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2009年国际大学生数学建模竞赛A题获奖论文
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2009年国际大学生数学建模竞赛A题获奖论文,希望对建模的同学有帮助!
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Team #4094
Round and Round We Go
February 9, 2009
Abstract
The study of traffic flow and control has been a fruitful area of mathematical research
for decades. Here we attempt to analyze and model the traffic flow that occurs in a
traffic circle. We use a powerful macroscopic approach developed by B. Piccoli that
uses network analysis and a wave-front tracking algorithm to produce powerful the-
oretical results with regards to right of way parameters in arbitrarily large networks
of traffic circles. We then follow the classical approach of Lighthill and Whitham
in order to model the effects of our proposed control system on the dynamics of a
traffic circle, employing the Runge-Kutta algorithm to process multiple solutions to
the famous conservation PDE with high-accuracy and speed. We found that prior-
itizing the right of way of the cars inside the circle optimizes the efficiency of the
circle, and developed a control system that responds to incoming traffic density i n
real time and keeps the outgoing flux at a maximum regardless of t he number of roads
leading to the junction. Next we developed a far more descriptive discrete model,
revising the older, standard car-following model, and reaffirmed our original control.
We conclude with simulations of our models on specific examples, a reflection of our
methods, and a technical description to a Traffic Engineer outlining how and when
to use our methods in traffic control development f or traffic circles.
1
Team # 4094 Page 2 of 15
Contents
1 Introduction 3
2 The Macroscopic Model without Control 4
3 Network Considerations 5
4 Macroscopic Model with Control 6
5 The Microscopic Model 9
6 Directions for a Traffic Engineer 12
6.1 Measuring Traffic Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Measuring Traffic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3 Programming the Co ntrol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.4 Multiple-Lane Traffic Circles . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Conclusion 14
List o f Figures
1 No Control Response to a One Peak Input . . . . . . . . . . . . . . . . . . . 4
2 Response Denisty and Input Flux for One Peak . . . . . . . . . . . . . . . . 7
3 Response Density and Input Flux for Square Wave . . . . . . . . . . . . . . 8
4 Monstrous Incoming Flux With and Without Control- 12 roads . . . . . . . 8
5 Microscopic Model Still Shots . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6 Density for different incoming fluxes and Radii . . . . . . . . . . . . . . . . 10
7 Typical Example of Mi cro-Model Density Results . . . . . . . . . . . . . . . 11
Team # 4094 Page 3 of 15
1 Introduction
This paper is concerned with modeling and optimizing traffic flow in a traffic circle. As in
past studies of traffic flow (i.e. [1], [3], [4], [7]), we rely heavily on the following conservation
law for cars:
ρ
t
+ f(ρ)
x
= 0, (1)
Where ρ : R
+
× [a, b] → [0, ρ
max
] is the density of cars (number of cars per unit distance),
and f : [0, ρ
max
] → R is the traffic flow or flux across a boundary (number of cars per
unit time). The flux and density are related by the following rule: f(ρ) = ρv(ρ). This
conservation assumption has been used in almost every macroscopic model for traffic flow
so far, and it makes for a very useful description of the dynamics involved. Here it is
appli ed to the idea of a traffic circle, which is an a lternative to a traditional traffic light
junction. We note here that no distinction is made in the paper between a traffic circle
and a roundabout, even though there are slight technical differences. The rest of our
assumptions we list here:
• No cars are created or destroyed, as described by (1)
• Traffic density within the roundabout is uniform.
• Traffic flux into the circle is constant within a sufficiently small time interval.
• Al l drivers that enter the traffic circle wil l exit without going around the loop multiple
times.
• The velocity is a function of ρ and it is zero at ρ
max
.
• The traffic from incoming roads is distributed on outgoing roads according to time-
varying coefficients.
• Drivers attempt to ma ximize flux when possible.
Assuming traffic flux to be constant during small time intervals may seem like a bit of a
stretch, but really it is just a condition that ensures rough continuity of incoming traffic
density. In fact, our model allows for a very large number of jump discontinuities, so
that the traffic control device can handle a piecewise constant wave of incoming traffic
density. The condition on the velocity is one given in most traffic models, and it tests out
empirically. We will b e employing a velocity equation derived empirically from [8] [3]:
v(ρ) =
v
s
, 0 ≤ ρ ≤ σ
β(
1
ρ
−
1
ρ
max
) , ρ > σ
(2)
Here, v
s
is the velocity a car would go if it was by itself, in other words it represents the
speed limit. The σ is what’s known as the “capacity” of a system; it is the density at which
the flux is maximum, and the β is a constant chosen to ma ke the function continuous.
Note that the function for velocity makes intuitive sense as well; as the density increases,
the velocity decreases, until it hits the maximum density, where it becomes zero.
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