Signalling blocklayout and strategy of trainoperation for saving energy in mass rapid transitsystems

B.R. Ke and N. Chen

Abstract: An approach for assisting the design of fixed-block signalling system of mass rapidtransit (MRT) systems by optimising the block layout and running speed code of each signallingblock between any two neighbouring stations is presented. The object is to achieve minimumenergy consumption with maximum train capacity. The apporach preadjusts the running speedrestriction of trains to avoid overspeeding on an alignment. At the same time it can also shave-offunnecessary speed changes to improve passenger comfort. The concept of equiblock signalling isused to assist this research. Considering the effect of gradients and the limits of minimum headwayof an ATO system and average train speed, a genetic algorithm is applied to determine the shortestlength, speed codes and positions of signalling blocks. Single train running dynamics betweenstations is simulated based on the signalling system of an MRT system with various aspects,positions and numbers of signalling blocks through acceleration and jerk limits. An heuristic searchis used to seek out the combination of train running speed code of each signalling block for energysaving.

List of symbols

as normal accelerationb constant to determine reduce

degree of d(t)bs normal decelerationE train energy consumption run-

ning between two stationsEn train energy consumption during

nth signalling blockEmin lowest train energy consumption

running between two stationsfi fitness value of ith geneF tractive effortg gradientgav,n average gradient of nth signalling

blockH practical headwayHspec minimum headway of specifica-

tionJ jerk restrictionli,j+1 braking distance from speed

code Vi to Vi+1

lij;jþ1 braking distance from speed

code Vi to Vi+1 at ith genelremainder remainder length after deducting

essential length of signallingblocks from distance betweentwo stations

L distance between stations

Lessential essential braking distance be-tween any neighbouring speedcodes

Ln length of nth signalling blockLi

n ending position of nth signallingblock of ith gene

N number of aspectspop population sizeP tractive powerr random number between �1

and 1R train running resistanceRindex equi-speed code with identical

speed code for each signallingblock to be closest to limits ofheadway and average speed butnot to exceed

Rindex_to_lower search range of individual blockfrom equispeed-code index tolower speed code

Rindex_to_upper search range of individual blockfrom equispeed-code indexupper speed code

Rlower lower limit of possible searchrange of speed code

Rupper upper limit possible search rangeof speed code

SB number of signalling blockstr response delay of ATO systemT evolved generationTmax maximum evolved generationv train running speedvan nth speed restriction of align-

mentvn train running speed code of nth

signalling blockVav practical average speedVav,spec minimum average speed of spe-

cification

The authors are with the Department of Electrical Engineering, NationalTaiwan University of Science and Technology, Taipei, Taiwan, Republic ofChina

r IEE, 2005

IEE Proceedings online no. 20045188

doi:10.1049/ip-epa:20045188

Paper first received 28th June and in revised form 14th December 2004

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 129

Vi ith speed code

V ij ; V

i0j jth speed code of ith gene before

and after crossover is performedW train weight

xij; x

i0j jth chromosome of ith gene

before and after crossover andmutation operation is executed

xi ith gene in population

xiUj upper limit of jth chromosome

of ith genea random number with range in

(�1, 1)d(T) formula for producing a random

number between �1 and 1;moreover, its value will gradu-ally be reduced when number ofevolved generation increases

Dt snapshot timeei

j jth chromosome of ith gene for

distributing proportion

1 Introduction

The first task is to consider the transportation capacity thataffects the specifications of a railway system. For designinga signalling system for mass rapid transit (MRT) systems,the minimum headway must be computed after the numberof passengers per hour and the kinds of vehicles aredetermined. Moreover, for a fixed-block signalling (FBS)system, aspects and speed codes will be devised to fit in withthe minimum headway and average train speed, andfurther, to obtain the proper positions of blocks. The finaltask is to perform operational tests and make any necessaryadjustments.

Gill and Goodman [1] proposed a computer-basedoptimisation technique for designing signalling systems ofMRT systems based on an equiblock principle. Using aformula for safe braking distance the essential and shortestblock length is confirmed. Chang and Du [2] applied agenetic algorithm (GA) in place of conventional methods toimprove an existing block design for equiblock signalling inan MRT system. Their primary objectives were to minimisethe headway between two neighbouring stations as far aspossible. Nevertheless, the time difference between thedesignated and practical minimum headway can be utilisedto reduce energy consumption by selecting the proper blocklayout and running-speed code at each block on thejourney.

How to minimise the headway has been discussed in [2],but the main object of this paper is to address thedesignated minimum headway for saving energy ratherthan simply the minimum headway. The designatedminimum headway can simultaneously satisfy requirementsof construction specification under normal conditions witha margin in minimum headway for service recovery. Thereare ways to reduce energy consumption of trains, forexample, by decreasing the maximum running speed andrunning the train under coasting mode [3–7]; however, theirobject is to minimise energy consumption under normalconditions rather than during peak train capacity. In thispaper the block-layout of signalling systems is designed andthe running speed code of each block determined tominimise energy consumption and satisfy the limitations ofminimum headway and average train speed when the MRTsystem operates during peak train capacity. Taking accountof the effects of signalling systems and gradients, the block-

layout of signalling systems consuming minimum energycan be devised using the least signalling blocks bycomparing energy consumption with various numbers ofblock sections.

