energies

Article Optimizing the Energy-Efficient Metro Train Timetable and Control Strategy in Off-Peak Hours with Uncertain Passenger Demands

Jia Feng 1,2,*, Xiamiao Li 1, Haidong Liu 2, Xing Gao 1 and Baohua Mao 2,*

1 School of Traffic & Transportation Engineering, Central South University, Changsha 410075, China; [email protected] (X.L.); [email protected] (X.G.) 2 Ministry of Education (MOE) Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, China; [email protected] * Correspondence: [email protected] (J.F.); [email protected] (B.M.)

Academic Editor: William Holderbaum Received: 6 January 2017; Accepted: 15 March 2017; Published: 29 March 2017

Abstract: How to reduce the energy consumption of metro trains by optimizing both the timetable and control strategy is a major focus. Due to the complexity and difficulty of the combinatorial operation problem, the commonly-used method to optimize the train operation problem is based on an unchanged dwelling time for all trains at a specific station. Here, we develop a simulation-based method to design an energy-efficient train control strategy under the optimized timetable constraints, which assign the dwelling time margin to the running time. This time margin is caused by dynamically uncertain passenger demands in off-peak hours. Firstly, we formulate a dwelling time calculation model to minimize the passenger boarding and alighting time. Secondly, we design an optimal train control strategy with fixed time and develop a time-based model to describe mass-belt train movement. Finally, based on this simulation module, we present numerical examples based on the real-world operation data from the Beijing metro Line 2, in which the energy consumption of one train can be reduced by 21.9%. These results support the usefulness of the proposed approach.

Keywords: metro; energy-saving; timetable; fixed running time; dwelling time

1. Introduction Metro systems play an important role to relieve urban traffic congestion in public transportation. A major current focus in metro systems is how to reduce the energy consumption. There are two main levels of train energy-efficient operation approaches. A recent emerging research interest is in the field of regenerative energy utilization [1–6], which focuses on developing a timetable including the dwelling time at stations and running time at sections (between two adjacent stations) in order to improve the utilization of regenerative energy by synchronizing the operations of accelerating and braking trains [7]. Compared with the upper level of timetable optimization, the lower level of energy-efficient control strategy design at sections has long attracted widespread attention [8–21] to calculate the speed profile with minimum tractive energy consumption under the timetable constraints [22]. The simulation-based method is commonly used to calculate the train traction energy consumption under complex track alignments [23–27] by energy-efficient driving [28–31]. The energy-efficient timetable and control strategy are closely related, and both of them play a key role in tractive energy consumption. The previous studies typically consider these two levels separately mainly because of the complexity of the combinatorial problem and the difficulty of applying the theory in practice. This has provided an incomplete view of metro system operation.

Energies 2017, 10, 436; doi:10.3390/en10040436 www.mdpi.com/journal/energies Energies 2017, 10, 436 2 of 20

Currently, there are few studies that have addressed the problem of energy saving considering both the timetable optimization and energy-efficient train control strategy. Ding et al. [32] formulated the train energy-efficient operation as a two-level optimization problem and designed a genetic algorithm to search for the optimal solution, in which the first level is designed to decide the appropriate coasting points for trains at sections, and the second level arranges the train running times at sections for minimizing the tractive energy consumption. Cucala et al. [33] designed a model for energy-efficient driving and timetables, in which the railway operator and administrator requirements are also included. Li and Lo [34] proposed an integrated energy-efficient operation model to jointly optimize the timetable and speed profile with minimum net energy consumption. Huang et al. [21] proposes an energy-efficient approach to reduce the traction energy by optimizing the train operation for multiple sections, considering both the trip time and driving strategy. Although a set of work has been done with a comprehensive view, more realistic work is needed to apply the optimal approach into practice. A timetable determines the dwelling time at stations and the running time at inter-stations for trains [35]. The dwelling time consists of three parts [36]: the time before the doors open, the period of time during passenger exchange and the time prior to departure after the doors have closed. For the doors, the open and close times are fixed; studies estimated the dwelling time by modelling passengers’ boarding and alighting process [37–42]. However, the number of passengers boarding and alighting trains in the off-peak hours is far less than that of the peak. Accordingly, the dwelling time actually for passenger exchange is much shorter than the planned one. The margin time between planned and integrant dwelling time can be assigned to the running time in order to get a more energy-efficient travel pattern. The current paper extends the previous research in the following aspects:

­ A more realistic model for the dynamic mathematical model of the metro train is formulated in this paper based on the assumption that a train is considered as a belt with uniformly-distributed mass instead of a mass point model [43]. ­ The timetable optimization model formulated in this paper allows trains to drive as the optimal control strategy with a fixed running time of the section contributed by the reduction of dwelling time at the station, which has a significant reduction in energy consumption (more than 20%). ­ Compared with [5], when we construct the train movement model, both the track gradients and the curves are taken into account in order to describe train energy consumption more accurately.

The rest of the paper is structured as follows. Section2 formulates both the timetable optimization module considering passenger boarding and alighting phase and the train operation simulation module in order to calculate the energy consumption of train runs with fixed time. Section3 describes the optimal function and the constraints of the real-time train energy-saving scheduling problem, meanwhile proposing a simulation-based solution approach for this problem. Section4 analyzes the sensitivity of the simulation. In Section5, the performance of the proposed model is evaluated via three case studies. Finally, conclusions and recommendations are provided in Section6.

2. Mathematical Formulations In this part, first, we construct an energy-saving optimal timetable module. We define the arrival and departure time of train j at station i as ai,j and di,j, respectively, and the corresponding arrival 0 0 and departure time in the timetable as a i,j and d i,j. The running time and the dwelling time of section i (from station i to station i + 1) are expressed as TR,i,j and TD,i,j, and according to the timetable, 0 0 the running time and the dwelling time are T R,i,j and T D,i,j. The definitions are shown in Figure1. Then, we develop a time-based train operation simulation module based on a control strategy with fixed running time. Energies 2017, 10, 436 3 of 20 Energies 2017, 10, 436 3 of 20

running direction

T T T T initial running Ri,1 Di, Ri, Di, +1 time speed TRi,1 TDi, TRi, TDi, +1 optimized running time

timetable dij-1, aij, aij, dij, aij+1, aij+1, dij+1,

station i-1 section i-1 station i section i station i+1

Figure 1.1. Schematic diagram of running time calculation.calculation.

