Download V. Tail Behaviors of the Models

Probability Models for Estimating the Probabilities of
Cascading Outages in High Voltage Transmission Network
Qiming Chen, Member, IEEE, Chuanwen Jiang, Member, IEEE,
Wenzheng Qiu, Member, IEEE, James D. McCalley Fellow, IEEE

Abstract — This paper discusses a number of probability
models for multiple transmission line outages in power systems,
including generalized Poisson model, negative binomial model
and exponentially accelerated model. These models are applied to
the multiple transmission outage data for a 20-year period for
North America.
The probabilities of the propagation of
transmission cascading outage are calculated. These probability
magnitudes can serve as indices for long term planning and can
also be used in short-term operational defense to such events.
Results from our research shows that all three models apparently
explain the occurrence probability of higher order outages very
well. However, the Exponentially Accelerated Model fits the
observed data and predicts the acceleration trends best. Strict
chi-squared fitness tests were done to compare the fitness among
these three models and the test results are consistent with what we
observe.
Index Terms—exponentially accelerated cascading, negative
binomial distribution, generalized Poisson distribution, power law,
high-order contingency, cascading, blackouts, rare events,
multiple transmission outages.
I. INTRODUCTION
T
he high order contingencies in power systems are of great
interest nowadays because of their potential to cause huge
losses and the advances in computing technology make the
on-line analysis of high order contingencies possible[1][2].
The word “high-order” here means loss of multiple elements
during a short time period in power systems. Such events are
usually of lower probability than N-1 events which means loss
of a single element in power systems. If power systems are
weakened due to losses of more than one transmission line,
what would be the probability that another transmission line
trips? It is difficult to calculate or estimate their probability due
to their few occurrences. However, it would be very useful to
predict the likelihood of those events. For example, it would
help engineers in the transmission and generation planning
process, where capital investments in new facilities must be
weighed against the extent to which those facilities reduce risk
associated with contingencies. This could also help system
operators to estimate and evaluate network security in
operations, for control-room decision-making. Here, preventive
actions, which cost money and are routinely taken in
anticipation of N-1 events, are not reasonable for a rare event,
Qiming Chen is a transmission planning engineer with PJM Interconnection
Valley Forge, PA 19403, USA. (e-mail: [email protected] ).
Chuanwen Jiang is with the Department of Electrical Engineering, Shanghai
Jiaotong University, Shanghai, 200030, China. (email: [email protected])
Wenzheng Qiu is a generation interconnection planning engineer with PJM
Interconnection Valley Forge, PA 19403, USA. (e-mail: [email protected] ).
James D. McCalley is professor of Electrical and Computer Engineering at
Iowa State University, Ames, IA 50011, USA. (e-mail: [email protected]).
since the certain cost of the preventive action cannot be
justified for the event that is so unlikely. Given that the number
of rare events is excessively large and it is neither possible nor
necessary to do analysis for all of them, to prioritize the events
becomes crucial for on line analysis. The best way to prioritize
event is by risk, which is the expected impact by definition.
However, considerable computation would be needed to find
out the impact. Another way to prioritize is by event
probability, assuming the impacts of events are of about the
same magnitude. The event with highest possibility will be
“computed” next in developing operational defense procedures.
Ways to estimate the probabilities of power system rare
events include fitting an existing probability model to historical
data, deriving the overall probability by system structure and
individual components; and using Monte Carlo simulation.
Dobson in [3] proposed the use of power law to model the
occurrences of large disturbances recorded in [4]. Later on, a
number of probability distributions, which are variants of
quasi-binomial distribution and generalized Poisson
distribution (GPD) [5], were proposed in [6]-[8]. Reference [9]
presents work done by importance sampling to expose hidden
failure. In this paper, a new model for forecasting the
probability of high order contingencies, exponentially
accelerated cascading (EAC), is proposed and it was compared
with negative binomial model [11] and generalized Poisson
model. There are different metrics in characterizing power
system rare events, including the number of customers
interrupted, power interrupted, energy not served, and the
number of elements lost (i.e. N-1, N-2,…, where N-K means the
lost of K transmission line in power system network). The latter
one is employed in our probability model because it better
conforms to planning and operating reliability criteria used in
the industry. For example, reliability standards performance
criteria are often categorized based on the number of elements
lost.
Although it is difficult to obtain the first hand statistical data
from the power industry, the survey in [12] provides a good
resource for academic research. All the conclusions and
models from this paper are based on the large amount of actual
statistical data gathered in [12]. This paper focuses on
analyzing the fitness of a number of general probability models
for the likelihood of transmission outages. It is not intended for
any specific class of transmission outages, nor does it point out
the methodology to identify and defend those contingencies.
Readers interested in this may refer to [1][2][13][14] for further
information.
Section II of this paper gives a preliminary analysis of the
transmission line outage for the past 20 years from 1965 to
1985. Section III describes three possible probability models:
negative binomial (NB), generalized Poisson distribution
(GPD)[1] and exponentially accelerated cascading (EAC). The
EAC model is developed in this paper specifically for
cascading transmission outages. Unlike the NB and GPD
models, it has not been found in any other statistical text.
Section III uses the maximum likelihood estimation to estimate
parameters for the three models. The tail behaviors of the
models are compared in Section V. Section VI uses a
chi-squared test to compare the fitness among the three models.
Section VII concludes and discusses.
II. TRANSMISSION OUTAGE STATISTICS
