Electrical tree growth and partial discharges analyzed by fractal and correlation dimensions

Abstract

Electrical treeing degradation is associated with partial discharge activity. Here we relate the growth of electrical trees with the correlation dimension of the reconstructed dynamic object obtained from the nonlinear time series of the partial discharges from the tree propagation, and with the box-counting fractal dimension of the resulting 3D structure of the trees. The growth of trees at 8, 10 and 12 kV has been analyzed; different PD dynamics were found, depending on both the voltage and the stage of growth. A lower fractal dimension is shown to be related to a lower correlation dimension. The results show that nonlinear time series analysis can be an alternative method of analyzing tree growth and associated partial discharges.

Analysis of Partial Discharges in Electrical Tree Growth Under Very Low Frequency (VLF) Excitation Through Pulse Sequence and Nonlinear Time Series Analysis

“…However, this was not a limitation, due to the use of enough windows or intervals for the study of the parameters' evolution during tree growth. Thus, several intervals of analysis were selected from the entire time series following a similar methodology previously used [11], [36]. Each interval was selected to have at least 10,000 PD events and at least 1,000 and 10 seconds for 0.1 and 50 Hz experiments respectively.…”

“…If this set describes a long term trajectory that has evolved to a bounded region of the space, it establishes an asymptotic limit represented by a bounded trajectory; this limit is usually called an attractor. The geometry of the attractor could be a very complex object with fractal properties; in such a case, we referred to that as a strange attractor [36]. To quantify the fractal properties of this attractor usually the correlation dimension is calculated [47].…”

“…In (4) Θ() is the Heaviside function and N is the total amount of points. The number C 2 ( ) can be interpreted as an average distribution of points of the set {x i } in small balls of radius [36]. Once C 2 is calculated, the correlation dimension is estimated by finding a range of where…”

“…A more general approach is to consider the PD phenomenon as a nonlinear dynamic process, and thus, to study it using the mathematical approach of nonlinear time series analysis (NLTSA) and chaos theory in order to characterize the PD behaviour and its sequence of discharges [29], [30]. NLTSA tools have been used in electrical treeing to analyze time series of PD and to describe the growth through nonlinear parameters [4], [11], [20], [35], [36]. In this approach, a similar type of graphical representation can be plotted, normally called "state-space trajectory" or "projection" [29].…”

“…In this approach, a similar type of graphical representation can be plotted, normally called "state-space trajectory" or "projection" [29]. On the other hand, the most used nonlinear parameter for characterization is the correlation dimension [20], [35], [36].…”