Fractal representation of the power daily demand based on topological properties of Julia sets

TL;DR

This paper introduces a novel fractal graphical method using Julia sets to visualize and analyze hourly power demand patterns, revealing insights into consumption behaviors throughout the day.

Contribution

It presents a new approach to represent power demand through Julia set fractals, linking topological properties to load consumption patterns.

Findings

Identified distinct fractal patterns for different power consumption levels

Observed similar behaviors in fractal diagrams across different hours

Facilitated understanding of daily load demand variations

Abstract

In a power system, the load demand considers two components such as the real power (P) because of resistive elements, and the reactive power (Q) because inductive or capacitive elements. This paper presents a graphical representation of the electric power demand based on the topological properties of the Julia Sets, with the purpose of observing the different graphic patterns and relationship with the hourly load consumptions. An algorithm that iterates complex numbers related to power is used to represent each fractal diagram of the load demand. The results show some representative patterns related to each value of the power consumption and similar behaviour in the fractal diagrams, which allows to understand consumption behaviours from the different hours of the day. This study allows to make a relation among the different consumptions of the day to create relationships that lead to the prediction of different behaviour patterns of the curves.

Tables

Table 1. Table 1: Daily power demand

$Hour$	$P$	$Q$	$S$	$P_{pu}$	$Q_{pu}$	$S_{pu}$
00:00:00	889	371	963	0.222	0.092	0.240
01:00:00	834	405	927	0.208	0.101	0.231
02:00:00	792	337	861	0.197	0.082	0.215
03:00:00	790	324	854	0.199	0.081	0.213
04:00:00	804	323	867	0.201	0.080	0.216
05:00:00	925	355	991	0.231	0.088	0.247
06:00:00	1041	482	1147	0.260	0.120	0.286
07:00:00	1105	556	1237	0.276	0.139	0.309
08:00:00	1191	610	1338	0.297	0.152	0.334
09:00:00	1256	704	1439	0.314	0.176	0.359
10:00:00	1309	744	1506	0.327	0.186	0.376
11:00:00	1366	775	1571	0.341	0.193	0.392
12:00:00	1385	793	1595	0.346	0.198	0.398
13:00:00	1356	774	1561	0.339	0.193	0.390
14:00:00	1337	759	1537	0.334	0.189	0.384
15:00:00	1350	774	1556	0.337	0.193	0.389
16:00:00	1336	773	1543	0.334	0.193	0.385
17:00:00	1312	749	1511	0.328	0.187	0.377
18:00:00	1287	687	1459	0.321	0.171	0.364
19:00:00	1420	683	1575	0.355	0.170	0.393
20:00:00	1389	660	1538	0.351	0.167	0.384
21:00:00	1311	605	1444	0.327	0.151	0.361
22:00:00	1175	544	1295	0.293	0.136	0.323
23:00:00	1030	489	1140	0.257	0.122	0.285

Equations

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

Full text

Fractal representation of the power daily demand based on topological properties of Julia sets.

Héctor A. Tabares-Ospina111Facultad de Ingeniería, Institución Universitaria Pascual Bravo, Medellín, Colombia. Email: h.tabares@pascualbravo.edu.co and John E. Candelo-Becerra222Facultad de Minas, Universidad Nacional de Colombia, Medellín, Colombia. Email: jecandelob@unal.edu.co

Abstract

In a power system, the load demand considers two components such as the real power ($P$) because of resistive elements, and the reactive power ($Q$) because inductive or capacitive elements. This paper presents a graphical representation of the electric power demand based on the topological properties of the Julia Sets, with the purpose of observing the different graphic patterns and relationship with the hourly load consumptions. An algorithm that iterates complex numbers related to power is used to represent each fractal diagram of the load demand. The results show some representative patterns related to each value of the power consumption and similar behaviour in the fractal diagrams, which allows to understand consumption behaviours from the different hours of the day. This study allows to make a relation among the different consumptions of the day to create relationships that lead to the prediction of different behaviour patterns of the curves.

