DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL
The Law of Ukraine "On Energy Efficiency" establishes that an energy audit is a systematic analysis of energy use and energy consumption within the boundaries determined by the nature and scope of energy audit work in order to determine, quantify and prepare a report on the possibi...
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Institute of Renewable Energy National Academy of Sciences of Ukraine
2025
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Vidnovluvana energetika| _version_ | 1871103915581243392 |
|---|---|
| author | Rozen , V. Rozen , P. |
| author_facet | Rozen , V. Rozen , P. |
| author_institution_txt_mv | [
{
"author": "V. Rozen ",
"institution": "Institute of General Energy of the National Academy of Sciences of Ukraine, Kyiv, Ukraine"
},
{
"author": "P. Rozen ",
"institution": "Institute of General Energy of the National Academy of Sciences of Ukraine, Kyiv, Ukraine"
}
] |
| author_sort | Rozen , V. |
| baseUrl_str | https://ve.org.ua/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T06:32:21Z |
| description | The Law of Ukraine "On Energy Efficiency" establishes that an energy audit is a systematic analysis of energy use and energy consumption within the boundaries determined by the nature and scope of energy audit work in order to determine, quantify and prepare a report on the possibilities of improving the energy efficiency.
When conducting an energy audit, it is important to determine not only the areas of work, but also the scope of work, which is determined by setting boundaries. Currently, the boundaries of energy audit are determined according to the experience and intuition of energy auditors.
The paper considers the issue of determining the boundaries of the energy audit scope in industry using the theory of intelligent systems. The importance of using pattern recognition and classification methods for solving problems of energy efficiency improvement is emphasized. It is proposed that classification methods be used to determine typical classes of consumers of fuel and energy resources, making it possible to determine the boundaries and conduct an energy audit. The automatic classification method – the hypersphere method - was chosen as the classification method. The input information for the description of production systems is determined, including indicators, metrics and parameters of energy efficiency.
An algorithm for the hypersphere method is developed and practical calculations are given for the conditions of the compressed air production system of a metallurgical enterprise in Zaporizhia. The characteristics of the equipment subject to priority inspection are provided. This algorithm can be used to solve a wide range of problems in the field of renewable energy. Ref. 12. Tab. 5.  |
| doi_str_mv | 10.36296/1819-8058.2025.2(81).67-74 |
| first_indexed | 2025-07-17T11:39:59Z |
| format | Article |
| fulltext |
67
Відновлювана енергетика. № 2/2025 | Комплексні проблеми енергетичних систем на основі НВДЕ
УДК 621.311 https://doi.org/10.36296/1819-8058.2025.2(81).67-74
DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK.
MATHEMATICAL MODEL
Received Mar. 11, 2025; accepted Jun. 27, 2025
Available online Jun. 30, 2025
Rozen V.1, Rozen P.2
Author for correspondence: Rozen Victor,
e-mail: v_p_rozen406@ukr.net
The Law of Ukraine "On Energy Efficiency" establishes that an
energy audit is a systematic analysis of energy use and energy
consumption within the boundaries determined by the nature and scope of energy audit work in order to deter-
mine, quantify and prepare a report on the possibilities of improving the energy efficiency.
When conducting an energy audit, it is important to determine not only the areas of work, but also the scope of
work, which is determined by setting boundaries. Currently, the boundaries of energy audit are determined ac-
cording to the experience and intuition of energy auditors.
The paper considers the issue of determining the boundaries of the energy audit scope in industry using the theory
of intelligent systems. The importance of using pattern recognition and classification methods for solving problems
of energy efficiency improvement is emphasized. It is proposed that classification methods be used to determine
typical classes of consumers of fuel and energy resources, making it possible to determine the boundaries and
conduct an energy audit. The automatic classification method – the hypersphere method - was chosen as the
classification method. The input information for the description of production systems is determined, including
indicators, metrics and parameters of energy efficiency.
An algorithm for the hypersphere method is developed and practical calculations are given for the conditions of
the compressed air production system of a metallurgical enterprise in Zaporizhia. The characteristics of the equip-
ment subject to priority inspection are provided. This algorithm can be used to solve a wide range of problems in
the field of renewable energy. Ref. 12. Tab. 5.
Keywords: classification, the consumer, fuel and energy resources.
