Energy-Efficient Train Operation Using Nature-Inspired Algorithmsdownloads.hindawi.com/journals/jat/2017/6173795.pdf · 2019. 7. 30. · Ford algorithm [30], reinforcement learning

Research ArticleEnergy-Efficient Train Operation Using Nature-InspiredAlgorithms

Kemal Keskin and Abdurrahman Karamancioglu

Department of Electrical and Electronics Engineering Eskisehir Osmangazi University 26480 Eskisehir Turkey

Correspondence should be addressed to Kemal Keskin kkeskinoguedutr

Received 27 July 2016 Revised 5 October 2016 Accepted 25 October 2016 Published 12 January 2017

Academic Editor Andrea DrsquoAriano

Copyright copy 2017 K Keskin and A Karamancioglu This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A train operation optimization by minimizing its traction energy subject to various constraints is carried out using nature-inspired evolutionary algorithms The optimization process results in switching points that initiate cruising and coasting phasesof the driving Due to nonlinear optimization formulation of the problem nature-inspired evolutionary search methods GeneticSimulated Annealing Firefly and Big Bang-Big Crunch algorithms were employed in this study As a case study a real-like train andtest track from a part of Eskisehir light rail network weremodeled Speed limitations various track alignments maximum allowabletrip time and changes in train mass were considered and punctuality was put into objective function as a penalty factor Resultshave shown that all three evolutionary methods generated effective and consistent solutions However it has also been shown thateach one has different accuracy and convergence characteristics

1 Introduction

Researchers have been focusing on new energy saving areasas energy becomes more expensive besides scarce availabilityand negative environmental effects in the process of itsgeneration and consumption One such area is the trainoperations process where there is a significant potentialto reduce energy consumption by optimizing operationstrategies Energy savings through improvements in drivingstrategies do not require any hardware modification oradditional manufacturing costs therefore it is the first choiceinmany cases In this manuscript we develop energy-efficientdriving strategies by choosing switching times among themotion modes of a railway vehicle In the present contextthe motion modes of a railway vehicle are traction cruisingcoasting and braking modes Finding the optimal switchingtimes involves solving a nonlinear optimization problemwithobjective function containing an integral and constraintsinvolving a differential equations set

In 1968 Ichikawa investigated train operation betweensuccessive stations in terms of minimizing energy consump-tion A method was proposed to decrease the complexityof the state variable problem and solve it afterwards [1]

In 1975 Hoang et al tried to reduce energy consumptionof train operation from a different viewpoint They notethat positioning of stations and the paths between themare already determined therefore remaining task wouldbe determining the ldquotunnel trace in the equivalent verticalplanerdquo They constructed an optimal control model andprovided a heuristic approach using a direct search algorithmThe proposed method was applied to a part of MontrealSubway [2] There is a line of research where Pontryaginrsquosmaximum principle was used to find optimal operationstrategy [3ndash5] In this line Howlett showed that an optimalstrategy for energy minimization should consist of accelera-tion cruising coasting and braking phases respectively [3]In this respect illustrative numerical examples of drivingstrategies for various speeds and gradients took place in[6 7] In 1997 Chang and Sim proposed a coasting controlstrategy based on genetic algorithm Punctuality and comfortwere taken into account in addition to energy consumptionThe proposed method has a strong mathematical basisand it converges in a significantly short time Also thisoptimized coast control strategy showed better performancein comparison with fuzzy ATO controller [8] In anotherstudy Howlett considered the problem using continuous and

HindawiJournal of Advanced TransportationVolume 2017 Article ID 6173795 12 pageshttpsdoiorg10115520176173795

2 Journal of Advanced Transportation

discrete control tools and used the Kuhn-Tucker equationsto analyze optimal switching points [9] Liu and Golovitcherproposed an analytical method based on maximum prin-ciple to find optimal input sequence and switching pointsDeveloped system was capable of finding optimal trajec-tory and suitable for real-time optimization due to itsanalytical nature [10] In 2004 Wong and Ho researchedto determine number of coasting and switching points bymeans of three various genetic algorithms [11] Acikbas andSoylemez developed simulation software which is suitable formultitrain and multiline tests They applied artificial neuralnetwork and genetic algorithm to model a part of metroline in Istanbul and compared these approaches in terms ofenergy consumption [12] In 2009 Howlett et al presenteda new analytical solution for tracks with steep uphill anddownhill sections to find optimal switching points [13] Kimand Chien investigated train operation with constraints oftime speed and energy consumption A simulation modelwas developed [14] and simulated annealing algorithm wasutilized to calculate optimal switching points and cruisingspeed It was clarified that it was possible to reduce energyconsumption by increasing the travel time [15] In anotherstudy optimization problem was handled by three differentmethods dynamic programming gradient methods andsequential quadratic programming [16] In [17] Sheu andLin focused on automatic train regulation (ATR) problemin the energy efficiency framework They developed a dualheuristic programming method which brought the ATRability of real-like adaptation In addition to controllingcoasting points and dwell time at stations scheduling opti-mization could increase energy efficiency performance Anenergy-efficient driving strategy and optimization methodfor manually driven high speed trains were developed [18]Energy-efficient strategies and timetable optimization werecombined together in [18ndash22] For subway systems recoveredenergy coming from regenerative braking can be used fortrain traction In this regard Yang et al developed schedulingrules that synchronized successive trains for braking andaccelerating Overlapping time was maximized by means ofinteger programming model and for timetable optimizationgenetic algorithm was utilized [23] Lu et al investigatedenergy-efficient train trajectory by means of dynamic speedcontrol Ant colony optimization genetic algorithm anddynamic programming algorithmswere used in searching theoptimal trajectory [24] Su et al proposed a new approachby combining optimal driving strategy with cooperative traincontrol The energy which came from regenerative brakingis used for traction of other trains [25] Similarly whileone research controlled headway time and dwell time toincrease energy savings from regenerative braking [26] theother one developed stop-skipping method [27] to decreasepassenger waiting time In another research a concept ofdynamic infrastructure occupation was presented to assessinfrastructure capacity under disturbed conditions as a com-plement to the established capacity indicator of scheduledinfrastructure occupation This new indicator is applied in acapacity assessment study of a Dutch railway corridor withdifferent signaling configurations under both scheduled anddisturbed traffic conditions [28] During the recent years

several other methods were applied to optimal train controlproblem such as fuzzy predictive control [29] Bellman-Ford algorithm [30] reinforcement learning [31] swarmoptimization [32] and NSGA-II algorithm [33]

In this study an energy-efficient train operation problemwas considered on a track with no steep sections Threeheuristic approaches Firefly Algorithm Genetic SimulatedAnnealing and Big Bang-Big Crunch were used to findswitching points for phases There is no known study ontrain operation optimization problem which employs oneof these algorithms In this study it was demonstrated thatthese three algorithms are appropriate to apply to the energy-efficient train operation optimization problem and a compar-ison between algorithms running times and optimality wasdiscussed Besides the fact that the train model and the trackwere real-like modeled the effect of number of passengers ontrain energy consumption and algorithmrsquos performance wereevaluated However the problem in case study was solvedsuccessfully for a complicated track with steep sections orspeed limitations more complex strategies are required (see[5 7 9 10 13])

In the next section we present the nonlinear optimizationformulation of the problem In Section 3 the evolutionarysolution methods used in this manuscript are introduced Inthe 4th section the solution methods are applied to a modelof locally existing real problem In the remaining part of themanuscript performances of the methods are discussed

2 Modeling the Motion

Themotion equations of train usingNewtonrsquos second law canbe written in the following form

119889119909119889119905 = V119889V119889119905 = 119865119905 minus 119865119887119898 minus 119877 minus 119877119892

