Traffic Flow Theory
& Simulation
S.P. Hoogendoorn
Lecture 1
Introduction
1Course 4821 - Introduction | 57
Photo by Wikipedia / CC BY SA
Photo by wikipedia / CC BY SA
2/4/12
Challenge the future
Delft
University of
Technology
Traffic Flow Theory & Simulation
An Introduction
Prof. Dr. Serg...
3Course 4821 - Introduction | 57
Introduction
  120.000 people to evacuate
  Evacuation time = 6 hour
  2,5 evacuees / car...
4Course 4821 - Introduction | 57
Introduction
  120.000 people to evacuate
  Evacuation time = 6 hour
  2,5 evacuees / car...
5Course 4821 - Introduction | 57
Introduction
Evacuation Walcheren in case of Flooding
6Course 4821 - Introduction | 57
Introduction
  It turns out that in simulation only 41.000 people survive!
  Consider ave...
7Course 4821 - Introduction | 57
Network traffic flow fundamentals
  Fundamental relation between network outflow (rate at...
8Course 4821 - Introduction | 57
Network traffic flow fundamentals
  Fundamental relation between network outflow (rate at...
9Course 4821 - Introduction | 57
Network traffic flow fundamentals
Demand and performance degradation
10Course 4821 - Introduction | 57
Network Performance Deterioration
  Important characteristic of traffic networks:
  Netw...
11Course 4821 - Introduction | 57
Introduction
  Traffic queuing phenomena: examples and empirics
  Modeling traffic conge...
12Course 4821 - Introduction | 57
1.
Traffic Congestion Phenomena
Empirical Features of Traffic Congestion
13Course 4821 - Introduction | 57
Historical perspective
Bruce Greenshields
Source: Unknown
14Course 4821 - Introduction | 57
First model of traffic congestion
  Relation between traffic density and traffic speed: ...
15Course 4821 - Introduction | 57
Fundamental diagrams
Different representations using q = k×u
q = Q k( )
= kU(k)
( )u U k...
16Course 4821 - Introduction | 57
Dynamic properties
Traffic congestion at bottleneck (on-ramp)
  Consider bottleneck due ...
17Course 4821 - Introduction | 57
Dynamic properties
Shockwave theory
  Predicting queue dynamics (queuing models, shockwa...
18Course 4821 - Introduction | 57
Dynamic features of road congestion
Capacity funnel, instability, wide moving jams
  Cap...
19Course 4821 - Introduction | 57
locatie(km)
tijd (u)
7 7.5 8 8.5 9 9.5
36
38
40
42
0
50
100
5
7
8
9
Capacity drop
Two ca...
20Course 4821 - Introduction | 57
Instability and wide moving jams
  In certain density regimes, traffic is highly unstabl...
21Course 4821 - Introduction | 57
Instability and wide moving jams
  Wide moving jams can exist for
hours and travel past ...
22Course 4821 - Introduction | 57
Pedestrian flow congestion
  Example of Jamarat bridge shows self-organized stop-go wave...
23Course 4821 - Introduction | 57
Pedestrian flow congestion
  Another wave example..
Start-stop waves in pedestrian flow
24Course 4821 - Introduction | 57
2.
Traffic Flow Modeling
Microscopic and macroscopic approaches to
describe flow dynamics
25Course 4821 - Introduction | 57
Modeling challenge
  Traffic flow is a result of human decision making and multi-actor
i...
26Course 4821 - Introduction | 57
Network Traffic Modeling
  Traffic conditions on the
road are end result of many
decisio...
27Course 4821 - Introduction | 57
Modeling approaches
  Two dimensions:
  Representation of traffic
  Behavioral rules, fl...
28Course 4821 - Introduction | 57
Modeling approaches
Microscopic and macroscopic approaches
Individual particles Continuu...
29Course 4821 - Introduction | 57
Microscopic models?
  Example of advanced micro-simulation model
Ability to described ma...
30Course 4821 - Introduction | 57
Modeling approaches
Microscopic and macroscopic approaches
Individual particles Continuu...
