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Transportation Research Part C 152 (2023) 104175
0968-090X/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Integrated optimization of electric bus scheduling and charging
planning incorporating exible charging and timetable
shifting strategies
Mengyuan Duan
a
,
b
, Feixiong Liao
b
,
*
, Geqi Qi
a
, Wei Guan
a
,
*
a
Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong
University, Beijing 100044, China
b
Urban Planning and Transportation Group, Eindhoven University of Technology, Eindhoven, the Netherlands
ARTICLE INFO
Keywords:
Bus scheduling
Charging planning
Timetable shifting
Peak power demand
Battery degradation
ABSTRACT
In a battery electric bus (BEB) network, buses are scheduled to perform timetabled trips while
satisfying time, energy consumption, charging, and operational constraints. Increasing research
efforts have been dedicated to the integrated optimization of multiple planning tasks to reduce
system costs. At a high integration level, this study determines the BEB scheduling and charging
planning with exible charging and timetable shifting strategies. We rst formulate an integrated
arc-based model to minimize the total costs considering the power grid pressure cost and sub-
sequently reformulate it into a two-stage model, for which we develop an effective solution
method. The rst stage minimizes the total operational costs including the eet, charging, and
battery degradation costs based on the column generation technique, and the second stage
minimizes the peak power demand through two timetable shifting strategies. It is found through
numerical experiments that the proposed integrated optimization model and solution method can
achieve signicant improvement in the utilization rate and reductions in the eet size, opera-
tional costs, and peak power demand compared to the two baseline models.
1. Introduction
It has been reported that the transportation sector accounts for about one-quarter of the global carbon dioxide emissions due to the
wide use of carbon-fueled vehicles (IEA, 2019), which is one of the main causes of climate change. This proportion is even higher in
developed economies, for example, one-third in the European Union (IEA, 2022). Therefore, it is necessary to nd substitute mobility
options to reduce emissions and secure a more sustainable environment. Electric mobility has proven to be instrumental in decar-
bonizing the transportation sector (Qu et al., 2022; Razeghi and Samuelsen, 2016). Amongst, electric public bus systems are considered
an important pillar that offers affordable and environmentally friendly mobility. Local governments in many countries implement a
series of policies, such as nancial incentives, customer subsidies, and taxation on petroleum, to promote the deployment of battery
electric buses (BEBs) (Nie et al., 2016). Despite the advantages of BEBs, they still face challenges in large-scale adoption (Ji et al., 2022;
Zhang et al., 2021a). The high capital investment, including the costs of embedded battery packs and auxiliary charging facilities,
makes BEBs less appealing than traditional diesel buses. The battery capacity of the BEBs gradually decreases during the charging and
* Corresponding authors.
E-mail addresses: f.liao@tue.nl (F. Liao), weig@bjtu.edu.cn (W. Guan).
Contents lists available at ScienceDirect
Transportation Research Part C
journal homepage: www.elsevier.com/locate/trc
https://doi.org/10.1016/j.trc.2023.104175
Received 10 December 2022; Received in revised form 19 March 2023; Accepted 15 May 2023
Transportation Research Part C 152 (2023) 104175
2
discharging cycles, resulting in the degradation of battery performance (Schoch et al., 2018; Zhang et al., 2019). Furthermore, due to
the complex planning processes for a BEB system, inefcient charging strategies and eet schedules are usually adopted by BEB op-
erators (Yao et al., 2020; Zhang et al., 2021b).
1.1. Literature review
In general, the planning tasks of bus systems include line designing, timetabling, bus scheduling, crew rostering, and charging
planning (if the buses are BEBs). In the existing studies of bus systems, timetabling and bus scheduling are two primary research
subjects. Timetabling is to determine the departure and arrival times of each trip in a bus network constrained by the given frequency
and level of service of each line. Bus scheduling is to assign buses to serve the timetabled trips on the condition that each trip is covered
by one bus. Because of the limited travel range of BEBs, the timetable needs reconstructions and more idle time for the charging
process, where charging planning makes decisions on when, how long, and how much electricity the bus is charged. As reviewed
below, these planning tasks have often been handled separately due to the high model complexity.
Typically, a timetable is determined based on the line characteristics and the passenger demands between origins and destinations.
With an emphasis on passenger services, the common objective of timetabling is to minimize passenger travel or transfer times (Wu
et al., 2016), while ignoring the subsequent planning tasks. For the BEB scheduling problem, route duration, route distance, and the
duration for replenishing energy at the charging points should be constrained due to the limitation of the driving range per charge.
Some studies assigned the BEBs to meet the condition that each timetabled trip is covered once, aiming to yield the least operational
costs. For example, given xed charging duration, Tang et al. (2019) proposed robust scheduling strategies for BEBs under stochastic
trafc conditions, and Rinaldi et al. (2020) focused on the optimal scheduling of a mixed BEB eet. These studies did not show any
charging exibility since the charging duration or volume is xed. Specically, each charging operation starts immediately upon
arrival at the charging station and has a determined charging duration or volume according to the “rst-in-rst-out” principle. Such an
inexible charging strategy cannot fully coordinate the demand of BEBs and the availability of charging resources. Assuming pre-
determined eet schedules, some studies were dedicated to the orderly and exible charging strategy. For example, Wang et al.
(2017) arranged the charging demand of BEBs to ensure recharging without any delays. He et al. (2020) optimized charging schedules
for a fast-charging system and determined when to charge a BEB. The obtained charging strategies are called “semi-exible charging”,
which indicates that the buses are charged to a pre-given electricity level while optimizing the sequence of charging operations. Unlike
their studies, Abdelwahed et al. (2020) optimized the charging schedules and determined where, when, and how long each BEB should
be charged while considering the time-dependent electricity prices in time-of-use (TOU) plans. This charging strategy is called “full-
exible charging”, as all the charging-related components need to be determined endogenously, such as the charging start time,
duration, and volume. However, the studies mentioned above adopted separated optimization for single planning tasks, which tends to
cause inefcient BEB planning outcomes (Schmid and Ehmke, 2015). Therefore, it is desirable to optimize these planning tasks
integratively to exploit the system’s capacity to the greatest extent and maximize its productivity and efciency (Ceder, 2001; Liu and
Ceder, 2018).
Integrated optimization has recently been a hotspot topic in the study of timetabling, bus scheduling, and charging planning.
Commonly, two interdependent planning tasks are addressed simultaneously, such as integrating timetabling with bus scheduling or
integrating bus scheduling with charging planning. The integrated optimization of timetabling and bus scheduling is only targeted at
conventional buses. For instance, Ibarra-Rojas et al. (2014) combined two integer linear programming models of timetabling and
vehicle scheduling to form a bi-objective integrated model. One objective is to maximize the number of passengers beneting from
well-coordinated transfers, and the other is to minimize the operational costs related to eet management. Schmid and Ehmke (2015)
aimed to minimize operational costs by balancing consecutive departures on a line according to predened departure time intervals. In
addition to the cost-efciency objective, they also considered the quality of a timetable from a passenger’s point of view. Liu and Ceder
(2018) developed a bi-level integer programming model to attain the optimality of timetable synchronization with vehicle scheduling.
The integration of three or more planning tasks is even more desirable but seems infeasible for most practical problems (Teng et al.,
2020).
Increasing research has integrated charging planning for BEBs into the aforementioned integrated optimization models developed
for non-BEBs. For instance, considering the limited driving range and charging requirement constraints, Liu and Ceder (2020) pro-
posed two formulations to minimize the total number of BEBs and chargers. One is based on the decit function theory, and the other is
an equivalent bi-objective integer programming model. Under travel range uncertainty, Li et al. (2021) addressed the multi-depot
vehicle location-routing-scheduling problem with two-stage stochastic programming. Some realistic operational conditions, such as
battery degradation, nonlinear charging prole, and peak power demand, were incorporated in different studies. To be specic, the
capacity and power of batteries in BEBs gradually fade due to the chemical and mechanical processes, and this degradation mechanism
always occurs owing to cyclic charging and discharging (Barr´
e et al., 2013; Pelletier et al., 2018); the nonlinear charging prole implies
varying charging efciency responding to different charging conditions (Liu et al., 2022; Zhang et al., 2021a); peak power demand has
a massive impact on the existing power system (Chen et al., 2018; Lopes et al., 2011; Vicini et al., 2012), which is triggered by a one-
time occurrence of charging demand and usually measured as the highest average power over a time interval (e.g., varying from 15 to
60 min) (He et al., 2020; Hledik, 2014; Qin et al., 2016). Pelletier et al. (2018) developed a comprehensive mathematical model that
incorporates a realistic charging process, time-dependent energy costs, battery degradation, grid restrictions, and facility-related
demand charges. They pointed out that optimizing the charging schedule contributes to keeping the facility-related charging de-
mand low at the expense of spreading out the charging activities throughout the day. Zhang et al. (2021) determined the BEB charging
strategies to minimize the total operational costs of a transit system. These studies reported that the incorporation of these charging
M. Duan et al.
Transportation Research Part C 152 (2023) 104175
3
conditions enhances realism and accounts for feasible eet size reductions and charging strategies. The integrated optimization
problems of any two of the above planning tasks have been demonstrated to be NP-hard (Ibarra-Rojas et al., 2014; Rinaldi et al., 2020).
The integrated optimization of bus scheduling, charging planning, and timetabling with the consideration of realistic charging con-
ditions would be even more challenging.
1.2. Focus of this study
As stated above, to the best of our knowledge, no study has investigated the bus scheduling and charging planning with timetable
shifting problem while considering the battery degradation, non-linear charging prole, and TOU plans. For comparative convenience,
we summarize the detailed characteristics of the relevant studies on the operational planning of BEBs in Table 1. As seen, the literature
is evolving to take into account realistic operational conditions such as battery degradation, non-linear charging prole, and TOU
plans. However, three notable literature gaps can be identied. First, the integrated optimization models of bus scheduling and
charging planning have not considered timetabling. As such, the existing studies fall short of investigating the inuence of charging
strategies and moderating the peak power demand by timetabling-related strategies, such as timetable shifting. Second, most existing
studies on bus scheduling and charging planning are based on xed or semi-exible charging strategies to obtain the sequence of
service trips. This treatment of charging simplies the modeling complexity but cannot capture the real operational conditions.
Abdelwahed et al. (2020) was the only one that considered full-exible charging, but their model focused on this single planning task.
Third, existing approaches for solving the scheduling problem focused on heuristics, branch-and-price, and optimization solver.
Although their problems can be solved well by these methods, they cannot solve the proposed integrated optimization problem well.
To ll these gaps, this study aims to coordinate bus scheduling, charging planning, and timetabling with full-exible charging as well
as incorporate realistic operational conditions in a BEB system.
Therefore, the main contributions of this study are presented as follows. First of all, we investigate an integrated optimization
model for timetable modications and joint bus and charging scheduling. This study thus far has the highest model integrity in the
literature. Second, with the consideration of full-exible charging, battery degradation, nonlinear charging prole and TOU plan, the
model aims to generate a joint eet and charging schedule to minimize the total costs, considering the power grid pressure cost. These
considerations can capture the real operational conditions. Third, we reformulate the arc-based model and develop a two-stage so-
lution method. This treatment separates the integrated objective into the operational costs and the peak power demand, which simplify
the complexity of the solution method.
The remainder of this paper is organized as follows. Section 2 presents the problem description. In Section 3, an optimization model
is proposed to characterize the integrated timetabling, BEB scheduling, and charging planning problem. Section 4 discusses the two-
stage solution method based on an arc-based model reformulation. Numerical experiments are conducted in Section 5 to verify the
effectiveness and computational efciency of the proposed model and solution method. Finally, Section 6 concludes the main con-
tributions and provides potential extensions for future work.
2. Problem description
In this section, we rst describe the setting of the research problem and then introduce the timetable shifting strategies in detail.
Table 1
Comparison of related references.
Reference Integrated
approach
Timetabling Battery
degradation
Nonlinear charging
prole
TOU
plans
Charging
exibility
Solution
algorithm
Li (2014) Y N N N N S B&P
Qin et al. (2016) N N N N Y S SIMULATION
Wang et al. (2017) N N N N N N CPLEX
Pelletier et al. (2018) Y N Y Y Y S CPLEX
Xu and Meng (2019) Y N N Y N S B&P, HA
Tang et al. (2019) N N N N N N B&P
Yao et al. (2020) Y N N N N S GA
Abdelwahed et al.
