Simulating the Effects of Noncrossing Block Sections Setting Rules on Capacity Loss of Double-Track Railway Line due to the Operation of out-of-Gauge Trains

Dispatchers often set noncrossing block sections (NCBSs) for railway out-of-gauge train (OGT) running on double-track railway line for safety reasons. In this paper, wewill investigate the best location, length, and number of noncrossing block sections to reduce railway capacity loss due to the operation of OGTs. Firstly, yielding, overtaking, stopping, starting, and other operation rules for OGTs running on double-track railway line were designed, and a simulation model based on cellular automata was further put forward. Then, an assessment model for double-track railway line capacity loss due to the operation of OGTs was set up. Some simulation experiments and the comparisons of these results were further given to achieve the optimal setting of NCBS for OGTs running on double-track railway line. In the case of NCBSs number minus one, capacity loss caused by the operation of OGTs can be reduced up to 15.2% in the upstream direction and 6.3% in the downstream direction. Also, the NCBSs should lie at the nearest block sections (BSs) to depot stations and the NCBSs lengths should be as less as possible.


Introduction
In recent years, freight railways are expected to experience increasing capacity constraints and a great variety of goods and vehicle types.Generally, trains and structures were built to vehicle gauges and structure gauges, respectively, and there are safe clearances between trains and infrastructures as discussed in Takao and Uruga [1].For common trains, there was a large safe gap between two crossing trains in double-track railway line.Railway out-of-gauge trains (OGTs) refer to the freight train vehicles that load goods with specific characteristics such as oversize, overweight, and being expensive.Usually, the OGT's loading gauge is beyond the vehicle or rolling stock gauge and even the structure gauge along its path as discussed in Edworthy [2], and railway administrations will perform a proving run on railway system to guarantee safety [3].When one OGT is running on one BS and comes across or meets with an opposing common train running in the parallel BS in double-track railway line, the gap space between these two running trains may no longer meet the safety requirement and thus serious conflicts may occur.To solve this problem, railway dispatchers often limit the train speeds.Also, they set some NCBSs for OGTs running; namely, they let opposing common trains stop at the intermediate stations in the parallel line so that OGTs will not come across with any opposing common trains in the NCBSs.Once the OGTs run out the NCBSs, the NCBSs turn into common BSs where common trains which run in one line can come across opposing train in the parallel line.Of course, the setting of NCBSs for OGTs inevitably interferes in the operation of common trains in the parallel line and further leads to the total transport capacity loss of double-track railway line.However, these goods are generally expensive and important construction, chemical engineering, and military equipment, and thus the owners of these goods are willing to pay high costs.Besides, the OGTs usually have higher freight ton kilometer than common trains do, which offsets the capacity loss due to OGTs running in some extent.In such cases, the transport of OGTs is necessary for both economic and political reasons.Appropriate rules for NCBSs setting problem (NCSP) are

Models for Various Trains Running on Double-Track Railway Line
In this section, we will present a simulation model for train movements on dedicated freight double-track railway line based on cellular automata theory.Based on the simulation model, a capacity loss assessment model will also be formulated.

