In this article, I present a diagnostic theory and implementation for soft short-circuits in lithium iron phosphate vehicle traction battery packs. My work is centred on the idea that a vehicle traction battery is not simply a collection of cells, but an interconnected electro-chemical system whose voltage information must be interpreted statistically. The main contribution is a full-dimensional statistical framework that couples an improved virtual voltage measurement scheme with an interpretable fault-indicator design and a parallel fault-localization mechanism. This framework explicitly deals with the intrinsic parameter inconsistency of lithium-ion cells and permits soft short-circuit detection at the millivolt level within a few minutes under realistic driving cycles.
I explain the limitations of existing model-driven and data-driven diagnostic methods, propose a robust measurement scheme for the vehicle traction battery, derive a fault detection indicator based on eigen-decomposition, and prove a quantitative relation between the fault-detection threshold and the original soft short-circuit signature. I also show that the same eigen-decomposition enables fault localization on the vehicle traction battery to run in parallel with detection, which greatly reduces false alarms and allows the detection threshold to be lowered without losing robustness. The method is validated on a compact vehicle traction battery pack consisting of fourteen series-connected lithium iron phosphate cells under the World Light Vehicle Test Cycle. Experimental results confirm that both momentary and cumulative soft short-circuits can be identified, localized, and continuously monitored over a wide state-of-charge window.
Background and Motivation
The electrification of road transport is widely considered to be an essential response to global environmental challenges. The vehicle traction battery has become the dominant energy carrier in battery-electric vehicles, and lithium-ion chemistry has the highest practical energy density among commercially available battery types. Nevertheless, rapid development of electric mobility has exposed a critical safety bottleneck: battery thermal runaway remains the leading cause of severe fire accidents in electric vehicles. A thermal runaway event in a vehicle traction battery is usually preceded by an internal or external short-circuit process. Once that short-circuit becomes strong, it may be too late to intervene. This is why timely detection of the early, less intense phase, often called a soft short-circuit, is regarded as one of the most valuable functions in a battery management system.
From the perspective of fault evolution, a soft short-circuit can appear in two different forms. The first is a momentary soft short-circuit, in which a local instantaneous electrical path, for example caused by cell deformation or separator damage, gives rise to a finite charge imbalance. The second is a cumulative soft short-circuit, which grows slowly as lithium dendrites penetrate the separator or as internal defects develop over many charge and discharge cycles. The cumulative form is particularly dangerous because its voltage signature is weak and may remain hidden inside normal cell-to-cell variations. I therefore classify soft short-circuit faults according to their temporal evolution and design the diagnostic method so that it can address both forms simultaneously.
The safety challenge is amplified by the electric-vehicle operating environment. A vehicle traction battery must deliver high current pulses during acceleration and accept regenerative current during braking. The current is never constant, and the state of charge varies widely. Under such dynamic loading, every battery cell in a series pack experiences the same macroscopic current, but unequal capacity, unequal resistance, and unequal polarization cause their terminal voltages to diverge slightly. This divergence is often innocuous, yet it creates a background voltage inconsistency that can mask the early abnormal signature of a soft short-circuit. I argue that any diagnostic method based on voltage measurement must explicitly suppress this intentional inconsistency before statistical fault detection can be reliable.
Figure-independent system reasoning shows that modern vehicle traction battery packs are assembled from many series-connected modules, and their management requires a distributed sensing architecture. The voltage of each cell is no more than a few volts, whereas the pack-level voltage can reach several hundred volts. Therefore, voltage sensing, balancing, and fault diagnosis are not independent tasks; they are integrated into a hierarchical battery management system. My design follows this hierarchy by placing the proposed statistical diagnostic layer immediately after the analogue front-end voltage measurement. The diagnostic layer then operates on standardized differential voltage streams rather than on raw terminal voltages.
Limitations of Existing Diagnostic Approaches
Existing fault-diagnosis research for the vehicle traction battery can be broadly separated into model-driven methods and data-driven methods. Model-driven methods are usually built on equivalent-circuit models, electrochemical models, thermal models, or coupled electro-thermal models. State estimators such as the extended Kalman filter, unscented Kalman filter, and H-infinity observers have been used to estimate the open-circuit voltage, state of charge, or internal resistance of an individual cell. The residual between the model output and the measured output is then used to detect an anomalous short-circuit resistance.
Model-driven methods have clear physical meaning. They can provide an equivalent short-circuit resistance and directly indicate the severity of a fault. However, they require an accurate model of the vehicle traction battery. The model must describe not only the nominal cell chemistry, but also the natural parameter spread among cells. In an actual vehicle traction battery, operating temperature, aging, and state-of-charge differ from cell to cell. It is difficult for a single model to capture all of those variations accurately. Moreover, many model-based estimators only recognize a fault after the state estimate has diverged sufficiently. The estimation delay may be long enough to prevent intervention during the early soft short-circuit phase.
Data-driven diagnostic methods, in contrast, do not require an explicit battery model. They use voltage, current, or temperature signals directly and seek to extract abnormal patterns from the data. Examples include Shannon entropy, sample entropy, local outlier factor, kernel principal component analysis, voltage cosine similarity, and independent component analysis. Data-driven methods can be fast and can detect faults that are difficult to separate with a first-principles model. Nevertheless, they have several weaknesses that have not been fully solved.
The first weakness is that the measurement stage rarely accounts for the intrinsic parameter inconsistency of the cells. Voltage differentials are computed between neighbouring cells, but if the cells naturally have different capacities or internal resistances, a non-faulty differential voltage channel will show a fluctuating baseline. That fluctuation can be mistaken for a fault signature, or it can hide a real but weak soft short-circuit signature. The second weakness is interpretability. Many statistical indicators are treated as black-box anomaly scores, and the user does not know how the chosen threshold corresponds to physical quantities such as short-circuit resistance or voltage deviation. The threshold is then selected by trial-and-error or by heuristics. This makes the diagnostic performance very dependent on the data quality and on prior knowledge. The third weakness is that localization is often performed only after detection has been completed, and it is usually treated as a secondary step. Consequently, detection cannot be corrected by localization information, and the full diagnostic system becomes less robust.
