=Paper=
{{Paper
|id=Vol-3942/S_11_Javorskyj
|storemode=property
|title=
Identification and Analysis of the Structure of Rhythmic Variability of Electricity Consumption Time Series for Estimating and Forecasting Energy Flows
|pdfUrl=https://ceur-ws.org/Vol-3942/S_11_Javorskyj.pdf
|volume=Vol-3942
|authors=Ihor Javorskyj,Valentyna Pleskach,Roman Yuzefovych,Oleh Lychak
}}
==
Identification and Analysis of the Structure of Rhythmic Variability of Electricity Consumption Time Series for Estimating and Forecasting Energy Flows
==
https://ceur-ws.org/Vol-3942/S_11_Javorskyj.pdf
Identification and Analysis of the Structure of
Rhythmic Variability of Electricity Consumption Time Series
for Estimating and Forecasting Energy Flows *
Ihor Javorskyj1,2, Valentyna Pleskach3,*, Roman Yuzefovych1,4, Oleh Lychak1
1
Karpenko Physico-Mechanical Institute of the NAS of Ukraine, 79060, 5 Naukova Str., Lviv, Ukraine
2
Bydgoszcz University of Science and Technology, 85796, 7 Al. Prof. S. Kaliskiego, Bydgoszcz, Poland
3
Taras Shevchenko National University of Kyiv, Volodymyrs'ka Str. 64/13, Kyiv, 01601, Ukraine
4
Lviv Polytechnic National University, 79013, 12 Bandera Str., Lviv, Ukraine
Abstract
This article is more of an overview, but it shows the importance of analysing the structure of the rhythmic
variation of the time series of electricity consumption. The integration of renewable energy sources and the
development of multi-directional smart grids present challenges for the stability and management of modern
energy networks. Unlike traditional unidirectional systems, smart grids incorporate feedback loops and distributed
energy generation, introducing high-frequency fluctuations that can propagate through the network, so This study
identifies and analyzes the rhythmic variability in electricity consumption time series using deterministic and
probabilistic approaches, including Fourier and spectral-correlation theories. An application of Periodically
Correlated Random Processes (PCRP) is proposed to model and forecast energy flows, offering a robust framework
for identifying hidden variability structures. The findings support enhanced decision-making for network
regulation and the optimization of smart grids architectures, ensuring resilience and reliability in energy
distribution.
Keywords1
Smart grids, renewable energy integration, rhythmic variability, PCRP, stochastic modeling, energy flow
forecasting
1. Introduction
The development of the Green transmission concept and the adoption within the EU of a number of
regulatory acts within the framework of the European Green Deal, which oblige to integrate into a single
energy network practically all available energy sources down to the level of individual households, the
integration of low-power renewable energy sources into common “global” consumption networks have
created serious challenges for the organization and management of the operation of such a network.
This kind of network, unlike previous, classical, essentially unidirectional networks of the type “from
power generation – through distribution networks – to consumers”, is transformed into a multi-stream
multi-directional network (Smart Grids) on the type of modern Internet data transmission networks.
This network itself is a generating multi-level distributed network, since it provides for reverse energy
flows, and each, even the smallest of its nodes is a separate dynamic system with feedback, i.e. it can also
potentially generate consumption-generation fluctuations. Minor fluctuations in consumption are not
critical in the case of a unidirectional system, although there have been significant accidents (blackouts)
that have led to major losses in power grids that arose as a result of minor “hidden” fluctuations. It should
be noted separately that a significant component of electricity consumption is chaotic, and known low-
frequency (daily, weekly, monthly) consumption cycles are known and non-critical, they are “global”, i.e.
they cover a significant part (or parts) of the unidirectional distribution network. In the case of a
generating multi-level network with feedback in nodes, short-term high-frequency fluctuations may occur,
primarily local, which can quickly lead to local oscillations and, as it was in the case of “unidirectional”
8th International Scientific and Practical Conference Applied Information Systems and Technologies in the Digital Society
AISTDS’2024, 2024, October 1, Kyiv, Ukraine
*Corresponding author.
