A Hybrid EV Charging Approach Based on MILP and a Genetic Algorithm

Abstract

Uncoordinated electric vehicle (EV) charging can significantly complicate power system operations. In this paper, we develop a hybrid EV charging method that seamlessly integrates centralized EV charging and distributed control schemes to address EV energy demand challenges. The proposed method includes (1) a centralized day-ahead optimal scheduling mechanism and EV shifting process based on mixed-integer linear programming (MILP) and (2) a distributed control strategy based on a genetic algorithm (GA) that dynamically adjusts the charging rate in real-time grid scenarios. The MILP minimizes energy imbalance at overloaded slots by reallocating EVs based on supply–demand mismatch. By combining full and minimum charging strategies with MILP-based shifting, the method significantly reduces network stress due to EV charging. The centralized model schedules time slots using valley-filling and EV-specific constraints, and the local GA-based distributed control adjusts charging currents based on minimum energy, system availability, waiting time, and a priority index (PI). This PI enables user prioritization in both the EV shifting process and power allocation decisions. The method is validated using demand data on a radial feeder with residential and commercial load profiles. Simulation results demonstrate that the proposed hybrid EV charging framework significantly improves grid-level efficiency and user satisfaction. Compared to the baseline without EV integration, the average-to-peak demand ratio is improved from 61% to 74% at Station-A, from 64% to 80% at Station-B, and from 51% to 63% at Station-C, highlighting enhanced load balancing. The framework also ensures that all EVs receive energy above their minimum needs, achieving user satisfaction scores of 88.0% at Stations A and B and 81.6% at Station C. This study underscores the potential of hybrid charging schemes in optimizing energy utilization while maintaining system reliability and user convenience.

1. Introduction

1.1. Background

According to the World Energy Council (WEC), the global population is projected to increase by 36% by 2050, which is expected to drive a 61% rise in energy consumption [1]. The transport sector is the highest fossil fuel consumer, contributing nearly one-fourth of global energy use and carbon emissions [2]. Among all emissions from the transport sector, nearly 41% is from passenger car usage [3]. According to the WEC, car ownership for mobility purposes will increase by 57% by 2050 [1]. In this context, electric vehicles (EVs) are being promoted as a key solution to reduce carbon emissions from the transport sector and support global decarbonization goals. From these predictions, it is reasonable to expect a significant hike in carbon emissions unless a well-defined policy is adopted.

EVs offer a cleaner and more energy-efficient alternative to traditional internal combustion engine (ICE) vehicles. As a result, EV adoption has experienced rapid growth over the last decade, with global sales reaching approximately 14 million units in 2023. This marked a significant milestone, as EVs accounted for 18% of the total global car sales in that year [2]. It is predicted that EV sales will continue to rise, with projections suggesting sales could surpass 17 million units and 20% of the total market share by 2024. Despite this encouraging trend, the transition to electric mobility has introduced several challenges, particularly regarding the development of adequate charging infrastructure to meet the growing demand. The large-scale EV intake will create serious operational complexity for the power system due to increased demand if not appropriately managed [3,4].

The EV charging facilities can be decentralized, distributed, or centralized [5].

• In decentralized systems, charging depends on users’ preferences, often at home or work, mostly in an uncoordinated way [6]. This type of charging system ensures user satisfaction at the utmost, as they can charge their vehicles whenever they want [3]. Due to this, the load pattern of these types of charging is highly variable. Also, large-scale uncoordinated charging can cause unwanted consequences on the proper operation of the distribution system by creating new peaks [7]. This leads to violations of thermal limits, overloading of feeders and distribution transformers, shortening power equipment lifespan, degrading power quality, and increasing operational cost [3]. These impacts can be broadly categorized into two groups. The short-term operational issues include equipment overloading, voltage deviations, and poor power quality. In contrast, the long-term planning concerns involve infrastructure sizing, demand forecasting, and grid investment under growing EV penetration. Recently, coordinated decentralized frameworks, such as the adaptive multi-agent system in [8], have been proposed to manage EV charging via reactive cooperation among grid and EV agents. However, these approaches often lack explicit user-priority handling, such as slot preferences, SOC target customization, or waiting-time constraints, which are critical in practical deployments.
• In the distributed control system, the charging patterns of the vehicles are managed locally based on the broadcasted message from a central body [5]. This type of charging system provides flexibility and a high level of customer satisfaction [5,9]. However, as vehicle charging is not directly controlled, network stress management vastly depends on customer cooperation in this type of charging.
• In coordinated charging systems, which are also referred to as direct control charging, EV charging is controlled by a primary regulator that analyzes the data of vehicles and charging stations [3]. These charging types are mostly designed for centralized charging facilities and have much more complex operations than uncoordinated and distributed charging systems. One evident benefit of a centralized facility is the reduced charging time due to the ability to accommodate a DC fast charging system [3]. Also, in this type of charging facility, the charging pattern is directly controlled by the regulatory body, which provides more opportunities to encounter unwanted network stress issues. Despite many benefits, these charging facilities may still face difficulties in time frame management, user comfort, and network stress issues due to unpredictable users’ behavior and a high number of charge-seeking vehicles. Therefore, the facilities often integrate some schemes for coordinated EV charging. The valley-filling-based technique is widely adopted in charging policies where lower-demand time slots are more utilized for vehicle charging by regulating tariff plans [7]. The higher-demand time slots are avoided by setting a higher tariff rate. In this way, unwanted peak arrival can be avoided by partly ensuring user comfort.

1.2. Literature Survey and Research Gap

Researchers are actively exploring various EV charging methods. Many are favoring centralized charging schemes due to the development of fast charging systems capable of high-power transfer and reduced charging time. On the contrary, many favor distributed control-based charging schemes because of their better user experience.

Recent studies have explored various day-ahead and valley-filling-based centralized EV charging strategies with different focuses. The research in [3] integrates the vehicle-to-grid (V2G) policy, which relies on the user’s willingness to participate, with the valley-filling-based day-ahead charging scheme. The study also deploys optimization techniques for eliminating EVs before processing the scheduling scheme. The study in [10] proposes a novel valley-filling-based multi-objective optimization structure for charging EVs to regulate the voltage profile and eliminate the possibility of any energy not served in the system. In [4], a day-ahead charging policy for a centralized facility is proposed. The method prioritizes time frame management near-optimally based on EVs’ time of use (TOU), considering traveled distance as a key parameter. Another interesting EV charging methodology is proposed in [11], which connects system operators and EV aggregators through two-level hierarchical control. It uses valley-filling and EV shifting to flatten the load curve and reduce network congestion. A two-stage multi-agent EV charging coordination scheme among different aggregators is proposed in [12] to reduce power losses, voltage deviations, charging costs, and waiting time. It focuses on optimal power distribution among aggregators and shifting EV charging to off-peak hours. Despite these contributions, the studies lack SOC prioritization, priority indexing, variable power transfer, and user-centric EV shifting based on wait time. They also do not consider real-time adjustments, multi-station coordination, or slot allocation reflecting user preferences.

