In recent years, the rapid advancement of electric vehicle technology has positioned it as a cornerstone of sustainable transportation globally. Among various types, extended-range electric vehicles (EREVs) have gained significant attention due to their ability to combine the benefits of pure electric drive with an auxiliary power unit, such as a fuel cell or internal combustion engine, thereby addressing range anxiety issues common in pure electric vehicle designs. In China, the electric vehicle market has experienced explosive growth, driven by government policies and increasing environmental awareness, making China EV a key player in the global automotive industry. However, optimizing energy management in these vehicles remains a critical challenge, as it directly impacts fuel efficiency, battery longevity, and overall performance. This paper focuses on developing an energy management optimization strategy for extended-range electric vehicles using the batch gradient descent method, aiming to minimize fuel consumption while maintaining desired driving dynamics.
The core of our approach lies in accurately modeling the EREV system and applying iterative optimization techniques to refine energy allocation. We begin by establishing the operational mechanisms of the electric vehicle, considering its dual power sources: a primary battery pack and a secondary fuel-based generator. This model incorporates vehicle dynamics, including forces such as rolling resistance, aerodynamic drag, and acceleration demands, to simulate real-world driving conditions. Subsequently, we define optimization objectives centered on reducing equivalent fuel consumption, which unifies electrical and fuel energy usage into a single metric. By employing batch gradient descent, we iteratively adjust energy distribution parameters based on the gradient of a loss function, ultimately converging to an optimal solution. This method is particularly suitable for handling large datasets typical of electric vehicle operations, such as those collected over extended driving cycles, ensuring robust and efficient energy management.
To elaborate on the EREV operational mechanism, we first consider the longitudinal dynamics of the electric vehicle. The total force required to propel the vehicle, \( F_{\text{rep}} \), is derived from fundamental principles of vehicle motion, accounting for various resistive forces. This can be expressed mathematically as:
$$ F_{\text{rep}} = F_{\text{roll}} + F_{\text{air}} + F_{\text{grade}} + F_{\text{acc}} $$
where \( F_{\text{roll}} \) represents the rolling resistance, calculated as \( F_{\text{roll}} = m g C_{\text{rr}} \), with \( m \) being the vehicle mass, \( g \) the gravitational acceleration, and \( C_{\text{rr}} \) the rolling resistance coefficient. The aerodynamic drag force, \( F_{\text{air}} \), is given by \( F_{\text{air}} = \frac{1}{2} \rho C_d A v^2 \), where \( \rho \) is the air density, \( C_d \) the drag coefficient, \( A \) the frontal area, and \( v \) the vehicle velocity. The grade resistance, \( F_{\text{grade}} \), accounts for slope effects as \( F_{\text{grade}} = m g \sin(\theta) \), with \( \theta \) being the incline angle. Finally, the acceleration resistance, \( F_{\text{acc}} \), is \( F_{\text{acc}} = m a \), where \( a \) is the acceleration. This comprehensive force model ensures that our electric vehicle simulation accurately reflects real-world driving scenarios, which is essential for effective energy management in the context of China EV developments.
Next, we model the powertrain components, starting with the electric motor. The output torque, \( T_{\text{out}} \), is related to the input torque, \( T_{\text{in}} \), and motor efficiency, \( \eta \), through the equation:
$$ T_{\text{out}} = T_{\text{in}} \eta $$
This accounts for energy losses during power transmission, which are critical in optimizing the overall efficiency of the electric vehicle. For the battery system, we track the state of charge (SOC) over time, which is vital for managing energy flow between the battery and the auxiliary power unit. The SOC at time \( t \) is defined as:
$$ \text{SOC}(t) = \text{SOC}(0) – \frac{1}{C} \int_0^t I(t) \, dt $$
where \( \text{SOC}(0) \) is the initial SOC, \( C \) the battery capacity, and \( I(t) \) the current (positive for charging, negative for discharging). This dynamic model allows us to simulate battery behavior under varying loads, a key aspect of energy management for extended-range electric vehicles in China, where driving patterns can be highly diverse.
With the EREV model established, we proceed to define the energy management optimization objectives. The primary goal is to minimize equivalent fuel consumption, which integrates both electrical energy from the battery and fuel energy from the auxiliary source into a unified cost function. This is represented as:
$$ \eta_{\text{eqv}} = \alpha \frac{E}{q_{\text{fuel}}} + \beta \frac{m_{H_2}}{q_{\text{fuel}}} $$
Here, \( E \) denotes the electrical energy consumed by the battery in joules, \( m_{H_2} \) the mass of hydrogen consumed by the fuel cell in kilograms, \( q_{\text{fuel}} \) the calorific value of fuel in joules per kilogram, and \( \alpha \) and \( \beta \) are conversion coefficients that depend on energy efficiency factors. This formulation enables us to treat the multi-objective problem of minimizing both fuel use and emissions as a single optimization task, which is crucial for enhancing the sustainability of electric vehicles in China. Additionally, we impose performance constraints to ensure practical viability, such as acceleration requirements (e.g., 0 to 30 km/h in under 10 seconds), battery SOC limits (maintained above 0.2 to prevent over-discharge), and speed tracking accuracy (deviation less than 3.2 km/h from target speed per second). These constraints are integrated into the optimization framework to balance energy efficiency with driving performance.
