Available online at www.sciencedirect.com
ScienceDirect Procedia - Social and Behavioral Sciences 138 (2014) 37 – 44
The 9th International Conference on Traffic & Transportation Studies (ICTTS’2014)
Calibrating and Modeling Expressway with Different Ramp Distances on Beijing 3rd Ring Road Xiao-mei Zhaoa,b,*, Dong-fan Xiea,b, Wei-wei Wanga,b, Nan Genga,b b
a School of Traffic and Transportation, Beijing Jiaotong University, No.3 Shangyuancun, Haidian District, Beijing 100044, P.R. China MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, No.3 Shangyuancun, Haidian District, Beijing 100044, P.R. China
Abstract As a typical bottleneck connecting freeway and common road, ramp is an important traffic congestion area. Various traffic phenomena near ramps can be described by the Cell Transmission Model ( CTM). However, real traffic data shows that traffic flow characteristics are far different for the ramp system with different ramp distances. We analyze traffic characteristics of ramp system with both an on-ramp and an off-ramp in Beijing 3rd ring road firstly. Then, it is found that ramp distance between the onramp and the off-ramp has a great impact on the capacity, due to the conflict of traffic flow between on-ramp and off-ramp. Finally the traffic conflict is considered in the merge cell and Capacity Drop CTM (CD-CTM) is proposed to embody the conflict. The simulation results suggest that CD-CTM fits the empirical data from Beijing 3rd ring road. © 2014 2014 The Elsevier Ltd. This is an open access Ltd. article under the CC BY-NC-ND license © Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of Beijing Jiaotong University (BJU), Systems Engineering Society of China (SESC). Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). Keywords: urban traffic flow; cell transmission model; ramp system; urban expressway
1. Introduction Traffic congestion is an inevitable problem in most large cities all over the world. In order to suppress traffic jams and increase the capacity of the road networks, the nature way is to build new infrastructures. Particularly, expressways are developed rapidly in some large cities of China, such as Beijing, Shanghai, and so on. Although expressways provide advantage service for traveling, ramps connecting expressways and local roads become new
* Corresponding author. Tel.: +86-10-51684265; fax: +86-10-51684240. E-mail address:
[email protected].
1877-0428 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). doi:10.1016/j.sbspro.2014.07.179
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bottlenecks, which usually induce congestions. To understand the nature of congestions, theoretical traffic flow models have been developed and calibrated with empirical data. CTM is a famous traffic flow model presented by Daganzo (1994,1995). In CTM, road is divided into homogenous cells. This model clearly describes the interaction between neighboring cells. It relies on a sending function depending only on the state of upstream cells, and on a receiving function depending on the state of downstream cells. CTM provides an intuitively appealing and easy approach to deterministically describe how the number of vehicles in consecutive cells of a freeway evolves over consecutive time intervals. So far, CTM has been extensively investigated and extended, e.g. LCTM (Lagged Cell Transmission Model) by Daganzo (1999), EL-CTM (Enhanced Lagged Cell Transmission Model) by Szeto (2009), and MCTM (Modified Cell Transmission Model) by Muñoz et al. (2004). Furthermore, many applications for ramp systems have been done by using CTM, e.g. Lebacque (1996); Jin and Zhang (2003); Gomes et al. (2003, 2006, 2008); Saidi et al. (2010). Since real traffic data is the basis for traffic flow modeling, both theoretical and practical methods are combined in this work to promote the reliability of the conclusions. To solve traffic problem near ramps in urban expressway, traffic flow characteristics must be studied firstly. In this work, the famous CTM model is used to depict the dynamics of traffic flow, due to its simplicity, high efficiency, and flexibility for extension. Above all, the most important reason is that CTM can reproduce most phenomena in real traffic, such as shock waves and so on. However, the expressway ramp system, together with the various driving behaviors in China, induces much more traffic flow sates, which cannot well be depicted by the original CTM. As a consequence, based on empirical traffic data obtained in Beijing 3rd ring road, the CTM is validated and calibrated for various systems with an on-ramp and an off-ramp. It is found that ramp distance has a great impact on capacity. Especially in small distance cases, the capacity drops off. For these system, the Capacity Drop CTM (CD-CTM) is proposed to reflect traffic conflict in the merge cell. The model is applied to Beijing 3rd road. The simulation results show that, both CD-CTM can depict traffic flow characteristics for the expressway ramp systems with different ramp distance. 