ABSTRACT
 Title of thesis: INTEGRATED ANALYSIS OF VERTICAL 
ALIGNMENTS AND SPEED PROFILES FOR RAIL 
TRANSIT ROUTES
 Shu-Ta Yeh, Master of Science, 2003
 Thesis directed by: Professor Paul M. Schonfeld
 Department of Civil and Environmental Engineering
 Frequent accelerations and decelerations for rail transit trains adversely 
affect in the major performance measures of travel time, tractive energy 
consumption and braking wear between stations. A dipped vertical alignment 
between rail transit stations provides trains some added advantages from gravity 
in accelerating as well as in decelerating. 
A deterministic simulation model based on basic kinematics and resistance 
relations has been developed to compute train motions and energy consumption 
on specified vertical alignments. A baseline case study and the sensitivity of 
results to parameter changes are analyzed. 
Vertical alignments and operating characteristics such as speeds and 
coasting distances are jointly optimized. Powell?s Method is used for numerically 
optimizing total costs and other objective functions. Without speed constraints, 
the optimized total cost savings exceed 6.5%. The optimized alignment depths 
and total cost savings decrease as speeds are constrained to lower values. 
The methods developed here may significantly improve rail transit 
systems.
INTEGRATED ANALYSIS OF VERTICAL ALIGNMENTS AND SPEED 
PROFILES FOR RAIL TRANSIT ROUTES
 By
 Shu-Ta Yeh
 Thesis submitted to the Faculty of the Graduate School of the
 University of Maryland, College Park in partial fulfillment
 of the requirements for the degree of
 Master of Science
 2003
 Advisory Committee:
 Professor Paul M. Schonfeld, Chair
 Professor Gang-Len Chang
 Professor Kelly Clifton

ii
 ACKNOWLEDGEMENT
 I would like to express my greatest appreciation to my advisor, Professor 
Paul Schonfeld, for his instruction and guidance for this research. Without his 
support, this thesis would not have been completed.
 I am grateful to other members of my advisory committee, Professor 
Gang-Len Chang and Kelly Clifton, for their guidance and comments on this 
research.
iii
 TABLE OF CONTENTS
 LIST OF TABLES...................................................................................................v
 LIST OF FIGURES ............................................................................................... vi
 Chapter 1. Introduction ............................................................................................1
 1.1 Problem Statement .......................................................................................1
 1.2 Study Objectives and Scopes .......................................................................2
 Chapter 2. Literature Review...................................................................................5
 2.1 Relevant Studies...........................................................................................5
 2.2 Studies in Cost Saving Models for Rail Transit System..............................6
 2.3 Kinetic and Empirical Engineering Equations.............................................7
 2.4 Railroad Simulation Models ........................................................................9
 2.5 Optimization of Total Cost ........................................................................11
 Chapter 3. Deterministic Simulation Model ..........................................................13
 3.1 Deterministic Simulation Model Assumptions..........................................13
 3.2 Structure of Deterministic Simulation Model............................................15
 3.3 Vertical Alignment Designs.......................................................................18
 3.4 Mathematical Relations for Deterministic Simulation Model ...................21
 3.5 Total Cost Evaluation ................................................................................28
 3.6 Description of Train Simulator Output ......................................................29
 Chapter 4. Evaluation of Results ...........................................................................30
 4.1 Comparison with Previous Model Results.................................................30
 4.2 Baseline Case Assumptions .......................................................................31
 4.3 Major Performance Measures Comparison of Baseline Cases ..................33
 4.4 Performance Profile Comparison...............................................................36
 4.4.1 Performance Profile Comparison for Case Set 1 ..............................37
 4.4.2 Performance Profile Comparison for Case Set 2 ..............................47
 4.5 Sensitivity Analysis ...................................................................................56
 4.5.1 Sensitivity Analysis Results..............................................................57
 4.5.2 Analysis of Sensitivity Results .........................................................59
 Chapter 5. Optimization of Simulation Results .....................................................66
 5.1 Optimization Methods ...............................................................................66
 5.2 Joint Optimization of Total Cost................................................................67
 5.3 Relations between Optimized Solutions and Alignment Depth ................71
 5.4 Sensitivity of Optimized Alignments to User Time Value and  
Energy Cost................................................................................................78
 5.5 Case Study .................................................................................................82
iv
 Chapter 6. Conclusions and Recommendations.....................................................84
 6.1 Summary of Research Results ...................................................................84
 6.2 Conclusions................................................................................................84
 6.3 Recommendations for Further Research....................................................86
 REFERENCES ......................................................................................................88
v
 LIST OF TABLES
 Table 3.1 Mathematical functions of vertical alignments.....................................19
 Table 4.1 Comparison with previous model results.............................................. 30
 Table 4.2 Parameters used in baseline cases.........................................................31
 Table 4.3 Simulation results with baseline parameters.........................................33
 Table 4.4 Selected baseline cases for simulation profile 
comparison. ..........................................................................................36
 Table 4.5 Travel time, tractive energy and braking energy for 
baseline case set 1 ................................................................................37
 Table 4.6 Existing conditions at begin-braking points for 
selected baseline cases .........................................................................45
 Table 4.7 Travel time, tractive energy and braking energy for 
baseline case set 2 ................................................................................47
 Table 4.8 Parameters for sensitivity analysis........................................................56
 Table 4.9 Vertical track alignments for sensitivity analysis .................................57
 Table 4.10 Sensitivity analysis for travel time......................................................57
 Table 4.11 Sensitivity analysis for tractive energy consumption .........................58
 Table 4.12 Sensitivity analysis for braking energy consumption .........................58
 Table 4.13 Begin-braking information for sensitivity analysis ............................63
 Table 5.1 Baseline values used in total cost function ...........................................66
 Table 5.2 Jointly optimized alignments, speed constraints and 
coasting distance ..................................................................................68
 Table 5.3 Jointly optimized alignments and coasting distance 
with a 75mph speed constraint.............................................................70
 Table 5.4 Jointly optimized alignments and coasting distance 
with a 55mph speed constraint.............................................................70
 Table 5.5 Underground alignments optimizations with a 59mph 
speed constraint....................................................................................83
 Table 5.6 Aboveground alignments optimizations with a 59mph 
speed constraint....................................................................................83
vi
 LIST OF FIGURES
 Figure 2.1 Dipped track profile used by Kim and Schonfeld (1997)......................5
 Figure 3.1 Flow chart of critical speed analysis ...................................................16
 Figure 3.2 Flow chart of train motion analysis .....................................................17
 Figure 3.3 Vertical track alignments between stations .........................................18
 Figure 4.1 Illustrations of vertical alignments ......................................................32
 Figure 4.2 Travel time-Maximum depth relations................................................34
 Figure 4.3 Tractive Energy-Maximum depth relations.........................................35
 Figure 4.4 Braking energy-Maximum depth relations..........................................35
 Figure 4.5 Location-Elevation relations for baseline case set 1 ...........................38
 Figure 4.6 Location-Gradient relations for baseline case set 1.............................38
 Figure 4.7 Location-Resistance relations for baseline case set 1..........................39
 Figure 4.8 Location-Acceleration relations for baseline case set 1 ......................40
 Figure 4.9 Location-Speed relations for baseline case set 1 .................................41
 Figure 4.10 Location-Time relations for baseline case set 1 ................................41
 Figure 4.11 Time-Acceleration relations for baseline case set 1 ..........................42
 Figure 4.12 Time-Speed relations for baseline case set 1.....................................43
 Figure 4.13 Time-Cumulative tractive energy relations for baseline case set 
1............................................................................................................44
 Figure 4.14 Time-Cumulative braking energy relations for baseline case set 
1............................................................................................................45
 Figure 4.15 Location-Elevation relations for baseline case set 2 .........................48
 Figure 4.16 Location-Gradient relations for baseline case set 2...........................48
 Figure 4.17 Location-Resistance relations for baseline case set 2........................49
 Figure 4.18 Location-Acceleration relations for baseline case set 2 ....................50
 Figure 4.19 Location-Speed relations for baseline case set 2 ...............................51
 Figure 4.20 Location-Time relations for baseline case set 2 ................................51
 Figure 4.21 Time-Acceleration relations for baseline case set 2 ..........................52
 Figure 4.22 Time-Speed relations for baseline case set 2.....................................53
 Figure 4.23 Time-Cumulative tractive energy relations for baseline case 
set 2 ......................................................................................................54
 Figure 4.24 Time-Cumulative braking energy relations for baseline case 
set 2 ......................................................................................................55
 Figure 4.25 Sensitivities of train speeds to parameter changes for 
alignment I ...........................................................................................64
 Figure 4.26 Sensitivities of train speeds to parameter changes for 
alignment II..........................................................................................64
 Figure 4.27 Sensitivities of train speeds to parameter changes for 
alignment III.........................................................................................65
 Figure 4.28 Sensitivities of train speeds to parameter changes for 
alignment IV ........................................................................................65
vii
 Figure 5.1 Depth-optimized total cost relations for 8500 ft station spacing.........72
 Figure 5.2 Depth-optimized total cost relations for 12500 ft station spacing.......72
 Figure 5.3 Depth-optimized total cost relations for 16500 ft station spacing.......73
 Figure 5.4 Alignment depth, speed constraint and total cost relations .................74
 Figure 5.5 Alignment depth, coasting distance and total cost relations with a 
75 mph speed constraint.......................................................................76
 Figure 5.6 Alignment depth, coasting distance and total cost relations with a 
55 mph speed constraint.......................................................................77
 Figure 5.7 Alignment depth, speed constraint and total cost relations when 
the user time value is 15$/psg-hr .........................................................79
 Figure 5.8 Alignment depth, speed constraint and total cost relations when 
the tractive energy cost is 0.3$/kwh.....................................................80
 Figure 5.9 Alignment depth, speed constraint and total cost relations when 
the braking energy cost is 0.2$/kwh ....................................................81
1
 Chapter 1. Introduction
 1.1 Problem Statement
 Rail transit systems are widely used to provide mass transportation in 
metropolitan areas. However, the frequent accelerations and decelerations for rail 
transit trains adversely affect in the major performance measures of travel time, 
propulsive energy consumption and braking wear between stations. If a dipped 
vertical track alignment between transit stations were applied, trains would take 
advantages from gravity in accelerating as well as in decelerating and reduce the 
disadvantages resulting from frequent accelerations and decelerations. With such 
a dipped transit track alignment, the following performance measures may be 
improved for a rail transit system: 
1. Travel time between stations and related users? and supplier?s costs.
 2. Energy consumption for propulsion.
 3. Braking wear and resulting maintenance costs.
 However, providing such a dipped vertical track alignment may lead to the 
following problems:
 1. Uncomfortable centrifugal acceleration in the vertical plane.
 2. Gradients exceeding the maximum climbing ability of trains.
 3. Increased construction cost.
2
 Problem 1 constrains the rate of gradient change to satisfy standee comfort 
and safety requirements. 
Problem 2 limits the maximum allowable gradient in order to start a train 
after an unexpected stop in a climbing section. 
Problem 3 may be negligible when a deep tunnel is used between stations. 
However, if an elevated or cut-and-cover section is applied, the increased 
construction cost to build dipped vertical track alignments may be quite 
substantial.
 1.2 Study Objectives and Scopes
 The primary objectives of this research are (1) To develop a deterministic 
simulator with basic train kinematic relations and engineering equations for 
distinguishing transit trains movements on various vertical alignments and 
operating parameters. (2) To use it to evaluate rail transit operating performance 
measures with various track alignments. (3) To use Powell?s Method and the 
deterministic simulator in optimizing alignments and operating characteristics.
 In simulation experiments, one-directional train movements on vertical 
track alignments between stations will be investigated. The deterministic 
simulator will compute train movements using relations of vehicle kinetics, 
resistances, tractive effort, power, propulsive energy consumption, and braking 
energy consumption. The research will be focused on evaluating the relations 
between vertical alignment designs, travel time, tractive energy, braking energy 
and total costs. 
3
 A baseline simulation experiment will be introduced in chapter 4 and then 
repeated with various parameter changes for sensitivity analysis. This section will 
concentrate on (1) Generating simulation results of train travel time, propulsion 
fuel consumption, and brake wear under various operating options and vertical 
track alignments, and (2) Comparing and analyzing the effect of operating option 
changes under vertical track alignments.
 Total costs functions are optimized in chapter 5 by using Powell?s Method 
with the deterministic simulator. The optimized results may be provided to rail 
transit planners, designers and operators as new cost saving methods for the future 
network designs. 
The major contributions different than previous studies are 
1. Detailed baseline case study and sensitivity analysis to parameter changes.
 2. Jointly optimization of vertical alignments and operating characteristics such 
as speeds and coasting distances
4
 This thesis will proceed through the following steps:
 1. Review of literature.
 2. Descriptions of the deterministic simulation model, which includes the 
structure of the model, input variables and control rules.
 3. Analysis of train performance measures of travel time, tractive energy 
consumption and braking wear between stations for various vertical track 
alignment designs with baseline parameters. After baseline analysis, a 
sensitivity analysis of changing in operating characteristics is investigated.
 4. Optimizations of alignment designs and operating parameters.
 5. Conclusions and recommendations.
5
 Chapter 2. Literature Review
 2.1 Relevant Studies
 The use of a dipped alignment in a rail transit system has not been 
extensively analyzed in the literature. One published study on using dipped 
profiles in rail transit was done by Kim and Schonfeld (1997). They have created 
a deterministic simulation model to analyze rail transit train movement, energy 
consumption and braking consumption on a dipped vertical track alignment. 
The vertical alignment used in this study is determined by two coefficients 
S, total parabolic curve length (which is equal to station spacing in this study), 
and 
, maximum depth halfway between two stations. The illustration and 
mathematical equations for that vertical alignment designs are shown in Figure 
2.1 
Figure 2.1 Dipped track profile used by Kim and Schonfeld (1997)
 


