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{{Paper
|id=Vol-3309/paper11
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|title=Mathematical model of the energy resource consumption process in the form of a random process with piecewise homogeneous components
|pdfUrl=https://ceur-ws.org/Vol-3309/paper11.pdf
|volume=Vol-3309
|authors=Leonid Scherbak,Iaroslav Lytvynenko,Serhii Kharchenko,Oleg Nazarevych,Volodymyr Hotovych
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==Mathematical model of the energy resource consumption process in the form of a random process with piecewise homogeneous components==
https://ceur-ws.org/Vol-3309/paper11.pdf
Mathematical model of the energy resource consumption
process in the form of a random process with piecewise
homogeneous components
Leonid Scherbaka, Iaroslav Lytvynenkob, Serhii Kharchenkoa, Oleg Nazarevychb, Volodymyr
Hotovychb
a
Institute of General Energy of the National Academy of Sciences of Ukraine, 172, Antonovycha Street, Kyiv,
03150, Ukraine
b
Ternopil Ivan Puluj National Technical University, 56, Ruska Street, Ternopil, 46001, Ukraine
Abstract
The paper is devoted to the development of a generalized mathematical model for the
description of random processes of energy resource consumption (electricity, water, natural
gas consumption). The model is presented in the form of a random process, which consists of
the sum of components: a trend (non-oscillatory component), several piecewise homogeneous
components of an oscillatory nature, and a stochastic component. To build this model, the
results of statistical processing of implementations of the energy resource consumption
processes of different scales and the nature of functioning of objects were used. At the same
time, the decomposition method of Singular Spectrum Analysis (SSA) was used to break down
realizations of random processes into components, and the Pelt method was used to select
homogeneity time intervals within the general observation interval. Polynomials of the third
order (trend component) and sinusoids (oscillating components) were used to approximate the
components of the energy resource consumption processes at homogeneity intervals. The
results of statistical processing show the adequacy of this model, and the selected process
components are physically justified.
Keywords
Energy resource consumption process, random process, SSA-Caterpillar method, Pelt method,
change points, trend, piecewise homogeneous components, stochastic component.
1. Introduction
The need for economy and optimal consumption of energy resources (electricity, water, natural gas
consumption) determine the relevance of the tasks of analysis, control, diagnosis and monitoring of the
relevant energy resource consumption processes. Today, such tasks are solved with the help of
specialized automated systems. The basis for the functioning of these systems is algorithmic software,
in particular, mathematical models of the energy resource consumption processes. At the same time,
the correctness and efficiency of the functioning of automated systems depend on the adequacy and
accuracy of mathematical models.
This paper is devoted to the development of a mathematical model of the energy resource
consumption processes in the form of a random process with piecewise homogeneous components,
which is a generalization of known models of resource consumption processes developed on the basis
of the random processes theory.
__________________________
ITTAP’2022: 2nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24, 2022,
Ternopil, Ukraine
EMAIL: prof_Scherbak@ukr.net (L. Scherbak); iaroslav.lytvynenko@gmail.com (I. Lytvynenko); nanoavia@ukr.net (S.
Kharchenko); taltek.te@gmail.com (O. Nazarevych), gotovych@gmail.com (V. Hotovych)
ORCID: 0000-0002-1536-4806 (L. Scherbak); 0000-0001-7311-4103 (I. Lytvynenko); 0000-0001-9808-7607
(S. Kharchenko); 0000-0002-8883-6157 (O. Nazarevych), 0000-0003-2143-6818 (V. Hotovych)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
2. Analysis of recent research
A review of the literature [7-14] showed that the processes of energy resource consumption
(electricity, water, natural gas consumption) by objects of different scales (house, settlement, district,
region, etc.) have the following common features:
1) They are formed as a result of the combined action of many individual consumers of different
capacities, the moments of turning on and off (resource consumption sessions), as well as the
duration of such sessions, are random (stochastic nature of the process);
2) They are periodic, because the activity of a person and society, in general, is periodic, with
different periods: hour, day, month, year, etc. (periodic nature of the process).
Therefore, it is quite justified to consider such processes as stochastic-periodic and, accordingly, to
apply the theory of random processes for their modeling, as well as the apparatus of mathematical
statistics for the practical study of their realizations.
In general, there are many scientific and technical publications dedicated to the construction and
application of mathematical models based on random processes. For example, in [1] the use of the
model of a periodic random process for a wide range of problems of stochastic process research is given.