My proposal addresses all three weaknesses within one coherent statistical framework. The virtual voltage measurement scheme attenuates the inconsistency-induced differential fluctuation. The fault indicator is derived from the eigen-decomposition of a standardized covariance matrix, and its threshold can be related quantitatively to the allowable fault voltage amplitude. The eigen-decomposition also gives eigenvector-based contributions that localize the faulty cell at the same time as the fault indicator is computed; therefore, detection and localization can correct each other in real time.
Electrochemistry and Measurement Preliminaries
Lithium-ion cells used in a vehicle traction battery are based on lithium-ion shuttle reactions between two insertion electrodes. In a lithium iron phosphate cell, the cathode is lithium iron phosphate and the anode is usually graphite. During charge, lithium ions deintercalate from the iron phosphate lattice, migrate through the electrolyte, and intercalate into the graphite. During discharge, the reverse process occurs. The terminal voltage of a cell under arbitrary load can be represented by an equivalent-circuit model consisting of an open-circuit voltage source in series with an ohmic resistance and one or more parallel resistance-capacitance networks. Although such a model is a simplification, it is sufficient to explain the voltage responses of a battery pack under dynamic current.
The open-circuit voltage of lithium iron phosphate is unique because it has a very flat plateau over most of the state-of-charge range. The plateau is an advantage for energy delivery but a challenge for many state-estimation algorithms because small voltage changes correspond to large state-of-charge changes. From a diagnostic point of view, the plateau also implies that the state-of-charge misalignment between cells creates only a small voltage misalignment in the middle part of the usable range. Near the bottom or top of charge, the open-circuit voltage curve falls or rises sharply. That transition creates a large voltage separation between cells and dominates differential voltage fluctuations.
I use the following model for the measured voltage of cell \(i\) at time instant \(k\). The open-circuit voltage \(V_{\mathrm{oc},i,k}\) is a nonlinear function of the cell state of charge, and the total resistance \(R_{s,i,k}\) is composed of an ohmic contribution, a charge-transfer polarization contribution, and a concentration-polarization contribution. The cell terminal voltage is written as
\[
V^{c}_{i,k}=V_{\mathrm{oc},i,k}-R_{s,i,k}I_{p,k},
\]
where \(I_{p,k}\) is the current flowing through the series-connected vehicle traction battery pack. When \(I_{p,k}\) is positive during charging, the terminal voltage is higher than the open-circuit voltage; when \(I_{p,k}\) is negative during discharging, the terminal voltage drops.
In a standard measurement topology, a voltage sensor is placed across each cell or across a group of parallel-connected cells. The measured differential voltage across two adjacent cells is often used to magnify the difference between their terminal voltages. The operation can be written as
\[
\Delta V_{i,k}=V_{i,k}-V_{i+1,k}
\]
for \(i=1,2,\ldots,n-1\), and, to preserve the cyclic position information around the vehicle traction battery, I also define
\[
\Delta V_{n,k}=V_{n,k}-V_{1,k}.
\]
The differential operation removes common-mode components that are shared by all cells, such as the slowly varying average voltage drift caused by temperature or by the overall state of charge. What remains is the difference between cells, which contains the information needed for state-of-charge inconsistency, resistance mismatch, and short-circuit diagnosis.
Inconsistency-Induced Differential Voltage Fluctuations
Any real vehicle traction battery is assembled from cells that are not perfectly identical. Manufacturing tolerances in electrode coating, electrolyte filling, separator alignment, and case assembly create small differences in capacity, impedance, and self-discharge rate. During pack operation, those differences interact with current, temperature, and aging. The result is that the terminal voltages of healthy cells are not identical, even when the cells are new and balanced.
The most important inconsistency sources are capacity, ohmic resistance, and polarization resistance. A capacity difference means that, for the same current integral, one cell will have a different state of charge from another. A resistance difference means that the ohmic voltage drop in response to a current pulse is different from cell to cell. A polarization difference means that the dynamic relaxation after a current change is different from cell to cell. These effects combine to form a differential voltage offset that varies over time and current amplitude.
To illustrate the consequence for a lithium iron phosphate vehicle traction battery, I consider the relationship between state of charge and open-circuit voltage. For states of charge above a low threshold and below a high threshold, the open-circuit voltage is almost flat. If two cells have slightly different capacities but have been aligned at a near-cut-off voltage, their state-of-charge offsets remain small and their open-circuit voltages remain close. If the cells have not been aligned, a small capacity offset can lead to a much larger open-circuit voltage offset at the steep parts of the open-circuit voltage curve. The resistance term also contributes to the differential voltage: when a dynamic current flows through the pack, the polarization resistances of the two cells produce different voltage drops, and the resulting differential voltage fluctuates with the current excitation.
This inconsistency-induced fluctuation is usually not dangerous by itself, but it is harmful to a fault diagnosis system. The reason is that the voltage signature of a soft short-circuit is also expressed as a deviation of one cell voltage from the expected pack-level pattern. If the normal inconsistency background is not suppressed, the fault deviation may be obscured. Even worse, the background inconsistency can create pseudo-fault signals during high-current pulses. In my methodology, I treat this issue explicitly and do not rely on the hope that cell manufacturing will become perfect enough to eliminate the problem.
Virtual Voltage Measurement Scheme
I propose a virtual voltage measurement scheme to reduce the impact of intrinsic cell inconsistency on the voltage signals of the vehicle traction battery. The virtual nature of the scheme means that no additional physical sensor is required. I only rearrange the measured data in the computation domain, not the physical wiring. Therefore, the method is easy to embed in the battery management system software of an existing vehicle traction battery.
