Mesoscopic Traffic Flow Model Considering Overtaking Requirements*

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1 Journal of Highway and Transponation Research and Development Vol. 9,No. 4 (2015)085 Mesoscopic Traffic Flow Model Considering Overtaking Requirements* LU Shou-feng (p ili )1**, WANG Jie(J:. )1, XUE Zhi-gui(M )2, LIU Xi-min(xU:JflJi)1 ( 1. School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Changsha Hunan , China; 2. Traffic Police Division of Public Security Bureau of Changsha city, Changsha Hunan , China) Abstract: Overtaking probability formula in the Prigogine - Herman traffic flow mesoscopic model is the function of traffic density, which is a linear function and does not consider overtaking requirements. In this paper, we use desired speed to improve the overtaking probability formula and propose a new overtaking probability formula that is nonlinear and a corresponding traffic flow mesoscopic model. The improved model can simultaneously consider traffic density and overtaking requirements, thus reflecting traffic flow operation realistically. We use the proposed model to simulate the diffusion process of vehicles from a high-density section to a low-density section. The example analyzes the difference between the linear overtaking probability formula and the parabolic Greenshield's overtaking probability formula. Results show that the linear overtaking probability formula evolves quickly and converges to three speed classes. The parabolic Greenshield's overtaking probability formula converges to six speed classes and can reflect speed distribution evolution reasonably. Key words: traffic engineering; gas kinetic theory; overtaking; desired speed; mesoscopic model; traffic flow 1 Introduction In low-density traffic, traffic flow involves vehicular movement wherein an individual has little interaction with other vehicles. By contrast, in high-density traffic, traffic flow corresponds to the movement of a platoon. This fact reveals the similarity between traffic flow and gas flow. PRIGOGINE and HERMAN [ l J are the pioneering researchers of the mesoscopic traffic flow model ( traffic kinetic model) and present a systematic study of this model in the monograph kinetic theory of vehicular traffzc. ( ). + ( ).. a mteract1on a adjustment (1) This equation describes the evolution of speed distribution function f ( x, v, t). The right side of formula ( 1 ) includes three terms, namely, the relaxation term, interaction term, and adjustment term : ( ). a (.il ) = ( af ) = at interation adjustment at relaxation f - _ fo, T (2) (1 -P) c( v -v)f, ( 3) =,\ (1 -P) c [ o ( v - v ) -f], ( 4 ) where, f( x, v, t) is the speed distribution function, and f ( x, v, t) dxdv is the probability of the number of vehicles. At time t, the position is between x and x + dx, and the speed is between v and v + dv. f0 ( x, v, t) is the desired speed distribution function,,\ is a parameter that is the function of density c, v is the average speed, and o is the Dirac function. The zero-order moment of speed distribution function f ( x, v, t ) is c ( x, t) = N r f( x, v, t) dv, which is the traffic density. The first-order moment of speed distribution function f ( x 'v 't) is ( x' t) = 1/ c ( x' t) r ef ( x ' v ' t ) dv ' which is the average speed. P is the passing probability, which is the function of density c, and P = 1 c Manuscript received June 19, 2015 Supported by the National Natural Science Foundation of China (No ) ; the Natural Science Foundation of Hunan Province (No ) ; and the Major Programs of Technology Bureau of Changsha City (No. Kl ) address: itslu@ 126. com

2 86 Journal of Highway and Transportation Research and Development the jam density. Tis the relaxation time, which is also a function of density c. In recent years, researchers in the field of applied mathematics, DELITALA and TOSIN C 2 l proposed a new mesoscopic traffic flow model based on mathematical kinetic theory, which remodels the acceleration and deceleration of the vehicle. The equations in both the Prigogine-Herman and Delitala-Tosin models are integral differential equations, which are difficult to solve. LU ( 3 4 l integrated cell transmission model with the Prigogine-Herman model and Delitala-Tosin model to establish a discrete traffic flow model, which is easy to solve. Traffic flow is composed of vehicle-driver units and is related to human factors ( e. g., self-driven ability). This definition shows the essential difference between traffic flow and the particles in Newtonian mechanics. Newtonian particles are isotropic, whereas traffic flow is anisotropic. Describing the self-driven ability in the