With the rapid growth of electric vehicle adoption globally, particularly in China EV markets, the demand for efficient charging infrastructure has become critical. The increasing number of electric vehicles poses significant challenges to power grids, especially during peak hours, leading to potential overloads, higher costs for users, and uneven utilization of charging stations. In China EV scenarios, where urbanization and energy transition are accelerating, optimizing the scheduling of electric vehicle charging is essential to balance user convenience, grid stability, and resource allocation. This paper addresses the large-scale charging scheduling problem for electric vehicles by proposing an Elite Enhanced Genetic Algorithm (EEGA) that integrates adaptive strategies and multi-objective optimization to minimize charging time, cost, charging pile utilization deviation, and grid load. The approach leverages mathematical modeling and simulation to provide near-optimal solutions, ensuring sustainable development in the electric vehicle ecosystem.
The proliferation of electric vehicles in China EV initiatives has underscored the need for intelligent charging management systems. Unlike conventional vehicles, electric vehicles require coordinated charging to avoid grid stress and maximize efficiency. Traditional methods often lead to suboptimal outcomes, such as prolonged waiting times or excessive costs. To overcome this, we formulate a multi-objective optimization model that considers various factors, including travel time, charging duration, and economic constraints. The model is solved using an enhanced genetic algorithm that incorporates elite retention and adaptive mechanisms to improve convergence and avoid local optima. By applying this framework, we demonstrate significant improvements in scheduling performance, contributing to the broader adoption of electric vehicles in smart city environments.
Mathematical Model for Electric Vehicle Charging Scheduling
The charging scheduling problem for electric vehicles involves multiple objectives that must be balanced to achieve optimal performance. We define four key objectives: total charging time, total charging cost, charging pile time utilization deviation rate, and grid load. Each objective is modeled mathematically, and constraints are applied to reflect real-world scenarios in China EV deployments. The model assumes a network with M electric vehicles and N charging piles, where each electric vehicle must be assigned to a single charging pile without conflicts.
Charging Time Objective
The total charging time for an electric vehicle includes the time to travel to the charging pile, waiting time at the pile, and the actual charging time. The objective is to minimize the sum of these times across all electric vehicles. Let \( t_{ij} \) represent the travel time for electric vehicle i to charging pile j, \( t_{\text{wait},ij} \) the waiting time, and \( t_{\text{charge},ij} \) the charging time. The total time \( t_{\text{total}} \) for one electric vehicle is given by:
$$ t_{\text{total}} = t_{ij} + t_{\text{wait},ij} + t_{\text{charge},ij} $$
For all electric vehicles, the minimization function is:
$$ \min F1 = \sum_{i=1}^{M} \sum_{j=1}^{N} S_{ij} (t_{ij} + t_{\text{wait},ij} + t_{\text{charge},ij}) $$
where \( S_{ij} \) is a binary variable indicating whether electric vehicle i is assigned to charging pile j. The travel time \( t_{ij} \) is computed as \( t_{ij} = d_{ij} / v_i \), with \( d_{ij} \) being the distance and \( v_i \) the average speed of electric vehicle i. The charging time \( t_{\text{charge},ij} \) depends on the battery capacity \( S_i^{\text{total}} \), current state of charge \( S_i^{\text{present}} \), energy consumption per km \( N_i \), and charging power \( W_j \):
$$ t_{\text{charge},ij} = \frac{ S_i^{\text{total}} – (S_i^{\text{present}} – d_{ij} \cdot N_i) }{ W_j } $$
Charging Cost Objective
The total charging cost for electric vehicles includes parking fees and electricity costs. The objective is to minimize the overall expense for users, which is crucial for promoting electric vehicle adoption in cost-sensitive regions like China EV markets. The cost function is defined as:
$$ \min F2 = P_{\text{cost}} + C_{\text{cost}} $$
where \( P_{\text{cost}} \) is the parking cost and \( C_{\text{cost}} \) is the charging cost. The parking cost accounts for the time spent parked and waiting:
$$ P_{\text{cost}} = \sum_{j=1}^{N} \sum_{i=1}^{M} S_{ij} \cdot C_{ij} \cdot P_{cj} + \sum_{j=1}^{N} \sum_{i=1}^{M} t_{\text{wait},ij} \cdot P_{cj} $$
Here, \( C_{ij} \) is a charging time matrix, and \( P_{cj} \) is the hourly parking fee at charging pile j. The charging cost is based on the electricity consumed:
$$ C_{\text{cost}} = \sum_{j=1}^{N} \sum_{i=1}^{M} S_{ij} \cdot H_{ij} \cdot t_{\text{charge},ij} $$
with \( H_{ij} \) representing the charging cost matrix per kWh.