This research study presents a new approach fordesigning signalling systems for MRT systems whileconsiders the effects of gradient between two neighbouringstations to save energy consumption of trains. Consideringthe minimum headway and average train speed, a geneticalgorithm is used to determine the shortest length, speedcodes and positions of signalling blocks. Furthermore, anheuristic search is used to seek out the combination of trainrunning speed code at each signalling block for energysaving.

For safety and equipment cost, the length of signallingblocks must be greater than the safe braking distance.Safety is the principal consideration for designing asignalling system. Even under normal conditions, it muststill have a sufficient braking distance. The designconsiderations are according to the requirements of theautomatic train operation (ATO) system and safe brakingdistance. According to the effect of gradient and therestriction of headway the length can be regulated. First, forconsistency of speed codes and simplifying the calculationof braking distances this paper creates the shortest lengthand speed codes of each signalling block by the principle ofequiblock signalling and the GA method. Secondly, thispaper regulates the running speed limit of trains in advanceto avoid overspeeding on an alignment; moreover, it shavesoff unnecessary speed change to improve passengercomfort. Thirdly, according to the given number of aspectsand signalling blocks, the positions of signalling blocks aredetermined by using a GA. And then according to theseresults, the running speed code of signalling blocks forenergy saving can be found for all possible positions ofsignalling blocks through an heuristic search. Comparingthe energy consumption for different positions of signallingblocks, the lowest one is the solution.

2 Mathematical modelling

For a FBS system the alignment parameters of a railway,such as gradient and curvature, the performance parametersof vehicles, such as acceleration/deceleration, traction force,braking force and jerk, the design parameters of thesignalling system, such as aspects, block positions and speedcodes, as well as the operation parameters of the systemsuch as headway, the number of passengers, and the time ofstation dwell, will influence the operation. It is toocomplicated to use formulations for calculating thedynamics of the railway system; therefore a computer isused to simulate the running result of vehicles in most ofthis research area. Based on a railway signalling system withdifferent aspects and speed codes, the train dynamics ofrunning between stations is simulated on a computer in thispaper. The result simulated is to select the aspects, speedcodes and block positions.

To meet the requirements of minimum headway andaverage train speed and to save energy consumption oftrains, the paper proposes a procedure to complete thedesign of block-layout of FBS systems and to resolve therunning speed code of each signalling block between twoneighbouring stations. The flowchart of this procedure isshown as Fig. 1 and explained as follows. First, the GA isused to solve the shortest length of signalling blocks andspeed codes for an equiblock signalling system. Secondly,unnecessary speed changes are avoided by adjusting linespeed for passenger comfort and energy saving. Thirdly,

130 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005

after inputting the number of aspects and signalling blocks,the GA is used again to determine the positions of signallingblocks. And then the heuristic search is used to decidethe optimal combination of running speed code ofeach signalling block for energy saving. The optimal results,including the optimal combination of running speedcode and position of each signalling block, can bedetermined by comparing the results of evolution repeat-edly. Finally, to compare the results of different numbers ofaspects and signalling blocks the optimal combination ofrunning speed code and position of each signalling blockcan be confirmed for the energy saving and restrictions ofminimum headway and average train speed between twoneighbouring stations.

Equation (1) shows the requirements of tractive effort torun at any location of a railway system. A moving trainmust overcome the resistance concerned with gradient, trainspeed and weight. The resistance of curvature and air is notlarge enough to affect the result for rapid transit systems.This paper adopts the formula of the Taipei Rapid TransitSystem. Regarding the power cost, the peak train power isconsidered during optimisation, but its influence on theresults is not obvious. The main factors affecting peak trainpower are the alignment, weight and acceleration. In thispaper the alignment and train weight one considered; alsothe train acceleration control is by PID control to meet thelimits of train velocity, acceleration and jerk. In (2) theconsumed power relates to the train speed as well; (3) showsthat all energy consumed during a journey is the sum of allthe products of snapshot time and its consumed power.

F ðg; vÞ ¼ as;W þ Rðg; vÞ ð1ÞP ðg; vÞ ¼ F ðg; vÞv ð2Þ

Eðg; vÞ ¼X

Pðg; vÞDt ð3Þ

The object of this paper is to minimise the energyconsumption for FBS transit railway systems. The majorfactors affecting energy consumption are the alignmentgradient and train running speed; therefore selectingdifferent signalling block-layouts and running speed codeswill reach a different result. But the requirements ofminimum headway and average train speed are influencedby the signalling block-layout and running speed code. Theheadway will drop when the train running speed increases,but the consumed energy and average speed will increase. Inaddition, the signalling block-layout also affects them.Therefore the objective function and constraint conditionscan be shown as follows:

objective function

Emin ¼ minXN

n¼1EnðLn; gav;n; vnÞ ð4Þ

constraint conditions

H � Hspec ð5Þ

Vav � Vav;spec ð6ÞAccording to the construction specifications of the TaipeiMass Transit System, Hspec and Vav,spec are two min and37km/h, respectively. The design procedure is discussed inthe following Sections.

3 Calculating shortest length of signalling blocksand speed codes in FBS systems

The concepts of equiblock signalling of an FBS systemmeans that they have the same braking distance betweentwo neighbouring speed codes. Its advantage is to reducethe headway to a minimum. However, if all the signallingblocks were designed to have the same shortest length, theequipment cost would be too much; therefore the methodcannot be adopted. This paper determines the block lengthby using the concept of equiblock signalling system andGA. The remainder of deducting the product of the numberof blocks and the shortest block length from the distancebetween two stations is distributed to each block using GA.Consequently the shortest block length and speed codes arecreated.

