Complex Traffic Dynamics on Freeways and in Urban Road ... · di Fisica af !w g Divn-l moscopaf!c...
Transcript of Complex Traffic Dynamics on Freeways and in Urban Road ... · di Fisica af !w g Divn-l moscopaf!c...
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Prof. Dr. rer. nat. Dirk Helbing, ETH Zurich, Computational Social Science with Anders Johansson, Martin Treiber, Arne Kesting, Stefan Lämmer, Martin Schönhof, and others
Complex Traffic Dynamics on Freeways and in Urban Road Networks
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§ Empirical findings § Time-dependent phenomena
§ Traffic flow modeling § Phase diagram of congested traffic states
§ Overcoming freeway congestion § Traffic light control
§ Further reading
Content
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§ Traffic sticks to lanes (quasi 1-dimensional) § Phase changes to stop-and-go traffic etc. § Behaves very different in different countries § In some countries, traffic is self-organized, here we have
top-down control § Traffic congestion during peak hours § Safety distance depends on speeds and relative velocities
Discussion: What Do You Know About Traffic?
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§ What’s the most efficient way to organize traffic? § Why the hype about roundabouts? § What’s the safest way to organize traffic? § What rules are universal, which ones are country-
specific? § How is environmental pollution related to traffic? § How will car2car communication change the operation of
traffic? § Do self-driving cars behave more rational than humans? § How resilient is traffic to the percentage of “bad drivers”?
Discussion: What Else Might Be Interesting?
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Counter-Intuitive Behavior of Traffic Flow
Perturbing traffic flows and, paradoxically, even decreasing them may sometimes cause congestion.
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Analytical Theory of Traffic Flows
OCT,SGW
FT
Q
(v
eh
icle
s/h
/la
ne
)up
!Q (vehicles/h/lane)
HCT
=Qm
ax
Qtot
!Q
=Q
-Q
c3out
Small Perturbation
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Speed-Density Relationship
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140 160
Density (veh/km/lane)
Spe
ed (k
m/h
)
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Flow-Density Relationship
0
500
1000
1500
2000
2500
3000
0 10 20 30 40 50 60 70 80 90 100
Density (veh/km/lane)
Vehi
cle
flow
(veh
/h/la
ne)
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Speed and Density as a Function of Time
0102030405060708090100110120
6:30 7:00 7:30 8:00 8:30 9:00 9:30 10:00
Speed (km/h)
Density (veh./km/lane)
Time (h)
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Time Gap Distribution
00.010.020.030.040.050.06
0 2 4 6 8 10 12
Rela
tive
Freq
uenc
y
(a)
Δt (s)
ρ=10-20 veh./km
left (f)right (f)both (f)
00.010.020.030.040.050.060.070.080.09
0 2 4 6 8 10 12Δt (s)
ρ=20-30 veh./km
(b)
freecongested
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Oscillatory Adaptation of Relative Speed and Distance
0
10
20
30
40
50
60
-4 -3 -2 -1 0 1 2 3 4 5
s (m
)
Δv (m/s)
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Wide Scattering of Congested Traffic Flows
0
500
1000
1500
2000
2500
3000
0 20 40 60 80 100 120 140 160
Density (veh/km/lane)
Flow
(veh
./h/la
ne)
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Capacity Drop
0500
10001500200025003000
0 20 40 60 80 100
Traf
fic F
low
Q (v
eh./h
/lane
)
Density ρ (veh./km/lane)
(a) Capacity Drop
ρcr
Qcr Left LaneρVsep
0
500
1000
1500
2000
2500
6:00 7:00 8:00 9:00 10:00 11:00 12:00
Traf
fic F
low
Q (v
eh./h
)
t (h)
(b)Discharge Flows
Right LaneLeft (Fast) Lane
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Elementary Traffic Patterns: Examples of Pinned Localized Clusters
The congested area remains localized as long as it does not reach the upstream end of the off-ramp.
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Wide Moving Jams (by B.S. Kerner et al.)
