Modelling the European Train Control System: Braking and Supervision Curves

EM504 DissertationA
Modelling the EuropeanTrain Control System
Braking and Supervision Curves
Thomas Gibson
201113465
2015/2016

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Abstract
In their present state, railways across the world use signaling systems based on principles derived from the
Victorian era. With these come a raft of difficulties due to the limitations of such systems, which ultimately result in
delays on the railways. The European Rail Traffic Management System (ERTMS) is an effort to overcome these
issues with a single standardized in cab signaling system implemented across Europe. Being developed by a
consortium of companies, this would be one of the largest developments in railway systems since the 19th century.
A key aspect of the system is the ability of an individual train to monitor its own speed and displacement relative to
a number of target locations. With this the train must generate a set of braking curves to ensure its safe operation.
Throughout this work, the ERTMS is explored in more detail with the aim of gaining a better understanding of what
is involved. More specifically, the braking curves will be fully investigated and modelled in the MATLAB and
Simulink environment, leading to the development of a robust model for analysing a train’s interactions along a
length of railway.

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i) Contents
1. Introduction.........................................................................................................................................................8
2. Objectives.........................................................................................................................................................10
3. Literature Review..............................................................................................................................................10
3.1. The Fundamentals of ERTMS .................................................................................................................10
3.1.1. ERTMS Levels.................................................................................................................................10
3.1.2. Trainborne Equipment and Train Characteristics ............................................................................12
3.2. ERTMS Modelling....................................................................................................................................15
3.3. Conclusions of Literature Review ............................................................................................................17
4. Technical Assessment......................................................................................................................................18
4.1. Emergency Brake Deceleration...............................................................................................................18
4.2. Service Brake Deceleration .....................................................................................................................20
4.3. Supervision Limits....................................................................................................................................20
4.3.1. Emergency Brake Intervention ........................................................................................................20
4.3.2. Service Brake Intervention...............................................................................................................21
4.3.3. Warning Supervision Limit ...............................................................................................................22
4.3.4. Permitted Speed ..............................................................................................................................22
4.3.5. Indication Supervision Limit .............................................................................................................22
4.3.6. Ceiling Supervision Limits................................................................................................................23
4.4. Review of Technical Understandings ......................................................................................................25
5. Modelling Procedure and Implementation........................................................................................................26
5.1. Input Data ................................................................................................................................................26
5.2. Mathematical Model.................................................................................................................................27
5.2.1. Assumptions ....................................................................................................................................28
5.2.2. Initialisation and Parameter Definitions ...........................................................................................28
5.2.3. Ceiling Supervision Limits................................................................................................................29
5.2.4. Accelerations ...................................................................................................................................29
5.2.5. Braking Curves ................................................................................................................................29
5.2.6. Mathematical Model Output.............................................................................................................30
5.3. Model Validation and Verification ............................................................................................................30
5.4. Extended Mathematical Model ................................................................................................................32
5.4.1. Target Identification .........................................................................................................................34
5.4.2. Braking Curve Calculation ...............................................................................................................35
5.4.3. Extended Model Output ...................................................................................................................36
5.5. Simulink Models.......................................................................................................................................37
5.5.1. Single Train Model...........................................................................................................................37
5.5.2. Single Train Model Output ...............................................................................................................38
5.5.3. Two Train Model (Moving Block) .....................................................................................................40
5.5.4. Two Train Model Output ..................................................................................................................41

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6. Discussion of Results .......................................................................................................................................44
6.1. Mathematical Model.................................................................................................................................44
6.2. Extended Mathematical Model ................................................................................................................45
6.2.1. Target Profiles..................................................................................................................................45
6.2.2. Temporary Speed Restrictions ........................................................................................................46
6.3. Single Train Simulink Model ....................................................................................................................46
6.3.1. Permitted Speed ..............................................................................................................................46
6.3.2. Brake Intervention............................................................................................................................48
6.4. Two Train Simulink Model .......................................................................................................................51
7. Conclusions and Future Work ..........................................................................................................................54
8. References .......................................................................................................................................................56
9. Appendix...........................................................................................................................................................58
9.1. Train Datasets..........................................................................................................................................58
9.1.1. TrainData1.xlsx................................................................................................................................58
9.2. Track Datasets.........................................................................................................................................59
9.2.1. TrackData1.xlsx ...............................................................................................................................59
9.2.2. TrackData2.xlsx ...............................................................................................................................60
9.2.3. TrackData3.xlsx ...............................................................................................................................61
9.2.4. TrackData4.xlsx ...............................................................................................................................62
9.2.5. TrackData5.xlsx ...............................................................................................................................63
9.2.6. TrackData6.xlsx ...............................................................................................................................64
9.2.7. TrackData7.xlsx ...............................................................................................................................65
9.3. Input Dataset MATLAB Scripts................................................................................................................66
9.3.1. Fixed Value Data [23] ......................................................................................................................66
9.3.2. National Value Data [23]..................................................................................................................67
9.3.3. Read in Track Data..........................................................................................................................68
9.4. MATLAB Code.........................................................................................................................................69
9.4.1. Mathematical Model – BrakingCurves.m.........................................................................................69
9.4.2. Extended Mathematical Model – Integrated_Model_1.m ................................................................74
9.4.3. Ceiling Supervision Limits – supervision_limits.m ...........................................................................79
9.4.4. Braking Curves – BrakingCurvesModel_V2.m ................................................................................80
9.4.5. Interpolation – Interpolate_1.m........................................................................................................82
9.4.6. Generate Target Braking Profiles – gen_target_profiles.m .............................................................83
9.4.7. Level 3 Model – Level_3_Model_1.m..............................................................................................88

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ii) Table of Figures
Figure 1: Railway Block Sections ..............................................................................................................................8
Figure 2: Standard British Signal Types [1] ...............................................................................................................9
Figure 3: ERTMS Level 1 [9] ...................................................................................................................................11
Figure 6: ERTMS System Architecture [12].............................................................................................................12
Figure 7: Driver Machine Interface Display [12] ......................................................................................................12
Figure 8: Traction Model [12]...................................................................................................................................13
Figure 9: Braking Model – Progressive Application of Brakes [12] .........................................................................13
Figure 10: Braking Model – Deceleration vs Speed [12].........................................................................................13
Figure 11: Dynamic Speed Monitoring by ETCS [12]..............................................................................................14
Figure 12: Braking Curves under Plain Line Operation [13] (System D refers to ERTMS level 2) .........................14
Figure 13: Braking Curve Inputs and Outputs [14] ..................................................................................................16
Figure 14: Generating the Test Configuration [14] ..................................................................................................16
Figure 15: RBC Execution Process [14] ..................................................................................................................17
Figure 16: ERTMS braking curves, where: I – Indication Point, P – Permitted Speed, W – Warning Point, SBI –
Service Brake Intervention, EBI – Emergency Brake Intervention, EBD – Emergency Brake Deceleration, SBD –
Service Brake Deceleration [23]. .............................................................................................................................18
Figure 17: Influence of Track Conditions on Abrake_emergency .....................................................................................19
Figure 18: Supervision limits for an EBD curve [23]................................................................................................21
Figure 19: Ceiling supervision limits [23] .................................................................................................................23
Figure 20: Definition of dVebi ....................................................................................................................................23
Figure 21: Mathematical model, BrakingCurves.m, flow diagram...........................................................................27
Figure 22: Mathematical model, BrakingCurves.m, output plot...............................................................................30
Figure 23: ERA Braking Curves Tool output plot, using the same data as for the MATLAB model. ......................31
Figure 24: : Extended mathematical model, Integrated_Model_1.m, flow diagram. ...............................................33
Figure 25: Theoretical target location for a LOA curve [23].....................................................................................34
Figure 26: Calculation of braking point. ...................................................................................................................35
Figure 27: Calculation of V_EBD1...........................................................................................................................35
Figure 28: a) The individual EBD curves and b) The combined EBD curve. ..........................................................36
Figure 29: Extended mathematical model, Integrated_Model_1.m, output plot for TrackData3.xlsx......................37
Figure 30: Overview of Simulink Single Train Model Integrated_Model_1_Sim.slx................................................38
Figure 31: Velocity profile output for the single train Simulink model......................................................................39
Figure 32: Train displacement output for the single train Simulink model...............................................................39
Figure 33: Two Train Simulink Model ......................................................................................................................40
Figure 34: Braking Curves Plot for TrackData7.xlsx ...............................................................................................41
Figure 35: Velocity and Displacement Profiles for Train 1. .....................................................................................42
Figure 36: Velocity and Displacement Profiles for Train 2. .....................................................................................42
Figure 37: Comparison of Velocity and Displacement Between Trains. .................................................................43
Figure 38: Comparison of EBD, EBI and P for a) dry rails and b) wet rails.............................................................44
Figure 39: Braking and supervision profiles for TrackData4.xlsx. ...........................................................................45
Figure 40: Braking and supervision profiles for TrackData5.xlsx with TSR of 60 km/hr set between 3 km and 4 km.
.................................................................................................................................................................................46
Figure 41: Speed profile for TrackData4.xlsx. .........................................................................................................47
Figure 42: Speed profile for TrackData5.xlsx with TSR of 60 km/hr set between 3 km and 4 km. .........................47
Figure 43: Displacement plot for TrackData5.xlsx with TSR of 60 km/hr set between 3 km and 4 km...................48
Figure 44: Demonstration of the Service Brake Intervention...................................................................................48
Figure 45: Supervision flags relative to Figure 44. ..................................................................................................49
Figure 46: Demonstration of the Emergency Brake Intervention. ...........................................................................49
Figure 47: Demonstration of EBI when approaching a target. ................................................................................50
Figure 48: a) Comparison of velocity profiles for the two trains, b) Comparison of the displacements of the two
trains, c) the distance maintained between the two trains towards the end of the simulation. ...............................51
Figure 49: Velocity and displacement profiles of the two trains when Train 1's EBI is triggered at 200 s. .............52

