美赛数学建模比赛论文模板

The Keep-Right-Except-To-Pass Rule

Summary

As for the first question, it provides a traffic rule of keep right except to pass, requiring us to verify its effectiveness. Firstly, we define one kind of traffic rule different from the rule of the keep right in order to solve the problem clearly; then, we build a Cellular automaton model and a Nasch model by collecting massive data; next, we make full use of the numerical simulation according to several influence factors of traffic flow; At last, by lots of analysis of graph we obtain, we indicate a conclusion as follow: when vehicle density is lower than 0.15, the rule of lane speed control is more effective in terms of the factor of safe in the light traffic; when vehicle density is greater than 0.15, so the rule of keep right except passing is more effective In the heavy traffic.

As for the second question, it requires us to testify that whether the conclusion we obtain in the first question is the same apply to the keep left rule. First of all, we build a stochastic multi-lane traffic model; from the view of the vehicle flow stress, we propose that the probability of moving to the right is 0.7and to the left otherwise by making full use of the Bernoulli process from the view of the ping-pong effect, the conclusion is that the choice of the changing lane is random. On the whole, the fundamental reason is the formation of the driving habit, so the conclusion is effective under the rule of keep left.

As for the third question, it requires us to demonstrate the effectiveness of the result advised in the first question under the intelligent vehicle control system. Firstly, taking the speed limits into consideration, we build a microscopic traffic simulator model for traffic simulation purposes. Then, we implement a METANET model for prediction state with the use of the MPC traffic controller. Afterwards, we certify that the dynamic speed control measure can improve the traffic flow .

Lastly neglecting the safe factor, combining the rule of keep right with the rule of dynamical speed control is the best solution to accelerate the traffic flow overall.

Key words：Cellular automaton model Bernoulli process Microscopic traffic simulator model The MPC traffic control

Content

Content (2)

1. Introduction (3)

2. Analysis of the problem (3)

3. Assumption (3)

4. Symbol Definition (3)

5. Models (3)

5.1 Building of the Cellular automaton model (3)

5.1.1 Verify the effectiveness of the keep right except to pass rule (4)

5.1.2 Numerical simulation results and discussion (5)

5.1.3 Conclusion (8)

5.2 The solving of second question (8)

5.2.1 The building of the stochastic multi-lane traffic model (8)

5.2.2 Conclusion (8)

5.3 Taking the an intelligent vehicle system into a account (8)

5.3.1 Introduction of the Intelligent Vehicle Highway Systems (9)

5.3.2 Control problem (9)

5.3.3 Results and analysis (9)

5.3.4 The comprehensive analysis of the result (9)

6. Improvement of the model (10)

6.1 strength and weakness (10)

6.1.1 Strength (10)

6.1.2 Weakness (10)

6.2 Improvement of the model (10)

7. Reference (12)

1. Introduction

As is known to all, it ’s essential for us to drive automobiles, thus the driving rules is crucial important. In many countries like USA, China, drivers obey the rules which called “The Keep-Right-Except-To-Pass (that is, when driving automobiles, the rule requires drivers to drive in the right-most unless they are passing another vehicle)”.

2. Analysis of the problem

For the first question, we decide to use the Cellular automaton to build models, then analyze the performance of this rule in light and heavy traffic. Firstly, we mainly use the vehicle density to distinguish the light and heavy traffic; secondly, we consider the traffic flow and safe as the represent variable which denotes the light or heavy traffic; thirdly, we build and analyze a Cellular automaton model; finally, we judge the rule through two different driving rules, and then draw conclusions.

3. Assumption

In order to streamline our model we have made several key assumptions

● The highway of double row three lanes that we study can represent

multi-lane freeways.

● The data that we refer to has certain representativeness and descriptive

● Operation condition of the highway not be influenced by blizzard or accidental factors ● Ignore the driver's own abnormal factors, such as drunk driving and fatigue driving ● The operation form of highway intelligent system that our analysis can reflect

intelligent system

● In the intelligent vehicle system, the result of the sampling data has high accuracy.

4. Symbol Definition

i The number of vehicles

t The time

5. Models

By analyzing the problem, we decided to propose a solution with building a cellular automaton model.

5.1 Building of the Cellular automaton model

Thanks to its simple rules and convenience for computer simulation, cellular automaton model has been widely used in the study of traffic flow in recent years.

Let )(t x i be the position of vehicle i at time t , )(t v i be the speed of vehicle i at time t ,

p be the random slowing down probability, and R be the proportion of trucks and buses, the distance between vehicle i and the front vehicle at time t is:

1)()(1--=-t x t x gap i i i , if the front vehicle is a small vehicle.

