The rapid advancement of the automotive industry has brought lithium-ion batteries to the forefront of energy storage solutions, particularly for electric vehicles (EVs), due to their high energy density, low self-discharge rate, and long cycle life. However, challenges such as prolonged charging times and significant temperature rise during charging hinder user experience and pose risks to battery longevity and safety. These limitations directly impact the widespread adoption of EVs. Therefore, developing an optimized charging strategy that balances speed and thermal management is critical for the advancement of EV technology. The most common method, Constant-Current Constant-Voltage (CCCV) charging, is simple but often leads to extended charging periods in the constant-voltage stage and undesirable heat generation. Multi-stage Constant Current Charging (MCCC) presents a promising alternative by strategically controlling the current in steps, offering potential improvements in both charging efficiency and battery health.
This work proposes a novel multi-stage constant current charging strategy for lithium-ion batteries, optimized using the Whale Optimization Algorithm (WOA). The core innovation lies in the use of a coupled electro-thermal model to accurately simulate battery behavior during charging, which then informs the optimization process. By constructing an objective function that jointly minimizes charging time and temperature rise, the WOA determines the optimal current for each charging stage. This methodology ensures that the proposed strategy is not based on arbitrary targets but is dynamically adapted to the internal electrical and thermal states of the lithium-ion battery.
Construction of the Coupled Electro-Thermal Model for Lithium-Ion Battery
Accurately modeling a lithium-ion battery during charging requires capturing both its electrical response and thermal dynamics, as these processes are intrinsically linked. We construct a coupled model by integrating an equivalent circuit model with a thermal model.
Electrical Model: Second-Order RC Equivalent Circuit
The electrical behavior of the lithium-ion battery is represented using a second-order RC equivalent circuit model, which offers a good balance between complexity and accuracy. This model includes an open-circuit voltage (OCV) source, an ohmic internal resistance, and two parallel RC branches representing the short-term and long-term polarization dynamics.
The governing equations for the model are derived from Kirchhoff’s laws. The terminal voltage $U_t$ is given by:
$$U_t = U_{OCV} + IR_0 + U_1 + U_2$$
The dynamics of the polarization voltages across the two RC pairs are described by:
$$
\frac{dU_1}{dt} = -\frac{U_1}{R_1 C_1} + \frac{I}{C_1}
$$
$$
\frac{dU_2}{dt} = -\frac{U_2}{R_2 C_2} + \frac{I}{C_2}
$$
Solving these differential equations yields the time-domain expressions for the polarization voltages:
$$
U_1 = IR_1 \left(1 – e^{-t/(R_1 C_1)}\right)
$$
$$
U_2 = IR_2 \left(1 – e^{-t/(R_2 C_2)}\right)
$$
The State of Charge (SOC) is estimated using the ampere-hour (Ah) integration method:
$$
SOC = SOC_0 + \frac{1}{C_n} \int_{0}^{t} I(\tau) d\tau
$$
where $SOC_0$ is the initial SOC, and $C_n$ is the nominal capacity of the lithium-ion battery.
Model parameter identification ($U_{OCV}$, $R_0$, $R_1$, $C_1$, $R_2$, $C_2$) was performed using the Hybrid Pulse Power Characterization (HPPC) test and a least-squares fitting algorithm. The identified parameters are functions of both SOC and temperature, as summarized in the table below, reflecting the nonlinear behavior of the lithium-ion battery.
| Parameter | Variation with SOC & Temperature |
|---|---|
| $U_{OCV}$ | Increases monotonically with SOC; slight variation with temperature. |
| $R_0$ | Higher at low and high SOC; decreases as temperature increases. |
| $R_1$, $R_2$ | Exhibit minima at mid-SOC; significantly lower at higher temperatures. |
| $C_1$, $C_2$ | Strongly dependent on SOC and temperature, representing kinetic and diffusion limitations. |
Thermal Model: Heat Generation and Temperature Prediction
The thermal dynamics of the lithium-ion battery are captured using a lumped-parameter model based on energy balance. The heat generation rate within the battery is calculated using the Bernardi equation, which accounts for both irreversible (joule) and reversible (entropic) heat:
$$
Q_p = I(U_t – U_{OCV}) + I T \frac{dU_{OCV}}{dT}
$$
where $Q_p$ is the total heat generation power. The first term, $I(U_t – U_{OCV})$, represents the irreversible Joule heat. The second term, $I T \frac{dU_{OCV}}{dT}$, represents the reversible reaction heat, where $\frac{dU_{OCV}}{dT}$ is the entropy coefficient, a key parameter that varies with the SOC of the lithium-ion battery.
The heat dissipation from the battery surface to the ambient environment is modeled as:
$$
Q_d = h S (T – T_0)
$$
where $h$ is the equivalent heat transfer coefficient, $S$ is the surface area, $T$ is the battery temperature, and $T_0$ is the ambient temperature.
