Development of a model to investigate the yaw behaviour of small horizontal axis wind turbines...

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M Narayana, G A Putrus, P S Leung, M Jovanovic and S McDonaldDevelopment of a model to investigate the yaw behaviour of small horizontal axis wind turbines

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Development of a model to investigate the yawbehaviour of small horizontal axis wind turbinesM Narayana1*, G A Putrus1, P S Leung1, M Jovanovic1, and S McDonald2

1School of Computing, Engineering and Information Sciences, Northumbria University, Newcastle upon Tyne, UK2Department of Electrical Technology, New and Renewable Energy Centre (NaREC), Blyth, UK

The manuscript was received on 29 January 2011 and was accepted after revision for publication on 9 August 2011.

DOI: 10.1177/0957650911421854

Abstract: Almost all small wind turbines have a tail vane to point the wind rotor into the winddirection. Various types of control mechanisms are used to protect the wind turbines from high-wind speeds. In this study, a generic model of a wind turbine is developed to investigate the yawoperation of small-scale horizontal axis wind turbines (HAWT) with a single tail vane. A tilt-upwind turbine is considered here as most of the other HAWT types can be simplified to a state ofthis model. In the tilted up wind turbines, the wind rotor is free to move around the horizontalaxis as well as the vertical one. When the wind speed exceeds the rated value, the wind rotor tiltsupwards, taking up a stable position with an inclination towards the direction of the wind, thuscontrolling the wind component responsible for power generation. This article describes thedevelopment of a mathematical model of small-scale HAWT using the D’Alembert’s principle toinvestigate the complex behaviour of the wind turbine under transient and steady-state oper-ating conditions. In this model, the blade element theory was adapted to predict the aerody-namic forces on the wind rotor at each rotor position. Combining the modified blade elementwith the momentum theories makes it possible to evaluate flow fields of induced velocitiesaround the wind rotor. Yaw movements are modelled by considering dynamic responses ofthe tail vane. Finally, simulation results are presented to illustrate the yaw behaviour of asmall wind turbine.

Keywords: D’Alembert’s principle, small-scale wind turbines, yaw behaviour

1 INTRODUCTION

The wind turbine must align itself in the direction of

the wind and has to adapt to any circumstances.

It also needs to protect itself from gusty winds

beyond the rated wind speed and this must happen

automatically. Larger wind turbines have computer-

driven control systems that operate servo-motors,

hydraulic motors, etc., to execute these functions.

Due to relatively limited investments in small wind

turbines, simple passive controls are needed wherever

possible. Passive yaw control mechanisms are being

commonly used in small wind turbines for both power

regulation and wind hunting. In this study, a mathe-

matical model was developed by considering a tilt-up

type small wind turbine built in with a single tail vane.

This system can be simplified to most of the other

horizontal axis wind turbine (HAWT) types built in

with single tail vanes. Figure 1 shows the kinetic

details of this system. Investigation of the kinetic

behaviour of tilt-up turbines is more complicated as

the wind rotor, the generator, and the counterweights

can be rotated freely around the pivot point in the

horizontal axis as well as around the vertical axis

(i.e. tower axis). The wind rotor is pivoted in a vertical

plane and it could be tilted upwards due to the wind

speed dependent thrust force of the wind rotor.

In this system, a counterbalancing weight or a

spring mechanism is required to produce the

*Corresponding author: School of Computing, Engineering and

Information Sciences, Northumbria University, Newcastle upon

Tyne, UK.

email: [email protected]

REVIEW ARTICLE 1

Proc. IMechE Vol. 000 Part A: J. Power and Energy



necessary restoring moment to hold the wind rotor in

the wind direction. At wind speeds above the rated

value, the wind rotor is tilted upward due to the resul-

tant torque of the thrust force of the wind rotor and

the restoring moment of counterbalancing weights

around the pivoted point. Normally in these systems,

a damper is used to control irregular vertical move-

ments [1]. This study intended to describe the devel-

opment of a mathematical model to investigate the

complex yaw behaviour of the small-scale wind

turbine.

2 MOMENTS DERIVED FROM D’ALEMBERT’SFORCES

Using the D’Alembert’s principle, a dynamic system

can be converted into a static system to be analysed

under static equilibrium conditions [2]. In this case,

for each and every point, forces of magnitude �mi.fi

are applied in the direction opposite to its accelera-

tion, where �mi is the mass and fi the acceleration at

the point i. D’Alembert’s forces on the wind rotor, the

generator, and the counterweight are shown in Fig. 1.

Orientations of the wind rotor and the tail vane are

illustrated in Fig. 2.

