Page 201 - Decision Making Applications in Modern Power Systems

164 Decision Making Applications in Modern Power Systems

recognition techniques, such as artificial neural network, PNN, and SVM,
have been applied, which is briefly discussed later.

6.3.1 Empirical mode decomposition
The main idea of EMD technique is to identify the oscillatory modes present
in the time scales defined by the interval between local extrema of the com-
posite signal [21,22]. The steps to get IMF from a distorted signal are as
follows:
Step 1. Find out local maxima and minima of the signal S(t).
Step 2. Interpolate between maxima to get upper envelope.
Step 3. Interpolate between minima to get lower envelope.
Step 4. Compute the mean of the upper and lower envelope m(t)by

e upper ðtÞ 1 e lower ðtÞ
mðtÞ 5 : ð6:38Þ
2
where e upper ðtÞ and e lower ðtÞ is the upper and lower envelopes of the signal
S(t).
Step 5. Extract
c 1 ðtÞ 5 SðtÞ 2 mðtÞ: ð6:39Þ
c 1 (t) is an IMF if it satisfies two conditions:
Condition 1: The number of local extrema of c 1 (t) is equal to or differ
from the number of zero crossing of c 1 (t) by one.
Condition 2: The average of c 1 (t) logically be zero.
If c 1 (t) does not fulfill, the above two conditions then repeat the steps
from 1 to 4 on c 1 (t) instead of S(t).
Step 6. Calculate residue, r 1 (t):
r 1 ðtÞ 5 SðtÞ 2 c 1 ðtÞ: ð6:40Þ

Step 7. If the value of residue, r 1 (t), exceeds the threshold error tolerance
value then repeat steps from 1 to 7 to obtain the next IMF and new
residue.
If n number of IMFs are obtained from an iterative manner, the original
signal can be reconstructed as

X
SðtÞ 5 c i ðtÞ 1 rðtÞ: ð6:41Þ
n

6.3.2 Hilbert transform

An analytic signal has a real part as well as an imaginary part. Magnitude of
the analytic signal gives the magnitude spectrum, and phase angle of the