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Figure 50: Velocity and displacement profiles of the two trains when Train 1's EBI is triggered at 400 s. .............53
Figure 51: Correction factors and A_brake_emergency for TrainData1 .................................................................58
Figure 52: Speed Profile for TrackData1 .................................................................................................................59
Figure 53: Gradient Profile for TrackData1..............................................................................................................59
Figure 60: Speed Profile for TrackData5.................................................................................................................63
iii) Table of Tables
Table 1: Abbreviations...............................................................................................................................................7
Table 2: Prefixes Used for ERTMS Variables [12] ..................................................................................................15
Table 3: Minimum and maximum ceiling supervision limit parameters as defined in Appendix A.3.1 of the System
Requirements Specification [23]..............................................................................................................................24
Table 4: Comparison of ERA Braking Curves Tool and MATLAB model results. ...................................................31
Table 5: Other parameters defined in TrainData1...................................................................................................58

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iv) Abbreviations
It is noted that the European Rail Agency provide a useful document which includes a glossary of terms and
abbreviations, this is available at [1]. This also provides the reference for a number of these terms.
Table 1: Abbreviations
Abbreviation Term
DMI Driver Machine Interface
DV
Difference Value between the Permitted Speed to e.g.
DV_EBImin Emergency Brake Intervention speed (minimum)
DV_EBImax Emergency Brake Intervention speed (maximum)
EBCL Emergency Brake Confidence Level
EBD Emergency Brake Deceleration curve
EBI Emergency Brake Intervention supervision limit
EOA End of Movement Authority
ERA European Railway Agency
ERTMS European Rail Traffic Management System
ETCS European Train Control System
EVC European Vital Computer
FLOI First Line of Intervention
GSM-R Global System for Mobile Communications for Railways
I Indication Supervision Limit
LOA Limit of Movement Authority
MA Movement Authority
MRSP Most Restrictive Speed Profile
P Permitted Speed Supervision Limit
RBC Radio Block Centre
SBD Service Brake Deceleration curve
SBI Service Brake Intervention supervision limit
SR Staff Responsible
SRS System Requirement Specification
SvL Supervised Location
TSR Temporary Speed Restriction
UNISIG UNIFE ETCS Working Group
W Warning Supervision Limit

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1. Introduction
Towards the end of the 19th century, as the Industrial Revolution came to an end, a rapid growth was observed in
the development of railways in the United Kingdom. These early railways had no systems for maintaining the
distances, or headways, between trains, with the driver relying on line of sight to assess the state of the line ahead.
As a consequence of this, along with driver inexperience, sub-standard brakes and limited cohesion between the
rails and the train’s wheels, there were a frequent number of collisions between trains. As the size and power of
steam locomotives increased, these collisions became more disastrous.
It was soon realised that some method of railway signalling was required as a method of preventing such collisions
from occurring. Various methods were trialled, with the earlier options relying on timings between trains at a fixed
location. For example, if a train left a station a second train would not be allowed to leave the same station on the
same piece of railway until ten minutes later. This system had a critical flaw in that if the first train were to break
down there was no way for the driver of the second train to know. Therefore, the number of rear end collisions
between trains remained high, particularly since railway locomotives were very unreliable at the time.
In an effort to combat this a new approach was implemented, maintaining headways in terms of distance rather
than time, as in Figure 1 below. This, known as block based signalling, formed the basis for the railway signalling
systems used worldwide today. By this system the railway is divided into block sections of a particular distance,
depending on the maximum permissible speed of traffic on that section of line and the required line capacity. To
prevent collisions, a second train is only allowed into a block section when it is proven that the block is empty. That
is, it has been confirmed that the first train has left the block, fully intact. A coloured light, or aspect, indicates the
state of the proceeding block to the train driver. A green aspect indicates that the line is clear and the train may
proceed at full speed. A yellow aspect, known as a signal at danger, indicates that the line ahead is clear, but the
driver must be prepared to stop at the next signal so must reduce their speed. A red aspect indicates that the block
ahead is occupied by another train. When a train passes a green or yellow aspect the signal turns to red. The red
signal will turn yellow when the train enters the next block. Early signals were controlled manually by signal box
operators. Now, however, the majority of these operate automatically. This approach to signalling became law on
all passenger railways following the Regulation of Railways Act in 1889.
Figure 1: Railway Block Sections
The original signalling systems used mechanical semaphore signals, as in the left of Figure 2. These were designed
to be fail-safe, that is in the result of mechanical failure the signal arm will fall to be horizontal indicating a stop. The
signals were operated through a system of pulleys and wires, connecting them to local signal boxes. Although
these signals are still used today in a number of locations, they proved to be very unreliable with frequent
mechanical failures, particularly in winter when cables would freeze, and in summer when cables would heat up
and elongate. The majority of signals have been replaced by coloured light aspects which have reduced the
frequency of mechanical failures. These signals can also be controlled remotely from centralised signalling centres,
leaving to the closure of a number of local signal boxes.
Signalling problems are a leading cause of delays to rail traffic. Since the system is fail safe if any issue arises then
no rail traffic is allowed to operate on the affected line. While coloured light signals have improved reliability
compared to semaphore signals there are still a number of areas where faults can occur, such as if cables are
damaged or stolen, or if there is a failure in the control system (which are mainly mechanical relay or computer
driven systems).
Occupied Block Occupied BlockBlock Block Block
Direction of Travel

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Figure 2: Standard British Signal Types [2]
These signalling systems are also very difficult and costly to uphold, requiring regular inspections and maintenance.
Considering the vast number of signals across the country this all adds up to a great expense, financially and in
terms of man-hours. Other issues with the current standard of railway signalling include limited line speed and
capacity. Since the train driver still relies on line of sight to see the trackside signals the line speed is limited to
ensure that the driver can see the signal in plentiful time to react. Also, since the railway is divided into physical
sections, the capacity (i.e. the number of trains allowed onto a length of railway at any one time) is limited. This is
a great issue in busy areas, such as around London, where the recent increase in the popularity of rail travel has
led to overcrowding on the trains operating in that area.
One means of overcoming these problems is through the implementation of in-cab signalling. This technology
would mean that physical lineside signals could be replaced by a display in the driver’s cab. The removal of physical
signals and the associated cabling would mean that the reliability of the system could be increased. In some areas
the railway line speed could also be increased, since the train driver would no longer need to rely on spotting the
lineside signals – the information is displayed to the driver at all times. Furthermore, the railway blocks could be
defined virtually rather than physically and thus the blocks could be made much smaller than they are today, without
the expense of adding new physical systems to the railways.
Modern day railways also implement several additional layers of safety, such as integrity checking and warning
systems. A train’s integrity is monitored by trackside equipment, such as track circuits or axle counters. These are
used to ensure that a train is complete when it leaves a block section, so that the block is proven to be clear.
Additional systems, such as the Train Protecting and Warning System (TPWS) will automatically stop a train if it
passes a red signal, in an event known as a Signal Passed at Danger (SPAD).
Historically, European railways have been divided by a number of different signalling systems between or
sometimes within each country. This makes cross border rail traffic very difficult as the trains must be equipped
with several different types of signalling and safety equipment. The European Railway Traffic Management System
(ERTMS) is an initiative aimed at standardising the European railways under a single cab-based signalling system
so that any rail traffic can travel easily between countries.
Semaphore Arm
Horizontal
Stop
Semaphore Arm
Raised
Proceed
Red Light
Aspect
Stop

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2. Objectives
The main objective for this dissertation was to conduct a study of the ERTMS, analysing the engineering principles
and engineering theory behind the system. A simplified model of the trainborne systems associated with the
ERTMS was to be produced in the MATLAB and Simulink simulation environment, as seen by the train and its
driver.
The model would be applied to a number of different track and train scenarios, considering real train data, including
emergency situations and the interaction between two trains.
The validity of the model was to be assessed through an extensive validation and verification procedure to prove
the integrity and safety of the system.
3. Literature Review
3.1. The Fundamentals of ERTMS
Before the system can be modelled a deep understanding of the technology is required as well as the companies
and technologies behind it. The system is being openly developed by a consortium of various companies, allowing
a vast amount of information to be readily available online. The main website to host this information is “ERTMS |
The European Railway Traffic Management System” [3] which includes a number of news articles and data sheets.
Overseeing the development of the system is a consortium of relevant companies known as UNISIG [4], an
associated member of UNIFE, the body which represents railway manufacturing in Europe [5]. UNISIG collaborates
closely with the European Railway Agency (ERA) to produce a set of standards that any company working with
ERTMS must conform to [6]. Network Rail is the company who owns and manages Britain’s railway network and
will oversee the implementation of ERTMS in the UK. More information is provided on their website [7] and on the
partner site “ERTMS Online” [8] including the developmental timeline and testing results.
All of the aforementioned sources host a great deal of information on ERTMS and since the system is being jointly
developed across a number of companies all the information is fairly consistent. A good summary is available in
the Network Rail document “Your guide to European Rail Traffic Management System (ERTMS)” [9]. The document
briefly summarises the main implications and advantages of the ERTMS, with the main difference being that the
current fixed lineside signals and speed signs will be removed with all information being displayed to the driver in
cab via the Driver Machine Interface (DMI). Train movements will be controlled by a regional control centre, rather
than the numerous signal boxes which currently each control a small section of track. Communications will be
transmitted wirelessly over the purpose-built GSM-R network between the control centre and the train and this
data, along with track data provided by devices mounted on the track known as Eurobalises, will be used by a
computer on-board the train to calculate the train’s movement authority. The system will also provide automatic
train protection which minimises the safety risk.
3.1.1. ERTMS Levels
The document discussed previously also introduces the two main components of ERTMS, the European Train
Control System (ETCS) and the GSM-R communications system, as well as the three widely aknowledged
application levels of ERTMS. Each level builds on the previous with increased capabilities and efficiency. The levels
are further defined in the documents “ERTMS Factsheet 3: ERTMS Levels” [10].
Under level one operation the train is fitted with all the required ERTMS equipment but the lineside equipment
reamins in place to support the safe operation of the train and maintatin distances betweer trains. The movement
authority given by the lineside equipment is repeated to the train by the lineside electronic unit (LEU) and track
balise and displayed on the DMI. This operation is shown below in Figure 3. The main advantages of this level are
the interoperability of trains between countries and equipment suppliers, and the improved safety as the on-board
equipment will calculate the required speed profile of the train and apply the brakes if the train over-speeds, as part
of the ETCS.

