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2013,25(5):725-730 DOI: 10.1016/S1001-6058(13)60418-7

Applying the maximum information principle to cell transmission model of tra- ffic flow* LIU Xi-min () Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China Traffic and Transportation Engineering College and Key Laboratory of Road Structure and Material of Commu- nication Ministry, Changsha University of Science and Technology, Changsha 410114, China, E-mail: roadtrafficuliu@126.com LU Shou-feng () Traffic and Transportation Engineering College, Changsha University of Science and Technology, Changsha 410114, China (Received October 31, 2012, Revised February 18, 2013) Abstract: This paper integrates the maximum information principle with the Cell Transmission Model (CTM) to formulate the velo- city distribution evolution of vehicle traffic flow. The proposed discrete traffic kinetic model uses the cell transmission model to cal- culate the macroscopic variables of the vehicle transmission, and the maximum information principle to examine the velocity distri- bution in each cell. The velocity distribution based on maximum information principle is solved by the Lagrange multiplier method. The advantage of the proposed model is that it can simultaneously calculate the hydrodynamic variables and velocity distribution at the cell level. An example shows how the proposed model works. The proposed model is a hybrid traffic simulation model, which can be used to understand the self-organization phenomena in traffic flows and predict the traffic evolution. Key words: kinetic traffic model, Cell Transmission Model (CTM), maximum information principle, traffic flow, velocity distribu- tion

Introduction The models for vehicular traffic flows can be di-

vided into macroscopic, mesoscopic and microscopic ones. For the macroscopic model, the related variables are directly the velocity, density and flow flux. For the mesoscopic model, the main concern is the velocity distribution. For the microscopic model, the first con- cern is the microscopic driving behavior. In the pre- sent paper, we focus ourselves on the mesoscopic model.

Until now, the approaches to the evolution equa- tions of velocity distribution can be summarized as

* Project supported by the National Natural Science Foun- dation of China (Grant No. 71071024), the Hunan Provincial Natural Science Foundation (Grant No.12JJ2025). Biography: LIU Xi-min (1980-), Female, Ph. D. Candidate, Lecturer Corresponding author: LU Shou-feng, E-mail: itslusf@gmail.com

three ones. The first approach is the Boltzmann-like treatments, which was initiated from a pioneered me- soscopic model. The second one is to use encounter rate and table of games, which is called the methods of discrete mathematical kinetic theory. The third one is to construct the lattice Boltzmann model of velocity distribution. The details can be referred to Ref.[1].

Vehicular traffic can be modeled as a system of interacting particles driven far from equilibrium. Using statistical physics methods to study vehicular traffic offers the possibility to examine various funda- mental aspects of this kind of truly non-equilibrium systems[2]. Because phase transitions, hysteresis effe- cts, and other nonlinear effects of synergetics dete- rmine spatiotemporal traffic pattern features, spatiote- mporal traffic phenomena may be considered an as- pect of synergetics[3,4]. Different classes of spatial- temporal patterns in traffic flows can be considered as different phases of the system. Kerner[5] modeled synergetic phenomena in spatial-temporal patterns as phase transition and defined a synchronized flow phase. Phase transitions in traffic flows on multilane

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roads were also studied in the framework of his three- phase traffic theory proposed by Kerner and Klenov[6,7].

Microscopic approach of synergetics has been applied to model jamming transition in traffic flows based on the Lorentz system. Synergetic scheme was proposed to describe the jamming transition in traffic flows, taking into account the internal fluctuations of characteristic acceleration/braking time[8]. On the basis of Ref.[8], the influence of the characteristic ac- celeration/braking time in the most probable headway deviation from its optimal value was studied, and headway deviation characterizing a phase transition was showed[9]. The homotopy perturbation method, the variational iteration method[10] and the differential transformation method[11] were used to give approxi- mations to the governing equations offered in Refs.[8,9], and these three methods could provide highly accurate analytical solutions.

