Based on the analysis of safety constraints, we further analyze the calculations for train delays under different path choices. As show in
Figure 5(a1,b1), at time
t, inbound train
i encounters a contingency and begins to slow down to a constant speed that below normal speed on the inbound track in segment
BC, and the track where train
i decelerates is highlighted with dash line in red color in
Figure 5. Meanwhile, the rest of trains (i.e., train
k and train
i + 1) travel normally on their dedicated tracks. At time
t, neither train
i + 1 nor train
k has passed node
B or node
C. Moreover, the outbound track in segment
BC is not occupied by any other trains. As time goes on, if inbound train
i + 1 chooses to travel on dedicated track in segment
BC, case 1 showed in
Figure 5a will be obtained; otherwise, we will obtain case 2 showed in
Figure 5b, where train
i + 1 uses the reverse direction track to overtake decelerating train
i in segment
BC.
In case 1, in order to meet C3, train
i + 1 may need to reduce its speed if train
i is delayed greatly by contingencies. The dash line in blue color in
Figure 5a highlights the track where train
i + 1 is delayed by train
i. In case 2, to satisfy C1 and C2, train
k and train
i may be delayed by train
i + 1.
Figure 6,
Figure 7 and
Figure 8 respectively illustrate the possible delay of each train in case 1 and case 2, where corresponding timetables are also presented. The dash lines in the timetables presented in
Figure 6,
Figure 7 and
Figure 8 denote the ideal trajectory of trains. In this paper, a train is called in the state of obstacle-travel situation when it has to decelerate due to potential train conflicts; otherwise, a train in free-travel situation is such a train that travels without potential conflicts with any other trains. Here, let
${Z}_{k}^{o}$ and
${Z}_{k}^{f}$ be the time duration
${d}_{k,B}-t$ for train
k in obstacle-travel situation and free-travel situation, respectively. Then, if train
i + 1 conflicts with train
k, train
k’s delay
${Z}_{k,i+1}$ caused by train
i + 1 equals to
${Z}_{k}^{o}-{Z}_{k}^{f}$. The same calculation also goes for train
i and train
i + 1.
Actually, the train delay in this paper can be categorized into two types: one is the train delay caused solely by contingencies (e.g., equipment failure, bad weather, etc.) and cannot be avoided or reduced by any rescheduling strategies, which is termed as fixed train delay (e.g., ${D}_{i,BC}$); the second train delay is cause by operational conflicts between trains, which is termed as conflict train delay. Specifically, under same conditions, different rescheduling strategies may lead to different conflict train delay. Further, we can evaluate the effectiveness of the rescheduling strategies according to the total conflict train delay. The less total conflict train delay is, the more effective rescheduling strategy can be. The following discussions present the detailed analysis on conflict train delay.
3.2.1. Case 1 Normal Train Traveling on the Dedicated Track
The possible delay that train
i + 1 may encounter when tracing behind train
i on dedicated track is illustrated in
Figure 6. In fact, if
${T}_{i+1,B}-({T}_{i,B}+{D}_{i,BC})\ge T$, train
i + 1 is able to travel on dedicated track without decelerating, meanwhile safety constraint C3 can always be satisfied because of the relatively large departure interval between train
i + 1 and train
i. Train
i’s sudden decelerating does not affect the normal operation of any other trains, thus train
i + 1’s switching track is unnecessary.
However, if
${T}_{i+1,B}-({T}_{i,B}+{D}_{i,BC})<T$, train
i + 1 have to decelerate in consideration of C3, then we have
Accordingly, the conflict train delay of train
i + 1 can be calculated by
Moreover, the operation of train i will not be affected by any other trains if train i + 1 travels on dedicated track, thus we have ${Z}_{i,i+1}=0$. Obviously, no outbound trains will be delayed since the outbound track in segment BC will not be utilized by any inbound trains in case 1.
Finally, the total conflict train delay here equals to that of train
i + 1, which can be calculated by
3.2.2. Case 2 Normal Train Traveling on Reverse Direction Track
If we reschedule inbound normal train i + 1 to travel on reverse direction track in segment BC, there are two kinds of potential conflicts between train i + 1 and other trains, leading to different delays. One is the head-on conflicts between train i + 1 and the outbound trains (i.e., train k); the second is the rear-end conflicts between train i + 1 and the inbound trains (i.e., train i).
To analyze these two types of conflicts in detail, we divide case 2 into two subcases, namely subcase 2.1 and subcase 2.2, which will be discussed respectively below.
