The wind field leaves its fingerprint on the rotor response. This fact can be exploited by using the rotor as a sensor: by looking at the rotor response, in the present case in terms of blade loads, one may infer the wind characteristics. This paper describes a wind state observer that estimates four wind parameters, namely the vertical and horizontal shears and the yaw and upflow misalignment angles, from out-of-plane and in-plane blade bending moments. The resulting observer provides on-rotor wind inflow characteristics that can be exploited for wind turbine and wind farm control. The proposed formulation is evaluated through extensive numerical simulations in turbulent and nonturbulent wind conditions using a high-fidelity aeroservoelastic model of a multi-MW wind turbine.
The wind blowing over a wind turbine rotor leaves its own specific fingerprint on the machine response. If this information is rich enough and if the wind turbine response can be measured (for example in terms of loads), then one may think of turning the rotor into a wind sensor and use it to infer the wind inflow.
Measurements of the rotor inflow during operation are attractive for a number of
reasons, as they may find a wide range of applications. For example, information on
the wind speed at the rotor disk is typically useful for wind turbine control, as
controller behavior is often scheduled as a function of wind speed. In addition,
knowledge of the wind direction with respect to the rotor is necessary not only to
maximize energy harvesting, but also because operating with excessive misalignment
increases loading. Wake redirection strategies
Unfortunately, high-quality information on wind inflow is generally difficult to obtain. Onboard wind turbines, wind speed is typically measured by cup or sonic anemometers, while direction is provided by wind vanes. These sensors invariably suffer from a number of disturbances, such as the presence of the nacelle, blade passing and wake-induced flow deformation. Although most of these effects can be mitigated by the use of calibrated transfer functions, filtering and ad hoc processing of the raw measurements, all these sensors provide only local information at the specific point in the flow where they are installed. For control applications, it is clear that rotor-equivalent information is generally more appropriate than local data because what determines the overall rotor response is what is felt by the whole rotor rather than what takes place at a specific point. Additionally, certain wind characteristics can only be defined over a rotor disk and do not have pointwise equivalents, for example shears. Met masts, being equipped with multiple wind sensors away from the rotor, do not suffer from some of these issues. However, the problem of mapping the information from a met mast to the rotor disk of a wind turbine is generally very difficult to solve, and it will clearly always be prone to possibly severe inaccuracies. With lidar (light detection and ranging), laser-based sensing technology is rapidly becoming a game changer, and other remote sensing solutions are also very promising. While their potential is clearly very real and will probably have a deep impact on wind energy technology, these devices are still not in widespread use, mostly because of cost, reliability, availability and lifetime issues.
In this scenario, wind sensing by using the rotor response seems to offer an attractive alternative. In fact, any wind property estimated from the rotor response will be nonlocal and rotor effective in contrast to local sensors. In addition, this approach provides measurements directly at the rotor disk, avoiding the need for mapping flow characteristics from one point to another.
The rotor-effective wind speed estimator
This idea was first generalized by
A much simpler approach was developed later in
Following the idea described in
The concept of the wind turbine as a wind sensor was recently extended to the
detection of wake impingement in
Motivated by the very promising validation results both in simulations and in the
field, the present paper extends and improves the formulation of
First, extensive numerical experiments have shown that the load–wind model on which the estimator is based must consider at least four wind states instead of two, i.e., the two yaw misalignment and upflow angles as well as the two horizontal and vertical shears. These four states, together with the mean rotor-equivalent speed, represent the lowest-order full approximation of the wind inflow at the rotor disk: the two angles give the orientation of the mean wind vector with the rotor axis, while mean speed and the two shears describe a tilted planar (or mixed linear–exponential, depending on the type of shears considered) inflow. All of these states leave significant signatures in the low-frequency response of the rotor. Therefore, failure to include one of them in the model will invariably create inaccuracies in the others.
Second, the paper shows that the estimators of these four states should be limited to
the use of the
Third, the paper compares both a linear and a nonlinear (quadratic) load–wind model. Both models are scheduled with respect to wind speed in order to account for the different characteristics of a wind turbine in its wind speed operating range. Numerical experiments show that the two are very similar in performance, with a small improvement in accuracy for the nonlinear model over the linear one.
