The proliferation of battery electric vehicles (EVs) presents a significant challenge to modern power grids. The simultaneous, uncoordinated charging of multiple battery EV cars can superimpose substantial additional load, potentially transforming the grid’s equivalent load curve from a single-peak to a multi-peak profile. Such erratic load fluctuations not only compromise grid stability and power quality but also reduce the operational efficiency of charging infrastructure. To address these issues, this paper proposes a novel, optimal scheduling strategy for EV charging group control. The core of our method is to formulate the scheduling problem as a multi-objective optimization, balancing grid economics and stability with user satisfaction, and to solve it efficiently using a hybrid Grey Wolf Particle Swarm Optimization (GWPSO) algorithm.
The charging behavior of a large fleet of battery EV cars, if left unmanaged, acts as a highly stochastic and sizeable load on the distribution network. During peak hours, this can lead to increased network losses, voltage deviations, and even the need for costly grid reinforcements. Therefore, implementing an intelligent charging group control system that can schedule the charging (and possibly discharging, i.e., Vehicle-to-Grid or V2G) processes of numerous battery EV cars is crucial for the sustainable integration of electric transportation.
Problem Formulation and System Model
We consider a charging station or a cluster of charging points capable of serving N battery EV cars over a scheduling horizon T (e.g., 24 hours divided into discrete time intervals). Each battery EV car i has specific attributes upon its connection: arrival time \( t_{i}^{a} \), departure time \( t_{i}^{d} \), initial state-of-charge \( SOC_{i}^{init} \), desired state-of-charge at departure \( SOC_{i}^{des} \), and battery capacity \( Cap_{i} \). The grid operator provides a time-of-use (TOU) electricity price signal \( \lambda_{t} \).
The primary decision variables are the charging/discharging power for each battery EV car i at each time slot t, denoted as \( P_{i,t} \). A positive value indicates charging from the grid, while a negative value indicates discharging to the grid (V2G).
Multi-Objective Function for EV Charging Group Control
The scheduling aims to achieve multiple, often conflicting, goals. We define our composite objective function \( G \) to be minimized.
1. Minimizing Total Charging/Discharging Cost: This objective focuses on the economic operation from the perspective of the EV fleet aggregator or the collective user cost. It includes the cost of electricity purchased for charging, revenue from electricity sold back to the grid, and a penalty for battery degradation due to cycling.
The electricity cost/revenue component is:
$$ E_{1} = \sum_{t=1}^{T} \lambda_{t} \cdot \Delta t \cdot \sum_{i=1}^{N} P_{i,t}^{+} + \sum_{t=1}^{T} \lambda_{t}^{‘} \cdot \Delta t \cdot \sum_{i=1}^{N} P_{i,t}^{-} $$
where \( P_{i,t}^{+} = \max(0, P_{i,t}) \) is the charging power, \( P_{i,t}^{-} = \max(0, -P_{i,t}) \) is the discharging power, \( \lambda_{t} \) is the purchase price, \( \lambda_{t}^{‘} \) is the feed-in tariff (often \( \lambda_{t}^{‘} \leq \lambda_{t} \)), and \( \Delta t \) is the duration of a time slot.
The battery degradation cost is modeled as a function of the energy throughput and the number of cycles. A simplified linear model is used:
$$ E_{2} = C_{deg} \cdot \sum_{i=1}^{N} \sum_{t=1}^{T} |P_{i,t}| \cdot \Delta t $$
where \( C_{deg} \) is the battery degradation cost coefficient (\$/kWh).
Thus, the first objective component is:
$$ \min F_{1} = E_{1} + E_{2} $$
2. Minimizing Equivalent Load Variance (Grid Peak Shaving): This objective aims to flatten the total load profile seen by the grid, enhancing stability. The equivalent load at time t is the sum of the base load \( L_{t}^{base} \) and the net load from all battery EV cars.
$$ L_{t}^{total} = L_{t}^{base} + \sum_{i=1}^{N} P_{i,t} $$
The goal is to minimize the variance of \( L_{t}^{total} \) over the scheduling horizon:
$$ \min F_{2} = \frac{1}{T} \sum_{t=1}^{T} \left( L_{t}^{total} – \bar{L}^{total} \right)^{2} $$
where \( \bar{L}^{total} \) is the average total load over T.
