An integrated micro-macro approach for high-speed railway energy-efficient timetabling problem

Transportation Research Part C 112 (2020) 88–115

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

An integrated micro-macro approach for high-speed railway energy-efficient timetabling problem

T

Yan Xua,b, Bin Jiac, Xiaopeng Lib, , Minghua Lid, Amir Ghiasie ⁎

a

College of Metropolitan Transportation, Beijing University of Technology, Beijing 100124, China Department of Civil and Environmental Engineering, University of South Florida, Tampa, FL 33620, USA Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Beijing Jiaotong University, Beijing 100044, China d Beijing Urban Construction Group Co., Ltd., Beijing 100088, China e Surface Transportation Division, Leidos, Inc., Reston, VA 20190, USA b c

ARTICLE INFO

ABSTRACT

Keywords: High-speed railway Train timetabling Energy saving Speed control Train trajectory

Energy efficiency of train operations is influenced largely by the speed control and the scheduled running time in the train timetable. In practice, the running time of a train is often determined in the train timetabling process at the macroscopic level while the energy-efficient speed control of a train on a segment is often determined at the microscopic level with the given timetable. They are usually optimized separately due to limited computational resources, which however may result in sub-optimal solutions. To address this issue, this paper proposes a novel integrated micro-macro approach for better incorporating train energy-efficient speed control into the railway timetabling process. Firstly, we formulated the integrated train timetabling and speed control optimization problem as a nonlinear mixed-integer programming model. Due to its complexity, we reformulate it on the basis of flow conservation theory in a space–time-speed (STS) network and solve the problem in two steps. In the first step, a set of pre-solved energyefficient train trajectory templates is generated by a segment-level optimization approach with consideration of train travel time, entry speed and exit speed to save computation time. In the second step, a near-optimum train energy-efficient timetable solution is found by a fast algorithm, which consists of the shortest generalized cost path algorithm, conflict detection and resolution algorithm, and calculation of dynamic headways between two successive trains. The numerical experiments demonstrate that the developed approach provides better outcomes than the benchmark case in terms of both train journey time and energy consumption.

1. Introduction The energy efficiency of transportation systems is critical to economic prosperity and sustainability of our society, particularly under gravitated climate change and increasing air pollutions (e.g. severe smog issues in China (Martínez Fernández et al., 2019)). High-speed railway (HSR) transportation, as one of the fastest expanding transportation modes in recent years, offers fast, safe and comfortable mobility services. Yet as compared with conventional trains, HSR is made possible only by bearing excessive energy consumption to overcome extra aerodynamic resistance at high speed. As a result, railway companies and researchers have investigated measures to decrease energy consumption to improve the

⁎

Corresponding author. E-mail address: [email protected] (X. Li).

https://doi.org/10.1016/j.trc.2020.01.008 Received 27 March 2019; Received in revised form 29 November 2019; Accepted 10 January 2020 0968-090X/ © 2020 Elsevier Ltd. All rights reserved.

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system’s sustainability and profitability, including developing energy-efficient rolling stock, deploying measures concerning auxiliary energy consumption, energy-efficient train control (EETC) (i.e., speed control in hereinafter) at the microscopic level, and energyefficient train timetabling (EETT) at the macroscopic level. Since EETC and EETT belong to operation planning optimization that can directly reduce train energy consumption without additional investment, they have been intensively studied in the literature (Scheepmaker et al., 2017). An energy-efficient train control can directly reduce the operation cost as well as providing a number of external benefits such as decreasing CO2 emissions and noise hindrance (Scheepmaker and Goverde, 2015). On the other hand, as trains with limited capacities receive increasingly growing demands, especially in Asian countries (Xu et al., 2015), their journey time (determined by optimally scheduling the arrival and departure times of the investigated trains and their safety headways during travel, i.e., train timetable) reliability also faces unprecedented challenges, while the impact of train speed control on energy savings highly depends on the timetable. These issues highlight the need for a trade-off between journey time reliability and energy efficiency. To address these issues, this paper aims to incorporate microscopic speed control into macroscopic train timetabling decisions for energy savings. 1.1. Literature review 1.1.1. Review of energy-efficient speed control We first examine existing studies on speed control at the microscopic level. Many research efforts aim at finding the optimal speed control strategies for a single train between two adjacent stops within a given time (Scheepmaker et al., 2017). Howlett et al. (1994) showed the coasting is the most important driving phase due to the short stop distances and pointed out that a potential energyefficient train trajectory for a train traveling on level track consists of maximum acceleration, cruising, coasting, and maximum braking in Howlett (2000). Higgins et al. (1996) proposed various optimal train speed control strategies for a train on a track with constant grades, followed by a study on optimal driving strategies on a track with a continuously varying gradient and a given travel time (Howlett and Cheng, 1997). Aradi et al. (2013) used a predictive optimization model for a train driver advisory system to generate the energy-efficient driving strategies with consideration of train traction and braking force characteristics, track profiles, and speed limits. Chevrier et al. (2013) investigated the calculation of alternative running times between two stations with varying gradients and speed limits. They proposed a bi-objective optimization model that balances the trade-off between minimizing running time and energy consumption with a heuristic Evolutionary Algorithm (EA). Su et al. (2013) considered the energy-efficient speed control problem on level tracks with the simple assumptions that the maximum traction, maximum braking and resistance forces are all constants and can be formulated analytically for all regimes based on the Pontryagin’s Maximum Principle (PMP). Su et al. (2014) extended the previous model with maximum traction, maximum braking and resistance forces as functions of speed. He et al. (2018) divided train route into discrete subsections and searched the speed at each end of each subsection by the genetic algorithm, then determined the corresponding speed profiles in each subsection based on geographical conditions and high-speed energy-efficient driving strategies. During train operation, perturbation is inevitable and it may cause delays to trains, thus computationally efficient algorithm for recalculating optimal energy-efficient driving strategies for trains in real-time is desired. To address this issue, Sicre et al. (2014) proposed a genetic algorithm with fuzzy parameters based on the accurate simulation of a train motion to calculate a new efficient manual driving strategy on high-speed trains in real-time when significant delays arise. Fernández-Rodríguez et al. (2018) proposed a new real-time algorithm for manual driving model for high-speed trains with aims to balance the running time, energy consumption, and the risk of delay in arrival, where the uncertainty in manual driving was modeled with means of fuzzy numbers. Montrone et al. (2018) formulated a mixed-integer linear model to find a real-time driving regime combination for a train to minimize energy consumption, where the possible driving regime combinations were obtained by splitting train routes into subsections and obtaining the regimes on these subsections by the PMP. Additionally, for the two-train energy-efficient train control problem, Albrecht et al. (2015) designed a heuristic procedure to find the optimal driving strategy that minimizes the total energy consumption and allows the trains from opposite directions to finish on time while adhering to the separation safety constraints. 1.1.2. Review of energy-efficient train timetabling Energy-efficient train timetabling (EETT) is the problem of finding a timetable for one or more trains on a railway corridor or network that maximizes the energy efficiency of train operations at the macroscopic tactical level. A handful of studies made plausible attempts in integrating speed profile optimization into the train timetabling decision process (Sicre et al., 2010; Cucala et al., 2012; Canca and Zarzo, 2017; Song et al., 2014; Wang and Goverde, 2019, 2016; Yang et al., 2015a,b; Zhou et al., 2017). One stream of energy-efficient timetabling is to find the optimal distribution of running time supplements that are beneficial for energy-efficient driving. On one hand, for a single train along its trip, Sicre et al. (2010) considered optimizing the running time distribution for a high-speed train service along its trip and proposed a method based on simulation and optimization model to minimize the energy consumption. In this work, a Pareto curve between running time and energy consumption per trip between two stops was firstly found by simulation. Then the available running time along its whole trip is determined by an optimization model. Their work was further extended to for high-speed trains in Cucala et al. (2012), which models uncertain delays as fuzzy numbers and punctuality constraints. Ding et al. (2011) proposed a two-level iterative optimization model to determine the energy-efficient driving strategy as well as the optimal distribution of the available running time supplement for a metro train. Scheepmaker and Goverde (2015) considered the energy-efficient speed control with varying gradients and speed limits and derived the PMP optimality conditions as well as developed a two-stage iterative algorithm that calculates the optimal cruising speed using Fibonacci search and the optimal coasting point for the given cruising speed using the bisection method to distributing running time supplements along a train trip. Wang and Goverde (2016) formulated the train trajectory optimization on along its trip as a multiple-phase optimal control 89

