1 the vulnerability of road networks under area-covering disruptions erik jenelius lars-göran...
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The vulnerability of road networks under area-covering disruptions
Erik JeneliusLars-Göran Mattsson
Div. of Transport and Location AnalysisDept. of Transport and Economics
Royal Institute of Technology (KTH)Stockholm, Sweden
INFORMS Annual Meeting 2008, Washington D.C., USA
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Background
• Road network a fundament of modern society
• Disruptions and closures can cause severe consequences for people and businesses
• Disruptive events may affect extended areas in space,e.g. extreme snowfall, hurricanes, floods, forest fires
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Background
• Past applied vulnerability studies focused on identifying important (critical, significant, vital) links
• Our aim: Study vulnerability to area-covering disruptions– Provide complement to single link failure analysis– Develop methodology for systematic analysis– Apply to large real-world road networks– Gain general insights
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Methodology
• Study area is covered with grid of equally shaped and sized cell
• Each cell represents spatial extent of disruptive event
• Event representation: All links intersecting cell are closed, remaining links unaffected
Hexagonal Square
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Methodology
• Multiple, displaced grids used to increase accuracy
• Advantages of grid approach: – No coverage bias: Each point in study area equally covered– Avoids combinatioral issues with multiple link failures– Easy to combine with frequency data
• Disadvantages:– Results depend on rotation
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Consequence model
• Indicator: Increase in travel time for users
• Constant, inelastic travel demand xij
• Initial link travel times from equilibrium assignment, no change during closure
• During disruption of cell, two possibilities:
1. No alternative routes
Unsatisfied demand, must delay tripuntil after closure
Total delay: 2)(
2 ijc
ij
xT
0 τ
τ
dept. time
delay/user
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Consequence model
2. Alternative routes
Users choose new shortest route, or if faster delay trip
Total delay:
. if2
, if2)(
2
cij
cijij
cij
ij
cij
ttx
tx
T
0 τ
τ
dept. time
delay/user
cijt
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Importance and exposure
• Cell importance: Total increase in travel time for all users when cell is disrupted
• Given collection of grids G and closure duration τ, Importance of cell c:
• Worst-case regional user exposure: Mean increase in travel time per user starting in region when most important cell for region is closed
i ij
cij GcTGc )(),|(I
ri ijij
ri ij
cij
Gc x
T
Gr
)(
max),|(UE wc
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Calculations
• Initial SP tree from start node using Dijkstra
• Remove link k in cell by setting long length L
• If k in SP tree, update tree under k
• If distance to node L: no alternative route
• Repeat for all links in cell
• Repeat for all cells in grids
• Repeat for all start nodes
• Calculation time independent of grid size
L
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Case study
• Swedish road network: 174,044 directed links, 8,764 centroids
• Three square cell sizes: 12.5 km, 25 km, 50 km
• 12 hour closure duration
Cell size # cells/grid # grids
12.5 km 64 x 128 4
25 km 32 x 64 4
50 km 16 x 32 16
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Cell importance12.5 km grid
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Cell importance25 km grid
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Cell importance50 km grid
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Cell importance
• Consequences as function of cell size
• Unsatisfied demand constitutes 97.6% - 99.3% of total increase in travel time
0
50000
100000
150000
200000
250000
300000
350000
0 10 20 30 40 50 60
Cell size (km)
Inc
rea
se
in t
rav
el t
ime
(v
eh
icle
ho
urs
) Mean
Std dev
Coeff var
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Worst-case county user exposure
• Exposure depends on concentrated travel demand, not network redundancy
• In most exposed county, more than 60% of demand unsatisfied
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Worst-casecell vs. link
• Area-covering disruption particularly worse in densely populated regions
• 12 of 21 counties: Worst-case link within worst-case cell
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Some insights
• Other factors behind vulnerability to area-covering disruptions compared to single link failures
• Vulnerability reduced through allocation of restoration resources rather than increasing redundancy
• Unsatisfied demand constitutes nearly all increase in travel time
– Unchanged link travel times may be reasonable assumption– Duration not significant for relative comparisons
• Results depend on link and demand location and regional partition
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Thank you!