美赛论文solution

Traffic Circle Detection

Problem analysis

Different cities and communities have different traffic circles. In order to make traffic more convenient and efficient, these traffic circles position stop signs or yield signs on every incoming road or position traffic lights there. We want to use mathematical method to determine how to choose the appropriate flow-control method for different traffic circles.

Traffic-control method

For different traffic conditions, we use different signposts. If the road number of vehicles flowing in the intersection is not too large, some stop signs or yield signs may be more fitted. For some complex traffic conditions, traffic lights may be better. In our model, we ignore other signposts and mainly talked about the time of traffic lights.

Traffic circle conditions

A circle may be a large one with many lanes or a small one with only one or two lanes. For different number of lanes, the volume varies. Besides, the number of entrances also affects the vehicle flow. At the same time, combined traffic circles should also be considered.

Green light period

For relatively complex traffic circles, traffic lights are essential. Our model should give the green light period at each entrance. Adjusting the control system with these data, we can get the optimal result.

Assumptions

Taking all restrictive conditions into consideration, we make several assumptions regarding the cases we deal with.

1.Considering that the yellow light interval is short, and to make our model simple,

we ignore it.

2.The passers-by and non-vehicles are stochastic and uncertain, so we ignore them

when we establish our model.

3.We assume that all drivers keep the traffic rule. They start up once the light

becomes green, that is, we ignore their reaction time.

4.Once the traffic circle is positioned with traffic lights, other objectives such as

yield signs and stop signs are not considered.

5.Traffic circles are not suitable for any road conditions, they are mainly applied to

branches or secondary truck road in urban area.

6.The entrances are not less than four, so we don’t consider the three entrances

cases.

7.Traffic is allowed to go in only one direction (right-of-way)

Description of the model

Three progressively related models are established to get the optimal result at a certain circle. First, we think the cases that the circle has four crossings, and they are symmetrical.

Original model :

Fig.1 schematic diagram of isolated traffic circle

As is stated above, we start our analysis from isolated traffic circle with four entrances （Fig.1）. While vehicles travel across the traffic circle, how best to reduce the delayed time is the key of the traffic control system. According to the optimization theory, we construct a mathematic model to calculate the minimum delayed time. To deal with this case, we think the number of vehicles in the whole signal period is constant.

Given T is the cycle time(including stoplight and green light), usually we let s r g T i i 100=+=(4,3,2,1=i ) Where

i g is the green light period at every entrance.(i=1,2,3,4);

i r is the stoplight period at one entrance.

As the longest time drivers could probably wait is i r , the shortest could probably be 0s, thus the average time of every driver should wait is

)4,3,2,1(2

2=-=i g T r i

i s

In one cycle time, the total number of waiting vehicles is T

g T U i

i -?

Where i U is the number of vehicles flowing in during one signal period.

So at one entrance the total time vehicles should wait is

)4,3,2,1(2

=-?-?

i r T T g T U i

i i The total delayed time of all entrances is

)4,3,2,1(2

4

1

=-?-?

=∑i r T T g T U D i

i i The constraint condition is

∑=4

1

i g c

Where c is the signal period .which determined by the green light period in the

traffic circle.

To make the total delayed time of all entrances minimum, we must look for the optimal green light period. We set the signal period c with the step of 5s from 100s to 140s. We use practical data in beijing ’s 10 highways and street roads as i U .

Our calculated results are as follows:

Table.2 Results of the simulation D with Signal periods

when the signal period vary in the given range. Changes in signal period have obvious effect on the total delayed time.

Unfortunately, when the signal period is 100s , 105s , 110s , we find the green light period is unreasonable. So the model we put forward is a rough one which is not very accurate. To get closer to reality, we change our algorithm and construct the mature model.

Mature model :

Using the Webster delay model, we can get

{

]})1(2/[)]1(2/[)1(9.022x q x x c d -?+--?=λλ,

(1)

Where

d is th

e average delayed time o

f travelin

g vehicle;

c is the cycle time of all the green light shine once;

λ is the ratio of green in all the time;

q is the flow rate;

x is the degree of saturation, that is, the ratio of actual flow with the traffic

capacity(N ).

So, the total delayed time at the traffic intersections can be described as

1n

i D d q

=∑, (2)

Where

i d states the average delayed time of the

i th entrance

TRLL (England) method believes that if vehicles are given with enough green light period, vehicles will transit the traffic circle smoothly. So, we just select the flow rate of one direction as our calculation data.