МАТЕМАТИЧНА МОДЕЛЬ ВИЗНАЧЕННЯ МЕЖ ОБСЯГУ РОБІТ З ЕНЕРГЕТИЧНОГО АУДИТУ
Отримано 11 бер. 2025 р.; рекомендовано до публікації 27 черв. 2025 р.
Доступно онлайн 30 черв. 2025 р.
Розен В. П.1, Розен П. В. 2
Автор для кореспонденції: Розен Віктор,
e-mail: v_p_rozen406@ukr.net
Законом України «Про енергетичну ефективність»
встановлено, що енергетичний аудит – це системати-
зований аналіз використання енергії та споживання ене-
ргії у межах, визначених характером та обсягом робіт з енергетичного аудиту з метою визначення,
кількісного вираження та підготовки звіту про можливості підвищення рівня енергоефективності.
Під час проведення енергетичного аудиту важливо визначити не тільки напрями роботи, а також об-
сяг роботи, який визначається встановленням меж. На теперішній час межі проведення енергетичного
аудиту визначаються відповідно досвіду та інтуїції енергоаудиторів.
У роботі розглядається питання визначення меж обсягу робіт з енергоаудиту виробничих систем з
використанням теорії інтелектуальних систем. Зазначено важливість використання методів розпі-
знавання образів і класифікації для вирішення задач підвищення рівня енергетичної ефективності.
1 Dr. of Sciences (Tech.), Professor
http://orcid.org/0000-0002-0440-4251
2 Postgraduate Student
http://orcid.org/0009-0003-7934-3398
1, 2 Institute of General Energy of the National
Academy of Sciences of Ukraine, Kyiv, Ukraine
1 д-р. техн. наук, професор
http://orcid.org/0000-0002-0440-4251
2 аспірант
http://orcid.org/0009-0003-7934-3398
1, 2 Інститут загальної енергетики Національної
Академії Наук України, Київ, Україна
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Запропоновано використовувати методи класифікації для визначення типових класів споживачів пали-
вно-енергетичних ресурсів, що дозволило визначити межі проведення енергетичного аудиту. В якості
методу класифікації було обрано метод автоматичної класифікації – метод гіперсфер. Визначена вхі-
дна інформація опису виробничих систем до якої входять індикатори, показники та параметри енерге-
тичної ефективності.
Розроблено алгоритм методу гіперсфер та наведені практичні розрахунки для умов системи виробни-
цтва стисненого повітря металургійного підприємства м. Запоріжжя. Надана характеристика обла-
днання, що підлягає першочерговому обстеженню. Цей алгоритм можна використовувати для вирі-
шення широкого спектру задач в сфері ВДЕ. Бібл. 12, Табл. 5.
Ключові слова: класифікація, споживач, паливно-енергетичні ресурси.
Introduction. Given its international commitments to cli-
mate change mitigation, Ukraine has been paying increas-
ing attention to energy efficiency and energy conservation
in recent years. Thus, in 2021, the Law of Ukraine “On En-
ergy Efficiency” (No. 1818-IX) was adopted. The law trans-
poses most of the provisions of Directive 2012/27/EU on
energy efficiency and Regulation (EU) 2017/1369.
Articles 10 “Energy Audits” and 11 “Energy Auditors” of this
Law define the legal framework for conducting energy au-
dits, as well as the qualifications of energy auditors.
As required by the Law, in 2023, the CMU adopted Resolu-
tion No. 1258 “On Approval of the Procedure and Require-
ments for Energy Audits of Processes and Transport”. The
provisions of Article 8(7) of Directive 2012/27/EU on the
content of energy audits were transposed into the CMU
Resolution.
In June 2024, amendments to the Law revised the approach
to energy audits, moving to a model based on energy con-
sumption. Currently, business entities with an average an-
nual energy consumption of 100 thousand MWh per year
or more are required to conduct an energy audit every four
years. Although this threshold is 36 times higher than that
in the EU, it is only the first step towards the introduction
of mandatory energy audits and lower thresholds are ex-
pected to be introduced in the coming years.
Since industrial enterprises will be obliged to conduct en-
ergy audits, this will increase production costs regardless of
whether the audit is conducted in-house or with the in-
volvement of a specialized audit company.
