GH Bladed Theory Manual

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GH Bladed

Theory Manual

Document No 282/BR/009

Classification Commercial in Confidence

Issue no. 11

Date July 2003

Author:

E A Bossanyi

Checked by:

D C Quarton

Approved by:

D C Quarton


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DISCLAIMER

Acceptance of this document by the client is on the basis that Garrad Hassan and

Partners Limited are not in any way to be held responsible for the application or use

made of the findings of the results from the analysis and that such responsibility

remains with the client.

Key To Document Classification

Strictly Confidential : Recipients only

Private and Confidential : For disclosure to individuals directly

concerned within the recipients

organisation

Commercial in Confidence : Not to be disclosed outside the recipients

organisation

GHP only : Not to be disclosed to non GHP staff

Clients Discretion : Distribution at the discretion of the client

subject to contractual agreement

Published : Available to the general public

2003 Garrad Hassan and Partners Limited


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Garrad Hassan and Partners Ltd Document: 282/BR/009 ISSUE:011 FINAL

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CONTENTS

1. Introduction 11.1 Purpose 11.2 Theoretical background 21.3 Support 31.4 Documentation 31.5 Acknowledgements 3

2. AERODYNAMICS 4

2.1 Combined blade element and momentum theory 42.1.1 Actuator disk model 42.1.2 Wake rotation 5

2.1.3 Blade element theory 62.1.4 Tip and hub loss models 8

2.2 Wake models 92.2.1 Equilibrium wake 92.2.2 Frozen wake 92.2.3 Dynamic wake 9

2.3 Steady stall 112.4 Dynamic stall 11

3. STRUCTURAL DYNAMICS 13

3.1 Modal analysis 133.1.1 Rotor modes 143.1.2 Tower modes 15

3.2 Equations of motion 163.2.1 Degrees of freedom 163.2.2 Formulation of equations of motion 163.2.3 Solution of the equations of motion 17

3.3 Calculation of structural loads 18

4. POWER TRAIN DYNAMICS 19

4.1 Drive train models 194.1.1 Locked speed model 194.1.2 Rigid shaft model 19

4.1.3 Flexible shaft model 194.2 Generator models 20

4.2.1 Fixed speed induction generator 204.2.2 Fixed speed induction generator: electrical model 214.2.3 Variable speed generator 224.2.4 Variable slip generator 23

4.3 Drive train mounting 244.4 Energy losses 244.5 The electrical network 25

5. CLOSED LOOP CONTROL 27

5.1 Introduction 275.2 The fixed speed pitch regulated controller 27


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5.2.1 Steady state parameters 285.2.2 Dynamic parameters 28

5.3 The variable speed stall regulated controller 285.3.1 Steady state parameters 28

5.3.2 Dynamic parameters 305.4 The variable speed pitch regulated controller 31

5.4.1 Steady state parameters 315.4.2 Dynamic parameters 32

5.5 Transducer models 335.6 Modelling the pitch actuator 335.7 The PI control algorithm 36

5.7.1 Gain scheduling 375.8 Control mode changes 385.9 Client-specific controllers 385.10 Signal noise and discretisation 39

6. SUPERVISORY CONTROL 40

6.1 Start-up 406.2 Normal stops 416.3 Emergency stops 416.4 Brake dynamics 426.5 Idling and parked simulations 426.6 Yaw control 42

6.6.1 Active yaw 426.6.2 Yaw dynamics 43

6.7 Teeter restraint 44

7. MODELLING THE WIND 457.1 Wind shear 46

7.1.1 Exponential model 467.1.2 Logarithmic model 46

7.2 Tower shadow 467.2.1 Potential flow model 467.2.2 Empirical model 477.2.3 Combined model 47

7.3 Upwind turbine wake 477.3.1 Eddy viscosity model of the upwind turbine wake 487.3.2 Turbulence in the wake 50

7.4 Time varying wind 517.4.1 Single point time history 517.4.2 3D turbulent wind 517.4.3 IEC transients 52

7.5 Three dimensional turbulence model 537.5.1 The basic von Karman model 537.5.2 The improved von Karman model 557.5.3 The Kaimal model 597.5.4 Compatibility with IEC 1400-1 597.5.5 Using 3d turbulent wind fields in simulations 59

8. MODELLING WAVES AND CURRENTS 61

8.1 Tower and Foundation Model 618.2 Wave Spectra 62


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8.2.1 JONSWAP / Pierson-Moskowitz Spectrum 628.2.2 User-defined Spectrum 62

8.3 Upper Frequency Limit 638.4 Wave Particle Kinematics 63

8.5 Wheeler Stretching 648.6 Simulation of Irregular Waves 648.7 Simulation of Regular Waves 668.8 Current Velocities 67

8.8.1 Near-Surface Current 688.8.2 Sub-Surface Current 688.8.3 Near-Shore Current 68

8.9 Total Velocities and Accelerations 698.10 Applied Forces 69

8.10.1 Relative Motion Form of Morisons Equation 698.10.2 Longitudinal Pressure Forces on Cylindrical Elements 69

9. POST-PROCESSING 71

9.1 Basic statistics 719.2 Fourier harmonics, and periodic and stochastic components 719.3 Extreme prediction 729.4 Spectral analysis 759.5 Probability, peak and level crossing analysis 759.6 Rainflow cycle counting and fatigue analysis 76

9.6.1 Rainflow cycle counting 769.6.2 Fatigue analysis 77

9.7 Annual energy yield 789.8 Ultimate loads 79

9.9 Flicker 79

10. References 80


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1. INTRODUCTION

1.1 Purpose

GH Bladed is an integrated software package for wind turbine performance and loading

calculations. It is intended for the following applications:

Preliminary wind turbine design

Detailed design and component specification

Certification of wind turbines

With its sophisticated graphical user interface, it allows the user to carry out the following

tasks in a straightforward way:

Specification of all wind turbine parameters, wind inputs and load cases.

Rapid calculation of steady-state performance characteristics, including:

Aerodynamic information Performance coefficients Power curves Steady operating loads Steady parked loads

Dynamic simulations covering the following cases: Normal running Start-up Normal and emergency shut-downs Idling Parked Dynamic power curve

Post-processing of results to obtain:

Basic statistics Periodic component analysis

Probability density, peak value and level crossing analysis Spectral analysis Cross-spectrum, coherence and transfer function analysis Rainflow cycle counting and fatigue analysis Combinations of variables Annual energy yield Ultimate loads (identification of worst cases) Flicker severity

Presentation: results may be presented graphically and can be combined into a word

processor compatible report.


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1.2 Theoretical background

The Garrad Hassan approach to the calculation of wind turbine performance and loading has

been developed over many years. The main aim of this development has been to produce

reliable tools for use in the design and certification of wind turbines.

The models and theoretical methods incorporated in GH Bladed have been extensively

validated against monitored data from a wide range of turbines of many different sizes and

configurations, including:

WEG MS-1, UK, 1991

Howden HWP300 and HWP330, USA, 1993

ECN 25m HAT, Netherlands, 1993 Newinco 500kW, Netherlands, 1993

Nordex 26m, Denmark, 1993

Nibe A, Denmark, 1993

Holec WPS30, Netherlands, 1993

Riva Calzoni M30, Italy, 1993

Nordtank 300kW, Denmark, 1994

WindMaster 750kW, Netherlands, 1994

Tjaereborg 2MW, Denmark, 1994

Zond Z-40, USA, 1994

Nordtank 500kW, UK, 1995

Vestas V27, Greece, 1995 Danwin 200kW, Sweden, 1995

Carter 300kW, UK, 1995

NedWind 50, 1MW, Netherlands, 1996

DESA, 300kW, Spain 1997

NTK 600, UK, 1998

West Medit, Italy, 1998

Nordex 1.3 MW, Germany, 1999

The Wind Turbine Company 350 kW, USA, 2000

Windtec 1.3 MW, Austria, 2000

WEG MS-4, 400 kW, UK, 2000

EHN 1.3 MW, Spain, 2001 Vestas 2MW, UK, 2001

Lagerwey 750 Netherlands, 2001

Vergnet 200, France 2001

This document describes the theoretical background to the various models and numerical

methods incorporated in GH Bladed.


