the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Determination of optimal wind turbine alignment into the wind and detection of alignment changes with SCADA data

### Niko Mittelmeier

### Martin Kühn

Upwind horizontal axis wind turbines need to be aligned with the main wind direction to maximize energy yield. Attempts have been made to improve the yaw alignment with advanced measurement equipment but most of these techniques introduce additional costs and rely on alignment tolerances with the rotor axis or the true north.

Turbines that are well aligned after commissioning may suffer an alignment degradation during their operational lifetime. Such changes need to be detected as soon as possible to minimize power losses. The objective of this paper is to propose a three-step methodology to improve turbine alignment and detect changes during operational lifetime with standard nacelle metrology (met) mast instruments (here: two cup anemometer and one wind vane).

In step one, a reference turbine and an external undisturbed reference wind signal, e.g., met mast or lidar are used to determine flow corrections for the nacelle wind direction instruments to obtain a turbine alignment with optimal power production. Secondly a nacelle wind speed correction enables the application of the previous step without additional external measurement equipment. Step three is a monitoring application and allows the detection of alignment changes on the wind direction measurement device by means of a flow equilibrium between the two anemometers behind the rotor.

The three steps are demonstrated at two 2 MW turbines together with a ground-based lidar. A first-order multilinear regression model gives sufficient
correction of the flow distortion behind the rotor for our purposes and two
wind vane alignment changes are detected with an accuracy of ±1.4^{∘} within 3 days of operation after the change is
introduced.

We show that standard turbine equipment is able to align a turbine with sufficient accuracy and changes to its alignment can be detected in a reasonably short time, which helps to minimize power losses.

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Modern large-utility-scale wind turbines are usually designed with an upwind rotor and therefore are active yaw controlled. The main reason for this design decision is the wind flow interference with the tower, which is less upwind of the structure. An active yaw control algorithm tries to align the rotor into the wind. The yaw system is slow compared to directional changes of the wind and immediate turbine reaction causes higher energy consumption of the yaw engines and wear of the components; therefore some degrees of yaw errors are accepted.

A systematic analysis of the impact of skewed flow on power production was
set up by Madsen (2000). He compared field measurements of a
100 kW rotor with aeroelastic simulation using standard blade element
momentum theory implemented in the HawC code and a new actuator disc
model of the HawC-3D code. He found good agreements between the new 3-D
model and the measurements. Pedersen et al. (2002) used the same data and explained the power loss following a
squared-cosine relationship with increasing yaw error larger than
30^{∘}.

The rotor disc area perpendicular to the wind direction is reduced by the cosine of the inflow angle, which explains the squared-cosine relationship of the power loss for large angles. Likewise the wind speed component perpendicular to the rotor also decreases by a cosine relation, and with the wind speed having a cubic impact on power a cubic-cosine relationship is more dominant for smaller angles (Burton et al., 2001; Mamidipudi et al., 2011). In addition to power losses, yaw errors also increase turbine loads (Wan et al., 2015; Damiani et al., 2018).

Achieving a good alignment of the turbine with the wind direction is difficult for the following reasons. On standard turbines, wind speed and wind direction measurements take place behind the rotor. This leads to flow distortion caused by the rotor blades and the nacelle. To overcome this problem, several techniques to measure the free-stream wind in front of the turbine have been presented in different publications. The authors (Mamidipudi et al., 2011; Fleming et al., 2014; Scholbrock et al., 2015; Brown and Oldroyd, 2015) propose using an upwind-looking nacelle-mounted lidar. Pedersen et al. (2008, 2014, 2015) and Demurtas et al. (2016) present a method to measure the free wind speed, direction and turbulence intensity with three rotating sonic anemometers on the spinner nose. Bottasso and Riboldi (2014) estimate yaw misalignment and vertical shear from cyclic blade loads.

But for all these suggested methods new hardware is involved, causing additional financial investments. Lidar solutions need multiple beams to provide a wind vector because one beam can only measure the line of site information. All necessary laser beams need free-flow conditions, and a homogeneous flow assumption is usually made. One beam being in the wake of neighboring turbines and the other beams being in free flow leads to the wrong interpretation of the wind direction (Kapp, 2017). Furthermore, according to the findings of McKay et al. (2013), the yaw alignment behavior changes for turbines with wind vanes mounted on the nacelle metrological mast when operating in the wake of neighboring turbines.

