Transcript Initial evidence for self-organized criticality in blackouts

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
power tail
probability
(log scale)
S -1
-S
e
blackout size S (log scale)
power tails have huge impact
on large blackout risk.
risk = probability x cost
NERC blackout data
15 years, 427 blackouts 1984-1998
(also sandpile data)
Probability
10-1
10-2
10-3
10-4
10-5
10-6
100
Sandpile avalanche
MWh lost
101
102
103
Event size
104
power tail in NERC data consistent with power
system operated near criticality
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
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
Extremes of loading
log-log plot
PDF
blackout size
VERY LOW
LOADING
independent failures;
pdf has exponential tail
TRANSITION ??
VERY HIGH
LOADING
PDF
blackout size
total blackout
with probability
one
Types of dependency in failure of
systems with many components
•
•
•
•
independent
common mode
common cause
cascading failure
CASCADE:
A probabilistic
loading-dependent model
of cascading failure
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
5 component example
5 component example
Normalize so that
initial load range is [0,1]
and failure threshold is 1
normalized initial disturbance d
D - (Lfail - Lmax)
d=
Lmax -Lmin
normalized load transfer p
p=
P
Lmax -Lmin
Formulas for probability of
r components fail for 0<d<1
r-1
n-r
n
( r ) d (rp+d) (1-rp-d) ; np+d<1
quasibinomial distribution; Consul 74
for np+d >1, extended quasibinomial:
• quasibinomial for smaller r
• zero for intermediate r
• remaining probability for r = n
average number of failures < r >
n=100 components
d
p
example of application:
modeling load increase
• Lmax = Lfail = 1
-1
L
• increase average load L
by increasing Lmin
-0
example of application:
• n = 100 components
• P = D = 0.005
0.005
• p=d=
1 - Lmin
probability distribution as
average load L increases
<r>
average # failures <r>
versus load L
p=d and n=100
100
80
60
40
20
0.5
0.6
0.7
0.8
0.9
L
example 2 of application:
back off Lmax ( n-1 criterion)
- Lfail = 1
k
- Lmax
- Lmin = 0
Increase average load leads to
change in d and p constant
f
1
p=0.00075, n=1000
0.1
0.01
d=0.25
d=0.0005
d=0.05
d=0.2
0.001
0.0001
0.00001
1
5
10
50 100
r
500 1000
GPD formulas for probability of
r components fail
r-1 -rl-
(rl+) e
/ r! ; nl+<n;r<n
; remaining probability for r = n.
For r<n agrees with
generalized Poisson distribution GPD
for nl+>n, extended GPD:
• GPD for smaller r
• zero for intermediate r
• remaining probability for r = n
probability distribution as
average load L increases
GPD model
f
1
L=0.82
0.1
0.01
L=0.747
0.001
0.0001
0.00001
L=0.2
1
2
5
10
20
50
100
r
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 =nd, l=np.
• 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 l=np=1.
OPA:
A power systems blackout
model including cascading
failure and self-organizing
dynamics
Why would power systems operate
near criticality??
• Near criticality, expected blackout size sharply
increases; increased risk of cascading failure.
Power Served
1.6 104
25
1.4 104
20
15
1.2 104
10
5
outages
0
1 104
1.2 104 1.4 104 1.6 104 1.8 104
Power Demand
Power Served
<Number of line outages>
30
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!
Ingredients of SOC in idealized
sandpile
• system state = local max gradients
• event = sand topples (cascade of events is an
avalanche)
1 addition of sand builds up sandpile
2 gravity pulls down sandpile
• Hence dynamic equilibrium with avalanches of all
sizes and long time correlations
system state
driving force
relaxing force
event
Power system
Sand pile
loading pattern
customer demand
response to blackout
limit flow or trip
gradient profile
adding sand
gravity
sand topples
Analogy between power system and sand pile
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
DC load flow model
(linear, no losses, real power only)
Power injections at buses P
generators have max power P
max
Line flows F
max
line flow limits + F
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
Response to blackout by engineers
For lines involved in the blackout,
increase line limit Fmax by a fixed percentage.
Also, when total generation margin drops below
threshold,increase generator power limit P max
at selected generators coordinated with line
limits.
Fast cascade dynamics
1 Start with daily flows and injections
2 Outage lines with given probability (initial
disturbance)
3 Use LP to redispatch
4 Outage lines overloaded in step 3 with
given probability
5 If outage goto 3, else stop
Objective: produce list of lines involved in
cascade consistent with system constraints
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
Model
1 day loop
Secular increase on demand
Random fluctuation of loads
Upgrade of lines after blackout
Possible random outage
If power shed,
it is a blackout
no
No outage
Is the total generation
margin below critical?
Yes
No
A new generator
build after n days
LP calculation
Are any overload lines?
1 minute
loop
Yes, test for outage
Line outage
Yes
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,…).
OPA model results include:
• self-organization to a dynamic equilibrium
• complicated critical point behaviors
Time evolution
• The system evolves to steady state.
• A measure of the state of the system is the average
fractional line loading.
Fij  Power flow between
70
60
50
40
30
20
10
0
2 104
4 104 6 104
Time (days)
8 104
20
15
10
5
6.06 10
4
4
6.07 10
Time (days)
200 days
Total overloads
1
0.9
0.8
0.7
0.6
0.5
0.4
0 100
nodes i and j
Total overloads
<M>
Fij
1
M =

Number lines Lines Fijmax
0
6.08 10 4
Steady state
Probabilty distribution
OPA/NERC results
102
101
100
10-1
10-2
10-4
NERC data
382-node Network
10-3
10-2
10-1
100
Load shed/ P ower served
Application of the OPA model
• The probability
distribution function of
blackout size for
different networks has a
similar functional form universality?
Probability distribution
102
101
100
IEEE 118
WSCC
N = 382
10-1
10-3
10-2
10-1
100
Load shed/ Power delivered
Effect of blackout mitigation
methods on pdf of blackout size
“obvious” methods can have
counterintuitive effects
Mitigation
• Require a certain minimum number of
transmission lines to overload before any
line outages can occur.
A minimum number of line
overloads before any line outages
Number of blackouts
• With no mitigation, there are
blackouts with line outages
ranging from zero up to 20.
• When we suppress outages
unless there are n > 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.
105
104
Base case
nmax = 10
nmax = 20
nmax = 30
103
102
101
100
0 10 20 30 40 50 60 70
Line outages
Forest fire mitigation
10 4
# of events of given size
Trees burned with efficient firefighters
Trees burned without firefighters
1000
100
10
1
0.1
20
40
60
Size of fire
80
100
Dynamics essential in evaluating blackout
mitigation methods
0.8
<M>
0.6
Suppress
n < 30
100
80
<M>
0.4
Standard
0.2
outages
0
-0.2
-0.4
-0.6
0
Number of line outages per blackout
• 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.
60
40
20
0
5000 10000 15000 20000
Time (days)
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
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!
Papers on this topic are available
from
http://eceserv0.ece.wisc.edu/~dobson/home.html