MLD-based Optimal Control Model for Train Traffic Control

Honggang Wang

1, a

, Qingru Zou

1, b

College of Traffic and Transportation, Chongqing Jiaotong University 400074, China

Keywords: Train Traffic Control, Mixed Logical Dynamic, Model Predictive Control.

Abstract: Trains are often delayed because of the unexpected events. How to eliminate trains delay is the key function

of the train traffic control system. For this problem, an optimal control model is presented in this paper based

on the mixed logical dynamic and the model predicative control theory under the situation that trains operation

are not disturbed seriously. The process of the train movement is studied and it is pointed out that the train

movement is a hybrid dynamic system because the evolution of the train operation status is driven by the

continuous part following the Newton’s laws of motion and lots of discrete events including the expected

events and unexpected ones. Based on the results, the train operation model is built using the mixed logical

dynamic theory without considering the junctions and assuming that the disturbances imposed on trains are

not serious. Based on the train operation model, the optimal control model for train traffic control is studied

using the model predictive control theory where the train operation model is used to predict the future status

of trains. Finally, the simulation shows that the model’s validity and correctness.

1 INTRODUCTION

Railway transport system is a dynamic system that

trains operate in an environment with many uncertain

factors such as natural environment and devices fault

etc. To guarantee trains safe and provide customers

good service, all trains must be monitored and

controlled by dispatchers. Generally, this task’s

implementations include 1) off-line scheduling when

all the train arrival times and departure times are

calculated before the train starts. The trains behave

exactly as they were planned. No unexpected event

happens and no new train can appear. 2) on-line

scheduling when the scheduling is performed during

the train traffic operation. Some trains have variable

delays, unexpected events happen, and new train

scheduling requests are required and accepted during

the operation.

For the on-line scheduling, these uncertain factors

usually lead to the train delay even stop in some

railway line. Because of the characteristics of the

railway transportation that the driver cannot change

the train routs at will, these uncertain factors result in

conflictions at junctions and trains delay which can

be propagated rapidly along the railway line. To

ensure trains safety, punctuality and provide better

services to passengers, railway transport departments

must control all trains. So it is becoming more and

more important daily task for the relative railway

transport departments to guaranty that all trains

behave according to the time table, especially with the

improving of trains operation speed and density in

recent years. In fact, how to guaranty the safety and

punctuality has been an important problem studied in

academic circle and railway industry since railway

transportation came into being. (Sundaravalli

Narayanaswami, Narayan Rangaraj, 2012) defined

the train traffic control as scheduling and

rescheduling. They defined the scheduling and

rescheduling as “an initial time allocation of

resources to meet demands in completing a task and

rescheduling is a later modification of such resource

allocations”. We prefer to refer the rescheduling as

train traffic control.

In reality, the uncertain factors can be classed into

two categories according to its seriousness. One is

very serious, for example, a strong earthquake and

railway line broken etc, which can force all trains to

stop. Another is not serious, for example, heavy rain,

strong wind etc, which can result in not all trains

stopping but conflictions and train delay. For the first

one, dispatchers often use the pre-defined scheme to

reduce the economic loss and ensure passengers safe.

However, when the second factors happen,

dispatchers usually change train route to solve

conflicts and determine a new timetable based on the

Wang, H. and Zou, Q.

MLD-based Optimal Control Model for Train Trafﬁc Control.

DOI: 10.5220/0010021800460053

In Proceedings of the International Symposium on Frontiers of Intelligent Transport System (FITS 2020), pages 46-53

ISBN: 978-989-758-465-7

original timetable according to real situations. The

aim is to restore the train operation order and ensure

the trains safety and punctuality finally. For the

adjustment of the train timetable problem, (Jih-Wen

Sheu, Wei-Song Lin, 2012) developed and evaluated

the Adaptive Optimal Control (AOC) method and the

evaluation shows that the AOC method is able to find

a near-optimal solution rapidly and accurately. For

conflicts and train delays considerably in time and

space during trains operation, in order to realistically

forecast and minimize delay propagation, (Andrea

D’Ariano and Marco Pranzo, 2008) decompose a

long time horizon into tractable intervals and use the

ROMA dispatching system to pro-actively detect and

globally solve conflicts on each time interval.

