Patent Publication Number: US-10782115-B2

Title: Detection of radial deformations of transformers

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of priority from U.S. Provisional Patent Application Ser. No. 62/382,789, filed on Sep. 2, 2016, and entitled “DETECTION OF RADIAL DEFORMATION OF TRANSFORMERS BY USING UHF STEPPED FREQUENCY BASED SAR IMAGING METHOD,” which is incorporated herein by reference in its entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure generally relates to online condition monitoring of transformers, and particularly to methods and devices for online detection of radial deformations of transformers. 
     BACKGROUND 
     Online monitoring of power system equipment, such as power transformers, plays an important role in improving reliability and efficiency of power systems. A transformer may be associated with different mechanical faults, which may be caused by mechanical forces that arise due to short circuits, explosions, and earthquakes, etc. For example, radial deformations may occur in a transformer winding due to forces that are incurred in a radial direction. The radial deformations may include one or more protuberances or fovea in the transformer winding. 
     Online detection of such mechanical defects protects a transformer from incurring serious damage and it allows for the transformer to be utilized for a longer service period. The mechanical faults in a transformer, along with aging, can over time impact winding insulation functionality, which in turn may cause short circuits and severe damage to the winding. In case of detection of a faulty transformer, the transformer may be taken out of service and it may be repaired urgently. This increases reliability of the power system. 
     Various methods have been disclosed in the art for monitoring transformer status, such as a transformer function method, low-voltage impulse test, ultra wideband sensor method, short circuit impedance method, and S-parameter-based method. However, these methods have various limitations such as not being able to provide a comprehensive online insight of the mechanical fault, for example, type, magnitude, and position of the mechanical fault. 
     There is, therefore a need in the art for a simple method and device for in-depth online detection of mechanical faults such as radial deformations in transformer windings. There is further a need in the art for a simple method and device capable of providing details of the faults without any interferences or misleading disturbances. 
     SUMMARY 
     This summary is intended to provide an overview of the subject matter of the present disclosure, and is not intended to identify essential elements or key elements of the subject matter, nor is it intended to be used to determine the scope of the claimed embodiments. The proper scope of the present disclosure may be ascertained from the claims set forth below in view of the detailed description below and the drawings. 
     In an exemplary embodiment consistent with the present disclosure, a method for detecting radial deformation in a winding of a transformer is disclosed. The method may include synthetic aperture radar (SAR) imaging of the winding using ultra high frequency (UHF) electromagnetic signals in a first instance of the winding to obtain a first image of the winding; SAR imaging of the winding using UHF electromagnetic signals in a second instance of the winding to obtain a second image of the winding; and comparing the first image of the winding and the second image of the winding to detect a radial deformation in the winding. The UHF electromagnetic signals may be transmitted as a plurality of successive sinusoidal signals, where frequencies of the successive sinusoidal signals gradually change from a first frequency to a second frequency. 
     In an exemplary embodiment, the SAR imaging may include: transmitting UHF electromagnetic signals by a first transceiver antenna and receiving reflected UHF electromagnetic signals from the winding by a second transceiver antenna. Transmitting of UHF electromagnetic signals and receiving of reflected UHF electromagnetic signals may be carried out in a series of scanning steps in which the first transceiver antenna and the second transceiver antenna may be gradually moved along a longitudinal axis of the winding from one scanning step to the next. The SAR imaging may further include calculating an amount of energy reflected from each point of the winding based on the received reflected UHF electromagnetic signals for each of the scanning steps. 
     According to an exemplary embodiment, comparing the first image and the second image may involve comparing the amount of energy reflected from each point of the winding in the first instance with the amount of energy reflected from each point of the winding in the second instance. According to an exemplary embodiment, calculating the amount of energy reflected from each point of the winding may be carried out utilizing Kirchhoff migration method. 
     According to an exemplary embodiment, the plurality of successive sinusoidal signals may be transmitted in a series of successive frequency steps, where in each frequency step, a sinusoidal signal may be transmitted and the frequency of each transmitted sinusoidal signal changes from one frequency step to the next with a predetermined frequency step. 
     According to an exemplary embodiment, in order to obtain a time domain UHF electromagnetic signal for each scanning step, an inverse fast Fourier transform (IFFT) may be performed on magnitudes and phases of the received reflections of the plurality of successive sinusoidal signals for all frequency steps in each scanning step. 
     In an exemplary embodiment, the method for detecting radial deformation in a winding of a transformer may further include comparing frequency of each transmitted sinusoidal signal in each frequency step with a frequency of a received reflection of the transmitted sinusoidal signal in that frequency step. 
     In an exemplary embodiment, the method for detecting radial deformation in a winding of a transformer may further include identifying a frequency step for which frequency of a transmitted sinusoidal signal differs from a frequency of a received reflection of the transmitted sinusoidal signal as a partial discharge-corrupted frequency step. 
