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Jointly optimize high-speed train route, timetable and trajectories: a unified modeling approach using space-time-speed grid networks Leishan Zhou School of Traffic and Transportation Beijing Jiaotong University Beijing, 100044, China Email:
[email protected] Xuesong Zhou School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ, 85281, USA Email:
[email protected] Tel.: +1 480 9655827 (Corresponding Author) Jinjin Tang School of Traffic and Transportation Beijing Jiaotong University Beijing, 100044, China Email:
[email protected] Lu Tong School of Traffic and Transportation Beijing Jiaotong University Beijing, 100044, China Email:
[email protected] Submitted for publication in Transportation Research Part B
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Abstract: This paper considers a high-speed rail corridor that requires high-fidelity scheduling of train speed for a large number of trains with both tight power supply and temporal capacity constraints. This research aims to systematically integrate problems of macroscopic train timetabling and microscopic train trajectory calculation that have been handled traditionally at different spatial and temporal resolutions in a sequential fashion. To fully exploit the benefit of joint optimization for these two closely related problems across representation regimes, we develop a unified modeling framework using three-dimensional space-timespeed grid networks to characterize both second-by-second train trajectory and segment-based timetables at different space and time resolutions. The discretized time lattices can approximately track the train position, speed and acceleration solution through properly defined spacing and modeling time intervals. Within a Lagrangian relaxation-based solution framework, we propose a dynamic programming solution algorithm to find the speed/acceleration profile solutions in a carefully discretized space-time-speed grid network with dualized train headway and power supply constraints. The proposed numerically tractable approach can better handle the non-linearity in solving the differential equations of motion, and systematically describe the complex connections between two problems. We further use a real-world case study in the Beijing-Shanghai high-speed rail corridor to demonstrate the effectiveness and computational efficiency of our proposed methods and algorithms. Keywords: train trajectory planning; train timetabling; energy consumption; space-time-speed network
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1 Introduction In general, high-speed rail offers a fast, safe and comfortable transportation mode with a high carrying capacity. Many new high-speed rail lines are being operated or designed to serve increasing intercity passenger flow along economic corridors throughout the world. On the other hand, due to its increased operating speed, high speed train units consume much higher energy than conventional locomotives to overcome extra aerodynamic resistance. For high speed rail operators, how to design time-efficient train timetables for passengers and energy-efficient train speed profiles under tight power supply and track capacity constraints becomes extremely important. Faced with service competitions from emerging fuelefficient automobiles and state-of-the-art aircrafts (Chester and Horvath, 2012), it is important for the rail industry to improve its own ridership in the multimodal intercity transportation market, reduce the overall rail system operating cost, and contribute to societal sustainability in a long run. In the last few decades, there has been a steady move towards the systematical use of computer simulation models and optimization tools to evaluate and further improve the energy use for single or a group of trains running on a corridor. The planners and dispatchers of high-speed trains need to consider an ever-widening range of analysis goals to archive, each with its own particular set of constraints and focus areas. To name a few, commonly used optimization criteria for train operations include time-optimal, energy-optimal or comfortableness. In a typical train timetabling application, the time-optimal schedule aims to minimize passenger in-vehicle travel time so as to meet the high mobility requirement from highspeed rail riders. However, demand-oriented timetable solutions might lead to much high energy consumption compared to energy-optimal schedules that typically have sophisticated designed acceleration and breaking profiles at various grade segments and speed-limit conditions. Clearly, no single optimization goal can purport to be best for all situations, but the commonly used sequential processing of train timetabling and speed control is difficult in its own right to achieve the full system benefit under tight resource constraints. We now start a systematic review of two related research areas, namely train timetabling and train speed control. Cordeau et al. (1998) offered a comprehensive survey on both train routing and scheduling problems, including many early literature with both constant and variable train speed at segment levels. Recent studies, to name a few, Tornquist and Persson (2007), D'Ariano et al. (2007, 2008), Corman et al. (2010), Meng and Zhou (2014), Samà et al. (2016), Fu and Dessouky (2016), have examined various important topics of train scheduling, for example, timetabling under a complex network or N-track condition, real time scheduling to mitigate delay propagation under various degrees of disturbances. Corman and Meng (2013) offered a systematic review on real-time train dispatching, rescheduling and disposition under stochastic and dynamic conditions. In order to describe many complex but practically important train operational constraints. Focusing on China railroad timetabling applications, early work by Zhou et al. (1998) applied a Discrete Event Dynamic System (DEDS) method for offline train scheduling applications. Medanic and Dorfman (2002) and Dorfman and Medanic (2004) discussed the time-efficient and energy-efficient scheduling methods using a general DEDS modeling framework for prioritizing train movement events for a group of trains running along a rail corridor. Early studies on energy-efficient train movement control (e.g., Milroy, 1980; Benjamin, 1989) typically focused on controlling single train speed under various track and geometric conditions, where a locomotive has several levels of traction control gears, each corresponding to different energy consumption rates. Cheng and Howlett (1992, 1993) studied critical speed thresholds to minimize the fuel consumption of individual trains on horizontal lines. Howlett (1996) proposed various optimal train speed control strategies for a train on segmented constant grades, followed by the study by Howlett and Cheng (1997) on optimal driving strategies on a track with continuously varying grades, and the paper by Cheng et al. (1999) on segmented constant gradient with speed-limiting. As one of the key studies in this area, Liu et al. (2003) proposed an analytical model for the energy-efficient train speed/acceleration control along long distance rail lines with various speed-limits and grade changes. They considered a continuously changing control variables to reduce the computational complexity associated with discretized resolutions. Wong and Ho (2003) studied the coast control strategies and related switch points of subway train movement using 3
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hierarchical genetic algorithms. Bai and Mao (2009) discussed the energy consumption performance with respect to different coasting distances under various practical considerations such as braking/lower speed restrictions and train speed uniformity. Aiming for a globally optimal strategy, the study by Howlett et al. (2009) calculates the critical switching points on a track with steep grades through a local energy minimization principle. Recent notable processes along this research line have been made by Bocharnikov et al. (2010), Domínguez et al. (2012), Rodrigo et al. (2013) to optimize energy through the fully utilized regenerative energy sources. Though a comprehensive perturbation analysis for local optimal points (satisfying necessary optimality conditions), the studies by Albrecht et al. (2011, 2013) show that the optimal switching points can be uniquely determined for each steep section of track. There has been a number of emerging studies about how to perform train timetabling and speed profile optimization simultaneously. To achieve a better energy performance for subway trains, Su et al. (2013) proposed an iterative sequential optimization model to take into account both train speed profiles and trip times. Li and Lo (2014a) proposed a nonlinear integer optimization model and a genetic algorithm-based solution method to jointly consider timetabling and speed control between switching points along a metro rail line. They also specifically considered (1) how to synchronize the acceleration and breaking events and (2) how to utilize regenerative energy for energy consumption minimization. A more recent paper by Li and Lo (2014b) developed a simplified but innovative analytical model that assumes two switching points per segment for dynamic train control and schedule adjustment, in order to further construct a linear approximation-based convex programming