(2020)
N N N N Y F CPLEX
He et al. (2020) N N N N Y S GAMS, CPLEX
Liu and Ceder (2020) Y N Y Y N N DFT, LM, AMFM
Li et al. (2021) Y N N N N S HA
Liu et al. (2021) N N N N Y N B&P, HA
Zhang et al. (2021a) Y N Y Y N S B&P, LCM
This study Y Y Y Y Y F Two-stage, CG
TT =Timetabling; Y =Yes; N =No; S =Semi-exible charging; F =Full-exible charging; B&P =branch-and-price; HA =heuristic algorithm; DFT =
decit function theory; LM =lexicographic method; AMFM =adjusted max-ow method; GA =genetic algorithm; LCM =label correcting method.
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Transportation Research Part C 152 (2023) 104175
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2.1. Research problem
The research problem focuses on BEB scheduling and charging planning with timetable shifting. The studied transit network can be
illustrated in Fig. 1, which is composed of a single depot where all buses are housed at night, one central charging station with a limited
number of candidate chargers located in the depot, several transit lines served by BEBs, and bus stops. The central depot charging
requires that a BEB returns to the depot to replenish their electricity during the idle time between two service trips. According to the
practice of BEB operations, the BEBs run round trips along the lines, and the start and end locations of the bus line, i.e., the bus stops of
departure and arrival, are called terminals. For example, Line1 is composed of a sequence of stops; once a daily operation begins, a BEB
departs from the depot and then performs the round trip based on terminal A. Due to the limited space capacity, a depot is usually
located in a different location from the terminals. Therefore, there is certain distance between the depot and the terminals. To ensure
that BEBs have enough electricity to perform the tasks in chronological order, they are not allowed to return to the depot except for
replenishing their electricity. This network structure with one central depot and multiple neighboring terminals applies broadly to
small-sized and medium-sized cities (Perumal et al., 2021) and has been widely used in the BEB literature.
The planning horizon T is discretized into equal time intervals as T= {0,1,2,⋯,|T|}, where | • | gives the cardinality of a set. The
sets of terminals and directed lines are denoted by S and L, respectively. Each line l∈L has its original timetable Tl
0, and the set of
timetabled trips is represented as Il= {1, ..., |Il|} in ascending order by their departure times. Thus, we have I= ∪l∈LIl and Il∩Il′=∅, l,
l′∈L,l∕= l′. Trip i is associated with a line li∈L, terminal si∈S, departure time di∈Tl
0, travel time tti, and electricity ei (in per-
centage) consumed by the trip. For consistency, the volume of electricity replenished at the charging station is also expressed in
percentage. The objective is to minimize the total costs, considering the power grid pressure cost. The three key components, including
modied timetable, eet schedule, and the corresponding charging schedule are elaborated below.
2.2. Timetable shifting strategy
The limited number of chargers at the central depot often leads to a high peak power demand (Liu and Ceder, 2018; Lin et al.,
2019). We adopt the timetable shifting strategy, which has been applied for timetable synchronization in public transport (Fonseca
et al., 2018), to adjust the start charging time to moderate the peak power demand. Large deviations from the original timetable would
not only perturb the regular timetabled services but also confuse passengers with accustomed trip plans. More importantly, they would
cause massive adjustments to the following scheduling tasks. To coordinate with the eet and charging schedules, the original
timetable should only be shifted within an acceptable range. We consider two types of shifting strategies, including block shifting and
differentiated shifting. Block shifting indicates that the timetabled trips of a line are shifted equally while differentiated shifting means
that the departure time of each timetabled trip makes a small shift within a time window. Essentially, block shifting is a special case of
differentiated shifting. In a mathematical way, a generic form is proposed to represent the shifting strategy. Specically, let di denote
the original departure time of trip i and Δi denote the shift of trip i. Note that Δi may be positive or negative, where a positive value
indicates the time is moved backward and a negative value indicates the time is moved forward. Shift Δi should be constrainted by a
time window, that is Δi∈ {di−−di,di+−di}, where d−
i and d+
i denote the earliest and latest departure times at the terminal,
respectively. When all trips in the timetable of one line are shifted equally, it indicates a block shifting strategy. To distinguish it from
differentiated shifting, Δl is introduced to denote the block shift of line l, where Δi=Δj=Δl,∀i,j∈Tl
0,l∈L, Δl∈ {d−
l,d+
l}, where d−
l
and d+
l represent the pre-determined minimum and maximum time shifts of line l, respectively.
After timetable shifting, the actual departure time of trip i is represented as di+Δl+Δi,∀i∈Tl
0,l∈L. Note that the departures of
the trips in the same line can never overtake each other, meaning that the sequence of departures remains. Similar to the formulations
in Fonseca et al.(2018), the shift window d−
i,d+
ifor trip i can be created based on the original timetables as
d−
i=di−di−di−1−1
2,∀i,i−1∈Il,l∈L(1a)
Fig. 1. An illustration of the BEB transit network.
M. Duan et al.
Transportation Research Part C 152 (2023) 104175
5
d+
i=di+di+1−di
2,∀i,i+1∈Il,l∈L(1b)
if i refers to the rst or the last trip of a line, we set d−
i=d+
i.
For illustration purposes, Fig. 2 shows the two timetable shifting strategies given the original timetable. Fig. 2(a) presents the block
shift Δl<0 for line l, indicating that all the timetabled trips of line l are moved forward |Δl|; Fig. 2(b) presents the shift time windows
{d−
i,d+
i}and {d−
i+1,d+
i+1}for trips i and i+1 (d−
i+1=d+
i+1), indicating that the departure times can be shifted within the time
windows.
3. Model formulation
This section presents the formulations of the research problem. First, we show the primary assumptions and notations of this study.
Second, the network structure and components used to model the problem are presented. Next, we discuss the constraints concerning
the ow conservation of BEBs, time window, energy consumption and replenishment, timetable shifting, and charging station ca-
pacity. Finally, we consider three parts of the operational costs and the peak power demand in the objective function. The total
operational costs include the eet cost, the charging cost, and the cost incurred by battery degradation.
3.1. Notations and assumptions
The primary notations used in the formulations are listed in Table 2, and the assumptions used throughout this paper are sum-
marized as follows.
Assumption 1.BEBs are homogeneous in terms of battery size and capacity. Each bus can only serve one line on an operation day, and the
battery capacity is sufcient to cover at least one service trip.
Assumption 2.There is only one charging station located in the depot where the physical distances between the chargers are neglected. A
charging operation of a BEB can only occur at the charging station and is carried out with one of the chargers.
Assumption 3.A BEB can be charged only after nishing a round trip at the terminal.
Assumption 4.The routes of the buses for all the origin–destination pairs are known and xed. Stochasticity in the road network is not
considered, and the running times of BEBs between any two consecutive stops remain the same as those in the original timetable.
Assumption 5.A BEB with an initial level of battery electricity leaves the depot and begins its service, and the electricity of buses running on
lines should always fall within the feasible range.
Assumptions 1 and 5 are consistent with the common practices of bus systems (He et al., 2019; An, 2020). Assumptions 2, 3 and 4
are adopted to simplify the problem, which are widely adopted in the literature related to the planning and operation of BEBs (Pelletier
et al., 2018; Chen et al., 2018; Liu et al., 2021; Uslu and Kaya, 2021; Zhang et al., 2021a). Although the fast-charging method occurs at
terminals in many studies (He et al., 2019; Wang et al., 2017), we make an assumption that the charging operations occur at the
charging station since they can be transformed in the same formulations by the method of reduction (i.e., setting the distances between
the depot and the terminals to 0). In addition, the consideration of time and energy consumption stochasticity is beyond the scope in
this paper.
3.2. Network of eet schedule and charging strategy
To address the research problem rigorously, we follow Li (2014) to construct a network graph denoted by G= (V,A), where V and A
Fig. 2. (a) Block shifting strategy; (b) Differentiated shifting strategy.
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Transportation Research Part C 152 (2023) 104175
6
are the node set and arc set, respectively. Node set V is partitioned as V=I∪ {Oo,Od,c}. In the network, single depot O is extended as
the source depot Oo where BEBs start daily operations, the sink depot Od where BEBs nish the daily operation, and the charging
station c where BEBs replenish their electricity, respectively. Node set I includes all timetabled trips, each of which refers to a period in
time, not a typical spatial point. Besides, we use Vo=I∪ {Oo}and Vd=I∪ {Od}to dene the arcs. The set of arcs A is partitioned as
A=Ao∪Ad∪AI, in which each arc (i,j) ∈ A describes the movement from node i∈Vo to another node j∈Vd. The set of arcs A should
Table 2
The primary notations.
Sets
T set of timesteps, |T|indicates the number of time intervals
Q set of power demand periods
S set of terminals
L set of lines in the BEB network
I set of timetabled trips
Il set of timetabled trips of line l
V set of all nodes, V=I∪ {Oo,Od,c}
Vo,Vd set of trip nodes with the source or sink depot node,Vo=I∪ {Oo},Vd=I∪ {Od}
A,AI set of all arcs or service trip arcs
Ao,Ad set of pull-out arcs or pull-in arcs
BEB system setup
O,c depot and the charging station
Oo,Od source and sink of the depot
R feasible eet schedule
Tl
0 original timetable of line l
I1
l rst trip of line l
li line of trip i
tti travel time of trip i [min]
si terminal of trip i
di scheduled departure time of trip i
ei proportion of electricity consumed by trip i [%]
h−
i,h+
i minimum and maximum headways [min]
di−,di+lower and upper bounds of shift time window of trip i [min]
dl
−,dl
+minimum and maximum times of block shifts of line l [min]
ρ
turnaround time [min]
Hinit initial SOC of a BEB at the beginning of daily operation [%]
Hlo,Hmax minimum and maximum electricity of a BEB [%]
E battery electricity of a BEB [kWh]
f(
χ
),f−1 implicit function of SOC for duration
χ
and the inverse function of f(
χ
)
ξ(H0,Hk)function of battery capacity fading rate corresponding to the SOC from H0 to Hk
d(H0,Hk)cost function incurred by battery degradation changing from H0 to Hk
τ
(h,m)deadhead travel time between locations h and m [min]
e(h,m)proportion of energy consumption for deadhead travel from location h to m [%]
τ
min minimum charging duration [min]
ce
t electricity price at timestep t [$/kWh]
cb xed cost of a BEB [$]
κ number of chargers
q power demand period, q∈Q
t1
q,t2
q start and end time of period q
Intermediate variables
Hi remaining SOC after completing trip i
Hc
i remaining SOC after completing trip i after arriving at the charging station before the charging starts
τ
a
i actual arrival time of trip i
cit binary variable: 1 if the bus is charging at timestep t after trip i; 0, otherwise
Δl,Δi block shift of line l and differentiated shift of trip i
ϑij binary variable: 1 if charging consecutively occurs after trips i and j by the same bus; 0, otherwise.
pt unit power of the charging station at timestep t [kW]
wit charging electricity at timestep t after trip i [%]
θq average power demand during demand period q [kW]
θ peak power demand of the charging station [kW]
τ
(i)charging duration when a BEB is charged after trip i [min]
w(i)proportion of electricity replenished when a BEB is charged after trip i [%]
Decision variables
μ
i binary variable: 1 if a charging operation happens after trip i; 0, otherwise
ait binary variable: 1 if the BEB is charged after trip i at timestep t; 0, otherwise
bit binary variable: 1 if the BEB has nished charging after trip i at t; 0, otherwise
xij binary variable: 1 if trips i and j are successively served by one bus; 0, otherwise
τ
d
i departure time of trip i
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also include deadhead trips without transporting passengers. The arc set Ao= {(i,j)|i∈Oo,j∈I}denotes the bus moving from the
depot to the departure terminal of trip j when its daily operation starts, called a pull-out arc shown in Fig. 3. Arc set Ad= {(i,j)i∈I,
j∈Od}, called pull-in arc, denotes the bus moving from the arrival terminal of trip i to the depot when its daily operation ends. Arc set
AI= {(i,j)|i,j∈I,i∕= j}denotes the bus moving from the arrival terminal of trip i to the departure terminal of trip j, called a service arc
for connecting two trips. Suppose the remaining energy is not sufcient for the next trip. In that case, the bus needs to replenish its
electricity at the charging station and then perform the next trip, which can be represented as {(i,c)→(c,j)|i,j∈I,i∕= j}and included in
the service arcs.
