Blade Optimization for Wind Turbines Wang Xudong1,2, Wen Zhong Shen1, Wei Jun Zhu1, Jens Nørkær Sørensen1 and Chen Jin2

1Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark

2State Key Laboratory of Mechanical Transmission, ChongQing University, Chongqing, China E-mail: shen@mek.dtu.dk; Phone: (45) 45 25 43 17; Fax: (45) 45 88 43 25.

Abstract In the development of new large megawatt size wind turbines, aerodynamic and structural optimizations have become interesting and important subjects for optimizing the energy yield of wind turbines. Optimization is usually based on maximizing the Annual Energy Production (AEP) for a given size of a wind turbine at a normal site. When the AEP is getting bigger, the cost of the turbine is increasing. So how to reduce the cost of a wind turbine per unit of energy is an important subject in modern wind turbine researches. In this paper a design tool for optimizing wind turbine blades is presented. The design model is based on an aero-elastic code that includes the structural dynamics of blades and the Blade-Element Momentum (BEM) theory. To model the main aero-elastic behaviour of a real wind turbine, the code employs 11 basic degrees of freedom (DOF) corresponding to 11 elastic structure equations. The objective of the optimization model is the minimum cost of energy which is calculated from the Annual Energy Production (AEP) and the cost of the whole rotor. The design variables are the blade shape parameters including chord, twist and relative thickness. To illustrate the optimization technique, three wind turbines of different size (the MEXICO 25 kW experimental rotor, the Tjæreborg 2MW wind turbine and the NREL 5MW virtual wind turbine) are applied. The results show that the optimization model can reduce the cost of energy of the original turbines, especially for large size wind turbines. Keywords: Blade optimization; Aerodynamics; Aeroelastics; Energy cost; Wind turbine design

Introduction In order to get wind energy more competitive to other energy sources, the first goal is to decrease the cost of energy from wind power. How to reduce the cost of a wind turbine per unit of energy is an important task in modern wind turbine researches. There are a number of recently published papers dealing with the optimization of wind turbines [1-3]. In the present work, we present a new method for optimizing the geometry of wind turbines with respect to maximizing the energy yield of a wind turbine. The method is based on combining an aere-elastic model containing 11 degrees of freedom with the Blade Element Momentum (BEM) technique. As an essential ingredient in the Blade Element Momentum (BEM) theory, the tip loss effect of rotors plays also an important role in the prediction of wind turbine performance. Recently, some classical tip-loss correction models that employ two-dimensional airfoil data were reviewed by Shen et al. [4] and found to be inconsistent. To overcome the inconsistency, a remedy was proposed with a function (the F1 function) that correlates the two-dimensional airfoil data to the force characteristics near the blade tip. This new tip correction model is used in the optimization.

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Aerodynamic/Aero-elastic Model The aerodynamic/aero-elastic code, used for computing the aerodynamic performance, such as output power, torque and thrust, is based on BEM theory and structural dynamic model of the blades and tower.

Aerodynamic model The aerodynamic model used here is based on the 1D-Blade Element Momentum theory (BEM) with the improved tip loss correction introduced in Shen et al. [5]. The principal points are summarized here.

Employing the momentum theory, the axial load and the torque are written as (1) and (2)

204 (1 )dT V aF aF rdrπρ= − (1)

304 (1 )dM V bF aF r drπρ= Ω − (2)

where F is the Prandtl tip loss function given as

12 (cos [exp( )]2 sinB R rF

r)

π φ− −

= − (3)

Employing the blade element theory, the axial load and the torque are written as

21

12 rel ndT B cV F C drρ= (4)

21

12 rel tdM B cV F C rdrρ= (5)

where are the 2D force coefficients and F( ,n tC C ) 1 is the correlation between the 2D force

coefficients and the 3D force coefficients on the blade. The F1 function is given as

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2 (cos [exp( )]2 sinB R rF g

r)

π φ− −

= − (6)

exp[ 0.125( 21)] 0.1g Bλ= − − + (7) Equating equation (1) to equation (4), and equation (2) to equation (5), the final formulas of the interference factors become

21 1

1

2 4 (1 )2(1 )

Y Y F Ya

FY+ − − +

=+

1 (8)

2

1(1 ) /(1 ) 1

baF Y a

=− − −

(9)

where 21 14 sin ( )nY F C Fφ σ= , 2 14 sin cos ( )tY F C Fφ φ σ= and (2 )Bc r=σ π . For more details,

the reader is referred to [5].

