the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# On wake modeling, wind-farm gradients, and AEP predictions at the Anholt wind farm

### Kurt Schaldemose Hansen

### Søren Ott

### Maarten Paul van der Laan

We investigate wake effects at the Anholt offshore wind farm in Denmark, which is a farm experiencing strong horizontal wind-speed gradients because of its size and proximity to land. Mesoscale model simulations are used to study the horizontal wind-speed gradients over the wind farm. From analysis of the mesoscale simulations and supervisory control and data acquisition (SCADA), we show that for westerly flow in particular, there is a clear horizontal wind-speed gradient over the wind farm. We also use the mesoscale simulations to derive the undisturbed inflow conditions that are coupled with three commonly used wake models: two engineering approaches (the Park and G. C. Larsen models) and a linearized Reynolds-averaged Navier–Stokes approach (Fuga). The effect of the horizontal wind-speed gradient on annual energy production estimates is not found to be critical compared to estimates from both the average undisturbed wind climate of all turbines' positions and the undisturbed wind climate of a position in the middle of the wind farm. However, annual energy production estimates can largely differ when using wind climates at positions that are strongly influenced by the horizontal wind-speed gradient. When looking at westerly flow wake cases, where the impact of the horizontal wind-speed gradient on the power of the undisturbed turbines is largest, the wake models agree with the SCADA fairly well; when looking at a southerly flow case, where the wake losses are highest, the wake models tend to underestimate the wake loss. With the mesoscale-wake model setup, we are also able to estimate the capacity factor of the wind farm rather well when compared to that derived from the SCADA. Finally, we estimate the uncertainty of the wake models by bootstrapping the SCADA. The models tend to underestimate the wake losses (the median relative model error is 8.75 %) and the engineering wake models are as uncertain as Fuga. These results are specific for this wind farm, the available dataset, and the derived inflow conditions.

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The Anholt wind farm is currently the fourth largest offshore wind farm in
the world power-wise. The layout of the Anholt wind farm was optimized to
minimize wake losses. The number of wind turbines (111), the wind-turbine
type, and the maximum allowed wind-farm area for turbine deployment (88 km^{2}) are examples of chosen constraints. The employed optimization tool
has a tendency to place most wind turbines at the edges of the wind-farm
area, while the remaining wind turbines are placed inside the wind farm with relatively large interspacing. For the particular case of Anholt, a number of
wind turbines were relocated from the optimized layout due to seabed that
turned out to be too soft (Nicolai Gayle Nygaard, personal communication, 2017).

So far the only reported studies on the wake effects of this wind farm are
those of Nygaard (2014), Nygaard et al. (2014), and van der Laan et al. (2017).
In the first, there is a comparison between the Park wake model
(Katic et al., 1986) and supervisory control and data acquisition (SCADA) for a row of turbines in the middle of the wind
farm for a given wind-direction and wind-speed range. The wake model
estimates the wake losses fairly well. The study also presents the results of
the Park model for other large offshore wind farms, clearly showing that this
wake model agrees with the SCADA for different inflow conditions rather well.
These are interesting findings because engineering wake models do not
generally include coupling with the vertical structure of the atmospheric
boundary layer; thus, they should tend to underpredict wake losses in large
offshore arrays (Stevens et al., 2016). However, the studies showing wake-model
underprediction in large offshore wind farms (Barthelmie et al., 2009)
analyze the wake observations using narrow wind direction sectors and do not
account for wind direction variability. In the study by Nygaard et al. (2014),
a comparison of two wake models, Park and the eddy viscosity model of
WindFarmer (GL Garrad Hassan, 2013), is performed against SCADA, revealing
that Park, with a wake-decay coefficient *k*=0.04, gives better results than
the model of WindFarmer with and without a large wind-farm correction. In the
study by van der Laan et al. (2017), the effect of the coastline on the wind farm
is investigated with a Reynolds-averaged Navier–Stokes (RANS) model, showing
that such a RANS setup is able to predict the horizontal wind-speed gradient
over the wind farm when compared to the SCADA and mesoscale model
simulations.