3)()(1--=-t x t x gap i i i , if the front vehicle is a truck or bus.

5.1.1 Verify the effectiveness of the keep right except to pass rule

In addition, according to the keep right except to pass rule, we define a new rule called: Control rules based on lane speed. The concrete explanation of the new rule as follow:

There is no special passing lane under this rule. The speed of the first lane (the far left lane) is 120–100km/h (including 100 km/h)；the speed of the second lane (the middle lane) is 100–80km8/h (including80km/h)；the speed of the third lane (the far right lane) is below 80km/ h. The speeds of lanes decrease from left to right.

● Lane changing rules based lane speed control

If vehicle on the high-speed lane meets control v v <, ),1)(min()(max v t v t gap i f i +≥, safe b i gap t gap ≥)(, the vehicle will turn into the adjacent right lane, and the speed of the vehicle after lane changing remains unchanged, where control v is the minimum speed of the corresponding lane.

● The application of the Nasch model evolution

Let d P be the lane changing probability (taking into account the actual situation that some drivers like driving in a certain lane, and will not take the initiative to change lanes), )(t gap f i indicates the distance between the vehicle and the nearest front vehicle, )(t gap b i indicates the distance between the vehicle and the nearest following vehicle. In this article, we assume that the minimum safe distance gap safe of lane changing equals to the maximum speed of the following vehicle in the adjacent lanes.

● Lane changing rules based on keeping right except to pass

In general, traffic flow going through a passing zone (Fig. 5.1.1) involves three processes: the diverging process (one traffic flow diverging into two flows), interacting process (interacting between the two flows), and merging process (the two flows merging into one)

[4].

Fig.5.1.1 Control plan of overtaking process

(1) If vehicle on the first lane (passing lane) meets ),1)(min()(max v t v t gap i f i +≥ and safe b i gap t gap ≥)(, the vehicle will turn into the second lane, the speed of the vehicle after lane changing remains unchanged.

5.1.2 Numerical simulation results and discussion

In order to facilitate the subsequent discussions, we define the space occupation rate as L N N p truck CAR ??+=3/)3(, where CAR N indicates the number of small vehicles on the driveway,truck N indicates the number of trucks and buses on the driveway, and L indicates the total length of the road. The vehicle flow volume Q is the number of vehicles passing a fixed point per unit time,T N Q T /=, where T N is the number of vehicles observed in time duration T .The average speed ∑∑?=T i

t i a v T N V 11)/1(, t i v is the speed of vehicle i at time t . Take overtaking ratio f p as the evaluation indicator of the safety of traffic flow, which is the ratio of the total number of overtaking and the number of vehicles observed. After 20,000 evolution steps, and averaging the last 2000 steps based on time, we have obtained the following experimental results. In order to eliminate the effect of randomicity, we take the systemic average of 20 samples [5].

Overtaking ratio of different control rule conditions

Because different control conditions of road will produce different overtaking ratio, so we first observe relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.

(a) Based on passing lane control (b) Based on speed control

Fig.5.1.3

Fig.5.1.3Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.

It can be seen from Fig. 5.1.3:

(1) when the vehicle density is less than 0.05, the overtaking ratio will continue to rise with the increase of vehicle density; when the vehicle density is larger than 0.05, the overtaking ratio will decrease with the increase of vehicle density; when density is greater than 0.12, due to the crowding, it will become difficult to overtake, so the overtaking ratio is almost 0.

(2) when the proportion of large vehicles is less than 0.5, the overtaking ratio will rise with the increase of large vehicles; when the proportion of large vehicles is about 0.5, the overtaking ratio will reach its peak value; when the proportion of large vehicles is larger than 0.5, the overtaking ratio will decrease with the increase of large vehicles, especially under lane-based control condition s the decline is very clear.

Concrete impact of under different control rules on overtaking ratio

Fig.5.1.4

Fig.5.1.4 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions. (Figures in left-hand indicate the passing lane control, figures in right-hand indicate the

speed control. 1f P is the overtaking ratio of small vehicles over large vehicles, 2f P is the overtaking ratio of

small vehicles over small vehicles, 3f P is the overtaking ratio of large vehicles over small vehicles, 4f P is the overtaking ratio of large vehicles over large vehicles.).

