Team Control Number. For office use only. For office use only. Problem Chosen. MCM/ICM Summary Sheet

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1 For office use only T1 T2 T3 T4 Team Control Number Problem Chosen C For office use only F1 F2 F3 F MCM/ICM Summary Sheet Traffic capacity is limited by the number of lanes of roads. Self-driving, cooperating cars have been proposed to increase capacity of highways without increasing lanes. In order to explore the influence of cooperating cars, we build a model to explain the effect of the number of lanes, peak and/or average traffic volume, and percentage of vehicles using self-driving, cooperating systems on traffic flow. To reveal the mechanism of different driving behavior of the microscope traffic flow, we simulate discrete events based on Cellular Automaton (CA). The simulation mainly includes three processes: car generating model, one lane model and lane changing model. For ordinary cars and self-driving cars, we define their safe distance, the probability of slowing down and the probability of changing lanes respectively. During the processes, we do statistics about traffic volume, density and speed. Control variable method is applied to measure the net effects of the number of lanes, volume and percentage of self-driving cars on the traffic flow. Take the road segment with three lanes in one direction, without self-driving cars, as the base case. In order to study the influence of different volumes on traffic operation, we analyze the change of velocity and density respectively as volume changes. The simulation shows that the traffic flow is in the stable state, so the maximum traffic flow represents the possible traffic capacity. As for the percentage of self-driving cars, volume-speed curves are drawn under different percentages. Possible capacity increases markedly when the mixed percentage of self-driving cars is at the range of [0.2, 0.3], [0.4, 0.5] and [0.8, 0.9]. So depending on the development of economy and technology, the introduction of self-driving cars should be executed for three steps, with the recommended percentage of self-driving cars being 0.3, 0.5 and 0.9. The effect of the number of lanes is studied based on three lanes, the most common situation. Then the lane number correction coefficient is introduced to calculate the possible traffic capacity of multilane highways. To analyze the traffic flow and integrate the factors, we introduce congestion index of a segment to evaluate the saturation of all the given segments. And the mean congestion index of the entire road is weighted according to the mileage of each segment. To test the sensibility of the model, we change the three coefficients. It is found that possible capacity doesn t change much along with the variation of probability of slowing down randomly and lane changing probability, except for safe distance. Key words: CA self-driving vehicles traffic capacity congestion index

2 Team # Page 1 of 22 How cooperating cars release congestion 1 Restatement of the Problem Traffic capacity is limited in many regions of the United States due to the number of lanes of roads. For example, in the Greater Seattle area drivers experience long delays during peak traffic hours because the volume of traffic exceeds the designed capacity of the road networks. Self-driving, cooperating cars have been proposed as a solution to increase capacity of highways without increasing number of lanes or roads. The behavior of these cars interacting with the existing traffic flow and each other is not well understood at this point. We are asked to establish a model of the effects on traffic flow of the number of lanes, peak and/or average traffic volume, and percentage of vehicles using self-driving, cooperating systems. The model should address cooperation between self-driving cars as well as the interaction between self-driving and non-self-driving vehicles. The model should then be applied to the data for the roads of interest, provided in the attached Excel spreadsheet. After building the model, the effects of allowing self-driving, cooperating cars on the roads has to be considered. And the model is applied to evaluate the conditions of the road segments listed above in Thurston, Pierce, King, and Snohomish counties. The analysis should include: How do the effects change as the percentage of self-driving cars increases from 10% to 50% to 90%? Do equilibria exist? Is there a tipping point where performance changes markedly? Under what conditions, if any, should lanes be dedicated to these cars? Does the analysis suggest any other policy changes? 2 Assumptions and Justifications There is some useful background information to be considered in the model: On average, 8% of the daily traffic volume occurs during peak travel hours. The nominal speed limit for all these roads is 60 miles per hour. Lane widths are the standard 12 feet. To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions: Standard passenger car is the only type of vehicle being considered, which is representative. The length of the car is 4.8 meters. Standard passenger cars occupy a large proportion of vehicles on the road. Study on interaction between standard passenger cars and self-driving cars is the basis for further study.