Fourth, experience has shown that angles (yaw misalignment and upflow) are significantly more difficult to estimate than shears. The paper explains the reason for this behavior from two different perspectives. From a mathematical point of view, an a priori analysis based on the singular value decomposition (SVD) demonstrates that angles have a lower level of observability than shears, implying that any small error or perturbation (in the model, in the measurements, in the numerical solution, etc.) will be significantly amplified during the model inversion necessary for the estimation of the wind states. From a physical point of view, this is also easily explained in terms of the sensitivity of angle of attack changes at the blade section to wind state changes. As angles of attack (and hence loads) change less in response to angle changes than to shear changes, angles are harder to estimate than shears when looking at rotor loads.
Finally, the paper demonstrates the performance of the estimator through extensive numerical simulations performed with a high-fidelity aeroservoelastic model of a multi-MW wind turbine. The numerical results illustrate the excellent ability of the proposed formulation to follow rapid fluctuations in shears. The same results also show very interesting behavior of the angle estimators. In fact, although angle estimates are indeed generally polluted by oscillations that depend on turbulence level, their mean errors are significantly low. An analysis that considers the probability distributions of wind speed and turbulence intensity at a given site shows that the expected average inaccuracy of the angle estimates is remarkably low, i.e., less than 1 degree. This means that angles, although apparently oscillatory on short time horizons, can be followed quite precisely in their mean value changes.
The paper is organized according to the following plan.
Section
The development of the proposed wind inflow observer is inspired by the idea of using the wind turbine as an anemometer. In this sense, wind is not only the source of energy to be harvested, but also one of the principal factors affecting the wind turbine response. Specifically, the present observer is based on the lowest load harmonics. Although other response indicators could be used in principle, for example accelerations, loads are considered in this work because they are now often measured onboard modern large wind turbines for enabling load feedback control, and load sensors will probably be standard equipment available on most future machines.
In order to understand the connection between blade loads and wind characteristics,
consider two different constant-in-time wind fields. A first wind field is
axially symmetric with respect to the rotation axis of the wind turbine rotor, while
the second is not in magnitude or direction. In the second – anisotropic –
case, differences in speed and/or direction over the rotor disk may be due to wind
shears (both vertical and horizontal) and/or misalignments with the wind direction
(due to both yawed flow and upflow caused by rotor uptilt, terrain orography, etc.).
In the axially symmetric case, the angle of attack experienced by the blade sections
during their azimuthal travel over the rotor disk will be constant; hence, the
resulting aerodynamic loads will also be constant. In the non-axially symmetric
case, any anisotropy in the wind will cause periodic fluctuations in the angle of
attack at the blade sections and hence periodic loads. The amplitude and phase of such
loads will depend on the wind field at the rotor disk and on the aeroelastic
characteristics of the rotor blades. Therefore, the amplitude and phase of the periodic
loads carry information on the wind anisotropy at the rotor disk. This fact can be
readily verified with simplified mathematical models of a rotating blade in an
anisotropic wind field, for example the classical flapping and lagging blade
model developed in
In this work, the wind field anisotropy is parameterized using four variables
(termed wind states in the following): the vertical shear exponent
The wind states are defined with respect to a nacelle-attached frame of reference
with origin at the hub made of three mutually orthogonal unit vectors
Definition of the four wind states used for parameterizing the wind field over the rotor disk.
Two different wind fields are considered in the following. In the
fully parameterized case, the wind field is completely defined at each instant
in
time by
Under the effects of a steady anisotropic wind, the response of a stable wind
turbine converges to a periodic motion. In such a regime, a generic blade load
Both out-of-plane (superscript
The formulation of a wind state observer necessitates a model expressing the
dependency of the loads on the wind conditions, in particular of the load
harmonics
Under a steady input
This approach leads to a white box model, i.e., a model using analytical
formulas to express relationships among the relevant variables based on physical
principles
In this paper, a third approach is used, which is entirely based on system
identification. In this case, the desired input–output relationship between loads
and wind states is considered as a black box
The data set for the identification of the black box model can be obtained either by
simulation or by measurements performed in the field. The former approach, which is
also the one that was used for the present work, is relatively simple because
in a simulation environment one can readily measure all necessary quantities
(loads and wind states). In contrast to this simplicity, it is clear that
any mismatch between the simulation model and reality will affect the quality
of the identified input–output model. While this is in principle a possible
drawback, one should not forget that the present approach only uses the very lowest
harmonics (typically only the
Another possible approach is to use field measurements. In this case the machine
should be equipped with load sensors, a met mast and a lidar or other flow
sensors to measure wind states. Each of these techniques implies its own hypothesis
(e.g., frozen turbulence in the case of flow measurements performed away from the
rotor disk), each is limited by its own specific inherent accuracy and each is
affected by errors and disturbances. While this approach is certainly possible and
was in fact successfully demonstrated in
Inspired by Eq. (
Matrix
Separating the effects of gravity from aerodynamic-induced loads allows for the correction of air density changes. This is important in practice because density, being dependent on temperature, undergoes significant fluctuations in the field, thereby affecting load measurements. The split of gravity-induced terms into constant and aerodynamically caused terms is also important, as it highlights the variability of the latter term with the operating condition.
