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JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 10, Issue 4, August 2010 Online English edition of the Chinese language journal
Cite this article as: J Transpn Sys Eng & IT, 2010, 10(4), 73−78.
Received date: Jun 18, 2009; Revised date: Jan 15, 2010; Accepted date: Mar 19, 2010 *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)60055-6
RESEARCH PAPER
Urban Traffic Flow Prediction with Variable Cell Transmission Model HU Xiaojian*, WANG Wei, SHENG Hui Transportation College, Southeast University, Nanjing 210096, China
Abstract: The real-time and reliable urban traffic flow prediction is the foundation of traffic management and control in urban traffic. A variable cell transmission model (VCTM) is proposed to improve the ability of the urban traffic flow prediction. Based on the analysis of the defects using the cell transmission model (CTM) to predict the urban traffic flow, two parameters are introduced into the VCTM according to the traffic flow conservation law. In the VCTM, three different types of cell connections are described by three different expressions including simple connection, merge connection, and diverge connection. Two parameters of the VCTM increase computational burden, but the cell length is adjustable which ensures that the VCTM can be widely utilized in the roads with different lengths. The simulation results indicate that the VCTM meets the requirements of the traffic flow prediction in the urban traffic. Key Words: traffic engineering; cell transmission model; urban road; traffic prediction
1 Introduction
In 1994, Daganzo put forward the cell transmission model (CTM)[1,2] that clearly described the physical effect of the queue and some dynamic characteristics of the traffic flow, including shock wave, queue form, queue dissipation, and the mutual effect. This model has superiority in describing the characteristics of the traffic flow of larger waves. The CTM fully considers the microscopic characteristics of the traffic flow and pays attention to the transmission between small groups of vehicles. Thus, the CTM can well describe the nonsaturated, saturated, and oversaturated traffic flow to meet the accuracy requirement of traffic flow prediction.
The advantages of the CTM are gradually being found in the field of traffic engineering, which brings a new idea for traffic flow study models and provides effective measures for the application of traffic engineering. Ref. [3] describes the simple network by the CTM, takes the minimum delay as the optimization objective, and applies the mixed-integer programming to obtain the optimal traffic control parameters. The results reveal the consistent conclusion with the traditional methods, which laid the basis for the application of the CTM in the traffic signal control field. Lo et al.[4] applied
the genetic algorithm (GA) to obtain the dynamic signal timing program in the situation of time-varying traffic based on the CTM. The results of the application indicate that this method is more effective than the TRANSYT. Chang and Laura[5,6] established a multiobjective traffic signal control strategy based on the improved CTM and obtained good control effects. Zeng[7] applied the CTM to formulate the traffic flow model in the urban road and optimizes the signal period, green ratio, and phase separation. The experimental results showed that the method got satisfactory results.
The CTM has been widely applied in the field of intelligent traffic management and control. However, most of these applications are on the highways, expressways, motor roads, or simple urban roads. The disadvantage of the CTM is poor adaptability in the complex urban road network.
2 Variable cell transmission model (VCTM)
As the lengths of urban roads are relatively short and nonunified, the CTM needs to establish different cell lengths according to the different lengths of urban roads. In addition, for determining the cell length, the length of the general section and the approach section of the road should be
HU Xiaojian et al. / J Transpn Sys Eng & IT, 2010, 10(4), 73−78
Fig. 1 Cells of the urban road
Fig. 2 Cells of the VCTM
Fig. 3 Cells of the CTM and the VCTM
considered. In the approach section, vehicles are not allowed to change lanes arbitrarily, thus the approaches in different directions should be defined as different cells, such as cells (9, L), (9, T), (9, R), as shown in Fig. 1. In the general section, vehicles change lanes freely, thus different lanes at the same location can be defined as a same cell, such as cells (5), (6), and (7), as shown in Fig. 1. When the lengths of the approach section and the general section are considered separately, it is difficult to build cells with the same length in the CTM such as cell (1) in Fig. 1.
The cell length built by the variable cell transmission model (VCTM) can be different, which is helpful to build the road network by the VCTM, as shown in Fig. 2.
The VCTM should meet the flow conservation law, namely “inflow−outflow=change in quantity”[8].
[ ] [ ]1( ) ( ) d ( 1) ( )i i i i iq t q t T k t k t l+− = + − (1) where qi(t) is the traffic flow rate from cell i−1 to i, when the time index advances from t to t+1, (veh/sec). ki(t) is the traffic flow density in cell i during tth time interval, (veh/m). dT is the time interval (sec). li is the length of cell i and not less than the distance traveled by a typical vehicle in the free stream velocity during the time interval Lc, (m).
