optimization of high-speed train control strategy for traction

Research ArticleOptimization of High-Speed Train Control Strategy forTraction Energy Saving Using an Improved Genetic Algorithm

Ruidan Su1 Qianrong Gu2 and Tao Wen1

1 College of Information Science and Engineering Northeastern University Shenyang 110004 China2 Service Science Research Center Shanghai Advanced Research Institute Chinese Academy of Sciences Shanghai 201203 China

Correspondence should be addressed to Ruidan Su suruidanhotmailcom

Received 27 March 2014 Accepted 9 April 2014 Published 4 May 2014

Academic Editor Young-Sik Jeong

Copyright copy 2014 Ruidan Su et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A parallel multipopulation genetic algorithm (PMPGA) is proposed to optimize the train control strategy which reduces the energyconsumption at a specified running timeThe paper considered not only energy consumption but also running time security andriding comfort Also an actual railway line (Beijing-Shanghai High-Speed Railway) parameter including the slop tunnel and curvewas applied for simulation Train traction property and braking property was explored detailed to ensure the accuracy of runningThe PMPGA was also compared with the standard genetic algorithm (SGA) the influence of the fitness function representationon the search results was also explored By running a series of simulations energy savings were found both qualitatively andquantitatively which were affected by applying cursing and coasting running status The paper compared the PMPGA with themultiobjective fuzzy optimization algorithm and differential evolution based algorithm and showed that PMPGA has achievedbetter result The method can be widely applied to related high-speed train

1 Introduction

Since October 1964 the worldrsquos first high-speed railway JapanTokaido Shinkansen was born high-speed railways startedthe rapid development Today most European countriesRussia Japan and China have constructed their complexhigh-speed railways networks Although the railway wasconsidered the most efficient way of travel compared toaircraft and auto vehicle it still consumes large amount ofenergy [1] in everyday running Researches showed that it stillhas large possibility tomake the train runmore efficiently [2ndash4]The reduction of energy consumption is also seen as one ofthe key objectives for the development of sustainablemobilityby use of high-speed train Research will lead to a decrease ofhuge energy consumption in everyday running of high-speedtrains Many scholars have been engaged in it

Yang et al [5] from Tongji University proposed a newenergy conservation track profile based on trigonometricfunction method in urban mass transit Simulation resultsshowed that it was effective in comparison with actual trackprofile Bocharnikov et al [6] applied a method for saving

energy consumption during a single-train journey by tradingoff reductions in energy against increases in running timein Bocharnikovrsquos research energy savings were found to beaffected by acceleration and braking rates and by runninga series of simulations in parallel with a genetic algorithmsearch method Chen et al [7] employed genetic algorithmsto optimize train scheduling The result showed that themethod can significantly reduce the maximum tractionpower Although these methods and algorithms were effec-tive they can only be applied inmass rapid transit (MRT) andlight rapid transit (LRT) systems Usually in MRT distancebetween two stations was short and the top running speedwas about 80ndash100 kmh In this case a train generally mustdecelerate in preparation for reaching the next station beforeit reaches the speed limit In Milroyrsquos doctoral dissertation[8] Aspects of Automatic Train Control it was proved that forshort distance train control represents three different motionregimes including acceleration coasting and braking Butlater in 1984 Howlett [9] proved that in long distancetrain running cruising was significant in minimizing energyconsumption Due to the difference between MRT LRT and

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 507308 7 pageshttpdxdoiorg1011552014507308

2 Journal of Applied Mathematics

high-speed trains these methods cannot be applied in high-speed trains for energy optimization

For high-speed trains energy saving and trains controloptimization were also studied by scholars Kawakami [10]from Central Japan Railway Company presents a dynamicpower saving strategy for Shinkansen traffic control theauthor made conclusion that predictive simulations in everylayer and target shooting operation of trains are the basisfor energy control With consideration of track gradient andspeed limits Cheng [11] summarized train control problemswith two different models tractionmechanical energymodel(TMEM) and traction energy model (TEM) in a long-haultrain Hwang [12] presented an approach to identify a fuzzycontrol model for determining an economic running patternfor a high-speed railway through an optimal compromisebetween trip time and energy consumption

In this paper taking the Beijing-Shanghai High-SpeedRailway as a case an improved PMPGA was applied tofind a perfect running with a specified running In thisresearch security stop precision and riding comfort wereconsidered and also the railway line parameter includes theslop tunnel and curve The result demonstrates that thePMPGA improved algorithm was better with the SGA andit has achieved conspicuous energy reduction

2 Train Traction Module

21 Train Traction Property Traction property curve is animportant curve demonstrating the relationship betweentrain traction effort and speed It was the most significantworkwhen a trainwas designed Figure 1 shows the schematicdiagram of traction property curve calculation

In Figure 1 there are three curves the top one is adhesion-limited braking force 119865max = 119891(V) the middle one is tractioneffort property 119865 = 119891(V) and the bottom one denoted as Wis the sum of resistances (eg bearing rolling air and graderesistance)119882 = 119891(V) Note that point A the cross of 119865max =119891(V) and119882 = 119891(V) correspond V

