Subway Energy-Efficient Management [1st ed.] 9789811577840, 9789811577857

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Uncertainty and Operations Research

Xiang Li Xin Yang

Subway Energy-Efficient Management

Uncertainty and Operations Research Editor-in-Chief Xiang Li, Beijing University of Chemical Technology, Beijing, China Series Editor Xiaofeng Xu, Economics and Management School, China University of Petroleum, Qingdao, Shandong, China

Decision analysis based on uncertain data is natural in many real-world applications, and sometimes such an analysis is inevitable. In the past years, researchers have proposed many efficient operations research models and methods, which have been widely applied to real-life problems, such as finance, management, manufacturing, supply chain, transportation, among others. This book series aims to provide a global forum for advancing the analysis, understanding, development, and practice of uncertainty theory and operations research for solving economic, engineering, management, and social problems.

More information about this series at http://www.springer.com/series/11709

Xiang Li Xin Yang •

Subway Energy-Efficient Management

123

Xiang Li School of Economics and Management Beijing University of Chemical Technology Beijing, China

Xin Yang State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing, China

ISSN 2195-996X ISSN 2195-9978 (electronic) Uncertainty and Operations Research ISBN 978-981-15-7784-0 ISBN 978-981-15-7785-7 (eBook) https://doi.org/10.1007/978-981-15-7785-7 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Subway is an electric passenger railway, which is operated either in underground tunnels or on elevated rails. In the past decades, the subway has received rapid development in China due to its high capacity, punctuality, and reliability. Up to June 2019, there have been 39 cities opening 174 urban rail transit lines, among which Shanghai operates the largest subway network, and Beijing operates the busiest subway network in the world. On December 30, 2017, Beijing subway Yanfang line started operation, the first domestically developed automated subway in China. For subway operations management, the optimal train control and scheduling methods have been extensively studied with various objectives, among which energy saving has attracted much attention from both researchers and practitioners on account of the rising energy prices and environmental concerns. The most popular energy-efficient management approaches include speed control and timetable optimization; the former is a type of commonly used methods while the latter is a class of emerging method, which respectively contribute to reducing traction energy consumption and improving regenerative energy absorption. Speed control approach optimizes the time-speed profile for trains at inter-station to minimize traction energy consumption. The literature on energy-efficient speed control can date back to the 1960s. In 1968, Ishikawa proposed the first optimal train control model to determine the most energy-efficient speed profile. After that, both theoretical analyses and heuristic algorithms have been given, among which Howlett and his Scheduling and Control Group made extraordinary contributions to laying the foundation for energy-efficient speed control theory or energy-efficient operation theory. Timetable optimization approach synchronizes the accelerating phases and braking phases of adjacent trains located in the same substation to maximize the regenerative energy absorption. Over the past years, a series of timetable optimization models have been formulated. For example, Ramos et al. (2007) first presented a timetabling problem that aims to maximize the overlapping time between accelerating actions and braking actions of adjacent trains, so that the accelerating trains can absorb the regenerative energy from braking trains as much v

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as possible. In 2013, Li and Yang measured the regenerative energy absorption as the integral of the minimum profile between traction energy and regenerative energy at the overlapping time and formulated an energy-efficient timetabling model. Essentially, timetable optimization and speed control are two closely related processes on energy saving. The timetabling process allocates the travel time among inter-stations, which significantly influences the traction energy consumption, while speed control process determines the accelerating time and braking time at inter-stations, which profoundly influences the regenerative energy absorption. Therefore, some researchers studied the integrated optimization on timetable and speed profile to minimize the net energy consumption, i.e., the difference between traction energy consumption and regenerative energy absorption. For example, Li and Lo (2014) proposed a mixed integer programming model and designed a genetic algorithm to minimize the net energy consumption, which integrally optimizes the accelerating time and braking time at inter-stations, dwell time, cycle time, and trip frequency. As extensions, the dynamic optimization methods and stochastic optimization methods on timetable and speed profile were also studied. The purpose of this book is to provide a powerful tool to handle the subway energy-efficient management problems. It provides a comprehensive presentation on train timetable optimization and speed control models with the objective of energy saving. The methods presented here are designed for but not limited to the subway system. It can be extended and applied to the timetabling and speed control for high-speed trains and other types of passenger trains. The book is suitable for researchers, engineers, and students in the fields of transportation science, management science, and so on. The readers will learn numerous new modeling ideas on reducing traction energy consumption and improving regenerative energy absorption and find this work a useful reference. Beijing, China May 2020

Xiang Li Xin Yang

Acknowledgment This work was supported by the National Natural Science Foundation of China (Nos. 71931001, 71722007, 71701013, 71621001), and Beijing Social Science Fund (No. 13JGC087).

Contents

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1 1 2 3 4 5 5 6 7

2 Energy-Efficient Speed Control . . . . . . . . . . . . . . . . . . 2.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Speed Control Model without Speed Limits 2.2.3 Optimal Control Model with Speed Limits . 2.3 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Timetabling with Overlapping Time Maximization . 3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . 3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Intermediate Variables . . . . . . . . . . . . . . 3.2.4 Decision Variables . . . . . . . . . . . . . . . . . 3.2.5 Model Assumptions . . . . . . . . . . . . . . . . 3.2.6 Cooperative Rules . . . . . . . . . . . . . . . . .

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1 Literature Overview . . . . . . . . . . . . . . . . . . . . 1.1 Energy-Efficient Speed Control . . . . . . . . . 1.1.1 Analytical Algorithms . . . . . . . . . . 1.1.2 Numerical Algorithms . . . . . . . . . . 1.2 Energy-Efficient Timetable Optimization . . 1.2.1 Overlapping Time Maximization . . 1.2.2 Regenerative Energy Maximization . 1.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2.7 Overlapping Time . . . . . . . . . . . . . . . . 3.2.8 Overlapping Time Maximization Model 3.3 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . 3.3.1 Representation Structure . . . . . . . . . . . . 3.3.2 Initialization . . . . . . . . . . . . . . . . . . . . 3.3.3 Evaluation Function . . . . . . . . . . . . . . . 3.3.4 Selection Process . . . . . . . . . . . . . . . . . 3.3.5 Crossover Operation . . . . . . . . . . . . . . 3.3.6 Mutation Operation . . . . . . . . . . . . . . . 3.3.7 General Procedure . . . . . . . . . . . . . . . . 3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Timetabling with Regenerative Energy Maximization . 4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . 4.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Model Assumptions . . . . . . . . . . . . . . . . . . 4.2.3 Timetabling Rules . . . . . . . . . . . . . . . . . . . 4.2.4 Objective Function . . . . . . . . . . . . . . . . . . . 4.2.5 Regenerative Energy Maximization Model . 4.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . 4.3.2 Allocation Algorithm . . . . . . . . . . . . . . . . . 4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Integrated Speed Control and Timetable Optimization . 5.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Net Energy Consumption . . . . . . . . . . . . . . . 5.2.3 Integrated Constraints . . . . . . . . . . . . . . . . . 5.2.4 Integrated Optimization Model . . . . . . . . . . . 5.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Representation Process . . . . . . . . . . . . . . . . . 5.3.2 Initialization Process . . . . . . . . . . . . . . . . . . 5.3.3 Evaluation Process . . . . . . . . . . . . . . . . . . . . 5.3.4 Selection Process . . . . . . . . . . . . . . . . . . . . . 5.3.5 Crossover Process . . . . . . . . . . . . . . . . . . . . 5.3.6 Mutation Process . . . . . . . . . . . . . . . . . . . . . 5.3.7 General Procedure . . . . . . . . . . . . . . . . . . . . 5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Dynamic Speed Control and Timetable Optimization . 6.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . 6.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Indices and Parameters . . . . . . . . . . . . . . . . 6.2.4 Decision Variables . . . . . . . . . . . . . . . . . . . 6.2.5 Net Energy Consumption . . . . . . . . . . . . . . 6.2.6 Constraints . . . . . . . . . . . . . . . . . . . . . . . . 6.2.7 Integrated Optimization Model . . . . . . . . . . 6.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Stochastic Speed Control and Timetable Optimization 7.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . 7.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Uncertainty of Train Mass . . . . . . . . . . . . . 7.2.3 System Constraints . . . . . . . . . . . . . . . . . . 7.2.4 Objective Function . . . . . . . . . . . . . . . . . . . 7.2.5 Stochastic Optimization Model . . . . . . . . . . 7.2.6 Formulation Complexity . . . . . . . . . . . . . . . 7.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Genetic Algorithm . . . . . . . . . . . . . . . . . . . 7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Literature Overview

In 2017, Yang et al. [49] presented a comprehensive survey on subway energyefficient management literature, in which speed control and timetable optimization are two mainly used subway energy-efficient management approaches: the former optimizes the speed profile at inter-stations to minimize the traction energy consumption and the latter synchronizes the accelerating actions and braking actions of trains to maximize regenerative energy absorption. Based on their work, this chapter mainly introduces the state-of-the-art on energy-efficient speed control, energy-efficient timetable optimization and their extensions on integrated optimization, dynamic optimization, and stochastic optimization approaches.

1.1 Energy-Efficient Speed Control Literature on train energy-efficient speed control study can date back to the 1960s. In 1968, Ishikawa [24] proposed an optimal control model to determine the train speed profile at inter-stations, which can be applied to both subway systems and general railway systems. The objective function is to minimize the electric energy consumption at inter-station. In 1980, Milroy [36] reformulated the problem as follows: ⎧  T ⎪ ⎪ min C(u, v) = u + (t)v(t)dt ⎪ ⎪ ⎪ ⎪ ⎨ s.t. v  (t) = u(t) 0− r (v(t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

v(0) = v(t) = 0 T 0 v(t)dt = S |u(t)| ≤ 1,

© Springer Nature Singapore Pte Ltd. 2020 X. Li and X. Yang, Subway Energy-Efficient Management, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-7785-7_1

(1.1)

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1 Literature Overview

which lays the foundation of the optimal train control theory. Although Milroy applied the model to general railway systems, Howlett et al. [20] proved that it was also suitable for subway systems. In this model, the objective function C(u, v) denotes the traction energy consumption, T is the travel time at inter-station determined by timetable, S is the length of inter-station, v(t) is the train speed, r (v) is the speed-dependent resistance applied on the train, u(t) is the unit force applied on the train and u + (t) is the positive part of u(t), i.e., u + (t) = max{u(t), 0}. The first constraint denotes the train motion equation, the second constraint denotes the values of boundary speed, the third constraint denotes that the travel distance should be equal to the length of inter-station and the fourth inequality normalizes the unit force applied on the train. Remark 1.1 In model (1.1), the state variable is time t. Note that we can also take position s as the state variable, then the energy-efficient speed control model can be rewritten as follows: ⎧  S ⎪ ⎪ min C(u, v) = u + (s)ds ⎪ ⎪ ⎪ 0 ⎪ ⎨ s.t. v  (s) = u(s) − r (v(s)) (1.2) v(0) = v(S) = 0 ⎪ ⎪  ⎪ S ⎪ ⎪ ⎪ 0 1/v(s)ds = T ⎩ |u(s)| ≤ 1, where the distance constraint is replaced with the time constraint. Over the past decades, these two basic models (1.1) and (1.2) have been extended to consider speed limits, traction efficiencies, variable gradients, variable curvatures, steep slopes, and other practical constraints and a large number of solution algorithms on these basic models and their extensions have been given, which can be grouped into analytical algorithms and numerical algorithms [43]. Generally speaking, analytical algorithms can obtain the optimal speed profiles, but only for some simplified models; while numerical algorithms are able to deal with complex models, they can only obtain sub-optimal solutions.

1.1.1 Analytical Algorithms The energy-efficient speed control problem was originally formulated as a continuous optimal control model. Asnis et al. [2] assumed that the unit force was a continuous control variable with uniform bounds, and took the Pontryagin maximum principle to find the necessary conditions for the optimal speed profile. To seek strict mathematical foundations, Howlett [17, 18] formulated the optimal speed control problem in an appropriate function space and concluded that the optimal speed profile exists and satisfies a Pontryagin-type criterion. Furthermore, Howlett [19] formulated the energy-efficient speed control problem as a finite dimensional constrained optimization model and took Pontryagin maximum principle to determine the optimal solution

1.1 Energy-Efficient Speed Control

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for the model (1.1), which produced the first theoretical confirmation that an optimal speed profile should use a maximum acceleration-cruising-coasting-maximum braking phase sequence. For taking the theory into practice, variable gradients, traction efficiencies, speed limits, and steep slopes were gradually considered in literature [1, 23, 30, 34]. For example, Liu and Golovitcher [34] gave an analytical solution to the energy-efficient speed control problem with variable gradients. Khmelnitsky [30] considered the operations scenario with variable gradients, speed-dependent traction efficiencies, and arbitrary speed limits. Howlett et al. [23] provided an analytical solution method with more than one steep slopes, which first divided the route into small segments such that each segment contains one steep slope, then solved the speed profile for each segment by using an optimal local principle. Albrecht et al. [1] proved that the optimal switching points are uniquely determined at each steep segment, and the global optimal speed profile is also unique. For the discrete optimal control model, the Scheduling and Control Group of the University of South Australia led by Howlett P.G. made outstanding contributions. For example, Howlett et al. [20] outlined the theoretical basis with a discrete control model for Metromiser system, which has been successfully applied to urban rail transits in Australia, Melbourne, Toronto, etc. Cheng and Howlett [4, 5] studied the energy-efficient speed control problem with discrete inputs for a track without varying gradients and speed limits. Pudney and Howlett [39] studied the problem with speed limits and proved that that speed limits were below the desired cruising speed on intervals where the speed must be held at the limits. Howlett and Cheng [21] solved the problem with continuously varying gradients. Furthermore, Cheng [6] tackled the problem with nonzero track gradients and speed limits, which was difficult to find an analytic solution because it was no longer possible to precisely follow arbitrary smooth speed limits. In 2000, Howlett [22] considered the problem with a generalized motion equation and concluded that the optimal strategy for discrete control could be used to approximate as closely as we want the optimal strategy obtained using continuous control. This conclusion means that both both the continuous control and discrete control models can apply on trains with continuous or discrete traction and braking forces.

1.1.2 Numerical Algorithms Since Howlett [19] has proved that the optimal speed profile consists of accelerating with maximum traction force, cruising, coasting, and decelerating with maximum braking force, some researchers proposed numerical algorithms to solve the optimal switching times or positions among different phases, which essentially transfers the optimal control problem to nonlinear optimization problem. In 1975, Hoang et al. [16] first studied the energy-efficient speed control problem using a numerical algorithm. Because of the extremely low calculation speed, the numerical algorithm was paid less attention in the following two decades. In recent years, with the development of computer performance and calculation theory, more and more studies

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1 Literature Overview

numerical algorithms are used to solve the energy-efficient speed control problem, such as genetic algorithm, gradient algorithm, dynamic programming and so on. For example, Han et al. [15] used a genetic algorithm to generate the speed profile for automatic operation trains along a track with non-constant gradient, curvature, and speed limits. Duarte and Sotomayor [13] proposed a gradient-restoration algorithm to determine the optimal speed trajectory together with the minimum energy consumption by a train for a round trip. Ke and Chen [27] designed a heuristic searching algorithm to solve the speed code and block layout of each signaling block between successive stations for mass rapid transit systems. Furthermore, Ke et al. [28] designed an ant colony optimization algorithm to find the energy-efficient speed profiles at inter-stations, which achieve a significant improvement to reduce the computational burden. In 2012, Ke et al. [29] presented a max-min ant system algorithm to optimize train speed profiles. The computation time reached a reasonable level, which made it possible to achieve online optimization. Dominguez et al. [9] designed a computer-aided procedure for subway automatic train operation (ATO) systems to select the optimal speed profile with minimum traction energy consumption. Dominguez et al. [10] considered the regenerative energy storage in substations and designed a genetic algorithm to find the optimal speed profile for minimizing energy consumption. The usage of onboard energy storage devices was considered by Dominguez et al. [11]. In 2014, Dominguez et al. [12] developed a multi-objective particle swarm optimization algorithm to obtain the consumption/time Pareto front based on the simulation of a train with a real ATO system. Lu et al. [35] proposed a position-based speed profile optimization model under the constraints of timetable, traction equipment characteristics, speed limits and gradients, and respectively, applied ant colony optimization algorithm, genetic algorithm, and dynamic programming algorithm to search the optimal speed profile. Su et al. [42] considered both the high-level speed profile and the low-level controller of ATO systems and designed a numerical algorithm to solve the energy-efficient speed control problem with a given trip time. In real subway systems, increasing coasting time is an efficient means of energy saving. Different methods to determine the optimal coasting points have been studied in the literature [3, 7, 44]. For example, Chang and Sim [3] applied the genetic algorithm to generate an optimal coasting control with a predetermined number of coasting points. Wong and Ho [44] presented the application of heuristic search methods to find the optimal coasting points for an inter-station with specified travel time. Chuang [7] proposed an artificial neural network to determine the optimal coasting speed of the train set to achieve the minimization of energy consumption and travel time.

1.2 Energy-Efficient Timetable Optimization Subway regenerative energy optimization is an emerging energy-efficient management approach, approach that optimizes train timetable such that the regenera-

1.2 Energy-Efficient Timetable Optimization

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tive energy from braking trains can be maximally utilized by accelerating trains. González-Gil et al. [25, 26] concluded that the studies focusing on improving the regenerative energy utilization by timetable optimization method have attracted more attention in recent years. In the first stage, researchers maximize the overlapping time among accelerating actions and braking actions of trains located in the same substations. In the second stage, researchers maximize the regenerative energy usage during the overlapping time.

1.2.1 Overlapping Time Maximization In 2007, Ramos et al. [40] presented a particular train timetabling model during offpeak hours, which aimed at maximizing the overlapping time between accelerating actions and decelerating actions of trains located in the same substations so that the regenerative energy from braking trains could be utilized by accelerating trains as much as possible. Peña-Alcaraz et al. [38] proposed to synchronize the operations of trains to reduce energy consumption from substations by maximizing the utilization of regenerative energy. The authors developed a power flow model to calculate the regeneration energy utilization factor at each unit of overlapping time and measure the regenerative energy utilization as the product of overlapping time and utilization factor. In 2013, Yang et al. [45] proposed a cooperative scheduling model to coordinate the accelerating actions and decelerating actions of adjacent trains, such that the regenerative energy from the braking trains can be immediately used by the accelerating trains. A simulation study based on the real data from the Beijing Metro Yizhuang Line showed that the cooperative scheduling model could significantly improve the overlapping time around 22%. Furthermore, Zhao et al. [50] proposed a two-objective timetabling model to maximize the overlapping time and minimize passenger travel time. They designed a simulated annealing algorithm to solve the model, and the results of numerical experiments showed that the overlapping time could be increased by 21.9%.

1.2.2 Regenerative Energy Maximization Nasri et al. [37] proposed a genetic algorithm to maximize the utilization of regenerative energy from braking trains. The effects of headway time among trains and dwell time at stations on the amount of regenerative energy utilization were studied and tested for the Tehran Subway system. Kim et al. [31] proposed a multi-objective mixed integer programming to minimize the peak energy and maximize regenerative energy utilization. They coordinated the train departure times at the terminal station but kept the planned travel time at inter-stations unchanged. Fournier et al. [14] developed an optimization model to maximize regenerative energy utilization by modifying dwell time at stations. A hybrid genetic/linear programming algorithm

6

1 Literature Overview

was implemented to solve this problem. Yang et al. [48] developed a timetabling approach based on real speed profiles to coordinate the arrivals and departures of trains located in the same substations, such that the regenerative energy can be more effectively absorbed by accelerating trains. In 2014, Yang et al. [46] proposed a timetable optimization model to coordinate up trains and down trains departing from and arriving in the same stations to improve the regenerative energy utilization and reduce the passenger waiting time. They conducted numerical examples based on the operation data from the Beijing Metro Yizhuang Line, and the results illustrated that the proposed model could save energy by 8.86% and reduce passenger waiting time by 3.22% in comparison with the current timetable. Zhao [51] developed a nonlinear integer programming model to maximize the regenerative energy utilization, and the results showed that the utilization could be improved by between 4% and 12% depending on the energy conversion rate.

1.3 Extensions Energy-efficient speed control approach focuses on optimizing the train’s speed profiles at inter-stations, which ignores the transmission of regenerative energy among trains. As a result, the obtained speed profile is optimal for a single train but may not be optimal for multiple trains. The energy-efficient timetable optimization approach synchronizes the operations of multiple trains to maximize the regenerative energy utilization but ignores the adjustment on speed profile. Therefore, in recent years, a few researchers studied the integrated optimization method jointly, optimizing the timetable and speed profile to minimize the net energy consumption, i.e., the difference between traction energy consumption and regenerative energy utilization, which could achieve better energy performance than separately optimize timetable or speed control. Ding et al. [8] formulated the energy-efficient speed control problem as a twolevel optimization model and designed a genetic algorithm to search for the optimal solution. The first level determined the appropriate coasting points at inter-stations, and the second level distributed the travel time among inter-stations to minimize the traction energy consumption. Su et al. [41] considered the timetable optimization and speed control for one train within one cycle operation. The objective was to minimize traction energy consumption. In order to achieve better performance on energy saving, Li and Lo [32] proposed an integrated optimization model and designed a genetic algorithm to search the timetable and speed profile with the minimum net energy consumption. They made comparisons among timetable optimization approach [45], speed control approach [41], and the proposed integrated optimization approach on net energy consumption. The results showed that when the headway is 90 s, the integrated optimization approach can reduce net energy consumption by 21.17% compared to the timetable optimization approach, and 6.35% compared to the speed control approach. All the results are based on the real-world operation data from Bejing Subway Yizhuang Line. Furthermore, Li and Lo [33] developed

1.3 Extensions

7

a dynamic integrated optimization approach with adaptive cycle time based on passenger demand. The results showed that the dynamic approach could reduce the net energy consumption by 7.86% compared with the integrated optimization approach [32]. Yang et al. [47] developed a multi-objective integrated optimization method to reduce net energy consumption and total travel time. They determined the timetable and speed profile by finding the optimal dwell time at stations and the maximum speed at inter-stations.

References 1. Albrecht AR, Howlett PG, Pudney PJ, Vu X (2013), Energy-efficient train control: From local convexity to global optimization and uniqueness, Automatica, 49(10): 3072–3078. 2. Asnis IA, Dmitruk AV, Osmolovskii NP (1985), Solution of the problem of the energetically optimal control of the motion of a train by the maximum principle, USSR Computational Mathematics and Mathematical Physics, 25(6): 37–44. 3. Chang C, Sim S (1997), Optimising train movements through coast control using genetic algorithms, IEE Proceedings-Electric Power Applications, 144(1): 65–73. 4. Cheng J, Howlett PG (1992), Application of critical velocities to the minimisation in the control of trains, Automatica, 28(1): 165–169. 5. Cheng J, Howlett PG (1993), A note on the calculation of optimal strategies for the minimization of fuel consumption in the control of trains, IEEE Transactions on Automatic Control, 38(11): 1730–1734. 6. Cheng J (1997), Analysis of Optimal Driving Strategies for Train Control Systems, University of South Australia, Adelaide, Australia: Ph.D.thesis. 7. Chuang HJ, Chen CS, Lin CH, Hsieh CH, Ho CY (2008), Design of optimal coasting speed for saving social cost in mass rapid transit systems, Proceedings of the Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, Nanjing, China, 2833–2839. 8. Ding Y, Liu H, Bai Y, Zhou F (2011), A two-level optimization model and algorithm for energyefficient urban train operation, Journal of Transportation Systems Engineering and Information Technology, 11(1): 96–101. 9. Domínguez M, Fernández-Cardador A, Cucala AP, Lukaszewicz P (2011), Optimal design of metro automatic train operation speed profiles for reducing energy consumption, Journal of Rail and Rapid Transit, 225(5): 463–474. 10. Domínguez M, Fernández-Cardador A, Cucala AP, Pecharromán RR (2012), Energy savings in metropolitan railway substations through regenerative energy recovery and optimal design of ATO speed profiles, IEEE Transactions on Automation Science and Engineering, 9(3): 496– 504. 11. Domínguez M, Fernández-Cardador A, Cucala AP, Blanquer J (2014), Efficient design of Automatic Train Operation speed profiles with on board energy storage devices, Computers in Railways XII, Rome, Italy, 509–520. 12. Domínguez M, Fernández-Cardador A, Cucala AP, Gonsalves T, Fernández A (2014), Multi objective particle swarm optimization algorithm for the design of efficient ATO speed profiles in metro lines, Engineering Applications of Artificial Intelligence, 29: 43–53. 13. Duarte MA, Sotomayor PX (1999), Minimum energy trajectories for subway systems, Optimal Control Applications and Methods, 20(6): 283–296. 14. Fournier D, Mulard D, Fages F (2012), Energy optimization of metro timetables: A hybrid approach, Proceedings of the 18th International Conference on Principles and Practice of Constraint Programming, Lecture Notes in Computer Science, Quebec City, Canada, 7–12.

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1 Literature Overview

15. Han S, Byen Y, Baek J, An T, Lee S, Park H (1999), An optimal automatic train operation (ATO) control using genetic algorithms (GA), TENCON 99. Proceedings of the IEEE Region 10 Conference, Cheju Island, South Korea, 360–362. 16. Hoang H, Polis M, Haurie A (1975), Reducing energy consumption through trajectory optimization for a metro network, IEEE Transactions on Automatic Control, 20(5): 590–595. 17. Howlett PG (1987a), Existence of an optimal strategy for the control of a train, South Australian Institute of Technology, School of Mathematics Report. 18. Howlett PG (1987b), Necessary conditions on an optimal strategy for the control of a train, South Australian Institute of Technology, School of Mathematics Report. 19. Howlett PG (1990), An optimal strategy for the control of a train, Journal of the Australian Mathematical Society Series B, 31: 454–471. 20. Howlett PG, Milroy IP, Pudney PJ (1994), Energy-efficient train control, Control Engineering Practice, 2(2): 193–200. 21. Howlett PG, Cheng J (1997), Optimal driving strategies for a train on a track with continuously varying gradient, Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 38(3): 388–410. 22. Howlett PG (2000), The optimal control of a train, Annals of Operations Research, 98: 65–87. 23. Howlett PG, Pudney PJ, Vu X (2009), Local energy minimization in optimal train control, Automatica, 45(11): 2692–2698. 24. Ishikawa I (1968), Application of optimization theory for bounded state variable problems to the operation of trains, Bulletino of JSME, 11(47): 857–865. 25. González-Gil A, Palacin R, Batty P (2013), Sustainable urban rail systems: Strategies and technologies for optimal management of regenerative braking energy, Energy Conversion and Management, 75: 374–388. 26. González-Gil A, Palacin R, Batty P, Powell JP (2014), A systems approach to reduce urban rail energy consumption, Energy Conversion and Management, 80: 509–524. 27. Ke B, Chen N (2005), Signalling blocklayout and strategy of train operation for saving energy in mass rapid transit systems, IEE Proceedings-Electric Power Application, 152(2): 129–140. 28. Ke B, Chen M, Lin C (2009), Block-layout design using max-min ant system for saving energy on mass rapid transit systems, IEEE Transactions on Intelligent Transportation Systems, 10(2): 226–235. 29. Ke B, Lin C, Yang C (2012), Optimisation of train energy-efficient operation for mass rapid transit systems, IET Intelligent Transport Systems, 6(1): 58–66. 30. Khmelnitsky E (2000), On an optimal control problem of train operation, IEEE Transactions on Automatic Control, 45(7): 1257–1266. 31. Kim KM, Kim KT, Han MS (2011), A model and approaches for synchronized energy saving in timetabling, Proceedings of the 9th World congress on railway research-WCRR, Lille, France. 32. Li X, Lo HK (2014a), An energy-effcient scheduling and speed control approach for metro rail operations, Transportation Research Part B: Methodological, 64: 73–89. 33. Li X, Lo HK (2014b), Energy minimization in dynamic train scheduling and control for metro rail operations, Transportation Research Part B: Methodological, 70: 269–284. 34. Liu R, Golovitcher I (2003), Energy-efficient operation of rail vehicles, Transportation Research Part A: Policy and Practice, 37(10): 917–932. 35. Lu S, Hillmansen S, Ho TK, Roberts C (2013), Single-train trajectory optimization, IEEE Transactions on Intelligent Transportation Systems, 14(2): 743–750. 36. Milroy IP (1980), Aspects of Automatic Train Control. Leicestershire, U.K.: Loughborough University. 37. Nasri A, Moghadam MF, Mokhtari H (2010), Timetable optimization for maximum usage of regenerative energy of braking in electrical railway systems, Proceedings of International Symposium on Power Electronics, Electrical Drives, Automation and Motion, 1218–1221. 38. Peña-Alcaraz M, Fernández A, Cucala AP, Ramos A, Pecharromán RR (2012), Optimal underground timetable design based on power flow for maximizing the use of regenerative-braking energy, Journal of Rail and Rapid Transit, 226(4): 397–408.

