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Performance comparison of analytical wake
models calibrated on a large offshore wind cluster
To cite this article: Diederik Van Binsbergen
et al
2024
J. Phys.: Conf. Ser.
2767 092059
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The Science of Making Torque from Wind (TORQUE 2024)
Journal of Physics: Conference Series 2767 (2024) 092059
IOP Publishing
doi:10.1088/1742-6596/2767/9/092059
1
Performance comparison of analytical wake models
calibrated on a large offshore wind cluster
Diederik van Binsbergen 1,2, Pieter-Jan Daems 2, Timothy
Verstraeten 2, Amir Nejad1and Jan Helsen2
1Department of Marine Technology, Norwegian University of Science and Technology
(NTNU), Otto Nielsens veg 10, 7052 Trondheim, Norway
2Department of Mechanical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, Brussels,
1050, Belgium
E-mail: dirk.w.van.binsbergen@ntnu.no
Abstract. This study benchmarks the performance of multiple analytical wake models using a
multi-level hyperparameter optimization framework for calibrating models with SCADA data in
the Belgian-Dutch offshore zone. The calibration targets wind coming the northwest (300 to 330
degrees) with wind speeds ranging from 7 to 9 m/s, a wind direction where wakes are clustered
across Belgian wind farms. Six wake models are evaluated, namely those described by Jensen
(1983), Bastankhah (2014), Niayifar (2016), Zong (2020), Nygaard (2020), and Pedersen (2022).
Relative and accumulated relative error between the calibrated wake models and SCADA data
are analyzed both on turbine, farm, and cluster level. Statistical moments of the residual error
between each model and SCADA data for individual turbines are presented in boxplots for each
wind farm and are analyzed using kernel density estimates. Additionally, the calibrated tuning
parameters are used to calculate wake losses, demonstrating a strong overlap across models.
The analysis reveals that the top-hat TurbOpark model shows the best performance, followed
by its Gaussian variant. The Gaussian models by Niayifar (2016) and Zong (2020), as well as the
top-hat model by Jensen (1983) all show relatively good performance, while the Gaussian model
by Bastankhah (2014) has the worst performance after calibration. While some models perform
better than other models, all models show similar trends post-calibration, indicating that with
proper calibration, any model can be viable, given its inherent limitations are recognized and
managed.
1. Introduction
With the growing size of offshore wind turbines and wind farms (WFs), understanding the impact
of wind turbine wakes on the expected yield and the levelized cost of energy has become crucial.
The propagation of the turbine wake results in power losses for downwind turbines and needs to
be considered in the yield assessment, controller design, and for layout optimization. The first
model for analyzing turbine wakes was introduced by Jensen in 1983 [1], characterized as a top-
hat model. This implies a uniform wind speed across the wake from the center to its boundary.
Additionally, this model assumes linear wake expansion. The uniform wind speed profile across
the rotor plane makes these types of models unrealistically sensitive for small changes in wind
direction. Furthermore, top-hat models underestimate the wake effect at the center of the wake,
while overestimating it at the edge of the wake. To address these shortcomings, a new model
employing a self-similar Gaussian distribution for the wake was introduced [2], aligning with
The Science of Making Torque from Wind (TORQUE 2024)
Journal of Physics: Conference Series 2767 (2024) 092059
IOP Publishing
doi:10.1088/1742-6596/2767/9/092059
2
the fundamental nature of turbulent plumes and wakes [3]. Subsequent models have built upon
these foundations, integrating elements from the Jensen model or the self-similar Gaussian model
[4, 5, 6, 7, 8, 9]. These newer analytical models, informed by a better understanding of wind
turbine wakes in different scenarios, have continued to evolve, enhancing the prediction accuracy
of wind turbine wakes using analytical models.
An improved understanding of wind turbines operating in larger farms and the increasing
size of turbines has led to significant advancements in analytical wake modeling. For instance,
it was discovered that earlier models tend to underestimate wake losses in far-wake regions, as
indicated by Ørsted in 2019 [10], often overestimating the resulting energy yield. This insight
prompted the development of new models, such as the TurbOPark / TurbOGauss [7, 9] models
and the Empiricial Gaussian model [11]. In parallel, progress in wake superposition techniques
led to the development of the Cumulative Wake model [8] and the model by Zong and Port´e-Agel
[6].
