Energy Conversion and Management 208 (2020) 112593
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Energy Conversion and Management
journal homepage: www.elsevier.com/locate/enconman
Comparative analysis and improvement of grid-based wind farm layout
optimization
T
Giovanni Gualtieri
National Research Council, Institute of Bioecomony (CNR-IBE), Via Caproni 8, 50145 Firenze, Italy
ARTICLE INFO
ABSTRACT
Keywords:
Wind farm layout optimization
Gridded layout
Literature case study
Wind turbine database
Levelized cost of energy
Self-organizing map
Among the main grid-based wind farm layout optimization studies addressed in the literature, 14 layouts have
been recomputed by selecting the levelized cost of energy as a primary objective function. Relying on 120 wind
turbine combinations, a previously developed optimization method targeting best turbine selection has then
been applied. All literature layouts were optimized, as capacity factors were (slightly) increased (78.89–80.90 to
83.02–83.07%), while levelized costs of energy were (significantly) reduced (130.37–370.42 to 54.01–142.64
$/MWh). This study concluded that neither the discrete nor the continuous optimization model can be recommended in all scenarios. In general, a capacity factor increase does not necessarily imply a decrease in
levelized cost of energy. The latter may be minimized by decreasing the overall wind farm capacity, the number
of turbines, or selecting turbines with lower rotor diameters or rated powers. By contrast, capacity factor may be
maximized by installing turbines with higher hub heights or lower rated speeds. Contradicting various findings,
using turbines with different rotor diameters, rated powers or hub heights is not recommended to minimize the
levelized cost of energy. Although addressed within several optimization studies, maximization of energy production is a misleading target, as involving the highest costs of energy.
1. Introduction
Wind farm layout optimization (WFLO) is commonly intended as
optimal positioning of wind turbines (WTs) in a wind farm (WF)
through minimization of power losses due to wake interferences between the WTs to maximize the annual energy yield (AEY) [1]. In
reality, this is a limiting definition, as AEY is not the only factor to take
into account [2]. Since WF planning is an economic project, AEY
maximization should be combined with the minimization of the cost of
energy [3].
1.1. Addressing wind farm layout optimization
When addressing the WFLO problem, two WF layout models are
generally applied [4]: (i) the continuous model, allowing WTs to be
positioned anywhere in the WF, subject to constraints; (ii) the discrete
model, only allowing WTs to be positioned at a finite number of locations. When applying the continuous model (i), WTs may either be irregularly placed (e.g. [5]), or positioned according to aligned (e.g. [6])
or staggered (e.g. [7]) arrays. In the discrete approach (ii), the domain
is divided into a discrete grid with WTs positioned in the centre of
specific cells (e.g. [8]). Depending on the layout scheme, the best WTs
positioning is pursued by seeking, respectively: (i) optimal downwind
and crosswind spacing between the WTs; (ii) most suitable grid cells to
place the WTs. According to various Authors (e.g. [4]), since enabling
to continuously vary the WTs placement, the continuous space search is
more capable of achieving global optimal solutions. Conversely, unless
when dealing with particularly dense grids (e.g. [9]), searching in a
discrete space generally leads to a significant WFLO computational
simplification [10]. An improved approach has been addressed by
various Authors (e.g. [11]) by combining both discrete and continuous
models into a single framework in an attempt to incorporate the advantages offered by each model: less model-solving complexity the
former, and more flexible WT positioning the latter. To this aim, Long
et al. [12] performed a thorough comparison of the advantages and
drawbacks in planning WF layout deriving from applying the discrete
vs. the continuous model.
The WFLO problem is non-convex (involving many optimal solutions) and cannot be solved using traditional optimization methods, but
requires heuristic optimization techniques, which were implemented by
most Authors [2]. Heuristic techniques include, e.g., Genetic Algorithms (GA) [13], Particle Swarm Optimization (PSO) [3], Evolutive
Algorithm (EA) [14], Viral System Algorithm (VSA) [8], or Extended
Pattern Search (EPS) [15]. GAs are used in more than 75% of WFLO
studies [4]. All these algorithms are strongly affected by the size of the
solution space. For example, if considering a gridded layout divided
E-mail address: giovanni.gualtieri@ibe.cnr.it.
https://doi.org/10.1016/j.enconman.2020.112593
Received 11 November 2019; Received in revised form 7 February 2020; Accepted 8 February 2020
0196-8904/ © 2020 Elsevier Ltd. All rights reserved.
Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
Nomenclature
LCoE
levelized cost of energy [$/MWh], Eq. (25)
horizontal and vertical WF full sizes [m]
Lh, Lv
overall number of generated WF layouts [–]
NLAY
NLIT
number of literature WF layouts [–]
NP, ND, NH number of combinations in the WF with WTs having
different Pr, D or Hhub [–]
NT, NTH number of WTs, and number of WT combinations [–]
NWF
number of WF layouts for each literature case study [–]
P
total wind power installed in the WF [kW]
Pe
electric power output from a WT [kW], Eqs. (8), (13)
Pr
WT rated power [kW]
wake radius downstream a WT [m], Eq. (1)
r0
rw
radius of wake expansion caused by an upstream WT [m],
Eq. (3)
v(z)
wind speed at generic height z AGL [m/s], Eq. (20)
v, vm
wind speed, and mean wind speed [m/s]
v0
free stream wind speed [m/s]
actual wind speed approaching the WT [m/s], Eq. (7)
vact
vhub
actual wind speed approaching the WT at Hhub [m/s]
vi, vr, vo WT cut-in, rated and cut-off wind speeds [m/s]
z, z1, z2 height AGL, and height at z1 and z2 AGL [m]
z0
site’s aerodynamic surface roughness length [m]
zref
wind reference height AGL [m]
site elevation ASL [m]
zsite
α
wake expansion coefficient of upstream WT [–], Eq. (4)
wind shear coefficient [–]
δ
wind speed deficit [–], Eqs. (5), (6)
Δh, Δv
horizontal and vertical WF grid cell sizes [m]
η
WF efficiency [%], Eqs. (12), (15)
Φ
WT design ratio [–], Eq. (21)
Ω
mean-to-rated wind speed [%], Eq. (22)
Abbreviations
AGL
AGL
AGL
PL
SOM
WF
WFLO
WT
above ground level
above ground level
above ground level
power law
self-organizing map
wind farm
wind farm layout optimization
wind turbine
Variables
a
A
AEY
CF
Ci
Cini
CT
D
Fsite
Ftot
Fwake
FWT
Hhub
WT axial induction factor [–], Eq. (2)
WT swept area [m2]
annual energy yield [MWh/y], Eq. (9)
capacity factor [%], Eq. (10)
annual cost at year i [$/kW/y]
initial capital cost [$]
WT thrust coefficient [–]
WT rotor diameter [m]
power losses of a single WT depending on the site type
[%], Eq. (17)
total power losses of a single WT experienced in the WF
[%], Eq. (18)
power losses of a single WT due to all wake interactions in
the WF [%], Eqs. (11), (14)
power losses of a single WT depending on the WT system
[%], Eq. (16)
WT hub height [m]
into Ncell cells where NWT WTs are to be installed, the number of possible solutions is: Ncell!/[NWT!(Ncell–NWT)!] [2].
The WFLO problem is also complicated by the turbulence loads
experienced by the WTs due to their mutual wake interactions in the
farm. These increase the fatigue degradation of WTs, thus shortening
their expected lifetime and consequently increasing the overall project’s
cost. Therefore, a proper balance between the maximization of energy
production and the minimization of structural loads should be targeted
[16]. On the other hand, both the positive and negative environmental
impacts involved by the installation of a grid-connected WF should not
be ignored. For example, Tao et al. [17] demonstrated that, since integration of the WF into the electrical network reduces air pollutant
emissions thus saving remarkable expenditures on pollutant penalty
costs, neglecting these costs undervalues the profit deriving from the
WF connection to the grid. By contrast, a relatively small amount of
economic benefit should be sacrificed for reducing the noise disturbance of the WF to neighbouring residents and animals.
approach in [18], i.e. the limited number of installed WTs, and thus the
waste of land resource [4]. These studies targeted the same goals of
Mosetti et al. [18], i.e. maximization of AEY [21] (also via maximization of WF array efficiency [22]), and/or minimization of a fitness
value, given by the ratio of WF costs to AEY [23]. Regardless, the following deficiencies should be pinpointed in all such studies due to the
use of: (i) the same WT model; (ii) a theoretical WT power curve; (iii) a
constant value for WT thrust coefficient (CT); (iv) a simplified objective
function (the fitness value), also relying upon (v) an overly-simplistic
cost model.
