The Optimal Location of the Electric Vehicle Infrastructure with

Heterogeneous Batteries in the Highways

Mohammed Bourzik

, Ahmed Elhilali Alaoui

Modelling and Scientific Computing Laboratory, Faculty of Science and Technology, Fez, Morocco

Euro-Mediterranean University of Fez, BP 51, Fez, Morocco

Keywords: Electric vehicles, Heterogeneous batteries, Long road, Mathematical programming.

Abstract: The dynamic wireless charging makes the possibility of charging electric vehicles without contact and while

it is in motion from the transmitters buried (segments and inverters) under the road. This technology is applied

for homogeneous buses by the Korean institute of advanced technology (KAIST), called online electric

vehicles (OLEV) (Jang, 2012). Our contribution in this work is to study the problem of locating wireless

charging infrastructure on a long route between origin O and destination S, with heterogeneous battery

vehicles. On the first side, each type of vehicle requires its allocation of segments in the road because of the

heterogeneity of batteries, which increases the number of recharge transmitters in the highway; for this

purpose, we search to minimize the infrastructure cost by reducing the number of segments and inverters. On

the other hand, the activity of a recharge segment may be helpful for one vehicle and useless for the other

since each vehicle type has its characteristics (autonomy, puissance, battery capacity). For this reason, we aim

to minimize the use of the recharge transmitters for each vehicle type. We propose to model the problem as a

mathematical problem and to solve it by CPLEX optimizer for limited instances.

1 INTRODUCTION

Since the previous years, the characteristics of

electric vehicle (EV) batteries have improved

considerably. For the same weight and similar

volume, the quantity of energy available increased,

which increases their autonomy, and it is sufficient to

travel on daily trips (for small to medium trips).

However, it is not enough for long journeys like the

highway. It is characterized by a long distance

between the origin and the destination, and the high

speed of vehicles on this kind of road drains their

battery very quickly.

Wireless charging (WC) is an effective solution

for long journeys; since it can charge the batteries

dynamically while the vehicle is in motion from a set

of inductive cables and inverters placed on the road

to meet the vehicle's charging requirements at all

times. This technology makes it possible to transfer

several tens of kilowatts over short distances (a few

tens of centimetres) with good efficiency and safety

for the human body if a correctly sized magnetic

shielding is present. The advantage would thus be to

recharge vehicles on the highway.

https://orcid.org/0000-0001-6735-8013

The WC system (WCS) is composed of EV and

power transmitters (see figure 1). An inverter

converts the DC power into high-frequency AC or

voltage. After converting the alternating magnetic

fields generated by an underground electric power

line to electric power, a pickup coil installed on the

bottom of the vehicle receives this electric power. The

received power goes through a rectifier and regulator

before arriving at the battery (Chun, 2014).

The WCS is an ideal solution for the highway, but

the high cost of this technology limits the use only

when the charge is needed. Many works presented the

problem of locating a minimum cost infrastructure of

electric vehicle induction charge. However,

researchers considered the vehicle's batteries to be

homogeneous. In reality, each vehicle battery has its

characteristic, so our contribution is to find an optimal

location of the electric vehicle infrastructure with

heterogeneous batteries in the highways. The

heterogeneity of vehicle batteries adds more

difficulty to search for places of lack of charge

because each type has its characteristics (range

power, consumption, charging rate). It requires a

mathematical study to find a minimum cost

244

Bourzik, M. and Elhilali Alaoui, A.

The Optimal Location of the Electric Vehicle Infrastructure with Heterogeneous Batteries in the Highways.

DOI: 10.5220/0010731900003101

In Proceedings of the 2nd International Conference on Big Data, Modelling and Machine Learning (BML 2021), pages 244-248

ISBN: 978-989-758-559-3

infrastructure that ensures the movement of each type

of vehicle without remaining out of load by the

minimum use of the segments for each vehicle type;

this is precisely our goal to this work.



Figure 1: Wireless charging system

2 LITERATURE REVIEW

Wireless charging (WC) allows charging the

vehicle’s battery without contact, known as Wireless

Power Transmitters (WPT), existing in two types.

