Multi-Train Energy Saving for Maximum Usage of Regenerative Energy by Dwell Time Optimization in Urban Rail Transit Using Genetic Algorithm

energies

Article Multi-Train Energy Saving for Maximum Usage of Regenerative Energy by Dwell Time Optimization in Urban Rail Transit Using Genetic Algorithm

Fei Lin *, Shihui Liu, Zhihong Yang, Yingying Zhao, Zhongping Yang and Hu Sun School of Electrical Engineering, Beijing Jiaotong University, No. 3 Shangyuancun, Beijing 100044, China; [email protected] (S.L.); [email protected] (Z.Y.); [email protected] (Y.Z.); [email protected] (Z.Y.); [email protected] (H.S.) * Correspondence: ﬂ[email protected]; Tel.: +86-136-4139-8863

Academic Editor: William Holderbaum Received: 2 February 2016; Accepted: 10 March 2016; Published: 17 March 2016 Abstract: With its large capacity, the total urban rail transit energy consumption is very high; thus, energy saving operations are quite meaningful. The effective use of regenerative braking energy is the mainstream method for improving the efﬁciency of energy saving. This paper examines the optimization of train dwell time and builds a multiple train operation model for energy conservation of a power supply system. By changing the dwell time, the braking energy can be absorbed and utilized by other traction trains as efﬁciently as possible. The application of genetic algorithms is proposed for the optimization, based on the current schedule. Next, to validate the correctness and effectiveness of the optimization, a real case is studied. Actual data from the Beijing subway Yizhuang Line are employed to perform the simulation, and the results indicate that the optimization method of the dwell time is effective.

Keywords: genetic algorithm; dwell time; energy saving; braking energy; multi-train; urban rail transit

1. Introduction Urban rail transit is an important part of urban public transport. With the characteristics of large capacity, high speed, security, punctuality, environmental protection, energy saving, and so on, urban rail transit is very popular and is especially suitable for large- and medium-sized cities [1]. China is the most populous country in the world and is very suitable for the development of urban rail transit. The first subway line in China was constructed in Beijing. Its planning began in 1953; the project was founded in 1965 and completed in 1969; and operations began in 1971. Today, the Beijing urban rail transit system is the second largest city subway system in the world. According to statistics from Beijing Subway, in the first three days of 2016, 16,367,700 passengers were transported, and 16,567 trains ran according to plan. In 2015, 2,832,000,000 passengers were transported, and a total of 2,324,486 trains ran according to plan [2]. Energy consumption of urban rail transit is much lower than buses or cars per passenger per km. However, the energy consumption of urban rail transit has a considerable scale with a large capacity. How to effectively reduce this energy consumption is a very significant matter. González-Gil [3] provided a comprehensive appraisal of the main practices, strategies and technologies currently available for minimizing the energy use of urban rail systems. The two main methods are energy-efficient driving and regenerative braking. The former has been well studied by many researchers [4–7], while the latter is a rather new approach [8–10]. The braking system of subway trains typically adopts regenerative braking as the main system and air brakes (friction brakes) as a supplement. When the vehicle uses regenerative braking, the motor works in the state of the generator,

Energies 2016, 9, 208; doi:10.3390/en9030208 www.mdpi.com/journal/energies Energies 2016, 9, 208 2 of 21 and the electric energy generated returns to the traction power grid. In an urban rail transit system, the distance between stations is relatively short; thus, the train must perform frequent starting and braking, and this aspect of the braking energy has a great potential for recycling. The regenerative braking energy can reach up to more than 30% of the traction energy. If the regenerative braking energy cannot be used, then it is converted to a resistance brake or air brake, and the braking energy is wasted in vain. The energy that is not used accounts for more than 40% of the total braking energy used. At present, the effective utilization of regenerative braking energy involves two main aspects of research [9]. One is using energy storage devices, such as flywheels [11–14], supercapacitors [15–21] and batteries [22–25]. New infrastructure requires a large amount of investment and is not easy to realize. The other method is operation optimization, which is more convenient. This method has also been studied by some researchers. Chen [26] optimized the timetable to reduce the maximum traction power. Kim et al. [27,28] developed a mixed integer programming model to minimize the maximum traction energy, which occurs when trains are running simultaneously. Gordeon [29] coordinated multi-trains to reduce the total traction energy. Ramos [30] coordinated a train’s acceleration and braking time to make full use of renewable energy. Yang et al. [31] increased the traction and braking overlap rate of different trains to optimize the timetable, thereby improving the utilization efficiency of regenerative braking. Zhao et al. [32] optimized schedules using the simulated annealing algorithm, by analyzing the relative position and running state of the adjacent train. The above researchers did not consider the effect of the traction power supply network. The following research considers the traction network model. Nasri et al. [33] optimized a timetable for maximum usage of regenerative braking energy and discussed the headway and stop time effect on the energy consumption. Albrecht et al. [34,35] researched the headway and stop time effect on the energy consumption based on a route in Berlin. In addition, there has also been other research with different parameter optimizations and methods [36–38]. In the present paper, the train dwell time is optimized to achieve the purpose of the use of regenerative braking energy. This paper is arranged as follows: Section2 outlines the overall situation. A multi-train system for urban rail transit is introduced. The constraint conditions of multi-train operation optimization, considering regenerative braking energy, are analyzed. The principle of dwell time optimization is introduced. In Section3, a train operation model and tractive power supply system model are established. The optimization goals of multi-train energy saving operation are settled. The calculation flow of multi-train operation is proposed. In Section4, the genetic algorithm optimization is described. A mathematical model is established, and the genetic algorithm is introduced briefly. The optimization method and calculation flowchart are proposed. Finally, the actual train data, line data and power supply system data of the Beijing Yizhuang subway line are employed for simulation, and the performance of the simulation system is tested.

2. Multi-Train System for Urban Rail Transit

2.1. Multi-Train System for Urban Rail Transit The multi-train operation optimization is based on the single-train operation model. The calculation of the operation of each train is constrained by the traction braking characteristics of the vehicle and line condition. Vehicle characteristics include traction performance of the vehicle, braking performance, performance of the train running resistance, vehicle formation, weight of the vehicle itself and load, etc. Line conditions include the ramp of the line, curve, speed limit, station spacing, etc. For multiple trains operating on the same route, the line conditions of a train running on the same line (uplink or downlink) are the same, and vehicle characteristics are generally the same, while the load of each train varies with the passenger ﬂow. The traction substation in the traction power supply system transfers the three-phase high voltage alternating current (AC) into low voltage direct current (DC) used by urban rail trains. The DC-side power supply system is the part of the whole traction power supply system that directly provides Energies 2016, 9, 208 3 of 21

powerEnergies for2016 the, 9, 2 train.08 It mainly includes the catenary (three rail), the feeder line, the track and the reflux3 of 21 line. The feeder line delivers direct current from the traction substation to the catenary; the train obtainscatenary the; the electric train energy obtains through the electric the direct energy contact through of the the current direct collector contact (pantograph of the current or collector collector shoe)(pantograph and the or catenary; collector and shoe) the and drive the traction catenary motor; and causesthe drive the traction train to motor operate. causes The the track train not to only operate. has theThe function track not of on a guidely has rail,the function but also thatof a guide of conduction rail, but reflux,also that leading of conduction the reflux reflux, from leading the track the into reflux the tractionfrom the substation track into throughthe traction the substation reflux line, through as shown the in reflux Figure line,1. as shown in Figure 1. Energies 2016, 9, 208 3 of 21

catenary; the train obtains the electric energy through the direct contact of the current collector (pantograph orTSS collector1 shoe) and the catenary; andTSS the2 drive traction motor causesTSS the3 train to operate. The track not only has the function of a guide rail, but also that of conduction reflux, leading the reflux from the track into the traction substation through the reflux line, as shown in Figure 1. traction uplink downlink return rail catenary

TSS1 TSS2 TSS3

traction uplink downlink return rail catenary

Figure 1. Schematic diagram of the urban rail transit power supply system. TSS, traction substation. Figure 1. Schematic diagram of the urban rail transit power supply system. TSS, traction substation.

