Mar 30, 2016 - Xian-Jun Shao and Wen-Lin He. Research Institute of State Grid Zhejiang Electric Power Company. Hangzhou, Zhejiang, 310014, China.
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Localization of Multiple Partial Discharge Sources in Air-Insulated Substation Using Probability-Based Algorithm Ming-Xiao Zhu, Yan-Bo Wang, Qing Liu, Jia-Ning Zhang, Jun-Bo Deng, Guan-Jun Zhang State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering Xi’an Jiaotong University. Xi’an, Shaanxi, 710049, China
Xian-Jun Shao and Wen-Lin He Research Institute of State Grid Zhejiang Electric Power Company Hangzhou, Zhejiang, 310014, China ABSTRACT Ultra-high-frequency (UHF) sensing technique has been introduced to detect and localize partial discharge (PD) sources in air-insulated substation (AIS). This paper presents a probability-based algorithm to localize multiple PD sources which may occur simultaneously in different power equipment. Assuming that the time difference of arrival (TDOA) between all pairs of antennas in a array are normally distributed, the probability density function (PDF) of PD source coordinates can be obtained by substituting the linearized form of time difference equations into PDFs of TDOAs. When large number of PD signals are recorded, the joint PDF (JPDF) can be calculated from the product of PDF of each TDOA. Then the PD coordinates to be solved are regarded as with highest probability, and can be solved by taking the derivative of JPDF. In the case of multiple PD sources, mixed UHF signals are separated by clustering the TDOA vectors with K Means clustering method. PD experiments are performed to test the presented algorithm, and the localization accuracy of proposed algorithm is compared with other typical methods such as Newton-Raphson, Particle Swarm Optimization and plane intersection method. The results indicate that the probability-based localization algorithm reasonably integrates the TDOAs of continuous signal sequence, which can effectively reduce the influence of TDOA estimation errors and improve the localization accuracy. Index Terms — Air-insulated substation, partial discharge, ultra-high frequency, localization, probability-based algorithm.
1 INTRODUCTION Partial discharge (PD) detection is a powerful technique for condition-based maintenance of high-voltage equipment, and provides valuable information for insulation condition diagnosis. Generally, online PD monitoring instruments are suitable for being assembled on high-cost apparatuses such as gas insulated system (GIS) and power transformer, and these instruments mainly focus on individual equipment [1, 2]. However, due to the economic consideration, power utilities may not be willing to adopt PD monitoring for other equipment such as circuit breaker, potential and current transformer and bushing etc. A radio-frequency sensing system is designed to monitor PD in the entire air-insulated substation (AIS), which is more cost-effective than the conventional online system [3, 4]. By using a four-antenna Manuscript received on 30 March 2016, in final form 3 September 2016, accepted 10 September 2016. Corresponding author: G.-J. Zhang.
array, the PD sources from different excited equipment can be found with a time-difference localization algorithm. To detect PD in the whole substation, the antenna array is generally mounted on a movable platform, in which the distance between antennas are relatively short. The experimental results indicate that the bearing of PD source can be determined with high-accuracy, whereas the range is generally estimated with high inaccuracy even the time differences are with minor error [3, 4]. The time-difference localization method consists of two major steps, namely, estimation of the time difference of arrival (TDOA) and solving the nonlinear time-difference equations [3-11]. The solution for time-difference equations have been extensively studied in localization of PD source in transformer. Since the equations cannot be solved with explicit expression, several kinds of solution methods such as Newton–Raphson iteration approach [3-5], space-grid-search algorithm [6, 7] and Chan direct solving method [8, 9] have
DOI: 10.1109/TDEI.2016.005964
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M.-X. Zhu, et al.: Localization of Multiple Partial Discharge Sources in Air-Insulated Substation Using Probability-Based Algorithm
been adopted. Moreover, the time-difference equations can be transformed to objective functions which ideally equal to zero, and the localization problem can be solved with optimization algorithms such as Particle Swarm Optimization (PSO) [10-12] and Genetic Algorithm [5] to find the coordinates with minimum objective function value. In [4], the bearing of PD source is calculated from the time delays between all pairs of antennas in the array, and the spread of range is calculated according to the uncertainty of TDOA equations. The majority of existing localization methods show some limitations such as the estimated range (distance) is generally with lower accuracy.
x x 2 y y 2 x x 2 y y 2 cT 1 1 2 2 12 2 2 2 2 x x1 y y1 x x3 y y3 cT13 (1) x x1 2 y y1 2 x x4 2 y y4 2 cT14 where c is the propagation speed of electromagnetic wave, which equals to the light speed. As can be seen, there are three equations with only two unknowns. Because of the nature of signals and background noise, the TDOAs are estimated with errors in practice, and the localization accuracy can be improved with the redundancy equation.