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model and solved by a pseudospectral method with consideration of general infrastructure features (varying gradients and speed limits), operational constraints and signal constraints. Then, they developed a multi-train trajectory optimization method for the energy-efficient timetabling problem by adjusting the running time allocation of given timetables in Wang and Goverde (2019). On the other hand, for the more trains on a railway corridor or network, Albrecht and Oettich (2002) used a simulation model to compute the energy consumption for each possible travel time of a train, and then calculated the optimal timetable with dynamic programming by taking passengers’ connection in mass rapid transit systems into account. Ghoseiri et al. (2004) developed a multiobjective optimization model for EETT, in which a Pareto frontier between energy consumption and passenger travel time was firstly found by the ε-constraint method and based on the obtained Pareto frontier detailed multi-objective optimization was performed. In their work, the fuel consumption cost was calculated by a function of average velocity between adjacent stations, and this approach for calculating the energy consumption also can be found in Li et al. (2013) and Yang et al. (2015a,b) for EETT in the railway system. Similarly, Canca and Zarzo (2017) adopted an empirical description of train energy consumption as a function of running times for EETT in rapid railway transit networks. Yang et al. (2012) considered energy consumption and travel time as the optimization objectives based on the coasting control method. Goverde et al. (2016) proposed a three-level framework for performance-based railway timetabling on a corridor, in which the energy-efficient speed profiles are optimized by determining the departure and arrival times at the intermediate stations within a given bandwidth. Zhou et al. (2017) formulated a model for joint optimization of highspeed train timetables and speed profiles with consideration of both tight power supply and temporal capacity constraints. More information about EETT in the railway system can be found in Martínez Fernández et al., (2019); Scheepmaker et al., (2017). Another stream is to synchronize timetable so that the regenerative braking energy can be used by nearby accelerating trains, especially in the metro system where has frequent braking and traction operations (Li and Lo, 2014a, 2014b; Gupta et al., 2016; Huang et al., 2017; Liu et al., 2018; Wang et al., 2018). For EETT in railway networks, Song et al. (2014) established traction calculation and minimum energy consumption calculation models with consideration of the regenerative braking for automatic train operation (ATO) strategy in multi-Train operations, and the simulation results indicated that RBE gradually augments with trains’ couples running on the line and that the RBE remains close to 30% when the number of operational trains is saturated. For EETT in the metro system, Peña-Alcaraz et al. (2012) developed a mathematical programming model based on the power flow model to synchronize metro systems by increasing running times instead of increasing dwell times. This is to minimize the total energy consumption by optimally using the regenerative braking energy of the trains. Li and Lo (2014a) presented a joint optimization model for the timetable and speed profile for the metro system, which included arrival/departure times, driving switching times, and arrival/maximum speeds at each segment as decision variables and was solved by a genetic algorithm. They further proposed a dynamic train scheduling and speed control framework for the metro system that considers passenger demands and regenerative energy utilization in Li and Lo (2014b). Yang et al. (2013) first described the problem in terms of a mathematical programming model by using headway and dwell time control, with at maximizing time overlaps of nearby accelerating and braking trains. In addition to energy savings by using regenerative braking, the authors also took into account passenger waiting time in Yang et al. (2014). Furthermore, they considered all trains in the same track interval of electricity supply and extended the time horizon to the whole day in Yang et al. (2015a,b) to synchronize the arrivals and departures of trains such that as much as possible regenerated energy can be used. Wang et al. (2018) developed a bi-objective timetable optimization model for metro EETT to reduce both passenger time and carbon emission of train operation taking regenerative braking energy into account. A linearly weighted compromise algorithm and a heuristic algorithm were designed to find the optimal headways and dwell times of trains. Mo et al. (2019) proposed an integer programming model with respect to energy consumption to generate a robust operational strategy. This strategy determines the arrival/departure time and the speed profile choice on each segment considering the influence of regenerative energy, speed profiles, physical environments and uncertain levels of passenger demand on energy consumption. Thus, it can be seen that regenerative braking is the most efficient for high frequent (e.g., metro-like) systems or when using high-voltage electricity systems. For highspeed railway networks considered in our paper, the frequency of trains is relatively low and the trains are already constrained by many network constraints. Therefore, adding such synchronization to trains on opposite directions of a double-track line makes the (NP-complete) timetabling (and traffic management) problem even more complicated. Hence, the regenerative braking is not taken into account in this paper despite its significance in research. 1.2. Main focuses of this study The existing studies on EETT in the railway system can be classified into two categories in terms of fuel consumption quantification. The first category assumes that the energy consumption on a railroad segment is a univariate function of the segment travel time or equivalently the average speed (Albrecht and Oettich, 2002; Ghoseiri et al., 2004; Ding et al., 2011; Li et al., 2013; Yang et al., 2015a,b;Canca and Zarzo, 2017). However, even for fixed travel time, the energy consumption of a train traversing a segment may vary across different speed profiles. For instance, a steady speed profile with fewer variations along the segment likely consumes less energy than one with sharp accelerations and decelerations, even if both profiles take the same time to traverse the segment. Additionally, this method cannot give a specific speed profile for trains. The second category instead considers fuel consumption as a function of the detailed speed profile on a segment, where the integration of instantaneous fuel consumption is determined by instantaneous speed, acceleration, etc. And the corresponding speed control profiles may be implemented directly (Sicre et al., 2010; Cucala et al., 2012; Scheepmaker and Goverde, 2015; Wang and Goverde, 2019, 2016; Zhou et al., 2017). For example, Zhou et al. (2017) made plausible efforts to consider detailed speed dynamics at a set of discrete-time points along a trip. While this category can capture more detailed train motions and have a more accurate estimation of fuel consumption, these models are often complex and difficult to solve relatively large-scale instances due to excessive spatial and temporal state variables. There lacks a modeling 90

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approach that captures speed variations yet has a relatively simple model structure and efficient solution performance. Further, most existing studies on train timetabling problem made relatively simple and conservative assumptions on train safety headways (Xu et al., 2017; Zhou et al., 2017). These works assumed that the departure headway, arrival headway and tracking headway between two consecutive trains at the same segments were constant to guarantee train operation safety, e.g., 3 min or larger. That may not be appropriate. On the one hand, a large headway may result in a waste of infrastructure resources. On the other hand, too small headway may increase rear-end collision risks. Thus, a dynamic headway is desired for improving the utilization of resources, especially along busy HSR lines. To address these issues, we develop a novel approach for solving EETT that much simplifies train speed control by compressing complex trajectory optimization into a few parameters without much loss of optimality and allows for flexible dynamic headway adjustment based on train states. The main contribution of this work is highlighted as follows: (1) A segment-level train trajectory optimization is developed. Different from the previous works focusing on optimizing detailed train trajectories, the proposed method does not need to control trajectory variables at every small time interval. Instead, it views a train’s trajectory on a relatively long segment (a track between two stations) as the basic unit. Note that on a railway segment, once the train’s travel time, entry speed and exit speed are given, the optimal trajectory shape is determined independently of train controls at other parts of the rail network. With this property, the proposed method pre-solves the optimal trajectory profiles for all types of segments for a range of travel time, entry and exit speeds. According to the optimal train control theory (Howlett, 2000), a potential energy-efficient train trajectory for a train traveling on a level track consists of maximum acceleration, cruising, coasting, and maximum braking. However, as pointed in Scheepmaker et al. (2017), the most important driving regime is cruising, which becomes even more important when the average distance between two stations increases in high-speed, regional, inter-city, and freight train systems. Hence, we assume that the segment-level optimization train trajectory in this study consists of a sequence of uniform acceleration - cruising - uniform braking phases. We set the maximum acceleration (deceleration) rate in the acceleration (braking) phases to ensure passenger comfort and also set a minimum deceleration rate in the braking phases by taking the low deceleration characteristic in coasting phases into account. After pre-solving the optimal trajectory for all segments and storing the results in lookup tables, the size of the integrated micro-macro approach for high-speed EETT problem is much reduced; i.e., the time scale of the basic links is expended from seconds to several minutes and their spatial scale is enlarged from tens of feet to tens of kilometers. This yields a much more compact model formulation and a much more efficient solution approach. (2) In this paper, the dynamic headways between two successive trains are considered in the process of conflict detection and resolution. As pointed out in Zhou et al. (2017), handling dynamic headway is a very challenging modeling issue, which is determined by trains’ travel states (i.e., dynamic train velocity), train control systems, etc. In this paper, since the optimal trajectory profiles of trains are pre-solved at the individual segment level, the dynamic headways are calculated according to the referred trains’ speed control profiles on the basis of the selected segment-level train trajectory. (3) With the pre-solved segment-level train trajectory sets, we find a near-optimum solution through a fast algorithm in the train timetable optimization process while meeting the headway constraints and flow balance constraints. The pre-solving technique allows us to significantly save computation time. The main differences between the relevant previous studies and this paper are listed as follows: (1) Ghoseiri et al. (2004), Li et al. (2013), Yang et al. (2015a,b) and Canca and Zarzo (2017) assumed that the energy consumption on a railroad segment is a univariate function of the segment travel time or equivalently the average speed. They found the optimal distribution of running time supplements beneficial for train energy-efficient speed control. In contrast, this paper proposes a novel approach for better incorporating detailed train energy-efficient speed control considering acceleration and deceleration phases into the railway timetabling process. (2) Sicre et al. (2010), Cucala et al. (2012) and Wang and Goverde (2016) focused on EETT for a single train. Whereas this paper mainly focuses on EETT for a mass of trains on a railway corridor that further enriches the technology in the railway transportation system. (3) Zhou et al. (2017) and Wang and Goverde (2019) considered detailed speed dynamics at a set of discrete-time points or position points along a trip. With this, it was complex and difficult to solve relatively large-scale instances due to excessive spatial and temporal state variables. Instead, this paper pre-solves segment-level train trajectory sets and develops a fast algorithm to find a near-optimum solution in the train timetable optimization process while meeting the headway constraints and flow balance constraints. The remainder of the paper is organized as follows. The next section defines the problem statement of this paper. Section 3 gives more details about the model formulation, including segment-level trajectory optimization, calculation of minimum headways and the mathematical model for the whole problem. Section 4 describes the proposed a fast algorithm to find the near-optimum solution for this problem. The case study is presented in Section 5, followed by conclusions in Section 6. 2. Problem statement This section states the investigated problem and the associated math notation. For the convenience of readers, the main notation 91

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Table 1 Notation in Section 2. Symbol

Definition

Set symbols N M K Km Nk

Set Set Set Set Set

Index symbols n m

of of of of of

consecutive stations, N : ={1, 2, , N } train types trains, K : ={1, 2, , K } trains of typem travel segment of train k , Nk : ={ok , , dk }

k, k

Station or segment index Train type index Train indexes

Parameters N K Ln qm

The total number of stations The total number of trains Length of segmentn Type m train mass

ok , dk EDTk , LATk max vmn a , sb , sc skn kn kn

origin station and destination station of train k Earliest departure time at the origin station ok and latest arrival time at the destination station dk for traink The maximum speed of train type m on segment n Travel distances in the three phases for train k on segment n , respectively

a b c f kn , fkn , f kn

Average tractive forces in the three phases for train k on segment n , respectively

min max mn , mn min DTkn

The planned dwell time for train k at station n

M Pn Sk ,n dd , aa kk , n kk , n max , max k k min k

Minimum and maximum travel times of train type m on segment n , respectively Sufficiently large constant The preceding segment for a train k on segment n , n Nk {Ok } Stop pattern of train k at stationn , i.e., if train k is planned to stop at stationn , Sk ,n = 1; otherwise 0 Departure and arrival headways between trains k and k on segment n , respectively Maximum acceleration and deceleration rates of train k , respectively Minimum deceleration rate of traink

ctk, cek

Cost coefficient of per unit of time and energy consumption for train k , respectively

Decision variables Akn , Dkn ukn , vkn , z kn akn , bkn, ckn okk , n

Entry and exit times for train k on segment n, respectively Entry and exit speeds for train k on segment n , ukn, vkn Vk , respectively Cruising speed for train k on segment n Duration times in the three phases for train k on segment n , respectively Travel sequence of train k and train k on segmentn , i.e., if train k is before the traink , okk , n = 1; otherwise 0

of this section is summarized in Table 1. We investigate an HSR corridor that is comprised of a set of discrete stations and segments connecting consecutive stations. We consider only level segments without grade in this study. We denote the set of stations as N : ={1, 2, , N } where N is the total number of stations and the index increases along the traffic direction. Apparently, there are N 1 segments, and we refer the segment connecting stations n and n + 1 as segment n , n N N . Let Ln denote the length of segment n , n N N . A set of k trains, denoted as K : ={1, 2, , K } , are running along this corridor. The trains are classified into a set of different types (e.g., based on their masses, speed limits, etc.) denoted by M . Let qm denote the mass of a typical type m train and Km denote the index set of type m trains, m M . Trains can load and unload passengers and overtake one another only at stations, while they follow each other with certain headway separations over each segment. We assume that all station capacities are sufficient and no delay occurs at track assignment within a station. Assume for each train k K , its origin station ok , destination station dk , earliest departure time EDTk (from ok ) and latest arrival time LATk (to dk ) are given. Let Nk denote the set of segments covered by train k . On each segment max n Nk , train k has a speed limit denoted by vmn . Similar to the traditional TTP at the macroscopic level, we intend to find the optimal space-time trajectory for train k , i.e., entry time Akn , and exit time Dkn at each visited segment n Nk , with an aim to minimize the total travel time under the safety headway constraints. Further, we intend to incorporate the speed control problem into the train timetable design process to minimize energy consumption at the microscopic level. Namely, we simultaneously seek a speed control profile of a train (see Fig. 1, which is an illustration of train speed profiles with uniform acceleration - cruising - uniform braking phases) that is compatible with the train timetable and also minimizes the energy consumption. We assume that a segment-level train trajectory in this study consists of a sequence of uniform acceleration (or braking) - cruising - uniform braking (or acceleration) phases. The specific move sequence for train k on segment n is essentially determined by the train’s entry speed ukn , exit speed vkn and travel time kn . Then for the three phases of train k on segment n , we denote their duration times as a c a c b b akn , bkn , ckn , respectively, their travel distances as skn , skn , skn , respectively, and their average tractive forces as f kn , fkn , f kn , respectively. The speed control profile of train k on segment n includes entry speed ukn , acceleration timeakn , cruising timebkn , deceleration 92

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Fig. 1. An illustration of train speed profiles with uniform acceleration - cruising - uniform braking phases.

time ckn , cruising speed z kn and exit speed vkn , which together with segment entry and exit times Akn , Dkn and sequence indicator okk , n between trains k and k compose the set of decision variables in the investigated problem:

x: ={Akn , Dkn , okk , n , ukn , akn , bkn , ckn, z kn , vkn | k

k

K, n

Nk }

Constraints At the macroscopic level, a train’s travel on a segment should be confined within a certain reasonable range due to schedule feasibility and energy efficiency: min mn

Dkn

max mn ,

Akn

k

Km, m

M, n

(1)

Nk,

where and are the maximum and minimum allowed travel times for type m train on segment n . min The dwell time for train k at station n should be no less than the minimum dwell time DTkn for boarding and alighting passengers: max mn

min mn

Akn

min DTkn ,

DkPn

k

K, n

(2)

Nk {ok }, Sk, n = 1.