Charging Pile Time Utilization Deviation Rate
To ensure balanced usage of charging infrastructure, we minimize the deviation in time utilization across charging piles. This is vital for preventing overcrowding at certain piles while others remain underused, a common issue in dense urban areas for China EV networks. The time utilization for a charging pile j is \( T_j = Q_j / W_j \), where \( Q_j \) is the available energy and \( W_j \) is the charging power. The average utilization \( \mu \) is:
$$ \mu = \frac{1}{N} \sum_{j=1}^{N} \frac{c_j}{T_j} $$
where \( c_j \) is the total charging time at pile j. The deviation rate is then:
$$ \min F3 = \frac{1}{N} \sum_{j=1}^{N} \left( \frac{c_j}{T_j} – \mu \right)^2 $$
Grid Load Objective
Minimizing the grid load helps prevent overloading and ensures stable power distribution, which is critical for large-scale electric vehicle integration. The average grid load is derived from the total energy consumed and the total charging time:
$$ C_{\text{time}} = \sum_{j=1}^{N} c_j $$
$$ Q_{\text{total}} = \sum_{j=1}^{N} c_j \cdot W_j $$
$$ \min F4 = \frac{Q_{\text{total}}}{C_{\text{time}}} $$
Constraints
The model includes several constraints to reflect practical limitations. First, each electric vehicle can only be assigned to one charging pile:
$$ \sum_{j=1}^{N} S_{ij} = 1 \quad \forall i $$
Second, the travel distance must not exceed the electric vehicle’s range based on its current charge:
$$ d_{ij} \leq \max d_i $$
where \( d_i = S_i^{\text{present}} / N_i \). Third, the charging power must be within operational limits:
$$ \min W_j \leq W_j \leq \max W_j $$
Multi-Objective Optimization Using Entropy-Weighted TOPSIS
To handle the multi-objective nature of the problem, we employ the entropy-weighted TOPSIS method, which combines the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) with entropy-based weighting. This approach eliminates dimensional differences and assigns weights based on the information entropy of each objective, ensuring a fair comparison. The steps are as follows:
- Normalize the decision matrix \( X = (x_{ij})_{m \times n} \), where m is the number of objectives and n is the number of solutions:
$$ r_{ij} = \frac{ \max\{x_i\} – x_{ij} }{ \max\{x_i\} – \min\{x_i\} } $$
- Calculate the feature proportion for each objective i in solution j:
$$ f_{ij} = \frac{r_{ij}}{\sum_{j=1}^{n} r_{ij}} $$
- Compute the information entropy for objective i:
$$ X_i = -(\log n)^{-1} \sum_{j=1}^{n} [f_{ij} \cdot \ln(f_{ij})] $$
- Determine the difference coefficient:
$$ b_i = 1 – X_i $$
- Obtain the weight coefficients \( \lambda_i \) such that \( \sum_{i=1}^{m} \lambda_i = 1 \):
$$ \lambda_i = \frac{b_i}{m – \sum_{i=1}^{m} X_i} $$
The overall evaluation function Y is then formulated as:
$$ \min Y = \lambda_1 \left( \frac{F1 – \min F1}{\max F1 – \min F1} \right)^2 + \lambda_2 \left( \frac{F2 – \min F2}{\max F2 – \min F2} \right)^2 + \lambda_3 \left( \frac{F3 – \min F3}{\max F3 – \min F3} \right)^2 + \lambda_4 \left( \frac{F4 – \min F4}{\max F4 – \min F4} \right)^2 $$
This function integrates all objectives into a single metric, facilitating the optimization process for electric vehicle charging scheduling.
Elite Enhanced Genetic Algorithm (EEGA) Design
The Elite Enhanced Genetic Algorithm (EEGA) is designed to solve the electric vehicle charging scheduling problem efficiently. It improves upon traditional genetic algorithms by incorporating adaptive strategies and elite reinforcement to enhance convergence and avoid local optima. The algorithm operates through several key steps: encoding, selection, crossover, mutation, and elite reinforcement, tailored specifically for the China EV context where scalability and speed are paramount.