3 Team # Page 2 of 22 All the self-driving cars are in the cooperating systems and can cooperate with each other. A self-driving car does not have a driver, and is instead guided by internal computers using internal or possibly external sensors. A cooperating car communicates and exchanges data with other cooperating self-driving cars as it decides what to do. Self-driving, cooperating cars are the future development trend. When the vehicles are generated, the initial percentage of self-driving, cooperating vehicles is evenly distributed on each lane. Self-driving vehicles and ordinary vehicles are mixed. Under normal circumstances, the proportion of vehicles of different types keep balanced in the lanes. If lanes are dedicated to self-driving, cooperating cars, suppose that the lane is designated for them from the inside. There is less interference in the inner lanes, so it can give full play to the advantage of self-driving s high capacity. The driver's driving level is medium, and his behavior is rational, while not considering the accidents. The majority of drivers are with moderate driving levels. This assumption ensures that the model is suitable for most drivers. 3 Notations Symbols T t v max Table 1 Notations Definition Length of a time step Number of the current time step The maximum speed that a vehicle can move in one time step l c x () jn t x j n+1 (t) v () jn t v j n+1 (t) P x jt y jt Gap safe,n Length of a single cell Location of the vehicle n on lane j at time t Location of the vehicle ahead of vehicle n on lane j at time t Speed of the vehicle n on lane j at time t Speed of the vehicle ahead of vehicle n on lane j at time t Percentage of self-driving cars in total traffic flow the generating probability of cars the generation control variable the category selection variable the required safety distance for the nth car

4 Team # Page 3 of 22 l n+1 b n τ n p change, i d n d n, other C p β i β L i the length of the front car the maximum deceleration speed of vehicle n the driver s reaction time of vehicle n The probability for a vehicle with type i ( i represents either conventional cars or self-driving cars) to change lane The number of empty cells between the car and front car on the same lane The number of empty cells between the car and the front car on the adjacent lane Possible capacity Congestion index of a segment Congestion index of the entire road The mileage of the segment 4 Model Overview Fig.1 Modeling process

5 Team # Page 4 of 22 5 CA Model with Multi-Type Cars Cellular Automaton, the main tool of microscopic traffic simulation research, has developed rapidly after being introduced into the transportation field. Each cell in the lattice grid takes a finite discrete state, following the same rules of action, and is updated synchronously according to the determined local rules. A large number of cells form the evolution of the state system through simple interactions. The basic structure of a single-lane model is as follows. Divide a lane into equal-sized grid points. Each grid has two states: empty or occupied by a vehicle. And each vehicle occupies exactly one grid. For the discretization of time, every changing step takes the same length of time clock. Take the vehicle s position, speed as state variables, which are limited and discrete. At each moment, the state is updated according to the defined rules, and it is repeated until the termination condition is satisfied. The main difference between the models is the updating rule. The stochastic model represented by the NaSch model is the main stream, which has been deeply researched and tested empirically. Multi-lane model is based on the single-lane model and the biggest change is the lane changing rules. Fig.2 The schematic diagram of cells occupied by cars 5.1 Initial parameter setting Use gray-scale map to display the simulation. The lattice is 17.5 feet long. Take a length of about 1 mile to simulate, and the number of grids is 300. Set matric A, matric V, matric Gap to store the states of grids. A represents the presence or absence of vehicles in each lattice: 0 means no car, 1 is an ordinary car, 2 is an autopilot; V represents the speed of the vehicle in the corresponding lattice; And Gap represents the distance between the vehicle in the lattice and the vehicle ahead of it. 5.2 Car generation model First of all, set the generating probability as P and the percentage of self-driving, cooperating cars as. For lane j, a random number x jt (x jt [0,1]) is generated first, which is the generation control variable. Whether to generate a car on lane i or not depends on the magnitude relationship of x jt and P. Supposing that a vehicle is generated on lane i and grid j, if grid j is empty, the car will be put in the grid; if grid j is occupied by another car, the newly generated car will be put on the grid behind grid i. Then to judge the type of the added car, a random number y jt (y jt [0,1]) is generated, which is the category selection variable. According to the magnitude relationship between y jt and, weather to add an ordinary car or a self-driving car

