A probabilistic loadingdependent model of cascading failure and possible implications for blackouts
ABSTRACT Catastrophic disruptions of large, interconnected infrastructure systems are often due to cascading failure. For example, large blackouts of electric power systems are typically caused by cascading failure of heavily loaded system components. We introduce the CASCADE model of cascading failure of a system with many identical components randomly loaded. An initial disturbance causes some components to fail by exceeding their loading limit. Failure of a component causes a fixed load increase for other components. As components fail, the system becomes more loaded and cascading failure of further components becomes likely. The probability distribution of the number of failed components is an extended quasibinomial distribution. Explicit formulas for the extended quasibinomial distribution are derived using a recursion. The CASCADE model in a restricted parameter range gives a new model yielding the quasibinomial distribution. Some qualitative behaviors of the extended quasibinomial distribution are illustrated, including regimes with power tails, exponential tails, and significant probabilities of total system failure.

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ABSTRACT: As the interconnection of power grids, failure of single component could propagate out, and because of the scale of the power system, it is almost impossible to analysis all failure patterns. Traditional methods for online cascading analysis could face almost unsolvable obstacles on computational burden and accuracy. This paper discusses the mechanisms of early warning of cascading blackouts. Researches have shown, from the complex system point of view, the existence of earlywarning signals to indicate for a wide class of systems whether a critical threshold is approaching. Instead of focusing on identifying the possible failure patterns that could lead to cascading blackouts, from the complex system point of view, early warning of cascading blackouts could be realized by identifying how far the system is to the critical state by simulation using fast computational techniques and a group of system indices, providing helpful aid for the operator.Power System Technology (POWERCON), 2012 IEEE International Conference on; 01/2012  [Show abstract] [Hide abstract]
ABSTRACT: Failure behavior is the state change process of product or part of a product which is relative to its environment, over time performance and can be detected from the outside. According to the different level of analysis, failure behavior analysis method can be divided into element failure behavior analysis method and system failure behavior analysis method. The formal reveals the variety failure mechanisms under the alone or coupled action of internal cause and external cause using coupling analysis method; the latter focuses on product failure performance law under the effect of variety failure mechanisms by means of failure propagation analysis or state analysis. This critical review from two aspects of element and system investigates and summarizes the current research status of failure behavior analysis method. The result shows that coupling analysis method has been mature at present, and there is plenty of supporting software for computeraided analysis. Failure propagation analysis method consists of graph theory based method, Petri Net method and complex network method. But the abovementioned methods are unilateral and isolated from each other. System failure behavior analysis method needs to synthetically use the current methods — coupling analysis, failure propagation analysis and state analysis method so that forms an analysis methodology which needs to clear the input and output of each method and improve the interface between application software. The comprehensive methodology that the failure behavior analysis follows will provide support to product reliability analysis and design improvement.01/2012;  SourceAvailable from: Junjian Qi
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ABSTRACT: In this paper, a blackout model that considers the slow process at the beginning of blackouts is proposed based on the improved OPA model. The model contains two layers of iteration. The inner iteration, which describes the fast dynamics of the system, simulates the power system cascading failure, including the tree contact and failure of lines caused by heating. Compared with the improved OPA model, the outer iteration, which describes the longterm slow dynamics of the system, adds the simulation of tree growth and utility vegetation management (UVM). Moreover, the proposed model also improves the simulation of protective relays and the dispatching center and makes them closer to practical conditions. The effectiveness of the proposed model is verified by the simulation results of Northeast Power Grid of China.Power and Energy Society General Meeting, 2012 IEEE; 07/2012
Page 1
Hawaii International Conference on System Sciences, January 2003, Hawaii. c ? 2003 IEEE.
A probabilistic loadingdependent model of cascading failure
and possible implications for blackouts
Ian Dobson
ECE Department
University of Wisconsin
Madison WI 53706 USA
dobson@engr.wisc.edu
Benjamin A. Carreras
Oak Ridge National
Laboratory
Oak Ridge TN 37831 USA
carrerasba@ornl.gov
David E. Newman
Physics Department
University of Alaska
Fairbanks AK 99775 USA
ffden@uaf.edu
Abstract
Catastrophic disruptions of large, interconnected infras
tructure systems are often due to cascading failure. For ex
ample, large blackouts of electric power systems are typi
cally caused by cascading failure of heavily loaded system
components. We introduce the CASCADE model of cascad
ing failure of a system with many identical components ran
domly loaded. An initial disturbance causes some compo
nents to fail by exceeding their loading limit. Failure of a
component causes a fixed load increase for other compo
nents. As components fail, the system becomes more loaded
and cascading failure of furthercomponentsbecomeslikely.
The probability distribution of the number of failed compo
nents is an extended quasibinomial distribution. Explicit
formulas for the extended quasibinomial distribution are
derived using a recursion. The CASCADE model in a re
stricted parameter range gives a new model yielding the
quasibinomial distribution. Some qualitative behaviors of
the extended quasibinomial distribution are illustrated, in
cluding regimes with power tails, exponential tails, and sig
nificant probabilities of total system failure.
