Highway Traffic Density Estimations

Informed by Velocity Measurements

Matt Wright and David Shulman23 June

2014

Outline

• Motivation

– Methods of flow measurement

• Computational Model Theory

– Cell Transmission Model (CTM)

– Godunov Discretization

– Ensemble Kalman Filter (EnKF)

• Algorithm Test Case

– Reference Implementation

– Results

• Future Work: Real-World Data Test

– Site Description

– Test Cases

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Motivation

• Current method of flow

measurement

– Inductive loops

– Limitations

• Additional Information

– GPS position data

– Calculated velocities

23 June 2014 3

Computational Model Theory

Cell Transmission Model (CTM)

– Discrete representation of linear traffic flow in space and time

– Longitudinally adjacent cells with mean densities

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𝜌𝑖𝑛+1 = 𝜌𝑖

𝑛 −∆𝑇

∆𝑋𝐺 𝜌𝑖

𝑛, 𝜌𝑖+1𝑛 − 𝐺 𝜌𝑖−1

𝑛 , 𝜌𝑖𝑛

ρi+2 ρi+3 ρi+4 …ρi+1ρi…

ΔX

ΔT

ρi+2 ρi+3 ρi+4 …ρi+1ρi…

n

n+1

Computational Model Theory

Godunov Discretization

– Determines flux between cells

– Evaluates minimum of upstream sending flow and downstream

receiving flow

General Form

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𝐺 𝜌1, 𝜌2 = 𝑚𝑖𝑛𝜌∈ 𝜌1,𝜌2 𝑄 𝜌 𝑖𝑓 𝜌1 ≤ 𝜌2𝑚𝑎𝑥𝜌∈ 𝜌2,𝜌1 𝑄 𝜌 𝑖𝑓 𝜌2 ≤ 𝜌1

𝐺 𝜌1, 𝜌2 =

𝑄 𝜌2 𝑖𝑓 𝜌𝑐 ≤ 𝜌2 ≤ 𝜌1𝑞𝑐 𝑖𝑓 𝜌2 ≤ 𝜌𝑐 ≤ 𝜌1𝑄 𝜌1 𝑖𝑓 𝜌2 ≤ 𝜌1 ≤ 𝜌𝑐min(𝑄 𝜌1 , 𝑄(𝜌2)) 𝑖𝑓 𝜌1 ≤ 𝜌2

Computational Model Theory

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𝑣𝑚𝑎𝑥 𝜌2 −𝜌22

𝜌𝑗𝑞𝑐

𝑣𝑚𝑎𝑥 𝜌1 −𝜌12

𝜌𝑗

𝐺 𝜌1, 𝜌2 =

𝑤𝑓 𝜌2 − 𝜌𝑗

𝑞𝑐

𝑣𝑚𝑎𝑥𝜌1

𝜌1, 𝜌2 ∈ 𝑤ℎ𝑖𝑡𝑒 𝑟𝑒𝑔𝑖𝑜𝑛

𝜌1, 𝜌2 ∈ 𝑙𝑖𝑔ℎ𝑡 𝑔𝑟𝑒𝑦 𝑟𝑒𝑔𝑖𝑜𝑛

𝜌1, 𝜌2 ∈ 𝑔𝑟𝑒𝑦 𝑟𝑒𝑔𝑖𝑜𝑛

Flux = Greenshields Daganzo-Newell Condition

Computational Model Theory

Incorporating Real-World Data

Greenshields

𝑣 = 𝑣𝑚𝑎𝑥 1 −𝜌

𝜌𝑗

Daganzo-Newell

𝑣 = 𝑣𝑚𝑎𝑥

𝑤𝑓 1 −𝜌𝑗

𝜌

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ρi+2 ρi+3 ρi+4 …ρi+1ρi…

? ? ?

Velocity Flux

Computational Model Theory

Ensemble Kalman Filtering (EnKF)

– Provides feedback to adjust model

– Estimates states in regions that have lack data

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𝜌𝑓𝑛 𝑘

𝑣𝑓𝑛 𝑘

= 𝑀𝜌𝑎𝑛−1 𝑘

𝑣𝑎𝑛−1 𝑘

+ η𝑛(𝑘)

𝜌𝑓𝑛

𝑣𝑓𝑛 =

1

𝐾

𝑘=1

𝐾𝜌𝑓𝑛 𝑘

𝑣𝑓𝑛 𝑘

𝑃𝑒𝑛𝑠,𝑓𝑛 =

1

𝐾 − 1

𝑘=1

𝐾𝜌𝑓𝑛 𝑘 − 𝜌𝑓

𝑛

𝑣𝑓𝑛 𝑘 − 𝑣𝑓

𝑛

𝜌𝑓𝑛 𝑘 − 𝜌𝑓

𝑛

𝑣𝑓𝑛 𝑘 − 𝑣𝑓

𝑛

𝑇

𝐺𝑒𝑛𝑠𝑛 = 𝑃𝑒𝑛𝑠,𝑓

𝑛 𝐻𝑛 𝑇(𝐻𝑛𝑃𝑒𝑛𝑠,𝑓𝑛 𝐻𝑛 𝑇 + 𝑅𝑛)−1

𝜌𝑎𝑛 𝑘

𝑣𝑎𝑛 𝑘

=𝜌𝑓𝑛 𝑘

𝑣𝑓𝑛 𝑘

+ 𝐺𝑒𝑛𝑠𝑛 𝜌𝑚𝑒𝑎𝑠

𝑛

𝑣𝑚𝑒𝑎𝑠𝑛 −𝐻𝑛

𝜌𝑓𝑛 𝑘

𝑣𝑓𝑛 𝑘

+ 𝑋𝑛(𝑘)

Model Summary

23 June 2014 9

𝜌𝑎𝑛−1 𝜌𝑓

𝑛

𝑣𝑓𝑛

𝜌𝑚𝑒𝑎𝑠𝑛

𝑣𝑚𝑒𝑎𝑠𝑛𝜌𝑎

𝑛

CTM

Velocity Function

Iterate in t

Kalman Filter

Algorithm Test Case

• Mathematica Reference

Implementation

• Algorithmically identical to

actual system for live

data

• Made for debugging

purposes

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x (km)

v (km/h)

ρ (veh/km)

Position

Time

Algorithm Test Case

V Coverage

ρ Coverage

5% 10% 35%

0%

5%

15%

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Future Work: Real Data Problem

• Site

– 11-mile stretch of I-880

– NB Lane only

• Data availability

– 10 stationary loop

detectors

• For comparison against

estimated density

– ~10000 probe points

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Future Work: Real Data Test Cases

Test Purpose

Sequentially remove loops, compare estimate to truth

Test probe data at filling in single-loop gaps

Stochastically remove several loops

Test error variance for particular loop removal

Include only loops upstream of congestion events

Test probe data’s usefulness at finding end of congestion events

• Several test cases

proposed/underway

• Repeated for varying amounts

of sampled probe points

• Estimation error can be

evaluated visually or

numerically

23 June 2014 13

Thank YouQuestions?

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