6 Team # Page 5 of 22 is determined. For lanes 1 to J, the updating of all the vehicle generating steps is performed in such a manner. The interval between two updates is one second. { x jt > P, ge jt = 1, generate a car on lane i at time t x jt < P, ge jt = 0, not generate a car on lane i at time t { y jt >, type jt = 2, generate a self driving car on lane i at time t y jt <, type jt = 1, generate an ordinary car on lane i at time t To find the effect of each factor on a wide range of traffic volumes, the generation rate is taken from 0.2 to 1, with an interval of As the simulation is a segment on a road, the vehicle will enter the road with a certain speed. The vehicle's initial speed is set to: auto-vehicles 5 and 4 for ordinary vehicles. In addition, the generation grid is pre-set long enough to prevent the congestion in front of the generated vehicle. In the simulation diagram, only 50 grids in front of the generation grid are displayed. After generation, calculate the current Gap by way of circulation. There are some rules set for the process (the current car s index is I and the front car is j): For a car without one in front: Gap (i, j) = max (m 1, Gap sa (1, 1) + 1) For the cars in the last row: Gap (i, end) = Gap sa (1, 1) Definition of safe distance This paper proposes the definition of safe distance based on the characteristics of the traffic flow. The calculation comprehensively considered the physical characteristics, driving speed, acceleration and deceleration performance of the automated and manual driving car, as well as the influence of reaction time on the driving behavior. And combined with the classic Gipps model, Safe Distance is introduced to improve the NaSch model. When the current car has an emergent braking, the vehicle must maintain a safe distance in order to avoid the occurrence of the rear-end collision with the vehicle. The safety distance Gap safe, refers to the distance between the front vehicle and the current vehicle. It is related to the braking performance of the current car and the car in front, and the driver's reaction time. According to the principle of safe distance, the following expression is obtained: Gap safe,n = x j n+1 (t) x jn (t) l n+1 = v jn (t)τ n + v jn(t) 2 2b n v j n+1(t) 2 2b n In the formula[1], Gap safe,n refers to the required safety distance for the nth car; x jn (t) is the position of the nth car at time t; x j n+1 (t) is the position of the car ahead of the nth car at time t; l n+1 is the length of the front car; b n is the maximum deceleration speed of the nth car; τ n refers to the driver s reaction time of the nth car; v jn (t) refers to the speed of the nth car at time t; v j n+1 (t) the speed of the car ahead of the nth car at time t.

7 Team # Page 6 of 22 Fig.3 The schematic diagram of safe distance Cooperating cars can react more quickly to incidents on the road. So the breaking performance of an ordinary car and a cooperating car differ greatly. After calculation and simplification, the safe distances between different objects are listed in the following form. Table 2 The safe distances between different objects Ordinary Ordinary Cooperating Cooperating Objects Ordinary Cooperating Ordinary Cooperating Safe distance (The distance and speed are both expressed with the number of grids.) 5.4 One lane model The cars on one lane move one after another. The safe distance that must be obeyed has been calculated on the above. The driver in an ordinary car or the control system in a cooperating car will take timely measures towards the change of the surrounding cars. Besides, the driving behavior changes due to the road conditions or the personal emotion of the driver. To better regulate the steps being taken to keep safety and avoid accidents, four measures are defined to describe the evolutionary process: (1) Acceleration rule When the car is running, the distance between the nth car and the car ahead may be larger than the required safety distance when the vehicle travels, i.e., Gap n > Gap safe, In order to satisfy the driver's travel for the higher expected speed, the car is in accordance with the following rules to accelerate driving: V(i, j) min ([V(i, j) + 1 vmax gap(i, j)]) (2) Deceleration rules When the distance between the nth car and its front car is less than the required safe distance when the vehicle is traveling, that is, Gap n <Gap safe, the vehicle will decelerate to ensure safe driving. The deceleration rule is: V(i, j) max (min ([V(i, j) 1 vmax gap(i, j)], 0) (3) Random Moderation Rules Taking into account the driver's uncertain driving behavior in the course of driving, introduce the random probability of slowing down R p in the evolution rules. A control variable R ij is generated to decide that weather to make random moderation or not.

