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2017年美国大学生数学建模竞赛O奖论文C-55585.pdf
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For office use only
T1
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Team Control Number
55585
Problem Chosen
C
For office use only
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2017
MCM/ICM
Summary Sheet
Highway Traffic Flow Model with Self-Driving Vehicles
Based on Cellular Automata
Summary
With the increasing lack of transportation capacity and the growth of self-driving vehicle(SDV)
industry, an evaluation should be made to find out the influence on traffic when more and more non-
self-driving-vehicles(NSDV) are replaced by SDVs while few studies were done on the interactions
between SDVs and NSDVs and the cooperations among SDVs themselves.
We choose cellular automata(CA) model to evaluate this problem after a careful study and com-
parison of different kinds of traffic flow models in the past few decades. In order to take the re-
lationships of SDVs and NSDVs into consideration, we improve the traditional CA model which
emphasizes on status and rules of changes, by redesigning these two factors. Before building a CA
model, discretization should be done first. By learning the average length, speed, acceleration of
running vehicles on highway and the reaction time of human beings, the size of a cell and the time
length of a turn are decided. After making assumptions and simplifying the problem, two inter-
related CA models are covered in this paper to simulate the changeable traffic: the Following Model
and the Multilane Traffic Model.
The Following Model is designed to simulate how a vehicle follows another in a single lane.
Rules for NSDVs and SDVs are different from each other: For an NSDV, the driver’s reaction time
and psychological characteristics are considered; For an SDV, the rules are based on the sharing of
information with other SDVs and the joint decision making. Specifically, we create a new conception
’SDV-Train’ to simulate the cooperations among SDVs.
The Multilane Traffic Model is based on the Following Model. In this model, besides following,
we try to find out when and how should a vehicle change a lane. Two main parameters are involved
in this model: Lane-Changing Motivation (LCM) and Lane-Changing Secuirty (LCS). LCM depends
on whether changing a lane can increase the speed and LCS shows the whether it is safe when lane-
changing. Only when both LCM and LCS are satisfied, may a vehicle change its lane. Details of
these two parameters vary between SDVs and NSDVs considering the huge difference between an
automatic control system and a human driver. A two-step turning method is specially made for this
model in correspondence with the real world.
After building and making improvements to the model, we write programs to simulate it and get
huge volumes of data. We analyze and visualize the data using Matlab, showing strong correlations
among three parameters: the average speed, the traffic flow and the percentage of the SDVs running
on the road. The increasing number of SDVs has great influence on the traffic flow which almost
triples when all the NSDVs are replaced by SDVs. Also, we find that a special lane for SDVs (SDV
Lane) should be built when the percentage reaches a certain level.
Based on the correlations we get in analysis, we apply our model to the Great Seattle area by
comparing the real data and the data we gain from simulations. We find that the lack of traffic
capacity in this area is huge. Although adding SDVs to the street can reduce this lack, it is not a cure.
We believe a comprehensive method should be applied in this area including setting a SDV Lane
and broadening highways in some particularly narrow parts.
Keywords: Traffic Flow Model; Self-Driving Vehicle; Cellular Automata
Contents
1 Introduction 1
2 Simplifications and Assumptions of the Problem 1
2.1 Features of the Highway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.2 Features of Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Special Features of NSDVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 Special Features of SDVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Choice and Basic Settings of the Model 3
3.1 Choice of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Basic Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Details of the Model 4
4.1 Following Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1.2 Following Rules for NSDVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.1.3 Following Rules for SDVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Multilane Traffic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2.2 General Rules of Changing Lanes . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.3 Lane-Changing Rules for NSDVs . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.4 Lane-Changing Rules for SDVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Analysis of the Results Obtained from the Model Simulation 13
5.1 Results of Following Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Results of the Multilane Traffic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 SDV Lane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Appliance of the Model 17
7 Sensitivity Analysis 18
7.1 Choice of the Parameters in LCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.2 Different Speed Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
8 Conclusions 19
9 Strengths and weaknesses 19
9.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10 A Letter 20
Appendices 22
Team # 55585
Team # 55585 Page 1 of 34
1 Introduction
Built in the 20th century, many highways were designed to meet the transportation demands
at that time. With the boom of population, urbanization and economy, the need of transportation
grows rapidly in the new century. Nowadays, highways in the Great Seattle area can no longer meet
people’s need and traffic delays can be seen everywhere during peak hours. However, at this time
building more roads or adding lanes in this area is extremely difficult and expensive. In order to
increase the capacity of highways without increasing the number of lanes or roads, allowing self-
driving vehicles(SDVs) to run on the road should be taken into consideration. A model is needed to
evaluate SDVs’ influence on the traffic flow.
We proposed to decompose this problem into three parts:
• Build a model that can simulate the traffic flow in different percentage of SDVs and non-self-
driving vehicles (NSDVs), number of lanes and traffic volume.
• Use the model to find the equilibria or tipping points and apply the model to the provided
data.
• Based on the data, decide whether there are some conditions where lanes should be dedicated
to SDVs and how the policy should be changed.
Firstly, we use cellular automata(CA) to simulate the traffic flow when there is only one lane.
This model is called the Following Model. In our model, we rule the way each cell behaves by
simplifying the behaviors of vehicles in real life, like when a vehicle will slow down or speed up.
We use different rules for SDVs and NSDVs in our model to simulate the cooperations among SDVs,
interactions between SDVs and NSDVs, unpredictability of human-beings and other factors.
