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Optimal relationship between power and design-driving loads for
wind turbine rotorsusing 1-D models
Lønbæk, Kenneth; Bak, Christian; Madsen, Jens I.; Dam,
Bjarke
Published in:Wind Energy Science
Link to article, DOI:10.5194/wes-5-155-2020
Publication date:2020
Document VersionPublisher's PDF, also known as Version of
record
Link back to DTU Orbit
Citation (APA):Lønbæk, K., Bak, C., Madsen, J. I., & Dam, B.
(2020). Optimal relationship between power and design-drivingloads
for wind turbine rotors using 1-D models. Wind Energy Science,
5(1), 155-170.https://doi.org/10.5194/wes-5-155-2020
https://doi.org/10.5194/wes-5-155-2020https://orbit.dtu.dk/en/publications/19869c4c-36ed-4260-b0a9-954f468792e9https://doi.org/10.5194/wes-5-155-2020
-
Wind Energ. Sci., 5, 155–170,
2020https://doi.org/10.5194/wes-5-155-2020© Author(s) 2020. This
work is distributed underthe Creative Commons Attribution 4.0
License.
Optimal relationship between power and design-drivingloads for
wind turbine rotors using 1-D models
Kenneth Loenbaek1,2, Christian Bak2, Jens I. Madsen1, and Bjarke
Dam11Suzlon Blade Science Center, Havneparken 1, 7100 Vejle,
Denmark
2Technical University of Denmark, Frederiksborgvej 399, 4000
Roskilde, Denmark
Correspondence: Kenneth Loenbaek
([email protected])
Received: 23 May 2019 – Discussion started: 8 July 2019Revised:
14 November 2019 – Accepted: 15 December 2019 – Published: 28
January 2020
Abstract. We investigate the optimal relationship between the
aerodynamic power, thrust loading and size of awind turbine rotor
when its design is constrained by a static aerodynamic load. Based
on 1-D axial momentumtheory, the captured power P̃ for a uniformly
loaded rotor can be expressed in terms of the rotor radius R andthe
rotor thrust coefficient CT. Common types of static design-driving
load constraints (DDLCs), e.g., limits onthe permissible
root-bending moment or tip deflection, may be generalized into a
form that also depends on CTand R. The developed model is based on
simple relations and makes explorations of overall parameters
possiblein the early stage of the rotor design process. Using these
relationships to maximize P̃ subject to a DDLC showsthat operating
the rotor at the Betz limit (maximum CP) does not lead to the
highest power capture. Rather, itis possible to improve performance
with a larger rotor radius and lower CT without violating the DDLC.
As anexample, a rotor design driven by a tip-deflection constraint
may achieve 1.9 % extra power capture P̃ comparedto the baseline
(Betz limit) rotor.
This method is extended to the optimization of rotors with
respect to annual energy production (AEP), inwhich the thrust
characteristics CT(V ) need to be determined together with R. This
results in a much higherrelative potential for improvement since
the constraint limit can be met over a larger range of wind speeds.
Forexample, a relative gain in AEP of+5.7 % is possible for a rotor
design constrained by tip deflections, comparedto a rotor designed
for optimal CP. The optimal solution for AEP leads to a thrust
curve with three distinctoperational regimes and so-called thrust
clipping.
1 Introduction
From the inception of the wind energy industry, it has been
aclear trend that rotor sizes have been increasing. However,
asdiscussed in Sieros et al. (2012), increasing the rotor size
isnot a clear way to decrease the cost of energy (CoE), since
therotor weight (closely related to rotor cost) will always
scalewith a larger exponent than the increase in power does. Itis,
therefore, argued that the lower CoE that has taken placeis mostly
due to improvements in technology. The turbineis structurally
designed to carry loads coming from aerody-namics (steady or
extreme) and the self-weight. Therefore,lowering the loads should
lead to a lighter blade. The steadyaerodynamic load is applied to
extract power, and increasing
the load leads to greater power production until the maxi-mum
power coefficient (max CP) is reached. Increasing theload should
lead to a heavier blade but it also leads to greaterpower
production. It goes to show that understanding therelationship
between loading, power production and struc-tural response is very
important for designing the most cost-effective turbine. This
follows a trend occurring in recentyears in which there is a belief
that wind turbine optimizationshould include a more holistic
approach, with concepts likemultidisciplinary design analysis and
optimization (MDAO)and systems engineering (Bottasso et al., 2012;
Zahle et al.,2015; Fleming et al., 2016; and Perez-Moreno et al.,
2016),where all of the parts of the turbine design that affect the
costshould be taken into account along with the overall
objective
Published by Copernicus Publications on behalf of the European
Academy of Wind Energy e.V.
-
156 K. Loenbaek et al.: Optimal relationship between power and
design-driving loads for wind turbine rotors
of minimizing the CoE. Some of these related works focusmore on
how the rotor loading affects the power and struc-tural response.
One of the concepts that comes out of this isthe so-called
low-induction rotor (LIR), in which the velocityinduction at the
rotor plane is lower than the value that max-imizes the power
coefficient. The concept was introduced byChaviaropoulos and Sieros
(2014) and it comes out of theoptimization of annual energy
production (AEP) by allow-ing the rotor to grow while constraining
the flap root bendingmoment to be the same as some baseline. They
state that themethod can increase AEP by 3.5 % with a 10 % increase
inthe rotor radius, thereby showing that the LIR can increaseAEP
while keeping the same flap root bending moment. Itagrees with
Kelley (2017) who allowed for a change in theradial loading,
resulting in a 5 % increase in AEP with a ra-dius increase of 11 %.
It was also investigated by Bottassoet al. (2015) who tested the
potential of using the LIR bothfor AEP improvements with load
constraints and as a cost-optimized rotor. They found the same
results as the previoustwo investigations; the LIR can improve AEP,
but when theyconsider the CoE they find that the LIR is not cost
effec-tive, meaning that the additional cost of extending the
bladeis not compensated by the increase in power. This conclu-sion
is opposed to the conclusion made by Buck and Garvey(2015b) who set
out to minimize the ratio between capitalexpenditures (CapEx) and
AEP. They arrive at LIR as theoptimal solution for minimizing
CapEx/AEP, which is takenas a measure of CoE. Overall it seems that
LIR can increaseAEP while keeping the same load as a non-LIR
baseline, butit is not clear if LIR is a cost-effective
solution.
Another concept that is relevant in the context of this pa-per
is thrust clipping (also known as peak shaving or forcecapping).
For turbines, it is often the case that the maximumthrust is
reached just before reaching the rated power, result-ing in a
so-called thrust peak. When using thrust clipping,this peak is
lowered at the cost of power. It is used withmany contemporary
turbines for load alleviation but is oftenadded as a feature after
the design process. Buck and Garvey(2015a) made a design study in
which they found that lower-ing the maximum thrust by 11 % leads to
a 9 % reduction inmaterial used, at the cost of 0.1 % less lifetime
energy, result-ing in an overall reduction of 0.2 % in the cost of
energy. Thisshows that including thrust clipping in the design
process canlead to a lower CoE.
