peter b andersen thesis

7/29/2019 Peter B Andersen Thesis

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Load Alleviation on Wind Turbine Blades usingVariable Airfoil Geometry (2D and 3D study)

M.Sc. Thesis

Technical University of DenmarkDepartment of Mechanical Engineering

Section of Fluid Mechanics

Peter B. AndersenJune 2005


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Preface

This report is the result of the work carried out by Peter B. Andersen, stu-dent at the Technical University of Denmark (DTU), in the fulfillment ofthe requirements for obtaining the degree Master of Science in Wind Energyat DTU.

The project has been developed during the period February, 2005 to June,2005 at Ris National Laboratory and the Fluid Mechanics Section of theMechanical Engineering Department (MEK) at DTU.

The supervision has been undertaken by Christian Bak, Mac Gaunaa andThomas Buhl from Ris and Niels K. Paulsen from IMM at DTU and JensN. Srensen from MEK at DTU.


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Abstract

The operating conditions of wind turbines make them subject to fluctuatingloads that create fatigue damage. Alleviating these loads would reduce theneeded materials or increase the lifespan of the blades.

2D modelAn aerodynamic model coupled with a rigid spring/damper model is used.A PID and a LQR control is implemented and investigated for a 2D airfoilsection in order to formulate a control strategy. Turbulent wind of 60 sec-onds is used. The standard deviation of the normal load was reduced by74% for the PID regulation when the flapwise deflection was used as statevarible for the control. For the LQR reduction was 82%. The PID controlwas chosen over the LQR because of simplicty and computational speed.

3D modelFlexible trailing edge (TE) flaps have been modeled on a 33 meter longV66 blade from Vestas. The structural blade model comprises a cantileverbeam with modal expansion of blade and camberline deformations. Theaerodynamic model includes a BEM model with various 3D corrections anda flap model which include effects of the wake history. The PID controlis implemented to control the flaps. Effects of system time lag, flap powerconsumption and signal noise is included. The equivalent flapwise bladeroot moment is reduced 61% using 7 meter TE flaps with infinite poweravailable for the flap actuators. TE flaps with 1% cross sectional blade massgives a reduction of 60% when flap actuators consume 100W/m maximum.The potential drops to 41% when signal noise is added to the control. Analgoritm has been proposed for collecting data which is less sensitive to noise.

Keywords: Wind Turbine, Load Alleviation, Fatigue Loads, Trailing EdgeFlaps, PID control, LQR control, Signal Noise.


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.

De betingelser som vindmller udsttes for gr dem sarbare for udmat-

telses laster. Ved at fjerne disse udmattelseslaster ville det vre muligt atreducere materiale forbruget eller forge en vinges levetid.

2D modelEn aerodynamisk model er kombineret med et stivlegemet fjeder system.En PID og LQR regulator er udviklet og undersgt for et 2D vingeprofilfor at formulere en kontrol strategi. En turbulent vindserie pa 60 sekunderer brugt. Standard afvigelsen pa normal kraften er reduceret 74% for PIDkontrollen nar den flapvise udbjning er brugt som tilstandsvariablen forkontrol systemet. For LQR kontrollen er reduktionen 82%. PID kontrollenblev valgt fremfor LQR da den er simplere at bruge og krver frre bereg-

ninger.

3D modelFleksible bagkant flapper er modelleret pa en 33 meter V66 vinge fra Vestas.Den strukturelle model bygger pa en fast indspndt bjlke hvor udbjningerneer beskrevet ved modal former for bjlken og bagkant flappen. Den aerody-namiske model inkluderer en BEM model med forskellige 3D korrektioner ogen flap model med klvands historik indbygget. PID kontrollen er indbyggettil at styre bagkant flapperne. Tidsforsinkelser, effekt forbrug til flappen ogsignal stj er inkluderet i modellen. Det kvivalente flapvise rod momenter reduceret 61% ved brug af 7 meter flapper, hvor der ikke er begrnsning

pa den effekt aktuatorene kan bruge. Bagkants flapper som vejer 1% af detsektionerne vejer giver en reduktion pa 60% nar flap aktuatorene maksi-malt ma bruge 100W/m. Potentialet falder til 41% nar stj signalet bliverinkluderet. En algoritme er foreslaet til opsamling af data, som er mindrestj flsom.


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Contents

1 Introduction 1

I 2D modelling 3

2 Method 3

3 Load considerations 7

4 Control 9

4.1 PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Result and discussion, part I 155.1 Determining state variable candidates . . . . . . . . . . . . . 17

5.2 PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.4 Choosing state variable, control and what to optimize for . . 22

5.5 Effects of changing the duration of the turbulent wind signal 22

6 Conclusion, part I 24

II 3D modelling 25

7 Method overview 25

7.1 Aerodynamic model . . . . . . . . . . . . . . . . . . . . . . . 26

7.2 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.3 Servo model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.4 Numerical considerations . . . . . . . . . . . . . . . . . . . . 38

8 Defining fatigue 39

9 PID control 40

9.1 Signal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.2 Tuning methods . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Results, part II 48

10.1 Model inspection . . . . . . . . . . . . . . . . . . . . . . . . . 49

10.2 Systems of flaps . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.3 Effects of damping, gravity, wind shear and tower effect . . . 56

10.4 Servo modeling, time delays and limited power . . . . . . . . 58

10.5 PID versus PI . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

10.6 S ignal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

10.7 Combining all effects . . . . . . . . . . . . . . . . . . . . . . . 64

11 Conclusion, Part II 66

12 References 70


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A Appendix part I 72

A.1 Calculating stiffness . . . . . . . . . . . . . . . . . . . . . . . 72A.2 Aerodynamic equations for flaps . . . . . . . . . . . . . . . . 73

A.3 Defining 2D load reductions . . . . . . . . . . . . . . . . . . . 75A.4 Laplace transformations . . . . . . . . . . . . . . . . . . . . . 76A.5 Linearizing equations for LQR . . . . . . . . . . . . . . . . . 77A.6 Potters algorithm for solving Riccatis equation . . . . . . . . 86A.7 Flap deflection series . . . . . . . . . . . . . . . . . . . . . . . 87

B Appendix part II - 3D modeling 89

B.1 Deriving mode shapes and eigenfrequencies for torsion . . . . 89B.2 Turbulent wind field at 15 stations . . . . . . . . . . . . . . . 91B.3 Calculating blade root moments . . . . . . . . . . . . . . . . . 92B.4 Aerodynamic power . . . . . . . . . . . . . . . . . . . . . . . 93B.5 Technical sensor data . . . . . . . . . . . . . . . . . . . . . . . 94B.6 Verifying MTM using Kp/Kd-sweep . . . . . . . . . . . . . . 96B.7 PSD percent reduction of flapwise root moment . . . . . . . . 97B.8 Final result in table form . . . . . . . . . . . . . . . . . . . . 98

C Appendix - Runge-Kutta algoritm 99

C.1 Firstorder Runge-Kutta . . . . . . . . . . . . . . . . . . . . . 99C.2 Second order Runge-Kutta . . . . . . . . . . . . . . . . . . . . 100


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Nomenclature

Roman symbols

A, A2 Helping variables used in the state space formulation of thegoverning equation for the LQR control method.

a A non-dim. position of the point where the 2D section isfastened. [2EAc 1]

ACC Abbreviation for Accelerometer.Ai, bi Flat plate expression coefficients for the aerodynamic re-

sponse functions.b Half sectional chord length (c/2). [m]

B, C Helping variables used in the state space formulation of thegoverning equation for the LQR control method.

c Sectional chord length. [m]

ctune determines the ratio of a signal coming from either the straingauge or the accelerometer.

CG Center (of) gravity, the vectorial point of attack for gravityon an airfoil section. [m]

Cn, Cx, Cy, C Damping coefficients for the 2D case [log%]

D Helping variables used in the state space formulation of thegoverning equation for the LQR control method.

D The damping matrix used for the 3D model. [8x8 matrixNs/m]

D Drag force. [N]Defl The degree of which a camberline mode shape is deflected.

DOF Degrees Of Freedom.E Helping variables used in the state space formulation of the

governing equation for the LQR control method.E Error, what the control algorithm seeks to minimize.EAx Point (of) elasticity (elastic axis). point where a normal

force (out of the plane) will not give rise to a bending of thesection. [m] (measured the LE in 2D and the c/4 in the 3Dmodel).

EI1 Bending stiffness about first principle axis [N m2]

EI2 Bending stiffness about second principle axis [N m2]

EM P External Modeling Principle

Epwr The error to be minimized in the PD regulator for estimatingthe needed power consumption for the flap. [deg]

EQF Equivalent flapwise root moment, found using a 60 secondturbulent wind series. [kNm]

EQE Equivalent edgewise root moment, found using a 60 secondturbulent wind series. [kNm]

F Helping variables used in the state space formulation of thegoverning equation for the LQR control method.

flap masses A percentage of the cross sectional mass than is used forbuilding the flap at that section. The mass for the servoactuator system is not included in this percentage. [%]

flapl chordwise length of flap [m]flapm mass of flap [m]fn, fx, fy, f Stiffness eigenfrequencies for the 2D case [Hz].


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G Helping variables used in the state space formulation of thegoverning equation for the LQR control method.

