Train Headway Models and Carrying Capacity of Super-Speed Maglev System∗

518

Shiwei HE∗∗, Rui SONG∗∗ and Tony EASTHAM∗∗∗

Train headway models are established by analyzing the operation of the Transrapid Super-speed Maglev System (TSMS). The variation in the minimum allowable headway for trains of diﬀerent speeds and consists is studied under various operational constraints. A potential Beijing – Shanghai Maglev line is used as an illustration to undertake capacity anal- yses with the model and methods. The example shows that the headway models for analyzing the carrying capacity of Maglev systems are very useful for the conﬁgurational design of this new transport system.

Key Words: Maglev, Headway, Modeling, Line Capacity

els of wheel-on-rail systems have been studied for dif- 1. Introduction ferent control schemes, such as fixed block and moving The super-speed maglev system Transrapid is a block(6). However, no detailed headway models for opera- guideway-bound transportation system for passengers and tional analysis of Maglev systems have yet been reported. high-value cargo traffic. It is one of the most significant in- The contents of this paper are as follows. First, TSMS novations in guided ground transport technology since the operation is analyzed and train following headway models construction of the first railroad. The non-contact tech- are set up. Second, the variation of minimum allowable nology of TSMS – controlled magnetics are used instead headway for trains of different consists is studied. Third, of mechanical components – overcomes for the first time carrying capacity of a Maglev system is calculated. The the limitations of wheel-on-rail technology. In operation, example of a Beijing – Shanghai Maglev line is used for Transrapid is faster and quieter than high speed rail sys- capacity analysis with the model and methods. Lastly, the tems. It is virtually impossible to derail and is comfort- main conclusions of this paper are presented. able at all speeds. The guideway of Transrapid requires 2. Dynamic Train Headway Models less space and can be flexibly aligned to accommodate lo- cal topographic variations. However, its cost-effectiveness 2. 1 Minimum headway criteria has yet to be proven. The first criterion for minimum headway of any The Transrapid system has been developed over 35 guided ground transportation system is that if a vehicle years(1) – (5). The first commercial Maglev line has been should stop suddenly for whatever reason, the following built at Pudong in Shanghai, China with German tech- vehicle must be able to come to a safe stop at an accept- nology. Also, the techno-economic attributes of Maglev able deceleration before reaching the location of the im- relative to high-speed wheel-on-rail technology are be- mobilized vehicle. For the case of a vehicle with a cruising ing evaluated for what is likely to be the first high-speed speed of 450 km/h, for example, the distance and time to inter-city line in China (Beijing – Shanghai). Train head- stop at a deceleration of 0.15 g are 5.2 km and 83 seconds, way is important both for safety and carrying capacity respectively. The minimum headway is, of course, speed when undertaking operational analysis. Headway mod- dependent, and other factors are likely to dictate a higher

∗ operational minimum headway. We now consider TSMS. Received 15th December, 2003 (No. 03-5152) 2. 2 Train following moving headway models for ∗∗ School of Traﬃc & Transportation, Northern Jiaotong TSMS University, Xizhimenwai, Beijing 100044, P.R. China. E-mail: [email protected] In the single-sided mode, one side of each power ∗∗∗ Department of Civil Engineering, Hong Kong University substation supplies energy to a section of the distributed of Science and Technology, Clear Water Bay, Kowloon, linear synchronous motor (LSM) armature in the guide- Hong Kong. E-mail: [email protected] way, while the adjacent sections are un-energised. The

