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An improved equation model for the train movement
Simulation Modelling Practice and Theory 15 (2007) 1156–1162 www.elsevier.com/locate/simpat
An improved equation model for the train movement KePing Li *, ZiYou Gao State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, PR China Received 6 July 2006; received in revised form 2 June 2007; accepted 23 July 2007 Available online 31 July 2007
Abstract In this paper, a new term is introduced into the equation of train movement, which is about the influences of the safety stopping distance and stations on the motion of train, when there are many trains on the line. The aim is that the improved equation can well describe the motion of train under the moving block condition. In order to test the improved model, we use a simulation analysis approach to solve a simplified form of the improved equation model. In the simulations, we investigate the space–time diagram for the railway traffic flow and the trajectories of the train movement. The numerical simulation results demonstrate that the improved model can well describe the dynamic behaviour of the train movement under the moving block condition. Some complex phenomena of train movement can be well reproduced but more work is required for a full simulation of the general model. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Train movement; Simulation analysis; Moving block
1. Introduction Train movement ‘‘depends’’ on the calculation of the speed and distance profile while a train is travelling from one point to another. In general, train movement is governed by many factors, such as the track geometry, control signalling, traction equipment, power supply system and speed restrictions. When we analyze and evaluate the theoretical control algorithm of train movement, the equation of the train movement is the basis for any computer simulation. In this field, one of the important problems is how to establish and solve the equation of the train movement. In the past decades, a number of studies have been done for the optimization and control of train movement. Most of them were focused on the solution to the optimal control of a single train. Petar and Guedial solved the problem of the energy optimal control of the motion of a train by the maximum principle [1]. Howlett and Cheng derived the equation of the motion of a train to determine the optimal control strategies for a train [2]. Ref. [3] proposed a solution algorithm to find an optimal driving strategy for train movement where discrete control is used and speed limits are imposed. Liu and Golovitcher outlined a method for the calcu*
Corresponding author. Tel.: +86 10 5168 2892. E-mail address: [email protected] (K. Li).
1569-190X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2007.07.006
K. Li, Z. Gao / Simulation Modelling Practice and Theory 15 (2007) 1156–1162
1157
lation of energy optimal control [4]. In Ref. [5], Howlett used a generalized equation of train movement to determine an optimal driving strategy, where the continuous control and discrete control are all discussed. Some other studies discussed the traction and braking applications. For example, Adinolfi et al. accessed the impact of the train braking techniques on the energy consumption [6]. However, all of them mentioned above did not consider the influences of other trains and stations in the equation of the train movement. Hence the calculation of train movement is isolated and may not dynamically capture the characteristic behaviour of a busy network. This limitation will strongly affect the computer simulation of railway traffic. In this work, we propose a new equation model, where the influence of other trains is considered especially in relation to the requirement that safe distances be maintained behind preceding trains that have stopped at a station. Using the proposed model, we hope that the computer simulation can well describe the reality of railway traffic. The paper is organized as follows. We introduce the principle of the moving block system in Section 2. In Section 3, we propose the improved equation model. The numerical and analytical results for a simplified form of the improved model are presented in Section 4. Finally, conclusions of this approach are presented. 2. Moving block system Moving block (MB) system is a special type of train signalling system [7]. Its advantages are that the line capacity can be increased, and the traffic fluidity and the energy efficiency can be improved. In practice, MB relies on continuous communication between train and wayside equipment, which can be used for control, protection and regulation of train movement. With the MB train control system, the minimum train separation to be defined solely by the dynamic performance of the train can be provided. On a MB equipped system, electronic communications between the control center and the train continuously control the trains, and make them maintain a safe stopping distance. At the same time, trains transmit continuously their current speeds and locations to the control center so that the control center knows the speeds and locations of all the trains in its area all the time. Using this information, controllers can optimize system performance and respond to events quickly and effectively. Fig. 1 indicates how two trains brake, and maintain a safety distance. From Fig. 1, we can see that as long as the following trains have sufficient braking capabilities, they can travel closely. One of the moving block schemes is the moving space block (MSB) [7], it is a simple scheme, in which the safety stopping distance between two successive trains is d min ¼ v2max =ð2bÞ þ SM;
ð1Þ
here vmax is the maximum speed, b denotes the deceleration rate of trains and SM is the safety margin distance. In general, the distance SM can be written as SM = svmax, where s is called the delay time. The delay time s is the driver and train equipment reaction time, which is an adjustable parameter. For the purposes of the present paper, we consider only the MSB scheme for our numerical simulations, here the delay time s is set to be s = 1. In automatic train control (ATC) system, the critical period occurs at the approach to station, where the preceding train must leave in time for the following train to travel into the station following its worst-case
speed
Braking Curve
Braking Curve
for Train 1 Train 1
for Train 2 Train 2
distance Fig. 1. Principle of the moving block system.
