energies
Article
A Simple Model for Wake-Induced Aerodynamic Interaction of
Wind Turbines
Esmail Mahmoodi 1 , Mohammad Khezri 2 , Arash Ebrahimi 3,4 , Uwe Ritschel 3,4, *, Leonardo P. Chamorro 5
and Ali Khanjari 6
1
2
3
4
5
6
*
Citation: Mahmoodi, E.; Khezri, M.;
Ebrahimi, A.; Ritschel, U.; Chamorro,
L.P.; Khanjari, A. A Simple Model for
Wake-Induced Aerodynamic
Interaction of Wind Turbines.
Energies 2023, 16, 5710. https://
doi.org/10.3390/en16155710
Academic Editors: Jao-Hwa Kuang,
Bing-Jean Lee, Ming-Hung Hsu,
Thin-Lin Horng and Chia-Cheng
Chao
Department of Mechanical Engineering of Biosystems, Shahrood University of Technology,
Shahrood 3619995161, Iran; esmail.mahmoodi@gmail.com
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 9177948974, Iran;
khezri.muhamad@gmail.com
Chair of Wind Energy Technology, Faculty of Mechanical Engineering and Marine Technologies, University of
Rostock, 18051 Rostock, Germany; arash.ebrahimi@uni-rostock.de
IWEN Energy Institute, 18119 Rostock, Germany
Department of Mechanical Science and Engineering, University of Illinois, Urbana, IL 61801, USA;
lpchamo@illinois.edu
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA;
ali.khanjari69@gmail.com
Correspondence: uwe.ritschel@uni-rostock.de
Abstract: Wind turbine aerodynamic interactions within wind farms lead to significant energy losses.
Optimizing the flow between turbines presents a promising solution to mitigate these losses. While
analytical models offer a fundamental approach to understanding aerodynamic interactions, further
development and refinement of these models are imperative. We propose a simplified analytical
model that combines the Gaussian wake model and the cylindrical vortex induction model to evaluate
the interaction between wake and induction zones in 3.5 MW wind turbines with 328 m spacing.
The model’s validation is conducted using field data from a nacelle-mounted LiDAR system on
the downstream turbine. The ‘Direction to Hub’ parameter facilitates a comparison between the
model predictions and LiDAR measurements at distances ranging from 50 m to 300 m along the rotor
axis. Overall, the results exhibit reasonable agreement in flow trends, albeit with discrepancies of
up to 15◦ in predicting peak interactions. These deviations are attributed to the single-hat Gaussian
shape of the wake model and the absence of wake expansion consideration, which can be revisited
to improve model fidelity. The ‘Direction to Hub’ parameter proves valuable for model validation
and LiDAR calibration, enabling a detailed flow analysis between turbines. This analytical modeling
approach holds promise for enhancing wind farm efficiency by advancing our understanding of
turbine interactions.
Keywords: wind energy; wake induced aerodynamic; LiDAR; wind flow interaction
Received: 1 July 2023
Revised: 24 July 2023
Accepted: 27 July 2023
1. Introduction
Published: 31 July 2023
The use of wind energy has significantly grown in recent decades [1]. Wind turbines
within wind farms are usually exposed to wake flows, resulting in an average power per
turbine lower than the power converted by an isolated unit. This power deficit can be
significant, and alignment and turbine distance are critical [2]. Various research efforts
have explored this phenomenon using both empirical and physics-based models.
Recent advancements in analytical wake models have focused on enhancing the accuracy of predicting wind turbine interactions. Physics-based models, like the pioneering
Jensen wake model [3], have been devised to simulate and predict complex wake interactions within wind farms. This model is based on mass conservation and represents
the wind turbine wake as a top hat-shaped distribution. However, recognizing the Gaussian distribution’s improved representation of the wake [4–6], Gao et al. [7] proposed the
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Energies 2023, 16, 5710. https://doi.org/10.3390/en16155710
https://www.mdpi.com/journal/energies
Energies 2023, 16, 5710
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Gaussian distribution of the Jensen mass conservation model. They further refined it by
incorporating ambient and added turbulence intensity using an empirical engineering
turbulence model, validating the outcomes through wind tunnel experiments. More recent
models, such as those developed by Bastankhah et al. [4] and Qian et al. [8], have integrated
additional parameters, such as yaw misalignment angles, to better capture real-world wind
farm conditions. Other models like the dynamic wake meandering model [9], the Super
Gaussian model [10], and its alternative forms [11], have also been introduced to enhance
wake simulation accuracy. Beyond the physics-based approaches, newer models such as
the curled wake model [12], its development for yaw conditions [13], subjected to veered
inflow [14,15], and CFD-based models, including the actuator line [16], actuator disk [17],
and large eddy simulation [18] models, have been developed to further improve the precision of wake modeling. Howland et al. [19] developed a physics-based, data-assisted
flow control model to predict the power-maximizing control strategy. The model considers
yaw misalignment angles that maximize the array power production within ±5◦ for most
wind directions. A thorough revision of the characteristics of the wake interacting with the
induction zone [20] between wind turbines is necessary to enable the robust prediction of
wind behavior on a wind farm and reduce power deficits at a lower cost.
Nacelle-mounted LiDARs are widely used to measure wind flow around wind turbines [21]. They allow estimating the inflow wind speed and turbulence intensity [22],
wind direction and vertical mean shear [23], power curve [24], turbine load carried in a
free flow condition [25], wake deficit, wake turbulence, wake meandering [26], and yaw
misalignment in the wake state [27], which are primarily used to control wind farms and to
evaluate theoretical and empirical models. Using LiDAR, Scholbrock et al. [28] conducted
field tests and described the fieldwork of a feed-forward collective pitch control and discussed its impact on speed regulation in above-rated winds. Giyanani et al. [29] proposed
an autoregressive model that included the blockage factor to estimate wind speeds using
LiDAR measurements accurately. Fleming et al. [30] performed field tests of wake steering
on a full-scale turbine, and the measurements were compared to predictions of a wind
farm control-oriented wake model. Adcock et al. [31] proposed an optimization framework
that leveraged high-fidelity field or simulation data to correct a lower-fidelity flow model.