31Course 4821 - Introduction | 57
Modeling approaches
Microscopic and macroscopic approaches
Individual particles Continuu...
32Course 4821 - Introduction | 57
Modeling approaches
Microscopic and macroscopic approaches
Individual particles Continuu...
33Course 4821 - Introduction | 57
3.
Example applications of theory and
models
34Course 4821 - Introduction | 57
Application of models
  Ex-ante studies: systematic comparison of alternatives during
di...
35Course 4821 - Introduction | 57
Application of models
  NOMAD model has been extensively
calibrated and validated
  NOMA...
36Course 4821 - Introduction | 57
Traffic State Estimation & Prediction
Applications of Kalman filters
Photo by fileradar....
37Course 4821 - Introduction | 57
Dynamic speed limits
  Algorithm ‘Specialist’ to suppress
start-stop waves on A12
  Appr...
38Course 4821 - Introduction | 57
4.
Course scope and overview
39Course 4821 - Introduction | 57
Scope of course CT4821
  Operational characteristics of traffic, so not:
  Activity choi...
40Course 4821 - Introduction | 57
Overview of flow variables (chapter 2)
Microscopic variables
(individual vehicles)
Macro...
41Course 4821 - Introduction | 57
Flow characteristics
  Microscopic characteristics (chapter 3)
  Arrival processes
  Hea...
42Course 4821 - Introduction | 57
Analytical tools
  Use fundamental knowledge for mathematical / numerical analysis
  Exa...
43Course 4821 - Introduction | 57
Course ct4821
  Lectures by Serge Hoogendoorn and Victor Knoop:
  Monday 8:45 – 10:30
  ...
44Course 4821 - Introduction | 57
5.
Traffic Flow Variables
45Course 4821 - Introduction | 57
Vehicle trajectories
  Positions xi(t) along roadway
of vehicle i at time t
  All micros...
46Course 4821 - Introduction | 57
Vehicle trajectories (2)
  Which are trajectories?
t
x (a) (b)
(c)
t0
47Course 4821 - Introduction | 57
Empirical vehicle trajectories
340 350 360 370 380 390 400 410
-450
-400
-350
-300
-250
...
48Course 4821 - Introduction | 57
Understanding trajectories
( ) ( )i i
d
v t x t
dt
=
( ) ( )
2
2i i
d
a t x t
dt
=
( ) (...
49Course 4821 - Introduction | 57
Applications of multiple trajectories
  Exercise with trajectories by drawing a number o...
50Course 4821 - Introduction | 57
Time headways
  Time headway hi: passage
time difference rear bumper
vehicle i-1 and i a...
51Course 4821 - Introduction | 57
Time headways (3)
  Headways are local variables (collected at cross-section x0)
  Mean ...
52Course 4821 - Introduction | 57
Distance headways
  Distance headway si:
difference between positions
vehicle i-1 and i ...
53Course 4821 - Introduction | 57
Distance headways (2)
  Distance headway: instantaneous microscopic variable
  Space-mea...
54Course 4821 - Introduction | 57
Distance headways (3)
55Course 4821 - Introduction | 57
Local, instantaneous and generalized
Local measurements
collected at x = 200
Instantaneo...
56Course 4821 - Introduction | 57
Macroscopic flow variables
  Traffic intensity q (local variable)
  Vehicle number passi...
57Course 4821 - Introduction | 57
Macroscopic flow variables (2)
  Traffic density k (instantaneous variable)
  Vehicle nu...
58Course 4821 - Introduction | 57
Mean speeds
59Course 4821 - Introduction | 57
Mean speeds
  Local mean speed / time mean speed (see next slide)
  speeds vi of vehicle...
60Course 4821 - Introduction | 57
Homogeneous & stationary variables
  Consider any variable z(t,x); z is:
  Stationary if...
61Course 4821 - Introduction | 57
Stationary flow conditions
62Course 4821 - Introduction | 57
Homogeneous flow conditions
63Course 4821 - Introduction | 57
Fundamental relation
  Consider traffic flow that is in stationary and homogeneous
  The...
