Wind Turbine Energy Cost Optimisation Using Various Power Models
DIVYA.P. S1, VIJILA MOSES2, MANOJ G*3, LYDIA. M4
1,2Department of Mathematics, Karunya Institute of Technology and Sciences, Karunya Nagar,
Coimbatore, Tamil Nadu, INDIA
3Department of ECE, Karunya Institute of Technology and Sciences, Karunya Nagar, Coimbatore,
Tamil Nadu, INDIA
4Department of Mechatronics, Sri Krishna College of Engineering and Technology, Coimbatore,
Tamil Nadu, INDIA
Abstract: - In modern times, the worldwide wind turbine installations have developed swiftly resulting
in the decrease of green gas emissions. Though wind is a free gift of nature, it is expensive to harness
this energy for useful applications like electricity generation. The cost of installation of the wind
turbine at a particular station does not depend only on the wind resource, but also on the structure of
the turbine and the energy conversion technology. The wind turbine Cost of Energy (CoE) is used to
estimate the payback time for the return on the investment made by the wind farm owners for the
turbine. Meticulous research is required to optimize the turbine CoE which will make wind a very
competent source of energy. In this article, in order to minimize the wind turbine CoE, the wind speed
is modelled using three different distributions namely, Dagum, Gamma and Weibull and the
evaluation of the turbine Annual Energy Production (AEP) is carried out. Mathematical functions
such as linear, quadratic and cubic have been used to model the wind power. For the cost analysis of
the turbine, the price model which was established by United States, National Renewable Energy
Laboratory (NREL) is employed. The comparative study of the proposed methodology have been
done for six different stations. The turbine CoE model is an element of two factors, the rated power Pr
of a turbine and the rated wind speed Vr of a turbine. Based on the results obtained, a broad
recommendation to reduce the turbine CoE is presented. This study enables us to figure out the
minimum turbine CoE among the three discussed mathematical distributions, the finest distribution
for wind speed modelling and the optimum mathematical function for wind power modelling. The
suitable size of the wind turbine also can be found by optimizing the rotor radius R of the turbine for
each data.
Key-Words: - Annual Energy Production, Cost of Energy, Dagum distribution, rated wind power, rated
wind speed.
Received: June 15, 2021. Revised: June 11, 2022. Accepted: July 21, 2022. Published: September 9, 2022.
1. Introduction
The selection of wind turbines based on wind
parameters determines the majority of the cost of
wind energy. In [3], author has briefly explained the
various types of wind turbines and also the elements
that most affect the cost. In [13], author has offered
an analysis of the wind power strategies in several
nations. Appropriate power strategies may improve
the installations of the wind turbine. The protection
expenses of offshore wind turbines through various
methods have been investigated [4]. The cost of
turbine in the offshore is extremely correlated to the
volume & the nacelle weight. It is more noteworthy
and useful to reduce the turbine CoE [12]. Quite a
lot of investigations have been made to reduce the
turbine CoE. In [2], a method for multidisciplinary
plan enhancement for offshore wind turbines
deliberated. They explored the tower design and the
rotor effects on the turbine CoE. In [11] the best
possible wind turbine for a particular station to catch
the most extreme power or else to decrease turbine
CoE with the utilization of self-sorting out maps
were chosen. An unhindered wind farm design
optimization technique was suggested to
concurrently enhance the turbines organization and
assortment [6]. A model have established
depending on plentiful evolutionary computing
procedures and blade component motion theory for
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Divya.P. S, Vijila Moses, Manoj G, Lydia. M
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the assortment of wind turbine [15]. A method for
calculating the impact of changes in the climate on
wind energy prices suggested and familiarized a
new Weibull transfer feature for characterizing the
environment indicator [7]. The selection of a cost-
effective wind turbine for a wind project is one of
the most critical tasks. In order to overcome that, a
system have been developed for measuring wind
turbines based on energy costs [1]. In [8], the CoE
of a turbine is demonstrated by the use of eight
variables: turbine rotor diameter, blade number, hub
height H, rotor speed, rated wind speed Vr, rated
wind power Pr, regulation type and generator type.
The method of optimization has become a
complicated one, because it has several variables. In
[14], the CoE model of a turbine is simplified to
four variables: R, Pr, tip-speed ratio (TSR) and H.
