Mathematics of Road Congestion Empirics, models and applications - PDF document

Mathematics of Road Congestion Empirics, models and applications Prof. Dr, Serge P. Hoogendoorn, Delft University of Technology 8/25/09 Technology Delft University of Challenge the future Introduction Network load and performance degradation • Consider average relation between number of vehicles in network (accumulation) and performance (number of vehicles flowing out of the network) • 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? 2 | 07 CWI Queuing Colloquium 1

Network traffic flow fundamentals Coarse model of network dynamics • Fundamental relation between network outflow (rate at which trip end) and accumulation Network outflow 1. Outflow increases 2. Outflow is constant 3. Outflow reduces Number of vehicles in network 3 | 07 CWI Queuing Colloquium Network traffic flow fundamentals Coarse model of network dynamics • Fundamental relation between network outflow (rate at which trip end) and accumulation Network outflow 1. Outflow increases 2. Outflow is constant 3. Outflow reduces Number of vehicles in network 4 | 07 CWI Queuing Colloquium 2

Network traffic flow fundamentals Demand and performance degradation 5 | 07 CWI Queuing Colloquium Network traffic flow fundamentals Implications for traffic network control 6 | 07 CWI Queuing Colloquium 3

Introduction Lecture overview • Traffic congestion phenomena: examples and empirics • Modeling traffic congestion in road networks • Model components of network models • Modeling principles and paradigms • Examples and case studies • Intermezzo Braess paradox • Model application examples • Traffic State Estimation and Prediction • Controlling congestion waves • Optimization of evacuation plans • Microscopic and macroscopic perspectives! 7 | 07 CWI Queuing Colloquium 1. Traffic Congestion Phenomena Empirical Features of Traffic Congestion 8 | 07 CWI Queuing Colloquium 4

Historical perspective Bruce Greenshields 9 | 07 CWI Queuing Colloquium First model of traffic congestion Fundamental diagram • Relation between traffic density and traffic speed: u = U(k) • Underlying behavioral principles? (density = 1/average distance) 10 | 07 CWI Queuing Colloquium 5

Fundamental diagrams Different representations using q = k × u 11 | 07 CWI Queuing Colloquium Dynamic properties Traffic congestion at bottleneck (on-ramp) • Traffic from on-ramp is generally able to merge onto mainstream • Resulting capacity (supply) is lower than demand • Queue occurs upstream of bottleneck and moves upstream as long as upstream demand > flow in queue (shockwave theory) 36 Driving direction 38 100 9 locatie (km) 40 8 50 7 42 5 0 7 7.5 8 8.5 9 9.5 tijd (u) Upstream traffic demand 12 | 07 CWI Queuing Colloquium 6

Dynamic features of road congestion Capacity funnel, instability, wide moving jams • Capacity funnel and capacity drop • Self-organisation of wide moving jams 13 | 07 CWI Queuing Colloquium Capacity funnel and capacity drop Free flow capacity and queue discharge rate • On-set of congestion often occurs downstream of bottleneck • Capacity before congestion > queue-discharge rate • Use of (slanted cumulative curves) clearly reveals this • N(t,x) = # vehicles passing location x at time t • Slope = flow 36 q 0 = 3700 − 3800 38 100 locatie (km) 9 − 3900 40 8 50 − 4000 7 42 5 N(t,x) 0 − 4100 7 7.5 8 8.5 9 9.5 tijd (u) − 4200 7 8 − 4300 ′ N ( t , x ) = N ( t , x ) − q 0 ⋅ t 9 − 4400 8 7.5 8 8.5 9 9.5 tijd (u) 14 | 07 CWI Queuing Colloquium 7

Instability and wide moving jams Emergence and dynamics of start-stop waves • 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 15 | 07 CWI Queuing Colloquium Instability and wide moving jams Emergence and dynamics of start-stop waves • 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 16 | 07 CWI Queuing Colloquium 8

