STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE
The application of statistical methods for the analysis of random processes to obtain quantitative estimates for the performance indicators of photovoltaic plants as a stochastic energy source is considered. The power generation process is represented by a set of daily random functions with their re...
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2024
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| author | Vasko, P. Mazurenko, I. Sysak , R. |
| author_facet | Vasko, P. Mazurenko, I. Sysak , R. |
| author_institution_txt_mv | [
{
"author": " P. Vasko",
"institution": "Institute of Renewable Energy, NAS of Ukraine, Kyiv, Ukraine"
},
{
"author": " I. Mazurenko",
"institution": "Institute of Renewable Energy, NAS of Ukraine, Kyiv, Ukraine"
},
{
"author": "R. Sysak ",
"institution": " Institute of Renewable Energy, NAS of Ukraine, Kyiv, Ukraine"
}
] |
| author_sort | Vasko, P. |
| baseUrl_str | https://ve.org.ua/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T06:32:20Z |
| description | The application of statistical methods for the analysis of random processes to obtain quantitative estimates for the performance indicators of photovoltaic plants as a stochastic energy source is considered. The power generation process is represented by a set of daily random functions with their respective trends and stochastic components. One hour was taken as the minimum stationari-ty interval of trend’s statistical characteristics. A time period of 16 years was studied to obtain statistically stable estimates. The meteorological database SARAH2 was used as a source of hourly data on the density of solar irradiance, ambient temperature and wind speed in the middle part of the Azov-Black Sea region of Ukraine. For mathematical modelling of power generation and electricity production processes, specialized software PVGIS version 5.2 of the European Commission was used. Hourly quantitative estimates of expected daily power generation trends for each month of the year, as well as their correlation functions, were computed. Algorithms were developed and levels of probabilistic assurance for hourly generated power were evaluated. Statistical estimates of the expected daily, monthly, and annual volumes of electricity production by photovoltaic power plants were studied, taking into account the meteorological conditions in the said region. |
| doi_str_mv | 10.36296/1819-8058.2024.2(77).105-116 |
| first_indexed | 2025-07-17T11:39:30Z |
| format | Article |
| fulltext |
105
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
УДК 621.311.25: 628.165 https://doi.org/10.36296/1819-8058.2024.2(77)105-116
STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A
SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE
Received Apr. 02, 2024; accepted Jun. 21, 2024
Available online Jul. 01, 2024
Vasko P.1, Mazurenko I.2, Sysak R.3
Author for correspondence: Vasko Petro,
e-mail: ivevasko@gmail.com
The application of statistical methods for the analysis of random
processes to obtain quantitative estimates for the performance
indicators of photovoltaic plants as a stochastic energy source is
considered. The power generation process is represented by a set
of daily random functions with their respective trends and stochastic components. One hour was taken as the
minimum stationari-ty interval of trend’s statistical characteristics. A time period of 16 years was studied to
obtain statistically stable estimates. The meteorological database SARAH2 was used as a source of hourly data
on the density of solar irradiance, ambient temperature and wind speed in the middle part of the Azov-Black
Sea region of Ukraine. For mathematical modelling of power generation and electricity production processes,
specialized software PVGIS version 5.2 of the European Commission was used. Hourly quantitative estimates of
expected daily power generation trends for each month of the year, as well as their correlation functions, were
computed. Algorithms were developed and levels of probabilistic assurance for hourly generated power were
evaluated. Statistical estimates of the expected daily, monthly, and annual volumes of electricity production by
photovoltaic power plants were studied, taking into account the meteorological conditions in the said region.
Bibl. 48, Tables 2, Fig. 8.
Key words: random function, electricity, probability, power, solar irradiance, statistics, photovoltaic plant.
СТАТИСТИЧНА ОЦІНКА ПОКАЗНИКІВ ФУНКЦІОНУВАННЯ ФОТОЕЛЕКТРИЧНИХ СТАНЦІЙ ЯК
ДЖЕРЕЛА ЕНЕРГІЇ ДЛЯ ОПРІСНЕННЯ ВОДИ В АЗОВО-ЧОРНОМОРСЬКОМУ РЕГІОНІ УКРАЇНИ
Отримано 02 квіт. 2024 р.; рекомендовано до публікації 21 чер. 2024 р.
Доступно онлайн 01 лип. 2024 р.
Васько П. Ф.1, Мазуренко І. Л.2, Сисак Р. М.3
Автор для кореспонденції: Васько Петро,
e-mail: ivevasko@gmail.com
Розглянуто застосування статистичних методів аналізу
випадкових процесів для отримання кількісних оцінок показ-
ників функціонування фотоелектричних станцій як стохас-
тичного джерела енергії. Процес генерування електроенер-
гії представлено сукупністю щоденних випадкових функцій
зі своїм трендом та стохастичною складовою. За мінімальний інтервал стаціонарності статисти-
чних характеристик тренду приймалась одна година. Досліджувався період часу тривалістю 16 років
для отримання статистично стійких оцінок. Використовувалась метеорологічна база даних SARAH2
з погодинною інформацією про щільність сонячного випромінювання, температуру навколишнього
середовища та швидкість вітру в середній частині Азово-Чорноморського регіону України. Для
1 Dr. of Tech. Sciences
https://orcid.org/0000-0001-8807-7173
2 Cand. of Tech. Sciences
https://orcid.org/0000-0002-0146-7396
3 Cand. of Tech. Sciences
https://orcid.org/0000-0003-4474-4776
1,2,3 Institute of Renewable Energy, NAS of
Ukraine, Kyiv, Ukraine
1 д-р техн. наук
https://orcid.org/0000-0001-8807-7173
2 канд. техн. наук
https://orcid.org/0000-0002-0146-7396
3 канд. техн. наук.