model. Through a discretized space-time network modeling framework proposed by Yang and Zhou (2014), the study by Yang et al. (2015) is intended to optimize energy-efficient train timetables through variable segment speed expressed as different segment travel times. Using a multitrain simulator, Zhao et al. (2015) recently developed a number of solution search algorithms, such as enhanced brute force, ant colony optimization, and genetic algorithm, to optimize multiple train trajectories, with joint goals of minimizing energy consumption and delay. Recently, Goverde et al. (2015) also recognized and highlights the importance of a consistent cross-resolution representation for macroscopic aggregated railroad network optimization, microscopic train timetabling and fine-turning train trajectory computation. Furthermore, Besinovic et al. (2015) proposed a hierarchical framework for robust timetable design, where feasible timetables are achieved through heuristic micro-macro interaction algorithm. Table 1 provides a comparison of major elements in integrated train timetabling and speed profile optimization. Table 1 Recent studies on integrated train timetabling and speed profile optimization. Major decision variable for timetabling Model Algorithm Publication Iterative algorithm to solve Trip time; speed between switching Nonlinear constraints macroscopic and Su et al. points and objective functions microscopic problems (2013) sequentially Arriving/departure time; speed between Nonlinear constraints Li and Lo Genetic algorithm switching point and objective functions (2014a) Arriving/departure time; speed between Convex optimization Li and Lo Analytical method switching point model (2014b) Discretized space-time Arriving/departure time; average Commercial optimization Yang et al. network to represent segment speed software GAMS (2015) variable speed Integer Linear Randomized multi-start Besinovic et Arriving/departure time; running time Programming (for greedy heuristic al. (2015) macroscopic problem) Discretized space-time-speed grid Lagrangian network, multi-commodity variables Linear constraints and Zhou et al. relaxation/Dynamic representing both timetable and speed objective functions (2015) programming profile characteristics
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Most of the existing studies on train speed control for energy-efficient operations focus on optimizing single train movement or a cluster of train speed profiles with simple overtaking relationships, and it is more challenging to further construct mathematically tractable models and computationally efficient algorithms for a general rail line with various number of switching points, especially on long rail segments with non-regular gradient changes. Improving both travel time and energy measures through a joint train timetabling and speed control optimization requires a broader understanding of complex constraints and operating processes at very different spatial and temporal regimes. In particular, the lack of a unified model representation for these two coupled problems prevent the models from working together in a seamless manner. As a practically important challenge for operating a large number of high-speed trains on an electrified high-speed rail corridor the roadside traction power supply facilities need to supply the voltage required by different trains on a second by second basis. The conversional train timetabling stage, operated typically at a minute by minute resolution, is unable to not only meet the overall energy supply constraints but also minimize the total power consumption for trains spatially distributed at the power supply segments of a corridor. From the train scheduling perspective, determining the order of trains at a station requires binary variables and if-then conditions if a formal integer programming model is constructed, and typically a branch-and-bound algorithm is then used inside a commercial optimization solver or in-house solver to find optimal solutions. The relatively long modeling time interval (e.g. min by min) used in timetabling is difficult to be connected to the high-fidelity spatial and temporal representation for the subsequent train movement control stage. Motivated by a wide range of trajectory optimization methods (but typically for single vehicle systems), such as nonlinear programming based formulation, dynamic programming based formulation, we carefully construct a space-time-speed network to seamlessly map the train trajectories to (1) the spacetime network for scheduling subject to time headway constraints, and then to (2) space-speed network for detailed train control that involves train position, speed and acceleration for accurate energy calculation. Through a carefully designed spatial and temporal discretization scheme, our research creates a cell-based network for flexibly representing a number of essential constraints for different train space-time-speed trajectories, and those constraints include (1) the spatial and temporal safety headway constraints for timetabling and moving block signal systems, (2) total energy consumption constraint across trains running on the same power supply segments but possibly on different tracks. It should be remarked that, one closely related study by Miyatake et al. (2009) solve train optimal speed control problem using Bellman’s dynamic programming (DP) method and described the discretized linearized state in a three-dimensional graph. Without an internally consistent three-dimensional space-time-speed network , it is difficult to connect train running control between different stages. Furthermore, we will fully take advantages of constructed space-time-speed network to develop a highly efficient solution algorithmic framework using dynamic programming and Lagrangian relaxation (LR) techniques. In particular, the DP algorithm can work well for the acyclic network with arcs always move forward along the time axis, and the LR method can dualize the coupling safety headway constraints and energy consumption constrains so that each train, in the decomposed subproblem, only needs to search its own space-time-speed network with generalized costs from LR multipliers. The reminder of this paper is organized as follows. The next section states the joint train timetable and speed profile optimization problem and provide the conceptual illustration of space-time-speed network. Section 3 develops a linear integer programming model and Lagrangian relaxation based Dynamic programming algorithm. Section 4 evaluates our model and algorithm with some real-world medium-scale and large-scale examples, and the final section concludes the paper.
2 Problem Statement and Conceptual Illustration 2.1 Problem statement This paper considers a high speed rail line as illustrated in Fig. 1, with a series of bi-directional track 5
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segments within a power supply district. The power supply district has an autotransformer in the middle, a section post on the left, and a traction/power substation on the right. Through contact wires, the power supply district covers three train stations and two train running segments. In this high-speed train corridor, we consider a moving block traffic signal system, where trains can follow each other on a track segment when satisfying minimum spacing headway requirements. For simplicity, only one type of trains are considered in the scheduling process, and different trains can pass each other only at station track sidings. The traction substation supplies power for all trains within its coverage territory. In general, an intercity high speed rail line is consist of a number of power supply districts, each typically ranging from 50 to 300 km. Given the origin and destination stations of trains with preferred departure time, skip-stopping plan and minimum dwell time, the problem under consideration needs to find optimal train schedules that can minimize a combined function of travel time and energy consumption subject to essential safety headway constraints and power supply resource constraints. Outbound
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Fig 2. High-speed train schedule
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The space-time diagram or the train string diagram in Fig. 2 shows a schedule with 5 trains numbered from 1 to 5. If we only consider the minimum safety headway ℎ between each pair of trains at stations, this schedule should be perceived as a feasible one without no station headway conflicts. However, as the power supply resource constraint applies to all trains running within the same power supply district, there exist extremely complex relationship and highly non-linear interaction between tractive effort and line voltage of a set of high-speed trains. Fig. 3 further shows a piecewise linear curve between maximum available tractive effort and line voltage supplied from a power substation high-speed trains. The tractive effort at wheel rim is a nonlinear function of train speed, acceleration, mass as well as aerodynamic and friction resistance under specific grade and curve conditions. If the assigned line voltage from the power supply system to a specific train is insufficient, then the available traction effort and the corresponding power could be reduced and further impact the tractive effort at wheel rim, resulting in a narrower acceleration range or lower maximum speed.