2. Traffic Characteristics of Ramp Systems in Beijing 3rd Ring Road In this paper, data collecting points are from Beijing 3rd ring road. These points are Zhao Gong Kou Bridge west, Hang Tian Bridge north, and Nong Zhan Bridge south, and labeled by Points 1 to 3 orderly. The inside circle is the inner ring, and the outside one is the outer ring. Points 1 are located in inner road, while points 2, 3 are in the outer road. Traffic data is obtained through Remote Traffic Microwave Sensor (RTMS) during 5:00-23:00 on May 13, 2010. All systems on these three observation points are ramp systems with an on-ramp and an off-ramp. We define the ramp distance as the distance between two adjacent ramps. The ramp distances on the points 3, 2, and 1 are distinct, and are respectively 95m, 220m, and 360m. The effect of ramp distance on traffic flow characteristics can be investigated in accordance with the real data detecting in these points. In Fig. 1(a), the flow-density diagrams are plotted according to real traffic data. These scatter points can be classified into three regions, free-flow region with low density, semi-stable region with medium density, and congestion region with high density. In the free-flow region, the flux increases linearly with the density. In the semistable region, the flux reaches the maximal value. In the congestion region, the flux decreases with density. By comparing with the flux-density diagram for different ramp distances, an interesting phenomenon can be found. In the system with ramp distance of 360m, scatter data points are mainly located in free-flow region, and more data points are located in large density region as ramp distance decreases. Furthermore, the maximum flow increases with ramp distance, but the corresponding critical densities are nearly the same.
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Xiao-mei Zhao et al. / Procedia - Social and Behavioral Sciences 138 (2014) 37 – 44 (a)
2000
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1800
average flow(vehicle/hour)
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1800
180
200
0
20
40
density(vehicle/kilometer)
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density(vehicle/kilometer)
Fig. 1. (a) Flux; (b) average flux versus density for various ramp distances
Fig. 2(a) shows that, as the ramp distance increases, more data distributes in free-flow region which coincides with the result in Fig. 1(a). In Fig. 2(b), the ramp system with a larger ramp distance has a trend with lager velocities. Fig. 2(c) displays standard deviation of the velocity to describe the velocity fluctuation. One can see that, the system with the ramp distance of 360m has a large velocity fluctuation. For the systems with ramp distances of 95m and 220m, the velocity fluctuations have no obvious difference.
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velocity(kilometer/hour)
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18 16 14 12 10 8 6 4 2 0
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Fig. 2. (a) Velocity; (b) average velocity; (c) standard deviation of velocity versus density for various ramp distances.
3. Data Calibration for Beijing 3rd Ring Road 3.1. Parameter calibration for CTM In CTM, four parameters need to be calibrated. They are free flow velocity, maximum flow, congestion backwave velocity, and jam density, respectively. x Free Flow Velocity v f Free flow velocity is the maximum velocity on road, where the interaction between vehicles can be neglected. So the average velocity at low density is set as free flow velocity. The average velocity, flow-density diagram and fundamental diagram of scatter point on Nong Zhan Bridge south are shown in Fig. 3. v f is also the slope of flowdensity curve in low density area in Fig. 3(b) and Fig. 3(c). x Maximum Flow qmax The maximal average flow is the capacity of traffic flow on roads. qmax is the maximum value in Fig. 3(b) and Fig. 3(c). x Congestion Back-wave Velocity w and Jam Density U jam The jam density and congestion back-wave velocity are estimated by using data during morning peak 8:00-9:00 by performing a least-squares fit for the flow-density relationship. During this time period, nearly all data is located on the congestion branch. w is set as absolute value of the slope of flow-density curve in high density area. The point where the high density region regression line crosses zero flow is considered as the jam density U jam . The fitted fundamental diagram of Nong Zhan Bridge south is labeled by solid lines in Fig. 3(c).
