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Design of single-track rail line for high-speed trains
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Rcr.-A Vol. 21A. NO. 1. Pp. 41-57,1987 Rintedin GreatBritain.
Trmpn.
DESIGN
OF A SINGLE-TRACK RAIL LINE FOR HIGH-SPEED TRAINSt E. R.
PETERSEN
and A. J.
TAYLOR
School of Business, Queen’s University, Kingston, Ontario, Canada (Received 7 March 1984; in revised form 20 May 1986) Abstmet-This paper presents a method for designing a single-track rail line for a reliable high-speed passenger train service. We first consider deterministic train performance and describe a general method for finding the best location and length of the passing tracks. The design is then modified to include slack, which is necessary to ensure on-time performance for trains encountering unexpected delay. An analysis of the robustness of the system to small delays is presented. Robustness to large train delays and slower traffic is incorporated through the provision of additional sidings. Finally, simulation results are presented which compare the performance of a single-track line to a fully double-tracked equivalent.
which illustrate and measure the robustness of a rail system, and compare the performance of single- and double-track configurations.
INTRODUCXION
In the early 198Os, VIA Rail, the Canadian passenger rail company, was considering a new high-speed
rail service for the heavily populated Eastern corridor, beginning with a link between the two major population centres, Toronto and Montreal. Part of this service would use existing double trackage, but part would require the construction of a new track. In most passenger services in the world, high-speed trains run on double track. This provides maximum flexibility for the train dispatchers and permits minimum interference between trains in opposite directions on any line. However, constructing double track versus single raises the capital cost by about C$6OO,OO0per kilometer of tine and VIA was interested in approximately 357 km of new line. The research question is, could a properly designed new track consisting of single-tracked rail line with passing sidings provide reliable high-speed service? If so, where should the sidings be placed, and how long should they be? A principal criterion for any such design is good on-time performance by the passenger trains, even when unanticipated delays are encountered. This paper presents a method for configuring track which focusses on the need for reliable service. We begin by assuming ideal train performance and describe a procedure for determining the ‘location and length of passing sidings. We then show that slack must be designed into the system to permit absorption of delays and maintenance of on-time performance, whether the line is single- or double-tracked. Finally, we present a method to permit a singletrackedsystemtocopewithlargetraindelaysorequip ment failures without causing deterioration in the performance of the remaining trains. We conclude with simulation demonstrations
TRACK CONFIGUKATXrNFOR IDKALJKKDTRAM8
In this section we assume a single train type with known cruising speed and acceleration characteristics. We further assume that no train will encounter any unanticipated ‘delays. Trains in both directions originate and terminate at terminals at the ends of the line, though intermediate station stops are allowed. The terminals are joined with a single-track line, with passing sidings to permit trains in opposite directions to pass one another. Location of pawing sidings For our ideal line, we wish to locate sidings to permit opposing trains to bypass each other, or meet, so that neither has to come to a full stop (called a “flying meet”). Assume that trains depart from both ends of the line at regular intervals h (for headway). Then the time between meets for any given train will in general be h/2. If the total transit time for a train in one direction (including all required stops and delays en route) is W, then the minimum number of meets it will encounter is M = int[2Wlh], where int[.] refers to the expression truncated to the next lowest integer. Figure 1 illustrates a time-distance (or “string”) diagram for a system in which W is about 413 of h, so M = 2. (Note that we could have two or three meets here for each train-normally we are interested in the minimum possible number.) Observe that if 2W is not an integer multiple of h, one has a certain amount of leeway within which to specify the relationship between train departures from either end of the line. In Fig. 1, given the times for the downbound trains, the upbound train schedule could be shifted within an interval (shown by the dashed lines for a single train j; each upbound tram would be shifted a corresponding amount) and still
tThe authors gratefully acknowledge the support of the Natural Science and Engineering Research Council of Canada and VIA Rail Canada Incorporated for their support of this study. 47
E. R.
48
F%IERSEN
and A. J. TAYLOR
DISTANCE
w
Lz&iG TIME
Fig. 1. Typical timc-distancc diagram for a rail line.
have two meets per train. The total leeway in the system is the sum of the gaps for any given train j between the arrival time of the last opposing train and j’s departure time and between the arrival time of j and the next opposing train’s departure time. This leeway, which is tk = (M + 1)h - 2W, can be allocated to either end of the line in any proportion by appropriate placing of the passing sidings and corresponding adjustment of the schedules to cause meets to take place exactly at the sidings. Observe in Fig. 1 that as the departure time of train j is shifted within the allowable leeway while still maintaining two meets, the location of the passing sidings will shift accordingly. To generalize this observation,
Fig. 2 plots distance along the line in transit time units on the vertical axis, and headway on the horizontal. To interpret the figure, observe for example that if the headway satisfies WI2 C h < 2Wl3, there will be at least three meets. The shaded areas on the figure show the possible siding locations yielding three meets as the departure time is varied over the permissible leeway, as in Fig. 1. Of course, once one passing siding is fixed, the location of the others will also then be established as well (there is one degree of freedom). The dashed lines in Fig. 2 give the siding location which allocates the leeway equally to both ends of the line. We draw your attention to the intimate relation-
W
Fig. 2. Locating passing tracks.
Hudwayn~Fmctim of Trmsit iii
49
Single-track rail tine for high-speed trains ship between the train interdeparture time h, the transit time W and the siding location. Given that the passing track has been set up for a particular train speed and frequency, if train frequency rises, then train speed must also rise for meets to occur in the same locations. Conversely, longer intertrain intervals imply slower speeds to have the meets occur at the same location. To illustrate, suppose we have an overall distance of 350 km, train speeds of 200 kmh and an intertrain headway of 1 hr. Then W is approximately 1.75 (apart from unavoidable delays we discuss later), and M = 3. If we set up a passing siding system for these parameters with sidings every 100 km, and later it was desired to alter the train frequency to 1 every 50 min, then speed would have to increase to 240 kmh to execute the meets correctly. The above discussion, and especially Fig. 2, assumed that trains travel at the same speed in each direction. This can be generalized to trains with unequal speeds or differing delays at meet points. For the minimum meet schedule, the sidings should be placed so that the sum of the times for an outbound and an inbound train to travel from the centre of one meet point to the centre of the next is equal to the headway (we will continue to assume equal headways in both directions as a natural requirement for vehicle balancing for such a train system). Thus, the track may be “tuned” with respect to siding location for any combination of required stops, differential speeds and any additional timetable slack which may be desired. Length of passing sidings #en trains in opposing directions meet, one is forced to move onto the passing siding, decelerating first to the maximum speed permissible over the switch (or “turnout speed”), while the other proceeds through with no delay. In keeping with our idealized perspective, we consider here only flying meets, so that the train taking the siding keeps moving if at all possible. Train acceleration and deceleration performance
DISTANCE
I
play a critical role in this analysis. In Petersen and Taylor (1982b), we show that an assumption of uniform acceleration and deceleration yields a good approximation to actual train performance for highspeed passenger trains, which implies we need only a single parameter to describe each. We choose the clasped time to reach cruising speed from a standing stark, T,, and the elapsed time to stop from cruising speed, T,, for this purpose. In this analysis, let s be the cruising speed, t be the turnout speed and p the length of the passing track. We assume that the minimum time gap between opposing trains on the same stretch of track (called the “safety headway”) is S. Figure 3 illustrates a flying meet with train i taking the siding and traversing it at the turnout speed t while train j continues on the main line at speed s. Simple geometry yields the minimum track length to execute a perfect flying meet as p0 = Z/[(l/s)
where the units of each quantity are assumed commensurable. The delay to the train taking the passing track is the time lost transiting or waiting on the track plus the turnout delay (the delay caused by the need to decelerate from s to t to enter the siding, and reaccelerate back to s after leaving it). Simple equations of motion yield the turnout delay of T(t) = OS(T,
+ T,)[l
- (t/s)]‘.