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braking curve so that trains travel at a safe distance. The period can be described by the minimum time headway which is defined as the time interval between successive trains passing the same site at a station. In this work, we discuss the situation where trains travel on a single railway line. Under this condition, the minimum time headway Tf is defined as follows [8]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vmax 2ðLd þ Lt Þ ; ð2Þ þ Tf ¼ s þ Td þ a b here a denotes the acceleration rate of trains, Td is the station dwell time, Lt is the length of the train and Ld is the distance that the preceding train has travelled from the station. When the length Lt and the distance Ld are omitted, i.e., Lt = 0, Ld = 0, the minimum time headway is written as vmax Tf ¼ s þ Td þ : ð3Þ b In our simulations, we record the minimum time headway between two successive trains at a station. In order to obtain the time Tf, the station dwell time Td is set to be large enough so that at a station, trains travel in close proximity. 3. The proposed model In the theory of railway traffic, the motion of a train can be described by the following equation [4]: dv=dt ¼ uf f ðvÞ ub bðvÞ wðvÞ gðxÞ;
ð4Þ
where uf is called the relative traction force which is defined as the proportion of the actual traction force to the maximum traction force, uf 2 [0, 1], f(v) is the specific maximum traction force, ub is called the relative braking force which is defined as the proportion of the actual baking force to the maximum braking force, ub 2 [0, 1], b(v) is the specific maximum braking force, w(v) is the specific resistance to motion, and g(x) is the specific external force caused by track grade and curve resistance. For simplicity, in this work, we only take into account the influences of the traction force and braking force. Under this constraint, Eq. (4) can be simply written as: dv=dt ¼ uf f ðvÞ ub bðvÞ:
ð5Þ
Although this is a gross simplification of the actual motion it is nevertheless adequate for the purposes of the MB simulation we will propose. Under the MB condition, train movement is mainly under the constraint of the safety stopping distance dmin. When the distance between two successive trains is smaller than the safety stopping distance, the following train would be forced to brake to a lower speed. This is an important characteristic behaviour of train movement under the MB condition. We propose that when we describe the motion of a train under the MB condition, the distance between the train and its preceding train should be considered in the equation of the motion of the train. This condition is imposed so that studies of train simulation, such as optimal control and scheduling of trains on a busy network, can properly describe the interaction between adjacent trains. In addition, the influence of the stations on train movement is also an important factor, which should be considered in the equation of the train movement. Based on the concepts mentioned above, we revise the equation of the train movement, i.e. Eq. (5). The revised equation is as follows: dv=dt ¼ uf f ðvÞ ub bðvÞ uh hðdðDxÞÞ;
ð6Þ
here uh is an adjustable parameter. In the revised equation, we introduce an additional term, i.e. the last term in the right hand side of the formula (6). The idea is that drivers adjust the train speeds according to the obtained values of d(Dx). When the current train i is behind the train i + 1, the terms xi and xi+1 are the locations of the trains and di(Dx) is the difference between the minimum distance and the actual distance. When the current train i is behind a station, and the station is occupied by the train i + 1, di(Dx) = dmin (xs xi), otherwise, d i ðDxÞ ¼ v2i =ð2bÞ ðxs xi Þ, xs represents the site of the station in front of the train i. In the proposed model, h(d(Dx)) is a hard limit function, which can be wrote as h(d(Dx)) = [1 + sign(d(Dx))]/2. This additional
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term means that when the distance between two successive trains is smaller than or equal to the safety stopping distance, the following train would be forced to brake to a lower speed. In general, when the distance between two successive trains is larger than the safety stopping distance, the following train is allowed to move with a larger speed. There are three terms in the right hand side of the formula (6). The first term and second term are dominated by the train speed, and the third term is dominated by the distance headway, which is related to the safety control of train movement. When a train travels on a railway line, sometimes it accelerates by the traction force, and sometimes it decelerates by the braking force. In the former case, uf > 0, ub = 0 and uh = 0. In the latter case, uf = 0, ub > 0 and uh > 0. We consider many trains moving on a single line which has a length L. These trains are numbered as 1, 2, . . ., from the last train. As discussed above, the equation of the motion of the train i is as follows: dvi =dt ¼ uf f ðvi Þ ub bðvi Þ uh hðd i ðDxÞÞ:
ð7Þ
By changing the time derivative to the difference in Eq. (7) can be further simplified to a discrete model, vi ðt þ dtÞ ¼ vi ðtÞ þ dtðuf f ðvi ðtÞÞ ub bðvi ðtÞÞ uh hðd i ðDxðtÞÞ;
ð8Þ
