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Train trajectory optimisation of ATO systems for metro lines
Article · November 2014
DOI: 10.1109/ITSC.2014.6957953
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Train Trajectory Optimisation of ATO Systems for Metro Lines *
Ning Zhao, Clive Roberts, Member, IEEE, Stuart Hillmansen, Paul Western, Lei Chen, Zhongbei Tian,
Tingyu Xin, Shuai Su
optimality [7]. Rachel Liu describes a Pontryagin maximum
principle application to find the optimal train movement
sequence to minimise energy usage [8]. However, most of the
research works do not consider the control process of ATO
systems, which automatically controls the train movements by
tracking the target speed trajectory [9-13]. If such a train
trajectory was implemented into an ATO system, the train
would have to switch the movement mode from motoring to
braking or coasting frequently, thus increasing energy usage.

Abstract— This paper describes an Enhanced Brute Force
Algorithm application to optimise train trajectory (driving
speed curve) for Automatic Train Operation (ATO) systems. A
multi-train simulator was developed specifically for the study. It
can be used to simulate the movement of railway vehicles and
calculate the detailed power system energy consumption with
different train trajectories when implemented on an AC or DC
powered railway line operating with multiple trains.
Results are presented using a practical train trajectory and
an optimal train trajectory with a full day timetable and
passenger flow on the Beijing Yizhuang Metro Line. Analysis of
the results shows that by using an optimal train trajectory, the
energy consumption around the power network can be
significantly reduced within a constrained journey time.
Furthermore, the results also show that the developed simulator
is able to facilitate the understanding of the railway traction and
power system, and that it provides guidance for adjusting the
service timetable and driving strategy to minimise energy usage.
This paper describes an Enhanced Brute Force Algorithm
application for calculating the optimal train trajectory for ATO
systems. A multi-train traction and power system energy
simulator was developed for this purpose, which is used to
simulate the train movements and calculate the detailed power
system energy consumption with a full day timetable and
passenger flow on the Beijing Yizhuang Metro Line. Both the
optimal train trajectory and a practical train trajectory are
implemented into the simulator for energy consumption
comparisons.
I. INTRODUCTION
The train trajectory, which is conventionally controlled by
the driver but, in modern train control systems, can be
calculated and implemented automatically, has a key effect on
energy consumption. As energy-efficient driving is becoming
a critical issue for railway operators due to rising energy prices
and environmental concerns, train trajectory optimisation is
increasingly important for modern railway design [1-4].
Researchers have set out a number of different approaches to
optimise train operation in recent literature. Hee-Soo Hwang
proposes a fuzzy control model to calculate an optimal
running pattern through a trade-off between journey time and
energy usage [5]. Bwo-Ren Ke proposes a MAX-MIN ant
system to optimise the train trajectory for mass rapid transit
systems with cable signalling systems [6]. Compared with
metaheuristics, exact search provides a more straightforward
approach and is guaranteed to find optimal solutions and prove
II. MULTI-TRAIN SIMULATOR DEVELOPMENT
The developed Multi-Train Simulator is designed to
achieve the following requirements:
 Simulate the detailed movement of railway vehicles
around an AC or DC powered railway network;
 Calculate voltages and currents in the power network
when trains are operating;
 Calculate the delivered power from the substations and
the power consumption at the railway vehicles;
 Analyse the overall energy consumed when specific
timetables are operated;
 Allow the modification of the behaviour of trains
within the simulation;
*Resrach supported by Beijing Laboratory of Urban Rail Transit and
Beijing Key Laboratory of Urban Rail Transit Automation and Control.
N. Zhao is with the Birmingham Centre for Railway Research and
Education, School of Electronic, Electrical and Computer Engineering,
University of Birmingham, Birmingham, UK (corresponding author to
provide phone: +44 (0)121 414 7522; fax: +44 (0)121 414 4291; e-mail:
n.zhao@bham.ac.uk).
C. Roberts, S. Hillmansen, P. Western, L. Chen, Z. Tian, T. Xin are with
the Birmingham Centre for Railway Research and Education, School of
Electronic, Electrical and Computer Engineering, University of Birmingham,
Birmingham, United Kingdom, B15 2TT, (e-mail: c.roberts.20@bham.ac.uk;
s.hillmansen@bham.ac.uk; p.weston@bham.ac.uk; l.chen.3@bham.ac.uk;
zxt279@bham.ac.uk; txx303@ bham.ac.uk).
S. Su is with the 2State Key Laboratory of Rail Traffic Control and Safety,
Beijing Jiao tong University, P. R. China (e-mail: 10111043@bjtu.edu.cn).
 Identify and quantify energy losses.
In order to develop the simulator, it is necessary to
consider the fundamental physics of train motion.
Lomonossoff’s Equations, which have previously been
described in earlier works, are used to solve the dynamic
movement equations [14-16]:
(1)
where Mtr is the effective mass; Ftr is the traction force effort;
Fgrad is the force due to the gradient; R is the vehicle resistance,
1
Preprint submitted to 17th International IEEE Conference on Intelligent
