Variable Neighborhood Search for the Electric Bus Charging Stations

Location Design Problem

Michal Koháni

and Stanislav Babčan

Department of Mathematical Methods and Operations Research, University of Zilina, Univerzitna 1, Zilina, Slovakia

Keywords: Variable Neighborhood Search, Location Problem, Electric Buses, Charging Infrastructure.

Abstract: In the paper we describe a mathematical model and propose a solution method for solving the electric bus

charging station’s location design problem. We formulate a location-scheduling mathematical model, where

the set of possible charging stations can be in terminals and depots and tours of all vehicles are known and

will be unchanged. To solve the problem, we propose solving method based on the Variable Neighbourhood

Search metaheuristic. Using the proposed method, we realised extensive numerical experiments on the test

datasets created from real operational data provided by the municipal transport operator in the city of Zilina.

1 INTRODUCTION

One of the main trends in transport sector nowadays

is using of alternative energy sources as a power for

vehicles due to the environmental aspects. Many

manufacturers are developing vehicles with

technologies like CNG, hybrid-fuel systems or full

battery vehicles. This trend also affects bus

manufacturers.

Many public transport providers are trying to

include these types of vehicles into their fleet. The

reasons for that can be environmental sustainability,

government grants that support renewable resources,

operating cost savings, or simply prestige in front of

public and customers. Using these types of vehicles

in service can be difficult due to the limitations that

these technologies have. Especially electric buses

have significant disadvantage compared to buses with

combustion systems and that is maximum range.

Most of the electric buses today has lower range than

traditional diesel buses, and thus they cannot fully

replace them. Therefore, providers must solve either

battery charging or battery swapping in daily

operating when using electric buses.

In battery charging technology, there are multiple

possibilities. Electric bus can be recharged at stops by

inductive road charging or with a plug-in technology

at terminal stops. In these types of charging, charging

speeds of chargers are rather fast. On the other hand,

https://orcid.org/0000-0002-9421-4899

overnight charging is mostly in depots and charging

speeds are lower because of durability of battery

capacitors. If providers of public transport want to

include electric buses into daily operations, they must

think of building charging infrastructure.

In this paper we are dealing with a problem of

electric bus charging stations location. We formulate

location-scheduling mathematical model, where the

set of possible charging stations can be in terminals

and depots and tours of all vehicles are known and

will be unchanged. To solve the problem, we propose

solving method based on the Variable

Neighbourhood Search metaheuristic. Using

proposed method, we realised extensive numerical

experiments on the test datasets created from real

operational data provided by the municipal transport

operator in the city of Zilina.

2 LITERATURE REVIEW

The topic of the charging infrastructure emerges in

the literature during last few years. Authors are

dealing mostly with the design of charging

infrastructure for electric cars or a fleet of cargo

vehicles. Some of the authors are also addressing the

problem of charging infrastructure for electric buses

considering the limitations related to this area.

Authors in paper (Xylia, 2017) are proposing

Koháni, M. and Bab

can, S.

Variable Neighborhood Search for the Electric Bus Charging Stations Location Design Problem.

DOI: 10.5220/0012398000003639

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 13th International Conference on Operations Research and Enterprise Systems (ICORES 2024), pages 317-324

ISBN: 978-989-758-681-1; ISSN: 2184-4372

317

complex design of the charging infrastructure.

Authors proposed linear model for the location of

charging stations in the urban area and tested it on the

data of Stockholm’s bus lines network. The

calculation was performed using a commercial

CPLEX IP solver. To solve the problem of charging

station location for electric buses, the modifications

of the mathematical models for designing of the

charging infrastructure for passenger electric vehicles

can be used. In the paper (Dickerman, 2010) authors

used a model that verifies the location of charging

stations generated by the genetic algorithm. Several

authors have been inspired by location problems

leading to solve the mixed integer programming

problem. In (Bauche, 2014) the demand for electric

vehicles charging in the city of Lyon was estimated

based on a traffic survey and a proper placement of

charging stations was found by the cost minimization.