Assuming a uniform temperature distribution (lumped capacitance model), the energy balance equation is:
$$
Q_p – Q_d = m C \frac{dT}{dt}
$$
where $m$ is the mass and $C$ is the specific heat capacity of the lithium-ion battery. The temperature rise over a charging period can be predicted by integrating this equation:
$$
\Delta T = \frac{1}{mC} \int_{t_0}^{t} \left[ I(U_t – U_{OCV}) + I T \frac{dU_{OCV}}{dT} – hS(T-T_0) \right] d\tau
$$
Model Coupling
The electro-thermal coupling is achieved by interconnecting the electrical and thermal models. The process begins with an input current profile. The electrical model computes the terminal voltage $U_t$ and the components $I(U_t – U_{OCV})$ and $U_{OCV}$ required by the thermal model. Simultaneously, the SOC estimation module updates the SOC, which is used to index the temperature- and SOC-dependent parameters ($R_0, R_1, C_1, R_2, C_2, dU_{OCV}/dT$) for both models. The thermal model then calculates the instantaneous heat generation $Q_p$ and predicts the battery temperature rise $\Delta T$. This updated temperature is fed back to the electrical model to adjust its parameters for the next time step, creating a dynamic, closed-loop simulation that accurately represents the behavior of the lithium-ion battery during charging.
Charging Strategy Optimization Framework
Experimental Basis and Multi-Stage Strategy Formulation
The experimental validation and parameter identification for the lithium-ion battery model were conducted on a standard battery test platform, comprising a cycler for current/voltage control, a thermal chamber for environmental regulation, and data acquisition systems. The target cell was a commercial 18650-type lithium-ion battery with a nominal capacity of 2000 mAh.
We formulate a five-stage constant current (MCCC) charging strategy. This approach replaces the single constant-current phase of CCCV with multiple descending constant-current steps. When the battery voltage reaches the cutoff voltage (e.g., 4.2 V) in one stage, the current immediately drops to a lower predetermined value for the next stage, and this process continues until the final stage is complete. The number of stages is chosen to balance performance gains with control complexity.
Definition of the Multi-Objective Optimization Problem
The primary goals are to minimize both the total charging time ($t_{end} – t_0$) and the maximum temperature rise ($T_{end} – T_0$) of the lithium-ion battery. These are often conflicting objectives; a faster charge typically generates more heat. Therefore, we construct a unified objective function $F$ that combines both normalized metrics:
$$
F = \alpha \cdot \frac{t_{end} – t_0}{t_{max} – t_0} + \beta \cdot \frac{T_{end} – T_0}{T_{max} – T_0}
$$
where $\alpha$ and $\beta$ are weighting coefficients satisfying $\alpha + \beta = 1$, $t_{max}$ is the maximum allowable charging time, and $T_{max}$ is the maximum safe temperature limit. For this study, we set $\alpha = \beta = 0.5$, assigning equal importance to speed and temperature. The optimization is subject to the following constraints for the lithium-ion battery:
| Variable | Constraint |
|---|---|
| Terminal Voltage, $U_t$ | 2.5 V ≤ $U_t$ ≤ 4.2 V |
| Charging Current, $I$ | 0 A ≤ $I$ ≤ 4 A (2C rate) |
| Stage Currents | $I_{k+1}$ ≤ $I_k$ for k=1,2,3,4 |
| Battery Temperature, $T$ | 25°C ≤ $T$ ≤ 50°C |
| Ambient Temperature, $T_0$ | 25°C |
The decision variables for the optimization are the five constant current values $[I_1, I_2, I_3, I_4, I_5]$ for the respective stages.
Whale Optimization Algorithm (WOA) for Parameter Search
The Whale Optimization Algorithm is a metaheuristic inspired by the bubble-net hunting behavior of humpback whales. It is effective for global optimization problems due to its balance between exploration and exploitation. We employ WOA to find the optimal current vector that minimizes the objective function $F$.
Step 1: Initialization. A population of whales (potential solutions) is randomly initialized within the current bounds. Each whale’s position vector represents a set of five charging currents: $\vec{X} = [I_1, I_2, I_3, I_4, I_5]$.
Step 2: Fitness Evaluation. For each whale (current set), the coupled electro-thermal model simulates the complete charging process of the lithium-ion battery. The model outputs the total charging time and final temperature rise, which are used to calculate the fitness value $F$ according to the defined objective function.
Step 3: Position Update (Simulating Whale Behavior). The algorithm iteratively updates whale positions using three mechanisms, controlled by parameters $a$, $A$, $C$, $l$, and a random number $p$.