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Figure 3: ERTMS Level 1 [10]
Level two builds on level one but with the removal of lineside signalling, as in Figure 4. The train’s movement
authority is passed directly to the train from a Radio Block Centre (RBC) via the GSM-R network and displayed on
the DMI. The balises are still required to transmit fixed messages to the train, such as its location, the speed limit,
or the track gradient. The train’s position is still recorded by traditional interlocking equipment (such as track circuits
or axle counters) and fed back to the RBC, therefore fixed ‘block’ sections are still required. This is currently the
most popular level amongst train and rail operating companies and it is proposed to be rolled out across the entire
rail network in the UK [9].
Once again level three builds upon level two. This, however, is still a conceptual level since the standards have not
yet been fully defined by the ERA. In general the distance between trains is regulated according to safety critical
train data, effectively creating moving blocks and reducing the requirement for fixed blocks (see Figure 5). This has
the advantage of increasing the line capacity and optimising train running speeds. There is no longer a need for
track circuits, axle counters or traditional interlocking equipment since the train integrity is checked on-board and
the train position is sent to the RBC via GSM-R.

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These levels are largely agreed upon by all companies involved in ERTMS and are defined technically in the
standards produced by the ERA [6]. While manufactures may make different solutions (for example Alstom [11]
produce a different product to ERTMS Solutions [12]) it is required that all systems are cross compatible to ensure
that trains can truly operate across borders.
3.1.2. Trainborne Equipment and Train Characteristics
The paper “European Rail Traffic Management System – An Overview” [13] further discusses these application
levels in agreeance with the previously considered documents. An additional level, level 0, is also introduced where
a train fitted with ERTMS equipment is run on a conventionally signalled line. i.e. the ERTMS equipment is ignored.
The architecture of the system is further defined in this paper (see Figure 6), identifying and describing the system’s
key components.
Figure 6: ERTMS System Architecture [13]
From the diagram above, signals are sent to and from the key management centre (or control centre) by the GSM-
R network. Data is generated by the interlocking equipment at the lineside and it fed to the train and the
management centre. The European Vital Computer (EVC) on-board the train uses this data to calculate the desired
train speed and braking point and displays the data on the DMI. The DMI display is also standardised by the ERA
specifications, therefore each company must produce their display to look similar to that in Figure 7.
Figure 7: Driver Machine Interface Display [13]

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Included on the DMI are a number of parameters that are required to be considered for simulation. These will
include the permitted speed (the maximum speed the train is allowed to travel), the actual speed (the real time
speed of the train), the target speed (the train speed calculated by the EVC) and the distance to target (the distance
to a theoretical stop signal for example). The speed warning indicator is illuminated when the train is travelling
faster than the permitted speed and gives the driver a warning to adjust their speed before the brakes are
automatically applied. The planning area may also be useful for simulation, displaying the oncoming line features
such as speed restrictions.
This paper also introduces the means of operation of the system in terms on dynamic speed monitoring, which will
be useful for modelling the system. The ERTMS equipment on-board the train monitors the train’s speed against
its position based on traction and deceleration models as presented in Figure 8 and Figure 9 below. These
represent the time delay in the acceleration cut off or braking command and execution. The braking capacity of the
train against speed is also shown in Figure 10, highlighting the function of constant values, deceleration = f(speed).
Of course the higher the train speed the slower the deceleration, or the higher the effort required to slow the train.
These parameters are specific to the individual train and would be uploaded into the ERTMS system as part of the
train data. Therefore, they will be inputs to the system model considered later.
Figure 8: Traction Model [13]
Figure 9: Braking Model – Progressive Application of Brakes [13]
Figure 10: Braking Model – Deceleration vs Speed [13]

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Under ETCS the train speed is supervised and automatically restricted where required, preventing the train from
travelling too fast or from running into occupied line sections. The speed monitoring process is described in Figure
11 below as a plot of braking distance against speed. The permitted value plot indicates the maximum allowed
speed on the line and drops to zero at a particular distance. This may be at a theoretical ‘red signal’ or station
platform (under ERTMS the red signal would be displayed on the DMI with no physical lineside signal). The actual
value is the train speed that will allow for normal, uninterrupted travel. The warning limit is the speed, greater than
the permitted speed, at which the warning light on the DMI (Figure 7) is illuminated. If the speed reaches the
service brake curve the trains service brakes are applied until the train returns to a safe speed. Should the
emergency brake value be reached the train’s emergency brakes are applied and the train would come to a halt.
Figure 11: Dynamic Speed Monitoring by ETCS [13]
The article “Performance Modelling For The National ERTMS Programme (NEP)” [14] largely agrees with these
principles and shows how these braking curves can be applied when approaching a red signal (R) (Figure 12). The
braking curves are compared to that a professional driver may apply under a conventional system. It can be seen
that when driving by lineside signals the driver starts braking much earlier (as soon as he/she sees the double
yellow (YY) signal) compared to the driver under ERTMS command. A much smoother curve is seen from the
ERTMS approach, thus the ERTMS system allows for much more efficient operation.
Figure 12: Braking Curves under Plain Line Operation [14]
(System D refers to ERTMS level 2)

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3.2. ERTMS Modelling
A number of modelling techniques have been applied to the system while developing the ERA’s specifications [6]
[13] [14] [15]. Therefore a standard ERTMS language has also been produced for the system variables being used,
as outlined in Table 2. These prefixes will be used for modelling accordingly.
Table 2: Prefixes Used for ERTMS Variables [13]
The textbook “Formal Methods Applied to Industrial Complex Systems” [15] introduces an in depth modelling
strategy used by the French railway company SNCF, based on the ERA specification. The same model is discussed
in the MathWorks video “Model-Based Approach for ERTMS Railway Wayside System Specification, Validation,
and Proof” [16]. Both these sources provide a similar background to ERTMS to that above and discuss more the
modelling procedure using similar variables prefixed by the letters in Table 2.
Since the models discussed here are of professional quality they are in much more detail than what is required for
this dissertation, for example, in the range of 350 sub system blocks were used in the modelling process over four
levels of refinement. This level of complexity would be difficult to achieve with the resources available so a more
simplified model will be produced.
In the modelling procedure it is discussed that the model must satisfy a number of conditions including the ability
for the user to trace and understand the model, to be testable and allow automated reasoning. Similar to [13] the
train’s speed and braking curves are discussed as inputs to the model, where the braking curves are calculated
based on a number of train and track characteristics, including:
 The train’s location (position and orientation)
 The train’s movement authority (MA)
 The most restrictive line speed profile (MRSP) of the train
 The deceleration time of the train
 The line gradients
 The targets determined by the above factors
 The train’s supervision limits and brake intervention curves
 The ESA Specifications [6]
Many of these factors are dependent on the distance travelled, estimated line speed and the current time. In some
cases, there may be more than one speed restriction on the line and under these circumstances it is critical that
the lowest speed is observed. This is also known as the most restrictive speed profile (MRSP) as mentioned above.
These factors are divided into three categories as in Figure 13 below. From these characteristics an example speed
profile and braking curve is produced.

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Figure 13: Braking Curve Inputs and Outputs [15]
In the plot above “EB1” (the solid line) represents the permitted speed and “EB0” (dashed line) represents the
braking curve of the train.
The ERA have produced an Excel tool that can be used to generate braking data for a train in the same way as
the on-board ETCS equipment would, capturing the train and trackside data. This is readily available to download
at [17] and may be useful for the modelling process. Train data is also readily available – Network Rail have
specified that the first lines in the UK to be fitted with ERTMS Level 2 are the Great Western and East Coast main
lines [7] with the new Hitachi Class 800 trains [18] (trainsets running on the East Coast are also to be known as
the Virgin Azuma [19]). The specification for these trains has been developed openly between the UK Government
and Hitachi so their formal data (including operating speeds, acceleration and braking curves) is available to the
public [20] [21].
Continuing the modelling process, [15] and [16] acknowledge that there is a need to interface with many various
styles of conventional interlocking equipment. The aim of their modelling process is to simulate the level 2 Radio
Block Centre (RBC) in Simulink and MATLAB software. Due to the discrepancies in the interlocking equipment they
chose to model a generic RBC core, with an interface to each interlocking.
For system testing the static and dynamic track and train input data was written in an Excel spreadsheet and
imported to MATLAB, as in Figure 14. The model was then animated and validated as shown in Figure 15.
Figure 14: Generating the Test Configuration [15]