As these trains share the same track infrastructure along the HSR corridor, every two successive trains should strictly comply with certain minimum headway constraints to guarantee safe operations as follows:

D kn

Dkn

dd kk , n

M·(1

okk , n ),

k, k

K, n

Nk

Nk ,

Akn

Akn

aa kk , n

M ·(1

okk , n),

k, k

K, n

Nk

Nk ,

Dkn

D kn

dd kk , n

M· okk , n ,

k, k

K, n

Nk

Nk ,

Akn

Akn

aa kk , n

M ·okk , n ,

k, k

K, n

Nk

Nk ,

(3)

where binary variable okk , n denotes whether train k is ahead of train k on segment n , M is a sufficiently large number that can switch on or off these constraints with the proper binary value of okk , n , and dd , aa are departure and arrival headways between trains k kk , n kk , n

and k on segment n , respectively. At the microscopic level, due to the speed limits, each train k ’s entry, cruising and exit speeds on segment n should satisfy

0

ukn, vkn, z kn

max vmn ,

k

Km, m

M, n

(4)

Nk,

max where vmn is the maximum allowed speed for type m train on segment n . Note that the average tractive force of a train in any phase is essentially determined by the speed variations and the resistance of this train. For example, the tractive force is normally put to zero in a decelerating phase, which lets a train slow down purely by the resistance force. Further, according to Newton's second law, the train tractive force accelerates this train in addition to overcoming the resistance in the acceleration phase, while it exactly cancels the resistance to maintain cruising speed z kn in the cruising phase. With this, the tractive force for train k on segment n can be calculated by the following equations. In the first phase, when ukn z kn , the train should first decelerate to reduce its speed from ukn to a cruising speed z kn , and thus the a tractive force f kn is normally put to zero, i.e., a f kn = 0, if ukn

z kn,

k

Km, m

M, n

(5)

Nk.

Since we set maximum and minimum deceleration rates in the braking phase to ensure the comfort of passengers and the compatibility of braking characteristics in the coasting phases, so that the range of duration time in the deceleration phase is ukn

zkn ukn z kn max , min k k

. That means the corresponding duration time in this situation is 93

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akn

ukn

z kn max , k

akn

ukn

if

z kn min , k

ukn

z kn ,

k

Km, m

M, n

Nk. (6)

On the other side, when ukn < z kn , the train should first accelerate to increase its speed from ukn to a cruising speed z kn , which means the train tractive force accelerates this train in addition to overcoming the resistance in this situation. For simplification, we use the average speed to calculate the basic resistance even though the instance speed is also compatible in our approach (see Appendix A). Then we have a f kn = qm · R 0 + R1·

ukn + z kn u + z kn + R2· kn 2 2

2

+ qm ·

z kn

ukn akn

,

ifukn < z kn,

k

Km, m

M, n

Nk .

(7)

where R 0 , R1, R2 are constant coefficient. Similarly, the corresponding duration time in this situation is

z kn ukn

akn

max k

,

ifukn < z kn ,

k

Km, m

M, n

Nk .

(8)

In the second phase, referred as the cruising phase, the train tractive force overcomes the basic resistance to maintain cruising b speed z kn , and the tractive force fkn can be calculated by b fkn = qm ·(R0 + R1· z kn + R2 ·z kn 2),

k

Km, m

M, n

(9)

Nk .

In the last phase, when z kn vkn , the train should decelerate from speed z kn to vkn to through (or stop at) station n + 1. The c in this situation can be calculated by tractive force f kn c f kn = 0, ifz kn

vkn,

k

Km, m

M, n

(10)

Nk.

With consideration of the maximum and minimum deceleration rates of the train, the corresponding duration time in the deceleration phase is

ckn

zkn

vkn max , k

ckn

zkn

if

vkn min , k

z kn

vkn,

k

Km, m

M, n

Nk. (11)

On the other side, when z kn < vkn , the train should finally accelerate to increase its speed from z kn to exit speed vkn to pass station c n + 1. The tractive force f kn in this situation can be calculated by c f kn = qm · R 0 + R1·

vkn + z kn v + z kn + R2 · kn 2 2

2

+ qm ·

z kn

vkn ckn

,

ifz kn < vkn,

k

Km, m

M, n

Nk .

(12)

And the corresponding duration time in the last phase is

ckn

vkn

z kn max k

,

ifukn < z kn ,

k

Km, m

M, n

Nk .

(13)

While the speed profile should also satisfy the travel time constraints

akn + bkn + ckn = Dkn

Akn ,

k

K, n

(14)

Nk

Further, a train’s speed profile on a segment should be consistent with the corresponding segment length: a b c skn + skn + skn = Ln , a skn :=

ukn + zkn · akn , 2

b skn : =z kn· bkn, c skn :=

k

k

vkn + z kn ·ckn, 2

k

K, n K, n

K, n k

Nk , Nk ,

Nk,

K, n

Nk .

(15)

Finally, the decision variables should be nonnegative:

x

(16)

0.

Objectives At the macroscopic level, the total train travel time can be formulated as:

ctk (Dkdk

ft (x): =

Akok )

(17)

k K

and at the microscopic level, the total energy consumption can be formulated as: 94

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Table 2 Notation in Section 3. Symbol

Definition

Set symbols T

Set of time ticks in the system, T : ={0, dT , 2dT ,

exit Vmn (u , )

Index symbols u, v t, s

Speed indices, u, v V and u, v often refer to the entry and exit speeds of a train on a segment, respectively Time indices, t , s T and often t < s

Tmn Vmn

Parameters T V

of of of of

, I T dT } , where I T : =round (T dT )

Set Set Set Set

V

speed ticks in the system, V : ={0, dV , 2dV , , I V dV } , where I V : =round (V dV ) all possible travel times for type m train on segment n speed levels for type m train on segment n , which is a subset of V all possible exit speeds that a type m train can reach given entry speed of u Vmn and travel time

Tmn on segmentn

Operation time horizon Highest speed limit across all segments Time discretization interval

dT

Speed discretization interval

dV

A possible travel time for the train through a segment Train travel state with entry speed u , exit speed v and travel time Energy consumption for a type m train on segment n with entry speed u The average speed of train k on segment n

[u, v, ] emn (u, v, ) mean vkn kn

h Tkkn ^ ([uv ]1 , [uv ]2 ) kkn ([uv ]1 ,

[uv ]2 )

Decision variables x kn (t , s, u, v ) ykn (t , s )

Vmn , exit speed v

Vmn and travel time

Tmn

A possible travel time for train k on segment n The entry time deviation between the two consecutive trains on the same segment The required minimum headway between the two consecutive train k and train k

Set of incompatible arcs between train k with travel state [u1, v1, 1: s1 t1], referred it as [uv ]1, and train k with travel state [u2, v 2, 2: s2 t2], referred it as [uv ]2 . kkn ([uv ]1 , [uv ]2 ) = {(u2, v2, 2, h) h < Tkkn ([uv ]1 , [uv ]2 )}

x kn (t , s, u, v ) = 1if STS traveling arc (n , n + 1, t , s, u, v ) is selected in the path of train k , 0 otherwise ykn (t , s ) = 1 if STS stopping arc (n , n , t , s, 0, 0) is selected for train k dwelling at station n , 0 otherwise

b

fe (x): = k K n Nk

a a b c c cek f kn ·skn + fkn ·skn + f kn · skn ,

(18)

where ctk and cek are the corresponding cost coefficients of travel time and fuel consumption for train k , respectively. Based on the above analysis, the considered problem can be formulated as a nonlinear multi-objective programming model as follow

minft (x) + fe (x) x

s . t . constraints (1)

(19)

(16),

Incorporating the speed control into the TTP as done in the above formulation is difficult to solve, especially when the dynamic minimum headway requirement between two successive trains is considered to improve the operation efficiency. Thus, instead of directly investigating the above formulation, we propose a more compact model in the next section that much simplifies the model structure and much expedites solution efficiency without compromising much optimality. 3. Simplified model formulation To solve the above-stated complex problem, Section 3.1 first constructs a discrete space–time-speed (STS) network representation of this problem, followed by the simplified model formulation for the integrated of train scheduling and speed control problem. Then Section 3.2 formulates a train segment-level trajectory optimization model for energy efficiency considering train entry and exit speeds and travel time at a segment. A method of enforcing the dynamic minimum headway requirement is explored on the basis of the selected trajectory of the considered trains in Section 3.3. The additional notations introduced in this section mainly due to the STS structure are summarized in Table 2. 3.1. Space-time-speed network construction In the STS network, the time horizon starts at time 0 and ends at time T . This time horizon is discretized into a sequence of time points with a small interval dT , and thus the timestamps are denoted as t = 0, dT , 2dT , , I T dT , where I T = round (T dT ) . We assume a train’s speed range is bounded by 0 from below and the highest speed limit V from above. The speed range is also discretized with interval dV , and the considered speed levels are v = 0, dV , 2dV , , I V dV , where I V = round (V dV ) . Furthermore, the space index is denoted by the corresponding station index, i.e., n = 1, 2, , N . Hence, given the set of physical stations N , the set of time intervals T and the set of velocity intervals V , we define the discrete STS network as G = (Q, A) where 95

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Q: ={(n, t , v ) n N, t T, v the following two types of arcs.

