Smart allocation and sizing of fast charging stations: a metaheuristic solution

ABSTRACT

The widespread adoption of electric vehicles (EVs) hinges significantly on effective network management, particularly the allocation and sizing of fast-charging stations (FCS) for EV users. Applying queuing theory based on M/M/c, the model seeks to identify the most effective number of chargers to minimise waiting time for electric vehicle users and ensure optimal utilisation of the FCS. The problem is formulated as a multi-optimisation problem where finding the optimal solution is done by implementing the non-dominated sorting genetic algorithm focusing on EV user satisfaction, voltage stability, and carbon dioxide emissions. Simulation results indicate that the model successfully enhances EV user satisfaction, environmental impact, and overall FCS utilisation. In a comparative analysis with state-of-the-art models, the proposed approach demonstrates a notable 40% improvement in EV user satisfaction and a significant 45% enhancement in FCS utilisation. This proves the effectiveness of the proposed model in optimising the performance of the network.

KEYWORDS:

• Fast charging station
• electrical vehicles
• network management
• queuing theory
• charging station utilisation
• CO₂ emissions

Nomenclature

Variables
$T_i^{\min }$	=	Minimum time an EV e takes to reach FCS
$T_e^c$.	=	Time to charge an EV, e
$D_e^i$	=	Distance between EV e and FCS i
$S_e$	=	EV e velocity/speed
$F_{\mathrm{acost}}^i$	=	Cost of accessing the FCS i
$L_{\mathrm{cost}}^i$	=	Cost of the land area where FCS I is on
$N_{\mathrm{ec}}^i$	=	Quantity of electric chargers, ec, in FCS i
$F_{\mathrm{Opcost}}^i$	=	Operation cost of FCS, i
$F_{\mathrm{Inscost}}^i$	=	Cost of installing FCS, i
$N^F$	=	Number of FCS
${\varphi }^i$	=	Utilisation rate of FCS, i
$N_{\mathrm{Bec}}^i$	=	Number of active ec
$P_0^i$	=	Probability a FCS i is idle
$P{L}_{\mathrm{base}}^i$	=	Base value of PL of FCS i
$P{L}_{\mathrm{extra}}^i$	=	Additional PL of FCS i
${\mathrm{\Lambda }}^i$	=	Voltage SI of FCS i
${\mathrm{\Lambda }}^{\mathrm{base}}$	=	Base voltage SI
$\left|V_b\right|$	=	Voltage scale at Bus
$R_b$	=	Resistance at b
$A_b$	=	Reactance at b
$C_{a,b}$	=	Conductance at a-b
${\theta }_b$	=	Voltage angle at b
$C{O}_e$	=	Amount of CO2 emission
$P_{\mathrm{rated}}$	=	Rated power of charger

Constants
$B_e^d$	=	Rate of EV battery discharge
$B_e^c$	=	Battery size
$E_{\mathrm{cost}}$	=	Electrical fees ($/kWh)
SOC	=	0.2
$L_{\mathrm{soc}}$	=	minimum bound of SOC
$U_{\mathrm{soc}}$	=	Maximum bound of SOC
$N_e$	=	Amount of EVs
$N_{\mathrm{ec}}^{\max }$	=	Limit number of e
$A_{\mathrm{ec}}$	=	Space amount for ec
$F_{\mathrm{cost}}$	=	Static fees for implementing FCS
$P_c$	=	Power in (kW)
$f_e$	=	Average emission factor of the EV fuel mix
$f_{\mathrm{engine}}$	=	Average emission factor of the fuel combustion engine
$F{E}_e$	=	Fuel economy of an EV e
$F{E}_{\mathrm{engine}}$	=	Fuel economy of the fuel combustion engine
${ϵ}_{\mathrm{grid}}$	=	Grid Power distribution Efficiency
${ϵ}_e$	=	EV charger efficiency
$\mu$	=	Service rate
$\lambda$	=	Arrival rate
${\varphi }_{\max }^i$	=	Maximum consumption rate of FCS
$W_t$	=	Maximum allowable waiting time
$w_k$	=	Weights of the function coefficients