(Joaquín Rodriguez, 2007) presented a constraint

programming model for the routing and scheduling of

trains running through a junction. The model can be

integrated into a decision support system used by

operators who make decisions to change train routes

or orders to avoid conflicts and delays and has been

applied to a set of problem instances. Preliminary

results show that the solution identified by the model

yields a significant improvement in performance

within an acceptable computation time. (Tiberiu

Letia, Mihai Hulea and Radu Miron, 2008) presented

a distributed method to schedule new trains where the

paths containing the block sections from one station

to another are dynamically allocated without leading

to deadlocks. In addition, some researchers made lots

of studies for the seriously disturbed railway

transportation (Francesco Corman, Andrea D'Ariano,

Ingo A. Hansen etc, 2011). In order to save energy,

the reference (Shigeto HIRAGURI, YujiHIRAO,

IkuoWATANABE etc. 2004) presents a train control

method based on trains movement and data

communication between railway sub-systems, which

can decrease some unexpected deceleration or stop

between stations.

The paper mainly focuses on the problem how to

restore the train operation order and reduce trains

delay. It is organized as: the Mixed Logic Dynamic

(MLD) is described briefly in section 2; In section 3,

the reason that train traffic control is a dynamic

system is given, and the MLD model for train traffic

control is given, which is used to describe the train

dynamic movement; In section 4, the optimal control

model for train traffic control is established based on

the model predictive control theory; In section 5, a

simulation result is given which shows the validity of

the model presented in this paper. In the last part, we

will point out the further studies.

2 MIXED LOGIC DYNAMIC

SYSTEM

In industries, there exist many systems having the

same characters. The systems are comprised of two

parts, the continuous part following the physical laws

and the discrete part. The two parts work together and

drive these systems evolution. These systems are

referred to Hybrid System (HS) or Hybrid Dynamic

System (HDS) which is attracting many researchers

interests (Y.Bavafa-Toosi, Christoph Blendinger,

Volker Mehrmann etc, 2008, M.J. Dorfman, J.

Medanic, 2004). In past, the continuous part and the

discrete part were studied separately. For example,

the discrete one is described using Petri Net,

Automata and Finite State Machine etc, meanwhile

the continuous part using difference equation.

However it is difficult to analysis the HS performance

if these two interactive parts are concerned

separately. In order to take the two parts into account

as a whole system, (Alberto Bemporad, Manfred

Morari, 1999) presented a framework for modeling

and controlling systems described by interdependent

physical laws, logic rules, and operating constraints.

This framework was denoted as mixed logical

dynamical (MLD) systems. Since the presence of the

MLD concept, the theory and its application were

studied by many researchers (Martin W. Braun and

Joanna Shear, 2010, Kazuaki Hirana, Tatsuya Suzuki

and Shigeru Okuma, 2002, Akira Kojima and Go

Tanaka, 2006). Here are given the procedures of

modeling a system in short, the detailed information

on MLD theory can be found in conference (Alberto

Bemporad, Manfred Morari, 1999) .

Step 1: Building the system’s mathematical model

following physical law under different situation

through analyzing the system.

Step 2: Building the logical proposition according

to the qualitative knowledge and constraints existed

in the model and transferring the logical proposition

into linear inequalities by introducing some logical

variables.

Step 3: Introducing some auxiliary variables

according to the relations between discrete events and

continuous variables and getting other inequalities.

Through combining the results from step 2 and the

results from step 3, a MLD model can be established.

Generally, the formulation of MLD model has the

style as below.

MLD-based Optimal Control Model for Train Trafﬁc Control

12 3

23145

(1) () () () ()

() () () () ()

() () () ()

t AxtButBtBzt

yt Cxt Dut D t Dzt

Et Ezt Eut Ext E



   

  













Where

()

is the state variable,

()yt

the output

variable,

()ut

the input control variable,

()t



the

logical variable,

()zt

the auxiliary continuous

variable and

(1,2,3)

Bi

(1,2,3)

Di

( 1,2,3,4,5)

Ei

are the constant coefficient

matrixes.

3 MLD MODEL FOR TRAIN

TRAFFIC CONTROL

3.1 Analysis of the Train Traffic

Control

The train motion must follow the physical law,

000 0

() () ()( ) ( )/2st st vt t t at t 

()

stands

for the train position at time

()vt

the train velocity

at time

is the acceleration. However, train

operation process must obey other rules which are

referred as discrete events, for example, trains must

decelerate when approaching a station or catching up

a strong cross wind, trains must change their

velocities when receiving dispatch information from

dispatchers. Obviously, the status evolution of trains

is driven by two parts, one is the continuous part

following physical law, another one is the discrete

events. So the train traffic control system is a typical

hybrid dynamic system.

In reality, the events imposed on trains can be

classed as two categories according to their

seriousness. One is serious, which forces trains to

stop for safety. Another is not serious, which only

forces trains to decelerate for safety. For the first one,

it’s difficult to build a mathematic model to describe

the train traffic control. Actually, what dispatchers

can do is to use predefined scheme to ensure the train

safety when the first one happens. For the second,

dispatchers can adjust the timetable to restore the

train operation order according to the real time

situation. Only the second events are considered in

this paper.