javor@pbs.edu.pl (I. Javorskyj); v.pleskach64@gmail.com (V. Pleskach); roman.yuzefovych@gmail.com (R. Yuzefovych);
olehlychak2003@yahoo.com (O. Lychak)
0000-0003-0243-6652 (I. Javorsky); 0000-0003-0552-0972 (V. Pleskach); 0000-0001-5546-453X (R. Yuzefovych);
0000-0001-5559-1969 (O. Lychak)
CEUR
Workshop
ceur-ws.org
ISSN 1613-0073
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
Proceedings
networks, spread over a short period of time to significant fragments of the entire “global” network. It is
clear that scenarios of this type are mostly probabilistic, but ignoring them, given the obvious ability of
active networks or electrical circuits with complex feedback loops to generate oscillations, including those
of a chaotic nature, is at least unreasonable. Therefore, identifying and analyzing the structure of the
hidden rhythmic variability of time series of energy consumption in networks can become a serious
safeguard against possible accidents on the one hand and an important tool for supporting network
regulation decisions and even a basis for optimizing its structure. This research is highly significant for the
future of energy development due to its alignment with key trends and challenges in the energy sector. It
is important is integration of renewable energy sources; smart grid evolution, decentralization of energy
systems. Future energy systems will heavily rely on renewable sources like solar and wind, which are
inherently variable and unpredictable.
By analyzing rhythmic patterns in energy consumption and generation, this research helps to integrate
these intermittent sources more seamlessly into the grid, ensuring stability and efficiency.
As energy systems evolve into smart grids, the complexity of managing bidirectional energy flows
increases. This research contributes to developing intelligent systems capable of predicting energy
demands and generation patterns, crucial for optimizing the operation of smart grids.
The shift towards decentralized energy systems, where individual households and small-scale producers
contribute energy, requires advanced forecasting and analysis tools.
Understanding consumption rhythms aids in managing these decentralized networks effectively,
minimizing losses and maximizing energy use.
Not to be overlooked also energy security and resilience, so the shift towards decentralized energy
systems, where individual households and small-scale producers contribute energy, requires advanced
forecasting and analysis tools and understanding consumption rhythms aids in managing these
decentralized networks effectively, minimizing losses and maximizing energy use. With the increasing
complexity of energy systems, the risk of disruptions and blackouts rises.
This research can provide tools to identify and mitigate hidden fluctuations and anomalies that could
compromise energy security.
Forecasting energy flows accurately allows for better planning and allocation of resources, reducing
waste and operational costs. This efficiency is critical for the sustainable growth of energy infrastructure
and keeping energy affordable.
Meeting global climate goals, such as those outlined in the Paris Agreement and European Green Deal,
requires innovative solutions for reducing carbon emissions.
By optimizing energy systems and integrating renewable resources, this research supports the
transition to low-carbon and sustainable energy models. The methods developed can drive technological
innovation in energy analytics, smart sensors, and IoT applications, laying the groundwork for future
advancements in energy management systems.
2. Approaches to the research of oscillatory processes
During studying of oscillatory processes two main attempts are conditionally distinguished. The first one
is grounded on the so-called phenomenological models, which is based on the principles, formulated by
Fourier: it is assumed that arbitrary oscillations can be combined from simple harmonic ones, that is, a
complex oscillatory process is interpreted as the result of superimposing of oscillations of individual
harmonic oscillators.
The second one is grounded on the construction of physical and mathematical models of the systems
that generate oscillations, and the analysis of their properties on this basis. Both of these attempts were
closely joined during their development. Despite the mathematical complexity of the analysis of system
oscillations, associated primarily with their nonlinearity and the dependence of parameters on time, this
methodology is successfully developing and important results have been obtained in it during the study of
the properties of oscillatory motion.
