Planning the most suitale trael speed for high frequency railway lines Alex Landex Technical Uniersity of Denmark, Centre for Traffic and Transport, Bygningstoret 1, 800 Kgs. Lyngy, Denmark, e-mail: al@ctt.dtu.dk Anders H. Kaas Atkins Denmark, Rail Transport Planning, Arne Jacosens Allé 17, 300 Køenhan S, Denmark, e-mail: anders.h.kaas@atkinsgloal.com Astract This paper presents a new method to calculate the most suitale trael speed for high frequency railway lines to achiee as much capacity as possile for congested railway lines. The method calculates the most suitale trael speed ased on the raking distance and information aout the interlocking system. Based on the raking distance it is possile to calculate the minimum headway time, and therey determine the uffer time when knowing the frequency. Hence the headway time can e diided into minimum headway time and uffer time. The uffer time is an indicator for the spare capacity of the railway line, and the more uffer time on the railway line, the etter punctuality and the etter possiilities to run more trains. Based on the descried method a case example from the suuran railway lines of Copenhagen will e shown. The case example shows that a reduction of the maximum trael speed y 6% in central Copenhagen can increase the capacity y 11%. The increased capacity will improe the punctuality of the trains in central Copenhagen een though some of the capacity will e used to run more trains through Copenhagen. Keywords Capacity, Headway, Buffer time, Simulation, Punctuality 1 Introduction High frequency railway lines often suffer from lack of capacity. These prolems can traditionally e soled y improing the infrastructure or undle the trains for etter utilization of the capacity. The capacity of high frequency railway lines depends on the minimum headway time of the trains. Calculations of headway times in rail systems in Scandinaia and in many OR-papers are often assumed independent of the trael speed. Normally, this assumption is reasonale, ut sometimes there is a ig difference in the headway time as a result of a slight change in the trael speed. The ig change in the headway time can e explained y the structure of the signal system with discrete lock sections, which can affect the trael speed. A train has to e ale to stop efore a restrictie signal. This means that the raking distance on a railway line with discrete lock sections has to e smaller than or equal to 1
the summarized length of free lock sections and a safety distance. A change in the trael speed changes the raking distance. Een a slight change in the raking distance can affect the headway distance consideraly, if an extra lock section is needed to ensure that the train is ale to stop efore a signal. In that way, the headway time can not always e assumed independent of the trael speed. Changes in the trael speed will not only affect the minimum headway time, ut also the uffer time etween the trains in the timetale due to a higher minimum headway time which will result in less uffer time. The uffer time ensures that a small delay of one train is not propagated to the following trains. Less uffer time increases the risk of propagation of delays. If the minimum headway time is getting larger than the planned or possile headway time (e.g. ecause of higher speed due to delays), the train will hae to reduce its trael speed. Therefore, it is important to e ale to plan the most suitale trael speed, especially for high frequency railway lines. Normally, the lengths of the lock sections of the railway line are adapted to the trael speed of the most common train type and speed of the rail line. This gies the disadantage that some trains might use too much capacity ecause of their train characteristics. Oer time the trains will also e replaced y new trains or the infrastructure will e upgraded, which can cause a higher utilization of the capacity, hence the lock lengths are not adapted to the new situation. This paper suggests a new method to change/optimize the trael speed to fit the lock lengths of the infrastructure. This makes it possile to gain more capacity on a railway line. 1.1 Paper ojectie This paper presents a new method to optimize the trael speed of trains, so that a etter utilization of the capacity can e achieed. The extra capacity achieed can e used for fewer delays and etter punctuality, more routes, or a comination of oth. Furthermore, the paper will test the method on a real case example from Copenhagen. 1. Paper outline The next section riefly descries the definitions and notations used in the following. Section 3 descries the method to calculate the most suitale trael speed step-y-step, starting with calculating the raking distance and then the trael speed of trains for oth discrete and continuous signalling systems. This proides the ackground for calculating the most suitale trael speed and the effects of deiations from the most suitale trael speed. Using the method deeloped in section 3, section 4 descries a case example of improed capacity on the suuran railways of Copenhagen. Here the capacity will e examined for scenarios with different trael speeds and different lengths of lock sections. The scenarios will e simulated in RailSys in order to carry out an ealuation. Against the ackground of section 3 and 4, section 5 concludes on the method and descries the perspecties. Definitions and Notation This paper uses terminology usually used in the railway literature. Howeer, since the terminology differs from country to country, an oeriew of the terminology used in this paper is proided in tale 1.