The trip chain indicates that a BEB starts from the depot, then performs several timetabled trips, replenishes its electricity between
the idle time of trips if the battery electricity is insufcient for the next service trip, and nally ends at the depot. The eet schedule
refers to the generation of trip chains, subject to the constraint that each timetabled trip is only performed once by one bus. One trip
chain consists of some nodes and arcs, and the trips are in ascending order regarding their departure times, satisfying energy con-
sumption and charging time constraints. We illustrate the feasible eet and charging schedules in Fig. 3(a). The time interval
Fig. 3. An illustrative example of the BEB trips: (a) network graph; (b) power demand comparation.
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8
associated with each node represents the period of the corresponding nodes, such as the trip node with the scheduled departure and
arrival times, the charging node with the start and end charging times, the state of charge (SOC) range before and after charging. This
feasible eet schedule R⊆R contains three trip chains, denoted as R= {r1,r2,r3}, namely, r1=Oo−i1−i3−c−i6−Od, r2=
Oo−i5−c−i8−Od, and r3=Oo−i2−i4−c−i7−Od, where c denotes charging. As for the charging schedule in Fig. 3(a), it includes
three charging operations expressed by a tuple
C=cj,ctj,etj,[soco,soct]|j=1,⋯,|I|, where cj is an indicator variable equal to 1 if a
charging operation happens after trip j and 0 otherwise, ctj denotes the start charging time, etj denotes the end charging time, [soco,soct]
indicates the SOC range before and after charging (%). This representation also implies the charging duration and volume of the
charging operation. The original charging schedule requires 2 chargers; by timetable shifting, the charging demand can be replenished
by 1 charger; the adjusted timetable and the optimized charging time are represented below nodes in Fig. 3(a). In addition, Fig. 3(b)
indicates the power demand comparison between without and with timetable shifting for the illustrative example. Setting 15 min as a
time interval, the peak power demand is 106.6 kW without shifting and 74.6 kW with shifting; the difference is 30% in this example,
which indicates an advantage of the timetable shifting strategy. To guarantee the physical turnaround of the bus, we consider an
average turnaround time
ρ
.
Related to the charging schedule, the charging volume needs to be determined, which is impacted by the charging duration and
other factors such as the TOU plans and the nonlinear charging prole. Let ce
t denote the electricity price in $/kWh associated with
timestep t∈T. As for the nonlinear charging prole, we use a constant current-constant voltage (CC-CV) scheme shown in Fig. 4(a),
commonly adopted by electric vehicles (Lam and Bauer, 2013; Schoch et al., 2018; Xu and Meng, 2019). During the constant current
(CC) phase, the charging current is held constant, and the battery’s terminal voltage increases until it reaches a certain maximum
value, which is specied by the manufacturer. The constant voltage (CV) phase starts subsequently, and the terminal voltage must be
maintained at the maximum value to avoid overcharging the battery. This scheme indicates that the variation of the battery’s SOC
cannot be assumed linear with the charging time (Pelletier et al., 2016). Similar to Montoya et al. (2017), we make a piecewise linear
function to formulate the variation of SOC and the charging duration. Let f(
χ
)be the charging function when soc0=0 and the battery is
charged for
χ
time intervals. As shown in Fig. 4(b), soc1,soc2,and soc3 represent the switch points of SOC in each piecewise part. Based
on soc0, we have
χ
0=f−1(soc0)and know in which piecewise part
χ
0+Δ
χ
is located. To put it simply, soc0 and Δ
χ
are the input
parameters of the piecewise function to obtain soc
χ
. Therefore, the nal SOC can be calculated given the piecewise function of a
charging prole, the SOC before visiting the charging station, and the charging duration.
3.3. Constraints
3.3.1. Flow conservation constraints
A eet schedule consists of several trip chains, and each trip chain performed by one bus is connected by the arcs from the pull-out
arc, then to the service arcs, and nally to the pull-in arc as illustrated in Fig. 3(a). To capture the relationship, xij is introduced as a
binary decision variable, where xij =1 means trips i and j are successively served by one bus and xij =0 otherwise. To reduce the
search space for the trip chain, infeasible or incompatible service arcs need to be removed from AI beforehand. For service trip arc (i,j),
let di and dj denote the scheduled departure time, respectively, where i,j∈Tl
0,l∈L. Considering the shifting strategy, the departure
time of trips will be modied possibly and shifted in a window d−
i,d+
i, where i∈Tl
0,l∈L, d−
i and d+
i denote the earliest departure
time of trip i and the latest departure time of trip j, which are calculated by Eqs. (1a) and (1b). If the earliest arrival time at the terminal
Fig. 4. Nonlinear charging prole of constant current-constant voltage (CC-CV).
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9
sj is greater than the latest departure time from the same terminal, service arc (i,j)should be eliminated. This condition indicates that
trip j can never be performed on time even if trip i arrives at the earliest, that is, d−
i+tti+
ρ
>d+
j, where d−
i+tti denotes the earliest
arrival time of trip i, d+
j denotes the latest departure time of trip j, and
ρ
is the bus turnaround time.
In practice, each timetabled trip is exactly included once in the eet schedule, as formulated in Eq. (2). To guarantee the continuity
of a trip chain, ow conservation between the connecting arcs needs to be satised as Eq. (3).
j:(i,j)∈A\Ao
xij =1,∀i∈I(2)
i:(i,j)∈Al\Ad
l
xij −
i:(j,i)∈Al\Ao
l
xji =0,∀j∈I(3)
3.3.2. Time window constraints
As stated above, this study considers exible charging with varying start charging time and duration. With minor shifts from the
scheduled departure time, we use
τ
d
i and
τ
a
i as non-negative integer variables to denote the actual departure and arrival times of trip i
respectively, which are limited by a shift time window of the original timetable. If the charging operation happens in the trip chain, the
continuity should also be guaranteed. We dene binary decision variable
μ
i as a charging indicator, where
μ
i=1 if a charging
operation happens after trip i and
μ
i=0 otherwise. In addition, given the exible charging and TOU plans, the charging schedule is
related to two binary cumulative time decision variables. If the bus is charged at a timestep t* after trip i, ait =1 at t≥t* and ait =0 at
other timesteps. Similarly, if the bus nishes charging at the timestep during t*+Δt after trip i, bit =1 at t≥t*+Δt and bit =0 at other
timesteps. Fig. 5 illustrates the relationship between ait and bit .
For service arc (i,j),
τ
a
i is affected by the timetable shifts and the turnaround time
ρ
, and thus should be equal to or earlier than
τ
d
j, as
formulated in Eq. (4). The start and end charging times are constrained by Eqs. (5a) and (5b), where
τ
(si,Oc)denotes the deadhead
travel time from the terminal to the charging station, and
τ
(Oc,sj)denotes the deadhead travel time from the charging station to the
departure terminal of trip j.
τ
a
i+
ρ
≤
τ
d
j+M1−xij,∀(i,j) ∈ AI(4)
τ
a
i+
τ
(si,Oc) − M(1−
μ
i) ≤
t∈T
tait −ai,t−1,∀i∈I(5a)
t∈T
tbit −bi,t−1+
τ
Oc,sj
μ
i≤
τ
d
j,∀(i,j)∈AI(5b)
where M indicates a large positive number.
The charging duration is bounded by the minimum and maximum charging durations (Wang et al., 2022). The minimum charging
duration is determined due to economic efciency, while the maximum charging duration is constrained by the battery capacity of the
BEB. Let
τ
(i)be the charging duration immediately after trip i, which can be calculated by ait and bit in Eq. (6a). The rst term on the
right side of the equation represents the end charging timestep, and the second term represents the start charging timestep.
Furthermore, let
τ
min denote the minimum charging duration and the actual charging duration is constrained by Eq. (6b), indicating
that
τ
(i)is positive if the charging operation happens after trip i, and
τ
(i) = 0 otherwise. Given the maximum charging duration |T|, the
upper charging duration is limited by the latter constraint.
τ
(i) =
t∈T
(ait −bit),∀i∈I(6a)
τ
min
μ
i≤
τ
(i) ≤ |T|
μ
i,∀i∈I(6b)
Finally, ait and bit delimit the exible charging duration as Eqs. (7a)-(7c). The coupling constraints between the charging operation
and the start and end charging times are shown in Eqs. (8a) and (8b).
ait ≥ai,t−1,∀i∈I,t∈T\1(7a)
Fig. 5. Illustration of two indicator variables.
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bit ≥bi,t−1,∀i∈I,t∈T\1(7b)
ait ≥bit,∀i∈I,t∈T(7c)
μ
i≤
t∈T
ait ≤ |T|
μ
i,∀i∈I(8a)
μ
i≤
t∈T
bit ≤ |T|
μ
i,∀i∈I(8b)
3.3.3. Energy consumption and replenishment constraints
The energy consumption constraints are used to guarantee the continuity of the trip chain. If the remaining electricity of a BEB does
not satisfy the next trip j, the bus needs to replenish energy at the charging station and then perform trip j. Eq. (9a) captures the
discharging dynamics between trips i and j, where Hi denotes the remaining SOC after completing trip i and Hc
i denotes the remaining
SOC of the bus after trip i but before charging happens. Similarly, if a charging operation happens after trip i, the energy consumption
and the replenishment should be considered. As described above, the charging electricity increases nonlinearly with the charging
duration and the initial electricity. Let w(i)be the proportion of charging electricity, which is determined by
τ
(i)and Hc
i. The pro-
portion of charging electricity is calculated as w(i) = f
τ
(i) +f−1Hc
i−Hc
i. The energy consumption and replenishment are con-
strained by Eqs. (9b) and (9c), where e(si,Oc)denotes the proportion of deadhead energy consumption from the terminal to the
charging station, and e(Oc,sj)denotes the energy consumption from the charging station to the arrival terminal. Noted that if a
charging operation immediately happens after trip i, Hc
i=Hi−e(si,Oc).
−M1−xij−M
μ
i≤Hi−ej−Hj≤M1−xij+M
μ
i,∀(i,j)∈AI(9a)
−M(1−
μ
i) ≤ Hi−e(si,Oc) − Hc
i≤M(1−
μ
i),∀i∈I(9b)
−M(1−
μ
i) ≤ Hc
i+w(i) − eOc,sj−ej−Hj≤M(1−
μ
i),∀(i,j) ∈ AI(9c)
Let Hmin and Hmax be the pre-dened minimum and maximum electricity and Hinit be the initial SOC of a BEB at the beginning of
the daily operations. To mitigate the range anxiety of the driver, the remaining electricity after trip i should be constrained as Eq. (10a)
following the natural process. Also, to avoid a BEB to visit the charging station frequently, the minimum level of electricity after
charging is constrained by Eq. (10b), indicating that the sum of Hc
i and w(i)is non-negative if and only if the charging operation
happens; Hc
i=0, otherwise.
Hmin ≤Hi≤Hmax,∀i∈I(10a)
μ
iHmin ≤Hc
i+w(i) ≤
μ
iHmax,∀i∈I(10b)
Finally, to ensure the integrity of the energy consumption of each bus when a daily operation starts or ends, the following con-
straints should be met. Specically, Eq. (11a) indicates that each bus departs with battery electricity Hinit from the depot, and Eq. (11b)
indicates that when the bus nishes its operation, the remaining SOC should be larger than the minimum value.
