Optimal Sizing of Electric Vehicle Charging Stacks Considering a Multiscenario Strategy and User Satisfaction

Abstract

The rapid growth of EVs relies on the development of supporting infrastructure, e.g., charging stations (CSs). The sizing problem of a CS typically involves minimizing the investment costs. Therefore, a flexible and precise sizing strategy is crucial. However, the existing methods suffer from the following issues: (1) they do not consider charging station deployments based on the charging stack; (2) existing sizing strategies based on smart charging technology consider a single scenario and fail to meet the demand for flexible operation under multiple scenarios in real-life situations. This paper proposes a novel CS sizing framework specific for charging stacks to overcome these challenges. Specifically, it first addresses the charging-stack-based CS sizing problem, and then it proposes the corresponding multiscenario constraints, i.e., exclusive and shared, for capacity-setting optimization. In addition, a novel quality of service (QoS) formulation is also proposed to better relate the user QoS levels to the CS sizing problem. Finally, it also explores the relationship between the investment costs and the total power of the needed charging stack under three business models. Extensive experiments show that the proposed framework can offer valuable guidance to CS operators in competitive environments.

1. Introduction

In the quest for smart and sustainable urban transportation, the shift to electric mobility is accelerating in many developed countries [1]. Industry reports indicate that electric vehicles constituted 2.6% of worldwide car sales and approximately 1% of the overall vehicle population in 2019, with more than 7.2 million EVs on roads. To facilitate this growth, a significant increase in charging infrastructure is needed, with projections pointing to a need for approximately 130 million private and 13 million public chargers by 2030 to satisfy the charging demands of EVs. As a result, multiple policy measures are being adopted to stimulate investment in this burgeoning sector. The greatest opportunity for reducing greenhouse gas emissions from EVs as opposed to conventional vehicles is during the usage phase, specifically when the vehicle is being charged. This reduction is maximized when the EV relies predominantly on renewable energy sources and the charging infrastructure is advanced with smart technology [2]. Consequently, there is a drive to develop sustainable commercial models for investment in CSs, prompting in-depth analysis that combines technological and economic factors. A key consideration in CS investment is determining the optimal size, which includes the quantity and variety of chargers to install. Additionally, progress in information and communication technologies enhances data acquisition, which is crucial for better integrating EVs into smart grids. The collected data and their aggregation facilitate the optimization of charging operations, known as smart charging [3,4]. Smart charging systems can enhance the operational efficiency of charging stations, meet the charging needs of various users, and ensure the reliability and safety of the charging process through efficient management and control strategies. These research areas are inter-related and collectively promote the development and application of intelligent charging technology. The literature distinguishes three primary research categories, where the first addresses planning and sizing problems for CSs, the second delves into intelligent charging stacks and the third considers the sizing problem of a CS while also accounting for the resulting QoS.

1.1. Charging Station Sizing

Several researchers have explored this topic by investigating various aspects of CS sizing strategies. For example, Simorgh et al. [5] modeled the problem of the optimal siting and sizing of EVCSs as an integer optimization problem with the objective of minimizing the investment cost, connection cost, total power loss cost and demand response cost; then, they solved the problem by using a particle swarm optimization algorithm. Luo et al. [6] considered several types of EV charging facilities (slow, medium and fast) with the aim of finding the optimal number of each type of EV charging facility. They also proposed a scenario-based two-stage optimization model and transformed it into mixed-integer second-order conic programming. Meng et al. [7] proposed an operational model for electric bus CSs, optimizing the allocation of charging facilities with an M/M/s/K queuing model to manage peak load and reduce electricity costs. Liu et al. [8] proposed a user equilibrium model for optimizing the placement and sizing of EVCSs to minimize investment costs, using a derivative-free optimization approach. Zhao et al. [9] proposed an optimization model for public EVCS capacity planning that maximized the fuzzy quality of service by considering queuing behavior, reliability and multiple charging options. Hayajneh et al. [10] developed a two-stage optimization model. In the first stage, a genetic algorithm is used to assign each EV to a specific energy consumption interval to minimize the total energy loss of all EVs. The second stage calculates the cumulative energy demand of the EVs assigned to each charging station based on the state of charge of the EV batteries. Then, the optimal number of fast and slow EV charging facilities at each station is determined through a linear optimization problem. Khaksari et al. [11] defined an optimization framework to solve the EVCS sizing problem. In their approach, the investment cost of the charging station operator is minimized while achieving a certain level of service quality for its customers. They also considered intelligent charging scenarios during operation and proposed a new formulation for the quality of service by using opportunity constraints.

These studies provide valuable insights into the sizing of charging stations. However, there is a need for further research into the specific requirements and considerations of intelligent charging stacks, which have adaptive power deployment capabilities and require optimized capacity allocation. The adaptability and flexibility of these charging devices present unique challenges and opportunities for sizing strategies, while the existing EVCS sizing studies do not consider this application scenario. Therefore, this study addresses this gap and provide insights into the optimal planning and allocation of CSs, which are specifically tailored for the scenario of intelligent charging stacks.

