Furthermore, on an operational level, when a disruption to the planned timetable occurs, one must reallocate the tracks in such a way that the negative impact of the disruption is minimized. Layover time is the time between the scheduled arrival and departure of a vehicle at a transit terminal station. The layover time at a terminal with tail tracks consists of several time elements, which are illustrated in Figure 3 , and the detail explanation of these elements was introduced by van Oort and van Nes [ 5 ].
Adequate buffer time at a terminal is always needed to absorb delays due to variability of train arrivals and terminal processes. This helps provide reliability of train departure times. However, too much buffer time reduces productivity, constrains terminal capacity, and may result in terminal congestion.
Theoretical capacity of a terminal is defined as [ 1 ]: The capacity of terminals and reversing points has a direct influence on the propagation of delays throughout the line, because the layover time scheduled at a terminal helps put trains back into sequence or on schedule.
We extended their model by adding crossovers or block sections to form multiple parallel tracks in a segment of high service demand. In this paper we denote a segment as a collection of one or multiple tracks between two points where a point merely is the connection between two segments. A track can only be occupied by one train at a time and the track can be either unidirectional i. That means, if two trains use the same track within a segment and thereby are separated by time, their paths are not considered to be in conflict, but in practice they could be.
The proposed model in this paper has handled this issue. Figure 4 shows the illustration of the terminal with two-tail tracks, and the terminal is divided into three segments. Segment 1 represents the platform tracks A and B , segment 2 represents double crossovers D and two main tracks C and E , and segment 3 represents tail tracks G and F. Segments 1 and 3 are double-tracked and segment 2 is triple-tracked.
It is also important to know in which direction s the track can be operated. A bidirectional track can be used in any direction but still only by one train at a time while a unidirectional only in one specific direction. Therefore, according to the operations practice of turning back, the directional rule of each track at all segments is shown in Figure 4. The temporary resource request by a train to occupy a segment is hereafter referred to as an event.
An event has initial start and end times for a track within the segment but then needs to be updated with new start and end times and possibly track name during operations. Figures 5 a and 5 b illustrate the platform and tail track occupation diagram and event allocation diagram. The event allocation diagram shows the resources for segments 1—3 and how the segments are allocated to the events. Each box represents a scheduled event and is numbered with respect to the train the event is associated with. Figure 6 shows the possible sequence of events of each train trip.
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For inbound trains, the events must start at track A in segment 1, and then select one track track C or track D in segment 2. If one train selects track C in segment 2, it must select track F in segment 3; otherwise it must select track D in segment 2 and track G in segment 3. Then the first event of the connecting trip, which is linked to the same TU, must select the same track in segment 3.
If an outbound train selects track F, it must select track D and track B subsequently, or if it selects track G, then it must select track E and track B.
The time taken to traverse segments 1 and 2 must equal the minimum required traversing time, which means that none of the trains can add extra stop time in segment 1 and segment 2. Every transit unit TU has six events and must follow a fixed sequence of segments. In modeling, we define two main train trips: In Figure 5 there are 8 train trips and therefore 4 TUs, for example, trip and trip are connected by TU01, and trip must share the same track with trip at segment 3. The conflicts are detected whenever different events try to occupy the same track at the same time, and these events require a gap time to separate them i.
The length required for the gap time is different depending on the moving direction of the conflicting events meeting at opposite directions or following at a same direction. Figure 7 a shows an example of events conflicting at track F, where train 2 arriving at track F must ensure that train 1 has left track F.
Figure 7 b shows an example of events conflicting at track D crossover , where train 4 arriving at track F through track D must ensure that train 3 has left track D. In this section, a Mixed integer programming MIP optimization model is developed for the problem of terminal capacity assessment with delay management. It is noted that the time units of following parameters and decision variables are all in seconds. The sets below are used for the mathematical model: Accordingly, the ordered set of events for TU01 in Figure 5 can be expressed as , ,.
The model uses the following parameters, which are assumed to be in integer values: The following decision variables are defined in the model: We consider objectives in the view of the following two aspects. Constraint 3 defines the start time of first event to be. Constraint 4 specifies that each train event is directly succeeded by the next one in the ordered set of events for the train trip. Constraint 5 defines the entering sequence of all arrival trains.
Constraint 6 ensures that the first entering train must depart first; namely, all the trains must obey the first in first out rule. At the segments that train can wait a long time; actual occupation time should be no less than the minimum occupation time; otherwise, actual occupation time must be equal to the minimum occupation time. These facts are depicted in constraints 7 and 8. Constraints 9 and 10 specify identical gaps between all arrival or departure trains at platform tracks. Constraint 11 enforces that every event must use exactly one track per relevant segment.
Constraints 12 and 13 ensure that the entering and departing trains at segment 1 must select track 1 and track 2, respectively. Constraint 14 ensures that if a train enters segment 3, it must select the same track ID of the foregoing event.
Constraints 15 and 16 ensure that if a train departs from track G, it must select track E, and if a train departs from track F, it must select track D. Constraints 17 through 20 specify that one event at a segment must end with the elapse of a required gap time before a next event may start on the same segment, if the events are using the same track of the segment. The length of the gap time depends on whether the conflicting trains meet or follow on each other. Four strategies, strategies 1 through 4, are considered in this study regarding the rule of turnbacking and delay management.
Strategies 1 and 2 are applied to quantify the maximal turnback capacity of the terminals with different operating methods, and strategies 3 and 4 are mainly applied to assess the delay propagation by different ways of tail track allocation. This strategy allows trains to select one fixed tail track. The strategy adopts a formulation including objective function 1 and constraints 3 though 20 in addition to the following constraint: This strategy allows trains to select any one of the tail tracks.
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Keith has over 40 years of experience in several senior manager and director roles across the UK rail industry and considerable overseas experience as a consultant. I am passionate about creating people focussed business strategies to drive operational results and I operate effectively in both the programme and business as usual environments.
Having joined the rail industry as a conductor, Mike brings first-hand knowledge and experience of a front-line operations and customer service role.