RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES
The complication of maintaining the power balance in the energy system under martial law conditions, ensuring the normal operating modes of power facility equipment, and maintaining the required quality of fossil energy resources used, among other factors, leads to an increase in harmful emissions i...
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General Energy Institute of the National Academy of Sciences of Ukraine
2026
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System Research in Energy| _version_ | 1871104453280530432 |
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| author | Kovtun, Svitlana Kuts, Volodymyr Kuts, Yurii Karaban, Artem |
| author_facet | Kovtun, Svitlana Kuts, Volodymyr Kuts, Yurii Karaban, Artem |
| author_institution_txt_mv | [
{
"author": "Svitlana Kovtun",
"institution": null
},
{
"author": "Volodymyr Kuts",
"institution": null
},
{
"author": "Yurii Kuts",
"institution": null
},
{
"author": "Artem Karaban",
"institution": null
}
] |
| author_sort | Kovtun, Svitlana |
| baseUrl_str | https://systemre.org/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T12:57:51Z |
| description | The complication of maintaining the power balance in the energy system under martial law conditions, ensuring the normal operating modes of power facility equipment, and maintaining the required quality of fossil energy resources used, among other factors, leads to an increase in harmful emissions into the atmosphere. Therefore, the development and improvement of environmental monitoring tools for energy facilities has become particularly relevant. In environmental monitoring systems, primary measurement information is generated by a system of sensors located around sources of pollutant emissions. One of the important tasks in the analysis of experimental data is the estimation of the distribution of pollutant concentrations in the vicinity of energy facilities using polar coordinates. To address this problem, modules for statistical analysis of circular data are included in the algorithmic and software support of monitoring systems. The task of verifying such modules requires the development of random angle generators capable of producing test samples of random angles with complex probability distributions. The article considers the methodological basis for solving problems of computer experiments with random angles, substantiates the use of multimodal probability distributions in problems of circular data analysis for atmospheric monitoring, proposes a random angle generator, and performs its verification using the example of generating a sample with a bimodal distribution. To evaluate the adequacy of the theoretical and empirical probability distributions of the random angle, the Kolmogorov criterion is used. The developed generator makes it possible to obtain samples of random angles with distributions defined analytically, by a table or graph (histogram), and can be used for testing algorithmic and software support in systems for statistical analysis of circular data. |
| doi_str_mv | 10.15407/srenergy2026.02.052 |
| first_indexed | 2026-05-30T01:00:11Z |
| format | Article |
| fulltext |
© Kovtun S., Kuts V., Kuts Yu., Karaban A., 2026
Це стаття відкритого доступу за ліцензією CC0 1.0 Universal
https://creativecommons.org/publicdomain/zero/1.0
52 ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86)
https://doi.org/10.15407/srenergy2026.02.052
UDC 519.24
Svitlana Kovtun1, Dr. Sci. (Engin.), Senior Researcher, https://orcid.org/0000-0002-6596-3460
Volodymyr Kuts1*, PhD (Engin.), https://orcid.org/0000-0002-1939-0032
Yurii Kuts¹,², Dr. Sci. (Engin.), Professor, https://orcid.org/0000-0002-8493-9474
Artem Karaban1, https://orcid.org/0009-0002-1430-1958
1General Energy Institute of NAS of Ukraine, 172 Antonovycha St., Kyiv, 03150, Ukraine;
2National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37
Beresteiskyi Ave., Kyiv, 03056, Ukraine
*Corresponding author: vladimir.kuts@live.com
_______________________________________________________________________________________
RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING
SYSTEM OF ENERGY FACILITIES
Abstract. The complication of maintaining the power balance in the energy system under martial law
conditions, ensuring the normal operating modes of power facility equipment, and maintaining the required
quality of fossil energy resources used, among other factors, leads to an increase in harmful emissions into
the atmosphere. Therefore, the development and improvement of environmental monitoring tools for energy
facilities has become particularly relevant. In environmental monitoring systems, primary measurement
information is generated by a system of sensors located around sources of pollutant emissions. One of the
important tasks in the analysis of experimental data is the estimation of the distribution of pollutant
concentrations in the vicinity of energy facilities using polar coordinates. To address this problem, modules
for statistical analysis of circular data are included in the algorithmic and software support of monitoring
systems. The task of verifying such modules requires the development of random angle generators capable
of producing test samples of random angles with complex probability distributions. The article considers
the methodological basis for solving problems of computer experiments with random angles, substantiates
the use of multimodal probability distributions in problems of circular data analysis for atmospheric
monitoring, proposes a random angle generator, and performs its verification using the example of
generating a sample with a bimodal distribution. To evaluate the adequacy of the theoretical and empirical
probability distributions of the random angle, the Kolmogorov criterion is used. The developed generator
makes it possible to obtain samples of random angles with distributions defined analytically, by a table or
graph (histogram), and can be used for testing algorithmic and software support in systems for statistical
analysis of circular data.
Keywords: random angle generator, computer experiment, trigonometric moments method, bimodal
distributions of random angles.
1. Introduction
In Ukraine, energy facilities are among the potentially hazardous sources of atmospheric pollution [1, 2].
In recent years, under the conditions of martial law, the issues of environmental safety and environmental
monitoring of energy facilities have become more acute and require special attention due to the increasing
difficulty of maintaining the power balance in the energy system, the complexity of ensuring normal operating
modes of energy facility equipment, and the need to maintain the appropriate quality of fossil energy resources,
among other factors. This leads to an increase in harmful pollutants in the combustion products of thermal
power plants. Therefore, the development and improvement of environmental monitoring tools for energy
facilities has become particularly relevant. A number of publications are devoted to various theoretical and
practical issues related to the development and implementation of environmental monitoring systems for
energy facilities in Ukraine, including [3–10].