Term ATC Block section Block occupation time Braking percentage Buffer time Delay Headway distance Headway time Punctuality Running time supplement Trael speed Tale 1: Short description of terminology Explanation Automatic Train Control (ATC) is a safety system ensuring a train to stop efore a restrictie (red) signal The length of track etween two lock signals, ca signals, or oth The time a lock section is occupied y a train The ratio of the raking force to total ehicle weight. The raking percentage expresses the raking force required for raking 1 ton The time difference etween actual headway and minimum allowale headway The aerage delay of the trains The distance etween the front ends of two consecutie trains moing along the same track in the same direction. The minimum headway distance is the shortest possile distance at a certain trael speed allowed y the signalling and/or safety system The time interal etween two trains or the (time) spacing of trains or the time interal etween the passing of the front ends of two consecutie ehicles or trains moing along the same lane or track in the same direction The part of trains arriing less than X minutes late The difference etween the planned running time and the minimum running time The speed of the train at a certain time The terms descried in tale 1 is further illustrated figure 1 Figure 1: Time definitions The notation in the paper will e as seen in tale. 3
Tale : Notation Explanation Symol Unit Braking retardation a r m/s Length of first lock section ehind train 1 B 1 m Length of lock section j ehind train 1 B j m Breaking percentage λ Braking ratio c Acceleration of graity g m/s Gradient i Length of train 1 L m Braking distance S m Optimal raking distance S,opt m Headway distance S h m Safety distance ehind a signal S s m The time it takes to achiee full raking force t s Buffer time t t s Headway time t h s Minimum headway time t h,min s Reaction time of the engine drier t r s Reaction time of the rakes t s s Total reaction time of rakes for calculation of t R s raking distance (t r + t s + ½ t ) Velocity/trael speed m/s The optimal elocity/trael speed opt m/s 3 Method This section descries how to calculate the raking distance for trains. The raking distance is a crucial parameter to calculate the possile trael speed for a train on a certain railway line. Knowing the raking distance and the possile trael speed it is possile to determine the most suitale trael speed using the new method descried in this section. Furthermore, this section will descrie the consequences of deiations from the most suitale trael speed. 3.1 Calculation of Braking Distance To e ale to calculate the most suitale speed to minimize the lock occupation time, it is necessary to know the possile raking distance at the current line section. In simple mechanics, it is possile to calculate the raking distance (S ) as a function of the speed () when the reaking starts and the raking retardation (a r ) of the train, S =. (1) a The reaking retardation (a r ) can e calculated in arious ways. In Denmark, the empiric Mindener formula is normally used [3] and [7], r 4
λ 6.1 λ + 61 a r = 0.061 1+ =. () 10 1000 λ descries the reaking percentage defined as the ratio of the raking force to total ehicle weight and expressing the raking force required for raking 1 t. λ has different alues for arious types of rakes and rolling stock and is normally found experimentally. Because equation () is empiric and λ is found y experiments, the formula takes all kinds of retardations including air resistance into account [7]. In this paper the wind speed will e neglected. Trains with anti-lock rakes are not allowed to otain a larger raking distance than 0% more than trains without anti-lock rakes according to approal y the International union of railways (UIC). Therefore, the equation () is rectified y 0% [7], 6.1 λ + 61 100 6.1 λ + 61 a r = =. (3) 1000 10 100 The influence of the gradient (i) can e found from the simple mechanics as an extension of the raking retardation. The extension of the raking retardation can for small gradients e assumed equialent to the product of the acceleration of graity (g) and the gradient (i) in percent and negatie at falls. Under normal circumstances trains do not rake sharply, ut only with a certain raking ratio (c). For instance the raking ratio for the Danish ATC system is 0.6 for train units and 0.7 for all other kinds of trains. Taking the raking ratio and the gradient into account the raking retardation of the train will e, a r 6.1 λ + 61 = c + g i. (4) 100 Comining the equations (1) and (4) it is possile to calculate the raking distance for trains without taking the reaction time of either the engine drier or the raking system into account, S = 6.1 λ + 61 c + g i 100. (5) The reaction time can e diided into the reaction time of the engine drier (t r ) and the reaction time of the rakes (t s ). The reaction time of the rakes can e further diided into the reaction time from the rakes are applied to the rakes start raking the train (t s ), and the time it takes from the train starts raking, until the rakes of the whole train are working with full raking force (t ). Depending on the type of rakes, the reaction times of the rakes can hae a large ariation. The reaction time of the engine drier (t r ), and the time from the rakes are applied to the rakes start raking the train (t s ) extends the raking distance proportionally with the speed. The time it takes from the rakes start working efore all rakes work with full raking force (t ) depends on the type of raking system. It can, howeer, e assumed that the raking force will increase linearly as descried in Tilli [7] and Andersson and Berg 5
