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Micro-Siting of Wind Turbines in an Optimal Wind Farm Area Using Teaching-Learning-Based Optimization Technique Micro-Siting of Wind Turbines in an Optimal Wind Farm Area Using Teaching-Learning-Based Optimization Technique

Authors:
  • Independent Researcher
  • COMSATS Institute of Informaton Technology, Wah Catt, Pakistan

Abstract and Figures

Nowadays, wind energy is receiving considerable attention due to its availability, low cost, and environment-friendly operation. Wind turbines are rarely placed individually but rather in the form of a wind farm with a group of several wind turbines. The purpose of this research is to perform studies on wind turbine farms in order to find the best distribution for wind turbines that maximizes the produced power, hence minimizing the wind farm area. Wind Farm Area Optimization (WFAO) is performed for optimal placement of wind turbines using elitist teaching–learning-based optimization (ETLBO) techniques. Three different scenarios of wind (first is fixed wind direction and constant speed, second is variable wind direction and constant speed, and third is variable wind direction and variable speed) are considered to find the optimal number of turbines and turbine positioning in a minimized squared land area that maximizes the power production while minimizing the total cost. Other research carried out in the past was to find the optimal placement of the wind turbines in a fixed squared land area of 2 km × 2 km. In the present study, WFAO–ETLBO algorithm has been implemented to get the optimal land area for the placement of the same number of turbines used in the past research. For Case 1, there is a significant reduction in land area by approximately 30.75%, 45.25%, and 51.75% for each wind scenario, respectively. For Case 2, the reductions in land area for three different wind scenarios are respectively 30.75%, 7.2%, and 7.2%. For Case 3, there is a reduction of 7.2% in land area for each wind scenario. It has been observed that the results obtained by the WFAO–ETLBO algorithm with a significant reduction in the land area along with optimal placement of wind turbines are better than the results obtained from the wind turbines placement in the fixed land area of 2 km × 2 km.
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Citation: Hussain, M.N.; Shaukat, N.;
Ahmad, A.; Abid, M.; Hashmi, A.;
Rajabi, Z.; Tariq, M.A.U.R.
Micro-Siting of Wind Turbines in an
Optimal Wind Farm Area Using
Teaching–Learning-Based
Optimization Technique.
Sustainability 2022,14, 8846.
https://doi.org/10.3390/su14148846
Academic Editors: Désiré
Rasolomampionona and
Klos Mariusz
Received: 31 May 2022
Accepted: 14 July 2022
Published: 19 July 2022
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sustainability
Article
Micro-Siting of Wind Turbines in an Optimal Wind Farm Area
Using Teaching–Learning-Based Optimization Technique
Muhammad Nabeel Hussain 1, Nadeem Shaukat 2,3, Ammar Ahmad 2, Muhammad Abid 4,5, Abrar Hashmi 6,
Zohreh Rajabi 7and Muhammad Atiq Ur Rehman Tariq 7, 8, *
1Department of Mechanical Engineering, Pakistan Institute of Engineering & Applied Sciences, Nilore,
Islamabad 45650, Pakistan; nabeel106d@gmail.com
2Center for Mathematical Sciences (CMS), Pakistan Institute of Engineering & Applied Sciences, Nilore,
Islamabad 46560, Pakistan; drnadeem_shaukat@pieas.edu.pk (N.S.); ammar_nustian@hotmail.com (A.A.)
3Department of Physics and Applied Mathematics (DPAM), Pakistan Institute of Engineering &
Applied Sciences, Nilore, Islamabad 45650, Pakistan
4Department of Mechanical Engineering, COMSATS University Islamabad, Wah Campus,
Wah Cantt 47040, Pakistan; drabid@ciitwah.edu.pk
5
Interdisciplinary Research Center, COMSATS University Islamabad, Wah Campus, Wah Cantt 47040, Pakistan
6Department of Electrical Engineering, Capital University and Technology, Islamabad 45750, Pakistan;
abrarhashmi313@gmail.com
7Institute for Sustainable Industries & Liveable Cities, Victoria University, P.O. Box 14428,
Melbourne, VIC 8001, Australia; zohreh.rajabi@live.vu.edu.au
8College of Engineering, IT & Environment, Charles Darwin University, Darwin, NT 0810, Australia
*Correspondence: atiq.tariq@yahoo.com
Abstract:
Nowadays, wind energy is receiving considerable attention due to its availability, low cost,
and environment-friendly operation. Wind turbines are rarely placed individually but rather in the
form of a wind farm with a group of several wind turbines. The purpose of this research is to perform
studies on wind turbine farms in order to find the best distribution for wind turbines that maximizes
the produced power, hence minimizing the wind farm area. Wind Farm Area Optimization (WFAO) is
performed for optimal placement of wind turbines using elitist teaching–learning-based optimization
(ETLBO) techniques. Three different scenarios of wind (first is fixed wind direction and constant
speed, second is variable wind direction and constant speed, and third is variable wind direction
and variable speed) are considered to find the optimal number of turbines and turbine positioning
in a minimized squared land area that maximizes the power production while minimizing the total
cost. Other research carried out in the past was to find the optimal placement of the wind turbines
in a fixed squared land area of 2
km ×
2
km
. In the present study, WFAO–ETLBO algorithm has
been implemented to get the optimal land area for the placement of the same number of turbines
used in the past research. For Case 1, there is a significant reduction in land area by approximately
30.75%, 45.25%, and 51.75% for each wind scenario, respectively. For Case 2, the reductions in land
area for three different wind scenarios are respectively 30.75%, 7.2%, and 7.2%. For Case 3, there is a
reduction of 7.2% in land area for each wind scenario. It has been observed that the results obtained
by the WFAO–ETLBO algorithm with a significant reduction in the land area along with optimal
placement of wind turbines are better than the results obtained from the wind turbines placement in
the fixed land area of 2 km ×2 km.
Keywords: wind turbine; micro-siting; wind farms; teaching–learning-based optimization; Jensen’s
wake modeling
1. Introduction
The computational intelligence techniques have been extensively used by numerous
studies for wind farm layout optimization purposes. Most of the contributions include
gradient-free methods, and most of them use soft computing. Soft computing includes
Sustainability 2022,14, 8846. https://doi.org/10.3390/su14148846 https://www.mdpi.com/journal/sustainability
Sustainability 2022,14, 8846 2 of 24
artificially intelligent techniques which are stochastic in nature and metaheuristic. The
effectiveness and the comparison studies of these soft computing techniques have been
shown in different studies by different researchers for the positioning optimization of
wind turbines. In order to place a large number of wind turbines in the wind farm, it is
needed to find the optimal placement of these turbines to get the maximum expected power
production at a minimizing cost.