If we let r y r q S =, ( r S is the saturation flow rate),

we can have the following formula:

2

2

'1

(1)0.92(1)

2()n r r r r

r r r r cy S y D q y y λλλ??-=+?

?--?

?

∑

(3)

To get the minimum 'D , we only need to let '0dD dt

=,

Thus, the optimal circle time 0(5)(1)

KL c Y +=-,

(4)

And

K is a coefficient, in real cases, 1.5K = L is the sum of the losing time

i Y y =∑. With 0()i

i y g c L Y

=

- (i g is the green light period), we can get the final conclusion.

However, if Y is too high, 0c will have a great deviation, we need to improve the problem-solving procedure.

We use the method of non-linear program, and try to find the minimum value of total delayed time. Modifying formula (3) at the condition of 4n =, we have

∑??????-+--=4

12

2)(2)

1(2)1(9.0r r r r r r r r y q y y c q D λλλ

(5)

The coefficient 0.9 does not influence the result, so we omit it. If we let i g express the green light period in i th entrance, then 1234c g g g g =+++. And in a

certain entrance, S is constant, which can only be obtained through survey. q can also be obtained through investigation. We use the data in a typical intersection in Nanjing, and replace saturation flow rate with saturation flow rate of natural turn, which is shown in the following table.

Table.3 the saturation flow rate of right turn of Nanjing intersection

When

1i =, 10.085y =, 111123.25/0.0342 pcu/s q y S pcu h =?==

22

1111123411111223422

1123411234

(1)123.25()2(1)2()0.546448()0.0000293103()

()0.085c y D q g g g g y q y g g g g g g g g g g g g g λλλ??-=+=+++?

??--??++++++-+

+++ When 2i =, 20.203y =, 222294.38/0.0818pcu/s q y S pcu h =?==

22

22

22123422222213422

2123421234

(1)294.35()2(1)2()0.627()0.00007

(()

0.203c y D q g g g g y q y g g g g g g g g g g g g g λλλ??-=+=+++?

??--??

++++++-+

+++ When 3i =, 30.166y =, 333257.3/0.0715pcu/s q y S pcu h =?==

22

33

33123433333212422

3123411234

(1)257.3()2(1)2()0.59952()0.0000535484

(()0.166c y D q g g g g y q y g g g g g g g g g g g g g λλλ??-=+=+++?

??--??

+++

+++-+

+++ When 4i =, 40.085y =, 444111.15/0.0309pcu/s q y S pcu h =?==

22

44

44123444444212422

4123411234

(1)111.15()2(1)2()0.546747()0.0000328846

(()

0.0855c y D q g g g g y q y g g g g g g g g g g g g g λλλ??-=+=+++?

??--??

++++++-+

+++ 12341

11234

23

231234123422

12232

1234232430.0036125

(){

(0.085)

0.02060450.013778

(0.203)(0.166)

1[399.689282.377256.241()

312.782443.213504.022D g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g =++++

-+++++

-+-++++++++

++++++++2441234406.268(430.055490.864677.836)]}

g g g g g ++++ (6)

Constraint condition :

To find the minimum value of the objective function, we need some constraint condition to help us solve the non-linear program.

Driver cannot endure queuing all the time and not being evacuated, so a new parameter should be introduced to describe the minimum effective green light period, which now is expressed as t min . We assume that the number of arriving vehicles obeys Poisson distribution. From the data above, we can get the average number of arriving vehicles, which we use as the expectation of Poisson distribution (i a ). Four tables were generated through investigation to show the relationship between the number of arriving vehicles and their possibilities.

For the first entrance, 11100 3.42361a q =?=,

min /t N S =, where N express the number of arriving vehicles at a certain cumulative probability.

Here, we let the probability equal 94.04%, then 6N =, min11/14.9t N S ==, so

114.9g ≥.

(7)

In a similar way, from Table.5, Table.6 and Table.7, let the probability separately be 96.03%, 96.89%, 96.19% , we can obtain that

min2min3min432.3s,27.9 s,=16.6 s t t t ==.

So

232.3g ≥ s

(8)

327.9g ≥s

(9)

416.6g ≥s

(10)

For any cases of traffic circle, the cycle time will have an inevitable influence on the average delayed time. If the cycle time is too short, the acceleration time will accounted for a higher proportion. As a result, the traffic control will be inefficient. On the other hand, assuming that circle time is too long, vehicles will pass through freely and make a waste of the great flow rate. Ordinarily, we think

25120s c s ≤≤

Now, we can solve objective function (6) with Constraint conditions (7) (8) (9) and (10).