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1.3 Support

GH Bladed is supplied with a one-year maintenance and support agreement, which can be

renewed for further periods. This support includes a hot-line help service by telephone, faxor e-mail:

Telephone: +44 (0)117 972 9900

Fax: +44 (0)117 972 9901

E-mail [email protected]

1.4 Documentation

In addition to this Theory Manual, there is also a GH Bladed User Manual which explains

how the code can be used.

1.5 Acknowledgements

GH Bladed was developed with assistance from the Commission of the European

Communities under the JOULE II programme, project no. JOU2-CT92-0198.


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2. AERODYNAMICS

The modelling of rotor aerodynamics provided by Bladed is based on the well establishedtreatment of combined blade element and momentum theory [2.1]. Two major extensions of this

theory are provided as options in the code to deal with the unsteady nature of the aerodynamics.

The first of these extensions allows a treatment of the dynamics of the wake and the second

provides a representation of dynamic stall through the use of a stall hysteresis model.

The theoretical background to the various aspects of the treatment of rotor aerodynamics

provided byBladedis given in the following sections.

2.1 Combined blade element and momentum theory

At the core of the aerodynamic model provided by Bladed is combined blade element and

momentum theory. The features of this treatment of rotor aerodynamics are described below.

2.1.1 Actuator disk model

To aid the understanding of combined blade element and momentum theory it is useful initially

to consider the rotor as an actuator disk. Although this model is very simple, it does provide

valuable insight into the aerodynamics of the rotor.

Wind turbines extract energy from the wind by producing a step change in static pressure across

the rotor-swept surface. As the air approaches the rotor it slows down gradually, resulting in an

increase in static pressure. The reduction in static pressure across the rotor disk results in the air

behind it being at sub atmospheric pressure. As the air proceeds downstream the pressure climbs

back to the atmospheric value resulting in a further slowing down of the wind. There is therefore

a reduction in the kinetic energy in the wind, some of which is converted into useful energy by

the turbine.

In the actuator disk model of the process described above, the wind velocity at the rotor disk Udis related to the upstream wind velocity Uo as follows:

U a Ud o= ( )1

The reduced wind velocity at the rotor disk is clearly determined by the magnitude of a,the axialflow induction factor or inflow factor.

By applying Bernoullis equation and assuming the flow to be uniform and incompressible, it

can be shown that the powerPextracted by the rotor is given by :

P AU a ao= 2 13 3 ( )

where is the air density and A the area of the rotor disk.

The thrust Tacting on the rotor disk can similarly be derived to give:


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2.1.3 Blade element theory

Combined blade element and momentum theory is an extension of the actuator disk theory

described above. The rotor blades are divided into a number of blade elements and the theory

outlined above used not for the rotor disk as a whole but for a series of annuli swept out by each

blade element and where each annulus is assumed to act in the same way as an independent

actuator disk. At each radial position the rate of change of axial and angular momentum are

equated with the thrust and torque produced by each blade element.

The thrust dTdeveloped by a blade element of length drlocated at a radius ris given by:

dT W C C cdr L D= +1

22 ( cos sin )

where W is the magnitude of the apparent wind speed vector at the blade element, is

known as the inflow angle and defines the direction of the apparent wind speed vectorrelative to the plane of rotation of the blade, c is the chord of the blade element and CL and

CD are the lift and drag coefficients respectively.

The lift and drag coefficients are defined for an aerofoil by:

C L V S L = / ( )1

22

and

C D V S D = / ( )1

22

whereL andD are the lift and drag forces, Sis the planform area of the aerofoil and V is the

wind velocity relative to the aerofoil.

The torque dQ developed by a blade element of length drlocated at a radius ris given by:

dQ W r C C cdr L D= 122 ( sin cos )

In order to solve for the axial and tangential flow induction factors appropriate to the radial

position of a particular blade element, the thrust and torque developed by the element are

equated to the rate of change of axial and angular momentum through the annulus swept out

by the element. Using expressions for the axial and angular momentum similar to thosederived for the actuator disk in Sections 2.1.1 and 2.1.2 above, the annular induction factors

may be expressed as follows:

a g g= +1 11/ ( )

and

a g g, / ( )= 2 21

where


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gBc

r

C C

FHL D1 22 4

= +

( cos sin )

sin

and

gBc

r

C C

FL D

22 4

=

( sin cos )

sin cos

HereB is the number of blades and Fis a factor to take account of tip and hub losses, refer

Section 2.1.4.

The parameterHis defined as follows:

for a H =0 3539 10. , .

for a Ha a

a a> =

+ +0 3539

4 1

0 6 0 61 0 79 2. ,

( )

( . . . )

In the situation where the axial induction factor a is greater than 0.5, the rotor is heavily

loaded and operating in what is referred to as the turbulent wake state. Under these

conditions the actuator disk theory presented in Section 2.1.1 is no longer valid and the

expression derived for the thrust coefficient:

C a aT = 4 1( )

must be replaced by the empirical expression:

C a aT = + +0 6 0 61 0 792. . .

The implementation of blade element theory in Bladed is based on a transition to the

empirical model for values of a greater than 0.3539 rather than 0.5. This strategy results in a

smoother transition between the models of the two flow states.

The equations presented above for a and a

can only be solved iteratively. The procedure

involves making an initial estimate of a and a, calculating the parameters g1 and g2 as

functions of a and a

, and then using the equations above to update the values of a and a

.This procedure continues until a and ahave converged on a solution. InBladedconvergence

is assumed to have occurred when:

a a tol k k 1

and

a a tol k k' ' 1

where tolis the value of aerodynamic tolerance specified by the user.


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2.1.4 Tip and hub loss models

The wake of the wind turbine rotor is made up of helical sheets of vorticity trailed from each

rotor blade. As a result the induced velocities at a fixed point on the rotor disk are not constant

with time, but fluctuate between the passage of each blade. The greater the pitch of the helicalsheets and the fewer the number of blades, the greater the amplitude of the variation of induced

velocities. The overall effect is to reduce the net momentum change and so reduce the net power

extracted. If the induction factor a is defined as being the value which applies at the instant a

blade passes a given point on the disk, then the average induction factor at that point, over the

course of one revolution will be aFt,, whereFt is a factor which is less than unity.

The circulation at the blade tips is reduced to zero by the wake vorticity in the same manner as at

the tips of an aircraft wing. At the tips, therefore the factor Ftbecomes zero. Because of the

analogy with the aircraft wing , where losses are caused by the vortices trailing from the tips, Ftis known as the tip loss factor.

Prandtl [2.2] put forward a method to deal with this effect in propeller theory. Reasoning that, inthe far wake, the helical vortex sheets could be replaced by solid disks, set at the same pitch as

the normal spacing between successive turns of the sheets, moving downstream with the speed

of the wake.

The flow velocity outside of the wake is the free stream value and so is faster than that of the

disks. At the edges of the disks the fast moving free stream flow weaves in and out between

them and in doing so causes the mean axial velocity between the disks to be higher than that of

the disks themselves, thus simulating the reduction in the change of momentum.

The factorFtcan be expressed in closed solution form:

F sdt

= 2

arccos[exp( )]

wheres is the distance of the radial station from the tip of the rotor blade and dis the distance

between successive helical sheets.

A similar loss takes place at the blade root where, as at the tip, the bound circulation must fall to

zero and therefore a vortex must be trailed into the wake, A separate hub loss factor Fh is

therefore calculated and the effective total loss factor at any station on the blade is then the

product of the two:

F F Ft h=

The combined tip and hub loss factor is incorporated in the equations of blade element

theory as indicated in Section 2.1.3 above.


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2.2 Wake models

2.2.1 Equilibrium wake

The use of blade element theory for time domain dynamic simulations of wind turbine

behaviour has traditionally been based on the assumption that the wake reacts instantaneously

to changes in blade loading. This treatment, known as an equilibrium wake model, involves a

re-calculation of the axial and tangential induction factors at each element of each rotor

blade, and at each time step of a dynamic simulation. Based on this treatment the induced

velocities along each blade are computed as instantaneous solutions to the particular flow

conditions and loading experienced by each element of each blade.