Engineering wake models are also often regarded as too simplistic for the
estimation of wake losses, yet they are those that are most used when
planning wind-farm layouts and for annual energy production (AEP)
estimations. This is because they can be easily implemented and optimized in
terms of computational performance. One cannot expect to characterize wakes
in detail with such models but for the estimation of power and energy
production means, they are sufficiently accurate when used properly
(Nygaard, 2014; Nygaard et al., 2014). Peña et al. (2014) show that the Park model is
able to predict the wake losses of the Horns Rev I wind farm in the North Sea
for different atmospheric stability conditions when using a
stability-dependent wake-decay coefficient. Peña et al. (2016) show that the
Park model is in good agreement with the Sexbierum cases in which two more
sophisticated wake models are also tested: a linearized RANS solution (Fuga)
and a nonlinear solution of the RANS equations that uses a modified
*k*-*ε* turbulence model. In the last two studies, the high
accuracy of the Park model is partly a result of accounting for the
variability in the wind direction (Gaumond et al., 2014). Since Fuga is a
computationally efficient wake model, whose results (in terms of wind-speed
deficits) are nearly equal to those of a nonlinear solution of the RANS
equations (Ott et al., 2011), we want to find out how different AEP and capacity
factor estimates are when compared to those of Park and of another wake model
that is a simple solution of the RANS equations, the G. C. Larsen model
(Larsen, 2009).

Wake models of all types have been mainly evaluated against offshore wind
farms that are well off the coast or where the effect of the land is assumed
to be minimal (Barthelmie et al., 2009; Réthoré et al., 2013; Stevens et al., 2016). The layout of the
Anholt wind farm offers the possibility of investigating the effect of land
proximity (∼ 20 km in the predominant wind direction) on the wind-farm
production. We are aware that the Anholt wind farm experiences strong
horizontal wind-speed gradients, which are translated into power gradients
for turbines that are not experiencing wakes (Damgaard, 2015). Another
example of the effect of the land on an offshore wind farm, in this case in
the Baltic Sea, is provided by Dörenkämper et al. (2015). The challenge is
therefore to find out how such gradients interfere with the wake losses and
how these affect the production and the AEP. This can be performed by simple
“coupling” of undisturbed^{1} wind climates at some (or all) turbines' positions,
in which the horizontal wind-speed gradient is embedded, with the wake
models. To the authors knowledge, there have not been attempts to study the
impact of the horizontal wind-speed gradient on wakes of wind farms using
engineering wake models yet, although there is an attempt to include
wind-direction gradients (Hasager et al., 2017). An obvious choice to derive the
wind climate is the use of a mesoscale model such as the Weather Research and
Forecasting (WRF) model (Skamarock et al., 2008), which is today often used
multi-purposely in the wind-energy community
(Hahmann et al., 2015; Platis et al., 2018; Storm and Basu, 2010). In the present work, we also want to
investigate the ability of WRF to model the horizontal wind-speed gradient
over the wind farm.

In this study, we first present (Sect. 2) a general background regarding the Anholt wind farm, the WRF mesoscale runs that we use to estimate the wind-farm climate, the wind-farm SCADA, the wake models, and the ways in which we account for the horizontal wind-speed gradient and estimate the wake models' uncertainty. Section 3 presents the results regarding the influence of the wind-speed gradient on flow cases and on the AEP, the results showing the evaluation of the wake models for two flow cases, and the analyses of the capacity factor, power loss, and model uncertainty. Finally, discussion and conclusions are given in the last two sections.

## 2.1 Definitions

We define the efficiency of the wind farm at a given wind speed *U* as

where *P*_{i} is the power of each individual turbine in the farm, *P*_{U} the power
of the turbine from the power curve at *U*, and *n*_{t} the number of turbines
in the wind farm.

We define the power loss of the wind farm as

where 〈〉 means ensemble average and *P*_{free} is the power of
the free-stream turbines (these are defined in Sect. 2.2.2).

We define the relative wake model error as

where the subscripts _{obs} and _{mod} refer to observations and model,
respectively.

## 2.2 Anholt wind farm

The Anholt wind farm is located in the Kattegat strait between Djursland and
the island of Anholt in Denmark (Fig. 1a). It
consists of 111 Siemens 3.6 MW-120 turbines with a hub height of 81.6 m and a
rotor diameter of 120 m (Fig. 1b). The smallest
distance between the turbines is 4.9 rotor diameters. The water has depths of
15–19 m, the wind farm area is 88 km^{2}, and full operation started in
summer 2013.

