Early Warning Method for Thermal Runaway High-Risk Cells Based on Nonlinear Mapping and Multidimensional Features

Abstract

In harsh marine environments, vessel energy storage systems (VESS) face elevated thermal runaway (TR) risk, yet practical early warning remains difficult because early voltage differences between TR high-risk cells and normal cells are often weak, warning thresholds vary across operating segments, and decisions relying on a single feature are prone to false or missed warnings. To overcome these difficulties, this study develops a four-part early warning strategy for TR high-risk cells in VESS. First, the original cell voltages are denoised through multiscale jump plus mode decomposition and Spearman correlation guided mode reconstruction to suppress irrelevant interference. Second, an improved Sigmoid nonlinear mapping is introduced to enhance subtle inter-cell voltage deviations and improve early separability. Third, sparse representation is used to construct a cell deviation score, and an adaptive threshold is employed to perform primary abnormal-cell screening under varying segment conditions. Finally, multidimensional mutual information value derived from voltage, temperature, and their rates of change is incorporated into a joint assessment methodology to further verify the abnormal state of flagged cells. Validation on 18 independent real operation cases comprising 2483 discharge segments shows that, across the evaluated TR high-risk cases, the shortest confirmed warning lead time achieved by the proposed strategy was 14 days. The proposed strategy also reduced false and missed warnings, outperformed the compared benchmark methods overall, and retained computational feasibility for onboard application in VESS.

1. Introduction

Electric vessels have shown remarkable potential in supporting China’s objectives of reaching carbon peaking by 2030 and carbon neutrality by 2060 [1,2]. Lithium-ion batteries are the preferred energy storage technology for electric vessels with their high energy density, extended lifespan, and low self-discharge rate [3]. Nevertheless, the marine environment presents specific challenges, including salt spray corrosion and high humidity [4,5]. The combination of these factors intensifies the mechanical, electrical, and thermal stresses on the cells, leading to faster degradation and an increased risk of thermal runaway (TR) [6,7]. Another major concern is the limited opportunity for inspection and maintenance during long voyages. When potential cell hazards are not detected early, TR may occur and threaten both crew and vessel safety [8,9]. In a recent marine-focused assessment, Bugryniec et al. [10] showed that thermal runaway in marine Li-ion battery energy storage systems poses more severe fire, explosion, and operational consequences than in many land-based applications, further highlighting the need for proactive warning strategies in vessel scenarios. Although TR early warning methods have been explored in land-based systems [11], the challenges in vessel applications remain insufficiently addressed. More advanced warning methods are therefore needed to ensure safety and support vessel electrification.

Traditional TR early warning methods primarily rely on signals such as smoke, temperature, venting sounds, and emitted gases [12,13,14,15]. However, these signals usually appear only after the cell condition has already deteriorated, meaning the cell is often close to TR [16]. As a result, such methods may not provide enough lead time for preventive intervention [17]. To address this, researchers focus on early warning technologies that can detect TR high-risk cells hours or days in advance, categorized into model-based, data-driven, and signal analysis-based methods [18].

Model-based methods detect abnormalities by establishing a mathematical model of the cell and comparing model outputs with measured signals or preset thresholds [19]. In general, these methods require relatively high sampling frequencies to support rapid state updating and parameter iteration. Yet in practice, vessel energy storage systems (VESS) face challenges from low sampling frequency and frequent information loss, which impair feature extraction and reduce model accuracy. Additionally, reliance on experimental data with predefined abuse scenarios makes it difficult to deliver reliable warnings in real-world conditions [20], making model-based approaches unsuitable for VESS.

By contrast, data-driven methods identify faults or anomalies by learning directly from large datasets, without requiring an exact mathematical model [21]. This reduces the burden of explicitly modeling complex electrochemical processes and can improve robustness to sampling precision and external disturbances. Fan et al. [22] developed a feature-augmented attentional autoencoder for early battery-pack fault detection and combined it with a local outlier factor module to reduce detection time and false warnings. Such a data-driven framework can improve sensitivity to incipient faults through latent-feature enhancement. However, it still depends on model training, hidden-feature construction, and additional algorithmic modules. That said, TR high-risk samples are typically scarce, which makes training more difficult and increases the risk of overfitting. Their computational burden, model complexity, and sometimes delayed warning response also constrain practical deployment in VESS [23].

Methods based on signal analysis frequently rely on time-domain and frequency-domain characteristics to detect abnormalities in cells, generating warnings when these features depart from expected distributions [24]. In their study, Yao et al. [25] applied the discrete wavelet transform to decompose and reconstruct the voltages, which helped mitigate noise interference and enhance the visibility of fault features, thereby contributing valuable input to the warning model. Their experimental results demonstrated that wavelet transformed voltage could significantly improve warning accuracy. Similarly, Li et al. [26] and Jiang et al. [27] employed empirical mode decomposition and variational mode decomposition (VMD), respectively, to eliminate noise interference in the voltage collected via a battery management system. However, the voltages in VESS are often characterized by a combination of oscillations and brief, sharp variations, which may limit the application of these decomposition methods due to issues such as mode-mixing and marginal effects [28]. In addition, the studies mentioned above involve screening and reconstructing each mode based on a specific frequency to extract fault characteristics. However, because the reactions associated with cell abnormalities may manifest across different frequency bands, filtering modes around only specific frequencies may inadvertently discard information relevant to those abnormalities [29]. Meanwhile, from a long-horizon warning perspective, the voltages of early-stage TR high-risk and normal cells can be highly similar. As a result, relying directly on denoised voltage may lead to missed warnings of potential risks.