1. Introduction

The Internet of Things (IoT) has emerged as a pivotal component in communication technology, particularly within the domain of the Internet of Vehicles (IoVs) (Asna et al. 2021). IoV can be conceptualised as a mesh network connecting vehicles, facilitating the sharing of crucial information about vehicles and traffic control. This collaborative sharing aims to enhance road safety for both drivers and passengers. Consequently, IoV plays a significant role in driving the realisation and development of autonomous driving technologies (Asna et al. 2021). In this context, electric vehicles (EVs) are recognised for their low carbon emissions, earning them global acclaim for their environmentally friendly attributes. As per the International Energy Agency, road vehicles constituted a significant portion of CO₂ emissions within the transportation sector, comprising 24% of the total global CO₂ emissions in the year 2020 (International Energy Agency 2023). In efforts to reduce greenhouse gas emissions, various initiatives have been undertaken to encourage the adoption of electric vehicle (EV) transportation. Notably, the UAE government has implemented measures such as providing free parking spaces and waiving renewal fees and registration fees for EVs. These incentives are designed to attract the usage of electric vehicles in lieu of traditional gasoline or diesel-powered cars (IRENA 2016). The improvement of network performance in vehicular networks has garnered significant attention with the aim of mitigating the limitations associated with Intelligent Transport Systems (ITS). Additionally, to address network issues linked to centralisation, blockchain technology (Wang, Cheng, and Li 2021) is employed to transform the network into a more decentralised and secure vehicular environment. As a result, the potential of blockchain-enabled smart contracts within the Internet of Vehicles (IoV) sector is increasingly acknowledged. Smart contracts, coded within an underlying blockchain architecture, inherently possess several desirable properties, including automation, decentralisation, immutability, and security. These self-executing programmes can be activated when specific conditions, as per predefined user criteria, are met. Smart contracts utilise computer hardware to process data, verify conditions, facilitate negotiations, and validate a contract. Smart contracts leverage computer hardware to execute various tasks, including processing data, verifying conditions, handling negotiations, and validating a contract (Kirli et al. 2022). Blockchain architectures primarily focus on data storage, whereas smart contracts are dedicated to managing contractual operations and transactions within the blockchain. The increasing adoption of electric vehicles (EVs) is poised to exert a substantial impact on the power distribution grid, particularly concerning the high-power demand associated with EVs. Consequently, there is a pressing need to establish a platform enabling charging point operators to efficiently handle EV users’ charging requests, ensuring that the charging requirements are met without surpassing the distribution grid capacity. Without careful consideration of this aspect, EV owners may encounter challenges such as unavailable charging stations and grid congestion due to the simultaneous charging of numerous EVs in the future. Blockchain technology holds significant promise in addressing challenges within the EV charging business. In Singh et al. (2021), the authors integrated blockchain technology into the Internet of Vehicles (IoV) and implemented smart contracts. This integration resulted in an increased trust value for network data and the implementation of effective incentive strategies. The application of smart contracts on the blockchain was specifically tailored to the urban traffic environment, incorporating the introduction of the concept of platoon selection. In Chen et al. (2019b), a smart contract has been implemented for the Electric Vehicle (EV) network, with the primary goal of preventing malicious and false transactions. While the smart contract autonomously carries out the contractual processes, it is imperative that it adheres to the terms and conditions of the company as well as national or international laws and regulations. In Amato et al. (2021), the authors introduced a formal model designed to verify the legal compliance of a smart contract utilised on the Internet of Things (IoT). This model can assess the legal conformity of a smart contract. In Chen et al. (2019a), to address the trust and efficiency issues inherent in traditional crowdsourcing incentive mechanisms, the authors devised a quality-driven auction incentive mechanism. This mechanism employs a smart contract to execute automatically, providing a solution to the challenges associated with traditional incentive approaches. By combining a smart contract and machine learning technology, in Xiong and Xiong (2019), the authors solved the problem of data availability using similarity learning in the blockchain system. Encouraging users to transition to Electric Vehicles (EVs) requires the underlying infrastructure to offer services comparable to traditional gas stations. It's crucial to note that electric charging takes more time than traditional refuelling, potentially leading to station congestion, prolonged waiting times, and user discomfort (Gusrialdi, Qu, and Simaan 2017). Studying the charging uncertainties of Electric Vehicle (EV), different queuing theories have been proposed, and the M/M/c model is extensively used in literature. Liang et al. (2014) proposed using M/M/c queue model for finding the optimal number of charging stations by focusing on different factors as traffic flow and road paths. Similarly, authors in Deb et al. (2019) proposed M/M/c queuing model aiming at predicting waiting times. Both studies (Deb et al. 2019; Liang et al. 2014) chosen a random value for the time of charging.

On the other hand, authors in Zhu et al. (2017) and Chen et al. (2017) concentrated on minimising waiting time and maximising user satisfaction by applying queuing theory to find the needed number of Charging Station (CS). However, these studies primarily focused on EV user satisfaction, neglecting the advantages of optimising CS utilisation during charging. In Simorgh, Doagou-Mojarrad, and Razmi (2020), an economic-based model was presented by applying the particle swarm optimisation to find the optimal CS locations and numbers; however, the system did not account for users’ satisfaction. Liu et al. (2018) proposed an optimisation model to enhance station consumption with ensuring EV user satisfaction with fast-charging services.

Many prior studies have optimised the size of Charging Station (CS) by primarily considering the consequences of charging status of EV on the grid power loss and users’ satisfaction. Some studies aimed to minimise waiting time by advocating for numerous fast chargers, potentially leading to underutilisation issues for Fast Charging Station (FCS) owners. Thus, optimising station capacity requires a delicate balance between the waiting time for EV users and CS utilisation. To the best of the author’s knowledge, no study has addressed the capacity of the FCS in relation to FCS consumption or utilisation on one hand and analysing the impact of FCS sizing and waiting time on the CO₂ emissions, on another hand.

