In this work, a new algorithm is presented to correct for pitch misalignment imbalances of wind turbine rotors. The method uses signals measured in the fixed frame of the machine, typically in the form of accelerations or loads. The amplitude of the one per revolution signal harmonic is used to quantify the imbalance, while its phase is used to locate the unbalanced blade(s). The near linearity of the unknown relationship between harmonic amplitude and pitch misalignment is used to derive a simple algorithm that iteratively rebalances the rotor. This operation is conducted while the machine is in operation, without the need for shutting it down. The method is not only applicable to the case of a single misaligned blade, but also to the generic case of multiple concurrent imbalances. Apart from the availability of acceleration or load sensors, the method requires the ability of the rotor blades to be commanded independently from one another, which is typically possible on many modern machines. The new method is demonstrated in a realistic simulation environment using an aeroservoelastic wind turbine model in a variety of wind and operating conditions.

The pitch system has the highest failure rate of all wind turbine components

Irrespective of the specific type of fault, a pitch imbalance will have as
a direct consequence not only a possible decrease in harvested energy but,
most importantly, also an increased level of vibrations and rotor speed
fluctuations

Currently, the downtime related to pitch failures is relatively high

Imbalance detection and correction techniques have been developed both in the
literature and in practical applications. For example,

Ad hoc controllers have also been formulated to correct for rotor imbalances

The analysis of signals such as loads and accelerations measured on the wind
turbine fixed frame provides a way to determine if a rotor is unbalanced.
In fact, it is well known that the amplitude of the 1P (once per revolution)
harmonic is an indicator of an unbalanced rotor. Recently, it was shown that
the phase of that same harmonic can be used to identify the unbalanced
blade(s)

In the present work, the same concept is used to automatically rebalance an
unbalanced rotor. In a nutshell, the method works as follows. First, an
unknown linear relationship is assumed between pitch setting of the blades
and the 1P amplitude of a signal measured in the fixed frame. Exploiting the
radial symmetry of a rotor, the coefficients of the linear relationship are
reduced to only two. In addition, this also has the effect of including the
phase information in the model, which eventually allows one to correctly
identify the pitch misalignment of each blade. Since the linear
imbalance–disturbance model is determined by two parameters, one single
additional measurement (in addition to the one performed on the currently
unbalanced configuration) is necessary to identify the unknown
imbalance–disturbance relationship. This is easily achieved by pitching the
blades by some amount and measuring the resulting 1P amplitude. Once the
linear relationship is known, it is trivial to compute the blade pitch offset
that, by zeroing the 1P amplitude, balances the rotor. To account for
possible small nonlinearities, the procedure can be iterated a few times as
necessary. A similar approach was presented in

The paper is organized as follows. Section

In a balanced rotor with

On the other hand, when an imbalance is present, other harmonic components can be detected in fixed-frame measurements, the most prominent typically being the 1P harmonic. Hence, detection and correction of rotor imbalances can be based on the analysis of the 1P harmonic measured in the fixed frame.

As an example, consider the measurement of nacelle fore–aft accelerations,
which are primarily caused by fluctuations in the rotor thrust. The thrust
force

Thrust force

The shear force of the generic

Assuming a periodic response, the harmonic amplitudes are the same for the
various blades; i.e.,

intermediate harmonics also pollute the spectrum, and

the phase of these harmonics indicates the unbalanced blade.

To better understand the effects of a pitch imbalance, the expression for the
aerodynamic contribution to the shear in a blade can be worked out
analytically. Following the approach of

Assuming a pitch misalignment

A word of caution is due in the interpretation of these analytical results. First of all, this analysis is based on the sole thrust force, while terms other than the cross-flow contribute to the 1P harmonic when considering yawing and nodding moments. In addition, the model is the simplest possible using one single degree of freedom and including various simplifications in the derivations. Nevertheless, the model is at least useful in qualitatively understanding the basic mechanisms by which fixed-frame vibrations are caused in an imbalanced rotor. After having served its purpose, the analytical model is dropped from the rest of the paper, the further developments of which are not based on it.

In this work, an imbalance–disturbance model is assumed in the form

Note that the assumed imbalance–disturbance model implies a linear relationship between the pitch misalignment of the blades and the 1P harmonic component of a measured response signal (acceleration or load) in the fixed frame. As shown later on, this assumption is not a limitation of the model because in fact the model can be iteratively identified as the rotor is rebalanced, thus effectively removing the linearity hypothesis. However, linearity is confirmed by the previously derived simple analytical model, and it is indeed generally also observed in extensive numerical simulations conducted by using state-of-the-art aeroelastic models.