If the track is shorter than pO, then the train must decelerate beyond t. Note that pI, the distance required to uniformly decelerate from f to 0 and reaccelerate back to t, can be written p, = [(TA + T,)t*]/2s. Then the meet delay for any passing track length L (again assuming that the siding train does not exceed
i
I
TIME
Fig. 3. Representatiye flying meet. Ror)21:1-D
+ (l/r)],
E. R. PETERSEN and A. J.
50
speed t on the passing track) can he written
TAnoR
length of the passing track is T,(L)
D(L) = 2s - 2Lls
= 2Ll(u,
= Lls + T(r),
+ T([2FLI(TA + T,)]"Z},0 < L CP,, = 2.5- 2Lls + T(r),
PI
= L(l/f
L >PO.
-
l/s) + T{f},
If we permit the train on the passing track to accelerate on entering the siding and then decelerate for the turnout at the far end, we will decrease the delay caused by the meet. At the same time, however, the minimum length of passing track required for a flying meet to occur is increased. Note first that only if sidings are longer than p. is there any reason to speed up. If the siding is longer than this, the train will accelerate for TA/(TA + TD) of the length of the passing track and decelerate over the remaining portion, unless it has reached the cruise speed s in the meantime. For this latter case, to reach s on the passing siding requires that the length be greater than p2, where p2
=
[(TA + T&?s][s2
- t2].
If L is less thanp,, then the maximum speed achieved is uo = [t’ + 2sLI(T,
+ T,)]“2.
Then the time required for the train to traverse the
0
2s = L,ls
L > p2.
+ Tp(L,).
The meet delay is then D(L)
= 2s - 2Lls + T(t),
L < Lm
= T,(L) - Lls + T(r),
L > L,.
Observe that for L > p2, the meet delay is constant at 2T{r}. Figure 4 demonstrates the above delays for a highperformance passenger train, illustrating the effect of passing track length on meet delays for a number of turnout speeds. The solid lines correspond to the train on the siding maintaining the turnout speed across the siding, and the dashed lines correspond to permitting the siding train to accelerate and decelerate on the passing track. Demonstration To illustrate the procedure with realistic data, we examine a dedicated passenger rail link from Montreal in the East through Ottawa and Kingston to
I
1 L&H
L < p2,
Again we can find the minimum length of passing track, L,,,, which permits a running meet with the train on the siding getting maximum benefit from acceleration, from solving
6--
0-l
+ t).
oF3RasspNG
TF&ic,
Fig. 4. Minimum flying meet delay.
En
51
Single-track rail line for high-speed trains
three flying meets, then, the average journey time for all trains, W, will be 117.95 + 1.24(3/2) =
Table 1. Train performance data Free running time Initial acceleration delay Final deceleration Intermediate stop accel./decel. Intermediate stoi, dwell times Transit time with no meets
lti.20 1.50 0.75 4.50 4.00 117.95
nlin min min min min min
Belleville at the Western end (where traffic would enter the existing double track system to Toronto). The line is 357.2 km long, which at 200 kmh yields a free running transit time of 107.2 minutes. The elapsed time to reach cruise speed is 3 minutes, and to stop from cruise speed is 1.5 minutes. Two intermediate station stops are planned, at Ottawa (141.7 km from Montreal) and Kingston (285.9 km from Montreal), with a station wait (dwell time) of 2 minutes at each. The train performance data are shown m Table. 1. Assuming a headway h (interdeparture time) of 60 minutes between trains in the same direction, then h is approximately l/2 of the transit time W. Figure i? shows that we could, in principle, operate the line with three meets, but this would permit almost no leeway in the schedule and meet delays would have to be very short to permit feasibility. As this is unrealistic, we will plan to have four meets, providing about 60 minutes’ leeway. One of the meets can be located at one of the intermediate station stops, so that only three passings sidings are required for the four meets. (It is impossible given the train performance and headway assumptions to have meets at both stops.) Assuming a turnout speed of 95 kmh, the minimum passing track length is 4 km. To allow for minor variations In train performance, we shall use 5 km passing tracks and assume that trains partially accelerate on the passing track, which gives an average delay to the train on the passing track of 1.2 minutes. With
119.81 minutes. We wish to locate the passing tracks so that the sum of the times required for the eastbound and the westbound train to pass from centre of each meet (the “loop” time) is equal to the headway time of 60 minutes. Assuming the Ottawa stop is one of the four meet points, we can then calculate the position of the other passing tracks as shown in Table 2. The first column in the table gives the distances from Montreal to the intermediate stations, Ottawa and Kingston, and the connecting point to the existing double track at Belleville. The next two columns tabulate the delays encountered by an eastbound (E/ B) train to Montreal and a westbound (W/B) train from Montreal to the centre of each passing track. Thus the W/B train is delayed 1.5 minutes accelerating to speed, 0.62 minutes decelerating to turnout speed and transiting to the midpoint, 0.62 minutes to decelerate and turn-out again and regain cruising speed, 1.75 minutes to the midpoint at Ottawa (0.75 minute’s deceleration time plus half the dwell time), and so on down the tine. As we mentioned, the loop time, or sum of transit times in both directions to go from centre to centre of meet points, must be equal to the headway of 60 minutes. Then, for example, the time for the trains to transit the distance between the first meet point and Ottawa (and be stopped in Ottawa) must equal 60 minus (0.62 + 1.75 + 2.5 + 0), or 55.13 minutes. At 200 kmh in each direction, this corresponds to a time of 27.57 minutes each, or 91.9 km. This, therefore, is the distance between Ottawa and the centre of the meet point between Ottawa and Montreal. We continue the analysis for the other meet points in an analogous fashion, locating the centre of each passing track and yielding the mileage column in Table 2. The relative times for the trains in
Table 2. Location of passing tracks with slack Delay to Lccation
Westbound
Montreal suburban Fit meet
1.5 0.62 0.62
Eastbound
Loop transit time
Length of lag at 200 km/h
’
Train times Mileage (km)
Westbound
0
0
94.8
47.0
47.0
123.15
- 29.63
Eastbound
53.15 53.13
88.55
53.13
88.55
46.26
77.1
Ottawa 141.7 Third meet
Kingston Fourth meet Belleville
! 1.7s 4.5 0 0 0.75 16.2
Headway slack: Montreal-25
0.62 0.62 4.5 1.7s 0.62 0.62 1.5 18.0
minutes; Belleville-26.4
230.25 285.9
307.35 357.2
minutes.
