Topological properties of fractal Julia sets related to the signs and
magnitudes of the real and reactive powers.

Héctor A. Tabares-Ospina111Facultad de Ingeniería, Institución Universitaria
Pascual Bravo, Medellín, Colombia. Email<EMAIL_ADDRESS>and
John E. Candelo-Becerra222Facultad de Minas, Universidad Nacional de Colombia,
Medellín, Colombia. Email<EMAIL_ADDRESS>

Abstract

In AC electrical systems, the power depends on the real power (P) due to
resistive elements and the reactive power (Q) due to the inductive and
capacitive elements, which are commonly studied by using phasor and scalar
methods. Thus, this paper focuses on applying the fractal Julia sets to
observe the topological properties related to the signs and magnitudes of the
real and reactive powers consumed or supplied by an electrical circuit. To
perform this, different power combinations were used to represent the fractal
diagrams with an algorithm that considers the mathematical model of Julia
sets. The study considers three type of loads: the first study considers the
change of real power when the reactive power is fixed; the second study deals
with the change of the reactive power when the real power is fixed; and
finally, the third study contemplates that both real and reactive powers
change. Furthermore, the fractal diagrams of the power in the four quadrants
of the complex plane are studied to identify the topological properties that
each sign and magnitude represent. A qualitative analysis of the diagrams
helps to identify that the complex power loads present some fractal graphic
patterns, with respect to the signs and magnitudes considered in the different
quadrants of the complex planes. The diagrams represented in the complex
planes save a relation in the forms and structure with other points studied,
concluding that the power is related to other figures in other quadrants.
Thus, this result allows a new study of the behavior of the power in an
electrical circuit, showing a clear relation of the different fractal diagrams
that the Julia sets obtained.

Keywords: Real power, reactive power, fractal geometry, Julia set, Mandelbrot
set, behavior patterns, power factor

##  1 Introduction

In 1975, the mathematic Benoit Mandelbrot was the responsible to define the
concept of fractal as a semi geometric element with a repetitive structure at
different scales [1], with characteristics of self-similarity as some nature
elements such as snowflakes, ferns, peacock feathers, and romanesco broccoli.
Fractal theory has been applied to various fields such as biology [2,3],
health sciences [4–8], stock markets [9], network communications [10–12], and
others. Fractal theory is one of the methods used to analyze data and obtain
relevant information in a highly complex problem. Thus, it has been used to
study the price of highly variable markets, which are not always explainable
from classical economic analyzes.

For example, in [9] the authors demonstrate that the current techniques have
some issues to explain the real market operation, and a better understanding
is achieved by using techniques as theory of chaos and fractals. In their
publication, the authors show how to apply the fractal behavior of stock
markets. Besides, the authors refer to multifractal analysis and multifractal
topology. The first describes the invariability of scaling properties of time
series and the second is a function of the Hölder exponents that characterize
the degree of irregularity of the signal and their most significant
parameters.

In [13], the authors discuss the basic principle of fractal theory and how to
use it to forecast the short-term electricity price. In the first instance,
the authors analyze the fractal characteristic of the electricity price,
confirming that price data have this property. In the second instance, a
fractal model is used to build a forecasting model, which offers a wide
application in determining the price of electricity in the markets.

Similarly, the authors of [14] demonstrate that the price of thermal coal has
multifractal features by using the concepts introduced by Mandelbrot-Bouchaud.
Hence, a quarterly fluctuation index (QFI) for thermal power coal price is
proposed to forecast the coal price caused by market fluctuation. This study
also provides a useful reference to understand the multifractal fluctuation
characteristics in other energy prices.

The fractal geometry analysis has been also applied to study the morphology
and the population growth of cities, and that the electricity demand is
related to the demography of cities. In [15] a multifractal analysis is used
to forecast the electricity demand, explaining that two fractals are found,
showing the behavior pattern of the power demand. Two concepts linked to
fractal geometry are fractal interpolation and extrapolation, related to the
resolution of a fractal-encoded image. In [16], an algorithm to forecast the
electric charge, in which fractal interpolation and extrapolation are also
involved. For the forecast data set, the average relative errors are only
2.303% and 2.296%. The result shows that the algorithm has advantages in
improving forecast accuracy.