GIt Torsional stiffness [N m2]

Gs

Transfer function.GX The generalized coordinates used in the 3D model.h Height above ground, used to calculate the wind shear. [m]Hdydx, .. Aerodynamic constant for deformable airfoilsHtower Height of tower. [m]Imm Mass moment of inertia around CG [Kgm2]Immflap Mass moment of inertia of the flap. [Kgm

2]IM P Internal Modeling PrincipleK The stiffness matrix used for the 3D model. [8x8 matrix

N/m]Kd Differential gain. [any]Ki Integral gain. [any]Kn, Kx, Ky, K Stiffness coefficients for the 2D case [N/m].Kp Proportional gain. [any]L Lift force. [N]LE Leading edgeM The mass matrix used for the 3D model. [8x8 matrix Kg]

M Torsional moment of a section. [Nm/m]Mreference Torsional moment using no flap. [Nm]M T M Multilayer Tuning Method, a gain tuning method used for

the 3D PID flap control.mi, m(z) Discrete and distributed mass along the V66 blade. [Kg/m]

mf(x) Non.dim. flap mass distribution.msf(x) Mode shape of flap in y direction at the chordwise positionx.

msi,x(z) Mode shape i for the blade in x direction at radial positionz.

msi,y(z) Mode shape i for the blade in y direction at radial positionz.

msj,(z) Torsional mode shape j for the blade at radial position z.N Normal force for a section [N/m]Nreference Normal force using no flap [N/m]ni Gaussian noise (used when simulating measuring data from

a strain gauge or an accelerometer).P The power supplied to the flap. [W]Paero Effect gain/loss due to the work performed by the aerody-

namic forces on the flap. [W]Pinertia Effect needed to balance the work done by inertial forces

while flapping. [W]P C Pressure center (the vectorial point of attack for the aero-

dynamic forces) [m]P Ii Constants used to model flap inertia forces.P(x, i) Sectional external force aligned with radial axis. [N]P(y, i) Sectional external force in the rotor plane. [N]

Pz,i Sectional external force perpendicular to the rotor plane.[N]


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Q The three-quarter equivalent quasi steady upwash.QC Effective equivalent three-quarter upwash.ri Radial position of section on blade. [m]R Length of V66 blade. [m]RHS Right Hand SideRM Percent reduction in RMS torsional moment compared to

reference (no control)[%]RN Percent reduction in RMS normal force compared to refer-

ence (no control)[%]RT Percent reduction in RMS transverse force compared to ref-

erence (no control)[%]s Non.dim. time used to model dynamics of finite thickness

airfoils. [2cr0 Ureldt]

SC Shear center. the point on the section where an in-planeforce will not rotate the section. [m] (measured from thec/4 in the 3D model).

SG Abbreviation for Strain Gauge.SN R Signal-to-noise ratio [dB]T Transverse force of a section [N/m]T E Trailing EdgeTreference Transverse force using no flap [N]u Real deflection. [m]uev,j A mode shape vector at the eigenfrequency r. [m]Vf Adjustment factor for the free wind speed, used when cal-

culating wind shear.V

relRelative wind velocity seen by airfoil section. [m/s]

Vrot Rotational velocity ri. [m/s]Vwind Free wind velocity. [m/s]x Edgewise deformation for 2D model. [m]XE Distance from PE to c/4. [m]xi State variables. [any]xi,ref Reference state variables, an optimum state that is desired.

[any]Xm Distance from CG to c/4. [m]Xs Distance from SC to c/4. [m]y Flapwise deformation for 2D model. [m]

yi Discrete flapwise deformation at iteration number i. [m]zi Aerodynamic state variable.


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Nomenclature

Greek symbols

Inflow angle for airfoil section also referred to as angle ofattack. [deg]

Deflection angle for flap, angle between camberline of flapand undeformed section. [deg]

j Logarithmic decrement for mode shape j. Rotational speed of blade. [rad/s]r Eigenvalue for dynamic system of equations. [rad/s]f Eigenvalue for dynamic system of equations. [Hz] Angle between relative wind seen by the blade and the ro-

tational plane. [deg] Angle between a line parallel to the rotor plane and the

chordline going through the EAx point. [deg] Torsional twist of section due to the aerodynamic moment.

[deg] Total torsional twist of section. (twist+pitch+defl) [deg]pitch The degree the whole blade is rotated (pitched), it is con-

stant for the entire blade. [deg]twist The designed and produced twist of the airfoil section, varies

section to section. [deg]0 twist + pitch Duration of a time window (e.g. used for the reference vari-

able defining E). [s]


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1 Introduction

1 Introduction

Wind turbine blades are subject to fast fluctuating loads during operation,due to turbulence, operating in yawed conditions, shear flow and the ef-fect of the tower. These fluctuating loads causes fatigue damage. Severalcomponents including the blades, are dimensioned according to the fatiguedamage done by these loads. Therefore, significant benefits can be obtainedif the fluctuating nature of these loads is minimized. The cost of the windturbine, and thereby the cost of energy generated per kWh, will potentiallybe reduced.

Birds are known for varying the geometry of the wings. The kestrel andthe common buzzard show great maneuverability in the air while hunting.They are often seen frozen in the air, despite the turbulence in the air.

Figure 1: The kestrel.

Nature is apparently able to keep the lift constant while hunting. Is itpossible for wind turbines to do the same? Recent work shows that it ispossible to alleviate load increments from yaw-errors, wind shear, gusts andturbulence by pitching the blades for MW (Mega-Watt) size wind turbines[20]. According to Larsen et al. [20] cyclic pitch can reduce the blade flapfatigue loads at the hub 15%, while it can be reduced 28% when using indi-

vidual pitch. It is believed that a further reduction is possible using TrailingEdge (TE) flaps. By enabling a TE flap to deform independently and quicklyalong the radial position of the blade, local fluctuations in the inflow is com-pensated for the deformation of the TE. Compared to independent pitchservo systems, the frequencies obtainable with TE flaps is expected to bemuch higher than the corresponding frequencies from a pitching blade. Sofar Basualdo [2] has carried out wind turbine investigations for a rigid TEflap mounted on a 2D profile. This study showed that the standard deviationof the airfoil position normal to the chord can be reduced 75% in a 2 secondperiod. Buhl et al. [5] showed it is possible to reduce the standard deviationof the normal force up to 85% for a 2D cross section using a flexible TE flap.

Previous analytical investigations into deflecting airfoils are mainly based onthe potential-flow aerodynamic model of Theodorsen [5], which include theaerodynamic effect of a flat TE flap on an airfoil. More advanced numerical


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Introduction 2

models deal with fully deformable camberlines instead of the flat plate [18],however, these panel code models are computationally costly. A new aero-dynamic model have been developed by Gaunaa [10] which can model flaps

with variable geometry with less demand on computational resources. Thismodel for the aerodynamic forces is used in the present work. The presentwork focuses on horizontal axis pitch regulated variable speed wind turbines(PRVS), where the flow is attached on the airfoil sections. This report isdivided in two parts.Part I, will investigate the potential of implementing a PID and a LQRcontrol on a 2D airfoil [22,23]. The airfoil is a RIS B1-18 airfoil developedfor PRVS turbines with a chord length of 1 meter and maximum heightof 18cm. The TE flap will be reducing fluctuations higher than 1P for a10MW wind turbine. A control strategy will be formulated. A criteria fordetermining load reductions will be considered. Finding a variable which

describes the state of the system well, will be needed when designing thecontrol. The control will have to be tuned once it is implemented. Differenttuning methods like Ziegler-Nichols for the PID control will be investigated[17]. The LQR control will be implemented and investigated for possibleuse. The LQR control makes it possible to combine state variables in waysthe PID can not. Parameter studies will be carried out to determine the per-cent reduction in standard deviation in the loads. Veers [28] turbulent windflow model will be used for the 2D section. The goal of the 2D investigationin part I, is to find the optimum control strategy for the 3D method. Anoptimum should be understood by a simple but effective way of controllingthe TE flap.

Part II, will focus on implementing a 3D modal for a V66 blade from Ves-tas. The structural model will be based on the cantilever beam theory withmodel expansion of mode shapes for blade and camberlines[15,30,31,32]. Anaerodynamic BEM model by Hansen [16] and the Gaunaa [10] aerodynamicmodel will be used to model the flaps. Dynamic inflow, effects of wind shear,tower shadowing, gravity, centrifugal stiffening and structural damping willbe part of the model. When the V66 blade is fully modeled, the effects offlapping will be added. First the actuators of the flaps will have infinitepower available, later a maximum of 100W per meter flap will be available.In order to operate the flaps, it is important to find an optimum ratio be-tween weight, actuator power and needed response time for the flaps. The

effects of signal noise and system delay will be included in the final stages ofthe simulation runs. The PID control will be used for the 3D model, flapswill be placed in the outer radial region of the blade. Up to six flaps withvarying dimensions will be modeled. It will be important to find optimumgains in the PID control for the 3D model, especially since more gains areneeded than for the 2D cross section. Parameter studies will be carried outto determine the reduction in equivalent flapwise blade root moment usingthe rain flow counting method [19]. Veers turbulent wind flow model will beused for the 3D model. The goal of the 3D investigation in part II, will beto find the potential of using TE flaps on a MW size wind turbine.

This study is part of the ADAPWING1

project developed at Ris.

