Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Traffic flow theory in macroscopic variables, fundamental diagrams and macroscopic traffic flow models.
Typology: Study notes
1 / 39
This page cannot be seen from the preview
Don't miss anything!
i
May 2002
Prof. L.H. Immers S. Logghe
FACULTY OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING SECTION TRAFFIC AND INFRASTRUCTURE KASTEELPARK ARENBERG 40, B-3001 HEVERLEE, BELGIUM
KATHOLIEKE UNIVERSITEIT LEUVEN
ii
This part of the course Basics of Traffic Engineering (H111) deals with the theory of traffic flow. This theory studies the dynamic properties of traffic on road sections.
We begin this course with a theoretical framework in which the characteristics of traffic flow are described at the microscopic level. We then examine a number of dynamic models that were formulated on the basis of empirical research. We conclude with a discussion of some recent observations on congestion.
The theories and models that will be discussed are developed on the basis of numerous observations on motorways. There is a difference between motorways and lower order roads such as provincial roads and urban streets. For the latter it are the intersections that dominate flow characteristics to a large degree. Traffic flow on intersections is the subject of a separate workshop on Signal-Controlled Intersections (H112).
This text is a second version. Remarks and suggestions continue to be appreciated.
Heverlee, May 2003
L.H. Immers S. Logghe
email: traffic@bwk.kuleuven.ac.be
tel: 0032-16- 0032-16-
address: Katholieke Universiteit Leuven Departement Burgerlijke Bouwkunde Sectie Verkeer en Infrastructuur Kasteelpark Arenberg 40 B-3001 Heverlee (Belgium)
Translated from the Dutch by Leni Hurley and Jim Stada (January 2005)
This chapter develops a theoretical framework in which the characteristics of traffic flows are described at the microscopic level. In a microscopic approach to traffic, each vehicle is examined separately.
Figure 1 A road with two vehicles along an x-axis and the same vehicles in a t-x co-ordinate system
On the left side in Figure 1, along a vertical X-axis, xα indicates the position of vehicle α
at time t 0. The vehicle in front of this vehicle is indicated by α+1. Since both vehicles travel across the road, their positions are time dependent. The right side of Figure 1 presents the vehicles in a t-x co-ordinate system.
The position of a vehicle through time is called a trajectory. In this course we use the rear point, the rear bumper of a vehicle, as the point of reference for the trajectory of that
vehicle. Figure 1 uses bold black lines to indicate the trajectories of vehicles α en α+1. The grey area represents the entire vehicle.
It is impossible for two trajectories to intersect when the vehicles travel on the same traffic lane. The speed vα of a vehicle if given by the derivative with respect to the
trajectory. The second derivative is the acceleration aα. Accelerating cars have positive values for ax and braking cars have negative values for aα.
A vehicle occupies a specific part of the road. This space occupancy or simply space sα consists of the physical length of the vehicle Lα and the distance dα kept by the driver to the vehicle in front, or:
Analogously to space, vehicles also use a certain segment of time which is called headway h. This headway time consists of the interval time or gap g and the occupancy o.
At constant speeds, or in general when acceleration is neglected, occupancy becomes:
The speed difference ∆ v is given by:
These variables can all be measured. Two aerial photographs taken in quick succession give us the positions, the speeds, the occupancies, the headways and the gaps. Using detection loops (that work on the magnetic-induction-principle) and detection cameras the speed, space, length and distance of vehicles can be measured fairly inexpensively.
Roads usually show a variety of vehicle types and drivers. We call the idealised traffic
state with only one type of road user homogeneous. A traffic state is stationary when it does not change over time. When this is the case, vehicles on homogeneous roads share the same speeds and trajectories are straight lines.
Figure 3 The measurement interval S 3
2.2 Density
Density is a typical variable from physics that was adopted by traffic science. Density k reflects the number of vehicles per kilometre of road. For a measurement interval at a
certain point in time, such as S 1 , k can be calculated over a road section with ∆X length as:
The index n indicates the number of vehicles at t 1 on the location interval ∆X. Total space
of the n vehicles can be set equal to ∆X, thus:
Figure 4 Location interval S 1
where the mean space occupancy in the interval S 1 is defined as:
Density k depends on the location, time and the measurement interval. We will, therefore, rewrite formula (2.1), in order to include these dependent factors in our notation. For the
location x 1 we take the centre of the measurement interval ∆X.
Density is traditionally expressed in vehicles per kilometre. Maximal density on a road fluctuates around 100 vehicles per kilometre per traffic lane.
