Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation R. Eddie Wilson, University of Bristol EPSRC Advanced Research Fellowship EP/E055567/1 http://www.enm.bris.ac.uk/staff/rew Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.1/25
Macroscopic Traffic Data M25 anticlockwise carriageway 1/4/2000 110 space (17km) vehicle trajectories stop-and-go waves average speed (km/h) 06:40 time 11:00 Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.2/25 0
Some facts and conclusions (I) Propagation of stop-and-go is (fairly) regular so can be captured by macroscopic deterministic models? v x Downstream interface does not spread (Kerner 90s) problem for LWR and I believe ARZ / Lebacque framework Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.3/25
Some facts and conclusions (II) Ignition of stop-and-go waves is irregular needs full noisiness of microscopic description (but predictions can only be probabilistic) Wavelength is much longer than vehicle separation how to capture the upscaling effect? General idea: identify families of models which are qualitatively ok and throw away models which are qualitatively inadequate IN FUTURE Fit models to microscopic data Use emergent macroscopic dynamics for predictions Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.4/25
Active Traffic Management system Aim, reduce: accidents, (variance of) journey times Queue Ahead warning systems Temporary speed limits Lane management Spacing of inductance loop pairs is in range 30m to 100m Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.5/25
ndividual Vehicle Data from ATM system x 10 4 5.4 85 98 104 104 93 86 89 107 5.4002 105 119 88 119 5.4004 5.4006 5.4008 5.401 103 104 117 87 89 98 101 107 88 89 101 117 100 88 93 107 107 101 89 93 108 108 88 99 88 113 109 94 119 90 113 104 111 119 5.4012 107 108 108 109 107 87 89 5.4014 107 107 109 109 111 101 117 5.4016 109 5.4018 5.402 107 91 105 1 2 3 108 91 111 1 2 3 111 87 114 1 2 3 117 109 111 1 2 3 90 116 114 109 1 2 3 113 1 2 3 Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.6/25
Zoom-view and future scope 89 location A location B Individual vehicle data gives 125 129 helicopter view (speeds km/h) time, 6 secs 96 117 113 108 89 95 119 113 111 Location B is 100m downstream of location A: note lane change Propose to reconstruct vehicle trajectories over 55 100m 1 week 1 2 3 lanes 1 2 3 lanes Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.7/25
Jam formation in simulations 200 simulation of Optimal Velocity model 180 160 dimensionless space 140 120 100 80 60 40 20 0 0 50 100 150 200 250 300 350 dimensionless time Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.8/25
Car-following models x Typical form 1100 00 11 00 110 1 00 110 1 01 01 01 x n+1 v n+1 00 11 00 11 00 11 000 111 00 11 0000 1111 00 11 v 01 00 110 1 n 00 11 00 110 1 01 01 01 01 01 01 01 01 x n h n x n 1 v n 1 ẋ n = v n, v n = f(h n,ḣn,v n ) and generalisations E.g. Bando model (1995) f = α {V (h n ) v n }, α > 0 V is Optimal Velocity or Speed-Headway function Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.9/25
Linear stability framework General car-following model v n = f(h n,ḣn,v n ), Equilbrium condition, there exists V (h) so that f(h, 0,V (h )) = 0 for all h > 0. Linearisation yields ṽ n = (D h f) h n + (Dḣf) hn + (D v f)ṽ n, with sensible sign constraints D h f, Dḣf 0 and D v f 0. Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.10/25
Linear stability, part 2 Re-arrangement ḣn = v n 1 v n gives h n = (D h f)( h n 1 h n ) + (Dḣf)( hn 1 hn ) + (D v f) hn. Then try exponential ansatz h n = real ( ce inθ e λt) θ is perturbation s discrete wavenumber real(λ) is growth rate to obtain quadratic { } λ 2 + (Dḣf)(1 e iθ ) (D v f) λ + (D h f)((1 e iθ ) = 0. Then derive results for λ(θ) in quite general terms (proofs omitted) Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.11/25
Technical details Short wavelength analysis, θ = π λ 2 + { 2(Dḣf) (D v f) } λ + 2D h f = 0 All coeffs positive, therefore stable roots Long wavelength analysis, θ > 0 small, λ = λ 1 θ + λ 2 θ 2 gives λ 1 = i(d h f)/(d v f) and λ 2 = (D { } hf) 1 (D v f) 3 2 (D vf) 2 (D h f) (Dḣf)(D v f) Can show neutral stability λ = iω for general θ is equivalent to λ 2 = 0. Therefore: need only analyse λ 2 Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.12/25
Onset from infinite wavelength 2 1 0 1 2 3 3 x 10 3 growth rate onset of instability with change in parameters 4 0 0.1 0.2 0.3 0.4 0.5 0.6 infinite wavelength discrete wavenumber Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.13/25
Onset at medium densities 0.1 long wavelength growth parameter 0.05 0 0.05 0.1 change in parameters 0.15 0 0.5 1 1.5 2 2.5 3 3.5 4 nondimensional headway Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.14/25