References

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39. Pudney PJ, Howlett PG (1994), Optimal driving strategy for a train journey with speed limits, Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 36(1): 38–49. 40. Ramos A, Pena M, Fernández-Cardador A, Cucala AP (2007), Mathematical programming approach to underground timetabling problem for maximizing time synchronization, Proceedings of International Conference on Industrial Engineering and Industrial Management, Madrid, Spain, 88–95. 41. Su S, Li X, Tang T, Gao ZY (2013), A subway train timetable optimization approach based on energy-efficient operation strategy, IEEE Transactions on Intelligent Transportation Systems, 14(2): 883–893. 42. Su S, Tang T, Chen L, Liu B (2015), Energy-efficient train control in urban rail transit systems, Journal of Rail and Rapid Transit, 229(4): 446–454. 43. Wang Y, Ning B, Cao F, Schutter BD, Van den Boom TJJ (2011), A survey on optimal trajectory planning for train operations, Proceedings of IEEE International Conference on Service Operations, Logistics, and Informatics, Beijing, China, pp. 589–594. 44. Wong KK, Ho TK (2004), Coast control for mass rapid transit railways with searching methods, IEE Proceedings-Electric Power Applications, 151(3): 365–376. 45. Yang X, Li X, Gao ZY, Wang H, Tang T (2013), A cooperative scheduling model for timetable optimization in subway systems, IEEE Transactions on Intelligent Transportation Systems, 14(1): 438–447. 46. Yang X, Ning B, Li X, Tang T (2014), A two-objective timetable optimization model in subway systems, IEEE Transactions on Intelligent Transportation Systems, 15(5): 1913–1921. 47. Yang X, Li X, Ning B, Tang T (2015), An optimization method for train scheduling with minimum energy consumption and travel Time in metro rail systems, Transportmetrica B: Transport Dynamics, 3(2): 79–98. 48. Yang X, Chen A, Li X, Ning B, Tang T (2015), An energy-efficient scheduling approach to improve the utilization of regenerative energy for metro rail systems, Transportation Research Part C: Emerging Technologies, 57: 13–29. 49. Yang X, Li X, Ning B, Tang T (2016), A survey on energy-efficient train operation for urban rail transit, IEEE Transactions on Intelligent Transportation Systems, 17(1): 2–13, 2016. 50. Zhao L, Li K, Su S (2013), A multi-objective timetable optimization model for subway systems, Proceedings of the 2013 International Conference on Electrical and Information Technologies for Rail Transportation, Changchun, China, 557–565. 51. Zhao L, Li K, Ye J, Xu X (2015), Optimizing the train timetable for a subway system, Journal of Rail and Rapid Transit, 229(8): 852–862.

Chapter 2

Energy-Efficient Speed Control

Energy-efficient speed control is an effective method to reduce traction energy consumption, which has attracted broad concerns from both researchers and practitioners in recent years. With given travel time at inter-station, there are generally multiple feasible speed profiles satisfying the time constraint, among which the energy-efficient speed profile is the one that minimizes the traction energy consumption. Over the past decades, there have been many studies on energy-efficient speed control models and algorithms [1–11]. In this chapter, we will introduce a basic model and algorithm given by Su et al. [12], focusing on the following aspects: (1) How to measure the traction energy consumption? (2) How to formulate the energy-efficient speed control model? (3) How to solve the energy-efficient speed profile?

2.1 Problem Description The energy-efficient speed control model can be formulated based on either position variable or time variable. In this chapter, we will formulate the model with position variable. Suppose that there is a train moving at inter-station with a given travel time. As shown in Fig. 2.1, there are generally multiple feasible speed profiles satisfying the travel time constraint, among which we need to solve the one that minimizes the traction energy consumption [12]. For formulating an intuitive, inspirational and illustrative model, we make the following assumptions on tracks and vehicle conditions: A21) The track is flat with no gradients and curvatures along the interstation; A22) The traction force, braking force, and resistance force are all constant.

© Springer Nature Singapore Pte Ltd. 2020 X. Li and X. Yang, Subway Energy-Efficient Management, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-7785-7_2

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2 Energy-Efficient Speed Control

Speed Speed profile 2 Speed profile 1

Speed profile 3

Position Fig. 2.1 An illustration on energy-efficient speed control problem

For the train energy-efficient speed control studies with position-dependent gradients, curvature and variable traction force, braking force and resistance force, the interested readers may refer to the literature [1, 9–11].

2.2 Model Formulation In this section, we formulate the train energy-efficient speed control model, which minimizes the traction energy consumption under the motion equation, time constraint, and speed limits. First, we prove the optimality conditions for the operation scenario without speed limits, such that the optimal speed profile can be obtained by solving a set of quadratic equations. Furthermore, we formulate a nonlinear optimization model for solving the optimal speed profile under the operation scenario with piece-wise speed limits.

2.2.1 Notations To better understand the train energy-efficient speed control models and algorithms, we first introduce the parameters and decision variables that will be used in this chapter. Note that the meaning of each symbol is used only in this chapter.

2.2 Model Formulation

2.2.1.1

Parameters s n v¯n tnmax tnmin pn kt (s) kb (s) F B R T Cn

2.2.1.2

13

Position variable, s ∈ [0, S] segment index, n = 1, 2, · · · , N speed limit at segment n the maximum allowable travel time at segment n the minimum allowable travel time at segment n starting position for the nth segment relative tractive force relative braking force the maximum unit traction force the maximum unit braking force unit resistance travel time at inter-station traction energy consumption at segment n

Decision Variables tn xn1 xn2 xn3 vn0 vn1 vn2 vn3 vn4

travel time at segment n cruising position at segment n coasting position at segment n braking position at segment n starting speed at segment n cruising speed at position xn1 coasting speed at position xn2 braking speed at position xn3 ending speed at segment n

Remark 2.1 Here xn1 is the switching position from the accelerating phase to the cruising phase, xn2 is the switching position from the cruising phase to the coasting phase, and xn3 is the switching position from the coasting phase to the braking phase. The values vn1 , vn2 , vn3 are the speeds at positions xn1 , xn2 , xn3 , respectively. For subway operations management, an inter-station generally has piece-wise speed limits for trains arising from the signaling and track conditions that make higher speeds hazardous [13, 14]. Here we divide each inter-station into N segments such that each segment has a constant speed limit v¯n (See Fig. 2.2). If we use pn to denote the starting position for the nth segment, then the nth segment can be expressed as [ pn , pn+1 ] with p N +1 = S. In what follows, we will first consider the energy-efficient speed control at segment without or with the constant speed limit, and then extend the results to the whole inter-station with piece-wise speed limits.

14

2 Energy-Efficient Speed Control Speed limit Segment 2

Segment 1

Segment N

0

S

Position

Fig. 2.2 An illustration on segment division at inter-station

2.2.2 Speed Control Model without Speed Limits First, we consider the speed control problem at segment n without any speed limit. If we use kt (s) to denote the relative traction force at position s with s ∈ [ pn , pn+1 ], the absolute traction force at position s is kt (s) × F, and the traction energy consumption per unit mass at the nth segment should be 

pn+1

Cn =

(kt (s) × F)ds,

(2.1)

pn

which will be taken as the objective function for the speed control model. Now we consider the constraint conditions. Denote kb (s) as the relative braking force at position s. According to the Work-Energy Theorem, if we handle train as a particle, the motion of the train can be described by the following differential equation kt (s) × F − kb (s) × B − R dvn (s) = , s ∈ [ pn , pn+1 ], ds vn (s)

(2.2)

where vn (s) is the speed profile with boundary conditions vn ( pn ) = vn0 , vn ( pn+1 ) = vn4 .

(2.3)

The trip time is the integral of 1/vn (s) over the interval [ pn , pn+1 ], which should be equal to the predetermined value tn , that is, 

pn+1

1/vn (s)ds = tn .

(2.4)

pn

Based on the above analyses, we have the following energy-efficient speed control model to minimize the traction energy consumption [12], that is,

2.2 Model Formulation

15

⎧  ⎪ ⎪ ⎨ min

pn+1

(kt (s) × F)ds

pn

s.t. Constraints (2.2), (2.3), (2.4) ⎪ ⎪ ⎩ kt (s), kb (s) ∈ [0, 1],

(2.5)

where the last constraint denotes the range of relative tractive force kt and relative braking force kb . Based on the Pontryagin maximum principle, Howlett [5] has proved that the traction energy consumption reaches the minimum with respect to the control variables kt and kb as follows: • • • •

Maximum accelerating phase: kt = 1, kb = 0; Cruising phase: kt ∈ [0, 1], kb = 0; Coasting phase: kt = 0, kb = 0; Maximum braking phase: kt = 0, kb = 1.

The results tell us that the optimal speed profile consists of maximum accelerating, cruising, coasting, and maximum braking phases, and is uniquely determined with the optimal switching positions/speeds among these phases. Remark 2.2 Thomas [15] gave a detailed explanation of the optimal phases: (i) The slower a train accelerates or brakes, the more time it needs to come to a standstill. In order to obtain the same trip time with a lower accelerating or braking rate, the train should accelerate to a higher speed, which consumes more energy. Therefore, the maximum accelerating and the maximum braking must be the most energy-efficient; (ii) Under most conditions, running resistance is positive such that partial acceleration is needed for keeping constant speed; (iii) During coasting without traction force and braking force, the train rolls forward and needs no energy. Thus, the earlier coasting can start, the more energy can be saved. Now we consider how to solve the optimal switching strategy, i.e., switching positions or switching speeds, among maximum accelerating, cruising, coasting and maximum braking phases. Theorem 1 For each segment with predetermined travel time tn , denote a feasible switching strategy as cruising position sn1 , coasting position sn2 and braking position sn3 . Then sn1 is increasing with respect to sn3 . Proof Suppose that there are two feasible speed profiles v(s) and w(s) with braking positions sn3 and wn3 , respectively. Without loss of generality, we assume sn3 ≤ wn3 . Let sn1 and wn1 be the corresponding cruising positions. If sn1 > wn1 , we have v(s) > w(s) with s ∈ [sn1 , sn3 ] and v(s) ≥ w(s) with s ∈ [ pn , sn1 ] ∪ [sn3 , pn+1 ] such that  tn =

pn+1 pn

 1/v(s)ds
v¯n , we have vn1 = v¯n . Proof If wn1 > v¯n , there should be a cruising phase at vn (s) for satisfying the travel time constraint. Since the traction energy consumption (2.6) is decreasing with respect to braking position sn3 and a higher cruising speed vn1 means a higher braking position sn3 under the travel time constraint, the minimum energy consumption will reach the maximum cruising speed with vn1 = v¯n . The proof is completed. Now we consider the solution for energy-efficient speed control model with speed limit v¯n . First, we obtain the optimal speed profile without speed limit by solving Eqs. (2.7) and (2.8), denoted as wn (s). If wn1 ≤ v¯n , wn (s) is the optimal solution with speed limits. Otherwise, if wn1 > v¯n , it follows from Theorem 3 that vn1 = v¯n and 2 v¯ 2 − vn0 . (2.10) sn1 = n 2(F − R)

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Fig. 2.4 An illustration on the optimal speed profile with speed limits

In this case, the energy-efficient speed profile should consist of maximum accelerating, cruising, coasting, and maximum braking phases (See Fig. 2.4). It follows from the travel time constraint and travel distance constraint that the coasting position sn2 and braking speed vn3 should satisfy the following optimality conditions ⎧ 2 2 v 2 − vn4 v¯ 2 − vn3 ⎪ ⎨ sn2 + n + n3 = pn+1 − pn 2R 2(B + R) s vn3 − vn4 v ¯ − v − s − vn3 v ¯ ⎪ n0 n2 n1 n ⎩ n + + = tn . + F−R v¯n R B+R

(2.11)

Furthermore, the optimal braking position sn3 can be determined as follows: sn3 = pn+1 − pn −

2 2 − vn4 vn3 . 2(B + R)

(2.12)

For the nth segment with speed limit v¯n , if the travel time is tn , starting speed is vn0 and ending speed is vn4 , we denote the optimal switching positions as sn1 , sn2 , sn3 , then the minimum traction energy consumption is Cn (tn , vn0 , vn4 ) = F × (sn1 − pn ) + R × (sn2 − sn1 ). The energy-efficient speed control problem at inter-station can be formulated as follows: ⎧ N ⎪ ⎪ ⎪ min Cn (tn , vn0 , vn4 ) ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ N ⎪ ⎨ s.t. tn = T (2.13) ⎪ ⎪ n=1 ⎪ ⎪ ⎪ 0 ≤ vn0 ≤ min{v¯n , v¯n−1 }, n = 1, 2, · · · , N ⎪ ⎪ ⎪ ⎪ 0 ≤ vn4 ≤ min{v¯n , v¯n+1 }, n = 1, 2, · · · , N ⎪ ⎩ tnmin (vn0 , vn4 ) ≤ tn ≤ tnmax (vn0 , vn4 ), n = 1, 2, · · · , N

2.2 Model Formulation

19

where v¯0 = v¯ N +1 = 0, which minimizes the total traction energy consumption by optimizing the travel time, starting speed, and ending speed at segments. In the last constraints, the minimum allowable travel time tnmin , and the maximum allowable travel time tnmax depend on both parameters v¯n , pn , pn+1 and decision variables vn0 , vn4 .

2.3 Genetic Algorithm Since the traction energy consumption Cn , the minimum travel time tnmin , and the maximum travel time tnmax in the model (2.13) are implicit, the gradient-based algorithms may not work such that we take the genetic algorithm to search the optimal solution, which has been successfully used to train energy-efficient speed control problems in the past literature [2–4]. A chromosome is defined as a feasible solution X = {(tn , vn0 , vn4 ), n = 1, 2, · · · , N }. Denote pop-si ze as the number of chromosomes at each generation. Without loss of generality, we set it as an even number. Generate pop-si ze chromosomes as the initialized population by running the following process pop-si ze times: (i) Randomly generate vn0 ∈ [0, min{v¯n , v¯n−1 }] and vn4 ∈ [0, min{v¯n , v¯n+1 }] for n = 1, 2, · · · , N ; (ii) Calculate the minimum allowable travel time tnmin (vn0 , vn4 ) and the maximum allowable travel time tnmax (vn0 , vn4 ) for n = 1, 2, · · · , N ; (iii) Randomly generate N real numbers tn satisfying the minimum, maximum and total travel time constraints. For each generation, a proportion of the current chromosomes are selected to breed the next generation. First, each chromosome is assigned a fitness so that its chance of being selected is related to this value. Here fitness is defined as the traction energy consumption. Second, reorder these pop-si ze chromosomes from good to bad such that X 1 has the lowest fitness, and X pop-si ze has the highest fitness. With given δ ∈ (0, 1), define a rank-based evaluation function E(X i ) = δ × (1 − δ)i−1 , i = 1, 2, · · · , pop-si ze. Finally, spin the roulette wheel pop-si ze times and select a chromosome each time. That is, generate a random number r ∈ [0, 1] and select the ith chromosome if E(X i−1 ) ≤ r ≤ E(X i ). Crossover is an important operation for generating new chromosomes. Divide the population into pop-si ze/2 pairs of parents (X 1 , X 2 ), (X 3 , X 4 ), · · · , (X pop-si ze−1 , X pop-si ze ). Here we illustrate the crossover operation by parents (X 1 , X 2 ). First, define a parameter pc to denote the crossover probability. Generate a random number r ∈ [0, 1], and perform the crossover operation if r < pc . Enumerate λ from 0.1 to

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2 Energy-Efficient Speed Control

0.9 with step 0.1 and generate a series of children Y (λ) = λ × X 1 + (1 − λ) × X 2 . Then we select the best two feasible chromosomes from the parents and children to replace X 1 and X 2 . Mutation is another important operation for updating the population. We perform the mutation operation for all chromosomes, and illustrate the mutation process by parent X . First, define a parameter pm as the mutation probability. Generate a random number r ∈ [0, 1], and perform the mutation operation if r < pm . Randomly generate two indices n 1 and n 2 with 1 ≤ n 1 , n 2 ≤ N , and n 1 = n 2 . Update the genes (tn , vn0 , vn4 ) at these two positions to generate a feasible child chromosome Y . Select a better chromosome from X and Y to the next generation. Following evaluation, selection, crossover and mutation operations, a new population of chromosomes is generated. GA will terminate after a given number of generations or the terminal condition is satisfied, e.g., the absolute deviation between the best-found fitness values of adjacent generations is less than a predetermined real number . We summarize the general procedure for the genetic algorithm as follows: Step 1 Initialize the terminal condition  = 0.05, population size pop-si ze and generation number Generation. Step 2 Randomly initialize pop-si ze chromosomes and calculate the corresponding fitness values. Step 3 Update the chromosomes by using selection, crossover, and mutation operations. Step 4 Repeat Step 3 until reaching the maximum generation number Generation or the terminal condition is satisfied. Step 5 Return the best-found chromosome and the fitness value.

2.4 Numerical Examples We present numerical examples based on the data from Beijing Subway Yizhuang Line in China. The line is opened on December 30, 2010, with 23.23 km long, contains 14 stations (Songjiazhuang, Xiaocun, Xiaohongmen, Jiugong, Yizhuangqiao, Wenhuayuan, Wanyuan, Rongjing, Rongchang, Tongjinan, Jinghai, Ciqunan, Ciqu, Yizhuang) and 13 inter-stations. The infrastructure data are described in Table 2.1, where the first column denotes the serial number for inter-station, the second column denotes the length of inter-station, the third column denotes the travel time, and the last column denotes the piece-wise speed limits. Note that each inter-station is divided into three segments by the piece-wise speed limits such that each segment has a constant speed limit. Other parameters are defined as follows: train mass M = 287, 080 kg, maximum traction force per unit mass F = 0.8 m/s2 , maximum braking force per unit mass B = 0.4 m/s2 and running resistance per unit mass R = 0.02 m/s 2 . The simulation is performed on a personal computer with a processor speed of 2.6 GHz and memory size 2 GB.

2.4 Numerical Examples

21

Table 2.1 The infrastructure data on Beijing Subway Yizhang Line Length (m) Travel time (s) Speed limit (m/s) 1 2 3 4 5 6 7 8 9 10 11 12 13

1332 1286 2086 2265 2331 1354 1280 1544 992 1975 2369 1349 2610

117 112 162 171 175 115 111 127 94 153 176 115 201

10/0-128, 20/128-1147, 14/1147-1332 13/0-130, 20/130-1085, 13/1085-1286 13/0-129, 20/129-1897, 11/1897-2086 13/0-128, 20/128-2073, 13/2073-2265 13/0-120, 20/120-2143, 12/2143-2331 14/0-130, 20/130-1168, 13/1168-1354 14/0-130, 20/130-1087, 13/1087-1280 14/0-130, 20/130-1350, 13/1350-1544 13/0-130, 20/130-796, 13/796-992 14/0-123, 20/123-1781, 13/1781-1975 14/0-128, 20/128-2165, 13/2165-2369 14/0-130, 20/130-1151, 13/1151-1349 13/0-133, 20/133-2300, 10/2300-2610

We solve the energy-efficient speed profiles at all inter-stations. The results are shown in Table 2.2, where the third column denotes the switching speeds at the second segment and the fourth column denotes the traction energy consumption along the whole inter-station. Since the travel time is near the minimum allowable value, the

Table 2.2 The energy-efficient speed profiles at inter-stations Inter-Station Switching speeds (m/s) 1 2 3 4 5 6 7 8 9 10 11 12 13

Songjiazhuang-Xiaocun Xiaocun-Xiaohongmen Xiaohongmen-Jiugong Jiugong-Yizhuangqiao Yizhuangqiao-Wenhuayuan Wenhuayuan-Wanyuan Wanyuan-Rongjing Rongjing-Rongchang Rongchang-Tongjinan Tongjinan-Jinghai Jinghai-Ciqunan Ciqunan-Ciqu Ciqu-Yizhuang

17.63, 17.63, 16.72 17.59, 17.59, 16.74 18.40, 18.40, 16.73 18.69, 18.69, 16.78 18.69, 18.69, 16.71 17.81, 17.81, 16.86 17.68, 17.68, 16.81 18.05, 18.05, 16.91 16.87, 16.87, 16.30 18.49, 18.49, 16.88 18.79, 18.79, 16.76 17.78, 17.78, 16.85 18.68, 18.68, 16.59

Energy (kWh) 12.74 12.69 13.95 14.30 14.32 12.95 12.77 13.32 11.67 13.96 14.45 12.93 14.62

22

2 Energy-Efficient Speed Control

optimal speed profile follows maximum accelerating-cruising along with speed limit strategy in the first segment and cruising along with speed limit-maximum braking strategy in the third segment. For the second segment with the highest speed limit and the longest distance, the optimal speed profile starts with speed v¯1 and end with speed v¯3 . The values of cruising speed vn1 , coasting speed vn2 , and braking speed vn3 are illustrated in Table 2.2. Taking the first inter-station from Songjiazhuang to Xiaocun for example, there are three segments with speed limits v¯1 = 10 m/s at interval [0, 128] m, v¯2 = 20 m/s at interval [128, 1147] and v¯3 = 14 m/s at interval [1147, 1332]. The travel time for train to pass the inter-station is T = 117 s. It is solved that the optimal switching speeds at the second segment is v21 = 17.63 m/s, v22 = 17.63 m/s and v23 = 16.72 m/s, which means that the cruising phase is not used, i.e., the optimal speed profile follows maximum accelerating-coasting-maximum braking strategy. The amount of traction energy consumption along the first inter-station is 12.74 kWh.

References 1. Albrecht AR, Howlett PG, Pudney PJ, Vu X (2013), Energy-efficient train control: From local convexity to global optimization and uniqueness, Automatica, 49(10): 3072-3078. 2. Chang C, Sim S (1997), Optimising train movements through coast control using genetic algorithms, IEE Proceedings-Electric Power Applications, 144(1): 65-73. 3. Domínguez M, Fernández-Cardador A, Cucala AP, Pecharromán RR (2012), Energy savings in metropolitan railway substations through regenerative energy recovery and optimal design of ATO speed profiles, IEEE Transactions on Automation Science and Engineering, 9(3): 496504. 4. Han S, Byen Y, Baek J, An T, Lee S, Park H (1999), An optimal automatic train operation (ATO) control using genetic algorithms (GA), Proceedings of the IEEE Region 10 Conference, Cheju Island, South Korea, 360-362. 5. Howlett PG (1990), An optimal strategy for the control of a train, Journal of the Australian Mathematical Society Series B, 31: 454-471. 6. Howlett PG, Pudney PJ (1995), Energy-efficient train control, New York, Springer-Verlag. 7. Howlett PG (1996), Optimal strategies for the control of a train, Automatica, 32(4): 519-532 8. Howlett PG (2000), The optimal control of a train, Annals of Operations Research, 98: 65-87. 9. Howlett PG, Pudney PJ, Vu X (2009), Local energy minimization in optimal train control, Automatica, 45(11): 2692-2698. 10. Khmelnitsky E (2000), On an optimal control problem of train operation, IEEE Transactions on Automatic Control, 45(7): 1257-1266. 11. Liu R, Golovitcher I (2003), Energy-efficient operation of rail vehicles, Transportation Research Part A: Policy and Practice, 37(10): 917-932. 12. Su S, Li X, Tang T, Gao ZY (2013), A subway train timetable optimization approach based on energy-efficient operation strategy, IEEE Transactions on Intelligent Transportation Systems, 14(2): 883-893.

References

23

13. Su S, Tang T, Roberts C (2015), A cooperative train control model for energy saving, IEEE Transactions on Intelligent Transportation Systems, 16(2): 622-631. 14. Su S, Tang T, Chen L, Liu B (2015), Energy-efficient train control in urban rail transit systems, Journal of Rail and Rapid Transit, 229(4): 446-454. 15. Thomas A (2008), Railway timetable and traffic, Eurailpress.

Chapter 3

Timetabling with Overlapping Time Maximization

Compared with other transport modes, urban rail transit has the highest energy efficiency. However, it is still one of the most energy-intensive industries due to the operation characteristics of high capacity and high frequency. Traditional energyefficient operation studies mainly focus on speed control for reducing traction energy consumption. Recently, researchers paid more attention to timetabling problems for improving the regenerative energy absorption among adjacent trains. In this chapter, we will introduce a timetabling method proposed by Yang et al. [1] to maximize the overlapping time between accelerating actions and braking actions of adjacent trains, such that the regenerative energy from braking trains can be maximally absorbed by accelerating trains. We mainly focus on the following three questions related to energy-efficient timetable optimization: (1) What is the regenerative energy in urban rail transit systems? (2) How can we use regenerative energy efficiently? (3) How to improve the regenerative energy utilization via timetable optimization?

