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Adaptive Passivity Based Individual Pitch Control for Wind Turbines in the Full LoadRegion

Sørensen, Kim L. ; Galeazzi, Roberto; Odgaard, Peter F.; Niemann, Hans Henrik; Poulsen, Niels Kjølstad

Published in:Proceedings of the 2014 American Control Conference

Link to article, DOI:10.1109/ACC.2014.6858651

Publication date:2014

Document VersionEarly version, also known as pre-print

Link back to DTU Orbit

Citation (APA):Sørensen, K. L., Galeazzi, R., Odgaard, P. F., Niemann, H. H., & Poulsen, N. K. (2014). Adaptive PassivityBased Individual Pitch Control for Wind Turbines in the Full Load Region. In Proceedings of the 2014 AmericanControl Conference (pp. 554-559). IEEE. https://doi.org/10.1109/ACC.2014.6858651

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Adaptive Passivity Based Individual Pitch Controlfor Wind Turbines in the Full Load Region

Kim L. Sørensen1,2 and Roberto Galeazzi2 and Peter F. Odgaard3 and Henrik Niemann2 and Niels K. Poulsen4

Abstract— This paper tackles the problem of power reg-ulation for wind turbines operating in the top region byan adaptive passivity based individual pitch control strategy.An adaptive nonlinear controller that ensures passivity ofthe mapping aerodynamic torque-regulation error is proposed,where the inclusion of gradient based adaptation laws allowsfor the on-line compensation of variations in the aerodynamictorque. The closed-loop equilibrium point of the regulationerror dynamics is shown to be UGAS (uniformly globallyasymptotically stable). Numerical simulations show that theproposed control strategy succeeds in regulating the poweroutput of the wind turbine despite fluctuations of the windfield due to wake and turbulence, without overloading the pitchactuators.

I. INTRODUCTION

In recent years the research and development of windenergy harvesting systems focused on optimizing differentaspects of the wind turbine in order to improve its Costof Energy (CoE). Increase the performance of the controlsystem is one of the ways to obtain such an enhancementby optimizing the wind turbine controller in a number ofways. The industrial state-of-the-art solutions rely on PIDcontrollers, which depend upon linearized models of the rotordynamics, and the strong non-linearities introduced by theaerodynamic torque are handled by gain scheduling. Standardapproaches to wind turbine control can be found in [12].

Nowadays, the state-of-the-art wind turbine is a threeblades variable speed pitch controlled wind turbine that froma control perspective has two control actuations, namely theblade pitch angle and the generator torque. The overall windturbine controller has two main control actions, leaving outyaw control, subsystem, and auxiliary controls. Each blade isactuated through a pitch servo, which can change the bladedeflection such that the aerodynamics of the wind turbineis modified. These three pitch actuators can be operatedwith different control references, and if so the controller iscalled an individual pitch controller (IPC); however this isnot the standard solution in which all three pitch actuatorsare fed with the same control reference. This control strategyis called collective pitch control (CPC). The other control

1Kim L. Sørensen is with the Centre for Autonomous Marine Operationsand Systems, Norwegian University of Science and Technology, Trondheim,Norway kim.sorensen@itk.ntnu.no

2Roberto Galeazzi and Henrik Niemann are with the Department ofElectrical Engineering, Technical University of Denmark, Kgs. Lyngby,Denmark {rg,hhn}@elektro.dtu.dk

3Peter F. Odgaard is with the Department of Electronic Systems, AalborgUniversity, Aalborg, Denmark pfo@es.aau.dk

4Niels K. Poulsen is with the Department of Applied Mathematicsand Computer Science, Technical University of Denmark, Kgs. Lyngby,Denmark nkpo@dtu.dk

actuation is the generator torque that can be commanded bythe electronic converters through which the generator is fullyor partly connected to the grid.

