Traffic Flow Theory
& Simulation
S.P. Hoogendoorn
Lecture 4
Shockwave theory
Shockwave theory I:
Introduction
Applications of the Fundamental Diagram
February 14, 2010
1
Vermelding onderdeel organisatie
Intro to shockwave analysis
Introduce application of fundamental diagram to shockwave
analysis with aim to understand importance of field location
Shockwave analysis:
Vehicles are conserved
Traffic acts according to the fundamental diagram (q = Q(k))
Predicts how inhomogeneous conditions change over time
FOSIM demonstration
Example 3 -> 2 lane drop and emerging shockwaves
(roadworks, incident, etc.)
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FOSIM example
Extremely short introduction to FOSIM
Build a simple network (8 km road with roadworks at
x = 5 km to 6 km)
Implement traffic demand
Assume 10% trucks
Suppose that upstream traffic flow > capacity of bottleneck
What will happen?
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Questions
Why does congestion occur
Macroscopically?
Microscopically?
Photo by My Europe
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Photo by Juicyrai on Flickr / CC BY NC
Microscopic description
Congestion at a bottleneck
Simplest is to compare the system to a (sort of) queuing system
Drivers arrive at a certain rate (demand) at specific time intervals
The n servers needs a minimum amount of time to process the
drivers (each lane is a server)
Service time T is a driver-specific (random) variable depending on
weather conditions, road and ambient conditions, etc.
(= minimum time headway of a driver)
Note that service time is directly related to car-following behavior
When another driver arrives when the server is still busy, he / she
has to wait a certain amount of time
Waiting time accumulates -> queuing occurs
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Macroscopic description
Compare traffic flow to a fluidic (or better: granular) flow though
a narrow bottleneck (hour-glass, funnel)
If traffic demand at a certain location is larger
than the supply (capacity) congestion will occur
Capacity is determined by number of lanes,
weather conditions, driver behavior, etc.
Excess demand is stored on the motorway
to be served in the next time period
Remainder: focus on macroscopic description
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Photo by My Europe
Questions
Why does congestion occur
Macroscopically?
Microscopically?
Where does congestion first occur?
Which traffic conditions (traffic phases) are encountered?
Where are these conditions encountered?
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Free flow
Capacity
Congestion
space
Free flow
time
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Definition of a shockwave
Consider over-saturated bottleneck
Boundaries between traffic regions are referred to as shockwaves
Shockwave can be very mild (e.g. platoon of high-speed vehicles
catching up to a platoon of slower driver vehicles)
Very significant shockwave (e.g. free flowing vehicles approaching queue
of stopped vehicles)
Traffic conditions change over space and time
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Definition of a shockwave2
Shockwave is thus a boundary in the space-time domain that
demarks a discontinuity in flow-density conditions
Example: growing / dissolving queue at a bottleneck
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Fundamental diagrams
What can we say about the FD at the different locations?
Some standard numbers (for Dutch motorways)
Capacity point ! n * 2200 pce/h/lane
Critical density ! n * 25 pce/km/lane
Jam-density ! n * 2200 pce/km/lane
Critical speed ! 85 km/h
Free speed: to be determined from speed limit
pce = person car equivalent
Exact numbers depend on specific characteristics of considered
location (traffic composition, road conditions, etc.)
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Emerging traffic states
space
End of
bottleneck
Bottleneck
location
Free-flow
Capacity
2
Congestion
Stationary shockwave
between capacity
and congested conditions
What is the flow
inside congestion?
And upstream of the
congestion?
time
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Emerging traffic states
space
End of
bottleneck
Free-flow
Q(k)
Capacity
4
Bottleneck
location
2
Congestion
k
Remainder:
focus on the
dynamics
of this
time
shock!
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Shockwave equations
Assume that flow-density relation Q(k) is known for all location x
on the road (i.e. different inside and outside bottleneck!)
Consider queue due to downstream bottleneck
Consider conditions in the queue
Flow q2 = Cb-n (capacity downstream bottleneck)
Speed, density follow from q2, i.e. u2 = U(q2) and k2 = q2/u2
Farther upstream of queue, we have conditions (k1,u1,q1)
Speed u1 > u2 " upstream vehicles will catch up with vehicles in
queue
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Shockwave equations2
Bottleneck
upstream cond.
(q1,k1,u1)
x
downstream cond.