Momentums of D’Alembert’s forces on the wind

rotor, the generator, and the counterweight system

about ‘O’ can be expressed as follows (Fig. 1)

M0 ¼X

�R0 þ �ri

� �� mi:

€�R0 þ€�ri

� �¼X

�mi�R0 þ �ri

� ��

€�R0 þ �R0 þ �ri

� �� €�ri

h i¼X

�mi�R0 �

€�R0 þ€�R0

X�miri|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

0

þ �R0 �X

�mi€�ri|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

0

þX

�mi �ri �€�ri

M0 ¼ m �R0 �€�R0 þ

dP _�ri � �mi �ri

� �h idt

ð1Þ

Now given that ð _� � _�, _�Þ, then _�R0 �_�ri. Velocity at

point ‘i’ ¼ _�R0 þ_�ri ¼ �Vi

; _�V��

i� _�ri

�� _� �r1ij j

�ri ¼ �r0 þ �r1i

ThereforeX_�ri � �mi �ri ¼

X�mi �r0 þ �r1ið Þ � �Vi

¼ �r0

X�miVi|fflffl{zfflffl}

0

þX

�mi �r1i � Vi

thenX

�mi �r1i � �Vi ¼ �I _�

Therefore

M0 ¼ m �R0 �€�R0 þ

dð�I _�Þ

dtð2Þ

The wind rotor, the generator, the counterweight

system, and the tail vane are free to move about k

axis. Therefore, the D’Alembert’s momentum of the

wind rotor about the k axis

Mk ¼k component of m �R0�€�R0þ

dð�I _�Þ

dt

� �ð3Þ

Similarly, the wind rotor and the counterweight

system are free to move about the j0 axis, and the cor-

responding the D’Alembert’s momentum can be

expressed as

Mj 0 ¼ j0 component of m �R0�€�R0þ

dð�I _�Þ

dt

� ð4Þ

Solutions of Mk and Mj 0 are described in the

Appendix 2.

Fig. 2 Orientation of the wind rotor and the tail vane

Fig. 1 Configuration of tilt-up type small wind turbineand D’Alembert’s forces

2 M Narayana, G A Putrus, P S Leung, M Jovanovic, and S McDonald




3 AERODYNAMIC FORCES ON THE WIND

ROTOR

In this study, the blade element theory is used to

determine forces and momentums by assuming a

blade composed of a number of aerodynamically

independent cross-sections having the same charac-

teristics of a blade at a proper angle of attack [3]. The

overall flow pattern through the wind rotor is pre-

sented in Fig. 3. At the yawed position, the wind

rotor blade experiences a higher relative velocity on

the advancing side than on the retreating one.

Therefore, the blade loads and the generated power

fluctuate.

The flow expansion angle is varying throughout

the length of each blade element. Its value is zero at

the wind rotor hub and maximum at the blade tip.

The flow expansion angle at the tip of the wind

rotor blades is estimated by the average axial induc-

tion factors (K0) of the wind rotor in normal flow. At a

given angle between wind velocity and wind rotor axis

(’), the expansion angle at the tip "(R) is given as fol-

lows [4, 5]

"ðRÞ ¼1

3

1� K 0ð Þ

1þ K 0ð Þ

� cos ’ ð5Þ

Then, assuming that the variation of flow expansion

angle is equal from hub to tip, the flow expansion

angle at radius r of the wind rotor is

"ðrÞ ¼r

R�

1

3�

1� K 0ð Þ

1þ K 0ð Þ

� � cos ’ ð6Þ

The flow components at the wind rotor blade

are shown in Fig. 4. For a particular wind rotor

position (�), the axial wind speed relative to the

blade element at a distance r from the wind rotor

axis can be written as (Fig. 4)

VaðrÞ ¼Va1ðrÞ þ Va2ðrÞ½ �

2¼ð1þ kÞ

2� Va1ðrÞ ð7Þ

According to Fig. 4, for a particular wind rotor

position (�), the circumferential wind speed Vr(r) in

the wind rotor plane relative to the blade element

at a distance r from the wind rotor axis can be

expressed as

Vr ðrÞ ¼Vr1ðrÞ þ Vr2ðrÞ½ �

2¼ð1þ hÞ

2� Vr1ðrÞ ð8Þ

where Va2ðrÞ ¼ k � Va1ðrÞ and Vr2ðrÞ ¼ h � Vr1ðrÞ.