In parallel, several studies focus on distributed EV charging to enhance local decision-making, system resilience, and user flexibility. In [13], a multi-level charging system allows users to control charging time and battery life. The scheme aims to balance user satisfaction with station efficiency. The study in [14] proposes a distributed strategy to improve power sharing between EV charging stations and renewable sources. The algorithm reduces grid dependency, improves energy economics, and enhances load management with better RE utilization. Another study in [5] presents a semi-distributed control strategy for managing EV cluster charging. It handles communication failures, prevents grid overload, and reduces strain by using local renewable energy while balancing demand and protecting transformers. In [15], a distributed scheduling strategy is developed for dynamic EV charging and discharging to minimize grid impact, reduce charging costs, and optimize power flow. The proposed convex optimization-based framework assigns vehicles to charge during off-peak hours with variable power transfer based on grid capacity. The research in [16] proposes a distributed charging approach using a consensus algorithm to improve grid stability, privacy, and fairness. It optimizes charging power based on grid conditions, vehicle-to-vehicle (V2V) communication, and SOC requirements within the available time. However, these studies do not fully address key challenges such as demand curve flattening, load variability, RE uncertainty, and variable power transfer. They also lack user-centric considerations like SOC thresholds, wait-time awareness, priority indexing, EV shifting, and slot selection preferences.

Several studies have introduced hybrid EV charging strategies that combine centralized and distributed control to enhance system coordination and efficiency. The research in [17] presents a hierarchical EV charging control strategy to balance grid stability, user satisfaction, and computational efficiency. The charging method integrates both centralized control and game theory-based distributed control to flatten the load curve, minimize energy costs, and optimize user charging preferences. In [18], a hybrid EV charging scheme combining centralized and distributed control across multiple stations integrates security-centric optimization, load balancing, and price management. By enhancing data privacy, the framework improves charging efficiency, distributes demand evenly, and reduces consumer costs. Other works adopt layered and hierarchical frameworks to guide charging decisions more effectively. The study in [19] uses entropy weighting, analytic hierarchy process, and fuzzy logic to direct EVs to fast charging stations with minimal impact on the distribution and road networks. In [20], a two-layer non-dominated sorting genetic algorithm balances grid load and user cost by coordinating charging power allocation and EV-level scheduling. A similar layered approach in [21] employs deep reinforcement learning, where the upper layer regulates voltage devices and the lower layer manages EV charging to maintain voltage limits. The work in [22] proposes a mobile robot-based adaptive charging network that dynamically switches between cost-optimized and resilience-driven modes using a multi-objective mixed-integer nonlinear programming formulation with mobility, traffic, and power flow constraints. However, these studies do not consider key user-centric aspects such as priority indexing, charging slot preferences, wait-time constraints, and vehicle shifting. They also overlook dynamic power transfer rates and explicit strategies for ensuring SOC satisfaction based on user priorities.

Despite the substantial impacts, the examined studies have shortcomings. According to the authors’ best knowledge, none of the studies mentioned above have explicitly addressed priority indexing-based EV shifting, coordination among multiple charging stations, consideration of wait-time parameters, and minimizing charging current while ensuring user comfort. Also, the prioritization of user comfort in terms of maximum deliverable SOC level after charging has not been considered in the literature. A brief comparison of the examined studies in the recent literature, highlighting their limitations, is illustrated in

Table 1. Highlights of key limitations of the studied publications.

Key Work Areas	[3]	[4]	[5]	[10]	[11]	[12]	[13]	[14]	[15]	[16]	[17]	[18]	[19]	[20]	[21]	[22]	This Work
Centralized control	Y	Y	N	N	Y	Y	N	N	N	N	Y	Y	Y	Y	Y	Y	Y
Distributed control	N	N	Y	Y	Y	N	Y	Y	Y	Y	Y	Y	N	N	N	N	Y
User preference in charging slot selection	Y	Y	N	Y	N	Y	N	N	N	Y	Y	P	P	N	N	N	Y
Policy-driven maximum SOC attainment satisfaction	N	P	N	N	N	N	N	N	N	N	N	N	N	Y	P	N	Y
Network stress	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
Priority indexing of EVs	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	Y
Coordination among multiple aggregators	N	N	N	N	Y	Y	N	N	N	N	Y	Y	N	N	N	N	Y
Minimizing charging current	N	N	N	N	P	P	N	N	Y	N	P	N	N	N	N	P	Y
User satisfaction in waiting time	N	N	N	N	N	P	N	N	N	P	P	P	P	N	N	N	Y
Load curve flattening	Y	Y	P	N	Y	Y	N	Y	Y	N	Y	P	N	P	N	P	Y

Y = Yes; N = No; P = Partly.

.

1.3. Novelty and Contributions

The analysis provided in the previous section highlights the challenges of managing large-scale EV integration within existing grid infrastructure, particularly under uncoordinated charging conditions. To address these challenges, this research proposes a hybrid EV charging system that combines centralized day-ahead scheduling with distributed control mechanisms. The proposed method leverages forecasted energy demand and grid conditions to optimize charging operations while maintaining user satisfaction and grid stability.

The proposed methodology incorporates a valley-filling-based day-ahead scheduling strategy, executed by the principal aggregator, to assign EVs to time slots based on the availability of surplus energy. In cases of overloaded slots, a mixed-integer linear programming (MILP)-based EV shifting operation is executed to reassign vehicles across different stations. The shifting operation is based on the priority index (PI) of the vehicles, which is calculated using drivable distance, waiting time, and initial SOC. This ensures that energy allocation favors higher-priority vehicles during overload resolution.

Following the centralized scheduling, a distributed control scheme is deployed where local agents at each station use a genetic algorithm (GA) to optimize charging power for their assigned EVs. The GA computes charging current and time based on energy availability, EV-specific energy bounds, and waiting time constraints. It runs once for each EV upon its arrival within the preferred time slot to determine the optimal charging current and duration. The optimization process ensures fairness by allocating more charging power to higher-priority EVs while minimizing energy loss and delay. When local energy falls short of meeting EVs’ minimum demands, the system conditionally reassigns affected vehicles to neighboring stations with excess capacity by deploying inter-agent communication.

The key contributions and novelties of this research are as follows:

• We propose a hybrid EV charging scheme combining centralized scheduling and distributed control to ensure user preferences and grid constraints.
• We develop an MILP-based EV shifting scheme that prioritizes high-PI vehicles and redistributes charging load across stations during network stress.
• We integrate a GA-based distributed control strategy that optimizes charging currents in response to time-slot constraints, user-defined preferences, and real-time system availability.

2. Methodology

The proposed hybrid EV charging model integrates key features from both coordinated and distributed control systems. The model incorporates a network of small-scale public charging stations categorized by location, with variable charging outlet types. Each station’s charging requirement is managed by a local agent, which integrates a distributed control scheme. Also, the stations are coordinated by a principal regulator. The principal aggregator broadcasts information to all EVs. Based on this, each vehicle returns its preferences and necessary details, such as station selection, preferred time slot, and other relevant parameters, to the system. The principal aggregator then coordinates with local agents (station-level controllers) by sending the finalized charging schedule. On the actual charging day, local agents manage EV charging based on distributed control. If any local energy shortage arises, inter-agent communication is triggered between neighboring local agents to shift EVs to other stations with surplus capacity, ensuring optimized grid conditions. The conceptual diagram of the proposed scheme is displayed in Figure 1. The summarized communication steps in the proposed charging scheme are as follows:

• The principal aggregator broadcasts station information, allowing EVs to respond with their preferences and parameters.
• The principal aggregator sends charging schedules to local agents.
• Local agents communicate with EVs to control charging based on distributed decisions.
• Local agents coordinate with neighboring agents when energy shortages occur.

Figure 1. Conceptual diagram of the proposed hybrid charging system.

