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Karl Stole-mail: [email protected]

Mark Balase-mail: [email protected]

Department of Aerospace Engineering Science,University of Colorado at Boulder,

Boulder CO

Full-State Feedback Control of aVariable-Speed Wind Turbine:A Comparison of Periodic andConstant GainsAn investigation of the performance of a model-based periodic gain controller issented for a two-bladed, variable-speed, horizontal-axis wind turbine. Performancbased on speed regulation using full-span collective blade pitch. The turbine is mowith five degrees-of-freedom; tower fore-aft bending, nacelle yaw, rotor position,flapwise bending of each blade. An attempt is made to quantify what model degrefreedom make the system most periodic, using Floquet modal properties. This justifiinclusion of yaw motion in the model. Optimal control ideas are adopted in the desigboth periodic and constant gain full-state feedback controllers, based on a lineaperiodic model. Upon comparison, no significant difference in performance is obsebetween the two types of control in speed regulation.@DOI: 10.1115/1.1412237#

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IntroductionIn the design of a wind turbine controller for a particular ope

ating condition, it is common to treat the turbine as a timinvariant system. As such, the variation in dynamic propertieseach blade rotates is ignored. Gain-scheduling techniques exthis basic principle by discretizing the expected operating reg~e.g., at various wind speeds! and switching between the constagains designed for each condition. For the design of these congains, there are various methods available. Proportional-integderivative ~PID! control is the most common technique becaulittle knowledge of the plant dynamics is necessary prior to impmentation@1#. As dynamics models of turbines are developemore information can be incorporated into model-based contlers, with methodical design procedures and increased capabOptimal control is one such method that has been investigrecently for wind turbines@2#. If the controller model is assumetime-invariant—as is invariably the case to take advantage oftensive control theory—this method again involves constant gaThe periodic nature of wind turbine dynamics, both from aeronamic loads and structural properties, suggests an improvemcan be made to overall performance if a periodic control law, wperiodic gains, is used. The goal of this paper is to makeassessment of the effectiveness of periodic control, as compto constant gain control, in speed regulation.

Model-based linear control design requires a suitably accudynamics model of the system. For structural dynamics, we mtake advantage of a utility that derives the equations of motiona simple wind turbine model explicitly. This code is referred toSymDyn in previous papers by the authors. In SymDyn, ribodies with single revolute joints and springs are used to moall flexible components—namely the tower, shaft, and blades.result is a relatively simple set of physical degrees-of-freedoThe equations of motion have been validated with ADAMS® @3#,a rigorously tested dynamics code, in another paper@4# and havebeen exercised in applications such as operating modal ana@5,6# and preliminary control studies@7#. For the analysis in thispaper, we use a simple 5-degree-of-freedom SymDyn model.

An explicit set of governing equations for aerodynamic loaon turbine blades is currently not available. Instead, to comp

Contributed by the Solar Energy Division of the THE AMERICAN SOCIETY OFMECHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR EN-ERGY ENGINEERING. Manuscript received by the ASME Solar Energy DivisioMarch, 2001; final revision July 2001. Associate Editor: D. Berg.

Copyright © 2Journal of Solar Energy Engineering

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the aeroelastic model we use aerodynamic subroutines from ADyn @8#, developed at the University of Utah. AeroDyn has becoupled with many other structural dynamics models, includYawDyn, FAST–AD, and ADAMS®. Details on how AeroDyn isused in this paper will be presented in a later section.

Optimal periodic control has seen rigorous mathematical trement in the last few decades@9# with application to helicoptercontrol @10# and more recently spacecraft orbit optimization@11#.Limited application of this theory has also been made to wturbines to reduce internal loads with active yaw control@12#. Thecontrol objective in this paper is to regulate turbine speed ifluctuating wind field near rated wind speed~region III or constantpower turbine operation!. We use a constant generator torque acontrol full-span collective blade pitch.