For a particular rotor position (�), axial and radial

relative flow components at the blade element (r) can

be expressed as

VaðrÞ ¼

�V1 � cos’� sin ’ � tan "ðrÞ � cos �ð Þ

� r �d’

dt� cos �

�

k þ 1

2

ð9Þ

Vr ðrÞ ¼ ! � r þ V1 � sin ’ � sin �ð Þ �h þ 1

2ð10Þ

’ ¼ cos�1 cos � � cos �ð Þ ð11Þ

Fig. 3 Overall flow pattern through the wind rotorFig. 4 Flow components at the blade element (r) for a

particular rotor position

Development of a model to investigate the yaw behaviour of small horizontal axis wind turbines 3




At r blade elements, the incidence angle �(r) for

a particular rotor position (�) is obtained by referring

to Fig. 4.

cot�ðrÞ ¼Vr ðrÞ

VaðrÞð12Þ

In the above equations, � represents the rotor posi-

tion in the swept circle. This angle is zero at the hor-

izontal position of the right-hand side as viewed from

upstream. Flow stream lines are expanded in the

downstream of the wind rotor.

The blade element theory is used to calculate the

elementary aerodynamic forces (i.e. lift dRl and drag

dRd) acting on the blade element. That is

dRl ¼1

2� � � Cl �W ðrÞ

2� l � dr ð13Þ

dRd ¼1

2� � � Cd �W ðrÞ

2� l � dr ð14Þ

For given r , �, _�, ’, _’ values, dFv r , �, _�, ’, _’� �

and

dFu r , �, _�, ’, _’� �

are the elementary axial and tangen-

tial components of the aerodynamic forces, respec-

tively, given by

dFv r , �, _�, ’, _’� �

¼ dRl � cos�ðrÞ þ dRd � sin �ðrÞ

ð15Þ

dFu r , �, _�, ’, _’� �

¼ dRl � sin�ðrÞ � dRd � cos�ðrÞ

ð16Þ

By substituting (13) and (14) into the above two

equations, one gets

dFvðr , �, _�, ’, _’Þ ¼1

2� � l �W 2 � dr � ðCl � cos�ðrÞ

þ Cd � sin �ðrÞÞ ð17Þ

dFuðr , �, _�, ’, _’Þ ¼1

2� � l �W 2 � dr � ðCl � sin�ðrÞ

þ Cd � cos�ðrÞÞ

ð18Þ

These forces can be resolved to obtain the

elementary axial thrust and torque on the rotor

blade ‘b’ as

Fb r , �, _�, ’, _’� �

¼ dFvðr , �, _�, ’, _’Þ and

Mb r , �, _�, ’, _’� �

¼ r � dFuðr , �, _�, ’, _’Þ

The resultant wind speed for the blade element is

derived as (Fig. 4)

W ðrÞ ¼VaðrÞ

sin�ðrÞð19Þ

Then, (17) and (18) can be simplified into the fol-

lowing forms

Fb r , �, _�, ’, _’� �

¼ 1=2 � � � l �VaðrÞ

sin�ðrÞ

� 2

� Cl � cos�ðrÞ þ Cd � sin�ðrÞð Þ � dr

ð20Þ

Mb r , �, _�,’, _’� �

¼ 1=2 � � � l � r �VaðrÞ

sin�ðrÞ

� 2

� ðCl � sin�ðrÞ þ Cd � cos�ðrÞÞ � dr

ð21Þ

The axial thrust and the aerodynamic moment on

the wind rotor can be represented by

FT ¼Xb

1

Xr¼R

r¼rðhubÞ

Fb r , �, _�, ’, ’� �

ð22Þ

TW ¼Xb

1

Xr¼R

r¼rðhubÞ

Mb r , �, _�, ’, _’� �

ð23Þ

The blade element theory evaluates aerodynamic

forces on a particular blade element for given

flow fields [3]. The momentum theory is applied to

the axial flow for wind rotor annulus of the corre-

sponding blade element [6], which makes it possible

to evaluate a field of induced velocity around the

wind rotor. For the yaw calculations, aerodynamic

forces of every blade elements of each wind

rotor blade are needed to be modelled by considering

the wind rotor position. In this calculation, the

momentum theory is applied to the 1/bth portion

of the annulus, where b is the number of blades.

Then, for a two-bladed wind rotor, the computational

domain covers a 180 portion of the wind rotor

disc. The induced flow of a skewed rotor not

being aligned with the rotor axis and its variation

across the rotor disc is not uniform [7]. To apply

modified blade element with momentum (BEM)

theory for a skewed wind rotor, no flow interactions

between adjacent rotor annuluses are assumed

for small yaw angles in this study [4]. For a parti-

cular wind rotor blade, the elementary thrust

force dFb r , �, _�, ’, _’� ��

and aerodynamic torque

Mb r , �, _�, ’, _’� �

¼ r � dFuðr , �, _�, ’, _’Þ� �

can be deter-

mined by considering the axial momentum of the

flow through 1/bth fraction of the annulus

Thrust ¼

rate of mass flow through

1=bth fraction of the annulus

with the thickness dr

264

375

�change in

the axial velocity

�





Then, the axial thrust on 1/bth fraction of the annu-

lus at radius r is given by

Fb r ,�, _�,’, _’� �

¼�r

b�:dr V1� r �

d’

dt� cos’