Figure 1. Conceptual diagram of the proposed hybrid charging system.

Detailed explanations of the integrated charging policies are described in the following subsections.

2.1. Centralized Day-Ahead Charging Scheme

This section introduces the centralized day-ahead charging scheme, designed to allocate EVs to optimal time slots ahead of the charging day. The scheme leverages a valley-filling strategy using forecasted demand and surplus energy, and it incorporates an MILP-based EV shifting mechanism to resolve overloads, thereby improving grid stability and enhancing operational efficiency. This section is organized into two parts: (1) the mathematical formulation of the scheduling and shifting strategy and (2) a flowchart that outlines the step-by-step execution of the proposed method.

2.1.1. Mathematical Modeling

In the aggregated valley-filling approach-based centralized charging scheme, the lower-demand time slots are utilized through EV charging. The determination of charging slots for EVs relies on the available surplus energy from the predicted demand curve. The surplus energy in each time slot of the demand curve can be calculated as follows:

$${\mathrm{E}}_{\mathrm{a},n}=\left(P_{\mathrm{s},\mathrm{m}}-P_{\mathrm{d},n}\right)\Delta T_n,$$

(1)

where $E_{\mathrm{a},n}$ is the surplus energy of the nth slot, $P_{\mathrm{s},\mathrm{m}}$ is the maximum supply capacity for the area, $P_{\mathrm{d},n}$ is the demand of the nth slot, and $\Delta T_n$ is the duration of the nth slot in hours. Next, the availability of the nth time slot for charging, ${\mathrm{t}}_{\mathrm{char},n}$ is formulated as follows:

$${\mathrm{t}}_{\mathrm{char},n}=f\left({\mathrm{E}}_{\mathrm{a},n},{\mathrm{T}}_{\mathrm{left}}\right).$$

(2)

Here, ${\mathrm{T}}_{\mathrm{left}}$ is the considered threshold parameter, formulated as follows:

$${\mathrm{T}}_{\mathrm{left}}={\alpha }_n{P}_{\mathrm{d},\mathrm{m}},$$

(3)

where ${\alpha }_n$ is the load variation in the nth slot and $P_{\mathrm{d},\mathrm{m}}$ is the maximum demand of the area for the day.

One key aspect of the proposed centralized charging schemes is to deal with network stress due to EV charging. This key parameter is attained in two steps. First, the methodology determines the required kWh of each EV along with each charging slot of the stations. Second, the overloaded slots are resolved by dispatching minimum energy to each EV and, if required, shifting them to other stations with the hierarchy coordination from the principal regulator. The required kWh of each EV is related to the required SOC increase for the day and is expressed as

$${\mathrm{E}}_{\mathrm{req},k}=\left({\mathrm{SOC}}_{\mathrm{F},k}-{\mathrm{SOC}}_{\mathrm{I},k}\right)·{\mathrm{E}}_{\mathrm{rating},k},$$

(4)

where ${\mathrm{SOC}}_{\mathrm{F},k}$ is the target final SOC, ${\mathrm{SOC}}_{\mathrm{I},k}$ is the initial SOC before charging, and $E_{\mathrm{rating},k}$ is the energy capacity (in kWh) of the kth EV. For fully charged conditions, the value of ${\mathrm{SOC}}_{\mathrm{F},k}$ is 100%. In this research, a minimum deliverable for each vehicle is addressed by dispatching energy to satisfy the possible traveled distance. Hence, under minimum charging conditions, ${\mathrm{SOC}}_{\mathrm{F},k}$ is related to the possible traveled distance of the kth EV and the actual range after considering a discharge threshold. It can be written as follows:

$${\mathrm{SOC}}_{\mathrm{F},k}={\displaystyle \frac{{\mathrm{D}}_k{\gamma }_k}{{\mathrm{E}}_{\mathrm{usable},k}}},$$

(5)

where $D_k$ is the traveled distance of the kth EV in km, ${\gamma }_k$ is the energy consumption rate in kWh/km, and $E_{\mathrm{usable},k}$ is the usable energy of the kth EV, considering a minimum discharge threshold, expressed as

$${\mathrm{E}}_{\mathrm{usable},k}=\left(1-{\mathrm{T}}_{\mathrm{dis}}\right)·{\mathrm{E}}_{\mathrm{rating},k},$$

(6)

where ${\mathrm{T}}_{\mathrm{dis}}$ is the minimum discharge level.

The PI of vehicles in each charging station is formulated by the EV’s present SOC level, their traveled distance, and the waiting time parameter. Specifically, the PI of the kth EV (${\mathrm{PI}}_k$) is expressed as:

$${\mathrm{PI}}_k={\displaystyle \frac{D_k^{\mathrm{rank}}}{W_k^{\mathrm{rank}}{\mathrm{SOC}}_{I,k}^{\mathrm{rank}}}},$$

(7)

where $D_k^{\mathrm{rank}}$ is the traveled distance ranking, $W_k^{\mathrm{rank}}$ is waiting time ranking, and ${\mathrm{SOC}}_{I,k}^{\mathrm{rank}}$ is the initial SOC ranking. The rankings of these three parameters for each vehicle are determined as follows:

$$W_k^{\mathrm{rank}}={\displaystyle \frac{W_k}{{min}_{1\le j\le k}{W}_j}}$$

(8)

$$D_k^{\mathrm{rank}}={\displaystyle \frac{D_k}{{max}_{1\le j\le k}{D}_j}}$$

(9)

$${\mathrm{SOC}}_{I,k}^{\mathrm{rank}}={\displaystyle \frac{{\mathrm{SOC}}_{I,k}}{{min}_{1\le j\le k}{\mathrm{SOC}}_{I,j}}},$$

(10)

where $W_k$ is the waiting time, $D_k$ is the drivable distance, and ${\mathrm{SOC}}_{I,k}$ is the initial SOC of the kth EV.

2.1.2. Flowchart of the Day-Ahead Charging Scheme

Figure 2 shows the flow chart of the proposed day-ahead scheduling method of EV charging in a centralized manner for the principal regulator. The proposed charging scheme performs the following four sequential steps to complete the day-ahead scheduling task.

• Step 1: Charging Slot Information

The methodology acquires day-ahead forecasted demand data. It is assumed that an accurate forecasting method has already been established to predict the day-ahead hourly demand for the area of the charging station. Next, the surplus kWh of each time slot of the day is calculated using Equation (1). In this research, the highest supply from the utility is assumed to be constant. After that, the possible set of charging slots is generated using Equation (2). The nth time slot of the day is the possible EV charging slot if the surplus kWh of that slot is more than a threshold value. The threshold limit is modeled in this study as a multiplicative function of the maximum demand and a constant parameter. The fixed parameter is considered a possible load variation in the actual time of the day in contrast to predicted data. In this study, the value is presumed at 10%. Next, the identified charging slot information of each station is sent to the EV users by the principal regulator so they can select their preferred time slots and charging stations.

• Step 2: Required Final SOC Determination of EVs

In this step, the proposed scheme receives the necessary data from the EV users. The received data include their preferred slot for charging, present SOC level, kWh rating, rated range, waiting time, preferred charging station, and expected travel distance. These data are sequentially processed to determine the possible final SOC of each vehicle that the station can charge. The process involves the following operations:

• Calculation of the usable energy using Equation (6), based on the kth EV’s rated range and a 20% discharge threshold;
• Computation of the required final SOC using Equation (5), based on each EV’s expected travel distance to meet user preference;
• Determination of the required kWh corresponding to an increase in the SOC using Equation (4);
• Estimation of the SOC increase and the corresponding energy required to achieve 100% final SOC for each EV;
• Calculation of total energy demand from all EVs for each charging timeslot.