The first type is the stationary WPT, which presents a

recharging mode from which the VE can be charged

by induction when it is in stop mode. The use of this

world is equivalent to the use of a charging cable,

except that stationary WPT technology has the

advantage of being practical and more secure for

more details see (Young, 2016). The second type is

dynamic WPT, which allows an EV to be charged by

induction while it is in motion. The Korean Advanced

Institute of Science and Technology KAIST has

installed this technology on its campus to allow its

buses to charge by induction while running called by

one line electric vehicle (OLEV)(Jang, 2012).

Ko, Y. D. and Jang Y. J (Ko, 2014) introduced the

concept of battery power that was instantly required.

They introduced a mathematical model that seeks to

minimize the cost of installing the power transmitters

and the cost of the battery according to its capacity.

They solved the model using the optimization

algorithm for particle swarms. Jeong et al. (Jeong

2014) have added the impact of battery charging and

discharge frequency on its life cycle.

Among the best-known works, we cite that of Young

Jae Jang (Jang, 2012), who proposed a mathematical

model to determine a compromise between the

battery capacity of an EV and the location of the

charge transmitters inductive on a fixed route a single

path. They assumed that the bus travel speed is preset,

and the batteries are identical. Liu and Song (Liu,

2017) studied the dynamic behaviour of this model

using a nonlinear model solved by genetic algorithms.

Young Jae Jang (Ko, 2014) compared the initial

investment costs of three types of charging systems.

The first type is stationary wireless charging (SWC),

in which charging happens when the vehicle is parked

or idle. The second type is quasi-dynamic wireless

charging (QWC), which allows the charging when the

car is moving slowly or in stop-and-go mode, and

dynamic wireless charging (DWC), in which vehicle

can charge even when it is in motion.

Nisrine Mouhrim et al. (Nisrine, 2018) do the

generation of the multiple paths. They considered a

multipath network between the origin and the

destination station. They sought a compromise

between the cost of installing the power transmitters

and the cost of the batteries, which are assumed

identical. Hassan Elbaz and Elhilali Alaoui Ahmed

(Hassane, 2020) search for a compromise between the

infrastructure cost and the battery capacity in a

multipath network, round-trip.

Xiaotong Sun et al. (Xiaotong Sun et al., 2020)

investigated the optimal deployment of static and

dynamic charging infrastructure considering the

interdependency between transportation and power

networks.

3 PROBLEMS AND MODELING

3.1 Problem Description and Objective

We consider a highway of origin O and destination S,

and a set of vehicles with heterogeneous batteries, and

we seek to satisfy by the least cost its needs of the

load during their journeys from O to the destination S

by the allocation of the power transmitters on the road

like a dynamic station (Fig. 2).

Figure 2: Highway with dynamic stations

We assume that the highway is divided into two

zones, the 1

road zone without stations, and the 2

that we will put the dynamic stations (Figure 2).

The 2

zone is subdivided into several congruent

segments, and we will consider each segment as a

potential transmitter. If the charging is needed, the

segment will be equipped with an inductive emitting

cable plus an inverter or will fit only with an emitting

cable, and if the loading is not needed, the segment

The Optimal Location of the Electric Vehicle Infrastructure with Heterogeneous Batteries in the Highways

245

will be inactive (see Fig. 4). We note that a single

inverter can power a limited series of successive

active segments.

Figure 3: Types of segments

We search firstly to determine the minimum of

active segments and inverters in the 2

zone to meet

the energy needs of vehicles.

On the other hand, each type of battery requires its

allocation of the segments because of the

heterogeneity of batteries; An active segment can be

useless for one vehicle type and profitable for the

other; for that, we search to minimize the number of

segments used by each vehicle type.

3.2 Mathematical Model

We seek to model the problem as a mathematical

problem with constraints. We present as follows a set

of notations, data, constraints, and objective function.

In this work, we assume that the fleet of vehicles has

the same speed and acceleration on the highway, and

the wireless charging system is capable of supplying

each type of vehicle.

3.2.1 Notations and Data

As we mentioned earlier, we discretized the highway

into segments of the same length; the first segment

will be denoted by 𝑂, the last segment by 𝑓, and the

other segments by 𝑔 with 𝑔 𝜖 𝑂,…,𝑓.

