The use of Markov chain model for rating location based turbine
performance
1,*GONGSIN ISAAC ESBOND, 2FUNMILAYO W. O. SAPORU
1Department of Statistics
University of Maiduguri
Borno State
NIGERIA
https://orcid.org/0000-0003-2448-5543
2Department of Statistics
University of Abuja
NIGERIA
*Correspondence author
Abstract: - It is shown how a 12-state Markov chain model can be used in rating turbine performance in a given
location. A 1.8 MW wind turbine exposed to wind speed in San Angelo, USA is used for illustration. The model fits
the data. As such, features of the model are used in providing indices for rating the performance of the turbine in
this location. Probability distribution of the wind speed in this location is introduced into each of the traditional
methods of computing average power output from the turbine power curve and expected extract-able power. The
estimates obtained are 937 kW and 826 kW, respectively.
Keywords/phrase: - Markov chain model, turbine performance rating, turbine power curve, wind energy, wind
speed data
Received: June 23, 2023. Revised: March 4, 2024. Accepted: April 4, 2024. Published: May 15, 2024.
1. Introduction
Since the mid-20th century to date, renewable power
sources have attracted so much attention, partly due to
the increasing power needs by the fast growing world
population in the face of fast depleting fossil fuel
sources, making fossil fuels consumption
unsustainable. The other part of the push for renewable
power sources is the concern over large volumes of
greenhouse gas (GHG) emission from fossil fuel
consumption, which is the key player in temperature
increases that causes global warming. These concerns
have made global renewable power supply to sore
geometrically over the last two decades. For instance,
renewable power contributed up to 26.2 % of global
electricity generation in 2018 and is projected to
increase to 45% by 2040 [21]. Similarly, the global
non-hydro renewable power generation capacity grew
by a record 184 GW in 2019, an increase of 20 GW
(12%) over the generation capacity of 2018 [1].
Wind power is a major player in the renewable power
market. Wind and hydroelectricity made up two-thirds
of the total renewable power generation in 2018 in the
27-member EU countries [22]. Wind power installed
capacity in 2018 exceeded 563 GW and accounted for
approximately 24% of the world’s total renewable
power generation capacity [20]. In 2020 wind power
generation capacity grew by 53 %, an increase of 93
GW of new installations [33]. The year 2021 saw
global grid connected increase in wind generated
energy of 94.3 GW, giving a cumulative total of 838.9
GW [30].
The intermittent nature of renewable power sources
poses a major challenge in the planning and operation
of their power systems. Hence, probability models
come in handy to characterize renewable power
sources in the face of this intermittency. A mixture
probability density provided the modeling capability
for determining uncertainty in loads on onshore and
offshore wind turbines [25]. There are many other
extensive literature on the use of parametric
probability models such as the Weibull, Raleigh,
lognormal, generalized extreme value, exponentiated-
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
1
Volume 4, 2024
epsilon and gamma distributions in the modeling of
renewable power potentials of many locations. See for
examples [12, 13, 23, 29].
One of the intermittent properties of wind speed as a
major renewable power contributor is that it changes
abruptly over short time period. It is hard to tell
whether a particular rate of wind speed will decrease,
remain same or increase within any time interval. The
stochastic nature of this intermittency can be captured
by Markov chain models. Consequently, time
homogeneous Markov chain models are used [19] in
the literature for modeling wind speed time series data
mainly for the purpose of forecasting. Markov chain
models are also useful in the determination of the
reliability indices of wind power systems [9]. A
Markov-based Back Propagation (BP) neural network
optimized by the Particle Swarm Optimization (PSO)
used for wind power prediction showed a non-
significant improvement over the traditional Markov
chain [31]. A semi-Markov model was also used to
evaluate the performance of wind turbine systems
[26].
In order to characterize wind power more accurately
forecast [2, 10, 15] and simulation [18, 24] are two
methods mainly used. Discrete time Markov chain has
been used for the generation of synthetic wind speed
and wind power time series [24, 27] as well as for
short term Wind Power Forecasting (WPF) with good
performance. Other methods for wind speed series can
be found in Mycielski Algorithm [5, 6].
These WPF methods use only onsite information in
forecasting targeted wind farms. Currently there is
ever-increasing number of wind farms over a region
prompting researchers to explore the spatio-temporal-
interdependence structure between wind farms in
improving WPF performance [28]. Towards the end
different artificial intelligence methods [3, 7], regime-
switching space-time methods [11], multichannel
adaptive filters [8], sparse vector autoregressive
(VAR)-based model [4, 34], and so on, have been
developed. The idea of spatial Markov chain used in
geo-statistical modeling has inspired the development
of a first order discrete spatio-temporal Markov chain
model for short term wind power forecasting (WPF)
[32].