(1)

where119909 and V are position and speed of the train respectively119865119905 is the tractive force 119865119887 is the braking force 119877 is the rollingresistance 119877119892 is the resistance caused by level change and119898 denotes mass of train In the sequel a train motion ona sequence of successive stations is considered We denotethe distance between stations 119894 and 119894 + 1 by 119883119894 allowedtravel time by 119879119894 and allowed maximum speed by 119881 Hencebetween stations 119894 and 119894 + 1 these parameters are restrictedwith following limits

0 le 119909 le 1198831198940 le 119905 le 1198791198940 le V le 119881(2)

Resistance of the train 119877 can be calculated by utilizing Davisequation [34]

119877 = 119860 + 119861V + 119862V2 (3)

where the coefficients 119860 119861 and 119862 correspond to massmechanical and air resistance respectivelyThese coefficients

Journal of Advanced Transportation 3

40

35

30

25

20

15

10

Trac

tive e

ffort

(kN

)

Speed (kmh)8647927264857650443236288216144720

Figure 1 Tractive effort speed graph for a tram with power 571 hp

vary depending on external forces and physical character-istics of train Level changes in track can be favorable eventhough they can function as a resistance against to the trainmotion For downhill part of track the contribution to theacceleration is positive and for uphill part of track it isnegative The resistance caused by gradient can be calculatedas follows

119877119892 = 119892 sin120572 (4)

where 119892 is gravitational acceleration and 120572 is the angle ofslope Tractive effort provides force to move train along therail line

Tractive effort is specifically defined according to trainrsquoscharacteristics It is restricted to certain limits due to adhesionbetween wheel and rail surfaces Tractive effort calculationmostly depends on engine power and current speed of train(see Figure 1) Maximum tractive effort is available at lowspeed For the beginning of motion maximum tractive effort(here it is 36 kN) is applied until train reaches 30 kmh speedOver this speed level power takes its maximum constantvalue and tractive effort changes inverse proportional tospeed It can be calculated by [35]

119865119905 = 2650120583119875V (5)

where120583 is the efficiency in convertingmotor power to tractiveforce 119875 is motor power V is the current speed of trainand 2650 is for unit conversion Using this equation averagepower can be calculated

119875 = 119865119905V2650120583 (6)

The total energy consumption is obtained by integrating thepower over time

119864 = int1198790119875119889119905 (7)

21 Train Operation Energy consumption of a train con-siderably depends on train operation An optimal trainoperation for a level track should consist of the following

70

60

50

40

30

20

10

Station 1Position (m)

MA CR CO BR

Station 2

Spee

d (k

mh

)

Figure 2Motion phasesMAmaximumacceleration CR cruisingCO coasting and BR braking

motion phases respectively maximum acceleration (MA)cruising (CR) coasting (CO) and braking (BR) [3] Anexample driving scenario between two successive stationsis shown in Figure 2 Next we provide more details on themotion phases below

211 Maximum Acceleration (MA) From beginning of thetravel till the start of the cruising phase maximum acceler-ation is applied to the train As mentioned earlier tractiveeffort is restrictedwith adhesion limitThemaximum tractiveeffort is calculated by

119865max = 119898119892120578 (8)

where 119898 is the mass of train 119892 is gravitational accelerationand 120578 is the friction constant Under an applied constantpower tractive effort stays constant until train comes to thecruising speed value

212 Cruising (CR) In this phase train continues its travelat a constant speed In order to hold the speed at constantvalue the applied tractive effort must equal the opposingforces to the train motion Uphill and downhill sections oftrack either contribute to or take away from the amount oftractive effort For some downhill sections there may be noneed for traction

213 Coasting (CO) In the coasting phase trainmoves alonga line under already obtained momentum and no tractionenergy is consumed This phase continues until train reachesthe safe stopping distance Safe stopping distance is a functionof remaining distance and current speed of train The safestopping distance is

119909ss = V2br2119886 (9)

where Vbr is current speed subject to braking 119886 is decel-eration and 119909ss is safe stopping distance On tracks withnonsteep constant gradient optimal braking speed whichdepends only on the uniquely defined cruising speed can becalculated by (9) This formula is efficient for level tracks


214 Braking (BR) In this phase constant force opposing thedirection of motion is applied to train Magnitude of brakingforce depends on train characteristics Since we just considertraction power for calculation of energy consumption inbraking phase it is assumed that there is no contribution tototal energy consumption

22 Total Energy Consumption and Optimization Totalenergy consumption for an optimal strategy can be decom-posed into its motion phases In the maximum accelerationphase the traction effort is fixed at its attainable maximumvalue therefore speed monotonically increases The end-point of this phase is denoted by 1198961 In the cruising phasespeed is fixed to a value Vcr which requires 119865119905 values to equalthe opposing resistance values This phase extends betweenthe points 1198961 and 1198962 The points 1198961 and 1198962 are called theswitching points In the subsequent phases (ie coasting andbraking phases) no energy is consumed due to zero tractionforceThus total energy consumption for an optimal strategythroughout successive stations is calculated by the followingequation

119864TOTAL = 119864MA + 119864CR (10)

where 119864MA and 119864CR denote energies consumed at the max-imum acceleration and cruising phases Equation (10) is validfor track with nonsteep sections It is desired to provideenergy-efficient travel while considering punctuality andcomfort A correct decision-making on the switching pointsbetween the phases has primary significance for the problemunder consideration since it determines the energy consump-tion Finding optimal switching points for a travel betweenstations 119894 and 119894 + 1 can be formulated as an optimizationproblem

min1198961 1198962

(int11989610119865119905 (119905) V (119905) 119889119905 + int1198962

1198961

119865119905 (119905) V119888119903119889119905)Subject to 119889V (119905)119889119905 = 119865119905 (119905) minus 119865119887 (119905)119898 minus 119877 (119905) minus 119877119892 (119905)

0 le 119905 le 119879119894119889119909 (119905)119889119905 = V (119905) 0 le 119905 le 119879119894119865119905 (119905) = 119865119905max 0 le 119905 le 1198961119865119905 (119905)= 5 + 00285Vcr + 00047V2cr + 119898119892120572

1198961 lt 119905 le 1198962119865119905 (119905) = 0 1198962 lt 119905 le 119879119894119865119887 (119905) = 0 0 lt 119905 le 1198963

119865119887 (119905) = 119865119887max 1198963 lt 119905 le 1198791198940 le 119865119905 le 119865119905max0 le 119865119887 le 119865119887maxV (0) = V (119879119894) = 00 le V (119905) le V (119879119894)119909 (0) = 0119909 (119879119894) = 119883119894

(11)Having formulated the energy optimization problem

above in the next section for its solution we present a reviewof three different evolutionary algorithms

3 Optimization Methods

In this section Genetic Simulated Annealing Firefly and BigBang-Big Crunch algorithms are reviewed briefly where theformer one is a hybrid algorithm and the latter two are stand-alone algorithms

31 Genetic Simulated Annealing Algorithm Genetic Algo-rithm (GA) and Simulated Annealing (SA) are two well-known tools for solving global optimization problems GA isan evolutionary searchmethod based on evolutionary theorySearch proceduremimics the natural genetics using operatorssuch as selection mutation and crossover Chromosomesrefer to candidate solutions and each of them is assigneda score with regard to fitness function New offspring aregenerated by applying genetic operators to chromosomesAfter several generations chromosomes which have betterscores are selected as optimal or suboptimal solution SA isanother nature-inspired optimization method which showsan analogy to physical annealing process in metallurgy Inthe physical process temperature is reduced gradually in thecooling phase of the heated material in order to preventdefects In the mathematical counterpart SA starts to searchfrom an initial point and next new candidate solutions aregenerated randomly by reducing temperature From newgenerations not only better solutions but also some worsesolutions are accepted with a certain probability Thus localminima can be avoided and the chance of finding optimumsolution is increased [36] The algorithms GA and SA havestand-alone features which can be used together to eliminateeach onersquos typical weaknesses GA employs the efficiencyof evolution theory such that new offspring have severalcharacteristics in common with its parent In this way thequality of solutions is maintained With the help of itsextensive search capability GA is practical for solving toughproblems However besides the uncertainty of computationaltime it can be incapable of avoiding local extrema in limitedtime as well [37] With the help of random search natureSA accepts worse solutions in addition to better ones with acertain rate It prevents being caught to a local extremum [38]


