Power Grid Stability in Small World Perspective

1
Power Grid Stability in Small World Perspective

• Charles Kim
• Department of Electrical and Computer Engineering
• Howard University
• September 25-27, 2006
• CRIS2006 Third International Conference on
  Critical Infrastructures

2
Power Grid and Dynamic Analysis

• Complex Network
• Long Distance Transmission
• Interconnection
• System Stability by State Equation (First order
  Differential equation) and Eigenvalue Analysis
  Matrix A
• Stable
• Unstable
• Planning Tool used as operational tool

3
Causes and Problems

• Major Blackouts
• WSCC 1996, Northeast 2003
• Common Causes
• Equipment Failure
• Vegetation Problem
• Human Error
• But No Single Cause
• Problems (according to report)
• No specific cause singled out
• Assumption and conditions in the dynamic analysis
• Relationships between network topology and system
  dynamics recognized but not realized

4
Another Angle Complementary Tool

• Topological analysis of power grid
• Investigation of relatedness with topology and
  cascading failure
• Random (or intentional) removal of nodes
  (generators, substation, etc) or transmission
  lines.
• Removal of the lines faulted in the actual
  failure in the order of event
• Topological Changes
• Providing an alternative operational (warning)
  tool for system operators
• Graphical Perspective of Blackouts and Major
  outages

5
Graph Theory

• Number of nodes (n)
• Size Number of edges (M)
• Degree (k)
• Critical Path Length (L)
• Shortest path distance between two nodes
• Clustering Coefficient (g)
• The degree to which neighboring nodes are
  connected to each other
• 3 types of network
• Regular
• Random
• Small World

6
Small World Network

• A small world graph is any graph with a
  relatively small L and a relatively large g.
• Small World Criteria
• L is close to Lrandom ln(n) / ln(k)
• g is much greater than grandom k / n
• Characteristics that make the small world
  phenomenon interesting
• The network is large
• The network is sparse people (or things) are
  connected to a small fraction of the total
  network
• The network is decentralized -- no single (or
  small ) of stars
• The network is highly clustered -- most
  friendship circles are overlapping
• globally significant changes can result from
  locally insignificant network change

7
Small World and Dynamics

• Topology affects dynamics
• Small world topology enhances signal propagation
• The dynamics are very non-linear -- with no clear
  pattern based on local connectivity.
• Diseases move more slowly in highly clustered
  graphs
• small local changes (shortcuts) can have dramatic
  global outcomes (disease diffusion)
• Infection of a whole population (an example)
• Regular Graph5 steps
• Random Graph3 steps
• Small World3 steps

8
Power Grid small world? - 14bus
9
Graph Analysis of Power Grids

• Larger networks are small world networks

10
Is it relevant?

• Some Recent Findings Suggestions
• The density of shortcut edges is an important
  factor in determining the probability of
  large-size epidemics, or failures.
• Networks with a very high density of shortcut
  edges exhibit primarily large-size failures.
• Networks with no shortcut edges tend to have only
  small-size failures.
• Thus, the presence of a few shortcut edges
  greatly increases the probability of large-size
  failures.
• Removing tie lines from power systems is
  obviously impractical, but monitoring and
  protection strategies could be employed to reduce
  the chance of disturbance propagation and
  cascading failures.
• Other Related Articles
• Model for Cascading Failures in Complex
  Networks PHYSICAL REVIEW E 69, 045104(R), (2004)
  
• Dynamics of Small World Networks and
  Vulnerability of the Electric Power Grid, 8th
  Symposium of Specialist in Electric Operational
  and Expansion Planning), Brazil, May 2002
• Cascade Control and Defense in Complex Networks
  Phys. Rev. Lett. 93, 098701(2004)
• Network Models Growth, Dynamics, and Failure
  Proceedings of the 34th Hawaii International
  Conference on System Sciences-2001
• Cascading Failure Analysis of Bulk Power System
  Using Small World Network Model 8th
  International Conference on Probabilistic Methods
  Applied to Power Systems, Iowa State University,
  Ames, Iowa, September, 2004

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WSPP Cascading Failures in 1996
12
Reconstruction of WSCC Faults
13
L and g Comparison Scenarios
14
Graphical Property Changes in the Scenarios
g

• the critical path lengths for the July outage
  scenarios show much higher than those of other
  scenarios including the no-outage scenario.
• a little increase in the path of the two August
  outage scenarios.

15
Conclusions and Remarks

• the preliminary results shown in this section do
  not directly answer the hypothesis of topological
  changes vs. cascading failures.
• The comparison of the scenarios is not complete.
  The sequential event (or line removal) and its
  effect to the critical path length and the
  clustering coefficient were not performed.
• Furthermore, the preliminary study was performed
  on reduced size grid of the WSCC grid.
• However, the preliminary results shed some
  insight in that they could relate the cascading
  outages to static topological measures, along
  with the dynamic indices that were traditionally
  used in a power operation modeling.
• further investigation is in need for the possible
  correlation of the topological measures to
  cascading outages.
• The basic method for this feasibility study is to
  graphically analyze all North American power
  grids that experienced major outages for a
  possible representation of a grid in terms of
  topology for its operational and stability
  status.