1. CASCADE model
Cascading failure is a standard cause of catastrophic fail
ure in large, interconnected infrastructure systems. For ex
ample, loadingdependent cascading failure occurs in large
blackouts of electric power transmission systems. The im
portance of these infrastructures to society motivates the
construction and study of models that capture salient fea
tures of cascading failure.
We define the CASCADE model of probabilistic cascad
ing failure with the following general features:
1. Multiple identical components, each of which has a
random initial load and an initial disturbance.
2. When a component overloads, it fails and transfers
some load to the other components.
Property 2 can cause cascading failure: a failure addition
ally loads other components and some of these other com
ponents may also fail, leading to a cascade of failure. The
components become progressively more loaded as the cas
cade proceeds. An initial version of CASCADE was used
to examine power transmission system critical loading and
power tails in probability distributions of blackout size [8].
1.1. Description of model
The CASCADE model has n identical components with
random initial loads. For each component the minimum ini
tial load is Lminand the maximum initial load is Lmax.
For j=1,2,...,n, component j has initial load Lj that is
a random variable uniformly distributed in [Lmin,Lmax].
L1,L2,···,Lnare independent.
Components fail when their load exceeds Lfail. When a
component fails, a fixed amount of load P is transferred to
each of the components.
To start the cascade, we assume an initial disturbance
that loads each component by an additional amount D.
Other components may then fail depending on their initial
loads Lj and the failure of any of these components will
distribute an additional load P ≥ 0 that can cause further
failures in a cascade. The CASCADE model can be defined
more precisely in algorithmic form:
Algorithm for CASCADE model
0. All n components are initially unfailed and have initial
loads L1,L2,···,Lndetermined as independent ran
dom variables uniformly distributed in [Lmin,Lmax].
1. Add the initial disturbance D to the load of component
j for each j = 1,...,n. Initialize iteration counter i to
one.
Page 2
2. Test each unfailed component for failure: For j =
1,...,n, if component j is unfailed and its load > Lfail
then component j fails. Suppose that micomponents
fail in this step.
3. If mi= 0, stop; the cascading process ends.
4. If mi > 0, then increment the component loads ac
cording to the number of failures mi: Add miP to the
load of component j for j = 1,...,n.
5. Increment iteration counter i and go to step 2
A simple example of the CASCADE model with 5 com
ponents producing a cascade is shown in Table 1. In this
cascade, two components fail in iteration 1, one compo
nent fails in iteration 2, and one component fails in iteration
3, and then the cascade ends with a total of 4 components
failed. That is, m1= 2, m2= 1, m3= 1, and then the cas
cade ends with a total of m1+ m2+ m3= 4 components
failed. Fig. 1 shows the succession of load increases in this
cascade labelled by their iteration number.
Table1.Componentloadsincreasinginacas
cade example with 5 components
initial load range [Lmin,Lmax] = [0.5,8.5]
failure load Lfail= 9.5
initial disturbance D = 3
load increment P = 1
iteration counter i
1234
8674
119 107
1311129
1412 1310
1513 1411
cascade ends with 1,3,2,4 failed
5
1
4
6
7
8
component number
initial random load Lj
initial disturbance D added
1 and 3 fail; 2P added
2 fails; P added
4 fails; P added
i
0
1
2
3
4
– integer loads used for convenience only.
– loads above Lfail= 9.5 indicate failure.
Since the cascade of failures is only determined by the
component loads relative to Lfail, an equivalent way to vi
sualize the cascade shows the component loads as if fixed at
their initial values and shifts the Lfailhorizontal line down
wards as shown in Fig. 2. In this point of view, the line at
first shifts downwards by the initial disturbance D and then
shifts down by miP in subsequent iterations. In Fig. 2, mi,
the previous number of component failures, conveniently
appears as the previous number of zeros that the line passes.
Consider the event that exactly r components fail in
a cascade governed by the algorithm for the CASCADE
12
component number
345
2
4
6
8
10
12
14
load
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
Lfail
Lmax
Lmin
Figure 1. Component loads increasing in cas
cade example with 5 components. 0 indicates
initial load and 1,2,3,4 indicate load at itera
tions 1,2,3,4 respectively. A component fails
when its load exceeds the Lfailhorizontal line.
12345
component number
load
0
0
0
0
0
D
2P
P
P
0
1
2
3
4
Figure 2. Alternative visualization of cascade
example with loads (zeros) shown as if fixed
at their initial values and the Lfailhorizontal
line shifting downwards. The line first shifts
down by D = 3 and subsequently shifts down
proportionally to the previous number of fail
ures (number of zeros the line just passed).
model. For convenience of description, we renumber the
components in the order they failed. Then an equivalent
description of the event that exactly r components fail is
that there are positive integers k and m1,m2,···,mksuch
that equations (1–6) on page 4 apply. We give some ex
amples to explain (2–6). Since component 1 failed on the
first iteration after its load L1was increased by the initial
disturbance D, L1+ D > Lfail. This is an example of
(2). After the first iteration, all the loads are increased by
m1P. Since the m1+ 1 component failed on the second
iteration, Lm1+1+ m1P + D > Lfail. Moreover, since
the m1+ 1 component did not fail on the first iteration,
Lm1+1+D ≤ Lfail. This is an example of (3). The compo
Page 3
nent r + 1 never fails and the load of component r + 1 was
increased by D + rP by the end of the cascade. Therefore
Lr+1+ D + rP ≤ Lfail. This is an example of (6).