4 8 12 15 4 8 12 15 4 8 12 15 4 8 12 15
Depart delay
12 11 11 11 12 11 12 12 12 11 12 10 12 11 12 10
delayed trams 10 27 65 82 0 5 12 12 0 14 57 64 0 13 53 64
late delayed trams 0 5 11 11 0 5 12 12 0 5 12 10 0 5 12 10
No.
other late trains
No.
3.3 2.92 4.1 6.2 1.5 2.4 3.5 5.1 0 2.02 3.19 4.91 0 0.85 2.84 4.46
Mean late arrival 33.3 93.4 311.0 593.9 15.4 67.8 275.2 499.2 0 38.4 220.2 363.5 0 15.4 184.3 330.2
Total lateness 1.06 2.57 3.60 0.32 0.77 1.91 2.77 0 0.44 1.52 2.42 0 0.17 1.28 2.20
0.70
System response ratio
18 21 18
10 25 13
18 22 16 10 19 29 22
<2 nun
23 18
9
21 16
12 8
18
9 19
17
9 22 17 17 15
:
9 5 17
6-8 min
18 15
4-6 min
10 5 22 15
2-4 in
Frequency of late trams
4
4
9 11
16
8-10 min
8
10
9 14
>lO min
tBased on four days running with 16 trains per day each way, two minutes’ schedule slack at each station, and about 10% of the trains subjected to the indicated departure delay (called “delayed” trams).
5 km sidings
Sidings
NO.
Table 3. Summarv of simulation statisticat
Single-track rail line for high-speed trains each direction is given in the last two columns of the same table. The above structure has assumed that trains perform precisely on schedule. To allow for variations in train performance, we may wish to include slack time in the schedule beyond the minimum running time requirements. Table 3 illustrates the repositioning of the sidings if we allow two minutes of this “schedule slack” for the arrivals at Ottawa and Kingston (a train would merely have an extended station stop if its slack was not needed). We reiterate that the above procedure is general and can be used to tune any line, even with trains operating at different speeds in each direction and with different delays (though we assume that all trains in a given direction operate at the same speed). ROBUSTNESS TO SMALL DELAYS The idealized system described in the previous section cannot tolerate any unexpected delays to a train if on-time performance is to be maintained. For the system to have a recovery capability, slack must be designed into the system. There are two sources of slack:
i. schedule slack: additional time built into the schedule to allow for lateness, and ii. train overspeed capacity: allowing the train to exceed normal cruising speed for a portion of its journey. If slack is available, then extra passing siding length (above that needed for the idealized design) increases the capability of the system to recover from a train delay. (That is, it lets us “use” the slack better.) However, it should be noted that if no slack is designed in, then even if the line is double-tracked there will be no capability to recover on-time performance if a train is delayed. To analyse the robustness of the system, we introduce the concept of active slack. This is the amount of delay a single train can incur and not cause any trains to be late arriving at their destination.
53
Active slack depends on! the position of the train along the line and the direction of travel. In the remainder of this section we shall analyse the active slack of a system. We will assume that the active slack is all schedule slack, since relying on train oversped capability to provide slack is not an efficient way to operate trains. Schedule slack analysis We assume that the line is divided into K legs, which are track segments between intermediate stations, and that leg k is assigned a schedule slack of w, for trains travelling in both directions. If a train arrives at an intermediate station without having used all of its slack in the preceding leg, it merely waits an additional time at the station corresponding to the unused slack time. That is, slack is not transferable between legs of the journey. The analysis of active slack will be explained using the simple three leg system illustrated in Fig. 5. The active slack, r,, for a train in the outbound direction (from A to D) is plotted as a function of train location along the line. Initially, the active slack is w,, the maximum time the inbound train can be delayed at the first passing track. Once the outbound train is past the first meet, its active slack jumps to
Min l 2w,, i wi‘I , \ i-l / which is 2w, in this example. The factor 2w, results becam the outbound train can be delayed by w, before it causes any delay to the inbound train. Once past the first scheduled stop at B, the active slack becomes
which is (w, + WJ in this example. Proceeding down the line, at the beginning
DIS
A a) lime-distance
Diagmm Fig. 5. Active slack on a rail he.
C B DISTANCE ALI)NG LINE
b) Active %ck
along Line
D
of the
E. R. PETERSEN and A. J. TAYLOR
54
m* leg the active slack is
Mn
(i w, i w), t-1
i-m
while after the next passing siding it becomes
Min
m-l 2 wi + 2w,, i wi . t-1 > ( i-l
From the form of this computation, note that if the schedule slack were the same on each leg, the active slack rises from a low point at the beginning of the line to a maximum at the middle and then declines again to a low point. This suggests that in constructing a train schedule, to achieve an even distribution of active slack across the line, the legs at the ends of the line should be assigned relatively more schedule slack than the central ones.
Extra passing track length Conventional wisdom among railroaders suggests that one way to make the schedule more robust to small variations in train performance is to have long passing tracks to prevent delays from cascading from one train to the next. The above analysis assumed the passing tracks were of minimal length required for the idealized case. Consider now what happens with extra passing track length. This is shown in Fig. 6. Let d be the extra length of passing track (in excess of that required for a flying meet. The meet point can now be shifted from M to M’, permitting the outbound trains to be delayed an additional time, t, =
d s)
where s is the cruise speed. The increase in slack is shown by the shaded areas in Fig. 6.
.W,
Passive slack Passive slack is the ability to decouple a delayed arriving train from successive departing trains in the reverse direction. The passive slack is the amount a train can be late before it affects the next departure. Passive slack is obtained by: i. headway slack, ii. double tracks at terminals. The leeway between trains, th, can be allocated (A&) to one end and ((1 - A)&) to the other end as discussed in the previous section on siding location. This passive slack can be increased by the presence of double-tracking at the ends of the line before the terminals. Simulation demonstrations To study the impact of varying the passing siding length on train delay and spillover to other trains, we begin with a base case described in Table 2 with passing sidings 5 km long. The train frequency is 16 trains per day each way, with one hour headway, beginning at 7 a.m. To study the robustness of the line to train delays, we delayed the departure of 10% of the trains on a random basis by different fixed amounts of 4,8, 12 and 15 minutes. For each of the alternatives we simulate four days running, or a total of 128 trains per run. Repeating this procedure for siding lengths of 5, 8, 12 and 15 km yielded the results summarized in Table 3. In this table, we define a quantity we call the system response ratio, which is the ratio of the total delay experienced by all trains divided by the delay imposed on the line by late train departure (i.e. the number of train delays times the departure delay above). The system response ratio thus forms a good measure of how well the line responds to a small delay. The impact of siding length on the response ratio is summarized in Fig. 7. Figure 7 shows that extra siding length has a strong
DISTANCE Diagram
AWNG
LINE
b) Active Slack along Line
Fig. 6. Active slack with extra passing track.
