Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set

Héctor A. Tabares-Ospina; Mauricio Osorio

doi:10.3934/dcdsb.2020008

Article Contents

2020, Volume 25, Issue 5: 1895-1905. Doi: 10.3934/dcdsb.2020008

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Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set

Héctor A. Tabares-Ospina^1, , and
Mauricio Osorio^2,

1.
Facultad de Ingeniería, Institución Universitaria Pascual Bravo, Calle 73 No. 73A - 226, Medellín, Colombia
2.
Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellín, Cra. 65 No. 59A - 110

^* Corresponding author: Hector A. Tabares-Ospina
^* Corresponding author: Hector A. Tabares-Ospina

Agencia de Educación Superior de Medellín (SAPIENCIA)

Received: December 31, 2018

Revised: April 30, 2019

Early access: December 2019

Published: May 2020

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Abstract

The present article proposes a new geometric space in the complex plane of the Mandelbrot set, framed in the diagram of orbits and attractors, to characterize the dynamics of the curves of the demand of daily electrical power, with the purpose of discovering other observations enabling the elevation of new theoretical approaches. The result shows a different method to evaluate the dynamics of the electric power demand curve, using fractal orbital diagrams. This method is a new contribution that extends universal knowledge about the dynamics of complex systems and fractal geometry. Finally, the reader is informed that the data series used in this article was used in a previous publication, but using a different fractal technique to describe its dynamics.

Keywords:

Mathematics Subject Classification: Primary: 62P30; Secondary: 97M50.

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Figure 1. Graphical representation of the Mandelbrot set

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Figure 2. Relationship between Mandelbrot set and orbital diagrams

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Figure 3. Algorithm with the steps used to obtain the fractal of the power demand

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Figure 4. The typical demand curves of active and reactive electric power

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Figure 5. Electric power demand curve plotted in the first quadrant of the complex plane of Mandelbrot set

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Figure 6. Representation of orbit diagram of power demand in the first quadrant of the complex plane of M set

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Table 1. Daily load demand represented by hour

Hour	$ P $	$ Q $	$ P_{pu} $	$ Q_{pu} $	$ NumOrbs $
$ 00:00:00 $	$ 889 $	$ 371 $	$ 0.222 $	$ 0.092 $	$ 5 $
$ 01:00:00 $	$ 834 $	$ 405 $	$ 0.287 $	$ 0.101 $	$ 5 $
$ 02:00:00 $	$ 792 $	$ 337 $	$ 0.197 $	$ 0.082 $	$ 5 $
$ 03:00:00 $	$ 790 $	$ 324 $	$ 0.199 $	$ 0.081 $	$ 5 $
$ 04:00:00 $	$ 804 $	$ 323 $	$ 0.201 $	$ 0.080 $	$ 3 $
$ 05:00:00 $	$ 925 $	$ 355 $	$ 0.231 $	$ 0.088 $	$ 5 $
$ 06:00:00 $	$ 1041 $	$ 482 $	$ 0.260 $	$ 0.120 $	$ 9 $
$ 07:00:00 $	$ 1105 $	$ 556 $	$ 0.276 $	$ 0.139 $	$ 9 $
$ 08:00:00 $	$ 1191 $	$ 610 $	$ 0.297 $	$ 0.152 $	$ 18 $
$ 09:00:00 $	$ 1256 $	$ 704 $	$ 0.314 $	$ 0.176 $	$ 30 $
$ 10:00:00 $	$ 1309 $	$ 744 $	$ 0.327 $	$ 0.186 $	$ 32 $
$ 11:00:00 $	$ 1366 $	$ 775 $	$ 0.341 $	$ 0.193 $	$ 50 $
$ 12:00:00 $	$ 1385 $	$ 793 $	$ 0.346 $	$ 0.198 $	$ 53 $
$ 13:00:00 $	$ 1356 $	$ 774 $	$ 0.339 $	$ 0.193 $	$ 44 $
$ 14:00:00 $	$ 1337 $	$ 759 $	$ 0.334 $	$ 0.189 $	$ 38 $
$ 15:00:00 $	$ 1350 $	$ 774 $	$ 0.337 $	$ 0.193 $	$ 41 $
$ 16:00:00 $	$ 1336 $	$ 773 $	$ 0.334 $	$ 0.193 $	$ 41 $
$ 17:00:00 $	$ 1312 $	$ 749 $	$ 0.328 $	$ 0.187 $	$ 41 $
$ 18:00:00 $	$ 1287 $	$ 687 $	$ 0.321 $	$ 0.171 $	$ 41 $
$ 19:00:00 $	$ 1420 $	$ 683 $	$ 0.355 $	$ 0.170 $	$ 89 $
$ 20:00:00 $	$ 1389 $	$ 660 $	$ 0.351 $	$ 0.167 $	$ 89 $
$ 21:00:00 $	$ 1311 $	$ 605 $	$ 0.327 $	$ 0.151 $	$ 41 $
$ 22:00:00 $	$ 1175 $	$ 544 $	$ 0.293 $	$ 0.136 $	$ 18 $
$ 23:00:00 $	$ 1030 $	$ 489 $	$ 0.257 $	$ 0.122 $	$ 14 $

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