# American Institute of Mathematical Sciences

May  2020, 25(5): 1895-1905. doi: 10.3934/dcdsb.2020008

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

 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

Agencia de Educación Superior de Medellín (SAPIENCIA)

Received  January 2019 Revised  May 2019 Published  May 2020 Early access  December 2019

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.

Citation: Héctor A. Tabares-Ospina, Mauricio Osorio. Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1895-1905. doi: 10.3934/dcdsb.2020008
##### References:
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show all references

##### References:
 [1] M. F. Barnsley, R. L. Devaney, B. B. Mandelbrot, H. O. Peitege, D. Saupe and R. F. Voss, The Science of Fractal Images, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3784-6.  Google Scholar [2] J. Borjon, Caos, Orden y Desorden en el Sistema Monetario y Financiero Internacional, Plaza y Valdes, 1$^{st}$ edition, New York, 2002. Google Scholar [3] T. A. Garcia, G. A. Tamura Ozaki, R. C. Castoldi, T. E. Koike, R. C. Trindade Camargo and J. C. S. Camargo Filho, Fractal dimension in the evaluation of different treatments of muscular injury in rats, Tissue Cell, 54 (2018), 120-126.  doi: 10.1016/j.tice.2018.08.014.  Google Scholar [4] H. R. Cui and L. Yang., Short-term electricity price forecast based on improved fractal theory, IEEE Int. Conf. Comput. Eng. Technol., 473 (2009), 347-351.  doi: 10.1109/ICCET.2009.73.  Google Scholar [5] J. Hernandez, S. Mejia and A. Gama, Fractal properties of biophysical models of pericellular brushes can be used to differentiate between cancerous and normal cervical epithelial cells, Colloids Surfaces B Biointerfaces, 170 (2018), 572-577.   Google Scholar [6] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar [7] R. Kumar and P. Chaubey, On the design of tree-type ultra wideband fractal Antenna for DS-CDMA system, J. Microwaves Optoelectron Electromagn Appl., 11 (2012), 107-121.   Google Scholar [8] G. Losa, Fractals and their contribution to biology and medicine, Medicographia, 34 (2012), 365-374.   Google Scholar [9] J. Ma and M. Zhai, Fractal and multi-fractal features of the broadband power line communication signals, Comput. Electr. Eng., 72 (2018), 566-576.   Google Scholar [10] B. B. Mandelbrot and R. L. Hudson, The (mis) Behaviour of Markets: A Fractal View of Risk, Ruin and Reward, Basic Books, New York, 2004.  Google Scholar [11] P. Moon, J. Muday, S. Raynor, J. Schirillo, C. Boydston, M. Fairbanks and R. P. Taylor, Fractal images induce fractal pupil dilations and constrictions, Int. J Psychophysiol, 93 (2014), 316-321.  doi: 10.1016/j.ijpsycho.2014.06.013.  Google Scholar [12] N. Popovic, M. Radunovic, J. Badnjar and T. Popovic, Fractal dimension and lacunarity analysis of retinal microvascular morphology in hypertension and diabetes, Microvascular Research, 118 (2018), 36-43.  doi: 10.1016/j.mvr.2018.02.006.  Google Scholar [13] V. Rodríguez, B. Prieto, H. Correa, M. Soracipa, P. Mendez, C. Bernal and et al., Nueva metodología de evaluación del Holter basada en los sistemas dinámicos y la geometría fractal: Confirmación de su aplicabilidad a nivel clínico, Rev. La Univ. Ind. Santander Salud., 48 (2016), 27-36.   