A telescope is mounted on each blade and is connected to a hub-based continuous-wave lidar with fibre optical cables. The lidar samples the inflow wind speed in front of the rotor plane at a rate of 5 Hz, and we intend to use the lidar measurements for control purposes. The lidar measurements are integrated into the system model according to Fig. , and we use a combination of large-eddy simulations and an aeroelastic simulation code to simulate and evaluate the lidar-based inflow measurements. Thus, lidar measurements are simulated in a realistic environment, where the effect of the induction zone and wind evolution, as well as the dynamic behaviour of the wind turbine, are taken into account. Moreover, the lidar simulator considers volumetric measurement, dynamics of the blade and tower, i.e. displacement, rotation, and linear velocity in 3-D space, and blade-rotation-induced velocity. Nevertheless, the rotational effect of the blade is not accounted for during the accumulation of a single measurement.

Figure  illustrates the coordinate systems and the telescope orientation. Here, the line-of-sight (LOS) wind speed measurement from blade i (ulos,i) is defined as 1ulos,i=∫ξminξmaxW(F,ξ)Vi(ξ)dξ∫ξminξmaxW(F,ξ)dξ, where Vi(ξ) is defined in Eq. () and W(F, ξ) is the lidar's weighting function, defined according to as 2W(F,ξ)=1ξ2+1-ξF2RR2, where RR is the Rayleigh range, set at 1573 m herein, as proposed by ; F is the focus distance and ξ is the range along the beam. The limits ξmin and ξmax, introduced in Eq. (), refer to the minimum and maximum range, respectively, along the beam. For a practical implementation of the lidar simulator, these values are chosen such that W(F,ξ)W(F,F) equals 0.02 at these limits. During discretization of Eq. (), the spatial resolution is set empirically at Δξ=0.1 m. A single-point measurement is given by 3Vi(ξ)=uh,i(ξ)vh,i(ξ)wh,i(ξ)-x˙t,h,iy˙t,h,iz˙t,h,iTℓx,h,iℓy,h,iℓz,h,i, where [uh,ivh,iwh,i]T is the wind speed vector along the laser beam expressed in the rotating hub coordinate system; [x˙t,h,iy˙t,h,iz˙t,h,i]T is the linear velocity vector of the blade segment where the telescope is mounted, expressed in the rotating hub frame of reference; and [ℓx,h,iℓy,h,iℓz,h,i]T is the unit vector of the laser beam in the rotating hub coordinate system. The aeroelastic simulation tool is capable of providing full kinematics information, i.e. positions, orientations, and linear and angular velocities, of any blade segment in the hub coordinate system.

During the inflow wind speed estimation, the velocity, displacement, and rotation of the blade segment are assumed to be known; therefore, the wind speed component parallel to the rotor shaft axis can be reconstructed as indicated in Eq. (). Without loss of generality, in the wind speed estimation, the weighting function of W(F,ξ) from Eq. () is neglected, and two assumptions are made: (1) the vh,i and wh,i components are zero, and (2) the mean wind velocity is parallel to the rotor axis; i.e. no tilt and no yaw misalignments are considered. Consequently, an estimate of the wind speed parallel to the rotor shaft axis (uh,est,i) is 4uh,est,i≈ulos,i+y˙t,h,iℓy,h,i+z˙t,h,iℓz,h,iℓx,h,i+x˙t,h,i. Nevertheless, such assumptions introduce errors into the lidar measurement that are presumed to exist in the identified uncertainty weight and, thus, are consequently considered during the controller development.

