-
t m
of Ca
Available online 18 July 2009
Annual Energy production
rizoy aan
velocity as well as at a variable velocity but at its maximum
power coefcient.
riticaldiam
rotational velocity at different wind speeds, it is possible to
havea power coefcient which is always at its maximum value.
To investigate this analysis thoroughly, and to evaluate
themathematical law on rotor rotational velocity, the authors
applied
accurate results. For the numerical stability of the
mathematicalmodel the greatest difculties are determining axial and
tangentialinduction factors, the lack of experimental measurements
on airfoillift and drag coefcients at high angles of attack, and
their three-dimensional representation. In order to take the
three-dimensionalrepresentation into account, the wind tunnel
experimentalmeasurements must be modied in order to consider radial
owalong the blades (centrifugal pumping [4]).
* Corresponding author.
Contents lists availab
Renewable
.e ls
Renewable Energy 35 (2010) 301306E-mail address:
[email protected] (M. Messina).parameters inuencing energy
production: rotor rotationvelocity, blade number, airfoil chord
distribution and longitu-dinal blade twist.
In this paper the inuence of rotor rotational velocity on
windturbine performance has been investigated. In particular, it
hasbeen observed that there exists amathematical law on the
variationof rotor rotational velocity with wind speed, which allows
the windturbine to always operate at its maximum power
coefcient.
It is well known that a wind turbine power coefcient presentsa
maximum value for a particular wind speed, which decreasesrapidly
for all other wind velocities. Vice versa, varying the rotor
Industry [4,612]. It enables the design of rotor blades by
uiddynamics, and the evaluation of wind turbine performance
(indesign and off-design conditions). Using this model, it is
possible todesign the rotor, choose the geometric characteristics
of the turbine(rotor diameter, aerodynamic airfoils, chord, pitch
and twist), andevaluate the forces acting on the blades, the torque
and power atthe rotor shaft. It is also possible to evaluate
turbine performancewith a wide range of wind velocities.
The BEM theory is based on the Glauert propeller theory
[12],modied for application to wind turbines. In recent years the
BEMtheory has been optimized and modied to provide increasingly1.
Introduction
Design parameter choice is cturbine performance. For any
xed0960-1481/$ see front matter 2009 Elsevier
Ltd.doi:10.1016/j.renene.2009.06.020This means that as wind
velocity varies there will always be a rotational velocity of the
turbine whichmaximises its coefcient. It would be sufcient
therefore to formulate the law governing the variation inrotational
velocity as it varied with wind velocity to arrive at a power
coefcient that is always the sameand its maximum.This work
demonstrates the methodology for determining the law governing the
rotational velocity of
the rotor and it highlights the advantages of a wind turbine
whose power coefcient is always atmaximum rather than very variable
in line with the variation of wind velocity.
2009 Elsevier Ltd. All rights reserved.
for optimising windeter there are various
a numerical model [3], based on BEM theory [1,2], and validate
itthrough experimental measurements [4].
The mathematical model based on BEM (Blade ElementMomentum)
theory is the most frequently used by Science andHorizontal axis
wind turbineMaximum power coefcientfurther increases in rotational
velocity a constant maximum power value was reached even as
windvelocity increased.Keywords:The mathematical code produced a
power coefciency curve which showed that notwithstandingTechnical
Note
Horizontal axis wind turbine working acoefcient continuously
R. Lanzafame, M. Messina*
Mechanical and Industrial Engineering Department, Faculty of
Engineering, University
a r t i c l e i n f o
Article history:Received 3 November 2008Accepted 14 June
2009
a b s t r a c t
The performance of a hocoefcient was evaluated bthen evaluated
for perform
journal homepage: wwwAll rights reserved.aximum power
tania, Viale A. Doria, 6-95125 Catania, Italy
ntal axis wind turbine continuously operating at its maximum
powercalculation code based on Blade Element Momentum (BEM) theory.