 

 
 


 

 
 
 +=
 <+=
 <=
 SxSforx
 S
 x
 S
 y
 S
 x
 Sforx
 S
 x
 S
 y
 S
 xforx
 S
 y
 6
 5122412
 6
 5
 62
 66
 6
 012
 2
 2
 2
 2
 2
 2
 
 
 
 x
 
y
 S
 1 2 3
6
 They have found that at the maximum allowable dip (using the maximum 
allowable gradient in alignments and a vertical dip halfway between stations 
equal to 1% of station spacing) the savings (compared with results on level track) 
exceed 9% for propulsive energy, 15% for braking energy, and 5% for travel time 
between rail transit stations. However, the previous study of Kim and Schonfeld 
has the following weaknesses:
 1. The resistance equation used in the study is too old and the air resistance 
coefficient pertains to trains operating outside tunnels.
 2. They assumed that the total parabolic curve length (one half-crest at both ends 
and one sag in the middle) is equal to station spacing. This assumption 
assumes no level platform section in either station and no level section in the 
middle between two stations.
 3. No total cost analysis and optimization is conducted in this study. The results 
for travel time, tractive energy and braking energy consumption are analyzed 
separately, without being integrated and jointly optimized.
 2.2 Studies in Cost Saving Models for Rail Transit System
 Several studies have been found on reducing operation cost for rail transit 
systems. Uher and Sathi (1983) use the Energy Management Model, a series of 
computer simulation programs developed by the Rail System Center of Carnegie-
 Mellon University, to smooth the peak-power demand component of energy use. 
The reductions of the peak-power demand make it easier for a transit agency to 
negotiate the electricity rate and reduce the required facility investment. 
7
 Vuchic (1981) and Profillidis (1995) addressed electrodynamic braking 
(also known as regenerative braking) system in their books. While applying this 
system during braking, deceleration is obtained by converting the electric traction 
motors into electric generators. The power generated from train kinetic energy 
may be returned to the power network or simply be released by an electric resistor 
as heat in to the air. 
The above two cost saving models can be used with dipped track profiles 
to take additional advantages from gravity in a rail transit system. However, due 
to the reduction in the required propulsive energy and braking energy for dipped 
track alignments, the additional cost savings from applying Energy Management 
Model or Regenerative Braking System together with dipped alignments will be 
lower than the additional cost savings from applying them with level tracks.
 2.3 Kinetic and Empirical Engineering Equations
 Vuchic?s (1981) textbook emphasizes a systematic description of the basic 
concepts, terms, and relationships of urban public transportation. This book 
presents detailed descriptions and analyses of different transit modes, with 
emphases corresponding to their technical/operational complexities and their 
relative roles in urban transportation. Moreover, this book presents basic 
equations of vehicle motion, a description of the forces acting on the vehicle: 
resistance and propulsion, a definition of adhesion then leads to expressions for 
vehicle acceleration and stopping and diagrams and equations for computations of 
station-to-station travel times. Hay (1982) has illustrated rail-engineering 
8
 equations such as rail tractive force output equation, energy consumption equation 
and rail resistance equations in his publication. The equations provided by Vuchic 
and Hay will be used in the deterministic simulator design. 
The American Railway Engineering Association (AREA) (1984) has 
provided almost all railway design guidelines used in current practice in its 
Manual for Railway Engineering. This manual requires that all changes in 
gradients should use vertical curve to connect and the length of a vertical curve is 
determined by the gradients to be connected. For high-speed main tracks, the rate 
of change should not be more than 0.05 % per station (of 100 ft) in sags and not 
more than 0.10 % per station in crest. For secondary main tracks, the rates of 
change may be twice those for high-speed main tracks. In 1926, W. J. Davis 
proposed an empirical formula for computing the ? Tractive Resistance of Electric 
Locomotives and Cars? moving on straight and level track. However, the 
increased dimensions and heavier loading of fright cars, the much higher 
operating speeds of fright and passenger trains, and changes in the track structure 
have made it desirable to modify the constants in the Davis equation. The 
modified Davis Equation approved by AREA (1984) is shown in Equation 3.16. 
Profillidis (1995) has provided train dynamic equations in his textbook. 
Various train traction and resistance relations developed and provided from 
organizations or agencies worldwide for various operating conditions are 
collected in this book. Section 12.2 of this book shows that the air resistance for 
trains operating in a tunnel with no openings is approximately three times the air 
resistance coefficient for a train running outside tunnels. 
9
 Wright and Ashford (1997) have provided vertical alignment design 
guidelines for railroads and rail transits. Railroad design is characterized by much 
smaller maximum grades and much longer vertical curves than are highways. The 
criterion for length of vertical curve is more critical for sags than for crests. 
Longer vertical curves are required for sag curves because of the tendency of 
undesirable slack to develop in the couplings as the cars in the front are slow 
down by the change of grade. Generally, much shorter curves are permitted along 
urban passenger rail lines than on mainline railroad tracks. 
The Washington Metropolitan Area Transit Authority (WMATA) (2001) 
provides design criteria in architectural, civil, utility, structure, mechanical, 
electrical, train control and communications, systems and trackwork for its rail 
rapid transit system (METRORAIL). Parabolic vertical curves are required in rail 
transit alignment designs. The requirements of minimum vertical curve lengths 
are shown in Section 3.3. Vehicle characteristics and operating parameters to be 
used in WMATA METRORAIL system are also provided in this manual.
 2.4 Railroad Simulation Models
 Kikuchi (1991) has developed a simplified simulation model for the 
operational analysis of a rail rapid transit train. The model simulates the 
movement of a train along a route and develops the relations of time vs. distance, 
time vs. speed and distance vs. speed. It was found that the model could fairly 
accurately simulate the relations between travel time and distance. 
10
 Due to the presence of non-linear and complex phenomena, Minciardi et 
al. (1994) have suggested the use of discrete event simulations for train movement 
problems. They use two simulators to describe rail transit movements and energy 
consumption. The first is a simulator that allows the performance analysis of 
given schedule; the second is an integrated system simulator that allows the 
electrical analysis of the network during the movement of the train along the 
track. 
Yu et al. (1994) have suggested an approach based on tabulated data to 
estimate traction motor efficiency. They use linear interpolation of an established 
data set to evaluate electric traction motors of a train moving at a specific speed 
and under a specific DC voltage. 
Kim and Schonfeld (1997) have developed a deterministic transit train 
simulator to analyze train performance on a dipped alignment. The simulator can 
compute train movements with input parameters including vehicle characteristics, 
control options and track alignment then provide performance measures of train 
movement and propulsive and braking energy consumption.
11
 2.5 Optimization of Total Cost
 Vuchic?s textbook (1981) provides cost components of various types of 
users? and supplier?s costs for analyzing the total cost of a rail transit system. The 
components include 1. Capital cost: right-of-way acquisition, permanent way, 
stations, track superstructure, power supply, controls and communications, 
vehicle, maintenance and storage, engineering and administration and include 
operating and 2. Maintenance costs: transportation, permanent-way maintenance, 
vehicle maintenances, power, general and administrative.  The total cost 
components to be used in this study are vehicle capital depreciation, power and 
maintenance cost and user in vehicle cost.
 Press et al. (1989) present many algorithms, including Golden Section 
Search for one-dimensional optimization and Powell?s Method for multi-
 dimensional optimization. Golden Section Search is a one-dimensional 
optimization method for minimizing an objective function that does not require 
pre-determined derivatives. It is also one bisection method developed to 
accelerate the convergence to optimized solutions. Powell?s Method is used in this 
study for jointly optimizing multiple variables. With Powell?s Method, one-
 dimensional optimizations (Golden Section Search in this study) for changing 
variables are repeatedly used until a combination of optimized values for which 
the objective function cannot be improved anymore is found. 
Inose and Ishikawa (1994) compare civil engineering costs vs. tunnel 
depth showing that for shield tunneling (deep tunneling), construction cost is 
basically independent to tunnel depth for sites with good geology. 
12
 For the values used in total cost analysis, the Energy Information 
Administration (2003) provides statistical information about nationwide electric 
retail prices in the U.S. The information of electric retail prices can be used in 
calculating tractive energy expense. The American Public Transportation 
Association (2003) provides average new vehicle costs for heavy rail in the 
United States. The Washington Metropolitan Area Transit Authority (WMATA) 
(2002) provides operating statistics for its METRORAIL system within the fiscal 
year ending June 2002. 
13
 Chapter 3. Deterministic Simulation Model
 The methodology used in this study is based on a deterministic simulation 
of train movements and energy consumption on a specified vertical track 
alignment for given control options. In this deterministic simulator, basic kinetic 
relations and experiential engineering equations are used to compute train 
movements and energy consumption. The results of travel time and energy 
consumption are included in a user and supplier cost function later to evaluate 
total effects of using dipped track alignment in a rail transit system. 
3.1 Deterministic Simulation Model Assumptions
 The following assumptions are considered in this deterministic simulation 
model:
 1. The vertical track profile is symmetric about a central axis.
 2. Parabolic curvatures are applied for the vertical curves while the gradient 
cannot exceed the maximum train climbing ability.
 3. Horizontal curvatures are negligible for this analysis.
 4. A concentrated mass is used to represent trains in motion.
 5. A train accelerates with its full power unless it exceeds the comfort-limited 
acceleration.
 6. The braking system can provide the maximum allowable comfort-limited 
deceleration rate.
14
 7. A train starts cruising after reaching its cruising speed and begins coasting 
after passing a prespecified begin-coasting point.
 Assumption 1 simplified the vertical alignment alternatives in this study. 
Based on this assumption, stations at both ends have to be equal in elevation.
 Assumption 2 avoids rapid gradient change along the vertical alignment 
that may cause high centrifugal acceleration and reduce passenger comfort. In 
addition, a limitation of maximum gradient is required because a train may have 
to start without any moment on a climbing section.
 Assumption 3 focuses this study only on the effects of vertical track 
alignments.
 Assumption 4 assumes that the entire train mass can be considered as 
concentrated at one point. Assuming all cars located on the same point with the 
same gradient may cause some acceptable error.
 Assumption 5 means that a train is assumed to use its full power to 
accelerate unless exceeding the comfort-limited acceleration.
 Assumption 6 means that a train can always reach its maximum (comfort-
 limited) deceleration rate that is constrained by standee comfort and available 
adhesive force for a non-emergency condition. 
Assumption 7 means that if cruising and coasting controls were applied in 
a simulation case, the start of cruising is controlled by train speed and the start of 
coasting is controlled by train position.
15
 3.2 Structure of Deterministic Simulation Model
 The structure of the deterministic simulator developed in this study 
consists of two parts:
 The first part of this deterministic simulator is designed to find out when a 
train should start braking in order to stop precisely in the next station. A critical 
speed vcritical(x) for a train in any point x between stations to start braking and stop 
precisely in the center of next station can be used to determine whether a train 
should brake or keep running. For any simulation step, if the train speed at its 
current location x exceeds the critical speed vcritical(x), the train should brake in this 
simulation step. To create the relations of critical speed vcritical(x) vs. position x , a 
small deterministic simulation that can analyze the movement from a stopped 
train to its previous stages is used here to determine the train dynamic during 
braking (Figure 3.1 is the critical speed analysis flow chart).
 The second part is a deterministic simulation that can compute train 
movements, tractive energy consumption and braking energy consumption 
between two stations and then stop at a precise point in the arrival station (Figure 
3.2 is the train motion analysis flow chart).
16
 Figure 3.1 Flow chart of critical speed analysis
 Comfort-limited braking 
rate (neg. value)
 Max
 Yes
 No
 INPUT DATA
 Vertical alignment
 Vehicle characteristics
 Remaining distance to stop   dremain i
 Critical speed vcritical(dremain i)
 Gradient G(dremain i)
 Resistance
 Adhesive force
 Available braking 
rate (neg. value)
 Feasible braking rate
 dremain i = station spacing?
 Gather Statistics
 Train location: xi
 Critical speed: )( icritical xv
 Stop
17
 Figure 3.2 Flow chart of train motion analysis
 INPUT DATA
 Vertical alignment
 Vehicle characteristics
 Cruise speed
 Point to begin coasting
 Critical Speed Analysis
 Critical speed at point x 
)(xV critical
 Details in Figure 3.1
 Train position: xi
 Train speed: )( ixv
 Gradient: )( ixG
 YesNo
 MaxMin
 Yes
 YesNo
 v(Xi)>vcritical(Xi) ?
 Feasible acceleration rate
 Propulsive force
 Adhesive force
 Resistance
 Comfort 
limit acc. 
Coasting? Cruising?
 Zero 
acceleration
 Available 
acceleration
 Feasible deceleration rate
 Resistance
 Adhesive force
 Comfort limit 
deceleration 
Available 
deceleration
 Propulsive / Braking Energy Consumption
 Arrived at
 Station?
 Gather Statistics
 Travel time
 Tractive energy consumption
 Braking energy consumption
 YesNo
 STOP
18
 3.3 Vertical Alignment Designs
 The vertical track alignments used in this study are generalized from that 
suggested by Kim and Schonfeld (1997). As shown in Figure 3.3, sections I and 
VII are two level platform sections; sections II, III, V and VI are two crest and 
two sag parabolic vertical curves with coefficients S and  in which S is the total 
length of the four parabolic curve sections and  is the maximum depth of the 
vertical alignment; section IV is a level bottom section.
 The mathematical functions for vertical track alignment are listed in Table 
3.1 where Lj presents the length of section j and xj; yj are horizontal and 
vertical distances origin from the left end of section j; Lpl is the platform length; 

 and S are coefficients for parabolic curves; Lbt is the level bottom length. The 
total station spacing Lss in this vertical track alignment is Lpl + S + Lbt.
 Figure 3.3 Vertical track alignments between stations
 Lpl/2 Lpl/2
 Li
 
yj
 xj
 Station 2Station 1
 S/6
 VIII
 Lbt S/3S/6 S/3
 II VIIV VIII
19
 Table 3.1 Mathematical functions of vertical alignments
 Section
 j
 Length
 Lj
 Elevation
 yj
 Tangent
 y?j
 Section 
description
 I
 2
 plL 0 0 Departing platform
 II 6
 S 2
 22
 12
 x
 S
   2224 xS 
   Descent crest
 III 3
 S
 3
 2
 32
 46
 x
 S
 x
 S
  