2.2. The Constraints of the Traction Power Supply System 2.2. The Constraints of the Traction Power Supply System Figure 1. Schematic diagram of the urban rail transit power supply system. TSS, traction substation. The constraints of the traction power supply system when the train is operating are mainly The constraints of the traction power supply system when the train is operating are mainly as as follows2.2. The[39 ]:Constraints of the Traction Power Supply System follows [39]: The constraints of the traction power supply system when the train is operating are mainly 1.1. ForFor urban urban rail rail transit transit traction traction power power supply supply systems, systems, due to due the fact to thatthe thefactDC that traction the DC substation traction as follows [39]: adoptssubstation the diodeadopts rectiﬁer, the diode the energyrectifier, has the unidirectional energy has unidirectional liquidity, and regenerative liquidity, and braking regenerative energy cannotbraking1. For ﬂow energy urban from cannot rail the transit substation flow traction from to powerthe the substation power supply supply systems, to the network. power due to thesupply fact thatnetwork. the DC traction 2. Feedersubstation loss. Due adopts to the the resistance diode rectifier, of thethe energytraction has network, unidirectional line liquidity,loss is inevitable, and regenerative and the line 2. Feederbraking loss. Dueenergy to cannot the resistance flow from the of thesubstation traction to the network, power supply line lossnetwork. is inevitable, and the line resistance changes with the change of the train operation position. resistance2. Feeder changes loss. Due with to the resistance change of of thethe traintraction operation network, position.line loss is inevitable, and the line 3. The train’s traction and regenerative braking curve are constrained by the traction network 3. The train’sresistance traction changes and with regenerative the change of brakingthe train operation curve are position. constrained by the traction network voltage, as shown in Figure 2, and the drop of traction network voltage, train tractive and voltage,3. The as train’s shown traction in Figure and regenerative2, and the braking drop of curve traction are constrained network by voltage, the traction train network tractive and regenerativevoltage, braking as shown capacity in Figure will 2, and be limited. the drop of traction network voltage, train tractive and regenerative braking capacity will be limited. regenerative braking capacity will be limited.

tractive U1>U2>U3 regenerative F tractive U1>U2>U3 B regenerative F forcfeorce B brabkriankging

U1 U1 UU11 U2 U3 U2 U2 U3 U2 U3 U3

v v FigureFigure 2. Tractive 2. Tractive and and regenerative regenerativev brakingbraking curve curve under under different different net voltages net voltages.. v

4. The Figure regenerative 2. Tractive braking and is regenerative restricted by braking the traction curve network under different voltage. Withnet voltages the increase. of 4. The regenerativetraction network braking voltage, is the restricted regenerative by braking the traction power is network limited. As voltage. shown in With Figure the 3, when increase of 4. tractionThe regenerativethe network network voltage, voltagebraking rises the is restricted to regenerative U1, the by train the braking limit traction motor power network current is limited. redu voltage.ces the As With output shown the of in increase the Figure 3 of, whentraction theregenerative network network voltage,braking voltage power; the rises regenerative when to U1, the thenetwork braking train voltage limit power reaches motor is limited. the current maximum As reduces shown limit thevoltage in Figure output U2, 3, of when the regenerativethe networkthen the braking regenerative voltage power; rises braking to when U1,is removed the the network train, and the limit voltage motor motor current reaches current is zero. the maximum reduces the limit output voltage of U2, the thenregenerative the regenerative braking brakingpower; when is removed, the network and the voltage motor reaches current the is zero. maximum limit voltage U2, then the regenerative braking is removed, and the motor current is zero. EnergiesEnergies2016 2016, 9, 2089, 208 4 of4 21of 21

motor current

-Iq maximum operation limiting voltage voltage

pantograph voltage

0 U1 U2 UMAX

Figure 3. Squeezing curve of regenerative braking. Figure 3. Squeezing curve of regenerative braking. 5. If there are energy storage devices (such as a super capacitor) in the line, they will affect the 5. chargingIf there are state energy of the storage energy storagedevices device.(such as For a super multi-train capacitor) operation, in the line, to accurately they will calculate affect the thecharging energy state consumption of the energy of the storage system device. and theFor utilizationmulti-train of operation, the regenerative to accurately braking calculate energy, the aenergy complete consumption simulation model of the of systemthe train and and tractionthe utilization power supply of the system regenerative is required. braking The model energy, ofa the complete traction simulation power supply model system of the includes train and the tractionmodel of power the substation, supply system the model is required. of the rectifierThe model unit, of the the resistance traction power model supply of the system line, the includes model the of themodel train, of the the substation, model of the the braking model of resistance,the rectifieretc unit,. the resistance model of the line, the model of the train, the model of the braking resistance, etc. 2.3. Principles of Dwell Time Optimization 2.3. Principles of Dwell Time Optimization Urban rail transit system station spacing is relatively short, usually 1.5–2 km; thus, it must be able toUrban quickly rail start transit or stop; system the station train’s tractionspacing is acceleration relatively short, is generally usually 0.9–1.0 1.5–2 m/skm; 2thus, and, it braking must be decelerationable to quickly is about start 1.0or stop; m/s 2the. Taking train’s thetraction Yizhuang acceleration Line as is an generally example, 0.9 it–1.0 takes m/s about2, and 39 braking s to acceleratedeceleration to 80 is km/h; about the 1.0 maximum m/s2. Taking power the needed Yizhuang for traction Line as is anabout example, 3.6 MW; it and takes traction about energy 39 s to consumptionaccelerate to is 80 28 km/h kW¨;h. the It maximum takes about power 27 s from need theed speedfor traction of 80 km/his about to 3.6 stop, MW the; and maximum traction power energy ofconsumption regenerative is braking 28 kW is·h approximately. It takes about 3.627 s MW; from the the produced speed of regenerative80 km/h to stop, braking the maximum energy is about power 16of kW regenerative¨ h; and train braking dwell is time approximately in the stop is 3.6 generally MW; the 30–35 produced s. A reasonableregenerative adjustment braking energy of the is dwell about time16 kW can·h improve; and train the dwell possibility time thatin the traction stop is conditions generally and30–35 braking s. A reasonable conditions adjustment occur at the of same the timedwell withtime different can improve trains, the thus possibility allowing that the traction trainconditions to absorb and greaterbraking regenerative conditions occur braking at the energy same andtime improve with different the regenerative trains, thus braking allowing energy the utilization traction ratetrain of to the absorb train. greater regenerative braking energyFigures and4 improveand5 are the powerregenerative curves braking of two trainsenergy running utilization in the rate same of the power train. range, and the blue and redFigure curvess 4 representand 5 are Trainthe power 1 and curves Train 2, of respectively. two trains running Train 2 startsin the 50same s after power Train range, 1. The and two the trainblue power and red curves curves are represent the same. Train In Figure 1 and4 ,T Trainrain 2, 2 respectively. operates at the Train first 2 station. starts 50 At s 118–135after Train s, then1. The thetwo traction train power and braking curves of are Train the 1same. and TrainIn Figure 2 overlap, 4, Train and 2 Trainoperates 1 can at usethe thefirst regenerative station. At 118 braking–135 s, energythen the of Traintraction 2;at and 179~198 braking s, Trainof Train 2’s traction1 and Train condition 2 overlap, and Trainand Train 1’s braking 1 can use condition the regenera overlap,tive andbraking the regenerative energy of Train braking 2; at energy 179~198 of Train s, Train 1 is partly2’s traction used bycondition Train 2; and at 210 Train s and 1’ later,s braking Train condition 1 stops, Trainoverlap, 2 brakes, and the and regenerative the regenerative braking braking energy energy of Train of Train 1 is partly 2 cannot used be used.by Train In Figure2; at 2105, thes and dwell later, timeTrain of 1 Train stops, 2 isTrain extended, 2 brakes and, and the the starting regenerative time of braking Train 2 inenergy the second of Train station 2 cannot is delayed. be used. It In can Figure be seen5, the that, dwell compared time of to Train Figure 2 is4 ,extended, in Figure5 and, the the train starting operating time condition of Train 2 overlapping in the second time station is extended is delayed. fromIt can 179~198 be seen s that, to 179~205 compared s, and to TrainFigure 2 4 uses, in Figure more regenerative 5, the train operati brakingng energy condition from overlapping Train 1, while time theis overlappingextended from time 179~198 of one train’ss to 179~205 traction s, conditionand Train and2 uses another more train’s regenerative braking braking condition energy is added from toTrain the 235~260 1, while s. the From overlapping Figures4 and time5 it of can one be seen train’s that traction the adjustment condition of and the another stop time train’s has a greatbraking impactcondition on the is added regenerative to the braking235~260 energy.s. From Figures 4 and 5, it can be seen that the adjustment of the stopTherefore, time has a based great onimpact theexisting on the regenerative schedule, reasonably braking energy. adjusting the stop time of the train can effectivelyTherefore, increase based the overlappingon the existing time schedule, of the train reasonably traction andadjusting braking, the so stop that time more of regenerative the train can brakingeffectively energy increase is utilized the ov byerlapping the traction time train of the nearby, train therebytraction improvingand braking, the so utilization that more efficiency regenerative of regenerativebraking energy braking is utilized energy by and the reducing traction thetrain system nearby, total thereby energy improving consumption. the utilization efficiency of regenerative braking energy and reducing the system total energy consumption. Energies 20162016,, 99,, 208208 55of of 2121 Energies 2016, 9, 208 5 of 21