In this paper, we investigate the relationship between theload,
power and structural response of wind turbine rotors.Simple
analytical models, based on 1-D aerodynamic mo-mentum theory and
Euler–Bernoulli beam theory, are intro-duced to establish the
first-order relationship between theseresponses. This provides a
useful framework for the initialrotor design, especially when
high-level design parameterssuch as the rotor radius need to be
fixed or there is a need tounderstand how load/structural responses
will change withrotor size. The effect on the power curve and the
relatedload/structural response with the variation in wind speeds
is
also investigated, which is useful for the initial design of
thehighly coupled aeroservoelastic rotor design problem.
The relatively simple models used in this paper do notcapture
the full complexity needed for detailed wind tur-bine rotor design
and should be considered a tool for early-stage rotor design and
overall exploration only. For exam-ple, the underlying theories (of
1-D aerodynamic momen-tum and Euler–Bernoulli beams) assume
steady-state con-ditions, while designs are often constrained by
load casesthat are linked with extreme, unsteady or non-normal
op-erational events, e.g., extreme turbulence, gusts,
emergencyshutdowns, subsystem faults or parked conditions. This is
alimitation of the model developed here, but if there is a
re-lation between the steady-state loads and the extreme
loads,which is very likely, then the results are still valid.
As mentioned before, the overall target for current
turbinedesign is to lower the CoE, but a cost model is not
used,which is also a limitation of this study. However, cost
modelsuse several assumptions made in the design process such asthe
price of components in the design or composite lay-up ofthe blades,
so a predicted cost will always be made with someuncertainty.
Instead, load constraints are considered, muchlike in the
above-mentioned LIR example. As was found byBottasso et al. (2015),
a constrained load might not lead to alower CoE. So, to accommodate
this, a constraint with a fixedmass is made, which is thought to be
a better approximationof a fixed cost.
This study is carried out in order to obtain an overviewof how
the rotor design is more fundamentally influenced bydifferent types
of aerodynamic loading. Thus, an issue likethe self-weight is
important for modern turbines but is notdirectly included in this
study; the static-mass moment espe-cially has an impact on
contemporary turbines. It could beincluded, but it was excluded to
keep the study as simple aspossible. Further discussion about the
limitations and possi-ble improvements of the study is given later
in Sect. 4.5.
2 Theory
This section will introduce the variables and the basic
rela-tionships used in this paper. It is split into two
subsections, inwhich Sect. 2.1 introduces aerodynamic variables,
equationsand the baseline rotor, while Sect 2.2 presents scaling
lawsused to formulate design-driving load constraints relative
tothe baseline rotor.
2.1 Aerodynamics
The theory underlying this Aerodynamics section is found
inSørensen (2016).
For wind turbine aerodynamics non-dimensional coeffi-cients are
often introduced and some of the common onesare for the rotor
thrust (CT) and power (CP).
Wind Energ. Sci., 5, 155–170, 2020
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K. Loenbaek et al.: Optimal relationship between power and
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Figure 1. Relationship between normalized rotor load CT andpower
coefficient CP from 1-dimensional momentum theory. Notethat around
the Betz limit a small change in CT does not lead to aproportional
change in CP; this is illustrated by 1CT and 1CP.
CT =T
12ρV
2πR2(1)
CP =P
12ρV
3πR2, (2)
where T and P are the rotor thrust and power, respectively;ρ is
the air density, V is the undisturbed flow speed and R isthe rotor
radius.
These definitions can be applied for any wind turbine ro-tor,
but in this paper, we will use a simplified relationship be-tween
CT and CP that is derived from classical 1-D momen-tum theory. This
implies an assumption of uniform aerody-namic loading across the
rotor plane. The classical equationsare often given in terms of the
axial induction (a), which isdefined as a = 1− Vrotor
V, where Vrotor is the axial flow speed
in the rotor plane. By combining the two classical momen-tum
theory expressions forCP(a) andCT(a) (Sørensen, 2016,p. 11, Eq.
3.8), the relationship between these coefficients isarrived at as
follows:
CT(a)= 4a(1− a)CP(a)= 4a(1− a)2
}⇒ CP (CT)= (1− a)CT
=12
(1+
√1−CT
)CT, CT ∈ [0,1], (3)
where a(CT) is found by inverting CT(a) and using the neg-ative
solution. A plot of CT vs. CP can be seen in Fig. 1.This CP(CT)
curve is monotonically decreasing in slopeand reaches a maximum of
CP = 16/27, corresponding tothe well-known Betz limit at CT = 8/9.
These monotonic-ity properties lead to the key observation that a
reductionin thrust (CT = 8/9−1CT) will not lead to a
proportionalchange in power (1CP). This motivates the investigation
inthis paper of the trade-off between power and loads.
2.1.1 Power capture and annual energyproduction (AEP)
One way to understand the power yield of a rotor is to con-sider
Eq. (2) as consisting of three separate terms as follows:
P =12ρV 3︸ ︷︷ ︸
Wind
·12πR2︸ ︷︷ ︸Size
·12CP︸︷︷︸
Coefficient
. (4)
“Wind” is the part of the equation that depends on the
windconditions, “size” is the part of the equation that depends
onthe rotor-swept area and “coefficient” is the part of the
equa-tion related to the power coefficient, representing the
capabil-ity of the rotor to extract power from the wind. The
combina-tion of Eqs. (2) and (3) provides an expression that
capturesthe last two terms, which are the only ones affected by
thedesign of the turbine; the result is as follows:
P̃(CT, R̃
)=
P
12ρV
3πR20
= CPR̃2=
12
(1+
√1−CT
)CTR̃
2, (5)
where R̃ equals R/R0, with R0 being the radius of the base-line
rotor. This equation will be referred to as the power cap-ture
equation. It shows that power can be changed by chang-ing either
the loading (CT) or the rotor radius (R). This willserve as the
basic equation when the power capture is opti-mized for a single
design point.
When considering turbine design over the range of oper-ational
conditions, annual energy production (AEP) is intro-duced as an
integral metric representing the energy producedper year given some
wind speed frequency distribution. Itcan be computed as the power
production (P ) weighted bythe probability density of wind speeds
(PDFwind) multipliedby the period of 1 year (Tyear) as follows:
AEP= Tyear12ρπR20
VCO∫VCI
P̃(CT(V ), R̃
)·V 3 ·PDFwind(V )dV. (6)
The wind speed probability distribution PDFwind will be
de-scribed with a Weibull distribution. VCI and VCO are the
windspeeds for cut in and cut out during wind turbine
operation.Here they are taken to be VCI = 3 m s−1 and VCO = 25 m
s−1,which are common numbers for modern wind turbines.