Using the Lingo software, we get the final result as follows:

114.9g s =;

232.3g s =; 327.9g s =;

416.6g s =;

And the total delayed time in a signal period is 2208.5s . If we let the green lights lighten at a certain clockwise order, it will avoid traffic jam in the circle and improve the traffic efficient.

Analysis of results

Analysis of the original model

We find that for different entrances, the green light period is determined by vehicle flow .The more cars travel through the entrance, the longer green light period will be.

Once the signal period c become longer, the total delayed time in the traffic circle will turn shorter. In other words, the traffic is more efficient, which is the goal of this model. However, considering the actual condition, if green light period is too long, too many cars in the traffic circle could lead to traffic jam. So we believe that the green light period will be limited in a certain range.

Analysis of the mature model

As is shown above, we realize that the final green light period which makes the objective function minimum equals their separate minimum value. With this conclusion, given any certain traffic circle, we can know S . If the flow rate q is also given, for a certain cumulative probability, we can get the number of traveling vehicles (N ), using min /t N S =, the optimal green light time (g ) can be figured out. When 1i =, 10.085y =, 10.0342 pcu/s q =, 114.9g s =. When 2i =, 20.203y =, 20.0818pcu/s q =, 232.3g s =. When 3i

=, 30.166y =, 30.0715pcu/s q =, 327.9g s =.

When 4i =, 40.085y =, 40.0309pcu/s q =, 416.6g s =.

A conclusion can be drawn that if an entrance has a larger flow rate, its green light period will be larger. That is to say, they are positively correlated. However, for further consideration, they are not linear positively correlated. This phenomenon corresponds with actual situations. There exists a balance point if a traffic circle is given. Knowing the flow rate, we can adjust green light period to best control the traffic. Besides, if there are more than 2 lanes, we prefer to set right turn lane. As these vehicles don ’t add the total delayed time and can increase the flow rate. Such method should be acceptable.

We generated several other sets of data, and simulated the tendency in a chart. Their relationship is presented as follows:

Fig.2 the relationship between flow rate and green light period We read the tendency clearly from the chart. However, considering that the Webster model applies only to situations when the actual traffic flow rate is not so close to its saturation flow, the tendency is unreliable when the flow rate is too large. Because of the limitation of time, we don’t discuss it in details.

Extension of the model

n-entrances model

We established the model with four entrances, and make a detailed description of it. Now, when the circle condition is more complex, for example, has n entrances, or the entrances are asymmetrical, this model can also be used because our objective function only relate to the number of entrance (n).

Combined intersection model

Fig 3 schematic diagram of combined intersections

What we considered above is under the condition that all the intersections are isolated. Now we extend our model to two interactional intersections （Fig 3）. The traffic schematic diagram is like fig.1

We assume system A and system B are two interactional circle systems and this combination is separated from other systems. Entrance 1A and 1B are common segment. Thus the flow of those unshared entrances still obeys Poisson distribution. We let the flow of these unshared roads be expressed as i A and )4,3,2(=i B i (Which is measurable in a certain road.). The saturation flow (S ) is also constant in one certain situation. To use the Webster delay model once again, we set the flow of every road is distributed to other road according to proportion (()i p B and ()i p A ). So we can get

1223344()()()()()()A p A Q A p A Q A p A Q A =++ 1223344()()()()()()B p B Q B p B Q B p B Q B =++

Where ()i Q B and ()i Q A express the flow of corresponding road. In actual situation,

()i p B is always obtainable. So the total delayed time at the combined intersection

D could be described as the function of i g , using the Webster delayed formula

∑?

?????-+--=4

12

2)(2)

1(2)1(9.0r r r r r r r r y q y y c q D λλλ

We use cumulative probability of Poisson distribution to restrict the minimum green light period. Under the condition of these constraint conditions, we can use the software Lingo to help us find the minimum of objective function. Thus we obtain the different green light period i g.

Considering traffic circle is mainly used to link crossings, it is difficult to obtain the actual data, so we just provide the idea instead of solving the problem with concrete data.

Stability analysis

We tested the effect of changing some base factors in the model. In viewing of the original model, we use an initial signal period of s

100, and gradually enlarge the signal period. The green light time i g changes slowly. When the signal period increase to 105, 110 and 115, i g changes only7.76%, 13.2% and 18.6%. The result suggests that longer signal period decide longer green light time period.

