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sustainability

Article

Locating Battery Swapping Stations for a Smart

e-Bus System

Joon Moon 1, Young Joo Kim 2, Taesu Cheong 1,* and Sang Hwa Song 3, *

1

School of Industrial Management Engineering, Korea University, Seoul 02841, Korea; nrbam123@korea.ac.kr

2Logistics System Research Team, Korea Railroad Research Institute, Uiwang 16105, Korea; osot@krri.re.kr

3Graduate School of Logistics, Incheon National University, Incheon 22012, Korea

*Correspondence: tcheong@korea.ac.kr (T.C.); songsh@inu.ac.kr (S.H.S.);

Tel.: +82-2-3290-3382 (T.C.); +82-32-835-8194 (S.H.S.)

Received: 15 December 2019; Accepted: 30 January 2020; Published: 5 February 2020

Abstract:

With the growing interest and popularity of electric vehicles (EVs), the electriﬁcation of

buses has been progressing recently. To achieve the seamless operation of electric buses (e-Buses) for

public transportation, some bus stations should play the role of battery swapping station due to the

limited travel range of e-Buses. In this study, we consider the problem of locating battery swapping

stations for e-Buses on a passenger bus trafﬁc network. For this purpose, we propose three integer

programming models (set-covering-based model, ﬂow-based model and path-based model) to model

the problem of minimizing the number of stations needed. The models are applied and tested on the

current bus routes in the Seoul metropolitan area of South Korea.

Keywords:

electric vehicle; electric bus; battery swapping; infrastructure optimization; sustainability

1. Introduction

As the concern over global warming and energy issues grows, electric vehicles are increasingly

considered as an alternative to traditional vehicles and have hence gained much attention from the

general public. With the increasing interest and popularity of electric vehicles (EVs), many countries

have developed electric buses and introduced them on a trial basis. In South Korea, the Seoul

metropolitan government has operated electric buses around signiﬁcant tourist destinations and has

been gradually expanding the network [

1

]. Jeju island, the largest island in Korea, is preparing for

pilot electric bus operation [

2

]. In addition to these attempts, academic research into the usefulness

and operation policies of electric vehicles is actively underway [

3

]. Outside Korea, China has grown its

market size of electric buses signiﬁcantly [

4

], and we can infer that the demand for e-Buses is increasing

continuously. Other major countries, such as the U.S, European countries, and India, have trends of

increasing demand and application of electric buses, and these trends are expected to continue in the

future [5].

There are several types of electric buses, as well as operation methods. The most typical ones

are the plug-in type, the wireless rechargeable type, the trolley bus and the battery exchange type.

Since each method has its special characteristics, it cannot be easily concluded which is the best.

However, depending on the type of electric bus, the factors and problems requiring consideration

vary widely. For example, the trolley bus needs wires for electric supply. Thus the constraint of

wires becomes an important consideration, and there are many studies treating issues related to wire

networks. Teoh et al. [

6

] cover routing network design and ﬂeet planning for Malaysia’s transportation

situation. On the other hand, for the battery charging system, efﬁcient route conﬁguration within a

given battery capacity is important since charging takes time and is usually only possible in depots.

Thus, regarding charging system problems, locating a depot or optimizing time scheduling given a

Sustainability 2020,12, 1142; doi:10.3390/su12031142 www.mdpi.com/journal/sustainability

Sustainability 2020,12, 1142 2 of 21

network conﬁguration are major concerns. In fact, some studies have focused on electric bus operations

and the problem of locating charging stations. Relatively recently, Rogge et al. [

7

] aimed to optimize

the charging schedule of electric buses to minimize overall costs, including vehicle investment, charger

investment, operational costs, and energy expenses. Liu and Wang [

8

] treated the problem of locating

several types of recharging facilities, especially wireless recharging facilities. There is also ongoing

research into operating electric buses based on battery swapping. In battery swapping cases, each

vehicle travels with a battery and can swap the battery at the designated places for exchange, so-called

battery swapping stations. The advantage of the battery swapping system is that it does not require

huge infrastructure as a trolleybus system does, and unlike the battery charging system, it is possible to

replace the battery quickly along the route. In the battery swapping system, locating battery swapping

stations is a crucial problem. Ko and Shim [

9

] deal with the issue of selecting the location of battery

exchange stations for the seamless operation of electric taxis, which operate with no ﬁxed routes.

Furthermore, several studies deal with the issues related to the scheduling of battery swapping when

operating electric vehicles or buses. For example, Chao and Xiaohong [

10

] and Kang et al. [

11

] dealt

with the time scheduling problem of battery swapping systems. Sarker et al. [

12

] proposed a numerical

model for selecting the location of battery swapping stations in terms of the supply of and demand

for electricity, and the scheduling of electrical supply. Our paper differs from those studies in that we

aim to determine both the location of swapping stations and the scheduling of battery swapping. We

approach battery swapping systems as being underlying problem situations. In doing so, we deal

with the problem of operating an electric bus in a public transit network with a battery swapping

system. To achieve this, we assume the installation of a so-called Quick Charger Machine (QCM) over

the public bus transit network (see Figure 1). We refer those who are interested in the QCM, to the web

page of reference [13].

Figure 1. Operational concept of a bus station with Quick Charger Machine (QCM) [13].

As Figure 1shows, QCM is a machine installed in a proper bus station and it transforms an

eligible bus station into a battery swapping station. Since the automated machine replaces the battery,

it is possible to replace the battery within a short time (say, within one minute) while passengers are

transferring. Inevitably, for the seamless operation of electric buses (e-Buses) in public transportation,

some existing bus stations should play the role of battery swapping stations due to the limited travel

range of e-Buses. Considering that the number of QCMs is related to the operational cost of the system,

Sustainability 2020,12, 1142 3 of 21

it is important to minimize the total number of QCMs installed. In this problem, we do not explicitly

consider the installation costs of these QCMs, which may differ by location.