 S
 x
 S
  412
 32  Descent sag
 IV btL 0 0 Level bottom
 V 3
 S 2
 52
 6
 x
 S
  5212 xS 
  Ascent sag
 VI 6
 S
 6
 2
 62
 412
 x
 S
 x
 S
 + 
 S
 x
 S
  424
 62 + Ascent crest
 VII
 2
 plL 0 0 Arriving platform
20
 According to the design manual of WMATA (2001), the maximum 
gradient should not exceed ?4% for rail rapid transit tracks. In this study, the 
maximum gradient is 4
/S and occurs at the end of section II and V. Therefore, 
the 
/S ratio in this study should not exceed 1% in any simulation case. 
For the vertical curve length Lvc, WMATA requires that while operating 
within the 75mph speed constraint, the minimum length of vertical curve shall 
determined by the formula: Lvc = (G1-G2) 100
 Where: Lvc = Length of curve, feet.
 (G1-G2) = Algebraic difference of grades connected by the vertical curve, 
percent.
 The absolute minimum length of vertical curve shall be 200 feet
 According to WMATA?s requirement, in this study, the formulation for 
minimum length requirement of vertical curvature with 75mph speed constraint 
can be rewritten with coefficients S and 
 as
 S
 S ?> 24000
21
 3.4 Mathematical Relations for Deterministic Simulation Model
 The following mathematical relations are used to determine train 
movement and energy consumption in the deterministic simulator.
 For any simulation step i, basic physical equations are applied for 
calculating train movement.
 s
 t
 ft
 d
 sft
 v
 ft
 x
 tvxdxx iiiiiiii
 +=+=+1 (3.1)
 s
 t
 sft
 v
 sft
 a
 sft
 v
 tavvvv iiiiiiii
 +=+=+ 21 (3.2)
 where xi = train location at step i; vi = train speed at step i; ai = train 
acceleration/deceleration at step i; di = distance increment at step i; vi = speed 
increment at step i; and t = time interval used in the deterministic simulation.
 The main relation used to determine a train?s acceleration is Newton?s 
second law
 lb
 m
 sft
 g
 lbw
 F
 m
 gF
 a 2
 
 = (3.3)
 where a = acceleration; F = force (unit: lbw); g = the acceleration due to gravity; 
and m = mass of a body . Because the accelerating force for a train in motion is 
the difference between its available tractive effort Eavailable and train resistance Rt, 
then available acceleration aavailable is
 lb
 m
 sft
 g
 lbw
 R
 lbw
 E
 sft
 a
 m
 gRE
 a ttavailableavailable
 t
 tavailable
 available 
 
 =
 
  22
 )( (3.4)
 where  = coefficient for rotating masses(usually 1.04~1.10(Vuchic 1981)); mt = 
train mass; Rt = train resistance 
22
 However, the available tractive effort Eavailable is limited by both tractive 
force Ft and adhesive force Fa on driving wheels. 
{ }
 lbw
 F
 lbw
 F
 lbw
 E
 FFE ataatavailable ,min= (3.5)
 The tractive force for rail vehicles is
 hmile
 V
 Hp
 P
 lbw
 F
 V
 PF tt 
 
 =
 375 (3.6)
 where P = power; V = speed;  = transmission efficiency (typically 0.82 (Hay 
1982)). Therefore, equation 3.6 can be rewritten as,  
hmile
 V
 Hp
 P
 lbw
 F
 V
 PF tt
 
 =
 308 (3.7)
 If a coasting operation (operating a train only with its moment, no traction from 
motors) is used for part of the travel between stations, the output tractive force 
should equal to zero during coasting operation.
 The adhesive force for steel wheels on rail is
 lbw
 F
 lbw
 F
 FF nana =
 ?? (3.8)
 where Fn = normal force = the fraction of gravity force perpendicular to rail 
surface plus centrifugal force; ? = adhesive coefficient. Figure 3.22 of Vuchic?s 
(1981) textbook shows that the relations between adhesive coefficient ? and train 
speed V for dry condition (in tunnel) are approximately equal to
 hmile
 VV =
 ?? 609.10015.03.0 (3.9)
23
 ( )
 22
 cos
 sft
 a
 radiansft
 g
 lb
 m
 lbw
 F
 g
 agm
 F lcentrifugatnlcentrifugatn
  +
 = (3.10)
 %100
 tan 1
 G
 radian
 G  = (3.11)
 where  = angel from level; ac = centrifugal acceleration; and G = percent of 
gradient.
 Centrifugal acceleration can be expressed as follows:
 ft
 r
 sft
 v
 sft
 a
 r
 v
 a ii
 lcentrifuga
 i
 i
 lcentrifuga 2
 2
 = (3.12)
 radianft
 d
 ft
 rd
 r iii
 i
 i
 i
 
 
 
 
 
 = (3.13)
 21 sft
 v
 s
 t
 ft
 d
 vtd iiii
 =  (3.14)
 %100
 tan
 100
 tan 111 iiiii
 G
 radian
 GG  =  (3.15)
 where ri = radius of vertical curvature at step i; and i = angle change during 
step i-1 to i
 In this study, train resistance is computed with the modified Davis 
Formula (AREA 1984). The unit resistance Ru (in pounds of force per short ton of 
vehicle weight) for a vehicle on a track without any horizontal curve is
 %
 2001.0206.0
 2 GnK
 hmile
 V
 ton
 w
 tonlbw
 RG
 wn
 VKV
 w
 R au
 a
 u +
 
 +++= (3.16)
24
 where w = weight per axle in short tons; K = aerodynamic resistance coefficient; 
V = vehicle speed; and na = number of axles per vehicle. The total resistance of a 
vehicle is the vehicle weight Wv multiplied by its unit resistance Ru.
 tonlbw
 R
 ton
 W
 lbw
 R
 RWR uvvuvv = (3.17)
 The total resistance of a train Rt is the summation of all individual vehicle 
resistances in a train. Thus, the total train resistance can be expressed as follows
 lbw
 R
 lbw
 R
 RR vt
 vall
 vt = (3.18)
 For reasons of both safety and comfort to standees in a rail transit vehicle, 
it is necessary to limit the longitudinal acceleration and deceleration rate a
 (usually ?0.1~0.15g). The maximum allowable acceleration or deceleration rate is 
formulated as
 2215.015.0 sft
 a
 sft
 ggag  (3.19)
 where g = acceleration due to gravity.
 Therefore, the maximum allowable acceleration rate amax that can be used for a 
transit vehicle with standees is
 2
 max
 2max 15.0 sft
 a
 sft
 gga  (3.20)
 An acceleration limit acruise used to prevent a train from exceeding the 
cruise speed in any simulation step is shown below
 s
 t
 sft
 v
 sft
 v
 sft
 a
 t
 vv
 a icruise
 cruiseicruise
 cruise
 i
 i
 
 
 
 = 2
 (3.21)
25
 where vcruise = cruise speed; vi = speed at step i; t = time interval used in 
simulation.
 Therefore, the feasible acceleration rate afeasible can be expressed as
 ( ) 22max22feasiblemaxfeasible ,,min sft
 a
 sft
 a
 sft
 a
 sft
 a
 aaaa cruiseavailablecruiseavailable= (3.22)
 For simulation steps during braking, the analysis process is very similar to 
steps used for accelerating. It was assumed earlier (Section 3.2) that the available 
braking force could reach the adhesive force limitation. Then the available 
braking rate bavailable is 
lb
 m
 sft
 g
 lbw
 R
 lbw
 F
 m
 gRFb tta
 t
 ta
 available 
 
 
 
 +=  2
 )( (3.23)
 The comfort-limited braking rate bmin is
 2
 min
 2min 15.0 sft
 b
 sft
 ggb  (3.24)
 Therefore, the feasible deceleration rate bfeasible can be expressed as 
(bavailable and bmin are negative values)
 ( ) 2min22feasibleminfeasible ,max sft
 b
 sft
 b
 sft
 bbbb availableavailable= (3.25)
 For any specific point x between stations, a critical speed vcritical(x) can be 
found from which a train can begin braking with full ability under non-emergency 
condition and stop precisely at the next station. These location-critical speed 
relations can be found by simulating backwards from a stopped train to its 
previous stages (see Figure 3.1). Whenever the critical speed vs. location table is 
26
 established, a 0-1 decision variable Ybraking(i) for determining whether a train at 
simulation step i  located at Xi with train speed v(Xi) should start braking can be 
specified as 
sft
 v
 sft
 vY
 vvY
 vvY
 XicriticalXiibraking
 XicriticalXiibraking
 XicriticalXiibraking )()()(
 )()()(
 )()()(
 ,0
 ,1
 

 

 
 <=
 =
 (3.26)
 where v(Xi) = train speed at simulation step i and located at point Xi ; vcritical(Xi) = 
critical speed at point Xi; Ybraking(i) = decision variables for a train to brake at 
simulation step i. If Ybraking(i) = 1, a train should brake at simulation step i. 
Therefore, for any simulation step i, the applied train 
acceleration/deceleration rate aapplied can be expressed as follows
 ( )
 2
 )(feasible
 2
 )(feasible)(
 2
 )(
 )(feasible)()(feasible)()( 1
 sft
 b
 sft
 aY
 sft
 a
 bYaYa
 iiibrakingiapplied
 iibrakingiibrakingiapplied
 
 +=
 (3.27)
 Energy consumption
 Once the applied train acceleration/deceleration rate is determined, the 
applied tractive effort Eapplied can be computed as
 lb
 m
 sft
 a
 lbw
 E
 maRE tappliedappliedtappliedtapplied +=
  2 (3.28)
 If Eapplied is positive, some tractive force from motors is required. If Eapplied 
is negative, some drag force from braking system is required. 
27
 The applied power Papplied used for generating the applied tractive effort 
Eapplied is
 hmile
 V
 lbw
 E
 hp
 PVE
 P appliedappliedappliedapplied 308
 
 = (3.29)
 The energy consumption e within time interval t is
 s
 t
 hp
 P
 kwh
 etPe appliedapplied
 =
 341.1
 1
 3600
 (3.30)
 If energy consumption ?e? is positive, some tractive energy consumption is 
recorded for this step. If a negative ?e? value is found here, braking energy 
consumption is recorded.
 Microsoft Excel 2000 is used in this study to compute train movements 
and energy consumption. For any data row i (simulation step i) in the spreadsheet, 
the performance measures of travel time, tractive energy consumption and 
braking wear between stations will be calculated with the processing flow as in 
Figures 3.1 and 3.2 with above mathematical relations. The major advantage for 
using spreadsheet in the calculation is that information for all simulation steps are 
stored and are easy to review train performance for any specific simulation steps. 
The disadvantage for using spreadsheet is that the computing time will be longer. 
The selected time interval used in the deterministic simulator is 0.01 second.
28
 3.5 Total Cost Evaluation
 In this study, the cost evaluation focus on costs for a one-directional 
operation between two stations. Thus, the part for user cost is user-in-vehicle cost. 
The supplier costs are the vehicle capital expenses (vehicle depreciation), energy 
expenses (energy consumption) and braking wear expenses plus the averaged 
construction cost for a one-directional operation between stations. The formulas 
for total cost between stations are
 $$$
 sut
 sut
 CCCCCC += (3.31)
 hrpsg
 v
 carpsg
 n
 traincar
 n
 hr
 TC
 vNnTC userpcuuserpcu = /$//$ (3.32)
 $/$/$/$$
 onconstructibrakingbtractivetvehiclecs
 onconstructibrakingbtractivetvehiclecs
 C
 kwh
 v
 kwh
 E
 kwh
 v
 kwh
 E
 hrvehicle
 v
 hr
 TnC
 CvEvEvTnC
 
