Cascading Failures in Power Grids - Analysis and Algorithms Saleh Soltan 1 , Dorian Mazaruic 2 , Gil Zussman 1 1 Electrical Engineering, Columbia University 2 INRIA Sophia Antipolis
Cascading Failures in Power Grids Power grids rely on physical infrastructure - Vulnerable to physical attacks/failures Failures may cascade An attack/failure will have a significant effect on many interdependent systems (communications, transportation, gas, water, etc.)
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Physical Attacks/Disasters EMP (Electromagnetic Pulse) attack Solar Flares - in 1989 the Hydro-Quebec system collapsed within 92 seconds leaving 6 Million customers without power Source: Report of the Commission to Assess the threat to the United States from Electromagnetic Pulse (EMP) Attack, 2008 Other natural disasters Physical attacks FERC, DOE, and DHS, Detailed Technical Report on EMP and Severe Solar Flare Threats to the U.S. Power Grid, 2010
Sniper Attack on a San Jose Substation, Apr. 2014 Source: Wall Street Journal
Cascading Failures - Related Work Report of the Commission to Assess the threat to the United States from Electromagnetic Pulse (EMP) Attack, 2008 Federal Energy Regulation Commission, Department of Energy, and Department of Homeland Security, Detailed Technical Report on EMP and Severe Solar Flare Threats to the U.S. Power Grid, Oct. 2010 Cascading failures in the power grid Dobson et al. (2001-2010), Hines et al. (2007-2010), Chassin and Posse (2005), Gao et al. (2011 ),… The N - k problem where the objective is to find the k links whose failures will cause the maximum damage: Bienstock et al. (2005, 2009) Interdiction problems: Bier et al. (2007), Salmeron et al. (2009 ), … Cascade control: Pfitzner et al. (2011 ), … Mostly do not consider computational aspects
Outline Background Power flows and cascading failures Real events, models, and simulations Impact of single line failures Pseudo-inverse of the admittance matrix and resistance distance Efficient algorithm for cascade evolution Vulnerability analysis
Power Grid Vulnerability and Cascading Failures Power flow follows the laws of physics Control is difficult It is difficult to “store packets” or “drop packets” Modeling is difficult Final report of the 2003 blackout – cause #1 was “inadequate system understanding” (stated at least 20 times) Power grids are subject to cascading failures: Initial failure event Transmission lines fail due to overloads Resulting in subsequent failures
Recent Major Blackout Event: San Diego, Sept. 2011 Blackout description (source: California Public Utility Commission)with the model
Real Cascade – San Diego Blackout 1350 CHINO 1300 SERRANO 15:38:38 SANTIAGO COACHELLA CITY 15:38:22 BLYTHE 15:38:21 1250 SAN ONOFRE 15:37:58 HASSAYAMPA SAN LUIS NILAND CANNON 15:37:56 1200 15:35:40 MISSION EL CENTRO 15:32:00 N. GILA IMPERIAL V. MIGUEL LA ROSITA 15:27:58 1150 TIJUANA 15:27:39 1100 2100 2200 2300 2400 2500 2600 2700 Failures “skip” over a few hops Does not agree with the epidemic/percolation models
Blackout in India, July 2012 The first 11 line outages leading to the India blackout on July 2012 (numbers show the order of outages) 6,7 10, 11 2 8,9 3 1 4,5 Functional Under Maintenance Tripped
Power Flow Equations - DC Approximation Exact solution to the AC model is infeasible 2 𝑗𝑘 − 𝑉 𝑗 𝑉 𝑘 𝑐 𝑗𝑘 sin 𝜄 𝑗𝑘 𝑔 𝑗𝑘 = 𝑉 𝑗 𝑘 𝑗𝑘 cos 𝜄 𝑗𝑘 − 𝑉 𝑗 𝑉 2 𝑐 𝑗𝑘 + 𝑉 𝑗 𝑉 𝑘 𝑗𝑘 sin 𝜄 𝑗𝑘 𝑅 𝑗𝑘 = −𝑉 𝑗 𝑘 𝑐 𝑗𝑘 cos𝜄 𝑗𝑘 − 𝑉 𝑗 𝑉 and 𝜄 𝑗𝑘 = 𝜄 𝑗 − 𝜄 𝑘 . We use DC approximation which is based on: 𝑘 𝑉 𝑗 ≡ 1, ∀𝑗 𝑘 𝑦 𝑗𝑘 sin 𝜄 𝑗𝑘 ≈ 𝜄 𝑗𝑘 𝑉 𝑗 = 1 𝑞.𝑣. for all 𝑗 𝑗 Pure reactive transmission lines – each line is characterized only by its reactance 𝑦 𝑗𝑘 = −1/𝑐 𝑗𝑘 𝑉 𝑗 , 𝜄 𝑗 , 𝑄 𝑗 , 𝑅 𝑗 Phase angle differences are “small”, implying that sin𝜄 𝑗𝑘 ≈ 𝜄 𝑗𝑘 Load Known as a reasonably good approximation Generator Frequently used for contingency analysis Do the assumptions hold during a cascade?