1.2. Intelligent Charging Stack

The electric vehicle industry has experienced significant growth and development over the past several years [12]. With the increasing number of electric vehicles, the construction and improvement of charging infrastructure has become an important issue. All-in-one charging stacks were once the mainstream choice, and centralized all charging equipment, control systems and cables in a single structure. However, with the expansion of the electric vehicle market and the diversification of user needs, some limitations of all-in-one charging stacks have gradually been revealed. First, due to its fixed structure and layout, the all-in-one charging post cannot flexibly respond to the needs of different regions and locations. For example, one area may need more charging equipment, while another area may need a more powerful charging station. This leads to limitations in the layout and scale adjustment of all-in-one charging stacks, which cannot effectively meet the needs of different regions. Second, there are some challenges in the troubleshooting and maintenance of all-in-one charging stacks. When a module fails, the entire charging stack may need to be removed from service, which can lead to increased outage time and maintenance costs [13]. The design of the one-piece charging stack may not be able to adapt to future technological upgrades and the requirements of new charging standards. This means that, over time, a one-piece charging stack may become obsolete and need to be replaced or undergo a large-scale technology update. For these reasons, in recent years, the domestic and international electric vehicle industry has begun to shift toward split charging stacks. Split charging stacks can be better adapted to the needs of different regions and locations through modularized design, providing a more flexible and scalable charging solution. In addition, the modular structure of the split charging stack also makes maintenance and upgrading more convenient and can be better adapted to future technological development and changes [14,15]. Overall, the move from a one-piece charging stack to a split charging stack at home and abroad is primarily intended to meet the growing market demand for electric vehicles, to provide a more flexible, scalable and reliable charging infrastructure and to adapt to future developments in electric vehicle technology.

From an operational point of view, first, the charging stack in split form can be modularized and combined according to demand, which is convenient for flexible layout and scale adjustment [16]. Operators can choose a single module or multiple modules to build charging stations according to the actual situation to adapt to the needs of different regions and places to diversify the risk of investment. Second, the charging stack in split form can be installed and expanded on demand. Operators can invest gradually according to market demand, reducing the risks associated with large one-time investments [17]. This helps to control costs and allows for further expansion once market demand stabilizes. Finally, split-form charging stacks typically have smaller unit capacities, which means that when a unit fails, it needs only to be repaired or replaced without affecting the operation of the entire charging station. This reduces maintenance costs and increases operational efficiency [18]. From the customer’s point of view, first, the split form of the charging stack allows the charging equipment to be installed in different locations and sites, such as parking lots, commercial areas, and residential areas. This means that it is easier for customers to find a charging device close to them and to charge quickly and conveniently. Second, even if one of the units fails, the other units can still operate normally, ensuring that customers can continue to use reliable charging equipment [19]. The scenario based on the charging stack can effectively reduce charging time and cost and can improve the charging efficiency and experience through the intelligent management and optimization of charging strategies. For the promotion of EVs and the increasing demand from users, the charging stack scenario has important promotion significance and market potential. However, in previous studies, charging stacks and intelligent management have not been integrated, which is also one of the main contributions of this paper.

1.3. QoS at Charging Stations

Service quality is a key performance indicator for assessing the operational capacity of charging stations, as it implicitly quantifies the acceptance of the proposed CS services from the end-user perspective [20]. Therefore, improving the service quality of charging stations is an important way to promote the sustainable development of the electric vehicle industry [21]. Some researchers tend to evaluate the performance of charging facilities, e.g., Xiao et al. [22] conducted a study evaluating the performance of charging facilities. They considered end-user satisfaction with QoS and the planning cost budget. Their aim was to determine the optimal locations of the CS, the optimal number of EV charging facilities installed at each CS and the optimal maximum allowable queue length and maximum capacity of each CS. This evaluation aimed to improve end-user satisfaction while considering the cost implications. Choi et al. [23] proposed a queuing model and two congestion control policies based on EV queue length thresholds. They conducted a comprehensive analysis of CS scheduling costs to determine the favorable conditions for applying each proposed QoS-aware congestion control policy. However, the authors did not jointly model the proposed QoS-aware schemes with the CS planning problem. Instead, they provided numerical results for specific case studies. Some studies have indirectly improved the QoS level of CS, such as [24], which proposed a pricing strategy and charging management method for electric vehicle auxiliary charging stations and electric vehicle users based on the Stackelberg game. After analysis, it can be concluded that the Stackelberg game model can not only improve the operating income of operators but also reduce the charging costs of EV users, making it a very effective and orderly charging strategy. Antoun et al. [25] refined the EVCS state of charge (SoC) by analyzing real usage data, introducing a dynamic SoC charging model and implementing a queuing system to minimize wait times. The approach assisted operators in optimizing station sizing to enhance customer service. Palani et al. [26] introduced a GEO-RBFNN hybrid method for optimizing the size and smart charging capabilities of an EVCS to enhance QoS. It minimized operational costs and ensured efficient service for EV owners through smart scheduling and probability-based task parameter approximation. Vandet et al. [27] developed optimizing the sizing of CSs by leveraging travel diary data and an information-sharing system, applying queuing theory to predict wait times and dynamically adjusting charging facilities to enhance charging service efficiency and user satisfaction.