Wind turbines typically operates in two general modes(not including low and high wind operation and additionalswitching modes): partial load and full load. In partial loadthe objective is to maximize the generated power. In thismode the pitch controllers are idle and the blades are kept atthe optimal pitch angle, while the rotor speed is controlled bythe generator torque. In full load the pitch actuator are usedto keep the rotor torque at the nominal value by controllingthe rotor speed, while at the same time the generator torqueis typically kept at its nominal value.

In the last decade nonlinear control theory has foundan interesting test bed in control design for wind turbinesystems, which certainly presents inherent challenges due tothe complex non-linearities introduced by the aerodynamictorque. The design of variable speed nonlinear controllershas received particular attention, also due to the fact thatthe nonlinear effects of the aerodynamic torque significantlysimplify when the wind turbine operates in the partial loadregion. Interesting examples can be found in [3], [4], [5],[8], [20], [22], and [24]. However in the full load region theaerodynamic torque lacks of an explicit representation as afunction of the tip speed ratio and the pitch angle; this hasrepresented a major challenge for the application of nonlinearcontrol methods such as e.g. feedback linearization, back-stepping, etc., that exploit such representation of the non-linearities to achieve regulation or tracking. Other controlstrategies as nonlinear model predictive control (e.g. [10],[15], [21]) and gain scheduling ([2] and references therein)have been proposed for the power regulation problem.

Since a wind turbine is an energy generating unit it wouldbe relevant to use passivity based methods to enforce regula-tion into the system. Passivity based control has been appliedfor a fixed pitch passive stall wind turbine [18] with theobjective to optimize power production using the generatortorque reference as control signal. This study showed aninteresting potential for using these methods in wind turbinecontrol. However an active pitch wind turbine is somewhatdifferent, so it may not be obvious how to extend the existingwork to this kind of wind turbines, and as well taking intoaccount full load control, where the pitch actuators are active.

This paper addresses the problem of power regulationin the full load region by means of a nonlinear individualpitch controller. Under the assumption of constant generatortorque, the control of the generated power is set up asregulation of the rotor angular velocity about its nominal

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value despite wind field fluctuations by means of the blades’pitch angles. First, feedback passivation [14] is appliedto establish a passive mapping between the aerodynamictorque and the rotor speed’s regulation error. Then dampinginjection is achieved through proportional feedback. Last, asimple control allocation strategy is applied to distribute thedemanded control effort to the individual pitch controllers.The lack of an explicit formulation of the power coefficientis overcome by enhancing the controller through gradientbased adaptation laws, which estimates the controller coef-ficients online. Simulation results confirm the effectivenessof the proposed control scheme, which fulfills the controlrequirements without overloading the pitch servos.

The theoretical tools utilized, namely feedback passiva-tion and gradient based adaptation laws, are certainly wellestablished; however the novelty of the paper resides in thecombination of these tool to provide an effective nonlinearcontrol architecture which achieves power output regulationin the full load region assuming no prior knowledge of theaerodynamic torque.

The paper is organized as follows. Section II presents thewind turbine dynamical model along with the wind fieldmodel. Section III is a thorough account of the passivity-based control design procedure, including online adapta-tion. Section IV presents the simulation results and theperformance assessment of the proposed controller. Last, theconcluding remarks found in Section V finalize the paper.

II. WIND TURBINE MODEL

The considered wind turbine is a horizontal axis three-bladed variable speed variable pitch system, which consistsof a wind turbine and a generator. In all aspects of modelingthe 3-bladed NREL 5MW offshore baseline wind turbine isutilized [13].

A. Wind field model

The wind field depends on numerous factors, such as geo-graphical position, season, terrain layout, tower dimensions,etc. The model presented here is an approximation of thewind field that impacts the wind turbine.