�

�

V1 � cos’� sin’ � tan"r � cos�ð Þ

�r �d’

dt� cos�

24

35 � ð1�k2Þ

ð24Þ

The aerodynamic torque

¼

rate of mass flow through

1=bth fraction of the

annulus with the thickness dr

264

375

�

change in

the radial

velocity

264

375� r

Then, aerodynamic torque due to 1/bth fraction of

the annulus at radius r is given by

Mb r , �, _�, ’, _’� �

¼� � r2

b� �:dr V1 � r �

d’

dt� cos ’

�

� ! � r þ V1 � sin ’ sin �ð Þ � 1þ kð Þ � h � 1ð Þ

ð25Þ

There are six equations, i.e. (9), (12), (20), (21),

(24), and (25) and six unknowns, i.e. VaðrÞ,�ðrÞ,ð

Fb r , �, _�, ’, _’� �

, Mb r , �, _�, ’, _’� �

, k, and hÞ. Therefore,

aerodynamic forces on each blade element can be

determined for either value of �, ’ (� and �), d�=dt ,

and d’=dt , when the aerofoil data (Cl and Cd curve

with �r) are known.

The wind velocity components and the respective

aerodynamic forces (Fs sin , Fs cos , and FT ) are

Fig. 5 Wind velocity components and aerodynamic forces applied on the wind rotor





shown in Fig. 5. Then, the side thrust force Fs

becomes

Fs ¼1

2� Cds � V

20 � � � As ð26Þ

where

V0 ¼ V1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos � � sin �ð Þ

2þ sin2 �

qð27Þ

For computational purposes, Fs is divided into two

components. Those components are Fs sin and

Fs cos in the wind rotor plane.

According to configuration of applied aerodynamic

forces on the wind rotor in Fig. 5, aerodynamic

moments on the wind rotor can be derived by con-

sidering equations (24), (25), and (26).

The aerodynamic moment of the wind rotor about

the k axis

MRk ¼Xb

1

Xr¼R

r¼rðhubÞ

rF r , �, _�,’, _’� �

� cos � � cos�

þTW � sin� þ x � sin � �ð Þ � Fs � cos

ð28Þ

and about the j 0 axis

MRj 0 ¼Xb

1

Xr¼R

r¼rðhubÞ

r � F r , �, _�, ’, _’� �

� sin �

þ x � cosð� �Þ � Fs � sin þ y � Ft

ð29Þ

To simplify the calculations in this study, the average

aerodynamic axial moment and thrust are deter-

mined by considering one rotation of the wind rotor

(as _� � _�, _�).

In tilt-up type small wind turbine systems, the wind

rotor part can be moved upward. Then kD _� can be

defined as the required damping component to

avoid irregular vertical movements of the wind rotor

part, consisting of the wind rotor, the generator, and

the counterweights. According to Fig. 5, the following

expression can be derived by considering the equilib-

rium of moments on the wind rotor part about the j 0

axis

kD _� þMj 0 ¼ MRj 0 þ X � sinð� �Þ �m � g ð30Þ

where kD is the damping factor for vertical

movements.

4 YAW MOTION OF THE WIND TURBINE

The contribution of wind rotor aerodynamics to

yaw dynamics is an extremely complex subject and

includes the effects of blade rigidity, horizontal and

vertical wind direction components, wind shear,

dynamic stall, and skewed wake effects [8]. In this

study, aerodynamic forces on the tail vanes were

calculated by considering the lift and drag forces by

lift and drag coefficients (Cl and Cd, respectively).

Relative wind speed to the tail vane and effective

angle of attack were considered here to determine

the dynamic forces on the tail vane. A velocity dia-

gram is shown in Fig. 6. It is assumed that lift and drag

coefficients for the tail vane can be approximated by

the standard ‘flat-plate equations’ [9].

The tail vane is some where in the downstream of

the wind rotor, so the change of the axial velocity

should be considered to evaluate the effective wind

velocity to the tail vane. The wind velocity at the

tail vane is greater than the downstream flow

value close to the wind rotor (V1 � k0), because it is in

combination with the normal wind flow; k 0 is the

average axial induction factor of the wind rotor,

which can be determined by the actuator disc theo-

rem [6]. In this study, suggested downstream axial

wind velocity at the tail vane is V1 � ð2k 0 � 1Þ for far

wake value [4].