Under the normal situation, according to the train

operation plan, each train leaves a station, accelerates

its speed to the maximum speed, decelerates its speed

when approaching the next station and stops or goes

through the station. Each train repeats these

procedures until it arrives at the terminal station. For

this situation, dispatchers only monitor all trains and

hardly control all trains. Under the abnormal

situation, i.e. train operation process is affected by

some events such as cross winds, conflictions at

junctions etc, trains must decelerate to some speed for

safety and then result in train delay. For this situation,

dispatchers must take measures to restore train

operation order and ensure the train safety. It is the

main task for dispatchers to make a new timetable (or

adjustment plan) to minimize or eliminate the train

delay in short time as soon as possible.

In addition, the discrete events imposed on trains

can be classified as two types: the predictive one and

the un-predictive one. The predictive events are the

events whose emerging time can be predictive or

calculated in advance such as trains entering station

event, trains leaving station events and dispatching

events etc. On the contrary, the un-predictive events

are the events whose emerging time cannot be

predictive or calculated in advance such as equipment

fault, strong wind and heavy rain etc.

Based on this analysis, we can draw a conclusion

that the essence of train traffic control is to minimize

or eliminate train delay resulting from the un-

predictive events through the predictive events.

3.2 MLD Model for Train Traffic

Control

(1) The Status Space Equation

According to the physical law, the status space

equation for train

(1) () ()

(1) () () ()/2

iii

iiii

vk vk akT

sk sk vkT akT

 

  







(1)

Where

is the sampling time interval,

()

the

acceleration of train

at time

()

the velocity

of train

at time

()

the position of train

time

The equation (1) can describe the train operation

without any disturbances, but it is not suitable for

train operation with disturbances. Let

() {0,1}

rk 

stand for the event imposed on the train

at time

() 1



means that the event happens, which

forces the train

to slow down its speed to a security

FITS 2020 - International Symposium on Frontiers of Intelligent Transport System

speed

. Using the logical variable

()

, the status

space equation (1) can be re-written as

( 1) ( ( ) ( ) )(1 ( )) ( )

(1) () () ()/2

iiiiig

iiii

vk vk akT rk rkv

sk sk vkT akT

   

  







(2)

The equation (2) can describe the train operation

with disturbances, however it does not include the

information on train entering station, train leaving

station and dispatching event.

Let

()



()



and

()



stand for the

entering station event, the leaving station event and

the running event at a constant speed separately,

where

() {0,1}





1, 2, 3j 

() 1





means

that the entering station event happens at time

otherwise the event doesn’t happen at time

. The

()



and

()



have the same meaning as the

()



. Let

()



()



and

()



stand for the

accelerating command event, the decelerating

command event and the uniform command event

separately, where

() {0,1}





1, 2, 3j 

() 1





means that dispatchers send an

accelerating command to the train

at time

otherwise means that dispatchers don’t send an

accelerating command to the train

at time

. The

()



and

()



are similar to the

()



. The

acceleration value of train

at time

()

described by these logic variables as below:

( ) ( ) 1

( ) 0 ( ) ( ) 1

( ) ( ) 1

iNA

akk

ak k k

ak k



















(3)

Where the values of

()



(

1, 2, 3j 

) are

determined by the onboard train control equipment

and track side signal equipment. The values of

()



(

1, 2, 3j 

) are determined by dispatchers according

to the real time situation. The variable

(

0a 

)

stands for the acceleration performance of trains.

()

ak a

means that train

is accelerating,

() 0



means that train

is running at a constant

speed and

()

ak a





means that train

decelerating.

Generally, the predictive event

()



and the

dispatch event

()



do not happen simultaneously

because a dispatcher cannot send an accelerating

command when he knows that the train is

accelerating. So, these logic variables

()



and

()



must satisfy the below constraint.

(() ())1

Nj Aj











(4)

According to the equation (4), the equation (3) can

be formulated as below.









11 22

() () () () ()

ii i i

iNA NA

ak k k a k k a

 

 

(5)

We substitute

()

in status space (2) with

equation (5) and get the status space equation as

below.

11 22

( 1) ( ) [( ( ) ( )) ( ( ) ( ))]

1() ()

(1) () ()

[( () ()) ( () ())] /2

ii i i

iiNANA

iig

iii

ii i i

NA N A

vk vk k k k k aT

rk rkv

sk sk vkT

kk kkaT

 

    



  

 











{}

(- )

(6)

(2) Constraints

According to the railway transport rules, the

departure time for a passenger train must be less than

the planned departure time, i.e. if train

is at the

station

, the departure time

must satisfy

,() 1

ir i r N A

kT t if s k st and





(7)

Where

denotes the departure time of train

station

denotes the position of station

in the

railway line,



is a minimal positive real.

