![Lagrangian formulation
Lagrangian Coordinates
Numerical solution An upwind scheme [less non-linear]
Traffic characteristics only move in the
same (downstream) direction
(increasing vehicle number instead of space)
MasterClass T&P / TIL 11 | 26
2.
New state estimator and application
MasterClass T&P / TIL 12 | 26](https://image.slidesharecdn.com/masterclassyuan-120117094413-phpapp01/85/PhD-research-Yuan-6-320.jpg)
![Application to Freeway Traffic
State Estimation [Essence]
The essence : to reproduce the freeway traffic conditions
based on limited measurement data
MasterClass T&P / TIL 15 | 26
Traffic state estimator based on EKF
Advantage in Lagrangian Coordinates
Exactness: ‘less non-linear’, more accurate, less
numerical diffusion
Implementation: more straightforward
Linearization: more accurate , ‘same’ sign (Differentiability)
A natural observation equation for floating car data
Challenge in Lagrangian Coordinates
Formulating proper observation models for spatially fixed
observations (Loop data) Solved!
Modelling network discontinuity (complex)
MasterClass T&P / TIL 16 | 26](https://image.slidesharecdn.com/masterclassyuan-120117094413-phpapp01/85/PhD-research-Yuan-8-320.jpg)
![Preliminary conclusion
• Lagrangian state estimation out-performs Eulerian state
estimation more accurate estimates.
• Both Eulerian & Lagrangian sensing data are well incorporated
• Promotes the application of EKF
(Solution to the mode-switching problem[upwind or downwind])
• Validates the (elementary) node models
MasterClass T&P / TIL 25 | 26
Further research directions
Developing more advanced Node Models and application
Comparing the performance of Lagrangian model with its
Eulerian counterpart at network levels (on/off ramp)
Using different combinations of data sources
Realistic data at network levels
Implementing the method in a real traffic network (A10)
MasterClass T&P / TIL 26 | 26](https://image.slidesharecdn.com/masterclassyuan-120117094413-phpapp01/85/PhD-research-Yuan-13-320.jpg)
PhD research (Yuan)
- 1. What does it mean to be a PhD? —Experience & Current Research 12-11-2010 Yufei Yuan, PhD Candidate Delft University of Technology MasterClass T&P / TIL Leading to Doctoral Research… Master — T & P Internship: Urban & Inter-Urban traffic control scenario management Master Thesis: Coordination of ramp metering control in motorway networks PhD Traffic state estimation of prediction for road network traffic control MasterClass T&P / TIL 2 | 26
- 2. What does it mean to be a PhD? Research e.g. (in my case) Traffic state estimation and prediction for road network control Publishing results Papers Conferences/Journals Give or follow courses/workshops (TUD or TRAIL) Contract projects with other parties (B.V. or Gov.) MasterClass T&P / TIL 3 | 26 0. Research scope and current research MasterClass T&P / TIL 4 | 26
- 3. Research Scope real traffic system real traffic system traffic traffic (a) state estimation traffic traffic actuators actuators • Monitoring / (b) state prediction sensors sensors state estimation (c) optimization State State State initial prediction estimation / estimation / • State estimation / state data fusion data fusion goals state prediction optimize • State prediction / DTM input: OD matrices, capacity constraints, measures optimization network specs, etc MasterClass T&P / TIL 5 | 26 Current research & results — Lagrangian Traffic State Estimation for Freeways Brief overview Eulerian/Lagrangian formulation of LWR (first-order traffic flow model) Lagrangian state estimator and application Empirical and simulation study Comparing with Eulerian case Conclusions and Further research MasterClass T&P / TIL 6 | 26
- 4. 1. Brief overview MasterClass T&P / TIL 7 | 26 Eulerian formulation of LWR Eulerian Coordinates Coordinates Variables Kinematic wave model No. of Vehices Fundametal diagrams (Daganzo, Smulders) Numerical solution (Mode switching) MasterClass T&P / TIL 8 | 26
