[American Institute of Aeronautics and Astronautics 48th AIAA Aerospace Sciences Meeting Including...

Blade Pitch Control with Preview Wind

Measurements �y

Jason Laksz Lucy Y. Pao x Alan Wright { Neil Kelley k Bonnie Jonkman ��

Light detection and ranging systems are able to measure conditions at a distance infront of wind turbines and are therefore suited to providing preview information of winddisturbances before they impact the turbine blades. In this study, preview-based distur-bance feedforward control is investigated for load mitigation both with and without theuse of multi-blade coordinate based controllers. Performance is evaluated assuming highlyidealized wind measurements that rotate with the blades and more realistic stationarymeasurements. The results obtained using idealized, \best case" measurements show thatexcellent performance gains are possible with reasonable pitch rates. However, the re-sults using more realistic wind measurements show that without further optimization ofthe controller and/or better processing of measurements, errors in determining the shearlocal to each blade can remove any advantage obtained by using preview-based feedforwardtechniques.

Nomenclature

R nominal blade radius [m],r0 radius at which individual wind measurements are made [m]w0 average (across the rotor disk) wind speed [m/sec],xr displacements that are inherently part of the blades [m],vr velocities corresponding to xr [m/sec],xar the vector of pitch actuator angles/states [deg],xt a vector of all other (�xed reference frame) turbine states,wsh the vector of �xed-frame wind perturbations used by FAST for linearization,w the vector of individual wind measurements, one-per-blade [m/sec],wmbc the vector of wind measurement MBC components (uniform, cosine and sine) [m/sec],cp the vector of pitch command inputs to the actuators [deg],p the vector of pitch angles achieved by the actuators [deg],pr the vector of pitch rates produced by the pitch actuators [deg/sec],mr the vector of blade root bending moments in the ap direction [kN-m],!g the generator speed [rpm],! the rotor speed [rad/sec],� the clock-wise angle of blade 1 from vertical [rad],�prev the angle of blade 1 expected after the elapsed preview time [rad],�prev the time delay between preview measurements of wind speed and their arrival at the turbine [sec].

�This work was supported in part by the US National Renewable Energy Laboratory and the US National Science Foundation(NSF Grant CMMI-0700877)yEmployees of the Midwest Research Institute under Contract No. DE-AC36-99GO10337 with the U.S. Dept. of Energy

have authored this work. The United States Government retains, and the publisher, by accepting the article for publication,acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish orreproduce the published form of this work, or allow others to do so, for the United States Government purposes.zDoctoral Candidate, Dept. of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO, Student

Member AIAA.xProfessor, Dept. of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO, Member AIAA.{Senior Engineer, NREL, Golden, Colorado, Member AIAA.kPrincipal Scientist, NREL, Golden, Colorado, Member AIAA.��Senior Scientist, NREL, Golden, Colorado, Member AIAA.

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American Institute of Aeronautics and Astronautics

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-251

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc.The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.All other rights are reserved by the copyright owner.

I. Introduction

The stochastic nature of the wind resource and the high initial capital cost and increasing, structural exibility of utility scale turbines motivate the adoption of advanced instrumentation and measurementtechnologies. One of the most attractive technologies is LIDAR (light detection and ranging) that has theability to make real-time measurements of wind conditions local to individual turbines. These types of mea-surements make it possible to employ disturbance feedforward techniques using actual wind measurementsinstead of employing wind estimates obtained from measurements of the turbine structural dynamics. Ina previous study,1 it was found that in non-turbulent conditions (but still time varying), controller perfor-mance is improved when using an aggregate measurement of wind such as horizontal hub height speed (HHS).However, in turbulent wind conditions, it was found that measurement of wind conditions local to each bladewere necessary to improve controller performance. Further, it was found that disturbance feedforward basedon HHS alone could actually be detrimental depending on the aggressiveness of the feedforward controller.Unfortunately, any improvement over and above that obtained using well-tuned feedback control came atthe expense of excessive pitch rates.

Figure 1. Preview control: the controller, in this case \FF(s)", has access to measurements of the disturbance \w(t)"T seconds before it reaches the turbine.

As shown in Figure 1, there is a variation of feedforward control known as preview2 that uses advanceknowledge of imminent disturbances (or commands). This technique is feasible using LIDAR technology sinceit is most easily con�gured to measure wind approaching the turbine rather than at the turbine. Hence, withLIDAR, there are measurements available of incoming wind perturbations before they impact the turbine.Additionally, depending on the LIDAR scanning pattern it is feasible to determine wind components ineither rotating or non-rotating reference frames as with multi-blade coordinates (MBC).3

In this study we explore the load mitigation performance of H1 MBC and non-MBC based previewcontrollers relative to an MBC, independent-pitch, feedback-only controller. The preview controllers haveaccess to time-advanced measurements of wind speed in addition to the feedback from generator speederrors and root bending perturbations that the feedback only controller uses. For comparison a PI collectivepitch controller using only generator speed error is also simulated. Simulations are performed using thesimulation code FAST4 developed at the National Renewable Energy Laboraty’s (NREL) National WindTechnology Center (NWTC) with a model of the three-bladed controls advanced research turbine (CART3).All simulations are done with a mean wind speed of 18 m/sec that places the CART3 squarely in aboverated (region 3) conditions where load mitigation is the primary control objective.

As explained in the next section, in ows for three di�erent meteorological conditions are generated usingNREL’s stochastic wind simulator Turbsim.5 These conditions are considered realistic for the NWTC site inGolden, CO and produce in ows that are non-uniform across the rotor plane with variational componentsabove and beyond what would be created by vertical and horizontal shear alone.

The preview measurements of wind variations are �rst obtained from three ideal measurements made at aradius equivalent to the 75% blade span in the vertical Y-Z plane at a distance equivalent to 0.45 sec previewin front of the turbine (8.1 m at a mean hub height wind speed of 18 m/sec). These ideal measurements

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are assumed to rotate in perfect unison with the instantaneous rotor speed so that the wind is measuredat the azimuth where the blades should arrive in an additional 0.45sec of travel based on the present rotorposition. Then the simulations are repeated using stationary wind measurements and interpolating windspeeds to the expected 0.45 sec blade positions assuming that the wind speed distribution is determinedby linear vertical and horizontal shear. As explained in Section II, these interpolations are straightforwardusing MBC coordinates.

In the following, we explain the wind measurement schemes implemented in our simulations. Section IIIof the paper introduces the version of preview control used in this study and explores the e�ect of pitch ratelimitation on achievable performance for various preview lengths. Then, in section IV the CART3 model andthe suite of controllers simulated are brie y summarized and followed by the section on simulation results.Except for two appendices, the paper concludes with a discussion of the results and plans for future work.Appendix VII explains the derivation of the turbine models used for controller design and the AppendixVIII details the design of the feedback controllers.

II. Wind Conditions and Measurements

Turbsim is used to produce thirty-one di�erent wind realizations for each of three meteorological condi-tions as summarized in Table 1. Each preview controller is simulated in each of the ninety-three resultingwind cases twice. One simulation uses idealized, best-case preview wind measurements and another simula-tion uses measurements that, as explained below, are inherently prone to error.