Analytical wake models typically include tuning parameters that require calibration. Tuning
these can be complicated, since the characteristics of wakes observed in the field often vary from
those in wind tunnel tests and are complex to measure.
Additionally, accurately quantifying model precision using SCADA data is challenging. The
difficulty arises from several factors, such as the stochastic nature of wind, sensor uncertainties,
natural fluctuations like diurnal cycles and annual cycles, atmospheric stability, varying site-
specific characteristics, and the fundamental assumptions that form the basis of most analytical
models, such as assuming a homogeneous and time-invariant flow field. Given these varied
sources of uncertainty, which contribute to the residual error between analytical models and
SCADA data, it is challenging to assess the accuracy of a specific analytical wake models solely
on SCADA data. This therefore highlights the importance of conducting comparative studies
between different wake models to fully understand their effectiveness.
2. Objectives
The main objective of this study is to conduct a comparative analysis of analytical wake models,
each previously calibrated on data from an offshore wind cluster with a capacity over 2.2 GW of
active power at rated capacity. The objectives are further divided into the following key areas:
(i) Quantifying model performance by analyzing the residual error for the calibrated wake
models on turbine, farm, and cluster level.
(ii) Quantifying uncertainty through examination of statistical moments.
(iii) Comparing predicted wake losses at both cluster and farm levels.
(iv) Evaluating top-hat and Gaussian profile models.
3. Methodology
In this work, the calibration of the wake models is performed on wind farms located in the
Belgian offshore zone, as shown in Figure 1. These farms are denoted from top-left to bottom-
right as WF1to WF5. While the Pywake framework is utilized within this study [12], the
methods are not restricted to this framework alone [13]. The calibration considers the wakes of
the entire Belgian-Dutch offshore zone, depicted in Figure 2.
The Belgian offshore zone, with a high power density of 9.5MW/km2and a rated capacity of
2.2GW, serves as an ideal case for cluster-wake studies. In contrast, the Dutch offshore zone has
a lower power density of 4.8MW/km2and a rated capacity of 1.5GW, totaling a rated capacity
of 3.7GW.
The analytical wake models selected for calibration are:
•The Jensen wake model (1983) [1, 14].
The Science of Making Torque from Wind (TORQUE 2024)
Journal of Physics: Conference Series 2767 (2024) 092059
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doi:10.1088/1742-6596/2767/9/092059
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•The Gaussian wake model by Bastankhah and Port´e-Agel (2014) [2].
•The Gaussian wake model by Niayifar and Port´e-Agel (2016 [4].
•The Gaussian wake model by Zong and Port´e-Agel (2020) [6].
•The top-hat version of the TurbOPark model by Nygaard et al. (2020) [7].
•The Gaussian version of the TurbOPark model by Pedersen et al. (2022) [9].
These models are further highlighted in Table 1. All models are optimized for a constant
turbulence intensity of 0.06 and a shear exponent of 0.12.
0 10 20 30
Easting [km]
0
10
20
30
Northing [km]
1
23
4
5
Figure 1. Wind farms within the Belgian
offshore zone used for the calibration of wake
models.
0 10 20 30
Easting [km]
0
10
20
30
Northing [km]
Figure 2. Wind farms within the
Belgian-Dutch offshore zone incorporated into
the wake calculation framework.
Table 1. Description of the analytical wake models considered and their respective sub
models, tuning parameters, and parameter bounds for calibration.