The condition (7) implies

() 1

ir NA ir

kst kTt

 



   

(8)

MLD-based Optimal Control Model for Train Trafﬁc Control

According to the MLD modeling theory, it can be

transferred to a inequality constraint form.

Let



be the minimal dwell time for train

station

, then the real time interval for train

station

must satisfy

ira ir

kT k T





(9)

Figure 1: Architecture of Optimization Control for Train

Traffic Control.

Where

ira

is the real arriving time for train

station

4 OPTIMAL CONTROL MODEL

FOR TRAIN TRAFFIC

CONTROL

The above model based on MLD theory can be used

to describe the process of train traffic control,

involving the trains running process, some uncertain

factors and the dispatcher’s activities. However, this

model cannot be used to optimize the train traffic

control problem. To get an optimal or sub-optimal

schedule solution, the Model Predictive Control

(MPC) is used. The architecture is illustrated as Fig.1

The Train Group module stands for the practical

trains, MLD module is the model for train traffic

control which is built above and the Controller is an

optimization controller. The Train Group module

outputs trains real trajectories, and the MLD module

trains future trajectories in some time interval. The

real trajectories and the future trajectories are fed

backed to the Controller module. The Controller

module will make an optimal solution or sub-optimal

solution according to the time table, the real

information from the Train Group module and the

predictive information from the MLD module.

4.1 Performance of the Optimization

There are many performances of optimization for

train traffic control, such as to minimize traffic delay,

to maximize traffic system throughput to fulfill train

timing requirements and to guaranty system safety.

The minimization traffic delay is used in this paper.

Let

aij

stands for the real arriving time of train

at station

dij

the real departure time of train

station

; Let

aij

and

dij

stands for the planned

arriving time and the planned departure time

respectively. The performance can be described as

below:

*2 *2

min ( ) ( )

aij aij dij dij

tt tt







4.2 Optimal Control Model for Train

Traffic Control

Based on the performance and the MLD model, the

optimal control model for train traffic control is

described as below:

*2 *2

min ( ) ( )

aijaij dijdij

tt tt







11 2 2

11 2

( 1) ( ) [( ( ) ( )) ( ( ) ( ))]

1() ()

(1) () ()[(() ())(()

( ))] / 2

,() 1

()

ii i i

iiNANA

iig

ii i

iiiNAN

ir i r N A

ir NA

vk vk k k k k aT

rk rkv

sk sk vkT k k k

kaT

kT t if s k st and

sk st

 

 





 

    



     















{}

(- )

ira ir

kT t

kT k T



 



5 SIMULATION

Taking a dispatch section including six stations and

5sections as an example, the railway line as below:

S1 S6S5S4S3S2

2000

4000

6000

8000

10000

q2 q3 q4 q5

Figure 2: Diagram of simulation railway line.

Where Sii  1,2, … ,6 Stands for stations, the

number (unit: meter) under the station is the station’s

position in the railway line, and qii  1,2, … ,5

stands for the corresponding section.

FITS 2020 - International Symposium on Frontiers of Intelligent Transport System

The parameters about trains are listed in the Table 1.

Table 1: Parameters of trains.

Parameters Value

Traction Acceleration

/ms

)

Breaking Acceleration

-2(

/ms

)

Normal speed in section

40(

/ms

)

limit speed in section

50(

/ms

)

Safe speed

20(

/ms

)

The length of station

50(

)

The minimal dwell time at station S2, S3, S4 and

S5 are listed in the Table 2 (Unit: second)

Table 2: The minimal dwell time at station.

Train ID S2 S3 S4 S5

H2 15 15 0 20

H4 15 15 0 20

H6 25 15 25 15

H8 10 15 15 15

H10 10 20 20 15

H12 15 15 15 20

The train operation plan is as figure 3.

During simulation, the Branch and Bound method

is used to solve the optimal control model. At last, the

adjusting process is illustrated as Fig 4 and the Fig.5

shows the real train operation situation.

Figure 3: Diagram of train operation plan.

Figure 4: Diagram of adjusting process.

MLD-based Optimal Control Model for Train Trafﬁc Control

Figure 5: Diagram of train adjustment results.

6 CONCLUSION

The essence of the train traffic control is to eliminate

or reduce the trains delays resulting from some

unexpected factors under the train safe, to ensure

trains behave according to train operation plan. Based

on this idea, the MLD model and optimal control

model are built in this paper taking in to account some

constraints such as dwell time, safe distance etc.

Finally, taking a dispatch section including six

stations and five sections as an example, the models

are verified on computer and the results shows it’s

validity and correctness.

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MLD-based Optimal Control Model for Train Trafﬁc Control