As shown in Figure 1, the actual running time can be obtained by adding the margin time0 As shown in Figure1, the actual running time can be obtained by adding the margin time ( ai,j–a i,j) (ai,j–a⁰i,j) to the initial running time, in which the margin time can be calculated by estimating the to the initial running time, in which the margin time can be calculated by estimating the passengers’ passengers’ boarding and alighting at the next station i + 1. boarding and alighting at the next station i + 1. For simplistic, we list the assumptions as follows: For simplistic, we list the assumptions as follows: (1) In this manuscript, steady demand within only two periods is considered: peak and off-peak. (1)(2) InIn this the manuscript,off-peak hours, steady there demand is no within congestion only two effect, periods indicating is considered: that all peakpassengers and off-peak. who are (2) waitingIn the off-peak on a specific hours, station there is platform no congestion can board effect, the indicating train. that all passengers who are waiting (3) Inon the a specific off-peak station hours platform we studied, can boardthe train the departure train. interval is a constant. (3)(4) TheIn the train’s off-peak departure hours wetimes studied, at each the station train are departure equal to interval the ones is of a constant. the initial timetable. (4)(5) InThe this train’s model, departure the motor times efficiency at each is station simplified. are equal to the ones of the initial timetable. (5) In this model, the motor efficiency is simplified. 2.1. Timetable Optimization Module 2.1. Timetable Optimization Module 2.1.1. Passenger Characteristics 2.1.1. Passenger Characteristics  The passenger arriving and alighting at the metro station for a given period htt0 , final i can be The passenger arriving and alighting at the metro station for a given period t0, t f inal can be modelled by a time-dependent origin-destination table [44–46]. modelled by a time-dependent origin-destination table [44–46]. ()()tt  1,1 1,I  κ1,1(t) ··· κ1,I (t) OD() t   (1)  . .. .  OD(t) =  . . .  (1) 0 II, (t ) 0 ··· κI,I (t) where κi,j(t) is the passenger arriving amount at station i at time t with destination station i’. whereAdditionally,κi,j(t) isthe the sum passenger of the amount arriving of amount arriving at passenger station i sat at time destinationt with destinationi’ for i' i station1, , Ii’. Additionally, the sum of the amount of arriving passengers at destination i’ for i0 ∈ {i + 1, . . . , I} represents the passenger arriving rate λi(t) at station i and time t. For a short time period, the represents the passenger arriving rate λ (t) at station i and time t. For a short time period, the passenger passenger arriving rate can be treated asi a uniform distribution [47]. arriving rate can be treated as a uniform distribution [47]. ⁰ When a train j dwells at a given station i according to the timetable at time 0a i,j, the passengers When a train j dwells at a given station i according to the timetable at time a i,j, the passengers waiting on the platform Pi,j begin to board the train. Additionally, the passengers Qi,j with trip waiting on the platform P begin to board the train. Additionally, the passengers Q with trip i,j i,j ⁰ destination i alight the train during the dwelling time, as well. The timetable dwelling time 0T D,i,j and destination i alight the train during the dwelling time, as well. The timetable dwelling time T D,i,j and actual dwelling time TD,i,j can be described as time window [a⁰i,j, d⁰i,j] and [ai,j, di,j], respectively. This actual dwelling time T can be described as time window [a0 , d0 ] and [a , d ], respectively. This D,i,j ⁰ i,j i,j i,j⁰ i,j train will depart station i at time d0 ij and drive to the station i + 1 at time 0a i+1,j with the in-vehicle train will depart station i at time d ij and drive to the station i + 1 at time a i+1,j with the in-vehicle passengers Oi,j. Note that our study is considering the off-peak hours, thus there are no passengers passengers O . Note that our study is considering the off-peak hours, thus there are no passengers that cannot boardi,j trains. In other words, the scenario in which the passengers have to wait for the that cannot board trains. In other words, the scenario in which the passengers have to wait for the next next train [48] will not happen. The number of in-vehicle passengers is an important influence factor train [48] will not happen. The number of in-vehicle passengers is an important influence factor of of train energy consumption. train energy consumption. & 00 ' Z aaTi, j+ T D,, i j Pi,j D,i,j ( t )dt , i , j Pi,j =ij, d 0 λ(t)dt , ∀i, j (2(2)) 0 ij,1 d i,j−1 i1 QPi,,, j i j i i' (3) i1

Energies 2017, 10, 436 4 of 20

i−1 Qi,j = ∑ Pi,j · γi,i0 (3) i=1 ( Pi,j − Qi,j, j = 1, 2, ···, J; i = 1 Oi,j = (4) Oi−1,j + Pi,j − Qi,j, j = 1, 2, ···, J; i = 2, ···, I where γi,i’ is the ratio describing the passengers boarding the train j at station i who alights at station i’. Thus, γi,i’ can be formulated by OD(t):

κi,i0 (t) γ 0 = (5) i,i I ∑ κi,i0 (t) i0=i+1

2.1.2. Dwelling Time Calculation Generally speaking, the dwelling time has two components [49]: (i) a fixed time for opening and closing doors; and (ii) door utilization time for boarding and alighting passengers. Thus, the dwelling time is deeply influenced by the passenger flow. According to the expression that Kim et al. [42] proposed, we propose a polynomial equation to min estimate the minimum dwelling time TD,i,j for train j at station i as Equation (6).

Tmin = 0.7021Q − 0.0068Q2 + 0.8417P − 0.0083P2 + 3.7953 · DOC D,i,j i,j i,j i,j i,j (6) −2.4495 · DOC2 + 1.0871 · DOC3 − 1.1385 where DOC is the degree of crowdedness [50]. Thus, the dwelling time TD,i,j shall satisfy the constraint as Equation (7).

h min ◦ i TD,i,j ∈ TD,i,j, TD,i,j (7)

2.2. Train Operation Simulation Module The train traction energy consumption is mainly affected by the running time in each segment [43,51]. When a train departs from station i before the departure time of the timetable, there will be more running time for the train to drive in the segment, which will lead to less traction energy consumption. In this section, we aim to analyze the energy consumption for trains reducing dwelling time and adding the time margins to the running time. Firstly, in order to calculate the traction energy consumption with time margin addition, we offer a means of optimizing train driving control strategies with a fixed time based on a time-saving pattern. Then, for the sake of accuracy, we describe the train operation model in a single rail segment with a belt with uniformly distributed mass instead of a mass point model.

2.2.1. Optimal Train Control Strategies with Fixed Time In our manuscript, there are six steps to obtain an energy-efficient strategy of one section.

(1) Generate a driving strategy of a time-saving pattern and the minimized running time of this section. (2) Calculate the margin time of the total section as the difference between the running time of the time-saving pattern and the one of the fixed time pattern. (3) Divide the section into several subsections by the changes of lines’ speed limits. Previous studies [5,18] have demonstrated that acceleration and coasting are both components of the energy-efficient strategy. Accordingly, the margin time should be allocated to deceleration subsections as much as possible. (4) Initialize the speed limit of each subsection, indicating that the speed limit shall reduce to a lower level from the original high one. Energies 2017, 10, 436 5 of 20

(5) With the limit of the given running time and initial speed of this subsection, calculate a driving Energiesstrategy 2017, 10 and, 436 output the actual running time. If the actual running time satisfies the error request,5 of 20 turn to the next subsection. (6)(6) OutputOutput thethe results,results, andand endend thethe simulationsimulation whenwhen allall sectionssections havehave beenbeen simulated.simulated.

The detailed illustration of Step (5) can be concluded as how to assign the dwelling time TD,i,j The detailed illustration of Step (5) can be concluded as how to assign the dwelling time TD,i,j margin of station i to the running time TR,i,j of section i. Generally speaking, when the train control margin of station i to the running time TR,i,j of section i. Generally speaking, when the train control strategiesstrategies in in each each section section are are given, given, both both the the maximum maximum traction traction and and coasting coasting phases phases areare energyenergy efficientefficient already.already. Thus,Thus, thethe marginmargin timetime shallshall bebe assignedassigned toto thethe brakingbraking phasephase inin orderorder toto obtainobtain moremore of of an an energy-saving energy-saving effect. effect. The The illustration illustration and and flowchart flowchart of of dwelling dwelling time time margin margin are are shown shown in Figuresin Figure2 ands 23 and, respectively. 3, respectively. It is important It is important to note to that note this that method this method also can also be used can to be the used situation to the insituation which the in which speed limitthe speed changes limit from changes high from to low. high to low. AsAs shown shown in in Figure Figure2, the2, theAO AOand andCO COare are the brakingthe braking and and coasting coasting curves, curves respectively,, respectively, and BXO and isBXO the coastingis the coasting and braking and braking curve wherecurve where the cut-off the cut point-off ispointX. is X.

uniform ) s / C B A Vmax m (

v

X coasting braking

Vmin

O

s (m)

FigureFigure 2. 2.Illustration Illustration of of dwelling dwelling time time margin margin assignment. assignment.

We define the time margin as Tm; the times of AO, CO, CA and BXO are respectively Tb, Tc, Tu We define the time margin as Tm; the times of AO, CO, CA and BXO are respectively Tb, Tc, Tu and Tc-b; moreover, the error of calculation is Terror. The main steps are as in Figure 3. and Tc-b; moreover, the error of calculation is Terror. The main steps are as in Figure3. BasedBased on on this this train train control control strategy strategy with fixed with time, fixed we time, can calculate we can the calculate train energy the consumption train energy withconsumption the approach with in th thee approach next section. in the next section.

