Research article 08 Jun 2021
Research article  08 Jun 2021
Modelbased design of a wavefeedforward control strategy in floating wind turbines
 ^{1}Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, Milan, 20156, Italy
 ^{2}Sowento GmbH, Donizettistraße 1A, 70195 Stuttgart, Germany
 ^{3}Delft Center for Systems and Control, Delft University of Technology, Delft, 2628 CD, the Netherlands
 ^{1}Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, Milan, 20156, Italy
 ^{2}Sowento GmbH, Donizettistraße 1A, 70195 Stuttgart, Germany
 ^{3}Delft Center for Systems and Control, Delft University of Technology, Delft, 2628 CD, the Netherlands
Correspondence: Alessandro Fontanella (alessandro.fontanella@polimi.it)
Hide author detailsCorrespondence: Alessandro Fontanella (alessandro.fontanella@polimi.it)
Floating wind turbines rely on feedbackonly control strategies to mitigate the negative effects of wave excitation. Improved power generation and lower fatigue loads can be achieved by including information about incoming waves in the turbine controller. In this paper, a wavefeedforward control strategy is developed and implemented in a 10 MW floating wind turbine. A linear model of the floating wind turbine is established and utilized to understand how wave excitation affects rotor speed and so power, as well as to show that collective pitch is suitable for reducing the effects of wave excitation. A feedforward controller is designed based on the inversion of the linear model, and a gainscheduling algorithm is proposed to adapt the feedforward action as wind speed changes. The performance of the novel wavefeedforward controller is examined first by means of linear analysis and then with nonlinear timedomain simulations in FAST. This paper proves that including some information about incoming waves in the turbine controller can play a crucial role in improving power quality and the turbine fatigue life. In particular, the proposed wavefeedforward control strategy achieves this goal complementing the industrystandard feedback pitch controller. Together with the wavefeedforward control strategy, this paper provides some insights about the response of floating wind turbines to collectivepitch control and waves, which could be useful in future controldesign studies.
Floating offshore wind turbines (FOWTs) are currently operated without any realtime information about ocean conditions. Industrystandard controllers are feedback (FB) only: the wind turbine controller reacts to the external disturbance of wind and waves as this occurs. One possibility for improving the current floating wind technology is to include realtime information about the marine environment in the turbine controller and to design new control logics based on that.
Concerning wind turbulence, feedforward (FF) control has recently drawn the attention of the research community, as it can effectively reduce fatigue loads and improve power production. Research has been mainly driven by improvements in the lidar (light detection and ranging) technology that enables measurement of the wind field upstream of the wind turbine. One of the first studies about lidarassisted control was carried out by Harris et al. (2006), which explored the potentialities of this new control strategy and found it can reduce blade fatigue loads by approximately 10 % in turbulent wind conditions. Since then, several control logics were developed based on the inclusion of lidar signals in the turbine controller, for example by Laks et al. (2010), Dunne et al. (2010, 2011, 2012), and Schlipf et al. (2013), demonstrating positive effects for the blade loads and the turbine components. There are very few studies about lidarassisted windFF control in floating wind turbines: in the paper of Schlipf et al. (2015), a collectivepitch FF controller is designed to reduce rotor speed oscillations caused by inflow turbulence. The lidar moves with the floating turbine, and measurements need to be corrected for that. The proposed feedback–feedforward (FBFF) controller, which takes into account the abovementioned movement of the lidar, can reduce power and rotor speed fluctuations by up to 80 % and tower, rotorshaft, and blade fatigue loads by 20 %, 7 %, and 9 % respectively.
Wave disturbance is responsible for a considerable fraction of dynamic excitation experienced by an FOWT. This was first shown in the work of Jonkman (2007), where the response of the NREL 5 MW turbine installed on land is compared to the same turbine mounted on the floating ITI Energy barge, in the presence of wind and waves. The rotorspeed excursions in the floating turbine are increased by 60 % because of the oscillations in wind speed caused by platform motion. Consequently, fluctuations in the generator power are larger as well. Tower shear forces and bending moments are increased. The analysis of Jonkman (2007) also shows that offshoretoonshore ratios decrease with decreasing severity of the wave conditions, suggesting that a large part of the increments is due to wave excitation. More specifically for the drivetrain, Nejad et al. (2015) assessed the loads in a 5 MW wind turbine mounted on four platform concepts for different wind and wave conditions. The analysis suggests an increment of the fatigue damage, which is mainly caused by the large waveinduced thrust force.
Applying the same idea behind windFF control to waves is therefore an attractive perspective, but the idea is largely unexplored. Raach et al. (2014) introduced a nonlinear model predictive control (NMPC) strategy that uses a perfect preview of a reduced wave disturbance to mitigate the turbine structural loads. Promising improvements over an industrystandard FB controller are shown, at the expense of a significant increase in the controller complexity. Moreover, most of the performance gain is seen for blade loads, which are caused by wind turbulence rather than by waves, so it appears the NMPC does not effectively counteract waves. Ma et al. (2018) developed and validated two algorithms for realtime forecasting of wave forces. Based on the predicted wave forces, a finitehorizon LQR controller is designed and applied to a tensionleg platform FOWT to minimize the towerbase foreaft bending moment, achieving mixed results. Al et al. (2020) introduced an inversionbased feedforward control strategy, showing it is an effective way of reducing waveinduced rotor speed oscillations.
The present paper further develops the concept of wave FF exploiting tools of modelbased control. The waveFF control strategy is enabled by an integrated model of the FOWT that captures its most relevant physics. Hence, this work proves the effectiveness of multidisciplinary analysis as a means to advance the current floating wind technology.
