Optimal relationship between power and design-driving loads ......In this paper, we investigate the relationship between the load, power and structural response of wind turbine rotors

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 Dam1

1Suzlon Blade Science Center, Havneparken 1, 7100 Vejle, Denmark2Technical 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 P 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 P 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 P 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 www.wind-energ-sci.net/5/155/2020/

K. Loenbaek et al.: Optimal relationship between power and design-driving loads for wind turbine rotors 157

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:

P(CT, R

)=

P

12ρV

3πR20

= CPR2=

12

(1+

√1−CT

)CTR

2, (5)

where R 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= Tyear12ρπR2

0

VCO∫VCI

P(CT(V ), R

)·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:

ÃEP(CT, R

)=

AEPTyearPrated

=AEP

Tyear12ρπR

20

1627V

30

=2716

VCO∫VCI

P(CT(V ), R

)· V 3·PDFwind(V )dV . (7)

V is a normalized wind speed given by V = V V0, whereV0 is the wind speed at which the turbine reaches the

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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 PDFwinddV is dimension-less. Throughout this paper ÃEP is calculated using adiscretization of the integral, which is computed us-

ing the trapezoidal rule given asVCO∫VCI

f (V ;CT, R)dV ≈

N∑i=1

f (Vi+1;CT,R)+f (Vi ;CT,R))2 1Vi , 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)

1P =CPR

2

CP,0R20− 1 (10)

1L=CTR

Lexp

CT,0RLexp0

− 1 (11)

1 ÃEP=ÃEPÃEP0− 1, (12)

where L 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 = 1

2ρV20 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

dr2EId2δ

dr2 =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



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

d2δ

dr2 (r = R)= 0,d3δ

dr3 (r = R)= 0︸︷︷︸Free tip

. (19)

The resulting displacement solution becomes

δ =11π120

V 2ρ

EIrCTR

5(

233

(EIr

EIt− 1

)r6+

111r5

−5

11

(EIr

EIt− 1

)r4+

1011

(23EIr

EIt−

53

)r3+

2011r2)

(20)

=11π120

V 2ρ

EIrCTR

5δshape

(r,EIr

EIt

), (21)

where the normalized radius (r ∈ [0,1]) has been introducedso that r = R · r . The polynomial shape of the deflection hasbeen collected in δshape. The maximum deflection occurs atthe blade tip (r = 1), which leads to a scaling relation andDDLC for tip deflection:

Scaling

δtip =11π120

V 2ρEIrCTR

5δshape

(r = 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



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 ≈ EH 2Bh2 (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

(r = 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,R

P =12

(1+

√1−CT

)CTR

2 (31)

subject toCT

CT,0RRexp ≤ 1, (32)

where the definition of R = 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.



3.2 AEP optimization

In contrast to the above mentioned optimization of powercapture, optimization with respect to AEP requires the de-termination of CT(V ), 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(V ),R

ÃEP=2716

VCO∫VCI

P(CT(V ), R

)· V 3·PDFwind(V )dV (33)

subject toV 2 CT(V )

CT,0RRexp ≤ 1; (DDLC)

2716 P

(CT(V ), R

)V 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 R(CT) can be usedto reformulate the problem from a constrained optimizationproblem in two variables to an unconstrained optimizationproblem in one variable.

P(CT, R

)=

12

(1+√

1−CT)CTR

2 (5)

R =(CT,0CT

) 1Rexp (30)

⇒ P (CT)=

C2 1Rexp

T,0

2

(1+

√1−CT

)C

1−2 1Rexp

T , (35)

with the optimization problem now as follows:

maximizeCT

P =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.

dP (CT)dCT

= 0⇒ (37)

CT =8(R2

exp− 3Rexp+ 2)

(3Rexp− 4

)2 . (38)

This unique solution is a maximum, which is apparentfrom 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

L=L

K0V20 R

Lexp0

= CTRLexp . (40)



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 (1P ).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 1L can be found interms of the baseline rotor as follows:

L= CTRLexp (40)

R =(CT,0CT

) 1Rexp (30)

L0 = CT,0RLexp0 = CT,0

⇒1L=

L

L0− 1=

(CT

CT,0

)1−LexpRexp− 1. (41)

Since it is known that CT ≤ CT,0 these conclusions follow:

Lexp <Rexp The load is lower than the baseline level.Lexp = Rexp The load is identical to the baseline level.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, namely thrust (Lexp =

2) and flap moment (Lexp = 3), it is seen that 1L is lowerthan the baseline, with1T =−6.6 % and1Mflap =−4.4 %.

But for loads where 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



Figure 5. Relative change in different rotor load parameters (1L) depending on DDLC. The scaling of loads have the form L= 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 (Vrated) 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 V 2) as follows:

2716

12

(1+√

1−CT)CTR

2V 3= 1

V 2 CTCT,0

RRexp = 1

⇒Vrated =

(1627

2(1+√

1−CT)CT

(CT

CT,0

) 2Rexp

) 13− 4

Rexp. (42)



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(R2

exp− 3Rexp+ 2)

(3Rexp− 4

)2 (38)

2. Vrated =

(1627

2(1+√

1−CT)CT

(CT

CT,0

) 2Rexp

) 13− 4

Rexp(42)

3. R =

(1

V 2rated

CT,0

CT

) 1Rexp

, (43)

With CT, Vrated and R, ÃEP 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(V ) and R. In this formulation,CT can be adjusted independently for each wind speed,which ideally can be achieved through blade pitch control.The relative radius R 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 R, thereby de-creasing the optimization problem to an unconstrained opti-mization in one variable (R). The CT function is given as

CT(V , R)=89

89 ≤ V

−2CT,0R−Rexp (max CP)

V −2CT,0R−Rexp 1≤ 27

1612

(1+√

1−CT)CTR

2 V 3 (constraint limit)

1= 2716

12

(1+√

1−CT)CTR

2 V 3 1> 2716

12

(1+√

1−CT)CTR

2 V 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 R. The optimization problem can bereformulated as



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.

maximizeR

ÃEP=

VCO∫VCI

P(CT(V , R), R

)· V 3·PDFwind(V )dV . (45)

The problem can be solved with most optimization solverssince the AEP can be computed explicitly if R 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



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 ∝ V

−2, andsince thrust is proportional to T ∝ CTV

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 R 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-



Figure 12. DDLC exponent (Rexp) vs. (a) relative change in radius (1R) and (b) relative change in AEP (1 ÃEP). 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 (1Lmax). The plot looks similar to Fig. 5 but 1Lmax 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 (1Lmax) does not compare the loads at each V but ratherthe max load for the baseline at V = 1 (rated wind speed) tothe max load for the optimized rotor for any V . 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-



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 R or R3 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-



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 % ∞ ∞


Opt. AEP Rexp 1AEP 1R (Lexp = 2) (Lexp = 3) (Lexp = 5)1T 1Mflap 1δtip

DDLC T 2 +69.7 % ∞ 0.0 % ∞ ∞


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.



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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Optimal relationship between power and design-driving loads ......In this paper, we investigate the relationship between the load, power and structural response of wind turbine rotors