Optimization of Train Plan for Urban Rail Transit in the Multi-Routing Mode

Journal of Modern Transportation Volume 19, Number 4, December 2011, Page 233-239 Journal homepage: jmt.swjtu.edu.cn DOI: 10.1007/BF03325763

Optimization of train plan for urban rail transit in the multi-routing mode

Lianbo DENG1*, Qiang ZENG1, Wei GAO1, Song BIN2

1. School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China 2. China Railway Siyuan Survey and Design Group Co. Ltd., Wuhan 430063, China

Abstract: The train plan of urban rail transit under multi-routing mode can be divided into three parts: train formation, train operation periods and corresponding train counts of each routing in each period. Based on the analysis of passenger’s general travel expenses and operator’s benefits, the constraints and objective functions are defined and the multi- objective optimization model for the train plan of urban rail transit is presented. Factors considered in the multi- objective optimization model include transport capacity, the requirements of traffic organization, corporation benefits, passenger demands, and passenger choice behavior under multi-train-routing mode. According to the characteristics of this model and practical planning experience, a three-phase solution was designed to gradually optimize the train formation, train counts as well as operation periods. The instance of Changsha Metro Line 2 validates the feasibility and efficiency of this approach. Key words: urban rail transit; multi-train-routing; train plan; multi-objective model; three-phase solution method © 2011 JMT. All rights reserved.

1. Introduction gramming. Bussieck et al. [8] treated the train plan as a train schedule (i.e. a line plan), and determined the ver years, the trains on urban railways in China number of trains connecting two terminal stations of a O have to stop at every station along a single long serving line in a fixed time interval. Claessens et al. [9] route. The daily train plan depends on the cross-section built an integer nonlinear model to maximize the num- flow in each operation period [1]. ber of direct travelers for the Dutch railway system. Sun In order to provide service efficiently and to decrease et al. [10] investigated the multi-train-routings of urban cost effectively, many cities in China have to take the rail, and proposed a two-phase solution for the maximal approaches such as train formation adjustment [2] and seat occupancy rate and the minimal count of train op- multi-routing operation [3] to make their rail transit eration periods. Deng et al. [4] considered single train networks more competitive. The traditional train plan of routing based on the analysis of passenger’s general urban rail transit has expanded into more advanced one, travel expenses, and established a multi-objective model which consists of train formation, train operation period with respect to transport capacities, transport organiza- and train frequency for each routing [4]. Among these tions, economic benefits and traffic demands. Refs. [11- components, the most important determinant of trans- 12] studied passenger’s choice behavior and flow as- port capacity is train formation. In the multi-train- signment in the urban rail transit network. Sang et al. routing mode, the train plan needs to consider not only [11] presented a graph theoretic framework for the pas- the heterogeneous distribution of passenger volumes in senger assignment problems that simultaneously en- the time-space, but also the passengers’ choices [5-6]. compassed the departure time and the route choice. Tian Chang et al. [7] presented a multi-objective optimiza- et al. [12] analyzed the equilibrium properties with in- tion model to minimize operation cost and total travel vehicle crowding effect and schedule delay cost in a time of passengers. Train stops, service frequency and many-to-one transit system. train count were optimized by fuzzy mathematic pro- Although the train plans of urban rail transit are es- sentially network problems. Unlike national railway

network, they are independent of each other to some ex- Received Aug. 29, 2011; revision accepted Nov. 17, 2011 *Corresponding author. E-mail: [email protected] (L.B. tent. In this paper, the train plan is targeted on one line DENG) with the consideration of the passenger choice. For sim- © 2011 JMT. All rights reserved plification, passengers transfer is ignored. doi: 10.3969/j.issn.2095-087X.2011.04.003