Single-track rail line for high-speed trains
15 MINUTE
5
DELAY
15
7 %
G”BL:‘TRACP
Fig. 7. Impact of slack on system response. impact on the ability of the line to recover from a train delay (as measured by the system response ratio) for delays in the range of 2 to 8 minutes to a train. A delay of more than 8 minutes causes a strong degradation in the line performance for all siding lengths (although the larger sidings perform somewhat better, as we would anticipate). We defer pursuing this point until the next section on robustness in the large. We also observe that, although performance improves with extra passing track length, most of the benefit has been gained with about 9% of the total line double-tracked, corresponding to 9 km sidings for flying meets. Again we reiterate that this section is only concerned with the line performance for small train delays. In Fig. 8 we have plotted the average journey time for all trains (again with about 10% being delayed on departure by the amounts indicated on the graph). This graph again shows that’considering only small delays (less than 8 minutes), with about 4% extra track (or 9% double-tracking on the line as a whole), we gained most of the potential benefits from increased siding length.
ule, the oncoming trains will likewise be subjected to large delays, and the disturbances will cascade through the schedule. This section describes how to modify the approach to deal with excessive disturbances, which may be of two forms; a large delay to a single high-speed passenger train, or the need to move a slow-speed train across the system (e.g. a depowered passenger train or a work train). We first note that with the system described so far, no excess capacity exists, and any extra trains cannot use any of the passing sidings without interfering with the high-speed passenger system. Noting that a high-speed train which has incurred a large
MINUTE
DELAY
ROBUSTNBSS TO LARGE DELAYS The previous section has described the provision of active slack to permit the rail system to recover from small train delays, on the order of 2 or 3 minutes. Now we examine the issue of larger train delays. Observe first that if we insist that a train incurring a large delay must execute its meets with oncoming trains at the original prespecified sched-
5
i
6 1; (3 X DOUBLE TRACK
lb
Fig. 8. Impact of extra siding length on journey time.
56
E. R. PETERSEN and A. J. TAYLOR
delay has the same characteristics as an extra train to be dispatched, we need some way to augment the system capacity, in such a way that we do not interfere with the primary high-speed traffic. The strategy we propose is to incorporate an additional set of sidings, which we call secondary sidings, to be used exclusively by the delayed passenger or extra trains. The primary high-speed traffic would have absolute priority over any extra trains, which would be diverted into a secondary siding to prevent interference with the primary schedule. The design issues then relate to the performance desired for trains operating on this secondary system. For example, we may wish to move the delayed train as quickly as possible, yet cause no interference to the primary traffic. This would suggest that the secondary system consist of long passing sidings, permitting flying meets with oncoming traffic. Although the performance of such a line would be good, the capital costs for the long secondary sidings would be high. On the other hand, we might wish to only clear the extra train from the system without causing interference at the lowest cost possible. This suggests short sidings in which the extra train would come to a halt, spaced so that the extra train can move along the track in the free time windows between primary high-speed train transits. Regardless of which option is chosen for the secondary system, the methods of siding location developed in Section 2 still apply. For example, choosing to stop the secondary traffic at the passing sidings which appropriate safety headways requires a system with a 1 km passing track about every 80 km. In particular, the track outlined in Table 3 requires siding at mileposts 33, 100, 179, 242 and 330 km from Montreal. If slow-speed work trains must also traverse the line, then the required additional capacity is obtained by an even closer spacing of secondary sidings. For example, to cope with a 75 kmh work train in the above system requires sidings spaced about 20 km apart. In the example, in addition to the already mentioned sidings, these are located at mileposts 16, 54, 79, 121, 142, 158, 200, 221, 264, 286, 307 and 347 km from Montreal. The performance of trains on this line is evaluated in the next section.
this comparison, the line:
we assumed the following traffic on
Passenger service: cruising speed of 200 kmh, with no overspeed 1 hour headways; 16 trains/day (no night trains) elapsed time to reach speed 3 min braking time 1.5 min schedule slack 2 min between stations 2 min station stops at Kingston and Ottawa 10% probability of train delays of 5 min at stations Express freight: cruising speed 120 kmh, 2 trains/day elapsed time to reach speed 10 min braking time 3 min
each way
Work trains: cruising speed 75 kmh, 1 train per day elapsed time to reach speed 10 min braking time 5 min Table 4 summarizes some relevant statistics from this comparison, showing that the single-track line handles the (dense) passenger traffic very well when compared with a double-tracked line. Because the freights and work train must keep clear of the passenger service on the single line, we observe that their performance is relatively poor on this line. Given the enormous capital cost-saving, however, this poor performance for the low priority traffic would appear to be acceptable. CONCLUSIONS
We have demonstrated a method for configuring a rail line which mainly carries a single class of traffic in each direction. Although we have used high-speed passenger service as our motivating example, the same precepts would apply to other lines such as monorail service, or dedicated rail lines from mines to loading points. For a given technology, a trade-off must be made between how much slack to design into the system, ensuring good on-time performance, against the in-
EVALUATION
A hypothetical line was designed using the principles outlined in this paper for the demonstration data outline in Section 2.3, which required about 50 km of passing siding for the 35 km line (or about 13% double track). A variety of simulation experiments were run using the simulation model described in Petersen and Taylor (1982a) which are too voluminous to report here, but are contained in Petersen and Taylor (1982b). Of particular interest, however, is how well this line performs when compared with a fully double-tracked equivalent line. To perform
Table 4. Comparison of single versus double track Variable Percent of passenger trains arriving late Average lateness for late passenger trains Express freight transit time Work train transit time
Single track (13% double) 16%
Double track 12.5%
4.3 min
2.6 min
6.31 hr
3.3 hr
11.2 hr
5.8 hr
Single-track rail line for high-speed trains
crease in the schedule time. As more slack is included, the scheduled transit time increases, but ontime reliability also increases. Particularly striking is the small amount of passing siding required. Various industry experts estimate that anywhere from 24% to 35% additional track is required, yet our method suggests only 13%, which has a substantial impact on the capital cost. Fundamental to the success of operating such a line, however, is the adoption of appropriate dispatch rules: reserve the primary passing sidings for the high-speed traffic, and once a high-speed train is substantially delayed, remove it from the high-priority traffic and force it onto the secondary siding sys-
57
tem (thereby causing it to accept further delays). Although this latter rule & contrary to the standard operating procedures 6f ino& dispatchers, and would be hard for them to accept, it is integral in achieving good performance with a single-tracked line of this nature. REFERENCES
Petersen E. R. and Taylor A. J. (1982a) A structured model
for rail line analysis and simulation. Transpn. Sci. 16, 192-206. Petersen E. R. and Taylor A. J. (1982b) Determinationof track capacity for high-speed passenger operations. CIGGT Technical Report Number 82-12.
