With the rapid advancement of new energy vehicles, the safety and reliability of power sources have become paramount. Among these, the lithium-ion battery stands as a critical component, and its fault diagnosis is essential for ensuring operational integrity. In this study, we propose a fault diagnosis technology for lithium-ion batteries based on an electro-thermal coupling model. This approach integrates the electrical and thermal characteristics of lithium-ion batteries to provide a comprehensive monitoring solution. The lithium-ion battery is a complex system where performance degradation or failures can lead to severe safety incidents, such as thermal runaway. Therefore, developing accurate fault diagnosis methods for lithium-ion batteries is crucial for enhancing the safety and longevity of energy storage systems. Our work focuses on combining model-based estimation with real-time data to detect anomalies in lithium-ion batteries early.
The electro-thermal coupling model is central to our methodology. It captures the dynamic interactions between electrical and thermal behaviors in lithium-ion batteries, which are often neglected in traditional models. For lithium-ion batteries, temperature variations significantly impact electrical parameters like internal resistance and capacity. Conversely, electrical operations, such as charging and discharging, generate heat that affects temperature. By coupling these aspects, our model offers a more precise representation of lithium-ion battery states. We utilize a second-order Thevenin equivalent circuit model for the electrical part and a lumped parameter thermal model for the thermal part. This integration allows for real-time tracking of voltage, current, and temperature in lithium-ion batteries.
The electrical model for lithium-ion batteries is derived from the second-order Thevenin equivalent circuit. It includes an open-circuit voltage source, ohmic resistance, and two RC networks to simulate polarization effects. The state-space equations are discretized for implementation. For a lithium-ion battery, the state of charge (SOC) is estimated using the ampere-hour integral method, while the terminal voltage is expressed as a function of SOC and internal states. The equations are as follows:
$$ SOC(t) = SOC_0 – \frac{\eta}{Q_n} \int_0^t I(\tau) d\tau \times 100\% $$
$$ U_0 = U_{OC} – I R_0 – U_1 – U_2 $$
$$ \frac{dU_1}{dt} = \frac{I}{C_1} – \frac{U_1}{R_1 C_1} $$
$$ \frac{dU_2}{dt} = \frac{I}{C_2} – \frac{U_2}{R_2 C_2} $$
After discretization, the state-space representation for the lithium-ion battery electrical model is:
$$ \begin{align*} SOC_{k+1} &= SOC_k – \frac{\Delta t}{Q_n} I_k \\ U_{1,k+1} &= U_{1,k} e^{-\frac{\Delta t}{\tau_{1,k}}} + I_k R_{1,k} (1 – e^{-\frac{\Delta t}{\tau_{1,k}}}) \\ U_{2,k+1} &= U_{2,k} e^{-\frac{\Delta t}{\tau_{2,k}}} + I_k R_{2,k} (1 – e^{-\frac{\Delta t}{\tau_{2,k}}}) \\ U_{0,k} &= U_{OC}(SOC_k) – I_k R_{0,k} – U_{1,k} – U_{2,k} \end{align*} $$
where $$ \tau_{i,k} = R_{i,k} C_{i,k} $$ for i = 1, 2. These equations form the basis for simulating the electrical behavior of lithium-ion batteries under various operating conditions.