My scheme contains two actions. The first action is an alignment operation in the time dimension. For a lithium iron phosphate battery, I select a near-cut-off voltage that lies below the main open-circuit voltage plateau. When the vehicle traction battery is discharged to this voltage during a calibration or maintenance procedure, the battery balancing system is activated to bring all cells to nearly the same voltage. Once this alignment is done, balancing is disabled and the pack is charged normally. During the subsequent discharge period, the cells track the same flat plateau region, and therefore the differential voltage caused by the capacity mismatch is much smaller than it would be without the alignment operation.
The second action is a rearrangement in the space dimension. Instead of using the physical channel order of the sensors, the battery management system can construct a virtual cell order by a reordering matrix \(A_{v}\). The virtual voltage vector is
\[
\tilde{V}_{k}=A_{v}V_{k},
\]
where \(A_{v}\) is a permutation matrix. The choice of the permutation matrix is not arbitrary. I choose it according to the correlations among cell voltages or according to the similarity of dynamic internal resistances. Cells with similar voltage trajectories should be placed close to each other in the virtual measurement order, so that their differential voltage remains small. Once the vehicle traction battery ages, the similarity relationships may change; therefore, the matrix \(A_{v}\) can be updated periodically from a historical dataset without requiring a hardware modification.
After the virtual reordering operation, the cyclic differential voltage vector is formed by the following linear transformation:
\[
\Delta \tilde{V}_{k}=D A_{v} V_{k},
\]
where \(D\) denotes the cyclic difference operation matrix. The cyclic differential vector has \(n\) entries for an \(n\)-cell pack, but its rank is only \(n-1\). In order to apply the subsequent covariance analysis without dropping useful localization information, I divide the cyclic differential vector into two subsets. Each subset contains \(n-1\) independent differential voltages and is formed by deleting one different entry of the cyclic vector. I denote the two resulting coefficient matrices by \(A_{d,1}A_{v}\) and \(A_{d,2}A_{v}\).
In the experimental comparison, I will show that the virtual measurement scheme reduces the standard deviations of most differential channels by roughly a factor of two to three. More importantly, the mean deviation of the standard deviations across the channels is reduced by almost one order of magnitude. This creates a much more consistent statistical baseline for fault detection.
Sliding-Window Standardization and Covariance Estimation
Fault diagnosis of a vehicle traction battery under real driving conditions cannot rely on a single time sample. Single-sample measurements are noisy, and a single abrupt drop in voltage can be caused by load transients as well as by a genuine fault. A sliding-window formulation is therefore preferable; it smooths high-frequency noise but preserves the gradual or sudden evolution of a short-circuit signature.
Let \(w\) denote the window width. At every time instant \(k\), the battery management system assembles the last \(w\) samples of the virtual differential voltage into a matrix. Each column of this matrix corresponds to a differential voltage channel, and each row corresponds to a sampling instant. I denote the resulting matrix as
\[
\bar{X}_{k} = \bigl(\tilde{X}_{k} – \mathbf{1}\mu_{0}^{T}\bigr)\Sigma_{0}^{-1},
\]
where \(\mu_{0}\) is the historical mean vector of the virtual differential voltage channels and \(\Sigma_{0}\) is the diagonal standard-deviation matrix computed under normal operating conditions. Standardization is important because different differential voltage channels may have slightly different noise levels. After standardization, every channel is expressed in units of its own standard deviation, and no channel is accidentally given more importance simply because its voltage range is larger.
I then compute the sample covariance matrix from the standardized matrix:
\[
C_{k}=\frac{1}{w-1}\bar{X}_{k}^{T}\bar{X}_{k}.
\]
For the virtual differential voltage vector of size \(n-1\), the covariance matrix has dimension \((n-1)\times(n-1)\). It is a real symmetric and positive semi-definite matrix, so it always admits an orthonormal eigen-decomposition. I write this decomposition as
\[
C_{k}=Q_{k}\Lambda_{k}Q_{k}^{T},
\]
where
\[
\Lambda_{k}=\operatorname{diag}\{\lambda_{1,k},\lambda_{2,k},\ldots,\lambda_{n-1,k}\}
\]
contains the eigenvalues in descending order, and \(Q_{k}\) is the orthogonal matrix whose columns are the corresponding eigenvectors.
Eigen-decomposition is often described as a purely algebraic tool, but it has a geometric interpretation that is very useful for fault diagnosis. The covariance matrix rotates the original differential voltage data into a new coordinate system defined by the eigenvectors. In this new coordinate system, the components of the standardized data are linearly uncorrelated. The eigenvalues quantify the variances along each new coordinate axis. Under normal operation, the eigenvalues are determined by the ordinary cell-to-cell voltage fluctuation. When a soft short-circuit occurs, the fault imposes an additional deviation on one cell, and that deviation is not aligned with the main eigenvectors of the normal covariance matrix. Therefore, one or more eigenvalues of the covariance matrix will increase. The eigen-decomposition thus makes the fault component visible without requiring a separate residual-generation model.
Fault Detection Indicator Design
In order to obtain a single scalar fault indicator that can be monitored by the battery management system, I define the indicator \(DI_{k}\) as the largest normalized eigenvalue deviation within the current sliding window:
\[
DI_{k}=\max_{j\in\{1,\ldots,n-1\}}
\left|\frac{\lambda_{j,k}-\mu_{\lambda,j}^{*}}{\sigma_{\lambda,j}^{*}}\right|,
\]
where \(\mu_{\lambda,j}^{*}\) and \(\sigma_{\lambda,j}^{*}\) denote the mean and standard deviation of the \(j\)-th eigenvalue stream in normal historical data. The superscript * always denotes a parameter obtained from normal operation. The use of the maximum over all eigenvalues means that the indicator is sensitive to any significant change in the multivariate covariance structure. It does not require the user to know which spatial direction will be affected by the fault before the fault occurs.
I select the largest normalized eigenvalue rather than the sum of all eigenvalues because the sum can hide a localized fault inside many unchanged eigenvalues. The use of normalization by the historical eigenvalue distribution is also deliberate. Different differential channels of a vehicle traction battery may not have exactly equal variance even after standardization; the eigenvalue statistics reflect the remaining multivariate structure. Normalizing them makes the detection criterion universal and allows the same threshold logic to be used for different driving cycles and slightly different battery packs without retuning from scratch.