where dt is an adjustable parameter that allows for the time lag when the railway traffic flow is varying. Here dt is set to be dt = 1. The difference equation model described by Eq. (8) is suitable for the computation simulation analysis since the time variable is discrete. In this work, we simulate the equation model under the open boundary condition. The open boundary condition is as follows. Firstly, when the section from the site 1 to the site Ls is empty, a train with the speed vmax is created. This train immediately moves according to the equation model. The parameter Ls is called the departure interval, it is larger than the safety stopping distance. Secondly, at the site L, trains simply move out of the system. In order to compare simulation results to field measurements, one iteration roughly corresponds to 1 s, and the length of a unit is about 5 m. This means, for example, that vmax = 10 units/update corresponds to vmax = 180 km/h. It should be pointed out that in the proposed equation model, the braking force includes two terms, i.e. ubb(v) and uhh(d(Dx)). The term ubb(v) is related to the speed control of single train where the influence of other trains and stations on the motion of the train are not considered, and the term uhh(d(Dx)) is related to the influences of other trains and stations. In our simulations, we only focus on the influences of the safety stopping distance and stations on the motion of train. So ub is set to be ub = 0. In many other studies, such as the optimal control of the train movement, the term ubb(v) is necessary. In other words, if we consider the optimal control of the train movement, the term ubb(v) must be considered. 4. Numerical simulations We carry out a computer simulation for the train movement described by Eq. (8). The computation approach is to iterate Eq. (8) under the open boundary condition. The main program is that at each time step, for all trains, we use the current speeds and positions of trains to calculate the speeds and sites of these trains at the next time step. In addition, vi must meet two conditions: (1) vi 6 vmax; (2) vi P 0. In the simulations, these two constraints can be completed by the following steps: (1) if vi > vmax, then vi = vmax; (2) if vi < 0, then vi = 0. In simulations, we focus on the case where the density of trains is high. Under such a condition, we can observe how trains change their speeds, and maintain the safety stopping distance between them. The traffic situation employed in this work is as follows. A number of trains are travelling on a single line with the length L = 1000 (L = 1000 units corresponds to L = 5000 m). The length of the simulation time is T = 1000. One station is designed at the middle of the system, i.e. the site l = 500. When trains arrive at the station, they need stop for a time Td, and then leave the station. The parameters ub and uh must be selected with suitable values so that the sum of the braking capabilities of two different sources do not exceed the total braking capabilities. Here the parameters ub and uh are set to be ub = 0 and uh = 1. In this case, we assume that the braking capabilities of train are within the scope of the total braking capabilities. uf is set to be uf = 0.5. In order to study the complex dynamic behaviour of the train movements, we investigate the space–time diagram of the railway traffic flow. The parameters vmax, Ls and Td are, respectively, set to be vmax = 10,
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Fig. 2. Local space–time diagram of the railway traffic flow.
Ls = 70 and Td = 5. Fig. 2 plots the space–time evolution of the railway traffic flow. Here the simulation (and for the other simulations too) makes a further simplification by assuming that f(v) = 1. The horizontal direction indicates the direction in which trains move ahead, and the vertical direction indicates time. In Fig. 2, the positions of trains are indicated by dots. From Fig. 2, we can find that the railway traffic flow is similar to that of road traffic. The main difference is that in railway traffic, the safety stopping distance must be maintained among the trains. All trains start from the departure site, i.e. the site l = 1, and then arrive at the station, i.e. the middle site l = 500. After the station dwell time Td, they leave the station. When they arrive at the arrival site, i.e. the site l = 1000, they leave the system. Before the station, the train delays form and propagate backward. These are the characteristic behaviours of the train movements on a single railway line under the MB condition. In reality, the dynamic behaviour of train movements and train traffic are complex near any station. When a train arrives at a station, it needs to stop to enable passengers to board and alight. In this case, if the station dwell time is larger than its planned time, the train delays possibly form and propagate backward. Fig. 3 shows the local space–time diagram, which displays the positions and speeds of the trains near the station. Here numbers represent the speeds of the trains and P denotes the site of the station. The values of the parameters vmax, Ls and Td are same as that used in Fig. 2. From Fig. 3, it can be clearly seen that the train c needs to stop at the station, it decelerates, at the following steps, the train d that is directly behind the train c decelerates, and then, 750
time
730
9
8
6
5
3 4
0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4
0.5 0 0 0 0 2.5 4
7 6.5
7.5
8
8.5
9
6 5 5.5 4.5 5 4 4.5 3.5 4 3 3.5 2.5 3 2 2.5 1.5 2 1 1.5 5 0.5 1 9 6 0 0.5 8.5 7 0 0 8 8 0 0 7.5 9 0 0 7 10 0 0 6.5 9.5 0 1 6 0 2 5.5 0 3 5 0 4 4.5 1 3.5 4 2 3 3.5 3 2.5 3 4 2 2.5 5 1.5 2 6 1 1.5 7 0.5 1 8 0 0.5 9 0 0 9 10 0 0 8.5 9.5 0 0 8 9 0 0 7.5 0 0 7 0 0 6.5 1 0 6 2 0 5.5 3 0 5 4 1.5 4.5 5 3.5 4 6 3 3.5 7 2.5 3 8 2 2.5 9 1.5 2 10 1 1.5 9.5 0.5 1 10
740
7
9.5
e
720
710
700
9.5
c
d
9.5
P 400
450
500
550
600
position Fig. 3. A diagram displaying the positions and speeds of trains near the station.