This problem can be partly viewed as a location routing problem (LRP) variant.The LRP considers

vehicle routing and related resources (e.g., facilities, customers and so on) to ﬁnd proper facility

locations, determine the optimal number of facilities or vehicles, and optimize the routing plan [

14

].

The LRP itself has many variants depending on the particular purpose or problem situation [

15

,

16

].

In this paper, the problem is to ﬁnd the number of facilities and their optimal locations. In the general

LRP, some nodes have demands, and vehicles route them to satisfy those demands. In this problem,

we can see the similarity with the LRP in the sense that the facilities considered are bus stations with a

QCM and the vehicles travel over the bus transit network while preventing battery shortage by visiting

some of the QCMs. There are many papers in literature (e.g., [

17

–

19

]) introducing the basic concept

of the LRP, its variants and solution approaches. Amaya et al. [

20

] study the operation of vehicles

and the location of depots with capacitated vehicles, a problem deﬁned as the capacitated arc routing

problem (CARP). The speciﬁc problem of that paper is that the service vehicle (SV) travels service arcs

in a graph, and since the SV has a ﬁnite capacity, to enable its continuous service, it is replenished

by a reﬁlling vehicle (RV). There is a depot to reﬁll the RV, and the paper tries to ﬁnd the optimal

location of the depot and optimal min-cost path. If we assume the reﬁlling vehicles are ﬁxed and

uncapacitated, this problem situation becomes similar to our assumed problem situation. Nevertheless,

our paper is completely different in that it considers the cost incurred in routing.

Xing et al. [21]

consider several depots in the CARP model. Bekta¸s and Elmasta¸s [

22

] deal with a similar problem

to ours, and ﬁnd the optimal locations for the depots, but consider just one main depot in the bus

route for fulﬁlling buses without any other reﬁll point.Yang and Sun [

23

] assume the problem of

delivering goods to customers from a depot using electric vehicles. The depot serves as both a supply

of goods and a battery swapping station. Within this setting, they select depot locations that minimize

overall shipping costs. Rogge et al. [

7

] propose optimal charging priority and charging location in the

battery swapping problem, and examine a similar situation to ours. However, this paper is different

to ours in that it deals with a centralized charging system, which means that battery charging occurs

only at the depot. Boccia et al. [

24

] deal with an interesting problem similar to ours in some respects.

They determine the location of the facility on the ﬂow network that maximizes the ﬂow through

the facilities. To summarize the above, our study signiﬁcantly differs from previous studies. In our

problem, each node (i.e., a bus station with QCM) can serve as a reﬁll point while most of the previous

works set a separate depot as the central reﬁll point.

This paper is organized as follows. Section 2presents the three different proposed mathematical

programming models for the concomitant model. In Section 3, the proposed models are applied

and tested on the current bus transit network in the Seoul metropolitan area of South Korea and

the experimental results are discussed, including comparisons of the models. Finally, we provide a

conclusion in Section 4.

2. Model

In this section, we present the mathematical programming models for swapping station

deployments for an e-Bus transit network. The deployment of a network of swapping stations

is essential given e-Buses’ limited travel range. This paper considers the problem of locating battery

swapping stations for electric buses on a bus transit network while appropriately addressing the battery

capacity of each bus in operation. Given the situation, the objective of our problem is to optimize

network performance. Speciﬁcally, we maximize the ﬂow covered by a predetermined number of

stations or minimize the number of stations needed to cover trafﬁc ﬂows. For this, three different

mixed-integer programming models are proposed to address the concomitant problem. These are the

set-covering-based model (Section 2.3.1), the ﬂow-based model (Section 2.3.2), and the path-based

model (Section 2.3.3).

Sustainability 2020,12, 1142 4 of 21

2.1. Assumption

Before presenting the mathematical models, we ﬁrst present the assumptions while maintaining

the essence of the problem. We note that all these assumptions are applied to the three models. First,

battery swapping can be achieved within the time period the passengers are transferring so that the

problem of interest is free from battery swapping time constraints. Second, if a bus route from a depot to

the end does not require battery swapping during a journey, we exclude this route from consideration.

Third, we also assume that a battery in the swapping station is always a fully charged battery; in other

words, there are no replacements with an incompletely charged battery. Lastly, considering QCM

assumptions, we ignore the possibility that there exist some stations where battery swapping facilities

cannot be installed due to geographical and spatial problems or administrative problems.

2.2. Notation

The sets, parameters, and decision variables used in the models are listed in Table 1:

Table 1. Mathematical notations used in this paper.

Sets and Parameters

N={1, 2, . . . , N}. Set of bus stops

A={(i,j)|i,j∈ N,j(6=i)is directly accessible from i(i.e., i→j)}. Set of arcs

Ns⊆ N Set of potential bus stops equipped with battery swapping facility (Quick Change Machine (QCM))

R={1, 2, . . . , R}. Set of bus routes

Do⊂ NsSet of origins (depots) for all bus routes

Dd⊂ NsSet of destinations (depots) for all bus routes

rTotal number of bus stops on route r∈ R

Dmax Maximum travel distance per charge

−→

Tr≡(nr

1

,

nr

2

,

. . .

,

nr

r)

. Ordered sequence of bus stops on route

r∈ R

where

nr

i∈ N

for all

i=

1, 2,

. . .

,

r

.

Note that (i)

(nr

i

,

nr

i+1)∈ A

for

i=

1,

. . .

,

r−

1 and (ii)

nr

i≺rnr

j

holds for any

i

and

j

such that

i<j

where the precedence relationship in the ordered sequence is denoted by ≺r.