All the reasoning is made with reference to a floating wind turbine, but it is deemed valid for any FOWT. The floating wind turbine of reference is based on an opensource concept and is defined in Sect. 2. The idea is to use tools of multivariable systems control to gain insight about the effects of waves on the FOWT response and assess which is the best control input (generator torque or collective pitch) to mitigate them. Then, this knowledge is leveraged to design a feedforward controller that reduces power fluctuations caused by waves. A controloriented linear model of the FOWT is required first for the multivariable analysis and later for the synthesis of the feedforward controller. The controloriented linear model is briefly introduced in Sect. 3. Section 4 deals with the input–output analysis. The feedforward controller is designed in Sect. 5. Again, linear analysis is utilized to assess the controller performance, which is shown to be highly dependent on the wind turbine operating point. Hence, a gainscheduling law is introduced to have the maximum performance in any wind condition. The wavefeedforward controller requires as input a preview of the incoming waves, which is obtained based on the algorithm presented in Sect. 6. In Sect. 7, the feedforward controller and the wave prediction algorithm are implemented in a nonlinear, mediumfidelity model of the floating wind turbine, and numerical simulations are carried out in realistic environmental conditions to evaluate the benefits of the feedforward control strategy. Section 8 draws the conclusion and gives some recommendations for future work.
This section defines the floating system that is considered in this study. The FOWT is formed by the DTU 10 MW (Bak et al., 2013) wind turbine and the INNWIND.EU TripleSpar platform (Azcona et al., 2017; Lemmer et al., 2020a). The characteristics of this FOWT concept are similar to those of current commercial projects and are publicly available.
The floating wind turbine is regulated with an industrystandard generatorspeed controller. In belowrated winds the controller maximizes the extracted power by keeping the bladepitch angle θ constant and varying the generator torque Q_{G} as a function of generator speed ω_{G} squared:
with ${k}_{\mathrm{G}}=\frac{\mathrm{1}}{\mathrm{2}}\mathit{\rho}\mathit{\pi}{R}^{\mathrm{5}}({C}_{\mathrm{p},\mathrm{max}}/{\mathit{\tau}}^{\mathrm{3}}{\mathit{\lambda}}_{\mathrm{opt}}^{\mathrm{3}})$, where ρ is the air density, R the rotor radius, and τ the transmission ratio. C_{p,max} is the maximum power coefficient, which is achieved for zero pitch angle and the optimal tipspeed ratio λ_{opt}.
In aboverated winds, the controller regulates the extracted power to its rated value, setting the generator torque to a constant value, equal to rated. Generatorspeed oscillations are directly reflected by the wind turbine power output. Rotor speed is regulated to its rated value ω_{G,r} by the collectivepitch controller (CPC), which reacts to the generatorspeed feedback as
where k_{P} and k_{I} are the proportional and integral gains, tuned following the modelbased approach of Fontanella et al. (2018) to achieve the maximum damping for the platform pitch mode and for the drivetrain mode. A gainscheduling factor is introduced to adjust the PI controller gains as wind speed varies. The generatorspeed feedback controller constitutes the baseline configuration against which the benefits of wave FF are assessed.
The waveFF control strategy we want to develop is modelbased, and its development requires a linear model of the floating wind turbine. The controldesign model is derived based on linear firstprinciple equations of the most important physics of the FOWT, rather than from the linearization of a higherorder model. The main features of the model are recalled below, while a detailed description is reported in the article of Fontanella et al. (2020). The model describes the global dynamics of the FOWT, neglecting the dynamics of single components. It considers the rigidbody platform motions and the rotor dynamics about a steadystate configuration (operating point) set by an average wind speed, platform motion, rotor speed, collective pitch, and generator torque. The inputs are generator torque and collective bladepitch angle, which are the main control variables for the FOWT, in addition to wind turbulence and wave elevation. The model equations are cast in statespace form and are valid only for small perturbations about the operating point.
The structural dynamics builds on the theory of multibody systems. The model considers the FOWT components rigid bodies; this simplification is deemed acceptable in a controloriented model, as the bandwidth of an FOWT controller is usually lower than the flexible modes of the tower, blades, and drivetrain. Moreover, the focus of the controloriented model is the coupled rotor–platform response induced by waves more than the dynamics of the flexible components.
Rotor aerodynamics are introduced into the model with a simplified approach. The aerodynamic model does not consider the single blade but computes the integral rotor forces. This simplification is valid because the FOWT global dynamics is determined by the integral rotor loads, rather than by the loads of the single blades. This assumption is reasonable for a reducedorder model of the FOWT, as noticed by Lemmer et al. (2020b). Only the rotor torque and thrust force are considered because they drive the global dynamics of the floating turbine: aerodynamic torque sets the wind turbine power production and thrust force the motion of the floating platform. Torque and thrust are modeled by means of the quasisteady approach, based on the derivatives of the torque and thrust curves of the wind turbine. The formulation of the controloriented model enables the inclusion of unsteady aerodynamic effects associated with the FOWT motion, which may have an influence on the platform response. In this respect, a similar approach to the one presented by Bayati et al. (2017) could be used.
3.1 Frequencydependent hydrodynamic loads
Hydrodynamic radiation and firstorder wave forces are modeled by means of lineartimeinvariant parametric models.
Frequencydependent radiation forces are approximated by a parametric model in statespace form, from the added mass and damping matrices of panel code precalculations. In this work, the frequency domain identification method of the MATLAB toolbox developed by Perez and Fossen (2009) is used, but other methods are available in the literature, for example the one by Janssen et al. (2014).