The CASCADE model is summarized in table 2.
Table 2. CASCADE model with load range
[Lmin,Lmax] and failure load Lfail
Lj
initial load of component j;
uniformly distributed in [Lmin,Lmax]
load increase at each
component when failure
initial disturbance
number of components
max component load
min component load
component failure load
P
D
n
Lmax
Lmin
Lfail
Table 3. Normalized CASCADE model with
load range [0,1] and failure load 1
formula
Lj−Lmin
Lmax−Lmin
?j
initial load of component j;
uniformly distributed in [0,1]
load increase at each
component when failure
initial disturbance
number of components
max component load
min component load
component failure load
p
P
Lmax−Lmin
D+Lmax−Lfail
Lmax−Lmin
d
n
1
0
1
1.2. Normalized model
Suppose that the initial disturbance D is changed to
D+K and that the failure load Lfailis changed to Lfail+K.
Adding K to the initial disturbance has the effect of increas
ing all the loads by K at the beginning of the cascade, but
since the failure load is also increased by K, the failure of
components is unchanged, and this holds throughout the en
tire cascading process. That is, the cascade of component
failures is unchanged, except that all loads and failure loads
are increased in value by K. This conclusion holds for both
positive and negative K. Thus we have
Principle 1 The cascade process is unchanged if the initial
disturbance D and the component failure load Lfailare in
cremented by the same amount K.
Principle 1 also follows from the expressions in (2–6) only
depending on the difference Lfail− D.
It is convenient to normalize the model in two steps:
The first step is to change the initial disturbance D to
D + Lmax− Lfailand change the failure load Lfailto
Lfail+ Lmax− Lfail= Lmax. According to Principle 1
with K = Lmax−Lfail, these changes have no effect on the
componentfailuresinthecascadingprocess. Thesechanges
are chosen to make Lfailcoincide with Lmax.
The second step defines the normalized initial load
?j=
Lj− Lmin
Lmax− Lmin
(12)
Then ?jis a random variable uniformly distributed on [0,1].
Moreover, the failure load is ?j= 1. Let
p =
P
Lmax− Lmin,d =D + Lmax− Lfail
Lmax− Lmin
(13)
Then p is the amount of load increase on any component
when one other component fails expressed as a fraction of
the load range Lmax− Lmin. Similarly d is the initial dis
turbance expressed as a fraction of the load range. The
CASCADE model and its normalized parameters are sum
marized in table 3. Also, the normalized conditions for r
components to fail are shown in (1) and (7–11).
2. Distribution of number of failed components
This section derives recursive and explicit formulas for
the probability distribution of the number of failed compo
nents.
Definition: f(r,d,p,n) is the probability that r compo
nents fail in the CASCADE model with normalized initial
disturbance d, normalized load transfer amount p, and n
components.
2.1. Cases d ≤ 0 and d ≥ 1
When the initial disturbance d ≤ 0, no components fail
and
?1
When the initial disturbance d ≥ 1, all n components
fail immediately and
?0
2.2. Deriving the recursion for 0 < d < 1
f(r,d,p,n) =
;r = 0
;0 < r ≤ n
0
?
,d ≤ 0
(14)
f(r,d,p,n) =
;0 ≤ r < n
;r = n
1
?
,d ≥ 1
(15)
This subsection assumes throughout that0 < d < 1. The
initial disturbance d causes immediate failure of the com
ponents that have initial load ? in (1 − d,1]. Therefore the
Page 4
Condition for r components to fail
r = m1+ m2+ ··· +mk with mi> 0, i = 1,2,···,k
Lfail− D < L1,···,Lm1
Lfail− D − m1P < Lm1+1,···,Lm1+m2≤ Lfail− D
Lfail− D − (m1+ m2)P < Lm1+m2+1,···,Lm1+m2+m3≤ Lfail− D − m1P
...
Lfail− D − (m1+ m2+ ··· + mk−1)P < Lm1+···+mk−1+1,···,Lr≤ Lfail− D − (m1+ m2+ ··· + mk−2)P (5)
Lr+1,···,Ln≤ Lfail− D − rP
(1)
(2)
(3)
(4)
(6)
Normalized version of (2–6):
1 − d < ?1,···,?m1
1 − d − m1p < ?m1+1,···,?m1+m2≤ 1 − d
1 − d − (m1+ m2)p < ?m1+m2+1,···,?m1+m2+m3≤ 1 − d − m1p
...