The following section of this paper describes the structural aaerodynamic models in more detail and explains how the periolinear model is derived. The remaining sections introduce methfor measuring periodicity, describe the control system desigand then present results comparing the constant and periodiccontrollers.

Model DescriptionThe modeled turbine is a downwind, free-yaw, variable spe

machine with two blades and no hub teeter. Only flap motioneach blade is considered, using a flap hinge and torsional sprinmodel the first flap-bending mode. Similarly, only fore-aft bening of the tower is modeled. The tower sections, shaft, and blsections are treated as rigid bodies. The result is 5 degreefreedom; tower fore-aft pitch~t!, nacelle yaw~g!, shaft rotation/azimuth position~c!, and flap of each blade (b1 ,b2). Thesedegrees-of-freedom are chosen to give a periodic yet low osystem to analyze. Properties are chosen to resemble the AWhorizontal-axis wind turbine. See Fig. 1 for an illustration aTable 1 for a list of the main geometric properties.

For the system described, the nonlinear equations of motionbe represented by the following vector equation.

fI ~qI ,qI ,qI ,LI !50I , (1)

where

qI 5@t g c b1 b2#T, the angular displacements,

qI 5dqIdt

, qI 5d2qIdt2

,,

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Fig. 1 SymDyn wind turbine model showing degrees-of-freedom and geometric parameters

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andL is the vector of applied loads, described shortly.

Aerodynamics. Aerodynamic loads are calculated by AerDyn subroutines at prescribed elements along each blade leusing blade-element theory. The element loads are then sumand applied to the structure at the blade flap hinges, consiswith the SymDyn configuration in Fig. 2. Models for dynaminflow and dynamic stall in AeroDyn are not incorporated.simple hub-referenced wind field is used, consisting of a unifowind field over the rotor swept area, modified by a vertical shfactor. A gust speed component adds to the mean wind speedis not modified by shear. The user’s manual for AeroDyn@8# pro-vides additional description details.

With two blades, there are a total of six forces and six momeA constant generator torque is also applied to the shaft, witreaction torque on the nacelle. By assuming a constant meanspeed and vertical shear factor, the aerodynamic loads becofunction of rotor orientation and velocity, blade pitch and wingust speed. This is summarized by the following load vecfunction.

LI 5LI ~qI ,qI ,w,u! (2)

wherew is the wind gust speed, andu is the collective blade pitch angle.

Linearized System. The aeroeleastic turbine model is nonliear in all its variables. Combining Eqs.~1! and ~2!:

fI ~qI ,qI ,qI ,LI ~qI ,qI ,w,u!!50I (3)

A linear state-space representation is obtained by first pertureach of the system variables about an operating point, as follo

qI 5qI op1DqI , qI 5qI op1DqI , qI 5qI op1DqI ,

Table 1 Geometric properties of the turbine model

Description Symbol Value

Hub height dhh 25 mHeight of tower fore-aft hinge dt 9.3 mYaw axis to flap hinge distance dn 2.8 mShaft to flap hinge distance dh 5.8 mBlade length - 13 mPrecone angle b0 7°

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MDqI 1CDqI 1KDqI 5EI Dw1HI Du (4)

whereDqI is the vector of perturbed angular displacements,Dw is the perturbed guest speed,Du is the perturbed blade pitch,

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Finding the Jacobian matrices for the structural dynam(] fI /]qI , . . . ,] fI /]LI ) is straightforward since the nonlinear eqution, f, is in symbolic form from SymDyn. Without a symbolirepresentation for aerodynamics, calculating the remaining Jbians (]LI /]qI , . . . ,]LI /]u) requires numerical differentiationCurve fitting is used to minimize numerical errors resulting frothis calculation.

The linear system parameters~M, C, K, EI , HI ! are functions ofthe operating point. For example, the operating azimuth posi(cop) is periodic in time, with period equal to one rotor revoltion. This results in a linearperiodic system. The operating conditions are further described in the next section.