1The ADAPWING project overview is available at www.risoe.dk/vea/adapwing


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3 Method

Part I

2D modelling

Modeling a 2D airfoil section, gives qualitative results in a simple way thanusing a full 3D model of a blade. An aerodynamic model coupled with arigid spring/damper model will be used. A PID and a LQR control will beimplemented and investigated for a 2D airfoil section in order to formulatea control strategy.

2 Method

The 2D aeroservoelastic system is modeled by coupling a 2D model for theaerodynamics of a deformable airfoil, a solid body spring system with the

forces from the deforming TE added, and a model for the control of thedeformable TE. The aerodynamic model is developed by Gaunaa [10] (theunsteady wake is modeled using indicial functions). The full 2D model is animplicit model requiring convergence for each time step. A description ofthe models are found in the following paragraphs together with descriptionsof the various assumptions of the models.

Structural model

The structural model consists of a spring model and a deformable TE model.The spring model is a rigid body suspended in linear springs and dampers.

This part is solved using a standard solver for the translatory ( x, y) androtational () directions. The differential equations governing the motionof the solid body is given below.

xm + CXx + KXx = FX + ml2cos( + o) + mlsin( + o)

(1)

ym + CYy + KYy = FY + ml2sin( + o) mlcos( + o)

(2)

(Imp + ml2) + C + K = M + Msolid + ml {xsin( + o) ycos( + o)}

(3)

where o marks the offset angle found as the angle between the x-axisand the line through the elastic axis and the center of gravity when theprofile is in the equilibrium state. The offset angle is 50, the mass of thesection is 40Kg/m, and the moment of inertia at the center of gravity (CG)is 2Kgm2. The moment of inertia around CG for the airfoil, is assumedconstant when flapping the tail flap. Eigenfrequencies of motion are 1,2and 10Hz corresponding to flapwise, edgewise and torsional direction. Thespring constants are given in table 24 in appendix A.1. An illustration ofthe spring system is shown in Figure 2.

The airfoil is fastened to the springs 30cm from LE at the location of

the elastic axis (EAx). The CG is behind EAx 35cm from LE. The pressurecenter (PC) is moving back and forth and can at different times be foundin front or behind both CG and EAx. The last part of the structural model


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Method 4

Figure 2: Structural spring model

computes the forces from the deformations of the flexible TE flap. Theinertial loads from the deformation of the airfoil are found by integratingthe forces along the flap given by the equations below

Nsolid =

0

.9cc (x) msf(x) dx (4)

Msolid = abNsolid +

0

.9ccx (x) msf(x) dx (5)

where (x) labels the mass distribution along the flap, which is nearlyconstant. The flap weighs 0.8Kg or 2% of the sectional mass. The modeshape of the TE flap is suggested by Troldborg [27] and shown in Figure 3.

Figure 3: The shape of the TE flap

These forces, which are not taken into account with the basic solid bodystructural model, are added to the spring model. The external forces FX, FYand M originates from the aerodynamics of the airfoil and the deformationof the TE.

FX = Tcos( + o) + (N + Nsolid)sin( + o) (6)

FY = Tsin( + o) + (N + Nsolid)cos( + o) (7)

The relative deformations of the airfoil shape is given by equation (8)using the flap mode shape msf.

y(x, t) =

Ndefli=1

msf(x)Defli(t) (8)


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5 Method

For a detailed description of the 2D model refer to Buhl et al [5].

Aerodynamic model

The Gaunaa model [10] which is implemented in this work use a camberlinewhich deforms dynamically. The model is computationally very efficient dueto the indicial function formulation of the problem in combination with thepossibility of evaluating certain elementary integrals connected to the deflec-tion shapes before the actual time integration. The aerodynamic model wascompared to a full Reynolds-Averaged Navier-Stokes simulations of the RisB1-18 airfoil. The conclusion of this comparison showed the steady forcesto be comparable using flap deflections at 100. The dynamics in responsefunction matched well using TE flap deflection for 50. The dynamics for aflat plate is different than one with finite thickness. Hansen et al [11] showedthat the dynamics of a thick plate could be modeled with great accuracy

using indicial response functions. The response function coefficients used,corresponds to the Ris B1-18 airfoil developed by Gaunaa[9].

(s) =Cl(s)

Cl()= 1 0.0821e0.0199s 0.1429e0.7817s 0.3939e0.1453s

(9)

s =2

c

t0

Vreldt (10)

The effects of using finite thickness airfoils is greater time-lag in the

response function. The basic output from the model is a pressure differencedistribution over the airfoil, once the shape of this profile is known otherquantities like forces and moments can be calculated. Keeping in mind thatthe camberline is deformed as illustrated in Figure 4.

Figure 4: Pressure distribution on camberline.

The integration is made possible by making sure the local forces arealigned and split up component wise. Toward the leading edge the pressuregoes toward infinity, this is used to find one of the contributions to theedgewise force. In appendix A.2 the full equations for the aerodynamic isgiven.

The assumptions in the model are infinitely high Reynolds number, at-tached flow which is the case for small Angle Of Attacks (AOA) and mod-erate TE flap deformations. No viscous effects are included in the model,which means the steady drag is zero. The camberline consists of two deflec-

tion shapes, the first deflection shape is the constant (Defl1 = 1) chamber-line shape corresponding to the RIS B1-18 airfoil. The derivatives for thefirst deflection shape are zero ( Defl1 = Defl1 = 0). The second deflection


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Method 6

shape is used for the last 10% of the camberline. This deflection shape isused to describe the deformable TE movement. The aerodynamic modelcould make use of an arbitrary number of deflection shapes. A change in

wind speed is experienced instantly by all points on the camberline.A detailed description of the aerodynamic model is given by Gaunaa[10].

Wind model

The density of the air is = 1.225kg/m3. The undisturbed wind is modeledusing a constant wind of 60m/s in x direction and a 60 second turbulentseries given by the Veers [28] model. The parameters for the Veers model isa turbulence intensity of 0.10, mean wind at 10m/s in one minute, a lengthscale of 600m, and a coherence decay factor of 12, which is the spectrumprescribed by the Danish Standard DS472 [8]. The turbulence is sampled

at 1000Hz. In this work it is assumed that a pitch servo handles all 1Pvariations, which is chosen to correspond to 6 seconds, appropriate for a10MW wind turbine. The variations faster than 1P come from turbulencein the wind and tower shadow effects etc. It is assumed we can decouple the1P effects from the wind signal and the structural response and only focuson minimizing fatigue for higher frequencies.


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7 Load considerations

3 Load considerations

Before looking into the control of the deformable TE, it is important toreflect on what to optimize for. Should the standard deviation in the normalor transverse loads be alleviated? It would be ideal to reduce fatigue fromtorsion, transverse and normal loads, but is it plausible? In this chaptersome considerations are outlined for each type of load.

Normal force

The airfoil has the least stiffness in the flapwise direction. Forces acting inthis direction are large due to the orientation of the lift. The combinationof small stiffness and large forces should give large deformations. Intuitivelythe normal force is expected to couple better with a flap deformation than

a transverse force, however, this remains to be investigated.

Transverse force

Although the aerodynamic model does not include the effect of viscous drag,contributions to the transverse force are found in the model. The distrib-ution of pressure on the deformed camberline yield a transverse force com-ponent. Furthermore the pressure goes toward infinity near the LE. Thisinfinite pressure is acting on the LE of the infinitesimal thickness, gives riseto a finite force component in edgewise direction in the unsteady case.

Stresses in fibers increase with the distance to the neutral axis. This

principle also applies to wind turbine blades, see Figure 5. When design-ing blades, the strength of the glue in the trailing edge is a critical area.The minimization of fatigue in the transverse load is therefore important.Gravity is expected to play a great role for the transverse load. Gravity isincluded for the 3D model in part II, but not investigated in the 2D case.

Figure 5: Illustration of neutral axis and distances to outer fiber

Torsion

The effects of torsion include buckling and whirling. It will be beyond thescope of this report to investigate these effects. It might be needed to look attorsion for a full blade with several airfoil sections coupled together. Some

aeroelastic codes neglect torsion based on the assumption that the blade hasinfinite stiffness in the torsional direction. It will be interesting to see howthe torsional deflection is affected by the control.


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Load considerations 8

Rated power

It is important that the control mechanism does not reduce the power out-put. For the 2D model the power output will be evaluated based on the

reduction of the average transverse loads, which will be regarded as an in-dicator of power output from the wind turbine.

Load reduction in 2D

The 2D model use the standard deviation as a measure of fatigue. Thereduction in standard deviation for the normal load will be labeled RN, thestandard deviation reduction for the transverse loads is termed RT, andfinally RM is used for torsion. RN, RT and RM are described and derivedin appendix A.3 (the 3D model apply the rain flow counting technique andtheory of equivalent load as a measure of fatigue).


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9 Control

4 Control

It takes several ingredients to make a good control. The control needs toknow what to look for in the system. The parameters the control monitorsare called state variables (xi). When the system is operational, the statevariable moves away from an optimum position (xi,ref) due to the externalforces exerted by the wind. A control mechanism will force the state variableback to the optimum position. The control mechanism is represented by theflap deflection angle () which will be used as the control parameter.

4.1 PID

The Proportional, Integral and Differential (PID) control is a widely usedmethod. It uses three gain terms to feed the control in a closed control loop.Figure 6 show a flow diagram for the PID regulator used in the 2D airfoil

case.

