**4. Proposed procedure**

184 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

The ability to estimate the probability density functions, based on training patterns, is fundamental to the use of Eq. (8). Frequently, a priori probabilities can be known or estimated, and the loss functions require subjective evaluation. However, if the probability densities of the categories patterns to be separate are unknown, and all that is known is a set of training patterns, then, these patterns provide the only clue to the estimation of that unknown probability density. A particular estimator that can be used is (Specht, 1990):

1 1 ( )( ) ( ) exp <sup>2</sup>

Where *i* is the pattern number, *m* is the total number of training patterns, *ai x* is the i-th

( ) *Af x* is simply the sum of small Gaussian distributions centered at each training sample.

The probabilistic neural network is basically a Bayesian classifier implemented in parallel. The PNN, as described by Specht (Specht, 1988), is based on estimation of probability density functions for the various classes established by the training patterns. A schematic diagram for a PNN is shown in Figure 5. The input layer *X* is responsible for connecting the input pattern to the radial basis layer. *X xx x* 1 2 ,,, *<sup>M</sup>* is a matrix containing the

*xx xx f x m*

*m T*

<sup>1</sup> <sup>2</sup>

*n i*

(2 )

 

*A n*

*<sup>A</sup>* , and

Fig. 5. Schematic diagram of a Probabilistic Neural Network

**3.2 Structure of the Probabilistic Neural Network** 

training pattern of category

vectors to be classified.

2

*ai ai*

(10)

is the smoothing factor. It should be noted that

The proposed procedure is shown schematically in Figure 6. The real data file contains phases A-B-C voltages and currents waveforms, as well as digital signals that indicate the statuses of protective devices, as relays and circuit breakers, acquired by DDR and IED installed in the electrical system substations. These raw data are coded in the COMTRADE format for power systems (IEEE Standard Common Format for Transient Data Exchange), (IEEE Std C37.111, 1999). So, to obtain the voltages and currents signals it is firstly necessary to decode the COMTRADE data, and select the desired waveforms to be analyzed.

Fig. 6. Schematic diagram representing the proposed processing procedure.

Before inputting the voltages waveforms to the processing stage, a pre-processing routine is accomplished to standardize the raw data due to the different voltage levels that are

Application of Wavelet Transform and Artificial Neural Network to

level 1 to level 3, and (e) level 3 approximation.

the previously selected decomposition level:

the number of samples; <sup>ˆ</sup> ( ) *<sup>s</sup> d n* is the new value of ( ) *<sup>s</sup> d n* ;

coefficients *s*.

Where:

The

Extract Power Quality Information from Voltage Oscillographic Signals in Electric Power Systems 187

Fig. 7. Signal decomposition in 3 levels. In (a) original signal. From (b) to (d) details from

The wavelet transform performance to detect disturbances in electrical signals is substantially improved if a procedure for reducing noise level is applied to the decomposition level coefficients to be used in the detection process. This feature is highlighted in (Yang et al, 1999; 2000 & 2001). So, to better characterize the disturbance location in the signal, it is applied the following algorithm presented in (Misiti et al, 2000), to

ˆ () () () () 0

*n N* 1,2, , is the number of the decomposition level *s*, ( ) *<sup>s</sup> d n* , coefficient and *N* is

*s sss s s*

*d n d n if d n and d n if d n*

*sss s s*

*d n if d n and d n*

0 ( )

coefficients considered, as proposed in (Santoso et al, 1997).

() () () 0

 

  (12)

*s s*

*<sup>s</sup>* is a threshold based on the maximum absolute value of the decomposition level

*<sup>s</sup>* value used was 10% of the maximum absolute value of the decomposition level

A voltage waveform containing voltage sag is shown in Figure 8(a). In (b) it is presented the details level used to detect the disturbance beginning and (c) presents new details values after the noise reduction algorithm is applied. In (c) it can be observed smaller coefficients magnitudes over the entire signal which improves the algorithm performance used to detect the disturbance.

encountered in the power system topology. In the case study presented here, the power transmission system presents 230 kV and 500 kV voltage levels. The standardization is performed by converting the phase voltages to per unit (p.u.) values considering the voltage peak value as base voltage.

## **4.1 Processing stage**

In the processing stage, the wavelet transform is applied to the voltage waveforms to obtaining signals patterns that characterize short duration voltage variations (SDVV) and transient variations (TV) due to system faults. These obtained patterns are used as inputs to two Probabilistic Neural Networks for SDVV classification (PNN1), as well as to classify the fault type that has occurred (PNN2). The classification results will form a database which can be used to evaluate power quality indices for the electrical system.