64Course 4821 - Introduction | 57
Which speed to use in q = k×u?
  Can we apply the fundamental relation q = ku for an
het...
65Course 4821 - Introduction | 57
Which speed to use in q = k×u?
  Can we apply the fundamental relation q = ku for an
het...
66Course 4821 - Introduction | 57
Difference arithmetic & space mean
  Example motorway data time-mean speed and space-mea...
67Course 4821 - Introduction | 57
What about the derived densities?
  Suppose we derive densities by k = q/u…
0 50 100 150...
68Course 4821 - Introduction | 57
Conclusion average speeds…
  Care has to be taken when using the fundamental relation q ...
69Course 4821 - Introduction | 57
Generalized definitions of flow (Edie)
  Generalized definition flow, mean speed, and de...
70Course 4821 - Introduction | 57
Generalized definition (Edie)
  Each vehicle j is present in rectangle for some period r...
71Course 4821 - Introduction | 57
Overview of variables
Local
measurements
Instantaneous
measurements
Generalized
definiti...
72Course 4821 - Introduction | 57
Furthermore...
  Homework:
  Read through preface and chapter 1
  Study remainder of cha...
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1. introduction

  1. 1. Traffic Flow Theory & Simulation S.P. Hoogendoorn Lecture 1 Introduction
  2. 2. 1Course 4821 - Introduction | 57 Photo by Wikipedia / CC BY SA Photo by wikipedia / CC BY SA
  3. 3. 2/4/12 Challenge the future Delft University of Technology Traffic Flow Theory & Simulation An Introduction Prof. Dr. Serge P. Hoogendoorn, Delft University of Technology Photo by wikipedia / CC BY SA
  4. 4. 3Course 4821 - Introduction | 57 Introduction   120.000 people to evacuate   Evacuation time = 6 hour   2,5 evacuees / car   A58: 2 lanes   N57, N254: 1 lane   Total available capacity?   Each lane about 2000 veh/h   Total capacity 8000 veh/h   How to calculate evacuation time Evacuation Walcheren in case of Flooding
  5. 5. 4Course 4821 - Introduction | 57 Introduction   120.000 people to evacuate   Evacuation time = 6 hour   2,5 evacuees / car   A58: 2 lanes   N57, N254: 1 lane   Total available capacity?   Each lane about 2000 veh/h   Total capacity 8000 veh/h   How to calculate evacuation time Evacuation Walcheren in case of Flooding 48000 8000 veh/h
  6. 6. 5Course 4821 - Introduction | 57 Introduction Evacuation Walcheren in case of Flooding
  7. 7. 6Course 4821 - Introduction | 57 Introduction   It turns out that in simulation only 41.000 people survive!   Consider average relation between number of vehicles in network (accumulation) and performance (number of vehicles completing their trip)   How does average performance (throughput, outflow) relate to accumulation of vehicles?   What would you expect based on analogy with other networks?   Think of a water pipe system where you increase water pressure   What happens? Network load and performance degradation
  8. 8. 7Course 4821 - Introduction | 57 Network traffic flow fundamentals   Fundamental relation between network outflow (rate at which trip end) and accumulation Coarse model of network dynamics Number of vehicles in network Network outflow 3. Outflow reduces 1. Outflow increases 2. Outflow is constant
  9. 9. 8Course 4821 - Introduction | 57 Network traffic flow fundamentals   Fundamental relation between network outflow (rate at which trip end) and accumulation Coarse model of network dynamics Number of vehicles in network Network outflow 3. Outflow reduces 1. Outflow increases 2. Outflow is constant
  10. 10. 9Course 4821 - Introduction | 57 Network traffic flow fundamentals Demand and performance degradation
  11. 11. 10Course 4821 - Introduction | 57 Network Performance Deterioration   Important characteristic of traffic networks:   Network production degenerates as number of vehicles surpasses the critical number of vehicles in the network   Expressed by the Macroscopic (or Network) Fundamental Diagram # vehicles in network (at a specific time) Production Actual production Two (and only two!) causes: 1. Capacity drop phenomenon 2. Blockages, congestion spillback and grid-lock
  12. 12. 11Course 4821 - Introduction | 57 Introduction   Traffic queuing phenomena: examples and empirics   Modeling traffic congestion in road networks   Model components of network models   Modeling principles and paradigms   Examples and case studies   Model application examples   Traffic State Estimation and Prediction   Controlling congestion waves   Microscopic and macroscopic perspectives! Lecture overview
