Modelling and Simulation for Train Movement Control Using Car-Following Strategy

Commun. Theor. Phys. 55 (2011) 29 34 Vol. 55, No. 1, January 15, 2011 Modelling and Simulation for Train Movement Control Using Car-Following Strategy LI Ke-Ping (Ó ), GAO Zi-You (Ô Ð), and TANG Tao (» ) State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China (Received April 13, 2010; revised manuscript received July 1, 2010) Abstract Based on optimal velocity car-following model, in this paper, we propose a new railway traffic model for describing the process of train movement control. In the proposed model, we give an improved form of the optimal velocity function V opt, which is considered as the desired velocity function for train movement control under different control conditions. In order to test the proposed model, we simulate and analyze the trajectories of train movements, moreover, discuss the relationship curves between the train allowable velocity and the site of objective point in detail. Analysis results indicate that the proposed model can well capture some realistic futures of train movement control. PACS numbers: 02.70.-c, 89.40.-a Key words: train movement control, railway traffic, car-following model 1 Introduction Train movement control system is an important part of railway signalling system. Its function is to supervise the train movement velocities according to the positions of trains, the safety distance and the station routes etc. In the engineering planning and design, when we analyze and evaluate the theoretical control algorithm of train movement, computer-based simulation is necessary. Recently, many results have been reported in this field. [1 5] A key step is how to determine the relationship curve between the velocity and distance under a specified traffic condition. In the past decades, traffic flow problems have attracted great attentions. A number of studies have been done for investigating the various aspects of traffic phenomena. [6] Car-following model is one of the successful traffic models for investigating the various aspects of traffic phenomena. It has some advantages, which are suitable for simulating train movement under different control conditions. For example, driver controls the acceleration or deceleration of vehicle so that he/she maintains the legal safety velocity according to the motions of other vehicles. Since car-following model uses realistic driver behavior and detailed vehicle characters, many car-following models have been proposed to simulate various traffic flows, which are observed in real traffic. [7 12] In this work, based on optimal velocity car-following model, we propose a new improved model for simulating train movement control. The relationship curves between velocity and distance are discuss in detail. Since the proposed simulation model is a kind of dynamics model, which can well capture the dynamics behaviors of trains under different traffic conditions. Moreover, it provides a powerful tool for analyzing and evaluating train movement control algorithm. To our knowledge, this work explicitly shows this effect for car-following model for the first time. The paper is organized as follows. In Sec. 2, we introduce the optimal velocity model. The principle of train movement control is introduced in Sec. 3. In Sec. 4, we outline the proposed model. The numerical and analytical results are presented in Sec. 5. Finally, conclusion of this approach is outlined. 2 Optimal Velocity Model In car-following model, each vehicle obeys the common equation of motion. Usually, it is assumed that each vehicle has the legal velocity. He/She controls the acceleration or deceleration so that he/she maintains the legal safety velocity according to the motion of his/her leading vehicle. In car-following theories, the basic philosophy for the n-th vehicle (n = 1, 2,...) can be summarized by the equation, [13 14] [Response] n [Stimulus] n. (1) Each driver responds to his/her surrounding traffic conditions only by changing the velocity of the vehicle. The stimulus may be composed of the velocity of the vehicle and the distance headway, etc. The nature of the stimulus can be the behavioral force or generalized force. [11] In carfollowing model, the driver maintains a safe distance from his/her leading vehicle by choosing own desired velocity. Classical car-following model was proposed by physicist Pipes. [7] In order to account for the time lag, Chandler suggested an improved car-following model, [8] ẍ n (t + T) = λ[ẋ n+1 (t) ẋ n (t)], (2) Supported by the National Natural Science Foundation of China under Grant Nos. 60634010 and 60776829 and the State Key Laboratory of Rail Traffic Control and Safety (Contract No. RCS2008ZZ001 and RCS2010ZZ001), Beijing Jiaotong University c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