3.1 Problem Description In vehicles with electric traction motors, the energy put into accelerating them and into moving them uphill can be reconverted into electricity by using the motors as generators during the braking process, which is known as the regenerative braking [2]. Roughly speaking, the regenerative braking is an energy recovery mechanism, which slows a vehicle down by converting its kinetic energy into electricity [1]. Subway system is an electric passenger railway in an urban area, in which the regenerative braking technique has been widely applied, including the London underground, Madrid metro, New York subway, Beijing subway, and so on. Generally speaking, regenerative braking can recover about 40% of the traction energy [3]. © Springer Nature Singapore Pte Ltd. 2020 X. Li and X. Yang, Subway Energy-Efficient Management, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-7785-7_3

25

26

3 Timetabling with Overlapping Time Maximization Substation

Substation Overhead contact line Substation energy

Regenerative energy

Substation energy Pantograph

Pantograph Accelerating train

Braking train

Track

Track

Station

Fig. 3.1 Illustration on regenerative energy flow Regenerative energy

Fed back into the overhead contact line

Consumed by train itself

Used by auxiliary systems

Absorbed by on-board storage devices

Wasted by heating resistors

Absorbed by wayside storage devices

Used to accelerate other trains

Fig. 3.2 Distribution of regenerative energy

The regenerative energy is primarily used to supply power for auxiliary systems of the braking train itself, such as the ventilation, air-conditioning, and lighting. If the train is equipped with onboard storage devices such as super-capacitors, some energy will be reserved by them. Note that the energy storage devices are generally of limited capacity and high cost, most trains operated in real-world systems are not equipped with them. The surplus energy accounting for most of the regenerative energy is transmitted backward along the conversion chain and fed back into the overhead contact line, which can be immediately used to accelerate adjacent trains located in the same power supply interval. The process of regenerative energy flowing from braking train to accelerating train is shown in Fig. 3.1. However, if any feedback energy cannot be utilized timely, for example, there are more trains braking but no trains accelerating at that time, it will be wasted by heating resistors or absorbed by wayside storage devices (if any) installed on the overhead contact line. The energy absorbed by storage devices can be reused later. To summarize, the distribution of regenerative energy is described in Fig. 3.2. Timetable optimization aims at determining the optimal departure and arrival times for trains at stations such that the regenerative energy can be maximally used to accelerate other trains. As the regenerative energy actually reached in operation

3.1 Problem Description

27

is only a part of the traction energy arising from the conversion losses, most of the regenerative energy can be absorbed by the adjacent trains if the timetable is optimally determined and conducted in practice. This chapter proposes a timetable optimization model, which synchronizes the accelerating and braking times of successive trains to maximize the overlapping time, such that the regenerative energy from braking trains can be simultaneously used by accelerating trains located in the same power supply interval.

3.2 Model Formulation The model aims to improve the overlapping time between accelerating operations and braking operations such that the regenerative energy can be fully utilized. On account of the periodicity, a timetable being efficient to a fixed couple of trains will be efficient for all trains.

3.2.1 Notations For a better understanding of the timetable optimization model, we first introduce the parameters and variables that will be used in this chapter. Note that the meaning of each symbol is used only in this chapter.

3.2.2 Parameters n i Tn tan tbn [l T , u T ] [lh , u h ] [ln , u n ] λ( j, k) O

station index, n = 1, 2, . . . , N ; train index, i = 1, 2, . . . , I ; travel time at inter-station n; accelerating time at inter-station n; braking time at inter-station n; total travel time window; headway window; dwell time window at station n; if inter-station j and inter-station k belong to the same power supply interval, λ( j, k) = 1; otherwise, λ( j, k) = 0; overlapping time between accelerating phase and braking phase

28

3 Timetabling with Overlapping Time Maximization

3.2.3 Intermediate Variables dni+ ani−

ending time stamp of accelerating phase of train i after departure from station n, which is defined as dni +tan ; beginning time stamp of braking phase of train i before arrival at station n, which is defined as ani −tbn−1 .

3.2.4 Decision Variables h dni ani xn

headway; time stamp of train i departing from station n; time stamp of train i arriving at station n; dwell time required at station n

3.2.5 Model Assumptions The following assumptions are considered in the formulation according to the realworld operation characteristics of subway systems. (A3-1) The regenerative energy generated by braking trains can feed back into the overhead contact line, and can be used immediately by accelerating trains located in the same power supply interval. In Fig. 3.3, the regenerative energy generated by train 2 can be used by train 1. In addition, we ignore the transmission losses of electricity since the transmission distance is short between two successive trains. (A3-2) A power supply interval is an electrical generation, transmission, and distribution system. In general, there are two kinds of power supply modes in the urban

Power supply section m

Power supply section m+1

Upline track train 1

train 2

train 3

Downline track

train 4

Overhead contact line

Fig. 3.3 An illustration on the power supply structure

Ground

3.2 Model Formulation

29

rail transit system, including single side feeding mode and both sides feeding mode. In this chapter, we consider the single side feeding mode, such that the regenerative energy from the upline can not be transmitted to the downline. In Fig. 3.3, the regenerative energy generated by train 2 in upline can not be used by train 4 in the downline. (A3-3) All trains running in the same direction share a standard timetable except a headway, which means that they are assigned the same dwell time at each station, and the same trip time at each inter-station. The only difference is that the later train is operated at a particular time later than the former one. A3-4: At all inter-stations, the accelerating time and braking time of trains are known parameters, which can be calculated by using the energy-efficient speed control algorithm.

3.2.6 Cooperative Rules For subway systems, the passenger demand generally varies significantly between peak hours and off-peak hours. Generally speaking, the operations management department increases the transport capacity by shortening the headway at peak hours and decreases the transport capacity by lengthening the headway at off-peak hours. In the following, we first propose the cooperative timetabling rules at peak hours scenario and off-peak hours scenario, respectively. Peak Hours Scenario: The operations management department sets a lower headway for handling the higher passenger demand. The cooperative timetabling rule is shown in Fig. 3.4: phase (a), train i accelerates to depart from station n and train i + 1 decelerates to stop at station n. If they belong to the same power supply section, train i can absorb the regenerative energy from train i + 1; after a headway operation, train i decelerates to stop at station n + 1, and train i + 1 accelerates to depart from station n. Then train i + 1 can use the regenerative energy from train i, which is illustrated by phase (b). Off-Peak Hours Scenario: The operations management department sets a higher headway for handling the lower passenger demand. The cooperative timetabling rule is shown in Fig. 3.5: phase (a), train i accelerates to depart from station n + 1 and train i + 1 decelerates to stop at station n. If they belong to the same power supply i+1

i

Phase (a) Station n

i+1

Phase (b) Fig. 3.4 Cooperative timetabling scenario at peak hours

i

Station n+1

30

3 Timetabling with Overlapping Time Maximization i+1

i

Phase (a) Station n

i+1

Station n+1

i

Station n+2

Phase (b) Fig. 3.5 Cooperative timetabling scenario at off-peak hours

section, train i can use the regenerative energy from train i + 1; after a headway operation, train i decelerates to stop at station n + 2, and train i + 1 accelerates to depart from station n. In this case, train i + 1 can absorb the regenerative energy from train i, which is illustrated by phase (b). Remark 3.1 Some studies have shown that the utilization of regenerative energy is very low if the headway was longer than 10 min. The reason is that the distance between successive trains is too long so that they generally locate in different power supply sections.

3.2.7 Overlapping Time Now we formulate the overlapping time between the accelerating phase and the braking phase of successive trains belonging to the same power supply section. For simplicity, we denote a = {ani , 1 ≤ i ≤ I, 1 ≤ n ≤ N }, d = {dni , 1 ≤ i ≤ I, 1 ≤ n ≤ N − 1}, and denote the overlapping time as O(a, d). In what follows, we will formulate the overlapping time at peak hours and off-peak hours, respectively.

3.2.7.1

Peak Hours Scenario

Once a couple of accelerating and braking trains belong to the same power supply section, we should maximize the overlapping time to improve the regenerative energy utilization. For peak hours scenario, the operations of train i and train i + 1 are described as follows (See Fig. 3.6): i denotes the process that train i runs from station n to • The duration from dni to an+1 station n + 1, which can be divided into accelerating [dni , dni+ ], cruising+coasting i− i− i ] and braking [an+1 , an+1 ]. [dni+ , an+1 i+1 i+1 • The duration from dn−1 to an denotes the process that train i + 1 runs from i+1 denotes the process that station n − 1 to station n, the duration from dni+1 to an+1 train i + 1 runs from station n to station n + 1 and the duration from ani+1 to dni+1 denotes the dwell time of train i + 1 at station n.

3.2 Model Formulation

31 Zone (a)

Speed (m/s)

Zone (b)

Train i − +1

+

+1 −1

+1 + −1

+1 −

+1

+1

Train i+1

+1

+1 +

+1 − +1

+1 +1

Time (s)

Fig. 3.6 Overlapping time at peak hours

• Zone (a) and zone (b) corresponds to phase (a) and phase (b) of Fig. 3.4, respectively. In zone (a), train i accelerates to depart from station n while train i + 1 decelerates to stop at station n. In zone (b), train i decelerates to stop at station n + 1 while train i + 1 accelerates to depart from station n. • The red box denotes the accelerating phase while the green one denotes the braking phase and the gray bar denotes the overlapping between accelerating phase and the braking phase. For each 1 ≤ i ≤ I − 1 and 1 ≤ n ≤ N − 1, the overlapping time at zone (a) can be expressed as follows: ⎧ 0, ⎪ ⎪ ⎨ n t ∧ (d i+ − a (i+1)− ), 1 Oin (a, d) = an−1 n i+1 n i t ∧ (an − dn ), ⎪ ⎪ ⎩ b 0,

if dni+ < an(i+1)− if an(i+1)− ≤ dni+ ≤ ani+1 if dni ≤ ani+1 < dni+ if ani+1 < dni .

(3.1)

The overlapping time at zone (b) can be expressed as follows: ⎧ 0, if dn(i+1)+ < ani− ⎪ ⎪ ⎨ n i− i t ∧ (d (i+1)+ − an+1 ) if ani− ≤ dn(i+1)+ ≤ an+1 2 (a, d) an−1 n i Oin i i i+1 (i+1)+ ⎪ tb ∧ (an+1 − dn ), if dn+1 ≤ an+1 < dn ⎪ ⎩ i 0, if an+1 < dni+1 .

(3.2)

In general, the total overlapping time of all trains at all stations is O(a, d) =

N −1 I −1    i=1 n=1

 1 2 Oin (a, d)λ(n − 1, n) + Oin (a, d) .

(3.3)

32

3 Timetabling with Overlapping Time Maximization Zone (a)

Speed (m/s)

Zone (b)

Train i +

+1 −1

+1 + −1

+1 −

+1

− +1

Train i+1

+1

+1

+1 +

+1 − +1

+1 +1

Time (s)

Fig. 3.7 Overlapping time at off-peak hours

At phase (a), train i runs from station n to station n + 1 while train i + 1 runs from station n − 1 to station n. So, we hereby introduce a parameter λ(n − 1, n) which takes value 1 if the two inter-stations belong to the same substation and n = 1; otherwise, it takes value 0.

3.2.7.2

Off-Peak Hours Scenario

At off-peak hours, the operations for train i and train i + 1 are shown in Fig. 3.7. The discussion is described as follows: i i to an+2 denotes the process that train i runs from sta• The duration from dn+1 i+ i , dn+1 ], cruistion n + 1 to station n + 2, which is divided into accelerating [dn+1 i− i− i i+ ing+coasting [dn , an+2 ] and braking [an+2 , an+2 ]. i+1 to ani+1 denotes the process in which train i + 1 runs from • The duration from dn−1 i+1 denotes the process in station n − 1 to station n, the duration from dni+1 to an+1 which train i + 1 runs from station n to station n + 1 and the duration from ani+1 to dni+1 denotes the dwell time of train i + 1 at station n. • Zone (a) and zone (b) corresponds to phase (a) and phase (b) of Fig. 3.5, respectively. In zone (a), train i accelerates to depart from station n + 1 while train i + 1 decelerates to stop at station n. In zone (b), train i decelerates to stop at station n + 2 while train i + 1 accelerates to depart from station n. • The red box denotes the traction phase while the green one denotes the braking phase and the gray bar denotes the overlapping between the accelerating time and the braking time.

For each 1 ≤ i ≤ I − 1 and 1 ≤ n ≤ N − 1, the overlapping time at zone (a) can be expressed as follows:

3.2 Model Formulation

33

⎧ 0, ⎪ ⎪ ⎨ n t ∧ (d i+ − a (i+1)− ), 1 Oin (a, d) = an−1 n+1i+1 n i ⎪ t ∧ (an − dn ), ⎪ ⎩ b 0,

i+ if dn+1 < an(i+1)− i+ (i+1)− if an ≤ dn+1 ≤ ani+1 i+ i i+1 if dn+1 ≤ an < dn+1 i i+1 if an < dn+1 .

(3.4)

The overlapping time at zone (b) can be expressed as follows: ⎧ 0, ⎪ ⎪ ⎨ n i− t ∧ (d (i+1)+ − an+2 ), 2 (a, d) = an−1 n i Oin i+1 ⎪ t ∧ (a − d ), n n+2 ⎪ ⎩ b 0,

i− if dn(i+1)+ < an+2 i− i (i+1)+ if an+2 ≤ dn ≤ an+2 i i+1 (i+1)+ if dn ≤ an+2 < dn i if an+2 < dni+1 .

(3.5)

In general, the total overlapping time of all trains at all stations is O(a, d) =

I −1  N −1  

 1 2 Oin (a, d)λ(n − 1, n + 1) + Oin (a, d)λ(n, n + 1) .

(3.6)

i=1 n=1

At phase (a), train i runs from station n + 1 to station n + 2 while train i + 1 runs from station n − 1 to station n, and we use λ(n − 1, n + 1) to denote whether these inter-stations belong to the same power supply section. At phase (b), train i runs from station n + 1 to station n + 2 while train i + 1 runs from station n to station n + 1, and we use λ(n, n + 1) to denote whether these inter-stations belong to the same power supply section.

3.2.8 Overlapping Time Maximization Model Based on the above analysis, we formulate a cooperative timetabling model by maximizing the overlapping time as follows: ⎧ max O(a, d) ⎪ ⎪ ⎪ ⎪ s.t. lh ≤ dni+1 − dni ≤ u h , 1 ≤ i ≤ I − 1, 1 ≤ n ≤ N − 1 ⎪ ⎪ ⎪ ⎪ ln ≤ dni − ani ≤ u n , 1 ≤ i ≤ I, 1 ≤ n ≤ N − 1 ⎨ i an+1 − dni = tn , 1 ≤ i ≤ I, 1 ≤ n ≤ N − 1 ⎪ ⎪ l T ≤ a iN − a1i ≤ u T , 1 ≤ i ≤ I ⎪ ⎪ ⎪ ⎪ ani ∈ Z, 1 ≤ i ≤ I, 1 ≤ n ≤ N ⎪ ⎪ ⎩ dni ∈ Z, 1 ≤ i ≤ I, 1 ≤ n ≤ N − 1.

(3.7)

The first constraint ensures that the headway should be in an appropriate range such that successive trains will not collide rear end; the second constraint limits the range of dwell time at stations while the upper and lower bounds are generally fixed according to the passenger demand; the third constraint defines the travel time at inter-stations; and the fourth constraint limits the total travel time along the line.

34

3 Timetabling with Overlapping Time Maximization Dwell time

+1 +1

+1 +1

Station n

Headway time

+1

+1

+1

Headway time

+1

Train i+1

Dwell time

Travel time

Train i

Station n+1

Fig. 3.8 Directed graph

In what follows, we employ the graph theory approach to simplify the proposed model. At peak hours, the operation of trains shown in Fig. 3.4 can be described as a direct graph, as demonstrated in Fig. 3.8. The arc length between node ani and node dni i denotes the dwell time xn = dni − ani at station n, the arc length between node an+1 and i i i node dn denotes the travel time tn = an+1 − dn and the arc length between node dni+1 and node dni denotes the headway h = dni+1 − dni . Define x = (x1 , x2 , . . . , x N −1 ), the objective function can be rewritten as follows: O(h, x) =

I −1  N −1  

 1 2 Oin (h, x)λ(n − 1, n) + Oin (h, x) ,

(3.8)

i=1 n=1 1 2 where Oin and Oin are, respectively, defined as follows:

⎧ 0, ⎪ ⎪ ⎨ n t ∧ (t n + tbn−1 − (h − xn )), 1 Oin (h, x) = an−1 a ⎪ tb ∧ (h − xn ), ⎪ ⎩ 0,

if h − xn > tan + tbn−1 if tan ≤ h − xn ≤ tan + tbn−1 if 0 ≤ h − xn < tan if h − xn < 0

(3.9)

and ⎧ 0, if tn − h > tan + tbn−1 ⎪ ⎪ ⎨ n n−1 n t ∧ (t + tb − (tn − h)), if tan ≤ tn − h ≤ tan + tbn−1 2 Oin (h, x) = an−1 a ⎪ if 0 ≤ tn − h < tan ⎪ tb ∧ (tn − h), ⎩ 0, if tn − h < 0.

(3.10)

Finally, the mathematical model can be rewritten as follows: ⎧ max O(h, x) ⎪ ⎪ ⎪ ⎪ ⎨ s.t. lh ≤ h ≤ u h ln ≤ x n ≤ u n , 1 ≤ n ≤ N − 1 ⎪ ⎪ h, xn ∈ Z, 1 ≤ n ≤ N − 1 ⎪ ⎪ N −1 ⎩ (xn + rn ) ≤ u T . l T ≤ n=1

(3.11)

3.2 Model Formulation

35

According to the analysis, we know that there are 2I (N − 1) decision variables in model (3.7), and there are only N decision variables in the model (3.11). It is obvious that the problem is simplified.

3.3 Genetic Algorithm Genetic Algorithm (GA) is a stochastic search method that mimics the process of natural evolution, which was first initiated by Holland [4] in 1975. Due to its extensive generality, strong robustness, high efficiency, and practical applicability, GA has been widely applied to the solution to integer programming problems [5–7], and the research on transportation system [8–10]. In this chapter, we apply GA to solve the overlapping time maximization model (3.11). GA usually starts from a population of randomly generated chromosomes. The number of individuals in the population is called population size. Each chromosome in the population is assigned a survival probability based on the value of the evaluation function, which is some measure of fitness. A new population will be generated by a selection process using some sampling mechanism based on the fitness values, and the crossover and mutation operations. The cycle from one population to the next one is called a generation. After performing the genetic system a given number of cycles, we decode the best chromosome into a solution [1].

3.3.1 Representation Structure We take the binary encoding approach. A solution (h,x1 ,…,x N −1 ) is represented by a chromosome C=(c1 ,c2 ,…,c N ), where gene ci represents the ith decision variable in binary form. Take X = (90, 30, . . . , 30), for example, the corresponding chromosome is shown in Fig. 3.9.

Fig. 3.9 A solution X is represented by a chromosome C

36

3 Timetabling with Overlapping Time Maximization

3.3.2 Initialization Define an integer pop-si ze as the size of the population, which generally depends on the nature of the problem, but typically contains hundreds of chromosomes. Randomly, initialize a point and check its feasibility. If it is feasible, then it will be accepted as a chromosome. If not, we regenerate a point randomly until a feasible one is obtained. We repeat this procedure pop-si ze times and generate the first population. Here a chromosome is feasible if its corresponding solution satisfies all constraints in model (11). The algorithm of initialization is described as follows: Step 1. Set i = 1; Step 2. Randomly generate N binary vectors c1 , c2 , . . . , c N such that C i =(c1 ,c2 ,…,c N ) is a feasible chromosome; Step 3. If i = pop_si ze, stop; otherwise, set i = i + 1 and go to Step 2.

3.3.3 Evaluation Function The evaluation function is used to measure the likelihood of reproduction for each chromosome. That is, the chromosomes with higher fitness will have more chance to produce offspring. For the overlapping time maximization model, we use the objective function in the model as the evaluation function Eval(C) = F(h, x).

(3.12)

For the regenerative energy maximization model, we take the rank-based evaluation method. First, we reorder the chromosomes from good to bad according to the objective values. Then for each α ∈ (0, 1), we define the rank-based evaluation function as follows: Eval(ci ) = α(1 − α)i−1 , i = 1, 2, . . . , pop-si ze. The rank-based evaluation algorithm is described as follows: Step 1. Step 2. Step 3. Step 4.

Initialize a real number α ∈ (0, 1) and set i = 1. Calculate the objective values for all chromosomes. Reorder these chromosomes according to their objective values. Calculate the evaluation value for the ith chromosome Eval(ci ) = α(1 − α)i−1 .

Step 5. If i < pop-si ze, set i = i + 1, and goto step 4. Please note that the maximal value of Eval(C) means the best individual.

(3.13)

3.3 Genetic Algorithm

37

3.3.4 Selection Process The selection of chromosomes is made by spinning the roulette wheel, which is a fitness-proportional selection. Each time one chromosome is selected for a new child population. Continuing this process pop_si ze times, we can get the next generation. Without loss of generality, assume that the chromosomes have been ordered according to the evaluation function values. Let p0 = 0 and pi = ij=1 Eval(C j ), i = 1, 2, . . . , pop_si ze. Then the selection algorithm is described as follows: Step 1. Step 2. Step 3. Step 4.

Set j = 1; Randomly generate a number r ∈ (0, p pop_si ze ]; Select the chromosome C i such that r ∈ ( pi−1 , pi ]; If j ≥ p pop_si ze , stop; otherwise, set j = j + 1 and go to Step 2.

3.3.5 Crossover Operation Assume that the probability of crossover operation is Pc . Randomly select two parent chromosomes C i and C j and randomly generate a real number r ∈ [0, 1]. If r < Pc , C i and C j produce two new chromosomes X and Y through crossover operator. For example, if the kth gene cik of the chromosome C i and the kth gene c jk of the chromosome C j are both defined by 8 bit binary variables cik = (a, b, c, d, e, f, g, h) and c jk = (a  , b , c , d  , e , f  , g  , h  ). We define xk = (a  , b , c , d  , e, f, g, h) and yk = (a, b, c, d, e , f  , g  , h  ). The crossover process is shown in Fig. 3.10. If X and Y are feasible, take them to replace their parents; otherwise, keep their parents. The crossover operation algorithm is described as follows: Step 1. Initialize a crossover probability Pc , and set i = 1. Step 2. Randomly generate a real number r from [0, 1]. If r ≤ Pc , set i = i + 1 and randomly select two chromosomes

Fig. 3.10 Crossover process

38

3 Timetabling with Overlapping Time Maximization

c = (c1 , c2 , . . . , cn ), c = (c1 , c2 , . . . , cn ). Step 3. Randomly generate an integer number k from {1, 2, . . . , n}. Define  , . . . , cn ). x = (c1 , . . . , ck , ck+1 , . . . , cn ), y = (c1 , . . . , ck , ck+1

If x and y are feasible chromosomes, take them to replace c and c . Step 4. If i ≤ pop_si ze, go to step 2.

3.3.6 Mutation Operation The mutation is another operator for updating the chromosomes. Let Pm be the probability of mutation. Randomly select a chromosome Ci as parent for mutation and randomly generate a real number s ∈ [0, 1]. If s < Pm , randomly select a gene of the chromosome, and randomly select a bit of the gene. If the selected bit is 0 (or 1), take 1 (or 0) to replace it. If the new chromosome is feasible, take it to replace its parent; otherwise, keep its parent. The mutation operation algorithm is described as follows: Step 1. Initialize a mutation probability Pm , and set i = 1. Step 2. Generate a random number r from [0, 1]. Step 3. If r ≤ Pm , set i = i + 1. Randomly generate a chromosome c as the parent, and update the chromosome with the mutation operator. Step 4. If i ≤ pop-si ze, go to step 2.

3.3.7 General Procedure The aforementioned process can be summarized in the following algorithm: Step 1. Initialize the number of population pop_si ze, max generations max_generation and i = 1. Step 2. Initialize pop_si ze feasible chromosomes as initial population. Step 3. Calculate the evaluation function values for all chromosomes in the population. Step 4. Produce the next generation through selection, crossover and mutation operations. Step 5. If i = max_generation, stop and return the best found solution; otherwise, set i = i + 1 and go to Step 3.

3.4 Numerical Examples

39

3.4 Numerical Examples In this section, we present some numerical examples to illustrate the efficiency of the proposed models and solution method based on the operation data from Beijing Yizhuang Subway Line of China, which contains 14 stations, 13 sections and 6 substations. Generally speaking, the distance between two successive substations is 2–4 km and each power supply section provides power for three stations except the third one which supplies power for four stations since the inter-station distances of the four stations are smaller. The results are compared with the powerful and efficient optimization software Matlab 7.11, under the same running environment: a Windows 7 platform of a personal computer with processor speed 2.4 GHz and memory size 2 GB. We get the current operation timetable of Beijing Yizhuang line from Beijing Mass Transit Railway Operation Corporation Limited. According to the current operation timetable, the total travel time is T = 2087 s and the arrival time and dwell time at each station are shown in Table 3.1. The trip time required on the nth inter-station i − ani − xni . The other parameters are listed can be easily calculated to be tn = an+1 n−1 n as follows: ta = 25 s, tb = 35 s, lh = 90 s, u h = 360 s, ln = (xn − 5) s and u n = (xn + 5) s. Note that we only consider the regenerative energy utilization between two successive trains in the following examples. In what follows, we generate two timetables for the Beijing Yizhuang line [1]. One is for the peak hours scenario in Example 3.1, and the other is for the off-peak hours scenario in Example 3.2. Example 3.3 illustrates availability. Example 3.4 presents the capacity of GA in large-scale instances. Example 3.1 In this example, we present the cooperative timetabling at peak hours. We set l T = u T = 2087 s, pop_si ze = 1000, Pc = 0.8, Pm = 0.5 and max_generation = 30. By performing GA, we solve the optimal headway h = 90 s

Table 3.1 The current operation timetable for the Beijing Yizhuang line Station Songjiazhuang Xiaocun Xiaohongmen Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s)

0 30 Yizhuangqiao 710 35 Rongchang 1246 30 Ciqu 1927 45

220 30 Wenhuayuan 835 30 Tongjinan 1440 30 Yizhuang 2087 –

358 30 Wanyuan 979 30 Jinghai 1620 30 – – –

Jiugong 545 30 Rongjing 1112 30 Ciqunan 1790 35 – – –

40

3 Timetabling with Overlapping Time Maximization

Table 3.2 The optimal timetable for the Beijing Yizhuang line at peak hours Station Songjiazhuang Xiaocun Xiaohongmen Jiugong Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s)

0 25 Yizhuangqiao 710 35 Rongchang 1241 35 Ciqu 1922 50

215 35 Wenhuayuan 835 25 Tongjinan 1440 25 Yizhuang 2087 –

358 25 Wanyuan 974 35 Jinghai 1615 35 – – –

540 35 Rongjing 1112 25 Ciqunan 1790 30 – – –

Table 3.3 The optimal timetable for the Beijing Yizhuang line at peak hours with the relaxed travel time Station Songjiazhuang Xiaocun Xiaohongmen Jiugong Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s)

0 25 Yizhuangqiao 700 30 Rongchang 1236 35 Ciqu 1917 50

215 25 Wenhuayuan 820 35 Tongjinan 1435 25 Yizhuang 2082 –

348 25 Wanyuan 969 35 Jinghai 1610 35 – – –

530 35 Rongjing 1107 25 Ciqunan 1785 30 – – –

and the optimal dwell time with CPU time 1.38 s, which is recorded in Table 3.2. The total overlapping time is 166 s. For the original timetable, we take h = 90 s, and the total overlapping time is calculated to be 136 s, which implies that our approach can increase the overlapping time by (166 − 136)/136 = 22.06%. In order to obtain more overlapping time, we relax the total travel time. Now, we change the value of l T and u T by setting l T = (T − 10) s, u T = (T + 10) s. The other parameters keep unchanged. We solve the optimal headway h = 90 s and the optimal dwell time, which is recorded in Table 3.3. The overlapping time increases to 171 s, and the travel time decreases to 2082 s. So our approach can increase the total overlapping time by (171 − 136)/136 = 25.74%.

