condition is smaller than that in video two, so when the three lanes combine their respective vehicle flow, the free lane in video one will hold heavier load than normal. Since the denser the flowing density K is, the lower the speed V will be, more confluence density in video one will result in lower flowing speed, which finally cause even severe congestion. Meanwhile, the longer time of congestion in video one also reflect the feature, just from our direct visual feelings.
To sum up, the impact on passing capability at cross sectional area caused by lane occupancy in video one is greater than that in video two, and the greater the flowing ratio is, the more serious of congestion will be.
3. Question three
3.1 Establishment of the model
Based on the analogy above, we use hydromechanics theory to set up our mathematical model. Hydromechanics simulation theory applies hydromechanics basic method to simulate consequence equation and then establish the consequence equation of vehicle flow. In the theory, we consider the change of flowing density as fluctuation of water waves, or say, car flow waves. When car flow changes with respect to traffic conditions and cause density to change, the car flow waves emerge and spread. Through analysis of the car flow waves, we find the relationship between traffic volume and traffic density, and velocity, and describe the process of congestion and elimination. [3] 3.1.1 Relationship among velocity, traffic volume and density
In the traffic flow theory, velocity V, traffic volume Q and traffic density K are three basic factors to show the traffic flow features.
Velocity is the reciprocal of the time within a certain distance, being the simplest index showing the operational effectiveness in the road. In the flow, every vehicle goes at their own speed. Supposing the speed obeys the Gaussian distribution and the average is the center, while other speed is shown by standard deviations.
The traffic volume Q is the vehicle number passing through a certain cross sectional area per unit time:
In the formula, T is time and N represents numbers of vehicles. The traffic density K is the vehicle number N in per unit length L:
The distance between the head of the two vehicles is called head-head length l, and within L, each head-head distance l is not equal, for which the average la is:
Take that into traffic density definition formula, we have:
Thus, the traffic density could be the reciprocal of head-head length.
The relationship of traffic density, traffic volume and velocity can be expressed in the formula as follows: [2]
Then we can discuss the relationship among volume Q, density K and velocity V.
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Provided with the speed of each car stay same:
Then the passing vehicle number N within period T of length L is:
Thus:
The formula above tells us that the traffic volume can be expressed by multiplying traffic density and velocity.
Regularly, the speed varies from cars to cars, so if we divide the vehicles into groups by speed magnitude, for each group:
Then the total traffic volume is:
In the formula, m is the number of groups, is the average speed. Therefore:
The basic relationship of the three fundamental factors is shown in figure 6:
Figure 6. the two-dimension and three-dimension of traffic volume, traffic density and velocity
First of all, the V-K relationship accords with the Greensheilds model, that is, the liner model. Supposing V-K is liner: (a, b are undefined coefficients); set K=0, V = Vf (free speed), and take it in the formula above, we get a = Vf; when the density reaches the biggest K=Kj (Kj is block density, meaning the traffic density when traffic volume is over saturation and the average velocity is down to zero), V=0, also taking them into V-K relationship equation and we have ; so we have relationship of V-K:
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Therefore we know the bigger the traffic density is, the lower the speed will be. For Q-K relationship, we can deduce it from V-K relationship:
The Q-K relationship fit the quadratic function, with its graph through the original point. We set Qm as
the max volume, Km being the max volume density, apparently, (Km，Qm) is the highest point of the function. When
, . Then when K < Km, it means non-congestion, correspondingly congestion
for K > Km, and when K = Kj, Q=0, no vehicle can pass theoretically.
Form the V-K and Q-K relationship, we deduce V-Q relationship:
Which is also fitting with quadratic function, and we define best traffic volume Vm for the V when Q is in its biggest. In realistic situation, V changes versus Q, so we should set Q as changing factor. When V > Qm, with the increase of Q, V goes down until both of them are zero.