4 4 列车TRA1IN1 T列车RAI2N2 3 列车TRA1IN1 T列车RAI2N2 3 2

W 2 M W 1

M 1

M W / e

r 0

M W w e

o 0 w p

o -1 停站时间dwell n p

列车功率 i 停站时间 a -1 dwell n

列车功率 time r i a T -2 time r

T -2 -3 -3 -4 再生制动能量不能被利用Regenerative braking -4 再生制动能量不能被利用Reneegregnye rcaatnivneo tb braek uinsegd 0 50 100 150 energy200 cannot be use250d 0 50 100opera运行时间tion tim150e /ss 200 250 opera运行时间tion time /ss

Figure 4 4.. Train power curves before dwell time modiﬁcation.modification. Figure 4. Train power curves before dwell time modification.

4 4 列车TRA1IN1 列车TRA2IN2 3 列车TRA1IN1 列车TRA2IN2 3 2 2

W 1 M W 1

MW 0 r

/ e 0 r w e

o -1 停站时间dwell w p

列车功率 o -1 停站时间dwell

n time p i

列车功率

a -2

n time r i a T -2 r

T -3 -3 Reg再生制动能量被利用enerative braking -4 Regenerative braking -4 ene再生制动能量被利用rgy can be used 0 50 100 150 energy can200 be used 250 0 50 100operat运行时间ion tim150e / ss 200 250 运行时间 s operation time / s

Figure 5.5. Train power curves after dwell time modiﬁcation.modification. Figure 5. Train power curves after dwell time modification. 3.3. Train Train Operation Operation and and Power Power Supply Supply System System M Modelingodeling 3. Train Operation and Power Supply System Modeling 3.1.3.1. TrainTrain OperationOperation ModelModel 3.1. Train Operation Model A multi-train multi-train operation operation model model is established is established based based on the on single-train the single operation-train operation model, in model, which thein which trainA multi operationthe -traintrain operation stateoperation equation state model and equation theis established constraint and the conditionsconstraint based on conditions that the need single to that-t berain satisﬁed,need operation to be such satisf model, asied the, dynamicinsuch which as the performancethe dynamic train operation performance of the state train ofequation and the the train lineand and conditions,the the constraint line conditions are conditions the same,, are the butthat same, theneed traction butto bethe satisf network tractionied, voltagesuchnetwork as constraintsthe voltage dynamic constraints on performance the operation on the ofoperation ofthe the train train of and the are the train added. line are c onditions Assumingadded. Assuming, are that the there same,that are there butN trains arethe Ntraction on trains the line,networkon the for line, trainvoltage forj, train the constraints train j, the operation train on operationthe modeloperation mustmodel of meetthe must train the meet followingare the added. following conditions: Assuming conditions: that there are N trains on theTrain line, kinematicskinematics for train j, model:themodel: train operation model must meet the following conditions: Train kinematics model: v  t ď t ď t . jvj t j0j0 t t jf j f xj “ v t t t (1) x$j   j j0 jf  (1) 0 ts´j0 ď t ď tsj f ¯ x j  0 ts t ts (1) &   j0 jf  0 ´tsj0  t ts jf ¯ %  EnergiesEnergies2016 ,20169, 208, 9, 208 6 of 216 of 21

  fj u j,,, U T j v j  r j x j v j  t j0  t  t jf  . fj uj, UTj, vj ´ rj xj, vj tj0 ď t ď tj f v j   (2) vj “ (2) $ 0 ` 0 ˘ ` ˘ ts´ tsďj0t ď tts ¯ ts jf  &  j0 j f ´ ¯ wherewherexj is x thej is the position position of% train of trainj at tj time;at t time;vj is v thej is the speed speed of train of trainj at tj attime; t time;uj is u thej is the traction traction or braking or braking controlcontrol parameters parameters of train of trainj; UTj j; isU theTj is pantographthe pantograph voltage voltage of train of trainj; fj is j the; fj is unit the mass unit tractivemass tractive force orforce brakingor braking force with force current with control current parameters control parametersfj, voltage fjU, voltageTj and speed UTj andvj; r speedj is the v sumj; rj ofis the running sum of the resistancerunning and resistance accessories and resistance accessories of resistance the unit mass of the of uni traint massj in the of currenttrain j in position the currentxj with position speed xvj withj; tj0 isspeed the starting vj; tj0 is time the starting of train jtime; tj f is of the train dwell j; tjf time is the of dwell train j ;timetsj0 isof the train start j; ts momentj0 is the ofstart the moment dwell time of the of traindwellj; andtimets ofj f trainis the j; end and moment tsjf is the of end the moment dwell time of the of traindwellj. time of train j. BeginningBeginning and and end end state state constraints constraints of the of train:the train:

x tj0x“ tj0xj0, x j 0,0v t vj0 t “ j 0 0 (3)(3) ` ˘ ` ˘ x t “ x v t “ j f x tjfj f , x jf,0 vj f t jf  0 (4)(4) ´ ¯ ´ ¯ x ptq “ xj f , v ptq “ 0 tsj0 ď t ď tsj f (5) x t  xjf,0 v t  ts j0  t  ts jf  (5) ´ ¯ Train dynamic performance constraint: Train dynamic performance constraint: ´ ď ď 1 11u j u 1  (6)(6) j

FigureFigure6 is the6 is schematic the schematic diagram diagram of uj , of where, uj, where when, when the value the ofvalueuj is of greater uj is greater than zero, thanfj is zero, positive,fj is positive, as the traction as the force;traction when force; the when value the of u valuej is less of than uj is zero, less than then fzero,j is negative, then fj is as negative, is the braking as is the force;braking when theforce; value when of uthej is value zero, thenof ujf jisis zero zero,, then and thefj is train zero is, and in the the coasting train isdrifting in the coasting condition, drifting as showncondition, in Equation as shown (7). in Equation (7).

force受力 f

maximum tractive 最大牵引力force fmax

-1 +1 u

maximum braking fmin 最大制动力force

FigureFigure 6. Deﬁnition 6. Definition of control of control parameter parameter u. u.

