CASCADING FAILURE

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CASCADING FAILURE
Ian Dobson ECE dept., University of Wisconsin
USA Ben Carreras Oak Ridge National Lab
USA David Newman Physics dept., University of
Alaska USA
Presentation at University of Liege March 2003
Funding in part from USA DOE CERTS and NSF is
gratefully acknowledged
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power tail
probability
(log scale)
-1
S
-S
e
blackout size S (log scale)
power tails have huge impact on large blackout
risk.
risk probability x cost
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NERC blackout data15 years, 427 blackouts
1984-1998 (also sandpile data)
power tail in NERC data consistent with power
system operated near criticality
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Cascading failure large blackouts

• dependent rare events many combinations hard
  to analyze or simulate
• mechanisms hidden failures, overloads,
  oscillations, transients, control or operator
  error, ... but all depend on loading

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Loading and cascading

• LOW LOAD- weak dependence- events nearly
  independent - exponential tails in blackout size
  pdf
• CRITICAL LOAD- power tails in blackout size pdf
• HIGHER LOAD- strong dependence- total blackout
  likely

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Extremes of loading
log-log plot
VERY LOW LOADING independent failures pdf has
exponential tail
PDF
blackout size
TRANSITION ??
VERY HIGH LOADING total blackout with
probability one
PDF
blackout size
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Types of dependency in failure of systems with
many components

• independent
• common mode
• common cause
• cascading failure

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CASCADEA probabilistic loading-dependent model
of cascading failure
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CASCADE model

• n identical components with random initial load
  uniform in Lmin, Lmax
• initial disturbance D adds load to each component
• component fails when its load exceeds threshold
  Lfail and then adds load P to every other
  component. Load transfer amount P measures
  component coupling, dependency
• iterate until no further failures

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5 component example
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5 component example
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Normalize so that initial load range is 0,1
and failure threshold is 1

• normalized initial disturbance d d
• normalized load transfer p
• p

D - (Lfail - Lmax) Lmax -Lmin
P Lmax -Lmin
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Formulas for probability of r components fail
for 0ltdlt1
r-1
n-r
)
(
d (rpd) (1-rp-d) npdlt1 quasibinomial
distribution Consul 74
n r

• for npd gt1, extended quasibinomial
• quasibinomial for smaller r
• zero for intermediate r
• remaining probability for r n

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average number of failures lt r gt n100 components
d
p
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example of applicationmodeling load increase
-

• Lmax Lfail 1
• increase average load L by increasing Lmin

1
L
-
-
0
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example of application

• n 100 components
• P D 0.005
• p d

0.005 1 - Lmin
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probability distribution asaverage load L
increases
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ltrgt
average failures ltrgt versus load L pd and n100
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example 2 of applicationback off Lmax ( n-1
criterion)
-
Lfail 1
k
-
Lmax
-
Lmin 0
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Increase average load leads to change in d and p
constant
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GPD formulas for probability of r components
fail
r-1
-rl-q

• (rlq) e / r! nlqltnrltn
• remaining probability for r n.
• For rltn agrees with
• generalized Poisson distribution GPD

• for nlqgtn, extended GPD
• GPD for smaller r
• zero for intermediate r
• remaining probability for r n

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probability distribution as average load L
increasesGPD model
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SUMMARY OF CASCADE

• features of loading-dependent cascading failure
  are captured in probabilistic model with analytic
  solution
• extended quasibinomial distribution with n,d,p
  approximated by GPD with qnd, lnp.
• distributions show exponential or power tails or
  high probability of total failure
• power tail and total failure regimes show greatly
  increased risk of catastrophic failure
• power tails when lnp1.

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OPAA power systems blackoutmodel including
cascading failure and self-organizing dynamics
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Why would power systems operate near criticality??

• Near criticality, expected blackout size sharply
  increases increased risk of cascading failure.