Table 1. Meteorological input parameters for TurbSim: a three dimensional wind vector is generated over a 31x31 pointgrid in the vertical Y-Z plane, centered so that it that encompasses the rotor disk. Over time the grid is sampled at20Hz for a total duration of 630 seconds. The wind pro�le within the grid is varied by the vertical stability parameterRitl and the mean friction velocity (shearing stress) u�D; a power law variation of the vertical wind speed pro�le isspeci�ed by the listed shear exponent �0.

The turbulent wind �elds produced by TurbSim contain variations in addition to shear, but for the sakeof clarity in the following discussion, we focus only on linear shear. A linear shear pro�le can be pictured asa perturbation away from uniform that is planar with some o�set and vertical and/or horizontal tilt.

Shear is typically quanti�ed in terms of the change in wind speed seen at the blade tips as a (unitless)fraction of the spatial average wind speed w0 across the rotor disk. If the horizontal wind speed is perturbedaway from the nominal w0 due to a uniform (spatially within the rotor plane) amount wu, as well as byhorizontal �h and vertical �v shear components, then the total perturbation �x at a location with horizontaland vertical coordinates (y; z) within the rotor disk is

�x(y; z) = �vw0z

R+ �hw0

y

R+ wu; (1)

where R is the radius of the rotor disk.We assume the use of three measurements from which an equivalent shear perturbation is deduced as

follows. Let � represent the clockwise angle of blade one (the convention used by FAST) from vertical andr0 represent the radius at which the wind speed is measured. Then the measurement position at radius r0along blade 1 is (y; z) = (�r0sin(�); r0 cos(�)) so that as a function of azimuth � the perturbation is

�x(�) = �vw0r0R

cos(�)��hw0r0R

sin(�) + wu (2)

= wccos(�) + wssin(�) + wu:

The MBC perturbations seen at the measurement radius r0,

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wmbc =

264wuwcws

375 =

2641 0 00 w0

r0R 0

0 0 �w0r0R

375264wu�v

�h

375 = M�1m2s wsh; (3)

are determined by the shear wTsh = [wu;�v;�h] encountered by the blades; Mm2s scales from MBC per-turbations to equivalent shear perturbations. The MBC components can be recovered from individualmeasurements [w1; w2; w3] = [�x(�);�x(� + 2�

3 );�x(� + 4�3 )] made along each blade at radius radius r0,

using the basic MBC transformation

wmbc =

264 13

13

13

23 cos(�) 2

3 cos(� + 2�3 ) 2

3 cos(� + 4�3 )

23 sin(�) 2

3 sin(� + 2�3 ) 2

3 sin(� + 4�3 )

375264w1

w2

w3

375 = T (�)�1 w: (4)

The convention for inverting T (�) turns out to be convenient for computing the MBC transformation of thelinearized turbine model, as outlined in Appendix VII.B. This relationship is exact if the perturbation isdue only to uniform and linear shear. Otherwise, this relationship may be viewed as a means of estimatingshear from individual measurements. Similarly, if the MBC perturbation wmbc is known, then the windperturbations measured at the blades when the rotor azimuth angle is � are given by the inverse MBCtransformation

w =

264w1

w2

w3

375 =

2641 cos(�) sin(�)1 cos(� + 2�

3 ) sin(� + 2�3 )

1 cos(� + 4�3 ) sin(� + 4�

3 )

375264wuwcws

375 = T (�) wmbc: (5)

A version of the FAST code is modi�ed to output the three wind measurements wT = [w1; w2; w3]at a distance ahead of the turbine which corresponds to 0.45 seconds at the average wind speed. Themeasurements are equally spaced around 2� radians at a radius equivalent to the 75% span of the blades.Since the FAST code is written to march the TurbSim wind �le past the turbine according to the averagewind speed for the �le and this is known a priori, the exact distance corresponding to the desired previewtime is known. In this way, it is possible to get very accurate measurements of the wind speed variationsthat will arrive at each blade.

In a stationary mode, the measurements are always taken with w1 at vertical (� = 0) and the othertwo locations spaced at multiples of 2�=3. Then, the MBC components are computed according to (4) with� = 0. In the non-MBC, preview controller, the blade positions expected after the preview time has elapsedare predicted according to

�prev = �rotor + ! � �prev; (6)

where ! is the present rotor speed and �prev is the desired preview time of 0.45 sec. The wind speeds thatthe blades will see at the preview position are estimated from the stationary measurements by interpolatingto the predicted blade locations using (5) with � = �prev. The MBC, preview controller uses the MBC com-ponents directly so the interpolation step is not necessary; after MBC transforming the individual stationarymeasurements, the results are fed directly to the controller (see Figure 6).

In a second, more accurate, rotating mode, the present blade locations are interpolated ahead to theirpositions (�prev) expected after the elapsed preview time and then the wind speeds [w1; w2; w3] are taken infront of the turbine from the expected blade positions. As these predicted positions rotate with the actualblade positions, these measurement locations in front of the turbine also rotate. In e�ect, the only errorin the resulting preview wind speeds for the rotating mode is due to the error in predicted blade locationwhich occurs because the rotor speed is not exactly constant. These more accurate measurements are thentransformed to their MBC components, or not, depending on whether the preview controller is MBC ornon-MBC based, respectively.

The result of these two measurement modes for a given simulation run are displayed in Figure 2 (a)wherein the actual wind speeds seen at 75% span are time-shifted to align with the preview measurements.

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The rotating measurements are shown in the top plot and are indistinguishable from the wind speeds actuallyencountered. The stationary preview measurements (center plot and the amplitude spectrum in the bottomplot) tend to have a low pass �ltering e�ect due to the MBC interpolation and produce a visible error. Mostof the error in the stationary approach is due to the fact that the wind conditions produce more than justshear variation across the rotor disk; the shear measured at the stationary locations can be signi�cantlydi�erent than that at the actual rotor positions.

Figure 2. Wind speeds at 75% span of one blade. (a) The top plot shows actual wind speeds and rotating measurements;center plot shows actual wind and speeds interpolated from stationary measurements. The bottom plot shows theamplitude spectrum of the wind seen at the blade and that of the interpolated measurements. (b) The histograms oferrors relative to actual wind speeds for rotating measurements and interpolations based on stationary measurements.

The maximum error (near 6 m/s) produced by the stationary-interpolation scheme is over an order ofmagnitude larger than that produced by the rotating measurements (< 0:5 m/s) as can be observed in thehistograms (Figure 2 (b)) of the prediction errors from data taken at the 75% span of one blade during 600sec of simulation time. Without utilizing some form of azimuth prediction (e.g., Kalman �ltering or someother model-based prediction), the rotating measurements are as good as we are able to produce at thistime. In fact, since in reality the wind speeds ahead of the turbine continue to evolve en route and thetime to arrival is not constant (as is implemented in FAST) the rotating measurements can be consideredunrealistically accurate. In future work, we will also consider the distortions and noise that are typical inLIDAR measurements. However, in this study, the goal for the rotating scheme is to better understand thebest possible outcome using preview control, while the stationary measurements serve as a �rst investigationinto the e�ect of errors in wind speed measurements.

