Solitary Density Waves for Improved Traffic Flow Model with Variable Brake Distances

Transcription

1 Commun. Theor. Phys. 57 ( Vol. 57, No., February 15, 01 Solitary Density Waves for Improved Traffic Flow Model with Variable Brake Distances ZHU Wen-Xing (ý 1, and YU Rui-Ling (Ù 1,, 1 School of Control Science and Engineering Shandong University, Jinan 50061, China Department of Computer Science and Technology, Dezhou University, Dezhou 5303, China (Received August 1, 011; revised manuscript received November 7, 011 Abstract Traffic flow model is improved by introducing variable brake distances with varying slopes. Stability of the traffic flow on a gradient is analyzed and the neutral stability condition is obtained. The KdV (Korteweg-de Vries equation is derived the use of nonlinear analysis and soliton solution is obtained in the meta-stable region. Solitary density waves are reproduced in the numerical simulations. It is found that as uniform headway is less than the safety distance solitary wave exhibits upward form, otherwise it exhibits downward form. In general the numerical results are in good agreement with the analytical results. PACS numbers: Cn, 0.60.Cb, Ln, a Key words: gradient highway, solitary waves, variable brake distances 1 Introduction Traffic jams have been a serious problem in modern society, which had attracted many physicists and scholars with different backgrounds. [1 35] The solitary density wave in traffic flow was discussed analytically and numerically in many literatures since 1993, the single-pulse density wave was found by Kerner and Konhauser [5] in numerical simulations of the hydrodynamic traffic model. In 1995, the KdV equation was derived from the hydrodynamic model by Kurtze and Hong [6] by the use of the nonlinear analysis method. A conclusion was drawn that the single-pulse density wave is the soliton. Subsequently, Komatsu and Sasa [7] derived the modified KdV equation from the car-following model and concluded that the density wave has the kink-antikink shape. In 1999, Muramatsu and Nagatani [4] investigated soliton density wave and kink-antikink density wave by using a car-following model of a one-dimensional traffic flow with open boundaries. The theoretical and numerical results are very interesting. In 008, Zhu and Dai [5] reproduced soliton and kink density waves with periodic boundary condition. They found that the solitons appear near the neutral stability line that it exhibits upward form when the initial headway is smaller than the safety distance, otherwise it exhibits downward form. In 009, Komada et al. [3] proposed a new car-following model with a gravitational force effect based on the optimal velocity model. They performed a series of simulations to investigate the traffic behaviors on a highway with sags. But the solitary density wave on a highway with gradients was not discussed. In this paper the solitary density waves will be discussed by the use of linear stability theory and nonlinear analysis method based on the improved traffic flow model with variable brake distances. The remainder of the paper is organized as follows: in Sec. the improved model is given and the stability condition and neutral stability condition are obtained. In Sec. 3 the soliton solution is obtained by the use of the nonlinear analysis. In Sec. 4 a simulation is conducted to reproduce the solitary waves. Section 5 is the conclusion. Improved Model and Its Stability We consider a situation that vehicles move ahead on a single lane gradient highway under a periodic boundary condition. Figure 1 shows an illustration of the gravitational force acting upon a vehicle moving on an uphill and downhill highway respectively. The slope of the gradient is represented by θ, the gravity is g and the mass of the vehicle is m. Then, a horizontal force mg sin θ acts upon a vehicle when a driver does not operate the brake. If the driver operates the brake, the external force would be reduced by the brake control. The extended model with an effect of the slope was Supported by the National Natural Science Foundation of China under Grant No , the Postdoctoral Science Foundation of China under Grant No , the Special Foundation for Postdoctoral Innovation Program of Shandong Province under Grant No , and the Scientific Project of Jinan City under Grant No Corresponding author, zhuwenxing@sdu.edu.cn yrl@dzu.edu.cn c 011 Chinese Physical Society and IOP Publishing Ltd

