美赛数学建模比赛论文模板

1、The Keep-Right-Except-To-Pass RuleSummaryAs 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

2、 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 lowe

3、r 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 whe

4、ther 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 u

5、se 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 re

6、quires 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

7、 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 accelerat

8、e the traffic flow overall.Key words ：Cellular automaton modelBernoulli processMicroscopic traffic simulatormodelThe MPC traffic control.ContentContent .21.Introduction .32.Analysis of the problem.33.Assumption .34.Symbol Definition.35.Models .35.1Building of the Cellular automaton model .35.1.1Veri

9、fy the effectiveness of the keep right except to pass rule.45.1.2Numerical simulation results and discussion.55.1.3Conclusion .85.2The solving of second question .85.2.1The building of the stochastic multi-lane traffic model.85.2.2Conclusion .85.3Taking the an intelligent vehicle system into a accou

10、nt .85.3.1Introduction of the Intelligent Vehicle Highway Systems.95.3.2Control problem .95.3.3Results and analysis .95.3.4The comprehensive analysis of the result .96. Improvement of the model.106.1 strength and weakness.106.1.1Strength .106.1.2Weakness .106.2Improvement of the model .107. Referenc

11、e.12.1. IntroductionAs 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 driv

12、e in the right-most unless they are passing another vehicle)”.2. Analysis of the problemFor 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

13、 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. Assumpt

14、ionIn order to streamline our model we have made several key assumptionsThe highway of double row three lanes that we study can represent multi-lane freeways.The data that we refer to has certain representativeness and descriptiveOperation condition of the highway not be influenced by blizzard or ac

15、cidental 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 systemIn the intelligent vehicle system, the result of the sampling data has high accuracy.4. Symbol Def

16、initioniThe number of vehiclestThe time5. ModelsBy analyzing the problem, we decided to propose a solution with building a cellular automaton model.5.1 Building of the Cellular automaton modelThanks to its simple rules and convenience for computer simulation, cellular automaton model has been widely

17、 used in the study of traffic flow in recent years.Let xi (t) be the position of vehiclei at time t , vi (t ) 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

18、:gapixi1 (t )xi (t) 1, if the front vehicle is a small vehicle.gapixi1 (t )xi (t) 3, if the front vehicle is a truck or bus.5.1.1 Verify the effectiveness of the keep right except to pass ruleIn addition, according to the keep right except to pass rule, we define a new rule called:Control rules base

19、d 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 120100km/h (including 100 km/h)； the speed of the second lane (the middle lane) is 10080km8/h (including80km/h) ；the speed of the t

20、hird lane (the far right lane) is below 80km/ h. The speeds of lanes decrease from left to right.Lane changing rules based lane speed controlIfvehicleonthehigh-speed lanemeetsvvcontrol ,gapif (t )min( vi (t)1, vmax ) ,gapib (t ) gapsafe , the vehicle will turn into the adjacent right lane, and the s

21、peed of the vehicle after lane changing remains unchanged, where vcontrol is the minimum speed of thecorresponding lane.The application of the Nasch model evolutionLet Pd be the lane changing probability (taking into account the actual situation that somedrivers like driving in a certain lane, and w

22、ill not take the initiative to change lanes), gapif (t ) indicates the distance between the vehicle and the nearest front vehicle,gapib (t) indicates thedistance between the vehicle and the nearest following vehicle. In this article, we assume that the minimum safe distance gap safe of lane changing

23、 equals to the maximum speed of the following vehicle in the adjacent lanes.Lane changing rules based on keeping right except to passIn 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), intera

24、cting 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 firstlane (passing lane) meets gapif (t )min( vi (t )1,vmax )andgapib (t )gapsafe , the vehicle will turn into the second lane,

25、the speed of the vehicle after lanechanging remains unchanged.5.1.2 Numerical simulation results and discussionIn order to facilitate the subsequent discussions, we define the space occupation rateas p(N CAR3Ntruck ) / 3L , where N CARindicates the number of small vehicles on thedriveway, N truckind

26、icates the number of trucks and buses on the driveway, and L indicates thetotal length of the road. The vehicle flow volumeQis the number of vehicles passing a fixedpoint per unit time, Q N T / T ,whereNTisthenumber of vehicles observed in timeTi, vit is the speed of vehicle i at time t .durationT .

27、The average speed Va(1/ NT )11vitTake overtaking ratiop f as the evaluation indicator of the safety of traffic flow, which is theratio 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 obtaine

28、d 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 relation

29、ships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.(a) Based on passing lane control(b) Based on speed controlFig.5.1.3Fig.5.1.3 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different contr

30、ol conditions.It can be seen fromFig. 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 den

31、sity 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

32、, 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 rul

33、es on overtaking ratioFig. 5.1.4Fig.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 thespeedcontrol.Pf 1 is the overtakingratio of

34、smallvehiclesover largevehicles,Pf 2 is the overtakingratio of.small vehicles over small vehicles,Pf 3 is the overtaking ratio of large vehicles over small vehicles,Pf 4 is theovertaking ratio of large vehicles over large vehicles.).It can be seen fromFig. 5.1.4:(1) The overtaking ratio of small veh

35、icles 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 surpasslow-speed large vehicles by the passing lane, while under speed control condition, small vehic

36、les 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 speedFig. 5.1.5 Relationships among vehicle density, proportion of large vehicles and average speed under different control condi

37、tions. (Figures in left-hand indicates passing lane control, figures in right-hand indicates speedcontrol.X a is the average speed of all the vehicles,X a1 is the average speed of all the small vehicles,X a 2 isthe average speed of all the buses and trucks.).It can be seen fromFig. 5.1.5:(1) The ave

38、rage speed will reduce with the increase of vehicle density and proportion of large vehicles.(2) When vehicle density is less than 0.15, X a , X a1 and X a 2 are almost the same under bothcontrol conditions.Effect of different control conditions on traffic flow.Fig. 5.1.6Fig. 5.1.6 Relationships amo

39、ng 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 indicatesthe traffic flow difference between the two conditions.It can be seen fromFig. 5.1.6：(1) When vehi

40、cle 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 ConclusionI

41、n 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 question5.2.1 The building of the stochastic multi-l

42、ane traffic model5.2.2 ConclusionOn 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 jamsituation, applying B(2, PR (x)results with a tendency of moving to the right lane

43、 for thisdriver. However in reality, drivers tend to find an emptier lane in a jam situation. For thisreason, we apply a Bernoulli process B( 2,0.7) where the probability of moving to the right is0.7and to the left otherwise, and the conclusion is under the rule of keep left except to pass, So, the

44、fundamental reason is the formation of the driving habit.5.3 Taking the an intelligent vehicle system into a accountFor 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

45、 perfect the performance of the freeway by lots of analysis.5.3.1 Introduction of the Intelligent Vehicle Highway SystemsWe 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

46、the states when the speed limits are applied in Fig. 5.3.1. We implement a METANET model for prediction purpose Control problemAs 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

47、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 analysisWhen the density is high, it is more difficult to control the traffic, since the mean speed might already be below the control speed. Therefore