Simulation Model Based on Cellular
Automata.Our research focuses on trains running at automatic blocking zone on two-direction and four-line double-track railway line.Based on cellular automata theory, we put forward the simulation model by including an actual line and virtual line for each direction.On the virtual line, a virtual intermediate station is set aside at the same location of the corresponding actual intermediate station, and both intermediate stations are used for temporary stop to realize overtaking or yielding between OGTs and common trains in the same direction and crossing between OGTs and opposing common trains in NCBSs.The virtual BSs without any train are further set aside at the same location of their actual BSs for consistency.Generally, there are two depot stations (district stations), namely, origin station and destination station.A set of BSs and intermediate stations lie between the origin and destination stations.Also, there are three assumptions for our research: (i) Each track or siding in railway stations can only host one train during certain time interval for security.Thus, all intermediate and depot stations are viewed as independent BSs, and each of these BSs is considered as a cellular station.
(ii) Each BS or cellular station can only host one train at each time, and both overtaking and yielding are realized at certain intermediate station, namely, cellular station.
(iii) All trains including OGTs run one by one and satisfy the minimum security distance constraint on doubletrack railway line.
Notations used in the simulation model are shown in Notations.
For train operation safety, the minimum security distance for two adjacent trains is decided by train speeds, BS lengths, and railway signaling indication system.Thus, minimum security cellular interval is also ensured by such minimum security time interval which is necessary to grant the space for completely blocking the train in case of emergency.Let  (1)   and  (2)   express running time in BS  for the preceding train and the following one, respectively.When the nearest preceding BS  + 1 at the same direction is empty after the preceding train leaves section , the minimum arriving time for the following train at section +1 is calculated by  min + (2)   .
The minimum security time interval  min is further ensured by −  (2)   +  (1)  +1 } ∀ ∈ {1, 2, . . .,  − 1} .(1) Furthermore, the minimum security cellular interval is calculated by ( As for train-following operations at automatic blocking zone on double-track railway line, the th train may be overtaken by other trains if V  max < V −1 max and V  max < V +1 max .When the preceding BS of a running OGT is a NCBS, if the minimum running time for OGT passing through the NCBS is shorter than that for the closest opposing train arriving at and passing through the parallel NCBS, the OGT goes on running without any stop; otherwise, the OGT should stop at its preceding intermediate station because of crossing operation.If the OGT is running in one direction in NCBS without stop, the opposing stopping trains in the intermediate stations in the parallel line should depart from the intermediate stations according to the relative positions between the OGTs and their stopped intermediate stations.Also, only if safety gap of crossing operations gap(,   , ) is less than certain fixed value  may the corresponding train be considered to stop at its nearest preceding station for safety reasons.
Trains should stop at certain cellular station for overtaking or crossing operation which occurred in NCBS, and thus train's running cannot be treated as a simple shift of cellular space between the actual line and the virtual one at the same direction.These are the main differences between train overtaking (yielding) and lane-changing in traditional two-lane cellular automata problem.Besides, overtaking and crossing operation time, minimum security time interval, relative positions between OGT and intermediate stations at different direction, instantaneous speed for various trains, and other parameters should be considered in the cellular automation (CA) model of NCSP.Rules of the CA model can be concluded into the following eight points.
(8) Position Renewing.New speed and positions for trains are calculated by (1) to (7) All the above rules are applicable to train running operations on double-track railway line from  moment to  + 1 moment, which is also the basis for the simulation model based on cellular automation in the paper.

Capacity Loss Assessment
Model.Let V  , V   ,   , and  denote average speed among all common trains, average speed and capacity loss weighing for train , and the total number of trains, respectively.The longer the distance between current train and its preceding train, the less the influence of its preceding train on the current train.Let   denote the weighing function about train distance Δ + (), and the function is a decreasing one.Combined with the simulation model for various trains running on double-track railway line, a capacity loss assessment model aiming to minimize the running time loss rate per train and maximize the train flow is formulated as where  1 and  2 denote train transfer coefficients including their important degrees of the running time loss rate per train and train flow.The measurement unit in ( 3) is one train.The speed transmission effect is presented in (4), which means the influence for the following train caused by its preceding train due to different speeds.Also, minimum security cellular interval is expressed by constraint ( 5), whereas (6) expresses train position renewing process.Constraint (7) ensures trains' velocities constraints.

Simulation Experiments and Comparisons
Taking the four-aspect automatic blocking zone on doubletrack railway line as simulation environment, we will analyze and compare the influence on railway capacity loss caused by OGT running with different setting locations, lengths, and numbers of NCBS so as to get better NCBSs setting rules for NCSP.There are four intermediate stations, two depot stations, and 100 BSs in the automatic blocking zone with a length of 100 000 meters.All intermediate and depot stations are viewed as independent BSs, and each of these BSs is considered as a cellular station.Thus, there are 106 cells and the length for each cell is 1 000 meters.
In the above simulation environment, we assume that update time interval is 20 seconds and trains depart from depot stations in decreasing order of the running speed.Moreover, all trains passing through the zone in certain period are shown in Table 1.