17
Figure 15: RBC Execution Process [15]
While these sources discuss the modelling procedure in an extensive depth and have provided a great deal of
useful information they show little of the actual model that was constructed or the results of animation. Also, where
sections of code are shown the code tends to be written in French causing a language barrier. In fact, it is generally
difficult to find information on modelling the system in Simulink, especially at a simplified level.
It is therefore useful to review literature on modelling conventional interlocking systems and adapting these models
to the ERTMS levels, similar to the implementation of ERTMS on existing railways. Although not using Simulink, in
[22] and [23] formal models of railway interlocking systems are discussed, introducing the theory of moving block
technology. The systems under consideration in these papers are well decomposed and also consider the safety
implications, a very important issue for railway systems. These papers also follow a detailed modelling approach
and provide a number of control references that may be converted to the MATLAB language.
3.3. Conclusions of Literature Review
Throughout this literature review a number of works have been analysed, including websites, textbooks and IEEE
journal articles. A great deal of information has been found on the theory behind ERTMS, outlining the reasons
behind the system and the various application levels of the system. In particular the ERTMS website [3] is very
useful thanks to the number of factsheets available.
More technical documents such as “European Rail Traffic Management System – An Overview” [13] have been
very valuable in adding detail to the information found online by fully decomposing the system as in Figure 6. In
this paper, as well as several others including [14] and [15], the braking and acceleration characteristic are
discussed at length. This, coupled with the technical information by Hitachi [20] and the Department for Transport
[21] should provide a good base for the modelling of a train under ERTMS control.
In general, it has been difficult to find information on modelling the system as a whole, particularly in the
Simulink/MATLAB environment. However some key works have been identified – for example the book by
Boulanger [15] discusses the modelling theory to a great depth. Although the systems here are to a much higher
degree of complexity than is required for this project, a number of key points were outlined that will be useful for
later simulation.
In addition, a greater understanding has been gained in reviewing these works allowing for a simplified ERTMS
model to be built. From this the project objective can be better defined:
 To produce accurate and reliable models to generate the train braking curves of the three ERTMS levels
in Simulink and MATLAB, as seen by the control centre and the train driver, and to compare these levels
to a conventional interlocking.
 Input ‘track’ data will be provided to the model in the form of an Excel spreadsheet, allowing for a number
of different track scenarios to be applied easily. This data will include line speed and track feature as
functions of train displacement.
 To model a train on the system based on the Hitachi Class 800. The ‘train’ model will use this data to
calculate the train’s braking curves and the desired speed as a function of displacement. A closed loop
control system would replicate the train driver in maintaining the desired speed. The interaction between
two trains will be assessed.
 To validate and verify the model, analysing the integrity and safety of the ERTMS system based on the
model. A number of fault conditions will be applied to ensure that safe operation is maintained.
 To assess the benefits of the system in terms of line speed, capacity and cost.

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4. Technical Assessment
As discussed previously, a critical part of the ERTMS is the generation of the dynamic braking curves. These
curves, as seen in Figure 16, are calculated by the train’s on-board equipment and are based on a set of
fundamental equations as outlined in chapter 3 of the ERTMS System Requirements Specification (SRS) [24].
These equations are discussed in detail through this section. The SRS also contains two important datasets, Fixed
Value Data and National Values, which are referred to throughout the description of these equations. These can
be found in Appendices A.3.1 and A.3.2 of chapter 3 of the SRS.
Figure 16: ERTMS braking curves, where: I – Indication Point, P – Permitted Speed, W – Warning Point, SBI –
Service Brake Intervention, EBI – Emergency Brake Intervention, EBD – Emergency Brake Deceleration, SBD –
Service Brake Deceleration [24].
To calculate the braking curves and desired velocities, the on-board equipment continuously monitors a list of target
locations. These targets may be of the following types:
 A decrease in the Most Restrictive Speed Profile (VMRSP), the maximum allowable velocity that the train is
allowed to travel as a function of distance along the railway.
 The End of Authority (EOA), the location to which the train is authorised to move, where the target speed
is zero.
 The Limit of Authority (LOA), the location to which the train is authorised to move, where the target speed
is non-zero.
 The location where the maximum distance allowed to run in Staff Responsible is reached, with a target
speed of zero. Staff Responsible is a mode of ERTMS where the train is allowed to move under the driver’s
own authority.
4.1. Emergency Brake Deceleration
The Emergency Brake Deceleration (EBD) curve is the fundamental basis for the calculation of most of the other
braking curves and is defined as the safe braking distance required to reach zero velocity by the target. The safe
deceleration is calculated by considering the guaranteed deceleration of the emergency brake system as well as
the acceleration/deceleration due to the line gradient.
EBD Curve SBD Curve
EBD Foot

19
The EBD curve, denoted dEBD in metres as a function of velocity, is calculated by the equation:
𝑑 𝐸𝐵𝐷(𝑉) =
𝑉2−𝑉𝑡𝑎𝑟𝑔𝑒𝑡
2
2𝐴 𝑠𝑎𝑓𝑒(𝑉,𝑑)
m (1)
Where V is the instantaneous velocity of the train (m/s), Vtarget is the train’s target velocity (nominally 0, m/s) and
Asafe(V,d) is the safe deceleration (m/s2) of the train as a function of velocity and displacement. This is a safety
critical value and takes into account all relevant train parameters, and is given by equation (2):
𝐴 𝑠𝑎𝑓𝑒(𝑉, 𝑑) = 𝐴 𝑏𝑟𝑎𝑘𝑒_𝑠𝑎𝑓𝑒(𝑉, 𝑑) + 𝐴 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡(𝑑) m/s2 (2)
Assuming normal adhesion conditions.
In equation (2) Abrake_safe(V,d) is the safe emergency brake deceleration and Agradient(d) is the
acceleration/deceleration due to gradient of the railway (both in m/s2). These are given by equations (3) and (4)
respectively:
𝐴 𝑏𝑟𝑎𝑘𝑒 𝑠𝑎𝑓𝑒
(𝑉, 𝑑) = 𝐾𝑑𝑟𝑦𝑟𝑠𝑡(𝑉, 𝑀 𝑁𝑉𝐸𝐵𝐶𝐿) ∗
(𝐾𝑤𝑒𝑡 𝑟𝑠𝑡(𝑉) + 𝑀 𝑁𝑉𝐴𝑉𝐴𝐷𝐻 ∗ (1 − 𝐾𝑤𝑒𝑡 𝑟𝑠𝑡(𝑉))) ∗ 𝐴 𝑏𝑟𝑎𝑘𝑒_𝑒𝑚𝑒𝑟𝑔𝑒𝑛𝑐𝑦(𝑉, 𝑑) m/s2 (3)
𝐴 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑔 ∗
𝑔𝑟𝑎𝑑
1000+10∗𝑀 𝑟𝑜𝑡𝑎𝑡𝑖𝑛𝑔_𝑚𝑎𝑥
m/s2 for uphill gradients
𝐴 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑔 ∗
𝑔𝑟𝑎𝑑
1000+10∗𝑀 𝑟𝑜𝑡𝑎𝑡𝑖𝑛𝑔_𝑚𝑖𝑛
m/s2 for downhill gradients (4)
The additional parameters in the above equations are defined as follows:
 Kdryrst (V,MNVEBCL) is the rolling stock correction factor and is defined as part of the train data. This is a
function of velocity and is relative to the confidence level on emergency brake safe deceleration, MNVEBCL,
a constant defined in the National Values.
 Kwerrst(V) is also a rolling stock correction factor and represents the decrease in available deceleration
when on wet rails, as compared to being on dry rails, due to the reduced cohesion between the train
wheels and the rails. This is also part of the train data.
 MNVAVADH is the weighting factor for the available wheel to rail adhesion, as defined in the National
Values.
 g is the acceleration due to gravity, 9.81 m/s2.
 grad is the gradient of the railway in ‰, where uphill gradients are represented by positive values.
 Mrotating_max and Mrotating_min are the maximum and minimum possible rotating masses respectively, as a
percentage of the total train weight and as defined in the Fixed Value Data.
Also referenced in equation (3) is Abrake_emergency(V,d), the emergency brake deceleration as a function of speed
and a function of the locations where the contribution of the train’s different brakes changes. For example, Figure
17 below shows that as a train reaches the target, and thus its speed has decreased, the regenerative brake can
no longer be used. Therefore, the emergency brake deceleration is reduced.
Figure 17: Influence of Track Conditions on Abrake_emergency