The wind field is modeled as wind speed velocity at agiven position in the rotor field. The model consists of fourdifferent contributions: mean wind vme; wind shear vsh; towershadow vto; wake vwa. The unified wind field model vw isgiven by

vw (t;∆r,θ) = vme(t)+ vsh(t;∆r,θ)

+ vto(t;∆r,θ)+ vwa(t;∆r,θ), (1)

where ∆r is the radial distance from the rotor center to anypoint on the length of the blade, and θ is the azimuthalangle (or angular coordinate). Wind shear and tower shadowwere modeled as in [6], therefore descriptive expressions forthose two contributions are omitted for brevity. The wakeand stochastic additions have been modeled as in [7], withthe exception that in this paper the intensity of the wakehas the shape of a Gaussian distribution. For a completemathematical description of the complete wind field the

60

80

100

−30−20−100102030

13

13.5

14

14.5

15

15.5

Lateral [m]Height [m]

Win

dspe

ed [m

/s]

13.2

13.4

13.6

13.8

14

14.2

14.4

14.6

14.8

15

Fig. 1. Complete wind field at a specific time instant.

interested reader is referred to [25]. The wind field at aspecific time instant is shown in Fig. 1.

It is assumed that the wind field can be split into threeequivalent wind signals, each of which acts on one of thethree blades, i.e.

vw,b j+1

(t;θ0 +

2π j3

)=

1n

n

∑i=1

vw,i

(t;∆ri,θ0 +

2π j3

), (2)

with i and n denoting the blade number and total number ofblades, respectively; where j = 0,1,2, and θ = θ0 representsthe azimuthal coordinate associated with the highest point inthe rotor field.

B. Turbine model

A conventional way of characterizing the aerodynamic ef-ficiency of a wind turbine is the power coefficient CP (λ ,β )

1,which is a function of the tip-speed-ratio λ and the pitchangle β . CP is given by the ratio between the aerodynamicpower Pr captured by the wind turbine and the poweravailable Pa in the wind area A swept by the blades, i.e.

CP (λ ,β ) =Pr

Pa. (3)

The available power is given by

Pa =12

ρAv3w, (4)

with ρ being the air density, vw denoting the wind speed,and A = πr2, where r is the blade tip radius. Rearranging(3) and inserting (4) the aerodynamic power is obtained as

Pr =12

ρACP (λ ,β )v3w, (5)

with λ , rω/vw, where ω is the rotor angular velocity. Theshape of the function CP depends on the geometry of thewind turbine system. An illustration of a typical CP surfaceobtained through experimental measurements ([9]) is shownin Fig. 2.

Applying Newton’s second law for rotation the windturbine dynamics is described by

Jrω = Ta(λ ,β ,vw)−NgTg, (6)

1The arguments of CP will be hereafter omitted for simplicity.

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where Jr and Ng are positive constants denoting the rotorinertia and gear ratio, respectively. Tg is the generator torquethat is assumed to be constant, whereas the aerodynamictorque Ta readily follows from (5) and the definition of λ

Ta =Pr

ω=

12

ρArCP (λ ,β )v2

w

λ. (7)

Equation (6) with (7) is the state equation, where β is thecontrollable input and ω is controlled output.

III. NONLINEAR ADAPTIVE INDIVIDUALPITCH CONTROLLER

The control objective is to operate the wind turbine in thefull load region, ensuring nominal power output P? despitewind field fluctuations. This objective is achieved through thedesign of an adaptive nonlinear control law, which first estab-lishes a passive mapping from the aerodynamic torque to therotor speed’s regulation error through feedback passivation[14, Chapter 14], and then ensures uniform global asymptoticstability (UGAS) of the origin of the error dynamics viadamping injection.

Due to the lack of an explicit formulation of the aerody-namic torque Ta, which is only available through the look-up table of the power coefficient CP, feedback passivationis achieved by including gradient based adaptation laws thatestimate on-line the controller coefficients.