Ref. [12] is a detailed resource for power system reliability
investigators considering the scale of the survey and the
difficulty of obtaining statistics on power systems from
different sources. The statistic data analyzed in this paper is the
total number of elements lost in each contingency in North
America from 1965 to 1985 [12], as indicated in TABLE I and
Fig. 1. The last two columns give a summary. According to
[12], the data reported in TABLE I, whenever an event involves
components of different voltage levels, it will be counted as one
instance only with a specific voltage level.
TABLE I
High order transmission outages statistics
An IEEE survey of US and Canadian overhead
transmission outages at 230kv and above, 1965-1985, [12]
Number of Contingences By Line Voltage Levels
Cont.
Type
Total No & Perc
230kv
345kv
500kv
765kv
N-1
3320
5807
721
295
10143
89.85%
N-2
303
577
35
36
951
8.42%
N-3
39
99
3
2
143
1.27%
N-4
18
16
0
2
36
0.32%
N-5
7
1
0
0
8
0.07%
N-6
0
1
0
1
2
0.02%
N-7
3
1
0
0
4
0.04%
N-8
2
0
0
0
2
0.02%
10
100
1
N-1
N-2
1000
10000
TABLE II
Conditional Probability of N-K contingencies
Cont.
Accru. No
Conditional
Prob (%)
Increasing
Rate
K ≥1
11289
K ≥2
1146
C1=10%
-
K ≥3
195
C2=17%
C1/C2=170.0%
K ≥4
52
C3=27%
C2/C3=158.8%
K ≥5
16
C4=31%
C3/C4=114.8%
K ≥6
8
C5=50%
C4/C5=161.3%
K ≥7
6
C6=75%
C5/C6=150.0%
K ≥8
2
C7=33%
-
-
-
The data is rearranged in TABLE II to obtain conditional
probabilities conveniently. The second column of TABLE II
counts the number of events with more than k lines outaged.
For example, there are a total of 11289 events that involve at
least one line outage, and 11289 is just the summation of the
number of all outages listed in TABLE I. 1146 is the total
number of events that involve at least two lines and so on. The
last column of TABLE II lists the estimated conditional
probability of N-K contingencies derived from the sixth column
of TABLE I. They are calculated by the formula as follows:
Pr( K  k  1)
Ck  Pr( K  k  1 K  k ) 
(1)
Pr( K  k )
The estimate of Pr(K  k  1 K  k ) can be found by simply
replacing Pr( K  k  1) and Pr( K  k  1) with the occurrence
frequencies of the events K ≥ i and K ≥ k+1, respectively. Fig. 2
illustrates the relationship between Ck and k. Fig. 2 shows that
the probability of any transmission outage that involves loss of
more than one line is 10%, a value significantly larger than that
commonly assumed. If North America power system loses 2
transmission lines and there is no other information considering
the timing, causes, location of the event, the probability that it
loses at least an additional line would be 17%, which is almost
doubled compared to 10%. If the system loses 6 lines, the
chance of losing another line is almost certain. This means that
if a system is already in a weakened condition, the probability
that it continues weakening would keep increasing until the
event develops into a system-wide blackout.
N-3
0%
10% 20% 30% 40% 50% 60% 70% 80%
N-4
N-5
N-6
N-7
N-8
Fig. 1. log{Pr(k)} v.s. N-k plot
Pr(K ≥ 2/ K ≥ 1)
Pr(K ≥ 3/ K ≥ 2)
Pr(K ≥ 4/ K ≥ 3)
Pr(K ≥ 5/ K ≥ 4)
Pr(K ≥ 6/ K ≥ 5)
Pr(K ≥ 7/ K ≥ 6)
Pr(K ≥ 8/ K ≥ 7)
Fig. 2. Pr(K ≥ k / K ≥ k) v.s. k
The increasing in conditional probability Ck with k is
understandable. The loss of one element immediately raises the
likelihood of losing another element, which has a similar effect,
and so on. A fault and the follow-up relay trip of the
component(s) cause transient oscillations throughout the power
system and make other protection devices more likely to
operate. The forced outage of one generator or line changes the
power flow pattern, and some circuits, being more loaded, may
trip either by proper or unintended protection operation. The
more severe the previous event is, the more likely an additional
event will follow. This tendency might be modeled statistically
using a number of probability distributions, such as Poisson
model, negative binomial model, power law, and GPD
[2]-[3][6]-[9].
Some caution must be taken when using the data in [12] to
draw a conclusion. First, a few utilities in the survey reported
their transmission contingencies by single line outages [12]. If
an event involved the loss of three lines, it was reported as three
different single outages. In order to prevent a multiple line
outage event from being counted as several single line outage
events, the survey processed the data provided by those
utilities. Outages reported by those utilities with identical
initiating timing, i.e., occurring within one minute, were
considered as one contingency with multiple line outages. All
other utilities reported outages by events, i.e., multiple
transmission outages in a single event were reported as one
instance of outage. Second, it appears that some huge
blackouts that outaged many lines, for example, the
Northeastern US blackout of 1965 [15], were not correctly
registered in the statistics, since the last stage of this event
obviously outaged more than 8 lines. In order to mitigate the
uncertainty in data error, all the outages that involve the loss of
more than 7 lines are grouped into a single category, i.e., K ≥ 7.
The number of events in this category is 4 + 2 = 6 and
constitutes about 0.06% of total observed events. In order to
mitigate the impact that would arise because of the possible
inaccuracy, all the discussion that follows will be based on this
treatment.
III. THREE DISCRETE PROBABILITY MODELS FOR
INTERDEPENDENT EVENTS
We have shown in [11] that both Poisson Model and Power
Law model are not desirable for transmission outages because
the former underestimates the interdependence among
transmission outages and the later overestimates. Generalized
Poisson model (GPD) is first proposed to model
over-dispersion [5] (variance greater than mean - note the
variance and mean are equal for Poisson). There is evidence
supporting the choice of GPD for the distribution of
transmission line outages on small test systems [4][7].
This section introduces a new model: exponentially
accelerated cascading model (EAC), which is specifically
proposed for cascading transmission outages. The cluster
model in [11], which is actually Negative Binomial model (NB)
and gives the best among the three models in [11] fitting for the
transmission outage statistics of TABLE I, will be re-discussed
as well.
A. Negative binomial distribution
The negative binomial distribution (NB) is an established
model for rare events such as car accidents [10][16], where
interdependence between events or over-dispersion in a data set
exists. We have presented in [11] that it is a better model for
transmission line outage than Power Law model and Poisson
model. The pdf (probability density function) of NB is given
by
( 1  k  1)   
Pr K  k  ,   