For control system design and modal analysis, the final steto transform the linear equations into familiar first-order staspace form.

xI 5AxI 1Bu1Bdud (5)

where

A5F 0535 I535

2M 21K 2M 21CG , the state matrix,

B5F 0535

M 21HI G , the control input matrix,

Bd5F 0535

M 21EI G , the wind input matrix,

xI 5@DqI DqI #T, the vector of perturbed states,

u5Du, the control input~blade pitch!, and

ud5Dw , the wind disturbance input~gust speed!.

Operating Point. At the operating point, the nonlinear vectoequation~6! must be satisfied. This is one of the assumptions uwhich the linearization is based.

fI ~qI op ,qI op ,qI op ,LI ~qI op ,qI op ,wop ,uop!!50I (6)

The operating point is consequently found by time simulationthe nonlinear system until a trim or steady-state solutionreached. In general, for a periodic system, this trim solution is aperiodic in time. In the cases studied, this variation was foundbe small and therefore averaged values are used for the operpoint.

qI op51

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qI op50I , except cop is constant (7)

qI op50I

In using an averaged trim solution for the operating point, subquent calculations are simplified and control by state regulahas a more meaningful objective, i.e. constant, not periodic,erence speed. However, Eq.~6! is no longer satisfied, which suggests that the linear model will behave less like the nonlinmodel.

Table 2 lists the chosen wind conditions~AeroDyn input param-eters!, blade pitch and constant generator torque to keep rspeed at a mean of 57.5 rpm. The resultant period of dynamicT,is 1.04 seconds. With a constant generator torque, power regtion is achieved through speed regulation only. Shown in F3–6 are selected components of the trim solution. Notice thatvariations about the mean values are relatively small, particulfor rotor speed. Also, the mean yaw angle is nonzero due toimbalance of aerodynamic loads caused by the vertical wind s~similar toP-factor in a propeller-powered aircraft!. Upon averag-ing the trim solution, the operating point is given by

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PeriodicityPeriodic dynamics in a wind turbine model arise from t

choice of structural degrees-of-freedom, the presence of graand the aerodynamic factors that are incorporated. With aeronamics, any time-invariant spatial variation in the wind fiecauses periodic loads as each blade samples changing wind vity with each revolution. The primary sources of once-prevolution load variations are vertical shear, horizontal shecause, in a steady-state condition, the rotor would be yawedposition such as to minimize load imbalance across the ro

Fig. 3 Tower fore-aft angle variation in the periodic trim solu-tion „mean shown dashed …

Fig. 4 Yaw angle variation in the periodic trim solution „meanshown dashed …

Table 2 Chosen operating conditions „presented in AeroDynnomenclature where possible …

Description Value

Mean wind speed V517 m/sWind direction d50°Vertical speed Vz50 m/sHorizontal shear factor HSHR50Vertical shear factor VSHR50.2Operating gust speed wop(5VG)50 m/sOperating blade pitch uop515°Generator torque Mg523.1 kNm

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Tower shadow affects only a small sector of the blade sweptand, therefore, would require compensation from a high bawidth controller, which may not be possible. Time-varying spavariations in the wind field can contribute to the once-prevolution periodicity of the system, depending on the frequecontent. However, since these variations are transient in nathey are best treated as disturbance inputs in the simulation m

In contrast, it is generally difficult to visualize the affect thdifferent structural degrees-of-freedom have on how periodicoverall dynamics are. There are a number of factors to consDegrees-of-freedom in the rotating frame, such as blade flaphub teeter, are affected by gravity loads, which are periodic inframe. This is somewhat less significant compared to the influethat the rotor has on degrees-of-freedom in the non-rotating fraFor example, the moment of inertia about the yaw axis chanperiodically with azimuth position of a two-bladed rotor. Thchange could be more than 100 percent depending on blade leand mass. Hence, yaw dynamics are greatly affected. Towernamics are similar except that the large tower and nacelle inetend to lessen the periodic influence of the rotor.