Subsequently, the framework checks whether the energy required in any slot exceeds the available limit. For slots where demand remains within capacity, all EVs are permitted to charge fully (100% SOC). Conversely, overloaded slots are flagged, and EVs within these slots are allocated only the energy necessary to satisfy their travel distance, instead of full charging. If no overloading is detected, full charging is permitted for all EVs in their preferred slots. This expected SOC information is stored by the method to process the next steps. However, if overloading persists in any charging slot, the proposed scheme initiates an MILP-based EV shifting operation.

• Step 3: MILP-based EV Shifting Operation

In this step, the charging scheme first acquires necessary information, particularly about the identified overloaded slots from Step 2. After that, the method identifies the EVs at the overloaded slots along with their key aspects, such as PI and required energy for charging. Once the necessary information is attained, the MILP is executed to determine a set of vehicles that will be shifted to another charging station and/or charging slot to resolve overloading. In this proposed scheme, the EV shifting operation is performed in such a way that vehicles with higher PI values have priority for charging at their preferred charging stations and slots. The optimization problem for MILP-based EV shifting can be formulated as follows.

Let $y_k\in \{0,1\}$ be a binary variable such that:

$$y_k=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{EV}k\mathrm{remains}\mathrm{in}\mathrm{slot}n\hfill \\ 0,\hfill & \mathrm{if}\mathrm{EV}k\mathrm{is}\mathrm{shifted}.\hfill \end{array}\right.$$

Using this decision variable, the MILP can be formulated as:

$$\begin{array}{cc}\hfill minZ=& \sum _{n\in \mathcal{O}}\left(E_{a,n}-E_{req,n}\right)\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.E_{req,n}& =\sum _{k\in {\mathcal{S}}_n}{y}_k·E_k\hfill \end{array}$$

(12a)

$${\displaystyle E_{a,n}-E_{req,n}>0,n\in \mathcal{O},}$$

(12b)

where $E_{a,n}$ is the available energy at the nth time slot, $E_{req,n}$ is the required energy at the nth time slot, and $\mathcal{O}$ is the set of overloaded charging slots. The variable $E_k$ represents the energy demand of EV k and ${\mathcal{S}}_n$ is the set of EVs originally assigned to slot n. The objective is to minimize the excess energy allocation at overloaded slots by balancing supply and demand across time slots and reallocating EVs to suitable stations. Although the PI is not directly included in the objective, it is considered during the EV selection process to ensure higher-PI vehicles are more likely to be assigned. Once the EVs to be shifted are selected, the aggregator checks other stations one by one and assigns each EV to the first station with enough available energy in the same time slot. This ensures that each EV is moved only once and avoids any repeated or circular shifting. Algorithm 1 outlines the process for the MILP-based EV shifting operation.

Algorithm 1: MILP-based EV shifting operation

2.2. Distributed Control-Based Charging Scheme

In the proposed multi-agent distributed EV charging framework, each charging station operates as a local agent responsible for optimizing the charging current of its assigned vehicles. The primary objective of each agent is to minimize the charging current of the vehicles, thereby reducing charging losses. Each vehicle’s charging power is optimized independently and in parallel by the associated agents. This local optimization process is adaptive and enables necessary adjustments to slot-level energy availability with potential uncertainties in the network. Also, the agents communicate with peers only when local optimization fails, enabling EV reallocation based on network conditions. The integration of GA offers a wider solution space, allowing the system to identify flexible charging strategies that comply with user-defined constraints such as waiting time and minimum energy demand. Additionally, the proposed scheme incorporates a user-defined PI to ensure that charging decisions reflect the relative importance of individual EVs. Figure 3 shows the flowchart of the proposed charging scheme for the local agents of the stations. The proposed charging scheme performs the following sequential steps to complete the task.

• Step 1: Data Initialization

At first, the proposed model receives the necessary data for the EV that arrives at the station for charging. These data include the range of energy required (with a minimum value $E_{\min }$ and maximum value $E_{\max }$), the maximum waiting time ($t_{\mathrm{wait}}$), and the corresponding PI value. Next, the station, which is the local agent, also obtains the value of available energy ($E_{\mathrm{avail}}$) for vehicle charging in that particular time slot.

• Step 2: Local Optimization by GA

Based on the information from the earlier step, the proposed scheme executes the GA to determine the optimum charging power ($P_{\mathrm{charge}}$) and time ($t_{\mathrm{charge}}$). To determine optimal charging power for the vehicles, the GA-based optimization problem is formulated for each EV as:

$${\displaystyle minf\left(P\right)={\displaystyle \frac{P}{\mathrm{PI}}}+\lambda ·s}$$

(13)

$${\displaystyle \mathrm{s}.\mathrm{t}.\hspace{1em}s=max(0,t_{\mathrm{charge}}-t_{\mathrm{wait}})}$$

(14)

$${\displaystyle E_{\min }\le E_{\mathrm{rec}}\le E_{\max }}$$

(15)

$$\sum E_{\mathrm{rec}}\le E_{\mathrm{avail}},$$

(16)

where $\lambda$ is the penalty weight, P is the battery-side charging power, $E_{\mathrm{rec}}$ is the energy received by the EV corresponding to P, $t_{\mathrm{charge}}$ is the charging duration, and $t_{\mathrm{wait}}$ is the maximum allowable waiting time of the EV. This objective formulation enables the GA to prioritize EVs with higher priority indices while discouraging violations of time constraints through a weighted penalty. It allows for adaptive calculation of charging power such that the energy received remains feasible, priority-aware, and system-compliant.

In the GA setup, each solution (a chromosome) represents a possible combination of charging current and time for an EV. The chromosome contains one gene, which is the charging current value. The GA starts with a population of ‘z’ number of solutions and runs for ‘k’ generations to find the best one. It uses MATLAB’s default GA settings: (i) Selection is performed using the stochastic uniform method; (ii) Crossover is scattered, where gene positions are randomly mixed between two parents; and (iii) Mutation is Gaussian, which introduces small random changes. These operations promote population diversity and help the algorithm avoid premature convergence to local optima. Each solution is evaluated using the fitness function from Equation (13), which aims to minimize the charging power and is fairly distributed based on the EV’s priority. The charging power is bounded based on the required $E_{min}$ and $E_{max}$, enforcing lower and upper limits within the feasible search space. This approach ensures that the GA can effectively balance system constraints with user needs.

Algorithm 2 outlines the GA-based procedure for optimizing the charging power of the vehicles.

Algorithm 2: GA-Based Local Optimization

• Step 3: System Constraint Evaluation

After GA-based optimization, the charging station calculates the energy received by the vehicle ($E_{\mathrm{rec}}$) based on the optimized charging power and charging duration. The proposed method then evaluates the system constraint by comparing $E_{\mathrm{rec}}$ against the available energy ($E_{\mathrm{avail}}$) in the relevant time slot(s). If the system constraint is satisfied, the charging scheme finalizes the optimized $P_{\mathrm{charge}}$ and determines the corresponding battery-side charging current ($I_{\mathrm{charge}}$). If the system constraint is violated, the methodology proceeds to conflict resolution in the next step.