It wasce and Annual Energy Production (AEP) at a constant standard
rotational
le at ScienceDirect
Energy
evier .com/locate/renene
-
ewa2. Mathematical code
The mathematical model for the uid dynamics wind turbinedesign
(and for the WT performance evaluation), developed ina previous
work [3], is based on Blade Element Momentum Theory.By applying
momentum and angular momentum conservationequations, the axial
force and torque acting on the blade sector isobtained (as given in
Equations (1) and (2)),
dN r2V20 1 a2sin2 f
Nb CLcos f CD sin f c dr1 (1)
dM r2V01 a
sinf$u r11 a0
cos fNb CLsin f CD cos f c r1dr1
(2)
and thus the torqueM at the rotor shaft is the summation of dM
forall the blade sectors.
Nomenclature
a axial induction factora0 tangential induction factorr1 blade
local radiusc airfoil chordci twist logarithmic polynomial
coefcientsai CL logarithmic polynomial coefcientsbi CD logarithmic
polynomial coefcientsc0 Weibull scale parametern rotor rotational
velocityws wind speedv wind velocityv mean wind velocityWT wind
turbineAEP annual energy productionBEM blade element momentumHAWT
horizontal axis wind turbineR1 wind rotor radius
R. Lanzafame, M. Messina / Ren302The wind turbine power is given
by PM*u, where u is therotor angular velocity.
The lift (CL) and drag (CD) coefcients for a given airfoil
areevaluated from wind tunnel measurements [6]. The
experimentalvalues were tted to obtain mathematical functions to
apply to thesimulation model. To t the experimental data to the
lift and dragcoefcients, a fth-order logarithmic polynomial
(shifted by 10degrees) was implemented for the Attached FlowRegime
and HighLift, Stall Development Regime Dynamic Stall (see [3] for
theaerodynamic regions denition), as shown in Equations (3) and
(4).
As described in [3], centrifugal pumping (3D aerodynamiceffects)
was taken into account (Equation (5)) with a slight incre-ment in
the CL experimental values in the early region of the FlatPlate,
Fully Stalled Regime.
CL X5i0
ailna 10 i (3)
CD X5i0
bilna 10 i (4)
Theaiandbi coefcientsweredeterminedbymeansof the least
squaremethod, tting experimental data for the S809 airfoil at Re
106.For the Flat Plate, Fully Stalled Regime the
mathematicalfunctions of Equations (5) and (6) were
implemented.
CL 2 CLmax$sin a$cos a (5)
CD CDmax$sin2 a (6)In Equation (7) CLmax and CDmax are
shown.
CLmax CLja 45 and CDmax CDja 90 (7)The numerical stability of
the mathematical code depends on
tangential and axial induction factors. In this code the
inductionfactors, reported in Equations (8)(10), were implemented
[3].
For a< 0.4:
a 14F sin2 f
c Nb 1
(8)
Re Reynolds numberV0 wind velocity far up streamN rotor normal
forceCL lift coefcientCD drag coefcientNb number of bladesM torqueF
Tip loss factorCP power coefcientCN normal force coefcientP powerK
Weibull shape parameterPw power of a wind machineEw energy from a
wind machineq twista angle of attack4 incoming ow direction angler
air densitylr local speed ratio
ble Energy 35 (2010) 3013062p r1CLcos f CD sin f
while for a 0.4 [7]:
a 18F 20 3CN50 36F 12F 3F 4
p36F 50 (9)
and
a0 12
1 4
l2ra1 a
s 1
!(10)
As reported in [4] and [8], F is the Prandtl Tip Loss Factor,
and isdened as:
F 2par cos
exp
Nb r1 R12 r1sin f
(11)
To verify the validity of the mathematical code, the
simulateddata was compared with the NREL data from the NASA-Ames
windtunnel tests [4]. The UAE Phase VI wind turbine has two
twistedblades, a variable chord along the blade, and a rotor
diameter of10 m [5]. The aerodynamic airfoil is the S809 and is
constant alongthe blades; the pitch is three degrees and rotational
velocity is72 r/min. Fig. 1 shows a comparison between the
simulated and
-
characteristics: two blade rotor, rotational velocity of 72
r/min,
models and validation of the code with experimental measure-
shows the power curve of the same turbine (Prated 10 kW at
wind
00.10.20.30.40.50.60.70.80.9
1
0.00 1.00 2.00 3.00 4.00 5.00 6.00r [m]
CHORD [m]
Fig. 3. Chord distribution.