V } denotes the set of nodes and A denotes the set of arcs connecting nodes in Q . Set A contains

(a) Traveling arc ((n, t , u), (n + 1, s, v )) A refers to an arc that starts from node (n, t , u) Q , meaning departing from station n at time t with entry speed u , and ends at node (n + 1, s, v ) Q , meaning arriving at station n + 1 at time s with exit speed v . The travel time on this arc is (s t ) ; (b) Stopping arc ((n, t , 0), (n, s, 0)) A refers to stopping at station n from time t to time s . The dwelling time duration (s t ) should be no less than the required dwell time. With the STS network, whether train k goes through traveling arc ((n, t , u), (n + 1, s, v )) A on segment n can be denoted by a binary variable xkn (t , s, u , v ) . If it is, we have xkn (t , s, u , v ) = 1; otherwise, xkn (t , s, u , v )=0. Additionally, variables Akn , Dkn , ukn , vkn in the above-stated formulation can be represented by the binary variables {xkn (t , s, u, v )} . Besides, to ensure flow balance in the STS network, let ykn (t , s ) denote whether train k selects stopping arc ((n, t , 0), (n, s, 0)) A at station n . If it is, we have ykn (t , s ) = 1; otherwise, ykn (t , s ) = 0 . With these representations, the above-stated complex problem can be accordingly reformulated as follows. Objective function Instead of multi-objective (17)–(18) in the original formulation, a total generalized cost of all the trains is adopted as the objective function, for the integrated train scheduling and speed control problem.

Z = min

m M

k Km

+ ctk· ykn (t , s )·(s

n Nk

t
exit v Vmn (u , s t )

u Vmn

xkn (t , s, u , v )(cek ·emn (u , v, s

t ) + ctk ·(s

t ))

t)

(20)

The objective function (20) includes the total generalized cost that is the weighted summation of the travel times and the energy consumption rates of all trains. For a stopping arc, the corresponding energy consumption is assumed to be zero and the generalized cost only consists of travel time cost, i.e., ctk·(s t ) . For traveling arc of a given train type m on segment n , the journey time and energy costs are ctk·(s t ) and ce, k ·emn (u , v, s t ) respectively. Since the train energy consumption formulation is nonlinear as stated in the original formulation, it is difficult to solve the train energy consumption directly. To tackle this challenge, we decompose the original problem and propose a segment-level trajectory optimization approach in Section 3.2, where constraints (1), (4), (5)–(15) in the original formulation are taken into account. Namely, when a type m train k travels on segment n with entry speed u , exit speed v and travel time (s t ) , the optimal train energy-saving trajectory, represented by akn , bkn , ckn, z kn , and corresponding energy consumption emn (u, v , s t ) can be get by the proposed approach in Section 3.2. Flow balance constraints in STS network In order to depict a feasible train path in the STS network, a set of flow balance constraints are formulated in this subsection. Before given these constraints, we denote the sets of the available exit times for a type m train on segment n with entry time t by smn (t ) , i.e., smn (t ) = {s T s t Tmn} , where Tmn refer to the set of possible travel times for type m train on segment n . For each individual train k, starting from its origin at the vertex (ok , EDTk , 0) and ending at its last sink vertex (dk , LATk , 0) , we consider the following flow balance constraints.

xkn (t , s, 0, v ) = 1,

k

Km, m

M , n = ok .

(21)

exit (0, s t ) t EDT k s smn (t ) v Vmn

) is the set of all possible exit speeds that a type m train can reach given entry speed of u Vmn and travel time where Tmn on segment n , and Vmn refer to the set of speed levels for a type m train on segment n , respectively. Constraints (21) indicate that one traveling arc must be selected at the first segmentok since train k starts from the origin station. Constraints (22) ensures that train k passes station n without stopping at this station. exit Vmn (u ,

xkPn (s , t , v , u) = s < t T v VmPn

xkn (t , s, u, v ) = 1,

u, t

T, k

Km, m

M, n

Nk

s > t T v Vmn

{ok }, Sk, n = 0.

(22)

Constraints (23) and (24) guarantee that train k stops at the intermediate station n as planned. Constraints (23) ensures the upstream flow balance when train k arrives at station n and constraints (24) guarantees the downstream flow balance when train k departs from stationn , respectively.

xkPn (s , t , v , 0) = s < t T v VmPn

ykn (s, t ) = s
ykn (t , s ) = 1,

t

T, k

Km, m

M, n

Nk

{ok}, Sk, n = 1.

s>t T

xkn (t , s, 0, v ) = 1,

t

T, k

Km, m

M, n

s > t T v Vmn

The flow balance at the last segment Dk is guaranteed by Constraints (25). 96

Nk {ok }, Sk, n = 1.

(23) (24)

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xkDk (t , s, u, 0) = 1,

k

Km, m

M , n = dk

(25)

t LAT k s smn (t ) u Vmn

The flow balance constraints (21)–(25) ensure each train can start from its origin station and arrive at its destination station in the STS network. A minimum generalized cost path algorithm for trains in the STS network will be presented in Section 4.2 based on the flow balance constraints as well as constraints (2). Constraints (26), corresponding to constraints (3) in the original formulation, ensures the safe headway between each pair of consecutive trains.

xkn (t1, s1, u1, v1) + x kn (t1 + h, t1 + h + 2, u2 , v2)

1,

n

{Nk Nk }, k

Km, [uv ]2

kkn ([uv

]1 , [uv ]2 ),

(26)

where kkn ([uv ]1 , [uv ]2 ) is the set of incompatible arcs between train k with travel state [u1, v1, 1: s1 t1], referred as [uv ]1, and train k with travel state [u2, v2, 2: s2 t2], referred as [uv ]2 ;i.e., kkn ([uv ]1 , [uv ]2 ) = {(u2, v2, 2, h) h < Tkkn ([uv ]1 , [uv ]2 )} , where h is the entry time deviation between the two consecutive trains on this segment and Tkkn ([uv ]1 , [uv ]2 ) is the required minimum headway between two consecutive trains k and k . Fig. 2 illustrates the STS network for only one train. In Fig. 2, a train departs from station n at time t with speed u and arrives at station n + 1 at time s with speed v without any stop. That means the train goes through traveling arc ((n, t , u), (n + 1, s, v )) on segment n . While traveling arc ((n + 1, s, v ), (n + 2, s + mmin on segment n + 1 and stopping arc (n + 1) + 2, 0)) min ((n + 2, s + mmin (n + 1) + 2, 0), (n + 2, s + m (n + 1) + 5, 0) ) are chosen by this train. Note that the infeasible STS vertexes for this train beyond the time and speed limits can be easily eliminated, which reduces the size of the STS network. For instance, since the referred train is planned to dwell at stationn + 2 , the vertexes with zero speed are available (illustrated as the green circles in Fig. 2) but not the vertexes with nonzero speed (illustrated as the gray circles in Fig. 2). Since it’s not easy to calculate the energy consumption emn (u, v , s t ) for a type m train on segment n with entry speed u , exit speed v and travel time (s t ), we will develop a segment-level trajectory optimization approach in Section 3.2. Further, a method of enforcing the dynamic minimum headway requirement is explored on the basis of the selected trajectory of the considered trains in Section 3.3 to deal with nonlinear constraints (26). 3.2. Segment-level trajectory optimization approach After the STS network is constructed, note that for each arc ((n, t , u), (n + 1, s, v )) , the entry speed, the exit speed and the travel time are fixed as u , v and s t . Then define the state of arc ((n, t , u), (n + 1, s, v )) as 3-tuple [u, v, : =s t ]. Note that the optimal trajectory for a train k on this segment is actually determined by its state [u, v, ] and its train types. Given travel state [ukn , vkn, kn] for mean = Ln kn . For operational convenience, in this study, we neglect train k traversing segment n , the average velocity is obtained as vkn gradients, curve resistances, tunnels, and speed limits, and assume that train travel with uniform acceleration and deceleration rates1. Thus all possible cases for train k traveling on segment n can be classified into five cases with different phase patterns as follow. mean mean mean mean , vkn < vkn , vkn vkn or ukn < vkn , then train k first accelerates to increase its speed from ukn to cruising Case 1. If ukn vkn speed z kn , then it cruises at a constant speed z kn for a certain time, and finally, it decelerates to vkn (as illustrated in Fig. 3(a)). mean mean mean mean , vkn vkn , vkn > vkn or ukn vkn , then train k slows down from velocity ukn to cruising speed z kn , then Case 2. If ukn > vkn it cruises at a constant speed z kn for a certain time, and finally, it accelerates to vkn (as illustrated in Fig. 3(b)). mean mean Case 3. If ukn < vkn and vkn > vkn , then the speed profile has three possible options: (a) train k accelerates from ukn to cruising speed z kn , then maintains this speed for certain time, and finally accelerates again to exit speed vkn (see Case 3-1 in Fig. 3(c)); (b) train k accelerates from speed ukn to the exit speed vkn , then maintains this velocity all the way, i.e., z kn = vkn (see Case 3–2 in Fig. 3(c)); or (c) train k cruises at a velocity ukn for a certain time, i.e., z kn = ukn , then accelerates to exit speed vkn right upon exiting the segment (see Case 3-3 in Fig. 3(c)). mean mean and vkn < vkn , then the speed profile has three possible options, just opposite to Case 3: (a) train k Case 4. If ukn > vkn decelerates from speed ukn to cruising speed z kn , then maintains this speed for a certain time, and finally decelerates again to exit speed vkn (see Case 4-1 in Fig. 3(d)); (b) train k decelerates from speed ukn to exit speed vkn , then maintains this velocity until exiting this segment, i.e., z kn = vkn (see Case 4-2 in Fig. 3(d)); or (c) train k cruises at a velocity ukn for a certain time, then decelerates to exit speed vkn right upon exiting this segment, i.e., z kn = ukn , (see Case 4-3 in Fig. 3(d)). mean mean mean = ukn = vkn during the whole trip on Case 5. If ukn = vkn and vkn = vkn , then train k simply cruises with velocity z kn = vkn this segment (i.e., Case 5 in Fig. 3(e)).

All cases in Fig. 3 shall satisfy the speed variation range (6), travel time constraints (14) and distance constraints (15). However, due to their phase pattern differences, different cases are subject to different subsets of the other constraints on train force and duration time:

• Case 1 is subject to constraints (7)–(11); 1 For more complex networks where this assumption needs to be relaxed, the general methodology remains the same except that the segment-level trajectory optimization needs to consider site dependent constraints of feasible trajectories. Thus the proposed model framework remains applicable.

97

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Fig. 2. A train-based STS network.

Fig. 3. The shapes of the speed profile with different entry and exit speeds.