1 − d − (m1+ m2+ ··· + mk−1)p < ?m1+···+mk−1+1,···,?r≤ 1 − d − (m1+ m2+ ··· + mk−2)p (10)
?r+1,···,?n≤ 1 − d − rp
(7)
(8)
(9)
(11)
probability of any component immediately failing is d and
the probability of any component not immediately failing is
1−d. Since the initial loads are independent, the probability
that r = 0 components fail is
f(0,d,p,n) = (1 − d)n
(16)
Also, in the case of one component (n = 1),
f(1,d,p,1) = d
(17)
For the rest of the subsection we assume that 1 ≤ r ≤ n.
If r ≥ 1 components fail, then a certain number of com
ponents k with 1 ≤ k ≤ r must have failed immediately
due to the initial disturbance d only. Let Ekbe the event
that k components fail immediately. Then, since the initial
component loads are independent,
?n
Since E1,E2,···,Er are mutually exclusive and collec
tively exhaustive events, the law of total probability gives
P[Ek] =
k
?
dk(1 − d)n−k
(18)
f(r,d,p,n) = P[r components fail]
r
?
We claim that
=
k=1
P[r components failEk]P[Ek]
(19)
P[r components failEk] = f(r − k,
kp
1 − d,
p
1 − d,n − k)
(20)
To establish claim (20), consider the n−k components that
did not immediately fail under the condition that Ek oc
curred. Since none of the n − k components failed imme
diately, their loads ? must lie in [0,1−d] and are uniformly
distributed in [0,1−d] (that is, the distribution conditioned
on Ekis uniform in [0,1 − d].) For the n − k components,
Lmin= 0 and Lmax= 1 − d.
AfterEk, eachofthen−k componentshashadaloadin
crease d from the initial disturbance and an additional load
increase kp from the immediate failure of k components.
Therefore, after Ek, the total initial disturbance for each of
the n − k components is D = kp + d.
To summarize, after Ek, the failure of the n − k initially
unfailed components is governed by the CASCADE model
with initial disturbance D = kp + d, load transfer P = p,
Lmin= 0, Lmax= 1−d, Lfail= 1, and n−k components.
Normalizing using (13) yields that the failure of the
n−k initially unfailed components is governed by the CAS
CADE model with normalized initial disturbance
malized load transfer
the probability that r components fail given Ekis the prob
ability that r − k of the n − k components fail and this
probability is given by (20).
Combining (18), (19), (20) yields the recursion
kp
1−d, nor
p
1−dand n−k components. Therefore
f(r,d,p,n) =
r
?
; 0 < r ≤ n,
k=1
?n
k
?
dk(1 − d)n−kf(r − k,
0 < d < 1
kp
1 − d,
p
1 − d,n − k)
(21)
Page 5
2.3. Extended quasibinomial formula for f
This subsection establishes the extended quasibinomial
formulas for the distribution f of the number of failed com
ponents. The special case of the quasibinomial distribution
is also discussed.
We summarize formulas (15–17) and recursion (21):
f(r,d,p,n) =
Equations
f(r,d,p,n) for all n ≥ 1 and d > 0. It is straightforward
to prove by induction that f(r,d,p,n) is a probability
distribution for all n ≥ 1 as detailed in Appendix A.
Consider the expressions
?n
;0 ≤ r ≤ (1 − d)/p, r < n
f(r,d,p,n) = 0 ;(1 − d)/p < r < n, r ≥ 0
n−1
?
Appendix B proves that (2729) satisfy (22−26) and hence
are explicit formulas for the extended quasibinomial proba
bility distribution.1
For np+d ≤ 1 the probability distribution given by (27
29) reduces to (27) alone:
?n
and this is the quasibinomial distribution introduced by
Consul[5]tomodelanurnprobleminwhichaplayermakes
strategic decisions. Thus (2729) is an extended quasibino
mial distribution in that its range of parameters is extended
to allow np + d > 1. We will see below that this extended
parameter range often describes regimes with a high proba
bility of all components failing.
0
1
; 0 ≤ r < n,
; r = n,
d ≥ 1
d ≥ 1
0 < d < 1
0 < d < 1
(22)
(23)
(24)
(25)
(1 − d)n; r = 0,
d
r
?
; r = 1, n = 1,
?
; r > 0, n > 1,
k=1
?n
k
dk(1 − d)n−kf(r − k,
kp
1 − d,
p
1 − d,n − k)
0 < d < 1
(26)
(22−25)and recursion(26) define
f(r,d,p,n) =
r
?
d(rp + d)r−1(1 − rp − d)n−r
(27)
(28)
f(n,d,p,n) = 1 −
s=0
f(s,d,p,n)
(29)
f(r,d,p,n) =
r
?
d(rp + d)r−1(1 − rp − d)n−r(30)
1We comment on how (2729) apply for np+d > 1 or n > (1−d)/p.
If (1−d)/p < n < 1+(1−d)/p, (27) applies for r < n, (28) does not
apply, and (29) gives the probability of r = n. If n > 1+(1−d)/p, (27)
appliesonlyforr < (1−d)/p, (28)giveszeroprobabilityfor(1−d)/p <
r < n, and (29) gives the probability of r = n.
