The Breakdown Phenomenon FS

1
The Breakdown Phenomenon F?S

• Its probabilistic nature and its consequences for
  subsequent congestion

TRAIL Physics of Traffic Course
Chris van Hinsbergen
2
Topics

• Spontaneous breakdown at onramps
• Induced breakdown at onramps
• Spontaneous breakdown at offramps
• Spontaneous breakdown away from bottlenecks
• Freeway capacities
• Synchronized flow Empirical features

3
Topics

• Spontaneous breakdown at onramps
• Induced breakdown at onramps
• Spontaneous breakdown at offramps
• Spontaneous breakdown away from bottlenecks
• Freeway capacities
• Synchronized flow Empirical features

4
Spontaneous breakdown at onramps
Page 271
v
120
80
40
t
0635
0640
0645
0650
0655
0700
upstream
onramp
downstream
qin
qsumqinqon
qon
5
Features of synchronised flow
v
qsynch
2400
qmax(free,emp)
qFS(B) or pre-discharge flow rate can be lower
than qmax(free,emp), due to probabilistic nature
of breakdown
6
Speed and flow patterns
Page 271
v
v
v
upstream
onramp
downstream
q
q
q
3000
3000
3000
2000
2000
2000
1000
1000
1000
t
t
t
7
Self-sustaining or not?
Page 273
v
120
80
40
1
2
3
t
0625
0630
0635
0640
0645
0650
q
Not self-sustaining
Self-sustaining
8
Self-sustaining or not?
Page 273

• If qupltqbottleneck upstream front cannot
  propagate upstream
• Stokes

9
Topics

• Spontaneous breakdown at onramps
• Induced breakdown at onramps
• Spontaneous breakdown at offramps
• Spontaneous breakdown away from bottlenecks
• Freeway capacities
• Synchronized flow Empirical features

10
Induced breakdown at onramps

• Wide moving jam passes
• Accelerating vehicles (still slow) from moving
  jam force vehicles at onramp to drive slowly
• In turn, slow vehicles on the onramp prevent
  acceleration of vehicles on main road

11
Induced breakdown due to WMJ
Page 283
v
v
v
120
120
120
80
80
80
40
40
40
t
t
t
upstream
onramp
downstream
q
q
q
3000
3000
3000
2000
2000
2000
1000
1000
1000
t
t
t
12
Induced breakdown due to MSP
Page 284
MSP
v
v
v
120
120
120
80
80
80
40
40
40
t
t
t
upstream
onramp
downstream
q
q
q
3000
3000
3000
2000
2000
2000
1000
1000
1000
t
t
t
13
Topics

• Spontaneous breakdown at onramps
• Induced breakdown at onramps
• Spontaneous breakdown at offramps
• Spontaneous breakdown away from bottlenecks
• Freeway capacities
• Synchronized flow Empirical features

14
Spontaneous breakdown at offramps

• If fraction of vehicles leaving main road is high
• F?S transition can occur due to lane changing
• Downstream front is at some distance of offramp
  (1,5km in the example)

15
Spontaneous breakdown at offramps
Page 287
v
v
v
120
120
120
80
80
80
Left lane(s)
40
40
40
Right lane
t
t
t
upstream 2
upstream1
offramp
q
q
q
3000
3000
3000
2000
2000
2000
1000
1000
1000
t
t
t
16
Topics

• Spontaneous breakdown at onramps
• Induced breakdown at onramps
• Spontaneous breakdown at offramps
• Spontaneous breakdown away from bottlenecks
• Freeway capacities
• Synchronized flow Empirical features

17
Breakdown away from bottlenecks

• Observed rarely
• Usually not self-sustaining
• Fronts can travel both downstream and upstream

18
Breakdown away from bottlenecks
Page 291
v
v
v
120
120
120
80
80
80
40
40
40
Spontaneous
Induced
Induced
t
t
t
upstream
location X
downstream

• Both downstream and upstream front travel
• Downstream and upstream SP are induced

19
Topics

• Spontaneous breakdown at onramps
• Induced breakdown at onramps
• Spontaneous breakdown at offramps
• Spontaneous breakdown away from bottlenecks
• Freeway capacities
• Synchronized flow Empirical features

20
F?S and pattern formation
Page 273
v

• 1 2 do not lead to pattern formation
• When studying capacities, only consider
  transitions where time that pattern exists gtgt
  averaging interval Tav

120
80
40
t
21
Freeway Capacity

• Avg. flow rate at which free flow remains at the
  bottleneck during a given Tav with probability
• PC(B) 1 PFS(B) lt 1
• There are an infinite number of freeway
  capacities, determined by two attributes
• Averaging interval Tav
• Probability PC(B)

22
Probability of F ? S transition
Page 275
PFS(B)

• Larger Tav, lower C
• Average freeway capacities over different control
  parameters (weather, etc.)

23
Probability of F ? S transition
PC(B)0 Maximum capacity
Page 275
PFS(B)
1,0

• For one set of qon and qin!

0,8
PC(B)0,67 Tav 1 min ? C2950 veh/h
0,6
0,4
0,2
0
0
2000
2400
2800
3200
PC(B)1Below capacity
qsum
PC(B)1 Minimum capacity
24
Freeway capacity for sets of qin qon
qin
Page 221
general pattern
SJ(B)
FS(B) PFS(B)1 ? maximum capacity
synchronized
flow
Fth(B) PFS(B)0 ? minimum capacity
free flow
qon
25
Freeway capacity for sets of qin qon
qin
PFS(B)
1,0
general pattern
0,8
0,6
SJ(B)
0,4
0,2
FFS(B)
synchronized
0
flow
0
2000
2400
2800
3200
qsum
q
Fth(B)
qmax(free,emp)
free flow
qon
0
?
26
Topics

• Spontaneous breakdown at onramps
• Induced breakdown at onramps
• Spontaneous breakdown at offramps
• Spontaneous breakdown away from bottlenecks
• Freeway capacities
• Synchronized flow Empirical features

27
Synchronized flow Empirical features

• Points in Flow/Density plane can move in all
  directions

q
28
Three types of Synchronized Flow
Page 295296 297

• Homogeneous state
• Homogeneous-in-speed state platooning
• Nonstationary inhomogeneous state

q
29
Flow/density waves

• Homogeneous-in-speed
• Flow/Density waves travel with positive velocity
  (downstream?)
• Different waves in different lanes
• Downstream waves are more likely at higher speeds
• Upstream waves are more likely at lower speeds

30
Synchronization across lanes

• Vehicle speeds can synchronize across different
  lanes
• This usually only occurs away from bottlenecks
• Especially in nonstationary and inhomogeneous
  state, speeds can vary a lot between lanes
• Often occurs in pinch regions where speeds are
  low
• However always tendency to synchronization

31
Overlap in density
Page 300
q
3000
0
0
?
?
32
?v between left lane and middle lane
Page 301

• Z-shaped form of dependence of ?v on density
• Middle branch cannot be observed because this is
  an unstable state
• Overlap decreases with lower speeds in SP

?v
40
30
20
10
?
10
20
30
40