Fault Diagnosis in Power Lines using Hilbert Transform and Fuzzy ...

Fault Diagnosis in Power Lines using Hilbert Transform and Fuzzy Classifier F. M. Rivera-Calle1* , L. I. Minchala-Ávila2* J. C. Montesdeoca-Contreras3* Carrera de Ingeniería Electrónica Universidad Politécnica Salesiana Cuenca-Ecuador 1 [email protected], 2 [email protected] 3 [email protected]

Abstract—Early detection of faults in power lines allows improve the service quality and therefore a reduction in high operating costs that a failure of this type implies. This paper describes a method used to determine the type of failure occurs in a three-phase over time, using tools as Hilbert transform and fuzzy classifier for successful detection is done. The algorithm developed uses each of the power lines phases which are analyzed in its angle of coverage and its variation in time, after this analysis the results classified by a classifier Fuzzy c-means. This classifier makes groups of fault data and no-fault data. The results show a high performance in classified values near to zero as correct. Index Terms—Fault Diagnosis, Power Networks, Hilbert Transform, Fuzzy Classifier.

I. I NTRODUCTION Power lines began to build from a little over 100 years ago, and since that time mankind has had important advantages compared to those who did not had this service. Due to the high complexity in the structure of existing power grids, growing according to population growth and its industry, the fact detect a fault and locate where it is really created, that has made it an important case study. Failure caused by physical electrical system, have large negative consequences for the safety of workers, the growth of industry, economic implications, alterations and damage to equipment among others. In algorithms for detection and diagnosis of faults in electrical networks have developed some using neural networks, Bayesian networks optimal theories among others which have had favorable results, but not the best. In the development of this research a signal analysis are done, drawing out one of the parameters that make up and making a special study according to result. Similar methods have been previously applied to less complex systems raised grid; but whose principle is rugged enough for the case study. Hilbert transform is used for extracting the analysis parameters of the source signal, mathematical methods for analysis and to this point corresponds data conditioning, then the Fuzzy c-means algorithm for data classification resulting from conditioning which leads to the detection and diagnosis of faults is used. Some previous work concerning the Hilbert transform being used as extraction method of the phase space for fault detection on power lines by Zhenxing Liu [8], who

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J. A. Morales-Garcia4# Carrera de Ingeniería Eléctrica Universidad Politécnica Salesiana Cuenca-Ecuador 4 [email protected]

use the Hilbert transform as an ideal tool for troubleshooting lag in induction motors squirrel cage type; failures are taken into consideration: breakthroughs rotor bars and eccentricity, this study is based on spectral analysis of the stator current with the Hilbert module initially performing a diagnostic approach Park vector and phase-shift analysis on each of the three phases making up the signal of the rotor. Another case study is the problem of detecting faults in power systems using Hilbert transform has already been explored in previous research by Chen Chunling [6], who applies different types of failures to an electrical signal and these can be recognized by using the Hilbert transform.The basic principle is the detection of shift of the characteristic signal which has a frequency of 50Hz. The model is able to recognize signals without failure, over voltage signals, signals with momentary voltage to ground, faulty voltage fluctuation, harmonic in the voltage signal, transient signals, obtaining accurate results without a lot of delays in the case of a single node. In this research analysis over a faulty node will be expanded within the network, these faults are present simultaneously so the complexity in the management of information increases because they will not have only 3 variables for analysis but rather 72. Additionally, being a network system for the cascade effect present false information; meaning that one or more nodes may give misleading information about its state of failure, so one of the challenges is to recognize if this information is true or false. As these other cases can be mentioned as the Limei Yan, Yang Y and Liu Y. In this context this document presents the following structure, the section II is a brief description of the Hilbert transform and classification system Fuzzy c-means, In the section III the implemented algorithm is presented, the section IV presents the experimental test and results, finally the section V the conclusions are presents.

II. M ATHEMATICAL P ROCESS

The next step is update to membership values, for this process the eq. 8 is used.