Initial control studies quickly revealed that the extent of peodicity of an open-loop system determines how effective periocontrol will be, compared to constant gain control. The followitwo tests are conceived to quantify periodicity and hence obthe most suitable structural model for control studies.

Test 1: Parameter VariationConsider the periodic system, with periodT, defined by

Fig. 5 Blade #1 flap variation in the periodic trim solution„mean shown dashed …

Fig. 6 Rotor speed variation in the periodic trim solution„mean shown dashed …

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xI 5A~ t !xI 1B~ t !u1Bd~ t !ud , (8)

where

A~ t1T!5A~ t !,

B~ t1T!5B~ t !, and

Bd~ t1T!5Bd~ t !.

We can partitionA, B, andBd to form

A~ t !5A01DA~ t !

B~ t !5B01DB~ t ! (9)

Bd~ t !5Bd01DBd~ t !

whereA0 , B0 , andBd0 contain the mean values ofA(t), B(t),andBd(t) over t P @0,T#. ThenDA(t), DB(t), andDBd(t) con-tain the periodic variations, themagnitudesof which, compared toA0 , B0 , andBd0 , would possibly give a measure of the degreeperiodicity. Thus, we may define the following quantities to mesureA(t), B(t), andBd(t), respectively.

a5maxtP@0,T#

iDA~ t !iiA0i

, where i•i is the matrix 2-norm

b5maxtP@0,T#

iDB~ t !iiB0i

, (10)

bd5maxtP@0,T#

iDBd~ t !iiBd0i

Note that this test considers only the magnitude of periodic vations in the system matrices. No phase information is takenaccount.

Test 2: Modal Variation.The stability of a periodic system can be determined usFloquet-Lyapunov theory@6,13#. Essentially, Floquet theory usea periodic coordinate transformation to change the original sysinto one with constant coefficients. A conventional eigenanalyis then used to calculate the system’scharacteristic exponents~often referred to as Floquet modes or Floquet exponents! as fol-lows.

lC~A~ t !!51

Tln~e.v.~F~T,0!!! (11)

where F(T,0) is the state transition matrix, satisfyingxI (T)5F(T,0)xI (0).

A periodic system is then defined as stable if all its characistic exponents$lc(A(t))% lie inside the left half of the complexplane, i.e., Re(lci),0, ; i .

The characteristic exponents of atime-invariant system aresimply equal to the conventional eigenvalues$l0(A)% of that sys-tem. This is certainly not true for a general periodic system. Hoever, one would expect that a weakly periodic system would hcharacteristic exponents close in value to the set of conventieigenvalues found by incrementing time over the period. That

$lc~A~ t !!%'$l0~A~ t0!!% ;t0P@0,T#. (12)

One would also expect that a strongly periodic system would hconventional eigenvalues that trace a larger region, within whthe characteristic exponents may or may not lie. The results ofmodal variation test are best viewed in the complex plane~Arganddiagram!. Note that this test considers only the periodic naturethe state matrix,A(t). The periodicity of the input matrices~B(t)andBd(t)! will also affect the performance of any controller.

Example Comparison. To illustrate the described tests wcompare two open-loop wind turbine models. The first has alldegrees-of-freedom of the model already defined, except w



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tower fore-aft pitch removed–leaving yaw, azimuth, and flangle of two blades. The second model includes tower forepitch but with yaw motion locked.

The results from the parameter variation test are shown in T3. They suggest that the second model has greater periodic vtions in the system matrices than the first model. This wouldply that model 2, with the fore-aft tower degree-of-freedom,more periodic than the model with free yaw motion.

Results from the modal variation test, shown in Figs. 7 anddraw the opposite conclusion. The first model has conventioeigenvalues that trace a much larger region in the complex plThe characteristic exponents of the second model in fact lie vclose to its conventional eigenvalues. This suggests that peridynamics play a more significant role in model 1, with the ya~not tower fore-aft! degree-of-freedom. On basic physical grounthis would appear to be the correct conclusion, since one woreason that the large tower and nacelle mass should maketower dynamics less sensitive to rotor position, unlike ydynamics.