119886 is greater than the Vmax

Now according to the curve traction effort property119865 = 119891(V)could be generated as

(119865V minus 119865V0)

V=(119865V1015840 minus 119865V0)

V10158400 le V le V1015840

119865V lowast V = 119865max lowast 119881max V1015840 le V le Vmax

(1)

In the above formula 119865V represents the traction force whenthe speed is VV1015840 is the speed on the intersection point ofconstantmoment segment and constant power segment119865maxrepresents traction force limitation

22 Train Resistance To ensure that the TE was able todrive the train with a speed the total resistances in thispaper defined as119882 must be known Total resistances includebasic resistance119882

0(axle friction resistance track resistance

rolling resistance journal resistance air force resistance andvibration resistance) and extra resistance 119882

119895 119882119895includes

grade resistance (119882119894) curve resistance (119882

119903) and tunnel

resistance (119882119904)

Fmax

998400 (998400) (998400) max

A

W = f()

Fmax = f()

1

2

3

a

Train speed (kmh)

Trac

tion

effor

tF(k

N)

F0

(F0 )

Figure 1 Diagram of traction property curve calculation

It [13] was found that speed was the main factor whicheffects the basic resistance and basis resistance can beexpressed by a quadratic equation formulated as follows

1205960= 119886 + 119887V + 119888 sdot V2 (2)

where the coefficients 119886 119887 and 119888 are dependent on axle loadnumber of axles cross-section of the train and shape of thetrain

According to [14] considering the train as a multiparticleobject we can have the 119908

119895(119909) as the following function

119908119895(119909) =

1

119871[sum 119894119894lowast 119897119894+ 600sum

119897119903119894

119877+sum(119908

119904lowast 119897119904)] (3)

where 119871 is the length of the train and 119894119894and 119897119894represent the

gradient and grade length 119877 119897119903119894are the curve radius and

length 119908119904119894 119897119904119894are the tunnel resistance and length

Then the motion equation and the 119886 V119894 and 119878

119894were

formulated as below

119886 =119889V119889119905=119865 minus 119861 minus (120596

119894+ 120596119903+ 120596119904+ 1205960)

119872 (1 + 120574)

119881119894= 119886Δ119905 + 119881

119894minus1

119878119894=119881119894+ 119881119894minus1

2Δ119905 + 119878

119894minus1

(4)

where119881119894was the speed of currentmoment V

119894minus1was the speed

of last moment 119886 was the acceleration of current moment 119904119894

was the distance of currentmoment from the first station and119904119894minus1

was the distance of the last moment from the first station

3 Traction Energy Module (TEM)

In order to achieve minimal energy consumption generallytrain control for running between stations including accel-eration cruising coasting and braking should be applied atappropriate time Golovitcher [15] and Khmelnitsky [16] ana-lyzed the trainmovement process with nonlinear constrained

Journal of Applied Mathematics 3

BA C D E FSpee

d lim

it (k

mh

)

Distance (km)ximinus1 xi

A = 120587r2

Figure 2 Diagram of different running status during one journey

differential equations and concluded that a maximum eco-nomic train running strategy should contain four statusesmaximum traction cursing coasting and maximum brak-ing For analyzing station-to-station travel time and distanceprofile it is essential to comprehend the description ofthe motion statuses and their mathematical expressions Inmaximum traction power is used to overcome gravity (ifclimbing) and the dynamic resistance so as to accelerateWhen cruising power is used to overcome the resistanceto maintain the train at the constant speed at this timethe acceleration is zero When coasting the running trainonly suffers from the force of resistance Applying coastingwhen the train runs between stations as much as possible isconsidered to be the most effective energy consumption wayWhen braking with regeneration technology fitted energycan be produced using the motor as a generator

A trainrsquos journey may have variables coast intervals(Figure 2) to achieve an optimal solution Figure 2 shows atrainrsquos status and changing point during a running betweentwo stations In the figure the points mean the followingA traction B cursing start point C coasting start point Dcoasting E cursing F braking

Now the aim is to find an optimal control strategy forminimal energy consumption in a round trip between twostations This problem can be seen as a double optimizationproblem

Traction energy module can be described as followsMake119883 the distance between two stations and travel time

was fixed 119879 [0 119879] can be divided as

0 = 1199050le 1199051le 1199052sdot sdot sdot le 119905

119899le 119905119899+1= 119905 (5)

where 1199050is the initial time and 119905

119899+1is the final time in the

time space [119905119896minus 119905119896+1] train travel distance is [119909

119896minus 119909119896+1] and

in [0 119909]

0 = 1199090le 1199091le 1199092le sdot sdot sdot le 119909

119899le 119909119899+1= 119909 (6)

Total energy consumed by the train can be defined as follows

min 119864 = int

119909119891

1199090

119906119891(119909) 119891 (V) 119889119909

st

119889119905

119889119909=1

V

V119889V119889119909=

119906119891(119909) 119891 (119909) minus 119906

119887(119909) 119887 (V)

119872119892minus 1199080(V) minus 119908

119895(119909)

700

600

500

400

300

200

100

00 50 100 150 200 250

Speed (kmh)