In this series of studies, QoS is primarily modeled based on the waiting time and queue length generated from queuing models. Therefore, there are also some shortcomings that need to be further addressed. First, they did not fully consider more user-centered factors. In addition to waiting time and queue length, other user expectations and needs for charging services, such as charging efficiency, charging facility availability, and charging speed, should also be considered. These factors can better meet users’ individual needs and improve user experience and satisfaction. Second, users’ satisfaction with charging services was not considered in the studies. Each user’s satisfaction with the charging service may be affected by various factors, and the user’s satisfaction assessment should also be considered in the modeling to obtain a comprehensive understanding of the user’s feedback and expectations of the service and to improve the service quality through improvement and optimization. To address these issues, one of our contributions is to employ user charging satisfaction as a QoS assessment factor for charging stations, thus linking user feedback and expectations of the service to the issue of charging station sizing. It is also a novel QoS evaluation method designed for charging stack scenarios.

1.4. Contributions

In the preceding literature review, previous studies on the CS sizing problem are examined, considering the projected charging demand and potential queuing models for QoS. Additionally, in a series of studies, intelligent charging solutions were proposed to optimize the operational objective of a CS, assuming a fixed infrastructure (EV charging facilities). The existing fixed-capacity strategies do not consider the scenario of intelligent charging stacks. However, intelligent charging can significantly change the daily energy demand profile of a CS. This interaction between intelligent charging and sizing decisions on the number and type of EV charging facilities has not been thoroughly explored in the literature. For example, if a CS has flexible charging capacity that can be shifted over time, it may require fewer EV charging facilities by optimizing EV charging times during operation. Thus, there remains a research gap in optimizing CS sizing decisions, accounting for cost, QoS and the impact of intelligent charging. At the same time, the focus on user experience and convenience is limited. While some researchers consider aspects such as waiting times and queue lengths, there is room for further exploration and optimization of user-centric factors. Improving the overall customer experience is crucial for promoting EV adoption and for fostering positive public sentiment toward charging infrastructure. This involves enhancing factors such as charging efficiency and user satisfaction.

This paper considers the CS in two different operational scenarios and uses the corresponding multiscenario constraints to formulate an objective function to minimize the investment cost (number and type of EV charging facilities) and the total power of the charging stack. It also proposes using EV charging satisfaction as a QoS evaluation tool for the CS to consider user experience and convenience. Due to the potential benefits of intelligent management and optimized charging strategies in reducing charging time and costs, improving charging efficiency and enhancing the user experience, it has accounted for the operational scenarios of charging stacks. This paper includes the total power of the charging stack as one of the optimization objectives. This has been overlooked in previous studies. It models the QoS based on the charging completion time of different EVs by substituting the EV charging satisfaction function to calculate their average satisfaction. It also models and analyzes the impact of the CS function under two different operational scenarios. It considers the parameters of the probability distributions to approximate the parameters of the incoming load and solve the sizing problem by using mixed-integer optimization. Finally, extensive simulations for a specific case study are performed to analyze the relationship between the QoS and investment costs under intelligent charging capabilities. The main contributions of this paper can be summarized as follows:

• An intelligent charging model is proposed based on the charging stack during the charging process, extending the solution to the charging station sizing problem.
• Multiscenario strategies specific for intelligent charging stacks are proposed, i.e., exclusive and shared operational scenarios. The total power of the charging stack is accounted for in the sizing of the charging station.
• A new CS QoS level formula is designed to map the user charging satisfaction to the EV charging satisfaction function. Based on the definitions of three different levels of business models, the corresponding QoS values are obtained.
• Simulation results are presented, showing the relationships among the QoS, the costs of installed EV charging facilities, the total charging stack power and the impact of intelligent charging (in a shared operating scenario) on reducing infrastructure costs during the sizing phase.

The remaining sections of this paper are structured as follows. Section 2 introduces the system model, and outlines the problem formulation. Section 3 provides details on the evaluation setup and presents the simulation results. Finally, Section 4 concludes this paper.

2. System Model

2.1. Notations

The notations used in the model development are described as follows:

• Index:

i 

Index of electric vehicle (EV);

j 

Index of EV charging facility;

t 

Index of slot.

• Set:

E

Set of simulated EV numbers;

J

Set of candidate EV charging facilities for the charging station;

T

Set of simulation time slots.

• Variables:

y_i,j,t 

1 if the charging of the i-th EV is at the j-th EV charging facility at the t-th time slot, and 0 otherwise;

N_c 

Number of charging station candidates for EV charging facilities;

y_i,j

1 if EV i is assigned to EV charging facility j, and 0 otherwise;

x_i,j,t

1 if EV i starts charging at EV charging facility j at time slot t, and 0 otherwise;

a_i 

Time slot for EV i to reach charging stations;

p_i,j

The charging power of EV i at EV charging facility j in the exclusive scenario;

p_i,j,t

The charging power of EV i at EV charging facility j at time slot t in the shared scenario;

n_j 

The maximum rated charging power provided by the j-th EV charging facility;

P_full

The total power of the charging stack;

l_i,j

Time span required for EV i to complete charging at EV charging facility j;

f_i 

The time slot at which EV i finishes charging at the station;

ε_i,t

1 if the i-th EV completes charging during the t-th time slot, and 0 otherwise;

F 

The average satisfaction of the CSs;

N_c 

The number of EV charging facilities.