Wake model Jensen
(1983)
Bastankhah
(2014)
Niayifar
(2016)
Zong
(2020)
Nygaard
(2020)
Pedersen
(2022)
Turbulence model - -
Crespo
Hernandez
(1987)
Crespo
Hernandez
(1987)
Integrated Integrated
Superposition model Squared
sum
Squared
sum
Linear
sum
Weighted
sum
Squared
sum
Linear
sum
Blockage model - - - - - -
Ground model - - - - - Mirror
Tuning parameters
Ωref
k = 0.04 k = 0.032 ka= 0.38
kb= 0.004
ka= 0.38
kb= 0.004 A = 0.6 A = 0.04
Parameter bounds
[Ωmin,Ωmax ]
k∈
[0.001, 0.2]
k∈
[0.001, 0.2]
ka∈
[0.001, 1.0]
kb∈
[0.001, 0.06]
ka∈
[0.001, 1.0]
kb∈
[0.001, 0.06]
A∈
[0.001, 1.0]
A∈
[0.001, 0.6]
The Science of Making Torque from Wind (TORQUE 2024)
Journal of Physics: Conference Series 2767 (2024) 092059
IOP Publishing
doi:10.1088/1742-6596/2767/9/092059
4
The time-series calibration procedure, as described in van Binsbergen et al. (2023) and van
Binsbergen et al. (2024) [13, 15], is performed on all wind farms depicted in Figure 1. The
analysis within this paper focuses on wind directions ranging between 300 and 330 degrees and
wind speeds between 7.0 and 9.0 m/s. As noted in van Binsbergen et al. (2024) [15], this
particular wind direction tends to experience a limited amount of heterogeneous inflow and has
demonstrated promising results with the Gaussian TurbOPark model by Pedersen et al. (2022)
[9]. The calibration procedure is performed on one year of SCADA (Supervisory Control and
Data Acquisition) data.
The optimization consists of three stages, all described in detail in van Binsbergen et al.
(2023) [13]. For clarification the cost function and the final optimization stage are shortly
described below. The cost function consists of two components, fand g.fpenalizes the error
on turbine level, while gpenalizes the error of the entire cluster, both described by Equations
1 and 2, respectively. Here, NW F indicates the number of wind farms, and NW T the number of
wind turbines for a specific farm i.Pmodel is the power derived from the wake model for farm
iand turbine j, and Pscada is the active power from SCADA data for farm iand turbine j.
U,ϕ, and Ωare the freestream wind speed, freestream wind direction, and the set of tuning
parameters for each wake model, respectively.
f(U, ϕ, Ω) = 1
PNW F
i=1 NW T,i
NW F
X
i=1
NW T,i
X
j=1
Pmodel
i,j (U, ϕ, Ω)−Pscada
i,j 2
(1)
g(U, ϕ, Ω) =
NW F
X
i=1
NW T,i
X
j=1
Pmodel
i,j (U, ϕ, Ω)−
NW F
X
i=1
NW T,i
X
j=1
Pscada
i,j
2
(2)
For each averaged 10-minute timestamp the following calibration procedure is performed:
•Stage 1: Optimization of wind speed.
•Stage 2: Optimization of wind speed and wind direction.
•Stage 3: Optimization of wind speed, wind direction and wake parameters.
For the first and the second stage a Quasi Monte Carlo (QMC) sampler by Bergstra and
Bengio [16] is used, exploring the search space of the initially estimated freestream wind speed
and wind direction. For the final stage the wind speed, wind direction and the wake parameters
are calibrated simultaneously by minimizing the cost-function in Equation 3 using a multivariate
Tree-Parzen Estimator (TPE) algorithm by Bergstra et al. [17]. The initial values for wind speed
and wind direction of the final calibration stage are based on results from the previous stage,
denoted as U∗and Φ∗, while aand bdetermine the weight between fand g, taken as a
b= 4.0,
prioritizing the error on turbine-level. The parameter space, ˜
Ω, is defined by the minimum
and maximum values of the parameters, Ωmin and Ωmax , respectively, from Table 1. While
the freestream wind speed and wind direction are not the primary target of the calibration,
neglecting them can lead to inaccurate results, since the estimates from SCADA do not always
reflect the true freestream wind conditions.
minimize a·f(U,Φ,Ω) + b·g(U,Φ,Ω) (3)
subject to 0.95U∗≤ U ≤ 1.05U∗(4)
and Φ∗−15 ≤Φ≤Φ∗+ 15 (5)
and Ω∈˜
Ω(6)
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doi:10.1088/1742-6596/2767/9/092059
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For further elaboration on the optimization, van Binsbergen et al. (2023) and van Binsbergen
et al. (2024) [13, 15] can be consulted.
To deal with power-curve mismatches, power-curve filtering is performed together with
filtering based on operational codes, such as alarm annotations, of the wind turbines. During
operational windows accompanied with low active power, the turbine is annotated as inactive,
since low active power corresponds with low thrust loads, thereby assuming it has a limited
effect on the internal wind farm flow field. The wake simulation is then performed without the
turbines with low active power. When the number of inactive turbines reaches over half of the
wind farm, the data is not considered for calibration. Data where underperformance and general
power-curve mismatches occur above the limit given to low active power is not considered for
calibration. Additional details on the filtering procedure can be found in van Binsbergen et al.
