selection of best suitable traffic model - … selection of best suitable traffic model axay s....

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SELECTION OF BEST SUITABLE TRAFFIC MODEL

Axay S. Shah, Jaydeepsinh P. ChudasamaM.Tech students, Transportation system Engg.

BVM Engineering CollegeVallabh Vidyangar

[email protected]

Dr. L.B.ZalaAssociate Professor, Civil Engineering Department


[email protected]

Dr.(Mrs.) T.A.DesaiProfessor & Head, Mathematics Department


[email protected]

Aakar N. RogheliaAssistant Professor, Mathematics Department


[email protected]

Abstract

Huge amount of literature has been produced on the relationships between the speed, flow and density of traffic and the factors which affects this relationship.Flow, Speed and Density are three attributes which describe uninterrupted traffic stream. The models by Greensheild(1934) and Greenberg(1959) are very well known and have been in application for quite sometime. The purpose of this research paper is to select an efficient and suitable traffic flow model to predict the space mean speed us, w.r.t mean free flow speed, uf, and the density, k. The correlation coefficient will then be calculated.

Keywords: traffic, model, regression analysis, correlation coefficient

1. Introduction

Traffic networks-consisting of highways , streets , and other kinds of roadways provide convenient and economical conveyance of passengers and goods. The basic activity in transportation is a trip, defined by its origin /destination, departure time/arrival time and travel route .This research is essentially about the relationship between speed and density on urban road of Vallabh Vidyanagar. The data for this study is collected from Anand - Vidyanagar road.Objective of this research is: a short section devoted to examining some problems and constraints of regression analysis,given that it is the most frequently used technique in this research. There have been two approaches in mathematical modeling of traffic flow. One approach, from a microscopic view, studies individual movements of vehicles and interactions between vehicle pairs. This approach considers driving behavior and vehicle pair dynamics. . The other approach studies the macroscopic features of traffic flows such as flow rate q, traffic density k and travel speed v. The basic relationship between the three variables is: q= kv. Macroscopic models are more suitable for modeling traffic flow in complex networks since less supporting data and computation are needed.In this paper macroscopic traffic flow models are executed both theoretically and numerically. Traffic

flows are classified according to traffic conditions, roadway conditions and traffic network structure. Different types of traffic flow are described by different models.

2. Literature Review

The problem based literature review for model is described below:

2.1 Greenshield Model

Macroscopic stream models represent how the behaviour of one parameter of trafiic flow changes with respect to another. Most important among them is the relation between speed and density. The first and most simple relation between them is proposed by Greenshield. Greenshield assumed a linear speed-density relationship as illustrated in figure to derive the model. The equation for this relationship is shown below.

Figure.1 Linear Speed-Density relationship

fs f

j

uu u k

k

(1)

where su is the space mean speed at density k,

fu is the mean free speed and kj is the jam density.

This equation (1) is often referred to as the

13-14 May 2011 B.V.M. Engineering College, V.V.Nagar,Gujarat,India

National Conference on Recent Trends in Engineering & Technology

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Greenshield's model. It indicates that when density becomes zero, speed approaches free flow speed .

2.2 Calibration of Greenshield’s Model

Inorder to use this model for any traffic stream, one shouid get the boundary values ,

especially free flow speed ( fu ) and jam density

( jk ). This has to be obtained by field survey and

this is called calibration process. Although it is difficult to determine exact free flow speed and jam density directly from the field , approximate values can be obtained from a number of speed and density observations and then fitting a linear equation between them. Let the linear equation be y a bx such that y is mean free speed uf

and x denotes the density k. Using linear

regression method , coefficients a and b can be solved as,

1 1 12

2

1 1

1

1

n n n

i i i ii i i

n n

i ii i

x y x yn

b

x xn

And

1 1

1 n n

i ii i

ba y x

n n

or a y bx

Where ix and iy are the samples , n is the number

of samples and x and y are the mean

of ix and iy respectively .

2.3 Greenberg’s Logarithmic Model

Greenberg assumed a logarithmic relation between speed and density. He proposed,

ln js

ku c

k (2)

This model has gained very good popularity because this model can be derived analytically. (This derivation is beyond the scope of this notes). However, main drawbacks of this model is that as density tends to zero, speed tends to infinity. This shows the inability of the model to predict the speeds at lower densities.

3. Data Collection :

To calibrate the existing models for speed data for traffic on Anand-Vidyanagar road , spot

speed data were collected. For collection of spot-speed, trap- length method with 30m trap length at Akshar Purshottam Chhatralaya site were adopted. The data collected is presented in Table-1 and Table-2 below:

Table -1 Speed and Density observationsGreensheild

Sr. No

Mean speed(us)

Frequency

Density (k)su

(kmph) (yi) NO. Vehicles/Km(xi)

Kmph

1 13.45 5 166.7 39.708

2 18.45 5 166.7 39.708

3 23.45 11 366.7 38.986

4 28.45 18 600.0 38.143

5 33.45 30 1000.0 36.699

6 38.45 53 1766.7 33.930

7 43.45 25 833.3 37.301

8 48.45 15 500.0 38.504

9 53.45 4 133.3 39.828

10 58.45 2 66.7 40.069

11 63.45 2 66.7 40.069

Table -2 Speed and Density observations Greenberg

Sr. No

Mean speed (us)

Frequency

(lnk )su

(kmph) (yi) NO. Vehicles/Km (xi)

Kmph

1 13.45 5 5.12 41.46

2 18.45 5 5.12 41.46

3 23.45 11 5.90 37.66

4 28.45 18 6.40 35.29

5 33.45 30 6.91 32.83

6 38.45 53 7.48 30.09

7 43.45 25 6.73 33.71

8 48.45 15 6.21 36.17

9 53.45 4 4.89 42.54

10 58.45 2 4.20 45.87

11 63.45 2 4.20 45.87



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4. Data Analysis :

The macroscopic models (1 & 2) is fitted using regression analysis in excel sheet. The model and graph have been presented in figure 2 and figure 3. The calibration coefficient is given in table 3:

Figure.2 Speed-Density relation, Greenshield

Figure.3 Speed-Density relation Greenberg

Table-3 : Correlation coeffieicnt of Greenberg and Greenshield model

Model Greenberg

(non-linear)

Greenshield

(non-linear)

2R 0.956 0.860

Conclusion

The coefficient 2R value of Greenberg

and Greensheild model is 0.956 and 0.860 respectively describe that the both models are good fit for speed and density parameters. As per the test of goodness-of-fit, the model that described the field data more adequately was the Greenberg

model, which showed the 2R value of a 0.956

is suitable, whereas the Greeenshield model is not considered suitable.

References

[1] B.D. Greenshields, A study of traffic capacity Proceedings of Highway Research Board, Vol.14[93].

[2] D.R.Drew. Deterministic aspects of freeway operations and control. Highway Research Record., Vol.99, 1965

[3] H. Greenberg, An analysis of traffic flow Operations Research, Vol. 7, july, 1959

[4] N.J.Garber and L.A.Hoel, Traffic and highway engineering, Brooks/Cole,2002.



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selection of best suitable traffic model - … selection of best suitable traffic model axay s....

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