mesoscopic traffic flow model is an important study focus. PRIGOGINE and HERMAN C 1 l introduced the relaxation term to describe the self-driven ability. The relaxation term describes the approaching process of vehicle speed towards the desired speed and reveals the difference between the vehicle-driver unit and gas particle. However, the relaxation term is a phenomenological description, i. e., it does not derive the mechanism of traffic flow operation. NELSON ( s J conducted an analytical derivation of the acceleration term by using the correlation model and mechanical model, which influenced the development of the Prigogine-Herman model and improved the relaxation term. PA VERI-FONTANA C6l extended the speed distribution function into the joint distribution function of speed and desired speed to improve the Prigogine-Herman model. However, the mathematical form of the joint distribution function is difficult to establish. LU C 7 l introduced the desired speed to improve the Delitala-Tosin model and obtained the conclusion that the average speed of traffic flow is not only related to density but also to the desired speed difference. RONG [ s J divided the overtaking process into four steps, which include desire, condition, behavior, and ending, and established the overtaking model. In the current paper, we first analyze the decision process of a driver in changing lanes and overtaking. We then introduce the desired speed to improve the overtaking probability formula of the Prigogine-Herman model. Finally, we establish a new mesoscop1c traffic flow model to describe traffic flow realistically. 2 The improved overtaking probability formula The decision process of lane changing is based on the perception of surrounding vehicles. The lane changing process is affected by driving desire and the traffic condition of adjacent lanes. Driving desire is the desired speed of the driver. In the lane changing process, each driver will think about three questions C 9 l : ( 1 ) Is there a desire to change lanes? (2) Is the present driving situation at the neighboring lane favorable? ( 3) Is movement to a neighboring lane possible? If all three questions are answered positively, the driver will changes lane. Lane changing can be divided into two types: lane changing from a fast lane to a slow lane and lane changing from a slow lane to a fast lane. The desire to change to a slow lane results from the actual need to move out of the way or the general order to keep right. The desire to change to a fast lane results from obstructions on the actual lane caused by a slow vehicle in front of the driver. The level of obstruction is a function of the difference between the actual speed of the front vehicle and the driver's own desired speed. The safety level of lane changing is determined by the space headway and speed difference. The lane changing probability formula of the Prigogine-Herman model is P=l -c/c P,c <c P. If c >c P, P =0, where c P is the jam density, and c/ c P is the normalized density, which is a dimensionless variable. This formula shows that overtaking probability is low when density is high. This formula analyzes overtaking from a phenomenological perspective and does not consider the relation between speed and desired speed. The relation between overtaking probability and density is linear. Figure 1 shows the relation between overtaking probability P and normalized density. When the density mcreases, the overtaking probability decreases linearly. However, this relationship does not show the relation between overtaking probability and critical density. Overtaking probability is not only related to density but also to the desired speed. When the difference be-

3 LU Shou-feng, et al: Mesoscopic Traffic Flow Model Considering Overtaking Requirements 87 p = ( 1 -_ _) x w - we ' P The relation between overtac P (t) king probability and normalized density is illustrated in figure 3. Fig. 1 Relation between overtaking probability and normalized density in the Prigogine-Herman model tween speed and desired speed is large, the overtaking requirement is large. Otherwise, the overtaking requirement is low. The improved overtaking probability formula is expressed as follows: P=(l-_ _)x w- v, (5) c P w where w is the desired speed. For formula ( 5 ), when the density is low, the speed of all vehicles will reach the desired speed w after some time. Furthermore, the overtaking probability P = 0 and traffic flow will reach a stable state. When the type of density is jam density, the overtaking probability is p =0. To show the relation between overtaking probability and density with regard to desired speed, we