Series C, Vol. 47, No. 2, 2004 JSME International Journal 519

headway will be used as the dynamic separation of two trains in the following analysis. 2. 3 Departing, arrival headway models When the trains pass an off-line station and do not stop, the minimum headway is similar to the train following moving headway as determined by Eqs. (1) – (4). If at Fig. 1 Minimum train separation for a single-sided energy sup- least one train stops at the station, the minimum headway ply system must be increased. 2. 3. 1 Departing headway models The headway for a train leaving an off-line station should at least meet Eqs. (5) and (6). f L +0.5L + L + L f 1 j l,q a s Iz = tu +t f + (5) v¯q f 1 f = + + + + + Lz Lu L f L j 0.5Ll,q La Ls (6) f 1 f 1 where Iz and Lz are headway and distance interval of the departing train relative to the leading train when the lead- Fig. 2 Minimum train separation for a double-sided energy ing train needs to leave at least one energy supply section supply system and the merging switch must be changed to allow the departing train to join the mainline; t f , tu, L f and Lu are times minimum separation of two Maglev trains is thus two of changing and confirming the integrity of the switch and LSM sections (see Fig. 1). For the double-sided mode, the corresponding distances advanced by the leading train; each power-substation can energise two sequential sec- and La is a safety distance. Also, Lg is length of switch; tions, and the minimum separation of two Maglev trains Lb is redundancy distance when a train stops;v ¯q is the av- becomes just one section of the propulsion system (see erage velocity of the leading train when passing through Fig. 2). Note that each section of the propulsion system f the switch in its mainline position; and L j is the distance may be subdivided and each sub-section switched sequen- between the center of the station and the furthest switch ffi tially to increase motor e ciency. in the departure direction. Suppose one station (together In order to assure safety with the two configurations with its deceleration and acceleration lanes) corresponds of energy supplying system, the minimum separation (Lz) to an energy supply section with length Ls,wehave: between Transrapid trains should meet Eqs. (1) and (3), f L = max(0.5Ls,Lg +0.5Ll,h + Lb)(7) respectively. The time interval (headway) (Iz) should meet j Eqs. (2) and (4). After passing through the merging switch, the leading and

following trains are on the same route. Thus, train depart- Lz = Lc + Lz,h + La +0.5Ll,q +0.5Ll,h +2Ls (1) + + + + ing headway should also meet Eqs. (1) – (4). Departing Lz,h La 0.5l,h 0.5Ll,q 2Ls f Iz = tc + (2) headway (I ) is the greater of the headway of Eqs. ((2) or v z h (4)) and (6). We have : Lz = Lc + Lz,h + La +0.5Ll,q +0.5Ll,h + Ls (3) I f = max{I ,I f 1} (8) Lz,h + La +0.5l,q +0.5Ll,h + Ls z z z Iz = tc + (4) vh The calculation method for train departing headway from where Lc and tc are the distance and time advanced by the a station yard is similar to that for intermediate stations. following train when the leading train transmits informa- When trains depart from yards of different lengths, safe tion to the following train, via the central control system. train headway is assured by considering switch activation f Lz,h is the dynamic braking distance of the following train. time and braking time before the mainline switch. L j is Ll,q and Ll,h are the lengths of the leading and following then calculated as: f trains, respectively. vh is the average velocity of the fol- = + + + L j max(0.5Ls,Lg 0.5Ll,h La lz,h)(9) lowing train in Lz. La is a safety distance, and Ls is the separation of adjacent power sub-stations. The result can be put into Eqs. (5) and (6) to obtain the The train dynamic headway models of Eqs. (1) – (4) appropriate minimum train departing headway. are basic conditions for assuring safe train operation. In 2. 3. 2 Arrival headway models The headway ff order to maximize line capacity, the following train should for a train stopping at an o -line station should meet maintain minimum dynamic headway. If headway is in- Eqs. (10) and (11). d creased, the leading train will not influence the follow- L +0.5Ll,h + La + Lz,h d1 = + + j ing train in normal operational conditions. This minimum Iz tu td (10) v¯h

