i
An overview of the system model is depicted in Fig. 1 where the receiver estimates the number of PD sources in power equipment for remote monitoring applications.
u2(t) uˆ1(t)
System model: We consider N unknown PD sources written as an N × 1 vector u of a collection of impulsive waveforms as T (1) u = u1 (t)u2 (t), . . . , uN (t) where ui (t) are complex-valued signals emitted by the ith PD sources. The receiver has M-dimensional observations x into an M × N complexvalued mixing channel H such that x = Hu
(2)
y=x+n
(3)
where n is an additive background noise generated by thermal noise and also ambient noise in a substation. Note that no precise knowledge is available of either the mixing channel or the sources. To recover discharge sources from observation, we need to find G, an inverse matrix such that ∗
uˆ = G y
(4)
where G*H = I and G* is the conjugate transpose of G. The number of PD sources can be estimated from the demixed signals by measuring the
uˆ 2(t)
sources u
h
MN (t
(t ) h M1
uN (t)
Introduction: Recent advances in wireless sensor networks in substations can provide significant improvements for protection, control, automation and monitoring. One example of remote monitoring applications is rapid insulation diagnosis in power equipment using wireless intelligent electronic devices (WIEDs). Partial discharge (PD) activities can cause irreversible damage and possible failure of electrical insulation systems. Insulation performances and lifetime can be evaluated by measuring the number of PD sources in power equipment. Their electromagnetic radiations can be detected by wireless devices for remote monitoring applications. The resulting signal is highly impulsive where the spectrum can cover very large frequency bands (above a few gigahertz) [1, 2]. This Letter proposes a technique for the fast identification of PD sources. This can be implemented in a low-cost WIED for remote monitoring applications. By using multiple antennas, PD sources can be separated based on blind source separation (BSS) techniques via generalised eigenvalue decomposition, presented in [3]. Furthermore, the presence of significant impulsive events produces heavy-tailed distribution where the excess kurtosis is greater than zero. Therefore, the number of PD sources can be estimated by measuring the excess kurtosis of the demixed signals at each antenna. To demonstrate the efficiency and performance of the proposed method, we use a generic and realistic impulsive noise model developed in [4], where impulsive waveforms are generated by autoregressive (AR) models and their parameters can be estimated from various measurement campaigns in substations. Discharge sources are modelled as a spatial Poisson point process (PPP). One can control physical parameters such as the density of PD sources and the average intensity of an impulsive component over background noise.
h11(t)
u1(t)
t)
A technique for the fast identification of partial discharge (PD) sources is proposed for the detection of mechanical failure or damage to insulation materials by using wireless remote control and monitoring systems in substations. An estimation of the number of PD sources can help to evaluate the insulation performance and lifetime of power equipment. Multiple PD sources can be generated during the operating voltage where their electromagnetic radiations are highly impulsive, nonGaussian noise and the resulting probability distribution function is heavy-tailed. Multiple PD sources can be estimated by their electromagnetic radiations via blind source separation (BSS) and measuring the excess kurtosis using low-cost wireless intelligent electronic devices. The efficiency and performance of the proposed method is demonstrated by simulating PD sources based on the spatial Poisson point process where the number of sources is a random variable not known by the receiver. Assuming non-white and decorrelated or non-Gaussian and independent sources, results show that the number of significant PD sources can be estimated with low error rate. Underdetermined problems in BSS can affect performances.