Transportation Systems. Received June 24, 2014.
CONFIDENTIAL. Limited circulation. For review only.
which can be calculated by the following equation with
constants a, b, c, known as the Davis equation [17]. The
constants are empirical and relative to the track and
aero-dynamic resistance.
( )
| |
typical movement modes for train motion can be obtained, as
shown in Figure 1. The traction system characteristics of a
Yizhuang Line vehicle are shown in Figure 2.
Speed
(2)
The forward tractive effort control signal will equal zero
when the power is shut down, i.e. in coasting mode or braking
mode. In that case, no energy usage will be included.
Speed limit
Motoring
mode
Cruising
mode
Coasting
mode
Braking
mode
̇
{
̇
( )
( )
( )
( )
(3)
Figure 1. Four modes of train movement.
A number of parameters should be configured into the
simulator. Figure 4 shows the system input and output
diagram. Fixed parameters that are not changed during the
modelling process are considered as constant values, such as
route data, power system characteristics and train traction
characteristics. Dynamic parameters, such as acceleration
rate, train weight and passenger flow, change from time to
time corresponding to train speed, schedule, gradient profile
and movement mode. An iterative technique is implemented
into the simulator to calculate rail traction and power system
solutions [18, 19].
where s is the train position; uf and ub are the control signal for
forwards effort and backwards braking effort; f(v) and b(v) are
the maximum tractive effort and braking effort at the current
speed; r(v) is the resistive force at the current vehicle speed;
g(s) is the force due to the gradient at the current position.
The boundary conditions are given as follows. The initial
conditions are:
( )
{
(4)
( )
III. TRAIN TRAJECTORY OPTIMISATION
and the final conditions are:
As shown in Figure 4, different train trajectories can be
applied into the simulator. In this study, an optimal train
trajectory application is considered. The result will be
compared with that of a practical train trajectory. Figure 5
shows the simulation flowchart.
( )
{
(5)
( )
Some constraints are imposed:
( )
{
[
]
[
]
As show in Figure 6, the journey is divided into a number
of sections (dot dash lines) in terms of the line speed limit
changes, altitude changes and distance. The train movement
mode in each section needs to be optimised in order to
minimise the overall energy consumption. Cruising mode
will not be considered in the optimisation as it is difficult to
track precisely using ATO systems.
(6)
where v_lim(s) is the train target speed or line speed limit
(whichever is smaller) at the current position s.
Based on Equation 1, Equation 2 and Equation 3, four
Figure 2. Traction system characteristic of a Yizhuang Line vehicle.
2
Preprint submitted to 17th International IEEE Conference on Intelligent
Transportation Systems. Received June 24, 2014.
CONFIDENTIAL. Limited circulation. For review only.
Input data
where n is the number of trains to be optimised; Trun is the
journey time; Tmax and Tmin are the maximum and minimum
permitted scheduled journey times; Erun is the train energy
usage, which can be calculated using the following equation:
( ) [ ( )] ( )
(9)
∫
Calculate the
movement of first train
Arrive at a
station
No
Yes
Final station
No
Yes
Read line speed,
gradients. Calculate
braking distance
New movement
authority
Dwell time
where f [v(t)] is the maximum tractive effort at the current
vehicle speed v(t).
No
In this optimisation study, each potential movement
sequence is assumed as a candidate solution and has an
individual solution domain. The complexity of the domain
depends on the assumed search boundary and search interval.
A conventional brute force search enumerates all possibilities
in the solution domain to find the optimum [21]. This rapidly
becomes impractical due to the processing time required for
complex problems [22]. The developed enhanced brute force
algorithm is able to address this problem by constraining the
solution domain [23, 24].
Key :
Yes
ti
Update movement
authority
Calculate the distance
to the limitation of
movement authority
- time
si
- train position
vi
- train speed
pi
Movement
authority
simulation
st
- train engine power
i
- station position
a ac - acceleration rate
a co - coasting rate
Calculate the distance
to the next station and
the next speed limit
change position
a cu - cruising rate
a br
- braking rate
(
Choose mode
Acceleration
Coasting
Cruising
(
Braking
)
Fixed parameters:
Record results to
arrays
Train control method
Route data:
 Line speed limits
 Network gradient
 Station location
 Network curvature
Output figures,
store results to
database
Figure 3. Simulation flowchart [20].
Train traction data:
 Traction power
 Regenerative power
 Resistance
(7)
(
)
∑
 Motor efficiency
(8)
(10)
(
) (11)
Simulation input:
Timetable (TA)
Train trajectory (TR)
 Target speed
 Coasting point
 Movement sequence
Simulation output:
Power system data:
 Rectifier characteristics
 Feeder cable resistances
 Traction return path resistance
 Conductance to ground
 Crossbonds resistance
 Network voltage range
The aim of the train trajectory optimisation in this study is
to find the most appropriate train movement sequence (TR1 to
TRi) to minimise energy usage within a constant journey time.
The cost function to be minimised is thus:
∑
∑
)
Multi-train
Simulator
Substation energy usage
Auxiliary system energy
usage
Train energy usage
Train operation time
Train schedule diagram
Dynamic parameters:
Acceleration rate