JSME International Journal Series C, Vol. 47, No. 2, 2004 520

d1 = + + d + + + Lz Lu Ld L j 0.5Ll,h La Lz,h (11) d1 d1 where Iz and Lz are the headway and distance interval of the arriving train when the leading train does not stop and the divergent switch position is changed to allow the arriving train onto the deceleration lane into the station; td, tu, Ld and Lu are times of changing and confirming the integrity of the divergent switch, and the corresponding distances advanced by the following train;v ¯h is the average d velocity when the following train passes the switch; L j is the distance between the furthest switch in the arrival direction and the center of the station. Suppose again that Fig. 3 Transrapid vehicle headway the station lane corresponds to one energy supply section with length Ls,wehave: of a leading train departing from a station and the follow- Ld = max(0.5L ,L +0.5L + L ) (12) j s g l,q b ing train passing that station will be discussed in 3.2.2. Before changing the divergent switch, the trains are on the 3. 2. 2 Arrival headway in intermediate stations same route. Thus, the arrival headway should also meet The distance Ld and the speed of a following train v are d j Eqs. (1) – (4). Train following arrival headway (Iz )isthe main factors that influence the arrival headway at interme- maximum of the headway of Eqs. ((2) or (4)) and (10). We diate stations. The headway of a leading train arriving at then have : or passing a station and the following train arriving at that d = { d1} d Iz max Iz,Iz (13) station is mainly influenced by L j . The headway of a leading train arriving at a station and the following train pass- 3. Analysis of Train Following Headway ing that station is mainly influenced by passing velocity v. Train following headway is influenced by many fac- As the latter headway is usually smaller than the former, tors such as propulsion system design, train length, speed it is used as the boundary headway. The relationship of limit and gradient. We have calculated train following headway and motor section length is shown in Table 2. headway for double line operation. All calculations are 3. 2. 3 Other kinds of headway in stations The based on vehicle parameters taken from Transrapid pub- model of train departure headway from identical yards of (7) f lished data . As a double-sided energy supplying system a sectional station (Iz ) is the same as that for departure- can have a shorter headway than the single-sided option, departure from an intermediate station. So the trend is we consider only the double-sided configuration for head- similar to Table 1. way and capacity calculations. The following departure headway from different 3. 1 Analysis of train following headway in sections yards is redefined by the headway of passing a convergent In this section, the relationship between the mainline switch. This headway is restricted by the switch chang- headway of Transrapid trains Iz, velocity v and the length ing headway and following headway. The following head- of energy supplying sections is shown in Fig. 3, as deter- way has been discussed previously. The switch changing mined by the methodology of this paper, with Transrapid headway is determined by the passing velocity of a lead- data. ing train. The velocity is related to yard collocation, the In Fig. 3, the allowable headway Iz is clearly speed distance between starting place of the leading train and the dependent. See for example, curve 2. When v is within convergent switch. 0km/h – 200 km/h, Iz drops quickly. When v is within The model of arrival headway for a sectional station d 200 km/h – 400 km/h, Iz drops more slowly. When v is (Iz ) is the same as that for arrival-arrival at an intermediate larger than 400 km/h, Iz increases gradually. Thus, there station. So, the headway is similar to Table 2. The bound- exists an optimal velocity corresponding to the minimum ary headway in stations is essentially the combination of Iz. train departure and arrival movements. Table 3 lists the 3. 2 Analysis of train following headway in stations headway of one such combination, when the leading train 3. 2. 1 Departing headway at intermediate stations departs from a station and the following train arrives at f From Eqs. (5) – (7), L j is an important factor that influ- that station. ences the headway for a train departing from an intermedi- 4. Capacity Analysis of TSMS ate station. Also, the allowable headway of two trains departing from a station is longer than that of a leading train Carrying capacity of a Maglev system on a double passing a station and a following train departing from that line is calculated for each direction (up or down). In one station. So, the boundary headway is that of two trains de- direction, carrying capacity is restricted by the minimum parting from a station, as shown in Table 1. The headway of the following: moving carrying capacity Ny, following