Also the firstorder wave excitation is introduced into the model with a parametric model in statespace form, which connects the wave elevation to the wave forces. This choice allows having the wave elevation, rather than wave forces, as input to the model. The wave excitation model is obtained based on the waveforce coefficients, which are usually computed at discrete frequencies through a panel code (e.g., WAMIT, 2020). In the present case, the parametric model is defined by means of system identification of the impulse response function of the force coefficients. This approach was first applied to floating turbines by Lemmer et al. (2020c), and it is currently available to model wave excitation in OpenFAST (Jonkman et al., 2018). Identification of the parametric model from frequency domain data (i.e., the force coefficients) is also a possibility, which was for example used by Al et al. (2020). The waveforce model of panel code data is noncausal, which means a force is developed before the wave reaches the center of the platform. The panel code data, in the form of impulse response functions, are causalized before system identification by introducing a time delay t_{d} (with t_{d}>0) to have zero response for negative times. The time delay is embedded inside the identified parametric model and, consequently, into the linear model of the FOWT. The response obtained from the linear model is delayed by a time t_{d} with respect to the input wave elevation.
An input–output analysis is carried out to gain insight into the FOWT response to the available controls, generator torque and collective pitch, and the wave disturbance. The analysis answers the question of which is the best combination of controls to reject the negative effects of waves. This information is used later to support the synthesis of the waveFF control strategy. Moreover, the analysis gives a picture of the FOWT dynamics that may prove to be also useful for other purposes.
The analysis starts from the controldesign model in a transfer function representation:
The system has two outputs, rotor speed and towertop motion, collected in $\widehat{\mathit{y}}=[{\mathit{\omega}}_{r},\phantom{\rule{0.33em}{0ex}}{x}_{\mathrm{tt}}{]}^{T}$; two control inputs, collective pitch and generator torque, collected in $\widehat{\mathit{u}}=[\mathit{\theta},\phantom{\rule{0.33em}{0ex}}{Q}_{\mathrm{g}}{]}^{T}$; and two disturbance inputs, variation from average of the hubheight wind speed and wave elevation, $\widehat{\mathit{d}}=[v,\phantom{\rule{0.33em}{0ex}}\mathit{\eta}{]}^{T}$. The model of Eq. (3) is used to compute the outputs' deviation from their steadystate value, due to a change in the control and disturbance inputs.
To facilitate the interpretation of input–output analysis results, the model of Eq. (3) needs to be scaled. This ensures that inputs and outputs are of the same importance. The scaled model is obtained by dividing any variable by its maximum expected (for disturbances) or allowed (for control inputs) change. The output, input, and disturbance scaling matrices (D_{y}, D_{u}, and D_{d} respectively) are
where ω_{0} is the rated rotor speed, Q_{g,0} the rated generator torque, and U the mean wind speed. The scaled model is
with
Given an input between 0 and 1, where 0 is no input and 1 is the maximum expected value, the outputs of the model of Eq. (5) take a value between 0 and 1, where 0 is no output and 1 corresponds to the maximum expected or allowed value for the output.
4.1 Control inputs
The model without disturbances $\widehat{\mathit{y}}=\widehat{\mathbf{G}}\widehat{\mathit{u}}$ is considered first. The transfer function matrix $\widehat{\mathbf{G}}$ has two couples of input and output directions, each with an associated gain. For any selected frequency, the directions and gains of matrix $\widehat{\mathbf{G}}$ are obtained from its singular value decomposition (SVD) (Levine, 1996):
The column vectors of $\mathbf{V}=[\stackrel{\mathrm{\u203e}}{\mathit{v}},\phantom{\rule{0.33em}{0ex}}\underset{\mathrm{\xaf}}{\mathit{v}}]$ are the input directions, the column vectors of $\mathbf{U}=[\stackrel{\mathrm{\u203e}}{\mathit{u}},\phantom{\rule{0.33em}{0ex}}\underset{\mathrm{\xaf}}{\mathit{u}}]$ the output directions, and the respective singular values are along the diagonal of $\mathbf{\Sigma}=\mathrm{diag}(\stackrel{\mathrm{\u203e}}{\mathit{\sigma}},\phantom{\rule{0.33em}{0ex}}\underset{\mathrm{\xaf}}{\mathit{\sigma}})$. When the input vector $\widehat{\mathit{u}}$ has the same direction as that of vector $\stackrel{\mathrm{\u203e}}{\mathit{v}}$, the output $\widehat{\mathit{y}}$ is along the direction $\stackrel{\mathrm{\u203e}}{\mathit{u}}$, the gain is equal to $\stackrel{\mathrm{\u203e}}{\mathit{\sigma}}$, and it is the largest possible for that frequency. The input produces the most effect on the output, and the directions of $\stackrel{\mathrm{\u203e}}{\mathit{v}}$ and $\stackrel{\mathrm{\u203e}}{\mathit{u}}$ are named the strongest. Conversely, when the input is directed as $\underset{\mathrm{\xaf}}{\mathit{v}}$, the gain is $\underset{\mathrm{\xaf}}{\mathit{\sigma}}$, and the input has the least effect on the output, which is along $\underset{\mathrm{\xaf}}{\mathit{u}}$. The directions of $\underset{\mathrm{\xaf}}{\mathit{v}}$ and $\underset{\mathrm{\xaf}}{\mathit{u}}$ are named the weakest.
The steadystate (i.e., zerofrequency) plant model of the FOWT in a 16 m/s wind is
The (1, 1) element of $\widehat{\mathbf{G}}$ is much larger than the (1, 2) element, so rotor speed is a lot more sensitive to a steadystate (i.e., very slow) change in collective pitch, the first input, than in generator torque, the second input. Collective pitch has an effect on both rotor speed, the first output, and towertop motion, the second output, in the same direction. If collective pitch is increased, the rotor is slowed because of the decreased aerodynamic torque, and the nacelle moves upwind because of the lower thrust force. The plant model is decomposed into its SVD:
The strongest and weakest input directions are obtained by different combinations of collective pitch and generator torque (e.g., the strongest is given by 0.997 for collective pitch and 0.082 for generator torque). The ratio between the gain in the strongest and weakest direction (i.e., the condition number) is $\mathrm{CN}=\stackrel{\mathrm{\u203e}}{\mathit{\sigma}}/\underset{\mathrm{\xaf}}{\mathit{\sigma}}=\mathrm{14.0}$. The system is said to be illconditioned. At a steady state, the input combinations with prevailing collective pitch have a much stronger effect on the FOWT than the input combinations with prevailing generator torque. The strongest and weakest output directions are given by different combinations of rotor speed and towertop motion. From $\stackrel{\mathrm{\u203e}}{\mathit{u}}=[\mathrm{0.916},\phantom{\rule{0.33em}{0ex}}\mathrm{0.401}{]}^{T}$, it is seen that the effect of the strongest input combination, an increase in collective pitch, is to slow down the rotor and to move the nacelle upwind. This is in agreement with the result of the inspection of $\widehat{\mathbf{G}}(j\mathit{\omega}=\mathrm{0})$ and makes sense from a physical point of view.