Hence, the effected wind velocity to the tail vane is

U1 ¼ V1 � 2k 0 � 1ð Þ ð31Þ

By considering the velocity diagram in Fig. 6, one can

establish the following relationships

Vvt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU1 þ lv � _� � sin �ð Þ

2þ lv � _� � cos �ð Þ

2q

ð32Þ

Effective angle attack (�0) to the tail vane is:

�0 ¼ tan�1 U1 � sin �þ lv � _�ð Þ

U1 � cos �

� ð33Þ

The flat-plate equations are [9]: Clv ¼ K0 � sin �0� cos �0

and Cdv ¼ K0 � sin2��0; K0 is a constant. Here, it is

assumed that K0¼ 2 [10].

Then, lift and drag forces (Rlv and Rdv) on the wind

vane can be calculated as follows

Rlv ¼1

2�� Clv �V

2vt �Av and Rdv ¼

1

2�� Cdv �Vvt �Av

Fu ¼Rlv �cos�0 þRdv � sin�0, and

Fv ¼Rlv � sin�0 �Rdv �cos�0:

Fig. 6 Aerodynamic forces on the tail vane and windvelocity diagram for the tail vane





Substituting the solution for Clv, Cdv, Rlv, and Rdv in

the expression for Fu

Fu ¼1

2�K0 �V

2vt �� Av � sin�0 � cos2 �0 þ sin3 �0

� �ð34Þ

Aerodynamic moment of tail vanes about the k

axis is

�lv �1

2K0 � V

2vt � � � Av � sin �0 � cos2 �0 þ sin3 �0

� �Then, the equation of the system angular motion is

of the form

Mkþ I1 � €��MRk ¼�lv �1

2�K0 �V

2vt �� Av

� sin�0 �cos2�0 þsin3�0� � ð35Þ

where Mk is the D’Alambert’s moment of the

wind rotor about the k axis and MRk the wind rotor

aerodynamic moment of about the same axis.

For small � values, equation (33) can be approxi-

mated as

�0 ¼ �þlv � _�

U1ð36Þ

Using this approximation, (35) can now be simpli-

fied as

MRk þ I1 � €��Mk ¼ �lv �1

2� K0 � V

2vt � � � A � �

0 ð37Þ

The above equation is similar to an alternative form

used to represent the yaw dynamics as derived by

Kristensen [10] for the dynamic response of wind

vane. Therefore, Kristensen’s equation verifies the

derivation of yaw dynamics by ‘flat-plate equations’.

Under the yawed condition, the reduction of axial

velocity is induced by the wake skewness. There are

several models to analyse the skewed wake effects of

the wind rotor [11]. According to the actuator disc

theory [6], the skewed wake expansion is in direction

of the downwind blade for yawed flow condition.

Large wake angle ( �j j) will occur due to the significant

reduction of axial velocity at the wind rotor, as a result

of the large coefficient of thrust (Ct). The skew angle is

determined by the ratio between the inflow velocity

and edgewise component of the wind velocity.

Previous studies have shown that the wake skewed

angle is always slightly greater than the angle between

wind velocity and wind rotor axis ( ’�� 5 �j j) [12]. To

evaluate the effects of wake skewness on the vane due

to the yawed and tilt-up wind rotor by flow analysis

methods is a complicated exercise. As yaw angles,

yaw rate, and thrust coefficient (Ct) variation are

small in this study, the skew angle can be assumed

to be equal to the yaw angle for all yaw positions. That

means, effect of the wake skewness on the tail vane is

neglected in this study. Therefore, it is assumed that

¼ 0. A schematic diagram of the yaw motion is

shown in Fig. 7.

5 DYNAMIC BEHAVIOUR OF THE WIND

TURBINE

Generated aerodynamic torque by the wind rotor

depends on the wind speed and the wind rotor speed.

Torque characteristics (Cm) of the wind rotor are deter-

mined by considering the geometrical parameters

(blade angles and chord lengths) and characteristics of

the wind rotor profiles [3].

If TW ðV1, _�Þ is the aerodynamic torque about the

wind rotor axis and TLð _�, electrical loadÞ the restoring

torque by the load (generator), which relies on the

rotational speed and electrical load, then

TW � TL ¼ �I�� €� ð38Þ

Bearing in mind that Mk and MRk are given by (3)

and (28) and Mj 0 and MRj 0 by (4) and (29), from (24),

(29), and (38), it is clear that if V1, �, _�, �, _�, and _�

are known at a particular time instant (t0), then the

corresponding €�, €�, and €� parameters could be iden-

tified. After a very small time interval (�t), the values of

�, _�, �, _�, and _� could be determined by assuming

linear relationships within this interval. Under these

conditions, it is possible to apply an iterative method

Fig. 7 Schematic diagram of the yaw motion





to investigate the variation of kinematics parameters

for a given initial known operating condition. In this

study, series of iterative calculations are used to

march the solutions.