The first studies of the properties of oscillations mostly used a deterministic approach, according to
which the time changes of the function, describing the oscillations occur according to certain established
dependencies, which actually represent their defining feature – repetition. Among the deterministic
oscillations two classes are distinguished: oscillations with limited power and oscillations with limited
energy [1, 2]. Among the first class, a special place is occupied by periodic and almost periodic oscillations,
the properties of which are analyzed using Fourier series. Oscillations belonging to the second class are
also called transient or vanishing. Methods of their analysis and processing are based on the theory of
Fourier integrals. Determination of Fourier coefficients or Fourier images that correspond to functions
given in one way or another is called harmonic analysis [3]. Harmonic analysis methods transfer the study
of oscillations from the time domain to the frequency domain, in which their properties are characterized
by the amplitude and phase spectra, as well as the power spectrum or energy spectrum. Note that the
power spectrum does not depend on the phase relations between the harmonic components that form the
oscillations, so it cannot be used to describe their shape in the time domain. The spectra of periodic and
almost periodic oscillations are discrete, and the spectra of vanishing oscillations are continuous. The first
ones do not belong to the class of oscillations with limited energy, so they cannot be represented by the
Fourier-Riemann integral. Combining the discrete and continuous cases allows obtaining a representation
in the form of the Fourier-Stieltjes integral, which can be used to describe the sum of periodic or almost
periodic oscillations and vanishing oscillations. However, non-vanishing oscillations with a continuous
spectrum remain outside the scope of such a model.
Mathematically rigorous theory of non-vanishing oscillations with a continuous spectrum was
developed by N. Wiener [4, 5] and was called by the author himself generalized harmonic analysis.
N. Wiener moved away from the analysis of individual oscillatory processes and come on to the study of
their correlation functions, defined by means of time averaging. The Fourier transform of such correlation
functions characterizes the power of the harmonic components of the oscillation. However, when directly
transferring such an approach to the analysis of real oscillatory processes, researchers encountered a
number of problems that manifested themselves as instability of the estimates of the spectra obtained
during the processing of experimental data. Similar problems appear when using a purely spectral
approach to the analysis of non-vanishing oscillations, introduced by A. Schuster [6–8] and called
periodogram analysis. Therefore, the instability of the results can be explained primarily, as noted in the
work of A. Einstein [9], by fluctuating time changes in the studied quantities. Fluctuating components can
be added to regular oscillations, modulate oscillations as a whole, as well as its individual components. In
many situations, the power of fluctuation changes is significant if compared to the power of regular
oscillations and even may exceed it. In this case methods for analyzing and processing of oscillations
should be developed on the basis of their models in the form of random processes. Such an approach is also
necessary because of it is the probabilistic characteristics of random processes that those properties of
oscillations, that are essential for determining their physical nature and describing the state of the objects
that generate them are mostly associated.
3. Random oscillations analysis methods
The theory of random processes and methods of their statistical analysis are developed based on their
division into classes.
The methods of stationary random processes and certain types of nonstationary ones are developed and
mostly used in practice: random processes with evolutionary nonstationarity, including locally stationary
ones, and random processes with the so-called rhythmic structure. The latter include periodically
nonstationary random processes and their generalizations: bi-, poly- and almost periodically nonstationary
random processes. In the second-order theory (also called spectral-correlation theory), the properties of a
random process are described by the mathematical expectation, correlation function and its Fourier
transform - spectral density. The foundations of the spectral-correlation theory of stationary random
processes were performed in the works of A.Ya. Khinchin [10, 11]. The theorem on the spectral
decomposition of the correlation function proved by him is actually another expression of S. Bochner's
theorem on the harmonic analysis of positive definite functions [12]. N. Wiener's condition on the
existence of the correlation function obtained by means of time averaging was replaced by A.Ya. Khinchin
with the condition of stationarity of the random process. In the case of ergodic stationary random
processes, A.Ya. Khinchin's theorem follows from N. Wiener's theorem, therefore the theorem on the
spectral decomposition of the correlation function is often called the Wiener-Khinchin theorem [13]. The
results obtained by A.Ya. Khinchin were developed by A.M. Kolmogorov [14, 15] and G. Kramer [16, 17],
who substantiated the harmonic decompositions of the random processes. Non-correlating of harmonics in
this expansion is a consequence of the stationarity of the random process. These results are actually a
transfer to random processes of the Fourier-Stiltjes transform, while the transition to the ordinary Fourier
integral in this case is impossible. The pioneer of empirical spectral analysis, i.e., the estimation of the
spectral density of a stationary random process from experimental data, is considered to be J. Tukey [18].
