Dynamic Origin-Destination Demand Flow Estimation under Congested

Traffic Conditions

Xuesong Zhou (Univ. of Utah)Chung-Cheng Lu (National Taipei University of Technology)

Kuilin Zhang (Argonne National Lab)

Presented at INFORMS 2011 Annual Meeting

1

Motivation

Why existing dynamic OD estimation methods are difficult to produce desirable results under congested conditions, when using link-flow proportions.

3 Difficulties in the past our new methods:

1. Partial derivatives with respect to path flow perturbation2. Single-level path flow estimation framework with gap

function term

2

Literature Review

• Bi-level framework – Yang et al. (1992); Florian and Chen (1995)

• Solution algorithm – Iterative estimation-assignment (IEA) algorithms– Sensitivity-analysis based algorithms (SAB)

3

Iterative Estimation-Assignment Method

• Upper level Constrained ordinary least-squares problem

s.t. non-negativity constraints Lower level:– Link flow proportion = Dynamic traffic assignment ( ))– Solution procedure

),,(),,(ˆ jihlp ),,( jid

tLl

tlji

jijitllc

cdpZ,

2),(

,,),,(),,(),,( ]ˆ[min

Dynamic OD Demand

Dynamic Traffic Assignment

T im e

Dem

andFlow Pattern

Dynamic OD Demand Estimation

T im e

Link

flow

Link Proportions

Measurement Equations cl,t=Σi,j,tp(l,t)(i,j,t) × di,j,t+

Difficulty in IEA Algorithms

Upper-level optimization model does not consider the dependence of link-flow proportions on the OD flows. = function(d)

5

Optimization problem

Traffic observations

Dynamic Network Loading/

DTA Simulator

Dynamic OD demand matrix

Convergence

Link-flow proportions

No

Yes

Consistent dynamic OD demand matrix

),,(),,(ˆ jitlp

Sensitivity-Analysis Based (SAB) Algorithms

• Approximate the derivatives through simulation • for each OD pair and each time interval in every iteration (Tavana, 2001)• Gradient approximation methods

– Simultaneous Perturbation Stochastic Approximation (SPSA) framework by Balakrishna et al. (2008); Cipriani et al. (2011)

• Difficulty: Computationally Intensive– Does not simultaneously achieve user equilibrium

and minimize measurement deviations

6

Difficulty 3: How to Utilize Density/Travel Time Measurements

Automatic Vehicle Identification Automatic Vehicle LocationLoop Detector Video Image Processing

Point Point-to-pointSemi-continuous path trajectory

Continuous path trajectory

Our Approach: Use Spatial Queue Model to evaluate partial

derivatives with respect to path flow perturbation

8Inspired by study by Ghali and Smith (1995)

Case 1: Partially Congested Link

9

Link inflow and outflow increase by 1 at two time stamps:entering time and end of queue duration, respectively.

Case 1: Partially Congested Link

10

Link density (number of vehicles) increases by 1 between two timestamps: entering time, end of queue duration.

Case 1: Partially Congested Link

11

The flows arriving between two time stamps experience the additional delay 1/c, because it takes 1/c to discharge this perturbation flow (similar to the results by Qian and Zhang 2011)

Case 2: Free-flow Conditions

12

Number of vehicles (i.e., link density) increases by 1 from entering time to leaving time.

Case 2: Free-flow Conditions

13

Link inflow and outflow increase by 1 at entering time and leaving time, respectively.

Case 2: Free-flow Conditions

14

Individual travel times are not changed (= free flow travel time, FFTT)

Case 3: Two Partially Congested Links

The perturbation flow on the second link starts at the end of queue duration of the first link; rather than the vehicle entering time on second link

15

Here!

Not Here!

Similar work by Shen, Nie and Zhang (2007) for path marginal cost analysis

Case 4: Queue Spillback

16

Individual extra delay depends on when the vehicle/perturbation flow joins in the queue.

Beyond A/D Curves: How to Model Queue Spillback?

• Forward and backward wave representation in Newell’s simplified kinematic model

17

Time axis

( )( )b

length bBWTT bw

( )( )f

length aFFTT av

Time t-1

Spac

e axi

s

Link b

Link a

A(b,t-1)

D(b,t-BWTT(b)-1)

Our Method to Overcome for Difficulty 1

• Derive analytical, local gradients of different measurement types, with respect to flow perturbation – link flow, density and travel time

• Valuable gradient information considers the dependences of link flow/density/travel time changes on OD flows

18

1. Path flow adjustment Min

(1) deviation between measured and estimated traffic states

(2) the deviation between aggregated path flows and target OD flows

S.T. dynamic user equilibrium (DUE) constraint

2. Aggregate path flows over all paths demand flows

Now move to Challenge 2:Path flow Estimation Framework

19

Demand flow target demand path flow target demand

20

Quick Review: Single-level OD Estimation

• Linear programming PFE by Sherali et al. (1994)

• Nonlinear programming PFE by Bell et al. (1997) on estimating stochastic UE path flows

• Nie and Zhang (2008): single-level formulation based on variational inequalities– Qian and Zhang (2011) further incorporated the travel time gradients

• Nie and Zhang (2010), and Shen and Wynter (2011) integrated the integral term in Beckmann’s UE formulation (1956) with the measurement deviations

Step 1: Lagrangian Reformulation

• Describe the DUE constraint based on a gap function– DUE Gap

• Dualize DUE constraint to the measurement deviation function with a non-negative (Lagrange) parameter – Measurement deviation function Z(r), including link flow,

density, and travel time

21

g(r, ) = wp{r(w,,p)[c(w,,p)(w,)]}.