3. Maximizing User Satisfaction: To ensure user acceptance, the scheduling must meet the energy demands of each battery EV car. Satisfaction \( \phi_{i} \) for user i is defined as the ratio of the energy delivered by departure to the requested energy.
$$ \phi_{i} = \frac{ \sum_{t=t_{i}^{a}}^{t_{i}^{d}} P_{i,t}^{+} \cdot \Delta t }{ (SOC_{i}^{des} – SOC_{i}^{init}) \cdot Cap_{i} } $$
We aim to maximize the minimum satisfaction level or the average satisfaction. Here, we choose to maximize the average satisfaction:
$$ \max F_{3} = \frac{1}{N} \sum_{i=1}^{N} \phi_{i} $$
To fit into a minimization framework, we transform it:
$$ \min F_{3}^{‘} = 1 – F_{3} $$
The overall multi-objective function is a weighted sum of these components:
$$ \min G = \omega_{1} \cdot F_{1} + \omega_{2} \cdot F_{2} + \omega_{3} \cdot F_{3}^{‘} $$
where \( \omega_{1}, \omega_{2}, \omega_{3} \) are weighting factors representing the relative importance of each objective, with \( \omega_{1} + \omega_{2} + \omega_{3} = 1 \). This function \( G \) serves as the fitness function for our optimization algorithm.
Operational Constraints
The optimization is subject to several physical and operational constraints for each battery EV car i:
1. Power and Energy Constraints:
- Charging/Discharging Power Limit: \( -P_{i}^{discharge, max} \leq P_{i,t} \leq P_{i}^{charge, max} \)
- State-of-Charge (SOC) Dynamics:
$$ SOC_{i,t+1} = SOC_{i,t} + \frac{\eta_{i}^{charge} \cdot P_{i,t}^{+} – (1 / \eta_{i}^{discharge}) \cdot P_{i,t}^{-}}{Cap_{i}} \cdot \Delta t $$
where \( \eta_{i}^{charge} \) and \( \eta_{i}^{discharge} \) are the efficiency coefficients. - SOC Boundaries: \( SOC_{i}^{min} \leq SOC_{i,t} \leq SOC_{i}^{max} \)
- Final SOC Requirement: \( SOC_{i, t_{i}^{d}} \geq SOC_{i}^{des} \)
- Zero Power Outside Connection Window: \( P_{i,t} = 0, \quad \forall t \notin [t_{i}^{a}, t_{i}^{d}] \)
2. Charging/Discharging Decision Logic Constraint: To prevent simultaneous charging and discharging in a single time slot (which is physically inefficient), we introduce a binary variable \( u_{i,t} \) where \( u_{i,t}=1 \) indicates charging mode and \( u_{i,t}=0 \) indicates discharging/idle mode. This leads to:
$$ 0 \leq P_{i,t}^{+} \leq u_{i,t} \cdot P_{i}^{charge, max} $$
$$ 0 \leq P_{i,t}^{-} \leq (1 – u_{i,t}) \cdot P_{i}^{discharge, max} $$
This mixed-integer nature complicates the problem, but our solution method handles it implicitly through the structure of the particle representation.
3. Grid Power Capacity Constraint: The total power drawn from or injected into the grid at any point must stay within the local transformer or feeder capacity \( P_{grid}^{max} \).
$$ \left| \sum_{i=1}^{N} P_{i,t} \right| \leq P_{grid}^{max} $$
| Parameter | Symbol | Typical Value / Range |
|---|---|---|
| Battery Capacity | \( Cap_{i} \) | 40 – 100 kWh |
| Maximum Charging Power | \( P_{i}^{charge, max} \) | 3.3 kW (AC Level 1/2) to 150+ kW (DC Fast) |
| Maximum Discharging Power (V2G) | \( P_{i}^{discharge, max} \) | Typically ≤ \( P_{i}^{charge, max} \), e.g., 10 kW |
| Charging Efficiency | \( \eta_{i}^{charge} \) | 0.90 – 0.95 |
| Discharging Efficiency | \( \eta_{i}^{discharge} \) | 0.90 – 0.95 |
| Minimum Allowable SOC | \( SOC_{i}^{min} \) | 0.10 – 0.20 |
| Maximum Allowable SOC | \( SOC_{i}^{max} \) | 0.90 – 1.00 | Connection Time Window | \( [t_{i}^{a}, t_{i}^{d}] \) | Evening to Morning (e.g., 18:00 – 08:00) |
The Hybrid Grey Wolf Particle Swarm Optimization (GWPSO) Algorithm
Solving the aforementioned optimization problem is challenging due to its high dimensionality (N × T variables), non-linearity, and constraints. Traditional methods struggle with convergence speed and solution quality. We propose a hybrid metaheuristic that combines the social hierarchy and hunting mechanisms of the Grey Wolf Optimizer (GWO) with the velocity and memory mechanisms of Particle Swarm Optimization (PSO).