When it comes to our mature model, what we concerned is the relationship between flow rate and green light period. Although the data we get is limited, we could still find that when the flow rate changes from 0.309 to 0.342, green light period changes only 11.4%. All of these mean that our model has good stability and it is reasonable.

Strengths and weaknesses

Strengths

1.We take full consideration of the practical background of the original problem.

2.Through analysis, we can learn that our models have well stability, which is

important for mathematical models.

3.We make several reasonable assumptions by neglecting minor factors, so our

models are practical. With these models, we can solve most physical issues.

4.Sufficient analysis was made, and we can have a deep understanding of this

problem.

5.At the end of the mature model, we clearly show the relationship between the

traffic flow rate and the green time period. Given a flow rate value of a certain entrance, we can get the optimal green light period.

Weaknesses

1.We ignore the yellow light, but in real cases, yellow light has effect on the traffic

flow, especially when the cycle time is short.

2.we have not consider any traffic jams or delayed time brought by accidental

factors, which can not be avoid in physical cases. However, these is a common problem

3.It is a fact that our data is obtained through investigation. We also have no

doubt that different road conditions have different result, so we must admit that there exists error.

4.The original model think that the longest time drivers could probably wait is the

stoplight time. However, we know that if the number of waiting vehicles is too large, they have to wait during the former stoplight time. This adds deviation to the result.

5.In the mature model, we used Webster model, but when it applies to situations

when the traffic flow in close to saturation flow.

References

●Programming method to optimize the time assignment at the traffic intersection

(Liu Ying. Li Yuewu)

Screening number 1008-844X(2002)01-0078-02

●Design of optimize the time assignment at the signalized intersections under

mixed traffic flows conditions (Zheng Changjiang. Liu Wei)

Screening number 1002-0268(2005)04-0116-04

●Design of roundabout crossings in the plane (Wang Yangzhen )

screening number 1004-4345(2006)02-0036-05

●Study on optimize the time assignment based on delay model

(Zhang Xiaocui. Chai Gan)

Screening number 1008-5696(2007)03-0082-03

●Journal of shanxi normal university

V ol.31 Sup apr.2003 1001-3857(2003)Sup.-0010-04

●Book: Study on optimal control algorithm of the traffic signal in urban areas

(Jian Lilin) P.33~34

Technical Summary

(To be presented to the Traffic Engineer)

Our mathematical models are established to solve the problem of controlling the traffic flow in, around and out of a circle. After a detailed analysis of practical problem and a series of reasonable assumptions, we finally find a method that can help to control the traffic.

Since the traffic jam often appeared at the entrance of intersections, we set up traffic lights in front of every entrance. Through changing the green light period and sequence, we relatively successfully make the total delayed time minimum. We assume that the radius of central part, the number and width of lanes etc. won ’t lead to traffic jam. We develop an algorithm which could find the relationship between green light period and the flow rate from every entrance.

Imagine there is a certain intersection, the inflow of vehicles may be influenced by the time of day and the official holiday. Luckily, the inflow is measurable and the saturation flow is also obtainable. Using our improved Webster delayed model, we could easily find the expression of total delayed time D . Through calculating, we reasonably let the green light period be min t (the minimum effective green light period according to the guarantee rate). Using the formula of cumulative probability

p k k

n

k =?-∑=0

!

^)exp(λλ; With the known expectation λ (λ is the number of vehicles in the certain time) and a wanted p (the cumulative probability of Poisson distribution), we get an n as the number of vehicles, thus the green time period that makes the total delayed time

minimum is

S

n

. Different time during a day and official holiday only have an effect on the flow rate of intersections. As this flow rate is easily measured, it is convenient for a traffic engineer to choose the optimal green light period.

In specific examples, if the entrances of intersections are misshapen or the number of entrances is above five, our model can still be used. According our algorithm, provided that our assumptions are satisfied, we can easily obtain the most suitable green light period according to a certain guarantee rate, no matter how many entrances there are or whether the intersection is isolated. Through investigation, the inflow of every entrance is easily known. Under the condition of unsaturated situation, the optimal green light period could be found.

In conclusion, the algorithm we provide is relatively accurate and has wide application. It can calculate the suitable green light period which makes the total delayed time minimum. It is therefore worthy to be put to practical use in the traffic-control system.