The SVD of the plant model $\widehat{\mathbf{G}}\left(j\mathit{\omega}\right)$ is computed for several frequencies up to 0.3 Hz and for seven operating points of wind speeds between 12 and 24 m/s. The top plot of Fig. 1 shows the magnitude of the two components of $\stackrel{\mathrm{\u203e}}{\mathit{u}}$, that is the fraction of collective pitch and generator torque in the strongest input combination; the middle plot shows the magnitude of the first and second component of $\stackrel{\mathrm{\u203e}}{\mathit{v}}$, the fraction of rotor speed and towertop motion in the strongest output direction; the bottom plot shows the corresponding singular value, the gain. Collective pitch is the most effective input, at any frequency and at any aboverated wind speed. Pitching blades affect both rotor speed and towertop motion, because the action of pitching the turbine blades modifies the aerodynamic torque and thrust at the same time. At the platform mode frequencies, the response is almost only towertop motion and the gain is increased: it takes a small collectivepitch action to move the nacelle, because the resulting rotor thrust variation excites the resonant response of the platform. Controlling rotor speed is hard. In the wave frequency range, the gain is decreased so it becomes more difficult to control the system, and rotor speed is easier to control than towertop motion.
In summary, collective pitch is the most effective control in aboverated winds. It has an effect on both rotor speed and towertop motion. In the frequency range where the wave is active, collective pitch becomes less effective, so it is harder to control the wind turbine.
4.2 Disturbances
The wind and waves disturbances are here considered separately. The direction of a disturbance is
where ${\widehat{\mathbf{g}}}_{d}$ is the appropriate column of ${\widehat{\mathbf{G}}}_{\mathrm{d}}$ (the first for wind, the second for wave). The disturbance condition number (DCN) is
where $\stackrel{\mathrm{\u203e}}{\mathit{\sigma}}(\cdot )$ is the maximum singular value. The DCN measures the control effort required to reject a given disturbance, relative to rejecting a disturbance with the same magnitude but aligned with the strongest output direction (i.e., the direction where controls are the most effective; Skogestad and Postlethwaite, 2005). The higher the DCN is, the harder it is to reject the disturbance with the available controls.
The effect of wind and wave disturbance in the frequency range up to 0.3 Hz is assessed in Fig. 2, considering seven operating points of wind speed between 12 and 24 m/s. Wind turbulence acts directly on the rotor causing a variation in the aerodynamic torque, which affects rotor speed. Wind turbulence also acts on the platform, through the rotor thrust, but this excitation mechanism is less effective than wave forcing. The wind disturbance is aligned to the rotor speed output direction. Collective pitch is very effective for controlling rotor speed, and rejecting the wave disturbance with collective pitch does not require a large effort. This is visualized by the DCN. The wave is aligned to towertop motion and partially shifts towards rotor speed for increasing frequency. Waves act on the platform but also excite the rotor response. The platform motion caused by waves produces a variation in the apparent wind speed, which affects rotor torque and then rotor speed. This mechanism of excitation is more effective above the platform pitch frequency. The analysis considers 0^{∘} waves, that do not excite lateral motions (sway, roll, yaw). For non0^{∘} waves, the response of these DOFs is also significant and contributes to oscillations of the wind inflow. The gain of the wave is maximal where the wave produces the largest platform motions, i.e., at the frequencies of platform modes, where the wave excites the FOWT in resonance, and above the platform pitch frequency, where the strength of wave forcing is the maximum. The wave disturbance is not aligned to the rotor speed output direction, and the DCN shows it is very hard to counteract the wave disturbance by means of wind turbine controls.
To sum up, waves effect rotor speed because waves drive the platform motion which result in an apparent wind speed at the rotor. The wave excitation is stronger at the platform frequencies, where the FOWT is excited in resonance, and above the platform pitch frequency. Moreover, it is quite hard to counteract the wave disturbance by means of controls available in the wind turbine. Lemmer et al. (2016) carried out a similar MIMO analysis for the same floating turbine system but based on a different linear model. They equally found that the wave has a significant effect on rotor speed and towertop motion and that it is difficult to counteract wave excitation by means of the control inputs available in the wind turbine. All this shows that new control strategies specific to FOWTs are needed to deal with waves. The wavefeedforward control strategy leverages the knowledge of the FOWT dynamics to improve the performance of the traditional wind turbine controller with respect to the mitigation of the wave effects.
The waveFF controller cancels the oscillations of rotor speed and hence of the turbine power output that are caused by waves. The additional collectivepitch command it produces is summed to the pitch signal of the existing generatorspeed FB controller and counteracts the variation in aerodynamic torque caused by the platform motion induced by waves. The FBFF control strategy is shown in Fig. 3.
For wave disturbance rejection, the reference signal r is zero and the closedloop rotor speed output ω is
where G_{s} is the collective pitch to rotorspeed plant, G_{d} the wavedisturbance model, K_{fb} the FB controller, K_{ff} the FF controller, and η the wave disturbance. In the modelinverse approach, the FF controller K_{ff} is designed to cancel the effect of η on ω; thus the controller transfer function is
K_{ff} is the transfer function between the input wave elevation measurement and the collectivepitch command. In general, G_{s}, G_{d}, and K_{ff} depend on the wind turbine operating condition and, so, on the mean wind speed.