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distance headway
150
100
50
2
4
6
8
10
train Fig. 4. Distribution of the distance headway for vmax = 10, Ls = 80 and Td = 5.
the train e that is behind the train d also decelerates. As the time proceeds, a number of trains that are before the station P are delayed. The delayed effect increases the number of trains near the station. This is the reason that delays form before the station. These results demonstrate that the proposed model can successfully capture the expected delay in train movements. Under the MB condition, one of the important characters of train movement is that the safety stopping distance between two successive trains must be maintained. Whenever the distance between two successive trains is smaller than the safety stopping distance, the following train must decelerate. In order to investigate the situation about the spatial organization of trains, we measure the distribution of the distance headway Dsi at a given time. Here Dsi is distance from the train i to the train i + 1. Fig. 4 shows the distribution of the distance headway at the time t = 1000 for uh = 1. Here the solid line denotes the measurement results, the dotted line denotes the safety stopping distance and the dash line denotes the safety margin distance. In Fig. 4, at the time t = 1000, there are many trains travelling on the single line. From Fig. 4, we can see that the measurement results are all larger than the safety margin distance, but some of them are smaller than the safety stopping distance. The simulation results indicate that if the density of trains is high, trains often decelerate to maintain the safety stopping distance.
35
minimum time headway
30 25 20 15 10 5
5
10
15
20
maximum speed Fig. 5. How the minimum time headway varies with the maximum speed for uh = 1 and Td = 5.
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The minimum time headway at the approach to station is critical and can be used to test the proposed model. For this purpose, we record the time headway of trains at the station and compare them to the theoretical results calculated by Eq. (3). Fig. 5 shows how the minimum time headway Tf varies with the maximum speed vmax. Here we make many measurements with different values of Ls, and then average them. In Fig. 5, the dotted line denotes the measurement values using the proposed model, and the solid line denotes the theoretical results. From Fig. 5, it is obvious that the simulation values of Tf using the proposed model are basically close to the theoretical values. Nevertheless, as mentioned above, many simplifications have been adopted in the proposed equation model. This reduces the accuracy of the proposed model. In Ref. [4] the train speed is subject to a constraint, i.e. v 6 V(x). Here V(x) is a step function of coordinate x, which is not related to the motion of its preceding train. Under the MB condition, we use the optimal approach proposed in Ref. [4] to control the motion of a train. When the condition v < V(x) is met, the train would accelerate. In this case, if the distance between this train and its preceding train is smaller than the safety stopping distance, this train must decelerate. The reason of the occurrence of the conflicting result is that under the MB condition, the control of a train is related to the site of its preceding train, however, this factor was not considered in Ref. [4]. In fact, the similar problems also exist in other optimal control approaches, such as Ref. [5]. In order to overcome these deficiencies, we directly introduce a new term into the equation of the motion of a train, i.e. the term uhh(d(Dx)) in Eq. (6). 5. Conclusions In conclusions, we introduce a new term into the equation of the motion of a train under the moving block condition. Using the iterative algorithm, we simulate the train movement described by Eq. (8). The numerical simulation results indicate that the proposed model can well describe the train movement under the MB condition. Not only the dynamic behaviours of the train movement can be described, but also some complex phenomena observed for real railway traffic, such as the train delays, can be reproduced. Our studies are based on a simplified form of the equation of motion outlined in Ref. [4] with an additional term added to capture the interaction between adjacent trains. Further similar studies should be done based on the general equation of motion but without the subsequent simplifications used here. In addition, the new term h introduced into the improving equation model has a simple expression. Further studies also include more realistic forms for the delay function h. We think that it is worthy of further study. Acknowledgement The project is supported by National Basic Research Program of China under Grant No 2006CB705500, the National Natural Science Foundation of China under Grant No 60634010, New Century Excellent Talents in University under Grant No NCET-06-0074, the Key Project of Chinese Ministry of Education under Grant No 107007 and the Science and Technology Foundation of Beijing Jiaotong University under Grant No 2004SM026. References [1] K. Petar, S. Guedial, Minimum-energy control of a traction motor, IEEE Trans. Automat. Contr. 17 (1972) 92–95. [2] P.G. Howlett, J. Cheng, Optimal driving strategies for a train on a track with continuously varying gradient, J. Aust. Math. Soc. B 38 (1997) 388–411. [3] J. Cheng, Y. Davydova, P. Howlett, P. Pudney, Optimal driving strategies for a train journey with non-zero track gradient and speed limits, IMA J. Math. Appl. Bus. Ind. 10 (1999) 89–115. [4] Rongfang (Rachel) Liu, Iakov M. Golovitcher, Energy-efficient operation of rail vehicles, Trans. Res. A 37 (2003) 917–932. [5] P. Howlett, The optimal control of a train, Ann. Oper. Res. 98 (2000) 65–87. [6] A. Adinolfi, R. Lamedica, C. Modesto, A. Prudenzi, S. Vimercati, Experimental assessment of energy saving duo to trains braking in an electrified subway line, IEEE Trans. Power Deliver. 13 (1998) 536–1542. [7] L.V. Pearson, Moving block signalling, Ph.D. thesis, Loughborough University of Technology, England, 1973. [8] R.J. Hill, Electrical railway traction: Part 4 – Signalling and interlocking, IEE Power Eng. J. (1995) 201.