It can be seen from Fig. 5.1.4:

(1) The overtaking ratio of small vehicles over large vehicles under passing lane control is much higher than that under speed control condition, which is because, under passing lane control condition, high-speed small vehicles have to surpass low-speed large vehicles by the passing lane, while under speed control condition, small vehicles are designed to travel on the high-speed lane, there is no low- speed vehicle in front, thus there is no need to overtake. ● Impact of different control rules on vehicle speed

Fig. 5.1.5 Relationships among vehicle density, proportion of large vehicles and average speed under different control conditions. (Figures in left-hand indicates passing lane control, figures in right-hand indicates speed control. a X is the average speed of all the vehicles, 1a X is the average speed of all the small vehicles, 2a X is the average speed of all the buses and trucks.).

It can be seen from Fig. 5.1.5:

(1) The average speed will reduce with the increase of vehicle density and proportion of large vehicles.

(2) When vehicle density is less than 0.15,a X ,1a X and 2a X are almost the same under both control conditions.

● Effect of different control conditions on traffic flow

Fig.5.1.6

Fig. 5.1.6 Relationships among vehicle density, proportion of large vehicles and traffic flow under different control conditions. (Figure a1 indicates passing lane control, figure a2 indicates speed control, and figure b indicates the traffic flow difference between the two conditions.

It can be seen from Fig. 5.1.6：

(1) When vehicle density is lower than 0.15 and the proportion of large vehicles is from 0.4 to 1, the traffic flow of the two control conditions are basically the same.

(2) Except that, the traffic flow under passing lane control condition is slightly larger than that of speed control condition.

5.1.3 Conclusion

In this paper, we have established three-lane model of different control conditions, studied the overtaking ratio, speed and traffic flow under different control conditions, vehicle density and proportion of large vehicles.

5.2 The solving of second question

5.2.1 The building of the stochastic multi-lane traffic model

5.2.2 Conclusion

On one hand, from the analysis of the model, in the case the stress is positive, we also consider the jam situation while making the decision. More specifically, if a driver is in a jam situation, applying ))(,2(x P B R results with a tendency of moving to the right lane for this driver. However in reality, drivers tend to find an emptier lane in a jam situation. For this reason, we apply a Bernoulli process )7.0,2(B where the probability of moving to the right is 0.7and to the left otherwise, and the conclusion is under the rule of keep left except to pass, So, the fundamental reason is the formation of the driving habit.

5.3 Taking the an intelligent vehicle system into a account

For the third question, if vehicle transportation on the same roadway was fully under the control of an intelligent system, we make some improvements for the solution proposed by us

to perfect the performance of the freeway by lots of analysis.

5.3.1 Introduction of the Intelligent Vehicle Highway Systems

We will use the microscopic traffic simulator model for traffic simulation purposes. The MPC traffic controller that is implemented in the Matlab needs a traffic model to predict the states when the speed limits are applied in Fig.5.3.1. We implement a METANET model for prediction purpose[14].

5.3.2 Control problem

As a constraint, the dynamic speed limits are given a maximum and minimum allowed value. The upper bound for the speed limits is 120 km/h, and the lower bound value is 40 km/h. For the calculation of the optimal control values, all speed limits are constrained to this range. When the optimal values are found, they are rounded to a multiplicity of 10 km/h, since this is more clear for human drivers, and also technically feasible without large investments.

5.3.3 Results and analysis

When the density is high, it is more difficult to control the traffic, since the mean speed might already be below the control speed. Therefore, simulations are done using densities at which the shock wave can dissolve without using control, and at densities where the shock wave remains. For each scenario, five simulations for three different cases are done, each with a duration of one hour. The results of the simulations are reported in Table5.1, 5.2, 5.3.

●Enforced speed limits

●Intelligent speed adaptation

For the ISA scenario, the desired free-flow speed is about 100% of the speed limit. The desired free-flow speed is modeled as a Gaussian distribution, with a mean value of 100% of the speed limit, and a standard deviation of 5% of the speed limit. Based on this percentage, the influence of the dynamic speed limits is expected to be good[19].

5.3.4 The comprehensive analysis of the result

From the analysis above, we indicate that adopting the intelligent speed control system can effectively decrease the travel times under the control of an intelligent system, in other words, the measures of dynamic speed control can improve the traffic flow.

Evidently, under the intelligent speed control system, the effect of the dynamic speed control measure is better than that under the lane speed control mentioned in the first problem. Because

of the application of the intelligent speed control system, it can provide the optimal speed limit in time. In addition, it can guarantee the safe condition with all kinds of detection device and the sensor under the intelligent speed system.

