1. Introduction
Lithium-ion batteries (LIBs) offer advantages such as high power density and long cycle life, making them widely used in applications like electric vehicles and grid energy storage [
1,
2,
3]. However, lithium-ion battery (LIB) systems are vulnerable to harsh external environments, operational stresses, and misuse, which can result in multiple faults that not only hasten degradation but also potentially trigger incidents such as thermal runaway [
4,
5]. These concerns have raised public apprehension about the safety of LIBs. Therefore, diagnosing various faults is crucial for enhancing the safety of LIBs.
Recent analyses conducted by various research organizations regarding thermal runaway events in LIBs have revealed that internal short circuits (ISCs) are a leading factor contributing to these incidents [
6,
7,
8]. Internal short circuits in LIBs are typically triggered by mechanical damage, electrical overload, and thermal stress [
9,
10,
11]. Mechanical abuses, such as collisions, crushes, and nail penetrations, can compromise the battery structure, causing severe deformation that leads to electrical connections between the positive and negative electrodes, thereby initiating an ISC [
12,
13]. Thermal abuse can lead to significant shrinkage of the separator, allowing contact between sections of the positive and negative electrodes, which in turn induces an ISC [
14,
15,
16]. Electrical abuse can lead to the deposition of metals such as lithium on the negative electrode, forming lithium dendrites. These dendrites may eventually penetrate the separator, causing an ISC. Moreover, manufacturing defects, such as metal particle contamination and electrode burrs introduced during battery production, can also result in ISCs [
17,
18,
19]. Internal short circuits often experience a gradual evolution phase before escalating into thermal runaway. At the initial ISC, i.e., micro short circuit (MSC), the equivalent short-circuit resistance (SR) remains high, leading to minimal changes to the battery’s parameters., which makes these faults difficult to detect [
20].
Current methods for diagnosing MSCs in LIB packs can be generally divided into detection and estimation techniques. Detection methods encompass various strategies that identify and locate MSC cells within the LIB pack by analyzing voltage correlations, state of charge (SOC) correlations, and the ranking of charging voltages. For instance, Lai et al. [
21] developed an early detection technique for internal short circuits based on SOC correlations. This method employs an extended Kalman filter to estimate the SOC for each cell in the battery pack. Subsequently, a moving window approach calculates the SOC correlation coefficients among neighboring cells. If the correlation coefficients between a specific cell and its adjacent cells on either side drop below a certain threshold, that cell is deemed to be in an MSC state. Additionally, Chang et al. [
22] proposed the constant assumption regarding the ranking of charging voltages among cells in a battery pack, suggesting that the voltage order of healthy cells remains consistent. In contrast, the presence of an MSC leads to variations in the charging voltage ranking of the affected cell, which facilitates the detection and localization of MSC cells.
Estimation methods include those based on the mean-difference model, charging characteristics, and Coulomb counting (CC). Zheng et al. [
23] employed a mean-difference model and extended the Kalman filtering algorithm to obtain the SOC deviations for each cell in the LIB pack. Decision tree analysis revealed that the SOC deviations of low-capacity cells in a series battery pack are correlated with the average SOC of the pack, whereas the SOC deviations of MSC cells are time-dependent. Mutual information was utilized to quantitatively assess the correlation between the SOC deviations of individual cells and the average SOC, enabling the differentiation between MSC cells and low-capacity cells. Ma et al. [
24] selected the median cell from the voltage ranking results as the representative cell for normal cells within the LIB pack. Linear fitting of the SOC difference curve before the inflection point was used to estimate short-circuit current (SC) and resistance. The results demonstrate that this method allows for the quantitative detection of MSC faults. Charging characteristic-based methods extract features from data collected during the constant current charging phase that represents the energy consumed by the SR. SC is then calculated by observing the rate of change of these features over time [
25]. CC-based methods utilize an equivalent circuit model (ECM) to establish state-space equations for the LIB. SC is integrated as a variable into these equations. The SOC and SC are then estimated using algorithms such as extended Kalman filtering and H
∞ nonlinear observers. The presence of an MSC is determined by examining the discrepancy between the SOC estimated by the observer and the SOC calculated by CC [
26]. Additionally, Shen et al. [
27] proposed an SR estimation method based on an ECM and adaptive filtering algorithm. Furthermore, Lei et al. [
28] introduced an MSC diagnostic method based on the maximum charging voltage variation between adjacent cycles. The key distinction between these two approaches lies in the fact that the former relies on the ECM of LIBs, while the latter, essentially a data-driven method, primarily estimates SC based on changes in voltage characteristics.