(2023) [13].
4. Results
The results section is divided into two subsections. The first focuses on uncertainty quantification
by analyzing the residual error on cluster, farm, and turbine level and by examining the statistical
moments for each wind turbine. This is followed by a comparison of predicted wake losses before
and after calibration of the tuning parameters.
4.1. Model and calibration performance
To fully assess the performance of the analytical wake models, both the relative error on cluster
level and turbine level needs to be analyzed, since the cost function consists of two components,
one penalizing the error of the entire cluster and one penalizing turbine errors. Therefore model
performance is evaluated by measuring relative and accumulated relative error between the
model predictions and SCADA data at each averaged 10-minute timestamp. This error metric
is comparable to the one used in Nygaard et al. (2022) [18], but not identical. The equations
defining these metrics, as referenced in Equations 7 and 8, are used to visualize the data in
Figure 3. The results are then represented as a distribution for the available data between 300
and 330 degrees and wind speeds between 7.0 and 9.0 m/s using a kernel density estimate.
In this context, Pcluster
rel and Pf arm
rel represent the relative error for the entire cluster and each
individual wind farm, respectively.
Pcluster
rel =PNW F
i=1 PNW T,i
j=1 Pmodel
i,j −Pscada
i,j
PNW F
i=1 PNW T,i
j=1 Pscada
i,j
(7)
Pfarm
rel =PNW T
j=1 Pmodel
j−Pscada
j
PNW T
j=1 Pscada
j
(8)
Figure 3 shows the kernel density estimate of the relative error in [%] between the analytical
models and SCADA data, while Tables 2 and 3 show the mean and the standard deviation
of the relative error in [%] between the analytical models and SCADA data, respectively. For
Figure 3 and Table 2, a value below 0 suggest that the model underestimates the energy yield
and overestimates the wake losses. Conversely, a value above 0 indicates an overestimation of
the yield and an underestimation of the wake losses.
Results indicate that, across the entire cluster, the Gaussian wake models by Niayifar [4]
and Zong [6], along with the top-hat TurbOPark model [7], exhibit lower mean and standard
deviations. On the other hand, the Jensen model [1], Gaussian wake model by Bastankhah [2],
and the Gaussian TurbOPark model [9] display relatively larger means and standard deviations.
However, it is important to note that the overall performance of the model is not fully represented
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Journal of Physics: Conference Series 2767 (2024) 092059
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doi:10.1088/1742-6596/2767/9/092059
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by the absolute error for the entire cluster or wind farm, since the cost-function used in the
calibration also considers turbine level errors.
20 10 0 10 20
Relative error [%], entire cluster
0.00
0.05
0.10
0.15
0.20
Density [-]
50 25 0 25 50
Relative error [%], wind farm 1
0.00
0.02
0.04
0.06
Density [-]
50 25 0 25 50
Relative error [%], wind farm 2
0.00
0.01
0.02
0.03
0.04
0.05
Density [-]
50 25 0 25 50
Relative error [%], wind farm 3
0.00
0.01
0.02
0.03
0.04
0.05
Density [-]
50 25 0 25 50
Relative error [%], wind farm 4
0.00
0.01
0.02
0.03
0.04
0.05
Density [-]
50 25 0 25 50
Relative error [%], wind farm 5
0.00
0.01
0.02
0.03
0.04
0.05
Density [-]
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Wake loss overestimation Wake loss underestimation
Figure 3. Relative error between calibrated analytical wake models and SCADA data.
Table 2. Mean relative error between calibrated analytical wake models and SCADA data.
Wake model Jensen
(1983)
Bastankhah
(2014)
Niayifar
(2015)
Zong
(2020)
Nygaard
(2020)
Pedersen
(2022)
Cluster -3 -3 0 -1 0 0
WF1-4 -6 -2 -1 0 -1
WF2-5 -8 3 2 2 0
WF3-7 -9 0 -2 -2 -2
WF45 7 1 0 6 8
WF52 1 0 0 -1 -4
To perform a more detailed analysis of the wake model performance, the error between
individual turbines is analyzed using Equations 9 and 10. Here, Pcluster
acc,wt and Pf arm
acc,wt represent
the accumulated absolute error for each turbine within the entire cluster and the accumulated
absolute error for each turbine per wind farm, respectively.