Start

Input parameterscost function

definition

Generate initialpopulation

Evaluation

Selectioncrossovermutation

Satisfystoppingcriteria Satisfy

stoppingcriteria

Output

Accept

Stop

No

Yes

No

No

Yes Yes

Yes

No

Update T

Generate sk+1from sk

Accept sk+1

k = k + 1

probability eminusΔT

Initialize s T k

E (sk+1) minus E(sk) lt 0

Figure 3 Flowchart of Genetic Simulated Annealing algorithm

Even though the SA can avoid local extrema its efficiencydepends on initial point Choosing inappropriate initial pointmay result in worse solutions and a long computational time

Genetic Simulated Annealing (GSA) is a combinationof GA and SA At the beginning of algorithm initializationparameters such as population size number of variableslower and upper bounds for each variable mutation andcrossover rates selection method annealing and temper-ature functions are defined Then GA part of algorithmis activated and stopping criteria are defined as a certainnumber of generations At the end of this part of algorithma suboptimal solution is generated The second part ofalgorithm employs SA with the initial solution from the firstpart Algorithm flowchart is given in Figure 3

GSA has been applied to many areas including jobscheduling [39] multiple project scheduling [40] discretetime-cost tradeoff [41] traveling salesman and error correct-ing code design [42] mixed-model assembly line sequencing[43] and train energy optimization [36] problems

32 Big Bang-Big CrunchMethod Big Bang-Big Crunch (BB-BC) is a global optimization method which is inspired bythe formation of the universe BB-BC method comprisestwo main phases big bang and big crunch At the bigbang phase individuals from initial population scatter alongthe search space randomly On that sense this phase ofalgorithmhas resemblance toGAAfter random initializationof population individuals take various places in search space

Random number generators are adjusted to certain values tohold newoffspring in the search spaceThen big crunch phasefollows the big bang phase An output point namely centerof mass is generated based on population data This crunchprocess can be formulated for aminimization problem as [44]

119904119888 = sum119870119894=1 (1119891119894) 119904119894sum119870119894=1 (1119891119894) (12)

where 119904119888 is the center of mass 119904119894 is the position vector forthe 119894th individual 119891119894 represents the fitness value of the 119894thindividual and119870 is the population size After the big crunchphase it is required to create new members which will beused in next iteration of big bang phase New populationis generated around the center of mass using followingformulation

119904new119894 = 119904119888 + 120590 (13)

where 119904new119894 stands for new populationrsquos 119894th individual and 120590 isstandard deviation coefficient Through (13) new individualscannot go out of search space Standard deviation coefficientis calculated by

120590 = 05119903 (119904max minus 119904min)1 + 119895ℎ (14)

where 119903 is a random number which is defined with normaldistribution 119895 is iteration number 119904max and 119904min are the upper


and lower limits for search space respectively ℎ is coefficientfor the contract of search space For subsequent iterations thecenter of mass is calculated again and big bang big crunchsteps are repeated until a stopping criterion is met Algorithmsteps can be given as follows

(1) Create a random initial population with119870members(2) Calculate the fitness function of every individual(3) Calculate center of mass using (12)(4) Create new candidates by using (13)(5) Return to step (2) until stopping criteria have been

met

Although BB-BC algorithm has been announced in recentyears it has been applied many areas including design ofspace trusses [45] nonlinear controller design [46] fuzzymodel inversion [47] damage detection [48] and energy-efficient motion control of train [49] problems

33 Firefly Algorithm Firefly Algorithm (FA) is a swarmintelligencemethod inspired by lightning behavior of firefliesIt was proposed by Yang in 2008 [50] FA mainly depends onthree significant ideas

(i) Fireflies have no gender Any of them can be attractedto other fireflies

(ii) Attractiveness is comparative to brightness Forinstance considering two flashing fireflies one whichhas less glitter will move towards to more glitterone When distance increases attractiveness andbrightness decrease expectedly If both fireflies arenot glittery enough to attract other one then randommovement occurs

(iii) The view of objective function defines the brightnessof a firefly It is possible to express brightness indifferent ways however a basic one may make use ofthe objective function of the relevant maximizationproblem

Two issues are worth attention for firefly algorithm lightintensity and attractiveness Essentially the light intensity119868(119889) can be defined using the inverse square law [50]

119868 (119889) = 1198681199041198892 (15)

where 119868119904 refers to the intensity at source and 119889 is the distancebetween fireflies Attractiveness is directly related to the lightintensity seen by neighbor fireflies Let 120573 be attractiveness ofa firefly it can be defined as

120573 = 12057301198901205741198892 (16)

where 1205730 denotes the attractiveness at 119889 = 0 and 120574 is lightabsorption coefficient The distance between two fireflies 119894and 119895 at points 119901119894 and 119901119895 can be defined as follows [50]

119889119894119895 = 10038171003817100381710038171003817119901119894 minus 11990111989510038171003817100381710038171003817 = radic 119897sum119896=1

(119901119894119896 minus 119901119895119896)2 (17)

Objective function 119891(119901) 119901 = (1199011 119901119897)119879Generate initial population 119901119894 (119894 = 1 2 119899)Determine light intensity 119868119894 at 119901119894 by 119891(119901119894)Define light absorption coefficient 120574While (119905 lt 119872119886119909119866119890119899119890119903119886119905119894119900119899) dofor 119894 = 1 119899 do

for 119895 = 1 119899 doif 119868119894 lt 119868119895 then

move firefly 119894 towards 119895end ifupdate attractiveness withdistance 119889 via 119890minus120574119889evaluate new solutions and update 119868119894

end forend forrank the fireflies and find thecurrent global best 119892lowast

end whilepostprocess results

Algorithm 1 Firefly algorithm [50]

where 119901119894119896 is the 119896th component of the spatial coordinate 119901119894of 119894th firefly sdot denotes the Euclidean norm and 119897 denotesthe number of components Also the movement of firefly 119894 tofirefly 119895 is determined by

119901119894 = 119901119894 + 1205730119890minus1205741198892119894119895 (119901119895 minus 119901119894) + 120572120576119894 (18)

where second term refers to attraction and the third termrepresents randomization and120572 is randomization parameterRegarding to the information given above algorithmrsquos pseudocode is shown in Algorithm 1

FA has been applied to many areas including learningrobot motion trajectories [51] heart disease prediction [52]and arterial cannula shape optimization [53] problems

4 A Case Study

This research focuses on energy optimization for an urbanrail transit system In this regard different searchingmethodsfor global optimization problem have been described in theprevious sections In order to verify the efficiency of proposedoptimization algorithms a case study and its results for eachmethod are given in this section

41 Case Study Background A particular segment of Eskise-hir Urban Rail Network was taken into account for thecase study and a real-like tram model was created withcharacteristics which are given in Table 1

The total length of test track is 3314m There are sevenstations where the train must stop (see Figure 4) Travelstarts at Osmangazi University station and ends at Stadyumstation Considering successive stations train motion canbe examined in partial tracks To interpret the figure as


