International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 1343194, 13 pages

http://dx.doi.org/10.1155/2016/1343194

## Application Research of the Sparse Representation of Eigenvector on the PD Positioning in the Transformer Oil

^{1}State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Baoding, Hebei 071003, China^{2}Hebei Provincial Key Laboratory of Power Transmission Equipment Security Defense, North China Electric Power University, Baoding, Hebei 071003, China^{3}State Grid Hebei Electric Power Research Institute, Shijiazhuang 050021, China^{4}Department of English, North China Electric Power University, Baoding, Hebei 071003, China

Received 8 April 2016; Revised 23 August 2016; Accepted 27 September 2016

Academic Editor: Elias Aboutanios

Copyright © 2016 Qing Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The partial discharge (PD) detection of electrical equipment is important for the safe operation of power system. The ultrasonic signal generated by the PD in the oil is a broadband signal. However, most methods of the array signal processing are used for the narrowband signal at present, and the effect of some methods for processing wideband signals is not satisfactory. Therefore, it is necessary to find new broadband signal processing methods to improve detection ability of the PD source. In this paper, the direction of arrival (DOA) estimation method based on sparse representation of eigenvector is proposed, and this method can further reduce the noise interference. Moreover, the simulation results show that this direction finding method is feasible for broadband signal and thus improve the following positioning accuracy of the three-array localization method. And experimental results verify that the direction finding method based on sparse representation of eigenvector is feasible for the ultrasonic array, which can achieve accurate estimation of direction of arrival and improve the following positioning accuracy. This can provide important guidance information for the equipment maintenance in the practical application.

#### 1. Introduction

Electrical equipment working status is directly related to the reliable operation of power system. And the practice has proved that the PD is the main reason for the high voltage electrical equipment insulation breakdown finally. In order to avoid accidents and timely find the potential danger, it is necessary for the electrical equipment partial discharge testing to ensure system reliability [1–6].

In the PD detection, an array sensor is used to collect ultrasonic signals generated by the PD. Then the array signal processing technology is used to complete the source direction of arrival (DOA) estimation and positioning. This method not only has strong anti-interference ability, but also has high positioning accuracy, and it has been widely used in many areas [7–9]. However, the main processing object of the traditional array signal processing methods is a narrowband signal, and the corresponding variety of space spectrum estimation (direction of arrival, DOA) methods that have high resolution and fast computing speed have been successfully applied. The electrical equipment ultrasonic signal generated by the PD in transformer oil is a typical broadband signal [10, 11], so the study on the DOA estimation algorithm that is suitable for wideband signal has extremely important significance.

The more classical wideband direction finding algorithm is mainly divided into two categories.

The first kind of method is incoherent subspace algorithm (ISM algorithm) [12, 13]. It is that a broadband signal is divided into a number of narrowband signals, and the average value is obtained after estimating the DOA of each narrowband signal. This method is a simple average of the narrowband signal processing results, which has a large amount of calculation, and cannot overcome the shortcomings of subspace algorithms adopted by the narrowband signal, such that it is easily affected by noise and sampling points and cannot solve the coherent sources. The second method is the coherent signal subspace algorithm (CSM algorithm) [14–17]. A focusing matrix is used to focus on all frequency components on a single reference frequency. Narrowband signal processing method is used to estimate the DOA of the covariance matrix after focusing, which reduces the correlation coefficient between signals, and can achieve the goal of coherent solution. Moreover, the existing CSM algorithm has to use the traditional narrowband signal processing method after focusing, which is still unable to avoid the disadvantages of subspace algorithms.

Mallat and Zhang in 1990s proposed the theory of signal sparse decomposition [18, 19]. It can be constructed by using different ways according to the specific signal form and different research purposes. Although, the signal is represented by a handful of basis functions, the information in the signal also focuses on these few basis functions, so it is more conducive to extract and explain the essential characteristics of signals. At present, the signal sparse decomposition has been widely used in signal noise reduction, compression, coding and image processing, and other fields [11]. In this paper, the sparse decomposition theory is applied to the PD signal DOA estimation. According to the array signal direction vectors, an overcomplete atom dictionary is established. The matching pursuit (MP) algorithm is used to choose the appropriate atoms, and the angle information contained in the atoms is the DOA of signal sources.

This work studies the PD positioning method in the transformer oil based on the sparse representation of eigenvectors. Taking a nine-element circular ultrasonic array sensor as an example, the mathematical model of ultrasonic array signals is given. Firstly, the broadband PD signals are received by an ultrasonic array sensor, and the covariance matrix of a single frequency is obtained by using RSS focusing method [20]. Then an eigenvector corresponding to the maximum eigenvalue is obtained through eigendecomposition of the covariance matrix obtained; the eigenvector is as the amount to be decomposed. According to the reference frequency and the steering vector form of an array signal, a step and step overcomplete dictionary is established, and thus the DOA estimation of the PD signal can be obtained by MP. Moreover, this method can further reduce the noise interference. Finally, according to the results, the PD source is located by using the three-array cross positioning principle. The simulation and experimental results show that the direction finding method based on sparse representation of eigenvectors can get higher accuracy of the DOA estimation results and improve the subsequent positioning precision.

#### 2. Broadband PD Ultrasonic Array Signal

##### 2.1. The Mathematical Model of Array Signal

The research results show that the ultrasonic frequency produced by the PD in transformer oil is mainly concentrated in the range of 50 kHz to 400 kHz, the center frequency is between 70 kHz and 200 kHz, and so the PD ultrasonic signal source is a typical broadband signal.

Assuming that a uniform array consists of* M* equally spaced elements and there is a space with broadband signals, the incident angle is, respectively, , and the signal received from the th element can be expressed aswhere is incident broadband signal; is additive noise; is time difference relative to the reference node when the th signal source is received by the th element.

The time shift theorem of Fourier transform is as follows: a signal is carried on Fourier transform after the signal has a time shift equal to that of the signal that has a phase delay after Fourier transform. If is the Fourier transform form of , that is,

then the Fourier transform form of is

For the signal received by the th element, both sides of (1) are analyzed based on Fourier transform:

The Fourier transform for elements can be written in matrix form, which is

And they can be written as

Among them, the steering vector matrix is

The signal direction matrix is different from narrowband direction matrix. Here, the frequency is the whole band of the signal, while the frequency is a single fixed value in a narrowband model.

When the signal is analyzed based on the discrete Fourier transform (DFT) with points, the frequencies are discrete points, and then (6) can be discrete as

The steering vector matrix iswhere is a steering vector:

##### 2.2. The Structure of the Circular Ultrasonic Array Sensor

The circular ultrasonic array sensor is composed of identical elements evenly distributed on the circumference with a radius of in the plane; the elements are arranged as shown in Figure 1 (e.g., taking nine element). The coordinate system of the sphere is used to express the DOA of the incident plane wave, and o is in the center of the array, which is the origin of the coordinate system. Consequently, it is taken as a reference point. In addition, when the incident signal direction is , azimuth is expressed as the angle between the -axis and a projection in the plane, and the projection is wired from the reference point to the source of the signal. The pitch angle is the angle between the -axis and the wired one that is from the reference point to the source of signal. Then the delay time in which the signal arrives at the th element relative to the reference element is