assume that the relation between speed and density satisfies Greenshield's linear formula, i. e., V = w(l -clc P ). We assume that all vehicles' speed is w and P = ( 1 -cl c P ) [ w -w ( 1 -cl c P ) JI w. The relation between overtaking probability and normalized density is illustrated as figure 2. Fig S 02 ll. [ O.IS o.os o o 0 0_1 0_ o.s Normalized desnity Relation between overtaking probability and normalized density in the proposed formula (speed-density relation is Greenshield's formula) When we assume that the relation between speed ' and density satisfies Underwood's model V = w e ;;;, we can assume that the desired speed of all vehicles is w and Fig. 3 Relation between overtaking probability and normalized density in the proposed formula (speed-density relation is Underwood's formula) Figures 2 and 3 show that the improved relation between overtaking probability and density is a nonlinear parabolic curve. If the density is low or high, the overtaking probability will be small because the interaction between vehicles is insignificant in low-density traffic and each vehicle easily reaches the desired speed. The interaction between vehicles is large in high-density traffic; thus, overtaking is difficult and the overtaking probability is low. The improved overtaking probability formula can better consider the overtaking requirements and determine whether space headway is allowed to overtake. If a vehicle cannot overtake, such a vehicle will interact with other vehicles and the probability of not passing will be 1 -P. The improved overtaking probability in formula ( 2) is integrated into formula ( 1 ) to obtain a new mesoscopic traffic flow model. The new mesoscopic traffic flow model uses the cell transmission model to obtain a discrete model that is easy to solve. For the solution method, refer to reference [ 3]. The cell transmission model proposed by DAGANZO [ioj is a method for solving the macroscopic traffic flow model, which is simple and easy to compute. The density updating formula of each cell is expressed as follows : K(t + Lit,x) = K(t,x) - [ O:+i - O:J, (6) where, K is the cell density, t is the time, x is the cell position, Q is the cell flow rate, Lit is the time step, and Lix is the space step. The cell variables correspond to the number of vehicles and speed distribution. The symbol meanings in figure 4 are the following: i -1 is the num-

4 88 Journal of Highway and Transportation Research and Development ber of a cell, and n, 1 ( t) is the number of vehicles in cell i - l. I( i -1, v, t) is the number of vehicles with speed v in i -1 cell at t. i -l i i + 1 n,_l(t),f(i-l,v,t) n,(t),f(i,v,t) n.,l(t),f(i +l,v,t) Fig. 4 Cell and cellular variables The speed distribution evolution in cell i is calculated by the following formula: l(i,v,t + 1) = [n(i,t)] f(i,v,t) + yi(t) l(i - l,v,t) + (1 -P)[I, nj (i,vj,t)]f(i,v,t) - (1 -P) n.(t)l(i,v,t) - v, v Ll(i,vj,t) -yi+1(t) l(i,v,t)] ( " 1 l)' (7) v,<v n i,t + if v < vmin' l I( i, v,t + 1) = [ y.( t )I( i - 1, v, t) ] (" l), n i,t + (8) where, vmin is the mm1mum speed and vmax is the maximum speed. At t + 1, the number of vehicles n( i, t + 1) in cell i is calculated by the conservation equation n ( i, t + 1) = n ( i, t) +Yi ( t) -y, + 1 ( t), in which y. ( t) and Yi+ 1 ( t ) is calculated by O: = min l S (!(. 1 ), R (!(. dtu) f. Yi ( t) is the number of vehicles flowing into cell i from the upstream cell at t. y, + 1 ( t) is the number of vehicles flowing into the downstream cell from cell i at t. The relaxation process is calculated by the exponential relaxation function: t I ( v, t + 1 ) = lo + [/( v, t) - lo ] exp ( - T), ( 9) where, lo is the desired speed distribution function, which describes the drivers' desire. Tis relaxation time. 3 Example analysis To analyze the performance of the discrete traffic flow mesoscopic model, we study the discontinuous initial density case. The initial density is set as follows: { 161 veh/km, x < 500 m, k(o,x) = ( 10) 39 veh/km, x > 500 m. The flow rate and density function is expressed as follows: q = { k, k < ' k k ' ( 11) The unit of density k is veh/km, the unit of flow rate q is veh/h, the unit of speed v is km/h, and the u nit of x is m. The flow rate and density formula are fitted by detector data. The jam density is veh/km, i. e., veh/ cell. The critical density is veh/km, i. e., veh/ cell. Initial condition: we simulate traffic flow on a road with length of m. Road density is 107 veh/km and 39 veh/km on the first and second 500 m of road, respectively. The speed distribution for these two densities is illustrated in figure 5. The discrete traffic flow model divides the speed domain