(1)

(2)

(3)

(4)

Curve 1 train traction propertyCurve 2 basic resistanceCurve 3 adhesion-limited braking force (wet)Curve 4 adhesion-limited braking force (dry)

Trac

tion

effor

t (kN

)

Figure 3 Train traction property and adhesion-limited brakingforce

119905 (1199090) = 0 119905 (119909

119891) = 119879 V (119909

0) = 0 V (119909

119891) = 0

V le 119881 (119909) 119906119891isin [0 1] 119906

119887isin [0 1]

(7)

where119864 is the energy consumption and119879 is a fixed timewhenthe train travels between two stations 119905(119909

0) is start time 119905(119909

119891)

is arrival time and V(1199090) and V(119909

119891) represent the start speed

and final speed it was obvious that V(1199090) and V(119909

119891) are equal

to 0 119906119891and 119906119887were coefficient of traction power and braking

Then the train control strategy set was 119878 = 119904119894 =

traction(T) cursing(CR) coasting(C)Braking(B) =

TCRCBFinally the train control matrix was defined as

119862 = [1198880 1198881 1198882 119888

119894 119888

119899minus1 119888119899] (8)

where 119888119894= [119909119894 119904119894] 119909119894is the position and 119904

119894is the control

strategy start at the position 119909119894 From Figure 2 we can see

that 119904119894isin S 119909

0= 0 and 119909

119899can be easily calculated by the

last braking process

4 Minimize the Energy Consumption withParallel Multipopulation Genetic Algorithm

The genetic algorithm (GA) [9 10] is a method for solvingboth constrained and unconstrained optimization problemsbased on natural selection the process that drives biologi-cal evolution The genetic algorithm repeatedly modifies apopulation of individual solutions [17] At each step the GA


Table 1 Experiment of PMPGA with different subpopulation group and gene length

Experiment 119873sp Gene length Group size 119875119888

Generation 119875119898

119875V

E1 3 50 100 07 300 0068 02E2 3 100 100 07 300 0068 02E3 6 50 100 07 150 0068 02

selects individuals at random from the current populationto be parents and uses them to produce the children for thenext generation Over successive generations however as weall know the standard GA has the premature convergencephenomenon and slow searching process In our researchwe apply PMPGA which is a simulation of gene isolationand gene migration in biological evolution process whereall populations are divided into many subpopulations withdifferent control Because the subpopulations have differentgene patterns and their genetic processes are independentthe global optimumand the fully search are guaranteed by thedifference in evolutionary directionTheoptimal individual isquoted by other subpopulations through migration operator

Finally considering the optimal object and the constraintconditions the PMPGA compute process can be described asbelow

41 Chromosome Take the sequence of train control strategywhich contains the sequence of train operating conditionsand the corresponding sequence of conversion locations forthe operation section as a chromosome The function is

119862 = [(1199090 1199040) (1199091 1199041) sdot sdot sdot (119909

119894 119904119894) sdot sdot sdot (119909

119897 119904119897)] (9)



strategy start at the position 119909119894 119904119894were discrete variables

which contain four control strategies [TCRCB] eachcontrol strategy corresponds to one energy consumptionformula 119909

119894uses the real number encoding 119897 is the length of

chromosome and it is also variable

42 Initial Population Population is constructed using chro-mosomes each chromosome represents a single solutionpoint in the problem space In our research donate individualmatric 119880(119880 isin 119862) was randomly created with different genelength Gene length means possible times of traction strategyduring a running Consider the distance between two sta-tionsWe assign themaximum gene length as119866119871max Each119880

1015840

was a control matrix and all created Us compose 119873 subpop-ulation group each subpopulation group is denoted as 119875 =1199011 1199012 1199013 119901

119896 and 119896 is the number of populations in a

subpopulation group In our research119873sp assigned as numberof subpopulation groups will be computed in parallel

43 Fitness Function Applying the individual which meansthe control matrix to the energy calculated formula we canget the object value The fitness evaluation is based on theminimization of the energy consumption which is definedas

Fit (119909) = 1

119862max + obj + 119862 (10)

Considering the fastest running strategy the maximumenergy consumption is about 4000 kwh we make coefficient119862max as 4000 and 119888 was 01

44 Standard of Convergence The convergence criterion iswhether the maximum evolutionary generation is reachedor the best individual remains unchanged among severalgenerations If the algorithm is not convergent then continueto the next operations otherwise searching process ends

Selection operation Roulette wheel selection first cal-culates each individual 119909

119894

1015840 corresponding proportion of itsfitness value to the total fitness value of the whole populationlabeled as 119901

119894 by

119901119894=

Fit (119909119894)

sum119873

119895=1Fit (119909

119895)

(11)

where 119894 = 1 2 119873 and 119873 is the size of population Thenthe operator repeats119873 times of selecting an individual fromthe current population to generate the new population Ineach time 119886 random real number 119902 uniformly scattered inthe range (0 1) is generated and the individual 119909

119896where 119896

satisfies (20) is selected

119896 = min119895 |119895minus1

sum

119894=1

119901119894le 119902 119895 = 1 2 119873 (12)