Understanding the effects of these two aspects on aerodynamic loading can help establish
correlations and coherence-based look-up tables with inflow-loading relationships for each
operational regime. Giyanani et al. [32] presented a correlation between LiDAR-based
wind speed and aerodynamic loading for three LiDAR measurement ranges below and
above the rated operation modes. Qu et al. [33] described data-driven yaw misalignment
calibrations of a wind turbine via LiDAR verification and proposed a yaw zero-point misalignment calibration method to improve wind direction signal accuracy. Chen et al. [34]
proposed a two-step Cholesky decomposition for factorizing coherence matrices in wind
field generation. Rinker [35] presented a preprocessing approach to convert LiDAR data
to PyConTurb-ready constraints. Couto et al. [36] explored the effects of wave-/windinduced oscillations on the power performance of a wind turbine operating on a WindFloat
floating system. Very recently, Russell et al. [37] conducted a comprehensive review of
LIDAR-assisted control strategies in wind turbines, emphasizing their capacity to decrease
structural loads, prolong turbine lifetimes, enhance power capture, and decrease the cost
of energy. They offered valuable suggestions for future research and implementing these
strategies in the industry.
It is worth highlighting the use of the LiDAR technique in understanding turbine
wake flows and the parameters affecting the technique. In particular, Iungo et al. [38]
performed field measurements on the wake of a 2 MW wind turbine using three highresolution scanning Doppler wind LiDARs. They showed significant turbulence and
velocity fluctuations in the near-wake region at the turbine’s top tip height, pointing out
potential fatigue loads for downstream turbines. Lundquist et al. [39] explored the Doppler
beam swinging (DBS) technique assumption of flow homogeneity, using it in complex
flows such as wind turbine wakes. The quantification of a stably stratified flow past a
Energies 2023, 16, 5710
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wind turbine revealed significant errors in the downwind observations, with streamwise
component errors reaching 30% of the inflow. DBS-based assessments of wake flow deficits
in the near wake could be relied on by three rotor diameters downwind, but interpretations
of field observations should consider the uncertainties caused by flow inhomogeneity.
Klaas et al. [40] conducted a non-dimensional, model-based parameter study to investigate
the influence of different factors on LiDAR measurement errors in complex terrains. They
separated the LiDAR errors into two components: flow curvature at the measurement
points and local speed-up effects. They also indicated that this approach allows for a
comprehensive interpretation of how the LiDAR’s half-cone opening angle impacts the
measurements and insight into the uncertainty associated with error estimations based on
the inflow and outflow angles at the measurement points.
This paper introduces a simple wake–induction interaction model, employing the
Gaussian distribution of the Jensen wake model. We highlight the significance of physicsbased models in understanding and predicting flow interactions between wind turbines
on a wind farm. We propose an evaluation parameter to validate our model based on
comparing model predictions with field measurements acquired from a nacelle-mounted
four-beam LiDAR. Using LiDAR measurements has proven valuable in validating and
refining flow models, and recent advancements in modeling and measurement techniques
have contributed to significant progress in wind farm optimization and power deficit
reduction. Section 2 details the field setup and methods, while Section 3 presents the
findings and analysis. Section 4 provides the main conclusions.
2. Materials and Methods
This study discusses a simple method for predicting longitudinal wind speeds in the
interaction zone between wake and induction. We combine a 2D Jensen wake model with a
Gaussian-shape distribution velocity deficit [41] and a cylindrical vortex induction zone
model [42,43]. While the Gaussian distribution of the Jensen wake model does not consider
the momentum balance, it provides a good prediction of longitudinal wind speed in a wake
based on the conservation of mass [4,7]. In addition, the cylindrical vortex induction zone
model used in this study can estimate wind speed in non-yaw actuator disc simulations
with relatively low errors [44].
2.1. Field Setup
We tested and validated the interaction model using data obtained from a nacellemounted LiDAR system equipped with four beams. The LiDAR system was installed on
a utility-scale wind farm comprising two 3.5 MW Siemens SG3.4-175 wind turbines. The
wind turbines are positioned near Rostock City in Germany, with an alignment towards the
north at 297 degrees and a distance of 328 m between them (Figure 1a). The arrangement
of the wind turbines indicates that, when the wind blows from 297 or 117 degrees north,
the wake effect caused by one turbine reduces the power production of the other turbine.
A four-beam LiDAR system was installed on the nacelle of wind turbine No. 2 (Figure 1b).
This LiDAR system enabled the measurement of wind parameters at 10 vertical planes
located upwind of turbine No. 2, ranging from 50 to 450 m (0, 75, 100, 150, 200, 250, 300, 350.
400, and 450 m) axial distances in front of the turbine (as illustrated in Figure 2a). The three
planes positioned at 350, 400, and 450 m distances were excluded from the investigation,
as they were outside the regions between the turbines. The measuring methodology, as
depicted in Figure 2a, facilitated the direct estimation of the upwind velocity deficit and
axial induction using this measurement system.
The measuring LiDAR sensor (Figure 2b) maintains constant α and β angles of 5 and
15 degrees, respectively. As a result, the angles ϑ and ϕ can be directly calculated based on
these fixed values.