between 0 km/h and 120 km/h into 24 intervals, and step is 5 km/h. The horizontal axis in figures 5 is the number of speed interval t'> 0.1 ] Fig. 5 Speed distribution at 107 veh/km Speed distribution at 39 vch/km '-fl1t = Speed distribution in the density of 107 veh/km and 39 veh/km The parameters are cell length x = 50 m, the time interval is t = 2 s, the relaxation time T is 300 s, the road length is m, and the number of cells is 20. Cells 0 and 9 are the boundary cells and these two cells have constant density at each time. The cell description is illustrated in figure 6. Cell 1-9 Cell Cell O Cell 19 high density low density Fig. 6 Cell description We simulate traffic flow evolution by using the linear overtaking probability formula and Greenshield's overtaking probability formula. The cell density evolution results are similar ( figure 7 ). The comparison of speed distribution evolution is illustrated in figures 8 and 9, which show the speed distribution evolution in cells 9 and 12, respectively. The horizontal axis in figures 8 and 9 is the number of speed interval. From the above figures, we can conclude that cell

5 ... LU Shou-feng, et al: Mesoscopic Traffic Flow Model Considering Overtaking Requirements 89 Fig. 8 Fig I;,..., a o.6 "c::i Jj o. s _...,_ r-so s p200 s - -1=100 s - +-1"'250 s _.._t=lso s -+-1"'300 s Fig. 7 8.; - : 1 0.2!. \ I z Cell evolution direction Cell density evolution... t=o s ---t=ios -t=70s o.a. _<=-./._-1;_... ' _--_ -_--.._.. - _- _-.. _ 0 s 10 ls (a) Simulation result of linear overtaking probability formula 0.9 I; 0.8 ;;;- 0.7!>! 0.6 ';; o s. o.!l 04.!l 1 0:3 (.... t=o s - t=40s - - t=l30 s \ o z.a :::. -_ -.:: ""'.:: ::: ::i. _::: 0 s 10 1s (b) Simulation result of Greenshield's overtaking probability formula Speed distribution evolution comparison in the ninth cell i i!l 0.7,t, 0.6 "c::i :i o.s r 1 1] 8 :1! \ t=o s -- t=io s - t=70s i 0.1 \ Z O _ 0 s b 0.8 > 0.6 1l i : 0.7.r, I 25 (a) Simulation result of the linear overtaking probability formula... t=o s -- t=40s -F'l30s ] \ 0.1 '- z 0 0 s (b) Simulation result ofgreenshield's overtaking probability formula Speed distribution evolution comparison in the 12th cell density evolution is the same for the linear overtaking probability formula and Greenshield's overtaking probability formula. For the cell speed distribution evolution, two models have large differences. For the linear overtaking probability formula, speed distribution evolves quickly and converges to speed classes 2, 3, and 4. Greenshield's overtaking probability formula can show the dynamic process of speed distribution. In this formula, speed distribution first evolves to low-speed groups and then to high-speed groups; this phenomenon describes the propagation process of vehicles from a high-density zone to a low-density zone and that speed distribution converges to speed classes 2, 3, 4, 5, 6, and 7. Thus, Greenshield's overtaking probability formula is more realistic than other formulas. 4 Conclusion and discussion This paper first analyzes the lane changing decision process and then analyzes the deficiency of the Prigogine Herman linear overtaking probability formula. By introducing the desired speed, we propose a new overtaking probability formula. We plot the overtaking probability curves for Greenshield's linear speed - density relation and Underwood's exponential speed - density relation. If the density is low or high, the overtaking probability will be small because the interaction between vehicles is insignificant in low-density traffic and each vehicle can easily reach the desired speed. The interaction between vehicles is large for high-density traffic; thus, overtaking is difficult and overtaking probability is low. Thus, the improved overtaking probability formula can better consider the overtaking requirements and determine whether space headway is allowed to overtake. The integration of the improved overtaking probability formula and discrete mesoscopic traffic flow model obtains a new mesoscopic traffic flow model. The example shows the difference between linear overtaking probability formula and Greenshield's overtaking probability formula. For the linear overtaking probability formula, speed distribution evolves quickly and converges to speed classes 2, 3, and 4. Greenshield's overtaking probability formula can show the dynamic process of speed distribution wherein speed distribution first evolves to low-speed groups and then to high-speed groups. This formula describes the propagation process of vehicles from a highdensity zone to a low-density zone and that speed distribution converges to speed classes 2, 3, 4, 5, 6, and 7.