In this paper, we will use a dimensionless measure for AEPwhich
is equivalent to the so-called capacity factor, definedas
follows:
˜AEP(CT, R̃
)=
AEPTyearPrated
=AEP
Tyear12ρπR
20
1627V
30
=2716
ṼCO∫ṼCI
P̃(CT(Ṽ ), R̃
)· Ṽ 3 ·PDFwind(Ṽ )dṼ . (7)
Ṽ is a normalized wind speed given by V = Ṽ V0, whereV0 is the
wind speed at which the turbine reaches the
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158 K. Loenbaek et al.: Optimal relationship between power and
design-driving loads for wind turbine rotors
rated power (Prated). Throughout this paper it is takento be V0
= 10 m s−1. It should further be noted thatPDFwinddV is
dimensionless and by nondimensionaliz-ing AEP it also follows that
PDFwinddṼ is dimension-less. Throughout this paper ˜AEP is
calculated using adiscretization of the integral, which is computed
us-
ing the trapezoidal rule given asṼCO∫̃VCI
f (Ṽ ;CT, R̃)dṼ ≈
N∑i=1
f (Ṽi+1;CT,R̃)+f (Ṽi ;CT,R̃))2 1Ṽi , where the discretization
(N )
was found to become insignificant for N = 200.
2.1.2 Baseline rotor
The work here aims to demonstrate an improved rotor perfor-mance
compared to a baseline design. This baseline design ischosen to be
a turbine operating at the Betz limit below therated wind speed and
keeping a constant power above therated power.
CT,0 =89≈ 0.889; CP,0 =
1627≈ 0.593 (8)
This choice of baseline mimics the typical practice of
de-signing wind turbines target operation at the maximum CPbelow
the rated power. In reality, turbines will not achieve amaximum CP
at CT = 8/9 since losses alter the relationshipbetween CT and CP,
but this does not change the fact that tur-bines are operated at
the point of the maximum CP. Figure 2shows the power and thrust
curves for the baseline rotor.
In this paper, all results are presented as the change in
per-formance relative to that of the baseline rotor. For this
rea-son, all of the relevant variables (denoted with a zero in
thesubscript) will be normalized by the corresponding baselinerotor
values.
1R =R
R0− 1 (9)
1P̃ =CPR
2
CP,0R20− 1 (10)
1L̃=CTR
Lexp
CT,0RLexp0
− 1 (11)
1 ˜AEP=˜AEP˜AEP0− 1, (12)
where L̃ as well as Lexp is a generalized load that is
intro-duced in Sect. 4.1 (Effects on loads), and it is written
herefor later reference.
2.2 Scale laws and constraints for design-driving loads
In this section, examples of static aerodynamic design-driving
loads (DDLs) will be presented. These examples are
not meant to be exhaustive but include several of the key
con-siderations that constrain the practical design of wind
tur-bine rotors. From the scaled loads, design-driving load
con-straints (DDLCs) are introduced, which limit loads so thatthese
do not exceed the levels of the baseline rotor. Based onthe DDL
examples, it is shown that DDLCs can be elegantlyput in a
generalized form.
2.2.1 Thrust (T )
Thrust typically does not limit the design of the rotor
itselfbut more likely is a constraint imposed from the design ofthe
tower and/or foundation. The thrust scaling and the asso-ciated
DDLC is given by
ScalingT = 12ρV
20 πR
2CT⇒
DDLC
(T )= TT0=
CTCT,0
(RR0
)2≤ 1.
(13)
2.2.2 Root flap bending moment (Mflap)
The root flap moment is the bending moment at the
rotationalcenter in the axial flow direction. To compute Mflap, the
1-Dmomentum theory relations for infinitesimal thrust (dT )
andmoment (dM) are integrated; they are first expressed as
dT =12ρV 2CT2πrdr (14)
dMflap = rdT , (15)
where r is the radius location of the infinitesimal load (r
∈[0,R]). The moment scaling and DDLC can be found as fol-lows:
ScalingMflap =
∫ R0 dMflap =
13 ρV
20 CTπR
3⇒DDLC(Mflap
)=
MflapMflap,0
=CTCT,0
(RR0
)3≤ 1.
(16)
As shown, Mflap scales with R3 so it grows faster than thepower,
which scales as R2. Mflap is important for the bladedesign since
the flap-wise aerodynamic loads need to betransferred via the blade
structure to the root of the blade.
2.2.3 Tip deflection (δtip)
Tip deflection is a common DDLC for contemporary utility-scale
turbines, where tip clearance between tower and blademay become
critical because of the relatively long and slen-der blades. To get
an idea of how tip-deflection scales withchanges in loading and
rotor radius Euler–Bernoulli beamtheory; (Bauchau and Craig, 2009,
p. 189, Eq. 5.40) is used.For the problem here, it takes the form
of
d2
dr2EI
d2δdr2=
dTdr=
12ρV 2CT2πr, (17)
where δ is the deflection in the flap-wise direction of theblade
at location r . EI is the stiffness of the blade at loca-tion r .
For modern turbines the stiffness decrease towards
Wind Energ. Sci., 5, 155–170, 2020
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K. Loenbaek et al.: Optimal relationship between power and
design-driving loads for wind turbine rotors 159
Figure 2. (a) The dimensionless power and thrust for the
baseline rotor as a function of wind speed. Overlaid (in blue) is
the Weibull windspeed frequency distribution used throughout
(IEC-class III: Vavg = 7.5; k = 2). (b) CT and CP as a functions of
wind speed. These curvesreflect how most turbines are operated
today, targeting the maximum power coefficient below the rated
power, which leads to a thrust peakjust before the rated power.
the tip of the blade. To get an estimate for the stiffness, it
isassumed that stiffness follows the size of the chord (EI ∝ c).The
chord is given by the equation in Sørensen (2016, p. 68,Eq. 5.26);
with an approximation for the outer part of theblade it can be
found that c ∝ R/r which means that EI ∝R/r . An approximate model
for EI that has EI ∝ R/r canbe made,
EI (r)=EIr
1+(EIrEIt− 1
)rR
, (18)
where EIr is the stiffness at the root and EIt is the
stiffnessat the tip of the blade. As mentioned above for wind
turbinesEIr >EIt.