The proposed indicator belongs to the class of full-dimensional statistical analytics because the covariance matrix includes all independent differential voltage channels simultaneously. Unlike a method that monitors one channel at a time, the covariance matrix uses the relationships between channels. This is especially important for a multiorgan vehicle traction battery system, because a short-circuit in one cell changes not only the voltage of that cell, but also the correlations between its adjacent differential channels.
Detectability Analysis and Threshold Interpretation
One of my main design goals is to make the threshold of the proposed fault indicator interpretable. A data-driven threshold that is simply a percentile of the historical indicator distribution may work in practice, but it does not tell the engineer what level of physical fault deviation can be detected. I therefore derive a quantitative relation between the fault indicator threshold and the short-circuit voltage deviation.
Suppose that a soft short-circuit appears on the \(l\)-th virtual differential channel. The standardized differential voltage matrix can be modelled as
\[
\bar{X}_{k}=\bar{X}_{k}^{*}+\bar{a}_{l}f_{sc,k},
\]
where \(\bar{a}_{l}\) is the standardized coefficient column corresponding to the fault channel and \(f_{sc,k}\) is the amplitude of the voltage deviation caused by the short-circuit. The covariance matrix then receives an additional rank-one contribution proportional to \(f_{sc,k}^{2}\). Taking the expected trace of the covariance matrix gives a direct relation between the sum of eigenvalues and the fault amplitude. Using the fact that the normalized eigenvalue norm is always greater than or equal to the arithmetic mean of the normalized eigenvalue shifts, I obtain the threshold condition
\[
\delta_{sc,i}\ge
\frac{t_{f}}{w}\,
\frac{\max_{l}\|\bar{a}_{l}\|^{2}}{\sum_{j=1}^{n-1}\sigma_{\lambda,j}^{*}}\,
f_{sc}^{2},
\]
where \(t_{f}\) is the number of consecutive samples in the sliding window affected by the fault, \(w\) is the window width, and \(i\in\{1,2\}\) corresponds to the two independent differential subsets. In this relation, \(\delta_{sc,i}\) is the fault-detection control limit, while \(f_{sc}\) is the original short-circuit voltage signature that the user wants to detect. The other quantities, such as \(\bar{a}_{l}\) and \(\sigma_{\lambda,j}^{*}\), are derived from the normal historical data.
The relation is central to threshold self-commissioning. If the vehicle manufacturer specifies that a soft short-circuit above a certain voltage deviation must be detected, the proposed method computes the required threshold directly from the statistics of the normal operation. Conversely, if the user sets a threshold for operational reasons, the corresponding minimum detectable fault amplitude can be calculated from the inverse relation. This removes much of the guesswork that is common in pure anomaly-detection methods and gives the fault-indicator threshold a clear physical meaning.
For a cumulative soft short-circuit, the voltage deviation \(f_{sc}\) grows gradually with time. The fault duration \(t_{f}\) inside the sliding window also grows as the fault becomes more persistent. Consequently, the fault contribution accumulates in the covariance matrix, making the proposed indicator progressively larger. This is exactly what is required to distinguish a slowly developing soft short-circuit from a temporary voltage disturbance that lasts only for one current pulse.
Threshold Self-Commissioning Strategy
The threshold self-commissioning is performed offline using a historical dataset that is known to be free of soft short-circuit faults. The historical dataset should cover the same driving cycle and the same state-of-charge range in which the diagnostic method will be deployed. The following steps are used in my strategy.
First, the virtual measurement scheme is applied to the historical voltage data. The reordering matrix is chosen from the correlation structure of the healthy pack, and the mean vector \(\mu_{0}\) and the standard deviation matrix \(\Sigma_{0}\) are estimated. The cyclic differential voltage vector is then computed and standardized.
Second, the covariance matrix and its eigenvalues are calculated over each sliding window of the historical data. This produces one time series for each eigenvalue. From these time series, I compute the reference means \(\mu_{\lambda,j}^{*}\) and the reference standard deviations \(\sigma_{\lambda,j}^{*}\). If the historical dataset contains a sufficiently large number of windows, these estimates are stable.
Third, I use the proposed quantitative threshold relation to convert a chosen allowable fault amplitude \(f_{sc,\text{allow}}\) into the control limits \(\delta_{sc,1}\) and \(\delta_{sc,2}\). Because the two differential subsets overlap, the final control limit is
\[
\delta_{d}=\max\{\delta_{sc,1},\delta_{sc,2}\}.
\]
In addition, I also compute a robustness-adjusted threshold with a significance level \(\alpha\) by using kernel density estimation on the historical \(DI_{k}\) sequence. The final threshold that is applied online may be the smaller of the physical threshold and the statistical threshold, subject to the constraint that false alarms remain acceptable. The major advantage of the physical threshold is that it can be set before any fault data are available, and it can be updated if the operating envelope of the vehicle traction battery changes.
I also recognize that voltage data streams of the vehicle traction battery can change with degradation. Therefore, the reference statistics do not have to be fixed for the entire life of the pack. The historical baseline can be rebuilt whenever the vehicle traction battery undergoes a major balancing operation, a battery swap, or a scheduled maintenance event. In a cloud-connected battery management system, the baseline can also be updated over-the-air.
Fault Localization via Eigenvector Contribution
For a vehicle traction battery composed of many cells, detection is not sufficient. The battery management system must also determine which cell is abnormal so that the driver or the maintenance service can take appropriate action. The eigen-decomposition that provides the fault detection indicator also provides eigenvectors that can be used for localization. I take advantage of this fact in my design.