An improved equation model for the train movement KePing Li *, ZiYou Gao State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, PR China Received 6 July 2006; received in revised form 2 June 2007; accepted 23 July 2007 Available online 31 July 2007
Abstract In this paper, a new term is introduced into the equation of train movement, which is about the influences of the safety stopping distance and stations on the motion of train, when there are many trains on the line. The aim is that the improved equation can well describe the motion of train under the moving block condition. In order to test the improved model, we use a simulation analysis approach to solve a simplified form of the improved equation model. In the simulations, we investigate the space–time diagram for the railway traffic flow and the trajectories of the train movement. The numerical simulation results demonstrate that the improved model can well describe the dynamic behaviour of the train movement under the moving block condition. Some complex phenomena of train movement can be well reproduced but more work is required for a full simulation of the general model. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Train movement; Simulation analysis; Moving block
1. Introduction Train movement ‘‘depends’’ on the calculation of the speed and distance profile while a train is travelling from one point to another. In general, train movement is governed by many factors, such as the track geometry, control signalling, traction equipment, power supply system and speed restrictions. When we analyze and evaluate the theoretical control algorithm of train movement, the equation of the train movement is the basis for any computer simulation. In this field, one of the important problems is how to establish and solve the equation of the train movement. In the past decades, a number of studies have been done for the optimization and control of train movement. Most of them were focused on the solution to the optimal control of a single train. Petar and Guedial solved the problem of the energy optimal control of the motion of a train by the maximum principle [1]. Howlett and Cheng derived the equation of the motion of a train to determine the optimal control strategies for a train [2]. Ref. [3] proposed a solution algorithm to find an optimal driving strategy for train movement where discrete control is used and speed limits are imposed. Liu and Golovitcher outlined a method for the calcu*
Corresponding author. Tel.: +86 10 5168 2892. E-mail address: [email protected] (K. Li).
1569-190X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2007.07.006
K. Li, Z. Gao / Simulation Modelling Practice and Theory 15 (2007) 1156–1162
1157
lation of energy optimal control [4]. In Ref. [5], Howlett used a generalized equation of train movement to determine an optimal driving strategy, where the continuous control and discrete control are all discussed. Some other studies discussed the traction and braking applications. For example, Adinolfi et al. accessed the impact of the train braking techniques on the energy consumption [6]. However, all of them mentioned above did not consider the influences of other trains and stations in the equation of the train movement. Hence the calculation of train movement is isolated and may not dynamically capture the characteristic behaviour of a busy network. This limitation will strongly affect the computer simulation of railway traffic. In this work, we propose a new equation model, where the influence of other trains is considered especially in relation to the requirement that safe distances be maintained behind preceding trains that have stopped at a station. Using the proposed model, we hope that the computer simulation can well describe the reality of railway traffic. The paper is organized as follows. We introduce the principle of the moving block system in Section 2. In Section 3, we propose the improved equation model. The numerical and analytical results for a simplified form of the improved model are presented in Section 4. Finally, conclusions of this approach are presented. 2. Moving block system Moving block (MB) system is a special type of train signalling system [7]. Its advantages are that the line capacity can be increased, and the traffic fluidity and the energy efficiency can be improved. In practice, MB relies on continuous communication between train and wayside equipment, which can be used for control, protection and regulation of train movement. With the MB train control system, the minimum train separation to be defined solely by the dynamic performance of the train can be provided. On a MB equipped system, electronic communications between the control center and the train continuously control the trains, and make them maintain a safe stopping distance. At the same time, trains transmit continuously their current speeds and locations to the control center so that the control center knows the speeds and locations of all the trains in its area all the time. Using this information, controllers can optimize system performance and respond to events quickly and effectively. Fig. 1 indicates how two trains brake, and maintain a safety distance. From Fig. 1, we can see that as long as the following trains have sufficient braking capabilities, they can travel closely. One of the moving block schemes is the moving space block (MSB) [7], it is a simple scheme, in which the safety stopping distance between two successive trains is d min ¼ v2max =ð2bÞ þ SM;
ð1Þ
here vmax is the maximum speed, b denotes the deceleration rate of trains and SM is the safety margin distance. In general, the distance SM can be written as SM = svmax, where s is called the delay time. The delay time s is the driver and train equipment reaction time, which is an adjustable parameter. For the purposes of the present paper, we consider only the MSB scheme for our numerical simulations, here the delay time s is set to be s = 1. In automatic train control (ATC) system, the critical period occurs at the approach to station, where the preceding train must leave in time for the following train to travel into the station following its worst-case
speed
Braking Curve
Braking Curve
for Train 1 Train 1
for Train 2 Train 2
distance Fig. 1. Principle of the moving block system.