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Fig 3. Relationship between pantograph-catenary voltage and traction effort rate at wheel rim (from the formal Ministry of China Railway)
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Using a particular example in Fig. 2, if the total power required, even running at different directions of track, is insufficient during at a particular time stamp, say at 20 min, it would lead to the slow running speed of these trains, and the delays for the arrival time at next stations. In a real-time scheduling application, if one train, say train 1, already has an initial delay. It has to carefully consider the required additional power to speed up in order to return back to the planned schedule as soon as possible. If excessive acceleration actions are taken simultaneously by multiple trains, how to fully synchronize the (spatially distributed) train movements and schedules, more specifically, starts and stops of trains, is an extremely complex process as it has to deal with the complicated nonlinear interconnections between multiple trains at very finer time and space resolutions.
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2.2 Continuous functional form for train motion To derive detailed energy consumption based on the kinematics of train movements, we now list a number of related differential train motion equations and the notations in Table 2. For notational convenience, we omit the train index. Symbol 𝑯 𝒕 𝒅(𝒕) 𝒗(𝒕) 𝒎 𝒂𝒄𝒄(𝒕) 𝒅𝒆𝒄(𝒕) 𝒘(𝒗(𝒕)) 𝒉(𝒅(𝒕))
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Table 2 Parameters used in the equation of motion Definition Set of time clocks for analysis, {1, … , 𝑇} Indices of time clocks, 𝑡 ∈ 𝐻 Distance along the track at time 𝑡 Speed of the train at time 𝑡 mass of a train Acceleration associated with train traction force at time 𝑡, that is the ratio of tractive force 𝐹 and mass 𝑚. Deceleration associated with train breaking force at time 𝑡 Resistive acceleration due to train motion at speed 𝑣(𝑡) Resistive acceleration due to the track gradient and curve at location 𝑑(𝑡)
Eq. (2) defines the vehicle speed 𝑣(𝑡) from distance change. Eq. (3) indicates that the resultant acceleration/deceleration is affected by train traction 𝑎𝑐𝑐(𝑡) or braking force 𝑑𝑒𝑐(𝑡), speed-related train motion resistance 𝑤(𝑣(𝑡)) and location-related track grade and curve resistance ℎ(𝑑(𝑡)). d𝑑(𝑡) = 𝑣(𝑡) (2) d𝑡 d𝑣(𝑡) d𝑡
= 𝑎𝑐𝑐(𝑡) − 𝑑𝑒𝑐(𝑡) − 𝑤(𝑣(𝑡)) − ℎ(𝑑(𝑡)) (3) The energy cost of the train journey is the total energy supplied to the train, which can be calculated 7
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from the work done by the traction force. The total energy cost or the total mechanical power at the wheelrail interface 𝐽 of the journey from time 1 to time 𝑇 is therefore formulated in Eq. (4). 𝑇 𝐽 = ∫𝑡=1 𝑚 ∙ 𝑎𝑐𝑐(𝑡) ∙ 𝑣(𝑡) d𝑡 (4) 2.3 Constructing space-time-speed grid network for joint train routing, timetabling and trajectory optimization as a path finding problem Before constructing the space time network representation, we now start discretizing the time horizon to a sequence of time intervals with a certain length, e.g. 1 second, the time index can be denoted as 𝑡 = 1, 2, … . , |𝐻|. A space-time network representation has been widely used in transportation route optimization, various scheduling applications, and general dynamic network flow modeling. Interested readers are referred to general survey papers by Powell et al. (1995), and recent development for time-dependent stochastic shortest path by Yang and Zhou (2014) and Meng and Zhou (2014). Typically, for a physical rail segment (𝑖, 𝑗) connecting a pair of physical nodes 𝑖 and 𝑗, we can create the corresponding time-expanded arcs (𝑖, 𝑗, 𝑡, 𝑠) where 𝑡 is entering time, 𝑠 is the exit time of a traveling arc; and 𝑠 − 𝑡 can be train running time for a certain train 𝑘 on link/segment (𝑖, 𝑗). For a waiting arc at node 𝑖, one can use (𝑖, 𝑖, 𝑡, 𝑡 + 1) to denote the waiting step from time 𝑡 to time 𝑡 + 1 at the same node 𝑖. As shown in Eq. (4) for calculating train energy consumption, one of the key elements is the acceleration 𝑎𝑐𝑐(𝑡) at different times. For a conventional space-time network, for a pair of space-time vertices (𝑖, 𝑡) and (𝑗, 𝑠), there are an infinite number of trajectories connecting these two points, each with the same speed but different possible acceleration/deceleration strategies. In this case, it is difficult to directly use the attributes at two adjacent points (𝑖, 𝑡) and (𝑗, 𝑠) to calculate the resulting train motion characteristics and energy consumption rates that highly depends on the train speed 𝑢 at each time clock. For a set of physical nodes/stations 𝑁 and a set of physical rail links 𝐿, this paper defines a directed STS network 𝐺 = (𝑄, 𝐴) with each STS vertex (𝑖, 𝑡, 𝑢) ∈ 𝑄 and each STS arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∈ 𝐴 that indicates the train movement from node 𝑖 to node 𝑗 at entering time 𝑡 with speed 𝑣 and exit time 𝑠 with speed 𝑢. The notations based on STS network are listed in Table 3. Table 3 Subscripts, parameters and variables used in mathematical formulations. Definition Set of trains Set of time stamps in the planning time horizon Set of nodes/stations Set of links Set of speed values Set of STS vertices Set of STS arcs Index of trains, 𝑘 ∈ 𝐾 Time indices, 𝑡, 𝑠, 𝜏 ∈ 𝐻 Indices of nodes, 𝑖, 𝑗 ∈ 𝑁 Time increment and space increment Index of physical link, (𝑖, 𝑗) ∈ 𝐿 Values of instantaneous speed, 𝑢, 𝑣 ∈ 𝑉 Indices of space-time-speed vertexes, (𝑖, 𝑡, 𝑢), (𝑗, 𝑠, 𝑣) ∈ 𝑄 Index of space-time-speed arcs indicating the train movement from node 𝑖 to node 𝑗 at entering time 𝑡 with speed 𝑣 and leaving time 𝑠 with speed 𝑢, arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑣, 𝑢) ∈ 𝐴 ̅̅̅̅̅(𝒊, 𝒋, 𝒕, 𝒔, 𝒖, 𝒗) Average acceleration associated with train traction force on STS arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) 𝒂𝒄𝒄 Mechanical energy consumption amount on traveling arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) 𝒆(𝒊, 𝒋, 𝒕, 𝒔, 𝒖, 𝒗) Symbol 𝑲 𝑯 𝑵 𝑳 𝑽 𝑸 𝑨 𝒌 𝒕, 𝒔, 𝝉 𝒊, 𝒋 ∆𝒕, ∆𝒅 (𝒊, 𝒋) 𝒖, 𝒗 (𝒊, 𝒕, 𝒖), (𝒋, 𝒔, 𝒗) (𝒊, 𝒋, 𝒕, 𝒔, 𝒖, 𝒗)
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To succeed in meeting a number of study objectives for joint scheduling of train schedule and movement, the construction of the STS network will be guided by the following set of principles. 1. Similar to cellular automata models for traffic flow simulation (Daganzo, 2006), the space along 8
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2. 3.
