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Understanding platoon dispersion is critical for the coordination of traffic signal control in an urban traffic network. Assuming that platoon speed follows a truncated normal distribution, ranging from minimum speed to maximum speed, this paper develops a piecewise density function that describes platoon dispersion characteristics as the platoon moves from an upstream to a downstream intersection. Based on this density function, the expected number of cars in the platoon that pass the downstream intersection, and the expected number of cars in the platoon that do not pass the downstream point are calculated. To facilitate coordination in a traffic signal control system, dispersion models for the front and the rear of the platoon are also derived. Finally, a numeric computation for the coordination of successive signals is presented to illustrate the validity of the proposed model.

At an intersection, lights change from red to green permitting drivers to proceed straight through the intersection. On urban roads these cars will be traveling at different speeds. While moving downstream, the platoon spreads out in a long segment and cars do not uniformly arrive at the next intersection; this is called platoon dispersion. As a platoon moves downstream from an upstream intersection at green phase end time, the cars in the platoon become segmented due to compression and splitting at the downstream intersection’s signal lights. It is obvious that using platoon dispersion theory to optimize signal timing plans for traffic signal control could effectively reduce the number of stops and thereby lead to a sharp decline in pollution emissions and fuel consumption.

However, platoon dispersion makes signal coordination more complicated [

As mentioned previously, Pacey’s model is the most successful combination of theoretical and experimental work on traffic platoons. Most current research based on this model assumes that platoon speed follows a normal distribution, spreading from negative infinity to positive infinity. This does not properly reflect the field situation. Grace and Potts [

To address the defects of Pacey’s model, the authors of this paper propose a more realistic platoon dispersion model, which assumes that the velocity of cars follows a truncated normal distribution, ranging from a minimum speed to a maximum speed [

In Pacey’s model, the cars in a platoon are assumed to move with unchanged speeds (i.e., it is ideally treated as the average speed of vehicles measured between adjacent intersections). It is assumed that (a) all the vehicles behind the stop lines uniformly start up after the signal turns from red to green, (b) a car’s speed is independent of its position in a platoon, and (c) there is no interaction between cars and a faster car can pass a slower one without hindrance.

A definite value is assigned to the probability that certain cars will have positive speed. Pacey’s research has proven that car speeds in a platoon are normally distributed with mean

Its calculation formula can be expressed as follows:

for

for

for

As the constant

Assuming that the start time of the upstream signal green phase

This paper studies the movement of a platoon from the beginning of a green phase, until it passes a downstream intersection. As illustrated in Figure

Spreading of platoon.

Let

Following (

Let

Using (

for

for

for

for

for

The platoon dispersion model proposed in this paper is described in the previous section. If

In general, only dispersion behavior at the front and rear of the platoon is important in traffic signal coordination. When adjacent intersections are not too far apart, fast cars at the rear of the platoon are unable to get to the front and the slower cars at the front do not have time to filter back to the rear. Hence, the front and rear of the platoons may be treated separately.

A good design of coordinated lights aims at reducing the number of platoon leaders stopped at the second intersection before its light turns green. The rear does not significantly affect the front, which helps in the mathematical analysis of the behavior of the front of the platoon. Using Pacey’s assumptions, the initial density function

The density of the front of the platoon past

As mentioned previously, cars at the front travel at a range of speeds

for

for

for

Another goal of signal coordination is to reduce the number of stragglers at the rear who miss the green phase. Assuming the upstream signal green time ends at time 0, the initial density function

Applying a similar limit process to (

The previous calculations show that the speed of vehicles in the rear of the platoon is less than

for

for

for

Form proposed model,

To verify the validity of the proposed model, this paper compares the difference between the front and rear platoon dispersion characteristics in the proposed model and in Pacey’s model. Furthermore, the effect of changing parameter

The difference in the density distribution function between proposed model and Pacey’s model decides that our model is more realistic than Pacey’s model. In order to prove that,

Comparison of the proposed model’s and Pacey’s density distribution function

The following can be concluded based on Figure

The speed density of both Pacey’s model and proposed model follows a normal distribution, which lead to a situation that there are more vehicles traveling around the mean speed, and fewer vehicles at higher or lower speeds. The speed distribution determines the platoon density distribution function. Therefore, the density in the middle of the platoon is higher than that in the front or rear of the platoon. As time passes, the platoon becomes more dispersed along the road, and the hump of the platoon density distribution function becomes less significant.

In the proposed model, the queuing vehicles traveling at different speed

In Pacey’s model, the spread of vehicles in the speed range

The difference in the speed density between proposed model and Pacey’s model decides that there are also more cars in the middle of the platoon and fewer cars in the two tails in the former, compared to the latter.

If the green phase of the downstream intersection is started at

The ratio to the maximum initial flow of the average number of cars having passed the downstream point at the front of platoon under

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

30 | |||||||||||

60 | |||||||||||

90 | |||||||||||

120 |

Note: from the proposed model/from Pacey’s model.

Table

It is assumed that the downstream signal green time ends at time

The ratio to the maximum initial flow of the average number of cars not having passed the downstream point at the rear of platoon under

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

30 | |||||||||||

60 | |||||||||||

90 | |||||||||||

120 |

Note: from the proposed model/from Pacey’s model.

The data in Table

It can be concluded from the previous results that

In addition,

Influence of

Influence of

Platoon dispersion is the foundation of coordinating traffic signal control in an urban traffic network. This paper proposes a new platoon dispersion model which assumes that speed density follows a truncated normal distribution. This addresses the main defect of Pacey’s model and matches the field situation. To calibrate proposed model, values of four parameters, namely, the average speed of vehicles, the standard deviation of speed, minimum speed, and maximum speed, are quantified. Using test data in Grace and Potts’s paper [

This work was supported by the High-Technology Research and Development Program of China (863 program) under contract no. 2007AA11Z201 and the National Science Foundation of China under contract no. 61174188. This support is gratefully acknowledged. However, all facts, conclusions, and opinions expressed here are solely the responsibility of the authors. The authors thank the anonymous referee for her/his helpful comments.