Since it is nearly impossible to guarantee that the whole model
identification and rebalancing procedure will be conducted in exactly the
same wind conditions, it is important to reduce the dependency of the model
on the operating point. To this end, the harmonic amplitude vector

To simplify the identification of the model coefficients, the radial symmetry
of the rotor can be exploited. Assuming a periodic response, the effects of a
misalignment in the second blade will be the same as those caused by a
misalignment in the first blade, but shifted by

The imbalance–disturbance model might be affected by proximity to resonant conditions or by the presence of vibration control algorithms implemented onboard the turbine control system. The first problem is readily addressed by avoiding identifying the model and rebalancing the machine in the proximity of resonant conditions, which is easily done since these are typically well known. The second problem might require switching off these additional control loops during identification and rebalancing, although no general statements are possible here and the situation would have to be analyzed in detail for any specific implementation of such algorithms.

Before computing the pitch adjustments that rebalance the rotor, one needs to
identify the unknown coefficients in Eq. (

At the beginning of the procedure, one has not yet adjusted the rotor pitch,
and hence

Now that Eq. (

By appending the zero-collective constraint to the imbalance–disturbance
model, one gets

Blades are now pitched by

Graphical representation of the rotor rebalancing algorithm.

Inspecting the values of the computed pitch adjustments

In this work, the proposed rebalancing procedure is demonstrated with the
help of aeroservoelastic simulations of a 3 MW horizontal axis wind turbine.
The machine, characterized by an 80 m hub height and a rotor diameter of 93 m,
has cut-in, rated and cut-out speeds equal to 3, 12.5 and 25 m s

Different combinations of initial pitch misalignments in the range

Accelerometers are placed on the machine main bearing, with the goal of measuring the fixed-frame response of the system, and they are simulated in the mathematical model including the effects of measurement noise. Various tests were conducted in order to identify an optimal accelerometer configuration. Typically, the best results were obtained when two accelerometers are located to the two sides of the main bearing and spaced as far as possible from each other. The two accelerometer signals are subtracted one from the other, yielding a differential measurement proportional to the yawing accelerations of the rotor.

The model described in Sect.

Figure

Cosine (squares) and sine (circles) 1P components of the main-bearing scaled differential acceleration as functions of pitch misalignment.

It is interesting to observe that the misalignment of each different blade leaves a unique fingerprint on the measured signal. This means that the linear model not only contains information on the severity of the misalignment, but also on where the misalignment is located.

Next, the performance of the proposed algorithm is tested in a variety of
different wind conditions. The model expressed by
Eq. (

Figure

Residual pitch misalignment as a function of the number of steps for given wind speed and turbulence intensity, but variable conditions according to series A through D.

In the figure, the abscissa represents the various steps of the procedure. At
the beginning (step 0), a 1P acceleration is measured in the fixed frame.
Next, one or more blades are randomly pitched (step 1), while keeping the
collective constant. In the resulting new configuration, a new 1P
acceleration is measured. Since this step is random, the unbalance of the
blades may worsen in this first step. The algorithm is now applied by first
identifying the model and then computing the pitch adjustment

The figure shows that the proposed algorithm is capable of rebalancing the rotor in a very small number of steps, typically ranging between three and four. It should be noted that during each one of these steps, the machine is operating in markedly different operating conditions, as described by the series reported in the Appendix. Notwithstanding these very significant operational changes, the procedure seems to be quite robust.

An important remark is due at this point. As wind conditions may change from one step to the next, in general it is not possible to guarantee that the imbalance will always diminish at each step of the algorithm. Indeed, some of the following numerical experiments show that the imbalance may occasionally increase. However, this happens only in the case of radical changes in wind conditions from one step to the next. It would be relatively straightforward to avoid such situations by implementing some simple logic in the procedure. For example, one might monitor the operating parameters and continue with rebalancing only when changes do not exceed a certain threshold. In addition, if one observes an increase in the 1P harmonic amplitude after a rebalancing step, then that step might be rejected and the blades could be pitched back to their previous setting. To consider a worst case scenario, in all numerical experiments presented here these simple precautions were not taken. Therefore, the algorithm was forced to continue irrespective of the severity of operating changes. Because of this, the results show occasional increases in the imbalance throughout the iterations. Nevertheless, these same results also show that the algorithm was always eventually able to successfully rebalance the rotor in a very small number of steps.

Figure

Residual pitch misalignment as a function of the number of steps for given turbulence intensity and variable wind speed, but variable conditions according to series E and F.

The effects of noise on the measurement of the accelerations driving the algorithm were then investigated. In fact, small imbalances induce only small 1P harmonics in the fixed frame so that the effects of noise on the measurements can be significant.

Measurement noise is modeled by adding a white Gaussian signal to the
accelerations measured on the multibody wind turbine model. Five different
signal-to-noise ratios (SNRs) are considered, namely SNR

In addition to acceleration noise, the study also considered the effects of
errors in the measurement of the average wind speed

To separate the effects of measurement noise from the stochastic disturbances caused by turbulence, series composed of 3 min long nonturbulent wind conditions are considered first.

