This article presents a comprehensive study on internal short circuit (ISC) fault diagnosis for an energy storage battery pack based on state-of-charge (SOC) observation. A second-order RC equivalent circuit model is established and model parameters are identified online via the forgetting factor recursive least squares algorithm. A variable-gain sliding mode observer is designed for accurate SOC estimation of lithium-ion batteries. Based on the observed SOC, several ISC fault detection strategies are proposed, including fixed-threshold, dynamic-threshold, and SOC-variation-based methods. The proposed methods are verified under FUDS and DST driving cycles and are further extended to series-connected battery packs. Extensive comparisons with traditional sliding mode observers, dual extended Kalman filtering, and CNN-LSTM demonstrate the superiority of the proposed approach in terms of accuracy and robustness. The results show that the proposed diagnostic framework can effectively detect and locate internal short circuit faults in energy storage battery systems, thereby significantly improving safety and reliability.
1. Introduction
With the rapid development of electric vehicles and renewable energy systems, lithium-ion batteries have become one of the most crucial energy storage devices. However, under complex operating conditions, batteries may eventually suffer from thermal runaway, which is often triggered by internal short circuit faults. Internal short circuit is considered a common link in the evolution from normal operation to thermal runaway. Therefore, it is urgent to develop battery management algorithms that can accurately estimate battery states and detect faults at an early stage. This work focuses on battery modeling, state estimation, and ISC fault diagnosis for an energy storage battery pack.
In this study, a second-order RC equivalent circuit model is selected as the basis for battery modeling. The model parameters are identified using the forgetting factor recursive least squares method. A variable-gain sliding mode observer is proposed for SOC estimation, which overcomes the severe chattering problem of the conventional sliding mode observer. The estimated SOC is then used to construct ISC fault diagnosis indicators. Several diagnostic strategies are developed and evaluated under different operating conditions. Finally, the proposed method is extended to a series-connected battery pack to enable fault detection and localization.
2. Lithium Battery Modeling and Parameter Identification
2.1 Battery Type Selection and Working Principle
Lithium-ion batteries are classified into several types based on the cathode material, including NCM, LFP, LCO, LTO, and LMO. In this work, the NCM ternary lithium battery is selected as the research object because of its high energy density, high voltage, and long cycle life, which are particularly suitable for electric vehicle applications.
The working principle of a lithium-ion battery relies on the repeated migration of lithium ions between the positive and negative electrodes. During charging, lithium ions move from the cathode through the electrolyte and separator to the anode, while electrons flow through the external circuit. During discharging, the reverse process occurs, releasing electrical energy to the load.
2.2 Second-Order RC Equivalent Circuit Model
The equivalent circuit model is widely used in battery management systems due to its simplicity and acceptable accuracy. In this work, the second-order RC model is adopted. The model consists of an ohmic resistance \(R_0\), two RC networks representing polarization effects, and an open-circuit voltage source \(U_{ocv}\). The electrical behavior of the model is described by the following equations:
$$
\begin{aligned}
\dot{U}_{P1} &= -\frac{1}{R_{P1} C_{P1}} U_{P1} + \frac{1}{C_{P1}} i \\
\dot{U}_{P2} &= -\frac{1}{R_{P2} C_{P2}} U_{P2} + \frac{1}{C_{P2}} i
\end{aligned}
$$
where \(U_{P1}\) and \(U_{P2}\) are the polarization voltages, \(R_{P1}\) and \(R_{P2}\) are the polarization resistances, and \(C_{P1}\) and \(C_{P2}\) are the polarization capacitances. The terminal voltage \(U_t\) is given by:
$$
U_t = U_{ocv} – U_{P1} – U_{P2} – i R_0
$$
2.3 Online Parameter Identification with Forgetting Factor Recursive Least Squares
To obtain the model parameters in real time, the forgetting factor recursive least squares (FFRLS) algorithm is employed. The algorithm updates the parameter vector \(\theta_k\) recursively as follows:
$$
\theta_k = \theta_{k-1} + K_{Ls,k} \left( y_k – \varphi_k^T \theta_{k-1} \right)
$$
The gain matrix \(K_{Ls,k}\) and covariance matrix \(P_{Ls,k}\) are computed by:
$$
K_{Ls,k} = \frac{P_{Ls,k-1} \varphi_k}{\lambda + \varphi_k^T P_{Ls,k-1} \varphi_k}
$$
$$
P_{Ls,k} = \frac{1}{\lambda} \left( I – K_{Ls,k} \varphi_k^T \right) P_{Ls,k-1}
$$
where \(\lambda\) is the forgetting factor. The relationship between the identified coefficients and the actual battery parameters is derived from the model transfer function. After discretization, the parameter vector is defined as:
$$
\theta_k = \left[ k_1, k_2, k_3, k_4, k_5 \right]^T
$$
and the battery parameters can be recovered from these coefficients.