StadyumVişnelik

350m

667m

437m

207m

204m

364m

293m

540m0

0

1

2

2

6

1

1

OsmangaziUniversity

Porsuk

Speed limit

A

B

StationsNegative slope

Positive slopeLevel

Atatuumlrk Bulvarı

15 kmh

Speed limit15 kmh

Goumlztepe

Buumlyuumlkdere

Figure 4 A part of Eskisehir light rail network subjected to test

Table 1 Train characteristics

Total mass 34000 kgMaximum motor power 571 hpNumber of cars 5 pcsMax speed limit 70 kmhCapacity 150 passengers

intended let us read the figure for the first three stationsAt the beginning train starts its motion from OsmangaziUniversity station and stops at Porsuk station The lengthof this part is 364m and there is 1 positive grade Thesecond part of total track is between Porsuk station and thefollowing first sharp curvature This part is 204m long withno gradient Train speed goes down to 15 kmh at the end ofthis part and keeps it at this level along the curvature Afterpassing the curvature new part begins between the curvatureand Buyukdere station Since the train comes from previouspart with 15 kmh constant speed it starts to accelerate from15 kmh in this part This partrsquos length is 207m and has 2positive slope

42 Operation Strategy Only the MA and CR phases con-tribute to the energy consumption of the train As no energyis consumed in CO phase increasing duration of CO phasein a strategy leads to drop in energy consumption Howeverthis affects the total travel time adversely Energy efficiency

Table 2 Estimated motion phases for the parts of track

Part of Track Length Estimated PhaseSequence

Osmangazi University ndashPorsuk 364m MA + CR + CO + BR

Porsuk ndash Curvature A 204m MA + CR + BRCurvature A ndash Buyukdere 207m MA + CR + BRBuyukdere ndash Goztepe 437m MA + CR + CO + BRGoztepe ndash Ataturk Bulvari 667m MA + CR + CO + BRAtaturk Bulvarı ndashCurvature B 293m MA + CR + BR

Curvature B - Visnelik 350m MA + CR + CO + BRVisnelik ndash Stadyum 540m MA + CR + CO + BR

should be provided by adhering to punctuality Thereforepunctuality takes place in the optimization scheme as a hardconstraint and no tradeoff is allowed between punctualityand energy consumption

An optimum trajectory for short distances does notconsist of CO phase [3] In this study the parts with under350m length is considered as a short distance Regarding thisa predicted motion phase sequence for each part of track isgiven in Table 2 Thus the search algorithms to be employeduse this grantedmotion phase sequences and this contributesefficiency of the search processes


Table 3 GSA parameter selection test

Test label Crossover rate Mutation rate Selectionfunction

Crossoverfunction

Annealingfunction

Temperaturefunction Energy cons

GSA 1 08 001 Roulette Single-point Boltzmann Boltzmann 510 kwhGSA 2 09 002 Tournament Two-point Boltzmann Boltzmann 519 kwhGSA 3 07 004 Roulette Intermediate Boltzmann Boltzmann 518 kwhGSA 4 08 001 Roulette Single-point Fast Exponential 512 kwhGSA 5 09 002 Tournament Two-point Fast Exponential 514 kwhGSA 6 07 004 Roulette Intermediate Fast Exponential 515 kwh

43 Optimization Parameters In train operation researcharea optimization of speed profile of a train has a challengingmathematical structure It is desired to find switching pointsfor certain motion phases to minimize energy consumptionby taking constraints on physical limitations time andcomfort into consideration It is important to note thatswitching motion phases from one to another is an NP-hardproblem [54] Since analytical approaches have limitationsin finding a solution to this problem evolutionary methodsbecome prominent instead [15]

For the train model under consideration to test the evo-lutionary optimization methods a simulator was developedin MATLAB environment It takes variable track alignmentsspeed and comfort limitations into consideration In this set-up output consists of speed position and time values andenergy consumption of train

In this research Genetic Simulated Annealing Fire-fly and Big Bang-Big Crunch algorithms were separatelyemployed to minimize energy consumption of a trainPerformances of the methods rely significantly on theirparameter settings The chosen parameters for each methodare presented below

431 Genetic Simulated Annealing Parameters This methodis a combination of two well-known algorithmsThe first oneGenetic algorithm (GA) is capable of finding suboptimalsolutions in short computational times Herewith at thebeginning of optimization GA was employed until it reachesa fitting generation Obtained solution was given to thesecond algorithm simulated annealing algorithm (SA) as aninitial solution For the GA part it is significant to determinenot only crossover and mutation rates but also selectionand crossover functions whereas temperature and annealingfunction are important parameters for second part of themethod

For satisfactory results GSA needs to have well-chosenparameter settings These settings are generally selected byrepeated trial and error To reduce the computational burdenin this process a simplified test track in our case 2000msingle track with various gradients and no curvature isused In the parameter setting process the costs obtained forvarious conditions are given in Table 3 Noting that the testlabeled GSA 1 has the best cost we use its settings for theactual problem with the test track shown in Figure 4 A briefsummary of the settings is as follows

(i) population size 75

(ii) crossover rate 08

(iii) mutation rate 001

(iv) selection function roulette

(v) crossover function single point

(vi) annealing function Boltzmann

432 Big Bang-Big Crunch Algorithm Parameters For BigBang-Big Crunch algorithm finding new solution candidatesis achieved by adding a random number to the center ofmassThis randomnumber value is chosen to be decreased asiteration number increases Parameters which belong to BigBang-Big Crunch algorithm are given as follows


(ii) initial point for each variable to be optimized averageof its attainable minimum and maximum values

(iii) random number 119903119896+1 = 119903119896 sdot 10minus4119873 where 119896 and119873 arethe iteration and generation numbers

433 Firefly Algorithm Parameters Attractiveness and lightabsorption coefficient are two significant parameters to deter-mine the speed of convergence and efficiency of firefly algo-rithm For the simulations to be carried out the algorithmparameters were heuristically chosen as follows


(ii) attractiveness 120573 02(iii) light absorption coefficient 120574 1(iv) randomization number 120572 05

44 Simulation Results In the case study we apply GSA FAandBB-BCalgorithms to solve the train speed trajectory opti-mization problem To display the performance robustness ofthe algorithms for the test track in Figure 4 the simulationswere performed for three different total travel times 345 secs350 secs and 360 secs Furthermore for the same purposetwo cases (with no passenger andwith passengers) were takeninto account


Table 4 Energy consumption (kwh) for different time limits (nopassenger)

Total travel time FA GSA BB-BC345 s 1045 1018 985350 s 1026 1000 978360 s 1016 984 939

441 Case I In this case where the train has no passengertrain starts its motion from Osmangazi University stationand travel ends at Stadyum station (see Figure 4) There arefive more stations between departure and arrival stationsTrain should stop at each of these stations For the sakeof simplicity in presentation dwell times are disregardedThe alteration of gradient through the test track is givennumerically in Figure 4 and graphically in Figure 5(a) Thereare two sharp curvatures on track where train speed needsto be limited At these points train speed is constrained to15 kmh Speed limits for the test track is shown on the speed-position graphics in Figure 5(b)

Simulations using GSA FA and BB-BC algorithms wereconducted with the parameters given in the previous subsec-tion Optimization results for total travel time of 350 secs aregiven in the form of speed trajectories in Figure 6

Interpreting the optimal speed trajectories in Figure 6it is noticed that between the first two stations all thealgorithms result in all the motion phases However betweenthe 2nd and 3rd stations BB-BC and FA algorithms result inno coasting phase and give only the MA CR and BR phasesFor this part the GSA proposes only the phases MA andBR A similar distinctive outcome by the GSA algorithm alsooccurs between Ataturk Bulvari and Visnelik stations whereit eliminates CR phase and apply only the MA CO and BRphases For the other parts the sequence of motion phasescomplies with those shown in Table 2 Operation strategyis controlled by determining speed levels for each phaseMaximum speeds of BB-BC GSA and FA solutions are56 kmh 63 kmh and 55 kmh respectivelyThe simulationsfor Case I are conducted for three different total travel timelimits and for each algorithm corresponding energy costs areshown in Table 4

