The rapid growth of the global economy has intensified the consumption of traditional fossil fuels and worsened environmental pollution. Among the major contributors, conventional internal combustion engine vehicles are responsible for a significant share of petroleum consumption and harmful emissions. To address these challenges, electric vehicles have emerged as a powerful alternative. Lithium-ion batteries, particularly lithium iron phosphate batteries, have become the preferred energy storage technology in electric vehicles because of their high energy density, long cycle life, low self-discharge rate, and favorable thermal stability. As a result, the traction battery pack is universally recognized as the core energy source of electric vehicles, and its safe and reliable operation is of utmost importance.
However, the widespread adoption of electric vehicles has been accompanied by safety incidents caused by thermal runaway of traction battery packs. Thermal runaway can lead to fire and even explosion, posing severe threats to human life and property. A short circuit is usually an inevitable intermediate stage between battery abuse and thermal runaway. If the short circuit can be detected at an early stage, especially at the stage known as soft short circuit, mitigation measures can be taken to prevent further deterioration. Therefore, effective soft short circuit fault diagnosis for traction battery packs is crucial for the safe deployment of electric vehicles.
The existing approaches for soft short circuit diagnosis can be broadly classified into model-driven methods and data-driven methods. Model-driven methods rely on equivalent circuit models, electrochemical models, thermal models, or coupled electro-thermal models. They usually estimate model parameters or states and generate residuals to detect faults. Although these methods provide physical insights and direct estimation of equivalent short circuit resistance, their accuracy depends heavily on model fidelity. Under realistic dynamic driving conditions, battery parameter uncertainties and cell inconsistencies reduce the reliability of model-based residuals. Moreover, they often require a sufficiently large residual before producing a reliable diagnosis, which limits their timeliness.
Data-driven methods have attracted increasing attention because they avoid explicit battery modeling. They directly process measured voltage, current, and temperature data to extract statistical features. Typical methods include entropy-based algorithms, correlation coefficients, principal component analysis, and various machine-learning techniques. Nevertheless, several fundamental issues remain. First, many data-driven methods do not fully consider the inherent parameter inconsistency among cells in a traction battery pack. This inconsistency produces voltage dispersion that can mask the small fault signature of a soft short circuit. Second, the fault detection indicators are often heuristically defined without a quantitative relationship between the indicator threshold and the physical fault amplitude. This lack of interpretability makes the detection performance dependent on prior knowledge and limits adaptive threshold setting. Third, in most existing diagnostic frameworks, fault localization is treated as a sub-task of fault detection, and localization usually cannot provide synchronous feedback to correct false alarms in detection.
In this paper, I present a complete soft short circuit fault diagnosis framework for traction battery packs based on explainable full-dimensional statistical analytics. My work improves the three essential stages of diagnosis: voltage measurement, fault detection, and fault localization. In the measurement stage, I propose a virtual voltage measurement scheme to suppress the unwanted differential voltage fluctuations caused by inherent cell inconsistencies. In the detection stage, I design a fault detection indicator based on eigen-decomposition of the covariance matrix of the standardized differential voltages. Through a rigorous detectability analysis, I derive a quantitative relationship between the indicator threshold and the original fault signature, thereby enabling self-commissioning of the control limit. In the localization stage, I evaluate the contribution of abnormal eigenvectors to identify the faulty cell. Because the eigen-decomposition simultaneously yields eigenvalues and eigenvectors, the proposed localization can operate in parallel with detection and mutually correct each other. Consequently, the method significantly reduces false alarm rates while improving diagnostic sensitivity.
Voltage Measurement and Analysis for Traction Battery Packs
A lithium-ion cell consists of a positive electrode, a negative electrode, a separator, electrolyte, and current collectors. During charging, lithium ions deintercalate from the positive electrode, migrate through the electrolyte, and intercalate into the negative electrode. During discharging, the reverse process takes place. For lithium iron phosphate cells, the open-circuit voltage demonstrates a very flat voltage plateau over most of the operating state-of-charge range. This plateau is a particularly important property exploited in my measurement design.