To calculate the spectral density, he used smoothed estimates of the correlation function. The smoothing
procedure itself, i.e., the use of the correlation function estimates of certain windows in calculating the
Fourier transform, made it possible to obtain valid estimates of the spectral density, and therefore led to
stable results of empirical spectral analysis. This method of nonparametric spectral analysis was called the
Blackman-Tukey method [19]. After smoothing, the estimates of the spectral density obtained using the
periodogram method of A. Schuster [6–8], which can be considered as a special case of the Blackman-
Tukey correlogram method. It differs from the latter only in the method of calculating the spectral density.
The mathematical expectation of a stationary random process determines the regular component of the
oscillation, and the correlation function characterizes the relationships between the values of the
oscillatory process at time points and depends only on the difference. The power spectral density describes
the distribution of the oscillation power in harmonics, therefore it is called the power spectral density. This
value, as already noted, is not affected by the phase relations between the harmonic components, therefore
it cannot characterize time structure of the oscillatory process, namely its repeatability, which, although in
an idealized form, is represented by deterministic models. If the oscillations contain regular time changes,
then within the framework of the stationary model they are analyzed in terms of the correlation function
and the power spectral density, which is not logical and creates methodological problems as well as
difficulties in interpreting the results of statistical processing of experimental data. This was first noted by
E. Slutsky [20], who showed that peak values of spectral density can be the result of a certain type of
relations of time series, but not only the presence of a regular periodic component. A natural resolve of
this situation is to isolate regular oscillations in the form of a time-varying mathematical expectation
function and to study it using deterministic function methods. The mathematical model of oscillation then
will be a random process nonstationary in terms of mathematical expectation. However, as the results of
real data processing show [20–30], in many cases nonstationarity is also inherent in higher order moment
functions, and then it is necessary to move on to the analysis of the properties of oscillations based on
their models in the form of periodically nonstationary random processes.
4. Methods of analysis of periodically nonstationary random processes
(PNRP)
The first step in the development of the theory of nonstationary random oscillations can be considered an
extension of the conditions for the existence of a harmonic decomposition, which were introduced by
M. Loev [31].
Random processes for which such a decomposition exists were called harmonized. In the general case,
individual harmonics of the decomposition are correlated, therefore their property in the frequency
domain is described by the two-frequency spectral density. The two-frequency spectral density takes on a
diagonal form, that is, becomes a function of one argument, in the case of a stationary random process,
when the harmonics of the decomposition are uncorrelated. When determining the two-frequency spectral
density from experimental data, a number of difficulties arise, therefore, in practice, for the analysis of
nonstationary random oscillations, a time-dependent spectral density, which is called a variable
(instantaneous) spectrum [32], as well as the spectral density of their stationary approximation, are often
used. The probabilistic characteristics of a stationary approximation are the time-averaged probabilistic
characteristics of a nonstationary random process. The correlation function of a stationary approximation
has all the properties of the correlation function of a stationary random process, so its Fourier transform is
called the power spectral density of the stationary approximation. This quantity describes the frequency
distribution of the time-averaged power of the oscillations.
A more complete and deep study of the structure of nonstationary random oscillations allows
specifying the type of time changes in their probabilistic characteristics. In this way, we arrive at models of
oscillations in the form of locally stationary random processes and random processes with a rhythmic
structure. The latter are called cyclostationary in literature [23, 29, 30]. We note that belonging to
processes with a rhythmic structure does not exclude the properties of local stationarity.