When an abnormal eigenvalue \(\lambda_{q,k}\) is found, I examine its associated eigenvector \(q_{q,k}\). The squared entries of this eigenvector show how strongly each differential channel participates in the abnormal variance direction. The contribution of the \(i\)-th differential channel is normalized by
\[
\eta_{i}=\frac{v_{q,k}(i)^{2}}{\sum_{m=1}^{n-1}v_{q,k}(m)^{2}},
\]
where \(v_{q,k}(i)\) is the \(i\)-th component of the eigenvector. The normalized contributions satisfy \(\sum_{i}\eta_{i}=1\), so each term can be interpreted as the fraction of the anomalous variance allocated to the corresponding differential channel.
For a series-connected vehicle traction battery, each individual cell is shared by two adjacent differential channels. A fault in cell \(i\) will therefore affect both the \((i-1)\)-th differential channel and the \(i\)-th differential channel. To combine their evidence, I compute a cell-level contribution value as
\[
\zeta_{i}=
\begin{cases}
0.5\,(\eta_{i}+\eta_{i+1}), & i=1,2,\dots,n-1,\\
0.5\,(\eta_{n}+\eta_{1}), & i=n,
\end{cases}
\]
with the understanding that the cyclic differential representation is used. The index with the maximum \(\zeta_{i}\) is identified as the suspected faulty cell. In some cases, the fault is so weak that only one differential channel shows a high contribution. The direct grouping logic from the cyclic topology still allows the corresponding cell to be identified.
The important aspect of this localization method is that it is not a separate computation running after detection. The eigenvalues and eigenvectors come from the same matrix decomposition, so fault detection and localization are naturally synchronized. If a large eigenvalue occurs because of a high-current pulse rather than a genuine short-circuit, the eigenvector is usually distributed over many differential channels rather than concentrated around one cell. I use this property in the decision rule by applying a contribution threshold \(\eta_{\mathrm{th}}\). A fault alarm is confirmed only when the maximum cell-level contribution exceeds this threshold. Pseudo-faults caused by large current transients generally do not satisfy this condition.
The proposed contribution threshold therefore acts as a logical AND operation between the detection statistic and the localization statistic. This makes the diagnostic system more robust than a method that makes its decision only on the basis of the magnitude of a scalar indicator. More importantly, because the localization information is available immediately, the threshold of the detection statistic can be reduced without causing a corresponding increase in false alarms. This allows the vehicle traction battery to be monitored more sensitively.
Complete Diagnostic Workflow
I summarize the complete diagnostic workflow as follows.
During the offline stage, the battery management system collects a normal historical dataset from the vehicle traction battery. It applies the virtual voltage measurement scheme, estimates the normalization parameters, calculates the historical eigenvalue statistics, and determines the initial control limits. It also selects the contribution threshold \(\eta_{\mathrm{th}}\) from the distribution of the eigenvector contributions. Because both thresholds are expressed in a common statistical language, the user can adjust the sensitivity of the system without having to re-label fault samples.
During the online stage, the system acquires the voltage and current measurements at the sampling rate of the analogue front end. Every new sample updates the sliding-window matrix. The standardized virtual differential matrix is computed, followed by the covariance matrix and its eigen-decomposition. The fault indicator \(DI_{k}\) is evaluated. If \(DI_{k}\) is below the control limit, the system continues to monitor. If \(DI_{k}\) is above the control limit, the eigenvector contribution is checked. If the contribution is concentrated enough, the fault alarm is issued and the faulty cell is identified. If the contribution is diffuse, the alarm is treated as a temporary outlier, and the monitoring process continues.
The online computation is efficient because consecutive sliding windows overlap strongly. I use the rank-one modification algorithm for the eigenvalue update, which reduces the computational burden compared to a full eigen-decomposition at every sample. This is important for implementation in an embedded battery management system where memory and computational resources are limited.
Experimental Platform and Fault Injection
I constructed a compact vehicle traction battery experimental platform to validate the proposed method. The platform consists of fourteen A123 ANR26650M1B lithium iron phosphate cells connected in series. The cells are welded together with nickel strips, which simulates the interconnection resistance present in a real battery pack. A commercial battery management system prototype is used for voltage acquisition and active balancing. The analogue front-end chip measures each cell voltage with high accuracy, and the microcontroller executes the statistical diagnostic algorithm. The data are recorded at a sampling frequency of ten hertz.
The battery specifications are listed in the following table.
| Parameter | Value |
|---|---|
| Nominal voltage | 3.3 V |
| Cell capacity | 2.5 Ah |
| Voltage range | 2.0–3.6 V |
| Internal resistance | about 6 mΩ |
| Standard charge rate | 1C |
| Maximum discharge rate | 2C in the experiment |
| Pack configuration | 14S1P |
I used a bidirectional DC power supply to load the pack according to the World Light Vehicle Test Cycle. The maximum current amplitude is about five amperes, which corresponds to 2C for a 2.5 Ah cell. The pack is operated over a state-of-charge range from about ninety-three percent to about ten percent. This wide state-of-charge window covers the flat open-circuit voltage plateau as well as the steep portions of the discharge curve.
To emulate a soft short-circuit, I used a time-varying resistance method. An externally controlled resistor is connected in parallel with the target cell through a fast relay. The equivalent resistance of the short-circuit path is denoted \(R_{\mathrm{sc}}\). When the resistance is infinitely large, the pack is in the normal condition. When the resistance decreases to a finite value, a small current bypasses the target cell and causes its terminal voltage to drift downwards relative to the healthy cells.
The fault amplitude is controlled by the effective short-circuit resistance. I used the following model to compute the expected terminal voltage of a faulty cell
\[
V^{\mathrm{ssc}}_{i,k+t}=
\frac{V^{\mathrm{c}}_{i,k+t}+\Delta V_{\mathrm{oc,est},k+t}}
{1+R_{s,i,k+t}/R_{\mathrm{sc},i,k+t}},
\]
where \(V^{\mathrm{c}}_{i,k+t}\) is the nominal cell voltage, \(\Delta V_{\mathrm{oc,est},k+t}\) is the estimated open-circuit voltage shift caused by the fault-induced charge loss, and \(R_{s,i,k+t}\) is the dynamic source resistance of the target cell. I validated the fault injection method by comparing the simulated voltage response with the experimentally measured response when a ten-ohm fixed fault resistor is switched into the circuit. The agreement between the simulated and measured data is sufficiently accurate for diagnostic evaluation.