This study's main contributions are the following:

1. Introducing a multi-objective network management model for a Fast Charging Station (FCS) considering different factors as users’ satisfaction by reducing FCS access time, enhancing FCS operational efficiency by maximising utilisation, improving distribution network technical performance by reducing power loss and enhancing voltage stability, and addressing green concerns by sequentially minimising CO₂ emissions. Proposing a queuing algorithm with the primary goal of maximising Fast Charging Station (FCS) utilisation and enhancing user satisfaction.

2. Optimising the network management methodology by applying the NSGA-II to find the optimal FCS locations, minimising the waiting time of EVs, minimising the CO₂ emissions and increase the voltage stability of the grid.

The rest of this paper is organised as follows: Section 2 provides a review of the literature, Section 3 outlines the network management scheme, and the queuing theory for FCS sizing mechanism, Section 4 involves the simulation, verification, and analysis of the proposed algorithm, and Section 5 summarises the article and its key concerns.

2. Literature review

Effective management of networks plays a crucial role in the Internet of Vehicles (IoVs), underscoring the importance of careful consideration regarding the allocation and sizing of Fast Charging Stations (FCS). Inadequate attention to these aspects during the integration of FCSs into the distribution network can lead to adverse effects on the power grid, resulting in increased power loss, voltage instability, and imbalances in demand and supply. Therefore, ensuring optimal FCS allocation and sizing is vital to prevent potential negative consequences on the power grid within the context of IoVs (Deb, Tammi, and Mahanta 2018).

Additionally, a shortage of Fast Charging Stations (FCS) can cause increased congestion at stations and diminished user satisfaction, mainly due to prolonged waiting times for charging. Addressing these challenges necessitates meticulous network management, involving the identification of optimal locations for constructing Charging Stations (CSs) and the efficient distribution of chargers. Authors in Ahmad et al. (2022) optimised Fast Charging Station (FCS) locations by minimising cost of implementing charging and minimising the cost of power loss costs by applying the hybrid grey wolf optimisation-particle swarm optimisation (PSO) algorithm. However, it's important to note that the study did not consider the capacity of the FCS, which is crucial for conducting a comprehensive analysis of power loss and station congestion.

Furthermore, the authors in Awasthi et al. (2017) introduced a Charging Station (CS) planning model driven by grid safety to enhance power quality parameters, specifically power loss and voltage profile, through a hybrid genetic algorithm-particle swarm optimisation (PSO) approach. However, the authors overlooked considerations related to Electric Vehicle (EV) user aspects and station congestion. Another study explored a model for coordinated FCS location and sizing within a distribution and transportation network, as detailed in Mainul Islam, Shareef, and Mohamed (2018). The proposed model employed the binary lightning search algorithm to address Charging Station (CS) problems, taking into consideration factors such as transportation costs, station installation costs, and substation loss costs. Nevertheless, the model did not account for station utilisation and waiting time aspects when optimising station capacity.

While some literature conducted a comprehensive examination of the impact of travel distance and waiting time on the electric ride-hailing fleet in Li, Liu, and Wang (2021), the authors did not assess how station utilisation affects congestion management. The authors in Asna et al. (2022) developed a model for Charging Station (CS) placement, considering multiple objective criteria, including travel cost, station cost, and power loss. The concept of mixed passenger–freight first-last mile (MiFLM) transport has emerged as a promising solution to mitigate the negative impacts of passenger transport and logistics in both urban and rural areas, enhancing the overall efficiency and financial sustainability of transport operations. Authors in Bruzzone, Cavallaro, and Nocera (2023) focus on the long-term energy and environmental implications of MiFLM transport on buses in a medium-sized European city. The findings suggest that transitioning to more sustainable fuels significantly enhances the energy performance of mobility. This transition is further supported by a policy-driven modal shift toward active modes of transportation, rather than relying solely on MiFLM solutions. While MiFLM can play a role in reducing energy consumption and pollutant emissions, its impact is more effective when combined with other measures, such as promoting sustainable fuels and encouraging modal shifts, ultimately leading to positive effects on externalities like congestion. Also, utilising energy system modelling of China's transportation sector, Du et al. (2021) proposed a study that constructs marginal emission abatement cost (MAC) curves. It explores various factors influencing the curve's shape in the energy system by analysing different forward-looking scenarios. The findings indicate that implementing a system-wide CO₂ tax (approximately $650/ton of CO₂) would be necessary to achieve emission reductions of up to 63% by 2030, compared to a scenario without a tax. Nakata et al. (2010) proposed an advanced approach to designing energy systems to address global warming and ensure energy security. They argue that conventional energy systems focus on high-efficiency technologies, while advanced systems require innovative design. The concept of a Low Carbon Society (LCS) is introduced as a multidimensional approach that aims to extensively restructure global energy networks. The authors suggest using energy-economic models to show feasible future solutions involving renewable resources, combined heat and power, and smart grid operations. They also mention recent trials of energy models related to waste-to-energy, clean coal, transportation, and rural development, highlighting the variety of technologies and linkages between supply and demand sides.