NCBSs Setting Location.
With the same train departing order in origin station and other conditions, the whole zone occupation time and travel speed (calculated by the total length of the whole automatic blocking zone dividing the sum of total running time and stop time) for various trains will change in different extent due to different locations of the NCBS.Assume that there is only one NCBS for OGT running and the length for each section between different stations is 20 000 meters.With above models, results for different NCBSs setting locations are shown in Table 2.
Table 2 shows that when there are no NCBSs in the up running direction, their average travel speeds fluctuate within the scope of [−5%, 5%] due to the random slowing down rule.Comparing blocking sections with NCBS and those without NCBS, we can find that the difference of zone occupation time is within the scope of [−5%, 5%], and the capacity loss is almost zero in the up running direction.But, for the down direction, when the NCBS is blocking section 1 or 5 (as for train speed-distance curve, please see Figures 1 and 5), the total zone occupation times and running speeds of the trains are the same as those without NCBS.Thus, the NCBS that is the nearest to a depot station has the minimum impact on double-track railway line capacity.When the NCBS is blocking section 2 in the left (see Figure 2), trains 26561 and 36531 in the down direction should stop and yield to OGT 70210 in the up direction, which increases their blocking zone occupation times.Due to the short stop and yielding time of trains 26561 and 36531, the trains' impact on running speed of OGT 70641 in the down direction is tiny and the OGT's total zone occupation time is almost the same as that without NCBS.Therefore, when the blocking section 2 in the left is the NCBS, double-track railway line capacity is less affected.When blocking section 3 or 4 in the left is NCBS (as for train speed-distance curve, please see Figures 3 and 4), the travel time of OGT 70641 and its blocking zone occupation time increase due to stop at intermediate stations for yielding to trains 82702 and 26512 in the up direction.The time interval for adjacent trains is 8 minutes.The capacity loss is 1.96 if NCBSs are considered (cf. Figure 3) and 2.12 trains if NCBSs are not considered (cf. Figure 4).If only a single NCBS is considered and the NCBS is blocking section 3 or 4 in the left, then the NCBS has the biggest impact on capacity loss which is almost 2 trains; if the NCBS is blocking section 2 in the left, then the NCBS's impact on capacity comes second.Because of NCBS, the travel times of faster trains are increased and the total zone occupation times are slightly further increased.If the NCBS is adjacent to a depot station such as blocking section 1 or 5 in the left, then NCBS has the smallest impact on capacity loss of double-track railway line.Different train speeds have different influence on capacity as discussed in [8,9,16].Based on the above analysis, we can conclude that the smaller the distance between the NCBS and the depot station, the smaller the impact for a NCBS on capacity loss caused by OGT; the closer the distance between the NCBSs with the middle of the blocking zone, the greater the impact for a NCBS on capacity loss caused by OGT.Thus, the NCBS setting location rule for    NCSP is that the NCBS setting location should be as near as possible to depot stations and OGT's synthetic outline should also be considered.

NCBSs Setting Length.
According to above conclusion in Section 3.1, we only choose blocking section 3 as the NCBS in the next experiment.Assume that the lengths of other BSs are fixed; we progressively increased the length of blocking section 3 in the left and then compare results from such experiment with or without NCBS.The results for different NCBSs setting length are shown in Table 3. Table 3 shows that there is no NCBS situation occurring in the up direction, and thus the zone occupation time has no obvious change.But, for the down direction, the blocking section 3 as the NCBS has no impact on railway capacity, when the NCBS length is 10000 m.The zone occupation time increases by 3.8%, 4.9%, and 6.7% and capacity losses are 0.83, 1.13, and 1.58 trains, respectively, when the blocking section 3 is NCBS with the length of 15000 m, 20000 m, and 25000 m, respectively.If the length of blocking section 3 is 30000 m without NCBS, the train speed-distance curve is shown in Figure 6.If the length of blocking section 3 is 30000 m without NCBS, the train speed-distance curve is shown in Figure 7.When the length of blocking section 3 increased from 15000 to 30000, the stop time of OGT 70641 at the intermediate station increases significantly (see Figures 8 and  9), which makes its zone occupation time increase by 15.1% and capacity loss is 3.75 trains caused by such case.In Figures 6 and 7, the distances reach 120 km, because the length of blocking section 3 in the left is changed from original 10 km to 30 km.
The above results show that a proper reduction of NCBSs setting length can raise the double-track railway line's capacity with NCBSs substantially, especially the sections with busy OGT running.Capacity loss due to speed differences can be compensated for by additional stations [13], and railway capacity on double-track railway line with OGT running can be released by increasing the intermediate station number and further shortening the NCBS length.Also, we can conclude that the capacity loss caused by OGT running is the biggest when the NCBSs setting location is in the middle of such double-track railway BSs.Moreover, the longer the  NCBSs setting length, the bigger the capacity loss caused by OGT running on double-track railway line.Thus, the NCBSs setting length rule for NCSP is that the NCBS length should be as short as possible, and we can choose shorter sections as NCBSs.

