With the rapid global proliferation of battery electric vehicles (BEVs), primarily driven by their energy-saving and environmental benefits, significant challenges have emerged for power grid operation. The uncoordinated, or disorderly, charging of a massive number of battery EV cars can severely strain grid infrastructure, leading to increased peak-to-valley load differences, compromised grid stability, degraded power quality, and elevated operational costs. Consequently, developing efficient orderly charging algorithms is paramount. These algorithms must intelligently schedule the charging time and power of each battery EV car to safeguard grid security, minimize user expenses, and enhance user satisfaction. This paper aims to address this multifaceted problem by constructing a comprehensive multi-objective optimization model and employing an advanced solution algorithm.
Existing research has explored various strategies, with early works focusing on time-of-use (TOU) pricing to incentivize off-peak charging. While effective in reducing user costs, such single-objective approaches often neglect critical aspects like grid load balancing and nuanced user experience. Subsequent studies introduced multi-objective optimization frameworks. However, the performance of conventional algorithms in terms of convergence speed and solution precision when dealing with this high-dimensional, constrained problem can be insufficient. To bridge these gaps, this paper proposes a multi-objective optimization model that simultaneously minimizes grid load fluctuation, minimizes total charging cost, and maximizes user satisfaction. An Improved Non-dominated Sorting Genetic Algorithm II (NSGA-II) is developed and applied to solve this model, striving to achieve a superior Pareto-optimal frontier that balances the interests of both the grid and the users.
Analysis of Battery EV Car Charging Characteristics
An accurate model of battery EV car charging characteristics is fundamental to formulating an effective scheduling strategy. These characteristics can be broadly categorized into power attributes and user behavior patterns.
The charging power of a battery EV car is determined by the vehicle’s onboard charger and the charging point. Typical AC charging piles deliver power in the range of 3.3 kW to 22 kW, while DC fast-charging stations can exceed 100 kW. For the purpose of orderly charging, we assume the charging power for each battery EV car is adjustable within a defined range, up to the maximum rated power of its connected charger, denoted as $P_{i}^{max}$ for vehicle $i$. This adjustability is a key control variable for our optimization.
User charging behavior is inherently stochastic. Three primary stochastic variables define a charging session for a battery EV car: plug-in time (start time), required charging duration, and required energy. Statistical analysis of empirical data reveals that the daily plug-in time for residential users often follows a probability distribution, such as a Normal or Log-Normal distribution. For instance, the plug-in time $t_{start,i}$ for user $i$ might be modeled as:
$$t_{start,i} \sim \mathcal{N}(\mu_{start}, \sigma_{start}^2)$$
where $\mu_{start}$ often aligns with the evening return-home time (e.g., 18:00) and $\sigma_{start}$ represents the variability. The required charging energy $E_{i}^{req}$ is related to the daily driving distance and the battery capacity $E_{b,i}$ of the battery EV car, while the desired plug-out time $t_{end,i}$ defines the available charging window. The relationship between required energy and power is:
$$E_{i}^{req} = \int_{t_{start,i}}^{t_{end,i}} P_i(t) \, dt$$
where $P_i(t)$ is the charging power of the i-th battery EV car at time $t$.
Formulation of the Multi-Objective Orderly Charging Model
The core of our approach is a mathematical model that formalizes the trade-offs between grid operation, user economics, and user experience.
Objective Functions
We define three conflicting objectives to be optimized simultaneously.
Objective 1: Minimize Grid Load Fluctuation. To promote grid stability and reduce stress on infrastructure, we aim to flatten the total load curve. This is quantified by minimizing the variance of the aggregate load over the scheduling horizon $T$ (e.g., 24 hours).
$$\min \; f_1 = \sigma^2 = \frac{1}{T} \sum_{t=1}^{T} (L_{total}(t) – \bar{L}_{total})^2$$
where $L_{total}(t)$ is the total grid load at time interval $t$, and $\bar{L}_{total}$ is the average load. The total load comprises the base load $L_{base}(t)$ and the aggregated load from all $N$ battery EV cars:
$$L_{total}(t) = L_{base}(t) + \sum_{i=1}^{N} P_i(t)$$
Objective 2: Minimize Total Charging Cost. From the users’ perspective, minimizing electricity cost is a primary concern. Under a TOU tariff structure, the total cost for all users is:
$$\min \; f_2 = C_{total} = \sum_{t=1}^{T} \sum_{i=1}^{N} [C_{TOU}(t) \cdot E_i(t)]$$
where $C_{TOU}(t)$ is the electricity price at time $t$, and $E_i(t) = P_i(t) \cdot \Delta t$ is the energy delivered to the i-th battery EV car during time interval $t$ of length $\Delta t$.