Although some articles in the literature considered a combination of critical factors such as waiting time, FCS costs, and power loss, few addressed the technical performance of the network grid, which significantly impacts overall network efficiency. In light of this, the present paper introduces a multi-objective network management model designed to be environmentally friendly. The model aims to enhance the convenience of Electric Vehicle (EV) users, optimise FCS economic benefits, improve network grid technical performance, and minimise carbon dioxide emissions.

3. Smart allocation and sizing of fast charging stations

Several critical factors are directly linked to the reduction of carbon emissions from electric vehicles (EVs). Firstly, the environmental impact is a key consideration, primarily determined by the carbon intensity of the electricity used to charge EVs. The efficiency of EVs is also pivotal, as higher efficiency translates to lower energy consumption and subsequently, reduced carbon emissions. Secondly, energy efficiency analysis is crucial, encompassing factors such as battery technology, vehicle design, and driving conditions. These elements significantly influence the amount of energy required to propel EVs, directly impacting their carbon footprint. Lastly, infrastructure planning plays a vital role, particularly in optimising the deployment of charging stations. Strategic placement of charging stations encourages EV adoption and facilitates more efficient use of EVs, thereby minimising carbon emissions. Collectively, addressing these factors is imperative for achieving significant reductions in the carbon emissions of EVs and promoting a sustainable transportation ecosystem. Hence, in the following section, such critical factors such as EV user convenience, FCS financial values, practical grid performance, green impact illustrated by CO₂ emissions, and queueing theory were considered for network management problem.

3.1. EV client convenience

EV users choose the route that is not congested to reach FCS or in other words, the route that provides less travel time than shortest distance. Hence, users’ satisfaction is revealed by how convenient is accessing the FCS. If the shortest distance path is congested, the EV will deplete its energy; hence, the least travel time path is considered. In this study, travel time (T_tr) is estimated by using the GPS Distance Matrix API to provide real distance and travel time information between EV location and FCS location pair while considering path traffic data (Google Maps Platform 2020). The predicted time that EVs takes to travel T_tr is shown in Equation (1). (1) $\mathrm{Ttr}=\left[\begin{array}{ccc}{T}_{11}\cdots & T_{12}\cdots & T_{1N^F}\\ ⋮& ⋮& ⋮\\ T_{N_{e}1}\cdots & T_{N_{e}2}\cdots & T_{N_e{N}^F}\end{array}\right]$(1)

In the matrix above, every row indicates the estimated travel time for an electric vehicle (EV) to reach all fast charging stations (FCS). This estimation is calculated using the Google Maps Distance Matrix API. This API is a Google Maps service that offers information on travel distance and time considering real-time road traffic data for a given origin-destination pair (Google Maps Platform 2020). Then, the chosen FCS will be the one that provides the minimum time among all FCS, which is represented as $T^e$, given by Equation (2). (2) $T^e=\underset{e}{min}(T_{e1}\dots T_{e{N}^F})$(2)

The State of Charge (SoC) (Li, Liu, and Wang 2021) for an electric vehicle's battery is a measure of the remaining capacity in the battery as a percentage of its total capacity. The formula to calculate the State of Charge is given in percentage, shown in Equation (3): (3) $\mathrm{SoC}\left(\text{\%}\right)={\displaystyle \frac{B_e^c-{\int }_{t0}^{t1}{B}_e^d\left(t\right)d\left(t\right)}{B_e^c}}$(3) Where $B_e^d\left(t\right)$ is the discharge rate of EV as function of time, and $B_e^c$ is the battery capacity. In practical scenarios, determining the remaining capacity of an Electric Vehicle (EV) battery can be intricate due to the dynamic nature of the discharge rate, $B_e^d.$ The discharge rate is subject to variations over time, influenced by factors such as driving conditions, temperature, and usage patterns. For a more accurate estimation, the integral of the discharge rate over time is utilised, capturing the cumulative discharge within a specific time interval. This approach offers a more nuanced comprehension of the comprehensive discharge dynamics, proving essential for precise capacity calculations.