8 Team # Page 7 of 22 Only when R ij > R p will vehicle i on lane j slow down. The speed of the car changes in accordance with the random slowing down rules. v n (t) max (v n (t) 1,0) In the probability matrix of slowing down randomly, the cooperating vehicles do not slow down, and set the probability of ordinary vehicles to slow down as So the matrix is Rp = [0.05 0]. (4) Location update Update the position of the car based on the evolutionary rules of the speed. The updating rule is expressed in the formula: x n (t) x n (t) + v n (t) 5.5 Lane changing model General rules In general, a time step will be divided into two sub time steps in the cellular automaton model for vehicles: in the first sub step, vehicles change lanes in accordance with lane changing rules; in the second sub step, vehicles update their traveling status in accordance with one lane rules. Usually the driver is driven by some motivation to change lanes, there are two situations: 1. Running condition of the adjacent road is better than the one running on at present. 2. Vehicles cannot drive at expected speed on the currant lane. Also, the change needs to meet two conditions: 1. Have the motivation of lane changing. 2. Safety condition: no collision with others when changing lanes. According to the rules above, Rickert et al. first proposed a set of lane changing rules as follows[2,3]: Rule 1: d n < l, where l = min (v n + 1, v max ). This rule is a motivation criterion: the distance between vehicle n and its front car is not enough for it to move at the expected speed, therefore, there will be a possibility to change lane. Rule 2: d n,other > l o, where l o is suggested to be the same as l. This rule is to check whether the condition on the adjacent lane is better than the current one. Rule 3: d n,back > l o,back, where l o, back is suggested to be v max. This rule ensures a safe distance between the current car and the back car on the target lane. Rule 4: rand() < p change, Even though all the motivation and safety condition are satisfied, vehicles merely change lanes at a certain probability. The probability of changing lanes will be 0.5 if the ordinary vehicle satisfies the requirement. But if the automatic driving vehicle satisfies the requirements, it will change lanes. So the lane changing matrix Rz = [0.5 1].

9 Team # Page 8 of Rules for ordinary cars For ordinary cars, vehicles interact with each other based on the judgment of driver. Each driver is only the information receiver of surroundings and will take action depending on their custom and experience. Therefore the process above includes a set of randomness. Also, drivers have longer and various reaction time, so there should be a larger space distance to ensure safety. The critical lane changing condition is shown as follows: Fig.4 The lane changing condition As the example shown in the figure above, car n with speed vn 4 cannot accelerate on lane j while lane j 1 has enough space. The number of empty cells between the back car and the current car is 10, which is larger than the minimum safety distance mentioned before. Therefore, both motivation condition dn min( vn 1, vmax ) and dn, other dn and safety condition d, n back d are satisfied. The probability for a driver to change lane safe is set 0.8. A random number between 0 and 1 will be generated and if it is less than 0.8, the driver will change lane Rules when there are self-driving cars The ordinary car driver cannot identify whether the adjacent car is ordinary or self-driving car. So their operating rules are not affected by the rate of self-driving cars. However, for self-driving cars which need less reaction time and can interact with each other, their operating rules will change a lot. There are four situations in total for self-driving cars to determine whether to change lanes or not. Situation 1: both the front and back car on the adjacent lane are ordinary cars. Under the circumstance, rules will not change at all because the safety distance to the back car is dependent on the performance of the back vehicle and the distance to the front car is dependent on the expected speed which remains the same as ordinary cars. Situation 2: the front car is an ordinary car while the back car is a self-driving car. In this case, the motivation condition will not change but the safety condition will change a little due to cooperation between self-driving cars. A self-driving car can send a request of lane changing to the back car on the adjacent lane. The car can either accept or reject the request. We can assume that lane changing request will all be accepted. The

10 Team # Page 9 of 22 back car can change its speed to cooperate with the current car s lane changing behavior. So the safety distance will be 5-1=4 and v v 1. back Situation 3: the front car is a self-driving car while the back car is an ordinary car. In this situation, the motivation condition will change a little but the safety condition will remain the same. Although neither the distance to the front car on the current lane nor the adjacent lane can meet the need to drive at expected speed, the front self-driving car can accelerate when receiving the request of lane changing to expand the distance to the current car. The motivation condition will change as back d d and v v and v 1 Gap. n n, other front max front front The formula ensures that the front car has the condition to accelerate. Situation 4: both the front and the back car are self-driving cars. In this case, both the motivation condition and the safety condition will change. How to change them respectively can refer to situation 3 and The models of the special lanes The special lanes refer to the lanes used merely for self-driving, cooperating cars. While the cooperating cars could occupy the ordinary lanes, those ordinary cars couldn t run on the special lanes. The communication between the cooperating cars becomes more convenient and more efficient. So the conditions of the traffic flow have some difference compared with those hybrid lanes. The concrete changes can be list as follows: Car-generating model The special lane adds a cooperating car with the probability P1 in one update. So the probability of the added car being a cooperating car equals to 1. Car-following model Once a lane is dedicated to self-driving cars, the headway of them on the lane will maintain a smaller fixed value so the capacity of the special lane will improve a lot. The safe distance has the minimum value, while the moving principle remains unchanged. Lane-change model The only change is that only self-driving cars are allowed to change lanes between ordinary lanes and special lanes. The lane-changing rules on ordinary lanes are the same as mentioned above. 6 Simulation Analysis 6.1 Indexes of traffic flow Typical indicators should be chosen to evaluate effects on traffic flow. Traffic volume, speed and density are three basic parameters of continuous traffic flow. Traffic volume Traffic volume refers to the number of traffic entities passing through a road, a section or a lane in a selected period of time.