Regarding the costs illustrated in Table 4 for everytotal travel time limit BB-BC demonstrates superior per-formance compared to GSA and FA solutions When BB-BC is employed energy consumption is reduced by 6 and334 compared to FA and GSA respectively Thus it can beconcluded that BB-BC has better cost performance comparedto the other two methods

442 Case II Train mass is a major factor affecting theenergy consumption adversely In this case optimal drivingstrategies are searched for the train loadedwith varying num-ber of passengers In this case certain number of passengersis assumed to get in the train at every station in order toevaluate the impact of passenger load An exemplary numberof passengers just before train departs the indicated stationare given in Table 5 Assuming the average mass of an adult

Table 5 Number of passengers at each station

Station Number of passengersOsmangazi University 0Porsuk 17Buyukdere 41Goztepe 54Ataturk Bulvari 97Visnelik 114

Table 6 Energy consumption (kwh) for different time limits (withpassenger)


Table 7 Average convergence results

FA GSA BB-BCConvergence (generation) 24 56 44

passenger is 86 kg [55] trainrsquos mass at the stations is showngraphically in Figure 7

Apart from the trainrsquos mass keeping Case I conditionsintact the speed trajectory corresponding to 350 secs totaltravel time is given in Figure 8

Regarding Figure 6 a likewise interpretation of Figure 8is possible Energy consumption corresponding to threedifferent total travel times is shown in Table 6

The BB-BC as in the previous case exhibits a betterperformance compared to the other two When BB-BC isemployed energy consumption is reduced by 584 and 3on average compared to FA and GSA respectively Althoughthere is an increment in train mass approximately by 28energy consumption increases by 11 The results show thatthe GSA and FA algorithms perform reasonably well underthe conditions where the train mass changes throughout thesimulation However the results also show that these twoalgorithms are outperformed by the BB-BC algorithm

45 Discussion Even though the heuristic optimizationmethods have common features they differ in each othernot only in terminology but also in algorithmic structureAll three methods are evolving population based methodswhere each member of a population is a solution candidateRandomness is significant for global optimization tools interms of exploring new solutions along the search spaceWiththe advantage of being a hybrid algorithmGSA employs bothGA and SA to satisfy randomness FA attributes randomnessto fireflyrsquos motion whereas BB-BC provides it as energydissipation

The results in Tables 4 and 6 were in terms of optimalcosts Table 7 illustrates convergence rate features of thealgorithms


800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik

Figure 5 Altitude (a) and speed limitation (b)

70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

Figure 6 Speed-time graph for all algorithms (no passenger)

From what we can observe from Table 7 FA convergedto a solution faster than the others However its providedsolution is mediocre compared to the others For the opti-mizations which have restrictions or have time problemscaused by slow simulation model and infrastructure FAalgorithm might provide practical solutions In spite ofslow convergence rate BB-BC generates the lowest energyconsumption Therefore for the optimizations which needmore efficient solution and have appropriate simulation envi-ronment BB-BC might be employed GSA provides bettersolutions compared to FA but it suffers from convergence

5 Conclusion

In this manuscript optimal train operation strategies aredeveloped using three nature-inspired metaheuristic algo-rithms Genetic Simulated Annealing Firefly and Big Bang-Big Crunch Their performances are tested via MATLAB

Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik

Figure 7 Train mass for each station

70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari

Figure 8 Speed-time graph for all algorithms (with passenger)


simulations for a local rail line under various test conditionsThe simulations take track alignments speed limitationsand train mass into consideration GSA FA and BB-BCsearching methods were compared for finding the optimalspeed trajectory Besides various track alignments and speedlimitations changes in train mass are also considered toprovide real-like model

Obtained results may be summarized as follows whenchosen appropriate parameters GSA method is influentialat providing solutions close to the optimal ones AlthoughFA converges to the solution in short times it still performsmediocre solutions All algorithms give consistent results forboth no passenger and with passenger conditions WhileBB-BC reaches the lowest cost solution it takes a signif-icant computational time The main contribution of thismanuscript is the illustration of successful applicability ofthree metaheuristic optimization methods to the optimaltrain operation problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the Eskisehir OsmangaziUniversity Scientific Research Foundation under Grant no2015-772

References

[1] K Ichikawa ldquoApplication of optimization theory for boundedstate variable problems to the operation of trainrdquo Bulletin ofJSME vol 11 no 47 pp 857ndash865 1968

[2] H H Hoang M P Polis and A Haurie ldquoReducing energyconsumption through trajectory optimization for a metronetworkrdquo IEEE Transactions on Automatic Control vol 20 no5 pp 590ndash595 1975

[3] P Howlett ldquoAn optimal strategy for the control of a trainrdquoAustralian Mathematical Society Journal Series B AppliedMathematics vol 31 no 4 pp 454ndash471 1990

[4] I PMilroyAspect of automatic train control [PhD dissertation]Loughborough University of Technology Sydney Australia1980

[5] E Khmelnitsky ldquoOn an optimal control problem of trainoperationrdquo IEEE Transactions on Automatic Control vol 45 no7 pp 1257ndash1266 2000

[6] C Jiaxin and P Howlett ldquoApplication of critical velocities tothe minimisation of fuel consumption in the control of trainsrdquoAutomatica vol 28 no 1 pp 165ndash169 1992

[7] P Howlett ldquoOptimal strategies for the control of a trainrdquoAutomatica vol 32 no 4 pp 519ndash532 1996

[8] C S Chang and S S Sim ldquoOptimising train movementsthrough coast control using genetic algorithmsrdquo IEEProceedingsmdashElectric Power Applications vol 144 no 1 p65 1997

[9] PHowlett ldquoTheoptimal control of a trainrdquoAnnals of OperationsResearch vol 98 pp 65ndash87 2000

[10] R Liu and I M Golovitcher ldquoEnergy-efficient operation of railvehiclesrdquo Transportation Research Part A Policy and Practicevol 37 no 10 pp 917ndash932 2003

[11] K K Wong and T K Ho ldquoDynamic coast control of trainmovement with genetic algorithmrdquo International Journal ofSystems Science vol 35 no 13-14 pp 835ndash846 2004

[12] S Acikbas and M T Soylemez ldquoCoasting point optimisationfor mass rail transit lines using artificial neural networks andgenetic algorithmsrdquo IET Electric Power Applications vol 2 no3 pp 172ndash182 2008

[13] P G Howlett P J Pudney and X Vu ldquoLocal energy minimiza-tion in optimal train controlrdquo Automatica vol 45 no 11 pp2692ndash2698 2009

[14] K Kim and S I-J Chien ldquoSimulation-based analysis of traincontrols under various track alignmentsrdquo Journal of Transporta-tion Engineering vol 136 no 11 pp 937ndash948 2010

[15] K Kim and S I-J Chien ldquoOptimal train operation forminimum energy consumption considering track alignmentspeed limit and schedule adherencerdquo Journal of TransportationEngineering vol 137 no 9 pp 665ndash674 2011

[16] MMiyatake andH Ko ldquoOptimization of train speed profile forminimum energy consumptionrdquo IEEJ Transactions on Electricaland Electronic Engineering vol 5 no 3 pp 263ndash269 2010

[17] J-W Sheu and W-S Lin ldquoEnergy-saving automatic train reg-ulation using dual heuristic programmingrdquo IEEE Transactionson Vehicular Technology vol 61 no 4 pp 1503ndash1514 2012

[18] A P Cucala A Fernandez C Sicre andMDomınguez ldquoFuzzyoptimal schedule of high speed train operation to minimizeenergy consumption with uncertain delays and driverrsquos behav-ioral responserdquo Engineering Applications of Artificial Intelli-gence vol 25 no 8 pp 1548ndash1557 2012