The thermal model for lithium-ion batteries is based on a lumped parameter approach, assuming uniform temperature distribution within the battery. It consists of two thermal states: core temperature and surface temperature. The heat generation rate in lithium-ion batteries is calculated from electrical losses, primarily due to irreversible heat from internal resistances. The governing equations are:
$$ C_c \frac{d(T_c – T_a)}{dt} = Q – \frac{T_c – T_s}{R_c} $$
$$ C_s \frac{d(T_s – T_a)}{dt} = \frac{T_c – T_s}{R_c} – \frac{T_s – T_a}{R_s} $$
Here, Q is the heat generation rate, which for lithium-ion batteries is approximated as $$ Q = I (U_0 – U_{OC}) $$, ignoring reversible heat effects for simplicity. Discretizing these equations yields:
$$ \begin{bmatrix} T_{ca,k+1} \\ T_{sa,k+1} \end{bmatrix} = \begin{bmatrix} 1 – \frac{\Delta t}{R_c C_c} & \frac{\Delta t}{R_c C_c} \\ \frac{\Delta t}{R_c C_s} & 1 – \frac{\Delta t}{R_c C_s} – \frac{\Delta t}{R_s C_s} \end{bmatrix} \begin{bmatrix} T_{ca,k} \\ T_{sa,k} \end{bmatrix} + \begin{bmatrix} \frac{\Delta t}{C_c} \\ 0 \end{bmatrix} Q_k $$
where $$ T_{ca} = T_c – T_a $$ and $$ T_{sa} = T_s – T_a $$. This thermal model is coupled with the electrical model to form the electro-thermal coupling model for lithium-ion batteries.
Parameter identification is critical for accurate modeling of lithium-ion batteries. We employ the Forgetting Factor Recursive Least Squares (FFRLS) algorithm to estimate time-varying parameters in real-time. This method is particularly suitable for lithium-ion batteries due to their nonlinear and dynamic nature. The FFRLS algorithm updates parameters based on recent data, reducing the influence of old measurements. The steps are as follows:
1. Initialize parameters and covariance matrix.
2. Compute gain vector: $$ H_k = \frac{W_{k-1} \phi_k}{\lambda + \phi_k^T W_{k-1} \phi_k} $$
3. Update parameter estimate: $$ \theta_k = \theta_{k-1} + H_k (h_k – \theta_{k-1} \phi_k^T) $$
4. Update covariance matrix: $$ P_k = \frac{1}{\lambda} (W_{k-1} – H_k \phi_k^T W_{k-1}) $$
For the electrical model of lithium-ion batteries, the regression vector and observation are defined as:
$$ \phi_{E,k} = [U_{0,k-1}, U_{0,k-2}, I_k, I_{k-1}, I_{k-2}] $$
$$ h_{E,k} = U_{0,k} $$
$$ \theta_{E,k} = [a_E, b_E, c_E, d_E, e_E]^T $$
The parameters such as $$ R_0 $$, $$ R_1 $$, $$ R_2 $$, $$ C_1 $$, and $$ C_2 $$ for lithium-ion batteries are derived from these estimates. Similarly, for the thermal model, we use FFRLS to identify thermal parameters like $$ R_c $$, $$ R_s $$, $$ C_c $$, and $$ C_s $$. The open-circuit voltage of lithium-ion batteries is fitted as a polynomial function of SOC at different temperatures, as shown in the table below.
| Temperature (°C) | P1 | P2 | P3 | P4 | P5 | P6 | P7 | R² |
|---|---|---|---|---|---|---|---|---|
| -20 | 1.0440 | 0.0474 | -3.1950 | 3.0420 | -0.9391 | 0.6312 | 3.4390 | 0.9996 |
| -10 | 2.8800 | -4.0670 | -1.8770 | 5.7390 | -2.9190 | 1.0570 | 3.4140 | 0.9999 |
| 0 | 4.0310 | -0.7750 | 1.1320 | 5.9560 | -4.0660 | 1.4710 | 3.3570 | 0.9999 |
| 10 | 14.5200 | -41.5600 | 42.5300 | -17.7000 | 2.0660 | 0.9573 | 3.3250 | 0.9996 |
| 25 | 11.3700 | -31.1700 | 27.2100 | -6.1020 | -2.6950 | 1.9910 | 3.2200 | 0.9997 |
The open-circuit voltage for lithium-ion batteries is expressed as $$ U_{OC} = P_1 SOC^6 + P_2 SOC^5 + P_3 SOC^4 + P_4 SOC^3 + P_5 SOC^2 + P_6 SOC + P_7 $$. This table summarizes the polynomial coefficients at various temperatures for lithium-ion batteries, ensuring accurate SOC estimation.