Actually, as Chowdhury et al. [3] pointed out, the WFLO problem
should address two synergistic issues: (i) optimal WT allocation, and (ii)
optimal WT selection. While several literature studies exist on heuristic
algorithms, merely representing best WTs allocators (e.g. [24]), a
minority of works focused on seeking the best-suited characteristics of
commercial WTs [3], as in most cases only one single model was used,
either when adopting the discrete [25] or the continuous [26] WFLO
model, or a combination of both [11]. In other words, rather than a
further WFLO assumption, WT characteristics should be treated as an
array of variables to be carefully analysed for detecting the unique
combination required to solve the WFLO problem. To this aim, for example, Hayat et al. [27] analysed the advantage of alternating 2- and 3blade WTs in a WF, while the benefits of vertically-staggered WFs were
explored placing small WTs between large WTs by Chatterjee and Peet
[28], as well as using different hub heights by Stanley et al. [29] or Wu
et al. [30]. A 3-variable analysis was addressed, e.g., by Mirghaed and
Roshandel [31], who targeted the minimization of WF costs through
variation of WT hub height, rotor diameter and rated power, while
parameters at WF-level such as the number of WTs and total installed
power were also taken into account by Pookpunt and Ongsakul [32].
Following this survey, it is clear that all key factors – both at WT-
1.2. Literature survey
Mosetti et al. [18] first addressed the optimal positioning of WTs in
a WF, applying a GA algorithm and Jensen’s wake model [19]. They
dealt with: (i) a flat onshore site; (ii) three ideal wind conditions; (iii)
the same (theoretical) WT model installed over the whole WF; (iv) a
square (2 km × 2 km) gridded layout with 200-m square cells [18].
WFLO was pursued by maximizing AEY and minimizing the installation
costs of WTs [2]. Following and in the framework of this seminal work,
several grid-based WFLO studies were addressed, either using the same
assumptions as in [18] (e.g. [20]), or slightly different ones that involved a denser grid up to 20 × 20 [1] or even 50 × 50 [9] cells. All of
these studies attempted to address a recognized drawback of the
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Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
and WF-level – should be concurrently analysed to actually perform a
WFLO comprehensive study.
studies and implemented in several WFLO software, including WAsP
[37] and WindPro [38].
According to Jensen’s model, the wake expands linearly behind a
WT and a constant wind speed deficit in the radial direction x occurs
[39]. If considering an upstream WTi having a rotor radius Ri, the radius
of the wake spreading downstream WTi is [14]:
1.3. Goals and contributions of the paper
The goals of the present study are the following:
(i) to comparatively analyse main grid-based WFLO studies addressed
in the literature after their recomputation using more real-world
settings;
(ii) to optimize them by applying a previously developed method [33]
which targets an optimal selection of commercial-scale WTs;
(iii) to provide insights into optimization functions and parameters that
mostly influence WFLO.
1
1
r0, i = Ri
ai
2ai
(1)
where ai is WTi axial induction factor [39]:
ai = 0.5(1
1
(2)
CT , i )
with CT,i the WTi thrust coefficient.
At radial distance xij of a downstream WTj, the radius of wake expansion due to WTi is [39]:
Literature studies where an optimal 10 × 10 (2 km × 2 km) gridbased layout was achieved against wind scenario ‘c’ defined in [18]
have been recomputed. According to goal (i), the first contribution of
the present paper is to overcome deficiencies (detailed in section 1.2)
affecting such studies, for which the following were never applied: (i)
120 combinations of commercial WTs rather than using a single WT
model; (ii) experimental (instead of theoretical) power curves for each
WT; (iii) a continuous function of wind speed rather than a constant
value for CT; (iv) the levelized cost of energy (LCoE) as an objective
function instead of the simple fitness value; (v) the thorough NREL cost
model rather than the simplified cost model proposed in [18].
Performing optimization of such studies by applying the method
developed in [33] (goal ‘ii’) enables to deliver the paper's second contribution: since the WTs best selection rather than the WTs best positioning is pursued, layout optimization – contrary to those studies – has
been performed by varying the type of WTs while retaining the number
of WTs and their placement as assumed. Since the WFLO method presented in [33] was previously applied to the continuous WF layout
model, a paper's further contribution is not only to test its applicability
to the discrete model, but also to make a straight comparison – since
application conditions are similar – between the discrete and the continuous WFLO model: this offers the opportunity to possibly determine
whether one approach is superior to the other.
An additional contribution while targeting goal (iii), is a multipurpose and a multi-variable analysis of grid-based WFLO studies that
have never been addressed before: to concurrently consider all possible
optimization functions (power production, WF efficiency, capacity
factor or cost of energy) and parameters that mostly influence WFLO –
both at WT- and WF-level – allows to genuinely achieve a comprehensive WFLO study.
rw, ij = r0, i +
(3)
i x ij
where αi is WTi-induced wake expansion coefficient [14]:
i
=
0.5
ln(Hhub, i z 0 )
(4)
with Hhub,i the hub height of WTi and z0 site’s roughness length.
A single wake between two WTs causes a velocity deficit given by
[39]:
ij
r0, i
rw, ij
= 2ai
2
Aoverlap, ij
Aj
(5)
where Aj is WTj swept area, and Aoverlap,ij the overlapping area between
Aj and WTi-induced wake area at xij. Aoverlap,ij can be calculated by
applying the method described in [39].
Summarizing, for a WTj subject to multiple wakes induced by Ni
upstream WTi, the total velocity deficit can be calculated by summing
up the Ni contributions provided by Eq. (5) [39]:
Ni
j
2
ij
=
Ni
=
2ai
i=1
i=1
r0, i
rw, ij
2
Aoverlap, ij
2
Aj
(6)
Ultimately, actual wind speed approaching the downstream WTj is
given by:
vact , j = v0 (1
(7)
j)
2.2. Wind power output and wind farm efficiency
2. Methods
The electric power Pe(v) generated by a real WT is defined as [40]:
Wake losses between the WTs were calculated by using Jensen’s
model, while all other losses were assessed through an analytic method.
A previously-developed curve as a function of wind speed was used for
CT, while the NREL model was applied for WF cost analysis. SOMs were
used to provide insights into WF layout optimization functions and
parameters that influence WFLO the most.
Pe (v ) =
m e Pm (v )
=
m e Cp P ( v )
=
1
Cp
2
m e
A v3 =
1
2
T
A v3
(8)
where Pm(v) is the mechanical power extractable from wind power P(v)
available for a WT with swept area A affected by a wind speed v [41], ρ
is site’s air density, Cp is the WT power coefficient, ηm and ηe are mechanical transmission and electric conversion efficiencies of the WT,
andηT = Cpηmηe is total power efficiency [42].
Annual energy yield (AEY) of a WT over a 1-year period (t =8760 h)
is [43]:
2.1. Wake losses: Jensen’s model
When a uniform free wind speed v0 hits a WT, a cone of slower and
more turbulent air develops behind the WT, causing the so-called ‘wake
effect’ [26]. Wake interactions between WTs are accounted for by the
wake models. Jensen's analytical wake model, although one of the
oldest, is still very effective in WFLO because of its simplicity [4].
