1. Traffic variables as function of space and time ! ! !

2. UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) roadside detection of traffic data direction of traffic flow measurement crosssection traffic flow q time gap Δt local speed v density k headway s momentary speed v mom

3. Sketch to explain the principle of moving averages for building traffic flow variables space coordinate Averaging frame reflects the correlation length Spatially and temporally varying traffic flow variables as moving mean values

4. Continuity equation UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK number of vehicles N = k·Δx temporal variation limit gives con- tinuity equation k t + (q(x+Δx)-q(x))/Δx = 0 k t + q x = 0

5. k t + q x = 0 expresses the law of conservation of a traffic stream and is known as the conservation or continuity equation. This equation has the same form as in fluid flow. If sinks or sources exist within the section of the roadway, then the conservation equation takes the more general form: k t + q x =g(x,t) where g(x,t) is the generation (dissipation) rate in vehicles per unit time per unit length. In practice, generation of cars is observed when flow is interrupted (such as at entrances, exits,or intersections).

6. k t + q x = 0 is a state equation that can be used to determine the flow at any section of the roadway. Its attractiveness is that it relates two fundamental dependent variables, density k and flow rate q, with the two independent ones (i.e., time t, and space x). Solution of the continuity equation is impossible without an additional equation or assumption. One possibility is to consider a momentum equation similar to Newtons law of motion.

7. Another option is the one adapted in the simple con- tinuum modeling. It simply states that flow q, is a function of density k, i.e., q = q(k)=kv(k). This, or equi- valently, v = v(k), is a very reasonable assumption, but it is only valid at equilibrium. For this reason the high order continuum models are, in principle, more appealing but in practice have problems to prove superior to the simple continuum alternative.

9. The so transformed continuity equation is a first order quasi-linear, partial differential equation which can be solved by the method of characteristics

10. Shock wave formation resulting from the solution of the conservation equation

11. The density k is constant along a family of curves called characteristics or waves; a wave represents the motion (propagation) of a change in flow and density along the roadway. The characteristics are straight lines emanating from the boundaries of the time- space domain.

12. The slope of the characteristics is: This implies that the characteristics have slope equal to the tangent of the flow-density curve at the point representing the flow conditions at the boundary from which the characteristic emanates. The density at any point x,t of the time space domain is found by drawing the proper characteristic through that point.

13. The characteristics carry the value of density (and flow) at the boundary from which they emanate. When two characteristic lines intersect, then density at this point should have two values which is physically unrealizable; this discrepancy is explained by the generation of shock waves. In short, when two characteristics intersect, a shock wave is generated and the characteristics terminate. A shock then represents a mathematical discontinuity (abrupt change) in k,q, or v.

14. The speed of the shock wave is where k d, q d represent downstream and k u, q u upstream flow conditions. In the flow concentration curve, the shock wave speed is represented by the slope of the line connecting the two flow conditions (i.e., upstream and downstream)

15. Macroscopic Traffic Flow Models Characteristics:  The dynamics of the traffic parameters of a complete cross section of the carriageway is considered (only rarely a differentiation for single lanes)  Traffic stream is approximated via continuous parameters:  Vehicle density k(x,t) or/and average speed v(x,t) Types of models: –Hydrodynamic first order models (only k(x,t)) Equations are comparable to those of hydrodynamics, density only –Hydrodynamic second order models (k(x,t) and v(x,t)) Equations are similar to those Navier-Stokes-equations, description based on densities and speeds –Gas kinetic models Vehicles take over the part of molecules (furnish microscopic explanation of hydrodynamic models)

16. Hydrodynamic Model of Lighthill, Whitham (1955) and Richards (1956) [LWR] Basis: Continuity Equation:  conservation of number of vehicles (1.1) (without off/on-ramps) Assumption: traffic flow reacts instantaneously to changes in density (1.2)q(x,t) = q( k(x,t) )inserted in (1.1) leads to (1.3) with speed (1.4) with fundamental diagram, e.g.:

17. LWR-Theory (1) the complicated expression: expresses just the following:

18. LWR-Theory(2) with the fundamental diagram Creates the most simple and the oldest theory on traffic flow! q slope: v free

19. –Free Traffic The driver is free chose his speed independently from the speed of other vehicles. Traffic states and traffic phenomena (I) –Bounded Traffic All Drivers are forced to adjust the speed of their vehicles to the speed of other vehicles Classification based on a Level-of-Service concept (Traffic flow of different levels of quality) –Partly Bounded Traffic At least from time the drivers are forced to adjust their speed to the speed of other drivers  Vehicle streams of different and changing length.

20. –Synchronised Traffic Like bounded traffic, but at low speed and an almost optimized (!) traffic flow Spontaneous jam formation (trajectories) Source: Treiterer 1967 –Spontaneous Formation of congestions/ „ Congestion coming out of nowhere “ congestions lacking any obvious reason (for instance accident, construction site) –Wide moving jams big compact high density congestions, moving against the driving direction Traffic stated and –phenomena (II) x (km)

21. Traffic states and –phenomena (III) Here you enter the congestion……and here you leave it But here is the origin of the congestion „Congestion coming out of nowhere“ Time (in seconds) Position (in meters) relocation

22. Congestion versus Synchronized Traffic

23. Fundamental diagram source: from Schadschneider 2001, Kerner 1997 MeasuredFlow J (vehicles / time unit) (also called Traffic volume q ) Parameters: Speed v Density k [vehicles / km]  Occupancy: relative density [%] Empirical Schematical (with „synchronised traffic“)

24. As analytical expression for a fundamental diagram including stable and unstable traffic flow description van Aerde proposed from queueing theoretical background the relation with c 1, c 2, c 3 positive constants and free flow speed v f Exercise : Fundamental Diagram after van Aerde

25. (a)What is the behavior of the van Aerde approach for v ≈v f and v ≈0 ? (b)(b) Derive the speed-volume relation v=v(q) from the van Aerde approach and confirm the asymptotic behavior from question (a) ？ (c)(c) How to calculate the maximum flow q=q max and the corresponding speed v qmax ? Does this speed depend on the constant c 3 ?