30 Communications in Theoretical Physics Vol. 55 here x n (t) is the position of vehicle n at time t, T is a response time lag, and λ is the sensitive coefficient. For a more realistic description, Newell presented the optimal velocity car-following model. [9] More than 30 years later, Bando et al. suggested a famous optimal velocity model, [10] ẍ n (t) = V opt ( x n (t)) ẋ n (t), (3) τ where τ is the relaxation time, and x n (t) = x n+1 (t) x n (t) is the distance headway of vehicle n at time t. V opt, called optimal velocity function, is taken as V opt ( x n (t)) = (v max /2){tanh[ x n (t) x c ] + tanh[x c ]}. According to the driving strategy described in formula (3), the n-th vehicle tends to maintain a safe velocity that depends on the relative position. Lately, many works have been done, which extended the optimal velocity model. But, most of them are focused on analyzing density wave characters from the point of physics. 3 Principle of Train Movement Control Train movement control system plays a central role in securing train movement safety and increasing transportation efficiency, which is regarded as the nerve center in railway transportation. In principle, train movement control is to determine the allowable velocity of the current train according to the positions of other trains, the necessary safety distance and the station routes etc. In the today s world, leading train movement control systems are the France U/T system, German LZB system, and Japan ATC system. The essential differences among these leading systems are the control mode and the information transmission mode between train and ground. Two types of the train movement control system have been developed, i.e., the fixed block and moving block systems. One of the important control mode is the continuous objective distance control mode. In the objective distance control mode, the velocity control model (or simulation model) is used to calculate the relationship curve between the train allowable velocity and the objective distance. Such a calculation is based on the position of objective point, the objective velocity and the train parameters, etc. In the fixed block system, the entrance of the occupied block in front of the current train is regarded as the objective point. According to the braking performance, the allowable velocity of the current train is calculated. In the moving block system, the position of the leading train s rear of the current train is regarded as the objective point. Usually, the train safety distance mainly includes two parts, i.e. the train braking distance and safety protection distance. In this paper, we mainly consider the continuous objective distance control mode. Fixed block system has been widely used on modern railway traffic for more than one century. With the fixed block system, the train movement is strongly related to the colors of the signalling lights. However, as signalling technology developed, moving block system gains considerable importance. With the moving block system, electronic communications between the control center and trains continuously control the trains, and make them maintain safety distance. 4 Proposed Model In optimal velocity model, V opt represents driver s desired velocity, which depends on the distance headway of vehicles. In order to describe various aspects of traffic phenomena, one has to postulate a specific function form of V opt. In general, as x n 0, V opt 0, and as x n, V opt is bounded. Along this line of approach, we propose a new form of V opt, which can be regarded as the driver s desired velocity in train movement control. The new function has the following form, V opt ( x n (t)) = ((v c v f )/2)[tanh(coe ( x n (t) sm)) + tanh(coe sm)] + v f, (4) where v c is the current limited maximum velocity, v t is the limited maximum velocity at objective point, and sm is the safety protection distance. coe is an adjustable parameter, which is used to change the slop of the function V opt. Figure 1 shows the figure of V opt for v c = 2, v t = 0, and sm = 4. In Fig. 1, s represents the braking distance. The site of the objective point is at the site of x = 0. Fig. 1 A plot of V opt. In moving block system, v c represents the maximum velocity v max, and v t is set to be v t = 0. In fixed block system, v c and v t have complex form. For example, we consider a three-aspect fixed block system. (i) when the color of the signalling light in front of the n-th train is red, this train is not allowed to pass through the site where the signalling light is located. In this case, v c = v r and v t = 0. Here v r is the limited velocity under the red light condition. (ii) when the color of signalling light in front of the n-th train is green, this train can travel with the maximum velocity. In this case, v c = v max and v t = v y. Here v y is the limited velocity under the yellow light condition. (iii)