−M(1−xoi) ≤ Hinit −e(Oo,si) − ei−Hi≤M(1−xoi),∀(o,i) ∈ Ao(11a)
Hi−esi,Od+M(1−xid) ≥ Hmin ,∀(i,d) ∈ Ad(11b)
3.3.4. Timetable shifting constraints
Based on the original timetable, the timetable shifting strategies adjust slightly departure times of the trips for better synchroni-
zation. The block shift Δl of timetable Tl
0 for line l is constrained by Eq. (12a), and the adjusted departure time should fall within the
bounds dened by the lower and upper time shifts, formulated as Eq. (12b). According to Section 2.2, the actual departure time after
adjustment is formulated as Eq. (13).
d−
l≤Δl≤d+
l,∀l∈L(12a)
d−
i≤Δi+di≤d+
i,∀i∈I(12b)
τ
d
i=di+Δl+Δi,∀i∈Il,l∈L(13)
In addition, the adjusted timetable should meet the headway constraints. The minimum headway is determined by the operator for
safety consideration and the maximum headway is usually related to the service level. The headway between sequenced trips i and i+1
is constrained by Eq. (14), where h−
i and h+
i represent the minimum and maximum headways, respectively.
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11
hi−≤
τ
d
i−
τ
d
i−1≤hi+,∀i∈Il\I1
l,l∈L(14)
3.3.5. Capacity constraints
The number of chargers is limited by the available space at the charging location (Abdelwahed et al., 2020). The number of buses
charged at any timestep should not exceed the number of chargers at the charging station. As shown in Fig. 5, ait −bit indicates whether
the bus is charging at timestep t.
μ
i(ait −bit) = 1 denotes that a charging operation occurs for trip i at timestep t. Each bus can serve a
maximum of one trip at any timestep, and a trip can only be served by one bus. Thus, the constraint of charging station capacity is
formulated as Eq. (15), where κ denotes the number of chargers at the depot.
i∈I
μ
i(ait −bit) ≤ κ,∀t∈T(15)
3.4. Objective function
3.4.1. Fleet cost
Fleet cost is dependent on the number of used BEBs. In general, it is related to the amortized cost, the depreciation cost, the cost of
insurance and maintenance, etc. (Zhang et al., 2021a). In the BEB network, the number of buses corresponds to the number of trip
chains in the eet schedule. With the denition of the pull-out arc indicator xoi, the eet cost associated with the number of buses can
be calculated by
FC =
i∈I
cbxoi (16)
where cb is the xed cost of one BEB.
3.4.2. Charging cost
The charging cost is dependent on the charging time, charging volume, and unit electricity price. As formulated above, the charging
volume is represented as w(i) = f
τ
(i) +f−1Hc
i−Hc
i. To calculate the charging cost, wit is introduced as the charging electricity at
timestep t after trip i. If a charging operation happens after trip i, the proportion of charging electricity is formulated as wit =
fΔt+f−1Hi,t−1−Hi,t−1, where Δt denotes the unit time interval, t indicates the charging timestep, and Hi,t−1 denotes the
electricity level at timestep t−1. Because of the implement of TOU plans, ce
t denotes the unit electricity price at timestep t. And the
charging cost can be calculated as Eq. (17), where E denotes the battery capacity of a BEB.
CC =
i∈I
t∈T
ce
twitE(17)
3.4.3. Battery degradation cost
The battery degradation cost is incurred by each cyclic discharging and charging process of the battery (Lam and Bauer, 2013). Let
∇ϑ= {(i,j)|i,j∈I}be the set of consecutively charging operations, which can be obtained from the charging schedule along the trip
chain. Let ϑij be a consecutively charging operation indicator, where ϑij =1 means two consecutively charging operations happen after
trips i and j served by the same bus, and ϑij =0 otherwise. According to Zhang et al. (2021), we show the discharging and charging of
electricity between the consecutively charging operations in Fig. 6. If ϑij =1, the SOC changes of one cyclic discharging and charging
Fig. 6. Illustration of discharging and charging cycle.
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process are represented as Hc
i+w(i),Hc
jand {Hc
j,Hc
j+w(j)}, where Hc
i and Hc
j denote the remaining SOC at the charging
station before charging, Hc
i+w(i)and Hc
j+w(j)denote the SOC after charging, respectively.
Let d(*)denote the charging or discharging degradation cost corresponding with the SOC changes, which is calculated by an
empirical model (Lam and Bauer, 2013) developed to describe the aging of batteries over each charge and discharge cycle. We provide
detailed calculations in the supplementary document. The total battery degradation cost is formulated as Eq. (18), where
d(Hc
i+w(i),Hc
j)and d(Hc
j,Hc
j+w(j)) represent the degradation costs corresponding with the SOC changes of battery discharging
from Hc
i+w(i)to Hc
j and charging from Hc
j to Hc
j+w(j), respectively.
DC =
(i,j)∈∇ϑ
ϑijdHc
i+w(i),Hc
j+dHc
j,Hc
j+w(j) (18)
3.4.4. Peak power demand
The peak power demand reects the stress of the power grid, which is a prescribed measurement over time intervals during the
billing period (Perumal et al., 2021). Let pt denote the unit power of the charging station at timestep t, represented as pt=i∈IwitE,
∀t∈T. Let
α
denote the length (in the unit of h) of a time interval for calculating the peak power demand; thus, a day (24 h) is divided
into 24/
α
demand periods. The set of demand periods is denoted by Q= {1,2,...,24/
α
}. For a demand period indexed by q∈Q, t1
q and
t2
q denote the start and end times, respectively. The average power demand θq of q is calculated as Eq. (19a) and the peak power
demand θ is dened as Eq. (19b).
θq=1
α
t∈[t1
q,t2
q]
pt,∀q∈Q(19a)
θ=maxθq∀q∈Q(19b)
Taken together, the objective function includes costs related to the eet, charging, battery degradation, and peak power demand.
Because of the different units,
σ
is introduced as a positive weight factor for θ by converting the pressure of the power grid to the
economic consequence. The BEB scheduling and charging planning problem can be formulated as an integrated optimization model as
Eq. (20). In fact,
σ
is a critical parameter indicating how the pressure of the power grid is integrated into the model. In a relevant study
with a similar objective function, He et. al (2020) set the value of
σ
xed as 12.765 $/kW. However, since it is hard to calibrate the
value for applications, we do not set a xed value of
σ
but develop an efcient method to solve it in the next section.
minC=FC +CC +DC +
σ
θ(20a)
s.t.
Eqs. (2)-(19)
xij,
μ
i,ait,bit ∈ {0,1},∀i,j∈I,t∈T,
τ
d
i∈Z,∀i∈I(20b)
Remark 1.Eq. (20) represents the common arc-based model including the planning task of the BEB schedule, whose variants can be found in,
for example, Li (2014), He et al. (2020), Rinaldi et al (2020), Zhang et al. (2021) and Zhou et al (2022). It should be pointed out that our
model is an extension of Zhang et al. (2021) and the primary differences are that the full-exible charging and timetable shifting strategies as
well as the objective to reduce peak power demand are considered in our study. These features undoubtedly add up to model complexity. Given
the high model complexity, an efcient solution method is needed to address a sizable BEB network.
4. Solution algorithm
To solve the above-formulated problem, we design a two-stage solution method, which is commonly adopted in the literature
(Carosi et al., 2019; Huang et al., 2021; Li et al., 2021). The objective function of the arc-based model is decomposed to separate the
operational costs and the peak power demand. This is done by assuming that for transit companies, minimizing the operational costs is
the primary goal and reducing the peak power demand is the secondary goal.Section 4.1 describes the solution method of the rst
stage, minimizing the total operational costs based on the column generation (CG) technique. Section 4.2 describes the timetable
shifting strategies for minimizing the peak power demand.
4.1. First-stage method based on CG
The problem of BEB scheduling and charging planning in the rst stage is NP-hard as the number of variables and constraints in the
formulation increases exponentially with the network size. We reconstruct a path-based formulation and use the CG technique to solve
the rst-stage problem. The CG technique has been demonstrated as effective in solving large-scale optimization problems (Steiner and
Irnich, 2018; Xu and Meng, 2019; Liu et al., 2021). The CG starts with a restricted master problem (RMP) with a subset of variables and
M. Duan et al.
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13
associated columns. After optimally solving the RMP, the dual solutions are used to nd the non-basic variables with the minimum
reduced costs implicitly, referred to as a pricing problem, which can be solved by a labeling method (Gao et al., 2022). If the reduced
minimum cost is negative, the variable and column corresponding to the reduced minimum cost are added to the RMP, and the next
iteration is triggered; otherwise, the optimal solution for this stage is claimed to be found in the current iteration. We use a larger
timestep in the rst-stage problem, while a smaller one in the second-stage problem ensures computational efciency without lossing
solution delity. The reasons are as follows. First, the size of the timestep affects the number of labels in the pricing problem, and as the
step size increases, the labels will grow exponentially (Xu and Meng, 2019; Lin et al., 2020). Second, the purpose of the rst stage is to
obtain the eet schedule and the charging volume, for which the step size has less effect on the objective value, as demonstrated in
Section 5.1.
4.1.1. Path-based formulation
We dene a binary variable λr, which equals 1 if path r is selected, and 0 otherwise. The path-based model can be formulated as
minC=
r∈R
Crλr(21a)
s.t.
r∈R
Kr
iλr=1,∀i∈I(21b)
r∈R
Ur
tλr≤κ,∀t∈T(21c)
λr∈ {0,1},∀r∈R(21d)
where C and Cr denote the total operational costs of path r, respectively; incidence variable Kr
i equals 1 if trip i is covered by path r,
and 0 otherwise; incidence variable Ur
t equals 1 if the BEB is charged at timestep t in path r, and 0 otherwise.
Cr=FCr+CCr+DCr
=cb+
i∈r
t∈T
ce
twitE+
(i,j)∈∇ϑr
ϑijdHc
i+w(i),Hc
j+dHc
j,Hc
j+w(j) (21e)
where path r is associated with one BEB; FCr denotes the cost of the BEB; CCr denotes the charging cost for the BEB; DCr denotes the
battery degradation cost, and ∇ϑr denotes the set of consecutively charging operations along path r.
For notational convenience, δ= {δi|i∈I}and β= {βt|t∈T}are used as the dual variables associated with Eqs. (21b) and (21c). By
relaxing Eq. (21d), we can use the CG technique to tackle the RMP and pricing problem iteratively. The CG technique starts with an
initial feasible solution R to solve the RMP and determine the dual variables. Next, we construct and minimize the reduced cost Θ(λr)to
solve the pricing problem. If Θ(λr) ≥ 0, no column can be added to R and the RMP has been solved with a proven optimal solution in
this stage; otherwise, variable λr obtained from the pricing problem is added to R for future iterations. The reduced cost function of
variables λr is expressed as
Θ(λr) = Cr−
i∈I
Kr
iδi−
t∈T
Ur
tβt(22)
As the CG technique starts with an initial solution, we introduce a simple heuristic method to construct a feasible path set, which is
usually adopted by operators (Abdelwahed et al., 2020; Liu et al., 2021). In the beginning, the departure time
τ
d
i is xed to the original
timetable Tl
0 for all i∈Il,l∈L, and the initial SOC of a BEB is equal to Hinit when departing from depot O. At the outset, a BEB serves
the trip with the earliest departure time. After this trip is completed, we check if the BEB can still serve another trip satisfying the
minimal battery level constraint. If so, we choose the trip with the earliest departure time from the remaining trip set; otherwise, we
choose available timesteps for charging the bus until the battery electricity reaches to Hinit. If no trip can be served anymore, we direct
the current trip to the depot, and the constructed trip chain constitutes an initial feasible RMP solution. This heuristic method is
repeated until all trips in the trip set I are served. As a consequence, a feasible solution is generated.
4.1.2. Pricing problem
It can be seen from Eq. (22) that checking Θ(λr) ≤ 0 corresponds to the problem of nding the minimum-cost paths in the network
G(V,A). The pricing problem aims to nd a minimum-cost path from source node Oo to destination node Od subject to time and
electricity constraints. Essentially, the constraints of the pricing problem are similar to the original arc-based model. The pricing
problem is to nd a feasible trip chain with a negative reduced cost, while the original arc-based model is to nd the optimal eet
schedules composed of several trip chains. The pricing problem can be formulated as
M. Duan et al.
Transportation Research Part C 152 (2023) 104175
14
Cr=min
r∈RΘ(λr) = min
r∈RCr−
i∈I
Kr
iδi−
t∈T
Ur
tβt(23a)
s.t.