In this paper, a probability-based localization algorithm which can effectively integrate the continuous signal sequence is presented. It is assumed that the TDOAs are normally distributed, the coordinates are calculated with the objective of maximizing the probability. Section 2 introduces the theory and derivation procedures of the proposed algorithm. In Section 3, several properties of the probability-based algorithm including the accuracy analysis and 50%probability area are summarized. The procedures of PD source localization are illustrated in Section 4 in detail. At last, the presented algorithm is tested with PD experiments while 2.2 changing the position of PD source.
2 PROBABILITY-BASED LOCALIZATION ALGORITHM The probability-based localization algorithm was firstly proposed in [13] as a passive localization method in radar electronic countermeasure. In this work, that algorithm is adopted to localize multiple PD sources in AIS, and its several properties are further demonstrated. 2.1 UHF SENSING SYSTEM IN AIS Figure 1 shows the schematic diagram of UHF sensing system for PD detection and localization in AIS. The antenna array and acquisition system are generally mounted on a movable platform, in which the antennas are with short distance to reduce the size of whole system. UHF signals radiated by PD are coupled with 4-antanna array, and directly sampled with high-speed digitization instrument such as oscilloscope or acquisition card. Owing to the high sampling rate, the arrival time of PD wave can be estimated with sub-ns accuracy. Assuming that the arrival time of four antennas are t1, t2, t3 and t4, the time difference of arrival (TDOA), e.g., T12=t1-t2, can be determined. The localization of PD source can be realized by solving the time-difference equations. Due to the relatively large space of AIS, the angle of arrival of UHF signals are in the region of ±5° from the plane of the array. For this reason, the height of the PD source can be neglected and the AIS is defined as a 2-D space [4]. If the antennas are installed at (xi, yi), i=1, 2, 3, 4, and the coordinate of PD source is (x, y), the time-difference equations are
Figure 1. Schematic diagram of PD source localization in AIS.
SOLVING TIME-DIFFERENCE EQUATIONS WITH PROBABILITY-BASED TECHNIQUE The error of TDOA is caused by many factors such as noise level, signal intensity and accuracy of TDOA estimation method, etc. From a statistical point of view, the resulting TDOA conforms normal distribution, and this assumption has been verified by some measurements [14]. Assuming that the estimated TDOA of one pair of antennas (#1 and #i, i=2, 3, 4) is T0, the probability density function (PDF) of TDOA T is defined as
T T0 2 p T (2) exp 2 T2 2 T where σT is the standard deviation of T. The PDF takes maximum value when T equals to the estimated TDOA. The PDF of PD source coordinates (x, y) can be obtained by substituting the linearized form of time difference equations into PDFs of TDOAs. Then the PD coordinates to be solved are regarded as with maximum probability, and can be solved by taking the derivative of PDF of (x, y). For simplification, each time-difference equation in equation (1) can be written as 1
f x, y T
(3)
2 2 2 2 f x, y x x1 y y1 x xi y yi / c (4) By using Taylor series expansion to equation (4) and neglect the high-order terms, the following equation can be obtained.