The FF controller transfer function obtained from Eq. (13) is shown in Fig. 4 for different operating conditions. There is a significant difference between the generic shape assumed by K_{ff} in belowrated and aboverated conditions. The amplitude is increased in belowrated winds because collective pitch is not effective for controlling rotor speed. Here, a variation in collective pitch produces a smaller variation in rotor torque than in aboverated winds. For this reason, it is decided to confine the action of the FF controller to the aboverated region: when the mean pitch angle falls below a threshold, the FF action is switched off in order to prevent excessive pitch actuator usage. The aboverated controller K_{ff} has a peak at the platform pitch natural frequency which is not present in belowrated winds. In aboverated winds, the platform pitch mode damping is decreased and wave excitation leads to a large response at this frequency. This causes significant oscillations of the nacelle, with consequently large variations in the apparent wind speed, and of aerodynamic torque. A high control effort is therefore required to balance the wind fluctuations.
The controlsynthesis procedure described above is valid for any platform typology. When a different platform is considered, the disturbance model changes, because forcing produced by waves depends on the platform geometry and the way waves interact with it. The FF transfer function K_{ff}, the product between the inverse of the plant model and the disturbance model, changes accordingly. The FF controller responds to waves and acts in the frequency range where most of the wave energy is. In this frequency range, the amplitude of K_{ff} is increased if the platform is more exposed to wave excitation, which means a larger control effort is required for wave load rejection. The platform modes are expected to change. However, these are usually outside the wave frequency range and have little influence on the waveFF action.
5.1 Disturbance rejection analysis
Considering the FBFF controller of Fig. 3 and the closedloop disturbance response of Eq. (12), the FB, the FF, and the FBFF sensitivity function is defined respectively as
The sensitivity function of the FBFF controller is computed for different aboverated wind speeds U_{i} to account for the different behavior of the wind turbine:
with ${U}_{i}=\mathrm{12},\mathrm{13},\mathrm{\dots},\mathrm{24}$ m/s and $\stackrel{\mathrm{\u203e}}{U}=\mathrm{16}$ m/s. The sensitivity function tells how the disturbance is propagated to the FOWT response through the FB, FF, and FBFF controllers. The lower the magnitude of the sensitivity function, the lesser the disturbance effect on the FOWT response. Figure 5 reports the sensitivity function of the FB and FBFF controllers and compares it to the typical power spectral density (PSD) of wind and waves (rescaled). Two curves are shown for the FB controller: one is with gains for the onshore DTU 10 MW turbine (Hansen and Henriksen, 2013) and one is with detuned gains. In the case of original gains, wind turbulence is inside the CPC bandwidth (0.074 Hz): at the controller cutoff frequency the wind spectrum is around 3 % of its maximum value. Wave loads are just above the cutoff frequency (how much above depends on the sea state). The FB controller with original gains rejects the wind disturbance but is ineffective against waves. In the case of detuned FB, the bandwidth is shorter (0.019 Hz): the wind spectrum is 18.6 % of its maximum value at the controller cutoff frequency. Moreover, the disturbance sensitivity in the controller bandwidth is increased as the rotorspeed tracking performance is degraded. Hence the controller is less effective against the wind disturbance. The effectiveness of CPC with detuned gains is decreased, but detuning is needed to make the floating system stable without modifying the structure of the FB controller. As it has been shown by Larsen and Hanson (2007), van der Veen et al. (2012), and Lackner (2009), when onshore tuning is utilized, CPC may lead to an unstable response of the platform pitch mode. Bandwidth of the FB controller, hence also its effectiveness against wind turbulence, could be increased by means of nonminimum phase zero compensation (Fischer, 2013) where pitch control is used in combination with dynamic generator torque. Another possibility is to replace the FB controller with a more complex multivariable controller, as done by Lemmer et al. (2016). Both techniques can be used in synergy with feedforward control to further improve the floating wind turbine response to environmental loads. Interestingly, the capacity of the FB controller to reject wave loads is not influenced much by detuning. Comparison of the sensitivity function for the FB and FBFF controller (notice that S_{fbff} is defined with respect to wave disturbance, whereas S_{fb} is valid for any disturbance) shows that sensitivity to waves is greatly reduced by addition of wave FF.
The disturbance rejection function is derived from the sensitivity function, and it directly relates the wave disturbance to the closedloop rotor speed. For the FB and the FBFF controllers, it is defined respectively as
The disturbance rejection function of the FB and FBFF controllers at a 16 m/s wind speed is shown in Fig. 6 (thick solid line and thin dotted line). The magnitude of T_{fb} is increased in correspondence to the platform pitch mode and at higher frequencies. The disturbance rejection function of the FBFF controller is computed for different aboverated wind speeds U_{i}:
with ${U}_{i}=\mathrm{12},\mathrm{13},\mathrm{\dots},\mathrm{24}$ m/s and S_{fbff}(U_{i}) already obtained in Eq. (15). The magnitude of T_{fbff}(U_{i}) is shown in Fig. 6. The disturbance rejection function, so also the performance of the FBFF controller, is sensitive to the mean wind speed. This is due to the rotor aerodynamics which changes for different operating conditions. The benefit of wave FF is maximal at 16 m/s, the operating point considered for model inversion; lower elsewhere; and minimal in 12 m/s wind. T_{fbff} is higher than T_{fb} around the platform pitch natural frequency. Combining the FB controller with the FF controller strengthens the coupling between platform pitch and rotor speed.
