Uncertainty and Operations Research

Xiang LiXin 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-worldapplications, 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, manage-ment, manufacturing, supply chain, transportation, among others. This book seriesaims to provide a global forum for advancing the analysis, understanding,development, and practice of uncertainty theory and operations research for solvingeconomic, engineering, management, and social problems.

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Xiang Li • Xin Yang

Subway Energy-EfficientManagement

123

Xiang LiSchool of Economics and ManagementBeijing University of Chemical TechnologyBeijing, China

Xin YangState Key Laboratory of Rail Traffic Controland SafetyBeijing Jiaotong UniversityBeijing, China

ISSN 2195-996X ISSN 2195-9978 (electronic)Uncertainty and Operations ResearchISBN 978-981-15-7784-0 ISBN 978-981-15-7785-7 (eBook)https://doi.org/10.1007/978-981-15-7785-7

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Preface

Subway is an electric passenger railway, which is operated either in undergroundtunnels or on elevated rails. In the past decades, the subway has received rapiddevelopment in China due to its high capacity, punctuality, and reliability. Up toJune 2019, there have been 39 cities opening 174 urban rail transit lines, amongwhich Shanghai operates the largest subway network, and Beijing operates thebusiest subway network in the world. On December 30, 2017, Beijing subwayYanfang line started operation, the first domestically developed automated subwayin China.

For subway operations management, the optimal train control and schedulingmethods have been extensively studied with various objectives, among whichenergy saving has attracted much attention from both researchers and practitionerson account of the rising energy prices and environmental concerns. The mostpopular energy-efficient management approaches include speed control and time-table optimization; the former is a type of commonly used methods while the latteris a class of emerging method, which respectively contribute to reducing tractionenergy consumption and improving regenerative energy absorption.

Speed control approach optimizes the time-speed profile for trains at inter-stationto minimize traction energy consumption. The literature on energy-efficient speedcontrol can date back to the 1960s. In 1968, Ishikawa proposed the first optimaltrain control model to determine the most energy-efficient speed profile. After that,both theoretical analyses and heuristic algorithms have been given, among whichHowlett and his Scheduling and Control Group made extraordinary contributionsto laying the foundation for energy-efficient speed control theory or energy-efficientoperation theory.

Timetable optimization approach synchronizes the accelerating phases andbraking phases of adjacent trains located in the same substation to maximize theregenerative energy absorption. Over the past years, a series of timetable opti-mization models have been formulated. For example, Ramos et al. (2007) firstpresented a timetabling problem that aims to maximize the overlapping timebetween accelerating actions and braking actions of adjacent trains, so that theaccelerating trains can absorb the regenerative energy from braking trains as much

v

as possible. In 2013, Li and Yang measured the regenerative energy absorption asthe integral of the minimum profile between traction energy and regenerativeenergy at the overlapping time and formulated an energy-efficient timetablingmodel.

Essentially, timetable optimization and speed control are two closely relatedprocesses on energy saving. The timetabling process allocates the travel timeamong inter-stations, which significantly influences the traction energy consump-tion, while speed control process determines the accelerating time and braking timeat inter-stations, which profoundly influences the regenerative energy absorption.Therefore, some researchers studied the integrated optimization on timetable andspeed profile to minimize the net energy consumption, i.e., the difference betweentraction energy consumption and regenerative energy absorption. For example, Liand Lo (2014) proposed a mixed integer programming model and designed agenetic algorithm to minimize the net energy consumption, which integrally opti-mizes the accelerating time and braking time at inter-stations, dwell time, cycletime, and trip frequency. As extensions, the dynamic optimization methods andstochastic optimization methods on timetable and speed profile were also studied.

The purpose of this book is to provide a powerful tool to handle the subwayenergy-efficient management problems. It provides a comprehensive presentationon train timetable optimization and speed control models with the objective ofenergy saving. The methods presented here are designed for but not limited to thesubway system. It can be extended and applied to the timetabling and speed controlfor high-speed trains and other types of passenger trains. The book is suitable forresearchers, engineers, and students in the fields of transportation science, man-agement science, and so on. The readers will learn numerous new modeling ideason reducing traction energy consumption and improving regenerative energyabsorption and find this work a useful reference.