To model the dynamic behavior of cells, equivalent circuit models are frequently employed. The simplest model is the Rint model, expressed as \(U = U_{oc} – I R_0\). The Thevenin model adds a parallel RC branch to describe polarization: \(U = U_{oc} – I R_0 – U_p\). More accurate second-order RC models include two RC branches representing electrochemical polarization and concentration polarization. However, the model parameters are affected by state-of-charge, temperature, and aging. In a traction battery pack, cells are connected in series and parallel to meet power and energy requirements. Inevitably, the individual cells have slight differences in capacity, internal resistance, and self-discharge rate. These differences originate from manufacturing tolerances, assembly conditions, and aging processes.
The inconsistency among cells is a critical issue for voltage-based fault diagnosis. When the same current flows through all series-connected cells, the cells with different capacities accumulate different state-of-charge values. Because the open-circuit voltage is a nonlinear function of state-of-charge, cells with different states of charge exhibit different terminal voltages. Moreover, the ohmic internal resistance and the polarization resistance vary with cell state and are not identical across cells. Consequently, the measured cell voltages contain an inconsistency-related voltage component that causes differential voltage fluctuations.
To extract the soft short circuit fault signature, I first perform a differential operation on the measured cell voltages. The standard measurement topology for \(n\) series-connected cells uses \(n\) voltage sensors. The \(i\)-th sensor measures the voltage across the \(i\)-th cell. The differential voltage vector is defined as
$$
\Delta V^s_k = A_d V^c_k \approx A_d \left( V_{oc,k} – I_k R_{s,k} \right)
$$
where \(A_d\) is the difference operation matrix of dimension \((n-1)\times n\), \(V_{oc,k}\) is the open-circuit voltage vector, \(R_{s,k}\) is the internal resistance vector, and \(I_k\) is the pack current at time step \(k\). For the difference operation, I adopt
$$
A_d = \begin{bmatrix}
1 & -1 & 0 & \cdots & 0 \\
0 & 1 & -1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1 & -1 \\
-1 & 0 & \cdots & 0 & 1
\end{bmatrix}
$$
One of the most challenging issues is that differential voltage signals computed from standard measurements still contain severe fluctuations caused by the cell-to-cell parameter inconsistency. Figure 2.9(a) in the original study shows that the dynamic internal resistance of the lithium iron phosphate cell exhibits a characteristic peak in the mid-state-of-charge range. When current flows through cells with slightly different dynamic resistances, an extra voltage difference \(\Delta V_{IR}\) appears across the differential channels. Besides, capacity mismatch causes the state-of-charge of different cells to drift apart. Because the open-circuit voltage curve has a flat plateau, the voltage difference due to state-of-charge mismatch is relatively small only if the cell voltages are initially aligned within the plateau region.
As the current profile in a practical traction battery pack changes rapidly, the differential voltage fluctuation due to parameter inconsistency can be large enough to obscure the soft short circuit signature. Therefore, I introduce a virtual voltage measurement scheme that suppresses these fluctuations from both temporal and spatial perspectives.
Virtual Voltage Measurement Scheme
Let the direct measurement vector be \(V^s = [V_1^s, V_2^s, \dots, V_n^s]^T\). The virtual measurement vector is obtained by a permutation and recombination operation:
$$
V^{sv} = A_v V^s
$$
where \(A_v\) is an \(n\times n\) virtual transformation matrix. The matrix is designed so that each virtual measurement is still a physical cell voltage, but the order of cells is re-arranged according to their internal resistance consistency. In practice, I create a new spatial order of cells so that cells with similar dynamic internal resistance are adjacent in the virtual measurement space. This re-arrangement reduces the differential voltage between adjacent virtual cells.
From the temporal perspective, I exploit the flat open-circuit voltage plateau of lithium iron phosphate cells. A near-cutoff-voltage alignment operation is performed once when the pack is close to the discharge cutoff. More precisely, when the pack voltage reaches a selected alignment voltage (e.g., 2.94 V per cell in my experiments), the battery equalizer is activated until the voltage of every cell is aligned near that point. After alignment, the equalizer is disabled for the rest of the discharge and subsequent charge-discharge cycles. Because of the flat plateau, the cell voltages remain well aligned over a wide state-of-charge window, and the differential voltage fluctuations are dramatically reduced.