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the rail corridor is discretized in increment 𝛥𝑑, so the location of any node 𝑖 can be denoted as 𝑛𝑖 ∙ 𝛥𝑑, where 𝑛𝑖 is an integer number. As a result, the distance between any node pair (𝑖, 𝑗) should be an integer multiple of 𝛥𝑑. The time along the planning horizon is discretized in increment 𝛥𝑡, the mapped time stamp for index 𝑡 is 𝑡 ∙ 𝛥𝑡. If we consider the high-speed train constantly accelerates/decelerates during 𝛥𝑡, the speed along the speed dimension is discretized in increment 𝛥𝑑/𝛥𝑡 . This property can meet the following proposition: for any arc (𝑖, 𝑗, 𝑡, 𝑡 + 1, 𝑢, 𝑣) between time index 𝑡 to time index 𝑡 + 1, when 𝑣 is a feasible value (as an integral multiple of 𝛥𝑑/𝛥𝑡), then the ending speed 𝑢 is uniquely determined as a feasible value as well. The upper bound speed at node 𝑖 for train 𝑘 is the minimum value of the maximum speed of train 𝑘 and speed limit at node 𝑖. For node 𝑖 which is not an intermediate stop for train 𝑘, the corresponding STS vertex (𝑖, 𝑡, 0) for any time 𝑡 and its connected arcs should be set to invalid for train 𝑘. That is the vertex and arc are eliminate from train 𝑘’s STS network and thus not usable by train 𝑘. For node 𝑖′ which is an intermediate stop for train 𝑘, we should eliminate any arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) satisfying 𝑖 < 𝑖′ < 𝑗 to avoid non-stopping train activity. There are three types of arcs: (i) traveling arcs (𝑖, 𝑗, 𝑡, 𝑡 + 1, 𝑢, 𝑣) moving from node 𝑖 to node 𝑗 with a unit time interval from 𝑡 to time 𝑡 + 1; (ii) waiting arc (𝑖, 𝑖, 𝑡, 𝑡 + 1, 0, 0) for node 𝑖 that is the origin or destination node of train 𝑘 ; (iii) dwell arc (𝑖, 𝑖, 𝑡, 𝑠, 0, 0) for node 𝑖 where 𝑖 is an intermediate stop for train 𝑘 and time duration (𝑠 − 𝑡) is the required stopping time. A traveling arc with a speed change from 𝑢 to 𝑣 is only feasible when the corresponding acceleration/deceleration value falls in the range between maximum acceleration determined by traction power, and maximum deceleration determined by braking force.
The proof for guiding principle 3 can be briefly described as follows. The travel distance between (𝑖, 𝑗) is (𝑛 𝑗 − 𝑛𝑖 ) × 𝛥𝑑 , the average speed is (𝑢 + 𝑣)/2 , so that (𝑢 + 𝑣)/2 ∙ 𝛥𝑡 = (𝑛 𝑗 − 𝑛𝑖 ) ∙ 𝛥𝑑 , that is 𝑢 = 2(𝑛 𝑗 − 𝑛𝑖 ) ∙ (𝛥𝑑/𝛥𝑡) + 𝑣. Since we have setup the space-time lattice with the space increment 𝛥𝑑 and the time increment 𝛥𝑡, we can easily show that if 𝑣 is an integer multiple of (𝛥𝑑/𝛥𝑡) then 𝑢 must definitely be an integer multiple of (𝛥𝑑/𝛥𝑡). By recognizing the above network construction principles, we now start with a few remarks on the STS grid network. By adding the speed dimension into the space-time network, where each vertex is now associated with a triplet of location, time and speed, the acceleration/deceleration rate between two vertexes can be uniquely determined. More importantly, from a state-space representation perspective (typically for dynamic programming), the STS network can clearly define the speed state and the associated the stop/moving status of the train movement. This enables modelers to flexibly impose the feasible arcs or arc transitions for different practical rules such as speed limits, maximum acceleration/deceleration rates as well as the required stopping stages at the origin, destination and intermediate stations. Specifically, an STS network is able to systematically incorporate many modeling methods based on (i) the space-time network used in the general area of train scheduling and (ii) the train speed profiles used in the area of optimal train control problem. For example, the space-time network based train routing and scheduling algorithm (e.g. Meng and Zhou, 2014) can be readily applied to the STS network representation, while the specific constraints on train motions and energy cost functional form (e.g. Yang et al., 2012) can be also easily adopted. The example in Fig. 5 depicts a simple train motion trajectory in an STS grid network, where a train departing at time 𝑡 from node 𝑖 with an initial speed 0 stops at node 𝑖 at time 𝑡 + 4. By using this STS network representation, we can also account for the location-specific attributes related to train running states, such as gradient resistances, curve resistances, speed limits, and power supply constraints. For instance, for varying speed limits at different railroad segments, one can simply eliminate the STS vertexes beyond the speed limits from the feasible solution domain, these unreachable vertexes will never be chosen during the 9
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train movement optimization process. Fig. 5, where green cycles represent the candidate STS vertexes while the gray ones represent the infeasible vertexes, illustrates the spatial speed limit requirement of 𝑣 = 2 (e.g. due to construction zone) at location 𝑖 + 2 and 𝑖 + 3, compared to the other segments with a normal speed limit of 𝑣 = 5. Velocity
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Candidate vertex
Non-candidate vertex
Train trajectory (selected arc)
Selected vertex
Fig 5. A train trajectory in an STS grid network (with a speed limit between nodes 𝑖 + 2 and 𝑖 + 3). Using a post-processing step, we can further map the (optimized or selected) three-dimensional train trajectory from the STS network to the two-dimensional driver-oriented train speed profiles (on speed-space plane) and scheduling-relevant train timetables (on space-time plane). As shown in Fig. 6, the projection of train trajectory, the space-speed network, clearly shows the acceleration, cruising and deceleration stages in the train speed profile for various sections of rail track, and the projection on the space-time plane indicates train timetables. Velocity
v+5
v+4 i+6 i+5 i+4 i+3 i+2 i+1 i
v+3
v+2 v+1 v t
t+1
t+2
t+3
t+4
Time
Space/Distance
15 16 17 18
Train trajectory (selected arc)
Projection
Fig 6. Projections of a train STS trajectory to the space-time plane and the speed-space plane
10
1 2
2.4 Energy consumption function for train traveling arcs By plugging in the corresponding values for STS traveling arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) for the continuous
3
presentation for speed change in Eq. (3), that is, d𝑡 ≅ 𝑠−𝑡 ; 𝑤(𝑣(𝑡)) ≅ 𝑤 ( can derive the average acceleration ̅̅̅̅̅(𝑖, 𝑎𝑐𝑐 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) in Eq. (5).