To address this, Tang et al. [30] employed linear mapping to proportionally amplify the voltage and quantified voltage differences by calculating the longitudinal outlier average (LOA), which distinguishes these similarities. However, in cases of substantial “similarity,” the linear mapping method fails to highlight local feature differences, which may result in inaccurate warning outcomes. Sun et al. [31] developed a sparse voltage data observer that quantifies the differences between normal and abnormal cells into anomaly scores, enabling warnings by tracking these variations. The approach, however, relies on a fixed sparse observer, which fails to adapt to characteristics that shift with environmental or operational conditions. Jiang et al. [32] employed the modes extracted from voltage mode decomposition to compute correlation coefficients between cells, using these correlation coefficients as features for early warning through the isolated forest algorithm. Further, Jiang et al. [33] combined the voltage characteristic parameters obtained from a sparse self-encoder and singular value decomposition to achieve early warning using the Chauvenet criterion and dynamic thresholding. Yu et al. [34] proposed an energy–entropy method combined with Z-scores for diagnosing voltage abnormality in battery systems. By introducing sliding and calculation windows, their method improved the timeliness of long-term voltage abnormality diagnosis and showed advantages over several entropy- and outlier-oriented alternatives. Nevertheless, the warning decision still relies mainly on voltage signal irregularity. Hong et al. [35] used voltage as a feature to generate warnings by quantifying dynamic voltage changes through multiscale entropy and voltage anomaly coefficients. However, depending solely on fixed thresholds and a single voltage feature for warning generation may result in insufficient accuracy across different scenarios.

Despite these advances, two methodological gaps remain for early warning under real VESS conditions. First, most existing studies improve only one stage of the warning process, such as signal denoising, feature amplification, anomaly scoring, or threshold setting, but seldom organize them into a coherent warning architecture for long-horizon, segment-varying data under real operation. Second, current methods often rely either on a single-feature screening result or on direct feature fusion, while lacking a staged confirmation mechanism that can preserve early sensitivity and simultaneously suppress single-feature-induced false warnings. These gaps limit both the adaptability and the practical credibility of warning decisions in long-horizon vessel operation.

Accordingly, this study develops an early warning strategy for TR high-risk cells in VESS by combining nonlinear voltage mapping, adaptive abnormal-cell screening, and joint assessment methodology. The main intended contributions are summarized as follows.

(1)

In the early stage of abnormal evolution, the voltage responses of TR high-risk cells and normal cells are often highly similar, which limits the effectiveness of direct warning based only on denoised voltage signals. To improve early separability, an improved Sigmoid nonlinear mapping method is introduced to amplify subtle inter-cell voltage deviations and enhance the sensitivity of voltage features to incipient abnormalities.

(2)

Because abnormal cell evolution is case-dependent and the corresponding screening boundary varies across segments and operating states, a cell abnormality screening method combining sparse representation with adaptive thresholding is developed. Sparse representation is used to construct the cell deviation score (DS) from reconstruction error, while the adaptive threshold is used to determine a segment-dependent screening boundary, thereby improving the adaptability of abnormal cell identification under real operating conditions.

(3)

Since voltage feature alone is insufficient to fully characterize the abnormal state of cells under complex operating conditions, a joint assessment methodology based on multidimensional mutual information value (MIV) is further established. By incorporating voltage, temperature, and their rates of change into multidimensional feature vectors, this methodology provides consistency confirmation for the cells identified in the primary screening stage, thereby reducing single-feature-induced false warnings and improving the credibility of the final warning decision.

The remainder of this paper is organized as follows: Section 2 describes voltage processing and the principles of the proposed warning strategy. Section 3 presents a preliminary description and analysis of the validation data. Section 4 discusses the warning results. Section 5 concludes the paper.

2. Data Processing and Proposed Early Warning Method

The proposed early warning strategy is designed as a four-part framework. Its overall logic is to first improve the quality of the raw voltage signal, then enhance the separability between TR high-risk cells and normal cells, next perform primary abnormal-cell screening under segment-dependent conditions, and finally refine the warning decision through multidimensional state assessment. In this way, signal processing, voltage feature enhancement, abnormal screening, and joint assessment methodology are organized into one complete warning process. The overall framework is illustrated in Figure 1.

(1)

Data processing: Only voltage, temperature, and their corresponding rates of change are retained as downstream analysis variables, while current and SOC are treated as information on partitioning data segments and are not included in the subsequent analysis. The original voltages are then decomposed using multiscale jump plus mode decomposition (MJMD), with a genetic algorithm used for parameter optimization. Denoised voltages are obtained by reconstructing the decomposed modes according to the Spearman-correlation criterion.

(2)

Voltage mapping and abnormal cell screening: An improved Sigmoid nonlinear mapping method is applied to the denoised voltages to enhance inter-cell differentiation. A sparse representation scheme is then used to construct a representative voltage basis, and each mapped voltage is reconstructed through a sparse linear combination. Based on the reconstruction error, a cell DS is calculated and combined with the adaptive threshold to screen abnormal cells.

(3)

Calculation of MIV: To capture both static and dynamic cell characteristics, four-dimensional feature vectors are constructed from voltage, temperature, and their corresponding rates of change. The MIV of each cell’s four-dimensional vector is then calculated, providing the basis for the subsequent joint assessment.

(4)

Joint assessment methodology: Based on the abnormal screening results and the MIV of each cell, a joint assessment method with three decision rules is established to further improve early warning accuracy.

2.1. Voltage Processing

VESS operates in complex environments, and voltage acquisition is particularly sensitive to sampling accuracy because the measured signal is at the millivolt level and can be strongly affected by disturbances. Accordingly, a voltage decomposition and reconstruction scheme, commonly used in related studies, is adopted to filter extraneous interference from the raw voltage signal. To address the limitations of conventional decomposition methods for non-stationary step-like voltages, MJMD is used to decompose the original voltage. The technique models the signal as a sum of $G$ adaptive amplitude, frequency-modulated modes and a sparse jump term. The decomposition effect is primarily influenced by the bandwidth penalty, jump weights, scaling factor, minimum jump amplitude threshold, and the number of modes $G$ [36]. To avoid unstable decomposition caused by manually selected parameters, a genetic algorithm is used to optimize the key hyperparameters, with the total variance explained ratio taken as the fitness function. The parameter settings are presented in

Table 1. Parameter settings for the MJMD method.

Parameters	Value
Bandwidth penalty	5000
jump weights	0.05
Scaling factor	50
Minimum jump amplitude threshold	0.45
Number of modes	4

.