(1) Subcase 2.1 Delay analysis of train i + 1 and train k
In the condition that ${T}_{k,B}-{T}_{i+1,B}\ge {H}_{i+1,k}^{da}$, train i + 1’s arrival time at node B is earlier than train k and the time interval is longer than time headway ${H}_{i+1,k}^{da}$, which means C1 can always be satisfied while train k traveling with normal speed on dedicated track.
However, if
$0<{T}_{k,B}-{T}_{i+1,B}<{H}_{i+1,k}^{da}$, timetable shown in
Figure 7b will be obtained, in this situation train
k’s arrival time is later than train
i + 1 but the time interval is shorter than
${H}_{i+1,k}^{da}$, train
k still needs to decelerate before reaching node
B. If
${T}_{k,B}-{T}_{i+1,B}\le 0$, timetable shown in
Figure 7c will be obtained, in which the planned arrival time for train
i + 1 at node
B is later than train
k. Therefore, in the condition that
${T}_{k,B}-{T}_{i+1,B}<{H}_{i+1,k}^{da}$, train
k needs to reduce its speed in segment
AB because of C1, and the dash line in blue color in
Figure 7a highlights the track where train
k travels below normal speed. Moreover, in both
Figure 7b,c, we have
${Z}_{k}^{o}={T}_{i+1,B}+{H}_{i+1,k}^{da}$ and
${Z}_{k}^{f}={T}_{k,B}$. Then, the conflict train delay of train
k caused by train
i + 1, in the condition that
${T}_{k,B}-{T}_{i+1,B}<{H}_{i+1,k}^{da}$, can be calculated by
Then, according the above-mentioned analysis in this subsection, the delay time of outbound train
k in case 2 can be calculated by
Obviously, there is no delay for train i + 1 in this subcase, i.e., ${Z}_{i+1,k}=0$.
(2) Subcase 2.2 Delay analysis of train i + 1 and train i
According to the discussion in
Section 3.2.1, the condition that
${T}_{i+1,B}-({T}_{i,B}+{D}_{i,BC})<T$ is true when train
i + 1 is rescheduled to travel on the reverse direction track. Specifically, we will consider to reschedule train
i + 1 to travel on reverse direction track only in the condition that train
i + 1 will experience time delay on dedicated track. Moreover, safety constraint C2 should be taken into account when train
i + 1 travels on reverse direction track.
For detailed analysis, we divide the condition that ${T}_{i+1,B}-({T}_{i,B}+{D}_{i,BC})<T$ into three scenarios, namely ${T}_{i,B}+{D}_{i,BC}-{T}_{i+1,B}\ge {H}_{i+1,i}^{aa}$ and $0<{T}_{i,B}+{D}_{i,BC}-{T}_{i+1,B}<{H}_{i+1,i}^{aa}$ and $0\le {T}_{i+1,B}-({T}_{i,B}+{D}_{i,BC})<T$, and each scenario will be discussed respectively. First, in the condition that ${T}_{i,B}+{D}_{i,BC}-{T}_{i+1,B}\ge {H}_{i+1,i}^{aa}$, the arrival time of train i + 1 at node B is earlier than train i and the time difference is no less than ${H}_{i+1,i}^{aa}$, thus, there is no conflict between train i + 1 and train i in case 2, then we have ${Z}_{i,i+1}=0$.
However, if
$0<{T}_{i,B}+{D}_{i,BC}-{T}_{i+1,B}<{H}_{i+1,i}^{aa}$ or
$0\le {T}_{i+1,B}-({T}_{i,B}+{D}_{i,BC})<T$ is true, train
i + 1 will conflict with train
i because of C2, which leads to conflict train delay of train
i. In
Figure 8a, the dash line in blue color highlights the track where train
i is delayed by train
i + 1 and the dash line in red color highlights the part of track where train
i decelerates due to contingencies. Both in
Figure 8b,c, we have
Then, conflict train delay of train
i caused by train
i + 1 can be calculated by
Based on the aforementioned analysis, the conflict train delay of train
i in case 2 can be calculated by
For train i + 1, no conflict train delay will be obtained in case 2, i.e., ${Z}_{i+1,i}=0$.
In conclusion, the total conflict train delay of train
i,
i + 1 and
k in the case that train
i + 1 chooses to travel on the reverse direction track can be calculated by
where
${Z}_{k,i+1}$ and
${Z}_{i,i+1}$ can be calculated by Equations (5) and (8), respectively.