The optimization problem was resolved by a

universal IP solver. A similar methodology was used

in works in (Cavadas, 2014), (Lam, 2014) and

(Ghamami, 2015). Solutions of proposed

optimization problems are usually verified using

simulation methods such as in (Sweda, 2011). When

designing a network of charging stations for electric

buses, we will also take advantage of solving similar

problems. In (Czimmermann, 2017) and (Kohani,

2017), authors have proposed the methodology and a

mathematical model for location of charging stations

for the fleet of electric vehicles using the location-

scheduling model which is solved using the IP solver.

We will also take advantage of solving different types

of location problems. In the paper (Janacek, 2008)

authors have designed an efficient algorithm for

solving an uncapacitated facility location problem.

The basic methodology for solving the problem of

designing public service systems using location

problems was described in (Janacek, 2012).

Verification of all proposed methods and solutions

will require extensive computational and simulation

experiments, utilizing experience in designing of

emergency medical stations network and simulation

verification of these suggestions described in

(Janosikova, 2017).

3 MATHEMATICAL

FORMULATIONS OF THE

PROBLEM

Our goal is to find optimal locations for charging

stations for electric buses. In our approach we have

focused on plug-in technology for charging vehicles.

Number of charging points at each located charging

station is part of the solution as well. The model was

described in (Vasilovsky, 2019).

3.1 Problem Definition

The model respects current configuration of

schedules and trips of buses. Possible locations of

charging stations can be terminal stops and depots. At

terminal stops, buses wait for a next trip, in depots

they stay overnight. Daily tour of bus is to serve

multiple trips of given schedule. Each trip is the set of

stops that vehicle must serve at specified time. After

the tour the bus returns to depot. Bus can have

different schedules for each day in week thus the

model simulates operating and charging of vehicles

for multiple days (usually one week).

Let I be the set of terminal stops and depots, where

charging station can be placed. Let D be the set of

days. Let V be the set of all vehicles (electric buses).

Daily schedule of vehicle v

∈

V consists of ordered

set of trips J

that vehicle serves during the day d

∈

Order of the trips is defined by the time sequence. Set

contains deadhead trips (transfers without

passengers between terminal stops) as well. Each trip

starts and ends at terminal stop or depot. Each trip

(except deadhead) has information about time of

arriving at the stop and the time of leaving the stop.

Arrival time at the terminal stop i

is the end time of

the trip j −1 and departure time from the terminal stop

is the start time of the trip j if i

= i

. Otherwise,

vehicle must serve deadhead trip between stop i

and

. Let’s say that vehicle’s j − 1 trip has end time at

7:00 a.m. at the terminal stop i

and the next trip j

starts at 7:20 a.m. at the terminal stop i

and i

6= i

Also let suppose that time needed for travelling

between i

and i

is 10 minutes. Vehicle can leave

terminal stop i

at any time between 7:00 a.m. and

7:10 a.m. to be at the terminal stop i

on time. For

simplicity we allow that vehicle can only leave the

stop at the following cases and that is immediately

leaving the stop after arriving or staying at the stop as

long as possible before needed transfer to the next

trip’s, i.e., at 7:00 a.m. or 7:10 a.m. in above example.

From the point of charging, vehicle can be charged

between 7:00 - 7:10 a.m. at the terminal stop i

or in

the time 7:10 - 7:20 a.m. at the terminal stop i

(if

there are charging stations placed at the stops i

and

). These cases are considered as the trips and set J

contains both. Because of that, additional set J

pvd

, J

pvd

⊂

contains trips j

∈

with first case i.e.” leaving

the stop after arriving” case. This set is used for

building constraints which choose from one of these

cases. The value of energy consumption of vehicle v

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

318

∈

V on the trip j

∈

during the day d

∈

D is

represented by b

jvd

. Let T

ijvd

is the set that represents

time interval that vehicle v

∈

V spends at the terminal

stop/depot i

∈

I before trip j

∈

on the day d

∈

In this model, we use discretization of time intervals.