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K. Li, Z. Gao / Simulation Modelling Practice and Theory 15 (2007) 1156–1162
braking curve so that trains travel at a safe distance. The period can be described by the minimum time headway which is defined as the time interval between successive trains passing the same site at a station. In this work, we discuss the situation where trains travel on a single railway line. Under this condition, the minimum time headway Tf is defined as follows [8]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vmax 2ðLd þ Lt Þ ; ð2Þ þ Tf ¼ s þ Td þ a b here a denotes the acceleration rate of trains, Td is the station dwell time, Lt is the length of the train and Ld is the distance that the preceding train has travelled from the station. When the length Lt and the distance Ld are omitted, i.e., Lt = 0, Ld = 0, the minimum time headway is written as vmax Tf ¼ s þ Td þ : ð3Þ b In our simulations, we record the minimum time headway between two successive trains at a station. In order to obtain the time Tf, the station dwell time Td is set to be large enough so that at a station, trains travel in close proximity. 3. The proposed model In the theory of railway traffic, the motion of a train can be described by the following equation [4]: dv=dt ¼ uf f ðvÞ ub bðvÞ wðvÞ gðxÞ;
ð4Þ
where uf is called the relative traction force which is defined as the proportion of the actual traction force to the maximum traction force, uf 2 [0, 1], f(v) is the specific maximum traction force, ub is called the relative braking force which is defined as the proportion of the actual baking force to the maximum braking force, ub 2 [0, 1], b(v) is the specific maximum braking force, w(v) is the specific resistance to motion, and g(x) is the specific external force caused by track grade and curve resistance. For simplicity, in this work, we only take into account the influences of the traction force and braking force. Under this constraint, Eq. (4) can be simply written as: dv=dt ¼ uf f ðvÞ ub bðvÞ:
ð5Þ
Although this is a gross simplification of the actual motion it is nevertheless adequate for the purposes of the MB simulation we will propose. Under the MB condition, train movement is mainly under the constraint of the safety stopping distance dmin. When the distance between two successive trains is smaller than the safety stopping distance, the following train would be forced to brake to a lower speed. This is an important characteristic behaviour of train movement under the MB condition. We propose that when we describe the motion of a train under the MB condition, the distance between the train and its preceding train should be considered in the equation of the motion of the train. This condition is imposed so that studies of train simulation, such as optimal control and scheduling of trains on a busy network, can properly describe the interaction between adjacent trains. In addition, the influence of the stations on train movement is also an important factor, which should be considered in the equation of the train movement. Based on the concepts mentioned above, we revise the equation of the train movement, i.e. Eq. (5). The revised equation is as follows: dv=dt ¼ uf f ðvÞ ub bðvÞ uh hðdðDxÞÞ;
ð6Þ
here uh is an adjustable parameter. In the revised equation, we introduce an additional term, i.e. the last term in the right hand side of the formula (6). The idea is that drivers adjust the train speeds according to the obtained values of d(Dx). When the current train i is behind the train i + 1, the terms xi and xi+1 are the locations of the trains and di(Dx) is the difference between the minimum distance and the actual distance. When the current train i is behind a station, and the station is occupied by the train i + 1, di(Dx) = dmin (xs xi), otherwise, d i ðDxÞ ¼ v2i =ð2bÞ ðxs xi Þ, xs represents the site of the station in front of the train i. In the proposed model, h(d(Dx)) is a hard limit function, which can be wrote as h(d(Dx)) = [1 + sign(d(Dx))]/2. This additional
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term means that when the distance between two successive trains is smaller than or equal to the safety stopping distance, the following train would be forced to brake to a lower speed. In general, when the distance between two successive trains is larger than the safety stopping distance, the following train is allowed to move with a larger speed. There are three terms in the right hand side of the formula (6). The first term and second term are dominated by the train speed, and the third term is dominated by the distance headway, which is related to the safety control of train movement. When a train travels on a railway line, sometimes it accelerates by the traction force, and sometimes it decelerates by the braking force. In the former case, uf > 0, ub = 0 and uh = 0. In the latter case, uf = 0, ub > 0 and uh > 0. We consider many trains moving on a single line which has a length L. These trains are numbered as 1, 2, . . ., from the last train. As discussed above, the equation of the motion of the train i is as follows: dvi =dt ¼ uf f ðvi Þ ub bðvi Þ uh hðd i ðDxÞÞ:
ð7Þ
By changing the time derivative to the difference in Eq. (7) can be further simplified to a discrete model, vi ðt þ dtÞ ¼ vi ðtÞ þ dtðuf f ðvi ðtÞÞ ub bðvi ðtÞÞ uh hðd i ðDxðtÞÞ;
ð8Þ