  13. 13. 12Course 4821 - Introduction | 57 1. Traffic Congestion Phenomena Empirical Features of Traffic Congestion
  14. 14. 13Course 4821 - Introduction | 57 Historical perspective Bruce Greenshields Source: Unknown
  15. 15. 14Course 4821 - Introduction | 57 First model of traffic congestion   Relation between traffic density and traffic speed: u = U(k)   Underlying behavioral principles? (density = 1/average distance) Fundamental diagram u = U(k) = u0 1 − k kjam ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟
  16. 16. 15Course 4821 - Introduction | 57 Fundamental diagrams Different representations using q = k×u q = Q k( ) = kU(k) ( )u U k= ( )u U q=
  17. 17. 16Course 4821 - Introduction | 57 Dynamic properties Traffic congestion at bottleneck (on-ramp)   Consider bottleneck due to on-ramp   Resulting capacity (capacity – ramp flow) is lower than demand   Queue occurs upstream of bottleneck and moves upstream as long as upstream demand > flow in queue (shockwave theory) locatie(km) tijd (u) 7 7.5 8 8.5 9 9.5 36 38 40 42 0 50 100 5 7 8 9 Driving direction Upstream traffic demand
  18. 18. 17Course 4821 - Introduction | 57 Dynamic properties Shockwave theory   Predicting queue dynamics (queuing models, shockwave theory)   Predicts dynamics of congestion using FD   Flow in queue = C – qon-ramp   Shock speed determined by: locatie(km) tijd (u) 7 7.5 8 8.5 9 9.5 36 38 40 42 0 50 100 5 7 8 9 Driving direction ω12 = Q(k2 ) − Q(k1 ) k2 − k1 Congestion as predicted by shockwave theory C − qon−ramp Upstream traffic demand qupstream
  19. 19. 18Course 4821 - Introduction | 57 Dynamic features of road congestion Capacity funnel, instability, wide moving jams   Capacity funnel (relaxation) and capacity drop   Self-organisation of wide moving jams
  20. 20. 19Course 4821 - Introduction | 57 locatie(km) tijd (u) 7 7.5 8 8.5 9 9.5 36 38 40 42 0 50 100 5 7 8 9 Capacity drop Two capacities   Free flow cap > queue-discharge rate   Use of (slanted cumulative curves) clearly reveals this   N(t,x) = #vehicles passing x until t   Slope = flow 7.5 8 8.5 9 9.5 −4400 −4300 −4200 −4100 −4000 −3900 −3800 tijd (u) N(t,x) q0 = 3700 7 8 9 ! !N (t,x) = N(t,x) " q0 #t Cqueue Cfreeflow Pbreakdown k k
  21. 21. 20Course 4821 - Introduction | 57 Instability and wide moving jams   In certain density regimes, traffic is highly unstable   So called ‘wide moving jams’ (start-stop waves) self-organize frequently (1-3 minutes) in these high density regions Emergence and dynamics of start-stop waves
  22. 22. 21Course 4821 - Introduction | 57 Instability and wide moving jams   Wide moving jams can exist for hours and travel past bottlenecks   Density in wide moving jam is very high (jam-density) and speed is low Emergence and dynamics of start-stop waves Start-stop golf met snelheid 18 km/u Gesynchroniseerd verkeer Stremming Stremming
  23. 23. 22Course 4821 - Introduction | 57 Pedestrian flow congestion   Example of Jamarat bridge shows self-organized stop-go waves in pedestrian traffic flows Start-stop waves in pedestrian flow Photo by wikipedia / CC BY SA
  24. 24. 23Course 4821 - Introduction | 57 Pedestrian flow congestion   Another wave example.. Start-stop waves in pedestrian flow
  25. 25. 24Course 4821 - Introduction | 57 2. Traffic Flow Modeling Microscopic and macroscopic approaches to describe flow dynamics
  26. 26. 25Course 4821 - Introduction | 57 Modeling challenge   Traffic flow is a result of human decision making and multi-actor interactions at different behavioral levels (driving, route choice, departure time choice, etc.)   Characteristics behavior (inter- and intra-driver heterogeneity)   Large diversity between driver and vehicle characteristics   Intra-driver diversity due to multitude of influencing factors, e.g. prevailing situation, context, external conditions, mood, emotions   The traffic flow theory does not exist (and will probably never exist): this is not Newtonian Physics or thermodynamics   Challenge is to develop theories and models that represent reality sufficiently accurate for the application at hand Traffic theory: not an exact science!