2.2.2.2.2.2.Train Train Movement Movement Simulation Simulation Models Models InIn this this paper, paper, the the train train model model is consideredis considered as aas mass a mass belt belt instead instead of a massof a mass point. point. Thus, Thus, the force the analysisforce analysis has to has be to reformulated. be reformulated. There There are are two two major major approaches approaches to to simulate simulate train train movement,movement, time-basedtime-based and and event-based event-based models models [52 [52].]. The The time-based time-based model model requires requires a a highly highly computational computational demanddemand asas aa significantsignificant amountamount ofof informationinformation hashas toto bebe produced produced duringduring everyevery update,update, inin whichwhich traintrain movementmovement isis evaluatedevaluated at each each inte interval.rval. The The full full details details of of train train movement movement are are needed needed when when we wewant want to to calculate calculate the the energy energy consumption consumption accurately. accurately. Thus, Thus, we we establish a time-based time-based model model to to describedescribe train train movement. movement. Firstly,Firstly, we we deal deal with with the the force force analysis analysis based based on on the the mass mass belt belt assumption. assumption. ThereThere are are three three kinds kinds ofof force force acting acting on on a train a train driving driving between between successive successive stations: stations: the tractionthe traction force forceTf, the T resistancef, the resistance force Rforcef and R thef and braking the braking force B forcef. Bf. TheThe tractiontraction forceforce TTf f cancan bebe represented represented asas aa functionfunction of of train train speed speed vv,, which which can can be be simplysimply calculatedcalculated if if the the locomotive locomotive traction traction curve curveis isobtained. obtained. TheThe specificspecific (per(per massmass unit)unit) tractiontraction forceforcet t((vv)) cancan be be represented represented as as a a function functionf (f(vv)) related related to to the the speed speedv v. . t()() v f v (8) t(v) = f (v) (8)

The resistance force Rf consists of the basic running resistance and the additional resistance. The specific (per mass unit) basic running resistance r(v) is generally calculated as a quadratic equation of train speed v:

2 r() v r0  r 1  v  r 2  v (9)

Energies 2017, 10, 436 6 of 20

The resistance force Rf consists of the basic running resistance and the additional resistance. The specific (per mass unit) basic running resistance r(v) is generally calculated as a quadratic equation of train speed v: Energies 2017, 10, 436 2 6 of 20 r(v) = r0 + r1 · v + r2 · v (9) wherewhere the the coefficients coefficientsr1 rand1 andr2 rare2 are related related to to the the train train mass mass and and the the interaction interaction between between tracks tracks and and traintrain wheels; wheels nevertheless,; nevertheless the, the coefficient coefficientr0 ris0 is related related to to the the aerodynamics aerodynamics of of the the trains. trains.

Start

No Tm  0 Yes

t  Tc () T u  T b

No tTm Yes k 1

Backstep n time steps from O to X (the point form coasting to braking )

Extend coasting curve from X to line Vmax to a point B

Calculate Tba

t  T () T  T Drive as curve CO c b ba b

No t  Tm  Terror

Yes No tTm Yes Take the midpoint of travel time of XO segment as new point X

kk1

End

FigureFigure 3. 3.Flowchart Flowchart of of dwelling dwelling time time margin margin assignment. assignment.

TheThe specific specific (per (per mass mass unit) unit) additional additional resistance resistancew(x) is causedw(x) is by caused the track by condition the track consisting condition consisting of unit gradient resistance wg, unit curvature resistance wr and unit tunnel resistance wt, of unit gradient resistance wg, unit curvature resistance wr and unit tunnel resistance wt, which can be shownwhich as can a function be show ofn as the a positionfunction ofof thethe trainpositionx: of the train x:

w() x wg  w r  w t (10) w(x) = wg + wr + wt (10) In this paper, we take the train as a belt with length S. Thus, when the train runs on a track with a continuously varying gradient, the specific (per mass unit) gradient resistance can be calculated as follows: 1 S w()() x  g x  s ds g  (11) M 0

where M is the train traction weight (containing the mass of both the train Mt and the loading passengers Mp) (kg); ρ is the mass per unit length (kg/m); g(x − s) is the gradient of position (x − s) (‰); S is the train length (m).

Energies 2017, 10, 436 7 of 20

In this paper, we take the train as a belt with length S. Thus, when the train runs on a track with aEnergies continuously 2017, 10, 436 varying gradient, the specific (per mass unit) gradient resistance can be calculated7 of 20 as follows: Take Figure 4 for example: when the train runsS to the position x where the front part (length is s2) 1 Z runs to the second slope with gradientw ( gx2) and = the ρtail· g of(x the− s )trainds (length is s1) still exists on the (11)first g M slope with gradient g1, the specific (per mass unit)0 gradient resistance wg(x) can be calculated as follows: where M is the train traction weight (containing the mass of both the train Mt and the loading passengers Mp) (kg); ρ is the mass per unit length1 ss12 (kg/m); g(x − s) is the gradient of position (x − s) ( ); S is the train length (m). wg ()() x  g x  s ds M  h Take Figure4 for example: when the train runs0 to the position x where the front part (length is s ) runs to the second slope with gradient gandg1  the s 1  tail  ofg 2 the s 2 train (length is s ) still exists on the 2  2 1 (12) first slope with gradient g1, the specific (per mass unit)()ss12 gradient resistance wg(x) can be calculated as follows: g1 s 1  g 2  s 2  s1+s2 1 R wg(x) = M ss12ρ · g(x − s)ds 0 ρ·g ·s +ρ·g ·s (12) Additionally, the specific (per mass unit)= gradient1 1 2 resistance2 wg(x’) can be calculated in a similar ρ(s +s ) way. 1 2 = g1·s1+g2·s2 s1+s2  1 ss12 Additionally, the specific (per massw()() x unit) gradient  g x resistance  s ds wg(x’) can be calculated in a g  similar way. M 0 s0 +s0 g12 2 s 1   g 3  s 2 w (x0) = 1 R ρ · g(x − s)ds (13) g M  0()ss12 0 0 gρ·g2 s·s1 +ρ g·g3 ·s s2 (13) = 2 10 0 3 2  ρ(s1+s2) 0ss12 0 g2·s1+g3·s2 = 0 0 s1+s2 ) m ( x' n o

i ' t

a s2

v ' g3 e

l s1 e

x

s2 s1 g2

g1  sg2 horizontal distance (m)  sg 1 Figure 4. Schematic diagram of specific (per mass unit) gradient resistance considering the train length. Figure 4. Schematic diagram of specific (per mass unit) gradient resistance considering the train length. When the train runs to position x, which is a part of a curve, the specific (per mass unit) curvature resistanceWhenw r thecan train be expressed runs to as position Equation x, which (14) considering is a part the of a train curve, length. the specific (per mass unit) curvature resistance wr can be expressed as Equation (14) considering the train length. 10.5·α ( ) ≥  L (x) , Lr x S w (x) = 10.5r  (14) r L (x) 600 r ,()Lr x S  R · S , Lr(x) < S  Lxr () wxr ()  (14) ◦ 600 Lx() where α is the angle of the curve ( ); Lr(x) is the lengthr of,() theL curve x S (m); R is the radius of the curve (m).  RS r

where α is the angle of the curve (°); Lr(x) is the length of the curve (m); R is the radius of the curve (m). The specific (per mass unit) tunnel resistance wt can be simply calculated by Equation (15).

wtt = 0.00013 L ( x ) (15)

where Lt(x) is the length of the tunnel (m).