Encoding
We use a matrix encoding scheme where each chromosome represents a scheduling strategy as an \( m \times n \) matrix. For a population of size z in generation s, the population \( Z_s = \{A_1, A_2, \dots, A_z\} \), where each individual \( A_i^{mn} \) is defined as:
$$ A_i^{mn} = \begin{pmatrix}
a_{i11} & a_{i12} & \cdots & a_{i1n} \\
a_{i21} & a_{i22} & \cdots & a_{i2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{im1} & a_{im2} & \cdots & a_{imn}
\end{pmatrix} $$
Here, \( a_{iuv} \) is a binary gene element indicating whether electric vehicle u is assigned to charging pile v. Each row must have exactly one 1 to satisfy the constraint that an electric vehicle charges at only one pile.
Selection Operator
Selection is performed using the roulette wheel method based on fitness, which is the inverse of the evaluation function Y: \( \text{Fit} = 1 / Y \). The probability of selecting individual i is:
$$ P = \frac{\text{Fit}(x_i)}{\sum_{i=1}^{N} \text{Fit}(x_i)} $$
This ensures that individuals with lower Y values (better solutions) have higher chances of selection, aligning with the goal of optimizing electric vehicle charging.
Crossover Operator
The crossover operator uses an adaptive probability \( G_c \) to balance exploration and exploitation. For two parent individuals \( A_i^{mn} \) and \( A_j^{mn} \), if a random number \( \delta \leq G_c \), crossover occurs by swapping randomly selected rows. The adaptive probability is defined as:
$$ G_c = \begin{cases}
K_1 \frac{f_l – f_{\min}}{f_{\text{avg}} – f_{\min}}, & \text{if } f_l < f_{\text{avg}} \\
K_2, & \text{otherwise}
\end{cases} $$
where \( K_1 \) and \( K_2 \) are constants in [0,1], \( f_{\min} \) is the minimum fitness in the population, \( f_l \) is the lower fitness of the two parents, and \( f_{\text{avg}} \) is the average fitness. This adaptivity helps maintain diversity while favoring fitter individuals in electric vehicle scheduling.
Mutation Operator
Mutation is applied with an adaptive probability \( G_m \) that varies with generations to initially increase diversity and later refine solutions. The mutation probability is given by:
$$ G_m = \alpha + \gamma \cdot \cos\left(-0.5\pi + \pi \frac{e}{E}\right), \quad 0 < \alpha + \gamma < 1 $$
where \( \alpha \) is the base mutation probability, \( \gamma \) is the maximum gain, e is the current generation, and E is the total generations. If a random number \( \delta \geq G_m \), mutation is performed by flipping bits in randomly selected rows, ensuring that each row still has exactly one 1. This random flip strategy enhances exploration in the search space for electric vehicle assignments.
Elite Reinforcement
After crossover and mutation, the new population \( Z_{s+1} \) (size z/2) is merged with the parent population \( Z_s \) to form a combined population \( Q_s \) of size 3z/2. The top p individuals with the lowest Y values are retained in \( Q’_s \). The best individual in \( Q’_s \) undergoes elite reinforcement, where random searches are conducted on its matrix rows to find improvements. If a better solution is found, it replaces the current elite individual. This step accelerates convergence and ensures high-quality solutions for electric vehicle charging scheduling.
Simulation Experiments and Results Analysis
To validate the proposed EEGA approach, we conducted simulations using MATLAB, considering a scenario with 100 electric vehicles and 20 charging piles, typical for urban China EV environments. The initial positions of electric vehicles and charging piles were randomly generated, and key parameters are summarized in the following tables. The EEGA was configured with a population size of 100, 300 generations, \( K_1 = K_2 = 1 \), maximum mutation probability of 0.5, and minimum mutation probability of 0.1.