Objective 3: Maximize User Satisfaction. User satisfaction is a composite metric reflecting charging convenience. We model it using fuzzy logic, considering two factors: waiting time and charging completeness. Satisfaction decreases if charging is delayed significantly after plug-in, and increases as the battery state-of-charge (SOC) approaches the user’s desired level by the plug-out time.
Let $\mu_{i}^{wait}(t)$ and $\mu_{i}^{soc}(t)$ be the membership functions (ranging [0,1]) for waiting time satisfaction and SOC satisfaction for the i-th battery EV car at time $t$, respectively. The aggregate satisfaction is:
$$\max \; f_3 = S_{total} = \sum_{i=1}^{N} ( \omega_{wait} \cdot \mu_{i}^{wait} + \omega_{soc} \cdot \mu_{i}^{soc} )$$
where $\omega_{wait}$ and $\omega_{soc}$ are weighting coefficients summing to 1. A sample linear membership function for SOC satisfaction could be:
$$\mu_{i}^{soc} = \begin{cases}
0 & \text{if } SOC_i(t_{end,i}) < SOC_i^{min,req} \\
\frac{SOC_i(t_{end,i}) – SOC_i^{min,req}}{SOC_i^{desired} – SOC_i^{min,req}} & \text{if } SOC_i^{min,req} \le SOC_i(t_{end,i}) \le SOC_i^{desired} \\
1 & \text{if } SOC_i(t_{end,i}) \ge SOC_i^{desired}
\end{cases}$$
Constraints
The optimization is subject to the following physical and user-imposed constraints for each battery EV car $i$:
1. Power Constraint: The charging power must be within the limits of the charger.
$$0 \le P_i(t) \le P_i^{max}, \quad \forall t, i$$
2. Time Window Constraint: Charging can only occur within the user’s available connection period.
$$P_i(t) = 0, \quad \forall t \notin [t_{start,i}, \; t_{end,i}], \quad \forall i$$
3. Battery Dynamics & Capacity Constraint: The SOC evolves with charging and must remain within safe limits.
$$SOC_i(t) = SOC_i(t-1) + \frac{P_i(t) \cdot \Delta t}{E_{b,i}}, \quad \forall t, i$$
$$SOC_i^{min} \le SOC_i(t) \le SOC_i^{max}, \quad \forall t, i$$
4. Energy Demand Constraint: The total energy delivered must meet or exceed the user’s minimum requirement by the departure time.
$$SOC_i(t_{end,i}) \cdot E_{b,i} \ge E_i^{req}, \quad \forall i$$
Solution via an Improved NSGA-II Algorithm
The formulated model is a complex, non-linear, constrained multi-objective optimization problem. We employ an enhanced version of the NSGA-II algorithm to find its Pareto-optimal solutions.
Overview of Standard NSGA-II
NSGA-II is a prominent elitist multi-objective evolutionary algorithm. Its key steps in each generation are: 1) Fast non-dominated sorting to rank individuals into Pareto fronts (F1, F2,…). 2) Crowding distance calculation to estimate the density of solutions around an individual within the same front. 3) Selection using a binary tournament based on rank and crowding distance. 4) Crossover and mutation to create offspring. 5) Elitism by combining parent and offspring populations and selecting the best N individuals for the next generation.
Proposed Algorithmic Enhancements
To better suit the orderly charging problem, we introduce two major improvements.
1. Adaptive Crossover and Mutation Probabilities: Fixed probabilities ($P_c$, $P_m$) limit performance. We adapt them based on an individual’s fitness rank. Individuals in lower (better) fronts benefit from lower mutation to fine-tune, while those in worse fronts benefit from higher crossover and mutation to explore.
Let $rank(i)$ be the non-dominated front number of individual $i$ (1 is best). The probabilities are adjusted as:
$$P_c(i) = P_{c,max} – (P_{c,max} – P_{c,min}) \cdot \frac{rank(i)-1}{N_{fronts}-1}$$
$$P_m(i) = P_{m,min} + (P_{m,max} – P_{m,min}) \cdot \frac{rank(i)-1}{N_{fronts}-1}$$
where $N_{fronts}$ is the total number of fronts in the current population.
2. Enhanced Elite Preservation with an External Archive: An external archive $A$ is maintained to store all non-dominated solutions found during the entire evolutionary process. This archive is updated at each generation. If its size exceeds a threshold $|A|_{max}$, pruning is performed based on crowding distance to maintain diversity. The selection operator for creating the mating pool now draws individuals both from the main population and the archive $A$, accelerating convergence towards the true Pareto front.
