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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!