With the equation for EI , Eq. (17) can be solved by indef-inite
integration, with the integration constants determinedfrom the
following boundary conditions:
δ(r = 0)= 0,dδdr
(r = 0)= 0︸ ︷︷ ︸Clamped root
d2δdr2
(r = R)= 0,d3δdr3
(r = R)= 0︸ ︷︷ ︸Free tip
. (19)
The resulting displacement solution becomes
δ =11π120
V 2ρ
EIrCTR
5(
233
(EIr
EIt− 1
)r̃6+
111r̃5
−5
11
(EIr
EIt− 1
)r̃4+
1011
(23EIr
EIt−
53
)r̃3+
2011r̃2)
(20)
=11π120
V 2ρ
EIrCTR
5δshape
(r̃,EIr
EIt
), (21)
where the normalized radius (r̃ ∈ [0,1]) has been introducedso
that r = R · r̃ . The polynomial shape of the deflection hasbeen
collected in δshape. The maximum deflection occurs atthe blade tip
(r̃ = 1), which leads to a scaling relation andDDLC for tip
deflection:
Scaling
δtip =11π120
V 2ρEIrCTR
5δshape
(r̃ = 1, EIr
EIt
)⇒
DDLC(δtip)=
δtipδtip,0=
CTCT,0
(RR0
)5≤ 1,
(22)
where it has been implicitly assumed that any change in
stiff-ness needs to follow
EIr
EIt=
EIr
EIr,0
(EIr,0
EIt,0+
267
)−
267, (23)
with the simplest way to satisfy this relation being thatEIr
=EIr,0, which gives EIrEIt =
EIr,0EIt,0
.
2.2.4 Tip deflection with constant mass
The final example of a DDL is also based on tip deflection
butincludes a condition to maintain a constant mass of the
load-carrying structure of the blade. To this end, the stylized
spar-cap layout depicted in Fig. 3 is assumed. This layout
consistsof two planks. The stiffness of a spar-cap structure with
ahomogeneous Young’s modulus (E) can be found from thestiffness of
the rectangle and the parallel axis theorem (see
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2020
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160 K. Loenbaek et al.: Optimal relationship between power and
design-driving loads for wind turbine rotors
Fig. 3 for the variable definitions) as follows:
Irect =Bh3
12
EI = 2E(Irect+A
(H−h
2
)2)A= Bh
EI = 2E
(Bh3
12+Bh
(H −h
2
)2)
= EH 2Bh
2
(h2
3H 2+
(1−
h
H
)2). (24)
For modern wind turbines h/H � 1, meaning that a
commonapproximation is
EI ≈ EH 2Bh
2. (25)
To compute the mass for such a structure it will be as-sumed
that plank height h and the plank width B are con-stant and the
change in EI comes from a decrease in build-ing height H . Then, if
h is decreased when R is increased,the following relationship needs
to be satisfied for the massof the planks to be constant (assuming
a constant mass den-sity),
Rh= R0h0. (26)
From there it follows that changes in the radius of the
rotorwill change the stiffness as
EI ≈ EH2Bh2 (25)
h=R0h0R
(26)
}EI ≈ E
H 2BR0h0
2R. (27)
Combining this equation with the tip deflection equation(Eq.
21), scaling and DDLC can be found as follows:
Scalingδtip =
11π120
V 2ρEIr
CTR5δshape
(r̃ = 1, EIr
EIt
)EI ≈ E
H 2BR0h02R
}
⇒
DDLC(δtip+mass
)=
CTCT,0
EIr,0EIr
(RR0
)5=
CTCT,0
(RR0
)6≤ 1,
(28)
with the use of the fact that changing h by the same mag-nitude
for the whole blade leads to EIr
EIt=
EIr,0EIt,0
and therebydoes not affect δshape. It should be noted that
choosing B tochange instead will lead to the same scaling but the
differenceis that changing the plank thickness might lead to
higher-order effects, although they are expected to be
insignificant.
2.2.5 Generalizing the constraint form
Considering the four DDLC examples presented above, thereappears
to be a pattern in the scaling relations that may bewritten as
follows:
Figure 3. Assumed spar-cap structure with dimensions: H is
thetotal build height, h is the space between planks and B is the
plankwidth.
CT
CT,0
(R
R0
)Rexp≤ 1, (29)
where Rexp is the exponent of R in the DDLC.If the constraint
limit is met, the following relationship can
be written
R = R0
(CT,0
CT
) 1Rexp
. (30)
3 Formulation of rotor design problems
Based on the performance and constraint relationships out-lined
in the previous section, this section will present theformulation
for rotor design as optimization problems. Twodifferent classes of
problems are introduced, namely power-capture optimization and AEP
optimization, where the latteris a generalization of the former
with the constraint depend-ing on the wind speed.
3.1 Power-capture optimization
The optimization problem can be stated as
maximizeCT,R̃
P̃ =12
(1+
√1−CT
)CTR̃
2 (31)
subject toCT
CT,0R̃Rexp ≤ 1, (32)
where the definition of R̃ = R/R0 has been used for
consis-tency. The solution for this optimization problem is
presentedin Sect. 4.1.
It should be noted that this optimization problem is similarto
the problem that is given by Chaviaropoulos and Sieros(2014) in
which they optimize while keeping Mflap. So theoptimization problem
in this paper is a generalization of theiroptimization problem.
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K. Loenbaek et al.: Optimal relationship between power and
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3.2 AEP optimization
In contrast to the above mentioned optimization of powercapture,
optimization with respect to AEP requires the de-termination of
CT(Ṽ ), so it involves a function opposed to ascalar value. It is
also necessary to set the rated power to con-stant value, while the
wind speed at which the rated power isreached is allowed to change.
The problem can be formulatedas
maximizeCT(Ṽ ),R̃
˜AEP=2716
ṼCO∫ṼCI
P̃(CT(Ṽ ), R̃
)· Ṽ 3 ·PDFwind(Ṽ )dṼ (33)
subject toṼ 2 CT(Ṽ )
CT,0R̃Rexp ≤ 1; (DDLC)
2716 P̃
(CT(Ṽ ), R̃
)Ṽ 3 ≤ 1 (rated power),
(34)
where the wind speed scaling has been added to the DDLC.
4 Results and discussion
This section discusses the solutions to the rotor design
opti-mization problems introduced in the previous section.
4.1 Optimizing for power capture
The constrained optimization problem maximizing powercapture, as
stated in Sect. 3, may be simplified based on theobservation that
optimum solutions will occur at the DDLconstraint limit. To
understand this, consider that the powercapture of a rotor with an
inactive constraint may always beimproved by scaling the rotor up
until the constraint is met.This is true irrespective of the DDLC
that determines therotor design. Hence, an explicit relation R̃(CT)
can be usedto reformulate the problem from a constrained
optimizationproblem in two variables to an unconstrained
optimizationproblem in one variable.
P̃(CT, R̃
)=
12
(1+√
1−CT)CTR̃
2 (5)
R̃ =(CT,0CT
) 1Rexp (30)
⇒ P̃ (CT)=
C2 1Rexp
T,0
2
(1+
√1−CT
)C
1−2 1Rexp
T , (35)
with the optimization problem now as follows:
maximizeCT
P̃ =C
2 1Rexp
T,0
2
(1+
√1−CT
)C
1−2 1Rexp
T . (36)
By differentiating the objective function (Eq. 35 with re-spect
to CT and finding its root, the optimal CT as a functionof Rexp is
arrived at.
dP̃ (CT)dCT
= 0⇒ (37)
CT =8(R2exp− 3Rexp+ 2
)(3Rexp− 4
)2 . (38)This unique solution is a maximum, which is
apparent
from the always-positive value of 1P in Fig. 4. This figureshows
the optimal solution for CT and CP, as well as the rela-tive change
in radius (1R) and power (1P ) compared to thebaseline rotor. In
the plots in Fig. 4a and c, CP is observed toapproach the dashed
baseline performance (Betz rotor) muchfaster than CT as Rexp
increases. This is a consequence of therelationship between CT and
CP (Fig. 1). Especially aroundthe Betz limit, the gradient is very
small, which means thatchanges in CT do not lead to proportional
changes in CP.Turning to the two plots in Fig. 4b and d, it is seen
that thelowerCP is more than compensated for by increasingR
sincethe relative change in power (1P ) is always positive.