Here, our focus is to use the time homogeneous
Markov chain model in modeling the performance of a
wind turbine in a specific location. We are interested
in showing how to characterize the salient features in
the power curve specific to a turbine using Markov
chain method. This will help system operators and
marketers make informed optimal decisions regarding
its likely performance, particularly when it concerns
investment in planning and development of wind farm
in a location.
2 Materials and Methods
2.1 Markov chain model
For the purpose of illustration we use the power curve
of a 1.8 MW wind turbine with 100 m rotor diameter
at 100 m turbine hub height and 1.225 kg/m3 air
density. The details are given in Table 1 below.
Table 1 Power curve of a 1.8 MW wind turbine
Wind speed
(m/s)
< 3
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12
> 20
Power, ,
(kW)
0
51
175
346
584
913
1313
1660
1784
1799
1800
0
Definition 1
Notice, from Table 1, that the turbine has cut-in and
cut-out wind speeds of 3 m/s and 20 m/s, respectively,
for which times the turbine power generation is zero.
Also, notice that the wind speed between 12 m/s and
20 m/s, inclusive, gives the optimal turbine
performance.
As interest lies in a model that can capture the
intermittent nature of the wind speed that give rise to
this power curve, we employ a Markov chain model.
Wind speed at specified time intervals are captured in
order to create a discrete state space for the model. It
should be noted that although wind speed is a
continuous real-valued random variable, a discrete
state space Markov model is chosen in order to make
the model mathematically tractable. Also the states
chosen are not sacrosanct but sufficiently reflective of
the change of status of the Markov chain for modeling
purpose. The construction of the states of the Markov
chain are presented in Table 2 below.
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
2
Volume 4, 2024
Table 2 States of the Markov chain
State
1
2
3
4
5
6
7
8
9
10
11
12
Wind speed
range (m/s)
< 3
3 – 4
4 – 5
5 – 6
6 – 7
7 – 8
8 – 9
9 - 10
10 – 11
11 – 12
12 – 20
> 20
Note: Upper bounds exclusive except state 11
The usual assumption of a Markov chain model
obtains. That is, there is Markov dependence and time
homogeneity of the Markov chain.
2.2 The problems
Pertinent questions that are of interest to wind power
developers, investors and manufacturers, specific to
the wind speed in a location for a given wind turbine,
are given below.
i. What is the downtime of the wind turbine? That is,
the estimate of the proportion of its idle time in the
long run due to intermittency of wind speed.
ii. What time duration will it take a wind turbine to
return to optimal, or near-optimal, power
generation having moved to an idle state or low
power generation state?
iii. What is the amount of power that a wind turbine
will be able to generate in the long run, in a given
location?
iv. What is the average extract-able power a given
turbine can generate in a given location?
The model can provide answers to these questions
thereby enabling interested persons make informed
decisions regarding the location-specific performance
of a wind turbine.
2.3 The Data
The wind speed time series data used in this study for
illustration are real-time average wind speed recorded
at every 5-minute interval from the 1st hour of January
1, 2010 to the 24th hour of December 31, 2012. They
are 315,360 data values of wind speed time series for a
location near San Angelo, Texas, with site ID 998289
on latitude oN and longitude oW.
The data were recorded at 100 m above ground level
with no missing value. They were obtained at
http://www.wind.nrel.gov.
The data were processed for analysis as follows. The
data were classified into 12 states of the Markov chain
according to Table 2 above. The transition matrix is of
the form
, , 1,2,...,12
ij
n i jN
1
where
ij
n
is the number of transitions from state to
state . This is given in Table 3 below.
Table 1 Transition matrix of the wind speed data
State
[,1]
[,2]
[,3]
[,4]
[,5]
[,6]
[,7]
[,8]
[,9]
[,10]
[,11]
[,12]
[1,]
23776
1004
47
8
7
1
1
1
0
1
0
0
[2,]
1011
17163
1540
49
6
3
5
2
0
0
0
0
[3,]
34
1532
23183
2050
57
17
7
1
0
0
0
0
[4,]
7
43
1997
27444
2536
61
16
6
1
0
0
0
[5,]
8
18
77
2449
31081
2876
56
9
2
1
0
0
[6,]
4
4
27
81
2790
32434
3024
67
14
4
0
0
[7,]
1
6
4
14
54
2938
31533
2980
62
11
10
0
[8,]
1
4
1
9
20
84
2884
28935
2459
42
17
0
[9,]
1
2
3
3
11
16
60
2360
21780
1784
62
0
[10,]
1
0
3
1
6
7
16
64
1690
14042
1188
0
[11,]
2
2
1
3
9
11
11
31
74
1132
20140
30
[12,]
0
1
0
0
0
0
0
0
0
0
29
71
2.4 Test of the time homogeneity of the model
The maximum likelihood estimator of the elements of
the one-step transition probability matrix of the
Markov chain can be obtained from Table 3, and is
computed from
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
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Volume 4, 2024
ˆˆ, 1,2,...,12
ij
p i jP
2
where
.