To demonstrate the advantage of MJMD, the decomposition result for a representative cell is compared with that of the commonly used VMD method under the same parameter settings. The mode mixing rate (MMR) is used as the evaluation metric, and the results are shown in Figure 2. To calculate the MMR, power spectral density analysis is first applied to each decomposed voltage mode to obtain its frequency energy distribution, defined as:

$$P_{g,k}(f)=\mathcal{F}\left\{u_{g,k,j}\right\}$$

(1)

where k is the cell number. j is the sampling time. $\mathcal{F}$ is the Fourier transform. $P_{g,k}(f)$ is the energy distribution of the mode function $u_{g,k,j}$ at frequency f.

The proportion of the dominant frequency energy $R_{g,k}$ in the mode can be expressed as:

$$R_{g,k}={\displaystyle \frac{{\int }_{f_{p,g,k}-\Delta f}^{f_{p,g,k}+\Delta f}{P}_{g,k}(f)df}{{\int }_0^{f_{\max }}{P}_{g,k}(f)df}}$$

(2)

where $f_{p,g,k}$ is the dominant frequency of the mode g. $\Delta f$ is the small frequency window, and $f_{\max }$ is the upper limit of the frequency range.

The average proportion of non-dominant frequency energy across all modes is calculated, yielding the overall MMR:

$$MMR={\displaystyle \frac{1}{G}}{{\displaystyle \sum }}_{g=1}^G\left(1-R_{g,k}\right)$$

(3)

IMF1–5 in Figure 2 represent the mode numbers under both methods. As shown in Figure 2, except for the DC mode (IMF1), MJMD yields a higher dominant-frequency energy share for each mode and a lower MMR than VMD, indicating weaker mode mixing.

To avoid information distortion caused by frequency-only mode screening, Spearman’s rank correlation coefficient is used to evaluate the correlation between each MJMD mode and the original voltage, so that interference components can be identified and removed. It has been shown that there is a monotonically decreasing Spearman correlation between each mode and the original signal obtained by VMD or similar methods. The least relevant of these modes is usually noise or irrelevant interference. In contrast, others contain physically meaningful information, and further culling will result in the loss of valid information about the signal [37,38]. Therefore, only the least correlated modes are removed, and the remaining modes are superimposed to obtain the denoised voltage. The method for calculating correlation is as follows:

$${\rho }_{g,k}=1-{\displaystyle \frac{6{\displaystyle \sum _{j=1}^J}{\left(R_{x,k,j}-R_{g,k,j}\right)}^2}{J\left(J^2-1\right)}}$$

(4)

where ${\rho }_{g,k}$ is the correlation coefficient. J is the length of the data segment. $R_{x,k,j}$ is the rank of the original voltage, and $R_{g,k,j}$ is the rank of the $u_{g,k,j}$.

Figure 3 shows that the adopted method effectively suppresses irrelevant interference in the original voltage, reduces noise influence, and preserves the intrinsic signal characteristics more clearly.

2.2. Voltage Mapping and Screening of Abnormal Cells

2.2.1. Voltage Mapping

Abnormal voltage variations in TR high-risk cells can be on the order of tens of millivolts, making the voltage differences between TR high-risk cells and normal cells often extremely small [39]. As a result, reliable early warning is difficult to achieve at the early stage if the denoised voltage is used directly. Therefore, the denoised voltages need to be mapped to enhance the distinguishability between TR high-risk cells and normal cells for subsequent evaluation. However, traditional linear mapping [30] cannot adaptively adjust its sensitivity across small differences and extreme values, which may make the mapped results either too conservative or overly reactive.

To address this issue, a nonlinear voltage mapping method based on an improved Sigmoid function is adopted. The improved Sigmoid mapping is used because its positive half-axis is smooth, bounded (0.5 to 1), and most sensitive when the input is close to zero. In other words, small inter-cell voltage differences can be amplified effectively in the output, which improves early-stage distinguishability. By contrast, when the voltage difference becomes large, the mapped value gradually approaches its upper bound, helping avoid overreaction and keeping the mapping stable. The specific principles are as follows.

The matrix form of the denoised voltage obtained after voltage processing can be expressed as:

$$\tilde{V}=\left[\begin{array}{cccc}{\tilde{x}}_{1,1}& {\tilde{x}}_{2,1}& \cdots & {\tilde{x}}_{k,1}\\ {\tilde{x}}_{1,2}& {\tilde{x}}_{2,2}& \cdots & {\tilde{x}}_{k,2}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{x}}_{1,j}& {\tilde{x}}_{2,j}& \cdots & {\tilde{x}}_{k,j}\end{array}\right]$$

(5)

where ${\tilde{x}}_{k,j}$ is the denoised voltage.

The proposed nonlinear mapping method is shown below:

$$z_{j,k}={\displaystyle \frac{1}{1+\exp \left[-{\alpha }_{map}{\left(1+\overline{{\tilde{x}}_j}-{\tilde{x}}_{k,j}\right)}^{{\gamma }_{map}}\right]}}$$

(6)

where $z_{j,k}$ is the mapped voltage. $\overline{{\tilde{x}}_j}$ is the reference voltage. ${\alpha }_{map}$ is the amplification coefficient, set to 2. ${\gamma }_{map}$ is the dynamic exponent. A brief explanation of Equation (6) is provided for clarity. Cell abnormalities are relatively rare, so most cell voltages remain within the normal range. To represent the overall voltage trend while reducing the influence of outliers, the reference voltage is taken as the mean across cells at the same sampling instant after removing the maximum and minimum values. In typical operation, $\overline{{\tilde{x}}_j}-{\tilde{x}}_{k,j}$ is usually stable and generally remains below 1, so the mapped output tends to stay in a relatively stable range. The added offset term (1) prevents zero-difference input and avoids excessive compression in the mapped voltage. Meanwhile, ${\alpha }_{map}$ controls the mapping sensitivity, whereas ${\gamma }_{map}$ primarily adjusts the dynamic response to voltage differences. The validation results in later sections show that the proposed mapping method performs better than the compared mapping methods in this study.

2.2.2. Abnormal Cell Screening Based on Sparse Representation

The mapped voltages provide a clearer representation of voltage feature distribution. To quantify how far a cell deviates from normal operation, a set of basis vectors is constructed using sparse representation theory to approximate the mapped voltage [40]. The reconstruction error is then used to quantify deviation and screen abnormal cells. The specific principle is as follows.