2.4 Parameter Identification Results under Different Forgetting Factors
The identification experiments are conducted on a 2.6 Ah NCM lithium battery under the FUDS and DST dynamic profiles at 25 °C. The forgetting factor \(\lambda\) plays a critical role in the identification performance. A larger \(\lambda\) results in smoother parameter estimates but slower tracking capability; a smaller \(\lambda\) leads to faster tracking but more severe fluctuations.
Table 2-1 compares the terminal voltage prediction errors under different forgetting factors.
| Forgetting Factor \(\lambda\) | Maximum Terminal Voltage Error (mV) | Parameter Stability |
|---|---|---|
| 0.99 | ~20 | High, smooth |
| 0.96 | ~15 | Medium, occasional spikes |
| 0.93 | ~15 | Low, severe spikes in \(R_{P2}\) |
To balance accuracy and stability, \(\lambda = 0.99\) is selected for subsequent SOC estimation and fault diagnosis.
The identified parameters under the DST profile are shown in Table 2-2 for different SOC levels.
| SOC (%) | \(R_0\) (Ω) | \(R_{P1}\) (Ω) | \(C_{P1}\) (F) | \(R_{P2}\) (Ω) | \(C_{P2}\) (F) |
|---|---|---|---|---|---|
| 100 | 0.00519 | 0.00239 | 23021.8 | 0.00014 | 24174.8 |
| 80 | 0.00570 | 0.00392 | 59805.0 | 0.00875 | 27604.4 |
| 60 | 0.00557 | 0.00922 | 26339.8 | 0.00054 | 24524.4 |
| 40 | 0.00600 | 0.00528 | 30514.1 | 0.00052 | 29757.7 |
| 20 | 0.00740 | 0.00795 | 25933.4 | 0.00240 | 76202.2 |
3. State of Charge Estimation of Energy Storage Battery
3.1 Sliding Mode Observer Design
Sliding mode observer (SMO) is a nonlinear state estimation method with strong robustness against model uncertainties and external disturbances. For the second-order RC battery model, a state-space representation is constructed with state variables \(U_{P1}\), \(U_{P2}\), and SOC. The sliding surface is defined as the terminal voltage estimation error:
$$
s = e_t = \hat{U}_t – U_t
$$
The sign function is used in the switching term. The observer structure is:
$$
\begin{aligned}
\dot{\hat{U}}_t &= f_1(\hat{x}) + L_1 \operatorname{sgn}(e_t) \\
\dot{\hat{U}}_{P1} &= f_2(\hat{x}) + L_2 \operatorname{sgn}(e_t) \\
\dot{\hat{U}}_{P2} &= f_3(\hat{x}) + L_3 \operatorname{sgn}(e_t) \\
\dot{\hat{SOC}} &= f_4(\hat{x}) + L_4 \operatorname{sgn}(e_t)
\end{aligned}
$$
To guarantee stability, the Lyapunov function is defined as:
$$
V = \frac{1}{2} e_t^2
$$
which requires \(\dot{V} < 0\). This imposes constraints on the observer gains \(L_i\).
3.2 Variable-Gain Sliding Mode Observer
The conventional SMO suffers from high-frequency chattering caused by the fixed switching gain. To solve this problem, a variable-gain sliding mode observer (VG-SMO) is proposed. The switching gains are designed as:
$$
L_i = \xi_i + h_i |e_t|
$$
where \(\xi_i\) is a small positive constant and \(h_i\) is a positive coefficient. This adaptive gain mechanism provides a large gain during the initial convergence phase to accelerate error reduction, and then automatically reduces the gain as the error becomes small, thereby suppressing chattering.
3.3 Simulation Results and Comparisons
The proposed VG-SMO is evaluated under the FUDS and DST cycles. The SOC reference is obtained by coulomb counting:
$$
SOC(t) = SOC(0) – \frac{\int_0^t i(\tau) d\tau}{Q_0}
$$
Figure 3-1 shows the SOC estimation results of the conventional SMO and the VG-SMO under FUDS.