The methodology employed to acquire and assess the LiDAR measurements and
evaluate the interaction model is discussed in Section 2.5. It describes the process of
Energies 2023, 16, 5710
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Energies 2023, 16, x
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estimating the ‘Direction to Hub’ parameter from LiDAR measurements and subsequently
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comparing it with the basic predictions.
a
b
b
a
LiDAR
LiDAR
Turbine 1
Turbine 1
Turbine
Turbine 22
Turbine22
Turbine
Figure
1.1.Experimental
farmlayout;
layout;
(b)
nacelle-mounted
LiDAR
onTurbine
Turbine
Figure
setup.
(a) Wind
Windfarm
layout;
(b)
nacelle-mounted
LiDAR
on
Figure
1.Experimental
Experimentalsetup.
setup. (a)
(a)
Wind
(b)
nacelle-mounted
LiDAR
on Turbine
2. 2. 2.
RWS
RWS
1 1
HigherBeams
Beams
Higher
a a
b
RWS
RWS
0 0
Z
Z
RWS3
RWS
3
LiDAR
LiDAR
RWS2
RWS
2
Measuring
Measuring
Planes
Planes
Lower Beams
Lower Beams
West (270◦)
West (270◦)
T1
T1
27◦
27◦
ϕϕ
α
α
Z
Z hh
ϑ
ϑ
Y
Y
β
β
X
X
T2
T2
Figure2.2.(a)
(a)Measuring
Measuring locations,
locations, beams,
LiDAR
in aligned
turbines;
(b) LiDAR
Figure
beams,and
andplanes
planesofofthe
the
LiDAR
in aligned
turbines;
(b) LiDAR
Figure
2. (a) and
Measuring
locations,
beams, and
planes
of the
LiDAR10in
aligned
turbines;
(b) every
LiDAR
coordinate
andparameters.
parameters.
LiDAR
sensor
computes
min
of average
datadata
every
coordinate
The LeoSphere
LeoSphere
LiDAR
sensor
computes
10
min
of average
coordinate
and
parameters.
The
LeoSphere
LiDAR
sensor
computes
10
min
of
average
data
every
1010min
2020to
to2022).
2022).
minfor
fortwo
two years
years (from
(from 2020
10 min for two years (from 2020 to 2022).
2.2. Two-Dimensional Gaussian Jensen Wake Model
The measuring LiDAR sensor (Figure 2b) maintains constant α and β angles of 5 and
In measuring
employing
the
basic
wake
predict
velocity
deficit
in
turbine
The
LiDAR
sensor
(Figure
2b)tomaintains
constant
α andcalculated
β the
angles
of
5 and
15 degrees,
respectively.
AsJensen
a result,
themodel
angles
ϑ
and φthe
can
be directly
based
wake,
it
is
assumed
that
the
wake
expands
radially
in
a
circular
pattern,
with
the
wake
15ondegrees,
respectively.
these fixed
values. As a result, the angles ϑ and φ can be directly calculated based
as the
axis of symmetry. The velocity deficit within the wake is then represented
on centerline
these
fixed
values.
The methodology employed to acquire and assess the LiDAR measurements and
as a Gaussian distribution of mass, where the highest velocity deficit at the wake’s cenThe methodology
to acquire
assess
thedescribes
LiDAR the
measurements
and
evaluate
the interactionemployed
model is discussed
in and
Section
2.5. It
process of estiterline gradually decreases as the distance from the centerline increases [4,7]. This mass
evaluate
interactiontomodel
is
discussedfrom
in Section
2.5.
It describes and
the process
of estimating
the
‘Direction
Hub’
parameter
LiDAR
measurements
subsequently
distribution is described by Equation (1), which accounts for the wake velocity deficit, the
mating
thefrom
‘Direction
Hub’
parameter
from LiDAR
measurements
and
subsequently
comparing
it with
the to
basic
predictions.
distance
the wake
centerline,
and the turbulence
intensity
of the ambient
flow
and is
comparing
it
with
the
basic
predictions.
used here.
!
√
2
2.2. Two-Dimensional
2 1Wake
− 1Model
− CT
U Gaussian Jensen
r
= 1−
exp −2 0
(1)
2.2. Two-Dimensional
Gaussian
Jensen(kWake
Model
0 x/R
k xvelocity
+R
In employing U
the
basic
model
deficit in the turbine
∞
+ 1)2to predict the
wake Jensen wake
wake,
it is assumed
the
wakewake
expands
radially
in a circular
pattern,
with
the turbine
wake
In employing
thethat
basic
Jensen
model
to predict
the velocity
deficit
in the
centerline
as
the
axis
of
symmetry.
The
velocity
deficit
within
the
wake
is
then
represented
wake, it is assumed that the wake expands radially in a circular pattern, with the wake
as a Gaussian
of mass,The
where
the highest
deficit
at the
wake’s
centercenterline
as thedistribution
axis of symmetry.
velocity
deficitvelocity
within the
wake
is then
represented
line
gradually
decreases
as
the
distance
from
the
centerline
increases
[4,7].
This
mass
disas a Gaussian distribution of mass, where the highest velocity deficit at the wake’s centertribution
is described
(1), which
accounts
for the
wake velocity
deficit,
line
gradually
decreasesbyasEquation
the distance
from the
centerline
increases
[4,7]. This
massthe
disdistance from the wake centerline, and the turbulence intensity of the ambient flow and is
Energies 2023, 16, 5710
5 of 13
where U∞ is the inflow wind velocity, U is the wake velocity, x is the downwind distance
from the wind turbine, r is the radial distance from the rotor axis, R is the rotor radius, CT
is the thrust coefficient, and k0 is the wake expansion rate [45] given in Equation (2), where
Ti is the reference turbulence intensity.
k0 = 0.3 Ti
(2)
2.3. Cylindrical Vortex Induction Model
Recently, Keane et al. [46] proposed an analytical method for the wind in the induction
zone. Note that the axial velocity decreases from its upwind value to the rotor plane [47].