Set T

is union of all T

ijvd

for a day d. Charging

speed of the charging station placed at stop/depot i

∈

I is E

in the kilowatt hour units (kWh).

Binary variable y

represents decision about (not)

locating of charging station at stop i

∈

I. It is set to 1

if charging station will be placed at the stop/depot i.

Otherwise it is set to 0.

Integer decision variable q

represents number of

charging points at the station i. Binary variable x

ijvtd

represents decision about charging vehicle. Variable

is set to 1 if vehicle v is charged before the trip j at the

stop i at the time t on the day d.

The continuous decision variable d

jvd

represents

amount of energy (kWh) stored in battery of the

vehicle v before trip j on the day d. Value of the

variable depends on amount of energy before trip j −

1, consumption of energy on the trip j−1 and amount

of the charged energy before trip j−1.

Binary decision variable z

represents decision

which deadhead case will be applied for vehicle v and

trips j and j + 1. If z

is set to 1, vehicle v can be

charged at the final stop of the trip j−1 (final stop of

the trip j−1 is same as start stop of the deadhead trip

j). If z

is set to 0, vehicle v can be charged at the final

stop of the the deadhead trip j (start stop of the trip

j+1).

3.2 Mathematical Model of the

Problem

𝑚i𝑛𝑞



∈

(1)

𝑞



≤𝑆𝑦



,∀𝑖 ∈ 𝐼

(2)

𝑑



= 𝑀 ,∀𝑣∈𝑉 (3)

𝑥



∈





∈

≤𝑞



,∀𝑑∈𝐷,𝑡∈𝑇,𝑖∈𝐼

(4)

𝑑



+ 𝑒



𝑥



∈



∈

≤𝑀 ,∀𝑑∈𝐷,𝑣

∈𝑉,

𝑗

∈

𝐽



(6)

𝑑



≤ 𝑑



− 𝑏



+ 𝑒



𝑥



∈



∈

,∀𝑑

∈𝐷,𝑣∈𝑉,

𝑗

∈

𝐽



−

{

}

(6)

𝑑



≤ 𝑑







+𝑒



𝑥







∈







∈

,∀𝑑

∈𝐷−

{

}

,𝑣∈𝑉,

𝚥

∈

𝐽



(7)

𝑥



∈



≤𝑆𝑧



∈

,∀𝑑 ∈ 𝐷,𝑣 ∈ 𝑉,

𝑗

∈

𝐽

𝑝



(8)

𝑥



∈



≤ 𝑆1−𝑧





∈

,∀𝑑

∈𝐷,𝑣∈𝑉,

𝑗

∈

𝐽

𝑝



(9)

𝑦



∈

{

0,1

}

,∀𝑖∈𝐼 (10)

𝑞



∈𝑍



,∀𝑖∈𝐼 (11)

𝑥



∈

{

0,1

}

,∀𝑑∈𝐷,𝑣∈𝑉,

𝑗

∈

𝐽



,𝑖∈𝐼,𝑡

∈𝑇



(12)

𝑑



∈𝑅



,∀𝑑∈𝐷,𝑣∈𝑉,

𝑗

∈

𝐽



(13)

𝑧



∈

{

0,1

}

,∀𝑑∈𝐷,∀𝑣∈𝑉,

𝑗

∈

𝐽

𝑝



(14)

The objective (1), is to minimize the total sum of

charging points of the electric infrastructure.

Constraint (2) ensure that charging points can be sited

at the location i only if charging station is placed at

the location i. Constraint (3) initializes battery state of

the vehicles to full battery capacity on the first day

before first trip j. Constraint (4) ensure that number

of vehicles charged at the station i at the same time t

must be lower or equal to number of charging points

at i. Constraint (5) prohibit to exceed battery

capacity. Constraint (6) ensure that battery is

charging and discharging according to trips and

charging at charging stations. Constraint (7) is similar

to constraint (6), except that constraint is applied for

last trip e

of the previous day d

∈

1 and first trip 1 of

the next day d. Constrains (8) and (9) ensure that

vehicle v is charged before or after deadhead trip j.