[1]. This gies the raking distance as, S = + ( t r + t s + ½ t ). 6.1 λ + 61 c + g i 100 (6) The raking distance has een seen calculated in arious ways differing from the expression in equation (6). A short oeriew of different formulas for calculating the raking distance can e found in e.g. Profillidis [5] and also Barney, Haley and Nikandros []. In this paper it is, howeer, chosen to use the expression in equation (6) while it is a commonly used empiric formula, and no etter formulations are known used internationally y the authors. Other ways of calculating the raking distance can also e used for the further calculations in the method descried in this paper. 3. Calculation of Trael Speed To e ale to calculate possile trael speeds for trains, it is necessary to know the allowed raking distance of the train. The possile trael speed () will according to equation (6) e, a r t Where : t a R r = t r + t s R ± + ½ t S + a 6.1 λ + 61 = c + g i 100 t R r. (7) When calculating the possile trael speeds for trains using equation (7), only the positie alue of the term in the square root is used. The allowed raking distance is determined y the signal system of the railway line. If there is a moing lock system on the line, the maximum allowed raking distance is the actual raking distance of the train plus a safety distance. Moing lock systems are still not common and not existing in Denmark. Instead traditional discrete lock systems are used and on the main lines supplied with either continuous or discrete ATC (with or without wiggly wire) and on the Copenhagen suuran railway lines the HKT (speed control and train stop) system, which is similar to the continuous ATC system. Discrete ATC The Danish ATC system is, from the point of origin, ased on discrete locks where the ATC information is only updated at alises placed close to the signals a so-called discrete ATC system. The Danish discrete ATC system is in many ways similar to the German PZB (Punkt Zug Beeinfussung) system. The headway distance (S h ) for the discrete ATC system can e measured as the sum of lock sections within the raking distance (S ) of a train and an extra lock section, a safety distance (S s ) after the red signal and the length of the train in front (L), as shown in figure. 6
Figure : Discrete locks and discrete ATC The headway distance can then e expressed as, n Sh B j + Ss + j= 1 L. (8) The sum of B j s descries the numer of lock sections in the raking distance plus an extra lock section. The raking distance can e expressed as, n S. (9) B j j= Besides the expression in equation (9), the raking distance depends on the ATC system. At present, the Danish ATC system can only look or 3 lock sections ahead, which limits the trael speed, hence the speed depends on the allowed raking distance. The future ETCS (European Train Control System) will e ale to look more than 3 lock sections ahead. Continuous ATC and HKT Line sections with a high rate of capacity utilization can e equipped with wiggly wire so that a continuous ATC system is achieed. The wiggly wire system is, as descried in Kaas [4], expensie to estalish and should therefore only e estalished at railway lines with a high rate of utilization or at ottlenecks. The wiggly wire makes it possile to update the ATC system along the line instead of only at alises. This gies the adantage of eing ale to speed up the train when a lock section ahead ecomes free, which improes the capacity. 7
Figure 3: Discrete locks and continuous ATC Since the wiggly wire can update the ATC system at all times, the headway distance will e smaller than for a discrete ATC system. The headway distance can e expressed as, S S + B + S L. (10) h 1 s + The raking distance can e expressed as in equation (6). If the raking distance is equal to the length of a whole numer of lock sections, the headway distances calculated in equations (8) and (10) are exactly alike. Like the Danish discrete ATC system, the Danish continuous ATC system can only look or 3 lock sections ahead. The future ETCS system can howeer look further ahead. On the suuran railway lines in Copenhagen, the HKT system is used. The HKT system is similar to the continuous ATC system. The HKT system can, howeer, look more than 3 lock sections ahead, ut the HKT system can only e programmed for up to 4 different speed limits depending on the numer of free lock sections ahead. 3.3 Calculating the most Suitale Trael Speed The most suitale trael speed can e defined in different ways. For the passengers, the most suitale trael speed is achieed when the total trael time (including waiting times etc.) is as short as possile. For a congested railway line, the most suitale trael speed is when the headway time is as short as possile ecause it results in the highest possile capacity. For the railway company, the most suitale trael speed will e a mix of the most suitale trael speed for the passengers and the shortest headway time on the congested railway line/lines. The headway time (t h ) depends on the headway distance (see equations (8) and (10)) and the lock occupation time, which is equialent to the trael speed of the train is, 8