Several pieces of research have been performed to optimize the placement of wind
turbines [
1
,
2
]. Mosetti et al. employed the genetic algorithm to find the optimal wind
turbine placement in wind farms [
1
]. Grady et al. challenged the results of Mosetti et al. and
claimed that the results produced by Mosetti et al. were not optimum [
2
]. Mittal et al. also
employed the genetic algorithm to find the optimal placement of wind turbines in wind
farms [
3
,
4
]. They produced results for three different scenarios for fixed wind direction
at constant speed but having a change in direction and variable wind speed and variable
direction with some preferred directions, respectively. They used Jensen’s analytical wake
model for modeling the wake effect, and their main objective was to minimize the value of
cost per unit of power [
5
]. They recommended that a sufficient number of generations were
not considered to arrive up to the optimum point. Rabia et al. proposed a method for wind
farm layout optimization by using definite point selection and a genetic algorithm that
can improve the wind farm’s output power by changing the wind farm’s dimensions with
an area size of 2
km ×
2
km
[
6
]. They rotated the square-shaped wind farm by 45 degrees
towards the uniform direction of the wind, and a definite point selection criterion was set
in order to face the upstream wind. These researches proposed the optimal number of
turbines along with their optimal placements in the wind farm of the fixed square land area
of 2
km ×
2
km
. Many have used gradient-based wind farm layout optimization solvers
consisting of combinatorial optimization and sequential linear or sequential quadratic
programming [
7
9
]. Until now, a number of contributions through the implementation
of intelligent evolutionary techniques have been made toward the optimal placement of
wind turbines in a given land area of a wind farm. The different sub-categories of the
soft computing techniques include Genetic algorithms with different hub heights [
10
], a
new mathematical programming approach to optimize wind farm layouts [
11
], viral-based
optimization algorithm [
12
], particle swarm optimization technique [
13
], mixed integer
linear programming [
14
], multi-population genetic algorithm [
15
], colony algorithm [
16
],
random and local search algorithm [
17
19
], and simulated annealing [
20
]. Optimizing
the layout of wind farm turbines using genetic algorithms has been performed in Tehran
province [
21
]. A new approach for power prediction has been implemented to analytically
model wind farms [22].
In order to maximize the wind energy capture, a model for wind turbine placement
based on the wind distribution has already been proposed [
23
]. The model calculated wake
loss based on wind turbine locations and wind direction. A multi-objective evolutionary
strategy algorithm has been developed to solve the transformed bi-criteria optimization
problem, which maximizes the expected energy output, as well as minimizes the constraint
violations. A proposed wind farm layout upgrades by adding different (in type and/or
hub height) commercial turbines to an existing farm have been introduced and optimized.
Three proposed upgraded layouts have been considered: internal grid, external grid, and
external unstructured. A genetic algorithm has been used for the optimization purpose,
and the manufacturer’s power curve and a general representation for thrust coefficient
were used in power and wake calculations, respectively. A simple field-based model has
been implemented while considering both offshore and onshore conditions [
24
,
25
]. WFLO
problem that gives total freedom to the wind farm area shape has been presented for
the first time while considering that increasing the degrees of freedom in the decision
space can lead to more efficient solutions in an optimization problem. Multi-objective
optimization with the power output (PO) and the electricity cable length (CL) as objective
functions in Horns Rev I (Denmark) via 13 different genetic algorithms: a traditionally
used algorithm, a newly developed algorithm, and 11 hybridizations resulting from the
Sustainability 2022,14, 8846 3 of 24
two has been presented. Turbine wakes and their interactions in the wind farm have
been computed through the in-house Gaussian wake model [
26
]. A distributed genetic
algorithm has been deployed to study the wind turbine’s layout in order to improve the
wake effect [
27
]. The optimal arrangement of the wind farm and the best values for the
hubs of its wind turbines in Manjil, Iran, have been proposed. Jensen’s wake model has
been used by considering wind regimes and geographic data of the considered area with
electricity generation costs along with the objective function, based on the Mosetti et al.
cost function, by implementing the particle swarm optimization (PSO) algorithm [
28
]. A
software design tool has been developed to examine the significance of wind turbine’s
layouts (M, straight and arch shapes) and spacing on the final energy yield with five times
rotor diameter distance between each turbine has been simulated and has resulted in 22.75,
22.87 and 21.997 GWh/year for the M shape, Straight line and Arch shape, respectively [
29
].
The relocation of several wind turbines has been transformed into a single-player
reinforcement learning problem, which is further handled by Monte-Carlo Tree Search
integrated into the evolutionary algorithm, improving the exploitation potential in the
adaptive genetic algorithm. The enhanced algorithmic exploitation used on the wind
farm in New Jersey resulted in a significant improvement and better performance when
compared to benchmark algorithms [
30
]. A novel coupling approach has been used
to optimize the layout of wind farms while taking into account inhomogeneous wave
loads on monopile-supported wind turbines. The basic objective is to position every
turbine in the best possible way, minimizing the wind farm’s overall wave load while
maintaining a good annual energy output [
31
]. The wake model, optimization technique,
and objective function, respectively, have been the PARK approach, genetic algorithm,
and Levelized Cost of Electricity (LCOE). A total of 36 Cartesian wind farm grids with
various resolutions were taken into account in order to find the best wind farm layouts.
Following the identification of the ideal layout, an economic analysis was carried out to
look at the effects of governmental incentives on the viability of the ideal wind farm [
32
].
Recent experimental, computational, and theoretical research initiatives that have helped
advance our comprehension and capacity to foresee the interactions of the turbulent
atmospheric boundary layer (ABL) flow with wind turbines and wind farms have been
reviewed [
33
]. The gradients required for optimization have been discovered by solving
the adjoint equations of the flow model at a cost that is independent of the number of
control variables, enabling the optimization of huge wind farms with numerous turbine
sites. For idealized test situations, gradient-based optimization of wind turbine placement
has been shown. These instances show new optimization heuristics, including rotational
symmetry, local speedups, and nonlinear wake curvature effects. On more complicated
wind rise shapes, layout optimization has also shown to be effective. This includes a full
annual energy production (AEP) layout optimization over 36 input directions and five
wind speed bins [
34
]. A novel approach for the optimization of wind farm layouts has been
published that enables the use of CFD models to precisely replicate wake effects and terrain-
induced flow characteristics. This methodology uses a gradient-based algorithm and an
adjoint method for gradient computations. In contrast to other studies, this methodology
has a general formulation to be used for a variety of wind farm layouts, wind resource
distributions, and topographic conditions. Benefits of an ideal wind farm design that
makes use of CFD models are shown for both idealized and practical scenarios, where it is
possible to achieve notable increases in annual energy production by strategically placing
turbines on difficult terrain [
35
]. By using an undirected graph to model the quadratic
integer formulation of the discretized layout design problem, a novel method has been
developed to quickly produce approximative optimal layouts to support infrastructure
design decisions. This method effectively captures the spatial dependencies of the design
parameters caused by wake interactions. The probabilistic inference on the undirected
graph uses sequential tree-reweighted message forwarding to approximate the turbine
siting. By comparing the method’s performance to a cutting-edge branch and cut algorithm
at various wind regime complexity levels and wind farm discretization resolutions, its
Sustainability 2022,14, 8846 4 of 24
efficacy has been evaluated [
36
]. In order to determine the best wind farm dimensions
where the most area may face the free stream velocity, an area rotation method has been
created. The placement of the turbines has been suggested using a novel technique known
as definite point selection (DPS), which may also be used to locate the wind farm’s zero
wake effect spots. DPS requires a minimum distance between neighboring turbines for
operation safety [37].