234 Lianbo DENG et al. / Optimization of train plan for urban rail transit in the multi-routing mode

2. Related concepts on passenger train plan tTb . Ht e The train plan of urban rail transit : determines Train routing set is expressed as Uusel ^ / ll, train formation and the number of trains for each routing and operation period. For simplicity, we only consider 1, ,Hull ;se , S` where sl and el are the two termi- the train plan in a sequence of 1, 2, ,H s . The number nals of routing ul . The train routing modes include: ad- of trains on routing u during T is d l , and train set jacent routing, nested routing, and the mixed mode l k k l (Fig. 1). For smooth train operation and passenger is Ddk ^`ktu 1, 2, , Hl ; 1, 2, , H . According to boarding, H can not be excessive. Among all the rout- u above definitions, : ^`bT,, D can denote the train ings, the ones running in all operation periods are basic plan of urban rail transit. routings whereas others running in peak periods are non-basic ones. Adjacent routings are basic. The long routing of the nested routing mode is a basic one while 3. Passenger’s general travel expenses the short is not. The train plan for the adjacent routing mode can be optimized on a single routing basis. Com- Passenger flow of urban rail transit fluctuates in paratively, the train plan of the nested routing mode [,]TTs e . It is relatively steady in certain duration, which should be optimized according to passenger’s choice. is called passenger travel period. [,]TTs e can be divided Further, the train plan of the mixed mode is a combina- into H 0 passenger travel periods. The passenger flow tion of both plans over two routing modes. The train t 0 plan optimization of the nested routing mode is the fo- from station i to station j in period Tk is f (,ijk , ), for cus in this paper. 0 ij,1,2,,, Hs kH 1, 2, ,t . The train frequency is unified in one passenger travel period, and one train operation period may include one or more passenger (a) Adjacent routing mode travel periods. If the passenger travel period T 0 is in- k1 cluded in train operation period T , the relationship k2 (b) Nested routing mode can be expressed as TT0 , for kH 1, 2, ,0 , and kk12 1 t

kH2 1, 2, ,t . The passenger’s general travel expense usually in- (c) Mixed mode cludes ticket fare, travel time and congestion cost.

Ticket fare is a monetary expenditure, which can be Fig. 1 Routing modes expressed as person-kilometers fare rate p . Individual ticket fare for passenger flow f (,ijk , ) is Let [,]TTs e denote the daily operation period of urban rail transit. The train formation is usually Pijk,, pwij , . unchanged. For the train plan : , b is the train formation length (i.e. number of carriages in a train), Travel time includes riding time and waiting time. and V is the average carrying capacity of carriages. Riding time is equal to the train running time between Considering the factors such as station capacity and the stations, which can be calculated by dividing the dis- track length, the feasible formation length is in a range tance wij , with the train speed J . The riding time of ªºbb, , where b and b are the shortest and the ¬¼ for passenger flow f (,ijk , ) is longest formation lengths respectively. According to the fluctuation of passenger flow in one wij , G ijk,, . day and the variation of train frequency in [,]TTs e , we J can define the train operation period Ti . Set Waiting time refers to the time period from a passen- ab TTtti ^`iii ,1,2,, Ht to be train operation ger’s arrival to departure. The waiting time is related to a b train frequency. When TT0 , the average waiting periods, where ti and ti denote the start and the end kk12 time of Ti respectively; Ht is the count of train time for the flow fijk(, ,1 ) is calculated by a operation periods under the conditions of tT1 s and

Journal of Modern Transportation 2011 19(4): 233-239 235

The length restriction of the train formation is V Tk Z(,ijk , ) 2 , 1 d l ¦ k2 bbbdd, and b is an integer. (2) uUueijll:(,) The train operation periods should satisfy where T is the period length of T ; V is a parameter k2 k2 ba related with the passenger flow distribution, and V is ttkk 1,1,2,,1; k Ht a equal to 0.5 for uniform passenger flow. tTks , k 1; (3) f (,ijkl , , ) is the volume of passengers which select b tTke , kH t . train routing u in f (,ijk , ). Within the passenger l The train frequency should be restricted by train 0 travel period Tk , the passenger volume on routing ul in headway W [13]. The upper limit of train frequency is the section (,ii 1) E is Tk tW ,1,,,iHkH St1 1,2,,. el i l '' ¦ dk (4) g(,ikl , ) ¦¦ f ( i , j , kl , ). lU '' eii(, 1) u ji 1 is l l

We introduce the congestion cost function The lowest train frequency for each routing varies. y gikl(, , ) to describe the passenger discomfort for For basic routings, such as long routings, its train frequency should be no less than the minimal operational routing ul and passenger flow period Tk . frequency. In another word, its train operation interval The time and congestion cost of one passenger should should be no less than the maximum operation interval be converted into monetary expenditure. From station i time W . As for non-basic routings, such as short rout- to station j in T 0 , the individual travel expense is 0 k ings, the train frequency should be zero or the train op-