Figure  illustrates the induction zone effect for laminar inflow. Note that the lidar measurement is affected by the rotor induction. The reduction depends on the position of the telescope along the blade radius (R) and the focus distance of the laser beam (F), where the wind speed measurement takes place. To account for this effect in the lidar-based inflow wind speed measurement, we construct a second-order polynomial function (f), whose inputs are chosen as rotor speed (ωr), blade pitch angle (βi), and blade root flapwise and edgewise moments (Mfw,i, Mew,i). Rotor speed and blade pitch angles are easily measured, and we assume that blade root flapwise and edgewise moment sensors are also available for implementing this method. Therefore, the estimated wind speed parallel to the rotor shaft axis (uh,est,i) is corrected as 5ucor,i=uh,est,i+Δuest,i, where 6u0-u(F,R)≈Δuest,i=fF,R,ωr,βi,Mfw,i,Mew,i. The second-order polynomial function (f) is fitted to the data extracted from 10 min large-eddy simulations with laminar inflow for mean wind speeds between 4 and 25 m s-1. u(F, R) is the wind speed at an upstream distance F from the blade and at a blade radial position of R, and u0 is taken from the same blade radial position of R but at an upstream distance of 3 times the rotor diameter (3-D).

To assess the performance efficiency of the blade-mounted lidar-based inflow wind speed measurement, we introduce a new signal called the blade-effective wind speed (ubeff,i), which is determined as the contribution of the inflow wind speed on each blade segment ui(r) to the flapwise blade root bending moment; the inflow wind speed refers to the longitudinal wind speed in the rotor axis direction. The contribution depends on the radial distance (r) and the local thrust coefficient (CT) of the blade segment as expressed by 7ubeff,i=∫RhubRtipCTr,ui(r)r2ui2(r)dr∫RhubRtipCTr,ui(r)r2dr. The local thrust coefficients are resolved from steady-state simulations for each blade segment from cut-in to cut-out wind speeds.

In the subsequent step, we introduce the multiblade coordinate transformation (MBC) that simplifies the controller design by transforming a time-varying system into a time-invariant system and decouples the individual pitch from the collective pitch control.

Figure  demonstrates the manner in which the feedforward controller is implemented. First, the measured inflow wind speed is transformed to the non-rotating frame of reference by applying MBC transformation (Tmbc(θ+ϕ)) in accordance with Eq. (), where θ denotes the azimuth angle. 8ucor,colucor,yawucor,tilt=Tmbc(θ+ϕ)ucor,2ucor,3ucor,1, where 9Tmbc(θ)=13131323cos⁡nhθ23cos⁡nhθ+2π323cos⁡nhθ+4π323sin⁡nhθ23sin⁡nhθ+2π323sin⁡nhθ+4π3. A phase shift (ϕ) is introduced into the transformation to consider that the measured inflow wind speed hits the wind turbine blade after this azimuth angle change. This value varies with respect to several parameters, including the selected focus distance, inflow wind speed, and rotor speed. The estimated wind speed parallel to the rotor shaft axis from blade 1 is used to determine the blade pitch control at blade 1; hence, the order of the estimated wind speeds parallel to the rotor shaft axis has changed to ucor,2, ucor,3, and ucor,1. Further, the control signals or the blade pitch angles (βcol, βyaw, βtilt) are determined by the feedforward controller (Kff,f). If the preview time provided by the lidar is greater than the time delay induced by the feedforward controller, an additional time delay (e-sTid) is introduced into the system. Finally, the delayed control signals (βcol,d, βyaw,d, and βtilt,d) are transformed to the rotating frame of reference using the pseudo-inverse of the MBC transformation (Tmbc+(θ)). The main structure of the feedforward individual pitch controller in Fig.  can be used in the feedforward trailing edge flap controller as well.

The MBC transformation plays a considerably important role because it can transform a frequency component of interest, such as 1P, 2P, or 3P , to a low-frequency component, named 0P. It is dependent on the selected value of nh in Eq. (). For example, 1P will be transformed to 0P when nh is specified as 1, and 2P will be transformed to 0P when nh is specified as 2.