(k )( 1 )     1 
k 1
  1 


1 
   
 1
(2)
where k  1, 2, ...
The mean and variance of NB are given by
E( K )    1
(3)
Var ( K )     2
Note the sample space of K is {1, 2, 3,…} instead of {0, 1, 2,
…} here, which is different from the usual way of defining NB.
We do so because we want the sample space to be consistent
with the number of lines outaged in power systems.
B. Generalized Poisson distribution (GPD) model
A generalized Poisson distribution is given by
   (k  1) 
k 2
Pr( K  k  , ) 
(k  1)!
where k  1, 2, 3 ..., and   0
exp    (k  1)  ,
(4)
where 1 > θ > 0 and λ > 0.
There is also a GPD defined for θ ≥ 1. However, that is not
discussed here in order to limit the scope of this paper. When θ
= 0, the GPD degrades to a regular Poisson distribution with
parameter λ.
The mean and variance of GPD are given by
E( K )   (1   ) 1  1
(5)
Var ( K )   (1   ) 3
(6)
For the same reason as we stated for negative binomial
distribution, we use {1, 2, 3…} instead of {0, 1, 2…} for the
sample space of K.
C. Exponentially Accelerated cascading model (EAC)
This model is based on the observation that there is an
increasing trend of conditional probabilities in TABLE II and
Fig. 2 approximately following a potential exponential
relationship.
Note that the ratios of the conditional
probabilities in the last column of TABLE II vary in the vicinity
of 150%. The exponential accelerated cascading (EAC) model
proposed here assumes that the probability of another or more
transmission outage(s) follows an exponential function of the
number of lines already lost in the system. Denote the
conditional probability Pr(K  k  1 K  k ) as Ck , then
Ck 1    Ck , where  is a constant and   1 (7)
Suppose after N-1 contingency happens, the probability that
one
or
more
line
outage
happens
with
probability p  C1  Pr(K  2 K  1) , then
Ck  p   k 1
(8)
Pr( K  k )  Pr( K  2 K  1)  Pr( K  3 K  2)  
 Pr( K  k ) Pr( K  k  1)
 C1  C2  Ck 1
  p   p 1   p 2    p k 1 
 p k 1 
by
differentiation
and
necessary to use the log form to process it in computer. Since
all the N-K contingencies with k≥ 7 in TABLE II are grouped
due to their potential inaccuracy, the log likelihood formula
here for the data set is given by
log L( 1 ,, m k1 ,, k n )
 n log Pr( K  k  ,, )  N log Pr( K  7  ,, )
k
1
m
7
1
m
k{1, 2 ,, 6}
(9)
Pr(K  k )  Pr(K  k )  Pr(K  k K  k )