When the volume density is lower than best traffic density, the flow is in its free passing state; before the traffic volume gets its top value, traffic volume increase with traffic density; when traffic density is close to max volume density, the flow starts to be blocked and queues exist, with speed come down; afterwards, with traffic volume going up, the vehicles queuing up increases continuously and the velocity and traffic volume decrease, finally to the state of nearly zero volume; the traffic is seriously in congestion. 3.1.2 Establishment of flow consequence equation
We make analogy of traffic flow and fluid flow, combining the deduce in question two, we have table as follows:
Physical quality Continuum Discrete element Variable factors Momentum State equation Consequence equation Supposing the time interruption between car passing cross section A and B is , distance is ; the inflow volume is q and density in is k, the outflow volume is .
According to substance conservation law:
That is:
Or:
Stretching to limitation we have:
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hydromechanics One-way uncompressed fluid Molecules Traffic flow One-way uncompressed car flow vehicles
The equation above tells that the density increase with the decrease of vehicle distance. Taking q=ku into the equation, then we get the consequence equation in traffic flow system:
3.1.3 Gathering and dispersing wave in vehicle flow
When vehicles are in queues facing the congestion caused by lane occupancy or similar situation, they will lower the speed and queue up as high-density queues; when congestion is over, those vehicles start to move to eliminate the block condition and form a mid-density car flow. The phenomenon that cars pass on the interface of two parts with different density is called vehicle flowing fluctuation. The interface of two parts with different density is called car flowing wave, and the speed of car flowing wave is called wave speed.
When vehicle flow transform form low density to higher one, the wave is called gathering wave, and when from high to lower, it is called dispersing wave.
Assuming the wave speed is W, volume in the two states is and , density in the two states is and . Supposing these factors of a motorcade change from to , from to , from (interval distance correspondingly) to . We set A as the first speed-changing point, with its time and position both are 0; B as the second, with time 0 and position x; then form time 0 to t, the distance changes , then we have:
Thus,
Due to , we have:
Therefore, we get the wave speed formula:
Assuming that the volume and density of both states is quite close, then the formula can turn to:
This is the wave speed formula in vehicle wave spreading process.
[3]
3.2 Solutions to the model
In order to analyze the relationship between length of queue affected by traffic accident and passing capability, accident duration, upstream traffic volume, we firstly set symbols for our studying objects – L for queue length, for passing capability at accidental cross sectional area, for upstream traffic volume, t for accident duration.
We set the measuring section from accidental cross sectional area to 120m upstream. In the gathering and dispersing wave equation, W is vehicle velocity, then we get the wave speed formula:
For inflow volume, combined by information provided in attachment 5 and analysis to video one above, we find that the inflow volume at upstream traffic light spot can be apparently divided into busy
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period and leisure period. Therefore, we make statistics of busy and leisure inflow density both at traffic light and the entrance of residential area for period of 60s, according to video one, and get the original queue density. For outflow, that is, the passing capability of accident cross sectional area, we count continuously for the entire process (appendix 2 shows the data), still using 60s as a period. Based on the average value 18.54pcu/min in question 2, we figure out that the outflow volume during half period is 9,27pcu.
Supposing the inflow density upstream for busy period is , the duration is ; for leisure period is and , then
Taking advantage of wave speed formula:
Considering the original queue length, the formula above can be transferred in to integral form:
This is the relationship function we want. Then we can calculate the parameters:
Within the 120m measuring section, based on information from video, we measure that the saturation vehicle unit is 36pcu. We can transform it to queue length coefficient:
The original condition of L is .
The inflow volume alternate at 30s half period, which shows like square wave, while the outflow volume remains nearly stable -- . Also, , , , so the passenger car unit accumulates and increase periodically.
For the jumping part in video, we count the unit right before and after that part and use the D-value as the increment value.
The result is shown in figure 7, in which the horizontal axis is the practical time in the monitor video, and the dotted line represents the jumping part.
滞留车辆排队长度随时间变化的关系图150
滞留车辆排队长度/m1005004244464850时刻/min52545658
13 Figure 7. changing of accumulated vehicle queue length versus time