Figure 6: The PID regulator

The control feedback is labeled F. Disturbance in the measurements islabeled xi and will be modeled for the 3D blade model. The proportionalterm adjust the feedback proportional to the state variable, and is given byequation (11). The integral gain term detects changes over time and ensuresthat the system does not drift away from a given optimum reference value.The integral term is given by equation (12). The differential term respondwell to changes. If no changes occur, the gain will be zero. The differential

term is given by equation (13).

up = Kp xi (11)

ui = Ki

t0

xidt (12)

ud = Kd xi (13)

Formulating the PID algorithm

The PID algorithm is taken from Ogatas [22,23] and Haugens [17] PID

formulation and it is used for tuning a given transfer function.

Gs(s) = Kp +Ki

s+ Kds (14)


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Control 10

Where Gs represents a transfer (output/input) function. Using an inverseLaplace-transformation, which is described in appendix A.4, the feedbackfunction F(xi) is given by equation (15).

F(xi) = Kp xi + Ki

t0

xidt + Kd xi (15)

Based on the load considerations, the horizontal, lateral and torsionaldeflections will be investigated to find a potential candidate as state variablefor the control. The PID formulation is given by equation (16) to (18)depending on the state variable in use.

Kpx x + Kix t

0x dt + Kdx x (16)

Kpy y + Kiy

t0

y dt + Kdy y (17)

Kp + Ki

t0

dt + Kd (18)

There are numerous PID formulations available in different textbooks.Some formulate the PID control on time series, some write up the differentialgains in abstract ways in order to combat signal noise. Often text bookdictates integrating the integral gain from t = 0 to t = [17,22,23]. In ourcase an infinite time window will yield the lifetime average deflection of the

wind turbine. It has been mentioned that this approach makes the systemmore resilient to fluctuating errors [17]. However, instead of forcing the TEflaps to keep the structural deflection close to a lifetime averaged deflection,it would be more natural to keep the deflection (xi,ref) close to an averagedeflection given by a finite time window ().

xi,ref =1

tt

xi(t)dt (19)

This is more natural as we seek to minimize 1P disturbances for 10MWwind turbines using = 6s. It would also make the control able to handle

average wind speeds changing from 5m/s one moment to 15m/s the next.An investigation into the effect of varying is carried out in the result sec-tion.

The PID control will utilize the Internal Modeling Principle (IMP). In-stead of using a state variable directly in the PID formulation, an errorfunction (E) will be used.

E xi xi,ref (20)

The principle of using state variable directly in a PID formulation, is

called applying the Explicit Modeling Principle (EMP), and is shown inequation (16),(17) and (18). The three gain terms in the PID will be rewrit-ten to two by applying the IMP.


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11 Control

F(xi) = Kp xi + Kit

0

xi dt + Kd xi

F(xi) Kp

xi +

Ki

Kp

tt

xi dt

+ Kd xi

F(xi) Kp

xi

1

tt

xi dt

+ Kd xi

F(xi) Kp (E) + Kd xi (21)

This should simplify the gain tuning process, once an optimal time aver-age window is found. The integral gain is hidden within the error formulationin our case, ensuring that the state variable does not drift away from the

optimum state.

Control strategy

Each state variable is assumed to correspond to a given load. The statevariables listed in table 1 will be investigated for the model.

Deflection orientation Kp, (Ki) Kd Correlated force

edgewise x x Transverse

flapwise y y Normal

torsional direction Torsional moment

Table 1: Chosen state variables and forces assumed to be correlated to thegiven state variable.

There are many reasons for choosing x,y and to represent the stateof the system. Most importantly, it is assumed that these state variablescan be measured with technology available today - like strain-gauges andaccelerometers. It is a general assumption that each state variable has aload which it correlates well with. Example: by minimizing the variance inflapwise deflection, it is assumed that the variance in normal force also willbe minimized.

An attempt will be made to introduce several state variables at the sametime, in order to investigate any possible additional reduction potentials.Combining equation (16), (17) and (18) is shown in equation (22).

(t) = Kpx(xxref)+Kdxx+Kpy(yyref)+Kdyy+Kp(ref)+Kd(22)

Setting a gain pair (Kpi,Kdi) to zero will cause the corresponding state

variable to have no effect in the PID control. It should be mentioned thatthe LQR regulator is better suited for combining state variables.


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Control 12

Tuning methods

Sweep

One way of finding optimal gains, is to sweep through an array of propor-tional and differential gains. This method will reveal an extrema in the flappotential while stepping through the discrete points in the Kp,Kd array.The question remains if the found extrema is the global extrema. Othergain tuning rules like Ziegler-Nichols, can be used to verify the location ofthe extrema. The sweep method becomes computationally ineffective whenthe size of the gain array increases.

Ziegler-Nichols

The Ziegler-Nichols tuning rule for PID controllers are based on increasingthe proportional gain until instability (keeping other gains zero). At the

critical point before instability the proportional gain Kcr and the criticalperiod Pcr is noted down. The Ziegler-Nichols tuning rule states that theproportional gain should be 0.6 times Kcr and the differential gain shouldbe 0.125 Pcr Kp and finally Ki is

2Pcr

Kp.


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13 Control

4.2 LQR

The LQR regulation is a control method that is based on minimizing somepredefined quadratic performance indices [22,23]. The governing equations

of the aeroservoelastic model is put on state space form in appendix A.5.

The state space formulation

The flap control system is represented by a state space formulation.

x = Ax + B (23)

where x represents the seven state variables shown in equation (A-40).The state variable will be evaluated from a given reference (xref) as in (24).

x = f(x, ) f

x(x x

ref) +

f

(

ref) (24)

The A is a constant Jacobian matrix and B is a vector. The B vector isderived in appendix A.5 together with the helping variables CYi, CTi andCadi. The Jacobian matrix is given by

fixi

= A =

0 1 0 0 0 0 0CY1 CY2 CY3 CY4 Cad1 Cad1 Cad1

0 0 0 1 0 0 0CT1 CT2 CT3 CT4 Cad2 Cad2 Cad2

0 B1Vb A1B1V

b A1V 0 B1V

b 0 0

0 B2V

b A2B3V

b A2V 0 0 B2V

b 00 B3Vb A3

B2Vb A3V 0 0 0

B3Vb

where Ai and bi are the flat plate coefficients for the aerodynamic re-sponse function for the RIS B1-18 profile.

Quadratic performance index

The quadratic performance index which has to be minimized is given byequation (25).

J =

0L(x, )dt (25)

where L(x, ) is a quadratic function of x and . This quadratic L functionwill yield the linear control law for finding see equation (26).

= KTx(t) (26)

where K is known as the feedback gain or optimal control vector. In order tofind the feedback gain a symmetric positive definite (or semi-definite) Q ma-

trix has to be introduced together with a R scalar. Q and R determines the

relative importance of the error and the expenditure of energy between theseven state variables and control variable () for the system. The quadraticL function utilizes Q and R for the performance index J, see equation (27).

J =

0xTQx + 2Rdt (27)


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Control 14

A strategy for finding the elements in Q will be described later. The idea

is to minimize J during operation. Ogata [23] explains how this problem canbe addressed using the reduced-matrix Riccati equation.

ATP + P A PBRBTP = Q (28)

which is solved for the helping (positive definite) P matrix using eitherthe LQR function of Matlab or Potters method see appendix A.6, wherematrix system is built based on Q,R,A and B and solved for eigenvectors and

eigenvalues. Equipped with this newly found helping matrix the feedbackgain vector becomes (29).

K =1

RBTP (29)

and finally with this feedback gain the optimal control signal is given by (26).

Notes on the usage of the LQR method, the method is often referredto as the optimal control method. It should be noted that the method isonly optimal for the given Q and R parameters in the quadratic L func-

tion. Furthermore the accuracy is built on the assumption that the systemis described correctly using the state space formulation. The LQR modelformulated in this chapter does not include noise, actuator slag, maximumactuation velocity and time lag in the control.

Tuning strategy

There are 28 independent variables in the Q and a R parameter. It can bedifficult to find the best combination of so many variables. To begin with theR parameter was set to one as a general rule, which mean the Q-elementswill have to compensate for the energy-error-ratio between internal statevariables and . First the off diagonals was set to zero and the diagonalsQii to a fraction of the min/max span of the corresponding state variableor a fraction of the mean state value shown in equation (30) and (31).

Qii =1

max(xi) min(xi)(30)

Qii = 1mean(xi)

(31)

The problem with using either of the equations (30) and (31) was di-vision by zero. The simple sweep technique was taken into use when theaerodynamic Q-coefficients was lumped together to one parameter. After alot of sweeps using diagonals and off-diagonals, two results was found. OneQij combination that maximized RN and one combination that maximizedRM. The result is shown in chapter 5.


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15 Result and discussion, part I


The constraints on the movement of the TE flap will be investigated firstly.The Ziegler-Nichols tuning rule will be investigated, primarily to get a secondopinion on where the optimum gains are, but also to speed up the tuningprocess. The influence of the length of the average time window will beinvestigated. An optimum state variable will be determined and a controlstrategy will be formulated, which will include the type of control to be used(PID or LQR). And finally the reduction potentials will be investigated.

Flap constraints

The goal was to find out what an ideal flap is, so a flap producer will knowwhat to aim for, if the full potential of the flap is to be achieved by the con-trol. The bullet points below lists the constraints used for the 2D PID model.