No. 1 Communications in Theoretical Physics 31 when the color of signalling light in front of the n-th train is yellow, this train can only be allowed to travel with the limited velocity. In this case, v c = v y and v t = v r. The dynamics behavior of train movement near station is complex. In fixed block system, if the station is occupied by one train, the color of the first signalling light in front of the station is red. In moving block system, the current train must keep the safety distance from the station. Otherwise, the current train can travel into the station directly. As the current train arrives at the station, there are two possible cases: (i) it stops at the station. After the station dwell time T d, it leaves the station; (ii) it passes through the station with a limited velocity. The optimal velocity model proposed in Ref. [10] is primitively used to investigate road traffic flow. The deceleration and acceleration of vehicle are limited between the region [ 3 m/s 2, 4 m/s 2 ]. But such a region in railway traffic is smaller. Moreover, the velocity relaxation time τ in railway traffic is larger than that in city traffic and freeway traffic (realistic velocity relaxation time is of the order of 10 s in city traffic and 40 s in freeway traffic). In order to use car-following model to simulate the train movement in railway traffic, here we further improve the optimal velocity model. The improved model is as follows, ẍ n (t) = c 1 [1 exp( c 2 τ)] V opt ( x n (t)) ẋ n (t). (5) τ In above equation, we introduce a function c 1 [1 exp( c 2 τ)]. This function ensures the improved equation model to meet two conditions: (i) the deceleration and acceleration of train are limited between a smaller region; (ii) collision can be avoided when the velocity relaxation time τ is of high order. Here c 1 and c 2 are adjustable parameters. The values of c 1 and c 2 are related to the maximum velocity v max. For a given v max, we need to choose suitable values of c 1 and c 2. The boundary condition used in this paper is open. Considering a single line track section with a length of L, the boundary condition is as follows. In fixed block system, when the color of the first signalling light is green, a train with the velocity v max is created. In moving block system, when the section from the site 1 to the site L s is empty, a train with the velocity v max is created. The newborn train immediately travels according to the equation model (5). At the site L, trains simply move out of the system. In order to compare simulation results to reality observations, one iteration roughly corresponds to 1 sec, and the length of a unit is about 1 m. This means, for example, that v max = 10 units/update corresponds to v max = 36 km/h. 5 Numerical Computations Using the optimal velocity car-following model, the analytical calculations are very difficult. Usually, the dynamical equation of car-following model is rewritten as a difference equation. Such a difference equation is solved using iteration method. Basic programme is that at each time step, for all trains, we use the current velocities and positions of trains to calculate the velocities and positions of these trains at next time step. In simulations, a system of L = 1000 is considered, which represents a single line track section. The number of the iteration time step is T s = 1000. The train safety distance is calculated by x c = vmax 2 /2b + sm. b represents the average braking rate. Here b and sm are respectively set to be b = 0.5 and sm = 5. The values of v max and τ are taken as v max = 10 and τ = 100. The station dwell time T d is set to be T d = 20. After sufficient transient, we begin to record the relevant data of traffic flow. 5.1 Moving Block System Fig. 2 (a) A diagram displaying the positions and velocities of the tracked train; (b) A diagram displaying the positions and times of the tracked train. With the moving block control system, one station is designed at the middle of the system. As a train arrives at the station, it needs to stop for a time interval T d, and then leaves the station. A number of simulations demonstrate that when the departure interval L s is low, some trains must decelerate and then stop behind their leading trains. Under such a condition, we can observe how trains change their velocities, and maintain the safety distance