Presuming that the nacelle-mounted measurement equipment is calibrated to ensure good alignment with the wind direction after commissioning or after a service interval, there are chances that the wind direction instrument is loosened, bent or warped over time and orientation drifts (Bromm et al., 2018).

Current methods to detect a change in the alignment of turbines are based on wind direction in situ comparison in case of redundancy of the device on the nacelle or by comparing wind direction measurements nearby (e.g., other turbines, met masts or lidars). The absolute wind directions of one or more wind vanes in the neighborhood are compared. This method works only if all true north alignments have been performed with high accuracy. Still there is a small risk of having the nacelle wind vane misalignment in the same magnitude but with different sign than the north misalignment; then both errors average out and the apparent wind direction will not be suspicious.

The purpose of this paper is to propose a new methodology to determine the best turbine alignment with standard nacelle-based equipment and monitor this alignment for the early detection of persistent yaw misalignment due to the vane falling out of calibration. We start with some basic information and definitions about wind turbine yaw systems and explain the experimental test setup and data handling in Sect. 2. In Sect. 3.1 we describe a methodology to improve the alignment of a reference wind turbine with an undisturbed reference wind speed measurement to obtain the maximum power production. In Sect. 3.2, we derive a model to correct the nacelle wind speed measurement behind the rotor in order to repeat the alignment improvement without an external wind measurement device (e.g., met mast or lidar). Section 3.3 describes a technique allowing standard turbine nacelle met mast equipment (e.g., two cup anemometers and one wind vane) to detect an alignment change of the wind direction measurement device during operation. Results and discussion of a demonstration case with a test wind turbine operating with different alignment offsets is provided in Sect. 4, followed by the conclusion in Sect. 5.

## 2.1 Yaw control

The true wind direction *ϑ* can be derived from the nacelle
position *ϕ* and the relative wind direction on the nacelle, referred to
as yaw error *γ*, with

The nacelle position *ϕ* is the angle between the rotor axis and the
marking of the true north as displayed in Fig. 1a. Turbines are aligned
to the true north after commissioning, but the equipment to measure the
rotation of the nacelle is often not accurate enough and leads to a drift of
the true north marking during operational lifetime
(Bromm et al., 2018). In consequence, old turbines in
the fleet may often not show a correct true wind direction since the wind
direction is derived from the sum of the yaw position and the yaw error
*γ* (Eq. 1). This is not a problem for the yaw controller using only
relative wind direction but for reanalysis purposes or sector management
these data need to be corrected.

A yaw controller needs to find a good balance between minimizing the average yaw error as soon as possible and staying within the design limits of the yaw component. To account for this, most controller algorithms can operate in different envelopes separated with a certain hysteresis. For instance, at lower wind speeds, at which in general higher turbulence intensities occur, larger yaw error angles are accepted for a longer period before the yaw system starts to move. At higher and more steady wind speeds, the accepted misalignments and periods are reduced to enable the turbine to react faster to wind direction changes. This behavior is site dependant and different settings for different locations can be favorable. With an increasing number of yaw position corrections and decreasing acceptance of yaw errors, the quality of the measured wind direction on the nacelle becomes more important.

## 2.2 Wind direction measured by nacelle anemometry

Most wind turbines are equipped with one, or in case of the need for
redundancy, a second wind direction measurement using a wind vane or sonic
anemometer to measure the orientation of the rotor with regards to the
incoming wind direction. In general the yaw controller is set up to adjust
the nacelle direction towards a specific measured average value *μ* from
the wind direction sensor. As the wind direction device is in the fixed
reference system on the nacelle, the average of the measured value *μ*
should converge to zero. In this way, negative values indicate inflow from
the left (looking upwind) and positive values from the right. To ensure
that the value 0^{∘} is measured when the wind direction device is in
line with the rotor axis, each instrument comes with a reference marking and
needs to be properly aligned with the reference orientation of the nacelle.
Some devices need to consider an assembly angle *φ*, when the
measured value zero is not in line with the rotor axis. In our example in
Fig. 1b, a good alignment is achieved for $\mathit{\phi}=\mathrm{0}{}^{\circ}$.

This task is challenging because manufacturing tolerances on the nacelle and its met tower can lead to some degrees of misalignment. Different optical techniques such as notch and bead sights or laser devices are used to measure the orientation of the instrument along the rotor axis of the turbine.