Tr=nr

1,nr

2, . . . , nr

r. Ordered set of bus stops on route r∈ R

Dr(α,β)Travel distance from bus stop α∈ N to bus stop β∈ N on route r∈ R

Nr

s=(Tr∩ Ns). Set of potential QCMs in Tr

Tr

α=nα≡nr

i,nr

i+1, . . . , nr

jo⊂ T r

. Maximal ordered subset (MOS) of

Tr

, starting from a potential QCM

α∈ Nr

ssuch that Dr(nr

i,nr

j)≤Dmax and Dr(nr

i,nr

j+1)>Dmax

•

For given

Tr

α=nα≡nr

i,nr

i+1, . . . , nr

j≡βo

,

Dr(α

,

β)

is the maximal distance that e-Bus can

drive without any battery swapping, assuming that a battery swap is performed at α∈ Ns.

Decision variables

yi1 if a QCM is located at bus stop i∈ Ns, and 0 otherwise

We remark that the eligibility of each bus stop for a QCM (

Ns

) can be conﬁrmed during the

preprocessing process by considering factors such as electrical grid infrastructure and the availability

of the construction space required for a QCM.

Sustainability 2020,12, 1142 5 of 21

2.3. Mathematical Programming Formulations

In this section, we present the mathematical programming formulations for the problem of

interest—i.e., set-covering formulation, ﬂow-based formulation and path-based formulation—and then

discuss the pros and cons of each formulation.In fact, we propose three types of formulations for

the same strategic decision making problem of locating QCMs in a bus transit network. We will see

that the set-covering formulation is efﬁcient in terms of computational performance but that it lacks

ﬂexibility when incorporating additional issues such as Quality of Service. To address this drawback,

ﬂow-based and path-based formulations are also proposed since their models can incorporate these

additional issues, as we will discuss in the subsequent sections.

2.3.1. Set-Covering Formulation

First, with the maximal ordered subset (MOS) deﬁned in Table 1, this problem can be basically

viewed as a set-covering problem. Figure 2illustrates the deﬁnition of MOS and the basic idea of

the set-covering formulation. If at least one QCM is included in all the MOSs for each bus route,

we can conﬁrm that no full battery discharge occurs during the bus’s traversal of that route, since

MOS is deﬁned as the maximal distance that an e-Bus can drive without needing a battery swap.

By incorporating the deﬁnition of MOSs, the following formulation can be proposed to minimize the

overall number of QCMs.

Figure 2. Basic idea of set-covering formulation.

Model (Set-covering model)

minimize ∑

α∈Ns

yα(1a)

subject to ∑

β∈T r

α∩Ns

yβ≥1, ∀r∈ R,∀α∈ N r

s(1b)

yα=1, ∀α∈ Do(1c)

yα∈ {0, 1},∀α∈ Ns(1d)

The objective function in Equation

(1a)

is to minimize the total number of QCM installations

over the e-Bus transit network. We note that, as for conventional facility location models, this

objective function can be easily modiﬁed to minimize QCM installations and annual maintenance costs.

Sustainability 2020,12, 1142 6 of 21

However, in this paper we limit our attention to minimizing the total number of QCM installations.

Constraint

(1b)

indicates that each MOS must contain at least one QCM, as discussed above. Constraint

(1c) guarantees that all the depots have QCM functionality.

We assume that

|Ns|

is large enough to warrant that

Tr

α∩ Ns6=φ

for all

r∈ R

,

α∈ N r

s

; otherwise,

the problem becomes infeasible. This formulation has a compact form, compared with the other

formulations below. However, the set-covering model only suggests the locations of QCMs among

potential bus station candidates for QCM installation with

y

variables. We will see that the other

proposed formulations simultaneously determine not only the QCM locations but also the battery

swapping schedule of each bus en route. Thus, with this formulation, it is difﬁcult to take Quality

of Service (QoS) into account because of the difﬁculty of analyzing the operational efﬁciency of each

bus and route due to limited information on where and how many times battery swapping occurs.

We discuss this aspect further in Section 3.5. The formulations introduced below complement these

perspectives and should be more informative.

2.3.2. Flow-Based Formulation

As noted earlier, the set-covering formulation in Section 2.3.1 does not explicitly provide detailed

locations of where to swap batteries for each bus (i.e., battery swapping scheduling for each bus en

route). Thus, we here suggest an alternative model which overcomes the limitation of the ﬁrst model.

This model is based on the idea of minimizing the number of QCM installations while satisfying all

the in- and out-ﬂows and the connectivity of all stations. For this, we ﬁrst introduce the additional

notations in Table 2and then introduce the ﬂow-based model.

Table 2. Additional notations for ﬂow-based formulation.

Sets and Parameters

Lr(nr

i)

Last bus stop where a QCM is installed in

Tr

nr

i

(i.e.,

Lr(nr

i) = nr

j

if the ordered set of

Tr

nr

i∩ Ns

with

respect to ≺ris equal to nnr

i, . . . , nr

jo).

Ir(α)≡(Tr

κ\{α})∩ N r

swhere Lr(κ) = α. Note that Ir(α) = φif α∈ Do.

Or(α)≡(Tr

α\{α})∩ N r

s. Note that Or(α) = φif α∈ Dd.

Decision variables

xr

ακ

1 if a bus on the route

r

performs a battery swapping at QCM

α

and the next immediate battery

swapping is performed at QCM κwhere α,κ∈ N r

sand α≺rκ, and 0 otherwise.