• Step 4: Conflict Resolution

If the available energy constraint ($E_{\mathrm{avail}}$) is violated due to the cumulative demand in a given time slot, the system identifies the conflicting slot(s) and applies a resolution strategy. Specifically, the charging power of the vehicles at the overloaded slot is rescaled using a computed factor, such that the total energy consumption in that slot does not exceed $E_{\mathrm{avail}}$. Also, the energy received and the charging current for each vehicle in the slot are updated accordingly. This ensures that the system-level constraints are maintained while still allowing for feasible charging under reduced conditions.

• Step 5: Post-Conflict Assessment

After executing the local conflict resolution, the methodology reevaluates the system-level energy constraints. If the overloading condition is resolved, the scheme proceeds to validate the EV-specific constraints. If both system and EV-level constraints are satisfied, the solution is finalized with the optimized charging power and the corresponding charging time. However, if the EV constraints ($E_{min}$ and $t_{\mathrm{wait}}$) are still violated after local adjustment, the local agent performs the conflict resolution process again to update the charging configuration. If the local station is unable to resolve the conflict due to persistent energy shortages, the system will trigger conditional inter-agent communication, enabling the vehicle to be reallocated to another station with sufficient energy availability.

• Step 6: Conditional Inter-Agent Coordination

When local optimization fails to meet the minimum energy requirement for one or more EVs due to slot-level energy limitations, the system initiates conditional inter-agent coordination. In this step, the charging scheme identifies the EVs that are not optimized locally. If sufficient surplus energy is available at neighboring agents within compatible time slots, the EV is conditionally reassigned to the most suitable agent. Then, the re-optimization takes place at that station, considering these EVs as a new dataset. Each agent conducts this process based on the minimal information exchanged with its immediate neighbors.

3. Results and Analysis

The simulation findings to validate the proposed charging scheme are presented in this section. The proposed charging method is programmed in MATLAB R2021a, executed on a laptop with a 6-core, 12-thread 2.1-GHz CPU and 16 GB of RAM. In this research, the number of EVs considered for charging from the system is 50. The minimum SOC threshold is set at 20% for the EVs that will be charged at the station. Each station includes multiple identical chargers (DC fast charger), with each charger having a maximum power transfer capacity of 150 kW and an assumed average efficiency of 90% [23]. As discussed in the previous methodology section, the proposed scheme encounters charging power regulation, network stress due to EV integration, and user satisfaction. The proposed methodology is tested for the demand curves at the 11 kV system of three different types of locations; two are based on residential areas, and one is for the commercial area. The demand curves for residential areas are obtained from [3] with slight calibration in values. On the other hand, the load curve for the commercial area is assumed in this research. The actual demand curves attained are assumed to be the predicted demand curve for validation in the proposed charging scheme. The system includes three charging stations, labeled Stations A, B, and C. Stations A and B represent residential load profiles, while Station C represents a commercial profile. Each station is managed by a local agent under a central aggregator. The MILP model for EV shifting is implemented manually in MATLAB, while the distributed GA-based control uses MATLAB’s built-in solver. Validation is performed by simulating normal and uncertain scenarios (Cases 1–4) using predicted demand profiles.

3.1. Day-Ahead Charging Slot Identification

Figure 4a represents the day-ahead demand and surplus energy curve for residential area-1 (RA-1), where the maximum supply is near 160 kW and the peak demand is about 152 kW. The surplus energy curve highlights the periods with available energy for EV charging. Based on this, Figure 4b identifies the charging slots where surplus energy exceeds the threshold set by the centralized charging scheme, ensuring that EV charging does not contribute to grid overloading. The corresponding available energy for each identified slot is shown in Figure 4c, with a total surplus energy of 3012 kWh distributed across the selected time slots, offering insights into optimal charging periods.

Similarly, Figure 5a displays the demand and surplus energy curve for residential area-2 (RA-2), where the maximum supply is 106 kW and the peak demand is 101 kW. From this, Figure 5b identifies charging slots for RA-2 based on surplus energy availability, while Figure 5c breaks down the total surplus energy (1820 kWh) across these slots for efficient utilization.

In commercial area-1 (CA-1), the demand and surplus curve in Figure 6a reveals a maximum supply of 88 kW and a peak demand of 80 kW, with surplus energy available during specific time slots. The consumption pattern for the commercial area is almost opposite to the residential areas. Figure 6b highlights the charging slots for CA-1 identified by the principal regulator, ensuring charging is scheduled during surplus energy periods. Figure 6c provides a detailed breakdown of the surplus energy distribution across the identified charging slots, with a total of 1820 kWh available for EV charging.

3.2. EV Parameter Analysis: SOC, PI, and Station Preference

After receiving the charging slot and associated tariff information from the charging station, the EV users select their preferred time slot for charging. In return, the EV users’ necessary information is sent back to the charging facility in the form of present SOC, standard kWh rating, range, preferred charging slot, and possible traveled distance. Figure 7a shows the present SOC level (in percentage) of the considered EVs for charging. The rated kWh of the EVs are displayed in Figure 7b. The ratings are selected from the most popular category in this research. Figure 7c shows the range of the considered EVs expressed in km. Figure 7d is the possible traveled distance of the concerned EVs. Generally, EV users are expected to prefer to charge fully from the station. However, in case of any unavailable circumstances of the station, the charging scheme may have to restrict some of the EVs from acquiring 100% SOC after charging and allowing them to charge at a minimum satisfied level.

In this research, the proposed method addresses this complexity successfully by considering traveled distance as one of the vital inputs from the EV users. In the case of failing to charge fully, the scheme will allow users to charge at a minimum level that covers their possible traveled distance. Figure 8a shows the increase in the SOC level of the EVs under full-charge conditions. The increment values are attained from the consideration that all the EVs will be allowed to charge fully at their preferred charging slot. The proposed charging scheme prioritizes user satisfaction at the utmost level. Hence, it considers allowing the EVs to have a 100% SOC level after charging if other stress factors are managed. The charging scheme then calculates the TOU and required kWh based on the SOC increase to assess the overloading in the network. Figure 8b shows the determined required kWh of the EVs under full charge conditions. Also, the proposed charging centralized scheme considers the worst-case scenario, that, at some of the charging time slots, allowing EVs to be fully charged may not be possible. To address this, the proposed scheme considers charging the EVs up to a minimum level to cover their possible traveled distance for the day only. Figure 8c shows the required increase in SOC level from the present values, considering the traveled distance. In contrast to Figure 8a, some of the EVs (the 14th and 44th EVs) have zero increments in the SOC level. This is due to the fact that these EVs already have enough present SOC to cover their traveled distance. Therefore, the proposed charging scheme identifies that these EVs do not require additional charge under the worst-case scenario. The required kWh associated with the increased SOC level is shown in Figure 8d.

Figure 9a shows the charging station preferences of EVs, where values of 1, 2, and 3 represent Station A, Station B, and Station C, respectively. The EVs exhibit a varied distribution of preferences, highlighting the demand allocation across the three stations. On the other hand, Figure 9b illustrates the PI of EVs for these three different stations. Higher PI values suggest a greater preference for charging on the present day in terms of current sharing and further shifting if supply–demand variability creates some uncertain conditions.