Power Flow Equations - DC Approximation A power flow is a solution (𝑔,𝜄) of: ,∀ 𝑣 ∈ 𝑊 𝑔 𝑣𝑤 = 𝑞 𝑣 𝑤∈𝑂 𝑣 𝑣𝑤 = 0,∀ 𝑣,𝑤 ∈ 𝐹 𝜄 𝑣 − 𝜄 𝑤 − 𝑦 𝑣𝑤 𝑔 Matrix form: 𝑤 𝐵Θ = 𝑄 𝐵 is the admittance matrix of the grid defined as: 0, 𝑣 ≠ 𝑤 𝑏𝑜𝑒 𝑣, 𝑤 ∉ 𝐹 − 1 , 𝑣 ≠ 𝑤 𝑏𝑜𝑒 𝑣, 𝑤 ∉ 𝐹 𝑣 𝑦 𝑣𝑤 𝑏 𝑣𝑤 = − 𝑏 𝑤𝑥 , 𝑣 = 𝑤 𝜄 𝑣 ,𝑞 𝑣 𝑥∈𝑂 𝑣 Load ( 𝑞 𝑣 < 0 ) If 𝐵 + is its pseudo-inverse Generator ( 𝑞 𝑣 > 0 ) Θ = 𝐵 + 𝑄
Line Outage Rule Different factors can be considered in modeling outage rules The main is thermal capacity 𝑣 𝑗𝑘 Simplistic approach: fail lines with 𝑔 𝑗𝑘 > 𝑣 𝑗𝑘 Not part of the power flow problem constraints More realistic policy: 20 Compute the moving average 15 𝑗𝑘 𝑔 𝑗𝑘 ≔ 𝛽 𝑔 𝑗𝑘 + 1 − 𝛽 𝑔 10 ( 0 ≤ 𝛽 ≤ 1 is a parameter) 5 Deterministic outage rule: 0 Fail lines with 𝑔 𝑗𝑘 > 𝑣 𝑗𝑘 1 2 3 4 5 6 Stochastic outage rules
Cascading Failure Model (Dobson et al.) 𝑣 13 = 1800 MW 𝑄 13 = 1400 MW 𝑄 13 = 3000 MW 1 𝑄 1 = 𝑔 1 = 2000 MW 𝑄 1 = 0 MW 3 𝑦 13 = 10 Ω 𝑄 3 = −𝑒 3 = −3000 MW 𝑄 3 = 0 MW Until no more lines fail do: Adjust the total demand to the total 2 supply within each component of 𝐻 Use the power flow model to 𝑄 2 = 0 MW 𝑄 2 = 𝑔 2 = 1000 MW compute the flows in 𝐻 Update the state of lines 𝜊 𝑗𝑘 Initial failure causes disconnection according to the new flows of load 3 from the generators in Remove the lines from 𝐻 according the rest of the network to a given outage rule 𝑃 As a result, line 2,3 becomes overloaded
Numerical Results (Bernstein et al., IEEE INFOCOM’ 14) Obtained from the GIS (Platts Geographic Information System) Substantial processing of the raw data Used a modified Western Interconnect system, to avoid exposing the vulnerability of the real grid 13,992 nodes (substations), 18,681 lines, and 1,920 power stations. 1,117 generators (red), 5,591 loads (green) Assumed that demand is proportional to the population size
Cascade Development – San Diego area N -Resilient, Factor of Safety K = 1.2
Cascade Development – San Diego area
Cascade Development – San Diego area
Cascade Development – San Diego area
Cascade Development – San Diego area
Cascade Development – San Diego area 0.33 N -Resilient, Factor of Safety K = 1.2 Yield = 0.33 For ( N- 1) - Resilient Yield = 0.35 For K = 2 Yield = 0.7 (Yield - the fraction of the demand which is satisfied at the end of the cascade)
Outline Background Power flows and cascading failures Real events, models, and simulations Impact of single line failures Pseudo-inverse of the admittance matrix and resistance distance Efficient algorithm for cascade evolution Vulnerability analysis
Metrics for Evaluating the Impact of a Single Failure 𝑔 𝑓′ 𝑓 = 4,5 𝑓 ′ = {1,5} 𝑔′ 𝑓 𝑔 𝑓 Flow Change after failure in the edge 𝑓 𝑓 Δ𝑔 𝑓 = 𝑔′ 𝑓 − 𝑔 Edge Flow Change Ratio 𝑇 𝑓,𝑓′ = Δ𝑔 𝑓 𝑔 𝑓 Mutual Edge Flow Change Ratio 𝑁 𝑓,𝑓′ = Δ𝑔 𝑓 𝑔 𝑓′
Graph Used in Simulations Western interconnection: 1708-edge connected subgraph of the U.S. Western interconnection Erdos-Renyi graph: A random graph where each edge appears with probability 𝑞 = 0.01 Watts and Strogatz graph: A small-world random graph where each node connects to 𝑙 = 4 other nodes and the probability of rewiring is 𝑞 = 0.1 Barabasi and Albert graph: A scale-free random graph where each new node connects to 𝑙 = 3 other nodes at each step following the preferential attachment mechanism
Effects of a Single Edge Failure Δ𝑔 Edge Flow Change Ratio , 𝑇 𝑓,𝑓′ = 𝑓 𝑓 𝑔 Very large changes in the flow Sometimes far from the failure There are edges with positive flow increase from zero, far from the initial edge failure
Updating the Pseudo-Inverse Objective: Compute the mutual edge flow change ratios Recall that Θ = 𝐵 + 𝑄 Method: Update the pseudo-inverse of the admittance matrix upon failure 𝑌 = 1,0,0,0,−1 𝑢 𝑗 𝑘 Admittance Matrix: 𝐵 Admittance Matrix: 𝐵′ 𝐵 ′ = (𝐵 + 𝑏 𝑗𝑘 𝑌𝑌 𝑢 ) Theorem: 𝐵 ′+ = 𝐵 + 𝑏 𝑗𝑘 𝑌𝑌 𝑢 + = 𝐵 + − 1 −1 +𝑌 𝑢 𝐵 + 𝑌 𝐵 + 𝑌𝑌 𝑢 𝐵 + 𝑏 𝑗𝑘 *A similar theorem independently proved by Ranjan et al., 2014
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