Optimal Sizing of Electric Vehicle Charging Stations Considering Urban Traffic Flow for Smart Cities

Abstract

Achieving high penetration of electric vehicles (EVs) is one of the objectives proposed by the scientific community to mitigate the negative environmental impact caused by conventional mobility. The limited autonomy and the excessive time to recharge the battery discourage the final consumer from opting for new environmentally friendly mobility alternatives. Consequently, it is essential to provide the urban road network with charging infrastructure (CI) to ensure that the demand generated by EV users is met. The types of terminals to be considered in charging stations (CS) are fast and ultra-fast. The high energy requirements in these types of terminals could stress the electrical systems, reducing the quality of service. To size and forecast the resources needed in CI, it is of great interest to model and predict the maximum number of EVs, in each hour, that each CS will have to serve according to the geographic area in which they are located. Our proposal is not based on an assumed number of vehicles to be served by each CS, but rather it is based on the analysis of vehicular traffic in geo-referenced areas, so that the load managers can design the topology of the CS. The maximum vehicular concentration is determined by some considerations such as the road system, direction of the route, length of the road segment, the intersections, and the economic zone to which it belongs. The topology of the road map is freely extracted from OpenStreetMap to know the latitude and longitude coordinates. Vehicular traffic will be modeled through the topology obtained from OpenStreetMap and other microscopic variables to understand the traffic engineering constraints. In addition, the Hungarian algorithm is used as a minimum cost decision tool to allocate demand to the CS by observing vehicular traffic as a cost variable. The multi commodity flow problem (MCFP) algorithm aims to make commodities flow through the road network with the minimum cost without exceeding the capacities of the road sections. Therefore, it is proposed to solve the transportation problem by directing the vehicular flow to the CS while minimizing the travel time. This situation will contribute significantly to the design of the topology of each CS, which will be studied in future research.

1. Introduction

The use of conventional transportation is a current topic of study due to the reduction of air quality by the emission of $C{O}_2$ noise, and visual contamination [1,2]. It possible to solve this problem; electric mobility has attracted a great deal of interest from the scientific community. However, new engineering challenges arise, such as the sizing of CS to recharge the batteries of EVs that travel on a road network. The sizing problem consists of deciding on the number and type of chargers to install in the CS. The introduction of EVs on a large scale represents a heavy load on the electrical distribution systems (EDS) since they demand high electrical energy in short periods in systems with fast and ultra-fast charging terminals. Thus, the massive introduction of EVs to the system may stress EDS, causing voltage instability. It is possible to reduce the stress on the electricity system; it is necessary to predict the demand to estimate the necessary infrastructure resources for optimal sizing of the CS. The need to limit the level of EV penetration and the optimal distribution of available energy among them to make sure that the stress on distribution grids is reduced is noted in [3]. Thus, these constraints directly impact the user experience, so that the potential EV consumer would not be able to find any incentive to buy non-conventional mobility alternatives. Therefore, it is essential to create charging infrastructures that ensure the quality of electric service and maximize the user experience by recharging batteries to promote the consumption of electric vehicles.

The essential difficulty facing this new sustainable mobility alternative is the most significant travel distance, which is limited by the battery capacity and the lack of CI for EVs [4,5,6]. As EVs represent an essential load in EDS and match the residential load profiles, the need to size the CS arises considering the different topologies in terms of trajectory and how these are related to vehicular traffic variables. In order to model the effects of the massive introduction of EVs in the electric distribution networks, it is imperative to estimate the most considerable demand generated by electric mobility in geo-referenced areas. Thus, this study will make it possible to analyze the possibility of introducing new load peaks in substations to avoid stress on the power system. If the power demanded by electric mobility exceeds the installed power of the substations, it is necessary to expand the substation’s capacity and the transmission lines of the distribution power system. Therefore, a complex problem is faced due to EV user satisfaction, battery capacity, EC occupancy, charging station topology, medium and low voltage substation chargeability, and the amount of power available from the grid. On the other hand, considering a single electrical system to cover the demand generated by electric mobility would be very costly and technically unfeasible. Therefore, by analyzing the most significant traffic density in a specific geo-referenced area, it is possible to design the topology of the CS and the primary energy sources that will give the electrical power needed for each CS.