(i,j)∈Ao
xij =1(23b)
(i,j)∈Ad
xij =1(23c)
Eqs. (3)-(11).
where Eq. (23a) minimizes the reduced cost of path r; the pull-in and pull-out from the depot are constrained by Eqs. (23b) and
(23c).
The pricing problem is an extension of the resource-constrained shortest path problem (Li, 2014) by allowing replenishment along a
path, for which the label correcting method is suitable for solving the problem. To implement the labeling method, each node i∈I is
assumed to be associated with multiple labels representing partial paths. The label extension at each node is due to the different
charging strategies, including whether a charging operation happens after this node, the start charging times, and the charging
duration. According to Liao (2016, 2019), we initially construct a three-dimensional matrix of 2 × |T| × |T|units for each node, where
each unit represents the potential label of this node. G is reconstructed as a space–time network with acyclic paths. Let i→j be a link of
G(i,j∈I), and we code any label k at node i as lk(i) = [ck,wk,sk,t
→k,tk], where ck and wk are the cost and current SOC of the corre-
sponding path, sk denotes the charging state with a value of 0 or 1, t
→k and tk are the start charging time and charging duration,
respectively. Also, pk, spk, and csk are used to record the current path, the SOC change at each node of the path, and the corresponding
charging strategies of this partial path, respectively. All these labels at node i are grouped into a set denoted by L(i). Taking arc (i,j)for
example to illustrate the label correcting process, suppose we have label k at node i represented by lk(i) = [ck,wk,sk,t
→k,tk]; then, a label
denoted by lu(i) = [cu,wu,su,t
→u,tu]at node j is generated if the feasibility and dominance checks are performed for link i→j, where
cu=ck+cj+cij, wu=wk+Hij +w(i), cij and Hij denote the arc cost and energy consumption respectively, su=1 or 0 is determined
by the electricity feasibility check, the values of t
→u and tu are determined by the time feasibility check. The dominance check is to
discard the labels at node j that are dominated by any other labels in terms of the cost and SOC.
The CG-based method is typically incurred with the tailing-off effect, indicating that the method reaches a plateau and becomes
slow to approach the optimum (Wang et al., 2018). To overcome this problem, we use the following stabilization technique (Addis
et al., 2012; L. Zhang et al., 2021) to speed up the convergence. Let Φ= {δ,β}denote the vector of a dual solution of the RMP and Φ
represent the update of Φ. Initially, we set Φ as vector 0, and a modied dual vector
Φ can be computed by
Φ=
ε
1•Φ+(1−
ε
1)Φ in the
iterative process, where
ε
1∈ [0,1]is a weighting parameter. Given
ε
1, the CG procedure is executed until no columns can be added to
the RMP or the objective function reduction of RMP is no greater than a priori dened threshold
ε
2 after a pre-dened number of
iterations. Then, we update Φ as
Φ and increase
ε
1 gradually for a new iteration of the above process. The CG procedure terminates
when
ε
1=1 and no columns can be added.
4.2. Second-stage method of timetable shifting
Timetable shifting affects the charging schedule but has little impact on the trip sequences served by the buses. As described in
Section 2.2, two timetable shifting strategies are applied. The second stage aims to minimize the peak power demand while controlling
the cost increase of the rst stage and modications to the timetable. The outputs of this stage include the adjusted departure times and
the start and end charging times. We formulate the model of this stage as
minΦ =θ+
ω
1
i∈I
τ
d
i−di(24)
s.t.
FC +DC +
i∈r
t∈T
ce
twitE≤
ω
2C(25)
Eqs. (4)-(6a), (12)-(15).
Different time shifts may induce the same peak power demand, and modications to the timetable are also presented in the
objective function. The rst term of Eq. (24) denotes the weighted peak power demand and the second term denotes the weighted value
of time shifts, where
ω
1 is a weight factor. Eq. (25) indicates that the total operational costs increase inuenced by the shifting strategy
should be within a certain range, where
ω
2 is the controlling factor. In addition, it should be noted that the eet and battery degra-
dation costs do not change as the eet schedule and the SOC changes are determined in the rst stage. Therefore, FC and DC in Eq. (25)
can be neglected. In other words, the timetable shifting strategy simply changes the charging cost CC.
M. Duan et al.
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15
In this stage, a smaller timestep size (e.g., 1 min) is adopted because a larger timestep makes no sense for the timetable modi-
cations. Nevertheless, a small timestep size signicantly increases the solution space, and it is difcult to solve directly by a solver (e.g.,
CPLEX, GUROBI). We design an iterative timetable shifting algorithm involving two shifting strategies, in which block shifting is rst
applied for all trips, followed by the differentiated shifting implemented many times for the subset trips with minor modications. In
other words, once the set of shifts in the blocking shifting is determined, the differentiated shifting strategy is implemented several
times until the peak power demand does not decrease in an acceptable time; then the next iteration of block shifting begins. The
timetable shifting algorithm involves a restricted model of Eq. (24), where variable Δl is pre-determined and variable
τ
d
i for the subset
trips are xed. With this variable xed strategy, the restricted model could be directly solved by a solver.
As for the selection of the subset of timetabled trips, based on Fonseca et al. (2018), we designed a random selection strategy,
denoted as Z(u), where u denotes the size of the selected trip set. We assume that any trip i∈I has an equal probability of being
selected. We set the selected size u as 0.3|I|. In each iteration of the second stage, the selected trips are denoted as subset I′. For each
i∈I′, the differentiated shifts Δi are set as 0, i.e., the departure time
τ
d
i is xed as di+Δl. For unselected trip i∈I\I′, Δi is the decision
variable and need to be solved in the second stage. Note that the selecting strategy are applied multiple times until the objective value
of the second stage remains the same comparing with the last selection. Obviously, u is related to the solution precision of the second
stage and this value is associated with the solution efciency.
The pseudo-code of the timetable shifting algorithm is described as follows. The input consists of a set of trips I, the original
timetable T0, eet schedule R, and charging strategy C1
R with specied charging durations, two stopping criteria (stopCriterion1 and
stopCriterion2), and a selection strategy Z(u). Both the two stopping criteria are based on the gaps between the solutions obtained in
two adjacent iterations. Specically, given Φ1 and Φ2 as the solution values of two adjacent iterations and gap threshold
ε
, if
(Φ2−Φ1)/Φ1≤
ε
, the iterative process ends and otherwise continues. The algorithm starts with solving Eq. (24) without timetable
modications; thus, departure time
τ
d
i is xed to T0 for all i∈I. An initial solution S0 is generated composed of the initial timetable
T0 and charging strategy C2
R including all start charging times. The iterative procedure is described in Lines 3–10, which runs until the
stopCriterion2 is met.
The iteration of block shifting starts in Line 4 by random selecting Δl satisfying Eq. (12a) and a new timetable T
η
1 is generated,
where
η
1 records the number of iterations. The iteration of differentiated shifting starts in Line 6 by applying trip selection strategy Z(u)
.A new solution is calculated in Line 7 by solving Eq. (24), with
τ
d
i xed to T
η
1 for all i∈I′. The obtained solution is at least as good as
the current best solution due to the stopping criteria. S* is constantly updated as the best-found solution and returned once the
stopCriterion2 is met.
Iterative Timetable Shifting Algorithm
Input: I, T0, R, C1
R, stopCriterion1, stopCriterion2,Z
Initialization:
1: S0=C2
R,T0,Φ0← solve Eq. (24) by a solver with Δl=0 for all l∈L,
τ
d
i xed to T0 for all i∈I
2:
η
1=1
3: While stopCriterion1 not satised do
4: T
η
1← random selected Δl for all ∀l∈L satisfying Eq. (12a)
5: While stopCriterion2 not satised do
6: I′← selecting trips with strategy Z
7: S
η
1=C2
R,T′
η
1,Φ
η
1← solve Eq. (24) by a solver with
τ
d
i xed to T
η
1 for all i∈I′
8: end while
9:
η
1=
η
1+1
10: end while
11: return S
η
1, Φ
η
1
The pseudo-code of the two-stage solution method is outlined below, and the owchart is depicted in Fig. 7. First, the method starts by
obtaining the initial solution with simple heuristic method. Subsequently, the two stages including CG technique and timetable shifting
performs until the stopCriterion0 is met. The stopCriterion0 of the whole algorithm is set as (C′
−C)/C<
ε
, where C′and C are the total
operational costs of two adjacent iterations in the CG procedure, and
ε
is the relative gap threshold.
Remark 2.Note that the solutions obtained by the CG technique may not be optimal due to the lack of an optimality procedure (e.g., branch
and bound) (Xu and Meng, 2019). However, this is acceptable because the result of the rst stage is further optimized in the second stage
through an iterative process. As the possible combinations of time shifts constitute a sizeable set, it is difcult to solve Eq. (24) directly by a
solver. The iterative timetable shifting algorithm and trip selection strategy ensure that a better solution is obtained within a limited computation
time. It should be also noted that the resolutions of time discretization in the two stages play a key role. Sensitivity tests are usually conducted to
M. Duan et al.
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16
determine the suitable resolutions for balancing computation efciency and solution quality.
Two-Stage Solution Method
1: Fix timetable T0 and generate the initial solution F0= {R0,CR0}by the simple heuristic method, then obtain the total operational costs C0 and peak power
demand θ0, presented in Section 4.1.1;
2: Set iteration counter
η
2=1
3: Obtain F1= {R1,CR1}and C1 using the CG procedure with input F0, presented in Section 4.1.1;
4: While stopCriterion0 not satised do
5: Obtain S
η
2=C2
R
η
2,T′
η
2and θ
η
2 using the iterative method of timetable shifting, presented in Section 4.2;
6:
η
2=
η
2+1;
7: Obtain F
η
2= {R
η
2,CR
η
2}and C
η
2 using the CG procedure with inputs R0, CR
η
2−1 and C2
R
η
2, presented in Section 4.1.1;
8: end while
return F
η
2, C
η
2
Remark 3.We present the complexity analyses for the proposed arc-based model and the two-stage solution method. In general, the
complexity of the models is closely related to the number of independent variables, such as the number of service trips, lines, and power demand
periods as well as the number of timesteps. The arc based-model is a MIP optimization problem, which is a NP-hard problem per se. The order of
magnitude is dictated by variables a, b, θq and x whose sizes are respectively |I| • |T|, |Q|, and |I|2, and by constraints (4), (5b), (7), (9a), (9c),
and (19a) whose sizes are |I|2, |I| • |T|and |Q|. To optimize the operational costs and reduce the peak power demand, we reduce the
computational complexity by reformulating the arc-based model into two-stage model. The rst stage model involves the column generation
Table 3
Setup of instances.
Instance batch Headway (min) No. of trips
IB1 20 16
IB2 15 21
IB3 12 26
IB4 10 31
IB5 8 38
IB6 5 61
Fig. 7. The owchart of the two-stage solution method.
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17
technique, resulting in a runtime complexity of O(L• |Il|2•T)by the recursive formulation in each iteration; the second stage model with the
timetable shifting strategy has a heavily reduced scale of variables, which is an integral programming problem and can be easily solved by the
heuristic rules and an optimization solver.
5. Numerical experiments
In this section, small instances are rst generated to evaluate the performance of the proposed two-stage solution method. To
further assess the effectiveness, a real-life case study with a similar setting in Zhang et al. (2021) is conducted. We explore the impact of
exible charging and timetable shifting strategies on optimal solutions to highlight their roles in reducing the total operational costs
and moderating peak power demand. All computations are performed on a PC with i7-9850U @ 2.60 GHz CPU and 8 GB RAM. The
programming language is Python.
5.1. Small instances
We generate a set of illustrative small-sized instances in a planning time horizon from 6:00 to 12:00 according to Zhou et al (2022).
These instances have different headways ranging from 5 to 20 min to assess the efciency of the proposed solution method as shown in
Table 3. We set the number of chargers to 1 for the convince of studying the utilization rate in such small cases. Supposed that the
energy consumption is 1.35 kWh/km (Gao et al., 2017) per round trip of 20 km in length. A BEB has a battery capacity of 120 kWh,
with which a full charge can empower around three trips. The timestep is set as 10 min, meaning that the start charging time and
charging duration should be an integer multiple of the timestep. Considering the battery discharges when not in use after a full charge
overnight, each BEB is assumed to have a SOC of 0.95 when it starts daily operations. The minimum charging duration is 10 min (Duan
et al., 2021); the minimum headway between any two timetabled trips is 5 min. We use the time-dependent electricity prices (Wang
et al., 2021) shown in Table 4. For safety concerns, the SOC of the BEB battery should be within the range [Hmin =0.2,Hmax =0.95].