To solve this optimization problem, we employ the batch gradient descent method, which iteratively updates energy allocation parameters by computing the gradient of the loss function over the entire dataset. The update rule is given by:
$$ \theta_{k+1} = \theta_k – \eta_0 \nabla_\theta L(\theta_k) $$
where \( \theta_k \) is the parameter vector at iteration \( k \), \( \eta_0 \) the learning rate controlling step size, and \( \nabla_\theta L(\theta_k) \) the gradient of the loss function \( L \) with respect to \( \theta \). The loss function, \( L \), is designed to encapsulate the equivalent fuel consumption and penalty terms for constraint violations, ensuring that the optimization aligns with our objectives for the electric vehicle. We initialize parameters randomly or based on prior knowledge and repeat the iteration until convergence criteria are met, such as gradient magnitude falling below a threshold or reaching a maximum number of iterations. This approach leverages the batch nature of the algorithm to handle the complexity of EREV energy management, making it well-suited for applications in China EV systems where data from diverse driving conditions are available.
In our experimental setup, we simulate a fuel cell-based EREV to validate the proposed optimization strategy. The vehicle prioritizes pure electric mode when battery SOC is high (e.g., SOC ≥ 70%) and switches to extended-range mode using the fuel cell when SOC drops below a threshold (e.g., SOC ≤ 30%). This configuration is common in advanced electric vehicle designs in China, aiming to maximize zero-emission driving while ensuring extended range. The key parameters for the EREV are summarized in the following table, which provides a foundation for our simulations:
| Component | Parameter | Value |
|---|---|---|
| Vehicle | Total Mass (kg) | 2000 |
| Fuel Cell | Rated Power (kW) | 80 |
| Fuel Cell | Maximum Power (kW) | 100 |
| Drive Motor | Maximum Power (kW) | 150 |
| Drive Motor | Maximum Torque (Nm) | 350 |
We utilize simulation tools like AVL Cruise and Simulink/Stateflow to create a virtual test platform, replicating real driving cycles with varying speeds, accelerations, and road gradients. The batch gradient descent algorithm is implemented programmatically to optimize energy分配 parameters, with the loss function incorporating equivalent fuel consumption and performance constraints. For instance, the gradient computation involves partial derivatives of the loss function with respect to control variables, such as the power split between the battery and fuel cell. This process is repeated over multiple epochs to refine the strategy, ensuring that the electric vehicle operates efficiently across different scenarios common in China EV applications.
The experimental results demonstrate significant improvements after applying the optimization. We compare key performance metrics before and after optimization, as shown in the table below, highlighting the effectiveness of our approach in enhancing the electric vehicle’s energy management:
| Optimization Variable / Performance Metric | Before Optimization | After Optimization |
|---|---|---|
| Range in Pure Electric Mode (km) | 120 | 135 |
| Range in Extended-Range Mode (km) | 350 | 400 |
| Total Range (km) | 470 | 535 |
| Equivalent Fuel Consumption (L/100 km) | 4.5 | 4.0 |
| Energy Recovery Efficiency (%) | 60 | 65 |
| Average Power Output Efficiency (%) | 82 | 87 |
As evident from the results, the optimized strategy increases the pure electric range by 15 km, attributed to more efficient battery energy utilization through refined control actions. The extended-range mode sees a 50 km boost in range, resulting from optimal power distribution between the fuel cell and battery, which minimizes wasteful energy dissipation. Overall, the total range improves by approximately 14%, showcasing the potential of this method to extend the driving distance of electric vehicles in China, where long-distance travel is often a concern. Moreover, equivalent fuel consumption decreases by 0.5 L/100 km, indicating better fuel economy and reduced operational costs. Energy recovery efficiency rises by 5%, due to enhanced regenerative braking strategies that capture kinetic energy during deceleration. The average power output efficiency also improves by 5%, reflecting smoother power delivery and reduced losses in the drivetrain. These gains underscore the practicality of batch gradient descent for real-world EREV applications, aligning with the growing emphasis on efficiency in the China EV sector.
Further analysis involves examining the convergence behavior of the batch gradient descent algorithm. The loss function typically decreases monotonically over iterations, with the gradient magnitude reducing as parameters approach optimal values. For example, in one simulation run, the loss decreased from an initial value of 10.5 to below 2.0 within 100 iterations, using a learning rate of \( \eta_0 = 0.01 \). This rapid convergence demonstrates the algorithm’s suitability for energy management tasks in electric vehicles, where real-time or near-real-time optimization may be required. Additionally, we tested the robustness under different driving cycles, such as urban and highway patterns common in China, and observed consistent performance improvements, validating the generalization capability of our approach.
In conclusion, this study successfully applies batch gradient descent to optimize energy management in extended-range electric vehicles, achieving notable enhancements in range, fuel efficiency, and overall performance. By integrating detailed vehicle dynamics modeling with iterative optimization, we address key challenges in the electric vehicle industry, particularly in the context of China EV advancements. The proposed method not only reduces energy consumption but also supports the transition toward sustainable transportation by maximizing the use of clean energy sources. Future work will focus on adapting the algorithm for real-time implementation, incorporating machine learning techniques to predict driving conditions, and expanding the framework to include other types of electric vehicles, such as plug-in hybrids or full battery-electric models. Through continuous refinement, we aim to contribute to the global evolution of electric vehicle technology, fostering economic and environmental benefits for societies worldwide.