 +++=
 (3.33)
 where Ct = total cost for a one-directional operation; Cu = user cost; Cs = supplier 
cost; T = travel time between stations; nc = number of cars per train; np = number 
of passengers per car; vuser = average in vehicle time value per passenger; vvehicle = 
vehicle depreciation per revenue hour; Et = tractive energy consumption; vtractive = 
unit tractive energy cost; Eb = braking energy consumption; vbraking = unit braking 
energy cost; Cconstruction = averaged construction cost for a one-directional 
operation between stations.  
29
 3.6 Description of Train Simulator Output
 After completing the calculation, the deterministic simulator is required to 
produce the following results
 1. Overall statistical information
 This part of report includes the statistics for a one-directional operation 
between stations. The results contain
 a. Review of input parameters: 
Track alignment, vehicle characteristics and operation options.
 b. Major performance measures: 
Total travel time; tractive energy consumption; and braking consumption.
 c. Cost components: 
Users-in-vehicle cost, vehicle capital expense, propulsive energy cost, 
braking cost and total cost
 d. Special event descriptions
 Points of begin cruising, coasting, and braking; maximum speed and first 
point to reach it; maximum and minimum centrifugal acceleration.  
2. Position-sorted and time-sorted simulation profiles
 Performance profiles between stations are provided in this part of reports. 
The profile data are sorted by either train locations or simulation times for the 
purpose of comparing and analyzing results between simulation cases. The 
information in performance profiles includes train location, travel time, speed, 
depth, gradient, tractive effort, resistance, applied acceleration, tractive energy 
and braking energy consumption and accumulation.
30
 Chapter 4. Evaluation of Results
 4.1 Comparison with Previous Model Results
 Numerical results from the deterministic simulator developed in this study 
were compared with results in Kim and Schonfeld?s (1997) study. A modified 
train deterministic simulator using the same train resistance and engineering 
equations as Kim and Schonfeld?s is developed to check the accuracy of the 
deterministic simulator used in this study.
 Table 3.2 shows comparison of results with the same inputs into 
simulators developed in this study and in Kim and Schonfeld?s study. The results 
show that the simulation model developed in this study is equivalent to the one 
developed by Kim and Schonfeld. The discrepancies between results may due to 
the selection of units. English units are used for simulator in this study while Kim 
and Schonfeld used SI units in their simulator.
 Table 4.1 Comparison with previous model results 
Travel Time
 (Sec.)
 Tractive Energy
 (KWH)
 Braking Energy
 (KWH)
 Baseline Assumptions
 Station Spacing: 3000m
 Max acceleration: 1.3m/s2
 Car power: 520 kw/car
 Unconstrained speed
 Car weight: 40 ton
 Kim & 
Schonfeld Yeh
 Kim & 
Schonfeld Yeh
 Kim & 
Schonfeld Yeh
 Baseline 119.0 118.8 75.6 73.9 58.5 57.2
 Acc.: 1.0 m/s2 125.7 125.3 72.1 71.1 56.3 55.0
 Pow.: 416 KW 125.5 125.4 66.9 65.8 50.9 50.40%
  
D
 ip
 Spd.: 120 km/hr 122.7 122.8 57.1 56.7 40.8 41.2
 Baseline 114.7 114.6 71.9 70.8 53.6 52.9
 Acc.: 1.0 m/s2 121.4 121.3 67.8 67.0 51.1 49.8
 Pow.: 416 KW 120.4 120.0 63.4 62.6 46.2 46.1
 0.
 5%
  
D
 ip
 Spd.: 120 km/hr 120.5 120.8 55.1 54.5 38.3 38.7
 Baseline 113.5 111.0 68.5 67.7 49.3 48.7
 Acc.: 1.0 m/s2 118.6 117.7 63.8 63.0 46.1 44.7
 Pow.: 416 KW 121.7 115.9 60.2 59.7 42.6 42.01%
  
D
 ip
 Spd.: 120 km/hr 119.5 118.9 55.1 54.5 35.2 38.4
31
 4.2 Baseline Case Assumptions
 The simulation is conducted for baseline operating parameters with 
various vertical alignments and then, repeated with certain parameter changes for 
sensitivity analysis.  
The evaluations of simulation results in this section focus on the analysis 
of baseline cases with specified parameters of vehicle characteristics, control 
parameters and station spacing under various vertical alignments. The design 
parameters used in this baseline evaluation are listed in Table 4.2. The numeral 
assumptions for vehicle characteristic and control options are generated from the 
values given by WMATA (2001) manual of design criteria. A high air resistance 
coefficient is used because Profillidis (1997) suggests that the air resistance 
coefficient for a train running in a tunnel with no openings is approximately three 
times the air resistance coefficient for a train outside tunnels.
 Table 4.2 Parameters used in baseline cases
 Parameters Value Units
 Station spacing Lss 12500 Feet
 Platform length Lpl 500 Feet
 Total curve length S Variable Feet
 Maximum depth 
 Variable FeetAli
 gn
 m
 en
 t
 Level bottom length Lbt Variable Feet
 Car power 520 KW
 Car weight 50 Ton
 Powered-axle/ Axle per car 4/4 -
 Number of cars per train 6 -
 Comfort-limited acceleration 4.827 (0.15g) ft/s2
 Cruise speed Unconstrained ft/s2
 Coasting distance No coasting Feet
 Si
 m
 u
 la
 tio
 n
  
A
 ss
 u
 m
 pt
 io
 n
 s
 Air resistance coefficient As in tunnels -
32
 Within baseline station spacing (12,500 ft), various alignment designs 
with combinations of parabolic curve coefficients S and 
 are analyzed. 
According to the definition of S and 
 from Section 3.4, total curve length S 
should be smaller or equal to total station spacing minus length of platform and 

/S ratio should not exceed 1%. The illustrations of vertical alignment designs 
with 12,500 feet station spacing are shown in Figure 4.1. Operating on alignments 
with relatively short curves, trains will dive to the lowest elevation in a short 
distance and then keep operating in the level section until they reach the climbing 
curves. However, due to the 1% constraint of 
/S ratio, alignments with short 
curves cannot have depths greater than 0.01S.
 Vertical alignments
 -140.00
 -120.00
 -100.00
 -80.00
 -60.00
 -40.00
 -20.00
 0.00
 0 2000 4000 6000 8000 10000 12000 14000
 Distance (ft)
 Depth (ft
 )
 (S,
)=(12000,0) (S,
)=(12000,60)
 (S,
)=(12000,120) (S,
)=(6000,60)
  Figure 4.1 Illustrations of vertical alignments
33
 4.3 Major Performance Measures Comparison of Baseline Cases
 27 vertical alignment designs with baseline operating parameters have 
been calculated in this section. The results for travel time T, cumulative tractive 
energy consumption Et and cumulative braking energy consumption Eb for the 
cases with various alignment designs are listed in Table 4.3.
 Table 4.3 Simulation results with baseline parameters
 Total Parabolic Curve Length S (ft)
 2000 4000 6000 8000 10000 12000Max. Depth
 
 (ft) Profile %* Profile %* Profile %* Profile %* Profile %* Profile %*
 0 T (sec) 149.46 0.00 149.46 0.00 149.46 0.00 149.46 0.00 149.46 0.00 149.46 0.00
 Et (kwh) 106.48 0.00 106.48 0.00 106.48 0.00 106.48 0.00 106.48 0.00 106.48 0.00
 Eb (kwh) 58.82 0.00 58.82 0.00 58.82 0.00 58.82 0.00 58.82 0.00 58.82 0.00
 20 T (sec) 146.10 -2.24 146.22 -2.16 146.87 -1.73 147.40 -1.38 147.29 -1.45 147.72 -1.16
 Et (kwh) 103.36 -2.93 104.10 -2.23 104.61 -1.75 104.92 -1.47 105.15 -1.25 105.34 -1.07
 Eb (kwh) 54.04 -8.12 54.63 -7.11 55.23 -6.10 55.67 -5.35 56.01 -4.77 56.38 -4.13
 40 T (sec) 143.47 -4.01 144.37 -3.40 144.65 -3.22 145.22 -2.84 145.68 -2.53
 Et (kwh) 101.72 -4.47 102.79 -3.47 103.48 -2.82 103.94 -2.39 104.31 -2.05
 Eb (kwh) 50.40 -14.32 51.60 -12.26 52.51 -10.72 53.27 -9.43 53.96 -8.26
 60 T (sec) 141.50 -5.33 142.42 -4.71 143.23 -4.17 144.04 -3.63
 Et (kwh) 101.02 -5.13 102.12 -4.10 102.81 -3.45 103.35 -2.95
 Eb (kwh) 47.92 -18.53 49.42 -15.97 50.58 -14.00 51.63 -12.22
 80 T (sec) 140.38 -6.07 141.51 -5.32 142.48 -4.67
 Et (kwh) 100.85 -5.29 101.77 -4.43 102.47 -3.77
 Eb (kwh) 46.39 -21.13 47.97 -18.45 49.35 -16.10
 100 T (sec) 140.06 -6.29 141.18 -5.54
 Et (kwh) 100.80 -5.33 101.65 -4.54
 Eb (kwh) 45.41 -22.79 47.13 -19.87
 120 T (sec) 139.69 -6.53
 Et (kwh) 100.91 -5.23
 Eb (kwh) 44.92 -23.62
 T: Travel time   Et: Tractive energy consumption   Eb: Braking energy consumption
 %* : Percentage Saving (compared to results of level track) 
34
 Figures 4.2 to 4.4 show relations between major performance measures of 
travel time, tractive energy consumption and braking wear between stations and 
various alignment designs. Compared with results of level track, the deepest 
alignment (S=12000 ft; 
=120 ft) has the greatest saving in travel time, tractive 
energy and braking energy. The maximum saving rates are 6.53% in travel time, 
5.23% in tractive energy and 23.62% in braking energy. In addition, the saving 
for cases with the same total curve length S but different in maximum depth  are 
approximately proportional to the value of . Moreover, when comparing results 
from cases with the same maximum depth 
, cases with shorter total parabolic 
curves S (trains dive to the lowest points in a shorter distance) have greater 
savings in travel time, tractive energy and braking energy.
 Travel Time-Max. Depth
 138
 140
 142
 144
 146
 148
 150
 152
 0 20 40 60 80 100 120 140
 Max. Depth (ft)
 Travel time (sec
 )
 S=2000 S=4000 S=6000 S=8000
 S=10000 S=12000
 Figure 4.2 Travel time-Maximum depth relations
35
 Tractive Energy-Max. Depth
 100
 101
 102
 103
 104
 105
 106
 107
 0 20 40 60 80 100 120 140
 Max. Depth (ft)
 Tr
 a
 c
 tiv
 e
  
En
 e
 rg
 y 
(K
 W
 H
 )
 S=2000 S=4000 S=6000 S=8000
 S=10000 S=12000
 Figure 4.3 Tractive Energy-Maximum depth relations
 Braking Energy-Max. Depth
 40
 42
 44
 46
 48
 50
 52
 54
 56
 58
 60
 0 20 40 60 80 100 120 140
 Max. Depth (ft)
 B
 ra
 ki
 n
 g 
En
 er
 gy
  
(K
 W
 H
 )
 S=2000 S=4000 S=6000 S=8000
 S=10000 S=12000
 Figure 4.4 Braking energy-Maximum depth relations
36
 4.4 Performance Profile Comparison 
Performance profiles between four selected alignment designs (Table 4.4; 
Figure 4.1) with baseline assumptions are discussed in this section. In addition, 
the selected baseline cases were classified into two sets based on two concepts. 
The first set of cases includes cases 1, 2, and 3 whose vertical alignments have the 
same total curve length S (12000 ft) but different maximum depths 
 (0 ft; 60 ft; 
120 ft). The second set of cases includes cases 1, 2, and 4, which contain one level 
track and two dipped alignments with the same value in maximum depth 
 (60ft) 
but different values of total curve length S (6000ft; 12000ft). 
Table 4.4 Selected baseline cases for simulation profile comparison.
 Cases Total curve 
length S (ft)
 Maximum 
depth 
 (ft)
 
 / S ratio
 1 - 0 0%
 2 12000 60 0.5%
 3 12000 120 1%
 4 6000 60 1%
 For performance profiles of each selected baseline case, location-sorted 
and time-sorted profile are provided here. The reason for using a location-sorted 
data is to observe and compare train movements under related geographic 
conditions between alignments. The reason for using time-sorted data is to 
observe and compare time-related information.
37
 4.4.1 Performance Profile Comparison for Case Set 1
 The first case set contains three alignments with the same total curve 
length S but different maximum depth 
 (0 ft; 60 ft; 120 ft). In this set of baseline 
cases, the savings for travel time, tractive energy and braking energy are listed in 
Table 4.5. For baseline cases in set 1, the savings in travel time, tractive energy 
and braking are approximately proportional to the value of maximum depth 
.  
Table 4.5 Travel time, tractive energy and braking energy for baseline case set 1
 Case1
 (S=12000; 
=0)
 Case 2
 (S=12000; 
=60)
 Case 3
 (S=12000; 
=120)
 Result Saving* Result Saving* Result Saving*
 Travel Time (sec) 149.46 0% 144.04 -3.63% 139.69 -6.54%
 Tractive E. (kwh) 106.48 0% 103.35 -2.94% 100.91 -5.23%
 Braking E. (kwh) 58.82 0% 51.63 -12.22% 44.92 -23.63%
 Saving* is the difference in percentage compared to results of level track 
Figures 4.5 and 4.6 show the relations between train locations and their 
elevation or gradient. The case with larger maximum depth 
 has greater change 
in both elevation and gradient. 
38
 Location-Elevation
 -140.00
 -120.00
 -100.00
 -80.00
 -60.00
 -40.00
 -20.00
 0.00
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 El
 ev
 at
 io
 n
  