An important component of environmental monitoring systems for energy facilities is the subsystem for
analyzing the spatial distribution of emissions of harmful substances into the atmosphere generated during the
https://orcid.org/0000-0002-6596-3460
https://orcid.org/0000-0002-1939-0032
https://orcid.org/0000-0002-8493-9474
ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86) 53
combustion of fossil fuels and the operation of auxiliary equipment at energy facilities. Such analysis makes
it possible not only to determine the directions most vulnerable from the standpoint of environmental safety
with respect to the spread of harmful substances in the air from the pollution source, but also to analyze the
dynamics of their accumulation and concentration, to make forecasts regarding possible exceedances of
permissible concentration limits of pollutants in areas adjacent to energy facilities, and to detect possible
unauthorized emissions of harmful substances into the atmosphere.
The processes of formation and dispersion of harmful substances in the atmosphere are probabilistic in
nature and depend on many factors, such as the quality of fossil fuels, their combustion regimes, wind patterns,
and the operating modes of auxiliary equipment, among others. In environmental monitoring systems, primary
measurement information about the state of the environment is generated by a system of sensors located around
sources of pollutant emissions. One of the key issues in the analysis of experimental data obtained from such
sensors is the estimation of the distribution of pollutant concentrations in the vicinity of energy facilities in
polar coordinates. This problem can be solved using methods of mathematical statistics designed for the
analysis of results of angular observations [11–13]. Therefore, the algorithmic and software support of
environmental monitoring systems should include modules for the statistical analysis of experimental data
with circular distributions. Certain aspects of the practical application of statistical methods for circular data
analysis are considered in [14]; a probabilistic model of a random angle (RA) was developed in [15]; and
studies in [16] investigated the application of the method of sample trigonometric moments to problems of
approximating probability distributions of measurement results of signal phase shifts.
The application of statistical methods for the analysis of circular data in problems of processing sensor
data governed by circular distributions has its own specific features and requires both additional analytical
studies and the development of appropriate algorithmic and software support, as well as its verification. It
should be noted that the concentration density distributions of harmful emissions around energy facilities, due
to the influence of various factors, have a complex multimodal structure and are poorly approximated by
known standard unimodal circular distributions. Therefore, solving the problem of verifying modules for the
statistical analysis of circular data requires the development of RA generators capable of generating test
sequences of RAs with complex multimodal probability distributions. In [8], the idea of using the inverse
function method to generate unimodal circular data for testing phase meters was explored. This idea requires
further development and confirmation of the possibility of its use for creating RA generators with multimodal
probability distributions.
The aim of this paper is to develop a software-based random angle generator with multimodal
distributions and to verify its performance.
2. Methodological Basis for Solving Computer Experiment Tasks with Random Angles in
Atmospheric Monitoring Systems of Energy Facilities
2.1. Tasks and Theoretical Foundations of Computer Experiments for Generating RAs
When performing atmospheric monitoring around energy facilities with pollution sources, primary
information about pollutants is obtained from a set of sensors 𝐶1–𝐶𝑛 arranged in a circle (perimeter) at a
considerable distance from the pollution source(s). An example of such a sensor system arrangement is shown
in Fig. 1a.
The measurement information ( ) 1jg , j , n g= , where g is the dimensionality of the parameter g,
from the sensors is transmitted to the data collection and processing unit. If this measurement information is
normalized according to the expression
1
n
*
j j j
j
g g g
=
= , a new set ( )1*
jg , j , n= is obtained, for which
1
1
1*
j
j
g
=
= . This provides a formal basis to consider the elements of
*
jg as elements of a circular histogram
(Fig. 1b) [11], obtained from the observation of random angles )0 2, (or ), − ), and to apply
54 ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86)
circular probability distributions for their analysis [11–13]. These distributions have their specific measures of
central tendency—circular mean, mode, and median—as well as measures of dispersion—circular variance
and circular standard deviation.
a b
Fig. 1. Example of the arrangement of atmospheric monitoring sensors around a pollutant emission source (a) and the
circular histogram in polar coordinates (b)
The inclusion of modules for processing circular data in the algorithmic and software support of air
monitoring systems, in turn, requires informational support from computer experiments with RAs. A
characteristic feature of RAs in atmospheric monitoring systems is that, due to various factors, the actual
circular distributions of pollutant concentrations often exhibit a multimodal structure. This means that the
corresponding probability density of RAs shows two or more peaks (modes), indicating data heterogeneity and
the presence of different subgroups within the dataset. Such heterogeneity may, for example, arise from
measurements obtained from different pollution sources or from changes in sampling conditions, such as shifts
in wind direction.
A computer experiment for generating RAs, including those with multimodal probability distributions,
is aimed at addressing the following main tasks: generating RA samples with a specified probability
distribution, determining the probability distribution of RAs from sample data, and evaluating the adequacy of
empirical and theoretical distributions (Fig. 2).
Fig. 2. Tasks and Theoretical Foundations of Computer Experiments for RA Generation
The cyclical nature of angular data makes it impossible to apply traditional methods used in the statistical
analysis of linear random variables (RVs). Since methods of circular statistics have not yet gained widespread
use in domestic scientific literature, the theoretical foundations necessary for understanding the proposed
methodology of computer experiments with RAs are briefly outlined below.