Based on the thorough literature analysis provided in Section 1, prior studies have
focused on the issues with wind farm layout optimization. However, the present study
focuses on the following objectives:
Minimization of the land area of wind farm
Maximization of the power production
Minimization of the total cost
This study is novel because it uses a teaching learning-based optimization technique
to offer a new optimization function for wind farm area optimization. The importance of
the issue is to draw attention to the economic benefits of area reduction before building
the wind turbines in a wind farm with a smaller footprint. The elitist teaching–learning-
based optimization (ETLBO) algorithm [
38
42
] is employed to perform wind farm area
optimization (WFAO) along with the optimal placement of the wind turbines. The WFAO–
ETLBO algorithm is used due to its certain advantages over other optimization techniques.
The Optimization procedure starts with the input of basic parameters of the wind turbine,
such as wind turbine specifications, total area and specific dimensions of the wind turbine,
wind scenarios, terrain characteristics, total number of wind turbines to be installed in
the wind farm. The area or dimensions of a wind farm are strongly influenced by the
available wind scenarios and terrain characteristics. These are value-added outcomes that
have not been identified so far. When using the proposed method, wind farms have been
selected to use the minimum width in the wind speed direction to achieve maximum power
generation at the lowest cost. The minimized area is found for the optimal placement of the
turbines and performed for three different wind scenarios having fixed wind direction and
constant speed, variable wind direction, and constant speed and variable wind direction
and variable speed, respectively, to achieve maximum energy production at the minimum
total cost. Results of WFAO–ETLBO are compared with the other studies and proposed that
the significant reduction in area can be made to place the same number of turbines with
comparable results obtained by other studies. It is seen that the results are more accurate
and advantageous than the others and are discussed in detail in the subsequent sections.
2. Materials and Methods
2.1. Jensen’s Wake Effect Modeling
Jensen’s wake model to find the optimal wind farm layout design in a minimized
land area is taken into consideration in the present study because previous studies have
extensively utilized the current wake model. Here, it is considered to be appropriate for
validation purpose. It was initially proposed by Mosetti et al. [2] and Grady et al. [3]. The
assumptions made in the initial studies are still being used in recent studies. The schematic
diagram of Jensen’s wake model is shown in Figure 1.
Where
α=tan θ
2
,
θ=
2
tan1
0.09437
=
10.787 and
x
is the wind downstream distance.
As wind turbines are to be placed at the origin of each cell, so xcan have only these values:
x= 5 d, 10 d, 15 d is to 45 d, where d is the distance between 1st and 10th turbine (if
there is turbine)
So, we will be only evaluating the u
u0on above values of x.
Sustainability 2022,14, 8846 5 of 24
Sustainability 2022, 14, 8846 5 of 25
Figure 1. Wake effect model of wind turbine.
Where 𝛼 = tan
, 𝜃=2tan0.09437=10.787 and 𝑥 is the wind downstream dis-
tance. As wind turbines are to be placed at the origin of each cell, so x can have only these
values:
x = 5 d, 10 d, 15 d is to 45 d, where d is the distance between 1st and 10th turbine (if
there is turbine)
So, we will be only evaluating the
on above values of 𝑥.
Expression for calculating the velocity of air behind the turbine after it has passed
through the turbine is given by N. O. Jensen’s wake model [4] in Equation (1),
𝑢
𝑢=1 2𝑎
󰇡1+𝛼(𝑥
𝑟)󰇢 (1)
where, 𝑎=upstream velocit
y
velocit
y
just behind the rotor
upstream velocit
y
This model predicts the velocity of air behind the turbine rotor at any distance 𝑥
from the turbine. This model predicts that the velocity of air is the smallest just behind the
rotor, and it starts recovering its velocity as it moves away from turbines. At a large dis-
tance from turbines, velocity fully recovers and becomes equal to the free stream velocity
[10]. A change in velocity with distance is plotted in Figure 2.
For calculating the velocity of air at a turbine experiencing multiple wakes, the re-
sultant velocity is calculated by assuming that sum of kinetic energy (K.E.) deficit at the
turbine being considered is equal to the K.E. deficit of mixed wake [11].
Figure 1. Wake effect model of wind turbine.
Expression for calculating the velocity of air behind the turbine after it has passed
through the turbine is given by N. O. Jensen’s wake model [4] in Equation (1),
u
u0
=
12a
1+αx
r12
(1)
where,
a=upstream velocity velocity just behind the rotor
upstream velocity
This model predicts the velocity of air behind the turbine rotor at any distance
x
from
the turbine. This model predicts that the velocity of air is the smallest just behind the rotor,
and it starts recovering its velocity as it moves away from turbines. At a large distance
from turbines, velocity fully recovers and becomes equal to the free stream velocity [
10
]. A
change in velocity with distance is plotted in Figure 2.
For calculating the velocity of air at a turbine experiencing multiple wakes, the re-
sultant velocity is calculated by assuming that sum of kinetic energy (K.E.) deficit at the
turbine being considered is equal to the K.E. deficit of mixed wake [11].
Sustainability 2022,14, 8846 6 of 24
Sustainability 2022, 14, 8846 6 of 25
Figure 2. Variation of wind speed after the turbine with respect to distance x.
2.2. Fitness Evaluation
2.2.1. Wind Farm Cost Estimation
Mosetti et al. presented an empirical relation to calculate cost/year dependent on
number of turbines in Equation (2a);
cost of 𝑁 turbines =𝑁 2
3+1
3𝑒.
(2a)
They assumed that above-modeled cost/year of a single turbine is one with a maxi-
mum reduction in the cost of 1/3 for each additional turbine.
𝑙𝑖𝑚
→2
3+1
3𝑒.
=2
3 (2b)
So, if the cost for each turbine is 1. For large number of turbines, cost reduces from 1
to
i.e., 1−
=
cost reduction can be performed for each additional turbine [11,12].
2.2.2. Wind Farm Power Estimation
Power for wind turbine is given in Equation (3a,3b);
𝑃=1
2𝑚󰇗𝑢=1
2(𝜌𝐴𝑢)𝑢 (3a)
𝑃=1
2𝜌𝐴𝑢 (3b)
𝜌 is considered to be constant here. 𝜌 is calculated according to general gas equation
at fixed atmospheric conditions given by Equation (4);
𝜌=𝑃
𝑅𝑇 (4)
𝜌=101325
287293=1.2 kgm
Using Equation (4), the ideal power 𝑃 can be estimated as;
Figure 2. Variation of wind speed after the turbine with respect to distance x.