Evidence that Jensen's model outperformed other wake models has
been provided at both onshore (e.g. [34]) and offshore (e.g. [35]) locations. Originally proposed by Jensen [19] and later developed by
Katic et al. [36] for studying WFLO, the model is applied within most
AEY = t
Pe (v ) f (v ) dv = t
0
m e Pm (v ) f
0
(v ) dv = t
0
1
2
T
A v 3f (v ) dv
(9)
Capacity factor CF is the ratio of AEY to the energy (Er) that the WT
could have produced if operated at its rated power over the same period
[44]:
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Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
CF =
AEY
Er
attempted to deal with this issue, proposing a CT continuous curve as a
function of Cp following a best-fit procedure. Very few WFLO studies
considered CT as a value varying with wind speed [2], as for example
the study performed by Ali et al. [45].
Based on CT experimental curves provided by manufacturers for 50
onshore commercial WTs [46], a further improvement has been proposed in [33] by applying a best-fit procedure, which returned the
following CT(v) curve:
(10)
Overall power losses a generic WTj undergoes in the WF caused by
all wake interactions are:
Pact , j
Fwake, j = 1
(11)
P0
where Pact,j and P0 represent the power productions corresponding to
vact,j and v0 in Eq. (7).
WF efficiency, returning the amount of energy extracted from total
available energy, is the ratio of total actual power output to total ideal
power output. For a WF including N WTs, it yields [44]:
N
=
Pact
=
Pideal
j=1
CT = (5.47581845
(12)
Mosetti et al. [18] introduced a WT theoretical model whose power
output as a function of wind speed resulted from applying the laws of
Betz and the momentum in the airflow passing through the WT swept
area [2]. Assuming ηT = 40%, ρ = 1.225 kg/m3, and D = 40 m, Eq. (8)
can thus be expressed through the following theoretical equation [14]:
The PL is the wind speed vertical extrapolation model most widely
used in wind energy applications (73.5%) because of its simplicity and
greatest accuracy [47]. Based on the PL, a known wind speed value at
height z1 is extrapolated to height z2 (i.e., WT hub height) by [47]:
with Pe(v) in kW. Eq. (13) was used in all grid-based WFLO literature
studies analysed.
In the special case of the WT theoretical model, a modified expression for Eq. (11) can be derived by using Eq. (13):
Pact , j
Fwake, j = 1
P0
= 1
v (z2 ) = v (z1)
(14)
Accordingly, combining Eq. (13) into Eq. (12) yields [14]:
N
P
= act =
Pideal
N
Pact , j
j=1
N P0
=
j =1
N
3
0.3 vact
,j
(15)
Specific WT parameters, derived from basic parameters, may be
calculated to better analyse WT characteristics, and thus assess how
their particular regulation may influence WT performances.
For example, the WT design ratio [42]:
For any WT, power losses other than wake losses can be quantified
by applying the method reported in [41]. Losses linked to gearbox,
generator, converter, and unavailability & repair are calculated as depending on the WT system; losses due to electric grid connection, icing/
soiling, and other generic factors are calculated as a function of the type
of site.
WT-related losses FWT are calculated as:
[(1
fgearbox ) (1
fgenerator ) (1
fconverter ) (1
by:
fgrid ) (1
fice ) (1
fother )]
This mean-to-rated wind speed Ω quantifies the percentage of the
site’s wind speed the WT exploits on average when operating at rated
conditions.
funav )]
2.7. Wind farm cost estimation
(17)
Summarizing, total losses Ftot affecting any WT in the farm are given
Ftot = 1
[(1
Fwake) (1
FWT ) (1
Fsite )]
(22)
= vm vr
(16)
[(1
(21)
= vr vi
quantifies the range of the region a WT operates between its vi and vr
values.
A further WT parameter is the ratio of the site’s mean wind speed to
the WT rated wind speed [33]:
while site-related losses Fsite are calculated as:
Fsite = 1
(20)
2.6. Specific wind turbine parameters
(0.3 v03 )
2.3. Turbine-related, site-related and total losses
FWT = 1
z2
z1
where v(z1) and v(z2) are wind speeds at heights z1 and z2, respectively,
and is the wind shear coefficient. Although is a function of various
parameters (e.g., atmospheric stability, terrain roughness, height range)
and experiences significant inter-daily and inter-annual variations [48],
when the site’s estimates are not available a rough value as a sole
function of the landscape can be used (e.g. [49]).
3
vact
,j
v03
(19)
2.5. Wind speed vertical profile
(13)
Pe (v ) = 0.3 v 3
1
Eq. (19) has been compared against the CT values averaged from the
considered 50 WTs, eventually returning the scores summarized in
Table S1 of the Supplementary material. Eq. (19) is particularly accurate when the WTs operate above their rated speed. Across the full wind
speed range (3–25 m/s), a mean bias of 0.007 m/s, root mean square
error of 0.027 m/s, and R2 of 0.993 are achieved.
Pact , j
N P0
10 6v 5.00641402 + 1.132584887)
An optimization function (the fitness value) was introduced to address the WFLO problem [50]:
(18)
(23)
Objective = Cost Pe
where Pe is the total yearly power output extracted from the WF and
Cost is the overall yearly cost of the WF, defined as [18]:
2.4. Turbine thrust coefficient
One of the major WT operational parameters, thrust coefficient CT is
also crucial for achieving an accurate wake analysis [7]. In the large
majority of WFLO studies (e.g. [32]), a constant value of 0.88 is assumed for CT. However, as remarked by Serrano et al. [2], a CT fixed
value may lead to significant errors in evaluating the wake effect. Relying on CT curves from 13 commercial WTs, Abdulrahman et al. [7]
Cost = N
2
1
+ e
3
3
0.00174 N2
(24)
with N the number of WTs installed in the WF.
As noted in section 1.2, both a rough objective function (Eq. 23) and
a simplified cost model (Eq. (24)) were used within most grid-based
WFLO studies (e.g. [23]). A more economically accurate approach was
4
Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
proposed, e.g., by Castro et al. [13], who suggested considering the
project’s net present value (NPV) as an optimization function since
taking into consideration crucial parameters such as the initial capital
cost, its discount rate and the WF full lifetime. In addition to NPV,
Shamshirband et al. [51] proposed that the project’s interest rate of
return (IRR) should also be considered. According to this perspective,
LCoE is probably the most reliable metric to be used for assessing the
economic performances of a WF project [3]. The LCoE is the ratio between lifetime costs and lifetime electricity production, both discounted
back to a common year through a discount rate reflecting the average
capital cost [52]. For a WF project, LCoE may be calculated as [41]:
LCoE =
Cini +
n
i=1
n
i=1
similar feature values are arranged close to each other on the map,
while dissimilar data result in neurons allocated on different map edges.
Therefore, the grid of neurons allows to easily detect the relationships
between the variables and, possibly, the cluster structure in the original
data [55].
The SOM algorithm performs an iterative training process where
any input neuron is connected to all output neurons. Once the single
winner neuron is identified, neurons are connected to adjacent neurons
by a neighbourhood function. Gaussian, Cut Gaussian, bubble, and
Mexican hat are the most commonly used neighbourhood functions.
Another two parameters that control this learning procedure are the
learning rate and the neighbourhood radius, both decreasing monotonically with the training steps [55].
Ci (1 + d )i
AEY (1 + d )i
(25)
3. Layouts from literature studies
with Cini the initial capital cost, Ci the annual cost at year i from installation, d [%] the investment’s interest rate, and n the operational
lifetime [years]. More accurate analytical models than Eq. (24) since
also taking Pr into account were proposed (e.g. [53]). The most reliable
cost model is likely the one developed at NREL by Fingersh et al. [54],
where the cost for each single WT component and subsystem are calculated. This model is fully detailed in Table 5 of [32], where: (i) Cini is
expressed as a function of R, Pr and Hhub by means of 23 equations; (ii)
Ci is expressed as a function of Pr and AEY based on 3 equations.
A total of 14 grid-based layouts developed in the literature have
been comparatively analysed (Fig. 1). Their characteristics and scores
are summarized in Table 1. All layouts have been built-up based on
Mosetti et al. [18], who proposed a square gridded domain divided into
10x10 possible WT locations. Assuming the installation of a theoretical
WT with D = 40 m, Hhub = 60 m, yielding a power output as in Eq.