Model (Flow-based model)

minimize ∑

α∈Ns

yα(2a)

subject to xr

ακ ≤yα,∀α,κ∈ Nswith α≺rκ,∀r∈ R (2b)

xr

ακ ≤yκ,∀α,κ∈ Nswith α≺rκ,∀r∈ R (2c)

∑

κ∈Or(α)

xr

ακ −∑

κ∈Ir(α)

xr

κα =bα,∀α,κ∈ Nswith α≺rκ,∀r∈ R (2d)

where bα=1 if α∈ Do,bα=−1 if α∈ Dd, and 0 otherwise

yα=1, ∀α∈ Do(2e)

yα∈ {0, 1},∀α∈ Ns(2f)

xr

ακ ∈ {0, 1},∀α,κ∈ Nswith α≺rκ,∀r∈ R (2g)

Sustainability 2020,12, 1142 7 of 21

Constraints

(2b)

and

(2c)

indicate that a QCM can only be installed when the variable

x

is selected,

which means that a bus in route

r

will use

α

and

κ

stations for swapping batteries. Constraint

(2d)

is

the connectivity constraint with the conditional parameter

bα

. Constraint

(2e)

indicates that all the

depots should also play the role of QCM, and Constraints (2f) and (2g) say that yαand xr

ακ are binary

variables. The advantage of this model is that we can determine not only the QCM locations but

also the detailed battery swapping schedules for each bus. Moreover, this model gives a potential

chance for insight into aspects of Quality of Service (QoS) since

x

variables suggest the detailed routing

schedule information for each bus so we can analyze the efﬁciency of each bus route. On the other

hand, a drawback of this model is that if a bus route has more QCMs than its minimally required

number, a bus on the route does not necessarily follow the suggested battery swapping schedules

(i.e., alternative plans exist, and this model cannot capture that aspect). Moreover, compared with

the set-covering formulation, the complexity of this formulation is signiﬁcantly increased in terms

of the number of decision variables and constraints because, with

n

stations, the total number of

x

variables is as much as

(n2−n)/

2, in the worst case. In addition, as we can see in the structure of the

formulation, the number of constraints also increases in proportion to the number of variables. This

implies that, depending on the number of entire stations of our problem or stations over the certain bus

transit network, the computational complexity of the model dramatically increases. Thus, it may be

necessary to come up with more efﬁcient modeling or solution approaches. In this regard, we propose

another formulation utilizing a column generation algorithm suitable for solving a large-scale problem,

as presented in the next section.

2.3.3. Path-Based Formulation

We now present a path-based formulation for this problem. The main difference between the

ﬂow-based model and the path-based model is that the decision variable

x

in the ﬂow-based model

indicates the previous and next stations for the battery swap. On the other hand, the path-based model

groups these

x

variables based on each route and deﬁnes the set of paths throughout a bus itinerary,

where each path corresponds to a feasible battery swapping schedules. In summary, a path is nothing

but one of a subset of QCM stations for battery swapping on each route. Table 3shows additional

notations for the path-based formulation.

Table 3. Additional notations for path-based formulation.

Sets and Parameters

P(r)

Set of paths for the route

r∈ R

where a path is a set of consecutive sub-routes where connecting

points correspond to QCMs

Nr(l)Set of stations which the path lfor the route rvisits.

Decision variables

xr

l1 if the path l∈ P(r)is selected, and 0 otherwise.

The path-based model can be formulated as follows:

Sustainability 2020,12, 1142 8 of 21

Model (Path-based model)

minimize ∑

α∈Ns

yα(3a)

subject to ∑

l∈P (r)

xr

l=1, ∀r∈ R (3b)

∑

l∈P (r)kα∈Nr(l)

xr

l≤yα,∀α∈ Ns,r∈ R (3c)

xr

l∈ {0, 1},yα∈ {0, 1},∀l∈ P (r),α∈ Ns,r∈ R (3d)

The

x

variables of the ﬂow-based model are translated into a path, which becomes much simpler

than before. Constraint (3b) means only one path should be chosen from among the possible paths

for each route, and Constraint (3c) indicates that if a path is selected, then the nodes included in

that path must be QCM stations. Finally, all the decision variables are binary variables, as indicated

by Constraint (3d). The problem is that, although this formulation looks simple, creating a path

set for each route and considering all these paths are not straightforward at all. With

n

nodes in a

bus route, the total number of possible paths can reach up to

n2

. Thus, we approach this problem

based on column generation techniques. Note that a column generation algorithm is useful when

dealing with problems featuring large numbers of variables because it avoids the enumeration of

all possible variables and instead only evaluates them as needed. We note that column generation

algorithms cannot be applied to all cases, but that they are applicable to this problem in two respects.

First, LP-relaxation is essential for application of the column generation algorithm, and because this

problem treats a kind of network-ﬂow problem, we can use LP-relaxation [

25

]. Moreover, as described

above, this problem has a large number of variables compared to the number of constraints.

According to the column generation algorithm, we divide this model into a master problem

MP

and

sub-problems

SP(r)

for each route

r∈ R

. We then iteratively solve the problem by repeatedly adding

the feasible paths from each sub-problem to the master problem using the column generation algorithm.

Model MP (Master problem of path-based model)

minimize ∑

α∈Ns

yα(4a)

subject to ∑

l∈P (r)

xr

l=1, ∀r∈ R (4b)

∑

l∈P (r)kα∈Nr(l)

xr

l≤yα,∀α∈ Ns,r∈ R (4c)

xr

l∈ {0, 1},yα∈ {0, 1},∀l∈ P (r),α∈ Ns,r∈ R (4d)

The master problem

MP

is basically the same as the path-based formulation (3), but starts with a

partial set of paths as an initial basis. Let

πr

and

µα,r

be the dual variables for Equations (4b) and (4c),

respectively. To solve the problem with these dual variables, the linear programming relaxation of the

master problem is solved. If the reduced cost for

x

variable,

−∑α∈Nr(l)µα,r−πr

, turns out to be

negative, then the corresponding column

xr

l

can be added to the restricted master problem. To ﬁnd a

candidate x-variable to add, we need to solve the following pricing sub-problem SP(r)for a route r:

Sustainability 2020,12, 1142 9 of 21

Model SP(r)(Subproblem of path-based model for route r)

minimize ∑

α∈Ns(r)