Locally stationary random processes are characterized by the fact that their probability characteristics
change significantly over a relatively large time interval, and this interval is much larger than the
correlation interval, which is determined by the behavior of the correlation function with respect to the
shift. At the same time, for any moment in time, the probability characteristics of such processes have the
properties of stationary random processes (hence the term local stationarity). Although the correlation
function is variable in time, it is a positive definite function for each, and the change in spectral density is
non-negative and is interpreted as an instantaneous power spectral density. Under such conditions, when
studying the probability structure of real fluctuations, statistical methods of stationary random processes
can be used. The main issue here is determining the length of the processing interval, since due to the
presence of two trends in the behavior of the evaluation efficiency characteristics with increasing length –
increasing bias and decreasing variance – there is a certain optimal length of the processed segment that
provides the minimum total processing error.
5. PCRP approach to analysis of the PNRS
A special place in the development of the theory and methods of analysis of stochastic oscillations belongs
to their models in the form of random processes with a rhythmic structure.
In the second-order theory, they are defined as processes with a periodic or almost periodic change in
time of the mathematical expectation, correlation function, spectral density. They are called periodically
correlated random processes (PCRP) and their generalization - almost periodically correlated random
processes (APCRP), as well as subclasses of the latter: bi- and poly-periodically correlated processes [22, 24,
25, 33, 34]. Research, based on models of nonstationary random processes belonging to these classes
naturally combines and develops deterministic and probabilistic approaches, which are based, respectively,
on the theory of periodic and almost periodic functions and the theory of stationary random processes.
PCRP models and their generalizations make it possible more adequately describe the structure of the
oscillatory process, covering as separate extreme cases the above-mentioned representations, as well as
models known in the literature, in which attempts have been made to combine the features of repeatability
and stochasticity: additive, multiplicative, their combinations, poly-harmonic, quadrature, etc. All this
creates a basis for the analysis of stochastic oscillations not only by special means characteristic of each
model, but also in terms common to all of them.
Random processes with periodic and almost periodic in time probabilistic characteristics were first
considered in the work of L.I. Koronkevych [35]. This was done when studying solutions of differential
equations with periodic and almost periodic coefficients and a force in the form of a stationary random
process. This work is referred to by E.G. Gladyshev [36], analyzing the properties of the Fourier
coefficients of the correlation function, their representations, and the issue of harmonization of these
classes of nonstationary random processes. E.G. Gladyshev first used the terms “periodically correlated”
and “almost periodically correlated” random processes. U. Bennett [37] and R.L. Stratonovich [38]
considered random processes with periodically varying characteristics as a models (V. Pleskach [39]),
suitable for describing of telecommunication signals and fluctuation oscillations in telecommunication
systems. The first author introduced the term “cyclostationary”, and the second – “periodically
nonstationary”. The latter term was also used in signal analysis in a later work by L. Franks [40]. The first
studies on the theory of estimating probabilistic characteristics of the PCRP were carried out by
L.I. Gudzenko [41, 42]. He obtained the conditions for the consistency of estimates of the Fourier
coefficients of the mathematical expectation and the correlation function, as well as estimates of these
characteristics themselves, which are based on the first estimates. The author showed the asymptotic
convergence of the estimates calculated in this way to those found by averaging the PCRP samples taken
over the correlation period. As L.I. Gudzenko noted, the validity of the latter estimates is a consequence of
the fact that this type of reference sequences are stationary and stationary related. The properties of
reference sequences were analyzed more fully in the work of Ya.P. Dragan [43]. Estimates of correlation
components, as well as estimates of spectral components of the PCRP constructed on their basis, were
studied by H. Hurd [44, 45].