Rcr.-A Vol. 21A. NO. 1. Pp. 41-57,1987 Rintedin GreatBritain.
Trmpn.
DESIGN
OF A SINGLE-TRACK RAIL LINE FOR HIGH-SPEED TRAINSt E. R.
PETERSEN
and A. J.
TAYLOR
School of Business, Queen’s University, Kingston, Ontario, Canada (Received 7 March 1984; in revised form 20 May 1986) Abstmet-This paper presents a method for designing a single-track rail line for a reliable high-speed passenger train service. We first consider deterministic train performance and describe a general method for finding the best location and length of the passing tracks. The design is then modified to include slack, which is necessary to ensure on-time performance for trains encountering unexpected delay. An analysis of the robustness of the system to small delays is presented. Robustness to large train delays and slower traffic is incorporated through the provision of additional sidings. Finally, simulation results are presented which compare the performance of a single-track line to a fully double-tracked equivalent.
which illustrate and measure the robustness of a rail system, and compare the performance of single- and double-track configurations.
INTRODUCXION
In the early 198Os, VIA Rail, the Canadian passenger rail company, was considering a new high-speed
rail service for the heavily populated Eastern corridor, beginning with a link between the two major population centres, Toronto and Montreal. Part of this service would use existing double trackage, but part would require the construction of a new track. In most passenger services in the world, high-speed trains run on double track. This provides maximum flexibility for the train dispatchers and permits minimum interference between trains in opposite directions on any line. However, constructing double track versus single raises the capital cost by about C$6OO,OO0per kilometer of tine and VIA was interested in approximately 357 km of new line. The research question is, could a properly designed new track consisting of single-tracked rail line with passing sidings provide reliable high-speed service? If so, where should the sidings be placed, and how long should they be? A principal criterion for any such design is good on-time performance by the passenger trains, even when unanticipated delays are encountered. This paper presents a method for configuring track which focusses on the need for reliable service. We begin by assuming ideal train performance and describe a procedure for determining the ‘location and length of passing sidings. We then show that slack must be designed into the system to permit absorption of delays and maintenance of on-time performance, whether the line is single- or double-tracked. Finally, we present a method to permit a singletrackedsystemtocopewithlargetraindelaysorequip ment failures without causing deterioration in the performance of the remaining trains. We conclude with simulation demonstrations
TRACK CONFIGUKATXrNFOR IDKALJKKDTRAM8
In this section we assume a single train type with known cruising speed and acceleration characteristics. We further assume that no train will encounter any unanticipated ‘delays. Trains in both directions originate and terminate at terminals at the ends of the line, though intermediate station stops are allowed. The terminals are joined with a single-track line, with passing sidings to permit trains in opposite directions to pass one another. Location of pawing sidings For our ideal line, we wish to locate sidings to permit opposing trains to bypass each other, or meet, so that neither has to come to a full stop (called a “flying meet”). Assume that trains depart from both ends of the line at regular intervals h (for headway). Then the time between meets for any given train will in general be h/2. If the total transit time for a train in one direction (including all required stops and delays en route) is W, then the minimum number of meets it will encounter is M = int[2Wlh], where int[.] refers to the expression truncated to the next lowest integer. Figure 1 illustrates a time-distance (or “string”) diagram for a system in which W is about 413 of h, so M = 2. (Note that we could have two or three meets here for each train-normally we are interested in the minimum possible number.) Observe that if 2W is not an integer multiple of h, one has a certain amount of leeway within which to specify the relationship between train departures from either end of the line. In Fig. 1, given the times for the downbound trains, the upbound train schedule could be shifted within an interval (shown by the dashed lines for a single train j; each upbound tram would be shifted a corresponding amount) and still
tThe authors gratefully acknowledge the support of the Natural Science and Engineering Research Council of Canada and VIA Rail Canada Incorporated for their support of this study. 47
E. R.
48
F%IERSEN
and A. J. TAYLOR
DISTANCE
w
Lz&iG TIME
Fig. 1. Typical timc-distancc diagram for a rail line.
have two meets per train. The total leeway in the system is the sum of the gaps for any given train j between the arrival time of the last opposing train and j’s departure time and between the arrival time of j and the next opposing train’s departure time. This leeway, which is tk = (M + 1)h - 2W, can be allocated to either end of the line in any proportion by appropriate placing of the passing sidings and corresponding adjustment of the schedules to cause meets to take place exactly at the sidings. Observe in Fig. 1 that as the departure time of train j is shifted within the allowable leeway while still maintaining two meets, the location of the passing sidings will shift accordingly. To generalize this observation,
Fig. 2 plots distance along the line in transit time units on the vertical axis, and headway on the horizontal. To interpret the figure, observe for example that if the headway satisfies WI2 C h < 2Wl3, there will be at least three meets. The shaded areas on the figure show the possible siding locations yielding three meets as the departure time is varied over the permissible leeway, as in Fig. 1. Of course, once one passing siding is fixed, the location of the others will also then be established as well (there is one degree of freedom). The dashed lines in Fig. 2 give the siding location which allocates the leeway equally to both ends of the line. We draw your attention to the intimate relation-
W
Fig. 2. Locating passing tracks.
Hudwayn~Fmctim of Trmsit iii
49
Single-track rail tine for high-speed trains ship between the train interdeparture time h, the transit time W and the siding location. Given that the passing track has been set up for a particular train speed and frequency, if train frequency rises, then train speed must also rise for meets to occur in the same locations. Conversely, longer intertrain intervals imply slower speeds to have the meets occur at the same location. To illustrate, suppose we have an overall distance of 350 km, train speeds of 200 kmh and an intertrain headway of 1 hr. Then W is approximately 1.75 (apart from unavoidable delays we discuss later), and M = 3. If we set up a passing siding system for these parameters with sidings every 100 km, and later it was desired to alter the train frequency to 1 every 50 min, then speed would have to increase to 240 kmh to execute the meets correctly. The above discussion, and especially Fig. 2, assumed that trains travel at the same speed in each direction. This can be generalized to trains with unequal speeds or differing delays at meet points. For the minimum meet schedule, the sidings should be placed so that the sum of the times for an outbound and an inbound train to travel from the centre of one meet point to the centre of the next is equal to the headway (we will continue to assume equal headways in both directions as a natural requirement for vehicle balancing for such a train system). Thus, the track may be “tuned” with respect to siding location for any combination of required stops, differential speeds and any additional timetable slack which may be desired. Length of passing sidings #en trains in opposing directions meet, one is forced to move onto the passing siding, decelerating first to the maximum speed permissible over the switch (or “turnout speed”), while the other proceeds through with no delay. In keeping with our idealized perspective, we consider here only flying meets, so that the train taking the siding keeps moving if at all possible. Train acceleration and deceleration performance
DISTANCE
I
play a critical role in this analysis. In Petersen and Taylor (1982b), we show that an assumption of uniform acceleration and deceleration yields a good approximation to actual train performance for highspeed passenger trains, which implies we need only a single parameter to describe each. We choose the clasped time to reach cruising speed from a standing stark, T,, and the elapsed time to stop from cruising speed, T,, for this purpose. In this analysis, let s be the cruising speed, t be the turnout speed and p the length of the passing track. We assume that the minimum time gap between opposing trains on the same stretch of track (called the “safety headway”) is S. Figure 3 illustrates a flying meet with train i taking the siding and traversing it at the turnout speed t while train j continues on the main line at speed s. Simple geometry yields the minimum track length to execute a perfect flying meet as p0 = Z/[(l/s)
where the units of each quantity are assumed commensurable. The delay to the train taking the passing track is the time lost transiting or waiting on the track plus the turnout delay (the delay caused by the need to decelerate from s to t to enter the siding, and reaccelerate back to s after leaving it). Simple equations of motion yield the turnout delay of T(t) = OS(T,
+ T,)[l
- (t/s)]‘.