     According to an exemplary embodiment, identifying a partial discharge-corrupted frequency step is carried out by a generalized likelihood ratio test and transmission of the sinusoidal signal for a partial discharge-corrupted frequency step may be repeated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawing figures depict one or more implementations in accord with the present teachings, by way of example only, not by way of limitation. In the figures, like reference numerals refer to the same or similar elements. 
         FIG. 1A  is a schematic representation of a synthetic aperture radar (SAR) imaging setup for detection of radial deformation in a transformer winding, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 1B  is a schematic representation of a side-view of a SAR imaging setup for detection of radial deformation in a transformer winding, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 1C  depicts an implementation of the Kirchhoff migration formula based on a discrete boundary condition function. 
         FIG. 2A  illustrates a method for detecting radial deformation in a transformer winding, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 2B  illustrates a method for ultra-high frequency synthetic aperture radar (UHF SAR) imaging process, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 2C  illustrates a method of detecting signals which contain a partial discharge signal, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 3  is a frequency vs. time graph showing an implementation of stepped-frequency method, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 4A  is a schematic representation of an intact transformer winding model, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 4B  is a color-coded map of an intact transformer winding model generated using the amounts of energy calculated by the Kirchhoff migration method, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 5A  is a schematic representation of a deformed transformer winding model, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 5B  is a color-coded map of an deformed transformer winding model generated using the amounts of energy calculated by the Kirchhoff migration method, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 5C  shows a partial-discharge corrupted color-coded map of an intact transformer winding model, consistent with one or more exemplary embodiments of the present disclosure. 
         FIG. 6  is a schematic functional block diagram of one embodiment of a system for detecting radial deformation in a transformer winding, consistent with one or more exemplary embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description is presented to enable a person skilled in the art to make and use the methods and devices disclosed in exemplary embodiments of the present disclosure. For purposes of explanation, specific nomenclature is set forth to provide a thorough understanding of the present disclosure. However, it will be apparent to one skilled in the art that these specific details are not required to practice the disclosed exemplary embodiments. Descriptions of specific exemplary embodiments are provided only as representative examples. Various modifications to the exemplary implementations will be readily apparent to one skilled in the art, and the general principles defined herein may be applied to other implementations and applications without departing from the scope of the present disclosure. The present disclosure is not intended to be limited to the implementations shown, but is to be accorded the widest possible scope consistent with the principles and features disclosed herein. 
     Synthetic aperture radar (SAR) imaging of power transformer windings using ultra-wideband (UWB) signals (i.e., UWB SAR imaging) may be used for detection of radial deformations in the transformer windings. In the exemplary UWB SAR imaging method, a Gaussian pulse that is modulated by a sinusoidal carrier may be transmitted by a transmitting antenna into an environment containing the transformer winding, and a reflected signal from the transformer winding may be received by a receiving antenna. 
       FIG. 1A  is a schematic representation of a SAR imaging setup for detection of radial deformation in a transformer winding  101  and  FIG. 1B  is a side-view of the same SAR imaging setup. Referring to  FIG. 1A , a first transceiver antenna  102  and a second transceiver antenna  103  may be placed at a specific distance from transformer winding  101 . For example, transceiver antennas  102 ,  103  may be installed on transformer tank (not illustrated in  FIG. 1A ). First transceiver antenna  102  may transmit a signal and second transceiver antenna  103  may receive a reflected signal from the surface of the transformer winding  101 . The reflected signal received by second transceiver antenna  103  may provide one-dimensional (1D) information about the transformer winding  101 . Referring to  FIGS. 1A and 1B , in order to create a two-dimensional (2D) image of the transformer winding  101 , first and second transceiver antennas  102 ,  103  may be moved in a stepwise manner along longitudinal axis  106  of the transformer winding with predetermined step sizes and at each step, the steps of sending a signal and receiving a reflected signal may be repeated to obtain 1D information for each step. Each step of sending a signal receiving a reflected signal may be referred to as a scanning step  104  and each scanning steps  104  may occur at and equal distance from one another. Once 1D information of the transformer winding is gathered for all scanning steps  104  along the height of the transformer winding  101 , then a 2D image of the transformer winding  101  may be generated. Number of scanning steps  104  along the height of the transformer winding  101  determines the resolution of the 2D image of the transformer winding  101 . 
     According to an exemplary embodiment, Kirchhoff migration method may be performed on the received signals in all scanning steps  104  to form a 2D image of the transformer winding  101 . Referring to  FIGS. 1A and 1B , a Cartesian coordinate system  105  may be defined with three mutually perpendicular axes X, Y, and Z. X and Y axes may be defined on a plane perpendicular to longitudinal axis  106  of transformer winding  101  and Z axis may be defined parallel to longitudinal axis  106  of the transformer winding  101 . The 2D image of the transformer winding  101  may be generated by calculating magnitude of the reflected signal for every point on transformer winding  101  on planes parallel to X-Y plane for all scanning steps  104 . 
     Final formulation of the Kirchhoff migration method for plotting the 2D image is as Equation (1) below: 
     