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2.2. Justification for Using the von Mises Circular Distribution in the Analysis of Circular Data
for Atmospheric Monitoring
In the statistical analysis of circular observation data, various circular probability distributions are used,
including the cardioid, von Mises, triangular, and wrapped distributions [11–13]. In [17], comprehensive
information is provided on all known wrapped distributions to date ‒ 45 distributions for continuous RAs and
10 distributions for discrete RAs. For each distribution, the main circular characteristics are indicated. In this
work, we used the von Mises circular distribution, which has certain advantages compared to others, including
a convenient analytical representation that simplifies statistical analysis and a satisfactory approximation of
the wrapped normal distribution. The wrapped normal distribution has the property that the convolution of two
wrapped normal distributions is itself a wrapped normal distribution [11–13]. These features of the von Mises
distribution provided the basis for its use in constructing RA generators with multimodal distributions.
The von Mises probability density function for a random angle is defined as:
( )
( )
( ) )
0
1
0 2 0
2
Мp | ,k exp k cos , , , , k
I k
= −
, (2.1)
where is the circular mean direction of the random RA, k is the concentration parameter of the distribution
around , θ is the value of the RA, and ( )0I k is the modified Bessel function of the first kind of order zero.
The von Mises circular distribution is unimodal and symmetric with respect to the point ( )mod2 within the
interval ( ), − + , it has two inflection points. The cumulative distribution function (CDF) of the von
Mises distribution does not have a simple analytical expression and is represented as an infinite series:
( )
( )
( )
( )
10
sin1 2
| , , , 0
2
М q
q
q
F k I k k
I k q
=
−
= +
. (2.2)
A multimodal circular probability density will be defined as a weighted sum of independent RAs
1s , s , S = [18] with a unimodal von Mises circular distribution (2.1).
( ) ( ) (
1
0 1
S
s М s s s
s
p A p | ,k , A ,
=
= , (2.3)
where 𝑆 is the number of independent RAs, sA is the weight coefficient, with
1
1
S
s
s
A
=
= . A common case of a
multimodal distribution is the bimodal distribution, which in general form can be represented as:
( ) ( ) ( ) ( )1 1 1 1 2 21М Мp A p | ,k A p | ,k = + − . (2.4)
The density of distribution (2.4) has five parameters: 1 2 1 2 1, , k , k , A . The parameter 1A is also called
the “mixing parameter,” and it represents the probability of occurrence of the RA 1 values in the mixture.
In bimodal distributions, the highest frequency of RA observations occurs at two distinct values. In the
case of two unequal modes, the larger is referred to as the primary mode, and the smaller as the secondary
mode. The antimode is the least frequent angle value between the modes. Bimodal distributions have a special
property: unlike unimodal distributions, the mean value can provide a more reliable estimate of the sample
than the median [19].
Bimodal distributions are encountered in the study of natural phenomena and in the analysis of various
human activities, including the study of geysers, traffic patterns, and daily water consumption [20].
56 ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86)
2.3. The Inverse Function Method as a Way to Generate RA Samples
The task of modeling vectors of sampled RA values with circular distributions using the inverse function
method can be generally formulated as follows [16]. A probability density function ( )f of a continuous RA
with domain )0 2, is given. It is necessary to generate a sample of size N1 of this RA, that is, a
sample ) 1 0 2i i, i , N , , = = .
The essence of the inverse function method for generating RA samples is that a sample of RA with a
probability distribution function ( )F , defined on the interval )0 2, , is obtained through a functional
transformation of a random variable with a uniform distribution over the interval 0 1a , of the form:
( )1F a−= , (2.5)
where ( )1F a−
– is the inverse function of ( )F function (it is assumed that ( )1F a−
exists).
Generating a sample using the inverse function method involves performing the following sequence
of steps:
1. Specify the probability density function of the RA , set the parameters of the distribution law,
define the sample size N , and determine the number of n <<N approximation points
(corresponding to the number of sensors in the monitoring system).
2. Determine the cumulative distribution function (CDF) of the RA as two sets of discrete,
corresponding angle values. )( )1 0 2j j, j , n, , = and the corresponding values of the
cumulative distribution function (CDF) ( ) )( )( )1 0 2j jF , , j , n, , = = .
3. Form the vector ( )1 0 1i ia , i ,N , a , = = of pseudorandom numbers with a uniform distribution
over the interval [0, 1].
4. Obtain the set of values of the inverse function ( ) ( ) ( )1 1 0 1i iF a , a a , i , N , a ,− = = . This step is
advisable to perform using approximate numerical methods.
5. Obtain a vector of size N1 of sampled RA values with distribution ( )F .
6. Construct the histogram and verify the agreement between the theoretical and experimental
distributions.
2.4. Estimation of RA Probability Density Using the Method of Sample Trigonometric Moments
In mathematical statistics, various methods are used to approximate the distributions of random
variables based on experimental data. Well-known approaches include approximating RV distributions using
Pearson curves [21] and Johnson curves [22]. In [16], the advantages of the trigonometric moments method
for approximating unimodal RA distributions were highlighted. In this paper, the effectiveness of the sample
trigonometric moments method for approximating multimodal RA distributions is examined.
The trigonometric moments method is based on an important property of the characteristic function of
a RA: it uniquely determines the probability density function of the RA [11‒13].