 Pr(K  k )  1  Pr(K  k  1 K  k )
 p k 1 
L(1 ,, m x1 ,, xn )
of
L(1 ,, m x1 ,, xn ) is sometimes extremely small, it is

( k 1)( k  2 )
2
( k 1)( k 2 )
2
that
1  p 
k 1
(10)
(12)
where nk is the number of N-k contingencies and N7 is the
number of N-K contingencies with K ≥ 7.
Equation (12) will be applied to all three models in this paper.
The sample spaces of the NB and GPD models shift from the
usual {0, 1, 2, …} to {1, 2, 3 …}, so N-1 events correspond to
the events K = 1 in NB and GPD models and so on .
TABLE III
A. Estimating the parameters of Negative Binomial model
Conditional Probability of N-K contingencies

Cont.
k
K≥k
Count
Cond.
Prob
(%)
Ck
Pr( K  k )
Pr( K  k )
1
11289
-
p 0
p0 0  1
p 0  0 (1  p )
2
1146
10%
p 1
p 0
p 1  0 (1  p )
3
195
17%
p 2
p2 1
p 2  (1  p 2 )
p 
3
p  (1  p )
4
52
27%
p
5
16
31%
p 4
p4 6
p 4  6 (1  p 4 )
50%
p
6
k≥7
8
6
75%
3
5
-
3
3
3
3
5
p 
10
p  (1  p )
p 
15
-
6
5
10

Substitute the pdf Pr k  ,  of NB in (2) into the log
5
In equation (8), Ck increases with k without bound. It could
go to infinity. However, Ck must be less than or equal to one
because it represents a probability. To solve this apparent
inconsistency, this model assumes that the exponential law only
valid up to k = 6 and p 5  1 . If a system loses 7 lines in a
sequence for one outage, the system is considered collapsed
and there is no need to count further lost lines for statistical
purpose.
Note in the above derivation, the condition K  1 is omitted,
since all the discussion here assumes that there is already a
contingency, that is, Pr( K  1)  1 .
likelihood formula (12) to get
k 1
( 1  k  1)      1 

Log L   N k log

 
k 1,...,6
(k )( 1 )     1      1 
1
1
k 1

( 1  k  1)      1  


 6  log 1  



1
1
1 

 k 1,...,6 (k )( )           
(13)
The MLE Estimation for Negative Binomial model is
equivalent to finding the  and  that maximize (13).
Numerical technique is needed to find the global maximum
(ˆ , ˆ) because there is no close form available for (ˆ , ˆ) .
Fig. 3 plots the contour graph of ( ,  ) for likelihood. With
the help of this plot, the optimal (ˆ , ˆ) is found out to be
(2.675, 0.1225).

0.128
0.126
(2.675, 0.1225)
0.124
0.122
IV. MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS
0.120
The log maximum likelihood estimation (MLE) [13] is given
by
log L(1 ,, m k1 ,, k n )   log Pr( k i 1 ,, m )
(11)
i{1, 2 ,,n}
 = (1, 2, … ,m) and x = (k1, k2, …, kn) are defined as the
distribution parameter vector and variable vector, respectively.
The  that maximizes L( | x) is called a MLE of the parameter
, denoted as ˆ . It should be noted that ˆ must be a global
maxima. Because it is easier to find the maxima of (11) than
0.118
0.116
2
2.2
2.4
2.6
2.8
3

Fig. 3. Contour plot of likelihood function (13): NB Model
3.2
B. Estimating the parameters of generalize Poisson model
The log likelihood formula for the GPD model is given by
Log L 

   (k  1) 
k 2
k 1,...,6
N k log
k!
Pr K  k   2.675 ,   1.1225  
exp    (k  1) 
k 2


   (k  1) 
 6  log 1  
exp    (k  1) 
(k  1)!
 k 1,...,6

(14)
The maximum likelihood estimate of θ is given by the
solution of (14), assuming no censored data, according to [1]
(k  2)( k  1)
c
ni
 n(k  1)  0

k 1
(k  1)  (k  k )
(15)


   (k  1)(1   )
However, (15) is only correct for uniform sample data with k
known for each of the observed events. Since the sample data
available is heterogeneous because of the incomplete
observablity of N-K events (K ≥ 7), it is not convenient to use
(15) to get the maximum (ˆ, ˆ) . Inspection o f the contour
graph in Fig. 4 yields the MLE of (ˆ, ˆ ) to be (0.108, 0.155).
  