A. Assumptions

The control law was developed under the following as-sumptions:

A1 The rotor speed ω is measured.A2 Three independent wind speed measurements vw,i

are available, one for each blade.A3 The wind speed is a finite energy, bounded signal,

i.e. vw(t) ∈L2∩L∞, and vw(t)> 0 ∀ t ≥ 0.A4 The generator torque is kept constant at its nominal

value, i.e. Tg = T ?g .

Concerning assumption A1, measuring ω is standardpractice in windmill systems. Moreover, knowledge of vw,imay be available through wind speed estimators such asthose described in [23], or alternatively through LIDARprofiling. Assumption A4 follows from the control objective

of constant nominal power output, achieved through set-pointregulation of the rotor velocity, i.e.

P = ωNgTg. (8)

B. Controller DesignThe design of the individual pitch controller is based on a

time-varying, linear in the parameters approximation of thepower coefficient

CP(λ ,β )≈ CP(λ ,β ) = α0(t)+α1(t)λ +α2(t)β , (9)

where α j(t), j = 0, . . . ,2, are unknown time-varying param-eters2, whose rate-of-change is assumed to be small, i.e.α j ≈ 0.

Substituting (9) into the aerodynamic torque (7), yields

Ta =12

ρArn

∑i=1

1nλi

CP,i(λi,βi)v2w,i, (10)

where the separation of the aerodynamic torque into threeequally weighted contributions follows the method suggestedin [19]. The system dynamics (6) then reads

ω =Φ

Jr

(n

∑i=1

1nλi

CP,i(λi,βi)v2w,i−

Ng

ΦT ?

g

), (11)

where Φ = ρAr/2, and the output measurement vector isy = [ω,vw,1,vw,2,vw,3]

T.The main contribution of the paper is the following.Proposition 1: Consider the regulation error dynamics

eω =−Φ

Jr

(n

∑i=1

1nλi

CP,i (λi,βi)v2w,i−

Ng

ΦT ?

g

), (12)

where eω(t) = ωd −ω(t) is the regulation error, and ωd isthe constant nominal set-point for the rotor speed. Then, theindividual pitch control law (i = 1, . . . ,N)

βi =1

α2,i

[−α0,i− α1,iλi +

λi

v2w,i

(Ng

ΦnT ?

g −uC,i

)], (13)

where uC,i is an additional control component to be chosen,together with the adaptation laws

˙α0,i =−Γiv2

w,ieω

λi, (14)

˙α1,i =−Γiv2w,ieω , (15)

˙α2,i =−ΓiProj

(v2

w,ieω βi

λi

), (16)

render the origin of (12) UGAS.Remark 2: The operator Proj(·) is the projection operator

[16, Lemma E.1], which is used to guarantee a bounded awayfrom zero estimate of the uncertain input gain for each pitchcontrol signal.

Proof: Consider the radially unbounded and positivedefinite function Lyapunov function candidate

V (eω , αp,i) =12

(Jr

Φe2

ω +n

∑i=1

1nΓi

α2p,i

), (17)

2The time argument will hereafter be omitted for simplicity.

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with p = 0,1,2 denoting the parameter number, Γi ∈R+ theadaptation rate for the individual pitch controller. The timederivative of V along the trajectories of the system reads

V =Jr

Φeω eω +

n

∑i=1

1nΓi

αp,i ˙αp,i

=− eω

(n

∑i=1

v2w,i

nλi(α0,i +α1,iλi + α2,iβi + α2,iβi)−

Ng

ΦnT ?

g

)

−n

∑i=1

1nΓi

(α0,i ˙α0,i + α1,i ˙α1,i + α2,i ˙α2,i

), (18)

where the dynamics of the estimation errors are given by˙αp,i = − ˙αp,i (since αp,i ≈ 0), with the estimation errorsdefined as αp,i = αp,i− αp,i.