Simplify Eq. (1):
( ) ( ) [ ]1d1 ( ) ( )i i i i
i
Tk t k t q t q tl ++ = + − (2)
Equation (2) shows the characteristics of the traffic flow in the cell, which shows that there is no significant difference between the VCTM and the CTM. Eq. (2) is more complex
and requires more computing, but the VCTM is more applicable to the urban road network via introducing the cell length and the cell density into the model. First, the cell length is more flexible and the length does not meet the limit that the cell lengths should be unified. Second, the cell density can be obtained by the traffic flow detection.
In the VCTM, vehicle transmission between cells is considered in two ways: the vehicle number from the upstream cell and the vehicle number into the downstream cell.
As shown in Fig. 3, according to the principles of the CTM, during the time interval dT, the vehicle number from the upstream cell i−1 is determined by the vehicle number of the gray section of cell i−1 and the maximum number of the traffic flow from cell i−1 to cell i; therefore, the vehicle number from the upstream cell i−1 is described as
1 1 11
( ) min ( ), ( )ci i i
i
LS t n t Q tl− − −−
⎧ ⎫= ⋅⎨ ⎬
⎩ ⎭ (3)
where Si−1(t) is the number of the outflow from cell i−1 (veh) during the tth time interval, ni−1(t) is the number of vehicles in the cell i−1, and Q i−1(t) is the maximum number of the traffic flow from cell i−1 to i during the tth time interval (veh).
The vehicle number into the downstream cell i is determined by the vehicle number of the gray section of cell i and the maximum number of the traffic flow from cell i−1 to cell i; therefore, the vehicle number into the downstream cell i is described as
[ ]( ) min ( ), ( ) ( )ci i i i
i
LR t Q t N t n tl
δ⎧ ⎫⋅= −⎨ ⎬
⎩ ⎭ (4)
where Ri(t) is the number of the inflow to cell i during the tth time interval, δ is the transmission coefficient of the jam, and Ni(t) is the maximum number of vehicles that can be presented in cell i.
According to Eqs. (3) and (4), the number of the vehicles’ transfer between cells is
[ ]11
( ) min ( ), ( ), ( ) ( )c ci i i i i
i i
L Ly t n t Q t N t n tl l
δ−
−
⎧ ⎫⋅= −⎨ ⎬
⎩ ⎭ (5)
Both sides of Eq. (5) are divided by the time interval:
[ ]11
( ) ( )min ( ), , ( ) ( )i c i ci i i
i i
y t L Q t Ln t N t n tdT dT l dT dT l
δ−
−
⎧ ⎫= −⎨ ⎬⋅ ⋅⎩ ⎭
(6)
HU Xiaojian et al. / J Transpn Sys Eng & IT, 2010, 10(4), 73−78
According to Eq. (1), Eq. (6) can be transformed as
[ ]max1
1
( ) min ( ), ( ), ( ) ( )i i i i ii i
v vq t n t q t N t n tl l
δ−
−
⎧ ⎫⋅= −⎨ ⎬
⎩ ⎭ (7)
where qimax(t) is the maximum of the traffic flow rate from cell
i−1 to i during the tth time interval (veh/sec) and v is the free stream velocity.
Equation (7) is simplified as ( ) ( ) ( ){ }max
1( ) min , ( ), jami i i i iq t v k t q t v k t k tδ− ⎡ ⎤= ⋅ −⎣ ⎦ (8)
where kijam(t) is the density of the traffic jam in cell i during tth
time interval. Equation (8) describes the relationship between the traffic
flow density and the traffic flow rate of the cell in the VCTM. Compared with the CTM, this equation reflects the process of transmission, queuing, and evacuation and the relationships among the traffic volume, the traffic speed, and the traffic density. The VCTM meets the requirement that the cell length be variable.
The cell length is introduced into the VCTM to ensure that the building cell is more flexible. The length does not meet the limit that the cell length should be unified, and the length should be no less than the distance traveled by a typical vehicle in the free stream velocity during the time interval.