The quantitative metrics MAE, RMSE, and MaxE are used for comparison. Table 3-1 summarizes the SOC estimation errors under the FUDS profile.
| Method | MAE (%) | RMSE (%) | MaxE (%) |
|---|---|---|---|
| Conventional SMO | 0.49 | 0.61 | 2.11 |
| Variable-Gain SMO | 0.37 | 0.43 | 0.85 |
Under the DST profile, the results are shown in Table 3-2.
| Method | MAE (%) | RMSE (%) | MaxE (%) |
|---|---|---|---|
| Conventional SMO | 0.40 | 0.61 | 1.52 |
| Variable-Gain SMO | 0.35 | 0.49 | 1.34 |
3.4 Comparison with DEKF and CNN-LSTM
To further validate the superiority of the proposed VG-SMO, comparisons are made with the dual extended Kalman filter (DEKF) and a hybrid CNN-LSTM neural network model. The DEKF is developed based on the second-order RC model, where two EKFs are used to jointly estimate the state vector and the parameter vector. The CNN-LSTM model takes voltage, current, and temperature data as inputs, with CNN extracting spatial features and LSTM capturing temporal dependencies.
The estimation errors under FUDS are listed in Table 3-3.
| Method | MAE (%) | RMSE (%) | MaxE (%) |
|---|---|---|---|
| Variable-Gain SMO | 0.37 | 0.43 | 0.85 |
| DEKF | 1.28 | 1.46 | 2.81 |
| CNN-LSTM | 1.40 | 1.56 | 3.70 |
Under DST, the errors are shown in Table 3-4.
| Method | MAE (%) | RMSE (%) | MaxE (%) |
|---|---|---|---|
| Variable-Gain SMO | 0.35 | 0.49 | 1.34 |
| DEKF | 0.46 | 0.73 | 1.94 |
| CNN-LSTM | 0.37 | 0.51 | 7.45 |
The results demonstrate that the proposed VG-SMO achieves the lowest MAE and RMSE in most cases, and significantly reduces the maximum estimation error, especially when compared with CNN-LSTM under the DST profile.
3.5 SOC Estimation for Series-Connected Battery Packs
For a battery pack, direct application of single-cell SOC estimation methods may suffer from high computational burden or low accuracy. In this work, a difference model approach is adopted. One cell is selected as the reference and its SOC is estimated using the VG-SMO. The SOC of the remaining cells is then inferred from the voltage differences between the reference cell and the other cells. This method greatly reduces the computational cost while maintaining acceptable accuracy.
Table 3-5 shows the SOC estimation errors for a four-series battery pack using the multi-cell method (independent SMO for each cell) and the difference model method.
| Cell | Multi-Cell MAE (%) | Multi-Cell RMSE (%) | Difference Model MAE (%) | Difference Model RMSE (%) |
|---|---|---|---|---|
| Cell1 | 1.60 | 2.05 | 1.60 | 2.05 |
| Cell2 | 2.06 | 2.63 | 1.81 | 2.67 |
| Cell3 | 1.83 | 2.30 | 1.60 | 1.97 |
| Cell4 | 1.82 | 2.51 | 1.60 | 1.96 |
The difference model achieves comparable accuracy with substantially lower computation, making it suitable for real-time applications in energy storage battery management systems.
4. Internal Short Circuit Fault Diagnosis
4.1 Fault Simulation of Internal Short Circuit
Internal short circuit is a critical failure mode that can lead to thermal runaway. In this work, the ISC fault is simulated by connecting a small resistor in parallel with the battery cell. This method is widely used because it is repeatable and easy to control. The ISC fault is injected between 5000 s and 6000 s in the simulation, representing a sudden short-circuit condition during the discharge process.
The terminal voltage under normal and faulty conditions is shown in Figure 4-1. It can be observed that the terminal voltage with ISC is slightly lower than the normal voltage, but still within the operating voltage range, which makes it difficult to detect by simple voltage limit checking.
To model the ISC fault, a parallel resistor \(R_{ISC}\) is added to the battery model. The equivalent circuit is modified accordingly.
4.2 Diagnosis Based on SOC Error with Fixed Threshold
When the battery is healthy, the SOC estimated by the VG-SMO closely tracks the reference SOC obtained by coulomb counting. When an ISC fault occurs, the voltage drop leads to an increasing deviation between the estimated SOC and the reference SOC. The diagnosis is performed by comparing the absolute SOC error with a threshold \(\delta\):
$$
SOC_{error} = |\hat{SOC} – SOC_{ref}| > \delta
$$
The threshold is set as:
$$
\delta = \eta \cdot MaxE
$$
where \(\eta\) is a safety factor (set to 1.5) and \(MaxE\) is the maximum SOC estimation error under healthy conditions (2%). Therefore, \(\delta = 3\%\).