The axial velocity downwind of the rotor may be expressed as U∞ − aU∞ , where a is the
axial induction factor, calculated from Equation (3) [48].
a=
1−
√
1 − CT
2
(3)
The largest velocity deficit occurs at the largest local thrust coefficient (CT ) [49]. Accordingly, the cylindrical vortex induction zone model by Branlard et al. [44] estimates
axial wind velocity in non-yaw actuator disk simulation based on Equations (4) and (5).
U
U∞
= 1−ε
induction
1−
√
1 − CT
2
R−r R − r + |R − r|
xk( x, r )
2
2
2
√
+
K k ( x, r ) +
k (0, r ), k ( x, r )
ε=
2| R − r |
R + r∏
2π rR
s
4rR
k( x, r ) =
( R + r )2 + x 2
(4)
(5)
(6)
where x is the upwind distance from the downwind rotor [43], and K and ∏ are the
complete first and third kinds of elliptical integrals [42].
2.4. A Simple Interaction Model
Energies 2023, 16, x
An integration method is proposed to establish a wake–induction interaction model
for two partially aligned wind turbines under different wind flow directions. The model
considers the effect of varying wind directions, as illustrated in Figure 3.
Upwind Turbine
Generates Wake
6 of
𝑟
𝑥
+𝜃
𝑑
𝑥´
r´
R
Downwind Turbine
Generates Induction
Figure
3. Basic
parameters
in partially
aligned
wind turbines.
Figure
3. Basic
parameters
in partially
aligned
wind turbines.
To integrate the wake and induction models, their respective coordinates must alig
Figure 3 illustrates the axial distance (𝑥) and radial distance (r) from the upwind roto
plane and axis, which are used in the wake model. Similarly, the axial distance (x′) an
radial distance (r′) from the downwind rotor’s plane and axis, which are employed in t
Energies 2023, 16, 5710
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To integrate the wake and induction models, their respective coordinates must align.
Figure 3 illustrates the axial distance (x) and radial distance (r) from the upwind rotor’s
plane and axis, which are used in the wake model. Similarly, the axial distance (x0 ) and
radial distance (r0 ) from the downwind rotor’s plane and axis, which are employed in the
induction model (Equations (5) and (6)), are also shown. A conversion can be applied to establish a correspondence between the two coordinate systems, allowing the transformation
of one set of coordinates to the other as
x 0 = x − dcos θ and r 0 = |r − dsin θ |
(7)
The distance between the two wind turbines is denoted as ‘d’, and the wind inflow
angle (θ) represents the direction of the wind flow in relation to the line connecting the
two towers. In this scenario, two synchronized variations of axial velocities occur between
the turbines: the wind velocity from the wake (as described in Equation (1)) and the
wind velocity from the induction (as described in Equation (4)). The interaction model
incorporates the normalized fraction of the wake’s axial velocity in the induction, as
illustrated in Equation (8). For the sake of simplicity, it is assumed that any change in the
thrust coefficient at the downwind rotor plane caused by the upwind wake is negligible.
U
U
U
=
(8)
U∞ w,i
U∞ wake U∞ induction
2.5. Evaluation Methodology
As outlined in Section 2.1, the test is conducted to measure wind characteristics
utilizing a nacelle-mounted LiDAR system installed on a wind turbine. Based on these
measurements, a simple model is set to predict the wind characteristics, as detailed in
Sections 2.2 and 2.3, and the interaction model in Section 2.4. However, to assess the
model’s accuracy, it is essential to consider the potential impact of installation errors and
other factors that could influence the measurements. Hence, in this section, we introduce
the ‘Direction to Hub’ parameter and describe the method used to estimate it from the
LiDAR measurements. This parameter incorporates the ‘shift angle’ parameter, which
denotes the angular error on the horizontal plane during LiDAR installation, as depicted in
Figure 4a. Despite adhering to stringent installation guidelines, errors may occur due to
structural deformations, calibration loss, and installation base issues, potentially leading to
measurement inaccuracies. To account for this, axial wind velocities for the LiDAR’s right
(UR ) and left (UL ) beams are calculated using Equation (9).
UR = RWS R /cos( β + α) , UL = RWSl /cos( β − α)
(9)
The radial (r) and axial distances (x) of the measuring points on the left and right
LiDAR beams are calculated as follows:
x R = dcos θ −
r R = dsin θ +
y
y
cos( β + α) , x L = dcos θ −
cos( β − α)
cos β
cos β
y
sin( β + α)
cos β
, r L = dsin θ −
(10)
y
sin( β − α)
cos β
where y is the axial distance of the measuring planes from the LiDAR head. The measurement planes are positioned at seven specific distances (50, 75, 100, 150, 200, 250, and 300 m)
in front of the LiDAR system. Within these planes, the LiDAR beams capture the radial
component of the wind speed at its four corners (as shown in Figure 4b).
cos 𝛽
Energies 2023, 16, 5710
cos 𝛽
where y is the axial distance of the measuring planes from the LiDAR head. The measu
ment planes are positioned at seven specific distances (50, 75, 100, 150, 200, 250, and 3
m) in front of the LiDAR system. Within these planes, the LiDAR beams capture
the rad
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component of the wind speed at its four corners (as shown in Figure 4b).