  27. 27. 26Course 4821 - Introduction | 57 Network Traffic Modeling   Traffic conditions on the road are end result of many decisions made by the traveler at different decision making levels   Depending on type of application different levels are in- or excluded in model   Focus on driving behavior and flow operations Model components and processes 0. Locatiekeuze 1. Ritkeuze 2. Bestemmingskeuze 3. Vervoerwijzekeuze 4. Routekeuze 5. Vertrektijdstipkeuze 6. Rijgedrag demand supply short term longer term Location choice Trip choice Destination choice Mode choice Route choice Departure time choice Driving behavior
  28. 28. 27Course 4821 - Introduction | 57 Modeling approaches   Two dimensions:   Representation of traffic   Behavioral rules, flow characteristics Microscopic and macroscopic approaches Individual particles Continuum Individual behavior Microscopic (simulation) models Gas-kinetic models (Boltzmann equations) Aggregate behavior Newell model, particle discretization models Queuing models Macroscopic flow models
  29. 29. 28Course 4821 - Introduction | 57 Modeling approaches Microscopic and macroscopic approaches Individual particles Continuum Individual behavior Microscopic (simulation) models Gas-kinetic models (Boltzmann equations) Aggregate behavior Newell model, particle discretization models Queuing models Macroscopic flow models Helly model: d dt vi (t + Tr ) = α ⋅ Δvi (t) + β ⋅ s* (vi (t)) − si (t)( )
  30. 30. 29Course 4821 - Introduction | 57 Microscopic models?   Example of advanced micro-simulation model Ability to described many flow phenomena Photo by Intechopen / CC BY
  31. 31. 30Course 4821 - Introduction | 57 Modeling approaches Microscopic and macroscopic approaches Individual particles Continuum Individual behavior Microscopic (simulation) models Gas-kinetic models (Boltzmann equations) Aggregate behavior Newell model, Bando model Queuing models Macroscopic flow models Bando model: d dt vi (t) = V(1 / si (t)) − vi (t) τ
  32. 32. 31Course 4821 - Introduction | 57 Modeling approaches Microscopic and macroscopic approaches Individual particles Continuum Individual behavior Microscopic (simulation) models Gas-kinetic models (Boltzmann equations) Aggregate behavior Newell model, particle discretization models Queuing models Macroscopic flow models kinematic wave model: ∂k ∂t + ∂q ∂x = r − s q = Q(k) ⎧ ⎨ ⎪ ⎩ ⎪
  33. 33. 32Course 4821 - Introduction | 57 Modeling approaches Microscopic and macroscopic approaches Individual particles Continuum Individual behavior Microscopic (simulation) models Gas-kinetic models (Boltzmann equations) Aggregate behavior Newell model, particle discretization models Queuing models Macroscopic flow models Prigogine-Herman model: ∂ρ ∂t + v ∂ρ ∂x + ∂ ∂v ρ Ω0 (v) − v τ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ∂ρ ∂t ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ INT
  34. 34. 33Course 4821 - Introduction | 57 3. Example applications of theory and models
  35. 35. 34Course 4821 - Introduction | 57 Application of models   Ex-ante studies: systematic comparison of alternatives during different phases of the design process   Road- and network design   Traffic Control / Management Strategies and Algorithms   Impact of traffic information   Evacuation planning   Impact of Driver Support Systems   Training and decision support for decision makers   Traffic state estimation and data fusion   Traffic state prediction   Model predictive control to optimize network utilization Different models, different applications
  36. 36. 35Course 4821 - Introduction | 57 Application of models   NOMAD model has been extensively calibrated and validated   NOMAD reproduces characteristics of pedestrian flow (fundamental diagram, self-organization)   Applications of model:   Assessing LOS transfer stations   Testing safety in case of emergency conditions (evacuations)   Testing alternative designs and Decision Support Tool   Hajj strategies and design Examples NOMAD Animatie by verkeerskunde.nl
