algorithm Validation with Real-World Vehicle Data
To rigorously evaluate the proposed Safety Entropy algorithm, I conducted validation tests using large-scale, real-world operational data. The first dataset comprised cloud-based telemetry from over 2,500 electric vehicles, collected in compliance with national monitoring standards. This dataset included 24 vehicles with confirmed historical battery safety incidents (e.g., thermal runaway, internal short circuits) and approximately 2,479 normal vehicles, creating a highly imbalanced test set realistic for field deployment.
The algorithm was applied to the historical data stream of each vehicle. For the 24 fault vehicles, the model successfully generated early warnings for 23 of them, corresponding to a detection accuracy of:
$$ P_T = \frac{n_T}{n_{sum}} = \frac{23}{24} \approx 95.83\% $$
where \(P_T\) is the true positive rate, \(n_T\) is the number of correctly identified fault vehicles, and \(n_{sum}\) is the total number of fault vehicles. Among the entire fleet of 2,503 vehicles, the algorithm produced 28 alerts. With 23 correct alerts and 5 alerts from normal vehicles, the false positive rate (FPR) was calculated as:
$$ P_W = \frac{n_W}{N_{sum}} = \frac{5}{2503} \approx 0.20\% $$
where \(P_W\) is the false positive rate, \(n_W\) is the number of false alarms, and \(N_{sum}\) is the total number of vehicles. This extremely low FPR is critical for operational acceptance, as frequent false alarms would lead to warning fatigue and system distrust.
Most significantly, the temporal analysis revealed that for the correctly identified fault vehicles, the Safety Entropy-based warning was triggered at least 7 days prior to the actual recorded safety incident. This represents a monumental improvement over traditional methods that typically offer warnings on the order of minutes or hours. The algorithm’s capability to highlight growing inconsistencies—such as a specific cell’s voltage beginning to diverge from the pack or a localized temperature hotspot—days in advance demonstrates its power for proactive risk mitigation in lithium-ion battery systems.
| Validation Case | Total Vehicles | Fault Vehicles | Correct Alerts | False Alerts | Accuracy | False Positive Rate | Key Advance |
|---|---|---|---|---|---|---|---|
| EV Fleet Cloud Data | 2,503 | 24 | 23 | 5 | ~95.8% | ~0.20% | >7-day early warning |
Extended Validation Using Charging Infrastructure Data
To further demonstrate the versatility of the Safety Entropy framework, a second validation was performed using data from the charging infrastructure side. This scenario is particularly relevant for fleet operators and public charging stations. I analyzed charging session records from over 500 public fast-charging piles over a four-month period. The available data, being less frequent than vehicle-borne telemetry, required a tailored feature set of 14 dimensions, including voltage/temperature profiles, incremental capacity (IC) curve characteristics, derived internal resistance, and charging behavior patterns.
In this context, the algorithm functioned as a screening tool during charging sessions. For example, one charging pile’s log showed two users with elevated Safety Entropy scores. Analysis of the feature contributions pointed to significant voltage inconsistency (0.15 V) for one user and severe temperature inconsistency (16°C) for another as the primary risk contributors. Such an assessment could trigger protective actions at the charger, such as limiting charging power or notifying the user/operator. Other piles showed users with persistently high but sub-threshold entropy values, often linked to excessively long charging sessions—a behavior indicative of potential battery management issues or underlying cell degradation in the lithium-ion battery. These users could be flagged for further inspection or advisory services.
This cross-validation proves that the Safety Entropy model is not confined to the vehicle BMS alone. It can be effectively deployed in distributed monitoring architectures, including cloud analytics platforms and smart charging stations, providing a multi-layered safety net for lithium-ion battery systems throughout their usage cycle.
Conclusions and Broader Implications
This research presents a comprehensive and novel framework for the early diagnosis of faults in lithium-ion battery systems by introducing the concept of Safety Entropy. Moving beyond the limitations of direct parameter thresholding, this approach provides a unified, probabilistic metric for quantifying thermal runaway and failure risk. The integration of a dynamically weighted multi-feature fusion mechanism ensures the diagnostic system remains sensitive and relevant across the entire lifespan of the lithium-ion battery, from initial deployment to advanced age. The validation results are compelling: the algorithm achieves high detection accuracy (~96%) with a remarkably low false alarm rate (~0.2%), and most importantly, it provides a critical early warning lead time of more than seven days prior to potential failure events.
The implications of this work extend far beyond the electric passenger vehicle sector, which served as the primary validation platform. The core principles of the Safety Entropy model and its adaptive weighting scheme are directly transferable to any application employing large-scale lithium-ion battery energy storage. This includes electric buses, railway rolling stock, maritime vessels, and stationary grid storage—all domains where safety and reliability are non-negotiable. By enabling truly proactive risk management, this framework provides a powerful tool for system designers, operators, and maintenance crews to enhance the safety pedigree of lithium-ion battery technology, thereby accelerating its safe integration into the broader fabric of sustainable transportation and energy infrastructure globally.