32 Communications in Theoretical Physics Vol. 55 among them. We track one train which departs at the time t = 500. Since the trains before the tracked train stop, the tracked train needs to stop at one site. Figure 2 plots the velocity variance of the tracked train for L s = 2x c and coe = 0.1. Here the parameters c 1 and c 2 are respectively set to be c 1 = 100 and c 2 = 0.001. Figure 2(a) shows how velocity varies with position, and Fig. 2(b) shows how velocity varies with time. In Fig. 2, the objective point of the tracked train is its leading train. From Fig. 2, we can clearly see the current allowable velocity of the tracked train at each site from the current site to the site of objective point. With moving block control system, along track section except for station, there are no signalling lights among trains. All trains determine their allowable velocities according to the positions of their leading trains. Here safety distance is an important factors, which must be maintained. For example, when the distance between two successive trains is smaller than the safety distance, the following train must decelerate. In order to test the proposed model, we record the distance headway s n at a given time. s n is defined as s n = x n+1 x n. Figure 4 shows the distribution of the distance headway at the time t = 1000. Here the solid line denotes the measurement results, and the dotted line denotes the safety protection distance. In Fig. 4, at the time t = 1000, there are 5 trains travelling on the single line. From Fig. 4, it is obvious that all the measurement results are larger than the safety protection distance. The simulation result indicates that the proposed model is an effectively tool for simulating train movement. Here the collisions between successive trains can be avoided. Fig. 3 A diagram displaying the positions and velocities of the tracked train for L s = 2x c. Fig. 5 Distribution of the distance headway for L s = x c. Fig. 4 Accelerations and decelerations of trains for L s = x c. The parameter coe used in formula (4) is an adjustable parameter. As mentioned in Sec. 4, it is used to change the slop of the function V opt. When one train travels from one station to other station, we can change the acceleration rate or deceleration rate by controlling the parameter coe. Figure 3 places the velocity variance of one tracked train for coe = 0.02. Compared Fig. 3 to Fig. 2(b), it is obvious that the train decreases gradually in Fig. 3. The deceleration shown in Fig. 3 is smaller than that shown in Fig. 2(a). In this paper, our motivation is to improve the existing car-following model, and make it suitable for describing train movement control in railway traffic. In order to reduce the regions of the accelerations and decelerations of trains, we introduce the function c 1 [1 exp( c 2 τ)]. By changing the values of c 1 and c 2, the decelerations and accelerations of trains can be limited between a smaller region. Figure 5 plots the decelerations and accelerations of all trains at all time. From Fig. 5, we can see that the accelerations of trains are between the region [0 m/s 2, 1 m/s 2 ], and the decelerations of trains are between the region [ 1 m/s 2, 0 m/s 2 ]. In general, the higher the values of c 1 and c 2 are, the larger the region limited the deceleration and acceleration are. In order to further study the dynamics behaviors of the train movements, we investigate the space-time diagram of the railway traffic flow for different values of the parameters c 1, c 2, Ls, and T d. Figure 6 plots the spacetime evolution of the railway traffic flow for coe = 0.1

No. 1 Communications in Theoretical Physics 33 and T d = 40. Here the horizontal direction indicates the direction in which trains move ahead, and the vertical direction indicates time. In Fig. 6, the positions of trains are indicated by dots. From Fig. 6, all trains start from the departure site (l = 1), and then arrive at the station (l = 500). After the station dwell time T d, they leave the station. When they arrive at the arrival site l = 1000, they leave the system. In Fig. 6(b), before the station, the train delays form and propagate backward. These are the characteristic behaviors of the train movements on a single railway line. that the proposed model can successfully capture the real delay in railway traffic. Fig. 7 A diagram displaying the positions and speeds of two tracked trains for c 1 = 120 and c 2 = 0.015. 