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

1 Introduction

Benoit Mandelbrot defined the concept of fractals as a semi-geometric element with a repetitive structure at different scales [1], with characteristics of self-similarity as seen in some natural formations 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, 5, 6, 7, 8], stock markets [9], network communications [10, 11, 12], and others. Fractal theory is one of the methods used to analyse data and obtain relevant information in highly complex problems. Thus, it has been used to study the price of highly variable markets, which are not always explainable from classical economic analysis.

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 reviewed there are no papers focused on the daily real and reactive power consumption based on the topological properties of the Julia sets. There are no graphical analyses that show the behaviour of the system by observing different fractal patterns. Most studies on fractals are focused on other types of applications such as medicine, biology, communications, electronics, leading to an important opportunity to perform the study on power systems. In addition, the characteristics of the real and reactive powers are not analysed in depth by applying fractal geometry, concluding that these techniques are not commonly used to study the different power consumption. Therefore, this paper studies a typical load demand curve of an electric power system with the fractal theory of Julia sets. The graphic study focuses on determining the characteristics that the fractal diagrams created from Julia sets related with the complex numbers of real and reactive powers and seeking for other graphical patterns of the load demand curve. For this reason, this work proposes the following hypothesis: The load demand curve has a clear fractal pattern obtained from the Julia sets, which allows to characterize the consumption behaviour. The main contribution of this article is related to the development of a new methodology, as a complement to those found in literature, which allows to characterize the daily load demand curve. This paper confirms that the density of the folds in the Julia sets reveal the transitivity in the electric power consumption registers, which is related to the irregularity of the hourly consumption. Eventually, the fractal topology that is obtained from the Julia sets reflects when the load is inductive or capacitive.

2 Research method

This methodology is based on constructing an algorithm that allows an analysis of the fractal diagrams applied to the typical load demand curve. Below, this section shows the general procedure and the algorithms implemented to obtain the Mandelbrot and Julia sets, which are useful tools to graph fractals from the complex numbers related to the load demand.

2.1 General procedure

Figure 1 presents a step-by-step procedure applied to graph the fractal diagrams from the load demand with the Mandelbrot and Julia sets. This figure shows that the first step ($P1$) is to convert the initial data to manage the procedure to the Mandelbrot and Julia algorithms. Next, the Mandelbrot algorithm is programmed according to the mathematical theory ($P2$) to generate a new data set. Besides, the Julia algorithm is also programmed to perform the generation of the new sets, based on the Mandelbrot set. With these data sets, it is possible to plot the different fractals ($P4$) which are then analysed to present the different results in this paper ($P5$) and the corresponding conclusions.

2.2 Load demand curve

The process begins by reading the typical load demand records of real and reactive defined for a $24$-hour period (see Table 1). The per unit values of the power demand are calculated with the following expression: $per\_unit\_power=actual\_power/base\_power$. In this case, the base power is $4000$ MVA. These data are used to plot the diagrams with the programmed algorithms, in which the lowest and highest consumption points are considered to evaluate the different fractal diagrams.

2.3

Algorithm to create the 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.

[TABLE]

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 2 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.

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.

2.4 Algorithm to create the Julia sets

At the beginning of the century $XX$, mathematics Gastón Julia and Pierre Fatuo, 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, in Figure 3, the complex number $C=0.30+0.21i$ lies within the $M$ set and produces Julia sets of connected points represented in black, and the points that go to infinity are represented in different colors according to the number of iterations necessary to escape.

However, as shown in Figure 4, the complex number $C=0.40+0.15i$ lies on the boundary of the $M$ set and produces a Julia set of points that are partially connected and distributed in different subgroups.

Finally, Figure 5 shows a complete Julia sets of the complex number $C=0.50+0.21i$, located outside of the $M$ set.

In summary, there are three types of Julia sets: the first set is formed by connected points that are obtained when the complex number $C$ is inside of the Mandelbrot set; the second set is formed by partially connected points that are obtained when the complex number $C$ is the boundary of the Mandelbrot set; and the third set is formed by non-connected points when the constant $C$ is outside of the Mandelbrot set, resulting in infinite collections of isolated points with no discernible pattern. An important relation between the Mandelbrot and Julia sets is given when 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, the Julia set $J(f_{C})$ is formed by non-connected points.