III. Preview Control

Preview control is most readily accomplished in discrete time where time delays can be modeled as �rstorder shifts at the sample rate. In this study we take the approach outlined in the article by Takaba2

and augment the linearized, discrete-time plant model with chains of delays as described in Appendix VII.This incorporates the delay between the measurement of wind speeds and their arrival at the turbine intothe linearized, turbine model in the most straightforward fashion. The contents/measurements stored inthe delay chains are then viewed as part of the state of the augmented plant as depicted in Figure 3 for astate-feedback controller. The state-feedback architecture provides a relatively uncomplicated case in whichthe e�ect of pitch rate limitations can be investigated.

In this con�guration, the wind measurements are able to e�ect actuation before the wind impacts theturbine so this is truly preview control. The design of the state (plant and preview) feedback gains can beaccomplished with any desired approach. However, without formulating some goal with respect to cancelingthe e�ect of the wind at the plant outputs or states, feedback from the preview states is not required (e.g.,as would be the case for pole placement). Here we minimize the H1 gain from the preview measurementswT (k) = [w1(k) w2(k) w3(k)] to the blade root bending moments mT

r (k) = [mr1(k) mr2(k) mr3(k)] with theadditional constraint that the maximum, vector pitch rate pTr (k) = [pr1(k) pr2(k) pr3(k)] satisfy

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Figure 3. state-feedback, preview controller: the linearized plant (a) is augmented with states to model the delaybetween measured wind perturbations and their arrival at the turbine. The complete, linear preview turbine modelalso incorporates a simple actuator (b) so that pitch rate can be explicitly included in the design of the state-feedbackgains.

maxk

�pr(k)T pr(k)

�� 2

e2p: (7)

The performance level e2p is a speci�ed, peak magnitude for the pitch rate vector pr(k) for all windperturbations w(k) with total square area bounded by unity

1Xk=0

w(k)Tw(k) � 1: (8)

As explained in the book by Skelton, Iwasaki, and Grigoriadis (SIG)6 and the article by Scherer, Gahinetand Chilali,7 there are linear, matrix inequality (LMI) conditions guaranteeing these objectives that can beposed in a form that allows linear programming techniques to be applied. Given a state space realizationfor the augmented system

x(k + 1) = Ap x(k) +Bw w(k) +Bc cp(k); x(0) = 0 (9)pr(k) = Cr x(k) +Drw w(k) +Drc cp(k)mr(k) = Cm x(k) +Dmw w(k) +Dmc cp(k);

a state-feedback gain K results in a closed loop satisfying (7) and having H1 gain less than e2e if there isa positive de�nite matrix P and a general matrix G satisfying the following two LMIs:

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264 P ? ?

0 Iw ?

CrP +DrcG Drw 2e2pIr

375 > 0; (10)

26664P ? ? ?

0 Iw ? ?

ApP +BcG Bw P ?

CmP +DmcG Dmw 0 2e2eIm

37775 > 0:

Here ? represents the transpose of the corresponding block in the lower part of the matrix while Iw, Irand Iw are identity matrices with dimensions compatible with w, pr, and mr, respectively. The second ofthese inequalities follows quite quickly starting from the discrete-time bounded real lemma8 while a completeproof of the �rst LMI is given in SIG.6 Once solutions P and G are found, a state-feedback gain is given byK = GP�1. These inequalities are linear in the variables P and G and solutions can be found using linearprogramming techniques.

We compute solutions and the minimum achievable e2e for these LMIs using YALMIP9 and SDPT310

for peak pitch rate magnitudes e2p from 7.7 to 19 deg/sec (slightly above the peak rate for the CART3).Again, these are constraints on the magnitude of the pitch rate vector, but we �nd that they correlate fairlywell with peak, individual pitch rates produced by the resulting closed-loop linear model in response to acollective step change in wind speed.

These results are displayed in Figure 4. The plot on the left shows the open and closed-loop maximumsingular value versus frequency, of the transfer function from wind perturbations to the blade root bendingmoments. Each closed-loop response shows the result using a di�erent amount of preview; the same generouspitch rate constraint e2p of 15 deg/sec was applied in optimizing each closed loop. As the preview time isincreased, the peak, over frequency, of the singular values decreases with the largest improvement havingtaken place by about 0.40 seconds of preview. Beyond 0.45 seconds of preview there is a diminishing return.This is also evident in the plot on the right which shows the H1 gain achieved versus preview time for afamily of pitch rate constraints. For moderate to aggressive (say � 12 deg/sec for the CART3) pitch rates,there is very little improvement beyond 0.4 sec. of preview. However, if the system is pitch rate limited (say� 8 deg/sec), then additional preview beyond 0.75 sec (the maximum used in this study) may provide stillfurther decreases in the peak frequency response{ though, the performance achieved is no where near thatobtained with higher pitch rates.

An important mitigating factor that has not been considered in this study is the frequency range inwhich reducing the response to wind energy is most bene�cial. In particular, no frequency weighting wasused in augmenting the turbine model. For the purposes of design, �ltering representative of the expectedfrequency content of wind disturbances could be added at the bending moment outputs in formulating thecost function. Then in implementation, these extra dynamics would be subsumed into the controller. Theimportant point being that in the absence of frequency weighting, the limit in achievable performance isessentially set by the high frequency gain from wind to blade root bending moment. If little wind energy isexpected at these frequencies, then de-emphasizing this range with output weighting could signi�cantly e�ectthe results{ particularly where pitch rate limitations are concerned. Better control (more attenuation) mightbe achieved at lower frequencies within the bandwidth of the pitch actuator with appropriate weighting.

IV. Overview of Controllers

The state-feedback controllers explored in the previous section are not realizable and simply implementingthe feedback gains via an observer would not give the same closed-loop frequency response as true statefeedback. For this reason, the controllers simulated are dynamic output feedback controllers, but essentially,the same preview augmentation technique is used in their design as explained in detail in Appendix VIII.Increasing the amount of preview entails increasing the order of the plant by the number delays required forthe desired preview length and then multiplying this again for each measurement that is previewed. Thelargest preview we were successfully able to design for was 0.45 seconds (9 delays) at a controller sample rateof 20Hz (0.05 sec/sample). With three preview wind measurements (one per blade) this translates into an

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Figure 4. Performance results for pitch rate constrained, state-feedback preview controllers. (a) Open and closed-loop,maximum singular-value response of blade root bending moment to wind perturbations for a maximum pitch rate of15 deg/sec for a range of preview times as indicated (in seconds) by the color bar inset. (b) Best wind-to- ap momentH1 gain versus preview time for various pitch rate constraints as indicated (in deg/sec) by the color bar inset.

additional 27 states in the plant model{ needless to say, these are high-order controllers. However, based onthe state-feedback study, this length of time should be su�cient to gauge the bene�ts available from previewcontrol.

