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Transportation Engineering 2010 Fall Semester

Fundamental Principles of Traffic Flow

Transportation System Analysis

• Given the transportation demand, what is the performance of the transportation network?

**Planning Engineering
**Traffic Demand

O/D Flows

User Cost/Time Nonuser Cost

Pollution Indirect Effect

**Demand
Analysis
**

Trip Generation Trip Distribution Mode Choice Route Choice

**Network Performance
**

Vehicle Driver Characteristics

Geometric Design Traffic Flow

Capacity Analysis Queueing

Temporal & Spatial Network Equilibrium

Time-Space Diagram

x

t

at time t1

at time t2

t2t1

Trajectory

Time-Space Diagram

• Location vs. Time • Offers a lot of useful information about

movement of the vehicles • Trajectory

– Curves that define a single position for every movement of time, x(t)

Time-Space Diagram

• From basic Physics – Vertical length = distance traveled – Horizontal length = time consumed – First derivative (Slope) = velocity = dx/dt – Second derivative = acceleration = d2x/dt2

Time-Space Diagram

• Possible vehicle trajectories?

Time-Space Diagram x

t

1

2

x

t

1

2

x

t

1

2

1 goes faster 1 is not moving and 2 is moving bankwards

1 is accelerating and 2 is decelerating

x

t

1 2 3Headway (sec) between 1 and 2

Spacing (m) between 2 and 3

Example

• Stations of a transit vehicle (bus or subway)

• If the distance between stations is not long enough the vehicle can’t reach its cruising speed

x

t

Theoretical trajectory without middle station

Cruising speed

Link

Stations

Delay for introducing middle station

Example

• Assume the acceleration and deceleration rates are constant • Total delay from the stop: v/2a+stopping time +v/2a

Cruising speed

v/2a v/2a

v/2a v/2a

v/a

v/a v²/a

Constructing Trajectories

• How to time-space diagram – Vehicle kinematics (solving ordinary

differential equations) – Empirical way

• Aerial surveys • Stationary observers (Traffic detectors) • Driver logs

Aerial Surveys • Take consecutive photographs to

the same road segment • Place them next to each other,

separated by the time interval between shots

• Draw lines across the different pictures

• following the location of the individual vehicles (these are the trajectories)

x

t Δt

Stationary Observers

• Measure the time at which every vehicle passes the observers • Place them next to each other, separated by the distance

intervals • Draw lines following the time of the individual vehicles (these

are the trajectories) Link

Stations (Detectors)

x

t

Driver Logs

• Drivers record the time and location along their trip • Plot the corresponding points • Draw lines following the points corresponding to the individual

vehicles (these are the trajectories)

x

t

Traffic Stream Definition

x

t

s

h

v

Traffic Stream Definition

x

t T

Traffic Stream Definition

∑

x

t

L

Traffic Stream Definition

• Space Mean Speed

∑

∑

∑

Vt=Arithmetic mean of vehicle speeds (산술평균)

Vs=Harmonic mean of vehicle speeds (조화평균)

x

t

L

T

Stationary Traffic Condition

x

t

L

T

v

v

s

h

Fundamental Diagram

• Flow-Density relationship – If density is 0, flow is 0 (No vehicle) – At a certain density, flow is 0 (Traffic Jam)

• Maximum jam density (kj) – As density increases from 0, flow increases

initially. – After the max flow point (qmax), flow

decreases as density increases.

Fundamental Diagram

• Example – k0=0 veh/km, q0=0 veh/h – k1=1, v1=vmax=vf , q1=vf x 1=vf – k2=10 (spacing = 100m), v2<vf ,

q2=v2 x 10>vf – k3=50 (spacing= 20m) →slow

v3« vf , q3=k3v3 →decreasing – k4=125 (spacing= 8m)= kj →jam

v4= 0, q4=0

Fundamental Diagram

Density

Flow

Speed

Flow

Speed

Density vf

qmax

kj qmax

kj vf Uncongested

Flow

Congested Flow

Greenshields Model

∙

2

2

Flow

Density

qmax

kj

4

k0

Real World Observation

• Triangular Model – Newell (1970’s)

Flow

Density

qmax

kj

Uncongested Congested

vf

kop

kop≈ kj /5

Shockwave

T T+1 T+5T+2 T+3 T+4

t

x

T+6

Speed of Shockwave

Uncongested (A)

Congested (J) (Queue)

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