(13), they assumed square grid cells with a size of 5D = 200 m, with an
overall layout extent therefore equal to 50D × 50D (2 km × 2 km).
Only layouts based on wind scenario ‘c’ defined in [18] have been
considered herein.
Mosetti et al. [18] achieved an optimal layout comprising 15 WTs
(Fig. 1a). Only Ulku & Alabas-Uslu [10], proposing a layout with WTs
positioned along the first and last rows (Fig. 1k), used the same number
of WTs, while all studies following [18] largely increased this number.
All Authors employed WTs with the same Hhub value (60 m), except
MirHassani & Yarahmadi [26], who also used WTs with Hhub = 78 m
(Fig. 1h), and a combination of 8 WTs with Hhub = 50 and 17 WTs with
2.8. Self-organizing map
The SOM is an unsupervised learning artificial neural network
model that operates a nonlinear projection from a large data space to a
small grid of neurons [55]. The original data space (input layer) is
projected to a grid of neurons (output layer) while maintaining the
topological and metric relationships in the original data. The SOM can
recognize groups of similar input variables so that neurons having
Fig. 1. Analysed literature gridded layouts (Δh = Δv = 200 m; Lh = Lv = 2 km): (a) Mosetti et al. [18]; (b) Grady et al. [50]; (c) Emami & Noghreh [56]; (d) Turner
et al. [57]; (e) Patel et al. [25]; (f) Parada et al. [1]; (g) MirHassani & Yarahmadi [26]; (h) MirHassani & Yarahmadi [26]; (i) MirHassani & Yarahmadi [26]; (j)
Abdelsalam & El-Shorbagy [20]; (k) Ulku & Alabas-Uslu, ‘a’ [10]; (l) Ulku & Alabas-Uslu, ‘b’ [10]; (m) Ulku & Alabas-Uslu, ‘c’ [10]; (n) Ulku & Alabas-Uslu, ‘d’ [10].
Hhub is 60 m for all layouts except 78 m (h) and 50 & 78 m (i). Wind conditions as in Fig. S1 apply for all layouts.
5
Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
Table 1
WFLO literature studies analysed and compared.a,b,c,d
Layout
Study
Ref.
Design
Hhub (m)
No WTs
η (%)
Pe (kW/y)
Fitness (×10−4)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
Mosetti et al.
Grady et al.
Emami & Noghreh
Turner et al.
Patel et al.
Parada et al.
MirHassani & Yarahmadi
MirHassani &Yarahmadi
MirHassani &Yarahmadi
Abdelsalam & El-Shorbagy
Ulku & Alabas-Uslu, ‘a’
Ulku & Alabas-Uslu, ,’b’
Ulku & Alabas-Uslu, ‘c’
Ulku & Alabas-Uslu, ‘d’
[18]
[50]
[56]
[57]
[25]
[1]
[26]
[26]
[26]
[20]
[10]
[10]
[10]
[10]
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
60
60
60
60
60
60
60
78
50 & 78
60
60
60
60
60
15
39
28
39
39
40
39
25
25
41
15
28
39
46
84
86.62
91
3,695
32,038
32,262
32,453
33,810
34,173
50,639
36,832
34,132
34,461
12,103
20,776
26,932
30,424
36.1
8.031
1a
1b
1c
1d
1e
1f
1g
1h
1i
1j
1k
1l
1m
1n
91.40
93.62
7.96
7.875
8.155
10.5
9.5
10.0
10.4
a
For each layout, original characteristics and scores are reported.
Values are missing where not reported in the works.
c
Eq. (15) used to calculate η.
d
Wind conditions as in Fig. S1 apply for all layouts; for normalization purposes, they also apply for layouts (g), (h) and (i), although in the original work
zref = 78 m was set [26].
b
Hhub = 78 m (Fig. 1i) being the only case with multiple Hhub values. As
a result of the optimization algorithms that aimed at maximizing the
downwind distance between the WTs, in layouts (e) and particularly (j)
the majority of WTs are positioned towards the outer boundaries of the
WF [25]. In layout (d) WTs are positioned according to a quite regularly
staggered design very similar to that of an array layout. Conversely,
layout (f) is scattered and irregular, with WTs mostly located in the
outermost zone of the WF, particularly on the prevailing wind directions (270–350°) [1]. The optimized layouts proposed by Ulku &
Alabas-Uslu [10], particularly (m) and (n), are structured in arrays with
WTs aligned along the grid columns. Apart from layout (a), the fitness
value – where available – exhibits a narrow range (7.875–10.5x10-4),
with layout (f) returning the lowest value.
Table 2
Assumptions and constraints for WF layouts.
Section
Site
Description
Location
Topography
Elevation
Roughness length
Wind shear coefficient
Air density
Parameter
zsite
z0
Reference height
Vertical profile
Assumptions and constraints, derived from the literature, apply for
both (i) recomputation and (ii) optimization of WFLO literature layouts
(Table 2).
Wind turbines
No. models
(i) layouts
recomputation
(ii) layouts
optimization
No. combinations
(i) layouts
recomputation
(ii) layouts
optimization
Thrust coefficient
4.1. Site characteristics and wind conditions
Site characteristics and wind conditions are basically the same as
those assumed in [33]. Following [49], =0.15 was considered herein.
Wind scenario ‘c’ defined in [18] – plotted in Fig. S1 of the supplementary material – has been assumed, which refers to a height (zref) of
60 m AGL. To normalize all compared studies, zref = 60 m was also
assumed for layouts (g), (h) and (i), although in the original work
zref = 78 m was set [26]. Wind values at zref are adjusted to the proper
Hhub of any WT by applying the PL (Eq. (20)) with = 0.15. Although a
theoretical one, this wind scenario is an accepted reference for most
grid-based WFLO studies (e.g. [58]). It was assumed herein to normalize comparison among all literature studies and to retain the scores
originally achieved therein. In any case, this wind scenario is the most
realistic among the three defined in [18]. On the other hand, refined
wind condition settings such as the one proposed by Haces-Fernandez
et al. [59], where wind speed and direction were segmented into bins to
create a 2-D matrix with all possible combinations, were not considered
since they are beyond the scope of this work.
1.225 kg/m3
zref
v(z)
multi-directional wind with
variable wind speed
(vm = 14 m/s, Fig. S1)
60 m AGL
varying based on PL (Eq. (20))
NT
NTH
a
6
1 (2)a
120
CT
Power curve
The WT database developed in the computation tool detailed in [60]
was used. This database has since been upgraded to include over 350
1
39
WF layout
Design
Size (Fig. 1)
horizontal full size
Lh
vertical full size
Lv
horizontal grid cell size Δh
vertical grid cell size
Δv
WT placement
WT minimum spacing
WT selection criteria
(i) layouts recomputation
Manufacturer and
model
(ii) layouts optimization
Manufacturer(s)
Model(s)
4.2. Wind turbines
onshore
flat
0 m ASL
0.30 m
0.15
ρ
Wind conditions
Wind case study
4. Assumptions and constraints
Value(s)
varying with wind speed (Eq.
(19))
experimental
gridded (row/column)
2000 m
2000 m
200 m
200 m
study-dependent (Fig. 1)
4 rotor diameters
the WT as in Table 4 (case ‘a’)
the same for the whole WF
the same WT along the row,
and varied along the column
Values in brackets refer to layout (i) in Table 1.
Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
onshore commercial models. WT characteristics were retrieved from
manufacturers’ websites as well as from online WT databases such as
Wind turbines models [46] and WindPower [61].
It is accepted (e.g. [32]) that minimum crosswind and particularly
downwind WT spacing should be observed in order to mitigate wake
interactions between WTs in the WF. This constraint should also be
considered for any wake model’s application. For example, Jeon et al.