Dynamic structural model The aerodynamic loads on wind turbine blades are calculated using the BEM theory. In order to accurately predict the aero-elastic characteristics of a real wind turbine, the structural dynamics of the wind turbine needs to be taken into account. The structural dynamics of the wind turbine is

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given by the following general dynamic equation [6]

0 0 0 gM x C x K x F+ + = (10)

where 0M is the mass matrix, is the damping matrix, 0C 0K is the stiffness matrix and gF is

the generalized force associated with external loads. The x represents a number of unknowns

associated with the degrees of freedom (DOF). For simplicity, degrees of freedom (DOF) in total are used to describe a three-bladed wind turbine as shown in Figure 1. For each wind turbine blade, the first three eigenmodes (the first and second flapwise and the first edgewise modes) are counted. The other two unknowns are the displacement in the axial direction of the rotor root and the azimuth angle of one blade. In summary, the unknowns can be written

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1 1 2 1 1 2 1 1 21 1 1 2 2 2 3 3 3( , , , , , , , , , ,f e f f e f f e f )x w x x x x x x x x xϕ= (11)

where is the axial displacement of the whole rotor; w ϕ is the azimuth displacement of the

blades; jix is the deflection coefficient of the j-th mode on the i-th blade. After knowing the loads

and the deformations, the vibration velocities and accelerations of each mode are calculated from the equation (10).

ϕ

w

,11

fx fx 21

ex11

0V

fx 22

ex12,1

2fx

fx 23

ex13

,13

fx

Figure 1: Definition of degrees of freedom (DOF) for a wind turbine

Since only the first three modes are considered, the total deformation of a blade is calculated from the linear combination of the three major modes

1 1 1 1 2 2( ) ( ) ( )f f e e f fu x u x x u x x u x= ⋅ + ⋅ + ⋅ (12)

Once the displacement is known, the velocity and the acceleration of the blade are found as follow

1 1 1 1 2 2( ) ( ) ( )f f e e f fu x u x x u x x u x= ⋅ + ⋅ + ⋅ (13)

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1 1 1 1 2 2( ) ( ) ( )f f e e f fu x u x x u x x u x= ⋅ + ⋅ + ⋅ (14)

The velocity u is the vibration velocity of the blade and will contribute to the relative velocity as

0rel rotV V V W u= − + − (15)

where is the wind speed, is the rotational speed and 0V rotV W is the induced velocity. In equation

(13), + is the blade velocity, hence the relative wind speed due to the blade motion has a

negative sign.

rotV u

Optimization Model The most important issue when performing optimizations is to locate all important parameters and a suitable object function. From earlier optimization studies of wind turbines [1-4], the most convenient way is to minimize the cost of energy based on AEP and the production cost of a turbine or a rotor. In order to achieve the goal, the key point is to estimate the cost of a wind turbine. Another important point is to choose a set of suitable design variables. In the following subsections, the cost model and the design variables used in present optimization model are presented.

Cost model As the costs from operation and maintenance often can be accounted as a small percentage of the capital cost, the reduction of the capital cost becomes an essential task for designing wind turbines. Moreover, a well designed wind turbine with a low cost of energy always has an aerodynamically efficient rotor. Therefore, the rotor design plays an important role for the whole design procedure of a wind turbine. In the current study, we restrict our objective to the cost from the rotor. Thus the objective function is defined as:

( ) rotorCf x COE

AEP= = (16)

where COE is the cost of energy of a wind turbine rotor and is the total cost for producing, transporting, erecting a wind turbine rotor. In the current study, the fixed part of the cost for a wind turbine rotor is chosen to be 0.1. Therefore the total cost of a rotor, , is a relative value defined as

rotorC

rotorb rotorC

(1 )rotor rotor rotor rotorC b b w= + − (17) where is the weight parameter of the rotor. In the present study, the weight parameter is calculated from the chord and mass distributions of the blades. Supposing that a blade can be divided into cross-sections, is estimated as

rotorw

n rotorw

,

1 ,

ni i opt

rotori tot i or

m cw

M c=

×=

×∑ (18)

where is the mass of the i-th cross-section of the blade; is the averaged chord of the im ,i optc

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i-th cross-section of the optimized blade; is the averaged chord of the i-th cross-section of

the original blade;