AccordingAccording to Equation to Equation (6) and(6) and the the relationship relationship between between traction traction force force and and network network voltage voltage (Figure 2), the relationship among train force fj (traction or braking force), control parameters uj, (Figure2), the relationship among train force fj (traction or braking force), control parameters uj, network voltage UTj and train speed vj can be obtained, as shown in Equation (7): network voltage UTj and train speed vj can be obtained, as shown in Equation (7):  uj f jmax  U T j,0 v j  u j  uj ¨ fjmax UTj, vj uj ě 0 fj uj,fVjTj u, v jj,, V“ T j v j    (7)(7) # uj ¨ fjmin UTj, vj uj ă 0 uj` f jmin U˘ T j,0 v j ` u j˘ ` ˘      ` ˘ ` ˘ where fjmax is the maximum traction force of train j with the current speed vj and voltage UTj; and fjmin where fjmax is the maximum traction force of train j with the current speed vj and voltage UTj; and fjmin is the maximum braking force of train j with the current speed vj and voltage UTj. is the maximum braking force of train j with the current speed vj and voltage UTj. Line speed limit constraint: Energies 2016, 9, 208 7 of 21 Energies 2016, 9, 208 7 of 21

0 v V x (8) Line speed limit constraint: j jmax  j  ď v ď V x where Vjmax is the maximum speed allowed0 ofj trainjmax j in thej current location xj. (8) where Vjmax is the maximum speed allowed of train j in` the˘ current location xj. 3.2. Traction Power Supply System Model 3.2. TractionIn a multi Power-t Supplyrain operation System Modelprocess, power taken from the catenary and position of every train is constantlyIn a multi-train changing. operation In terms process, of circuit power theory, taken the traction from the power catenary supply and system position is ofa complex every train time- is constantlyvarying network changing.; the power In terms supply of circuit system theory, state the is related traction to power location supply x(t), speed system v(t) is, a pantograph complex time-varyingvoltage U(t) network; and train the control power parameters supply system u(t) of state each is relatedtrain, as to well location as thex(t) number, speed ofv(t) circuit, pantograph equations voltageand theU(t) trainand number train control and location parameterss. u(t) of each train, as well as the number of circuit equations and theWith train a number normal and bilateral locations. power supply, the traction substation bus link uplink and downlink catenaryWith a (third normal rail) bilateral and, similarly, power supply, uplink the and traction downlink substation rail are bus linked link through uplink and the downlink return line. catenaryThe substation (third rail)s of the and, traction similarly, network uplink all andcontribute. downlink Due rail to th aree parallel linked operation through theof the return uplink line. and Thedownlink substations trains, of the it is traction necessary network to consider all contribute. the interaction Due to between the parallel the upper operation and oflower the uplinkcatenary and and downlinktrack when trains, establishing it is necessary the mathematical to consider the model interaction of the betweenDC side [ the40]. upper and lower catenary and track when establishing the mathematical model of the DC side [40]. 3.2.1. Power Substation Model with Rectifier Units 3.2.1. Power Substation Model with Rectifier Units The external characteristic of the rectifier unit is that the DC output voltage curve changes with theThe load external current characteristic or short-circuit of the current. rectifier The unit rectifier is that unit the DCin the output DC traction voltage curvepower changes supply withsystem thesimulation load current is often or short-circuit treated as current.a constant The ideal rectifier voltage unit source in the with DC tractioninternal powerresistance, supply as systemshown in simulationFigure 7. is often treated as a constant ideal voltage source with internal resistance, as shown in Figure7.

TSS1 TSS2 TSS3 traction downline RTSS return Tss Vo VTSS upline third rail rail ETSS

Figure 7. Schematic diagram of rectiﬁer units. Figure 7. Schematic diagram of rectifier units.

The TSS is the traction substation; RTSS is the substation equivalent resistance; ETSS is the The TSS is the traction substation; RTSS is the substation equivalent resistance; ETSS is the equivalent voltage source of the substation; and VTSS is the substation busbar voltage. In fact, the externalequivalent characteristic voltage sourceof the rectifier of the unitsubstation is nonlinear.; and V InTSS this is paper,the substation the output busbar characteristic voltage. curve In fact, ofthe the external two parallel characteristic 12-pulse rectifierof the rectifier circuits unit is equivalent is nonlinear. to the In externalthis paper, characteristic the output curve characteristic of the 24-pulsecurve of rectifier the two circuit, parallel and 12 the-pulse substation rectifier external circuits characteristic is equivalent curve to the adopts external the characteristic multi-linear modelcurve of proposedthe 24-pulse in [41 rectifier]. circuit, and the substation external characteristic curve adopts the multi-linear model proposed in [41]. 3.2.2. Train Model 3.2.2. Train Model The train power taken from the network is often dependent of the network voltage, and the currentThe is greatly train power influenced taken by from the networkthe network voltage; is often thus, dependent the power of source the network model isvoltage, used for and the the traincurrent model is ingreatly the traction influenced power by supply the network system. voltage The speed,; thus position, the power and powersource ofmodel each is train used at eachfor the momenttrain model can be in obtained the traction using power the trainsupply operation system. model.The speed, As the position train isand based power on theof each power train source at each model,moment the circuitcan be equations obtained areusing a nonlinear the train equationoperation set model. and must As the be solvedtrain is using based the on iterative the power method. source Whenmodel, train the traction circuit occurs, equations the power are a nonlinear value is positive, equation which set and means must that be electrical solved using power the must iterati beve obtainedmethod. from When the train DC net; traction when occurs, the train the is coasting,power value the poweris positive, used which is train means auxiliary that power electrical obtained power must be obtained from the DC net; when the train is coasting, the power used is train auxiliary power Energies 2016, 9, 208 8 of 21 Energies 2016, 9, 208 8 of 21 fromobtained the DCfrom net the power; DC net and power; during and regenerativeduring regenerat braking,ive braking, the power the power is negative, is negative, which which means means that regenerativethat regenerative braking braking power power flows flows from from the trainthe train to the to powerthe power grid grid [42]. [42].

3.2.3.3.2.3. TractionTraction NetworkNetwork ModelModel TheThe lineline resistanceresistance ofof thethe tractiontraction networknetwork changeschanges withwith thethe changechange ofof thethe locationlocation ofof thethe traintrain andand thethe positionposition betweenbetween trains,trains, whichwhich mainlymainly includesincludes thethe lineline resistanceresistance betweenbetween thethe traintrain andand thethe substation,substation, thethe lineline resistanceresistance between trains and the leakage resistance of the track to the ground. TheThe leakageleakage resistanceresistance isis generatedgenerated duedue toto thethe factfact thatthat thethe mostmost tractiontraction currentcurrent ﬂowsflows backback toto thethe tractiontraction substationsubstation along the rail line,line, asas partpart ofof itit leaksleaks intointo thethe ground,ground, thenthen ﬂowsflows backback toto thethe railrail andand then back to the traction traction substation. substation. When When calculating calculating the the equivalent equivalent resistance, resistance, it itis isnecessary necessary to toconsider consider the the leakage leakage resistance. resistance. According According to to the the uniform uniform transmission transmission line line theory, theory, the accurate accurate equivalentequivalent ππ circuitscircuits [[4433,,4444]] areare usedused forfor eacheach sectionsection ofof thethe lineline tracks.tracks. ConsideringConsidering thethe precisionprecision ofof thethe modeling,modeling, thethe upperupper andand lowerlower railsrails shouldshould bebe modeledmodeled separately,separately, as as shown shown in in Figure Figure8 .8.

substation up conductor rail substation substation substation + + + + down conductor rail

Rtss Rtss Rtss Rtss Vtss Vtss Vtss train Vtss Etss Etss Etss Etss - - - -

leak resistance to ground

FigureFigure 8.8. Equivalent circuitcircuit ofof thethe DCDC powerpower supplysupply system.system.

3.3.3.3. Optimization GoalGoal ofof Multi-TrainMulti-Train EnergyEnergy SavingSaving OperationOperation

3.3.1.3.3.1. CalculationCalculation ofof EnergyEnergy ConsumptionConsumption ParametersParameters AccordingAccording toto the the circuit circuit solution solution of of the the traction traction power power supply supply system, system, the dynamicthe dynamic electrical electrical data, suchdata, assuch voltage, as voltage, current current of each of train each and train substation, and substation, can be obtained,can be obtained, and according and according to the voltage to the andvoltage current and integral,current integral, the train the and train substation’s and substation energy’s consumption energy consumption values can values be obtained. can be obtained. WhenWhen thethe traintrain currentcurrent isis positive,positive, itit isis inin thethe tractiontraction state,state, andand thethe powerpower integrationintegration ofof thethe correspondingcorresponding timetime isis thethe traintrain tractiontraction energyenergy consumption.consumption. TT T T Etra Ptra dt  I tra  U train dt (9) Etra “ 00Ptradt “ Itra ¨ Utraindt (9) ż0 ż0 where Ptra is the train traction power; Itra is the train traction current; Utrain is the train pantograph wherevoltage;Ptra andis T the is trainthe total traction running power; time.Itra is the train traction current; Utrain is the train pantograph voltage;When and theT is train the total current running is negative, time. it is in the regenerative braking state, and the power integrationWhen the of the train corresponding current is negative, time is it isthe in train the regenerative regenerative braking braking state, energy. and the power integration of the corresponding time is the train regenerative braking energy. TT E P dt  I  U dt (10) regT00 regT reg train Ereg “ Pregdt “ Ireg ¨ Utraindt (10) where Prega is the train regenerative brakingż0 power; andż0 Ireg is the train regenerative braking current. According to the calculated results of the substation, the power integration at the time substation where Prega is the train regenerative braking power; and Ireg is the train regenerative braking current. current is positive, which means that the output energy is the substation traction energy consumption within the corresponding time.