According to the Law of Ukraine “On Energy Efficiency”, an
energy audit is a systematic analysis of energy use and en-
ergy consumption within the limits determined by the na-
ture and scope of energy audit work in order to identify,
quantify and prepare a report on opportunities to improve
energy efficiency. The same definition is given by the stand-
ard DSTU ISO 50002-2016 Energy audit. Requirements and
guidelines for their conduct. That is, the nature and scope
of energy audit work are the determining factors that affect
everything, especially the duration of the work and its cost.
Thus, when conducting an energy audit, it is very important
to correctly determine the nature and scope of energy au-
dit work.
According to DSTU ISO 50002-2016, the energy audit scope
of work is the extent of energy uses and related activities to
be included in the audit, as defined by the organization in
consultation with the energy auditor. This scope can include
several boundaries.
The energy audit scope of work is determined by its bound-
aries. The energy audit scope of work can be determined by
energy and economic indicators and parameters, including:
− specific energy consumption per unit of output;
− energy intensity of production;
− energy saving potential;
− parameters of the energy consumption mode;
− volume of energy consumed;
− capacity of energy consumers;
− technical specification of energy consuming equipment;
− installed capacity of energy consumption;
− economic indicators;
− payment for the consumed energy;
− energy efficiency indicators, etc.
Usually, the boundaries for dividing consumers into large,
medium and small are set almost arbitrarily by developers
of energy audit methodologies or government agencies.
The problem of determining the boundaries of consumer
division according to the established criteria is encountered
when solving the following tasks:
− division of enterprises by the volume of energy con-
sumed to develop specific energy consumption rates;
− determining the boundaries of external and internal en-
ergy audit;
− coordination of energy consumption with the real con-
ditions during the implementation of renewables [1];
− determining the optimal power value according to the
criterion of costs for the construction of renewables;
− analyzing the modes of electricity consumption of elec-
tricity supply systems [2];
− assessing the reliability of a combined power supply sys-
tem using renewables;
− determining the criteria for the required power gener-
ated by solar and wind power plants [2];
− assessing the mathematical expectation of power im-
balances of the combined power system; [1];
69
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− determining restrictions on the operating mode of the
backup energy source [1];
− determining the criteria: power, current efficiency for
solving the problem of assessing the efficiency of bio-
mass mini-boilers within an administrative unit or heat-
generating organization.
It should also be noted that traditional energy sources are
the main source of greenhouse gas emissions that cause
global warming. The use of renewables helps to reduce
these emissions and reduce air and water pollution. To as-
sess the effectiveness of renewables implementation, in-
formation received during a preliminary energy audit of fa-
cilities of different hierarchical levels is used. It is important
to reduce the cost of determining the amount of energy
and economic information, which depends on the scope of
the energy audit.
The purpose of this paper is to increase the validity of de-
cision-making regarding the boundaries of the scope of
work during an energy audit by applying a mathematical
method of classifying fuel and energy resource consumers
based on the hypersphere method.
To achieve this goal, the following tasks were solved:
Determination of factors influencing energy efficiency that
the decision maker sets.
Development of mathematical support for determining
typical classes of consumers by energy and economic indi-
cators and parameters.
The study's object is to establish the energy audit scope of
work by examining energy and economic indicators and pa-
rameters of industrial enterprise energy consumption.
When conducting an energy audit, the correct solution to
this issue is to direct the energy audit team's efforts to au-
diting just such a number of consumers as to improve the
quality of the energy audit without increasing its duration.
Therefore, it is important to establish which energy-con-
suming equipment in the company's workshops should be
the focus of the auditor's attention in the first place.
Materials and results of the study. Today, the practice of
conducting energy audits of industrial enterprises has be-
come widespread. There are various methods for conduct-
ing energy audits [3, 4]. Some audit companies have be-
come highly specialized in certain types of energy carriers
and types of consumers. However, at present, energy audi-
tors determine the class of the most energy-intensive con-
sumers only based on their own experience and do not use
mathematical methods that could simplify the procedure
for selecting consumer classes.