3.4 Numerical Examples

41

Fig. 3.11 The sensitivity analysis on accelerating time tan

Fig. 3.12 The sensitivity analysis on braking time tbn−1

Now we carry out the sensitivity analyses on accelerating time tan and braking time tbn−1 to justify the proposed formulation’s efficiency. First, we, respectively, set tbn−1 = 25 s, 30 s, 35 s, and 40 s, and let tan vary from 20 s to 30 s. Second, we fix tan = 20 s, 25 s, 30 s and 35 s, and let tbn−1 vary from 30 s to 40 s. The results are illustrated in Figs. 3.11 and 3.12, which show that the best found values vary smoothly [1]. Example 3.2 In this example, we consider the cooperative scheduling in off-peak hours scenario. We set l T = u T = 2087 s, pop_si ze = 200, Pc = 0.8, Pm = 0.5 and max_generation = 30. We solve the optimal headway h = 213 s and the optimal dwell time, which is recorded in Table 3.4. It is calculated that the total overlapping time is 91 s. For the original timetable, we take h = 213 s, and it is calculated that

42

3 Timetabling with Overlapping Time Maximization

Table 3.4 The optimal timetable for the Beijing Yizhuang line at off-peak hours Station Songjiazhuang Xiaocun Xiaohongmen Jiugong Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s) Station Arrival time (s) Dwell time (s)

0 27 Yizhuangqiao 699 31 Rongchang 1239 33 Ciqu 1928 44

217 27 Wenhuayuan 820 33 Tongjinan 1436 32 Yizhuang 2087 –

Table 3.5 The best found values by GA 1st 2nd Overlapping time (s) Overlapping time (s)

352 26 Wanyuan 967 33 Jinghai 1618 30 – – –

535 29 Rongjing 1103 32 Ciqunan 1788 38 – – –

3rd

4th

5th

165

166

165

166

166

6th 161

7th 166

8th 166

9th 166

10th 164

the total overlapping time is 79 s, which implies that our approach can increase the overlapping time by (91 − 79)/79 = 15.19%. Example 3.3 In order to illustrate the availability of GA, we set l T = u T = 2087 s and perform the proposed algorithm ten times with pop_si ze = 1000, Pc = 0.8, Pm = 0.5, and max_generation = 30. The computation results are recorded in Table 3.5 [1]. The exact overlapping time calculated by Matlab 7.11 is 166 s and the average error is 0.54%, which demonstrates that GA is available. Example 3.4 In order to present the capacity of GA on large-size instances, we randomly generate ten instances as follows: randomly generate the number of stations from 30 to 50 with the MATLAB function rand(); randomly generate the distances between successive stations from 1 to 2 km; randomly generate the dwell time from {30, 35, 45}. The trip time is obtained based on the experienced formula tn = 99.37 − 0.02664sn + 0.00002254sn2 , where sn devotes the distance between station n and station n + 1. The computation results and the CPU time are recorded in Table 3.6. It shows that GA can obtain a good solution (with average relative error 0.26%) within a short time (with maximum computing time 8.47 s).

References

43

Table 3.6 The results of some large-size instances Number of stations 45 35 CPU time (s) Optimal value (s) Best found value (s) Relative error (s) Number of stations CPU time (s) Optimal value (s) Best found value (s) Relative error (s)

6.36 1045 1040 0.48% 49 6.59 1123 1118 0.45%

7.21 834 829 0.60% 40 7.10 872 872 0

40

43

47

6.85 915 915 0 32 7.93 694 694 0

6.83 997 997 0 32 8.47 703 703 0

6.57 1109 1105 0.36% 35 7.94 737 732 0.68%

References 1. Yang X, Li X, Gao ZY, Wang H, Tang T (2013), A cooperative scheduling model for timetable optimization in subway systems, IEEE Transactions on Intelligent Transportation Systems, 14(1): 438–447. 2. Hasegawa I, Uchida S (1999), Braking systems, Japan Railway and Transport Review, 20: 52–59. 3. Gunselmann W (2005), Technologies for increased energy efficiency in railway systems, Proceeding of European Conference on Power Electronics and Applications, Erlangen, Germany, 1–10. 4. Holland JH (1975), Adaptation in Natural and Artificial Systems, Ann Arbor: University of Michigan Press. 5. Özgüven C, Yavuz Y, Özbakır L (2012), Mixed integer goal programming models for the flexible job-shop scheduling problems with separable and non-separable sequence dependent setup times, Applied Mathematical Modelling, 36(2): 846–858. 6. Lin W, Wang C (2004), An enhanced 0-1 mixed-integer LP formulation for traffic signal control, IEEE Transactions on Intelligent Transportation Systems, 5(4): 238–245. 7. Wu Y, Close TJ, Lonardi S (2011), Accurate construction of consensus genetic maps via integer linear programming, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 8(2): 381–394. 8. Mesbah M, Sarvi M, Currie G (2011), Optimization of transit priority in the transportation network using a genetic algorithm, IEEE Transactions on Intelligent Transportation Systems, 12(3): 908–919. 9. Abe K, Konno M, Furuta M (2004), Identification of unknown parameters in the dynamic analysis of a subway track by genetic algorithms, Boundary elements XXVI, 19: 229–238. 10. Sánchez-Medina JJ, Galán-Moreno MJ, Rubio-Royo E (2010), Traffic signal optimization in ‘La Almozara’ district in Saragossa under congestion conditions, using genetic algorithms, traffic microsimulation, and cluster computing, IEEE Transactions on Intelligent Transportation Systems, 11(1): 132–141.

Chapter 4

Timetabling with Regenerative Energy Maximization

As mentioned in Chap. 1, regenerative braking is an energy recovery mechanism used in subway systems to recover the traction energy during braking into electricity. In order to maximize the regenerative energy utilization, Yang et al. [1] presented a timetable optimization approach to coordinate the arrivals and departures of all trains located in the same electricity supply interval so that the energy regenerated from braking trains can be more effectively utilized to accelerate trains. Based on the literature [1], this chapter aims to enhance the regenerative energy utilization by making minor adjustments of the dwell times to the current timetable while using the real-world speed profiles and keeping the cycle time and the number of trains unchanged. We mainly focus on the following three questions: (1) How to measure the regenerative energy using kinetic equations? (2) How to coordinate all trains located in the same electricity supply interval? (3) How to formulate the timetabling model with regenerative energy maximization?

4.1 Problem Description This chapter states a timetable optimization problem to improve the utilization of regenerative energy for subway systems. Consider a directed subway system G 0 = (N , E), where N is a finite set of stations and E is a finite set of sections between adjacent stations. We use (n, n + 1) ∈ E to denote the inter-station between station n to station n + 1. In Fig. 4.1, if segments AB and CD represent station n and station n + 1, respectively, then segment BC represents inter-station (n, n + 1).

© Springer Nature Singapore Pte Ltd. 2020 X. Li and X. Yang, Subway Energy-Efficient Management, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-7785-7_4

45

46

4 Timetabling with Regenerative Energy Maximization Inter-station (n, n+1) B

A

D

C

Station n

Station n+1

Fig. 4.1 Illustration of subway stations and inter-stations Speed (m/s) Coasting Accelerating

Braking Time (s)

Fig. 4.2 Description of the general process of a train running at a inter-station

According to the real-world operation scenarios of subway systems, trains depart from the first station and run along the down direction to the terminal station N . Then they turn around to station N + 1 and run on the up direction back to station 2N . The down direction and up direction form one running cycle. Trains change the running direction again at the first station and get ready for another running cycle. The running process of trains is cyclic.

4.2 Model Formulation 4.2.1 Notations The symbols are listed below for the convenience of formulating the problem under consideration, and some of them are indicated in Fig. 4.2.

4.2 Model Formulation

4.2.1.1

Indices and Parameters I i N n Ns s psb pse m h G g T

C Nc t c s(n,n+1) t(n,n+1) v(n,n+1) (t) a t(n,n+1) c t(n,n+1) b t(n,n+1) tt η1 η2 β ln un

4.2.1.2

47

number of trains train index, i = 1, 2, . . . , I number of stations in the same direction station index, n = 1, 2, . . . , 2N number of electricity supply intervals electricity supply interval index, s = 1, 2, . . . , Ns starting position of electricity supply interval s ending position of electricity supply interval s train mass time headway gradient gravitational acceleration daily operation time, i.e., the period from the time when the first train is put into operation to the time when the last train comes back to the depot cycle time C = h I , i.e., the period that a train runs from station 1 to station 2N or one running cycle number of cycles time index, t = 1, 2, . . . , T cycle index, c = 1, 2, . . . , Nc length of inter-station (n, n + 1) running time at inter-station (n, n + 1) speed profile for trains at inter-station (n, n + 1), t ∈ (0, t(n,n+1) ] accelerating time at inter-station (n, n + 1) coasting time at inter-station (n, n + 1) braking time at inter-station (n, n + 1) turnaround time at the first station and the terminal station conversion factor from electricity to kinetic energy conversion factor from kinetic energy to electricity transmission loss coefficient on recovery energy minimum dwell time at station n maximum dwell time at station n

Decision and Intermediate Variables xn vic(n,n+1) (t) pic(n,n+1) (t) λ(i, t, s) 1 tic(n,n+1) 2 tic(n,n+1) 3 tic(n,n+1) 4 tic(n,n+1)

dwell time at station n speed of train i at inter-station (n, n + 1) of cycle c at time t position of train i at inter-station (n, n + 1) of cycle c at time t it takes value 1 if train i is located in electricity supply interval s at time t; otherwise, it takes value 0 time that train i departs from station n of cycle c switching time of train i from accelerating to coasting at inter-station (n, n + 1) of cycle c switching time of train i from coasting to braking at inter-station (n, n + 1) of cycle c time that train i arrives at station n + 1 of cycle c

48

4 Timetabling with Regenerative Energy Maximization

4.2.2 Model Assumptions The following assumptions are used in the model for simplifying the formulation process. (A4-1) The regenerative energy feeds back into the overhead contact line and can be immediately utilized to accelerate trains. If the feedback energy cannot be used timely, it will be wasted by heating resistors installed on the overhead contact line. (A4-2) Through analysis of the real-world speed profiles, the general process of a train running at each inter-station can be divided into three working phases shown in Fig. 4.2: accelerating, coasting and braking [2, 3]; and the period of each working phase is fixed with known running time. (A4-3) The conversion factors between electricity and kinetic energy and the transmission loss coefficient on regenerative energy are considered constant parameters. These values were obtained from real-world engineering experience.

4.2.3 Timetabling Rules For analyzing the effect of train timetable on the utilization of regenerative energy, timetabling rules are discussed here. By coordinating the arrivals and departures of trains located in the same electricity supply interval, the energy regenerated by braking trains can be better utilized to accelerate trains [4]. For example, as shown in Fig. 4.3, the area between two successive substations denotes an electricity supply interval, the solid arrows denote the electricity from substations and the dashed arrows denote the regenerative energy flow. Braking trains i − 1 and j regenerate the electricity and feed back to the overhead contact line, which can be absorbed by trains i and j + 1 for acceleration. We coordinate the time that train i departs from station n − 1, the time that train i − 1 arrives at station n + 1, the time that train j + 1 departs from station 2N − n and the time that train j arrives at station 2N − n + 2 to increase the overlap time among accelerating trains i and j + 1 and braking trains i − 1 and j, such that the utilization of regenerative energy can be improved.

4.2.4 Objective Function The objective of the model is to minimize the total energy consumption for all trains servicing on the whole line, i.e., the difference between the required energy for accelerating trains and the utilization of the regenerative energy. For simplicity, we denote x = {xn | n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1}. We assume that the first train departs from station 1 at time zero.

4.2 Model Formulation

49

Substation

Substation

Overhead contact line Train i Station n-1

Train i-1 Track

Station n+1

Station n

Overhead contact line Train j

Train j+1 Track

Station 2N-n+2

Station 2N-n

Station 2N-n+1

Fig. 4.3 Illustration of timetabling rule with the utilization of regenerative energy

For each 1 ≤ i ≤ I and 1 ≤ c ≤ Nc , train i departs from the first station of cycle c at 1 (x) = (c − 1)C + (i − 1)h, (4.1) tic(1,2) and for each 2 ≤ n ≤ N − 1, train i departs from station n of cycle c at 1 tic(n,n+1) (x) = (c − 1)C + (i − 1)h +

n−1 

t(k,k+1) +

k=1

n 

xk .

(4.2)

k=2

After train i arrives at terminal station N shown in phase (a) in Fig. 4.4, it needs some extra time to turn around. The turnaround process shown in Fig. 4.4 is described as follows: • At phase (b), train i slowly pulls into the turnaround track. • At phase (c), train i stops on the turnaround track and changes the running direction. • At phase (d), train i slowly pulls into station N + 1 and finishes the turnaround process. Turnaround is completed by the automatic train operation (ATO) system, and the turnaround time is generally set as a constant tt . Train i departs from station N + 1 of cycle c at 1 tic(N +1,N +2) (x) = (c − 1)C + (i − 1)h +

N −1  k=1

t(k,k+1) +

N −1 

x k + tt ,

k=2

and for each N + 2 ≤ n ≤ 2N − 1, train i departs from station n of cycle c at

(4.3)

50

4 Timetabling with Regenerative Energy Maximization Phase a)

Phase b)

Train Station N

Station N

Station N+1

Turnaround track

Station N+1

Turnaround track

Station N+1

Phase c)

Station N

Train

Phase d)

Station N Train Turnaround track

Turnaround track

Train Station N+1

Fig. 4.4 Description of the turnaround process

1 tic(n,n+1) (x) = (c − 1)C + (i − 1)h +

N −1 

t(k,k+1) +

k=1

+

n−1  k=N +1

t(k,k+1) +

N −1 

x k + tt

k=2 n 

(4.4) xk .

k=N +2

For each 1 ≤ i ≤ I , 1 ≤ c ≤ Nc , 1 ≤ n ≤ N − 1 and N + 1 ≤ n ≤ 2N − 1, the switching point from accelerating to coasting, the switching point from coasting to braking and the arrival time to station n + 1 are ⎧ 2 a 1 tic(n,n+1) (x) = tic(n,n+1) (x) + t(n,n+1) ⎪ ⎪ ⎪ ⎨ c 3 2 tic(n,n+1) (x) = tic(n,n+1) (x) + t(n,n+1) ⎪ ⎪ ⎪ ⎩ 4 b 3 tic(n,n+1) (x) = tic(n,n+1) (x) + t(n,n+1) .

(4.5)

For each 1 ≤ i ≤ I , 1 ≤ c ≤ Nc , 1 ≤ n ≤ N − 1 and N + 1 ≤ n ≤ 2N − 1, based on the real-world speed profile v(n,n+1) at inter-station (n, n + 1), the speed for train i at inter-station (n, n + 1) of cycle c at time t is

4.2 Model Formulation

51 1 vic(n,n+1) (x, t) = v(n,n+1) (t − tic(n,n+1) ).

(4.6)

For each 1 ≤ i ≤ I , 1 ≤ c ≤ Nc and 1 ≤ n ≤ N − 1, the position for train i at inter-station (n, n + 1) of cycle c at time t is pic(n,n+1) (x, t) =

t 

vic(n,n+1) (x, o),

(4.7)

1 o=tic(1,2)

and for each N + 2 ≤ n ≤ 2N − 1, the position for train i at inter-station (n, n + 1) of cycle c at time t is pic(n,n+1) (x, t) =

N −1 

s(n,n+1) −

n=1

t 

vic(n,n+1) (x, o).

(4.8)

1 o=tic(N +1,N +2)

1 2 For each tic(n,n+1) ≤ t < tic(n,n+1) with 1 ≤ i ≤ I , 1 ≤ c ≤ Nc , 1 ≤ n ≤ N − 1 and N + 1 ≤ n ≤ 2N − 1, the required electricity for accelerating train i at time unit [t, t + 1] is 2 2 (x, t + 1) − vic(n,n+1) (x, t))/2 f ic(n,n+1) (x, t) = [(vic(n,n+1)

+ gG( pic(n,n+1) (x, t))vic(n,n+1) (x, t)]m/η1 .

(4.9)

It is important to note that the required electricity for accelerating a train at any time unit is variable, but the total required electricity for a train between stations is a constant because the driving strategy between stations is fixed. The required electricity for train i at inter-station (n, n + 1) of cycle c is a t(n,n+1)

eic(n,n+1) =



2 2 [(v(n,n+1) (t + 1) − v(n,n+1) (t))/2

t=1

(4.10)

+ gG( p(n,n+1) (t))v(n,n+1) (t)]m/η1 . Thus, the total required electricity on the whole line throughout the operation time can be formulated as follows:  N −1  Nc I  2N −1    Ef = eic(n,n+1) + eic(n,n+1) . (4.11) i=1 c=1

n=1

n=N +1

3 4 ≤ t < tic(n,n+1) with 1 ≤ i ≤ I , 1 ≤ c ≤ Nc , 1 ≤ n ≤ N − 1 For each tic(n,n+1) and N + 1 ≤ n ≤ 2N − 1, the energy regenerated from braking train i at time unit [t, t + 1] is

52

4 Timetabling with Regenerative Energy Maximization 2 2 wic(n,n+1) (x, t) = [(vic(n,n+1) (x, t) − vic(n,n+1) (x, t + 1))/2

+ gG( pic(n,n+1) (x, t))vic(n,n+1) (x, t)]mη2 (1 − β).

(4.12)

If there are accelerating trains located in the same electricity supply interval, the regenerative energy will be absorbed by these trains. Otherwise, it will be wasted by the heating resistors installed on the overhead contact line. Thus, the total utilization of regenerative energy on the whole line throughout the operation time can be formulated as follows: I

Ns T  I    min wic(n,n+1) (x, t)λ(i, t, s), f ic(n,n+1) (x, t)λ(i, t, s) , Er (x) = t=1 s=1

i=1

i=1

(4.13) where λ(i, t, s) denotes whether train i is located in the electricity supply interval s at time t or not, i.e., λ(i, t, s) =

1, if psb ≤ pic(n,n+1) (x, t) < pse

(4.14)

0, otherwise.

The total energy consumption for all trains servicing on the whole line throughout the operation time is the difference between the required electricity E f and the utilization of regenerative energy Er , that is, E(x) = E f − Er (x).

(4.15)

Based on the real-world speed profiles, we can accurately evaluate the braking kinetic energy variations and use them in the calculation of the regenerative energy by multiplying the braking kinetic energy variations with the conversion factor. The use of real-world speed profiles significantly simplifies the modeling of train movements between stations in the timetable optimization formulation, which allows the conversion factor from kinetic energy to electrical energy without the need to explicitly model the details of kinetic energy.

4.2.5 Regenerative Energy Maximization Model The energy-efficient scheduling problem is formulated as the following integer optimization model

4.2 Model Formulation

53

⎧ ⎪ ⎪ min E(x2 , x3 , . . . , x N −1 , x N +2 , x N +3 , . . . , x2N −1 ) ⎪ ⎪ ⎪ ⎪ ⎪ N −1 2N −1 N −1 2N −1 ⎪     ⎪ ⎪ ⎨ s.t. C = xn + xn + t(n,n+1) + t(n,n+1) + 2tt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

n=2

n=N +2

n=1

n=N +1

(4.16)

ln ≤ xn ≤ u n , n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1 xn ∈ Z, n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1.

The first constraint ensures that the cycle time remains unchanged to satisfy the cyclicity of timetable, the second constraint ensures that the dwell time satisfies the upper and lower boundary constraints according to the passenger demand at each station and the last set of constraints ensure that the decision variables are all integer numbers. Note that in real-world subway systems, the dwell times are bounded by the maximum and minimum values, which implicitly satisfy the passenger demand at the station. This dramatically simplifies the complexity of the timetable optimization and increases in the practical aspect of real-world implementation. These upper and lower bound values were determined based on the actual passengers of boarding and alighting at the station.

4.3 Solution Method As the objective function of the proposed model is neither convex nor continuous (i.e., nonlinear and non-smooth), the traditional optimization method such as branchand-bound algorithm [5] and Newton algorithm [6] may fail to find a appropriate solution. We design a genetic algorithm [1] and an allocation algorithm [7] to solve the nonlinear integer programming model. Genetic algorithm is suitable for solving complicated objective functions, but it may have difficulty in satisfying the constraint set. To improve the efficiency of genetic algorithm, first, we ignore the equality constraint such that we can find an approximate solution fast. Then based on the approximate solution, we can obtain a proper solution by using the allocation algorithm.

4.3.1 Genetic Algorithm In this chapter, we apply the genetic algorithm to solve the following optimization model without the equality constraint.

54

4 Timetabling with Regenerative Energy Maximization

Start

Randomly create initial population

End

Output the best found approximate solution

Calculate the evaluation function values for all chromosomes

Yes

Stop?

Produce the next generation by selection, crossover, mutation

No

Fig. 4.5 Flow chart of the genetic algorithm

⎧ min E(x2 , x3 , . . . , x N −1 , x N +2 , x N +3 , . . . , x2N −1 ) ⎪ ⎪ ⎪ ⎨ s.t. ln ≤ xn ≤ u n , n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1 ⎪ ⎪ ⎪ ⎩ xn ∈ Z, n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1.

(4.17)

The genetic algorithm usually starts with an initial set of randomly generated feasible solutions, encoded as chromosomes called a population. A new population of chromosomes is generated following evaluation, selection, and mutation operations. The genetic algorithm will terminate after a given number of iterations of the above steps. The detailed procedure of the genetic algorithm is provided in the Appendix. We provide here a flow chart of the genetic algorithm procedure in Fig. 4.5, and briefly summarize the main steps as follows: Step 1. Initialize parameters: population size pop_si ze, initial crossover probability Pc , initial mutation probability Pm and max_generation. Set generation index i = 1. Step 2. Initialize pop_si ze feasible chromosomes as initial population. Step 3. Calculate the evaluation function values for all chromosomes. Step 4. Select the chromosomes by spinning the roulette wheel. Step 5. Produce the next generation through crossover and mutation operations. Step 6. If i = max_generation, stop and return the best found approximate solution. Otherwise, set i = i + 1, and go to step 3.

4.3.2 Allocation Algorithm The approximate solution solved by GA may not satisfy the first constraint of the model (see Eq. (4.16)). We define T = C−  N −1  2N −1 N −1 2N −1     xn + xn + t(n,n+1) + t(n,n+1) + 2tt . n=2

n=N +2

n=1

n=N +1

(4.18)

4.3 Solution Method

55 Start Initialize a large enough positive number δ and a number ε = 0

Set ΔT = ΔT - ε

Input the approximate solution, and calculate the discrepant time ΔT

Yes

Stop? No Yes

ΔT > 0 ?

Set ε = 1

No

Output the good solution

Set ε = -1

Update the approximate solution

End

Fig. 4.6 Flow chart of the allocation algorithm

Here we design an allocation algorithm to allocate the discrepant time T for the approximate solution, such that we obtain a satisfactory solution satisfying all constraints. For simplicity, we denote ⎧ g {yk | k = 1, 2, . . . , 2N − 4} = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {xn | n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ {lkg | k = 1, 2, . . . , 2N − 4} = ⎪ ⎪ {1n | n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1} ⎪ ⎪ ⎪ ⎪ ⎪ g ⎪ ⎪ ⎪ {u k | k = 1, 2, . . . , 2N − 4} = ⎪ ⎪ ⎪ ⎩ {u n | n = 2, 3, . . . , N − 1, N + 2, . . . , 2N − 1},

(4.19)

and we define the approximate solution as y a = {yka | k = 1, 2, . . . , 2N − 4} and a g good solution as x = {yk | k = 1, 2, . . . , 2N − 4}. The procedure for allocation algorithm is depicted in Fig. 4.6, and summarized as follows: Step 1. Step 2. Step 3. Step 4. Step 5.

Initialize a large enough positive number δ and a number ε = 0. Input the approximate solution y a , and calculate the discrepant time T . Set T = T − ε. If T = 0, stop and return the good solution x = y a . If T > 0, set ε = 1. Otherwise, set ε = −1.

56

4 Timetabling with Regenerative Energy Maximization

Step 6. Set k = 1, i = 1 and f i = δ. Step 7. Set yka = yka + ε. g g Step 8. If yka ≤ u k or yka ≥ lk , calculate f k = E( y a ). Otherwise, set f k = δ and k = k + 1, and go to step 7. Step 9. If f k < f i , update i = k and f i = f k . Otherwise, keep f i . Step 10. Set yka = yka − ε and k = k + 1. Step 11. If k > 2N − 4, set yia = yia + ε and go to step 3. Otherwise, go to step 7.