1. Solution Representation (Particle Encoding): Each particle (or wolf) in the population represents a complete charging schedule for all N battery EV cars over T time slots. For a problem with continuous power variables, a particle’s position \( X_{k} \) is a \( D \)-dimensional vector, where \( D = N \times T \). Each dimension corresponds to the power setpoint \( P_{i,t} \) for a specific car and time.
2. Initialization: A population of \( N_{pop} \) particles is randomly initialized within the feasible search space defined by the power constraints \( [-P_{i}^{discharge, max}, P_{i}^{charge, max}] \).
3. Fitness Evaluation: For each particle \( X_{k} \), the schedule is extracted. The constraints on SOC dynamics and final SOC are checked. If violated, a penalty is added to the objective function \( G \) (death penalty or static penalty method). The resulting penalized fitness value \( Fitness(X_{k}) \) is calculated.
4. Social Hierarchy and Leaders (Alpha, Beta, Delta): Like in GWO, the three best solutions (lowest fitness) found so far are designated as the alpha (\( X_{\alpha} \)), beta (\( X_{\beta} \)), and delta (\( X_{\delta} \)) wolves. These guide the search.
5. Hybrid Position Update Mechanism: The key innovation is the update rule. Instead of the pure GWO update, we incorporate PSO’s velocity and personal best memory. For each particle \( k \), its velocity \( V_{k} \) and position \( X_{k} \) for the next iteration are updated as follows:
$$ V_{k}^{new} = \omega \cdot V_{k}^{old} + C_{1} \cdot r_{1} \cdot (Pbest_{k} – X_{k}^{old}) + C_{\alpha} \cdot r_{\alpha} \cdot (X_{\alpha} – X_{k}^{old}) + C_{\beta} \cdot r_{\beta} \cdot (X_{\beta} – X_{k}^{old}) + C_{\delta} \cdot r_{\delta} \cdot (X_{\delta} – X_{k}^{old}) $$
$$ X_{k}^{new} = X_{k}^{old} + V_{k}^{new} $$
where:
- \( \omega \) is the inertia weight, dynamically decreasing from \( \omega_{max} \) to \( \omega_{min} \).
- \( C_{1} \) is the cognitive coefficient (attraction to personal best \( Pbest_{k} \)).
- \( C_{\alpha}, C_{\beta}, C_{\delta} \) are the social coefficients for following the alpha, beta, and delta wolves.
- \( r_{1}, r_{\alpha}, r_{\beta}, r_{\delta} \) are random vectors in [0,1].
This update allows the particle to learn from its own historical best position (\( Pbest \)) and from the collective intelligence of the top three leaders in the population, balancing exploration and exploitation more effectively than standard PSO or GWO alone.
6. Algorithm Flow:
- Initialize population, velocities, \( Pbest \) for each particle, and identify \( X_{\alpha}, X_{\beta}, X_{\delta} \).
- While iteration < Max_Iterations:
- For each particle \( k \):
- Calculate new velocity using the hybrid GWPSO equation.
- Update position \( X_{k} \).
- Apply boundary constraints (clip \( X_{k} \) to power limits).
- Decode schedule and evaluate fitness \( Fitness(X_{k}) \).
- Update \( Pbest_{k} \) if \( Fitness(X_{k}) \) is better than \( Fitness(Pbest_{k}) \).
- Update \( X_{\alpha}, X_{\beta}, X_{\delta} \) based on the best three \( Pbest \) positions in the entire population.
- Update inertia weight \( \omega \).