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4.2. Service Brake Deceleration
The Service Brake Deceleration (SBD) is the expected distance required to come to a halt at the target using the
train’s service brakes. The expected deceleration is not safety critical so does not take into account the same
correction factors as the EBD curve. Therefore, the SBD curve often appears much steeper than the EBD curve,
as seen in Figure 16 above.
Similar to the EBD, the SBD is calculated as follows:
𝑑 𝑆𝐵𝐷(𝑉) =
𝑉2−𝑉𝑡𝑎𝑟𝑔𝑒𝑡
2
2𝐴 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑(𝑉,𝑑)
m (5)
Where Aexpected(V,d) is the anticipated deceleration of the train under normal operation, given by:
𝐴 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑(𝑉, 𝑑) = 𝐴 𝑏𝑟𝑎𝑘𝑒_𝑠𝑒𝑟𝑣𝑖𝑐𝑒(𝑉, 𝑑) + 𝐴 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡(𝑑) m/s2 (6)
Similar to Abrake_emergency(V,d), Abrake_service(V,d) is the deceleration of the service brake as a function of speed, relative
to the locations where the contribution of the train’s different brakes changes.
4.3. Supervision Limits
The other curves displayed in Figure 16 above are known as supervision limits and will be discussed further in this
section. These limits, calculated by the on-board equipment, are used for a number of reasons such as providing
a comfortable speed profile without excessive accelerations, decelerations, or wear on the train. The curves also
assist the driver in maintain the correct speed to avoid emergency brake intervention, and to ensure that the train
adheres to the appropriate track limitations (in terms of distance or speed).
4.3.1. Emergency Brake Intervention
Possibly the most critical of these supervision limits is the Emergency Brake Intervention (EBI). This defines the
location at which the train’s emergency brakes will be applied automatically to ensure that the train remains within
the imposed speed or distance limitations (i.e. to prevent the train from over speeding or surpassing its limit of
authority). This limit will be reached if the driver fails to adhere to the recommended braking profile and ensures
that the train is brought to a halt if the driver is unable to apply the brakes manually.
The EBI curve is calculated as follows:
𝑑 𝐸𝐵𝐼(𝑉𝑒𝑠𝑡) = 𝑑 𝐸𝐵𝐷(𝑉𝑏𝑒𝑐) − 𝐷𝑏𝑒𝑐 m (7)
Where Vest is the estimated velocity of the train, and Vbec and Dbec are the compensated speed and distance
travelled, respectively, during the time elapsed between the EBI command and the full application of the train’s
emergency brakes. These correction factors are calculated using equations 8 and 9:
𝑉𝑏𝑒𝑐 = 𝑚𝑎𝑥{(𝑉𝑒𝑠𝑡 + 𝑉𝑑𝑒𝑙𝑡𝑎0 + 𝑉𝑑𝑒𝑙𝑡𝑎1), 𝑉𝑡𝑎𝑟𝑔𝑒𝑡} + 𝑉𝑑𝑒𝑙𝑡𝑎2 m/s (8)
𝐷𝑏𝑒𝑐 = 𝑚𝑎𝑥 {(𝑉𝑒𝑠𝑡 + 𝑉𝑑𝑒𝑙𝑡𝑎0 +
𝑉 𝑑𝑒𝑙𝑡𝑎1
2
) , 𝑉𝑡𝑎𝑟𝑔𝑒𝑡} 𝑇𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛
+ (𝑚𝑎𝑥{(𝑉𝑒𝑠𝑡 + 𝑉𝑑𝑒𝑙𝑡𝑎0 + 𝑉𝑑𝑒𝑙𝑡𝑎1), 𝑉𝑡𝑎𝑟𝑔𝑒𝑡} +
𝑉 𝑑𝑒𝑙𝑡𝑎2
2
) 𝑇𝑏𝑒𝑟𝑒𝑚 m/s2 (9)
Where the additional parameters (also seen in Figure 18) are defined as:
 Vdelta0 is the speed under reading amount (also denoted Vura) used to compensate for measurement
inaccuracies in the measured velocity of the train.
 Vdelta1 and Ttraction are the respective speed and time compensations for the time delay between the EBI
command and the cut off of the train’s tractive effort (release of the throttle).
 Vdelta2 and Tberem are the respective speed and time compensations for the remaining time following the
traction cut-off up to full braking effort being reached.

21
The traction time, Ttraction, and remaining time, Tberem, are defined by the equations:
𝑇𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 = 𝑚𝑎𝑥 {(𝑇𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑐𝑢𝑡−𝑜𝑓𝑓
− (𝑇 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 + 𝑇𝑏𝑠2)) , 0} s (10)
𝑇𝑏𝑒𝑟𝑒𝑚 = 𝑚𝑎𝑥{(𝑇𝑏𝑒 − 𝑇𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛), 0} s (11)
Ttraction_cut-off is the time taken between the cut off command being issued (which is triggered when the warning limit
is passed) and the tractive effort cut off. Twarning is the time taken between the warning curve and the Service Brake
Intervention (SBI), as defined in the fixed value data, while Tbs2 it the time from the SBI to the EBI, defined in the
train data. Therefore, Twarning + Tbs2 accounts for the total time occurred between the warning limit and the EBI.
Tbe is the total time elapsed between the EBI and the EBD, also defined in the train data.
Furthermore, Vdelta1 and Vdelta2 are defined in equations 12 and 13 below:
𝑉𝑑𝑒𝑙𝑡𝑎1 = 𝐴 𝑒𝑠𝑡1 𝑇𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 m/s (12)
𝑉𝑑𝑒𝑙𝑡𝑎2 = 𝐴 𝑒𝑠𝑡2 𝑇𝑏𝑒𝑟𝑒𝑚 m/s (13)
Where Aest1 and Aest2 are the respective estimated accelerations during Ttraction and Tberem, as measured at the time
of calculation.
Figure 18: Supervision limits for an EBD curve [24].
4.3.2. Service Brake Intervention
The Service Brake Intervention (SBI) is used in a similar way to the EBI, only to apply the service brakes instead
of the emergency brakes, with the intention of slowing the train down before the EBI is reached. This function may
not be available on all trains, depending on the configuration of the ETCS on-board equipment. Using the service
brakes in this way can mitigate the frequent use of the emergency brakes, reducing wear on the rolling stock and
track experienced through emergency braking.
The SBI may be defined in two different ways; If the target is an end of authority (EOA) the supervision limit (dented
SBI1) is calculated over the time between the SBI and the SBD curves, Tbs1. Otherwise, for an EBD based target
the supervision limit (SBI2) will be calculated over the time between the SBI and the EBI, Tbs2.
The values of Tbs1 and Tbs2 are defined based on the service brake feedback and are measured, where available,
from the train’s main brake pipe and brake cylinder pressures. This feedback allows for the distances between the
SBI and the SBD or EBD to be minimised. An algorithm for the calculation of these values is presented in detail in
Appendix A.3.10 of the SRS [24] but has been omitted from this work for simplicity. If the service brake feedback
is not available, these parameters are set to Tbs, the time required for the service brakes to build up to full braking
effort, as defined in the train data. Furthermore, if the SBI is unavailable, these parameters are set to zero in order
to achieve the maximum performance.

22
The two SBI curves are calculated as:
𝑑 𝑆𝐵𝐼1(𝑉𝑒𝑠𝑡) = 𝑑 𝑆𝐵𝐷(𝑉𝑒𝑠𝑡) − 𝑉𝑒𝑠𝑡 𝑇𝑏𝑠1 m (14)
𝑑 𝑆𝐵𝐼2(𝑉𝑒𝑠𝑡) = 𝑑 𝐸𝐵𝐼(𝑉𝑒𝑠𝑡) − 𝑉𝑒𝑠𝑡 𝑇𝑏𝑠2 m (15)
It is noted that if either the EBI or SBI are triggered the train will be brought to a halt autonomously by the on-board
equipment. Once the train has stopped, the driver must take steps to reset these safety systems and prove that he
or she is in fact fit to proceed with the train. Any time that these limits are surpassed will be subject to later
investigation into the cause of the incident (such as mechanical failure or diver negligence). The more severe
incidents may be investigated by the Government’s Rail Accident Investigation Branch [25].
4.3.3. Warning Supervision Limit
The Warning supervision limit (W) is the location where an indication appears to the driver (normally a light on the
driver machine interface), informing the driver to apply the brakes before reaching the SBI or EBI, with the intention
or avoiding intervention from the on-board equipment. Similar to the SBI, the W limit is calculated in different ways
for the EOA or EBD based targets:
𝑑 𝑊(𝑉𝑒𝑠𝑡) = 𝑑 𝑆𝐵𝐼1(𝑉𝑒𝑠𝑡) − 𝑉𝑒𝑠𝑡 𝑇 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 m for the EOA (16)
𝑑 𝑊(𝑉𝑒𝑠𝑡) = 𝑑 𝑆𝐵𝐼2(𝑉𝑒𝑠𝑡) − 𝑉𝑒𝑠𝑡 𝑇 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 m for an EBD based target (17)
As mentioned above, Twarning is the time taken between W and the SBI, as defined in the fixed value data.
4.3.4. Permitted Speed
The Permitted speed supervision limit (P) is the recommended speed displayed to the driver, observing the speed
restrictions imposed by the track, and allowing for a comfortable means of braking for the driver while avoiding
excessive brake wear and to limit wasted traction energy. Again, these are calculated differently for the EOA and
EBD curves:
𝑑 𝑃(𝑉𝑒𝑠𝑡) = 𝑑 𝑆𝐵𝐼1(𝑉𝑒𝑠𝑡) − 𝑉𝑒𝑠𝑡 𝑇𝑑𝑟𝑖𝑣𝑒𝑟 m for the EOA (18)
𝑑 𝑃(𝑉𝑒𝑠𝑡) = 𝑑 𝑆𝐵𝐼2(𝑉𝑒𝑠𝑡) − 𝑉𝑒𝑠𝑡 𝑇𝑑𝑟𝑖𝑣𝑒𝑟 m for an EBD based target (19)
In the above equations, Tdriver is the driver’s reaction time between P and the SBI, as defined in the fixed values.
4.3.5. Indication Supervision Limit
The Indication supervision limit (I) is the location where an indication will be presented to the driver on the DMI, to
inform the driver that they are approaching a target and must prepare to take action. The indication point is
calculated as a distance from the permitted supervision limit, i.e.
𝑑𝐼(𝑉𝑒𝑠𝑡) = 𝑑 𝑃(𝑉𝑒𝑠𝑡) − 𝑉𝑒𝑠𝑡 𝑇𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑖𝑜𝑛 m (20)
Where 𝑇𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑖𝑜𝑛 = 𝑚𝑎𝑥{(0.8 ∗ 𝑇𝑏𝑠), 5} s (21)
Tindication is the time elapsed between I and P and is a function of the service brake build up time, Tbs. This is intended
to improve the performance of the system. Where Tbs is small, Tindication is set to a constant of 5 seconds.