Thus, a heuristic model is proposed using Hungarian and MCFP algorithms based on linear problems. Previous work has solved [7] the optimal place of charging stations based on linear problems; for such a reason, the present article assumes that the locations of charging stations are the optimal locations and spend efforts in directing the vehicular flow to the different CS at the lowest cost [8,9,10]. The microscopic model simulates vehicular density in a geo-referenced area and understands vehicular flow’s basic variables. The essential traffic flow variables are flow (or volume), speed, and density. Flow is defined as the number of vehicles per unit of time, distance per unit of time is speed, and several vehicles per unit of distance are defined as vehicle density [11,12]. These variables will allow us to relate the different nodes by applying graph theory as an essential tool to formulate the different equations in an optimization problem [13,14,15]. An essential aspect being considered in the microscopic analysis is that they are helpful in situations of free traffic, i.e., the movement of the vehicle on the traffic road does not take into account the behavior of the driver, who decides the characteristics of such movement at any given moment. The analysis of vehicle traffic will allow us to look at the physical characteristics of the studied area, the characteristics of the vehicles, and variables such as vehicle spacing and the average speed at which an electric vehicle travels on a road map. Spacing refers to the spatial advance between vehicles. The speed and spacing between vehicles are modeled with a random vector under the discrete Poisson distribution [16,17,18].

Sizing and allocating resources to CS minimizes the capital cost and thus allows for testing the use of existing electrical systems or offering incentives for the construction of new distribution systems, in order to maximize the user experience with the construction of CIs with fast and ultra-fast charging terminals. The reviewed literature shows studies that address CS sizing under expected demand without discussing the optimization of EC sizing solutions. This work aims to develop a strategy to decide the most significant vehicle density and how this is directed towards the multiple CS options with the shortest travel paths. Furthermore, they decide the number and types of chargers based on an assumed number of vehicles that need to be recharged. Figure 1 shows the conceptualization of the problem to be solved. The contributions of this article are as follows.

• A theoretical sizing model is introduced to develop charging station infrastructures for electric vehicles in a heterogeneous transportation system considering the paths; this application will allow us to foresee the highest and lowest demands in periods, thus allowing the sizing of the charging infrastructure with the minimal cost.
• The algorithm is designed to reduce vehicle paths using a geo-referenced vehicle flow routing system considering the topology of the road map.
• The organization and management of road vehicle traffic are solved with the Hungarian algorithm and multi-commodity flow problem under linear programming considerations.
• The proposed academic model provides industrial application potential for charging station infrastructure providers in smart cities for competitive environments.

Finally, this article is organized as follows: Section 2 discusses the requirements for charging infrastructure. Section 3 discusses the relevance of geo-referenced path analysis and then details the problem formulation. Section 4 presents the results. Finally, Section 5 concludes this article.

2. EV Charging Station Infrastructure

The importance of building charging infrastructure for electric mobility, considering vehicular traffic in geo-referenced areas, depends on the sizing of the charging station capacity and how the traffic is distributed on the road map. The problem of sizing a recharging station usually consists of minimizing the cost of the capital investment in recharging systems, provided that CS can satisfy a fixed number of recharging requests [19]. A queuing theory-based CS sizing algorithm that benefits EV users, cost savings for CI, better station utilization, and reduced impact on power systems and the environment is proposed in [20]. Furthermore, it explains that grid energy loss and traffic congestion are vital factors that affect the location and sizing of CS. Thus, the primary considerations for the building of CIs in this article are vehicle density in geo-referenced scenarios and the routing of traffic to the various charging stations to optimally size and allocate resources to ensure that the demand generated by electrical mobility is satisfied. Therefore, three areas of research are related to the development of this article: demand assignment to charging stations, shortest trip trajectories, and sizing of operational capacity in EV charging stations. There are microscopic and macroscopic models that relate different traffic variables such as volume, speed, interval density, and spacing [21,22,23]. Consequently, to understand the different elements of vehicular traffic and how they relate to each other in a geo-referenced area, the microscopic model is used. The results generated from a microscopic analysis are valuable to characterize the most significant vehicle density in urban areas. As a result, transportation evaluation strategies are a severe problem to be solved because they involve geographic elements, vehicle distribution, and traffic variability subject to human behavior that make it impossible to predict with precision the path of a vehicle’s patterns. The trajectory and demand allocation have been considered decision variables for the least-cost routing of vehicular traffic and the optimal sizing of the charging station. In [24], the path of vehicles is considered based on realistic trip patterns; however, considering a rigid trip pattern reduces much of the size of the analysis space, i.e., a starting and ending point is considered, reducing the sample size where multiple users participating in the road network can recharge their batteries. Therefore, this article studies the most significant vehicular density that a zone could experience to give the necessary resources for CS distributed in the study and to guarantee the possibility of recharging users who use the road network.