In [2], the model of a random process is presented as the sum of a stationary random process and the
value of a mathematical expectation that changes under the influence of an indicator function. Also,
there are well-known approaches associated with the use of periodically correlated random processes
[3, 4], Markov processes [5, 6], etc.
In this paper, the set of random processes of energy consumption is considered as a subset of a large
and numerous set of random processes.
Among the well-known publications, a significant number of publications are devoted to the problem
of modeling energy consumption processes. In particular: in [7, 8], the mechanism of formation of the
energy resource consumption process (specifically, natural gas consumption) as a result of the action of
many consumers of different power and with random consumption sessions in time is substantiated,
and it is also proposed to use a conditional linear periodic random process as a model; in [9] a model is
proposed as the sum of a deterministic trend and a piecewise stochastic-periodic process, which takes
into account the change in the mode of operation of the consumer network over a long observation
interval; [10] proposed a model of the power consumption process in the form of a piecewise
homogeneous periodic random process, which includes a set of homogeneous components (periodic
with a period of 24 hours) of random processes as a vector periodic process; in [11] a mathematical
model of the electric load of organizations at an interval of one day was developed and substantiated in
the form of a piecewise homogeneous random process with uncorrelated values (white noise in the
broad sense); in [12] a model of electricity consumption is proposed as a multi-component random
process with moments of disorder (a sudden change in process dynamics), which consists of a trend, a
piecewise-periodic random process, and a piecewise-stationary random process; in [13] a
cyclostationary conditional linear random process is proposed as a model of the random process of
water consumption; in [14], the model of the gas consumption process is presented in the form of the
sum of the trend, cyclic random process and stochastic residual, which also takes into account the
segment cyclical structure of the process.
In these publications, partial cases of the energy resource consumption processes were considered
as the subsets of a large set of stochastically periodic random processes. This paper proposes a
comprehensive generalized model that combines both deterministic and stochastic components of the
energy resource consumption process.
3. Main part
In general, the research process of random processes of energy resource consumption can be
described as shown in Fig. 1.
Here, under the formation of an energy resource, we mean a certain network (topology) of spatially
separated consumers of such resources as electricity, water, and natural gas, and this network itself is
called a stochastic spatio-temporal environment of consumption (city, village, district, region, country,
organization, etc.).
Stochastic Hardware and
Energy spatio- software for Results
resource temporal energy of
formation environment of resource research
consumption consumption
research
Figure 1: The general scheme of the formation of the process of energy resources consumption and
their research
Taking into account the nature of energy resource consumption processes, and the results of
considered studies [6 – 14], in general, the model of the energy resource consumption process can be
represented by the expression (1):
(, t ) = A(t ) + B(t ) + (, t ), (1)
where A(t ) is a deterministic component (trend), B (t ) is a deterministic, periodic component with a
certain period of oscillation, Z (, t ) is a stochastic component.
The proposed mathematical model is an additive sum of components. As a rule, the trend component
has the character of a smooth curve without sharp changes (oscillations). As a component B (t ) , there
can be several components with different periods and amplitudes of oscillations (the specific number
of components is determined in the process of statistical processing based on the experience of the
researcher). In this case, the oscillating component B (t ) will be the sum of these components and
model (1) will take the following form:
n
(, t ) = A(t ) + Bi (t ) + (, t ) . (2)
i =1
The stochastic component of the model Z (, t ) is a set of random factors that affect the object of
research. In the literature dealing with the theoretical foundations of the SSA method [15], this
component is often called the stochastic residual.
The obtained results of statistical processing, which make it possible to decompose the
implementation of a random process into components and find change points (time moments of rapid
or even sudden changes in the dynamics of the process), make it possible to specify the general model
and present it in the form:
(, t ) = Aj (t ) I (t , t j ) + B j (t ) I (t , t j ) + j (, t ) I (t , t j ), , t [0, T ],
n n n
j =1 j =1 j =1
j −1 j )
t , t , j = 1, n − 1 (3)
t j =[0, T ], t j =
n
, t j ti = for i j, t j .
t ,t , j = n
j −1 j
j =1
Here, the interval of observation t 0, T is divided by change points (distortion points)
t j , j = 1, n into n intervals. The belonging of a specific fragment of a process component to the j -th
interval is given by the indicator function:
1, t t j
I (t , t j ) = , j = 1, n. (4)
0, t t j
At each of the n intervals, the trend and fluctuation components are described by some deterministic
functions. At the same time, the trend and fluctuating components have a deterministic character on
several observation subintervals, and together these observation intervals form the overall observation
interval t 0, T of the implementation of a specific energy resource consumption process.