### 2.2.1 SCADA

We have access to 10 min means of SCADA for the period from 1 January 2013 to 30 June 2015. Data include nacelle wind speed, yaw position, pitch angle, rotor speed, power reference, air temperature, rotor inflow speed, and active power. We also produce a filtered SCADA dataset by identifying periods when each turbine was grid connected and produced power during the entire 10 min period. The dataset excludes periods when any turbine was either parked or idling, those with starting and stopping events, and where power was curtailed or boosted. We find turbine nos. 1, 36, 65, and 68 to be boosted with power values 5 % above the rated value. The result is a time series of 7440 10 min values starting in July 2013 until December 2014.

### 2.2.2 Inflow conditions

Due to the lack of undisturbed mast measurements in the SCADA, we derive the inflow conditions from the filtered SCADA dataset. We estimate an “equivalent” wind speed based on either the 10 min SCADA's power or pitch angle values in combination with the manufacturer's power curve or the average pitch curve extracted from the SCADA. The inflow reference wind speed is computed as the average equivalent wind speed for groups of four undisturbed turbines as shown in Table 1. A group of four turbines is used to robustly estimate the inflow wind speed and 10 different sectors are needed to avoid the influence of Djursland and the island of Anholt. The inflow reference wind direction is computed as an average yaw position for pairs of undisturbed wind turbines listed in Table 1. The yaw position calibration is performed as in Rodrigo and Moriarty (2015). The turbines that we use to derive the inflow conditions are shown in Fig. 1b.

## 2.3 Wind-farm climate

We perform simulations of the wind climate over a region covering the Anholt
wind farm using the WRF version 3.5.1 model. Simulations are carried out on
an outer grid with horizontal spacing of 18 km × 18 km (121 × 87 grid points),
a first nested domain of 6 km × 6 km (280 × 178 grid points), and a second nest with its center in the middle of Jutland,
Denmark, of 2 km × 2 km (427 × 304 grid points). The simulations
use 41 vertical levels from the ground to about 20 km. The lowest 12 levels
are within the 1000 m of the surface with the first level at ∼ 14 m.
Initial boundary conditions and fields for grid nudging come from the
European Centre for Medium-Range Forecasts ERA-Interim Reanalysis
(Dee et al., 2011) at 0.7^{∘} × 0.7^{∘} resolution. Other choices in
the model setup are standard and commonly used in the modeling community.
Further details regarding the simulations are provided in Peña and Hahmann (2017).
Figure 2 shows the Anholt wind climate at hub height
at a WRF grid point in the middle of the wind farm based on the WRF hourly
outputs for 2014 (the model is run for 1982–2015). The model output is
logarithmically interpolated to hub height. Most winds come from the west,
south-southwest, and southeast directions and winds between 5 and 15 m s^{−1} are the most
frequent (the all-sector mean wind speed is 9.23 m s^{−1} at hub height).

## 2.4 Wake models

We use three different wake models: the Park wake model with the
commonly used offshore value of *k*=0.04, the G. C. Larsen model
(Larsen, 2009), and Fuga (Ott et al., 2011). The first two are engineering
wake models and Fuga is a linearized flow solver of the steady-state RANS
equations using an actuator-disk approach. For the two engineering wake
models, the local wake deficits *δ*_{i} are superposed to compute the
speed deficit at the *n*th turbine. This is performed in two different ways:
linearly ${\sum}_{i=\mathrm{1}}^{n}{\mathit{\delta}}_{i}$ and as a quadratic sum
${\left({\sum}_{i=\mathrm{1}}^{n}{\mathit{\delta}}_{i}^{\mathrm{2}}\right)}^{\mathrm{1}/\mathrm{2}}$.

Due to the high computational efficiency of these wake models, we can easily
perform wake analyses over given wind-speed and wind-direction ranges and
AEP-like calculations using the values in the time series (no need for
distributions). For the latter calculations, we create lookup tables (LUTs)
for each wake model, which contain the total wind-farm power output for
specific undisturbed wind directions and wind speeds. Figure 3 shows a comparison of the efficiency of the
wind farm (Eq. 1) predicted by the wake models. All wake
models show the highest wake losses at the directions in which most wind
turbines are aligned, i.e., at ∼ 160 and 340^{∘}, and 45 and 235^{∘}.
At 5 m s^{−1}, the Park linear model generally shows the highest wake
losses followed by Larsen linear and Fuga models (within the direction in
which
turbines are most aligned). At 5 and 10 m s^{−1}, *η* ∼ 0.9 for all
wake models excluding the most aligned directions, with the Larsen quadratic and
Park linear models showing the highest and lowest efficiencies,
respectively.