( ) ( ) )
1
0 , 0, 2
2π
iq
q
q
p f e
−
=−
= , (2.6)
where 1i = − , ( )0qf – the trigonometric moment of order 𝑞of a RA relative to the zero direction (i.e., the
central trigonometric moments of the RA)
Equation (2.6) represents the expansion of the function ( )p into a Fourier series, whose coefficients
are the trigonometric moments of order q.
The characteristic function of a RA , is generally defined as a complex-valued sequence of its
trigonometric moments of order 0 1 2q , , ,...= relative to the zero direction.
ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86) 57
( ) ( )
2
0
, 0 1 2 iq
qf exp in e dF q , , ,...
= = = M , (2.7)
where M – mathematical expectation operator.
The trigonometric moment qf is expressed in Cartesian and polar coordinate systems as:
( )q q q q qf a ib exp i= + = , (2.8)
where q qa , b is a sequence of real numbers (the sequence of cosine and sine moments) is defined as:
( ) ( ) ( ) ( ) ( ) ( )
2 2
0 0
cos cos , sin sinq qa M q q dF b M q q dF
= = = = , (2.9)
and the real numbers q qa , b and ,q q are related to each other by the following relations:
( )2 2 ,q q q q qa b Arg f = + = , (2.10)
As a result of performing an angular measurement experiment with RAs having a probability density
function ( )p , a sample of size 𝑁 is obtained: ( ) )1,... ,... , 0, 2i N i . For this sample, the sample
trigonometric moments of order 𝑞 relative to an arbitrary direction )0, 2 are available for calculation.
ˆ ( )( )
1
1ˆ ˆˆ ˆ( ) ( ) ( ) ( ) qi
N
imiq
q q q q
i
f e a ib r e
N
−
=
= = + = , (2.11)
where
( ) ( ) ( ) ( )
1 1
1 1ˆˆ cos , sin
N N
q i q i
i i
a q b q
N N= =
= − = − . (2.12)
The sample trigonometric function is defined as a complex-valued sequence of sample trigonometric
moments ( )( )0 0 1 2qf̂ , q , , , ....= .
By replacing in equation (2.6) the functions ( )0qf with the corresponding sample central trigonometric
moments ( )0qf̂ , and limiting the series to terms with indices 1 1Q , Q− we obtain an expression suitable for
determining an estimate of the empirical probability density function of the RA,
( ) ( ) ( ) )
1 1
1 1
1 1ˆˆ ˆ ˆ0 1 2 cos sin , 0, 2
2π 2π
Q Q
iq
q q q
q Q q
p f e a q a q−
=− =
= = + +
. (2.13)
Equation (2.13) defines the algorithm for calculating the empirical probability density function of the
RA based on the results of their observations or on the angular histogram obtained in a simulation experiment.
2.5. Assessment of the Adequacy of Theoretical and Empirical Probability Distributions of RA
At the final stage of the experiment for generating a sample of pseudorandom angles, the adequacy of
the empirical probability distribution of the random angle ( )p̂ , is evaluated. This is carried out by
determining the absolute error, which is calculated as the difference between the specified probability density
function (2.3) and the empirical one (2.13) (i.e., approximated from experimental data or from the histogram
obtained during the computer experiment).
( ) ( ) ( ) )ˆ , 0, 2p p p = − . (2.15)
For the function ( )p the following statistics are available: the maximum and minimum values, the
mean value, the variance, and the standard deviation.
58 ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86)
3. Implementation of the Computer Experiment for RA Generation and Analysis of the Obtained
Results
3.1. General Scheme of the Computer Experiment
The procedure for conducting the experiment includes the following set of interrelated stages [8]:
1. Formulation of the experimental task and initial data, including the determination of the form of the
probability distribution of the RA, its parameters, the sample size, and the number of sensors.
2. Generation of an RA sample using the inverse function method according to Section 2.2.
3. Estimation of the empirical distribution based on the obtained RA sample.
4. Assessment of the adequacy of the theoretical and empirical RA distributions.
5. Visualization and analysis of the obtained results.
3.2. Formulation of the Random Angle Sample Generation Problem and Its Implementation
According to the general methodology for conducting a computer experiment for RA generation
discussed in Section 3.1, a software implementation of a pseudorandom angle generator with bimodal
distributions was developed. The generator was tested by generating an RA sample with a specified distribution
and comparing it with the empirical distribution obtained from the RA sample. The research task was
formulated as follows.
It is necessary, using the inverse function method, to develop a software RA generator with a bimodal
distribution in the MATLAB environment [18] and to study the generation process by obtaining a sample of
RAs with a bimodal distribution formed by two von Mises distributions. The sample size is N=1000, and the
parameters of the bimodal distribution are as follows: circular mean directions – 1 0,5 = rad and 2 3 =
rad; concentration parameters – 1 5k = і 1 3k = ; and mixing parameter – 1 0 3A ,= . The process of generating
the RA sample should be illustrated at all stages by presenting graphs of the theoretical and empirical
probability density functions of the RA, the direct and inverse cumulative distribution functions, a graphical
representation of the sample, and both linear and circular histograms of the obtained RA sample.
3.3. Results of Testing the Software Implementation of the Pseudorandom Angle Generator
Below are the results of the simulation experiment for generating a RA sample based on the specified
input data. Figure 3 shows the graphs of the given (theoretical) distributions: (a) the densities of the two von
Mises distributions obtained according to (2.1); (b) the multimodal probability density function according to
(2.4).
Fig. 3. Graphs of the von Mises probability densities for the RA components (a) and the multimodal circular
probability density function (b)
The modes of the bimodal distribution coincide with the modes of the unimodal distributions, with the
primary mode at 2 3 = read, the secondary mode – 1 0,5 = rad, and the antimode at 1.54a rad.
ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86) 59
Figure 4a shows the cumulative distribution function (CDF) of the von Mises distribution ( )F , and
Figure 4b shows its inverse function ( )1F a−
. The function ( )F was obtained by integrating the probability
density function ( )p , using the trapezoidal method.
Fig. 4. Graphs of the cumulative multimodal distribution ( )F (a) and its inverse distribution ( )1F a−
(b)
Figure 4b illustrates the idea of generating a RA sample using the inverse function method, showing the
determination of a single sample element )0, 2j , я corresponding to a pseudorandom number 0, 1ja
with a uniform distribution.
The entire RA sample of size N=1000 is shown in Figure 5a, and the corresponding linear histogram is
shown in Figure 5b. The histogram was constructed by uniformly dividing the entire range of angle values into
20 intervals, each with a width of 0.1 0.314 rad. A distinctive feature of the software implementation of the
RA generator is that the sample generation itself is carried out using the spline function, which is applied with
a nonstandard ordering of arguments as follows:
- the vector of probability function values ( )F of size n;
- the vector of angle values for which the function ( )F of size n is evaluated;
- the vector of pseudorandom numbers with a uniform distribution of size N >> n.
Such an implementation of the inverse function method does not require the analytical determination of
the inverse function for generating the RA sample.
Fig. 5. Sample of random angles with a bimodal distribution (a) and the corresponding linear histogram (b)
A more natural and illustrative way to represent circular data is a circular histogram (polar wedge
diagram) [5]. Such a diagram for the obtained sample is shown in Fig. 6a.
In this diagram, the observation results are represented by sectors with a common vertex at point O. The
angles of the sectors correspond to the selected class intervals of size ( 0.1 ), while the radius vectors are equal
to the heights of the bars in the linear histogram. In the diagram shown in Fig. 6a, the angle values are plotted
counterclockwise from the Ox axis.
60 ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86)
Fig 6. Circular histogram of the RA sample (a) and its representation as a 3D graph (b)
At the final stage of the experiment, the correspondence between the distribution of the generated sample
and the theoretical distribution was verified. For this purpose, the empirical probability density of the bimodal
RA was first determined from the obtained angle sample according to equation (2.13). The resulting graph of
this density is shown in Fig. 7a.
The graph of the absolute error, determined according to equation (2.15) as the difference between the
specified (theoretical, a priori) and the empirical (a posteriori) probability density functions of the RA, is shown
in Fig. 7b. This graph demonstrates that the absolute value of the error does not exceed 0.005 over the entire
interval of angles )0, 2 , which indicates a good agreement between the theoretical and experimental bimodal
RA distributions.
In addition, the adequacy of the theoretical and experimental distributions was evaluated using the
Kolmogorov criterion according to the methodology described in [8]. The tested hypotheses are formulated in
terms of probability distribution functions. The null and alternative hypotheses regarding the distributions are
defined as follows:
( ) ( )
( ) ( )
0 : ,
: ,
e
a e
H F F
H F F
=
where ( )eF – the empirical distribution obtained from the generated RA sample.
Fig. 7. Graph of the empirical probability density function ( )p̂ and its absolute error p
For the performed simulation experiment with the bimodal distribution, the following results were
obtained:
- the maximum value of the absolute difference between the experimental ( )eF and theoretical ( )F
probability density functions of the RA was 0.00528D = ;
- the critical value of the probability parameter in the Kolmogorov test was –
0.00528 1000 0.167D N = = ;
- for the chosen significance level 0.05 = the critical value of the parameter о 1.358 = was
determined;
Overall, it was concluded that the distributions are adequate, since the inequality о0.167 1.358 =
is satisfied.
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3.4. Discussion of the Obtained Results
Modern statistical methods for processing circular data have significant potential for use in the energy
sector, particularly in environmental monitoring systems for energy facilities to determine the spatial
distribution of harmful substances in the near-surface layers of the atmosphere around these objects. Such
systems require the verification of their algorithmic and software components, which in turn involves the
development of RA generators capable of producing test sequences with complex probability distributions.
The methodology for generating RA samples proposed in this paper is based on a combination of the
inverse function method and the trigonometric moments method, known from the section of mathematical
statistics devoted to the analysis of circular data. The software generator developed on this basis is capable of
producing high-quality samples over a wide range of RA sample sizes. Its applicability for generating samples
with multimodal angular distributions has been experimentally demonstrated. The effectiveness of the
generator was verified using the task of generating an RA sample with a bimodal distribution, defined as a
weighted sum of two independent RAs with von Mises distributions. In the conducted experiment, the absolute
error, calculated as the difference between the theoretical and empirical probability density functions of the
RA, did not exceed:
for a sample of size N=1000 the value was 0.005p , and for a sample of size N=100 it was 0.03p .
In the example presented, the trigonometric moments method plays a supplementary role and is used
only to assess the adequacy of the RA generator. However, this method can also have independent value in
tasks involving the evaluation of angular measurement results, particularly as a method for constructing
interval estimates of angular observations when the experimental data size is limited, making it impossible to
perform formal hypothesis testing on the distributions of circular data.
4. Conclusions
The task of verifying the algorithmic and software components of systems for processing circular data
requires the development of RA generators capable of producing test samples with complex probability
distributions. Such a need particularly arises in atmospheric monitoring systems around energy facilities,
which allow determining the spatial distributions of harmful substances from pollution sources. This paper
examines the methodological basis for generating RA samples with multimodal circular distributions, which
is based on a combination of the inverse function method for RA sample generation and the sample
trigonometric moments method. This approach allows obtaining empirical circular distributions from sample
data and verifying the adequacy of the generated data.