The sizing of the charging stations relates to the technology of the terminals required to charge the battery of the electric vehicles on the road networks. Likewise, these technologies are approved and refer to the capacity of the batteries’ charging times: in fact, the range of travel of an electric vehicle on a road network with 100% of the battery charged. The average range of an EV for the class by European homologation criteria L7e-C is 123 km with an average battery capacity of 15 kW. In addition, the automotive mass must be less than or equal to 400 kg, excluding the mass of the batteries for electric vehicles. One more detail is that L7e-C corresponds to four-wheeled micro cars.

Table 1. European homologation class L7e-C features.

Electrical	Autonomy	Terminal	Motor	Maximum	Regenerative	Category Average
Vehicle	(km)	Voltage (V)	(kW)	Speed (km/h)	System	Dimensions (m)
Tazzari Zero	200	230	15–25	90	✓
Renault Twizy	100	230	17	80	✓	Height < 2.5
Audi Urban	73	230–400	15	100	-	Width < 1.5
Peugeot BB1	120	230	15	60	-	Length < 3.7

shows some characteristics of vehicles designed for urban use of type-approval class L7e-C.

Therefore, the decision problem is to direct the flow of vehicles to charge stations, reduce trip times, allocate users to each charging station, and generate efficient dispatch policies [25,26]. Moreover, the increased penetration of unconventional vehicles in electrical distribution systems needs particular attention because the use of high power in short periods could result in voltage instability, causing the reduction of the quality, safety, and continuity of the electrical service [27]. Therefore, the charging infrastructure for batteries in electric vehicles plays an essential role in motivating consumption and developing this new unconventional mobility alternative. The importance of sizing CS with optimizing criteria for electric mobility lies in the fact that the electricity generated cannot be stored, as is the case with conventional service stations, until users have access to the electricity available at the charging stations.

In [28,29,30], the authors propose modeling mixed vehicular traffic where conventional and unconventional mobility participates; furthermore, it is suggested that multiple charging terminals be considered according to homologation according to the requirements needed by each user. Several variables such as limited capacity, path chance, trip time, flow conservation, charging time, spatio-temporal collaboration, terminal types, and traffic events are considered in [31,32,33], and the main aspects to obtain robust solutions that respond to the different situations where unconventional mobility practices exist are introduced. There are three types of terminals that have a direct relationship with EV charging time. Type I and II terminals are applicable in residential areas for slow and semi-fast charging, respectively, while type III terminals, corresponding to ultra-fast charging, apply exclusively to charging infrastructures in the public sector. Each type of terminal to recharge batteries in electric vehicles has its topologies and specific technical features. For example, in ultra-fast charging terminals, the input voltage is DC, while in slow charging terminals, AC is used. Therefore, the optimal dimensions of the charging stations are critical so that they can meet the hourly demand for battery recharging. In [34,35], methods for clustering and the construction of minimum spanning trees using graph theory as optimization tools under a linear programming formulation are proposed.

Current contributions in the literature have been focused on multi-objective models to solve the charging station planning problem to incentivize the mass introduction of electric vehicles. However, microscopic analysis has not been considered to study the density of acceptable vehicular traffic in an area under study. Neither has it considered the multiple options that an electric vehicle can use to recharge its battery. These EV battery recharging options will be limited by distances and the availability of roads to design paths that involve reduced vehicle density, thus optimizing trip times to the charging stations. Another fundamental contribution is that by using free data downloaded from the OpenStreetMap platform, the case study can be recreated to add to the analysis of vehicular traffic using the geographic characteristics of the road network, topology, and lengths of the different road cross-sections.