The xed cost of employing one BEB is 16.5 $/day, a linear charging prole is adopted for slowing charging at night, and the pa-
rameters for the battery degradation are adopted from Zhang et al. (2021). The peak power demand is in two or three order of
magnitude while the sum of time shifts is in one order of magnitude. Thus, we set
ω
1 to 0.001, with the aim of optimizing the peak
power demand while choosing the solution with the smallest time shifts. As shown in Fig. 4, the nonlinear charging prole in CV phase
is approximated by a piecewise linear function under fast charging for daily operations.
ω
2 is the controlling factor, and set as 1.05.
Thus, we adopt the charging prole of fast charging, which is expressed as Eq. (26), indicating that the full charging time for a depleted
battery is 1.5 h.
f(
χ
) =
0.8
χ
,
χ
∈ [0,1)
0.8+0.5(
χ
−1),
χ
∈ [1,1.1)
0.85 +0.375(
χ
−1.1),
χ
∈ [1.1,1.5]
(26)
where
χ
is the charging duration (h) and f(
χ
)denotes the charging SOC.
We implement the simple heuristic method (Section 4.1.1) as the rst baseline to compare the optimal schedule obtained by the
two-stage method. According to the battery level, the simple heuristic method always chooses to replenish energy or serve the trip with
the earliest departure time. To further validate the effectiveness of our model and method, the model suggested by Zhang et al. (2021)
is also solved by the CG technique for comparison as the second baseline. In their model, the charging strategy is semi-exible. Once a
bus visits the charging station, it will be charged to a pre-given electricity level while optimizing the sequence of charging operations at
the depot.
For the sake of convenience, we denote these three methods respectively as P1 (simple heuristic method and xed charging
strategy), P2 (CG method with semi-exible charging strategy), and P3 (our model and two-stage method). For the charging in P1, the
start charging time is not exible and the batteries are charged to a specied electricity level; for P2, the start charging time is exible
and the charging duration is xed given the remaining electricity volume; and for P3, both the start charging time and charging
duration (related to electricity volume) are exible. The comparations of the results are presented in Fig. 8. The gure shows that the
total operational costs and the cost components including eet, charging, and battery degradation costs increase as the number of trips
increases, as expected. The average utilization rate of BEBs uctuates with the number of trips, where the average utilization rate is
derived by the service time of transporting passengers divided by the total operational time. In addition, this gure shows the result of
the peak power demand, which is determined by the use of the single charger with an upper bound value of 96 kW.
Compared with the results of P1 and P2, P3 obtains additional total operational costs savings by 17–21% and 7–14% for the six
Table 4
Price of the electricity of the day.
Electricity price ($/kWh) Time of the day
0.049 22 pm – 7 am
0.121 7 pm – 9 am, 11 am – 13 pm, 16 pm – 18 pm, 20 pm – 22 pm
0.172 18 pm – 20 pm
0.173 9 am −11 am, 13 pm – 16 pm
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instances (Fig. 8(a)), respectively, indicating that the proposed model and solution method can produce better eet and charging
schedules. Compared to the eet cost obtained by P1 and P2, the costs of P3 are reduced by approximately 31% and 16% respectively
(Fig. 8(b)), implying a more compact eet schedule. This result is corroborated by a 45% and 20% increase in the utilization rate of
BEBs (Fig. 8(c)), respectively. Nevertheless, the charging cost associated with P3 is higher than the results of P1 and P2 (Fig. 8(d)). The
results can be explained by the reduction in eet size but increases in the number of service trips per BEB and electricity consumption
for the operations. As for battery degradation, we know that larger SOC variations cause faster battery degradations (L. Zhang et al.,
2021; Zhou et al., 2022). As expected, the battery degradation cost from P3 is less than those of P1 and P2 (Fig. 8(e)) because the full-
exible charging strategy in P3 avoids larger SOC variations. Similar to the charging cost, we notice that the battery degradation costs
are approximately the same for IB5 and IB6 for the three methods because of the setting of only one charger. Finally, in Fig. 8(f), we see
that P1 has the highest peak power demands in the six instances, whereas the peak power demands of P3 are not always lower than
those of P2. To demonstrate that the number of used chargers impacts on peak power demand, we further investigate a large case study
in Section 5.2.
To evaluate the effects of the length of the timestep, we test the computation time and the efciency using the six instances. The
results are presented in Table 5. Columns 1 and 2 present the problem instance; columns 3 and 4 show the computational results of the
two-stage solution method, including the total operational costs and the peak power demand; columns 5 and 6 are the running times to
obtain the optimal solution (T_CPU Time) and spending on solving the pricing problem (PP_CPU Time), respectively. With different
timesteps for the same instance, the optimal costs appear approximately stable, indicating that the optimal costs are to a small extent
inuenced by timesteps in these cases. According to the CPU times, we observe that the computation efciency is largely and nega-
tively affected by the number of trips and positively affected by the timesteps. In other words, longer computation times are associated
(c) Utilization rate (d) Charging cost
(e) Battery degradation cost (f) Peak power demand
(b) Fixed fleet cost
Charging cost ($)
Total operational costs ($)
0
100
200
300
400
500
600
700
800
900
INB1 INB2 INB3 INB4 INB5 INB6
0
50
100
150
200
250
INB1 INB2 INB3 INB4 INB5 INB6
Battery degradation ($)
0
20
40
60
80
100
120
INB1 INB2 INB3 INB4 INB5 INB6
Peak power demand (kW)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
INB1 INB2 INB3 INB4 INB5 INB6
Utilization rate
(a) Total operational costs
0
100
200
300
400
500
600
INB1 INB2 INB3 INB4 INB5 INB6
P1 P2 P3
0
20
40
60
80
100
120
INB1 INB2 INB3 INB4 INB5 INB6
Fixed fleet cost ($)
0
100
200
300
400
500
600
700
800
900
INB1 INB2 INB3 INB4 INB5 INB6
0
50
100
150
200
250
INB1 INB2 INB3 INB4 INB5 INB6
0
20
40
60
80
100
120
INB1 INB2 INB3 INB4 INB5 INB6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
INB1 INB2 INB3 INB4 INB5 INB6
Utilization rate
0
100
200
300
400
500
600
INB1 INB2 INB3 INB4 INB5 INB6
0
20
40
60
80
100
120
INB1 INB2 INB3 INB4 INB5 INB6
Total operational costs ($)
Fig. 8. Comparison of the results between P1, P2, and P3.
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19
with more trips and shorter timesteps. In addition, it is found that the size of the timestep also has a signicant impact on the peak
power demand in the instances. When the timestep is larger, it means more bias in the peak power demand. Summarily, when the
timestep is 10, there is no signicant difference in the accuracy from a smaller timestep, but the computation time is signicantly less.
To balance computation efciency and accuracy, the timestep of 10 min is adopted in the large case study below.
5.2. Real BEB network
In this subsection, we further demonstrate the effectiveness of the proposed model and solution method with a real-world BEB
network from Zhang et al. (2021), of which the original timetable has 210 trips as listed in the supplementary document (Table S1).
The network has one depot, two terminals, and six lines (numbered from 1 to 6), as depicted in Fig. 9. The lengths of the six lines are 10
km, 12.5 km, 30 km, 17.5 km, 20 km, and 22.5 km, respectively. Lines 1, 2, 4, 5 and 6 are unidirectional, while line 3 is circular. We
consider the round trip travel times (terminal to terminal) are 69 min, 86 min, 103 min, 120 min, 138 min, and 155 min, respectively.
That is to say, for lines 1, 2, 4, 5 and 6, the travel time of a trip doubles the one-way travel time; for line 3, the travel time of a trip is
equal to the one-way travel time. The BEBs are medium-duty buses equipped with 162 kWh batteries and the planning time horizon is
set for the whole day including daily operations, electricity replenishment, and overnight charging. The number of fast chargers for
electricity replenishment is 15, and a sufcient number of slow chargers with a charging power of 50 kW is provided for overnight
charging. Other parameters are consistent with those for the small instances.
Table 5
Comparison of cost and runtimes in experiments of different timesteps.
Batch Timestep (min) Total operational costs ($) Peak power demand (kW) T_CPU Time (s) PP_CPU Time (s)
INB1 2 156.13 89.6 365 16
5 156.44 89.6 233 7
10 154.88 89.6 250 7
15 158.41 96 197 7
INB2 2 202.72 76.8 926 31
5 205.9 83.2 365 10
10 204.68 83.2 342 9
15 206.95 64 381 8
INB3 2 260.52 96 1232 37
5 261.31 89.6 455 13
10 262.83 83.2 480 11
15 258.69 96 439 11
INB4 2 307.86 96 2111 57
5 306.4 96 744 15
10 306.32 96 713 13
15 306.43 96 631 13
INB5 2 387.22 96 3935 79
5 391.34 83.2 1212 22
10 394.1 96 845 17
15 396.84 96 845 16
INB6 2 662.82 89.6 14,924 229
5 663.28 89.6 3658 50
10 658.67 76.8 2362 33
15 660.46 96 1727 31
Fig. 9. Geographical distribution of selected BEB lines.
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The optimal solution can be obtained by the two-stage solution method within 5 h, 2.5 h for each stage. As this is a deterministic and
planning problem that does not require real-time responses, we consider it feasible. The optimized eet schedule and charging op-
erations are shown in Table 6, which includes 45 BEBs and 104 charging operations. The charging details including the start charging
time, end charging time and charging volume, and the optimized timetable is shown in the supplementary document (Table S2). The
total operational costs are $2207.09, including a eet cost of $726, a charging cost of $979.28, and a battery degradation cost of
$501.81. The optimized peak power demand is 855.36 kW, and the utilization rate is 0.571. The block shifts are [-5, −5, 1, 5, 2, 1] in
minutes for lines 1 to 6, where a negative and a positive number represent moving forward and backward for the whole line,
respectively. Also, some differentiated shifts in the timetable are made for 17 trips, as shown in the supplementary document
(Table S3).
We compare the results of P3 with those obtained from P1 and P2, as reported in Fig. 10. Compared to P1, P3 has a 22% reduction in
the total operational costs, 20% reduction in eet cost, 1% savings in charging cost, and 45% savings in the battery degradation cost.
Thus, it can be concluded that our model and solution method have signicant advantages over P1, which is usually adopted by the
BEB operators. Compared to P2, it is worth noting that P3 has a 7.1% of total operationals cost savings but a 17% increase in the battery
degradation cost. The result indicates that the full-exible charging strategy involves less total operational costs, while it accelerates
the battery aging process and results in higher costs for battery capacity fading. Further, we can see in Fig. 11 that the peak power
demand is reduced by 17% and the used number of chargers is reduced by 2. This makes sense because more buses being charged at the
Table 6
Fleet schedule and the charging operations of the six lines.
Line NO. BEB NO. Trip chains Line NO. BEB NO. Trip chains
1 1 1–6-11-c-18–26-c-32-c-38 4 23 124–129-c-135-c-140-c-146
1 2 2–7-12–16-c-20-c-27-c-34 4 24 125–130-c-136-c-141-c-147
1 3 3–8-13-c-17–23-c-28–33 4 25 126–131-c-137-c-142-c-148
1 4 4–9-14-c-22–25-c-30-c-35 4 26 127–132-c-143-c-149
1 5 5–10-15-c-19–24-c-29-c-35 4 27 128–134-c-139-c-145
1 6 21–31-37 4 28 133–138-c-144
2 7 39–44-51-c-60-c-66-c-72 5 30 151-c-157-c-163-c-169
2 8 40–47-c-53–62-c-68 5 29 154-c-160-c-166-c-172
2 9 41–46-c-54–57-c-64-c-70 5 31 155-c-161-c-167
2 10 42-c-49–52-c-56–59-c-65-c-71 5 32 152-c-158-c-164-c-170
2 11 43–48-c-55–61-c-67 5 33 153-c-159-c-165-c-171
2 12 45–50-c-58-c-63-c-69 5 34 150-c-156-c-162-c-168
3 13 73-c-83–90-c-99-c-108-c-117 6 35 173-c-182-c-192-c-202-c-210
3 14 74–101-116 6 36 174-c-189-c-201
3 15 75–82-c-92-c-110 6 37 175-c-185-c-194-c-204
3 16 76–84-c-93-c-106-c-119 6 38 176-c-186-c-195-c-207
3 17 77–87-96–103-112–121 6 39 177-c-187-c-196-c-206
3 18 78–85-c-94-c-102-c-111-c-120 6 40 178-c-188-c-198-c-208
3 19 79–86-c-95-c-105-c-115 6 41 179-c-197
3 20 80-c-89–100-c-109-c-118 6 42 180-c-190-c-200-c-209
3 21 81–88-c-97-c-104-c-113-c-122 6 43 181-c-191-c-205
3 22 91–98-c-107–114-c-123 6 44 183-c-193-c-203
6 45 184-c-199
Notes: c indicates a charging operation.