When maximizing power capture for a given thrust(Rexp = 2;
dashed vertical blue line in Fig. 4), it is found thatCT→ 0
and1R→∞while1P → 50 %, which was foundby investigating the behavior
of the limit value when Rexp→2. Since 1R→∞ is not of much practical
interest, furtherexplanation is not given here. Alternatively, the
maximumpower for a given flap root moment (Rexp = 3; orange linein
Fig. 4) may be achieved by increasing the rotor radius by11.6 %
compared to the baseline design (maximum CP). Thecorresponding
relative increase in power 1P is 7.6 %. Fi-nally, designs
constrained by tip deflection (Rexp = 5; greenline in Fig. 4) allow
the relative power 1P to increase by1.90 % with a relative change
in radius1R of 2.30 %. A tablewith the results for the increase in
power capture (1P ) andradius (1R) for four designs (Rexp = 2, 3,
5, 6) can be seen inFig. 6. In conclusion, rotors with a static
aerodynamic DDLCshould not be designed for the maximum CP, as more
powercan be generated by rotors with a lower CT and a larger
ra-dius R, without violating the relevant DDLC.
Effect on loads
Even though meeting the constraint limits means that thechosen
DDL will be the same as the baseline, it is interestingto know what
happens to loads that scale differently than theDDL. As an example,
if the DDLC is Mflap (Rexp = 3) it isa given that it will not
change relative to the baseline, but itcould be interesting to know
what happens to T and δtip.
To investigate it we will introduce a generalized load (L)as a
measure of how a load scale.
L=K0V20 CTR
Lexp , (39)
where K0 is a scaling constant and Lexp is the generalizedload
exponent. The generalized load equation can be madenon-dimensional
with
L̃=L
K0V20 R
Lexp0
= CTR̃Lexp . (40)
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162 K. Loenbaek et al.: Optimal relationship between power and
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Figure 4. (a) Optimal CT as a function of the constraint R
exponent (Rexp). (c) Rexp vs. CP; notice that the optimal CP curve
has a steeperslope and hugs the baseline closer than CT. (b) Rexp
vs. relative change in radius 1R. (d) Rexp vs. relative change in
power capture (1P̃ ).Despite the similar shape of the curves, a
difference between the two is that 1P (Rexp→ 2)= 50 %, while
1R(Rexp→ 2)→∞. Thevertical lines represent each of the example
constraints (∗ DDLC: design-driving load constraint).
The difference between Lexp and Rexp is that Rexp resultsin a
design, whereas Lexp is a load for a design. Take a de-sign made
for tip deflection (Rexp = 5) as an example, thenLexp = 3 will
describe the Mflap load for that design.
An equation for the relative change 1L̃ can be found interms of
the baseline rotor as follows:
L̃= CTR̃Lexp (40)
R̃ =(CT,0CT
) 1Rexp (30)
L̃0 = CT,0R̃Lexp0 = CT,0
⇒1L̃=
L̃
L̃0− 1=
(CT
CT,0
)1−LexpRexp− 1. (41)
Since it is known that CT ≤ CT,0 these conclusions follow:
Lexp Rexp The load is larger than the baseline level.
This agrees with Fig. 5, which illustrates the effect of
designconstraints (DDLCs) on different loads. For example,
con-sider tip deflection (Rexp = 5; DDLC(δtip); the dashed
greenline in Fig. 5). Looking at the solid green line (Lexp = 5)
itis seen that the relative change in L is zero as expected.
Nowlooking at the loads with Lexp Rexp the loads are increased.
Ifthere was a load that scaled like Lexp = 6 the load wouldbe
increased by 1L(Lexp=6) =+2.3 %. Furthermore, Fig. 5shows that the
relative decrease in load is always most pro-nounced for the thrust
(Lexp = 2), with the biggest impact oc-curring around Rexp ≈ 2.5.
All of the relative change curveshave distinct minima but at the
same time are characterizedby large plateaus of relatively small
change. Another obser-vation is how quickly the curves grow for
Lexp >Rexp. TakeDDLC(Mflap) as an example; in this case 1δtip
=+24.5 %and 1L(Lexp=6) =+38.9 %. The relative change in loads
be-comes smaller as Rexp increases. A sketch with a zoomed-inview
of the tip and a table with the values can be seen inFig. 6.
4.2 Low-induction rotor
The concept in this section was mentioned in the Introduc-tion
since it has had some attention over the recent years.
Thelow-induction rotors (LIR) are rotors designed with a loweraxial
induction a than the level that maximizes CP. The con-cept is, to a
certain degree, analogous with optimization ofrotors for power
capture.
To investigate such an LIR design, it was chosen to fix theCT
value below the rated power in order for it to be the sameas for
the power-capture optimization for a given Rexp. If theradius was
set to the same value as for power capture, it willresult in the
constraint limit not being met since the turbine
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Figure 5. Relative change in different rotor load parameters
(1L̃) depending on DDLC. The scaling of loads have the form L̃=
CTRLexp ;e.g., Lexp = 2 scales as the rotor thrust T and Lexp = 5
scales as the tip deflection δtip. Each curve depicts how a load
parameter wouldchange depending on the design-driving constraint.
As an example, consider a design limited by tip deflection
DDLC(δtip), i.e., Rexp = 5,which matches the dashed green line. Tip
deflection meets the requirements, while thrust (T ) is lowered by
6.6 % and flap moment Mflap by4.4 %.
Figure 6. Sketch of a turbine with the load/structural response
outlined. The zoomed-in figure shows the radius increase (1R) and
the changein tip deflection (1δtip) for two different DDLCs (bold
black line is the baseline). The table shows the relative change in
power, radius andload/structural response for different DDLCs. Rexp
= 2 is a thrust constraint design, Rexp = 3 is a flap moment
constraint design, Rexp = 5is a tip-deflection constraint design
and Rexp = 6 is the tip deflection+constant mass constraint
design.
reaches the rated power earlier. SinceCT is fixed and the
con-straint limit needs to be met, the wind speed at which the
tur-bine reaches the rated power (Ṽrated) can be found. It is
foundthrough the normalized power (the integrant of Eq. 7
withoutthe PDFwind) and the constraint limit with wind speed
scaling(Eq. 30 multiplied with Ṽ 2) as follows:
2716
12
(1+√
1−CT)CTR̃
2Ṽ 3 = 1
Ṽ 2 CTCT,0
R̃Rexp = 1
⇒Ṽrated =
(1627
2(1+√
1−CT)CT
(CT
CT,0
) 2Rexp
) 13− 4
Rexp. (42)
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164 K. Loenbaek et al.: Optimal relationship between power and
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Figure 7. Power and thrust curves for a low-induction rotor
(solid lines), designed using the present method with the DDLC
exponentRexp = 3, which corresponds to an Mflap constraint. The
dashed line is the baseline rotor optimized for a max CP.