The Kolmogorov criterion was adapted to evaluate the quality of the RA generator by testing the
adequacy of the specified theoretical distribution against the empirical probability density distribution obtained
from the generated data.
A software implementation of an RA generator with a multimodal distribution has been developed. The
generator was experimentally tested by generating samples with a bimodal distribution formed from two RAs
with von Mises distributions. The developed software implementation utilizes features of the MATLAB
environment, which enabled the realization of the inverse function method without requiring its analytical
determination.
The developed RA software generator can be valuable both for conducting computer experiments with
random angular data and for integration into software designed to process circularly distributed data.
Further research will focus on investigating methods for estimating sample statistics of bimodal
distributions.
References
1. Zaporozhets, A., Babak, V., Sverdlova, A., Shcherbak, L., & Kuts, Yu. (2022) Review of the state of air pollution
by energy objects in Ukraine. System Research in Energy, № 1-2(71), 42‒52.
https://doi.org/10.15407/srenergy2022.02.042
2. Zaporozhets, A., Babak, V., Isaienko, V., & Babikova, K. (2020). Analysis of the air pollution monitoring system
in Ukraine. Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, 298 (pp.85–110).
Springer, Cham. https//doi.org/10.1007/978-3-030-48583-2_6
http://dx.doi.org/10.15407/srenergy2022.02.042
https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=8xMuKuoAAAAJ&citation_for_view=8xMuKuoAAAAJ:eq2jaN3J8jMC
https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=8xMuKuoAAAAJ&citation_for_view=8xMuKuoAAAAJ:eq2jaN3J8jMC
62 ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86)
3. Zaporozhets, A., Redko, O., Babak, V., Eremenko, V., & Mokiychuk, V. (2018). Method of indirect measurement
of oxygen concentration in the air. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, 5, 105–114.
https://doi.org/10.292002/nvngu/2018-5/14
4. Isaienko, V., Zaporozhets, A., Babikova, K. Gulevets, D., & Savchenko, S. (2020). Review of methods and means
of monitoring the air pollution. Proceedings of the National Aviation University, 80(3), 61–70.
https://doi.org/10.18372/2306-1472.80.14275
5. Zaporozhets, A. (2020) Overview of quadrocopters for energy and ecological monitoring. In V. Babak, V. Isaienko,
A. Zaporozhets (Eds.), Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, 298
(pp. 15–36). Springer, Cham. https:/doi.org/10.1007/978-3-030-48583-2_2
6. Zaporozhets, A.O., & Khaidurov, V.V. (2020). Mathematical models of inverse problems for finding the main
characteristics of air pollution sources. Water, Air, & Soil Pollution, 231, 563. https://doi.org/10.1007/s11270-020-
04933-z
7. Zaporozhets, A.O. (2021). Correlation analysis between the components of energy balance and pollutant emissions.
Water, Air, & Soil Pollution, 232, 114. https://doi.org/10.1007/s11270-021-05048-9
8. Babak, V.P., Babak, S.V., Eremenko, V.S. Kuts, Yu.V., Myslovych, M.V., Scherbak, L.M., & Zaporozhets, A.O.
(2021). Models and Measures in Measurements and Monitoring, 266. Springer International Publishing.
https://doi.org/10.1007/978-3-030-70783-5
9. Zaporozhets, A., & Kuts, Yu., (2025). Identification of Air Pollution Sources. Noise signals, Modelling and
Analyses, 567 (pp. 197–222). doi.org/10.1007/978-3-031-71093-3_7
10. Babak, V., Babak, S., Eremenko, V., Kuts, Yu., & Zaporozhets, A. (2025). Information-Measuring Systems. Theory
and Application. Studies in Systems, Decision and Control, 592. Springer, Cham. doi.org/10.1007/978-3-031-
89406-0
11. Mardia, K. (2000). Directional Statistics. John Willey & Sons.
12. Fisher, N. (2000). Statistical Analysis of Circular Data. Cambridge University Press.
13. Jammalamadaka, S. Rao. (2001). Topics in circular statistics. Singapore: World Scientific Publishing Co. Pte. Ltd.
14. Kuts, Yu., & Shcherbak, L. (2009). Statistical phase measurement. Ternopil [in Ukrainian].
15. Mei, Z., Kuts, Y., Kochan, O., Lysenko, I., Levchenko, O., & Vlakh-Vyhrynovska, H. (2022). Using Signal Phase
in Computerized Systems of Non-destructive Testing. Measurement Science Review, 22(1), 32–43.
https://doi.org/10.2478/msr-2022-0004
16. Kuts, Yu. (2015). A study of the method of selective trigonometric moments in problems of approximating angular
data distributions. Information Processing Systems, 6(131), 111–115 [in Ukrainian].
17. Bell, W., & Nadarajah, S. (2024). A Review of Wrapped Distributions for Circular Data. Mathematics, 12(16), 2440.
https://doi.org/10.3390/math12162440
18. Multimodal distribution. Retrieved March 12, 2026, from https://en.wikipedia.org/wiki/Multimodal_distribution
19. Mosteller, F., & Tukey, J. W. (1977). Data Analysis and Regression: A Second Course in Statistics. Reading, Mass:
Addison-Wesley. ISBN 0-201-04854-X.