5.2 Fixed Block System Fig. 8 A diagram displaying the positions and velocities of the tracked train. Fig. 6 Local space-time diagram of the railway traffic flow for c 1 = 120 and c 2 = 0.015. (a) L s = 5x c; (b) L s = 2x c. Under MB condition, the dynamics behaviors of train movements are complex. Between successive trains, the safety stopping distance must be maintained. Especially, as the density of trains on a same railway line is higher, trains need continuously accelerate or brake. In this case, the train delays possibly form and propagate backward. Figure 7 shows the local space-time diagram which displays the positions and speeds of two trains. Here numbers represent the speeds of the trains. The values of the parameters c 1, c 2, and Ls are same as that used in Fig. 6. From Fig. 7, it can be clearly seen that the train A decelerates, at the following steps, the train B that is directly behind the train A decelerates. As the time proceeds, a number of trains are delayed. This is the reason that train delays form before the station. These results demonstrate With the fixed block control system, the considered tack section is divided into two blocks. Three signalling lights are respectively located at the sites l = 1, l = 500, and l = 1000. The parameters c 1 and c 2 are set to be c 1 = 25 and c 2 = 0.01. The limited velocities v y and v r are taken as v y = 6 and v r = 2. We track one train which departs at the time t = 565. Figure 8 plots the velocity variance of the tracked train for coe = 0.1. Here the objective points are the signalling lights in front of the tracked train. In Fig. 8, the color of the signalling light, which is located at the site l = 500 is green, and the color of the signalling light which is located at the site l = 1000 is yellow. From Fig. 8, we can see the deceleration process of the tracked train. In other words, we can determine the allowable velocity at each site from the current site to the site of the objective point. When the color of the signalling light in front of the tracked train is green, the tracked train departures from the site l = 1. In this case, the tracked train travels with the maximum velocity v max. Near the block boundary (i.e. the site l = 500), the

34 Communications in Theoretical Physics Vol. 55 tracked train decelerates gradually, and then travels into the second block. Within the second block, the tracked train travels with the limited velocity v y = 6. The simulation result indicates that the proposed model can well capture the dynamic behavior of train movement under fixed block control condition. Fig. 9 Decelerations of the tracked train. Figure 9 plots the deceleration, which corresponds to the Fig. 8. From Fig. 9, we can see that the deceleration of the track rain is between the region [0 m/s 2, 1 m/s 2 ]. At first, the deceleration of the tracked train is zero. In this case, the tracked train travels with the maximum velocity v max. Then, the deceleration of the tracked train is negative. In this case, the tracked train travels into the second block by reducing its velocity. Finally, the deceleration of the tracked train is zero again. In this case, the tracked train travels with the limited velocity v y. This means that the proposed model can reproduce some typical futures of train movement control. 6 Conclusions In conclusions, an improved car-following model is used to model and simulate the train movement control in railway traffic. By changing the time derivative, the proposed equation model can be solved by computer programming. Here, we simulate the train movement under different block system conditions. The numerical simulation results indicate that the proposed model can well describe the train movement control. Under a specified traffic condition, for each train, we can determine its allowable velocity at each site from its current site to the site of its objective point. In practical train movement control system, train movement is under many factors restriction. The allowable velocities of trains are affected by the constraints of track geometry, traction equipment, and station route etc. In the proposed model, these factors are not considered. This results in that the determination of the allowable velocities of trains is simplified. But we believe that the proposed model will open new perspectives for analyzing and evaluating train movement control. References [1] K. Petar and S. Guedial, IEEE Transaction on Automatic Control 17 (1972) 92. [2] P.G. Howlett and J. Cheng, J. Australian Mathematical Society, Series B 38 (1997) 388. [3] J. Cheng, Y. Davydova, P. Howlett, and P. Pudney, IMA Journal of Mathematics Applied in Business and Industry 10 (1999) 89. [4] Rongfang (Rachel) Liu and Iakov M. Golovitcher, Transportation Research Part A 37 (2003) 917. [5] P. Howlett, Annals of Operations Research 98 (2000) 65. [6] D. Chowdhury, L. Santen, and A. Schadschneider, Physics Reports 329 (2000) 199. [7] L.A. Pipes, J. Appl. Phys. 24 (1953) 274. [8] R.E. Chandler, R. Herman, and E.W. Montroll, Oper. Res. 6 (1958). [9] G.F. Newell, Oper. Res. 9 (1961) 209. [10] M. Bando, K. Hasebe, and A. Nakayama, Phys. Rev. E 51 (1995) 1035. [11] D. Helbing and B. Tilch, Phys. Rev. E 58 (1998) 133. [12] M. Treiber, A. Hennecke, and D. Helbing, Phys. Rev. E 62 (2000) 1805. [13] R. Herman and K. Gardels, Sci. Am. 209 (1963) 35. [14] D.C. Gazis, Science 157 (1967) 273.