It is obvious that in the roulette wheel selection the fitterindividuals have a greater chance of survival than the greaterones

Crossover Uniform crossover operator the crossover oper-ator works as follows After the two ldquoparentsrdquo are drawneach corresponding pair of coordinates exchanges its valuesindependently with the same probability 0 lt 119903 lt 1 asfollows

119883119905+1

1= 119903119883119905

1+ (1 minus 119903)119883

119905

2

119883119905+1

2= 119903119883119905

2+ (1 minus 119903)119883

119905

1

(13)

In formula (13) 1198831199051 119883119905

2represent the gene of parents and

119883119905+1

1 119883119905+1

2represent the next generation

Mutation Operator Using random number generator to gen-erate a number between 0 and 1 if it is less than the probabilityofmutation119901

119898 chromosomes domutation Severalmutation

positions are rolled randomlyIn order to find the best solution we define different

gene length and different number of subpopulation groupsfor confrontation By SGA the population size is 100 gene


0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitness

Fitn

ess

Generation0 40 80 120 160 200 240 280

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Fitness

Generation0 40 80 120 160 200 240 280

(b)

Figure 4 Evolutionary curve of standard GA (left) and PMPGA E1 (right)

Table 2 Basic train information

Motor cartrailernumber of cars 14216Number of axles 64Train weight 8956 (t)Outpower (kw) 615lowast16Voltage rating (v) 3000Current rating (A) 230Highest running speed (kmh) 380Cursing speed (kmh) 350

length is 50 the maximum evolutionary generation is 300and Pc = 07 Pm = 0068 and Pv = 02 The specifiedrunning time is 20mins The adjustment coefficient A ofrunning performance index function is 36 The update timeinterval is 1 s for multiparticle train simulator (see Table 1)

By PMPGA we try 3 groups of experiments as below andthe update time interval is 1 s formultiparticle train simulator

5 Case Study and Simulation

In this project we use c to develop a simulation envi-ronment Then the improved train control strategy can beverified and compared with the previous one The trains runin the Beijing-Shanghai High-Speed Railway from Beijingto Langfang the line length is 1305121 km and the distancebetween Beijing and Langfang is 595 km Reality line param-eters including grade tunnel curve and speed restriction areall considered in the simulation

Train traction property basic train information andreality line parameters were showed in Figure 3 Table 2 andTable 3

From the simulation result Figure 4 shows that withstandard GA themaximum fitness rises much faster after the140th generation and even faster at the 220th generation afterabout the 240th generation the fitness reaches the maximum

value and becomes stable after that Compared with the E1the maximum fitness rises sharply at the 75th generationand becomes stable from the 120th generation The resultshows that the parallel multipopulation GA has the speedof convergence and the precision is considerably improvedalso it avoids the premature convergence phenomenon ofsingle-population evolutionary algorithm and maintains theevolutionary stability of the best individuals

For experiment E1 (Figure 4 right) and E2 (Figure 5left) we can see that the gene length was extended to 100which does not cause any improvement Both curves reachthe maximum value and become stable at about the 120thgeneration From the result of E1 the gene length 50 is enoughfor the control strategy between two stations

For experiment E3 when 119873sp was extended from 3 to6 gene length was set as 50 and generation was set as 150The speed of convergence was improved At about the 85thgeneration the curves become stable and reach themaximumvalue

When applying the control strategy to the simulationsystem we got the following result

From Figure 6 we can see that the running strategy wasapplied to save energy consumption and cursing and coast-ing strategy were also applied in appropriate time Runningresults were compared in Table 4

We can see that when running time from Beijing toLangfang was 1610158403210158401015840 when applying the fastest strategyenergy consumption is 39577 kwh When running time wasset extended to 2010158400010158401015840 energy consumption was reducedto about 32524 kwh and 32472 kwh which save 1782 and1795 compared with the fastest running time

In order to verify the efficiency of the PMPGA wecompared it with another optimal algorithm one is fromYanXH who proposed an algorithm based on differentialevolution [18] and the other one is from WangDC whoproposed a multiobjective fuzzy optimization [19] We setup module apply the algorithm at the same train and same


0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation0 40 80 120 160 200 240 280

Fitness

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation

Fitness

0 20 40 60 80 100 120 140

(b)

Figure 5 Evolutionary curve of E2 (left) and E3 (right)

2000

1500

1000

500

0

minus500

200

150

100

50

0

minus50

450

400

350

300

250

200

150

100

50

30

20

10

80 130 160 280 330

F F H H

00 00 00 00

460

minus13550 690

10 10

266400 1100194 minus63

2100 85067950

521550

minus621950 3200

251200

121450

minus122300

351200 1450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

350

17 18 19 20 21 22

00

(km

h)

(min

)

(A)