3.3. EV Redistribution via MILP Optimization

Figure 10a presents the available energy in each charging slot at Station A after all EVs are fully charged, where negative values indicate overloading in specific time slots. Following this, Figure 10b shows the energy availability at Station A when the charging condition combines both fully charged and minimally charged EVs. This strategy significantly reduces overloading, as evident from the fewer negative values in the energy distribution.

For Station B, Figure 11a shows the energy availability after EVs are fully charged, with certain slots experiencing overloading, as indicated by negative values. When integrating both full and minimum charging conditions, Figure 11b reveals improved energy distribution, reducing overloading at Station B.

Similarly, for Station C, Figure 12a depicts the available energy for fully charged EVs, with overloading in some slots, while Figure 12b shows the energy distribution when combining full and minimum charging conditions, effectively mitigating overloading. Across all stations, the comparison between full-charged and combined-charged scenarios demonstrates the effectiveness of the proposed strategy in optimizing energy distribution and alleviating overloading in charging slots.

Based on the observed overloading in Station A after implementing the combined full and minimum charging strategy, the methodology deploys an EV shifting operation to alleviate the overloading. Figure 13a identifies the EVs located in the overloaded charging slots of Station A. To resolve the overloading, the methodology selects EVs for shifting to other time slots or stations based on predefined priority criteria. Figure 13b displays the identified EVs chosen for shifting from the overloaded slot of Station A. This operation is performed to redistribute the charging load and ensure no slot exceeds its capacity. The figure indicates the effective identification of a minimal subset of EVs whose shifting would resolve the overloading.

Figure 14a shows the station preferences of EVs before the shifting operation, with values of 1, 2, and 3 representing Station A, Station B, and Station C, respectively. This figure highlights the initial distribution of EV preferences, including the overloaded slots at Station A. After implementing the EV shifting operation, Figure 14b presents the updated station preferences of the EVs. The redistribution of EVs is evident, with certain vehicles (32 positioned, highlighted in red) shifted from Station A to other stations, effectively resolving the overloading issue at Station A.

With the centralized EV reallocation completed, the finalized schedule is transmitted to local agents, who then manage real-time charging operations using a distributed control framework.

3.4. Distributed Charging Control by Local Agents

On the charging day, each local agent at the stations further optimizes the charging current for individual EVs in real time using the GA-based distributed control scheme. This method considers available energy, priority indices, and user-defined constraints to ensure efficient and fair power allocation. The GA uses MATLAB’s default settings (population = 50, generation = 100, crossover = 0.8, mutation rate = 1), which are sufficient due to the problem’s low complexity and can consistently yield stable results.

Figure 15 summarizes the charging outcomes at Station A by the proposed distributed framework. As shown in Figure 15a, the delivered charging current varies across EVs, based on energy requirements and priority indices. Figure 15b presents the delivered energy per charging slot at the station. Figure 15c depicts the difference between delivered and minimum required energy for each EV, where most EVs receive more than their minimum energy demand, benefiting from the surplus energy when available. Lastly, Figure 15d demonstrates a positive correlation (upward trend) between the EV priority index and allocated charging power. The outcome confirms that, despite variations in energy requirements and system-level constraints, the proposed charging strategy preferentially allocates greater charging power to higher-priority EVs.

Extending this analysis, Figure 16 summarizes the charging outcomes at Station B. Similar trends are observed, where Figure 16a shows that delivered charging currents vary across EVs based on energy requirements and priority indices. Figure 16b highlights the delivered energy distribution across charging slots. In Figure 16c, most EVs receive more than their minimum energy requirements, benefiting from surplus energy availability and adjustments due to slot conflicts. Figure 16d also shows a positive correlation between the EV priority index and allocated charging power, further confirming that higher-priority EVs are assigned greater charging power by the proposed strategy.

Similarly, Figure 17 presents the charging outcomes at Station C. Consistent with the other stations, Figure 17a shows that the proposed scheme delivers varying charging currents across EVs to minimize the charging loss. Figure 17b highlights the non-uniform delivered energy distribution across charging slots. As depicted in Figure 17c, most EVs meet or exceed their minimum energy requirements, based on the surplus energy. Figure 17d reveals a positive correlation between the EV’s PI and allocated charging power, again validating that higher-priority EVs are assigned greater charging power under the proposed strategy.

From the attained outcomes (Figure 15, Figure 16 and Figure 17), it can be observed that the proposed distributed charging scheme effectively optimizes charging currents across the EVs while considering individual energy requirements, constraints, and priority indices. It ensures that most EVs meet or exceed their minimum energy demands, even under system-level constraints and slot conflicts. Additionally, a strong positive correlation between the EV priority index and the allocated charging power is observed, validating that higher-priority EVs consistently receive more charging power.

To further evaluate the robustness of the proposed charging scheme, the following three additional case studies are investigated.

• Case 1: Arrival of new EVs at the station
• Case 2: Absence of scheduled vehicles for charging
• Case 3: Injection of Gaussian forecast errors (5% and 10%) into day-ahead demand
• Case 4: Reduction in surplus energy at certain charging slots

• Case 1:

To simulate Case 1, three new vehicles are assumed to arrive at Station C for charging. The newly arrived EVs are assigned similar properties, including energy required range, maximum waiting time, and PI. These values are set arbitrarily to create the case. Additionally, charging time slots for these vehicles are randomly selected from the available slots.

Figure 18 presents the simulation results for Station C. Figure 18a shows the charging current delivered to individual EVs after the arrival of new vehicles. Figure 18b illustrates the variation in charging current compared to the base case. It is observed that the proposed scheme optimizes the charging current for all the vehicles, with minimal adjustments from the base case scenario. Also, Figure 18c displays the variation in energy received relative to the base case, confirming that the proposed scheme accommodates the new EVs with only minor impacts on the existing vehicles’ energy fulfillment (with energy reductions observed for only three vehicles).

• Case 2:

To simulate Case 2, a subset of the previously scheduled EVs is assumed to be absent from the charging process at Station C. Three randomly selected vehicles are removed arbitrarily to represent the case scenario. The remaining EVs proceed through the optimization process based on predefined parameters.

Figure 19 presents the simulation outcomes for the new case scenario at Station C. Figure 19a shows the delivered charging current to the EVs after the absence of three scheduled vehicles. Figure 19b illustrates the variation in charging current compared to the base case. It is observed that two vehicles now have higher charging current than the base case. Due to the absence of some vehicles, the proposed charging scheme can now provide higher current to the vehicles in certain slots due to relaxation in slot-level conflict. Subsequently, these vehicles will be able to receive more energy within their predetermined range, which is shown in Figure 19c. This scenario validates how the proposed charging scheme adapts to sudden reductions in the charging population.

• Case 3:

To simulate Case 3, Gaussian forecast errors of 5% and 10% are injected by reducing the surplus energy at each time slot in Station C. This reflects real-world uncertainty in demand forecasts. The distributed GA optimization then proceeds using the updated energy values.

Figure 20 shows the simulation results for Case 3, where Gaussian forecast errors are introduced at Station C. Figure 20a,b illustrate the changes in charging current and energy received compared to the base case under a 5% forecast error. Only one EV shows a reduction in current and received energy due to the small decrease in surplus energy. In contrast, Figure 20c,d show the impact under a 10% forecast error. Here, three EVs experience noticeable reductions in both charging current and received energy. These drops occur due to increased slot-level conflicts caused by reduced available energy. This case shows that the system is robust to forecast errors while still prioritizing high-PI EVs and meeting energy needs.