3.2. FCS accessing cost

The cost for accessing a FCS for an EV, e, is given by $F_{\mathrm{acost}}^e$, which considers energy consumed, $E_e^c,$ and energy cost in Equation (4). (4) $F_{\mathrm{acost}}^e=E_{\mathrm{cost}}\times E_e^c$(4)

The energy consumed is given by Equation (5): (5) $E_e^c=T_e^c\times P_c\times {ϵ}_e^{-1}$(5)

The objective function, $f_1$, represents the total FCS accessing cost by electrical vehicle, e, is given by (6). (6) $f_1=\sum _{e\in N_e}{F}_{\mathrm{acost}}^e$(6)

3.3. FCS economic cost

Installation costs and operational costs of FCS are two key factors for determining their optimal number, where it is important to minimise their relative values. The approximate installation cost is composed of technical and land area costs. The technical cost encompasses expenses related to the fundamental machinery and equipment essential for establishing a Charging Station (CS). Meanwhile, the land area cost varies based on the chosen site for FCS placement. In this study, an area of 30 m² is assumed for installing EV charger with a distance of 1.5 m between the electrical chargers. Consequently, the overall installation cost of one FCS can be expressed as indicated in Equation (7): (7) $F_{\mathrm{Inscost}}^i=F_{\mathrm{cost}}+(A_{\mathrm{ec}}{L}_{\mathrm{cost}}^i+F_{\mathrm{acost}}^i{P}_{\mathrm{rated}}{N}_{\mathrm{ec}}^i)$(7) Where $P_{\mathrm{rated}}$ is the average of battery energy in (kWh) to the charging time in hours. The operational cost $F_{\mathrm{Opcost}}^i$, is related to the expected power demand at each charging station i, and the electricity cost per unit. The expected power P_i is correlated with the FCS noted power and number of active FCS, ${\text{β}}^i$. Hence, P_i is given by Equation (8) and $F_{\mathrm{Opcost}}^i$ is given by Equation (9) (8) $\mathrm{Pi}=P_{\mathrm{rated}}\times {\text{β}}^i$(8) (9) $F_{\mathrm{Opcost}}^i=\mathrm{Pi}\times E_{\mathrm{cost}}$(9) where ${\text{β}}^i$ denotes the expected number of busy FCS in charging station i.

Therefore, the objective function $f_2$ for minimising the total busy charging station costs is given by Equation (10). (10) $f_2=\sum _{i=1}^{N_{\mathrm{Bec}}^i}(F_{\mathrm{Inscost}}^i+F_{\mathrm{Opcost}}^i)$(10)

3.4. Grid technical performance

The implementation of FCSs needs careful attention for grid power transfer or power distribution in the system. Fast charging has direct effect on the grid power and VS (Etezadi-Amoli, Choma, and Stefani 2019; Rupa and Ganesh 2014; Yunus, De La Parra, and Reza 2011). Charging the EVs usually is done by obtaining power from the electrical grid which adds more demand on the system and thus affecting the overall system VS. This study focuses on two major performance metrics, the power loss index (PLI) and VSI aiming at reducing the variation in load patterns associated with EVs.

3.4.1. Power loss index (PLI)

The sum of all power losses on all branches reflects the PL of the distribution as given by Equation (11). (11) $\mathrm{PLoss}=\sum _{\begin{array}{c}i,j=1,i\ne j\\ \end{array}}^n{G}_{i,j}(V_i^2+V_j^2-2V_i{V}_j\cos ({\theta }_i-{\theta }_j\left)\right)$(11) where $G_{i,j}$ is the conductance of branch i-j, n is the number of bus lines, ${\theta }_i$ is the voltage angle of bus i. When the EV utilises the FCS, PL will occur, thus the PLI is the ratio of PL occurred after installing the FCS, $P_{\mathrm{Loss}}^{\mathrm{FCS}}$ to the PL before installing the FCS, $P_{\mathrm{Loss}}^{\mathrm{base}}$. Therefore, the objective function for minimising the active power loss, denoted by ${\mathrm{f}}_3$, is given by Equation (12). (12) $f_3=P_{\mathrm{Loss}}^{\mathrm{FCS}}/P_{\mathrm{Loss}}^{\mathrm{base}}$(12)

3.4.2. Voltage stability index (VSI)

Voltage stability index ($\mathrm{\Lambda }$) is a tool used to assess and monitor the health of the distribution system, making sure it can handle the additional load from EV charging without compromising the quality of power delivery. Voltage stability in the context of a fast charger for electric vehicles refers to the ability of each bus in the system to maintain a stable and consistent voltage level during the charging process. Stable voltage is crucial for ensuring the safe and efficient charging of electric vehicles. The assessment often involves comparing actual voltage levels to specified or desired voltage values. One approach to assessing voltage stability involves calculating the voltage deviation or percentage deviation before and after disturbance occurrence. $\mathrm{\Lambda }$ measured at bus j is given by Equation (13) (13) ${\mathrm{\Lambda }}_j=\left[\left(\mathrm{Vmb}-\mathrm{Vma}\right)/\mathrm{Vmb}\right]\times 100$(13) Where Vmb, Vma are the measured voltage at a specific point in the electrical system before the disturbance has occurred and after the disturbance has occurred, respectively. The objective function, $f_4$, is defined in Equation (14). (14) $f_4=1-\sum _{j=2}^{\mathrm{Nbus}}{\mathrm{\Lambda }}_j/{\mathrm{\Lambda }}^{\mathrm{base}}$(14) $\mathrm{s}.\mathrm{t}.V_i^{\min }\le \mathrm{Vi},\mathrm{Vj}\le V_i^{\max }$