• Case 2 is subject to constraints (5)–(6), (9), (12)–(13); • Case 3 has three possible speed control profiles: • Case 3-1 is subject to constraints (7)–(9) and (12)–(13) • Case 3-2 is subject to constraints (7)–(11) • Case 3-3 is subject to constraints (5)–(6), (9) and (12)–(13); • Case 4 is subject to constraints (5)–(6), (9)–(11); • Case 5 is subject to constraints (9). Note that once travel state [u, v, ] is given, then it is easy to identify the case that fits the travel state. If it is Case 1, 2 or 5, the associated constraints specified above together actually will determine a unique trajectory satisfying given travel state [u, v, ] although there is a mass of trajectories satisfying the travel state constraint. For instance, the low acceleration rate at high cruising speed and the high acceleration rate at a low cruising speed (represented by t1 and t2 in Fig. 3 respectively) may both satisfy the given travel state [u, v, ] in Case 1, but we select the optimal one as our segment-level train trajectory since the energy consumption is one of our optimization objectives. While, if it is Case 3 or 4, the constraints are variational in each subcase. Since Case 3 has different subcases, the trajectory in the subcase with the minimum energy consumption will be selected as the optimal trajectory for this case. The same selection applies to Case 4 as well. It can be fed into the corresponding case to solve a unique trajectory satisfying 98

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Fig. 4. An example of two conflict-free train trajectories.

the travel state and case specific constraints on a segment. Through the proposed segment-level trajectory optimization approach, the time scale of the basic links can be expended from seconds to several minutes and their spatial scale is enlarged from tens of feet to tens of kilometers for the considered problem. For computational-efficiency purposes, we tabulate a mapping relationship between available travel states on each segment and their corresponding minimum energy consumption by CP optimizer in CPLEX in advance. 3.3. Dynamic minimum headway requirement The minimum headway requirement between two consecutive trains (e.g., departure, arrival, and meet headways) is the time separation that prevents trains from having track conflicts with each other, see the constraints (3) in the original formulation in Section 2. Most existing train scheduling optimization models treat these minimum headway requirements as fixed given values (e.g., 2 or 3 min for departure/arrival headways) regardless of the referred trains’ trajectories. Since the headways between a pair of consecutive trains highly depend on these two train trajectories, while the optimal trajectory for a given travel state can be obtained through the above proposed segment-level trajectory optimization approach in Section 3.2, dynamic minimum headway requirement could be considered in this paper, that are formulated as nonlinear constraints (26). However, it’s difficult to deal with nonlinear constraints (26). Thus, we present a method in Algorithm 1 to obtain the dynamic minimum headway requirement between two consecutive trains. Fig. 4 illustrates an example of the minimum headway requirement between two trains, where uvtpreceding = [84, 84, 12] and uvtfollowing = [0, 0, 16]. As shown in Fig. 4(a), when the headway time set istH = 1 min , the two train trajectories will conflict at t = 732 with headway distance 1998 m since the safety space headway (which is used to guarantee safety operations between two successive trains in the moving block signaling system) on this segment is set as 2 km . Thus, when the headway time increases to 2min , the minimum headway distance is 5467 m at t = 862 (see Fig. 4(b)), which satisfies the safety space headway requirement. Thus, the minimum headway requirement between the two trains is set as 2 min . Algorithm 1 Calculation of the minimum headway requirement. Step 1. Find the trains’ speed control profiles according to their travel states and calculate their trajectories based on the speed control profiles. Set the headway tH = 1 min . Step 2. Compare the train trajectories, go to Step 3. Step 3. If the two trajectories meet the safety space headway on the whole segment, then set the minimum headway as tH and quit; otherwise, set tH : =tH + 1 and go to Step 2.

4. Solution algorithm Although many solution algorithms have been proposed on both linear and integer train timetabling optimization problems, most existing methods rely on commercial optimization solvers that could take a significant amount of computation time and memory space to solve a real-world problem. This fact is even more severe for the proposed STS network due to the multidimensional feature of this network that makes the feasible region extremely large. Hence, to overcome this issue, this section presents a fast algorithm to efficiently solve the proposed train timetabling optimization model that generates a conflict-free timetable for a set of trains on the HSR corridor. The additional notations introduced in this section mainly due to the proposed algorithm are summarized in Table 3. 99

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Table 3 Relevant subscripts and parameters for conflict detection and resolution algorithms. Symbol

Definition

ah , fo a _ah, f _fo An, ah , Dn, ah

Preceding and following train indices of two consecutive trains on a segment, respectively Train a _ah is before train ah and train f _fo is after train fo indices, respectively Entry and exit times for train ah on segment n after rescheduling, respectively Fixed entry and exit times for train ah on segment n to respect safety constraints that cannot be changed at current, respectively

fix Anfix , ah , Dn, ah

Ani ,ah , Dni , ah

etotal, total ETn, k, LTn, k FTn,ah l Qn

tHni,(ah, fo) tHn(ah,fo) Tint un,ah , vn, ah k ,(n, t , u) err i n, ah n, ah max min n, ah , n, ah

Initial entry and exit times for train ah on segment n , respectively Total energy consumption and journey time for all trains, respectively Earliest and latest departure vertex at station n for train k , respectively Available travel time set for train ah on segment n with current entry and exit speed Train serial number on a segment Real-time departure sequence of trains on segmentn Initial minimum departure headway requirement between the trains ah and fo from stationn Minimum departure headway requirement between the trains ah and fo from station n after rescheduling Departure time intervals at the origin station Entry and exit speeds for train ah on segment n after rescheduling, respectively Label cost of the vertex (n , t , u) for traink Time deviation between the minimum required departure headway and actual departure headway Initial travel time for train ah on segment n Travel time for train ah on segment n after rescheduling Maximum and minimum available travel times for train ah on segment n , respectively

4.1. Solution frameworks To guarantee safe train operations, it is essential to consider the headway requirement constraints (26) in the train timetabling problems. However, these constraints are nonlinear and very complex. Therefore, we first formulate a new relaxed problem (P2) that consists of all the constraints in (P1) except constraints (26). Without consideration of the headway requirement constraints (26) and trains are assumed travel in free-travel situations, then we find that the unique shortest path with the minimum generalized cost for each train from original station to the destination station specifies the ideal train timetable, which serves as the lower bound to the problem (P1). Since train safe headway requirement constraints (26) are relaxed, there might be some conflicts between trains in this ideal train timetable. Thus, we develop a fast algorithm to detect and resolve these train conflicts. Let Qn be the real-time sequence of trains on segment n , which is dynamic in the process of conflict resolution. Qn plays an important role in the process of conflict detection and resolution since we detect the conflict from the first to the last train on each segment. First of all, we set the train sequence according to their entry times. When more than one train has the same entry time, the train sequence is determined by their exit times on the segment. Basically, the trains with the sooner exit times are put at the front list of the sequence. Moreover, if their departure and exit times are exactly identical, we keep the same sequence as it was on the previous segment n 1. This conflict detection and resolution framework are shown in Algorithm 2 and the solution framework is shown in Fig. 5. Algorithm 2 Conflict detection and resolution framework. Step 1. Initialization. Set the sequence of trains on each segment according to their preferred entry time. Step 2. Ideal timetable. Use Algorithm 3 to find the path with the minimum generalized cost for each train. Step 3. Set n: =1, l: =1. Step 4. Determine train queue Qn . Step 5. Conflict detection. Detect train conflict between the train Qn (l) and its following train Qn (l + 1) on segment n . If a conflict is found, go to Step 6; otherwise, go to Step 7. Step 6. Conflict resolution. Set train Qn (l) as the preceding train ah and train Qn (l + 1) as following train fo , respectively. If the train ah and fo violate the headway constraint, employ Algorithm 4 to eliminate the headway conflict and then go to Step 7; otherwise, if overtaken completed at the current station, employ Algorithm 5 to eliminate the overtaking conflict, else if overtaken completed at the front station, employ Algorithm 6 to eliminate the overtaking conflict, and then go to Step 7. Step 7. Set l: =l + 1. Ifl < K 1, go to Step 5; otherwise go to Step 8. Step 8. Set n: =n + 1. If n < N 1 , set l: =1 and go to Step 4; otherwise, go to Step 9. Step 9. Check the feasibility from the first segment to the last segment. If no conflict found, terminate and output results; otherwise, jump to the segment n where conflict found, and go to Step 4.

4.2. The shortest generalized cost path algorithm A special feature of the STS network is that several available traveling arcs may exist since there are various running times and exit speeds for each train at the same entry vertex. However, the presented Algorithm 3 in this sub-section can handle this feature by finding the optimal traveling arcs. Let k,(n, t , u) denote the label cost of the vertex (n, t , u) for train k , and ETn, k and LTn, k denote the 100

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Fig. 5. Solution framework.

earliest and latest departure vertex at station n for train k , respectively. Algorithm 3 The algorithm for the path with the minimum generalized cost. Input: The STS network G = (Q, A) , the desired earliest departure and last sink vertices for each train k K . Output: The path with the minimum generalized cost in the STS network for each traink K . Step 1. Initialization. Set the label cost of each vertex (n , t , u) to infinity, i.e., k ,(n, t, u) : = , k K , the label cost of start vertex of each train to 0, and its predecessor vertex to ( 1, 1, 1) ; Step 2. Label correcting process for each train For each train k K do For each segment n N do Find the first and last vertex with the label cost less than For t

[ETn,k , LTn, k ] do

at station n , and save the value on the time dimension, denoted by ETn, k and LTn,k , respectively.

max ] do For u [0, vmn If k ,(n, t, u) < Then

If the train is planned to dwell at the front station Then For each available travel state of train k at segment n do Revise the label cost of the sink vertex and record its preceding vertex if the cost of the sink vertex reduces considering the stop arcs. Else the train proceeds without stopping at the front station Then For each available travel state of train k at segment n do Revise the label cost of the sink vertex and record its preceding vertex if the label cost of the sink vertex reduces. End End // for each available vertex End // for each entry speed End // for each time End // for each segment End // for each train Step 3. Return STS train trajectories that correspond to the minimum generalized cost For each train k K do Find the super sink vertex with the shortest label cost and backtrace to generate space-time-speed trajectories. Save the entry time Akn , exit time Dkn , entry speed ukn , and exit speed vkn for each train k on segment n . End // for each train

Fig. 6 illustrates three paths for traink from station 1 to –station 3. After the label correcting process, each vertex yields a generalized cost, and vertex (3, 33, 0) obtains the minimum generalized cost of 10,296 with ctk = 200 and cek = 1. With this vertex, the total travel time is total = 33 min and the total energy consumption is etotal = 3696.4 kw·h . Following Step 3 in Algorithm 3, one can easily reverse the backward path and find the path with the minimum generalized cost as sequence (33, 3, 0) → (16, 2, 16) → (0, 1, 0) (see the red solid line in Fig. 6). By Algorithm 3, we are also able to find several paths for this train from the origin to the destination station by changing the ctk and cek values. For instance, the segment-level trajectory optimization problem becomes a minimum time path problem when 101

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Fig. 6. A simple example of three paths for traink from station 1–3.