• Attributes:

n_j 

EV charging facility j rate;

Q_i 

Charging energy demand for EV i at the CS;

μ 

The threshold to ensure the charging speed of EVs in charging;

F_i(f_i)

Charging satisfaction of EV i;

$T_d^i$ 

Expected departure time slot of EV i;

$T_u^i$ 

Maximum acceptable delay time slot for EV i from the expected departure time slot;

U_p 

The additional cost per kW of the charging stack;

U_c 

The cost per EV charging facility;

U_cp

The fixed cost of the charging stack.

The scenarios (exclusive and shared) in the paper are modeled based on real-world application scenarios. In reality, some CSs use all-in-one charging stacks, which typically have a fixed power output and are less flexible. This inflexibility often leads to inefficiencies in power utilization and higher operational costs. Other CSs use split charging stacks, which can provide flexible power distribution and automatically adjust the output power of each interface according to the number of connected electric vehicles. Two operational scenarios of charging power are shown in Figure 1.

As shown in Section 1.2, an intelligent charging stack is different from a traditional integrated charging station because it effectively reduces operational costs and improves power utilization. However, the previous fixed-capacity strategy did not consider the charging stack as an optimization objective. Therefore, we designed this multiscenario strategy specifically for the characteristics of the charging stack, including the exclusive model and the shared model. The modeling steps for these two strategies are introduced separately.

2.2. Modeling in an Exclusive Operating Scenario

The exclusive operation scenario follows the first-come first-service (FCFS) strategy, where the charging stack serves the first arriving EV first. When an EV arrives, if there is unused power remaining in the charging stack and the remaining power is greater than the maximum rated power of the EV charging facility, the charging stack immediately starts the charging task at the maximum rated power of the EV charging facility. The total charging power of all EVs within the same time slot cannot exceed the total power capacity of the charging stack. In other words, when the charging stack reaches its maximum power load, any new charging requests are rejected, and the new EVs join the queue and wait for the previous EVs to finish charging. Additionally, when the charging stack reaches its maximum power load and a new EV arrives, the charging request is also rejected and the new EV joins the queue, waiting for the EV in front of it to complete charging. Finally, when the remaining power of the charging stack reaches the maximum rated power of an EV, the EVs in the queue join the charging task in the order that they arrived.

Specifically, the scenario constraints for the exclusive operation scenario described above are as follows: first, the number of EVs charging in the same time slot cannot exceed $N_c$, and the corresponding constraint can be obtained as follows:

$$\sum _{i\in E}\sum _{j\in J}{y}_{i,j,t}\le N_c,\forall t\in T.$$

(1)

In addition, to ensure that there is at most one EV i charging at EV charging facility j at time slot t, constraint (2) is added, which states that

$$\sum _{i\in E}{y}_{i,j,t}\le 1,\forall t\in T,j\in J.$$

(2)

For the allocations of EVs to EV charging facilities, we must consider that each EV can be charged with only one EV charging facility, which gives us constraint (3):

$$\sum _{j\in J}{y}_{i,j}=1,\forall i\in E.$$

(3)

To ensure that each EV has only one moment to start charging, constraint (4) can be obtained:

$$\sum _{t\in T}\sum _{j\in J}{x}_{i,j,t}=1,\forall i\in E.$$

(4)

At the same time, the values of the variables $y_{i,j}$ can be determined by constraint (5), i.e., it can guarantee that the variables $x_{i,j,t}$ and $y_{i,j}$ have a value of 1 on the same charging facility j:

$$y_{i,j}=\sum _{t\in T}{x}_{i,j,t},\forall i\in E,j\in J.$$

(5)

According to the FCFS strategy, constraint (6) is obtained:

$$\sum _{t\in T}\sum _{j\in J}{x}_{u,j,t}\le x_{p,j,t},\forall u,p\in E:a_u<a_p,$$

(6)

where $a_u$ and $a_p$ denote the arrival times of EV u and EV p, respectively.

By combining constraints (1)–(6), the charging behavior of the EV and the charging behavior after queuing can be carried out in the order of the arrival time at the charging station. At the same time, EVs cannot be charged until they arrive, which means that

$$\sum _{j\in J}\sum _{t\in T}{x}_{i,j,t}\ge a_i,\forall i\in E.$$

(7)

During the charging phase, the value of the charging power variable $p_{i,j}$ is determined for EV i at EV charging facility j via $y_{i,j}$, namely, constraint (8):

$$p_{i,j}=\left\{\begin{array}{cc}{\eta }_j,\hfill & y_{i,j}=1\hfill \\ 0,\hfill & \mathrm{other}\hfill \end{array}\right.,\forall i\in E,j\in J.$$

(8)

The total power of the simultaneously charging EVs cannot exceed the total power of the charging stack $P_{\mathrm{full}}$, i.e., constraint (9):

$$\sum _{i\in E}\sum _{j\in J}{y}_{i,j,t}·p_{i,j}\le P_{\mathrm{full}},\forall t\in T.$$

(9)

Then, the time span $l_{i,j}$ needed for EV i to complete charging at EV charging facility j is

$$l_{i,j}=\left\{\begin{array}{cc}{\displaystyle \frac{Q_i}{p_{i,j}}},\hfill & p_{i,j}\ne 0\hfill \\ 0,\hfill & \mathrm{other}\hfill \end{array}\right.,\forall i\in E,j\in J,$$

(10)

where $Q_i$ is the charging energy demand needed by the i-th EV when it arrives at the charging station.