Cross-sectionalview

Longitudinal

Curve

view

SpeedElectric current

Speed limitPower

R400

L125

R460

L461

R800

L1000

R9000 R4500

L1000

R10000

L549

R4000

L4719

R3500

L653 L526

Figure 6 Result of normal specified time running strategy

Table 3 Railways line parameters and units

Distance Gradient Altitude Slopelength

Curveposition

Curveradius

Curvelength Station Speed

limitTunnelposition

Bridgeposition Others

km permil m m km m m km kmh km km mdash

Table 4 Comparison of running results

Rail line Section length Running strategy Time set Actual running time Energy consumptionBeijing-Langfang 595 km Fastest mdash 16min 32 s 39577 kwhBeijing-Langfang with SGA 595 km Specified time 20min 00 s 19min 59 s 32524 kwhBeijing-Langfang with PMPGA 595 km Specified time with GA 20min 00 s 20min 00 s 32472 kwh


Table 5 Experiment confrontation with other algorithms

Experiment Section length Running strategy Time set Actual running time Energy consumptionBeijing-Langfang withPMPGA 595 km Specified time with GA 20min 00 s 20min 00 s 32472 kwh

Beijing-LangfangE5 595 km Differential evolution 20min 00 s 20min 00 s 33629 kwh

Beijing-LangfangE6 595 km Fuzzy optimization 20min 00 s 19min 59 s 34021 kwh

railway lines and get the following results In Table 5 wedefine Yanrsquos experiment as E5 and Wangrsquos as E6 The resultshows that with Yanrsquos algorithm the train was run with abetter accuracy in time and E6 is worse But E5 and E6rsquosexperiments show that the energy consumption was about356 and 477 more than the PMPGA result It is provedthat the PMPGA algorithm is better with the fuzzy controloptimization and algorithm based on differential evolution

6 Conclusion

When a train running schedule is fixed security stop preci-sion and riding comfortmust be satisfiedWe can save energyconsumption by optimizing the control strategy In this papera SGA and PMPGA were applied to find a perfect runningbased on a specified time By taking the Beijing-ShanghaiHigh-Speed Railway (Beijing-Langfang section) as a case theresult demonstrates that the SGA and PMPGA were able toreduce energy consumption but the improved PMPGA hashigher speed to convergence and has achieved conspicuousenergy reduction also PMPGA has achieved better resultcompared with the multiobjective fuzzy optimization algo-rithm and differential evolution based algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V Prakash Bhardwaj and Nitin ldquoOn the minimization ofcrosstalk conflicts in a destination based modified omeganetworkrdquo Journal of Information Processing Systems vol 9 no2 pp 301ndash314 2013

[2] J Hui Chong C Kyun Ng N Kamariah Noordin and B MohdAli ldquoDynamic transmit antenna shuffling scheme for MIMOwireless communication systemsrdquo Journal of Convergence vol4 no 1 2013

[3] S Masoumi R Tabatabaei M-R Feizi-Derakhshi and KTabatabaei ldquoA new parallel algorithm for frequent pattern min-ingrdquo Journal of Computational Intelligence and Electronic Sys-tems vol 2 no 1 pp 55ndash59 2013

[4] H Kumar Gupta P K Singhal G Sharma and D PatidarldquoRectenna system design in L-band (1-2 GHz) 1 3 GHz for wire-less power transmissionrdquo Journal of Computational Intelligenceand Electronic Systems vol 1 no 2 pp 149ndash153 2012

[5] L Yang YHu and L Sun ldquoEnergy-saving track profile of urbanmass transitrdquo Journal of Tongji University vol 40 no 2 pp 235ndash240 2012

[6] Y V Bocharnikov A M Tobias C Roberts S Hillmansenand C J Goodman ldquoOptimal driving strategy for tractionenergy saving on DC suburban railwaysrdquo IET Electric PowerApplications vol 1 no 5 pp 675ndash682 2007

[7] J-F Chen R-L Lin and Y-C Liu ldquoOptimization of an MRTtrain schedule reducing maximum traction power by usinggenetic algorithmsrdquo IEEE Transactions on Power Systems vol20 no 3 pp 1366ndash1372 2005

[8] I P Milroy Aspects of automatic train control [PhD thesis]Loughborough University 1980

[9] PGHowlett ldquoExistence of an optimal strategy for the control ofa trainrdquo School of Mathematics Report 3 University of SouthAustralida 1988

[10] T Kawakami ldquoIntegration of heterogeneous systemsrdquo in Pro-ceedings of the Fourth International Symposium on AutonomousDecentralized Systems pp 316ndash322 1993

[11] J-X Cheng ldquoModeling the energy-saving train control prob-lems with a long-haul trainrdquo Journal of System Simulation vol11 no 4 1999

[12] H S Hwang ldquoControl strategy for optimal compromisebetween trip time and energy consumption in a high-speed rail-wayrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 28 no 6 pp 791ndash802 1998

[13] T Songbai ldquoStudy on the running resistance of Quasi-highspeed passenger trainsrdquo Science of China Railways vol 18 no1 1997

[14] Z Zhongyang and S Zhongyang ldquoAnalysis of additionalresistance calculation considering the length of the train anddiscuss of the curve additional resistance clause in the TractionRegulationsrdquo Railway Locomotive amp Car vol 2 2000

[15] I Golovitcher ldquoAn analytical method for optimum train controlcomputationrdquo Izvestiya Vuzov Seriya Electrome Chanica no 3pp 59ndash66 1986