2 30 Communications in Theoretical Physics Vol. 57 formulated as follows: d χ n (t { dt = a where V ( χ n dχ n(t dt }, (1 V ( x n = v f c max[tanh( x n x u,b (θ + tanh(x u,b (θ] v g,u,max[tanh( x n x u,b (θ + tanh(x u,b (θ], ( for a highway with an uphill gradient, and V ( x n = v f c max[tanh( x n x d,b (θ + tanh(x d,b (θ] + v g,d,max[tanh( x n x d,b (θ + tanh(x d,b (θ], (3 for a highway with a downhill gradient, where x n (t is the position of vehicle n at time t, x n (t = x n+1 (t x n (t is the headway of vehicle n at time t, a is the sensitivity of the driver, which equals to 1/τ and τ is the delay time of the driver, V ( x n is the optimal velocity function of vehicle n at time t, v fc max is the maximal velocity on the highway without any slope, v g,u,max and v g,d,max are the maximal reduced and enhanced velocity on uphill and downhill gradient which are formulated as v g,u,max = v g,d,max = mg sin θ γ, (4 where γ is a friction coefficient, x u,b (θ and x d,b (θ are the brake distances for vehicles on an uphill and downhill gradient respectively, which vary with the slopes. Brake distance models are given as follows x u,b (θ = x c (1 α sin θ, (5 x d,b (θ = x c (1 + β sin θ, (6 where x c is a safety distance on a highway without any slope, α, β are constants, for simplicity, the values are taken α = β = 1. Fig. 1 Illustration of the gravitational force acting upon a vehicle on a slope: uphill and downhill. These two models reflect the relationship between a slope and a brake distance. With the increase of a slope the brake distance of a vehicle increases on a downhill gradient and decreases on an uphill gradient. When a slope θ takes zero x u,b = x d,b = x c and the whole model has the same form as Bando s optimal velocity model. In order to study the stability of the traffic flow on an uphill/downhill highway we assume that N vehicles move homogeneously on a circular lane with a length L and a slope θ. We choose the slope 0 θ 30, which is just for theoretical analysis. Without loss of the generality we take v g,u,max = v g,d,max = sin θ, v f,max =. If let V 0 ( x n = tanh( x n x(θ + tanh(x(θ (7 where x(θ = x u,b (θ/x d,b (θ. Then the improved optimal velocity function with a slope effect as ( ± sin θ V ( x n = V 0 ( x n. (8 Equation (1 can be rewritten as d x n (t {( ± sin θ dt = a V 0 ( x i dx n(t dt }. (9 The asymmetric forward difference was used to rewrite the above equation ( ± sin θ x n (t + τ x n (t + τ τ V 0 ( x n = 0. (10 The initial solution of N-vehicles system is given as follows: ( ± sin θ x (0 n (t = hn + V 0 (h, (11 where x (0 n (t is the initial position of the n-th vehicle at time t. h is the average headway of the system and h = L/N. n is the number of the investigated vehicles in the N-vehicle system. V 0 (h is the optimal velocity function. Assume that a disturbance y n (t = e ikn+zt (k = 0, 1,,..., N 1, y n 1, z = u + iv (u and v are real is added into uniform traffic flow: x n (t = x (0 n (t + y n(t. (1 After neglecting the higher-order terms of y n (t and inserting Eq. (1 into Eq. (10 we obtain the linearized equation as ( ± sin θ y n (t + τ = y n (t + τ + V 0 (h y n(t. (13 The solution of the Eq. (13 is related to z which satisfies with ( ± sin θ (e zτ 1e zτ V 0 (h(eik 1 = 0. (14