Pcluster
acc,wt =PNW F
i=1 PNW T,i
j=1 |Pmodel
i,j −Pscada
i,j |
PNW F
i=1 PNW T,i
j=1 Pscada
i,j
(9)
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Table 3. Standard deviation of relative error between calibrated analytical wake models and
SCADA data.
Wake model Jensen
(1983)
Bastankhah
(2014)
Niayifar
(2015)
Zong
(2020)
Nygaard
(2020)
Pedersen
(2022)
Cluster 5 5 3 3 4 4
WF113 13 11 11 12 11
WF218 18 15 15 14 16
WF319 27 17 15 17 19
WF415 17 17 21 12 14
WF511 11 12 12 12 16
Pfarm
acc,wt =PNW T
j=1 |Pmodel
j−Pscada
j|
PNW T
j=1 Pscada
j
(10)
Figure 4 shows the kernel density estimate of the accumulated error for individual turbines in
[%], while Table 4 presents the median values of the accumulated errors for individual turbines.
The Gaussian wake model by Bastankhah [2] shows the highest accumulated error for each
WF and the entire cluster, followed by the Jensen model [1]. It can be observed that the top-
hat TurbOPark model [7] shows the best performance regarding this metric, followed by the
Gaussian TurbOPark model [18].
0 25 50 75
Accumulated error [%], entire cluster
0.00
0.01
0.02
0.03
0.04
0.05
Density [-]
0 20 40 60
Accumulated error [%], wind farm 1
0.00
0.01
0.02
0.03
0.04
0.05
Density [-]
0 50 100
Accumulated error [%], wind farm 2
0.00
0.01
0.02
0.03
0.04
Density [-]
0 50 100
Accumulated error [%], wind farm 3
0.00
0.01
0.02
0.03
0.04
Density [-]
0 50 100
Accumulated error [%], wind farm 4
0.00
0.01
0.02
0.03
0.04
Density [-]
0 50 100
Accumulated error [%], wind farm 5
0.00
0.01
0.02
0.03
0.04
Density [-]
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Figure 4. Accumulated relative error between calibrated analytical wake models and SCADA
data.
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Table 4. Median accumulated relative error between calibrated analytical wake models and
SCADA data.
Wake model Jensen
(1983)
Bastankhah
(2014)
Niayifar
(2015)
Zong
(2020)
Nygaard
(2020)
Pedersen
(2022)
Cluster 37 47 33 33 30 33
WF126 30 24 23 22 23
WF241 51 36 37 31 34
WF340 48 34 35 32 34
WF443 55 41 41 32 36
WF540 52 38 37 34 38
Figures 5, 6, 7, and 8 support the previously made observations. Here, for each wind
turbine the moments of the normalized error between calibrated wake model and SCADA data
(Pwt
rel =Pmodel−Pscada
Pscada ) are calculated based on the time series data. The distribution of the
moments for each wind farm are then depicted in boxplots. Again, a mean lower than 0 indicates
that the model underestimates the energy yield and overestimates the wake losses. Conversely, a
mean higher than 0 indicates an overestimation of the yield and an underestimation of the wake
losses. Figure 5 shows that the Bastankhah model [2] exhibits the most consistent deviation
from a zero mean. Additionally, a trend is noticeable with the Niayifar and Zong models [4, 6],
where, further downstream, the spread in mean deviation increases. Moreover, the Jensen, top-
hat TurbOPark, and Gaussian TurbOPark models [1, 7, 9] show better consistency with fewer
fluctuations in their mean values further downstream.
Entire cluster Wind farm 1 Wind farm 2 Wind farm 3 Wind farm 4 Wind farm 5
80
60
40
20
0
20
Mean [%]
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Figure 5. Time-series mean of the normalized error between model and SCADA for
each wind turbine within the entire cluster and each wind farm.
In Figure 6, the analysis of the standard deviation between calibrated wake model and
measurements shows that the TurbOPark model [7] outperforms others, followed by the Gaussian
TurbOPark [9], the Jensen [1], Niayifar [4], and Zong [6] wake models. Consistently, the
Bastankhah model [2] shows the highest standard deviations. Additionally, an increase in
standard deviation further downstream is observed.