2.2. Fitness Evaluation
2.2.1. Wind Farm Cost Estimation
Mosetti et al. presented an empirical relation to calculate cost/year dependent on
number of turbines in Equation (2a);
cos t of Nturbines =N2
3+1
3e0.00174N2(2a)
They assumed that above-modeled cost/year of a single turbine is one with a maxi-
mum reduction in the cost of 1/3 for each additional turbine.
lim
N2
3+1
3e0.00174N2=2
3(2b)
So, if the cost for each turbine is 1. For large number of turbines, cost reduces from 1
to 2
3i.e., 1 2
3=1
3cost reduction can be performed for each additional turbine [11,12].
2.2.2. Wind Farm Power Estimation
Power for wind turbine is given in Equation (3a,3b);
P=1
2
.
mu2=1
2(ρAu)u2(3a)
P=1
2ρAu3(3b)
ρ
is considered to be constant here.
ρ
is calculated according to general gas equation
at fixed atmospheric conditions given by Equation (4);
ρ=P
RT (4)
ρ=101325
287 293 =1.2 kgm3
Sustainability 2022,14, 8846 7 of 24
Using Equation (4), the ideal power Pideal can be estimated as;
Pideal =1
21.2 π
4402u3=754u3watts (5)
Assuming efficiency of turbine to be 40%. The actual power P produced is calculated
by using Equation (6a);
Efficiency =P
Pideal
=1.2 kgm3(6a)
P=0.4 754
1000 u3KW =0.3(uwmwe)3KW (6b)
where
uwmwe
is the velocity of the turbine with multiple wake effect. Thus, Equation (6b)
can be written as;
P(uwmwe)3(6c)
Equation (6c) shows that power produced by the turbine is directly proportional to
the cube of velocity [13,23].
2.2.3. Evaluation of Fitness Function
The main objective of the present study is the generation of optimal layout of the
turbines at such positions in a minimized wind farm land area so as to produce maximum
power while minimizing the cost. Therefore, the objective of the layout optimization
problem can be stated mathematically as;
objective function =minimizeTotal Cost
Total Power ×Area
=minimize N2
3+1
3e0.00174N2
0.3(uwmwe)3×xy!(7)
where
x
and
y
are the lengths and width of the wind farm. In case of square land area;
x=y.
As in the current study, we are already talking about the economic aspect of the wind
farms without considering the available budget in the formulation. There are primarily two
types of objective functions used in the optimization formulas in the previous outdated
studies (cost per unit power and one over power). They do not, however, have any financial
or technological restrictions that might effectively regulate a budget limitation or a turbine
number. In the present objective function formulation, the economic aspect is obvious with
the consideration of land area.
2.2.4. Calculation of Efficiency
Efficiency of the wind farm is the amount of energy extracted from the total energy
of the wind farm without considering the effect of wake. It should not be confused with
the efficiency of the wind turbine. It estimates the actual power produced from the wind
farm compared to the power produced from the same number of turbines. The efficiency of
the turbine is considered as the aerodynamic efficiency of the rotor or blade of the turbine.
It is a measure of the energy extracted from the wind through blades. Mathematically,
efficiency (
η
) of the wind farm installed with N number of turbines is estimated by the
ratio of the total power with multiple wake effects, i.e.,
Ptot,wmwe
to the total power without
wake effects, i.e., Ptot, wowe;
η=Ptot,wmwe
Ptot,wowe (8)
Sustainability 2022,14, 8846 8 of 24
In case of Scenario-II, with constant speed and variable direction, the formulations for
the total power with multiple wake effects (
Pwmwe
) and without considering wake effects
(Pwowe) are respectively given as;
Pwmwe =
NT
k=1
36
θi=1
0.3 f(θi1)10,ju03v
uk
3;θi== 0, 10, 20 . . . 350 (9)
Pwowe =NT
36
θi=1
0.3 f(θi1)10,ju03;θi== 0, 10, 20 . . . 350 (10a)
where;
f0=f10 =f20 =. . . =f350 =1
36 (wind ocurrence probability @ each angle)
36
θi=1
f(θi1)10,j=1
So,
Pwowe =NT0.3 u03(10b)
While in case of Scenario-III, with variable speed and variable direction, the formula-
tions for the total power with multiple wake effects (
Pwmwe
) and without considering wake
effects (Pwowe) are respectively given as;
Pwmwe =
NT
k=1
j
36
θi=1
0.3 f(θi1)10,juj3v
uk
3(11)
Pwowe =NT
j
36
θi=1
0.3 f(θi1)10,juj3(12)
where;
θi=0, 10, 20 . . . 350
j=8, 12, 17 (wind speed values)
k=1, 2, 3 . . . NT(Number o f Turbines).
2.3. Elitist Teaching–Learning-Based Optimization Algorithm
The problem of wind farm area optimization along with placement optimization
is solved using teaching–learning-based optimization technique with elitism. Teaching–
Learning-based optimization algorithm (TLBO) is proposed by the Indian researcher R.
Venkata Rao [
41
]. This optimization algorithm mimics the teaching and learning in the
environment of classroom to improve the average students of classroom. The whole
learning procedure of teacher’s and learner’s phase will continue until the convergence
criteria is met [
42
]. The pseudo code for the TLBO algorithm with elitism is given in
Figure 3below;
Sustainability 2022,14, 8846 9 of 24
Sustainability 2022, 14, 8846 9 of 25
Pseudo-code Teaching-learning-based optimization algorithm
1: Generate initial population of wind farm layouts as initial learners L
2: Calculate the mean of each learner in the population (Lmean)
3: Compute the fitness value 𝑓 of each learner in the population
4: Identify the best fitness (Lteacher)
5: Keep assigned number of elite solutions
6: Generate a new population based on the Lteacher, Lmean and TF.
7: for i = 1 : 1 : No. of learners in the population L
8: 𝑇 = 𝑟𝑎𝑛𝑑[1,2]
9: 𝐿(𝑛𝑒𝑤) = 𝐿(𝑜𝑙𝑑) + 𝑟𝑎𝑛𝑑(𝑖) (𝐿 𝑇 ∗ 𝐿)
10: end for
11: Update population of learner L by comparing old population 𝐿(𝑜𝑙𝑑) and new
population 𝐿(𝑛𝑒𝑤)
12: for i = 1 : 1 : No. of learners in the population L
13: if (𝑳𝒊(𝒐𝒍𝒅) < 𝑳𝒊(𝒏𝒆𝒘))
14: 𝐿= 𝐿(𝑛𝑒𝑤)
15: else
16: 𝐿= 𝐿(𝑜𝑙𝑑)
17: end if
18: end for
19: Randomly select two learners as Li and Lj from the population and improve their
fitness.
