In recent years, the rapid development of electric vehicles has placed high demands on the performance of lithium-ion batteries, particularly in terms of fast charging capabilities. However, fast charging often leads to lithium plating on the anode, which poses significant safety risks and accelerates battery degradation. As a researcher in this field, I have been focusing on developing strategies to enable safe and efficient fast charging for lithium-ion batteries. This article presents my work on a dual-loop control scheme based on an electrode equivalent circuit model (EECM) to prevent lithium deposition during fast charging. The goal is to achieve rapid charging while maintaining the anode potential above the lithium plating threshold (0 V vs. Li/Li+), thereby ensuring battery safety and longevity.
The core of my approach involves real-time monitoring and control of the anode potential using a closed-loop observer and a charging current controller. By integrating model-based estimations with feedback control, I aim to optimize charging currents dynamically, minimizing charging time without compromising battery health. In this article, I will detail the modeling, parameter identification, control design, and experimental validation of this strategy. Throughout, I will emphasize the importance of the electrode equivalent circuit model in capturing internal battery states, which is critical for preventing lithium deposition in lithium-ion batteries.
To begin, I established an electrode equivalent circuit model for lithium-ion batteries, which extends traditional equivalent circuit models by representing the cathode and anode separately. This model allows for the estimation of internal electrode potentials, particularly the anode potential, which is key to detecting lithium plating risks. The model consists of two Thevenin circuits in series, one for the cathode and one for the anode, as shown in the following equations:
The terminal voltage of the lithium-ion battery is given by:
$$U_t = U_{oc} – I R_0 – U_1^{ca} – U_1^{an}$$
where \(U_t\) is the terminal voltage, \(U_{oc}\) is the open-circuit voltage, \(I\) is the current (positive for discharge), \(R_0\) is the ohmic resistance, and \(U_1^{ca}\) and \(U_1^{an}\) are the polarization voltages of the cathode and anode, respectively. The dynamics of the polarization voltages are described by:
$$\frac{dU_1^{ca}}{dt} = -\frac{U_1^{ca}}{R_1^{ca} C_1^{ca}} + \frac{I}{C_1^{ca}}$$
$$\frac{dU_1^{an}}{dt} = -\frac{U_1^{an}}{R_1^{an} C_1^{an}} + \frac{I}{C_1^{an}}$$
Here, \(R_1^{ca}\) and \(C_1^{ca}\) are the polarization resistance and capacitance for the cathode, and \(R_1^{an}\) and \(C_1^{an}\) are for the anode. The anode potential, which must be kept above 0 V vs. Li/Li+ to avoid lithium deposition, can be expressed as:
$$U_t^{an} = U_{oc}^{an} + I R_0^{an} + U_1^{an}$$
where \(U_{oc}^{an}\) is the anode open-circuit voltage. This model forms the basis for my state estimation and control strategy.
Parameter identification for the electrode equivalent circuit model is crucial for accurate simulations. I conducted experiments on a three-electrode lithium-ion battery with a capacity of 0.6 Ah, using a lithium metal reference electrode. The parameters, including ohmic resistances, polarization resistances, and time constants for both electrodes, were identified under various state-of-charge (SOC) conditions. The results are summarized in the table below, which shows how these parameters vary with SOC. This variability highlights the need for adaptive control in fast charging strategies for lithium-ion batteries.