Maximum deflection (5deg) Maximum deflection speed of flap (100deg/s) Maximum deflection acc. of flap (4000deg/s2)

The flap constraint for the derivative deflections where chosen based onwhat the flap needed for a 10 second turbulent signal. Figure 43 and Fig-ure 44 in appendix A.7 shows deformations and derivatives needed whileoptimizing RN and RM. The chosen values should be seen as a best casescenario. For the 3D model in part II, different masses will be applied tothe flap and limited power will be available for the servo engine of the TEflap actuator, which will set more realistic constraints on the movement ofthe flap.

Reference time window

The parameter study is conducted for the PID regulator. The study islimited to the flapwise direction. Small will make the reference value (yref)change rapidly and vice versa. For each of the six values, both a Kp,Kdgain sweep is conducted and the Ziegler-Nichols tuning rule shown in table2 is used to find the optimum gains.

period [ms] Kcr Pcr Kp Kd50 -830 0.4s -500 -25

100 -670 0.5s -400 -25

1000 -450 0.7s -270 -25

6000 -375 1.0s -225 -28

10000 -375 1.0s -225 -28

15000 -450 0.7s -270 -25

Table 2: Suggested PD gains using Ziegler-Nichols tuning rule

The tuning method does not promise to find the absolute best gains

for the state variable, however, the gains suggested by Ziegler-Nichols areusually quite accurate as seen in Figure 7 and Figure 8. The flap constraintsshould not be applied when determining Kcr and Pcr.


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Result and discussion, part I 16

Figure 7: Showing RN. Using six different time integral values. The star* indicates the gains suggested by Ziegler-Nichols and x illustrates theoptimum Kp,Kd gains found using the sweep method.


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[ms] 50 100 1000 6000 10000 15000

max(RN) [%] 58 59 67 74 74 68

Table 3: Optimum RN using the PID regulation for different values.

Looking at optimizing RN, the optimum duration for the averaging timewindow () is 6 to 10 seconds, according to table 3. As the wind signal wasdesigned for a 10MW wind turbine where the pitching mechanism shouldhandle disturbances above 6 seconds. It seems perfectly valid to implementthe 6 second averaging time window for the integral term. The Ziegler-Nicholas tuning rule is compared to the simple gain sweeps in Figure 7. Thegains found using the simple sweep method, is nearly identical to the gainssuggested by Ziegler-Nicholas. Please note the magnitude of the reductionpotential, the variance in normal load is decreased 74%.

5.1 Determining state variable candidates

One of the main objectives is to determine which state variable(s) to beused. The control can include numerous different state variables, especiallyfor the LQR control. It is important to find an optimum way of control-ling the flaps using as little parameters as possible, otherwise the controlwill become expensive and complex. Deflections in flapwise, edgewise andtorsional directions are considered.

PID control using y

Optimizing for RN using y as state variable gives RN at 75% shown in Figure8, and 45% for RT, keeping in mind the effects of gravity is not included.However, RM showed -400% which indicates four times larger variance inthe torsional moment than the uncontrolled case. Notice in Figure 9 howtorsion fluctuates much more for the controlled case.

The wind signal used is the 60 second turbulent wind. The optimumcontrol gains in Figure 8 appears to be homogeneous, there are no suddendrop in RN along the Kp, Kd gains. It would leave a very narrow marginfor errors in the control, if a dramatic drop in RN was seen from one gainto the next. The load series for the first 10 seconds are shown in Figure 9where the gains are set to Kp=-200 and Kd=-40, which according to Figure

8 should be gains with high RN potential. Figure 9 shows a steady normalforce and a more fluctuating torsion. A drop in the mean transverse force(T) indicates a drop in power production. Using y as a state variable, themean value of T remains constant. This indicates that the wind turbinekeeps a constant power production despite flapping.

PID control using x

The result of using x as state variable while optimizing for RT indicatesno RT potential. During gain sweeps the best Kp,Kd gains are at origowhere RT is 0%. This undermines using x as a state variable for optimizing

RT. For the normal load the highest potential in RN is 8% (Kp=250 andKd=10), however, due to effects of coupeling between normal and transverseforces, the RT drops to -20%, which is shown in Figure 10. Using x as a


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Figure 8: RN values. Gain sweep using y as state variable for the PIDregulator.

Figure 9: Load responses for the PID regulation optimizing RN using y

as state variable with gains Kp=-200/Kd=-40. (top) Normal force. (mid)Transverse force. (bottom) Torsion. The red curve is the uncontrolled caseand the blue the controlled.


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state variable seems of little use for the PID control, even when the effectsof gravity is not included.

Figure 10: RT values. Gain sweep using x as state variable for the PIDregulator.


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PID control using

The result of using as state variable while optimizing for RM indicates aRM potential of 40% as shown in Figure 11. It should be noted that RN

drops -25% and RT drops -15%. The load series for nearly 2 seconds areshown in the figure, where the gains are set to Kp=3000 and Kd=130. TheFigure shows a steady torsion and a more fluctuating normal force whileusing these gains for the PID control. Power output (T) is not affected byregulation.

PID control using x,y, combined

Numerous different combinations of state variables was attempted usingequation (22) as base. The gain tuning became complex and the potentialdid not seem to justify the effort. There seemed to be a drop in the mean

tangential force (T), which indicates loosing power for the wind turbine.Some of the results of the combined state variables is shown in table 4.

5.2 PID

The most representative results for the PID regulation is shown in table 4.

Kpx Kdx Kpy Kdy Kp Kd RN RT RM < T >

- - -225 -28 - - 75% 45% -400% 0%

- - - - 3000 130 -25% -15% 40% 0%

- - -225 -28 3000 130 15% 10% -250% 2%

- - -150 -10 2900 120 30% 25% -160% 6%200 8 -225 -28 - - 70% 50% -420% 0%

250 8 - - - - 25% 20% -300% 0%

- - -225 -28 78 0.8 72% 50% -350% 0%

Table 4: Tabulated results for PID regulators. Column < T > shows themean percentage reduction in the transverse force.

Using y as state variable seems like a good choice, it promises good resultsfor RN and RT. Generally it is impossible to gain a reduction for both forcesand moment at the same time. There is a choice to be made, either youchose to minimize the variance in torsion or the forces. Maximizing RNmeans minimizing RM and vice versa. Attempting to minimize both seemsto make the flap into an parasuite, where the flap attempts to use the reverseeffect used by the gondolere in Venice who pushed forward the boat, herethe dynamics of the flapping will make the wind turbine loose power. Thestatic drag is not part of the model, however Buhl et al [5] showed that thedynamic drag correspond to a loss in mean transverse force of 0.38%.

5.3 LQR

The best and most representative results for the LQR regulation is shownin table 5.

The LQR control seems to give good results, the best RN possible was82% and 45% for RT remember no gravity is part of the model. Againan increase in RN is at the expense of RM. What is interesting with the


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Figure 11: (top) RM values. Gain sweep using as state variable for thePID regulator. (bottom) Real time simulation of moment and normal forcewith and without optimum Kp, Kd parameters for the PID regulation.Setting the gain pair (Kp,Kd) equal to zero gives the uncontrolled case.


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Q11 Q22 Q33 Q44 Q12 Q23 Qxx RN RT RM T

0.1 0.001 0.01 0.75 -450 0 0.001 82% 45% -375% 3%

0 0 140 60 0 10 0.001 -30% -25% 20% 1%

Table 5: Tabulated results for LQR regulator. Column T shows how muchthe power output is reduced in percent.

LQR is the unique ability to couple the state variables. The disadvantage ofusing the LQR regulator is the complexity. The model is far more complexcompared to the PID model, especially when it comes to tuning the gainsfor a full blade with multiple cross sections.

5.4 Choosing state variable, control and what to optimizefor

Based on the results from the PID and the LQR control the flapwise bladedeflection (y) will be chosen as the state variable, the goal will be to reducefatigue in the flapwise direction. For both the PID and the LQR controlthe torsional variance increases dramatically. Instead of spending time onreducing this increase for the 2D section, the problem will be postponedto part II of the report for the full blade. The PID control will be chosenbecause it is simpler and computationally less demanding.

5.5 Effects of changing the duration of the turbulent windsignal

When tuning gains, it is important that the gains does not become toospecific for the particular turbulent wind series. The following graph showsthat it is possible to achieve larger reduction potentials if the turbulent windsignal is small, however, the graph also shows that the potential goes towarda constant value for times series larger than 40-50 seconds.

Figure 12: Turbulent wind series with 10,20,30,40,50 and 60 seconds dura-

tion. PID control is used with y as state parameter.

Using different turbulent signals with equal length (using differed RAND-


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SEED in Pascal), did not make a difference in the reduction potential.


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Conclusion, part I 24

6 Conclusion, part I

The objective for part I, was to implement and formulate a control strategyand to find optimum state variable(s) for the control. A simple and effectivecontrol strategy have been formulated using the PID regulator. The flapwisedeflection is chosen to describe the state of the system used by the PIDcontrol. For the 2D airfoil, the optimum gains for the PID control canbe found using Ziegler-Nichols tuning rules. The variances in normal loadhave been reduced 75% and for the transverse load a reduction of 45% wasobtained. The problem related to the 400% increase in torsional variance ispostponed to part II of the report. Finally decreasing the duration of theturbulent wind signal indicated that a 60 second turbulent wind series, willgive reliable results and not over predict the reduction potential.