where dt is an adjustable parameter that allows for the time lag when the railway traffic flow is varying. Here dt is set to be dt = 1. The difference equation model described by Eq. (8) is suitable for the computation simulation analysis since the time variable is discrete. In this work, we simulate the equation model under the open boundary condition. The open boundary condition is as follows. Firstly, when the section from the site 1 to the site Ls is empty, a train with the speed vmax is created. This train immediately moves according to the equation model. The parameter Ls is called the departure interval, it is larger than the safety stopping distance. Secondly, at the site L, trains simply move out of the system. In order to compare simulation results to field measurements, one iteration roughly corresponds to 1 s, and the length of a unit is about 5 m. This means, for example, that vmax = 10 units/update corresponds to vmax = 180 km/h. It should be pointed out that in the proposed equation model, the braking force includes two terms, i.e. ubb(v) and uhh(d(Dx)). The term ubb(v) is related to the speed control of single train where the influence of other trains and stations on the motion of the train are not considered, and the term uhh(d(Dx)) is related to the influences of other trains and stations. In our simulations, we only focus on the influences of the safety stopping distance and stations on the motion of train. So ub is set to be ub = 0. In many other studies, such as the optimal control of the train movement, the term ubb(v) is necessary. In other words, if we consider the optimal control of the train movement, the term ubb(v) must be considered. 4. Numerical simulations We carry out a computer simulation for the train movement described by Eq. (8). The computation approach is to iterate Eq. (8) under the open boundary condition. The main program is that at each time step, for all trains, we use the current speeds and positions of trains to calculate the speeds and sites of these trains at the next time step. In addition, vi must meet two conditions: (1) vi 6 vmax; (2) vi P 0. In the simulations, these two constraints can be completed by the following steps: (1) if vi > vmax, then vi = vmax; (2) if vi < 0, then vi = 0. In simulations, we focus on the case where the density of trains is high. Under such a condition, we can observe how trains change their speeds, and maintain the safety stopping distance between them. The traffic situation employed in this work is as follows. A number of trains are travelling on a single line with the length L = 1000 (L = 1000 units corresponds to L = 5000 m). The length of the simulation time is T = 1000. One station is designed at the middle of the system, i.e. the site l = 500. When trains arrive at the station, they need stop for a time Td, and then leave the station. The parameters ub and uh must be selected with suitable values so that the sum of the braking capabilities of two different sources do not exceed the total braking capabilities. Here the parameters ub and uh are set to be ub = 0 and uh = 1. In this case, we assume that the braking capabilities of train are within the scope of the total braking capabilities. uf is set to be uf = 0.5. In order to study the complex dynamic behaviour of the train movements, we investigate the space–time diagram of the railway traffic flow. The parameters vmax, Ls and Td are, respectively, set to be vmax = 10,
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Fig. 2. Local space–time diagram of the railway traffic flow.
Ls = 70 and Td = 5. Fig. 2 plots the space–time evolution of the railway traffic flow. Here the simulation (and for the other simulations too) makes a further simplification by assuming that f(v) = 1. The horizontal direction indicates the direction in which trains move ahead, and the vertical direction indicates time. In Fig. 2, the positions of trains are indicated by dots. From Fig. 2, we can find that the railway traffic flow is similar to that of road traffic. The main difference is that in railway traffic, the safety stopping distance must be maintained among the trains. All trains start from the departure site, i.e. the site l = 1, and then arrive at the station, i.e. the middle site l = 500. After the station dwell time Td, they leave the station. When they arrive at the arrival site, i.e. the site l = 1000, they leave the system. Before the station, the train delays form and propagate backward. These are the characteristic behaviours of the train movements on a single railway line under the MB condition. In reality, the dynamic behaviour of train movements and train traffic are complex near any station. When a train arrives at a station, it needs to stop to enable passengers to board and alight. In this case, if the station dwell time is larger than its planned time, the train delays possibly form and propagate backward. Fig. 3 shows the local space–time diagram, which displays the positions and speeds of the trains near the station. Here numbers represent the speeds of the trains and P denotes the site of the station. The values of the parameters vmax, Ls and Td are same as that used in Fig. 2. From Fig. 3, it can be clearly seen that the train c needs to stop at the station, it decelerates, at the following steps, the train d that is directly behind the train c decelerates, and then, 750
time
730
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0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4
0.5 0 0 0 0 2.5 4
7 6.5
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9
6 5 5.5 4.5 5 4 4.5 3.5 4 3 3.5 2.5 3 2 2.5 1.5 2 1 1.5 5 0.5 1 9 6 0 0.5 8.5 7 0 0 8 8 0 0 7.5 9 0 0 7 10 0 0 6.5 9.5 0 1 6 0 2 5.5 0 3 5 0 4 4.5 1 3.5 4 2 3 3.5 3 2.5 3 4 2 2.5 5 1.5 2 6 1 1.5 7 0.5 1 8 0 0.5 9 0 0 9 10 0 0 8.5 9.5 0 0 8 9 0 0 7.5 0 0 7 0 0 6.5 1 0 6 2 0 5.5 3 0 5 4 1.5 4.5 5 3.5 4 6 3 3.5 7 2.5 3 8 2 2.5 9 1.5 2 10 1 1.5 9.5 0.5 1 10
740
7
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e
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9.5
c
d
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P 400
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position Fig. 3. A diagram displaying the positions and speeds of trains near the station.