The density definition in (2.4) is confined to a certain point in time. The next step is to generalise this definition If we multiply numerator and denominator of (2.4) by the infinitely small time interval dt around t1, density becomes:
The denominator of (2.5) now becomes equal to the area of the measurement interval S 1. The numerator reflects the total time spent by all vehicles in the measurement interval S 1.
In the same way we define the density at location x , at time t and for a measurement interval S as:
By way of illustration: Density according to definition (2.7) for x 2 , t 2 in the measurement interval S 2 , as illustrated once more in Figure 5:
Figure 5 Time interval S 2
the area of the measurement interval no longer appears in definition (2.14):
Totaltimespentby vehiclesinS
Totaldistanccoveredby vehiclesinS ( ,, )
k xt S
qxt S u xtS
In another form this definition of the mean speed is also called the fundamental relation of traffic flow theory:
This relation irrevocably links flow rate, density and mean speed. Knowing two of these variables immediately leads to the remaining third variable.
We calculate the mean speed for the measurement intervals S 1 and S 2 as follows:
For the location interval S 1 the density is given by (2.5) and the flow rate by (2.13). The mean speed for these n vehicles in the interval S 1 at location x 1 and point in time t 1 then becomes:
We get the mean speed for a location interval by averaging the speeds of all of the vehicles in this interval.
For the time interval S 2 , density was calculated in (2.8) and flow rate in (2.9). The mean speed for m vehicles then becomes:
This shows that the mean speed over a time interval is the harmonic mean of the individual speeds.
If we take the normal arithmetical average of the individual vehicle speeds in a time interval we get the time-mean speed ut , as defined in (2.18):
This time- mean speed ut differs from the mean speed u and does therefore, NOT comply with the fundamental relation (2.15).
The difference between the mean speed and the time-mean speed is illustrated by the example below:
Figure 6 Motorway with two traffic lanes.
Consider a long road with two traffic lanes, where all vehicles on the right traffic lane travel at 60 km/h and the vehicles on the left lane at 120 km/h. All the vehicles on the first traffic lane that passed a detector during a 1 minute time interval can be found on a 1 kilometre long road section. For the left traffic lane, the length of this road section equals 2 kilometres. Thus, when the time- mean speed is assessed, faster cars are considered over a much longer road section than slower cars. When we calculate mean speed, and also when we calculate density, the length of the road section used is the same for fast and slow cars. Therefore, the proportion of fast vehicles is overestimated when calculating time- mean speed thus making it always larger than or equal to the mean speed.
Example problem: Assume for that 1200 vehicles/hour pass on both traffic lanes in the example above. What are the density, the flow rate, the mean speed and the time- mean speed on this road?
Solution: q = 2400 vehicles/hour k = 30 vehicles/km u = 80 km/hour ut = 90 km/hour
Analogously we can also define the space-mean speed ux for a location interval as the mean of the speeds of all vehicles in this location interval or:
Equation (2.16) shows that the spa ce mean speed equals the mean speed as defined in (2.14).
Thus we distinguish three definitions: the mean speed u , the time-mean speed ut and the space-mean speed ux. Here u always equals ux and the fundamental relation applies to these definitions. The time-mean speed ut is different and does NOT comply with the fundamental relation.
The previous chapter defined three macroscopic variables: flow rate q , density k and mean speed u. Because of the fundamental relation q = k.u (2.15) there are only two independent variables. This chapter introduces an empirical relation between the two remaining independent variables. We do this by assuming stationary (flow rates do not change along the road and over time) and homogeneously composed traffic flow (all vehicles are equal). This means that we can simplify the notation somewhat because the dependence on location, time and measurement interval no longer applies in a stationary flow.
3.1 Observations.
On a three-lane motorway we measured the flow rate q and the mean speed u during time intervals of one minute. Each observation, therefore, gives a value for the mean speed u and a value for the flow rate q. Figure 7 shows the different observation points in a q-u diagram.
Figure 7 Observation points in a q-u diagram
We calculate the density k (= q / u ) for each observation. This means that the points of observation can also be plotted in a k-q diagram (Figure 8) and a k-u diagram (Figure 9).
Figure 8 Observation points in a k-q diagram.
Figure 9 Observation points in a k-u diagram.
The observations were carried out on an actual motorway where traffic is not homogeneous: there is a variety of vehicle types and drivers behave in a variety of ways.
Nor is real traffic stationary: vehicles accelerate and decelerate continuously. Abstracting from the inhomogeneous and non-stationary characteristics, we can describe the
empirical characteristics of traffic using an equilibrium relation that we can present in the form of the three diagrams shown above.
road, the speed restrictions in operation at any particular time and the weather. At free speed, flow rate and density will be close to zero.