While not proven analytically, the modal variation test is blieved to provide the more justifiable measure of periodicity. Itsuggested that phase variations in the system matrices causparameter variation test to be less useful, as in the example gAlso, the modal variation test is based on system characterithat more closely relate to control design attributes, namely,bility and dynamic response. A numerical control system studythe two four degrees-of-freedom models~results not provided forbrevity! showed that periodic control performs best when the ydegree-of-freedom is included. This further supports the hypoesis that the modal variation test is the more accurate for demining periodicity.

The purpose of providing a test for periodicity is to findstructural model that cannot be controlled effectively when treaas a time-invariant system. The two-bladed, free-yaw turbmodel displays the greatest amount of periodicity that the authhave found. The original 5-degree-of-freedom model descri

Fig. 7 Modal Variation Test for Model 1 „with yaw degree offreedom …

Table 3 Results of the parameter variation test

Model 1 Model 2

a 0.30 0.37b 0.0055 0.0092bd 0.030 0.029



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early is of this class. The remainder of this paper is devotedstudying the response of this turbine to constant gain and pericontrollers.

Control System DesignTo simplify the study, full-state feedback is used without a st

estimator. It is also assumed that the wind gust variation is knoby the controller at all times. The design of a wind and stestimator is not difficult, but requires that the system be obseable through measurement of rotor speed alone~or possibly withother signals!. It is known that wind and state estimation cannimprove on full-state feedback control. Whether state estimaaffects periodic control more than constant gain control is ananswered question at this time.

The following subsections describe the design of the consgain and periodic gain controllers.

Constant Gain Controller. Since the linear state-space reresentation is periodic, a suitable time-invariant orfrozensystemis obtained by choosing some representative time within theriod (t fP@0,T#) and calculating the state matrices as follows.

xI 5AfxI 1Bfu1Bd fud (13)

where

Af5A~ t f !,

Bf5B~ t f !, and Bd f5Bd~ t f !.

The value oft f is chosen to correspond to a rotor azimuth angle90° ~blades horizontal!. This selection is made on the basis ofparametric control study, by comparing system performancevarious freezing azimuth angles. Using an angle of 90° resultthe best controller when acting on the linear periodic plant. Inestingly, it is possible to destabilize the linear periodic plant whparticular freezing angles are chosen~e.g., 135°!.

Optimal control via linear quadratic regulation~LQR! is em-ployed to calculate the constant feedback gains. This is a stanprocedure for time-invariant systems, detailed in many texts sas @14#. The basic theory is briefly described below as an intduction to the periodic case.

LQR design involves finding the linear feedback gains thminimize the quadratic cost function,

J5E0

`

~xI TQxI 1uI TRuI !dt (14)

Fig. 8 Modal Variation Test for Model 2 „with tower degree offreedom …



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Table 4 Control performance results

Q0

EOG Wind Case Sampled Wind Case

Constant Gain Control Periodic Gain Control Constant Gain Control Periodic Gain ContrSpeed RMS

@rpm2#ADC

@deg/s#Speed RMS

@rpm2#ADC

@deg/s#Speed RMS

@rpm2#ADC

@deg/s#Speed RMS

@rpm2#ADC

@deg/s#

1 0.137 1.87 0.136 1.83 0.104 1.45 0.104 1.4410 0.0459 2.01 0.0452 1.98 0.0359 1.56 0.0352 1.5100 0.0148 2.00 0.0141 1.99 0.0109 1.61 0.0102 1.6

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whereQ is the symmetric, positive semidefinite weighting on tstates, andR is the symmetric, positive definite weighting on thinput, uI 5GfxI .