The TSR is an operational parameter, while the
other three are physical properties of the turbine.
The S-type curve, a widely used method for
representing the output power of the turbine is being
modified [9]. In this approach, the CoE of a turbine
is simplified to a function of three variables: rotor
diameter, turbine capacity and H. A mathematical
model is developed to minimize the CoE using only
two variables: Vr and Pr. As a result, CoE
minimization is progressively decreased from eight
variables to only two [5].
2. Types and Operating regions of
Wind Turbines
The capacity of the Wind Turbine (WT) to
serve loads dependably and economically in the
context of the inherent uncertainty associated
with wind as a resource presents a problem for
power engineers. Modelling the reaction of
WTs to changes in wind speed and network
frequency
is a crucial first step in overcoming this
obstacle. The power curve of a wind turbine,
which shows the relationship between power
output and observed wind speed, is the
traditional method of evaluating a wind
turbine's performance. A wind turbine is made
up of five main components and numerous
auxiliary ones. The tower, rotor, nacelle,
generator, and base or foundation make up the
key components. A wind turbine cannot operate
without all of these, as depicted in Fig.1. The
electrical network whose frequency impacts the
machine's slip is directly connected to fixed-
speed WTs using squirrel cage induction
generators. Fixed-speed WTs therefore respond
to grid frequency disruptions in a way that
lessens the disturbance. WTs with variable
speeds are easier to control. For instance,
maximum power point tracking (MPPT) used to
maximize the amount of wind energy captured
causes higher variability in WTG active power
output due to changes in wind speed at low
wind speeds.
There are various wind turbine models in
use these days. The rotation of the rotor shaft, the
mode of operation, and the power rating of wind
turbines are used to rate them. Based on the
rotation, there are two types of wind turbines,
namely vertical-axis and horizontal-axis wind
turbines. According to the operation of the turbine it
is classified as fixed and variable-speed wind
turbines. Based on its power level, it is categorized
as a small turbine with less than 100 kW of power,
moderate turbine with power between 100 kW – 1
MW and a massive turbine of power above 1 MW.
There will be little power in area I since the wind
speed is less than the cut-in speed (Vc), and the
turbine will be in backup mode. Wind speed in
region II will be higher than Vc but lower than the
rated Fig1. A wind turbine's operational zones
speed (Vr), causing the turbine in this region to
produce extreme power. Whereas the wind speed
in the region III is above Vr and below the cut-out
speed (Vf) and thus the turbine's output power is
limited to the rated power (Pr). In region IV, the
wind speed will be greater than Vf, so the turbine
will be shut down to avoid damage. By optimizing
turbine output in area II, we can achieve maximum
turbine yield power. Thus, the power (P) of the wind
turbine in various operating regions is:
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(1)
where V is the wind speed.
3. Cost Minimization Methodology
The objective of this study is to give a scientific way
to limit the CoE of a turbine with the use of various
probability distributions and to find the appropriate
distribution to minimize the CoE for the particular
region. The turbine CoE is the ratio of the total
turbine cost and the turbine Annual Energy
Production (AEP). Six variables namely R, Pr, H,
Vc, Vr and Vf were there in the turbine CoE function.
frcr
frcr
VVVHPAEP
VVVHPRCost
CoE ,,,,
,,,,,
(2)
This was simplified and reduced to an element of
two variables, Pr and Vr by Chen et al. (2018).
Hence in the process of cost minimization, the Pr
varies from 1 – 3 MW with the increment of 0.1
MW and the Vr varies from 8 – 16 m/s with the
increment of 0.5 m/s.
Thus
),( rr VPfCoE
),(
),(
rr
rr VPAEP
VPCost
The NREL cost model [10] is used to calculate
the overall cost of a wind turbine.
Cost = ICC x FCR + AOE
(3)
where ICC - Initial Capital Cost,
FCR - Fixed Charge Rate of the turbine
AOE - Annual Operating Expense of the turbine.
All these values are obtained from NREL.
It must be clarified that the cost of the wind turbine
is the average annual cost over the intended lifetime
of the wind turbine.
The ICC is the total of the Balance-of-
Station (BoS) and wind turbine system cost, which
is comprised of numerous subsystems, containing
electronic, electrical, and mechanical control
systems, as well as some supplementary systems.