Shift Angle
α
β−𝛼
β+𝛼
y (v)
V
x (u)
HW
RWSL
U
Dh
β-α
RWSR
Wake Induction
Zone
b
a
Figure
(a) Shift
angle
installation
error,and
and (b)
(b) estimation
estimation ofof
Direction
to Hub
in the
Figure
4. (a)4.Shift
angle
installation
error,
Direction
to Hub
in nacellethe nacelle-ba
based
LiDAR.
LiDAR.
The axial and tangential wind velocity components in Equations (11) and (12) are
The axialfrom
andthetangential
components
in Equations
(11)
and (12)
obtained
projection wind
of the velocity
radial wind
velocity measured
by LiDAR
along
the directions.
obtained
from the projection of the radial wind velocity measured by LiDAR along
directions.
RWS L
RWS R
RWS R
tan( β + α)
U=
−
+
(11)
cos𝑅𝑊𝑆
+ α) tan( β + α) +
tan
β
−
α
cos
( β − α) cos( β𝑅𝑊𝑆
(
)
( β + α) 𝑅𝑊𝑆
tan 𝛽 + 𝛼
𝑈=
−
𝛽 + 𝛼 + tan 𝛽 − 𝛼 + cos 𝛽 + 𝛼
cos 𝛽 − RWS
𝛼 L cos 𝛽 +
𝛼 R tan
RWS
1
V=
cos( β − α)
𝑅𝑊𝑆
−
cos( β + α)
𝑅𝑊𝑆
tan( β + α) + tan( β − α)
(
(12)
1
The wind direction
𝑉 = relative to the−rotor axis at the hub height, referred to as Direction
(
cos 𝛽 −using
𝛼 the
cosaxial
𝛽 +and
𝛼 tangential
tan 𝛽 +wind
𝛼 +velocity
tan 𝛽 components.
−𝛼
to Hub (Dh ), can be determined
Equation (13) provides the Dh , considering the correction for the negative impact of the
The wind direction relative to the rotor axis at the hub height, referred to as Direct
shift angle in the formulation.
using the
axial and tangential wind
velocity componen
to Hub (𝐷 ), can be determined
V
U
−
U
L
R
Equation (13) provides
the correction
for the negative impact
of
−1 𝐷 , considering
Dh = tanthe
= tan−1
(13)
U
U
tan
β
+
α
U
tan
β
−
α
(
)+
(
)
L
R
shift angle in the formulation.
3. Results and Discussion
𝐷ℎ = tan
𝑉
= tan
𝑈 −𝑈
(
Figure 5a displays the power
wind
on tan
the examined
𝑈 production𝑈oftan
𝛽 turbines
+ 𝛼 +𝑈
𝛽 − 𝛼 wind
farm, revealing that the turbines have a rated power of 3.3 MW and a rated wind speed of
approximately 10 m/s. Due to the relative proximity of the turbines, with a distance less
than commonly
recommended, the wake effects are considerably pronounced along the
3. Results
and Discussion
aligned direction. When the wind blows from around 297 or 117 degrees north, one turbine
Figure 5a reduced
displays
the power
ofgenerated
wind turbines
on turbine.
the examined
wi
experiences
production
due toproduction
the wake effect
by the other
This
farm,effect
revealing
that
the turbines
have
a rated
of 3.3 MW
and
a rated
wind speed
is clearly
illustrated
in Figure
5b, where
thepower
wake generated
by the
upwind
turbine
(T2) interacts with the rotor of the downwind turbine (T1), resulting in power loss in the
downstream turbine along the aligned direction (117 degrees in Figure 5b).
than commonly
recommended,
the wake in
effects
are considerably
pronounced
along theby the
turbine.
This effect
is clearly illustrated
Figure
5b, where the
wake generated
alignedturbine
direction.
When
the wind
blows
from around
297 or 117 degrees
north,
turupwind
(T2)
interacts
with
the rotor
of the downwind
turbine
(T1),one
resulting
in
bine
experiences
reduced
production
due
to
the
wake
effect
generated
by
the
other
turpower loss in the downstream turbine along the aligned direction (117 degrees in Figure
bine. This effect is clearly illustrated in Figure 5b, where the wake generated by the up5b).
wind turbine (T2) interacts with the rotor of the downwind turbine (T1), resulting in
8 of 13
power loss in the downstream turbine along the aligned direction (117 degrees in Figure
5b).
Energies 2023, 16, 5710
b
b
Power Deficit in the
Waked Turbine
a
Power Deficit in the
Waked Turbine
a
Figure 5. Power production: (a) power—wind speed curve of turbine No. 1, and (b) angular
distribution
of theproduction:
power production
of turbine
No.
1 exhibiting
aNo.
power
deficit
around
the wind
Figure 5.
power—wind
speed
curve
of turbine
1,
and
angular
distriFigure
5. Power
Power production:(a)(a)
power—wind
speed
curve
of turbine
No.
1,(b)
and
(b) angular
inflow
angle
ofpower
117 degrees
north.
bution
of the
production
of turbine No. 1 exhibiting a power deficit around the wind inflow
distribution of the power production of turbine No. 1 exhibiting a power deficit around the wind
angle of 117 degrees north.
inflow angle of 117 degrees north.
Figure 6 illustrates the flow’s contours; the velocity deficit generated by the upwind
Figure
illustrates
the flow’s
flow’s
contours;
the velocity
velocity
deficit
generated
by the
the upwind
upwind
66 illustrates
the
the
deficit
by
turbineFigure
is depicted
in Figure
6a.contours;
This velocity
deficit
is generated
further affected
due to the
turbine
is
depicted
in
Figure
6a.
This
velocity
deficit
is
further
affected
due
to
the
inducturbine isofdepicted
in Figure 6a.
This velocity
is further
affected
to the induction
induction
the downwind
turbine
(Figuredeficit
6b). As
a result
of thisdue
interaction,
the flow is
tion
ofdownwind
the downwind
turbine
(Figure
6b).