TT E P dt  U  I dt (11) ss 00ss s s Energies 2016, 9, 208 9 of 21

According to the calculated results of the substation, the power integration at the time substation current is positive, which means that the output energy is the substation traction energy consumption within the corresponding time. T T Ess “ Pssdt “ Us ¨ Isdt (11) ż0 ż0 where ESS is the substation energy consumption; PSS is the substation power; US is the substation voltage; and IS is the substation current. In this paper, the model of the braking resistor is added to the actual simulation to calculate the unemployed regenerative braking energy. The braking resistance absorption power integration is the regenerative braking energy, which is not being used at the running time.

T T Ereg_unuse “ Pbr_resdt “ Us ¨ Ibr_resdt (12) ż0 ż0 where Ereg_unused is the braking resistance absorption energy; Pbr_res is the braking resistance absorption power; and Ibr_res is the braking resistance absorption current.

3.3.2. Optimization Goal of Multi-Train Operation When operating multi-trains, considering the utilization of regenerative braking energy, the energy consumption of the system consists of the following:

Ess “ Etra ´ Ereg_used ` Eloss (13) ÿ ÿ ÿ ÿ Ereg_used “ Ereg ´ Ereg_unuse (14) ÿ ÿ ÿ where ESS is the total energy consumption of all substations; Etra is the total traction energy consumption of all trains; E is the system total used regenerative braking energy; E is ř reg_used ř loss the total traction power supply network line loss; E is the system total unused regenerative ř reg_unuse ř braking energy; and E is the total regenerative braking energy of all trains. reg ř As can be seen, the effective use of multi-train regenerative braking energy can reduce the system’s ř total substation energy consumption. The multi-train operation optimization aims to optimize the substation total energy consumption, as shown in Equation (15).

M J “ Ess (15) S“ ÿ1 where J is the substation total energy consumption; and M is the number of traction substations. To more accurately measure the use of the regenerative braking energy before and after optimization, we deﬁne the regenerative braking energy utilization, namely the ratios of total regenerative braking energy used by the trains and total regenerative braking energy generated by the trains, as shown in Equation (16):

Ereg ´ Ereg_unuse η “ ˆ 100% (16) reg-used E ř řreg ř where ηreg´used is the regenerative braking energy utilization.

3.4. Calculation Flow of Multi-Train Operation

3.4.1. Calculation of Train Operation In this paper, the discussion of multi-train operation is built on the assumption that each of the trains is strictly restrained by schedule, and the models are simpliﬁed: the former means assuming the Energies 2016, 9, 208 10 of 21 operation curves of each train running in the same direction are the same; and the latter means not considering the effect of traction network voltage to train traction and braking performance, assuming that the current network voltage can meet the needs of the train traction and braking performance. As shown in Figure9, dwell time, running time, train date and line data are the input data of single train operation calculation. Additionally, the time-power curve and the time-distance curve obtainedEnergies 2016 by, 9, single-train 208 operation calculation are the input data of multi-train operation simulation.10 of 21

power traction system energy parameter simulation regenerative headway model braking of energy timetable synchronization traction time power substation dwell supply energy time time-power system consumption running curve single unutilized time train time-distance regenerative operation braking train data curve calculation energy

line data

Figure 9. Flowchart of multiple train operation calculation.

3.4.2.3.4.2. Multi-TrainMulti-Train OperationOperation SimulationSimulation First, the simulation model of the traction power supply system is built according to the actual parameters ofof the the line. line. Via Via the the position, position, power power and and other other information information of the of uplink the uplink and downlink and downlink trains attrains each at moment each moment obtained obtained by single-train by single running-train running calculation, calculation, and based and on based the constraint on the constraint of headway of andheadway synchronization and synchronization time according time to according the schedule, to the the sch time,edule, position the time, and power position of andmultiple power train of operationmultiple train are obtained, operation to are be usedobtained, as the to input be used of the as tractionthe input power of the supply traction system power simulation supply system model. Assimulation shown in model. Figure 9As, according shown to in theFigure input 9 multi-train, according operation to the input data, the multi structure-train current operation traction data, networkthe structure equivalent current circuit traction in thenetwork model equivalent of the traction circuit power in the supply model system, of the traction we can determinepower supply the electricalsystem, we parameters can determine of the tractionthe electrical network parameters and obtain of the total traction traction network energy and consumption obtain the total and regenerativetraction energy braking consumption energy of and the regenerative multi-train operation, braking energy as well of as the the multi total-train energy operati consumptionon, as well and as unusedthe total regenerative energy consumption braking energy and unused of the regenerative traction power braking system energy in a given of the period traction of time.power system in a given period of time. 3.4.3. Overall Process of Calculation 3.4.3.Based Overall on Process MATLAB, of Calculation this paper next carries out the programming of the multi-train operation simulationBased model,on MATLAB, and the this basic paper framework next carries and out data the structure programming of the model of the are multi shown-train in operation Figure9. Figuresimulation9 is the model, summary and the of thebasic content framework of Sections and data3.4.1 structureand 3.4.2 .of the model are shown in Figure 9. Figure 9 is the summary of the content of Sections 2.4.1 and 2.4.2. 4. Genetic Algorithm Optimization 4. GeneticBased Algorithm on the mathematical Optimization model of the multi-train operation established above, the dwell time inﬂuenceBased on on the the regenerative mathematical braking model energy of the utilization multi-train and operation total energy established consumption above, of the the dwell system time is researched.influence on The the energyregenerative consumption braking modelenergy based utilization on the and dwell total time energy is established consumption and of is the optimized system withis researched. genetic algorithms. The energy Simulationconsumption and model veriﬁcation based on are the performed dwell time based is established on the actual and is data optimized of the Yizhuangwith genetic Line. algorithms. Simulation and verification are performed based on the actual data of the Yizhuang Line. 4.1. Mathematical Model

4.1. MathematicalThe research Model of the dwell time is based on the assumption that the headway, synchronization time and inter-station travel time are all ﬁxed. The research of the dwell time is based on the assumption that the headway, synchronization time and inter-station travel time are all fixed. Figure 10 is the schematic diagram of the dwell time modification. Assuming that the number of stations in the line is N, then the dwell time at each station of the line are: t1, t2, …, tN, where tN is the dwell time at the N-th station. The total energy consumption of the system is different with different dwell time combinations; thus, the problem can be abstracted as a combinatorial optimization problem of dwell time and energy consumption, such as shown in Equations (17) and (18). Energies 2016, 9, 208 11 of 21

Figure 10 is the schematic diagram of the dwell time modiﬁcation. Assuming that the number of stations in the line is N, then the dwell time at each station of the line are: t1, t2, ... , tN, where tN is the dwell time at the N-th station. The total energy consumption of the system is different with different dwell time combinations; thus, the problem can be abstracted as a combinatorial optimization problem Energies 2016, 9, 208 11 of 21 of dwell time and energy consumption, such as shown in Equations (17) and (18).

t1 t2 …… tN E1 operation站间运行时间 time 停站时间dwell N个车站N between stops time stations t1 t2 …… tN E2

FigureFigure 10. 10. SchematicSchematic diagram diagram of of dwell dwell time time modification. modiﬁcation.