Cijkl ,,, Pijk ,, eration interval should not be less than W 0 , or equal 0. j1 The lowest train frequency for each routing is denoted E ªºijk,, (,,) ijk ygikl (,,),c ¬¼GZ ¦ as follows: iic T where E refers to the average time value of passengers. k tW ,,lU l 01 dk (5) T 4. Multi-objective optimization model k tW ordlUl 0, , l 02k dk Within the passenger travel period T 0 , the train load k1 where U1 and U 2 are the sets of basic and non-basic factor varies at different sections of train routings. From routings. the perspective of transport capacity and operation cost, Each individual passenger will select affordable the load factor is controlled in the sections of passenger trains. Thus, the passenger flow distribution will reach flow peak to ensure reasonable overall load factor. The user equilibrium state. It must be noted that, there is al- lower limit of the train load factor is denoted by M0 ways stochastic in passenger choice behavior. All in all, ranging from 0.7 to 0.9, and upper limit is denoted by the passenger flow of urban rail transit is stochastic equilibrium. M1 ranging from 1.2 to 1.5. The minimal and maximal As passenger volume fluctuates with time periods, passengers volume of the section are respectively M Vb 0 the passenger choice behavior should be analyzed for ** 0 and M1Vb. Let (,ii 1) be the peak section in T , and k1 each period. We use logit distribution to describe the the corresponding peak passenger flow be passenger choice behavior, the probability that passen- *0 If the length of ger flow f (,ijk ,1 ) selects routing ul is: gi(, k111 ) max(,)^` gik k 1,2,, Ht . 0

0 0 0 period T is T , then for TTkk , the train load fac- expOCi ( , jk ,10 , l ) k1 k1 12 M(,ijkl , , ) . 10 (6) tor in flow peak section satisfies ¦expOCi ( , jk ,1 , l ) lU

00 MM01Vbdkk T Vbdkk T where O is the utility parameter. 21ddgi(,* k ) 21. (1) 1 The general travel expense of all passengers should TTkk 22 be minimized:

236 Lianbo DENG et al. / Optimization of train plan for urban rail transit in the multi-routing mode

0 Ht For the service providers, their primary benefits come minZ1 ¦¦¦¦ fijkl(,,,) Cijkl ,,, , lUk 1 iSjS from ticket income. The train type in the urban rail transit is homogeneous and the ticket fare has no distinction. where the ticket fare and travel time can be viewed as If the target passenger flow is relatively stable, the ticket constants, and the passenger travel expense related with income is constant. Let per-train-kilometer cost be cT , the train plan is expressed as and per-car-kilometer cost be cu , then the objective j1 function of the operation cost is * C i,,, jkl E ¬¼ªºZ i ,, jk¦ y gi (,,).c kl iic HHtu l minZ2Tu ¦¦ (cbcdwse )kll , . (8) Thus, the objective function Z1 is simplified as kl 11

0 Ht Maximizing the average load factor is also an impor- minZc fijkl(,,,) C* ijkl ,,, . 1 ¦¦¦¦ (7) tant target of the train plan optimization: lUk 1 iSjS

je i Twij,(,,,) fijkl'' k2 ¦¦ 1 HHHtSS1 1 ji'' 11 i (9) max Z . 3 ¦¦¦¦¦ 0 l lUbVw sll, e k 111kT: 0 T i ji Td 2 1 kk12 kk12