PHASE VI
0
2
4
6
8
10
12
14
16
5 10 15 20 25Wind Speed [m/s]
Po
wer [kW
]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Cp
[-]
Exp. Power Simulated PowerExp. Cp Simulated Cp
Fig. 1. PHASE VI wind turbine performances. Numerical code
validation. Comparisonbetween simulated and experimental data.
R. Lanzafame, M. Messina / Renewable Energy 35 (2010) 301306
303ments are all reported in [3].The power coefcient reported in
Fig. 4 shows typical behaviourexternal diameter of 10 m, twist and
chord as per Figs. 2 and 3,aerodynamic cross-section and prole as
per aerodynamic proleS809. The power coefcient and power produced
by this turbine arereported in Figs. 4 and 5. In particular, the
twist was determined soas to maximise AEP (23.5 MWh/year) at a site
with a modest winddistribution (Fig. 6).
An ample description of the calculation code,
mathematicalexperimental data showing that there is good agreement
betweenthe experimental and simulated power and power coefcient
(Cp).
3. Numerical simulations
Numerical simulations were carried out to evaluate windturbine
performance where the rotational velocity is constant andthen
compare it with that of the same turbine with a rotationalvelocity
varying as wind velocity varies.
3.1. Reference wind turbine: n const
Using a calculation code ne-tuned by the authors [3],
theperformances were evaluated of a wind turbine with the
followingwith amax value of 0.37 at awind velocity of 6.2 m/s,
whereas Fig. 5
0
4
8
12
16
0 1 2 3 4 5 6
Tw
ist [d
eg
]
Blade radius [m]
twist [deg]Pol.Log.
Fig. 2. Twist for max AEP.velocity 11 m/s).
3.2. Wind turbine performance at variable rotational velocity:ns
const
Having evaluated wind turbine performance at constant
rota-tional velocity with AEPmaximised, it was then compared to
that ofthe same turbine at variable rotational velocity.
Using the mathematical code reported in [3], the various
powercurves at varying rotational velocities were calculated (Fig.
7).
Note how at low velocities turbine performance is better at
lowwind velocities. Fig. 7 shows that power is greater and
cut-invelocity is ever lower as rotational velocity decreases (this
allowsturbine start-up at ever lower wind velocities). Vice versa,
lowrotational velocities have the disadvantage of peaking at low
power(Prated).
To exploit the low rotational velocities advantages as well
asthose at high velocities (high values of rated power), the rotor
mustturn at varying speeds with the variation of wind velocity. It
istherefore necessary to work out the mathematical law
whichdescribes this.
This in turn means that the rotor should always be working
atmaximum efciency.
To work out the law, the various Cp (power coefcients) need tobe
evaluated for the various rotor rotational velocities.
Once the power curves at the different velocities have
beennoted, the power coefcient can be evaluated by Eq. (12):
Cp P12rV
30pR
21
(12)0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 10 15 20 25 30
Cp
Wind Speed [m/s]
Fig. 4. Power coefcient for max AEP at n const.
-
At various rotor velocities, the behaviours of the various
powercurves cp can be plotted against wind velocity (Fig. 8).
Note how as rotation velocities vary the power coefcient
curveproduces the same maximum Cp even as wind velocity
increases.