https://orcid.org/0000-0003-4474-4776
1,2,3 Інститут відновлюваної енергетики НАН
України, м. Київ, Україна
106
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
математичного моделювання процесів генерування потужності та виробництва електроенергії за-
стосовувалось спеціалізоване програмне забезпе-чення PVGIS версії 5.2 Європейської Комісії. Отри-
мано очікувані погодинні кількісні оцінки параметрів щоденних трендів генерування потужності для
кожного місяця року та їх кореляційні функції. Розроблено алгоритми та визначено ймовір-нісні рівні
забезпеченості щогодинної генерованої потужності. Досліджено статистичні оцінки очікуваних ден-
них, місячних та річних обсягів виробництва електроенергії фотоелектростанціями з урахуванням
метеорологічних умов регіону. Бібл. 48, табл. 2, рис. 8.
Ключові слова: випадкова функція, електроенергія, імовірність, потужність, сонячне випромінювання,
статистика, фотоелектрична станція.
Introduction. The importance of the problem of water de-
salination on the industrial scale in the Azov-Black Sea re-
gion of Ukraine is caused, first of all, by the need to prepare
fresh water for the electrolytic production of “green” hy-
drogen, in accordance with Ukraine’s participation in the
implementation of the European program “2x40 GW Green
Hydrogen Initiative” [1, 2]. The program envisages the con-
struction of 10 GW capacity of electrolysers on the territory
of Ukraine for the production of low-emission hydrogen in
the amount of 1.65 million tons per year. That, in turn, will
stipulate the demand for prepared fresh water of at least
24 million m3/year [3]. It’s important to note that the needs
for fresh water in Azov-Black Sea region of the country are
not limited only to the implementation of the “green” hy-
drogen production program. The use of water desalination
technologies is relevant both in industry and the private
sector for the preparation of drinking water, for agriculture
and desalination of mine waters [4 – 8]. The choice of an
acceptable desalination technology depends on the com-
position and content of salts in the feed water and also the
requirements to desalinated water’s properties [9 – 12].
Each technology is characterized by the corresponding
costs and amounts of energy consumed to produce a unit
of fresh water (kW∙h/m3, kJ/kg, etc). Energy supply for tech-
nological processes can be based on either fossil fuels or
renewable sources [13 – 15]. The southern territories of the
country have the highest potential of wind energy and solar
irradiance. A number of powerful wind power plants (WPP)
and photovoltaic power plants (PVP) have already been
constructed and commissioned there [16]. The use of en-
ergy from WPPs and PVPs has no alternative for the pro-
duction of “green” hydrogen, taking into account the stage
of desalinated water preparation for the operation of elec-
trolysers. The general view of a high-output PVP on the ter-
ritory of Ukraine is shown in Fig. 1 [17].
This paper aims at investigating indicators of the PVP’s sto-
chastic power generation process to achieve the maximum
integration of its energy into the technological schemes of
water desalination in the climatic conditions of the Azov-
Black Sea region of Ukraine. The stochastic changes of the
generated power are caused by the influence of cloudiness,
air temperature, and wind speed at the PVP’s location [18].
However, the energy equipment of desalination facilities
operates under a stable power supply, thus requiring coor-
dination in time of the power consumption and generation
processes [19 – 21]. The search for acceptable compromise
solutions regarding the implementation of rational modes
of energy generation and consumption requires the deter-
mination of equipment load diagrams of the desalination
technological scheme and statistical estimations of the sto-
chastic process of electrical energy generation at the PVP.
Fig. 1. General view of a high-output PVP [17]
107
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
The measurements of the PVP’s generated power during
several individual days, which are shown in Fig. 2, indicate
that it is appropriate to use the principles of random pro-
cesses theory [22 – 24] to analyse the variability of the
PVP’s operation indicators (power and volume of electric-
ity production). Therefore, the stochastic process of PVP’s
power generation during long time intervals can be repre-
sented by a set of daily random functions; statistical
methods for the analysis of such processes are covered in
[25 – 28].
Problem statement. The main task of this study is to deter-
mine the quantitative statistical estimates of the parameters
of power generation stochastic processes and the PVP’s elec-
tricity production in the climatic conditions of the Azov-Black
Sea region of Ukraine. The set of daily random functions of
PVP power generation, that is necessary for achieving statis-
tically stable results, will be formed using mathematical
modelling of the photo-voltaic modules’ operation modes
and information about the changes in meteorological factors
at the plant’s location over a long time [18, 29, 30].