In summary, current estimation methods calculate SC based on the SOC differences over time and further estimate SR. However, these methods have several limitations. Firstly, existing these approaches require the ECMs. The ECMs used for LIBs frequently do not accurately represent the internal reaction processes, as their reliability is significantly affected by the parameters of the model. As a result, the accuracy of SR estimation is constrained by the inherent precision of the model and is vulnerable to changes in these parameters. Secondly, these methods necessitate the calculation of SC and SR for each cell within the LIB pack [
27]. In real-world scenarios, only a minimal number of cells within a LIB pack develop MSCs, accounting for an insignificant percentage of the overall cell count. Consequently, the indiscriminate calculation of SC and SR for every cell is inefficient and burdensome, especially in large LIB packs. Therefore, enhancing the accuracy and computational efficiency of MSC fault diagnosis in LIB packs is a critical issue that needs to be addressed.
To address the issues present in current research, this paper proposes an early ISC diagnosis method for LIB packs based on incremental capacity (IC) and dynamic time warping (DTW) distance. First, the terminal voltages of all cells in the LIB pack are ranked to obtain the median terminal voltage, from which the median IC curve is derived. This median IC curve serves as a reference standard representing the state of healthy cells within the LIB pack. Next, the DTW distance between each cell’s IC curve and the median IC curve is calculated. Cells with DTW distances exceeding a predefined threshold are diagnosed with MSC faults. For the detected MSC cells, the SC and SR are quantitatively calculated based on the variation in the end-of-charge voltage. Experimental results demonstrate that the proposed method effectively detects MSC cells within the LIB pack and accurately computes their SR, thereby quantifying the evolution and severity of MSCs.
2. Diagnostic Method
The overall structure of the proposed diagnostics is displayed in
Figure 1. As shown, this method encompasses three primary stages: First, the terminal voltages of all cells within the LIB pack are arranged in ascending order to determine the median terminal voltage. This median value is then utilized to derive the median IC curve, which serves as a reference standard representing the condition of healthy cells in the battery pack. Following this, the DTW distance between each cell’s IC curve and the median IC curve is computed. Cells that exhibit DTW distances surpassing a specified threshold are identified as MSC cells. Lastly, for the MSC cells identified, the SC and SR are estimated based on the changes observed in the maximum charging voltage (MCV).
2.1. Extraction of Incremental Capacity Curves
As shown in
Figure 1, once the median terminal voltage of the cells in a LIB pack is established, the corresponding median IC curve must be derived. Furthermore, the IC curve for each individual cell should be calculated based on its terminal voltage. The IC curve is typically determined using the following equation:
where
Q represents the battery’s charging capacity, and
V denotes the terminal voltage under constant current charging. Equation (1) illustrates the calculation of IC based on equal voltage intervals Δ
V, where
Q2 −
Q1 represents the change in charging capacity within the voltage interval Δ
V. In this study, the IC is calculated using the equal voltage interval method. To effectively capture the characteristics of the IC curve, Δ
V is set to 1 mV.
The IC curve obtained using the aforementioned numerical differentiation method is susceptible to measurement noise. Therefore, it is essential to apply an appropriate filter to achieve a smooth IC curve. In this study, a moving average filter is employed for smoothing the IC curve. Given a time-varying signal
s contaminated by noise, the moving average filter can be designed as follows:
where
sr represents the value of the original signal
s at time
r,
denotes the corresponding filtered value, and 2
Np + 1 is the window size of the moving average filter, where
Np is an integer. The parameter
l indicates the time lag, and
sr−l is the value of the original signal at time
r −
l. Based on the reference [
27],
Np is set to 15 in this study.