23
4.3.6. Ceiling Supervision Limits
The above analysis has all been carried out as speed monitoring when braking to a target. However, the ETCS
system still monitors the train’s speed when not approaching a target, to ensure that the train’s velocity does not
exceed the limit specified for the section of railway. Similar to above, the permitted speed, warning limit, SBI and
EBI are monitored, relative to the most restrictive speed profile of the railway. These appear as constant values
against distance where the permitted speed is constant, as in Figure 19, and are known as ceiling supervision
limits.
Figure 19: Ceiling supervision limits [24]
The train would nominally be travelling at or below the permitted speed, P. It the train begins to accelerate (through
any factors such as driver negligence, negative gradient or mechanical interruptions), the speed of the train may
increase past the warning limit, and an indication will appear on the DMI allowing the driver to rectify the error. If
no action is taken the speed may increase past the SBI curve (where available) or the EBI curve, causing the trains
service or emergency brakes to be applied, bringing the train to a halt.
To calculate the ceiling supervision limits a number of parameters are defined in the fixed value data. These
outline the maximum and minimum values for each curve, as outlined in Figure 20 and Table 3.
Figure 20: Definition of dVebi

24
Table 3: Minimum and maximum ceiling supervision limit parameters as defined in Appendix A.3.1 of the System
Requirements Specification [24].
Parameter Value (km/hr) Definition
dVebi_min 7.5 The minimum speed difference between P and EBI supervision limits.
dVebi_max 15 The maximum speed difference between P and EBI supervision limits.
Vebi_min 110
The value of permitted speed where dVebi starts to increase towards
dVebi_max
Vebi_max 210 The value of permitted speed where dVebi reaches dVebi_max
dVsbi_min 5.5 The minimum speed difference between P and SBI supervision limits.
dVsbi_max 10 The maximum speed difference between P and SBI supervision limits.
Vsbi_min 110
The value of permitted speed where dVsbi starts to increase towards
dVsbi_max
Vsbi_max 210 The value of permitted speed where dVsbi reaches dVsbi_max
dVwarning_min 4 The minimum speed difference between P and W supervision limits.
dVwarning_max 5 The maximum speed difference between P and W supervision limits.
Vwarning_min 110
The value of permitted speed where dVwarning starts to increase
towards dVwarning_max
Vwarning_max 140 The value of permitted speed where dVwarning reaches dVwarning_max
The ceiling supervision limits are thus calculated as follows:
Emergency Brake Intervention
When VMRSP > Vebi_min:
𝑑𝑉𝑒𝑏𝑖 = 𝑚𝑖𝑛 {(𝑑𝑉𝑒𝑏𝑖 𝑚𝑖𝑛
+ 𝐶𝑒𝑏𝑖(𝑉 𝑀𝑅𝑆𝑃 − 𝑉𝑒𝑏𝑖 𝑚𝑖𝑛
)) , 𝑑𝑉𝑒𝑏𝑖 𝑚𝑎𝑥
} km/hr (22)
Where: 𝐶𝑒𝑏𝑖 =
𝑑𝑉 𝑒𝑏𝑖 𝑚𝑎𝑥−𝑑𝑉 𝑒𝑏𝑖 𝑚𝑖𝑛
𝑉 𝑒𝑏𝑖 𝑚𝑎𝑥−𝑉 𝑒𝑏𝑖 𝑚𝑖𝑛
(23)
When VMRSP ≤ Vebi_min: 𝑑𝑉𝑒𝑏𝑖 = 𝑑𝑉𝑒𝑏𝑖 𝑚𝑖𝑛
km/hr (24)
Service Brake Intervention
When VMRSP > Vsbi_min:
𝑑𝑉𝑠𝑏𝑖 = 𝑚𝑖𝑛 {(𝑑𝑉𝑠𝑏𝑖 𝑚𝑖𝑛
+ 𝐶𝑠𝑏𝑖(𝑉 𝑀𝑅𝑆𝑃 − 𝑉𝑠𝑏𝑖 𝑚𝑖𝑛
)) , 𝑑𝑉𝑠𝑏𝑖 𝑚𝑎𝑥
} km/hr (25)
Where: 𝐶𝑠𝑏𝑖 =
𝑑𝑉 𝑠𝑏𝑖 𝑚𝑎𝑥−𝑑𝑉 𝑠𝑏𝑖 𝑚𝑖𝑛
𝑉 𝑠𝑏𝑖 𝑚𝑎𝑥−𝑉 𝑠𝑏𝑖 𝑚𝑖𝑛
(26)
When VMRSP ≤ Vsbi_min: 𝑑𝑉𝑠𝑏𝑖 = 𝑑𝑉𝑠𝑏𝑖 𝑚𝑖𝑛
km/hr (27)

25
Warning Supervision Limit
When VMRSP > Vwarning_min:
𝑑𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 = 𝑚𝑖𝑛 {(𝑑𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑖𝑛
+ 𝐶 𝑤𝑎𝑟𝑛𝑖𝑛𝑔(𝑉 𝑀𝑅𝑆𝑃 − 𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑖𝑛
)) , 𝑑𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑎𝑥
} km/hr (28)
Where: 𝐶 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 =
𝑑𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑎𝑥−𝑑𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑖𝑛
𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑎𝑥−𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑖𝑛
(29)
When VMRSP ≤ Vwarning_min: 𝑑𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 = 𝑑𝑉 𝑤𝑎𝑟𝑛𝑖𝑛𝑔 𝑚𝑖𝑛
km/hr (30)
4.4. Review of Technical Understandings
Throughout this section the ETCS System Requirements Specifications have been studied, particularly Chapter 3
[24]. From this, the fundamental equations behind the calculation of the ETCS braking curves and supervision limits
have been found. These equations will be invaluable during the modelling process.
Of course, this has only been a top level analysis, the SRS provides a number of other equations defining different
aspects of the train’s performance and various scenario definitions which have been omitted at this level.

26
5. Modelling Procedure and Implementation
MATLAB and Simulink are a pair of powerful software packages produced by The MathWorks which can be used
to model and analyse dynamic systems. MATLAB is suited to modelling mathematical equations through typed
code, while Simulink is based on a graphical user interface (GUI) where systems can be built in block diagram
format. Both packages can interact seamlessly, thus MATLAB can be used to initialise and run a Simulink model,
and then post-process the results.
Both packages contain user-friendly editors and provide a robust test environment. Extensive support is also
available online, making these packages ideal for the modelling stage of this project.
As discussed previously, the aim of this report was to produce braking curve models for the three levels of ERTMS.
Train data was provided based on the Hitachi Class 800 since it is readily available [20], and these trains will be
the first major mainline services to run with ETCS technologies in the UK. The model was tested under a number
of scenarios as discussed throughout this chapter.
5.1. Input Data
As mentioned previously, train data was based on the new Class 800 trainsets. For simplicity, standardised track
data was defined for the project. However, the model was designed to make it easy for actual track data to be
applied at a later date.
The track and train datasets were defined in Microsoft Excel and then imported into the MATLAB environment. This
allowed for the datasets to be easily manipulated without going through the MATLAB software. A number of
different datasets for each category could be built in separate Excel files, allowing a number of scenarios to be
defined. These datasets are outlined in the Appendices 9.1 and 9.2.
The train data is similar to that that would be loaded into the train’s on-board computer during installation. Certain
characteristics may be edited by the driver (such as the correction factors) if the train’s rolling stock was ever to
change. It is noted that only one train dataset was used for this project since the focus was based on the different
track scenarios. However, the model is configured so that any future work would allow for different train datasets
to be easily applied. The dataset takes some of the Class 800 characteristics, however correction factors were
based on those discussed in [26] to allow for easy validation of the model.
The track data was modelled to represent the data transferred to a train as it crosses a balise. Therefore, it only
contains the data for a block of finite distance (nominally 5 km). This however could easily be updated for future
use to include longer sections and more detail. For easy comparisons between datasets the gradient was
maintained at -10 ‰ for each set. This is also representative of the numerical examples given in [26], used for
model validation.
MATLAB scripts were written for the fixed value and national value datasets defined in appendices A.3.1 and A.3.2
of the SRS. These scripts are presented in Appendices 9.3.1 (Fixed_Value_Data.m) and 9.3.2
(National_Value_Data.m) of this document. A script was also written, ReadTrackData.m, to import the track and
train data from the Excel spreadsheets, and is seen in Appendix 9.3.3. Within this file, the command xlsread is
used to import the Excel data, which is then sorted into its relevant categories. These scripts can be referenced by
any future model, saving the need to re-write these commands and definitions for each model.
Any other input data, where required, was written into the model script. This included data such as the train’s initial
displacement and velocity. All velocities were given in km/hr, as in the ERA standards. This had to be taken into
account when carrying out any calculations.