(−µα,ryr

α−πr)(5a)

subject to xr

ακ ≤yr

α,∀(α,κ)∈ E(r)(5b)

xr

ακ ≤yr

κ,∀(α,κ)∈ E(r)(5c)

∑

(α,κ)∈E (r)

xr

ακ −∑

(κ,α)∈E (r)

xr

κα =bα,∀α∈ Ns(r)(5d)

xr

ακ ∈ {0, 1},yr

α∈ {0, 1},∀(α,κ)∈ E (r),α∈ Ns(r)(5e)

In the sub-problem

SP(r)

,

bα=

1 if

α∈ Do

,

bα=−

1 if

α∈ Dd

, and 0 otherwise. Note that

the path generation sub-problem

SP(r)

is a variant of the shortest path problem. Unlike a typical

shortest path problem,

SP(r)

minimizes node costs instead of edge costs. It is trying to minimize

the sum of the costs incurred at the nodes visited. The problem

SP(r)

can be easily transformed to

an edge-cost-minimizing shortest path problem. Since the arc weights in the transformed model are

non-negative (the dual variables for Equation (4c),

µα,r

, are negative), the sub-problem

SP(r)

can be

solved efﬁciently by Dijkstra’s algorithm.

The objective function (5a) indicates the reduced cost of the master problem, and thus if the

objective function of

SP(r)

is non-negative, it means that we have found a candidate

x

-variable

with a negative reduced cost. Constraints (5b) and (5c) show the activating condition of

y

, since the

master problem

MP

is a minimization problem so that a variable with negative reduced costs could

further decrease the objective function of

MP

. When the sub-problem cannot ﬁnd any solution with a

negative objective function value, we can conclude that all the necessary variables have been added to

the restricted master problem. The algorithm is then terminated after deriving the optimal solution of

the master problem.

3. Case Study on a Smart e-Bus System

In this section, we run numerical experiments with the three proposed mathematical programming

models in Section 2and evaluate them with the actual bus transit network data of the Seoul

metropolitan area, South Korea. All the bus transit network data used in this experiment are available

at [

26

]. When it comes to the driving range of the e-Bus, the electric bus under consideration can drive

up to 60 km with a fully charged battery, according to [

13

]. All the experiments were performed using

an Intel(R) Core(TM) i7-4770 CPU at 3.40 GHZ with 32 GB memory.

Figure 3shows the summary statistics and a plot of the entire dataset we use. We use this dataset

from Seoul, South Korea, to evaluate the performance of the proposed mathematical programming

models. In doing so, we vary the size of the experimental dataset to better understand the tested

formulations in terms of computational time and quality of solution as dataset size increases. When it

comes to experimental dataset generation, several factors are considered. Since the battery capacity is

60 km per charge, we prioritize including in each dataset the bus routes longer than 60 km, which we

refer to as the primary routes. Indeed, the experimental datasets are generated with those primary

routes, and the non-primary routes added as necessary. The same suite of generated datasets is then

used to evaluate the numerical performance of each proposed mathematical programming approach.

We note that the numerical experiment is conducted 10 times with same conditions for each dataset

and each mathematical programming approach.

Sustainability 2020,12, 1142 10 of 21

Figure 3. Public bus transit network with stations in the Seoul metropolitan area, South Korea.

3.1. Experiments with Set-Covering Model

We ﬁrst present the experimental results from the set-covering formulation and Table 4and

Figure 4show the summary of the results.

Table 4. Experimental results of the set-covering model.

# of routes

(# of Stations)

Optimal # of

QCM Stations

Avg. CPU Time

(sec)

Max Gap between

CPU Times (sec)

# of

Variables

# of

Constraints

5 (441) 12 0.086 0.012 441 595

10 (822) 20 0.301 0.076 822 1185

20 (1611) 37 0.708 0.071 1611 2422

30 (2286) 53 0.968 0.066 2286 3631

40 (2681) 65 1.326 0.411 2681 4710

50 (3058) 73 2.001 0.444 3058 5822

60 (3597) 82 2.493 0.558 3597 7074

70 (3929) 102 2.561 0.329 3929 7880

80 (4235) 121 2.848 0.186 4235 8537

90 (4435) 137 3.034 0.11 4435 9282

100 (4732) 152 3.378 0.121 4732 10,287

200 (7008) 256 5.614 0.233 7008 18,194

300 (8059) 315 7.725 1.17 8059 25,352

400 (8937) 382 9.246 0.377 8937 31,492

500 (10,934) 549 9.87 0.4 10,934 34,908

600 (12,650) 721 9.793 0.459 12,650 37,799

635 (13,191) 782 10.313 1.357 13,191 38,636

Sustainability 2020,12, 1142 11 of 21

Figure 4. Graphical summary of the experimental results of the set-covering model.

As the graphs in Figure 4show, we observe that there exists a linear relationship between dataset

size and the number of decision variables as well as the number of constraints. We remark that, in the

set-covering model, the variable

y

is the only decision variable, and the number of constraints is

almost equal to the size of the dataset. Moreover, for each route

r

, there are at most

|N r

s| −

1 number

of MOSs. Thus, despite the complexity of the model structure, at least the number of variables and

constraints does not increase exponentially, and we can learn that empirically the computational time

of the model also increases linearly. This model seems quite easy to use experimentally to solve this

problem, at least for our dataset. However, this also shows the limitation of the set-covering model of

not giving intuition about where to swap a battery.

Figure 5shows plots of the experiment results of four test instances. Most of the stations are

centralized in the downtown, especially in Figure 5d. This is because the location is decided by the

route, not the distance between the QCM stations. Considering this point, if there are too many

centralized stations, a penalty could be imposed on the distance between neighborhood stations as an

extra constraint. We left this as a further study.

