Enhanced Power System Security Assessment Through Intelligent Decision Trees - Complex System Modelling and Control Through Intelligent Soft Computations - page 351

parameters and loads. The boundary region (where both acceptable and unaccept-
able performances occur) begins approximately at around 11,500 11,700 MW and

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ends at around 12,500 12,700 MW. Therefore, these simulation results verify the

ability of the proposed stage-I method to estimate the boundary region in the multi-
dimensional operating parameter state space at a highly reduced computing
requirements (i.e., about 20 CPF computations, compared to 24,000 simulations for
Fig. 11 ).

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5.2.2 Stage-II: Importance Sampling

Many MCS studies in the past have assumed a multivariate normal distribution of
load data (Wan et al. 2000 ). But in this study, importance sampling is performed on
actual empirical non-parametric distribution obtained from the projected historical
data of loads. Figure 12 shows three marginal load distributions among the 640 load
vectors that make up the multivariate historical data. It is seen that the multivariate
distribution is made up of marginal distributions that are not exactly normal, but by
visual inspection some looks close to normal, some uniform, some discrete and so
on. So a multivariate Normality assumption will give misleading results.
Furthermore, these marginal distributions are not independent to model them
separately as a group of normal, uniform and discrete distributions respectively; but
they are mutually correlated, and the sampling method must preserve their inter-
dependencies or correlations while sampling. So considering both the non-parametric
nature of the marginal distributions and their mutual correlations, the whole sampling
task becomes very challenging. Therefore, as mentioned in the Sect. 4.2.2 , copulas
are used that could ef

ciently work with multiple non-parametric marginal distri-
butions and their mutual correlation (rank correlation) to produce correlated multi-
variate random vectors from original multivariate distribution de ned by empirical

historical data.
After identifying the boundary region limits, the empirical multivariate distri-
bution of boundary region f 1 f1(x) is begotten from historical data by

filtering the

records within the identi ed boundary limits. When p = 1 in Eq. ( 8 ), we have

complete sampling bias towards the boundary region f 1 f1(x). The inter-dependencies
between various individual loads are captured in the sampling process that use

Fig. 12 Some sample marginal distributions from historical load data