The unknown matrix of coefficients
As previously noted, the input–output model should be scheduled in terms of the wind
speed
Behavior of two load sensitivities as functions of wind speed
Samples of the wind states and associated loads are now collected at
An example of the typical behavior of the model coefficients is given in
Fig.
The assumption of linearity in the input–output
relationship (Eq.
Comparison between measured and predicted harmonics for the linear
model (dashed thick lines) and the nonlinear model of order 2 (solid thin lines).
Normalized
In this work, an aeroservoelastic simulation model is used to represent the
dynamic behavior of a wind turbine in all different scenarios of interest. The model
represents a horizontal axis wind turbine with a rotor of 93 m diameter with an
uptilt of 4.5
The aeroservoelastic model of the machine is developed using the finite-element
multi-body code
Comparison between measured and predicted harmonics for the
nonlinear model of order 2 (solid thin lines). Normalized out-of-plane
To test the performance of the linear and nonlinear models, the wind turbine was
simulated in a variety of different operating conditions. Fully parameterized steady
winds were generated at
From the full range of tests performed, Fig.
The figure shows that both models are capable of capturing the relevant behavior of
the harmonics with respect to wind states. The relationships appear to be linear,
with only very minor nonlinearities. These analyses also graphically illustrate the
sensitivity of the loads with respect to the wind parameters. As expected, even
though all parameters have a certain effect on all loads, cosine harmonics are
mainly influenced by the couple
In contrast, the
The previous analysis performed in steady wind conditions showed that the
To investigate these effects, a 10 min simulation was performed at a 5 m s
Comparison in turbulent wind conditions between measured harmonics
(thick solid line) and harmonics predicted with a second-order nonlinear model
(thin solid line).
By looking at Fig. 5a, it appears
that there is an excellent match between the predictions and measurements for the
in-plane
In contrast, Fig. 5b shows completely different
behavior of the measurements and predictions for the
The problem of computing an estimate
The generalized least-squares estimate of
Wind states observed using the linear model for different steady
inflow conditions at 5 m s
Wind state observed with the nonlinear model for different steady
inflow conditions at 5 m s
For the nonlinear model (Eq.
The estimator (Eq.
Figure
The match improves with the use of the nonlinear model, as reported in
Fig.
The observability of the wind parameters is analyzed next. As one can easily imagine, the level of accuracy of the estimates strongly depends on the sensitivity of the moments with respect to the to-be-estimated parameters and the noise in the measurements.
Assuming a linear model, the real (unknown) wind state vector
The covariance Cov
This analysis also provides information on the dependence of loads on wind states.
In fact, one can easily show that
The same analysis can be applied to the nonlinear case by linearizing
Eq. (
The a priori analysis was applied to the identified input–output model. Three
different values of the noise covariance
Expected standard deviation of the wind state estimates as
a function of wind speed.
For the first case, matrices
The first and second columns of
To interpret matrix
As a side observation, also note the symmetry between the couples
Table
Expected variance of wind state estimates based on the a priori analysis.
Finally, Fig.
Similar results not shown here for the sake of brevity were obtained with the nonlinear model.
Given the behavior of the linear and nonlinear observers and the results of the
SVD-based a priori observability analysis, the following conclusions can be
made.
In general, it should be possible to estimate both shears with satisfactory
precision, as their errors are moderate even for significant measurement noise
levels. It is expected that the estimation of both yaw misalignment and upflow angle
will be more significantly affected by measurement noise. Because of this, the
estimation of these angles should be accompanied by a suitable filtering
action in order to remove fast fluctuations. This also implies that these
angles can only be estimated on longer time horizons than in the case of
shears. The observation accuracy should increase with increasing wind speed. The nonlinear model appears to be more accurate than the linear model for the
estimation of yaw misalignment and upflow angles. However, shears
also seem to be captured well by the linear model.