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25 Method overview

Part II

3D modelling

Inspired by the big potential seen in the 2D model of Part I using the PIDregulator, the potential of varying flexible TE flaps will be investigated for afull blade. First an overview of the model will be given, then a few numericalconsiderations, next the PID controls will be defined and finally the resultswill be presented.

7 Method overview

As in the 2D modeling, the aeroservoelastic code consists of three parts, anaerodynamic, a structural and a control part. A modified version of theunsteady Blade Element Momentum (BEM) model described by Hansen[16] will be used. The flap aerodynamics and chamberline dynamics will betaken directly from part I.

The blade is modeled as a cantilever beam using modal expansion de-scribed by ye [30]. When the mode shapes are linearly combined, theydescribe the overall deflection of the blade. None of the mode shapes cou-ple, they are all assumed orthogonal. The orientation of the coordinatesystem is shown in Figure 13.

Figure 13: Orientation of coordinates for the blade.


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Method overview 26

It is assumed, that Installing the TE flap will not affect the stiffness,eigenfrequencies and mode shapes of the blade. The mass of the flap isassumed much smaller than the mass of the blade. Flap-masses at 1% and

2% of the airfoil cross section weight will be used. The tower and the mainshaft is assumed stiff. For simplicity tilt, cone and yaw angle are assumed tobe zero. Blade pitching will be used to keep the flow from separating alongthe TE flaps. Although V66 is a pitch regulated, variable speed machine,the blade is rotated at a constant 19.8 Revolutions Per Minute (RPM).Structural damping, wind shear, tower shadowing and gravity will also beintroduced. Effects of buckling and whirling are not considered.

A PID regulation will be installed for each flap using the flapwise de-flection as state variable. Power consumption for the flap actuators will bemodeled, as will time delays in the response of the control system. Gaussiandistributed errors will be added to the state variables to simulate signal noise

in the measuring systems used by the control. The model use profile datafrom a 33 meter long Vestas V66 blade, for which airfoil and structural datahave been supplied by Vestas.

7.1 Aerodynamic model

The BEM model is widely used for aeroelastic modeling. Natively, the BEMtheory assumes an infinite number of blades and ignores the effects of flowin spanwise direction. Tip effects are included for the tip of the wing byintroducing Prandtls tip-loss factor [12], this makes the code valid for a

finite number of blades. To predict quasi-static loads, the AOA () mustbe found. This requires knowledge of induced velocity (W) and the relativewind velocity. When the blade is loaded, the induced velocity is generateddue to the spanwise derivative of the loads and the rotational nature of theproblem. The relative velocity is given by equation (32), as the sum of thewind, rotation, induced velocity and speed of the deflection of the wing.

Vrel = Vwind + Vrot + W + Vdefl (32)

Vrot is given by the rotational speed () times the radial position of the

section. In some BEM codes, an iterative loop is used to find W, however,the explicit modeling principle is used in the present model.

Steady aerodynamics

Lift and drag coefficient tables are used to find the quasi-static loads for theblade sections. The coefficients for 10o to 20o are shown in Figure14.

The AOA () is the angle between the relative wind and the rotationalplane minus twist, pitch and deflection angle. In order to find the quasi-

stationary wake, the induced velocity is taken from Bramwells [4] derivationof Glauerts relation between thrust and the induced velocity, see equation(33).


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27 Method overview

Figure 14: Cd (bottom), Cl(top) as a function of in degrees.

Wn = n W = T2A |V|

(33)V = |Vwind + n(n W)| (34)n = (0, 0, 1)T (35)

The equation is derived from momentum theory, which dictates the in-duced velocity far down the wake to be two times the induced velocity at therotor plane. The velocity component normal to the rotor plane is labeledWn. An illustration is shown in Figure 15.

Figure 15: The wake behind a rotor disc seen from above.

It is assumed that only the lift contributes to the induced velocity, andthat the induced velocity acts in the opposite direction of the lift, see Figure16 (left). The angle between the relative wind (Vrel) and the rotational planeis termed .

The area that is covered by one blade for an annular strip is given byequation (36). An illustration is shown in Figure 16 (right).

dA =

1

B 2zdz (36)Ultimately the normal and tangential induced velocities are given by

equation (37).


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Method overview 28

Figure 16: (left) Orientation of local induced velocity, (right) Annular strip.

Wn

=Lcosdz

22zdzB F |Vwind + n(n W)|=

BLcos

4zF |Vwind + n(n W)|

Wt =BLsin

4zF|Vwind + n(n W)|(37)

F =2

cos1(ef) (38)

f =B

2

R z

z sin

where F is Prandtls tip-loss factor.

Unsteady aerodynamics

Wind turbine rotors are designed for maximum power extraction, whichleads to highly loaded discs compared to helicopter blades and air planes.The values of the induction factors (a) are around 1/3, which stems from thewake behind the wind. The unsteadiness of the wake arises when performingcollective motions like pitching the blade. The effects of the wake are builtup over time, and contradicts the rapid changing induced velocity predictedby the steady state formulation of Bramwell [4], which is derived on theassumption of a fully developed steady stake wake. The TUDk [24] inflowmodel is implemented to model the dynamics of the wake behind the rotors,and shown in the equations below.

yW + 1dyW

dt= xW + k1

dxWdt

(39)

zW + 2dzW

dt= yW (40)

1 =1.1

1 1.3a

R

Vwind(41)

2 = (0.39 0.26z2

R2)1 (42)

This amounts to one short and one longer time scale for the decay. The

time constants are used to model the dynamics of the near and far wakedynamics.


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29 Method overview

As for part I, dynamic effects of flapping are introduced by the Gaunaa[10] aerodynamic model. The flaps are included as part of an airfoil crosssections.

The Gaunaa model include the steady force coefficients for an airfoilsection, however, these coefficients are already given by the steady aerody-namics of the blade. The steady aerodynamics is taken out of the Gaunaaaerodynamic model by performing two calculations instead of one for eachairfoil section which has a flap associated. The extra flap calculation con-tains the steady aerodynamics, which is subtracted from the full sectionalaerodynamic model, see equation (43) to (45).

L =1

2V2cCl + Lflap Lflap,static (43)

D =

1

2 V

2

cCd + Dflap Dflap,static (44)M = Mflap (45)

where Lflap,static and Dflap,static corresponds to the lift and drag forceson a section where the flap is locked in = 00. The Lflap, Dflap is fulllift and drag contributions from a section which flaps. The difference inload between flap and flap-static, correspond to the dynamics of flappingpredicted from the 2D unsteady aerodynamic model. This difference in loadis added to the external forces for the section. The torsional calculations istaken fully from the Gaunaa [10] model.

The dynamic response of a thick flap is modeled using the indicial re-

sponse function. Coefficients for a RIS B1-18 profile and coefficients for aNACA 0015 airfoil is investigated. The dynamic response of the NACA 0015profile is faster than a RIS B1-18 profile, and not far from the dynamicresponse of a flat plate given by Theodorsen[25].

NACA(s) =Cl(s)

Cl()= 1 0.1297e0.0322s 0.4014e0.2222s (46)

B118(s) =Cl(s)

Cl()= 1 0.0821e0.0199s 0.1429e0.7817s 0.3939e0.1453s

(47)

s =2c

t

0Vreldt (48)

The dynamic response of a Vestas V66 blade is not known, the fasterNACA 0015 profile is used to model the thickness contribution to the dy-namic response. The difference between using the RIS B1-18 profile andthe NACA 0015 profile can not be seen in the result. The NACA 0015 profile,is therefore considered valid to use for modeling the thickness contributionin the dynamic response of a V66 blade.

Since the aerodynamic model is derived under the assumption of at-tached flow, a given pitch angle of the blade is used to assure that the flow

does not reach the area of separation. As no stall is assumed, no dynamicstall is included in the model. The pitched blade will reduce the poweroutput of the wind turbine.


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Method overview 30

7.2 Structural model

Rotor blades for wind turbines are slender structures which can effectivelybe modeled as cantilever beams, see Figure 17. The material is assumed

isotropic, deflections are assumed to obey the Bernoulli Euler slender beamtheory. The external forces for each section (Ni,Ti,Mi) are dictated by theaerodynamic forces and fictitious forces.

Figure 17: Cantilever beam.

In order to make dynamic calculations of the blade, eight shape functionsare used to describe the deflections, thus the model has eight Degrees OfFreedom (DOF). The model by Hansen [12] and ye [30] is used to determinethe shape functions and the eigenfrequencies of the blade. Eigenfrequenciesare shown in table 6.

DOF 1f 1e 2f 3f 2e 4f 1 2

Eig.frq[Hz](f) 1.04 1.87 2.81 5.56 6.07 9.21 9.99 16.4

Eig.frq[rads ](r) 6.56 11.8 17.7 34.9 38.1 57.9 62.8 103

Table 6: Eigenfrequencies and damping for the V66 blade.

The first and second edgewise modes (e1, e2) model edgewise blade de-flections. The first, second, third and fourth flapwise modes (1f,2f,3f,4f)model flapwise deflections and finally the first and second torsional modes(1, 2) gives the torsional deflection. The total deflection of a single modeshape is normalized to unity in 3D-space then projected to the xz- and yz-plane of the blade. The distribution of a mode shape in the xz- versusthe yz-plane depend on blade pitching, twisting, and torsional deflection.