where μi is the mean of the ith demixed signals. In the presence of significant impulsive signals, the instantaneous amplitude distribution is heavy-tailed where κi > 0. Thus, the estimated number of PD sources Nˆ is given by summing κi greater than a given threshold Th > 0 to be defined in this Letter (ki . Th ) (6) Nˆ =
1N (
M. Au✉, B.L. Agba and F. Gagnon
excess kurtosis given by the fourth moment about the mean minus three E (ˆui (t) − mi )4 ki = (5) 2 − 3 E (ˆui (t) − mi )2
h
Fast identification of partial discharge sources using blind source separation and kurtosis
BSS
Nb sources estimation
Nˆ
)
mixing H
observation y
demixing G
Fig. 1 Overview of system model
BSS via generalised eigenvalue decomposition: PD phenomena are stochastic processes where the pulse height, event and the spectrum depend on various physical parameters such as electric field intensity, the free electrons rate, aging mechanism etc. [5]. Hence, in the presence of multiple PD sources, it is reasonable to consider that sources are independent or at least decorrelated. Moreover, according to the physical characteristic of radiated radio frequency signals from PD activity measured in [1, 2], one can assume that PDs are non-Gaussian. In measurement campaigns in substations, these impulsive signals have transient behaviour with damped oscillation. Therefore, sources may be non-white and/or non-stationary processes. From [3], two conditions are sufficient for source separation via generalised eigenvalue decomposition. (i) if sources are independent or decorrelated, the covariance matrix Ry written by Ry = E[yy∗ ] = Rx + Rn = HRu H∗ + Rn
(7)
where Ru and Rn are diagonal. Therefore, Rx is also diagonal. Assuming the background noise is modelled as a circular complex Gaussian noise, Rn = σ 2I. (ii) If PD sources are non-Gaussian, non-stationary or nonwhite, there exists Qu which has the same diagonalisation property such that Qx = HQu H∗
(8)
Qy = Qx + Qn
(9)
From (3) and (8), we write where Qn is also diagonalisable. From these two conditions, generalised eigenvalue decomposition can be used for source separation [3]. Indeed, in (7) and (9), by multiplying them by G, and also (9) is multiplied by −1 L = Rx Q−1 x + Rn Qn , we have Ry G = Qy GL
(10)
From statistical assumptions, Q can have various forms. By assuming that sources are: † non-stationary and decorrelated, we have [6] Qy, 1 = Ry = E[yy∗ ] † non-white and decorrelated, we have [7] Qy, 2 = Ry (t) = E y(t)y∗ (t + t)
ELECTRONICS LETTERS 10th December 2015 Vol. 51 No. 25 pp. 2132–2134
(11)
(12)
(13)
where y is the conjugate of y and Tr(Ry) is the trace of Ry. The inverse matrix G is given by the generalised eigenvalue of the matrices Ry and Qy. Simulation and results: The performance of the proposed technique is demonstrated by Monte Carlo simulation where PD sources are simulated based on the spatial PPP and the number of sources is a random variable not known by the receiver. The average number of sources per unit volume or surface is given by λs. For a large observation time, sources emit impulsive discharges where their events follow a temporal PPP. The average number of emissions per unit time per sources is given by λe. From [2, 4], transient impulsive signals from discharge activity are simulated by using AR models such that a single discrete-time impulse ut is written as p fi ut−i + 1t (14) ut =
values of Γ, where discharges are more significant, the receiver can estimate the exact number of PD sources when Qy,2 is applied. Fig. 3 shows the performance of the receiver for a given number of observations where only Qy,2 is applied. The probability of error Pe is high when the number of observations is lower than the number of PD sources (M < N ). The system is said to be underdetermined because BSS cannot recover more than M sources. Nevertheless, there are at least M PD sources. When the number of observations is greater than the number of PD sources (M > N ), the system is overdetermined and the probability of error Pe decreases drastically because BSS can recover the exact number of PD sources. However, a very large number of observations do not provide better performance. 100
10−1 probability of error, Pe
where τ is a time delay. In this Letter, we consider that τ = 1 sample. † non-Gaussian and independent, we have [8] Qy, 3 = E[y∗ yyy∗ ] − Ry Tr(Ry ) − E yyT E[yy∗ ] − Ry Ry
i=1
where coefficients {ϕi} are parameters of the model which can be obtained from measurements in various substations and ɛt is a Gaussian noise process where the variance is unconditional, non-constant over time, in order to take into account the impulse duration. The pulse height is an exponential random variable where the average intensity of the impulsive component over background noise is given by G = s2u /s2n . In this Letter, the receiver operates for an average number of PD sources of λs = 4 sources per unit volume or surface. For an observation time of 5 × 104 samples, the average number of emissions is λs = 5 discharges per unit time per sources. We use the second order of the AR model where p = 2 in (14) and parameters are obtained from measurements in various substations. 100