As a result
of
this interaction,
the
flow
is disof
the
turbine
(Figure
6b).
As
a
result
of
this
interaction,
the
flow
is
distributed
distributed
between the two turbines, as shown in Figure 6c.
tributed between the two turbines, as shown in Figure 6c.
between the two turbines, as shown in Figure 6c.
T1 T1
a
a
T1 T1
bb
cc
Wind
Wind
Inflow
Inflow
angle: 15 ͦ
angle: 15 ͦ
T2
T2
T2interaction model,
Figure 6. (a) Wake model, (b) inductionT2model, and (c) the wake–induction
shown for the inflow angle of 15 degrees.
Figure
6.6.(a)
model,(b)(b)
induction
model,
(c) the wake–induction
interaction
Figure
(a)Wake
Wake model,
induction
model,
and (c)and
the wake–induction
interaction model,
shownmodel,
shown
forinflow
the inflow
angle
of 15 degrees.
for the
angle of
15 degrees.
The wake–induction interaction model, encompassing a wide range of inflow angles
and accounting
for all potential
partialmodel,
wakes,encompassing
is subsequently
applied
to evaluate
the
twoThe
wake–induction
interaction
a wide
range
of inflow
angles
The
wake–induction
interaction
model,
encompassing
a wide
range
of inflow
angles
utility-scale
wind
turbine
array.
Figure
7
displays
contours
illustrating
the
turbine
interand accounting for all potential partial wakes, is subsequently applied to evaluate the
and
accounting
for
all
potential
partial
wakes,
is
subsequently
applied
to
evaluate
the
actions observed wind
from aturbine
top view
of the
LiDAR
laser beams.
Figure
7a depictsthe
theturbine
impact twotwo-utility-scale
array.
Figure
7 displays
contours
illustrating
utility-scale
wind turbine array. Figure 7 displays contours illustrating the turbine
interactions observed from a top view of the LiDAR laser beams. Figure 7a depicts the
interactions
observed
a topfrom
viewthe
of aligned
the LiDAR
laser
beams.
Figure
7a depicts the
impact of downwindfrom
induction
turbine,
while
Figure
7b,c represent
scenarios involving different inflow directions.
Energies 2023, 16, x
9 of 13
of downwind induction from the aligned turbine, while Figure 7b,c represent scenarios
involving different inflow directions.
of downwind induction from the aligned turbine, while Figure 7b,c represent scenarios
9 of 13
(x/D)
involving different inflow directions.
Energies 2023, 16, 5710
b
(y/D) (y/D)
a
(x/D)
250
300
b
a
c
300
c
200
250
150
200
100
150
75
50
100
75
50
Figure 7. Contours: wake and induction interaction model between the turbines at the inflow angle
𝜃: (a) 0 degrees, (b) −7 degrees, and (c) 15 degrees, all with the shift angle α = 4.4°. Black dots are
Figure
7.locations
and
interaction
model
between
theturbines
turbinesatfrom
at
the
inflow
angle
LiDAR
atwake
the
measuring
planes
with an
indicated
axial
distance
the
downstream
Figure
7.Contours:
Contours:
wake
andinduction
induction
interaction
model
between
the
the
inflow
angle
𝜃:
(a)
0
degrees,
(b)
−7
degrees,
and
(c)
15
degrees,
all
with
the
shift
angle
α
=
4.4°.
Black
dots
turbine
meters).
θ: (a) 0(in
degrees,
(b) −7 degrees, and (c) 15 degrees, all with the shift angle α = 4.4◦ . Black dots areare
LiDAR locations at the measuring planes with an indicated axial distance from the downstream
LiDAR locations at the measuring planes with an indicated axial distance from the downstream
turbine
(in LiDAR
meters). data obtained from various overlapping angles of the two turbines can
The
turbine (in meters).
30
25
25
20
20
15
15
10
10
5
5
0
0
-5
-5
-10
-10
-15
-15
240
240
a
a
50m
75m
50m
100m
75m
100m
150m
150m
200m
200m
250m
250m
300m
300m
35
35
Direction to Hub (Degree)
Direction to Hub (Degree)
Direction
Hub
(Degree)
Direction
to to
Hub
(Degree)
35
35
30
be compared with the model to assess their agreement. In Figure 7, the black lines repreThe
LiDAR
data
fromvarious
variousoverlapping
overlapping
angles
of the
turbines
can
The
LiDARbeams,
data obtained
obtained
angles
of the
two two
turbines
canthe
be
sent the
LiDAR
and thefrom
black dots represent
the
measurement
points on
measbecompared
comparedwith
with
the
model
to
assess
their
agreement.
In
Figure
7,
the
black
lines
reprethe model to assess their agreement. In Figure 7, the black lines represent
uring planes (Figure 7b). The parameter d/D = 2.3 corresponds to the tower-to-tower dissent
LiDAR
beams,
the black
represent
the measurement
the measthethe
LiDAR
beams,
andand
the black
dots dots
represent
the measurement
pointspoints
on the on
measuring
tance on the wind farm.
planes
(Figure
7b). The
d/D = 2.3
to the tower-to-tower
distance on
uring
planes
(Figure
7b).parameter
The parameter
d/Dcorresponds
= 2.3 corresponds
to the tower-to-tower
dis8wind
shows
a comparison between the model and the measured data. The calcuthe Figure
wind
farm.
tance
on the
farm.
latedFigure
Dh
using
Equation
(13) exhibits
a similar
trend
tothe
the
measured
data.
However,
Figure
shows
comparison
between
the
and
measured
data.
The
calcu8 8shows
aacomparison
between
the model
model
and
the
measured
data.