Sustainability 2020,12, 1142 12 of 21

Figure 5. Plots of the optimal QCM installations obtained from the set-covering model.

3.2. Experiments with Flow-Based Model

We now discuss the experimental results from the ﬂow-based model. In the ﬂow-based model

experiment, we cannot get a solution in case of more than 20 routes due to the computational memory

becoming full, as presented in Table 5.

Table 5. Experimental results of the ﬂow-based model.

# of routes

(# of Stations)

Optimal # of

QCM Stations

Avg. CPU Time

(sec)

Max Gap between

CPU Times (sec)

# of

Variables

# of

Constraints

5 (441) 12 16.835 0.202 972,846 240,434

10 (822) 21 35.891 1.223 2,205,658 681,564

15 (1230) 28 122.667 1.658 6,757,662 1,816,809

18 (1552) 32 687.125 5.805

10,757,662

4,816,809

20 (1611) 37 2356.156 15.425

34,541,551

12,426,973

30 (2286) - - - - -

40 (2681) - - - - -

50 (3058) - - - - -

The computational time increases exponentially as the size of the data set increases (see Figure 6a),

as does the number of decision variables and constraints (see Figure 6c,d). However, as stated earlier,

the advantage of using this model is that we can obtain the battery swapping schedule for each route

with xvariables.

Sustainability 2020,12, 1142 13 of 21

Figure 6. Graphical summary of the experimental results of the ﬂow-based model.

Figure 7shows the same results as with the set-covering model in terms of the optimal QCM

locations. However, due to the limitation of the test size, the plots of the result are also restricted to

data sizes, 10 (Figure 7a) and 20 (Figure 7b) only. From the result, note that we can conﬁrm the same

QCM installations as the set-covering model.

Figure 7. Plots of the optimal QCM installations obtained from the ﬂow-based model.

3.3. Experiments with Path-Based Model

We ﬁnally present the experimental results from the path-based formulation and Table 6and

Figure 8summarize the results.

Sustainability 2020,12, 1142 14 of 21

Table 6. Experimental results of the path-based model.

# of Routes

(# of Stations)

Optimal # of

QCM Stations

Avg. CPU Time

(sec)

Max Gap between

CPU Times (sec)

# of

Variables

# of

Constraints

Avg. # of

Iterations

5 (441) 12 15.781 0.042 515 202 86

10 (822) 21 60.418 0.084 956 422 215

15 (1230) 28 285.118 1.584 1355 569 348

20 (1611) 37 1296.621 3.584 1950 744 586

30 (2286) 53 2874.516 6.871 3125 968 863

40 (2681) 65 4225.889 10.668 4857 1268 1153

50 (3058) 73 5826.146 15.139 22,618 1605 1868

60 (3597) 82 7361.411 23.584 30,520 2841 2735

70 (3929) 102 8165.156 43.458 37,211 3869 3868

80 (4235) 121 10,041.453 167.165 43,584 5412 5066

90 (4435) 137 14,025.584 304.545 51,251 8169 6166

100 (4732) 152 23,218.975 518.107 58,518 9887 7259

200 (7008) 256 81,540.145 682.618 135,685 34,815 25,015

300 (8059) 315 345,154.587 822.123 209,871 65,036 68,121

Figure 8. Graphical summary of the experimental results of the path-based model.

The computational results on the data set indicate that the proposed approach works to ﬁnd the

optimal solution of every problem instance under consideration. As Figure 8a shows, the computational

time tends to increase exponentially, but not as rapidly as the ﬂow-based model, as also shown by

comparing Figure 6a and Figure 8a. The path-based model overcomes the limitation of the result of

the ﬂow-based model. Note that the ﬂow-based model cannot derive the solution when the dataset

becomes too large. Thus, the path-based model more effectively manages the computer memory

compared than ﬂow-based one. This is due to the column generation algorithm’s characteristic of

solving the problem iteratively, and not using all the columns.

Figure 9shows plots of results from the path-based model experiment. We note that the outcome

is consistent with previous models.

Sustainability 2020,12, 1142 15 of 21

Figure 9. Plots of the optimal QCM installations obtained from the path-based model.

3.4. Model Comparison

In this section, we compare the results discussed in Sections 3.1–3.3 and summarize pros and cons

of each model.

Table 7presents a comparison of the three proposed models. The column ‘dataset’ indicates the

used subsets in the experiment. For each model, the ﬁrst column indicates the computational time,

and the next column indicates the optimal number of QCM installations for each problem instance.

The result conﬁrms that every model we propose yields the same optimal solution. The set-covering

model clearly has a strong point and weak point. While it offers limited information, only the location

of QCM installations, it suggests the result superiorly fast compared to other models. Both the

ﬂow-based model and path-based model are useful in that they can additionally offer information

about where to swap batteries. In terms of computation time, they fall short of the set-covering

model. However, the path-based model shows better performance than the ﬂow-based one. Figure 10

incorporates and compares the previous experiments’ computational time results graphically.

Figure 10. Comparison of model computational times.

Sustainability 2020,12, 1142 16 of 21

Table 7. Comparison of the computational results of the three models.