Constraints (10 - 14) are obligatory constraints.

4 VARIABLE NEIGHBORHOOD

Variable Neighborhood Search (VNS) is a

metaheuristic used to solve combinatorial and

nonlinear optimization tasks. Its principle consists in

systematically changing the structure of the

neighborhood while searching for a solution. VNS

can be implemented in a variety of ways, where

slightly different steps and strategies can be used

compared to basic VNS, but the principle of changing

the neighborhood remains. In this work, we will deal

with basic VNS (Mladenovic, 1997)

4.1 VNS

Changes to the structure of the neighborhood in VNS

are based on the following principless:

Variable Neighborhood Search for the Electric Bus Charging Stations Location Design Problem

319

• A local optimality criterion in one neighborhood

may not be a local optimality criterion for

another neighborhood.

• The global optimality criterion is a local

optimality criterion for all neighborhoods.

• Empirical evidence shows that for most tasks

the local optimality criteria are relatively close

to each other.

VNS is described by three basic steps:

• Finding an initial solution (Shaking procedure)

where the algorithm try to avoid a local

minimum in the given area.

• Local search, where the current solution could

be improved by using simple heuristics

(replacement heuristics, insertion heuristics...).

• Changing of neighborhoods, where the

algorithm can continue searching in the next

neighborhood or start searching the

neighborhoods from the first one.

4.2 Solution Approach

4.2.1 Creating a Starting Solution

The finding of initial solution for the VNS algorithm

in our paper was implemented in two ways:

In the first method, a so-called "simulation test

run" was launched, where charging stations and

charging points were created in the event that an

electric bus arrived at a stop and its battery capacity

dropped below 70% and no charging point was

available for charging, at that stop there was a new

charging station built (if there was no charging station

at the stop) or another charging point was added to the

stop (if a charging station was already built at the

stop).

In the second method, the number of charging

stations to be built was determined At each stop it was

decided whether a charging station would be built on

it or not based on probability. If a charging station

was built at a given stop, the number of charging

points in the range of one to five was generated for it.

Subsequently, the solution created in this way was

subjected to verification of the admissibility of the

solution using simulation. If the solution was

admissible then this solution was used as the starting

solution. Otherwise, the random solution generation

process had to be repeated.

The advantage of the first method of generating

the initial solution was the speed of the solution

generation, because only one simulation run was

always enough to generate an acceptable solution. In

the second method, a situation could arise that the

generated solution might not be admissible and the

whole process could be repeated several times, which

significantly increased the time for creating the initial

solution.

The disadvantage of the first method was the same

starting point for VNS algorithm, and thus the process

of searching for admissible solutions was limited only

to the set of admissible solutions that can be reached

from this single solution. In the second method, it is

assumed that due to the different starting points for

the VNS algorithm, we were able to search a larger

set of admissible solutions than in the first case.

4.2.2 Local Search

The local search process in the current neighborhood

of the solution was implemented by randomly

selecting a solution from the current neighborhood of

the solution to avoid getting stuck in the local

optimum and to examine a larger number of

admissible solutions. Acceptance of the searched

solution as the new best solution of the solved

problem was realized in the case that the searched

solution was admissible, where the verification of the

admissibility of this solution was tested using

simulation. When searching each neighborhood, the

neighborhood was defined in a way where all

solutions in the current neighborhood were better in

terms of the objective function than the best solution

found so far.