Based on the arrival time $a_i$ of EV i at the charging station and the charging time span $l_{i,j}$ needed by EV i, the time slot $f_i$ for EV i to complete the charging task is obtained:

$$f_i=a_i+\sum _{j\in J}{l}_{i,j},\forall i\in E.$$

(11)

2.3. Modeling in a Shared Operation Scenario

The shared operation scenario is similar to the exclusive use scenario, as it follows a first-come, first-served strategy. When an electric vehicle (EV) arrives, if the remaining power of the charging station is unused and greater than the maximum rated power of the EV, the charging task starts immediately at the maximum rated power of the EV. Additionally, the total power of all the charging EVs cannot exceed the total power of the charging station.

However, the main difference between the shared and exclusive use scenarios lies in the way a new EV is handled when the remaining power of the charging station is insufficient to meet its maximum charging power. In the shared scenario, the new EV is still allowed to join the charging process. In this case, the charging station intelligently allocates a portion of the power from other charging EVs to accommodate the newly arrived EV. This operation strategy aims to provide better service to customers by maximizing user satisfaction and prioritizing first-arriving customers. Consequently, a portion of the charging power of the first arriving customers is sacrificed to allow more customers to participate in the charging task.

As new EVs continue to arrive, this shared strategy continuously reduces the charging power of the EVs already in the charging process, enabling the new EVs to continue joining the charging task until a specific threshold is reached. To ensure that the charging EVs maintain a certain charging rate, the newly arriving EVs are placed in a queue. The aforementioned intelligent judgment and allocation processes are sequentially repeated when an EV completes its charging.

The specific modeling is as follows: some of the constraint Equations (1)–(7) and (9) in the exclusive sharing formula are also applicable to the shared operation scenario, and mainly differ in the power allocation constraints. Under the shared operation scenario, some modifications to Equation (8) are made. First, the sharing strategy threshold is defined as $\mu \in \left[0,1\right]$, and its value is between 0 and 1, which represents the maximum acceptance level of sharing; at this time, the actual charging power $p_{i,j,t}$ of EV i at time slot t is

$$p_{i,j,t}\le {\eta }_j,\forall i\in E,t\in T.$$

(12)

$$p_{i,j,t}\le {\displaystyle \frac{P_{\mathrm{full}}}{{\sum }_{i\in E}{\sum }_{j\in J}{y}_{i,j,t}·{\eta }_j}}{\eta }_j,\forall i\in E,t\in T.$$

(13)

At the same time, the upper limit of the number of EVs in total charging should meet the threshold to ensure the charging speed of EVs in charging, i.e., constraint (14):

$${\displaystyle \frac{P_{\mathrm{full}}}{{\sum }_{i\in E}{\sum }_{j\in J}{y}_{i,j,t}·{\eta }_j}}\ge \mu ,\forall i\in E.$$

(14)

By constraint (15), the correspondence between the actual charging power $p_{i,j,t}$ and $y_{i,j,t}$ can be ensured:

$$p_{i,j,t}\left\{\begin{array}{cc}>0,\hfill & y_{i,j,t}=1\hfill \\ =0,\hfill & y_{i,j,t}=0\hfill \end{array}\right.,\forall i\in E,j\in J,t\in T.$$

(15)

EVs can be charged only when they reach the $a_i$ charging point.

$$p_{i,j,t}=0,\forall t\in T<a_i,i\in E,j\in J.$$

(16)

Let us again define the binary variable ${\epsilon }_{i,t}\in \left\{0,1\right\}$ to denote whether EV i finishes charging at time slot t. Then, the binary variable ${\epsilon }_{i,t}$ satisfies constraint (17):

$$\sum _{j\in J}\sum _{\tau =1}^t{p}_{i,j,\tau }\ge {\epsilon }_{i,t}·Q_i,\forall i\in E,t\in T.$$

(17)

The binary variable ${\epsilon }_{i,t}$ also has only one exact moment of completion in time, i.e.,

$$\sum _{t\in T}{\epsilon }_{i,t}=1,\forall i\in E.$$

(18)

Then, the time span $f_i$ for EV i to complete the charging task is

$$f_i=\sum _{t\in T}{\epsilon }_{i,t}·t,\forall i\in E.$$

(19)

In summary, the modeling process of exclusive and shared operation scenarios is completed, overcoming the drawbacks of previous optimization methods that consider a single scenario only and being more in line with the actual needs of real-life station investment. At the same time, the original intention of introducing exclusive operation scenarios is to have a comparative benchmark for comparing the station investment cost and total power of charging stacks in shared operation scenarios (considering intelligent charging).