[16] E Khmelnitsky ldquoOn an optimal control problem of trainoperationrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 45 no 7 pp 1257ndash12662000

[17] B Singh and D Krishan Lobiyal ldquoA novel energy-aware clusterhead selection based on particle swarm optimization for wire-less sensor networksrdquoHuman-Centric Computing and Informa-tion Sciences vol 2 article 13 2012

[18] X H Yan B G Cai and B Ning ldquoResearch on multi-objectivehigh-speed train operation optimization based on differentialevolutionrdquo Journal of the China Railway Society vol 35 no 92013

[19] D C Wang K P Li and X Li ldquoMulti-objective energy-savingtrain schedulingmodel based on fuzzy optimization algorithmrdquoScience Technology and Engineering vol 12 no 12 2012

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high-speed trains these methods cannot be applied in high-speed trains for energy optimization

For high-speed trains energy saving and trains controloptimization were also studied by scholars Kawakami [10]from Central Japan Railway Company presents a dynamicpower saving strategy for Shinkansen traffic control theauthor made conclusion that predictive simulations in everylayer and target shooting operation of trains are the basisfor energy control With consideration of track gradient andspeed limits Cheng [11] summarized train control problemswith two different models tractionmechanical energymodel(TMEM) and traction energy model (TEM) in a long-haultrain Hwang [12] presented an approach to identify a fuzzycontrol model for determining an economic running patternfor a high-speed railway through an optimal compromisebetween trip time and energy consumption

In this paper taking the Beijing-Shanghai High-SpeedRailway as a case an improved PMPGA was applied tofind a perfect running with a specified running In thisresearch security stop precision and riding comfort wereconsidered and also the railway line parameter includes theslop tunnel and curve The result demonstrates that thePMPGA improved algorithm was better with the SGA andit has achieved conspicuous energy reduction

2 Train Traction Module

21 Train Traction Property Traction property curve is animportant curve demonstrating the relationship betweentrain traction effort and speed It was the most significantworkwhen a trainwas designed Figure 1 shows the schematicdiagram of traction property curve calculation

In Figure 1 there are three curves the top one is adhesion-limited braking force 119865max = 119891(V) the middle one is tractioneffort property 119865 = 119891(V) and the bottom one denoted as Wis the sum of resistances (eg bearing rolling air and graderesistance)119882 = 119891(V) Note that point A the cross of 119865max =119891(V) and119882 = 119891(V) correspond V

119886 is greater than the Vmax

Now according to the curve traction effort property119865 = 119891(V)could be generated as

(119865V minus 119865V0)

V=(119865V1015840 minus 119865V0)

V10158400 le V le V1015840

119865V lowast V = 119865max lowast 119881max V1015840 le V le Vmax

(1)

In the above formula 119865V represents the traction force whenthe speed is VV1015840 is the speed on the intersection point ofconstantmoment segment and constant power segment119865maxrepresents traction force limitation

22 Train Resistance To ensure that the TE was able todrive the train with a speed the total resistances in thispaper defined as119882 must be known Total resistances includebasic resistance119882

0(axle friction resistance track resistance

rolling resistance journal resistance air force resistance andvibration resistance) and extra resistance 119882

119895 119882119895includes

grade resistance (119882119894) curve resistance (119882

119903) and tunnel

resistance (119882119904)

Fmax

998400 (998400) (998400) max

A

W = f()

Fmax = f()

1

2

3

a

Train speed (kmh)

Trac

tion

effor

tF(k

N)

F0

(F0 )

Figure 1 Diagram of traction property curve calculation

It [13] was found that speed was the main factor whicheffects the basic resistance and basis resistance can beexpressed by a quadratic equation formulated as follows

1205960= 119886 + 119887V + 119888 sdot V2 (2)

where the coefficients 119886 119887 and 119888 are dependent on axle loadnumber of axles cross-section of the train and shape of thetrain

According to [14] considering the train as a multiparticleobject we can have the 119908

119895(119909) as the following function

119908119895(119909) =

1

119871[sum 119894119894lowast 119897119894+ 600sum

119897119903119894

119877+sum(119908

119904lowast 119897119904)] (3)

where 119871 is the length of the train and 119894119894and 119897119894represent the

gradient and grade length 119877 119897119903119894are the curve radius and

length 119908119904119894 119897119904119894are the tunnel resistance and length

Then the motion equation and the 119886 V119894 and 119878

119894were

formulated as below

119886 =119889V119889119905=119865 minus 119861 minus (120596

119894+ 120596119903+ 120596119904+ 1205960)

119872 (1 + 120574)

119881119894= 119886Δ119905 + 119881

119894minus1

119878119894=119881119894+ 119881119894minus1

2Δ119905 + 119878

119894minus1

(4)

where119881119894was the speed of currentmoment V

119894minus1was the speed

of last moment 119886 was the acceleration of current moment 119904119894

was the distance of currentmoment from the first station and119904119894minus1

was the distance of the last moment from the first station

3 Traction Energy Module (TEM)

In order to achieve minimal energy consumption generallytrain control for running between stations including accel-eration cruising coasting and braking should be applied atappropriate time Golovitcher [15] and Khmelnitsky [16] ana-lyzed the trainmovement process with nonlinear constrained