First, principal component analysis (PCA) is applied to the mapped voltage matrix [41], and the first m principal component voltages in $z_k$ are extracted to form the initial basis set $D$, as expressed:

$$D=\left[z_1,z_2,\dots ,z_m\right]$$

(7)

Each basis vector in the initial set can characterize the mapped voltage. For a mapped voltage of cell k, the sparse representation is written as:

$$z_k\approx D\cdot {\alpha }_{sr,k}$$

(8)

where ${\alpha }_{sr,k}$ is the sparse coefficient vector, indicating the sparse linear combination of the voltage within the set.

To obtain the optimal ${\alpha }_{sr,k}$, the Lasso regression algorithm is used to control the balance between reconstruction error and sparsity by optimizing the objective function as shown below:

$$\underset{{\alpha }_{sr,k}}{\min } {\left|z_k-D{\alpha }_{sr,k}\right|}^2+{\lambda }_{sr}{\alpha }_{sr,k}$$

(9)

where ${\lambda }_{sr}$ is the regularization parameter used to balance reconstruction error and sparsity, $e_{sr,k}$ is the reconstruction error, defined as:

$$e_{sr,k}=z_k-D{\alpha }_{sr,k}$$

(10)

The basis set $D$ is then updated iteratively using the reconstruction error and sparse coefficients according to:

$$D\leftarrow D+{\eta }_{sr}\cdot e_{sr,k}\cdot {\alpha }_{sr,k}$$

(11)

where ${\eta }_{sr}$ is the update step size, which controls the speed of the set’s updates.

The updates stop when set $D$ reaches the maximum number of iterations or satisfies the following convergence condition:

$$D-D_{pre}\le D_{td}$$

(12)

where $D_{pre}$ is the set from the previous iteration, and $D_{td}$ is the convergence threshold of the set.

Finally, the deviation of each cell’s reconstruction error from the average reconstruction error of all cells is defined as the cell deviation score (DS), as follows:

$$D{S}_k={\overline{e}}_{sr,k}-{\displaystyle \frac{1}{K}}\sum _{k=1}^K{\overline{e}}_{sr,k}$$

(13)

where $D{S}_k$ is DS of the cell k. ${\overline{e}}_{sr,k}$ is the average value of the reconstruction error of cell k for all sampling points.

To improve the adaptability of abnormal cell screening in real operating scenarios, an adaptive threshold setting method is introduced. A cell is marked as abnormal when its DS exceeds the threshold, which is calculated as follows:

$$V_{th}={\theta }_{DS}\cdot {\mu }_{DS}+(1-{\theta }_{DS})\cdot ({\zeta }_{DS}\cdot \max (D{S}_k)+(1-{\zeta }_{DS})\cdot ({\mu }_{DS}+{\beta }_{DS}\cdot {\sigma }_{DS}))$$

(14)

where $V_{th}$ is the threshold. ${\theta }_{DS}$ is the weight coefficient used to balance the mean and subsequent terms, set to 0.3. ${\xi }_{DS}$ is the weight coefficient for balancing the maximum value and statistical features, set to 0.5. ${\mu }_{DS}$ is the mean of all DS. ${\sigma }_{DS}$ is the standard deviation of all DS; ${\beta }_{DS}$ is the amplification factor of the standard deviation, set to 12; and $\max (D{S}_k)$ is the maximum DS for all cells in this segment. Unlike a fixed threshold, this threshold is determined from the DS statistics of the current segment. As a result, the screening boundary can vary with the local data distribution, which improves the adaptability of abnormal-cell screening under different operating conditions and segment states.

2.3. Evaluation of Cell State Based on MIV

If TR high-risk cells are identified only from DS-based abnormal screening, false warnings or missed warnings may occur because the decision relies on a single voltage feature. To improve final warning accuracy, the correlation among multidimensional features across cells is further quantified using a mutual information method [42], which provides an additional assessment of each cell state. Since the mean characteristics of all cells within the same data segment can represent normal behavior, the mutual information between each cell and the mean feature vector is calculated to identify abnormal cell behavior. Voltage and temperature are fundamental indicators of cell operating state and thermal behavior. However, static voltage and temperature alone may not adequately capture the dynamic evolution of early anomalies. Therefore, the voltage change rate and temperature change rate are further introduced to construct a four-dimensional feature set for MIV calculation. By contrast, current and SOC, although used in the initial data processing stage for segment partitioning, are not included in the joint assessment because, within the same segment, they mainly characterize the shared operating background rather than cell-specific inconsistency and thus provide limited additional discriminatory information for abnormal cell identification.

For cell k, the four-dimensional feature vector and the corresponding mean vector of sampling point j within each segment can be defined as:

$$\begin{array}{c}{x}_{j,k}=[z_{j,k},T_{j,k},\Delta z_{j,k},\Delta T_{j,k}]\\ {\overline{x}}_j=[{\overline{z}}_j,{\overline{T}}_j,{\overline{\Delta z}}_j,{\overline{\Delta T}}_j]\end{array}$$

(15)

where $\Delta z_{j,k}$ is the rate of change of the mapped voltage. $T_{j,k}$ and $\Delta T_{j,k}$ are temperature and rate of change of temperature, respectively. ${\overline{z}}_j$ is the average of the mapped voltage. ${\overline{T}}_j$ is the average of temperature. ${\overline{\Delta z}}_j$ is the average of the mapped voltage rate of change. ${\overline{\Delta T}}_j$ is the average of the rate of temperature change.

Because the differenced features and the original series have different lengths, the first sample of the original series is discarded so that all feature dimensions remain length-consistent before MIV calculation.

Before MIV is calculated, the multidimensional feature sequences of the two variable sets are grouped to estimate their probability distributions. Because segment lengths vary, the Freedman–Diaconis rule is used to determine the number of groups adaptively, as follows:

$$B_d=⌈{\displaystyle \frac{\max (X_d)-\min (X_d)}{2\cdot IQR(X_d)/N^{1/3}}}⌉,\hspace{1em}d=1,2,3,4$$

(16)

where $X_d$ is the complete set of data in the d-th dimension. $IQR(X_d)$ is the quartile spacing of dimension d. $B_d$ is the number of groups. $N=J\times K$ is the total number of samples. $\lceil .\rceil$ denotes upward rounding.