[19] S Su X Li T Tang and Z Gao ldquoA subway train timetableoptimization approach based on energy-efficient operationstrategyrdquo IEEE Transactions on Intelligent Transportation Sys-tems vol 14 no 2 pp 883ndash893 2013

[20] C Sicre A P Cucala A Fernandez and P LukaszewiczldquoModeling and optimizing energy-efficient manual driving onhigh-speed linesrdquo IEEJ Transactions on Electrical and ElectronicEngineering vol 7 no 6 pp 633ndash640 2012

[21] L Yang K Li Z Gao and X Li ldquoOptimizing trains movementon a railway networkrdquo Omega vol 40 no 5 pp 619ndash633 2012

[22] S Su T Tang and C Roberts ldquoA cooperative train controlmodel for energy savingrdquo IEEE Transactions on IntelligentTransportation Systems vol 16 no 2 pp 622ndash631 2015

[23] X Yang X Li Z Gao H Wang and T Tang ldquoA cooperativescheduling model for timetable optimization in subway sys-temsrdquo IEEE Transactions on Intelligent Transportation Systemsvol 14 no 1 pp 438ndash447 2013

[24] S Lu S Hillmansen T K Ho and C Roberts ldquoSingle-train trajectory optimizationrdquo IEEE Transactions on IntelligentTransportation Systems vol 14 no 2 pp 743ndash750 2013

[25] S Su T Tang X Li and Z Gao ldquoOptimization of multitrainoperations in a subway systemrdquo IEEE Transactions on IntelligentTransportation Systems vol 15 no 2 pp 673ndash684 2014

[26] X Yang B Ning X Li and T Tang ldquoA two-objective timetableoptimization model in subway systemsrdquo IEEE Transactions onIntelligent Transportation Systems vol 15 no 5 pp 1913ndash19212014

[27] Y Wang B De Schutter T J J van den Boom B Ningand T Tang ldquoEfficient Bilevel approach for urban rail transitoperation with stop-skippingrdquo IEEE Transactions on IntelligentTransportation Systems vol 15 no 6 pp 2658ndash2670 2014


[28] R M P Goverde F Corman and A DrsquoAriano ldquoRailway linecapacity consumption of different railway signalling systemsunder scheduled and disturbed conditionsrdquo Journal of RailTransport PlanningampManagement vol 3 no 3 pp 78ndash94 2013

[29] Y Bai T K Ho B Mao Y Ding and S Chen ldquoEnergy-efficient locomotive operation for Chinese mainline railwaysby fuzzy predictive controlrdquo IEEE Transactions on IntelligentTransportation Systems vol 15 no 3 pp 938ndash948 2014

[30] S Lu P Weston S Hillmansen H B Gooi and C RobertsldquoIncreasing the regenerative braking energy for railway vehi-clesrdquo IEEE Transactions on Intelligent Transportation Systemsvol 15 no 6 pp 2506ndash2515 2014

[31] J Yin D Chen and L Li ldquoIntelligent train operation algorithmsfor subway by expert system and reinforcement learningrdquo IEEETransactions on Intelligent Transportation Systems vol 15 no 6pp 2561ndash2571 2014

[32] M Domınguez A Fernandez-Cardador A P Cucala TGonsalves and A Fernandez ldquoMulti objective particle swarmoptimization algorithm for the design of efficient ATO speedprofiles in metro linesrdquo Engineering Applications of ArtificialIntelligence vol 29 pp 43ndash53 2014

[33] W Carvajal-Carreno A P Cucala andA Fernandez-CardadorldquoOptimal design of energy-efficient ATO CBTC driving formetro lines based on NSGA-II with fuzzy parametersrdquo Engi-neering Applications of Artificial Intelligence vol 36 pp 164ndash1772014

[34] T Davis Transportation Energy Data Book Center for Trans-portation Analysis Washington DC USA 20th edition 2000

[35] WW Hay Railroad Engineering JohnWiley amp Sons New YorkNY USA 2nd edition 1982

[36] K Keskin and A Karamancioglu ldquoA hybrid optimizationalgorithm for energy efficient train operationrdquo in Proceedingsof the International Symposium on Innovations in IntelligentSysTems and Applications (INISTA rsquo15) pp 1ndash6 Madrid SpainSeptember 2015

[37] A Chen T Jiang Z Chen and Y Zhang ldquoA genetic andsimulated annealing combined algorithm for optimization ofwideband antennamatching networksrdquo International Journal ofAntennas and Propagation vol 2012 Article ID 251624 6 pages2012

[38] J Wang Q Duan Y Jiang and X Zhu ldquoA new algorithm forgrid independent task schedule genetic simulated annealingrdquoin Proceedings of the 2010 World Automation Congress (WACrsquo10) pp 165ndash171 Kobe Japan September 2010

[39] L Wang and D Zheng ldquoAn effective hybrid optimizationstrategy for job-shop scheduling problemsrdquoComputers ampOper-ations Research vol 28 no 6 pp 585ndash596 2001

[40] P-H Chen and S M Shahandashti ldquoHybrid of genetic algo-rithm and simulated annealing for multiple project schedulingwith multiple resource constraintsrdquo Automation in Construc-tion vol 18 no 4 pp 434ndash443 2009

[41] R Sonmez and O H Bettemir ldquoA hybrid genetic algorithm forthe discrete time-cost trade-off problemrdquo Expert Systems withApplications vol 39 no 13 pp 11428ndash11434 2012

[42] H Chen N S Flann and D W Watson ldquoParallel geneticsimulated annealing a massively parallel SIMD algorithmrdquoIEEETransactions on Parallel andDistributed Systems vol 9 no2 pp 126ndash136 1998

[43] M Rabbani S Sadri N Manavizadeh and H Rafiei ldquoAnovel bi-level hierarchy towards available-to-promise inmixed-model assembly line sequencing problemsrdquo Engineering Opti-mization vol 47 no 7 pp 947ndash962 2015

[44] O K Erol and I Eksin ldquoA new optimization method BigBangmdashBig Crunchrdquo Advances in Engineering Software vol 37no 2 pp 106ndash111 2006

[45] C V Camp ldquoDesign of space trusses using big bang-big crunchoptimizationrdquo Journal of Structural Engineering vol 133 no 7pp 999ndash1008 2007

[46] M Dogan and Y Istefanopulos ldquoOptimal nonlinear controllerdesign for flexible robot manipulators with adaptive internalmodelrdquo IETControlTheoryampApplications vol 1 no 3 pp 770ndash778 2007

[47] T Kumbasar I Eksin M Guzelkaya E Yesil I Eksin andE Yesil ldquoBig bang big crunch optimization method basedfuzzy model inversionrdquo in MICAI 2008 Advances in ArtificialIntelligence 7th Mexican International Conference on ArtificialIntelligence Atizapan de Zaragoza Mexico October 27ndash31 2008Proceedings vol 5317 of Lecture Notes in Computer Science pp732ndash740 Springer Berlin Germany 2008

[48] Z Tabrizian E Afshari G G Amiri M H Ali Beigy and SM P Nejad ldquoA new damage detection method big Bang-BigCrunch (BB-BC) algorithmrdquo Shock and Vibration vol 20 no4 pp 633ndash648 2013

[49] K Keskin and A Karamancioglu ldquoEnergy efficient motioncontrol for a light rail vehicle using the big bang big crunchalgorithmrdquo in Proceedings of the 14th IFAC Symposium onControl in Transportation Systems (CTS rsquo16) pp 442ndash446Istanbul Turkey May 2016

[50] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress London UK 2008

[51] M Mitic and Z Miljkovic ldquoBio-inspired approach to learningrobotmotion trajectories and visual control commandsrdquo ExpertSystems with Applications vol 42 no 5 pp 2624ndash2637 2015

[52] N C Long P Meesad and H Unger ldquoA highly accurate fireflybased algorithm for heart disease predictionrdquo Expert Systemswith Applications vol 42 no 21 pp 8221ndash8231 2015