In most cases the quadratic cost,J, has no physical significancebut rather provides a means to trade-off competing design reqments; state regulation versus control usage. The optimal full-sfeedback gains,Gf , are calculated from the solution of the Algebraic Riccati Equation~ARE!,

AfTPf1PfAf1Q2

1

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TPf50. (15)

Then

Gf521

RBf

TPf .

The objective is to regulate rotor speed, soQ is chosen to have theform

Q5Q0 diag@0, 0, 1, 0, 0, 0, 0, 1, 0, 0#,

a diagonal matrix with nonzero entries corresponding to tstates; azimuth error~Dc! and rotor speed error (Dc). Q0 is ascalar weighting, which together withR provides the means tochange the control objective. Increasing values inQ0 ~with Rconstant! produces a controller that attempts to regulate sperror, with little regard to the amount of control input~blade pitch!used.

Provided we have (Af ,Bf) stabilizable, the closed-loop system

xI 5AcxI , where Ac5Af1BfGf , (16)

is guaranteed to be stable for thefrozen system. For the windturbine system under investigation (Af ,Bf) is controllable.

To account for wind gust input,ud , the control input is aug-mented by a disturbance gain.

u5GfxI 1Gd fud (17)

From Eqs.~13! and ~16!, the closed loop system is then

xI 5AcxI 1~BfGd f1Bd f!ud . (18)

The disturbance gain is calculated to minimize, in a least-squsense, the effect of wind gust on the system. So we use,

Gd f52Bf1Bd f ,

whereBf15Bf

T(BfBfT)21, the pseudoinverse ofBf .

Periodic Controller. Optimal control for periodic systems inot as straightforward as for time-invariant systems. Howetheory dealing with this subject has been developed~e.g.,@9#! andis readily applicable. Minimizing the quadratic cost function~Eq.~14!! for a periodic system requires finding a solution to the Priodic Riccati Equation~PRE!.

H P~ t !1A~T!TP~ t !1P~ t !A~ t !1Q21

RP~ t !B~ t !B~ t !TP~ t !50

P~ tfinal!5Pfinal(19)

Then the optimal linear gains are



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The PRE is a differential matrix equation and can be integrabackward in time from the final condition,Pfinal , until conver-gence to a symmetric, positive definite, and periodic solutiP(t). The final condition is chosen to be the solution of the AR~15!, Pfinal5Pf .

The closed-loop periodic system is stable provided (A(t),B(t))is stabilizable. That is, all unstable characteristic exponentsA(t) are reachable. Such is the case for the wind turbine mostudied.~For a test of reachability, see@9#.!

Similar to the constant gain case, the disturbance gain~nowperiodic! can be calculated to minimize the effect of wind gust

Gd~ t !52B~ t !1Bd~ t ! (20)

ResultsTwo turbulent wind conditions are chosen to test the perf

mance of the two controllers. In both conditions a constant mwind speed of 17 m/s is used~with other properties as per Table 2!but with different gust speed profiles. Recall that, while the mewind speed is modified by the vertical shear factor, the gust spis not. The first wind case is an IEC standard one-year extreoperating gust~EOG-1 yr! as described in Ref.@15#. The gustspeed starts at zero and reaches a maximum of 9 m/s at 10onds, as shown in Fig. 9~combined with the mean wind speed!.The second wind case is based on actual sampled wind data tat the National Wind Technology Center. It is the more turbuleprofile with fluctuations of gust speed between27 m/s and 7 m/s,as shown in Fig. 10.

A comparison is made between the periodic and constantcontrollers, as designed in the previous section. The control inweighting,R, is fixed at unity and the state weighting,Q0 , is runthrough 1, 10, and 100. The nonlinear aeroelastic model ofturbine is used for simulation. For reference, the open-loopsponse of rotor speed for the two wind cases is given in Figsand 12. This illustrates how well the system responds withoucontroller present. Controller performance is measured bymetrics, RMS speed error and actuator duty cycle~ADC!. Actua-tor duty cycle is related to pitching rate and measures the tangle pitched by the blades divided by the total simulation timGenerally, the better the speed regulation~lower RMS speed er-ror! the higher the control usage~higher ADC!. This is the typicaldesign trade-off.