Fig. 10. Comparison of the cost components between solution methods of P1, P2, and P3.
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Fig. 11. Comparison of the peak power demand and the number of used chargers by P1, P2, and P3.
Fig. 12. Comparison of the total operational costs and cost components for the three charging strategies.
Fig. 13. Comparison of the peak power demand and number of used chargers for the three charging strategies.
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same time create a higher peak power demand; in other words, reducing the peak power demand is associated with decreasing the
number of used chargers. These results in Figs. 10 and 11 conrm that the integrated optimization model and the two-stage solution
method together lead to a more efcient eet and charging schedules, and also have a signicant advantage in further reducing the
peak power demand and decreasing the number of chargers.
The following analysis further illustrates the advantages of the full-exible charging strategy. Fig. 12 shows the comparison derived
from the proposed two-stage solution method with the three charging strategies. As expected, the full-exible charging strategy results
in the lowest total operational costs, the lowest eet size, and the highest utilization rate of BEBs. A exible charging strategy increases
the charging opportunity, allows for more trips to be performed, and requires fewer BEBs to nish the trip service, thereby resulting in
lower total operational costs and an increased utilization rate. In addition, Fig. 13 shows that the exible charging strategy uses fewer
chargers as well as a smaller peak power demand. The disadvantage is that the full-exible charging strategy results in a greater battery
degradation cost compared to the semi-exible charging strategy because the higher SOC variation or average SOC causes faster
battery capacity degradation (Zhang et al., 2019).
For illustration purposes, the SOC curves of the three charging strategies of line 2 are presented in Fig. 14, where we set the initial
and maximum battery electricity as 95%. Fig. 14(a) shows that the BEBs visit the charging station if the remaining electricity level is
too low for the next trip. Obviously, the xed charging strategy leads to high SOC between 30 and 95% and hence more battery
degradations compared with the other two charging strategies. Thus, it is an inferior charging strategy because it does not only occupy
more vehicles but also has a serious impact on battery life. Fig. 14(b) and (c) show the SOC curves for the semi-exible charging
strategy and the full-exible charging strategy, respectively. They both take into account the number of available chargers, TOU plans,
the current electricity level, and the power load on the grid. Comparing these two gures, we can see that the semi-exible charging
strategy maintains high SOCs between 70 and 95% in most cases, while the full-exible charging strategy keeps the SOC between 20
and 65%. Based on the nding that a higher average SOC level results in a faster battery capacity degradation (Zhang et al., 2019), the
full-exible charging strategy with an average of 42.5% SOC seems to outperform the semi-exible counterpart with 82.5% average
SOC. However, it is not true in terms of the higher battery degradation cost associated with the full-exible charging strategy. It is
because, with the full-exible charging strategy, the battery SOC varies considerably, including an initial discharge approximately
from 95% to 30% or a nal charging from 30% to 100% at the end of the operation as seen in Fig. 14(c). Finally, it can also be seen that
because of the limited resources (e.g., chargers and power grid) or the search for better charging timing due to the TOU plans, BEBs in
the full-exible charging strategy are enabled to wait for a short time before starting to charge and choose an appropriate charging
duration. Overall, the full-exible charging strategy leads to the more efcient eet and charging schedules.
To examine the effects of the timetable shifting strategy, we set up a scenario without the consideration of timetable shifting for
comparison. As seen in Table 7, regardless of the solution method, the timetable shifting strategy leads to a 17–26% decrease in peak
power demand and a reduction in the number of chargers, although the total operational costs increase slightly by 0.3–0.7%. The
increase in the total operational costs is mainly due to the increase in charging costs as a result of the TOU plans, while the charging
volumes remain the same. To exclude the inuence of the TOU plans on the total operational costs, we set a xed electricity price of
0.15$/kWh in the test. It is found that the total operational costs remain unchanged after applying the timetable shifting strategy but
the peak power demand decreased considerably by 26% compared with not applying the strategy. Using the full-exible charging
strategy and keeping other setups the same, Fig. 15 shows the daily power demand for central depot charging before and after applying
the strategy, demonstrating that the timetable shifting strategy has a signicant impact on reducing the grid load.
Through the above numerical analyses, we conclude that the proposed model and two-stage solution method perform well in
generating the eet and charging schedules. The results show that the full-exible charging and timetable shifting strategies signif-
icantly inuence the total operational costs and the peak power demand. For the operators of BEB networks, the design of the
timetable, the eet schedule, and the charging strategy are not only crucial for reductions in the eet size and number of chargers but
also important for the relief of the power grid pressure. We recommend that the operators should apply the exible charging strategy
and adopt the timetable shifting strategies including block shifting and differentiated shifting strategies. In this way, the total oper-
ational costs can be saved considerably and the power grid pressure can be relieved, associated with a decrease in number of used
chargers.
6. Conclusions and future work
This paper investigates the optimal eet and charging schedules with the consideration of timetable shifting, battery degradation,
nonlinear charging prole, and TOU plans. We propose an integrated optimization model with high complexity and reformulate the
arc-based model into a two-stage model. We further develop a two-stage solution method involving the CG technique and iterative
adjustments of timetable shifts to nd optimal solutions. Numerical experiments considering small instances and a real-world case are
performed to demonstrate the efciency and effectiveness of the model and solution method. The results show that the proposed model
and method can achieve signicant reductions in the total operational costs and peak power demand compared to two baselines. To
highlight the inuence of the full-exible charging and timetable shifting strategies, sensitivity analyses are conducted. The results
show that the strategies contribute to the peak power demand reduction and the increased utilization rate of BEBs. These results
conrm the outperformances of the proposed eet schedule and charging planning model that incorporates the two useful strategies.
In future work, we would make more efforts to improve our work. First, we will further investigate the conjecture that using a larger
capacity battery could reduce the battery degradation cost caused by the full-exible charging strategy. Second, the inuence of
timetable shifting on passenger demand and preferences (Liao et al., 2013, 2020) should be considered because timetable shifts may
lead to a change in waiting time for passengers and further decrease their travel accessibility (Qin and Liao, 2021, 2022). Third, some
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Fig. 14. SOC curves of the three charging strategies.
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realistic conditions, such as the more bus types, variable charging power, wireless charging strategy, and multiple depots in the BEB
network should be considerd. Fourth, the coefcients weighing the importance of different components in the objective functions
should be calibrated before real-world applications.
CRediT authorship contribution statement
Mengyuan Duan: Conceptualization, Methodology, Validation, Writing – original draft. Feixiong Liao: Conceptualization,
Methodology, Supervision, Writing – review & editing. Geqi Qi: Conceptualization, Supervision. Wei Guan: Conceptualization, Re-
sources, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to
inuence the work reported in this paper.
Data availability
No data was used for the research described in the article.
Acknowledgements
This work is jointly supported by the Fundamental Research Funds for the Central Universities (No. 2019JBZ003), the National
Natural Science Foundation of China (No.72288101, No.72271127, No.71801134), and the Dutch Research Council. The rst author is
grateful for the nancial support from the China Scholarship Council (CSC).
Appendix A. Supplementary material
Supplementary data to this article can be found online at https://doi.org/10.1016/j.trc.2023.104175.
Table 7
Results without and with the timetable shifting strategy.
Without timetable shifting strategy With timetable shifting strategy
C CC θ C CC θ
P1 2816.46 983.12 1233.63 ↑ 0.3% ↑ 0.8% ↓ 17%
P2 2358.74 1121.42 1328.94 ↑ 0.7% ↑ 1.6% ↓ 23%
P3 2196.85 969.05 1149.12 ↑ 0.5% ↑ 1% ↓ 26%
Notes: C =total operational costs; CC =charging cost; θ =peak power demand; ↑ =increase; ↓ =decrease.
Fig. 15. Comparison of the power demand with and without the timetable shifting strategy.
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References
Abdelwahed, A., van den Berg, P.L., Brandt, T., Collins, J., Ketter, W., 2020. Evaluating and optimizing opportunity fast-charging schedules in transit battery electric
bus networks. Transp. Sci. 54, 1601–1615. https://doi.org/10.1287/trsc.2020.0982.
Addis, B., Carello, G., Ceselli, A., 2012. Exactly solving a two-level location problem with modular node capacities. Networks 59, 161–180. https://doi.org/10.1002/
net.20486.
An, K., 2020. Battery electric bus infrastructure planning under demand uncertainty. Transp. Res. Part C Emerg. Technol. 111, 572–587. https://doi.org/10.1016/j.
trc.2020.01.009.
Barr´
e, A., Deguilhem, B., Grolleau, S., G´
erard, M., Suard, F., Riu, D., 2013. A review on lithium-ion battery ageing mechanisms and estimations for automotive
applications. J. Power Sources 241, 680–689. https://doi.org/10.1016/j.jpowsour.2013.05.040.
Carosi, S., Frangioni, A., Galli, L., Girardi, L., Vallese, G., 2019. A matheuristic for integrated timetabling and vehicle scheduling. Transp. Res. Part B Methodol. 127,
99–124. https://doi.org/10.1016/j.trb.2019.07.004.
Ceder, A., 2001. Efcient Timetabling and Vehicle Scheduling for Public Transport. Comput. Sched. Public Transp. 37–52 https://doi.org/10.1007/978-3-642-56423-
9_3.
Chen, H., Hu, Z., Zhang, H., Luo, H., 2018. Coordinated charging and discharging strategies for plug-in electric bus fast charging station with energy storage system.
IET Gener. Transm. Distrib. 12, 2019–2028. https://doi.org/10.1049/iet-gtd.2017.0636.
Duan, M., Qi, G., Guan, W., Lu, C., Gong, C., 2021. Reforming mixed operation schedule for electric buses and traditional fuel buses by an optimal framework. IET
Intell. Transp. Syst. 15, 1287–1303. https://doi.org/10.1049/itr2.12098.
Fonseca, J.P., van der Hurk, E., Roberti, R., Larsen, A., 2018. A matheuristic for transfer synchronization through integrated timetabling and vehicle scheduling.
Transp. Res. Part B Methodol. 109, 128–149. https://doi.org/10.1016/j.trb.2018.01.012.
Gao, Z., Lin, Z., LaClair, T.J., Liu, C., Li, J.M., Birky, A.K., Ward, J., 2017. Battery capacity and recharging needs for electric buses in city transit service. Energy 122,
588–600. https://doi.org/10.1016/j.energy.2017.01.101.
Gao, Y., Xia, J., D’Ariano, A., Yang, L., 2022. Weekly rolling stock planning in Chinese high-speed rail networks. Transp. Res. Part B Methodol. 158, 295–322. https://
doi.org/10.1016/j.trb.2022.02.005.
He, Y., Song, Z., Liu, Z., 2019. Fast-charging station deployment for battery electric bus systems considering electricity demand charges. Sustain. Cities Soc. 48
https://doi.org/10.1016/j.scs.2019.101530.
He, Y., Liu, Z., Song, Z., 2020. Optimal charging scheduling and management for a fast-charging battery electric bus system. Transp. Res. Part E Logist. Transp. Rev.
142 https://doi.org/10.1016/j.tre.2020.102056.
Hledik, R., 2014. Rediscovering Residential Demand Charges. Electr. J. 27, 82–96. https://doi.org/10.1016/j.tej.2014.07.003.