3 No. Communications in Theoretical Physics 303 is Assume that the solution form of the above equation z = z 1 (ik + z (ik + (15 By expanding Eq. (13 we obtain z 1 = V 0(h, ( ± sin θ V z = 0 (h ( ( ± sin θ 1 3 V 0 (hτ. (16 The stability condition is obtained by the stability criterion V 0 (h < 3τ( ± sin θ. (17 The neutral stability condition is also obtained V 0 (h = 3τ( ± sin θ. (18 The neutral stability lines (solid lines and coexisting curves (dotted lines for different slopes of a gradient highway are plotted in Fig. where one can observe that the area below the neutral stability line is an unstable region, the area above the coexisting curve is a stable region and the area between the neutral stability line and coexisting curve is a meta-stable region. Patterns (a and (b show the neutral stability curves and coexisting curves for uphill and downhill situations on highway respectively. A metastable region is pointed out with arrow in pattern (a and (b for the slope θ = 0. In the uphill situations the stable area becomes larger and larger with the increase of the slopes. While in the downhill situations the stable area increases with the decrease of the slopes. So the stability of the traffic flow varies with the slopes of a gradient. Furthermore, the unstable area moves from low density region to the high density region with the increase of the slopes in an uphill situation and the contrary motion of the unstable area with the increase of the slopes is observed in a downhill situation. In addition one can observe that the apexes of the neutral stability curves in pattern (a and (b are the critical points (h c, a c, which are corresponding to different slopes θ. Obviously, h c = 4 m, 3.3 m,.6 m, m in pattern (a and h c = 4 m, 4.7 m, 5.4 m, 6 m in pattern (b are corresponding to θ = 0, 10, 0, 30 respectively. 3 Nonlinear Analysis and Soliton Solution In Fig., the areas between the solid lines and dotted lines are the meta-stable regions for different slopes of gradient highway. If a perturbation is added to the traffic flow in the meta-stable region the solitary waves would appear. Now let us analyze the improved model by using nonlinear analysis method. Fig. Neutral stability curves (solid lines and coexisting curves (dotted lines for different slopes in two situations: patterns (a and (b are corresponding to uphill and downhill situation respectively. For convenience, Eq. (10 is rewritten as ( ± sin θ x n (t + τ x n (t + τ τ (V 0 ( x n+1 (t V 0 ( x n (t = 0. (19 In order to derive the equation describing the collective motion on coarse-grained scales, [10,6] the slowly varying behaviors at long wavelength are considered. For 0 < ε 1, we define the slow variables X and T for space variable n and time variable t. X = ε(n + bt, and T = ε 3 t, (0 where b is a constant to be determined and the headway is set as x n (t = h + ε R(X, T. (1 Inserting Eq. (0 and Eq. (1 into Eq. (19 and performing Taylor expansions to sixth order of ε, we obtain the following expression ε 3[ b ± sin θ ] V 0 X R + ε 4[ 3b τ ± sinθ V 0 ] X! 4 R + ε5[ T R + 7b3 τ X 3 3! R ± sin θ V X R ± sin θ V 0 4 XR ]

4 304 Communications in Theoretical Physics Vol ε 6[ ( 15b 4 τ 3 3bτ T X R + ± sin θ V 4! 48 where X = X, 0 4 X R ± sin θ 8 T = T, X T = X T, V 0 = dv 0( x d x V 0 X R] + o(ε 6 = 0, ( x = h, V 0 = d V 0 ( x x = h. d x Near the neutral stability lines h = h s, V 0(h s = 0, V 0 (h s > 0. By taking b = ( ± sin θ/v 0(h s, τ/τ s = 1 ε and τ s = /3V 0 ( ± sin θ the third-order and fourth-order of ε is eliminated from Eq. (. If the o(ε 6 is ignored then we obtain the KdV equation ( ± sin θ T R V ( ± sin θ XR V 0 R XR = 0.(3 Perform the following transformations ( ± sin θ T = V 0 54 ρ 3 T, (4 ( ± sin θ X = V 0 54 ρ 3 X, (5 R = V 0 ( ± sin θ ρ R, (6 where ρ is a negative constant. We have T R 3 X R R X R = 0. (7 The soliton solution of the headway is obtained x n (t = h + 3V 0 (1 V 0 τ sech { 7 (1 τ τ s 4 τ s [ ( n + τ ( ± sinθ ]} V τ s 0t. (8 4 Numerical Simulations Computer simulations are carried out to reproduce the solitary waves in the meta-stable regions. Assume that initially N vehicles distribute homogenously on a single lane gradient highway with a slope θ and length L under periodic boundary condition. Vehicles move on the road and update their positions and velocities according to the Newton s motion equation on the highway. The slopes of uphill and downhill highway are identical. For simplicity we take v g,u,max = sinθ and v g,d,max = sin θ. When θ = 0, 14.5, 30, v g,u,max = v g,d,max = 0, 0.5, 0.5 m/s respectively. The corresponding safety distances are taken as h c = 4, 3, m for an uphill situation and h c = 4, 5, 6 m for a downhill situation. The simulation duration is taken as s long enough for the vehicles to reach the steady state. The sensitivity and the time step are taken as a = 0.5, t = 0.1 s respectively. For an uphill situation we choose the following initial condition: (i θ = 0, h c = 4 m, h = 5.3 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h 0.5(n = N/, x n (0 = h + 0.5(n = N/ + 1, N = 00; (ii θ = 0, h c = 4 m, h =.678 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h + 0.5(n = N/, x n (0 = h 0.5(n = N/ + 1, N = 00; (iii θ = 14.5, h c = 3 m, h = 4.43 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h 0.5(n = N/, x n (0 = h + 0.5(n = N/ + 1, N = 00; (iv θ = 14.5, h c = 3 m, h = m, x n (0 = h(n N/, n N/ + 1, x n (0 = h + 0.5(n = N/, x n (0 = h 0.5(n = N/ + 1, N = 00; (v θ = 30, h c = m, h = m, x n (0 = h(n N/, n N/ + 1, x n (0 = h 0.5(n = N/, x n (0 = h + 0.5(n = N/ + 1, N = 00; (vi θ = 30, h c = m, h = 0.85 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h + 0.5(n = N/, x n (0 = h 0.5(n = N/ + 1, N = 00, where h c is the safety distance, which is related to the slope, h is the homogeneous headway, x n (0 is the headway of the n-th vehicle at time 0 and N is the total number of vehicles in the system. The simulation results are plotted in Figs. 3 and 4 which include 6 patterns (a (a*, (b (b*, and (c (c*. Figure 3 shows the headway profiles of the solitary waves at t = s on an uphill gradient highway with different slopes θ = 0, 14.5, 30 corresponding to patterns (a (a*, (b (b*, (c (c* respectively. Figure 4 exhibits space-time evolution of solitary waves after t = s on an uphill gradient highway with different slopes θ = 0, 14.5, 30. From these patterns one can easily observe that the solitary wave appears in varying region for different slopes of gradient highway. With the increase of the slope the uniform headway decreases, which may lead to solitary wave with a small perturbation. Moreover the solitary wave appears in upward form as h < h c, and downward form as h > h c, which are the same as the conclusion in Ref. [5]. The solitary waves propagate backward observed in Fig. 4. For a downhill situation when θ = 0 the initial settings are the same as those in the uphill situation. So we choose the following initial condition when θ 0 : (i θ = 14.5, h c = 5 m, h = 6.39 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h 0.5(n = N/, x n (0 = h + 0.5(n = N/ + 1, N = 00; (ii θ = 14.5, h c = 5 m, h = 3.61 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h + 0.5(n = N/, x n (0 = h 0.5(n = N/ + 1, N = 00; (iii θ = 30, h c = 6 m, h = 6.45 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h 0.5(n = N/, x n (0 = h + 0.5(n = N/ + 1, N = 00;

