On the Nature of Traffic Instabilities. Martin Treiber. What Traffic Data and Mathematics Can Tell us. TU Dresden. TGF 2009, Shanghai.

Transkript

1 On the Nature of Traffic Instabilities What Traffic Data and Mathematics Can Tell us TU Dresden TGF 2009, Shanghai 1 1

2 Overview Part I: Analyzing traffic data around the world - Qualitative aspects: Recurring spatiotemporal patterns - Quantifying the data: Spatial growth rates, wavelengths, and propagation velocities Part II: Modeling and simulating the phenomena - Some stability concepts in closed and open systems - From the stability properties to the traffic patterns Part III: Analytical Math - Can the convective instability possibly explain some of the observations? 2 2

3 Visualizing Spatiotemporal Patterns using the Adaptive Smoothing Method Independent smoothing in space and time dimension Adaptive smoothing 3 3

4 Testing the ASM: M 42 North (England) Original data ASM, =1 km 4 4

5 A9 S (Germany, near Munich): Oscillating Congested Patterns TSG OCT Downstream front: Fixed or moving upstream with velocity c Upstream front: Noncharacteristic velocity Internal structures: Moving with velocity c as well Amplitude of internal structures grows when moving upstream Frequency grows with severety of bottleneck 5 5

6 M 42 (England): Triggered Stop & Go Waves 6 6

7 M 25 (England): Isolated Stop&Go Waves 7 7

8 A5S (Germany): Homogeneous Jam, Oscillating Congestions, and Triggered Stop&Go Waves OCT HCT Calculating upstream jam front propagation according to LWR TSG 8 8

9 M 42: England has Compact Oscillating Congested Patterns as well Nov 19, 2008 Jul 30,

10 the Same Applies for America x OR 217 (Orlando, USA) V(t) - <V> (Diploma thesis of Benjamin Zielke) 10 10

11 Composite Pattern (A9, Holland) Dec 12,

12 Decomposing the Composite Pattern (A8-East, Germany) A8-East, section containing the Irschenberg Uphill bottleneck Laneclosing bottleneck 12 12

13 Interactions Between Bottlenecks and Congested Traffic (A5S, Germany) German A5South Extended, Stop&Go Localized, moving (MLC) Localized, fixed (PLC) 13 13

14 Also Localized Patterns Can Interact (A5 South, Germany) Localized, fixed (PLC) Localized, moving (MLC) 14 14

15 A5 South: All Congestions Before Noon => Interactive traffic-data mining tool S, morning, any: 3 main bottlenecks weak, strong, strong S, Select MLC => only top bottleneck emits them N, afternoon, any => different patterns compared to S N, All days, all patterns all day, moving => moving bottlneck N, accident => jams caused by accidents 15 15

16 A5 North: All Congestions After Noon 16 16

17 A5 South: All Homogeneous Patterns 17 17

18 Position (km), Part IB: Quantitative Empirical Analysis: Propagation Speed, Wavelength, and Growth Rate 18 18

19 Position (km) Determining the Spatial Growth Rate V(t) - <V> 19 19

20 First Results FT Growth rate = c * (spatial rate) HCT OCT TSG MLC Characteristic period: minutes TSG HCT OCT MLC TSG OCT FT (Measure of the bottleneck strength) 20 20

21 Part II: Modeling and Simulating the Phenomena The Intelligent-Driver Model (IDM) Equations of motion: Dynamic desired distance 21 21

22 Modeling the Driving Style In the IDM, the driving style is characterized by Parameter Typical Value Desired velocity v0 120 km/h Desired headway T 1.5 s Desired acceleration a m/s2 Desired deceleration b 2.0 m/s2 Minimum gap s0 2m Gap contribution s1 5m 22 22

23 Simulating the closed system 23 23

24 Some String Stability Concepts Localized perturbation: One car accelerates or decelerates w/o reason Linear string instability: Growth rate Re(λ(k))>0 for some wavenumber k, or Nonlinear instability (metastability): Convective (meta-)stability: Linearly (meta-)stable, and 0 and nonzero integral value for ε above 24 24

25 Stability Classes as a Function of Density Class 1: Maximum flow is unstable Class 2: Maximum flow is (meta-)stable, but (convective) instabilities at higher densities Class 3: Unconditionally stable Subclasses a/b: Whether restabilization (b), or not (a) at high densities Simulate Class 1/2/3: a=0.3/0.6/2 Class a-b:s1=0/14 ρ=28/km, 45/km, 80/km 25 25

26 Macrosimulations (GKT Model) The concept of stability classes applies to macromodels as well 26 26

27 From a Ring Road to an Open System Absolute Threshold to Convective Ring road linear convective instability: instability: instability: a=0.9 m/s^2 a=0.7 m/s^2 a=0.8 m/s^2 Open system (bottleneck) 27 27

28 Transition Closed to Open Systems from the Flow-Density Perspective Bottleneck The bottleneck strength is the new control variable determining the density immediately upstream of the bottleneck! 28 28

29 Reproducing the Observed Basic Patterns 29 29

30 Using the Intelligent Driver Model Simulate this! The parameters correspond to stability class 2b 30 30

31 Arranging the Patterns according to Traffic Inflow and Bottleneck Strength German A5 near Frankfurt 31 31

32 Same Analysis for a Wide Variety of Other Models (Example: Triggered Stop&Go Waves) 32 32

33 Simulating Complex Patterns Combinations of bottlenecks: Jam-bottleneck and Jam-Jam interaction Two bottlenecks and interacting congested patterns Offramp-onramp combination 33 33

34 The Role of Fluctuations GKT model IDM Stationary +Variable part No variable part 34 34

35 Part III: Analytical Analysis of the Convective Instability: Model classes A general class of ODE microscopic models: Equilibrium: A general class of PDE macroscopic models (in most cases, D=0) Equilibrium: The pressure P depends on the velocity variance θ: 35 35

36 Nonlocal macroscopic models The derivatives of the acceleration function A( ) can also be replaced by nonlocal terms, such as in the GKT model Such models often have more favourable mathematical properties (no infinite characteristic velocities, ) Equilibrium: Nonlocality: Example GKT model: s a=vt+1/ρmax 36 36

37 Preparation: Linear Instability The complex growth rate and linear stability conditions can be easily obtained by Fourier mode expansion of the deviations Stability criterion Microscopic models: where Macroscopic models: Expansion Stability criterion Nonlocal models: 37 37

38 Convective Instability Solution to a δ- Perturbation: maximum growth: Maximum growth rate and associated frequency: Group maximum growth rate: Complex maximum growth rate: where Convective Instability Criterion: 38 38

39 Testing the Analytic Formula Against a Microscopic Model Analytic perturbation field for the convective threshold and the corresponding numerical simulation 39 39

40 and Against a Macroscopic Model Analytic perturbation field for the macroscopic GKT convective threshold and the corresponding numerical simulation 40 40

41 Further Research: Analytic Response to Sustained δ- Korrelated Noise Sustained noise (e.g. onramp) Position (km), sustained noise Spatial growth scale of amplitudes A(x,t) Growth scale 41 41

42 Conclusions A comprehensive empirical study revealed qualitative universalities, and quantitative variations relating to different driver behavior A wide range of models (macro, micro, including CA) reproduced the qualitative features when simulated correctly (open system, calibrated parameters, including the measurement). Three-phase models are unnecessarily complicated The concept of a convective instability together with sustained local noise may even lead to an ab-initio analytical understanding of the congestion patterns 42 42

43 Fakultät Verkehrswissenschaften Friedrich List Thank you for your attention! and Arne Kesting, TU Dresden Technische Universität Dresden MOVING THE WORLD.