V (km/h)
I1I2
I3
I1I2
I3
I1I2
I3
TimeTime
(a) (b)
Moving Jams Moving JamsFree Flow
Synchroni-zed Flow
Q (v
eh./h
)
Location (km)
Location (km)
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Elementary Traffic Patterns: Homogeneous Congested Traffic
Homogeneous congested traffic requires large bottleneck strengths exceeding about 700 vehicles per kilometer and lane, which usually occur after serious accidents only. That is, if cases of accidents are excluded from the data set, one cannot find homogeneous congested traffic (if street capacities are properly dimensioned).
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The underlying dynamics of this transition is a “boomerang effect”
The boomerang effect was observed in 18 out of 245 cases of traffic breakdowns.
Transitions from Free to Congested Traffic
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§ “Traffic is a fluid medium.” § Relevant variables:
§ Conservation of vehicles § Continuity equation:
Fluid-Dynamic Traffic Equations
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§ “Flow Q and density ρ are empirically correlated.”
§ Fundamental diagram Qe(ρ)
Specification of the Flow as a Function of Density
Lighthill-Whitham(-Richards) (LWR) model (describes a conservation of the number of vehicles)
derivative plays an important role
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Kinematic Waves
0
2
4
6
8
10
0 2 4 6 8 10
05
10 0
5
10
0
50
100
Density
Location Time
Loca
tion
Time
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Fundamental Diagram
0500
10001500200025003000
0 20 40 60 80 100 120 140 160Equi
libriu
m F
low
(veh
./h)
Density (vehicles/km)
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Speed-Distance Relationship in the Bando-Sugiyama Car-Following Model
v0
dd0
1/ρ( jam , 0)
1/ρ v0( ),out
00
v
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Unstable and Metastable Traffic States
! ! !
"!c
(b)
(a)
(c)
!c1 !c2 !c3 !c4 !max
!c1 !c2 !c3 !c4 !max
!c1 !c2 !c3 !c4 !max
#
#
#
"!(x) "!(x)
x x0 0
Diagram of states of traffic flow
States of homogeneous traffic flow:
Types of structures in traffic flow:Complex sequences of traffic
jams or of "dipole-layers"
Sta-ble
State
Sta-ble
State
"Metastable" State
"Metastable" State
Unstable State
Localized Structures
Nonlocalized Structures
"Dipole Layers"
Traffic Jams
xx x
Source: B.S. Kerner
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Breakdown of Traffic due to a Supercritical Reduction of Traffic Flow
Perturbing traffic flows and, paradoxically, even decreasing them may sometimes cause congestion.
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Congested Traffic States Simulated with a Macroscopic Traffic Model
Phys. Rev. Lett. 82, 4360 (1999).
Similar congested traffic states are found for several other traffic models, including “microscopic” car-following models.
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Schematic Instability Diagram
Grey areas = metastable regimes, where result depends on perturbation amplitude
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Theoretical Phase Diagram of Traffic States and Universality Classes
Phase diagrams are not only important to understand the conditions under which certain traffic states emerge. They also allow one to categorize the more than 100 traffic models into different universality classes. From the universality class that reproduces the empirically observed stylized facts, one may choose any representative, e.g. the simplest or the most accurate one, depending on the purpose.
OCT,SGW
FT
Q
(v
eh
icle
s/h
/la
ne
)up
!Q (vehicles/h/lane)
HCT
=Qm
ax
Qtot
!Q
=Q
-Q
c3out
Small Perturbation
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The Outflow Depends on the Weather Conditions
1400
1500
1600
1700
1800
1900
2000
0 10 20 30 40
Out
flow
Qou
t (ve
h/h/
lane
)
Range of Visibility (km)
D
DD
DD
D D
D
D
DD
D
D
DD
D
DW
W
(w)W
W
W
(w)(w)
(w)(w)
(w)
W
W
WW
W: wet road(w): affected by showers
D: dry road
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Empirical Phase Diagram for Scaled Flows
A scaling by the outflow, that varies from day to day, gives a clearer picture.
