OD Matrix Estimation for Urban Expressway

JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 10, Issue 2, April 2010 Online English edition of the Chinese language journal

Cite this article as: J Transpn Sys Eng & IT, 2010, 10(2), 83 87.

Received date: Jul 10, 2009; Revised date: Oct 27, 2009; Accepted date: Nov 10, 2009 *Corresponding author. E-mail: [email protected] Copyright © 2010, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(09)60035-0

RESEARCH PAPER

OD Matrix Estimation for Urban Expressway WANG Yuanqing*, YANG Liu, GENG Yin, ZHENG Mengmeng School of Highway, Chang’an University, Xi’an 710064, China

Abstract: OD information of urban expressway is the basis of analyzing the traffic operation and performance. A proper OD estimation method is capable of reducing the traffic surveys pressure and avoid traffic jams during survey and unreal OD information. This paper proposed a systematic method of OD estimation including traffic zone division in the form of the entrance and exit ramps of one direction on expressway, and established the seed OD matrix of high quality using the Gravity and Frator method. It estimated the OD matrix in the OD estimation module in TransCAD software, which was applied in the Inner Ring Road in Guangzhou city. The results showed that the relative errors of the traffic volumes on the links and ramps, the average trip distance and the traffic mileage are small with the seed OD matrix calculation and OD matrix estimation, which manifests that the estimated OD matrix is accurate and reliable and the proposed method is appropriate for the urban expressway traffic analysis.

Key Words: urban traffic; urban expressway; OD matrix estimation; traffic zone division; seed OD matrix

1 Introduction

Currently the urban expressway systems of metropolises in China have taken shape. Urban expressways are the main frame of the road system, especially for automobiles, entirely closed and access controlled. As the artery of the urban transportation and economy, urban expressways connect the main roads and the radial roads, gathering and distributing the traffic in central area and connecting the urban and suburban areas. The traffic flow features in trip distance and space on the expressway are the important basis for reducing traffic jams, ensuring the traffic service level in the city center and evaluating the transportation investment performance. Therefore, the origin–destination (OD) information of the expressway is of much importance.

The expressway capacity is about 2000pcu/lane/hour. If the survey on OD information is conducted by the roadside, the traffic jams and the changes on vehicle routing choice will be caused, and the OD information will be distorted. Because the expressway has many accesses and heavy traffics, it needs a lot of photographing equipment and much data processing work to get the OD information by the vehicle number plate survey[1,2]. Consequently, the OD matrix estimation technique

has the advantages of cost saving, is much convenient, and has high practical value compared with the traditional methods mentioned above. Wei et al. suggested that a high-quality seed OD matrix could play an important role in improving the accuracy of the OD matrix estimation[3,4]. When there is no historical data, it can be used to calculate the traffic volume between zones. To avoid the above problems that will cause difficulty in the traffic flow analysis on the expressway, this paper proposed a systematic method for OD matrix estimation to improve the accuracy and reliability of the matrix estimation result. This method makes use of the traffic volume data obtained by the existing traffic detectors at the ramps and the sections of the expressway, divide the traffic zones according to the ramp service areas to estimate their traffic generation and attraction, and establishes the seed OD matrix using the gravity and double-constraint Fratar method.

This method has been applied and tested in the Inner Ring Road Project in Guangzhou based on the TransCAD software platform. The result shows that this method can achieve a higher fitting precision with lower cost on traffic data collection, and the key issue of establishing seed OD matrix for the OD matrix estimation of the urban expressway has been solved, through the integration of related techniques.

WANG Yuanqing et al. / J Transpn Sys Eng & IT, 2010, 10(2), 83 87

2 Basis of expressway OD estimation: traffic zone division

All the urban expressways are connected with other roads in cities, and all vehicles enter and exit through ramps, so the ramps can be the objects of the traffic zone division, and the traffic zones can be divided according to ramp service areas.

The traffic zones are divided by three ways: first, the area served by a pair of adjacent entrance and exit ramps in the same direction of the expressway as a traffic zone; second, the area served by a pair of entrance and exit ramps in two directions of the expressway as a traffic zone; third, the area served by a single ramp, either an entrance or an exit ramp, as a traffic zone. Whichever approach is adopted, the traffic volumes of the entrance ramp and exit ramp can be summed as the traffic generation volumes Pi of the zone i and traffic attraction volumes Ai of the zone i, respectively.

The traffic generation and attraction volumes of each zone by surveying traffic volume data or using the traffic detectors on the ramps can be easily obtained using this method of dividing the traffic zone by ramps compared with traditional methods, in which traffic zones are divided according to the administrative districts, land-use planning, or natural separation. This approach provides the base data for the seed OD matrix calculation and the verification in the next step.