Figure 4-2 shows the SOC estimation and diagnosis result.
In the simulation, the SOC error exceeds the threshold 171 seconds after the fault occurrence, successfully detecting the ISC fault.
4.3 Diagnosis Based on Dynamic Threshold
In practice, the reference SOC obtained by coulomb counting may be inaccurate due to unknown initial SOC or battery aging. When the initial SOC value has a 15% deviation, the fixed-threshold method fails to detect the fault because the SOC error is already large even in the healthy state.
To address this issue, a dynamic threshold is proposed:
$$
\delta(T) = \frac{1}{T} \sum_{t=1}^{T} |\hat{SOC}(t) – SOC_{ref}(t)| \pm \eta \cdot MaxE
$$
This dynamic threshold adapts to the accumulated estimation error and allows the fault to be detected even when the reference SOC has a constant bias.
Figure 4-3 shows the detection result with the dynamic threshold under a 15% initial SOC error.
4.4 Diagnosis Based on SOC Variation
When no reliable reference SOC is available, the SOC variation method is used. The instantaneous SOC variation is defined as:
$$
\Delta SOC(t) = \hat{SOC}(t) – \hat{SOC}(t-a)
$$
where \(a\) is the window size of a recursive moving window. Choosing a proper \(a\) amplifies the fault characteristics. In this work, \(a = 50\) seconds is selected.
For a healthy battery, \(\Delta SOC\) remains in the range \([-6\%, 0\%]\). When an ISC fault occurs, \(\Delta SOC\) first drops below \(-6\%\), and later exceeds \(0\%\) due to the two-point calculation. Therefore, two thresholds are used:
$$
\Delta SOC < -6\% \quad \text{or} \quad \Delta SOC > 0\%
$$
Figure 4-4 shows the detection result under the no-reference-SOC scenario.
The fault is successfully detected at 5221 s and 6281 s, which provides an early warning for thermal runaway prevention.
4.5 Battery Pack Internal Short Circuit Fault Diagnosis and Localization
The SOC variation method is extended to a five-series-one-parallel (5S1P) battery pack. The battery pack model is built in Simulink, where each cell is represented by the second-order RC model. The cell Cell-03 is deliberately set to experience an ISC fault during the simulation. The VG-SMO is applied to each cell using the measured current and voltage data.
The SOC estimates are shown in Figure 4-5. The detection results are summarized in Table 4-1.
| Cell | \(\Delta SOC\) in healthy range? | Fault detected? |
|---|---|---|
| Cell-01 | Yes | No |
| Cell-02 | Yes | No |
| Cell-03 | No | Yes |
| Cell-04 | Yes | No |
| Cell-05 | Yes | No |
The results clearly indicate that only Cell-03 is identified as faulty, which is consistent with the simulation setup. This demonstrates that the proposed method can not only detect the ISC fault but also locate the faulty cell in a series-connected energy storage battery pack.
5. Conclusion and Future Work
This paper presents a systematic study on SOC estimation and internal short circuit fault diagnosis for an energy storage battery pack. The main conclusions are as follows:
(1) A second-order RC equivalent circuit model is established for the lithium-ion battery. The FFRLS algorithm with a forgetting factor of 0.99 provides stable and accurate parameter identification results under FUDS and DST conditions.
(2) A variable-gain sliding mode observer is proposed for SOC estimation. The variable-gain mechanism reduces chattering and improves estimation accuracy. Compared with the conventional SMO, DEKF, and CNN-LSTM methods, the proposed VG-SMO achieves lower MAE, RMSE, and MaxE under both FUDS and DST profiles.
(3) The VG-SMO is combined with a difference model for battery pack SOC estimation, which achieves comparable accuracy to the multi-cell method with significantly lower computational cost.
(4) ISC fault diagnosis strategies based on SOC observation are developed. The fixed-threshold method works when an accurate reference SOC is available. The dynamic-threshold method handles the case of biased reference SOC. The SOC variation method with a recursive moving window is effective when no reference SOC is available. The method is successfully extended to a 5S1P battery pack for fault detection and localization.
Future work will focus on validating the proposed methods under different temperatures and aging conditions, and on developing fault isolation techniques to distinguish ISC from other battery faults. The ultimate goal is to enhance the safety and reliability of energy storage battery systems in real-world applications.