,i orc

totM is the total mass of the blade. The power curve is determined from the BEM method. In order to compute the AEP, it is

necessary to combine the power curve with the probability density of a wind (i.e. the Weibull distribution). The function defining the probability density can be written in the following form

11( ) exp( ( ) ) exp( ( ) )ki

i iV V

f V V V ki

A A+

+< < = − − − (19)

where A is the scale parameter, is the shape factor and is the wind speed. In the current study, the shape factor is chosen to be k=2 corresponding to the Rayleigh distribution.

k V

If a wind turbine operates about 8700 hours per year, its AEP can be evaluated as 1

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1( ( ) ( )) ( ) 87002

N

i i i ii

AEP P V P V f V V V−

+=

= + × < < ×∑ 1+ (20)

where is the power at the wind speed of . ( )iP V iV

Design variables and constraints To obtain a reliable optimization of a wind turbine blade, the geometry of the blade needs to be represented as much as possible. This requires a great number of design variables. On the other hand, the selection of more design variables in the optimization procedure requires more calculation time. The design variables are often chosen to be the parameters controlling rotor shape, airfoil characteristics and regulate rotational speed and pitch angle. The rotor shape is controlled by the rotor diameter, chord, twist, relative thickness and shell thickness. The airfoil characteristics are the lift and drag dependency on the angle of attack. Based on a general chord distribution, a cubic polynomial is used to control the chord distribution. Because of the multiple distribution characteristics, a spline function is used to control the distributions of twist angle and relative thickness. The constraints of the design variables are

min maxi i iX X X≤ ≤

(21)

3,2,1=i

where is the lower limit and is the upper limit. miniX maxiX

As a usual procedure for optimization problems, we have one objective function and multiple constraints. To achieve the optimization, the fmincon function in Matlab is used.

Optimization Results

MEXICO experimental rotor As a first consideration of the optimization, the MEXICO rotor is chosen. The rotor has 3 blades with a radius of 2.25 m. In the BEM computations, 20 uniformly distributed blade elements are used. Since the part of a blade near the root mostly connects to the hub of the rotor and does not produce much energy, the optimization design is performed from the position at a radius of 0.45 m to the tip of the blade. In the optimization process, the lower limits for chord, twist angle and relative thickness are 0 m, 0° and 18%, respectively and the upper limits are 0.24 m, 20° and 100%, respectively. To reduce the computation time, 4 points along the blade are used to control

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the shape of the blade. The rotor diameter, the rotational speed of the rotor and the airfoil shapes remain unchanged.

Using the aerodynamic/aero-elastic code the torque and the thrust of the MEXICO rotor are calculated. Chord, twist angle and relative thickness are required to be decreasing from the starting point to the tip of the blade. With these constraints, the optimization computation converges after about 18 iterations. In Figures 2 and 3, the chord and twist angle distributions of the original and optimized MEXICO rotors are shown. From Figure 2, we can see that the chord keeps almost the original distribution on the most part of the blade except that it reduces a lot in the region near the blade tip. The reason is that this part of the blade does not contribute too much to the power and thus is not required to have a thick chord. The change in the twist angle (Figure 3) is not significant because of the constraint applied on the maximal thrust. The relative thickness distribution doesn’t change during the present optimization. The AEP of the optimized rotor is reduced about 0.8% whereas the cost of the optimized rotor has been reduced about 1.9%. Thus the cost of energy for MEXICO rotor can be reduced about 1.15%.

Figure 2: The chord distribution of the original and optimized MEXICO rotors.

Figure 3: The twist angle distribution of the original and optimized MEXICO rotors.

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Tjæreborg 2MW rotor As a second test case of the optimizations, the Tjæreborg 2MW rotor is chosen. The rotor has 3 blades with a radius of 30.56 m. In the BEM computations, 20 uniformly distributed blade elements are used. For the same reason that the part of a blade near the root mostly connects to the hub of the rotor and does not produce much energy, the optimization design is performed from a radial position at a radius of 6.46 m to the tip of the blade. In the optimization process, the lower limits for chord, twist angle and relative thickness are 0 m, 0° and 12.2%, respectively and the upper limits are 3.3m, 8° and 100%, respectively. To reduce the computation time, 4 points along the blade are used to control the shape of the blade. The rotor diameter, the rotational speed of the rotor and the airfoil shapes remain unchanged.