After applying the virtual transformation, the transformed differential voltage vector becomes
$$
\Delta V^{sv}_k = A_d A_v V^c_k
$$
To support further analysis, this transformed differential vector is separated into two sub-vectors to satisfy the rank requirement. The sub-matrices \(A_{d,1}\) and \(A_{d,2}\), together with \(A_v\), define the coefficient matrices \(\bar{A}_1 = \Sigma_0^{-1} A_{d,1} A_v\) and \(\bar{A}_2 = \Sigma_0^{-1} A_{d,2} A_v\), where \(\Sigma_0\) is the standardization diagonal matrix obtained from historical normal data. The grouping of differential channels is performed to ensure that every physical cell is observed with equal sensitivity in both sub-vectors.
Soft Short Circuit Fault Detection Indicator
With the virtual measurement scheme in place, I form a sliding-window data matrix from the virtual differential voltage vector:
$$
X_k = \begin{bmatrix}
x_{k-w+1,1} & x_{k-w+1,2} & \cdots & x_{k-w+1,n-1}\\
x_{k-w+2,1} & x_{k-w+2,2} & \cdots & x_{k-w+2,n-1}\\
\vdots & \vdots & \ddots & \vdots\\
x_{k,1} & x_{k,2} & \cdots & x_{k,n-1}
\end{bmatrix}
$$
where \(w\) is the window width. Each column of \(X_k\) corresponds to a differential channel. To eliminate the effects of different voltage levels and scales, the matrix is standardized using the historical mean and standard deviation:
$$
\bar{X}_k = \left( X_k – X_s \right) \Sigma_0^{-1}
$$
Here, \(X_s\) is the matrix of the historical reference voltage offsets. The covariance matrix of the standardized differential measurements is computed by
$$
C_k = \frac{1}{w} \bar{X}_k^T \bar{X}_k
$$
The covariance matrix \(C_k\) is real and positive semidefinite. Eigen-decomposition yields
$$
C_k = V_k \Lambda_k V_k^T
$$
where \(\Lambda_k = \mathrm{diag}\{\lambda_{1,k}, \lambda_{2,k}, \dots, \lambda_{n-1,k}\}\) is the eigenvalue matrix and \(V_k\) is the orthonormal eigenvector matrix. The eigenvectors define a new orthogonal coordinate system in which the data variances are independent. A soft short circuit fault introduces an abnormal voltage component along one or more directions, resulting in a significant increase in the corresponding eigenvalue. I therefore define the fault detection indicator as the normalized maximum eigenvalue deviation:
$$
DI_k = \max_{1 \le j \le n-1} \frac{\lambda_{j,k} – \mu_{\lambda_j}^{*}}{\sigma_{\lambda_j}^{*}}
$$
where \(\mu_{\lambda_j}^{*}\) and \(\sigma_{\lambda_j}^{*}\) denote the mean and standard deviation of the eigenvalue \(\lambda_j\) under normal operation. In practice, the indicator is computed for each of the two sub-vectors. The detection statistic activates a fault alarm when \(DI_k\) exceeds the control limit.
Detectability Analysis and Control Limit Self-Commissioning
A major contribution of this paper is the explicit quantitative relation between the fault detection threshold and the physical fault amplitude. In the presence of a soft short circuit on the \(l\)-th virtual channel, the standardized differential measurement matrix can be expressed as
$$
\bar{X}_k = \bar{X}_k^{normal} + \frac{1}{\sigma_{x_l}^{*}} f_{sc,k} \bar{a}_l^T
$$
where \(f_{sc,k}\) is the fault vector in the sliding window and \(\bar{a}_l\) is the \(l\)-th column of the standardized coefficient matrix. By applying the trace operator to the covariance matrix and using the commutativity of the trace and expectation, I can relate the eigenvalues to the fault signature. After algebraic manipulation, the control limit must satisfy
$$
\delta_{sc} \ge \sigma_{\lambda_j^{*}}^{-1} \frac{t_f}{w} f_{sc}^2 \max_{l \in [1,n-1]} \left( \bar{a}_l^T \bar{a}_l \right)
$$
where \(t_f\) is the fault duration within the sliding window and \(f_{sc}\) is the equivalent fault voltage amplitude. Because the standardization has already normalized each differential channel, the term \(\bar{a}_l^T \bar{a}_l = (\sigma_{x_l}^{*})^{-2}\) is the inverse of the variance of the \(l\)-th differential channel. Thus, for at least one of the two sub-matrices, the maximum of this term must be considered. This analysis demonstrates that the detection threshold is directly proportional to the squared fault voltage amplitude and inversely proportional to the differential channel variance. Hence, the virtual voltage measurement scheme, which significantly reduces the differential channel variance, is essential for achieving high detection sensitivity.