[34] demonstrated that the errors in predicting velocity deficit by
Jensen’s wake model can be accepted for a downstream WT spacing
above 3.75D. Therefore, consistently with Pookpunt and Ongsakul
[32], a minimum downwind distance of 4D was assumed between adjacent cells in the gridded layout: since Δh = Δv = 200 m, this constraint yielded a maximum D = 50 m (Table 2). Furthermore, a
minimum Pr = 200 kW was set for withdrawing the very small WT
models. Eventually, a total of NT = 39 WTs have been used, extracted
from the WT database after setting D ≤ 50 m and Pr ≥ 200 kW. Their
characteristics are reported in Table 3. Notably, for any WT, experimental power curves have been used rather than approximated (e.g.
[32]), best-fitted (e.g. [3]), or theoretically-constructed (e.g. [18] and
all following grid-based WFLO studies) power curves. Since WTs with
multiple hub heights were handled as distinct WTs, the overall number
of WT combinations was actually NTH = 120.
2002–2020 inflation rate. For WT-related losses, the fx percentages
depending on WT system, derived from [41] and given in Table S2,
were used in Eq. (16); for site-specific losses, fgrid = 2%, fice = 2% and
fother = 0% were set in Eq. (17). To calculate LCoE, d = 5% and
n = 20 years were set in Eq. (25). This procedure generates two outputs: (i) a summary spreadsheet (with NWF records), listing characteristics and scores of all generated layouts; (ii) NWF spreadsheets, one for
each layout, reporting characteristics and scores of each WT in the
farm. Summarizing, NWF = 1,758 layouts are generated and analysed
by the WFLO procedure for each of NLIT literature layouts. Since
NLIT
=
14,
the
overall
number
of
layouts
is:
NLAY = NWF*NLIT = 1,758*14 = 24,612.
A far more simplified WFLO procedure is activated when recomputing the original literature studies as, for each case, only one
single WT (at most with two Hhub values) is imported from the WT
database, and only one layout (NWF = 1) is generated and then analysed: in this case, the number of processed layouts (NLAY = NWF*NLIT)
is 14.
6. Results and discussion
The considered literature layouts have been recomputed and then
optimized. In doing so, real WT models (exhibiting experimental power
curves supplied by manufacturers) have been considered instead of the
theoretical WT model (featuring the theoretical power curve defined in
Eq. (13)) originally used in each of those studies. For the same reason,
the more general Eq. (12) rather than Eq. (15) was used to calculate WF
efficiency. Key factors that influence layout optimization the most have
also been investigated.
4.3. Wind farm layouts
WF layouts are presented in Fig. 1. When literature studies are recomputed, a single WT model is used. When layouts are optimized, the
same WT selection criterion proposed in [33] is followed, which is
detailed in the following. Once each WT is retrieved from the database,
all possible WT combinations obtained by varying model, D and Hhub
are implemented. For any layout instance, a common manufacturer is
chosen for all WTs to be installed in the farm. WTs are the same for each
row, while they are varied from one row to the other. The particular
case where a unique WT model is installed throughout the farm is
considered, too.
6.1. Recomputation of literature layouts
The optimal layouts achieved in the original studies (Table 1) have
been recomputed by considering the same application conditions
(Table 2). In doing so, the Nordex N43-600 wt has been selected,
proving to be the model within the WT database that best-fits the
characteristics of the theoretical WT defined in [18]. WT characteristics
are detailed as (a) in Table 4, while the WT power curve is shown in Fig.
S3a. Although not the same model as in [18], this WT offers the advantage of having two Hhub values in addition to 60 m (i.e. 50 and 78 m)
which enable to also run the layouts (h) and (i) proposed in [26].
Table 5 summarizes the scores achieved after recomputing all literature layouts by employing the Nordex N43-600 wt It is apparent
that, irrespective of the number of installed WTs, the overall WF capacity and corresponding energy production, all layouts return very
similar scores of η and CF, as shown by their very narrow ranges:
98.81–99.61% (η), and 78.89–80.90% (CF). By contrast, a very wide
range is returned for Pe (7,101–21,773 kW/y) and LCoE
(130.37–370.42 $/MWh). Overall, it should be noted that – except for
layout (a) – significantly higher power output was achieved in layouts
optimized in the original studies (Table 1). These findings support the
hypothesis that the heuristic algorithms implemented within each study
likely optimized all such layouts while targeting both Pe and CF maximization rather than LCoE minimization.
Layout (a) returns the lowest LCoE (130.37 $/MWh), while layout
5. Wind farm layout optimization procedure
The applied WFLO procedure (Fig. S2) consists of two stepwise
Fortran modules: a layout generator, and a layout analyser. The same
procedure is used to perform both layout recomputation and optimization. This procedure is a variation of the one developed in [33].
In the most general case (i.e., the optimization approach), based on
the WT selection criteria, the layout generator ingests from the database
all WTs with D ≤ 50 m and Pr ≥ 200 kW. Each layout in Fig. 1 is then
loaded to enable allocation of each WT accordingly. After processing all
available WT combinations, the module generates NWF layouts, each
including sequential code, database identifier and coordinates of all
allocated WTs. During the second stage, all previously generated layouts are ingested by the layout analyser, with characteristics of any WT
retrieved from the database. The layout analyser performs the computation of CT(v), vertical extrapolation of wind speed, Jensen’s model
and corresponding losses/efficiency, losses linked to the site and WT,
energy output scores, and cost model. The NREL cost model [54] was
used to calculate LCoE after setting [$] as the currency and the average
Table 3
Summary of characteristics of all processed onshore WTs.a
Score
Median value
Range
a
Rated power
Rotor diameter
Hub height
Wind speeds
Pr (kW)
D (m)
Hhub (m)
Cut-in
vi (m/s)
Rated
vr (m/s)
Cut-off
vo (m/s)
Φ
600
220–900
43
26–50
50
30–78
3.5
2.5–5.0
14.0
11.0–18.0
25.0
20.0–28.0
4.3
3.1–5.3
Total WT models: NT = 39. Total WT combinations (sorted by Hhub): NTH = 120.
7
Design ratio
Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
Table 4
Characteristics of WTs used or achieved while running all WFLO literature layouts.a
Approach
(case)
Application condition
(i) layouts recomputation
(a)
Recomputing
(ii) layouts optimization
(b)
Minimizing LCoE
(c)
a
b
Maximizing CF
Model
Layout(s)
Pr (kW)
D (m)
Hhub (m)
vi (m/s)
vr (m/s)
vo (m/s)
Systemb
Nordex
N43-600
All
600
43
50
60
78
3
13.5
25
A
Vergnet
GEV 26/220
Windtec 650
(a)
220
26
60
3.5
17
20
A
(a)
600
50
71.5
3.5
11
20
C
Power curves for each WT are plotted in Fig. S3.
Coding for WT systems is reported in Table S2 [41].
(h) the highest CF (80.90%). The number of installed WTs significantly
affects the results, as shown by the near matching scores from 28- wt
layouts (c) and (l), as well as those from 39- wt layouts (b), (d), (e), (g)
and (m). Scores from 15- wt layout (k) are very similar to those from
15- wt layout (a), too, particularly as minimum Pe and LCoE are basically the same, and values of all other indicators are only slightly worse.
Pe scores are also a strict function of the number of installed WTs.
Contrary to suggestions by MirHassani and Yarahmadi [26] that using
WTs with different Hhub leads to an increase in the power output, with
Pe = 11,962 kW/y layout (i) does not improve performances of the
layout (h), which has the same design but WTs with identical Hhub.
From the scores in Table 5, it is apparent that LCoE is a much more
straightforward economic indicator than the fitness value, because not
only does it exhibit a wider variation range than the latter (Table 1), but
it also completely reverses the ranking of (a) from the worst (Table 1) to
the best (Table 5) layout.
Significantly improved η values are achieved with respect to the
original applications (Table 1, where reported), where, however, Jensen’s wake model was applied using CT = 0.88. To assess the differences against the present study where the CT(v) approach is implemented, the CT = 0.88 option has been also analysed herein (see
Table S3). It is apparent that η values are lower (overall ranging
94.69–98.18%), thus indicating that using CT = 0.88 is more conservative than using CT(v). The impacts are however quite marginal on
Pe (7,025–21,512 kW/y), CF (77.94–80.50%), and LCoE
(130.50–371.29 $/MWh).