BA C D E FSpee

d lim

it (k

mh

)


A = 120587r2








119899le 119905119899+1= 119905 (5)




119896minus 119909119896+1] and

in [0 119909]


119899le 119909119899+1= 119909 (6)


min 119864 = int

119909119891

1199090

119906119891(119909) 119891 (V) 119889119909

st

119889119905

119889119909=1

V

V119889V119889119909=

119906119891(119909) 119891 (119909) minus 119906

119887(119909) 119887 (V)

119872119892minus 1199080(V) minus 119908

119895(119909)

700

600

500

400

300

200

100

00 50 100 150 200 250

Speed (kmh)

(1)

(2)

(3)

(4)


Trac

tion

effor

t (kN

)


119905 (1199090) = 0 119905 (119909

119891) = 119879 V (119909

0) = 0 V (119909

119891) = 0

V le 119881 (119909) 119906119891isin [0 1] 119906

119887isin [0 1]

(7)


0) is start time 119905(119909

119891)




119891) are equal





119862 = [1198880 1198881 1198882 119888

119894 119888

119899minus1 119888119899] (8)




that 119904119894isin S 119909

0= 0 and 119909









119875V

E1 3 50 100 07 300 0068 02E2 3 100 100 07 300 0068 02E3 6 50 100 07 150 0068 02




119862 = [(1199090 1199040) (1199091 1199041) sdot sdot sdot (119909

119894 119904119894) sdot sdot sdot (119909

119897 119904119897)] (9)








1015840





Fit (119909) = 1

119862max + obj + 119862 (10)




119894


119894 by

119901119894=

Fit (119909119894)

sum119873

119895=1Fit (119909

119895)

(11)


119896where 119896


119896 = min119895 |119895minus1

sum

119894=1

119901119894le 119902 119895 = 1 2 119873 (12)



119883119905+1

1= 119903119883119905

1+ (1 minus 119903)119883

119905

2

119883119905+1

2= 119903119883119905

2+ (1 minus 119903)119883

119905

1

(13)

In formula (13) 1198831199051 119883119905


119883119905+1

1 119883119905+1







0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitness

Fitn

ess

Generation0 40 80 120 160 200 240 280

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Fitness

Generation0 40 80 120 160 200 240 280

(b)


















0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation0 40 80 120 160 200 240 280

Fitness

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation

Fitness

0 20 40 60 80 100 120 140

(b)


2000

1500

1000

500

0

minus500

200

150

100

50

0

minus50

450

400

350

300

250

200

150

100

50

30

20

10

80 130 160 280 330

F F H H

00 00 00 00

460

minus13550 690

10 10

266400 1100194 minus63

2100 85067950

521550

minus621950 3200

251200

121450

minus122300

351200 1450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

350

17 18 19 20 21 22

00

(km

h)

(min

)

(A)

Cross-sectionalview

Longitudinal

Curve

view


Speed limitPower

R400

L125

R460

L461

R800

L1000

R9000 R4500

L1000

R10000

L549

R4000

L4719

R3500

L653 L526




Curveposition

Curveradius


limitTunnelposition











6 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










BA C D E FSpee

d lim

it (k

mh

)


A = 120587r2








119899le 119905119899+1= 119905 (5)




119896minus 119909119896+1] and

in [0 119909]


119899le 119909119899+1= 119909 (6)


min 119864 = int

119909119891

1199090

119906119891(119909) 119891 (V) 119889119909

st

119889119905

119889119909=1

V

V119889V119889119909=

119906119891(119909) 119891 (119909) minus 119906

119887(119909) 119887 (V)

119872119892minus 1199080(V) minus 119908

119895(119909)

700

600

500

400

300

200

100

00 50 100 150 200 250

Speed (kmh)

(1)

(2)

(3)

(4)


Trac

tion

effor

t (kN

)


119905 (1199090) = 0 119905 (119909

119891) = 119879 V (119909

0) = 0 V (119909

119891) = 0

V le 119881 (119909) 119906119891isin [0 1] 119906

119887isin [0 1]

(7)


0) is start time 119905(119909

119891)




119891) are equal





119862 = [1198880 1198881 1198882 119888

119894 119888

119899minus1 119888119899] (8)




that 119904119894isin S 119909

0= 0 and 119909









119875V

E1 3 50 100 07 300 0068 02E2 3 100 100 07 300 0068 02E3 6 50 100 07 150 0068 02




119862 = [(1199090 1199040) (1199091 1199041) sdot sdot sdot (119909

119894 119904119894) sdot sdot sdot (119909

119897 119904119897)] (9)








1015840





Fit (119909) = 1

119862max + obj + 119862 (10)




119894


119894 by

119901119894=

Fit (119909119894)

sum119873

119895=1Fit (119909

119895)

(11)


119896where 119896


119896 = min119895 |119895minus1

sum

119894=1

119901119894le 119902 119895 = 1 2 119873 (12)