(ft
 )
 Case 1 Case 2 Case 3
 Figure 4.5 Location-Elevation relations for baseline case set 1
 Location-Gradient
 -5
 -4
 -3
 -2
 -1
 0
 1
 2
 3
 4
 5
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Gradient (%
 )
 Case 1 Case 2 Case 3
 Figure 4.6 Location-Gradient relations for baseline case set 1
39
 Figure 4.7 is the location-resistance diagram. For trains operating on 
dipped alignments, the train resistances are affected rapidly by the changes of 
gradient.
 Location-Resistance
 -20000
 -10000
 0
 10000
 20000
 30000
 40000
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 R
 es
 is
 ta
 n
 ce
  
(lb
 w
 )
 Case 1 Case 2 Case 3
 Figure 4.7 Location-Resistance relations for baseline case set 1
 Figure 4.8 is the location-acceleration diagram. On a dipped alignment, 
due to the low resistance in the first half (0~6250 ft), descending trains can 
operate with slightly higher acceleration rate than trains on level track. 
40
 Location-Acceleration
 -6
 -4
 -2
 0
 2
 4
 6
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 A
 cc
 .
 (ft
 /s
 ^
 2)
 Case 1 Case 2 Case 3
 Figure 4.8 Location-Acceleration relations for baseline case set 1
 Figures 4.9 and 4.10 show location-speed and location-time relations. The 
speed gaps between case 1, case 2 and case 3 increase with the train movements 
in the descending section (first half). In the climbing section (second half), the 
train speed gaps between cases decrease with the train movements. The greatest 
speed differences between the three cases occur around halfway between stations 
(lowest points for dipped alignments). Due to high operating speeds, travel times 
between stations are short for dipped alignments. 
41
 Location-Speed
 0
 20
 40
 60
 80
 100
 120
 140
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Sp
 ee
 d 
(ft
 /s
 )
 Case 1 Case 2 Case 3
 Figure 4.9 Location-Speed relations for baseline case set 1
 Location-Time
 0
 20
 40
 60
 80
 100
 120
 140
 160
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Ti
 m
 e 
(se
 c)
 Case 1 Case 2 Case 3
 Figure 4.10 Location-Time relations for baseline case set 1
42
 Figure 4.11 is the time-acceleration diagram. This figure shows that after 4 
seconds (approximately 40 feet in travel distance) the traction motors cannot 
provide acceleration higher than comfort-limited acceleration (0.15g). After that, 
the acceleration rates for three cases are influenced by the gradient.
 Time-Acceleration
 -6
 -4
 -2
 0
 2
 4
 6
 0 50 100 150
 Time (sec)
 Ac
 ce
 lra
 tio
 n
  
(ft
 /s
 ^
 2)
 Case 1 Case 2 Case 3
 Figure 4.11 Time-Acceleration relations for baseline case set 1
43
 Figure 4.12 shows the relations between time and speed. The areas under 
trajectory lines are total station spacing. Because the station spacing for the three 
baseline cases are equal, the trajectory line with higher peak value (speed) can 
reach the x-axis faster (trains arrive next station in shorter time).
 Time-Speed
 0
 20
 40
 60
 80
 100
 120
 140
 0 20 40 60 80 100 120 140 160
 Time (sec)
 Sp
 ee
 d 
(ft
 /s
 )
 Case 1 Case 2 Case 3
 Figure 4.12 Time-Speed relations for baseline case set 1
44
 Figure 4.13 shows the relations between time and cumulative tractive 
energy. Because no speed constraints are applied in baseline study, trains in all 
cases operate with their full power until starting to brake. Thus, the energy 
consumption per unit time is equal for trains in all cases and trains start braking 
earlier consume less tractive energy between stations.
 Time-Cu. Tractive Energy
 0
 20
 40
 60
 80
 100
 120
 0 20 40 60 80 100 120 140 160
 Time (sec)
 Cu
 . 
Tr
 ac
 tiv
 e 
En
 er
 gy
  
(kw
 h)
 Case 1 Case 2 Case 3
 Figure 4.13 Time-Cumulative tractive energy relations for baseline case set 1
 Figure 4.14 shows the relations between travel time and cumulative 
braking energy. Table 4.6 shows the existing conditions at the begin-braking 
points for case 1 to case 4. In Table 4.6, the summation of train kinetic and 
potential energy for cases with deeper alignments is low. This is because cases 
with deeper alignments operate in shorter travel time and acquire less kinetic 
45
 energy from traction motors. Moreover, due to the high operating speed for trains 
on dipped alignment, they waste more kinetic energy in air resistance and require 
less breaking energy to stop.
 Time-Cu. Braking Energy
 0
 10
 20
 30
 40
 50
 60
 70
 0 20 40 60 80 100 120 140 160
 Time (sec)
 Cu
 .
  
B
 ra
 ki
 n
 g 
En
 er
 gy
  
(kw
 h)
 Case 1 Case 2 Case 3
 Figure 4.14 Time-Cumulative braking energy relations for baseline case set 1
 Table 4.6 Existing conditions at begin-braking points for selected baseline cases 
Begin-Braking 
Point (ft)
 Travel Time 
(sec)
 Elevation
 (ft)
 Speed
 (ft/s)
 Kinetic plus Potential 
Energy (lb-ft2/s2)
 Case 1 10966 124.9 0 118.90 7069*mt*
 Case 2 11176 121.3 -5.8 112.47 6139* mt*
 Case 3 11348 118.5 -8.1 105.45 5297* mt*
 Case 4 11161 118.6 -23.4 113.69 5707* mt*
 Note: mt* = Train mass in pounds
46
 The energy conservation law can be used to explain why the savings for 
cases with the same parabolic curve length S are roughly proportional to the 
maximum depth 
. By the energy conservation law, if the air friction is negligible, 
all released potential energy from vertical drop should transform to kinetic 
energy. The energy conservation equation is listed below 
( ) ( )2020 2
 1
 2
 1 VmVVmghm ttt +=    or    2
 842 200 ghVVV
 ++
 = (4.1)
 where V is the speed increment caused by vertical drop
 V0 is initial speed
 mt is train mass
 g is acceleration due to gravity
 h is vertical drop
 For the above equation, if the initial speed V0 is much greater than zero, 
the speed increment V will be approximately proportional to the vertical drop h. 
The alignment equations in Table 3.1 show that the vertical drops h for cases with 
the same total curve length S are proportional to the coefficient 
 (maximum 
depth). Therefore, the speed increment V for dipped alignment is also 
approximately proportional to the maximum depth 
 and the reduction of total 
travel time will be also roughly proportional to the maximum depth 
 (Figure 4.9). 
Moreover, because the tractive energy consumption strongly depends on travel 
time and the braking energy consumption strongly depends on tractive energy 
consumption, the resulting tractive energy consumption and braking energy 
consumption should also be approximately proportional to the maximum depth 
.
47
 4.4.2 Performance Profile Comparison for Case Set 2
 The second set of baseline cases contains one level track and two dipped 
track alignments with the same maximum depth 
 (60ft) but different total curve 
length S (12000ft; 6000ft). In this set of baseline cases, the savings for travel 
time, tractive energy and braking energy are listed in Table 4.7. For the two 
dipped alignments, the saving rates increase as the total curve length S decreases.  
Table 4.7 Travel time, tractive energy and braking energy for baseline case set 2
 Case1
 (S=12000; 
=0)
 Case 2
 (S=12000; 
=60)
 Case 4
 (S=6000; 
=60)
 Profile Saving* Profile Saving* Profile Saving*
 Travel Time (sec) 149.45 0% 143.83 -3.76% 141.19 -5.53%
 Tractive E. (kwh) 106.48 0% 103.47 -2.82% 101.87 -4.33%
 Braking E. (kwh) 58.82 0% 51.71 -12.10% 48.62 -17.35%
 Saving* is the difference in percentage compared to results of level track
 Figures 4.15 and 4.16 show the relations between train locations and their 
elevation or gradient for baseline case set 2. In this set, trains in case 2 dives/rises
 to/from the maximum depth (60 ft) with a smaller gradient change (within ? 2%) 
and over a longer distance (250~6250 ft; 6250~12250 ft) while trains in case 4 
dives/rises to/from the maximum depth (60 ft) with a greater gradient change 
(within ? 4 %) and over a shorter distance (250~3250 ft; 9250~12250 ft).  
48
 Location-Elevation
 -70.00
 -60.00
 -50.00
 -40.00
 -30.00
 -20.00
 -10.00
 0.00
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 El
 ev
 at
 io
 n
  
(ft
 )
 Case 1 Case 2 Case 4
 Figure 4.15 Location-Elevation relations for baseline case set 2
 Location-Gradient
 -4
 -3
 -2
 -1
 0
 1
 2
 3
 4
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 G
 ra
 di
 en
 t (%
 )
 Case 1 Case 2 Case 4
 Figure 4.16 Location-Gradient relations for baseline case set 2
49
 Figure 4.17 shows the location-resistance relations for baseline case set 2. 
The train resistances on dipped vertical track alignments are influenced by the 
track gradient. 
Location-Resistance
 -20000
 -10000
 0
 10000
 20000
 30000
 40000
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 R
 es
 is
 ta
 n
 ce
  
(lb
 w
 )
 Case 1 Case 2 Case 4
 Figure 4.17 Location-Resistance relations for baseline case set 2
 Figure 4.18 shows the location-acceleration relation. Trains in case 1 on 
level track accelerate smoothly until starting brake. Trains in case 2 on long curve 
accelerate with slightly faster than trains in case 1 for the first half (descending) 
section and slightly slower than trains in case 1 for the second half (climbing) 
section. Trains in case 4 accelerate with high rate within its descending section 
(0~3250 ft) and operates in high speed without deceleration within the level 
section (3250~9250 ft).
50
 Location-Acceleration
 -6
 -4
 -2
 0
 2
 4
 6
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 A
 cc
 .
 (ft
 /s
 ^
 2)
 Case 1 Case 2 Case 4
 Figure 4.18 Location-Acceleration relations for baseline case set 2
 Figures 4.19 and 4.20 show location-speed and location-time diagrams. In 
general, trains operating on steeper vertical alignments gather more advantages in 
accelerating to high speed early and have less travel time between stations.
51
 Location-Speed
 0
 20
 40
 60
 80
 100
 120
 140
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Sp
 ee
 d 
(ft
 /s
 )
 Case 1 Case 2 Case 4
 Figure 4.19 Location-Speed relations for baseline case set 2
 Location-Time
 0
 20
 40
 60
 80
 100
 120
 140
 160
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Ti
 m
 e 
(se
 c)
 Case 1 Case 2 Case 4
 Figure 4.20 Location-Time relations for baseline case set 2
52
 Figures 4.21 and 4.22 show time-acceleration and time-speed relations. 
From the two diagrams, trains in case 4 accelerate at a higher rate at the beginning 
and begins decelerating earlier than trains in case 2. 
Time-Acceleration
 -6
 -4
 -2
 0
 2
 4
 6
 0 50 100 150
 Time (sec)
 Ac
 ce
 lra
 tio
 n
  
(ft
 /s
 ^
 2)
 Case 1 Case 2 Case 4
 Figure 4.21 Time-Acceleration relations for baseline case set 2
53
 Time-Speed
 0
 20
 40
 60
 80
 100
 120
 140
 0 20 40 60 80 100 120 140 160
 Time (sec)
 Sp
 ee
 d 
(ft
 /s
 )
 Case 1 Case 2 Case 4
 Figure 4.22 Time-Speed relations for baseline case set 2
54
 Figure 4.23 shows the time-cumulative tractive energy relations. As 
mentioned earlier, trains reaching their begin-raking points faster consume less 
tractive energy. In Table 4.6, trains in case 4 are the earliest reaching begin-
 braking point and consume the least tractive energy.  
Time-Cu. Tractive Energy
 0
 20
 40
 60
 80
 100
 120
 0 20 40 60 80 100 120 140 160
 Time (sec)
 Cu
 .
  
Tr
 ac
 tiv
 e 
En
 er
 gy
  
(kw
 h)
 Case 1 Case 2 Case 4
 Figure 4.23 Time-Cumulative tractive energy relations for baseline case set 2
55
 Figure 4.24 shows the relations between travel time and cumulative 
braking energy. In the results of Table 4.6, trains in case 4 need less braking 
energy than trains in case 2 because at the begin-braking point the initial speeds 
for the two dipped alignments are almost the same but elevation for trains in case 
4 are lower than trains in case 2.   
Time-Cu. Braking Energy
 0
 10
 20
 30
 40
 50
 60
 70
 0 20 40 60 80 100 120 140 160
 Time (sec)
 Cu
 .
  