ForFor NN stations,stations, assuming assuming that that the the dwell dwell time time of train, tstopstop, ,has has k kcombinations,combinations, then then tstop_itstop_i is theis the i- thi-th dwell dwell time time combination. combination. tstop_i “ tt1, t2, ¨ ¨ ¨ ti ¨ ¨ ¨ , tN´1, tNu (17) tttttt ,, , , (17) stop_12 i i N 1 N  k combinations of dwell time have corresponding system total consumption Ei: k combinations of dwell time have corresponding system total consumption Ei: Ei “ f tstop_i (18) Eftistopi _ (18) ` ˘ For urban rail transit, there are typically 10–15 stations. Assuming that the dwell time of eachFor station urban is rail adjusted transit, by there´5 s–+5are typically s on the 10–15 basis stations. of the original Assuming schedule, that the there dwell are time 1010–1015 of each stationcombinations is adjusted of dwell by −5 time.s–+5 s The on the number basis of combinationsthe original schedule, is very large,there are too 1010–1015 large to be combinations enumerated; ofthus, dwell an optimizationtime. The number algorithm of combinations is required foris ve thery solution. large, too In large a variety to be of enumerated; heuristic algorithms, thus, an optimizationgenetic algorithms algorithm operate is required directly for on thethe structuresolution. ofIn thea variety object, of with heuristic no derivation algorithms, or functiongenetic algorithms operate directly on the structure of the object, with no derivation or function continuity continuity limit, thus allowing the use of a very complex ﬁtness function (also known as the target limit, thus allowing the use of a very complex fitness function (also known as the target function), function), and the variable range of variables can be restricted. Randomization optimization methods and the variable range of variables can be restricted. Randomization optimization methods can can automatically obtain and guide the optimization search space and do no need to identify the rules; automatically obtain and guide the optimization search space and do no need to identify the rules; thus, they are more suitable for optimization in which time is the variable. Therefore, this paper uses a thus, they are more suitable for optimization in which time is the variable. Therefore, this paper uses genetic algorithm to optimize the dwell time. a genetic algorithm to optimize the dwell time. 4.2. Introduction of Genetic Algorithm 4.2. Introduction of Genetic Algorithm A genetic algorithm is a random search algorithm, referencing biological natural selection andA genetic genetic mechanisms. algorithm is Differenta random from search traditional algorithm, algorithms, referencing genetic biological algorithms natural searchselection from and a geneticrandomly-generated mechanisms. initial Different solution, from and traditional through the selection,algorithms, crossover genetic and algorithms mutation operationssearch from iterate a randomly-generatedstep by step to produce initial new solution, solutions. and Each through individual the selection, in a group crossover is a solution, and mutation called chromosomes. operations iterateThe merits step ofby chromosomes step to produce are evaluated new solutions. by the fitnessEach individual function. Accordingin a group to is the a fitnesssolution, value, called we chromosomes.select a certain numberThe merits of outstanding of chromosomes individuals are eval fromuated the by previous the fitness generation, function. to formAccording the following to the fitnessgroup throughvalue, we crossover select a andcertain mutation number operations. of outstanding After severalindividuals generations from the of previous evolution, generation, the algorithm to formconverges the following to the optimal group chromosome, through crossoveri.e., the and final mu solutiontation ofoperations. the problem. After several generations of evolution, the algorithm converges to the optimal chromosome, i.e., the final solution of the problem. 4.3. Optimization Method 4.3. Optimization Method Different combinations of dwell time have different impacts on the energy consumption of the powerDifferent supply combinations system. By means of dwell of thetime genetic have diffe algorithmrent impacts to encode on the each energy dwell consumption time, the optimal of the powercombination supply of system. the dwell By time means can of be the solved. genetic algorithm to encode each dwell time, the optimal combination of the dwell time can be solved. Assuming that the train headway and synchronization time of the uplink and downlink train are fixed at the same period, the train operation curves in the same direction are the same (i.e., the operation curves of the uplink trains are the same, and the operation curves of the downlink trains are the same), and the dwell times of the uplink and downlink trains at the same station are the same, as shown in Equation (19).

ttttttdown  12,, i , N 1 , N  (19) ttttttup  N,, N121 i ,, Energies 2016, 9, 208 12 of 21

Assuming that the train headway and synchronization time of the uplink and downlink train are ﬁxed at the same period, the train operation curves in the same direction are the same (i.e., the operation curves of the uplink trains are the same, and the operation curves of the downlink trains are the same), and the dwell times of the uplink and downlink trains at the same station are the same, as shown in Equation (19). Energies 2016, 9, 208 t “ tt , t , ¨ ¨ ¨ t ¨ ¨ ¨ , t , t u 12 of 21 down 1 2 i N´1 N (19) tup “ ttN, tN´1, ¨ ¨ ¨ ti ¨ ¨ ¨ , t2, t1u where N is the number of stations; ti is the dwell time of the i-th station; tdown is the dwell time combinationwhere N is theof downlink number ofline; stations; and tupt isi is the the dwell dwell time time combination of the i-th of station; the uplinktdown line.is the dwell time combinationBased on of the downlink above assumptions, line; and tup iswe the can dwell build time a dwell combination time optimization of the uplink model line. based on the geneticBased algorithm. on the above assumptions, we can build a dwell time optimization model based on the genetic algorithm. 4.3.1. Encoding 4.3.1. Encoding The optimization parameter is the train dwell time; thus, the collection of the train dwell time at each The station optimization is encoded parameter as a chromosome, is the train and dwell each time; station thus, dwell the collection time is of the the gene train forming dwell time the chromosome.at each station Using is encoded real numbers as a chromosome, to encode the and dwell each time, station the dwell schematic time isdiagram the gene of formingdwell time the enchromosome.coding is shown Using in Figure real numbers 11. to encode the dwell time, the schematic diagram of dwell time encoding is shown in Figure 11.

车站序号serial number of stations 1 2 3 …… n 停站时间dwell time 30 35 27 …… 36 染色体chromosome

基因gene

Figure 11. Schematic diagram of dwell time encoding. Figure 11. Schematic diagram of dwell time encoding.

InIn an an urban urban rail rail transit transit system, system, the the length length of of the the train train dwell dwell time time is is related related to to the the number number of of passengers,passengers, the number of train doors and other factors [ 4545]. In In order order to to ensure ensure the boarding/alighting boarding/alighting needsneeds of of passengers passengers during during actual actual operation, operation, this this paper paper optimizes optimizes the dwell the dwell time based time on based the existing on the existingtrain schedule train schedule in a certain in a certain range. range. Therefore, Therefore, the encoding the encoding must must restrain restrain the size the ofsize the of gene,the gene, e.g., e.g.the, dwellthe dwell time time is required is required within within 25–45 25 s– for45 eachs for station,each station, or adjusted or adjusted within within´5 s–+5 −5 s – on+5 thes on basis the basisof the of original the original dwell dwell time, time, to ensure to ensure the feasibilitythe feasibility of the of dwellthe dwell time time after after optimization. optimization. Using Using the thegenetic genetic algorithm, algorithm, the initialthe initial population populat ision generated, is generated, andfor and each for chromosome,each chromosome the genes, the mustgenes satisfy must satisfyEquation Equation (20). (20). tlow_limt ď ti ď tup_limt (20) tlow_limt ti t up_limt (20) where tup_limit is the upper limit of dwell time; and tlow_limit is the lower limit of dwell time. where tup_limit is the upper limit of dwell time; and tlow_limit is the lower limit of dwell time. 4.3.2. Generating the Initial Population 4.3.2. AccordingGenerating to the the Initial genetic Population algorithm, the initial population of the combination of different dwell timesAccording is randomly to the generated, genetic asalgorithm, shown in the Figure initial 12 population. of the combination of different dwell times is randomly generated, as shown in Figure 12. EnergiesEnergies 20162016,, 99,, 2 20808 13 of 21

serial车站序号 number of stations

1 2 3 …… n 停站时间dwell time 30 35 27 …… 36

染色体chromosome 30 35 27 …… 36 1

30 35 27 …… 36 2 种群population ……

30 35 27 …… 36 m

Figure 12. Initial population of dwell time. Figure 12. Initial population of dwell time. 4.3.3. Fitness Function 4.3.3. Fitness Function The optimization of the train dwell time affects the utilization of the regenerative braking energy whenThe the optimization multi-train isof operating. the train dwell The electrictime affects energy the usedutilization by the of train the regenerative originates from braking the tractionenergy whensubstation, the multi thus-train the optimizationis operating. objectiveThe electric function energy is used the totalby the energy train consumptionoriginates from of the the traction traction substation,power supply thus system. the optimiza The negativetion objective value of function the energy is the consumption total energy of consumption the ﬁtness function of the istraction shown powerin Equation supply (21). system. The negative value of the energy consumption of the fitness function is shown in Equation (21). T

Fit_function “ ´T Psdt (21) Fit_function =-ż P dt 0 s (21) 0 where Ps is the total power of all substations; and T is the total operation time. where Ps is the total power of all substations; and T is the total operation time. 4.4. Calculation Flowchart 4.4. CalculationFigure 13 isFlowchart the algorithm ﬂowchart of the train dwell time optimization using a genetic algorithm.