Consequently, the interest of this article is to build a linear programming model that incorporates a study of free traffic situations that allows us to test the most significant vehicle density concerning paths in geo-referenced areas in order to direct the vehicular flow to the available charging stations. Predicting the most significant number of vehicles that could circulate in a geo-referenced area and understanding how they are distributed in trajectories, the designer of the charging infrastructure could use the proposed model to make the best decision about the topological design of the CS. Therefore, the study developed in this article has a high potential for the development of service providers for the creation of charging infrastructures, and the analysis is based on the study of geo-referenced traffic in terms of trajectories, looking at the features of the road network to decide the input variables for the linear optimization model. Finally, 

Table 2. Summary: comparison of the results with other methods.

Author, Year	Allocation Sizing	Density Traffic	Urban Study	Heterogeneous Vehicles	Scalable	Microscopic Traffic Simulation	Other Characteristics
Gan et al., 2020 [36]	✓	✖	✖	✖	✓	✖	Max profit service, nonlinear integer problem (NLIP)
Inga et al., 2019 [37]	✓	✖	✓	✖	✓	✖	Min cost flow Multicommodity network
Demir et al., 2019 [16]	✖	✖	✖	✖	✓	✖	Min distance travel Deterministic, Integer Linear Programming (ILP)
Azadi et al., 2019 [38]	✖	✓	✖	✓	✖	✖	Routes, Mixed-Integer Linear Programming (MILP) Multicommodity network
Bevrani et al., 2019 [39]	✖	✖	✓	✓	✖	✖	Nonlinear programming (NLP) Multicommodity network
Campaña et al., 2019 [7]	✓	✓	✓	✖	✖	✖	Min congestion road Segmentation ILP Neighborhood
Liu et al., 2019 [40]	✖	✓	✖	✓	✖	✖	Max flow Fuzzy multi-objective
Ghasemi et al., 2019 [41]	✓	✓	✖	✖	✖	✖	Min shortages Multi-objective particle swarm optimization (PSO)
Proposed work (OSCSI)	✓	✓	✓	✓	✓	✓	Multicommodity Flow, Revised simplex, MILP, NP-Hard, Hungarian method

shows the taxonomy of the related studies, where one can see a summary of the problem statement, the objective function, and the main constraints analyzed with which they discussed the problem.

3. Framework and Model

This article proposes a theoretical model to give resources and solve the transportation problem using linear programming in heterogeneous scenarios for CI implementation. The main aim is to allocate resources and guide the flow of demand to the charging station using the most extraordinary transit capacity to cut travel times from a node where demand originates to a supply station.

3.1. Path Analysis with Graph Theory Using Free Data

The study of traffic is highly complex since it involves people’s behavior and their particular mobilization needs. However, in this article, the purpose is to test the most critical scenario, which is when there is a high traffic density, for the optimal sizing of CS. Sizing for charging stations is directly proportional to the number of vehicles served by each charging point during peak demand periods; this data allows for building and designing with technical criteria the best topology required for each charging station distributed in the study area. The CS topology refers to the required number of fast chargers and ultra-fast chargers, and even battery replacement could be considered as an alternative to cut the times needed in a supply station for electrical mobility. Consequently, the purpose of this article is to build a linear programming model that includes a study of free traffic scenarios that allows us to test the most significant vehicle density in terms of trajectories in geo-referenced areas, in order to be able to direct the vehicular flow to the available charging stations in the case study.

Another essential detail in this article is that this model works with data available on a freeway as obtained from OpenStreetMap. In addition, OpenStreetMap will allow us to download essential data of latitude and longitude coordinates together with their mapping to use them in real places, allowing us to execute dimension processes in charging infrastructures from free data obtained from the web. Therefore, since the maximum number of vehicles that could enter a CS is predicted, IC designers would not have to start from an assumed number of vehicles, as has been done so far, to design the CS topology.

A non-convex combinatorial problem is used to solve the multi-commodity flow problem, including the Hungarian algorithm. The traffic rates and traffic directed to CS. Multiple commodities are generated with the optimal allocation. Thus, the following necessary constraints must be considered: (i) commodities, (ii) travel demand, and (iii) capacity of crossing between two intersections. The architecture of the proposed method is shown in Figure 2.

