2017年美国大学生数学建模竞赛O奖论文C-55585.pdf资源-CSDN文库

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根据给定文件的信息，我们可以提炼出以下相关的IT与数学建模领域的知识点： ### 一、背景介绍 #### 美国大学生数学建模竞赛（MCM/ICM）美国大学生数学建模竞赛（Mathematical Contest in Modeling/Interdisciplinary Contest in Modeling）是一项国际性的竞赛活动，旨在培养学生的数学建模能力、团队合作能力和解决实际问题的能力。该竞赛每年由美国数学及其应用联合会（COMAP）主办，吸引了全球众多高校的学生参与。 ### 二、论文概述 #### 题目：高速公路交通流模型与自动驾驶车辆本文主要探讨了在自动驾驶车辆(Self-Driving Vehicle, SDV)逐渐普及的情况下，如何通过建立交通流模型来评估其对现有非自动驾驶车辆(Non-Self-Driving Vehicle, NSDV)的影响。随着自动驾驶技术的发展，越来越多的NSDV将被SDV取代，这将对整个交通系统产生重大影响。目前关于SDV与NSDV之间的相互作用以及SDV之间的协作的研究较少，因此本文提出了一种基于元胞自动机(Cellular Automata, CA)模型的方法来解决这一问题。 ### 三、研究方法与模型构建 #### 元胞自动机(CA)模型元胞自动机是一种用于模拟复杂系统行为的数学模型，它通过简单的规则在各个离散的时间步上更新每个元胞的状态。CA模型在交通流模拟领域得到了广泛的应用，因为它们能够有效地模拟道路上车辆的运动特性。 - **模型改进**： - 在传统CA模型的基础上，本文重新设计了状态和变化规则，以便更好地反映SDV和NSDV之间的关系。 - 通过对现实世界中高速公路车辆平均长度、速度、加速度以及人类反应时间的学习，确定了元胞大小和时间步长。 #### 模型构建步骤 1. **离散化**：首先进行空间和时间上的离散化处理。 2. **假设简化**：为了便于模型的构建，需要做出一定的假设和简化处理。 3. **模型构建**： - **跟随模型**：模拟单车道上一辆车跟随另一辆车的行为。对于NSDV，考虑驾驶员的反应时间和心理因素；对于SDV，则基于与其他SDV的信息共享和联合决策制定规则。 - **多车道交通模型**：基于跟随模型，进一步考虑车道变换的情况。模型中包含两个主要参数：车道变换倾向(Lane-Changing Propensity)和车道变换概率(Lane-Changing Probability)。 ### 四、模型特点与创新点 #### SDV-Train概念为模拟SDV之间的协作，文中提出了一个新概念——“SDV-Train”，即模拟多个SDV作为一个整体进行行驶，以提高道路的通行效率和安全性。 ### 五、结论与展望通过建立上述两种CA模型，本文不仅分析了SDV与NSDV之间复杂的交互作用，还探讨了SDV之间的合作机制。这种模型不仅可以帮助我们更好地理解未来交通系统的运作方式，还可以为相关政策制定者提供有价值的参考依据，从而促进自动驾驶技术的安全发展和社会应用。该篇获奖论文通过深入研究并建立有效的交通流模型，为自动驾驶车辆在未来的交通系统中的应用提供了重要的理论基础和技术支持。

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For ofﬁce use only

Team Control Number

55585

Problem Chosen

For ofﬁce use only

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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