The asymmetry in optimization is quantified by analyzing the skewness for each wind turbine,
as shown in Figure 7. For the upstream wind farm, a positive skew indicates a frequent
underestimation of wake effects, possibly due to heterogeneous inflow or blockage, which are both
not considered within the framework. Further downstream the models tend to skew negatively,
The Science of Making Torque from Wind (TORQUE 2024)
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Entire cluster Wind farm 1 Wind farm 2 Wind farm 3 Wind farm 4 Wind farm 5
20
40
60
80
100
Standard Deviation [%]
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Figure 6. Time-series standard deviation of the normalized error between model
and SCADA for each wind turbine within the entire cluster and each wind farm.
suggesting an overestimation of the wake effect.
Entire cluster Wind farm 1 Wind farm 2 Wind farm 3 Wind farm 4 Wind farm 5
2
1
0
1
2
Skewness [-]
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Figure 7. Time-series skewness of the normalized error between model and SCADA
for each wind turbine within the entire cluster and each wind farm.
Kurtosis results are presented in Figure 8. A higher kurtosis corresponds to heavier
distributed tails, but it is important to interpret kurtosis together with variance, since a low
variance combined with high kurtosis implies that extremes are not as far from the mean as
they would be as with high variance. Wind farm 1, with the lowest standard deviation, shows
the highest kurtosis. Additionally, a trend can be observed where wake models with a higher
standard deviation exhibit lower kurtosis. This trend cannot be seen between the wind farms,
where the kurtosis slowly increases form wind farm 2 to wind farm 5.
When assessing key indicators for model performance in this deep-array scenario, namely
by analyzing statistical moments and the relative and accumulated relative error, the top-hat
TurbOPark model [7] shows the best performance, followed by the Gaussian TurbOPark model
[9], especially further downstream. Interestingly, the performance of the Jensen model [1] does
not decrease further downstream, unlike that from the Niayifar and Zong models [4, 6], which
initially perform relatively well but show decreased performance further downstream. Across all
scenarios, the Bastankhah model [2] consistently shows the worst performance.
Given the similarities between both TurbO models, apart from the radial wind speed profile,
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Entire cluster Wind farm 1 Wind farm 2 Wind farm 3 Wind farm 4 Wind farm 5
0
5
10
15
Kurtosis [-]
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Figure 8. Time-series kurtosis of the normalized error between model and SCADA
for each wind turbine within the entire cluster and each wind farm.
one hypothesis for the better performance of the TurbOPark model [7] is its overestimation of the
wake deficit at its edges. The wake model assumes no turbine yaw misalignment, which can be
present to some extent and can steer the wake away from a perpendicular direction to the wind.
The absence of modeling yaw misalignment can lead to an underestimation of the wake effects
when partial wake overlap of downwind turbines occurs. The top-hat models characteristic
of overestimating the wake at its edges can indirectly mitigate this issue, compensating for
the potential underestimation by Gaussian models. Furthermore, wake meandering can be
considered as cause for the better performance of the TurbOPark model [7]. Braunbehrens
and Segalini (2019) [19] suggest addressing wake meandering by widening the modeled wake,
a phenomenon a top-hat model may indirectly account for more effectively than the Gaussian
wake models.
4.2. Wake loss prediction
The calibrated tuning parameters, calculated using the optimization framework, are now used to
compare wake losses before and after calibration for each analytical wake model. The calibrated
values are determined as the median from the bin corresponding wind directions between 300
and 330 degrees and wind speeds from 7 to 9 m/s. Subsequently, wake losses at a wind speed of
8 m/s and a wind directions ranging from 300 to 330 degrees are calculated for the entire cluster
and each individual wind farm.
The comparative results, shown in Figure 9, illustrate how the calibration framework affects
expected wake losses. It reduces the predicted wake losses for the Gaussian TurbOPark model
[9] and increases the wake losses for the Jensen [1], Bastankhah [2], and TurbOPark [7] models.