The capacity of a road is equal to the maximum flow rate qc. The maximum flow rate of qc has an associated capacity speed of uc and a capacity density of kc. The diagram shows that the capacity speed uc lies below the maximum speed uf.
3.3 Mathematical models for the fundamental diagrams
In this section we present mathematical expressions for the equilibrium relations given by the fundamental diagrams. We examine the original diagram of Greenshield and the triangular diagram.
Greenshield drew up a first formulation that was based on a small number of slightly questionable measurements. In this formulation the relation in the k-u diagram is assumed to be linear, leading to parabolic relations for the remaining diagrams (see Error! Reference source not found. ).
Figure 11 The fundamental diagrams according to Greenshield
In Greenshield's diagrams, the capacity speed uc is half the maximum speed uf. The
capacity density kc in this model is half the maximum density kj. This formulation is a rough simplification of observed traffic behaviour, but is still frequently used because of its simplicity and for historical reasons. The equilibrium function in the k-u diagram can be written as::
Applying the fundamental relation gives the other relations ( Qe(k) and Ue(q) ). Note that the relation Ue(q) is not a function!
A second much-used formulation assumes that the fundamental k-q diagram is triangular in shape. This simple diagram has many advantages in dynamic traffic modelling, as will be discussed in Chapter 4.
In this equilibrium relation the mean speed equals the maximum speed for all traffic
states that have densities smaller than the capacity density. The branch of the triangle that links the capacity state with the saturated state, has a negative constant slope w. Figure 12 represents this triangular diagram.
Figure 12 The fundamental diagrams when a triangular k-q diagram applies
For cell i at time tj we can now write the following formulation of the state:
or:
Taking the limit with respect to the time step and letting cell length approach zero results in the following partial differential equation representing the conservation law of traffic :
We add another assumption to this conservation law: All possible dynamic traffic states comply with the stationary fundamental diagrams. This means that although traffic states on roads can change over time, they still comply with the fundamental diagrams at each moment and at every location. Therefore the successive traffic states 'move' as it were across the bold black lines in the fundamental diagrams.
This assumption allows us to write the flow rate in function of density as follows:
Inserting (4.4) in (4.3) and applying the chain rule gives a partial differential equation that only contains partial derivatives with respect to density.
In (4.5) the expression z(x,t) represents the volume of traffic that enters the road at time t and location x (a negative value for exiting traffic) and dQe(k)/dk , or in short Qe’(k) represents the derivative of the fundamental k-q function. In the subsequent derivation we assume a concave fundamental diagram which means that Qe’(k) will always decrease for increasing densities.
Using the fundamental diagram in the traffic conservation law led to the first dynamic traffic model in the 1950s. This model was named after the people who first proposed it: the LWR- model (Lighthill, Whitham, Richards). Several schemes were developed to numerically solve this equation with the help of a computer in order to obtain a traffic model that could be applied to practical situations. In the following section we will study this equation in an analytical way in order to gain some insight into some of the dynamic characteristics of a traffic stream.
4.2 Characteristics
The partial differential equation (4.5) is known in mathematical analysis as the "Burgers equation". It can be solved analytically with the help of given boundary conditions. If we apply the equation to a road without feeder- and exit lanes and if, for the sake of convenience, we assume that Qe’(k) equals c , the conservation equation (4.5) can be simplified to:
Solving this equation means finding the traffic density on this road in function of time and location. The solution to this equation is given by:
where F is an arbitrary function. By inserting (4.7) in (4.6) we can verify that (4.7) does indeed solve the partial differential equation. The solution implies that when x-ct is constant, density also remains constant. This means that all points on a straight line with slope c have the same density.
Example : For a point on the x-axis ( x = x 0 and t = 0) equation (4.7) gives: k(x 0 , 0 ) = F(x 0 ). At ( x 0 +ct, t ) density k(x 0 +ct, t) also equals F(x 0 ). Thus all points on the straight line with slope c through (x 0 , 0 ) have the same density equal to k(x 0 ,0 ).
If we know the value of the density at a point, we can draw a straight line through that point with slope c. The density then remains constant along this line. Such a straight line is known as a solution line or characteristic.
We sketch the t-x diagram in Figure 14a. Assume that the initial value in x 0 equals k 0. A straight line with slope c can then be drawn through x 0 along which density also equals k 0.
Figure 14 (a) the t-x diagram and (b) the k-q fundamental diagram
In actual fact the value of c equals Qe’(k 0 ). This is the derivative of the fundamental diagram function for k 0. In other words, c equals the slope of the tangent to the fundamental k-q diagram in k 0. We can now draw the k-q diagram to scale with the t-x