6.2. Optimization of literature layouts
For each literature layout, a total of NWF = 1758 layout instances
have been generated, ultimately allowing the detection of only one
solution that satisfies the selected optimization function. Table 6 shows
the comparative results of layouts that minimize LCoE, while Table 7
shows those of layouts that maximize CF.
All recomputed literature layouts (Table 5) have been optimized by
the present optimization method, as both LCoE values have been reduced and CF values increased. While LCoE reduction is substantial
(130.37–370.42 to 54.01–142.64 $/MWh, Table 6), CF increase is
marginal (78.89–80.90 to 83.02–83.07%, Table 7). It is worth noting
that for all literature case studies and both optimization conditions, all
optimal layouts have been achieved by using identical WTs over the
whole WF: this means that a unique combination for Pr, D and Hhub
(NP = ND = NH = 1) applies. Furthermore, for all case studies, the
same WT model proved to minimize LCoE, and the same WT model to
maximize CF. As shown in Table 4, LCoE minimization is accomplished
by using the Vergnet GEV 26/220 wt (whose power curve is plotted in
Fig. S3b), while CF maximization is achieved by using the Windtec
650 wt (power curve in Fig. S3c). With respect to layouts recomputation, LCoE minimization is accomplished by reducing Pr and D, and
concurrently increasing vr in the selected WT (Table 4), while CF
maximization is accomplished – for the same Pr – by increasing D and
decreasing vr . These outcomes agree with suggestions, e.g., by Abdulrahman and Wood [7], that small-medium sized WTs with low hub
height should be installed to minimize LCoE.
Table 5
Comparative results of WF literature layouts recomputed using a single WT model.a,b,c,d
Layout
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
Ref., and Hhub
Mosetti et al. [18], 60 m
Grady et al. [50], 60 m
Emami & Noghreh [56], 60 m
Turner et al. [57], 60 m
Patel et al. [25], 60 m
Parada et al. [1], 60 m
MirHassani & Yarahmadi [26], 60 m
MirHassani & Yarahmadi [26], 78 m
MirHassani & Yarahmadi [26], 50 & 78 m
Abdelsalam & El-Shorbagy [20], 60 m
Ulku & Alabas-Uslu [10], ‘a’, 60 m
Ulku & Alabas-Uslu [10], ‘b’, 60 m
Ulku & Alabas-Uslu [10], ‘c’, 60 m
Ulku & Alabas-Uslu [10], ‘d’, 60 m
Cumulated
Mean
Cumulated
Mean
Cumulated
No WTs
P (MW)
η (%)
Pe (kW/y)
AEY (MWh/y)
CF (%)
LCoE $/MWh)
15
39
28
39
39
40
39
25
25
41
15
28
39
46
9
23.4
16.8
23.4
23.4
24
23.4
15
15
24.6
9
16.8
23.4
27.6
99.61
99.21
99.35
99.23
99.08
99.13
99.04
99.41
99.53
99.05
98.90
98.81
98.95
98.82
7116
18,480
13,273
18,481
18,473
18,950
18,471
12,135
11,962
19,419
7,101
13,253
18,466
21,773
62,335
161,884
116,271
161,897
161,826
165,998
161,806
106,301
104,784
170,112
62,209
116,094
161,766
190,732
79.06
78.97
79.01
78.98
78.95
78.96
78.94
80.90
79.74
78.94
78.91
78.89
78.92
78.89
130.37
316.16
231.00
316.16
316.18
323.92
316.19
207.38
207.70
331.67
130.43
231.08
316.21
370.42
a
Characteristics of each WF layout are depicted in Fig. 1.
Eq. (12) used to calculate η.
c
Wind conditions as in Fig. S1 apply for all layouts; for normalization purposes, they also apply for layouts (g), (h) and (i), although in the original work
zref = 78 m was set [26].
d
The same WT (Table 4, case ‘a’) is used for all WFLO studies.
b
8
Energy Conversion and Management 208 (2020) 112593
G. Gualtieri
Table 6
Comparative results of WF literature layouts optimized by LCoE minimization.a,b,c,d
Layout
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
a
b
c
d
Ref., and Hhub
Mosetti et al. [18], 60 m
Grady et al. [50], 60 m
Emami & Noghreh [56], 60 m
Turner et al. [57], 60 m
Patel et al. [25], 60 m
Parada et al. [1], 60 m
MirHassani & Yarahmadi [26], 60 m
MirHassani & Yarahmadi [26], 78 m
MirHassani & Yarahmadi [26], 50 & 78 m
Abdelsalam & El-Shorbagy [20], 60 m
Ulku & Alabas-Uslu [10], ‘a’, 60 m
Ulku & Alabas-Uslu [10], ‘b’, 60 m
Ulku & Alabas-Uslu [10], ‘c’, 60 m
Ulku & Alabas-Uslu [10], ‘d’, 60 m
Cumulated
Mean
Cumul.
Mean
Cumul.
No WTs
P (MW)
η (%)
AEY (MWh/y)
CF (%)
LCoE ($/MWh)
15
39
28
39
39
40
39
25
25
41
15
28
39
46
3.30
8.58
6.16
8.58
8.58
8.80
8.58
5.50
5.50
9.02
3.30
6.16
8.58
10.12
99.63
99.22
99.37
99.25
99.10
99.14
99.05
99.36
99.39
99.07
98.90
98.82
98.96
98.83
21,928
56,945
40,902
56,950
56,924
58,392
56,917
36,519
36,522
59,839
21,882
40,835
56,902
67,090
75.86
75.76
75.80
75.77
75.74
75.75
75.73
75.80
75.80
75.73
75.69
75.67
75.71
75.68
54.01
122.61
91.17
122.60
122.62
125.47
122.62
82.59
82.59
128.33
54.05
91.20
122.63
142.64
Mean
Cumul.
Mean
Cumul.
Characteristics of each WF layout are depicted in Fig. 1.
Eq. (12) used to calculate η.
Wind conditions as in Fig. S1 apply for all layouts.
The same WT (Table 4, case ‘b’), installed over the whole WF, achieved to optimize all layouts.
Table 7
Comparative results of WF literature layouts optimized by CF maximization.a,b,c,d
Layout
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
a
b
c
d
Ref., and Hhub
Mosetti et al. [18], 60 m
Grady et al. [50], 60 m
Emami & Noghreh [56], 60 m
Turner et al. [57], 60 m
Patel et al. [25], 60 m
Parada et al. [1], 60 m
MirHassani & Yarahmadi [26], 60 m
MirHassani & Yarahmadi [26], 78 m
MirHassani & Yarahmadi [26], 50 & 78 m
Abdelsalam & El-Shorbagy [20], 60 m
Ulku & Alabas-Uslu [10], ‘a’, 60 m
Ulku & Alabas-Uslu [10], ‘b’, 60 m
Ulku & Alabas-Uslu [10], ‘c’, 60 m
Ulku & Alabas-Uslu [10], ‘d’, 60 m
Cumulated
No WTs
P (MW)
η (%)
AEY (MWh/y)
CF (%)
LCoE ($/MWh)
15
39
28
39
39
40
39
25
25
41
15
28
39
46
9.00
23.40
16.80
23.40
23.40
24.00
23.40
15.00
15.00
24.60
9.00
16.80
23.40
27.60
99.63
99.25
99.38
99.28
99.14
99.18
99.10
99.38
99.41
99.11
98.98
98.90
99.02
98.90
65,494
170,232
122,231
170,236
170,216
174,587
170,211
109,134
109,137
178,941
65,459
122,182
170,200
200,729
83.07
83.05
83.06
83.05
83.04
83.04
83.04
83.06
83.06
83.04
83.03
83.02
83.03
83.02
125.03
309.33
224.86
309.33
309.33
317.01
309.34
201.82
201.82
324.69
125.04
224.87
309.34
363.10
Characteristics of each WF layout are depicted in Fig. 1.
Eq. (12) used to calculate η.