Figure 2 illustrates in a simplified manner how the
flow behind a rotor is skewed as function of the aerodynamic thrust and
torque and other disturbances attached to the nacelle roof. A wind direction
measurement *μ* obtained from behind the rotor needs to be corrected with
a flow deflection angle *δ*(*U*_{∞}Ω), which can be
assumed constant or take into account its relationship of the wind speed
*U*_{∞} or rotor speed Ω
(Kragh et al., 2013).

A well-aligned and flow-corrected wind direction measurement will then provide the measure of the yaw error with

where *μ* is the output of the wind direction instrument, *δ*(*U*_{∞}Ω) the flow deflection function and *φ*
the assembly angle. Because of the cyclical flow distortion of the rotating
blades, the wind direction measurement *μ* fluctuates a lot and an
averaging in the range of 10 s to 1 min is mandatory for the use of Eq. (2).

The yaw error vanishes if the flow is orientated perpendicular to the rotor
plane since the flow deflection *δ* is compensated for by the well-aligned
($\mathit{\phi}=\mathrm{0}{}^{\circ}$) but skewed measurement *μ*. Negative *γ*
values indicate flow from the left and positive values from the right. A
positive flow deflection angle *δ* provokes the turbine to face more
to the right when looking upwind.

A methodology to estimate the flow deflection function *δ*(*U*_{∞}Ω) based on a comparison between the true wind direction
measurement at the turbine and a nearby met mast or lidar includes further
stumbling caused by the accuracy of the true north alignment. An accurate
true north alignment of a met mast wind direction instrument at hub height
is difficult. The international standard for power curve verification
testing based on nacelle anemometry (IEC 61400-12-2, 2013)
provides a table with components of the combined uncertainty for a wind
direction signal from the turbine and a met mast adding up to approximately
±6^{∘}.

Figure 3 shows an example of the yaw error *γ* plotted against the wind speed corrected with a constant flow deflection
angle *δ*. The solid points with the black border represent wind speed
bin medians and the solid line is the overall mean. In this example the
median of the error varies depending on the mean wind speed between
−4 and +2^{∘}. Thus, the assumption of a constant flow
deflection angle *δ* has to accept a range of misalignments of at
least ±3^{∘}. Kragh et al. (2013) suggest improving this behavior with a linear correction as a
function of the rotor speed since the aerodynamic torque and the induced
wake rotation are correlated with it.

A simple method to correct the wind speed measurement on the nacelle, to be
representative for the free-flow wind speed in front of the turbine, is a
nacelle transfer function as described in IEC 61400-12-2 (2013).
This linear model uses a slope *β*_{1} and an offset *β*_{0} in
the form

Pedersen et al. (2013) introduce more advanced correction functions especially for spinner anemometry, including the induction factor as a function of the wind speed.

## 2.3 Experimental setup and data handling

A site with Senvion MM92 turbines (92 m rotor diameter, 2 MW rated power)
has been chosen to validate the new alignment method. The two test turbines
are installed on flat terrain in northeastern Germany. Figure 4 displays the
layout of the test setup and a frequency distribution of the wind speed by
wind direction of the recorded data. The turbines are 4.1 rotor diameters
apart from each other approximately perpendicular to the prevailing wind
direction. Meteorological data are recorded by a ground-based Leosphere
Windcube V2 lidar as averages of 1 min periods in a distance between 2.5
and 3.1 rotor diameters. In addition to the standard supervisory control and data acquisition (SCADA) data with 10 min average
periods, turbine data with a resolution of 1 Hz are available. Lidar and
turbine data are synchronized with GPS time and merged as averages of 1 min
periods. Downtimes and wind directions outside the free-flow sector are
filtered but yaw activities during the 1 min period are neglected. The
wind speed is corrected for air density as suggested in IEC
61400-12-1 (2017). Turbine WT1 has aerodynamically modified blades and is
used to demonstrate the method of the alignment for best power production. To
verify the monitoring ability of the method, two changes are made at Turbine
WT2 with standard blades. At the first change the wind vane is mechanically
aligned with notch and bead sights to ensure an assembly angle *φ*
equal to zero and the flow deflection angle *δ* is set to 0^{∘}.
The second change adds 7^{∘} to the flow deflection angle *δ* for
the rest of the test period.

## 2.4 Dynamic effects and time averaging

The standard wind turbine and environment condition data in the wind energy community are averages of 10 min periods. For the purpose of detecting the best operational orientation and to detect misalignments, it is beneficial to use a shorter averaging period. First, this gives more data points in total, and second, less averaging means less information gets lost.