However, in the present model the torsional deflection is not affecting theprojection of the mode shapes in the xz- and yz-plane. Eight generalizedcoordinates (GX) are used to quantify the oscillation in each DOF. Mul-tiplying GX with the corresponding mode shape and adding them up willyield the overall movement in flapwise-, edgewise- and torsional direction.The eight mode shapes are shown in Figure 18 they assume blade pitchingof 50. According to ye the model was not build for high eigenfrequencies,however, assuming adequately small time steps, the high eigenfrequencies oftorsion will be modeled.


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31 Method overview

Figure 18: Mode shape (flapwise-yz) 1f, 2f, 3f, 4f, (edgewise-xz) 1e, 2e.Torsional mode shape 1, 2.


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Method overview 32

The torsional mode shapes and eigenfrequencies are derived and de-scribed in B.1. The structural data used is shown in table 7. The datais also available in the work of Dayton [7] and as exercise notes [14]. The

bending and torsional stiffnesses are needed when calculating the eight modeshapes and corresponding eigenfrequencies.

r[m]

c[m]

twist[deg]

tc %

M[Kgm]

ImmKgm

m2GI

MNm2EI1

MNm2EI2

MN m2XE[m]

Xm[m]

0.85 1.420 0.00 99.00 596 381 1806 2890 2900 0 0

2 1.420 0.00 99.00 219 140 551 937 938 0 0

3.2 1.420 16.66 99.00 232 148 168 754 767 0 0

7.2 2.765 16.66 30.00 158 139 72.9 226 486 .046 .152

10.2 2.510 11.97 25.00 138 123 41.9 113 463 .054 .158

13.2 2.255 8.60 23.00 119 102 25.5 68.9 343 .062 .162

16.2 2.000 6.01 21.00 99.1 82.7 15.3 39.8 196 .054 .167

19.2 1.745 3.98 19.00 83.7 50.2 8.12 20.3 106 .045 .15422.2 1.490 2.40 17.00 69.3 26.8 3.89 9.34 50.8 .051 .146

25.2 1.235 1.20 15.00 56.3 12.6 1.57 3.6 28.5 .053 .124

28.2 0.980 0.40 14.25 33.1 2.17 0.44 .935 12.9 .046 .1

30.2 0.810 0.10 13.00 19.8 1.23 0.28 .247 6.54 .043 .104

31.2 0.686 0.05 12.00 15.4 .931 0.20 .114 4.06 .059 .112

32.2 0.514 0.02 12.00 10.8 .137 0.18 .046 .963 .071 .127

32.9 0.108 .0025 11.13 8.84 .0434 0.12 .021 .232 .076 .14

Table 7: Blade geometry, mass and stiffnesses. The table include pointsof elasticity (XE) and mass centers (Xm), which are used to determine the

overturning moment of a section.

The dynamic blade model is given by ye [30], also described by Hansen[15]. It uses the theory of virtual work for infinitesimal deflections describedby ye [31]. The governing equation of motion for the blade, make useof eight Generalized Coordinates (GX) one for each DOF. The motion isdescribed by a second order system shown in equation (49).

M GX+ D GX+ K GX = GF (49)

The mass matrix (M), damping matrix (D) and stiffness matrix (K)

are all diagonal matrices. If RHS of equation (49) was constant (or a func-tion of time only) it would be possible to solve the equations as a numberof single DOF equations including frequency response methods. However,when aerodynamic loads are involved the equations will still be coupled onRHS. This forces a numerical solution in a time domain for the coupled setof equations. This system is solved using an explicit Runge-Kutta methodfound in Hansen [15]. The diagonal elements in generalized mass matrix arefound using equation (50).

GMi = R

0

m(z) ms2i,x(z) + ms2i,y(z) dzGMj =

R0

Imm(z) ms2j,(z) dz (50)


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33 Method overview

Index i indicate the four flapwise and two edgewise mode shapes, j indi-cate the two torsional mode shapes. Mode shapes projected in edgewise(xz-plane) direction are labeled msi,x and flapwise(yz) msi,y. Torsional mode

shapes are labeled msj,. These elements make up the diagonal of M.

M =

GM1f 0 0 0 0 0 0 00 GM1e 0 0 0 0 0 00 0 GM2f 0 0 0 0 00 0 0 GM3f 0 0 0 00 0 0 0 GM2e 0 0 00 0 0 0 0 GM4f 0 00 0 0 0 0 0 GM1 00 0 0 0 0 0 0 GM2

The generalized stiffness diagonal elements for the stiffness matrix arecalculated as

GKj = 2r,jGMj , (51)

where the eigenfrequency r,j is measured in radians per second. The diag-onal elements of the generalized damping matrix are found using equation(52)

GDj = r,jGMjj

, (52)

where j is the logarithmic decrement, keeping in mind that the formulationof equation 52 assumes small decrements. The structural damping coeffi-

cients for the V66 blade is not available. The damping coefficients in table 8are taken from measurements performed on a LM blade length 19.1 metersby Hansen [13].

DOF 1f 1e 2f 3f 2e 4f 1 2

Damping[log%]() 1.8 3.6 2.0 2.4 5.6 3.2 5.9 6.0

Table 8: Structural damping provided by [13].

These structural damping coefficients are assumed to be larger than thecoefficients for a 33 meter long V66 blade, because the LM blade is 13 meter

shorter. The effect of using larger damping coefficients should give a greaterdifference between using and not using structural damping. The effect ofusing structural damping is shown in the result chapter.

The centrifugal stiffening is implemented in the model, as described byye [32]. The distributed mass (mi) found in radial position (ri) from thenacelle is experiencing the centrifugal force (Fz) described by equation (53).

Fz,i = mi2ri (53)

The force is split in cross sectional contributions and added to equation(49) on the Right Hand Side (RHS) as an external force.

Gravity plays a major role when designing blades. The body forces of thesemassive structures gives rise to large oscillations in root moments whichcause fatigue. The layers of laminates in the TE are subject to large stress


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Method overview 34

concentrations, especially when the blade is rotating. Compression stressesare generally considered most harmful. The fibers in the laminates can sus-tain tension much better. This makes it interesting to include the effect of

gravity in the model and see how this affects the performance of the flapsfocusing on the equivalent root moments of the blade.

7.3 Servo model

In part I, it was assumed that the aeroservoelastic model had an infiniteamount of power available for the flap actuator. The model will now beextended to include a constraint on how much power the flap actuator willbe allowed to consume, this should give a more realistic flap response. Theneeded power for balancing inertia (Pinertia) and aerodynamic (Paero) ef-fects while flapping is given by Gaunaa [10]. The forces of inertia prohibitsthe flaps from responding instantly. Both during acceleration and decel-

eration the control will have to avoid overshooting the needed deflectionangle. Overshooting can cause instability. New parameters includes a realdeflection angle (real) always lagging behind the needed deflection angle(target). Arbitrary flap-masses of 1%, 2% or 5% of the cross sectional masswhere chosen and shown together with the chordwise flap length in Figure19.

Figure 19: (left) the mass of one meter flap. (right) the chord wise lengthof the flap is given as 10% of the chord length (c) of the section.

The flap has not been designed yet and different materials have beensuggested for the flap (eg. Piezoelastic material) so trying out different flap-masses was needed. The flaps are not designed to absorb the spanwise TEstresses on the wing. This design criteria should minimize the needed massfor the flap. Based on a guess of what the flap might look like, an arbitraryflap mass distribution was made and shown in Figure 20. The flap deflectionshape used is identical to the one used in part I.

The non dimensional flap mass distribution is labeled mf. A flap isassociated to one of the 15 cross section along the blade. It is assumed, thatthe effect of a flap starts and ends at the half distance between two sections.Example: flapping section 10 at 25.2m is felt from 23.7m to 26.7m on theblade (hence the spanwise length of the flap is 3m) see Figure 21.

Knowing the needed deflection angle target a needed power (P) for theflap actuator will be given by the local PID power regulator explained later.The following power equation has to be fulfilled by the model

P + Pinertia + Paero = 0 (54)The aerodynamic power (Paero) comes from the dynamics of the air,

which has to be moved, when flapping. Paero is given by Gaunaa [10] and


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35 Method overview

Figure 20: The mass distribution along all flaps, summing up the discretepoints equal unity.

Figure 21: The blade and the flaps


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Method overview 36

shown in appendix B.4. The inertia term (Pinertia) includes the neededpower to accelerate and decelerate the flap when flapping. The inertia isisolated from equation (54) and the new flap moment is given by Gaunaa in

equation (55).

Meff = Pinertia + (y + ab) P I1flapmflapl P I2bflapmflapl (55)

where the weight of the flap for the section is labeled flapm and thechordwise length of the flap is called flapl. The new flap deflection is foundusing an ODE solver on equation (56)

P I3flapmflapl = Meff (56)

The flap-masses and the flap mass distribution are used in the power-

and flap control algorithm. Basic structural constants (P Ii) are calculatedbased on the non dimensional mass distribution of the flap (mf) and theflap mode shape deflection (msf).

P I1 =

c0.9c

msf(x) mf(x) dx (57)

P I2 =

c0.9c

msf(x) mf(x) x dx (58)

P I3 =

c0.9c

ms2f(x) mf(x) dx (59)

A flow chart with some of the basic parts of the full model is shown inFigure 22.