The impact of these adaptive strategies on the algorithm’s search behavior is summarized below:
| Individual Fitness Rank (Front) | Assigned $P_c$ | Assigned $P_m$ | Intended Search Emphasis |
|---|---|---|---|
| High (Worse Fronts, e.g., >Front 3) | High (~0.9) | High (~0.1) | Global exploration, diversification, escaping local optima. |
| Medium (Middle Fronts) | Medium (~0.75) | Medium (~0.055) | Balanced exploration and exploitation. |
| Low (Best Fronts, e.g., Front 1) | Low (~0.6) | Low (~0.01) | Local exploitation, fine-tuning of promising solutions. |
Algorithm Flowchart for Orderly Charging
1. Initialization: Generate an initial population of size $N_{pop}$. Each chromosome encodes the charging power schedules $P_i(t)$ for all $N$ battery EV cars, typically using real-valued or integer representation.
2. Fitness Evaluation: Decode each chromosome, calculate the three objective function values ($f_1, f_2, f_3$) and constraint violation degrees.
3. Constraint Handling & Non-dominated Sorting: Apply penalty functions or constrained-domination principles. Perform fast non-dominated sort on the feasible/penalized population.
4. Crowding Distance Assignment: Calculate crowding distance for individuals in each front.
5. Selection, Crossover, Mutation: Perform tournament selection. Apply simulated binary crossover (SBX) and polynomial mutation using the adaptive probabilities $P_c(i)$ and $P_m(i)$ to create the offspring population.
6. External Archive Update: Merge the current archive with the new non-dominated individuals from the combined parent-offspring population. Prune the archive if necessary.
7. New Population Formation: Combine parent and offspring populations. Select the best $N_{pop}$ individuals based on front rank and crowding distance to form the next generation.
8. Termination Check: Repeat steps 2-7 until the maximum generation $G_{max}$ is reached. Output the final external archive as the approximated Pareto-optimal set.
Simulation Design and Results Analysis
Parameter Settings and Scenario
We simulate a residential community with 100 battery EV cars over a 24-hour scheduling period (96 intervals of 15 minutes). Key parameters are listed in the table below.
| Parameter Category | Symbol / Description | Value / Setting |
|---|---|---|
| Grid & Load | Base Load Profile $L_{base}(t)$ | Historical data from a residential feeder |
| Scheduling Horizon $T$ | 24 hours (96 intervals) | |
| Time Interval $\Delta t$ | 15 minutes | |
| Battery EV Car | Number of Battery EV Cars $N$ | 100 |
| Battery Capacity $E_b$ | Uniform [50, 80] kWh | |
| Initial SOC $SOC(t_{start})$ | Uniform [0.2, 0.4] | |
| Desired Final SOC $SOC^{desired}$ | 1.0 | |
| Max Charging Power $P^{max}$ | 7 kW (AC Level 2) | |
| User Behavior | Plug-in Time $t_{start}$ | Normal Dist. $ \mathcal{N}$(18:00, (2h)$^2$) |
| Plug-out Time $t_{end}$ | $t_{start}$ + Uniform [6, 10] hours | |
| Required Energy $E^{req}$ | To reach $SOC^{desired}$ from initial SOC | |
| Pricing | Peak Price | 1.2 CNY/kWh (12:00-18:00, 20:00-22:00) |
| Off-Peak Price | 0.8 CNY/kWh (08:00-12:00, 18:00-20:00, 22:00-24:00) | |
| Valley Price | 0.3 CNY/kWh (00:00-08:00) | |
| Algorithm | Population Size $N_{pop}$ | 200 |
| Maximum Generations $G_{max}$ | 300 | |
| Adaptive $P_c$ / $P_m$ | $P_{c,max}=0.9, P_{c,min}=0.6$ $P_{m,max}=0.1, P_{m,min}=0.01$ |
|
| Archive Size $|A|_{max}$ | 100 |
Performance Evaluation and Comparative Analysis
We compare the results of our proposed Multi-Objective Orderly Charging (MOOC) algorithm using Improved NSGA-II against three benchmarks: 1) Uncontrolled Charging (UC): Each battery EV car starts charging at its plug-in time at maximum power until its demand is met. 2) Single-Objective: Min Load Variance (SO-LV): A genetic algorithm optimizing only $f_1$. 3) Single-Objective: Min Cost (SO-C): A genetic algorithm optimizing only $f_2$.