Gill and Goodman [1] incorporate the response delay ofan ATO system, the jerk limit of the train and the constantacceleration during the ATO-system response delay toformulate the braking distance between two speed codes.This paper uses this formula and GA to create the shortestblock length and speed codes through an individualprogram. The minimum block length is shown by thefollowing formula [1]:

Ii;jþ1 ¼Vi tr þas

Jþ bs

2Jþ astr

bsþ a2

s

2bsJ

� �þ Viþ1

bs

2J

� �þ 1

2bs

ðV 2i � V 2

iþ1Þ þ1

2ast2r þ

astr

Jþ 1

3

a3s

J2

þ 1

2

a2s bs

J2þ asbstr

Jð7Þ

Genetic algorithms differ greatly from the traditionalapproach [8]. For example, genetic algorithms considermany points in the searching space at one time; thereforethey have a better chance of obtaining a global optimalvalue, and of avoiding sinking into local optimal values.Genetic algorithms are suitable for any kind of objectivefunction, and do not have complex mathematical require-

input related systemparameters between two stations

start

setting number of aspects andsignalling blocks

create shortest length andspeed codes of signalling blocks

decide the positions ofblock-layout of signalling

system by using GA

yes

no

end

search optimal combination ofspeed codes for saving energy

by using heuristic search

continue toevolve

regulate the range of line speed

Fig. 1 Flowchart of design of block-layout and its running speedcodes

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 131

ments. They use probability to find the direction of search,and hence are suitable for all kinds of optimisationproblems.

Equations (8) and (9) are expressions of objectivefunction and constraint condition for creating the shortestblock length and its set of speed codes by means of GA; (8)is the objective requirement for finding a set of speed codesto meet the requirement of the shortest braking distance,while (9) is the constraint condition for ensuring that speedcodes {V1, V2, y, VN} are in descending order.

objective requirement

l1;2 ¼ l2;3 ¼ � � � ¼ lN�1;N ð8ÞConstraint condition

V14V24 � � �4VN ð9ÞThe steps of this process to achieve the objective require-ment using GA are explained as follows.

3.1 RepresentationThe representation is performed in real numbers becausethere are too many figures for a binary encoding. Theadvantages of real-number encoding are to save thecalculation time of the computer, to reduce the error oftransformation, and to increase the precision of speedcodes.

3.2 Initialising populationThe population is the space of possible solutions forsearching speed codes of FBS based on equiblock principlewith different number of aspects and maximum runningspeed. Genes in the population indicate candidate combina-tions of speed codes. In a gene each chromosome that isproduced by random number generator to create an initialpopulation exhibits a speed code; therefore its number willincrease or decrease depending on the number of aspects.

Notice should be taken of the order of speed codes with theconstraint condition of (9); if the contents of chromosomesshow that speed codes are out of descending order, theywould be cancelled and reproduced. An N-aspect systemthat produces k genes can be described as follows:

x1 ¼ ½x11; x12; � � � ; x1N � ¼ V 11 ; V

12 ; � � � ; V 1

N

� �x1 ¼ ½x21; x22; . . . ; x2N � ¼ V 2

1 ; V22 ; � � � V 2

N

� �...

xpop ¼ ½xpop1 ; xpop

2 ; � � � ; xpopN � ¼ V pop

1 ; V pop2 ; � � � ; V pop

N

� �ð10Þ

The sequence of genes will be permuted in accordance withtheir fitness values after creating initial population. Theobject is to reach the same braking distance betweenneighbouring speed codes. The fitness of the kth gene withN-aspects is given by

fk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN�1i¼1

XN�1j¼iþ1

lki;jþ1 � lk

j;jþ1

� �2vuut ð11Þ

Table 1: Braking distances under four kind of aspects

No. Ofaspects

Speed codes (km/h) Brakingdistances(m)

1 2 3 4 5 6 7

4 80.00 68.88 52.04 0.00 � � � 165.66

5 80.00 72.82 62.96 47.81 0.00 � � 144.66

6 80.00 75.01 68.44 59.34 45.23 0.00 � 132.50

7 80.00 76.37 71.71 65.55 56.95 43.52 0.00 124.74

System parameters: maximum line speed 80 km/h; constant acceleration/decleration +/�1.0m/s2; jerk limit +/�1.0m/s3; response delay ofATO system 1.0s

station 1 station 2

distance

line speed

break points

speed limits

braking distancefrom va2 to va3

direction oftravel

BSRBPSSRBPva1 va2 va3

va3

va2

va1

Fig. 2 Determining BSRBP and SSRBP

station 1 station 2

distance

line speed

break points

speed limits

va1

direction oftravel

BSRBP 1

braking distance from va2 to va3

braking distance from va1 to va2after moving

braking distance from va1 to va2before moving

BSRBP 2va2 va3

va3

va1

va2

Fig. 3 Solving braking distance overlaps between two BSRBPs

132 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005

3.3 Crossover operationThe object of crossover operation is to produce new geneswith different chromosomes. A gene shows a combinationof speed codes and chromosomes reveal speed codes amonga gene. New genes will become a possible solution to satisfyall requirements. The candidate genes for performing thecrossover operation in the population are randomlydetermined. First, a random number produced betweenzero and one by a random number generator is assigned toeach gene and compared with the crossover probability.Secondly, the crossover operation of a gene is performedwhen the given random number is greater than crossoverprobability. All genes that conformed to this condition mustbe picked from the population to perform the crossoverwith each other. Finally, the execution of the crossoveroperation is aimed at the individual chromosome at thesame position in genes. The calculation is based on adirection-based operation [8] and its formula is given by