## 2.5 Accounting for the wind-farm gradient

One way to account for the effect of the horizontal wind-speed gradient within a wind farm, which is not the result of wake effects themselves, on the wind-farm power output is by estimating the wake losses using the undisturbed wind speed and direction at each individual turbine position for each time realization as inflow condition instead of using a single undisturbed wind speed and direction as it is commonly performed. At each turbine position, we will therefore have both a time series of velocity deficits (and thus power values) because of the change with time of inflow conditions and a time series, with a number of members equal to the number of turbines in the farm, of velocity deficits for each inflow condition experienced by each turbine for each time realization. Then, the wind-farm power time series, as an example, can be estimated by averaging the power resulting from all inflow conditions for the same time realization (for the Anholt case this means 111 conditions) and then averaging the results of all turbines. This is hereafter known as a gradient-based analysis. The wind and inflow at each turbine must be undisturbed and so mesoscale model simulations over the wind-farm area (without the wind farm) are an obvious option to estimate the wind climate at each turbine position.

Due to the very high efficiency of the Park model (in a MATLAB script it takes milliseconds to perform one simulation of Anholt for a single inflow wind speed and direction), when using the WRF hourly time series, we can perform 111 simulations (i.e., 111 different inflow conditions that are interpolated from the WRF grid into the turbine positions) in a couple of seconds. Thus, we can perform a gradient-based AEP analysis with hourly WRF winds in just few hours. It is important to note that we can perform traditional (i.e., with a single inflow condition per time realization) AEP calculations with all wake models much faster using pre-computed LUTs.

## 2.6 Uncertainty estimation

We quantify the uncertainty of the wake models using a nonparametric
circular-block bootstrap similar to the approach of Nygaard (2015). The
idea is to “wrap” the power-output time series (from both measurements and
simulations) of the wind farm around a circle. Blocks of the time series with
a given size, which is here selected according to Politis and White (2004) based on
the wind-speed time series, are then randomly sampled. The number of sampled
blocks is given by the total size of the time series and the block size. The
number of bootstrap replications should be large enough to ensure a close-to-zero Monte Carlo error. By bootstrapping the power-output time series, we can
estimate the bootstrapped *PL* (Eq. 2) and so estimate a
distribution of *ϵ* (Eq. 3). Details and code
implementations of a number of bootstrapping techniques can be found in
Sheppard (2014).

The analysis of the influence of the horizontal wind-speed gradient in Sect. 3.1 is performed with the WRF model outputs for 2014 and the filtered SCADA dataset. For AEP estimations (Sect. 3.1.1), we only use WRF model outputs. The westerly flow case in Sect. 3.1.2 uses the filtered SCADA dataset, as well as the south flow case in Sect. 3.1.3, and the WRF model outputs. For the capacity factor calculations in Sect. 3.2, we use all the SCADA results available for 2014 and the WRF model outputs for the same year. The analyses of the power loss and model uncertainty in Sect. 3.3 and 3.4 are performed on the filtered SCADA.

## 3.1 Influence of the wind-farm gradient

Figure 4 shows the mean horizontal wind-speed gradient
at hub height in and surrounding the Anholt wind farm based on simulations
from the WRF model for the year 2014. The left frame shows the average for
all wind speeds and directions and the right frame the average for all wind
speeds and directions within 270 ± 30^{∘}, which have been filtered using
the simulated wind direction at hub height at the position of turbine 15. The
influence of Djursland (see Fig. 1a) on the wind
at the farm is clear even for the omnidirectional case. The impact of
Djursland is much stronger when looking at westerly winds so we could expect
an impact on the results of wake models when the flow is particulary from
these directions. The horizontal wind-speed gradient is mainly due to the
roughness effect of the land surrounding the wind farm (van der Laan et al., 2017).
Although it is not shown, the island of Anholt east of the farm also has an
impact on the wind speed at the wind farm for northeasterly flow but this is
not as strong as that of Djursland for westerly flow. For westerly winds
(270 ± 30^{∘}), the WRF-simulated average hub-height wind-speed
difference between turbine nos. 1 and 30 is 0.62 m s^{−1}, whereas for
easterly winds (90 ± 30^{∘}) it is 0.12 m s^{−1} between turbine nos. 86 and 111.