Fig. 2. Set of daily random functions of PVP’s power generation
Mathematical modelling of power generation processes
and outputs of the PVP and processing of the results. The
main components of the high-output PVP’s electrical circuit
include photovoltaic modules (PVM), photovoltaic batter-
ies, inverters, internal power transmission lines, internal
transformer substations, a diagnostic and monitoring sys-
tem, a control system, output transformer substations and
power transmission lines for connection to the industrial
power system or power consumer [31, 32]. PVM is the
smallest component that can be connected to the electric
circuit of the PVP. The nominal power of modern industrial
PVMs is usually less than 600 W, therefore PVPs have a dis-
tributed structure with a large number of components. So,
in particular, a typical electrical scheme of a 50 MW PVP
includes approx. 200,000 PVMs, 2,200 inverters, and 40 in-
ternal transformer substations (0.4/10 kV).
During operation, some part of the components fails and
needs to be replaced, losses of electrical energy in the equip-
ment and transmission lines exist, and there is a degradation
of photovoltaic properties of PVMs. All this together decreases
the power efficiency of the PVP. Degradation of PVM’s photo-
voltaic properties is caused by temperature changes cyclicity,
the influence of the ultraviolet spectrum of solar radiation and
hot spots on photocells, the cracks appearing on the module
surfaces and sealing failures [33, 34]. PVM manufacturers pro-
vide warranties regarding yearly degradation level, that de-
pends on the type of modules and can change in the range of
(0.25...0.8)% per operation year [35].
Considering the above and theoretical principles of photo-
voltaic energy [18, 29, 30, 32], let’s write down the initial
expressions for determining the generated power of the
PVP at an arbitrary moment in time (t) as follows:
( ) ( ) ( ) ( ) ( )( )tPNktktNktPNtP nedgns ,, = , (1)
( ) ( ) ( ) ( )tkftItP nrn = ,,, 1 , (2)
( ) ( ) ( ) ( )( )tvtQtPftk nn ,,2= , (3)
where ( )tPs – power of the PVP; N – number of PVMs in
the PVP; ( )tPn – power of the PVM; gk – equipment avail-
ability factor of the PVP; dk – degradation factor of the
PVM; ek – factor of electrical power losses in the compo-
nents of power plant (internal electrical joins, invertors,
transformer substations, tie lines); rI – solar energy flow
density; , – geographical coordinates at the PVP loca-
tion; φ – tilt-angle between the PVM surface and the hori-
zontal plane; α – azimuth of the PVM surface normal rela-
tive to the South; nk – PVM efficiency ratio; v,Q – air
temperature and wind speed at the PVP’s location.
Set of equations (1) – (3) allows for mathematical modelling
of PVP power depending on the climatic conditions at the
construction site, spatial arrangement and electrical per-
formance of the PVM, and the power plant design.
Climatic conditions are characterized by the solar energy
flow density on a clear day, the presence of cloudiness, air
PS(t,
1)
05:00
20:00
ti
ti+k
t, hrs
PS(t, 2)
PS(t, j)
t, hrs
t, hrs
108
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
temperature and wind speed. Density values rI depend on
geographic coordinates, meteorological factors, and time of
day. Daily trend of the rI value changes is uniquely deter-
mined by the trajectory of the solar irradiance flow relative
to the PVM surface [36], and the stochastic component – by
the rate of change in cloud cover, ambient temperature, and
wind speed. In order to obtain statistically stable estimates
of values ( ) ,,tIr , it is required to possess the results of
meteorological observations at long time intervals [37]. The
necessary information for the territory of the Azov-Black Sea
region of Ukraine can be found in the international data-
bases of NASA SSE (NASA Surface meteorology and Solar En-
ergy) and PVGIS (Photovoltaic Geographical Information Sys-
tem) of the European Commission. Further research will use
the PVGIS database [38 – 40], which provides weather infor-
mation on solar irradiance, air temperature, and wind speed
at a height of 10 meters above the Earth’s surface for 16
years: from 2005 to 2020, inclusive. In accordance with
methodological recommendations on the applied analysis of
hydro-meteorological factors [41], the use of a period of 16
consecutive years is sufficient to determine trend fluctua-
tions and random features of the process. PVGIS software
version 5.2 [42] implements hourly calculations of PVM
power with arbitrary orientation of its surface, taking into ac-
count ambient temperature and wind cooling. Therefore, ap-
plication of the mentioned software allows for obtaining
quantitative values of expressions (1) – (3), which, after data
systematization, can be represented by a set of daily random
functions of the generated power ( )itPs , , as it is shown in
Fig 2, and random sequence of values (time series) of the
power at fixed moments of day time tj for the set of random
functions:
( ) YititP jjs ...,,2,1,24...,,2,1,, == , (4)
where Y – number of observations of the daily random
functions of the PVP’s generated power.
Sequential values of the time series, in contrast to simple sta-
tistical samplings, can be mutually dependent for the closely
located observations. The strength of the mutual depend-
ency is determined by physical properties of the process and
by discretization step of the sample time moments.
We will perform the statistical processing of the defined
above random process of the PVP power generation, ac-
cording to the periodical changes of the meteorological fac-
tors [37, 41] and the capabilities of the specified software,
for the hourly, daily, and monthly time intervals during the
16-year period of consecutive years using the mathemati-
cal principles from [24, 25, 43].