(12). The jth chromosomes (xi0j and xiþ10

j Þ of new ith and

(i+1)th genes are calculated by the sum of themselves (xi0j

and xiþ10j ) and the product of the difference between xi

j and

xiþ1j and a random number a. Because the range of random

number a is between �1 and 1, it is possible that newchromosomes are greater or smaller than the originalchromosomes after executing this operation. The result willbecome a part of offspring

xi0j ¼ xi

j þ aðxi � xiþ1J Þ

xiþ10j ¼ xiþ1

j þ aðxjþ1j � xi

jÞ

(ð12Þ

3.4 Mutation operationThe process of mutation operation is similar to thecrossover operation. A random number between zero andone is assigned to each gene and its chromosomes in thepopulation, and compared with the mutation probability.The mutation operation of a chromosome is performedwhen the given random numbers for the chromosome andits gene are simultaneously greater than the mutationprobability. The calculation formula is given as (13) and(14) [18]:

xi0j ¼ x0j þ ðx0jU � x0jÞdðT Þ ð13Þ

dðT Þ ¼ rð1� TTmaxÞb ð14Þ

station 1 station 2

distance

line speed

break points

speed limitsva5va1 va3

direction oftravel

braking distancefrom va3 to va5

va2SSRBP SSRBP BSRBP SSRBP

va5

va4

va3

va2

va1

Fig. 4 Solving braking distance of BSRBP and position of SSRBP overlap

station 1 station 2

distance

line speed

break points

speed Limits

braking distance

SSRBPSSRBP

direction oftravel

BSRBP SSRBP

before regulation65km/h 75km/h 80km/h 65km/h 80km/h

after regulation65km/h 75km/h 65km/h 80km/h

400m

214.84m400m

200m 200m 400m 685m

685m585.16m

gradients

(m)1.0% −0.75% 1.5%

300 1328 1885 720

−0.8%

927

0.5% −0.5%

1528

− 1.0% −0.5%

0 600 700

Fig. 5 Comparison of pre-and post-regulated speed restrictions of alignment

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 133

The result of mutation operation for the jth chromosome xij

of ith gene are calculated by the sum of itself xij and the

product of the difference between xiUj and xi

j and d(T). Therange of d(T) gradually reduces as the evolved generationincreases and the reduced degree depends on the constant b.In other words, the range of mutation operation graduallyreduces to speed up convergence to a better solution whenapproaching the end of evolution. For conforming to thedescending order of chromosomes in a gene, the upper limit

xiUj of a chromosome xi

j (speed code V ij ) must be equal to its

previous chromosome xij�1 (speed code V i

j�1). Because the

range of possible optimal values is extremely concentratedin terms of the nature of objective requirements, it is noteasy to escape from the local optimal value when it sinks.The method for solving this problem is to reduce the rangeof the mutation operation to increase both the possibilitiesof escaping form the local optimum value and entering theglobal optimum value; therefore the number of timesconvergence to different local optimal solutions will bepossibly more than once.

3.5 Selection operationWhen the selection operation is executed the enlargedpopulation will include genes and the offspring produced bythe crossover and mutation operations. Moreover, the genesof the offspring will be inserted into the original populationin descending order of fitness. The genes of the enlargedpopulation over the number of original population will becancelled and the previous others will constitute a newpopulation.

3.6 Ending evolutionThe evolution terminates when the fitness of an optimalresult is small enough or the maximum time arrives.

Table 1 shows the shortest braking distance and its set ofspeed codes obtained by using GA with differenct aspects.

4 Adjustment of speed limits on alignment

Too many changes of speed limits on an alignment willcertainly cause difficulties in the planning and design ofsignalling systems. The approach to set speed restrictionbreak points (SRBP) in this paper preadjusts the runningspeed of trains to avoid overspeeding on an alignment;moreover, it cancels unnecessary speed changes to improvepassenger comfort and save energy. There are two kinds ofSRBP: braking (BSRBP) and speed-raising (SSRBP). TheBSRBP takes place when the speed restrictions of alignmentfall; therefore it must consider the braking distance andother factors, such as the response delay of the ATO systemand the train jerk limit. The SSRBP is situated at the borderof two line-speed sections where the change of line speed is

from low to high. The steps for adjustment of speedrestrictions on an alignment are as follows:

4.1 Step 1: Determine BSRBP and SSRBPThe BSRBP and SSRBP and sought from station 2 tostation 1 in Fig. 2. When the speed restrictions of alignmentfall, such as from va2 to va3, subtract the braking distancefrom the starting point of the lower line speed va3 and settleon that point to be a BSRBP. When the speed restrictions

Table 2: List of speed limit sections before and after regulation

Before regulation After regulation

No Startpoint (m)

End point(m)

Speedlimit (km/h)

Length(m)

No Startpoint (m)

End point(m)

Speedlimit (km/h)

Length(m)

1 0 400 65 400 1 0 400 65 400

2 400 600 75 200 2 400 614.84 75 214.84

3 600 800 80 200 3 614.84 1200 65 585.16

4 800 1200 65 400 4 1200 1885 80 658

5 1200 1885 80 685

deduct essential length from distancebetween two stations and distributeremainder to each signalling block

start

continue to evolve

yes

no

end

initialise population

determine optimal combination ofrunning speed code of each signalling block

using heuristic search

sort genes in populationaccording to their energy consumption

perform crossover operation andput results into offspring

perform mutation operation andput results into offspring

Fig. 6 Flowchart of finding optimal combination of block-layoutand running speed code

134 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005

of alignment rise, such as from va1 to va2, settle on thestarting point of the higher line speed va2 to be a SSRBP.