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Forces shaping power transmission

• Load increase (2 per year) and increase in bulk
  power transfers, economics
• Engineering

• new controls and equipment
• upgrade weakest parts

these engineering forces are part of the
dynamics!
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Ingredients of SOC in idealized sandpile

• system state local max gradients
• event sand topples (cascade of events is an
  avalanche)
• addition of sand builds up sandpile
• gravity pulls down sandpile
• Hence dynamic equilibrium with avalanches of all
  sizes and long time correlations

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Analogy between power system and sand pile
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OPA model Summary

• transmission system modeled with DC load flow and
  LP dispatch
• random initial disturbances and probabilistic
  cascading line outages and overloads
• underlying load growth load variations
• engineering responses to blackouts upgrade lines
  involved in blackouts upgrade generation

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DC load flow model(linear, no losses, real power
only)
Power injections at buses P
max
generators have max power P
Line flows F
max
line flow limits F
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Slow and fast timescales

• SLOW load growth and responses to blackouts.
  (days to years)slow dynamics indexed by days
• FAST cascading events.(minutes to hours)fast
  dynamics happen at daily peak load timing
  neglected

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Response to blackout by engineers
For lines involved in the blackout,
max
increase line limit F by a fixed
percentage.
Also, when total generation margin drops below
threshold,increase generator power limit P at
selected generators coordinated with line limits.
max
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Fast cascade dynamics

• Start with daily flows and injections
• Outage lines with given probability (initial
  disturbance)
• Use LP to redispatch
• Outage lines overloaded in step 3 with given
  probability
• If outage goto 3, else stop

Objective produce list of lines involved in
cascade consistent with system constraints
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Conventional LP redispatch to satisfy limits
Minimize change in generation and loads (load
change weighted x 100) subject to
overall power balance line flow limits load
shedding positive and less than total
load generation positive and less than generator
limit
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Model
Is the total generation margin below critical?
1 day loop
Yes
Secular increase on demand Random fluctuation of
loads Upgrade of lines after blackout Possible
random outage
No
A new generator build after n days
LP calculation
If power shed, it is a blackout
Are any overload lines?
1 minute loop
Yes, test for outage
no
Yes
No outage
Line outage
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Possible Approaches to Modeling Blackout Dynamics
Complexity (nonlinear dynamics, interdependences)
Model detail (increase details in the
models, structure of networks,)
OPA model
By incorporating the complex behavior, the OPA
approach aims to extract universal features
(power tails,).
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OPA model results include

• self-organization to a dynamic equilibrium
• complicated critical point behaviors

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Time evolution

• The system evolves to steady state.
• A measure of the state of the system is the
  average fractional line loading.

200 days
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Steady state
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OPA/NERC results
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Application of the OPA model

• The probability distribution function of blackout
  size for different networks has a similar
  functional form - universality?

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Effect of blackout mitigation methods on pdf of
blackout size
obvious methods can have counterintuitive
effects
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Mitigation

• Require a certain minimum number of transmission
  lines to overload before any line outages can
  occur.

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A minimum number of line overloads before any
line outages

• With no mitigation, there are blackouts with line
  outages ranging from zero up to 20.
• When we suppress outages unless there are n gt
  nmax overloaded lines, there is an increase in
  the number of large blackouts.
• The overall result is only a reduction of 15 of
  the total number of blackouts.
• this reduction may not yield overall benefit to
  consumers.

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Forest fire mitigation
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Dynamics essential in evaluating blackout
mitigation methods

• Suppose power system organizes itself to near
  criticality
• We try a mitigation method requiring 30 lines to
  overload before outages occur.
• Method effective in short time scale. In long
  time scale very large blackouts occur.

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KEY POINTS

• NERC data suggests power tails and power system
  operated near criticality
• power tails imply significant risk of large
  blackouts and nonstandard risk analysis
• cascading loading-dependent failure
• engineering improvements and economic forces can
  drive to criticality
• in mitigating blackout risk, sensible approaches
  can have unintended consequences

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BIG PICTURE

• Substantial risk of large blackouts caused by
  cascading events need to address a huge number
  of rare interactions
• Where is the edge for high risk of cascading
  failure? How do we detect this in designing
  complex engineering systems?
• Risk analysis and blackout mitigation based on
  entire pdf, including high risk large blackouts.
• Developing understanding and methods is better
  than the direct experimental approach of waiting
  for large blackouts to happen!

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Papers on this topic are available from
http//eceserv0.ece.wisc.edu/dobson/home.html