For a given rated wind speed the rotor radius can be foundusing
the following steps:
1. CT =8(R2exp− 3Rexp+ 2
)(3Rexp− 4
)2 (38)2. Ṽrated =
(1627
2(1+√
1−CT)CT
(CT
CT,0
) 2Rexp
) 13− 4
Rexp(42)
3. R̃ =
(1
Ṽ 2rated
CT,0
CT
) 1Rexp
, (43)
With CT, Ṽrated and R̃, ˜AEP can be computed using Eq. (7).The
LIR is illustrated by the examples in Figs. 7 and 8
where the present analysis framework has been applied
withconstraints pertaining to flap moments (Rexp = 3) and tip
de-flections (Rexp = 5).
In both cases, the resulting power curves are slightly abovethe
equivalent baseline ones, and the thrust peaks are reducedcompared
to the baseline. The relative change in AEP re-sults in a smaller
change than the change in power at the de-sign point. For the case
with DDLC(Mflap), 1AEP= 6.0 %while the power capture increased by
1P = 7.6 %. The cor-responding improvements for a
tip-deflection-constrained ro-tor, DDLC(δtip), are 1AEP= 1.2 % and
1P = 1.9 %. Thelower relative improvement for the LIR is related to
theamount of the power that is produced below the rated power.The
results for the LIR are summarized in Fig. 9 with a ta-ble and a
sketch showing the relative changes in AEP, radius,thrust,
root-flap moment and tip deflection for four differentdesigns (Rexp
= 2, 3, 5, 6). From Fig. 9 the thrust constraintdesign (DDLC(T
);Rexp = 2) is seen to have diverging valuesfor1R,1Mflap and1δtip.
As was the case for power-captureoptimization these results are
found from investigating the re-sult of the limit in which Rexp→ 2.
Even though the result
of 1R→∞ is interesting, the corresponding consequenceof 1Mflap→∞
makes this infeasible for practical use, sothis will not be studied
further here.
4.3 AEP-optimized rotor
As mentioned in Sect. 3, the variables considered for
op-timization of AEP are CT(Ṽ ) and R̃. In this formulation,CT can
be adjusted independently for each wind speed,which ideally can be
achieved through blade pitch control.The relative radius R̃ couples
the rotor operation across allwind speeds, as it is necessarily
constant. Based on initialstudies, the optimizer targets solutions
with three distinct op-erational ranges, which, ordered by wind
speed, are as fol-lows:
– operation with maximum power coefficient (max CP);
– operation at constraint limit (constant thrust T ); and
– operation at the rated power.
This can be used to make CT a function of R̃, thereby
de-creasing the optimization problem to an unconstrained
opti-mization in one variable (R̃). The CT function is given as
CT(Ṽ , R̃)=89
89 ≤ Ṽ
−2CT,0R̃−Rexp (max CP)
Ṽ −2CT,0R̃−Rexp 1≤ 2716
12
(1+√
1−CT)CTR̃
2 Ṽ 3 (constraint limit)
1= 271612
(1+√
1−CT)CTR̃
2 Ṽ 3 1> 271612
(1+√
1−CT)CTR̃
2 Ṽ 3 (rated power),
(44)
where the last equation needs to be solved to getCT; the
solu-tion is a third-order polynomial, which is more easily
solvednumerically.
The only free parameter that needs to be determined tofind the
optimal AEP is R̃. The optimization problem can bereformulated
as
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K. Loenbaek et al.: Optimal relationship between power and
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Figure 8. Power and thrust curves for rotor with the DDLC
exponent Rexp = 5 (solid lines), corresponding to a δtip
constraint. The dashedline is the baseline rotor optimized for max
CP.
Figure 9. Sketch of a turbine with the load/structural response
outlined. The zoomed-in figure shows the radius increase (1R) and
the changein tip deflection (1δtip) for two different DDLCs (bold
black line is the baseline). The table shows the relative change in
power, radius andload/structural response for different DDLCs. Rexp
= 2 is a thrust constraint design, Rexp = 3 is a flap moment
constraint design, Rexp = 5is a tip-deflection constraint design
and Rexp = 6 is the tip deflection+constant mass constraint
design.
maximizeR̃
˜AEP=
ṼCO∫ṼCI
P̃(CT(Ṽ , R̃), R̃
)· Ṽ 3 ·PDFwind(Ṽ )dṼ . (45)
The problem can be solved with most optimization solverssince
the AEP can be computed explicitly if R̃ is given. Theoptimization
problem was solved with the L-BFGS-B algo-
rithm described in Zhu et al. (1997) though the use of
SciPy(Millman and Aivazis, 2011).
Examples of the resultant power and thrust curves can beseen in
Figs. 10 and 11, for DDLC(Mflap) and DDLC(δtip),respectively.
Looking at Fig. 10 (Rexp = 3) it is clear thatthe power and thrust
curves have changed quite substan-tially, compared to the baseline
Betz rotor (dashed curves).The thrust curve does not have a sharp
peak anymore butrather a flat plateau. As mentioned in the
Introduction this
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166 K. Loenbaek et al.: Optimal relationship between power and
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Figure 10. Power and thrust curve for an AEP-optimized rotor
(solid lines) where the DDLC exponent is Rexp = 3, which is
equivalent to aconstraint on Mflap. The dashed line is the baseline
rotor optimized for the max CP below the rated power.
Figure 11. Power and thrust curve for an AEP-optimized rotor
(solid lines) where the DDLC exponent is Rexp = 5, which is
equivalent to aconstraint on δtip. The dashed line is the baseline
rotor optimized for the max CP below the rated power.
is often referred to as thrust clipping. It comes from theDDLC
equation (Eq. 44) which shows that CT ∝ Ṽ −2, andsince thrust is
proportional to T ∝ CTṼ 2, it means that thethrust is constant. As
mentioned, the region where the ro-tor is thrust clipped is also
where the DDLC is active, soopposed to the baseline and LIR rotor,
the DDLC is activeover a larger range of V . The larger range of V
is also partlywhy 1R = 44.6 %, which is a huge increase. As a
result, italso leads to a large increase, with 1AEP= 19.9 %. This
isa very large change in R̃ and the feasibility of such a de-sign
is doubtful. As it is shown later, the change in maximumloads (see
Fig. 13) leads to a significant change in loads withLexp
>Rexp.
