Enhanced Power System Security Assessment Through Intelligent Decision Trees - Complex System Modelling and Control Through Intelligent Soft Computations - page 344

1if Y t

\

h ð y Þ¼ IY ð t Þ¼ ð 4 Þ

\

0ifY t

and hence,

Z 1

X

n

P ð Y t Þ¼ h ð y Þ f ð y Þ dy ¼ Eh ð ð Y Þ Þ¼ h ð y i Þ ð 5 Þ

\

i ¼ 1

1

The above expectation function gives crude Monte Carlo estimation (Rubinstein
1981 ), where yi are Monte Carlo samples taken from the distribution f(y), the post-
contingency performance index probability distribution. This estimation has a
variance associated with it, as the quantity h(yi) i ) varies with yi. i . Importance sampling
attempts to reduce the variance of the crude Monte Carlo estimator by changing the
distribution from which the actual sampling is carried out. Suppose it is possible to

find a distribution g(y) such that it is proportional to h(y) f(y), then the variance of
estimation can be reduced by reformulating the expectation function as,

Z 1

X

n

g Þ

g ð y Þ

ð y h ð Y Þ f ð Y Þ h ð y i Þ y i Þ

g ð y i Þ

f ð

ð Þ¼ ð Þ ð Þ ¼ ¼ ð 6 Þ

P Y \ t h y f y dy E

g ð Y Þ

i ¼ 1

1

where y i are Monte Carlo samples drawn from the distribution g(y). This ensures the
quantity is almost equal for all yi. i . In effect, by choosing the

sampling distribution g(y) this way, the probability mass is redistributed according
to the relative importance of y, measured by the function |h(y)| f(y) (Ripley 1987 ).

f hy ðÞ fy ðÞ= gy ðÞg

4.2.2 Proposed Efficient Sample Generation

The property of importance sampling to bias the sampling using an importance
function g(y) towards an area of interest, as discussed above is used to generate
in uential operating conditions from operational state space, X. The joint proba-

bility distribution of the operational parameter space f(x) can be obtained from
historical data (Rencher 1995 ). Once we have a priori information about f(x), stage-
I operation provides the region in X through which the boundary most likely occurs
and therefore identi

fl

es approximately the x-space in which we want to bias the
sample generation. The region of interest for sampling is de ned using the indicator

function h(X), where S is the boundary region.

1ifY ð X Þ2 S

h ð X Þ¼ IX ð 2 S Þ¼ ð 7 Þ

0ifY ð X Þ 62 S

, as shown in Fig. 8 .
The importance function or the sampling distribution g(x) can be constructed

In a univariate case, we can de ne it as S ¼f x : x 1 x x 2 g