4.2 Step 2: Check whether break pointsoverlap between BSRBP and SSRBP or twoBSRBPs from station 2 to station 1

4.2.1 Braking distances overlap between twoBSRBPs: In Fig. 3 the area of braking distance from va2

to va3 overlaps the one from va1 to va2. The method ofsolving this problem is to move the end-point of brakingdistance of the BSRBP 2, which is farther than station 2, tothe starting point of braking distance of the BSRBP 1,which in nearer than station 2.

4.2.2 Braking distance of BSRBP and posi-tion of SSRBP overlap: In Fig. 4, within the brakingdistance of a BSRBP whose position is at the starting pointof line speed va5, a SSRBP occurs. Trains running on therailway system are unable to brake immediately to linespeed va5 when accelerating to va4, thus the SSRBP withinthe section for line speed limit va4, is cancelled.

4.3 Step 3: According to method ofdeciding SRBP, speed restriction sectionsare created from station 2 to station 1Figure 5 compares alignment before and after regulationbetween station 1 and station 2. The distance is 1885mbetween stations. The start and end points, line speed andlength of each speed restriction section are shown asTable 2. There are five speed restriction sections beforeregulation and four after regulation. During regulation thestart point of the fourth section is moved forward for 800mto 614.84m because of BSRBP. The third section is tooshort with a length of 14.84m and hence is merged intosecond.

5 Exploring optimal combination of position andrunning speed code of each signalling block by usi-ng GA for saving energy

After adjustment of line speed this paper explores theoptimal combination of position and running speed code ofeach signalling block for saving energy using GA accordingto the various aspects and signalling blocks. Its result mustmeet the requirements of minimum headway and averagetrain speed. The number of aspects and signalling blocks arerelated to each other. Increasing the number of signallingblocks are related to each other. Increasing the number ofsignalling blocks within a fixed distance, the number ofaspects can be theoretically advance for reducing theheadway. This advantage is not obvious when the responsedelay of an ATO system is taken into account, and mayeven have the opposite effect. At the same time theequipment cost will increase. Decreasing the number ofsignalling blocks can diminish the cost, but will make theheadway longer. Consequently it is necessary to select aproper number of aspects and signalling blocks for savingenergy and satisfying the requirement of minimum headwayand average train speed.

After setting a reasonable number of aspects andsignalling blocks, the product of the shortest length andthe number of signalling blocks is deducted from thedistance between two stations. The remainder is distributedto each signalling block by using GA for saving energy. Theevolutional process using GA, including the representation,initialisation of population, crossover, mutation and selec-tion is the same as the one in Section 1. The process whose

flowchart is shown in Fig. 6 is illustrated in the followingsteps.

5.1 Step 1: Content of chromosomesrandomly produced to create initialpopulationOnce the position of platform blocks are confirmed thenumber of chromosomes of each gene will be less than thenumber of signalling blocks between the two stations byone. Genes in a population indicate different signallingblock-layouts. The population with pop genes for a realnumber encoding in a FBS system with SB signalling blocksbetween two stations can be described by the followingequation:

x1 ¼ ½x11; x12; � � � ; x1SB�1� ¼ e11; e12; . . . ; e1SB�1

� �x2 ¼ ½x21; x22; � � � ; x2SB�1� ¼ e21; e

22; � � � ; e3SB�1

� �...

xpop ¼ ½xpop1 ; xpop

2 ; � � � ; xpopSB�1� ¼ epop

1 ; epop2 ; � � � ; epop

SB�1� �

ð15Þ

5.2 Step 2: After deducting essential lengthof signalling blocks from distance betweentwo stations, remainder is distributed toeach signalling blockThe essential length Lessential of each signalling block fordifferent aspects is obtained by the braking distancebetween any two neighbouring speed codes in Table 1.The distribution method is according to the proportion ofeach random number to the sum of total random numberwithin a gene. The distribution result of each gene accordingto (15) can be described as follows:

lremainder ¼ L� SB� Lessential ð16Þ

Lin ¼ Li

n�1 þ lremainder �ei

nPSB�1

j¼1ei

j

1 � n � SB ð17Þ

where L0¼ 0 and LSB¼L.