In the literature, there are not papers related to the study of the signs and
magnitudes of real and reactive powers by using Mandelbrot and Julia sets.
Besides, the characteristics of the real and reactive powers are not deeply
analyzed applying the fractal geometry, concluding that these techniques are
not commonly used to study the different behaviors of the power consumption
and supply. For this reason, this work focuses on performing a detailed
analysis of the characteristics of the real and reactive powers based on the
Julia and Mandelbrot sets. Particularly in this work, the real and reactive
powers are changed to represent the different Julia sets that allow performing
observations to determine a qualitative study. With the purpose to
characterize the real and reactive power with the Julia sets as to observe
other behavior with the fractal geometry such as: fractal graphic pattern,
which requires a new interpretation to identify the different signs. Thus, the
following hypothesis is tested: the signs and magnitudes of the real and
reactive electrical powers show some related fractal topology patterns in the
Julia set, which allows to identify the behavior of the electricity
consumption and supply.

The rest of this document is organized as follows. Section 2 includes the
brief explanation of the theory of the Mandelbrot sets and Section 3 presents
the theory of Julia sets. Section 4 presents the method applied in this
research and the algorithm used to identify the topological properties of the
different Julia sets of real and reactive power in electric systems. Section 5
presents the results and discussion of the most relevant examples of signs and
magnitudes of real and reactive powers. Finally, the main conclusions of this
research work are summarized.

###  1.1 Mandelbrot set

Mandelbrot set, denoted as $M=\\{$ $c\in C/J_{c}\\}$, represents the sets of
complex numbers $C$ obtained after iterating the from the initial point
$Z_{n}$ and the selected constant $C$ as shown (1), the results form a diagram
with connected points remaining bounded in an absolute value. One property of
$M$ is that the points are connected, although in some zones of the diagram it
seems that the set is fragmented. The iteration of the function generates a
set of numbers called orbits. The results of the iteration of those points out
of the boundary set tend to infinity.

$Z_{n+1}=F\left(Z_{n}\right)=Z_{n}^{2}+C$ (1)

From the term $C$, a successive recursion is performed with $Z_{0}=0$ as the
initial term. If this successive recursion is dimensioned, then the term $C$
belongs to the Mandelbrot set; if not, then they are excluded. Therefore,
Figure 1 shows the Mandelbrot set with points in the black zone that are
called the prisoners, while the points in other colors are the escapists and
they represent the velocity that they escape to infinite.

Figure 1: Representation of Mandelbrot diagram

---

From this figure, the number -1 is inside of the set while the number 1 is
outside of the set. In the Mandelbrot set, the fractal is the border and the
dimension of Hausdorff is unknown. If the image is enlarged near the edge of
the set, many areas the Mandelbrot set are represented in the same form.
Besides, different types of Julia sets are distributed in different regions of
the Mandelbrot set. Whether a complex number appears with a greater value than
2 in the 0 orbit, then the orbit tends to infinity.

The pseudocode of the algorithm that is used to represent the Mandelbrot set
is presented as follows:

Start

For each point $C$ in the complex plane do:

Fix $Z_{0}=0$

For $t=1$ to $t_{\max}$ do:

Calculate $Z_{t}=Z_{t}^{2}+C$

If $|Z_{t}|>2$ then

Break

End if

If $t<t_{\max}$ then

Draw $C$ in white (the point does not belong to the set)

Else if $t=t_{\max}$ then

Draw $C$ in black (as the point does belong to the set)

End if

End For

End

In this research, the presented algorithm has been used to obtain the
Mandelbrot set and the diagram that represent it. Some points related to the
real and reactive powers with the respective signs are studied in the
Mandelbrot set and related to those points created for the Julia sets as
explained in the following sections.

###  1.2 Julia sets

The mathematics Gastón Julia and Pierre Fatuo, at the beginning of the century
$XX$, developed a fractal sets that are obtained by iterating complex numbers.
The Julia sets of a holomorphic function $f$ is constituted by those points
that under the iteration of $f$ have a chaotic behavior and each point of the
set forms a different set $f$ that is then denoted by $J(f)$. The Fatou set
consists of the points that have a stable behavior when they are iterated. The
Fatou set of a holomorphic function $f$ is denoted by $F(f)$ and it is a
complement of $J(f)$. An important family of the Julia sets is obtained from
the simple quadratic functions, for example
$Z_{n+1}=F\left(Z_{n}\right)=Z_{n}^{2}+C$, where $C$ is a complex number. The
values obtained from this function are denoted the $J_{c}$, with points of $Z$
obtained from the parameter $C$ that belong to the Julia sets. Other points
obtained during the iteration are excluded from the Julia sets as they tend to
infinite.

