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Structural optimization procedure of a composite wind turbine blade for reducing both material cost and blade weight a
b
Weifei Hu , Dohyun Park & DongHoon Choi
c
a
Mechanical Engineering, University of Iowa, Iowa City, USA
b
Graduate School of Mechanical Engineering, Hanyang University, Seoul, South Korea
c
The Center of Innovative Design Optimization Technology, Hanyang University, Seoul, South Korea Version of record first published: 28 Jan 2013.
To cite this article: Weifei Hu , Dohyun Park & DongHoon Choi (2013): Structural optimization procedure of a composite wind turbine blade for reducing both material cost and blade weight, Engineering Optimization, DOI:10.1080/0305215X.2012.743533 To link to this article: http://dx.doi.org/10.1080/0305215X.2012.743533
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Engineering Optimization, 2013 http://dx.doi.org/10.1080/0305215X.2012.743533
Structural optimization procedure of a composite wind turbine blade for reducing both material cost and blade weight Downloaded by [University of Iowa Libraries] at 07:27 19 February 2013
Weifei Hua , Dohyun Parkb and DongHoon Choic * a Mechanical
Engineering, University of Iowa, Iowa City, USA; b Graduate School of Mechanical Engineering and Center of Innovative Design Optimization Technology, Hanyang University, Seoul, South Korea c The
(Received 12 May 2012; final version received 28 September 2012) A composite blade structure for a 2 MW horizontal axis wind turbine is optimally designed. Design requirements are simultaneously minimizing material cost and blade weight while satisfying the constraints on stress ratio, tip deflection, fatigue life and laminate layup requirements. The stress ratio and tip deflection under extreme gust loads and the fatigue life under a stochastic normal wind load are evaluated. A blade element wind load model is proposed to explain the wind pressure difference due to blade height change during rotor rotation. For fatigue life evaluation, the stress result of an implicit nonlinear dynamic analysis under a time-varying fluctuating wind is converted to the histograms of mean and amplitude of maximum stress ratio using the rainflow counting algorithm Miner’s rule is employed to predict the fatigue life. After integrating and automating the whole analysis procedure an evolutionary algorithm is used to solve the discrete optimization problem. Keywords: composite materials; evolutionary algorithm; fatigue life prediction; horizontal axis wind turbine blade; structural optimization
1.
Introduction
In order to reduce electrical energy production costs, the size of commercial wind turbines has grown considerably during the past decade. Currently, the largest wind turbine installed is the Enercon E-126, which is officially rated at 6 MW and has a rotor diameter of 126 m. However, as the size of the wind turbine rotor increases, the structural performance, durability and dynamic stability requirements tend to become more and more challenging to meet. The two main structural performance requirements for wind turbines are quoted by Grujicic et al. (2010) as: • sufficient flapwise bending strength to withstand highly rare extreme static and dynamic loading conditions (e.g. 50-year return-period gust or a short-term extreme operating gust), • sufficient flapwise bending stiffness to ensure that a minimal clearance is maintained between blade tip and the turbine tower at all times during wind turbine operation. To satisfy the above two extreme operating conditions, several methods have been proposed to evaluate the extreme structural performances of horizontal axis wind turbine (HAWT) blades. *Corresponding author. Email:
[email protected]
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Veers and Butterfield (2001) investigated the load uncertainty in the extreme load estimation for wind turbines. Todoroki and Kawakami (2007), Hu et al. (2012) and Grujicic et al. (2009, 2010) analysed the blade in the horizontal position only and applied a uniform wind pressure to the fixed blade. However, this loading condition is too simplified to reflect the height dependency of wind speed as described in the International Electrotechnical Commission (IEC) Standard (2005). As the rotor rotates, the blade experiences different wind loads at different locations owing to the wind shear effect in the vertical direction. To represent the flapwise bending moment at the blade root, Ronold and Larsen (2000) proposed a polynomial expansion, which required a significant number of measured wind climate data and flapwise bending moment records. Another probabilistic approach, called the environmental contour method (Saranyasoontorn and Manuel 2006), was proposed to improve the extreme design load estimates. However, it is usually difficult to obtain a large number of measured wind data or blade stress/deflection responses in the blade design stage. For modelling a blade loaded by height-dependent wind loads in the design stage, a new blade element wind load model, called the BEWiL model, is proposed in this study.An extreme operating gust (EOG) according to IEC extreme wind conditions is applied to the BEWiL model. As a result, different blade elements are subject to different extreme dynamic wind pressures. Then, structural analyses under BEWiL are performed to evaluate the flapwise bending strength and stiffness of the blade. Wind turbines, by their very nature, are