3.2. Target Location Problem for Multi-Commodity Flow

Programming in the analysis of geo-distributed data is proposed as a multi-commodity flow target location problem. The commodities can be sent from its resources; MCFP aims to locate its targets so that the multi-commodity flow is optimized in some direction.

Let $G(V,E)$ be a directed graph where V represents the vertices and E the edges. The vertex represents the intersections where the demand for the commodity d and the CS are generated; moreover, V is the form of length $n+m$. The scalar n indicates the number of crosses and m the number of CS. The edges represent the different directions a path will take from a vertex $u_i$ to a vertex $v_j$. The set of edges is $e_{i,j}$ or $e_{j,i}$, where $einE$ and $i,j$ indicate the direction of flow that originates the optimal trajectories for each commodity set $\left(K\right)$ composed of origin ($u_k)$, destination ($v_k$), and demand ($d_k$). Therefore, $V(u,v)$, $E(e_i,e_j)$, and $k=(u_k,v_k,d_k)\in K$. At each cross section of length ($l)$, there is a number of vehicles $\left(\alpha \right)$ that generates a traffic rate $\left(q\right)$. The vector of concentration of vehicles $\left(X\right)$ will be given by the relation between q and the average speed $VEL$ at which the vehicles circulate in a time interval $t_s$. The length of the goods vector K will be directly proportional to the number of crossings it generates with $x\ne 0$, where $x\in X$ and represents the partial concentration of vehicles at each node. The equations that define what has been discussed so far are presented below.

$$\begin{array}{cc}{\alpha }_{i,j}=\frac{l_{i,j}+sp}{sp}\ast \lambda \hfill & \left[veh\right]\hfill \end{array}$$

(1)

$$\begin{array}{cc}{q}_{i,j}=\frac{{\alpha }_{i,j}}{t_s}\hfill & \left[\frac{veh}{h}\right]\hfill \end{array}$$

(2)

$$\begin{array}{cc}x=\frac{q_{i,j}}{VEL}\hfill & \left[\frac{veh}{km}\right]\hfill \end{array}$$

(3)

With (1), calculate the largest number of vehicles in a cross-section of the road $\left(l\right)$, considering the spacing between vehicles $\left(sp\right)$. $\lambda$ is the annual increase rate of the vehicle park. The rate or frequency of circulation of a vehicle in a given time interval is calculated with (2). Finally, (3) calculates the concentration of vehicles that will be received as a flow of goods over each node. A discreet Poisson probability distribution is used to model the stochastic of vehicle traffic. Therefore, the MCFP can be formulated as follows:

$$\begin{array}{c}\underset{X}{min}{f}^{T}X\hfill \end{array}$$

(4)

$$\begin{array}{c}A\ast X\le b\hfill \end{array}$$

(5)

$$\begin{array}{c}{A}_{eq}\ast X=b_{eq}\hfill \end{array}$$

(6)

$$\begin{array}{c}{b}_l\le X\le b_u\hfill \end{array}$$

(7)

The objective function presents with (4) and the constraint set with (5)–(7). f, b, and b are all constant vectors. A and $A_{eq}$ are constant matrices. The limits $b_l$ and $b_u$ can include constants or infinity. Finally, X is the vector of vehicle concentration variables. The MCFP variables are expressed as (8).

$$\begin{array}{c}{x}_{k,e}^{flow}=\mathrm{Flow}\mathrm{of}\mathrm{Commodity}k\mathrm{at}\mathrm{Edge}e,\forall k\in K,\forall e\in E\hfill \end{array}$$

(8)

The bounds define the feasibility region for the MCFP variables, as expressed in (9) and (10).

$$\begin{array}{cc}{x}_{k,e}^{flow}\ge 0;\hfill & \forall k\in K,\forall e\in E\hfill \end{array}$$

(9)

$$\begin{array}{cc}z\hfill & \ge 0\hfill \end{array}$$

(10)

Let us say that percent injected flow (PIF) describes the relationship between the flow of a commodity injected into the network and the demand for the commodity as expressed in (11). The objective of the MCFP can be described as maximizing the lowest PIF of all commodities as expressed in (12).

$$\begin{array}{cc}{z}_k\hfill & =\frac{{\sum }_{v\in V}{x}_{k,(s_k,v)}^{flow}}{d_k}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\mathrm{Objective}\hfill & :\mathrm{Maximize}min\{z_k:k\in K\}\hfill \end{array}$$