To determine the classes of consumers that are similar ac-
cording to the established criterion, when solving the issue
of determining the energy audit scope of work, the meth-
ods of pattern recognition and classification theory are
most suitable [5, 8, 12]. The use of statistical analysis
method is based on the assumption of homogeneity of the
population of energy consumers under study. However, a
real statistical population is almost always internally differ-
entiated, which corresponds to the hypothesis of class
compactness [8]. This hypothesis makes it possible to solve
the problem of dividing the population of energy consum-
ers under study into homogeneous groups (classes, clus-
ters, taxa) according to certain criteria.
Classification is the division of elements of a given set into
subsets of more similar objects [8]. To determine the en-
ergy audit scope of work, we will classify electricity con-
sumers using the hypersphere method by ordering them
into a variational series.
The hypersphere method was first applied in the electric
power industry to classify daily electrical load schedules of
industrial enterprises in [10] and later to solve the problem
of zoning administrative units of the Volyn region according
to their energy security status [11]. The hypersphere
method belongs to the automatic classification methods,
which allows excluding the decision maker from the classi-
fication process, but it does not reveal the internal struc-
ture of classes, such as hierarchical classification methods
[11].
Let us represent the consumers of energy as a set 𝑋, whose
elements are given by the numerical attributes 𝑝:
(1)
A class of energy consumers is a subset 𝜔𝑖 of similar con-
sumers. As a measure of similarity (closeness) between two
energy consumers, we will use the Euclidean distance be-
tween them 𝑑𝑖𝑗:
(2)
Euclidean distance is the most common proximity measure
used in object classification tasks [8, 10].
Having ordered all the elements 𝑑𝑖𝑗 of the matrix in ascend-
ing order, let's construct a series of variations:
. (3)
If there are classes of consumers, the distances 𝑑(𝑠) be-
tween homogeneous pairs of energy consumers will be
concentrated on the left side of the variational series (3),
and the distances between heterogeneous energy consum-
ers – on the right side. To study the variational series of en-
ergy consumers, we will construct an auxiliary numerical
sequence of the following form:
(4)
where
. (5)
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It should be noted that there may be several types of en-
ergy consumers at an enterprise with the same level of clas-
sification criteria. This can lead to uncertainty when analyz-
ing the auxiliary numerical sequence, since the distances
between them 𝑑𝑖𝑗 = 0. To avoid this, the level of energy
intensity of these consumers relative to each other must be
changed slightly (by 0.1%).
To further clarify the structure of energy consumers, we
will study the behavior of two auxiliary variational series
obtained by sifting the elements of the auxiliary numerical
sequence.
The algorithm for forming the auxiliary variational series is
as follows:
Let's find the Euclidean distances from the first energy con-
sumer 𝑥11 to all the others.
In the numerical sequence 𝑑12, 𝑑13, … , 𝑑1𝑛 find the min-
imum and maximum elements – 𝑑𝑚𝑖𝑛 and 𝑑𝑚𝑎𝑥 .
Repeat this operation for each energy consumer 𝑥𝑖 and
form two sets of distances from the results obtained
and
.
Most elements of the set 𝐷̃𝑚𝑖𝑛 correspond to homogene-
ous pairs of energy consumers, if the elements of the set
are available in time 𝐷̃𝑚𝑎𝑥 can correspond to both homo-
geneous and heterogeneous pairs of energy consumers.
The resulting sets 𝐷̃𝑚𝑖𝑛 and 𝐷̃𝑚𝑎𝑥 contain at least three
equal distances corresponding to the same pair of energy
consumers 𝑥𝑖, 𝑥𝑗 and 𝑥𝑗, 𝑥𝑖. By excluding one of these dis-
tances from the sets 𝐷̃𝑚𝑖𝑛 and 𝐷̃𝑚𝑎𝑥 and arranging the re-
maining ones in ascending order, we get new sets – varia-
tional series of distances:
; (6)
; (7)
whose elements correspond to different pairs of energy
consumers.
The behavior of the variational series 𝐷̃𝑚𝑖𝑛 and 𝐷̃𝑚𝑎𝑥 can
be used to determine the structure of energy consumer
groups at the enterprise. If the variational series 𝐷̃𝑚𝑖𝑛 has
no peaks, then the enterprise's energy consumers are ho-
mogeneous according to the chosen criterion for determin-
ing the energy audit scope of work or they can be divided
into closely spaced classes with constant density. If this se-
ries contains at least one local minimum, then the compa-
ny's energy consumers can be divided into classes with dif-
ferent densities. Among them may be those that contain
one element each – individual consumers that differ greatly
from others in terms of energy intensity, the number of
which can be determined.