Wind conditions as in Fig. S1 apply for all layouts.
The same WT (Table 4, case ‘c’), installed over the whole WF, achieved to optimize all layouts.
Table 6 shows that layout (a) (termed “L1”) returns the lowest LCoE
value (54.01 $/MWh), yet only slightly lower than layout (k) which
achieves a score very close to this. This value is consistent with the
global average LCoE value (60 $/MWh) reported for 2017-updated
onshore wind projects by IRENA [52]. Using observed wind data and
applying the NREL cost model, at the real site of Khaf in Iran (40-m
vm = 10.5 m/s) Mirghaed and Roshandel [31] designed a square
(1.73 km × 1.73 km) layout with crosswind/downwind WT distances
of 3D × 6D: notably, they obtained an LCoE value of 55 $/MWh very
close to the minimum values (54.01–54.05 $/MWh) obtained in the
current study (40-m vm = 13.2 m/s). Conversely, current LCoE minima
are larger than those (21–31 $/MWh) calculated by Rahbari et al. [5] at
a real site in Teheran based on observed wind data (10-m vm = 6.38 m/
s), where they designed four different rectangular layouts (sized
1.5 km × 2.5 km) with WTs irregularly allocated. Current LCoE scores
are either lower or higher than minimum LCoE value (103.46 $/MWh)
achieved in the previous WFLO study, based on the continuous model
and run under quite similar application conditions (i.e. same site features, layout shape and size, and wind conditions) [33]. This indicates
that the discrete model can perform either better or worse than the
continuous model and that, consistent with reviews, e.g., by Serrano
et al. [2] or Shakoor et al. [4], a superior model cannot be clearly ascertained.
The CF values maximized in Table 7 match among all the considered
studies (83.02–83.07%), thus reinforcing the hypothesis raised in section 6.1 that optimization of such literature layouts tends to a common
value, corresponding to maximum CF. Regardless, layout (a) (termed
“L2”) also proves to be the best solution for maximizing CF. Comparing
Tables 6 and 7, it is apparent that the LCoE minimization target combines with satisfactory scores of CF (75.67–75.86%, Table 6), while this
is not the case for CF maximization, which overall is not associated to
particularly profitable scores of LCoE (125.03–363.10 $/MWh,
Table 7). Current CF values are lower than the maximum CF value
(92.17%) achieved in the previous WFLO study based on the continuous
model [33]: although indirectly, this outcome is consistent with findings by Long et al. [12] that the continuous model is better than the
discrete model when maximizing the power output is set as WFLO
target.
Characteristics and performances of WTs installed in layouts optimized by LCoE minimization and CF maximization are detailed,
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G. Gualtieri
respectively, in Table S4 (L1) and Table S5 (L2), while their placement
within the WFs is depicted in Fig. S5. It is apparent that L1 and L2
correspond to the same layout (a), except that they employ two different WT models. As reported in Table 4, such WT models belong to
systems A and C as defined in Table S2, i.e. those accounting for the
lowest overall WT-specific losses. If considering characteristics of all
available WTs (Table 3), LCoE minimization (Table S4) is achieved
using a WT exhibiting the lowest Pr and D, and almost maximum vr.
These outcomes agree with suggestions by Chen et al. [40] that higher
vr values are required by sites with higher vm and result in lower WTrelated LCoE, and that larger WTs do not necessarily reduce LCoE. The
values of Ω (84.29–84.47%) indicate that on average the WT operates
below its rated speed. By contrast, CF maximization (Table S5) is
achieved through a WT of medium Pr, with minimum vr and Φ, and Hhub
close to the maximum (Table 3). Corresponding Ω values
(130.73–131.00%) indicate that to a remarkable extent (over 30%) the
WT operates above its rated speed, thus largely exceeding the threshold
over which the WF overall cost increase is adequately compensated for
by the revenues deriving from energy production.
Cini/Pr (Fig. S4n), and CF (Fig. S4p), meaning that – for any WT combination in the WF – all these parameters are insensitive to the selected
layout design. This outcome is particularly meaningful for CF, as it
confirms that whatever the chosen literature layout, this score does not
appreciably change. Conversely, patterns are layout-dependent for P
(Fig. S4d), total losses (Fig. S4m), AEY (Fig. S4o), and LCoE (Fig. S4q),
meaning that choosing a specific layout dramatically affects WF power
production and particularly the project’s economic viability. 15- wt
layouts (a) and (k) exhibit a similar pattern for P, AEY, and LCoE, with
main discrepancies apparent in η. For layouts (k) and (l) there is a relevant frequency of layout instances built-up with a twofold combination of Pr (Fig. S4a) and D (Fig. S4b), while for layout (k) this also
involves Hhub values (Fig. S4c). Layout (a) proves to be the best solution
overall for maximizing η, minimizing total losses, and – most importantly – minimizing LCoE. By contrast, layout (a) returns the lowest
overall AEY values. It is worth noting that the plots of AEY and LCoE
can almost perfectly be overlapped. In terms of total power losses (Fig.
S4m), wake losses are almost negligible (Fig. S4k), particularly if
compared to losses linked to the site and WT (Fig. S4l). Furthermore, in
the majority of layout instances, WTs operate in wind regimes below
their rated speed (Fig. S4j).
After lumping all literature layout instances together, a correlation
analysis between all parameters has been performed (Table S6). Since
no correlation exists between CF and LCoE (r = −0.03), increasing CF
can either reduce or increase LCoE. As shown by Figs. S2o and S2q, AEY
and LCoE are strongly correlated (r = 0.99): therefore, for a high wind
potential site, WFLO is surprisingly achieved by reducing the total energy production. LCoE is also strongly correlated to P (r = 1.00), thus
indicating that the overall installed power rather than the mere number
of WTs (r = 0.84) should be reduced for reducing LCoE. Pr affects LCoE
more than vr (r of 0.51 vs. –0.02), which contrasts with suggestions by
Chen et al. [40] that vr is more influential on LCoE than Pr. Distributions
of Ci and AEY are strongly correlated as well (r = 1.00), meaning that
annual expenditures increase with WF energy production. Total losses
6.3. Insights into wind farm layout optimization key factors
Optimization functions and key factors that mostly affect WFLO
have been investigated after applying kernel density plots, correlation
analysis and self-organizing maps to main WFLO parameters.
6.3.1. Kernel density and correlation analysis
All layout instances generated per literature study have been analysed in terms of the distribution of main WFLO parameters. The result
of this analysis is summarized through the kernel density plots shown in
Fig. S4, drawn by using the “sm.density.compare” tool [62] implemented in the “sm” R package [63]. For all literature layouts, quite
similar distributions may be observed for Pr (Fig. S4e), D (Fig. S4f), Hhub
(Fig. S4g), vr (Fig. S4h), Φ (Fig. S4i), Ω (Fig. S4j), Fsite & FWT (Fig. S4l),
Fig. 2. SOM component planes obtained while optimizing literature layouts for the following parameters: No. WTs, NP, ND, NH, P, Pr, D, Hhub, vr, Φ, Ω, η, Fsite & FWT,
Ftot, Cini/Pr, AEY, CF, and LCoE. Cumulated values are plotted for No. WTs, P and LCoE, while WF-averaged values for all other parameters. A total of 1,758 layouts by
14 literature case studies (NLAY = NWF*NLIT = 24,612) were processed.
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G. Gualtieri
are strongly correlated to site- and WT-related losses (r = 0.96), and
poorly correlated to η, and thus to array losses (r = 0.33): since sitespecific losses do not change, WT-related losses are the component that
primarily affects total losses, meaning that the system the selected WT
belongs to (see Table S2) is pivotal in affecting WF overall losses. On the
other hand, a significant correlation (r = 0.86) was ascertained between Ω and CF, yet it was not as strong (e.g., r > 0.95) as to imply
that Ω gives the same level of information as CF.