- Encircling Prey: When $|A| < 1$ and $p < 0.5$, whales move towards the current best solution $\vec{X}^*$.
$$ \vec{D} = |\vec{C} \cdot \vec{X}^*(t) – \vec{X}(t)| $$
$$ \vec{X}(t+1) = \vec{X}^*(t) – \vec{A} \cdot \vec{D} $$ - Bubble-net Attacking (Spiral Update): When $p \ge 0.5$, whales approach the best solution in a spiral path.
$$ \vec{D’} = |\vec{X}^*(t) – \vec{X}(t)| $$
$$ \vec{X}(t+1) = \vec{D’} \cdot e^{bl} \cdot \cos(2\pi l) + \vec{X}^*(t) $$ - Search for Prey (Exploration): When $|A| \ge 1$ and $p < 0.5$, whales move randomly to explore the search space, where $\vec{X}_{rand}$ is a random whale’s position.
$$ \vec{D} = |\vec{C} \cdot \vec{X}_{rand} – \vec{X}| $$
$$ \vec{X}(t+1) = \vec{X}_{rand} – \vec{A} \cdot \vec{D} $$
The coefficients are updated each iteration: $\vec{A} = 2\vec{a} \cdot \vec{r_1} – \vec{a}$, $\vec{C} = 2 \cdot \vec{r_2}$, and $\vec{a}$ decreases linearly from 2 to 0 over the iterations. Here, $l$ is a random number in [-1,1], $b$ is a constant, and $r_1$, $r_2$ are random vectors in [0,1].
Step 4: Convergence. Steps 2 and 3 repeat until the maximum number of iterations is reached or the fitness value converges. The position of the best whale $\vec{X}^*$ is then output as the optimal charging current profile for the lithium-ion battery.
| WOA Parameter | Value |
|---|---|
| Population Size | 200 |
| Number of Dimensions | 5 (one per charging stage) |
| Maximum Iterations | 200 |
| Spiral Constant ($b$) | 1 |
Optimization Results and Comparative Analysis
Optimal Charging Currents from WOA
The Whale Optimization Algorithm demonstrated efficient convergence. The best fitness value stabilized after approximately 61 iterations, indicating that the algorithm successfully located a high-quality optimum. The optimal five-stage constant current profile determined by the WOA for the specified lithium-ion battery is as follows:
| Charging Stage (k) | Optimal Current ($I_k$) | Current in C-rate |
|---|---|---|
| 1 | 2.982 A | 1.491 C |
| 2 | 1.658 A | 0.829 C |
| 3 | 1.258 A | 0.629 C |
| 4 | 0.642 A | 0.321 C |
| 5 | 0.506 A | 0.253 C |
Performance Comparison with Standard CCCV Charging
The proposed WOA-optimized MCCC strategy was compared against a standard CCCV strategy. The CCCV strategy was set with a constant-current phase at 1.491 C (matching the first stage of MCCC) until the voltage limit, followed by a constant-voltage phase until the current tapered to 0.1 C.
The simulation results based on the coupled model reveal significant advantages of the optimized strategy for the lithium-ion battery:
- Charging Time: The total charging time for the WOA-MCCC strategy was 4,494 seconds. In contrast, the CCCV strategy required 5,047 seconds to fully charge the lithium-ion battery. This represents a 10.96% reduction in charging time achieved by the optimized multi-stage approach.
- Temperature Rise: During the initial high-current phase, both strategies produced similar heat, leading to comparable temperature increases. However, as the CCCV strategy entered the lengthy constant-voltage phase, heat continued to accumulate. The WOA-MCCC strategy, by stepping down the current more aggressively, effectively managed heat generation in the later stages. The maximum temperature rise for the WOA-MCCC strategy was approximately 2.12% lower than that of the CCCV strategy.
The voltage and current profiles clearly show the step-down nature of MCCC versus the tapering current of CCCV after the voltage peak. The thermal profile confirms that the optimized strategy successfully limits the final temperature of the lithium-ion battery.
| Performance Metric | Standard CCCV Strategy | WOA-Optimized MCCC Strategy | Improvement |
|---|---|---|---|
| Total Charging Time | 5,047 s | 4,494 s | -553 s (-10.96%) |
| Maximum Temperature Rise | ΔT_CCCV | ΔT_MCCC | ~2.12% reduction |
| Key Mechanism | Single CC phase + long CV phase | Multiple descending CC phases | Eliminates inefficient CV tail |
Conclusion
In this work, we have successfully developed and validated an optimized multi-stage constant current charging strategy for lithium-ion batteries using the Whale Optimization Algorithm. The foundation of this approach is a high-fidelity coupled electro-thermal model that dynamically simulates the electrical and thermal behavior of the lithium-ion battery during the charging process. By formulating a dual-objective function targeting both minimal charging time and temperature rise, and employing the robust search capabilities of WOA, we derived an optimal set of descending charging currents.
The comparative analysis unequivocally demonstrates the superiority of the proposed WOA-MCCC strategy over the conventional CCCV method. The optimization yielded a 10.96% faster charging speed while simultaneously achieving a 2.12% reduction in the maximum battery temperature rise. This result highlights the effectiveness of the method in breaking the typical trade-off between speed and thermal load for lithium-ion batteries. The proposed framework, integrating accurate physical modeling with advanced metaheuristic optimization, provides a powerful and adaptable tool for designing efficient and safe charging protocols, contributing to enhanced performance and longevity of lithium-ion batteries in electric vehicle applications.