5.2 Gain scheduling
The FOWT dynamics (i.e., the response for a given input) depends on the mean wind speed. The turbine is more sensitive to variations in bladepitch angle in high winds, and this is visualized in the input–output analysis of Fig. 1. As shown in Fig. 2, sensitivity to waves remains constant and does not depend significantly on wind speed. The control effort required to counteract a given wave is different depending on wind speed, because the wind turbine responds in a different way to bladepitch angle variation. Intuitively, the pitch action required to reject the effects of a given wave is lower in high winds. To have the maximum possible reduction in the wave disturbance, the FF controller needs to consider how the FOWT dynamics is modified with operating conditions, and a gainscheduling strategy is introduced for this purpose.
Based on the procedure introduced above, a linear model of the FOWT is computed for several aboverated wind speeds, and, by means of Eq. (13), an FF controller is obtained for each of them. The transfer function of the FF controllers is shown in Fig. 7. From visual inspection of the figure, it is evident that the effort required to cancel the wave disturbance is maximal in nearrated winds and decreases in high winds. If the FF controller obtained from the 16 m/s model is used at any wind speed, the FF action would be less than ideal for wind speeds between rated and 16 m/s, and higher than ideal for greater wind speeds, leading to a decreased performance, as highlighted by the disturbance rejection analysis of Fig. 6.
Figure 7 also reveals the shape of the FF controller does not change much with wind speed except for the static gain. Based on this consideration, the performance of the FF controller is improved by adjusting the static gain based on the actual turbine operating condition. In other words, a single FF controller is computed for the 16 m/s condition and the static gain is modified as wind speed changes to reflect the changed dynamics of the FOWT. The actual collectivepitch angle is chosen as the scheduling variable. The gainscheduling law is obtained fitting a quadratic function to the DC gain of the K_{ff}(jω) computed for different aboverated winds. The scheduled FF controller is
where p_{2},p_{1}, and p_{0} are the coefficients of the quadratic bestfit function; β is collective pitch; and ${K}_{\mathrm{ff}}\left(\stackrel{\mathrm{\u203e}}{U}\right)$ is the FF controller for the 16 m/s wind speed.
In Fig. 7, the scheduled FF controllers ${K}_{\mathrm{ff}}\left(\mathit{\beta}\right)={c}_{\mathrm{ff}}\left(\mathit{\beta}\right){K}_{\mathrm{ff}}\left(\stackrel{\mathrm{\u203e}}{U}\right)$ are compared to the modelinversion FF controllers obtained from the evaluation of Eq. (13) for different aboverated wind speeds. The scheduled controller is a good approximation of the ideal case. The proposed scheduling strategy leaves the phase of K_{ff}(jω) unchanged, but this is acceptable since the phase does not change much with wind speed.
The disturbance rejection function of the FBFF controller with scheduling is obtained by replacing ${K}_{\mathrm{ff}}\left(\stackrel{\mathrm{\u203e}}{U}\right)$ with K_{ff}(β) in Eqs. (15) and (17), and it is shown in Fig. 8. The disturbance rejection in the wave frequency range is lower for any wind speed, as the controller action is adjusted based on the wind turbine operating condition.
The FF controller for implementation is obtained as in Eq. (18). The order of the transfer function ${K}_{\mathrm{ff}}\left(\stackrel{\mathrm{\u203e}}{U}\right)$ is too high for practical usage; a reducedorder approximation is utilized in place of the original transfer function. The lowpassfiltered collectivepitch angle measurement is used for scheduling.
The transfer function of the FF controller has an intrinsic delay of t_{d}. A suitable wave elevation measurement is required to compensate for the intrinsic delay of the FF controller: it is required to know the wave t_{d} before it arrives at the platform. The wave prediction is obtained from a measurement of the surface elevation at a point at a distance l upstream of the platform. The measurement is propagated downstream in space and forward in time.
The wave elevation at two points along the wave propagation direction is related by the frequency response function:
where k is the wave number,and, for gravity waves on deep water, k is approximately ${\mathit{\omega}}^{\mathrm{2}}/g$. The frequency response function that relates the upstream wave measurement to the wave prediction at the FOWT is
For a given distance l and a preview time of t_{d}, H(jω) behaves as a pure negativedelay operator only for $\mathit{\omega}>g{t}_{\mathrm{d}}/l={\mathit{\omega}}_{t}$. The wave spectral components with a frequency greater than ω_{t} are successfully predicted. Prediction of the wave components with a lower frequency is not possible because the wave arrives at the platform location in a time shorter than the preview time t_{d}. It is possible to predict the lowerfrequency harmonics by measuring the wave elevation far upstream of the FOWT or by decreasing the preview time.
For realtime control purposes, the wave prediction model of Eq. (20) is implemented as it is shown in Fig. 9. The wave elevation η is continuously measured in l meters upstream of the floating wind turbine with a sample rate T_{s}. At any time instant t_{k}, the discrete Fourier transform (DFT) of the last n samples is computed to obtain the complex spectrum y. The elementwise product between y and the transfer function H, evaluated at n discrete frequencies ${\mathit{\omega}}_{i}=(\mathrm{2}\mathit{\pi}i/n{T}_{\mathrm{s}})$, with $i=\mathrm{0},\mathrm{\dots},n\mathrm{1}$, gives the spectrum of the predicted wave elevation at platform location $\widehat{y}$. The inverse DFT of $\widehat{y}$ gives $\widehat{\mathit{\eta}}$, which is the wave elevation that is going to be experienced by floating wind turbine $d={t}_{\mathrm{d}}/{T}_{\mathrm{s}}$ time samples ahead in time.
Several technologies are available to measure the surface elevation. Some examples are waverider buoy, radar, or airborne or satellite technology. The radar technology is particularly attractive because it scans a large area, it detects waves far from its location (up to 4 km), and it is capable of fully autonomous operation. The Xband radar, commonly used by ships for navigation, has received a lot of attention as a remote wave sensor. Images of the wave field are obtained from the radar as radar beams are reflected and shadowed by the crests of the wave fronts. An example of this technology is the wave monitoring system WaMoS II introduced by Ziemer and Dittmer (1994) and at the base of the realtime wave prediction system developed by Reichert et al. (2010) within the On board Wave and Motion Estimator (OWME). A methodology based on 2D FFT is proposed by Naaijen and Wijaya (2014) to obtain a directional phaseresolved prediction of the wave elevation from radar data (additional information about the directional energy spectrum is required, e.g., from a wave buoy). A similar measurement could be used in waveFF control.