       
         
           
             
               
                 
                   
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     For purposes of the present disclosure, in Equation (1), U denotes 2D image function, which determines the amount of energy reflected from each point of the surface of the transformer winding. A matrix of image function values may be obtained, where each element of the matrix represents the amount of energy reflected from a corresponding point on the surface of the transformer winding. 
       FIG. 1C  depicts an implementation of the Kirchhoff migration formula based on a discrete boundary condition function. Transceiver antenna is located at point Q (kd M , 0) and an arbitrary point of the surface of the transformer winding is for example located at P (id 0 , jd 0 ). Magnitude of image function at arbitrary point, P (id 0 , jd 0 ), may be obtained by equation (1). T s  denotes scanning sample time, d M  denotes distance between two adjacent scanning steps, kd M  is coordinate of k-th scanning step on Z axis, d 0  denotes spatial point, M denotes number of image pixels on Z-axis, G denotes number of image pixels on X-axis, k is total number of scanned points, f denotes value of the field boundary, and c denotes wave speed. 
     In an exemplary embodiment, a 2D image of the transformer winding may be obtained by plotting an image function, U in a color-map scale. In the color-map scale plots, the color of a point or pixel indicates the magnitude of the image function at that point. According to an embodiment, an image correlation method may be utilized to detect radial deformations in the transformer winding. In the image correlation method, 2D images of the transformer winding at a given time during service life of the winding is compared to a 2D image of the winding before installation of the transformer, which will be described in detail later in the present disclosure. Color-map scale plots may serve as visual aids in the image correlation method. 
     According to an exemplary embodiment, in order to detect radial deformations in the transformer winding, values of the image function at a given time during service life of the winding may be compared to values of the image function before installation of the transformer, where any difference between the two values may be an indicator of a radial deformation. 
     It should be noted, transformer tanks are made of conductive materials, and electromagnetic waves cannot penetrate through the conductive materials. Therefore, according to an exemplary embodiment, in order to perform SAR imaging, a window may be installed on the transformer tank, for example, by cutting some part of the transformer tank and covering it by an insulator window. Then, the transceiver antennas may be placed in front of the insulator window. The electromagnetic waves may easily pass through the insulator, reaching the transformer winding, and reflecting back to the transceiver antennas. 
     Exemplary systems and methods disclosed herein may be directed to detection of radial deformations in power transformer windings by SAR imaging of power transformer windings using UHF electromagnetic signals (i.e., UHF SAR imaging) instead of UWB signals. 
     UHF SAR imaging may be used for detection of partial discharge in the transformer windings. Detection of both partial discharge and radial deformations may be carried out with two sets of antennas, where a first set of antennas may be utilized for UHF SAR imaging of power transformer windings in order to detect partial discharge, and a second set of antennas may be utilized for UWB SAR imaging of power transformer windings in order to detect radial deformations. Alternatively, a large antenna may be utilized for transmitting both UHF and UWB signals, which may be associated with limitations in online detection of partial discharge and radial deformations. Detection of both partial discharge and radial deformations, simultaneously, may be possible with only one set of antennas using UHF SAR imaging of power transformer windings. However, simultaneous detection of both partial discharge and radial deformations with only one set of antennas using UHF SAR imaging of power transformer windings may be lead to generation of inaccurate or incorrect two-dimensional images of transformer windings due to possible partial discharge occurrence. In other words, partial discharge occurrence during UHF SAR imaging may cause distortion in two-dimensional image of transformer windings. 
     According to exemplary embodiments of the present disclosure, in order to avoid partial discharge interference in generating two dimensional images of transformer windings by UHF SAR imaging, a UHF stepped-frequency method may be utilized for detection of radial deformations, and a generalized likelihood ratio test (GLRT) may be adapted and applied to detect partial discharge occurrence during UHF SAR imaging. 
       FIG. 2A  illustrates a method  200  for detecting radial deformation in a transformer winding, according to one or more exemplary embodiments of the present disclosure. In an exemplary embodiment, method  200  may include a first step  201  of UHF SAR imaging of the winding in a first instance to obtain a first image of the winding; a second step  202  of UHF SAR imaging of the winding in a second instance to obtain a second image of the winding; and a third step  203  of comparing the first image and the second image in order to detect a radial deformation in the winding. For purposes of this disclosure, the first instance may refer to a state where there are no radial deformations in the winding (i.e. intact winding state) or before the initial use of the windings and the second instance may refer to a state of the winding at any given time during service life of the transformer. For example, the second image may be obtained every day or every week. Alternatively, the first and state may refer to different periods of time, a first period of time when an image is captured and a second period of time when an image is captured. 
     Referring to  FIG. 2A , the first step  201  and second step  202  of the method  200  may involve UHF SAR imaging of the transformer winding in a signal range of 0.3 to 3 GHz. An embodiment of a UHF SAR imaging process that may be performed in the first step  201  and the second step  202  of method  200  is illustrated in  FIG. 2B . Referring to  FIG. 2B , a UHF SAR imaging process  210  may include a step  211  of transmitting UHF electromagnetic signals by a first transceiver antenna; a step  212  of receiving reflected UHF electromagnetic signals from the winding by a second transceiver; and a step  213  of calculating an amount of energy reflected from each point of the surface of the winding based on the received reflected UHF electromagnetic signals. According to an exemplary embodiment, a stepped-frequency method may be utilized for transmitting UHF electromagnetic signals by the first transceiver antenna. In the stepped frequency method, a transceiver antenna may transmit a range of sinusoidal signals instead of a train of Gaussian pulses. These sinusoidal signals may be successive sinusoidal signals, where the frequency of the signals changes from one sinusoidal signal to the next. 