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˄b˅
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qmax
10 0
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average flow(vehicle/hour)
average velocity(kilometer/hour)
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density(vehicle/kilometer)
density(vehcle/kilometer)
40
60
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160
ρ1 ρ2 density(vehicle/kilometer)
180
200
ρjam
220
Fig. 3. (a) Velocity-density; (b) average flow-density; (c) fundamental diagram of scatter data on Nong Zhan Bridge south
x Calibration Results The parameter calibration method mentioned above is applied to the 3rd ring road in Beijing. Aiming at different ramp distances, calibration results are shown in Table 1. Table 1. Parameters of CTM for different ramp distance cases.
U jam
vf
qmax
w
(meter)
(kilometer/hour)
(vehicle/hour)
(kilometer/hour)
(vehicle/kilometer)
95
47.73
1432.38
11.12
192.46
220
50.13
1523.89
10.38
209.39
360
51.19
1690.18
11.32
207.68
ramp distance
From Table 1, we can see that, the case with the larger ramp distance has the larger maximal flows. These results are in accordance with the traffic flow data analyzing in section 2. 3.2. Cell Transmission Model x Link Model In CTM, road sections are divided into a series of homogenous cells. It is assumed that the flow and the density comply with the trapezoidal fundamental diagram, as shown in Fig.4, where U jam , qmax , v f , w respectively denote the jam density, maximum flow, free-flow velocity, and velocity of the backward shock wave. Then, flow-density relationship can be expressed as,
q min{U v f , qmax , w( U jam U )}
(1)
flow
U jam
1/ v f 1/ w
qmax
vf
w U jam
density
Fig. 4. The trapezoidal fundamental diagram of CTM
The basic concept of CTM is to divide the road into homogenous cells (as shown in Fig. 5) and time into intervals, such that the cell length is equal to the distance traveled by free flowing traffic in one time interval. Then the flows are related to the current conditions at time t as indicated in Eq. (2). The vehicle number at time t+1 equals its vehicle number at time t, plus the inflow vehicles and minus the outflow vehicles, as in Eq. (3).
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i
i
Q
i
flow direction
Fig. 5. Sketch map for link model of CTM
min{ni 1 (t ), Qi (t ), w[ Ni (t ) ni (t )] / v f }
fi (t )
ni (t 1)
ni (t ) fi (t ) fi 1 (t )
(2) (3)
where, fi (t ) is the inflow of cell i for time interval t, ni (t ) is number of vehicles in cell i for time interval t, Qi (t ) is the maximal allowable inflow of cell i for time interval t, N i (t ) is the maximal number of vehicles allowable in cell i for time interval t. In Eq. (2)., the minimum value of the first two terms in the right side is the number of vehicles that cell i-1 can send for time interval t, and is denoted as Si 1 (t ) . Similarly, the minimum value of the last two terms is equivalent to vehicles that cell i can receive for time interval t, and is represented as Ri (t ) . Eq. (2). can thus be rewritten as:
fi (t )
min{Si 1 (t ), Ri (t )}
(4)
x Merge Model Fig. 6 shows vehicles of upstream cell i on mainline and vehicles of on-ramp cell i’ merging into downstream cell i+1. Assuming that, the total inflow of cell i+1 at time t is fin (t ) , and the inflow of cell i+1 from cell i and cell i’ are fi (t ) , fi ' (t ) respectively. pi (t ) , pi ' (t ) are the inflow percentage of fi (t ) and fi ' (t ) to fin (t ) . The outflow of cell i and cell i’ are calculated as follows,
fi (t )
fin (t ) pi (t ) , fi ' (t ) i
1
fin (t ) pi ' (t )
i+1
i’
(5)
main road
Q
flow direction
1
on-ramp
Fig. 6. Sketch of the merge junction
For the merge junction, the rules proposed by Jin, W. L. and Zhang, H. M. (2003) are applied in this paper. In this model, a demand-based distribution scheme is used as follows, pi (t )