Energies 2017, 10, 436 8 of 20

The specific (per mass unit) tunnel resistance wt can be simply calculated by Equation (15).

wt = 0.00013 · Lt(x) (15) where Lt(x) is the length of the tunnel (m). The braking force Bf can be calculated by a function of train speed v; moreover, the specific (per mass unit) braking force b(v) can be shown as a function h(v) related to the speed v.

b(v) = h(v) (16)

Thus, the specific force c can be calculated by Equation (17).

c = t(v) − r(v) − w(x) − b(v) (17)

2.2.3. Energy Consumption Calculation A large amount of previous studies [5,15,18,21,53–55] have demonstrated that the energy-efficient driving strategies of each section will be maximum traction (MT), coasting (CO) and maximum braking (MB). Additionally, in each phase, train movement is as in Equation (18).

vk+1 = vk + ak · σ

vk ≤ Vmax 1 2 lk+1 = lk + (vk · σ + 2 ak · σ )/3.6 (18)

Ek+1 = Ek + ek+1

ek = tk(v) · (lk − lk−1) where vk and vk+1 are the initial speeds of the k-th and (k + 1)-th time step, respectively (km/h); Vmax is the limit speed (km/h); ak is the accelerated speed at the k-th time step (km/(h·s)); σ is one time step (s); lk and lk+1 are the positions of the k-th and (k + 1)-th time step, respectively (m); Ek and Ek+1 are the accumulative energy consumptions of the k-th and (k + 1)-th time step, respectively (kWh); ek is the energy consumption of the k-th time step (kWh); tk(v) is the traction force of the k-th time step (N). Thus, the total energy consumption Etotal can be expressed as follows:

J I K Etotal = ∑ ∑ ∑ ek (19) j=1 i=1 k=1

3. Optimal Model and Solution Method

3.1. Optimal Model The real running time and dwelling time can be expressed as follows:

TD,i,j = di,j − ai,j (20)

TR,i,j = ai+1,j − di,j (21) We assume that the trains arriving time at each station are equal to the ones of the initial timetable, in order to ensure safe and reliable operation. Thus, the real arrive time of each train j shall satisfy Equation (22). ◦ ◦ ai+1,j − ai,j = ai+1,j − ai,j (22) Energies 2017, 10, 436 9 of 20

For the train timetable optimal problem in a metro line, the objective model can be expressed as h i Equation (23) by applying a weighted sum strategy within a given period t0, t f inal :

Fobj = minEtotal (23)

The constraints mainly include the running time constraints, dwelling time constraints, passenger demand constraints and train operation constraints, shown as (2)–(4), (7), (18) and (20)–(22).

3.2.Energies Solution 2017, Method10, 436 9 of 20

TheThe timetable timetable optimization optimization model model with with the the objective objective function function (23) (23) and and Constraints Constraints (2)–(4), (2)–(4), (7), (7), (18),(18), (20)–(22) (20)–(22) is is a a nonlinear nonlinear non-convex non-convex problem. problem. The The complexity complexity of of this this problem problem is is due due to to three three points:points: (1) (1) describing describing the the multivariable multivariable optimal optimal control control strategy strategy of of a a mass-belt mass-belt train train model model with with fixed fixed time;time (2); (2) solving solving the the optimal optimal objectives objectives with with stochastic stochastic characters; characters and; and (3) (3) dealing dealing with with the the nonlinear nonlinear constraints.constraints. A A simulation-based simulation-based solution solution approach approach is is developed developed to to solve solve this this timetable timetable optimization optimization model,model and, and the the framework framework is is shown shown in in Figure Figure5. 5.

start

passengers' characteristics Timetable Optimization Module

boarding and alighting passengers c a l c u la te t h e d w e l li n g ti m e limit of station i+1 of station i+1

calculate the running time for section i

train dataset Train Operation Simulation Module

track dataset optimal train control strategies with fixed-time ii == ii++11

mass-belt train model simulation

force models calculate the energy consumption

time-based motion model yes i ≤ I-1 yes nnoo output

end

FigureFigure 5. 5Flowchart. Flowchart of of the the simulation-based simulation-based algorithm. algorithm.

As shown in Figure 5, the main steps are as follows. As shown in Figure5, the main steps are as follows.

(1) Calculate the real-time dwelling time margin Tm for station i, ∀ i = 1, 2, …, I. ∀ (1)(2) CalculateA simulation the real-time model is dwelling then run timein section margin i forTm afor train station j withi, thei = fixed 1, 2, .time . . , I .margin calculated in (2) A(7 simulation), where the model train is control then run strategies in section cani for be a trainobtainedj with with the fixedthe algorithm time margin presented calculated in Section in (7), where2.1.2. the train control strategies can be obtained with the algorithm presented in Section 2.1.2. (3) Calculate the minimize energy consumption Ei,j by using the mass-belt train motion model and the force models according to the condition of section i. (4) The simulation process will be executed repeatedly until train j arrives at terminal station I.

4. Sensitivity Analysis of the Simulation In this section, three numerical examples are established to analyze the sensitivity of the radius curve, gradient, length of grade and speed limit, in order to identify the efficiency and effort of the proposed simulation. Then, we perform a systemic analysis of the accuracy between simulation and measurement values.

4.1. Sensitivity Analysis

Energies 2017, 10, 436 10 of 20

(3) Calculate the minimize energy consumption Ei,j by using the mass-belt train motion model and the force models according to the condition of section i. (4) The simulation process will be executed repeatedly until train j arrives at terminal station I.

4. Sensitivity Analysis of the Simulation In this section, three numerical examples are established to analyze the sensitivity of the radius curve, gradient, length of grade and speed limit, in order to identify the efficiency and effort of the proposed simulation. Then, we perform a systemic analysis of the accuracy between simulation and Energiesmeasurement 2017, 10, 436 values. 10 of 20

4.1. SensitivityWe study Analysisthe metro train energy-saving strategy with the same simulator as our study; some sensitivityWe study analysis the metrocan be trainshown energy-saving as follows with strategy the parameters: with the same the train simulator traction as weight our study; is 150. some0 t; thesensitivity length of analysis the train can is be110 shown m. as follows with the parameters: the train traction weight is 150.0 t; the length of the train is 110 m. 4.1.1. The Curve Sensitivity Analysis 4.1.1. The Curve Sensitivity Analysis Parameter values used in the simulation have been listed: the curve length is 500 m; the curve radiusParameter is increasing values from used 10 in0 them to simulation 600 m; and have the been condition listed: theof nocurve curve length is defined is 500 m; as the “∞ curve”. In addition,radius is increasingthere are three from speed 100 m limits to 600: 40 m; km/h, and the50 k conditionm/h and 6 of0 k nom/h. curve The is calculation defined as result “∞”. is In shown addition, in Figurethere are6. three speed limits: 40 km/h, 50 km/h and 60 km/h. The calculation result is shown in Figure6.

12 speed limit 40 km/h 11 speed limit 50 km/h 10 speed limit 60 km/h 9 8 7 6 5 4 energy consumption (kWh) energy consumption 3 2 0 100 200 300 400 500 600 700 800 curve (m)

Figure 6. Traction energy consumption curve on different radius curves. Figure 6. Traction energy consumption curve on different radius curves.

AsAs shown shown in in Figure Figure 66,, thethe curvescurves ofof energyenergy consumptionconsumption indicateindicate similarsimilar trends.trends.

(i)(i) WhenWhen the the radius radius curve curve is is larger larger than than 50 5000 m m,, the the energy energy consumption consumption is is close close to to the the same same one one of of nono radius radius curve. curve. (ii)(ii) WhenWhen the the radius radius curve curve is is smaller smaller than than 30 3000 m m,, the the energy energy consumption consumption is is larger. larger.

4.1.24.1.2.. The The Gradient Gradient Sensitivity Sensitivity Analysis Analysis ParameterParameter values values used used in in the the simulation simulation have have been been listed: listed: the the slope slope length length is is 30 3000 m m;; the the speed speed limitlimit is is 5 500 kkm/h.m/h. The The energy energy consumption consumption of of the the ascending ascending and and falling falling gradient gradient are are respectively respectively shownshown as as Figure Figure 77aa,b.,b.

12 7 6 10 5 8 4 6 3 4 2

2 1 energy consumption (kWh) energy consumption energy consumption (kWh) energy consumption 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 gradient (-‰) gradient (‰) (a) (b)

Figure 7. Traction energy consumption curve under different gradient. (a) Ascending gradient; and (b) falling gradient.

Energies 2017, 10, 436 10 of 20

We study the metro train energy-saving strategy with the same simulator as our study; some sensitivity analysis can be shown as follows with the parameters: the train traction weight is 150.0 t; the length of the train is 110 m.

4.1.1. The Curve Sensitivity Analysis Parameter values used in the simulation have been listed: the curve length is 500 m; the curve radius is increasing from 100 m to 600 m; and the condition of no curve is defined as “∞”. In addition, there are three speed limits: 40 km/h, 50 km/h and 60 km/h. The calculation result is shown in Figure 6.