We will determine the daily trend ( )jsm tP of the random
function of power generation for a particular month of the
year (m) by computing the set of its average values at fixed
time moments tj during the entire period of observation:
( ) ( )
=
=
mmYD
i
jsm
mm
jsm itP
YD
tP
1
,
1
, 24...,,2,1=jt , (5)
where m – sequential number of the month in the year, Ym
– number of the same months m within the observation pe-
riod, Dm – number of days in the month m, ( )itP jsm , – ran-
dom sequence of the values of power at the fixed day time
moments tj for month m.
The average values ( )jsm tP characterize a center of varia-
tion of the random process of power generation at the fixed
time moments of the specified month of the year. The var-
iation of the random power generation value around the
center is characterized by standard deviation σpm:
( ) ( ) ( )( )
= =
−
−
=
m mY
k
D
i
jsmjsm
mm
jpm tPitP
YD
t
1 1
22 ,
1
1
, 24...,,2,1=jt , (6)
The strength of interdependency of the random se-
quence members ( )itP jsm , is determined by normalized
correlation function ( ),jm tr :
( ) ( ) ( )jpmjmjm ttRtr 2,, = ,
( ) ( ) ( )( ) ( ) ( )( )
−
−+−
−−
=
mmYD
i
jsmjsmjsmjsm
mm
jm tPitPtPitP
YD
tR ,,
1
1
, , (7)
24...,,2,1=jt , ...,4,3,2,1= .
The measure of correlation relationship is the correlation
interval d, that characterizes an average number of days
between intersections j and (j + d) of the random sequence,
beyond which the values of the generated power can be
considered as practically independent random variables.
For the considered task, the value of correlation interval
can be understood as the expected duration of continuous
total cloudiness. Quantitatively, the value of correlation co-
efficient ( )jm td at fixed day time moments tj for month m
will be determined from the following condition:
( ) 02.0, jm tr for ( )jm td . (8)
Probability distribution of the random sequence is an im-
portant statistical property of the considered quantity.
However, for the task of integration of electrical energy
from renewable sources into technological schemes of wa-
ter desalination, it is more important to know the duration
of power generation [44]. Therefore, in further analysis, we
will use the statistical characteristic of the probabilistic as-
surance of the achievable levels of the generated power
( )Ptz jpm , . We will understand the mathematical essence
of variable z as the probability for the random value of
power to be greater than some given value P. Calculation
of the probabilistic assurance will be performed with the
109
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
use of random sequence ( )itP jsm , . Let’s divide the range
of smP variation into G intervals of the same width, moreo-
ver, let ( )gtP jsm , be a middle of interval g, where
g = 1, 2, …, G. Based on the sequence ( )itP jsm , , let’s build
ordered list of its values and then count a number
( )gtn jpm , of samples that fall in every interval. Depend-
ency ( )Ptz jpm , is computed as follows:
( ) ( )
( )
−=
PgtPGg
jpm
mm
jpm
jsm
gtn
DY
Ptz
,,1
,
1
1, , [p.u.], mDzz = p.u.days . (9)
Estimation of the expected amount of PVP electrical en-
ergy production is computed for monthly and yearly
time intervals based on the daily trend of the random
function of power generation for a particular month of
year (5):
( )
=
=
mm
m
YD
i Tj
jsm
mm
sm itP
YD
E
1
,
1
, (10)
=
12
1
sms EE , (11)
where smE , sE – expected monthly and yearly amount of
PVP electrical energy production; mT – interval of energy
generation in month m.
Dispersion of monthly amount of electrical power produc-
tion (σem) around the corresponding average values is de-
termined as:
( )( )
=
−
−
=
mmYD
i
smsm
mm
em EiE
YD 1
22
1
1
, ( ) ( )
=
mTj
jsmsm itPiE , . (12)
Calculation of the probabilistic assurance of the daily elec-
trical energy production amount during a month will be
performed according to the described algorithm for the
generated power:
( ) ( )
( )
−=
EgEGg
em
mm
em
sm
gn
DY
Ez
,1
1
1 , [p.u.], mDzz = p.u.days , (13)
where ( )gEsm – middle of interval g built for the random
sequence ( )iEsm , ( )gnem – number of samples that fall in
the interval g.
Numerical experiment and analysis of results. Computa-
tions were performed for a hypothetical PVP with installed
capacity of 1 kW, located in the central part of the Azov-Black
Sea region of Ukraine. For this value of the installed capacity,
all obtained quantitative results will be equivalent to the in-
dicators of a plant with arbitrary power represented in p.u.,
and therefore can be easily scaled for a designed PVP with
desired capacity. The following provisions and assumptions
were used to conduct the numerical experiment:
− computations are performed for the first operation year
of the PVP, without consideration of the degradation of
electrical properties of the photovoltaic modules;
− total power losses in the PVP, including the reliability of
the components, are equal to 14%;
− crystalline silicon photovoltaic panels are used;
− lower limit for the sensitivity of the modules to luminos-
ity is estimated around 150 W/m2;
− the panels mounting is fixed-tilt, with tilt angle corre-
sponding to the geographical latitude of the central part
of the region 47° (longitude 35°), azimuth – 0°;
− database of solar irradiance – PVGIS-SARAH2 with
hourly data for the time period of 2005-2020.