B
 ra
 ki
 n
 g 
En
 er
 gy
  
(kw
 h)
 Case 1 Case 2 Case 4
 Figure 4.24 Time-Cumulative braking energy relations for baseline case set 2
 In conclusion, for simulation cases with the same maximum depth 
 but 
different total curve length S, trains in cases with shorter S have advantages in 
accelerating to high speed early to reduce total travel time and tractive energy 
consumption. Moreover, because of the late rise in alignment for cases with short 
total curve length S, the elevations for trains at begin-braking point are lower and 
result in lower braking energy consumption. 
56
 4.5 Sensitivity Analysis
 The purpose of this section is to analyze the sensitivity of results to 
parameter changes listed in Table 4.8. For each instance of sensitivity analysis, 
the value of a selected parameter will be replaced by the new value in Table 4.8 
while other operating parameters will remain at their baseline values.
 Table 4.8 Parameters for sensitivity analysis
 Baseline 
assumptions
 Sensitivity 
analysis
 Unit
 Acceleration constraint 4.827 3.218 ft/s2
 Power per car 520 400 KW
 Speed constraint No constraint 75 MPH
 Dist. begin coasting No coasting 4000 Feet
 Air resis. coeff. 3X as Davis Eq. As Davis Eq. -
 Car weight 50 40 Ton
 Station spacing 12500 8500 Feet
57
 4.5.1 Sensitivity Analysis Results
 In this sensitivity analysis, four vertical track alignment designs are 
investigated as in Table 4.9. The simulation results for sensitivity analysis are 
shown in Tables 4.10-4.12.
 Table 4.9 Vertical track alignments for sensitivity analysis
 Total curve length S (ft) 
/S ratio Level bottom (ft)
 Alignment I - 0 -
 Alignment II Lss-Lpl 0.5% 0
 Alignment III Lss-Lpl 1% 0
 Alignment IV ? (Lss-Lpl) 1% ? (Lss-Lpl)
 Lss: Station spacing     Lpl: Length of platform
 Table 4.10 Sensitivity analysis for travel time
 Alignment I Alignment II Alignment III Alignment IVTravel time
 (Sec) Profile Saving Profile Saving Profile Saving Profile Saving
 Baseline 
assumptions 149.60 0 144.22
 -5.38
 -3.60% 140.04
 -9.56
 -6.39% 141.83
 -7.76
 -5.19%
 Acc. Const: 
3.218 ft/s2 157.91 0 152.34
 -5.57
 -3.53% 147.34
 -10.56
 -6.69% 149.90
 -8.01
 -5.07%
 Power per car:
 400 KW 160.31 0 154.18
 -6.13
 -3.83% 148.92
 -11.39
 -7.10% 150.90
 -9.41
 -5.87%
 Speed const:
 75 MPH 151.27 0 147.20
 -4.07
 -2.69% 147.24
 -4.02
 -2.66% 145.03
 -6.23
 -4.12%
 Coasting Dist:
 4000 feet 151.92 0 147.19
 -4.73
 -3.11% 143.86
 -8.06
 -5.31% 144.28
 -7.64
 -5.03%
 Air res. coeff:
 As Davis Eq. 143.47 0 137.37
 -6.10
 -4.25% 132.74
 -10.73
 -7.48% 134.91
 -8.56
 -5.97%
 Car weight:
 40 ton 143.46 0 139.17
 -4.29
 -2.99% 135.59
 -7.87
 -5.49% 137.30
 -6.16
 -4.30%
 Station spacing
 8500 feet 114.84 0 111.19
 -3.65
 -3.18% 107.89
 -6.95
 -6.05% 109.34
 -5.50
 -4.79%
 *Saving is the difference in value or in percentage compared with case with the 
same assumptions but operating on level track (Alignment I)  
58
 Table 4.11 Sensitivity analysis for tractive energy consumption
 Alignment I Alignment II Alignment III Alignment IVTractive Energy 
(KWH) Profile Saving Profile Saving Profile Saving Profile Saving
 Baseline 
assumptions 106.48 - 103.34
 -3.14
 -2.95% 100.91
 -5.57
 -5.23% 101.02
 -5.46
 -5.13%
 Acc. Const: 
3.218 ft/s2 101.72 - 97.99
 -3.74
 -3.67% 95.43
 -6.29
 -6.18% 94.49
 -7.24
 -7.11%
 Power per car:
 400 KW 91.31 - 87.97
 -3.34
 -3.65% 85.59
 -5.71
 -6.26% 85.95
 -5.36
 -5.87%
 Speed const:
 75 MPH 97.01 - 96.12
 -0.89
 -0.91% 88.34
 -8.66
 -8.93% 91.13
 -5.88
 -6.06%
 Coasting Dist:
 4000 feet 88.14 - 83.15
 -4.99
 -5.66% 79.00
 -9.13
 -10.36% 81.55
 -6.59
 -7.47%
 Air res. coeff:
 As Davis Eq. 96.88 - 93.11
 -3.77
 -3.89% 90.00
 -6.88
 -7.10% 90.73
 -6.15
 -6.35%
 Car weight:
 40 ton 100.01 - 97.84
 -2.16
 -2.16% 96.15
 -3.85
 -3.85% 95.86
 -4.15
 -4.15%
 Station spacing
 8500 feet 78.21 - 75.75
 -2.45
 -3.14% 73.67
 -4.53
 -5.80% 73.84
 -4.36
 -5.58%
 *Saving is the difference in value or in percentage compared with case with the 
same assumptions but operating on level track (Alignment I)  
Table 4.12 Sensitivity analysis for braking energy consumption
 Alignment I Alignment II Alignment III Alignment IVBraking Energy 
(KWH) Profile Saving Profile Saving Profile Saving Profile Saving
 Baseline 
assumptions 58.83 - 51.65
 -7.18
 -12.21% 44.95
 -13.88
 -23.59% 47.95
 -10.88
 -18.50%
 Acc. Const: 
3.218 ft/s2 56.05 - 47.99
 -8.06
 -14.38% 40.84
 -15.21
 -27.14% 43.32
 -12.73
 -22.71%
 Power per car:
 400 KW 50.12 - 42.96
 -7.16
 -14.28% 36.36
 -13.76
 -27.46% 39.52
 -10.60
 -21.15%
 Speed const:
 75 MPH 51.00 - 47.64
 -3.36
 -6.60% 40.23
 -10.77
 -21.12% 41.56
 -9.44
 -18.51%
 Coasting Dist:
 4000 feet 42.58 - 33.86
 -8.72
 -20.47% 25.75
 -16.83
 -39.53% 30.82
 -11.76
 -27.61%
 Air res. coeff:
 As Davis Eq. 75.62 - 69.88
 -5.74
 -7.59% 64.77
 -10.85
 -14.34% 66.94
 -8.68
 -11.48%
 Car weight:
 40 ton 48.93 - 43.10
 -5.84
 -11.93% 37.57
 -11.36
 -23.22% 39.86
 -9.07
 -18.54%
 Station spacing
 8500 feet 51.19 - 46.59
 -4.59
 -8.98% 42.23
 -8.95
 -17.49% 44.00
 -7.19
 -14.05%
 *Saving is the difference in value or in percentage compared with case with the 
same assumptions but operating on level track (Alignment I)  
59
 4.5.2 Analysis of Sensitivity Results
 Figures 4.25-4.28 show the speed-location relations for sensitivity analysis 
on various vertical alignments. Table 4.13 shows the existing conditions at the 
begin-braking points for sensitivity analysis. The four speed diagrams and the 
begin-braking condition table show changes in travel time, tractive energy 
consumption and braking energy consumption in the sensitivity analysis.
 Effect of Maximum Allowable Acceleration
 While operating with a low acceleration constraint, the saving for travel 
time and braking energy increases slightly. The change in maximum allowable 
acceleration rate result in only a small change because the acceleration constraint 
only effects small fraction of time in total simulation period (only for the first and 
last few seconds of acceleration and deceleration) (Figures 4.25-4.28). Because of 
the long braking distances due to the low deceleration rate, the begin-braking 
points for dipped alignments will be located lower and result more braking energy 
saving from potential energy.
60
 Effects of Power and Vehicle Weight
 Greater percentage savings are observed for low power vehicles on dipped 
alignments. Moreover, when vehicle weight decreases, the saving on dipped 
alignments decreases.
 In theory, the speed increment V from released potential energy is
 2
 842 200 ghVVV
 ++
 = (4.1)
 If the power/weight ratio increases, the train speed on level track will 
increases too. Thus, because of the higher initial speeds V0 for trains on level 
alignment, the speed increments V from potential energy while using dipped 
alignment will be lower (Figures 4.25-4.28) and result in fewer saving in travel 
time and tractive energy consumption.
 Effect of Speed Constraint
 With a low speed limit, the savings in travel time, tractive energy 
consumption and braking energy consumption are lower than without speed limit 
and are no longer proportional to the maximum depth 
 for cases with the same 
total curve length S. Trains reaching the speed limit within a descending section 
cannot transform potential energy into kinetic energy by accelerating to higher 
speeds. 
61
 Effect of Train Coasting
 Coasting can increase the percentage savings in tractive energy and 
braking energy. In Tables 4.10-4.12, when coasting, the percentage of saving in 
travel time drops slightly but savings in tractive energy and braking energy 
consumption are almost doubled. Coasting does not significantly increase travel 
time is because the non-power operating only results in small speed drop for the 
last few seconds (Figures 4.25-4.28). Trains cut off their traction power early and 
save more tractive energy by using coasting. Table 4.13 shows that compared 
with coasting on level track, trains coasting on a dipped alignment decelerate 
more before starting to brake and require less braking energy to stop. 
Effect of Air Friction
 An air resistance coefficient used for trains operating outside tunnels is 
applied in this sensitivity analysis. Low air friction increases the travel time and 
the tractive energy consumption benefits from using dipped alignments. 
Therefore, in this sensitivity analysis, low air friction amplifies (Figures 4.25-
 4.28) the train speed increments on dipped alignments and results more saving in 
travel time. Moreover, cutting off traction motors early also reduces the tractive 
energy consumption. However, the low air friction results in lower speed 
reductions during braking and less advantage in helping to stop trains. 
62
 Effect of Station Spacing
 The percentage savings for travel time and tractive energy consumption 
are not significant changed by applying different station spacings. However, for 
cases with dipped alignments and long station spacing, the initial speed at begin-
 braking point should be high. Thus, trains on long and dipped alignment waste 
more kinetic energy in air friction during braking and require less braking energy 
consumption. 
63
 Table 4.13 Begin-braking information for sensitivity analysis
 Changing 
Variables
 Begin-
 Braking 
Point (ft)
 Travel 
Time 
(sec)
 Elev.
 (ft)
 Speed
 (ft/s)
 Kinetic plus 
Potential 
Energy (lb-
 ft2/s2)
 Baseline assum. 10966 124.9 0 118.90 7069*mt
 Max. acc. :0.1g 10346 122.0 0 117.74 6932*mt
 Pow/car:400kw 11246 138.5 0 109.07 5948*mt
 Spd cons:75mph 11221 128.7 0 110.00 6050*mt
 Coasting:4000 ft 11462 131.5 0 99.93 4993*mt
 1x air resistance 10357 113.8 0 132.62 8794*mt
 Weig/car:40 ton 10863 117.9 0 122.74 7533*mt
 Alignment 
I
 Stat Spa:8500 ft 7216 92.3 0 110.22 6074*mt
 Baseline assum. 11176 121.3 -5.8 112.47 6139*mt
 Max. acc. :0.1g 10510 117.6 -15.1 113.15 5915*mt
 Pow/car:400kw 11432 133.5 -3.3 101.51 5044*mt
 Spd cons:75mph 11299 125.5 -4.5 107.49 5631*mt
 Coasting:4000 ft 11673 129.0 -1.7 89.31 3935*mt
 1x air resistance 10549 109.5 -14.5 130.78 8087*mt
 Weig/car:40 ton 11084 115.5 -6.8 116.28 6542*mt
 Alignment 
II
 Stat Spa:8500 ft 7316 89.5 -6.5 106.87 5500*mt
 Baseline assum. 11348 118.5 -8.1 105.45 5297*mt
 Max. acc. :0.1g 10708 114.7 -23.8 107.38 5001*mt
 Pow/car:400kw 11593 130.0 -4.3 93.58 4239*mt
 Spd cons:75mph 11540 127.9 -5.0 96.30 4474*mt
 Coasting:4000 ft 11876 128.1 -1.4 77.60 2966*mt
 1x air resistance 10714 105.9 -23.6 128.20 7458*mt
 Weig/car:40 ton 11266 113.5 -9.7 109.13 5644*mt
 Alignment 
III
 Stat Spa:8500 ft 7398 87.1 -10.9 103.15 4969*mt
 Baseline assum. 11161 118.6 -23.5 113.69 5707*mt
 Max. acc. :0.1g 10319 113.6 -48.6 118.47 5455*mt
 Pow/car:400kw 11455 130.5 -12.6 100.43 4637*mt
 Spd cons:75mph 11387 124.0 -14.9 103.66 4893*mt
 Coasting:4000 ft 11722 126.7 -5.6 86.64 3574*mt
 1x air resistance 10509 106.7 -44.1 135.28 7730*mt
 Weig/car:40 ton 11058 113.2 -27.3 117.96 6078*mt
 Alignment 
IV
 Stat Spa:8500 ft 7253 87.3 -24.9 109.72 5218*mt
 Note: mt = Train mass in pounds
64
 Speed Profile: Alignment I
 0
 20
 40
 60
 80
 100
 120
 140
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Sp
 ee
 d 
(ft
 /s
 ec
 )
 Baseline 0.1g 400kw 75mph
 4000coasting 1x resis 40ton
 Figure 4.25 Sensitivities of train speeds to parameter changes for alignment I
 Speed Profile: Alignment II
 0
 20
 40
 60
 80
 100
 120
 140
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Sp
 ee
 d 
(ft
 /s
 ec
 )
 Baseline 0.1g 400kw 75mph
 4000coasting 1x resis 40ton
 Figure 4.26 Sensitivities of train speeds to parameter changes for alignment II
65
 Speed Profile: Alignment III
 0
 20
 40
 60
 80
 100
 120
 140
 160
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Sp
 ee
 d 
(ft
 /s
 ec
 )
 Baseline 0.1g 400kw 75mph
 4000coasting 1x resis 40ton
 Figure 4.27 Sensitivities of train speeds to parameter changes for alignment III
 Speed Profile: Alignment IV
 0
 20
 40
 60
 80
 100
 120
 140
 160
 0 2000 4000 6000 8000 10000 12000 14000
 Location (ft)
 Sp
 ee
 d 
(ft
 /s
 ec
 )
 Baseline 0.1g 400kw 75mph
 4000coasting 1x resis 40ton
 Figure 4.28 Sensitivities of train speeds to parameter changes for alignment IV
66
 Chapter 5. Optimization of Simulation Results
 5.1 Optimization Methods
 The total cost function Ct in this study is only for a one-directional 
operation between stations. As mentioned in Section 3.6, total cost Ct for a one-
 directional operation between stations includes user-in-vehicle cost, vehicle 
capital expenses (vehicle depreciation), tractive energy expenses, braking wear 
expenses and construction cost. 
Because we assume the construction cost increments is not increased for 
dipped alignments. It can be simplified as zero for any alignment design in these 
total cost evaluations and comparisons. However, if construction cost was depth-
 related, the benefits of using dipped alignment in rail transit system could, of 
course, decrease. Table 5.1 shows the baseline values used in the total cost 
function for the following analysis.
 Table 5.1 Baseline values used in total cost function
 Variable Definition Value Units
 np Average passengers per car 50 Psg/car
 vuser Value of user time 5 $/psg-hr
 vvehicle Vehicle depreciation per revenue hour 50 $/car-hr
 Cconstruction
 Average construction cost for an one-
 directional operation between stations 0
 $/one-
 directional 
operation
 vtractive Unit tractive energy cost 0.15 $/kwh
 vbraking Unit braking energy cost 0.1 $/kwh
67
 The baseline case and sensitivity study in Chapter 4 show that the 
simulation results are smooth and continuous with respect to all simulation 
parameters. Therefore, Golden Section Search in one-dimensional optimizations 
and Powell?s Method in multi-dimensional optimization can be used with the 
deterministic simulator to minimize the total cost function.
 5.2 Joint Optimization of Total Cost
  Three station spacings (16,500 ft; 12500 ft; 8500 ft) are selected in the 
joint optimization studies. For all jointly optimizing problems, excepting 
changing variables, all other variables will use baseline assumed values as 
mentioned in Section 4.1.
68
 Table 5.2 shows jointly optimized solutions for the following changing 
variables: alignment coefficients S and 
 (for dipped alignment only), speed 
constraint Vcruise and coasting distance dcoasting. The optimizations demonstrate the 
following points:
 1. The optimized speed limits are equal to their maximum speeds. Therefore, 
unconstrained speeds are the optimized solutions for speed constraints.
 2.  Without a speed constraint, the percentage savings for using optimized 
dipped alignments are about 6.7%
 3. Without a speed constraint, the optimized alignment depths are about 0.75% 
of station spacings.
 4.  Without a speed constraint, the 
/S ratio for optimized alignment is about 1% 
(the maximum allowable 
/S value)
 Table 5.2 Jointly optimized alignments, speed constraints and coasting distance
 Station Spacing:
 8,500 ft
 Station Spacing:
 12,500 ft
 Station Spacing:
 16,500 ft
 Alignment Level Dipped Level Dipped Level Dipped
 S (ft)
 - 6595 - 10241 - 11915
 