Recently, the macroscopic approach of synerge- tics has been used to model complex social systems. The integration of the macroscopic approach of syne- rgetics and the continuity equation was used to model residential mobility macroscopically[12]. In this paper, we attempt to use the maximum information principle, a macroscopic approach of synergetics, to calculate the velocity distribution. The feature of this method is to use the macroscopic variables of traffic flow to de- rive the velocity distribution, without modeling the microscopic vehicle interactions. 1. Synergetics

Synergetics is initiated by Haken[13] in 1969, which dealt with complex systems, i.e., systems com- posed of many individual parts that are able to sponta- neously form spatial, temporal or functional structures by means of self-organization. Synergetics has formed two theoretical branches: microscopic or mesoscopic approach and macroscopic approach. For the former approach, the concepts of instability, order parameters and slaving are used, which can be cast into a rigorous mathematical form, and one could show the emerge- nce of structures and concomitantly of new qualities at a macroscopic level. For the latter approach[14], the maximum information principle is used, which is an analogy with thermodynamics. This approach treats complex systems by means of macroscopically obse- rved quantities, and then determines the microscopic structure of the processes which give rise to the ma- croscopic structure. The maximum information princi- ple claims that the probability distribution is the most possible probability distribution when the information is maximized. In this paper, we use the macroscopic approach to study the velocity distribution evolution of traffic flow.

2. Cell transmission model The Lighthill-Whitham-Richards (LWR) model

is a first-order hydrodynamic model of traffic flows, and a macroscopic approach that provides good appro- ximation of traffic flow evolution in realistic networks. Many numerical methods have been developed to solve related problems with the LWR model. One approach is to solve the Riemann problem by applying the Godunov method. Another approach is to use the demand and supply functions[15,16], which turns out to be a variant of the Godunov method. The third app- roach is the wave tracking resolution scheme[17]. The Godunov discretization scheme is efficient as it has been proved that the flow is constant during a time step. The transmission flow can be easily calculated using the following formula

1= min{ ( ), ( )}t t tx x xQ S K R K

where S and R are the demand and supply functions, respectively defined by

( ) = ( )ES K Q k if criticalK K ,

max( ) =S K Q if criticalK K

max( ) =R K Q if criticalK K ,

( ) = ( )ER K Q k if criticalK K With the transmission flow, we can write the density updating formula as

+1( + , ) = ( , ) ( )t tx x

tK t t x K t x Q Qx

The Cell Transmission Model (CTM) transforms the differential equations in the LWR model into simple difference equations. In the CTM, a road is di- vided into homogeneous and interconnected segments, referred to as cells, and piecewise linear relationships are assumed between flow and density at the cell level. 3. Proposed traffic kinetic model

Let flow flux and density for traffic flow be given. We wish we could derive the probability distri- bution of speed. In other words, we start from the ma- croscopic world and wish to draw conclusions about the microscopic world. In synergetics, a measure for the amount of information is connected with the number of possible events (realizations). There is an overwhelming probability of finding that the realized

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probability distribution has the greatest possibilities, and thus the greatest amount of information. This pri- nciple has been proved to be fundamental for applica- tion to realistic systems in physics, chemistry and bio- logy. Thus, maximum information of traffic flow means the corresponding speed probability distribu- tion has the greatest possibilities to occur.

The objective function is to maximize the amount of information. An expression for the per information is defined as

=1= ln

n

B i ii

S K P P (1) where iP is the relative frequency of the occurrence of possible speed.

The constraints are

=1= = =

tnx

i i ti x

QQPV VK (2)

=1= 1

n

ii

P (3) in which i is the discrete velocity class, txQ the trans- mission flow, and txK the cell density, which are cal- culated by the CTM.

In the process of solution, the Lagrang multiplier method is used. We multiply the left hand side of Eq.(2) by 1 , and the left hand side of Eq.(3) by ( 1) , and take the sum of the resulting expressions. We then substract this sum from 1BK S

. The factor 1

BK amounts to a certain normalization of 1 , .

1=1 =1

1 ( 1)n n

i i ii iB

S P PVK

(4) where

=1= ln

n

B i ii

S K P P Differentiating it with respect to iP , and setting the resulting expression equal to zero, we obtain

1ln 1 ( 1) = 0i iP V and then

1= exp( )i iP V (5)

Inserting Eq.(5) into Eq.(3) leads to

1=1

e = exp( )n

ii

V (6) which allows us to determine once 1 is determi- ned.