The obtained functional dependencies of the daily trend, as
well as its standard deviation, for all months of the year, are
shown in Fig. 3. They have a convex shape, which is a direct
consequence of the daily cycle of solar activity. According
to the obtained results, the process of annual power gen-
eration can be represented by three typical periods with
corresponding quantitative indicators: “winter” period
from November to February, “spring-summer” period from
April to September, and “transient period” including March
and October. In the “winter” period, it is possible to gener-
ate power for (7-8) hours a day. The power trend values are
low compared to the installed capacity of the PVP, and sig-
nificant variability of the generation process is observed.
These peculiarities are caused by significant cloudiness var-
iability, which is typical for this period of the year. For the
“spring-summer” period, the daily duration of generation is
expected to be within (11-12) hours with significant power
and low variability. In the most advantageous hours, the
maximum value of the power trend reaches the level of
(0.60-0.64) kW, which is comparable to the nominal value
of the generated power of 0.86 kW (according to the con-
ditions of the numerical experiment, described above). The
obtained power ratios indicate a low probability of cloudi-
ness at noon in the specified months.
Estimates of the normalized correlation function were com-
puted for each working hour of each month. The amount of
obtained information is too large for a detailed presenta-
tion in this publication, therefore we will only present the
results for the middle of the daylight hours in the typical
months of the above-mentioned periods of the year
(Fig. 4). According to the calculated dependencies, the in-
terruption of PVP power generation is expected in August,
due to continuous cloudiness, for at least 3 consecutive
days. For other months, the interruption of generation can
be observed for more than a week.
Probabilistic assurance of achievable levels of generated
power was calculated for each working hour of each
month, too. As an example, the obtained functional de-
pendencies for two different hours of the day of typical
months are presented in Fig. 5, where the selected points
on the curves correspond to the average values of the daily
trend in the PVP power output for the corresponding
month at the indicated hours.
110
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
Fig. 3. Daily trend and standard deviation of the random power generation function
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р σ
january
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
july
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
february
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
august
0
0.2
0.4
0.6
07:00 09:00 11:00 15:00 17:00 13:00 ti, hrs
P, σ,
kW
Р σ
march
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
september
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
april
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
october
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
may
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р σ
november
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р
σ
june
0
0.2
0.4
0.6
07:00 09:00 11:00 13:00 15:00 17:00 ti, hrs
P, σ,
kW
Р σ
december
111
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
Fig. 4. Dependencies of the normalized correlation function for typical months at 12 o’clock
Fig. 5. Curves of probabilistic assurance of PVP power for 9:00 a.m. and 12:00 a.m. of some months
The results of power generation around (12:00 a.m. – 1:00
p.m.) of the day characterize the area of maximum values
of the monthly trends in the plant’s energy efficiency
(Fig. 3). Therefore, the levels of probabilistic assurance of
generated power, determined for this moment in time, will
represent achievable values for each month during the
year. The information shown in Fig. 5 makes it possible to
determine achievable quantitative values of indicators for
typical months of the above-mentioned periods of the year.
For example, in August at 12:00 a.m., the generation will
exceed the monthly daily average value for 23 days, in April
– 18 days, in October – 17 days, in January – 8 days. Simi-
larly, the probabilistic assurance of other generation out-
puts can be determined. In particular, power generation of
more than 0.5 kW will be observed for 6 days in January, 17
days in October, 21 days in April, and 27 days in August. For
the 9:00 a.m., power generation of more than 0.3 kW will
be observed for 1 day in January, 15 days in October, 18
days in April, and 27 days in August.
According to the obtained results (Fig. 5), the maximum power
value is observed in April and reaches 0.92 kW, whereas the
maximum value in August is 0.82 kW. This situation is caused
by the influence of the ambient temperature and cooling of
the photovoltaic panels by wind on the efficiency of the solar
irradiance photoelectric conversion process.
The amount of electricity production is determined by the pa-
rameters of the daily power trend and the duration of genera-
tion in the specified time intervals. In this work, we analyzed
daily, monthly and annual volumes of production. Indicators
of the PVP’s daily electricity production for different months
of the year are given in Table 1 and partially visualized in Fig. 6
in the form of annual trends of the average value and standard
deviation. In the winter period, the process of electricity
-0.2
0
0.2
0.4
0.6
0.8
rm(ti, τ)
0 1 2 3 4 5 6 7 τ, days
august october juanuary april
0
0.2
0.4
0.6
0.8
Р,
kW
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 z, days
12:00
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
09:00
z, days
Р,
kW
112
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
production is characterized by significant variability, when the
value of standard deviation reaches the level of the average
value, which is due to the presence of cloudiness in this period
of time. The electricity output in the winter season is almost
three times smaller than in the summer months. In particular,
the average value of the daily electricity production in August
is about 4.8 kW∙h/day, while in January it is only 1.4 kW∙h/day.
The total duration of electricity generation in August is 372
hours, in April – 360 hours, in October – 300 hours, in January
– 248 hours.