Minr, , L(r, , ) = z(r) + [g(r, ) ]

Step 2: Gradient Based Algorithm

Individual gradients with respect to path flow adjustment

22

Adjust path flow on each path based on generalized gradient/Cost

Calculated based on the spatial queue model

Flowchart of the Algorithm

23

Path flow adjustment based on all gradients

Our Contribution for Challenge 2• New path flow-based optimization model for jointly

solving the complex OD demand estimation and UE DTA problems

• Simultaneous route and departure time user equilibrium (SRDUE) problem with elastic demand

• Final solution is a set of path flow patterns satisfying “tolled user equilibrium” (Lawphongpanich and Hearn, 2004)

24

Numerical Experiment No.1

Path FFTT (min)

Capacity (vhc/hr)

Assigned Flow (vhc/hr)

Travel Time (min)

Path 1 20 3000 5400 56

Path 2 30 3000 2600 56

25

Congested two-link Corridor: Total capacity = 6000 vhc/hourTotal demand = 8000 vhc/hour

Upper Bound and Lower Bound of Objective Function

26

0 1 2 3 4 5 6 7 8 9 10 11300000

320000

340000

360000

380000

400000

420000

d=7334

d=7335d=7336 d=7338 d=7347

d=7350d=7353 d=7436

Upper BoundLower Bound

Lagrangian multipler

Obj

ectiv

e fu

nctio

n

Path Flow Volume Convergence Pattern

27

0 2 4 6 8 10 12 14 16 18 200

1000

2000

3000

4000

5000

6000

7000

8000

9000

Ground-truth total demand

Estimated total demand

Ground-truth flow on route 1

Estimated Flow on route 1

Estimated Flow on route 2

Ground-truth flow on route 1

Iteration

Flow

Vol

ume

Path Travel Time Convergence Pattern

28

0 5 10 15 2040

45

50

55

60

65

70

Estimated travel time on route 1Estimated travel time on route 2User equilibrium travel time

Iteration

Tra

vel T

ime

(min

)

Robustness of Our Algorithm under Different Input Conditions

Information Availability Estimation ResultVolume observations on path 1 only

Volume observations on path 2 only

Error-free target demand,8000vhc/hr

Error-free travel time on path 1

Flow on path 1

Flow on path 2

Total estimated demand

Equilibrium travel time (min)

X 5051.7 2367.8 7419.5 53.7X 4967.7 2311.8 7279.4 53.1

X X 5011.8 2341.2 7353.0 53.4X X X 5387.9 2592.0 7979.9 55.9X X X X 5401.1 2600.7 8001.8 56.0

29

Experiment 2• A 2-mile section of I-210 Westbound, located in Los

Angeles, CA

30

33.049 ml 32.199 ml

761342

718206

764146

717107

32.019 ml

717669

716604 717668

30.999 ml

717664a

on on onoff

Sensor ID

Postmile

b c d

e f h i

0 10 20 30 40 50 60 70 80 90 100 110 120 1300

200400600800

100012001400160018002000

Density (vhc/ml/lane)

Flow

vol

ue (v

hc/h

our/

lane

)

This is a Congested Corridor…

31

6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM0

102030405060708090

D C

B

Time

Spee

d (m

ph)

33.049 ml 32.199 ml

761342

718206

764146

717107

32.019 ml

717669

716604 717668

30.999 ml

717664a

on on onoff

Sensor ID

Postmile

b c d

e f h i

Observed Lane Volume vs. Estimated lane Volume on Entrance Link

32

6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM0

200400600800

100012001400160018002000

Observed lane volume

Estimated lane volume

Time

Lan

e vo

lum

e (v

hc/h

our)

Observed vs. Estimated Speed on Link from Off-ramp h to Station c

33

6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM0

10

20

30

40

50

60

70

Observed speedEstimated speed

Time

Spee

d (m

ph)

Preliminary Experiment: A Real-world Traffic Network

34

858 nodes2,000 links208 zones

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

Iterations of Path Flow Adjustment

Link

Den

sity

Est

imat

ion

Err

or

(veh

icle

s/m

iles/

lane

)

Conclusions

1. Single-level, time-dependent OD demand estimation formulation, without using link proportions

2. A Lagrangian relaxation solution framework

3. Gradient-projection-based path flow adjustment process

4. Derive theoretically sound partial derivatives of link flow, density and travel time with respect to path flow perturbations

35

36

Historical OD demand

Path flow decomposition

Traffic Link CountOccupancy profileSpeed profileBluetooth records

Path flow vector 1

Path flow vector 2

Path flow vector 2

…

Measurement deviation based rapid gradient calculation generation

Gradient-based path flow adjustment

Gap function-based equilibration New path flow vectors

Convergence detection