For example, Figure 2 shows a simulation performed to obtain the Julia sets
for two complex numbers. Figure 2(a) shows the Julia sets calculated by
iterating the function $Z_{n+1}=F\left(Z_{n}\right)=Z_{n}^{2}+C$ when
$C=C1=-0.33+0.57i$ and Figure 2(b) shows the Julia sets by iterating the same
function when $C=C2=0.44+0.15i$. The results show that the complex number
$C1=-0.33+0.57i$ represented in Figure 2(a) produces a Julia set of points in
black and other points with different colors tend to the infinity according to
the number of iterations they require to obtain the result. However, the
complex number $C2=0.44+0.15i$ represented in Figure 2(b) produces a Julia set
with non-connected points.

Figure 2: Julia sets for two complex numbers (a) $C1=-0.33+0.57i$ | (b) $C2=0.44+0.15i$
---|---

An important relation between the Mandelbrot and the Julia sets is given when
the point $C$ belongs to the Mandelbrot set, then the Julia set $J(f_{c})$
obtain a series of points that are connected. On the other side, when the
point does not belong to the Mandelbrot set, then the Julia set $J(f_{c})$ is
formed by non-connected points. Therefore, there are two main classes of Julia
sets: points that form a unique piece and points that represent many pieces in
the complex plane.

One property of the Mandelbrot set is that the different types of Julia sets
are distributed in different regions of the set $M$. For all the above, it is
concluded that in Figure 2, $C1$ is in the set of $M$, and $C2$ is not in the
set. In general, any point within the $M$ cardioid or its boundary, the Julia
set of the $J(f_{c})$ has points that are connected. The most interesting $C$
values are those near the border of the Mandelbrot set, because it is there
when the set $J(f_{c})$ passes, suddenly, from connected points to non-
connected points.

##  2 Procedure implemented in the research

As the electrical power can be expressed as a complex expression given by
$S=P+jQ$, the combination values with the respective signs and magnitudes of
the real and reactive powers are studied through the representation of the
Julia sets, obtaining fractal geometric patterns that allow a qualitative
analysis. To validate the hypothesis, the procedure of this research was
developed to evaluate certain properties of the electrical power by using the
fractal geometry.

The first part of the procedure starts with the analysis of the real and
reactive power in the complex plane of the electric power. The following cases
are considered in the study:

  1. a)

The real power $P>0$ and the reactive power $Q>0$ corresponds to the first
complex plane.

  2. b)

The real power $P<0$ and the reactive power $Q>0$ corresponds to the second
complex plane.

  3. c)

The real power $P<0$ and the reactive power $Q<0$ corresponds to the third
complex plane.

  4. d)

The real power $P>0$ and the reactive power $Q<0$ corresponds to the fourth
complex plane.

In the second part, an algorithm was programmed to perform the numerical
experiments on the Mandelbrot and Julia sets by using the real and reactive
power with different power factors. To test the system, the most
representative Julia sets were executed.

As the $M$ set has infinite values in the quotation, the third part of this
work dedicates on determining two quotation points. The limits of $M$ in the
positive axis $(x,y)$ are represented in the diagrams and the values of $P$
and $Q$ are scale in the set of $M$ with values $C_{x}=0.25$ and $C_{y}=0.63$.

In order to validate the work hypothesis, the fourth part consisted of
operating the program to find a pattern from the fractal geometries, depending
on the different signs of the real and reactive powers. In the fifth part,
Julia sets were classified, according to their symmetry and degree of
irregularity in their borders, selecting the most interesting by their degree
of dispersion. Finally, the Julia sets related to each complex plane are found
and the conclusions are obtained.

##  3 Results and analysis

The following figures present the diagrams created with the Julia sets
generated from the real and reactive components of the electrical power. These
generated fractal geometries do not give a quantitative information of the
fact under study; however, the study of the fractals can be performed by a
qualitative analysis, whose interpretation is based on the experience.