2.4. Service Definition Based on Satisfaction with EV Charging

The satisfaction distribution function is set to reflect the service level of the charging station. According to the charging needs of each user and the expected charging completion time, the satisfaction level is the difference between the charging completion time and the expected charging completion time. Therefore, satisfying the different needs of different users at different charging times is also an important part of the charging station sizing optimization model [28].

In real-world scenarios, charging satisfaction increases as the actual charging completion time approaches the user’s expected completion time. Following this rule, EV users prefer to go to a charging station that can meet their expected charging time to complete charging. Charging satisfaction can therefore be described by a probability distribution function, as shown in Figure 2. The probability distribution function is set to Equation (20):

$$F_i\left(f_i\right)=\left\{\begin{array}{cc}1,\hfill & f_i<T_d^i\hfill \\ 1-{\displaystyle \frac{1}{1+e^{-0.8\times \left(f_i-\frac{T_d^i+T_u^i}{2}\right)}}},\hfill & T_d^i<f_i<T_u^i\hfill \\ 0,\hfill & f_i>T_u^i\hfill \end{array}\right.,\forall i\in E.$$

(20)

In Equation (20), $f_i$ is the charging completion time of EV i, $F_i\left(f_i\right)$ is the satisfaction of EV i and $T_d^i$ and $T_u^i$ are the critical values of satisfaction. When $f_i$ is less than $T_d^i$, satisfaction is 100%, and when $f_i$ is greater than $T_u^i$, satisfaction is 0. In the range of $[T_d^i,T_u^i]$, satisfaction shows an S-shaped decrease as $f_i$ increases.

In addition, the average satisfaction $\overline{F}$ of the charging stations is defined to satisfy constraint (21) as an indicator to assess the quality of service of the charging stations.

$$\overline{F}={\displaystyle \frac{{\sum }_{i\in E}{F}_i\left(f_i\right)}{EVs}}\ge Q,$$

(21)

where Q represents the lower limit of satisfaction with the charging station, defined in the range from 0 to 100%.

To make the above satisfaction function (20) for EV charging solvable, Equation (20) is transformed into Equation (22) by introducing three binary auxiliary variables $y_1$-$y_3$.

$$F_i\left(f_i\right)=y_1+\left(1-{\displaystyle \frac{1}{1+e^{-0.8\times (f_i-\frac{T_d^i+T_u^i}{2})}}}\right)\times y_2.$$

(22)

The following constraints are used to relax Equation (22):

$$y_1+y_2+y_3=1.$$

(23)

$$f_i\le T_d^i\times y_1+T_u^i\times y_2+M\times y_3.$$

(24)

$$f_i>T_d^i\times y_2+T_u^i\times y_3.$$

(25)

$$y_1,y_2,y_3\in \{0,1\}.$$

(26)

where M is a large constant.

Based on the service definition of EV charging satisfaction, three different business models are provided for the investment costs of CSs built by operators (the number of charging facilities and the total power of the charging stack), namely “Low commercial demand”,”Medium commercial demand” and “High commercial demand”. The specific definitions are detailed in Section 3.1.

2.5. Objective

The cost of charging stations is measured by various components, including initial installation expenses, ongoing maintenance costs, electricity consumption fees and any additional infrastructure requirements. Accurately assessing these factors is essential for estimating the overall cost of charging stations, ensuring cost-effectiveness and operational efficiency. To ensure the stability and representativeness of the final results, we have adopted a total of K groups of samples, representing the final results through the method of averaging.

The cost of charging stations in various operational scenarios is expressed as follows:

$${\displaystyle \frac{{\sum }_{k\in K}({min}_{P_{\mathrm{full}},N_c}{U}_{\mathrm{cp}}+P_{\mathrm{full}}^{\left(k\right)}·U_p+N_c^{\left(k\right)}·U_c)}{K}},$$

(27)

where $U_{\mathrm{cp}}$ is the fixed cost of the charging stack, $U_p$ is the additional cost per kW of the charging stack and $U_c$ is the cost per EV charging facility.

The constraints in the exclusive scenario are defined by

Equations (1)–(11), while the service constraints are defined by Equations (21)–(26).

Equations (1)–(7), (9) and (12)–(19) are utilized for the constraints in the shared scenario, while Equations (21)–(26) are employed for the service constraints.

By calculating the fixed cost of the charging stack and the cost per EV charging facility in a specific scenario, an optimal configuration for the charging station sizing task based on the charging stack can be obtained. This optimal configuration includes the total power of the charging stack $P_{\mathrm{full}}$ and the number of EV charging facilities $N_c$.

3. Simulation and Results

Many studies [29,30] have shown that the travel behavior of EV users follows a certain normal distribution. Specifically, these researchers have found that EV users have certain normal distribution characteristics in terms of travel distance, travel time and travel speed.