T f x0 , y0
f x
x x0 x x0
f y
y y0 y y0
(5)
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Substituting equation (5) into equation (2), we can obtain the PDF of coordinates (x, y)
2 a1 1 f 2 T2 x x x0 2 1 f exp , p T F x y 1 a2 2 T 2 T2 y y y0 2 f (6) f x x0 y y0 f x0 , y0 T0 f 1 f y y y x x x (10) a3 2 , F x y T x x x0 y y y0 2 T2 a 1 f D As can be seen from equation (6), F(x, y) is a quadratic 4 T2 x x x0 polynomial, and can be written with the following format. 1 f D a5 2 y 2 2 (7) F x, y a1 x a2 y a3 xy a4 x a5 y a6 T y y0 The computing formulas of the coefficients in equation (7) f f x0 y0 , T0 is the where D f x0 , y0 T0 are described in Section 2.3. x x x0 y y y0 Assuming that the TDOAs T12, T13 and T14 are independent, the joint PDF (JPDF) of one record, namely estimated TDOA, and (x0, y0) is the initial coordinate. The σT the JPDF of T12, T13 and T14, is the product of PDF of each can be eliminated in equation (9). The partial derivatives are TDOA. Here, one record represents the 4-channel data derived from equation (4). Take the time-difference equation radiated by once discharge. Large number of PD signals of antennas #1 and #i for example, the partial derivatives can can be acquired during the experiments, and the JPDF of be calculated as multiple records is the product of PDF of each record. As x xi x x1 f 1 can be seen from equation (6), the JPDF is also an 2 2 2 2 x c exponential function, and its exponent FJ(x, y) can be x x y y x x y y i i 1 1 (11) calculated as the sum of PDF exponent of each TDOA: f 1 y y y y i 1 M 2 2 2 2 j j j y c FJ x, y F12 x, y F13 x, y F14 x, y x xi y yi x x1 y y1 (8) j 1 0
0
3 PROPERTIES OF PROBABILITY-BASED ALGORITHM
A1 x 2 A2 y 2 A3 xy A4 x A5 y A6
where M is the number of records in one measurement, F1ij x, y is the PDF exponent of T1i for jth record, i=2, 3, 4, j=1, 2, …, M. As can be seen from equation (8), the FJ (x, y) is also a quadratic polynomial. The coefficients A1, A2, A3, A4 and A5 are the sum of corresponding coefficients of all TDOAs. The PD source coordinate (x, y) to be solved is regarded as with the highest probability. Then the localization problem is transformed into solving the (x, y) with maximum FJ(x, y). By setting the derivative of FJ (x, y) to 0, the PD coordinate (x, y) can be solved as 2 A2 A4 A3 A5 x A2 4 A A 3 1 2 A A A 2 3 A4 y 1 5 2 A3 4 A1 A2
(9)
2.3 COEFFICIENTS OF PDF EXPONENT The coefficients of F(x, y) in equation (7) can be calculated from equation (6) as
In this Section, several properties of the probability-based algorithm including the accuracy analysis and 50%probability area is summarized. 3.1 LOCALIZATION ACCURACY ANALYSIS TDOA error is the main factor causing the inaccuracy of localization algorithm. The TDOA error is reflected in the parameter D of coefficients a4 and a5. It is found that the probability-based algorithm can converge in a few recursions, and the initial coordinate (x0, y0) of following recursions are with little difference. Assuming that the initial coordinates of all recursions are identical, the coefficient A4 of M signals can be calculated as M 1 f f f A4 2 x0 y0 f x0 , y0 T0 T x x x0 i 1 x x x0 y y y0 M
M
i 1
i 1
where T0 Tr ni , Tr is the true TDOA, and ni is the TDOA error of ith signal. Since TDOA is normally distributed, the sum of ni is close to zero, and the combined effect of noise is weaken. In other words, the presented algorithm integrates the TDOAs of continuous signal sequence, which can effectively improve the localization accuracy.
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3.2 50% PROBABILITY LOCALIZATION AREA The probability distribution of coordinate (x, y) is another valuable information to determine the location of PD source, and is derived in this section. Assuming that the localization result is x , y , the first-order terms in equation (8) can be
eliminated by coordinate translation. F x, y A1 x12 A2 y12 A3 x1 y1 A6
(12)
x1 x x where . y1 y y By using coordinate rotation, the x1y1 term can be eliminated, as shown in equation (13). F x, y A1x22 A2 y22 A6 (13)
x1 x2 cos y2 sin
A3 1 arctan is the 2 A1 A2 rotation angle. The coefficients in equation (13) are A1 0.5 A1 A2 b
where
y1 x2 sin y2 cos
,
A2 0.5 A1 A2 b
A1 A2
b
2
(14)
A32
Substituting equation (13) into equation (6), then the JPDF function of PD coordinate can be written as 1 x2 y 2 1 p T exp( A6) exp 22 22 (15) 2 T 2 x y 1 1 x A A2 b 2 A1 1 . where 1 1 y 2 A A A b 1 2 2 As can be seen from equation (15), (x2, y2) follows a twodimensional normal distribution, and x and y are their
standard deviations, respectively. The contour with equal probability is an ellipse with the following equation x22
2 x
y22
y2
s2
(16)
where s is a constant value. If s2 equals to 1.386, the probability of PD source located in this ellipse is 50%, and the area of the ellipse is 1.386 x y . The smaller the 50%probability ellipse is, the more accurate the localization result is. From above analysis, the localization results of probabilitybased algorithm consist of PD coordinate and the 50%probability ellipse. The derivation of equations (12)-(16) are based on the assumption that TDOA is normally distributed, and 50%probability ellipse is calculated from statistical analysis of recorded signals. For this reason, the 50%-probability ellipse can more accurately represent the PD source distribution when large number of signals are considered in the algorithm. If the TDOA deviate from normal distribution, the proposed ellipse cannot well reflect the actual 50%-probability area.