The train frequency may vary with the passenger 0 Ht flow. However, if the train frequency is frequently minZb D ¦¦¦¦ fijkl(,,,) Cijkl ,,, changed, the organization of train operation will be dis- lUk 1 iSjS (11) HHtu turbed. Thus the train frequency should keep steady in l 1(),.D ¦¦ cbcdwseTukll an operation period as long as possible. And the number kl 11 of the train operation periods should be minimized, i.e. The formation length that corresponds to the minimal minZ4 Ht . (10) evaluation function Zb is optimal. The objective functions (7)-(10) and the constraints When the train formation length is b, the train fre- (1)-(6) constitute the multi-objective optimization model quency can be determined by section flow for the train plan of urban rail transit. * ¦ g(,ikl2 ,) kT: 0 T 5. Solution 1 kk12 in the train operation period kk22(1,2,,). Ht Con- The established optimization model can not be solved straint (1) determines a range of d l for formation k2 directly because it is discontinuous, non-differentiable, length b. Considering the objective function (9), we ob- non-convex and mixed of multi-objective. For this rea- tain d l : son, we work out a three-phase solving strategy: k2 In the first phase, the train formation is optimized. In the second phase, the train operation period is sub- minddVbdVbll maxd 1MM , l ^ kk22^` 10 k 2 stituted with passenger travel period, and the train count gi(,* k ,) l d for each routing is determined. ¦ 2 (12) kT: 0 T In the third phase, the strategy of merging train op- 1 kk12 eration periods is designed, and the train frequency is maxdVbdllMM , 1 Vb . ^`kk2210 ` adjusted accordingly. To simplify calculation, each passenger travel period 5.1. Determining train formation works as one train operation period, and all the trains run from the origin stations to the terminal stations on The objective functions (7) and (8) are added with the the urban rail lines. In this way, the solution of forma- introduction of D , the weight factor of the passenger tion length can be simplified to meet passenger demands. travel expense. The evaluation function of train forma- Every operation period has the train count tion length is dk , kH 1, 2, ,t . The algorithm is described in details below.

Journal of Modern Transportation 2011 19(4): 233-239 237

Algorithm 1 ll12 Step 4 Let dddddddkk ccc/2, k ccc/2,

and assign passenger flow on the routings according to Step 1 Initialization. Let the initial train formation logit distribution to obtain g(,ikl , ). length bb , the best train formation solution bb* Step 5 According to the passenger flow distribution, and its evaluation function Z * f . Each passenger determine the train count on the short routing by Eq. (12). 0 travel period work as train operation period, TTkk , l2 l2 l2 If dddk t ccc/2, let ddcc k ; otherwise let ddc k , kH 1, 2, ,0 . t and go to Step 3. Step 2 If bb! , stop. Step 3 Determine the train count d in train operation 5.3. Merging train operation periods k period T . Then examine constraint (4), if d ! 60 /W , k k In order for minimization, the operation periods of dk 60 /W ; examine constraint (5), if dk 60 /W 0 , similar train counts are merged. The train frequency can evaluate the difference of train counts, dk 60 /W 0 .

Step 4 Calculate the objective function value Z by k * * * d (11). If Z Z , then Z Z, and bb . bb 1, go to Okt kH 1, 2, , . T Step 2. k The train count difference in adjacent periods is dis- 5.2. Determining the train frequency tinguished by the following criterion:

In this stage, the passenger travel period is still used 2 GOOO ªºll l,2,3,,.kH as train operation period; thus every train operation pe- kk,1¦ ¬¼ k k1 k 1 t lU riod has the same passenger flow intensity. To reduce train operation cost, trains operate on the short routing The operation periods which have minimal G kk,1 will to meet the peak passenger flow. be merged in the first place. The algorithm is expressed F fi(,,)* l k l2 ¦ 2 as follows: ijeij,: , es ll , e 22 is the maximum passenger flow on the short routing. Algorithm 3 The largest train count on the short routing dU l2 is k2 Step 1 Calculate Ok for each train operation period,

ll l mindU22 max dUd 1MM Vb , d Vb where kH 1, 2, ,t . ^ kk22^` 10 k 2 (13) Step 2 Calculate criterion function 'G ^ kk,1 k FdUVbdUVbdmaxll22MM , 1 . lkk222^`10 ` 2,3, , Ht ` of adjacent periods. Thus, the train count on routing l is ddU l2 , and on Step 3 If 'I , stop; otherwise, sort the elements in 1 kk22 * l2 routing l is dU . set ' and select the minimal G kk,1 . 2 k2 Passengers who choose the short routing can choose Step 4 Assume kkand 1 are objective periods for the long routing at the same time. An extreme case is merging, determine the train counts on different routings that the long routing is the sole selection, i.e. the train by Eq. (12) and make passenger assignment according counts on long and short routings are d and 0. k2 to logit distribution. Then the feasibility of merging pe- The real train count falls between the two extremes, riods is judged by constraint (1). If it is feasible, which are determined by the selection behavior of pas- merge kkand 1 and calculate the new train count senger flow: l * dk 1 , then go to Step 1; otherwise, ''G \,kk,1 i.e., to

