In the pursuit of sustainable energy solutions, li ion battery technology has emerged as a cornerstone for electric vehicles and portable electronics. However, the demand for rapid charging poses significant challenges, particularly the risk of lithium plating on the anode, which can lead to safety hazards and reduced lifespan. As a researcher in this field, I have focused on developing a fast charging strategy that mitigates lithium plating while minimizing charging time. This article presents a dual-loop control scheme based on an electrode equivalent circuit model (EECM), integrating an anode potential observer and a charging current controller to enable safe and efficient fast charging for li ion battery systems. The approach leverages real-time state estimation and adaptive current regulation, ensuring that the anode potential remains above the lithium plating threshold (0 V vs. Li/Li+). Through extensive simulations and experimental validations, I demonstrate that this strategy significantly reduces charging time compared to conventional methods, all while preventing lithium deposition. The following sections detail the model development, control design, and verification, with an emphasis on formulas and tables to summarize key findings. By addressing the critical issue of lithium plating, this work contributes to advancing li ion battery technology for high-performance applications.
The rapid adoption of electric vehicles has intensified the need for fast charging capabilities in li ion battery systems. Traditional charging methods, such as constant current-constant voltage (CC-CV), often prioritize safety over speed, leading to prolonged charging times. In contrast, aggressive fast charging can induce lithium plating, where lithium ions deposit as metallic lithium on the anode surface instead of intercalating into the graphite structure. This phenomenon not only accelerates capacity fade but also raises safety concerns due to potential internal short circuits. Therefore, my research aims to balance charging speed and safety by monitoring and controlling the anode potential in real-time. The electrode equivalent circuit model serves as the foundation for this strategy, providing insights into internal electrode states without the computational burden of full electrochemical models. In this article, I describe how I developed and optimized this model, implemented a closed-loop observer for anode potential estimation, and designed a current controller with feedforward compensation to avoid overshoot. The results show that the proposed approach achieves a 46% reduction in charging time to 80% state of charge (SOC) compared to 1C CC-CV charging, while maintaining the anode potential above 4.8 mV, effectively suppressing lithium plating. This work underscores the importance of model-based control in enhancing li ion battery performance and reliability.
To begin, I established an electrode equivalent circuit model for the li ion battery, which extends conventional equivalent circuit models by representing the cathode and anode separately. This allows for the estimation of electrode-specific potentials, crucial for 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 battery terminal voltage \(U_t\) is given by:
$$U_t = U_{oc} – U_1^{ca} – U_1^{an} – I R_0$$
where \(U_{oc}\) is the open-circuit voltage, \(U_1^{ca}\) and \(U_1^{an}\) are the polarization voltages of the cathode and anode, respectively, \(I\) is the current (positive for discharge), and \(R_0\) is the ohmic resistance. The electrode potentials are expressed as:
$$U_t^{ca} = U_{oc}^{ca} – I R_0^{ca} – U_1^{ca}$$
$$U_t^{an} = U_{oc}^{an} + I R_0^{an} + U_1^{an}$$
with \(U_{oc} = U_{oc}^{ca} – U_{oc}^{an}\). The dynamics of the polarization voltages are governed by:
$$\frac{dU_1^{ca}}{dt} = \frac{I}{C_1^{ca}} – \frac{U_1^{ca}}{R_1^{ca} C_1^{ca}}$$
$$\frac{dU_1^{an}}{dt} = \frac{I}{C_1^{an}} – \frac{U_1^{an}}{R_1^{an} C_1^{an}}$$
In discrete-time form, with sampling interval \(\Delta t\), these can be written as:
$$U_{1,k+1}^{ca} = U_{1,k}^{ca} e^{-\frac{\Delta t}{\tau^{ca}}} + I_k R_1^{ca} \left(1 – e^{-\frac{\Delta t}{\tau^{ca}}}\right)$$
$$U_{1,k+1}^{an} = U_{1,k}^{an} e^{-\frac{\Delta t}{\tau^{an}}} + I_k R_1^{an} \left(1 – e^{-\frac{\Delta t}{\tau^{an}}}\right)$$
where \(\tau^{ca} = R_1^{ca} C_1^{ca}\) and \(\tau^{an} = R_1^{an} C_1^{an}\) are time constants. This model enables the simulation of battery behavior under various charging conditions, but it requires accurate parameter identification. I performed offline parameter identification using a three-electrode li ion battery setup, which included a reference electrode to measure electrode potentials directly. The battery had a capacity of 0.6 Ah and was tested at 20°C. The open-circuit voltages for the cathode, anode, and full cell were obtained through low-rate charging (0.05C), and the results were fitted to polynomial functions:
$$U_{oc}^{*} = P_1 SOC^7 + P_2 SOC^6 + \cdots + P_8$$
where \(U_{oc}^{*}\) represents \(U_{oc}\), \(U_{oc}^{ca}\), or \(U_{oc}^{an}\), and \(P_i\) are coefficients. The resistance and time constant parameters were identified from variable-current charging tests, as summarized in Table 1. This parameter set forms the basis for the real-time observer and controller design.