0 0.00050.001
p
0.0015 0.002
0.2
0
0.2
0.4
0.6
0.8
1
d
0
1000
Figure 3. Average number < r > of compo
nents failed as a function of p and d for
n = 1000. Linesarecontoursofconstant<r>.
White indicates < 100 failures and black indi
cates > 900 failures.
Consul [5] has derived the mean of the quasibinomial
distribution (30) as
nd
n−1
?
r=0
(n − 1)!
(n − r − 1)!pr
(31)
3. Examples of CASCADE distribution
This section shows examples of the qualitative behavior
of the CASCADE distribution for n = 1000 components
and how modeling choices lead to parameterizations of the
normalized CASCADE model.
3.1. Average number of failures
One way to summarize the behavior of the CASCADE
distribution as the parameters vary is to examine the average
number <r> of components failed. A contour plot of <r>
as a function of parameters d and p is shown in Fig. 3. For
the case n = 1000 considered in Fig. 3, the line np+d = 1
joins (p,d) = (1/n,0) = (0.001,0) to (p,d) = (0,1).
The region of Fig. 3 to the right of the line np + d = 1
and above the line d = 0 is mostly black due to the high
probability of all components failing making the average
number of failures near 1000 (but note that the corner of the
Page 6
black region near (p,d) = (0.001,0) is rounded so that a
neighborhood of (p,d) = (0.001,0) is white).
In the region to the left of the line np+d = 1 and above
the line d = 0, the quasibinomial distribution (30) applies
and the average number < r > of failures is governed by
(31). In particular, <r> is proportional to d.
The following subsections examine the extended quasib
inomial distribution in more detail along lines in Fig. 3 such
as p constant and d varying, or d constant and p varying.
3.2. Binomial distribution for p = 0
One limiting case of the CASCADE model occurs when
p = 0 and the distribution becomes binomial:
?n
It is well known that for large n and small d, (32) is well
approximated by the Gaussian distribution with mean nd
and variance nd(1 − d).
3.3. Behavior for constant small d
f(r,d,0,n) =
r
?
dr(1 − d)n−r
(32)
We assume that the normalized initial disturbance is
fixed at the small value d = 0.0001 and examine how the
distribution changes as p increases. For p = 0, the distri
bution is binomial as explained above. The distribution as
p increases from zero is shown in Fig. 4. The distribution
for p = 0.0001 has a tail slightly heavier than binomial but
still approximately exponential. The tail becomes heavier
as p increases and the distribution for p = 0.001 has an ap
proximate power tail over a range of r. Note that p = 0.001
approximately satisfies the condition np + d = 1. The dis
tribution for p = 0.002 has an approximately exponential
tail for small r, zero probability of intermediate r, and a
probability of 0.08 of all 1000 components failing. (If an
intermediate number of components fail, then the cascade
always proceeds to all 1000 components failing.)
A previous, restricted version2of CASCADE obtained
qualitatively similar results by assuming p = d and increas
ing p. These results were qualitatively similar to simulation
results obtained by increasing loading in models of black
outs caused by cascading failure in electric power transmis
sion systems [8].
Increasing the normalized power transfer p may be
thought of as strengthening the component interaction that
causes cascading failure. The progression in the pdf as p in
creases from exponential tail to power tail and then to an ex
ponential tail together with a significant probability of total
failure is of interest. When the power tail or the significant
2The CASCADE model of [8] in effect assumed P = D and Lmax=
Lfail= 1. Average loading L = (1+Lmin)/2 was increased by increas
ing Lmin, leading to increasing p(L) = d(L) = D/(2 − 2L).
probability of total failure occur, they have an important im
pact on the risk of cascading failure [3].
The exponent of the power tail can be approximated as
follows. Suppose that np + d ≤ 1. By writing λ = np and
θ = nd, and letting n → ∞ and p → 0 and d → 0 in such a
way that λ = np and θ = nd are fixed, it can be shown [6]
using Stirling’s formula that the quasibinomial distribution
f(r,d,p,n) may be approximated by
g(r,θ,λ) = θ(rλ + θ)r−1e−rλ−θ
r!
(33)
which is the generalized Poisson distribution [6].
Now for r >> 1, use Stirling’s formula to get
g(r,θ,λ) ≈
θ
(rλ + θ)√2πrλre(λ−1)(−r−θ
λ)
In the limit, the condition np + d = 1 becomes λ = 1 and
setting λ = 1 gives
g(r,θ,λ) ≈
θ
(r + θ)√2πr
(34)
which has gradient on a plot of logg against logr of −1.5+
θ
r+θ→ −1.5 as r → ∞.
15 1050 1005001000r
0.00001
0.0001
0.001
0.01
0.1
1f
d=0.0001, n=1000
p=0.002
p=0.0001
p=0.001
Figure 4. pdf for constant small d and varying
p. Probability of no failures is 0.90 for all p.
Lines are drawn through the plotted points.
p = 0.002 has isolated point at (1000,0.08).