1
   
( 1  k  1)

(k )( 1 )
k 1
  1 


1 
   
 1
(17)
   (k  1) 
k 2
Pr( K  k   0.108 ,  0.155 ) 

(k  1)!
 exp    (k  1) 
(18)
Pr( N  k p  0.10295 ,   1.515 )  p k 1 
( k 1)( k 2 )
2
1  p 
k 1
(19)
1.56
β
1.55
1.54
0.10295, 1.515
1.53
1.52
1.51
θ
1.5
1.49
0.18
0.17
1.48
0.1
(0.108, 0.155)
0.1005 0.101 0.1015 0.102 0.1025 0.103 0.1035 0.104 0.1045 0.105
p
0.16
Fig. 5. Contour plot of likelihood function (16) (EAC Model)
0.15
The three models above are evaluated for k = {1,2,3,4,5,6}
and k ≥ 7, and the results are shown in TABLE IV. These
results are also plotted in Fig. 6. By inspecting Fig. 6, it can
conclude that all three models reasonably predict the observed
data as all of them fits very well for k=1,…,6. A careful
examination of TABLE IV shows that for the N-K events with k
> 6, the EAC model predicts the occurrence of these extreme
events far more accurate than the other two models. EAC gives
0.00061 for the observed 0.00053, while the other two give
3.53E-05 and 9.16E-05, which are 10 times lower. Fig. 6 uses
log-scale for probabilities, so it shrinks the apparent difference
between three models. The strict statistic index χ2 is employed
in Section VI to do further comparison for the three models.
0.14
0.13
0.12
0.1
0.102
0.104
0.106
0.108
0.11
0.112
λ
Fig. 4. Contour plot of likelihood function
(14): GPD Model
C. Estimating the parameters for exponentially accelerated
cascading model
For the EAC model.
LogL( p,  n1 , n2 ,, n6 , N 7 )
  nk Log  p ( k 1) ( k 1)( k 2 ) 2 1  p k 1  6 log p 71 ( 71)( 72 ) 2 
6
TABLE IV
Comparing the fitness of three different probability models for the distribution
of observed multiple line outages
Cont.
k
Count
Observed
NB
GPD
EAC
N-1
1
10143
0.8985
0.8995
0.8976
0.8971
N-2
2
951
0.08424
0.08299
0.0830
0.08689
N-3
3
143
0.01267
0.01407
0.01487
0.01226
N-4
4
36
0.003189
0.00275
0.00334
0.00244
N-5
5
8
0.00071
0.00057
0.000842
0.00062
N-6
6
2
0.00018
0.000124
0.000228
0.00013
6
0.00053
3.52E-05
9.16E-05
0.00061
k 1
6
 6 (k  1)( k  2) 
  nk (k  1) Log( p )  
nk  Log ( ) 
k 1

2
 k 1

 n
6
k 1
k
log(1  p
k 1
)  36 log( p)  90 log( )
(16)
Inspecting contour graph Fig. 5 yields ( pˆ , ˆ ) = (0.10295,
1.515).
N-K,K > 6
1
Observed
Probability
Exponential Accelerated Cascading

 24.7%
   1
  exp(1   )  36.1%
(21)
(22)
Negative Binomial
0.1
Generalized Poisson Model
40%
35%
Pr( K ≥ k+1 / K ≥ k )
0.01
0.001
Asymptotic Line P=36.1%
30%
25%
20%
Assuming an N-2 contingency, the probability
that a next contingency happens increased to 19%
15%
10%
Assuming an N-1 contingency, the probability
that a next contingency happens is 10%
5%
0%
0.0001
1
3
2
4
5
6
7
8
9 10
1
i
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
Number of Line Out (k)
Fig. 6. The log-log plot of PDF for NB, GPD and EAC models
Fig. 8. Propagation of cascading sequence accelerates: GDP Model
V. TAIL BEHAVIORS OF THE MODELS
With the three models discussed in the last section, the
conditional probabilities can be obtained by
1   Pr( K  j )
Pr( K  k  1)
j k 1
Pr( K  k  1 K  k ) 