To self-commission the control limit, I first define a physically meaningful fault amplitude \(f_{sc}\). For a soft short circuit fault with equivalent resistance \(R_{sc}\), the fault amplitude generated at the virtual differential channel can be approximated by
$$
f_{sc} \approx \frac{\partial \Delta V^{sv}}{\partial t} \Delta t + \Delta V_{ocv}^{sv}(t_f)
$$
In practice, the value of \(f_{sc}\) can be evaluated from the allowed equivalent short circuit resistance, the open-circuit voltage change due to the fault-induced state-of-charge shift, and the polarization response. Once the desired minimum detectable fault amplitude is determined, the control limit \(\delta_d\) is obtained from the inequality above. Conversely, if a control limit is given, the minimum detectable fault amplitude can be calculated using the same formula. Thus, the method allows flexible tuning of the fault detection threshold with clear physical meaning.
Fault Localization through Eigenvector Contribution
When a soft short circuit is detected, the faulty cell must be identified as quickly as possible. Because the eigenvalues and eigenvectors are obtained simultaneously from the same eigen-decomposition, the localization information is available at no extra major computational cost. For an abnormal eigenvalue \(\lambda_{j,k}\), its corresponding eigenvector \(v_{j,k}\) indicates the direction of the anomaly in the standardized voltage space. I define the contribution of the \(i\)-th differential channel to this abnormal eigenvalue as the square of the \(i\)-th element of the eigenvector:
$$
\eta^{(i)} = \left( v_{j,k}^{(i)} \right)^2
$$
Owing to the orthogonality of the eigenvectors, these contributions satisfy
$$
\sum_{i=1}^{n-1} \eta^{(i)} = 1
$$
The contribution vector is then mapped back to the original cell locations by combining the contributions from the two differential sub-vectors. Let \(\eta_{i,t}\) be the contribution of the \(i\)-th physical cell at time \(t\). The maximum contribution is searched over all cells. The faulty cell is determined when the largest element satisfies the following conditions:
$$
\begin{cases}
\eta_{i,t} + \eta_{j,t} \ge \eta_{th}\\
\eta_{i,t} \ge \eta_{th} / 4, & \eta_{j,t} \ge \eta_{th} / 4
\end{cases}
$$
where \(\eta_{th}\) is the contribution threshold. The indices \(i\) and \(j\) correspond to the cells on both sides of the virtual differential channel that crosses the fault. If \(j = n\) when \(i = 1\), or if \(j = i-1\) for \(1 < i \le n\), then the fault is located at the \(i\)-th cell.
One important advantage of combining detection and localization is the ability to correct false alarms. When the traction battery pack operates under high current transients, even a well-balanced pack may exhibit temporary differential voltage spikes. These spikes can increase the largest eigenvalue and trigger the detection indicator, producing a false alarm. However, under normal operation, the eigenvector contributions are evenly distributed among all channels. In contrast, a real soft short circuit fault creates a highly localized contribution on the faulty channel. By requiring the contribution to exceed a certain threshold, I suppress the false alarms caused by current-driven voltage fluctuations. Therefore, the localization step acts as an independent verification of the detection result.