2.2. Detection of MSC Cells Based on DTW Distance
This study employs DTW to measure the similarity between each cell’s IC curve and the median IC curve, thereby detecting MSC cells within the LIB pack. DTW is a robust and highly accurate method for measuring the similarity of data sequences. By warping the time axis of the data sequences, DTW aligns data points between sequences, allowing for a more precise morphological similarity assessment, even for sequences of unequal lengths [
29].
We denote the median IC curve as
Cmed and the IC curve of an individual cell as
C:
where
M and
N represent the lengths of the median IC curve and the individual cell IC curve, respectively. The distance matrix
A is constructed based on the Euclidean distance between every pair of points on the two curves and is calculated as follows:
where
dij denotes the Euclidean distance between the
i-th sample point
cmed (
i) on the median IC curve and the
j-th sample point
c(
j) on the individual cell IC curve. The calculation formula is given as follows:
The DTW distance
D(
Cmed,
C) is defined as follows: it is the minimum cumulative distance value obtained by finding the optimal warping path
Wbest in the distance matrix
A that aligns
Cmed and
C. The optimal warping path
Wbest can be defined by the following formula:
Thus, the mathematical definition of DTW can be expressed as:
where
Wk represents the position of the warping path
W in the distance matrix
A. To ensure the uniqueness of the solution for the DTW mathematical definition, three constraints are imposed on Equation (8):
- (1)
Boundary Condition: The boundary points of the optimal warping path Wbest are fixed, specifically W1 = (1, 1) and Wk = (M, N).
- (2)
Monotonicity: This condition ensures the search direction for Wbest. Specifically, for a given Wk = (i, j) and Wk+1 = (i∗, j∗), it must hold that i∗ ≥ i and j∗ ≥ j.
- (3)
Continuity: The search for Wbest can only proceed to adjacent points. Specifically, for a given Wk = (i, j) and Wk+1 = (i∗, j∗), it must hold that i∗ ≤ i + 1 and j∗ ≤ j + 1.
Given the three constraints, the solution to Equation (8) can be obtained using dynamic programming. The state transition equation for dynamic programming is as follows:
Based on the above process, the DTW distance between each cell’s IC curve and the median IC curve can be computed. Cells with a DTW distance below a certain threshold are considered normal. In contrast, cells with a DTW distance exceeding the threshold are diagnosed as experiencing an MSC.
2.3. Estimation of SR Based on Maximum Charging Voltage Variation
For an MSC cell, this study further estimates the SC and SR to quantitatively assess the severity and progression of the MSC. This assessment aids the battery management system in implementing targeted countermeasures.
As depicted in
Figure 2, we examine the charging voltage of an MSC cell over two consecutive cycles. The figure clearly indicates that the MCV
2 during the second cycle (Cycle 2) is significantly lower than the MCV
1 recorded in the first cycle (Cycle 1). This decline in voltage is attributed to energy being lost due to the SR. If the MSC cell is removed from the pack after the second charging cycle and charged separately until its MCV matches that of Cycle 1, as indicated by the green dashed line in
Figure 2, the additional time taken to reach this state, represented as Δt, is referred to as the remaining charging time (RCC) for the MSC cell.
In the real-world operation of energy storage LIB packs, removing an MSC cell for independent recharging to get the RCC Δt is impractical. As a result, Δt is theoretically unknown. However, it can be noted that if we project the maximum charging voltage (MCV2) from the next cycle onto the voltage curve of the prior cycle and denote the corresponding point in time as t, the time interval from t to the end of the previous cycle equals the RCC Δt. Hence, the RCC Δt for an MSC cell in the current cycle can be estimated by mapping the current cycle’s MCV onto the charging voltage curve from the preceding cycle.
Once the RCC Δt in the current cycle has been determined, the energy loss during this cycle can be calculated. This is done by multiplying Δt by the charging current of the LIB pack. The resulting energy loss, caused by the SR, is expressed as follows:
where
Qloss,p denotes the energy loss of the MSC cell at the end of the
p-th charging cycle, and
I represents the charging current. The rate of change of energy loss with respect to time corresponds to the SC of the MSC cell. Thus, we have the following equation:
where
IMSC represents the MSC current,
Qloss,p denotes the estimated energy loss of the MSC cell at the end of the
p-th charging cycle,
Qloss,p−1 denotes the estimated energy loss at the end of the (
p − 1)-th charging cycle,
Tp is the time at the end of the
p-th charging cycle, and
Tp−1 is the time at the end of the (
p − 1)-th charging cycle.