[53] K Tesch andKKaczorowska ldquoArterial cannula shape optimiza-tion by means of the rotational firefly algorithmrdquo EngineeringOptimization vol 48 no 3 pp 497ndash518 2016

[54] V Xuan Analysis of Necessary Conditions for the OptimalControl of a Train University of South Australia 2006

[55] Federal Aviation Administration Aircraft Weight and BalanceControl Flights Standards Service Washington DC USA2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



discrete control tools and used the Kuhn-Tucker equationsto analyze optimal switching points [9] Liu and Golovitcherproposed an analytical method based on maximum prin-ciple to find optimal input sequence and switching pointsDeveloped system was capable of finding optimal trajec-tory and suitable for real-time optimization due to itsanalytical nature [10] In 2004 Wong and Ho researchedto determine number of coasting and switching points bymeans of three various genetic algorithms [11] Acikbas andSoylemez developed simulation software which is suitable formultitrain and multiline tests They applied artificial neuralnetwork and genetic algorithm to model a part of metroline in Istanbul and compared these approaches in terms ofenergy consumption [12] In 2009 Howlett et al presenteda new analytical solution for tracks with steep uphill anddownhill sections to find optimal switching points [13] Kimand Chien investigated train operation with constraints oftime speed and energy consumption A simulation modelwas developed [14] and simulated annealing algorithm wasutilized to calculate optimal switching points and cruisingspeed It was clarified that it was possible to reduce energyconsumption by increasing the travel time [15] In anotherstudy optimization problem was handled by three differentmethods dynamic programming gradient methods andsequential quadratic programming [16] In [17] Sheu andLin focused on automatic train regulation (ATR) problemin the energy efficiency framework They developed a dualheuristic programming method which brought the ATRability of real-like adaptation In addition to controllingcoasting points and dwell time at stations scheduling opti-mization could increase energy efficiency performance Anenergy-efficient driving strategy and optimization methodfor manually driven high speed trains were developed [18]Energy-efficient strategies and timetable optimization werecombined together in [18ndash22] For subway systems recoveredenergy coming from regenerative braking can be used fortrain traction In this regard Yang et al developed schedulingrules that synchronized successive trains for braking andaccelerating Overlapping time was maximized by means ofinteger programming model and for timetable optimizationgenetic algorithm was utilized [23] Lu et al investigatedenergy-efficient train trajectory by means of dynamic speedcontrol Ant colony optimization genetic algorithm anddynamic programming algorithmswere used in searching theoptimal trajectory [24] Su et al proposed a new approachby combining optimal driving strategy with cooperative traincontrol The energy which came from regenerative brakingis used for traction of other trains [25] Similarly whileone research controlled headway time and dwell time toincrease energy savings from regenerative braking [26] theother one developed stop-skipping method [27] to decreasepassenger waiting time In another research a concept ofdynamic infrastructure occupation was presented to assessinfrastructure capacity under disturbed conditions as a com-plement to the established capacity indicator of scheduledinfrastructure occupation This new indicator is applied in acapacity assessment study of a Dutch railway corridor withdifferent signaling configurations under both scheduled anddisturbed traffic conditions [28] During the recent years

several other methods were applied to optimal train controlproblem such as fuzzy predictive control [29] Bellman-Ford algorithm [30] reinforcement learning [31] swarmoptimization [32] and NSGA-II algorithm [33]

In this study an energy-efficient train operation problemwas considered on a track with no steep sections Threeheuristic approaches Firefly Algorithm Genetic SimulatedAnnealing and Big Bang-Big Crunch were used to findswitching points for phases There is no known study ontrain operation optimization problem which employs oneof these algorithms In this study it was demonstrated thatthese three algorithms are appropriate to apply to the energy-efficient train operation optimization problem and a compar-ison between algorithms running times and optimality wasdiscussed Besides the fact that the train model and the trackwere real-like modeled the effect of number of passengers ontrain energy consumption and algorithmrsquos performance wereevaluated However the problem in case study was solvedsuccessfully for a complicated track with steep sections orspeed limitations more complex strategies are required (see[5 7 9 10 13])

In the next section we present the nonlinear optimizationformulation of the problem In Section 3 the evolutionarysolution methods used in this manuscript are introduced Inthe 4th section the solution methods are applied to a modelof locally existing real problem In the remaining part of themanuscript performances of the methods are discussed

2 Modeling the Motion

Themotion equations of train usingNewtonrsquos second law canbe written in the following form

119889119909119889119905 = V119889V119889119905 = 119865119905 minus 119865119887119898 minus 119877 minus 119877119892

(1)

where119909 and V are position and speed of the train respectively119865119905 is the tractive force 119865119887 is the braking force 119877 is the rollingresistance 119877119892 is the resistance caused by level change and119898 denotes mass of train In the sequel a train motion ona sequence of successive stations is considered We denotethe distance between stations 119894 and 119894 + 1 by 119883119894 allowedtravel time by 119879119894 and allowed maximum speed by 119881 Hencebetween stations 119894 and 119894 + 1 these parameters are restrictedwith following limits

0 le 119909 le 1198831198940 le 119905 le 1198791198940 le V le 119881(2)

Resistance of the train 119877 can be calculated by utilizing Davisequation [34]

119877 = 119860 + 119861V + 119862V2 (3)

where the coefficients 119860 119861 and 119862 correspond to massmechanical and air resistance respectivelyThese coefficients


40

35

30

25

20

15

10

Trac

tive e

ffort

(kN

)

Speed (kmh)8647927264857650443236288216144720



119877119892 = 119892 sin120572 (4)



119865119905 = 2650120583119875V (5)


119875 = 119865119905V2650120583 (6)


119864 = int1198790119875119889119905 (7)


70

60

50

40

30

20

10


MA CR CO BR

Station 2

Spee

d (k

mh

)




119865max = 119898119892120578 (8)




119909ss = V2br2119886 (9)





119864TOTAL = 119864MA + 119864CR (10)


min1198961 1198962

(int11989610119865119905 (119905) V (119905) 119889119905 + int1198962

1198961



1198961 lt 119905 le 1198962119865119905 (119905) = 0 1198962 lt 119905 le 119879119894119865119887 (119905) = 0 0 lt 119905 le 1198963








Start


definition


Evaluation



stoppingcriteria

Output

Accept

Stop

No

Yes

No

No

Yes Yes

Yes

No

Update T


Accept sk+1

k = k + 1


Initialize s T k








119904119888 = sum119870119894=1 (1119891119894) 119904119894sum119870119894=1 (1119891119894) (12)


119904new119894 = 119904119888 + 120590 (13)


120590 = 05119903 (119904max minus 119904min)1 + 119895ℎ (14)





met







119868 (119889) = 1198681199041198892 (15)


120573 = 12057301198901205741198892 (16)


119889119894119895 = 10038171003817100381710038171003817119901119894 minus 11990111989510038171003817100381710038171003817 = radic 119897sum119896=1

(119901119894119896 minus 119901119895119896)2 (17)


for 119895 = 1 119899 doif 119868119894 lt 119868119895 then






119901119894 = 119901119894 + 1205730119890minus1205741198892119894119895 (119901119895 minus 119901119894) + 120572120576119894 (18)



4 A Case Study





StadyumVişnelik

350m

667m

437m

207m

204m

364m

293m

540m0

0

1

2

2

6

1

1

OsmangaziUniversity

Porsuk

Speed limit

A

B


Positive slopeLevel

Atatuumlrk Bulvarı

15 kmh

Speed limit15 kmh

Goumlztepe

Buumlyuumlkdere
















Crossoverfunction

Annealingfunction











































800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










40

35

30

25

20

15

10

Trac

tive e

ffort

(kN

)

