Cascading Failures in Power Grids - Analysis and Algorithms Saleh - - PowerPoint PPT Presentation
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
1 Electrical Engineering, Columbia University 2 INRIA Sophia Antipolis
Power grids rely on physical infrastructure - Vulnerable to
Failures may cascade An attack/failure will have a significant effect on many
EMP (Electromagnetic Pulse) attack Solar Flares - in 1989 the Hydro-Quebec
Other natural disasters Physical attacks
Source: Report of the Commission to Assess the threat to the United States from Electromagnetic Pulse (EMP) Attack, 2008 FERC, DOE, and DHS, Detailed Technical Report on EMP and Severe Solar Flare Threats to the U.S. Power Grid, 2010
Report of the Commission to Assess the threat to the United States
Federal Energy Regulation Commission, Department
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
Power flow follows the laws of physics Control is difficult
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:
2100 2200 2300 2400 2500 2600 2700 1100 1150 1200 1250 1300 1350 HASSAYAMPA
LA ROSITA TIJUANA IMPERIAL V. MIGUEL EL CENTRO NILAND BLYTHE SAN ONOFRE CANNON SAN LUIS MISSION SANTIAGO SERRANO CHINO COACHELLA CITY 15:27:39 15:27:58 15:32:00 15:35:40 15:37:56 15:37:58 15:38:21 15:38:22 15:38:38
Failures “skip” over a few hops Does not agree with the epidemic/percolation models
1 2 3 4,5 6,7 8,9 10, 11
The first 11 line outages leading to the India blackout on July
Exact solution to the AC model is infeasible
𝑗𝑘 = 𝑉𝑗 2𝑗𝑘 − 𝑉𝑗𝑉 𝑘𝑗𝑘 cos 𝜄𝑗𝑘 − 𝑉𝑗𝑉 𝑘𝑐𝑗𝑘 sin 𝜄𝑗𝑘
2𝑐𝑗𝑘 + 𝑉𝑗𝑉 𝑘𝑐𝑗𝑘 cos𝜄𝑗𝑘 − 𝑉𝑗𝑉 𝑘𝑗𝑘 sin 𝜄𝑗𝑘
𝑘.
We use DC approximation which is based on:
𝑉𝑗 = 1 𝑞.𝑣. for all 𝑗 Pure reactive transmission lines – each line is characterized only by its reactance 𝑦𝑗𝑘 = −1/𝑐𝑗𝑘 Phase angle differences are “small”, implying that sin𝜄𝑗𝑘 ≈ 𝜄𝑗𝑘
Known as a reasonably good approximation Frequently used for contingency analysis
Do the assumptions hold during a cascade? 𝑘 𝑉𝑗 ≡ 1, ∀𝑗 𝑦𝑗𝑘 sin 𝜄𝑗𝑘 ≈ 𝜄𝑗𝑘 𝑗 𝑘 Load Generator 𝑉𝑗, 𝜄𝑗, 𝑄𝑗, 𝑅𝑗
A power flow is a solution (𝑔,𝜄) of:
𝑣𝑤 = 𝑞𝑣 𝑤∈𝑂 𝑣
𝑣𝑤 = 0,∀ 𝑣,𝑤 ∈ 𝐹
Matrix form:
𝑥∈𝑂 𝑣
If 𝐵+ is its pseudo-inverse
𝑣 𝑤 Load (𝑞𝑣 < 0) Generator (𝑞𝑣 > 0) 𝜄𝑣,𝑞𝑣
Different factors can be considered in modeling outage rules
The main is thermal capacity 𝑣𝑗𝑘
Simplistic approach: fail lines with 𝑔
𝑗𝑘 > 𝑣𝑗𝑘
More realistic policy:
𝑗𝑘 ≔ 𝛽 𝑔 𝑗𝑘 + 1 − 𝛽 𝑔 𝑗𝑘
(0 ≤ 𝛽 ≤ 1 is a parameter)
Deterministic outage rule:
𝑗𝑘 > 𝑣𝑗𝑘
Stochastic outage rules
5 10 15 20 1 2 3 4 5 6
𝑄1 = 𝑔
1 = 2000 MW
𝑄2 = 𝑔
2 = 1000 MW
𝑄13 = 1400 MW 𝑄3 = −𝑒3 = −3000 MW 𝑦13 = 10 Ω 1 3 2 𝑣13 = 1800 MW 𝑄13 = 3000 MW 𝑄3 = 0 MW 𝑄1 = 0 MW 𝑄2 = 0 MW
Until no more lines
fail do: Adjust the total demand to the total supply within each component of 𝐻 Use the power flow model to compute the flows in 𝐻 Update the state of lines 𝜊𝑗𝑘 according to the new flows Remove the lines from 𝐻 according to a given outage rule 𝑃 Initial failure causes disconnection