ˆij
ij
i
n
pn
and
12
.
1
i ij
j
nn
.
The result is given as matrix below.
To test for the assumption of time homogeneity, the
entire dataset is divided into three subsamples, one for
each of the three years (2010, 2011, 2012) for which the
data were generated. The transition probability matrix
for each of these years are similarly estimated and
given as , and below.
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
4
Volume 4, 2024
To test for the time homogeneity of the model, the
following notations are adopted.
Let be the matrix of the number of one step
transitions at the time with elements,
t ij
n
which are the number of one step transitions from state
to state , ; and
.ti
n
is the row total
of .
Let denote the matrix of one step transition
probabilities at time , with elements
t ij
p
. Then the
estimates of
t ij
p
is given by
.
ˆ, 1,2,3; , 1,2,...,12
t ij
t ij
ti
n
p t i j
n
3
The hypothesis for the test is given by
0ˆ
: ; , 1,2,...,12
t ij ij
H p p i j
against
1ˆ
: ; 1,2,3
t ij ij
H p p t
And the test statistic is given by
2
3 12 12
.
1 1 1
ˆˆ
ˆ
t ij ij
ti
t i j ij
pp
Qn
p
4
Q
is asymptotically chi-square distributed with
12td
degrees of freedom, where
12 12
11
ij
ij
dd
is
the number of positive entries,
0
ij
n
(that is,
1
ij
d
if
0
ij
n
, and
0
ij
d
, otherwise).
Here,
3t
and
112d
, hence the degrees of freedom
for the test is
300
. The computed value of
330.72Q
with a p-value of
0.107
, suggesting that
the null hypothesis of time homogeneity should not be
rejected.
3 Results and discussion
3.1 Computation of measures for assessing
turbine performance
Some of the properties of Markov chain can be used in
assessing turbine performance particularly when
directed in answering the questions posed in section 3.
These are computed as follows.
3.1.1 Limiting probability distribution vector and
mean recurrence times
The limiting probability distribution vector gives the
proportion of times spent in each of the states of the
Markov chain in the long run.
Let
x
denote this vector. Then
ˆ
x xP
5
provides estimates for the elements of this vector.
These estimates are given below.
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
5
Volume 4, 2024
The inverse of the elements of this vector gives the
mean recurrence time vector denoted by
r
. The
estimate of
r
is given below.
By definition, the mean recurrence time gives the
length of time it takes to return to a state
j
S
for the first
time given it started in
j
S
.
3.1.2 Expected first passage times
Definition 2
If an ergodic Markov chain starts from state
i
S
, the
expected (or average) number of steps to reach state
j
S
for the first time is called the expected first passage (or
visit) time to state
j
S
.
It should be noted that when
ij
, the expected first
passage time becomes the mean recurrence time. The
mean (or expected) first passage time matrix,
M
, for an
ergodic Markov chain is determined from the
fundamental matrix,
F
. This is given [17] by
1
F I P X
6
where
I
is an identity matrix,
P
is the one-step
transition probability matrix for the Markov chain and
X
is the matrix whose rows are the vectors of the
limiting state transition probability distribution.
Then the
,th
ij
component of
M
is given by
jj ij
ij
j
ff
mx
7
where
ij
f
is the
,th
ij
component of
F
and
j
x
is the
th
j
component of limiting state transition probability
vector,
x
.
The estimated values for the matrix of expected first
passage times are computed from equations (6 and 7)
and tabulated below.