In Fig. 5a we extract the values from Fig. 4 at each turbine position by linearly interpolating the WRF winds to the turbine positions. For the omnidirectional case, the horizontal wind-speed gradient is lower than for westerly winds, as expected, and for both cases the strongest gradient is observed for the first row of turbines (1–30), which are those closer to Djursland.

Figure 5b shows SCADA-derived and
WRF-simulated average wind speeds at hub height for turbine nos. 1–30 for a
number of westerly flow cases. We select filtered SCADA based on the inflow
conditions described in Sect. 2.2.2 within the
wind-speed range 5–10 m s^{−1} and use the manufacturer's power curve to
derive each turbine's wind speed from the power output. For the comparison,
we extract the WRF-simulated winds by averaging the horizontal wind-speed
components on the corresponding free-stream turbines for each direction range
as given in Table 1. We also select WRF-simulated
winds within the same wind-speed range 5–10 m s^{−1}. It is observed that
the horizontal wind-speed gradient for westerly winds depends on the
particular direction. The strongest simulated and observed gradients are
found at 265 ± 5^{∘}, with the winds at turbine nos. 1–15 being lower than
those at turbine nos. 15–30. Generally, the simulated gradient agrees with
the observations fairly well, except for the range 295 ± 5^{∘}, in which the
SCADA shows the highest winds at the southern turbines. This can be an effect
of the topography on the turbines, which is not captured by WRF. It could
also be a wind-farm wall effect (Mitraszewski et al., 2012). A similar effect
(not shown) is observed when analyzing the SCADA-derived wind speeds of the
turbines south of each row for a direction of 80–90^{∘}: the wind speed
at turbine no. 1 is about 6 % higher than that at turbine no. 86.

### 3.1.1 Annual energy production

The difference in AEP when accounting for the wind-farm gradient information
and when assuming a horizontally homogenous wind field^{2} is lower than 1 % when using the 2014 hourly WRF
wind fields combined with the wake models (“average wind field” column in
Table 2). This is because, in general for this wind climate,
there are positive or negative errors in the production estimations that are
balanced during the year. The highest difference is observed for the WRF–Fuga
setup, in which the estimation using the “average wind” does not balance for
the low energy yield of the turbines in the south of the farm and the high
energy yield of those in the north as it does for the other WRF-wake model
setups.

The difference in the AEP estimation by accounting for the wind-speed
gradient and that by using the wind climate of turbine no. 1, which is the
position with the lowest average wind speed, is larger than 1 % for the
engineering wake models. Such a difference is rather large considering that the
AEP of the wind farm is ∼ 1889.3 GW h when averaging all models' AEP
estimations using the wind-gradient information. The same exercise using the
information of turbine no. 54 (in the middle of the farm) results in
differences very close to those using the average wind field. Using the
information of turbine no. 65 (at the top of the farm), the difference is
also large but positive as expected. For the Anholt wind farm and its wind
climate, in particular, these results show that although accounting for the
wind-farm gradient is important, it does not largely change the AEP
estimations compared to those based on a one-point wind climate, unless the
latter is not close to the average wind climate within the wind-farm area.
For comparison purposes (e.g., with the results in Fig. 5a) the yearly average wind speed of the
“homogenous” wind is 9.21 m s^{−1}.

### 3.1.2 Westerly flow cases

Given the impact of the horizontal wind-speed gradient on the AEP estimations (Sect. 3.1.1), it is relevant to study the wake losses under westerly flow conditions. Figure 6a shows, for 2014, the average WRF–Park quadratic power of each turbine in the wind farm when filtering for westerly wind directions (using the WRF-simulated wind climate at turbine no. 15), both accounting for the wind-speed gradient, as described in Sect. 2.5, and assuming a homogenous wind field (the average of the wind climates at each turbine). For a broad wind-direction range, both results are nearly identical and only small differences at specific turbines (up to 27.2 kW) are found when the wind-direction range is reduced; in this latter case we use the range that shows the largest gradients in Fig. 5b. It is important to note that, although it is not seen, the normalized average power of turbines 1–30 for the two “gradient” cases in Fig. 6a is slightly lower than 1 as expected.