The controllers are designed using a linearized turbine model from FAST that includes degrees of freedomfor the generator inertia, drive train compliance and the �rst ap mode in each blade as summarized in Table2. The wind conditions are set to to be uniform across the rotor plane at 18 m/sec and the generator torqueis set to rated, and FAST computes the steady-state response of the turbine and then linearizes the plantat multiple rotor positions. As explained in appendices VII and VIII, the average linear turbine model isfurther augmented so that the resulting controllers provide asymptotic rejection of 1P bending moments andintegral control of rotor speed. To provide a baseline for comparison, we also simulate non-preview collectivepitch (CP) and MBC independent pitch (IP) controllers as shown in Figure 5. Frequency responses of thebaseline controller SISO loops can be viewed in Appendix VIII.

Table 2. 3 Bladed Controls Advanced Research Turbine

The preview controllers are distinguished by the characteristics that all controller inputs and outputsconnect with the same controller-dynamics and that the turbine model is augmented with extra states toincorporate the preview delay. The combined feedforward-feedback compensation is designed simultaneouslyusing the standard, generalized H1 framework.11 This results in a controller of the same order as thecomplete, augmented plant as depicted in Figures 6 (a) and (b), and no e�ort to reduce the controller orderpost design is made in this study.

FAST computes linearizations of the turbine model with respect to pitch and perturbations wsh in windshear, but as shown in Figure 6, the wind measurements are used by the controller either directly as in

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Table 3. Controller summary

(a) or immediately after the application of the MBC tranformation as in (b) without computation of shear.This is possible since prior to design, the coe�cients in the linearizaed model as computed by FAST withrespect to shear perturbations are pre-scaled in accordance with (3) or with further application of the MBCtransformation so that the controllers can be based on the direct measurements [w1; w2; w3] of wind speedor their MBC transformed counter parts [wu; wc; ws]. Details are given in Appendix VII.

The preview controllers are optimized so that there are two controllers for each of the MBC and non-MBCpreview cases with moderate and slightly aggressive rms pitch rates as summarized in Table 3. Includingthe baseline controllers, this makes a total of six designs used in performance simulations.

For comparison, the maximum singular value of the transfer function from wind perturbation to bladeroot bending moment are shown in Figure 7 for the MBC controllers on the left and the non-MBC controllerson the right. The intention was to have comparable control e�ort (as characterized by rms pitch rate) betweenthe MBC and non-MBC cases, but the MBC controllers were inadvertently weighted for slightly better speedregulation. The non-MBC designs contain augmented dynamics to reject 1P bending moments and this isevident in the notch at 0.7Hz (41.7 rpm). The MBC designs were augmented to achieve similar rejection, butbecause of the MBC transformation, these controllers see 1P variation as a DC disturbance in the verticaland horizontal components and so they are designed/augmented to have asymptotic rejection at 0 Hz. Notein both MBC and non-MBC cases, the amount of attenuation in the peak gain, over all frequencies, is betterwith larger pitch rates and that the peaks achieved are comparable between MBC and non-MBC previewcontrollers. The baseline, feedback-only controllers have signi�cantly higher peaks.

V. Simulation Results

The six controllers are simulated in each of the 93 wind realizations obtained from TurbSim. In everycase, all degrees of freedom for the turbine model provided by FAST were enabled even though the controllerswere designed for a reduced set of DOFs (see Table 2). The preview controllers are simulated twice in eachwind condition. One simulation uses the rotating measurement scheme and the second uses wind speedsinterpolated from stationary measurements. Recall that when using stationary measurements, after MBCtransforming the individual, stationary measurements the MBC controller uses the results directly whilethe results must be interpolated for use with the non-MBC controller. It turns out that the inaccuracyinherent in the stationary measurements still e�ects MBC and non-MBC controllers with the same sort ofdegradation in performance. The measurements need to re ect conditions local to the blades; the shear (orequivalent MBC) components experienced by the blades may be signi�cantly di�erent than those measuredat the stationary locations.

Box plots of the results for pitch rate, generator power, and blade-root bending moment in the apdirection are displayed in Figure 8 using data from all 93 simulations. The box indicates the median (notchcenter), mean (red belt), �rst quartiles (box ends), and the whiskers denote 5% and 95% and the dotsshow outliers. When using rotating measurements, the preview controllers provide signi�cant performance

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Figure 5. Baseline controllers. The collective pitch controller provides integral control on rotor speed error. The IPcontrollers provide asymptotic rejection of 1P bending moments. In implementation the MBC PI controllers are absentduring the CP simulations while all PI blocks are in place during simulation of the MBC IP controller.

improvements when compared with the feedback-only collective pitch (CP) and independent pitch (IP) base-line controllers. However, when using stationary measurements, the performance of the preview controllersis degraded to below that of the IP feedback-only controller.

Damage equivalent loads (DELs) are computed for blade root bending moment and tower base bendingmoment in the fore-aft and side-side directions. RMS tower sway is also computed in the fore-aft and side-side directions. The percent change for each controller relative to the IP feedback-only baseline controller isindicated in the bar charts of Figure 9; the �rst controller on the left in both charts is the CP feedback onlycontroller which has the same performance independent of the preview measurement technique (as does thebaseline IP controller). All the metrics are de�ned so that lower is always better.

As long as measurements accurately re ect conditions local to the blades (rotating measurements, leftchart), relative to baseline, preview control provides signi�cant improvement in the root ap moment (darkblue) for which the controllers were designed to mitigate. In particular, the use of preview control decreasesloading by more than 20%. When there are errors in wind measurements (stationary measurements, rightchart), the preview performance in the ap moment metric increases loading by over 10% in some cases.

The controllers were not designed to mitigate tower motion, but in the case of tower side-side motion,blade ap appears to be correlated in such a way that mitigation of root loads also reduces side-side loading.Similar correlations may also be in play between ap and fore-aft motion, but if so, it is probably obscuredby the relationship between speed regulation and fore-aft loading. With respect to tower loads in the fore-aftdirection (green bars), the use of preview can be signi�cantly worse than feedback only.

Overall we can say that the potential performance improvements with preview control are signi�cant.However, in order to realize these advantages, during design, careful consideration of the shortcomings inthe measurement system and the expected frequency content of the wind must be taken into account. Asexplained in the overview of the controllers, in this study, no special consideration was made with respect toeither of these factors{ this is a topic for future work.