4 PROCEDURES OF PD SOURCE LOCALIZATION The procedures of localization method is presented in this Section. The TDOAs between antennas are estimated with the cross-correlation method, then the localization results are determined with the probability-based algorithm. 4.1 TDOA ESTIMATION Since the antenna array is generally with small size and no power equipment is located between antennas to affect the propagation of electromagnetic waves, the UHF signals of four channels are generally with similar shapes. Under this circumstance, the cross-correlation method, based on similarity of signals, is more suitable for TDOA estimation. The TDOA between two signals is determined in two steps: 1) The approximate arrival time is found using a simple thresholding algorithm, then the signals are windowed at both side of the position identified by thresholding method to avoid the influence of multipath propagation. 2) The TDOAs are more accurately estimated as the lag time that maximize the cross-correlation function between windowed UHF signals. The accuracy of TDOA is improved by interpolating the cross-correlation function with a higher sampling rate. The final TDOA is calculated as the summation of that determined by thresholding and cross-correlation methods. The complete procedures of the cross-correlation method can be found in [3]. 4.2
LOCALIZATION OF PD SOURCE USING PROBABILITY-BASED ALGORITHM Since PD usually occurs stably in a period of time, a number of signals can be acquired, and the localization accuracy can be improved by statistically analyze TDOAs of continuous signal sequence instead of single PD event. The probability-based algorithm provides an effective approach for this purpose: the coefficients of a latest detected PD can be directly added to that of former records according to equation (8), then the PD coordinates can be solved with equation (9). The TDOAs of continuous signals are comprehensively considered by using the joint probability distribution. According to the above-mentioned guideline, the probability-based algorithm can be implemented in a recursive way, which updates the localization results whenever a new record is available. In one recursion, a trigger–coefficient summation–coordinates recalculation procedure is repeated, and the flowchart is shown in Figure 2. Before using the probability-based algorithm, an appropriate initial location (x0, y0) is determined with plane intersection method which is described in detail in Appendix. For the first detected signal, the coefficient array A1=[a1, a2, a3, a4, a5] can be calculated by substituting initial coordinates (x0, y0) and estimated TDOA T0 into equations (10) and (11). To clearly illustrate the localization procedure, total coefficient array (TCA) [A1, A2, A3, A4, A5] is defined as the summation of coefficient arrays of all recorded signals. Take the second signal for example, the TCA is calculated as At2=A1+A2. Once the acquisition system
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is triggered and a new PD signal is recorded, e.g., the n PD is acquired, the TCA is calculated as Atn=Atn-1+An. Then the coefficients in TCA are substituted into equation (9) to calculate the PD coordinates. During the recursion, the initial coordinates for nth recorded PD are set as the localization results of n-1th iteration. Once a new PD is detected, the localization coordinates are calculated and can be refreshed in the software interface.
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classes, a new class is created. If the latest detected signal belongs to class k, the coefficient array is added to the TCA of class k, then the localization results of class k is recalculated according to equation (9) and refreshed in the software interface. It can be seen that the proposed separation method can perfectly work with the probability-based algorithm, and achieve the localization of multiple PD sources. As can be seen from equation (17), the separation results are dependent on the threshold value ε. As described above, the accuracy of TDOAs are dependent on various factors, and the estimation error is generally in the range of a few tens of a nanosecond. In the case of one PD source, lower ε values, e.g., ε