In this study, we focus on identifying the uncertainty weight that can be used during the feedback–feedforward individual and collective pitch control development with an objective to mitigate the 1P loads at the flapwise blade root bending moments and to enhance the rotor speed tracking. This indicates that by considering nh as 1, the measured inflow wind speeds are transformed to the non-rotating frame of reference in Eq. (), where the uncertainty weight identification is conducted. Further, the same methodology can be applied to identify the uncertainty weight for higher-harmonics control by selecting a larger integer value of nh.

We have already mentioned that the measured inflow wind speeds are transformed to the non-rotating frame of reference by applying the MBC transformation. In order to assess the performance efficiency of the blade-mounted lidar-based inflow wind speed measurement, the blade effective wind speeds are also transferred to the non-rotating frame using the MBC transformation as follows 10ubef,colubef,yawubef,tilt=Tmbc(θ)ubef,1ubef,2ubef,3, where Tmbc(θ) is defined in Eq. ().

We use the blade-mounted telescopes to measure the disturbance, or the inflow wind speed in this case. Afterward, the three measurements are transformed into the non-rotating frame of reference where they are used as inputs to the feedforward individual and collective pitch controllers. Figure  illustrates the disturbance rejection controller set-up with uncertainty. Each block in the figure represents a three-input and three-output system with a three-by-three matrix transfer function.

The control development is aimed at achieving disturbance rejection up to a certain frequency with measurement uncertainties. In other words, we want to find a controller that satisfies Eq. () for a chosen performance weight Wp. 11∥WpSfbSff,p∥∞<1, where the frequency-dependent feedback (Sfb) and feedforward sensitivity (Sff,p) functions with additive uncertainty are given by 12Sfb=I+Gwt,fKfb,f-1,13Sff,p=I+Gwt,fKff,fGn,f+ΔℓWℓGd,f-1, and 14Δℓ=δℓ,1000δℓ,2000δℓ,3∈C3×3,∥Δℓ∥∞≤1. This equation highlights the importance of knowing the frequency-dependent uncertainty weight Wℓ(jω) in advance, so as to ensure that the closed-loop system is stable and that the objective in Eq. () is satisfied for all perturbations (||Δℓ||∞≤1). For control development, the frequency dependent uncertainty weight of Wℓ(jω) and the nominal disturbance measurement model of Gn,f(jω) are missing; we discuss how they can be identified in the next subsections and later illustrate the process for the reference cases in Sect. .

Remark: only one objective is introduced in Eq. (); nevertheless, other objectives can be added, such as penalizing the control signal magnitude at high frequencies .

We employ black box system identification to establish the transfer functions (Gℓ) from the blade effective wind speeds (ubeff) to the corrected lidar-based inflow wind speeds (ucor) in the non-rotating (fixed) frame of reference 15ucor,f=Gℓubeff,f with 16Gℓ=Gℓ,col000Gℓ,yaw000Gℓ,tilt∈C3×3. The system identification is performed via the ssest function from with a 15th-order state-space model, which can capture all the relevant information. The order of the state-space model is found empirically through analysis of the Hankel singular values.

We separately identify the nominal disturbance measurement model (Gn,k(jω)) and the uncertainty weight (wℓ,k(jω)), where k∈{col, yaw, tilt}, as a fifth-order minimum phase filter for each of the inputs in such a way as to satisfy the following inequalities 17Gn,k(jω)<Gℓ,k(jω),∀ω and 18Gn,k(jω)+wℓ,k(jω)>Gℓ,k(jω),∀ω, leading to the diagonal nominal disturbance measurement model matrix of 19Gn=Gn,col000Gn,yaw000Gn,tilt and uncertainty weight matrix of 20Wℓ=wℓ,col000wℓ,yaw000wℓ,tilt. The order of the transfer functions are determined empirically during the analysis of the data. Lower orders could be selected as well; however, these would lead to higher uncertainties at high frequency.