Si (t ) / [ Si (t ) Si ' (t )] , pi ' (t )
1 pi (t )
(6)
x Diverge Model For a diverge situation in Fig. 6, vehicles in mainline upstream cell i-1 enter into mainline downstream cell i and off-ramp cell i’. Assume that total outflow of cell i-1 for time interval t is fout (t ) , and inflow of cell i and cell i’ from cell i-1 are fi (t ) , fi ' (t ) respectively. hi (t ) and hi ' (t ) are percentage of fi (t ) and fi ' (t ) to fout (t ) . fi (t ) and fi ' (t ) are calculated as follows,
fi (t )
fout (t )hi (t ) , fi ' (t )
fout (t )hi ' (t )
(7)
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1
i-1
Q
i
main road
i’
flow direction
1
off-ramp
Fig. 6. Sketch of the diverge section
4. Modified Cell Transmission Model for Systems with Different Ramp Distances According to the discussions in sections 2 and 3, the ramp distance has great influences on traffic flow. In this section, a modified CTM model will be proposed to consider the effect of the ramp distance. 4.1. Modified CTM for various ramp distances
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220 m
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95 m
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flow standard deviation(vehicle/hour)
x Simulation Results with CTM The distances for observation points 3, 2, 1 are 95 m, 220 m and 360 m respectively. Using calibrated parameters in section 3, the three systems are simulated with CTM, as shown in Fig. 7. The simulation data shows a lager deviation than the empirical data in medium density area. CTM can’t depict real traffic phenomenon well in this density area, and thus needs to be modified. 360 m
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0
density(vehicle/kilometer)
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20
ρ1
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Fig. 7. The standard deviation of flow under CTM
x CD-CTM (Capacity Drop CTM) Conflicts may frequently appear at the merge cell, due to the interaction between the flows from upstream mainline and on-ramp. As a result, the capacity of merge cell decreases. Assuming that the capacity of merge cell Q 'i (t ) decreases linearly with the inflow of on-ramp, Q 'i (t ) can be expressed as follows:
Qi '(t ) f i (t )
(1 y )Qi (t ) Qramp (t )
ri ' (t ) Qi (t )
min{ni 1 (t ), Qi '(t ), w[ N i (t ) ni (t )] / v f }
(8) (9)
where Qramp (t ) is the maximal allowable outflow on on-ramp at time t, y is the percentage of minimum capacity of merge cell to its original value, and ri ' (t ) is the real inflow from on-ramp into mainline. In Eq. (8)., if ri ' (t ) 0 , then Qi '(t ) Qi (t ) ; if ri ' (t ) Qramp (t ) , then Qi '(t ) yQi (t ) . In other words, if no on-ramp vehicle enters mainline, Qramp (t ) doesn’t change, otherwise it decreases. This model is called as CD-CTM (Capacity Drop CTM). x Simulation Results with CD-CTM The systems with the ramp distance of 95m, 220m, 360m are closest to empirical data when y is set respectively as 88.5%, 92.5% and 96.5%. Thus, y is set as 88.5%, 92.5% and 96.5% respectively for the three systems in the following. The simulations are carried out by both CTM and CD-CTM for the three road systems, as shown in Fig. 8
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and Table 2. From Fig. 8 and Table 2, it is found that in medium density area, the standard deviations in CD-CTM decrease greatly. Thus, CD-CTM can reflect real traffic phenomenon better than CTM. 95 m
(b)
350
CTM CD-CTM
300 250 200 150 100 50 0
0
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U1
40
60
U2
80
100
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flow standard deviation(vehicle/hour)
flow standard deviation(vehicle/hour)
400
density(vehicle/kilometer)
(c)
220 m
400
CTM CD-CTM
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U1
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flow standard deviation(vehicle/hour)