     Referring to  FIG. 2B , the step  211  and the step  212  may be carried out in a series of scanning steps. With further reference to  FIGS. 1A and 1B , during the stepped frequency method, at each scanning step  104 , first transceiver antenna  102  may send the successive sinusoidal signals with frequencies sweeping from the lowest frequency (f L ) to the highest frequency (f H ) with a predetermined frequency increment step (Δf) and at each frequency, the reflected signal may be picked up by second transceiver antenna  103 . 
       FIG. 3  is a frequency vs. time graph  300  showing an implementation of stepped-frequency method. Referring to  FIGS. 1A, 1B, and 3 , at a first scanning step  301  (similar to any of scanning steps  104 ), a sinusoidal pulse with a frequency  302  of f L  is transmitted by transceiver antenna  102  and received by transceiver antenna  103 . Then, a second sinusoidal pulse with a frequency  303  of f L +Δf is transmitted by transceiver antenna  102  and received by transceiver antenna  103 . The frequency is increased in this manner step by step until the frequency of the transmitted sinusoidal pulse reaches a frequency  304  of f H . Then, a time domain UHF electromagnetic signal for scanning step  301  may be obtained by performing an inverse fast Fourier transform (IFFT) on the magnitudes and phases of the gathered signals. After that, transceiver antennas  102  and  103  may be moved to a next scanning step  305  in order to obtain a time domain signal for scanning step  305 . This procedure is repeated for all scanning steps  104 . 
     Referring to  FIG. 2B , the step  213  may include applying the Kirchhoff migration method on the received time domain UHF electromagnetic signals in order to calculate an amount of energy reflected from each point of the surface of the winding. According to an embodiment, the final formulation of the Kirchhoff migration method for calculating the amount of energy reflected from each point of the surface of the winding may be presented as Equation (1), which was described in detail in preceding sections of the present disclosure. 
     Utilizing the stepped-frequency method as described above may help eliminate distorting effects of partial discharge occurrence on detection of radial deformation. In a UHF SAR imaging method, occurrence of partial discharge may distort the 2D image of the transformer winding, and in case of a partial discharge occurrence, the SAR imaging process must be repeated. In contrast, in a UHF SAR imaging process performed by a stepped-frequency method, if a partial discharge occurs during transmitting and receiving sinusoidal pulses, it is not necessary to repeat the entire process of transmitting and receiving all the frequencies. The process should only be repeated for the frequency or frequencies, at which partial discharge has occurred. A new measurement for these frequencies can replace the previous measurements, in order to construct a partial discharge-free image of the transformer winding. 
     It is necessary to detect signals which contain a partial discharge signal. Emitted signal of a partial discharge has a wide range of frequencies in UHF band. If there is a difference between frequencies of transmitted and received signals in a frequency step of the stepped-frequency method, it may indicate that partial discharge has happened at that frequency step, which means transmitting and receiving of signals must be repeated for that frequency. In other words, in instances when transmitted and received signals only differ in magnitude and phase, no partial discharge has occurred. However, in instances, when transmitted and received signals differ in magnitude, phase, and frequency, partial discharge has occurred. 
       FIG. 2C  illustrates a method  220  of detecting signals which contain a partial discharge signal. The method  220  may include a step  221  of transmitting a plurality of successive sinusoidal signals in a series of frequency steps; a step  222  of comparing frequency of each transmitted sinusoidal signal in each frequency step with a frequency of a received reflection of the transmitted sinusoidal signal in that frequency step; a step  223  of identifying a frequency step for which frequency of a transmitted sinusoidal signal differs from a frequency of a received reflection of the transmitted sinusoidal signal as a partial discharge-corrupted frequency step; and a step  224  of repeating transmission of the sinusoidal signal for the partial discharge-corrupted frequency step. 
     Referring to  FIG. 2C , in the step  222  and the step  223 , a generalized likelihood ratio test (GLRT) may be adapted and applied to detect partial discharge occurrence during UHF SAR imaging. In stepped-frequency method, a sinusoidal signal is transmitted and the reflection of a nearby object is recorded. If no partial discharge occurs during this process, the received signal contains a sinusoidal wave with a known frequency but unknown amplitude and phase. The FFT of this signal has a peak at the transmitted signal frequency and other FFT bins contain noise. On the other hand, if a partial discharge occurs, partial discharge signal is added to the received signal components and because of the wide-band nature of the partial discharge signal, it resembles a higher level of background noise. Referring to  FIG. 2C , in the step  222  and the step  223 , partial discharge occurrence may be detected by omitting the FFT bin that contains the received signal and then comparing sum of square of the amplitude of other FFT bins with a predetermined threshold. If the summation is greater than the threshold, then it is assumed that a partial discharge has occurred, but if the summation is smaller than the threshold, then it is assumed that the received signal only contains the transmitted signal with noise. 
     In an exemplary embodiment, in a first hypothesis (H 0 ), it may be assumed that there is no partial discharge in the received signal. If it is true, the k th  FFT bin contains the transmitted signal in frequency-domain, and other FFT bins contain the background noise. Therefore, if for purposes of the present disclosure, r n  and ae jθ  are received and transmitted signals, respectively, under H 0 , the probability density functions of the FFT bins are presented in Equations (2) and (3) as follows: 
     