| Parameter | Cathode Value | Anode Value |
|---|---|---|
| \(R_0\) (Ω) | 0.03–0.12 | 0.03–0.12 |
| \(R_1\) (Ω) | 0.1–0.3 | 0.1–0.3 |
| \(\tau\) (s) | 200–800 | 200–800 |
The core of my fast charging strategy is a dual-loop system comprising an anode potential closed-loop observer and a charging current closed-loop controller. The observer uses an extended Kalman filter (EKF) to estimate the anode potential in real-time, while the controller adjusts the charging current based on this estimate to keep the anode potential above a safe threshold. This integrated approach ensures that the li ion battery is charged rapidly without inducing lithium plating. I designed the EKF with the state vector \(\mathbf{x}_k = [SOC_k, U_t^{an}_k, U_{1,k}^{ca}, U_{1,k}^{an}]^T\), input \(u_k = I_k\), and measurement \(y_k = U_t_k\). The state transition and measurement equations are derived from the EECM, and the EKF iteratively updates the estimates to minimize error. The key steps involve linearization using Jacobian matrices \(\mathbf{A}_k\) and \(\mathbf{C}_k\):
$$\mathbf{A}_k = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & e^{-\frac{\Delta t}{\tau^{ca}}} & 0 \\
0 & 0 & 0 & e^{-\frac{\Delta t}{\tau^{an}}}
\end{bmatrix}, \quad \mathbf{C}_k = \left[ -\frac{dU_{oc}}{dSOC} \quad 0 \quad -1 \quad -1 \right]$$
This observer provides accurate estimates of the anode potential, with average errors below 5 mV across various operating conditions, as shown in Table 2. The robustness of the observer is critical for reliable fast charging of li ion battery systems.
| Charging Condition | Anode Potential Avg. Error (mV) | Max Error (mV) |
|---|---|---|
| 0.05C Constant Current | 0.54 | 4.11 |
| 1C CC-CV | 2.95 | 7.73 |
| Variable Current | 0.47 | 5.66 |
| Multi-stage CC-CV | 2.02 | 8.87 |
The charging current controller is designed to regulate the anode potential by adjusting the current based on the observer’s estimates. The control law is derived from the discrete-time model: at time step \(k\), the anode potential is \(U_t^{an}_k = U_{oc}^{an}_k + I_k R_0^{an} + U_{1,k}^{an}\). To achieve a desired setpoint \(U_0^{an}\) (e.g., 10 mV above the lithium plating threshold), I compute the optimal current for the next step as:
$$I_{c,k+1} = \frac{U_0^{an} – U_{oc}^{an}_{k+1} – U_{1,k}^{an}}{R_0^{an}}$$
However, to prevent overshoot and ensure smooth convergence, I incorporated a feedforward compensation method. This involves modifying the setpoint with an exponential decay function:
$$U_0^{an’} = U_0^{an} + A e^{-\frac{t}{\tau_c}}$$
where \(A\) is the initial compensation amplitude (set to 0.25 V based on empirical data) and \(\tau_c\) is a time constant determined from the estimated time to reach the threshold. This compensation allows the anode potential to approach the setpoint gradually, avoiding abrupt changes that could trigger lithium plating. The controller’s performance is evaluated through simulations, showing that the anode potential stabilizes within ±1 mV of the setpoint in 26 seconds, with a steady-state error below 0.05 mV. This precise control is essential for maintaining the health and safety of li ion battery during fast charging.