11 Team # Page 10 of 22 Q = N T In the formula, N = the number of vehicles passing a certain section of the road during the time period T (pcu); T = time period of investigation (h); The average space velocity On a given road segment, the average travel speed of all vehicles is the harmonic mean of the vehicle speed which can be used to evaluate analyze the state of the traffic flow for a given road space. In the formula, n V L / t / n nl / t s i i i 1 i 1 V s = space average speed (km / h); L = length of the road (km); t i = the time (h) required for the i-th vehicle to cross the road segment; n = total number of vehicles observed. Traffic density Traffic density refers to the density of vehicles on a lane, that is, the number of vehicles on a lane per unit length at a certain moment, also called the traffic density. k n / l V / Vs In the formula, k = density; n = number of vehicles on the road segment; l = length of the road. n Relationship between the three indicators above has been studied by many scholars. Greenshields presented a linear relationship model between speed and density. In the formula, u f =free flow speed k j =congestion density u u (1 k / k ) We have the relationship between volume and speed and density: q f j ku. Through simple derivation, relationship between volume and speed and relationship between volume and density can be obtained respectively as follows:

12 Team # Page 11 of 22 2 u q k j ( u ) u uk f q ku f k These relationships are in the form of quadratic curve. With the increase of speed, volume increases first and then decreases. There is a maximum volume. When speed is half of the free flow speed as well as density is half of the congestion density, traffic volume will reach its maximum, the correspond volume as this point is possible traffic capacity which refers to the capacity of a lane or a section of a road on real road and traffic conditions. The approximate shape of the flow velocity curve is as follows. f j 2 Fig.5 The relationship diagram of q-v The left side of the maximum flow is unsaturated flow while the right side of the maximum flow is saturated flow. Unsaturated flow is relatively stable and is the main objection to analyze traffic flow. 6.2 Data collection For data collection, at the end of the calculation of speed, the sum of the number of all the vehicles and the speed of the vehicle at each end point is counted. Among them, define the speed of the grids without cars as 0. After the completion of the cycle, count the collected data of speed. Because there is no vehicle on the road section at the initial time, we take a certain time after the traffic flow become stable to do statistics. Thus the flow of the segments, density and the average speed under certain conditions will be obtained. 6.3 Effects of traffic volume on traffic flow We use the control variable method to study the influence of various factors on traffic flow. First, consider the base case which meets conditions as follows:

13 Team # Page 12 of 22 The road has three lanes in one direction; All the cars are ordinary cars; Volume is the mere variable to be changed to evaluate its net effect. In order to find out the effect of a large range of traffic volume on various factors, the generation rate ( ) varies from 0.05 to 1 with a step size of The basic volume-speed curve and volume-density curve can be obtained as follows: Fig.6 velocity-volume curve Fig.7 density-volume curve From the diagram above, as volume increases, velocity decreases at first and then volume will keep at a certain level and the speed decreases continuously, the curve is in the part of steady flow, so the maximum volume is the possible traffic capacity and the value is reliable. Similarly, as shown in the diagram of density and volume, the curve is on the lower density side, which ensures the traffic steady, the maximum volume is the possible traffic capacity. The results show the simulation process is consistent with the reality. 6.4 Effects of percentage of self-driving car on traffic flow Allowing a certain rate of self-driving cars to enter the road is an effective way to increase the capacity without adding lanes. However, what percentage of self-driving cars involved in the traffic flow will bring the most effectiveness is the question we concerned. At each percentage, we can obtain a curve of volume and speed and each one has a maximum volume called possible traffic capacity. Draw a scatter plot of relationship between possible traffic capacity and percentage of self-driving cars, taking the possible capacity as the vertical axis and the percentage of self-driving cars as the horizontal axis. We can see from the scatter diagram when the maximum traffic capacity can be obtained and where the possible traffic capacity increases significantly.