12 speed limit 40 km/h 11 speed limit 50 km/h 10 speed limit 60 km/h 9 8 7 6 5 4 energy consumption (kWh) energy consumption 3 2 0 100 200 300 400 500 600 700 800 curve (m)

Figure 6. Traction energy consumption curve on different radius curves.

As shown in Figure 6, the curves of energy consumption indicate similar trends. (i) When the radius curve is larger than 500 m, the energy consumption is close to the same one of no radius curve. (ii) When the radius curve is smaller than 300 m, the energy consumption is larger.

4.1.2. The Gradient Sensitivity Analysis Parameter values used in the simulation have been listed: the slope length is 300 m; the speed limit is 50 km/h. The energy consumption of the ascending and falling gradient are respectively Energiesshown2017 as Figure, 10, 436 7a,b. 11 of 20

12 7 6 10 5 8 4 6 3 4 2

2 1 energy consumption (kWh) energy consumption energy consumption (kWh) energy consumption 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 gradient (-‰) gradient (‰) (a) (b)

Figure 7. Traction energy consumption curve under different gradient. (a) Ascending gradient; and EnergiesFigure 2017, 7.10Traction, 436 energy consumption curve under different gradient. (a) Ascending gradient; and11 of 20 (b) falling gradient. (b) falling gradient. As shown in Figure 7, the energy consumption changes linearly with the slope changes with the

sameAs grade shown length in Figure and speed7, the limit. energy consumption changes linearly with the slope changes with the same grade length and speed limit. 4.1.3. The Length of Grade and Speed Limit Sensitivity Analysis 4.1.3.There The Length are three of Grade kinds andof grade Speed section Limit Sensitivitylengths, 60 Analysis0, 400 and 200 m, respectively with the same gradientThere ofare 30‰. three The kinds traction of gradeenergy section consumption lengths, of 600, the 400train and from 200 the m, base respectively to the top with of the the slope same is gradientshown in of Figure 30 . The8. traction energy consumption of the train from the base to the top of the slope is shown in Figureh8.

20 length of grade is 600 m ) 18

16 length of grade is 400 m kW·h ( 14 length of grade is 200 m 12 10

8 consumption 6

4 Energy 2 0 10 30 50 70 90 110 speed limit (km/h)

FigureFigure 8. 8Traction. Traction energy energy consumption consumption ofof differentdifferent lengthslengths of grades grades and and speed speed limits limits..

AsAs shownshown inin FigureFigure8 8,, the the curves curves of of energy energy consumption consumption indicate indicate similar similar trends. trends. (1) All curves of energy consumption can be divided into three sections: low speed section, middle (1) All curves of energy consumption can be divided into three sections: low speed section, middle speed section and high speed section. speed section and high speed section. (2) Based on the 30‰ gradient, the middle sections are respectively 40–90 km/h, 40–80 km/h and (2) Based on the 30 gradient, the middle sections are respectively 40–90 km/h, 40–80 km/h and 40–70 km/h for the lengths of 600, 400 and 200 m. 40–70 km/h for the lengths of 600, 400 and 200 m. (3) In each speed section,h with the increase of speed, the energy consumption increases linearly. (3) In each speed section, with the increase of speed, the energy consumption increases linearly. 4.2. Accuracy Analysis 4.2. Accuracy Analysis There are three kinds of measurement data, AW0 (194.00 t), AW2 (279.68 t) and AW3 (303.20 t), indicatingThere arethe threeenergy kinds consumption of measurement with different data, AW load0 (194.00 factors. t), In AW each2 (279.68 kind of t) andmeasurement AW3 (303.20 data, t), indicatingaccording to the whether energy consumptionregenerative braking with different energy loadis used factors., these In still each can kind be divided of measurement into two types data,: the absolute error ε and the percent error δ can be calculated as Equations (24) and (25), respectively.

 xa (24)

 xa  100%   100%  (25) aa

where x is the simulation value (kWh) and a is the measurement value (kWh). Both the measurement and simulation values are shown in Table 1. Accordingly, we calculate the absolute error and the percent error.

Energies 2017, 10, 436 12 of 20 according to whether regenerative braking energy is used, these still can be divided into two types: the absolute error ε and the percent error δ can be calculated as Equations (24) and (25), respectively.

ε = |x − a| (24)

ε x − a δ = 100% × = 100% × (25) |a| a where x is the simulation value (kWh) and a is the measurement value (kWh). Energies 2017, 10, 436 12 of 20 Both the measurement and simulation values are shown in Table1. Accordingly, we calculate the absolute error and the percent error. Table 1. Traction energy consumption measurement values (kWh).

The Kinds ofTable Energy 1. TractionConsumption energy consumptionMeasurement measurement Simulatio values (kWh).n ε δ Traction energy consumption 375 361.64 13.36 3.56% AW0 The Kinds of Energy Consumption Measurement Simulation ε δ 100% regenerative braking energy 206 215.50 9.50 4.61% Traction energy consumption 375 361.64 13.36 3.56% AW 0 100%Traction regenerative energy consumption braking energy 206514 215.50530.35 9.5016.35 4.61%3.18% AW2 100%Traction regenerative energy braking consumption energy 514305 530.35308.04 16.353.04 3.18%0.99% AW2 100%Traction regenerative energy consumption braking energy 305549 308.04566.10 3.0417.10 0.99%3.11% AW3 Traction energy consumption 549 566.10 17.10 3.11% AW 100% regenerative braking energy 335 331.08 3.92 1.17% 3 100% regenerative braking energy 335 331.08 3.92 1.17% As shown in Table 1, the values of the percent error are all smaller than 5%, indicating that the accuracyAs shown of the insimulation Table1, the model values proposed of the percentis good, error and can are allbe applied smaller to than the 5%, actual indicating operation that of thethe accuracysubway system of the simulationenergy-saving model strategy proposed analysis. is good, and can be applied to the actual operation of the subway system energy-saving strategy analysis. 5. Numerical Examples 5. Numerical Examples In this section, three numerical examples are established to identify the efficiency and effort of the proposedIn this section, approach three for numerical the metro examplestimetable areoptimization established problem. to identify Firstly, the efficiencywe assess andthe different effort of thetrain proposed energy-efficient approach performance for the metro with timetable different optimization running time problem.s within Firstly, a specified we assess section, the aiming different to traindemonstrate energy-efficient the effectiveness performance of the with train different control running strategy times with withinfixed time. a specified Secondly, section, we utilize aiming the to demonstrateproposed train the movement effectiveness model of theconsidering train control both strategythe track with gradients fixed and time. the Secondly, curves to we calculate utilize the proposedtotal energy train consumption movement model of Line considering 2 (inner both ring) the of trackthe Beijing gradients metro and the system. curves For to calculate this case, the a totalcomparison energy consumption among the results of Line 2by (inner different ring) of running the Beijing and metro dwelling system. time Fors is this given case, to a comparison verify the amongperformance the results of the by energy different saving. running Last and, but dwelling not least, times we is demonstrate given to verify how the to performance use the proposed of the energyapproaches saving. to Last,reduce but energy not least, consumption we demonstrate by optimizing how to use the the train proposed timetable approaches problem to onreduce the energyreal-world consumption Beijing Line by optimizing2 (inner ring) the train with timetable uncertain problem and dynamic on the real-world parameters Beijing (i.e., Line passengers’ 2 (inner ring)time-dependent with uncertain origin and-destination dynamic demands), parameters which (i.e., passengers’are all taken time-dependent from historical detected origin-destination operation demands),data. Note which that the are timetable, all taken from train historical and track detected conditions operation are all data. collected Note that from the the timetable, Beijing Mass train andTransit track Railway conditions Operation are all Corporation collected from (BMTROC the Beijing, Beijing, Mass China Transit) and Railway not open Operation to public. Corporation The train (BMTROC,mass is 194.0 Beijing,0 t, and China) the gross and notload open hauled to public. is 279.6 The8 t train(assume mass the is 194.00passenger’s t, and weight the gross is load60 kg). hauled The istrain 279.68 structure t (assume is shown the passenger’s in Figure 9. weight is 60 kg). The train structure is shown in Figure9.

