Tracks of Power Quality Transients in High Order Statistics Spaces Agust´ ın Ag¨ uera P´ erez ∗ , Juan Jos´ e Gonz´ alez de la Rosa ∗ , Jos´ e Carlos Palomares Salas ∗ , Aurora Gil de Castro † , Antonio Moreno-Mu˜ noz † ∗ Research Group PAIDI-TIC-168 University of C´ adiz. Area of Electronics, Av. Ram´ on Puyol S/N. E-11202-Algeciras-C´ adiz-Spain Email: agustin.aguera@uca.es † Research Group PAIDI-TIC-168 University of C´ ordoba, Area of Electronics, Campus de Rabanales, Leonardo da Vinci building, E-14071 C´ ordoba, Spain Abstract—This paper deals with the automatic classification of power quality transients according to their amplitudes and frequencies, and following the geometrical pattern previously established via higher-order statistical measurements. The clus- tering is achieved thanks to the third and fourth-order features associated to these electrical anomalies, which in turn are coupled to the 50-Hz power-line. The main contribution of the paper is the novel finding that the maxima and the minima of these higher-order cumulants distribute according families of curves of constant frequency or constant amplitude. A couple of extremes (min-max) of the higher-order statistical estimator belongs to a couple of curves (constant-frequency and constant-amplitude). The random grouping along each curve reveals the a priori hidden geometry, in turn linked to the subjacent electrical anomaly. The 2D (amplitude-frequency) regular surface grid in the input space experiments a transformation to both output spaces, which is developed via the higher-order non-linear mapping. I. I NTRODUCTION Companies are putting a lot of efforts in innovation and new technologies to monitor and control Power Quality (PQ) anomalies because today’s equipment, and automated man- ufacturing devices, are highly sensitive to the power line signal’s imperfections (PQ events), making the production cost excessive. Malfunctioning not only has to be thereby detected, but also predicted and undoubtedly diagnosed, to identify the cause and prevent the system from a similar shock. This would be reflected a posteriori in an enhancement of the industrial production [1], [2]. Recent works are bringing a higher-order statistics (HOS) based strategy, dealing with PQ analysis [3], [4], and other fields of Science and Technology [5], [6], [7]. They are based in the following premise. Without perturbation, the 50-Hz of the voltage waveform exhibits a constant statistical behavior (stationarity), generally Gaussian. Deviations can be detected and characterized via HOS; non-Gaussian processes need at least 3  and 4 ℎ -order statistical characterization in order to be completely characterized, because 2  -order moments and cumulants are not capable of differentiate non-Gaussian events. Concretely, the problem of differentiating between a tran- sient of long duration named oscillatory (within a signal period) and a short duration transient, or impulsive transient (25 per cent of a cycle), has been outcome under controlled conditions in [8], and the idea of differentiating between healthy signals and signals with transients was pointed out and accomplished in [9]. This problem was previously described in [10] and matches the HOS category, in the following sense. The short transient could also bring the 50-Hz voltage to zero instantly and, generally affects the sinusoid dramatically. By the contrary, the long-duration transient could be considered as a modulating signal (the 50-Hz signal is the carrier), and is associated to load charges [10]. Consequently, given a signal, and once the analysis has been done, we can suggest the PQ type phenomena, and perhaps the origin of the fault in the energy distribution system. Thus, from the basis of these works, this paper conveys the idea of the inverse problem in PQ analysis. That is, given a set features, to guess the rest of the electrical anomaly’s characteristics and, occasionally, to target the faulty element in the energy plant. The present work shows that there is a relationship between the frequency and the amplitude of a concrete transient and its associated higher-order features. This is demonstrated via a mathematical transformation which maps the 2D (amplitude- frequency) input space into two output 2D spaces which relate higher-order features to the frequency and the amplitude. Concretely, the main contribution of the paper is the novel finding that the maxima and the minima of the higher-order cumulants distribute along these families of curves of constant frequency or constant amplitude. A couple of extremes (min- max) of the higher-order statistical estimator belongs to a couple of curves (constant-frequency and constant-amplitude), therefore the image position is given by the cross point of the curves. The random grouping along each curve reveals the a priori hidden geometry, in turn linked to the subjacent electrical anomaly. The paper is structured as follows. The following Section II explains the fundamentals and the importance for PQ monitor- ing. Higher-Order Statistics are outlined then in Section III. Finally, results are presented in Section IV and conclusions are drawn in Section V. II. POWER QUALITY CHARACTERIZATION Categorization of electrical transients based on waveform shapes and their underlying causes (or events) has been studied 978-1-4244-8807-0/11/$26.00 ©2011 IEEE 44