Based on the calculated SC, the SR can be further determined using the following formula:
where
RMSC denotes the short-circuit resistance and
UM represents the average voltage between the end-of-charge moments of two consecutive cycles. A smaller short-circuit resistance indicates a higher power dissipation due to the short circuit, which increases the likelihood of thermal runaway.
3. Experimental Section
Current techniques for triggering short circuits include mechanical damage, such as puncturing or crushing cells, temperature-sensitive materials into the battery, or promoting dendrite growth through overcharging or over-discharging. However, these methods mainly simulate late-stage ISC faults, which often result in rapid cell failure or explosion, offering limited control and repeatability. An alternative approach is to connect an external resistor in parallel with the battery terminals, which allows for a more flexible simulation of different short-circuit stages by adjusting the resistance. Since the heat generated by MSCs is minimal and effectively managed by the thermal system, thermal effects can be disregarded. MSC faults here are simulated by connecting an external resistor in parallel with the cells, enabling better control over the initiation and progression of MSCs, as well as improving both repeatability and control over the fault evolution process.
The experimental platform is illustrated in
Figure 3. All tests were carried out in a temperature-controlled environment at 25 degrees Celsius. The LIB testing system was operated via a host computer, which facilitated the execution of charge and discharge cycles on the batteries. During the experiments, data was sampled at a frequency of 1 Hz. Additionally,
Figure 3 includes a photograph of the series-connected battery pack situated inside the thermostatic chamber during the charge–discharge process. The experimental battery pack comprised eight cylindrical 21,700 LIBs arranged in series, with detailed specifications for each cell provided in
Table 1.
In the experiment, the LIB pack was charged at a constant current rate of 0.5 C. The charging process was halted once the maximum terminal voltage of any individual cell reached the cutoff value of 4.2 V to avoid overcharging. Following the charging phase, the battery pack was subjected to a dynamic stress test (DST) discharge. Discharging was stopped when the minimum terminal voltage of any cell dropped to the cutoff value of 2.75 V, preventing over-discharge. In total, 14 charge–discharge cycles were completed. Starting from the third cycle, different SRs were connected in series and then placed in parallel across the terminals of Cells 4 and 8 to simulate MSC evolution. By activating various switches, different magnitudes of SR were paralleled across the battery terminals to mimic the initiation and progression of MSCs. A lower resistance corresponds to a higher SC, indicating a more severe fault. The specific SRs applied during each cycle are listed in
Table 2.
5. Conclusions
This paper introduces a method for detecting and quantitatively evaluating MSC faults in energy storage LIB packs, utilizing IC analysis alongside DTW algorithms. The median IC curve, derived from the sorted terminal voltages of the cells in the LIB pack, serves as a reference point to define the condition of normal cells within the pack. The DTW algorithm quantitatively assesses the similarity between the IC curve of each cell and the median IC curve, facilitating the identification of MSC cells. For the identified MSC cells, the SR is estimated based on variations in charging voltage. The outcomes of fault detection and quantitative evaluation indicate that the median IC curve consistently corresponds with the IC curves of normal cells, effectively representing their condition within the LIB pack. The highest relative error in the SR estimation recorded is 4.26%. Since the proposed approach does not necessitate the development of a battery model and employs a detection-first, quantification-later strategy, it enhances both applicability and computational efficiency. Future research will aim to validate the effectiveness and practicality of this method across various battery material systems and under different environmental temperatures.
It should be noted that the current version of the proposed method requires a complete IC curve. In electric vehicle applications, the initial SOC may not be 0% due to variations in user charging habits, making it impossible to obtain a full IC curve in such cases. A potential solution to this issue is to extract key features from the IC curve, such as peak values, peak positions, and peak areas, to form a feature vector that can be used for fault detection in place of the complete IC curve. This will be a primary focus of our future research. Moreover, since temperature and initial SOC influence the shape and trend of IC curves, they also affect the selection of the threshold. In practical applications, extensive offline testing under different temperature and initial SOC conditions can be conducted to determine a threshold that is robust to these variations, ensuring diagnostic accuracy. This will be another key focus of our future work.