(a)
360 m
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60
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80
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density(vehicle/kilometer)
density(vehicle/kilometer)
Fig. 8. The standard deviation of flow under both CTM and CD-CTM
Table 2. The average standard deviation of flow at different density regions. density region low region
medium region
high region
CTM(vehicle/hour) when ramp distance is 95 m
52.08
126.09
47.48
CD-CTM(vehicle/hour) when ramp distance is 95 m
51.33
65.52
50.42
CTM(vehicle/hour) when ramp distance is 220 m
62.13
135.17
68.74
CD-CTM(vehicle/hour) when ramp distance is 220 m
61.63
59.69
62.72
CTM(vehicle/hour) when ramp distance is 360 m
63.96
114.34
51.45
CD-CTM(vehicle/hour) when ramp distance is 360 m
63.97
38.63
38.74
standard deviation
5. Conclusions Different ramp cases in Beijing 3rd ring road display various traffic phenomena. Thus, traffic flow data is firstly analyzed in the system with different ramps. It is found that small ramp distance between ramps causes a capacity drop. CTM parameters are calibrated, but the standard deviation in the medium density region is far different from the real data. Then, based on data analysis, CTM is modified to fit different ramp cases. For the system with different ramp distance, vehicles’ conflict at the merge cell is considered and CD-CTM (Capacity Drop CTM) is proposed. It suggests that, CD-CTM can decrease the deviation of simulation flow in medium density area. As a conclusion, CD-CTM can better depict traffic phenomena near ramps. Acknowledgements This work is funded by the National High Technology Research and Development Program (2011AA110303), and the Fundamental Research Funds for the Central Universities (2013JBM050). References Daganzo, C. F. (1994). The Cell Transmission Model: A Dynamic Representation of Highway Traffic Consistent with the Hydrodynamic Theory. Transportation Research Part B:Methodological, 28, 269-287. Daganzo, C. F. (1995). The cell transmission model: Part II: Network traffic. Transportation Research Part B: Methodological, 29, 79-93. Daganzo, C. F. (1999). The Lagged Cell-Transmission Model. The 14th International Symposium on Transportation and Traffic Theory. Jerusalem, Israel, July 20-23, 81-104.
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Gomes, G., & Horowitz, R. (2003). A Study of Two On-ramp Metering Schemes for Congested Freeways. The American Control Conference. Denver, United States, June 4-6, 5, 3756-3761. Gomes, G., & Horowitz, R. (2006). Optimal Freeway Ramp Metering Using the Asymmetric Cell Transmission Model. Transportation Research Part C: Emerging Technologies, 14, 244-262. Gomes, G., Horowitz, R., Kurzhanskiy, A. A., Varaiya, P., & Kwon, J. (2008). Behavior of Cell Transmission Model and Effectiveness of Ramp Metering. Transportation Research Part C: Emerging Technologies, 16, 485-513. Jin, W. L., & Zhang, H. M. (2003). On The Distribution Schemes for Determining Flows Through a Merge. Transportation Research Part B: Methodological, 37(6): 521-540. Lebacque, J. P. (1996). The Godunov Scheme and What it Means for First Order Traffic Flow Models. The 13th International Symposium on Transportation and Traffic Theory. Lyon, France, July 24-26, 647-677. Muñoz, L., Sun, X. T., Sun, D. F., Gomes, G., & Horowitz, R. (2004). Methodological Calibration of the Cell Transmission Model. American Control Conference. Boston, United States, June 30-July 2, 1: 798-803. Saidi, S., Kattan, L., & Harschnitz, C. (2010). A Proactive Travel Time Based Ramp Metering Algorithm for an Isolated Ramp. The 12th Word Conference on Transportation Research. Lisbon, Portugal, July 11-15, 1-21. Szeto, W. Y. (2009). The Enhanced Lagged Cell Transmission Model for Dynamic Traffic Assignment. Transportation Research Record, 2085, 76-85.