If the track is shorter than pO, then the train must decelerate beyond t. Note that pI, the distance required to uniformly decelerate from f to 0 and reaccelerate back to t, can be written p, = [(TA + T,)t*]/2s. Then the meet delay for any passing track length L (again assuming that the siding train does not exceed
i
I
TIME
Fig. 3. Representatiye flying meet. Ror)21:1-D
+ (l/r)],
E. R. PETERSEN and A. J.
50
speed t on the passing track) can he written
TAnoR
length of the passing track is T,(L)
D(L) = 2s - 2Lls
= 2Ll(u,
= Lls + T(r),
+ T([2FLI(TA + T,)]"Z},0 < L CP,, = 2.5- 2Lls + T(r),
PI
= L(l/f
L >PO.
-
l/s) + T{f},
If we permit the train on the passing track to accelerate on entering the siding and then decelerate for the turnout at the far end, we will decrease the delay caused by the meet. At the same time, however, the minimum length of passing track required for a flying meet to occur is increased. Note first that only if sidings are longer than p. is there any reason to speed up. If the siding is longer than this, the train will accelerate for TA/(TA + TD) of the length of the passing track and decelerate over the remaining portion, unless it has reached the cruise speed s in the meantime. For this latter case, to reach s on the passing siding requires that the length be greater than p2, where p2
=
[(TA + T&?s][s2
- t2].
If L is less thanp,, then the maximum speed achieved is uo = [t’ + 2sLI(T,
+ T,)]“2.
Then the time required for the train to traverse the
0
2s = L,ls
L > p2.
+ Tp(L,).
The meet delay is then D(L)
= 2s - 2Lls + T(t),
L < Lm
= T,(L) - Lls + T(r),
L > L,.
Observe that for L > p2, the meet delay is constant at 2T{r}. Figure 4 demonstrates the above delays for a highperformance passenger train, illustrating the effect of passing track length on meet delays for a number of turnout speeds. The solid lines correspond to the train on the siding maintaining the turnout speed across the siding, and the dashed lines correspond to permitting the siding train to accelerate and decelerate on the passing track. Demonstration To illustrate the procedure with realistic data, we examine a dedicated passenger rail link from Montreal in the East through Ottawa and Kingston to
I
1 L&H
L < p2,
Again we can find the minimum length of passing track, L,,,, which permits a running meet with the train on the siding getting maximum benefit from acceleration, from solving
6--
0-l
+ t).
oF3RasspNG
TF&ic,
Fig. 4. Minimum flying meet delay.
En
51
Single-track rail line for high-speed trains
three flying meets, then, the average journey time for all trains, W, will be 117.95 + 1.24(3/2) =
Table 1. Train performance data Free running time Initial acceleration delay Final deceleration Intermediate stop accel./decel. Intermediate stoi, dwell times Transit time with no meets
lti.20 1.50 0.75 4.50 4.00 117.95
nlin min min min min min
Belleville at the Western end (where traffic would enter the existing double track system to Toronto). The line is 357.2 km long, which at 200 kmh yields a free running transit time of 107.2 minutes. The elapsed time to reach cruise speed is 3 minutes, and to stop from cruise speed is 1.5 minutes. Two intermediate station stops are planned, at Ottawa (141.7 km from Montreal) and Kingston (285.9 km from Montreal), with a station wait (dwell time) of 2 minutes at each. The train performance data are shown m Table. 1. Assuming a headway h (interdeparture time) of 60 minutes between trains in the same direction, then h is approximately l/2 of the transit time W. Figure i? shows that we could, in principle, operate the line with three meets, but this would permit almost no leeway in the schedule and meet delays would have to be very short to permit feasibility. As this is unrealistic, we will plan to have four meets, providing about 60 minutes’ leeway. One of the meets can be located at one of the intermediate station stops, so that only three passings sidings are required for the four meets. (It is impossible given the train performance and headway assumptions to have meets at both stops.) Assuming a turnout speed of 95 kmh, the minimum passing track length is 4 km. To allow for minor variations In train performance, we shall use 5 km passing tracks and assume that trains partially accelerate on the passing track, which gives an average delay to the train on the passing track of 1.2 minutes. With
119.81 minutes. We wish to locate the passing tracks so that the sum of the times required for the eastbound and the westbound train to pass from centre of each meet (the “loop” time) is equal to the headway time of 60 minutes. Assuming the Ottawa stop is one of the four meet points, we can then calculate the position of the other passing tracks as shown in Table 2. The first column in the table gives the distances from Montreal to the intermediate stations, Ottawa and Kingston, and the connecting point to the existing double track at Belleville. The next two columns tabulate the delays encountered by an eastbound (E/ B) train to Montreal and a westbound (W/B) train from Montreal to the centre of each passing track. Thus the W/B train is delayed 1.5 minutes accelerating to speed, 0.62 minutes decelerating to turnout speed and transiting to the midpoint, 0.62 minutes to decelerate and turn-out again and regain cruising speed, 1.75 minutes to the midpoint at Ottawa (0.75 minute’s deceleration time plus half the dwell time), and so on down the tine. As we mentioned, the loop time, or sum of transit times in both directions to go from centre to centre of meet points, must be equal to the headway of 60 minutes. Then, for example, the time for the trains to transit the distance between the first meet point and Ottawa (and be stopped in Ottawa) must equal 60 minus (0.62 + 1.75 + 2.5 + 0), or 55.13 minutes. At 200 kmh in each direction, this corresponds to a time of 27.57 minutes each, or 91.9 km. This, therefore, is the distance between Ottawa and the centre of the meet point between Ottawa and Montreal. We continue the analysis for the other meet points in an analogous fashion, locating the centre of each passing track and yielding the mileage column in Table 2. The relative times for the trains in
Table 2. Location of passing tracks with slack Delay to Lccation
Westbound
Montreal suburban Fit meet
1.5 0.62 0.62
Eastbound
Loop transit time
Length of lag at 200 km/h
’
Train times Mileage (km)
Westbound
0
0
94.8
47.0
47.0
123.15
- 29.63
Eastbound
53.15 53.13
88.55
53.13
88.55
46.26
77.1
Ottawa 141.7 Third meet
Kingston Fourth meet Belleville
! 1.7s 4.5 0 0 0.75 16.2
Headway slack: Montreal-25
0.62 0.62 4.5 1.7s 0.62 0.62 1.5 18.0
minutes; Belleville-26.4
230.25 285.9
307.35 357.2
minutes.