I considered two fault categories. The first is a momentary soft short-circuit, which is created by switching the fault resistor from a very high value to a finite low value at one instant and leaving it in operation for a fixed period. The second is a cumulative soft short-circuit, which is created by varying the fault resistor smoothly over time according to a logistic-like transition:
\[
R_{\mathrm{sc}}(t)=R_{\mathrm{sc,min}}+
\frac{R_{\mathrm{sc,max}}-R_{\mathrm{sc,min}}}
{1+\exp\bigl(\gamma(t-t_{0})\bigr)}.
\]
This creates a realistic gradual increase of the fault current and therefore mimics the slow evolution of a lithium dendrite short-circuit. In my experiments, the fault resistance is set to about \(11.8\ \Omega\) at the most severe instant. The resulting voltage deviation is about \(10.7\ \mathrm{mV}\), which is small enough to challenge the detection capability of conventional data-driven methods.
The following table lists the relevant resistance components of the experiment.
| Resistance component | Typical value |
|---|---|
| Static internal resistance of a cell | about 6 mΩ |
| Nickel-strip connection resistance | about 6.6 mΩ |
| Filtered dynamic impedance | 8–17 mΩ |
| Fault connection resistance | about 7 mΩ |
| Steady-state polarization resistance | about 8 mΩ |
Experimental Validation of the Virtual Measurement Scheme
I now compare three different voltage measurement conditions to validate the virtual measurement scheme. The first condition uses a standard non-virtual configuration after the pack is charged under the standard constant-current constant-voltage procedure. The second condition uses a near-cut-off voltage alignment operation before charging, but keeps the original physical channel order. The third condition uses the full virtual voltage measurement scheme, including the near-cut-off voltage alignment and the optimized virtual channel reordering.
The differential voltage curves under these three conditions differ substantially. In the first condition, several differential channels show a large slowly drifting offset that follows the filtered dynamic internal-resistance signature. This offset is not caused by any fault; it is caused by the capacity and resistance mismatch of the cells. In the second condition, the offset is greatly attenuated because the cells are aligned at the same open-circuit voltage plateau. In the third condition, the residual differential fluctuation is reduced further because the virtual order places healthy cells with similar dynamic behavior adjacent to each other.
I quantify the improvement using two metrics: the standard deviation of each differential channel, and the mean deviation of these standard deviations across all channels. A smaller mean deviation means that the normal operating condition is more uniform across all differential voltage channels, which makes a single global fault threshold more representative.
| Differential channel | Condition 1 SD (mV) | Condition 2 SD (mV) | Condition 3 SD (mV) |
|---|---|---|---|
| 1 | 3.380 | 0.742 | 0.923 |
| 2 | 4.073 | 1.030 | 0.742 |
| 3 | 4.284 | 1.152 | 0.837 |
| 4 | 3.269 | 0.990 | 0.816 |
| 5 | 2.318 | 0.997 | 0.811 |
| 6 | 2.181 | 1.435 | 0.999 |
| 7 | 0.932 | 1.078 | 0.828 |
| 8 | 1.088 | 0.732 | 1.014 |
| 9 | 2.911 | 1.208 | 0.995 |
| 10 | 2.314 | 0.927 | 0.893 |
| 11 | 4.761 | 1.029 | 1.122 |
| 12 | 3.367 | 0.634 | 0.814 |
| 13 | 2.975 | 0.749 | 0.997 |
| Mean deviation | 0.880 | 0.170 | 0.094 |
The table confirms that the full virtual measurement scheme produces the smallest mean deviation. The third condition reduces the mean deviation to only about 10.6 percent of the first condition. The largest channel standard deviation is also reduced from \(4.761\ \mathrm{mV}\) to \(1.122\ \mathrm{mV}\). This consistency is very useful for the threshold relation derived previously, because the denominator of the threshold relation depends on the historical eigenvalue statistics of the normal datasets. A smaller inconsistency-induced variance produces a smaller denominator and therefore permits a lower detection threshold for the same allowable fault amplitude.
I also note that the virtual measurement scheme is robust over the entire tested state-of-charge window. The initial state of charge is about 93 percent and the final state of charge is about 10 percent. This covers more than 83 percent of the usable capacity of the vehicle traction battery. Even at the lowest state of charge, where the open-circuit voltage curve begins to fall sharply, the differential channels remain sufficiently consistent because the near-cut-off alignment was performed in the same low-voltage region.
Soft Short-Circuit Detection Results
I injected soft short-circuit faults into the eleventh cell of the vehicle traction battery pack. The target cell is labelled cell 11 in the physical assembly. I used the fault resistance value described earlier so that the resulting voltage deviation is comparable to a mild internal short-circuit. The proposed diagnostic system operates with a sliding window width of 100 samples. The control limit is self-commissioned from the normal historical dataset using the interpretable threshold relation.
The following table compares the diagnostic performance of the proposed method with two recent data-driven methods: a voltage cosine similarity method and a voltage-correlation independent component analysis method. I include both the momentary and cumulative fault scenarios.
| Fault scenario | Method | Detection time (s) | Sustained alarm | Continuous alarm after fault removal | Fault detection rate (%) |
|---|---|---|---|---|---|
| Momentary SSC | Voltage cosine similarity | 80.1 | No | No | about 0.00015 |
| ICA-based voltage correlation | 9.9 | No | Yes | about 0.029 | |
| Proposed method | 109.6 | Yes | Yes | 98.4 | |
| Cumulative SSC | Voltage cosine similarity | No detection | No | No | 0 |
| ICA-based voltage correlation | 649.8 | No | No | 0 | |
| Proposed method | 171.3 | Yes | Yes | 97.6 |
For the momentary soft short-circuit, all three methods are able to trigger a first alarm. The ICA-based voltage correlation method is the fastest, with a detection time of 9.9 seconds, and the voltage cosine similarity method detects the fault in 80.1 seconds. The proposed method detects the fault in 109.6 seconds. However, the conventional methods do not maintain a sustained alarm over the whole fault interval. Their indicators return below the threshold while the fault is still present, because the transient voltage feature that caused the first alarm decays once the polarization voltage reaches a new quasi-steady state. The proposed method, by contrast, continues to produce an elevated fault indicator as long as the soft short-circuit exists. This sustained alarm behavior is particularly important for a safety system, because a short-circuit cannot be considered harmless simply because its voltage derivative vanishes.