       
         
           
             
               
                 
                   
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     In equations (2) and (3), σ 0  and σ 0   2  denote mean and variance of the background noise, respectively. Under an alternative hypothesis (H I ), it is assumed that all FFT bins are corrupted by partial discharge signal and therefore the probability density functions of the FFT bins are presented in Equations (4) and (5), as follows: 
     
       
         
           
             
               
                 
                   
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                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     N 
                     , 
                     
                       n 
                       ≠ 
                       k 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
     In equations (3) and (4), σ 1  and σ 1   2  denote mean and variance of the partial discharge signal, respectively. According to Neyman-Person lemma, the likelihood function of the test should be compared with a threshold to decide between two hypotheses. The likelihood function of the received signals, L(r 1 , . . . , r N ) may be generated by dividing the probability density functions under alternative and first hypotheses, as presented by Equation (6) below: 
     
       
         
           
             
               
                 
                   
                     L 
                     ⁡ 
                     
                       ( 
                       
                         
                           r 
                           1 
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           r 
                           N 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∏ 
                         
                           n 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               
                                 r 
                                 n 
                               
                               | 
                               
                                 H 
                                 1 
                               
                             
                             ) 
                           
                         
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               
                                 r 
                                 n 
                               
                               | 
                               
                                 H 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               
                                 r 
                                 n 
                               
                               | 
                               
                                 H 
                                 1 
                               
                             
                             ) 
                           
                         
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               
                                 r 
                                 n 
                               
                               | 
                               
                                 H 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                       × 
                       
                         
                           ∏ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             
                               n 
                               ≠ 
                               k 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   n 
                                 
                                 | 
                                 
                                   H 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   n 
                                 
                                 | 
                                 
                                   H 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     -&gt; 
                     
                       L 
                       ⁢ 
                       
                         ( 
                         
                           
                             r 
                             1 
                           
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             r 
                             N 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             σ 
                             0 
                             2 
                           
                           
                             
                               σ 
                               0 
                               2 
                             
                             + 
                             
                               σ 
                               1 
                               2 
                             
                           
                         
                         ) 
                       
                       
                         N 
                         2 
                       
                     
                     × 
                     
                       ( 
                       
                         
                           
                             
                                
                               
                                 
                                   r 
                                   k 
                                 
                                 - 
                                 
                                   ae 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                               
                                
                             
                             2 
                           
                           
                             2 
                             ⁢ 
                             
                               σ 
                               0 
                               2 
                             
                           
                         
                         - 
                         
                           
                             
                                
                               
                                 
                                   r 
                                   k 
                                 
                                 - 
                                 
                                   ae 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                               
                                
                             
                             2 
                           
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   σ 
                                   0 
                                   2 
                                 
                                 + 
                                 
                                   σ 
                                   1 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                     × 
                     
                       
                         ∏ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           
                             n 
                             ≠ 
                             k 
                           
                         
                         N 
                       
                       ⁢ 
                       
                         exp 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                    
                                   
                                     r 
                                     n 
                                   
                                    
                                 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                   σ 
                                   0 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 
                                    
                                   
                                     r 
                                     n 
                                   
                                    
                                 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       σ 
                                       0 
                                       2 
                                     
                                     + 
                                     
                                       σ 
                                       1 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     -&gt; 
                     
                       L 
                       ⁢ 
                       
                         ( 
                         
                           
                             r 
                             1 
                           
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             r 
                             N 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             σ 
                             0 
                             2 
                           
                           
                             
                               σ 
                               0 
                               2 
                             
                             + 
                             
                               σ 
                               1 
                               2 
                             
                           
                         
                         ) 
                       
                       
                         N 
                         2 
                       
                     
                     × 
                     
                       exp 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               σ 
                               1 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                 
                                   σ 
                                   0 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       σ 
                                       0 
                                       2 
                                     
                                     + 
                                     
                                       σ 
                                       1 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                    
                                   
                                     
                                       r 
                                       k 
                                     
                                     - 
                                     
                                       ae 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     , 
                                     
                                       n 
                                       ≠ 
                                       k 
                                     
                                   
                                   N 
                                 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       r 
                                       n 
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
     Therefore, the decision is made as follows: 
     
       
         
           
             
               
                 
                   
                     L 
                     ⁡ 
                     
                       ( 
                       
                         
                           r 
                           1 
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           r 
                           N 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     
                       
                         ≶ 
                         
                           H 
                           1 
                         
                       
                       ⁢ 
                       η 
                     
                     
                       H 
                       0 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
     In Equation (9), η denotes the decision threshold, the value of which may be determined in accordance with the acceptable probability of false alarm (P fa ). Regarding the monotonically increasing phenomenon of the exponential function, the decision strategy may be rewritten as Equation (10) below: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                              
                             