For cell k, the group number corresponding to dimension d at time j can be expressed as:

$$p_{d,j}=binid\left(x_{k,d,j}\right),q_{d,j}=binid\left({\overline{x}}_{d,j}\right),\hspace{1em}{p}_d=1,2,\dots ,B_d$$

(17)

where $binid(\cdot )$ is the group number of the queried value in dimension d. $p_{d,j}$ is the group number corresponding to the d-th dimension of cell k at moment j. $q_{d,j}$ is the group number corresponding to the d-th dimension of the mean vector at moment j.

Then, the marginal frequency and marginal probability of a single dimension can be defined as:

$$bin\left(x_{k,d}^{(p_d)}\right)=\sum _{j=1}^J\mathbb{I}\left[p_{d,j}=p_d\right]$$

(18)

$$p(x_{k,d}^{(p_d)})={\displaystyle \frac{bin(x_{k,d}^{(p_d)})}{J}}$$

(19)

where $bin(x_{k,d}^{(p_d)})$ is the number of samples in which cell k is grouped in the d-th dimension with group number $p_d$. $\mathbb{I}\left[\cdot \right]$ is a conditional function that takes one if the condition is proper and zero otherwise. $p(x_{k,d}^{(p_d)})$ is the corresponding marginal probability of cell k in dimension d.

Similarly, the marginal probability of the mean vector in dimension d can be expressed as:

$$p({\overline{x}}_d^{(q_d)})={\displaystyle \frac{bin({\overline{x}}_d^{(q_d)})}{J}}$$

(20)

Further, the joint frequency and joint probability of cell k and the mean vector in the four dimensions can be expressed as:

$$bin(p,q)=\sum _{j=1}^J\prod _{d=1}^4\mathbb{I}[p_{d,j}=p_d]\mathbb{I}[q_{d,j}=q_d]$$

(21)

$$p(p,q)={\displaystyle \frac{bin(p,q)}{J}}$$

(22)

where $p=(p_1,p_2,p_3,p_4)$, $q=(q_1,q_2,q_3,q_4)$ denote the grouping number combination of cell k and the mean vector in four dimensions, respectively. $bin(p,q)$ is the number of samples that simultaneously satisfy $p_{d,j}=p_d,q_{d,j}=q_d$ in J. $p(p,q)$ is the four-dimensional joint probability.

Then, the MIV of the cell k and the mean vector can be expressed as:

$$I_k=\sum _{p_1=1}^{B_1}\sum _{p_2=1}^{B_2}\sum _{p_3=1}^{B_3}\sum _{q_4=1}^{B_4}\sum _{q_1=1}^{B_1}\sum _{q_2=1}^{B_2}\sum _{q_3=1}^{B_3}\sum _{q_4=1}^{B_4}p(p,q)\cdot \log {\displaystyle \frac{p(p,q)}{{\displaystyle \prod _{d=1}^4}\left[p(x_{k,d}^{(p_d)})\cdot ({\overline{x}}_d^{(q_d)})\right]}}$$

(23)

Define the cell whose MIV is lower than the set threshold as the MIV abnormal cell, and the threshold can be set as follows:

$$T_{MIV}={\alpha }_{MIV}\cdot {\mu }_{MIV}+(1-{\alpha }_{MIV})\cdot \min (I_k)$$

(24)

where $T_{MIV}$ is the threshold. ${\alpha }_{MIV}$ is the adjustment factor, set to 2.5. ${\mu }_{MIV}$ is the mean of MIV.

2.4. Joint Assessment Methodology

To improve the credibility of the final warning decision, DS and MIV are not treated as two parallel warnings. Instead, they are organized into a staged decision mechanism in which DS is used for primary screening, while MIV is used for multidimensional consistency confirmation of cells that have already shown sustained or recurrent DS abnormalities. The joint assessment rules are as follows:

Rule 1 (persistent high-risk identification). If a cell is marked as DS-abnormal in three consecutive segments, its abnormal behavior is considered persistently elevated, and the cell is directly identified as a TR high-risk cell.

Rule 2a (potential-risk identification). If a cell is detected as DS-abnormal in two consecutive segments, it is marked as a potential TR high-risk cell.

Rule 2b (recurrent potential-risk identification). If a cell has been marked as DS-abnormal in two historical segments and is detected as DS-abnormal again in the following segment, it is also marked as a potential TR high-risk cell.

Rule 3 (DS-MIV joint confirmation). If a cell is determined to be a potential TR high-risk cell in a segment and the MIV for that segment also shows an anomaly, the cell is finalized as a TR high-risk cell.

3. Data Description and Primary Analysis

In practice, the electrochemical reactions in the cell discharge process become more complex, aging effects are more pronounced, and local faults are more readily amplified under dynamic load conditions, providing a more accurate representation of the cell’s health status. Therefore, to validate the proposed strategy, historical data from 18 independent operation cases under different conditions were selected (including two TR cases, seven risky cases, and the rest as normal cases), totaling 2483 discharge segments, where the data from one complete discharge cycle constitute one discharge segment. For clarity, one TR case, one risk case, and two normal cases labeled as #1, #2, #3, and #4 were selected for detailed description. The data sampling interval is 10 s. The cathode material of the cell is nickel-cobalt-manganese, while the anode material is graphite, with a nominal voltage range of 2.7–4.2 V.

3.1. Initial Analysis of TR Case

The data for TR case #1 spans 88 days, in which cell 23 was identified as the TR cell. Figure 4 shows the denoised and mapped voltages on the day TR occurred. From Figure 4a, cell 23 did not exhibit any notable voltage deviations before TR. However, a sharp decline in voltage was observed at the 2746th sampling point that day. In traditional voltage-based evaluation, cell 23 would be categorized as safe before the TR event, highlighting the potential for missed warnings. A comparison of Figure 4a,b indicates that, following the application of the proposed mapping method, the differences between TR high-risk cell and normal cells are significantly amplified. This outcome demonstrates the efficacy of the proposed method.