4 8 12 15 4 8 12 15 4 8 12 15 4 8 12 15
Depart delay
12 11 11 11 12 11 12 12 12 11 12 10 12 11 12 10
delayed trams 10 27 65 82 0 5 12 12 0 14 57 64 0 13 53 64
late delayed trams 0 5 11 11 0 5 12 12 0 5 12 10 0 5 12 10
No.
other late trains
No.
3.3 2.92 4.1 6.2 1.5 2.4 3.5 5.1 0 2.02 3.19 4.91 0 0.85 2.84 4.46
Mean late arrival 33.3 93.4 311.0 593.9 15.4 67.8 275.2 499.2 0 38.4 220.2 363.5 0 15.4 184.3 330.2
Total lateness 1.06 2.57 3.60 0.32 0.77 1.91 2.77 0 0.44 1.52 2.42 0 0.17 1.28 2.20
0.70
System response ratio
18 21 18
10 25 13
18 22 16 10 19 29 22
<2 nun
23 18
9
21 16
12 8
18
9 19
17
9 22 17 17 15
:
9 5 17
6-8 min
18 15
4-6 min
10 5 22 15
2-4 in
Frequency of late trams
4
4
9 11
16
8-10 min
8
10
9 14
>lO min
tBased on four days running with 16 trains per day each way, two minutes’ schedule slack at each station, and about 10% of the trains subjected to the indicated departure delay (called “delayed” trams).
5 km sidings
Sidings
NO.
Table 3. Summarv of simulation statisticat
Single-track rail line for high-speed trains each direction is given in the last two columns of the same table. The above structure has assumed that trains perform precisely on schedule. To allow for variations in train performance, we may wish to include slack time in the schedule beyond the minimum running time requirements. Table 3 illustrates the repositioning of the sidings if we allow two minutes of this “schedule slack” for the arrivals at Ottawa and Kingston (a train would merely have an extended station stop if its slack was not needed). We reiterate that the above procedure is general and can be used to tune any line, even with trains operating at different speeds in each direction and with different delays (though we assume that all trains in a given direction operate at the same speed). ROBUSTNESS TO SMALL DELAYS The idealized system described in the previous section cannot tolerate any unexpected delays to a train if on-time performance is to be maintained. For the system to have a recovery capability, slack must be designed into the system. There are two sources of slack:
i. schedule slack: additional time built into the schedule to allow for lateness, and ii. train overspeed capacity: allowing the train to exceed normal cruising speed for a portion of its journey. If slack is available, then extra passing siding length (above that needed for the idealized design) increases the capability of the system to recover from a train delay. (That is, it lets us “use” the slack better.) However, it should be noted that if no slack is designed in, then even if the line is double-tracked there will be no capability to recover on-time performance if a train is delayed. To analyse the robustness of the system, we introduce the concept of active slack. This is the amount of delay a single train can incur and not cause any trains to be late arriving at their destination.
53
Active slack depends on! the position of the train along the line and the direction of travel. In the remainder of this section we shall analyse the active slack of a system. We will assume that the active slack is all schedule slack, since relying on train oversped capability to provide slack is not an efficient way to operate trains. Schedule slack analysis We assume that the line is divided into K legs, which are track segments between intermediate stations, and that leg k is assigned a schedule slack of w, for trains travelling in both directions. If a train arrives at an intermediate station without having used all of its slack in the preceding leg, it merely waits an additional time at the station corresponding to the unused slack time. That is, slack is not transferable between legs of the journey. The analysis of active slack will be explained using the simple three leg system illustrated in Fig. 5. The active slack, r,, for a train in the outbound direction (from A to D) is plotted as a function of train location along the line. Initially, the active slack is w,, the maximum time the inbound train can be delayed at the first passing track. Once the outbound train is past the first meet, its active slack jumps to
Min l 2w,, i wi‘I , \ i-l / which is 2w, in this example. The factor 2w, results becam the outbound train can be delayed by w, before it causes any delay to the inbound train. Once past the first scheduled stop at B, the active slack becomes
which is (w, + WJ in this example. Proceeding down the line, at the beginning
DIS
A a) lime-distance
Diagmm Fig. 5. Active slack on a rail he.
C B DISTANCE ALI)NG LINE
b) Active %ck
along Line
D
of the
E. R. PETERSEN and A. J. TAYLOR
54
m* leg the active slack is
Mn
(i w, i w), t-1
i-m
while after the next passing siding it becomes
Min
m-l 2 wi + 2w,, i wi . t-1 > ( i-l
From the form of this computation, note that if the schedule slack were the same on each leg, the active slack rises from a low point at the beginning of the line to a maximum at the middle and then declines again to a low point. This suggests that in constructing a train schedule, to achieve an even distribution of active slack across the line, the legs at the ends of the line should be assigned relatively more schedule slack than the central ones.
Extra passing track length Conventional wisdom among railroaders suggests that one way to make the schedule more robust to small variations in train performance is to have long passing tracks to prevent delays from cascading from one train to the next. The above analysis assumed the passing tracks were of minimal length required for the idealized case. Consider now what happens with extra passing track length. This is shown in Fig. 6. Let d be the extra length of passing track (in excess of that required for a flying meet. The meet point can now be shifted from M to M’, permitting the outbound trains to be delayed an additional time, t, =
d s)
where s is the cruise speed. The increase in slack is shown by the shaded areas in Fig. 6.
.W,
Passive slack Passive slack is the ability to decouple a delayed arriving train from successive departing trains in the reverse direction. The passive slack is the amount a train can be late before it affects the next departure. Passive slack is obtained by: i. headway slack, ii. double tracks at terminals. The leeway between trains, th, can be allocated (A&) to one end and ((1 - A)&) to the other end as discussed in the previous section on siding location. This passive slack can be increased by the presence of double-tracking at the ends of the line before the terminals. Simulation demonstrations To study the impact of varying the passing siding length on train delay and spillover to other trains, we begin with a base case described in Table 2 with passing sidings 5 km long. The train frequency is 16 trains per day each way, with one hour headway, beginning at 7 a.m. To study the robustness of the line to train delays, we delayed the departure of 10% of the trains on a random basis by different fixed amounts of 4,8, 12 and 15 minutes. For each of the alternatives we simulate four days running, or a total of 128 trains per run. Repeating this procedure for siding lengths of 5, 8, 12 and 15 km yielded the results summarized in Table 3. In this table, we define a quantity we call the system response ratio, which is the ratio of the total delay experienced by all trains divided by the delay imposed on the line by late train departure (i.e. the number of train delays times the departure delay above). The system response ratio thus forms a good measure of how well the line responds to a small delay. The impact of siding length on the response ratio is summarized in Fig. 7. Figure 7 shows that extra siding length has a strong
DISTANCE Diagram
AWNG
LINE
b) Active Slack along Line
Fig. 6. Active slack with extra passing track.