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
d𝑣(𝑡)
𝑣−𝑢
𝑢+𝑣 ); 2
ℎ(𝑑(𝑡)) ≅
ℎ(𝑖)+ℎ(𝑗) , we 2
ℎ(𝑖)+ℎ(𝑗)
̅̅̅̅̅(𝑖, 𝑎𝑐𝑐 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) = 𝑚𝑎𝑥 {( 𝑠−𝑡 + 𝑤 ( 2 ) + ) , 0} (5) 2 According to Eq. (4), we can compute the mechanical energy consumption 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) on STS arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) of each lattice in Eq. (6) (as an integral from time t to time s). 𝑢+𝑣 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) = 𝑚 × ̅̅̅̅̅(𝑖, 𝑎𝑐𝑐 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) × 2 × (𝑠 − 𝑡) (6) For train waiting and stopping arcs, the corresponding energy consumptions are assumed to zero. For travel arcs with deceleration, we assume zero energy consumed. For a more sophisticated train operating mode with regenerative energy sources, one can further determine a negative value for 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) to feed the energy from braking back to the power supply system.
3 Model and algorithm for joint train trajectory and scheduling optimization 3.1 Model The optimization model aims to incorporate train speed control into the timetable design process considering realistic operating constraints. Table 4 lists notations used in the optimization model. Symbols 𝒐𝒌 , 𝒅𝒌 𝑬𝑫𝑻𝒌 , 𝑳𝑨𝑻𝒌 𝑷 𝒑 𝑹𝒑 𝑮𝒑,𝒕 𝒉 𝑳′ (𝒊′, 𝒋′) 𝝋(𝒊′, 𝒋′, 𝝉) 𝒄𝒊,𝒋,𝒕,𝒔,𝒖,𝒗 𝒙𝒊,𝒋,𝒕,𝒔,𝒖,𝒗 (𝒌)
22 23 24 25 26 27 28 29 30 31 32 33 34 35
𝑢+𝑣
𝑣−𝑢
Table 4 Notations used in the joint train control and scheduling optimization model Definition Origin and destination of train 𝑘 Earliest departure time and latest arrival time of train 𝑘 Set of power supply districts Index of power supply district, 𝑝 ∈ 𝑃 Power supply capacity per time unit of power supply district 𝑝 Set of vertices in power supply district 𝑝 at time 𝑡 Safety time headway Set of basic track segments Index of basic track segment, (𝑖′, 𝑗′) ∈ 𝐿′ Set of incompatible STS vertices of basic track segment (𝑖′, 𝑗′) at time 𝜏 Cost on arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) =1 if the STS arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) is used in the tour of train 𝑘 =0 otherwise
Objective function The joint train control and scheduling problem aims to find the speed profiles and timetables under specific optimization goals (e.g. minimizing energy consumption). Within the STS network framework, the objective function to minimize the total cost of all high speed trains is stated in Eq. (7), where the sevendimensional binary variable 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) indicates if the STS arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) is used by train 𝑘 and 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 is the cost of the corresponding arc. min Z = ∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝐴(𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘)) (7) For an energy-optimal objective, we discuss in this paper, 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 is identical to the mechanical power consumption 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) defined in Section 3. When performing a travel time-optimal objective function, 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 = 0 for the waiting arcs at the origin 𝑜𝑘 and destination node 𝑑𝑘 and 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 = 𝑠 − 𝑡 otherwise. STS flow balance constraint 11
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42
To depict a feasible train tour in the STS network, a set of flow balance constraints is formulated as follows. For each individual train 𝑘, its feasible flow, starting from the super source at vertex (𝑜𝑘 , 𝐸𝐷𝑇𝑘 , 0) and ending at the super sink vertex (𝑑𝑘 , 𝐿𝐴𝑇𝑘 , 0), need to be satisfied at all vertices. We can state the flow balance constraints formally as follows. 1, 𝑖 = 𝑜𝑘 , 𝑡 = 𝐸𝐷𝑇𝑘 , 𝑣 = 0 ∑(𝑗,𝑠,𝑣)∈𝑄 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) − ∑(𝑗,𝑠,𝑣)∈𝑄 𝑥𝑗,𝑖,𝑠,𝑡,𝑣,𝑢 (𝑘) = {−1, 𝑖 = 𝑑𝑘 , 𝑡 = 𝐿𝐴𝑇𝑘 , 𝑣 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘 ∈ 𝐾 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (8) Power supply constraint Due to the capacity limitation of transformers illustrated in Section 2.2, in power supply district 𝑝 the total mechanical power required by trains cannot exceed the power supply capacity 𝑅𝑝 at any time. Let vertex set 𝐺𝑝,𝑡 represent all STS vertices in power supply district 𝑝 at time 𝑡, and we obtain the power supply constraints in Eq. (9). ∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝐺𝑝,𝑡 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) ≤ 𝑅𝑝 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑝 ∈ 𝑃, 𝑡 ∈ 𝐻 (9) Train safety constraint (time headway) As moving block or quasi-moving block signaling system is typically equipped in the high-speed rail lines, a time headway ℎ must be enforced for all vehicle trajectories. Defining the incompatible vertices set 𝜑(𝑖 ′ , 𝑗 ′ , 𝜏) , the time headway-related train safety constraints as follows. That is only one train can pass/occupy basic track segment (𝑖 ′ , 𝑗 ′ ) through time range [𝜏, 𝜏 + ℎ] for each time stamp 𝜏. It should be made clear that because of the subtle train speed difference and the relatively short distances of the basic track segments we do not consider train overtaking scenarios. ∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝜑(𝑖′ ,𝑗′ ,𝜏) 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) ≤ 1 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝑖 ′ , 𝑗 ′ ) ∈ 𝐿′, 𝜏 ∈ 𝐻 (10) Binary variable constraint There is also binary definitional constraints for variables 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘). 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) ∈ {0, 1} 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘 ∈ 𝐾, (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∈ 𝐴
(11)