Results are presented in Table 4. Larger values ofQ0 increasethe magnitude of the gains, with the expected outcome ofproved speed regulation—true for both controllers and both wcases. As a comparison, the RMS speed errors of the open-rotor speed responses~Figs. 11 and 12! are 7.4 rpm2 and 10.2rpm2 respectively. Closed-loop responses for the periodic contler with Q05100 are shown in Figs. 13 and 14. Correspondblade pitch commands are plotted in Figs. 15 and 16.

Based on the results in Table 4, there is no significant permance difference between the periodic gain and constantcontrollers. This is true for both the slowly varying EOG standawind case and the more erratic sampled wind. One advantage



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the periodic controller has is that stability is guaranteed when uwith a linear periodic plant, even at high gains. This is a funmental result of periodic optimal control theory. When applieda nonlinear plant~as in this study!, while stability is not guaran-teed, one could expect that the periodic controller is less likelydestabilize the system. The biggest disadvantage of this contris the additional issues of implementation. Periodic controlquires knowledge of the rotor azimuth position at all times, a sthat is certainly not observable by measuring rotor speed al

Fig. 9 Combined mean ¿gust wind speed data from the IECEOG-1 yr standard „at hub-height …

Fig. 11 Rotor speed open-loop response to the EOG windcase

Fig. 13 Rotor speed closed-loop response for both controllers„EOG wind, Q 0Ä100…



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Therefore additional hardware, such as an optical sensor, wbe required on the wind turbine to continuously measure angposition of the shaft. A constant gain control system is morebust in the sense that there is no additional hardware requiredcould fail.

To find a more favorable application for periodic control, it mabe necessary to investigate a different control objective. One rois with Independent Blade Control~IBC! as studied in the heli-

Fig. 10 Combined mean ¿gust wind speed for the sampledwind case „at hub-height …

Fig. 12 Rotor speed open-loop response to the sampled windcase

Fig. 14 Rotor speed closed-loop response for both controllers„sampled wind, Q 0Ä100…



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copter field@10#. While this does not change the natural dynamof the model~i.e., the state matrix,A(t)!, it does allow increasedcontrol capability~i.e., the input matrixB(t) increases in dimen-sion!. This will make the system inherently more periodic, butwhat degree is uncertain. Speed regulation may not benefit grbut other objectives, such as yaw control or load mitigation, mnow be possible.

In this study, the nonlinearity issues of the turbine havebeen addressed. Realistic wind conditions may force the turbinoperate in a region far beyond those for which the controllerbeen designed. Common gain-scheduling techniques could bplied for these cases. For the periodic control law it wouldnecessary for gain transitions to be smooth regardless of theriod of the gains~e.g., when the operating rotor speed is change!.With the use of azimuth position as the synchronizing variableissue should not be problematic.

Since all states of the wind turbine are used in the discuscontrollers, a state-estimator is needed for practical implemetion. For periodic control, one could design a periodic staestimator based on the periodic model of the system. Forconstant gain controller, one would have to choose a suitatime-invariant turbine model. It is unclear at this stage the degof suboptimality that will result in either case.

ConclusionsAn attempt is made to quantify the characteristic of periodic

with some success in the form of a modal variation test. Sucmeasure provides the control designer a method of predicwhether periodic control will be more effective than using costant gains. A two-bladed, free-yaw turbine model displays signcant periodicity and thus appears to be a promising candidateperiodic control.

Optimal control theory is applied to build both constant gaand periodic gain controllers, designed with the objective of relating speed with collective blade pitch. No significant perfomance difference between the two types of controllers is obseusing two different wind gust input cases and different gain mnitudes. Given the additional complexities of design and impmentation, the marginal benefits of periodic control do not appto justify its use for speed regulation. The continued future of tcontrol method may require a more sophisticated actuamethod, such as independent blade pitch, or a different objecfunction, such as mitigating tower or blade loads.