To validate the overall fast charging strategy, I conducted experiments on a three-electrode li ion battery under different conditions. The proposed strategy was compared with conventional 1C CC-CV charging, focusing on charging time and anode potential behavior. The results, summarized in Table 3, demonstrate that my strategy reduces the time to reach 80% SOC by approximately 46%, from 48.2 minutes to 26 minutes, while keeping the minimum anode potential at 4.8 mV, well above the lithium plating threshold. Furthermore, post-cycling analysis of batteries charged with this strategy showed no visible lithium deposition on the anode surface, confirming the effectiveness of the approach. In contrast, batteries subjected to 1.85C constant current charging exhibited significant lithium plating, highlighting the risks of uncontrolled fast charging. These findings underscore the importance of model-based control in optimizing li ion battery charging protocols.
| Charging Strategy | Time to 80% SOC (min) | Min Anode Potential (mV) | Lithium Plating Observed |
|---|---|---|---|
| Proposed Strategy (4C max) | 26 | 4.8 | No |
| Proposed Strategy (3C max) | 26.7 | 4.9 | No |
| 1C CC-CV | 48.2 | N/A | N/A |
| 1.85C Constant Current | ~26 | Below 0 | Yes |
The implementation of this fast charging strategy involves several key formulas and algorithms. For instance, the EKF algorithm iterates through prediction and update steps. The prediction step is:
$$\hat{\mathbf{x}}_k^- = f(\hat{\mathbf{x}}_{k-1}, u_{k-1})$$
$$\mathbf{P}_k^- = \mathbf{A}_{k-1} \mathbf{P}_{k-1} \mathbf{A}_{k-1}^T + \mathbf{Q}_{k-1}$$
and the update step is:
$$\mathbf{K}_k = \mathbf{P}_k^- \mathbf{C}_k^T (\mathbf{C}_k \mathbf{P}_k^- \mathbf{C}_k^T + \mathbf{R}_k)^{-1}$$
$$\hat{\mathbf{x}}_k = \hat{\mathbf{x}}_k^- + \mathbf{K}_k (y_k – g(\hat{\mathbf{x}}_k^-, u_k))$$
$$\mathbf{P}_k = (\mathbf{I} – \mathbf{K}_k \mathbf{C}_k) \mathbf{P}_k^-$$
where \(\mathbf{Q}\) and \(\mathbf{R}\) are noise covariance matrices. For the current controller, the compensated setpoint is used in the current calculation, ensuring that the anode potential dynamics are accounted for. This integrated system demonstrates how advanced control theory can be applied to li ion battery management, enabling faster charging without compromising safety. The use of the electrode equivalent circuit model simplifies the computational complexity while providing sufficient accuracy for real-time applications. As li ion battery technology evolves, such model-based approaches will be crucial for meeting the demands of electric vehicles and grid storage systems.
In conclusion, I have developed and validated a fast charging strategy for li ion battery that effectively prevents lithium plating through real-time anode potential control. The strategy combines an electrode equivalent circuit model with an EKF-based observer and a feedforward-compensated current controller, forming a dual-loop system that adapts to battery states during charging. Experimental results show significant improvements in charging speed and safety, with a 46% reduction in charging time and no evidence of lithium deposition. This work highlights the potential of model-based control to enhance li ion battery performance, addressing a critical barrier to widespread electric vehicle adoption. Future research could focus on extending this approach to different battery chemistries and integrating thermal management for holistic fast charging solutions. By continuing to innovate in li ion battery technology, we can unlock faster, safer, and more reliable energy storage for a sustainable future.
Throughout this study, the importance of accurate modeling and control for li ion battery systems has been emphasized. The electrode equivalent circuit model provided a balance between complexity and practicality, enabling real-time implementation without sacrificing essential dynamics. The observer and controller designs leveraged this model to achieve precise regulation of the anode potential, a key factor in preventing lithium plating. The successful validation under various conditions underscores the robustness of the approach. As the demand for fast charging grows, such strategies will become increasingly vital for maximizing the lifespan and safety of li ion battery in applications ranging from consumer electronics to electric vehicles. I believe that the insights gained from this work will contribute to ongoing efforts to optimize li ion battery performance and accelerate the transition to clean energy.