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“General Pattern” and “Pinch Effect”
According to Kerner, in the “generalized pattern”, synchronized traffic upstream of a bottleneck breads wide moving jams based on the “pinch effect”. That is, upstream of a section with “synchronized” congested traffic close to a bottleneck, a so-called “pinch region” gives spontaneously birth to narrow vehicle clusters. These perturbations should be growing while traveling further upstream. Eventually, wide moving jams form by the merging or disappearance of narrow jams. Once formed, wide jams suppress the occurrence of new narrow jams in between.
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(Intermittent) Activation of an Off-Ramp Bottleneck
Milder form of congested traffic upstream of off-ramp expected, e.g. OCT or SGW instead of HCT. Appearance would correspond to “generalized pattern”.
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Wide Scattering as Effect of Heterogeneous Traffic
The jam line with variable parameters can explain the observations quantitatively! Scattering and stochasticity do not contradict models with a fundamental diagram, just models with identical driver-vehicle units.
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“Synchronized Traffic” Considering Cars+Trucks
Homogeneous-in-speed states
Time series Fundamental diagram
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Homogeneous-In-Speed States
0500
1000150020002500300035004000
0 20 40 60 80 100
Traf
fic F
low
Q (v
eh./h
/lane
)
Density l (veh./km/lane)
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An Analytical !eory of Tra"c Flow
The European Physical Journal
A selection of articles by Dirk Helbing reprinted from The European Physical Journal B
!e European Physical Journal B
A selection of articles by Dirk Helbing
your physics journal
EPJ .org
Printed on acid free paper
Società Italianadi Fisica
An Analytical Theory of Traf!c Flow
D. HelbingDerivation of non-local macroscopic traf!c equations and consistent traf!c pressures from microscopic car-following modelsDOI: 10.1140/epjb/e2009-00192-5
D. Helbing and A.F. JohanssonOn the controversy around Daganzo's requiem for and Aw-Rascle's resurrection of second-order traf!c "ow modelsDOI: 10.1140/epjb/e2009-00182-7
D. Helbing and M. MoussaidAnalytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis of a simple traf!c "ow modelDOI: 10.1140/epjb/e2009-00042-6
D. Helbing, M. Treiber, A. Kesting and M. SchönhofTheoretical vs. empirical classi!cation and prediction of congested traf!c statesDOI: 10.1140/epjb/e2009-00140-5
M. Treiber and D. HelbingHamilton-like statistics in one dimensional driven dissipative many-particle systemsDOI: 10.1140/epjb/e2009-00121-8
D. Helbing and B. TilchA power law for the duration of high-"ow states and its interpretation from a heterogeneous traf!c "ow perspectiveDOI: 10.1140/epjb/e2009-00092-8
D. HelbingDerivation of a fundamental diagram for urban traf!c "owDOI: 10.1140/epjb/e2009-00093-7
D. Helbing and A. MazloumianOperation regimes and slower-is-faster effect in the control of traf!c intersectionsDOI: 10.1140/epjb/e2009-00213-5
A selection
of articles by D
irk Helb
ing rep
rinted
from T
he E
uropean
Physical Journ
al B
Traffic Physics
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Living Costs and Public Transport Determine Travel Time Budget Take Home Messages § The fundamental diagram describes the flow-density and average
speed-density relationships as well as the speed of kinematic waves and shock waves (jam fronts)
§ The understanding of the emergence of “phantom traffic jams” and other congestion patterns requires a model with delayed adaptation and unstable as well as metastable density areas
§ The instability of traffic flow comes with a capacity drop § There is a dynamic self-organization of characteristic traffic constants
such as the outflow from congested traffic and the speed of the downstream jam front
§ To derive the phase diagram of traffic states, one must also consider the characteristic constants of traffic flow, i.e. the outflow from congested traffic and dissolution speed of traffic jams
§ While the moment of traffic breakdowns is only probabilistically predictable, travel times before and after are
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Cooperative Driving Based on Autonomous Vehicle Interactions
“Attention: Traffic jam”
In: Transportation Research Record (2007)
§ On-board data acquisition („perception“)
§ Inter-vehicle communication
§ Cooperative traffic state determination (“cognition”)
§ Adaptive choice of driving strategy (“decision-making”)
§ Driver information
§ Traffic assistance (higher stability and capacity of traffic flow)
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Design of Traffic State Adaptive Cruise Control Invent-VLA: Intelligent Adaptive Cruise Control (IACC) for the avoidance of traffic breakdowns and a faster recovery from congested traffic
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Enhancing Traffic Performance by Adaptive Cruise Control
§ Traffic breakdowns delayed § Faster recovery to free traffic § High impact on travel times § Reduced fuel consumption
and emissions
ρ(veh./km/lane)
10% Equipped Vehicles
20% Equipped Vehicles
“Mechanism design”
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Jam Front Detection – Intervehicle Communication
Congested traffic
Change: Position,Time
Free traffic
Message core: Floating car (ACC)
Inter-Vehicle-Communication (V2V):
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Traffic-Adaptive Driving Strategy for ACC
Color of trajectories ≡
ACC operating state:
- Free traffic - Approaching to jam - Jam - Downstream jam front
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Statistics of Message Transmission
Distance between communicating vehicles exponentially distributed → Distributions for T1, T2, and T3, e.g.