27
5.2. Mathematical Model
The first stage of the modelling procedure was to transfer the equations discussed through section 4 into the
MATLAB environment, allowing for the calculation of the braking curves to a target. This model, named
BrakingCurves.m, is described in code in Appendix 9.4.1 and in the flow diagram in Figure 21 below. This model
was defined to consider only one target and therefore may not work effectively with all track datasets.
Figure 21: Mathematical model, BrakingCurves.m, flow diagram.
Initialise
Run
Fixed_Value_Data Fixed_Value_Data
Run
National_Value_Data
National_Value_Data
Define TrackData
and TrainData files
Run
ReadTrackData
ReadTrackData
TrackData.xlsx
TrainData.xlsx
Define Other Constants:
V_initial, V_delta0,
A_est1, A_est2, T_bs1, T_bs2,
T_traction, T_berem,
T_indication, V_delta1, V_delta2
Calculate Ceiling Supervision
Limits:
V_ebi
V_sbi
V_warning
Calculate Accelerations:
A_brake_safe
A_gradient
A_safe
A_expected
Calculate Braking Curves for
each value of V:
EBD, SBD, EBI, SBI1, SBI2,
W, P, I, FLOI
Plot Braking Curves and Ceiling
Limits.
Fixed Value Data
National Value Data
Track and
Train Data
Main Script
Auxiliary Script
Microsoft Excel File

28
5.2.1. Assumptions
To simplify the modelling process a number of assumptions were made, including:
 The train approached the target at a steady speed, and decelerated smoothly and consistently.
 At no point would the train accelerate under this model.
 The track had a consistent gradient, and was wet (a correction factor for wet rails is available in the National
Data (MNVAVADH), for this model it was kept at 0).
 There was zero speed reading inaccuracy (the ERTMS specification Subset-041 defines a speed
measurement inaccuracy of up to 2.98 %, for this model it was kept at 0).
 There was zero position reading inaccuracy (the ERTMS specification Subset-041 defines a position
measurement inaccuracy of up to 5 m + 5 %, for this model it was kept at 0).
 The traction cut-off time, Ttraction cut-off, was equal to zero, with the traction cut-off interface unavailable.
 The nominal rotating mass was unknown.
 The track had no reduced adhesion or brake inhibition profiles.
5.2.2. Initialisation and Parameter Definitions
The first stage of the model was to define all of the input parameters, constants and variables required for
calculation. Initially, the Fixed Value Data and National Value Data sets were collected through jumping to their
respective scripts, as discussed above and in Appendix 9.3. MATLAB allows for the running of one script from
within another by simply specifying the script name. For example, placing the line “Fixed_Value_Data” within the
BrakingCurves.m file will also run the Fixed_Value_Data.m file, populating the fixed value data in the MATLAB
workspace. A similar approach was used when importing the track and train datasets, using the ReadTrackData.m
script file. An alternative approach may have been to define functions instead of the auxiliary scripts. Functions,
however, require all inputs and outputs to be specified each time they are used. While this may improve the
functionality of the code, it would have led to long and complicated commands, particularly where a large number
of inputs or outputs were required.
It is noted though that using these scripts in this way requires that a consistent naming convention is adopted for
all variables, since these are taken directly from the global MATLAB workspace, rather than remaining local to a
function. Care was taken to ensure that any variables were not being overwritten unintentionally, particularly where
loops occurred in the code.
Following the import of data, other parameters were defined. This included the curve type (EOA or LOA), the target
type (EOA or EBD) and the target velocity (nominally zero). These would all be required to specify which
calculations were to be made for the supervision limits later in the model. The initial velocity was also set, normally
at 160 km/hr (100 mph), a fairly standard value for mainline operation.
Further to this, the error values and correction factors were defined. For simplicity, Vdelta0, Aest1 and Aest2 were all
set to zero, although they could be edited for any further work. Similarly, Tbs1 and Tbs2 were set to Tbs, which was
defined as part of the train input data
Ttraction, Tberem and Tindication were defined at this point using equations 10, 11 and 21 as defined previously. Vdelta1
and Vdelta2 were also calculated here using equations 12 and 13. Further to this, the vectors for each variable to be
calculated were pre-defined as zeros. This was to help optimise the code and reduce the time taken for the model
to execute, and could have been optimised further by setting the actual lengths of the arrays to the required values.
It is noted that throughout the model code, for i = 1:1 loops were used to break the code into sections. MATLAB
allows for the contents of such loops to be hidden, making the code easier to work on when only considering a
single section at a time.

29
5.2.3. Ceiling Supervision Limits
The first stage of the main calculations was to find the ceiling supervision limits using equations 22 to 30. To help
clarify the code, and since they depend entirely on the fixed value data, Cebi, Csbi and Cwarning were calculated in the
Fixed_Value_Data.m script. If-else loops were implemented to calculate the correct limits relative to VMRSP as
highlighted in section 4.3.6.
5.2.4. Accelerations
As was previously discussed, the ERTMS calculations use a number of different accelerations, including Asafe,
Abrake_safe, Agradient and Aexpected, as defined by equations 2, 3, 4 and 6 respectively. Abrake_safe and Agradient were
functions of velocity and displacement respectively, so were each calculated within a for loop, ensuring that a value
for each was calculated for each value of their respective parameter.
Similarly, Asafe and Aexpected were functions of both velocity and displacement, so were calculated through two for
loops (one for each velocity and displacement). Through doing this, a lookup table was effectively established for
each of the accelerations which could be referenced in later calculations using displacement or velocity.
5.2.5. Braking Curves
Once the acceleration values had been defined, the model could go on to calculate the braking curves relative to
a target. Similar to above, the curves were calculated within a for loop over the range of velocities between the
target speed and the maximum speed available from the train data. This allowed for the braking distances to be
calculated at each value of velocity, which were stored in arrays using the velocity as the index value.
When calculating the EBD and SBD curves (using equations 1 and 5) the acceleration values would normally be a
function of velocity and displacement, allowing the effects of a varying gradient to be taken into account. However,
for simplicity, and since the gradient was kept constant the accelerations were referenced against the varying
velocity but a constant displacement, for example:
EBD(v) = (((V(v)/3.6)^2)-((V_Target/3.6)^2))/(2*A_safe(v,1));
This meant that one less loop was required in the code for each equation, reducing the time required for the model
to execute. The reference, however, was still included for any future developments of the model.
Note that in the above equation (and others throughout the code) the velocity is divided by 3.6 in order to convert
from km/hr to m/s.
Following the calculations of the EBD and SBD, the other curves were calculated. The equations for the SBI1,
SBI2, W and P differed from those discussed previously in that the subtraction was been changed to an addition,
for example, equations 16 and 18 become:
W(v) = SBI1(v) + (V(v)/3.6)*T_warning;
P(v) = SBI1(v) + (V(v)/3.6)*T_driver;
This is down to the means by which these curves were being calculated, since they were being considered from
the target towards the train, with the target being at zero displacement. The equations, however, are designed to
be calculated from the train to the curve, with the front of the train being zero displacement.
An additional curve was introduced here also, known as the First Line of Intervention (FLOI), and was the first
location at which the on-board equipment would intervene through the application of the train’s brakes. This was
calculated as the maximum distance from the target of the SBI1, SBI2 or EBI curves.

30
5.2.6. Mathematical Model Output
After calculating the numerical data, the model produced an output plot containing all of the curves and their
respective ceiling supervision limits, allowing the user to easily analyse the output data. The plotting tools use the
line:
set(gca,'xdir','reverse')
This reversed the direction of the x axis so that the plot read as if the train was approaching the target from the left.
Figure 22 below shows an example output plot. Due to the quantity of data the plot as a whole can be quite difficult
to analyse (particularly if colour is unavailable). Throughout the rest of the report the output data will be manipulated
to be easier to read.
From the plot below, it is seen that a train approaching the target will receive an indication of the target 2 km from
the target location. At around 1.8 km the permitted speed displayed to the driver will begin to decrease, instructing
the driver to begin braking. The curves displayed follow the same order as those discussed in Figure 16 previously.
It is also noted that the SBI2 curve cannot be seen since it is fully covered by the FLOI. The results can also be
applied to trains travelling at different initial velocities. For example, a train travelling at 100 km/hr will receive an
indication at around 1 km from the target.
Figure 22: Mathematical model, BrakingCurves.m, output plot.
5.3. Model Validation and Verification
The basic model discussed above provided the framework for all further modelling throughout this project. Thus it
was of paramount importance to ensure that the model was valid.
Since the model was based on the equations defined by the ERA it could be assumed that the model was largely
valid, so long as these equations were used correctly. These were the same equations that will go into the actual
equipment on-board the trains, so must be valid for this system. Furthermore, the equations for the SBD and EBD
stem from the general equation of acceleration, an equation that has been proven valid through daily use over
hundreds of years.
A useful tool for verifying the functionality of the model was the ERA Braking Curves Tool (V3.0) [17]. As mentioned
in the literature review, this is a Microsoft Excel based platform designed to calculate the braking curves in the
same way as the model described above. The tool allows for similar input data to be entered, including the initial
train speed, target speed, the rolling stock correction factors and the various accelerations, equivalent to the data

31
being input to the MATLAB model. Running the braking curves tool yields the results in Figure 23, which upon
visual inspection can be seen to closely reflect those shown previously in Figure 22.
Furthermore, Table 4 below compares quantitatively the output data from the two models, for a train travelling at
160 km/hr. It can be seen that the two sets of results are identical. The ERA tool has undergone a thorough
validation process and is used daily by professionals in the industry. Therefore, by these results, it can be assumed
that the MATLAB model is valid.
This was further confirmed by repeating the two models under a number of different track and train scenarios. With
each simulation the two models yielded equivalent results.
It is noted that the Excel spreadsheet is in a protected format, the equations used are hidden and cannot be viewed
or adjusted. Therefore, the work done on the MATLAB model can confirm that the Excel model uses the same
equations in a similar way. Thus, the MATLAB model is a more open representation of the braking curves tool,
allowing the user to observe how the equations are actually calculated.
Figure 23: ERA Braking Curves Tool output plot, using the same data as for the MATLAB model.
Table 4: Comparison of ERA Braking Curves Tool and MATLAB model results.
Model
Distance from target (m) at 160 km/hr
Indication Permitted Warning FLOI EBI EBD SBI1 SBI2 SBD
EXCEL 1998.10 1775.88 1686.99 1598.10 1442.55 1286.99 1248.31 1598.10 1092.75
MATLAB 1998.10 1775.89 1686.99 1598.10 1442.55 1286.99 1248.31 1598.10 1092.75
0
20
40
60
80
100
120
140
160
180
200
05001000150020002500
Speed(km/h)
Distance from target (m)
EBD
SBD
EBI
SBI1
SBI2
FLOI
Warning
Permitted
Indication