20. Baptista, I. S.C., Dash, S., Arsh, A. M., Kandavalli, V., Scandolo, C. M., Sanders, B. C., & Ribeiro, A. S. (2025).
Bimodality in E. coli gene expression: Sources and robustness to genome-wide stresses. PLOS Computational
Biology, 21(2), e1012817. https://doi.org/10.1371/journal.pcbi.1012817
21. Pearson, E.S., & Kendall, M. (1987). Studies in the History of Statistics and Probability. London: Oxford, University
Press.
22. Hahn, G. J., & Shapiro, S. S. (1994). Statistical Models in Engineering. Hoboken, NJ: Wiley-Interscience.
23. Matrices and Arrays – MATLAB & Simulink. Retrieved March 12, 2026, from www.mathworks.com.
ГЕНЕРАТОР ВИПАДКОВИХ КУТІВ У СИСТЕМІ
МОНІТОРИНГУ АТМОСФЕРИ НАВКОЛО ОБ’ЄКТІВ
ЕНЕРГЕТИКИ
Світлана Ковтун1, д-р техн. наук, ст. досл., https://orcid.org/0000-0002-6596-3460
Володимир Куц1*, канд. техн. наук., https://orcid.org/0000-0002-1939-0032
Юрій Куц1,2, д-р техн. наук, професор, https://orcid.org/0000-0002-8493-9474
Артем Карабан1, https://orcid.org/0009-0002-1430-1958
1Інститут загальної енергетики НАН України, вул. Антоновича, 172, Київ, 03150, Україна;
2Національний технічний університет України «Київський політехнічний інститут імені
Ігоря Сікорського», Берестейський просп., 37, Київ, 03056, Україна
*Автор-кореспондент: vladimir.kuts@live.com
Анотація. В умовах воєнного стану ускладнюється підтримання балансу потужності в
енергосистемі, штатних режимів роботи обладнання об’єктів енергетики, дотримання
https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=8xMuKuoAAAAJ&cstart=20&pagesize=80&citation_for_view=8xMuKuoAAAAJ:g5m5HwL7SMYC
https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=8xMuKuoAAAAJ&cstart=20&pagesize=80&citation_for_view=8xMuKuoAAAAJ:g5m5HwL7SMYC
https://doi.org/10.292002/nvngu/2018-5/14
https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=8xMuKuoAAAAJ&cstart=20&pagesize=80&citation_for_view=8xMuKuoAAAAJ:J-pR_7NvFogC
https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=8xMuKuoAAAAJ&cstart=20&pagesize=80&citation_for_view=8xMuKuoAAAAJ:J-pR_7NvFogC
https://link.springer.com/book/10.1007/978-3-031-44347-3#author-1-4
https://scholar.google.com/citations?view_op=view_citation&hl=uk&user=8xMuKuoAAAAJ&citation_for_view=8xMuKuoAAAAJ:KUbvn5osdkgC
https://link.springer.com/chapter/10.1007/978-3-031-71093-3_7
https://doi.org/10.3390/math12162440
https://en.wikipedia.org/wiki/ISBN_(identifier)
https://en.wikipedia.org/wiki/Special:BookSources/0-201-04854-X
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11825099
https://doi.org/10.1371/journal.pcbi.1012817
https://www.amazon.com/Gerald-J-Hahn/e/B001HMLFWM/ref=dp_byline_cont_book_1
https://www.amazon.com/Samuel-S-Shapiro/e/B002884W5Q/ref=dp_byline_cont_book_2
https://en.wikipedia.org/wiki/Hoboken,_New_Jersey
ISSN 2786-7633. Системні дослідження в енергетиці. 2026. 2(86) 63
відповідної якості використовуваних викопних енергоресурсів тощо, що призводить до підвищення
викидів шкідливих речовин в атмосферу. Тому питання розвитку та удосконалення засобів
моніторингу довкілля об’єктів енергетики набуває особливої актуальності. В системах
екологічного моніторингу первинна вимірювальна інформація формується системою сенсорів,
розміщених навколо джерел забруднюючих речовин. Одним з важливих завдань аналізу
експериментальних даних є оцінювання розподілу концентрації забруднюючих речовин в околі
об’єктів енергетики у полярних координатах. Для його вирішення до складу алгоритмічно-
програмного забезпечення систем моніторингу включають модулі статистичного аналізу кругових
даних. Завдання верифікації таких модулів потребує створення генераторів випадкових кутів, які
були б спроможні формувати тестові вибірки випадкових кутів зі складними розподілами
ймовірностей. У статті розглянуто методологічний базис розв’язання завдань комп’ютерних
експериментів з випадковими кутами, обґрунтовано використання полімодальних розподілів
ймовірностей в задачах аналізу кругових даних моніторингу атмосфери, запропоновано генератор
випадкових кутів та виконано його верифікацію на прикладі генерування вибірки з бімодальним
розподілом. Для оцінювання адекватності теоретичного та емпіричного розподілів ймовірностей
випадкового кута використано критерій Колмогорова. Розроблений генератор дає змогу
отримувати вибірки випадкових кутів з розподілами, які задані аналітично, таблично чи графічно
(гістограмою), та може бути використаний для тестування алоритмічно-програмного
забезпечення в системах статистичного аналізу кругових даних.
Ключові слова: генератор випадкових кутів, комп’ютерний експеримент, метод тригонометричних
моментів, бімодальні розподіли випадкових кутів.