Experimental Verification
To validate the proposed soft short circuit fault diagnosis method, I constructed a compact traction battery pack consisting of 14 series-connected lithium iron phosphate cells. The cell specifications are summarized in the following table:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Nominal voltage | 3.3 V | Nominal capacity | 2.5 Ah |
| Internal resistance | 6 mΩ | Standard charge rate | 1C |
| Voltage range | [2.0 V, 3.6 V] | Maximum discharge rate | 28C |
The experimental platform includes a commercial battery-management-system prototype with an MC33771 analog front-end and a TMS320F28075 controller. The sampling frequency is 10 Hz. A bidirectional DC power supply was configured to charge and discharge the pack according to the World Light Vehicle Test Cycle. The maximum current amplitude was set to approximately 5 A, corresponding to 2C for the pack. The experimental setup is shown in the original publication.
In the experiments, the soft short circuit condition was simulated by connecting a variable resistance in parallel with one target cell. This configuration can represent both momentary and cumulative internal short circuit evolution. The instantaneous fault is modeled by suddenly switching the resistance from a very large value to a fixed small value. The cumulative soft short circuit is modeled by an exponentially decreasing resistance according to
$$
R_{sc}(t) = R_{sc,\min} + \frac{K}{1 + e^{-r(t – t_0)}}
$$
where \(K = R_{sc,\max} – R_{sc,\min}\) and \(r\) is the evolution rate. Both fault types produce a voltage deviation of a few millivolts on the corresponding differential channel. The fault was injected on cell 11, and the equivalent minimum short circuit resistance \(R_{sc,\min}\) was 11.8 Ω. The corresponding fault voltage amplitude \(f_{sc}\) was 10.7 mV. The detection threshold according to the confidence level \(\alpha=1-10^{-6}\) was computed as \(\delta_d = 15.0414\).
Comparison of Voltage Measurement Schemes
I first validated the effectiveness of the virtual voltage measurement scheme. Three different cases were compared: (1) standard charging to 100% SOC without alignment and with the original cell order; (2) near-cutoff alignment but with the original cell order; (3) near-cutoff alignment combined with virtual spatial re-ordering. The state-of-charge range covered 93.01% down to 9.66% during multiple driving cycles. The standard deviations of the differential voltage in each differential channel are listed in the table below:
| Cell channel | Standard deviation (mV) | Cell channel | Standard deviation (mV) | ||||
|---|---|---|---|---|---|---|---|
| Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | ||
| 1 | 3.380 | 0.742 | 0.923 | 8 | 1.088 | 0.732 | 1.014 |
| 2 | 4.073 | 1.030 | 0.742 | 9 | 2.911 | 1.208 | 0.995 |
| 3 | 4.284 | 1.152 | 0.837 | 10 | 2.314 | 0.927 | 0.893 |
| 4 | 3.269 | 0.990 | 0.816 | 11 | 4.761 | 1.029 | 1.122 |
| 5 | 2.318 | 0.997 | 0.811 | 12 | 3.367 | 0.634 | 0.814 |
| 6 | 2.181 | 1.435 | 0.999 | 13 | 2.975 | 0.749 | 0.997 |
| 7 | 0.932 | 1.078 | 0.828 | MD | 0.880 | 0.170 | 0.094 |
Case 1 shows significantly larger standard deviations, with the mean deviation being 0.880 mV. Case 2 reduces the mean deviation to 0.170 mV, demonstrating the benefit of the near-cutoff alignment. Case 3 further reduces the mean deviation to 0.094 mV, a reduction of 45% compared with Case 2. The maximum standard deviation in Case 3 is also 22% lower than that in Case 2. These results confirm that the proposed virtual voltage measurement scheme greatly improves the consistency of the differential voltage data, which is essential for detecting weak soft short circuit signals.
Soft Short Circuit Fault Diagnosis Results
The proposed method was compared with two recent data-driven techniques: the voltage cosine similarity method and the voltage-correlation-independent component analysis method. The comparison was performed for both instantaneous and cumulative soft short circuit faults. The following table summarizes the diagnostic results:
| Fault type | Method | Detection time (s) | Continuous alarm | Alarm after fault removal | Faulty cell identified | Fault detection rate (%) |
|---|---|---|---|---|---|---|
| Instantaneous SSC | ICA-VC | 9.9 | No | Yes | 11 | 0.0294 |
| VCS | 80.1 | No | No | 11 | 0.00015 | |
| Proposed method | 109.6 | Yes | Yes | 11 | 98.4 | |
| Cumulative SSC | ICA-VC | 649.8 | No | No | – | 0 |
| VCS | – | No | No | – | 0 | |
| Proposed method | 171.3 | Yes | Yes | 11 | 97.6 |
From the table, the proposed method diagnoses both instantaneous and cumulative soft short circuit faults within a few minutes. It provides continuous alarms during the fault injection period and continues to alarm after the fault is removed because the cumulative effect of the soft short circuit cannot be immediately eliminated. In contrast, the two benchmark methods only detect the instantaneous fault in certain windows, and they fail to produce reliable alarms for cumulative soft short circuit faults. The proposed method thus exhibits superior diagnostic capability for small fault amplitudes that are typical of early-stage soft short circuits in traction battery packs.