For fault diagnosis in lithium-ion batteries, we simulate faults by injecting virtual currents into the battery model. This approach mimics internal short circuits or other anomalies in lithium-ion batteries without physical modifications. The fault detection process involves state estimation using an Adaptive Extended Kalman Filter (AEKF). The AEKF enhances traditional EKF by adaptively adjusting noise covariance matrices, making it robust for nonlinear systems like lithium-ion batteries. The state estimation for lithium-ion batteries includes both electrical and thermal states. The AEKF steps are:
1. Initialize state vector and covariance matrix.
2. Predict state: $$ \hat{x}_{k|k-1} = A_{k-1} \hat{x}_{k-1|k-1} + B_{k-1} u_{k-1} $$
3. Predict error covariance: $$ P_{k|k-1} = A_{k-1} P_{k-1|k-1} A_{k-1}^T + Q^Z_{k-1} $$
4. Compute Kalman gain: $$ K_k = \frac{P_{k|k-1} C_k^T}{C_k P_{k|k-1} C_k^T + R_{k-1}} $$
5. Update state estimate: $$ \hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k (y_k – \hat{y}_{k|k-1}) $$
6. Update error covariance: $$ P_{k|k} = (I – K_k C_k) P_{k|k-1} $$
7. Adapt noise covariances: $$ d_k = \frac{1 – b}{1 – b^k}, \quad e_k = y_k – \hat{y}_{k|k-1} $$, $$ Q^Z_k = (1 – d_k) Q^Z_{k-1} + d_k K_k e_k e_k^T K_k^T $$, $$ R_k = (1 – d_k) R_{k-1} + d_k (e_k e_k^T – C_k P_{k|k} C_k^T) $$
For lithium-ion batteries, the state vector includes SOC, polarization voltages, core temperature, and surface temperature. The measurement vector includes terminal voltage and surface temperature. The system matrices are derived from the electro-thermal coupling model. To evaluate faults in lithium-ion batteries, we compute the Root Mean Square Error (RMSE) between measured and estimated values:
$$ RMSE = \sqrt{ \frac{1}{n} \sum_{i=1}^n (y_r(i) – y_p(i))^2 } $$
where $$ y_r $$ is the measured value and $$ y_p $$ is the estimated value for lithium-ion batteries. A threshold-based approach is used for fault classification in lithium-ion batteries, as shown in the table below.
| Fault Level | Voltage RMSE (V) | Surface Temperature RMSE (K) |
|---|---|---|
| Normal | < 0.01 | < 0.07 |
| Mild Fault | [0.01, 0.05) | [0.07, 0.50) |
| Severe Fault | ≥ 0.05 | ≥ 0.50 |
This table defines fault levels for lithium-ion batteries based on RMSE values, enabling automated diagnosis.
We validate our fault diagnosis method for lithium-ion batteries through simulation under Urban Dynamometer Driving Schedule (UDDS) conditions. The lithium-ion battery is subjected to dynamic current profiles, and faults are simulated by injecting virtual currents. For example, in a series of three lithium-ion batteries, we inject $$ I_0 = 0.2I $$ into Battery #1 to simulate a mild fault, $$ I_0 = I $$ into Battery #2 for a severe fault, and no injection into Battery #3 as normal. The AEKF estimates states for each lithium-ion battery, and RMSE is calculated. The results demonstrate that our method effectively detects faults in lithium-ion batteries. The RMSE values for the lithium-ion battery pack are summarized below.
| Battery | Voltage RMSE (V) | Surface Temperature RMSE (K) |
|---|---|---|
| #1 | 0.0176 | 0.1066 |
| #2 | 0.0715 | 0.7972 |
| #3 | 0.0015 | 0.0086 |
According to the fault classification table, Battery #1 is in mild fault, Battery #2 in severe fault, and Battery #3 is normal for lithium-ion batteries. This shows the efficacy of our electro-thermal coupling model-based approach for lithium-ion battery fault diagnosis.