6 SIMULATION STUDY

Realistic computer simulations are effective for

analysis and evaluation of system performance

and represent a necessary introductory step to

practical implementation or expensive field testing.

In this study, dynamic computer models for each

component of a small-scale wind energy conversion

system (SS-WECS) were developed in MATLAB/

SIMULINK. Minor factors such as dynamic defor-

mations of the wind rotor blades and consequence

of power transmission shaft are neglected in this

study. A small wind turbine system consists of a

wind rotor, a generator (normally a permanent

magnet generator), a yaw mechanism, a rectifier,

and a battery bank. In SS-WECSs, the wind rotor

and the generator are directly coupled to each

other. The wind rotor and the electrical generator

have different torque–speed characteristics and

these operate at the same rotational speed. Yaw

behaviour of the small wind turbine produced in

NERDC, Sri Lanka, was investigated using the

developed mathematical model. Specifications of

the NERDC small wind turbine are given in

Appendix 3. The MATLAB computer model was

used to simulate the system by applying 5 step

ramp of yaw error for 5 m/s wind speed.

Simulation results for different rotor speeds are

shown in Fig. 8. This shows that higher frequency

yaw oscillations with higher damping occur at low

rotational speeds. The sample time for this

simulation is set to 10 ms. The results show that at

low rotational speeds, the WECS with a yaw error

quickly reaches its zero yaw error position.

This means that for a step change of yaw error,

high yaw rates occur at low rotational speeds.

Therefore, the system quickly attains its stable con-

dition at low rotational speeds. In this study, simula-

tions are further elaborated for different wind speeds

and rotational speeds with varying parameters of

the tail vane such as area (Av) and momentum of iner-

tia (I1). Also, geometrical parameters of the tail vane

affect the dynamic response of the system. According

to the simulation results, the system incorporated

with larger tail vane area, higher tail boom length,

and lower moment of inertia of tail vane is more

stable. This study assists to optimize the configura-

tion of the wind turbine by taking into account

material usage and strength of material aspects of

the components.

For a complete SS-WECS model, the yaw mecha-

nism, the wind rotor, and the generator models need

to be combined together. A comprehensive SS-WECS

model developed in the MATLAB/SIMULINK envi-

ronment, which can be used to model dynamic sys-

tems and simulations, is presented in Fig. 9. In-built

permanent magnet synchronous machine and uni-

versal bridge rectifier blocks in Simpowersystems

tool box of the standard SIMULINK library were

utilized in this study [13]. The permanent magnet

synchronous machine block is based on a state-

space d–q model where the restoring torque is

derived by d and q generator currents [14]. Input

parameters of the model are d and q voltages and

aerodynamic torque (Ta). A conversion between

(d, q) and (a, b, c) coordinates can be done in a

usual manner using the well-known Park transforma-

tion [15]. Operations of a single-vane system for a

constant output DC voltage (40 V) were simulated

over a period of 8.3 h with measured time series

wind data of 1 s sample time at Blyth, UK. Figure 10

shows the yaw rate versus rotational speed and

wind speed. Inputs of the model are time series

wind data (speeds and directions) and outputs time

series data of rotational speeds, yaw errors, yaw rates,

and electrical power outputs. Results of this simula-

tion study have been experimentally verified by

the field test data generated at the University of

Newcastle, Australia [16].

The gyroscopic effect of wind rotor due to yaw

motion is denoted by the variations of D’Alembert’s

moments around j 0 axis (Mj). In tilt-up type small

wind turbines, these moments should be restored

by the counterweights to avoid unnecessary vertical

movements in yaw condition. According to the sim-

ulation, rapid vertical and horizontal movements

Time (s)

0 5 10 15 20 25 30 35 40 45 50 -3

-2

-1

0

1

2

3

Yaw

rate

(D

egre

e/s

)

Rotational speed = 20rad/s



Fig. 8 Dynamic yaw rate characteristicsof the system





take place at high yaw rates and high wind speeds as

Mj is being varied beyond the restoring limits of the

counterbalancing weights. This is the main drawback

of the tilt-up type control mechanism.

7 CONCLUSIONS

A comprehensive mathematical model to represent

the behaviour of small-scale wind turbines was devel-

oped and presented in this article. This model can be

used to optimize geometrical parameters of the wind

turbine system and mass of the counterweight in

order to obtain maximum power output from the

wind rotor in varying wind conditions. Yaw move-

ments were modelled by considering dynamic

responses of the wind vane and comparing this with

the alternative equation for the yaw dynamics. The

effective wind velocity at the tail vane was evaluated

by considering the axial velocity change in the

downstream of wind rotor.