5.3 Step 3: Using heuristic search todetermine optimal combination of runningspeed code of signalling blocks for savingenergy with different block-layoutsThe energy consumption of trains is a function of thegradient and running speed code of each signalling block.The energy consumption of a single train is defined as thefitness function of GA for achieving the goal of energysaving. Moreover, the gradient of alignment is used to assistin deciding the acceleration or deceleration of trains forreducing the search path. When a new gene is produced itsoptimal combination of running speed code of eachsignalling block is found by using the heuristic searchdescribed in Fig. 7. In other words, it is necessary to re-enforce operation simulations of a single train betweenstations based on the same block-layout. Genes arerearranged by the results of energy consumption indescending order. The flowchart of all processes in step 3is shown as Fig. 7 and is explained as follows:

(i) According to the descending speed codes, the operationsimulations for a single train will be implemented forfinding the Rindex that is an identical speed code for eachsignalling block to be closest but not exceed the limits.These limits include the headway and average train speed.Next, selecting the upper and lower range of speed codes to

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 135

be Rindex_to_upper and Rindex_to_lower, reduce the search space.Rindex_to_upper and Rindex_to_lower are exploited to determineRupper and Rlower shown as (18) and then improper speedcodes are cancelled within each signalling block because ofexceeding the line speed.

Rupper ¼Rindex � Rindex to upper

Rlower ¼Rindex þ Rindex to lowerð18Þ

(ii) Calculating the average gradient of each signalling blockis to be the basis of determining the running speed code.The train decelerates or cruises for saving energy on the

uphill area; therefore the search range of running speedcode must be from Rindex to Rlower. The search range mustbe form Rupper to Rindex in the downhill area.

(iii) The operation simulations of single train are performedfor calculating the energy consumption, headway andaverage speed in accordance with all combinations ofreasonable running speed code of each block. The optimalcombination of running speed code is found for savingenergy from this signalling block-layout.

To continue the example of Fig. 5, the system parametersare the same as Table 1 and the number of aspects and

start

end

input Rindex_to_upperand Rindex_to_lower

Rupper = Rindex − Rindex_to_upperRlower = Rindex + Rindex_to_lower

modify possible search range of running speed code ofeach block according to average gradient

no

no

yes

yes

calculate the average gradient of eachsignalling block

find identical speed code Rindex for eachsignalling block to be closest to limits of

headway and average speed but not to exceed

decide possible search range of runningspeed code of each signalling block

find possible combinations of running speed code one byone, and perform operation simulation of single train tocalculate energy consumption, headway and average

speed

satisfy limitsof headway and average

speed and consumelower energy

replace the previous resultwith optimal combination

continue to evolve

Fig. 7 Flowchart of finding optimal running speed code

Table 3: Content of a gene in initial population and its derived data

Block number 1 2 3 4 5 6 7 8 9 10

Content ofchromosomes

0.1421 0.1935 0.1679 0.1346 0.0494 0.1405 0.0535 0.0316 0.0868

Block lengths (m) 203.2 224.4 213.9 200.1 165.0 202.6 166.7 157.7 180.4 171.0

End point of blocks(m)

203.2 427.6 641.5 841.6 1006.6 1209.2 1375.9 1533.6 1714.0 1885.0

Speed limits (km/h) 65 65 65 65 65 65 80 80 80 80

Average gradients(%)

1.0 0.0047 �0.3137 0.6626 0.0176 �0.5 �0.6436 �0.9823 �0.5 �.05

Optimal combina-tion of runningspeed code (km/h)

47.81 47.81 62.96 47.81 47.81 62.96 62.96 62.96 62.96 62.96

136 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005

signalling blocks are set to be 5 and 10, respectively.The dwell time is 20 seconds. The limits of headway andaverage train speed are 120 seconds and 40km/h. Theshortest block length and speed codes are acquired fromTable 1; 144.66m and {80, 72.88, 62.96, 47.81, 0km/h},respectively. The content of one gene with nine chromo-somes in the initial population and derived data are shownin Table 3. The data include positions, average gradientsand the optimal combination of running speed codefor saving energy in each signalling block. The contents ofchromosomes have been normalised. The remainder, afterdeducting the essential length from the distance between

two stations, is 442.1m. It is distributed to each signallingblock in accordance with the proportion of the contentof each chromosome to the sum of them. The length ofeach block is calculated from the result of distributionplus the shortest block length and is then changed intomileage. The length of the last signalling block that isnamed as the platform signalling block is the length ofthe train plus 30m. Signalling block 10 has a fixed length(171m); therefore there are nine chromosomes in a gene.After calculating the average gradient of each block thepossible search range is determined. Both Rindex_to_upper andRindex_to_lower are assumed to be one. Because the identical

Rindex

speed code /

actual speed, km/h

signalling blocks

9 1086 754321

Rindex_to_upper

Rindex_to_under

average

gradients(%)0.0 %

1 / 80.0

5 / 0.0

4 / 47.81

3 / 62.96

2 / 72.88

speed Limits

(km/h) 0.0km/h

80.0km/h

1.0 %

Fig. 8 Possible search range of running speed code

Table 4: Optimal results for different number of aspects and signaling blocks

Aspects 5 5 5 5 5 6 6 6 6 6

No. ofblocks

8 9 10 11 12 8 9 10 11 12

End point (m)/ 1 494.3/4 162.5/4 186.6/4 207.5/4 185.5/4 356.8/4 418.34/5 189.1/5 179.4/5 172.3/5

running speed code 2 659.3/3 419.6/4 413.7/4 369.6/4 344.4/4 547.1/4 719.8/4 418.1/5 418.3/5 394.1/5

3 804.0/4 655.8/3 665.6/3 528.6/3 509.9/3 771.7/4 865.27/4 699.5/4 570.8/4 543.6/4