In essence, each separate subinterval is a time interval within which the topology of energy resource
consumers functions in a somewhat stable mode and the cumulative effect of various factors on this
topology (weather factors, length of day and night, economic factors, etc.) remains unchanged. A
change in these factors leads to the transition of the network of energy resource consumers to a different
mode of consumption.
To confirm the adequacy of the model (3), the work carried out statistical processing of several
implementations of the energy resource consumption processes. This statistical processing consists of
two stages:
1. Decomposition of the process implementation into individual components.
2. Change points detection and using them to divide process components into fragments that
correspond to separate intervals of process homogeneity.
In this study, singular spectrum analysis (SSA), also known as the Caterpillar method [15, 16], was
used to decompose realizations of energy consumption processes. Unlike other known methods
(autoregression and integrated moving average (ARIMA) [17], group method of data
handling (GMDH) [18], principal component method (PCA) [19], empirical mode decomposition
method (EMD) [20], wavelet analysis [21], etc.) the SSA method is characterized by relative simplicity
(it mainly consists in working with numerical matrices) and interactivity. The interactivity of the SSA
method is that the researcher can choose the degree of detail of decomposition (the number of
components that will be obtained as a result of decomposition) in an interactive mode. Some prior
experience in using this method is required at this stage of statistical processing. Some
recommendations for the practical application of the method are given in [15].
The SSA method takes as input the implementation of a random process obtained as a result of a
measurement experiment with a certain accumulation step, as a time-ordered sequence of counts (time
series). The purpose of the method is to decompose the time series into a set of components (trend,
periodic oscillatory components, stochastic noise component), which in total give the original series.
That is, the researcher a priori assumes the possibility of such decomposition as a consequence of the
nature of some real phenomenon, which is represented by the registered implementation of a random
process. At the same time, the reliability of such components can be explained by the fact that:
• a trend is a smooth curve of a non-oscillating type, which shows the general nature of the change
in the dynamics of the process over time and is considered a certain constant component of the process;
• fluctuating components are present in the structure of the process due to the periodicity of the
real phenomenon under investigation (a part of human society limited in space and time);
• the presence of a stochastic component in the structure of the process is caused by a large
number of stochastic factors forming the process, as a rule, of low intensity. In essence, this component
is the sum of these factors.
In the second stage of statistical processing, various statistical methods can be used to search for
change points. They can be, for example, binary segmentation [22], the method of neighbouring
segments [23], and the Brodsky-Darkhovsky method [24]. A more detailed analysis of various methods
of finding points of disturbance (classification and comparison of the efficiency of their application in
various fields: medicine, climate change, analysis of human activity, etc.) can be found in [25 – 27].
In this work, the Pelt method [28, 29] was used to find change points. This is an a posteriori method
that belongs to the family of methods for finding the optimum of the likelihood function and is
characterized by high computational efficiency. This method was chosen because of its interactivity, as
it allows the researcher to specify a different number of change points to be found, as well as to specify
the statistical characteristic behind which the distortion occurs (in particular, the mean and variance).
The change points detection process was based on a sudden change in the variance of the stochastic
component obtained as a result of decomposition at the previous stage. After that, the points of distortion
were generalized to other components of the energy resource consumption process. That is, it was
considered that other components have the same distortion points.
The paper researches several implementations of energy resource consumption processes that
characterize objects of different scales. In particular, statistical processing of data on electricity
consumption (building № 1 of the Ivan Puluj National Technical University in 2016), natural gas
consumption (Ternopil in 2009) and water consumption (100 apartment buildings in Ternopil in 2009)
were carried out. Some of the obtained results are presented in fig. 2 – 7. In particular, fig. 2 shows the
results of the decomposition of the time series of the electric energy consumption process. For clarity
in fig. 2, a) graphs of the process itself and its trend component are combined.
Figure 2: The results of the decomposition of the time series of the electric energy consumption
process using the SSA method: a) implementation and trend (component №1 of decomposition
results); b)-g) components №2-7 of the decomposition of an oscillatory nature; h) component №8 of
decomposition results (stochastic “noise”)
Fig. 3 shows the results of dividing the power consumption process and its component into
homogeneity intervals, which are represented by vertical lines on the graphs. Analogous results of
statistical processing of implementations of natural gas consumption and water consumption processes
are shown in Figs. 4 and 5, respectively.
