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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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