4.2.3 Change of Neighborhood

After completing the search of the current

neighborhood for the best solution found so far, it was

necessary to decide on the next neighborhood to be

searched. If admissible solution was found in the

current neighborhood, this solution was accepted as

the best solution found so far and the search of the

neighborhood continued in the first neighborhood of

the defined neighborhood structure. Otherwise, the

search continued in the next neighborhood of the

neighborhood structure. If a situation arose in which

the algorithm did not find an acceptable solution even

in the last neighborhood of the defined neighborhood

structure, the VNS algorithm ended its operation.

4.2.4 Structure of the Neighborhoods

The neighborhood structure for the VNS algorithm in

our approach consists of five neighborhoods. The

basic structure of the neighborhood is organized in

such a way that when moving to the next

neighborhood, this neighborhood is more extensive in

terms of the number of solutions in the given

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

320

neighborhood and more computationally demanding

when choosing the solution from the current

neighborhood.

The Neighborhood Induced by the Charging Point

Withdrawal Operation

This neighborhood contains solutions that can be

reached by removing any charging point from the

best-found solution. When removing charging point,

the charging station must not be cancelled, which

means that this point could only be removed from

stations where there is more than one charging point.

Neighborhood Induced by Canceling of the

Charging Station

This neighborhood contains solutions defined by

canceling the station from the set of built charging

stations of the best-found solution, which means that

the number of charging points at the cancelled

charging station was set to zero.

Neighborhood Induced by Operations of Building

a Charging Point and Cancelling the Station

In this neighborhood, there are solutions that can be

reached by cancelling the charging station and at the

same time adding a new charging point to an already

built charging station from the set of built stations of

the best solution found so far.

Neighborhood Induced by the Operation of

Building a new Charging Station and Cancelling

Another Charging Station

This neighborhood contains solutions that were

defined by building a new charging station at a stop

where a charging station has not yet been built and at

the same time cancelling a charging station from the

set of built stations. In the case of cancelled station,

the condition that the given station had two or more

charging points had to be met. The value of the

objective function of the solution where the station

with only one charging point would be cancelled is

the same as the value of the objective function of the

best-found solution.

Neighborhood Induced by the Operation of

Exchanging Two Charging Stations

In this neighborhood, there are solutions that were

created by exchanging the number of charging points

between the two built charging stations of the best-

found solution. To make such a solution more

advantageous than the best solution found so far, one

charging point was removed from one of these

stations after the process of changing the charging

points, which caused a decrease in the value of the

objective function compared to the value of the

objective function of the best solution found so far.

Removing a point could also lead to the cancellation

of the charging station itself.

4.2.5 Verification of Feasibility of the

Solution

When generating a starting solution or examining the

solution in the currently searched neighborhood, it

was necessary to verify whether these solutions are

feasible. A feasible solution for the problem of

charging station placement is one where the charging

station placement covers the requirements of all shifts

performed during the week.

To verify the admissibility of the solution, a

heuristic simulation approach based on the discrete

event simulation of the operation of electric buses

during the week was created. Individual events were

inserted into the event calendar, which was

implemented using a priority queue, where the

priority for each event was the time at which the given

event should occur. When selecting currently

processed events, the event with the lowest

occurrence time was selected. During the simulation,

four types of events occurred:

Arrival of the Electric Bus at the Stop

In this event, the battery capacity of the electric bus

was reduced due to the length of the route it travelled

before reaching the relevant stop associated with the

event. After each simulation spet, there was a check

of the current battery capacity, where if the current

capacity dropped to negative values, it meant that the

electric bus would not be able to reach next stop. In

this case, the simulation ended its operation, and the

currently tested solution was marked as infeasible

solution. If the battery capacity did not drop below

zero value, further events were planned. If the

charging station was built at the current stop and free

charging point was available for charging, the start

and end of electric bus charging events were added to

the event calendar. The times of occurrence of these

events were planned considering the currently

missing battery capacity and the next connection that

has electric bus to perform. If the stop of the next trip

was different from the current stop, events for the

start and end of the execution of the manipulation

move to the starting stop of the next trip were added

to the event calendar. If the electric bus had no other

trip scheduled after the current trip, the electric bus

ended its operation at the depot for the given day.