20: if (𝒇(𝑳𝒊)<𝒇(𝑳𝒋))
21: 𝐿(𝑛𝑒𝑤)= 𝐿(𝑜𝑙𝑑)+𝑟𝑎𝑛𝑑(𝑖)∗(𝐿−𝐿)
22: else
23: 𝐿(𝑛𝑒𝑤)= 𝐿(𝑜𝑙𝑑)+𝑟𝑎𝑛𝑑(𝑖)∗(𝐿−𝐿)
24: end if
25: Calculate the fitness values 𝑓(𝐿(𝑛𝑒𝑤)) 𝑎𝑛𝑑 𝑓(𝐿(𝑛𝑒𝑤))
26: Replace the worst solution with the elite solution
27: Check for termination criteria on the maximum difference between two succes-
sive iterations;
28: if (𝐦𝐚𝐱𝒇𝑳𝒊(𝒏𝒆𝒘)−𝒇𝑳𝒊𝟏(𝒏𝒆𝒘); ∀ 𝒊= 𝟏, 𝟐, 𝟑, ,𝑵 <𝜺𝟏)
29: if 󰇡󰇥󰇻𝒇󰇡𝑳𝒋(𝒏𝒆𝒘)󰇢−𝒇󰇡𝑳𝒋𝟏(𝒏𝒆𝒘)󰇢󰇻 ; ∀ 𝒊 =𝟏, 𝟐, 𝟑, ,𝑵󰇦 <𝜺𝟐󰇢
30: Reached Stop
31: else
32: Go to line 3 and Compute the fitness value 𝑓 of each learner in the popu-
lation
33: end if
34: else
35: Go to line 3 and Compute the fitness value 𝑓 of each learner in the population
36: end if
37: Return the optimal layout 𝐿 with minimum fitness
𝑓
Figure 3. Pseudo-code of the Teaching-learning-based optimization algorithm [40].
2.4. Different Wind Scenarios
2.4.1. Scenario-I: Uni-Directional Wind with the Identical Velocity
In this case, the simplest of all, wind blows from only one direction with constant
wind speed of 12 ms. In this case, it is easy to find an optimal placement of a certain
number of wind turbines in a given land area. As the direction of the wake is also unidi-
rectional, so the maximum number of turbines can be placed in line to get the maximum
power output. Most the turbines can be placed at the initial boundary and the end
Figure 3. Pseudo-code of the Teaching-learning-based optimization algorithm [40].
2.4. Different Wind Scenarios
2.4.1. Scenario-I: Uni-Directional Wind with the Identical Velocity
In this case, the simplest of all, wind blows from only one direction with constant wind
speed of 12
ms1
. In this case, it is easy to find an optimal placement of a certain number
of wind turbines in a given land area. As the direction of the wake is also unidirectional, so
the maximum number of turbines can be placed in line to get the maximum power output.
Most the turbines can be placed at the initial boundary and the end boundary. Rest of
the turbines can be placed randomly with the minimum effect of wake. In unidirectional
Sustainability 2022,14, 8846 10 of 24
case, the computational expense will also be very low to find the optimal solution rapidly.
Following wind rose shown in Figure 4a shows the frequency distribution for Scenario-I.
Sustainability 2022, 14, 8846 10 of 25
boundary. Rest of the turbines can be placed randomly with the minimum effect of wake.
In unidirectional case, the computational expense will also be very low to find the optimal
solution rapidly. Following wind rose shown in Figure 4a shows the frequency distribu-
tion for Scenario-I.
(a).Wind Distribution of Scenario-I
(b).Wind Distribution of Scenario-II
(c).Wind Distribution of Scenario-III
Figure 4. Frequency Distribution for (a) Wind Scenario-I; (b) Wind Scenario-II; (c) Wind Scenario-
III.
2.4.2. Scenario-II: Multi-Directional Wind with the Identical Velocity
In this case, wind blows at constant speed, but it may occur from any direction. The
circle is divided into 36 segments. Each segment has equal probability of wind occurrence
with speed of 12 ms. The wind rose shown in Figure 4b shows the frequency distribu-
tion for Scenario-II.
Wind Distribution (Scenario I)
12 m/s
Wind Distribution (Scenario II)
12 m/s
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340
Probability of Occurance
Angle(degrees)
Wind distribution of Scenario-III
8ms-1
12ms-1
17ms-1
Figure 4.
Frequency Distribution for (
a
) Wind Scenario-I; (
b
) Wind Scenario-II; (
c
) Wind Scenario-III.
2.4.2. Scenario-II: Multi-Directional Wind with the Identical Velocity
In this case, wind blows at constant speed, but it may occur from any direction. The
circle is divided into 36 segments. Each segment has equal probability of wind occurrence
with speed of 12
ms1
. The wind rose shown in Figure 4b shows the frequency distribution
for Scenario-II.
2.4.3. Scenario-III: Multi-Directional Wind with Variable Wind Velocity
Third case is relatively a general case. Here three wind speeds of 8
ms1
, 12
ms1
and
17
ms1
are considered. These three speeds have different probability of occurrence
Sustainability 2022,14, 8846 11 of 24
from different sides. In real wind scenarios, wind speed does not have fixed value; rather,
there are wind speed groups. The wind rose shown in Figure 4c shows the frequency
distribution for Scenario-III.
3. Results
Three different wind scenarios are considered to find the optimal area for placement
of turbines with the same number of turbines used in previous research and to maximize
the expected power production while minimizing the total cost. These scenarios used to
test the algorithm are based on the same settings of 2
km ×
2
km
land area, a 60 m hub
height, a 40 m radius, a constant thrust coefficient, and three different wind scenarios
used by three different researchers [
1
3
]. The studies of those three different researchers
on the same scenarios using a coarse grid with five times rotor diameter (i.e., 200 m) as
the center-to-center distance between two adjacent wind turbines and fixed land area are
compared with the results obtained from WFAO–ETLBO algorithm. These settings were
the need for comparison with previous literature. WFAO–ETLBO produces good results
with minimum fitness while producing maximum power and efficiency with a significant
reduction in land area as compared to Mosetti et al., Grady et al., and Mittal et al. with less
number of iterations as well. The results of all the three studies performed on the same
problems considered in this work are discussed in the subsequent sections.
3.1. Case 1 vs. WFAO–ETLBO
The Case 1 considers the results obtained by Mosetti et al. for all the three different
scenarios used in this study. Mosetti et al. used 26 turbines in for Scenario-I, 19 turbines for
Scenario-II, and 15 turbines for Scenario-III to find the optimal placement of wind turbines
in a fixed land area of 2 km ×2 km. The layout proposed was not symmetrical [1].