In the localization stage, the contribution matrix was derived and evaluated. The proposed method successfully identified cell 11 as the faulty cell. The fault contribution index was maintained above the preset threshold for the entire fault duration. Because the eigen-decomposition is performed simultaneously for all eigenvalues and eigenvectors, the localization was achieved in parallel with detection and reinforced the detection results.
Effect of Lowering the Detection Control Limit
One of the unique properties of the proposed approach is that the parallel localization can correct false alarms. When the detection control limit \(\delta_d\) is reduced, the theoretical and experimental detectable fault voltages diverge once \(\delta_d\) falls below 8.122. This discrepancy reveals that localization plays a dominant role in suppressing false alarms. I tested the performance by lowering the confidence level to \(\alpha = 0.998\), which effectively reduces the control limit. The experimental results show that the instantaneous soft short circuit was detected 101.8 s earlier, while the cumulative soft short circuit was detected 106 s earlier, without triggering any false alarms. The overall detection time was reduced by at least 61.9% compared with the original high-confidence setting.
This experiment confirms that the proposed fault localization mechanism, based on eigenvector contribution evaluation, serves as an effective synchronous correction for fault detection. Consequently, the detection threshold can be set more aggressively to improve early-stage fault sensitivity without sacrificing reliability.
Summary and Conclusions
In this paper, I have presented an interpretable full-dimensional statistical diagnostic framework for soft short circuit faults in traction battery packs. The contributions are threefold. First, I developed a virtual voltage measurement scheme that mitigates the adverse influence of inherent cell parameter inconsistency. By aligning cells near the cutoff voltage in time and re-ordering the measurement channels in space, the differential voltage fluctuations are dramatically reduced. The standard deviation of each differential channel and the mean deviation of all channels were greatly reduced, enabling weak fault signatures to be preserved.
Second, I designed a fault detection indicator based on the eigen-decomposition of the covariance matrix. The detectability analysis establishes a quantitative relationship between the detection threshold and the original fault amplitude. Therefore, the control limit can be self-commissioned from the desired physical fault severity, giving the method greater explainability and adaptability than conventional data-driven approaches.
Third, I formulated a fault localization strategy based on eigenvector contribution evaluation. Since the eigen-decomposition yields the eigenvalues and eigenvectors at the same time, the localization can be executed in parallel with detection and provides synchronous correction to reduce the false alarm rate. This property permits the control limit to be lowered, significantly accelerating the detection of both instantaneous and cumulative soft short circuit faults.
The experimental validation on a 14-cell lithium iron phosphate traction battery pack under World Light Vehicle Test Cycle conditions demonstrated that the proposed method reliably detects millivolt-level soft short circuit faults within minutes. Comparisons with two state-of-the-art data-driven methods confirmed that only the proposed method provides continuous and accurate alarms for both instantaneous and cumulative soft short circuits. The measured fault detection rate exceeded 97% in both scenarios. The virtual voltage measurement scheme proved to be a critical prerequisite for achieving this level of performance.
Future work will explore the diagnosis of soft short circuit faults in traction battery packs under more aggressive current profiles, such as high-power pulse operation in hybrid electric vehicles. The current framework will require robust measures to separate the current-induced voltage dispersion from the fault-induced voltage signature. In addition, extension to ternary lithium-ion batteries, which do not exhibit the flat open-circuit voltage plateau of lithium iron phosphate cells, will be investigated by adapting the temporal alignment strategy and spatial transformation matrix.