integrated aero-structural optimization design of pre … · integrated aero-structural...

Journal of Mechanical Science and Technology 30 (11) (2016) 5103~5113

www.springerlink.com/content/1738-494x(Print)/1976-3824(Online) DOI 10.1007/s12206-016-1028-2

Integrated aero-structural optimization design of pre-bend wind turbine blades†

Xiaofeng Guo1,*, Xiaoli Fu1, Huichao Shang1 and Jin Chen2 1School of Mechanical Science & Engineering, Zhongyuan University of Technology, Zhengzhou, Henan, 450007, China

2State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400030, China

(Manuscript Received January 26, 2016; Revised May 3, 2016; Accepted June 22, 2016)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In the optimization design of a pre-bend wind turbine blade, there is a coupling relationship between blade aerodynamic shape and

structural layup. The evaluation index of a wind turbine blade not only shows on conventional ones, such as Annual energy production (AEP), cost, and quality, but also includes the size of the loads on the hub or tower. Hence, the design of pre-bend wind turbine blades is a true multi-objective engineering task. To make the integrative optimization design of the pre-bend blade, new methods for the blade’s pre-bend profile design and structural analysis for the blade sections were presented, under dangerous working conditions, and consider-ing the fundamental control characteristics of the wind turbine, an integrated aerodynamic-structural design technique for pre-bend blades was developed based on the Multi-objective particle swarm optimization algorithm (MOPSO). By using the optimization method, a three-dimensional Pareto-optimal set, which can satisfy different matching requirements from overall design of a wind turbine, was ob-tained. The most suitable solution was chosen from the Pareto-optimal set and compared with the original 1.5 MW blade. The results show that the optimized blade have better performance in every aspect, which verifies the feasibility of this new method for the design of pre-bend wind turbine blades.

Keywords: Wind turbine; Pre-bend blade; Aero-structural design; MOPSO ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

With the increased diameter of wind turbine rotors, pre-bend design needs to be performed on large wind turbine blades to increase the allowable tip-deflection and reduce the blade weight. The study of pre-bend wind turbine blades can learn from the existing research results on non-pre-bend blades, which have been further studied from two aspects: Aerodynamic shape and structural layup.

In terms of the blade shape design, some researchers opti-mally designed the chord length and twist angle distributions of the blades so as to improve the power coefficient or Annual energy production (AEP) [1-3]. Sale, from the National Re-newable Energy Laboratory (NREL) of America, adopted the Bezier curve to express the distribution of blade chord lengths and twist angles [4], and developed a tool named HARP_Opt for the shape optimization of wind turbine blades. With a cer-tain structural layup, Xudong et al. constructed a structural dynamical model of the whole wind turbine to calculate the bending moment of blade root and the thrust on the tower [5]; they optimized the blade shape by taking the blade root loads as constraint conditions.

From the perspective of the blade structural analysis and op-timization design, Grifth and Ashwill presented a structural design model for the large wind turbine blades [6], and in their work a 10 MW wind turbine blade was optimized by using an analogy method. Jin et al. optimized the laminate thicknesses of each blade section by adopting a fluid-solid coupling method [7]; by using the method, they effectively reduced the mass of the blade on the premise of ensuring the blade struc-tural strength. Moreover, others have also made significant contributions to the blade aerodynamic and structural design [8-11].

The traditional optimization design of non-pre-bend blades as shown above is realized by using a serial design method in which the aerodynamic shape and structural layout were opti-mized independently. Therefore, it cost much time and effort. In recent years, some researchers have realized the coupling design relation between the aerodynamic shape and structural design and studied the non-pre-bend blades by using multidis-ciplinary or multi-objective optimization design method. With COE (Cost of energy) as the design objective, Ashuri et al. optimized a 5 MW wind turbine blade initially developed by the NREL of USA with the Multidisciplinary feasible optimi-zation algorithm (MDF) [7]. As a result, the cost of per kilo-watt hour was reduced by 2.3 %. Fischer et al. realized the optimization design of blades by locally changing the airfoils,

*Corresponding author. Tel.: +86 13526580367, Fax.: +86 371 62506618 E-mail address: [email protected]

† Recommended by Associate Editor Beomkeun Kim © KSME & Springer 2016

5104 X. Guo et al. / Journal of Mechanical Science and Technology 30 (11) (2016) 5103~5113

blade shapes and structural layout of an existing blade by us-ing the multi-objective optimization algorithm [12]. However, these optimization designs mainly focus on non-pre-bend blades, and currently fewer studies of integrated aerodynamic-structural optimization design on the pre-bend wind turbine blades have been reported.

In addition, to design a blade which has better compatibility with the wind turbine generator system, the steady control characteristics of blades not only have significant influence on the calculation of the AEP and ultimate loads of wind turbines, but also are the basis for designing the dynamic controller of wind turbines. Therefore, in the optimization design of blades, steady control strategies are supposed to be designed for each set of design variables.