The BoS cost contains infrastructure costs such as
framework, roads, licenses, electrical wiring,
installation and transportation costs. Table 1
illustrates the detailed initial capital cost of the
turbine. The cost of each element or infrastructure
depends on the rotor radius of the turbine (R), the
rated power (Pr) of the turbine and the height of the
hub (H).
Table 1. ICC of a wind turbine [10]
The turbine's AOE includes land-buying costs,
construction, maintenance and replacement costs.
These costs are determined by the rated turbine
power or the turbine's AEP. The overall turbine
cost is calculated using the following factors: R,
H, Pr, and annual turbine energy production.
Table 2 shows the specifics of each expenses.
Table 2. AOE of a wind turbine [10]
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The AEP for a turbine is calculated as follows:
)1(8760
ave
PAEP
(4)
where µ - total turbine losses represented by a
constant 0.17 and the mean turbine output power
Pave, is calculated as
0
)( dVVPfP
ave
(5)
From the equation (1)
f
r
r
c
V
V
r
V
V
ave dVVfPdVVfVPP )()()(
(6)
3.1 Operating Procedure of Proposed
Methodology
The operating procedure of minimization of turbine
cost of energy is explained in the stepwise algorithm
as given below:
Step 1: Input the Scale and Shape parameter values
of the data.
Step 2: Set the variable range for rated wind power
Pr and rated wind speed Vr as
max
,rr PP
,
maxmin ,rr VV
.
Step 3: Set the incremental step (m, n) for Pr, Vr.
Step 4: Initialize Pr=m(i)+
min
r
P
with i=0
Step 5: Initialize Vr=n(j)+
min
r
V
with j=0
Step 6: If
max
rr PP
&
max
rr VV
, evaluate the Cost,
AEP, and CoE else go to Step 5 with the
increment j=j+1
Step 7: If
max
rr VV
, then go to Step 4 with the
increment i=i+1.
Step 8: If
max
rr PP
, then print the minimum CoE
and optimal Pr and Vr.
3.2 Rotor Radius of a Turbine
The turbine's rotor radius R is a function rated wind
power and rated wind speed.
3
2
rgfmfpr
rVC
P
R
(7)
where
- Density of the air (1.225 kg/m3)
pr
C
- The blade's aerodynamic
efficiency (0.45)
mf
- Efficiency of the gearbox (0.96)
gf
- Efficiency of the generator (0.97)
3.3 Wind Speed Models
The Pave has a significant part in reducing
turbine CoE. In this article, the Dagum, Gamma,
and Weibull distributions were used to simulate
wind speed data from six distinct sites.
Table 3. Probability density functions (pdf)
and parameters of the examined
distributions
In the equation (5), replacing the pdf listed
in Table 3 we get Pave for the Dagum, Gamma,
and Weibull distributions respectively.
3.4. Wind Power Models
The turbine yield power between the regions Vc and
Vr is characterized by a mathematical equation of a
polynomial function, a logistic four-parameter
function, or a logistic five-parameter function in
cost minimization analysis. The output power of the
turbine is defined in this work using polynomial
equations of linear, quadratic, and cubic models.
Linear model. The linear model is fairly
straightforward, requiring only the variables Vc, Vr,
and Pr. In region II, the turbine yield power will
increase linearly as the wind speed increases. The
linear power model formulation is provided in
equation (8).
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r
cr
cP
VV
VV
VP
)(
(8)
By putting the wind power of linear model (8) in
equation (6), we get the linearly modelled mean
turbine output power.
dVVfPdVVfP
VV
VV
P
f
r
r
c
V
V
r
V
V
r
cr
c
ave )()(
f
r
r
c
V
V
r
v
v
c
cr
rdVVfPdVVfVV
VV
P)()()(
(9)
Quadratic model. The wind turbine yield power in
quadratic function is presumed to be proportionate
to square of the wind speed in the region II. To
describe the power using quadratic model, the
values needed are the Vc, Vr and Pr.