                               
                                 r 
                                 k 
                               
                               - 
                               
                                 ae 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 1 
                               
                               , 
                               
                                 n 
                                 ≠ 
                                 k 
                               
                             
                             N 
                           
                           ⁢ 
                           
                             
                                
                               
                                 r 
                                 n 
                               
                                
                             
                             2 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         ≶ 
                         
                           H 
                           1 
                         
                       
                       
                         H 
                         0 
                       
                     
                     ⁢ 
                     
                       
                         
                           2 
                           ⁢ 
                           
                             
                               σ 
                               0 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   σ 
                                   0 
                                   2 
                                 
                                 + 
                                 
                                   σ 
                                   1 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         
                           σ 
                           1 
                           2 
                         
                       
                       × 
                       
                         ln 
                         ⁡ 
                         
                           ( 
                           
                             
                               η 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       σ 
                                       0 
                                       2 
                                     
                                     + 
                                     
                                       σ 
                                       1 
                                       2 
                                     
                                   
                                   
                                     σ 
                                     1 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             
                               N 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     η 
                     1 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
     The problem with this decision is that a, θ, and σ 1  are unknown. Referring to above decision strategy in Equation (10), the value of σ 1  is not important because it only affects the threshold value (η 1 ). However, the threshold value may be directly computed considering the acceptable P fa . Both a and θ values directly affect the decision strategy. 
     In generalized likelihood ratio test (GLRT), the unknown parameters may be estimated using maximum likelihood (ML) method. The ML estimation of ae −jθ  is equal to r k . Therefore, 
     
       
         
           
             
               
                 
                   
                     E 
                     ⁢ 
                     
                       { 
                       
                         ae 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       } 
                     
                   
                   = 
                   
                     r 
                     k 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                            
                           
                             
                               r 
                               k 
                             
                             - 
                             
                               r 
                               k 
                             
                           
                            
                         
                         2 
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             
                               n 
                               ≠ 
                               k 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           
                              
                             
                               r 
                               n 
                             
                              
                           
                           2 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       ≶ 
                       
                         H 
                         1 
                       
                     
                     
                       H 
                       0 
                     
                   
                   ⁢ 
                   
                     η 
                     1 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     12 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           = 
                           1 
                         
                         , 
                         
                           n 
                           ≠ 
                           k 
                         
                       
                       N 
                     
                     ⁢ 
                     
                       
                          
                         
                           r 
                           n 
                         
                          
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
                       ≶ 
                       
                         H 
                         1 
                       
                     
                     
                       H 
                       0 
                     
                   
                   ⁢ 
                   
                     η 
                     1 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
     This inequality in Equation (13) shows that, to make a decision about the presence or absence of a partial discharge, first the FFT bin that contains the received signal should be omitted. Then, the sum of the square of the amplitude of other FFT bins should be compared with a predetermined threshold. If the summation is greater than the threshold, then it may be assumed that a PD has occurred, but if the summation is smaller than the threshold, then it may be assumed that the received signal only contains the transmitted signal with noise. 
     As mentioned earlier, in order to have an acceptable P fa , a suitable decision threshold should be set. The false alarm happens, if there is no partial discharge, but the decision may be that a partial discharge has occurred. If around 3000 frequency steps are generated before each image construction and we set the P fa  equal to 10 −7 , it means that only one out of 3.3×10 3  images may contain an error which is difficult to detect. 
     Regarding above-mentioned strategy, η 1  should be selected properly, so that the probability of Σ n=1,n≠k   N |r n | 2 &gt;η 1  be equal to 10 −7  under no partial discharge occurrence, where r n  samples are independent complex Gaussian noise with zero mean value and known variance σ 0   2 . Therefore, Σ n=1,n≠k   N |r n | 2 /σ 0   2  is a Chi-square variable with 2×(N−1) degrees of freedom. Therefore, the probability of Σ n=1,n≠k   N |r n | 2  crossing η 1  is equal to: 
     
       
         
           
             
               
                 
                   
                     P 
                     fa 
                   
                   = 
                   
                     1 
                     - 
                     
                       
                         1 
                         
                           
                             ( 
                             
                               N 
                               - 
                               2 
                             
                             ) 
                           
                           ! 
                         
                       
                       ⁢ 
                       
                         γ 
                         ⁡ 
                         
                           ( 
                           
                             
                               N 
                               - 
                               1 
                             
                             , 
                             
                               
                                 η 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                   σ 
                                   0 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
     Here γ is lower incomplete gamma function and is defined as follows: 
                     γ   ⁡     (     s   ,   x     )       =       ∫   0   x     ⁢       t     s   -   1       ⁢     e     -   t       ⁢   dt               Equation   ⁢           ⁢     (   15   )                 
Therefore,
 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       
                         
                           η 
                           1 
                         
                         
                           2 
                           ⁢ 
                           
                             σ 
                             0 
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         t 
                         
                           N 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         e 
                         
                           - 
                           t 
                         
                       
                       ⁢ 
                       dt 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           N 
                           - 
                           2 
                         
                         ) 
                       
                       ! 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           P 
                           fa 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     16 
                     ) 
                   