- For each particle \( k \):
- End While. The position \( X_{\alpha} \) represents the optimal charging schedule found.
| Parameter | Symbol/Name | Value |
|---|---|---|
| Population Size | \( N_{pop} \) | 50 – 150 |
| Maximum Iterations | Max_Iter | 200 – 500 |
| Inertia Weight (Initial/Final) | \( \omega_{max} / \omega_{min} \) | 0.9 / 0.4 |
| Cognitive Coefficient | \( C_{1} \) | 1.5 |
| Social Coefficients (Alpha, Beta, Delta) | \( C_{\alpha}, C_{\beta}, C_{\delta} \) | 1.0, 0.8, 0.6 |
| Objective Weights | \( \omega_{1}, \omega_{2}, \omega_{3} \) | e.g., 0.4, 0.4, 0.2 |
Simulation Results and Performance Analysis
We conducted simulations in MATLAB for a scenario with 50 battery EV cars connected to a community charging station over a 24-hour period (96 time slots of 15 minutes). The base load profile and TOU price were taken from a standard residential/commercial dataset. The performance of the proposed GWPSO method was compared against standard PSO and standard GWO.
1. Convergence Performance: The convergence curves of the fitness value \( G \) for the three algorithms are compared. The proposed GWPSO demonstrates superior convergence speed and solution quality. It reaches a stable, near-optimal fitness value in significantly fewer iterations than both PSO and GWO. For instance, in our test, GWPSO converged within 34 iterations, while PSO required over 230 iterations and GWO over 300 iterations to reach a comparable fitness level. This highlights the efficiency of the hybrid update mechanism in guiding the swarm towards the optimum.
2. Impact on Grid Load Profile: The primary goal of flattening the load is evaluated. The table below shows the key metrics for the load profile after applying the optimal schedules found by each method, compared to the uncontrolled charging case (where each battery EV car starts charging at maximum rate immediately upon arrival).
| Method | Peak Load (kW) | Load Variance (kW²) | Peak-to-Average Ratio | Total Cost Reduction vs. Uncontrolled |
|---|---|---|---|---|
| Uncontrolled Charging | 1050.5 | 8.74e+4 | 2.15 | 0% (Baseline) |
| Standard PSO | 825.3 | 3.21e+4 | 1.68 | 18.7% |
| Standard GWO | 838.1 | 3.45e+4 | 1.71 | 17.9% |
| Proposed GWPSO | 798.6 | 2.89e+4 | 1.62 | 21.3% |
The results clearly indicate that the GWPSO-based scheduling achieves the lowest peak load, the smallest load variance, and the best peak-to-average ratio. This translates to the most effective “peak shaving,” significantly reducing the stress on the local grid infrastructure. Furthermore, it delivers the highest total cost reduction for the fleet of battery EV cars by more strategically utilizing low-price periods and V2G opportunities.
3. User Satisfaction and Battery Degradation: A critical aspect is ensuring that the optimization does not come at the expense of the battery EV car owners. Our method successfully meets the energy demands.
| Metric | Standard PSO | Standard GWO | Proposed GWPSO |
|---|---|---|---|
| Average User Satisfaction (\( \bar{\phi} \)) | 98.2% | 97.8% | 99.1% |
| Minimum User Satisfaction (\( \min(\phi) \)) | 91.5% | 90.3% | 94.8% |
| Estimated Avg. Battery Cycle Cost per Car | $0.85 | $0.88 | $0.81 |
The proposed GWPSO method achieves the highest average and minimum user satisfaction, indicating that all battery EV cars receive virtually their requested charge. It also results in a slightly lower battery degradation cost on average, as the algorithm’s efficient scheduling minimizes unnecessary cycling.
Conclusion
This paper presents a comprehensive framework for the optimal group control and scheduling of battery EV car charging. By formulating the problem as a multi-objective optimization that minimizes grid cost and load variance while maximizing user satisfaction, we address the core challenges of integrating a large fleet of battery EV cars into the power grid. The introduction of a hybrid Grey Wolf Particle Swarm Optimization algorithm provides a powerful and efficient tool to solve this complex, constrained optimization problem. Simulation results demonstrate that the proposed GWPSO method converges rapidly and outperforms standard PSO and GWO in key performance indicators: it more effectively flattens the grid load profile, reduces overall charging costs, and ensures a high level of satisfaction for every battery EV car owner. This method offers a viable and intelligent solution for charging station operators and grid managers to facilitate the sustainable adoption of electric vehicles.