 (ft) 0.0 62.3 0.0 101.1 0.0 117.0
 Vcruise (mph) 64.2 73.4 73.7 83.3 79.1 87.1Op
 tim
 iz
 ed
  
dcoasting (ft) 3777 3842 4954 4759 5951 4498
 Vmax (mph) 64.2 73.4 73.7 83.3 79.1 87.0
 Lvmax (ft) 3697 3983 6592 6724 9481 10487
 T (sec) 120.34 113.07 155.03 145.90 188.72 175.49
 Et (kwh) 52.26 50.23 77.65 73.33 101.14 102.83
 Eb (kwh) 28.92 22.53 34.84 22.23 36.96 25.04
 Ct ($) 70.90 66.32 92.65 86.17 113.23 105.67
 Ct  Saving ($) - -4.58 - -6.47 - -7.55
 Pe
 rfo
 rm
 an
 ce
  
m
 ea
 su
 re
 s
 Ct  Saving (%) - -6.46 - -6.99 - -6.67
69
 Tables 5.3 and 5.4 show the Jointly optimized solutions for alignment 
coefficients S and 
 (for dipped alignment only), and coasting distance dcoasting
 while speed constraints are pre-specified (75 mph & 55 mph). Tables 5.3 and 5.4 
show the following points for optimized solutions with low speed constraints:
 1. For optimized alignments with a low speed constraint, trains accelerate to the 
speed constraint approximately at the point (Lvmax) where the descending 
curves are finished (Lpl/2 + S/2).
 2. Optimized parabolic curve designs for cases with the same speed constraint 
but different station spacings are approximately equivalent to each other. 
Moreover, the optimized total cost savings are approximately equal for cases 
with the same speed constraints but different station spacings (Tables 5.3 and 
5.4 show $5.4 total cost savings for a one-directional operation with a 75mph 
speed constraint and $2.1 with a 55mph speed constraint).
 3. The optimized total cost savings decreases as the speed limit decreases. 
4. The optimized 
/S ratios are also approximately equal to 1.0 (the maximum 
allowable 
/S value) for operating with speed constraints.
70
 Table 5.3 Jointly optimized alignments and coasting distance with a 75mph speed 
constraint
 Station Spacing:
 8,500 ft
 Station Spacing:
 12,500 ft
 Station Spacing:
 16,500 ft
 Alignment Level Dipped Level Dipped Level Dipped
 S (ft)
 - 7459 - 7654 - 8453
 
 (ft) 0.0 72.0 0.0 74.5 0.0 81.6
 O
 pt
 im
 iz
 ed
 dcoasting (ft) 4393 4223 5641 3687 5009 3402
 Vmax (mph) 66.0 75.0 74.4 75.0 75.0 75.0
 Lvmax (ft) 4106 3946 6858 3799 7146 3581
 T (sec) 120.37 113.78 155.95 149.01 190.75 185.07
 Et (kwh) 52.31 48.28 75.38 69.48 98.28 89.62
 Eb (kwh) 28.92 20.72 33.23 22.32 35.96 23.26
 Ct ($) 70.92 66.20 92.61 87.16 113.71 108.30
 Ct  Saving ($) - -4.72 - -5.45 - -5.41
 Pe
 rfo
 rm
 an
 ce
  
m
 ea
 su
 re
 s
 Ct  Saving (%) - -6.65 - -5.88 - -4.76
 Table 5.4 Jointly optimized alignments and coasting distance with a 55mph speed 
constraint
 Station Spacing:
 8,500 ft
 Station Spacing:
 12,500 ft
 Station Spacing:
 16,500 ft
 Alignment Level Dipped Level Dipped Level Dipped
 S (ft)
 - 2906 - 3115 - 2954
 
 (ft) 0.0 26.9 0.0 29.5 0.0 27.4
 O
 pt
 im
 iz
 ed
 dcoasting (ft) 2174 2610 2570 1783 2880 2440
 Vmax (mph) 55.0 55.0 55.0 55.0 55.0 55.0
 Lvmax (ft) 2119 1474 2119 1437 2119 1466
 T (sec) 127.60 127.42 177.73 176.15 227.68 226.41
 Et (kwh) 44.43 36.28 54.21 48.72 64.21 58.31
 Eb (kwh) 24.33 16.17 23.42 17.49 22.70 16.43
 Ct ($) 72.90 70.77 99.34 97.13 125.74 123.59
 Ct  Saving ($) - -2.13 - -2.21 - -2.15
 Pe
 rfo
 rm
 an
 ce
  
m
 ea
 su
 re
 s
 Ct  Saving (%) - -2.92 - -2.22 - -1.71
71
  5.3 Relations between Optimized Solutions and Alignment Depth
 For some sites with poor geology (i.e. high ground water level), alignment 
designs are limited within certain depths. This section is focused on the relations 
between optimized total costs and specified alignment depths with various speed 
constraints. Simulation cases with three station spacings (8500 ft, 12500 ft, 16500 
ft) and three speed constraints (unconstrained, 75 mph, 55 mph) with depth 
changes are optimized. The changing variables in this study are total curve length 
S and coasting distances dcoasting. Figures 5.1-5.3 show the optimized total cost vs. 
alignment depth relations. The following points are found in the optimization 
results:
 1. Without speed constraints, the optimized total costs decrease as the alignment 
depth increases.
 2. Without speed constraints, the marginal savings in total cost for unit increase 
in alignment depth of shallow tunnels are much higher than the marginal 
savings of deep alignments. Figures 5.1-5.3 show no significant total cost 
saving can be found by increasing tunnel depth beyond 60 ft.
 3. For any constrained speed, an optimized alignment depth can be found. The 
optimized alignment depth decreases as the speed constraint decreases.
72
 Depth vs. Optimized Total Cost 
(8500 ft Station Spacing)
 65.00
 66.00
 67.00
 68.00
 69.00
 70.00
 71.00
 72.00
 73.00
 74.00
 0 20 40 60 80 100
 Alignment Depth (ft)
 T.
 C.
  
fo
 r 
a
  
s
 in
 gl
 e
 -
 di
 re
 c
 tio
 n
  
o
 pe
 ra
 tio
 n
  
($)
 No speed constraint 75 mph speed constraint
 55 mph speed constraint
 Figure 5.1 Depth-optimized total cost relations for 8500 ft station spacing
 Depth vs. Optimized Total Cost 
(12500 ft Station Spacing)
 84.00
 86.00
 88.00
 90.00
 92.00
 94.00
 96.00
 98.00
 100.00
 102.00
 104.00
 0 20 40 60 80 100 120 140
 Alignment Depth (ft)
 T.
 C.
  
fo
 r 
a 
si
 n
 gl
 e-
 di
 re
 ct
 io
 n
  
o
 pe
 ra
 tio
 n
  
($)
 No speed constraint 75 mph speed constraint
 55 mph speed constraint
 Figure 5.2 Depth-optimized total cost relations for 12500 ft station spacing 
73
 Depth vs. Optimized Total Cost 
(16500 ft Station Spacing)
 100.00
 105.00
 110.00
 115.00
 120.00
 125.00
 130.00
 135.00
 0 50 100 150 200
 Alignment Depth (ft)
 T.
 C.
  
fo
 r 
a 
si
 n
 gl
 e-
 di
 re
 ct
 io
 n
  
o
 pe
 ra
 tio
 n
  
($)
 No speed constraint 75 mph speed constraint
 55 mph speed constraint
 Figure 5.3 Depth-optimized total cost relations for 16500 ft station spacing  
The three-dimensional relations between alignment depths, speed 
constraints and total costs for one-directional operations between 12,500 station 
spacing are shown in Figure 5.4. Combinations of 20 speed constraints and 25 
alignment depths are optimized in this section. Figure 5.4 shows that optimized 
alignment depths decrease as speed constraints decrease. With a specified depth, 
Figure 5.4 shows that, the total costs decrease as speed constraints increase.  
74
 Figure 5.4 Alignment depth, speed constraint and total cost relations
 0
 1
 0
 2
 0
 3
 0
 4
 0
 5
 0
 6
 0
 7
 0
 8
 0
 9
 0
 1
 0
 0
 1
 1
 0
 1
 2
 0
 50
 52
 54
 56
 58
 60
 62
 64
 66
 68
 70
 72
 74
 76
 78
 80
 82
 84
 86
 88
 Alignment Depth (ft)
 S
 p
 e
 e
 d
  
C
 o
 n
 s
 tr
 a
 in
 t 
(m
 p
 h
 )
 Total cost for one-
 directional operations ($)
 Station spacing: 12,500 ft
 
/S ratio: 1%
 Coasting Dist.: Optimized
 Other Assumptions: Baseline
 87
 91
 95
 99
 101
75
 Figures 5.5 and 5.6 show the relations between alignment depths, coasting 
distances and total costs for 75 mph and 55mph speed constraints. The diagrams 
show that the sensitivity of total costs to coasting distances on shallow tunnels 
(depth < 60 ft) are very small. Moreover, the sensitivities of total cost to 
alignment depth and coasting distance around the optimized point are not 
significant in both dimensions. However, a very sensitive area is located at upper 
right corner is because trains operating with long coasting distance on and deep 
alignments will cause significant speed drops and have disadvantages in travel 
time between stations. The blank upper-left areas of Figure 5.6 are cases where 
trains cannot reach the next station after long coasting in steep climbing sections. 
76
 Figure 5.5 Alignment depth, coasting distance and total cost relations with a 75 
mph speed constraint
 0 10 20 30 40 50 60 70 80 90 10
 0
 11
 0
 12
 0
 0
 200
 400
 600
 800
 1000
 1200
 1400
 1600
 1800
 2000
 2200
 2400
 2600
 2800
 3000
 3200
 3400
 3600
 3800
 4000
 4200
 4400
 4600
 4800
 5000
 5200
 5400
 5600
 5800
 6000
 6200
 6400
 6600
 6800
 7000
 Alignment Depth (ft)
 Co
 a
 st
 in
 g 
D
 is
 ta
 n
 c
 e
  