(12)

The capacity constraint can be summarized as the sum of the flow on any edge; therefore, it must be less than or equal to the capacity of the edge as presented in (13). The net flow for commodity k of edge e is represented in (14).

$$\begin{array}{cc}{\displaystyle c_e\ge \sum _{k\in K}{x}_{k,e}^{flow};}\hfill & \forall e\in E\hfill \end{array}$$

(13)

$$\begin{array}{cc}Ae{q}_{(k,uv)}=\hfill & \sum _{w\in V}{x}_{k,(uv,w)}^{flow}-\sum _{w\in V}{x}_{k,(w,uv)}^{flow}\hfill \end{array}$$

(14)

$$\begin{array}{cc}be{q}_{(k,uv)}=\hfill & 0,\forall k\in K,\forall uv\in \{w\in V:w\ne u_k,v_k\}\hfill \\ be{q}_{(k,uv)}=\hfill & d_k,\forall k\in K,\forall uv\in \{w\in V:w=u_k\}\hfill \\ be{q}_{(k,uv)}=\hfill & -d_k,\forall k\in K,\forall uv\in \{w\in V:w=v_k\}\hfill \end{array}$$

(15)

With (15), the conservation constraints are defined. The flow network describes a commodity through the sum of flows leaving and entering a node. The conservation constraint can be defined as a flow network that would be equal to zero if the node is not a source of a commodity; however, if the node is a source, the network flows would be equal to $d_k$. The network flow objective must be equal to $-d_k$. Finally, Algorithm 1 is used to solve the problem defined in this article. In step 1, demand is assigned to each CE considering vehicular traffic as a cost variable. Furthermore, in step 2, the commodities are each represented as a 3-tuple containing an origin node, a destination node, and a demand. The objective function is defined in step 3. The constraint and conservation equations are defined with steps 4 and 5. Finally, in steps 6 and 7, the problem is solved, and the results are interpreted, respectively.

Algorithm 1 Optimal Sizing for Charging Stations Infrastructure (OSCSI).

Input: Case study connectivity matrix = get$(.osm)$ $G_0$ = digraph(connectivity matrix) Output: Global solution $G_{solve}$; Variable:$u_k,v_k,d_k,k,q,w,z,x_{K(u_k,v_k,d_k)}^{flow}$, $cos{t}_{q,W}=get$(connectivity matrix); Step 1: Allocation $[u_k,v_k,x]=$ Hungarian$\left(cos{t}_{q,x}\right)$; Step 2: Commodities $K=[u_k,v_k,d_k]$; $K\left(find\right(K\left(3\right)=0\left)\right)=\left[\right]$; Step 3: Objective function $z_k=\frac{{\sum }_{v\in V}{x}_{k,(u_k,v)}^{flow}}{d_k}$ $A(k,e)=z_k$; $A(k,z+1)=1$ $f(z+1)=-1$; Define the objective function Step 4: Capacity constraints $A_{(e,k)}^{flow}=1$; $b_{(e,k)}^{flow}=$$G_0$.Edges.Weight Step 5: Conservation constraints $temp={\sum }_{w\in V}{x}_{k,(uv,w)}^{flow}-{\sum }_{w\in V}{x}_{k,(w,uv)}^{flow}$ $Ae{q}_{(k,uv)}=temp$ ; $Ae{q}_{(uv,k)}=-temp$ $be{q}_{(k,uv)}=0,\forall k\in K,\forall uv\in \{w\in V:w\ne u_k,v_k\}$ $be{q}_{(k,uv)}=d_k,\forall k\in K,\forall uv\in \{w\in V:w=u_k\}$ $be{q}_{(k,uv)}=-d_k,\forall k\in K,\forall uv\in \{w\in V:w=v_k\}$ Step 6: Dual Simplex solver $\left[x\right]$ = $linprog$( f, A, b, Aeq, beq) Step 7: Return $\left[G_{solve}\right]$ = digraph($x,u_k,v_k$)

4. Analysis of Results

The microscopic study presented with Figure 3 allows for understanding three fundamental variables of vehicular traffic, speed, flow (or volume), and vehicular concentration, considering the topology of the road map, the speed at which a vehicle travels, and frequency with which a vehicle passes through a point. Thus, the microscopic analysis will make it possible to describe vehicular traffic in terms of the road map in geo-referenced areas, bringing the problem closer to reality. Data obtained from the microscopic model concerning vehicular concentration are entered into the MCFP mathematical model proposed in this article.