Set-Covering Model Flow-Based Model Path-Based Model

# of Routes

(# of Stations)

CPU Time

(sec)

Optimal # of

QCM Stations

CPU Time

(sec)

Optimal # of

QCM Stations

CPU Time

(sec)

Optimal # of

QCM Stations

5 (441) 0.086 12 16.835 12 15.781 12

10 (822) 0.301 21 35.891 21 60.418 21

15 (1230) 0.514 28 122.667 28 285.118 28

20 (1611) 0.708 37 2356.156 37 1296.621 37

30 (2286) 0.968 53 - - 2874.516 53

40 (2681) 1.326 65 - - 4225.889 65

50 (3058) 2.001 73 - - 5826.146 73

60 (3597) 2.493 82 - - 7361.411 82

70 (3929) 2.561 102 - - 8165.156 102

80 (4235) 2.848 121 - - 10,041.453 121

90 (4435) 3.034 137 - - 14,025.584 137

100 (4732) 3.378 152 - - 23,218.975 152

200 (7008) 5.614 256 - - 81,540.145 256

300 (8059) 7.725 315 - - 345,154.587 315

400 (8937) 9.246 382 - - - -

500 (10,934) 9.87 549 - - - -

600 (12,650) 9.793 721 - - - -

635 (13,191) 10.313 782 - - - -

3.5. Quality of Service Analysis for Electric Buses

In this section, we further discuss how QoS, which was brieﬂy mentioned in Section 2.3.1, can be

incorporated in the proposed models. There are several aspects needing consideration when making

strategic decisions on QCM locations in public transit networks. One example might be how many

e-Bus routes should be associated with each QCM station. While the proposed QCM only requires

a short time for each battery swap, this service time can reduce the potential for serving multiple

buses simultaneously at one station. Thus, properly dispersing these ﬂocking vehicles across a battery

swapping queue could help to provide a better and more seamless service over the operation of the

entire bus transit system. Figure 11 shows the imbalance of the service rate measured by the number

of bus routes scheduled for battery swapping per one QCM station when the experimental dataset

with 100 routes is used.

Figure 11. The number of bus routes each QCM station serves.

Sustainability 2020,12, 1142 17 of 21

Figure 11 presents QCM station distribution in terms of the number of bus routes they service for

battery swapping. For example, while one QCM station serves 24 bus routes, 60 QCM stations serve

only one. Considering this, limiting the maximum number of bus routes served by a QCM station,

would allow QCM service rates to be appropriately balanced.

From the computational point of view, the set-covering model has outperformed the ﬂow-based

and path-based models. This section will determine how the battery swapping schedule of each

bus route can be utilized in the QoS analysis and demonstrate how the ﬂow-based and path-based

models can consider such QoS aspects while the set-covering model cannot. Note that the results

of the set-covering model show that while we can derive the number of bus routes including each

QCM station, we cannot directly incorporate constraints to limit the number of bus routes that each

QCM station serves. On the other hand, the ﬂow-based and path-based models contain information

about which QCM station each bus route should stop at for battery swapping within the

x

variables.

This is because these models support constraints explicitly limiting the number of bus routes passing

through each QCM station and can derive the exact number of routes each QCM station should serve

for battery swapping. In this regard, we limit the number of bus routes stopping at each QCM station

in order to disperse battery swapping servicing and hence reduce the chance of having to wait for

battery swapping at any station. We remark that in the path-based model this constraint can be easily

added through the

x

variables, and we name the new model the extended path-based model. We

can also add similar constraints to the ﬂow-based model. The additional notation (Table 8) and the

formulation of the extended path-based formulation are given below:

Table 8. Additional notation for the extended path-based formulation.

Additional Parameter

γMaximally allowable number of bus routes stopping at a QCM station for battery swapping

Model MPe(Master problem of the extended path-based model)

minimize (4a)

subject to (4b),(4c),(4d)

∑

r∈R

∑

l∈P(r)kα∈Nr(l)

xr

l≤γ∀α∈ Ns(6a)

Constraint (6a) limits the number of bus routes stopping at each QCM station

α∈ Ns

. By simply

adding the constraint, we can indirectly control the battery swapping demand at each QCM station

and disperse the trafﬁc. However, we would like to note that the additional constraint (6a) cannot

be directly incorporated at the initial stage of the column generation procedure of the path-based

model. If the value of

γ

in the constraint is not sufﬁciently large, the path-based model can become

infeasible at the initial stage of the column generation procedure because of an insufﬁcient number of

columns available. Thus, in order to incorporate the constraint and implement the column generation

procedure, we should either add constraint (6a) at a later stage after a number of iterations or reduce

the value of

γ

gradually over the iterations. In the numerical experiment, we add the additional

constraints at the later stage of the column generation procedure. To evaluate the effects of adding

the QoS constraints into the path-based model, we compare the results with the original path-based

model. Table 9shows the numerical test results.

Sustainability 2020,12, 1142 18 of 21

Table 9.

Computational result with(out) Quality of Service (QoS) constraints in the path-based model.

Original Path-Covering Model Extended Path-Based Model

# of Routes

(# of Stations)

# of QCM

Stations

Maximum

Flock (γ)

Variance of

Flocking

CPU

Time (sec)

# of QCM

Stations

Maximum

Flock (γ)

Variance of

Flocking

CPU

Time (sec)

5 (441) 12 4 1.401 15.781 14 2 0.863 16.115

10 (822) 21 7 1.987 60.418 22 4 1.478 69.054

15 (1230) 28 9 2.081 285.118 29 5 1.975 308.487

20 (1611) 37 9 1.993 1296.621 39 5 1.072 1583.142

30 (2286) 53 9 1.974 2874.516 55 5 1.882 3593.145

40 (2681) 65 9 2.013 4225.889 67 5 1.342 5451.396

50 (3058) 73 9 1.955 5826.146 76 5 1.277 9828.708

60 (3597) 82 11 2.154 7361.411 85 7 1.270 10,453.203

70 (3929) 102 15 3.377 8165.156 104 8 1.483 13,717.462

80 (4235) 121 21 3.670 10,041.453 125 10 2.495 12,903.267

90 (4435) 137 24 3.823 14,025.584 141 10 2.478 19,032.717

100 (4732) 152 25 4.284 23,218.975 158 12 2.876 34,317.645

Since dataset size limits the conﬁgurability of the

γ

value, the

γ

value is adaptively set according

to it. After adding the QoS constraints, we can observe the intuitive result that the number of the QoS

installations is more than with the original path-based model. The ‘maximum ﬂock’ column indicates

the maximum number of buses coming to a QCM station for battery swapping, and the ’variance of

ﬂocking’ column indicates the distribution of the number of buses scheduled to swap their batteries

at each QCM station. Table 9indicates that, as the number of QCM installations increases and the

number of bus routes stopping at a QCM station for battery swapping is restricted, the overall ﬂocking

variance decreases.