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Zurich‘s Laudable Goals
§ Better mobility § Less pollution
§ Less noise § Less traffic
The Noble Goals of Traffic Planning
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Adaptive Traffic Light Control § for complex street networks § for traffic disruptions (building sites, accidents, etc.) § for particular events (Olympic games, pop concerts, etc.)
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§ Directed links are homogenous road sections § Traffic dynamics: congestion, queues
§ Nodes are connectors between road sections § Junctions: merging, diverging
§ Intersections § Traffic lights: control, optimization
§ Traffic assignment § Route choice, destination flows
Road Network as Directed Graph
Local Rules, Decentralization, Self organization
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§ Propagation velocity c(ρ)
§ Free traffic: § c(ρ) = V0
§ Congested traffic: § c(ρ) ≈ -15 km/h (universal)
Triangular Flow-Density Diagram
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A Queueing-Theoretical Traffic Model The continuity equation for the vehicle density ρ(x,t) at place x and time t in road section i is
Terms Source =∂
∂+∂
∂xtxQ
ttx i ),(),(ρ
We assume the fundamental flow-density relation
⎪⎩
⎪⎨⎧
−=<=
=otherwise if
jamcong
crfree
TQVQ
Qi
iii /)/1()(
)()(
0
ρρρρρρρ
ρ
0iV = free speed on road section i T = safe time gap jamρ = jam density
The number Ni of vehicles in section i changes according to
)()()()()()(11 tQtQtQtQtQ
dttdN
iiiiii deprampdepdeparr −+=−= −−
arriQ = arrival rate of vehicles depiQ = departure rate of vehicles
Treatment of ramp flows at downstream section ends: )()()(1 tQtQtQ iii
rampdeparr +=+
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A Queueing-Theoretical Traffic Model The traffic-state dependent departure rate is given by
⎪⎩
⎪⎨
⎧
=−−≠≠=≠−
=
++
+
+
1)( if)()(0)( and 1)( if)(0)( and 1)( if)(
)(
1rampcong
1dep
1cap
1freearr
dep
tStQTtQtStStQtStSTtQ
tQ
iiii
iii
iiii
i
The capacity of congested road section is: ]0),(,)(),(max[)()( 1 tQQIItQtQItQ iiiiii Δ−−= + out
rampout
cap
Ii = number of lanes outQ = (1- ρcr/ρjam)/T = outflow per lane from congested traffic
The maximum capacity in free traffic is: ]0),(,)(),(max[)( 0
10 tQVIItQVItQ iiiiiiii Δ−−= + cr
rampcr
max ρρ
Definition of free (Si = 0), fully congested (Si = 1) and partially congested (Si = 2) traffic states:
⎪⎩
⎪⎨
⎧−≤−−=−<−−=
=otherwise2
)()( and )( if1)()( and 0)( if0
)( arrcongdep
maxfreearr
dttQTdttQLtldttQTdttQtl
tS iiiii
iiii
i
Li = length of road section i, li = length of congested road section
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A Queueing-Theoretical Traffic Model
Growth of the length li of congested traffic according to shock wave theory:
)/)/)]([(()/)/)((()/)]([()/)((
0
0
iiiiiiiiii
iiiiiii
IVtlLtQIctltQVtlLtQctltQ
dtdl
−−−−−−−−−= arrfreedepcong
arrdep
ρρ
jamcong
free
ρρρ
)1()(
,/)( 0
iii
iiii
TQQVQQ−=
=
with densities
Delay-differential equation for the travel time Ti on section i, when entered at time t:
1))(()()(1
))(()()( 11 −
++=−
+= −−
tTtQtQtQ
tTtQtQ
dttdT
ii
ii
ii
iidep
rampdep
dep
arr
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§ Movement of congestion
§ Number of vehicles
§ Travel time
Network Links: Homogenous Road Sections
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§ Side Conditions
§ Conservation
§ Non-negativity
§ Upper boundary § Branching
§ Goal function
Network Nodes: Connectors
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§ 1 to 1:
§ 1 to n: Diverging with branch weight αij
§ n to 1: Merging
Network Nodes: Special Cases
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§ Traffic light § Additional side condition
§ Intersection § Is only defined by a set of
mutually excluding traffic lights § Each intersection point gives
one more side condition
Intersections: Modelling
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Interdependence of Subsequent Intersections