Table 4 Matrix of expected first passage times
State
,
,
,
,
,
,
,
,
,
,
,
,
,
13
36
57
84
115
153
201
260
345
472
666
11161
,
260
16
33
62
94
132
180
240
325
451
645
11141
,
406
156
12
36
69
108
156
216
302
428
622
11118
,
501
255
107
10
39
79
128
188
274
400
594
11090
,
565
320
176
77
9
45
95
156
241
368
562
11057
,
613
368
225
129
59
8
54
115
200
327
521
11017
,
648
404
262
167
101
45
8
64
151
278
472
10967
,
673
429
287
193
125
74
35
9
90
218
413
10909
,
691
448
307
213
146
96
60
31
12
133
330
10826
,
705
462
321
228
161
113
78
52
31
19
208
10704
,
715
472
332
239
173
126
93
69
54
41
15
10496
,
703
460
325
237
174
129
99
78
66
58
25
3125
3.1.3 Expected Wind Power Generation
Wind power is the amount of power derivable from the
wind. Theoretically, only 59% of the power in the wind
is extract-able [14]. However, in practical applications,
the extract-able capacity of a wind turbine is 30% of the
power in the wind [16]. Consequently, the 1.8 MW
wind turbine power rating is not attainable in any given
location. Like the power curve given in Table 1, it is
more useful as a manufacturer’s index of specification
of the performance of a wind turbine.
The intermittent nature of the wind is a known factor
that compounds wind power generation. This has been
aptly described by the Markov chain model for the
location here and its salient features captured in the
limiting probability distribution vector,
x
, of the
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
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Volume 4, 2024
Markov chain. It might be pertinent to state that
x
gives the limiting probability of finding the chain in
state
j
S
(
1,2,...,12j
). Notice the correspondence
between the wind speed and the states of the Markov
chain as shown in Table 2 and the correspondence
between the wind speed and the power from the turbine
as shown in Table 1. Hence we can input the values of
x
into the calculation of power generated by a turbine
in a specified location, and consequently, make it more
informative to marketers, developers and buyers. This
computation is new and is described below. Here, we
estimate the wind power generated from a turbine using
this new method in two ways.
i Based on the 1.8 MW turbine power curve
The power curve output is given in Table 1. We create a
new row to allow multiplication with the corresponding
limiting probability values. This allows the
characteristics in the intermittent nature of the wind in
this location to be captured in the power curve. This is
shown in the table below.
Table 5 Location specific power curve of a 1.8 MW win turbine
Wind
< 3
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12
> 20
speed (m/s)
Power,
0
51
175
346
584
913
1313
1660
1784
1799
1800
0
j
Pe
(kW)
j
x
.07886
.06278
.08533
.10189
.11602
.12192
.11924
.1092
.08263
.05389
.06792
.00032
Hence, the expected power that can be generated from
the power curve for this location is given by
12
1
jj
j
PE x Pe
8
937kW
0.937MW
ii Based on extract-able power
Theoretically, the power in the wind is computed as
3
1
2
w
P C Au
9
where
is the air density,
C
is the capacity factor
that gives the practical amount of power a wind turbine
can generate,
A
is the swept area of the wind turbine
blade and
u
is the average wind speed. As explained
earlier, there is a limited amount of power a turbine can
generate from the wind. Allowing for the wind
characteristics in this location as reflected by x, the
expected extract-able power is given by
12 3
1
1
2
w f j j
j
EP C Ax u
10
where
j
u
is the mid-interval of the states of the Markov
model. We assume
1 12 0uu
m/s,
11 12u
m/s,
3
1.225 /kg m
,
0.3
f
C
and .
Substituting these values into equation (10) gives the
estimate of the extract-able power of
826
w
EP kW
0.826MW
3.2 Discussion
The test results show that a discrete state space discrete
time Markov chain model fits the wind speed data. This
enables the use of the features of this model to answer
questions that may be of interest to a wind turbine
marketer, investor or manager.
The transition probability matrix shows that
whenever the Markov chain is in any of the twelve
states, it has a higher chance of remaining there and a
very small chance of moving out to an immediate
neighborhood of the given state. It is also very
noticeable that it is almost impossible to move from any
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
7
Volume 4, 2024
of the states to any state farthest away in few
transitions. This aptly describes the intermittent nature
of the wind speed that can be experienced by a turbine
in this location. That is, the wind speed variability is
highly likely to be confined to within the speed limits
defining a specified state and at every transition time,
there is a very little chance of a drastic change in wind
speed that is characteristic of a state farthest away from
the given state. This is clearly exhibited in the
simulated transitions presented for 10,000 realizations
in Fig. 1. Two ideas are quickly identified from Fig. 1;
(1) transitions from any state gravitates to the central
states and their immediate neighborhoods, (2) the cut-
out state (state 12) is rarely visited. These information
corroborates the results of the matrix of first passage
times in Table 4.
The results from the limiting probability distribution
show that the turbine works for about 92 % of the time
and is down about 8 % of the time. The cut-in speed
(state 1) is mainly responsible for 7.8 % of the down-
time while the cut-out speed (state 12) is responsible for
only 0.03 %. Note that at down-times, the power
generated by the turbine is zero. States with the highest
proportion of working times are states 5, 6, 7 and 8.