119883119905+1

1= 119903119883119905

1+ (1 minus 119903)119883

119905

2

119883119905+1

2= 119903119883119905

2+ (1 minus 119903)119883

119905

1

(13)

In formula (13) 1198831199051 119883119905


119883119905+1

1 119883119905+1







0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitness

Fitn

ess

Generation0 40 80 120 160 200 240 280

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Fitness

Generation0 40 80 120 160 200 240 280

(b)


















0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation0 40 80 120 160 200 240 280

Fitness

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation

Fitness

0 20 40 60 80 100 120 140

(b)


2000

1500

1000

500

0

minus500

200

150

100

50

0

minus50

450

400

350

300

250

200

150

100

50

30

20

10

80 130 160 280 330

F F H H

00 00 00 00

460

minus13550 690

10 10

266400 1100194 minus63

2100 85067950

521550

minus621950 3200

251200

121450

minus122300

351200 1450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

350

17 18 19 20 21 22

00

(km

h)

(min

)

(A)

Cross-sectionalview

Longitudinal

Curve

view


Speed limitPower

R400

L125

R460

L461

R800

L1000

R9000 R4500

L1000

R10000

L549

R4000

L4719

R3500

L653 L526




Curveposition

Curveradius


limitTunnelposition











6 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119875V

E1 3 50 100 07 300 0068 02E2 3 100 100 07 300 0068 02E3 6 50 100 07 150 0068 02




119862 = [(1199090 1199040) (1199091 1199041) sdot sdot sdot (119909

119894 119904119894) sdot sdot sdot (119909

119897 119904119897)] (9)








1015840





Fit (119909) = 1

119862max + obj + 119862 (10)




119894


119894 by

119901119894=

Fit (119909119894)

sum119873

119895=1Fit (119909

119895)

(11)


119896where 119896


119896 = min119895 |119895minus1

sum

119894=1

119901119894le 119902 119895 = 1 2 119873 (12)



119883119905+1

1= 119903119883119905

1+ (1 minus 119903)119883

119905

2

119883119905+1

2= 119903119883119905

2+ (1 minus 119903)119883

119905

1

(13)

In formula (13) 1198831199051 119883119905


119883119905+1

1 119883119905+1







0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitness

Fitn

ess

Generation0 40 80 120 160 200 240 280

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Fitness

Generation0 40 80 120 160 200 240 280

(b)


















0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation0 40 80 120 160 200 240 280

Fitness

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation

Fitness

0 20 40 60 80 100 120 140

(b)


2000

1500

1000

500

0

minus500

200

150

100

50

0

minus50

450

400

350

300

250

200

150

100

50

30

20

10

80 130 160 280 330

F F H H

00 00 00 00

460

minus13550 690

10 10

266400 1100194 minus63

2100 85067950

521550

minus621950 3200

251200

121450

minus122300

351200 1450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

350

17 18 19 20 21 22

00

(km

h)

(min

)

(A)

Cross-sectionalview

Longitudinal

Curve

view


Speed limitPower

R400

L125

R460

L461

R800

L1000

R9000 R4500

L1000

R10000

L549

R4000

L4719

R3500

L653 L526




Curveposition

Curveradius


limitTunnelposition











6 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitness

Fitn

ess

Generation0 40 80 120 160 200 240 280

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Fitness

Generation0 40 80 120 160 200 240 280

(b)


















0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation0 40 80 120 160 200 240 280

Fitness

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation

Fitness

0 20 40 60 80 100 120 140

(b)


2000

1500

1000

500

0

minus500

200

150

100

50

0

minus50

450

400

350

300

250

200

150

100

50

30

20

10

80 130 160 280 330

F F H H

00 00 00 00

460

minus13550 690

10 10

266400 1100194 minus63

2100 85067950

521550

minus621950 3200

251200

121450

minus122300

351200 1450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

350

17 18 19 20 21 22

00

(km

h)

(min

)

(A)

Cross-sectionalview

Longitudinal

Curve

view


Speed limitPower

R400

L125

R460

L461

R800

L1000

R9000 R4500

L1000

R10000

L549

R4000

L4719

R3500

L653 L526




Curveposition

Curveradius


limitTunnelposition











6 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation0 40 80 120 160 200 240 280

Fitness

(a)

0000140

0000138

0000136

0000134

0000132

0000130

0000128

0000126

Fitn

ess

Generation

Fitness

0 20 40 60 80 100 120 140

(b)


2000

1500

1000

500

0

minus500

200

150

100

50

0

minus50

450

400

350

300

250

200

150

100

50

30

20

10

80 130 160 280 330

F F H H

00 00 00 00

460

minus13550 690

10 10

266400 1100194 minus63

2100 85067950

521550

minus621950 3200

251200

121450

minus122300

351200 1450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

350

17 18 19 20 21 22

00

(km

h)

(min

)

(A)

Cross-sectionalview

Longitudinal

Curve

view


Speed limitPower

R400

L125

R460

L461

R800

L1000

R9000 R4500

L1000

R10000

L549

R4000

L4719

R3500

L653 L526




Curveposition

Curveradius


limitTunnelposition











6 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















6 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









optimization of high-speed train control strategy for traction

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