r
cr
cP
VV
VV
VP 22
2
2
)(
(10)
By replacing the wind power from quadratic model
(9) into (5), we get the mean turbine output power,
which is quadratically modelled as
dVVfPdVVfP
VV
VV
P
f
r
r
c
V
V
r
V
V
r
cr
c
ave )()(
22
2
2
f
r
r
c
V
V
r
v
v
c
cr
rdVVfPdVVfVV
VV
P)()()( 2
2
22
(11)
Cubic model. The turbine yield power in cubic
expression, is expected to be proportionate to cube
of the wind speed, which indicates the turbine
proficiency is presumed to be a constant. The
cubic power model is given as
r
cr
cP
VV
VV
VP 33
3
3
)(
(12)
Substituting the cubic power model (11) in equation
(5), yields the cubically modelled mean turbine
output power.
dVVfPdVVfP
VV
VV
P
f
r
r
c
V
V
r
V
V
r
cr
c
ave )()(
33
3
3
f
r
r
c
V
V
r
v
v
c
cr
rdVVfPdVVfVV
VV
P)()()( 3
3
33
(13)
4. Results and Discussion
The findings of this study are deliberated
further down.
4.1 Data description
To offer a concrete presentation concerning
the above conferred approaches, real time data sets
have been considered from the U.S. National
Renewable Energy Laboratory and evaluated in this
paper. October – December 2006 hourly data were
used from six separate wind farms. The descriptive
statistics of the tested data are given in the Table 4.
Table 4. Descriptive statistics of the wind speed
data in various stations
In the process of turbine CoE minimization,
the shape and scale parameters for every distribution
at a given station are provided as inputs. As a result,
the parameters are calculated using the Maximum
Likelihood Estimate (see Table 5).
Table 5. Parameters of six data
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The minimum CoE has been identified, as well as
the optimized Pr and Vr, by using the scale and
shape parameters as inputs and altering the Vr and
Pr. Table 6 shows the minimum CoE for the six
stations that were modelled using the Dagum,
Gamma, and Weibull distributions with three
different methodologies.
Table 6. Minimum Cost of Energy
The best turbine rotor radius R is found by
combining the Pr and Vr to optimize the individual
data. For each data, the best Pr and Vr are found,
lowering the CoE. The turbine rotor radius for every
data set is calculated using these Pr and Vr, as shown
in Table 7. As a result, the proposed method is
useful for determining the appropriate size of wind
turbine for each station.
Table 7. Rotor Radius of the Turbine
Data
Optimized Pr and Vr
Rotor Radius
R (m)
Pr (MW)
Vr (m/s)
1
1
11
30.52
2
1.3
12.5
28.72
3
1.4
13
28.11
4
1.4
13
28.11
5
1.4
13
28.11
6
1.4
13
28.11
Fig.5.
Results of
minimized
Turbine CoE for six stations by modelling the wind
speed with Dagum distribution & wind power with
linear function
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From the Table 6, it is observed that among the
three linear, quadratic and cubic wind power
models, the minimum CoE occurred when using
linear model. Also among the three distributions,
the least CoE attained by modelling the wind speed
through Dagum distribution. The minimum CoE
comparison for the different wind power models
have been presented in the Figures 2, 3 and 4. From
the Table 6 and Figures 2, 3 and 4, it is observed
that for all the discussed wind power models, the
minimum CoE occurred while modelling the wind
speed with the Dagum distribution. Figure 5 depicts
a three dimensional (3D) map of the minimum
turbine CoE for all of the data discussed.
5. Conclusion
This article presents a mathematical strategy for
reducing the CoE of wind turbines. The Dagum,
Gamma, and Weibull distributions were used to
model the observed wind speed data in order to
minimise the CoE, while the linear, quadratic, and
cubic functions were used to represent the wind
power. The study is based on information gathered
from six separate stations. Utilizing the three
statistical distributions, comparative research was
conducted to estimate the minimum CoE. The
results of statistical distributions used to simulate
wind speed give the lowest turbine CoE for the
Dagum distribution. In accordance with the
mathematical calculations, the smallest CoE resulted
from modelling the power using a linear function.
Overall, this study demonstrates that by modelling
wind speed with the Dagum distribution and wind
power with a linear function, the turbine CoE can be
decreased. The suggested method also establishes
the best turbine rotor radius for each station. The
optimal turbine size for generating the most energy
at the lowest cost is thus identified.
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(Attribution 4.0 International, CC BY 4.0)
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