Table 1. Indicators of daily electricity production of PVP by months of the year
Indicators
Months
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Emax , kW∙h/day 5.200 5.998 6.559 6.748 6.525 5.993 5.887 6.079 6.140 5.834 5.189 4.688
E , kW∙h/day 1.407 2.059 3.443 4.251 4.506 4.488 4.604 4.793 4.257 3.171 1.928 1.275
σ , kW∙h/day 1.36 1.754 1.876 1.745 1.329 1.13 0.979 1.063 1.472 1.783 1.554 1.255
Tm, hours 248 254 310 360 372 390 403 372 330 300 269 248
Fig. 6. Trends in the PVP’s daily electricity outputs by months of the year
The obtained functional dependencies of the probability of
the daily volume of electric energy production at a monthly
time interval are presented in Fig. 7 for the most representa-
tive months of the respective seasons. The maximum possi-
ble value of the daily production is observed in the month of
April, the same as in the case of power generation. The total
duration of generation on the level of average daily produc-
tion or above is expected in August for 20 days, in April – 17
days, in October – 16 days, in January – 10 days. For example,
a daily volume of electric energy production of more than 4
kW∙h/day will be observed in August for 25 days, in April – 18
days, in October – 12 days, in January – 2 days. Exceeding the
daily production volume of 5 kW∙h/day will be observed in
August for 18 days, in April – 13 days, in October – 6 days, in
January – for less than one day. Probabilistic assurance of
other values of the daily amount of electricity production for
any month of the year can be determined in a similar way
based on the data in Table 2.
The results of calculation studies with respect to the annual
electricity production are presented in Fig. 8.
The highest electricity production within the investi-
gated time interval was observed in 2020 – nearly
1,300 kW∙h/year, and the lowest in 2006 – 1,120 kW∙h/year.
Consequently, the value of the capacity factor of this PVP
changes in the range (0.128 - 0.148) p.u. There are signs of
climatic cyclicity in the amount of electricity production with
a duration of (3-5) years.
Fig. 7. Probabilistic assurance of the daily volumes of electrical energy production for monthly time interval
0
1.0
2.0
3.0
4.0
5.0
6.0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Е, kW h/day
кВт·год/день
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
σ
kW∙h/day
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
z, days
113
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
Table 2. Probabilistic assurance of the PVP’s daily volumes of electrical energy by months, days
z,
days
Е, kW·h/day
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
0 5.200 5.998 6.559 6.748 6.525 5.993 5.887 6.079 6.140 5.834 5.189 4.688
2 4.182 5.296 6.160 6.254 5.981 5.684 5.605 5.748 5.779 5.468 4.447 3.934
4 3.512 4.769 5.872 6.082 5.851 5.576 5.520 5.674 5.613 5.258 4.119 3.189
6 2.822 3.763 5.578 5.973 5.742 5.477 5.438 5.601 5.521 5.062 3.817 2.399
8 1.939 2.893 5.177 5.791 5.616 5.358 5.341 5.525 5.399 4.793 3.248 1.869
10 1.421 2.199 4.576 5.533 5.460 5.214 5.237 5.460 5.295 4.550 2.659 1.389
12 1.157 1.752 4.127 5.257 5.237 5.045 5.131 5.409 5.143 4.159 2.011 1.039
14 0.843 1.371 3.748 4.934 4.975 4.882 5.009 5.291 4.942 3.677 1.551 0.767
16 0.718 1.033 3.294 4.536 4.748 4.720 4.836 5.159 4.623 3.235 1.138 0.650
18 0.615 0.825 2.813 3.878 4.477 4.539 4.668 4.967 4.300 2.913 0.795 0.518
20 0.529 0.654 2.447 3.468 4.243 4.256 4.425 4.728 3.954 2.323 0.687 0.418
22 0.472 0.547 1.984 2.997 3.865 3.899 4.258 4.492 3.429 1.788 0.548 0.372
24 0.382 0.442 1.580 2.250 3.573 3.590 4.008 4.224 2.858 1.218 0.457 0.317
26 0.305 0.360 1.152 1.770 3.163 3.045 3.647 3.828 2.201 0.767 0.357 0.269
28 0.251 0.205 0.871 1.225 2.478 2.237 3.198 3.366 1.392 0.592 0.267 0.225
30 0.186 - 0.636 0 1.354 0 2.246 1.869 0 0.430 0 0.162
Fig. 8. Annual volumes of electricity production of the PVP within the studied time interval
Discussion. The statistically estimated energy indicators for
the PVP’s power generation process are given for a plant
with the fixed-tilt mounting of PVMs and their orientation
to the South. The tilt angle was equivalent to the geograph-
ical latitude at the installation location (φ). That slope was
justified by estimation of the annual volume of electrical
energy production with variable angle in the range from
(φ – 20) to (φ + 20). Increasing the tilt angle leads to a de-
crease in production volume, therefore this range of angle
change was not studied. When the slope decreases, an in-
crease in volumes is observed. However, at the edge of the
specified range the increase is as small as approx. 5%, due
to weaker cooling of the structure by wind and consequent
increase in the PVM’s temperature. As a result, the output
of electrical energy decreases in autumn-winter period,
which causes significant variability in the production
amount throughout the year. The stochastic component of
the generated output remain virtually unchanged within
the specified range of changes in the tilt angle. An increase
in the PVM’s temperature contributes to faster degrada-
tion of their energy properties. It is worth noting that a spe-
cific value of PVM tilt angle for a particular plant will also
depend on conditions for cleaning the surface of the mod-
ules from dust, which improve when the angle increases.
These conditions vary depending on the location of the sta-
tion and its power. Therefore, the quantitative results pre-
sented in the publication can be used in feasibility studies
for PVP construction projects in the Azov-Black Sea region,
and then refined at the stage of operational design by ap-
plying specific input data and using specialized commercial
software.