Single-track rail line for high-speed trains
15 MINUTE
5
DELAY
15
7 %
G”BL:‘TRACP
Fig. 7. Impact of slack on system response. impact on the ability of the line to recover from a train delay (as measured by the system response ratio) for delays in the range of 2 to 8 minutes to a train. A delay of more than 8 minutes causes a strong degradation in the line performance for all siding lengths (although the larger sidings perform somewhat better, as we would anticipate). We defer pursuing this point until the next section on robustness in the large. We also observe that, although performance improves with extra passing track length, most of the benefit has been gained with about 9% of the total line double-tracked, corresponding to 9 km sidings for flying meets. Again we reiterate that this section is only concerned with the line performance for small train delays. In Fig. 8 we have plotted the average journey time for all trains (again with about 10% being delayed on departure by the amounts indicated on the graph). This graph again shows that’considering only small delays (less than 8 minutes), with about 4% extra track (or 9% double-tracking on the line as a whole), we gained most of the potential benefits from increased siding length.
ule, the oncoming trains will likewise be subjected to large delays, and the disturbances will cascade through the schedule. This section describes how to modify the approach to deal with excessive disturbances, which may be of two forms; a large delay to a single high-speed passenger train, or the need to move a slow-speed train across the system (e.g. a depowered passenger train or a work train). We first note that with the system described so far, no excess capacity exists, and any extra trains cannot use any of the passing sidings without interfering with the high-speed passenger system. Noting that a high-speed train which has incurred a large
MINUTE
DELAY
ROBUSTNBSS TO LARGE DELAYS The previous section has described the provision of active slack to permit the rail system to recover from small train delays, on the order of 2 or 3 minutes. Now we examine the issue of larger train delays. Observe first that if we insist that a train incurring a large delay must execute its meets with oncoming trains at the original prespecified sched-
5
i
6 1; (3 X DOUBLE TRACK
lb
Fig. 8. Impact of extra siding length on journey time.
56
E. R. PETERSEN and A. J. TAYLOR
delay has the same characteristics as an extra train to be dispatched, we need some way to augment the system capacity, in such a way that we do not interfere with the primary high-speed traffic. The strategy we propose is to incorporate an additional set of sidings, which we call secondary sidings, to be used exclusively by the delayed passenger or extra trains. The primary high-speed traffic would have absolute priority over any extra trains, which would be diverted into a secondary siding to prevent interference with the primary schedule. The design issues then relate to the performance desired for trains operating on this secondary system. For example, we may wish to move the delayed train as quickly as possible, yet cause no interference to the primary traffic. This would suggest that the secondary system consist of long passing sidings, permitting flying meets with oncoming traffic. Although the performance of such a line would be good, the capital costs for the long secondary sidings would be high. On the other hand, we might wish to only clear the extra train from the system without causing interference at the lowest cost possible. This suggests short sidings in which the extra train would come to a halt, spaced so that the extra train can move along the track in the free time windows between primary high-speed train transits. Regardless of which option is chosen for the secondary system, the methods of siding location developed in Section 2 still apply. For example, choosing to stop the secondary traffic at the passing sidings which appropriate safety headways requires a system with a 1 km passing track about every 80 km. In particular, the track outlined in Table 3 requires siding at mileposts 33, 100, 179, 242 and 330 km from Montreal. If slow-speed work trains must also traverse the line, then the required additional capacity is obtained by an even closer spacing of secondary sidings. For example, to cope with a 75 kmh work train in the above system requires sidings spaced about 20 km apart. In the example, in addition to the already mentioned sidings, these are located at mileposts 16, 54, 79, 121, 142, 158, 200, 221, 264, 286, 307 and 347 km from Montreal. The performance of trains on this line is evaluated in the next section.
this comparison, the line:
we assumed the following traffic on
Passenger service: cruising speed of 200 kmh, with no overspeed 1 hour headways; 16 trains/day (no night trains) elapsed time to reach speed 3 min braking time 1.5 min schedule slack 2 min between stations 2 min station stops at Kingston and Ottawa 10% probability of train delays of 5 min at stations Express freight: cruising speed 120 kmh, 2 trains/day elapsed time to reach speed 10 min braking time 3 min
each way
Work trains: cruising speed 75 kmh, 1 train per day elapsed time to reach speed 10 min braking time 5 min Table 4 summarizes some relevant statistics from this comparison, showing that the single-track line handles the (dense) passenger traffic very well when compared with a double-tracked line. Because the freights and work train must keep clear of the passenger service on the single line, we observe that their performance is relatively poor on this line. Given the enormous capital cost-saving, however, this poor performance for the low priority traffic would appear to be acceptable. CONCLUSIONS
We have demonstrated a method for configuring a rail line which mainly carries a single class of traffic in each direction. Although we have used high-speed passenger service as our motivating example, the same precepts would apply to other lines such as monorail service, or dedicated rail lines from mines to loading points. For a given technology, a trade-off must be made between how much slack to design into the system, ensuring good on-time performance, against the in-
EVALUATION
A hypothetical line was designed using the principles outlined in this paper for the demonstration data outline in Section 2.3, which required about 50 km of passing siding for the 35 km line (or about 13% double track). A variety of simulation experiments were run using the simulation model described in Petersen and Taylor (1982a) which are too voluminous to report here, but are contained in Petersen and Taylor (1982b). Of particular interest, however, is how well this line performs when compared with a fully double-tracked equivalent line. To perform
Table 4. Comparison of single versus double track Variable Percent of passenger trains arriving late Average lateness for late passenger trains Express freight transit time Work train transit time
Single track (13% double) 16%
Double track 12.5%
4.3 min
2.6 min
6.31 hr
3.3 hr
11.2 hr
5.8 hr
Single-track rail line for high-speed trains
crease in the schedule time. As more slack is included, the scheduled transit time increases, but ontime reliability also increases. Particularly striking is the small amount of passing siding required. Various industry experts estimate that anywhere from 24% to 35% additional track is required, yet our method suggests only 13%, which has a substantial impact on the capital cost. Fundamental to the success of operating such a line, however, is the adoption of appropriate dispatch rules: reserve the primary passing sidings for the high-speed traffic, and once a high-speed train is substantially delayed, remove it from the high-priority traffic and force it onto the secondary siding sys-
57
tem (thereby causing it to accept further delays). Although this latter rule & contrary to the standard operating procedures 6f ino& dispatchers, and would be hard for them to accept, it is integral in achieving good performance with a single-tracked line of this nature. REFERENCES
Petersen E. R. and Taylor A. J. (1982a) A structured model
for rail line analysis and simulation. Transpn. Sci. 16, 192-206. Petersen E. R. and Taylor A. J. (1982b) Determinationof track capacity for high-speed passenger operations. CIGGT Technical Report Number 82-12.