To optimally design the aerodynamic shape and structural layout of the pre-bend wind turbine blades concurrently, and make the optimized blade have better performance on every aspect, the main works of this paper are as follows: A new design method for the blade pre-bend curves is presented. Based on the structure mechanics theory of composite materi-als, the method for calculating the normal and shear stresses of each blade cross-section was derived so as to analyze the structural strength of the blade effectively. To simulate the ultimate design loads of the blade accurately, the combined working conditions were presented for the blade optimization. Considering the steady control characteristics of the wind turbine, a multi-objective optimization design model for the aero-structural optimization design of the pre-bend blade was established. By applying a 1.5 MW blade as an example, the integrated optimization design was conducted for the blade aerodynamic shape and structural layout, the results and evaluations were also presented.

2. Model of the pre-bend blade

2.1 Aerodynamic shape

The whole blade can be stretched from many blade sections, and the profile of each section can be transformed from a stranded airfoil; this transformation process can be expressed as shown in Eq. (1) and Fig. 1. The airfoil 1 is a standard air-foil. By converting its coordinate origin to the pitch axial cen-ter, airfoil 2 is obtained. Then, according to the local chord length, airfoil 2 is transformed based on the equal ratio so as to acquire the airfoil 3. Afterwards, airfoil 4 can be acquired by rotating the airfoil 3 around the coordinate origin based on the local twist angle. To obtain the airfoil 5, the airfoil 4 needs to be pre-curved. The obtained airfoil 5 is still vertical to the z axis. To ensure the finally converted airfoil profile being ver-tical to the pre-bend curve, airfoil 5 is twisted to acquire the final blade section profile 6.

0 0 0 1 2 3 4 5[ , , ,1] [ , , ,1]i i iX Y Z X Y Z T T T T T= × × × × × (1)

where [X0, Y0, Z0] is the coordinate of the basic airfoil; [Xi, Yi, Zi] shows the coordinate of the blade section after each trans-

formation, while T1, T2, T3, T4 and T5 are the transformation matrixes; YB is the vertical distance from the alignment point to the z axis; α is the pre-bend twist angle of the airfoil.

Given span-wise distributions of the chord length, twist an-gle and relative thickness, the shape of the pre-bend wind turbine blade can be determined.

2.2 Structural layout

The wind turbine blade is mainly paved with five kinds of materials, including uniaxial fiberglass fabric (uniaxial lam-ina), biaxial fiberglass fabric (biaxial lamina), tri-axial fiber-glass fabric (triaxial lamina), blade surface foam (panel foam) and web foam (web foam). The structure of a typical blade section is shown in Fig. 2.

The upper and lower surfaces of the blade have similar laminate layout. From the leading edge to the trailing edge, the blade surface can be divided into six regions, including Lead-

-1 -0.5 0 0.5 1 1.5 2-0.5

0

0.5

1

x

y

12345

(a) Transformation from airfoil 1 to 5

(b) Transformation from airfoil 5 to 6

Fig. 1. Coordinate transformation of the blade section profile.

Fig. 2. Structural model of the wind turbine blade.

X. Guo et al. / Journal of Mechanical Science and Technology 30 (11) (2016) 5103~5113 5105

ing edge reinforce (LER), Leading edge panel (LEP), girder cap (CAP), Trailing edge panel (TEP), Trailing edge uniaxial reinforce (TEUD) and Trailing edge reinforce (TER). In addi-tion, there are left web (LWEB) and right web (RWEB) be-tween the upper and lower surfaces, which are used to support the upper and lower surfaces and transfer shear loads. In these regions, the laminate layout of the LEP and the TEP are shown in Fig. 2(c), while those of the CAP and TEUD are presented in Fig. 2(d), and those of the LER and TER are shown in Fig. 2(e), and webs are displayed in Fig. 2(f).

3. Blade loads and strength analysis

3.1 Blade loads calculation

In the whole service life, wind turbine blades have to ex-perience various complex working conditions. Therefore, precisely calculating the ultimate loads of each blade section is necessary for the optimization design of a wind turbine blade. Currently, the tools for calculating and analyzing the blade loads mainly include GH-Bladed, FAST and HAWC2. Among them, the GH-Bladed software has been widely rec-ognized in the wind turbine industry, and has favorable second development interfaces. Accordingly, we adopted the GH-Bladed software (version 4.2) to calculate the blade loads in the optimization design.

Some researchers take the steady operational loads to check the blade structural strength in the optimization [7, 13], where the blades are assumed to be working at the rated wind speed, and the speed and direction of the wind are not changed. In this method, the blade load is calculated conveniently and quickly; however, this merely considers the single special case among various wind turbine operation conditions. Obviously, the obtained design loads are much less than the ultimate blade loads in the whole service life, and also cannot reflect the structure dynamics of the blade.