The analysis in Figure 3 shows the predicted vehicle concentration over 20 years with a theoretical annual rate of increase of $\lambda =5.4\%$. During the first 10 years, the growth of the vehicle park is modeled with a constant average speed of 15 km/h, from which about 4816 vehicles are obtained that will circulate at most during one hour in the most critical condition. The critical condition occurs when $sp$ is minimum, given that a section of track could accommodate the largest capacity of vehicles given its length. A random vector with Poisson distribution will be used in this article, where $0.03\le sp\le 2.5$ km will be used. Figure 3 in year 11 includes microscopic analysis with three types of average traffic speeds $10,15$ and 25 km/h to understand vehicle behavior as a function of traffic speed. Speed has an inverse effect on vehicular concentration, i.e., the less vehicular freedom of movement in a geo-referenced area indicates that there is a high vehicular concentration, and the greater the vehicular freedom of movement, the lower the vehicular concentration. This is because speed is inversely proportional to the concentration of vehicles in a time range. Thus, high vehicular concentration in a geo-referenced area is modeled when the travel speed is the least, and $sp$ is the least. Therefore, the functionality of the microscopic model, implemented in the heuristic, is demonstrated for the analysis of vehicular traffic considering features in geo-referenced areas, allowing the problem to approach reality.

Figure 4 shows the first conditions where the topology of the road map is defined considering the high vehicle stochastic and the most significant number of vehicles under Poisson random distribution that can circulate in a cross-section of the road map. Furthermore, Figure 4 shows the CS labeled with nodes 55 to 61 in green color, and nodes 1–54 are the points where the demand originates. Remember that the aim is to allocate resources to charging stations for electric vehicles at the least cost to meet the demand generated as a function of the most significant vehicle density in a geo-referenced area. Thus, each intersection is allocated a demand based on the concentration of vehicles, which is analyzed with the microscopic model mentioned in previous paragraphs. The main aim is to obtain the smallest path for the supply of electric power from a demand node to a supply node by looking at the most significant and most negligible limits of vehicular flow present in a cross-section of the road map. The demand is generated by the electric vehicles that need to recharge their batteries, and the power supply to the batteries will be provided by the charging stations distributed in the study area.

As there are multiple demands and a restricted number of CS, the commodities originate from the charging stations to the demand nodes, and the path approached by the algorithm will be the optimal one, considered the existing flow of vehicles in each time interval analyzed. The objective function of the problem is directly related to the flow of vehicles X; this will allow for cutting, as much as possible, the trip time from a demand node to an energy supply center for battery recharging in electric mobility. The demand generated at each intersection is related to K and will be the variable that will allow sizing the CS based on the vehicular flow directed to each charging station.

The hourly distribution of vehicles that will circulate in the case study is presented in Figure 5. In Figure 5, there is a total of vehicles without classifying what type of vehicle each is according to its primary energy, that is, we only know that at the hour 07:00 H, about 384 vehicles are circulating, without identifying how many of them are combustion or electric, buses, or trucks. Thus, an essential piece of advice for CI sizing is to test the percentage penetration of electric vehicles in the different regions where a case study is conducted. In the present case study, we assume that 100% of the number of vehicles distributed on an hourly basis in Figure 5 are electric. This consideration will allow us to size the necessary resources to cover the demand for battery recharging considering the 100% accessibility of electric vehicles in the vehicle park that participates in the study area. For this case study, it has been assumed that only light vehicles move in this area, and we consider that all of them are electric vehicles of class L7e-C with an average power of 15 kW for the sizing of their engine and that they are exclusively for urban use with an average range of 100 km and a battery capacity of 15 kWh. Figure 5 reveals another vital detail: the empirical criteria used to assume the most significant number of vehicles that will use the road map, $\mu +2\sigma$, where $\mu$ corresponds to the average value of the demands over 24 h, and $\sigma$ corresponds to the standard deviation. This criterion can be helpful to foresee the most significant power of the electric substation that will give the electric resource to the charging infrastructure for electric vehicles ensuring that the demand for battery recharging is covered in high demand hours from 06:00 H to 20:00 H. Therefore, the least energy demand considered in the worst scenario is 6.4 MWh.