Clearly in this interpretation of blade element theory the axial and tangential induced

velocities at a particular blade element vary with time and are not constant within the annulus

swept out by the element.

The equilibrium wake treatment of blade element theory is the most computationally

demanding of the three treatments described here.

2.2.2 Frozen wake

In the frozen wake model, the axial and tangential induced velocities are computed using

blade element theory for a uniform wind field at the mean hub height wind speed of the

simulated wind conditions. The induced velocities, computed according to the mean, uniform

flow conditions, are then assumed to be fixed, or frozen in time. The induced velocities

vary from one element to the next along the blade but are constant within the annulus swept

out by the element. As a consequence each blade experiences the same radial distribution ofinduced flow..

It is important to note that it is the axial and tangential induced velocities aUo and ar and

not the induction factors a and awhich are frozen in time.

2.2.3 Dynamic wake

As described above, the equilibrium wake model assumes that the wake and therefore the

induced velocity flow field react instantaneously to changes in blade loading. On the other

hand, the frozen wake model assumes that induced flow field is completely independent of

changes in incident wind conditions and blade loading. In reality neither of these treatments

is strictly correct. Changes in blade loading change the vorticity that is trailed into the rotorwake and the full effect of these changes takes a finite time to change the induced flow field.

The dynamics associated with this process is commonly referred to as dynamic inflow.

The study of dynamic inflow was initiated nearly 40 years ago in the context of helicopter

aerodynamics. In brief, the theory provides a means of describing the dynamic dependence of

the induced flow field at the rotor upon the loading that it experiences. The dynamic inflow

model used within Bladedis based on the work of Pitt and Peters [2.3] which has received

substantial validation in the helicopter field, see for example Gaonkar et al [2.4].

The Pitt and Peters model was originally developed for an actuator disk with assumptions

made concerning the distribution of inflow across the disc. In Bladedthe model is applied at

blade element or actuator annuli level since this avoids any assumptions about the

distribution of inflow across the disc.


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For a blade element, bounded by radiiR1 and R2 , and subject to uniform axial flow at a wind

speed Uo, the elemental thrust,dT, can be expressed as:

amUamU2dT Aoo &+=

where m is the mass flow through the annulus, mA is the apparent mass acted upon by the

annulus and a is the axial induction factor.

The mass flow through the annular element is given by:

dA)a1(Um o =

where dA is the cross-sectional area of the annulus.

For a disc of radius R the apparent mass upon which it acts is given approximately by

potential theory, Tuckerman, [2.5]:

3A R3

8m =

Therefore the thrust coefficient associated with the annulus can be derived to give:

a)RR(

)RR(

U3

16)a1(a4C

21

22

31

32

o

T &

+=

This differential equation can therefore be used to replace the blade element and momentum

theory equation for the calculation of axial inflow. The equation is integrated at each timestep to give time dependent values of inflow for each blade element on each blade. The

tangential inflow is obtained in the usual manner and so depends on the time dependent axial

value. It is evident that the equation introduces a time lag into the calculation of inflow which

is dependent on the radial station.

It is probable that the values of time lag for each blade element calculated in this manner will

under-estimate somewhat the effects of dynamic inflow, as each element is treated

independently with no consideration of the three dimensional nature of the wake or the

possibly dominant effect of the tip vortex. The treatment is, however, consistent with blade

element theory and provides a simple, computationally inexpensive and reasonably reliable

method of modelling the dynamics of the rotor wake and induced velocity flow field.


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2.3 Steady stall

The representation and to some extent the general understanding of aerodynamic stall on arotating wind turbine blade remain rather poor. This is a rather extraordinary situation in

view of the importance of stall regulation to the industry.

Stall delay on the inboard sections of rotor blades, due to the three dimensionality of the

incident flow field, has been widely confirmed by measurements at both model and full scale.

A number of semi-empirical models [2.6, 2.7] have been developed for correcting two

dimensional aerofoil data to account for stall delay. Although such models are used for the

design analysis of stall regulated rotors, their general validity for use with a wide range of

aerofoil sections and rotor configurations remains, at present, rather poor. As a consequence

Bladeddoes not incorporate models for the modification of aerofoil data to deal with stall

delay, but the user is clearly able to apply whatever correction of the aerofoil data he believesis appropriate prior to its input to the code.

2.4 Dynamic stall

Stall and its consequences are fundamentally important to the design and operation of most

aerodynamic devices. Most conventional aeronautical applications avoid stall by operating

well below the static stall angle of any aerofoils used. Helicopters and stall regulated

wind turbines do however operate in regimes where at least part of their rotor blades are in

stall. Indeed stall regulated wind turbines rely on the stalling behaviour of aerofoils to limit

maximum power output from the rotor in high winds.

A certain degree of unsteadiness always accompanies the turbulent flow over an aerofoil

at high angles of attack. The stall of a lifting surface undergoing unsteady motion is more

complex than static stall.

On an oscillating aerofoil, where the incidence is increasing rapidly, the onset of the stall can

be delayed to an incidence considerably in excess of the static stall angle. When dynamic stall

does occur, however, it is usually more severe than static stall. The attendant aerodynamic

forces and moments exhibit large hysteresis with respect to the instantaneous angle of

attack, especially if the oscillation is about a mean angle close to the static stall angle. This

represents an important contrast to the quasi-steady case, for which the flow field adjustsimmediately, and uniquely, to each change in incidence.

Many methods of predicting the dynamic stall of aerofoil sections have been developed,

principally for use in the helicopter industry.

The model adopted for inclusion of unsteady behaviour of aerofoils is that due to

Beddoes [2.8]. The Beddoes model was developed for use in helicopter rotor performance

calculations and has been formulated over a number of years with particular reference to

dynamic wind tunnel testing of aerofoil sections used on helicopter rotors. It has been used

successfully by Harris [2.9] and Galbraith et al [2.10] in the prediction of the behaviour of

vertical axis wind turbines.


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The model used withinBladedis a development of the Beddoes model which has been validated

against measurements from several stall regulated wind turbines. The model utilises the

following elements of the method described in [2.8] to calculate the unsteady lift coefficient

The indicial response functions for modelling of attached flow The time lagged Kirchoff formulation for the modelling of trailing edge separation and

vortex lift

The use of the model of leading edge separation has been found to be inappropriate for use on

horizontal axis wind turbines where the aerofoil characteristics are dominated by progressive

trailing edge stall.

The time lag in the development of trailing edge separation is a user defined parameter within

the model implemented inBladed. This time lag encompasses the delay in the response of the

pressure distribution and boundary layer to the time varying angle of attack. The magnitude of

the time lag is directly related to the level of hysteresis in the lift coefficient.

The drag and pitching moment coefficients are calculated using the quasi-steady input data along

with the effective unsteady angle of attack determined during the calculation of the lift

coefficient.


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3. STRUCTURAL DYNAMICS

In the early days of the industry, wind turbine design was undertaken on the basis of quasi-static aerodynamic calculations with the effects of structural dynamics either ignored

completely or included through the use of estimated dynamic magnification factors. From the

late 1970s research workers began to consider more reliable methods of dynamic analysis

and two basic approaches were considered: finite element representations and modal analysis.

The traditional use of standard, commercial finite element analysis codes for dealing with

problems of structural dynamics is problematic in the case of wind turbines. This is because

of the gross movement of one component of the structure, the rotor, with respect to another,

the tower. Standard finite element packages are only used to consider structures in which

motion occurs about a mean undisplaced position and for this reason the finite element

models of wind turbines which have been developed have been specially constructed to dealwith the problem.

The form of wind turbine dynamic modelling most commonly used as the basis of design

calculations is that involving a modal representation. This approach, borrowed from the

helicopter industry, has the major advantage that it offers a reliable representation of the

dynamics of a wind turbine with relatively few degrees of freedom. The number and type of

modal degrees of freedom used to represent the dynamics of a particular wind turbine will

clearly depend on the configuration and structural properties of the machine.