Since the horizontal wind-speed gradient does not seem to strongly impact the
wake behavior for broad wind-direction ranges, we compare the SCADA that
have been wind-speed and direction filtered with the wake models in Fig. 6b. The inflow conditions are derived from
the SCADA (see Table 1) and are used to run the
wake models. After filtering for wind speed and
direction (5–10 m s^{−1} and 270 ± 30^{∘}), 735 10 min cases are left. In this case the power
values are not normalized with the power of a unique turbine, as they are for
the plot in the top frame. Instead, we use the undisturbed turbine that is
closest to that from where we are extracting the power from. This aids to levelize
the SCADA mainly at turbine nos. 1–30. The wake models generally agree with
the SCADA, particularly Fuga, and along with this the engineering wake models'
variants using the linear sum of wake deficits generally show the highest wake
losses. For turbine nos. 31–60, where the wind farm experiences
single and double wakes mostly, the SCADA are between the models' results.
For turbine nos. 66–111, where multiple wakes occur, Larsen quadratic highly
underestimates the wake and the linear variants and Fuga seems to generally
agree better with the SCADA. However, the comparison is not completely fair
with the wake models because the reference power is not always higher than or
equal to that of the individual turbines when these are supposed to be in the
wake of a turbine. For example, in the case of turbine no. 31, we use turbine no. 3
as reference and in ∼ 19 % of the cases with the inflow conditions
analyzed in Fig. 6b, *P*_{3}<*P*_{31}.

### 3.1.3 Southerly flow case

Figure 7 illustrates the wake loss for the
north–south row in the middle of the wind farm (turbine nos. 45–65)
filtering for inflow conditions (9 ± 0.5 m s^{−1} and 168.7 ± 15^{∘},
which is the direction in which turbine nos. 45 and 46 are aligned) that are
derived from the SCADA of turbine nos. 45 and 66–68 (Table 1). After filtering for
wind speed and direction, 26 10 min cases are left. As expected from the results in Fig. 6b, for this multiple wake case, the models
using the linear variant agree better with the SCADA than those using the
quadratic variant when going deeper in the row. The Park quadratic model
predicts the wake loss of the first three turbines rather well but
underpredicts it when moving deeper in the row. The results from Fuga are
between the engineering model's variants.

Because the differences between SCADA and models in Fig. 7 are relatively large and the amount of 10 min
periods for the southerly flow case are 26 only, we also perform
actuator-disk RANS simulations in EllipSys3D (Sørensen, 2003) using a
modified *k*-*ε* turbulence model (van der Laan et al., 2015). The
results of the RANS model are very close to those of Fuga and Larsen linear,
also underestimating the wake loss. We can only speculate that for this
particular case, the high wake loss from the SCADA is due to atmospheric
conditions, in particular from periods under a rather stable atmosphere, that
we are not accounting for in the simulations. However, we do not have useful
observations to directly derive stability. We have atmospheric stability
measures from the WRF simulations but instantaneous WRF stability measures
are highly uncertain (Peña and Hahmann, 2012). Nygaard (2014) shows the same
case using another SCADA period and the wake losses are ∼ 10 % lower
than those we observe.

## 3.2 Capacity factor

Being able to estimate the AEP (Sect. 3.1.1) is important but it is more interesting to find out whether we are able to predict it, in our particular case, with the combined mesoscale-wake setup. For the exercise, the capacity factor is a better choice than the AEP since we can compare Anholt with other offshore wind farms.

We use all the SCADA data that are available for 2014. Theoretically, there should be 52 560 10 min samples for this year. However, the number of samples per turbine available in the SCADA varies and is never the theoretical one; the turbine with the highest number of samples is no. 7 (51 648) and that with the lowest is no. 77 (49 512). The average availability, taking into account all turbines, of observed samples is 98.10 %. Table 3 shows the observed and estimated capacity factors, which are predicted by the WRF-wake model setup and that account for both the wind-farm gradient and the observed average availability of samples.