Vehicles are considered to have a heterogeneous

battery, let 𝜑 be the set of batteries and 𝛼 the index of

a type of vehicles with 𝛼𝜖 0,…,

𝜑

.

The infrastructure cost is composed of the cost of

the recharge segments and the price of the inverters.

Let 𝐶



and 𝐶



respectively the unit cost of an

active segment without an inverter and the unit cost

per inverter.

The starting point of each vehicle type is 𝑂, we not

that 𝑡



the arrival time of the fleet at the starting point

of segment 𝑂. As mentioned earlier, the fleet of

vehicles has the same speed and acceleration on the

highway, so it will have the same time to arrive at the

starting point of each 𝑔

segment, which we note

by 𝑡



. We consider the following notations:

 𝐼





: The battery capacity of vehicle 𝛼

 𝑇



𝑡: The energy supply rate of vehicle 𝛼

 𝐷





𝑡



: The energy consumption rate of

vehicle 𝛼

 𝐼





𝑎𝑛𝑑 𝐼





: Are the lower and upper limits

of the battery level, respectively. These

values have the following relationship.

𝐼





𝐼





𝛿 and 𝐼





𝐼





𝛽 with

𝛿,𝛽 ∈



0,1



 𝑁



: The maximum number of the active

segment in each series that can use one

inverter

3.2.2 Decision Variables

The highway is subdivided into several congruent

segments, and we search the minimum number of

active segments and inverters into the road, that we

define two decision variables 𝑋



𝑎𝑛𝑑 𝑍



such as:

𝑋





1 𝑖𝑓 𝑡ℎ𝑒 𝑔



𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑖𝑠 𝑎𝑐𝑡𝑖𝑣𝑒

0 otherwise

𝑍





1 𝑖𝑓 𝑡ℎ𝑒 𝑔



𝑠𝑒𝑔𝑚𝑒𝑛𝑡 ℎ𝑎𝑠 𝑎𝑛 𝑖𝑛𝑣𝑒𝑟𝑡𝑒𝑟

0 otherwise

Each type of vehicle has its character because of

the heterogeneity of the batteries, so the use of the

segments is different from a kind of vehicle to

another; we then define the decision variable 𝑦





such

as:

𝑦







1 𝑖𝑓 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝛼 𝑢𝑠𝑒𝑠 𝑡ℎ𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑔

0 otherwise

Let 𝐼





𝑡



be the amount of energy in the battery 𝛼

at time 𝑡.

3.2.3 Constraints

1. We assume that vehicles start at the beginning of

segment 𝑂 with the maximum load 𝐼





𝐼





𝑡





𝐼





∀𝛼=0,…,|𝜑|

2. The remaining charge at the beginning of each

segment must always be greater than the battery’s

minimum capacity

𝐼



𝑡



  









𝐷





𝑡



𝑇



𝑡 𝒚

𝒈

𝜶

𝑑𝑡

𝐼





∀𝑔0,..,𝑓1 ; ∀𝛼0,…,|𝜑|

3. Updates the remaining charge at the beginning of

each segment.

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246

𝐼



𝑡



min  𝐼





; 𝐼



𝑡

















𝐷





𝑡





𝑇



𝑡 𝒚

𝒈

𝜶

𝑑𝑡, ∀𝑔0,…,𝑓1 ∀𝛼=0,…,|𝜑|

4. A segment g may be used by vehicle 𝛼 only if it is

active.

𝑦





𝑋



∀𝛼0,…,|𝜑|, ∀𝑔0,..,𝑓

5. A segment 𝑔 is active if it is used by at least one

type of vehicle.

𝑋





∑

𝑦





, ∀𝑔0,..,𝑓

6. A segment 𝑔 cannot have an inverter unless it is

active.

𝑍



 𝑋



∀𝑔0,..,𝑓

7. The start of each series of active segments must

contain one inverter, and one inverter can only

supply at most 𝑁



active segments.