Sh t h =. (11) The difference in the minimum headway distance etween the discrete and the continuous ATC system can e seen in figure 4. Figure 4: Minimum headway time as a function of the trael speed and the lock sections in the minimum headway distance Figure 4 is a conceptual figure showing that the minimum headway time in the discrete ATC system in the est case is equal to the headway time for the continuous ATC system. The headway time is the same, hence equations (8) and (10) are equialent when the length of the raking distance has exactly the same length as a whole numer of lock sections. The optimal trael speed is when the minimum headway time is as short as possile. When the trael speed is elow the optimal trael speed the minimum headway time can e reduced y speeding up since the lock occupation time is too long. At trael speeds aoe the optimal trael speed, the raking distance has ecome too long, so that the lock sections are resered for too long time. It is not possile to hae trael speeds which require looking more lock sections ahead than the ATC system allows. As earlier mentioned, there are numerous formulas for calculating the raking distance, see Profillidis [5] and Barney, Haley and Nikarandos [] for some other formulas. Common to all the formulas is the shape of the lines in figure 4. This indicates that the methods of calculating the headway time are the same and only the parameters ary. For the continuous ATC system, the trael speed giing the shortest headway time can e found as the gloal minimum. The gloal minimum can, since it is a continuous function, easily e found y an ordinary differential equation, 9
t h,min = + 6.1 λ + 61 c + g i 100 t ' h,min opt t h,min S = + B1 + S ( t + t + ½ t ) r + L + B + S + L 1 B1 + Ss + L = = 0 6.1 λ + 61 c + g i 100 6.1 λ + 61 = c + g i B1 + Ss + L 100 s s 1 s t h,min. (1) Equation (1) is a continuous function which means that a slight change in the optimal speed will not hae a significant impact on the minimum headway time, see figure 4. Therefore, it is often seen that other factors than the minimum headway distance or minimum headway time and therey the capacity often will e the determining factor for choosing the trael speed. For railway lines with a discrete ATC system (or without any ATC system), the trael speed has a greater impact on the minimum headway distance than when haing a continuous ATC system (cf. figure 4). Een a slight increase in the trael speed can cause a much longer minimum headway time, hence the raking distance occupies an extra lock section. When the raking distance (still) is inside the span of a lock section, the minimum headway time will decrease with increased trael speed. The optimal trael speed then occurs, when the minimum headway time for the discrete ATC system is close to the minimum headway time achieed with the continuous ATC system. The local minima with the discrete ATC system can e found y setting the raking distance equal to the length of a whole numer of lock sections, n S = B. (13),opt j= Equation (13) comined with equations (8) and (11) results in local minimum headway times which can e descried as, j 10
n j= B j = S,opt t h,min = n j= B + B + S + L Where :. (14) = + 6.1 λ + 61 c + g i 100 j 1 s ( t + t + ½ t ) r s The possiility of looking more than to 4 lock sections ahead reduces the possiility of optimizing the trael speed, hence it will not e possile to speed up in case of delays. Therefore, the planned optimal trael speed is often lower than the speed calculated in equation (14). Deiations from the Most Suitale Trael Speed The method for calculating the most suitale trael speed also proides the possiility of calculating the consequences of deiations from the most suitale trael speed. Knowing the planned headway time etween two susequent trains and the realized headway time at different trael speeds, also the uffer times are known for different trael speeds, hence the headway time can e diided into minimum headway time and uffer time, t = t + t. (15) h h,min t Information aout the minimum headway time and the planned headway time for a congested or high frequency railway line can e seen on figure 5. Figure 5: Minimum headway time as function of the speed and the planned headway time 11