32
5.4. Extended Mathematical Model
The model seen previously (as with the ERA Braking Curves Tool) was only capable of calculating the braking
curves for a single target at any given time. Therefore, the next stage in the modelling process was to extend the
model so that repeated targets could be monitored, in a similar way to the target monitoring in a train’s on-board
equipment. The extended model, named Integrated_Model_1.m, used most of the groundwork set out by the
mathematical model used previously. The code developed for this model is displayed in Appendix 9.4.2 and a flow
chart is presented in Figure 24 below. The same assumptions were made as in section 5.2.1, only for this model
the train was allowed to accelerate if the most restrictive speed profile was to increase.
The model was designed to mimic the functionality of the train’s on-board equipment. As it travels along the railway,
a train would move from one block section to another, passing over a balise which transfers a data package to the
train. This package contains the track data for the block section, and was represented by the import of the track
data Excel file to the model. The file provided the train with a list of track features, such as the most restrictive
speed profile, from which the train could generate a set of targets. Then, using these targets, the on-board
equipment would calculate the appropriate speed profile for the train, presenting the information on the DMI.
The main difference between the train and the model presented here is that the train is travelling from block to
block, whereas this model considered only one block at a time. The train would also continuously monitor its targets
and update the braking distances relative to the track conditions, whereas in this model the profiles were all
calculated at the start of the block section.
The model effectively reviewed the track data input and identified any targets (mainly changes in the VMRSP). Then,
the braking profile for each target was generated, using the same principles as in the model discussed previously.
Following this, the minimum of each profile was calculated along the length of the block section.
This model was also developed to take Temporary Speed Restrictions (TSRs) into account. A TSR may occur
where a section of the railway has experienced minor damage, or where engineering works are being carried out
on the railway nearby. The ERTMS system would allow for TSRs to be set remotely by the signalling centre,
informing the train’s on-board equipment over the GSM-R network. For this model, a section was included for TSRs
to be defined, taking into account their location and the speed values. These were then seen by the model as
additional targets.
For improved clarity, the model script was further broken down into auxiliary scripts; supervision_limits.m and
BrakingCurvesModel_V2.m as seen in Appendices 9.4.3 and 9.4.4 respectively. These were taken directly from
the BrakingCurves.m model discussed previously but were broken down for increased flexibility when transitioning
between models.
Previously, the braking curves were developed as distance values as functions of train velocity. However, for further
analysis it was more useful to have velocity data in terms of distance from the curve. i.e. the previous output data
described the location that the train driver had to start braking, relative to the speed of the train. For this model, the
acceptable speed of the train relative to its distance from the target was required.
To achieve this mathematically would have presented a number of issues, largely since the equations for EBD and
SBD would be of a higher order in terms of velocity. Therefore, as a solution, an additional script was developed,
Interpolate_1.m, as seen in Appendix 9.4.5. This script applied MATLAB’s interp1 function to each of the braking
curves (considered over the range of velocities between 0 and the maximum velocity stated in the train input data).
In doing so, the datasets were effectively converted to velocities as functions of displacement which could then be
stored in an array acting as a lookup table.

33
Figure 24: : Extended mathematical model, Integrated_Model_1.m, flow diagram.
Initialise
Run
Fixed_Value_Data Fixed_Value_Data
Run
National_Value_Data
National_Value_Data
Define TrackData
and TrainData files
Run
ReadTrackData
ReadTrackData
TrackData.xlsx
TrainData.xlsx
Define Other Constants:
V_initial, V_delta0,
A_est1, A_est2, T_bs1, T_bs2,
T_traction, T_berem,
T_indication, V_delta1, V_delta2
Fixed Value Data
National Value Data
Track and
Train Data
Main Script
Auxiliary Script
Microsoft Excel File
Determine Temporary Speed
Restrictions (if any) and their
locations.
Calculate Target
Locations
Run
supervision_limits
supervision_limits
Run
BrakingCurvesModel_V2
National_Value_Data
Ceiling supervision
limits
Generalised braking
curves Run
Interpolate_1
Interpolate_1
Calculate braking
profiles
Speed Dependent
Data
Repeat for
each target
Plot Braking Profiles
Calculate Minimum
Profiles

34
5.4.1. Target Identification
As described previously, the model used the BrakingCurvesModel_V2.m and Interpolate_1.m files to generate a
standard set of braking curves for the block section. This assumed that the gradient in that block section remained
constant. These curves were then applied to each target in turn, allowing the braking profiles for each target to be
calculated as follows:
First, the initial and target supervision limits were calculated from the ceiling supervision limits, which had been
calculated for the length of the block section by the supervision_limits.m file. The curve type was also identified,
being an EOA if the target speed was zero, or a LOA otherwise. These would be used later in the model to
determine which type of curves to produce.
The target type was then considered; if this was an increase in speed the target was effectively ignored and the
profiles were set to their respective ceiling supervision limits. If this was the case, the EBD, SBD, SBI1 and I profiles
were all set to the maximum velocity in the train input data. This was to effectively remove them from the output
plot since they were only required in areas where the train was braking.
Conversely, if the target was a decrease in speed further analysis was carried out. First, the theoretical location of
the target was found (as the location where the EBD curve reaches 0 km/hr); if the target was an EOA the theoretical
target was simply at the target location. However, if the target was a LOA, the theoretical target was beyond the
actual target location, as in Figure 25.
Figure 25: Theoretical target location for a LOA curve [24].
For a LOA, the EBD curve was defined as that which crossed through the EBI ceiling supervision limit (for beyond
the target) at the target location. Therefore, the distance to the theoretical target location could be calculated as
the distance to the actual target location, d1, plus the braking distance of the EBD between Vebi and 0 km/hr, d2.
Following this, and still for the particular target, the braking distance was found from the indication curve, I, using
the initial speed as the reference point. The braking location was then also found (the location where the driver
would first have to take action) by subtracting the braking distance from the theoretical target location, as in Figure
26 below.
The model used the braking point and the theoretical target as limits when calculating the braking curves. For
example, before the braking point the curves were all set to their respective ceiling limits relative to the VMRSP before
the target. Similarly, after the theoretical target was passed the curves would all be set to their respective ceiling
limits relative to the VMRSP beyond the target. For values between the braking point and the theoretical target the
curves would be calculated.
Theoretical target locationd1 d2Train

35
Figure 26: Calculation of braking point.
5.4.2. Braking Curve Calculation
The code to generate the braking profiles for each target was originally written in the main body of the script.
However, with further development of additional models this section was extracted and made into a new auxiliary
script, named gen_target_profiles.m (see Appendix 9.4.6). This way, it was ensured that any adjustments made to
the code was applied across all models using it.
For each target, the model considered each value of distance along the length of the block section. For the values
of distance within the braking area, the model would assess the value of each of the braking or supervision velocity
curves at each value of distance relative to the individual target. The step size used when assessing along the
length of the railway was defined by the track input data, for this model a value of 10 m was used. For example, in
TrackData3.xlsx (Appendix 9.2.3), the first target was a speed decrease from 160 km/hr to 100 km/hr occurring at
2 km from the start of the block section. The model calculated the various braking profiles at 10 m steps up to and
beyond this target (before repeating for the second target which occurs at 4 km).
During this process, the VEBD value for each target at each value of distance was calculated using the line:
V_EBD1 = V_EBD(EBD_d ==...
min(max(EBD_d),round(d_target_theory(target_number) - d_line(d))));
This set the EBD curve equal to the value of the generic curve created by the Interpolate_1.m file seen previously,
relative to the target. If the distance to the target was greater than the length of the EBD reference array (EBD_d),
the VEBD value was set to the maximum reference value. Figure 27 below provides a visual understanding of this.
Figure 27: Calculation of V_EBD1.
Theoretical target
location
Actual target
location
Braking
Point
Indicatio
n
Curve, I
EBD
EBI Ceiling Supervision Limit
Permitted Speed
Braking Distance
Speed
DistanceTrain
d_target_theory
Actual target
location
EBD
EBI Ceiling Supervision Limit
Permitted Speed
max(EBD_d)
Speed
DistanceTrain
d_line →
V_EBD1
d_target_theory – d_line

36
Furthermore, if VEBD was calculated to be less than the permitted velocity (i.e. after the target has been passed)
the VEBD from this location onwards was set to the maximum velocity.
The values of VEBI, VSBI2, VP, and VW were calculated in a similar manner, but also observed the constraints of the
ceiling supervision limits before and after the target location, relative to the curve type. The VSBD and VSBI1 curves
were also calculated in the same way but only if the target was an EOA, according to the SRS. If the target was a
LOA, these were set to the maximum reference velocity as previously discussed.
The indication curve, VI was also calculated by the same method. This curve, however, was set to the maximum
reference velocity before the braking area and after the target location had been passed, to simulate the switching
on and off on the indication light on the DMI.
Finally, once each of the profiles were generated, the model would find the minimum of each profile. For example,
Figure 28.a. shows the VEBD traces for each of the targets in TrackData3.xlsx alongside the VMRSP, while Figure
28.b. presents these on a single plot. It is worth noting that the horizontal and vertical sections of the EBD curves
would not be included in real on-board systems, but are included here to simplify the computation of the curves.
Through calculating these curves in this way it can be ensured that any overlapping curves can be accounted for.
Figure 28: a) The individual EBD curves and b) The combined EBD curve.
5.4.3. Extended Model Output
As with the previous model, the extended model would produce output plots of the braking and supervision profiles,
as in Figure 29. However, these plots displayed the data over the length of the block section, rather than the
distance to target as seen previously (In the real system, the distance to target would be found and displayed to
the driver by subtracting the train’s location within the block section from the target location. Also, where the ceiling
supervision limits were previously overlaid onto the plot, they are now part of the profile datasets. Thus the plot is
much more clear than that seen previously.
a.
b.

Modelling the European Train Control System: Braking and Supervision Curves

Modelling the European Train Control System: Braking and Supervision Curves

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