ctk = 1, cek = 0 , and it turns into an energy consumption minimization problem when ctk = 0, cek = 1. While the total travel time is total = 28 min and the total energy consumption is etotal = 5350.9 kw·h in the minimum time path, and total = 40 min and etotal = 2724 kw·h in the minimum energy consumption path. The corresponding train paths are shown as black and blue dash lines in Fig. 6, respectively. Comparing these three paths, we can see that the integrated train timetabling and speed control optimization model can balance the trade-off between passengers’ travel cost (i.e., trip time) and train operation cost (i.e., energy saving). 4.3. Conflict detection and resolution algorithms Detection and resolution of conflicts between train services is a critical issue in our train timetabling approach. After finding the path with the minimum generalized cost for each train, we perform the conflict detection algorithm, and once a conflict is found, we run the conflict resolution algorithm. The conflict detection algorithm verifies the feasibility of the macroscopic timetable by checking the absence of track conflicts between two consecutive trains. That is if the real departure headway between two considered trains is smaller than the dynamic minimum headway requirement between them, we define it as a conflict. Since we consider one-way train timetabling problem on a double-track railway corridor, two types of conflicts are considered, (1) headway conflict, where two consecutive trains do not meet the minimum headway constraints (see Fig. 7(a)–(c)); (2) overtaking conflict that is defined as the situation where the following train is about to catch up with its preceding train on a segment (see Fig. 7(d)). In general, train conflicts can be resolved by shifting the train departure times from stations or stretching the running times on segment (Bešinović et al., 2016). In particular, there are two ways to resolve an overtaking conflict, (a) let the train overtake at the current station, or (b) let the train overtake at the front station. Some primary resolution principles are listed as follow: (i) In headway conflict resolution, the preceding train has a higher priority and thus the following train with lower-priority should be rescheduled to meet the minimum headway requirement. (ii) In overtaking conflict resolution, the following train has a higher priority and thus the preceding train with lower-priority should be rescheduled to resolve the conflict. (iii) In all strategies, we first consider rescheduling the arrival and exit times of trains while keeping their entry and exit speeds to

Fig. 7. Illustration of train conflict types, (a)–(c) headway conflict; (d) overtaking conflict. 102

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Fig. 8. Illustration for headway conflict resolution.

ensure passenger comfort and train operational continuity. We do not reschedule their speeds unless it is necessary. (iv) Shifting the entry time of trains may cause other conflicts at the previous segment. Thus, stretching train running time is preferred over any changes to the entry time. 4.3.1. Headway conflict resolution strategies Fig. 8 illustrates the strategies for headway conflict resolution, where the two consecutive trains claim to use the same segment n . In this situation, the following train with lower-priority should be rescheduled. The detail of the headway conflict resolution algorithm is described in Algorithm 4. In above Fig. 8, b = max (A (A a)) means finding a maximum element but no greater than real number a in set A; b = min (A (A a)) means finding a minimum element but no smaller than real number a in set A, the same meaning in the following context. Algorithm 4 Headway conflict resolution. Step 1. Calculate the initial minimum departure headways of the two trains tHni,(ah, fo) according to their travel state, time deviation err . If their current departure headway is no less than 1 min and the travel time of train fo can be stretched, go to Step 2; otherwise, go to Step 3. Step 2. HR1: Increase the travel time for train fo gradually but not exceeds the maximum available travel time (see Fig. 8(b)). Once the two considered trains meet the minimum departure headways requirement, go to Step 5. If the travel time of train fo reaches to maximum travel time but without conflict resolution, go to Step 3. Step 3. Adjusting the departure time of train fo from station n . Set the entry time An, fo = Ani , fo + err . If train fo is planned to stop at station n , go Step 3.1; otherwise, go to Step 3.2. Step 3.1. HR2-1: Holding the initial travel time for train fo on segment n , i.e., n 1, fo

= max FTn

1, fo

FTn

1, fo

i n 1, fo

+

err

i n, fo ,

(see Fig. 8(c)) go to Step 4.

103

and set the travel time of train fo on segment n

1 as

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Fig. 9. Strategies for train overtaking completed at the current station. Step 3.2. HR2-2: If

n 1, fo

= An, fo

Ani

1, fo

is available for train fo traveling on segment n

1 , we set

n 1, fo

for train fo traveling on segment n

1 , go to Step 4;

otherwise, train fo is rescheduled to stop at station n , i.e., vfo,n 1 = ufo, n = 0 , and do (1) firstly, set the entry time for train fo on segment n as An, fo = Ani , fo + err ; then, set travel time for train fo traveling on segment n -1 as

An, fo Ani 1,fo )), nmax1, fo and the exit time on segment n 1 is Dn 1, fo = Ani 1, fo + n 1, fo ; n 1, fo = min min (FTn 1, fo (FTn 1, fo (2) select a feasible travel time for train fo on segment n as n, fo = min min (FTn, fo (FTn,fo Dni , fo Dn 1, fo )), nmax , fo (see Fig. 8(d)); (3) update entry/exit time of train fo on segment n according to the new minimum departure headways requirement tHn(ah,fo) and then go to Step 4. Step 4. Check the feasibility on segment n 1 . If a conflict occurs on segment n-1, then jump to segment n 1 to execute the conflict detection and resolution algorithm; otherwise, go to Step 5. Step 5. Update the entry /exit time at the rest of the segment of this train according to the time variation.

4.3.2. Overtaking completed at the current station strategies Fig. 9 illustrates the strategies for train overtaken at the current station, where train fo is about to catch up with its ahead train ah on segment n . In this situation, the ahead train with lower-priority should be rescheduled. The detailed algorithm for train overtaken completed at the current station is described in Algorithm 5. Algorithm 5 Overtaken complete at the current station Step 1. If n > 1 and train ah is not planned to stop at station n , go to Step 2; otherwise, go Step 7. Step 2. Train ah is rescheduled to stop at station n (i.e., vn 1,ah = un, ah = 0 ), and set n 1,ah = min (FTn

1, ah (FTn 1, ah

Dni

1, ah

Ani

1, ah ))

(see Fig. 9(b)), go to

Step 3. Step 3. Check the feasibility between train ah and train fo on segment n 1 (see Fig. 9(b)). If a conflict occurs, then execute Algorithm 4 and go to Step 4; otherwise, go to Step 7. Step 4. If entry time of train fo on segment n 1 is not changed, i.e., An 1, fo = Ani 1,fo , go to Step 5; otherwise, jump to segment n 2 to execute Algorithm 4.

Step 5. Check the feasibility between train fo and train f _fo on segment n 1 (see Fig. 9(b)). If a conflict occurs, then execute Algorithm 4 and go to Step 6; otherwise, go to Step 7. Step 6. If entry time of train f _fo on segment n 1 is not changed, i.e., An 1, f _fo = Ani 1,f _fo , go to Step 7; otherwise, jump to segment n 2 to execute Algorithm 4.

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Fig. 10. Illustration for train overtaking completed at the front station. Algorithm 5 Overtaken complete at the current station Step 7. Update entry time An, fo = Dn

1, fo

+ DTnmin , fo and set travel time

n, fo

= min (FTn, fo (FTn, fo

Dni , fo

An,fo )) for train fo on segment n (see Fig. 9(c)). If no

conflict occurs between the train a _ah and train fo on segment n , go to Step 9; otherwise, execute Algorithm 4 and go to Step 8. Step 8. If entry time of train fo on segment n is not changed, go to Step 9; otherwise, jump to segment n 1 to execute Algorithm 4. Step 9. Set entry time An, ah =max (Dn

1, ah

+ DTnmin , ah, An, fo + 1, and then set travel time

n, ah

= min (min (FTn, ah (FTn,ah

segment n (see Fig. 9(c)), go to Step 10. Step 10. Update the entry /exit time at the rest of the segment of these two trains according to the time variation.

Dni ,ah

An, ah )),

max n, ah )

for train ah on

4.3.3. Overtaking completed at the front station strategies In our work, we define that overtaken is completed at the front station when the ahead train is planned to dwell at the front station and it arrives at the front station no later than 2 min comparing to its following train. Fig. 10 illustrates these strategies for overtaken completed at the front station, where train fo is about to catch up with its ahead train ah on segment n . In this situation, the following train has higher priority and the ahead train should be rescheduled primarily. The detailed algorithm for train overtaken completed at the front station is described in Algorithm 6. Algorithm 6 Overtaken complete at the front station i Step 1. Set exit time Dn, ah = max (Dnfix , ah , Dn, a _ah + 1, Dn, fo

1) and travel time as

n, ah

= min (FTn, ah (FTn, ah

Dn,ah

Ani ,a _ah )) for train ah on segment n , go to

Step 2. Step 2. Check the feasibility between train ah and train fo on segment n (see Fig. 10(b)). If a conflict occurs, execute Algorithm 4 to resolve headway conflict, go to Step 3; otherwise, go to Step 4. Step 3. If entry time of train fo on segment n is not changed, i.e., An, fo = Ani , fo , go to Step 4; otherwise, jump to segment n 1 to execute Algorithm 4. Step 4. Set entry time An + 1, ah = max (Dn,ah + DTnmin + 1, ah, An + 1, fo + 1) and set the travel time for train ah on segment n + 1, and set the travel time

n + 1, fo

n + 1, ah

= min (min (FTn + 1, fo (FTn + 1, fo

= min (min (FTn + 1, ah (FTn+ 1, ah

Dni + 1, fo

An + 1, fo )),

Fig. 10(c)), go to Step 5. Step 5. Update the entry /exit time at the rest of the segment of these two trains according to the time variation.

105

max n + 1, fo )

Dni + 1,ah

An + 1, ah )),

max n + 1, ah)

for train fo on segment n + 1 (see

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Table 4 Mapping relationships among link travel states, control profiles, and energy consumption. u (m s )

v (m s )

0 0 0 0 0 40 40 40 84 84 84 84

0 0 0 0 0 68 68 68 0 0 0 84

(min ) 12 16 18 19 20 18 19 20 18 19 20 12

v mean (m s )

v c (m s )

83 63 56 53 50 56 53 50 56 53 50 83

– 80 75 60 60 44 53 50 60 54 51 83

a

(s )

– 200 188 150 150 12 120 81 120 148 156 5

b

(s )

– 540 520 860 800 26 972 1074 752 820 896 240

(s )

e (kw·h )

– 220 372 130 250 1042 48 45 208 140 148 475

– 2442.83 2023.25 1696.11 1603.68 1650.42 1547.48 1455.82 1158.36 1009.50 923.06 2595.48

c

5. Case studies This section aims to demonstrate the effectiveness and efficiency of our proposed approach by conducting numerical examples on a hypothetical small-scale network and a large-scale real-world railway corridor in China, i.e., Beijing-Shanghai HSR. All experiments are performed on a personal computer with an Intel(R) Core(TM) i7 with 2.70 GHz CPU and 8.00 GB memory, and the algorithms are coded in MATLAB on the Windows 10 OS. 5.1. Data preparation 5.1.1. Stop scheme In practice, the train stop plans are often pre-designed based on the predicted passengers’ demands and the required operational efficiency levels. There are two main methods presented for the train stop plan as follow, (1) depend on the stop times required at each station Yang et al. (2016) and (2) depend on the origin-destination (OD) of the passengers Yue et al., (2016). Since the train stop plan is not the primary focus in our study, for simplicity we use the first method to produce the train stop patterns. 5.1.2. Mapping energy consumption As analyzed in Section 3.2, the mapping relationship between available travel states on each segment and their corresponding minimum energy consumption e is desired to be tabulated in advance for computational-efficiency purposes. Table 4 shows several examples of the mapping relationship among the link travel state[u, v, ] and its speed control profiles a, b, c and v c , and corresponding minimum energy consumption e. In this table, v mean and v c denote the average and cruising speeds for train traveling on the segment, respectively, a, b, and c denote the duration times in the three phases of train travel process on the segment, respectively, and symbol ‘-’ means that no solution is found. In these examples, the length of the segment is set as 60 km , and the possible travel times range from 12 to 20 min corresponding to the average velocity between 50 and 83.33 m s (i.e., 180 and 300 km h ). From Table 4, it can be easily seen that shorter travel time would result in more energy consumption, which is as we expected. For instance, if u = 40 m s and v = 68 m s , the minimum energy consumption is 1455.82 kw·h when = 20 min , but that increases to 1650.42 kw·h when = 18 min . Further, no solution is found for travel state [0, 0, 12] because 12 min is too short for a train to traverse the whole segment since more time is needed to accelerate from speed 0 and decelerate to speed 0.