VI. Conclusions and Future Work

In this study preview controllers have been designed and compared against baseline collective pitch andnon-preview independent pitch controllers for load mitigation in above rated conditons. The controllers wereaugmented to provide integral control of rotor/generator speed and asymptotic rejection of 1P variationsin blade root bending moments. This biased the controllers to mitigate these speci�c, narrow band distur-bances, but other than that, no weighting or foreknowledge of wind frequency content or of measurement

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Figure 6. Dynamic output feedback, preview controllers. (a) the non-MBC controller is augmented with 1P dynamicsto reject 1P variation in bending moment and integral control on rotor speed error. (b) The MBC controller sees1P variation as DC levels on the vertical and horizontal components of the MBC tranformed bending moments andhence it uses integral augmentation only. All measurements that inherently rotate with the blades (including rotatingmeasurements of the wind or stationary wind measurements that are interpolate to wind conditions at the rotatingblade positions) are transformed before being passed to the MBC controller. Similarly, the MBC controller pitchoutput components must be converted to individual/rotating pitch commands for each pitch actuator with an inverseMBC transform.

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Figure 7. Dynamic output feedback, preview controllers. The MBC preview controllers (a) have gains that go to zeroat DC while the non-MBC controllers (b) have a notch at the 1P frequency of 0.7 Hz. The baseline controllers havesigni�cantly higher peak frequency responses.

characteristics was brought to bear in their design.The results show that preview control can provide signi�cant improvements in load mitigation. However,

the amount of improvement is directly a�ected by available pitch rate and the accuracy of wind measurements.In turbulent conditions where shear is not uniform, the use of preview measurements that do not re ectconditions local to each blade can actually reduce performance below that attained using control based onlyon feedback of turbine measurements. It appears that the amount of deterioration will be dependent onthe aggressiveness of the controller and the frequency range where measurements provide the best accuracy.Even the simple interpolation scheme of the stationary measurements used in this study still retained someaccuracy at lower frequencies in turbulent conditions.

Originally, we thought that MBC might provide some insulation against measurement errors since it isbased on non-rotating components of blade variables. In particular, there was hope that it might work wellwith the much simpler stationary measurements, but results show that this is not the case. In turbulentconditions, the \local" shear or equivalent MBC components of the wind can vary signi�cantly as the bladesare positioned within the rotor disk at di�erent azimuths. Simulation results show that without furtherconsideration, handling of measurements, and/or optimization of the controller to account for the errorsthat simpler measurement schemes may produce, the advantages of wind measurements can be greatlydiminished. When conditions become turbulent enough it may be better to stop using preview control.This turned out to be the case for both MBC and non-MBC controllers. In more uniform, less turbulentconditions, however, we still expect that the use of measurements simpler than a rotating scheme can bebene�cial for either MBC or non-MBC controllers. And, when wind variation is largely due to linear shearand uniform perturbations, MBC provides a signi�cant advantage in that the rejection of 1P variation inbending moments can be accomplished independent of rotor speed and with simpler controller dynamics.

The sensitivities to measurement error in turbulent conditions might be mitigated by more e�ort in threeareas. First, the controller should be optimized taking into account the expected spectral content of winddisturbances. Second, if it is known that the wind measurements have poor accuracy within certain (e.g.,high) frequency ranges, then at least the feedforward section of the controller response needs to be mutedat these frequencies to avoid actuation in response to noise. And third, the controller e�ort and, if possible,measurements need to be matched to the available actuator bandwidth. This study only took into accountactuator limitations by weighting pitch rate during optimization so that reasonable pitch rates were attained.Other than speci�c augmentations to achieve rejection of DC speed errors and 1P bending moments, thecontrollers were not optimized to address any speci�c frequency range. As a result, the amount of attenuationachieved was essentially determined by the response of the turbine at higher frequencies. We expect thatbetter performance and perhaps less sensitivity to measurement error, can be attained by weighting the

12 of 24


Figure 8. Data for all 93 simulations. Pitch rates (top left) clearly show that the preview-measurement techniquehas a signi�cant e�ect on results; other metrics show that in turbulent conditions, the accurate, rotating previewmeasurements produce superior results over feedback-only CP and IP controllers. Average generator power (lower left),RMS blade root bending moments (top right), and peak bending moments (lower right) all show that preview controllerperformance is superior with rotating measurements but degraded to below the performance of the IP controller whenmeasurements do not accurately re ect conditions local to the blades (as is the case with the stationary-interpolationtechnique). However, in general, the preview using the stationary-interpolation measurement technique is still betterthan collective pith CP.

performance objective to de-emphasize higher frequencies. If the majority of wind energy is within thebandwidth of the actuator, then in the presence of measurement errors it may still be possible to retainmuch of the performance gain achieved with ideal measurements.

VII. Appendix: Turbine Models

The starting point for obtaining a turbine model is simulation and linearization using FAST at ratedspeed and torque. The salient features of the CART3 are summarized in Table 2. The wind speed is setto 18 m/sec uniform and FAST simulates the turbine to �nd periodic operating conditions; then FASTlinearizes the turbine at a number of equally spaced rotor azimuths. We then use these linearizations toobtain an average state space model as described in the following sections.

VII.A. Appendix: Non-MBC Turbine Model

A linearized state space model of the turbine

_xp = AF (�)xp + [BFsh(�) BFp(�)]

"wsh

p

#; (11)"

mr

!g

#=

"CFm(�)CFg(�)

#xp +

"DFmsh(�) DFmc(�)DFgsh(�) DFgc(�)

#"wsh

p

#:

is obtained from FAST at rotor azimuth � where xTp = [xTt ; xTr ; v

Tr ] contains states representing the turbine

�xed-frame degrees of freedom (DOF) and blade displacements and velocities. We augment this system with

13 of 24


Figure 9. Performance relative to feedback-only, independent pitch (IP); the collective pitch, feedback-only controller(CP) is shown on far left of each plot. Bars indicate percent improvement (decrease) in metrics relative to IP. Leftplot shows results using the accurate rotating measurements and the right plot shows results using the stationary-interpolated measurements; the CP performance doesn’t change because it does not use preview measurements (nordoes the IP baseline). When using rotating measurements, the preview controllers have signi�cant performance im-provements over using feedback-only IP with the exception of the RMS velocities in the fore-aft direction of thenon-MBC preview controllers (green); the damage equivalent load in blade root ap moment (dark blue) is reduced byover 20% relative to feedback only for all preview controllers. When using the stationary measurements (right plot),the performance of the preview controllers indicates a sensitivity to measurement error; the lower pitch rate MBC andthe higher pitch non-MBC controllers show the biggest degradation in performance.

�rst order actuator models

_xar = Aar xar +Bac cp; (12)"p

pr

#=

"Cap

Capr

#xar +

"Dap

Dapr

#cp

where (see Figure 3)

Aar = �30 � I3�3; Bac = 30 � I3�3; (13)Cap = I3�3; Dap = 0 � I3�3;

Capr = �30 � I3�3; Dapr = 30 � I3�3;

to get a composite state space system

_x =

"AF (�) BFp(�) Cap

0 Aar

#x+

"BFsh(�) BFp(�) Dap

0 Bar

#"wsh

cp

#; (14)264!gmr

pr

375 =

264CFg(�) DFgp(�) CapCFm(�) DFmp(�) Cap

0 Capr

375x+

264DFgsh(�) DFgp(�)Dap

DFmsh(�) DFmp(�)Dap

0 Dapr

375"wshcp

#;

or more simply

_x = A(�)x+ [Bsh(�) Bc(�)]

"wsh

cp

#; (15)264!gmr

pr

375 =

264Cg(�)Cm(�)Cr(�)

375x+

264Dgsh(�) Dgc(�)Dmsh(�) Dmc(�)Drsh(�) Drc(�)

375"wshcp

#;

14 of 24


where xT = [xTt ; xTr ; v

Tr ; x

Tar].