Regardless of whether a suitable application can be foundperiodic control it can always serve as a baseline in the desigexcellent constant gains. If periodic control is used, and a fau

Fig. 15 Blade pitch control input for both controllers „EOGwind, Q 0Ä1000…



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detected in the synchronization hardware~e.g., azimuth sensor!,an optimized set of constant gains would be an approprbackup.

AcknowledgmentsThis work was supported by the National Renewable Ene

Laboratory ~NREL!, under contract number XCX-9-29204-04The authors wish to thank Gunjit Bir of NREL for his knowledgable advice on rotating system dynamics.

References@1# Hand, M., and Balas, M., 2000, ‘‘Systematic Controller Design Methodolo

for Variable-Speed Wind Turbines,’’ J. Wind Eng.,24, pp. 169–187.@2# Bassanyi, E. A., 2000, ‘‘Developments in Closed Loop Controller Design

Wind Turbines,’’Proc. of 19th ASME Wind Energy Symp.,Reno NV, pp. 64–74.

@3# Mechanical Dynamics, 1998,Using ADAMS/Solver (v9.1), Mechanical Dy-namics, Inc., Ann Arbor MI.

@4# Stol, K., and Bir, G., 2000, ‘‘Validation of a Symbolic Wind Turbine StructurDynamics Model,’’Proc. of 19th ASME Wind Energy Symp., Reno NV, pp.41–48.

@5# Stol, K., Bir, G., and Balas, M., 1999, ‘‘Linearized Dynamics and OperatiModes of a Simple Wind Turbine Model,’’Proc. of 18th ASME Wind EnergySymp., Reno NV, pp. 135–142.

@6# Bir, G., and Stol, K., 2000, ‘‘Modal Analysis of a Teetered-Rotor Wind Turbinusing the Floquet Approach,’’Proc. of 19th ASME Wind Energy Symp., RenoNV, pp. 23–33.

@7# Stol, K., Rigney, B., and Balas, M., 2000, ‘‘Disturbance Accommodating Cotrol of a Variable-Speed Turbine using a Symbolic Dynamics StructuModel,’’ Proc. of 19th ASME Wind Energy Symp., Reno NV, pp. 84–90.

@8# Hansen, A. C., 1996,Users Guide to the Wind Turbine Dynamics ComputPrograms YawDyn and AeroDyn for ADAMS, Mech. Eng. Dept., Univ. ofUtah, Salt Lake City UT.

@9# Bittanti, S., Laub, A.J., and Willems, J.C.~eds.!, 1991,The Riccati Equation,Springer Verlag, Berlin, pp. 127–162.

@10# McKillip, R., 1984, ‘‘Periodic Control of the Individual-Blade-Control Heli-copter Rotor,’’ Ph.D. thesis, MIT, Cambridge MA.

@11# Jensen, K. E., Fahroo, F., and Ross, I. M. 1998, ‘‘Application of OptimPeriodic Control Theory to the Orbit Reboost Problem,’’Proc. of AAS/AIAASpace Flight Mechanics Meeting, Univelt, Inc., San Diego CA, pp. 935–945.

@12# Ekelund, T., 2000, ‘‘Yaw Control for Reduction of Structural Dynamic Loain Wind Turbines,’’ J. Wind. Eng. Ind. Aerodyn.,85, pp. 241–262.

@13# Johnson, W. J., 1980,Helicopter Theory, Princeton University Press, PrincetoNJ, pp. 369–377.

@14# Kwakernaak, H., and Sivan, R., 1972,Linear Optimal Control Systems, WileyInterscience, New York NY.

@15# International Electrotechnical Commission~TC88!, 1999, ‘‘Wind TurbineGenerator Systems—Part 1: Safety Requirements,’’ 2nd Ed. IEC 6140Geneva.

Fig. 16 Blade pitch control input for both controllers „sampledwind, Q 0Ä1000…



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