Fig. 11. Stop patterns for 10 trains on the railway corridor. 106

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Table 5 Stop times corresponding passenger demands at each station. Station

1

2

3

4

5

6

7

8

9

10

Stop times demand

10

2

4

2

5

2

5

3

4

10

5.2. Small-scale case study In this small-scale example, we consider a one-way double-track railway corridor with 10 stations (see Fig. 11(a)). Stations 1 and 10 are the origin and destination stations for all trains, respectively. For simplicity, all sections between two consecutive stations are set to 60 km and all trains are assumed to be identical with the maximum possible speed of 84 m s (i.e., 302.4 km h ). Note that more sets of trains can be easily added. Moreover, if a train is planned to stop at an intermediate station, the required minimum dwell time is set as 2 min . Further, we assume that the stop times at each station are given as shown in Table 5, and the corresponding stop patterns are shown in Fig. 11(b), where the solid dot “●” represents that the train is planned to stop, and the hollow dot “○” indicates that the train does not have to stop at this station. 5.2.1. Influence of time and energy consumption cost coefficients In this section, we focus on investigating the influence of time and energy consumption cost coefficients on train timetable performance. We assume that the time increment in the STS network is always 1 min to be consistent with the time unit in real-world passenger train timetables. Further, the space dimension is discretized according to the locations of the stations along the railway corridor. On the speed dimension, the available entry or exit speeds are assumed to vary from 0 to 84 m s with the increment of dV = 4 m s , and thus we have 22 speed levels in this STS network. Moreover, the departure intervals are set as Tint = 10 min in the following cases. Additionally, the value of time and the value of energy consumption are assumed identical across all trains, thus the coefficients of time and energy consumption ctk, cek , k K in the objective can be replaced by ct and ce , respectively, in the following experiments. Fig. 12 shows the timetables for all trains with the minimum energy consumption (i.e., ct = 0, ce = 1) and with the minimum journey time (i.e., ct = 1, ce = 0 ). In Fig. 12(b), the dash lines represent the feasible arcs for train movements along with their trips after the conflict resolution. In the train timetable shown in Fig. 12(a), the total energy consumption of 10 trains is etotal = 1.20 × 105 kw·h and the total travel time of 10 trains is total = 1854 min , while etotal = 2.32 × 105 kw·h and total = 1287 min in Fig. 12(b). The latter timetable saves 567 min of the train journey time but the energy consumption is almost twice comparing to the former timetable, which makes it an undesired choice for the railway operators. On the other side, the average velocity of 10 trains is 194 km h in the former timetable, which is relatively too slow due to the passengers’ expectations of the HSR corridor (note that the operational velocity on Beijing – Shanghai HSR corridor varies between 200 and 274 km h ). Therefore, to balance the tradeoff between energy saving and journey time, it is very important to find reasonable values for time and energy consumption cost coefficients. Fig. 13 shows the variations of total travel time and energy consumption with different values of time cost coefficient. As it is shown in this figure, the total travel time decreases with the time cost coefficient increasing, but the total energy consumption significantly increases. In Yang et al. (2012), a trade-off timetable between the two conflicting objectives of journey time and energy consumption was determined by weight factors. Similarly, we also apply reasonable time and energy cost coefficients to balance the trade-off between the total travel time and energy consumption. From Fig. 13, we can see that the trade-off can be generated when ct = 200 and ce = 1 in this small-scale example. In this condition, the energy consumption is 1.65 × 105 kw·h , which indicates that the energy consumption reduces by 37.5% compared with that in the minimum time consumption timetable and increases by 28.9% compared with that in the minimum energy consumption timetable. While, in this trade-off timetable, the total journey time is

Fig. 12. Timetables for 10 trains on the small-scale railway corridor, where (a) is the train timetable with minimum energy consumption and (b) is the train timetable with minimum total travel time. 107

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Fig. 13. The time and energy consumption with different time cost coefficients (ce = 1).

1519 min, which reduces by 18.07% compared with that in the minimum energy consumption timetable and increases by 18.03% compared with the minimum time consumption timetable. Further, the average velocity of 10 trains is 215 km h , a reasonable value for the HSR train. Moreover, it is worth mentioning that the operators can also make a decision on the time and energy cost coefficients depending on their preference based on Fig. 13. 5.2.2. Influence of speed increment This subsection focuses on investigating the influence of speed increment on train timetable performance since the speed dimension could be discretized into different levels with parameter dV . For instance, if dV = 2 m s , the number of speed levels is 43 (i.e., {0, 2, 4, , 84} ). In the following cases, the departure interval at the origin station is set as Tint = 10 min and the coefficient of time cost ranges from 100 to 300. Fig. 14 illustrates the total travel time, total energy consumption, total generalized cost and train average velocity of all trains at different speed levels.

Fig. 14. Total travel time, total energy consumption, generalized cost of all trains and train average velocity at different speed levels. 108

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Table 6 Distances and travel times of each interstation section. Section

Distance (km)

Minimum travel time (min)

Maximum travel time (min)

Beijing South- Langfang Langfang-Tianjin South Tianjin South-Cangzhou West Cangzhou West-Dezhou East Dezhou East-Jinan West Jinan West-Taian Taian-Qufu East Qufu East-Tengzhou East Tengzhou East-Zaozhuang Zaozhuang-Xuzhou East Xuzhou East-Suzhou East Suzhou East-Bengbu South Bengbu South-Dingyuan Dingyuan-Chuzhou Chuzhou S-Nanjing South Nanjing South-Zhenjiang South Zhenjiang South-Danyang North Danyang North-Changzhou North Changzhou North-Wuxi East Wuxi East-Suzhou North Suzhou North-Kunshan South Kunshan South- Shanghai Hongqiao

59 72 88 108 92 43 71 56 36 63 79 77 53 62 59 69 25 32 57 26 32 43

12 15 18 22 19 9 15 12 8 13 16 16 11 13 12 14 5 7 12 6 7 9

19 24 29 36 30 14 23 18 12 21 26 25 17 20 19 23 9 11 19 9 11 14

As shown in Fig. 14, with the same time cost coefficient, there is little difference between total travel time, total energy consumption and train average speed with different speed increments, while the total generalized cost is almost the same with different speed increments. Additionally, these differences with different speed increments reduce with the increase of the time cost coefficient. For instance, when ct = 100 and ce = 1, the total travel time is total = 1810 min , total energy consumption is etotal = 1.24 × 105 kw·h and the average speed isv¯ = 198.9 km h with the speed increment of dV = 2 m s . While total = 1800 min , etotal = 1.25 × 105 kw·h , and v¯ = 200.00 km h with dV = 4 m s , and total = 1797 min , etotal = 1.26 × 105 kw·h and v¯ = 200.33 km h with dV = 6 m s . When ct = 200 and ce = 1, the total travel time total = 1519 min , total energy consumption etotal = 1.65 × 105 kw·h and average speed v¯ = 236.99 km h with the speed increment 2 m s and 4 m s . While total = 1506 min , etotal = 1.68 × 105 kw·h and v¯ = 239.04 km h with the speed increment of 6 m s . However, the number of vertices rapidly increases with the number of speed levels. As a result, the search region becomes larger and the solution time becomes longer. For instance, the average solution times for dV = 2 m s and dV = 6 m s are 30 and 1.4 s, respectively. Furthermore, there are more optional travel states due to the speed increment is too small, which directly result in more time to produce the mapping energy consumption. For instance, for a segment with 60 km , the number of travel states are 43 × 43 × 9 = 16, 641 (where the travel times range from 12 to 20 min) and 15 × 15 × 9 = 2, 205 with dV = 2 m s and dV = 6 m s , respectively. Therefore, to strike a good balance between the solution time and solution accuracy, we set the speed increment as 6 m s in the following large-scale experiments. 5.3. Large-scale experiments on Beijing-Shanghai HSR corridor This subsection applies the proposed train timetabling framework to the Beijing-Shanghai HSR corridor in China. With a total length of 1312 km , this HSR corridor consists of 23 stations and 22 sections. The distances and travel times between stations are listed in Table 6, where the maximum speed of trains is 300 km h . The stop times corresponding to the passenger demands at each station are listed in Table 7. Further, Fig. 15 shows the stop pattern for each train generated by the first method in Section 5.1.1. 5.3.1. Influence of the time cost and energy cost coefficients In this subsection, we investigate the influence of the time and energy consumption cost coefficients on the train timetable performance. In these instances, the departure interval is set as 15 min . Fig. 16 shows the effect of the time cost coefficient on the Table 7 Stop times corresponding passenger demands at each station. Station

Beijing South

Langfang

Tianjin South

Cangzhou West

Dezhou East

Jinan West

Taian

Qufu East

Stop times Station Stop times Station Stop times

40 Tengzhou East 4 Zhenjiang South 10

2 Zaozhuang 5 Danyang North 5

4 Xuzhou East 7 Changzhou North 8

4 Suzhou East 8 Wuxi East 10

5 Bengbu South 10 Suzhou North 10

20 Dingyuan 10 Kunshan South 8

5 Chuzhou S 10 Shanghai Hongqiao 40

5 Nanjing South 40

109

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Fig. 15. Stop patterns for 40 trains on the Beijing-Shanghai HSR line.

Fig. 16. Average travel time, average velocity and energy consumption with different time cost coefficients (ce = 1).

average travel time, the average velocity and the total energy consumption. From Fig. 16, we can see that the average velocity of trains and the energy consumption increases with the value of time cost coefficient, but the average travel time reduces with this parameter. In this experiment, a trade-off timetable can be obtained when ct = 180 and ce = 1. In this condition, the average velocity of 40 trains is 220 km h (which is a reasonable value for HSR trains as the operational velocity on the Beijing – Shanghai HSR corridor varies between 200 and 274 km h ). While, the total energy consumption is 1574.15 × 103 kw·h , the average travel time of 40 trains is 5.91 h. Therefore, we set the coefficients to ct = 180 and ce = 1 to balance the trade-off between the train journey time and the energy consumption in the following experiments. Fig. 17 shows the train timetable for 40 trains on the Beijing-Shanghai HSR corridor as well as two examples for bunching trains. As shown in Fig. 17(a), train 31 overtakes train 30 at Tengzhou E station and then these two trains get very close to each other. Moreover, trains 34 and 35 also get very close to each other from Danyang N to Chuzhou S stations. Note that as it is shown in 110

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Fig. 17. (a) is train timetable of 40 trains on the Beijing-Shanghai HSR corridor, and (b) are two examples for bunching trains (ct = 180, ce = 1).