WFAO–ETLBO algorithm is used to solve the same scenarios considered by
Mosetti et al.
with a different number of turbines. Table 1shows the comparisons of results for all the
wind scenarios obtained by Mosetti et al. with those obtained from WFAO–ETLBO with
fine grid meshing. The total power produced is 2.07%, 9.74%, and 9.97% lower than those
quoted by Mosetti et al. for all the scenarios, respectively. The relative change (R.C.) in
efficiency is 2.07%, 9.74%, and 9.26%, respectively, for all the scenarios. From Table 1it
can also be noted that the algorithm is converged in a significantly smaller number of
iterations as compared to Mosetti. Thus, the computational expense of WFAO–ETLBO is
expressively small in comparison with the soft computing technique used by the previous
studies. Figure 5represents difference plots for each learner to ensure the convergence of the
WFAO–ETLBO algorithm and show the convergence of the land area for optimal placement
of wind turbines in the case of all the scenarios, respectively. The most optimal layouts
generated with fine grid meshing for Scenario-I, Scenario-II, and Scenario-III, respectively,
are shown in Figure 6, along with the original layouts proposed by Mosetti et al. The
same number of turbines are placed at the optimal locations with a significant reduction of
30.75%, 45.25%, and 51.75% in the land area for Scenario-I, Scenario-II, and Scenario-III,
respectively, while producing comparable power for each scenario.
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Table 1. Comparisons of results obtained by Mosetti et al. [1] and WFAO–ETLBO.
Scenario-I Scenario-II Scenario-III
Mosetti et al. WFAO–ETLBO Mosetti et al. WFAO–ETLBO Mosetti et al. WFAO–ETLBO
Number of
turbines 26 26 19 19 15 15
Number of
individuals 200 100 200 100 200 100
Fitness value 0.001619 0.0015868 0.0017371 0.0019232 0.00099405 0.0010977
Total Power
(KW/year)
(R.C. (%))
12,352 12,607.9
(2.07) 9244 8343.51
(9.74) 13,460 12,116.91
(9.97)
Efficiency (%)
(R.C. (%)) 91.65 93.54
(2.07) 93.851 84.71
(9.74) 94.62 85.85
(9.26)
Converged
number of
iterations
400 75 350 15 400 15
Area Used (m) 2000 ×2000 1385 ×1385 2000 ×2000 1095 ×1095 2000 ×2000 966 ×966
% Reduction in
Area ~ 30.75 ~ 45.25 ~ 51.75
Simulation
Time (s) Not reported 480.2 Not reported 708.7 Not reported 534.5
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Figure 5. Difference plots for each learner and convergence plots for optimal length for all Scenarios using WFAO–ETLBO.
Scenario I: 26 number of turbines Scenario II: 19 number of turbines Scenario III: 15 number of turbines
Difference plot for each learner
Convergence plot for optimal length
0 20 40 60 80 100
-0.0010
-0.0005
0.0000
0.0005
0.0010
Differences
Number of l earners
Iteration = 1
Iteration = 18
Iteration = 37
Iteration = 56
Iteration = 75
020406080100
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
Differences
Number of learners
Iteration = 1
Iteration = 3
Iteration = 7
Iteration = 11
Iteration = 15
0 20406080100
-0.0006
-0.0004
-0.0002
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
Differences
Number of learners
Iteration = 1
Iteration = 3
Iteration = 7
Iteration = 11
Iteration = 15
0 20 40 60 80 100
1400
1600
1800
2000
2200
2400
2600
Number of learn ers
Iteration =1
Iteration =18
Iteration =37
Iteration =56
Iteration =75
020406080100
1000
1200
1400
1600
1800
2000
2200
Optimal length
Number of learners
Iteration = 1
Iteration = 3
Iteration = 7
Iteration = 11
Iteration = 15
0 20406080100
1000
1200
1400
1600
1800
2000
2200
Optimal length
Number of learners
Iteration = 1
Iteration = 3
Iteration = 7
Iteration = 11
Iteration = 15
Figure 5. Difference plots for each learner and convergence plots for optimal length for all Scenarios using WFAO–ETLBO.
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Figure 6. Proposed optimal placements in a minimized land area using Mosetti et al. [1] vs. WFAO–ETLBO.
Figure 6. Proposed optimal placements in a minimized land area using Mosetti et al. [1] vs. WFAO–ETLBO.
Sustainability 2022,14, 8846 15 of 24
3.2. Case 2 vs. WFAO–ETLBO
The Case 2 includes the same optimization problems considered by Grady et al. with
a fixed land area of 2
km ×
2
km
, considering 30 turbines for Scenario-I, 39 turbines for
Scenario-II, and 39 turbines for Scenario-III for the optimal micro-siting of the wind farm.
The results reported by Grady et al. are symmetrical because he optimized only one column
and translated the results to all the columns. The symmetrical configuration has an objective
function value lower than that of Mosetti et al. [2].
WFAO–ETLBO algorithm is also used to solve the same scenarios considered by
Grady et al.
Table 2shows the comparisons of results for all the wind scenarios obtained
by Grady et al. and WFAO–ETLBO with fine grid meshing. The total power produced
is 2.37%, 1.05%, and 3.02% lower than those quoted by Grady et al. for all the scenarios,
respectively. The relative change (R.C.) in efficiency is 2.37%, 1.05%, and 2.65%, respectively,
for all the scenarios. From Table 2, it can also be noted that the algorithm is converged in
a significantly smaller number of iterations as compared to Grady’s. Figure 7represents
difference plots for each learner to ensure the convergence of the WFAO–ETLBO algorithm
and show the convergence of the land area for optimal placement of wind turbines in the
case of all the scenarios, respectively. The most optimal layouts generated with fine grid
meshing for Scenario-I, Scenario-II, and Scenario-III, respectively, are shown in Figure 8,
along with the original layouts proposed by Grady et al. The same number of turbines are
placed at the optimal locations with a significant reduction of 30.75%, 7.2%, and 7.2% in
the land area for Scenario-I, Scenario-II, and Scenario-III, respectively, while producing
comparable power for each scenario.
Table 2. Comparisons of results obtained by Grady et al. [2] and WFAO–ETLBO.
Scenario-I Scenario-II Scenario-III
Grady et al. WFAO–ETLBO Grady et al. WFAO–ETLBO Grady et al. WFAO–ETLBO
Number of
turbines 30 30 39 39 39 39
Number of
individuals 600 100 600 100 600 100
Fitness value 0.0015436 0.0015812 0.0015666 0.0015800 0.0008403 0.0008665
Total Power
(KW/year)
(R.C. (%))
14,310 13,969.59
(2.37) 17,220 17,039.24
(1.05) 32,038 31,068.72
(3.02)
Efficiency (%)
(R.C. (%)) 92.015 89.83
(2.37) 85.174 84.28
(1.05) 86.619 84.32
(2.65)
Converged
number of
iterations
1203 15 3000 20 1000 30
Area Used (m) 2000 ×2000 1385 ×1385 2000 ×2000 1856 ×1856 2000 ×2000 1856 ×1856
% Reduction in
Area ~ 30.75 ~ 7.2 ~ 7.2
Simulation
Time (s) Not reported 134.86 Not reported 4978.29 Not reported 4101.44
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Figure 7. Difference plots for each learner and convergence plots for optimal length for all Scenarios using
WFAO–ETLBO.