The detection rate is another important metric. I define the detection rate as the percentage of time samples during the fault interval in which the diagnostic system correctly indicates the fault. In the momentary soft short-circuit test, the proposed method achieves a detection rate of 98.4 percent, whereas the two comparison methods have detection rates close to zero. The low detection rates of the comparison methods indicate that they detect only a short transient at the beginning of the fault, after which their indicators return to normal.
The cumulative soft short-circuit scenario is much more difficult. In this scenario, the equivalent short-circuit resistance decreases gradually rather than instantly, so the voltage deviation evolves slowly. The voltage cosine similarity method does not produce a clear alarm at all, and the ICA-based voltage correlation method detects the fault only after 649.8 seconds. The proposed method detects the cumulative fault after 171.3 seconds, which is faster than the correlation method by a large margin, and it maintains a detection rate of 97.6 percent. These results show that the proposed fault indicator, aided by the improved virtual measurement baseline, is especially effective when the fault signature accumulates slowly inside the sliding window.
Localization and Correction Behavior
In the experimental fault-injection tests, the proposed method not only detects the fault but also localizes it to cell 11. The localization information is obtained from the eigenvector contributions. For the duration of the experiment, the localization output remains concentrated around the same physical cell. A separate localization test with no fault injection shows that the contribution values are much more evenly spread over the pack, especially during high-current transient events. This confirms that the eigenvector contribution threshold is an effective way to reject pseudo-faults that arise from current-induced differential voltage fluctuations.
The proposed method also produces a continuous alarm after the fault is removed. This is an important difference from the conventional methods. When a permanent soft short-circuit has been present for some time, the faulty cell has lost additional charge relative to the healthy cells, and its state of charge has shifted downward. Even after the external short-circuit path is disconnected, the open-circuit voltage of the faulty cell is lower than that of the healthy cells. The residual state-of-charge imbalance cannot be removed instantly by a balancing system. Therefore, the symptom of the fault persists for a long time. In the experimental results, the proposed fault indicator continues to rise after the fault resistance returns to infinity, which alerts the operator to the fact that the pack is not yet normal. This “memory” effect is highly valuable for fleet management and battery maintenance.
To further illustrate the contribution of localization to detection, I examine the performance of the proposed method when the fault detection control limit is lowered. Without localization, lowering the control limit would increase the number of false alarms caused by transient current peaks. With the contribution threshold, those false alarms can be rejected because their anomalous eigenvectors are distributed over many cells. Therefore, the control limit can be safely lowered, which improves the detection speed.
In the experiment, I reduce the confidence level used for the statistical threshold so that the detection threshold decreases. The result is that the momentary soft short-circuit is detected about 101.8 seconds earlier than in the initial threshold setting. The cumulative soft short-circuit is detected about 106 seconds earlier. No unintended false alarms are triggered. This shows that the parallel design of detection and localization can significantly improve the speed of diagnosis without sacrificing reliability.
The benefit of localization becomes even more valuable when the vehicle traction battery is large. In a pack with dozens of series-connected cells, a high-current pulse creates a global differential voltage distortion that may cause the detection indicator to increase. In the eigenvector space, however, this distortion is often distributed over many cells. The contribution threshold can reject the global disturbance. A genuine soft short-circuit, by contrast, is physically localized, and its eigenvector contribution is concentrated on one or two differential channels. The localization stage therefore provides a powerful physical constraint that reduces false alarms in real driving.
Discussion of the Threshold Adjustment
The quantitative relation between the threshold and the fault amplitude leads to a practical curve that I validate experimentally. The curve shows the theoretical minimum detectable fault voltage as a function of the detection threshold. I compare this theoretical curve with the experimentally measured minimum detectable fault voltage. For high threshold values, the theoretical and experimental curves agree well. When the threshold falls below a certain value, the experimental curve becomes flatter than the theoretical curve. The reason is that the localization correction becomes the dominant source of robustness, so the detection threshold can be reduced further without producing false alarms.
The difference between the theoretical and experimental curves is at most about 6.5 percent. This difference is caused by the fact that some high-current operating conditions create a multi-channel anomaly that is not exactly the same as the single-channel assumption used in the theoretical derivation. Nevertheless, the difference is small enough that the self-commissioning procedure remains accurate for practical use.
I also discuss the choice of the allowable soft short-circuit amplitude. In a commercial vehicle traction battery, the allowable amplitude depends on the operating conditions and on the tolerance of the vehicle manufacturer. A smaller allowable amplitude leads to a lower detection threshold and an earlier fault alarm. The advantage of the proposed formulation is that the threshold is no longer an arbitrary statistical percentile. It is directly tied to a measurable physical quantity, namely the voltage deviation of the faulty cell relative to the healthy pack. This makes the diagnostic requirement easier to specify, verify, and certify.
Practical Implementation Considerations
The proposed method is intended to run inside the battery management system of an electric vehicle. Several practical considerations must be taken into account. The first is sampling rate. I used ten hertz in the experiment, which is consistent with many commercial battery management systems. The method also works with lower sampling rates, although the detection time may change because the sliding window needs to cover the same physical time span. The second is memory. The covariance matrix has size \(n-1\), where \(n\) is the number of series-connected cells or virtual cell positions. For a battery pack with 96 or 120 cells, the covariance matrix is in the order of one hundred by one hundred, which is manageable with modern microcontrollers.