Fig. 17(b), these two pairs of trains keep certain headway that satisfies the safety operation requirements. 5.3.2. Influence of departure interval This subsection investigates the influence of the train departure interval on the performance of the train timetable. The departure interval varies from 5 to 30 min , and the corresponding results are shown in Table 8. In this table, Z is the total generalized costs of all trains, t , e and Err are the travel time, the energy consumption and the generalized cost deviations from the ideal train timetable, and that are etotal = 1574.15 × 103 kw·h and total = 14203 min in the ideal train timetable respectively. As shown in Table 8, train travel time deviations t decreases with the departure intervals. For instance, the travel time deviation is 89 min when Tint = 5min , while that is only 12 min when Tint = 12min . That is because the smaller departure intervals means the higher train density on the railway line, and there may be more conflicts between trains that needed to be eliminated. On the contrary, the energy consumption can reduce with the train travel time increasing. For instance, the energy consumption can reduce 2067.13 kw· h and 1172.31 kw· h from the ideal timetable when Tint = 5min and Tint = 12min respectively. That is because it needs more energy to pull the train to complete the journey in a shorter time. Further, there is no conflict occurs when the departure interval is greater than 15 min . However, a greater departure interval results in less transportation capacity, thus train operators 111

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Table 8 The results for different departure intervals. Tint (min ) 5 8 10 12 15 20 25 30

total (min )

14,292 14,229 14,221 14,215 14,206 14,203 14,203 14,203

etotal (103 kw·h )

Z

1572.08 1573.17 1572.10 1572.98 1573.83 1574.15 1574.15 1574.15

4144642.86 4134394.84 4131878.01 4131677.68 4130914.61 4130689.99 4130689.99 4130689.99

t (min )

e (kw·h )

89 26 18 12 3 0 0 0

−2067.13 −975.15 −2051.98 −1172.31 −315.38 0.00 0.00 0.00

v mean (km h )

Err (%)

218.64 219.61 219.73 219.82 219.96 220.01 220.01 220.01

0.34 0.09 0.03 0.02 0.01 0.00 0.00 0.00

benefit from a relatively low departure interval. Therefore, to determine an acceptable departure interval for train operations, one should simultaneously take passenger demands, operation efficiency, and other major factors into account. 5.3.3. Influence of minimum headway requirement This subsection investigates the influence of the minimum headway requirement on the performance of train timetable. As the most existing train scheduling optimization models done, we set the minimum headway requirements as tH = 3min to demonstrate the advantages of the dynamic headway requirement. Table 9 shows the comparison results with fixed headway requirements. From Table 9, we can see that train travel time deviations with fixed minimum headway requirements are significantly greater than that with dynamic minimum headway requirement under the same departure time intervals. For instance, the train travel time deviations is 272 min when tH = 3min and that is only 89 min when dynamic minimum headway requirement is considered. It can be seen that the dynamic minimum headway requirement considered in our approach is beneficial to shorten train travel time. Since the train running time is getting longer and the corresponding energy consumption is reduced. For instance, when Tint = 5min , the energy consumption deviations from the ideal train timetable are −9552.10 kw· h and −2067.13 kw· h with fixed minimum headway requirements and dynamic minimum headway requirement respectively. Overall, the generalized cost deviations from the ideal train timetable Err with fixed minimum headway requirements are greater than that with dynamic minimum headway requirements. 5.3.4. Influence of train departure order In this subsection, we explore the effect of the train departure order on the performance of the train timetable. In these experiments, the departure interval is set as 15min , and the train departure orders are generated randomly, excluding scenario 1, which is the train departure order considered in above subsections, i.e., trains departing from trains 1 to 40 sequentially. Table 10 shows the train timetable results with different train departure orders. As shown in Table 10, the delay is only 3 min in scenario 1, but increases to 31 min in scenarios 5 and 9. This is because the trains with fewer stops on the Beijing-Shanghai HSR corridor depart first and the following trains are not able to easily catch up in scenario 1, thus the interactions between the trains in this scenario are less than that in other scenarios. Despite this, the average velocity and the total energy consumption are almost the same among different scenarios, because they are mainly influenced by the coefficients of time and energy costs. Thus, with given cost coefficients and characters of passenger demand, we can find an acceptable departure order that results in lower delays. In general, as the travel path for each train in the ideal timetable are optimal on the basis of pre-solved the optimal train trajectory sets, and the feasible timetable is getting by a greedy-heuristic-based algorithm (i.e., conflict detection and resolution algorithms in Section 4.3) on the basis of the ideal timetable, we can get a good solution for the high-speed railway energy-efficient timetabling optimization problem. As the comparison results shown in Table 8 and Table 10, the gap between the generalized cost in the feasible timetable and that in the ideal train timetable is 0–0.34%, that is small enough to be acceptable, so we summary that we can find the near-optimal timetable. 5.3.5. Influence of the number of trains This subsection investigates the efficiency of the proposed train timetabling framework. In the following experiments, the departure interval is set as 15min and the stop patterns for additional trains are assumed to repeating those of trains 1 to 40 in a cyclic Table 9 The comparison results with fixed headway requirements. Tint (min )

t (min ) 5 8 10 12 15

Dynamic minimum headway requirement

Fixed minimum headway requirements (tH = 3 min )

272 114 52 32 14

e (kw·h ) −9552.10 −5806.64 −2386.54 −1124.85 −823.585

Err (%) 0.95 0.36 0.17 0.11 0.04

112

t (min ) 89 26 18 12 3

e (kw·h ) −2067.13 −975.15 −2051.98 −1172.31 −315.38

Err (%) 0.01 0.10 0.11 0.06 0.10

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Table 10 The results for different departure orders. Scenario 1 2 3 4 5 6 7 8 9 10

total (min)

14,206 14,223 14,217 14,223 14,234 14,216 14,225 14,233 14,234 14,229

etotal (103 kw·h )

Z

1573.83 1574.57 1576.28 1573.08 1572.76 1573.68 1574.60 1572.92 1575.09 1574.68

4130914.61 4134711.46 4135344.26 4133217.81 4134878.79 4132557.17 4135096.54 4134856.29 4137212.79 4135901.16

t (min ) 3 20 14 20 31 13 22 30 31 26

e (kw·h ) −315.38 421.48 2134.27 −1072.17 −1391.19 −472.82 446.55 −1233.70 942.80 531.17

v mean (km/h)

Err (%)

219.96 219.70 219.79 219.70 219.53 219.81 219.67 219.55 219.53 219.61

0.01 0.10 0.11 0.06 0.10 0.05 0.11 0.10 0.16 0.13

manner. For instance, if there are 60 trains, the stop patterns of the first 40 trains are set as T1 to T40 and that of the last 20 trains is set as T1 to T20. Fig. 18 shows the solution times for different numbers of trains. The results show that the solution time almost linearly increases with the number of trains. However, as train timetable scheduling is not real-time operation management, the linear solution time increase scale is acceptable in producing an energy-efficient timetable. 6. Conclusion As the impact of train speed control on energy savings highly depends on the timetable, in this paper, we propose a novel approach to incorporate microscopic speed control into macroscopic EETT decisions for energy savings. We firstly formulate the integrated train timetabling and speed control optimization problem as a nonlinear mixed-integer programming model, with two conflicting objectives of energy consumption and journey time. Due to its complexity, we re-formulate it on the basis of flow conservation in a space–time-speed (STS) network considering their weighted sum as the objective function. To solve the problem efficiently, the train speed profiles are pre-solved by a segment-level trajectory optimization approach. This approach considers train travel time, entry speed and exit speed through a segment. Near-optimum train energy-efficient timetable solutions are found by a fast algorithm, which consists of the shortest generalized cost path algorithm, conflict detection and resolution algorithm, and dynamic minimum headway calculation. The results of the numerical experiments show that the pre-solving technique allows us to significantly save computation time and the fast algorithm can efficiently produce a good train timetable where the relative difference with the ideal timetable is acceptable. The results indicate that the total journey time and energy consumption may have a certain degree of fluctuation with different values of time and energy cost coefficients, thus the train operators can make the decision on the value of the two cost coefficients depending on their preference. The results also demonstrate that the dynamic minimum time headway requirement is beneficial to shorten train travel time compared with the fixed time headway requirement. Moreover, the train departure time intervals and the departure order influence the performance of the train timetable. Therefore, to determine these operation parameters, the passenger demand, the operation cost, and other important factors should simultaneously be taken into account. We consider two main directions for future research. First, the unit increment in time dimension can be shortened to improve the utilization of infrastructure, e.g., a time increment of 5 s as in Goverde et al. (2016), where the block theory is applied. Second, in this paper, we ignore a number of factors that can influence the train trajectory results including train traction and braking

Fig. 18. The solution time with the different numbers of trains. 113

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characteristics, line resistance with varying gradient profiles, and varying speed limits. It is interesting to consider these factors in constructing more realistic train trajectories. Additionally, since the trains are fed by means of AC electrical system, where allows the energy to flow from the utility grid to trains (passing through substations) but also to flow from trains to the utility grid, consideration of regenerative energy is an also meaningful research topic to save energy consumption in high-speed railway system. CRediT authorship contribution statement Yan Xu: Conceptualization, Methodology, Software, Writing - original draft. Bin Jia: Project administration. Xiaopeng Li: Supervision. Minghua Li: Software, Data curation. Amir Ghiasi: Writing - review & editing. Acknowledgment The research is supported in part by the U.S. National Science Foundation through Grants CMMI#1558887, the National Natural Science Foundation of China (Nos. 71901008, 71601006), the Beijing Postdoctoral Science Foundation (Nos. ZZ2019-110, 201822005) and the China Postdoctoral Science Foundation (Nos. 2019M650413, 2018M641138). Appendix A The derivation process for calculating energy consumption overcoming resistance Er .

Er = = = =

Ta r f (t ) v (t ) dt 0 Ta (R0 + R1· v (t ) + R2 ·(v (t )) 2) v (t ) dt 0 Ta R0 v (t ) dt + R1·(v (t ))2dt + R2 ·(v (t ))3dt 0 Ta 0

R0 v (t ) dt + R1·(v (t ))2dt + R2 ·(v (t ))3dt 1

1

3

1

= R0 ut + 2 R0 at 2 + R1 u2t + R1 aut 2 + 3 R1 a2t 3 + R2 u3t + 2 R2 u2at 2 + R2 ua2t 3 + 4 R2 a3t 4 )

Ta 0

.

where Ta, f r (t ), v (t ) are the acceleration time, instantaneous basic resistance and instantaneous velocity respectively; u ,a, t is the origin speed, acceleration rate, and time step respectively. z u When Ta = akn , u = ukn , a = kna kn , we have kn

1

Er = ·R 0 ukn akn + 2 R 0 (z kn 3

3 2 + R2 ukn akn + 2 R2 ukn (z kn

ukn ) akn + R1 ukn2akn + R1 ukn (z kn ukn ) akn + R2 ukn (z kn

1

ukn ) akn + 3 R1 (z kn 1

ukn ) 2akn + 4 R2 (z kn

ukn )2akn

ukn )3akn .

Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.trc.2020.01.008.

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