To more precisely simulate the blade ultimate loads in the optimization, we chose several extremely dangerous working conditions (combined conditions for short hereinafter), and calculated the ultimate loads of each blade section by using the GH-Bladed software combine with Matlab programming. Afterwards, these loads were used to check the structural strength of the blade. Based on the GL2010 specification [13], the combined conditions are shown in Table 1. These condi-tions have generality as they are the most extreme ones in the working environment of wind turbines.

3.2 Structural analysis

Fig. 3 demonstrates the reference coordinate for blade sec-tion structural analyzing. In the coordinate system of X-Y, the X axis overlapped with the chord axis. Pc and Sc are the pitch axis center and the shear center of the blade section, sepa-rately; Mx, My and Mz present the ultimate bending moments, while Nx, Ny and Nz are the ultimate shear forces, and these loads can be calculated by using GH-Bladed software. The

ultimate flap moment M1 and the ultimate edgewise bending moment M2 of the blade section can be obtained by converting Mx and My to those in the coordinate X′-Y′ system. After calcu-lating the ultimate design loads on each blade section, we analyzed the structural strength of each blade section based on the structural mechanics of composite material [14, 15].

3.2.1 Axial normal stress

In the X′-Y′ coordinate the axial normal stresses on each blade section can be calculated by using Eqs. (2) and (3).

1 2

1 2

( , )[ ] [ ] [ ]

zM M Nx y y xEI EI EA

e = - + , (2)

zz ( , ) ( , )x y E x ys e= (3)

where [EI1], [EI1] and [EA] are the flap stiffness, edgewise stiffness and axial stiffness of each section, respectively, while E shows the tensile elastic modulus of the laminate.

3.2.2 Shear stress

The wind turbine blade is a typical box beam with multiple closed cells. Based on the shear flow theory, suppose that each closed cell is vertically cut apart along the incision shown in

Table 1. Basic parameters of extreme working conditions for wind turbine blade design [13].

DLC V0 (m/s) △V(m/s) Vend

(m/s) Yaw

error(deg) Comments

Dlc1.3a Dlc1.3b Dlc1.3c

11 11 11

15 15 15

25 25 25

-10 0 10

ECD rise time 10s.

Dlc1.3d Dlc1.3e Dlc1.3f

20 20 20

15 15 15

25 25 25

-10 0 10

ECD rise time 10s.


11 11 11

10 10 10

11 11 11

-10 0 10

EOG50, gust period14s.

Dlc1.6d Dlc1.6e Dlc1.6f

20 20 20

10 10 10

20 20 20

-10 0 10

EOG50, gust period14s.


37.5 37.5 37.5

Three dimensional von Karman turbulent wind

conditions.

-20 0 20

EWM50,rotor idling, blades

feathered.

Fig. 3. Reference coordinate of the blade section.


Fig. 4 simultaneously; an open profile system is formed. Un-der the effect of the shear forces Nx and Ny, the shear flow q of the open blade section can be obtained through Eq. (4).

y x

x yx y

N Nq S SJ J

= +% , (4)

where Sx and Jx present the static moment and inertia moment of the blade section profile to the x axis from the free edge to the calculated point, respectively; Sy and Jy are the static mo-ment and inertia moment of the blade section profile to the y axis separately. The shear flow q% is as large as the static moment and is irregularly distributed along the blade section profile.

After being introduced, the effect of shear flow q is bal-anced with the shear forces Nx and Ny, but the blade section does not satisfy the condition of moment equilibrium. To en-sure the continuous deformation at the incision, that is, the open section is not deformed under the effect of the torque, unknown shear flows q01, q02 and q03 are assumed to be ap-plied to the incisions of each cell. According to the theorem of conjugate shearing stress, corresponding shear flows are sup-posed to be produced on the contours of each closed cell, as presented in Fig. 4. The shear flows on the closed cells I, II and III are analyzed through Eqs. (5)-(7).

1 01 02q q q q= + +% , (5)

2 01 02 03q q q q q= + + +% , (6)

3 02 03q q q q= + +% . (7) Eq. (4) can be considered as the shear flow q produced on

the open sections when shear forces Nx and Ny are acting on the shear center. q01, q02 and q03 can be regarded as the shear flows generated under the joint effect of the torque con-structed by transferring the shear forces Nx and Ny from the imaged shear center to the practical one and the torque suf-fered by the original section. The shear force acting on each closed cell section is formed by the superposition of these two force conditions.

To determine the unknown shear flows q01, q02 and q03, a moment equilibrium equation is listed with the shear center of the blade section as the moment center.

1 01 2 02 3 032 2 2o sM q ds A q A q A qr= + + +ò % , (8)

where A1, A2 and A3 are the closed areas of the three closed cells; Mo is the sum of the torque of Nx and Ny to the shear center and calculated Mz.

Suppose the section is in a normal shape, the relative twist angles of each cell under the effect of external loads are iden-tical and all equal to the relative twist angle of the whole blade section. On this basis, three compatibility equations of tor-sional deformation can be constructed.