At present, largely because of the very extensive computer processing requirements

associated with the use of finite element models, the state of the art in the context of wind

turbine dynamic modelling for design analysis is based squarely on the use of limited degreeof freedom modal models. The representation of wind turbine structural dynamics within

Bladedis based on a modal model.

3.1 Modal analysis

Because of the rotation of the blades of a wind turbine relative to the tower support structure,

the equations of motion which describe its dynamics contain terms with periodic coefficients.

This periodicity means that the computation of the modal properties of an operating wind

turbine as a complete structural entity is not possible using the standard eigen-analysisoffered by commercial finite element codes.

One solution to this problem is to make use of Floquet analysis to determine the modal

properties of the periodic system. However, the mode shapes obtained by such calculations

are complex and not directly useful for a forced response analysis.

An alternative solution is based on the use of component mode synthesis. Here the modal

properties of the rotating and non-rotating components of the wind turbine are computed

independently. The component modes are then coupled by an appropriate formulation of the

equations of motion of the wind turbine in the forced response analysis. This approach has

been adopted forBladed.


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3.1.1 Rotor modes

The vibration of the tapered and twisted blades of a wind turbine rotor is a complex

phenomenon. A classical method of representing the vibration is by means of the orthogonal,

uncoupled normal modes of the structure. Each mode is defined in terms of the followingparameters:

Modal frequency, i

Modal damping coefficient, i

Mode shape, i r( )

where the subscript i indicates properties related to the ith mode.

The modal frequencies and mode shapes of the rotor are calculated based on the followinginformation:

The mass distribution along the blade.

The mass distribution is defined as the local mass density (kg/m) at each radial station in

addition to the magnitude and location of any discrete, lumped masses.

The bending stiffnesses along the blade.

The bending stiffnesses are defined in local flapwise and edgewise directions at each radial

station.

The twist angle distribution along the blade.

The mode shapes are computed in the rotor in-plane and out-of-plane directions and hence

the flapwise and edgewise stiffnesses at each radial station are resolved through the local

twist angle.

The blade pitch and setting angles.

The mode shapes are computed in the rotor in-plane and out-of-plane directions and hence

the flapwise and edgewise stiffnesses at each radial station are resolved through the blade

pitch and setting angles. The user ofBladedmay select a series of different pitch angles for

which the modal analysis is carried out. During subsequent dynamic simulations, the modal

frequencies appropriate to the instantaneous blade pitch angle are therefore obtained by linear

interpolation of the results of the modal analyses.

The presence or otherwise of a hub teeter hinge for a two bladed rotor.

For a two-bladed rotor the hub can be rigid or teetered. The presence of a teeter hinge will

introduce asymmetric rotor modes involving out-of-plane rotation of the rotor about the teeter

hinge.

The presence or otherwise of a flap hinge for a one-bladed rotor.

For a one-bladed rotor the hub can be rigid or have a flap hinge. The presence of a flap hinge

will introduce rotor modes involving out-of-plane rotation of the rotor about the teeter hinge.

The counter-weight mass and moment of inertia about the flap hinge for a one-bladed rotor.

Whether the hub can rotate.


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Rotation of the hub will affect the frequencies and mode shapes of the in-plane rotor modes.

With the shaft brake engaged and the rotor locked in position, the in-plane modes will

include both symmetric and asymmetric cantilever-type modes. With the rotor free to rotate,

the cantilever-type asymmetric modes will be replaced by asymmetric modes involving

rotation about the rotor shaft.

The rotational speed of the rotor.

The frequencies and mode shapes of both in-plane and out-of-plane modes will be dependent

on the rotational speed of the rotor. This dependence is explained by the additional bending

stiffness developed because of centrifugal loads acting on the deflected rotor blades. The user

ofBladedmay select different rotational speeds for which the modal analysis is carried out.

During subsequent dynamic simulations, the modal frequencies appropriate to the

instantaneous rotational speed are therefore obtained by quadratic interpolation of the results

of the modal analyses.

The frequencies and mode shapes of the rotor modes are computed from the eigen-values andeigen-vectors of a finite element representation of the rotor structure. The finite element

model of the rotor is based on the use of two-dimensional beam elements to describe the mass

and stiffness properties of the rotor blades.

The outputs from the modal analysis of the rotor are the modal frequencies and mode shapes

defined in the rotor in-plane and out-of-plane directions. The modal damping coefficients are

an input defined by the user and may be used to represent structural damping.

3.1.2 Tower modes

The representation of the bending dynamics of the tower is based on the modal degrees of

freedom in the fore-aft and side-side directions of motion. As for the rotor, the tower modesare defined in terms of their modal frequency, modal damping and mode shape.

The modal frequencies and mode shapes of the tower are calculated based on the following

information:

The mass distribution along the tower.

The mass distribution is defined as the local mass density (kg/m) at each tower station height

in addition to the magnitude and location of any discrete, lumped masses.

The bending stiffness along the tower.

The tower is assumed to be axisymmetric with the bending stiffness therefore independent of

bending direction.

The mass, inertia and stiffness properties of the tower foundation.

The influence of the foundation mass and stiffness properties on the tower bending modes

may be taken into account. The model takes account of motion of the foundation mass and

inertia against both translational and rotational stiffnesses.

The mass and inertia of the nacelle and rotor

For calculation of the tower modes, the nacelle and rotor are modelled as lumped mass and

inertia located at the nacelle centre of gravity and rotor hub respectively. For one and two-

bladed rotors, the influence of the rotor inertia on the tower modal characteristics depends on

the rotor azimuth and this may therefore be defined by the user. The variation of the towermodal frequencies with rotor azimuth is normally small and the assumption of a single rotor


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azimuthal position for the modal analysis is therefore a reasonable approximation. The user

can, of course, determine the extent of the azimuthal variation in the tower modal frequencies

by undertaking the modal analysis at a series of different rotor azimuths.

The frequencies and mode shapes of the tower modes are computed from the eigen-valuesand eigen-vectors of a finite element representation of the tower structure. The finite element

model of the tower is based on the use of two-dimensional beam elements to describe the

mass and stiffness properties of the tower.

The outputs from the modal analysis of the tower are the modal frequencies and mode shapes

defined in the fore-aft and side-side directions. The modal damping coefficients are an input

defined by the user and may be used to represent structural damping.

3.2 Equations of motion

Because of the complexity of the coupling of the modal degrees of freedom of the rotating

and non-rotating components, the algebraic manipulation involved in the derivation of the

equations of motion for a wind turbine is a complicated problem. In the case of the dynamic

model within Bladed, the derivation has been carried out using energy principles and

Lagrange equations by means of a computer algebra package.

3.2.1 Degrees of freedom

The degrees of freedom involved in the equations of motion for the structural dynamic model

forBladedare as follows:

Rotor out of plane including teeter, maximum six modes

Rotor in-plane, maximum six modes

Nacelle yaw

Tower fore-aft, maximum three modes

Tower side-side, maximum three modes

In addition, a sophisticated representation of the power train dynamics is offered as described

in Section 4 of this manual.

3.2.2 Formulation of equations of motion

The equation of motion for a single modal degree of freedom, assuming no coupling withother degrees of freedom, is as follows:

&& & /q q q F M i i i i i i i+ + =22

where:

qi is the time dependent modal displacement,

M m r r dri irotor= ( ) ( )2

is the modal mass,


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and:

F f r r dri i

rotor

=

( ) ( ) is the modal force.

Heref(r)is the distributed force over the rotor or tower component.

The modal degrees of freedom are, of course, coupled and the formulation of the equations of

motion withinBladed is as follows:

[ ]&& [ ]& [ ]M q C q K q F+ + =

where [M], [C]and [K]are the modal mass, damping and stiffness matrices, q is the vector

of modal displacements and F the vector of modal forces. The system matrices are full due to

the coupling of the degrees of freedom and contain periodic coefficients because of the time

dependent interaction of the dynamics of the rotor and tower.

Because of their complexity, the equations of motion are not presented in this manual. The

following key comments are, however, provided:

Although the equations of motion are based on a linear modal treatment of the structural

dynamics, the model does contain non-linear terms associated primarily with gyroscopiccoupling.