###  3.1 Julia sets for the real power

In order to simplify the analysis, the study is performed first for the change
of real power with the power factor equal to $1$ or the reactive power equals
to $zero$. This study relates to a resistive circuit where the voltage $(V)$
is in phase with the current. Thus, Figure 3 presents the transformation of
the geometric fractal patterns according to the change of real power. These
figures are plotted in the complex plane, in which the $x$ axis corresponds to
the real component $(Re)$ and the $y$ axis corresponds to the imaginary
component $(Img)$. Figure 3(a) shows the fractal when real and reactive powers
are equal to $cero$, which obtains as a result a circle with radius equal to
$1$. Figure 3(b) presents the fractal when $P=D_{x}/2$, showing a
transformation of the closed fractal curve and evidencing contraction in its
symmetry axes. Figure 3(c) shows a fractal when $P=D_{x}$, showing that the
Julia set is close to the limit of the set $M$ or to separate the points.
Figure 3(d) shows that when $P>D_{x}$ or $P$ exceeds the limit of the set $M$,
the Julia set presents several points that are not connected or explode in a
point cloud.

Figure 3: Diagrams created with the Julia sets when the real power changes and the reactive power is equal to cero (a) $P=0,Q=0$ | (b) $P=\frac{D_{x}}{2},Q=0$ | (c) $P=D_{x},Q=0$ | (d) $P>D_{x},Q=0$
---|---|---|---

Therefore, the Julia sets form a simple closed curve that transforms to a
fractal form with the following relations: the upper and lower half-planes are
reflections of each other, while the right and left half planes are
reflections of each other. Besides, the points of the upper and lower half-
planes are symmetrical with respect to the real axis of the complex plane,
while the points of the right and left half planes are symmetrical with
respect to the imaginary axis of the complex plane.

In conclusion, reflective symmetry is presented with respect to the origin
point and it is confirmed that there exists a fractal geometric pattern as a
function of $P$. The relation found in this research is because the origin
Mandelbrot set is symmetric with respect to the real axis of the complex
plane. With respect to the fractal curve in the limits of the set, it reveals
the approximation that has the magnitude of the real power with respect to the
limit of the set of $M$. If the real power exceeds the limit of $M$ on the x
axis, the Julia set is not connected and presents several points as exploding.

###  3.2 Julia sets for the reactive power

Now, the fractal geometry obtained as a function of reactive power is studied.
This case represents a motor or generator without load, which considers only
reactive power for the magnetization of its coils, which is also evidenced by
a low power factor. Thus, Figure 4 shows the transformation of the fractal
geometry as a function of the change of the reactive power $Q$. Figure 4(a)
shows the fractal geometry when $P$ and $Q$ are equal to $zero$, which
represents is a circle of radius equal to $1$. Figure 4(b) shows the fractal
geometry when $D_{y}/2$, which is transformed into a closed fractal curve.
Figure 4(c) shows the fractal geometry when $Q=D_{y}$, presenting the Julia
set next to its separation because $Q$ is above the limit of $M$. Figure 4(d)
shows the fractal geometry when $Q>D_{y}$, which exceeds the limit of the set
of $M$, and the points of the Julia sets are not connected, with separated
zones.

Figure 4: Diagrams obtained with the Julia sets in function of the reactive power Q (a) $P=0,Q=0$ | (b) $P=0,Q=\frac{D_{y}}{2}$ | (c) $P=0,Q=D_{y}$ | (d) $P=0,Q>D_{y}$
---|---|---|---

In this case, there are different fractal patterns when the reactive power is
studied, showing the inverse reflective symmetry with respect to the origin.
In Figures 4(a), (b), and (c), the Julia set is a simple closed curve that
becomes a fractal curve. However, unlike the relationships shown in 3(a), (b)
and (c), in this case the half-planes keep inverted reflections of one
another. The relationship between the quadrants is the following: the upper
and lower half-planes are reflections inverted one from the other, while the
right and left half planes are reflections inverted one from the other. The
points of the upper and lower half planes are symmetrical inverted with
respect to the real axis of the complex plane, while the points of the right
and left half planes are symmetrical inverted with respect to the imaginary
axis of the complex plane.