3.4. Effect of reducing maximum load
One way to try to mitigate cascading failure in the CAS
CADE model is to reduce the maximum component load
Lmaxbelow Lfail. Let K = Lfail− Lmaxbe the amount
of the maximum load reduction. Then the system has no
failures for any initial disturbance D smaller than K. The
normalized amount of maximum load reduction is
k =
K
Lmax− Lmin
(35)
Page 7
and we will observe the effect of increasing k from zero.
The normalized initial disturbance d(k) and the normal
ized power transfer p(k) are now functions of k. In particu
lar, according to (13),
d(k) =
D − K
Lmax− K − Lmin=
P
Lmax− K − Lmin=
D − (Lmax− Lmin)k
(Lmax− Lmin)(1 − k)
P
(Lmax− Lmin)(1 − k)
p(k) =
Hence
d(k) =d(0) − k
1 − k
,p(k) =
p(0)
1 − k
(36)
and the probability of r components failing is
f(r,d(k),p(k),n) = f(r,d(0) − k
1 − k
,p(0)
1 − k,n)
(37)
This shows the effect of reducing the maximum load. If
k > d(0) then d(k) < 0 and there are no failures (14). For
small k, d(k) ≈ d(0) − k(1 − d(0)) so that the normalized
initial disturbance is decreased. However, the normalized
load transfer amount p(k) =
is shown in Fig. 5. The probability of no failures (r = 0;
p(0)
1−kincreases. An example
15 1050 1005001000r
0.00001
0.0001
0.001
0.01
0.1
1f
d(0)=0.001 p(0)=0.00075, n=1000
k=0
k=0.0009
Figure 5. Effect on pdf of increasing k from 0.
not shown in Fig. 5) is 0.37 for k = 0 and 0.90 for k =
0.0009. In this case the mitigation increases the probability
of no failures and decreases the probability of failures over
a range of r ≥ 1. However, the increase in p(k) makes the
tail of the distribution slightly heavier.
3.5. Effect of changing average load
We assume that the initial disturbance D = 0 and exam
ine the effect of changing the initial load range by changing
the average initial load L defined by
L =1
2(Lmax+ Lmin)
(38)
The range of initial load W = Lmax− Lmin. Then the
initial load range can be parameterized by L and W:
[Lmin,Lmax] = [L − W/2,L + W/2]
We choose to let W be constant and examine the effect
of varying L. Then, according to (13), the normalized initial
disturbance d(L) becomes an affine function of L and the
power transfer p is a constant:
(39)
d(L) =L + W/2 − Lfail
W
,p =P
W
(40)
The probability of r components failing is
f(r,d(L),p,n) = f(r,L + W/2 − Lfail
W
,P
W,n)
(41)
If L < Lfail− W/2, then d(L) < 0 and there are no
failures (14). If L > Lfail+ W/2, then d > 1 and all
components fail (15).
Now consider Lfail− W/2 ≤ L ≤ Lfail+ W/2 so that
0 ≤ d(L) ≤ 1. In particular, we choose Lfail= 1 and
W = 0.2 so that 0.9 ≤ L ≤ 1.1 and choose p = 0.00075
and n = 1000. Fig. 6 shows the pdf for d = 0.0005, 0.05,
0.2, 0.25 and, respectively, L = 0.9001, 0.91, 0.94, 0.95.
For this case, np + d = 1 occurs at d = 0.25 and the pdf
quickly becomes all components fail with probability one
for d > 0.25. In the range 0 ≤ d ≤ 0.25 the quasibinomial
formulas (30) and (31) apply and hence the mean number
of failed components increases linearly with d and L.
151050 1005001000r
0.00001
0.0001
0.001
0.01
0.1
1f
p=0.00075, n=1000
d=0.0005 d=0.05d=0.2
d=0.25
Figure 6. pdf for increasing d.
4. Application to blackouts
CASCADE is much too simple to represent with realism
the detailed aspects of a large scale electric power system.
However, it is plausible that the general features of CAS
CADE can be present in large blackouts involving cascad
ing failure of transmission lines and generation.
Page 8
There are many ways in which power system component
failures interact, including via the protection system, redis
tribution of power flow, dynamics, and operator or planning
errors. However, in accordance with the CASCADE model,
it is generally true that these interactions are loading de
pendent and failure of one component tends to stress other
components and make their failure more likely. If we focus
on cascading failure with interactions due to line overloads
and outages via redistribution of power flow (DC power
flow with LP dispatch), then results from the OPA model
[1, 2, 7] suggest that a variety of operational regimes are
possible [1, 2, 4]. Some of these regimes yield distributions
of blackout sizes that are qualitatively similar to those ob
served in CASCADE in the examples of Section 3.