4 984.4/3 809.7/4 833.5/4 674.4/3 668.5/3 1018.4/4 1020.0/4 854.3/4 738.8/4 695.0/4

5 1228.9/3 1042.0/3 1012.0/4 824.0/4 815.8/4 1449.0/4 1158.9/4 1024.5/4 898.2/4 836.5/5

6 1373.6/3 1384.7/3 1170.0/3 1024.8/4 975.0/4 1581.5/3 1401.1/4 1210.6/4 1080.2/4 991.5/5

7 1714.0/3 1529.3/3 1332.8/3 1181.4/3 1119.7/3 1714.0/3 1562.0/4 1359.4/4 1222.3/4 1125.6/4

8 1885.0/3 1714.0/3 1509.3/3 1373.0/3 1264.8/3 1885.0/3 1714.0/4 1515.5/4 1379.2/4 1268.2/4

9 1885.0/4 1714.0/3 1527.9/3 1416.3/3 1885.0/4 1714.0/4 1524.0/4 1412.4/4

10 1885.0/3 1714.0/3 1569.3/3 1885.0/4 1714.0/4 1555.5/4

11 1885.0/3 1714.0/3 1885.0/4 1714.0/4

12 1885.0/3 1885.0/4

Energy consumption(MJ)

101.7 102.5 98.7 101.6 107.0 109.4 101.1 101.1 101.1 101.1

Headway (s) 119.5 119.9 111.4 112.8 100.5 139.2 118.4 118.8 116.5 114.3

Average speed(km/h)

43.3 43.6 42.7 42.9 43.2 45.0 42.6 42.7 42.7 41.4

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 137

speed code Rindex is searched out and its value is three,Rupper and Runder are calculated by (16) to betwo and four, respectively. After deleting the speed codesexceeding the line speed, the possible search range ofrunning speed code of each signalling block is shownin Fig. 8. In this Figure two line speed limits are 65 and80km/h, respectively. The search range of heuristic search

is determined by the average gradient of each signallingblock and indicated by the shadow area. Finally, theoptimal combination of running speed code for savingenergy is found out in this block-layout, and its headwayand average train speed are 110.67 s and 42.502km/h,respectively.

5.4 Step 4: Executing crossover andmutation operations between differentgenes produces new offspringThe formulations of crossover and mutation operationare the same as (12)–(14) for producing the offspring. Whenexecuting the crossover operation each gene in thepopulation is assigned a random number to compare witha given crossover probability. If the assigned number isgreater than crossover probability, the crossover operationwill be executed and its results put into the offspring.Similarly, if the random assigned to each gene is greaterthan the mutation probability, the results of mutationoperation will be put into the offspring.

O04 O05

direction of

travel

distance

speed limits

0.0 365.74mileage of signallingblocks (m)

gradients(m) 0.0%

537.46 733.39 918.75 1128.47 1286.27 1551.12 1722.12

80.0 km/h

line speed running speed limits

47.8 km/h

63.0km/h

72.9 km/h

72.9 km/h

72.9 km/h

63.0 km/h 47.8

km/h47.8 km/h

3.0%

− 3.0%

a

0

20

40

60

80

spee

d, k

m/h

speed of trainspeed limit of alignmentspeed limit of operation

b

− 1

− 0.5

0

0.5

1

accelerationjerk

c

acce

lera

tion,

m/s

2je

rk, m

/s3

0 200 400 600 800 1000 1200 1400 1600

− 10

− 5

0

5

10

pow

er, M

W

dlocation , m

power consumption

Fig. 10 Optimal result of signalling block-layout and running speed code with five aspects from stations O04 to O05a Block-layout of signalling systemb Location-speed of train and -speed limit of operation and alignment curvec Location-acceleration and -jerk of traind Location-power consumption curve

98.00

100.00

102.00

104.00

106.00

108.00

generations

ener

gy c

onsu

mpt

ion,

MJ

average values optimal values

111 3 5 7 9

Fig. 9 Energy consumption of each generation after optimisation

138 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005

5.5 Step 5: Previous steps are continuallyrepeated until evolution is completeThe work from steps 2 to 4 will be performed for the genesof an offspring. Their results will be compared with theinitial population. Better ones will be reserved for a newpopulation until the evolution is finished unless the givenevolution time arrives.

Similarly, to continue the example of Fig. 5 with thesame parameters, the crossover and mutation probabilitiesare set to be 0.8 and 0.4, respectively. The lowestand average energy consumption for each gene in thepopulation is shown as Fig. 9. The difference betweenthe lowest and average values reduces gradually. Thismeans that both of them converge to the same value.After evolving ten times, the end-points of each signallingblock derived from the optimal result are {186.5, 424.8,662.4, 842.4, 1008.8, 1190.4, 1371.1, 1540.5, 1714.0,1885.0m} from start station to end station. Underthe limits of headway and average speed, theoptimal combination of running speed code for savingenergy is {4, 4, 3, 4, 4, 3, 3, 3, 3, 3}. In other words, the

running speed of each signalling block is {47.81, 47.81,62.96, 47.81, 47.81, 62.96, 62.96, 62.96, 62.96, 62.96km/h}.Its results are shown as Table 4 to determine the positionand running speed code of each signalling block withdifferent numbers of aspects and signalling blocks. It isobvious that the energy consumption for a block-layoutwith five aspects and ten signalling blocks is the lowest andthat the requirements of headway and average train speedare satisfied.