3.2. Initial Analysis of Non-TR Cases

Risk case:

The data for risk case #2 spans 90 days, in which cell 80 was flagged as a TR high-risk cell. Figure 5 shows the denoised and mapped voltages on day 52. As seen in Figure 5, cell 80 exhibits only subtle deviations, and these deviations become more noticeable toward the late-discharge stage as the voltage level drops. A plausible explanation is that polarization effects and inter-cell differences in internal resistance are more pronounced under low-voltage conditions, making the resistance contribution to the terminal voltage easier to observe. Still, the resulting differences are small and not easy to quantify from the original voltage alone. After the proposed mapping, the cell voltage separation is represented more clearly, which makes subsequent screening more tractable.

Normal case:

The data for normal case #3 spans 102 days. Figure 6 shows the denoised and mapped voltages on day 15. As illustrated in Figure 6, no clear abnormal voltage variation is observed in this segment. Although some fluctuations are present, they are more plausibly linked to abrupt load changes and cell-to-cell external characteristics rather than to a persistent fault signature. Consistent with this, the mapped voltages remain stable and broadly consistent across cells.

4. Analysis and Discussion of the Warning Results

4.1. Threshold Analysis

To verify the reasonableness of the abnormal screening threshold, nine abnormal cases and nine normal cases were selected, and the distribution of DS thresholds for all segments was analyzed. In Figure 7, the X-axis represents the case number, and the Y-axis represents the value normalized by the threshold maximum. Across both normal and abnormal cases, the medians and the upper/lower bounds show no obvious shared pattern. Even within the same case, the thresholds can vary markedly from segment to segment, as reflected by the spread in their maxima and minima. Taken together, this suggests that voltage characteristics do not follow a fixed regularity, likely because operating conditions differ across cases. Under such variability, a fixed threshold scheme is prone to missed warnings, whereas the proposed approach adapts the screening threshold to the operating condition of each case and, in doing so, mitigates inter-case differences.

4.2. Warning Results of TR Cases

Figure 8 summarizes the normalized DS of each cell in case #1 at different lead times before TR. The X-axis represents the number of days before TR, the Y-axis is the cell number, and the Z-axis is the result of DS normalized by the threshold for that segment. The red points in Figure 8 highlight cells flagged as abnormal. Cell 23 was repeatedly flagged and labeled as a potential TR high-risk cell on days 18, 15, and 11 before TR, which then triggered evaluation using MIV. The corresponding MIV values are listed in

Table 2. MIV of the TR case over 9 days.

Date	Abnormal ID	Minimum MIV	Threshold
Day 18	None	0.6955	0.6908
Day 17	23	0.9806	0.99
Day 15	25/23	0.916	0.9736
Day 11	23	0.9541	0.9574
Day 10	None	0.7731	0.6174
Day 8	None	0.6931	0.6892
Day 7	None	0.6931	0.6895
Day 6	None	0.9805	0.9773
Day 0	23	0.9217	0.9233

, and the MIV distributions for three representative days are shown in Figure 9. Notably, on day 11, cell 23 exhibits an abnormal MIV, and based on the joint assessment rule, this supports classifying cell 23 as a TR high-risk cell. Moreover, Figure 8 shows consecutive abnormal detections on days 8, 7, and 6 before TR, which further strengthens the final confirmation.

4.3. Warning Results of Non-TR Cases

Risk case:

Figure 10 presents the normalized DS of risk case #2 for nine representative days within a month. As shown in Figure 10, cell 80 is flagged as abnormal on days 2 and 13. However, because the abnormality is not detected three times consecutively, this evidence alone is insufficient to classify it directly as a TR high-risk cell. The picture changes on day 23, when cell 80 is flagged again and is therefore marked as a potential TR high-risk cell for joint assessment.

Figure 11 shows the MIV results for days 2, 13, and 23, while

Table 3. MIV of risk case over 9 days.

Date	Abnormal ID	Minimum MIV	Threshold
Day 1	None	0.9159	0.9149
Day 2	None	0.8769	0.8207
Day 4	None	0.6686	0.6672
Day 7	None	0.8769	0.8207
Day 13	None	0.845	0.7757
Day 17	None	0.6093	0.609
Day 19	None	0.6658	0.6657
Day 23	80	0.8745	0.9155
Day 31	None	0.6368	0.6367

lists the MIV values across the same 9 days. As shown in Figure 11 and Table 3, cell 80 shows an abnormal MIV on day 23, and the joint rule then supports classifying it as a TR high-risk cell. In this sense, the method still provides adequate early warning even when the anomalous signals are intermittent rather than continuous. It is also worth noting that MIV varies across segments, and an “MIV anomaly” is determined by its relation to the threshold rather than by the absolute magnitude alone. Finally, both TR and risk cases include segments where no abnormalities are detected. This is plausibly related to polarization effects or shifts in dominant reaction pathways under changing operating conditions [43,44], which may temporarily weaken or mask the outward signature of abnormal behavior. Under higher loads, polarization can amplify internal-resistance fluctuations and make abnormality more visible; under lighter loads, this effect can ease, and the cell may appear closer to “normal” until conditions change again.

Normal case:

Figure 12 shows the normalized DS of each cell in normal case #3 over 9 days within a month. As seen in Figure 12, cell 68 consistently exhibits the highest DS, yet all DS values remain below 1, indicating that no abnormal cells are detected by the DS screening.

Table 4. MIV of normal case over 9 days.

Date	Abnormal ID	Minimum MIV	Threshold
Day 1	None	0.8876	0.8845
Day 6	None	1.003	0.9927
Day 10	None	0.6960	0.6957
Day 13	31/43	1.0456	1.0641
Day 17	None	1.0687	1.0513
Day 19	None	1.0249	1.0168
Day 22	None	1.0104	1.001
Day 26	None	1.0061	1.0047
Day 30	67	0.7148	0.7418

reports the corresponding MIV over the same nine-day period. Although some cells exhibit abnormal MIV in certain segments, the DS of those cells remains normal. Under the joint assessment rule, these cells are therefore treated as normal, which helps avoid false warnings driven by a single feature.