The ideal case would be to measure, with a telescope, the exact inflow wind speed hitting the rotor blades to result in a nominal disturbance measurement transfer function with a gain of 1 over the entire frequency range. However, not only the inflow condition, but also the telescope parameters influence the nominal disturbance measurement model and the measurement uncertainty weight. In Sect.  we identify these transfer functions (Gn,k and wℓ,k), which can then be used for control development and analysis. Furthermore, we analyse how much the low-frequency gains of Gℓ deviate from 1 for several cases.

We neglect the cross-coupling between the yaw and tilt components in the system identification, but these are considered in the wind turbine and disturbance transfer functions in line with , so that the cross-coupling between the yaw and tilt components is included in the controller development.

Preview time plays an important role in the development of feedforward control. It must be larger than or equal to the time delay introduced by the feedforward controller and actuator dynamics. It is preferable for it to be equal, but a larger value is acceptable, as an additional time delay can be easily introduced into the feedforward controller, as shown in Fig. . To determine the optimal preview time for a given focus distance, we evaluate the cross-correlation between the blade effective (ubeff,k) and the corrected inflow (ucor,k) wind speeds, with k∈{col, yaw, tilt}, and we choose the index of the peak value as the available preview time.

We introduce a cost function which is based only on the coherence (γk2) between the blade effective (ubeff,k) and the corrected inflow (ucor,k) wind speeds, with k∈{col, yaw, tilt}: 21Jlp(f)=∑kJlp,k(f)=∑kγk2(f). By evaluating Jlp for the discrete set of sampled lidar and telescope parameters, the maximum of the objective function results in the optimal telescope parameters within the discrete set of sampled lidar and telescope parameters. In this way, we are able to judge the initially chosen telescope parameters.

To perform an analysis of the MBC transformation, we create three generic wind speed measurement signals with 22ucor,i=u0+∑j=161j3sin⁡j2πf0t+(i-1)2π3+π6, where u0, i, f0, and t are the offset or undisturbed inflow wind speed, blade index, 1P frequency, and time, respectively. Here, we considered harmonics of up to 6P (j=1 … 6). Figure  shows an example time series of the generated signals.

Figure  presents the power spectral densities of the wind speed measurement obtained from the first blade (ucor,1) and the collective (ucol), yaw (uyaw), and tilt (utilt) components after the MBC transformation, which is applied to the generic wind speed measurement signals (ucor,1, ucor,2, ucor,3). The figure highlights the MBC transformation keeping only 0P, 3P, and multiples of 3P. As describe, the frequency (f) in the non-rotating frame of reference arises from f±f0 from the rotating frame of reference; e.g. the 3P in the non-rotating frame of reference arises from the 2P and 4P contributions in the rotating frame of reference.

Several cases may illustrate the transfer of the measurement errors from the rotating to the non-rotating reference frame. First, we should consider the effect of over- or underestimation of the measured wind speed with one of the blade-mounted lidar systems, due to, e.g., different radial positions of the telescope along the blade radii or one of the telescopes having a different orientation, which reduces the DC offset (u0 in Eq. ) for one of the three generic signals. Next, the signals are transformed into the non-rotating frame of reference, which can be compared to the case where all the DC offsets are maintained for each of the three signals at the same level.

As Fig.  highlights, an undesired peak appears at 1P in the yaw and tilt components in the non-rotating frame of reference, due to the presence of asymmetries in the signals in the rotating frame of reference .

Second, in addition to the reduction in the DC offset for one of the signals, a 1∘ phase shift is added to the 1P harmonic to the same signal in the rotating frame of reference, which represents the case, for example, where one of the blade-mounted lidar focus distances differs from the other two. Figure  reveals that after applying the MBC transformation to the three generic signals, undesired higher-harmonic peaks arise in the non-rotating frame of reference. Interestingly, the phase shift that is introduced to one of the signals in the rotating frame of reference results in different higher harmonics in the components in the non-rotating frame of reference, e.g. a peak observed at 1P of the collective component and both 1P at 2P of the tilt and yaw components.