Speed (kmh)8647927264857650443236288216144720



119877119892 = 119892 sin120572 (4)



119865119905 = 2650120583119875V (5)


119875 = 119865119905V2650120583 (6)


119864 = int1198790119875119889119905 (7)


70

60

50

40

30

20

10


MA CR CO BR

Station 2

Spee

d (k

mh

)




119865max = 119898119892120578 (8)




119909ss = V2br2119886 (9)





119864TOTAL = 119864MA + 119864CR (10)


min1198961 1198962

(int11989610119865119905 (119905) V (119905) 119889119905 + int1198962

1198961



1198961 lt 119905 le 1198962119865119905 (119905) = 0 1198962 lt 119905 le 119879119894119865119887 (119905) = 0 0 lt 119905 le 1198963








Start


definition


Evaluation



stoppingcriteria

Output

Accept

Stop

No

Yes

No

No

Yes Yes

Yes

No

Update T


Accept sk+1

k = k + 1


Initialize s T k








119904119888 = sum119870119894=1 (1119891119894) 119904119894sum119870119894=1 (1119891119894) (12)


119904new119894 = 119904119888 + 120590 (13)


120590 = 05119903 (119904max minus 119904min)1 + 119895ℎ (14)





met







119868 (119889) = 1198681199041198892 (15)


120573 = 12057301198901205741198892 (16)


119889119894119895 = 10038171003817100381710038171003817119901119894 minus 11990111989510038171003817100381710038171003817 = radic 119897sum119896=1

(119901119894119896 minus 119901119895119896)2 (17)


for 119895 = 1 119899 doif 119868119894 lt 119868119895 then






119901119894 = 119901119894 + 1205730119890minus1205741198892119894119895 (119901119895 minus 119901119894) + 120572120576119894 (18)



4 A Case Study





StadyumVişnelik

350m

667m

437m

207m

204m

364m

293m

540m0

0

1

2

2

6

1

1

OsmangaziUniversity

Porsuk

Speed limit

A

B


Positive slopeLevel

Atatuumlrk Bulvarı

15 kmh

Speed limit15 kmh

Goumlztepe

Buumlyuumlkdere
















Crossoverfunction

Annealingfunction











































800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119864TOTAL = 119864MA + 119864CR (10)


min1198961 1198962

(int11989610119865119905 (119905) V (119905) 119889119905 + int1198962

1198961



1198961 lt 119905 le 1198962119865119905 (119905) = 0 1198962 lt 119905 le 119879119894119865119887 (119905) = 0 0 lt 119905 le 1198963








Start


definition


Evaluation



stoppingcriteria

Output

Accept

Stop

No

Yes

No

No

Yes Yes

Yes

No

Update T


Accept sk+1

k = k + 1


Initialize s T k








119904119888 = sum119870119894=1 (1119891119894) 119904119894sum119870119894=1 (1119891119894) (12)


119904new119894 = 119904119888 + 120590 (13)


120590 = 05119903 (119904max minus 119904min)1 + 119895ℎ (14)





met







119868 (119889) = 1198681199041198892 (15)


120573 = 12057301198901205741198892 (16)


119889119894119895 = 10038171003817100381710038171003817119901119894 minus 11990111989510038171003817100381710038171003817 = radic 119897sum119896=1

(119901119894119896 minus 119901119895119896)2 (17)


for 119895 = 1 119899 doif 119868119894 lt 119868119895 then






119901119894 = 119901119894 + 1205730119890minus1205741198892119894119895 (119901119895 minus 119901119894) + 120572120576119894 (18)



4 A Case Study





StadyumVişnelik

350m

667m

437m

207m

204m

364m

293m

540m0

0

1

2

2

6

1

1

OsmangaziUniversity

Porsuk

Speed limit

A

B


Positive slopeLevel

Atatuumlrk Bulvarı

15 kmh

Speed limit15 kmh

Goumlztepe

Buumlyuumlkdere
















Crossoverfunction

Annealingfunction











































800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start


definition


Evaluation



stoppingcriteria

Output

Accept

Stop

No

Yes

No

No

Yes Yes

Yes

No

Update T


Accept sk+1

k = k + 1


Initialize s T k








119904119888 = sum119870119894=1 (1119891119894) 119904119894sum119870119894=1 (1119891119894) (12)


119904new119894 = 119904119888 + 120590 (13)


120590 = 05119903 (119904max minus 119904min)1 + 119895ℎ (14)





met







119868 (119889) = 1198681199041198892 (15)


120573 = 12057301198901205741198892 (16)


119889119894119895 = 10038171003817100381710038171003817119901119894 minus 11990111989510038171003817100381710038171003817 = radic 119897sum119896=1

(119901119894119896 minus 119901119895119896)2 (17)


for 119895 = 1 119899 doif 119868119894 lt 119868119895 then






119901119894 = 119901119894 + 1205730119890minus1205741198892119894119895 (119901119895 minus 119901119894) + 120572120576119894 (18)



4 A Case Study





StadyumVişnelik

350m

667m

437m

207m

204m

364m

293m

540m0

0

1

2

2

6

1

1

OsmangaziUniversity

Porsuk

Speed limit

A

B


Positive slopeLevel

Atatuumlrk Bulvarı

15 kmh

Speed limit15 kmh

Goumlztepe

Buumlyuumlkdere
















Crossoverfunction

Annealingfunction











































800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












met







119868 (119889) = 1198681199041198892 (15)


120573 = 12057301198901205741198892 (16)


119889119894119895 = 10038171003817100381710038171003817119901119894 minus 11990111989510038171003817100381710038171003817 = radic 119897sum119896=1

(119901119894119896 minus 119901119895119896)2 (17)


for 119895 = 1 119899 doif 119868119894 lt 119868119895 then






119901119894 = 119901119894 + 1205730119890minus1205741198892119894119895 (119901119895 minus 119901119894) + 120572120576119894 (18)



4 A Case Study





StadyumVişnelik

350m

667m

437m

207m

204m

364m

293m

540m0

0

1

2

2

6

1

1

OsmangaziUniversity

Porsuk

Speed limit

A

B


Positive slopeLevel

Atatuumlrk Bulvarı

15 kmh

Speed limit15 kmh

Goumlztepe

Buumlyuumlkdere
















Crossoverfunction

Annealingfunction











































800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










StadyumVişnelik

350m

667m

437m

207m

204m

364m

293m

540m0

0

1

2

2

6

1

1

OsmangaziUniversity

Porsuk

Speed limit

A

B


Positive slopeLevel

Atatuumlrk Bulvarı

15 kmh

Speed limit15 kmh

Goumlztepe

Buumlyuumlkdere
















Crossoverfunction

Annealingfunction











































800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Crossoverfunction

Annealingfunction











































800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










800

790

780

770

760

750

740

80

70

60

50

40

30

20

10Spee

d lim

it (k

mh

)A

ltitu

de (m

)

Position (m)33142649199913028653640

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

(a)

(b)

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500



5 Conclusion


Trai

n w

eigh

t (to

nnes

)

Position (m)33142649199913028653640

60

Stad

yum

Osm

anga

ziU

nive

rsity

Pors

uk

Atat

uumlrk

Bulv

arı

Goumlz

tepe

Buumlyuuml

kder

e

55

50

45

40

35

30

Vişn

elik


70

60

50

40

30

20

10

0

Time (sec)

Spee

d (k

mh

)

BBBCGSAFirefly

350300250200150100500

OG

U

Pors

uk

Buyu

kder

e

Goz

tepe

Atat

urk

Visn

elik

Stad

yum

Bulv

ari





Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Competing Interests


Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Energy-Efficient Train Operation Using Nature-Inspired Algorithmsdownloads.hindawi.com/journals/jat/2017/6173795.pdf · 2019. 7. 30. · Ford algorithm [30], reinforcement learning