A. Hilbert Transform The Hilbert transform provides a phase shift of ± π2 , is g (t) a signal whose Fourier transform is G (f ) or G (ω) the Hilbert transform of g (t) is denoted by gˆ (t), see eq. 1, Hilbert transform is a linear operator. ˆ 1 ∞ g (τ ) 1 = dτ (1) gˆ (t) = g (t) πt π −∞ t − τ This equation defines an improper integral, since for t = τ the integral has a uniqueness or indeterminacy. To avoid this problem the integral is calculated symmetrically, see eq. 2. ˆ

∞

−∞

g (τ ) dτ = lim →0 t−τ

ˆ

t−

−∞

g (τ ) dτ + t−τ

ˆ

∞

t+

g (τ ) dτ t−τ



(2) Inverse Hilbert transform can be calculated by eq. 3, gˆ (t) and g (t) are part of pair transform of Hilbert. ˆ 1 ∞ gˆ (t) g (t) = − dτ (3) π −∞ t − τ

μci (x) =

k



j=l

1 x−vi 2 x−vj 2

1  m−1

1 ≤ i ≤ k,

xX (8)

Fuzzy c-Means updates prototypes and iterative functions using the above equations to converge, see eq. 9.  Θ(t+1) − Θ(t) ≤ 

(9)

Where Θ =is the matrix of membership values in t iteration  =is a threshold to be determined by the user C. Composition of Power Grid It’s called Network Power Verification (REC), at the network model in this research study. This REC is composed by 24 parent nodes, 20 generators with different voltage, 21 static loads, 15 dynamic loads, and 68 switches as shown in figure 1.

The Hilbert Transform definition it follows that gˆ (t) can be 1 , see interpreted as the convolution of g (t)with the signal πt eq. 4. gˆ (t) = g (t)

1 πt

(4)

B. Fuzzy c-Means The Fuzzy c-Means is used to recognition of failures in power grid, the function J1 target a normal function c-Means can be extended in two ways: 1) Fuzzy membership functions incorporated into the formula. 2) An additional parameter m is introduced as exponential weight in fuzzy membership Then the extended objective function is showen in eq. 5.

Jm (P, V ) =

k  

(μci (xk ))

m

 xk − vi 2

(5)

i=l xk X

Where, P are partitions of data set X formed by C1 , C2 , C3 , . . . Ck . The parameter m is the factor that determines the extent to which members of a partial cluster affect the result cluster. Fuzzy c-Means tries to find a good partition search for prototypes vi to minimize function Jm . This methodology requires search a membership function μci to minimize Jm . The matrix begins with a restriction showed by eq. 6. P 

μci = 1,

∀

i = 1, ..., X

(6)

i=1

The centers vi of i class are give by eq. 7.  m (μci (x)) x 1≤i≤k vi = x X m x X (μci (x))

(7)

Figure 1. The electricity grid distribution

The REC information is available, as the three lines (phases) power of each node, the frequency of the signal lines phase is always 60 Hz. This network was designed for the detection and diagnosis of faults for the first time using hybrid systems. III. M ETHODOLOGY As the parameter on which it works is the coverage angle, the Hilbert transform is used to obtain it. Hilbert transform is applied to obtain the analytic signal using eq. 10. H (s (t)) = SI (t) + jSQ (t)

(10)

Where, the real part is described by eq. 11, and the imaginary part are described by eq. 12. SI (t) = s (t) cos (2πω0 t)

(11)

SQ (t) = s (t) sin (2πω0 t)

(12)

The coverage angle signal in time domain is determined by eq. 13.  θ (t) = ∠ (SI (t) + jSQ (t)) = arctan

SQ (t) SI (t)

y (angle) = wt



Where θ = Actual angle, m = Actual angle and t = Instant of measurement.

(13) (14)

The figure 2 shows angle response using the Hilbert transform when eq. 13 is used, as result is eq. 14. In this test a sampling time of 5000 ms is used.

Figure 4. Angle of difference.

The figure 3 shows the three-phases of the 24 nodes. For a good clearance the figure 5 shows the angle difference for three-phases at node 5.

Figure 2. Angle response using the Hilbert transform.

The figure 3 shows a angle in time function, the slope od this function can be calculate by eq. 15 and eq. 16. m=

y 1887 + 1.571 = = 0.3777 x 5000

m=w=

2 ∗ π ∗ 60 [hz] 2πf = 1000 1000

(15)

(16) Figure 5. Angle difference at node 5.

In this process a unit used is [ms] then the divisor by 1000 are made in eq. 16.

To the angle difference in the time domain for 24 nodes, has been derived. In this way has been obtained the largest deviation in proposed point. The figure 6 represents the derivative function of figure 4, shows the largest variation for phases at 4000 ms.