| Electric Vehicle ID | Battery Capacity (kWh) | Current Charge (kWh) | Average Speed (km/h) | Energy Consumption (kWh/km) |
|---|---|---|---|---|
| 1 | 45 | 8 | 60 | 0.18 |
| 2 | 50 | 7 | 55 | 0.21 |
| 3 | 45 | 9 | 65 | 0.20 |
| 4 | 45 | 6 | 70 | 0.17 |
| 5 | 45 | 5 | 75 | 0.16 |
| 6 | 45 | 10 | 50 | 0.23 |
| 7 | 40 | 8 | 80 | 0.15 |
| 8 | 50 | 5 | 60 | 0.21 |
| 9 | 45 | 6 | 70 | 0.18 |
| 10 | 50 | 9 | 55 | 0.20 |
| Charging Pile ID | Parking Fee (USD/h) | Charging Cost (USD/kWh) | Charging Power (kW) |
|---|---|---|---|
| 1 | 5 | 0.98 | 50 |
| 2 | 4 | 1.20 | 60 |
| 3 | 6 | 1.25 | 50 |
| 4 | 5 | 1.30 | 60 |
| 5 | 6 | 0.90 | 50 |
| 6 | 3 | 1.30 | 70 |
| 7 | 4 | 1.20 | 60 |
| 8 | 5 | 1.40 | 60 |
| 9 | 4 | 1.10 | 50 |
| 10 | 3 | 1.30 | 70 |
Using the entropy-weighted TOPSIS method, the weights for each objective were calculated as shown in Table 3, based on the variability and importance of each metric in the China EV context.
| Objective | Weight (\( \lambda_i \)) |
|---|---|
| F1 (Charging Time) | 0.2559 |
| F2 (Charging Cost) | 0.2856 |
| F3 (Utilization Deviation) | 0.2023 |
| F4 (Grid Load) | 0.2562 |
We compared the EEGA with unscheduled charging (where electric vehicles choose the nearest pile), a traditional genetic algorithm (GA), and a Partheno-Genetic Algorithm (PGA) from literature. The results, averaged over 30 independent runs, are summarized in Table 4. The EEGA achieved the best performance across all metrics, demonstrating its effectiveness for electric vehicle charging optimization.
| Algorithm | Total Time (h) | Total Cost (USD) | Utilization Deviation (%) | Grid Load (kW) | Function Evaluations |
|---|---|---|---|---|---|
| Unscheduled | 350.2 | 7400.50 | 12.5 | 62.1000 | N/A |
| GA | 335.6 | 7256.30 | 8.83 | 59.0354 | 8900 |
| PGA | 329.0 | 7130.90 | 6.21 | 57.9865 | 11000 |
| EEGA | 321.5 | 7092.10 | 4.93 | 55.0312 | 5500 |
The distribution of electric vehicles across charging piles was more balanced with EEGA, as shown in Table 5, where each pile served approximately 5 vehicles, compared to uneven distributions in other methods. This balance is crucial for efficient resource use in China EV networks.
| Charging Pile ID | Unscheduled | GA | PGA | EEGA |
|---|---|---|---|---|
| 1 | 0 | 6 | 5 | 5 |
| 2 | 0 | 4 | 4 | 4 |
| 3 | 10 | 3 | 3 | 5 |
| 4 | 5 | 7 | 6 | 6 |
| 5 | 7 | 1 | 4 | 4 |
| 6 | 15 | 4 | 5 | 5 |
| 7 | 1 | 7 | 5 | 4 |
| 8 | 6 | 5 | 5 | 5 |
| 9 | 15 | 4 | 4 | 4 |
| 10 | 6 | 5 | 7 | 5 |
The convergence behavior of EEGA, compared to GA and PGA, is illustrated by the evaluation function Y over generations. EEGA reached a lower Y value faster, indicating superior convergence and solution quality for electric vehicle scheduling. Specifically, EEGA reduced charging time by approximately 4.2%, cost by 2.3%, utilization deviation by 4.4%, and grid load by 6.8% relative to GA, highlighting its potential for real-world China EV applications.
Conclusion
In this paper, we presented an Elite Enhanced Genetic Algorithm (EEGA) for optimizing electric vehicle charging scheduling, addressing key challenges in China EV ecosystems. By integrating adaptive crossover and mutation with elite reinforcement, EEGA effectively minimizes charging time, cost, charging pile utilization deviation, and grid load. The multi-objective model, solved using entropy-weighted TOPSIS, provides a comprehensive framework for balancing diverse requirements. Simulation results demonstrate that EEGA outperforms traditional methods, offering significant improvements in efficiency and reliability. This approach not only enhances user experience by reducing wait times and expenses but also supports grid stability and equitable resource use. Future work could explore dynamic pricing models and integration with renewable energy sources to further advance electric vehicle charging solutions in smart cities.