01 021 1 1 21

01 022 1 2 22

12 ( ) ( ) ( ) ( ) ( ) ( )1

2 ( ) ( ) ( ) ( ) ( ) ( )

eff eff eff

eff eff eff

q ds dsds q qA G s t s G s t s G s t s


q

q

-

-

é ù= + -ê ú

ê úë ûé

= - +êêë

ò ò ò

ò ò ò

%

%

Ñ Ñ

Ñ Ñ

03 2 3

02 033 2 3 33

)( ) ( )

1 .2 ( ) ( ) ( ) ( ) ( ) ( )

eff

eff eff eff

dsqG s t s


q

-

-

ìïïïïïí ùï - úï úûï é ùï = - +ê úï ê úë ûî

ò

ò ò ò%

Ñ Ñ

(9)

By combining Eqs. (8) and (9), q01, q02 and q03 as well as the

twist deformation angle θ of the blade section can be obtained. Geff(s) and t(s) are the equivalent shear modulus and the thickness of different laminated plates around the blade sec-tion, separately.

After obtaining the shear flows of q ,q01, q02 and q03, the total shear forces at each point of the blade section profile can be obtained; the blade section can be seen as made of many laminate plates around the blade section profile, and at each laminate, the mid-plane strains can be calculated by using the Classical laminate theory, which is shown in Eq. (10).

{ } { }* *

* *3é ù

= ê úë û

* 0

* *Α BN ε

M kB D, (10)

where *N and *M are the force and moment vectors acting on the composite laminate; *A , *B and *D are regulariza-tion stiffness matrix; the vector 0ε contains in-plane strain

0 ,xe 0ye and 0

xyg and *k contains out-plane curvature ,xk yk and y .xk

The lamina strains of each ply can be calculated by using Eq. (11).

0

0y y

xy0

.xx xy

xy xy

kz k

k

eee eg g

ì ü ì ü ì üï ï ï ï ï ï= +í ý í ý í ýï ï ï ï ï ïî þ î þ î þ

(11)

Then, the stress for each individual ply can be calculated

with Eq. (12).

Fig. 4. Shear flow on the blade section.


.x xy y

xy xy

s es et g

ì ü ì üï ï ï ï= é ùí ý í ýë ûï ï ï ïî þ î þ

C (12)

4. Pre-bend profile design

The deflection of the blade is changeful in the actual work-ing conditions as shown in Table 1 or GL2010, and thus it’s difficult to study the difference between the blade pre-bend shape and its actual deflections. For convenience, we studied the pre-bend theory at an ideal working condition, which is named steady operational conditions in the software GH-Bladed. The steady operational condition is an ideal working condition; it means the speed and direction of the wind are not changed, the blade is in a state of balance, and it has a fixed deflection shape. We analyzed the steady operational condi-tions of a 1.5 MW pre-bend wind turbine blade at wind speeds of 6.5 m/s and 11 m/s, respectively. The deformation shapes of the blade under these typical conditions are shown in Fig. 5.

As shown in Fig. 5, when the wind speed is 6.5 m/s, the pitch axis of the deformed blade is basically overlapped with the horizontal axis. When the wind reaches the rated speed of 11 m/s, the blade tip-deflection is up to 4.257 m. By studying the deformation curves of the blade, following inference can be obtained based on the Euler- Bernoulli beam theory. At 6.5 m/s, the deformation shape of a non-pre-bend blade which has the same chord length, twist distribution and structural layout with the original pre-bend blade, is supposed to be approxi-mately symmetric with the shape of the pre-bend blade along the horizontal axis. That is, the mirror image line of this non-pre-bend blade at a typical wind speed along the horizontal axis can be considered as the basis for designing the pre-bend profile of the blade.

To verify this inference, we established a non-pre-bend blade which has similar chord, twist distribution and layer structure with the original 1.5 MW pre-bend blade, as shown in Fig. 6. When the wind speed is 6.5 m/s, the blade tip-deflection of the non-pre-bend blade is 1.75 m, which is basi-cally identical with the original pre-bend value of the blade tip,

1.72 m. In addition, the deformation curve of the middle and posterior portion of the blade (R/2~R where R is the length of the blade in the wind rotor plane) is basically consistent with the mirror image line of the original pre-bend curve along the horizontal axis. At the root of the blade, the deformation curve of the non-pre-bend blade presents some difference with the mirror image line of the original pre-bend profile. However, based on practical experience, since the front R/3 section of the blades is generally not need to be pre-bend, on the whole, the deformation curve of the non-pre-bend blade is signifi-cantly similar to the mirror image of the original pre-bend profile.

Therefore, we propose a new method for expressing the pre-bend profile of a wind turbine blade.