𝑍



1𝑋



 𝑋



;∀𝑔0,..,𝑁



1

𝑍



1𝑍



𝑋















 𝑋



2

∀𝑔𝑁𝑖𝑛1,..,𝑓

─ For (1) if the segment 𝑔 is active and its

predecessor is active in this case, the segment 𝑔

does not have an inverter because it belongs to an

active series of length less than or equal to 𝑁



otherwise, if a segment 𝑔 is active and its

predecessor is inactive in this case the segment 𝑔

will have an inverter.

─ For (2), a g-segment can only contain one inverter:

If g is the start of an active series (i.e., g is an active

segment and its predecessor is idle). Or if g is

prefixed with a longer active string than 𝑁



segments.

3.2.4 Objective Function

We seek to minimize the cost of the infrastructure by

reducing the number of active segments and inverters

on the road (3). On the other hand, we search to

minimize the segments used by each vehicle type (4).

Therefore, we have two objectives to optimize.

𝑴𝒊𝒏 𝑪

𝒔𝒈𝒕

𝑿

𝒈





 𝑪

𝒊𝒏𝒗

𝒁

𝒈





 3

𝑴𝒊𝒏  𝒚

𝒈

𝜶

𝒇

𝒈𝟎𝜶𝝐𝝋







4 PROBLEMS AND SOLVING

Our objective in this work is to build at the lowest cost

a wireless charging infrastructure that ensures the

movement of the heterogeneous fleet of vehicles

without remaining out of charge by the minimum use

of the segments for each vehicle type. The problem is

transformed into an equivalent linear programming

problem and solved with a CPLEX optimizer for a

limited instance.

4.1 Transport Network Data

The instances used in this example serve only to

validate the model because the CEPLEX optimizer

can only solve a limited instances problem. It solves

our problem with four types of vehicles and 30

segments. Tables 1 and 2 contain the energy supply

rate, the energy consumption rate of each vehicle

type, and the other data.

Table 1: Vehicles data

Battery

Capacity

𝐼





The energy

supply rate

The energy

consumption

rate

Vehicle 1 8,8 4.2 1,8

Vehicle 2 32,2 3 1

Vehicle 3 18,4 5 4

Vehicle 4 25 5.5 0,9

Table 2: Other data

Notation Description Value

𝐶



The unit cost per

inverter

3000

𝐶



The unit cost of an

active segment without

an inverter

800

𝑁



The maximum number

of the segment in each

series can use one

inverter

𝛿 The lower limit

coefficients

0.2

𝛽 The upper limit

coefficients

0.8

The Optimal Location of the Electric Vehicle Infrastructure with Heterogeneous Batteries in the Highways

247

4.2 Results

We solve the problem for each one of the objectives

because CPLEX can only solve mono-objective

problems. For the first objective, we search to

minimize the infrastructure cost, and for the second

objective, we search to minimize the set of segments

used by each vehicle type. Table 3 contains the results

for each objective.

Table 3: Results for each objective

Minimizing

the 1

jective

Minimizing

the 2

jective

The number of segments

used

y the vehicle 1

18 12

The number of segments

used

y the vehicle 2

7 6

The number of segments

used

y the vehicle 3

23 21

The number of segments

used

y the vehicle 4

8 2

The infrastructure cost

(The cost of segments + the

cos

of inverters)

36400

41000

The uses number of

segments by vehicles

 𝑦











When we search to minimize the infrastructure

cost, each type of vehicle will necessarily use the

activated segments to not remain in breach of the load

even if they are useless for some vehicles, which

increases the use of the segments.

When we minimized the number of segments used

by the vehicles, the cost of the infrastructure increases

because each type of vehicle uses the segments when

the load is needed, which increases the number of

active segments because the locations where the

charge is required are different from one vehicle to

another.

5 CONCLUSION

This work presented a mathematical model that aims

to find an optimal location for the wireless charging

infrastructure on the highway to ensure the movement

of a heterogeneous fleet of vehicles without

remaining out of charge by the minimum use of

segments for each vehicle type. We solved the

problem by CEPLEX optimizer for each objective

function with limited instances. The next step is to

find a compromise between the two objectives, so we

will use a metaheuristic as a method of resolution.

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