Based on the overall analysis of all 18 evaluated cases, the proposed strategy achieved a minimum confirmed warning lead time of 14 days in the identified TR-risk cases. To interpret this overall lead time in a more practical vessel context, four representative routes were selected to reflect typical offshore and oceanic operations, as shown in

Table 5. Several typical routes and voyages.

Number	Route	Type	Day
[45]	Sweden-Finland	Bulk	≤5
[46]	Korea-Japan	Container	≤5
[47]	Rotterdam-New York	Container	7
[48]	Around the Mediterranean Sea	Cruise ship	≤10

. All four voyages are shorter than 10 days and cover a range of vessel types. Under these conditions, even the shortest warning window offered by the proposed strategy can span the full voyage and still leave a safety margin, supporting its practical feasibility.

We also assessed whether the algorithm can run fast enough on typical onboard hardware. Using an Intel i7-12700H processor, we selected the segment with the largest data volume and executed the full warning pipeline ten times; the average runtime was about 70 s. Prior work suggests a 10 min assessment window for vessel power-system condition monitoring [49]. Relative to that window, the proposed strategy can refresh warnings roughly every seven sampling intervals, indicating that real-time deployment is feasible even under constrained onboard computing resources.

4.4. Comparative Analysis

4.4.1. Effectiveness Analysis of Voltage Signal Processing

To validate the effectiveness of the voltage process, data from risk case #2 were used as a representative example. Figure 13 shows anomaly screening results obtained from the original voltage, with all other settings kept unchanged. The original voltage allows cell 80 to be flagged on days 2 and 23, but it fails to capture the abnormal state on day 13. By contrast, using the denoised voltage identifies cell 80 reliably on day 13 while preserving detections on days 2 and 23, as shown in Figure 10. This comparison suggests that denoising improves screening stability and reduces missed detections caused by noise or transient interference.

4.4.2. Comparison of Anomaly Screening Results of Three Mapping Methods

Based on the second day of the risk case #2, we compared the anomaly screening results of the proposed mapping, unmapped, and linear mapping methods, as shown in Figure 14. The X-axis denotes the cell index, and the Y-axis represents DS normalized by the threshold. The red marker indicates the abnormal cell. Without mapping, the normalized DS is the smallest and fails to identify cell 80. Linear mapping increases the DS (closer to 1) but still does not trigger an anomaly. In contrast, the proposed nonlinear mapping successfully flags cell 80 as abnormal. Overall, this comparison indicates that the proposed mapping reduces missed warnings relative to both unmapped and linearly mapped voltages.

4.4.3. Sensitivity Analysis of Key Parameters

To further justify the parameter settings used in the proposed strategy, sensitivity analyses were conducted for two influential parameters, namely the mapping coefficient ${\alpha }_{map}$ and the adaptive threshold coefficient ${\beta }_{DS}$. These two parameters directly govern the local amplification strength in the voltage mapping stage and the strictness of the DS screening threshold, respectively, and therefore have a direct impact on the final warning behavior. The experimental results are shown in

Table 6. Sensitivity analysis of the mapping parameter.

${\mathit{\alpha }}_{\mathit{m}\mathit{a}\mathit{p}}$	Case#1: Earliest Warning Time Within One Month	Case#1: Number of Cells with Missed Warning	Case#1: The Number of Days Within a Month with Detected DS Abnormalities	Case#2: Earliest Warning Time Within One Month	Case#2: Number of Cells with Missed Warning	Case#2: The Number of Days Within a Month with Detected DS Abnormalities
2	Day 4	0	8	Day 23	0	3
5	None	1	None	None	1	2
7	None	1	None	None	1	0

and

Table 7. Sensitivity analysis of the threshold parameter.

${\mathit{\beta }}_{\mathit{D}\mathit{S}}$	Case#1: Earliest Warning Time Within One Month	Case#1: Number of Cells with Missed Warning	Case#1: The Number of Days Within a Month with Detected DS Abnormalities	Case#2: Earliest Warning Time Within One Month	Case#2: Number of Cells with Missed Warning	Case#2: The Number of Days Within a Month with Detected DS Abnormalities
10	Day 16	0	9	Day 23	0	4
12	Day 20	0	8	Day 23	0	3
14	None	1	2	None	1	2

.

For ${\alpha }_{map}$, the results show that ${\alpha }_{map}=2$ provides effective warning performance in both the representative risk and TR cases, with valid warning lead time and no missed-warning cells. In contrast, when ${\alpha }_{map}$ is increased to 5 or 7, the warning capability deteriorates markedly and missed warnings occur in both cases. This indicates that moderate nonlinear amplification is beneficial for highlighting subtle early-stage inter-cell voltage differences, whereas excessively strong mapping may weaken the discriminability required for subsequent sparse-representation-based screening. Therefore, among the tested values, ${\alpha }_{map}=2$ was retained in this study.

For ${\beta }_{DS}$, both ${\beta }_{DS}=10$ and ${\beta }_{DS}=12$ maintain effective warning capability in the tested cases, while ${\beta }_{DS}=14$ makes the screening overly conservative and leads to missed warnings. In particular, a smaller ${\beta }_{DS}$ tends to trigger earlier warnings and yields more days identified as DS-abnormal, whereas a larger ${\beta }_{DS}=14$ suppresses the warning response. These results suggest that ${\beta }_{DS}$ mainly controls the balance between warning sensitivity and conservativeness. Considering this trade-off, ${\beta }_{DS}=12$ was retained as a moderate default setting in the present work.

Overall, the sensitivity analysis confirms that the selected parameter values are not arbitrary. Instead, they were chosen based on their warning performance within the tested ranges. The results also indicate that the proposed strategy remains effective under moderate parameter variation, while the threshold related parameter ${\beta }_{DS}$ should be calibrated carefully because it directly affects the warning boundary.