The waveFF control strategy is evaluated by means of numerical simulations in the servoaerohydroelastic code FAST (Jonkman and Marshall, 2005). The FAST model has 7 DOFs: platform motions (surge, sway, heave, roll, pitch, and yaw) and the rotor rotation. The drivetrain as well as the tower and blades is rigid. The hydrodynamic model is based on linear potential flow theory with viscous effects. The radiation and the firstorder wave forces are computed prior to the simulation based on the same WAMIT data that are used to build the controldesign model. The calculation of the frequencydependent radiation loads is based on the convolution integral of the retardation function matrix. Secondorder wave loads are modeled by means of the approximation technique introduced by Newman (1974).
The waveFF control strategy considered for the verification is displayed in Fig. 10. Three cases are considered: a baseline case with only FB control, the FBFF control without gain scheduling, and the FBFF control with gain scheduling. In the simulations, an ideal upstream wave measurement is used and the accuracy of the wave measurement system (e.g., radar) is not taken into account. Results are therefore indicative of the upper performance limit of waveFF control.
7.1 Environmental conditions
Three realistic turbulent wind and irregular wave combinations (see Table 1), representative of an offshore site with moderateseverity metocean conditions, are selected for assessing the performance gains of the waveFF control strategy. The reference offshore site is part of the Gulf of Maine (North Atlantic Ocean), about 25 km southwest of the island of Monhegan 65 km east of Portland, and the mean water depth is 130 m. Metocean data for the site are reported by Gonzalez et al. (2015). Three aboverated winds are considered, which are defined by the parameters of Table 1. For any condition, the wave was defined according to the Pierson–Moskowitz spectrum. The significant height and peak period were selected as the most probable combination of values for the assigned wind speed. Wind and waves are aligned to the 0^{∘} direction.
7.2 Wave prediction
The wave prediction algorithm presented in Sect. 6 is tested in the metocean conditions corresponding to the 16 m/s mean wind speed case of Table 1. The wave elevation is sampled every 0.1 s at a distance of 200 m upstream of the FOWT. The wave elevation at the platform location is computed based on the last 1000 samples, and a preview time of 7.5 s is requested. The wave elevation preview is compared to the wave at the platform in Fig. 11. The overall quality of the estimate is good. The PSD of the two signals reveals that the largest error is introduced into the lowfrequency harmonics. The error is due to the intrinsic characteristics of the transfer function on which the wave prediction algorithm is based. For the present case, the transfer function correctly predicts the wave harmonics above a threshold frequency of 0.058 Hz.
7.3 Steady wind
The effect of the FF control strategy is first demonstrated considering a steady wind without shear. With this assumption, the wave is the only disturbance acting on the FOWT. Sample time series of the rotor speed and bladepitch command for a 22 m/s wind speed case are shown in Fig. 12. The amplitude of rotor speed oscillations caused by the wave disturbance is reduced with FBFF control with respect to the FB case, and this is achieved at the expense of an increased pitch activity. The pitch effort required by the scheduled FBFF is less than that required without scheduling for a comparable disturbance rejection performance.
7.4 Turbulent wind
The FBFF control is evaluated in more realistic power production conditions. Turbulent wind fields were generated in TurbSim (Kelley and Jonkman, 2007) with the Kaimal spectrum, a powerlaw profile with exponent 0.14, and turbulence intensity selected for each mean wind speed according to IEC 61400 (class IC turbine). For every condition of Table 1, six independent wind–wave realizations, each 10 min long, were considered, as recommended by the IEC 61400 standard (IEC, 2005) to obtain statistically significant data. An initial presimulation time of 1000 s was included at the beginning of each simulation and cut out from results to exclude initial transients.
Sample time series of rotor speed and bladepitch angle for the 22 m/s case are shown in Fig. 13. As visible looking at the FB case, the largest fraction of rotor speed oscillations is due to wind turbulence. This is in agreement with the MIMO disturbance analysis of Sect. 4. The FF control reduces the part of rotor speed oscillations caused by waves, but it does not compensate for the effect of wind turbulence. The pitch actuation is increased with any FBFF compared to the FB case.
Power spectral density (PSD) of rotor speed and pitch angle command is computed from the aggregated time series of the six seeds relative to the same operating condition. Results for the 22 m/s case are reported in Fig. 14. FBFF clearly reduces the energy content of rotor speed associated with wave excitation (wave range 0.05–0.2 Hz) but has a negligible effect at a lower frequency, where rotor speed oscillations are in large part due to wind turbulence. The FF action results in an additional bladepitch command whose energy content is concentrated in the wave range. Rejection of wave disturbance is slightly improved by the addition of scheduling (FBFF + Sched.), which also requires a lower bladepitch effort compared with FBFF.