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distance headway
150
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train Fig. 4. Distribution of the distance headway for vmax = 10, Ls = 80 and Td = 5.
the train e that is behind the train d also decelerates. As the time proceeds, a number of trains that are before the station P are delayed. The delayed effect increases the number of trains near the station. This is the reason that delays form before the station. These results demonstrate that the proposed model can successfully capture the expected delay in train movements. Under the MB condition, one of the important characters of train movement is that the safety stopping distance between two successive trains must be maintained. Whenever the distance between two successive trains is smaller than the safety stopping distance, the following train must decelerate. In order to investigate the situation about the spatial organization of trains, we measure the distribution of the distance headway Dsi at a given time. Here Dsi is distance from the train i to the train i + 1. Fig. 4 shows the distribution of the distance headway at the time t = 1000 for uh = 1. Here the solid line denotes the measurement results, the dotted line denotes the safety stopping distance and the dash line denotes the safety margin distance. In Fig. 4, at the time t = 1000, there are many trains travelling on the single line. From Fig. 4, we can see that the measurement results are all larger than the safety margin distance, but some of them are smaller than the safety stopping distance. The simulation results indicate that if the density of trains is high, trains often decelerate to maintain the safety stopping distance.
35
minimum time headway
30 25 20 15 10 5
5
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maximum speed Fig. 5. How the minimum time headway varies with the maximum speed for uh = 1 and Td = 5.
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The minimum time headway at the approach to station is critical and can be used to test the proposed model. For this purpose, we record the time headway of trains at the station and compare them to the theoretical results calculated by Eq. (3). Fig. 5 shows how the minimum time headway Tf varies with the maximum speed vmax. Here we make many measurements with different values of Ls, and then average them. In Fig. 5, the dotted line denotes the measurement values using the proposed model, and the solid line denotes the theoretical results. From Fig. 5, it is obvious that the simulation values of Tf using the proposed model are basically close to the theoretical values. Nevertheless, as mentioned above, many simplifications have been adopted in the proposed equation model. This reduces the accuracy of the proposed model. In Ref. [4] the train speed is subject to a constraint, i.e. v 6 V(x). Here V(x) is a step function of coordinate x, which is not related to the motion of its preceding train. Under the MB condition, we use the optimal approach proposed in Ref. [4] to control the motion of a train. When the condition v < V(x) is met, the train would accelerate. In this case, if the distance between this train and its preceding train is smaller than the safety stopping distance, this train must decelerate. The reason of the occurrence of the conflicting result is that under the MB condition, the control of a train is related to the site of its preceding train, however, this factor was not considered in Ref. [4]. In fact, the similar problems also exist in other optimal control approaches, such as Ref. [5]. In order to overcome these deficiencies, we directly introduce a new term into the equation of the motion of a train, i.e. the term uhh(d(Dx)) in Eq. (6). 5. Conclusions In conclusions, we introduce a new term into the equation of the motion of a train under the moving block condition. Using the iterative algorithm, we simulate the train movement described by Eq. (8). The numerical simulation results indicate that the proposed model can well describe the train movement under the MB condition. Not only the dynamic behaviours of the train movement can be described, but also some complex phenomena observed for real railway traffic, such as the train delays, can be reproduced. Our studies are based on a simplified form of the equation of motion outlined in Ref. [4] with an additional term added to capture the interaction between adjacent trains. Further similar studies should be done based on the general equation of motion but without the subsequent simplifications used here. In addition, the new term h introduced into the improving equation model has a simple expression. Further studies also include more realistic forms for the delay function h. We think that it is worthy of further study. Acknowledgement The project is supported by National Basic Research Program of China under Grant No 2006CB705500, the National Natural Science Foundation of China under Grant No 60634010, New Century Excellent Talents in University under Grant No NCET-06-0074, the Key Project of Chinese Ministry of Education under Grant No 107007 and the Science and Technology Foundation of Beijing Jiaotong University under Grant No 2004SM026. References [1] K. Petar, S. Guedial, Minimum-energy control of a traction motor, IEEE Trans. Automat. Contr. 17 (1972) 92–95. [2] P.G. Howlett, J. Cheng, Optimal driving strategies for a train on a track with continuously varying gradient, J. Aust. Math. Soc. B 38 (1997) 388–411. [3] J. Cheng, Y. Davydova, P. Howlett, P. Pudney, Optimal driving strategies for a train journey with non-zero track gradient and speed limits, IMA J. Math. Appl. Bus. Ind. 10 (1999) 89–115. [4] Rongfang (Rachel) Liu, Iakov M. Golovitcher, Energy-efficient operation of rail vehicles, Trans. Res. A 37 (2003) 917–932. [5] P. Howlett, The optimal control of a train, Ann. Oper. Res. 98 (2000) 65–87. [6] A. Adinolfi, R. Lamedica, C. Modesto, A. Prudenzi, S. Vimercati, Experimental assessment of energy saving duo to trains braking in an electrified subway line, IEEE Trans. Power Deliver. 13 (1998) 536–1542. [7] L.V. Pearson, Moving block signalling, Ph.D. thesis, Loughborough University of Technology, England, 1973. [8] R.J. Hill, Electrical railway traction: Part 4 – Signalling and interlocking, IEE Power Eng. J. (1995) 201.