In our demonstration in Sect. 4, we will use data averages of 1 min
periods as lidar data of this sampling rate are available.
Figure 5 provides the explanation why higher
sampling rate improves the evaluation of the best turbine alignment. The
blue dashed line is a density distribution of all available 10 min data
samples as a function of the yaw error. The turbine controller will turn the
nacelle towards 0^{∘} when certain conditions are fulfilled. This
leads to a high data density around 0^{∘}. High yaw errors (> 15^{∘}) are accepted from a load perspective for short
periods only. Averages of 10 min periods may contain such events, but the
mean value is closer to 0^{∘}. The effect of power loss for extreme
yaw errors is much higher (red solid line: power loss as a function of the
yaw error) and therefore it is beneficial to look at data with a higher
sampling rate to better capture this effect.

There are now two variables which are key for a turbine with a wind direction
device behind the rotor to operate at its optimum power production. First the
flow deflection angle *δ*(*U*_{∞},Ω)
and second the assembly angle *φ*. The three steps in the following
sections propose new methodologies to determine optimal values for these
variables and a way to monitor and detect changes.

## 3.1 Best turbine alignment

The best turbine alignment maximizes power production. This is not automatically the case when the turbine faces orthogonal wind at hub height. The change of wind direction with height is called wind veer and leads to different angles of attack depending on the rotor angle of the blades. Our basic hypothesis to optimally align the turbine includes deriving the effective yaw error from a polynomial fit of the active power or the rotor speed as a function of the yaw error.

The active power measurement or respectively the rotor speed needs to be
plotted against the yaw error *γ* (Fig. 6)
for small wind speed bins. The power or respectively the rotor speed drops
with increasing yaw error *γ*. A second-order polynomial fit with the
method of least squares gives a convex curve. The orientation of the best
production is revealed at its maximum. A deviation of the average yaw error
from zero is used to correct the flow deflection angle for the corresponding
wind speed bin (WSbin). Figure 6 also demonstrates
the advantage of using data with a higher resolution than the standard 10 min
SCADA data. Both data sets are simulated distributions of the variable
of interest with a mean yaw error of 0^{∘} and the standard
deviations derived from Fig. 5. The simulation
time is 1000 min. The 10 min averages (a) provoke less curvature at the
polynomial fit, which may lead to small deviations in the mean yaw error
interpretation. The higher data resolution (b) provides a more accurate
result, closer to the simulated 0^{∘} yaw error.

For this technique, it is essential to have a high-quality reference of the wind speed. In the selected analysis sector, the wind speed measurement must not be influenced by the wind direction nor the way the turbine is aligned with the wind. This type of measurement can be obtained from a met mast or lidar in simple and flat terrain with no interference from neighboring turbines, which is ideally the case for prototype measurement.

## 3.2 Flow correction for nacelle instruments

The determination of the best turbine alignment, as described in the previous section, works only with a wind speed measurement which is not influenced by the yaw error itself. Therefore it is of great value to be able to correct the flow distortion at the nacelle instruments to avoid the necessity of additional external devices such as lidar, met mast or spinner anemometers.

Figure 7 displays a typical nacelle met mast of a turbine with two cup anemometers for redundant wind speed measurements and one wind vane for the control of the yaw system as it is used for our two reference turbines. All devices are protected with a lightning rod on top and aviation lights installed on the bottom. For this configuration the yaw error influence on the wind speed measurements is plotted in Fig. 8. The wind speed difference between the lidar in the undisturbed sector and each nacelle cup anemometer is plotted against the yaw error for different wind speed bins of the lidar.

The colored lines are linear regressions with the method of least squares
representing the respective wind speed bin. Both anemometers show a negative
slope, which indicates a wind speed drop for positive yaw errors. A change of
the yaw error *γ* directly changes the angle of attack at the rotor
blade, with its maximum effect at vertical blade positions.