The rotor teeter degree of freedom is provided through the first out-of-plane mode and the

equation of motion includes representation of mechanical damping, stiffness and pre-load

restraints as specified by the user.

The equation of motion for the nacelle yaw degree of freedom is based on the inertia of

the wind turbine about the yaw axis with mechanical restraints provided through yaw

damping and stiffness as specified by the user.

The aeroelasticity of the wind turbine is taken into account in the equations of motion byconsideration of the interaction of the total structural velocity vector with the wind

velocity vector at each element along the rotor blades. The total structural velocity vector

at each element on the rotor blades is composed of the appropriate summation of the

velocities associated with each structural degree of freedom. In addition to the feedback of

the structural velocities into the rotor blade aerodynamics, the structural displacement

associated with the rotor teeter and nacelle yaw is also taken into account.

3.2.3 Solution of the equations of motion

The equations of motion are solved by time-marching integration of the differential equations

using a variable step size, fourth order Runge Kutta integrator.


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3.3 Calculation of structural loads

The structural loads acting on the rotor, power train and tower are computed by the

appropriate summation of the applied aerodynamic loads and the inertial loads. The inertialloads are calculated by integration of the mass properties and the total acceleration vector at

each station. The total acceleration vector includes modal, centrifugal, Coriolis and

gravitational components.


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4. POWER TRAIN DYNAMICS

The power train dynamics define the rotational degrees of freedom associated with the drivetrain, including drive train mountings, and the dynamics of the electrical generator. The drive

train consists of a low speed shaft, gearbox and high speed shaft. Direct drive generators can

also be modelled.

4.1 Drive train models

4.1.1 Locked speed model

The simplest drive train model which is available is the locked speed model, which allows no

degrees of freedom for the power train. The rotor is therefore assumed to rotate at anabsolutely constant speed, and the aerodynamic torque is assumed to be exactly balanced by

the generator reaction torque at every instant. Clearly this model is unsuitable for start-up

and shut-down simulations, but it is useful for quick, preliminary calculations of loads and

performance before the drive train and generator have been fully characterised.

4.1.2 Rigid shaft model

The rigid shaft model is obtained by selecting the dynamic drive train model with no shaft

torsional flexibility. It allows a single rotational degree of freedom for the rotor and

generator. It can be used for all calculations and is recommended if the torsional stiffness of

the drive train is high. The acceleration of the generator and rotor are calculated from the

torque imbalance divided by the combined inertia of the rotor and generator, makingallowance for the gearbox ratio. Direct drive generators are modelled simply by setting the

gearbox ratio to 1. The torque imbalance is essentially the difference between the

aerodynamic torque and the generator reaction torque and any applied brake torque, taking

the gearbox ratio into account. However, this is corrected to account for the inertial effect of

blade deflection due to any edgewise blade vibration modes. To use the rigid shaft model, a

model of the generator must also be provided, so that the generator reaction torque is defined.

During a parked simulation, or once the brake has brought the rotor to rest during a stopping

simulation, the actual brake torque balances the aerodynamic torque exactly (making

allowance for the gearbox ratio if the brake is on the high speed shaft) and there is no further

rotation. However, if the aerodynamic torque increases to overcome the maximum or applied

brake torque, the brake starts to slip and rotation recommences.

The rigid drive train model may be used in combination with flexible drive train mountings.

In this case the equations of motion are more complex - see Section 4.3.

4.1.3 Flexible shaft model

The flexible shaft model is obtained by selecting the dynamic drive train model with torsional

flexibility in one or both shafts. It allows separate degrees of freedom for the rotation of the

turbine rotor and the generator rotor. The torsional flexibility of the low speed and high

speed shafts may be specified independently. As with the rigid shaft model, a model of the

generator must be provided so that the generator reaction torque is specified.


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The turbine rotor is accelerated by the torque imbalance between the aerodynamic torque

(adjusted for the effect of edgewise modes as explained in Section 4.1.2) and the low speed

shaft torque. The generator rotor is accelerated by the imbalance between high speed shaft

torque and generator reaction torque. The shaft torques are calculated from the shaft twist,

together with any applied brake torque contributions depending on the location of the brake,which may be specified as being at either end of either the low or high speed shaft.

During a parked simulation, or once the brake disk has come to rest during a stopping

simulation, the equations of motion change depending on the brake location. If the brake is

immediately adjacent to the rotor or generator then there is no further rotation of that

component, but the other component continues to move and oscillates against the torsional

flexibility of the shafts. If the brake is adjacent to the gearbox and both shafts are flexible,

then both rotor and generator will oscillate. However, if the torque at the brake disk

increases to overcome the maximum or applied brake torque, then the brake starts to slip

again.

The flexible drive train model may be used in combination with flexible drive train

mountings. In this case the equations of motion are more complex - see Section 4.3.

It should be pointed out that while the flexible shaft model provides greater accuracy in the

prediction of loads, there is potential for one of the drive drain vibrational modes to be of

relatively high frequency, depending on the generator inertia and shaft stiffnesses. The

presence of this high frequency mode could result in slower simulations.

4.2 Generator models

The generator characteristics must be provided if either the rigid or flexible shaft drive train

model is specified. Three generator models are available:

A directly-connected induction generator model (for constant speed turbines),

A variable speed generator model (for variable speed turbines), and

A variable slip generator model (providing limited range variable speed above rated)

4.2.1 Fixed speed induction generator

This model represents an induction generator directly connected to the grid. Its

characteristics are defined by the slip slopehand the short-circuit transient time constant .

The air-gap or generator reaction torque Q is then defined by the following differential

equation:

& [ ( ) ]Q h Q= 1 0

where is the actual generator speed and 0 is the generator synchronous or no-load speed.

The slip slope is calculated as

h

Pr

r r= ( )0


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where r is the generator speed at rated power outputPr , given by r=0 (1 + S/100)where

Sis the rated slip in %, and is the full load efficiency of the generator.

4.2.2 Fixed speed induction generator: electrical model

A more complete model of the directly-connected induction generator is also available in

Bladed. This model requires the equivalent circuit parameters of the generator to be supplied

(at the operating temperature, rather than the cold values), along with the number of pole

pairs, the voltage and the network frequency. It is also possible to model power factor

correction capacitors and auxiliary loads such as turbine ancillary equipment. The equivalent

circuit configuration is shown in Figure 4.1.

xr

xm

Rr/s R

s

xs

C

Ra

Xa

Figure 4.1: Equivalent circuit model of induction generator

The equivalent circuit parameters should be given for a star-connected generator. If the

generator is delta-connected, the resistances and reactances should be divided by 3 to convert

to the equivalent star-connected configuration.

The voltage should be given as rms line volts. To convert peak voltage to rms, divide by 2.

To convert phase volts to line volts, multiply by 3.

Since this model necessarily includes electrical losses in the generator and ancillary

equipment, it is not possible to specify any additional electrical losses, although mechanical

losses may be specified - see Section 4.4.

Four different models of the electrical dynamics of the system illustrated in Figure 4.1 are

provided:

Steady state

1st order

2nd order

4th order

The steady state model simply calculates the steady-state currents and voltages in Figure 4.1

at each instant. The 1st order model introduces a first order lag into the relationship between

the slip (s) and the effective rotor resistance (Rr/s), using the short-circuit transient time

constant given by [4.1]:

= X X x

X R

s r m

s r s

2

where Xs = xs+xm, Xr= xr+xm, and s is the grid frequency in rad/s.

Rs = Stator resistancexs = Stator reactanceRr= Rotor resistancexr= Rotor reactancexm = Mutual reactance

C = Power factor correctionRa = Auxiliary load resistanceXa = Auxiliary load reactances = slip


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The 2nd order model represents the generator as a voltage source behind a transientreactance X = Xs - xm

2/Xr, ignoring stator flux transients:

is (rs + jX) = vs -

where is and vs are the stator current and terminal voltage respectively. The dynamics of the

rotor flux linkage rmay be written as

1

1

s

r r r r s

r i js( )

&+

= +

where s is the fractional slip speed (positive for generating) and i r is the rotor current. This

can be re-written in terms of the induced voltage using

rm

r

jx

X

=

to give

Tr jX

r jXjs T j

X X

r jXvs s

s

ss

s

s0 0& =

+

+

+

+

where

TX

r

r

s r

0 = .