Inserting the control law (13) into (18) yields

V =− eω

n

∑i=1

v2w,i

nλi(α0,i + α1,iλ + α2,iβi)+ eω

n

∑i=1

1n

uC,i

−n

∑i=1

1nΓi

(α0,i ˙α0,i + α1,i ˙α1,i + α2,i ˙α2,i

). (19)

Rearranging (19) as

V =− 1n

n

∑i=1

[α0,i

(eω v2

w,i

λi+

˙α0,i

Γi

)+ α1,i

(eω v2

w,i +˙α1,i

Γi

)

+ α2,i

(eω v2

w,iβi

λi+

˙α2,i

Γi

)− eω uC,i

], (20)

and inserting the adaptation laws (14)-(16), results in

V = eω

n

∑i=1

1n

uC,i. (21)

Defining the output as y = eω , the system with input u =1

2n ρAr ∑ni=1 uC,i and output y is passive with the storage

function V . Moreover, the system is zero-state observable.Setting uC,i , −κd,ieω with κd,i > 0, the equilibrium pointof (12), (14)-(16) is concluded to be uniformly stable, i.e. eω

and αp,i are bounded. Further, since eω = ωd −ω , and ωdis constant, then ω is bounded. To show that the regulationerror converges asymptotically to zero the second derivativeof V is calculated

V =−2eω eω

n

∑i=1

1n

κd,i. (22)

The closed loop error dynamics is obtained by inserting (13)into (12)

eω =− Φ

Jrn

n

∑i=1

[v2

w,i

λi(α0,i + α1,iλi + α2,iβi)−κd,ieω

](23)

which shows that eω is uniformly bounded (note that thesignals vw,i are bounded according to assumption A3), henceV is bounded, implying that V is uniformly continuous.Application of Barbalat’s lemma [14] proves that eω → 0as t → ∞, therefore, ω uniformly asymptotically convergesto ωd .

TABLE IPARAMETER VALUES AT OPERATING POINT.

vme [m/s] ω [rad/s] λ [−] β [◦] CP,d [−] Tg [kNm]

15 1.2671 5.4485 11.0064 0.1814 40.6806

Remark 3: The control input

u =1

2nρAr

n

∑i=1

uC,i (24)

is the total aerodynamic torque that is needed in order toachieve regulation of eω . Hence setting the control signalsto uC,i ,−κd,ieω can be seen as a simple control allocationstrategy, that defines which share of the total aerodynamictorque u is delivered by each pitch controller. Moreover, theproportional feedback uC,i are injecting damping into thesystem.

IV. PERFORMANCE ASSESSMENT

A. Operating point

The system is designed to operate in the full load regioncharacterizzed by wind speeds above 11.4m/s for this partic-ular wind turbine [13]. To initialize the nonlinear simulationmodel at an operating point the following procedure isfollowed. First a mean wind speed vme is selected, then therotor speed ω is obtained from [13], and the value of thetip-speed-ratio λ follows from its definition. The generatortorque Tg is obtained by

Tg = T ?g =

P?

Ngω, (25)

where P? = 5MW. CP,d is determined by

CP,d =P?

Pa, (26)

where Pa is found from (4). Last, the pitch angle β isdetermined through simulations.

Note that to obtain β the initial estimates of α0,i, α1,i,and α2,i have been determined through simulation. The meanwind speed selected for simulations reflects an initial condi-tion related to a configuration for nominal power generation.The parameters obtained are found in Table I.

B. Simulation Results

To assess the asymptotic regulation capabilities of thedesigned adaptive passivity-based individual pitch controller,the system is tested on a deterministic simulation environ-ment where only step changes in the mean wind vme(t)are considered. Subsequently, the controller is tested on astochastic simulation environment, by applying the windfield model presented in Section II-A, for the purpose ofevaluating the performance under more realistic operationalconditions.

The responses to the deterministic simulation, driven bya low-pass filtered square wave wind signal centered around

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0 50 100 150 20012

14

16

18W

indSpeed[ms]

vw1vw2vw3

0 50 100 150 200

4.995

5.015.025.03

Simulat ion T ime [s ]

Power[M

W]

PP !