Departure of the Electric Bus From the Stop

The processing of this event meant the departure of

the electric bus from the stop associated with this

event.

Variable Neighborhood Search for the Electric Bus Charging Stations Location Design Problem

321

Start of Electric Bus Charging

The processing of this event caused one charging

point to be occupied at the charging station at the stop

associated with the event.

End of Electric Bus Charging

When processing these events, the current battery

capacity of the electric bus was increased in relation

to the time during which it was connected to the

charging point. After increasing the current battery

capacity of the electric bus, one charging point was

released at the charging station of the stop connected

to the event.

5 NUMERICAL EXPERIMENTS

The implementation was carried out in the IntelliJ

IDEA Community Edition 2020.3.2 development

environment. The language used for the

implementation was Java using JDK version number

15. The experiments were performed on our personal

computer. Computer parameters:

• Processor: Intel(R) Core(TM) i5-7300HQ CPU @

2.50GHz 2.50 GHz

• Installed RAM: 8.00 GB (Usable memory: 7.87 GB)

• Operating system: Windows 10 Home

To test the implementation of the VNS metaheuristic,

it was necessary to choose the type of electric bus that

will be used during the experiments. The Urbino 8.9

electric model from the Solaris brand was used as the

type of electric bus. The parameters specified by the

manufacturer for this electric bus are:

• Battery capacity: 140 kWh

• Energy consumption: 0.8 kWh/km

• Charging speed (plug-in): 1.33 kWh/min

• Charging speed - depot: 0.4 kWh/min

5.1 Test Scenarios

Three scenarios were proposed for testing, which

represent the operation of the electric bus in different

climatic conditions. All scenarios are created from

real operation data provided by the DPMZ –

transportation company.

The first scenario represents the operation of the

electric bus during the spring months. During these

months, significant effects of heat or winter are not

frequent, therefore the parameters of the electric bus

were kept as specified by the manufacturer.

Another scenario represents the operation of the

electric bus during the winter months. Because of low

temperatures on the battery, the capacity of the

electric bus was reduced by 25%. Also in the winter

months, it is necessary to heat the interior of the bus,

which represents additional battery consumption,

therefore the energy consumption per kilometre was

increased by 35%.

The last scenario represents the operation of the

electric bus during the summer months. During these

months, the battery consumption is burdened by the

operation of the interior air conditioning, therefore we

also increased the energy consumption per kilometre

by 35%.

For each of these scenarios, three variants of the

type of electric bus used were used. In the first

variant, the parameters of the electric bus were set to

their default values, in the second, the battery

capacity of the electric bus was increased by one

third, and in the third variant, the battery capacity was

reduced by one third.

5.2 Numerical Experiments

Experiments were carried out testing different

configurations of the VNS metaheuristic in terms of

the number of neighborhoods it searches and in terms

of the order of neighborhoods in which the search is

performed. During the execution of the experiments,

metaheuristics was run several times due to the

stochastic nature of the selection of elements in

individual neighborhoods.

5.2.1 Testing the VNS Configuration

In this subsection, the individual experiments

performed with different configurations of the VNS

algorithm are explained. For each experiment, the

value of the objective function, the number of

charging stations and charging poits and the time

required to perform the experiment with the given

configuration are given. The effectiveness of

individual experiments was evaluated according to

the averages of the objective function values and the

time required for the calculation.

In all tables the column denoted as “stat” is the

number of locations where the charging station will

be built, column denoted as “Pts.” Represents the

number of charging points at all stations, “Obj.” is the

value of the objective function and “comp. time” is

the computation time in second.

5.2.2 Basic Configuration

For the basic configuration, all neighborhoods were

used in the order in which they were listed in section 4.

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322

Table 1: Results for Basic configuration.

Scenario Stat. Pts. Obj.