In conclusion, our study presents a robust fault diagnosis technology for lithium-ion batteries using an electro-thermal coupling model. The integration of electrical and thermal dynamics in lithium-ion batteries allows for accurate state estimation and early fault detection. The use of FFRLS for parameter identification and AEKF for state estimation enhances the reliability of monitoring lithium-ion batteries. Simulation results under UDDS conditions confirm that our method can diagnose faults in lithium-ion batteries with high precision. Future work may involve experimental validation with real lithium-ion battery packs and extension to battery management systems. Overall, this contribution advances the safety and efficiency of lithium-ion batteries in applications like electric vehicles and energy storage.
The electro-thermal coupling model for lithium-ion batteries offers several advantages. First, it accounts for temperature effects on electrical parameters, which is crucial for lithium-ion batteries operating in varying environments. Second, the model’s adaptability through online parameter identification ensures that changes in lithium-ion battery health are captured over time. Third, the fault diagnosis scheme provides a quantitative measure via RMSE, facilitating automated alerts for lithium-ion battery systems. We emphasize that regular monitoring of lithium-ion batteries using such models can prevent catastrophic failures and extend lifespan.
Moreover, the mathematical formulations for lithium-ion batteries are designed to be computationally efficient, making them suitable for embedded systems. The state-space representations for lithium-ion batteries are linearized where necessary, but the AEKF handles nonlinearities effectively. For instance, the open-circuit voltage of lithium-ion batteries is a nonlinear function of SOC, but the filter accommodates this through Jacobian matrices. The thermal model for lithium-ion batteries, though simplified, captures essential heat transfer phenomena without excessive complexity.
In practice, implementing this fault diagnosis for lithium-ion batteries requires sensors for voltage, current, and temperature. The algorithm processes these inputs in real-time to update the model and estimate states. For lithium-ion batteries in large packs, individual cell monitoring can be achieved by applying the method to each cell independently. This decentralized approach ensures that faults in any single lithium-ion battery are detected promptly.
To further illustrate, consider the parameter identification process for lithium-ion batteries. The FFRLS algorithm continuously adjusts parameters based on incoming data. For example, as a lithium-ion battery ages, its internal resistance may increase. The FFRLS will track this change, and the model will reflect the updated resistance for accurate fault diagnosis. This adaptability is key for long-term monitoring of lithium-ion batteries.
Another aspect is the simulation of faults in lithium-ion batteries. By injecting virtual currents, we can test the diagnosis algorithm under various fault scenarios without damaging actual lithium-ion batteries. This is cost-effective and safe for development. The virtual current method mimics internal short circuits, which are common faults in lithium-ion batteries due to manufacturing defects or abuse.
The AEKF’s adaptive feature is particularly beneficial for lithium-ion batteries because measurement noise can vary with operating conditions. For instance, temperature sensor noise may increase at high temperatures in lithium-ion batteries. The AEKF automatically adjusts the noise covariance, maintaining estimation accuracy. This robustness makes the method suitable for real-world applications where lithium-ion batteries face dynamic environments.
In summary, our fault diagnosis technology for lithium-ion batteries leverages an electro-thermal coupling model to provide comprehensive monitoring. The lithium-ion battery is modeled as a coupled system, and advanced algorithms are used for parameter identification and state estimation. The results demonstrate successful fault detection, highlighting the potential for enhancing the safety and reliability of lithium-ion batteries. We believe this approach will contribute significantly to the field of energy storage and electric mobility, where lithium-ion batteries play a pivotal role.
For future improvements, we plan to incorporate more detailed thermal models for lithium-ion batteries, such as distributed parameter models, to capture temperature gradients. Additionally, machine learning techniques could be integrated to predict fault progression in lithium-ion batteries. However, the current model-based approach offers a solid foundation for real-time fault diagnosis of lithium-ion batteries. We encourage further research to optimize the computational load for large-scale lithium-ion battery systems.
Ultimately, the goal is to ensure the safe operation of lithium-ion batteries across various applications. By employing electro-thermal coupling models and adaptive filters, we can achieve early warning systems that prevent failures and extend the service life of lithium-ion batteries. This work underscores the importance of interdisciplinary approaches in advancing battery technology, particularly for lithium-ion batteries, which are at the heart of the renewable energy transition.