The
calcu- a
shift
angle
is
observed
in
the
measurements,
referred
to
as
zero-up
shift,
which
is
approxlated
Dh
using
Equation
(13)
exhibits
a
similar
trend
to
the
measured
data.
However,
a
shift
lated Dh using Equation (13) exhibits a similar trend to the measured data. However,
a
angle
is
observed
in
the
measurements,
referred
to
as
zero-up
shift,
which
is
approximately
imately
4.5
degrees
in
this
case
(Figure
8a).
Since
this
parameter
is
included
in
the
model
shift angle is observed in the measurements, referred to as zero-up shift, which is approx4.5 degrees
init this
case
(Figure
thisSince
parameter
is data
included
the
model
formulation,
is also
reflected
in Since
the 8a).
model’s
output
(Figure
8b).
Itinisformulaimportant
imately
4.5 degrees
in this
case 8a).
(Figure
this parameter
isin
included
the
model to
tion,
it
is
also
reflected
in
the
model’s
output
data
(Figure
8b).
It
is
important
to
note
that can
note
that theitmodel
be symmetrical
without
any
shift
which
formulation,
is alsodiagram
reflected should
in the model’s
output data
(Figure
8b).
It isangle,
important
to
the model diagram should be symmetrical without any shift angle, which can help identify
help
identify
installation
errors.
Additionally,
the
asymmetrical
distribution
of
Dh
calcunote that the model diagram should be symmetrical without any shift angle, which can
installation errors. Additionally, the asymmetrical distribution of Dh calculated from the
lated
from the
measurements
both sides of
tower-to-tower
connecting
line
may be
help
identify
installation
errors.on
Additionally,
thethe
asymmetrical
distribution
of Dh
calcumeasurements on both sides of the tower-to-tower connecting line may be attributed to the
attributed
to
the
rotation
direction
of
the
rotor.
lated
from
the
measurements
on
both
sides
of
the
tower-to-tower
connecting
line
may
be
rotation direction of the rotor.
attributed to the rotation direction of the rotor.
25
25
b
b
15
15
5
5
50m
50m75m
75m100m
150m
100m
150m
200m
200m
250m
250m
300m
300m
-5
-5
-15
-15
260
280
300
320
260
280
300
320
Inflow
Angle (Degree
(Degree--North)
North)
Inflow Angle
-25
-25
240
240
260 280280 300300 320 320
260
Inflow
Angle
(Degree
- North)
Inflow
Angle
(Degree
- North)
Figure 8. Direction to Hub. (a) LiDAR and (b) model.
Figure
8. Direction
DirectiontotoHub.
Hub.(a)
(a)LiDAR
LiDARand
and
model.
Figure 8.
(b)(b)
model.
Energies 2023, 16, x
Energies 2023, 16, 5710
10 of 13
10 of 13
A direct
direct comparison
comparison of
of the
the Direction
Direction to
to Hub
Hub (Dh)
(Dh) is
is shown
shown in
in Figure
Figure 9,
9, indicating
indicating
A
that
the
measurement
data
asymmetry
is
more
pronounced
than
the
calculations.
Despite
that the measurement data asymmetry is more pronounced than the calculations. Despite
correcting
the
shift
angle,
the
model
exhibits
more
symmetric
features.
However,
is
correcting the shift angle, the model exhibits more symmetric features. However,there
there
a
noticeable
discrepancy,
particularly
on
the
left
side
of
the
inflow
angle.
The
model’s
is a noticeable discrepancy, particularly on the left side of the inflow angle. The model’s
prediction of
of the
the location
location of
of the
the Dh
Dh peaks
peaks isis not
not accurately
accurately estimated,
estimated, with
with an
an average
average
prediction
difference of
of approximately
approximately 15
is is
attributed
to
difference
15 degrees
degrees towards
towardsthe
thecenter
centerline.
line.This
Thiserror
error
attributed
the
simplicity
of
the
wake
model,
as
it
is
a
single-hat
Gaussian-shaped
wake
model
and
to the simplicity of the wake model, as it is a single-hat Gaussian-shaped wake model
doesdoes
not include
the effect
of the
in the
wake,
which
leads
to significant
difand
not include
the effect
of nacelle
the nacelle
in near
the near
wake,
which
leads
to significant
ferences
between
the
proposed
model
and
the
actual
data,
especially
in
the
near
wake
differences between the proposed model and the actual data, especially in
near wake
regions(greater
(greaterdistances
distancesfrom
fromthe
thedown
down
turbine).
Another
potential
factor
contributing
regions
turbine).
Another
potential
factor
contributing
to
to this
error
is the
omission
of accurate
an accurate
wake
expansion.
this
error
is the
omission
of an
wake
expansion.
45
50 m
35
35
15
25
15
5
45
100 m
35
Dh (degree)
25
Dh (degree)
25
Dh (degree)
45
75 m
25
15
5
15
5
5
-5
-5
-5
-5
-15
-15
-15
-15
-25
-25
-60 -45 -30 -15 0
15 30 45 60
θ (degree)
15 30 45 60
25
25
15
15
5
5
-5
-15
-15
-15
-25
-25
-25
15 30 45 60
θ (degree)
Model
15
5
-5
-60 -45 -30 -15 0
15 30 45 60
θ (degree)
300 m
35
Dh (degree)
Dh (degree)
25
-60 -45 -30 -15 0
45
250 m
35
200 m
15 30 45 60
θ (degree)
45
35
-25
-60 -45 -30 -15 0
θ (degree)
45
Dh (degree)
-25
-60 -45 -30 -15 0
150 m
35
Dh (degree)
45
Measurement
-5
-60 -45 -30 -15 0
15 30 45 60
θ (degree)
1×σ (STDV)
-60 -45 -30 -15 0
15 30 45 60
θ (degree)
Figure 9.