3.2 Problem decomposition through to a set of single-train optimization problems through Lagrangian relaxation Compared with classic mathematic models: the shortest path problem, our joint train control and scheduling optimization problem has two additional side constraint (9) and (10). Instead of solving this problem directly, we relax these difficult constraints by incorporated them into the objective function with power supply multipliers 𝜋𝑝,𝑡 and safety multipliers 𝜌𝑖′ ,𝑗′ ,𝜏 following the Lagrangian relaxation procedure. The Lagrangian relaxation problem derived in Eq. (12) can be treated as a generalized least cost/shortest path ′ (𝑘) is problem in a time-expended network (Fischer and Helmberg, 2014). The dualized cost 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 defined in Eq. (13). min Z′ = ∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝐴 (𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘)) + ∑𝑝∈𝑃 ∑𝑡∈𝐻 [𝜋𝑝,𝑡 (∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝐺𝑝,𝑡 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) − 𝑅𝑝 )] + ∑(𝑖 ′ ,𝑗 ′ )∈𝐿′ ∑𝜏∈𝐻 𝜌𝑖 ′ ,𝑗 ′ ,𝜏 (∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝜑(𝑖 ′ ,𝑗 ′ ,𝜏) 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) − 1) ′ (𝑘) ⋅ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) − ∑𝑝∈𝑃 ∑𝑡∈𝐻 𝜋𝑝,𝑡 ∙ 𝑅𝑝 − ∑(𝑖 ′ ,𝑗 ′ )∈𝐿′ ∑𝜏∈𝐻 𝜌𝑖 ′ ,𝑗 ′ ,𝜏 = ∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝐴 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣
′ (𝑘) = 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣
(12)
𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘)
𝑓𝑜𝑟 (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∉ 𝐺𝑝,𝑡 , 𝜑(𝑖 ′ , 𝑗 ′ , 𝜏)
(𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) + 𝜋𝑝,𝑡 ∙ 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣)) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘)
𝑓𝑜𝑟 (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∉ 𝜑(𝑖 ′ , 𝑗 ′ , 𝜏)
(𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) + 𝜌𝑖 ′ ,𝑗 ′ ,𝜏 ) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘)
𝑓𝑜𝑟 (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∉ 𝐺𝑝,𝑡
{ (𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) + 𝜋𝑝,𝑡 ∙ 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) + 𝜌𝑖 ′ ,𝑗 ′ ,𝜏 ) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘) 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (13)
43 12
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3.3 Dynamic programming algorithm As the proposed STS network has three dimensions, the searching space of the joint timetabling and train control problem becomes extremely large. In order to find the optimal solution in reasonable time, it is important to design an efficient algorithm. In discrete optimization problems the progress of finding optimal objects over time is frequently modeled by shortest path problems in time expanded networks, but longer time spans or finer time discretization quickly lead to problem sizes that are intractable in practice. In convex relaxations the arising shortest paths often lie in a narrow corridor inside these networks. Motivated by this observation, we develop a general dynamic graph generation framework in order to control the networks’ sizes even for infinite time horizons. It can be applied whenever objects need to be routed through a traffic or production network with coupling capacity constraints and with a preference for early paths. Without sacrificing any information compared to the full model, it includes a few additional time steps on top of the latest arcs currently in use. This “frontier” of the graphs can be extended automatically as required by solution processes such as column generation or Lagrangian relaxation. The corresponding algorithm is efficiently implementable and linear in the arcs of the non-time-expanded network with a factor depending on the basic time offsets of these arcs. We give some bounds on the required additional size in important special cases and illustrate the benefits of this technique on real world instances of a large scale train timetabling problem. Input: STS network 𝐺 = (𝑄, 𝐴) built specifically for train 𝑘 Desired earliest departure vertex (𝑜𝑘 , 𝐸𝐷𝑇𝑘 , 0) and latest arrival vertex (𝑑𝑘 , 𝐿𝐴𝑇𝑘 , 0) of each train ∈ 𝐾. Output: The least cost train trajectory in a STS network for each train 𝑘 ∈ 𝐾 (i.e. train running speed profile and timetable) Step 1. Initialization Set label cost on each vertex 𝜆𝑘 (𝑖, 𝑡, 𝑢) to +∞ and set its predecessor 𝑝𝑟𝑒𝑘 (𝑖, 𝑡, 𝑢) to (−1, −1, −1) for each train 𝑘 ∈ 𝐾 Set label cost on vertex 𝜆𝑘 (𝑜𝑘 , 𝐸𝐷𝑇𝑘 , 0) = 0 for each train 𝑘 ∈ 𝐾 Set the values of Lagrangian multipliers: 𝜋𝑝,𝑡 = 0 for all 𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇, 𝜌𝑖′ ,𝑗′ ,𝜏 = 0 for all (𝑖 ′ , 𝑗 ′ ) ∈ 𝐿′ , 𝜏 ∈ 𝐻 Set Lagrangian iteration number 𝑞 = 0 Step 2. Label updating in dynamic programming For train 𝑘 ∈ 𝐾 do For 𝑡 = 𝐸𝐷𝑇𝑘 to 𝐿𝐴𝑇𝑘 do For each vertex (𝑖, 𝑡, 𝑢) ∈ 𝑉 do For each arc (𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∈ 𝐴 do ′ (𝑘) < 𝜆𝑘 (𝑗, 𝑠, 𝑣) If 𝜆𝑘 (𝑖, 𝑡, 𝑢) + 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 ′ (𝑘) Then 𝜆𝑘 (𝑗, 𝑠, 𝑣) = 𝜆𝑘 (𝑖, 𝑡, 𝑢) + 𝑐𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 𝑝𝑟𝑒𝑘 (𝑗, 𝑠, 𝑣) = (𝑖, 𝑡, 𝑢) End // for each arc End // for each vertex End // for each time End // for each train Calculate objective function values of the problem, which correspond to the upper bound (UB), lower bound (LB), respectively
13
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Step 3. Obtain the least cost STS train trajectory For train 𝑘 ∈ 𝐾 do Find the super sink vertex (𝑑𝑘 , 𝐿𝐴𝑇𝑘 , 0) and back trace to generate space time trajectory End // for each train Step 4. Updating Lagrangian multipliers based on subgradient calculation For each power supply district 𝑝 ∈ 𝑃 at time 𝑡 ∈ 𝐻 do 𝜋𝑝,𝑡 = 𝜋𝑝,𝑡 + 𝜃 𝑞 ([∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝐺𝑝,𝑡 𝑒(𝑖, 𝑗, 𝑡, 𝑠, 𝑢, 𝑣) ∙ 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘)] − 𝑅𝑝 ) End // for each power supply district at each time For each basic physical track segment (𝑖′, 𝑗’) ∈ 𝐿′ at time 𝜏 ∈ 𝐻 𝜌𝑖′ ,𝑗′ ,𝜏 = 𝜌𝑖′ ,𝑗′ ,𝜏 + 𝜃 𝑞 ([∑𝑘∈𝐾 ∑(𝑖,𝑗,𝑡,𝑠,𝑢,𝑣)∈𝜑(𝑖′ ,𝑗′ ,𝜏) 𝑥𝑖,𝑗,𝑡,𝑠,𝑢,𝑣 (𝑘)] − 1) do End // for each basic physical track segment at each time (𝜃 𝑞 is the Lagrangian subgradient step size at iteration 𝑞 (Fisher, 1985; Ahuja, 1993) Step 5: Termination condition test If 𝑞 is greater than a predetermined maximum value, terminate the algorithm; otherwise 𝑞 = 𝑞 + 1 and go back to Step 2. It should be particularly remarked that, typically, high-dimensional networks are complex to build, and storing such a network externally as an algorithm input leads to prohibitive memory space requirements. However, instead of explicitly storing the network, inside the multiple loops of the DP step 2, our efficient implementation can simply use a set of if-then rules or conditions to filter out the inaccessible STS arcs, and the DP optimality checking is only applied for (enumerated) accessible and feasible arcs. By doing so, a much simpler set of programming codes can be easily implemented and tested, compared to the typical network-prebuilding first and routing second approach. 