The 4th order model is a full d-q (direct and quadrature) axis representation of the generatorwhich uses Parks transformation [4.2] to model the 3-phase windings of the generator as an

equivalent set of two windings in quadrature [4.3]. Using complex notation to represent the

direct and quadrature components of currents and voltages as the real and imaginary parts of

a single complex quantity, we can obtain

x x x d

dt

i

i

x r jx s x r jx xr s

x r jx x s x r jx x s

i

i

x

xvs r m

s

s

r

r s m m r m

m s m s s r s r

s

r

r

m

s

=

+ + + +

+ +

+

2 2 1 1

1 1

( ) ( )

( ) ( )

where all the currents and voltages are now complex.

Where speed of simulation is more important than accuracy, one of the lower order models

should be used. The 4th order model should be used for the greatest accuracy, although in

many circumstances the lower order models give very similar results. The lower order

models do not give an accurate representation of start-up transients, however.

4.2.3 Variable speed generator

This model should be used for a variable speed turbine incorporating a frequency converter to

decouple the generator speed from the grid frequency. The variable speed drive, consisting

of both the generator and frequency converter, is modelled as a whole. A modern variable

speed drive is capable of accepting a torque demand and responding to this within a very

short time to give the desired torque at the generator air-gap, irrespective of the generator

speed (as long as it is within specified limits). A first order lag model is provided for thisresponse:


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QQ

sg

d

e

=+( )1

where Qd is the demanded torque, Qg is the air-gap torque, and e is the time constant of thefirst order lag. Note that the use of a small time constant may result in slower simulations. If

the time constant is very small, specifying a zero time constant will speed up the simulations,

without much effect on accuracy.

A variable speed turbine requires a controller to generate an appropriate torque demand, such

that the turbine speed is regulated appropriately. Details of the control models which are

available withBladedcan be found in Section 5.

The minimum and maximum generator torque must be specified. Motoring may occur if a

negative minimum torque is specified.

The phase angle between current and voltage, and hence the power factor, is specified, on the

assumption that, in effect, both active and reactive power flows into the network are being

controlled with the same time constant as the torque, and that the frequency converter

controller is programmed to maintain constant power factor.

An option for drive train damping feedback is provided. This represents additional

functionality which may be available in the frequency converter controller which adds a term

derived from measured generator speed onto the incoming torque demand. This term is

defined as a transfer function acting on the measured speed. The transfer function is supplied

as a ratio of polynomials in the Laplace operator, s. Thus the equation for the air-gap torque

Qgbecomes

QQ

s

Num s

Den sg

d

e

g=+

+( )

( )

( )1

where Num(s) and Den(s) are polynomials. The transfer function would normally be some

kind of tuned bandpass filter designed to provide some damping for drive train torsional

vibrations, which in the case of variable speed operation may otherwise be very lightly

damped, sometimes causing severe gearbox loads.

4.2.4 Variable slip generator

A variable slip generator is essentially an induction generator with a variable resistance in

series with the rotor circuit [4.3, 4.4]. Below rated power, it acts just like a fixed speed

induction generator, so the same parameters are required as described in Section 4.2.1.

Above rated, the variable slip generator uses a fast-switching controller to regulate the rotor

current, and hence the air-gap torque, so the generator actually behaves just like a variable

speed system, albeit with a limited speed range. The same parameters as for a variable speed

system must therefore also be supplied (see Section 4.2.3), with the exception of the phase

angle since power factor control is not available in this case.

Alternatively, a full electrical model of the variable slip generator is available. The generator

is modelled as in Section 4.2.2, and the rotor current controller is modelled as a continuous-time PI controller which adjusts the rotor resistance between the defined limits (with


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integrator desaturation on the limits), in response to the difference between the actual and

demanded rotor current. The steady-state relationship between torque and rotor current is

computed at the start of the simulation, so that the torque demand can be converted to a rotor

current demand. The scheme is shown in Figure 4.2.

Figure 4.2: Variable slip generator rotor current controller

4.3 Drive train mounting

If desired, torsional flexibility may be specified either in the gearbox mounting or between

the pallet or bedplate and the tower top. This option is only allowed if either the stiff or

flexible drive train model is specified, and it adds an additional rotational degree of freedom.

In either case, the torsional stiffness and damping of the mounting is specified, with the axis

of rotation assumed to coincide with the rotor shaft. The moment of inertia of the moving

components about the low speed shaft axis must also be specified. In the case of a flexible

gearbox mounting, this is the moment of inertia of the gearbox casing. In the case of a

flexible pallet mounting, it is the moment of inertia of the gearbox casing, the generator

stator, the moving pallet and any other components rigidly fixed to it.

If either form of mounting is specified, the direction of rotation of the generator shaft willaffect some of the internal drive train loads. If the low speed and high speed shafts rotate in

opposite directions, specify a negative gearbox ratio in the drive train model. The effect of

any offset between the low speed shaft and high speed shaft axes is ignored.

Any shaft brake is assumed to be rigidly mounted on the pallet. Thus any motion once the

brake disk has stopped turning depends on the type of drive train mounting as well as on the

position of the brake on the low or high speed shaft. For example if there is a soft pallet

mounting, then there will still be some oscillation of the rotor after the brake disk has stopped

even if both shafts are stiff.

As in the case of the flexible shaft drive train model, it should be pointed out that whilemodelling the effect of flexible mountings provides greater accuracy in the prediction of

loads, there is potential for one or two of the resulting drive train vibrational modes to be of

relatively high frequency, depending on the various moments of inertia and shaft and

mounting stiffnesses. The presence of high frequency modes could result in slower

simulations.

4.4 Energy losses

Power train energy losses are modelled as a combination of mechanical losses and electrical

losses in the generator (including the frequency converter in the case of variable speedturbines).

Torque

demand

Current

demand

PI with

limits

1

|I|Rotor

resistanc

Measured current |I|


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Mechanical losses in the gearbox and/or shaft bearings are modelled as either a loss torque or

a power loss, which may be constant, or interpolated linearly from a look-up table. This may

be a look-up table against rotor speed, gearbox torque or shaft power, or a two-dimensional

look-up table against rotor speed and either shaft torque or power. Mechanical lossesmodelled in terms of power are inappropriate if calculations are to be carried out at low or

zero rotational speeds, e.g. for starts, stops, idling and parked calculations. In these cases, the

losses are better expressed in terms of torque.

The electrical losses may specified by one of two methods:

Linear model: This requires a no-load lossLNand an efficiency , where the electrical power

outputPe is related to the generator shaft input powerPsby:

Pe = (Ps - LN)

Look-up table: The power lossL(Ps) is specified as a function of generator shaft input power

Psby means of a look-up table. The electrical power outputPe is given by:

Pe = Ps - L(Ps)

Linear interpolation is used between points on the look-up table.

Note that if a full electrical model of the generator is used, additional electrical losses in this

form cannot be specified since the generator model implicitly includes all electrical losses.

4.5 The electrical network

Provided either the detailed electrical model of the induction generator or the variable speed

generator model is used, so that electrical currents and voltages are calculated, and reactive

power as well as active power, then the characteristics of the network to which the turbine is

connected may also be supplied. As well as allowing the voltage variations, and hence the

flicker, at various points on the network to be calculated, the presence of the network may

also, in the case of the directly connected induction generator, influence the dynamic

response of the generator itself particularly on a weak network.

The network is modelled as a connection, with defined impedance, to the point of commoncoupling (PCC in Figure 4.2) and a further connection, also with defined impedance, to an

infinite busbar. Further turbines may be connected at the point of common coupling. These

additional turbines are each assumed to be identical to the turbine being modelled, including

the impedance of the connection to the point of common coupling. However they are

modelled as static rather dynamic, with current and phase angle constant during the

simulation. The initial conditions are calculated with the assumption that all turbines are in

an identical state, and the other turbines then remain in the same state throughout. Thus the

steady state voltage rise due to all the turbines at the point of common coupling will be taken

into account in calculating the performance of the turbine whose performance is being

simulated .