  37. 37. 36Course 4821 - Introduction | 57 Traffic State Estimation & Prediction Applications of Kalman filters Photo by fileradar.nl
  38. 38. 37Course 4821 - Introduction | 57 Dynamic speed limits   Algorithm ‘Specialist’ to suppress start-stop waves on A12   Approach is based on reduced flow (capacity drop) downstream of wave   Reduce inflow sufficiently by speed- limits upstream of wave Using Traffic Flow Theory to improve traffic flow ! q k k x 2 2 3 3 1 1
  39. 39. 38Course 4821 - Introduction | 57 4. Course scope and overview
  40. 40. 39Course 4821 - Introduction | 57 Scope of course CT4821   Operational characteristics of traffic, so not:   Activity choice and scheduling   Route choice and destination choice   Departure time or (preferred) arrival time choice   Traffic flow theory does not exclude any transportation mode!   Primary focus in this course will be of road traffic (cars) with occasion side-step to other modes (pedestrian flows)   Distinction between   Macroscopic and microscopic (and something in the middle)   Flow variables, (descriptive) flow characteristics and analytical tools (mathematical modelling and simulation)
  41. 41. 40Course 4821 - Introduction | 57 Overview of flow variables (chapter 2) Microscopic variables (individual vehicles) Macroscopic variables (traffic flows) Local Time headway Flow / volume / intensity Local mean speed Instantaneous Distance headway Density Space mean speed Generalized Trajectory Path speed Mean path speed Mean travel time
  42. 42. 41Course 4821 - Introduction | 57 Flow characteristics   Microscopic characteristics (chapter 3)   Arrival processes   Headway models / headway distribution models   Critical gap distributions   Macroscopic characteristics (chapters 4 and 5)   Fundamental diagram   Shockwaves and non-equilibrium flow properties
  43. 43. 42Course 4821 - Introduction | 57 Analytical tools   Use fundamental knowledge for mathematical / numerical analysis   Examples macroscopic tools   Capacity analysis (chapter 6)   Deterministic and stochastic queuing models (chapter 7)   Shockwave analysis (chapter 8)   Macroscopic flow models (chapter 9)   Macroscopic simulation models (13 and 15)   Examples microscopic tools   Car-following models (chapter 11)   Gap-acceptance models (chapter 12)   Traffic simulation (chapter 13 and 14) NOMAD Animatie by verkeerskunde.nl
  44. 44. 43Course 4821 - Introduction | 57 Course ct4821   Lectures by Serge Hoogendoorn and Victor Knoop:   Monday 8:45 – 10:30   Tuesday 8:45 – 10:30   Mandatory assignment (Hoogendoorn, Knoop + PhD’s):   Wednesday 13:45 – 17:30 (starting in week 1)   New Data analysis and FOSIM practicum (week 1-7)   Reports on assignment   Description assignment posted on blackboard beginning of next week   Course material:   Parts of the reader (blackboard)   Assignments (blackboard)
  45. 45. 44Course 4821 - Introduction | 57 5. Traffic Flow Variables
  46. 46. 45Course 4821 - Introduction | 57 Vehicle trajectories   Positions xi(t) along roadway of vehicle i at time t   All microscopic and macroscopic characteristics can be determined from trajectories!   In reality, trajectory information is rarely available   Nevertheless, trajectories are the most important unit of analysis is traffic flow theory
  47. 47. 46Course 4821 - Introduction | 57 Vehicle trajectories (2)   Which are trajectories? t x (a) (b) (c) t0
  48. 48. 47Course 4821 - Introduction | 57 Empirical vehicle trajectories 340 350 360 370 380 390 400 410 -450 -400 -350 -300 -250 -200 time (s) position(m)   Vehicle trajectories determined using remote sensing data (from helicopter) at site Everdingen near Utrecht   Dots show vehicle position per 2.5 s   Unique dataset
  49. 49. 48Course 4821 - Introduction | 57 Understanding trajectories ( ) ( )i i d v t x t dt = ( ) ( ) 2 2i i d a t x t dt = ( ) ( ) 3 3i i d t x t dt γ =