The two-frequency spectral representation of the correlation function was considered by A. Papoulis
[46], S.M. Rytov [47], H. Ogura [48]. The latter obtained a representation of the PCRP through stationary
random processes, which are selecting using frequency shift and linear bandpass filtering.
Such stationary components have a finite spectrum. Mathematically rigorous representations through
stationary random processes with a finite spectrum for PCRPs harmonized according to M. Loev were
analyzed by H. Hurd [49, 50]. V. Gardner and L. Franks [51] systematized a number of properties of PCRPs,
and also considered the problem of optimal linear filtering of PCRPs using filters with periodically varying
parameters.
The basics of the spectral-correlation theory of PCRP with limited average power were developed by
Ya.P. Dragan [22, 25, 33, 34, 52, 53]. He was the first to establish a representation for PCRP obtained using
a filter with periodically varying parameters in terms of stationary random processes with a infinite
spectrum [54]. Ya.P. Dragan [53, 54] outlined the class of D-harmonized random processes and showed
that this class of processes and the class of processes with limited average power are equivalent. For D-
harmonized PCRP, transformation was obtained in the general case through stationary random processes,
and a one-to-one correspondence between transformation with finite and infinite spectra was established.
A number of works by W. Gardner and colleagues [23, 28, 55–57] were motivated by
telecommunications problems. W. Gardner [57] developed a non-probabilistic approach to the analysis of
so-called cyclostationary processes, which can be considered a further development of N. Wiener's
generalized harmonic analysis. On the initiative of W. Gardner, a monograph [23] was published in 1994,
which collected the main results in the field of PCRP analysis and its applications obtained by well-known
specialists in this field. This monograph mainly highlights the results, published in the English-language
literature. The results of research by a wider range of specialists, including those from Ukraine and the
former Soviet Union, are briefly reviewed in [28]. This work also provides an extensive list of literature
(786 titles) on the theory, statistics of PCRP and generalizations, as well as applications in various fields of
science and technology. It should be noted that the first studies by scientists from Eastern Europe in the
English-language literature were characterized in the well-known monograph by A.M. Yaglom [58].
Among the latest monographic publications is the book by H. Hurd and A. Miami [29], devoted to the
analysis and statistics of periodically correlated random sequences and their applications. It develops the
results obtained in the initial works of E.G. Gladyshev [40, 59], Ya.P. Dragan [25, 33].
One of the first works on research on the use of PKRP methods for the analysis of real stochastic
oscillations, along with the already mentioned work of L.I. Koronkevych [35] in the field of mechanics,
was the work of L.I. Gudzenko on the study of the properties of fluctuations in self-oscillating systems.
S.M. Rytov [47] indicated the use of PCRP for the study of cyclic magnetization reversal noise,
O.F. Romanenko and T.O. Sergeyev [60] noted the possibility of describing by these processes the
turbulent flow of water adjacent to the ship's propeller, temporal changes in electricity consumption, and
physiological phenomena. V.I. Kolesnikova and A.S. Monin [61] emphasized the expediency of studying
the seasonal and daily variability of meteorological phenomena from the PCRP perspective, which is
constrained, in the authors' opinion, by insufficient elaboration of theoretical and methodological issues.
Ya.P. Dragan, in collaboration with K.S. Voychyshyn [62], and then independently [63, 64], considered
the general properties of the stochastic rhythmic model within the framework of the PCRP model, which is
interpreted as the result of a regular in the probabilistic sense repetition of cycles - an interconnected
sequence of phases in the development of the system under study. These works formulated the main
requirements for the rhythmic model: a description of its stochasticity, decomposition into harmonics and
repeatability of properties. K.S. Voychyshyn [65] made the first attempts to analyze the daily rhythmicity
of some geophysical processes based on such a model, and Ya.P. Dragan and I.M. Javorskyj applied this
approach to the study of sea wind waves [25, 66, 67]. In the first case, the rhythmicity analysis was carried
out at the level of the time dependence of the estimates of the mathematical expectation and dispersion, in
the second case, estimates of correlation characteristics were included in the analysis. It should be noted
that even earlier, the PCRP model, even in a somewhat broader aspect (with the involvement of histograms
and estimates of correlation functions), was used to study the daily changes of air temperature and
humidity, as well as soil temperature [68–73]. P.Ya. Groisman considered the correlation properties of
precipitation series within the framework of PCRP [74]. Other properties PCRP are considered in [75, 76].