6.3.1), the SOMs of Ftot basically match with those of Fsite & FWT. Since
the SOMs of η confirm that wake losses are almost negligible, it is also
confirmed that WF total losses are mainly driven by losses depending on
WT characteristics: therefore, the selection of WTs returning the lowest
losses (see Table S2) should be prioritised. In any case, reducing total
losses positively affects CF but not LCoE, as CF increases do not necessarily result in LCoE decreases.
The highest energy production may be obtained by the most expensive layouts, i.e. those that maximize the overall WF installed capacity and employ WTs with the highest values of Pr, D and Hhub.
However, AEY maximization is economically disadvantageous as they
correspond to the highest values of LCoE. This means that maximizing
the overall WF power output, as pursued within several studies (e.g.
[57]), is a misleading WFLO goal.
Summarizing, the SOMs proved useful in improving the multivariable investigations addressed in the past literature either dealing
with WFLO (e.g. [31]), or WT optimal site matching (e.g. [40]). They
are also particularly helpful when WFLO problems involving multiple
objective functions (as performed, e.g., by Abdulrahman and Wood [7])
are addressed. Since they are capable of representing the relationships
among main WF layout parameters through a single, compact plot, the
SOMs allow to achieve a straightforward WFLO pattern, thus efficiently
coping with the non-convex, highly complex, multi-purpose and multivariable nature of the WFLO problem.
6.3.2. The self-organizing maps
To more thoroughly investigate WFLO key factors, an advanced
analysis than both kernel density and correlation analysis (Section
6.3.1) was accomplished through the use of SOMs. A dataset including
NWF = 1758 layout instances multiplied by NLIT = 14 case studies, for a
total of NLAY = NWF*NLIT = 24,612 layouts, was used. The following 18
variables were selected as component planes of the SOMs: No. WTs, NP,
ND, NH, P, Pr, D, Hhub, vr, Φ, Ω, η, Fsite & FWT, Ftot, Cini/Pr, AEY, CF, and
LCoE. Ci was not considered since its distribution is the same as AEY
(r = 1.00, Table S6). The following final SOM structure was achieved
for the above 18 variables: (i) an input layer, including 24,612 neurons
corresponding to all layout instances; (ii) an output layer, including 100
neurons, represented through 10x10 gridded hexagons. Before SOM
training, all variables’ values were normalized such that their mean was
0 and their variance was 1 [64]. The SOMs were trained following the
requirements of the Kohonen’s algorithm [55]: thus, 60,000 iterations
(over 500 times the number of gridded neurons) were used, along with
initial values of 0.9 (close to 1) for learning rate and 7 units (higher
than grid’s radius) for neighbourhood radius. The Mexican hat was
selected as neighbourhood function since it results in a larger variables’
variation range than the most widely used Gaussian function, while the
goodness of SOM was assessed using the Kaski and Lagus error metrics
[65], that combine quantization and topographic errors. The graphical
output of SOM application, performed by using the Living For SOM tool
(http://livingforsom.com), is shown in Fig. 2.
In SOM topology of LCoE, lower-to-higher neurons are arranged
bottom to top, with minimum and maximum values located centrally. A
similar topology is exhibited by SOMs of P and AEY, and – to a lesser
extent – Pr, D, and No. WTs: this means that LCoE may be minimized by
decreasing the overall WF installed capacity, the energy production,
rated powers, rotor diameters or the number of WTs. These scores
confirm findings achieved in section 6.2 that lie behind the optimization of layout L1 (Table S4). The association of LCoE with NP, ND and NH
is weaker, as shown by their different SOM topologies: LCoE minima are
associated to lower but not minimum values of NP, ND and NH, which
are located in the centre of their respective maps. In any case, this
means that LCoE could be reduced by avoiding to install WTs with
multiple Pr, D or Hhub (as accomplished for layout L1). This finding
seems to contradict those from various Authors, suggesting that it is
more profitable to suitably combine different WTs rather than installing
identical WTs, particularly having different D [66] or different Hhub
[58]. Since the WT combinations in the farm reduce to just one for each
WT per manufacturer, WT selection criteria set in section 4.3 are dramatically simplified.
Notably, CF neurons do not arrange in a mirrored fashion with respect to LCoE neurons, suggesting that maximizing CF does not imply
minimizing LCoE: confirming the null correlation between CF and LCoE
(section 6.3.1), a CF increase may either result in an LCoE increase or
decrease. Rather, SOM clustering of CF shows a neurons arrangement
quite similar to the ones of Hhub, Φ, Ω, and (mirrored) vr, suggesting that
an increase (decrease for vr) in such variables leads to an increase in CF.
These results were anticipated in section 6.2 when analysing the
characteristics of optimal layout L2 (Table S5). As for the SOM of Ω, it is
confirmed that values close to 100% do increase CF, but also involve an
overall energy waste, since resulting in an energy over-production (see
AEY) not adequately compensated by a corresponding revenue (see
LCoE). Consistent with findings from the correlation analysis (section
7. Conclusions
A total of 14 grid-based layouts addressed in the literature and run
based on wind scenario ‘c’ defined in Mosetti et al. [18] have been
recomputed. Several application improvements have been performed to
the original settings, including the use of: (i) commercial-scale WTs
instead of a theoretical WT; (ii) experimental instead of theoretical WT
power curves; (iii) CT varying with wind speed rather than the 0.88
constant value; (iv) LCoE as an objective function rather than the simple
fitness value; (v) the detailed NREL cost model in place of the overlysimplistic cost model defined in [18]. All layouts returned very similar
scores of farm efficiency (98.81–99.61%) and CF (78.89–80.90%), but
widely varying scores of LCoE (130.37–370.42 $/MWh): this indicates
that CF maximization rather than LCoE minimization was the actual
target pursued by the heuristic algorithms implemented within each
WFLO study. Results indicated the 15- wt layout by Mosetti et al. [18]
as the optimal in minimizing LCoE (130.37 $/MWh), and the 25- wt
layout by MirHassani and Yarahmadi [26] as the optimal in maximizing
CF (80.90%). As such, results achieved from this layout recomputation
proved to be a novelty in themselves, as they were quite different from
those returned from the original studies, which adopted energy yield
maximization and/or fitness value minimization as an objective function.
In the present study, relying on 39 commercial WTs, a previously
developed optimization method [33] was applied which targets optimal
WT selection while retaining the original number of WTs and their
placement as an assumption. Previously applied to the continuous WF
layout model, the method has also been successfully applied to the
discrete layout model. For each literature case study, a total of 1,758
layout instances were generated, overall resulting in 24,612 layouts.
Method’s application proved to optimize all literature layouts, as CF
values were (slightly) increased (78.89–80.90 to 83.02–83.07%), while
LCoE values were (significantly) reduced (130.37–370.42 to
54.01–142.64 $/MWh). It is worth noting that the layout by Mosetti
et al. [18] proved to be the best both in minimizing LCoE (54.01
$/MWh) and in maximizing CF (83.07%). Furthermore, for all literature case studies and both optimization conditions, all optimal layouts
were achieved by using identical WTs: this means that only one combination of rotor diameter, rated power and hub height applies over the
whole WF. For all case studies, the same WT model (Vergnet GEV 26/
220) minimized LCoE, and the same WT model (Windtec 650)
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G. Gualtieri
maximized CF. LCoE scores were either better or worse than those
achieved in the previous WFLO study, implementing the continuous
model and run under similar application conditions [33]: therefore, a
model – be it discrete or continuous –cannot be recommended since
neither clearly outperforms the other.
A thorough analysis has been performed to investigate optimization
functions and the parameters that mostly influence WFLO. Although
specifically applying to high wind potential sites where mid-sized WTs
are supposed to be installed, various conclusions can be made:
Acknowledgments
No funds were received to support the research conducted in this
paper.
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://
doi.org/10.1016/j.enconman.2020.112593.
• the two optimization conditions are not associated, as a CF increase
does not necessarily imply an LCoE decrease;
• LCoE can be minimized by decreasing farm overall capacity or the
•
•
•
•
•
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CRediT authorship contribution statement
Giovanni
Gualtieri:
Conceptualization,
Data
curation,
Investigation, Methodology, Supervision, Validation, Visualization,
Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this paper.
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