Pr( K  k )
1   Pr( K  i)
i k
(20)
Fig. 7 shows that there is an increasing accelerating trend for
the probability of occurrence of next events with the number of
lines already lost. However, the acceleration rate decreases
continuously and stabilizes at 24.7%.
The conditional probability for the GPD model can be
calculated the same way as the NB model. Fig. 8 shows that the
GPD model is similar to NB model in that both of them give the
accelerating trend of cascading transmission line outages and
the acceleration rate converges as k becomes large, except that
the GPD model converges to probability 36.1%, which is much
more larger than the value 24.7% in NB.
Fig. 9 plots the conditional probabilities in (20) for the EAC
model. The values approximate the actually observed values
properly, with the observed values less than the EAC model for
k ≤ 3 and greater for k ≥ 4. When k ≥ 7, it can be judged from
the increasing trend that the subsequent event happens with
certainty, a prediction supported by the statistics in TABLE I.
Fig. 10 plots all the conditional probability from the three
ideal models and the actual data together in one graph. All
three models match the actual data closely when k≤3. When
k≥4, the NB and GPD models start to deviate from the observed
values and the EAC model keeps following.
0.9
Exponential Accelerated Cascading (EAC)
0.8
Observed conditional probability
0.7
0.6
0.5
0.4
30%
P(K ≥ k+1 / K ≥ k)
25%
Asymptotic Line P=24.7%
0.3
0.2
20%
Assuming an N-4 contingency, the probability
0.1
15%
that a next contingency happens is more than 30%
Assuming an N-2 contingency, the probability
that a next contingency happens increased to 17%
10%
0
0
1
2
3
4
5
6
Number Lines in Outage (k)
Assuming an N-1 contingency, the probability
that a next contingency happens is 10%
5%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Number of Line Out (k)
Fig. 7. Propagation of cascading sequence accelerates: NB Models
The asymptotic conditional probabilities for the NB and
GPD models are given by (21) and (22) respectively.
Fig. 9. Propagation of cascading sequence accelerates: EAC Model
7
decomposed into 5 exclusive sets. They are S1={1}, S2={2},
S3={3}, S4={4}, S5={5, 6, 7, … }. The reason to group them
this way is that for all Si and all three models tested,
P(XSk)Nk (where Nk is the number of samples that fall in set
Sk) is greater than 5%, which is suggested for the credibility of
the fitness test.
100%
Conditional Probability (K ≥ k+1|K ≥ k)
EAC
90%
Observed
80%
GPD
NB
70%
60%
TABLE V
50%
2-Test results for NB, GPD, and EAC models
40%
NB
30%
k
nk
10%
1
0%
2
2
3
4
5
6
7
8
Number Lines in Outage (k)
9
10
11
Fig. 10. Comparing the propagations of cascading
VI. FITNESS TEST OF THREE DIFFERENT PROBABILITY MODELS
The discussion in Section IV and V provides qualitative
evidence that the EAC model fits the data better than NB and
GPD models. In this section, the chi-squared test is applied to
provide quantitative evidence. The chi-squared test, based on
the Pearson theorem [17], is widely used in statistics to test the
fitness of a probability model to sample data. Suppose a certain
random trial has k possible outcomes, the probability that each
trial results in the kth outcome is pk, k=1, 2, 3, …, m, where
pk=1. If n trials are performed, and the kth outcome results Nk
times, then the multivariate distribution of Nk is
Pr( N1  n1 ,, N m  nm p1 ,, pm ) 
n!
p1n p2n  pmn
n1!n2 ! nm !
1
m
2
(23)
n
m
with
p
m
k
 n and
k 1
k
1
k 1
Pearson theorem: Suppose the parameters of a polynomial
distribution has the pdf as in (23), and define
 N
m
2 
 npk  np k
2
k
(24)
k 1
when n, 2 follows the chi-squared distribution 2 (k-1).
One can see from (24) that the statistic 2 is an index for how
much the samples deviate from the polynomial distribution to
be tested. The larger the statistics  2 is, the larger the deviation
is. In order to apply the Pearson theorem, we need to convert
the distribution we are going to test into a polynomial
distribution. Since the sample space of the three distributions is
{1, 2, 3, …}, and it is an infinite set, it can be grouped into m
exclusive sets denoted as Sk=1, 2, 3, …, m. Suppose K is a
random variable and its pdf is Pr(K=k), k{1, 2, 3, ...}. A total
of n samples of K are drawn from pdf f (k). Count and denote
the number of samples that are members of the set Sk. as Nk.
Denote pk as Pr(KSk). Then the random variables Nk, (k=1, 2,
… , m) follow the polynomial distribution of (23). The
statistics 2 defined in (24) follow 2(m-1) distribution. If 2 is
too large, then it is reasonable to doubt the fitness of the tested
model with respect to the data. In this test the sample space is
EAC
pk
pkn
pk
pkn
pk
pkn
10143
89.946%
10154.93
89.763%
10134.22
89.705%
10127.69
951
8.299%
936.95
8.300%
937.06
8.689%
981.02
3
143
1.407%
158.85
1.487%
167.92
1.226%
138.45
4
36
0.275%
31.02
0.334%
37.71
0.244%
27.50
≥5
17
0.073%
8.25
0.116%
13.10
0.136%
15.33
20%
1
GPD