3.1. Simulation Setup

K is set to 1000. A horizon T consisting of 24 equal time slots and 40 EVs is considered unless otherwise stated. Unless otherwise specified, the common threshold value $\mu$ and the service level of the charging station are set by default to 0.5 and to medium commercial demand. For vehicle types, probability weights (0.3, 0.4, 0.2, 0.1) are used to randomly generate the type distributions of private cars, taxis, minivans and large trucks [5]. Each type of EV has a corresponding battery capacity and energy consumption rate. Assumptions are made regarding the arrival time, departure time, distance traveled and charging energy demand of each electric vehicle (EV) at the charging station. First, the arrival time ($a_i$) of each EV is randomly generated by using a truncated normal distribution $N(\mu ,{\sigma }_2)$, where $\mu =8$ and ${\sigma }_2=2$. Second, the expected latest departure time ($d_i$) of each EV is randomly generated using a truncated normal distribution $N(\mu ,{\sigma }_2)$, where $\mu =17$ and ${\sigma }_2=4$. Third, each EV travels a certain distance ($p_i$) before arriving at the charging station. The data for the distance traveled are randomly generated by using a truncated normal distribution $N(\mu ,{\sigma }_2)$, where $\mu =30$ km and ${\sigma }_2=14$. Fourth, the average speed ($v_i$) of each vehicle is also randomly generated using a truncated normal distribution $N(\mu ,{\sigma }_2)$, where $\mu =50$ km/h and ${\sigma }_2=30$. Finally, the charging energy demand ($Q_i$) of each EV is generated based on the energy consumption rate (${\delta }_i$), travel distance ($p_i$), average speed ($v_i$) and battery capacity ($b_i$). These assumptions allow for the random generation of arrival times, departure times, distance traveled, average speeds and charging energy demand ($Q_i$) for each EV at the charging station. The charging energy demand of each EV is also generated by using a truncated distribution. The EV types and their characteristics are summarized in

Table 1. Types and specifications of EVs.

	Private Car	Taxi	Minivan	Large Truck
Battery Capacity $b_i$	75 kWh	120 kWh	170 kWh	250 kWh
Energy consumption rate ${\delta }_i$	0.35	0.40	0.55	0.85

.

$$Q_i=min\left\{b_i,{\delta }_i{\displaystyle \frac{p_i}{v_i}}\right\}.$$

(28)

For the choices of CS for charging facilities, according to the charging efficiency provided by the electric vehicle supply equipment (EVSE), the charging levels are divided into three categories. Level 1 charging is suitable for standard domestic environments such as garages when using a 120 V AC EV charging facility. Level 2 charging requires a 240 V socket. The level 3 EV charging facility is generally suitable for high-demand charging scenarios. The power output and price of each EV charging facility are shown in

Table 2. EV charging facility levels and costs.

EV Charging Facility Levels	EV Charging Facility Rate ${\mathit{\eta }}_{\mathit{j}}$	EV Charging Facility Cost ${\mathit{m}}_{\mathit{j}}$
level 1	2 kw	1000 $
level 2	4 kw	2000 $
level 3	19 kw	3000 $

. Regarding the service definition of charging stations, three Q-values that satisfy constraint (21) are used to evaluate the service levels.

• Low commercial demand (Q = 0): In this business model, there is no guarantee for when the charging of the electric vehicle is completed. The only guarantee is that each user’s energy needs are met within time frame T.
• Medium commercial demand (Q = 50%): This business model can ensure that the overall satisfaction of charging electric vehicles at the charging station is greater than 50%, which means that the majority of users are satisfied with the service quality of the charging station at 50%, which can also be considered a moderate service level.
• High commercial demand (Q = 90%): In this business model, each arriving EV i should satisfy its energy demand $Q_i$ at a time close to its lower limit $T_d^i$, which is equivalent to satisfying its charging demand immediately upon arrival at the station. In practice, this is equivalent to expensive customer service.

The number of candidate EV charging facilities $n_1$, $n_2$ and $n_3$ for the three EV charging facility types is set as ${\displaystyle \frac{{\sum }_{I\in E}{Q}_i}{{\eta }_j}}·T$, where ${\eta }_j$ is 2, 4 and 19, respectively. Therefore, the set J is 0 to $n_1+n_2+n_3$. A decision variable $q_j$ is then passed to indicate whether to press-fit this candidate EV charging facility, so $N_c\ast U_C$ in the objective function is ${\sum }_{j\in J}{q}_j·m_j$. Additionally, it is assumed that $U_P$ is 500 $. Finally, since the fixed cost of the charging stack refers to the equipment price, ignoring it does not affect the final results of the experiment.

The case of N EVs with the various quantities mentioned above was modeled in Python 3.8 and solved by Gurobi 10.0.

3.2. Simulation Results and Analysis

First, the relationship between station investment costs, the total power required for the charging stack and the proportion of charging facility levels is explored in the various operating scenarios. Second, in the context of exclusive operation scenarios, the relationship between station investment costs and the needed total power of the charging stack is examined for three different business models. These findings are useful for operators because they can determine the optimal operational strategies to meet user demand and reduce costs. Finally, the impact of the shared threshold $\mu$ on station investment costs is investigated. Next, let us analyze the experimental results through different sections.