3.5. Environmental impact

Electric vehicles (EVs) are becoming increasingly popular as a viable alternative to traditional internal combustion engine vehicles, largely because of their potential environmental advantages. EVs emit zero tailpipe emissions, which helps reduce air pollution and lower the overall carbon footprint compared to conventional vehicles. The environmental impact of EVs is determined by various factors, including the amount of CO₂ emitted, which is influenced by the average emission factor of the fuel mix used to generate electricity for EVs (measured in kgCO₂/kWh). The fuel economy of an EV, measured in kilowatt-hours per 100 miles or kilometres, also plays a crucial role in assessing its environmental impact. Additionally, the minimum time required for an EV to travel to a charging station, denoted by $T_i^{\min }$, is a key consideration. The environmental impact also takes into account the efficiency of the grid, denoted by ${ϵ}_{\mathrm{grid}}$, which reflects the energy losses during the transmission of electricity from the generation point to the EV charging point. Furthermore, the efficiency of the EV charger, denoted by ${ϵ}_e$, is considered, accounting for losses associated with the AC/DC converter in the charger. Therefore, the environmental impact can be calculated using Equation (15). (15) $E^e\mathrm{impact}={\displaystyle \frac{f_e\times F{E}_e\times T_i^{\min }}{{ϵ}_{e\times {ϵ}_{\mathrm{grid}}}}}$(15) Where $f_e$is the average emission factor of the EV fuel mix, FE_e is the fuel economy of the EV, and $T_i^{\min }$ is the minimum time that an EV needs to reach FCS. Therefore, the objective function, f5, for minimising the CO₂ emissions while charging an EV in a CS is defined by Equation (16) (16) $f5=\sum _{e=1}^{N_e}{E}_{\mathrm{impact}}^e$(16)

The CO₂ emissions caused by fuel combustion engine vehicles is calculated with regards to the same distance covered by EV as in Equation (17): (17) $E_{\mathrm{impact}}^{\mathrm{engine}}=f_{\mathrm{engine}}\times \mathrm{\Psi }\times T_i^{\min }/F{E}_{\mathrm{engine}}$(17) where $f_{\mathrm{engine}}$ denotes the average emission factor of an internal combustion engine, indicating the amount of CO₂ emitted per gallon of fuel burned. Ψ represents the upstream emission factor, which accounts for additional emissions associated with the production and processing of fuel and $F{E}_{\mathrm{engine}}$ represents the fuel economy of the internal combustion engine. Hence, the amount of saved CO₂ emission is given by Equation (18). (18) $E_{\mathrm{impact}}^{\mathrm{saved}}=E_{\mathrm{impact}}^{\mathrm{engine}}-E_{\mathrm{impact}}^e$(18)

3.6. Queueing theory for FCS sizing model

FCS are modelled by queuing system in which EV arrives at a station occurs randomly and is independent of each other, thus follows an exponential distribution. When all charging stations are busy, this is frustrating to users since more delay will be encountered and more congestion. Furthermore, the arrival rate to a FCS is predicted based on estimates of EV traffic flow, the initial State of Charge (SOC) of the battery set to 20%, maximum battery capacity, and battery discharge rate. The use of an exponential distribution implies a constant probability of EV arrivals over time, and this probability is determined by the predicted factors mentioned above. Since the time for each EV to charge is different depending on SOC, battery capacity, we assume the charging time to be uncertain. Hence, the EV arrival and charging process are considered to follow the Poisson process. Thus, the proposed scheme can be modelled by M/M/c to determine the optimal number of chargers, that reduces the delay and enhance station consumption. The charging time of an electric vehicle (EV) depends on several factors, including the battery capacity of the vehicle, the charging power of the charging station, and the initial state of charge (SoC) of the battery. The basic formula to calculate the charging time, $T_e^c,$ is given by Equation (19). (19) $T_e^c={\displaystyle \frac{B_e^c\times \left(1-\mathrm{SoC}\right)}{P_c}}$(19)

The mean service rate (μ) represents the number of electric vehicles (EVs) that can be charged during time limit. Equation (20) calculates the minimum number of chargers, $N_{\mathrm{ec}}^i$, required in the i^th FCS. (20) $N_{\mathrm{Bec}}^i={\displaystyle \frac{{\lambda }^i}{N_e}}$(20)

The tolerable waiting time of EV at FCS is by Equation (21): (21) $W_t=\frac{\mathrm{Li}}{{\varphi }^i\mu N_{\mathrm{Bec}}^i}$(21) Where L_i is the number of EV in a queue, is given by (22): (22) $\mathrm{Li}={\displaystyle \frac{\mathrm{P}0\times {\frac{{\lambda }^i}{\mu }}^{N_{\mathrm{ec}}}}{(N_{\mathrm{ec}-1)!}\times {\left(1-{\varphi }^i\right)}^2}}$(22) and P₀ is the probability that a charging station is empty, is given by (23) (23) $\mathrm{P}0={\left[1+\sum _{n=1}^{N_{\mathrm{ec}}-1}{\displaystyle \frac{{\frac{{\lambda }^i}{\mu }}^n}{n!}}\right]}^{-1}$(23)