In conclusion, it is confirmed that there exists another fractal geometric
pattern for the reactive power $Q$, which show the inverted reflective
symmetry with respect to the origin point is presented. The fractal geometry
in the limits reveals the approximation that the magnitude of the reactive
power has with respect to the limit of the set of $M$. If the reactive power
exceeds the limit of $M$ on the y axis, the Julia sets becomes not connected,
exploding in point cloud.

###  3.3 Julia sets of the real and reactive power values

Next, the fractal diagrams obtained from the changes of real and reactive
powers are presented. Therefore, the following figures show fractals for the
four quadrants. To perform the tests and show the results clearly, all values
were considered with the values $P\pm 0.22$, $Q\pm 0.22$.

####  3.3.1 Analysis on the first quadrant

Figure 5 shows the fractal diagrams related to the Mandelbrot and Julia sets
when the real and reactive power is located in the first quadrant of the
complex plane. Figure 5(a) shows the real and reactive powers with an
inductive element. Figure 5(b) shows the Mandelbrot diagram created by the
algorithm that iterates the complex number created by the real and reactive
power $(P=0.22,Q=0.22)$. Besides, a point in yellow is marked in this figure
to adjust the real and reactive power to the diagram. Figure 5(c) shows the
Julia set that is created from the point selected from the Mandelbrot diagram
and related to the real and reactive power studied. In general, when $P$ and
$Q$ are positive and located within the set $M$, then the equivalent set
$J(f_{c})$ is connected and its graph is a closed curve with inverted
reflective symmetry with respect to the origin point, and with dendrites
oriented in a clockwise direction.

Figure 5: Mandelbrot and Julia sets for a real and reactive power in the first complex plane (a) first quadrant | (b) Mandelbrot diagram | (c) Julia diagram
---|---|---

####  3.3.2 Analysis on the fourth quadrant

Figure 6 shows the fractal diagrams related to the Mandelbrot and Julia sets
when the power is at the fourth quadrant of the complex plane. Figure 6(a)
shows the real and reactive power with a capacitive element
($P=0.22,Q=-0.22$). Figure 6(b) shows the Mandelbrot diagram created by the
algorithm that iterates the complex number created by the real and reactive
power. Besides, a point in yellow is marked in this figure to adjust the real
and reactive power to the diagram. Figure 6(c) shows the Julia set that is
created from the point selected from the Mandelbrot diagram and related to the
real and reactive power studied.

Figure 6: Mandelbrot and Julia sets for a real and reactive power in the fourth complex plane (a) fourth | (b) Mandelbrot diagram | (c) Julia diagram
---|---|---

Figures 5 and 6 show that the forms have the same relation but they are
symmetrically inverted with respect to the origin and the equivalent set
$J(f_{c})$ is connected and with closed curve, with dendrites oriented anti-
clockwise.

####  3.3.3 Analysis on the second quadrant

Figure 7 shows the fractal diagrams related to the Mandelbrot and Julia sets
when the power is at the second quadrant of the complex plane. Figure 7(a)
shows the real power supplied to the circuit with a capacitive reactive power
$(P=-0.22,Q=0.22)$. Figure 7(b) shows the Mandelbrot diagram created by the
algorithm that iterates the complex number created from the power of the
electric circuit. Besides, a point in yellow is marked in this figure to
adjust the real and reactive power to the diagram. Figure 7(c) shows the Julia
set that is created from the point selected from the Mandelbrot diagram and
related to the real and reactive powers studied. When this values are in $M$,
the equivalent set $J(f_{c})$ is connected and the figure forms a closed
fractal different to the other two cases, with apparent clockwise direction
with respect to the real axis of the complex plane.

Figure 7: Mandelbrot and Julia sets for a real and reactive power in the second complex plane (a) second quadrant | (b) Mandelbrot diagram | (c) Julia diagram
---|---|---

####  3.3.4 Analysis on the third quadrant

Figure 8 shows a new fractal diagrams related to the Mandelbrot and Julia sets
when the power is at the third quadrant of the complex plane. Figure 8(a)
shows the real power supplied to the circuit with an inductive reactive power
$(P=-0.22,Q=-0.22)$. Figure 8(b) shows the Mandelbrot diagram created by the
algorithm that iterates the complex number created from the power of the
electric circuit. Besides, a point in yellow is marked in this figure to
adjust the real and reactive power to the diagram. Figure 8(c) shows the
diagram created with the Julia set from the point selected in the Mandelbrot
diagram and related to the real and reactive power studied. Figure 8 presents
a fractal geometry of Julia set with inverted symmetry with respect to Figure
7. When the power is in the third quadrant and inside the main bulb of the set
of $M$, the equivalent set $J(f_{c})$ is connected and its graph is a closed
curve, which appears in a clockwise rotation with respect to the real axis of
the complex plane.