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 5, $C1$ is in the set of M and $C2$ is not in the set. In general, for any point within the $M$ cardioid or its boundary, the Julia set of $J(f_{C})$ has points that are connected. The most interesting $C$ values are those near the boundary of the Mandelbrot set because the points can be transformed from connected points to non-connected points.

2.5 Algorithm to study the fractals of power demand

In order to obtain the results of the fractal topology patterns that represent the real and reactive power demand curves, the procedure shown in Figure 6 was followed.

The initial process stars reading data of the power base and the real and reactive powers, followed by the calculation of each per unit value. In this case, the $P$ and $Q$ curves with respect to the time are plotted to represent the power demand during the day. Next, the $M$ set is calculated and used to plot the fractal diagram. Now, the real and reactive powers of each point in the load demand curve are scaled into the $M$ set and used to obtain the $J$ sets. Then, the $J$ sets are plotted into fractal diagrams to analyse qualitatively their geometries.

3 Results and analysis

Figure 7 presents the typical demand curves plotted with the data of Table 1 and Figure 8 presents the power demand plotted in the first quadrant of the complex plane. As real and reactive powers are positive, they represent a load consumption related to inductive elements. Under these conditions, the three most interesting values of the power consumption are selected such as the lowest consumption at 3:00, the highest consumption at 19:00, and the approximate average consumption at 09:00. Other hours of the day represent diagrams that are forms between the values as shown in the following results in this section.

Figure 9 shows the fractal generated for each point of Figure 8. These fractals are created by performing iterations of the complex numbers obtained from the daily load demand.

All Julia sets plotted closed curves of connected points and represent the fractal topology of inductive power loads that belong to the first quadrant of the complex plane. The curves are transformed into fractal curves, where the semi plane save inverted reflections each other and with quadrants symmetrically inverted with respect to the origin. It is also true that, when the set of Julia is connected, the point $C$ does not reach the boundary of the M set, and do not generate periodicity of the Julia set.

Now, with respect to the topological properties of the Julia sets registered in the lower demand period $[00:00-05:00]$, the equivalent fractal topology is presented at $3:00$ with an inverted reflective symmetry. Another feature of the sub-period is the practical invariance in the load demand, which is evident in the unmoving folding at the boundary of the fractal curve. At $05:00$, the load demand begins to increase, generating greater fractal folds at the boundaries of the Julia set. At $09:00$, the significant increase in the load demand is evidenced by the increase in the fractal folds of the Julia set. From $12:00$ to $17:00$, an average load demand is maintained, representing insignificant changes in the boundary of the Julia set. The point of greatest interest occurs at $19:00$, when the greatest load demand, generating a topology with dense fractal folds at the boundary related to the properties of the peak load demand. From $19:00$ to $24:00$, the load demand decreases and the boundary of Julia set softens.

4 Conclusions

The paper presented a graphical representation of the power demand based on the topological properties of the Julia Sets, with the purpose of observing the different graphic patterns and relationship with each consumption in a daily load demand curve. An algorithm that iterates complex numbers of real and reactive powers is used to represent each fractal diagram of the consumption.

It is concluded that the load demand curve presents a clear fractal topology pattern of the Julia set and the following observations were obtained:

• •

A new way of visualizing the state of the power demand curves is performed by using the fractal diagrams of Julia set.

• •

The fractal topology of the Julia sets related to the properties of power demand does not give a quantitative but qualitative geometry information as results of studying the different images.

• •

The topology of the Julia set reveals that the real and reactive powers studied for the load demand curve, which is related to an inductive load, belongs to the first quadrant of the complex plane.

• •

The density of folds obtained from the Julia set is related to the proximity of the demand for real and reactive powers to the boundary of the Mandelbrot set.

• •

From the electrical point of view, the densities of folds are related to: (a) power consumption through the different hours and (b) the combination of the real and reactive power magnitudes during the day.

The load demand curves studied in this article produce Julia sets with connected points because the points are within the Mandelbrot set. The fractals found with the Julia sets evidence that the load demand curves relate to the steady-state operation, which represents the zone of predictable values. After repeating the simulation for different real and reactive powers within the first quadrant of the complex plane, they produce the Julia set with symmetrically inverted fractal curves with respect to the origin.

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

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.