In this analysis, the D’Alembert’s principle was

used to derive the governing equations of the

Fig. 9 SIMULINK wind turbine model

Fig. 10 Yaw rate versus rotational speed and wind speed





system, including gyroscopic and centrifugal/

centripetal effects. The D’Alembert’s principle pro-

vides a method of analysing dynamic systems (at

any degree of complexity) with a very simple funda-

mental approach. The wind speed applies basic exter-

nal forces to the tilt-up type wind turbine.

Aerodynamic forces on the wind rotor and the tail

vane are determined separately. Here, the rotational

speed of the wind rotor ( _�) was assumed to be very

high compared with the other angular movements

( _� and _�) of the system. In this study, the modified

blade element and momentum theories are used to

estimate the aerodynamics forces on the wind rotor.

The combination of the modified blade element and

momentum theories makes it possible to evaluate a

field of induced velocities around the skewed wind

rotor. In axial flow simulations, the yaw calculations

are needed to model only the aerodynamics of a

single blade with the other blades experiencing the

same load and flow pattern 1/b revolutions later.

FUNDING

This research received no specific grant from any

funding agency in the public, commercial, or not-

for-profit sectors.

� Authors 2011

REFERENCES

1 Narayana, M. Gyroscopic effect of small scale tilt uphorizontal axis wind turbine. In World renewableenergy congress, Brighton, 1–7 July 2000, pp. 2312–2315.

2 Lanczos, C. The variational principles of mechanics,Vol. 12, edition 4, 1970 (Dover Publication, Mineola,New York).

3 Gourieres, D. L. Wind power plants theory and design,1982 (Pergamon Press, Oxford).

4 Ackerman, M. C. Yaw modelling of small wind tur-bines. J. Wind Engng Ind. Aerodyn., 1992, 39, 1–9.

5 Magnusson, M. Near-wake behaviour of windturbines. J. Wind Engng Ind. Aerodyn., 1999, 80,147–167.

6 Burton, T., Sharpe, D., Jenkins, N., and Bossanyi, E.Wind energy handbook, 2001 (John Wiley & Sons, Ltd,Chichester).

7 Coleman, R. P., Feingold, A. M., and Stempin, C. W.Evaluation of the induced velocity field of an idealizedhelicopter rotor, Langley Memorial AeronauticalLaboratory, Langley Field, 1945.

8 Vermeer, L. J., Sørensen, J. N., and Crespo, A. Windturbine wake aerodynamics. Prog. Aerosp. Sci., 2003,39, 467–510.

9 Anderson, J. D. Fundamentals of aerodynamics,edition 3, 2001 (McGraw-Hill Science, New York).

10 Kristensen, L. Cups, props and vanes. TechnicalReport Riso-R-766 (EN), Riso National Laboratory,Roskilde, 1994.

11 Xu, G. and Sankar, L. N. Effects of transition, turbu-lence and yaw on the performance of horizontalaxis wind turbines. In Proceedings of the 38thAerospace sciences meeting and exhibition, Reno,Nevada, 10–13 January 2000.

12 Haans, W., Sant, T., Kuik, G. V., and Bussel, G. V.Measurement and modelling of tip vortexpath in the wake of a HAWT under yawed flowconditions. In Proceedings of the 43rd AIAAaerospace sciences meeting and exhibition, Reno,Nevada, 10–13 January 2005.

13 MathWorks SimPowerSystemsTM reference, Hydro-Quebec and the MathWorks, Natick, Massachusetts,2010.

14 Grenier, D., Dessaint, L. A., Akhrif, O.,Bonnassieux, Y., and Le Pioufle, B. Experimentalnonlinear torque control of a permanent-magnetsynchronous motor using saliency. IEEE Trans. Ind.Electron., 1997, 44, 680–687.

15 Leonhard, W. Controls of electrical drives, edition 3,2001 (Springer, Berlin, Heidelberg, New York).

16 Wright, A. K. and Wood, D. H. Yaw rate, rotor speedand gyroscopic loads on a small horizontal axis windturbine. Wind Engng, 2007, 31, 197–209.