(q2,k2,u2)
u1
Bottleneck
capacity
Vehicles
in queue
Shockwave
u2
Vehicles
upstream
u2
u1
t
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Shockwave equations3
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Shockwave equations4
Relative speed traffic flow region 1 with respect to S: u1 #12
Thus flow out of region 1 into shock S equals (explanation)
Relative speed traffic flow region 2 with respect to S: u2 #12
Thus flow into region 2 out of the shock must be
Conservation of vehicles over the shock (shock does not destroy
or generate vehicles)
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Shockwave equations5
Shockwave speed #12 thus becomes
The speed of the shock equals the ratio of
jump of the flow over the shock S and
jump in the density over the shock S
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Shockwave equations6
Bottleneck
x
Jump in flow
Vehicles
in queue
Shockwave
2
u2
u1
u2
Vehicles
upstream
Jump in density
u1
t
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Shockwave equations7
Remarks:
If k2 > k1 sign shockwave speed negative if q1 > q2 (backward
forming shockwave)
If k2 > k1 sign shockwave speed positive if q1 < q2 (forward
recovery shockwave)
If k2 > k1 sign shockwave speed zero if q1 = q2 (backward
stationary)
Classification of shockwaves
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Final remarks
Shockwave theory is applicable when
Q(k) is known for all location x
Initial conditions are known
Boundary conditions (at x1 AND x2) are known
Shockwaves occur when
Spatial / temporal discontinuities in speed-flow curve
(expressed Q(k,x)), e.g. recurrent bottleneck
Spatial discontinuities in initial conditions
Temporal discontinuities in boundary conditions at x1 (or x2)
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Applications of shockwave theory
Temporary over-saturation
Traffic lights
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Vermelding onderdeel organisatie
Shockwaves at a bottleneck
Temporary over-saturation of a bottleneck
Traffic demand (upstream)
q2
q1
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t1
t2
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Application of shockwave analysis
Three simple steps to applying shockwave theory:
1. Determine the Q(k) curve for all locations x
2. Determine the following external conditions:
initial states (t = t0)
boundary states (inflow, outflow restrictions, moving
bottleneck).
present in the x-t plane and the q-k plane
3. Determine the boundaries between the states (=shockwaves)
and determine their dynamics
4. Check for any ommisions you may have made (are regions with
different states separated by a shockwave?)
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Shockwaves at bottleneck2
x
2
3
1
3 capacity of
bottleneck
t1
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t2
25
Exercise temporary blockade
Fundamental diagram
without capacity drop
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Which shocks emerge
Duration of disturbance?
Draw a couple trajectories
What if q1 = 2200?
26
Studies of the fundamental diagram
Need for complete diagram or only a part of it? Will it in general
be possible to determine a complete diagram at a cross-section?
Is the road section homogeneous? Yes: observations at a single
cross-section. No: road characteristics are variable over the
section and a method such as MO might be suitable
Mind the period of analysis:
too short (1 minute): random
fluctuations much influence;
too long (1 hour): stationarity cannot
be guaranteed (mix different regimes)
Estimate parameters of the model chosen
using (non-linear) regression analysis
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Studies of the fundamental diagram2
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Many models will fit
your data
Hints for choosing
models?
Simplest model
possible
(parsimony)
Interpretation of
parameters
Theoretical
considerations
28
Studies of the fundamental diagram3
Demonstration
FOSIM
Fundamental
diagram
determined from
real-life data, by
assuming
stationary
periods
Dependent on
measurement
location
Flow per lane
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Studies of the fundamental diagram4
Estimation of free flow capacity using fundamental diagram
Approach 1:
Fit a model q(k) to available data
Consider point dq/dk = 0
Generally not applicable to motorway traffic because
dq/dk = 0 does not hold at capacity
Approach 2:
Assume fixed value for the critical density kc
Estimate only free-flow branch of the diagram
More for comparative analysis
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Studies of the fundamental diagram5
Application example: effect of roadway lighting on capacity
Two and three lane motorway
Before after study
Difficulties due to different conditions (not only ambient conditions
change)
See e.g. site SB daylight before after
Effect lighting on capacity approx 2.5% (2 lane) or 1.6% (3 lane)
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Studies of the fundamental diagram6
Effect on rain on capacity / fundamental diagram
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Studies of the fundamental diagram7
Effect on rain on capacity / fundamental diagram
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Studies of the fundamental diagram8
Estimating queue discharge rate
Only in case of observations of oversaturated bottleneck
Three measurement locations (ideally)
Upstream of bottle-neck (does overloading occur?)
Downstream of bottle-neck (is traffic flow free?)
At the bottle-neck (intensities are capacity measurements if
traffic state upstream is congested and the state downstream
is free)
Flow at all three points equal (stationary conditions) and at
capacity; use downstream point
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Summary of lecture
Fundamental diagram for a lane and a cross-section
Shockwave equations
Application of shockwave analysis
shockwave at bottleneck
Establishing a fundamental diagram from field observations
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Shockwaves signalized intersections
B
D
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Shockwave classification
6 types of shockwaves
Which situations do
they represent?
Examples?
Rear stationary
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Shockwave classification2
1. Frontal stationary: head of a queue in case of stationary /
temporary bottleneck
2. Forward forming: moving bottleneck (slow vehicle moving in
direction of the flow given limited passing opportunities)
3. Backward recovery: dissolving
queue in case of stationary or
temporary bottleneck
(demand l.t. supply); forming
or dissolving queue for
moving bottleneck
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Shockwave classification3
1. Forward recovery: removal of temporary bottleneck (e.g.
clearance of incident, opening of bridge, signalized intersection)
2. Backward forming: forming queue in case of stationary,
temporary, or moving bottleneck* (demand g.t. supply);
3. Rear stationary: tail of queue
in case recurrent congestion
when demand is approximately
equal to the supply
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Flow into schockwave
Consider a shockwave moving with speed
Flow into the shockwave = flow observed by moving observer
travelling with speed of shockwave
Number of vehicles
x
observed on S =
+ Veh. passing x0 during T
- Vechicles on X at t1
x0
T
t1
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t
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