1. 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

2. power tails have huge impact
probability
(log scale) -1 S e -S
blackout size S (log scale)
power tails have huge impact
on large blackout risk.
risk = probability x cost

3. NERC blackout data 15 years, 427 blackouts 1984-1998 (also sandpile data)
power tail in NERC data consistent with power system operated near criticality

4. 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

5. 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

6. Extremes of loading TRANSITION ?? VERY LOW 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

7. Types of dependency in failure of systems with many components
independent
common mode
common cause
cascading failure

8. CASCADE: A probabilistic loading-dependent model of cascading failure

9. 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

10. 5 component example

11. 5 component example

12. 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

13. Formulas for probability of r components fail for 0<d<1
n-r ( )
d (rp+d) (1-rp-d) ; np+d<1
quasibinomial distribution; Consul 74 n r
for np+d >1, extended quasibinomial:
quasibinomial for smaller r
zero for intermediate r
remaining probability for r = n

14. average number of failures < r >
n=100 components d p

15. example of application: modeling load increase-
Lmax = Lfail = 1
increase average load L by increasing Lmin 1 L - -

16. example of application:
n = 100 components
P = D = 0.005
p = d =
0.005
1 - Lmin

17. probability distribution as average load L increases

18. <r>
average # failures <r>
versus load L
p=d and n=100

19. example 2 of application: back off Lmax ( n-1 criterion)
Lfail = 1 k -
Lmax -
Lmin = 0

20. Increase average load leads to change in d and p constant

21. GPD formulas for probability of r components fail
-rl-q
(rl+q) e / r! ; nl+q<n;r<n
; remaining probability for r = n.
For r<n agrees with
generalized Poisson distribution GPD
for nl+q>n, extended GPD:
GPD for smaller r
zero for intermediate r
remaining probability for r = n

22. probability distribution as average load L increases GPD model

23. 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 q=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.

24. OPA: A power systems blackout model including cascading failure and self-organizing dynamics

25. Why would power systems operate near criticality??
Near criticality, expected blackout size sharply increases; increased risk of cascading failure.

26. 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!

27. 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

28. Analogy between power system and sand pile

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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 no
Yes, test for outage
Yes
No outage
Line outage

36. 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,…).

37. OPA model results include:
self-organization to a dynamic equilibrium
complicated critical point behaviors

38. Time evolution The system evolves to steady state.
A measure of the state of the system is the average fractional line loading.
200 days

39. Steady state

40. OPA/NERC results

41. Application of the OPA model
The probability distribution function of blackout size for different networks has a similar functional form - universality?

42. Effect of blackout mitigation methods on pdf of blackout size
“obvious” methods can have
counterintuitive effects

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

44. 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 > 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.

45. Forest fire mitigation

46. 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.

47. 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

48. 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!

49. Papers on this topic are available from http://eceserv0. ece. wisc