Huang, K., Liao, F., Gao, Z., 2021. An integrated model of energy-efcient timetabling of the urban rail transit system with multiple interconnected lines. Transp. Res.
Part C Emerg. Technol. 129 https://doi.org/10.1016/j.trc.2021.103171.
Ibarra-Rojas, O.J., Giesen, R., Rios-Solis, Y.A., 2014. An integrated approach for timetabling and vehicle scheduling problems to analyze the trade-off between level of
service and operating costs of transit networks. Transp. Res. Part B Methodol. 70, 35–46. https://doi.org/10.1016/J.TRB.2014.08.010.
IEA, 2019. Global EV outlook 2019: Scaling up the transition to electric mobility.
IEA, 2022. Global Energy Review: CO2 Emissions in 2021 Global emissions rebound sharply to highest ever level. Iea 1–14.
Ji, J., Bie, Y., Zeng, Z., Wang, L., 2022. Trip energy consumption estimation for electric buses. Commun. Transp. Res. 2, 100069 https://doi.org/10.1016/J.
COMMTR.2022.100069.
Lam, L., Bauer, P., 2013. Practical capacity fading model for Li-ion battery cells in electric vehicles. IEEE Trans. Power Electron. 28, 5910–5918. https://doi.org/
10.1109/TPEL.2012.2235083.
Li, J.Q., 2014. Transit bus scheduling with limited energy. Transp. Sci. 48, 521–539. https://doi.org/10.1287/trsc.2013.0468.
Li, L., Lo, H.K., Huang, W., Xiao, F., 2021. Mixed bus eet location-routing-scheduling under range uncertainty. Transp. Res. Part B Methodol. 146, 155–179. https://
doi.org/10.1016/j.trb.2021.02.005.
Liao, F., 2016. Modeling duration choice in space–time multi-state supernetworks for individual activity-travel scheduling. Transp. Res. Part C Emerg. Technol. 69,
16–35. https://doi.org/10.1016/j.trc.2016.05.011.
Liao, F., 2019. Joint travel problem in space–time multi-state supernetworks. Transportation (Amst). 46, 1319–1343. https://doi.org/10.1007/s11116-017-9835-6.
Liao, F., Arentze, T., Timmermans, H., 2013. Incorporating space–time constraints and activity-travel time proles in a multi-state supernetwork approach to
individual activity-travel scheduling. Transp. Res. Part B Methodol. 55, 41–58. https://doi.org/10.1016/j.trb.2013.05.002.
Liao, F., Tian, Q., Arentze, T., Huang, H.J., Timmermans, H., 2020. Travel preferences of multimodal transport systems in emerging markets: The case of Beijing.
Transp. Res. Part A Policy Pract. 138, 250–266. https://doi.org/10.1016/j.tra.2020.05.026.
Lin, D.Y., Juan, C.J., Chang, C.C., 2020. A Branch-and-Price-and-Cut Algorithm for the Integrated Scheduling and Rostering Problem of Bus Drivers. J. Adv. Transp.
2020 https://doi.org/10.1155/2020/3153201.
Lin, Y., Zhang, K., Shen, Z.J.M., Ye, B., Miao, L., 2019. Multistage large-scale charging station planning for electric buses considering transportation network and
power grid. Transp. Res. Part C Emerg. Technol. 107, 423–443. https://doi.org/10.1016/j.trc.2019.08.009.
Liu, T., Ceder, A., 2018. Integrated public transport timetable synchronization and vehicle scheduling with demand assignment: A bi-objective bi-level model using
decit function approach. Transp. Res. Part B Methodol. 117, 935–955. https://doi.org/10.1016/j.trb.2017.08.024.
Liu, T., Ceder, A., 2020. Battery-electric transit vehicle scheduling with optimal number of stationary chargers. Transp. Res. Part C Emerg. Technol. 114, 118–139.
https://doi.org/10.1016/j.trc.2020.02.009.
Liu, K., Gao, H., Liang, Z., Zhao, M., Li, C., 2021. Optimal charging strategy for large-scale electric buses considering resource constraints. Transp. Res. Part D Transp.
Environ. 99, 103009 https://doi.org/10.1016/j.trd.2021.103009.
Liu, Y., Wang, L., Zeng, Z., Bie, Y., 2022. Optimal charging plan for electric bus considering time-of-day electricity tariff. J. Intell. Connect. Veh. 5, 123–137. https://
doi.org/10.1108/JICV-04-2022-0008.
Lopes, J.A.P., Soares, F.J., Almeida, P.M.R., 2011. Integration of electric vehicles in the electric power system. Proc. IEEE 99, 168–183. https://doi.org/10.1109/
JPROC.2010.2066250.
Montoya, A., Gu´
eret, C., Mendoza, J.E., Villegas, J.G., 2017. The electric vehicle routing problem with nonlinear charging function. Transp. Res. Part B Methodol. 103,
87–110. https://doi.org/10.1016/j.trb.2017.02.004.
Nie, Y., Ghamami, M., Zockaie, A., Xiao, F., 2016. Optimization of incentive polices for plug-in electric vehicles. Transp. Res. Part B Methodol. 84, 103–123. https://
doi.org/10.1016/j.trb.2015.12.011.
Pelletier, S., Jabali, O., Laporte, G., 2016. Goods distribution with electric vehicles: Review and research perspectives. Transp. Sci. 50, 3–22. https://doi.org/10.1287/
trsc.2015.0646.
Pelletier, S., Jabali, O., Laporte, G., 2018. Charge scheduling for electric freight vehicles. Transp. Res. Part B Methodol. 115, 246–269. https://doi.org/10.1016/j.
trb.2018.07.010.
Perumal, S.S.G., Lusby, R.M., Larsen, J., 2021. Electric bus planning & scheduling: A review of related problems and methodologies. Eur. J. Oper. Res. 301, 395–413.
https://doi.org/10.1016/j.ejor.2021.10.058.
Qin, N., Gusrialdi, A., Paul Brooker, R., T-Raissi, A.,, 2016. Numerical analysis of electric bus fast charging strategies for demand charge reduction. Transp. Res. Part A
Policy Pract. 94, 386–396. https://doi.org/10.1016/j.tra.2016.09.014.
Qin, J., Liao, F., 2021. Space–time prism in multimodal supernetwork-Part 1: Methodology. Commun. Transp. Res. 1, 100016 https://doi.org/10.1016/j.
commtr.2021.100016.
Qin, J., Liao, F., 2022. Space–time prisms in multimodal supernetwork-Part 2: Application for analyses of accessibility and equality. Commun. Transp. Res. 2, 100063
https://doi.org/10.1016/j.commtr.2022.100063.
M. Duan et al.
Transportation Research Part C 152 (2023) 104175
26
Qu, X., Zhong, L., Zeng, Z., Tu, H., Li, X., 2022. Automation and connectivity of electric vehicles: Energy boon or bane? Cell Reports Phys. Sci. 3, 101002 https://doi.
org/10.1016/J.XCRP.2022.101002.
Razeghi, G., Samuelsen, S., 2016. Impacts of plug-in electric vehicles in a balancing area. Appl. Energy 183, 1142–1156. https://doi.org/10.1016/J.
APENERGY.2016.09.063.
Rinaldi, M., Picarelli, E., D’Ariano, A., Viti, F., 2020. Mixed-eet single-terminal bus scheduling problem: Modelling, solution scheme and potential applications.
Omega 96, 102070. https://doi.org/10.1016/j.omega.2019.05.006.
Schmid, V., Ehmke, J.F., 2015. Integrated timetabling and vehicle scheduling with balanced departure times. OR Spectr. 37, 903–928. https://doi.org/10.1007/
s00291-015-0398-7.
Schoch, J., Gaerttner, J., Schuller, A., Setzer, T., 2018. Enhancing electric vehicle sustainability through battery life optimal charging. Transp. Res. Part B Methodol.
112, 1–18. https://doi.org/10.1016/j.trb.2018.03.016.
Steiner, K., Irnich, S., 2018. Schedule-based integrated intercity bus line planning via branch-and-cut. Transp. Sci. 52, 882–897. https://doi.org/10.1287/
trsc.2017.0763.
Tang, X., Lin, X., He, F., 2019. Robust scheduling strategies of electric buses under stochastic trafc conditions. Transp. Res. Part C Emerg. Technol. 105, 163–182.
https://doi.org/10.1016/J.TRC.2019.05.032.
Teng, J., Chen, T., Fan, W., “David”,, 2020. Integrated Approach to Vehicle Scheduling and Bus Timetabling for an Electric Bus Line. J. Transp. Eng. Part A Syst. 146,
04019073. https://doi.org/10.1061/jtepbs.0000306.
Uslu, T., Kaya, O., 2021. Location and capacity decisions for electric bus charging stations considering waiting times. Transp. Res. Part D Transp. Environ. 90, 102645
https://doi.org/10.1016/J.TRD.2020.102645.
Vicini, R., Micheloud, O., Kumar, H., Kwasinski, A., 2012. Transformer and home energy management systems to lessen electrical vehicle impact on the grid. IET
Gener. Transm. Distrib. 6, 1202–1208.
Wang, G., Fang, Z., Xie, X., Wang, S., Sun, H., Zhang, F., Liu, Y., Zhang, D., 2021. Pricing-aware Real-time Charging Scheduling and Charging Station Expansion for
Large-scale Electric Buses. ACM Trans. Intell. Syst. Technol. 12, 1–26. https://doi.org/10.1145/3428080.
Wang, Y., Huang, Y., Xu, J., Barclay, N., 2017. Optimal recharging scheduling for urban electric buses: A case study in Davis. Transp. Res. Part E Logist. Transp. Rev.
100, 115–132. https://doi.org/10.1016/j.tre.2017.01.001.
Wang, Y., Liao, F., Lu, C., 2022. Integrated optimization of charger deployment and eet scheduling for battery electric buses. Transp. Res. Part D Transp. Environ.
109, 103382 https://doi.org/10.1016/J.TRD.2022.103382.
Wang, K., Zhen, L., Wang, S., Laporte, G., 2018. Column generation for the integrated berth allocation, quay crane assignment, and yard assignment problem. Transp.
Sci. 52, 812–834. https://doi.org/10.1287/trsc.2018.0822.
Wu, Y., Yang, H., Tang, J., Yu, Y., 2016. Multi-objective re-synchronizing of bus timetable: Model, complexity and solution. Transp. Res. Part C Emerg. Technol. 67,
149–168. https://doi.org/10.1016/j.trc.2016.02.007.
Xu, M., Meng, Q., 2019. Fleet sizing for one-way electric carsharing services considering dynamic vehicle relocation and nonlinear charging prole. Transp. Res. Part
B Methodol. 128, 23–49. https://doi.org/10.1016/j.trb.2019.07.016.
Yao, E., Liu, T., Lu, T., Yang, Y., 2020. Optimization of electric vehicle scheduling with multiple vehicle types in public transport. Sustain. Cities Soc. 52, 101862
https://doi.org/10.1016/j.scs.2019.101862.
Zhang, L., Wang, S., Qu, X., 2021a. Optimal electric bus eet scheduling considering battery degradation and non-linear charging prole. Transp. Res. Part E Logist.
Transp. Rev. 154, 102445 https://doi.org/10.1016/J.TRE.2021.102445.
Zhang, Y., Xiong, R., He, H., Qu, X., Pecht, M., 2019. State of charge-dependent aging mechanisms in graphite/Li(NiCoAl)O2 cells: Capacity loss modeling and
remaining useful life prediction. Appl. Energy 255, 113818. https://doi.org/10.1016/J.APENERGY.2019.113818.
Zhang, W., Zhao, H., Xu, M., 2021b. Optimal operating strategy of short turning lines for the battery electric bus system. Commun. Transp. Res. 1, 100023 https://doi.
org/10.1016/J.COMMTR.2021.100023.
Zhou, Y., Meng, Q., Ong, G.P., 2022. Electric bus charging scheduling for a single public transport route considering nonlinear charging prole and battery
degradation effect. Transp. Res. Part B Methodol. 159, 49–75. https://doi.org/10.1016/j.trb.2022.03.002.
M. Duan et al.
