Modal Analysis of Weighted Networks: Case studies

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Modal Analysis of Weighted Networks Case
studies for MANMADE project E.
Gutiérrez European Laboratory for structural
Assessment Institute for the Protection and the
Security of the Citizen (IPSC) Joint Research
Centre, European Commission, Ispra,
Italy ONCE-CS Open Network of Centers of
Excellence in Complex Systems Skopje 7-9 May 2007
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European Laboratory for structural Assessment
JRC-Ispra-Italy
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Analysis of Networked Systems Energy
Distribution

• Manmade networks are large, interconnected
  systems coupled by non-trivial dependencies
  (technical, natural phenomena social).
• How to analyse energy networks at the
  macro-scale?
• Large networks are not easily tractable with
  standard causal mathematical methods??Graph
  theoretic methods, dynamical systems..
• Networks can be defined in terms of the
  topological properties of their graphs.
• What can be said of their failure rates?
• What can be said of the vulnerability of the
  whole?

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Some approaches to diagnose status of networked
systems

• Statistical analysis
• How stressed are they?
• Blackout statistics (Gaussian vs. Power laws
  imply widely differing expectancies for
  widespread failures).
• Does system have memory of past events (Hurst
  Coefficient).
• Volatility
• Topological features
• Network architecture?
• Sensitivity of network links to different failure
  distributions.
• Network monitoring optimisation
• Identify the minimum set of locations to
  completely measure domain?
• Topological distribution of past events.

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Resilience of networks Case study of error and
attack tolerance of a sector of the European
electricity HV grid

• Study fragmentation of networks as a function of
  failure scenarios
• Random malfunctions.
• Malfunction of most-connected nodes.
• Malfunction of busiest nodes/lines
• How to identify potentially busy nodes and
  lines from a large network (assume poor knowledge
  of flow data).

.first, we need to convert the grid map into a
network graph
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Generate/obtain table of grid HV connections
Weighted connectivity matrix
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Spectra
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Graph of network
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The weighted graph representation of the
electricity grid sector chosen for analysis
219 vertices 309 HV lines
219 Possible flow modes
Visualization using Pajek Network analysis
program.
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Connectivity matrix can be used to study
magnitude and distribution of potential flows
(modal analysis)? high flux corridors.
Example Mode 217 connects node 63 strongly to
nodes 93,96 and 98. What happens if node 63 is
eliminated? Along which lines does mode flow
through?

• Topological properties of graphs (diameter,
  distance, cluster size) can be measured as a
  function of node/line elimination.

A and B are clearly different. How? How much, and
how does it affect vulnerability?
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Similarities
HV grid system
Structural mechanics

• Vertices
• (transformers, substations)
• Edges
• (HV lines)
• Connectivity matrix
• (power line interconnections)
• Laplacian
• name??
• Spectral properties
• ??
• Catastrophic Failures
• Frequency instabilities, blackouts

• Vertices
• (beam-to-column joints)
• Edges
• (beams, columns, slabs)
• Connectivity matrix
• (building design)
• Laplacian
• (Stiffness matrix)
• Spectral properties
• (natural frequency, vibration modes)
• Catastrophic Failures
• collapse, earthquakes..

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Some ideas from earthquake engineering to rank
network nodes

• Earthquake resistance of structures involves
  interaction of complex deflection modes.
• Need to associate modes to structural details
  (beams, columns).
• Use ABSSUM measure absolute sum of modal
  combination (passim earthquake engineering
  literature, Rosenblueth 1951)

Summation to a Grid node

• Here, we also suggest a combination of modal
  participation factors for load lines.

Summation to a Grid line
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The modal representation of transmission now
looks like this.
Modal flow intensity represented in grey-scale.
Size of vertex proportional to modal weighting.
Busy nodes and lines are proportional to size
of vertex and thickness lines.
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How are modal contributions distributed among the
nodes?
Where are high risk (ageing, new, exposed,
congested) assets?
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The order of most connected (Dgt5) nodes does
not necessarily agree with the order given by
the modal participation factors
  Table 3 Lines with highest/lowest cumulative
modal participation
Table 2 Nodes with highest/lowest cumulative
modal participation factors.
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Study fragmentation of network to selective and
random strategies
Original Grid Cluster
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..response of Network to selective and random
strategies
What happens to largest cluster
What happens to smaller clusters
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Elimination of nodes/lines results in network
partitions. The network may break up
into blackout islands.
A main cluster partition
Q. Which is best way of generating isolated
islands to keep system going?
Multiple partitions
The optimal partitioning strategy depends on
the topology of the network.strategies may be
calculated (malicious) or imposed by nature
(seismic, wind/snow storm..).
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Conjectures

• Fragmentation of HV grid network conforms to
  scale-free type.
• Resilient to random error failures (Albert et al.
  Nature, 406 2000, Barabassi et al. Science 286,
  1999).
• Susceptible to concerted attacks or chance errors
  in key highly connected nodes.
• But, highly-connected nodes are widely
  distributed? high-level strategy or extreme
  widespread natural event required to damage a
  substantial number in one go.
• Busy, high flow, corridors are not necessarily
  the most connected, but their elimination can
  fragment the system just as fast.
• High flow corridors are closely packed?easier to
  target?? natural hazard event does have to span
  wide geography.

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We have analysed time-series of blackouts. What
about their topology, do they coincide with any
special nodes or lines we have identified?
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Some other useful tools from graph theory

• If network topology strongly influences
  vulnerability, can it help us to define the best
  monitoring sites?? Graph dominating sets (GDS)
  for electrical circuits may be obtained by using
  Ohms Law and Kirchoffs Law, and graph-theoretic
  methods.

Examples
Which is the GDS for this network and how is it
related to its spectrum?
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Next steps, problems

• Extend analysis to other energy networks (gas and
  oil, distribution) and then,
• Develop network interdependency and coupling
  terms,
• Time-scale dependencies
• Operational dependencies
• Dynamical complexity (modeling non-linear
  processes at and between nodes requires influence
  models),
• Node diversity (substations, power plants..),
• Map blackouts to network topology,
• Flows,

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Modal (cluster) analysis can also be used to
study flow exchanges
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Imports Exports of Electricity
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Intensity of Power Exchanges in EU
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We hope to answer some in the MANMADE project

• The scope of the project concerns the network of
  networks that comprise Europes critical
  infrastructure concentrating primarily on energy
  supply, emergency response systems and subsidiary
  key infrastructures.
• The aim of the project is to assemble, develop
  and apply complementary mathematical methods to
  analyse large, man-made multi-element
  infrastructure systems that exhibit, so-called,
  complex behaviour.

www.manadenet.eu , EU funding under the DG RTD
NEST programme. Contract number 043363.
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Study generic network behaviour such as

• instabilities and collapse both structural
  (catastrophic failure of network components),
  functional (electricity grid blackouts, supply
  chain),
• volatility and memory (spot electricity pricing),
  
• feedback (influence on congestion in networks)
• inter-network coupling (e.g. vulnerability of
  interconnected networks to unexpected failures)

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MANMADE Project Partners
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• Thank you for your attention