Based on the Following Model we built, we put separate parrel lanes together and add new
rules to simulate the traffic flow on a multilane highway. This is the Multilane Traffic Model. After
simplifying the behaviors of real vehicles’ changing lanes, we make rules on when and how a cell
move across lanes. Both the motivation and the safety concern are considered. Furthermore, we
make special rules to simulate human behaviors and cooperations among SDVs including the form
of a chain of SDVs called the SDV-Train.
Secondly, using real-life parameters, we run the CA model and get a large number of data. By
analyzing the data, we find several interesting features of the mixed traffic flow. The correlations
among the average speed, the traffic flow and the percentage of SDVs are strong. These three pa-
rameters influence each other in their own way. When there are many lanes, the situation changes
and more interesting phenomena are found including the relationships between the efficiency of
each line and the percentage of SDVs. After comparison, we find out when and how to build a
special lane for SDVs (SDV Lane).
Thirdly, we compared our data with real data in the Great Seattle area. We find that there is
indeed a great lack of traffic capacity in this area. After changing NSDVs to SDVs, the traffic capacity
increases and even triples but we believe the traffic situation in this area is still not abundant. More
methods including broaden a few parts of the current highway and setting a SDV Lane should be
taken into consideration.
2 Simplifications and Assumptions of the Problem
2.1 Features of the Highway
1. Straight Road
A highway in this model should be straight or its degree of curvature can be ignored [1]. A
vehicle’s speed and other conditions does not change because of the shape of the road.
Team # 55585 Page 2 of 34
2. The number of lanes should remain constant for a long period.
3. The highway is in good condition and the traffic flow is not affected by the rough road.
4. No pedestrian, animal or any form of obstacle can be found on the highway so the traffic flow
would not be blocked.
5. Weather changes and illumination difference during different time is not taken into consid-
eration.
6. Rules for SDV Lane
• If a SDV Lane is set, all SDVs should run in this line while none of the NSDVs may run in
it.
• Only one SDV Lane can be set in our model considering the width of the highway is
limited.
• The SDV Lane will be placed on the edge of the road to prevent a separation of lanes for
NSDVs.
7. The width of each lane is 12 feet.
8. The speed limit of all highways is 60mph.
2.2 Features of Vehicles
1. In this model, only average length, speed, acceleration and other features of vehicles are used.
Although there is a great diversity among different vehicles, this difference can be ignored and
the result of the model would not be greatly affected.
2. All vehicles obey traffic laws. The violation of traffic laws does great harm to the driver’s
health, public security as well as the speed of total traffic flow. As a result, these circumstances
would not occur in our model:
• a vehicle exceeds the speed limit;
• a vehicle stops for no reason;
• a vehicle runs in an emergency lane or on the shoulder for no reason;
• a vehicle changes its lane but
– its turning light is not turned on in advance;
– the vehicle behind it shows its intention to change the lane;
• two vehicles run side by side in one lane;
• the distance between two vehicles in the same lane is too short.
3. Traffic accident is not taken into consideration. Details of its influence will be discussed in
Section 9.2.
4. Average daily traffic flow is used in this model. Traffic flow varies in different days, so an
average level should be used to simplify this model.
5. Horn is not used in our model. On highway, the influence of a horn is limited because the
distance between two vehicles is too long for a complicated sound signal to be heard and
understood clearly.
Team # 55585 Page 3 of 34
2.3 Special Features of NSDVs
1. Uncertainty of the estimation of distance and speed
Compared with computer systems, human drivers are more likely to make mistakes, especially
when it comes to the evaluation of distance and speed. Therefore, human drivers tend to slow
down the vehicle and choose not to change a lane when they cannot estimate the distance
clearly even when other vehicles are out of the minimum safe distance.
2. Longer reaction time
Human drivers need more time to decelerate and start their vehicle compared with self-driving
ones especially on the highway [2].
2.4 Special Features of SDVs
1. Short reaction time
SDVs are controlled by computer systems which run fast and can stay active for the whole
time. With modern technology, it is easy for an SDV to perceive the outside world and make
reactions accordingly in a short time.
2. Cooperations among SDVs
The system of an SDV is connected to the Internet, therefore information of all the SDVs on a
road is shared. With more information, a network decision-making system can be built which
is mentioned in previous studies [3].
3. Mature technology
The technology of self-driving is mature enough and no malfunction of SDVs’ guiding, driving
and decision-making system is taken into consideration in this model. If an accident does
happen to an SDV’s system, this SDV should be labeled as an NSDV.
4. Interactions between an NSDV and an SDV
As discussed in Section 3, for a human driver, there is no difference between an SDV and an
NSDV that he or she encounters on the street, for SDVs would make less mistakes than NSDVs
and would not cause more trouble to human drivers.
3 Choice and Basic Settings of the Model
3.1 Choice of the Model
In the past few decades, with the development of transportation, a great variety of models simu-
lating the traffic were built and improved, among which Continuous Medium model and CA model
are the most popular ones. Lighthill and Whitham firstly put forward the concept of continuous
medium model, while shortly afterwards Richards also put forward it independently, therefore it is
also named LWR model [4]. LWR model mainly focuses on the macroscopic homogeneity and sta-
bility of the traffic flow. However, in this problem, the cooperations between SDVs and the reactions
between an SDV and an NSDV must be further discussed, which makes it hard for us to adapt the
LWR model, because LWR model neglects the interactions between different particles in the flow.
After careful comparison, we find that traditional CA model with modifications can be used in this
problem.
In real life, the action of a vehicle taken by both human drivers and computer systems depends
on the status of the vehicle itself and the surrounding traffic, which is similar to the rules of CA,
which is originally discovered in the 1940s by Stanislaw Ulam and John von Neumann [5]. The basic
idea of CA is that it starts with a set of cells with status. A simple set of rules are created that the
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