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Power Grid Stability in Small World Perspective

Charles Kim

Department of Electrical and Computer EngineeringHoward University

September 25-27, 2006CRIS2006 Third International Conference on Critical Infrastructures

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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

BuAxdtdx

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Causes and ProblemsMajor Blackouts

WSCC 1996, Northeast 2003Common 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

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Another Angle Complementary Tool

Topological analysis of power grid Investigation of relatedness with topology and

cascading failureRandom (or intentional) removal of nodes

(generators, substation, etc) or transmission lines.Removal of the lines faulted in the actual failure in

the order of eventTopological Changes

Providing an alternative operational (warning) tool for system operators

Graphical Perspective of Blackouts and Major outages

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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 () The degree to which

neighboring nodes are connected to each other

3 types of network Regular Random Small World

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“Small World” Network A small world graph is any graph with a relatively small L and

a relatively large . Small World Criteria: L is close to Lrandom {~ ln(n) / ln(k)} is much greater than random {~ 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

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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 Graph:5 steps

Random Graph:3 steps

Small World:3 steps

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Power Grid: small world? - 14bus

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Graph Analysis of Power Grids

Larger networks are small world networks

JUNE 1, 2001 1 o f 1

Amy R. TeelSystem Planning,

Te chnical Operations

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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 Articles1. “Model for Cascading Failures in Complex Networks” PHYSICAL REVIEW E 69,

045104(R), (2004) 2. “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

3. “Cascade Control and Defense in Complex Networks” Phys. Rev. Lett. 93, 098701(2004)

4. “Network Models: Growth, Dynamics, and Failure” Proceedings of the 34 th Hawaii International Conference on System Sciences-2001

5. “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

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Reconstruction of WSCC Faults

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L and Comparison – Scenarios

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Graphical Property Changes in the Scenarios

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.

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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.