Google Scholar [14] G. Salvó and M. N. Piacquadio, Multifractal analysis of electricity demand as a tool for spatial forecasting, Energy Sustain. Dev., 38 (2017), 67-76.  doi: 10.1016/j.esd.2017.02.005.  Google Scholar [15] S. H. Strogatz, Nonlinear Dynamics and Chaos. With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015.   Google Scholar [16] H. A. Tabares-Ospina and J. E. Candelo-Becerra, Topological properties of fractal Julia sets related to the signs of real and reactive electric powers, Fractals, 10 (2019), 1950066, 11 pp. doi: 10.1142/S0218348X1950066X.  Google Scholar [17] H. A. Tabares-Ospina, J. E. Candelo-Becerra and F. E. Hoyos Velasco, Fractal representation of the power demand based on topological properties of Julia sets, Int. J. Electr. Comput. Eng., 9 (2019), http://dx.doi.org/10.11591/ijece.v9i4.pp. doi: 10.11591/ijece.v9i4.pp2831-2839.  Google Scholar [18] D. D. Ye, M. F. Dai, Y. Sun and W. Y. Su, Average weighted receiving time on the non-homogeneous double-weighted fractal networks, Phys. A, 473 (2017), 390-402.  doi: 10.1016/j.physa.2017.01.013.  Google Scholar [19] M. Y. Zhai, A new method for short-term load forecasting based on fractal interpretation and wavelet analysis, Int. J. Electr. Power Energy Syst., 69 (2015), 241-245.  doi: 10.1016/j.ijepes.2014.12.087.  Google Scholar [20] Z. Zhao, J. Zhu and B. Xia, Multi-fractal fluctuation features of thermal power coal price in China, Energy, 117 (2016), 10-18.   Google Scholar [21] H.A. Tabares-Ospina, F. Angulo, M. Osorio, A new methodology to analyze the dynamic of daily power demand with attractors into the Mandelbrot set, Fractals, 28 (2020) 2050003 1-9. DOI: 10.1142/S0218348X1950066X. Google Scholar
Graphical representation of the Mandelbrot set
Relationship between Mandelbrot set and orbital diagrams
Algorithm with the steps used to obtain the fractal of the power demand
The typical demand curves of active and reactive electric power
Electric power demand curve plotted in the first quadrant of the complex plane of Mandelbrot set
Representation of orbit diagram of power demand in the first quadrant of the complex plane of M set
Daily load demand represented by hour
 Hour $P$ $Q$ $P_{pu}$ $Q_{pu}$ $NumOrbs$ $00:00:00$ $889$ $371$ $0.222$ $0.092$ $5$ $01:00:00$ $834$ $405$ $0.287$ $0.101$ $5$ $02:00:00$ $792$ $337$ $0.197$ $0.082$ $5$ $03:00:00$ $790$ $324$ $0.199$ $0.081$ $5$ $04:00:00$ $804$ $323$ $0.201$ $0.080$ $3$ $05:00:00$ $925$ $355$ $0.231$ $0.088$ $5$ $06:00:00$ $1041$ $482$ $0.260$ $0.120$ $9$ $07:00:00$ $1105$ $556$ $0.276$ $0.139$ $9$ $08:00:00$ $1191$ $610$ $0.297$ $0.152$ $18$ $09:00:00$ $1256$ $704$ $0.314$ $0.176$ $30$ $10:00:00$ $1309$ $744$ $0.327$ $0.186$ $32$ $11:00:00$ $1366$ $775$ $0.341$ $0.193$ $50$ $12:00:00$ $1385$ $793$ $0.346$ $0.198$ $53$ $13:00:00$ $1356$ $774$ $0.339$ $0.193$ $44$ $14:00:00$ $1337$ $759$ $0.334$ $0.189$ $38$ $15:00:00$ $1350$ $774$ $0.337$ $0.193$ $41$ $16:00:00$ $1336$ $773$ $0.334$ $0.193$ $41$ $17:00:00$ $1312$ $749$ $0.328$ $0.187$ $41$ $18:00:00$ $1287$ $687$ $0.321$ $0.171$ $41$ $19:00:00$ $1420$ $683$ $0.355$ $0.170$ $89$ $20:00:00$ $1389$ $660$ $0.351$ $0.167$ $89$ $21:00:00$ $1311$ $605$ $0.327$ $0.151$ $41$ $22:00:00$ $1175$ $544$ $0.293$ $0.136$ $18$ $23:00:00$ $1030$ $489$ $0.257$ $0.122$ $14$
 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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