A more realistic design for modern turbines is found inFig. 11
(Rexp = 5). Here the changes are fewer but still sig-nificant with
1R = 10.7 % and1AEP= 5.8 %. It shows thesame shape as the
thrust-clipped curve, but now it is over asmaller range of V . As
mentioned in the Introduction, thrustclipping was also found by
Buck and Garvey (2015a) to be abeneficial way to lower CoE.
In Fig. 12 the relative change in R and AEP can be seen asa
function of the DDLC R exponent. The plot both containsthe result
for the AEP-optimized rotor (AEP opt.; solid blackline) and the
low-induction rotor (LIR opt.; dashed–dottedgray line). The
difference between the two is significant, es-pecially for 1AEP.
The results for the AEP-optimized ro-
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K. Loenbaek et al.: Optimal relationship between power and
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Figure 12. DDLC exponent (Rexp) vs. (a) relative change in
radius (1R) and (b) relative change in AEP (1 ˜AEP). The plot
contains bothof the changes for the cases of the low-induction
rotor (LIR opt.; dashed–dotted black line) and the AEP-optimized
rotor (AEP opt.; blacksolid). The changes in both AEP and radius
are much larger for the AEP-optimized rotor.
Figure 13. DDLC R exponent (Rexp) vs. relative maximum load
(1L̃max). The plot looks similar to Fig. 5 but 1L̃max is the change
inmaximum loading. As an example, when thrust (T ) is −30.8 % for
Rexp = 3 it means that the maximum thrust (for any wind speed)
is30.8 % lower than the maximum thrust for the baseline (which
happens just before the rated wind speed). Notice that the range
for the y scaleis much larger in this plot than for the
power-capture-optimized rotor. The potential reduction is more, but
it comes with the consequencethat Lexp >Rexp grows faster even
for high values of Rexp.
tor are summarized in Fig. 14 with a table and a sketchthat
shows the relative changes. As was the case for power-capture
optimization and LIR optimization, some values di-verge when Rexp→
2, and the results are found by investi-gating this limit. But
since it has no practical value, furtherexplanation is omitted
here.
Effect on loads
In Fig. 13 a plot of the relative change in maximum loadsas a
function of the DDLC R exponent. The relative max
load (1L̃max) does not compare the loads at each Ṽ but
ratherthe max load for the baseline at Ṽ = 1 (rated wind speed)
tothe max load for the optimized rotor for any Ṽ . The plot inFig.
13 is similar to the plot in Fig. 5 with the difference beingthat
it is the relative change in maximum loads, independentof wind
speed at which it occurred. Comparing the two plots,one should note
the range for the y scale in the two plots, withFig. 13 having the
larger range. It also means that the relativechange in the loads
for the AEP-optimized rotor experiencesa larger relative change.
But it also has the consequence thatloads with Lexp >Rexp grow
faster, especially for larger val-
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168 K. Loenbaek et al.: Optimal relationship between power and
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Figure 14. Sketch of a turbine with the load/structural response
outlined. The zoomed-in figure shows the radius increase (1R) and
thechange in tip deflection (1δtip) for two different DDLCs (bold
black line is the baseline). The table shows the relative change in
power,radius and load/structural response for different DDLCs. Rexp
= 2 is a thrust constraint design, Rexp = 3 is a flap moment
constraint design,Rexp = 5 is a tip deflection constraint design
and Rexp = 6 is tip deflection+constant mass constraint design.
ues of Rexp (> 5). A summary of the AEP-optimized rotorcan be
seen in Fig. 14, where a table of four different designs(Rexp = 2,
3, 5, 6) shows the relative change in AEP, radius,thrust, root-flap
moment and tip deflection.
4.4 Summary of findings
In Table 1 the tables shown in Figs. 6, 9 and 14 are
summa-rized. It compares the different optimizations to each
other.
As seen from the tables, the largest increase in1P/AEP isfound
using AEP optimization, which also leads to the largestincrease in
rotor radius (1R). It also shows that using thrustclipping seems to
be a better operational strategy than lowinduction, as the
design-driving constraint can be met over alarger range of wind
speeds and low induction is only neededaround maximum thrust and
not at low wind speeds.
In all three optimization cases, the optimization of the de-sign
with thrust constraint (DDLC(T ); Rexp = 2) leads todivergent
values for 1R and the loads. In all cases the re-sult is found by
investigating the behavior of the limit whenRexp→ 2. Since this is
not thought to be of much practicalvalue, the details are not
provided here.
4.5 Limitation of the study and possible improvements
The study shows that for a rotor constraint by a static
aero-dynamic DDL there is a benefit to lowering the loading
andincreasing the rotor size in terms of power/AEP. But, as itwas
found by Bottasso et al. (2015), having a rotor with the
same load constraint and increasing the radius does not meanthat
the cost is the same or that it is cost optimal. They foundthat the
increase in AEP did not compensate for the addedcost from
increasing the rotor radius. This problem of costvs. benefit is not
directly addressed in this paper, but by theDDLC δtip+mass, a
constraint in which the mass is kept con-stant. It is thought to be
a better approximation for a rotorwith a fixed price – but this
assumption needs to be tested.
Another issue that is not taken into account in this study isthe
influence of the turbines self-weight. As was found bySieros et al.
(2012) the self-weight becomes more impor-tant for larger rotors.
To accommodate for the added mass,a penalty could be added which
should scale as R̃ or R̃3 fortop head mass and static blade mass
moment, respectively.As discussed above, there could also be a
constraint imple-mented that will keep the mass or the mass moment
in theoptimization. Again this is a limitation of the study.
The fidelity of the models is also a limitation. Even though1-D
aerodynamic momentum theory is a common approxi-mation to do for
first-order studies in rotor design, it is wellknown that the
constantly loaded rotor is not possible to re-alize, and when
losses are included the constantly loaded ro-tor is not the optimal
solution anymore. At the same time, ifit was possible to decrease
the load at the tip more than atthe root, it would lead to less tip
deflection than a constantlyloaded rotor with a similar CT.
Extending the model to beable to handle radial load distribution is
one way of addingdetail to the model that could lead to even larger
improve-
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K. Loenbaek et al.: Optimal relationship between power and
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Table 1. Overview of the optimization results from optimizing
power capture (Opt. PC), low-induction rotor (Opt. LIR) and annual
energyproduction (Opt. AEP).