4.4 Numerical Examples We present two numerical examples to illustrate the effectiveness of the proposed model and solution method based on the real-life operation data from Beijing Yizhuang Subway Line [1]. We obtained the current operation data of Beijing Yizhuang Subway Line from the Beijing Mass Transit Railway Operation Corporation Limited [8]. The length, current running time, accelerating time, and braking time of sections are provided in Table 4.1; the current dwell time, minimum dwell time, and maximum dwell time at stations are provided in Table 4.2; and the remaining parameters are listed in Table 4.3. The gradients are minimal in Beijing Yizhuang Subway Line, so we consider the gradients zero in the examples. It is not suitable for subway lines where their track slopes are high. As the timetable is cyclic and for simplicity, we only consider the operation time as the period that trains finish two cycles, i.e., T = 2C = 9144 s. The operation time is approximately equal to the period of morning peak hours (from 7:00 am to 9:30 am) for the Beijing Yizhuang subway line. Remark 4.1 Abbreviation of stations name: Songjiazhuang (SJZ), Xiaocun (XC), Xiaohongmen(XHM), Jiugong (JG), Yizhuangqiao (YZQ), Wenhuayuan (WHY), Wanyuan (WY), Rongjing (RJ), Rongchang (RC), Tongjinan (TJ), Jinghai (JH), Ciqunan (CQN), Ciqu (CQ), Yizhuang (YZ). In the following examples, we generate the optimal timetable for the Beijing Yizhuang Subway Line in Example 4.1. We compare the energy-efficient scheduling approach and the previous cooperative scheduling approach on the utilization of regenerative energy in Example 4.2. The solution algorithm is performed at a Windows 7 platform of a personal computer with a processor frequency of 2.4 GHz and memory size of 2 GB. Example 4.1 We perform the GA procedure with pop_si ze = 200, max_generation = 60, Pc = 0.6 and Pm = 0.15 to find an approximate solution first. Based on the approximate solution, we perform the allocation algorithm to obtain a good solution that satisfies all the constraints. The approximate and good solutions are reported in Table 4.4. From the results, we can see that the total dwell time of the approximate solution is 748 s or 22 s less than the current total dwell time of 770 s. We distribute the 22 s to appropriate stations (station 2, station 6, and station

4.4 Numerical Examples

57

Table 4.1 Length, current running time, accelerating time and braking time of sections for the Beijing Yizhuang Subway Line Inter-station Length (m) Running time (s) Accelerating (s) Braking (s) SJZ-XC XC-XHM XHM-JG JG-YZQ YZQ-WHY WHY-WY WY-RJ RJ-RC RC-TJN TJN-JH JH-CQN CQN-CQ CQ-YZ YZ-CQ CQ-CQN CQN-JH JH-TJN TJN-RC RC-RJ RJ-WY WY-WHY WHY-YZQ YZQ-JG JG-XHM XHM-XC XC-SJZ

2631 1275 2366 1982 993 1538 1280 1354 2338 2265 2086 1286 1334 1334 1286 2086 2265 2338 1354 1280 1538 993 1982 2366 1275 2631

190 108 157 135 90 114 103 104 164 150 140 102 105 110 100 141 150 162 103 101 111 90 135 157 105 195

186 104 153 131 86 110 99 100 160 146 136 98 101 106 96 137 146 158 99 97 107 86 131 153 101 191

194 112 161 139 94 118 107 108 168 154 144 106 109 114 104 145 154 166 107 105 115 94 139 161 109 199

27) using the allocation algorithm, and we obtain the good solution with 770 s total dwell time. With a good solution, we obtain the optimal timetable presented in Table 4.5, and also shown in Fig. 4.7 in comparison with the current timetable. The total energy consumptions for the two timetables are 13,089 kW·h for the optimal timetable and 14,069 kW·h for the current timetable. The results show that the energy consumption is reduced by (14,069 − 13,089)/14,069 = 6.97%. According to the statistics provided by the Beijing Mass Transit Railway Operation Corporation Limited, about 18,675,955 kW·h of electricity is consumed each year for the Beijing Yizhuang Subway Line. The average industrial rate in Beijing is 0.81 CNY (or 0.13 USD), which implies that it can save about 18,675,955*6.97%*0.81=1,054,388 CNY (or 169,223

58

4 Timetabling with Regenerative Energy Maximization

Table 4.2 Current dwell time for the Beijing Yizhuang Subway Line Down SJZ XC XHM JG YZQ station

WHY

WY

Dwell – time (s) Minimum – time (s) Maximum – time (s)

30

30

30

35

30

30

20

20

20

25

20

20

40

40

40

45

40

40

Down Station

RC

TJN

JH

CQN

CQ

YZ

Dwell 30 time (s) Minimum 20 time (s) Maximum 40 time (s)

30

30

30

35

45

–

20

20

20

25

35

–

40

40

40

45

55

–

Up station YZ

CQ

CQN

JH

TJN

RC

RJ

Dwell – time (s) Minimum – time (s) Maximum – time (s)

45

35

30

30

30

30

35

25

20

20

20

20

55

45

40

40

40

40

Up Station WY

WHY

YZQ

JG

XHM

XC

SJZ

Dwell 30 time (s) Minimum 20 time (s) Maximum 40 time (s)

30

35

30

30

30

–

20

25

20

20

20

–

40

45

40

40

40

–

RJ

Table 4.3 Value and unit of some parameters Parameter I N Ns m h Value Unit

18 –

14 –

5 –

311800 kg

254 s

C

Nc

4572 2 s –

tt

η1

η2

β

240 s

0.7 –

0.8 –

0.05 –

USD) each year by our approach. The detailed calculation results are reported in Table 4.6. The comparison of energy consumption between the current and optimal timetables is shown in Fig. 4.8. Example 4.2 In Chap. 3, we proposed a cooperative scheduling (CS) approach to improve regenerative energy utilization by maximizing the overlapping time. For

4.4 Numerical Examples

59

Table 4.4 Approximate and good solutions of the model Station n 2 3 4 5 6 7 8 Approximate 31 Good xn 40 Station n

29 29

20 20

35 35

23 26

38 38

26 26

9

10

11

12

13

31 31

23 23

24 24

30 30

48 48

16

17

18

19

20

21

22

23

24

25

26

27

Approximate 49 Good xn 49

40 40

29 29

21 21

30 30

22 22

36 36

36 36

39 39

23 23

35 35

30 40

Table 4.5 Optimal timetable for the Beijing Yizhuang Subway Line Down SJZ XC XHM JG YZQ station

WHY

WY

Dwell time (s) Depart time (s)

–

40

29

20

35

26

38

0

230

367

544

714

830

982

Down station

RJ

RC

TJN

JH

CQN

CQ

YZ

Dwell time (s) Depart time (s)

26

31

23

24

30

48

–

1111

1246

1433

1607

1777

1927

–

Up station YZ

CQ

CQN

JH

TJN

RC

RJ

Dwell time (s) Depart time (s)

–

49

40

29

21

30

22

2272

2431

2571

2741

2912

3104

3229

Up station WY

WHY

YZQ

JG

XHM

XC

SJZ

Dwell time (s) Depart time (s)

36

36

39

23

35

40

–

3366

3513

3642

3800

3992

4137

4572

simplicity, they only considered adjacent trains located in the same electricity supply interval. In this chapter, we propose a more energy-efficient scheduling (EES) approach to better utilize the regenerative energy by considering all trains located in the same electricity supply interval and extending the operation time to the whole day. This example compares the utilization of regenerative energy and total energy

60

4 Timetabling with Regenerative Energy Maximization

Fig. 4.7 Comparison of the current and optimal timetables for one cycle Table 4.6 Energy consumption comparison between the current and optimal timetables Required electricity Utilization Total energy consumption Current timetable Optimal timetable Saved energy Saved money

15,292 kW·h 15,292 kW·h Daily 3566 kW·h 2888 CNY

1223 kW·h 2203 kW·h Monthly 106,990 kW·h 86,662 CNY

14,069 kW·h 13,089 kW·h Yearly 1,301,714 kW·h 1,054,388 CNY

consumption between the CS and EES approaches. The results reported in Table 4.7 show that the EES approach can improve the utilization of regenerative energy by (2203 − 1618)/1618 = 36.16% and further reduces the total energy consumption by (13,674 − 13,089)/13,674 = 4.28% compared to that of the CS approach.

References

61

Fig. 4.8 Comparison of energy consumption between the current and optimal timetables for one cycle Table 4.7 Energy consumption comparison between the CS approach and EES approach Regenerative energy Total energy Energy-saving (%) (kW·h) consumption (kW·h) CS approach EES approach

1618 2203

13,674 13,089

2.81 6.97

References 1. Yang X, Chen A, Li X, Ning B, Tang T (2015), An energy-efficient scheduling approach to improve the utilization of regenerative energy for metro systems, Transportation Research Part C: Emerging Technologies, 57: 13–29. 2. Howlett PG, Pudney PJ (1995), Energy-efficient train control, Adv. Ind. Control. New York, NY, USA: Springer-Verlag. 3. Howlett PG (1996), Optimal strategies for the control of a train, Automatica, 32(4): 519–532. 4. Yang X, Li X, Gao ZY, Wang H, Tang T (2013), A cooperative scheduling model for timetable optimization in subway systems, IEEE Transactions on Intelligent Transportation Systems, 14(1): 438–447.

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4 Timetabling with Regenerative Energy Maximization

5. Narendra PM, Fukunaga K (1977), A branch and bound algorithm for feature subset selection, IEEE Transactions on computers, 9: 917–922. 6. Roger A (1981), Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem, IEEE Transactions on Antennas and Propagation, 29(2): 232–238. 7. Yang X, Chen A, Gao ZY, Tang T (2019), An energy-efficient rescheduling approach under delay perturbations for subway systems, Transportmetrica B: Transport Dynamics, 7(1): 386–400. 8. Zhang L (2014), The operation data for the Beijing Metro Yizhuang Line, Technical Report, Beijing Mass Transit Railway Operation Corporation Limited (in Chinese).

Chapter 5

Integrated Speed Control and Timetable Optimization

Subway energy-efficient management mainly consists of a speed control approach and a timetable optimization approach. The former controls speed profile for a single train at inter-stations to minimize the traction energy consumption under the travel time constraint and the latter synchronizes the accelerating time and braking time of multiple trains to maximize the regenerative energy utilization among them. Note that the travel time at the speed control model is determined by timetable while the accelerating time and braking time at the timetable optimization model is determined by speed profile. To achieve a better performance on net energy consumption, i.e., the difference between traction energy consumption and regenerative energy utilization, this chapter introduces an integrated optimization approach (Li and Lo [2]) to jointly determine speed profile and train timetable. We mainly focus on the following three questions: (1) How to model the net energy consumption? (2) How to formulate the integrated speed control and timetable optimization model? (3) How to solve the optimal timetable and speed profile?

5.1 Problem Description Net energy consumption is the electricity absorbed by trains from the substations. In Fig. 5.1, the black flow denotes the traction energy consumption, the dark gray flow denotes the regenerative energy utilization, and the light gray flow denotes the net energy consumption. In this chapter, we propose an integrated optimization approach to minimize the net energy consumption by determining the speed profile at inter-stations and timetable along the line. First, quantitative analyses on traction energy consumption and regenerative energy utilization are given. Second, an energyefficient timetabling and speed control model is formulated to minimize the net energy consumption.

© Springer Nature Singapore Pte Ltd. 2020 X. Li and X. Yang, Subway Energy-Efficient Management, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-7785-7_5

63

64

5 Integrated Speed Control and Timetable Optimization Electric power network Substation Conversion loss

Diode Transmission loss

Transmission loss

Overhead contact line

Net energy

Regenerative energy Conversion loss

Yes

Traction energy

V < MTV

Accelerating train

Braking train

No Resistor

Track

Fig. 5.1 An illustration on regenerative energy absorption Segment

2N

Station

2N-1

2

3

Inter-station

2N-2

……

……

2N-3

……

N-4

N-3

N-2

Inter-station

N+4

N+3

N+2

N-1

N

Station

Station Turnaround

Turnaround Station Segment 2N

1

N+1

N

Fig. 5.2 An illustration on segment division

5.2 Model Formulation Generally speaking, subway line consists of stations, turnaround stations, and interstations. For inter-stations, the piece-wise speed limits are generally set for trains arising from curvature, signaling and track conditions that make higher speeds hazardous. For addressing the piece-wise speed limits, we divide the line into multiple segments such that each segment has a constant speed limit (see Fig. 5.2). First, we assemble the turnaround stations, stations and inter-stations from both up direction and down direction as a cycle. Furthermore, we divide each inter-station into several segments such that each segment has a constant speed limit. Finally, we handle the stations and turnaround stations as special segments with speed limit zero. Except for the two turnaround stations, each physical segment is counted twice by distinguishing up direction and down direction. Therefore, the total number of segments is an even number, denoted as 2N , where segment N and segment 2N are turnaround stations.

5.2 Model Formulation

65

5.2.1 Notations For a better understanding of the integrated optimization approach, the commonly used notations are first introduced, which include indices, parameters, decision variables and intermediate variables. Note that the meaning of each symbol is used only in this chapter, that is, a symbol may have a different meaning in other chapters.

5.2.1.1

Indices and Parameters t k i n j wn wn v¯n pn vkin h F B Rn αn βn γn θ1 θ2 θ3 τ σ M δ(n, j)

5.2.1.2

time index, t = 1, 2, . . . , T train index, k = 1, 2, . . . , K cycle index, i = 1, 2, . . . , I segment index, n = 1, 2, . . . , 2N substation index, j = 1, 2, . . . , J the maximum allowable dwell time at station n the minimum allowable dwell time at station n speed limit at segment n starting position of segment n speed profile for train k at the nth segment of cycle i headway the maximum traction force per unit mass the maximum braking force per unit mass the running resistance per unit mass for trains at segment n acceleration rate for trains at accelerating phase of segment n acceleration rate for trains at coasting phase of segment n acceleration rate for trains at braking phase of segment n conversion factor from electricity to kinetic energy conversion factor from kinetic energy to electricity transmission loss factor of regenerative energy cycle time the required time for turnaround operation train mass take value 1 if segment n is in substation j; take value 0, otherwise

Decision Variables mn qn akin dkin

maximum speed for trains at segment n arriving speed for trains at segment n arriving time for train k at the nth segment of cycle i departing time for train k at the nth segment of cycle i.

66

5.2.1.3

5 Integrated Speed Control and Timetable Optimization

Intermediate Variables ckin bkin

coasting time for train k at the nth segment of cycle i braking time for train k at the nth segment of cycle i

Remark 5.1 Suppose that traction force F, braking force B, and running resistance Rn are all constant at segments, then the acceleration rates at accelerating, coasting and braking phases are also constant satisfying αn = F − Rn , βn = Rn and γn = B + Rn , where resistance arises from line gradient, curvature, mechanical friction and aerodynamic friction. Remark 5.2 For subway operations management, trains follow the same operation strategy except for a headway difference. Headway among successive trains is determined by the passenger demand D, train capacity l, and its utilization rate κ. If we use ϕ to denote the trip frequency, i.e., the number of service trains per hour, we have D = l × κ × ϕ. It follows from h = 3600/ϕ that the headway is h = 3600 × l × κ/D. Generally speaking, the train capacity and its utilization rate are known parameters. Therefore, if passenger demand keeps steady, the headway can be handled as a known parameter. In Chap. 2, we have concluded that the energy-efficient speed profile should consist of maximum accelerating, cruising, coasting, and maximum decelerating. In practice, since the resistance is significantly less than the traction force and braking force, we combine cruising into the coasting phase such that the energy-efficient speed profile only contains accelerating, coasting, and braking phases. Then for each segment with a nonzero speed limit, we first draw trains to the maximum speed m n with acceleration rate αn , then coast for a while with deceleration rate βn , and finally brake with deceleration rate γn . The decision variables are these switching time and speeds among different phases and segments, denoted as follows: x = {akin , dkin , qn , m n | 1 ≤ k ≤ K , 1 ≤ i ≤ I, 1 ≤ n ≤ 2N }, which are illustrated by Fig. 5.3. For each 1 ≤ k ≤ K , we have dkin = aki(n+1) for all 1 ≤ i ≤ I and 1 ≤ n < 2N , and dki2N = ak(i+1)1 for all 1 ≤ i < I . That is, the arriving time at segment n + 1 is equal to the departing time at segment n, and the starting time at cycle i + 1 is equal to the ending time at cycle i. For each decision vector x, we consider the speed profile for train k at segment n of the ith cycle with 1 ≤ k ≤ K , 1 ≤ i ≤ I and 1 ≤ n ≤ 2N . The argument breaks down into two cases. If segment n is a station or turnaround station, it follows from v¯n = 0 that (5.1) vkin (x, t) = 0, ∀akin ≤ t ≤ dkin . Otherwise, if segment n is located in an inter-station with nonzero speed limit v¯n > 0, arriving time akin , arriving speed pn , maximum speed m n and departing time dkin , it

5.2 Model Formulation

67

Speed Segment n

Segment n+1 Segment n+1

Station

Time

Fig. 5.3 An illustration on decision variables

follows from the continuity of speed profile that coasting time ckin and braking time bkin should satisfy the following equations qn + αn (ckin − akin ) = m n , m n − βn (bkin − ckin ) = qn+1 + γn (dkin − bkin ), which implies that they could be expressed in terms of decision variables as follows: ckin = akin + (m n − qn )/αn , bkin = (γn dkin − m n + qn+1 − βn ckin )/(γn − βn ). Then according to the train motion equations, we have ⎧ qn + αn (t − akin ), if akin ≤ t < ckin ⎪ ⎪ ⎪ ⎨ vkin (x, t) = m n − βn (t − ckin ) , if ckin ≤ t < bkin ⎪ ⎪ ⎪ ⎩ qn+1 + γn (dkin − t), if bkin ≤ t ≤ dkin .

(5.2)

5.2.2 Net Energy Consumption Now we analyze the net energy consumption for trains during the daily operation period, i.e., the difference between the traction energy consumption and the regenerative energy utilization. First, we analyze the traction energy consumption for accelerating trains. For each akin ≤ t < ckin with 1 ≤ n ≤ 2N , 1 ≤ i ≤ I and 1 ≤ k ≤ K , the electricity required for accelerating train k at time unit [t, t + 1] of the ith cycle is f kin (x, t) = M × F × vkin (x, t)/θ1 , where M is the train mass, F is the unit traction force, vkin (x, t) is the train speed, and θ1 is the conversion factor from electricity to kinetic energy. Since there is no traction energy consumption during coasting and braking phases, f kin (x, t) takes value 0 when ckin ≤ t < dkin . Taking summation on f kin (x, t)

68

5 Integrated Speed Control and Timetable Optimization

with respect to t, n, i and k, the total traction energy consumption for all trains at all cycles is I  2N d K  kin −1  E T (x) = f kin (x, t). (5.3) k=1 i=1 n=1 t=akin

Second, we analyze the regenerative energy utilization among braking trains and accelerating trains. Note that the regenerative energy from braking trains can only be absorbed by trains that are accelerating and located in the same substations. For each bkin ≤ t < dkin with 1 ≤ n ≤ 2N , 1 ≤ i ≤ I and 1 ≤ k ≤ K , according to the mechanical power equation, the electricity regenerated from train k at time interval [t, t + 1] of the ith cycle is gkin (x, t) = M × B × vkin (x, t) × θ2 × (1 − θ3 ), where B is the unit braking force, θ2 is the conversion factor from kinetic energy to regenerative electricity, and θ3 is the transmission loss factor of the regenerative energy at the overhead contact line. Since there is no energy regenerated during accelerating and coasting phases, gkin (x, t) takes value 0 when akin ≤ t < bkin . If there are other trains accelerating in the same substation with train k, the regenerative energy will be transmitted to these trains via the overhead contact line. Otherwise, it will be dissipated and wasted by the onboard resistor. Divide the daily operation time interval [0, T ] = [0, τ ] ∪ [τ , 2τ ] ∪ · · · ∪ [(I − 1)τ , I τ ] ∪ [I τ , T ]. First, we analyze the regenerative energy utilization at subinterval [0, τ ]. In this case, trains either operate in their first cycle or stop at the depot. Denote n kt as the operation segment for train k at time t. If 0 ≤ t < ak11 , we have n kt = 0, i.e., train k stops at the depot. Otherwise, if ak1n ≤ t < dk1n , we have n kt = n. For each 1 ≤ t < τ and 1 ≤ j ≤ J , the regenerative energy utilization for trains in substation j at [t, t + 1] is  G 1t j (x) = min

K 

f k1n kt (x, t)δ(n kt , j),

k=1

K 

 gk1n kt (x, t)δ(n kt , j) ,

k=1

where f k10 (x, t) = gk10 (x, t) = 0 for all 1 ≤ k ≤ K . Taking summation with respect to j and t, the regenerative energy utilization during sub-interval [0, τ ] can be formulated as follows: τ −1  J  G 1t j (x). G 1 (x) = t=0 j=1

Furthermore, for each 1 ≤ i ≤ I − 1, we analyze the regenerative energy utilization at sub-interval [i × τ , (i + 1) × τ ]. For each i × τ ≤ t ≤ i × τ + h, except the first train enters its (i + 1)th cycle, the other trains are all in their ith cycle. Then we have n 1t = n if a1(i+1)n ≤ t < d1(i+1)n and n kt = n if akin ≤ t < dkin for all 1 ≤ n ≤ N and 2 ≤ k ≤ K . The regenerative energy utilization for trains in substation j at interval [t, t + 1] is

5.2 Model Formulation

69

G (i+1)t j (x) = min

 K 

gkin kt (x, t)δ(n kt , j)+g1(i+1)n 1t (x, t)δ(n 1t , j),

k=2 K 

 f kin kt (x, t)δ(n kt , j)+ f 1(i+1)n 1t (x, t)δ(n 1t , j)

k=2

for each 1 ≤ j ≤ J . Taking summation on G (i+1)t j (x) with t and j, we get the regenerative energy utilization for trains at [i × τ , i × τ + h]. For each [i × τ + k × h, i × τ + (k + 1) × h] with 1 ≤ k < K , considering the (k + 1)th train as the first train, it is found that the regenerative energy utilization at this time horizon is the same as at [i × τ , i × τ + h]. Therefore, the regenerative energy utilization during sub-interval [i × τ , (i + 1) × τ ] is G i+1 (x) = K ×

J iτ +h−1  t=iτ

G (i+1)t j (x).

j=1

Finally, we analyze the regenerative energy utilization at sub-interval [I × τ , T ]. In this case, trains are either in their I th cycle or at the depot. For each 1 ≤ k ≤ K and 1 ≤ n ≤ 2N , it is easy to prove that n kt = n if ak I n ≤ t < dk I n and n kt = 0 if dk I (2N ) ≤ t ≤ T . The regenerative energy utilization during this period can be formulated as follows:  K  T −1  K J    G I +1 (x) = min gk I n kt (x, t)δ(n kt , j), f k I n kt (x, t)δ(n kt , j) , t=I τ j=1

k=1

k=1

where gk I 0 (x, t) = f k I 0 (x, t) = 0 for all 1 ≤ k ≤ K . Based on the above analyses, the total regenerative energy utilization for all trains at the whole operation period [0, T ] is E R (x) = G 1 (x) + (I − 1) × G 2 (x) + G I +1 (x). In conclusion, the net energy consumption, i.e., the difference between the traction energy consumption and regenerative energy utilization, is E(x) = E T (x) − E R (x).

(5.4)

5.2.3 Integrated Constraints During the daily operation period [0, T ], i.e., from the time when the first train is put into operation to the time when the last train returns to the depot, all trains run the same number of cycles. The number of trains K and cycle time τ should satisfy the following constraints d K I (2N ) − a111 − T ≤ , (5.5)

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τ = K h, I τ + (K − 1)h = d K I (2N ) − a111 .

(5.6)

According to the periodicity of subway operation, the arriving time and departing time for trains at stations should satisfy the following constraints a(k+1)in − akin = h, ∀1 ≤ k ≤ K − 1, 1 ≤ i ≤ I, 1 ≤ n ≤ 2N

(5.7)

dki(2N ) − aki1 = τ , ∀1 ≤ k ≤ K , 1 ≤ i ≤ I.

(5.8)

The turnaround time σ required for trains changing from up/down direction to down/up direction is constant. Since this operation only occurs at turnaround stations, i.e., n = N and n = 2N , we have dki N − aki N = σ, dki(2N ) − aki(2N ) = σ, ∀1 ≤ k ≤ K , 1 ≤ i ≤ I.

(5.9)

If we use v¯n to denote the speed limit for trains at the nth segment, the boundary speeds qn , qn+1 and the maximum speed m n should satisfy 0 ≤ qn , m n , qn+1 ≤ v¯n , ∀1 ≤ n ≤ 2N ,

(5.10)

where q2N +1 = 0. For any segment 1 ≤ n ≤ 2N with a positive speed limit, the boundary speeds qn and qn+1 should satisfy the following constraints 2 2 v¯ 2 − qn+1 v¯ 2 − qn+1 v¯ 2 − qn2 v¯n2 − qn2 + n ≤ pn+1 − pn ≤ n + n . 2αn 2γn 2αn 2βn

(5.11)

Furthermore, for each segment 1 ≤ n ≤ 2N with boundary speeds qn and qn+1 satisfying conditions (5.11), the minimum travel time is obtained by accelerating to v¯n and then coasting and braking to segment n + 1 with speed qn+1 and the maximum travel time is obtained by accelerating to a certain speed and then coasting to segment n + 1 with speed qn+1 . Therefore, the travel time for trains at segment n should satisfy the following constraints ⎧

 2 ⎪ ⎪ qn+1 αn + βn qn2 qn qn+1 ⎪ ⎪ d − a ≤ + + 2( p − p ) − − ⎪ kin kin n+1 n ⎪ α β α β α βn ⎪ n n n n n ⎪ ⎪ ⎪ ⎪ ⎨ v¯n − qn v¯n qn+1 dkin − akin ≥ + − ⎪ αn βn γn ⎪ ⎪ ⎪ ⎪

⎪  ⎪ 2 ⎪ qn+1 γn − βn αn + βn 2 qn2 ⎪ ⎪ ⎪ − v ¯ − − − 2( p − p ) . n+1 n ⎩ βn γn αn βn n αn γn (5.12) Finally, with given boundary speeds qn and qn+1 satisfying conditions (5.11) and switching time akin and dkin satisfying (5.12), the braking speed z n and the maximum

5.2 Model Formulation

71

speed m n should satisfy the travel distance constraint 2 z 2 − qn+1 m 2n − qn2 m 2 − z n2 + n + n = pn+1 − pn , 2αn 2βn 2γn

(5.13)

and the travel time constraint m n − qn m n − zn z n − qn+1 + + = dkin − akin . αn βn γn

(5.14)

The dwell time at stations should be in some predetermined time windows according to the passenger demand and operation time for operating screen door. In general, for each 1 ≤ k ≤ K , 1 ≤ i ≤ I and station n, we have w kin ≤ dkin − akin ≤ w kin .

(5.15)

For each 1 ≤ k ≤ K , 1 ≤ i ≤ I and 1 ≤ n ≤ 2N , the switching time should be integer numbers, that is, (5.16) akin , dkin ∈ {0, 1, 2, . . . , T }.

5.2.4 Integrated Optimization Model Based on above analyses, we propose the following integrated speed control and timetable optimization model 

min E(x) s.t. Constraints (5.5)−(5.16),

(5.17)

which minimizes the net energy consumption under the integrated constraints.

5.3 Solution Method In the past decades, GA has obtained considerable success in providing satisfactory solutions to subway operations management problems [1–5]. In this section, we introduce a GA [2] to solve the integrated speed control and timetable optimization model (5.17), which contains the representation process, initialization process, evaluation process, selection process, crossover process, and mutation process.

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5.3.1 Representation Process First, we need to construct an one-to-one mapping between solution space and chromosome space to simplify the crossover and mutation operations. Here a chromosome is defined as a N × 5-dimensional matrix X , in which X (n, 1) = a11n , X (n, 2) = qn , X (n, 3) = m n , X (n, 4) = qn+1 and X (n, 5) = d11n for all 1 ≤ n ≤ 2N . If X is a chromosome, the cycle time is τ = X (2N , 5) − X (1, 1), and the number of trains is K = τ / h. Furthermore, according to the periodicity, for each 1 ≤ k ≤ K , 1 ≤ i ≤ I and 1 ≤ n ≤ 2N , we have akin = X (n, 1) + (k − 1)h + (i − 1)τ and dkin = X (n, 5) + (k − 1)h + (i − 1)τ .

5.3.2 Initialization Process For each 1 ≤ n ≤ 2N , we first randomly generate the arriving and departing speeds X (n, 2) and X (n, 4) satisfying (5.10) and (5.11). Furthermore, we randomly generate the arriving and departing time X (n, 1) and X (n, 5) according to the following three cases: (i) If segment n is a station, then X (n, 1) and X (n, 5) should satisfy the dwell time constraint (5.15); (ii) If segment n is a turnaround station, then X (n, 1) and X (n, 5) should satisfy constraint (5.9); (iii) If segment n is located in an inter-station with a nonzero speed limit, then X (n, 1) and X (n, 5) should satisfy constraint (5.12). Finally, we calculate the maximum speed X (n, 3) satisfying (5.13) and (5.14). Based on these values, we define the number of service trains as K = (X (2N , 5) − X (1, 1))/ h. Then we update X according to the following iterative process: Randomly select a segment n except the turnaround stations such that X (n, 5) − X (n, 1) − 1 is a feasible operation time; Set X (k, 5) = X (k, 5) − 1 for all k ≥ n and X (k, 1) = X (k, 1) − 1 for all k > n; Repeat this procedure X (2N , 5) − X (1, 1) − K h times. After this process, we have X (2N , 5) − X (1, 1) = K h. Determine the cycle time as τ = K h and the number of cycles as follows:   + 1, if T >  × τ +  I = , if T ≤  × τ + , where  = (T − (K − 1)h)/τ  and  = τ /2. Define pop-si ze as the size of population. Based on the above process, randomly generate pop-si ze chromosomes as the initialized population.

