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
23 June 2014 2
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
23 June 2014 4
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
23 June 2014 5
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
23 June 2014 6
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
23 June 2014 7
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
23 June 2014 8
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
23 June 2014 10
x (km)
v (km/h)
ρ (veh/km)
Position
Time
Algorithm Test Case
V Coverage
ρ Coverage
5% 10% 35%
0%
5%
15%
23 June 2014 11
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
23 June 2014 12
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?
23 June 2014 14
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