NCBSs Setting Number.
With certain trains and departing orders, we can see that the capacity impact is great once there is a NCBS for OGT running.If the NCBS number is increased in continuous way, then what would be the impact of different NCBS number for OGT running on capacity?Based on the simulation experiment of only one NCBS (blocking section 1 in the left), we successively add a blocking section as NCBS, and the results for different NCBSs setting numbers are shown in Table 4.
(1) Figure 10 shows train speed-distance curve without NCBS, while Figures 11-13 show train speed-distance curves with different NCBSs setting number.Specifically, the NCBS is blocking section 1 in the left in Figure 11; the NCBSs are blocking sections 1 and 2 in the left in Figure 12, and the NCBSs are blocking sections 1, 2, and 3 in the left in Figure 13.Compared with above three situations, all trains in the up direction have no stop or yielding behaviors; thus their zone occupation times are nearly the same as those without NCBSs (see Table 3), and capacity loss is not obvious.For the down direction, it is completely different.There is no OGT stop or yielding behavior in Figure 11 and capacity loss is almost zero.But, for two NCBSs, trains 26561 and 36531 stop and yield to OGT 70210 running in opposite direction at the second intermediate station in the left; thus the zone occupation time in the down direction increased by 3.8% and capacity loss is 0.83 trains.As for three NCBSs, trains 26561, 36531, and OGT 70641 stop and then yield to OGT 70210 running in opposite direction at the second and third intermediate stations in the left, respectively; thus the zone occupation time in the down direction increased by 12.2% and capacity loss is further 2.67 trains.
(2) Figure 14 shows train speed-distance curve with four NCBSs, such as blocking sections 1, 2, 3, and 4 in the left.For the up direction, there is only OGT 70210 stopping and then yielding to OGT 70641 running in the opposite direction at the third intermediate station in the left, and its zone occupation time increased by 4.3% and capacity loss is 0.79 trains.For the down direction, both trains 26561 and 36531 stop and yield to OGT 70210 running in the opposite direction at the second intermediate station in the left, and OGT 70641 stops and then yields to trains 82702 and 26512 in opposite direction at the fourth intermediate station in the left.Moreover, the zone occupation time in the down direction rises up 14.2% and capacity loss is up to 3.08 trains.
(3) Figure 15 shows train-distance curve with five NCBSs when the NCBSs are blocking sections 1, 2, 3, 4, and 5 in the left.As for the up direction, train 82702 and OGT 70210 stop and yield to train 70641 running in the opposite direction at the fourth and third intermediate stations in the left, respectively.Also, the zone occupation time in the up direction increases by 5.6% and capacity loss is 1.04 trains.Meanwhile, for the down direction, both trains 26561 and 36531 stop and yield to OGT 70210 running in the up direction at the second intermediate station in the left, while OGT 70641 stops and yields to train 26512 running in the up direction at the fourth intermediate station in the left.Therefore, the zone occupation time in the down direction reaches up to 12040 s (see Table 4), it increases by 15.1%, and capacity loss is 3.29 trains.
From Figures 12-15, the total stop/yielding numbers are 2, 3, 4, and 5, respectively, while the NCBS numbers are 2, 3, 4, and 5 accordingly.Thus, there is an interesting relationship between the stop/yielding numbers and the numbers of NCBSs.On the above basis, we can conclude that the more the number of NCBSs, the bigger the capacity loss caused by  OGT running; and the number of stop/yielding operations increases with the number of NCBSs in a linear way.Thus, the NCBSs setting rule for NCSP is that the number of NCBSs should be the least, and we cannot set other unnecessary NCBSs except for safety reasons.