Pedestrian flow congestion Start-stop waves in pedestrian flow • Example of Jamarat bridge shows self-organized stop-go waves in pedestrian traffic flows 17 | 07 CWI Queuing Colloquium Pedestrian flow congestion Other self-organization phenomena • Self-organization is common in pedestrian flows • E.g. bi-directional pedestrian flows show dynamic lane formation 18 | 07 CWI Queuing Colloquium 9

2. Traffic Flow Modeling Microscopic and macroscopic approaches to describe flow dynamics 19 | 07 CWI Queuing Colloquium Modeling challenge Traffic theory: not an exact science! • Traffic flow is a result of human decision making and 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 20 | 07 CWI Queuing Colloquium 10

Network Traffic Modeling Model components and processes • Traffic conditions on the Location choice road are end result of many decisions made by the Trip choice longer traveler at different term decision making levels Destination choice Mode choice • Depending on type of application different levels Route choice short are in- or excluded in model term Departure time choice demand • Focus on driving behavior and flow operations supply Driving behavior 21 | 07 CWI Queuing Colloquium Modeling approaches Microscopic and macroscopic approaches • Two dimensions: • Representation of traffic • Behavioral rules, flow characteristics Individual particles Continuum Individual Microscopic Gas-kinetic models behavior (simulation) models (Boltzmann equations) Aggregate Newell model, particle Macroscopic flow models behavior discretization models 22 | 07 CWI Queuing Colloquium 11

Microscopic modeling example NOMAD Pedestrian Flow Simulation NOMAD is a microscopic continuum model derived by applying  optimal control theory / dynamic game theory Model entails 3 behavioral levels:  Strategic level: activity scheduling  and global route choice Tactical level: local route choice  decision-making Operational level: walking, waiting,  executing activities 23 | 07 CWI Queuing Colloquium NOMAD Walker model NOMAD Pedestrian Flow Simulation  Model describes acceleration vector a (t)  Distinction between  Physical interactions normal force  Long-range interactions friction a (t) = a physical (t) + a control (t)  Physical interactions describe normal forces and tangential forces (friction) when pedestrians touch  Long-range interactions (control model) are derived by applying dynamic game theory 24 | 07 CWI Queuing Colloquium 12

Control model NOMAD Pedestrian Flow Simulation  Control model describes pedestrian interactions  Main behavioral assumptions (based on psychological research):  Pedestrian can be described as optimal, predictive controllers who make short-term predictions of the prevailing conditions, including the anticipated behavior of the other pedestrians  Pedestrians minimize ‘costs’ due to distance between pedestrians, deviations from desired speed and direction, and acceleration  Costs are discounted over time, yielding costs: ⎡ ⎤ || r q − r || ∞ − e −η t 1 1 ⎢ ( v 0 − v ) T ( v 0 − v ) + c 2 ⎥ ∑ ∫ R 0 J = a T a + c 1 e ⎢ ⎥ 2 2 q ⎢ ⎥ t ⎣ ⎦  Pedestrians are largely anisotropic particles 25 | 07 CWI Queuing Colloquium Pedestrian control cycle NOMAD Pedestrian Flow Simulation Controls applied by other pedestrians Pedestrian traffic Observable system Control flow charac- applied by teristics pedestrian p Pedestrian p Control response State estimation Observation model model model 26 | 07 CWI Queuing Colloquium 13

Pedestrian control cycle NOMAD Pedestrian Flow Simulation Chosen optimal p’s estimate of control current state Candidate for optimal control Determination Choose control State prediction walking pay-off weights Walking strategy 27 | 07 CWI Queuing Colloquium Control model NOMAD Pedestrian Flow Simulation  Simplifying assumption: a q = 0  After specification of state dynamics and cost function J; resulting problem is a simple optimal control problem  Applying Pontryagin’s minimum principle yields acceleration term: 28 | 07 CWI Queuing Colloquium 14