Each contributes slightly above 10 % of the total
working time of the turbine in the long run.
The model allows only 5 minutes of dueling in any state
before a change in status. That is, a transition time in 5
minutes only is allowed. The result of the mean
recurrence times shows that when in state 1 (cut-in
speed) it takes 13 transitions, or 65 minutes, before the
next return to state 1. That is, when down because of
cut-in speed (state 1), it works for about 65 minutes
before the next cut-in speed down-time. Whereas when
down because of a cut-out speed (state 12), it takes 10.9
days before experiencing the next cut-out down time. It
is worthy of note that state 12 has the highest value of
mean recurrence time while states 6 and 7 have the
lowest, 40 minutes each. That is, state 12 is the least
visited state and states 6 and 7 are the most visited
states, as seen in Fig. 1. Consequently, power output of
the turbine is mostly at the levels provided by the wind
speeds for the states 6 and 7. These are 111.3 kW and
156.56 kW, respectively.
The results of the first passage times show a clear
pattern; lowest times are recorded from any given state
to its immediate neighborhood states. The values
become higher the farther away the states get to the
given state. Again this clearly depicts the intermittent
nature of the wind speed in this location as mentioned
earlier.
Table 6 below shows clearly the working times of the
turbine before any down time. It is clear from this table
that the turbine will work for an average of 2.05 days
starting from any of the states 2 to 12 before a first
experience of a down-time due to cut-in speed. On the
other hand, it takes an average of about 38 days,
starting from any of the states 1 to 11, before a first
experience of a cut-out speed down-time.
Fig. 1 Realizations of Transitions between Wind Speed Classifications
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
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Volume 4, 2024
Table 6 First passage times to the two down-time states
Two methods for power generated by a 1.8 MW turbine
in the given location were computed and given below.
1. The expected power that can be generated from the
power curve using the added information of the
expected probability distribution of the wind speed
in the location. This is 937 kW.
2. The expected extract-able power which is 826 kW.
Again this involves using the estimated probability
distribution of the wind speed in the location for its
computation.
Both can be used as indices for rating turbine
performance in a given location. The extract-able power
is a more realistic estimate of power.
4 Conclusion
In this study, it is shown how a 12 states Markov chain
can be used to model the wind speed exposed to a
turbine in a given location so as to use the features of
the model in rating the turbine performance. A 1.8 MW
wind turbine exposed to wind speed data at a location
near San Angelo, Texas, USA is used for illustration.
The model is compatible with the wind speed data in
this location. Features of the model show that the
turbine will, in the long run, work for 92 % of the time
and experience 8 % down time. On the average it takes
2.05 days of working starting from any of its states
before a first experience of a down-time due to a cut-in
wind speed and 38 days of working before a first
experience of a cut-out wind speed down-time. The
turbine power outputs are most times characterized by
wind speed in the range for states 6 and 7. That is,
111.31 kW and 156.56 kW, respectively.
Two estimates of the power generation potentials of the
turbine in this location are obtained. They both
introduced the long-run probability distribution of the
wind speed experienced by the turbine in the traditional
method of obtaining such estimates. This is new. These
estimates are:
i. The expected power that can be generated from the
turbine power curve; 937 kW.
ii. The expected extract-able power; 826 kW.
Taking an extract-able power of 826 kW as an hourly
average, a turbine working at this capacity is able to
generate 594,720 kWh of electric power in a 30 days a
month basis. This is sufficient to power 685 average US
homes per month on a consumption rate of 867 kWh
per home. Consequently, a wind farm with 100 of such
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
9
Volume 4, 2024
turbines can power 68,500 average US homes in a
month.
These ratings of turbine performance are valuable
indices that can provide informed policy decisions for
any interested investors, marketers and wind turbine
management planners in this location.
The model has provided an understanding of the pattern
of transition of the wind speed within the states
described by the 1.8 MW turbine using discrete Markov
chain. The major assumption was that the wind speeds
are stationary. A study assuming the wind speeds to be
non-stationary will be appropriate, especially to
compare with the procedures in this research, to guide
better decisions. In addition, the combination of this
modeling procedure, or others, with artificial
intelligence could provide mode informative ideas.
These are suggested as future research areas.
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International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
11
Volume 4, 2024
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.1
Gongsin Isaac Esbond,
Funmilayo W. O. Saporu
E-ISSN: 2769-2477
12
Volume 4, 2024
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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