The traffic outflow rate si−1(t) (veh/sec) and the traffic outflow rate ri(t) (veh/sec) can be expressed as follows:
( ){ }max1 1( ) min , ( )i i is t vk t q t− −= (9)
( ) ( ){ }max( ) min ( ), jami i i ir t q t v k t k tδ ⎡ ⎤= −⎣ ⎦ (10)
In the form of merge connection, during the tth time interval, the traffic flow rates from cells (i−1, U) and (i−1, D) into cell i are expressed as follows:
( ) ( ) ( ){ } ( ) ( )( ) ( )
1, 1, 1, 1, 1,,
1, 1, 1,
mid ( ), ( ), ( )( )
( ) ( )i U i i D i U i i i U i D
i Ui U i i U i D
s t r t s t p t r t r t s t s tq t
s t r t s t s t− − − − −
− − −
⎧ − ≤ +⎪= ⎨> +⎪⎩
(11) ( ) ( ) ( ){ } ( ) ( )
( ) ( )1, 1, 1, 1, 1,
,1, 1, 1,
mid ( ), ( ), ( )( )
( ) ( )i D i i U i D i i i U i D
i Di D i i U i D
s t r t s t p t r t r t s t s tq t
s t r t s t s t− − − − −
− − −
⎧ − ≤ +⎪= ⎨> +⎪⎩
(12) where pi−1,U is the ratio of the vehicle number from cell (i−1, U) into cell i; pi−1, D is the ratio of the vehicle number from cell (i−1, D) into cell i, pi−1,U+pi−1,D=1; mid is the median value of the three values.
In the form of diverge connection, during the tth time interval, the traffic flow rates from cell i−1 into cells (i, U) and (i, D) are expressed as follows:
( ) ( ){ }1 , , , ,( ) min ( ), ,i i i U i U i D i Dq t s t r t r tβ β−= (13)
( ), ,( )i U i U iq t q tβ′ = (14)
( ), ,( )i D i D iq t q tβ′ = (15)
where βi, U is the ratio of the vehicle number from cell i−1 into cell (i, U) and βi, D is the ratio of the vehicle number from cell i−1 into cell (i, D), βi, U+βi, D=1.
3 Simulation analysis
This simulation considers an urban road to be the object, as shown in Fig. 4. The VCTM is used to predict the traffic flow from west to east in the road. This road is composed of the approach section and the general section. The length of the general section is approximately 287 m and the approach section is 81 m. The full length is approximately 368 m. The general section has two lanes from west to east direction, and the lanes are expanded to three lanes at the approach section.
To test the accuracy and reliability of the traffic flow prediction based on the VCTM, it is necessary to detect the real-time traffic flow in every cell and analyze the predicted data compared with the detected data. However, the detection equipment can only detect the overall traffic flow of the road because of the restriction on the distribution of detections in the traffic network. Therefore, VISSIM is taken as the traffic flow simulation platform to analyze the traffic flow prediction. Although VISSIM has defects such as differences between the simulated traffic flow and the actual traffic flow, VISSIM is widely adopted to simulate the traffic flow.
VISSIM generates the traffic flow of the simulated road and detects the traffic flow data of each cell. The traffic flow of each cell in every time interval is predicted based on the VCTM, and then, the prediction results are compared with the traffic flow data of each cell obtained by VISSIM to analyze the accuracy and reliability of the VCTM in the traffic flow prediction.
Fig. 4 Simulated road diagram
HU Xiaojian et al. / J Transpn Sys Eng & IT, 2010, 10(4), 73−78
Table 1 Cells’ parameters of simulated road
Cells Length (m) Maximum of density (veh/m)
Maximum of rate (veh/s)
(1) 77 0.4 1.2
(2) 70 0.4 1.2
(3) 70 0.4 1.2
(4) 70 0.4 1.2
(5, L) 81 0.2 0.6
(5, T) 81 0.4 1.2
(∞), (o) - ∞ ∞
Table 2 Analysis of Forecast cell density (700 veh/h)
Cells Mean of
simulated density
Mean of predicted density
Mean error
Equal coefficient
(1) 0.0161 0.0146 0.0014 1
(2) 0.0177 0.0150 0.0027 0.948
(3) 0.0176 0.0155 0.0021 0.864
(4) 0.0157 0.0151 0.0006 0.891
(5, L) 0.0562 0.0483 0.0079 0.863
(5, T) 0.0150 0.0297 –0.0147 0.833
Table 3 Analysis of forecast cell density (1250 veh/h)
Cells Mean of
simulated density
Mean of predicted density
Mean error Equal coefficient
(1) 0.0289 0.0250 0.0039 0.856
(2) 0.0310 0.0258 0.0052 0.866
(3) 0.0316 0.0268 0.0049 0.872
(4) 0.0297 0.0267 0.0030 0.851
(5, L) 0.1131 0.1088 0.0031 0.916
(5, T) 0.0329 0.0588 –0.0260 0.952
Assume that the free stream velocity is 50 km/h, the time interval is 5 s, and the minimum of the cell length is 69 m. Because of the different characteristics of the traffic flow in the approach section and the general section, the cells are built separately. Because it is unable to find a suitable length of the cell to satisfy the limit of the CTM, that is, the lengths of all cells should be unified and should be no less than the minimum of the cell length, the CTM cannot be applied to forecast the traffic flow of this simulated road. The VCTM is suitable to forecast the traffic flow of this simulated road because the VCTM ensures the building of cells the length of which is not unified. As shown in Table 1, seven cells and two virtual cells are established separately in the simulated road. The cells in the general section are (1), (2), (3), and (4); the cell in the left approach is (5, L); the cell in the straight approach is (5, T); the starting cell and the terminal cell of virtual cells are (o) and (∞), respectively. The parameters of the cells are listed in Table 1.