1. Grid Load Flattening: The primary grid-side benefit is load profile smoothing. The table below quantifies the impact.
| Charging Strategy | Peak Load (kW) | Peak Increase over Base (%) | Peak-to-Valley Difference (kW) | Load Variance Reduction vs. UC (%) |
|---|---|---|---|---|
| Base Load (No EVs) | 850 | 0% | 420 | – |
| Uncontrolled Charging (UC) | 1340 | 57.6% | 780 | 0% (Baseline) |
| SO-LV (Min Variance) | 1050 | 23.5% | 460 | 65.1% |
| SO-C (Min Cost) | 1100 | 29.4% | 520 | 55.6% |
| Proposed MOOC | 1020 | 20.0% | 430 | 71.8% |
The proposed MOOC algorithm achieves the most favorable grid performance among all coordinated strategies. It nearly maintains the original peak-to-valley difference of the base load, demonstrating excellent flattening. It outperforms the single-objective SO-LV strategy because the multi-objective framework avoids extreme solutions that might achieve marginally better variance at the expense of unacceptable cost or satisfaction, leading to a more practical and balanced load profile.
2. Economic and Satisfaction Analysis: From the user perspective, we evaluate average cost and satisfaction per battery EV car. For the MOOC algorithm, we select a balanced solution from the Pareto front with moderate values for all three objectives.
| Charging Strategy | Avg. Cost per Vehicle (CNY) | Cost Saving vs. UC (%) | Avg. Satisfaction Index [0,1] |
|---|---|---|---|
| Uncontrolled Charging (UC) | 38.5 | 0% | 0.72 |
| SO-LV (Min Variance) | 32.1 | 16.6% | 0.65 |
| SO-C (Min Cost) | 24.8 | 35.6% | 0.58 |
| Proposed MOOC (Balanced Solution) | 28.9 | 24.9% | 0.81 |
The SO-C strategy yields the lowest cost by aggressively shifting all charging to the valley period, but this comes at the cost of low user satisfaction, as many battery EV cars experience long waiting times. The proposed MOOC strategy strikes an effective balance, providing significant cost savings (24.9%) compared to unmanaged charging while simultaneously achieving the highest average user satisfaction. This is because the satisfaction objective actively manages waiting times and charging completeness, leading to a more acceptable schedule for users of battery EV cars.
3. Algorithm Performance Comparison: We compare the convergence and diversity of the standard NSGA-II and our Improved NSGA-II. The performance is evaluated using the Hypervolume (HV) indicator, which measures the volume of the objective space dominated by the Pareto front approximation (higher is better).
| Generation | Hypervolume (HV) – Standard NSGA-II | Hypervolume (HV) – Improved NSGA-II | Improvement (%) |
|---|---|---|---|
| 50 | 0.415 | 0.462 | 11.3 |
| 150 | 0.582 | 0.648 | 11.3 |
| 300 (Final) | 0.685 | 0.752 | 9.8 |
The Improved NSGA-II consistently achieves a higher Hypervolume throughout the evolution, confirming faster convergence and a superior final Pareto front that covers a larger dominated volume. The adaptive operators and external archive effectively enhance search capability and solution quality for this orderly charging problem.
Conclusion and Future Work
This research presented a comprehensive framework for the orderly charging of battery electric vehicles. A tri-objective optimization model was formulated, explicitly considering the minimization of grid load variance, the minimization of total user charging cost, and the maximization of composite user satisfaction. The model incorporates realistic constraints related to charger power, user time windows, and battery EV car battery dynamics. To solve this complex problem, an Improved NSGA-II algorithm was developed, featuring adaptive genetic operators and an enhanced elite preservation mechanism with an external archive.
Simulation results demonstrated the clear superiority of the proposed multi-objective approach. It effectively flattens the grid load curve, reducing the peak-to-valley difference significantly compared to uncontrolled charging and even outperforming a single-objective load variance minimization strategy by finding a more balanced solution. Simultaneously, it delivers substantial economic benefits to users of battery EV cars while achieving the highest level of user satisfaction among all coordinated strategies. The improvements to the NSGA-II algorithm were validated, showing better convergence and diversity performance.
Future work will focus on several extensions. Firstly, integrating distributed energy resources (DERs) like rooftop solar PV and stationary storage into the optimization model could enable a fully coordinated “Vehicle-to-Grid” (V2G) or “Vehicle-to-Home” (V2H) ecosystem. Secondly, moving from day-ahead scheduling to real-time control that responds to dynamic electricity prices and unexpected grid contingencies is a critical next step. Finally, the user behavior model can be refined using more granular data, and large-scale simulations involving thousands of battery EV cars across diverse geographic regions will be conducted to further stress-test the algorithm’s scalability and robustness.