( ) 0, / 3( ) [( / 3) / (2 / 3)] *a

max

b x x Rb x x R R b

= £ì üï ïí ý

= -ï ïî þ, (13)

where a is the index of the power function for pre-bend the profile; bmax presents the pre-bend value of the blade tip, while b(x) shows the pre-bend offset of each section along the blade span-wise. The bending deformation appears from about R/3 of the blade from the blade root. Based on the characteristics of the power function, the pre-bend profile presented in Eq. (13) is tangent to the horizontal axis (pitch axis) at the starting point of the pre-bend process. Therefore, it is ensured that the blade can realize smooth transition in the pre-bend transition section. Different pre-bend profiles can be acquired by chang-ing the values of a and bmax in Eq. (13).

The deformation curve of the non-pre-bend blade at a wind speed of 6.5 m/s is fitted by using Eq. (13), thus obtaining the fitting parameters a and bmax of 2.16 and 1.79 separately. The acquired mirror image line of the fitting curve along the x axis is the designed pre-bend profile. Comparing the pre-bend profile of the original 1.5 MW blade with the newly designed one, it can be found that the maximum relative deviation at

0 5 10 15 20 25 30 35 40-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Blade span-wise length(m)

The

blad

e de

form

atio

n(m

)

The original blade shapeThe deformed blade shape(6.5m/s)The deformed blade shape(11m/s)

Fig. 5. Comparison of blade shapes under typical wind speeds.

0 5 10 15 20 25 30 35 40-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


The

blad

e de

form

atio

n(m

)

Pre-bend curve of the original bladeDeformed shape of un-pre-bend bladeFitting curve of the deformed shapeThe mirror image of the fitting curve

Fig. 6. Generation of the pre-bend curve of the blade.


each point along the blade span-wise is 4.1 %, which is in a small range. Therefore, the feasibility and correctness of the method for designing this pre-bend profile are validated. 5. The optimization model

5.1 Parametric expressions of blades

5.1.1 Positions of standard airfoils The DU airfoil series are used in the blade optimization de-

sign [7]; the position of standard airfoils along the blade span-wise is shown in Table 2.

5.1.2 Parametric design of blade shapes

The shapes of every blade section depend on the distribu-tion of chord lengths, twist angles and relative thicknesses along the blade span-wise. In this paper, multi-order Bezier curve is used to express these distributions. Thereunto, the chord lengths distribution is expressed by using the tenth-order Bezier curve which uses the ordinates of the fourth to the ninth control points as design variables as c1~c6. The diameter of blade root is set to 1.8 m, and the chord length distribution of first half part of the blade is determined by c1. The distribution of twist angles from the maximum chord position of the blade to the blade tip is presented by the sixth-order Bezier curve, as setting the twist angle of blade root to 12°, and the twist angle of the position with a dis-tance of 1m to the blade tip is 0. We took the second to the fifth control points as design variables which are shown as β1~β4 [4].

5.1.3 Parametric design of blade structural layout

By researching the structural layout of the wind turbine blades, as shown in Fig. 2, we took the thickness of uniaxial lamina of CAP, triaxial lamina of the whole blade surface, and uniaxial lamina of TEUD as design variables.

To use a limited number of key variables to express the change of lamina thickness of the whole blade, a newly struc-tural design model is established in this paper, which is shown in Fig. 7. Cmax is the position of the maximum chord length (with a 35 % relative thickness) along the blade span-wise; Lc

is the length of a blade segment where the blade sections have the same and maximum thickness of uniaxial lamina in the CAP along the blade span-wise, and the length is set as one-third of the blade length; the distance of the segment’s left end

to the point of Cmax is Lc/4. While for the TEUD, there is also a blade segment which has a constant and maximum thickness of uniaxial lamina along the blade span-wise, and the length of this segment is Lc/3, and the left end of this segment has the same x-coordinate with the CAP’s. For the triaxial laminas of the whole blade surface along the blade span-wise, the thick-ness at the blade handle is fixed, while that from the end of the blade handle to the position of maximum chord length in the first half of the blade expressed by two straight lines inter-sected at point K1 , and it has a distance of Lc/4 to the Cmax. In this paper, the Lc is set as 0.35 times of the blade length, and the maximum thicknesses of uniaxial lamina in the CAP and TEUD are set as th1 and th2, respectively; that of the thickness at the turning point K1 of the triaxial lamina in the blade sur-face is set as th3. In addition, we set two intersection points K2

and K3 for the uniaxial lamina thickness of the CAP and TEUD; their thicknesses were set as th4 and th5, respectively. In this way, the structural layout of the whole blade is deter-mined.

5.1.4 Pre-bend profile

The pre-bend profile of the blade is designed by using the method proposed in Sec. 4. If the shape and the structural layout of the designed blade are known, the deformation of the un-pre-bend blade, which has the same chord and twist distri-bution, structural layout is calculated under steady-state opera-tion loads by using the GH-Bladed software. Then, the defor-mation shape is fitted using Eq. (13). The mirror image line of the obtained fitting curve to the x axis is the pre-bend profile of the designed blade.