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Figure 4.2: The network model

R1 + jX1 R2 + jX2

Infinite

busbar

Windfarm

interconnection

impedance

Network

connection

impedance

Other turbines

(if re uired)

Wind

turbine

PCC


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5. CLOSED LOOP CONTROL

5.1 Introduction

Closed loop control may be used during normal running of the turbine to control the blade

pitch angle and, for variable speed turbines, the rotor speed. Four different controller types

are provided:

1. Fixed speed stall regulated. The generator is directly connected to a constant frequencygrid, and there is no active aerodynamic control during normal power production.

2. Fixed speed pitch regulated. The generator is directly connected to a constant frequencygrid, and pitch control is used to regulate power in high winds.

3. Variable speed stall regulated. A frequency converter decouples the generator from thegrid, allowing the rotor speed to be varied by controlling the generator reaction torque. In

high winds, this speed control capability is used to slow the rotor down until aerodynamic

stall limits the power to the desired level.

4. Variable speed pitch regulated. A frequency converter decouples the generator from thegrid, allowing the rotor speed to be varied by controlling the generator reaction torque. In

high winds, the torque is held at the rated level and pitch control is used to regulate the

rotor speed and hence also the power.

For a constant speed stall regulated turbine no parameters need be defined as there is no

control action. In the other cases the control action will determine the steady state operatingpoint of the turbine as well as its dynamic response. For steady state calculations it is only

necessary to specify those parameters which define the operating curve of the turbine. For

dynamic calculations, further parameters are used to define the dynamics of the closed loop

control. The parameters required are defined further in the following sections.

Note that all closed loop control data are defined relative to the high speed shaft.

5.2 The fixed speed pitch regulated controller

This controller is applicable to a turbine with a directly-connected generator which uses blade

pitch control to regulate power in high winds. It is applicable to full or partial span pitch

control, as well as to other forms of aerodynamic control such as flaps or ailerons. In the

latter case, the pitch angle can be taken to refer to the deployment angle of the flap or aileron.

From the optimum position, the blades may pitch in either direction to reduce the

aerodynamic torque. If feathering pitch action is selected, the pitchable part of the blade

moves to reduce its angle of attack as the wind speed (and hence the power) increases. If

stalling pitch action is selected, it moves in the opposite direction to stall the blade as the

wind speed increases. In the feathering case, the minimum pitch angle defines the pitch

setting below rated, while in the stalling case the maximum pitch angle is used below rated,

and the pitch decreases towards the minimum value (usually a negative pitch angle) aboverated.


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Figure 5.1 shows schematically the elements of the fixed speed pitch regulated control loopwhich are modelled.

5.2.1 Steady state parameters

In order to define the steady-state operating curve, it is necessary to define the power set-

point and the minimum and maximum pitch angle settings, as well as the direction of pitching

as described above. The correct pitch angle can then be calculated in order to achieve the set-

point power at any given steady wind speed.

5.2.2 Dynamic parameters

To calculate the dynamic behaviour of the control loop, it is necessary to specify the dynamic

response of the power transducer and the pitch actuator, as well as the actual algorithm usedby the controller to calculate a pitch demand in response to the measured power signal.

Section 5.5 describes the available transducer and actuator models, while Section 5.6

describes the PI algorithm which is used by the controller.

5.3 The variable speed stall regulated controller

This controller model is appropriate to variable speed turbines which employ a frequency

converter to decouple the generator speed from the fixed frequency of the grid, and which do

not use pitch control to limit the power above rated wind speed. Instead, the generator

reaction torque is controlled so as to slow the rotor down into stall in high wind speeds. The

control loop is shown schematically in Figure 5.2.


The steady-state operating curve can be described with reference to a torque-speed graph as

in Figure 5.3. The allowable speed range in the steady state is from S1 to S2. In low winds it

is possible to maximise energy capture by following a constant tip speed ratio load line which

corresponds to operation at the maximum power coefficient. This load line is a quadratic

curve on the torque-speed plane, shown by the line BG in Figure 5.3. Alternatively a look-up

table may be specified. If there is a minimum allowed operating speed S1, then it is no

longer possible to follow this curve in very low winds, and the turbine is then operated at

nominally constant speed along the line AB shown in the figure. Similarly in high wind

speeds, once the maximum operating speed S4 is reached, then once again it is necessary to

Figure 5.1: The fixed speed pitch regulated control loop

Turbine

Pitch

actuator

ControllePower

transducer

Wind Electric

power

Measured

power

Pitch

demandBlade pitch

Power

set-point


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depart from the optimum load line by operating at nominally constant speed along the line

GH.

Once maximum power is reached at point H, it is necessary to slow the rotor speed down into

stall, along the constant power line HI. If high rotational speeds are allowed, it is of course

possible for the line GH to collapse so that the constant power line and the constant tip speed

Figure 5.2: The variable speed stall regulated control loop

Figure 5.3: Variable speed stall regulated operating curve

TurbineWind

Speed

transducer

Generator

speed

Measured

speed

Generator

torque

demand

Controlle

Desired

power,

torque,

speed

Power

transducerElectrical

power

Measured

power


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ratio line meet at point J.

Clearly the parameters needed to specify the steady state operating curve are:

The minimum speed, S1

The maximum speed in constant tip speed ratio mode, S4 The maximum steady-state operating speed. This is usually S4, but could conceivably be

higher in the case of a turbine whose characteristics are such that as the wind speed

increases, the above rated operating point moves from H to I, then drops back to H, and

then carries on (towards J) in very high winds. This situation is somewhat unlikely

however, because if rotational speeds beyond S4 are permitted in very high winds, there is

little reason not to increase S4 and allow the same high rotor speeds in lower winds.)

The above rated power set-point, corresponding to the line HI. This is defined in terms of

shaft power. Electrical power will of course be lower if electrical losses are modelled.

The parameterK which defines the constant tip speed ratio line BG. This is given by:

K = R5

Cp() / 23

G3

where

= air density

R = rotor radius

= desired tip speed ratio

Cp() = Power coefficient at tip speed ratio

G = gearbox ratio

Then when the generator torque demand is set to K2

where is the measured generator

speed, this ensures that in the steady state the turbine will maintain tip speed ratio and the

corresponding power coefficient Cp(). Note that power train losses may vary with rotationalspeed, in which case the optimum rotor speed is not necessarily that which results in the

maximum aerodynamic power coefficient.

As an alternative to the parameter K, a look-up table may be specified giving generator

torque as a function of speed.



response of both power and speed transducers, as well as the actual algorithm used by the

controller to calculate a generator torque demand in response to the measured power and

speed signals. Section 5.5 describes the available transducer and actuator models.

Two closed loop control loops are used for the generator torque control, as shown in Figure

5.4. An inner control loop calculates a generator torque demand as a function of generator

speed error, while an outer loop calculates a generator speed demand as a function of power

error. Both control loops use PI controllers, as described in Section 5.6.

Below rated, the speed set-point switches between S1 and S4. In low winds it is at S1, and

the torque demand output is limited to a maximumvalue given by the optimal tip speed ratio

curve BG. This causes the operating point to track the trajectory ABG. In higher winds, the

set-point changes to S4, and the torque demand output is limited to a minimumvalue given by

the optimal tip speed ratio curve, causing the operating point to track the trajectory BGH.

Once the torque reaches QR, the outer control loop causes the speed set-point to reduce along

HI, and the inner loop tracks this varying speed demand.


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5.4 The variable speed pitch regulated controller

This controller model is appropriate to variable speed turbines, which employ a frequency

converter to decouple the generator speed from the fixed frequency of the grid, and which use

pitch control to limit the power above rated wind speed. The control loop is shown

schematically in Figure 5.5.


The steady-state operating curve can be described with reference to the torque-speed graph

shown in Figure 5.6. Below rated, i.e. from point A to point H, the operating curve is exactly

as in the stall regulated variable speed case described in Section 5.3.1, Figure 5.3. Above

rated however, the blade pitch is adjusted to maintain the chosen operating point, designated

L. Effectively, changing the pitch alters the lines of constant wind speed, forcing them to

pass through the desired operating point.