14 Team # Page 13 of 22 Fig.8 velocity-volume curve Fig.9 velocity-volume curve under different percentages under different percentages As shown in the diagram above, each percentage has a corresponding curve. At the same speed or the same density, the overall trend of flow increases with the increase of percentage of self-driving cars. Draw a line chart on the relationship of possible traffic volume and percentage of self-driving cars as follows: Fig.10 possible capacity-percentage curve As shown in the line chart above, there is no equilibrium in the curve, in other words, possible traffic volume increases as percentage of self-driving cars increases, but the growth rate varies at different point. When the percentage is 10%, the possible traffic capacity only increases a little. When the percentage is 50%, possible traffic volume increases nearly 1000pcu/h. When the percentage is 90%, possible traffic volume increases about 2500pcu/h. The change degree indicates that possible traffic capacity does not increase linearly with the percentage of self-driving cars. It grows faster in the second half. We can get the growth rate at different point by derivation, but there is error in curve fitting. So the growth rate will be obtained from raw data. growth rate Q Q max j max i = 0.1*( j i )

15 Team # Page 14 of 22 Then the figure of growth rate and percentage of self-driving cars can be obtained as follows: Fig.11 growth rate-percentage curve The value of growth rate represents the marginal revenue (MR) of percentage of self-driving cars. As shown in the diagram, the growth rate increase markedly when the percentage of self-driving cars is 0.2~0.3, 0.4~0.5 and 0.8~0.9. The proportion of self-driving cars input into the system depends on the principle of marginal cost pricing, that is, when the marginal cost and marginal revenue equal, the maximum profit can be achieved. However, the marginal cost (MC) is unknown. Due to the cost and technical level at present, governor can take the three phases of the introduction policies. In the first phase, 30% of self-driving cars can be introduced. In the second phase, 50% of self-driving cars can be introduced. In the third phase, 90% of self-driving cars can be introduced. The corresponding possible traffic capacities are 3564pcu/h, 4028pcu/h and 5500pcu/h on road with three lanes. 6.5 Effects of the number of lanes on traffic flow If we assume that one lane has a certain capacity, it is obvious that more lanes will bring about larger capacity. But the degree of lane number s influence on the traffic capacity is not easy to obtain, and it cannot be simply multiplied. As the number of lanes is a discrete variable that contains only a finite number of values, we can do a simulation for every value. Because the usual highway contains 3 lanes in each direction, we take highway with 3 lanes as the base case. Highway with 2, 4, 5 lanes in one direction are the cases to be compared with that with 3 lanes. To simplify the process, percentage of self-driving car varies from 0 to 1and step length is 0.1. Volume-speed curve and volume-density curve will be obtained respectively under the circumstances that lane number is 2,3,4,5. Then with each lane number, there will be eleven volume-speed diagrams corresponding to the 11 different percentages of self-driving cars. Each curve has a maximum volume and relative percentage of self-driving cars. There are 11 groups of data containing possible traffic

16 Team # Page 15 of 22 volume and number of lanes. Let qi / q 3 be the expansion coefficient f i, then we get eleven f i, draw a line chart on the relationship of the expansion coefficient and the percentage of self-driving cars, taking the expansion coefficient as the vertical axis variable and the percentage of self-driving cars as the horizontal axis variable. Result is as follows: Table 3 The expansion coefficient expansion coefficient percentage 2 lanes 4 lanes 5 lanes average expansion coefficient lanes 4lanes 5lanes Fig.12 The expansion coefficient From the line chart above, the expansion coefficient shows some fluctuations as percentage of self-driving increases, and there is no obvious change trend. Expansion coefficient is not correlated with the proportion of self-driving cars. Take the average expansion coefficient value as f2 0.66, f4=1.32, f which represents possible traffic capacity changing when the numbers of lanes are two, four or five compared to three.