Looking at an airfoil cross section at the height of the cup anemometers
(Fig. 9), a negative yaw error (wind from the left looking upwind) decreases
the angle of attack and leads to a lower thrust coefficient *c*_{T}
as derived from the actuator disc theory with the axial induction factor *a* (Burton et al.,
2011):

The flow behind the rotor can be split into an axial and a tangential part
(Fig. 9). These two components are dependent on the wind speed and the rotor
speed and the flow deflection function *δ* can be described with

In consequence the wind speed at the nacelle met mast is a function of the
yaw error *γ* and with increasing wind speeds, when the maximum rotor
speed is reached, the magnitude of this effects changes. In order to be able
to apply the method of Sect. 3.1 with the nacelle wind speed device, we
correct the influence of the yaw error on the wind speed by introducing the
following first-order multilinear regression model with the two variables being
wind speed *v* and yaw error *γ*:

where the dependant variable *U*_{∞} is the free-flow wind speed in
front of the rotor, the first predictor variable *v* is the cup anemometer
wind speed, the second predictor variable *γ* is the yaw error, *β*_{1} and *β*_{2} are estimated regression coefficients, and *ε* is the residual.
With

and

we get

The coefficients are determined from the measured data of length *n* with
the method of least squares (ReliaSoft Corporation, 2015)
described with

with the design matrix **X**
and its transposed matrix **X**^{′}. *U*_{∞} are
the wind speed measurements in front of the rotor from the met mast or
lidar, for example. The fitted model provides a new wind speed ${\widehat{U}}_{\mathrm{\infty}}$, which
is much less influenced by the yaw error and can therefore be used in the
methodology described in Sect. 3.1.

## 3.3 Monitoring the flow equilibrium

The aforementioned methods to determine corrections for the nacelle anemometer and wind vane are only suitable in sectors without disturbance from neighboring turbines or obstacles. But most turbines are placed in wind farms and therefore a robust method to monitor and detect orientation changes to the wind direction measurement device is needed. Mandatory precondition of the following methodology is a nacelle met mast layout similar to the one in Fig. 7. The two cup anemometers need to be placed orthogonal and symmetrical to the rotor axis and need to be of the same kind. It is not necessary to have an individual wind tunnel calibration, but the operational characteristic has to be similar. A sensitivity study on the calibration uncertainty is given in Sect. 4.4.

The wind speed difference Δ between the two cup anemometers
*v*_{1} and *v*_{2} before the correction as described in Sect. 3.2 is
dependent on the yaw error *γ* and its magnitude changes with the
wind speed. The streamlines of a skewed flow behind the rotor (Fig. 2a)
pass different radial blade positions and with the amount of the tangential
flow component *a*^{′} greater than zero the wind speed difference
Δ changes for different wind speed bins. For the flow
characteristic drawn in Fig. 2b, the flow streamlines behind the rotor
are parallel to the rotor axis at the position of the met mast. Now the wind
vane is in line with the rotor axis and measures exactly the vane assembly
angle *μ*=*φ* (in our example 0^{∘}). We call this condition
flow equilibrium since the wind speed difference Δ is
approximately constant for different wind speed bins. The flow streamlines,
measured at the two cup anemometers *v*_{1} and *v*_{2}, pass
approximately the same radial rotor blade position and the tangential flow
component *a*^{′} is equal to zero (${a}^{\prime}=\mathrm{0}$). In contrast to Sect. 3.2
it is now necessary to use a single linear regression model without the yaw
error term *β*_{2}. This additional correction would lead to more or
less constant wind speed differences Δ for different wind
speed bins.

Let *i* and *j* be two different wind speed bins, then the linear regression models
can be described with

and

The linear regression model *j* is put on a level with the model *i*, which
can be resolved to

With *n* wind speed bins each linear regression line increases the number of
intersections *c* as follows:

From the number of intersections ${\mathit{\mu}}_{\mathrm{1}},\phantom{\rule{0.125em}{0ex}}\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{\mathit{\mu}}_{c}$ the minimum, maximum and standard deviation can provide a confidence interval. Changes to the alignment of the wind direction device can be detected by repeating this calculation in regular intervals and comparing the behavior of resulting flow equilibrium over time.

The methodology described in Sect. 3 is now tested at the two test
turbines from Sect. 2.3. Both turbines are equipped with a met mast layout
as shown in Fig. 7 and the data are recorded with consideration of a constant
flow deflection angle *δ*.

## 4.1 Best turbine alignment

A 3-month period of 1 Hz data from Turbine WT1 is aggregated to averages
of 1 min and merged with the lidar measurements. The data are filtered
for a sector with free-inflow conditions and grouped into bins of 1 ms^{−1}.

Figures 10 and 11 are an example of a well-aligned turbine with a constant flow deflection
angle *δ*. The turbine's active power and rotor speed are both
normalized with its rated value and are plotted against the yaw error
*γ*. The transparent dots represent averages of 1 min. The
solid dots are yaw error bin averages (bin size 1^{∘}) and for each
wind speed bin, differentiated with colors, a second-order polynomial fit
with the method of least squares provides a convex curve. The maximum of
this fitted curve is highlighted with a vertical bar and represents the
orientation of the turbine with the highest production.