2
11.88
5.15
3.90
(m, r)
(5, 2)
(5, 2)
(5, 2)
m-r-1
2
2
2
1- quantile
0.066%
13.80%
27.85%
The Pearson theorem assumes all parameters for the
distribution to be tested are known. If there is any unknown
parameter so that pi’s are just estimates, the degrees of freedom
of the chi-squared distribution need to be reduced by one for
each estimated parameter. The rule is if there are a total of r
estimated parameters, the degrees of freedom of the
chi-squared distribution are reduced by r to become m-r-1. For
the GPD model, parameter  and θ are estimates, so the
chi-squared distribution has degrees of freedom 5−2−1=2. The
NB and EAC models also have two estimated parameters, so
the degrees of freedom for both of them are 2 too. The test
result is summarized in TABLE V. The last row of the table
shows the probability of getting a sample deviation larger than
observed, assuming the sample comes from the corresponding
probability model. The EAC is far more fit than the other two.
VII. CONCLUSION AND DISCUSSION
A new model (EAC model) was proposed in this paper to
estimate the probabilities of high-order transmission line
contingencies. Two other possible models: NB and GPD were
also discussed in the paper. Comparison has been done among
these three models. All three models show that there is an
accelerating trend in the spreading transmission outages after
the initiating outage. The EAC model provides the best fits
among all three models. This model is simple and easier to
understand than the other two yet gives better prediction for the
observed outage data. Its two parameters p, which represents
the probability of occurrence of N-2 given an N-1 event at the
start of a cascading transmission blackout, and β, which
quantifies the increasing rate of the conditional probability, i.e.,
Pr(N-K-1/N-K), are estimated to be 10.3% and 1.515. That
means that around 10% of all transmission outages involve
more than one line and after the first outage, the cascading
outage happens with probability increasing by a factor 1.515
for each additional line lost.
All three models can be employed to evaluate and compare
large power systems’ resilience to cascading events. For
example, if the survey data is provided with information
regarding the locations of contingencies, we can model eastern
and western interconnection separately and compare the model
parameters for the two systems. On the other hand, if the
survey data is provided with time stamps for each contingency,
we can model the first and the last ten-year period separately.
This way, we can find out if the reliability of power systems in
US and Canada has been improved.
System Protection Schemes (SPS)[20], which involve a wide
range of automatic mitigation actions, such as under frequency
load-shedding, under voltage load-shedding, and controlled
islanding, are installed in a number of large power systems in
the last ten years. They are designed to prevent large blackouts
and should reduce the number of cascading outages. After they
are in operation for many years and with sufficient statistics, the
models in this paper can be applied to the collected statistics of
the system with and without SPS to find out if the tendency of
having a cascading blackout is reduced.
Risk and decision analysis, which is of great interests to
utilities as power industries are moving toward a more
competitive environment, depend on accurate estimate of
failure probabilities of power system components. The results
of this work will enhance decision making at both the planning
[21] and operational [22] level by giving quantitative
probabilities for high order contingencies. In particular,
operational procedures for defending against large outages are
of great interest to authors[2][14], and the proposed models in
this paper are candidates to aid in allocating computational
resources as they are used on-line to develop defense strategies
as real-time conditions change.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
VIII. ACKNOWLEDGMENT
The authors thank Professor Ian Dobson of University of
Wisconsin, Madison for useful discussions.
[22]
International Conference on System Sciences, Kauai, Hawaii, January
2006
I. Dobson, B.A. Carreras, D.E. Newman, “Branching process models for
the exponentially increasing portions of cascading failure blackouts,”
Thirty-eighth Hawaii International Conference on System Sciences,
Hawaii, January 2005
A. G. Phadke and J. S. Thorp, “Expose hidden failures to prevent
cascading outages in power systems,” IEEE Comput. Appl. Power, vol. 9,
no. 3, pp. 20–23, Jul. 1996
W.A. Thompson, Jr., Point Process Models with Applications to Safety
and Reliability, Chapman and Hall, 1988
Qiming Chen and James D. McCalley, “A Cluster Distribution as a Model
for Estimating High-order Event Probabilities in Power Systems,”
Probability In The Engineering and Information Sciences, Vol. 19, .Issue
04, 2005, pp489 – 505