The different expected accuracy in the estimation of shears and angles can be given
an even more intuitive explanation. Consider the blade section depicted in
Fig.
Effects of shear and misalignment changes on sectional angle of attack.
A change in shear will be seen by the blade section mainly as a change in
This is also easily shown by considering that the inflow angle is
Variation in the inflow angle
After having verified in the previous sections that blade load harmonics carry enough information to infer wind states in steady conditions, attention is now turned to the dynamic problem. The nonturbulent case is considered first by using fully parameterized wind fields with variable-in-time wind states. Next, the turbulent case is considered by using wind fields modeled by the Kaimal method for different constant mean wind states. Finally, turbulent conditions with variable-in-time mean quantities are considered.
Ideal nonturbulent and fully parameterized wind fields with time-varying wind
states were generated according to Eq. (
Figure 11a and b show the results obtained at 4 and
9 m s
Wind state observations in nonturbulent wind conditions with variable wind
parameters at 4
Wind state observations in turbulent wind conditions at 7 m s
Wind state observations in turbulent wind conditions at 19 m s
Standard deviation of the estimation error of yaw misalignment
Different turbulent wind fields were generated using the
As wind states are inferred from blade loads, which in turn depend on the wind
conditions at the location occupied by each single blade at each time instant,
an alternative way of computing the reference wind conditions was also used. In this
second implementation, wind parameters were computed by fitting the wind state
parameterization expressed by Eqs. (
Figures
These results suggest several possible considerations.
First, the estimates of both shears
The good behavior of the shear estimates suggests the possible use of a faster
filter in order to reduce the estimation delay. For example, the delay can be
reduced to only 4 s by using a filter cut-out frequency of 0.17 Hz, which
corresponds to 1.2 times the rotor frequency at 5 m s
An estimation of the angles
The general lower quality of the estimates for the angles was previously explained
by the a priori analysis, and it is clearly illustrated a posteriori by the
simulation results shown here. Various sources of error may ultimately be
responsible for the oscillations in the estimates shown by the plots, including
unmodeled dynamics, rapid pitch motions or variable rotor speed. It is interesting
to recall that the steady model (Eq.
It should also be remarked that an additional source of uncertainty is the ground truth. In fact, the presence of turbulent eddies in the flow implies that the wind field cannot be exactly parameterized by the assumed wind states. Hence, the reference quantities plotted here should also be considered only as indicative proxies of the actual wind states.
The observation errors were further analyzed from a statistical standpoint by generating five different 10 min turbulent wind field realizations and computing means and standard deviations. To eliminate the effects of the delay caused by the filter, which would have prevented any instantaneous comparison between the reference and observed quantities, reference wind states were processed with the same filter used for the moment harmonics.
Figure
As expected, the standard deviation increases with the TI level. Moreover, in regions II
and II
Shear errors also remain low at very high TI levels, as
illustrated by Fig. 14b and d, indicating that
fast, good-quality shear estimates are possible. In fact, for example, the standard
deviation of
The evaluation of the observer performance for the angles deserves special
attention. Looking at the yaw misalignment in Fig. 14a for regions II, II
Figure
Mean estimation error of yaw misalignment
The previous examples have shown that observed angles are typically affected by
spurious oscillations for the reasons explained by the a priori analysis. The same
examples, however, have also shown that mean values are typically well captured and
that the amplitude of oscillations is related to TI. This seems to indicate that
fast accurate observations of angles are generally not possible, while observations
on longer time windows might still be relatively accurate. By simple inspection
of the temporal responses, it is, however, not easy to get a clear idea of the actual
precision of the observers in turbulent conditions. In order to provide a more
meaningful indication of the observer accuracy, the “lifetime” standard deviation
of the observed states is evaluated in this section. This is computed by weighting
the results at each wind speed and TI with the corresponding probability
distributions at a given site. To this end, measurements taken at the offshore
platform FINO1
Minimum value, 10th, 25th, 50th,
75th and 90th TI percentiles as
functions of wind speed at 80 m above the water line at FINO1 from September
2003 to August 2007. Data taken from
Figure
Next, a shifted Weibull probability density function (PDF)
TI PDF
Wind-speed-specific standard deviation of the observation error for angles
Given the probability density function of the observation error
Figure
Finally, the lifetime standard deviations are reported in
Table
Lifetime
standard deviation and 2
The fact that the mean estimation errors of the angles, especially for yaw misalignment, are limited suggests the use of a moving average in order to lower the error standard deviation. This way one may capture the slower variations in the means while filtering out the faster oscillations. The resulting estimates can be used for slower control actions, for example yaw control, or for the slow scale monitoring of parameters of interest.