In Figs. 10 and 11 the vertical lines for the different wind speed bins vary between
−2 and +2^{∘}. The wind speed bins 6 and 8 ms^{−1} are well aligned but for the other bins a small misalignment is
observed. A flow deflection function based on wind speed or rotor speed
could improve this behavior. A rough assessment of the impact on the annual
energy production (AEP) using a constant offset and a wind distribution with
Weibull parameter *A*=8 and *k*=2 and applying the cos(*γ*_{i})^{3} function as losses to the power of the respective wind speed
bins results in an overall loss of ∼ 0.1 % AEP.

At lower wind speeds (6 and 7 ms^{−1}), higher turbulence causes
more wind direction changes and to reduce wear of the yaw components, higher
yaw errors are accepted. The range of power per bin increases with higher
wind speeds because of the cubic relationship. In consequence the polynomial
fit becomes more curved with higher wind speeds except for the wind speed bin 9 ms^{−1} in which rated power is already achieved in some gusts. The solid
dots, representing the yaw error bin average, follow the polynomial curve
fit closely where enough data are available, but for the outer ridges the
scatter increases.

The second-order polynomial fit through each wind speed bin is more curved
for the rotor speed compared to the active power. This behavior helps to
allocate the best alignment, except for the wind speed bin 9 ms^{−1}
in which the maximum rotor speed is already reached.
Figure 12 shows the power as a function of the
rotor speed and helps to understand why using the rotor speed as an
independent variable provides improved fitting curves. Both variables are
normalized with its maximum value. The transparent dots are averages of 1 min
periods. The different colors indicate the wind speed bin (bin width of
1 ms^{−1}). The straight colored lines are linear regressions with the
method of least squares. For lower wind speed bins (e.g., 6 ± 1 ms^{−1}), rotor speed has a larger variation compared to the active power
and larger variation leads to a more curved fitting function. The variation
in the rotor speed decreases with increasing wind speed bin but the
relationship visualized with the linear regressions in Fig. 12 indicates an
increased slope, which again helps to obtain a fitted curve with more
bending.

## 4.2 Flow correction for nacelle instruments

The same aggregated and filtered data set from the previous section is now used to fit the first-order multilinear regression model described in Sect. 3.2. The model results in

and respectively with *v*_{2}

where $\widehat{v}$ is the undisturbed wind speed from the fitted model, *v*_{1}
and *v*_{2} are the wind speeds of the respective cup anemometers, and *γ* is the yaw error. The model's capability to correct the flow behind the rotor
is demonstrated by a comparison of Fig. 8 with Fig. 13. Both figures show
the wind speed difference between the lidar and the nacelle wind speeds
plotted against the yaw error. The transparent dots are averages of 1 min
periods and the horizontal lines are linear regressions with the method of
least squares for the respective wind speed bins.

Figure 8 indicates a strong influence of the yaw
error on the wind speed measurement at the nacelle instruments (with
standard correction Eq. 3), leading to an underpredicted wind speed for
inflow angles from the right looking upwind (positive yaw errors). In Fig. 8a
the wind speed drop from a −30 to a 30^{∘} yaw error is
in the range of 0.4 to 1 ms^{−1} and in Fig. 8b in a range from 1.4
to 2.6 ms^{−1}. With the new correction proposed in Sect. 3.2
this relationship becomes significantly reduced. The fitted model provides a
wind speed much less dependent on the yaw error. In Fig. 13 (a and b) the
wind speed drop from a −30 to 30^{∘} yaw error is in the
range of 0.03 to 0.5 ms^{−1}. Likewise the bias is reduced from
0.97 to 0.08 ms^{−1} for ${\widehat{v}}_{\mathrm{1}}$ and 0.67 to
0.09 ms^{−1} for ${\widehat{v}}_{\mathrm{2}}$. Figure 14
demonstrates the best turbine alignment method with wind speed bins of 1 ms^{−1} from the nacelle anemometers without the correction (a) and with
the correction (b) of the first-order multilinear regression model. As
demonstrated in the previous section this turbine is aligned with a constant
flow deflection angle *δ* but this fact can only be verified with the
new flow correction of nacelle instruments.