On the whole, taking all the analysis from the first problem to the end into a account, when it is in light traffic, we can neglect the factor of safe with the help of the intelligent speed control system.

Thus, under the state of the light traffic, we propose a new conclusion different from that in the first problem: the rule of keep right except to pass is more effective than that of lane speed control.

And when it is in the heavy traffic, for sparing no effort to improve the operation efficiency of the freeway, we combine the dynamical speed control measure with the rule of keep right except to pass, drawing a conclusion that the application of the dynamical speed control can improve the performance of the freeway.

What we should highlight is that we can make some different speed limit as for different section of road or different size of vehicle with the application of the Intelligent Vehicle Highway Systems.

In fact, that how the freeway traffic operate is extremely complex, thereby, with the application of the Intelligent Vehicle Highway Systems, by adjusting our solution originally, we make it still effective to freeway traffic.

6. Improvement of the model

6.1 strength and weakness

6.1.1 Strength

●it is easy for computer simulating and can be modified flexibly to consider actual traffic

conditions ,moreover a large number of images make the model more visual.

●The result is effectively achieved all of the goals we set initially, meantime the conclusion is

more persuasive because of we used the Bernoulli equation.

●We can get more accurate result as we apply Matlab.

6.1.2 Weakness

●The relationship between traffic flow and safety is not comprehensively analysis.

●Due to there are many traffic factors, we are only studied some of the factors, thus our

model need further improved.

6.2 Improvement of the model

While we compare models under two kinds of traffic rules, thereby we come to the efficiency of driving on the right to improve traffic flow in some circumstance. Due to the rules of comparing is too less, the conclusion is inadequate. In order to improve the accuracy, We

further put forward a kinds of traffic rules: speed limit on different type of cars.

The possibility of happening traffic accident for some vehicles is larger, and it also brings hidden safe troubles. So we need to consider separately about different or specific vehicle types from the angle of the speed limiting in order to reduce the occurrence of traffic accidents, the highway speed limit signs is in Fig.6.1.

Fig.6.1

Advantages of the improving model are that it is useful to improve the running condition safety of specific type of vehicle while considering the difference of different types of vehicles. However, we found that the rules may be reduce the road traffic flow through the analysis. In the implementation it should be at the

85

V speed of each model as the main reference basis. In

recent years, the

85

V of some researchers for the typical countries from Table 6.1[ 21]:

Author Country Model

Ottesen and

Krammes2000

America LC

DC

L

DC

V

C

?

-

-

-

=01

.0

012

.0

57

.1

44

.

102

85

Andueza2000 Venezuela

]

.

[

308

.9

486

.7

)

/

894

(

)

/

2795

(

25

.

98

85

curve

horizontal

L

DC

Ra

R

V

T

+

+

-

-

=

]

.

[tan

819

.

27

)

/

3032

(

69

.

100

85

gent

L

R

V

T

+

-

=

Jessen2001 America

]

[

00239

.0

614

.0

279

.0

80

.

86

1

85

LSD

ADT

G

V

V

P

-

-

+

=

]

[

00212

.0

432

.0

10

.

72

85

NLSD

ADT

V

V

P

-

+

=

Donnell2001 America

2

2

)

2

(

85

00724

.0

40

.1

0140

.0

4.

78

T

L

G

R

V-

-

+

=

2

2

)3(

85

008369

.0

48

.1

0176

.0

1.

75

T

L

G

R

V-

-

+

=

2

2

)

4

(

85

00810

.0

69

.1

0176

.0

5.

74

T

L

G

R

V-

-

+

=

2

2

)

5

(

85

00934

.0

08

.2

1.

83

T

L

G

V-

-

=

BucchiA.BiasuzziK.

And SimoneA.2005 Italy

DC

V124

.0

164

.

66

85

-

=

DC

E

V4.0

46

.3

366

.

55

85

-

-

=

Meanwhile, there are other vehicles driving rules such as speed limit in adverse weather conditions. This rule can improve the safety factor of the vehicle to some extent. At the same time, it limits the speed at the different levels.

7. Reference

[1] M. Rickert, K. Nagel, M. Schreckenberg, A. Latour, Two lane traf?c simulations using

cellular automata, Physica A 231 (1996) 534–550.

[20] J.T. Fokkema, Lakshmi Dhevi, Tamil Nadu Traf?c Management and Control in

Intelligent Vehicle Highway Systems,18(2009).

[21] Yang Li, New Variable Speed Control Approach for Freeway. (2011) 1-66