                 
               
             
           
         
       
     
     This equation should be numerically calculated to find a suitable value of η 1  for a predetermined P fa . 
     With further reference to  FIG. 2C , once a partial discharge-corrupted frequency step is identified by for example the GLRT as described in detail herein, the transmission of the sinusoidal signal for that partial discharge-corrupted frequency step may be repeated in order to obtain a partial discharge-free frequency step. 
     EXAMPLE 
     In this example, an implementation of UHF SAR imaging with a stepped frequency method is described. For purposes of this disclosure, two simplified models of a transformer winding are provided. One intact transformer winding without any kind of deformation is provided which is referred to herein as intact model. A second model is prepared based on the intact model, on which a radial deformation is simulated. The second model is referred to herein as deformed model. 
     Referring to  FIG. 4A , a schematic representation of the intact model  400  is illustrated. The intact model  400  includes 14 Plexiglas disks, such as Plexiglas disk  401  and Plexiglas disk  402 . The Plexiglas disks are covered by copper strips and a thin plastic layer is disposed between the Plexiglas disks in order to separate the Plexiglas disks. For example, plastic layer  403  is placed between Plexiglas disk  401  and Plexiglas disk  402 . Diameter of each Plexiglas disk is about 600 mm and height of each Plexiglas disk is about 20 mm with a space of approximately 5 mm between the Plexiglas disks. Total height of the intact model  400  is 370 cm and distance  404  between transceiver antennas  405  and the intact model is approximately 350 mm. 
     UHF SAR imaging of the intact model  400  was carried out by a stepped frequency method as was described in detail in preceding sections of the present disclosure. In a first scanning step  406 , UHF electromagnetic signals were transmitted as a plurality of successive sinusoidal signals and received reflections of the plurality of sinusoidal signals from the intact model  400  were gathered. The signals were transmitted and received by two transceiver antennas  405 . A time domain reflected UHF electromagnetic signal for the first scanning step  406  was obtained by performing an inverse fast Fourier transform (IFFT) on the magnitudes and phases of the gathered reflected sinusoidal signals. Then, step by step, the two transceiver antennas  405  were moved along a scanning path  407  from one scanning step to the next, where transmitting and receiving the sinusoidal signals were performed for each scanning step along the scanning path  407 . In this example, 33 scanning steps were used for UHF SAR imaging of the intact model  400 , from the first scanning step  405  to a last scanning step  408 . After receiving the reflected UHF electromagnetic signals for all 33 scanning steps, an amount of energy reflected from each point of the surface of the intact model  400  was calculated using the Kirchhoff migration method as was presented in Equation (1). The calculated energies may either be presented as a matrix of values or the calculated energies may be presented as a color-coded image.  FIG. 4B  is a color-coded map of the intact model  400  generated using the amounts of energy calculated by the Kirchhoff migration method. In these color-coded maps, color of points represent the amount of energy reflected from a corresponding point on the surface of the intact model  400 . Scale  410  indicates the amount of energy reflected from the surface of the winding divided by the maximum reflected energy. 
     Referring to  FIG. 5A , a schematic representation of the deformed model  500  is illustrated. The deformed model  500  may be similar to the intact model  400 , except a deformation  409  in the form of a box with a height of approximately 6 cm, a length of approximately 4 cm and a depth of 2.5 cm, which is placed on the winding model. 
     UHF SAR imaging of the deformed model  500  was also carried out by a stepped frequency method as was described in detail in preceding sections of the present disclosure. In a first scanning step  406 , UHF electromagnetic signals were transmitted as a plurality of successive sinusoidal signals and received reflections of the plurality of sinusoidal signals from the deformed model  500  were gathered. The signals were transmitted and received by two transceiver antennas  405 . A time domain reflected UHF electromagnetic signal for the first scanning step  406  was obtained by performing an inverse fast Fourier transform (IFFT) on the gathered reflected sinusoidal signals. Then, step by step, the two transceiver antennas  405  were moved along a scanning path  407  from one scanning step to the next, where transmitting and receiving the sinusoidal signals were performed for each scanning step along the scanning path  407 . In this example, 33 scanning steps were used for UHF SAR imaging of the deformed model  500 , from the first scanning step  405  to a last scanning step  408 . After receiving the reflected UHF electromagnetic signals for all 33 scanning steps, an amount of energy reflected from each point of the surface of the deformed model  500  was calculated using the Kirchhoff migration method as was presented in Equation (1). The calculated energies may either be presented as a matrix of values or the calculated energies may be presented as a color-coded image.  FIG. 5B  is a color-coded map of the deformed model  500  generated using the amounts of energy calculated by the Kirchhoff migration method. In these color-coded maps, color of points represent the amount of energy reflected from a corresponding point on the surface of the deformed model  500 . 
     Referring to  FIG. 4B , spots  411  indicate a large amount of energy reflection, which is a sign of the presence of the winding surface in the environment, in which the electromagnetic waves are transmitted. Referring to  FIG. 5B , spots  411 ′ appear larger when compared to the spots  411  of the intact model, which means, a larger amount of energy is reflected from some points of the winding surface, which in turn may be an indicator of the deformation on the surface of the winding. As mentioned herein,  FIGS. 4B and 5B  are obtained utilizing the stepped frequency UHF SAR imaging method pursuant to an embodiment of the resent disclosure in order to eliminate the distortions that may be caused by partial discharge occurrence. For purposes of clarity,  FIG. 5C  shows a partial-discharge corrupted color-coded map of the intact model  400  generated without utilizing the stepped-frequency method. As can be seen in  FIG. 5C , the image is distorted and it may lead to a wrong decision that the winding is deformed. 