The torque and the thrust of the 2MW Tjæreborg rotor are calculated using the aerodynamic/aero-elastic code. Chord, twist angle and relative thickness are required to be decreasing from the starting point to the tip of the blade. With these constraints, the optimization computation converges after about 30 iterations. In Figures 4 and 5, the chord and twist angle distributions of the original and optimized Tjæreborg rotors are shown. From Figure 4, it is seen that the optimized blade has a much smaller value of chord in the region between 10 m and 23 m than the original rotor which has a linear chord distribution. At a radius of 15 m, the chord reduction reaches a maximum value of about 16 %. A likely explanation for this reduction is that both the axial and tangential forces in this zone are relatively small and only contributes slightly to the whole power production. From the position at a radius of 23 m to the position at a radius of 28 m of the blade, the chord keeps the original distribution. This is because the axial and tangential forces on this part of the blade contribute significantly to the power. Again the chord decreases significantly in the region near the tip like the case for the MEXICO rotor. The change in the twist angle is not very significant because of the constraint on the maximal thrust from which a bigger thrust would shorten the blade life and increase the cost. The AEP of the optimized rotor is reduced about 4% whereas the cost of the optimized rotor has been reduced about 7.1%. Thus the cost of energy for Tjæreborg rotor can be reduced about 3.4%.

Figure 4: The chord distribution of the original and optimized Tjæreborg 2MW rotors.

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Figure 5: The twist angle distribution of the original and optimized Tjæreborg 2MW rotors.

NREL 5MW virtual wind turbine As a final consideration of the optimizations, the NREL 5MW virtual rotor is chosen. The rotor is equiped with 3 blades of 63 m [11]. In the BEM computations, 40 uniformly distributed blade elements are used. The optimization design is performed from the position at a radius of 15 m to the tip of the blade. In the optimization process, the lower limits for chord, twist angle and relative thickness are 0 m, 0° and 18%, respectively and the upper limits are 5m, 15° and 100%, respectively. To reduce the computation time, 4 points along the blade are used to control the shape of the blade. The rotor diameter, the rotational speed of the rotor and the airfoil shapes remain unchanged as in the previous cases.

As in the previous case, the torque and the thrust of the 5MW NREL rotor are calculated using the aerodynamic/aero-elastic code. Chord, twist angle and relative thickness are required to be decreasing from the starting point to the tip of the blade. With these constraints, the optimization computation converges after about 36 iterations. In Figures 6 and 7, the chord and twist angle distributions of the original and optimized virtual NREL rotor are shown. From Figure 6, it is seen that the optimized blade has a much smaller chord value in the region between 30 m and 48 m than the original rotor which has an almost linear chord distribution. The chord reduction reaches a maximum value of about 8.2% at a radius of 40 m. The reason for this reduction is that both the axial and tangential forces in this zone are relative small. Between 48 m and 60 m on the blade, the optimized rotor has a slight smaller chord than the original rotor. From Figure 7, it is seen that the twist angle also is reduced significantly in the region between 25 m and 35 m. The change of twist angle is not very significant on the rest of the blade. The relative thickness distribution does not change during the actual optimization. The AEP of the optimized rotor is only reduced about 0.1% whereas the cost of the optimized rotor has been reduced about 2.7%. Thus the cost of energy for NREL virtual rotor can be reduced about 2.6%.

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Figure 6: The chord distribution of the original and optimized NREL 5MW virtual rotors.

Figure 7: The twist angle distribution of the original and optimized NREL 5MW virtual rotors.

Conclusions In this paper, an optimization model for wind turbine rotors, based on the structural dynamics of blades and a refined BEM theory with new tip loss correction has been developed. In the optimization, the objective of the model is the minimum cost of energy, which defined as the ratio between the annual energy production (AEP) and the cost of the rotor. The design variables are the shape parameters of the blade included chord, twist and relative thickness. Three different rotors with rated powers of 25kW, 2MW and 5MW respectively have been optimized by using the present optimization model. Through the optimizations, the overall cost of energy for each rotor was reduced about 1.15%, 3.4% and 2.6%, respectively.

Acknowledgements This work is supported by National Natural Science Foundation of China (No: 50775227) and Natural Science Foundation of ChongQing (No: CSTC, 2008BC3029). We would also thank

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China Scholarship Council for affording the scholarship. The European Commission’s fifth Framework Program (MEXICO, contract: ENK6-CT2000-00309) is acknowledged for the experimental data. The authors wish to give acknowledgements to Stig Øye and Martin Hansen for help with the FLEX code.

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