The only change we wish to make in this model for non-MBC design purposes is to have input coe�-cients which are relative to equivalent blade local wind perturbations (and will have corresponding previewmeasurements). Recall (Section II eq.’s (3) and (4)) that if the wind perturbation consists of only uniformand shear components, then a transformation from the equivalent individual perturbations to the shearperturbations used by FAST is given by

wsh = Mm2sT (�)�1w; (16)

where the conversion Mm2s from MBC to shear perturbations and the basic MBC transform T (�)�1 arede�ned in equations (3) and (4), respectively. Now, a state space model with input coe�cients relative toperturbations in individual wind measurements is obtained by substituting (16) into (11) to obtain

_x = A(�)x+ [Bsh(�) Bc(�)]

"Mm2sT (�)�1 0

0 I

#"w

cp

#; (17)264!gmr

pr

375 =

264Cg(�)Cm(�)Cr(�)

375x+


375"Mm2sT (�)�1 00 I

#"w

cp

#;

or simply

_x = A(�)x+ [Bw(�) Bc(�)]

"w

cp

#; (18)264!gmr

pr

375 =

264Cg(�)Cm(�)Cr(�)

375x+

264Dgw(�) Dgc(�)Dmw(�) Dmc(�)Drw(�) Drc(�)

375"wcp

#:

Then, an average state space system is obtained from the complete set of linearizations (18) at N azimuthangles �i by computing, for example

A =1N

NXi=0

A(�i): (19)

VII.B. Appendix: MBC Turbine Model

The MBC transform is used on each degree of freedom (DOF) that rotates with the turbine blades as afunction of azimuth �. Loosely speaking, the transform computes the cos and sin coe�cients of the 1Pvariation in the rotating variables. The basic transform is de�ned in terms of displacements; each DOFthat rotates with or is inherently part of a blade has a displacement/state xi. The goal is to express thedi�erential equation (15) for the linear system in states/coordinates that do not rotate with the blades. Thisis accomplished by expressing the rotating states as a function of their MBC/non-rotating counter parts.The rotating state xr is given by the inverse MBC transform of corresponding non-rotating componentsxTnr = [xu; xc; xs] according to

xr = T (�)xnr: (20)

Each DOF in a mechanical system also has an associated velocity vi = _xi which requires computation of

_xr = _T (�)! xnr + T (�) _xnr: (21)

15 of 24


where ! = _� is the rotor speed. So, the second-order transformation to the rotating DOFs is given by"xr

vr

#=

"T (�) 0_T (�)! T (�)

#"xnr

vnr

#(22)

where vnr = _xnr and

_T (�) =

2640 � sin(�) cos(�)0 � sin(� + 2�

3 ) cos(� + 2�3 )

0 � sin(� + 4�3 ) cos(� + 4�

3 )

375 : (23)

This transformation must be replicated for each rotating degree of freedom. For the sake of simplicity, weassume there is only one rotating DOF in the turbine model, but we add the three, parallel, �rst-ordermodels of the pitch actuators that rotate with the blades. Including these as part of the rotating systemrequires that we modify the transformation to

264 xrvrxar

375 =

264 T (�) 0 0_T (�)! T (�) 0

0 0 T (�)

375264 xnrbnr

xanr

375 ; (24)

where xar contains the pitch angles produced by the three actuators and xanr contains the uniform, cosineand sine components of the pitch in the non-rotating frame. This completes the transformation betweennon-rotating and rotating states. Now, recalling that all the other (non-rotating) turbine states are lumpedinto the vector xt, the complete (non-dynamic) MBC state transformation is given by

26664xt

xr

vr

xar

37775 =

26664I 0 0 00 T (�) 0 00 _T (�)! T (�) 00 0 0 T (�)

3777526664xt

xnr

vnr

xanr

37775 (25)

x = M(�)x̂:

We now make use of (25) to derive the state space representation of the linear di�erential equation innon-rotating/MBC coordinates. This is accomplished by �rst taking the time derivative of both sides of(25),

(26)26664_xt_xr_vr_xar

37775 =

266640 0 0 00 _T (�)! 0 00 �T (�)!2 + _T (�)! _! _T (�)! 00 0 0 _T (�)!

3777526664xt

xnr

vnr

xanr

37775+

26664I 0 0 00 T (�) 0 00 _T (�)! T (�) 00 0 0 T (�)

3777526664

_xt_xnr_vnr_xanr

37775

=

266640 0 0 00 _T (�)! 0 00 �T (�)!2 2 _T (�)! 00 0 0 _T (�)!

3777526664xt

xnr

vnr

xanr

37775+

26664I 0 0 00 T (�) 0 00 0 T (�) 00 0 0 T (�)

3777526664

_xt_xnr_vnr_xanr

37775_x = �M(�)x̂+Md(�)

d

dtx̂

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where

�T (�) =

2640 � cos(�) � sin(�)0 � cos(� + 2�

3 ) � sin(� + 2�3 )

0 � cos(� + 4�3 ) � sin(� + 4�

3 )

375 ; (27)

and simpli�cations (to get the diagonal matrix Md(�)) are obtained by assuming _! = 0 and utilizing the factthat _xnr = vnr. Equating this result with the right hand side of (15), substituting in (25) for x and solvingfor the time derivative of x̂ gives

d

dtx̂ = Md(�)�1(A(�)M(�)� �M(�))x̂+M�1

d (�)[Bsh(�) Bc(�)]

"wsh

cpnr

#; (28)

d

dtx̂ = Md(�)�1(A(�)M(�)� �M(�))x̂+M�1

d (�)[Bsh(�) Bc(�)]

"Mm2s 0

0 T (�)

#"wmbc

cpnr

#;

= AMBC(�)x̂+ [BwMBC(�) BcMBC(�)]

"wmbc

cpnr

#

where we have used the conversion Mm2s from MBC wind components to shear wind components that wasde�ned in (3). In a similar fashion for the outputs, we obtain

(29)264 !g

mrnr

prnr

375 =

264I 0 00 T (�)�1 00 0 T (�)�1

375264Cg(�)Cm(�)Cr(�)

375M(�)x̂

+

264I 0 00 T (�)�1 00 0 T (�)�1


375"Mm2s 00 T (�)

#"wmbc

cpnr

#264 !g

mrnr

prnr

375 =

264CgMBC(�)CmMBC(�)CrMBC(�)

375 x̂+

264DgwMBC(�) DgcMBC(�)DmwMBC(�) DmcMBC(�)DrwMBC(�) DrcMBC(�)

375"wmbccpnr

#:

An average MBC state space system is obtained from the complete set of linearizations in the same manneras for the non-MBC case, e.g.,

AMBC =1N

NXi=0

AMBC(�i): (30)

VIII. Appendix: Controller Design

Time delay is modeled in the most straight forward fashion in discrete time so the starting point for designis conversion of the models in previous sections to discrete time. We use a zero-order hold equivalent12 andtake the liberty of using the same variable names as in the continuous time realization. Then, the controllersare designed by �rst augmenting the discrete-time, linear model with delays matching the desired previewtime so that the time span between the measurement of wind and its arrival at the turbine is incorporatedinto the model. The model is further augmented so that the closed loop has the desired asymptotic propertieswith respect to the blade root bending moments and rotor speed. Plant outputs are selected for performanceobjectives as required for the standard H1 framework for use with the robust control toolbox in Matlabr.The following sections brie y document the construction of the augmented, generalized plant and its weightedversion for H1 design.