Train weight
Passenger flow
Figure 4. Flowchart of the developed simulator.
Figure 5. A full train schedule diagram.
3
Preprint submitted to 17th International IEEE Conference on Intelligent
Transportation Systems. Received June 24, 2014.
CONFIDENTIAL. Limited circulation. For review only.
TR1 TR2
TR3
TR4
TR5
TR6
TR7
schedule diagram obtained from the simulator are shown in
TABLE I and Figure 5 respectively.
TR8 TR9 TR10 TR11
The detailed energy consumption calculated from the
simulator is shown in TABLE II. It can be seen that optimal
operation can save energy usage by up to 16% with the same
timetable. The energy transmission efficiency from the
substations to the railway vehicles is approximately 96%. The
auxiliary systems, such as lighting, heating and air
conditioning, cost 1% of the substation energy usage.
TABLE I. SCHEDULED TIMETABLE OF BEIJING YIZHUANG
METRO LINE.
Figure 6. An example of a Beijing Yizhuang Metro Line train trajectory
optimisation.
The method initially calculates an estimated movement
sequence (TRest) based on the practical train trajectory and
constant journey time (Tcon), as shown in Equation 10. The
method then only considers the candidate solutions which are
close in value to the estimated solution, thereby reducing the
solution domain, as shown in Equation 11. The computing
time is therefore significantly decreased. The optimal and
practical train trajectories are shown in Figure 7. It can be
observed that the actual train switches its movement mode
from motoring to braking or coasting frequently as the ATO
system aims to track the constant cruising speeds. The optimal
train trajectory is able to avoid such situations, thereby
reducing energy usage.
Station
Station
location,
m
Dwell
time, s
Yizhuang railway
station
0
Ciqu
Operation time, s
Up
direction
Down
direction
40
0
0
1334
45
104
103
Ciqu South
2620
35
248
248
Jinghailu
4706
30
423
421
Tongjinanlu
6971
30
601
599
Rongchangdongjie
9309
30
790
791
Rongjingdongjie
10663
30
923
923
Wanyuanjie
11943
30
1052
1052
Yizhuang Park
13481
30
1194
1194
Yizhuangqiao
14474
35
1309
1309
Jiugong
16456
30
1477
1478
Xiaohongmen
18822
30
1660
1664
Xiaocun
20097
30
1793
1800
Songjiazhuang
22728
30
2016
2019
TABLE II. DETAILED ENERGY CONSUMPTION.
Parameters
Metro operation time,
hours
Train traction energy
consumption, kWh
Auxiliary system energy
consumption, kWh
Substation energy
consumption, kWh
Figure 7. Beijing Yizhuang Metro Line practical and optimal train trajectory
(up direction)
IV. SIMULATION AND OPTIMISATION RESULTS
Transmission loss, kWh
The simulation is designed using the optimal and the
practical train trajectories with a full day timetable and
passenger flow on the Beijing Yizhuang Metro Line, which is
an urban railway line connecting Beijing city centre and
Yizhuang district. The scheduled timetable and the full train
Energy transmission
efficiency
Total passenger flow,
thousands of people
Operation with
practical train
trajectory
Operation with
optimal train
trajectory
17.7
17.7
83771
70884 (-15%)
823
823
88188
73955 (-16%)
3594
3071 (-15%)
96.0%
95.8%
1128
1128
4
Preprint submitted to 17th International IEEE Conference on Intelligent
Transportation Systems. Received June 24, 2014.
CONFIDENTIAL. Limited circulation. For review only.
As shown in Figure 8, the network energy usage continues
to increase from when the first train departs the first station
until the line is fully loaded at 7:00. Between 9:00 to 16:00
(off-peak time), the energy usage reduces by half, as the train
service interval increases significantly. The last train departs
the first station at 22:05. After that, the number of the trains in
the network reduces, and thus the energy consumption also
reduces. Figure 9 shows the network energy usage per
passenger. It can be observed that the number remains
relatively stable between 7:00 and 21:00, which indicates that
the train service interval plan fits the passenger flow demand.
and power system, and furthermore it provides
guidance for adjusting the service timetable and
driving strategy to minimise energy usage;

It was found that increasing the service interval before
7:00 and after 21:00 on the Beijing Yizhuang Metro
Line is able to improve energy efficiency, thereby
reducing the energy consumption significantly;

The choice of train trajectory employed on a line has a
direct impact on improving train performance,
thereby reducing the energy usage;

The enhanced brute force algorithm was shown to be
able to provide a good performance to achieve
optimal results.
ACKNOWLEDGMENT
This research is jointly supported by Beijing Laboratory of
Urban Rail Transit and Beijing Key Laboratory of Urban Rail
Transit Automation and Control.
REFERENCES
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[3]
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:0
0
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00
-2
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[9]
V. CONCLUSION
[10]
This paper describes a train trajectory optimisation
application to minimise energy consumption within a constant
journey time. A multi-train simulator was developed to
simulate the train movement and calculate the power system
energy consumption around the Beijing Yizhuang Metro Line
with a full day timetable and passenger flow. This study has
the following key findings:

[11]
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It was identified that the developed simulator is able
to facilitate the understanding of the railway traction
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