Дата першого надходження статті до журналу: 16.04.2026
Дата прийняття статті до друку після рецензування: 27.05.2026
Дата публікації (оприлюднення): 30.05.2026
|
| id | systemreorg-article-958 |
| institution | System Research in Energy |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:24:20Z |
| publishDate | 2026 |
| publisher | General Energy Institute of the National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | systemreorg/f2/2427b023e17067208816caa11e7a1ff2.pdf |
| spelling | systemreorg-article-9582026-07-18T12:57:51Z RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES Генератор випадкових кутів у системі моніторингу атмосфери навколо об’єктів енергетики Kovtun, Svitlana Kuts, Volodymyr Kuts, Yurii Karaban, Artem random angle generator, computer experiment, trigonometric moments method, bimodal distributions of random angles. : генератор випадкових кутів, комп’ютерний експеримент, метод тригонометричних моментів, бімодальні розподіли випадкових кутів. The complication of maintaining the power balance in the energy system under martial law conditions, ensuring the normal operating modes of power facility equipment, and maintaining the required quality of fossil energy resources used, among other factors, leads to an increase in harmful emissions into the atmosphere. Therefore, the development and improvement of environmental monitoring tools for energy facilities has become particularly relevant. In environmental monitoring systems, primary measurement information is generated by a system of sensors located around sources of pollutant emissions. One of the important tasks in the analysis of experimental data is the estimation of the distribution of pollutant concentrations in the vicinity of energy facilities using polar coordinates. To address this problem, modules for statistical analysis of circular data are included in the algorithmic and software support of monitoring systems. The task of verifying such modules requires the development of random angle generators capable of producing test samples of random angles with complex probability distributions. The article considers the methodological basis for solving problems of computer experiments with random angles, substantiates the use of multimodal probability distributions in problems of circular data analysis for atmospheric monitoring, proposes a random angle generator, and performs its verification using the example of generating a sample with a bimodal distribution. To evaluate the adequacy of the theoretical and empirical probability distributions of the random angle, the Kolmogorov criterion is used. The developed generator makes it possible to obtain samples of random angles with distributions defined analytically, by a table or graph (histogram), and can be used for testing algorithmic and software support in systems for statistical analysis of circular data. В умовах воєнного стану ускладнюється підтримання балансу потужності в енергосистемі, штатних режимів роботи обладнання об’єктів енергетики, дотримання відповідної якості використовуваних викопних енергоресурсів тощо, що призводить до підвищення викидів шкідливих речовин в атмосферу. Тому питання розвитку та удосконалення засобів моніторингу довкілля об’єктів енергетики набуває особливої актуальності. В системах екологічного моніторингу первинна вимірювальна інформація формується системою сенсорів, розміщених навколо джерел забруднюючих речовин. Одним з важливих завдань аналізу експериментальних даних є оцінювання розподілу концентрації забруднюючих речовин в околі об’єктів енергетики у полярних координатах. Для його вирішення до складу алгоритмічно-програмного забезпечення систем моніторингу включають модулі статистичного аналізу кругових даних. Завдання верифікації таких модулів потребує створення генераторів випадкових кутів, які були б спроможні формувати тестові вибірки випадкових кутів зі складними розподілами ймовірностей. У статті розглянуто методологічний базис розв’язання завдань комп’ютерних експериментів з випадковими кутами, обґрунтовано використання полімодальних розподілів ймовірностей в задачах аналізу кругових даних моніторингу атмосфери, запропоновано генератор випадкових кутів та виконано його верифікацію на прикладі генерування вибірки з бімодальним розподілом. Для оцінювання адекватності теоретичного та емпіричного розподілів ймовірностей випадкового кута використано критерій Колмогорова. Розроблений генератор дає змогу отримувати вибірки випадкових кутів з розподілами, які задані аналітично, таблично чи графічно (гістограмою), та може бути використаний для тестування алоритмічно-програмного забезпечення в системах статистичного аналізу кругових даних. General Energy Institute of the National Academy of Sciences of Ukraine 2026-05-30 Article Article application/pdf https://systemre.org/index.php/journal/article/view/958 10.15407/srenergy2026.02.052 System Research in Energy; No. 2 (86) (2026): System Research in Energy; 52-63 Системні дослідження в енергетиці; № 2 (86) (2026): Системні дослідження в енергетиці; 52-63 2786-7102 2786-7633 en https://systemre.org/index.php/journal/article/view/958/843 Copyright (c) 2026 Svitlana Kovtun, Volodymyr Kuts, Yurii Kuts, Artem Karaban https://creativecommons.org/publicdomain/zero/1.0 |
| spellingShingle | random angle generator computer experiment trigonometric moments method bimodal distributions of random angles. Kovtun, Svitlana Kuts, Volodymyr Kuts, Yurii Karaban, Artem RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES |
| title | RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES |
| title_alt | Генератор випадкових кутів у системі моніторингу атмосфери навколо об’єктів енергетики |
| title_full | RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES |
| title_fullStr | RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES |
| title_full_unstemmed | RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES |
| title_short | RANDOM ANGLE GENERATOR IN THE ATMOSPHERIC MONITORING SYSTEM OF ENERGY FACILITIES |
| title_sort | random angle generator in the atmospheric monitoring system of energy facilities |
| topic | random angle generator computer experiment trigonometric moments method bimodal distributions of random angles. |
| topic_facet | random angle generator computer experiment trigonometric moments method bimodal distributions of random angles. : генератор випадкових кутів комп’ютерний експеримент метод тригонометричних моментів бімодальні розподіли випадкових кутів. |
| url | https://systemre.org/index.php/journal/article/view/958 |
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