Wave FF is designed to reduce the effects of wave disturbance and improve rotor speed regulation. However, it is expected to also affect the structural loads for the turbine components. Fatigue loads for each operating condition are evaluated in terms of damageequivalent loads (DELs) computed with MLife (Hayman and Buhl, 2012). Wave FF is also expected to affect platform motions, and this is quantified by the standard deviation of platform surge, roll, and pitch. DELs and standard deviations are computed for every operating condition based on the aggregated time series of six seeds. Variation in DELs and standard deviations with FBFF + Sched. compared with FB is examined in Fig. 15 for the three load cases. The lowspeed shaft (LSS) torque DEL is reduced by up to 16 %. The LSS torque depends on the aerodynamic and generator torque. The wave FF reduces the aerodynamic torque oscillations caused by waves and so the fatigue loads for the LSS. The lower dynamics of LSS torque is reflected in the platform roll motion which is reduced as well. The highest LSStorque DEL reduction is achieved in high winds when waves are the strongest. Blade pitch is increased because of the additional waveFF command, and the increment is proportional to the strength of waves. As demonstrated in the input–output analysis of Sect. 4, aerodynamic torque and rotor thrust are both affected by blade pitching. The modification of the thrust force induced by wave FF impacts the alongwind platform motions and the tower foreaft loads. Platform surge and pitch motions are increased, with consequently higher fatigue loads for the mooring system and lower loads for the tower. It is counterintuitive, but larger motions imply lower tower loads. As noticed by Fleming et al. (2016), a smooth motion of the nacelle releases the energy introduced by waves, which therefore is not put into tower bending.
This paper investigated a modelinversion feedforward control strategy for mitigation of wave excitation in floating offshore wind turbines. A linear controldesign model is utilized to carry out an MIMO analysis of the floating wind turbine. Collective pitch is more effective than generator torque for controlling rotor speed in aboverated winds. Above the platform natural frequencies, wave equally affects rotor and platform motions, with the same strength of wind turbulence. Based on linear analysis, a modelinversion feedforward controller is designed for canceling the waveinduced rotorspeed (and generatorpower) oscillations using collective pitch. The feedforward controller is added to an industrystandard feedback controller, and the performance improvement is demonstrated by means of linear analysis. A gainscheduling algorithm is devised to improve the controller performance by adapting the feedforward action as the wind turbine operating condition changes. The control strategy is finally verified by means of timedomain simulations in a nonlinear aeroservohydroelastic model. It is found that feedforward control can reduce the standard deviation of rotor speed by up to 2 %. It also has a positive side effect on the fatigue loads of several wind turbine components: the shaft torsion is reduced by up to 16 % and the towerbase foreaft bending by up to 5 %. Platform motions are slightly increased, and this is reflected in the mooring line loads. The bladepitch actuator usage is increased. WaveFF control improves the dynamic response of the floating turbine without requiring the replacement of the industrystandard feedback controller. A wave measurement and forecast system must be implemented, but this is feasible to obtain with technologies already used in the maritime industry. The extra cost of the wave measurement system and fatigue of the bladepitch actuators are likely to be offset by the lower cost of the turbine generator and tower, which can be redesigned in light of the reduced overspeed and fatigue loads respectively.
The following suggestions should be considered in future work about wavebased and wavefeedforward control in floating wind turbines:

The wavefeedforward controller is sensitive to the accuracy of wave elevation prediction and to the fidelity of the wavedisturbance model. The focus of this work is about development of the wavefeedforward control strategy and did not address the topic of uncertainties. In the present work, uncertainties in the wave measurement are only due to the preview algorithm and, as it is shown, the prediction error is small. Larger errors are expected when using a realistic measurement of upstream waves. Model uncertainties are mostly related to identification of the waveexcitation model. The model we consider here has been assessed against a mediumfidelity model in a previous work (Fontanella et al., 2020) and was deemed sufficiently accurate for control design. Indepth analysis of the controller robustness with respect to model and measurement uncertainties should be addressed in future works. Implementing the waveFF control in FOWTs based on different platform typologies may reduce uncertainty about the benefits of this control strategy.

The proposed feedforward controller is linear and compensates only for firstorder wave loads. Recent numerical and experimental studies, for example the one of Roald et al. (2013), prove that secondorder wave loads have a noticeable effect on a floating turbine response. Thus, a possible research suggestion is to investigate nonlinear controllers and to include secondorder hydrodynamics in the controller design.

In the controldesign model, rotor aerodynamics are modeled based on the quasisteady theory. Thus, the controller obtained from the model does not account for unsteady aerodynamic effects, which may be significant for the response in the upper wave frequency range (Mancini et al., 2020). It is therefore suggested to develop a controloriented model of the unsteady rotor aerodynamics and to include it in the controldesign model, so to investigate how unsteadiness affects the response of the controlled FOWT.

In the case at hand, the feedforward controller is designed to regulate rotor speed, and the reduction in tower loads is obtained as a positive side effect. A large fraction of tower loads is caused by waves, so it is advisable to use wave information to reduce tower fatigue loads.

The wave prediction model may find application in several controlrelated tasks that are not envisioned here. Waves drive the rigidbody motion of the floating turbine, and this is likely to affect the turbine wake (Wise and Bachynski, 2020). Wave prediction may be included in future floating wind farm control strategies.

Singleinput singleoutput feedback controllers remain the default choice in floating wind turbines, and advanced controllers are still far from being used in commercial projects. Tighter relationships between industry and academia are advisable to promote the adoption of advanced control strategies.
MATLAB codes and simulation results of this article can be obtained by contacting the authors.
AF and MA developed the wavefeedforward control methodology in all its aspects. MB and JWvW supervised the research activity, mentoring AF and MA. Finally, AF prepared the manuscript of this article with contributions from all coauthors.
The authors declare that they have no conflict of interest.
This paper was edited by Amir R. Nejad and reviewed by Farid Khazaeli Moghadam and one anonymous referee.
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 Abstract
 Introduction
 Definition of a reference floating wind turbine
 The controldesign model
 FOWT response to controls, wind, and waves
 The wavefeedforward control strategy
 Wave measurement and prediction
 Results
 Conclusions
 Code and data availability
 Author contributions
 Competing interests
 Review statement
 References
 Abstract
 Introduction
 Definition of a reference floating wind turbine
 The controldesign model
 FOWT response to controls, wind, and waves
 The wavefeedforward control strategy
 Wave measurement and prediction
 Results
 Conclusions
 Code and data availability
 Author contributions
 Competing interests
 Review statement
 References