5.2 The design variables

The design variables are shown in Table 3; the blade chord length and twist distribution are expressed with Bezier curves; c1~c6 are control point values of blade chord length distribu-tion; β1~β4 are the control point values of twist angle distribu-

Table 2. Distribution of airfoils along the blade span-wise. μ = r/R Thicknesses Airfoils Boundary conditions

1/R 100 % Round airfoils Re = 2.0×106, Ma = 0.15

0.16 40 % DU400 Re = 3.5×106, Ma = 0.15

0.21 35 % DU350 Re = 4.0×106, Ma = 0.15

0.30 30 % DU300 Re = 5.0×106, Ma = 0.15

0.45 25 % DU250 Re = 5.0×106, Ma = 0.15

0.66 21 % DU210 Re = 5.0×106, Ma = 0.15

0.89 18 % NACA64618 Re = 3.5×106, Ma = 0.15

0 5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

Cmax

Lc/4Lc

Lc/3

Lc/4K1

K2

K3


Lam

ina

thic

knes

s(m

)

Triaxial lamina of blade surfaceUniaxial lamina of CAPUniaxial lamina of TEUD

Fig. 7. The structural design model of the blade optimization.


tion, and the detailed description of the control points can also be seen in Sec. 5.1 and Ref. [4]. In addition, th1~ th5 are con-trol point values for blade structural design. 5.3 The objective function

We took the maximization of the AEP, the minimization of the blade mass and the minimization of the thrust to the hub as the design objective, which is shown as follows.

1( ) /intF X AEP AEP= (14)

2 ( ) massF X Blade= (15)

3( ) hubF X Thrust= (16)

1 2 3( ) min( ( ), ( ), ( ))F X F X F X F X= (17)

where X is the vector of design variables as shown in Table 3; AEP is the annual energy production of the wind turbine; Blademass is the overall mass of the blade; Thrusthub shows the ultimate thrust on the hub. To make the change of F1(X) have the same tendency with F2(X) and F3(X), the AEPint is set as 2E+07 in F1(X), which is larger enough than the possible value of a 1.5 MW wind turbine.

5.4 Constraint conditions

(1) The structural strength of the blade is checked based on the criterion that the triaxial lamina stress on the outermost layer of the blade does not fail. The failure criterion is ex-pressed as Eqs. (18) and (19).

s s£ é ùë û , (18)

t t£ é ùë û , (19)

where s and t are the normal and shear stress calculated for each position of every blade section, respectively, and they can be obtained by Eq. (12); sé ùë û and té ùë û show the allow-able tension and shear stress of the triaxial composite material, and their values are set as 65.5 and 77.4 MPa [6].

(2) To avoid resonance, the first-order flapwise frequency ω1f and the first-order edgewise frequency ω1e of the designed blade are required not to be overlapped with rotational fre-quency of the wind turbine, which is three-times the rotation frequency of the wind turbine rotor.

1 13 ; 3f eP Pw w¹ ¹ , (20)

60rp n= , (21)

where nr is the speed of the wind turbine rotor. In this paper, the speed of the rotor is set in a range of 12.14 rpm~21.43 rpm with the transmission ratio of the gearbox as 70. Thus, the rotation frequency of the rotor P is in a range of 0.20 Hz~ 0.36 Hz.

(3) To avoid the blade tip crushing against the tower, the maximum blade tip-deflection has to be limited. In the sta-tionary state, the distance between blade tip to the tower can be expressed as Eq. (22) [13].

hub tilt cone tower,tipsin( ) / 2stX X R da a= + - - (22)

where Xhub is the distance from the hub center to the axis of the tower, R is the length of the blade, αtitle is the rotor tile angle, αcone is the pre-cone angle, dtower,tip is the diameter of the tower where overlap with the blade-tip. For the original 1.5 MW blade, the stX is 10.3 m, and the maximum blade-tip deflection ,maxtipX under all the working conditions is 60 % of stX ; thus, in the optimization, the constraint of the blade deflection is expressed as Eq. (23).

0.6 .tip,max stX X£ (23)

5.5 Optimization algorithm

Because the standard particle swarm algorithm has high precision and fast convergence rate, thus, we took a self-making MOPSO program by combining Pareto strategy with standard PSO algorithm for the blade optimization.

For the objective function 1 2 3min( ( ), ( ), ( ))f x f x f x , if x* is one of the feasible solution, it needs to meet the following two conditions simultaneously: If and only if there is no feasible solution x, ( ) ( *), {1,2,3}i if x f x i£ Î and there is at least one

( ), {1,2,3}jf x j Î , make ( )jf x < ( *)jf x . As there are multi-ple optimal solutions for the multi-objective optimization, which form a Pareto optimal set, designers can select the most suitable one from this set according to the specific needs.