1. FigureThe first 13 generation is the algorithm of the population flowchart is generated of the randomly,train dwell and time the m optimizationindividuals generatedusing a geneticcorrespond algorithm. to the collection of dwell time, where m is the population quantity. 2. Combining the dwell time in Step 1 and train operation data, the input direct current supply 1 The first generation of the population is generated randomly, and the m individuals generated model is used to establish the corresponding circuits to calculate the electric and energy correspond to the collection of dwell time, where m is the population quantity. consumption data of the trains and power substation. 2 Combining the dwell time in Step 1 and train operation data, the input direct current supply 3. According to the fitness function, Equation (20) is used calculate the fitness value of model is used to establish the corresponding circuits to calculate the electric and energy each individual. consumption data of the trains and power substation. 34. AccordingAccording to the the fitness fitness value function, calculated Equation in Step (20) 3, we is perform used calculate genetic manipulation, the fitness value such of as eachselection, individual. cross, mutation, and so on, to generate a new population. 45. AccordingWe determine to the whether fitness the value termination calculated conditions in Step 3, are we met, perform and if genetic they are manipulation, not satisfied, thesuch new as selection,generation cross, of population mutation, and data so will on, be to inputgenerate into a Stepnew 2population. to continue the calculation; if they are 5 Wesatisfied, determine then the whether calculation the termination and output results conditions are terminated. are met, and if they are not satisfied, the new generation of population data will be input into Step 2 to continue the calculation; if they are satisfied, then the calculation and output results are terminated. Energies 2016, 9, 208 14 of 21 Energies 2016, 9, 208 14 of 21

multiple train simulation ini初始种群tial population 多列车运行仿真停站时间dwell time

m多列车运行仿真ultiple train simulation 单列车运行模型single train model

c适应度函数计算alculate fitness function 牵引供电仿真模型traction power supply simulation model

populati种群进化on evolution population 种群进化 selection evolution 选择

交叉cross meet the N 满足终止termination con条件ditions m突变utation Y

结束end new新种群 population

Figure 13.13. Flowchart of the dwell time optimization algorithm.algorithm.

5.5. Dwell Dwell Time Time Optimization Optimization Based Based on on Yizhuang Yizhuang Subway Subway Line Line The traction power supply system of the Beijing subway Yizhuang Line adopts the distributed power supply mode, with a total of 12 traction substations. T Thehe tractiontraction substation adopts the 24-pulse24-pulse rectiﬁer,rectifier, asas describeddescribed in in detail detail in in Section Section 3.2.1 2.2.1.. Regenerative Regenerative braking braking absorbing absorbing resistors resistors are setare in set each in ofeach the of traction the traction substation. substation. Using Using the thirdthe third rail powerrail power supply supply mode, mode, the the rated rated voltage voltage is DC is DC 750 750 V. V. TheThe t trainrain equivalent equivalent mass mass of of Beijing Beijing Yizhuang Yizhuang subway subway line line w withith different different load load cases cases is listed is listed in inTable Table 1. 1In. InTable Table 1, 1AW, AW represents represents the the train train passenger passenger capacity. capacity. AW0 AW0 represents represents no passengers. no passengers. AW2 AW2represents represents rated rated passenger passenger capacity. capacity. AW3 AW3 represents represents overload overload passenger passenger capacity. capacity. M M represents motor car, T represents trailertrailer car and Tc representsrepresents trailertrailer withwith a cab.cab. AW3 isis thethe normalnormal operation condition of Beijing Yizhuang Yizhuang subway subway line. line. The simulation data are AW3 level. level. Train Train and line parameters are listed in TableTable2 2.. A A single single motor motor rated rated power power is is 180 180 kW. kW. There There are are four four motors. motors.

TableTable 1. Train equivalent mass of Beijing Yizhuang subwaysubway lineline withwith differentdifferent loadload cases.cases.

TrainTrain Equivalent Equivalent Mass (t)Mass (t) Tc1 Tc1 M1 M1 TT M3M3 M2M2 Tc2 AW0AW0 3333 35 35 2828 3535 3535 3333 AW2AW2 46.5646.56 50.24 50.24 43.24 43.2450.24 50.24 50.2450.24 46.56 AW3 50.4 54.5 47.5 54.5 54.5 50.4 AW3 50.4 54.5 47.5 54.5 54.5 50.4

Table 2. Train and line parameters. Table 2. Train and line parameters. ParametersParameters ValueValue ParametersParameters ValueValue NominalNominal voltage voltage 750 V750 V EfficiencyEfficiency of of inverter inverter 0.930.93 MaximumMaximum voltage voltage 900 V900 V EfficiencyEfficiency of of motor motor 0.9150.915 MinimumMinimum voltage voltage 680 V680 V EfficiencyEfficiency of of gear gear 0.9750.975 Power-factor of load 0.85 Motor rated power 180 kW*4 RegenerationPower limit-factor voltage of U1/U2load 900 V/9500.85 V MaximumMotor rated acceleration power 1801 m/s kW*42 RegenerationMaximum limit speed voltage U1/U2 80900 km/h V/950 V MaximumMaximum deceleration acceleration 11 m/s m/s2 2 ResistanceMaximum of traction networkspeed 50.007 80Ω/km km/h Maximum Resistance deceleration of rail 0.0091 Ωm/s/km2 Resistance of traction network 50.007 Ω/km Resistance of rail 0.009 Ω/km Energies 2016, 9, 208 15 of 21 Energies 2016, 9, 208 15 of 21

The Yizhuang subway line has a total of 14 stations. According to the current actual operating schedules,The Yizhuang the dwell subway times of line each has station a total are of listed 14 stations. in Table According 3. The dwell to the time current throughout actual the operating day of eachschedules, train is the the dwellsame, timesand the of dwell each stationtime of arethe listedsame sta intion Table of3 .up The trains dwell and time down throughout trains is the the same day. of each train is the same, and the dwell time of the same station of up trains and down trains is the same. Table 3. Dwell times of Yizhuang subway line. Table 3. Dwell times of Yizhuang subway line. Serial Dwell Dwell Station Name Station Name Number Time (s) Time (s) Serial Dwell Dwell 1 SongjiazhuangStation Name / YizhuangStation Railway Name Station / Number Time (s) Time (s) 2 Xiaocun 30 Ciqu 35 3 1Xiaohongmen Songjiazhuang 30 / YizhuangCiqu Railway South Station / 35 4 2Jiugong Xiaocun 30 30 Jinghailu Ciqu 35 30 3 Xiaohongmen 30 Ciqu South 35 5 Yizhuangqiao 35 Tongjinanlu 30 4 Jiugong 30 Jinghailu 30 6 Yizhuang Culture Park 30 Rongchandongjie 30 5 Yizhuangqiao 35 Tongjinanlu 30 7 6 YizhuangWanyuanjie Culture Park30 30 RongchandongjieRongjingdongjie 30 30 8 7Rongjingdongjie Wanyuanjie 30 30 RongjingdongjieWanyuanjie 30 30 9 8Rongchandongjie Rongjingdongjie 30 30 Yizhuang Wanyuanjie Culture Park 30 30 10 9Tongjinanlu Rongchandongjie 30 30 YizhuangYizhuangqiao Culture Park 30 35 11 10Jinghailu Tongjinanlu 30 30 YizhuangqiaoJiugong 35 30 12 11Ciqu Jinghailu South 35 30 Xiaohongmen Jiugong 30 30 13 12 CiquCiqu South35 35 XiaohongmenXiaocun 30 30 14 13Yizhuang Railway Ciqu Station 35/ Songjiazhuang Xiaocun 30 / 14 Yizhuang Railway Station / Songjiazhuang / Using the genetic algorithm to encode the 12 dwell times, the upper and lower bounds of the dwellUsing time of the each genetic train algorithm are bound to to encode the −5– the+10 12s range dwell of times, the original the upper stop and time. lower In the bounds simulation, of the thedwell headway time of is each 5.5 trainmin; the are boundsynchronization to the ´5–+10 time sof range the up of train the original and the stop down time. train In is the 3.5 simulation, min; and thethe headway train operation is 5.5 min; data the use synchronization automatic train time operation of the up ( trainATO) and actual the down operation train data. is 3.5 min;The geneticand the parametertrain operation settings data are use shown automatic in Table train operation4. Figure (ATO)14 is anactual evolution operation of data. the search The genetic for the parameter optimal solutionsettings arein the shown algorithm in Table calculation4. Figure process.14 is an evolution of the search for the optimal solution in the algorithm calculation process. Table 4. Simulation parameters. Table 4. Simulation parameters. Simulation Parameters Value SimulationPopulation Parameters size Value 40 Maximum numberPopulation of generations size 40 100 MaximumGeneration number gap of generations 100 0.95 Generation gap 0.95 CrossoverCrossover probability probability 0.7 0.7 MutationMutation probability probability 0.01 0.01