5.3 Solution Method

73

5.3.3 Evaluation Process For each chromosome X , its fitness value is defined as the corresponding net energy consumption. Without loss of generality, assume that these chromosomes are arranged from good to bad such that X 1 has the lowest fitness, and X pop-si ze has the highest fitness. The evaluation process assigns each chromosome a probability of reproduction so that its chance of being selected is proportional to its fitness relative to the other chromosomes. That is, the chromosomes with higher fitness values will have more chance to produce offspring. For each α ∈ (0, 1), define a rank-based evaluation function Eval(X i ) = α(1 − α)i−1 , i = 1, 2, . . . , pop-si ze.

5.3.4 Selection Process During each successive generation, a proportion of the existing population is selected to breed a new generation. The selection process is based on spinning the roulette wheel pop-si ze times, and selecting a chromosome each time, which is summarized as follows: Calculate the reproduction probability r0 = 0, ri =

i 

Eval(X j ), i = 1, 2, . . . , pop-si ze;

j=1

Randomly generate a real number r from (0, r pop-si ze ] and select chromosome X i if ri−1 < r ≤ ri ; Repeat this process pop-si ze times and obtain pop-si ze chromosomes. Essentially, roulette wheel is a fitness-proportional selection, where fitter chromosomes are typically more likely to be selected.

5.3.5 Crossover Process Crossover is one of the mainly used operations for generating the next population. First, we define a parameter pc to denote the crossover probability. Repeat the following process pop-si ze times: Generate a random number r from [0, 1], and select the chromosome X i if r < pc . Note that not all chromosomes can be selected, but different chromosomes have an equal chance to be selected. Without loss of generality, assume that chromosomes X 1 , X 2 , . . . , X m are selected, and m is an even number. Divide them into the following pairs: (X 1 , X 2 ), (X 3 , X 4 ), . . . , (X m−1 , X m ).

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5 Integrated Speed Control and Timetable Optimization

We illustrate the crossover operation based on the first pair (X 1 , X 2 ). Generate a random number λ from the unit interval [0, 1], then the crossover operation on X 1 and X 2 will produce two children Y1 and Y2 as follows: Y1 = λX 1 + (1 − λ)X 2 , Y2 = (1 − λ)X 1 + λX 2 . Note that the children may be unfeasible. We redo the crossover operation until two feasible children are obtained, or a number of cycles are finished. Then we select the best two chromosomes from the parents and children to replace the parents X 1 and X 2.

5.3.6 Mutation Process The mutation is another important operation for updating the chromosomes. We define a parameter pm to denote the mutation probability and randomly select some chromosomes as parents in a similar way to the process of selecting parents for crossover operation. For each selected parent X , we randomly generate an index n except n = N and n = 2N . If segment n is a station, we randomly generate a dwell time w from [w11n , w 11n ], set X (k, 5) = X (k, 5) − (X (k, 5) − X (k, 1) − w) for k ≥ n and X (k, 1) = X (k, 1) − (X (k, 5) − X (k, 1) − w) for k > n. If segment n has nonzero speed limit, we randomly generate the boundary speeds qn , qn+1 , and the switching time a11n , d11n under the constraints (5.10), (5.11) and (5.12). Set X (n, 2) = qn , X (n, 4) = qn+1 , and 

X (k, 5) = X (k, 5) − X (n, 5) + X (n, 1) + d11n − a11n , if k ≥ n, X (k, 1) = X (k, 1) − X (n, 5) + X (n, 1) + d11n − a11n , if k > n.

Calculate the maximum speed X (n, 3) satisfying constraints (5.13) and (5.14).

5.3.7 General Procedure Following evaluation operation, selection operation, crossover operation and mutation operation, a new population of chromosomes is generated. GA will terminate after a given number of iterations of the above steps. We summarize the general procedure for GA as follows: Step 1. Step 2. Step 3. Step 4.

Randomly initialize pop-si ze chromosomes. Calculate the fitness values for all chromosomes. Calculate the evaluation values for all chromosomes. Select the chromosomes by spinning the roulette wheel.

5.3 Solution Method

75

Table 5.1 The practical timetable of Beijing Subway Yizhuang Line Station Songjiazhuang Xiaocun Xiaohongmen Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time

0 30 Yizhuangqiao 607 35 Rongchang 1203 30 Ciqu 1802 45 Ciqu 2491 45 Rongchang 3095 30 Yizhuangqiao 3686 35 Songjiazhuang 4290 30

135 30 Wenhuayuan 800 30 Tongjinan 1320 30 Yizhuang 2034 40 Ciqunan 2628 35 Rongjing 3237 30 Jiugong 3867 30 Turnaround 4320 180

264 30 Wanyuan 932 30 Jinghai 1482 30 Turnaround 2074 180 Jinghai 2816 30 Wanyuan 3366 30 Xiaohongmen 4034 30

Jiugong 431 30 Rongjing 1061 30 Ciqunan 1665 35 Yizhuang 2254 40 Tongjinan 2978 30 Wenhuayuan 3498 30 Xiaocun 4163 30

Step 5. Update the chromosomes by crossover and mutation operations. Step 6. Repeat the second to fifth steps Generation times. Step 7. Return the best-found chromosome and decode it to a solution.

5.4 Numerical Examples In order to illustrate the efficiency of the integrated speed control and timetable optimization model, we conduct numerical experiments on Beijing Subway Yizhuang line. The current timetable is shown in Table 5.1, and the data on speed limit is described in Table 2.1. The whole line is divided into 108 segments, including 78 segments in inter-stations, 28 stations, and 2 turnaround stations. Parameters are listed as follows: conversion factor from electricity to kinetic energy θ1 = 0.9, conversion factor from kinetic energy to regenerative electricity θ2 = 0.6, transmission loss

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Table 5.2 The optimal timetable of Beijing Subway Yizhuang line Station Songjiazhuang Xiaocun Xiaohongmen Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time Station Arrival time Dwell time

0 27 Yizhuangqiao 696 32 Rongchang 1378 26 Ciqu 2051 49 Ciqu 2770 44 Rongchang 3423 28 Yizhuangqiao 4073 32 Songjiazhuang 4742 28

135 30 Wenhuayuan 918 28 Tongjinan 1500 27 Yizhuang 2320 37 Ciqunan 2929 31 Rongjing 3579 28 Jiugong 4273 32 Turnaround 4770 180

278 26 Wanyuan 1069 27 Jinghai 1688 27 Turnaround 2357 180 Jinghai 3129 33 Wanyuan 3715 33 Xiaohongmen 4453 27

Jiugong 492 27 Rongjing 1210 33 Ciqunan 1904 34 Yizhuang 2537 37 Tongjinan 3300 27 Wenhuayuan 3863 29 Xiaocun 4596 29

factor of regenerative electricity θ3 = 0.1, acceleration rate at accelerating phase α = 0.8 m/s2 , deceleration rate at coasting phase β = 0.02 m/s2 , deceleration rate at braking phase γ = 0.4 m/s2 , train mass M = 287,080 kg, turnaround time σ = 180 s and operation time T = 64800 s. The GA is coded in Matlab 7.1 under the running environment: a Windows 7 platform of a personal computer with a processor speed of 2.4 GHz and memory size 2 GB. When GA is performed, we set pop-si ze = 40, pc = 0.8, pm = 0.5 and Generation = 70. First, we illustrate the efficiency of the integrated speed control and timetable optimization model. Based on the practical timetable, we take h = 90 s, K = 50, I = 13 and τ = 4500 s, which results in net energy consumption 203, 850 kWh. On the other hand, we solve the integrated optimization model by performing the genetic algorithm. After 70 iterations, the best-found solution (see Table 5.2) shows that 55 trains should be put into operation, and each train needs to run 12 cycles with 4950 s for each cycle. Compared with the currently used timetable, five more trains are used which reduces the number of cycles from 13 to 12 for each train. Most importantly,

5.4 Numerical Examples

77

Table 5.3 Comparisons among integrated optimization, timetable optimization and speed control approaches Headway Model Traction Energy Regenerative Net Energy Energy 90

120

150

180

240

300

Integrated Timetabling Speed Control Integrated Timetabling Speed Control Integrated Timetabling Speed Control Integrated Timetabling Speed Control Integrated Timetabling Speed Control Integrated Timetabling Speed Control

195,090 257,830 191,020 146,670 177,900 142,980 115,180 144,320 113,510 101,120 116,700 99,996 83,264 95,721 82,886 68,109 70,838 66,536

45,520 68,100 31,310 23,180 32,250 13,000 14,200 16,090 12,180 3,630 10,280 2,044 2,320 3,314 1,504 5,294 5,573 3,690

149,570 189,730 159,720 123,490 145,650 129,980 100,980 128,230 101,330 97,485 106,420 97,950 80,943 92,407 81,383 62,815 65,265 62,846

it lengthens the cycle time from 4500 s to 4950 s, which can significantly reduce the traction energy consumption, although a minor influence on the regenerative energy utilization may happen to arise from the reduction on the synchronization of accelerating and braking actions. The net energy consumption of the best-found solution is 149, 570 kWh, which implies that the integrated optimization approach achieves energy conservation by (203,850−149,570)/203,850 = 26.63%. Since the total travel time increases by 450 s for each cycle, compared with the currently used timetable, about 17 s is added for each travel at inter-station. However, it is generally acceptable since most passengers take short travels. Second, we make comparisons among speed control approach, timetable optimization approach and integrated optimization approach on traction energy, regenerative energy and net energy consumption. For the speed control approach, we first solve the speed profile with minimum traction energy consumption for a single train. Then we extend the speed profile to all trains at all cycles according to the periodicity of subway operation and calculate the regenerative energy utilization and the net energy consumption. For the timetable optimization model, we first solve the timetable with maximum regenerative energy utilization. Then we calculate the energy-efficient speed profile for trains at inter-stations and finally calculate the net energy consumption. Taking the headway from 90 s to 300 s, the computation

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results are recorded by Table 5.3, where the third column denotes the traction energy consumption, the fourth column denotes the regenerative energy utilization and the fifth column denotes the net energy consumption. It is concluded that the integrated optimization approach can significantly cut down the net energy consumption. For example, when h = 90 s, the integrated optimization model reduces net energy consumption by 21.17% compared with the timetable optimization model, and 6.35% compared with the speed control model.

References 1. Li X, Lo HK (2014), An energy-effcient scheduling and speed control approach for metro rail operations, Transportation Research Part B, 64: 73–89. 2. Ding Y, Liu HD, Bai Y, Zhou FM (2011), A two-level optimization model and algorithm for energy-efficient urban train operation, Journal of Transportation Systems Engineering and Information Technology, 11(1): 96–101. 3. Mazloumi E, Mesbah M, Ceder A, Moridpour S, Currie G (2012), Efficient transit schedule design of timing points: A comparison of ant colony and genetic algorithms, Transportation Research Part B, 46(1): 217–234. 4. Tuyttens D, Fei HY, Mezmaz M, Jalwan J (2013), Simulation-based genetic algorithm towards an energy-efficient railway traffic control, Mathematical Problems in Engineering, Article ID 805410, 1–12. 5. Yang X, Li X, Gao ZY, Wang H, Tang T (2013), A cooperative scheduling model for timetable optimization in subway systems, IEEE Transactions on Intelligent Transportation Systems, 14(1): 438–447.

Chapter 6

Dynamic Speed Control and Timetable Optimization

For subway systems, since the passenger demands frequently change in daily operation hours, the train timetable and speed profile could be correspondingly adjusted for lowering the net energy consumption. Generally speaking, for operation hours with low passenger demand, we could increase the cycle time to reduce traction energy consumption. Once the headway and cycle time at the next cycle is changed according to the predicated passenger demand, we need to adjust the timetable and speed profile accordingly. In this chapter, we introduce a dynamic speed control and timetable optimization approach proposed by Li and Lo [4], which mainly focuses on the following three questions: (1) How to measure the regenerative energy storage? (2) How to optimize the reference timetable and speed profile using an online algorithm? (3) How to dynamically optimize the net energy consumption?

6.1 Problem Description As the development of passenger demand forecasting technique, a promising research direction on subway energy-efficient management is dynamic speed control and timetable optimization, which dynamically adjusts the speed profile and timetable up to the change of passenger demand such that the net energy consumption can be further reduced. As the first attempt, Li and Lo [4] proposed a dynamic timetabling and speed control framework, which consists of passenger demand forecasting, speed control and timetable optimization, and automatic train control (See Fig. 6.1). Here we mainly concern the speed control and timetable optimization method, focusing on traction energy consumption and regenerative energy storage. We divide the whole line into multiple segments such that each segment has a constant speed limit and gradient, where stations and turnaround stations are considered as special segments with speed limit and gradient zero (See Fig. 5.2). We assemble all segments from the up direction and down direction as a cycle and name the required time for completing the operations at all segments as cycle time. As © Springer Nature Singapore Pte Ltd. 2020 X. Li and X. Yang, Subway Energy-Efficient Management, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-7785-7_6

79

80

6 Dynamic Speed Control and Timetable Optimization Passenger demand forecasting algorithm

Speed control & timetabling algorithm

Automatic train operation algorithm

Passenger demand

Reference timetable & speed profile

Online speed profile

Temporary delay

Cycle time & headway time The current cycle

The next cycle

Fig. 6.1 An illustration on dynamic speed control and timetabling

we have introduced in Chap. 5, the headway time and cycle time are determined by the train capacity and passenger demands. Generally speaking, the train capacity is constant while the passenger demands change frequently during daily operation hours, hence the headway time and cycle time could be adjusted correspondingly to lower the net energy consumption. For example, if the passenger demand is low, we can increase the cycle time for reducing the traction energy consumption. The traditional subway timetabling approaches divide the daily operation time horizon into peak hours and non-peak hours and take different values of headway time and cycle time at peak hours and off-peak hours. However, since there are still significant variations on passenger demands within the peak-hours (off-peak hours), we should divide the daily operation time horizon into finer intervals for carrying out delicacy operation management. In order to solve this problem, Li and Lo [4] proposed a dynamic train timetabling and speed control framework. First, the passenger demand is predicted, and the headway time and cycle time for trains at the next cycle are determined. Second, the reference timetable and speed profile for trains at the next cycle are optimized subjects to the constraints on headway time and cycle time. For each segment, we assume that the speed profiles at interstations consist of accelerating, cruising, and braking phases, in which coasting phase is omitted for speeding up the solution process. Since the running resistance at cruising phase is very small compared with the traction force and braking force, this simplification is acceptable for obtaining a reference timetable and speed profile. Finally, the automatic train operation system [5, 6] is used to operate trains with real conditions based on the reference timetable and speed profile. Note that the dynamic train timetabling and speed control approach essentially divides the daily operation into multiple cycle operations, and trains take different cycle operation strategies depending on the forecasted passenger demands.

6.2 Model Formulation

81

6.2 Model Formulation In this section, we formulate a new integrated timetable optimization and speed control model, in which the regenerative energy storage is considered. That is, we assume that all regenerative energies from braking trains are stored at the onboard capacitors and used at accelerating actions later (Fig. 6.2).

6.2.1 Assumptions The following assumptions are given to simplify the modeling and solution processes such that a satisfactory timetable and speed profile can be obtained within a short computing time [4]. • The energy-efficient speed profile at each segment should consist of accelerating, cruising, coasting, and braking. However, since the running resistance is far less than the traction force and braking force (See Fig. 6.3), the energy conservation at the coasting phase is very limited. In addition, it is slightly easier to manage the mathematics with a constant cruising speed than with decreasing coasting speed, so we omit the coasting phase and assume that the speed profile takes the accelerating-cruising-braking phase sequence. • In practice, the maximum traction force and the maximum braking force both depend on the train’s speed. Take the maximum traction force, for example. Since the maximum power c is generally constant, we have dv/dt = c/v, which implies that a higher speed v results in a lower traction force [2]. However, the speeddependent traction force and braking force leads to complicated formulae on

Electric power network Substation Diode Conversion loss Transmission loss Overhead contact line Traction energy consumption

Regenerative energy

Regenerative energy Capacitor

Track

Fig. 6.2 An illustration on regenerative energy storage

Capacitor

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6 Dynamic Speed Control and Timetable Optimization

Fig. 6.3 The running resistance curve of a real subway line (Li and Lo [4])

accelerating time, braking time, traction energy consumption, and regenerative electricity such that longer computation time is needed on the timetable and speed profile optimization. Therefore, in order to make a quick calculation on the reference timetable and speed profile, we assume that the maximum traction force and braking force are all constant, which is also the assumption on the early studies of optimal train control theory. • The conversion from kinetic energy to electricity is a very complicated process, which is influenced by many factors such as the lower and upper limits of voltage in overhead contact line, flows of charge-discharge of the capacitors, and so on. Therefore, the conversion factors between electricity and kinetic energy are generally uncertain quantities [3]. Here the constant conversion factors (e.g., the mean values of these uncertain quantities) are used for simplifying the solution process. Note that the function of this section is to obtain a reference timetable and speed profile for the automatic train operation systems. Even if we include the coasting phase in speed control, use a speed-dependent traction force and braking force, take variable conversion factors between electricity and kinetic energy and consider all other real operation conditions, the obtained optimal timetable and speed profile still need to be adjusted as the influences of delays or disturbances at the next cycle. In other words, the effort on considering all practical conditions is meaningless in most cases. What is worse, it will increase the computing time and a sophisticated algorithm will be needed.

6.2 Model Formulation

83

6.2.2 Notations For a better understanding of dynamic timetable optimization and speed control model, the notations are first introduced, including indices, parameters, and decision variables.

6.2.3 Indices and Parameters n k D ς κ h F B θ1 θ2 σ M C gn αn βn ln un vn wn wn

segment index, n = 1, 2, . . . , 2N train index, k = 1, 2, . . . , K passenger demand per hour train capacity train capacity utilization rate headway time the maximum traction force per unit mass the maximum braking force per unit mass conversion factor from electricity to kinetic energy conversion factor from kinetic energy to regenerative electricity turnaround time train mass cycle time, i.e., the period required to complete one cycle for one train gradient of segment n acceleration rate at segment n during accelerating phase, αn = F − gn deceleration rate at segment n during braking phase, βn = B + gn length of segment n speed limit at segment n speed profile at segment n the maximum operation time at segment n the minimum operation time at segment n

6.2.4 Decision Variables an cn bn dn

arriving time at segment n for the first train switching time from accelerating to cruising at segment n for the first train switching time from cruising to braking at segment n for the first train departuring time from segment n for the first train

Subway systems have a cyclic timetable and speed profile, which means that trains follow each other with a fixed headway h, and take the same dwelling time at stations and the same speed profile at inter-stations. For each 1 ≤ k ≤ K and 1 ≤ n ≤ 2N ,

84

we have

6 Dynamic Speed Control and Timetable Optimization

⎧ akn = an + (k − 1) × h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ckn = cn + (k − 1) × h ⎪ ⎪ bkn = bn + (k − 1) × h ⎪ ⎪ ⎪ ⎪ ⎩ dkn = dn + (k − 1) × h,

(6.1)

where akn , ckn , bkn , dkn are the switching times for train k at the nth segment. Cycle time C is determined by the passenger demand D, the train capacity ς and its utilization rate κ. If we use K to denote the number of service trains per cycle, it should satisfy D = ς × κ × K × (3600/C). Furthermore, it follows from C = K × h and h ≥ h min that the cycle time is C = K × max {3600 × ς × κ/D, h min } , and the headway time among adjacent trains is h = max {3600 × ς × κ/D, h min } , where h min is the minimum headway time for ensuring the operation safety. Train capacity and its utilization rate are both constant in practice. Therefore, once we obtain the predicted passenger demand for the next cycle, the cycle time and headway time can be determined soon.

6.2.5 Net Energy Consumption For dynamic train timetabling and speed control approach, the most critical issue is how to timely adjust the reference timetable and speed profile corresponding to the updated headway time and cycle time. Here we approximate the net energy consumption in the integrated optimization model into a convex function by using the linearization method, such that we can use the Kuhn-Tucker conditions to solve the reference timetable and speed profile. For each 1 ≤ n < 2N , we have dn = an+1 , d N = a N + σ and d2N = a2N + σ. That is, once a train leaves the current segment, it enters the next segment automatically, which means that the decision variables {d1 , d2 , . . . , d2N } can be expressed by {a1 , a2 , . . . , a2N }. Furthermore, we will solve the switching points {c1 , c2 , . . . , c2N } and {b1 , b2 , . . . , b2N } based on {a1 , a2 , . . . , a2N }. Then the decision variables may be simplified to a = {a1 , a2 , . . . , a2N }. Divide the collection of segments into subsets N1 = {1 ≤ n ≤ 2N | u n < u n−1 ∨ u n+1 } and N2 = {1 ≤ n ≤ 2N | u n ≥ u n−1 ∨ u n+1 }. For each n ∈ N1 , since trains have a lower speed at this segment, we will control the speed to minimize the running time for ensuring operation efficiency. The argument breaks down into three cases:

6.2 Model Formulation

85

• If u n < u n−1 and u n < u n+1 , we set vn ≡ u n , which implies that cn = an and bn = an+1 . • If u n+1 > u n > u n−1 with 2αn ln ≥ u 2n − u 2n−1 , we first draw train from u n−1 to u n with acceleration rate αn , then cruise a while to keep speed u n . In this case, we have cn = an + (u n − u n−1 )/αn and bn = an+1 . • If u n+1 < u n < u n−1 with 2βn ln ≥ u 2n − u 2n+1 , we first cruise a while to keep speed u n , then brake train from u n to u n+1 with deceleration rate βn . In this case, we have cn = an and bn = an+1 − (u n − u n+1 )/βn . In general, the switching speeds are cn = an + max{u n − u n−1 , 0}/αn and bn = an+1 − max{u n − u n+1 , 0}/βn . Remark 6.1 For each segment n ∈ N1 , if u n+1 > u n > u n−1 and 2αn ln < u 2n − u 2n−1 , the speed limits u n is essentially meaningless for trains. In this case, we will combine segments n and n + 1 with common speed limit u n+1 . Similarly, if u n+1 < u n < u n−1 and 2βn ln < u 2n − u 2n+1 , we will combine segments n and n − 1 with common speed limit u n−1 . On the other hand, for each segment n ∈ N2 , since trains have a higher speed at this segment, we will control the speed by using the maximum accelerating, cruising and maximum braking phase sequence. That is, we first draw trains from u n−1 to a certain speed wn with acceleration rate αn , then cruise a while and brake from speed wn to u n+1 with deceleration rate βn . According to the motion equation, with given arrival time an and departure time an+1 , the accelerating time cn − an , cruising time bn − cn and braking time dn − bn should satisfy the following nonlinear equations: ⎧ 2    wn − u 2n−1 /2αn + (bn − cn )wn + wn2 − u 2n+1 /2βn = ln ⎪ ⎪ ⎪ ⎨ (cn − an ) + (bn − cn ) + (dn − bn ) = an+1 − an ⎪ ⎪ ⎪ ⎩ u n−1 + αn (cn − an ) = u n+1 + βn (dn − bn ),

(6.2)

where wn = u n−1 + αn (cn − an ) is the cruising speed. The first equation means that the total running distance is ln , the second equation means that the total running time is an+1 − an , and the last equation means that both the ending speed of accelerating and starting speed of braking are equal to the cruising speed. Based on these equations, the cruising time satisfies (bn − cn )2 = (an+1 − an + u n−1 /αn + u n+1 /βn )2 − (1/αn + 1/βn )(u 2n−1 /αn + u 2n+1 /βn + 2ln ). For simplifying the solution process on integrated optimization model, we approximate the cruising time as a linear function of an+1 − an , i.e., bn − cn → (1 − pn )(an+1 − an ) + qn . Taking it into (6.2), the cruising time and braking time are solved based on an and an+1 as follows: ⎧ βn u n+1 − u n−1 ⎪ ⎪ ⎪ ⎨ cn = an + αn + βn ( pn (an+1 − an ) − qn ) + αn + βn ⎪ αn u n−1 − u n+1 ⎪ ⎪ ( pn (an+1 − an ) − qn ) − . ⎩ bn = an+1 − αn + βn αn + βn

(6.3)

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6 Dynamic Speed Control and Timetable Optimization

Remark 6.2 The discussion on the linear approximation of cruising time can be found in Li and Lo [4]. Since trains take the same timetable and speed profile except for a headway difference, they consume the same quantity of net energy consumption for each cycle operation, i.e., the difference between the traction energy consumption for accelerating trains and the storage regenerative energy from braking trains. First, we analyze the energy consumption for accelerating trains. Let a = {an | n = 1, 2, . . . , 2N } be a decision vector. For each an ≤ t < t + t ≤ cn with 1 ≤ n ≤ 2N , according to the mechanical power equation, the required electricity for drawing one train at time interval [t, t + t] is M F(u n−1 + αn (t − an ))t/θ1 , where M is the train mass, F is the maximum traction force per unit mass, and θ1 is the conversion factor from electricity to kinetic energy. Taking integration with respect to time t, the traction energy consumption during the accelerating phase is 

cn

M F(u n−1 + αn (t − an ))/θ1 dt

an

= [αn (cn − an )2 + 2u n−1 (cn − an )]M F/(2θ1 ). Furthermore, taking summation on n, the total traction energy consumption for accelerating one train at all segments is 2N

[αn (cn − an )2 + 2u n−1 (cn − an )]M F/(2θ1 ).

(6.4)

n=1

Second, we analyze the regenerative energy storage from braking trains. For each bn ≤ t < t + t ≤ dn with 1 ≤ n ≤ 2N , according to the mechanical power equation, the electrical energy regenerated at time interval [t, t + t] is M B(u n+1 + βn (dn − t))tθ2 , where B is the maximum braking force per unit mass, θ2 is the conversion factor from kinetic energy to regenerative electricity. Taking integration on time t, the regenerative energy storage is 

dn

M B(u n+1 + βn (dn − t))θ2 dt

bn

= [βn (dn − bn )2 + 2u n+1 (dn − bn )]M Bθ2 /2. Furthermore, taking summation on n, the total regenerative energy storage of one train at all segments is 2N n=1

[βn (dn − bn )2 + 2u n+1 (dn − bn )]M Bθ2 /2.

(6.5)

6.2 Model Formulation

87

Finally, it follows from (6.3) and dn = an+1 that the difference between traction energy consumption (6.4) and regenerative energy storage (6.5) is 2N

λ2n (an+1 − an )2 + λ1n (an+1 − an ) + λ0n .

(6.6)

n=1

Denote ξn = u n+1 − u n−1 − βn qn and ηn = u n−1 − u n+1 − αn qn , parameters λ2n , λ1n and λ0n are defined as follows: λ2n =

αn βn (βn F/θ1 − αn Bθ2 )M pn2 , 2(αn + βn )2

(αn u n+1 + βn u n−1 − αn βn qn )(βn F/θ1 − αn Bθ2 )M pn , (αn + βn )2  



αn ξn2 2u n−1 ξn M F βn ηn2 2u n+1 ηn M Bθ2 . = + − + (αn + βn )2 αn + βn 2θ1 (αn + βn )2 αn + βn 2 λ1n =

λ0n

6.2.6 Constraints Now we analyze the constraints for optimizing timetable and speed profiles. First, we consider the cycle time constraint, i.e., the sum of running times at inter-stations, dwelling times at stations, and operating times at turnaround stations should be equal to the cycle time. Denoting a2N +1 = d2N , we have 2N (an+1 − an ) = C.