Obtained from the GIS (Platts Geographic Information System) Substantial processing of the raw data Used a modified Western Interconnect system, to avoid exposing
13,992 nodes (substations),
1,117 generators (red),
Assumed that demand is
N-Resilient, Factor of Safety K = 1.2
Flow Change after failure in the edge 𝑓
𝑓 = 𝑔′𝑓 − 𝑔 𝑓
Edge Flow Change Ratio
𝑓
𝑓
Mutual Edge Flow Change Ratio
𝑓
𝑓′
𝑓 = 4,5 𝑓′ = {1,5}
𝑓
𝑓′
Western interconnection: 1708-edge connected subgraph of the
Erdos-Renyi graph: A random graph where each edge appears
Watts and Strogatz graph: A small-world random graph where
Barabasi and Albert graph: A scale-free random graph where
Edge Flow Change Ratio , 𝑇𝑓,𝑓′ =
Δ𝑔
𝑓
𝑔
𝑓
Very large changes in the flow Sometimes far from the failure There are edges with positive flow increase from zero, far from
Objective: Compute the mutual edge flow change ratios Recall that Θ = 𝐵+𝑄 Method: Update the pseudo-inverse of the admittance matrix
*A similar theorem independently proved by Ranjan et al., 2014
𝑌 = 1,0,0,0,−1 𝑢 𝑗 𝑘
1 𝑏𝑗𝑘
−1+𝑌𝑢𝐵+𝑌 𝐵+𝑌𝑌𝑢𝐵+
Resistance Distance between nodes 𝑗 and 𝑘
+ + 𝑏𝑘𝑘 + − 2𝑏𝑗𝑘 +
The resistance distance between two nodes is a measure of their
All graphs have 1374 nodes All the edges have reactances equal to 1
𝑠 𝑗, 𝑘 𝑘 𝑗
Mutual Edge Flow Change Ratio
𝑓
𝑓′
Using pseudo-inverse of the admittance matrix, 𝑓 = 𝑗, 𝑘 , 𝑓′ = {𝑞,𝑟}
Mutual Edge flow change ratios (𝑁𝑓,𝑓′) are independent of the
Initial Failure
All graphs have 1374 nodes and each point represents the average
𝐵 = 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 −1 2 𝐵+ = 0.4 −0.2 −0.2 0.4 −0.2 −0.2 −0.2 0.4 −0.2 −0.2 −0.2 0.4 −0.2 −0.2 0.4 𝐵′ = 1 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 1 𝐵′+ = 1.2 0.4 −0.2 −0.6 −0.8 0.4 0.6 −0.4 −0.6 −0.2 0.4 −0.2 −0.6 −0.4 0.6 0.4 −0.8 −0.6 −0.2 0.4 .2
𝐵′ = 1 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 1 𝐵′+ = 1.2 0.4 −0.2 −0.6 −0.8 0.4 0.6 −0.4 −0.6 −0.2 0.4 −0.2 −0.6 −0.4 0.6 0.4 −0.8 −0.6 −0.2 0.4 .2
𝐵′′ = 1 −1 −1 1 1 −1 −1 2 −1 −1 1 𝑏′23
−1 − 2𝑏′23 + + 𝑏′22 + + 𝑏′33 + = 0
𝐵2
′+ − 𝐵3 ′+ = 0.6
0.6 −0.4 −0.4 −0.4
The Pseudo-inverse Based Algorithm identifies the evolution of
𝑢 ∗ 𝑊 2)
Compared to the classical algorithm which runs in 𝑃(𝑢 𝑊 3) If 𝑢 = |𝐺
𝑢 ∗| (one edge fails at each round), our algorithm performs
The color of each point represents the yield value of a cascade whose epicenter is at that point
Yield: The ratio between the demand supplied at stabilization and
The minimum Yield problem is NP-hard Based on our results, after failure on an edge {𝑞, 𝑟}, the flow
𝑗𝑘 ≤ 𝑠 𝑞,𝑟 1−𝑠 𝑞,𝑟 𝑔 𝑞𝑟
Edges with large 𝑠 𝑞,𝑟 𝑔
𝑞𝑟 have large impact on flow changes
The algorithm removes edges with large 𝑠 𝑞, 𝑟 𝑔
𝑞𝑟
The yield at stabilization
Cascade propagation models differ from the classical
Studied properties of the admittance matrix of the grid Derived analytical techniques for studying the impact of a single
Developed an efficient algorithm to identify the evolution of the
Developed a simple heuristic to detect the most vulnerable edges Using the resistance distance and the pseudo-inverse of