Fig. 8 shows the flow chart for optimally designing the

Table 3. Design variables and description. No. Design parameters/unit Symbols Ranges

1 Blade length /(m) R 38~39

2 Chord length of control point 4/(mm) c1 3.6~3.9






8 Twist angle of control point 3/(mm) β1 9.0~10.0



11 Twist angle of control point 6 /(mm) β4 -0.2~0.2

12 Maximum thickness of uniaxial lamina in CAP /(mm) th1 35~65

13 Maximum thickness of uniaxial lamina in TEUD /(mm) th2 10~25

14 Thickness of triaxial lamina in the whole blade surface at point K1/(mm) th3 0~8

15 Thickness of uniaxial lamina in CAP at point K2/(mm) th4 15~40

16 Thickness of uniaxial lamina in TEUD at point K3/(mm) th5 5~12


blade, the velocity and position of the ith particle can be ex-pressed by using Eqs. (24) and (25).

( ) ( )1

1 1 2 1- -k k k k k ki i i i g iw c r p x c r p xu u+ = + + (24)

1 1k k ki i ix x u+ += + (25)

where w，c1，c2，r1，r2 are weight coefficients, +1,k k

i ix x and +1,k k

i iu u are the position and velocity of the ith particle in the kth generation and (k+1)th; pi, pg the the individual and global best solutions, respectively.

The external repository (ERP) for storing best solutions is constructed. For the particles updated in each iteration, Pareto decision is conducted between the current one and each solu-tion in ERP. The particles in the ERP are sorted by the crowd-ing distance value; the pg is random selected from the top of the 10 % particles in the sorted queue, and the size of the ERP is kept not to exceed the maximum value [16]. Through con-tinuously maintaining and updating the ERP, the best solu-tions can finally be obtained [17, 18]. The constraint condition is conducted through the penalty function method [17]; the iteration is finished as the minimum crowding distance value is less than the tolerance and the number of the solutions is reached to the maximum size of the ERP.

In the optimization design of blades, due to the change of the shapes and structural layout of blades, static control pa-rameters for the new designing blade need to be designed in the GH-Bladed software so as to make the blade matching with the generator system and accurately calculate the ultimate loads. These parameters include the optimal tip-speed ratio λopt

and steady-state control parameter kopt [19]. In the design of the static control parameters for the wind turbine, the pitch angle of the blade at the rated wind speed is set as 0°. Under such circumstance, λopt can be programmed and selected ac-cording to the following principles, which aim to find larger maximum power coefficient CPmax, flatter top of the CP-λ curve and smaller λopt. Afterwards, kopt is calculated based on the following equations [18].

5

3 3

1 1( , )2opt P opt

opt

K R CG

pr l bl

= (26)

2opt gT K w= (27)

where T and ωg are the torque and speed of the electric genera-tor, respectively; R and β are the blade length and the pitch angle, separately, while G is the transmission ratio of the gearbox.

Before reaching the rated wind speed, the controlling torque of the wind turbine when the blade works at the optimal λopt is calculated according to Eqs. (26) and (27). When the blade does not work at the optimal λopt, the speed of the electric gen-erator of the wind turbine is at the boundary condition of the maximum or minimum speed owing to the limitation of the speed range of the electric generator [19]. At this time, the controlling torque of the electric generator could be obtained according to Eqs. (28) and (29).

= g RGv

wl , (28)

25

3 3

1 ( , )2

gPT R C

Gw

pr l bl

= . (29)

After reaching the rated wind speed, the contr olling torque

of the electric generator is equal to the rated torque of the wind turbine.

Thus, the sub process in the blade optimization algorithm shown in Fig. 8 can be described with Fig. 9; its main func-tions are generation of blade pre-bend curve and wind turbine

Fig. 8. Flow chart of the optimization algorithm.

Fig. 9. Flow chart of the sub process.


basic control strategy, calculating AEP, Blademass, blade-tip deflection and ultimate loads.

6. Optimization results

The parameters used in the optimization algorithm were presented as follows: the inertia weight w was set as 0.9; the learning factors C1 and C2 were both set as 0.5; r1 and r2 were both set as random value between 0 and 1; the dimension of the variables and the size of the population were valued as 16 and 40; the maximum time of iteration in this algorithm was set as 200. After 15.2 hours of calculation, the Pareto optimal front was obtained as shown in Fig. 10.

The optimal set contains 100 optimized blades basically dis-tributed on the panel of the optimal front, suggesting a favor-able optimization effect. Three optimal solutions, named blades A, B and C, are selected from the optimal set. The blades A and B have the minimum and the maximum hub thrust among all the optimal solutions, respectively; while the blade C shows the largest AEP on the premise of not increas-ing the maximum hub thrust than the original 1.5 MW blade. Table 4 displays the comparison of the performance parame-ters of these three blades with those of the original blade.