(6.7)

n=1

Second, according to the passenger demands at stations and the operation time for opening/closing screen door, the dwelling times for trains at stations should satisfy a time window constraint. In addition, according to the requirements on operation efficiency and robustness, the running times for trains at segments should also take values at a predetermined time window. In general, we have the following time window constraints w n ≤ an+1 − an ≤ wn , 1 ≤ n < 2N .

(6.8)

Remark 6.3 Denote N = {1 ≤ n ≤ 2N | 0 < u n < u n−1 ∨ u n+1 } ∪ {N , 2N } as the collection of segments with fixed operating time. The calculations on wn and wn break down into four cases. Case 1. If u n−1 < u n < u n+1 and 2αn ln ≥ u 2n − u 2n−1 , we first draw train from speed u n−1 to u n with acceleration rate αn , then cruise a while to keep speed u n .

88

6 Dynamic Speed Control and Timetable Optimization

In this case, the running time at this segment is an+1 − an =

u n − u n−1 ln − (u 2n − u 2n−1 )/2αn + αn un

and we take wn = w n = (u n − u n−1 )/αn + ln /u n − (u 2n − u 2n−1 )/2αn u n . Case 2. If u n−1 > u n > u n+1 and 2βn ln ≥ u 2n − u 2n+1 , we first cruise a while to keep speed u n , then brake train from speed u n to u n+1 with deceleration rate βn . In this case, the running time is an+1 − an =

u n − u n+1 ln − (u 2n − u 2n+1 )/2βn + βn un

and we take wn = w n = (u n − u n+1 )/βn + ln /u n − (u 2n − u 2n+1 )/2βn u n . Case 3. If u n > 0, u n < u n−1 and u n < u n+1 , we have vn = u n . In this case, the running time is an+1 − an = ln /u n and we take w n = w n = ln /u n . Case 4. If n = N or n = 2N , we have an+1 − an = σ and take w N = w N = w 2N = w 2N = σ.

6.2.7 Integrated Optimization Model Based on the above analyses, the optimal timetable and speed profile should have lower traction energy consumption and higher regenerative energy storage. Therefore, we propose the following integrated optimization model ⎧ 2N ⎪ ⎪ ⎪ min λ2n (an+1 − an )2 + λ1n (an+1 − an ) + λ0n ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ⎨ 2N ⎪ ⎪ s.t. (an+1 − an ) = C ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ⎪ ⎩ w n ≤ an+1 − an ≤ w n , n = 1, 2, . . . , 2N ,

(6.9)

which minimizes the net energy consumption subject to the cycle time and time window constraints. Since λ2n , λ1n , λ0n are given parameters for all n = 1, 2, . . . , 2N , the objective is a quadratic function. Furthermore, since the constraints are all linear, model (6.9) is a convex optimization problem.

6.3 Solution Method

89

6.3 Solution Method Since model (6.9) is a convex optimization problem, it has one and only one optimal solution. If we use x to denote the optimal solution, according to the Kuhn-Tucker conditions [1], we have ⎧ 2λ2n (xn+1 − xn ) + λ1n + θ + φn − ψn = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ φ (x −x −w )=0 n

n+1

n

n

⎪ ⎪ ψn (wn − (xn+1 − xn )) = 0 ⎪ ⎪ ⎪ ⎪ ⎩ φn , ψn ≥ 0, 1 ≤ n ≤ N ,

(6.10)

where θ, φn and ψn are Lagrangian multipliers. For each n ∈ N, since w n = w n , we have xn+1 − xn = wn (or w n ). Now we solve (6.10) for n ∈ / N. The basic process is shown as follows: calculate φn with known θ → calculate ψn with known θ → calculate xn+1 − xn with known θ → solve the value of θ. First, we calculate the value of φn with known θ. The argument breaks down into two cases. • If 2λ2n w n + λ1n + θ < 0, we have φn > 0. Otherwise, if φn = 0, we have 2λ2n (xn+1 − xn ) + λ1n + θ + φn − ψn ≤ 2λ2n w n + λ1n + θ < 0, which is in contradiction with the first equation of (6.10). Taking φn > 0 into (6.10), we have xn+1 − xn = w n , ψn = 0 and φn = −(λ1n + θ) − 2λ2n w n . • If 2λ2n w n + λ1n + θ ≥ 0, we have φn = 0. Otherwise, if φn > 0, we have xn+1 − xn = w n and ψn = 0, which is in contradiction with 2λ2n (xn+1 − xn ) + λ1n + θ + φn − ψn = 0. In general, we get φn = max{−(λ1n + θ) − 2λ2n w n , 0}. Second, we calculate the value of ψn with known θ. Similarly, the argument also breaks down into two cases. • If 2λ2n w n + λ1n + θ > 0, we have ψn > 0. Otherwise, if ψn = 0, we have 2λ2n (xn+1 − xn ) + λ1n + θ + φn − ψn ≥ 2λ2n w n + λ1n + θ > 0, which is in contradiction with the first equation of (6.10). Taking ψn > 0 into (6.10), we have xn+1 − xn = w n , φn = 0 and ψn = 2λ2n wn + λ1n + θ. • If 2λ2n w n + λ1n + θ ≤ 0, we have ψn = 0. Otherwise, if ψn > 0, we have xn+1 − xn = w n and φn = 0, which is in contradiction with 2λ2n (xn+1 − xn ) + λ1n + θ + φn − ψn = 0. In general, we get ψn = max{2λ2n w n + λ1n + θ, 0}. Third, we calculate the operating time xn+1 − xn with known θ. It follows from the first equation of (6.10) that xn+1 − xn =

ψ n − φn λ1n + θ − . 2λ2n 2λ2n

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6 Dynamic Speed Control and Timetable Optimization

Based on the above analyses, taking φn = max{−(λ1n + θ) − 2λ2n w n , 0} and ψn = max{2λ2n w n + λ1n + θ, 0}, we have



λ1n + θ λ1n + θ λ1n + θ xn+1 − x=n max w n + , 0 + min w n + ,0 − 2λ2n 2λ2n 2λ2n



λ1n + θ λ1n + θ = min wn + , 0 + max − , wn 2λ2n 2λ2n



λ1n + θ λ1n + θ = min max − , w n , w n + max wn + ,0 2λ2n 2λ2n

λ1n + θ (6.11) , wn , wn . = min max − 2λ2n Finally, we solve the Lagrangian multiplier θ. Take (6.11) into cycle time constraint (6.7). It follows from w 1 + w 2 + · · · + w n ≤ C ≤ w 1 + w 2 + · · · + wn that there exists at least one θ such that

λ1n + θ min max − , wn , wn + wn = C. (6.12) g(θ) = 2λ2n n ∈N / n∈N Since g is a decreasing function with respect to θ, we solve θ by using the bisection method. After we solve the value of θ, we can get the optimal operating times by taking θ back into formula (6.11).

6.4 Numerical Examples This section presents some numerical experiments based on the actual operation data of Beijing Subway Yizhuang Line of China, which shows that the integrated approach can reduce the net energy consumption around 11% than the practical timetable. Furthermore, with given passenger demand sequence at off-peak hours, the dynamic timetabling and speed control approach with adaptive cycle time can reduce the net energy consumption around 7% than the static approach with fixed cycle time. The Yizhuang line has a daily ridership of approximately 180,000 in 2014, and the whole line is divided into 122 segments, including 92 segments located in interstations, 28 stations, and 2 turnaround stations. The practical timetable is shown in Table 6.1, and the data on speed limit is described in Table 6.2. Take the interstation from SJ to XC for example. It is divided into 5 segments [0, 150], [150, 480], [480, 1161], [1161, 2501], and [2501, 2631], which have speed limits 13.89 m/s, 23.61 m/s, 18.06 m/s, 23.61 m/s, and 16.67 m/s, respectively. Parameters are listed

6.4 Numerical Examples

91

Table 6.1 The practical timetable of Yizhuang Line in 2014 Station Songjiazhuang (SJ) Xiaocun (XC)

Xiaohongmen (XH)

Arrival Dwell

0 30

220 30

358 30

Station

Jiugong (JG)

Yizhuangqiao (YQ)

Wenhuayuan (WH)

Arrival Dwell

545 30

710 35

835 30

Station

Wanyuan (WY)

Rongjing (RJ)

Rongchang (RC)

Arrival Dwell

979 30

1112 30

1246 30

Station

Tongjinan (TJ)

Jinghai (JH)

Ciqunan (CN)

Arrival Dwell

1440 30

1620 30

1790 35

Station

Ciqu (CQ)

Yizhuang (YZ)

Turnaround

Arrival Dwell

1927 45

2077 30

2107 180

Station

Yizhuang (YZ)

Ciqu (CQ)

Ciqunan (CN)

Arrival Dwell

2287 30

2427 45

2572 35

Station

Jinghai (JH)

Tongjinan (TJ)

Rongchang (RC)

Arrival Dwell

2748 30

2928 30

3120 30

Station

Rongjing (RJ)

Wanyuan (WY)

Wenhuayuan (WH)

Arrival Dwell

3253 30

3384 30

3525 30

Station

Yizhuangqiao (YQ)

Jiugong (JG)

Xiaohongmen (XH)

Arrival Dwell

3645 35

3815 30

4002 30

Station

Xiaocun (XC)

Songjiazhuang (SJ)

Turnaround

Arrival Dwell

4137 30

4362 30

4392 180

as follows: conversion factor from electricity to kinetic energy θ1 = 0.7, conversion factor from kinetic energy to regenerative electricity θ2 = 0.5, acceleration rate αn = 0.5 m/s2 , deceleration rate βn = 0.8 m/s2 , train mass M = 287080 kg and turnaround time σ = 180 s. The following examples are simulation studies based on the actual operation data of the Beijing Subway Yizhuang Line. First, we compare the practical timetable with the optimal timetable on the theoretical energy calculations. Second, we compare the dynamic integrated optimization approach with the static integrated optimization approach in the scenario with variable passenger demands.

92

6 Dynamic Speed Control and Timetable Optimization

Table 6.2 The practical speed limit of Yizhuang Line in 2014 Piece-wise speed limit (position/speed limit/ …/position) SJ-XC XC-XH XH-JG JG-YQ YQ-WH WH-WY WY-RJ RJ-RC RC-TJ TJ-JH JH-CN CN-CQ CQ-YZ

0/13.89/150/23.61/480/18.06/1161/23.61/2501/16.67/2631 2631/16.67/2643/23.61/2797/20.83/3534/23.61/3780/16.67/3905 3905/16.67/3918/23.61/5808/20.83/6141/16.67/6271 6271/16.67/6281/23.61/8122/16.67/8254 8254/16.67/8265/23.61/9116/16.67/9246 9246/16.67/9259/23.61/10655/16.67/10785 10785/16.67/10797/23.61/11933/16.67/12065 12065/16.67/12077/23.61/13289/16.67/13419 13419/16.67/13431/23.61/14649/19.44/15426/23.61/15624/16.67/15756 15756/16.67/15768/23.61/17891/16.67/18021 18021/16.67/18033/23.61/19982/16.67/20107 20107/16.67/20120/23.61/21264/16.67/21394 21394/16.67/21406/23.61/22569/16.67/22728

Example 6.1 This example compares the practical timetable with the optimal timetable of the integrated optimization model with the same cycle time C = 4572 s. Taking γ1 = 1.10 and γ2 = 1.20, the optimal timetable is shown in Table 6.3. The net energy consumption is 334.72 kWh, and the computation time is 0.014 s. Compared with the practical timetable, which has a net energy consumption of 376.80 kWh, the optimal timetable can achieve energy conservation by (376.80−334.72)/376.80=11.17%. One reason is that the optimal timetable generally increases the running times at shorter inter-stations (e.g., YQ-WH, WY-RJ, RJ-RC, CN-CQ, CQ-YZ) by reducing the running times at longer inter-stations (e.g., SJ-XC, XH-JG, RC-TJ, TJ-JH) such that trains have a lower speed at each inter-station. Example 6.2 In this example, we consider the following scenario for comparing the dynamic and static scheduling approaches: there are five cycles at off-peak hours with passenger demand sequence {8011, 7942, 7874, 7807, 7742}. Take γ1 = 1.12 and γ2 = 1.27. The cycle times and energy consumptions are shown in Table 6.4. For the dynamic approach, we calculate the minimum energy consumption with variable cycle time. For the static approach, we calculate the energy consumption with a fixed cycle time of 4494 s, i.e., the minimum cycle time, for satisfying the passenger demands. According to Table 6.4, it is shown that the dynamic approach can save energy by 7.86% compared with the static approach. On the other hand, if we take the average cycle time of 4572 s for the static approach, then it consumes almost the same amount of energy with the dynamic approach. However, it will leave 7942 − 7874 = 68 passengers not served at the first cycle, and 8011 − 7874 = 137 passengers not served in the second cycle. In general, compared with the static approach, the dynamic approach can satisfy the variable passenger demand with lower energy consumption.

6.4 Numerical Examples

93

Table 6.3 Running time distributions at optimal timetable and practical timetable Inter-station The practical running time (s) The optimal running time (s) SJ-XC XC-XH XH-JG JG-YQ YQ-WH WH-WY WY-RJ RJ-RC RC-TJ TJ-JH JH-CN CN-CQ CQ-YZ YZ-CQ CQ-CN CN-JH JH-TJ TJ-RC RC-RJ RJ-WY WY-WH WH-YQ YQ-JG JG-XH XH-XC XC-SJ

190.00 108.00 157.00 135.00 90.00 114.00 103.00 104.00 164.00 150.00 140.00 102.00 105.00 110.00 100.00 141.00 150.00 162.00 103.00 101.00 111.00 90.00 135.00 157.00 105.00 195.00

175.03 95.01 147.44 134.59 96.46 122.27 111.10 114.86 151.81 147.73 139.39 111.45 113.84 113.84 111.45 139.39 147.73 151.81 114.86 111.10 122.27 96.46 134.59 147.44 95.01 175.04

Table 6.4 Comparison between dynamic approach and static approach Cycle 1 Cycle 2 Cycle 3 Cycle 4 Passenger demand Cycle time Energy: dynamic Energy: static C = 4494 s Energy: static C = 4572 s

Cycle 5

8011

7942

7874

7807

7742

4494 374.12

4533 356.59

4572 343.07

4611 330.54

4650 319.25

374.12

374.12

374.12

374.12

374.12

343.07

343.07

343.07

343.07

343.07

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6 Dynamic Speed Control and Timetable Optimization

References 1. Li X, Lo HK (2014), Energy minimization in dynamic train scheduling and control for metro rail operations, Transportation Research Part B, 70: 269-284. 2. Pascoe R, Eichorn T (2009), What is communication-based train control? IEEE Vehicular Technology Magazine, 4(4): 16–21. 3. Wang H, Yu F, Zhu L, Tang T, Ning B (2014), Finite-state Markov modeling for wireless channels in tunnel communication-based train control systems, IEEE Transactions on Intelligent Transportation Systems, 15(3): 1083-1090. 4. Howlett PG, Pudney PJ (1995), Energy-Efficient Train Control, New York, Springer-Verlag. 5. Khmelnitsky E (2000), On an optimal control problem of train operation, IEEE Transactions on Automatic Control, 45(7): 1257–1266. 6. Boyd S, Vandenberghe L (2004), Convex Optimization, Cambridge University Press.

Chapter 7

Stochastic Speed Control and Timetable Optimization

The speed control and timetable optimization approach for subway systems has recently attracted more attention because of its exemplary achievements in energy conservation. However, most studies often ignore the spatial and temporal uncertainties of train mass. This chapter introduces a stochastic subway timetable optimization and speed control model proposed by Yang et al. [1] to minimize the total traction energy consumption, where these real-world operating conditions are explicitly considered in the model formulation and solution algorithm. Based on literature [1], we mainly answer the following three questions: (1) What is the spatial and temporal uncertainties of train mass? (2) How to formulate a two-phase stochastic programming model? (3) How to design a simulation-based genetic algorithm procedure incorporated with the optimal train control algorithm to find the optimal solution?

7.1 Problem Description In some real-world subway systems, subway managers divide the daily operation into three periods: peak hours, off-peak hours, and night hours based on the daily passenger demand profile. Trains operated during each period are typically assigned with a period-specific timetable and speed profile. However, the three assigned timetables and speed profiles are closely related. To facilitate operations and avoid faults during switching, dwell time at the same station, running time at the same inter-station, and turnaround time are all kept the same during the three operational periods. Further, the speed profile used by all trains at the same inter-station is also kept the same. Only the headways during the three operational periods are altered to accommodate the different passenger demand profiles. That is to say, the operations of subway trains are considered as cyclic and periodic (i.e., similar operational considerations are also adopted in Li and Lo [2]; Li and Lo [3]; Niu and Zhou [4]). We only need © Springer Nature Singapore Pte Ltd. 2020 X. Li and X. Yang, Subway Energy-Efficient Management, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-7785-7_7

95

96

7 Stochastic Speed Control and Timetable Optimization

to generate a single set of dwell time, running time, and speed profile, and three different headways to determine three sets of timetable and speed profiles for peak hours, off-peak hours, and night hours, respectively. As train mass is uncertain at different inter-stations and different operational periods, this chapter develops a stochastic optimization model considering the train mass at each inter-station as a stochastic variable based on distinct probabilities for three operational periods to determine the optimal timetable and speed profile. Note that this is not the same as three different deterministic optimization models for the three operational periods. Otherwise, we cannot ensure using the same dwell time, running time, and speed profile for the three operational periods with different headways to accommodate different passenger demand patterns. In the stochastic model, the timetable and speed profile of one train completing one cycle trip are optimized to minimize the total traction energy consumption. Once we obtain the optimal timetable and speed profile of one train completing one cycle trip, we can use the obtained dwell time, running time, and speed profile to determine the optimal timetable and speed profile of all trains throughout the overall operational period with assigned headways.

7.2 Model Formulation In this section, we formulate a stochastic speed control and timetable optimization model, in which the spatial and temporal uncertainties of train mass due to the stochastic passenger volume are considered.

7.2.1 Notations For a better understanding of stochastic speed control and timetable optimization model, the notations are first introduced, and some of them are indicated in Fig. 7.1. di1

Station n

ai(n+1) Inter-station (n, n+1)

Station n+1

Station 2N Time

di(n+1)

ai(2N)

Fig. 7.1 Illustration of some notations in a subway line

aiN

Time Station N

Turnaround

Turnaround

Station 1

din

ain

Station N+1

di(N+1)=aiN+tt

7.2 Model Formulation

7.2.1.1

Indices and Parameters i n acin acin acin dcin dcin dcin tcn tcn tt C v Xn x gn (x) Vn (x) F(v) B(v) r (v) m τ1n τ2n τ3n

7.2.1.2

97

train index, i = 1, 2, · · · , I station index, n = 1, 2, · · · , 2N Current time stamp of train i arriving at station n upper bound of time stamp of train i arriving at station n lower bound of time stamp of train i arriving at station n current time stamp of train i departing from station n upper bound of time stamp of train i departing from station n lower bound of time stamp of train i departing from station n upper bound of running time at inter-station (n, n + 1) lower bound of running time at inter-station (n, n + 1) turnaround time at the first station and the terminal station total travel time, i.e., the period that a train from departure at station 1 to arrival time at station 2N train speed index length of inter-station (n, n + 1) length unit gradient force per length unit x at inter-station (n, n + 1) speed restriction per length unit x at inter-station (n, n + 1) maximum traction force maximum braking force basic running resistance empty vehicle mass passengers mass on the train at inter-station (n, n + 1) during peak hours passengers mass on the train at inter-station (n, n + 1) during off-peak hours passengers mass on the train at inter-station (n, n + 1) during night hours

Variables ξn din ain k F (x, ξn ) k B (x, ξn ) vn (x, ξn ) tn (x, ξn )

stochastic train mass at inter-station (n, n + 1) optimal time stamp of train i departing from station n optimal time stamp of train i arriving at station n output rate of traction force per length unit x, k F (x, ξn ) ∈ [0, 1] output rate of braking force per length unit x, k B (x, ξn ) ∈ [0, 1] train speed per length unit x at inter-station (n, n + 1) running time per length unit x at inter-station (n, n + 1)

Remark 7.1 Note that ai(n+1) − din denotes the running time at inter-station (n, n + 1); din − ain denotes the dwell time at station n; and k F (x, ξn ), k B (x, ξn ), vn (x, ξn ), tn (x, ξn ) and ai(n+1) − din can determine the optimal speed profile at inter-station (n, n + 1).

98

7 Stochastic Speed Control and Timetable Optimization

7.2.2 Uncertainty of Train Mass Train mass covers the empty vehicle mass and passenger mass on the train. The empty vehicle mass is a fixed value. However, the passenger mass on the train is spatially and temporally uncertain, since the passenger volumes are not only spatially uncertain across different inter-stations but also temporally uncertain across different operational periods. We can see it in Fig. 7.2. We analyze the data from Beijing Yizhuang Subway Line as follows [1]. The daily operational period of this line is from 5:00 am to 11:00 pm. The peak hours are from 6:30 am to 9:30 am and from 4:30 pm to 7:30 pm, the night hours are from 9:30 pm to 11:00 pm, and the remaining operation time is off-peak hours. The average numbers of passengers on the train at each inter-station during different operational periods are shown in Fig. 7.2. Note that the Songjiazhuang Station (last station of the Yizhuang Line) is located in the city center of Beijing that connects to the Beijing subway network; hence, the passenger volume increases along the up direction to the Songjiazhuang Station. In addition, the Yizhuang Station had not yet opened to the public on March 23, 2014; hence, there are only hired staffs (for counting the number of passengers) on the train between the Yizhuang and Ciqu stations. As described in the Problem Statement, we want to generate a set of dwell time, running time, and speed profile suitable for the overall operational period. We use p1n , p2n , and p3n to denote the probabilities for peak hours, off-peak hours, and night hours. Thus, the train mass at inter-station (n, n + 1) can be defined as a discrete stochastic variable, i.e.,

Fig. 7.2 Average numbers of passengers on the train at each inter-station

7.2 Model Formulation

99

⎧ ⎪ ⎪ m + τ1n , with probability p1n , ⎪ ⎨ ξn = m + τ2n , with probability p2n , ⎪ ⎪ ⎪ ⎩ m + τ3n , with probability p3n ,

(7.1)

where m is the empty vehicle mass; τ1n , τ2n , and τ3n are the masses of passengers on the train at inter-station (n, n + 1) during peak hours, off-peak hours, and night hours, respectively; p1n , p2n , and p3n are positive real numbers between (0, 1) satisfying p1n + p2n + p3n = 1. As the train mass at each inter-station is a stochastic variable, there are 2N -2 different stochastic variables in the formulation.

7.2.3 System Constraints The intention of the integrated optimization model is to determine the optimal timetable and speed profile. The optimality of the timetable and speed profile is interdependent (i.e., both sets of decision variables are driven by the objective function given below). Therefore, we divide the modeling process into two phases: Given the speed profile, the first phase determines the timetable by scheduling the arrival and departure times for each station to obtain the dwell time at each station and the running time at each inter-station and estimates the total traction energy consumption for one cycle trip (i.e., departing from station 1 arriving at station 2N ); the second phase determines the speed profile for each inter-station with the scheduled arrival and departure times from the timetable and calculates the traction energy consumption for each inter-station. Remark 2 Train mass at each inter-station is a stochastic variable based on distinct probabilities for peak hours, off-peak hours, and night hours. Therefore, the calculation of the traction energy consumption for each inter-station is a stochastic evaluation.

7.2.3.1

Schedule Phase

For the schedule phase, the model should satisfy the following constraints: (1) Departure time constraints For each n ∈ [1, 2N − 1], the optimal departure time for train i at station n should be an integer and valued between the lower and upper bounds of departure time for train i at station n, i.e., din ∈ {0, 1, 2, · · · , C}, dcin ≤ din ≤ dcin , ∀ n ∈ [1, N − 1] ∪ [N + 1, 2N − 1].

(7.2)

100

7 Stochastic Speed Control and Timetable Optimization

(2) Arrival time constraints For each n ∈ [2, 2N ], the optimal arrival time for train i at station n should be an integer and valued between the lower and upper bounds of arrival time for train i at station n, i.e., ain ∈ {0, 1, 2, · · · , C}, acin ≤ ain ≤ acin , ∀ n ∈ [2, N ] ∪ [N + 2, 2N ]. (7.3) (3) Running time constraints For each n ∈ [1, N − 1] ∪ [N + 1, 2N − 1], the optimal running time for train i at inter-station (n, n + 1) should be valued between the lower and upper bounds of the current running time for train i at inter-station (n, n + 1), i.e., tcn ≤ |ai(n+1) − din | ≤ tcn , ∀ n ∈ [1, N − 1] ∪ [N + 1, 2N − 1].

(7.4)

(4) Dwell time constraints For each n ∈ [2, N − 1] ∪ [N + 2, 2N − 1], the dwell time for train i at station n should be equal to the current dwell time, i.e., din − ain = dcin − acin , ∀ n ∈ [2, N − 1] ∪ [N + 2, 2N − 1].

(7.5)

(5) Turnaround time constraints The turnaround time in the optimal timetable should be equal to the current turnaround time, i.e., (7.6) di(N +1) − ai N = tt , where the dwell time at stations N and N + 1 is included in the turnaround time. (6) Total travel time constraints The total travel time for one cycle in the optimal timetable should be equal to the current total travel time for one cycle, i.e., ai(2N ) − di1 = C.

7.2.3.2

(7.7)

Speed Profile Phase

Given an inter-station (n, n + 1) with length X n and scheduled running time Tn = ai(n+1) − din , denote tn (x, ξn ), vn (x, ξn ), gn (x), and Vn (x) as the running time, train speed, gradient force, and speed restriction per length unit x, respectively. For each n ∈ [1, N − 1] ∪ [N + 1, 2N − 1], the model in the speed profile phase should satisfy the following constraints: (7) Speed restriction constraints The train speed should be no more than the speed restriction per length unit x and should be 0 at the starting and ending points, i.e.,

7.2 Model Formulation

101

E[vn (0, ξn )] = 0, E[vn (X n , ξn )] = 0, ∀ n ∈ [1, N − 1] ∪ [N + 1, 2N − 1], (7.8) E[vn (x, ξn )] ≤ Vn (x), ∀ x ∈ (0, X n ), n ∈ [1, N − 1] ∪ [N + 1, 2N − 1]. (7.9) (8) Complete running time constraints The accumulated running time per length unit x at inter-station (n, n + 1) should be equal to the scheduled running time at inter-station (n, n + 1), i.e.,  E[

Xn

tn (x, ξn )dx] = Tn , ∀ n ∈ [1, N − 1] ∪ [N + 1, 2N − 1].