Power

set-point

Figure 5.5: The variable speed pitch regulated control loop

TurbineWind

PI

controllerPI

controller

Speed

transducerGenerator

speed

Measured

speed

Generator

torque

demand

Controlle

Desired

torque

and speed

Measured power

Measured speed

Pitch

actuator

Generator torque demand

Blade pitch

Figure 5.4: Stall regulated variable speed control loops

Pitch

demand

Speed

demand


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Once rated torque is reached at point H, the torque demand is kept constant for all higher

wind speeds, and pitch control regulates the rotor speed. A small (optional) margin is

allowed between points H (where the torque reaches maximum) and L (where pitch control

begins) to prevent excessive mode switching between below and above rated control modes.

However, this margin may not be required, in which case points H and L coincide. As with

the stall regulated controller, the line GH may collapse to a point if desired.

Clearly the parameters needed to specify the steady state operating curve are:

The minimum speed, S1

The maximum speed in constant tip speed ratio mode, S4

The speed set-point above rated (S5). This may be the same as S4. The maximum steady-state operating speed. This is normally the same as S5.

The above rated torque set-point, QR.

The parameter K which defines the constant tip speed ratio line BG, or a look-up table.

This is as defined in Section 5.3.1.



response of the speed transducer and the pitch actuator, as well as the actual algorithm used

by the controller to calculate the pitch and generator torque demands in response to the

measured speed signal. Section 5.5 describes the available transducer and actuator models.

Figure 5.6: Variable speed pitch regulated operating curve


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Figure 5.7 shows the control loops used to generate pitch and torque demands. The torque

demand loop is active below rated, and the pitch demand loop above rated. Section 5.6

describes the PI algorithm which is used by both loops.

Below rated, the speed set-point switches between S1 and S4. In low winds it is at S1, andthe torque demand output is limited to a maximumvalue given by the optimal tip speed ratio

curve BG. This causes the operating point to track the trajectory ABG. In higher winds, the

set-point changes to S4, and the torque demand output is limited to a minimumvalue given by

the optimal tip speed ratio curve, causing the operating point to track the trajectory BGH, and

a maximum value of QR. When point H is reached the torque remains constant, with the

pitch control loop becoming active when the speed exceeds S5.

5.5 Transducer models

First order lag models are provided in Bladed to represent the dynamics of the power

transducer and the generator speed transducer. The first order lag model is represented by

& ( )yT

x y= 1

wherex is the input andy is the output. The input is the actual power or speed and the output

is the measured power or speed, as input to the controller.

5.6 Modelling the pitch actuator

The pitch actuator may be modelled as either a pitch position or pitch rate actuator, and either

active or passive dynamics may be specified.

The simplest model is a passive actuator, with the relationship between the input and the

output represented by a transfer function. For the pitch position actuator, the input is the

pitch demand generated by the controller and the output is the actual pitch angle of the

blades. For the pitch rate actuator, the input is the pitch rate demand generated by the

controller and the output is the actual pitch rate at which the blades move. The transfer

Speed

set-pointPI

controller

PI

controllerMeasured speed

Blade pitch

Generator torque demand

Figure 5.7: Pitch regulated variable speed control loops

Above rated

Below rated


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function may be a first order lag, a second order response, or a general transfer function, up

to 8th order.

The first order lag model is represented by

& ( )yT

x y= 1

wherex is the input andy is the output. The second-order model is represented by

&& & ( )y y x y+ = 2 2!

where is the bandwidth and ! the damping factor. The general transfer function model is

represented by numerator and denominator polynomials in the Laplace operator.

For detailed calculations, especially to understand the loads on the pitch actuator itself and

the duty which will be required of it, it is possible to enter a more detailed model. This can

take into account any internal closed loop dynamics in the actuator, and also the pitch motion

resulting from the actuator torque acting on the pitching inertia, with or against the

aerodynamic pitch moment and the pitch bearing friction. The bearing friction itself depends

critically on the loading at the pitch bearing.

Figure 5.8 shows the various options for controlling the pitch angle, starting from either a

pitch position demand or a pitch rate demand. The pitch position demand may optionally be

processed through a ramp control, shown in Figure 5.9, which smooths the step changes in

demand generated by a discrete controller by applying rate and/or acceleration limits. Then

the pitch position demand can act either through passive dynamics to generate a pitchposition, or through a PID controller on pitch error to generate a pitch rate demand. Rate

limits are applied to the output, with instantaneous integrator desaturation to prevent wind-up

in the PID case. Thus the pitch rate demand may come either from here or directly from the

controller. This rate demand can act either through passive dynamics to generate a pitch rate,

or through a PID controller on pitch rate error to generate an actuator torque demand. In the

latter case, the pitch actuator passive dynamics then generate an actual actuator torque, which

acts against bearing friction and any aerodynamic pitching moment to accelerate the pitching

inertia of the blades and the actuator itself. An optional first order filter on each PID input

allows step changes in demand from the controller to be smoothed, and instantaneous

integrator desaturation prevents wind-up when the torque limits are reached.

Both PID controllers include a filter on the differential term to prevent excessive high

frequency gain. Also there is a choice of derivative action, such that the derivative gain may

be applied either to the feedback (i.e. the measured position or rate), the error signal, or the

demand. The latter case represents a feed-forward term in the controller.

If passive pitch rate dynamics are selected, the response will be subject to acceleration limits

calculated from the aerodynamic pitching moment, bearing friction and the actuator toque

limits acting on the pitching inertia. If the total pitching inertia is zero, no limits will be

applied.

The pitch bearing sliding friction torque is modelled as the sum of four terms: a constant, a

term proportional to the bending moment at the bearing, and a terms proportional to the axialand radial forces on the bearing. Sometimes the actuator cannot overcome the applied


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torques and the pitch motion will stick. Before it can move again, the break-out or stiction

torque must be overcome. This is modelled as an additional contribution to the friction

torque while the pitch is not moving. This additional contribution is specified as a constant

torque, plus a term proportional to the sliding friction torque.

PID controller

Measured

pitch

position

Passive

dynamics

Pitch position

demand from

controller

Measured

pitch rate

Pitch rate

demand from

controller

Actual pitch

position

+-

Pitch rate

demand

Passive

dynamics

PID controller

Bearing

loads

Actuator

torque

demand

Passive

dynamics

Actuator

torque

Bearing

friction

+

-

Pitching

inertia

Pitch rate

Acceleration

limits

Pitching

inertia

Actuator

torque

limits

Pitching

moment

Ramp control

Figure 5.8: Pitch actuator options


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0

0.2

0.4

0.6

0.8

1

1.2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

Timesteps

Demand

Raw demand

Rate limit

Acceleration limit

Rate & acceleration limits

The ramp is re-started each timestep. If the ramp is not completed by the end of the timestep

and an acceleration limit is specified, the slope at the start of the next timestep will be non-

zero.

Figure 5.9: Ramp control for pitch actuator position demand

5.7 The PI control algorithm

All the closed loop control algorithms described above use PI controllers to calculate the

outputy (pitch, torque or speed demand) from the inputx (power or speed error). The basic

PI algorithm can be expressed as

& &y K x K xp i= +

whereKp andKi represent the proportional and integral gains. The ratioKp/Ki is also known

as the integral time constant. Calculation of appropriate values for the gains is a specialisttask, which should take into account the dynamics of the wind turbine together with the

aerodynamic characteristics and principal forcing frequencies, and should aim to achieve

stable control at all operating points and a suitable trade-off between accuracy of tracking the

set-point and the degree of actuator activity.

Straightforward implementation of the above equation leads to the problem of integrator

wind-up if the output y is subject to limits, as is the case here. This means that the raw

output calculated as above continues to change as a result of the integral (Ki) term even

though the actual output is being constrained to a limit. When the direction of movement ofy

changes, it will then take a long time before it comes back to the limit so that the final

(constrained) output starts to change. This is avoided in the continuous-time implementation

of the PI controller by an additional term -"y/Td in the above equation, where "y is the


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GH Bladed Theory Manual

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