If the variational series 𝐷̃𝑚𝑎𝑥 has no peaks, then the enter-
prise's energy consumers are homogeneous according to
the selected criterion or consist of closely spaced classes. If
this series has at least one significant local minimum, then
the enterprise's energy consumers can be divided into 𝑘
classes, in which the maximum distances of heterogeneous
pairs have a different order.
Let's form new sets of elements that will be used to classify
consumers using the hypersphere method. Having ex-
cluded from the sets 𝐷̃𝑚𝑖𝑛 all elements for which there is
a correlation:
(8)
and adding them to the set 𝐷̃𝑚𝑎𝑥, we will get new sets –
variational series:
(9)
(10)
Then we can assume that almost all elements of the set are
generated by homogeneous pairs of energy consumers.
The subset of pairs of energy consumers, each of which
generates an element of variational series (12), is denoted
by 𝐿𝑚𝑖𝑛:
(11)
The state of energy consumers in the hyperspace is set by
a point, the location of which is determined by the value of
indicators or parameters of the criteria that determine the
depth of the energy audit.
The hypersphere method has the following algorithm. First,
the compaction points of the general group of energy con-
sumers are found, which are initially taken as the centers of
the classes,
At the same time, a preliminary classification is performed.
The compaction points are found using the method of hy-
perspheres moving in the plane. The movement of hy-
perspheres starts from different, so-called, reference
points and ends at the compression points. Energy consum-
ers that are switched on during the movement of one hy-
persphere are considered to be consumers of the same
class. If several hyperspheres stop at the same point, all en-
ergy consumers that are switched on by them during move-
ment are combined into one class. Thus, we get as many
classes as the number of compaction points detected by
the hypersphere. The resulting number of classes depends
on the size of the hypersphere radius. The determination of
the optimal value of this radius 𝑑𝑜𝑝𝑡 follows from the vari-
ational series analysis data (4). The issue of determining
𝑑𝑜𝑝𝑡 is considered in [7, 9, 10].
To determine 𝑑𝑜𝑝𝑡 we first find the average value of 𝑑̅ of
variational series (4):
, (12)
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Then the value of the radius of the hypersphere is as fol-
lows:
𝑑𝑜𝑝𝑡 = 𝑚𝑎𝑥(𝑑̅, 𝑑𝑚𝑖𝑛
(𝑢) ) + 𝑑𝑚𝑖𝑛
(1)
(13)
where 𝑑𝑚𝑖𝑛
(𝑢)
, 𝑑𝑚𝑖𝑛
(1)
are the last and first members of the
variational series.
Then, the preliminary classification is refined, taking into ac-
count the possibility of classes with non-empty intersection.
Model calculations of the distribution of energy-consum-
ing equipment into classes by installed capacity. Let us
consider the distribution of energy-consuming equipment
by the criterion of the installed capacity of such consumers
as electric motors. The compressed air production system
is the most energy-intensive (in terms of electricity con-
sumption) at iron works in Zaporizhzhia. Air compressor
units operate 24 hours a day as the production cycle is con-
tinuous, so the distribution can be performed both by in-
stalled capacity and electricity consumption.
The compressed air production system uses electrical
equipment, the technical characteristics of which are given
in Table 1.
Table 1. Technical specification of the equipment of the compressed air system of an iron works
No. Air compressor Capacity, m3/min Pressure, bar Power of the electric drive, MW
Air compressor workshop No.1
1 K-250-61-5 250 9 1.600
2 K-250-61-1 250 9 1.600
3 K-250-61-5 250 9 1.600
4 TsTK-275/9 250 9 1.600
5 K-500-61-5 500 9 3.150
6 K-500-61-5 500 9 3.150
Air compressor workshop No.2
1 K-250-61-1 250 9 1.750
2 K-250-61-1 250 9 1.750
3 K-250-61-1 250 9 1.750
4 K-250-61-1 250 9 1.750
5 K-1500-62-2 1,500 7 10.000
6 K-1500-62-2 1,500 7 10.000
Air compressor workshop No.3
1 K-250-61-1 250 9 1.500
2 K-250-61-1 250 9 1.500
3 K-250-61-1 250 9 1.500
4 K-250-61-1 250 9 1.500
Air compressor workshop No.4
1 K-345-91-1 345 12.5 3.500
2 K-250-61-5 250 9 1.600
Air compressor station of the oxygen production shop
1 K-1700-61-1 1,700 7 10.000
2 K-1500-62-2 1,500 7 9.000
3 K-1700-61-1 1,700 7 10.000
4 K-1500-62-2 1,500 7 9.000
5 K-1500-62-2 1,500 7 10.000
6 K-1500-62-2 1,500 7 10.000
7 K-1500-62-2 1,500 7 10.000
According to Table 1, we compile the general group of energy consumers:
Let's find the value of the Euclidean distance
between each two consumers.