17 of 24


VIII.A. Appendix: Non-MBC Preview Controller

Assume that the continuous-time state-space system from the previous section has been converted to discretetime to obtain

x(k) = Ax(k) + [Bw Bc]

"wt(k)cp(k)

#; (31)264!g(k)

mr(k)pr(k)

375 =

264CgCmCr

375x(k) +

264Dgw Dgc

Dmw Dmc

Drw Drc

375"wt(k)cp(k)

#:

Here the wind input has been renamed as wTt = [wt1; wt2; wt3] to emphasize that this is the wind at the75% span of each turbine blade; wT = [w1; w2; w3] is now reserved for the measurements out in front of theturbine. The �rst step in preview control design is to augment this state-space model with delay chains toincorporate, as part of the turbine model itself, the time span between measurement of wind w and its arrivalat the turbine wt. Say, for example, the amount of preview is three samples, then a state-space realizationof this delay is

xwi(k + 1) =

2640 0 01 0 00 1 0

375xwi(k) +

264100

375wi(k); (32)

wti(k) = [0 0 1] xwi(k);

where we use one delay chain per measurement/blade (i = 1; 2; 3). When these realizations are put inparallel, a composite system with state xTW = [xTw1; x

Tw2; x

Tw3] is denoted as

xW (k + 1) = AD xW (k) +BD w(k) (33)wt(k) = CD xW (k):

Now, assuming the wind does not change en route to the turbine, an accurate model which includes thedelay between measurement and arrival is

"x(k + 1)xW (k + 1)

#=

"A BwCD

0 AD

#"x(k)xW (k)

#+

"0 Bc

BD 0

#"w(k)cp(k)

#; (34)26664

!g(k)mr(k)pr(k)w(k)

37775 =

26664Cg DgwCD

Cm DmwCD

Cr DrwCD

0 0

37775"x(k)xW (k)

#+

266640 Dgc

0 Dmc

0 Drc

I 0

37775"w(k)cp(k)

#:

Note that the vector of wind measurements w(k) have been fed through as an additional system output.This is done because the preview controller has access to this vector as a \feedback" measurement and it isessentially treated the same as the other feedback signals in the H1 design process.

It is desired to have integral control on generator speed error and rejection of 1P variations in the bladeroot bending moments. So, the average model is augmented with an integral/accumulator of perturbations!g(k) in generator speed

xs(k + 1) = xs(k) + !g(k) (35)!gs(k) = xs(k);

18 of 24


and also with second-order dynamics that oscillate at the 1P frequency !1p. A realization of a 1P oscillatoris

_x1p =

"0 �!1p

!1p 0

#x1p +

"01

#u; (36)"

ys1p

yc1p

#=

"1 00 1

#x1p:

Both states of the 1P dynamics are made available as outputs; the controller will have access to both whilethe ys1p output will be made part of the cost. Denote the discrete-time, parallel combination of three ofthese systems as

x1P (k + 1) = A1px1P (k) +B1p mr(k) (37)"mrs(k)mrc(k)

#=

"C1ps

C1pc

#x1P (k);

where the input vector mr(k) is the turbine root bending moments and mrs and mrc hold the ys1p and yc1poutputs, respectively, for the three parallel systems. Now, with these new outputs and additional dynamics,the augmented turbine model is given by

26664x(k + 1)xW (k + 1)xs(k + 1)x1P (k + 1)

37775 =

26664A BwCD 0 00 AD 0 0Cg 0 1 0

B1p Cm B1p DmwCD 0 A1p

3777526664x(k)xW (k)xs(k)x1P (k)

37775+

266640 Bc

BD 00 Dgc

0 B1p Dmc

37775"w(k)cp(k)

#; (38)

266666666664

!g(k)!gs(k)mr(k)mrs(k)mrc(k)pr(k)w(k)

377777777775=

266666666664

Cg DgwCD 0 00 0 1 0Cm DmwCD 0 00 0 0 C1ps

0 0 0 C1pc

Cr DrwCD 0 00 0 0 0

377777777775

26664x(k)xW (k)xs(k)x1P (k)

37775+

266666666664

0 Dgc

0 00 Dmc

0 00 00 Drc

I 0

377777777775"w(k)cp(k)

#:

Finally, for the sake of completeness, we document the �nal two steps in the process of generalizing theaverage model for H1 design. The plant outputs (with the exception of mrc and w) are duplicated andserve as the cost objective, while the original set of outputs (with the exception of the pitch rate pr) aremade available for feedback to the controller.

All the performance outputs are weighted by additional scalar factors except for the generator speedoutput which requires additional handling. Dynamics are added to this output to emphasize the drive trainresonance at frequency !dt and thereby assure su�cient damping of this mode. We use a dynamic weightgiven by

Wdt(s) =s=(0:2�2) + 1

(s=!dt)2 + s=!dt=5 + 1(39)

that has gain increasing from unity at 0.2Hz with a 14dB peak (a Q of 5) at the drive train resonance andthen a 20dB/dec roll o� there after. Denote a discrete-time realization of this weighting function by

xdt(k + 1) = Adtxdt(k) +Bdt!g(k) (40)!gw(k) = Cdtxdt(k)

19 of 24


and let [Kdt;Ks;Km;Kpr] denote additional scalar weights for the generator speed, integral of generatorspeed, and pitch rate, respectively. De�ne the vector of performance outputs as yp and the vector offeedback outputs as yf . Then, the �nal form of the generalized turbine model is

2666664x(k + 1)xW (k + 1)xs(k + 1)x1P (k + 1)xdt(k + 1)

3777775 =

2666664A BwCD 0 0 00 AD 0 0 0Cg DgwCD 1 0 0

B1pCm B1pDmwCD 0 A1p 0BdtCg BdtDgwCD 0 0 Adt

3777775

2666664x(k)xW (k)xs(k)x1P (k)xdt(k)

3777775+

26666640 Bc

BD 00 Dgc

0 B1pDmc

0 BdtDgc

3777775"w(k)cp(k)

#;

(41)

"yp(k)yf (k)

#,

266666666666666666664

2666664!gw(k)!gs(k)mr(k)mrs(k)pr(k)