The best modern methods of analysis and demodulation of signals, representing PNRS are based on the
application of the Hilbert transform and the construction of an analytical signal. It is necessary due to
complex mutual amplitude- and phase- modulations between signal components. A significant advantage
of applying the Hilbert transform to the analysis of diagnostic signals is the ability to analyze all types of
carrier's modulations: amplitude, frequency and phase, as well as their effective separation [77, 78].
However, frequent neglecting of known significant limitations in the application of the Hilbert transform
to the processing of modulated signals formulated by the Bedrosian and Nuttall theorems [79, 80] leads to
errors in detecting defects and significant errors in their classification as well as assessment of the degree
of development. To avoid such troubles, the decomposition of the PNRS by empirical methods into a
number of narrow-band so-called “eigenmode functions” and the application of the Hilbert transform to
analyze the structure of each of these modes separately is used [Feldman, Huang]. The most widely used
are various versions of the empirical method of decomposition (EMD) (so-called Hilbert-Huang method)
[78, 81]. This empirical method involves a cyclic procedure of signal analysis with the selection and
subtraction of “average modes” from the signal until the rest bacame purely stochastic. The difference
between the various implementations of the method is mainly reduced to the use of different stopping
criteria for mode selection and methods of signal interpolation between peaks (except for the original cubic
spline method proposed by Huang). In the works [82-84], the correlation structure of the PNRS was
analyzed theoretically for different types of carrier signal modulations. Based on this study, a model for
such signal was proposed and a rigorous theoretical analysis of its application under the conditions of
various types of carrier modulations was carried out. A method for narrow-band filtration and
demodulating of the quadratures of the modulating components was developed and verified. Its
effectiveness for real diagnostic vibration signal data processing was demonstrated. In particular, it was
proposed to estimate the degree of defect development in the system under investigation based on higher-
order joint correlations between different signal components. For this purpose, a map of correlations
between different signal components was constructed. The dependence of defect development degree
estimates on signal filteration parameters was shown and conditions for avoiding the leakage effect during
signal processing were established.
6. Conclusion
Use of the PCRP model and its generalizations for the analysis of stochastic oscillations is based on a data
processing methodology grounded on the theory of evaluating the entire complex of probabilistic
characteristics of a given class of nonstationary random processes.
The development of such a methodology and its use for studying the structure of variability of
periodically nonstationary random oscillations in electrical networks will provide an important basis for
the development and reliable operation of modern “smart power grids”.
The research is crucial due to its relevance in modern energy systems, especially with the shift toward
smarter and more sustainable grids. The transition from unidirectional traditional energy systems to
bidirectional, multi-level smart grids has introduced complexities. Analyzing rhythmic variations helps in
understanding and optimizing these flows, minimizing inefficiencies and disruptions. Smart grids, with
their feedback loops, are more susceptible to high-frequency and localized fluctuations that can propagate
and cause system-wide failures. Identifying hidden patterns in energy consumption can provide early
warnings for such risks. Understanding the rhythmic variability of electricity consumption enables better
forecasting and regulation, which is critical for balancing supply and demand, so supports the stability and
reliability of power systems, especially as renewable energy sources are intermittent. Accurate modeling
and forecasting of energy flows reduce operational costs, improve resource allocation, and enhance the
economic feasibility of renewable energy such projects.
The study bridges deterministic and probabilistic approaches, offering a robust methodology to analyze
and forecast energy flows in modern power systems, thereby addressing critical challenges in energy
management and sustainability.
Declaration on Generative AI
The authors have not employed any Generative AI tools.
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