The calculation of quantitative statistical estimates of the
PVP’s energy indicators was based on the determination of
the daily trend in power generation and its random compo-
nent for each month of the year separately using an
1000
1050
1100
1150
1200
1250
1300
2
0
0
5
2
0
0
6
2
0
0
7
2
0
0
8
2
0
0
9
2
0
1
0
2
0
2
0
2
0
1
1
2
0
1
2
2
0
1
3
2
0
1
4
2
0
1
5
2
0
1
6
2
0
1
7
2
0
1
8
2
0
1
9
Е, kW h/year
114
Відновлювана енергетика. №2/2024 | Гідро-воднева енергетика
observation interval of 16 years. The duration of the sta-
tionarity interval was equal to one hour according to the
used database of historical data. Therefore, the obtained
characteristics of the probabilistic assurance of power lev-
els and correlation functions at monthly time intervals are
characterized by significant reliability and allow quantita-
tive assessment of the electrical energy production pro-
cess. For example, the duration of interruption in PVP
power generation, due to continuous cloudiness, is ex-
pected in August for at least three consecutive days (fig. 4).
In January, this interval is at least eight days in a row. Since
the maximum value of daily electricity production in August
is almost three times greater than in January (fig. 5), it can
be assumed that the minimum capacity of the PVP electric-
ity storage must exceed, at least three times, the largest
daily generation volume in the annual time interval. It is
possible to use pumped hydro energy storage on land [45],
sea [46], or underground [47] to store the energy of a large
PVP. However, it is worth noting that determining the
amount of energy storage in each specific project requires
comprehensive consideration of water storage, monitoring
and control systems, operating modes, and equipment
costs to achieve competitive technical and economic per-
formance indicators of the facility [48].
Conclusions
1. The process of energy generation by a photovoltaic plant
is represented by a set of daily random functions. Algo-
rithms have been developed for calculating expected
trends and their stochastic components, normalized corre-
lation functions, probabilistic assurance of different levels
of power and volumes of electrical energy production. Cal-
culation studies were performed for a hypothetical power
plant with installed capacity of photovoltaic modules of 1
kW that allows further scaling of the results to arbitrary
power. To obtain statistically stable estimates, a 16-year
time interval from 2005 to 2020 was studied. As the source
of information, meteorological database SARAH2 was used,
containing hourly information on the density of solar irra-
diance, ambient temperature, and wind speed in the mid-
dle part of the Azov-Black Sea region of Ukraine. The ob-
tained quantitative results can be used for feasibility
studies of the integration of PVP energy into the technolog-
ical schemes of water desalination in this region. The re-
sults can be further specified at the stage of operational de-
sign by applying specific input data and using specialized
commercial software.
2. In the most advantageous hours of the summer period,
the maximum value of the power trend reaches the level of
(0.60-0.64) kW, and the corresponding average daily power
value is 0.4 kW. Algorithms have been developed for calcu-
lating the probability of exceeding the various required lev-
els of the power and volumes of electricity production by
the PVP in each month and each hour. In particular, at noon
(12:00 a.m.), power generation of more than 0.4 kW will be
observed: in August – 28 days, in April – 22 days, in October –
18 days, in January – 7 days. In the morning, at 9:00 a.m.,
the duration of generation of the average daily power value
of more than 0.4 kW can be expected: in August – 18 days,
in April – 15 days, in October – 11 days, in January – 0 days.
In August, the interruption of PVP’s power generation,
caused by continuous cloudiness, is expected for at least 3
consecutive days. For the months of the autumn-winter pe-
riod, the interruption of generation, due to continuous
cloudiness, can be observed for more than a week.
3. Statistical estimates of the PVP’s expected daily, monthly
and annual volumes of electricity production were ob-
tained taking into account the meteorological conditions of
the region. The achievable annual value of the capacity fac-
tor of the PVP, provided that the generated energy is fully
integrated into the technological schemes of water desali-
nation, can be expected in the range of (0.128 – 0.148) p.u.
In the winter period, the process of electricity generation
demonstrates significant variability. The value of standard
deviation reaches the level of the average value, which is
caused by the presence of cloudiness in this period of time.
The volume of electricity production in the winter months
is almost three times smaller than in the summer months.
In particular, the average value of the daily amount of elec-
tricity produced by the hypothetical PVP in August is about
4.8 kW∙h/day, and in January – only 1.4 kW∙h/day.
It is determined that the minimal capacity of the PVP stor-
age system, required for the stable and uninterruptible
power supply of the desalination facility, must be at least
three times as big as the maximum daily volume of energy
production of the plant evaluated in the yearly time inter-
val.
The signs of climatic cyclicity in the amount of electricity
production with a duration of (3-5) years were detected.
4. Successful integration of PVP electricity into technologi-
cal schemes of water desalination requires detailed tech-
nical and economic optimization of the entire technological
complex composed of the desalination facility, power
plant, energy and water storages, monitoring and control
system, as well as the cost of equipment, to achieve com-
petitive technical and economic performance indicators.