3.4 Reduction of searching space The early studies of the single train control prove that for one journey the energy efficient driving strategy typically requires four stages, in the order of full power traction, speed holding/cruising, coasting and full braking (Pudney and Howlett, 1994). We now analyze the complexity of the dynamic programming algorithm of the label updating procedure. In this multi-loop framework the algorithm first performs train selection operations |𝐾| times, and for each train it searches along the time, space and speed axes. Therefore, an apparent estimation for the worst computational time could be 𝑂([|𝐻| × |𝑁| × |𝑉|]2 ) for each train. We need to recognize the following fact to further precisely estimate the complexity. The structure of STS transportation network leads to a very limited outgoing degree for each STS node. For instance, the change of time index from 𝑡 to 𝑠 along its outgoing arcs is simply defined as a constant number of options, namely, second by second travel time and dwell time. The change of space index and speed index is defined through the same way. As a result, instead of having a complicated quadratic form that considers all three dimension change, the complexity in the real-life example should be 𝑂(|𝐻| × |𝐸|), where |𝐸| is the maximum outgoing degree of the STS nodes. To further reduce the search space, one can adopt shooting heuristics, which has been widely used in the vehicle trajectory numerical optimization field. Interested readers on shooting heuristics are referred to the classical survey by Von Stryk and Bulirsch (1992), a dissertation by Tang (2012) on a full-scale simulation-based platform, and a recent study by Zhou et al. 2016 for emerging automated vehicle trajectory optimization applications.
4 Numerical experiments 4.1 An illustrative example We first illustrate the proposed space-time-speed grid network concept using a simple example on a railroad section without taking account of the power supply constraints. Two trains travel from node 0 to node 775. 14
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As discussed in Section 3.4, there are generally five types of STS arcs, i.e. full power traction, cruising, coasting and full braking arcs as well as waiting arcs. Given a constant speed limit of 7 units, different types of arcs are built by following the STS network construction principles and listed in Table 5. During the acceleration process, a train travels 230 distance units and takes 42 time units to acceleration from standstill to up to the maximum speed. The full power traction arcs (0,230, 𝑡, 𝑡 + 42, 0,7) are built at node 0 for each time 𝑡 , whose cost equals to the energy consumption for acceleration. 2) Cruising arcs (𝑖, 𝑖 + 8, 𝑡, 𝑡 + 1, 7,7) describe the speed-holding stage where a train moves 8 distance units within 1 time unit at the maximum speed. The arc cost equals to the energy consumed against resistance. 3) Train coasting occurs before the braking. Its corresponding arcs, e.g. (𝑖, 𝑖 + 106, 𝑡, 𝑡 + 12, 7,6), are constructed at each node. 4) Full braking arcs allow trains to stop at their destinations. The time and distance taken varies due the initial braking speed. Moreover, as the mechanical power is not required in the last three types of arcs, their arc costs are all zero. Fig. 7 provides the results based on the constructed STS network. Fig. 7-(a) demonstrates two train STS trajectories, with the projected train timetables in Fig. 7-(b), and projected in Fig. 7-(c). Table 5 Sample values associated with STS arcs in the illustrative example Arc Arc type 𝑖 𝑗 𝑡 𝑠 𝑢 Full power traction 0 230 𝑡 𝑡 + 42 0 Cruising 𝑖 𝑖+8 𝑡 𝑡+1 7 Coasting 𝑖 𝑖 + 106 𝑡 𝑡 + 12 7 1509 1550 𝑡 𝑡 + 11 6 Full braking 1550 𝑡 7 0 0 𝑡 𝑡+1 0 Waiting 1550 1550 𝑡 𝑡+1 0
𝑣 7 7 6 0 0 0 0
Arc cost 1772 24 0 0 0
Speed
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Space Time
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(a) Train STS trajectories
Time
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(c) Train speed profiles
(b) Train timetables
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Fig 7. Train STS trajectories, timetables and speed profiles for the illustrative example (units: uniformed speed steps, time steps, and space steps)
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4.2 Examples of Beijing-Shanghai high-speed rail corridor In the following experiments focusing on the Beijing-Shanghai (Jinghu) high-speed rail corridor, shown in Fig. 8, which starts its commercial train services on June 30, 2011 and has served more than 200 million passengers till April, 2014 (He et al., 2015). We first apply our proposed STS-oriented solution algorithms to construct lower bound solutions, which could still have possible conflicts associated with dualized capacity resource constraints after multiple iterations of Lagranginan relaxation solution algorithm. The feasible upper solutions are then generated from the possible close-to-optimal lower bound solutions, by applying a set of combined simulation-shooting heuristics described by Tang (2012). Fig. 9 shows a simplified traction characteristic and the running resistance of trains to be used in our test cases, with the speed limit of 300 km/h and the power supply capacity of 50MVA for each power supply district illustrated in Fig. 1.