Green wave
Non-optimal phase shift
Different cycle times
Stochastic switching
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Self-Organized Oscillations at Bottlenecks and Synchronization § Pressure-oriented, autonomous,
distributed signal control: § Major serving direction alternates, as
in pedestrian flows at intersections § Irregular oscillations, but
‘synchronized’ § In huge street networks:
§ ‘Synchronization’ of traffic lights due to vehicle streams spreads over large areas
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Travel time minimization, “homo economicus”
Central control, “benevolent dictator”
Same, but other-regarding coordination with neighbors
Comparing 3 Ways to Organize a Complex System
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Top-down regulation
Clearing longest queue
Stefan Lämmer and Dirk Helbing
Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control
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Top-down regulation
Clearing longest queue
Selfish optimization,
self-organization
Stefan Lämmer and Dirk Helbing
Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control
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Top-down regulation
Clearing longest queue
Selfish optimization,
self-organization
Adam Smith’s invisible hand works
Stefan Lämmer and Dirk Helbing
Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control
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Top-down regulation
Clearing longest queue
Selfish optimization,
self-organization
Stefan Lämmer and Dirk Helbing
Adam Smith’s invisible hand fails
Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control
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Top-down regulation
Clearing longest queue
Selfish optimization,
self-organization
Other-regarding optimization,
self-regulation
Stefan Lämmer and Dirk Helbing
Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control
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Top-down regulation
Clearing longest queue
Selfish optimization,
self-organization
Other-regarding optimization,
self-regulation
Stefan Lämmer and Dirk Helbing
Invisible hand works
Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control
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§ Advantage of centralized control is large-scale coordination
§ Disadvantages are due to § vulnerability of the network
§ information overload
§ wrong selection of control parameters
§ delays in adaptive feedback control
§ Decentralized control can perform better in complex systems with heterogeneous elements, large degree of fluctuations, and short-term predictability, because of greater flexibility to local conditions and greater robustness to perturbations
Centralized Control and Its Limits
Leve
l of g
oal
achi
evem
ent
Level of autonomy of control
(Windt, Böse, Philipp, 2006)
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Living Costs and Public Transport Determine Travel Time Budget Take Home Messages § Traffic flow can be improved using self-organization
concepts based on local interactions § Traffic assistance systems, based on adaptive cruise
control (ACC) systems using radar sensors, reduce time gaps and increase acceleration in a traffic-state-dependent way to increase capacity and stabilize the flow
§ Self-organized oscillations of pedestrian flows at bottlenecks have inspired the principle of a self-organizing traffic light control (self-control, self-regulation)
§ The self-control combines local minimization of waiting times at each intersection with a immediate clearing of queues, which would otherwise create spill-over effects
§ Self-organization outperforms top-down control in case of largely variable, hardly predictable complex systems
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§ D. Helbing (2001) Traffic and related self-driven many-particle systems. Reviews of Modern Physics 73, 1067-1141.