9. Comparison
Comparison of
of Direction
Direction to
to Hub
Hub (Dh)
(Dh) at
atvarious
various locations
locations (from
(fromFigure
Figure5).
5).
Figure
The asymmetric
asymmetric feature
feature in
in the
the wake
wake region
region can
can be
be attributed
attributed to
to the
the rotation
rotation direction
direction
The
of
of the
the turbine
turbine blades,
blades, which
which is
is not
not taken
taken into
into account
account in
in the
the proposed
proposed model.
model. ItIt combines
combines
the
the Gaussian-distributed
Gaussian-distributedJensen
Jensenwake
wakemodel
modeland
andaa cylindrical
cylindrical vortex
vortex induction
induction model
model and
and
is
is designed
designed to
to provide
provide aa fast and cost-effective
cost-effective analytical approach
approach for approximating
approximating flow
flow
interactions
thethe
simplicity
of this
model
does
not not
capture
the
interactionsin
inthe
theturbine
turbinearray.
array.However,
However,
simplicity
of this
model
does
capture
specific
effects
of
turbine
blade
rotation,
which
may
contribute
to
the
observed
asymmetry
the specific effects of turbine blade rotation, which may contribute to the observed asymin
the wake
metry
in theregion.
wake region.
Figures
Figures 88 and
and 99 highlight
highlight the
the discrepancy
discrepancy between
between the
the model
model and
and the
the measurements
measurements
in
value of
of the
the Dh
Dhpeaks.
peaks.The
Thedifference
differenceininpeak
peakvalues
valuescan
can
attributed
in the
the maximum
maximum value
bebe
attributed
to
to
weaker
influence
of induction
compared
to wake
the wake
effect,
as both
effects
are
thethe
weaker
influence
of induction
compared
to the
effect,
as both
effects
are symsymmetric
respect
the downstream
turbine.
As the
shift
angle increases,
the
metric withwith
respect
to thetodownstream
turbine.
As the shift
angle
increases,
the induction
induction
effect
becomes
more
pronounced.
The
difference
in
peaks
between
the
model
effect becomes more pronounced. The difference in peaks between the model and the
and
the measurements
diminishes
as thefrom
distance
from the
upwind
turbineparticularly
increases,
measurements
diminishes
as the distance
the upwind
turbine
increases,
particularly
noticeably
on
the
left
side
of
the
inflow
angle.
noticeably on the left side of the inflow angle.
As
As discussed,
discussed, most
most outliers
outliers were
were observed
observed in
in the
the left-half
left-half region
region of
of the
the inflow
inflow angle.
angle.
To
improve
the
accuracy
and
reduce
these
outliers,
further
research
is
needed.
To improve the accuracy and reduce these outliers, further research is needed. This
This could
could
involve
involve developing
developing more
more advanced
advanced wake
wake and
and induction
induction models,
models, considering
considering more
more accuaccurate
rate wake
wake expansion
expansion effects,
effects, and
and incorporating
incorporating correction
correction functions
functions such
such as
as double-head
double-head
Gaussian
Gaussian functions
functions in
in the
the interaction
interactionformulation.
formulation.
Energies 2023, 16, 5710
11 of 13
4. Conclusions
This study presents a simple wake–induction interaction model evaluated using field
data obtained from a utility-scale wind turbine system equipped with a nacelle-mounted
four-beam LiDAR. The model provided reasonable estimating of the wake–induction interaction between the two turbines using the Direction to Hub (Dh) parameter. However, the
actual wake profile exhibited asymmetry within the limited range of the angles covered by
the experimental data, which was not fully captured by the proposed model with a maximum Dh angle of 245 degrees (Figure 8). Despite this limitation, Dh proved to be a sensitive
parameter for evaluating theoretical models and could also be used for horizontal position
calibrations of nacelle-mounted LiDARs. To improve the model, it is recommended to
consider wake expansion correction and explore alternative wake or induction formulations
to enhance the accuracy of peak value estimations for Dh. Also, implementing correction
functions can help mitigate prediction outliers in the wake–induction interaction model,
thus improving its applicability for wind farm flow simulation and control. To sum up,
this study presents a basic methodology for measuring and predicting wind characteristics
using LiDAR measurements. By developing and validating a simple model based on these
measurements, we showcased the potential to enhance the efficiency of wind turbine arrays.
The precise estimations of wind characteristics hold significant importance for optimizing
wind turbine operations and reducing the cost of wind energy production.
Author Contributions: Conceptualization, E.M., M.K., U.R., A.E. and L.P.C.; Methodology, E.M.;
Validation, A.E. and U.R.; Formal analysis, M.K.; Investigation, E.M.; Resources, A.E. and U.R.; Data
curation, M.K.; Writing—original draft, E.M.; Writing—review & editing, E.M., M.K., L.P.C. and A.K.;
Visualization, M.K.; Supervision, E.M.; Funding acquisition, U.R. All authors have read and agreed
to the published version of the manuscript.
Funding: The researcher of this project was finantially supported by the University of Rostock and
the Shahrood University of Technology, independently.
Data Availability Statement: The data presented in this study are restricted but may be available by
agreement between parties. Please contact the corresponding author.
Acknowledgments: We want to thank the Wind Energy Technology Institute, University of Rostock,
the Wind-Projekt Company as the owner of the LiDAR system for providing the experimental
data for this project, the Siemens Gamesa Company for their consultants, Center for International
Scientific Studies and Collaborations (CISSC) of Iran for their support, and M.Kamandi for the
valuable discussions.
Conflicts of Interest: The authors declare no conflict of interest.
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