To test whether it is indeed possible to follow changes in the mean, large changes
in yaw misalignment were simulated. Turbulent wind fields were generated with
Yaw misalignment for an 8
For the very low TI levels shown in Fig. 19a, both the mean and instantaneous values of yaw misalignment can be sufficiently well captured even without the use of a filter. With increasing turbulence, spurious oscillations of the estimates mask the mean wind direction change. However, it appears that the use of a moving average is capable of eliminating the faster fluctuations, revealing the presence of a change in wind direction. Clearly, higher values of turbulence require longer filtering windows with consequently longer time delays. This delayed detection is, however, compatible with the usually rather slow and conservative approach used for yaw control in which the actual realignment of the machine is performed only when a wind direction change of some significant entity has been observed for a sufficiently long window of time, usually many tens of seconds.
As a final remark, the nonlinear observer appears to perform slightly better than the linear one, as it is more easily visible for low turbulence conditions.
This paper has presented a method to estimate the wind inflow at the rotor disk of
an operating wind turbine. The proposed method uses the low-frequency response of
the wind turbine limited to the
An input–output model was formulated to represent the relationship between wind states and load harmonics. The model was treated as a black box with unknown coefficients estimated by using the simulated response of a wind turbine implemented in a high-fidelity aeroservoelastic model. The input–output relationship was then inverted in a least-squares sense in order to provide estimates of the wind states when fed with measured load harmonics. The statistical properties of the model and, in turn, the observability of the wind states were analyzed using the SVD. This a priori analysis highlighted the different nature of the problem of estimating shears and angles, the former being characterized by a higher level of observability than the latter. Finally, the proposed observer was analyzed in a wide range of operating conditions in turbulent wind fields with different characteristics.
From the results of the present study, the following conclusions can be made.
The behavior of the blade out-of-plane and in-plane load harmonics at
It is not advisable to include in the model harmonics that are higher than
Wind states can be estimated from An a priori observability analysis shows that the accuracy of the shears is
generally superior to that of the angles. This is not because of a limit
to the present specific formulation, but it is due to the intrinsic
sensitivity of angle of attack changes to wind state changes, which is
different for angles and shears. Extensive simulations in turbulent conditions have shown that the
mean value of the estimation error is generally significantly low for
all states. For example, the mean yaw error is about 0.5 Standard deviations of the shears are generally very low even for
high TI levels, implying that the observer is capable of following fast
shear fluctuations with good precision. Standard deviations for angles are significantly higher due to their
overall lower observability. In general, angle estimates are polluted
by rapid spurious oscillations due to the amplification of errors
through the inverted estimation model. This implies that one cannot
generally follow rapid variations in the angles, and only observations on
longer timescales are possible. Although polluted by fluctuations, on average even the angle estimates are
of good quality thanks to their small mean errors. An analysis, conducted by taking into account the probability distributions
of both wind speed and TI at the offshore FINO1 platform in the German Bight,
has shown that the expected standard deviation of the estimation error in the
angles is below 1 It was shown that, by filtering the estimated yaw misalignment with a moving
average, one may track with good accuracy significant mean changes in the wind
direction even for very high TI, indicating the possible use of this estimate
to drive the wind turbine yaw control system.
The proposed formulation should be extended to consider the possible presence of an individual pitch control (IPC) strategy. This can be done by also including in the load–wind model the presence of a term depending on pitch load harmonics. As these quantities are known, they represent further inputs that do not change the overall approach, although the model will have additional coefficients that need to be identified. This extension of the formulation has already been tested, and it will be described in a forthcoming publication.
Data can be obtained upon request from the authors (carlo.bottasso@tum.de.).
The authors declare that they have no conflict of interest.This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding program Open Access Publishing. Edited by: Sandrine Aubrun Reviewed by: two anonymous referees