So, the maximum waiting time of an EV is the ratio of the energy consumed to the total distance (Km) between EV and FCS, given by (24) (24) $W_{\max }=\frac{E_e^c}{D_e^i}$(24)

The optimal number of FCS is explained in flowchart in Figure 1. Hence, the multi-objective function can be expressed as in Equation (25). (25) $\mathrm{F}=\omega 1f_1+\omega 2f_2+\omega 3f_3+\omega 4f_4+\omega 5f_5$(25)

Figure 1. Flowchart for Optimal number of charging stations.

The weights can be calculated using rank order centroid ROC (Liu and Tan 2004) by Equation (26). (26) $\omega \mathrm{i}={\displaystyle \frac{1}{5}\times \sum _{i=1}^5{\displaystyle \frac{1}{i}}}$(26)

Non dominated sorting GA (NSGA-II) is used to solve the optimisation problem. Detailed description of NSGA-II is found in Deb et al. (2002). The flow chart of the problem is found in Figure 2. The algorithm begins by generating an initial population randomly, with each individual containing candidate Charging Stations (CSs). Each individual represents the decision variable N. After selecting the CS locations, the algorithm evaluates the objective function values and determines the station capacity using the queuing algorithm illustrated in Figure 2. The algorithm then computes individual fitness values for NSGA-II using the multi-objective function defined in Equations (24) and (25). In NSGA-II, a process of non-dominated sorting is conducted, wherein the crowding distance of individuals is calculated and sorted based on this distance. Subsequently, the algorithm performs selection, crossover, and mutation processes to obtain the new population set in both optimisation algorithms. However, in NSGA-II, an elitist preservation strategy is employed in addition to these procedures to create the next generation. The objective function evaluation process continues until an optimal solution is achieved. NSGA-II provides a set of Pareto solutions instead of a single solution, with the output corresponding to the optimal CS location.

Figure 2. Flowchart for solving optimisation problem.

4. Results and discussion

The simulation parameters, including the EV distribution, charging station locations, and grid characteristics, were carefully selected to reflect real-world conditions. The suggested algorithm for the optimal Charging Station model underwent simulation using NSGA-II within the MATLAB environment. The simulation results were then scrutinised to assess the effects of CS installation in the designated area, considering aspects such as FCS accessing cost, economic advantages, technical performance of the power grid, and environmental impact. Table 1 presents the EV parameter.

Table 1. Simulation parameters value.

Parameter	Value
No. of EVs	100
Rate of EV battery discharge (kW/h)	5.20
Battery capacity(kWh)	25
CO2 emission factor of fuel mix (kgCO2/kWh)	0.46
Fuel economy of EV(kWh/100Km)	28
Grid efficiency(%)	93.5
Charger efficiency of EV	95.5
CO2 emission factor of gasoline (kg/gallon)	8.75
Upstream factor	1.25
Fuel economy of engine	36
Maximum no. of iteration	100
Population size	200
Mutation probability	0.05
Crossover probability	0.6

In Figure 3, the frequency distribution of Electric Vehicles (EVs) concerning the time required to reach Charging Stations (CSs) is compared with existing methods (Asna et al. 2022; Li, Liu, and Wang 2021). The histogram highlights that 67% of EVs can reach CSs within a 15-min timeframe, while only 5% of EVs exceed 30 min to access CSs, consequently leading to reduced travel costs. The proposed algorithm outperforms (Asna et al. 2022; Li, Liu, and Wang 2021) by 40% and 45%, respectively.

Figure 3. FCS access time.