6 90 Journal of Highway and Transportation Research and Development Thus, Greenshield's overtaking probability formula is more realistic than other formulas. The suggestion for further study of the Prigogine Herman model should tackle instantaneous acceleration Traffic Kinetic Model-integrating the Lagged Cell Transmission and Continuous Traffic Kinetic Models [ J]. Transportation Research Part C : Emerging Technologies, 2011, 19 ( 2) : [4] LU Shou-feng, LIU Cai-hong, LIU Xi-min. A New Traffic and deceleration assumptions. The Prigogine-Herman Kinetic Model for Heterogeneous Condition [ J]. International Journal of Non - Linear Mechanics, 2013, 55: 1 - model includes the interaction process and relaxation process. In the interaction process, speed distribution e volves from high speed to low speed. The maximum speed is 120 km/h, the minimum speed is 0 km/h, and [ 5] 9. NELSON P. A Kinetic Model of Vehicular Traffic and Its Associated Bimodal Equilibrium Solutions [ J J. Transport the simulation step is 2 s. Thus, the maximum deceleration Theory and Statistical Physics, 1995, 24 ( 112/3 ) : is ml s2 In the relaxation process, speed dis tribution evolves from low speed to high speed. In this paper, speed distribution adopts an exponential function [6] PAVERI-FONTANA S L. On Boltzmann-like Treatments for Traffic Flow; A Critical Review of the Basic Model and an Alternative Proposal for Dilute Traffic Analysis [ J J. to reach the desired speed distribution and the formula is Transportation Research, 1975, 9 (4): f(v,t+l) =/0+[/(v,t)-/0] exp (-t/t). The maximum [7] LU Shou-feng, LIU Cai-hong, LIU Xi-min. A Traffic Kinetic acceleration is m/s2 The maximum accelera tion and deceleration are not real because of the instantaneous acceleration and deceleration assumption of the Prigogine-Herman model. In the future, we will further improve the instantaneous acceleration and deceleration assumption. References [8] [9] Model Considering Desired Speed [ J]. Journal of Transportation Systems Engineering and Information Technology, 2013, 13 (2): RONG Jian, LIU Shi-jie, SHAO Chang-qiao, et al. Application of Overtaking Model in Two-lane Highway Simulation System [ J]. Journal of Highway and Transportation Research and Development, 2007, 24 ( 11) : (in Chinese) WIEDEMANN R, REITER U. Microscopic Traffic Simulation: [ l J [2] PRIGOGINE I, HERMAN R. Kinetic Theory of Vehicular Traffic [ M]. Amsterdam: Elsevier Press, D ELIT ALA M, TOSIN A. Mathematical Modeling of Vehicular the Simulation System Mission, Background and Ac tual State [ R J. Brussels: CEC European Managers, Traffic: A Discrete Kinetic Theory Approach [ J]. Mathematical Models and Methods in Applied Sciences, 2007' 17 (6): [ 10 J DAGANZO C F. The Cell Transmission Model: A Dynamic Representation of Highway Traffic Consistent With the Hydrodynamic Theory [ J J. Transportation Research Part [3] LU Shou-feng, DAI Shi-qiang, LIU Xi-min. A Discrete B, 1994, 28 ( 4) : (Chinese version's doi: /j. issn , vol. 32, pp , 2015)

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