The uncorrected data in Fig. 14a gives the impression that the turbine is
misaligned by round about 10^{∘} or more to the left. With the new
flow correction of nacelle instruments applied, the estimated yaw errors are
within ±3^{∘}, centered around 0^{∘}. This variation
is deemed acceptable for using a constant flow deflection angle. The
correction of the wind speed bias also leads to lower rotor speed values per
wind speed bin.

Theoretically, the model (Eqs. 13 and 14) could now be transferred to turbines with the same flow characteristics, namely same rotor blade type and layout (e.g., sensor types and positions) of the nacelle metrology mast. A validation of this statement was not possible with the test setup due to different rotor blade modifications at the two turbines. An outlook for further improvement is a self-learning algorithm, which could be capable of using different flow deflection angles in different shear and veer conditions.

## 4.3 Monitoring the flow equilibrium

The flow equilibrium as described in Sect. 3.3 is calculated for two
different periods of 3 days each. In Fig. 15 data with averages of 1 min
are
used to calculate the difference between the two nacelle anemometers as a
function of the vane measurement *μ* for wind speed bins of 1 ms^{−1}.
For each wind speed bin a linear regression model is fitted and the points
of intersections between the straight lines are marked with thin black
vertical lines. The median of all vertical lines is a bold red bar and is
used to read the vane assembly angle *φ*. From the first period
displayed in Fig. 15a to the second period plotted in Fig. 15b the flow deflection
angle *δ* is changed in the control software from 0 to
7^{∘}. In Fig. 15a the flow equilibrium result is approximately
5.3^{∘} and in Fig. 15b it returns to a value of 0.7^{∘}. The
change of the flow equilibrium yields 6^{∘}. It is very likely that
a software change is not fully linearly translated into the real flow
conditions and the flow equilibrium provides a standard deviation which
covers the correct range of the expected 7^{∘} change. Therefore it
is assumed that the change is detected and the results are well within
sufficient accuracy.

The flow equilibrium method needs a measurement period with enough data in
at least two wind speed bins to ensure that for each of these wind speed bins a
linear regression model can be built and the 95 % confidence interval
does not overlap the regression line of the next bin. The intersection
between the models indicates the vane measurement *μ*=*φ* when the
flow is parallel to the rotor axis at the location of the anemometers. In
the demonstration case (Fig. 15) 3 days with averages of 1 min
periods is sufficient, leading to a data count per wind speed bin of 171–625 measurements. With more than two wind speed bins we obtain further
intersections. They can be used to provide a confidence interval. In our
example with four wind speed bins we obtain six intersections (Eq. 12). We use the median for the final result to reduce the influence of
outliers.

It is now of interest to see if the proposed monitoring method is capable of
detecting the two changes at Turbine WT2. Figure 16 demonstrates one possible
monitoring application with the flow equilibrium method. For the total
presented period only data with averages of 10 min are available. Each dot
represents an evaluation performed as for one of the plots in Fig. 15 with
1 month of data. The error bars indicate 1 standard deviation
added to or subtracted from the median. The horizontal transparent grey ribbon is a
90 % confidence interval (1.28 times standard deviation) of all points in
the plot. The vertical transparent rectangles mark the months in which the
changes are conducted. For the 2 months with interception the result is not
displayed because the data set consists of two different alignments. The
first 2 months were operated with a yaw error of approximately
−2^{∘}. In November the wind vane was mechanically aligned to achieve
$\mathit{\phi}=\mathrm{0}{}^{\circ}$ and the constant flow deflection angle was set to
$\mathit{\delta}=\mathrm{0}{}^{\circ}$. Since the second intervention in April the turbine is
operating with $\mathit{\delta}=\mathrm{7}{}^{\circ}$, which is seen as the best constant offset
for the optimal aligned turbine. A flow equilibrium (black dot) within the
range of ±1^{∘} is seen as accurate enough as the mean standard
deviation of all error bars is ±1.4^{∘}.

## 4.4 Practical guidance

In the following section we would like to give a brief summary of guidance
and experience about the use of the presented method. In general, at least
one of the anemometers is heated to avoid icing. In those instances, one
wind speed measurement will be biased in the case of icing. An ideal setup needs
two heated wind speed measurement devices. In our example, Turbine WT2 has
only one heated anemometer and therefore the measured temperature is
filtered to always be above 8 ^{∘}C.

A smaller wind speed bin size of for instance 0.5 ms^{−1} leads to a
higher number of intersections but reduces the number of measurements and
this leads to higher variations in the linear regression. In our experience
we obtained the best results with a bin size of 1 ms^{−1}.