Let

1=1

= exp( )n

ii

Z V (7) Then e = Z or = ln Z . Inserting Eq.(5) into Eq.(2) gives

1=1

e exp( ) =n

i ii

V V V (8)

Because

1=11

= exp( )( )n

i ii

Z V V ,

Eq.(8) can be rewritten as

1

1= ZVZ

(9)

Inserting Eq.(5) into Eq.(1) yields

1=1

= ln[exp( )] =n

B i ii

S K P V

1=1

( )n

B i ii

K P V ,

max1 1

=1= ( + ) = +

n

i iiB

S P V VK

(10) For Eq.(9),

1=11

1=1

1 1= = exp( ) =exp( )

n

i ini

ii

Z QV V VZ V

yielding

1 1=1 =1

exp( ) = exp( )n n

i i ii i

QV V V

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Taking the logarithm on its both sides

1 1=1 =1

exp( ) = ln exp( )n n

i i ii i

QV V V ,

1 1=1 =1

( + ln ) = ( )n n

i i ii i

QV V V yields

=11

=1

ln=

1

n

ii

n

ii

V

Q V

(11)

Thus, according to Eq.(11) we can calculate 1 , then Z and . According to Eq.(5), we can calculate iP . According to Eq.(10), we can calculate maxS . 4. Example

We simulate the speed distribution evolution for congestion spreading on the road with 1 km length using the proposed model. We discretize the road into cells with the same length of 50 m. The number of cell is twenty. The direction of traffic flow is from cell 1 to cell 20. The cell discretization is illustrated in Fig.1.

Fig.1 Cell discretization

The initial density on the road is discontinuous as follows:

( = 0, ) = 107 veh/kmK t x , 500 mx ,

( = 0, ) = 39 veh/kmK t x , 500 mx The cell density is 5.35 veh/cell as x 500 m, and the cell density is 1.95 veh/cell as x 500 m. The initial condition shows that the first 500 m section has higher density, and traffic congestion will spread to the downstream.

The volume-density relation is

= 339.45 + 57.7q k , 58.8k ,

= 4656.55 15.66q k , 58.8k where k is the density, q is the flow flax and x is

the position. The jam density is set as 224.3 veh/km, i.e., 11.2 veh/cell.

The speed ranges from 1 km/h to 120 km/h, which are discretized into 24 classes, and the discre- tization step is 5 km/h. The cell 1 and cell 20 are re- spectively set as the source cell and the destination cell, in which the density and speed distributions re- main unchanged. The number of vehicles distribution in the cell 1 and cell 20 are illustrated in Fig.2. The initial number of vehicles distribution from the cell 2 to the cell 10 is the same as the number of vehicles distribution in the cell 1. The initial number of vehi- cles distribution from the cell 11 to the cell 19 is the same as the number of vehicles distribution in the cell 20.

Fig.2 The number of vehicles distribution

Fig.3 Cell density evolution

Fig.4 Speed distribution at equilibrium

The vehicle transmission between the cells is cal-

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culated by the CTM. The speed distribution evolution from the cell 2 to the cell 19 is calculated by the tra- ffic kinetic model proposed in this paper. The time step is taken as 2 s. The obtained cell density evolu- tion is illustrated in Fig.3. Discontinuous initial den- sity evolves to an equilibrium one after some time and the density is almost the same for each cell. When tra- ffic flow evolves to the equilibrium state, the equili- brium speed distribution is the same in each cell, which is illustrated in Fig.4. The evolution of the Lagrange multiplier and the maximum information are respectively illustrated in Figs.5 and 6. The maxi- mum information in cells, i.e., from the cell 2 to the cell 10, in which the initial density is high and then gradually decreases as the order number of cell den- sity decreases. However, the maximum information for cells, i.e., from the cell 11 to the cell 19, in which the initial density is low, does not increase as the cell order increases.

Fig.5 Evolution of in cells from 1 to 18

Fig.6 Evolution of maximum information maxS in cells from 1

to 18 5. Conclusion

In this paper, we have proposed a discrete traffic kinetic model which integrates the CTM with the maximum information principle. The presented model can deduce the speed distribution based on the macro- scopic average speed, which is a better attempt to de- duce the mesoscopic state from the macroscopic varia- bles. The model provides a way to study the relation

between information and traffic flow situation. An example shows the relation between maximum infor- mation and the density. As the density is higher than the critical density, the maximum information of speed distribution decreases with decreasing density. As the density is lower than the critical density, the maximum information of speed distribution remains almost unchanged for different densities. Acknowledgement

This work was supported by the Open Fund of Key Laboratory of Road Structure and Material of Ministry of Transport, Changsha University of Scie- nce and Technology (Grant No. kfj100206). References [1] LU S....