Figure 12 compares the results shown in Figure 11 with the corresponding results from the

extended model. As Figure 12 shows, no QCM station serves more than

γ

bus routes. This restriction

allows the ﬂocking phenomenon to be resolved without signiﬁcantly increasing the optimal number of

QCM installations, implying that severe ﬂocking would occur at only a few stations. We can conclude

that the extended model helps to effectively redistribute the demand for battery swapping away from

overused stations. We conduct further experiments with the extended path-based model to examine

how sensitive the results are to

γ

. Table 10 shows the variation in the optimal number of QCM stations

and the ﬂocking variance according to γ.

Figure 12. The detailed QoS result for 100 routes.

Sustainability 2020,12, 1142 19 of 21

Table 10. Computational result as the γvalue changes.

Extended Path-Based Model Experimented with 60 Routes

γ# of QCM Stations Maximum Flock Variance of Flocking CPU Time (sec)

6 infeasible - - -

7 85 7 1.270 10,453.204

8 84 8 1.621 9358.721

9 84 9 1.819 8715.780

10 82 10 2.084 7521.184

11 (no restict) 82 11 2.154 7361.411

Table 10 shows the trade-off between the optimal number of QCM stations and ﬂocking variance

as the value of

γ

varies. Flocking can be mitigated by changing the

γ

value without too much

degradation in the optimal number of QCM installations. We also note that it is possible for the

extended model to become infeasible if

γ

becomes too small. In summary, we demonstrate that,

unlike the set-covering model, the ﬂow-based or path-based model can address various considerations,

such as QoS, and that, by controlling the number of bus routes stopping at a station, the additional

QoS constraints can help to reduce the chances of high demand for battery swapping at individual

QCM stations.

4. Conclusions and Future Research

In this paper, we studied the effective operation of electric buses, assuming the battery swapping

system, in which batteries can be swapped in bus stations by using special equipment called quick

charger machines. The purpose of this study is to minimize the total number of QCM installations

in existing bus stations over the urban bus transit network while providing the seamless operation

of a public bus service in a metropolitan area. To address this problem, we suggest three different

mathematical models based on mixed-integer programming. The ﬁrst model, the set-covering-based

formulation, has fast computational times since it is formulated as a low complexity model.

The deﬁciency of this model is that it does not suggest a detailed schedule of where to swap batteries

for each bus; it only provides the locations of the QCM stations. On the other hand, the ﬂow-based

formulation and path-based formulation do provide this important additional information, unlike the

set-covering model. However, these models are highly computationally complex, as shown by the

experiment results with actual data. The column generation algorithm used in the path-based model

makes this model better balanced in terms of computational time and completeness of information

than the set-covering model or ﬂow-based model. In Section 3, we check the performance of each

model and their validity for this problem with the actual bus transit network data from the Seoul

metropolitan area. Above all, in Section 3.5, as a QoS analysis, we introduce additional constraints to

the path-based model so that the demand for battery servicing at QCM stations is distributed more

evenly over the entire bus transit network and demonstrate that the approach is effective. Most of all,

we show the improved ﬂexibility and scalability of the ﬂow-based and path-based models compared to

the set-covering model. This paper contributes signiﬁcantly to the understanding of how to introduce

an electric vehicle to an urban area. The varied models can solve this problem and give insight into

further studies.

For future work, we can generalize the assumptions used in our problem. First, we assume that

all buses depart from the depot at full battery capacity. However, if each bus rotates its route several

times, then the initial or ﬁnal condition of the battery can be different every time. By taking this

into account, the improved model can reﬂect the initial battery level. Moreover, considering the cost

of installing each QCM on an individual basis can be meaningful since here we assume the cost of

installation is the same for all stations, and, therefore, only focus on minimizing the total number

of QCMs. For another problem approach, we can ﬁrst solve the set-covering model and then solve

another scheduling model. The set-covering model only suggests the optimal location of QCM stations

but not the speciﬁc scheduling, and the result of the ﬂow-based model and path-based model are also

Sustainability 2020,12, 1142 20 of 21

restricted by their computational complexity. Thus, proposing additional scheduling problems that

utilize the QCM location results of the set-covering model could be another methodological approach,

although one that would not guarantee an optimal solution. It is apparent from numerical experiments

with real-world data that a heuristic algorithm may be necessary to solve these problems efﬁciently.

Finally, once the strategic decision to install QCMs has been made, operational decisions, such as those

concerning battery pack charging/discharging based on usage and the availability of battery packs

eligible for battery swapping, need to be addressed.

Author Contributions:

Conceptualization, T.C. and S.H.S.; methodology, J.M., Y.J.K., T.C. and S.H.S.; validation,

T.C. and S.H.S.; formal analysis, J.M.; investigation, J.M. and Y.J.K.; data curation, J.M. and Y.J.K.; writing—original

draft preparation, J.M. and T.C.; writing—review and editing, Y.J.K., T.C. and S.H.S.; supervision, T.C. and S.H.S.;

project administration, T.C. and S.H.S.; funding acquisition, T.C. All authors have read and agreed to the published

version of the manuscript.

Funding:

This research was supported by Basic Science Research Program through the National Research

Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2018R1D1A1B07047651) and also

supported by Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government

(MOTIE) (P0008691, The Competency Development Program for Industry Specialist).

Acknowledgments:

We would like to acknowledge four anonymous reviewers for their constructive and

helpful comments.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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