  50. 50. 49Course 4821 - Introduction | 57 Applications of multiple trajectories   Exercise with trajectories by drawing a number of them for different situations   Acceleration, deceleration,   Period of constant speed,   Stopped vehicles   Etc.   See syllabus and exercises for problem solving using trajectories:   Tandem problem   Cargo ship problem   Also in reader: discussion on vehicle kinematics described acceleration ai(t) as a function of the different forces acting upon the vehicle
  51. 51. 50Course 4821 - Introduction | 57 Time headways   Time headway hi: passage time difference rear bumper vehicle i-1 and i at cross- section x (easy to measure)
  52. 52. 51Course 4821 - Introduction | 57 Time headways (3)   Headways are local variables (collected at cross-section x0)   Mean headway for certain period T of the n vehicles that have passed x0   Exercise: express the mean headway for the cross-section as a function of the mean headways H1 and H2 per lane 1 1 n ii T h h n n= = =∑
  53. 53. 52Course 4821 - Introduction | 57 Distance headways   Distance headway si: difference between positions vehicle i-1 and i at time t (difficult to measure!)   Gross distance headway and net distance headway
  54. 54. 53Course 4821 - Introduction | 57 Distance headways (2)   Distance headway: instantaneous microscopic variable   Space-mean distance headway of m vehicles at time instant t for roadway of length X 1 1 m ii X s s m m= = =∑
  55. 55. 54Course 4821 - Introduction | 57 Distance headways (3)
  56. 56. 55Course 4821 - Introduction | 57 Local, instantaneous and generalized Local measurements collected at x = 200 Instantaneous measurements collected at t = 20 Generalized measurements determined for time-space region
  57. 57. 56Course 4821 - Introduction | 57 Macroscopic flow variables   Traffic intensity q (local variable)   Vehicle number passing cross-section x0 per unit time (hour, 15 min, 5 min)   If n vehicles pass during T, q is defined by:   Referred to as flow, volume (US)   How can flow be measured? 1 1 1 1 1n n ii ii n n q T hh h n = = = = = = ∑ ∑
  58. 58. 57Course 4821 - Introduction | 57 Macroscopic flow variables (2)   Traffic density k (instantaneous variable)   Vehicle number present per unit roadway length (1 km, 1 m) at instant t   If m vehicles present on X, k is defined by   Also referred to as concentration   How can density be measured?   Now how about speeds? 1 1 1 1 1m m ii ii m m k X ss s m = = = = = = ∑ ∑
  59. 59. 58Course 4821 - Introduction | 57 Mean speeds
  60. 60. 59Course 4821 - Introduction | 57 Mean speeds   Local mean speed / time mean speed (see next slide)   speeds vi of vehicles passing a cross-section x during period T   Instantaneous / space mean speed (next slide)   speed vj of vehicles present at road section at given moment t   We can show that under special circumstances we can compute space-mean speeds from local measurements 1 1 n L ii u v n = = ∑ 1 1 m M jj u v m = = ∑ 1 1 1 1 (harmonic average local speeds) n M i i u n v − = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑
  61. 61. 60Course 4821 - Introduction | 57 Homogeneous & stationary variables   Consider any variable z(t,x); z is:   Stationary if z(t,x) = z(x)   Homogeneous if z(t,x) = z(t)
  62. 62. 61Course 4821 - Introduction | 57 Stationary flow conditions
  63. 63. 62Course 4821 - Introduction | 57 Homogeneous flow conditions
  64. 64. 63Course 4821 - Introduction | 57 Fundamental relation   Consider traffic flow that is in stationary and homogeneous   Then the so-called fundamental relation holds   Assume intensity q, density k and that all drive with speed u q ku= Number of vehicles passing cross-section at X equals the number of vehicles that is on the road at t = 0, i.e. kX = qT With T=X/u we get fundamental relation
  65. 65. 64Course 4821 - Introduction | 57 Which speed to use in q = k×u?   Can we apply the fundamental relation q = ku for an heterogeneous driver population?   Yes -> the trick is to divided the traffic stream into homogeneous groups j of drivers moving at the same speed uj   Now, what can we say about the speed that we need to use?