The vehicular traffic concentration and the different trajectories are shown in Figure 6. It has been considered that the hour of highest demand occurs at 07:00 H (see Figure 5) in executing the algorithm, thus resulting in assigning a CS to each vehicular flow from the demand node to the consumption node in periods of high vehicular traffic concentration. Linear programming is based on the observance of constraints that limit the problem, and globally optimal solutions are obtained. An interesting detail that can be verified with Figure 6 is the observance of the conservation constraints, where it is satisfied that the supply provided by the CS satisfies the demand, achieving equality between the source (charging station) and the demand (need for battery recharging).

Below, with

Table 3. Partial demand for each charging station at hour 07:00 H.

ID	Vehicular	# Vehicles
	Concentration
55	−2.87	40
56	−2.95	41
57	−3.67	50
58	−3.38	47
59	−3.72	52
60	−3.26	46
61	−3.71	52
Total	23.56	328

Notes. # equals number.

, a summary from Figure 6 is shown for the reader’s convenience.

Table 3 shows the number of vehicles to be served by each charging station. The standard deviation of the number of vehicles to be covered by each CS at 07:00 H is five vehicles, which means that the variability of the number of vehicles between charging stations corresponds to only 10.38%. As a result, given that there is no significant variability, it can be concluded that the mathematical model proposed in this article not only assigns the resource with a lower cost but is also efficient in balancing the demand in the CI.

Therefore, in Figure 6, the planner can be supported to plan the sizing of each CS by analyzing the vehicular flow paths. Thus, the designer may be able to make engineering decisions considering real geo-located scenarios. On the other hand, the length of the accumulated roadway of each cross-section is another consideration: in this case, 13.98 km of the road network. This cumulative length parameter, when multiplied by the traffic concentration with units $\frac{\mathrm{Veh}}{\mathrm{km}}$, will allow us to calculate the number of vehicles that a CS should be able to serve (see Table 3). As a result, a linear model can give globally optimal solutions by observing the problem’s constraints. Future work will study the management of recharging at CS to increase customer satisfaction and control the power delivered to the EV batteries to guarantee the protection of the electrical distribution networks.

5. Conclusions

This article presents theoretical models for sizing and allocating resources to CS infrastructures for EVs in a heterogeneous transport system, considering trajectories and restrictions of capacity in each section of the road. This application will allow us to predict the most significant and most minor demands in hourly periods, thus allowing us to size the charging infrastructure with the least cost. Considering actual vehicular traffic variables in geo-referenced scenarios has allowed us to bring the problem closer to a real scenario. Microscopic analysis has made it possible to consider traffic variables, obtaining results that show the most significant vehicle density generated according to the topology of the scenario. Thus, the results obtained guarantee reliability for decision making in the different stages of engineering for the sizing of CI. Furthermore, with the microscopic analysis, it has been possible to understand vehicular traffic behavior and how it affects the resource allocation processes in vehicle systems in smart cities. Moreover, this algorithm is designed to reduce vehicle trip length with a geo-referenced vehicular flow routing system; the topology of the road network and its physical characteristics obtained free of charge from the OpenStreetMap platform are considered. Therefore, a novel approach is proposed in this article, considering the spatial study of the area under analysis, i.e., how the vehicle’s speed, the physical dimensions of the vehicle, the lengths from intersection i to j, and the spacing between vehicles interact. On the other hand, the organization and management of vehicular traffic on the road to allocate resources to charging stations is solved using the Hungarian algorithm and the multi-commodity flow problem. Linear programming is used to solve the problem, where a scalable model is proposed to deal with the increase of electric vehicles and possible changes in hourly patterns of mobility. As a result, the proposed academic model has a potential industrial application for infrastructure providers of charging stations in smart cities in developing countries. Finally, the study of vehicular traffic should be extended to macroscopic models designed to describe vehicles’ general behavior when the flow has constant interruptions. A further detail that is not considered is the efficient use of CS to manage CS use.