Equilibrium curves 2 1.8 1.6 speed 2 1.8 1.6 speed 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.2 headway 0 0 1 2 3 4 5 0.4 0.2 density 0 0 0.5 1 1.5 2 2.5 0.7 0.6 flow 0.5 0.4 0.3 0.2 0.1 density no observations due to sensing method 0 0 0.5 1 1.5 2 2.5 Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.15/25
Other types of linear (in)stability Notional experiment in semi-infinite column of vehicles where second vehicle is instantaneously perturbed out of equilibrium Linearised dynamics of nth vehicle h n + [ (Dḣf) (D v f) ] hn +(D h f) h n = (D h f) h n 1 +(Dḣf) hn 1 Solve resonant oscillators inductively, large t h n (t) tn 1 (n 1)! [ λ(dḣf) + (D h f) 2λ + (Dḣf) (D v f) ] n 1 e λt where λ is stable platoon eigenvalue Use moving absolute space frame t = nh /(c + v ) and Stirling s formula to define growth wedge Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.16/25
Problems (?) with linear instability Setting a reduced speed limit to induce mid-range density and increase flow does not induce flow breakdown Stop-and-go waves almost always ignite at merges or other large amplitude externalities These problems may explain the continuing adherance to one-phase PDE models, be they first order like LWR or second order like ARZ/Lebacque Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.17/25
Introduction to bifurcation theory Loss of stability of uniform flow is via a Hopf bifurcation, of which there are two types: supercritical subcritical norm stable jam unstable jam stable unstable stable unstable parameter supercritical: stable periodic solutions are born subcritical: unstable periodic solutions are born, branch bends back so what is dynamics? Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.18/25
Introduction to bifurcation theory Loss of stability of uniform flow is via a Hopf bifurcation, of which there are two types: supercritical subcritical stable jam norm stable jam unstable jam stable unstable stable unstable parameter Subcritical bifurcation with cyclic fold gives jump to large ampitude traffic jam solution plus region of bistability Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.18/25
Computational results Application of numerical parameter continuation tools to analyse stop-and-go waves on the ring road 0.5 α two traffic jams Hopf (k = 1) fold (k = 1) Hopf (k > 1) fold (k > 1) stopping v amp k = 1 k = 2 0.4 0.3 0.2 k = 3 k = 4 0.1 collision h 0 0 1 2 3 h 4 REW, Krauskopf and Orosz, also group of Gasser Large perturbations (lane changes at merges?) cause jump to jammed state Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.19/25
Search for new dynamics This explanation still requires uniform flow to be unstable in some parameter regime. Is a fix possible? Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.20/25
Search for new dynamics This explanation still requires uniform flow to be unstable in some parameter regime. Is a fix possible? Design bifurcation diagram: stable jam norm unstable jam always stable uniform flow headway Ongoing work v n = α(ḣn)f (V (h n ) v n ) Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.20/25
Alternative: travelling wave analysis Computationally wasteful (and perhaps inappropriate) to analyse wave structures via bifurcations of periodic orbits of large systems of ODEs/DDEs Instead: travelling wave analysis. Two methods: Weakly nonlinear continuum limit (Kim, Lee, Lee): ρ t + (ρv) x = 0, see TGF 01 } [ v t + vv x = α {ˆV (ρ) v + α ˆV (ρ) ρ x 2ρ + v ] xx 6ρ 2, Single advance/delay equation, derived from h n 1 (t) = h n (t + τ), v n 1 (t) = v n (t + τ) substitution in car-following model (ongoing work with Tony Humphries, McGill) Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.21/25
PSfrag Travelling wave phase diagram See TGF 01 R C L C R C L C L C R C R C R C CR R C LR LC RC ρ + L C R C L C R C L C L C L C L C CL RL L C R C L C R C R C R C ρ Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.22/25
Recent discrete computation (stable) 4 Solutions on (h,h + ) plane, τ d =0 α=2.2 3.5 3 2.5 h + 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 h Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.23/25
Recent discrete computation (unstable) 4 Solutions on (h,h + ) plane, τ d =0 α=1 3.5 3 2.5 h + 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 h Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.24/25
Broad conclusions For the car-following community: Still some work to do in understanding fully pattern mechanisms at the nonlinear level and on the infinite line. Fitting models to new sources of microsopic data. For the PDE community: Vanilla versions of LWR/ARZ/Lebacque do not qualitatively replicate data or what car-following models do generically (even at the linear level). This needs a fix NB global existence results will become ugly / difficult. Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation p.25/25