- 5. Eulerian Coordinates Lagrangian Coordinates How about Lagrangian coordinates? Rencent Studies… Leonhard Euler Joseph Louis Lagrange MasterClass T&P / TIL 9 | 26 Lagrangian formulation of LWR Lagrangian Coordinates Coordinates Variables Kinematic wave model Position of Vehices Fundametal relations Numerical solution An upwind scheme…(Next) MasterClass T&P / TIL 10 | 26
- 6. Lagrangian formulation Lagrangian Coordinates Numerical solution An upwind scheme [less non-linear] Traffic characteristics only move in the same (downstream) direction (increasing vehicle number instead of space) MasterClass T&P / TIL 11 | 26 2. New state estimator and application MasterClass T&P / TIL 12 | 26
- 7. Traffic state estimation based on EKF Eulerian Coordinates Discretized LWR model Process model ρ ti+1 − ρ ti qti − qti −1 + =0 Δt Δx Fundametal relations Observation (measurement) model However, the process model is highly non-linear, hard to solve; mode-switching, large error (wrong sign)… MasterClass T&P / TIL 13 | 26 A new model-based EKF state estimator Lagrangian Coordinates (Explicit) Discretized Lagrangian model Process model s ti+1 − s ti vti − vti −1 + =0 Δt Δn Fundametal relations Observation (measurement) model Both Eulerian and Lagrangian sensing data are considered MasterClass T&P / TIL 14 | 26
- 8. Application to Freeway Traffic State Estimation [Essence] The essence : to reproduce the freeway traffic conditions based on limited measurement data MasterClass T&P / TIL 15 | 26 Traffic state estimator based on EKF Advantage in Lagrangian Coordinates Exactness: ‘less non-linear’, more accurate, less numerical diffusion Implementation: more straightforward Linearization: more accurate , ‘same’ sign (Differentiability) A natural observation equation for floating car data Challenge in Lagrangian Coordinates Formulating proper observation models for spatially fixed observations (Loop data) Solved! Modelling network discontinuity (complex) MasterClass T&P / TIL 16 | 26
- 9. Challenge: Network Discontinuity MasterClass T&P / TIL 17 | 26 3. Empirical and simulation study MasterClass T&P / TIL 18 | 26
- 10. Empirical Study comparing with Eulerian Case Upstream in-flow known M42 motorway in UK Full individual data Same (speed) observations 1. Lagrangian: FCD 2. Eulerian: Loops 3. Ground truth data Study area: downstream of onramp MasterClass T&P / TIL 19 | 26 Empirical Study Figure: RMSE comparison between two methods for 8 simulation runs of scenario 200m-loop. Blue(E) Red(L) The most important observation: in all scenarios the Lagrangian state estimator out-performs its Eulerian counterpart by up to 24% for density and 75% for speed. MasterClass T&P / TIL 20 | 26
- 11. Empirical Study Figure: Snapshots of a small region from the whole x-t speed map for the Eulerian estimation (Left) and the Lagrangian estimation (right) Rectangles: discritized (calculation) cells Curved lines: trajectories of vehicle groups MasterClass T&P / TIL 21 | 26 Simulation Study with Network Discontinuity Off-Ramp On-Ramp Origin Driving direction Destination Inflow Von-Neumann out-flow condition Upstream in-flow known On-ramp & off-ramp flow known MasterClass T&P / TIL 22 | 26
- 12. Simulation Study with Network Discontinuity Node models in Lagrangian state estimator To do: Further compared with Eulerian approach FOSIM synthetic data realistic MasterClass T&P / TIL 23 | 26 4. Preliminary conclusion and further research MasterClass T&P / TIL 24 | 26
- 13. Preliminary conclusion • Lagrangian state estimation out-performs Eulerian state estimation more accurate estimates. • Both Eulerian & Lagrangian sensing data are well incorporated • Promotes the application of EKF (Solution to the mode-switching problem[upwind or downwind]) • Validates the (elementary) node models MasterClass T&P / TIL 25 | 26 Further research directions Developing more advanced Node Models and application Comparing the performance of Lagrangian model with its Eulerian counterpart at network levels (on/off ramp) Using different combinations of data sources Realistic data at network levels Implementing the method in a real traffic network (A10) MasterClass T&P / TIL 26 | 26