| Parameter | Value Range (across SOC) | Unit |
|---|---|---|
| \(R_0^{ca}\) (Cathode Ohmic Resistance) | 0.01 – 0.05 | Ω |
| \(R_0^{an}\) (Anode Ohmic Resistance) | 0.005 – 0.03 | Ω |
| \(R_1^{ca}\) (Cathode Polarization Resistance) | 0.02 – 0.1 | Ω |
| \(R_1^{an}\) (Anode Polarization Resistance) | 0.01 – 0.08 | Ω |
| \(\tau^{ca}\) (Cathode Time Constant) | 10 – 100 | s |
| \(\tau^{an}\) (Anode Time Constant) | 5 – 80 | s |
Based on this model, I designed a dual-loop control scheme consisting of an anode potential closed-loop observer and a charging current closed-loop controller. The observer uses an extended Kalman filter (EKF) algorithm to estimate the anode potential in real-time, while the controller adjusts the charging current to keep the anode potential near a safe threshold. The EKF algorithm iteratively updates state estimates, including SOC and anode potential, by minimizing errors between measured and predicted terminal voltages. The state vector is defined as:
$$\mathbf{x}_k = [\text{SOC}_k, U_k^{an}, U_{1,k}^{ca}, U_{1,k}^{an}]^T$$
with the system input being the current \(I_k\) and the output being the terminal voltage \(U_{t,k}\). The state transition and measurement equations are linearized using Taylor series expansion, and the Kalman gain is computed to correct estimates. This approach ensures robust estimation even under varying operating conditions for lithium-ion batteries.
The charging current controller is designed to regulate the anode potential by dynamically adjusting the charging current. The control law is derived from the discretized model equations. At time step \(k\), the anode potential is given by:
$$U_k^{an} = U_{oc}^{an}(SOC_k) + I_k R_0^{an} + U_{1,k}^{an}$$
To avoid lithium deposition, I set a threshold \(U_0^{an}\) (e.g., 0.01 V) and aim to control the current such that the anode potential remains above this threshold. The controller computes the optimal current for the next time step as:
$$I_{c,k+1} = \frac{U_0^{an} – U_{oc}^{an}(SOC_{k+1}) – U_{1,k}^{an}}{R_0^{an}}$$
where \(SOC_{k+1}\) is predicted using the coulomb counting method. However, to prevent overshoot and oscillations, I introduced a feedforward compensation method. This involves modifying the threshold with an exponential decay function:
$$U_0^{an’} = U_0^{an} + A e^{-t/\tau}$$
where \(A\) and \(\tau\) are tuning parameters. This compensation ensures that the anode potential approaches the threshold smoothly, without dropping below it, which is critical for preventing lithium plating in lithium-ion batteries during fast charging.
To validate the effectiveness of my proposed strategy, I conducted extensive simulations and experiments. The anode potential observer was tested under various charging profiles, including constant current, constant current-constant voltage (CC-CV), and variable current conditions. The results demonstrated high estimation accuracy, with average errors below 5 mV and maximum errors under 10 mV for the anode potential. This precision is essential for reliable control in fast charging applications for lithium-ion batteries.
The charging current controller was evaluated under different maximum current limits and anode potential thresholds. The table below summarizes the performance metrics, including settling time and steady-state error, for various configurations. These results show that the controller can quickly bring the anode potential to the desired range and maintain it with minimal error, ensuring safe operation of lithium-ion batteries.
| Maximum Current | Anode Potential Threshold (V) | Settling Time (s) | Steady-State Error (mV) |
|---|---|---|---|
| 4C | 0.01 | 16 | 0.03 |
| 3C | 0.01 | 22 | 0.03 |
| 2C | 0.01 | 23 | 0.02 |
| 4C | 0.02 | 18 | 0.03 |
| 3C | 0.02 | 26 | 0.03 |
| 2C | 0.02 | 26 | 0.02 |
In experimental tests, I compared my fast charging strategy with a conventional 1C CC-CV method. Using my approach with a maximum current of 4C and an anode potential threshold of 0.01 V, the charging time to reach 80% SOC was reduced by approximately 46%, from 48.2 minutes to 26 minutes. Throughout the charging process, the anode potential remained above 4.8 mV, well above the lithium plating threshold. Post-cycling analysis of the lithium-ion battery showed no signs of lithium deposition on the anode, confirming the effectiveness of the strategy in preventing degradation.