(ft)
 Total cost for one-
 directional operations ($)
 Station spacing: 12,500 ft
 
/S ratio: 1%
 Speed constraint: 75 mph
 Other Assumptions: Baseline
 88
 90
 92
 90
77
 Figure 5.6 Alignment depth, coasting distance and total cost relations with a 55 
mph speed constraint
 0 10 20 30 40 50 60 70 80 90 10
 0
 11
 0
 12
 0
 0
 200
 400
 600
 800
 1000
 1200
 1400
 1600
 1800
 2000
 2200
 2400
 2600
 2800
 3000
 3200
 3400
 3600
 3800
 4000
 4200
 4400
 4600
 4800
 5000
 5200
 5400
 5600
 5800
 6000
 6200
 6400
 6600
 6800
 7000
 Alignment Depth (ft)
 Co
 a
 s
 tin
 g 
D
 is
 ta
 n
 c
 e
  
(ft
 )
 Total cost for one-
 directional operations ($)
 Station spacing: 12,500 ft
 
/S ratio: 1%
 Speed constraint: 55 mph
 Other Assumptions: Baseline
 98
 102
 100
 100
78
 5.4 Sensitivity of Optimized Alignments to User Time Value and Energy Cost
 Expected changes of important parameters such as urban development and 
increases in energy costs within the design life of tunnels and difficulties of 
modifying alignments, the sensitivity of optimized alignment designs to changes 
in such parameters should be explored in order to avoid unsuitable alignments at 
later times. Figures 5.7 - 5.9 are three-dimensional relations between alignment 
depths, speed constraints and total costs when the passenger time value, tractive 
energy cost and braking energy cost are increased significantly from their baseline 
values. Compared with Figure 5.4, Figure 5.7-5.9 shows that the sensitivity of 
optimized alignments to increased passenger time value, tractive energy cost and 
braking energy cost are not significant. Therefore, the optimized alignments will 
not change significantly due to such changes in costs.
79
 Figure 5.7 Alignment depth, speed constraint and total cost relations when the 
user time value is 15$/psg-hr
 0
 1
 0
 2
 0
 3
 0
 4
 0
 5
 0
 6
 0
 7
 0
 8
 0
 9
 0
 1
 0
 0
 1
 1
 0
 1
 2
 0
 50
 52
 54
 56
 58
 60
 62
 64
 66
 68
 70
 72
 74
 76
 78
 80
 82
 84
 86
 88
 Alignment depth (ft)
 S
 p
 e
 e
 d
  
c
 o
 n
 s
 tr
 a
 in
 t 
(m
 p
 h
 )
 Total cost for a one-
 directional operation ($)
 Station spacing: 12,500 ft
 
/S ratio: 1%
 Coasting Dist.: Optimized
 Value of user time: 15$/psg-h
 Other Assumptions: Baseline
 206212218
 218
 227
80
 Figure 5.8 Alignment depth, speed constraint and total cost relations when the 
tractive energy cost is 0.3$/kwh 
0
 1
 0
 2
 0
 3
 0
 4
 0
 5
 0
 6
 0
 7
 0
 8
 0
 9
 0
 1
 0
 0
 1
 1
 0
 1
 2
 0
 50
 52
 54
 56
 58
 60
 62
 64
 66
 68
 70
 72
 74
 76
 78
 80
 82
 84
 86
 88
 Alignment depth (ft)
 S
 p
 e
 e
 d
  
c
 o
 n
 s
 tr
 a
 in
 t 
(m
 p
 h
 )
 Total cost for one-
 directional operations ($)
 Station spacing: 12,500 ft
 
/S ratio: 1%
 Coasting Dist.: Optimized
 Tractive energy cost: 0.3$/kwh
 Other Assumptions: Baseline
 101
 98
 101
 104
81
 Figure 5.9 Alignment depth, speed constraint and total cost relations when the 
braking energy cost is 0.2$/kwh
 0
 1
 0
 2
 0
 3
 0
 4
 0
 5
 0
 6
 0
 7
 0
 8
 0
 9
 0
 1
 0
 0
 1
 1
 0
 1
 2
 0
 50
 52
 54
 56
 58
 60
 62
 64
 66
 68
 70
 72
 74
 76
 78
 80
 82
 84
 86
 88
 Alignment depth (ft)
 S
 p
 e
 e
 d
  
c
 o
 n
 s
 tr
 a
 in
 t 
(m
 p
 h
 )
 Total cost for one-
 directional operations ($)
 Station spacing: 12,500 ft
 
/S ratio: 1%
 Coasting Dist.: Optimized
 Braking energy cost: $0.2/kwh
 Other Assumptions: Baseline
 9194
 94
 91
82
 5.5 Case Study
 Benefits of using dipped track profiles between stations are estimated in 
this section with operating characteristics the same as WMATA METRORAIL 
system. The manual of design criteria (WMATA, 2001) and the annual financial 
report of WMATA (2002) show that the speed constrain of METRORAIL system 
is designed for operating safely under 75 mph. Moreover, METRORAIL trains 
are currently operated under 59 mph (95kph). Therefore, the use of dipped 
alignments with 59 mph speed constraint will be optimized. Moreover, because 
Figure 5.4 shows about $2 total cost saving for every one-directional operation 
between stations by using 20ft-depth alignment, a 20ft (3 stories) elevated station 
with an approaching ascending curve and a departing descending curve can be 
considered possible in economy and will be investigated in this case study (no 
consideration of construction costs in this study). Table 5.5 shows optimized
 results for operating with a 59 mph speed constraint inside tunnels (high air 
resistance). With optimized alignment (38ft-depth), A $2.80 saving in total cost 
for one one-directional operation between stations can be expected. Table 5.6 
shows optimized solutions for operating with 59 mph speed constraint and a 20ft 
elevated station (S=2000ft, 
=20ft) above ground (low air resistance). A $2.22 
total cost saving can be found by using elevated station sections. According to 
WMATA METRORAIL timetable (2003) for red line operations, there are 46,000 
train operations per direction per year between METRORAIL red line stations. 
Therefore, the estimated annual total cost savings for a 38ft-depth double track 
83
 tunnel between stations is $257,600 and annual total cost savings for a 20ft 
elevated double track station is $204,240. 
Table 5.5 Underground alignments optimizations with a 59mph speed constraint
 Underground Station Spacing:8,500 ft
 Station Spacing:
 12,500 ft
 Station Spacing:
 16,500 ft
 Alignment Level Dipped Level Dipped Level Dipped
 S (ft) - 3864 - 3763 - 3982
 
 (ft) 0.0 38.6 0.0 37.6 0.0 39.8
 O
 pt
 im
 iz
 ed
  
dcoasting (ft) 2939 2786 2927 2353 2939 2701
 Vmax (mph) 59 59 59 59 59 59
 Lvmax (ft) 2709 1721 2709 1733 2709 1711
 T (sec) 123.72 122.89 169.94 168.32 216.17 215.36
 Et (kwh) 47.88 38.52 60.19 52.37 72.44 63.08
 Eb (kwh) 26.19 16.51 26.23 17.84 26.19 16.50
 Ct ($) 71.66 68.87 96.62 93.80 121.57 118.79
 Ct  Saving ($) - -2.79 - -2.82 - -2.78
 Pe
 rfo
 rm
 an
 ce
  
m
 ea
 su
 re
 s
 Ct  Saving (%) - -3.89 - -2.92 - -2.29
 Table 5.6 Aboveground alignments optimizations with a 59mph speed constraint
 Aboveground Station Spacing:8,500 ft
 Station Spacing:
 12,500 ft
 Station Spacing:
 16,500 ft
 Alignment Level Dipped Level Dipped Level Dipped
 S (ft) - 2000 - 2000 - 2000
 
 (ft) 0 20 0 20 0 20
 O
 pt
 im
 iz
 ed
  
dcoasting (ft) 2634 2532 2639 2318 2634 2532
 Vmax (mph) 59 59 59 59 59 59
 Lvmax (ft) 2427 2012 2427 2012 2427 2012
 T (sec) 122.14 120.32 168.38 166.44 214.59 212.77
 Et (kwh) 40.20 35.01 45.31 40.40 50.43 45.24
 Eb (kwh) 30.63 25.27 30.62 25.53 30.63 25.27
 Ct ($) 70.16 67.94 94.05 91.83 117.92 115.70
 Ct  Saving ($) - -2.22 - -2.22 - -2.22
 Pe
 rfo
 rm
 an
 ce
  
m
 ea
 su
 re
 s
 Ct  Saving (%) - -3.17 - -2.36 - -1.89
84
 Chapter 6. Conclusions and Recommendations
 6.1 Summary of Research Results
 The primary results of this research include:
 1. Development of a deterministic simulation model to investigate the rail transit 
vehicle movement and energy consumption under several control options and 
track alignment designs.
 2. Analysis of the effects of using dipped vertical track alignments with baseline 
control options.
 3. Sensitivity analysis of simulation parameters under various vertical track 
alignment designs.
 4. Jointly optimizing of alignment designs and operating parameters.
 6.2 Conclusions
 1. Using dipped vertical alignments can improve the travel time, tractive energy 
and braking energy. Under the baseline parameters, the maximum observed 
saving for cases with 12,500 feet station spacing are 6.53% in travel time, 
5.23% in tractive energy consumption and 23.62% in braking energy 
consumption.
 2. For simulation cases with the same total curve length S and no speed 
constraints, the savings in travel time, tractive energy and braking energy are 
approximately proportional to the maximum depth 
.
85
 3. Among simulation cases with fixed maximum depth 
, those with shorter total 
curve length S have greater savings in travel time, tractive energy and braking 
energy.
 4. If the vehicle power-weight ratios decrease, the savings for using dipped 
alignments in rail transit system increase.
 5. While operating outside tunnels (with low air friction), the savings for using 
dipped alignment in travel time and tractive energy consumption are greater.
 6. Operating with constrained speeds limit the benefits from dipped alignments. 
Trains reaching speed limit within descending section cannot transform 
potential energy into kinetic energy by accelerating to higher speeds.
 7. Without a speed constraint, using optimized dipped alignments can save 6.7% 
in total cost for a one-directional operation.
 8. The optimized total cost savings in dollars are approximately equal for cases 
with the same low speed constraints but different station spacings.
 9. With a 75 mph speed constraint, an optimized 75 feet depth alignment can 
save $5.4 in total cost for a one-directional operation.
 10. With a 55 mph speed constraint, an optimized 27 feet depth alignment can 
save $2.1 in total cost for a one-directional operation.
 11. The optimized total cost savings decrease as the speed constraints decrease.
 12. The 
/S ratios for optimized alignments are approximately equal to 1%.
 13. For optimized alignments with low speed constraint, trains reach the speed 
constraint at approximately the same point (Lvmax) where the descending 
curves are finished (Lpl/2 + S/2).
86
 14. Without speed constraints, the marginal savings in total cost for unit increase 
in alignment depth of shallow tunnels are much higher than the marginal 
savings of deep alignments.
 15. The sensitivities of total cost to alignment depths and coasting distances 
around the optimized point are not significant in both dimensions.
 16. The sensitivities of total costs to coasting distances on shallow tunnels are not 
significant.
 17. The sensitivity of optimized alignment designs to changes of user?s value, 
energy price and braking energy cost is not significant.
 18. With METRORAIL operating characteristics, the estimated annual total cost 
savings for a 38ft-depth double track tunnel between stations is $257,600 and 
annual total cost savings for a 20ft elevated double track station is $204,240.
 6.3 Recommendations for Further Research
 The following extensions to this work are suggested for further studies:
 1. Studies of non-symmetric alignment designs
 Study of stations in different level is preferred. Moreover, the vertical 
curves used in dipped alignment may separate into descending and climbing 
curves with different curvature designs to fit the best alignments in helping 
accelerating and in decelerating. 
87
 2. Optimizing total costs with depth-related information
 In this study, the construction cost increments are assumed as zero while 
using deep tunnels. However, in real world, the uses of dipped alignments will 
cause some additional costs in constructions or in administrations. A total cost 
analysis and optimization that includes depth-related costs is preferred in the 
future study.
 3. Study of above ground operations
 This study shows that the marginal benefits of alignment depth changes in 
shallow alignments are good. Therefore, instead of using ?dipped tunnel between 
stations?, using of ?elevated stations in both ends of above ground tracks? may be 
studied more in the future.
 4. Jointly optimizing of station spacings
 Integrated study with station spacing may be introduced in the future 
studies. The jointly optimizing in multi-stations problems require more analyses 
in passenger accessing time, waiting time and station dwelling time.  
5. Jointly optimizing with other design and control decisions  
Joint analysis in vertical alignment design and other decisions like vehicle 
design are recommended in future studies. 
88
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