Figure

Residual pitch misalignment as a function of the number of
steps for different SNRs in constant uniform inflow at 11 m s

Figure

It appears that the method very effectively reduces the initial misalignments. Indeed, results show a very modest effect of SNR, except for the lowest value of 5 dB that seems to take a bit longer to converge. The apparently surprising lack of sensitivity to SNR can be explained by the changing yaw misalignment within the steps. Indeed, as shown in Eq. (8b), the 1P harmonic measured in the fixed frame is related to the presence of a cross-flow component. Therefore, a bit of misalignment of the rotor axis with respect to the wind vector eases rebalancing because it makes the effects of an unbalance more prominent and therefore less affected by noise.

Residual pitch misalignment as a function of the number of steps for different SNRs for variable nonturbulent inflow (series G).

Figure

It is also interesting to observe that convergence is actually faster in
turbulent conditions (Fig.

Residual pitch misalignment as a function of the number of
steps for different SNRs with turbulent inflow at TI

A large number of tests performed in additional operating conditions and SNR values confirm the findings reported herein. Clearly, one should choose a sensor with the highest SNR possible in the frequency range of interest. However, these results suggest that even fairly limited values of SNR should typically be sufficient for the algorithm to completely rebalance a rotor in turbulent and varying wind conditions.

This paper has described a new method to detect and correct pitch imbalances in wind turbine rotors. The method uses a measured signal in the fixed frame, typically in the form of accelerations or loads. The signal is demodulated to extract the 1P harmonic, which is then related to the misalignment of the blades by a linear model. By exploiting the axial symmetry of the rotor, the phase of the signal is used to detect which blades are unbalanced. The use of the rotor axial symmetry has the additional effect of reducing the number of free parameters in the model to only two.

The model parameters are readily identified by measuring the signal and computing its harmonics at two different pitch settings, something that is easily achieved by simply pitching the blades by a small chosen amount. The procedure can be performed while the machine is in operation, without shutting it down. The method also works if measurements are taken at different operating conditions, which is indeed inevitable in the field. Once the model has been identified, its inversion readily yields the pitch adjustments of the various blades that rebalance the rotor. If, after rebalancing, some remaining 1P harmonic is detected, the whole procedure can be repeated, thereby eliminating the effects of possible small nonlinearities in the imbalance–disturbance relationship. The whole approach has fairly minimal requirements, as it only assumes the availability of a sensor of sufficient accuracy and bandwidth to detect the 1P harmonic to the desired precision and the ability to command the pitch setting of each blade independently from the others.

Extensive numerical simulations were conducted with the proposed procedure using a detailed aeroservoelastic model of a multi-MW wind turbine. The analysis considered realistic scenarios in which measurements and rebalancing were performed in operating conditions characterized by varying air density, wind speed, yaw misalignment, upflow, shear and turbulence intensity. The simulation environment also considered the modeling of measurement noise and disturbances.

Based on the results presented herein, the following conclusions may be
drawn.

The relationship between pitch imbalance and 1P fixed-frame harmonics appears to be linear and unique depending on the location of the misalignment. This allows one to not only quantify the severity of the imbalance, but also the unbalanced blade(s).

In realistic wind conditions, i.e., with turbulent wind and variable air density, speed, vertical shear and wind rotor angles, the proposed algorithm successfully rebalances the rotor typically within four iterations. To account for possible changes in the mean value of wind speed and/or density, the simple scaling of the 1P input by the dynamic pressure was sufficient to guarantee a good performance in all tested conditions.

Given the relatively small magnitude of the signals that are
generated by small misalignments of the blades, one might expect that particular attention
should to be paid to the selection of the installed sensors. However,
results have shown that measurements are rather insensitive to SNR. Indeed, values of SNR

Good results were obtained by using observation windows of 10 min. Although longer time windows might appear to be beneficial to smooth out fluctuations due to turbulence and noise, one should also consider that long time windows might also imply significant changes not only in the operating conditions, but also in rotor speed, which should also be duly accounted for.

Notwithstanding the very promising results obtained here in a simulation environment, a demonstration in the field remains indispensable to prove the actual effectiveness and applicability of the proposed method in practice. Finally, future studies should consider the case of simultaneous aerodynamic and mass imbalances.

Data can be provided upon request. Please contact the corresponding author Carlo L. Bottasso (carlo.bottasso@tum.de).

The following tables report the values of the relevant operational and wind parameters used for the verification of the rebalancing algorithm.

Series A. Initial blade misalignment:

Series B. Initial blade misalignment:

Series C. Initial blade misalignment:

Series D. Initial blade misalignment:

Series E. Initial blade misalignment:

Series F. Initial blade misalignment:

Series G. Initial blade misalignment:

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: Joachim Peinke Reviewed by: two anonymous referees