Tc M T M M Tc

Figure 9 9.. TThehe structure structure of a metro train consisting of three motor cars and three trailer cars.

5.1.5.1. A Case of One Single Section Section’s’s Energy Consumption This example considers one specified section of Line 2 (inner ring) named Fuchengmen Station– This example considers one specified section of Line 2 (inner ring) named Fuchengmen Fuxingmen Station, the length of which is 1832 m, and the gradients are described in Table 2. In this Station–Fuxingmen Station, the length of which is 1832 m, and the gradients are described in Table2. case study, the initial running time is 137 s, and the fixed running time changes from 139 s to 153 s, In this case study, the initial running time is 137 s, and the fixed running time changes from 139 s to increasing by 2 s. For the sake of simplicity, we assume that the train mass is 194.00 t. The results of energy consumption and train speed profiles are calculated with each running time, which is shown in Table 3 and Figure 10. As shown in Table 3, the energy consumption descends with the increase of the fixed running time, with a sharp decrease at the beginning (5.9% optimal rate when the running time is 139 s) and a gradual decline (the optimal rates are no more than 1.0%) when the fixed running time is much larger than that of the initial timetable. From Figure 10, we can see that the coasting speed profiles have a smoothly changing trend when the gradient condition changes, indicating that the utilization of a uniformly-distributed mass-belt model to describe train operation progress is reasonable and practical. Moreover, the speed profile with a longer fixed running time is much lower than that with a shorter time. These results indicate the usefulness of the train control strategy with fixed time.

Energies 2017, 10, 436 13 of 20

153 s, increasing by 2 s. For the sake of simplicity, we assume that the train mass is 194.00 t. The results of energy consumption and train speed profiles are calculated with each running time, which is shown in Table3 and Figure 10. As shown in Table3, the energy consumption descends with the increase of the fixed running time, with a sharp decrease at the beginning (5.9% optimal rate when the running time is 139 s) and a gradual decline (the optimal rates are no more than 1.0%) when the fixed running time is much larger than that of the initial timetable. From Figure 10, we can see that the coasting speed profiles have a smoothly changing trend when the gradient condition changes, indicating that the utilization of a uniformly-distributed mass-belt model to describe train operation progress is reasonable and practical. Moreover, the speed profile with a longer fixed running time is much lower than that with a shorter time. These results indicate the usefulness of the train control strategy with fixed time. Energies 2017, 10, 436 13 of 20 Table 2. Gradients of section Fuchengmen Station-Fuxingmen Station. Table 2. Gradients of section Fuchengmen Station-Fuxingmen Station.

SegmentSegment Grade Grade (‰) ( ) LengthLength (m) (m) 1 13.0 3.0h 1207.71207.7 2 23.0 3.0 140.3140.3 3 3 −2.0− 2.0 218.0218.0 4 4 −8.0− 8.0 120.0120.0 5 5 −3.0− 3.0 121.0121.0 6 64.8 4.8 25.0 25.0 Total Total- - 1832.01832.0

Table 3.3. Energy consumption of section Fuchengmen Station–FuxingmenStation–Fuxingmen Station.Station.

Fixed Running Time (s) 137 139 141 143 145 147 149 151 153 Fixed Running Time (s) 137 139 141 143 145 147 149 151 153 Energy consumption 22.99 21.63 21.58 21.45 21.29 21.19 21.15 20.97 20.92 Energy(kWh) consumption (kWh) 22.99 21.63 21.58 21.45 21.29 21.19 21.15 20.97 20.92 Optimal rate - 5.9% 0.2% 0.6% 0.7% 0.5% 0.2% 0.9% 0.2% Optimal rate - 5.9% 0.2% 0.6% 0.7% 0.5% 0.2% 0.9% 0.2%

Fuchengmen Station Fuxingmen Station

train running direction

speed limit

initial running time running time +16s

3.0 3.0 2.0 8.0 3.0 4.8 1480 140 218 120 121 168

Figure 10.10. TheThe speed profilesprofiles with different fixedfixed running time. The line and the dotted line are the speed profilesprofiles withwith runningrunning timetime ofof 137137 ss andand 153153 s,s, respectively.respectively.

5.2.5.2. The Case of the Whole Line In thethe traintrain energy-efficient energy-efficient operation operation progress, progress, both both the the train train and trackand track condition condition have beenhave given.been However,given. However, the number the of number on-board of passengers on-board willpassengers strongly will affect strongly the train affect operation the performancetrain operation by changingperformance the train by changing traction mass.the train In this traction case study, mass. we In identify this case how study, much we the identify number how of passengers much the onnumber board of will passengers affect the on train board traction will affect energy the consumption train traction byenergy calculating consumption the energy by calculating consumption the energy consumption for a train running the whole of Beijing metro Line 2. When the train traction mass increases from 194.00 t to 268.88 t, the energy consumption with each section adding a fixed running time (10 s) is shown in Table 4. Moreover, the energy consumption is shown in Figure 11 when both the traction mass increases from 194.00 t to 268.88 t and each section running time increases form 2 s to 12 s. From Table 4, we can conclude that the energy consumption precisely increases with the increase of traction mass. The results illustrate that the energy consumption shows an even lower increasing rate (18.5% = (384.18 − 324.31)/324.31) than the one of traction mass (38.6% = (268.88 − 194.00)/194.00). It seems probable that the presented train operation performance is all optimized by the fixed time train control, which contributes to the low energy consumption.

Energies 2017, 10, 436 14 of 20

for a train running the whole of Beijing metro Line 2. When the train traction mass increases from 194.00 t to 268.88 t, the energy consumption with each section adding a fixed running time (10 s) is shown in Table4. Moreover, the energy consumption is shown in Figure 11 when both the traction mass increases from 194.00 t to 268.88 t and each section running time increases form 2 s to 12 s. From Table4, we can conclude that the energy consumption precisely increases with the increase of traction mass. The results illustrate that the energy consumption shows an even lower increasing rate (18.5% = (384.18 − 324.31)/324.31) than the one of traction mass (38.6% = (268.88 − 194.00)/194.00). It seems probable that the presented train operation performance is all optimized by the fixed time train control, which contributes to the low energy consumption. Energies 2017, 10, 436 14 of 20 Table 4. Energy consumption with different train traction mass. Table 4. Energy consumption with different train traction mass.

TractionTraction Mass (t) Mass (t) 194.00 194.00195.44 195.44196.88 196.88 198.32 198.32 199.76 199.76 201.20 202.64202.64 204.08204.08 205.52205.52 Energy consumption (kWh) 324.31 325.05 325.71 327.15 327.92 329.85 330.76 332.23 333.17 Energy consumption (kWh) 324.31 325.05 325.71 327.15 327.92 329.85 330.76 332.23 333.17 TractionTraction mass (t) mass (t) 206.96 206.96208.40 208.40209.84 209.84 211.28 211.28 212.72 212.72 214.16 215.60215.60 217.04217.04 218.48218.48 Energy Energyconsumption consumption (kWh) (kWh)333.79 333.79335.55 335.55336.28 336.28 337.73 337.73 338.41 338.41 339.82 340.76340.76 342.4 342.4 343.08 343.08 TractionTraction mass (t) mass (t) 219.92 219.92221.36 221.36222.80 222.80 224.24 224.24 225.68 225.68 227.12 228.56228.56 230.00230.00 231.44231.44 Energy Energyconsumption consumption (kWh) (kWh)343.99 343.99345.35 345.35346.46 346.46 348.15 348.15 348.86 348.86 350.13 351.07351.07 352.83352.83 353.43353.43 TractionTraction mass (t) mass (t) 232.88 232.88234.32 234.32235.76 235.76 237.20 237.20 238.64 238.64 240.08 241.52241.52 242.96242.96 244.40244.40 Energy Energyconsumption consumption (kWh) (kWh)354.24 354.24356.09 356.09356.79 356.79 358.51 358.51 359.17 359.17 360.83 361.54361.54 362.59362.59 363.97363.97 Traction mass (t) 245.84 247.28 248.72 250.16 251.60 253.04 254.48 255.92 257.36 Traction mass (t) 245.84 247.28 248.72 250.16 251.60 253.04 254.48 255.92 257.36 Energy Energyconsumption consumption (kWh) (kWh)364.88 364.88 366.5 366.5 367.15 367.15 368.85 368.85 369.43 369.43 371.31 371.98371.98 372.93372.93 374.83374.83 Traction mass (t) 258.80 260.24 261.68 263.12 264.56 266.00 267.44 268.88 Traction mass (t) 258.80 260.24 261.68 263.12 264.56 266.00 267.44 268.88 Energy consumption (kWh) 375.61 377.71 378.31 380.27 380.96 382.56 383.25 384.18 Energy consumption (kWh) 375.61 377.71 378.31 380.27 380.96 382.56 383.25 384.18