Beijing, China Xiang LiMay 2020 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).

vi Preface

Contents

1 Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Energy-Efficient Speed Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Analytical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Energy-Efficient Timetable Optimization . . . . . . . . . . . . . . . . . . . 41.2.1 Overlapping Time Maximization . . . . . . . . . . . . . . . . . . . 51.2.2 Regenerative Energy Maximization . . . . . . . . . . . . . . . . . . 5

1.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Energy-Efficient Speed Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Speed Control Model without Speed Limits . . . . . . . . . . . 142.2.3 Optimal Control Model with Speed Limits . . . . . . . . . . . . 17

2.3 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Timetabling with Overlapping Time Maximization . . . . . . . . . . . . . . 253.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.3 Intermediate Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.4 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.5 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.6 Cooperative Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

vii

3.2.7 Overlapping Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.8 Overlapping Time Maximization Model . . . . . . . . . . . . . . 33

3.3 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 Representation Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.3 Evaluation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.4 Selection Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.5 Crossover Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.6 Mutation Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.7 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Timetabling with Regenerative Energy Maximization . . . . . . . . . . . . 454.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.2 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.3 Timetabling Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.4 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.5 Regenerative Energy Maximization Model . . . . . . . . . . . . 52

4.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.2 Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Integrated Speed Control and Timetable Optimization . . . . . . . . . . . 635.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.2 Net Energy Consumption . . . . . . . . . . . . . . . . . . . . . . . . . 675.2.3 Integrated Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.4 Integrated Optimization Model . . . . . . . . . . . . . . . . . . . . . 71

5.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3.1 Representation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.2 Initialization Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.3 Evaluation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.4 Selection Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.5 Crossover Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.6 Mutation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3.7 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

viii Contents

6 Dynamic Speed Control and Timetable Optimization . . . . . . . . . . . . 796.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2.3 Indices and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2.4 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2.5 Net Energy Consumption . . . . . . . . . . . . . . . . . . . . . . . . . 846.2.6 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2.7 Integrated Optimization Model . . . . . . . . . . . . . . . . . . . . . 88

6.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7 Stochastic Speed Control and Timetable Optimization . . . . . . . . . . . 957.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.2.2 Uncertainty of Train Mass . . . . . . . . . . . . . . . . . . . . . . . . 987.2.3 System Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.2.4 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2.5 Stochastic Optimization Model . . . . . . . . . . . . . . . . . . . . . 1037.2.6 Formulation Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.3.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.3.2 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Contents ix

Chapter 1Literature Overview

In 2017, Yang et al. [49] presented a comprehensive survey on subway energy-efficient management literature, in which speed control and timetable optimizationare two mainly used subway energy-efficient management approaches: the formeroptimizes the speed profile at inter-stations tominimize the traction energy consump-tion and the latter synchronizes the accelerating actions and braking actions of trainstomaximize regenerative energy absorption. Based on their work, this chaptermainlyintroduces the state-of-the-art on energy-efficient speed control, energy-efficienttimetable optimization and their extensions on integrated optimization, dynamic opti-mization, and stochastic optimization approaches.

1.1 Energy-Efficient Speed Control

Literature on train energy-efficient speed control study can date back to the 1960s. In1968, Ishikawa [24] proposed an optimal control model to determine the train speedprofile at inter-stations, which can be applied to both subway systems and generalrailway systems. The objective function is to minimize the electric energy consump-tion at inter-station. In 1980, Milroy [36] reformulated the problem as follows:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

min C(u, v) =∫ T

0u+(t)v(t)dt

s.t. v′(t) = u(t) − r(v(t))v(0) = v(t) = 0∫ T0 v(t)dt = S

|u(t)| ≤ 1,

(1.1)

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

1

2 1 Literature Overview

which lays the foundation of the optimal train control theory. Although Milroyapplied the model to general railway systems, Howlett et al. [20] proved that itwas 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 deter-mined by timetable, S is the length of inter-station, v(t) is the train speed, r(v) is thespeed-dependent resistance applied on the train, u(t) is the unit force applied on thetrain 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 constraintdenotes the values of boundary speed, the third constraint denotes that the traveldistance should be equal to the length of inter-station and the fourth inequality nor-malizes 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 takeposition s as the state variable, then the energy-efficient speed control model can berewritten as follows: ⎧

⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

min C(u, v) =∫ S

0u+(s)ds

s.t. v′(s) = u(s) − r(v(s))v(0) = v(S) = 0∫ S0 1/v(s)ds = T

|u(s)| ≤ 1,

(1.2)

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 extendedto consider speed limits, traction efficiencies, variable gradients, variable curvatures,steep slopes, and other practical constraints and a large number of solution algorithmson these basicmodels and their extensions have beengiven,which canbegrouped intoanalytical algorithms and numerical algorithms [43]. Generally speaking, analyticalalgorithms can obtain the optimal speed profiles, but only for some simplifiedmodels;while numerical algorithms are able to deal with complex models, they can onlyobtain sub-optimal solutions.

1.1.1 Analytical Algorithms

The energy-efficient speed control problemwas originally formulated as a continuousoptimal control model. Asnis et al. [2] assumed that the unit force was a continuouscontrol variable with uniform bounds, and took the Pontryagin maximum principleto find the necessary conditions for the optimal speed profile. To seek strict mathe-matical foundations, Howlett [17, 18] formulated the optimal speed control problemin an appropriate function space and concluded that the optimal speed profile existsand satisfies a Pontryagin-type criterion. Furthermore, Howlett [19] formulated theenergy-efficient speed control problem as a finite dimensional constrained optimiza-tionmodel and took Pontryaginmaximumprinciple to determine the optimal solution

1.1 Energy-Efficient Speed Control 3

for the model (1.1), which produced the first theoretical confirmation that an optimalspeed profile should use a maximum acceleration-cruising-coasting-maximum brak-ing phase sequence. For taking the theory into practice, variable gradients, tractionefficiencies, speed limits, and steep slopes were gradually considered in literature[1, 23, 30, 34]. For example, Liu and Golovitcher [34] gave an analytical solution tothe energy-efficient speed control problemwith variable gradients. Khmelnitsky [30]considered the operations scenario with variable gradients, speed-dependent tractionefficiencies, and arbitrary speed limits. Howlett et al. [23] provided an analyticalsolution method with more than one steep slopes, which first divided the route intosmall segments such that each segment contains one steep slope, then solved thespeed 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 steepsegment, and the global optimal speed profile is also unique.

For the discrete optimal control model, the Scheduling and Control Group of theUniversity of South Australia led by Howlett P.G. made outstanding contributions.For example, Howlett et al. [20] outlined the theoretical basis with a discrete controlmodel for Metromiser system, which has been successfully applied to urban railtransits in Australia, Melbourne, Toronto, etc. Cheng and Howlett [4, 5] studiedthe energy-efficient speed control problem with discrete inputs for a track withoutvarying gradients and speed limits. Pudney andHowlett [39] studied the problemwithspeed limits and proved that that speed limits were below the desired cruising speedon intervalswhere the speedmust be held at the limits. Howlett andCheng [21] solvedthe problem with continuously varying gradients. Furthermore, Cheng [6] tackledthe problemwith nonzero track gradients and speed limits, which was difficult to findan analytic solution because it was no longer possible to precisely follow arbitrarysmooth speed limits. In 2000,Howlett [22] considered the problemwith a generalizedmotion equation and concluded that the optimal strategy for discrete control couldbe used to approximate as closely as we want the optimal strategy obtained usingcontinuous control. This conclusion means that both both the continuous control anddiscrete control models can apply on trains with continuous or discrete traction andbraking forces.

1.1.2 Numerical Algorithms

Since Howlett [19] has proved that the optimal speed profile consists of acceleratingwith maximum traction force, cruising, coasting, and decelerating with maximumbraking force, some researchers proposed numerical algorithms to solve the opti-mal switching times or positions among different phases, which essentially transfersthe optimal control problem to nonlinear optimization problem. In 1975, Hoang etal. [16] first studied the energy-efficient speed control problem using a numericalalgorithm. Because of the extremely low calculation speed, the numerical algorithmwas paid less attention in the following two decades. In recent years, with the devel-opment of computer performance and calculation theory, more and more studies