3 Seed OD matrix calculation

The seed OD matrix is of great importance, as it is the basis of the OD matrix estimation process. Also it reflects the traffic distribution structure; therefore, Wei et al. suggested that a high-quality seed OD matrix could significantly improve the accuracy of the matrix estimation results. When there is no historical OD information except the traffic generation and attraction volumes of each zone, traffic distribution methods can be used to calculate the traffic volume between zones[5].Among the commonly used calculation models, the gravity model can ensure more practical results since the attracted strength and impedance between the traffic zones have been taken into consideration, so the seed OD matrix can be established by using the gravity model. Travel time or travel distance can be used as the impedance parameters in the gravity model, and the traffic volumes between zone i and zone j are shown as Eq. (1).

i jij

ij

P At K

R (1)

where tij is the traffic volume between zone i and zone j; Pi is the observed traffic generation volume of the zone i; Aj is the observed traffic attraction volume of the zone j; Rij is the distance or free-flow travel time between zone i and zone j; k,

, are the model parameters. If the relative errors between the generation and attraction

volumes drawn from OD matrix through the gravity model and those from survey on the ramps, it is necessary to modify the seed OD matrix with double-constraint Fratar method till the accurate results are achieved. This method is usually used in the OD matrix calculation, so only a brief discussion is introduced below, and the detail information can be found in Refs. [5] and [6]. .

(0)iP and (0)

jA are the traffic generation and attraction volumes, respectively, after using the gravity model to estimate the seed OD matrix. Define the difference coefficients between generation and the observed volumes, and attraction and observed volumes are as follows:

(0)(0)i

pii

PFP

, (0)(0)

jaj

j

AF

A (2)

Therefore, the revised traffic volumes between zone i and zone j are:

(1) (0) (0)( , )ij ij pi ajt t f F F (3)

where

(0)(0)

(0) (0) (0) (0) (0) (0)

12 ( ) ( )

i j ji

i j ij j j ij i ij i

P A APfP A t A A t P P

Summing the traffic distributions, we can obtain (1) (1)

i ijj

P t and(1) (1)j ij

iA t , if (1)

iP . If (1)jA and Pi, Aj are not

same and if the error has exceeded the acceptable upper limit standard, then we use (1)

iP and (1)jA to replace the

(0)iP and (0)

iA in the Eq.(2), we continue to the second iteration until the coefficient values in the Eq. (4) are close to 1, and the corresponding ( )k

ijt is the initial traffic volume between zone iand zone j in the seed OD matrix for OD matrix estimation.

( )( )

k ipi k

i

PFP

, ( )( )

jkaj k

j

AF

A (4)

4 OD matrix estimation theory and process based on TransCAD platform

Assume that Va be the volume of the road section a, Tij be the traffic volume between zone i and zone j, and a

ijP be the traffic proportion through the road section a, between zone iand zone j. Obviously,

(0 1)a aa ij ij ij

ijV T P P (5)

The above equation is the basic equation that makes matrix estimation using the traffic volume on the expressway. As long as there is enough traffic volume on the road sections, the value of Tij can be obtained by solving simultaneous linear equations. The randomness of the traffic volume from the survey on the road section is taken into account in the OD matrix estimation module of the TransCAD software. Thus, any of the traffic assignment methods can be used to realize the matrix estimation function through multiple iterations between the traffic assignment and the matrix estimation[1,3].


Fig. 1 Expressway OD matrix estimation process

Fig. 2 Road network in TransCAD software

Table 1 Estimated Peak hour OD matrix and seed OD matrix

Estimated OD matrix

All-or-nothing assignment

method

User equilibrium assignment

method

Seed OD matrix

Matrix volume (pcu) 48642.04 49024.76 49991

Relative error –2.698% –1.933% -

OD matrix estimation module requires the input data including the traffic volumes observed on the road sections, sectional expressway capacity, free-flow time, seed OD matrix, the traffic assignment method and the corresponding impedance function parameters. Among them, the traffic volumes observed on the road sections, sectional expressway capacity and free flow time are contained in the line layer data file, and they are the basic data of the traffic assignment.

Using the traffic zone division method and the gravity and double-constraint Fratar methods, the seed OD matrix can be

established; then, the OD matrix estimation and traffic assignment were made in the TransCAD; whether the relative errors reached the standard was judged, and the optimized OD matrix was obtained.

5 Calibration indexes

Calibration is usually done by comparing observed traffic volume with the model fitting results. After the traffic zones are divided based on the ramps, more calibrations can be done on the traffic generation and attraction volume of each zone, comparing the traffic volumes in the columns and rows with those surveyed on the ramps to analyze the errors. Moreover, the reliability and accuracy of the estimated matrix can also be checked through the users’ average trip distances or mileages of the vehicles on the urban expressway.

6 Case study

The Inner Ring Road in Guangzhou is a closed elevated expressway ring road built over the original road in the center of the city, which is one of the express ring roads of large-scale, full-feature, and well-equipment express ring roads in China at present. Fig. 2 shows the abstract Inner Ring Road network in TransCAD transportation planning software. The area, served by a pair of adjacent entrance and exit ramps in the same direction of the expressway, is regarded as a traffic zone in traffic zone division method, and thus 52 traffic zones are totally divided. The data come from the traffic volumes on the road sections and ramps in 12 hours of a weekday in 2006.