Figure 7. Difference plots for each learner and convergence plots for optimal length for all Scenarios using WFAO–ETLBO.
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Figure 8. Proposed optimal placements in a minimized land area using Grady et al. [2] vs. WFAO–ETLBO.
Figure 8. Proposed optimal placements in a minimized land area using Grady et al. [2] vs. WFAO–ETLBO.
Sustainability 2022,14, 8846 18 of 24
3.3. Case 3 vs. WFAO–ETLBO
The Case 3 includes the same wind scenarios considered by Mittal et al.
Mittal et al.
used 44 fixed turbines for Scenario-I, 38 turbines for Scenario-II, and 41 turbines for Scenario-
III placed in a fixed land area of 2
km ×
2
km
to find the optimal locations of the wind
turbines [3].
WFAO–ETLBO algorithm is finally used to solve the same scenarios considered by
Mittal et al. Table 3shows the comparisons of results for all the wind scenarios obtained
by Mittal et al. with those obtained from WFAO–ETLBO with fine grid meshing. The
total power produced is 5.36%, 2.71%, and 2.34% lower than those quoted by Mittal et al.
for all the scenarios, respectively. The relative change (R.C.) in efficiency is 5.77%, 2.71%,
and 3.32%, respectively, for all the scenarios. From Table 3, it can also be noted that the
algorithm is converged in a significantly smaller number of iterations as compared to
Mittal. Thus, the computational expense of WFAO–ETLBO is expressively very small in
comparison with the soft computing technique used in past studies. Figure 9represents
difference plots for each learner to ensure the convergence of the WFAO–ETLBO algorithm
and show the convergence of the land area for optimal placement of wind turbines in the
case of all the scenarios, respectively. The most optimal layouts generated with fine grid
meshing for Scenario-I, Scenario-II, and Scenario-III, respectively, are shown in Figure 10,
along with the original layouts proposed by Mittal et al. The same number of turbines are
placed at the optimal locations with a significant reduction of 7.2% in the land area for all
the scenarios while producing comparable power for each scenario.
Table 3. Comparisons of results obtained by Mittal et al. [3] and WFAO–ETLBO.
Scenario-I Scenario-II Scenario-III
Mittal et al. WFAO–ETLBO Mittal et al. WFAO–ETLBO Mittal et al. WFAO–ETLBO
Number of
turbines 44 44 38 38 41 41
Number of
individuals Not reported 100 Not reported 100 Not reported 100
Fitness value 0.0013602 0.0015101 0.0015273 0.0015699 0.00084379 0.0008641
Total Power
(KW/year)
(R.C. (%))
21.936 20,758.66
(5.36) 17,259 16,790.89
(2.71) 33,262 32,482.32
(2.34)
Efficiency (%)
(R.C. (%)) 96.17 90.62
(5.77) 87.612 85.24
(2.71) 86.729 83.85
(3.32)
Converged
number of
iterations
Not reported 23 Not reported 15 Not reported 20
Area Used (m) 2000 ×2000 1856 ×1856 2000 ×2000 1856 ×1856 2000 ×2000 1856 ×1856
% Reduction in
Area ~ 7.2 ~ 7.2 ~ 7.2
Simulation
Time (s) Not reported 94.51 Not reported 1950.44 Not reported 3002.15
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Figure 9. Difference plots for each learner and convergence plots for optimal length for all Scenarios using WFAO–ETLBO.
Scenario I: 44 number of turbines Scenario II: 38 number of turbines Scenario III: 41 number of turbines
Difference plot for each learner
Convergence plot for optimal length
0 20 40 60 80 100
-0.0008
-0.0006
-0.0004
-0.0002
0.0000
0.0002
0.0004
0.0006
0.0008
Differences
Number of Learners
Iteration = 1
Iteration = 5
Iteration = 11
Iteration = 17
Iteration = 23
0 20 4 0 60 80 100
-0.0004
-0.0002
0.0000
0.0002
0.0004
0.0006
Differences
Number of learners
Iteration = 1
Iteration = 3
Iteration = 7
Iteration = 11
Iteration = 15
0 20406080100
-0.0004
-0.0003
-0.0002
-0.0001
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
Differences
Number of learners
Iteration = 1
Iteration = 7
Iteration = 15
Iteration = 22
Iteration = 30
0 20406080100
1800
1900
2000
2100
2200
2300
2400
2500
Optimal length
Number of Learners
Iteration = 1
Iteration = 5
Iteration = 11
Iteration = 17
Iteration = 23
0 20406080100
1800
1900
2000
2100
2200
2300
2400
Optimal length
Number of learners
Iteration = 1
Iteration = 3
Iteration = 7
Iteration = 11
Iteration = 15
0 20406080100
1800
2000
2200
2400
2600
2800
3000
Optimal length
Number of learners
Iteration = 1
Iteration = 7
Iteration = 15
Iteration = 22
Iteration = 30
Figure 9. Difference plots for each learner and convergence plots for optimal length for all Scenarios using WFAO–ETLBO.
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Figure 10. Proposed optimal placements in a minimized land area using Mittal et al. [3] vs. WFAO-ETLB.
Figure 10. Proposed optimal placements in a minimized land area using Mittal et al. [3] vs. WFAO-ETLB.
Sustainability 2022,14, 8846 21 of 24
In Figure 11 below, it can be observed that the optimum point of efficiency is at
26 turbines
for all the nine cases with a different number of turbines used in this study,
where efficiency is higher than other approaches, and then it reduces swiftly again, but
after 41 turbines it again started to increase. From Figure 12, it can be clearly observed that
the optimum point of total power is at 41 turbines for all the nine cases with a different
number of turbines used in this study, where total power is higher than other approaches,
and then it reduces swiftly again before and after the 41 turbines. It can be concluded
that the generalization of the number of turbines is not possible, for which the proposed
methodology gives the best result.
Sustainability 2022, 14, 8846 21 of 25
In Figure 11 below, it can be observed that the optimum point of efficiency is at 26
turbines for all the nine cases with a different number of turbines used in this study, where
efficiency is higher than other approaches, and then it reduces swiftly again, but after 41
turbines it again started to increase. From Figure 12, it can be clearly observed that the
optimum point of total power is at 41 turbines for all the nine cases with a different num-
ber of turbines used in this study, where total power is higher than other approaches, and
then it reduces swiftly again before and after the 41 turbines. It can be concluded that the
generalization of the number of turbines is not possible, for which the proposed method-
ology gives the best result.