6 Results

Figures 10 and 11 show the optimal result of signallingblock-layout and running speed code of eachsignalling block between stations O04 and O05 on theHsinchuang line of the Taipei Rapid Trasit Systemproduced by using the proposed approach in thispaper. The example is common to mass rapid transitsystems. The only line speed is 80km/h. There are downhilland uphill areas in the train starting and stoppingstages, respectively. This design advantage is to assist

power consumption

0 200 400 600 800 1000 1200 1400 1600− 10

− 5

0

5

10

d

pow

er, M

W

location,m

acce

lera

tion,

m/s

2

c

−1

− 0.5

0

0.5

1

jerk

, m/s

3

accelerationjerk

speed of trainspeer limit of alignmentspeed limit of operation

0 200 400 600 800 1000 1200 1400 16000

20

40

60

80

spee

d, k

m/h

b

O04 O05

direction oftravel

distance

speed limits

0.0 379.50mileage of signallingblocks (m)

gradients

(m)0.0%

527.21 698.20 838.18 1124.79 1257.74 1551.12 1722.12

80.0 km/h

lines peed running speed limits

68.4 km/h75.0km/h

75.0km/h

80.0km/h

80.0 km/h 75.0km/h

75.0 km/h68.4 km/h

3.0%

− 3.0%

a

Fig. 11 Optimal result of signalling block-layout and running speed code with six aspects from stations O04 to O05a Block-layout of signalling systemb Location-speed of train and -speed limit of operation and alignment curvec Location-acceleration and -jerk of traind Location-power consumption curve

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 139

train operation in accelerating and decelerating. Fig. 10ashows the optimal block-layout of five-aspect signallingsystem with eight signalling blocks and different runningspeed code on each signalling block for the lowestenergy consumption. The curves of train speed and speedlimit of alignment and operation are shown as Fig. 10b.Train acceleration is by PID control to meet therequirements of train velocity, acceleration and jerk, andtheir results are shown as Fig. 10b and 10d. Figure 10cshows the power consumption of a train running betweenstations. The headway and train average speed meet thespecification requirements and are 119.51 s and 42.16km/h,respectively. The optimal result of six-aspect signallingsystem is shown as Fig. 11 which also has eight signallingblocks. The headway and train average speed are 118.12 sand 47.62km/h, respectively. To compare their energyconsumption and peak power, the first is lower than the lastabout 3.1 and 16.05%, respectively. This approach is notonly common to mass rapid transit systems, but alsosuitable for alignments with many changes of speed limitsand gradients.

7 Conclusions

An approach has been proposed for optimising theblock-layout of FBS systems and determining therunning speed cod of each signalling block betweentwo neighbouring stations. It has emphasised minimumenergy consumption for each running train and thelowest number of signalling blocks for equipmentcost under the constraint of minimum headway andaverage train speed with train maximum capacity. Its mainobject is to address the designated minimum headway forsaving energy rather than to obtain the practice minimumheadway.

Train acceleration control is by PID control to meet thelimits of train velocity, acceleration and jerk. The methodsof equiblock signalling and GA are introduced to the design

of signalling block-layout. The fitness of GA is determinedby performing the single train movement simulation tocalculate the energy consumption, headway and averagetrain speed. Due to preadjusting the train running speed tomeet the line speed and determining the running speed codeby considering the average gradient of each signalling block,this approach also demonstrates its advantage when therailway system has many changes of speed limits andgradients.

A methodical approach is important for simplifyingthe design of block-layout of singalling systems.The computer-based technique proposed assists the designof block-layout for saving energy. It eases the design,reduces the time taken for design, and makes possible futureapplications. For example, the technique could be used todetermine running speed codes under normal conditionsand service recovery.

8 References

1 Gill, D.C., and Goodman, C.J.: ‘Computer-based optimisationtechniques for mass transit railway signalling design’, IEE Proc.- B,1992, 139, (3)

2 Chang, C.S., and Du, D.: ‘Improved optimisation method using geneticalgorithm for mass transit signalling block-layout design’, IEE Proc.-Electr. Power Appl., 1998, 145, (3)

3 Chang, C.S., and Sim, S.S.: ‘Optimising train movements through coastcontrol using genetic algorithm’, IEE Proc.- Electr. Power Appl., 1997,144, (1)

4 Hwang, H.-S.: ‘Control strategy for optimal compromise between triptime and energy consumption in a high-speed railway’, IEEE Trans,Syst., Man Cybern. –Part A, 1998, 28, (6)

5 Chang, C.S., Xu, D.Y., and Quek, H.B.: ‘Pareto-optimal setbased multiobjective tuning of fuzzy automatic train operationfor mass transit system’, IEE Proc.- Electr. Power Appl., 1999,146, (5)

6 Chang, C.S., and Xu, D.Y: ‘Differential evolution based tuning offuzzy automatic train operation for mass rapid transit system’, IEEProc.- Electr. Power Appl., 2000, 147, (3)

7 Khmelnitsky, E.: ‘On an Optimal control problem of train operation’,IEEE Trans. Autom. Contral, 2000, 45, (7)

8 Gen, M., and Cheng, R.: ‘Genetic algorithms and Engineering design’(Wiley, 1997)

140 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005