The information in figure 5 shows that it is possile to run the train at (almost) all speeds if the railway line is equipped with continuous ATC. If the railway line is only equipped with a discrete ATC system it is not possile to run the train at certain trael speeds, hence the minimum headway time is longer than the planned headway time. This can also e seen y examining the uffer time of the train, since it is not possile to run a train with a negatie uffer time as seen in figure 6. Figure 6: Minimum headway time and uffer time as function of the trael speed 4 Results The preious section descried how to calculate the raking distance. Using the raking distance and information aout the infrastructure (lock lengths and signalling system), it is possile to calculate the trael speed giing the shortest minimum headway time. The shortest possile minimum headway time gies more uffer time, and y that more capacity, which can e used for running more trains, achiee a etter punctuality, or a comination of oth. The calculated optimal trael speed will not necessarily e the est trael speed in practice since the possiility of catching up delays or ensure a short trael time are also important factors. In the following a case example from the suuran railways of Copenhagen will e presented. In the case example, it will e shown how the method descried in this paper can e used to achiee more capacity and a etter punctuality on a congested railway line. In the case example present safety procedures and lock lengths of the suuran railways of Copenhagen are taken into account. 4.1 A case example The Copenhagen S-train system suffers from lack of capacity since all the lines, except from the cross line, use the same railway line through central Copenhagen, as shown in figure 7. In the morning peek hour, 10 S-train routes run through central Copenhagen 1
using a scheduled serice in which trains operate at equal and fixed time interals of 0 minutes. Figure 7: The Copenhagen S-train system year 005 Due to the intense traffic in central Copenhagen with trains eery second minute in the morning peek hours, een small delays are easily spread to the whole system. In the future it is expected that more routes will run through Copenhagen. To run any more trains through Copenhagen with a satisfactory delay distriution it is necessary to increase the capacity. The most congested part of the railway line in Copenhagen is an approximately 3 km long section from the central station (Køenhan H) to Østerport ia Vesterport and Nørreport (cf. figure 7). Today, oth new and old trains are running on the railway line. Per January 006, only new trains will e running on the line. When all trains on railway line are new it will e possile to optimize the trael speed according to the lock sections to achiee more capacity. Today, the maximum trael speed etween the central station 13
and Østerport is 80 km/h which (with the current lock sections) results in a minimum headway time of 114 seconds. By reducing the maximum trael speed to 60 km/h it is possile to reduce the minimum headway time to 101 seconds, see tale 3. Tale 3: Optimization of trael speed on the suuran railway line in central Copenhagen Maximum trael speed Minimum headway time Minimum running time 80 km/h 114 seconds 97 seconds 60 km/h 101 seconds 315 seconds As shown in tale 3, it is possile to reduce the minimum headway time y 13 seconds or aout 11% just y reducing the trael speed aout 6% on a congested railway line. The reduced minimum headway time results in more capacity which can e used for fewer delays and etter punctuality, more routes, or a comination of oth. If a further decrease in the minimum headway time is requested it is necessary to change the length of the existing lock sections. Changing the length of the existing lock sections can reduce the minimum headway time y an extra 8-9 seconds, as shown in tale 4. Tale 4: Optimization of lock sections on the S-train line in central Copenhagen Scenario Minimum headway time (theoretically) Old S-trains, existing lock lengths 116 seconds north ound 114 seconds south ound New S-trains, existing lock lengths 10 seconds north ound 101 seconds south ound New S-trains, improed lock lengths 93 seconds north ound 93 seconds south ound Further improement of the minimum headway time is not possile, unless changes in the present security procedures are accepted. Simulation of the changes To ealuate the effects of reducing the trael speed a simulation in RailSys has een carried out. In the simulations only the section etween the central station (Køenhan H) and Østerport has een ealuated. 85% of the trains hae een inducted initial entry delays of 0- minutes (equally distriuted) and the remaining 15% of the trains hae een inducted initial entry delays of -5 minutes (equally distriuted) at Køenhan H and Østerport. The results of the different scenarios are shown in tale 5. Tale 5: Aerage delay and punctuality (less then ½ minutes delayed) at different cominations of infrastructure/train types and numer of train routes Scenario 10 trains per 0 minutes 11 trains per 0 minutes 1 trains per 0 minutes Minimum running time Old S-trains, existing NOT NOT 74 sec./65% lock lengths POSSIBLE POSSIBLE 97 seconds New S-trains, existing NOT 1 sec./91% 4 sec./89% lock lengths POSSIBLE 315 seconds New S-trains, improed lock lengths 0 sec./91% 3 sec./89% 7 sec./88% 315 seconds 14