eliminate G * from ' , then go to Step 3. Algorithm 2 kk,1

0 0 Step 1 TTkk , kH 1, 2, ,t . 6. Case analysis

l2 l2 Step 2 Calculate dU k by Eq. (13), ddUdccc k ,0. We take Changsha Metro Line 2 in the initial opera- Step 3 If ddccc , train count on the long routing is tion stage as an example [3]. This is a 28-station line, dd c, and that on the short routing is c stop. k d , starting at Zhenqiao Road Station and ending at

238 Lianbo DENG et al. / Optimization of train plan for urban rail transit in the multi-routing mode

Guangda Station. In addition to the long routing (34.02 ters of train operation are set as follows: km) between the start and the end, trains can also oper- cT 60 CNY/(train·km), cu 10 CNY/(car·km). Set ate on the short routing (21.36 km) between Donglei 0 E 30 CNY/h, T 6, T 24 and H 18 h, mean- Road Station and New Changsha Station. V 220 per- s e t ing the passenger travel period is one hour. son/car, b 4 and b 9 cars, 2 min and 15 0 W W 0 For TT , the congestion cost function of the sec- kk12 min, M 0.95 , M 1.4, D 0.5. And the cost parame- 0 1 tion (,ii 1) on routing l is

0, Tgikl(,,) d bVTd0 l , kk211 k2 ° ygikl (,1 , ) ® Tgikl(, , ) k2 1 40l °0.15( ),(,,)Tgiklkk1 ! bVTdk. bV T0 d l 212 ¯° kk12

With the computer of 1 GB RAM and 1.7 GHz CPU, cost and the passenger travel expense. The train opera- the optimization process spent about 2 sec. Fig. 2 shows tion cost and average passenger travel expense under the relationship between the evaluation function (11) different values of D are calculated (see Fig. 5). With and train formation length. The dotted line shows that the decease of D , the passenger travel expense declines, the formation length of 4 cars does not meeting trans- and the train operation cost rises. In general, the value of portation capacity. The best formation length is 6, and D is in the range of 0.4–0.6. the corresponding value of evaluation function is 6 809 232.5 CNY. 18 Long-routing Short-routing 80 15

78 12

CNY) 76 5 9 74 6 72 Number of trains 3 70

68 0 Evaluation function (× 10 07 09 10 13 15 17 19 21 23

66 ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ 22 06 08 09 12 14 16 18 20 456789 Period Train formation length (car) Fig. 3 Train counts in different periods Fig. 2 Solving train formation length 30 As shown in the Fig. 3, among the total 158 trains, 18 trains run on the long routings, 11 trains on the short 25 Long-routing routings at the morning peak (from 7 to 8 a.m.), 14 Short-routing trains run on the long routings, and 4 trains on the short 20 routings at the evening peak (from 17 to 18 p.m.). The operation benefit is 2 340 275 CNY, and the average 15 general travel expense for one passenger is 14.27 CNY. 10 As indicated in Fig. 4, when train operation periods are ofNumber trains merged, the train count decreases to 155. Operation bene- 5 fit rises to 2 364 770 CNY with a little increase in average passenger general travel expense, i.e., 14.41 CNY. The 0 average train load factor is 60%. A relative low load 07 08 11 12 12 15 16 17 18 20 24 ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ ˉ 06 07 08 11 12 13 15 16 17 18 factor is because of the unbalanced passenger spatial 20 distribution. Period In the model, the weight factor of the passenger travel expense D plays a role in balancing the train operation Fig. 4 Final train counts after merging

Journal of Modern Transportation 2011 19(4): 233-239 239

15.5

Operation cost 1.35 15.1 Travel expenses per passenger 14.7 1.25 14.3 1.15 13.9

Travel expenses (CNY) 13.5 1.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Operation cost (billion CNY) cost (billion Operation Weight factor

Fig. 5 Effect of weight factor on train plan

Changsha Metro Line 2, Wuhan: China Railway Siyuan 7. Conclusions Survey and Design Group Co. Ltd., 2009 (in Chinese). [4] L.B. Deng, Q. Zeng, W. Gao, et al., Optimization

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