Since we are performing a single-attribute distribution (a
special case of (2)), the distance between two consumers is
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calculated by the formula:
For instance, ,
.
Based on the results of these calculations, we develop the
set 𝐷 (3). The values of the elements of the set 𝐷 are given
in Table 2.
Table 2. Values of the elements of the set 𝑫
d11 0.001 0.001 … … … 8.4 8.4 8.4
0.001 d22 0.001 … … … 8.4 8.4 8.4
0.001 0.001 d33 … … … 8.4 8.4 8.4
… … … … … … … … …
… … … … … … … … …
8.4 8.4 8.4 … … … d23 23 0.001 0.001
8.4 8.4 8.4 … … … 0.001 d24 24 0.001
8.4 8.4 8.4 … … … 0.001 0.001 d25 25
According to Table 2 and according to (4), we construct the
main variational series 𝑑(𝑠). The results of the calculations
are summarized in Table 3. To avoid uncertainty, we change
the value of distances equal to “0” to 0.001.
Table 3. Main variational series 𝒅(𝒔)
d(s)={ 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
… … … … … … … … … …
1.90 1.90 2.00 2.00 2.00 2.00 5.50 5.50 5.85 5.85
5.85 5.85 6.50 6.50 6.50 6.50 6.50 6.50 6.50 6.85
6.85 6.85 6.85 685 6.85 6.85 6.85 6.85 6.85 6.85
… … … … … … … … … …
8.50 8.50 8.50 8.50 8.50 8.50 8.50 8.50 8.50 8.50
8.50 8.50 8.50 8.50 8.50 8.50 8.50 8.50 8.50 8.50 }
To study the main variational series d(s), using (4) and (5), we develop an auxiliary numeric sequence 𝑟(𝑖) (Table 4).
Table 4. Auxiliary numeric sequence 𝒓(𝒊)
r(i)={ 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.05 1.00 1.00 1.00 2.75 1.00 1.06 1.00 1.00
1.00 1.11 1.00 1.00 1.00 1.00 1.00 1.00 1.05 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00}
Since the main variation series does not have statistically
significant local minima, we will continue the study of the
general group of energy consumers by examining the
behavior of two auxiliary variation series 𝐷̃𝑚𝑖𝑛, 𝐷̃𝑚𝑎𝑥. The
auxiliary variation series are obtained according to (6) and
(7).
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Відновлювана енергетика. № 2/2025 | Комплексні проблеми енергетичних систем на основі НВДЕ
To select classes, we use the hypersphere method.
Let's find the average value 𝑑̅ of auxiliary variational series
(4):
By the formula 𝑑𝑜𝑝𝑡 = 𝑚𝑎𝑥(𝑑̅, 𝑑𝑚𝑖𝑛
(𝑢) ) + 𝑑𝑚𝑖𝑛
(1)
determine
the radius of the hypersphere:
As reference points, let's choose a pair of consumers of the
general group with the largest distance 𝑑(𝑠) = 8.5.