Figure 8: Mandelbrot and Julia sets for a power that is in the third complex plane (a) third quadrant | (b) Mandelbrot diagram | (c) Julia diagram
---|---|---

Figures 7 and 8 showed that the forms have the same relation, but they are
symmetrically inverted with respect to the origin. Again, a qualitative
interpretation is that the capacitive and inductive electric power corresponds
to two types of electrical phenomena contrary to each other. These two figures
showed the behavior of the power with fractal diagrams: the first figure
represented the reactive power related to capacitive elements as the current
leads the voltage; and the second figure represented the reactive power
related to inductive elements as the current lags the voltage.

##  4 Conclusions

This paper used the fractal Julia sets to observe the topological properties
related to the signs and magnitudes of the real and reactive powers consumed
or supplied by an electrical circuit. The method consisted in considering
different power combinations to represent the fractal diagrams with an
algorithm that iterates the mathematical expressions of the Julia sets. The
study considered the change of real power when the reactive power is fixed,
the change of the reactive power when the real power is fixed, and the change
of both real and reactive powers. Furthermore, the fractal diagrams of the
power in the four quadrants of the complex plane were studied to identify the
topological properties that each sign and magnitude represent.

The study of the fractal diagrams helped to identify that the real and
reactive powers represents clear fractal patterns with the Julia sets. When
the reactive power is greater than zero and the real power changes, the Julia
set presents an inverted symmetry with respect to the origin point. Besides,
when the real and reactive powers are equal to zero, the fractal diagram is a
circle with a radius of one in the complex plane. As a function of real power,
two Julia sets are generated with different fractal patterns that characterize
the signs and magnitudes. The characteristics of the Julia sets depend on the
values of the initial point $Z_{n}$. Additionally, these are groups with a
chaotic behavior, because their fractal curve cannot be predicted. When
numerical analysis is repeated for different real and reactive powers, they
produce different fractal curves that are symmetrically inverted with respect
to the origin, as explained in this work.

##  5 Acknowledgement

This work was supported by the Agencia de Educación Superior de Medellín
(Sapiencia), under the specific agreement celebrated with the Institución
Universitaria Pascual Bravo. The project is part of the Energy System
Doctorate Program and the Department of Electrical Energy and Automation of
the Universidad Nacional de Colombia, Sede Medellín, Facultad de Minas.

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##  7 Bibliography of authors

Figure 9: Authors |
---|---
First Author’s | Second author’s

Firts Author’s: Héctor A. Tabares-Ospina: received his Bs. degree in
Electrical Engineering in 1997 and his Master in Systems Engineering in 2005
from Universidad Nacional de Colombia. He is now studing doctoral studies. He
is an Assistant Professor of Institución Universitaria Pascual Bravo. His
research interests include: Fractal geometry, artificial intelligence,
operation and control of power systems; and smart grids. He is a Junior
Researcher in Colciencias and member of the Research Group - GIIEN, at
Institución Universitaria Pascual Bravo. https://orcid.org/0000-0003-2841-6262

Second Author’s:John E. Candelo-Becerra: received his Bs. degree in Electrical
Engineering in 2002 and his PhD in Engineering with emphasis in Electrical
Engineering in 2009 from Universidad del Valle, Cali - Colombia. His
employment experiences include the Empresa de Energía del Pacífico EPSA,
Universidad del Norte, and Universidad Nacional de Colombia - Sede Medellín.
He is now an Assistant Professor of the Universidad Nacional de Colombia -
Sede Medellín, Colombia. His research interests include: engineering
education; planning, operation and control of power systems; artificial
intelligence; and smart grids. He is a Senior Researcher in Colciencias and
member of the Applied Technologies Research Group - GITA, at the Universidad
Nacional de Colombia. https://orcid.org/0000-0002-9784-9494.