Comp.

time [s]

Standard

battery

capacity

Spring 8 8 240 000 492

Winter 10 16 318 000 720

Summer 9 11 276 000 392

Increased

battery

capacity

Spring 8 8 240 000 593

Winter 10 11 303 000 425

Summer 8 8 240 000 275

Decreased

battery

capacity

Spring 10 13 309 000 284

Winter 11 19 354 000 500

Summer 10 14 312 000 493

Table 2: Results for three operations.

Scenario Stat. Pts. Obj.

Comp.

time [s]

Standard

battery

capacity

Spring 8 9 243 000 148

Winter 10 15 315 000 254

Summer 10 12 306 000 198

Increased

battery

capacity

Spring 8 9 243 000 186

Winter 9 13 282 000 201

Summer 8 8 240 000 175

Decreased

battery

capacity

Spring 11 12 333 000 184

Winter 12 19 381 000 234

Summer 11 15 342 000 253

Table 3: Results for four operations.

Scenario Stat. Pts. Obj.

Comp.

time [s]

Standard

battery

capacity

Spring 8 9 243 000 144

Winter 10 16 318 000 239

Summer 10 12 306 000 163

Increased

battery

capacity

Spring 7 10 219 000 135

Winter 11 12 333 000 155

Summer 8 8 240 000 155

Decreased

battery

capacity

Spring 10 12 306 000 168

Winter 12 18 378 000 265

Summer 11 17 348 000 281

5.2.3 Changing the Number of Operations

Inducing Searched Neighborhoods

In the following experiment, algorithm

configurations with a lower number of operations

inducing individual neighborhoods were tested. In the

first experiment, a configuration with the first three

operations inducing neighborhood was used, in the

second with the first four operations.

5.2.4 Changing the Order of Searched

Neighborhoods

In this experiment, configurations were designed that

contain all the neighborhoods of the basic

configuration, but the order of searching these

neighborhoods was changed. In this experiment, the

order of searched neighborhoods was opposite to that

of the basic configuration.

Table 4: Results for opposite order of operations.

Scenario Stat. Pts. Obj.

Comp.

time [s]

Standard

battery

capacity

Spring 9 13 282 000 452

Winter 11 20 357 000 516

Summer 10 17 321 000 461

Increased

battery

capacity

Spring 7 13 228 000 391

Winter 9 20 303 000 517

Summer 9 18 297 000 453

Decreased

battery

capacity

Spring 11 18 351 000 521

Winter 12 27 405 000 689

Summer 10 19 327 000 537

5.3 Discussion

Configurations with three and four operations

inducing individual neighborhoods gave the worst

results in only two cases, even though they worked

with a smaller number of admissible solutions

searched. Since the neighborhood order of these

configurations was taken from the order of the basic

configuration, we can assume that the order of

searched neighborhoods has a significant impact on

the solutions that the algorithm can produce.

The configuration that searched neighborhoods in

the reverse order of the base configuration did the

worst, where the solution value it was able to deliver

for individual scenarios represented the worst value

delivered among all configurations in most scenarios.

Variable Neighborhood Search for the Electric Bus Charging Stations Location Design Problem

323

6 CONCLUSIONS

In the paper we described the mathematical model

and propose solution method for solving of the

electric bus charging station’s location design

problem. We formulated location-scheduling

mathematical model, where the set of possible

charging stations can be in terminals and depots and

tours of all vehicles are known and will be

unchanged. To solve the problem, we proposed

solving method based on the VNS metaheuristic.

Using proposed method, we realised extensive

numerical experiments on the test datasets created

from real operational data provided by the municipal

transport operator in the city of Zilina. From the

numerical experiments we can see that choosing to

search the neighborhoods from simpler to more

complex ones (basic configuration) is a better strategy

than searching the neighborhood s from more

complex to simpler ones.

ACKNOWLEDGEMENTS

This work was supported by the research grant VEGA

1/0654/22 Cost-effective design of combined

charging infrastructure and efficient operation of

electric vehicles in public transport in sustainable

cities and regions.

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