NCBSs Setting with Continuous
Way. OGT often has to stop and yield to other trains in the opposite direction when the gap space between two trains on double-track line no longer meets the safety requirement, which has great influence on the capacity of those sections, especially when there are continuous NCBSs within the blocking zone.In order to quantize the impact of the NCBSs with continuous way on railway capacity, we simulate all possible situations for such case and then analyze their results in a horizontal comparison way.Also, results are shown in detail in Table 5.
From Table 5, we will analyze the impact of the continuous way of NCBSs setting on double-track railway line capacity in the following ten setting ways.
(S1) First is continuous NCBSs including blocking sections 1 and 2 in the left.There are four types, Cases 1, 5, 8, and 10, whose train speed-distance curves are shown in Figures 12-15.The average zone occupation times for all these cases are 9145 s and 11645 s in the up and down directions, respectively.Compared with the OGT running without NCBS, the zone occupation time increases by 2.8% and capacity loss is 0.51 trains in the up direction, while the zone occupation time increases by 11.3% and capacity loss is 2.47 trains in the down direction.
(S2) Second is continuous NCBSs including blocking sections 2 and 3 in the left.There are six types, Cases 2, 5, 6, 8, 9, and 10, whose train speed-distance curves are shown in Figures 16,13,17,14,18,and 15.The average zone occupation times for such six cases are 9204 s and 11834 s in the up and down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 3.4% and capacity loss is 0.63 trains in the up direction, while the zone occupation time increases by 13.1% and capacity loss is 2.86 trains in the down direction.
(S3) Third is continuous NCBSs including blocking sections 3 and 4 in the left.There are six types, Case 3 and Cases 6∼10, whose train speed-distance curves are shown in Figures 19, 17, 20, 14, 18, and 15.The average zone occupation times for such six cases are 9290 s and 11910 s in the up and      down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 3.3% and capacity loss is 0.61 trains in the up direction, while the zone occupation time increases by 12.7% and capacity loss is 2.76 trains in the down direction.(S5) Fifth is continuous NCBSs including blocking sections 1, 2, and 3 in the left.There are three types, Cases 5, 8, and 10, whose train speed-distance curves are shown in Figures 13, 14, and 15.The average zone occupation times for all these cases are 9227 s and 11907 s in the up and down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 3.7% and capacity loss is 0.68 trains in the up direction, while the zone occupation time increases by 13.8% and capacity loss is 3.01 trains in the down direction.
(S6) Sixth is continuous NCBSs including blocking sections 2, 3, and 4 in the left.There are four types, Cases 6, 8, 9, and 10, whose train speed-distance curves are shown in Figures 17,14,18,and 15.The average zone occupation times for all these cases are 9320 s and 11960 s in the up and down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 4.7% and capacity loss is 0.88 trains in the up direction, while the zone occupation time increases by 14.3% and capacity loss is 3.13 trains in the down direction.
(S7) Seventh is continuous NCBSs including blocking sections 3, 4, and 5 in the left.There are three types, Cases 7, 9, and 10, whose train speed-distance curves are shown in Figures 20, 18, and 15.The average zone occupation times for all these cases are 9274 s and 11860 s in the up and down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 4.2% and capacity loss is 0.78 trains in the up direction, while the zone occupation time increases by 13.4% and capacity loss is 2.92 trains in the down direction.
(S8) Eighth is continuous NCBSs including blocking sections 1, 2, 3, and 4 in the left.There are only two types, Case 8 and Case 10, whose train speed-distance curves are shown in Figures 14 and 15.The average zone occupation times for such two cases are 9340 s and 11990 s in the up and down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 4.9% and capacity loss is 0.92 trains in the up direction, while the zone occupation time increases by 14.6% and capacity loss is 3.19 trains in the down direction.
(S9) Ninth is continuous NCBSs including blocking sections 2, 3, 4, and 5 in the left.There are only two types, Case 8 and Case 10, whose train speed-distance curves are shown in Figures 18 and 15.The average zone occupation times for such two cases are 9330 s and 11970 s in the up and down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 4.8% and capacity loss is 0.90 trains in the up direction, while the zone occupation time increases by 14.4% and capacity loss is 3.15 trains in the down direction.
(S10) Tenth is continuous NCBSs including all BSs.The train speed-distance curves are shown in Figure 15.The zone occupation times for such case are 9400 s and 12040 s in the up and down directions, respectively.Compared with OGT running without NCBS, the zone occupation time increases by 5.6% and capacity loss is 1.04 trains in the up direction, while the zone occupation time increases by 15.1% and capacity loss is 3.29 trains in the down direction.
From above analysis, we can conclude that different continuous NCBSs setting ways have different impacts on railway capacity caused by OGT running, as shown in Figure 22.
In Figure 22, the horizontal axis presents above 10 NCBSs setting ways from Case 1 to Case 10, and the vertical one expresses the capacity loss (unit: one train).From Figure 22, we can know that the capacity loss may be at peak among all these ten ways with the continuous NCBSs setting ways including Case 3, Case 6, Case 8, and Case 10.Moreover, for Case 3, Case 6, Case 8, and Case 10, their NCBSs all include sections 3 and 4 in the left.Meanwhile, compared with other positions settings of NCBSs, section 3 in the left has the biggest impact on railway capacity from Section 3.1.Therefore, if there appears continuous NCBSs setting way, the elimination of section 3 in the left as a NCBS has the greatest impact on capacity.Besides, capacity loss in the up direction is less than that in the down direction.The reason is
2) OGT Stopping and Yielding Rule.If gap(,   , ) ≤  and [gap(,   , ) +   − gap(,   , )] * V  max <   V  max , OGT  should pass through the NCBS without any stop at its preceding cellular station   , but its nearest preceding train at the opposing direction must stop at the cellular   only if gap(,   , ) ≤   .If gap(,   , ) ≤  and [gap(,   , ) +   − gap(,   , )] * V  max ≥   V  max , OGT  should stop at the cellular station   at a probability of   , but its nearest preceding train at the opposing direction must pass through the section without any stop at the cellular   .

Table 1 :
Trains passing through given blocking zone within certain time interval.

Table 2 :
Results for different NCBSs setting locations.

Table 3 :
Results for different NCBSs setting length.

Table 4 :
Results for different NCBSs setting number.

Table 5 :
Results of NCBSs setting with different continuous way.