550 500 480 500 460 440 450 420

400 400 380

350 360 energy consumption (kWh) consumption energy 340 300 0 190 2 200 210 4 220 6 230 8 240 running time margin (s) 250 traction mass (t) 10 260 12 270

Figure 11.11. TheThe energy energy consumption consumption of Beijingof Beijing metro metro Line Line 2 with 2 differentwith different traction traction masses andmasses running and runningtime margins. time margins.

ItIt is found in in Figure Figure 11 that when when the running time time margin margin of each station station is is improved improved from from 0 s toto 6 6 s, s, the the energy energy consumption of of one one train train with with 26 268.888.88 t traction mass mass decreases decreases much more more quickly fromfrom aboutabout 505.93505.93 kWh kWh to to approximately approximately 466.85 466.85 kWh. kWh. Moreover, Moreover, the effect the effect of the of running the running time margin time marginlonger than longer 6 s onthan the 6 energys on the consumption energy consumptio slows downn slows smoothly. down In smoothly. contrast, theIn contrast, energy consumption the energy consumptionis evidently linearly is evidently increased linearly with increased the increasing with the traction increasing mass. traction Such an mass. increase Such is acceleratedan increase byis acceleratedmore traction by forcemore ortraction a longer force duration or a longer of traction duration applied of traction to achieve applied the to sameachieve train thespeed same train for a speedheavier for train a heavier [43]. train [43].

5.3. A Real-World Case Study We consider a real-world case study over the Beijing metro Line 2 (inner ring), which is a loop line consisting of 18 stations and 18 sections with a total length of 23.6 km. In daily operations, the planned cycle time is 2640 s; the minimal headway is hmin = 120 s; and the maximal headway is hmax = 420 s. More details can be found in Table 5. In this study, we use the real-world passenger demand data collected by the smart card dataset from BMTROC on a weekday of April 2014. Due to the page limitations, we show the number of arriving and departing passengers at each station of Line 2 in Figure 12. It is obvious that the passenger demands are significantly heterogeneous for different stations and different hours, which is corroborated by previous studies [56,57]. In the numerical experiments, we consider the time window from 10:00 to 12:00 in off-peak hours, during which a total of 16 trains are operated. Based on the passenger demand data, we calculate the results of the energy consumption of each section as represented in Figure 13.

Energies 2017, 10, 436 15 of 20

5.3. A Real-World Case Study We consider a real-world case study over the Beijing metro Line 2 (inner ring), which is a loop line consisting of 18 stations and 18 sections with a total length of 23.6 km. In daily operations, the planned cycle time is 2640 s; the minimal headway is hmin = 120 s; and the maximal headway is hmax = 420 s. More details can be found in Table5. In this study, we use the real-world passenger demand data collected by the smart card dataset from BMTROC on a weekday of April 2014. Due to the page limitations, we show the number of arriving and departing passengers at each station of Line 2 in Figure 12. It is obvious that the passenger demands are significantly heterogeneous for different stations and different hours, which is corroborated by previous studies [56,57]. In the numerical experiments, we consider the time window from 10:00 to 12:00 in off-peak hours, during which a total of 16 trains are operated. Based on the passenger demand data, we calculate the results of the energy consumption of each section as represented in Figure 13. Energies 2017, 10, 436 15 of 20 Table 5. Basic operation data of Beijing Metro Line 2 (inner ring). Table 5. Basic operation data of Beijing Metro Line 2 (inner ring).

Running DwellingDwelling Section Section Length Length (m) (m) * * Time (s) TimeTime (s) (s) Passengers Jianguomen Station–Chaoyangmen Station Station 1763 1763 123 123 45 45 166,945 166,945 Chaoyangmen Station–Dongsishitiao Station 1027 1027 88 30 30 157,873 Dongsishitiao Station–Dongzhimen Station 824 824 78 50 50 148,313 Dongzhimen Station–Lama Temple Station 2228 2228 174 45 45 114,565 Lama Temple Station–Andingmen Station 794 794 74 30 30 143,292 Andingmen Station–Guloudajie Station 1237 1237 98 50 50 145,613 Guloudajie Station–Jishuitan Station 1766 129 50 160,098 Guloudajie Station–Jishuitan Station 1766 129 50 160,098 Jishuitan Station–Xizhimen Station 1899 166 60 161,824 Jishuitan Station–Xizhimen Station 1899 166 60 161,824 Xizhimen Station–Chegongzhuang Station 909 87 45 156,004 ChegongzhuangXizhimen Station–Chegongzhuang Station–Fuchengmen Station Station 909 960 8587 30 45 159,932156,004 ChegongzhuangFuchengmen Station–Fuxingmen Station–Fuchengmen Station Station 9601832 137 85 50 30 157,867159,932 FuxingmenFuchengmen Station–Changchunjie Station–Fuxingmen Station Station 1832 1234 115137 30 50 109,896157,867 ChangchunjieFuxingmen Station–Changchunjie Station–Xuanwumen Station Station 1234 929 115 85 30 30 110,013109,896 ChangchunjieXuanwumen Station–Xuanwumen Station–Hepingmen StationStation 929 851 8285 30 30 154,568110,013 XuanwumenHepingmen Station–Hepingmen Station–Qianmen Station Station 8511171 9582 30 30 155,703154,568 QianmenHepingmen Station–Chongwenmen Station–Qianmen Station Station 1171 1634 123 95 45 30 157,306155,703 ChongwenmenQianmen Station–Chongwenmen Station–Beijing Railway Station Station 1634 1023 112123 60 45 136,104157,306 ChongwenmenBeijing Railway Station–Beijing Station–Jianguomen Railway Station Station 1023 945 101112 60 60 133,743136,104 Beijing Railway Station–Jianguomen* Source from Station http://www.bjsubway.com/station/zjgls/# 945 101 . 60 133,743 * Source from http://www.bjsubway.com/station/zjgls/#.

9000 6000 8000 5000 7000 8000 4000 5000 6000 7000 5000 3000 4000 6000 4000 5000 2000 3000 3000

2000 4000 1000 2000 1000 3000 0 2000 1000 0 Chegongzhuang Station 1000 Fuxingmen Station Chegongzhuang Station 0 Fuxingmen Station Hepingmen Station 00:00-00:30 Hepingmen Station 0 Beijing Railw ay Station 19:00-19:30 Beijing Railway Station 00:00-00:30 Dongsishitiao Station 14:00-14:30 Dongsishitiao Station 19:00-19:30 14:00-14:30 Andingmen Station 09:00-09:30 Andingmen Station 09:00-09:30 Xizhimen Station 04:00-04:30 Xizhimen Station 04:00-04:30 0 0 Figure 12. PassengerPassenger ( (leftleft)) arrive arrive and and ( (rightright)) departure departure flow flow of Line 2 on 17 April 2014.

35 before optimation 30 after optimation

25

20

15

10 energy consumption(kWh)

5

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 sections

Figure 13. Energy consumption of each section before and after the optimization.

Energies 2017, 10, 436 15 of 20

Table 5. Basic operation data of Beijing Metro Line 2 (inner ring).