The method is very sensitive to the instruments used. We only obtain good results when the same type of anemometer is used and the nacelle met mast layout is similar to Fig. 7. A mix of cup and sonic anemometer leads to wrong intersections. This opens the question about the calibration accuracy for the used anemometers. In our test case, the cup anemometers installed use the same factory provided calibration factors. Structural tolerances can lead to small differences in measurement results. For power curve verification tests, the calibration factors are usually determined in a wind tunnel for each device separately. To understand the impact of such tolerances, we have performed a sensitivity study. The evaluation in Fig. 15 is recalculated and the measurements of anemometer 1 are modified by adding an offset and in a second step by changing the slope of the linear correction function (Eq. 3).

Adding an offset on one anemometer changes only the ordinate, but the flow equilibrium delivers the same results. Different operational characteristics such as different slopes are more critical. In Fig. 17 the flow equilibrium is plotted versus the slope of the linear correction function of anemometer 1. The result from Fig. 15 is visible at the slope equal to one. With increasing slope, the flow equilibrium moves to the left. A negative yaw error would be expected.

A one percent slope difference between the anemometers leads to an error in
the flow equilibrium of approximately 1.8^{∘}. In case of poor
anemometer calibration, the assembly angle may be derived with a bias, but
the monitoring application of the value is still possible. A change in
operational characteristic over the life time of the anemometer has not been
considered.

The method is robust against swapping anemometer 1 and 2. The slope of the linear regression line changes sign, but the result for the flow equilibrium is the same.

With this new method, two mayor advantages are achieved, compared to the marked solutions mentioned in the introduction. Firstly, the turbine alignment is not relying on multiple wind direction measurements which reduces uncertainties caused by the challenge of finding true north. And secondly the permanent monitoring application works without the need of additional hardware which reduces the costs for operators.

This paper presented and demonstrated a three-step methodology to determine the best turbine alignment with standard nacelle-based equipment and monitor this alignment for an early detection of persistent yaw misalignment due to the vane falling out of calibration.

Step one uses a reference device (e.g., met mast or lidar) to find a
correction for the wind vane to achieve the optimal power production. A
constant flow deflection angle will accept yaw errors within an accuracy of
±3^{∘}, a correction as a function of wind speed or rotor
speed could reduce the average yaw error. With a higher sampling rate of averages of 1 min periods instead of the standard 10 min
average SCADA data, more extreme yaw errors can be recorded and the effect
of power and rotor speed reduction is more pronounced. The use of the rotor
speed instead of the power itself helps to get more curvature at the
polynomial fit which is used to detect the alignment with the highest power
performance. The result is independent of errors easily introduced by true
north alignments of any wind direction measurement device and the
optimization aims for the maximum power production which is not necessarily
the case when the wind is perpendicular to the rotor plane at hub height.

Step two presents a first-order multilinear regression model to correct the flow distortion behind the rotor and enables standard turbine equipment to apply the methodology of step one without any external measurement device. Without such correction, the wind speed measurement at the nacelle is strongly influenced by the yaw error and leads to an over prediction for negative yaw errors or an under prediction for positive yaw errors.

Finally in step three the assembly angle of the wind vane is determined from
SCADA data by means of a flow equilibrium using two cup anemometers of the
same kind on a nacelle met mast. This is of great value as it can be used
for monitoring purposes during lifetime operation with no need of additional
equipment. Requirement for the metrology mast of the turbine are two heated
anemometers of the same kind and one wind direction device. The operational
characteristic of the two anemometers has to be the same. An offset between
the two devices does not change the result, but a linear slope of more than
1 % leads to an offset in the derived flow equilibrium of approximately
1.8^{∘}.

In our demonstration of the monitoring methodology, two orientation changes
could be detected after three days of operation with an accuracy of ±1.4^{∘}. We could show, that standard turbine equipment is able to
align a turbine with sufficient accuracy and changes to its alignment can be
detected in a reasonable short time which helps to minimize losses.

Data are not available to the public.

The authors declare that they have no conflict of interest.

We would like to thank Senvion GmbH for providing the data and time to
investigate this topic. Moreover we are very grateful for all colleagues
who helped and contributed with fruitful discussions. Furthermore,
acknowledgement is given to the R Core Team for developing the
open-source language R (R_Core_Team, 2015).

Edited by: Gerard J. W. van Bussel

Reviewed by: two anonymous referees

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