Figs. 11 and 12 present the comparison of the chord length and twist angles distributions of these three blades with that of the original blade. To avoid the occurrence of flutter in the blade tip, the twist angle of this section has a sudden increase which can be seen in Fig. 12. As can be seen from Fig. 11 and Table 4, compared with the original blade, the blade A pre-sents an obviously reduced blade length and chord lengths;

meanwhile, its thrust to the hub and AEP reduces by 6.50 % and 5.94 %, respectively; the blade length, thrust to the hub and the AEP of the blade B increase by 2.2 m, 3.46 % and 4.75 %; on the premise of not increasing the thrust to the hub, the AEP of the blade C increases by 1.66 %. Comprehensively considering all of the performance indexes, we took blade C as the optimally designed blade.

Fig. 13 illustrates the comparison of the designed lamina

Table 4. Comparison of the performance parameters of the optimized blades with those of the original blade.

Blades AEP /(kW·h)

Largest thrust /(kN)

Blade mass /(kg)

Maximum chord /(m)

Blade length /(m)

Original 4.21e+06 237.1 5,620.5 3.00 38.0

A 3.96e+06 221.7 5,512.2 2.64 37.3

B 4.41e+06 245.3 5,889.4 3.11 40.2

C 4.28e+06 236.9 5,568.2 2.96 38.6

4.6

4.8

5

5.2

5.4 4600 4800 50005200 5400 5600

5800 6000

2

2.2

2.4

2.6

2.8

x 105

Original blade

B

A

C

F2(X)

F1(X)

F 3(X)

Fig. 10. Figure of the optimal solution set.

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5


Chor

d le

ngth

(m)

The original bladeBlade ABlade BBlade C

Fig. 11. Chord length distributions of the optimized blades.

5 10 15 20 25 30 35 40 45-2

0

2

4

6

8

10

12

14


Twis

t ang

le(°

)

The original bladeBlade ABlade BBlade C

Fig. 12. Twist angle distributions of the optimized blades.

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Original triaxial laminaOptimized Triaxial lamina

Original CAP uniaxial laminaOptimized CAP uniaxial laminaOriginal TEUD uniaxial laminaOptimized TEUD uniaxial lamina


Lam

ina

thic

knes

s(m

)

Fig. 13. Comparison of lamina thickness of the blades.


thickness of the optimally designed blade (the blade C) with that of the original blade. As shown in the figure, compared with the original blade, the maximum thicknesses of the uni-axial lamina in CAP and the TEUD of the blade C reduce by 7.36 % and 6.65 %. In contrast, the thickness of the triaxial lamina at the key point K1 in the first half of the blade has a little increase. The first-order flapwise frequency and the first-order edgewise frequency of the optimized blade satisfy the design requirements, showing that the optimized blade has favorable structural dynamics performance.

The stresses of the optimally designed blade were analyzed and calculated. It was found that the section which is 5.4 m to the blade root surface presents the largest shear stress of 65.3 MPa, while the section which is 6.8 m to the blade root sur-face shows the largest tensile stress, namely, 77.4 MPa, as presented in Fig. 14. Neither the maximum shear stress nor the tensile stress of the blade is larger than the allowable safety values of 65.5 MPa and 77.4 MPa, indicating that the opti-mized blade meets the design requirement of the strength. Before the optimization, the maximum tip pre-bend value bmax

of the original blade was 1.72 m, while that bmax of the opti-mized blade C was 1.76 m, and the exponential parameter a of the pre-bend profile was 2.08. In addition, the maximum tip-deflection of the original blade at steady wind speed of 11 m/s was 5.649 m, while that of the optimized blade C was 5.752 m. The above comparisons show that though the optimized blade is longer, the thickness of the uniaxial lamina in CAP section

reduces, while for the increase of bmax, the overall mass and loads of the optimized blade are guaranteed to be basically identical with those of the original blade.

7. Conclusions

The methods of pre-bend curve design and blade structural analysis were derived for pre-bend blade optimization. Cou-pling a MATLAB program and the GH-Bladed software leads to a powerful tool suitable for simulating the blade ultimate design loads under combined working conditions. A multi-objective optimization technique which can realize the inte-grated design of the aerodynamic shape and composite layout of the pre-bend blades is presented. Results obtained on the 1.5 MW blade show that the 100 blades in the Pareto-optimal set have different appearance and performance, so designers can select the most suitable blade according the matching requirement of the wind turbine.

With the increase of the blade length, the twisting deflection is becoming more and more obvious, especially for wind tur-bines working in lower wind speed conditions. The twisting deflection may have complicated influences on the blade aerodynamic, loads and structural performance. Hence, the next step could be the blade optimization design considering aero-elastic deflection.

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (No: 51175526) and Fundamental research project of Education Department of Henan province (No: 16A460041).

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x(m)

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(a) Normal stresses on the section of 6.8 m to the blade root surface

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Xiaofeng Guo received his Ph.D. in Mechanical Engineering at Chongqing University in 2015. He is a Lecturer in the School of Mechanical Science & Engineering at Zhongyuan University of Technology.

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