Opt. PC Rexp 1P 1R (Lexp = 2) (Lexp = 3) (Lexp = 5)1T 1Mflap
1δtip
DDLC T 2 +50.0 % ∞ 0.0 % ∞ ∞DDLC Mflap 3 +7.6 % +11.6 % −10.4 %
0.0 % +24.5 %DDLC δtip 5 +1.9 % +2.3 % −6.6 % −4.4 % 0.0 %DDLC
δtip+mass 6 +1.2 % +1.4 % −5.5 % −4.2 % −1.4 %
Opt. LIR Rexp 1AEP 1R (Lexp = 2) (Lexp = 3) (Lexp = 5)1T 1Mflap
1δtip
DDLC T 2 +49.7 % ∞ 0.0 % ∞ ∞DDLC Mflap 3 +6.0 % +14.9 % −12.9 %
0.0 % +31.9 %DDLC δtip 5 +1.2 % +2.6 % −7.5 % −5.1 % 0.0 %DDLC
δtip+mass 6 +0.7 % +1.6 % −6.2 % −4.7 % −1.6 %
Opt. AEP Rexp 1AEP 1R (Lexp = 2) (Lexp = 3) (Lexp = 5)1T 1Mflap
1δtip
DDLC T 2 +69.7 % ∞ 0.0 % ∞ ∞DDLC Mflap 3 +19.9 % +44.6 % −30.8 %
0.0 % +109.0 %DDLC δtip 5 +5.7 % +10.6 % −26.2 % −18.3 % 0.0 %DDLC
δtip+mass 6 +3.9 % +7.0 % −23.8 % −18.4 % −6.6 %
ments. It could be done through the use of blade elementmomentum
(BEM) theory.
For modern turbine design, it is often the case that
thestructural design is determined by the aeroelastic extremeloads,
such as extreme turbulence or gusts. With the simplic-ity of the
models in this study, this is not taken into con-sideration. But if
the extreme load happens in normal oper-ation there will likely be
a direct relationship between thesteady and extreme loads, meaning
that a decrease in steadyloads will also lead to a decrease in the
extreme load. This isan assumption that should be tested in future
work. If thedesign-driving load is happening in nonoperational
condi-tions, e.g., extreme wind in parked conditions, grid loss
orsubcomponent failure, then the analysis tool cannot be di-rectly
applied.
5 Conclusions
A first-order model framework for the analysis of windturbine
rotors was developed based on aerodynamic 1-D momentum theory and
Euler–Bernoulli beam theory.This framework introduces the concept
of design-drivingload (DDL) for which a generalized form has been
devel-oped in which loads only differ by a scaling exponent
Rexp,e.g., thrust scales as Rexp = 2, root-flap moment as Rexp =
3and tip deflection as Rexp = 5. Despite the simplicity of
themodel, this study has shown important trends in how to de-sign
rotors for maximum power capture. It has been shownthat the
potential increase in power capture is very dependenton the
relevant constraint, e.g., thrust as the constraining load
compared to the more restrictive tip deflection. Furthermore,it
was concluded that the best way to design a rotor for in-creased
power capture using aeroelastic considerations is notto maximize CP
but rather to relax CP and operate at lowerloading (lower CT). How
much one should relax CP dependson the chosen design-driving
constraint (Rexp). The resultsfor optimizing for power capture are
summarized in Table 1(Opt. PC).
The optimization of power capture determines the bestpossible
design for a given wind speed. By considering theannual energy
production (AEP), an optimal design acrossthe range of operational
wind speeds can be found for agiven wind speed frequency
distribution. Optimal AEP wasconsidered with two different
approaches, namely the low-induction rotor (LIR) and full AEP
optimization. For LIR,theCT value below the rated power was set to
the value foundfrom power-capture optimization for the chosen Rexp.
Thenthe radius was increased compared to the
power-capture-optimized rotor, since it will reach the rated power
earlierwith the same rotor size. A summary of the results can
beseen in Table 1 (Opt. LIR).
For the full AEP optimization, CT was allowed to take onany
positive value below the Betz limit (0≤ CT ≤ 8/9) forall wind
speeds. The optimal AEP is obtained for a rotor thatoperates in
three distinct operational regimes:
– operation with maximum power coefficient (max CP);
– operation at constraint limit (constant thrust T ); and
– operation at the rated power.
www.wind-energ-sci.net/5/155/2020/ Wind Energ. Sci., 5, 155–170,
2020
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170 K. Loenbaek et al.: Optimal relationship between power and
design-driving loads for wind turbine rotors
The results from the optimization are summarized in Table 1(Opt.
AEP). It shows significantly larger relative improve-ments in
power/energy compared to power-capture- and LIR-optimized rotors.
This comes at the cost of a larger increasein rotor radius. In the
range where the optimum turbine op-erates at the constraint limit,
the thrust curve is clipped (in amanner also known as peak shaving
or force capping). This isa control feature used for many
contemporary turbines, so itis interesting that this study,
independent of this knowledge,shows that thrust clipping is a very
efficient way to increaseenergy capture while observing certain
load constraints. It isalso the main reason behind the relatively
large possible im-provements in AEP, as the constraint limit is met
over a largerrange of wind speeds.
In spite of relatively crude model assumptions made, thispaper
provides profound insight into the trends of rotor de-sign for
maximum power/energy, e.g., the use of thrust clip-ping. As wind
turbine rotors continue to develop towardslarger diameters with
slender (more flexible) blades, thetype of design-driving load
constraint also evolves. With thepresent model framework, the
conceptual implications of thisdevelopment become clearer; an
increase in AEP of up to5.7 % is possible compared to a traditional
CP-optimized ro-tor – without changing technology, using bend-twist
couplingor other advanced features. Finally, this work has
demon-strated an approach to formulate an optimization
objectivethat couples power and load/structural response though
thepower-capture optimization. This approach may be extendedinto
less crude model frameworks, e.g., by introducing radialvariations
in rotor loading.
Data availability. No data sets were used in this article.
Author contributions. KL came up with the concept and mainidea,
as well as made the analysis. All authors interpreted the
resultsand made suggestions for improvements. Also, some modeling
haschanged based on discussions between the authors. KL prepared
thepaper with revisions from all coauthors.
Competing interests. The authors declare that they have no
con-flict of interest.
Acknowledgements. We would like to thank Innovation FundDenmark
for funding the industrial PhD project that this article is apart
of. We would like to thank all employees at Suzlon Blade Sci-ence
Center for being a great source of motivation with their interestin
the results. We would like to thank all people at DTU Risø whocame
to us with valuable inputs.
Financial support. This research has been supported by the
In-novation Fund Denmark (grant no. 7038-00053B).
Review statement. This paper was edited by Mingming Zhangand
reviewed by two anonymous referees.
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AbstractIntroductionTheoryAerodynamicsPower capture and annual
energy production (AEP)Baseline rotor
Scale laws and constraints for design-driving loadsThrust
(T)Root flap bending moment (Mflap)Tip deflection (tip)Tip
deflection with constant massGeneralizing the constraint form
Formulation of rotor design problemsPower-capture
optimizationAEP optimization
Results and discussionOptimizing for power captureLow-induction
rotorAEP-optimized rotorSummary of findingsLimitation of the study
and possible improvements
ConclusionsData availabilityAuthor contributionsCompeting
interestsAcknowledgementsFinancial supportReview
statementReferences