5 No. Communications in Theoretical Physics 305 (iv θ = 30, h c = 6 m, h = 4.55 m, x n (0 = h(n N/, n N/ + 1, x n (0 = h + 0.5(n = N/, x n (0 = h 0.5(n = N/ + 1, N = 00, where the parameters have the same meanings as those in the uphill situation. Fig. 3 Headway profiles of the solitary waves at on uphill gradient highway with different slopes corresponding to patterns (a (a*, (b (b*, (c (c* respectively. Fig. 4 Space-time evolution of solitary waves after on uphill gradient highway with different slopes corresponding to patterns (a (a*, (b (b*, (c (c* respectively. The simulation results for the downhill situation are plotted in Figs. 5 and 6, which show headway profiles of the solitary waves at t = s and space-time evolution of solitary waves after t = s on a downhill gradient highway with different slopes θ = 0, 14.5, 30. From these patterns one can observe that the uniform headway increases with the increase of the slopes, which is opposite to the varying tendency in an uphill situation. The solitary wave appears in upward form as h < h c and downward form as h > h c, which are the same as those in an uphill situation. And the solitary waves propagate backward, which can be observed in Fig. 6.

6 306 Communications in Theoretical Physics Vol. 57 Fig. 5 Headway profiles of the solitary waves at on downhill gradient highway with different slopes corresponding to patterns (a (a*, (b (b*, (c (c* respectively. Fig. 6 Space-time evolution of solitary waves after on downhill gradient highway with different slopes corresponding to patterns (a (a*, (b (b*, (c (c* respectively. 5 Conclusions Traffic flow model is improved by introducing variable brake distance in describing the motion of the vehicles on a gradient highway. Stability of the new model is analyzed by the use of linear stability theory and the stability and neutral stability conditions are obtained at the same time. Nonlinear analysis method is taken to derive the KdV equation and the soliton solution is found near the neutral stability lines in the meta-stable region. A series of simulations are conducted to simulate solitary waves with different slopes of a gradient highway. It is found that solitary wave appears in the different forms with different initial settings. References [1] L.A. Pipes, J. Appl. Phys. 4 ( [] M.J. Lighthill and G.B. Whitham, Proc. Roy. Soc. A 9 ( [3] P.I. Richards, Oper. Res. 4 ( [4] G.F. Newell, J. Oper. Res. Soc. 9 ( [5] B.S. Kerner and P. Konhauser, Phys. Rev. E 48 (

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