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| id | veorgua-article-461 |
| institution | Vidnovluvana energetika |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-07-19T01:13:32Z |
| publishDate | 2024 |
| publisher | Institute of Renewable Energy National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | veorgua/17/bae6194569199ccf5ba3661f39f21317.pdf |
| spelling | veorgua-article-4612026-07-18T06:32:20Z STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE СТАТИСТИЧНА ОЦІНКА ПОКАЗНИКІВ ФУНКЦІОНУВАННЯ ФОТОЕЛЕКТРИЧНИХ СТАНЦІЙ ЯК ДЖЕРЕЛА ЕНЕРГІЇ ДЛЯ ОПРІСНЕННЯ ВОДИ В АЗОВО-ЧОРНОМОРСЬКОМУ РЕГІОНІ УКРАЇНИ Vasko, P. Mazurenko, I. Sysak , R. random function, electricity, probability, power, solar irradiance, statistics, photovoltaic plant. випадкова функція, електроенергія, імовірність, потужність, сонячне випромінювання, статис-тика, фотоелектрична станція. The application of statistical methods for the analysis of random processes to obtain quantitative estimates for the performance indicators of photovoltaic plants as a stochastic energy source is considered. The power generation process is represented by a set of daily random functions with their respective trends and stochastic components. One hour was taken as the minimum stationari-ty interval of trend’s statistical characteristics. A time period of 16 years was studied to obtain statistically stable estimates. The meteorological database SARAH2 was used as a source of hourly data on the density of solar irradiance, ambient temperature and wind speed in the middle part of the Azov-Black Sea region of Ukraine. For mathematical modelling of power generation and electricity production processes, specialized software PVGIS version 5.2 of the European Commission was used. Hourly quantitative estimates of expected daily power generation trends for each month of the year, as well as their correlation functions, were computed. Algorithms were developed and levels of probabilistic assurance for hourly generated power were evaluated. Statistical estimates of the expected daily, monthly, and annual volumes of electricity production by photovoltaic power plants were studied, taking into account the meteorological conditions in the said region. Розглянуто застосування статистичних методів аналізу випадкових процесів для отримання кількісних оцінок показників функціонування фотоелектричних станцій як стохастичного джерела енергії. Процес генерування електроенергії представлено сукупністю щоденних випадкових функцій зі своїм трендом та стохастичною складовою. За мінімальний інтервал стаціонарності статистичних характеристик тренду приймалась одна година. Досліджувався період часу тривалістю 16 років для отримання статистично стійких оцінок. Використовувалась метеорологічна база даних SARAH2 з погодинною інформацією про щільність сонячного випромінювання, температуру навколишнього середовища та швидкість вітру в середній частині Азово-Чорноморського регіону України. Для математичного моделювання процесів генерування потужності та виробництва електроенергії застосовувалось спеціалізоване програмне забезпе-чення PVGIS версії 5.2 Європейської Комісії. Отримано очікувані погодинні кількісні оцінки параметрів щоденних трендів генерування потужності для кожного місяця року та їх кореляційні функції. Розроблено алгоритми та визначено ймовір-нісні рівні забезпеченості щогодинної генерованої потужності. Досліджено статистичні оцінки очікуваних денних, місячних та річних обсягів виробництва електроенергії фотоелектростанціями з урахуванням метеорологічних умов регіону. Бібл. 48, табл. 2, рис. 8. Institute of Renewable Energy National Academy of Sciences of Ukraine 2024-07-01 Article Article application/pdf https://ve.org.ua/index.php/journal/article/view/461 10.36296/1819-8058.2024.2(77).105-116 Vidnovluvana energetika ; No. 2(77) (2024): Scientific and applied Journal renewable energy ; 105-116 Возобновляемая энергетика; ##issue.no## 2(77) (2024): Scientific and applied Journal renewable energy ; 105-116 Відновлювана енергетика; № 2(77) (2024): Науково-прикладний журнал Відновлювана енергетика; 105-116 2664-8172 1819-8058 10.36296/1819-8058.2024.2(77) uk https://ve.org.ua/index.php/journal/article/view/461/370 Copyright (c) 2024 P. Vasko, I. Mazurenko, R. Sysak https://creativecommons.org/licenses/by-nc-nd/4.0 |
| spellingShingle | random function electricity probability power solar irradiance statistics photovoltaic plant. Vasko, P. Mazurenko, I. Sysak , R. STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE |
| title | STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE |
| title_alt | СТАТИСТИЧНА ОЦІНКА ПОКАЗНИКІВ ФУНКЦІОНУВАННЯ ФОТОЕЛЕКТРИЧНИХ СТАНЦІЙ ЯК ДЖЕРЕЛА ЕНЕРГІЇ ДЛЯ ОПРІСНЕННЯ ВОДИ В АЗОВО-ЧОРНОМОРСЬКОМУ РЕГІОНІ УКРАЇНИ |
| title_full | STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE |
| title_fullStr | STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE |
| title_full_unstemmed | STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE |
| title_short | STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE |
| title_sort | statistical evaluation of performance indicators of photovoltaic plants as a source of energy for water desalination in the azov-black sea region of ukraine |
| topic | random function electricity probability power solar irradiance statistics photovoltaic plant. |
| topic_facet | random function electricity probability power solar irradiance statistics photovoltaic plant. випадкова функція електроенергія імовірність потужність сонячне випромінювання статис-тика фотоелектрична станція. |
| url | https://ve.org.ua/index.php/journal/article/view/461 |
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