It is clear that we can estimate the observed capacity factor using the WRF-wake model setup fairly well. However, it is important to note that wind turbines are not always working and underperform when compared to the manufacturer's power curve. The predicted AEP or capacity factor of a combined mesoscale-wake model is typically higher than the observed value; however, we want to know the capacity factor of a wind farm regardless of the operating conditions.

## 3.3 Power loss

Table 3 also shows the wind farm *PL* based on the
SCADA's 7440 10 min values and using Eq. (2) with the
inflow conditions as defined in Table 1. The
results for the wake models are computed interpolating the models' LUTs with
the same inflow conditions derived from the SCADA. All models, except for
Park linear, predict lower *PL*s than the SCADA; Park quadratic,
Larsen linear, and Fuga slightly underestimating the wake loss.

One way to show that the estimations of power of the free-stream turbines are sound is to compare the manufacturer power curve with the SCADA-derived power (averaging the power of the turbines in Table 1) and SCADA-derived inflow wind speed. This is illustrated in Fig. 8a, where we show the power curve of the turbine and the SCADA-derived values (no interpolation is made). Figure 8b shows a similar comparison but in this case we derive the gross wind-farm power (i.e., 111 times the power of the free-stream turbines) and that derived from the power curve at the estimated free wind speed. Both figures show that our definition of the free-stream turbines is sound (no evident wake effects are observed) and that the turbines do follow the manufacturer's power curve.

However, this does not give us an idea about the validity of the
SCADA-derived inflow conditions for the turbines that are far from those we
use to derive the inflow conditions. By filtering the SCADA-derived inflow
conditions for westerly flow (270 ± 30^{∘}), so that no wakes are observed
for turbine nos. 1–30, we can derive power curves for the turbines at the
beginning and end of that row (i.e., nos. 1 and 30) and compare them to,
for example,
the manufacturer's power curve. As expected, the power curves for turbines nos. 1 and 30 are below and above the manufacturer's power curve, the difference
for turbine no. 1 being as high as 500 kW, which is the turbine with the lowest
average wind speed according to the WRF simulations (Fig. 5a). Within the wind-speed range where we
observe such differences in power, the difference in wind speed is about 1 m s^{−1}.

## 3.4 Model uncertainty

Also based on the SCADA's 7440 10 min values, we find an optimal block length
for the circular bootstrap of 242 samples. On average, such sample length
corresponds to about 10 days, which is long enough to capture the correlation
between samples. We use 10 000 bootstrap replications and find that, for example,
*ϵ* for the Park quadratic model stabilizes after 2000 replications.
Figure 9 shows the distribution of *ϵ* for all
models where positive *ϵ* values denote a model that overestimates the
power (underestimates the wake loss), whereas negative *ϵ* values denote a
model that underestimates the power (overestimates the wake loss).

For the particular case of the Anholt wind farm and for the filtered SCADA
used in the analysis, Larsen linear has the distribution with lowest bias and
the second largest *σ* value (after Park linear), whereas
Larsen quadratic has the highest bias and lowest *σ* values. The results
for Park quadratic and Fuga are similar, both bias and *σ*. Park linear,
as expected due to the previous results, is the only model that systematically
overestimates the wake loss. If we could extrapolate these results to an AEP
analysis, we would expect non-conservative AEP estimations (except for
Park linear), with Park quadratic, Fuga, and Larsen linear being slightly
optimistic and Larsen quadratic too optimistic.

It is important to note that some of our results depend on the methods we use
to derive the undisturbed inflow conditions of the wind farm. We show that
for power analyses of individual turbines, whose inflow conditions are
greatly affected by the horizontal wind-speed gradient (like turbine nos. 1
or 30), this is an important matter (see Fig. 8a).
For this particular wind farm and wind climate, the differences between the
undisturbed inflow conditions derived from turbines in the middle of the long
rows and the inflow conditions derived from turbines to either side of the
rows compensate for the overall wind-farm long-term analyses (e.g., AEP and
capacity factor). One way to further analyze the impact of different inflow
conditions is to derive them for each individual undisturbed turbine. We can
then potentially perform analyses (flow cases, power loss, and capacity
factor) in a similar fashion as that we use for accounting for the horizontal
wind-speed gradient^{3} and validate our findings.