R. Adler, S. Daniel, C. Heising, M. Lauby, R. Ludorf, T. White, “An
IEEE survey of US and Canadian overhead transmission outages at 230
kV and above”, IEEE Trans. on Power Delivery, Vol. 9, Issue 1, pp 21
-39, Jan. 1994
Mili, L., Qui, Q. and Phadke, A.G. (2004) ‘Risk assessment of
catastrophic failures in electric power systems’, Int. J. Critical
Infrastructures, Vol. 1, No. 1, pp.38–63
Qiming Chen, Kun Zhu James D. McCalley, “Dynamic decision-event
trees for rapid response to unfolding events in bulk transmission
systems,” Power Tech Proceedings, 2001 IEEE Porto Volume 2, 10-13
Sept. 2001
Joseph C. Swidler; David S. Black; Charles, R. Ross; Lawrence J.
O’Connor, Jr., Report to the President By the Federal Power Commission
On the Power Failure in the Northeastern United States and the Province
Of Ontario on November 9-10, 1965
Miaou, S.-P., and Lum, H., "Modeling Vehicle Accidents and Highway
Geometric Design Relationships," Accident Analysis and Prevention,
25(6): 689-709, 1993
J. Neter, M. H. Kutner, C. J. Nachtsheim, W. Wasserman, Applied Linear
Statistical Models (4th edition), R. D. Irwin, 1996
D. C. Elizondo, J. de La Ree, A. G. Phadke, and S. Horowitz, “Hidden
failures in protection systems and their impact on wide-area
disturbances,” in Proc. IEEE Power Eng. Soc. Winter Meeting, vol. 2, 28
Jan.–1 Feb. 2001, pp. 710–714
Jun Zhang; Jian Pu; McCalley, J.D.; Stern, H.; Gallus, W.A., Jr.; “A
Bayesian approach for short-term transmission line thermal overload risk
assessment,” IEEE Transactions on Power Delivery, Vol. 17, Issue 3,
July 2002, pp770 - 778
CIGRE, task Force 38.02.19, “System protection schemes in power
networks,” CIGRE SCTF 38.02.19, 2000
Miranda, V.; Proenca, L.M.; “Why risk analysis outperforms probabilistic
choice as the effective decision support paradigm for power system
planning,” IEEE Transactions on Power Systems, Vol. 13, Issue 2, May
1998 pp:643 - 648
M. Ni; J. McCalley; V. Vittal; and T. Tayyib; “On-line risk-based security
assessment,” IEEE Transactions on Power Systems, Vol. 18., No. 1,
February, 2003, pp 258-265
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Qiming Chen, James D. McCalley, “Identifying High Risk N-k
Contingencies for Online Security Assessment” IEEE Transactions on
Power Systems, Vol. 20, Issue 2, May 2005, pp 823 – 834
Qiming Chen, "The probability, identification and prevention of rare
events in power systems," Ph.D. dissertation, Dept. Electrical and
Computer. Eng., Iowa State Univ., Ames, 2004
B.A. Carreras, D.E. Newman, I. Dobson, A.B. Poole, “Evidence for
self-organized criticality in a time series of electric power system
blackouts,” IEEE Transactions on Circuits and Systems I, Vol. 51, no 9,
Sept. 2004, pp 1733 - 1740
The Disturbance Analysis Working Group (DAWG), [on-line]
Website:http://www.nerc.com/dawg/welcome.html
Consul, P. C. Generalized Poisson Distributions: Properties and
Applications, Marcel Dekker, New York, 1989
I. Dobson, B.A. Carreras, D.E. Newman, “A branching process
approximation to cascading load-dependent system failure,”
Thirty-seventh Hawaii International Conference on System Sciences,
Hawaii, January 2004
I. Dobson, K.R. Wierzbicki, B.A. Carreras, V.E. Lynch, D.E. Newman,
“An estimator of propagation of cascading failure,” Thirty-ninth Hawaii
Qiming Chen (M’04) received the B.S. and M.S. from
Huazhong University of Science and Technology, Wuhan,
China in 1995 and 1998 respectively. He received the
Ph.D. degree in electrical engineering, Iowa State
University, Ames, in 2004.
He joined PJM
Interconnection, Eagleville, PA, as a Transmission
Planning Engineer in 2003.
Chuanwen Jiang (M’04) is an associate professor of the
School of Electric Power Engineering of Shanghai Jiaotong
University, P.R.China. He got his M.S. and Ph.D. degrees
in Huazhong University of Science and Technology and
accomplished his postdoctoral research in the School of
Electric Power Engineering of Shanghai Jiaotong
University. He is now in the research of reservoir dispatch,
load forecast in power system and electric power market.
Wenzheng Qiu (M’04) received the B.S.(1995) and
M.S.(1998) from Huazhong University of Science and
Technology, Wuhan, China in 1995 and 1998. She worked
for North East Power Research Institute from 1998 to 2000.
She received the Ph.D. degree in electrical engineering,
Iowa State University, Ames, in 2004. Currently, She
works in PJM Interconnection, Eagleville, PA, as a
Transmission Planning Engineer.
James D. McCalley (F’04) received the B.S., M.S., and
Ph.D. degrees in electrical engineering from Georgia
Institute of Technology, Atlanta, in 1982, 1986, and 1992,
respectively. He was with Pacific Gas & Electric
Company, San Francisco, CA, from 1985 to 1990 as a
Transmission Planning Engineer. He is now a Professor of
Electrical and Computer Engineering at Iowa State
University, Ames, where he has been employed since
1992. He is a registered professional engineer in California.