3.2.1. Results Analysis in Multiple Scenarios

The impact of the various models on CS investment costs is investigated, and Figure 3 shows the results for medium commercial demand. As shown in Figure 3, the shared model has lower investment costs than the exclusive model. This is because, in the shared operation scenario, an intelligent charging strategy is used to ensure that each electric vehicle has a specific charging rate (satisfying constraint (14)) and allow more vehicles to charge. In this way, it can effectively reduce operating costs by using low-power charging facilities. Furthermore, this difference increases as the number of EVs increases, indicating that it is indeed important to consider different CS charging strategies in the task of CS sizing, especially as the popularity of EVs increases.

In Figure 4, by comparing the power requirements of the two models under different numbers of EVs, it can be seen that the shared scenario has a more flexible power allocation, which not only reduces resource waste but also improves the user charging experience. Moreover, this approach further reduces investment costs and is highly adaptable to potential future increases in the rated charging power of EVs. Additionally, under the same total power of the charging station, a shared operation scenario can serve approximately 10 more EVs than can a dedicated operation scenario.

Figure 5 shows the allocation ratios of different EV charging facilities under different EVs in an exclusive scenario, while Figure 6 shows the results in a shared scenario. In the exclusive scenario, the CS mainly uses level 1 and level 2 EV charging facilities, while, in the shared scenario, due to its flexible power distribution, the CS mainly uses level 3 EV charging facilities. In addition, due to the inability to freely allocate power in the exclusive scenario, more EV charging facilities may be needed to charge EVs. Finally, it is found that, regardless of the scenario, the algorithm tends to select some level 1 and level 2 EV charging facilities in most cases. This is because having more EV charging facilities still has benefits, as it provides more options for the initial allocation of EV charging facilities in EVs, thereby improving the user’s charging experience.

3.2.2. Results Analysis for Different Business Models

Notably, an additional charger category was incorporated into the simulations: ultrafast chargers, which offer a charging power of 120 kW at a cost of USD 50,000. As indicated in Table 1 and Table 2, these chargers are capable of charging taxis, minivans and large trucks more quickly than other charger types. Intuitively, one might consider deploying some of them for their speed. However, the simulation results indicate that these ultrafast chargers were not necessary for meeting the specified commercial demand; hence, they were not utilized in the installations (consequently, they are not included in the figures).

Figure 7 and Figure 8 compare the exclusive investment cost and the total configured power for the three considered commercial demand scenarios. The cost of building a station in the case of high commercial demand is almost 1.5 times greater than that in the case of low commercial demand. This business model can consider building a station in a location with extremely high user charging demand; otherwise, medium commercial demand can be considered with relatively low additional costs, and is able to meet the charging requirements of most users. Moreover, as commercial demand increases, so does the total power needed to configure it.

3.2.3. Results Analysis with Different $\mu$

Figure 9 examines the relationship between the shared threshold $\mu$ and the station investment cost. The results show that the value of the shared threshold $\mu$ is not necessarily higher, as the station investment cost is related to the user’s charging satisfaction function. When users share more power, the charging completion time is longer, which leads to higher station investment costs to achieve the corresponding business demand model. This situation is more pronounced when $\mu$ is greater than 0.7.

4. Conclusions

In this paper, the sizing problem of a charging station (CS) is considered, i.e., deciding the amounts and types of EV charging facilities to be installed as well as the total power of the charging stack. The ability of a CS to control electric vehicle charging in two different operating scenario (exclusive and shared) models is considered, enhancing the size optimization model of the literature. The sizing problem is formulated as a cost-minimization problem, with the inclusion of chance constraints for quality of service (QoS). In particular, the QoS level of a CS is defined by a satisfaction function that reflects users’ satisfaction with the charging completion time. The level is determined by comparing the average charging satisfaction with a predefined threshold.

For the case study presented, our simulation results verified substantial differences in the choice of EV charging facility type, depending on which operating scenario is available to the CS. Using intelligent charging strategies can effectively improve the utilization of charging facilities and can flexibly allocate the total power of the charging stack while ensuring a certain level of QoS. In a shared operation scenario, under the same total power of the charging stack, more vehicles can be provided with charging services, which reduces operating costs. Compared with dedicated operation, shared operation uses intelligent power allocation technology to flexibly adjust resource usage, achieving greater resource utilization. Additionally, shared operation can pursue maximum revenue by sacrificing some power, further reducing operating costs. These measures improve the utilization of charging facilities and customer satisfaction, bringing significant advantages to shared operation. Finally, the motivation for considering the CS operating scenario in the CS sizing problem was verified experimentally since the shared operating scenario was shown to reduce the infrastructure cost, and the effect is more significant at large charging stations.

The model only takes into account the number and type of EV charging facilities and the total power of the charging stack as the total investment cost, without considering factors such as geographical conditions, traffic congestion and the power grid. In addition, the model’s consideration of user behavior patterns is based on probabilistic distribution models, which do not fully reflect the complexity of the real world. The next phase of research will consider incorporating more factors into the model for a more in-depth analysis. It will also consider optimizing the configuration based on actual user behaviors.