4.4.4. Validity Analysis of Multidimensional Features

To demonstrate the effectiveness of the proposed joint assessment method, case #4 was selected as an example. Figure 15 illustrates the DS over the 3 days when the case was flagged as abnormal, along with the corresponding MIV on day 57. As shown in Figure 15a, cell 28 was flagged for the third time on day 57; we therefore labeled it as a potential TR high-risk cell. However, further evaluation is required in conjunction with the MIV to confirm the risk classification when employing the proposed joint assessment method. According to Figure 15b, although cell 28 has the lowest MIV on day 57, it does not cross the threshold, and the cell is therefore judged as normal.

Across all cases, we observe that voltage alone is often sufficient to provide early warning for TR high-risk cells. Still, normal cases can occasionally produce voltage-driven anomalies in certain segments. Re-evaluating flagged cells with multidimensional features is therefore a practical way to suppress false warnings.

4.4.5. Comparison of Early Warning Results with Other Methods

To verify the superiority of the proposed strategy, one segment from each of cases #1, #2, and #3 was selected to compare with the warning effect of the traditional entropy and the LOA methods. As shown in Figure 16, the three columns correspond to the first day of the normal case, the second day of the risk case, and the first 18 days of the TR, respectively. The red point indicates the abnormal cell.

From Figure 16a–c, abnormal cells are detected in the normal and TR cases, but cell 80 is not effectively identified in the risk case. Even though the maximum LOA tends to increase as the risk level rises, a fixed threshold can still lead to under-reporting when early differences between TR high-risk and normal cells are subtle. From Figure 16d–f, the entropy-based approach shows the opposite failure mode: it raises early warnings for cells 7 and 67 in the normal case, yet it fails to trigger warnings for the risk and TR cases. In comparison, from Figure 16g–i, the proposed method provides consistent early warning across the three risk levels without introducing false or missed warnings in these cases.

To fully validate the superiority of the proposed method, the overall warning outcomes of the proposed method and other methods are further summarized in

Table 8. Comparison of overall warning performance for the proposed and benchmark methods across all evaluated cases.

Method	Total Number of Cells	Number of True TR High-Risk Cells	Number of Cells Ever Identified as High-Risk	Cumulative Number of False Warnings
Proposed	276	2	2	2
LOA	276	2	10	34
Entropy-based	276	2	40	141
Variational Autoencoder	276	2	32	74
Isolated Forest	276	2	48	225
Support Vector Machine	276	2	45	337

. The proposed strategy identifies the two true TR high-risk cells while producing the fewest false warnings. By contrast, although the LOA, entropy-based, variational autoencoder, isolated forest, and support vector machine methods detect more abnormal candidates, they also introduce substantially more false warnings. This result indicates that, under the highly imbalanced real data considered in this study, increasing the number of flagged cells does not necessarily improve practical warning quality. From a safety-and-operations perspective, missed warnings may directly increase safety risk, whereas frequent false warnings can substantially raise operational and maintenance (O&M) costs [50]. Given the higher difficulty and expense of onboard O&M in VESS, the proposed strategy offers a more practical balance between effective TR risk identification and operational burden.

The above findings are also consistent with, and further extend, the conclusions reported in prior studies. Existing signal-processing and decomposition studies have shown that voltage denoising can improve the visibility of abnormal features, but denoised voltage alone may still be insufficient when the early-stage responses of TR high-risk cells and normal cells remain highly similar [25,26,27]. Similarly, previous warning methods based on linear mapping, sparse observers, dynamic thresholding, or voltage entropy have demonstrated the value of enhancing voltage sensitivity or screening abnormal behavior [30,31,33,35], yet their performance may still be limited when operating conditions vary across segments or when the warning decision relies mainly on a single feature. In contrast, the present results indicate that the combination of nonlinear voltage enhancement, adaptive abnormal-cell screening, and joint assessment is better suited to long-horizon real VESS operation. In particular, the proposed strategy not only preserves early warning capability, but also better suppresses false warnings.

5. Conclusions

To address early warning of TR high-risk cells in complex marine environments, this study develops an early warning strategy that integrates voltage denoising, nonlinear voltage mapping, adaptive abnormal-cell screening, and joint assessment methodology. Based on the verification and comparative analysis of real operation cases, the following conclusions can be drawn.

(1)

The MJMD denoising scheme, together with Spearman correlation selection, effectively suppresses irrelevant voltage interference while preserving useful signal information. This provides a more reliable basis for the subsequent warning process.

(2)

The improved Sigmoid nonlinear mapping makes subtle early-stage differences between TR high-risk cells and normal cells more visible. In doing so, it addresses the limited local sensitivity of linear mappings and helps reduce false warnings when the original voltage differences are extremely small.

(3)

Sparse representation combined with adaptive thresholding improves adaptability across degradation stages and operating conditions. Compared with fixed-threshold approaches, it better controls both false warnings and missed warnings, which in turn strengthens the strategy’s generalizability.

(4)

By incorporating both static and dynamic cell characteristics, the MIV-based joint assessment provides an additional check beyond voltage alone. This improves the reliability of high-risk cell identification and helps suppress false warnings that can arise from single-feature screening.

The experimental results show that, across all 18 evaluated cases, the proposed strategy achieved a minimum confirmed warning lead time of 14 days in the identified TR-risk cases, which satisfies the warning-time requirement of vessel applications. Across the evaluated cases, the method shows consistently better overall performance than the compared traditional methods. Meanwhile, the computational cost remains moderate, supporting real-time deployment on typical onboard computing platforms. Taken together, the study offers a practical and deployable solution for TR early warning in VESS, with clear implications for improving vessel safety and reducing operational burden.

Although the present validation was performed on operating cases with the battery chemistry considered in this study, the proposed strategy is constructed from measurable cell-level signals and relative inter-cell deviation characteristics rather than from chemistry-specific electrochemical modeling. This gives the method potential for application to other marine secondary battery systems with comparable monitoring conditions. At the same time, differences in voltage response, thermal behavior, degradation evolution, and signal quality across battery chemistries and operating environments may affect the appropriate settings of the mapping, screening, and joint assessment modules. Future research will therefore focus on extending the validation of the proposed strategy to broader battery chemistries and vessel operating conditions, while further refining parameter calibration and onboard verification to improve its cross-scenario applicability and engineering practicality.