VSB Power Line Fault Detection

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The signals a, b, and c in Figure 35 are smooth signals which are phase-shifted by a certain degree, while the signal d is a rough signal. The various type of fractals like Petrosian, Katz, and Detrended Fluctuation Analysis (DFA) calculated on the above signals indicate that:

• The difference between the fractal values is less between the signals a, b, and c.
• The difference between the fractal values is more between the signals a, b, c, and signal d.

A similar kind of analysis is performed on the current medium voltage power line signals to measure the ‘roughness’ of the signal with and without the presence of the PD pattern.

As mentioned above, the following fractal dimension values are obtained for each signal:

1. Petrosian Fractal Dimension
2. Katz Fractal Dimension
3. Detrended Fluctuation Analysis (DFA)

These fractal values are calculated on the DWT denoised signal.

Petrosian Fractal Dimension

Petrosian Fractal Dimension (FD) is used to provide a fast computation of FD of a signal by translating the signal into a binary sequence.

Here a binary sequence is created based on the difference between the consecutive samples of the signal. If the difference between the consecutive samples is greater than the standard deviation of the signal then the respective sequence value is made 1 else it is made 0.

Once the binary sequence is created, the Petrosian FD is calculated using the formula shown in Figure 36.

Figure 36: Petrosian FD Formula

The code shown below performs the above-mentioned operation. Code reference

Katz Fractal Dimension

Katz FD is computed directly from the waveform thereby eliminating the preprocessing step of creating a binary sequence as in the case of Petrosian FD.

The formula to calculate the Katz FD is shown in Figure 37.

Figure 37: Katz FD Formula

The code shown below calculates the Katz FD for a given signal. Code reference

Detrended Fluctuation Analysis (DFA)

The fractal dimension of the signal using DFA is calculated in the following way:

1. Divide the signal into windows of size w.
2. Fit a polynomial curve of degree 2 in each window such that the root of squared errors in each window is minimized.
3. Calculate the average error over the entire signal for that window size
4. The error is directly proportional to the window size was shown in Figure 38.

Figure 38: Detrended Fluctuation Analysis

5. Solving the equation in Figure 38 gives the Fractal Dimension (D) as shown in Figure 39.

Figure 39: Fractal Dimension calculation using DFA

Here the slope of the plot of ln(E) vs. ln(w) gives the Fractal Dimension.

The code shown below calculates FD using DFA. Code reference

Higuchi Fractal Dimension

The Higuchi FD is calculated as shown in Figure 40.

Figure 40: Image Source

The code shown below calculates the Higuchi FD. Code reference

Entropy features

In information theory, the entropy of a random variable is the average level of “information”, “surprise”, or “uncertainty” inherent in the variable’s possible outcomes. — Wikipedia

It is also known as Shannon’s Entropy and is defined as shown in Figure 41.

Figure 41: Shannon’s Entropy

This paper discusses some of the drawbacks of Shannon’s Entropy, which are mentioned below:

• It neglects the temporal relationships between the values of the time series.
• It requires some prior knowledge about the system. The PDF of the time series under analysis should be provided beforehand.
• It is best designed to deal with linear systems and poorly describe non-linear chaotic systems.

Following entropy measures were formulated to overcome the limitations of Shannon’s entropy measure for time series data.

1. Permutation entropy
2. SVD entropy

Permutation entropy

The permutation entropy of a time series data is calculated using the formula shown in Figure 42.

Figure 42: Permutation Entropy

The permutation entropy is calculated by creating a set of sub-series (vectors) of data from the given time series data (population). The sub-series are obtained by sampling the time series data based on the following two parameters:

1. Order — Defines the length (number of samples) of the sub-series.
2. Delay — Defines the number of values in the time series data (population) to skip while sampling for the sub-series. A delay of 1 refers to a sample of consecutive values in the time series data.

Upon creating a set of sub-series, the values of each sub-series of data are sorted and their respective original index values are obtained, i.e., ‘numpy.argsort’ is performed on each sub-series of the data. This is referred to as π as per the equation shown in Figure 42.

The illustration of the calculation of the permutation entropy is shown in Figure 43.

Figure 43: Permutation Entropy calculation example

The code to calculate the permutation entropy for a given time series data is shown below. Code reference.

SVD entropy

Here SVD refers to Singular Value Decomposition. The entropy of the time series data is calculated using the singular values of the matrix formed from the sub-series of the data.

The SVD entropy is calculated using the formula shown in Figure 44.

Figure 44: SVD Entropy

Similar to permutation entropy, a set of sub-series vectors are formed from the time-series data using the order and delay parameters. The created sub-series vectors are then arranged in a matrix form of size (number of sub-series vector) x (order). The singular values of this matrix are obtained using SVD. This is referred to as σ in the equation shown in Figure 44.

The illustration of the calculation of SVD entropy on time series data is shown in Figure 45.

Figure 45: SVD Entropy calculation example

The code to calculate the SVD entropy for a given time series data is shown below. Code reference.

Final Model

The final binary classification is performed using a Deep Learning (DL) model comprising of:

• Bi-directional LSTM
• Bi-directional GRU
• Attention layer
• Dense layer

The architecture of the DL model is as shown in Figure 46.

Figure 46: Final Deep Learning Model

Bi-directional LSTM

Long Short Term Memory (LSTM) is used when the input is a sequence, mainly to capture the relationship between a given sequence and its past. A uni-directional LSTM captures the relationship of the current sequence with its past sequence while a bi-directional LSTM captures the relationship of the current sequence with its future sequence as well.

Figure 47 illustrates how a cell of a uni-directional LSTM looks like. Kindly refer to this blog to understand more about LSTMs.

Figure 47: LSTM cell. Image Source

The statistical features that were discussed earlier which are of dimension (160×57) are fed as input to the Bi-directional LSTM. Here,

• 160 is the length of the input sequence
• 57 is the dimensionality of each input sequence

A total of 128 Bi-directional LSTM units are used in the first layer. The illustration of the same is shown in Figure 48.

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