probability of error, Pe
10−1
10−2
Qy, 1 10−3
Qy, 2 Qy, 3
10−4 −20
−10
0
10
20
GdB
Fig. 2 Probability of error against ΓdB. On basis of PPP model: average PD source density λs = 4 sources per unit volume or surface, average emission density λe = 5 discharges per unit time per sources, NB and observation M = 20 > N
For BSS, we use these three forms of Qy to recover PD sources where their capabilities to estimate the correct number of sources are compared via their probability of error Pe (the estimated number of PD sources is not correct). To limit the probability of false alarm, the estimated number of sources Nˆ is determined when the threshold Th = 1 according to (6). In this condition, since the background noise is modelled as a Gaussian noise, the excess kurtosis is zero. Hence, the probability of false alarm is Pfa = 0. The mixing channel H is an M × N symmetric circular complex Gaussian noise. In this Letter the probability of error Pe is obtained from 15000 Monte Carlo simulations. The performance of the receiver is plotted in Fig. 2 where the probability of error Pe is given for different values of Γ and different forms of Qy. The number of PD sources is a random variable which is not known by and not available to the receiver. The number of observations M = 20 is always greater than the number of sources (M > N). By applying Qy,2 and Qy,3 for BSS, the probability of error Pe decreases when the average intensity of the impulsive component is higher compared with the background noise. However, Pe is high and nearly constant when Qy,1 is applied. As a result, the number of PD sources can be determined with low probability of error if we assume non-white and decorrelated or non-Gaussian and independent sources. For high
10−2 M=2 M=5 M=7
10−3
M = 10 M = 25 10−4
M = 35 −20
−10
0
10
20
GdB
Fig. 3 Probability of error against ΓdB with various NB and observation M. On basis of PPP model: average PD source density λs = 4 sources per unit volume or surface, average emission density λe = 5 discharges per unit time per sources
Conclusion: A fast identification of PD sources is proposed where the number of sources can be estimated in power equipment for remote monitoring applications using low-cost WIED. The technique is based on BSS via generalised eigenvalue decomposition and the number of sources can be estimated by measuring the excess kurtosis. If we assume non-white and decorrelated or non-Gaussian and independent sources, the exact number of PD sources can be identified with low probability of error specially when discharges are significant. Performances can be affected by the underdetermined problem in BSS. This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/) Submitted: 26 August 2015 E-first: 12 November 2015 doi: 10.1049/el.2015.2957 One or more of the Figures in this Letter are available in colour online. M. Au, B.L. Agba and F. Gagnon (Department of Electrical Engineering, École de Technologie Supérieure, Montréal, Québec, Canada) ✉ E-mail:
[email protected] References 1 Au, M., Gagnon, F., and Agba, B.L.: ‘An experimental characterization of substation impulsive noise for a RF channel model’, PIERS Proc., 2013, 1, pp. 1371–1376 2 Au, M., Agba, B.L., and Gagnon, F.: ‘A model of electromagnetic interferences induced by corona discharges for wireless channels in substation environments’, IEEE Trans. Electromagn. Compat., 2015, 57, (3), pp. 522–531 3 Parra, L., and Sajda, P.: ‘Blind source separation via generalized eigenvalue decomposition’, J. Mach. Learn. Res., 2003, 4, pp. 1261–1269 4 Au, M., Agba, B.L., and Gagnon, F.: ‘Analysis of transient impulsive noise in a Poisson field of interferers for wireless channel in substation environments’, Arxiv, 2015. Available at http://www.arxiv.org/abs/ 1504.06880 5 Van Brunt, R.J.: ‘Stochastic properties of partial-discharge phenomena’, IEEE Trans. Electr. Insul., 1991, 26, (5), pp. 902–947 6 Parra, L., and Spence, C.: ‘Convolutive blind separation of non-stationary sources’, IEEE Speech Audio Process., 2000, 8, (3), pp. 320–327 7 Weinstein, E., Feder, M., and Oppenheim, A.V.: ‘Multi-channel signal separation by decorrelation’, IEEE Speech Audio Process., 1993, 1, (4), pp. 405–413 8 Cardoso, J.F., and Souloumiac, A.: ‘Blind beamforming for non-Gaussian signals’, Proc. IEEE F (Radar Signal Process.), 1993, 140, (6), pp. 362–370
ELECTRONICS LETTERS 10th December 2015 Vol. 51 No. 25 pp. 2132–2134