Fig. 3. Rotor speed regulation in deterministic simulation environment-1. Wind field signal for each blade, 2. Nominal power and power outputresponse.

15m/s and with an amplitude of ±1m/s, is shown in Figs. 3-4. Clearly, the proposed control scheme achieves smooth andprecise asymptotic rotor speed regulation, thereby maintain-ing power output at its nominal value, despite the completelack of knowledge of the aerodynamic torque.

The stochastic simulation is driven by the wind fieldpresented in Section II-A, and the responses are displayedin Figs. 5-7. The closed loop system shows good set-pointregulation capabilities of the desired rotor speed with a meansquared regulation error of 3.37 ·10−7rad2/s2, and it fulfillsthe control objective with a standard deviation for the poweroutput of 2.29kW. It is also worth noting that the controlobjective is fulfilled with a control effort in line with thesystem capabilities with a maximum pitch rate of 1.18◦/s,which is considerable lower than the pitch rate saturationlimit of±10◦/s, hence the control scheme could be applicablefor larger scale wind turbines with slower pitch actuators.The pitch rates are displayed in the center plot of fig. 6.Finally, the parameter estimates display similar behavior foreach blade, and as seen in Figure 7 assumes values in closeproximity to their initial guesses.

V. CONCLUSION

Maximizing power output while alleviating structuralstress for wind turbines operating in the full load regionis essential to prolong wind turbine longevity and decreaseoperation and maintenance costs. This paper has proposeda nonlinear adaptive passivity-based individual pitch controlstrategy for the control of a variable speed variable pitchwind turbine operating in the full load region. Due to thelack of an explicit analytical expression for the nonlinearpower coefficient, the designed controller was augmentedwith gradient based adaptation laws to estimate the controllercoefficients online.

It was shown that the proposed controller guaranteesuniform global asymptotic stability of the rotor speed’sregulation error. Deterministic and stochastic simulations

0 50 100 150 2005

10

15

PitchAngle

[!]

! 1! 2! 3

0 50 100 150 2001.261.2651.271.275

Simulat ion Time [s ]

Roto

rSpeed[r

ad

s]

!!d

0 50 100 150 200ï1

0

1

PitchRate

s[! s]

! 1

! 2

! 3

Fig. 4. Rotor speed regulation in deterministic simulation environment- 1.Pitch signals for each blade, 2. Pitch rate signals for each blade, 3. Desiredand actual rotor speed.

0 50 100 150 20013

14

15

16

WindSpeed[ms]

vw1vw2vw3

0 50 100 150 2004.99

4.995

5

5.005

5.01

Simulat ion T ime [s ]

Power[M

W]

PP !

Fig. 5. Rotor speed regulation in stochastic simulation environment- 1.Wind field signal for each blade, 2. Nominal power and power outputresponse.

confirmed the expected performance: despite model uncer-tainties and a largely fluctuating wind field the rotor speedis promptly and successfully regulated to its nominal value,which in turn guarantees the nominal electrical power output.Further the control objective was fulfilled with a controleffort well within the capabilities of the system.

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0 50 100 150 2008

10

12

PitchAngle

[!]

! 1! 2! 3

0 50 100 150 200

1.266

1.268

1.27

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Roto

rSpeed[r

ad

s]

!!d

0 50 100 150 200ï10

0

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0 50 100 150 200ï0.1

ï0.099

ï0.098

!0

! 0 , 1! 0 , 2! 0 , 3

0 50 100 150 2000.21

0.22

0.23

!1

! 1 , 1! 1 , 2! 1 , 3

0 50 100 150 200ï0.095

ï0.09

ï0.085

!2

Simulat ion Time [s ]

! 2 , 1! 2 , 2! 2 , 3

Fig. 7. Rotor speed regulation in stochastic simulation environment- 1.Pitch signal for each blade, 2-4. Parameter estimates for adaptive controllaws.

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