subject to a great number of cyclic loads. A turbine with approximately 30 rpm operating for 4000 hours per year would experience more than 108 cycles over a 20-year lifetime (Manwell, McGowan, and Rogers 2009). Many researchers have investigated fatigue life prediction regarding blades. Grujicic et al. (2009) applied average wind velocity plus a time-varying component of wind-induced loading to a blade finite element model and integrated the rainflow counting algorithm, the Goodman diagram and Miner’s rule to predict the fatigue life. Mahri and Rouabah (2002) employed statistically distributed wind speed for a site in use. Have (1992) gave the final definition of two standardized fatigue loading sequences for wind turbine blades. Rajadurai et al. (2008) adopted a static load and a linear S-N curve to calculate allowable cycles corresponding to the peak working stress. Other kinds of fatigue load have also been studied (Shokrieh and Rafiee 2006; Kong, Bang, and Sugiyama 2005; Nada and Flay 1999; Hahn et al. 2002; Ronold and Christensen 2001). These fatigue life prediction methods are all based on the cumulative damage theory called Miner’s rule. For the wind turbine blade fatigue analyses above, two categories of wind load were used: long-term statistical experimental loads and short-term computer-based simulative loads. In this study, a dynamic short-term (10 minutes) fluctuating wind load is generated according to Law’s time-varying wind load model (Law, Bu, and Zhu 2005) using MATLAB. For aerodynamic wind load calculation, state-of-the-art practical methods, e.g. blade element momentum (BEM) and Navier–Stokes solvers are surveyed by Hansen et al. (2006) A quadratic relationship between wind pressure and wind speed is applied in this article for simplicity. After an implicit nonlinear analysis under the generated fluctuating load, the dynamic stress result is extracted and used to predict the fatigue life based on the rainflow counting algorithm and Miner’s rule. Based on the flapwise bending stress and tip deflection analyses under BEWiL and the fatigue life prediction described above, the thicknesses, fibre angles and material type of composite layers are to be optimally determined in order to simultaneously minimize the material cost and weight of a HAWT blade while satisfying the design constraints on stress ratio, tip deflection and fatigue life in this study. The remainder of this article includes four sections. In Section 2, a BEWiL model is proposed and applied to the structural analysis under an EOG load. Section 3 describes the fatigue life prediction of this study. A discrete optimization problem for the structural design of a 2 MW wind turbine composite blade is formulated and solved in Section 4. Finally, results and conclusions are given in Section 5 and 6, respectively.
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2.
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Structural analysis under an extreme operating gust load
To evaluate whether or not the current design satisfies the constraints on sufficient flapwise bending strength and minimal clearance between the blade tip and turbine tower, an EOG is applied on the blade with BEWiL modelling. The blade configuration and the properties of the composite materials will be described below. A blade finite element model is built and the structural responses are analysed using SAMCEF (SAMTECH 2007).
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2.1.
Blade modelling
In general, HAWT blades are made of glass fibre reinforced plastic (GFRP) owing to its light weight, high strength and stiffness and superior fatigue and corrosion properties. However, as blade dimensions become larger, GFRP may not be appropriate for the alternative blade designs, e.g. slender planform or thin foils, which leads to the employment of lighter and stronger materials such as carbon fibre reinforced plastic (CFRP). Since the material cost of CFRP is much higher than that of GFRP, the arrangement of GFRP and CFRP in composite layers has a significant effect on material cost. The unit volume cost of CFRP was reported to be 10 times more expensive than that of GFRP by Todoroki and Kawakami (2007). The configuration of the CFRP/GFRP hybrid blade structure is shown in Figure 1. The blade is 40 m long and has a 2 MW rated power. The blade is composed of three parts: root, skin and shear web. The number of layers in the root, skin and shear web is 20, 20 and 10, respectively. The shear web prevents shear deflection and has the biggest influence on the bending modes of the blade. The twist distribution of the airfoils along the blade is ignored for simplicity. In this study, the materials corresponding to root, skin and shear web are specified as CFRP, GFRP, and hybrid of CFRP and GFRP, respectively. CFRP is assigned for the root since it is the part with high stresses, and GFRP for the skin since it is the part
Figure 1. Configuration of the blade model: (a) shading; (b) wireframe. GFRP = glass fibre reinforced plastic; CFRP=carbon fibre reinforced plastic.
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Table 1.
Material properties.
E (GPa) ν σxt (MPa) σxc (MPa) σyt (MPa) σyc (MPa) σzt (MPa) σzc (MPa) σxy (MPa) σyz (MPa) σxz (MPa) ρ (kg m−3 ) Cost (1 m−3 )
GFRP
CFRP
56 0.17 1447 1447 51 206 51 206 93 68 68 1500 1
90 0.042 1500 1500 40 246 40 246 93 68 68 1250 10
Note: GFRP = glass fibre reinforced plastic; CFRP = carbon fibre reinforced plastic.