For instance, 𝑥̅13
(0)
= 1.5 та 𝑥̅20
(0)
= 10. Let’s draw two hy-
perspheres of radius with centers at
points 𝑥̅13
(0)
, 𝑥̅20
(0)
and define subsets of points 𝑀13
(0)
,𝑀20
(0)
:
𝑀13
(0)
= {
1.6; 1.6; 1.6; 1.6; 3.15; 3.15; 1.75; 1.75;
1.75; 1.75; 1.5; 1.5; 1.5; 1.5; 3.5; 1.6
} ;
𝑀20
(0)
= {10; 10; 10; 9; 10; 9; 10; 10; 10},
Let’s determine the centers of compression (average val-
ues) of the subsets 𝑀13
(0)
, 𝑀20
(0)
:
𝑥̅13
(1)
= 1.925, 𝑥̅20
(1)
= 9.89
Similarly, we define the subsets 𝑀13
(1)
, 𝑀20
(1)
. To do this, we
will use the centers of compression as reference points
𝑥̅13
(1)
= 1.925,𝑥̅20
(1)
= 9.89 and draw two hyperspheres of
radius with centers at points 𝑥̅13
(1)
and 𝑥̅20
(1)
.
Let’s determine subsets of points M13
(1)
, M20
(1)
:
𝑀13
(1)
= {
1.6; 1.6; 1.6; 1.6; 3.15; 3.15; 1.75; 1.75;
1.75; 1.75; 1.5; 1.5; 1.5; 1.5; 3.5; 1.6
} ;
𝑀20
(1)
= {10; 10; 10; 9; 10; 9; 10; 10; 10}.
and find their centers of compression 𝑥̅13
(2)
= 1.925, 𝑥̅20
(2)
=
9.89.
We verify the condition that the distances between the
centers of compression of the sets 𝑀13
(0)
, 𝑀13
(1)
and 𝑀20
(0)
,
𝑀20
(1)
respectively, are less than
𝜀
2
=
𝑑(1)
2
= 0 . Since
𝑑(𝑥̅13
(1)
, 𝑥̅13
(0)) = 0 and 𝑑(𝑥̅20
(1)
, 𝑥̅20
(0)) = 0 , we finally select
subsets of points:
𝑀13
(1)
= {
1.6; 1.6; 1.6; 1.6; 3.15; 3.15; 1.75; 1.75;
1.75; 1.75; 1.5; 1.5; 1.5; 1.5; 3.5; 1.6
} ;
𝑀20
(1)
= {10; 10; 10; 9; 10; 9; 10; 10; 10}.
After selecting two subsets, we find the difference of the
sets 𝐴1 = 𝐴0 ∖ ⋃ 𝑀𝑖
2
𝑖=1 , 𝐴0 = 𝑋 . Because A1 = 0, the first
stage is over. As a result of the calculations, the following
preliminary classes were obtained M1 = 𝑀13
(1)
та M2 = 𝑀20
(1)
.
Thus, we identify two classes of electrical equipment –
compressors – one of which is characterized by a higher in-
stalled capacity than the other class. This class determines
the energy audit scope of work and includes compressors
K-1500-62-2 and K-1700-61-1 with a capacity of 9 and 10
MW, which are subject to a priority audit.
Conclusions. As a result of the research, the following have
been developed:
1. An algorithm for classifying energy consumers by various
factors, based on the use of the hyper-sphere method,
which allows determining the scope of work during an en-
ergy audit.
2. Methodology for determining the boundaries of energy
audit, which allows focusing on certain consumers with the
greatest energy saving potential, which will save time and
improve the quality of energy audits.
3. This methodology can be used for a wide range of tasks
in the field of renewable energy, starting from the imple-
mentation stage and ending with an energy audit of exist-
ing generation.