Series C, Vol. 47, No. 2, 2004 JSME International Journal 521

Table 1 Headway of two trains departing from station

Table 2 Headway of two trains arriving at a station

Table 3 Headway of one train departing and another train arriving at a station

Table 4 Parameters, headway and maximum carrying capacity

(8) departure carrying capacity in station N f , and following port completed in 2000 . Cases 1, 2 are forecast data for arrival carrying capacity in station Nd.Wehave: 2010. Cases 3, 4 are forecast data for 2020. The difference is that cases 1, 3 only consider flows within the Beijing – NU(D) = min{N ,N ,N } (14) y f d Shanghai corridor, while cases 2, 4 include the additional U(D) U(D) C = αγCnU /β (15) flows passing through Beijing – Shanghai. This data set includes the increased passenger flows generated by the where NU(D) is the maximum section carrying capacity of f introduction of a Beijing – Shanghai high-speed line. We the up (down) line. N = T ÷ I , N = T ÷ I , N = T ÷ Id. T y z f z d z use this data for comparisons with the carrying capacity of is the operating time per day, T = 1440−T (T is main- w w TSMS we obtained for different operating conditions. tenance time). CU(D) is the operational passenger load ca- In consideration of the increased passenger demands pacity of the up (down) line. Cn is the number of seats in 2010 and 2020, some possible solutions are illustrated in n vehicular sections per train. α is an experiential co- in Fig. 4. The x-axis shows the line sections between Bei- efficient which is the ratio between operational capacity jing and Shanghai. The y-axis shows the forecast per- and maximum capacity, β is a coefficient which is the ra- direction passenger flows in each section. Five operational tio between operational headway and minimum headway, possibilities are listed in the figure. For example, “10 – γ is an experiential coefficient which is the average ratio 5 min” means that each Transrapid train has 10 cars and between actual passengers and seats capacity. each train operates with a minimum headway of 5 min. Using the above headway models of TSMS, carrying Similarly for “4 – 8 min”. “10 – 8 min”, “10 – 10 min” and capacity of parallel train schedules can be obtained. The “6 – 10 min”. parameters, headway and carrying capacity are shown in From Fig. 4, the 2010 demand of case 1 for the Bei- Table 4. jing – Nanjing section can be met with 6-car trains oper- We use four scenarios for forecast passenger OD data ating at 10 min headway. If the headway is subsequently (shown in Fig. 4), from the China – Japan co-research re-

JSME International Journal Series C, Vol. 47, No. 2, 2004 522

5. Conclusions By analyzing the operational characteristics of the Transrapid Super-speed Maglev System (TSMS), the dynamic train headway models are set up. The headway and line capacity for trains of different consists has been studied with the models. The example of a Beijing – Shanghai Maglev line has been used for capacity analysis. The analysis demonstrates that the headway models and the meth- Fig. 4 Comparison of demand and capacity for different train ods for analyzing the carrying capacity are very useful configurations for a Beijing – Shanghai high-speed line for operational design of this new ground transportation mode. reduced to 8 min, the projected demand for 2020 of case 3 References can be met. The section from Nanjing – Shanghai has the heaviest ( 1 ) Center for Transportation Research Argonne National projected traffic in the Beijing – Shanghai corridor. From Laboratory, Survey of Foreign Maglev Systems, Uni- Fig. 4, the 2010 demand of case 1 can be served by 10-car versity of Chicago, (1992). trains operating at 8 – 10 min headway. With a subsequent ( 2 ) EU-Study, Transrapid for Central and Eastern Europe, (1999). enhanced operation and a headway reduction to 5 – 8 min, ( 3 ) Proceedings MAGLEV’98, 15th International Confer- the 2010 projected passenger demand of case 2 can be sat- ence on Magnetically Levitated System and Linear isfied. Drives, Yamanashi, Japan, (1998). The diurnal peaking in travel demand should also be ( 4 ) Proceedings MAGLEV’2000, 16th International Con- considered. Suppose the ratio of OD demands at high- ference on Magnetically Levitated System. Rio de est peak time (within a 2 hour interval) to total daily flow Janeiro, Brazil, (2000). remains 1:4.5 – 1:6 by previous analysis(9). Considering ( 5 ) Transrapid International, Guideway System Overview, the Nanjing – Shanghai with heaviest peak hour traffic, the (1998). ( 6 ) Song, R., He, S. and Zhu, S., Train Headway Models 2010 demand of case 2 can be served by 10-car trains op- and Carrying Capacity of Intelligent Railway System, erating at less than 8 min headway. To meet the 2020 de- International Conference on Traffic and Transportation mand of case 3, it appears that headway will need to be re- Studies, Beijing, China, (1998), 840–849. duced to less than 5 min. It seems impossible for Maglev ( 7 ) Transrapid International, High-Tech for “Flying on the high-speed trains to satisfy much higher capacity demands Ground”, (2001). like case 4 with current technical and financial conditions. ( 8 ) China Academy of Railway Sciences, Research Re- Having determined the headway and consist combi- port on Forecasting of Passenger Demand in Beijing – nations needed to meet projected passenger demands, this Shanghai High-speed Railway, (2000). ( 9 ) China Academy of Railway Sciences, Research Re- information can now be fed back to determine constraints port on Operations Design in Beijing – Shanghai High- on section length of the LSM propulsion system by refer- speed Railway, (2001). ence to Tables 1 – 3.

Series C, Vol. 47, No. 2, 2004 JSME International Journal