Fig. 8 shows how the wind turbine can work at maximum ef-
and 11.
value of Prated (in this case 20 kW). These values can then be
readagainst wind speed in Fig. 12. For every pair of (n, w ) values
(n:
(therefore power coefcients steeply decreasing from theirmaximum
as wind speed varies, Fig. 4), and the second operating at
0
2
4
6
8
10
12
1 3 5 7 9 11 13 15 17 19 21 23 25
Po
wer [kW
]
Wind Speed [m/s]
Fig. 5. Power curve for max AEP at n const.
0
20
40
60
80
100
120
140
160
180
200
Po
we
r [k
W]
30405060708090100110120130140150160170180190200
Power curves
n [r/min]
n
R. Lanzafame, M. Messina / Renewable Energy 35 (2010)
301306304To avoid excessive rotor loads, the power generated
(Prated,Fig. 11) should be limited, always acting on the rotor
velocity.
4. Regulating power
Once a maximum value for the rated power (Prated) is
estab-lished, the power is kept constant (straight line in Fig. 11)
, alwaysacting on the rotation velocity.ciency by varying the
rotational velocity as wind velocity changes.Fig. 9 shows the link
between the rotational velocity at which
the rotor ought to rotate and varying wind velocities.In order
for the turbine towork atmaximumefciency (Fig. 9), the
relationship between rotational velocity and wind velocity is
linear.Varying the rotor velocity produces the dashed lines in
Figs. 10Fig. 6. Wind Speed probability density function.s
rotational velocity; ws: wind speed) Fig. 10 can provide the
corre-sponding power coefcients.
The dashed line in Fig. 13 shows the working points of the
windturbine.
5. Comparing annual energy production
In managing a wind farm, or more simply a single rotor, what
ismost important is to maximise annual energy production so as
torepay the investment.
To measure the annual energy production (AEP), the two
windturbines were compared, one operating at constant velocityFor
example, if the rated power is 20 kW, the law of rotorvelocity is
represented by the straight line in Fig.12. From point A toB,
turbine efciency is constant (at maximum); from point B to Cturbine
power is limited.
To deduce the law of rotor velocity during power
regulation(phase BC), it is sufcient to read the various values of
rotationvelocity on the power curves which intersect with the
constant
0 5 10 15 20 25Wind Speed [m/s]
Fig. 7. Power curves at varying rotational velocities of the
rotor.varying rotor velocities (power coefcient always maximum up
tothe rated power).
The comparison refers to the wind distribution in Fig. 6.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 5 10 15 20 25 30
Cp
Wind Speed [m/s]
POWER COEFFICIENT
30405060708090100110120130140150160170180190200
n [r/min]
n
Fig. 8. Power coefcient at varying rotor velocities.
-
020406080
100120140160180
0 2 4 6 8 10 12 14ro
tatio
nal velo
city [r/m
in
]
wind speed [m/s]
Cp max
Fig. 9. The relationship between turbine rotor velocity and wind
velocity.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
50
100
150
200
250
0 2 4 6 8 10 12 14 16 18
Cp
ro
ta
tio
na
l v
elo
city
[rp
m]
n [rpm]Cp
A
B
C
R. Lanzafame, M. Messina / Renewable Energy 35 (2010) 301306
305Fig. 10. Effective cp curve.Clearly, AEP will increase directly
with the rise in Prated, so as tomake the comparison signicant,
Prated is limited to 10 kW (so toothe reference turbine Fig. 5) and
its highlighted advantages willonly derive from low rotor
velocities, not high ones.
The bold line in Fig. 14 shows the new power curve for
variablerotational velocities. Prated is set at 10 kW and at low
wind speedsthe closeness of the power curve to the Betz limit curve
means theturbine is working at high power coefcients. Fig. 14 also
shows thewind speed range in termsof probability density (pdf)
byhistogram.
What was described about the power coefcient can also beseen in
Fig. 15. Note how the power coefcient remains atmaximum for low
wind speeds, whereas once Prated is reached, itdecreases in order
to regulate the power produced.