The punctuality increases, hence more uffer time or capacity is achieed. The increased capacity, or uffer time, can e used to run een more train routes through Copenhagen. Een though more train routes are running through Copenhagen, the simulations show that the delays and punctuality will improe compared with the situation of today, see tale 5. The results of the simulation in tale 5 show that an optimization of the trael speed can improe the capacity of a congested railway line consideraly. Changing the trael speed is therefore a cheep way to improe the capacity of a ottleneck. The extra capacity can e used to run more trains, improe the delays and punctuality or a mix of oth. 5 Conclusions and Perspecties This paper has descried how to calculate the raking distance using the standard Mindener formula. Haing knowledge aout different signalling systems, safety systems, and the length of the lock sections, it is possile to use the deeloped method to calculate the most suitale trael speed for a high frequency railway line. The paper has shown that een slight changes in the trael speed can hae a large impact on the capacity at railway lines with a discrete ATC system. The large impact on the capacity is due to large changes in the minimum headway time; hence the raking distance requires an extra lock section to e ale to rake efore a restrictie (red) signal. Contrary to the discrete ATC system, slight changes in the trael speed with continuous ATC system or HKT system do not hae equally large impacts on the headway time and the capacity. Therefore, high frequency railway lines and ottlenecks on the infrastructure should hae a wiggly wire and a continuous ATC system to improe the capacity. Changing the maximum trael speed on a congested railway line can hae a ig impact on the capacity. The capacity can also e improed een though the railway line is equipped with wiggly wire and a signalling system similar to a continuous ATC system. The paper has shown that extending the trael time y 6% on a small section of the suuran railways of Copenhagen, which has a HKT system similar to the continuous ATC, the minimum headway time can e reduced y aout 11% in central Copenhagen. By upgrading the signalling system it is possile to improe the minimum headway time y another 9%. Changing the trael speed on a railway line in central Copenhagen, which acts as a ottleneck for the entire suuran railways of Copenhagen, it is possile to run more trains or improe the delays and punctuality for the entire suuran railways. The deeloped method of optimizing the trael speed to the lock lengths has through simulations shown to e a powerful tool to gain as much capacity as possile on an existing railway line. The simulations hae furthermore shown that it is possile to run more trains with a etter punctuality when the trael speed is adjusted to the lock lengths. Simulations are a difficult way of examining the effects of changes in the trael speed, an easier way to ealuate the effects on the capacity is to use the method descried in the UIC capacity leaflet [8]. In the future the deeloped method can e used on railway lines which are ottlenecks in the railway system. This can e done y changing the running time supplements in the timetale, so that some of the time supplements reallocate etween the open line and the stations, which has een descried in Rudolph [6]. With a further deelopment and implementation of the method it will e possile to use the method in the planning of timetales. The method can e used to plan the most suitale trael speed for the trains, and in this way e used to calculate the optimal running time supplements for the railway line or the specific train. 15
In order to improe the descried method to calculate optimal running time supplements a stochastic element is required to determine the proaility distriution of delays. Haing this stochastic element imedded in the method it will e possile to calculate the optimal comination of trael speed, running time supplements, and uffer times. In this way the aerage delay and punctuality of the railway system can e improed, ut it requires the right stochastic description of the risk of delays, as descried in Vromans [9]. References [1] Andersson, E., Berg, M. Railway systems and rolling stock, Vol., Kungl Tekniska Högskolan, Stockholm, 1999 (in Swedish) [] Barney, D., Haley, D., Nikandros,G., Calculating Train Braking Distance, Conferences in Research and Practice in Information Technology, ol. 3, 001 [3] Kaas, A.H., Methods for calculating capacity on railway lines, Technical Uniersity of Denmark, 1998 (in Danish) [4] Kaas, A.H., Optimization of lock sections on railways with ATC-systems, In: Lohmann-Hansen, A. (ed.), Trafikdage på AUC 95, pp. 185-19, Transportrådet, Aalorg, 1995 (in Danish) [5] Profillidis, V.A., Railway Engineering, pp. 31-34, Section of Transportation Democritus Thrace Uniersity Greece, Aeury Technical, Aldershot, 1995 [6] Rudolph, R., Entwicklung on Strategien zur optimierten Anordnung und Dimensionierung on Zeitzuschlägen im Eisenahnetrie, Uniersität Hannoer, Eurailpress Tetzlaff-Hestra Gmh & Co., Hamurg, 004 [7] Tilli, J.S., ATC on Danish railway lines, Technical Uniersity of Denmark, 1991 (in Danish) [8] UIC leaflet 406, Capacity, 1 st edition, UIC International Union of Railways, France, 004 [9] Vromans, M.J.C.M., Kroon, L.G., Stochastic Optimization of Railway Timetales, TRIAL Research School, Delft, 004 16