The third consideration is computation. A full eigen-decomposition at every sample can be expensive. I therefore use a recursive rank-one eigenvalue update that exploits the substantial overlap between consecutive sliding windows. The update reduces the repeated calculation burden while preserving the required numerical accuracy. This recursive algorithm is well suited to the real-time implementation of the proposed diagnostic method for the vehicle traction battery.
The fourth consideration is adaptation to aging. The voltage correlations among cells change as the vehicle traction battery ages. A diagnostic method that uses fixed historical statistics may eventually become too sensitive or too insensitive. In my framework, the baseline statistics and the virtual reordering matrix can be updated periodically. The update can be triggered by the completion of a balancing operation or by a service event. Because the update is performed in the same mathematical framework as the original calibration, no additional algorithmic infrastructure is needed.
The fifth consideration is the computation of the current state of charge. The proposed method does not directly require the state of charge in the real-time detection step, because it compares simultaneous voltage differences among cells. The state-of-charge information is used only during the offline calibration to define the valid operating range. This makes the method less sensitive to state-of-charge estimation error, which is often one of the weakest parts of a battery management system.
Application to Other Cell Chemistries
Although the experimental validation is performed with lithium iron phosphate cells, the proposed statistical framework can be extended to other lithium-ion chemistries with appropriate modifications. The virtual voltage measurement scheme includes a near-cut-off voltage alignment action that is tailored to the flat open-circuit voltage plateau of lithium iron phosphate. For a cell chemistry with a sloped open-circuit voltage curve, such as a nickel-manganese-cobalt cathode, the alignment action may be less effective because a small state-of-charge mismatch always produces a corresponding voltage mismatch. In that case, the measurement baseline can be improved by more frequent balancing operations or by a model-assisted voltage correction before the differential voltages are computed.
However, the eigen-decomposition-based fault detection indicator and the eigenvector-based localization method are chemistry-independent. They operate on the standardized covariance matrix of the measured voltage differences. If the normal baseline is established from a healthy dataset of the specific pack, the same algorithm can be used for lithium iron phosphate, nickel-manganese-cobalt, or other lithium-ion chemistries. The main change is the calibration procedure and the selection of the allowable fault amplitude.
I also note that the proposed method does not depend on a specific battery equivalent-circuit model. This is an advantage over model-based methods because the equivalent-circuit parameters of a lithium-ion cell change with state of charge, temperature, current rate, and aging. The statistical method does not require real-time estimation of all those parameters. It uses the historical data as its own implicit model. As long as the normal operating pattern remains statistically representative, the method can adapt to the current state of the battery.
Advantages and Limitations
The advantages of the proposed method can be summarized as follows. First, the virtual measurement scheme suppresses the intrinsic inconsistency-induced differential voltage fluctuation, which is an often-neglected source of error in the vehicle traction battery. Second, the fault detection indicator is interpretable: its threshold is quantitatively related to the allowable fault voltage amplitude. Third, the eigenvector-based localization runs in parallel with detection and corrects false alarms, which allows the detection threshold to be lowered. Fourth, the method can detect both momentary and cumulative soft short-circuits with high detection rates. Fifth, it maintains a persistent alarm after a fault is removed, which provides important information about the residual state-of-charge imbalance.
There are also limitations. The virtual measurement scheme was designed with the open-circuit voltage plateau of lithium iron phosphate in mind. For a vehicle traction battery with a strongly sloped open-circuit voltage curve, extra correction measures may be necessary. The fault injection method used in the experimental validation simulates an external parallel resistance, which is an approximation of a real internal short-circuit. Internal short-circuits have complex interactions with the separator, electrolyte, and electrode materials; however, from the voltage measurement point of view, the parallel-resistance model remains the most widely used approximation in the literature. Finally, the proposed method relies on the quality of the normal historical dataset. A severely unbalanced or damaged vehicle traction battery must be rebalanced or repaired before the baseline is established; otherwise, the baseline will contain some abnormal characteristics and the threshold relation will be biased.
Conclusion
In this article, I have proposed an interpretable full-dimensional statistical analytics method for soft short-circuit fault diagnosis in the vehicle traction battery. The method covers all stages of the diagnostic chain: measurement, detection, localization, and threshold commissioning. I first presented a virtual voltage measurement scheme that reduces the effects of inherent parameter inconsistency in lithium iron phosphate battery packs. The scheme uses a near-cut-off voltage alignment action combined with a virtual channel reordering operation. This reduces the differential voltage standard deviation and the mean deviation of the standard deviations, thereby providing a cleaner input for the subsequent detector.
I then designed a fault detection indicator based on the eigen-decomposition of the standardized differential voltage covariance matrix. The indicator uses the largest normalized eigenvalue deviation as the fault statistic. Through a detectability analysis, I established a quantitative relation between the indicator threshold and the original soft short-circuit voltage signature. This relation enables threshold self-commissioning with clear physical meaning, rather than relying on a black-box anomaly threshold. I also derived an eigenvector contribution measure that localizes the faulty cell in the series pack. Because the eigenvalues and eigenvectors of the covariance matrix are obtained from the same linear algebra operation, detection and localization can be performed simultaneously. The localization information can be used to reject false alarms caused by current-induced differential voltage fluctuations, and the detection threshold can be reduced to improve detection speed.
The experimental validation on a fourteen-cell vehicle traction battery pack under the World Light Vehicle Test Cycle demonstrates the practical value of the method. The proposed approach detects and localizes a soft short-circuit with a fault resistance of about 11.8 ohms and a voltage signature of about 10.7 millivolts within a few minutes. In the considered state-of-charge range, the detection rates for momentary and cumulative soft short-circuits exceed 97 percent. The virtual measurement scheme reduces normal differential voltage inconsistency by almost one order of magnitude. The localization-based threshold reduction shortens the detection time by more than 60 percent without triggering false alarms. This combination of physical interpretability, robustness, and sensitivity makes the method well suited for embedding in the next-generation battery management system of electric vehicles.