The examples of Section 3 also illustrate how different
models and parameterizations can be defined in the CAS
CADE model and then normalized so that the formulas for
the CASCADE distribution apply. For example, the maxi
mum load reduction in Section 3.4 roughly models some of
the overall features of applying an nk security criterion to
a system with cascading failure. (If we regard initial com
ponent failures as providing an initial disturbance to n re
maining components, then the reduction in maximum load
roughly approximates in CASCADE the effect of the nk
criterion in that the system can survive a certain number k of
initial component failures proportional to K. The analysis
in Section 3.4 considers only the n remaining components
where n = nk.)
The CASCADE model can be used to test ideas about
cascading failure in complicated contexts such as large
power system blackouts. Indeed, as sketched in section 3.3,
a restricted version of the CASCADE model has already
been used to show power tail behavior qualitatively similar
to that obtained in blackout models [8]. Further application
of CASCADE to better understand cascading failure in the
OPA blackout model [1, 2, 4] and the power system itself is
promising future work.
We have started to explore the relation of CASCADE
to fiber bundle models of material failure [9]. Instead of
regarding initial loads Lj as random and the failure load
Lfailas fixed, one can regard Lfailas random and Lj as
fixed. This makes CASCADE more comparable to a model
of a single cascade of failure in an increasingly loaded
fiber bundle. Global load sharing fiber bundle models
redistribute the load of a failed fiber equally to all unfailed
components so that, in contrast to CASCADE, the amount
of load transferred to each unfailed component tends to
increase as the cascade proceeds. However, Kloster et al.
hold load transfer fixed when analyzing a single cascade of
failure and obtain the generalized Poisson distribution (33)
in equation (11) of [9].
5. Conclusions
We define and explain the CASCADE model of loading
dependent cascading failure of a system of identical compo
nents. Formulas for the pdf of the number of failed compo
nents are derived using a recursion. These formulas quan
tify the effect on the pdf of the initial disturbance and the
amount of load transfer when a component fails. Thus fea
tures of loadingdependent cascading failure are captured in
a probabilistic model with an analytic solution.
The pdf of the number of failed components is an ex
tended quasibinomial distribution; that is, a quasibinomial
distribution with an extended parameter range. CASCADE
appears to be a new model that yields as a special case the
quasibinomial distribution. The recursion offers a simple
way to derive the distribution that avoids complicated alge
bra or combinatorics.
The tail of the pdf can range from exponential to an ap
proximately power tail, or there can be a high probability
of total failure (all components fail). Compared to systems
with no cascading dependency, the power tail and total fail
ure regimes show greatly increased probabilities of most of
the components failing. Indeed under these regimes, the
risk of catastrophic failure from cascading rare events can
be comparable to or in excess of the risk of the more fre
quent smaller disruptions. (Although catastrophic failures
are rarer, they have huge costs and their risk is the product
of frequency and cost.) For a more detailed discussion of
this claim in the context of blackout risk due to power tail
regimes in distributions of blackout size see [3].
For future work, we look forward to systematic descrip
tion of CASCADE model properties and its application to
understand cascading failure in large interconnected com
plex systems. We hope that CASCADE can contribute to
the goal of approximate quantification of the risks of catas
trophic infrastructure failure due to cascading rare events
and the mitigation of these risks.
6. Acknowledgements
I. Dobson and B.A. Carreras gratefully acknowledge
coordination of this work by the Consortium for Elec
tric Reliability Technology Solutions and funding in part
by the Assistant Secretary for Energy Efficiency and Re
newable Energy, Office of Power Technologies, Transmis
sion Reliability Program of the U.S. Department of En
ergy under contract 9908935 and Interagency Agreement
DEA1099EE35075 with the National Science Foundation.
I. Dobson, D.E. Newman, and B.A. Carreras gratefully ac
knowledge support in part from NSF grants ECS0214369
and ECS0216053. Part of this research has been carried
out at Oak Ridge National Laboratory, managed by UT
Battelle, LLC, for the U.S. Department of Energy under
Page 9
contract number DEAC0500OR22725.
References
[1] B.A. Carreras, V.E. Lynch, M. L. Sachtjen, I. Dobson, D.E. New
man, Modeling blackout dynamics in power transmission networks
withsimplestructure, 34thHawaiiIntl.Conf.SystemSciences, Maui,
Hawaii, Jan. 2001.
[2] B.A. Carreras, V.E. Lynch, I. Dobson, D.E. Newman, Dynamics, crit
icality and selforganization in a model for blackouts in power trans
mission systems, 35th Hawaii Intl. Conf. System Sciences, Hawaii,
Jan. 2002.
[3] B.A. Carreras, V.E. Lynch, D.E. Newman, I. Dobson, Blackout
mitigation assessment in power transmission systems, 36th Hawaii
Intl. Conf. System Sciences, Hawaii, Jan. 2003.
[4] B.A. Carreras, V.E. Lynch, I. Dobson, D.E. Newman, Critical points
and transitions in an electric power transmission model for cascading
failure blackouts, to appear in Chaos, scheduled for December 2002.
[5] P.C. Consul, A simple urn model dependent upon predetermined
strategy, Sankhy¯ a: The Indian Journal of Statistics, Series B, vol. 36,
Pt. 4, 1974, pp. 391399.