The success of this work hinges on the accurate modeling of lithium-ion batteries using the electrode equivalent circuit model. By capturing the internal dynamics of both electrodes, the model enables precise state estimation and control. The integration of EKF for observer design and feedforward compensation for current control results in a robust system that can adapt to changing battery conditions. This is particularly important for lithium-ion batteries in electric vehicles, where operating conditions vary widely.
In conclusion, I have developed and validated a dual-loop fast charging strategy for lithium-ion batteries that effectively prevents lithium deposition. The strategy combines an anode potential observer based on an electrode equivalent circuit model with a charging current controller that uses feedforward compensation. This approach ensures safe and efficient fast charging, with significant reductions in charging time and no observable lithium plating. Future work will focus on extending this strategy to account for temperature effects and aging in lithium-ion batteries, further enhancing its practicality for real-world applications.
Throughout this research, the importance of advanced modeling and control techniques for lithium-ion batteries has been underscored. As the demand for faster charging grows, such strategies will be crucial in balancing speed with safety and longevity. The electrode equivalent circuit model provides a valuable framework for understanding and manipulating internal battery states, paving the way for smarter battery management systems. I believe that continued innovation in this area will drive the adoption of electric vehicles and support the transition to sustainable energy.
To further illustrate the key equations and relationships, I summarize the core mathematical model below. The electrode equivalent circuit model for lithium-ion batteries is represented by the following set of equations, which form the foundation for the control strategy:
$$U_t = U_{oc}^{ca} – U_{oc}^{an} – I(R_0^{ca} + R_0^{an}) – U_1^{ca} – U_1^{an}$$
$$\frac{dU_1^{ca}}{dt} = -\frac{1}{\tau^{ca}} U_1^{ca} + \frac{I}{C_1^{ca}}$$
$$\frac{dU_1^{an}}{dt} = -\frac{1}{\tau^{an}} U_1^{an} + \frac{I}{C_1^{an}}$$
$$U_t^{an} = U_{oc}^{an} + I R_0^{an} + U_1^{an}$$
where \(\tau^{ca} = R_1^{ca} C_1^{ca}\) and \(\tau^{an} = R_1^{an} C_1^{an}\). The EKF algorithm linearizes these equations around the current state estimate, with the Jacobian matrices computed as:
$$\mathbf{A}_k = \begin{bmatrix}
1 & 0 & 0 & 0 \\
\frac{\partial U_{oc}^{an}}{\partial \text{SOC}} & 0 & 0 & 1 \\
0 & 0 & e^{-\Delta t / \tau^{ca}} & 0 \\
0 & 0 & 0 & e^{-\Delta t / \tau^{an}}
\end{bmatrix}$$
$$\mathbf{C}_k = \left[ -\frac{\partial U_{oc}}{\partial \text{SOC}} \quad -1 \quad -1 \quad -1 \right]$$
These matrices are used in the prediction and update steps of the EKF, enabling real-time estimation of the anode potential in lithium-ion batteries.
The charging current control law, with feedforward compensation, ensures that the anode potential \(U_k^{an}\) tracks the modified threshold \(U_0^{an’}\). The control input is computed as:
$$I_{c,k+1} = \frac{U_0^{an’} – U_{oc}^{an}(SOC_{k+1}) – U_{1,k}^{an}}{R_0^{an}}$$
where \(U_0^{an’} = U_0^{an} + A e^{-t/\tau}\). The parameters \(A\) and \(\tau\) are tuned based on the expected charging time to prevent overshoot. This control law is implemented in a closed-loop fashion, with the observer providing the necessary state estimates.
In summary, my work demonstrates that through careful modeling and control design, it is possible to achieve fast charging for lithium-ion batteries without compromising safety. The electrode equivalent circuit model offers a practical way to monitor internal states, while the dual-loop control scheme ensures precise regulation. As lithium-ion batteries continue to evolve, such approaches will be integral to unlocking their full potential in fast-charging applications.