Figure 3. Angle response using the Hilbert transform.

The figure 4 shows the “angle of difference”, it is defined by eq. ∠dif f erence = θ = −mt

(17)

Figure 6. Derivative function of angle difference.

The figure 7 shows the comparative between derivative of three phases.

Figure 9. Angle response using the Hilbert transform

Figure 7. Comparative between derivative of three phases.

As figure 5 is somewhat complex due to the amount of data stored, for example below the three lines of a single node, in order to have greater clarity in the explanation will be made only. Figure 8 depicts the stage at node 3 which contains study and failures between line A and line B. In detailing carefully figure 6 a dotted line that is not within the overall result of the derivative (data without failure) is observed. These points correspond to a transient signal is to start the data file any scenario. As this information is untrue, the next step is to remove the first 50 data which for the moment, it is assumed that there will have failed, and this is where the transition is located.

It has been seen that the data is called faulty belong to two groups; Well, they really correspond to a single group, but it produces a reflection on the opposite octant. To avoid duplicate information only take the data from the first octant, which will help in the time of clustering method using Fuzzy c-means.

Figure 10. Angle response using the Hilbert transform

To make a better assessment of how each point is part of one of these groups, the tool data classification Fuzzy c-Means is used.

Figure 8. Angle response using the Hilbert transform

If the information shown again in figure 9. This figure is obtained, three groups, one with the highest concentration data (center) and two with a lower concentration (side) data are clearly observed. As the derivative of the difference of a signal that no trouble will be approximately zero, then the data source and near flawless data cataloged. For the case study data without failure will be the highest concentration since its value is closer to zero. The more remote data correspond to data with a much greater difference to zero and will, according to the definition, data fails.

Figure 11. Angle response using the Hilbert transform

The Fuzzy c-Means function is a technique in which a data set is organized into a number n of groups (clusters), where each data in the data set is assigned a degree of membership to a group U. For example, a figure that approaches the center of a group (cluster) will have a high degree of membership and a fact that this far from the center will have a much lower degree of belonging. This value of membership varies between 1 and 0, the value 1 means it is a member of the group, while the value 0 indicates that it is not, values that fall between these two will indicate which are partially members of either group. In the algorithm of Fuzzy c-Means, for each iteration, a function object is minimized to find the best location of groups.

Table I T IME TO START FAULTS IN DIFFERENT ENVIRONMENTS

Test 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Estimate time to start fault Fault time (ms) Real Fault time (ms) 951 1000 2985 3000 3954 4000 492 500 1979 2000 178 200 3453 3500 2082 2100 4179 4200 790 800 3176 3200 360 400 1192 1200 3281 3300

(ms) 49 15 46 8 21 22 47 18 21 10 24 40 8 19

IV. T EST AND R ESULTS The result of the algorithm developed for the mains failure AC lines Node 2, Node 4 line C to ground node 18 lines 19 lines AB and Node C to Earth is shown.

V. C ONCLUSIONS In the process of detection and diagnosis is important to consider the design of information and classification; conditioning success will depend on the classification results. The use of Hilbert transform and Fuzzy c-means algorithm, provides results showing the superior effectiveness than others works done with other techniques in the same area. Hilbert algorithm - Fuzzy c-Means has an effectiveness of 100% for all cases of failure regardless of number of simultaneous failures that occur in the network. The algorithm Hilbert Fuzzy c-Means has a very good response to signals with noise amplitude, it shown in the mathematical process. The system achieves handle noise up to ±30% of the variation in amplitude of the response signal without errors. R EFERENCES

Figure 12. Result Hilbert algorithm Fuzzy c-Means

Failures: 2: fault lines A-C 4: fault line C-GND 18: fault lines A-B 19: fault line C-GND. Hilbert algorithm Fuzzy c-means has allows to provide an estimate of the instant the fault started; this time is only an estimate since the Hilbert transform requires some time to wound to its stable state. The results of this test were analyzed according to the number of failures, the corresponding fault is initiated at time 4000ms. Each of the above table results were verified with the original setting and a deviation range between 8 and 50 ms over the original onset of failure was obtained. Some of the times of the original presence of the fault Vs estimated time for the algorithm Hilbert Fuzzy c-Means will be shown.

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