Figure 11. Behaviour of efficiency with increasing number of turbines.
Figure 12. Behaviour of total power with increasing number of turbines.
85.85
84.71
93.54
89.83
84.32
84.28
85.24
83.85
90.62
94.62
93.85
91.65
92.02
86.62
85.18
87.61
86.73
96.17
15 19 26 30 39 39 38 41 44
EFFICIENCY
NUMBER OF TURBINES
Efficiency by WFAO-ETLBO Reference Efficiency
12,116.91
8,343.51
12,607
13,969
31,068.72
17,039
16,790.89
32,482.32
20,758.66
13,460
9,241
12,352
14,310
32,038
17,220
17,259
32,262
21,936
15 19 26 30 39 39 38 41 44
TOTAL POWER (KW/YEAR)
NUMBER OF TURBINES
Total Power by WFAO-ETLBO Reference Total Power
Figure 11. Behaviour of efficiency with increasing number of turbines.
Sustainability 2022, 14, 8846 21 of 25
In Figure 11 below, it can be observed that the optimum point of efficiency is at 26
turbines for all the nine cases with a different number of turbines used in this study, where
efficiency is higher than other approaches, and then it reduces swiftly again, but after 41
turbines it again started to increase. From Figure 12, it can be clearly observed that the
optimum point of total power is at 41 turbines for all the nine cases with a different num-
ber of turbines used in this study, where total power is higher than other approaches, and
then it reduces swiftly again before and after the 41 turbines. It can be concluded that the
generalization of the number of turbines is not possible, for which the proposed method-
ology gives the best result.
Figure 11. Behaviour of efficiency with increasing number of turbines.
Figure 12. Behaviour of total power with increasing number of turbines.
85.85
84.71
93.54
89.83
84.32
84.28
85.24
83.85
90.62
94.62
93.85
91.65
92.02
86.62
85.18
87.61
86.73
96.17
15 19 26 30 39 39 38 41 44
EFFICIENCY
NUMBER OF TURBINES
Efficiency by WFAO-ETLBO Reference Efficiency
12,116.91
8,343.51
12,607
13,969
31,068.72
17,039
16,790.89
32,482.32
20,758.66
13,460
9,241
12,352
14,310
32,038
17,220
17,259
32,262
21,936
15 19 26 30 39 39 38 41 44
TOTAL POWER (KW/YEAR)
NUMBER OF TURBINES
Total Power by WFAO-ETLBO Reference Total Power
Figure 12. Behaviour of total power with increasing number of turbines.
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4. Conclusions
The main objective of this study was to perform Wind Farm Area Optimization
(WFAO) for optimal placement of wind turbines using the Elitist Teaching Learning-based
Optimization (ETLBO) method. Three different wind scenarios (Scenario-I: fixed wind
direction and constant velocity, Scenario-II: variable wind direction and constant velocity,
Scenario-III: variable wind direction and variable velocity) were considered in order to
minimize the square land area with a fixed number of wind turbines while maximizing the
power generation with minimum cost. The results of WFAO–ETLBO have been compared
with the results of three researchers who conducted the optimization studies by considering
the fixed land area of 2
km ×
2
km
. Same wind scenarios have been considered by these
three researchers. In the first case (Mosetti et al.), there is a significant reduction in land
area for all the scenarios. For Scenario-I, there is an increase of 2.07% in both power and
efficiency. While in the case of Scenario-II and Scenario-III, the relative change in power, as
well as inefficiency, is 9.74% and 9.26%, respectively, which is less compared to the 45.25%
and 51.75% reduction in land area, respectively. In the second case (Grady et al.), there is a
significant reduction in land area for Scenario-I as compared to Scenario-II and Scenario-III.
For all the three scenarios, the relative change in power, as well as inefficiency, is 2.37%,
1.05%, and 3.02%, respectively, which is again relatively less as compared to the 30.75%,
7.2%, and 7.2% reduction in land area respectively. In the third case (Mittal et al.), there
is a reasonable reduction of 7.2% in land area for all the scenarios. The relative change in
power, as well as inefficiency, is 5.77%, 2.71%, and 3.32%, respectively, which is again less
compared to the 7.2% reduction in land area. Based on the above comparison, the relative
reduction in power and efficiency are less than the relative reduction in land area. For
all three cases, the ETLBO method has proposed the optimized wind farm layout with a
reduced land area with a maximum power output decrease of 9.97% while saving a lot of
computational effort.
For design efforts, multiple factors need to be optimized, and each optimization
requires significant effort. For example, the third scenario is close to a realistic scenario with
variable wind speed and variable direction for which computational cost matters. Therefore,
savings in computational efforts of the proposed approach without compromising the
design quality is worth mentioning. Therefore, it can be concluded that the ETLBO method
can be efficiently applied for wind farm area optimization and for optimal placements of
wind turbines with less computational cost as compared to past studies. Another aspect
includes the economic advantage of a reduction in land cost due to the reduction in land
area with the same number of wind turbines. This impact will be discussed and will be
considered in more detail in the upcoming research on the realistic wind farm installed at
the Jhampir, Sindh, Pakistan, using more recent and sophisticated wake models along with
the new formulations of the objective function by taking into account the available number
of turbines and the available budget.
Author Contributions:
N.S. and M.A. conceived of the presented idea. M.N.H. and N.S. developed
the theory and performed the computations. A.H., A.A., Z.R. and M.A.U.R.T. verified the strategy
implemented. M.A.U.R.T. and A.H. encouraged M.N.H. to investigate the teaching—learning based
algorithm for the micro-siting of wind turbines in a minimized land area and N.S. supervised the
findings of this work. All authors equally contributed in writing the initial and final draft of the
manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Informed Consent Statement: Not applicabale.
Data Availability Statement: Not applicable.
Acknowledgments:
The authors would like to sincerely appreciate the provision of relevant data by
Alternative Energy Development Board (AEDB) in Pakistan.
Conflicts of Interest: The authors declare no conflict of interest.
Sustainability 2022,14, 8846 23 of 24
Nomenclature
WFAO Wind farm area optimization
ETLBO Elitist Teaching–Learning-Based Optimization
u0undisturbed/freestream wind speed
ainterference coefficient/induction/perturbation coefficient
rrrotor radius
r1downstream rotor radius
drotor diameter
xwind downstream distance
θwake spread angle
αentrainment constant
K.E. kinetic energy
Nnumber of turbines
Pactual power of wind turbine
Pideal ideal power of wind turbine
ρdensity
uwmwe velocity of the turbine with multiple wake effect
ηefficiency of wind farm
Pwmwe power with multiple wake effects
Pwowe power without wake effects
Lmean Mean of a learner
Lteacher Best learner identified as a teacher
TFTeaching factor
rand Uniformly distributed random
Liith learner
Ljjth learner
f(L)Fitness value of learner
εconvergence criteria
R.C. relative change
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