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|
| id | veorgua-article-527 |
| institution | Vidnovluvana energetika |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-07-19T01:15:47Z |
| publishDate | 2025 |
| publisher | Institute of Renewable Energy National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | veorgua/55/9082374fbec0e3e7f2185598a0f21755.pdf |
| spelling | veorgua-article-5272026-07-18T06:32:21Z DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL МАТЕМАТИЧНА МОДЕЛЬ ВИЗНАЧЕННЯ МЕЖ ОБСЯГУ РОБІТ З ЕНЕРГЕТИЧНОГО АУДИТУ Rozen , V. Rozen , P. classification, the consumer, fuel and energy resources. класифікація, споживач, паливно-енергетичні ресурси. The Law of Ukraine "On Energy Efficiency" establishes that an energy audit is a systematic analysis of energy use and energy consumption within the boundaries determined by the nature and scope of energy audit work in order to determine, quantify and prepare a report on the possibilities of improving the energy efficiency. When conducting an energy audit, it is important to determine not only the areas of work, but also the scope of work, which is determined by setting boundaries. Currently, the boundaries of energy audit are determined according to the experience and intuition of energy auditors. The paper considers the issue of determining the boundaries of the energy audit scope in industry using the theory of intelligent systems. The importance of using pattern recognition and classification methods for solving problems of energy efficiency improvement is emphasized. It is proposed that classification methods be used to determine typical classes of consumers of fuel and energy resources, making it possible to determine the boundaries and conduct an energy audit. The automatic classification method – the hypersphere method - was chosen as the classification method. The input information for the description of production systems is determined, including indicators, metrics and parameters of energy efficiency. An algorithm for the hypersphere method is developed and practical calculations are given for the conditions of the compressed air production system of a metallurgical enterprise in Zaporizhia. The characteristics of the equipment subject to priority inspection are provided. This algorithm can be used to solve a wide range of problems in the field of renewable energy. Ref. 12. Tab. 5.  Законом України «Про енергетичну ефективність» встановлено, що енергетичний аудит – це систематизований аналіз використання енергії та споживання енергії у межах, визначених характером та обсягом робіт з енергетичного аудиту з метою визначення, кількісного вираження та підготовки звіту про можливості підвищення рівня енергоефективності. Під час проведення енергетичного аудиту важливо визначити не тільки напрями роботи, а також обсяг роботи, який визначається встановленням меж. На теперішній час межі проведення енергетичного аудиту визначаються відповідно досвіду та інтуїції енергоаудиторів. У роботі розглядається питання визначення меж обсягу робіт з енергоаудиту виробничих систем з використанням теорії інтелектуальних систем. Зазначено важливість використання методів розпізнавання образів і класифікації для вирішення задач підвищення рівня енергетичної ефективності. Запропоновано використовувати методи класифікації для визначення типових класів споживачів паливно-енергетичних ресурсів, що дозволило визначити межі проведення енергетичного аудиту. В якості методу класифікації було обрано метод автоматичної класифікації – метод гіперсфер. Визначена вхідна інформація опису виробничих систем до якої входять індикатори, показники та параметри енергетичної ефективності. Розроблено алгоритм методу гіперсфер та наведені практичні розрахунки для умов системи виробництва стисненого повітря металургійного підприємства м. Запоріжжя. Надана характеристика обладнання, що підлягає першочерговому обстеженню. Цей алгоритм можна використовувати для вирішення широкого спектру задач в сфері ВДЕ. Бібл. 12, Табл. 5. Institute of Renewable Energy National Academy of Sciences of Ukraine 2025-06-30 Article Article application/pdf https://ve.org.ua/index.php/journal/article/view/527 10.36296/1819-8058.2025.2(81).67-74 Vidnovluvana energetika ; No. 2(81) (2025): Scientific and applied Journal renewable energy ; 67-74 Возобновляемая энергетика; ##issue.no## 2(81) (2025): Scientific and applied Journal renewable energy ; 67-74 Відновлювана енергетика; № 2(81) (2025): Науково-прикладний журнал Відновлювана енергетика; 67-74 2664-8172 1819-8058 10.36296/1819-8058.2025.2(81) uk https://ve.org.ua/index.php/journal/article/view/527/434 Copyright (c) 2025 V. Rozen , P. Rozen https://creativecommons.org/licenses/by-nc-nd/4.0 |
| spellingShingle | classification the consumer fuel and energy resources. Rozen , V. Rozen , P. DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL |
| title | DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL |
| title_alt | МАТЕМАТИЧНА МОДЕЛЬ ВИЗНАЧЕННЯ МЕЖ ОБСЯГУ РОБІТ З ЕНЕРГЕТИЧНОГО АУДИТУ |
| title_full | DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL |
| title_fullStr | DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL |
| title_full_unstemmed | DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL |
| title_short | DETERMINING THE BOUNDARIES OF THE ENERGY AUDIT SCOPE OF WORK. MATHEMATICAL MODEL |
| title_sort | determining the boundaries of the energy audit scope of work. mathematical model |
| topic | classification the consumer fuel and energy resources. |
| topic_facet | classification the consumer fuel and energy resources. класифікація споживач паливно-енергетичні ресурси. |
| url | https://ve.org.ua/index.php/journal/article/view/527 |
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