37777752666666664

!g(k)!gs(k)mr(k)mrs(k)mrc(k)w(k)

3777777775

377777777777777777775

=

266666666666666666664

0 0 0 0 KdtCdt

0 0 Ks 0 0KmCm KmDmwCD 0 0 0

0 0 0 C1ps 0KprCr KprDrwCD 0 0 0Cg DgwCD 0 0 00 0 1 0 0Cm DmwCD 0 0 00 0 0 C1ps 00 0 0 C1pc 00 0 0 0 0

377777777777777777775

2666664x(k)xW (k)xs(k)x1P (k)xdt(k)

3777775+

266666666666666666664

0 00 00 KmDmc

0 00 KprDrc

0 Dgc

0 00 Dmc

0 00 0I 0

377777777777777777775

"w(k)cp(k)

#:

H1 optimization �nds a dynamic, stabilizing, output-feedback controller

cp(k) = G � yf (k); (42)

that minimizes

J(G) =P1k=0 yp(k)T yp(k)P1k=0 w(k)Tw(k)

: (43)

It can be shown that this is equivalent to minimizing the peak over all frequencies of the maximumsingular value of the closed-loop transfer function from w to z.13 We use the robust control toolbox inMatlabr to �nd a near optimum controller for two di�erent values of the pitch rate weight Kpr and therebyget controllers that have di�erent RMS pitch rates.

Figure 10 shows the open-loop plant and weighted performance outputs. The augmented responses showthat we require asymptotic rejection of DC speed at the generator (top plot) and 1P bending moments inthe blade root ap (center plot); the gain in the performance/cost at these frequencies is in�nite. The lowerplot shows that the weight applied to pitch is biased towards high frequencies and dominates (in terms ofmagnitude) the performance outputs. The blue \Pr W1" and red \Pr W2" weights result in RMS pitchrates of about 7 deg/sec and 11 deg/sec respectively. The response with respect to the other blade localwind inputs is the same; although it is not apparent without seeing the response of blade 2 to disturbancesat blade 1, the wind-to-bending moment responses are largely uncoupled.

VIII.B. Appendix: MBC Preview Controller

The procedure for design of the MBC preview controller is nearly identical to that used in the non-MBCcase, except of course, that the starting point is the linearized, MBC turbine model (28) and (29). There are

20 of 24


Figure 10. Non-MBC Performance Output Weighting{ in all cases, larger response implies more weight implies betterattenuation in closed loop. The top two plots show response to blade 1 local wind disturbances: top plot shows weightingused on the generator speed output; center plots show the weighting used on blade root bending moments. The bottomplot shows the weighting on the pitch actuator outputs in response to pitch commands.

two additional signi�cant caveats. The �rst is in implementation (see Figure 6(b)) where the all the signalsgoing into the controller need (if they are not already) to be transformed to non-rotating coordinates viaT�1(�) while the outputs of the controller, need to be transformed back to rotating coordinates with T (�).The second is that in the non-rotating frame, 1P variations appear as DC amplitudes in the cos and sincomponents. The upshot being that in order to have asymptotic rejection of 1P bending moments, we onlyneed to augment the system with integrators on the cos and sin components as compared with the use ofthree second-order oscillating systems on the individual bending moments as in the non-MBC case. The useof MBC coordinates a�ords another signi�cant advantage in this regard in that the rejection of 1P variationsis accomplished independent of rotor speed. In the non-MBC case, the plant was augmented with dynamicswhich only oscillate at exactly !1p and therefore only provide asymptotic rejection at that frequency. Whenrotor speed is not at the expected !1p, then the ability of the non-MBC controller to attenuate 1P variationis diminished. Use of the MBC transform, which determines the amplitude (and phase) of the 1P variationspatially as a function of rotor position instead of speed, circumvents this problem.

Again, the discrete-time, average, MBC turbine (+actuator) model is augmented with delays equalto the desired preview. However, in this case, the input to the delays is a measure of the non-rotating(MBC transformed) components of the wind in front of the turbine. The turbine model is also augmented

21 of 24


with integrators on the cos and sin components of blade root bending moment, and the same performanceweighting on the generator speed output (this output is in the non-rotating frame so it is the same in MBCand non-MBC cases).

We spare the reader from another derivation that would be nearly identical to the one presented forgeneralizing the non-MBC turbine. Instead, we simply state that the generalized MBC turbine model can beobtained from (41) by replacing non-MBC matrices with their MBC counterparts, removing the mrc output,and replacing the state-space realization fA1p; B1p; C1pg for three parallel, second-order 1P oscillators withthe simpler realization fI2x2; I2x2; I2x2g for two parallel accumulators. This serves to augment the MBCmodel with accumulators on perturbations in the cosine and sine components of the non-rotating bendingmoment.

Figure 11 shows the plant open loop and weighted performance outputs except for pitch rate which isdisplayed in Figure 12. Unlike the non-MBC plot in Figure 10 which shows the response from only one input,we show the response from all MBC inputs to all outputs. This serves to illustrate that the collective channelis signi�cantly di�erent than the vertical and horizontal channels and to show that the system is to a largeextent diagonal. That is, the collective output is very insensitive to vertical and horizontal perturbations(and vise-versa); the vertical and horizontal outputs are uncoupled except at frequencies above 1 Hz. Thesame observations with respect to coupling can be made upon viewing the MBC pitch model response inFigure 12. As was the case for the non-MBC design, the blue \W1" and red \W2" weights result in RMSpitch rates of about 7 deg/sec and 11 deg/sec respectively.

VIII.C. Appendix: Non-Preview Baseline Controllers

We conclude with bode plots of the non-preview controller loops. These are very conservative PI controllersdesigned with bandwidths that hopefully minimize the amount of bending moment generated while regulatingspeed and also prevent the vertical and horizontal MBC loops from coupling.

Figure 11. MBC Performance Output Weighting{ in all cases, larger response implies more weight implies betterattenuation in closed loop. The top row shows generator speed response; the lower three rows show MBC collective,vertical, and horizontal bending moment component responses respectively. Note that in the lower 3-by-3 set of plots,the diagonal responses are larger than the o� diagonal except for cross coupling of vertical and horizontal shear atfrequencies above 1 Hz.

22 of 24


References

1J. H. Laks, L. Y. Pao, and A. Wright, \Combined feedforward/feedback control of wind turbines to reduce blade apbending moments," in Proceedings AIAA/ASME Wind Energy Symposium, Orlando, FL, Jan. 2009, pp. 82{86.

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Figure 12. Pitch rate weighting{ in all cases, larger response implies more weight implies less pitch in closed loop.Pitch rate is weighted so that it dominates the cost except at DC.

Figure 13. Baseline loop responses. (a) Collective pitch to generator speed; the addition of a notch guaranteesattenuation of drive train resonance. (b) The vertical-to-vertical and horizontal-to-horizontal loops are identical; thecross over is very conservative to insure gain is less than 1 where vertical and horizontal channels couple (at frequenciesabove 1 Hz).

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