We also estimate the power loss and the uncertainty of the wake models based on a rather discontinuous and short filtered SCADA dataset. Therefore, our results might be biased and caution must be taken when generalizing our findings. A clear example is that related to the model uncertainty, where we find that most wake models underestimate the wake losses. With a longer dataset, the biases can change (and models might start to produce conservative results) but the relative position of the models will most probably be maintained, Park linear and Larsen quadratic being the most conservative and most optimistic models, respectively. If the same models are evaluated with SCADA from other wind farms, the biases will most probably change.

We show that our WRF-wake model setup is able to rather accurately predict the capacity factor of the Anholt wind farm. Anholt is the offshore wind farm with the highest all-life capacity factor in Denmark (49.4 %) and the highest in the world for a wind farm older than 2 years, outperforming Horns Rev II, which has, in principle, more favorable wind conditions. One of the reasons for this is the Anholt wind-farm layout, which highly minimizes the wake losses.

The results for the two flow cases illustrate what we already expected; Park linear shows the highest and Larsen quadratic the lowest wake deficits. This is mainly because of the values we choose for the wake decay coefficient. It is important to note that we can obtain similar wake deficits with both the Park linear and Park quadratic models when tuning the wake decays. Physically, it makes more sense to linearly sum the wake deficits but the quadratic approach is normally used due to a historical general good match of model predictions with observed power deficits, for the values normally suggested for the wake decay (0.04–0.05 for offshore conditions). The RANS model shows similar values to Fuga, as expected due to the similarity of the models' physics, both showing a better comparison to the SCADA for the two flow cases than the traditional Park quadratic model, also as expected.

For the Anholt wind farm, we show from both the SCADA and WRF model simulations that for a number of wind directions, there is a clear influence of the land on the free-stream wind speed at the positions of the turbines closer to the coast. However, for AEP calculations for which we run three different wake models using mesoscale model outputs as inflow conditions, accounting for the horizontal wind-speed gradient (also derived from the mesoscale model results) does not have a large impact on the results when compared to AEP calculations based on, first, a wind climate that is the average of all wind climates at the turbines' positions and, second, a wind climate correspondent to a position in the middle of the wind farm. It does, however, differ from the calculation using a wind climate that is strongly influenced by the horizontal wind-speed gradient particularly for the engineering wake models.

We look at two flow wake cases with two different engineering wake models and some of its variants and a linearized RANS model. The first case corresponds to westerly winds, for which the influence of the horizontal wind-speed gradient is largest. Here the wake models, and Fuga in particular, agree with the SCADA fairly well. The second case corresponds to southerly winds, for which the wake losses are highest. Here, the wake models tend to underestimate the wake deficit when compared to the SCADA. This is also translated into a wake-model tendency to underestimate the observed power loss, on average 0.31 % less than that derived from the SCADA.

Using our mesoscale-wake model setup, we find that the estimated capacity factors are 0.27–4.60 % biased when compared to those computed from the SCADA. Finally, using inflow conditions derived from the SCADA and by circularly block bootstrapping these, we estimate the relative error of the wake models. We find that these models tend to underestimate the wake losses, except for one wake model variant. The engineering wake models are found to be as good as the linearized RANS Fuga model. However, these are results that are wind farm and SCADA specific and that depend on the definition of inflow conditions; therefore similar analyses need to be reproduced at different wind farms, using more SCADA and different methods to derive the inflow conditions.

The Anholt SCADA can be made available by Ørsted upon request to Miriam Marchante Jiménez (mirji@orsted.dk). The WRF data can be made available by DTU Wind Energy upon request to Andrea N. Hahmann (ahah@dtu.dk).

The authors declare that they have no conflict of interest.

This article is part of the special issue “Wind Energy Science Conference 2017”. It is a result of the Wind Energy Science Conference 2017, Lyngby, Copenhagen, Denmark, 26–29 June 2017.

We would like to thank Ørsted and partners for providing
the SCADA. Also, we thank Charlotte B. Hasager for promoting and leading the Anholt
wind-farm internal project at DTU Wind Energy and Patrick Volker for making the
mesoscale model simulation outputs easily accessible. Finally, we would like
to thank the three anonymous reviewers and Nicolai Gayle Nygaard for their
comments on the paper.

Edited by: Julie
Lundquist

Reviewed by: three anonymous referees

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