Table 2 presents a comparison of the performance between the proposed Charging Station (CS) model and existing models (Asna et al. 2022; Li, Liu, and Wang 2021). The existing models, as shown in the table, opted for a fewer number of chargers in each station, neglecting the issue of charging station congestion. For instance, algorithms (Asna et al. 2022; Li, Liu, and Wang 2021) selected 2 and 4 chargers in Station No.12 to accommodate 12 EVs and 8 EVs, respectively. This resulted in prolonged queues, with users experiencing a waiting time of 45.52 and 36.4 mins, respectively, which is an unfavourable waiting time for most EV users. A similar challenge is observed in Station No.18, where 3 chargers serve 12 EVs and 6 EVs, respectively, leading to difficulties in service provision. Consequently, the queue length becomes unmanageable, causing the charging system to be unstable. The results indicate that the proposed FCS sizing algorithm outperforms those of Li, Liu, and Wang (2021) and Asna et al. (2022). Additionally, the power loss incurred in Li, Liu, and Wang (2021) and Asna et al. (2022) methods is higher than in the proposed model by 40% and 45%, respectively, despite the lesser number of chargers in each station compared to the proposed model. This difference arises because the proposed method estimates FCS load based on the concept of busy chargers, whereas the existing method considers all chargers when calculating power demand in each station. In summary, the comparative study affirms the effectiveness of the proposed FCS planning model in determining optimal FCS placement and sizing, benefiting EV users, station operators, and distribution network performance. These results can serve as a foundation for decision-makers facing diverse scenarios when implementing FCSs. Furthermore, the study model facilitates detailed analyses of the effects of various parameters, such as EV arrival rate and utilisation rate, on FCS placement design from the perspectives of FCS economy, technical grid performance, and environmental considerations. In terms of the impact of charging station (CS) sizing on CO₂ emissions, it is evident that a limited number of CS stations results in increased travel distances for electric vehicles (EVs), consequently raising CO₂ emissions. The proposed algorithm demonstrates superior performance in this regard by selecting the optimal number and locations of CSs. Another notable observation is that while all three algorithms identify FCS 14 as the optimal location, the waiting times vary: 4.83 min for the proposed algorithm, 12.31 min for Algorithm (Asna et al. 2022), and 57.2 min for Algorithm (Li, Liu, and Wang 2021). This disparity in waiting times directly correlates with higher CO₂ emissions, underscoring the impact of waiting times, CS sizing, and optimal location on the environment. In regards to carbon footprints revealed by amount of CO₂ emitted when using traditional gasoline engine, the proposed algorithm proved to be ecofriendly by saving 52% of CO₂ emissions in compared to traditional gasoline engine vehicles; whereas, algorithm (Asna et al. 2022) have saved 45.33% of CO₂ emissions, and algorithm (Li, Liu, and Wang 2021) have saved 43.01% of CO₂ emissions. It is important to note that when considering the energy efficiency of the grid in an electric vehicle (EV) system, the value of the carbon footprint can vary significantly. The carbon footprint of an EV is directly influenced by the source of electricity used for charging. If the grid supplying the electricity is more energy-efficient and relies on cleaner energy sources such as renewables (e.g. solar, wind, hydroelectric), the carbon footprint of the EV will be lower. Conversely, if the grid relies heavily on fossil fuels (e.g. coal, natural gas), which have higher carbon emissions per unit of energy produced, the carbon footprint of the EV will be higher. Therefore, improving the energy efficiency of the grid and increasing the share of renewable energy sources can lead to a reduction in the carbon footprint of EVs. This is another factor to be considered in the expansion of the current study for future work that is to consider the grid to harvest energy from the solar panels.

Table 2. Comparison with state-of-the-arts.

Network Management	Optimal Location	Optimal Number	Arrival Rate (EV/Hr)	Station Utilisation (%)	Waiting Time [min]	Total CO2 Emission	Carbon Footprints	Extra PowerLoss (kW)
Proposed	1	5	16	90.2	8.51	88.76	173.69	30.82
	11	5	16	83.3	3.90
	12	12	45	86	1.21
	14	8	28	84	4.83
Algorithm (Asna et al. 2022)	1	3	15	72.4	25.31	190.82	420.88	43.148
	2	3	12	74.5	28.22
	10	2	12	70.1	44.54
	11	5	10	73.2	22.53
	12	2	12	72.6	45.52
	14	1	1	68.3	12.31
	18	3	12	79.4	14.71
Algorithm (Li, Liu, and Wang 2021)	1	7	3	54.12	11.52	245.37	570.42	44.689
	2	7	3	49.98	5.72
	5	3	1	51.6	1.56
	7	2	5	50.4	6.24
	10	1	1	43.44	32.5
	12	4	8	44.7	36.4
	14	2	5	42.06	57.2
	16	1	1	43.92	28.6
	18	3	6	43.56	58.5

5. Conclusion

In this study, we introduced a multi-objective planning model that considers various factors, including economic benefits, technical merits, and environmental impact, for the optimal allocation of Fast Charging Stations (FCSs). To enhance cost savings and meet EV user expectations regarding acceptable waiting times for recharging, we employed a novel queuing algorithm to determine the FCS sizing. To validate the capabilities of our study, we conducted a comparative analysis with previous works, affirming that queuing-based charging station capacity planning is crucial for ensuring adequate service to EV users while maximising the efficient use of charging facilities. The proposed algorithm showed notable enhancement decrease in CO₂ emissions in comparison with the other two algorithms, since optimal location and sizing of charging stations have direct effect on the CO₂ emissions. As the EV waits more time to charge or the more distance it travels to reach the CS, the more CO₂ emissions can be caused. Expanding the current study to specify different scenarios for uncertainties in EV systems is a promising direction for future work. These scenarios could include variations in factors such as driving conditions, charging infrastructure, battery technology, stakeholders’ acceptance, and energy sources. By considering these scenarios, one can better evaluate the energy performance of EVs under various conditions, which is crucial for understanding their real-world impact and improving their efficiency and sustainability.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data used to support the findings of this study are available from the corresponding author upon request.