In the gravity model, the free-flow time between two zones and the empirical value of the average speed of the vehicle on the urban express road can be calculated from the distance matrix provided by TransCAD software (distance matrix is divided by average speed of the vehicle, and the unit of the free-flow time is hour). The value of the parameter of and is 1. If the value of the parameter K is 1, the peak hour generation and attraction volume of each zone is around 100 times larger than the actual ramp volume; therefore, the value of the parameter K is set to be 10–2. After the double-constraint Fratar method was used in iteration for four times, the total peak hour seed OD matrix volume is 49991 pcu. All-or-nothing assignment method and user equilibrium assignment method were used in the TransCAD matrix estimation module, and the matrix estimated results are 48642.04 pcu and 49024.76 pcu, respectively, and the relative errors compared with the seed OD matrix are respectively –2.698% and –1.933%, respectively. Table 1 shows the matrix volume comparison results and the relative errors. Table 2 shows the relative error comparisons of the traffic assignment on the road section and traffic generation and attraction. Fig. 3 shows value changes of root-mean-square-error (RMSE).

Yuexiu District Liwan District

Haizhu District


Table 2 Calibration results of the assignment and the traffic generations and the attractions (%)

All-or-nothing assignment method User equilibrium assignment method Statistics Relative error of the

road section volume Relative error of the traffic

generation and attraction volumeRelative error of the road

section volume Relative error of the

traffic generation and attraction volume

Minimum 0.00 0.03 0.00 0.00

Maximum 8.40 8.10 1.73 10.01

Mean 0.62 1.27 0.22 6.11

Standard error of average 0.09 0.18 0.03 1.84

Standard deviation 1.26 1.71 0.35 17.39

Variance 0.02 0.03 0.01 3.02

0

500

1000

1500

2000

0 20 40 60 80 100 120 140 160

Iteration times

RM

SE

User equilibrium assignment methodAll or nothing assignment method

Fig. 3 Value of RMSE gradually decreased as iteration times increased

The statistical analysis in Table 2 shows that the relative errors of the road section volume and the generation and attraction volume are all small and are in the acceptable error range. Fig. 3 shows that after the first iteration using all-or-nothing assignment method, the value of the RMSE decreased from 2031.65 to 508.34, and then, the value of RMSE gradually decreased as the iteration times increased. After the first iteration using the user equilibrium method, the value of the RMSE decreased from 200.98 to 106.61, and then the value of RMSE gradually decreased as the iteration times increase. In this case study, after using the all-or-nothing assignment method and the user equilibrium method, the two results only have a little difference because the expressway is ring shaped, and the access is controlled. The traffic zones are divided based on the ramps, and the traffic volumes between two zones are assigned on the shortest path of the expressway between the two zones. It needs to be verified whether both the assignment methods are effective to analyze other urban expressways.

Also the average trip distance can be used to check the reliability of the OD matrix. The average distance is 3.95 km, which is obtained from the peak hour vehicle mileage (192.163×103 pcu·km) divided by the estimated matrix volume (the vehicle mileage is obtained from the OD matrix estimation result in TransCAD). Compared with the average trip distance of 4.45 km on the Inner Ring Road in recent

years (2000-2005) in reference 7, the relative error is -12.36%. The empirical value of the proportion of the peak hour traffic volume to the all-day volume is 8%–8.5% (the average value of 8% and 8.5% is used in this paper). Thus, the estimated vehicle mileage of 12 hours is 233×104 pcu·km. Compared with the vehicle mileage 214×104 pcu·km of 12 hours one day in 2006 in reference 8, the relative error is 8.8%. The results manifest that the estimated matrix is almost consistent with the actual surveys, and the relative errors are within the acceptable range.

7 Conclusions

Traffic zone division is the premise and basis for the OD matrix estimation, which directly influences the accuracy of the estimated matrix. And a high-quality seed OD matrix can improve the reliability and accuracy of the matrix estimation results. The case study of the Inner Ring Road Project in Guangzhou illustrated that the accuracy and reliability of the estimated OD matrix can be improved by using the proposed traffic zone division method (taking the area served by a pair of adjacent entrance and exit ramps in the same direction of the expressway as a traffic zone) and the seed OD matrix calculation method (using the gravity and double-constraint Fratar method). The method proposed in this paper is very simple, convenient and practical, which can provide the basic data for the operation, economic and social performance evaluation of the urban expressway.

Acknowledgement

This research was funded by the Science and Technology Project of Transportation Ministry of China (Nos. 2006-312-812-120, 2006-319-812-140), and the authors express their thanks to all the persons who have contributed to this study.

References

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[2] Liu J Q. Application of OD matrix estimation in regional comprehensive transportation. Comprehensive Transportation, 2006, 6: 60–63.

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[4] Guo G Y. Study on the OD matrix estimation from link volumes. Shanghai Highway, 1999, 2: 41–44.

[5] Lu H P. Traffic Planning Theory and Method (2nd Edition), Beijing: Tsinghua University Press, 2006.

[6] Wang W, Guo X C. Traffic Engineering, Nanjing: Southeast University Press, 2003.

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OD Matrix Estimation for Urban Expressway