       FIG. 6  is a schematic functional block diagram of a system  600  for detecting radial deformation in a transformer winding, consistent with one more exemplary embodiments of the present disclosure. As provided herein, the system  600  may include a data processing unit  601 , a data acquisition unit  602 , a first transceiver antenna  603  that may be similar to the first transceiver antenna  102  of  FIG. 1A , and a second transceiver antenna  604  that may be similar to the first transceiver antenna  103  of  FIG. 1A . 
     Referring to  FIG. 6 , in one exemplary embodiment, the data processing unit  601  may include a memory  605  and a processor  606 . Processor  606  may be a general-purpose computer processor or a specialized processor designed specifically for use with the data processing unit  601 . Processor  606  is coupled with the memory  605  to permit storage of data and software that are to be manipulated by commands to the processor  606 . 
     The data processing unit  601  may be operatively connected to the first transceiver antenna and the memory may include executable instructions encoded thereon, such that upon execution by the processor  606 , the data processing unit  601  may urge the first transceiver antenna  603  to transmit UHF electromagnetic signals using the stepped-frequency method of this disclosure in a series of scanning steps as was described in detail in preceding sections of the present disclosure. The second transceiver antenna  604  may receive the reflection of the transmitted signals by the first transceiver antenna  603  for each scanning step. The data acquisition unit  602  gathers the received signals for all scanning steps. Magnitudes and phases of the gathered signals may be stored on the memory  605 . The memory  605  may further include executable instructions, such that upon execution by the processor  606 , the processor  606  performs IFFT on the magnitude and phases of the gathered signals in order to obtain a time domain reflected UHF electromagnetic signal for each scanning step. The processor  606  may further be configured to calculate an amount of energy reflected from the surface of the transformer based on the obtained time domain reflected UHF electromagnetic signals of all scanning steps. 
     While the foregoing has described what are considered to be the best mode and/or other examples, it is understood that various modifications may be made therein and that the subject matter disclosed herein may be implemented in various forms and examples, and that the teachings may be applied in numerous applications, only some of which have been described herein. It is intended by the following claims to claim any and all applications, modifications and variations that fall within the true scope of the present teachings. 
     Unless otherwise stated, all measurements, values, ratings, positions, magnitudes, sizes, and other specifications that are set forth in this specification, including in the claims that follow, are approximate, not exact. They are intended to have a reasonable range that is consistent with the functions to which they relate and with what is customary in the art to which they pertain. 
     The scope of protection is limited solely by the claims that now follow. That scope is intended and should be interpreted to be as broad as is consistent with the ordinary meaning of the language that is used in the claims when interpreted in light of this specification and the prosecution history that follows and to encompass all structural and functional equivalents. Notwithstanding, none of the claims are intended to embrace subject matter that fails to satisfy the requirement of Sections 101, 102, or 103 of the Patent Act, nor should they be interpreted in such a way. Any unintended embracement of such subject matter is hereby disclaimed. 
     Except as stated immediately above, nothing that has been stated or illustrated is intended or should be interpreted to cause a dedication of any component, step, feature, object, benefit, advantage, or equivalent to the public, regardless of whether it is or is not recited in the claims. 
     It will be understood that the terms and expressions used herein have the ordinary meaning as is accorded to such terms and expressions with respect to their corresponding respective areas of inquiry and study except where specific meanings have otherwise been set forth herein. Relational terms such as first and second and the like may be used solely to distinguish one entity or action from another without necessarily requiring or implying any actual such relationship or order between such entities or actions. The terms “comprises,” “comprising,” or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. An element proceeded by “a” or “an” does not, without further constraints, preclude the existence of additional identical elements in the process, method, article, or apparatus that comprises the element. 
     The Abstract of the Disclosure is provided to allow the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in various implementations. This is for purposes of streamlining the disclosure, and is not to be interpreted as reflecting an intention that the claimed implementations require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed implementation. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separately claimed subject matter. 
     While various implementations have been described, the description is intended to be exemplary, rather than limiting and it will be apparent to those of ordinary skill in the art that many more implementations and implementations are possible that are within the scope of the implementations. Although many possible combinations of features are shown in the accompanying figures and discussed in this detailed description, many other combinations of the disclosed features are possible. Any feature of any implementation may be used in combination with or substituted for any other feature or element in any other implementation unless specifically restricted. Therefore, it will be understood that any of the features shown and/or discussed in the present disclosure may be implemented together in any suitable combination. Accordingly, the implementations are not to be restricted except in light of the attached claims and their equivalents. Also, various modifications and changes may be made within the scope of the attached claims.