Running Dwelling Section Section Length (m) * Time (s) Time (s) Passengers Jianguomen Station–Chaoyangmen Station 1763 123 45 166,945 Chaoyangmen Station–Dongsishitiao Station 1027 88 30 157,873 Dongsishitiao Station–Dongzhimen Station 824 78 50 148,313 Dongzhimen Station–Lama Temple Station 2228 174 45 114,565 Lama Temple Station–Andingmen Station 794 74 30 143,292 Andingmen Station–Guloudajie Station 1237 98 50 145,613 Guloudajie Station–Jishuitan Station 1766 129 50 160,098 Jishuitan Station–Xizhimen Station 1899 166 60 161,824 Xizhimen Station–Chegongzhuang Station 909 87 45 156,004 Chegongzhuang Station–Fuchengmen Station 960 85 30 159,932 Fuchengmen Station–Fuxingmen Station 1832 137 50 157,867 Fuxingmen Station–Changchunjie Station 1234 115 30 109,896 Changchunjie Station–Xuanwumen Station 929 85 30 110,013 Xuanwumen Station–Hepingmen Station 851 82 30 154,568 Hepingmen Station–Qianmen Station 1171 95 30 155,703 Qianmen Station–Chongwenmen Station 1634 123 45 157,306 Chongwenmen Station–Beijing Railway Station 1023 112 60 136,104 Beijing Railway Station–Jianguomen Station 945 101 60 133,743 * Source from http://www.bjsubway.com/station/zjgls/#.

9000 6000 8000 5000 7000 8000 4000 5000 6000 7000 5000 3000 4000 6000 4000 5000 2000 3000 3000

2000 4000 1000 2000 1000 3000 0 2000 1000 0 Chegongzhuang Station 1000 Fuxingmen Station Chegongzhuang Station 0 Fuxingmen Station Hepingmen Station 00:00-00:30 Hepingmen Station 0 Beijing Railw ay Station 19:00-19:30 Beijing Railway Station 00:00-00:30 Dongsishitiao Station 14:00-14:30 Dongsishitiao Station 19:00-19:30 14:00-14:30 Andingmen Station 09:00-09:30 Andingmen Station 09:00-09:30 Xizhimen Station 04:00-04:30 Xizhimen Station 04:00-04:30 0 0 Energies 2017, 10, 436 16 of 20 Figure 12. Passenger (left) arrive and (right) departure flow of Line 2 on 17 April 2014.

35 before optimation 30 after optimation

25

20

15

10 energy consumption(kWh)

5

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 sections

FigureFigure 13. EnergyEnergy consumption consumption of each section before and after the optimization.

In Figure 13, we compare the energy consumption of each section of Line 2 before and after the optimization; it is clearly shows that the energy consumption is obviously decreased after the utilization of the optimal model. As detailed in Figure 13, there are some sections with appreciable energy savings, such as Sections 3, 5, 9, 13 and 14, which are the sections with shorter lengths. Take Section 5 for example: the results illustrate that the energy consumption will show a great decrease from about 21.9 kWh for before optimal to 11.9 kWh for after optimal. In other words, the energy consumption is nearly halved when the running time of this section increases nine seconds. However, this kind of great decrease will not happen in all sections, caused by associated reasons, such as dwelling time margin, gradient condition of the section, and so on. On the whole, the total energy consumption of one train decreases from about 407.3 kWh down to 317.9 kWh. This energy savings rate (21.9%) is much higher than those considering regenerative energy utilization (8.86% in [4], 5.12% in [5] and 8% in [6]) considering the usage of regenerative energy.

6. Conclusions Based on a dwelling time calculation approach on the basis of representatively dynamic changes of the passenger flow in different time intervals of its daily operation, a metro train timetable and control strategy optimization model is newly developed in this research to reduce the traction energy consumption during off-peak hours. A mass-belt train movement simulation model provides a way of calculating the traction energy consumption with fixed running time considering both the basic running resistance and the additional resistance (such as resistance force caused by track gradients and curves). It has been confirmed that the proposed train simulation model is able to effectively obtain a reasonable driving control strategy with satisfactory optimal energy-saving results. The case studies with the application of the proposed approach show that the newly-developed model is capable of rationally reducing the train traction energy consumption on the basis of meeting the boarding and alighting demand of passengers on the platform. This enables the quick capture of the dwelling time at a station with uncertain and dynamic passenger time-dependent demands, which leads to a longer running time and a lower energy consumption. Furthermore, this approach can be combined with a real-time monitoring of the passengers on station platforms in order to contribute to an off-peak energy-efficient control system. For the sake of simplicity, we assume that the train arrival time of stations does not change; in other words, the dwelling time margin of station i only can be added to the running time of section i. Therefore, many other different kinds of timetable change assumptions need to be studied with much more optimal scenarios simulated in future research to further validate the results of this research. Energies 2017, 10, 436 17 of 20

Furthermore, the comparative analyses of the waiting time value of passengers who arrive at the station during the dwelling time margin and that have to wait for the next train also ought to be made in the future to enrich this work. The good energy-saving effort of this proposed model is obtained on the basis of sacrificing the travel time of a part of the passengers.

Acknowledgments: The research described in this paper was substantially supported by projects (71621001, U1334207) from the National Natural Science Foundation of China, the China Postdoctoral Science Foundation funded project (2015M582347) and The Postdoctoral Science Foundation of Central South University. The authors would like to thank the anonymous referees for their helpful comments and valuable suggestions, which improved the content and composition substantially. Author Contributions: Jia Feng contributed to the conception of the study. Xiamiao Li contributed significantly to the analysis and manuscript preparation. Haidong Liu made outstanding contributions to the simulation approach. Xing Gao helped to perform the analysis with useful discussions. Baohua Mao provided the metro line, train and passenger data. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

Passenger Flow

λi(t) The passenger arriving rate at station i and time t κi,i’(t) The passenger arriving amount at station i at time t with destination station i’ Pi,j Number of passengers waiting on the platform who board the train j at station i Qi,j Number of passengers alight the train j with trip destination i Oi,j Number of in-vehicle passengers of train j driving from station i to station i + 1 γi,i’ The ratio describing the passengers boarding the train j at station i who alighting at station i’ DOC The degree of crowdedness

Train Timetable

0 a i,j The arrive time for a train j arriving at station i according to the timetable 0 d ij The departure time for a train j leaving station i according to the timetable ai,j The actual arrive time for a train j arriving at station i di,j The actual departure time for a train j leaving station i 0 T D,i,j The dwelling time of a train j at station i according to the timetable TD,i,j The actual dwelling time of a train j at station i min TD,i,j The minimum dwelling time for train j at station i TR,i,j The running time of a train j at section i, which is defined as the one between the stations i and i + 1 Train Control Strategies with Fixed-Time

Tm The dwelling time margin Tb The braking time from point A to point O Tc The coasting time from point A to point O Tu The uniform time from point C to point A Tc-b The coasting-braking time from point B to point O passes point X Terror The calculation error Energy Consumption

Tf The traction force t(v) The specific (per mass unit) traction force Rf The resistance force r(v) The specific (per mass unit) basic running resistance r0, r1, r2 The coefficients of basic running resistance Energies 2017, 10, 436 18 of 20 w(x) The specific (per mass unit) additional resistance wg(x) The unit gradient resistance wr(x) The unit curvature resistance wt The unit tunnel resistance M The train traction weight Mt The mass of the train Mp The mass of loading passengers ρ The mass per unit length g(x − s) The gradient of position (x − s) S The train length α The angle of the curve Lr(x) The length of the curve R The radius of the curve Bf The braking force b(v) The specific (per mass unit) braking force c The specific force vk The initial speeds of k-th time step vk+1 The initial speeds of (k + 1)-th time step Vmax The limit speed ak The accelerated speed at k-th time step σ One time step lk The positions of k-th time step lk+1 The positions of (k + 1)-th time step Ek The accumulative energy consumption of k-th time step Ek+1 The accumulative energy consumption of (k + 1)-th time step ek The energy consumption of k-th time step Etotal The total energy consumption

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