Fig. 11. Effective power curve.wind speed [m/s]
Fig. 12. Trends of rotational velocity and power coefcient
during regulation.With the turbine at variable rotor velocity, the
annual energyproduced is 26.5 MWh/year. This means an increase of
12.77% overa wind turbine operating at n const.
In short, taking account of structural aspects, it would
bepossible to raise Prated, and further increase AEP as in Table
1.
0
2
4
6
8
10
12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
1 3 5 7 9 11 13 15 17 19 21 23 25
Po
wer, P
w, [kW
]
Freq
uen
cy H
isto
gram
Wind Speed [m/s]
pdfPower, Pw, [kW]Betz limit [kW]
Fig. 14. Power curve at variable rotor velocity ns const.
Fig. 13. Effective power curve during regulation.
-
On the other hand, according to a law of the rotor
velocityrotation as a function of wind speed, it is possible to
make a hori-zontal axis wind turbine work continuously at its
maximum powercoefcient.
To determine this law, a calculation code based on BladeElement
Momentum theory which had already been validated byexperimental
measurements in scientic literature and numeroussimulations, was
produced to create a power curve and powercoefcient as the
rotational velocity of a wind rotor varied.
Finally, a comparison was made between the annual energyproduced
(AEP) by a wind turbine at constant rotational velocityand the same
turbine at variable velocities. The comparison high-lighted an
increase in AEP of about 13% for the turbine working atvariable
velocity.
References
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Renewable EnergyNovember 2007;32(14):2291305. Elsevier Science,
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[4] Lindenburg C. Investigation into rotor blade aerodynamics.
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0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 5 10 15 20 25 30
Cp
Wind Speed [m/s]
Fig. 15. Power coefcient at variable rotor velocity ns
const.
Table 1Percentage increase in AEP as rated power (comparison
with ns const) increases.
% Prated Prated [kW] AEP [MWh/year] % AEP
0 10 26.5 0
R. Lanzafame, M. Messina / Renewable Energy 35 (2010)
3013063066. Conclusions
20 12 28.8 8.6840 14 30.4 14.7160 16 31.9 20.3780 18 33.4
26.03This work produced a methodology which allows a horizontalaxis
wind turbine to work continuously at its maximum
powercoefcient.
A wind turbine operating at constant rotational velocity hasa
maximum power coefcient for a given wind velocity whichdecreases as
wind speed decreases.[5] Hansen MOL. Documentation of code and
airfoil data used for the NREL 10-mwind turbine. ROTABEM-DTU;
November 2000.
[6] Jonkman JM. Modeling of the UAE wind turbine for renement of
FAST_AD.NREL/TP-500-34755; December 2003.
[7] Buhl L. Jr. A new empirical relationship between thrust
coefcient andinduction factor for the turbulent windmill state.
Technical Report NREL/TP-500-36834; August 2005.
[8] Corten GP. Flow separation on wind turbine blades. Ph.D.
Thesis, UtrechtUniversity; January 2001.
[9] Meyer CJ, Kroger DG. Numerical simulation of the ow eld in
the vicinity ofan axial fan blade. International Journal of
Numerical Methods in Fluids2001;36:94769.
[10] Moriarty PJ, and Hansen AC. AeroDyn theory manual,
Technical Report NREL/TP-500-36881; January 2005.
[11] Sphera DA, editor. Wind turbine technology: fundamental
concepts of windturbine engineering, 1998.
[12] Glauert H. The elements of airfoil and airscrew theory.
Cambridge UniversityPress; 1926.
Horizontal axis wind turbine working at maximum power
coefficient continuouslyIntroductionMathematical codeNumerical
simulationsReference wind turbine: n=constWind turbine performance
at variable rotational velocity: nneconst
Regulating powerComparing annual energy
productionConclusionsReferences