[6] P.C. Consul, Generalized Poisson distributions, Dekker, NY 1989.
[7] I. Dobson, B.A. Carreras, V. Lynch, D.E. Newman, An initial model
for complex dynamics in electric power system blackouts, 34th
Hawaii Intl. Conf. System Sciences, Maui, Hawaii, Jan. 2001.
[8] I. Dobson, J. Chen, J.S. Thorp, B.A. Carreras, D.E. Newman, Exam
ining criticality of blackouts in power system models with cascading
events, 35th Hawaii Intl. Conf. System Sciences, Hawaii, Jan. 2002
[9] M. Kloster, A. Hansen, P.C. Hemmer, Burst avalanches in solv
able models of fibrous materials, Physical Review E, vol. 56, no. 3,
September 1997.
A. f is a probability distribution
This appendix proves by induction that f defined by
(22−26) defines a probability distribution for n ≥ 1, d > 0.
If d ≥ 1, then (22), (23) define a probability distribution.
Now assume 0 < d < 1. f nonnegative in (22–25) and
recursion (26) imply that f is nonnegative for all n ≥ 1.
Inthecasen = 1, (24), (25)give?1
according to (24) and (26),
?
n
?
= (1 − d)n+
n
?
=
k
r=0f(r,d,p,1) = 1.
If n > 1 and?m
n
f(r,d,p,n) = (1 − d)n+
?n
r=0f(r,d,m,p) = 1 for m < n, then,
r=0
r=1
r
?
k=1
k
?
dk(1 − d)n−kf(r − k,
kp
1 − d,
p
1 − d,n − k)
k=1
n
?
n
?
r=k
?n
?n
k
?
?
dk(1 − d)n−kf(r − k,
kp
1 − d,
p
1 − d,n − k)
k=0
dk(1 − d)n−k= 1
In the special case np + d ≤ 1, f reduces to the qua
sibinomial distribution (30) and we see that the recursive
definition of f allows a proof that (30) is a probability dis
tribution that is more elementary than the proof in [5].
B. Extended quasibinomial formulas satisfy
the recursion
This appendix proves that extended quasibinomial for
mulas (27–29) satisfy recursion (22−26) for d > 0.
If d > 1, (27) does not apply (because d > 1 and r <
(1 − d)/p ⇒ r < 0) and (28), (29) satisfy (22), (23).
If d = 1, (27) (for r = 0) and (28) (for 0 < r < n)
verify (22). Then (29) verifies (23).
Now assume 0 < d < 1. For r = 0, (27) satisfies (24).
For r = 1 and n = 1, (29) satisfies (25). Moreover, for
r > 0 and n > 1, we claim that (27–29) satisfies (26).
To prove this claim, we first assume that 0 < r ≤ (1 −
d)/p and n > 1. Since
r ≤1 − d
p
⇒ r − k ≤
1 −
kp
1−d
p
1−d
,
(42)
(27) applies to all the instances of f in the right hand side
of (26) so that these instances of f may be written as
f(r − k,
?n − k
kp
1 − d,
?
k = 1,2,···,r
p
1 − d,n − k) =
kp
1 − d
r − k
?rp
1 − d
?r−k−1?
1 −
rp
1 − d
?n−r
(43)
(Note that (r − k)
the right hand side of (26) to get
?n
?n − k
=
r
k=1
?n
?n
According to (27), the left hand side of (26) also evaluates
to (44).
Now we assume that (1 − d)/p < r < n. Since
1 − d
p
1−d
p
1−d+
kp
1−d=
rp
1−d.) Substitute (43) into
r
?
k=1
k
?
dk(1 − d)n−k
?
r
?
(1 − rp − d)n−r
r − k
?n
kp
1 − d
?r
?rp
?k
1 − d
rdk(rp)r−k(1 − rp − d)n−r
?r − 1
?r−k−1?
1 −
rp
1 − d
?n−r
?
?
?
k
=
r
r
?
k=1
k − 1
?
dk(rp)r−k
=
r
(1 − rp − d)n−rd(rp + d)r−1
(44)
< r < n ⇒
1 −
kp
1−d
p
< r − k < n − k,
Page 10
(28) implies that
f(r − k,
kp
1 − d,
p
1 − d,n − k) = 0,k = 1,2,···,r
and hence (26) is verified.
Now we assume that r = n. Since r = n ⇒ r − k =
n − k, (29) implies that
kp
1 − d,
1 −
s=0
f(n − k,
p
1 − d,n − k) =
f(s,
1 − d,
n−k−1
?
kp
p
1 − d,n − k),k = 1,2,···,r
Substitution into the right hand side of (26) gives
1 − (1 − d)n−
n−1
?
s=1
s
?
k=1
?n
k
?
dk(1 − d)n−k
f(s − k,
kp
1 − d,
p
1 − d,n − k) = 1 −
n−1
?
s=0
f(s,d,p,n)(45)
wherethelaststepusestheresultestablishedabovethat(27)
satisfies (26). According to (29), the left hand side of (26)
also evaluates to (45).
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