Beijing South Langfang Tianjin South Cangzhou West Dezhou East Jinan West Taian Qufu East Tengzhou East Zaozhuang West Xuzhou East Suzhou East Bengbu South Dingyuan
Zhenjiang South Danyang North Chuzhou East South Nanjing Changzhou North Wuxi East Suzhou North Shanghai Hongqiao Kunshan South
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Fig 8. Beijing-Shanghai high speed railroad corridor 600 550 18400KN (Full power)
500 450
13200KN (75% Full power)
F[KN]
400 350
3.0%
300 250
8000KN (50% Full power) 2.0%
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100
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50 0
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Running resistance (gradient = 0%)
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v [km/h]
Fig 9. Traction characteristics of high-speed trains
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The first set of experiment covers a partial corridor from Beijing North station to Jinan West station, with a distance of 419 km. We first extracted 18 trains from the real-life high-speed railroad timetable, starting from 7:00AM. In this medium scale case, we have to carefully select the discretization scheme to avoid the computational overhead due to increased dimensions. In particular, the constructed STS grid has 10 seconds, 100 meters and 43km/h, respectively for discretized time, space/distance and speed steps. The entire LR optimization algorithm takes about 40GB Ram and about 30 CPU min (without parallelization) to compute 10 iterations. Fig. 10 plots the solution in STS grid network and corresponding STS trajectories, timetables and speed profiles after 10 iterations. Fig. 11 shows the converted upper bound solution, through the simulator developed by Tang (2012), in terms of the speed profile, the cumulative travel time and acceleration rate.
Fig 10. Train STS trajectories, timetables and speed profiles of iteration 10
Fig 11. Speed profile, the cumulative travel time and acceleration rate along the time axis, obtained from full-scale train trajectory simulation In the following, we now consider the entire Beijing-shanghai high-speed rail line, which is powered by 26 power substations, with the corresponding real-world timetable shown in Fig. 12, which specifies the train 17
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normal running time per segment and required dwell times for our optimization algorithm. By applying single-shooting heuristics and a number of other commonly used heuristic path search algorithms such as A*, beam search and rolling horizon-based decomposition, we are able to optimize the STS trajectories by synchronizing train departure and dwell time. The overall CPU time for this heuristic-oriented algorithm (Tang, 2012) is about 2 hours for a total of 188 trains. Fig. 13 shows one of the optimized real-world timetable, while the overall total energy consumption is reduced from 6169852.5 kWh to 5429470 kWh (about 87.5%). Under different speed limit conditions, Table 6 details the per-train optimization changes in terms of end-to-end travel time (i.e. traveling speed) and used energy.
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Fig 12. A real-world timetable of Beijing-Shanghai high-speed rail line
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Fig 13. One optimized train schedule solution for Beijing-Shanghai high-speed rail line
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Table 6 Per-train measure of effectiveness for entire corridor test case under different speed limit conditions Before Optimization Optimization Average Energy consumption Energy consumption Speed speed(km/h) per train per kilometer Average speed(km/h) per train per limit(km/h) (kwh) kilometer(kwh) 200 178.5 35.8 163.6 30.8 220 192.7 40.1 182.3 33.9 240 223.1 43.1 212.8 37.4 260 244.3 46.45 242.5 41.7 280 263.7 50.2 251.6 46.6 300 286.6 62.8 270.8 55.1
5 Conclusions In this paper, we proposed a joint train trajectory optimization method to reduce system-wide energy consumptions with sufficient flexibility within the schedule constraints. The space-time-speed network framework is established to characterize both train timetables and speed profiles at different space and time resolutions. Through a dynamic programming based algorithm, the approach is tested on Beijing-Shanghai high-speed rail and the energy consumption obviously declines during the peak hours. It should be remarked, to our limited knowledge, there are very few prior studies using continuous space-time-time constructs to illustrate the importance of tracking vehicle location, movements, speed jointly. For example, Kuijpers et al. (2011) proposed kinetic space-time prisms to consider acceleration limits on the top of classical spacetime prisms. Zhao et al. (2015) used an interesting 3 dimensional continuous space-time-speed representation to graphically illustrate the detailed time-step-based vehicle movement simulation. In our research, by fully discretizing the speed levels and synchronizing the entering and leaving speed levels u and v according to the space-time grid network, we are able to enable an extremely feasible highdimensional search path algorithm for routing, timetabling and trajectory optimization applications. This carefully network construction method nicely links two domain problems together: (i) discrete optimization for train scheduling and (ii) continuous optimization for trajectory planning. Our constructed STS grid network can be also viewed as a sophisticated adaption of the broader state-spacetime network-based modeling framework proposed by Mahmoudi and Zhou (2016) for the vehicle routing problem with vehicle carrying capacity and time window constraints. In our future research, we are interested in different effective heuristics to reduce the search space. An integrated optimization processing should be considered under stochastic disturbances, real time and other more realistic and complicated train operating scenarios. Currently, our research teams are also working on extending STS modeling framework to better control and smooth speed changes in emerging applications of autonomous vehicle routing, platooning in highway and urban arterial networks (Li et al. 2015; Lu et al. 2016 and Wei et al. 2016).
Acknowledgements The authors would like to thank Dr. Junhai Chen’s valuable help and comments for this paper. The second author is partially funded by U.S. Department of Energy’s (DOE) Advanced Research Projects Agency – Energy (ARPA-E), “Traveler Response Architecture using Advanced Novel Signaling for Network Efficiency in Transportation (TRANSNET)”, partially funded by National Science Foundation – United States under Grant No. CMMI 1538105 “Collaborative Research: Improving Spatial Observability of Dynamic Traffic Systems through Active Mobile Sensor Networks and Crowdsourced Data. The third author is partially supported by China Ministry of Transport Construction Project in Science and Technology 2015318223010. 19
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