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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

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2 Power Grid and Dynamic Analysis zComplex Network zLong Distance Transmission zInterconnection zSystem Stability by State Equation (First order Differential equation) and Eigenvalue Analysis: Matrix A yStable yUnstable zPlanning Tool used as operational tool

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3 Causes and Problems zMajor Blackouts yWSCC 1996, Northeast 2003 zCommon Causes yEquipment Failure yVegetation Problem yHuman Error yBut No Single Cause zProblems (according to report) yNo specific cause singled out yAssumption and conditions in the dynamic analysis yRelationships between network topology and system dynamics recognized but not realized

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4 Another Angle Complementary Tool yTopological analysis of power grid yInvestigation of relatedness with topology and cascading failure xRandom (or intentional) removal of nodes (generators, substation, etc) or transmission lines. xRemoval of the lines faulted in the actual failure in the order of event xTopological Changes yProviding an alternative operational (warning) tool for system operators yGraphical Perspective of Blackouts and Major outages

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5 Graph Theory zNumber of nodes (n) zSize – Number of edges (M) zDegree (k) zCritical Path Length (L) yShortest path distance between two nodes Clustering Coefficient ( ) yThe degree to which neighboring nodes are connected to each other z3 types of network yRegular yRandom ySmall World

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6 “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 L random {~ ln(n) / ln(k)} is much greater than random {~ k / n} zCharacteristics that make the small world phenomenon interesting: yThe network is large yThe network is sparse – people (or things) are connected to a small fraction of the total network yThe network is decentralized -- no single (or small #) of stars yThe network is highly clustered -- most friendship circles are overlapping yglobally significant changes can result from locally insignificant network change

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7 Small World and Dynamics zTopology affects dynamics zSmall world topology enhances signal propagation zThe dynamics are very non-linear -- with no clear pattern based on local connectivity. zDiseases move more slowly in highly clustered graphs zsmall local changes (shortcuts) can have dramatic global outcomes (disease diffusion) zInfection of a whole population (an example) yRegular Graph:5 steps yRandom Graph:3 steps ySmall World:3 steps

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8 Power Grid: small world? - 14bus

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9 Graph Analysis of Power Grids zLarger networks are small world networks

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10 Is it relevant? Some Recent Findings & Suggestions y The density of shortcut edges is an important factor in determining the probability of large-size epidemics, or failures. y Networks with a very high density of shortcut edges exhibit primarily large- size failures. y Networks with no shortcut edges tend to have only small-size failures. y Thus, the presence of a few shortcut edges greatly increases the probability of large-size failures. y 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. z Other Related Articles 1.“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”, 8 th 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” 8 th International Conference on Probabilistic Methods Applied to Power Systems, Iowa State University, Ames, Iowa, September, 2004

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11 WSPP Cascading Failures in 1996

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12 Reconstruction of WSCC Faults

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13 L and Comparison – Scenarios

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14 Graphical Property Changes in the Scenarios z the critical path lengths for the July outage scenarios show much higher than those of other scenarios including the no-outage scenario. z a little increase in the path of the two August outage scenarios.

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15 Conclusions and Remarks z the preliminary results shown in this section do not directly answer the hypothesis of topological changes vs. cascading failures. z 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. z Furthermore, the preliminary study was performed on reduced size grid of the WSCC grid. z 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. z further investigation is in need for the possible correlation of the topological measures to cascading outages. z 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.

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