  66. 66. 65Course 4821 - Introduction | 57 Which speed to use in q = k×u?   Can we apply the fundamental relation q = ku for an heterogeneous driver population?   For one group j (vehicles having equals speeds) we have   The total flow q simply equals   For the total density we have   Let u = q/k, then   If we consider qj = 1, then…   In sum: the fundamental relation q = kuM may only be used for space-mean speeds (harmonic mean of individual vehicle speeds)!!! j j jq k u= j j jq q k u= =∑ ∑ jk k= ∑ ( )/ / / ⎛ ⎞ ⎜ ⎟= = = = ⎜ ⎟ ⎝ ⎠ ∑∑ ∑ ∑ ∑ ∑ j j jj j j M j j j j j q u uk u q u u k q u q u 1 1/ M j u u u = = ∑ ∑
  67. 67. 66Course 4821 - Introduction | 57 Difference arithmetic & space mean   Example motorway data time-mean speed and space-mean speed 0 20 40 60 80 100 120 0 20 40 60 80 100 120 time mean speed spacemeanspeed
  68. 68. 67Course 4821 - Introduction | 57 What about the derived densities?   Suppose we derive densities by k = q/u… 0 50 100 150 200 0 50 100 150 200 density using time mean speed densityusingspacemeanspeed
  69. 69. 68Course 4821 - Introduction | 57 Conclusion average speeds…   Care has to be taken when using the fundamental relation q = kuM that the correct average speed is used   Correct speed is to be used, but is not always available from data!   Dutch monitoring system collects average speeds, but which?   Same applies to UK and other European countries (except France)   Exercise:   Try to calculate what using the wrong mean speeds means for travel time computations
  70. 70. 69Course 4821 - Introduction | 57 Generalized definitions of flow (Edie)   Generalized definition flow, mean speed, and density in time- space plane   Consider rectangle T × X   Each vehicle i travels distance di   Define performance   P defines ‘total distance traveled’   Define generalized flow   Let X « 1, then di ≤ X ii P d= ∑ / := = ∑ id XP q XT T nX n q XT T = =
  71. 71. 70Course 4821 - Introduction | 57 Generalized definition (Edie)   Each vehicle j is present in rectangle for some period rj   Defined total travel time   Generalized definition of density   When T « 1, then rj ≤ T   Definition of the mean speed jj R r= ∑ / := = ∑ jr TR k XT X mT m k XT X = = = = = ∑ ∑ ii G jj dP q u R k r
  72. 72. 71Course 4821 - Introduction | 57 Overview of variables Local measurements Instantaneous measurements Generalized definition (Edie) Variable Cross-section x Period T Section X Time instant t Section X Period T Flow q (veh/h) Density k (veh/km) Mean speed u (km/h) 1n q T h = = q ku= ii d q XT = ∑ q k u = 1n k X s = = jj r k XT = ∑ ( )1/ L ii n u v = ∑ jj v u n = ∑ q u k =
  73. 73. 72Course 4821 - Introduction | 57 Furthermore...   Homework:   Read through preface and chapter 1   Study remainder of chapter 2 (in particular: moving observer, and observation methods)
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