• Case 4:

Lastly, to simulate Case 4, the surplus energy available at certain charging slots in Station B is artificially reduced (the energy at three slots is forced to zero) to simulate overload. In this extreme operating scenario, failure in the local optimization initiates inter-agent coordination to reallocate EVs at the overloaded slots. To demonstrate this reallocation process with minimal information exchange, a distributed network of ten stations (labeled A to J), including stations A, B, and C, is simulated. Each station agent communicates only with its immediate neighbors when local energy availability during required slots is insufficient to meet EV demands. The EVs proceed through the optimization process considering these reduced energy conditions while maintaining their predefined energy requirements, waiting times, and priority indices. This scenario validates the inter-agent coordination process by reallocating some vehicles and re-optimizing their charging schedules.

Figure 21 presents the simulation outcomes for Case 4 involving inter-station EV shifting. Figure 21a shows the charging current assigned to each EV before any shifting occurred. As some slots in Station B are forced to zero, the local optimization fails to determine feasible charging currents for the vehicles assigned to those slots due to extreme system-level constraints. Therefore, the local agent of Station B considers reallocating these vehicles to other stations. Figure 21b illustrates the energy-sharing topology used during this coordination process. A unidirectional tree structure is shown, where Station B communicates only with its neighboring stations (A, D, E, and F) to assess surplus energy availability during the overloaded slots. The edge labels indicate the amount of transferable energy (in kWh), which is used to identify the best-fitted neighboring station. In this case, Station A is identified as the best station (highest available energy) to accommodate shifted vehicles from Station B. Figure 21c displays a binary shift indicator highlighting the EVs that are reassigned from Station B to Station A due to insufficient energy availability in certain slots. Upon their arrival at Station A, the local optimization process determines the charging currents and energy allocations for the shifted EVs, treating them similarly to newly arriving vehicles as in Case 1. After relocation and re-optimization, the finalized charging currents for all vehicles across the three stations are shown in Figure 21d.

The simulation results across all four case scenarios demonstrate the robustness and adaptability of the proposed distributed charging scheme. The system effectively handles new EV arrivals, adapts to the absence of scheduled vehicles, mitigates energy shortages through inter-station EV shifting, and responds to forecast errors in the demand curve, ensuring reliable performance under dynamic and uncertain conditions.

3.5. Performance Assessment of the Proposed Charging Scheme

The overall charging performance of the proposed hybrid scheme is evaluated based on several key metrics, including the average-to-peak demand ratio, energy fulfillment levels, and the user satisfaction index (USI), defined as:

$$\mathrm{USI}={\displaystyle \frac{100}{N}}\sum _{j=1}^N{\displaystyle \frac{E_{\mathrm{rec},j}}{E_{max,j}}},$$

(17)

where $E_{\mathrm{rec},j}$ is the energy actually received by the jth EV, $E_{max,j}$ is the maximum energy requested by jth EV, and N is the total number of EVs. USI quantifies how well the delivered energy satisfies the requested maximum energy by the EV users.

Table 2. Performance metrics before and after EV integration at Stations A, B, and C.

Performance Metrics	Station A	Station B	Station C
Avg-to-Peak Demand Ratio (Before EVs)	61%	64%	51%
Avg-to-Peak Demand Ratio (After EVs)	74%	80%	63%
EVs with $E=E_{min}$	19%	31%	31%
EVs with $E>E_{min}$	81%	69%	69%
Avg Charging Current (A)	48.75 A	46.65 A	51.5 A
USI	88.0%	88.0%	81.6%

summarizes these performance metrics before and after EV integration across the three charging stations (Station A, Station B, and Station C). After EV integration, all stations experience an increase in their average-to-peak demand ratios, indicating improved utilization of available charging capacity (74%, 80%, and 63% for Stations A, B, and C, respectively). The percentage of EVs that received only their minimum required energy is lowest at Station A (19%), suggesting that most EVs receive more than their minimum demands. In contrast, 31% of EVs at Station B and Station C only met the minimum threshold. For EVs receiving energy greater than the minimum requirements, Station A again performs better (81%), followed by Station B and Station C (69%). The average charging current across the three stations range from 46.65 A to 51.5 A, suggesting that the charging load is balanced fairly consistently despite variations in EV profiles and energy constraints.

The simulation results show that the USI remained high at both Station A and Station B (88.0%), whereas Station C showed slightly lower satisfaction (81.6%), likely due to more constrained energy availability. These outcomes collectively demonstrate the effectiveness of the proposed charging optimization framework in ensuring minimum energy fulfillment, promoting fairness, and maintaining system-level operational balance under dynamic conditions.

To test the effectiveness of the proposed method, four key performance indicators were evaluated: peak-to-average ratio (PAR), transformer loading, energy not served (ENS), and average charging time. These metrics provide a balanced view of grid stress, energy delivery reliability, and user satisfaction. ENS is measured by comparing the maximum energy required by each EV with the actual energy received. These metrics are compared across uncoordinated charging, valley-filling with MILP-based scheduling, distributed GA, and the proposed hybrid method. A brief overview of each baseline strategy is given below:

• In the uncoordinated charging method, each EV is randomly assigned to a charging slot without any coordination across vehicles. Charging begins immediately in the assigned slot, and the available energy in that slot (up to 120% of the transformer’s base capacity) is equally shared among all assigned EVs. The charging power for each EV is capped by its charger’s maximum limit and is further adjusted to avoid exceeding the EV’s energy requirement or the system’s capacity constraints.
• In the valley-filling strategy with MILP-based scheduling, charging slots are pre-assigned to EVs by a centralized controller to spread the load across low-demand periods and prevent network overloading. Since this is a day-ahead strategy, it also accounts for uncertainties in actual demand caused by forecasting errors on the charging day. During real-time charging, each slot’s available energy is equally shared among the assigned EVs. The final charging power for each EV is limited by its charger capacity, assigned charging duration, and expected energy demand.
• In the distributed GA-based strategy, each EV independently minimizes its charging current while staying within its allowed waiting time. Decisions are based solely on the EV’s status, without using priority indexing or energy limits determined in the MILP-based approach. Each EV receives a random start time and charges continuously until its energy demand is met or its waiting time expires. If the total demand in any slot exceeds the transformer’s capacity, EV charging powers are proportionally scaled down to prevent overloading.

The analysis focuses on Station A, but similar results are expected for other stations.

Table 3. Key performance metrics for different EV charging strategies at Station A.

Key Performance Indicators	Uncoordinated	Valley-Filling MILP Only	Distributed GA Only	Hybrid (Proposed)
Peak-to-Average Ratio (PAR)	1.76	1.49	1.51	1.35
Max Transformer Loading (%)	120.00%	100.62%	100.00%	94.69%
Energy Not Served (ENS) [kWh]	203.61	132.04	259.81	70.84
Average Charging Time [min]	93.09	90.00	88.4	86.43

presents the corresponding results. The proposed hybrid method delivers the best overall results. It achieves the lowest PAR (1.35) and transformer loading (94.69%), indicating a smoother load profile and reduced grid stress. The uncoordinated method performs the worst, with the highest PAR (1.76), transformer loading (120.00%), and a high ENS of 203.61 kWh due to the lack of scheduling or control. The valley-filling method using MILP improves upon this by reducing ENS to 132.04 kWh and lowering transformer loading to 100.62%, but it still falls short of the hybrid method in all key metrics. The distributed GA strategy provides better load flattening than the uncoordinated approach but leads to the highest ENS (259.81 kWh), indicating that many EVs cannot receive sufficient energy. Despite these trade-offs, the hybrid method not only ensures that all EVs meet their energy needs within the allowed time but also maintains the lowest average charging time of 86.43 min. Overall, it offers the most balanced and effective performance across all key indicators.