(7.10)

0

(9) State variable constraints Based on the optimal train control theory, the general equations of motion are E[

1 dtn (x, ξn ) − ] = 0, ∀ n ∈ [1, N − 1] ∪ [N + 1, 2N − 1], (7.11) dx vn (x, ξn )

dvn (x, ξn ) − dx k F (x, ξn )F[vn (x, ξn )] + k B (x, ξn )B[vn (x, ξn )] + r [vn (x, ξn )] + gn (x) ] = 0, ξn vn (x, ξn ) ∀ n ∈ [1, N − 1] ∪ [N + 1, 2N − 1], (7.12) where the maximum traction force F(v) is non-negative; the maximum braking force B(v) is non-positive; the basic running resistance r (v) is negative; the gradient force gn (x) can be positive, zero, or negative. Equations (7.11) and (7.12) denote the time state and speed state per length unit. The complexity of evaluating these equations is much increased since the maximum traction force, maximum braking force, and basic running resistance are all variable according to the train speed variation. Take the trains operated in the Beijing Yizhuang Subway Line as an example; these real-world operating conditions are graphically depicted in Fig. 7.3 along with the formulas. E[

7.2.4 Objective Function The operations of subway trains are considered as periodic by the following two points: First, trips from one train at different cycles are distributed with the same dwell time at stations and the same speed profile at inter-stations; second, trips from different trains are distributed with the same dwell time at stations and the same speed profile at inter-stations. For simplicity, we consider the total traction energy consumption of one train completing one cycle trip (i.e., departing from station 1 arriving at station 2N ) for calculating the objective function. Therefore, the total

102

7 Stochastic Speed Control and Timetable Optimization

Fig. 7.3 Basic running resistance and maximum traction and braking forces for trains operated in the Beijing Yizhuang Subway Line

traction energy consumption of all trains throughout the overall operational period can be simply summed. We first describe the traction energy consumption of one train at one inter-station. For each n ∈ [1, N − 1] ∪ [N + 1, 2N − 1], the traction energy consumption of train i at inter-station (n, n + 1) is formulated as follows:  J (din , ai(n+1) , ξn ) =

Xn

k F (x, ξn )F[vn (x, ξn )] dx,

(7.13)

0

where x is per length unit; X n is the length of inter-station n; F[vn (x, ξn )] is the maximum traction force per length unit x (see Fig. 7.3b); k F (x, ξn ) is the output rate of traction force per length unit x; k F (x, ξn )F[vn (x, ξn )] is the final output traction force on train i per length unit x. The traction energy consumption is obtained using the final output traction force to integrate the inter-station length. Remark 7.2 Equation (7.13) shows that the traction energy consumption of train i at each inter-station contains an independently distributed discrete random variable. The total traction energy consumption of train i completing one cycle trip is the sum of the traction energy consumption of train i at each inter-station. Therefore, it contains 2N − 2 independently distributed discrete random variables. For simplicity, we denote d = {din |n = 1, 2, · · · , 2N − 1}, a = {ain |n = 2, 3, · · · , 2N } and ξ = {ξn |n = 1, 2, · · · , N − 1, N + 1, · · · , 2N − 1}. Based on the lin-

7.2 Model Formulation

103

earity of the expected value, the expected value of the total traction energy consumption of train i completing one cycle trip (i.e., departing from station 1 arriving at station 2N ) is the sum of the expected value of traction energy consumption of train i at each inter-station. It is formulated as follows: E[JTotal (d, a, ξ)] = E[

N −1 

N −1 

J (din , ai(n+1) , ξn )]

n=N +1

n=1

=

2N −1 

J (din , ai(n+1) , ξn ) +

E[J (din , ai(n+1) , ξn )] +

2N −1 

E[J (din , ai(n+1) , ξn )].

n=N +1

n=1

(7.14) This final objective function explicitly considers the uncertain train mass and the variable traction force.

7.2.5 Stochastic Optimization Model Based on the above analysis, the integrated timetable and speed profile optimization problem with uncertain train mass is formulated as a two-phase stochastic programming model. Since the objective function contains stochastic variables, we take the expected value criterion to minimize the average total traction energy consumption value. The two-phase stochastic programming model is formulated as follows: ⎧ min E[JTotal (d, a, ξ)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s.t. Constraints (7.2) − (7.7) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ where the expected value J (din , ai(n+1) , ξn ) of the JTotal (d, a, ξ) ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(7.15)

is computed by  E[J (din , ai(n+1) , ξn )] = min E{

Xn

k F (x, ξn )F[vn (x, ξn )] dx}

0

s.t. Constraints (7.8) − (7.12).

We can see that the first phase determines the timetable by scheduling the arrival and departure times (i.e., a and d) constrained by the speed profile (i.e., k F (x, ξn ), k B (x, ξn ), vn (x, ξn ), and tn (x, ξn )), and the second phase determines the speed profile constrained by the scheduled arrival and departure times.

104

7 Stochastic Speed Control and Timetable Optimization

7.2.6 Formulation Complexity Solving the two-phase stochastic programs usually has a number of difficulties, and we analyze the complexity of the formulated two-phase stochastic programming model. For the first phase, we need to determine the timetable by scheduling the arrival and departure times for each station, and the number of all possible arrival and departure times combinations is  = [(acin − acin + 1)(dcin − dcin + 1)]2N −2 .

(7.16)

In the numerical examples (to be shown in Sect. 7.5), the number  is 8126 , which, clearly, is a considerably large number. Hence, it is challenging to find the optimal arrival and departure times from all these possible combinations. For the second phase, we need to determine an energy-efficient speed profile for each inter-station with the scheduled arrival and departure times from the timetable. It has two levels of difficulties: (1) the traction energy consumption and energyefficient speed profile are nonlinear relative to the arrival and departure times due to the variable traction force, braking force, and basic running resistance shown in Fig. 7.3; and (2) we need to repeat a number of times to solve the stochastic program for obtaining the expected value considering the uncertain train mass. Obviously, the formulated two-phase stochastic programming model is very complex, and it is very difficult to find the optimal solution using the classical optimization methods such as branch-and-bound algorithm and Newton algorithm.

7.3 Solution Method Evolutionary algorithms, particularly the genetic algorithm, have shown to be effective in solving complex problems. Due to its extensive generality, strong robustness, high efficiency, and practical applicability, genetic algorithm has been widely adopted as a numerical method for solving many subway optimization problems [5]. In this section, we first use Fig. 7.4 to show the whole structure of the solution method to the two-phase stochastic program. Then we design a simulation-based genetic algorithm procedure shown in Fig. 7.5 to find the optimal solution [1]. The simulation-based genetic algorithm procedure mainly includes two modules, i.e., simulation module and genetic algorithm module. The simulation module is designed based on the optimal train control algorithm to determine the energyefficient speed profile and to calculate the expected value of traction energy consumption for each inter-station. The genetic algorithm module (including the representation, selection, crossover, and mutation operations) is adopted to determine the timetable by scheduling the arrival and departure times for each station.

7.3 Solution Method

105

First phase: Scheduling Timetable

Start

Arrival time at each station

Update

Departure time at each station

Distribute

No

Best?

Yes

End

Expected value of total tractive energy consumption

Feedback

Second phase: Speed profile Running time for inter-station (1, 2)

Energy-efficient speed profile for inter-station (1, 2)

Expected value of tractive energy consumption for inter-station (1, 2)

Running time for inter-station (N-1, N)

Energy-efficient speed profile for inter-station (N-1, N)

Expected value of tractive energy consumption for inter-station (N-1, N)

Running time for inter-station (N+1, N+2)

Energy-efficient speed profile for inter-station (N+1, N+2)

Expected value of tractive energy consumption for inter-station (N+1, N+2)

Running time for inter-station (2N-1, 2N)

Energy-efficient speed profile for inter-station (2N-1, 2N)

Expected value of tractive energy consumption for inter-station (2N-1, 2N)

Fig. 7.4 Structure of the solution method

7.3.1 Simulation Simulation module presents the detailed procedure to obtain the energy-efficient speed profile for subway inter-station with a given length and running time. Proposition 7.1 (Existence of an Optimal Speed Profile) Let Tn ≥ Tnmin (Tnmin denotes the minimum running time at inter-station (n, n + 1)). There exists a control scheme k F = k F (x, ξn ) and k B = k B (x, ξn ), a corresponding speed sequence v = vn (x, ξn ), and a time sequence t = tn (x, ξn ) satisfying the differential constraints vv  = k F /v + k B + r (v) + g and t  = 1/v, the initial conditions v(0) = 0 and v(X n ) = 0 and time constraints t (0) = 0 and t (X n ) = 0 in such a way that the X cost J = 0 n k F /v dx is minimized. Outline of Proof The Hamiltonian function is defined as follows: H=

β − kF α + 2 [k F + k B v + r (v)v + gv], v v

where the adjoint variables α = α(x) and β = β(x) are solutions to the differential equations α =

β − kF α + 3 [2k F + k B v + r (v)v − r  (v)v 2 + gv] and β  = 0. 2 v v

106

7 Stochastic Speed Control and Timetable Optimization

1. Define maximum number of generations max_generation (index: j1) 2. Define population size pop_size (index: j2) 3. Define number of inter-stations of 2N-2 (index: n) 4. Generate initial population

j1 = 1

j 1 = j1 + 1

j2 = 1 j 2 = j2 + 1 n=1 n=n+1 Simulation module Yes n > 2N-2

GA module 1. Representation 2. Selection 3. Crossover 4. Mutation

No

Yes j2 > pop_size

No

Yes j1 > max_generation

No

Yes Report final solutions

Fig. 7.5 Flowchart of the simulation-based genetic algorithm procedure

We wish to maximize the Hamiltonian subject to the constraints k F ∈ [0, 1] and k B ∈ [0, 1]. Thus, we define a Lagrangian function H = H + ρk F + σ(1 − k F ) + τ k B + ω(1 − k B ), where ρ, σ, τ and ω are the Lagrange multipliers. Maximizing H subject to the given constraints, there exists five possible phases of the optimal control (Howlett and Pudney [6]): (1) maximum acceleration: α > v, k F = 1 and k B = 0; (2) speed-holding: α = v, 0 ≤ k F ≤ 1 and k B = 0;

7.3 Solution Method

107

(3) coasting: 0 < α < v, k F = 0 and k B = 0; (4) partial braking: α = 0, k F = 0 and 0 ≤ k B ≤ 1; (5) maximum braking: α < 0, k F = 0 and k B = 1. At the same time, Howlett and Pudney [6] also proved that the speed-holding phase only occurs in long inter-stations, and the partial braking phase cannot occur. The inter-stations in subway systems are often with short travel distance and approximatively constant gradient track (i.e., gradients only have small changes within an inter-station as the inter-station length is short). Howlett and Pudney [6], Albrecht et al. [7], and Albrecht et al. [8] have shown that the most energy-efficient control strategy for a subway inter-station only contains the maximum acceleration, coasting, and maximum braking. Remark 7.3 For the optimal train control strategy, the initial results by Milroy et al. [9] suggested that an energy-efficient speed profile should consist of three phases: maximum acceleration, coasting, and maximum braking. At this early stage, it was not understood that the speed-holding phase should be used in some conditions. Then Lee et al. [10] concluded the speed-holding phase should be included after the maximum acceleration phase when the running time is sufficient. This optimal control strategy is widely used in general railway systems (Howlett [11]). Later, the Transport Control Group at the South Australian Institute of Technology developed the urban railmiser system (Howlett and Pudney [6]) and showed that the speed-holding phase only occurs in long inter-stations. That is to say, the most energy-efficient control strategy only contains the maximum acceleration, coasting and maximum braking for a typical subway journey inter-station with short travel distance. For the key principles of optimal train control strategies, readers can refer to the recent comprehensive review by Albrecht et al. [7, 8]. Given an inter-station (n, n + 1) with length X n and running time Tn , we divide the inter-station into many units (e.g.., 1 m). For simplicity, we denote vn (x, ξn ) = {vn (x, ξn )|x = 1, 2, · · · , X n }. We define X 1 as the switching point from the maximum acceleration phase to the coasting phase, and X 2 as the switching point from the coasting phase to the maximum braking phase. The procedure for obtaining the energy-efficient speed sequences of the inter-station (n, n + 1) is described as follows: Step 1. Set a matrix U = 0 and a real number W = 0. Step 2. Generate ξn based on the probabilities. Step 3. Set X 1 = 0 and X 2 = X n . Step 4. Set X 1 = X 1 + 1, X 2 = X 2 − 1, x = 0 and vn (x, ξn ). Step 5. For 0 ≤ x < X 1 , generate vn2 (x + 1, ξn ) − vn2 (x, ξn ) = 2(F[vn (x, ξn )] + r [vn (x, ξn )] + gn (x)). For X 1 ≤ x < X 2 , generate vn2 (x + 1, ξn ) − vn2 (x, ξn ) = 2(r [vn (x, ξn )] + gn (x)). For X 2 ≤ x < X n , generate vn2 (x + 1, ξn ) − vn2 (x, ξn ) = 2(B[vn (x, ξn )] + r [vn (x, ξn )] + gn (x)). ξn ) = 1/vn (x, ξn ). Step 6. For 0 < x ≤ X n , calculate tn (x, n Step 7. If vn (X n , ξn ) = 0 and Tn = x=X x=1 tn (x, ξn ), obtain vn (x, ξn ) and J (din , ai(n+1) , ξn ). Otherwise, go to step 4.

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7 Stochastic Speed Control and Timetable Optimization

di1 di2

di(2N-1) ai2 ai3

ai(2N) Decision variables

l1

l2

l2N-2 Chromosome L

Fig. 7.6 Structure of a chromosome

Step 8. Calculate U = U + vn (x, ξn ) and W = W + J (din , ai(n+1) , ξn ). Step 9. Repeat step 2 to step 8 for Y times, where Y is a sufficiently large positive integer. Step 10. Return vn (x, ξn ) = U/Y and E[J (din , ai(n+1) , ξn )] = W/Y .

7.3.2 Genetic Algorithm Genetic algorithm usually starts with an initial set of randomly generated feasible solutions encoded as chromosomes called a population. The decision variables, i.e., departure time d = {din |n = 1, 2, · · · , 2N − 1} and arrival time a = {ain |n = 2, 3, · · · , 2N }, constitute a chromosome L = {ln |n = 1, 2, · · · , 2N − 2} (see Fig. 7.6). A new population of chromosomes is generated following the evaluation, selection, crossover, and mutation operations. Genetic algorithm will terminate after a given number of iterations of the above steps. The process can be summarized as follows: Step 1. Initialize parameters: population size pop_si ze, crossover probability Pc , mutation probability Pm and max_generation. Set generation index j1 = 1. Step 2. Initialize pop_si ze feasible chromosomes as the initial population. Step 3. Calculate the evaluation function values for all chromosomes. Step 4. Select the chromosomes by spinning the roulette wheel. Step 5. Produce the next generation through crossover and mutation operations. Step 6. If j1 = max_generation, return the best found solution. Otherwise, set j1 = j1 + 1, and go to step 3.

7.4 Numerical Examples In this section, we present two numerical examples based on the operation data (e.g.., speed restrictions, gradients, vehicle traction, and braking forces, basic running resistance, passengers demand, etc.) from the Beijing Yizhuang Subway Line to examine the practicability of the developed two-phase stochastic programming model to a real-world subway line [1].

7.4 Numerical Examples

109

We first provide the real-world operation data of the Beijing Yizhuang Subway Line and then present two cases to illustrate the practicability of the developed model. Example 7.1 shows the optimal timetable and speed profile of the developed model using the provided real-world operation data and makes a comparison with the current timetable and speed profile. Example 7.2 makes a comparison between the previous deterministic integrated optimization model (Li and Lo [2]) and the developed stochastic integrated optimization model (considering the uncertainty of train mass and the variability of traction force, braking force, and basic running resistance). Remark 5 Abbreviation of stations name: Songjiazhuang (SJZ), Xiaocun (XC), Xiaohongmen(XHM), Jiugong (JG), Yizhuangqiao (YZQ), Wenhuayuan (WHY), Wanyuan (WY), Rongjing (RJ), Rongchang (RC), Tongjinan (TJ), Jinghai (JH), Ciqunan (CQN), Ciqu (CQ), Yizhuang (YZ), Lower bound of the running time (LR), Upper bound of the running time (UR). The current length, running time, the bounds of running time for each interstation, and the dwell time for each station are provided in Table 7.1. The empty vehicle mass is 199,000 kg, and the masses of passengers on the train at each interstation during different periods are provided in Table 7.2. The parameters in the basic running resistance, maximum traction force and maximum braking force formulas are λ1 = 3779.90 N, λ2 = 1.57, λ3 = 8.52 ∗ 10−4 , λ4 = 310, 000 N, λ5 = 86, 228 N, λ6 = 960.00, λ7 = 260, 000 N, λ8 = 6265.30, vα = 10.00 m/s, vmax = 22.22 m/s and vβ = 16.67 m/s. Example 7.1 This example provides the results of the developed two-phase stochastic programming model and makes a comparison with the current operating timetable and speed profile to illustrate the effectiveness of the developed model on energy conservation. We test the behavior of the simulation-based genetic algorithm procedure on a real case study by running for three times with pop_si ze = 30, max_generation = 120, Pc = 0.6 and Pm = 0.15 on a Windows 8.1 platform of personal computer with a processor frequency of 2.4 GHz and memory size of 8 GB. We perform the solution procedure with pop_si ze = 30, max_generation = 60, Pc = 0.6 and Pm = 0.15 in the following cases. We obtain the optimal timetable presented in Table 7.3 and also shown in Fig. 7.7 in comparison with the current timetable, and the optimal speed profile shown in Fig. 7.8 in comparison with the current speed profile. Based on the optimal timetable and speed profile, the total traction energy consumptions of one train for completing one cycle trip during peak hours, off-peak hours, and night hours for the optimal timetable and speed profile are 210.84 kW·h, 200.28 kW·h, and 191.62 kW·h, respectively. They are also compared with those of the current timetable and speed profile in Table 7.4. The results show that the total traction energy consumptions are reduced by 10.66%, 9.94% and 9.13% during peak hours, off-peak hours, and night hours, respectively. As shown in Figs. 7.7 and 7.8, the maximum train speed, acceleration rate, and deceleration rate of the optimal speed profile at each inter-station are larger than

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7 Stochastic Speed Control and Timetable Optimization

Table 7.1 Current length, current running time, the bounds of running time and dwell time for the Beijing Yizhuang Subway Line Inter-station Length (m) Running time LR (s) UR (s) Dwell time (s) (s) Up direction YZ-CQ CQ-CQN CQN-JH JH-TJN TJN-RC RC-RJ RJ-WY WY-WHY WHY-YZQ YZQ-JG JG-XHM XHM-XC XC-SJZ Down direction SJZ-XC XC-XHM XHM-JG JG-YZQ YZQ-WHY WHY-WY WY-RJ RJ-RC RC-TJN TJN-JH JH-CQN CQN-CQ CQ-YZ

1334 1286 2086 2265 2338 1354 1280 1538 993 1982 2366 1275 2631

110 100 141 150 162 103 101 111 90 135 157 105 195

106 96 137 146 158 99 97 107 86 131 153 101 191

114 104 145 154 166 107 105 115 94 139 161 109 199

45 35 30 30 30 30 30 30 35 30 30 30 –

2631 1275 2366 1982 993 1538 1280 1354 2338 2265 2086 1286 1334

190 108 157 135 90 114 103 104 164 150 140 102 105

186 104 153 131 86 110 99 100 160 146 136 98 101

194 112 161 139 94 118 107 108 168 154 144 106 109

30 30 30 35 30 30 30 30 30 30 35 45 –

those of the current speed profile. It is because the optimal speed profile chooses larger traction and braking forces to reduce the traction energy consumption for each inter-station. Moreover, the optimal timetable increases the running time at interstations (e.g.., XHM-XC, XC-SJZ, SJZ-XC, XC-XHM) with a larger train mass while decreases the running time at inter-stations (e.g.., YZ-CQ, CQ-CQN, CQN-CQ, CQYZ) with a smaller train mass, such that the total traction energy consumption can be further reduced. These changes in the timetable and speed profile, albeit minor in

7.4 Numerical Examples

111

Table 7.2 Masses of passengers on the train at each inter-station during different periods (unit: kg) Inter-station Peak hours Off-peak hours Night hours Up direction YZ-CQ CQ-CQN CQN-JH JH-TJN TJN-RC RC-RJ RJ-WY WY-WHY WHY-YZQ YZQ-JG JG-XHM XHM-XC XC-SJZ Down direction SJZ-XC XC-XHM XHM-JG JG-YZQ YZQ-WHY WHY-WY WY-RJ RJ-RC RC-TJN TJN-JH JH-CQN CQN-CQ CQ-YZ

1680 6630 7620 14460 28140 33990 35550 38070 42750 47910 57960 59100 59910

1680 5010 5640 8820 18330 22590 25890 28590 30120 33210 36360 37290 39810

1680 2940 3000 4620 8160 10260 11880 12840 14280 15600 16320 16800 17280

69150 65790 64290 56670 51900 47850 38940 32310 20640 9840 6210 5760 1680

46170 43740 42090 35040 31890 30120 25890 23610 17460 8790 5940 5790 1680

34680 33120 31260 24720 21540 19680 18120 16620 13680 6300 4800 4560 1680

Figs. 7.7 and 7.8, can achieve an average of around 10% saving on the total traction energy consumption. The results show that, in comparison with the current timetable and speed profile, the optimal timetable and speed profile: (1) reduces the maximum train speed at each inter-station by choosing larger traction and braking forces, such that the traction energy consumption for each inter-station can be reduced; and (2) increases the running time at inter-stations with a larger train mass while decreases the running time at inter-stations with a smaller train mass, such that the total traction energy consumption can be further reduced.

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7 Stochastic Speed Control and Timetable Optimization

Table 7.3 Optimal timetable for one cycle for the Beijing Yizhuang Subway Line Up station YZ CQ CQN JH TJN RC Arrival – 107 250 423 600 794 time (s) Departure 0 152 285 453 630 824 time (s) Up Station WY WHY YZQ JG XHM XC Arrival 1055 1194 1316 1489 1679 1817 time (s) Departure 1085 1224 1351 1519 1709 1847 time (s) Down SJZ XC XHM JG YZQ WHY station Arrival – 2480 2621 2811 2978 3105 time (s) Departure 2286 2510 2651 2841 3013 3135 time (s) Down RJ RC TJN JH CQN CQ Station Arrival 3384 3517 3709 3886 4053 4186 time (s) Departure 3414 3547 3739 3916 4088 4231 time (s)

RJ 926 956 SJZ 2046 – WY 3250 3280 YZ 4332 4572

Example 7.2 This example makes a comparison between the integrated optimization model (Li and Lo [2]) and the proposed stochastic integrated optimization model. For convenience, the two models are, respectively, named as the deterministic model and the stochastic model for the comparison. For the deterministic model, we solve the optimal timetable and speed profile with the constant train mass (287,080 kg), traction force (235,406 N), braking force (109,090 N), and basic running resistance (5742 N) (i.e., these variables take the same values in all inter-stations and during all periods). The calculation is based on the method provided in Li and Lo [2] by removing the calculation of regenerative braking energy. For the stochastic model, we solve the optimal timetable and speed profile with the uncertain train mass and variable traction force, braking force, and basic running resistance from the real-world operation data. The obtained total traction energy consumptions during different operational periods for the two models (see the last row in Table 7.5) show that the developed stochastic model reduces the total traction energy consumptions by 3.35%, 3.12%, and 3.04% during peak hours, off-peak hours, and night hours, respectively, in comparison with the deterministic model. The obtained traction energy consumption for each inter-station during different operational periods for the two models are presented in Table 7.5 and shown in Fig. 7.9. We can see that although the traction energy consumptions for the inter-

7.4 Numerical Examples

Fig. 7.7 Comparison of the current and optimal timetables for one cycle

Fig. 7.8 Comparison of the current and optimal speed profiles for one cycle

113

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Table 7.4 Total traction energy consumptions of one train completing one cycle trip during different periods Peak hours Off-peak hours Night hours Current timetable and 236.01 kW·h speed profile Optimal timetable and 210.84 kW·h speed profile Energy saving rate 10.66%

222.38 kW·h

210.87 kW·h

200.28 kW·h

191.62 kW·h

9.94%

9.13%

Table 7.5 Comparisons on traction energy consumption (Unit: kW·h) between the deterministic and stochastic models Deterministic model Stochastic model Inter-station Peak Off-peak Night Peak Off-peak Night YZ-CQ CQ-CQN CQN-JH JH-TJ TJ-RC RC-RJ RJ-WY WY-WHY WHY-YZQ YZQ-JG JG-XHM XHM-XC XC-SJZ SJZ-XC XC-XHM XHM-JG JG-YZQ YZQ-WHY WHY-WY WY-RJ RJ-RC RC-TJ TJ-JH JH-CQN CQN-CQ CQ-YZ

6.6693 6.5244 7.5216 9.1275 8.9313 8.1041 7.7645 8.7056 5.9267 10.0860 12.0441 7.0129 8.4333 10.5099 7.3760 10.8283 11.5280 6.4479 9.1292 7.4857 8.2123 8.7448 8.9211 9.7614 6.0210 6.3260

6.6693 6.4525 7.4479 8.8095 8.5625 7.6059 7.3350 8.2284 5.4590 9.3524 11.0350 6.2434 7.7468 9.6850 6.6742 9.8069 10.5150 5.9549 8.4098 6.9683 7.9090 8.6575 9.0053 9.8459 6.0212 6.3260

6.6693 6.3775 7.3681 8.6082 8.1898 7.0994 6.8737 7.5815 5.1222 8.5850 10.0950 5.6131 7.0433 9.2464 6.4243 9.2731 9.9442 5.6383 8.0355 6.7325 7.6776 8.4595 8.8755 9.6944 6.0776 6.3260

6.7870 6.5244 7.5216 9.2254 8.4466 7.9298 7.7645 8.7056 5.2775 9.2944 11.2526 6.1893 7.9406 9.8643 6.7029 10.1000 10.7514 5.8863 8.6220 6.9832 8.0975 8.7448 9.1642 9.9821 6.3967 6.6901

6.7870 6.4525 7.4479 9.0057 8.0780 7.4299 7.3350 8.2284 5.0236 8.6048 10.2894 5.5486 7.2520 9.1515 6.1335 9.1236 9.8345 5.3931 7.9568 6.5895 7.7942 8.6575 9.1020 9.9776 6.3969 6.6901

6.7870 6.3775 7.4234 8.8049 7.7610 6.9219 6.8737 7.5815 4.6136 7.9420 9.5392 4.9826 6.6742 8.7129 5.7462 8.6374 9.4072 5.1455 7.5244 6.3537 7.4472 8.4595 8.9724 9.9151 6.3274 6.6901

7.4 Numerical Examples

115

Fig. 7.9 Comparisons on traction energy consumption between the deterministic and stochastic models

stations with a smaller train mass (e.g.., CQN-CQ and CQ-YZ, see Fig. 7.9) of the stochastic model are a bit increased in comparison with the deterministic model, the total traction energy consumption is reduced. It is because the stochastic model decreases the running time at inter-stations with a smaller train mass and allots the saved time to the inter-stations with a larger train mass. The results show that, in comparison with the deterministic model, the developed stochastic model: (1) can reduce the total traction energy consumption during peak hours, off-peak hours, and night hours; and (2) decreases the running time at interstations with a smaller train mass and reallocates the saved time to the inter-stations with a larger train mass, such that the total traction energy consumption can be reduced.

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