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Table5 shows the constraint conditions of dwell time and the results of the dwell time optimization in the simulation. It can be seen that the dwell time after the optimization is only increased by 29 s, while the whole running time of the Yizhuang Line is about 34 min; thus, the impact on the running time is minimal.

Table 5. Dwell time of Yizhuang subway line before and after optimization.

Optimization Scope (s) Difference before Serial Dwell Post-Optimization Station Name and after Time (s) Lower Upper Time (s) Number Optimization Limit Limit 1 Songjiazhuang / / / / / 2 Xiaocun 30 25 40 35 5 3 Xiaohongmen 30 25 40 35 5 4 Jiugong 30 25 40 32 2 5 Yizhuangqiao 35 30 45 36 1 6 Yizhuang Culture Park 30 25 40 34 4 7 Wanyuanjie 30 25 40 35 5 8 Rongjingdongjie 30 25 40 33 3 9 Rongchandongjie 30 25 40 34 4 10 Tongjinanlu 30 25 40 34 4 11 Jinghailu 30 25 40 35 5 12 Ciqu South 35 30 45 31 ´4 13 Ciqu 35 30 45 30 ´5 14 Yizhuang Railway Station / / / / / The total time 375 404 29

Table6 shows the energy consumption changes before and after dwell time optimization. It can be seen that the system total energy consumption is decreased from 1084 down to 995 kW¨ h, a decrease of 8%; meanwhile, the utilization rate of regenerative braking is improved from 71%–79%, for an improvement of 8%. Based on the genetic algorithm, the dwell time optimization effect is significant; the total energy consumption is reduced; and the utilization rate of regenerative braking is improved. Assuming that the train operation curve is unchanged in simulation, namely the train running time between stations is fixed, only the dwell time is optimized. As for the train, because the operation curve is the same, the train traction energy consumption and regenerative braking energy are the same before and after optimization. The fact that the energy consumption of the train traction and regenerative braking energy before and after optimization have a certain difference, as shown in Table6, is a result of the simulation results when compared to the multiple trains running results in fixed 15 min; before and after optimization, the dwell time of the train changes; the input power of train operation is different in the same time.

Table 6. Energy consumption after dwell time optimization.

Before After Rate of Energy Consumption (kW¨ h) Optimization Optimization Change Train traction energy consumption 1674 1655 Train regenerative braking energy 930 922 Substation total energy consumption 1084 995 ´8% Braking resistance absorption energy 270 195 The line losses energy 70 67 Utilization rate of regenerative braking 71% 79% 8%

Figures 15 and 16 are the total output power of the traction substation and the absorption power of braking resistance before and after dwell time optimization, respectively. For the convenience of comparison, in the ﬁgure, the power curve is ﬁlled with color; the envelope curve is the power value; the area surrounded by the envelope is the value of energy consumption. The blue curve is the total power of the substation, and the red curve is the total power absorbed by the braking resistance, i.e., the regenerative braking energy cannot be used. By comparison, it is clear that the optimized substation Energies 2016, 9, 208 17 of 21

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FigureFigure 21. 21. PowerPower of ofSongjiazhuang’s Songjiazhuang’s braking braking resistance resistance before before and and after after dwell dwell time time optimization optimization.. Figure 21. Power of Songjiazhuang’s braking resistance before and after dwell time optimization. 6. Conclusions Conclusions 6. Conclusions In thisthis paper,paper, thethe basicbasic principlesprinciples ofof traintrain energyenergy savingsaving optimizationoptimization based on dwell time are introduced,In this paper, and the the optimizationoptimization basic principles model of basedtrain energy on dwelldwell saving timetime optimization isis established.established. based With on the dwell systemsystem time totaltotal are energyintroduced, consumption consumption and the minimumoptimization minimum as as themodel the objective objective based function on function dwell and time and based is based established. on the on multi-train the Withmulti operationthe-train system operation model total andenergymodel power and consumption power system system simulation minimum simulation model, as the model, while objective usingwhile afunctionusing genetic a genetic andalgorithm, based algorithm, theon optimizationthe the multi optimization-train model operation model of the dwellmodelof the time and dwell ispower solved. time system is In solved. addition, simulation In takingaddition, model, the taking Yizhuang while theusing Line Yizhuang a asgenetic an example, Line algorithm, as based an the example, on optimization the Yizhuang based onmodel Line the currentofYizhuang the dwell dwell Line time current and is solved. underdwell thetime In condition addition, and under of taking thethe train condition the running Yizhuang of the time Linetrain increasing asrunning an example,by time 29 s, increasing the based regenerative on by the 29 brakingYizhuangs, the regenerative rate Line of thecurrent system braking dwell is increased ratetime ofan d the byunder system8%, andthe condition is the increased total traction of the by train 8%, energy andrunning consumption the time total increasing traction is reduced energy by by29 8%;s,consumption the thus, regenerative the total is reduced traction braking by energy rate 8%; ofthus consumption the, the system total is traction reduced increased energy by by 8%. 8%, consumption The and results the total show is reduced traction that using byenergy this 8%. methodconsumptionThe results is practical show is reducedthat and using effective. by this 8%; method thus, theis practical total traction and effective. energy consumption is reduced by 8%. The results show that using this method is practical and effective. Acknowledgments: This research is supported by the National National Natural Science Foundation of China under Grant 5157701. Acknowledgments:Grant 5157701. This research is supported by the National Natural Science Foundation of China under AuthorGrant 5157701 Contributions:. Fei Lin contributed to the conception of the study and the algorithm. Shihui Liu, ZhihongAuthor Contributions: Yang and Yingying Fei Lin contributed Zhao contributed to the conception significantly of the to study the analysisand the algorithm and manuscript. Shihui Liu, preparation. Zhihong ZhongpingAuthorYang and Contributions: Yingying Yang helped Zhao Fei contributed to Lin perform contributed significantly the analysisto the conception to with the analysis constructive of the and study manuscript discussions. and the algorithm preparation Hu Sun. Shihui provided. Zhongping Liu, Zhihong the Yang line andYanghelped vehicle and to Yingyingperform data. the Zhao analysis contributed with constructive significantly discussi to the analysisons. Hu andSun manuscript provided the preparation line and vehicle. Zhongping data. Yang helped to perform the analysis with constructive discussions. Hu Sun provided the line and vehicle data. ConflictsConflicts ofof Interest:Interest: TheThe authorsauthors declaredeclare nono conflictconflict ofof interest.interest. Conflicts of Interest: The authors declare no conflict of interest. References References 1. Barrero, R.; Van Mierlo,Mierlo, J.; Tackoen, X.X. EnergyEnergy savingssavings inin publicpublic transport.transport. IEEE Veh. Technol. Mag. 2008,, 3,,

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