Dynamic traffic models

Transcription

1 Agenda Dynamic traffic models Joakim Ekström Johan Janson Olstam Andreas Tapani Introduction to dynamic traffic modeling Dynamic traffic assignment Link and network dynamics Contram A commercial dynamic modeling software Introduction to Contram computer exercise Time dynamic Traffic conditions vary in time (Dynamic route choice) Travel demand vary in time and depends on the current level of congestion (Dynamic departure time choice) Link dynamic In static models the V/D-function is used to describe the relationship between speed and flow Not realistic for oversaturated traffic condition A more realistic modeling of traffic when demand excides capacity Macroscopic flow simulation Queue modeling Spillback Static models Long term infrastructure planning Short term traffic infrastructure planning When congestion can be assumed to be low or in applications where a simpler modeling of congestion effects is enough For large scale networks (regions) Dynamic models Short term traffic infrastructure planning (Off-line) In applications where it is important to capture effects due to congestion. i.e. environmental analysis, congestion charging, signal setting schemes Operative use (On-line) Traffic information Rerouting Estimation of air quality

2 Static models inability to model peak demand. demand Static models inability to model peak demand. The static case demand Time of day Time of day Static models inability to model peak demand. The dynamic case demand Static models inability to model peak spreading demand Time of day Time of day

3 Static models inability to model peak spreading The static case by using elastic demand demand Static models inability to model peak spreading The dynamic case by using retiming model demand Time of day Time of day Static models inability to model congestion In Static models Volume/Delay All demand is assumed to be served In a dynamic model More advance link models Route choice will depend on the current level of congestion The arrival time will depend on the level of congestion in the network Dynamic User Equilibrium Static User Equilibrium Every user minimizes his or her travel time All used routes in a relation have the same travel time and no unused route has a lower travel time Dynamic User Equilibrium Same as for the static case but depends on how we assume the road users choose routes Predictive Traffic conditions in dynamic network are predictable and travelers can get the information on the current and future states of the network, they make their route choice decisions according to future traffic conditions and the predicted travel time. This kind of user equilibrium is called predictive user equilibrium. Reactive Traffic conditions in dynamic network are not predictable, travelers can only get the information on the current states of the network. They try to minimize their individual travel time by continuously updating their route choices according to the current states of the network. This kind of user equilibrium is called reactive user equilibrium.

4 Dynamic Traffic Assignment - Network loading Depending on what properties we require the dynamic model to incorporate it can be possible to formulate the model as a mathematical program (Johan) similar to the static user equilibrium problem. Even if we can formulate the problem it will be very difficult to solve, i.e. to find the flows which will result in a dynamic user equilibrium. Use of approximate solution procedure (heuristics). This process is usually described as network loading. How to load the demand onto the network, and propagate the flow throughout the network, in order to (hopefully) reach a dynamic user equilibrium (Andreas) Dynamic Traffic Assignment Route choice in DTA-models Problem classes DTA formulation/solution approaches Dynamic system optimal (DSO) Reactive dynamic user-optimal (RDOU or RDUE) Predictive dynamic user-optimal (PDOU or PDUE) Mathematical programming Optimal control Variational inequality Heuristics/Simulation

5 Cf. Static Assignment Variational Inequality formulation SUE

6 Example Method of Successive Averages (MSA) 1. Load into the network 2. Determine auxiliary paths 3. Calculate as: If ( ) ( ) x = 3 t x = x = 2 t x = Repeat until convergence criteria is satisfied If Cf. Frank-Wolfe for the static case MSA: Small example 1. Update travel times (Network loading) 2. Direction finding find auxiliary paths 3. Line search - step length calculation 4. Update flows 5. Convergence test 1 OD-pair (1-2) 2 links and 2 paths f 121t f 122t 2 time periods with the demand [ ] d 12t = 2000,1000 For simplicity, lets use the notation f f, f f and d = d 121t = 1t 122t = 2t 12t t

7 Variations of the DTA problem Driver route choice model Multiple user classes Variable/Elastic demand Departure time choice Rerouting Real-time/On-line Some DTA-programs Ex. Assignment in DYNAMEQ DYNAMEQ DYNASMART DYNAMIT CONTRAM

8 Summary Difficult to solve Math. Programming or Variational Inequality formulations of real size DTA-problems But Math. Programming and Variational Inequality formulations gives theoretical insights of the DTAproblem Most commercial programs uses heuristics like MSA Convergence not assured Not possible to derive mathematical properties Dynamic Network Loading Dynamic Network Loading Modeling approaches Assigment model (Travel choice) Travel times OD flows Network loading model (Traffic model)

9 Today s focus: Macroscopic Analytical modeling approaches The Dynamic Network Loading problem (DNL): Determine network traffic dynamics given OD flows Given that route choice is determined, DNL decomposes into Link dynamics Nodal conservation Link dynamics Given: The physical characteristics of a link Entry and exit capacities Length Free-flow speed Initial and boundary conditions E.g. an empty road Determine how traffic moves from entry to exit node Desirable properties A generic link model Non-negativity FIFO Traffic entering the link earlier is expected to exit earlier Flow conservation Link traffic volume = cumulative inflow cumulative outflow Causality Outflow and travel time depend on inflow at or before the corresponding time of entry but not after Minimum travel time There exists a minimum travel time > 0 Finite clearing time There exists a maximum travel time Capacity The exit flow cannot exceed the links capacity

10 How to obtain exit flow or traversal time Delay function model: Four link models Exit flow model (Linear) delay function model Exit function model: Point-Queue model Cell transmission model FIFO conditions? A numerical example (Nie & Zhang, 2005) Discretization

11 Exit flow model (Linear) delay function model Calibration: Calibration: Double counting effect Point queue model Cell transmission model Number of queued vehicles at the exit node Calibration: Numerical implementation of the LWR-model

12 Cell transmission model (cont d) Calibration: In summary Assignment model (Travel choice) Travel times OD flows Given route choice Network loading model (Traffic model) Dynamic Network loading: - Link dynamics - Nodal conservation FIFO? Model types: -Exit function -Delay function -Point queue -LWR (Cell transmission) In practice?