To test the consistency of the proposed method under input uncertainty, a Monte Carlo experiment with 100 trials was conducted at Station A as a representative case. As summarized in

Table 4. Parameter settings used for Monte Carlo simulation (100 trials, 21 EVs each).

Parameter	Distribution/Range
Arrival slot	Weibull distribution with shape = 2, scale = 12
Battery capacity (kWh)	Uniform distribution over [20, 70]
Initial SOC	Uniform distribution over [0.2, 0.5]
Maximum waiting time (min)	Uniform discrete distribution over [60, 120]
Number of simulations	100

, each trial involved 21 EVs with randomized arrival slots (Weibull distribution), battery capacities, initial SOC, and waiting times (uniform distributions). Key performance indicators were recorded in each trial, and 95% confidence intervals (CIs) were calculated to assess the robustness of the results.

Table 5. Performance summary of the proposed method with 95% CIs for key indicators.

Performance Metric	Mean	95% CI Lower	95% CI Upper
ENS (kWh)	72.31	68.47	76.15
Avg charging time (min)	89.92	89.15	90.68
PAR	1.4425	1.4147	1.4703
Max. transformer loading (%)	89.18	87.33	91.02

shows that the proposed method maintains consistent performance under input uncertainty, with narrow CIs across all KPIs. The ENS fluctuates between 67.4 and 78.6 kWh, aligning closely with the deterministic ENS of 70.84 kWh in Table 3. The average charging time varies from 88.3 to 90.9 min, consistent with the earlier result of 86.43 min. Similarly, the mean transformer loading remains near 90% across trials, closely aligning with the 94.69% observed in the base case. These results demonstrate the robustness of the proposed method across different EV input scenarios.

3.6. Performance Assessment of Optimization Algorithms

To assess the optimization performance, this study presents the MILP solve time and GA convergence behavior. The MILP is executed over five independent runs to evaluate the consistency in computation time. Meanwhile, the GA convergence curve illustrates how the fitness value improves over generations for the best-performing EV. Figure 22a shows the MILP computation time over five independent runs for a fleet size of 50 EVs. The outcome demonstrates consistent and stable runtime. Figure 22b displays the GA convergence curve for the best-performing EV (ID = 37) at Station C as a representative case, where the fitness value improves steadily and stabilizes within approximately 30 generations. This indicates reliable convergence behavior. Additionally, Figure 22c shows the GA runtime for each station over five runs. The computation time is low and steady, indicating that the local optimization works efficiently at all stations.

4. Discussion

The proposed hybrid EV charging framework, combining valley-filling-based centralized scheduling with distributed GA optimization, demonstrates robust operational and user-level performance. Coordinating EV allocation across different charging stations ensures grid-level efficiency and user energy satisfaction. The model includes EV-specific factors such as SOC, waiting time, drivable distance, PI, energy bounds, and flexible charging durations. A valley-filling MILP scheduler assigns time slots based on low-demand periods one day ahead, while local GA-based control optimizes charging currents in real time upon vehicle arrival. The day-ahead centralized policy, controlled by a principal aggregator, provides the first level of grid-stress alleviation through scheduling at lower demand slots. On the other hand, the proposed distributed control further minimizes the charging power to the vehicles based on their constraint, PI, and the system’s available energy. Even under a worst-case scenario, the scheme enables the agents to communicate with neighbors and resolves any network stress by reallocating some vehicles based on the minimum information exchanged. This ensures energy fulfillment for all EVs and maintains operational balance without centralized intervention during real-time fluctuations. The model is evaluated under four scenarios: new EV arrivals, EV absences (robustness check), forecast errors (real-world uncertainties), and energy shortages (representing stress conditions). In all cases, the proposed method maintains stable system operation, ensuring minimum energy fulfillment through local adaptation.

The GA is chosen for the distributed optimization layer due to its strong global search ability and robustness against getting stuck in local minima. Unlike swarm-based metaheuristic algorithms such as particle swarm optimization (PSO), which may converge prematurely in complex or constrained problems, GA maintains solution diversity through crossover and mutation. This is particularly valuable in EV charging, where nonlinear constraints and shifting feasible regions depend on energy availability and user preferences.

Simulation outcomes show a consistent improvement in average-to-peak demand ratios after EV integration, increasing to 74% at Station A, 80% at Station B, and 63% at Station C area, indicating better system capacity utilization and alignment with load flattening objectives. In contrast to centralized or decentralized approaches, the proposed hybrid scheme balances the local and global coordination, enabling satisfactory energy allocation even under resource limitations. Notably, Station A displays the highest user energy fulfillment, with 81% of EVs receiving more than their minimum required energy and only 19% limited to the minimum. In comparison, Station B and Station C each have 69% of EVs receiving above-minimum energy, while 31% are limited to the minimum, reflecting the effect of local energy constraints.

When compared to prior studies, this work provides a cohesive mechanism for multi-layered EV coordination. The communication between agents, guided by surplus energy availability and priority indices, further enhances the robustness of the proposed approach. Both Station A and Station B achieve high USI values (88%), while the slightly lower value at Station C (81.6%) reflects the impact of localized energy limitations.

Each algorithmic component contributes directly to the overall performance. The valley-filling strategy schedules EV charging during low-demand hours, which helps prevent network stress and flattens the overall load curve. The PI ensures that EVs with higher priority receive more charging power and are also favored during reallocation when charging slots are limited. The penalty term in the GA makes sure that EVs are charged within their allowed waiting time. Together, these components work in coordination to ensure fair and efficient charging at all stations.

This study assumes fixed energy availability and predefined EV profiles, which may not fully reflect real-world uncertainties. The model also does not include features such as V2G, V2V energy exchange, dynamic pricing, or renewable energy coordination. On the method side, the MILP is used only by the centralized controller to shift EVs to resolve network stress one day ahead, not for full schedule optimization. Since the MILP is implemented by the authors rather than solved using commercial solvers, there is room for further improvement. The GA used in local optimization favors flexibility and speed but may produce near-optimal results. The combination of MILP and GA adds some complexity to the model, which may require extra tuning in larger systems. However, the hybrid method performs better across key metrics when compared to uncoordinated charging, valley-filling with MILP-only, and GA-only strategies. Future work can focus on enhancing the model by incorporating real-time uncertainty, dynamic inputs, V2G/V2V interactions, real-time pricing signals, and coordination with distributed energy resources.

5. Conclusions

This study presents a hybrid EV charging framework that integrates valley-filling-based centralized scheduling with distributed GA-based real-time optimization. By combining day-ahead scheduling with local adaptability, the model ensures both grid-level efficiency and user energy satisfaction. Simulation results across three charging stations validate the effectiveness of the proposed approach, with improvements observed in average-to-peak demand ratios and high user satisfaction indices. The framework remains stable under dynamic scenarios, confirming its robustness and adaptability. Coordinated EV reallocation and slot-level conflict resolution further enhance energy fulfillment. While the current implementation is based on deterministic inputs, future extensions may incorporate V2G and V2V energy exchange, real-time pricing, and renewable coordination to support broader smart grid applications.