The reference case we use in this investigation is based on the NREL 5 MW generic wind turbine . We use an actuator line model through the coupling between the FASTv7 aeroelastic simulation code and PALM (Parallelized Large-Eddy Simulation Model) as explained by . The operating conditions correspond to a hub-height mean wind speed of 13.06 m s-1, which is above the rated value of 11.4 m s-1. Furthermore, the 10 min simulation results in a turbulence intensity of 8.5 % and a wind shear corresponding to a power law description with an exponent of approximately 0.12. The baseline controller of the wind turbine ensures that the generator speed is kept at 1173.7 rpm , thereby resulting in a mean rotor speed (ωr) of 11.74 rpm and further leading to a 1P frequency of f0=0.195 Hz.

For an analysis of the induction zone effect, we set the range of the focus distance and telescope position along the blade radius at F∈[10, 40 m] and R∈[20, 60 m], based on a previous investigation . The range of the other input variables are determined by the results of simulations with laminar inflow and power law wind shear with coefficients of 0.1, 0.2, and 0.3. An approximation of the induction zone effect introduces some uncertainties into the measurement, but they are included in the identified uncertainty weight.

stress that an elevated peak around the crossover frequency (just below the 1P frequency) of the feedback–feedforward controller sensitivity function leads to increased loads across the wind turbine components. Here, the crossover frequency of the controller is defined where the sensitivity function first crosses -3 dB from below. Uncertainties impose limitations on the achievable performance ; e.g. the peak of the sensitivity function may increase due to uncertainties in the system. Therefore, it is important to analyse how the lidar measurement uncertainty is affected by, e.g., mounting misalignment of the telescope on the blade or cases where the focus distance or position of the telescope along the blade span differs from the optimal parameters. Identifying the lidar measurement uncertainty as a frequency-dependent minimum-phase filter enables the inclusion of such parameters in the control development, allowing an analysis of its impact on the stability and performance of the closed-loop system. As we explain in detail in Sect. , a straightforward solution to determine the telescope and lidar parameters, such as focus distance, telescope position along the blade radius, and telescope orientation on the blade, is to assume that the blades are rigid, that the rotor speed and pitch angle are constant, and that Taylor's frozen turbulence hypothesis holds . We perform large-eddy simulation (LES) in the subsequent sections to examine the usefulness and limitations of these assumptions and further analyse the uncertainties in the blade-mounted lidar measurement as well as the measurement sensitivity with respect to lidar and telescope parameter changes. The investigated cases are described in Sect. – and summarized in Table . Section  describes how the measurement uncertainties are affected when one or two telescopes are aligned differently than the others. First, we assume that the orientation angle misalignment is unknown. Second, we assume that this orientation angle misalignment can be identified, so that the lidar-based inflow wind speed measurement can be corrected.

For each case, first the transfer functions (Gℓ,k) from the blade effective wind speeds (ubeff,k) to the corrected lidar-based inflow wind speeds (ucor,k) are identified. Next, the nominal disturbance measurement models (Gn,k) and the uncertainty weights (wℓ,k) for each of the inputs are estimated to satisfy Eqs. () and (). Figure  provides a summary of the identified DC gain upper (Gn,k+wℓ,k) and lower (Gn,k) bounds of the transfer functions (Gℓ,k) from the blade effective wind speeds (ubeff,k) to the corrected lidar-based inflow wind speeds (ucor,k).

We would like to act only below the 1P (0.195 Hz) frequency; therefore, below this frequency, it is desired that the gain of Gn,k is 1 and that the measurement uncertainty is small but still covers the worst case. A higher percentage of measurement uncertainty can be tolerated at frequencies above 1P by designing the feedforward controller accordingly, e.g. a model inversion-based feedforward controller with a low-pass filter with a crossover frequency below 1P. With Fig. , we show how wide variation in the DC gain of Gℓ,k agreed with the identified nominal disturbance measurement models and the additive uncertainty weights.