Figure 3: The division into homogeneity intervals: a) implementation of the electric energy
consumption process; b) trend component; c) component with an oscillation period of one day; d)
stochastic component
Figure 4: The division into homogeneity intervals: a) implementation of the natural gas consumption
process; b) trend component; c) component with an oscillation period of one day; d) stochastic
component
Figure 5: The division into homogeneity intervals: a) implementation of the water consumption
process; b) trend component; c) component with an oscillation period of one day; d) stochastic
component
Figs. 6 and 7 show some of the results of components approximation of the energy resource
consumption processes. The approximation of the trend component was carried out using the polyfit
and polyval functions (the method of least squares) of the Matlab software environment, while the
Curve Fitting Toolbox package of the Matlab environment was used to approximate the fluctuating
components (by finding the coefficients of the Fourier series of the approximating curve).
Figure 6: Approximation on the first interval of homogeneity of the components of the electric energy
consumption process: a) trend component; b) component with an oscillation period of one day
Figure 7: Approximation on the second interval of homogeneity of the components of the natural gas
consumption process: a) trend component; b) component with an oscillation period of one day
In particular, for the electric energy consumption process on the first homogeneity interval, the trend
component is approximated by a polynomial of the third degree
a1 (t ) = 21612t 3 − 0, 0722t 2 + 9, 7314t + 1, 0221, t 1,185 hours, and the oscillating component
with an oscillation period of one day is approximated by a sinusoid
b1 (t ) = −0, 4316 − 5898cos(0.2617t ) − 2202sin(0.2617t ), t 1,185 hours.
For the water consumption process on the second homogeneity interval, the trend component is
approximated by a polynomial of the third degree
a2 (t ) = 1, 7259*10 t − 0, 0607t + 121, 0737t − 9447, t 748,1286 hours, and the oscillating
−6 3 2
component with an oscillation period of one day is approximated by a sinusoid
b2 (t ) = −0,5403 − 2596cos(0.5235t ) − 5873sin(0.5235t ), t 748,1286 hours.
A similar approximation can be carried out for all fluctuating components and the trend component
of the decomposition of the energy resource consumption process implementation on all homogeneity
intervals.
4. Discussion of obtained results
The physical interpretation of change points consists in the transition of the network of energy
consumers from one established mode of consumption to another, for example, in the transition between
seasons (spring, summer, etc.), which affect the consumer network. In this case, homogeneity intervals
are time intervals within which the functioning of the network of energy resource consumers is
unchanged (it is within a certain mode).
On the other hand, the components that we get as a result of the decomposition of the implementation
of the energy resource consumption process are also interesting. The trend shows the general dynamics
of the development of the process and is some generalized, integral sum of the functioning of all
consumers who are included in the network of energy resource consumers. Each of the oscillating
components has a constant period of oscillation and its presence as part of a random process is explained
by the periodicity of human activity. For example, in the composition of all implementations of the
energy resource consumption processes developed in this work, there is a component with an oscillation
period of one day (component № 2 of decomposition), i.e., 24 counts for an accumulation interval of
one hour. The stochastic component is the sum of all random factors that affect the users of the energy
resource, and therefore it was used to search for change points during the statistical processing carried
out in the paper.
Naturally, each network of energy resource consumers has its own individual characteristics.
Therefore, the task of concretizing the model proposed in this paper (decomposition with a certain level
of detail (the number of detected components), change points detection, approximation of deterministic
components, etc.) is one of the tasks that are solved during the study of the corresponding random
process as a characteristic of a specific network of energy consumers.
5. Conclusions
The paper proposes a model of the energy resource consumption process, which generalizes known
stochastic models of energy resource consumption processes and combines both deterministic and
additive components. This is consistent with the stochastically periodic nature of processes of this type.
In order to confirm the adequacy of the proposed model, statistical processing of implementations of
resource consumption processes was carried out, which characterize the topologies of consumers with
different scales and modes of operation. The SSA and Pelt statistical methods were used for the
corresponding processing, although it is also possible to use other, similar methods. The highlighted
components of energy resource consumption processes are physically justified.
Prospects for further research include clarifying the nature of the stochastic component of the model
(for example, the distribution on different intervals of homogeneity), which depend on the specific
implementation of the energy resource consumption process being studied, as well as simulation
modeling of both individual components of the model and the implementation of the energy resource
consumption process as a sum of components as a whole.
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