Wind model

The turbulent wind used in the aerodynamic model is taken from Veers [28].The 3D turbulent wind has a 10m/s average wind speed with a turbulenceintensity of 0.1. A spectral length scale of L=600 was used to indicate thefrequencies with most spectral energy in the wind, which is the spectrumprescribed by the Danish Standard DS472 [8]. Based on a constant rota-tional speed of 19.8 RPM, 15 rotationally sampled time series have been

saved. Each of the 15 series correspond to a cross section on the blade. Theturbulent wind signal for each of the 15 stations is seen in appendix B.2.On top of the turbulent wind, the model gives the option to also includethe effects of tower shadowing and wind shear. The effect of wind shear,is included in the model. The wind shear is implemented as a simple 1Dmodel, using 0.25 as the exponential factor. The tower shadowing and windshear model is taken directly from the description given by Hansen [16].


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37 Method overview

Figure 22: Floatchart diagram of the full model


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Method overview 38

7.4 Numerical considerations

If mode shapes with high eigenfrequencies are used, the time steps have tobe sufficiently small. If too large time steps are used, the higher frequencies

will be mistaken for low order oscillations, illustrated in Figure 23.

Figure 23: An illustration of using too large time steps when higher orderfrequencies appear as incorrect lower order oscillations.

This is known as aliasing. Table 9 lists the resolution of each mode shapeusing the time step of 0.001 second, which is used throughout this work.

DOF 1f 1e 2f 3f 2e 4f 1 2

Eig.frq[Hz] 1.04 1.87 2.81 5.56 6.07 9.21 9.99 16.4

time steps/dt 961 534 355 179 164 108 100 60

Table 9: Number of calculated time steps for one full oscillating cycle in agiven mode shape.

Finding an optimum ratio between time step resolution and calculationspeed is important. It has not been investigated into the depth it deservesin this report, however, it is assumed that 60 simulated time steps (worstcase) should be sufficient for our purposes. When simulating the effects ofthe aerodynamic forces on the flaps, it can be argued that 0.001s is too largea time step. A preliminary study was carried out using time step of 0.0001s,apart from taken 10 times longer to finish, the result was exactly the sameas when using time steps of 0.001 second.


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39 Defining fatigue

8 Defining fatigue

The equivalent load is used as a measure of fatigue for a given structure.The term equivalent root moment refers to a time series of the root momenttransformed into a single equivalent load, in this case an equivalent momentat the root of the blade. This method reduces the complexity of comparingtwo time series to comparing two equivalent moments. The abbreviationEQF will be used for equivalent flapwise root moment for a 60s turbulenttime series, consequently EQE will be used for the edgewise direction.

Figure 24: Rain flow counting

The process of finding the equivalent loads are outlined below. Firstthe time series is processed finding local extrema. The amplitude from oneextrema to the subsequent are measured and grouped in a number of am-

plitude ranges, as shown in Figure 24 for a 40 second flapwise root momentsignal. A single low frequency amplitude can contain many higher frequencyamplitudes. All of this is sorted out by the use of the rain flow countingmethod described by Kensche [19]. Based on the rainflow count of ampli-tude ranges, it is possible to find an equivalent load (S0) for a time series.The theory of making a load series equivalent to a single load is used, seeequation (60).

S0 =

1

N0

i

Ni Smi

1m

(60)

N0 is the total number of ranges, Ni the count of amplitudes in therange, Si the range. The m is a structural constant. For glass-reinforcedplastics the value m=10 is usually used.


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PID control 40

9 PID control

In part I, a PID control found an optimum deflection angle at any giventime, by using a few basic constraints the movement of the flap was keptwithin a reasonable range. In chapter 7.3, the servo model makes it possibleto supplied a flap actuator with power and thereby accelerate the flap. Nowthe task is to device a control, which can utilize this new ability to supplypower to a flap and thereby making it deflect. The new PID control for theflaps will be implemented in two stages. See Figure 25.

Figure 25: Schematic illustration of the two stage PID flap-control.

Figure 25 illustrates how the new power control extension (stage 2) isintegrated with the original PID control (stage 1). The original PID control

will continue to find the best deflection for the flap, while the new PIDcontrol will attempt to reach this flap deflection by supplying an amount ofpower to the flap actuator.

PID stage one

At stage one, an optimum deflection angle (target) is found. This angleis found using a PID control algorithm that minimizes the fluctuation inflapwise deflection (y). The PID control algorithm for stage one is given byequation 61.

target = Kp

y 1t

ty dt

+ Kd yt (61)

In part I, the 10MW wind turbine had a 1P -integration period of 6seconds. This turbine is 1.5MW so has been reduced to 3 seconds whichis somewhat close to 1P.

PID stage two

The control of the second stage must keep the real angle close to targetusing another PID regulator. The second stage model predicts the powerneeded by a flap-actuator. The goal of this PID regulator, is to ensure that

the TE deflection angle (target), is reached fast, using little power and notcausing instability or overshoot. To estimate the needed power at a giventime, the error (Ppwr) which has to be minimized is defined in equation (62).


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41 PID control

Epwr real target (62)

This error will be used to find the best proportional, integral and differ-

ential gains for a simple feedback regulator given by equation (63).

P(Epwr) = Kp Epwr + Ki

Epwrdt + KddEpwr

dt(63)

The result of a PID power control using 2% flap-mass is shown in Fig-ure 26. Keep in mind that a good power regulation means ensuring a fasttransition from one deflection angle to another while limiting power con-sumption. A small Pascal program was constructed to find optimum powercontrol gains for various needed step responses in target. The integral gainfor the power control had no effect and was taken out of the control.

Figure 26: (top) Step response in target from 00 to 50 for the sections 9 to

14. (bottom) the supplied power provided by the PID power regulator.

Indicial function representation

In order to speed up the calculations the inertial and aerodynamic effects offlapping have been formulated as indicial functions instead of using an ODEsolver. An indicial function uses the exponential solution to the first orderdifferential equation see equation (64)

real = real,oldet + target 1 e

t

(64)By combining these exponential functions the effects of inertia and the

aerodynamic loads can be calculated in much faster way only requiring in-formation about the prior time step, and tuned time coefficients ( ). An


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PID control 42

example of using exponential fitting instead of solving the ODE, is shownin Figure 27.

Figure 27: The exponential fits to the PID-power-control for a step responsein needed flap angle from 00 to 50.

9.1 Signal noise

A measured signal, always have a certain quality. A measured deflection of15cm might be 14cm. The rate of deflection used in the differential gain, iseven more sensitive to noise. Often the disturbance from an external sourcewill show up in the signal. The theory of signal-to-noise ratios (SNR) dealswith this problem, one way of defining this ratio is shown in equation (65),where logarithm base 10 is used.

SN R 20log rms signalrms noise

(65)

The noise can be offset oriented or variance oriented. Only the vari-ance oriented noise will be simulated. The Gaussian random distributionis used to simulate the variance. The Gaussian noise distribution pollutesthe simulated deflection used by the PID control. It can be necessary toremove the disturbances using filters. Most cleaning methods require a se-ries of measurements before being able to determine what the real signal is.This cleaning takes time, which may not be available if the full potential ofthe control is to be obtained. It is important to investigate noise, especiallysince the differential gain is widely used. The size of the error is given by the

range of what is being measured. The Full Scale (FS) range is in this workdefined as the range of flapwise deflection max(y) min(y) found in the 60second series. Although the range will be smaller for wind speed at 10m/s,


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43 PID control

the FS range is assumed constant at 2m throughout this work. Example:A strain gauge with 0.5% (FS) accuracy is used to measure flapwise deflec-tions. The standard deviation is 2m x 0.5% = 1cm. If measurements of a 2

meter deflected blade is repeated 1000 times, the result will be like Figure28. Measuring zero meter flapwise blade deflection will show the same noisedistribution as measuring two meters.

Figure 28: Simulated signal noise for measuring the y deflection.

In real life situations, the signal may not be Gaussian, there might behigher order peaks which will have to be filtered out in the real signal. Otherload components are likely to be picked up by the strain gauge, keeping inmind the blade is made of anisotropic material. However, the simulation of

signal noise with a gaussian random distribution is a good first approxima-tion when investigating the effects of noise.

Strain gauges

In order to measure the deflection of a wing the surface strain, measuredwith a strain gauge can be used. The strain gauge would be applied on thesurface of the wing. The strain would then have to be transformed to someform of flapwise displacements at the various cross sections along the blade.

Figure 29: Typical strain cycle showing non-linearity, hysteresis, and zeroshifting

Loading and unloading a blade is expected to deviate much from theideal linear strain/stress curve because of non-isotropic structural effects,see Figure 29. There are several considerations concerning the effect of


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PID control 44

hysteresis, and zero shifting which have to be dealt with, however, this isnot the scope of this work. According to Thomsen et al. [26] the accuracyof a strain gauge will deteriorate over time. A zero drift of 100m/m in

a few weeks is likely to happen. The company Chartillion has created adigital force tester (TCD500 Series - based on strain gauges), and claim tobe able to measure strains with full scale accuracy better than 0.25%. Thearea of measurements is enormous, and there are many aspects which havenot been included in this rough signal noise estimation.

Accelerometers

The accelerometer can measure the acceleration in the flapwise deflection(y). According to the internet page Metra Mess- und FrequenztechnikRadebeul by Manfred Weber [29] accelerometers should not be expected

to have accuracies less than 5%. In appendix Figure 48 and 49 the technicalspecification for a vibrational se

peter b andersen thesis

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