§ D. Helbing and M. Schreckenberg (1999) Cellular automata simulating experimental properties of traffic flows. Physical Review E 59, R2505-R2508.
§ D. Helbing, A. Hennecke, and M. Treiber (1999) Phase diagram of traffic states in the presence of inhomogeneities. Physical Review Letters 82, 4360-4363.
§ D. Helbing (2005) Traffic flow. In: Alwyn Scott (ed.) Encyclopedia of Nonlinear Science (Routledge, New York).
Further Reading
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§ D. Helbing and K. Nagel (2004) The physics of traffic and regional development. Contemporary Physics 45(5), 405-426.
§ D. Helbing, A. Hennecke, V. Shvetsov, and M. Treiber (2002) Micro- and macro-simulation of freeway traffic. Mathematical and Computer Modelling 35(5/6), 517-547.
§ D. Helbing (2001) Traffic and related self-driven many-particle systems. Reviews of Modern Physics 73, 1067-1141.
Further Reading
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§ D. Helbing, A. Hennecke, V. Shvetsov, and M. Treiber (2001) MASTER: Macroscopic traffic simulation based on a gas-kinetic, non-local traffic model. Transportation Research B 35, 183-211.
§ C. Leduc, K. Padberg-Gehle, V. Varga, D.Helbing, S. Diez, and J. Howard (2012) Molecular crowding creates traffic jams of kinesin motors on microtubules. PNAS 16, 6100-6105
§ D. Helbing and A. Mazloumian (2009) Operation regimes and slower-is-faster effect in the control of traffic intersections. European Physical Journal B 70(2), 257–274.
Further Reading
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§ D. Helbing (2009) Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models. European Physical Journal B 69(4), 539-548.
§ D. Helbing and A. Johansson (2009) On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models. European Physical Journal B 69(4), 549-562.
§ D. Helbing and M. Moussaid (2009) Analytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis of a simple traffic flow model. European Physical Journal B 69(4), 571-581.
Further Reading
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§ D. Helbing, M. Treiber, A. Kesting, and M. Schönhof (2009) Theoretical vs. empirical classification and prediction of congested traffic states. European Physical Journal B 69(4), 583-598.
§ D. Helbing and B. Tilch (2009) A power law for the duration of high-flow states and its interpretation from a heterogeneous traffic flow perspective. European Physical Journal B 68(4), 577-586.
§ M. Treiber and D. Helbing (2009) Hamilton-like statistics in onedimensional driven dissipative many-particle systems. European Physical Journal B 68(4), 607-618.
Further Reading
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§ D. Helbing (2009) Derivation of a fundamental diagram for urban traffic flow. European Physical Journal B 70(2), 229-241.
§ S. Lämmer and D. Helbing (2008) Self-control of traffic lights and vehicle flows in urban road networks. JSTAT P04019.
§ M. Treiber, A. Kesting, and D. Helbing (2007) Influence of reaction times and anticipation on stability of vehicular traffic flow. Transportation Research Record 1999, 23-29.
§ D. Helbing, J. Siegmeier, and S. Lämmer (2007) Self-organized network flows. Networks and Heterogeneous Media 2(2), 193-210.
Further Reading
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§ M. Krbalek and D. Helbing (2004) Determination of interaction potentials in freeway traffic from steady-state statistics. Physica A 333, 370-378.
§ D. Helbing (2003) A section-based queueing-theoretical traffic model for congestion and travel time analysis in networks. Journal of Physics A: Mathematical and General 36, L593-L598.
§ R. Kölbl and D. Helbing (2003) Energy laws in human travel behaviour. New Journal of Physics 5, 48.1-48.12.
§ V. Shvetsov and D. Helbing (1999) Macroscopic dynamics of multilane traffic. Physical Review E 59, 6328-6339.
Further Reading
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§ M. Treiber, A. Hennecke, and D. Helbing (1999) Derivation, properties, and simulation of a gas-kinetic-based, non-local traffic model. Physical Review E 59, 239-253.
§ D. Helbing (1996) Gas-kinetic derivation of Navier-Stokes-like traffic equations. Physical Review E 53, 2366-2381.
Further Reading