APPENDIX 1

Notation

As affected wind rotor area for the side thrust

Av area of tail vane

Cd drag coefficient for blade element

Cds drag coefficient for side thrust force of wind

rotor

Cdv drag coefficient for tail vane

Cl lift coefficient for blade element

Clv lift coefficient for tail vane

Cp power coefficient

Ct thrust coefficient

CG centre of gravity of wind rotor, generator,

and counterweights

CW counterweight

dr width of the blade element

Fs side thrust of wind rotor

h radial induction factor

I moment of inertia of wind rotor about wind

rotor axis

I1 moment of inertia of tail vane about the k

axis

k axial induction factor

K0 average axial induction factors

K0 constant for the ‘flat-plate’ equation

kD the damping factor for vertical movements

lv distance to the aerodynamic centre of the

tail vane from k axis





m mass of rotor, generator, and counterweight

r distance from the centre of the wind rotor

R radius of wind rotor

TG restoring momentum of generator

TW axial aerodynamic momentum on the wind

rotor

U1 effected wind velocity to the tail vane

V1 velocity of wind

Va(r) axial wind velocity at the rotor plane relative

to the blade element (r)

Va1(r) axial inflow wind velocity relative to the

blade element (r)

Va2(r) axial outflow wind velocity relative to the

blade element (r)

Vr(r) radial wind velocity at the rotor plane rela-

tive to the blade element (r)

Vr1(r) radial inflow wind velocity relative to the

blade element (r)

Vr2(r) radial outflow wind velocity relative to the

blade element (r)

W(r) wind velocity at the rotor plane in the plane

of blade element (r)

W1(r) inflow wind velocity in the plane of blade

element (r)

W2(r) outflow wind velocity in the plane of blade

element (r)

� yaw angle

angle between O–CG and k 0 axis (Fig. 2)

"(r) flow expansion angle at a distance r from the

rotor axis

� position angle of the wind rotor_� or ! angular velocity of the wind rotor

�0 tip-speed ratio

�(r) incidence angle for blade element (r)

’ angle between wind velocity and wind rotor

axis

� density of air

� wake angle

� angular velocity of the wake behind the

rotor in the rotor plane

APPENDIX 2

According to Fig. 2

�R ¼ a � sinð� �Þ � cos �i þ a � sinð� �Þ � sin�j

þ a � cosð� �Þk

a¼ R�� ; �I¼ �I

�� : cos�:cos�:iþcos�:sin�:jþsin�:kh i

;

�I�I�� ¼unit vector along the rotor axis:

Mk ¼ k component of m �R� €�Rþdð�I _�Þ

dt

� �

Mk

�� ¼ �m

�a � sin � �ð Þ � cos �:A

� a � sin � �ð Þ sin �:B

� I�� €� � sin � þ _� � _� � cos ��

M k ) D0Alembert0s momentum about the k axis:

Mj 0 ¼ j0 component of m �R� €�Rþdð�I _�Þ

dt

�

Mj 0

�� ¼ sin ��

a � sin � �ð Þ � sin�

� €� � sin � �ð Þ � �2 � cos � �ð Þ� �� A:a � cos � �ð Þ

��m � cos �

�a: sin � �ð Þ

� cos�:a €� � sin � �ð Þ � �2 � cos � �ð Þ� �

� B:a � cos � �Þð��m þ C

Mj 0 ) D0Alembert0s momentum about the j 0 axis:

where

A¼d2 a � sin ��ð Þ � sin�½ �

dt 2¼ a½� €� �cos ��ð Þ

� sin�� _�2 � sin ��ð Þ � sin�� _� � _� �cos ��ð Þ

�cos�þ €� � sinð��Þ �cos�� _� � _� �cosð��Þ

�cos�� _�2 � sinð��Þ � sin��

B ¼d2 a � sin � �ð Þ � cos�½ �

dt 2

¼ a½� €� � cos � �ð Þ � cos�� _�2 � sin � �ð Þ

� cos�þ _� � _� � sin� � cos � �ð Þ � €� � sin � �ð Þ

� sin�þ _� � _� � cos � �ð Þ � sin�� _�2

� sin � �ð Þ � cos��

C ¼ � �I�� sin� �

€� � cos � � cos �

þ _� � _�ð� sin�Þ � cos �

þ _� � _� � ð� sin �Þ � cos �

264

375

þ �I�� cos� �

€� � cos � � sin�

þ _� � _� � ð� sin �Þ � sin�

þ _� � _� � cos � � cos �

264

375

APPENDIX 3

Wind rotor

Radius of wind rotor: 1.105 m

Blade profile: NACA4415

Number of blades: 2

Moment of inertia (I): 9.77 kg.m 2

Tail vane

Vane area (Av): 0.12 m2





Distance to the centre of mass of the tail vane from k

axis (lv): 1.2 m

Moment of inertia around k 0 axis (I1): 7 kg.m2

Mass of wind rotor (M)¼ 25 kg, drag coefficient for

side thrust force of rotor (Cds)¼ 0.2, affected rotor

area for the side thrust (As)¼ 0.2 m2

Generator

Phase resistance (Rph): 1.25 �

Phase inductance (Lph): 0.003 075 H

Flux linkage by established by magnets: 0.10 692 V.s

Number of pole pairs (p): 6





Development of a model to investigate the yaw behaviour of small horizontal axis wind turbines...

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