The signal cycles of both intersections are 60 s, the green time of the straight direction flow is 25 s, and the left-turn direction flow is 10 s. The ratio of the vehicles between the left-turn direction flow and the straight direction flow is 3:7. The traffic flow of the simulated road is simulated as the nonsaturated flow (700 veh/h) and the saturated flow (1250 veh/h). The traffic flows are real-time predicted by the VCTM. The prediction results are shown in Tables 2 and 3.
The mean errors of cells (1), (2), (3), and (4) are approximately 0, and the prediction results are good. It is obvious that the mean errors of cells (5, L) and (5, T) are relatively large and become larger with the increase of the traffic flow in the simulated road. In two situations, the prediction data of cell (5, L) are slightly smaller than the simulated data and the prediction data of cell (5, T) are slightly larger. The main reasons are as follows: first, the travel directions of simulated vehicles are stochastic, thus the ratio of the vehicles between the left-turn direction flow and the straight direction flow is not strictly equal to 3:7 during a short period of simulation, but the ratio is strictly equal to 3:7 in the VCTM during any periods of simulation. Second, vehicles are not allowed to change lanes arbitrary in the approach section, and the VCTM strictly meets this requirement, but the simulation allows the phenomenon of changing lanes in the approach section, which is more consistent with the actual situation of travel vehicles.
According to the mean errors and the equal coefficients of Tables 2 and 3, it is obvious that the mean error increases and the accuracy of the prediction decreases with the increase of traffic flow. However, the mean errors are almost under 0.01, and the equal coefficients are above 0.85. The prediction results are satisfactory, although the prediction errors of a few cells are slightly worse than the ideal value, which is acceptable.
4 Conclusions
(1) According to the flow conservation law, the cell length and the cell density are introduced into the VCTM to ensure that the lengths of cells are not unified, which is helpful to build cells. The VCTM is suitable for different kinds of urban roads, and the VCTM can decrease the number of cells to improve the operation efficiency.
(2) The simulation results indicate that the VCTM can build the cell with variable length for the traffic flow prediction, and the accuracy between the prediction data and the simulation data is good, which ensures that the traffic flow prediction by the VCTM meets the requirements of the prediction in the urban network.
(3) On the basis of Eq. (3), the prediction of the VCTM is that the distribution of the traffic flow in the cell should be uniform. From the macro point of view, the distribution is approximately uniform because of the short length of the cell.
HU Xiaojian et al. / J Transpn Sys Eng & IT, 2010, 10(4), 73−78
However, it is necessary to further study the distribution in the VCTM to improve the accuracy of the traffic flow prediction.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (No. 50738001) and the Post-Doctoral Research Funding Schemes of Jiangsu Province of China (No. 0902004B).
References
[1] Daganzo C F. The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transportation Research Part B, 1994, 28(4): 269–287.
[2] Daganzo C F. The cell transmission model, part II: Network
traffic. Transportation Research Part B, 1995, 29(2): 79–93.
[3] Lo H K. A novel traffic signal control formulation. Transportation Research Part A, 1999, 33(6): 433–448.
[4] Lo H K, Chang E, Chan Y C. Dynamic network traffic control. Transportation Research Part A, 2001, 35(8): 721–744.
[5] Chang J L. Developments of Multi-Criteria Signal Timing Strategies for Critical Intersections, University of Miami, 2003.
[6] Laura M M. Macroscopic Modeling and Identification of Freeway Traffic Flow, University of California, 2004.
[7] Zen J Q, Wang J J, Tang L, et al. CTM-RH based line control for urban traffic lights. Journal of China University of Science and Technology, 2006, 36(11): 1232–1236.
[8] Wang W, Guo X C. Traffic Engineering, Nanjing, China: Southeast University Press, 2000.