17 Team # Page 16 of Whether to set special lanes or not Whether to set dedicated lanes or not is determined by way of with-and-without comparison method. Possible traffic capacity is the indicator to judge. Number of lanes to set is determined by the percentage of self-driving cars. Different percentage is corresponding with different optimal number of lanes. Each number from 0 to the maximum number of lanes should be tested to find the optimal dedicated number of lanes at each percentage of self-driving cars. The maximum number of lanes is three. Take one special lane as an example: Fig.13 The ratio of capacity with or without one special lane From the diagram above, possible traffic capacity with a special lane is greater than that without a special lane when percentage of self-driving cars is 40% or 80%. So the special lane should be set at these percentages. 6.7 Conclusion From analysis above, we can draw the conclusion on every indicator as follows: 1. From the simulation results, the relationships between volume and velocity as well as volume and density are in the region of stable traffic flow. The maximum volume is possible traffic capacity. 2. Capacity of two, four and five lanes in one direction is 0.66, 1.32, 1.66 times respectively than the capacity of three lanes in one direction, f 0.66, f 1.32, f Capacity increases continuously as percentage of self-driving cars increases. But the growth rate is different at different percentage. When the percentage is 0.3, 0.5 and 0.9, the capacity increases markedly. The suggested percentage of self-driving cars is 0.3, 0.5 and 0.9, in which case possible traffic capacities are 3564pcu/h, 4028pcu/h and 5500pcu/h. 4. One lane should be dedicated to self-driving cars when the percentage of self-driving cars is 40% or 80%, the main method is with-without-comparison method.

18 Team # Page 17 of Model Application Governors most concerned matter is the percentage of self-driving cars. When the optimal percentage of self-driving car is applied, how much capacity will be increased and how well the road will operate is the matter cause for concern.. To evaluate the running state of the road, we introduce the congestion index to measure the saturation of the degree. Congestion index as: Vi i C pi i of a segment i is defined Where i is the congestion index of segment i V i is the volume of segment i C pi is the possible traffic capacity of segment i Each segment has a congestion index i, the mean congestion index of the entire road can be weighted according to the mileage of each segment. = L i L i i Where L i is the mileage of segment i. There are two different intensities of traffic flow needed to be considered: peak hour volume and average volume. There are differences between the numbers of lanes of each segment. The basic possible traffic volume is obtained on the condition that number of lanes is 3. Therefore, segments which have less or more than 3 lanes should be multiplied by the corresponding correction factor on the basis of the basic possible traffic capacity. The basic possible traffic capacity in the case study is under the circumstance that the number of lanes is three and the percentage of self-driving cars is 0.9 to determine whether the traffic will operate better with the maximum reasonable percentage.

19 Team # Page 18 of 22 The result is as follows: Table 4 The congestion index at peak hours Route_ID intersection congestion index Olympia 0.64 Rte 101 intersection 0.81 SR 510 intersection 0.83 SR 512 intersection 0.89 SR 16 Intersection 0.90 I705 intersection 1.22 SR167 Joins 0.91 I405 intersection 1.16 I90 intersection 1.11 SR 520 intersection 1.19 I405 intersection 0.86 Start of I90 in downtown Seattle 0.34 Intersection with I Intersection with I Intersection with I Intersection with SR Intersection with I Intersection with SR Intersection with I Intersection with I When the percentage of self-driving cars is 90%, there are 4 congested sections on interstates 5 and 4 congested sections on interstates 405. The section between SR167 intersection and I90 intersection has the largest congestion index. 90% is a high mixed percentage. If the congestion situation still remains oversaturated, building more lanes should be considered. All of the congestion indexes at non-peak hours of different segments are less than 1. This indicates that introducing 90% self-driving cars can greatly release the congestion in non-peak periods.

20 Team # Page 19 of Sensitivity Analysis There are a few parameters set in the CA model including safe distance, probability of slowing down randomly and lane changing probability. This part will test the influence of parameters above as they increase or decrease. The usual method is to increase or decrease 10% of the value of the parameters to observe the change in possible traffic capacity. Change in possible traffic capacity under the recommended percentage of self-driving cars is mostly concerned, so the sensitive analysis will focus on it. The middle column of the tables below is the current value of the parameters while the left and right columns are the smaller and larger parameters respectively. safe distance From the table below we can draw the conclusion that safe distance between different types of cars largely affects possible traffic capacity for it determines minimum headway between vehicles. Table 5 The sensibility analysis of Gap safe α Gap safe ( ) ( ) ( ) probability of slowing down randomly From the table below, the conclusion can be drawn that possible traffic capacity will not be strongly affected by the probability of slowing down randomly. Table 6 The sensibility analysis of R p α R p lane changing probability From the table below, we can draw the conclusion that lane changing probability will not greatly affect possible traffic capacity. Table 6 The sensibility analysis of P change P change α