Interestingly, for the entire cluster, the Niayifar [4] and Zong [6] models showed little change
with calibration, however, a subtle shift is observed where the upstream losses are slightly
more underestimated, while the downstream losses are more overestimated with the calibrated
tuning parameters. The performance mismatch between the Bastankhah model (2014) and other
models is more pronounced, likely due to the limitations in accurately accounting for wake losses
at varied turbine spacings. This is apparent in wind farm 5, where the calculated wake losses
by the Bastankhah model [2] deviate from those of other models for all wind directions. Similar
discrepancies are observed in wind farms 3 and 4, particularly with wind coming from the Dutch
zone, characterized by larger turbine spacings in comparison to the Belgian zone. A similar, but
less evident, pattern is seen for the Jensen model [1].
The Science of Making Torque from Wind (TORQUE 2024)
Journal of Physics: Conference Series 2767 (2024) 092059
IOP Publishing
doi:10.1088/1742-6596/2767/9/092059
11
0.0
0.2
0.4
0.6
0.8
1.0
Normalized
wake losses with reference
tuning parameters [-]
Entire cluster Wind farm 1 Wind farm 2 Wind farm 3 Wind farm 4 Wind farm 5
305 315 325
Wind direction [ ]
0.0
0.2
0.4
0.6
0.8
1.0
Normalized
wake losses with calibrated
tuning parameters [-]
305 315 325
Wind direction [ ]
305 315 325
Wind direction [ ]
305 315 325
Wind direction [ ]
305 315 325
Wind direction [ ]
305 315 325
Wind direction [ ]
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Jensen (1983)
Gaussian, Bastankhah (2014)
Gaussian, Niayifar (2015)
Gaussian, Zong (2020)
TurbOPark (2020)
Gaussian TurbOPark (2022)
Figure 9. Comparison of normalized wake effects: On the top are the predicted normalized
wake losses using reference tuning parameters, and below are the normalized wake losses using
calibrated tuning parameters. As the basis for normalization within each column the lowest
and highest wake loss values from either reference or calibrated parameters are used.
5. Conclusion
A multi-level hyperparameter optimization framework is utilized to calibrate analytical wake
models using SCADA data from five wind farms in the Belgian-Dutch offshore zone. The
calibration is performed on wind coming from north-west (300 - 330 degrees) and wind speeds
ranging from 7 to 9 m/s, a direction identified as promising for calibration due to its limited
inflow heterogeneity and large amount of clustered wind farms [15]. A constant turbulence
intensity of 0.06 and a shear exponent of 0.12 are assumed, without considering blockage models.
Six models are compared in this study, namely the analytical wake models of Jensen (1983),
Bastankhah (2014), Niayifar (2016), Zong (2020), Nygaard (2020) and Pedersen (2022). The
relative and accumulated relative error between the measurement data and the model predictions
are analyzed both per wind farm and for the entire cluster, as well among individual turbines.
Statistical moments for each wind turbine are calculated and presented as boxplots per wind
farm.
In general, the TurbOPark model (2020) shows the best overall performance, followed by the
Gaussian TurbOPark model (2022), Niayifar (2016) model, and Zong (2020) model. Where the
Niayifar (2016) and Zong (2020) models perform relatively well close to the upstream turbines,
further downstream their performance decreases. This is opposite to the top-hat and Gaussian
TurbOPark models (2020,2022) and the Jensen model (1983). The Bastankhah (2014) model
performs poorest of all models.
The Science of Making Torque from Wind (TORQUE 2024)
Journal of Physics: Conference Series 2767 (2024) 092059
IOP Publishing
doi:10.1088/1742-6596/2767/9/092059
12
Using calibrated tuning parameters, new wake losses are calculated for the specified wind
conditions. The performance shortfall of the Bastankhah (2014) model appears to be primarily
linked to variable and larger turbine spacings, particularly noticeable in Wind Farm 5 and
under conditions of more northerly directed winds, where larger turbine spacings are present.
Conversely, the TurbOPark models shows their effectiveness in these scenarios, underscoring its
development purpose.
It’s important to note that despite varying levels of accuracy, all models show similar trends
in wake losses after calibration. This suggests that with proper calibration, any model can be
effective, provided its fundamental limitations are understood and accounted for.
Acknowledgments
The authors would like to acknowledge the support of VLAIO in the context of the Blauwe
Cluster project Cloud4Wake. The authors would also like to acknowledge the Energy Transition
Funds for their support of the Poseidon project. This research was supported by funding
from the Flemish Government under the “Onderzoeksprogramma Artifici¨ele Intelligentie (AI)
Vlaanderen” programme.
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