21 Team # Page 20 of 22 In conclusion, our model has good stability. The efficient way to improve capacity is to decrease headway between vehicles, which is the exact reason why to introduce self-driving cars. 9. Strengths and Weaknesses 9.1 Strengths The cellular automata model is established to simulate the discrete events, and performs excellently to study the complex dynamical systems. In the model, different types of vehicles are analyzed respectively to analyze the behaviors of various vehicles in a variety of conditions which is accord with real situation. Control variable method is applied to analyze net effect of each variable. Mean traffic congestion indicator of the whole road is introduced to evaluate the macroscopic road operation state. The model is universal and can be applied to all segments. 9.2 Weaknesses For there are many uncertain factors in reality, it is difficult to use the analytical model to describe, so only the simulation model is given. Due to the lack of data of different traffic volume in different directions in the intersection, the influence of the intersection is not considered. The selection of some parameters is based on the experience and the subjective cognition of the driver's behavior. No strict argument is applied. MATLAB produces pseudo-random numbers, so there exists a certain degree of deviation. References [1] QIU Xiao-ping, MA Li-na, ZHOU Xiao-xia. Study on manual-autopilot mixed traffic flow based on safe distance [J]. Journal of Transportation Systems Engineering and Information, 2016, 16(4): [2] Jia Bin, Gao Ziyou, Li Keping, Models and simulations of traffic system based on the theory of cellular automaton[m]. Beijing: Science Press [3] LIU Xiao-ming, WANG Xiu-ying. Research on behavior model of cellular automaton lane changing based on information interaction [J].Application Research of Computers, 2010, 27(10):

22 Team # Page 21 of 22 A Letter to the Governor We know that you are deeply concerned about the lack of traffic capacity at highway 5, 90, 405, 520. We analysis the problem by establishing a model and find some solutions. Therefore, some recommendations will be provided. Traffic demand and supply are two major aspects of the transportation system. Seeking the balance between them is the main goal of facility construction and operation management. Under the conditions where the demand is mainly rigid, it is considered to increase the amount of supply. The average annual traffic volume of vehicles is analyzed, and it is found that the number of lanes is generally insufficient. From preliminary data statistics, it is found that the number of road sections with the peak hour traffic demand of more than 1800 (front time remained at 2 seconds, and this time is considered the minimum distance of man driving) reached 102. So it is particularly important to expand road capacity. Under the conditions of high land prices and the tension of land use, it is not a very economical and reasonable program to rely solely on road construction and increasing the number of lanes. The emergence of new technologies makes it possible to increase capacity without increasing the number of lanes. Self-driving, cooperating vehicles have shorter response time than human drivers. And vehicles in the cooperating systems can interact with each other to increase overall throughput and driving stability markedly than manual driving, while reducing randomization interference to traffic capacity. So autopilot vehicles will be the future development trend of the car. However, with the current development of technologies, it is impossible to replace all the vehicles with autopilot vehicles. Under this condition, it is not right to increase the percentage of self-driving cars blindly. On the one hand, the cost constraints; On the other hand, in the process of driving, the autopilot vehicles tend to maintain a smaller headway, to a certain extent, decreasing the possibility of other cars to change lanes to that lane and get an advantage of speed, causing a weakening effect on the overall capacity of the peer. After data analysis, the possible capacity increases markedly when the mixed percentage of self-driving cars is at the range of [0.2, 0.3], [0.4, 0.5] and [0.8, 0.9]. So depending on the development of economy and technologies, the encouraging and controlling policy towards self-driving cars can be executed with three steps, with the percentage of self-driving cars being 0.3, 0.5 and 0.9. After the congestion evaluation, it is found that interstates 5 and 405 have congestion indexes close to or over 1 in the peak hours of traffic. Through the analysis of the segments between the various intersections, congestion index of segment between SR167 and 190 on interstate 405 was found to be 1.54 revealing that the traffic is extremely crowded. The sections can t meet the demand even in the use of the best mixing ratio. The only measure can be taken to satisfy the extra demand is to increase the lanes on the road.