with relatively low stresses and large material volume. The materials used for the shear web are the hybrid of CFRP and GFRP to make a balance between cost and strength. Material properties of GFRP and CFRP are shown in Table 1. Geometric and meshed models are built in SAMCEF. 2.2. Structural analysis using the BEWiL model To obtain the extreme stress and tip deflection, an EOG was applied on the blade surface (IEC 2005). If the blade satisfies the maximum stress and the tip deflection constraints under such an extreme condition, the blade is likely to be reliable under all actual conditions. The main relationship among time, height and wind speed is defined in Equation (1): V (z, t) =
V (z) − 0.37Vgust sin(3π t/Text )(1 − cos(2π t/Text )) for 0 ≤ t ≤ Text V (z) otherwise
(1)
where V (z) denotes the average wind speed as a function of height z above the ground and the power law exponent α shall be assumed to be 0.2; Text = 10.5 s; Vhub is the wind speed at hub height zhub ; and Vgust is the hub height gust magnitude that will be given for the standard wind turbine classes. V (z) and Vgust are defined by Equations (2) and (3), respectively: V (z) = Vhub (z/zhub )α Vgust = Min 1.35(Vel − Vhub ), 3.3
σ1 1 + 0.1(D/1 )
(2) (3)
where σ1 is the turbulence standard deviation; 1 is the turbulence scale parameter; D is the rotor diameter; and Ve1 is the extreme wind speed with a recurrence period of 1 year, for a steady extreme wind model. Ve1 is relative to Ve50 , which is the extreme wind speed with a recurrence period of 50 years. The formulations of these symbols are given in Equations (4)–(7): σ1 = Iref (0.75Vhub + b) 0.7zhub zhub ≤ 60m 1 = 42m zhub > 60m
(4) (5)
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Figure 2. A rotating rotor with divided blades.
Vel (z) = 0.8Ve50 (z) z 0.11 Ve50 (z) = 1.4Vref zhub
(6) (7)
where b = 5.6 m s−1 and Iref and Vref are, respectively, the turbulence intensity and reference wind speed, the values of which are determined by the category of wind turbine. Here the blade could be mounted on a class IA rotor, thus Iref = 0.16 and Vref = 50 m s−1 (IEC 2005). When the rotor rotates, the blade height changes accordingly. To eliminate the height z, rotor speed ω is introduced in Equation (8). The blade is divided into 20 different elements, as shown in Figure 2. It is assumed that the wind applied on one blade element is uniform but differs from any others. Even for the same blade element, the wind applied will change as the rotor rotates. As a result, the whole blade is subject to ‘rotational and dynamic’ wind loads, although the blade is fixed. This is named BEWiL modelling. The final wind pressure P is calculated by Equation (9), where ρ, V and CP represent the air density, wind speed and pressure coefficient, respectively: 2π ω z = zhub + r sin t (8) 60 P=
1 2 ρV Cp 2
(9)
Two typical wind pressure results obtained by the single wind load analysis and BEWiL are compared in Figure 3. Using the BEWiL modelling, it is clearly observed that the wind pressure fluctuation on the blade tip is larger than that on the root. The single EOG wind pressure and BEWiL EOGs were separately applied on two blade models with the same initial design. Figure 4 shows that the most peaks of both tip deflection and maximum stress ratio of the BEWiL model are
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Figure 3. Wind pressure comparison: (a) single extreme operating gust (EOG)wind pressure; (b) EOG wind pressures of different elements.
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Figure 4. Results comparison between models loaded by single extreme operating gust (EOG) and blade element wind load (BEWiL) EOGs: (a) tip deflection; (b) maximum stress ratio.
appreciably larger than those of a blade model loaded by a single EOG, although the fluctuations of tip deflection and maximum stress ratio are similar. The largest tip deflection using the BEWiL model is found to be 1724 mm, which is 14.17% larger than the 1510 mm obtained using the single EOG loaded blade model. In addition the maximum stress ratio using the BEWiL model is 0.4183, which is 14.35% larger than 0.3658 obtained using the single EOG loaded blade model. Figure 5 shows a typical blade deflection, and the maximum stresses are located in the root.
3.
Fatigue life prediction
For fatigue life prediction, a 10 minute fluctuating wind is simulated during the normal operation of the wind turbine. After a dynamic nonlinear implicit analysis, the stress ratio R (R is the ratio of applied stress and allowable strength) data is extracted for fatigue analysis. Here, it is assumed that the S-N curve is a linear line in a loglinear basis. The final fatigue life is predicted based on the rainflow counting algorithm and Miner’s rule.
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Figure 5.
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Blade deflection and stress distribution in the root.
3.1. Timevarying fluctuating wind It is generally fairly straightforward to quantify the fatigue strength of the structural materials (GFRP or CFRP composites in the case of the wind turbine blades) under constant-amplitude loading conditions. However, the real wind load applied on the blade is a time-varying windinduced loading, which is irregular and stochastic, and the associated load history affects the component fatigue life in complex ways. Thus, a time-varying wind-induced loading, which consists of an average wind speed and a time-varying fluctuating wind speed, is used for fatigue life prediction. This wind speed at height z above the ground, v(z, t), can be expressed as Equation (10) (Law, Bu, and Zhu 2005): v(z, t) = v¯ (z) + vˆ (z, t)
(10)
where v¯ (z) and vˆ (z, t) denote the average wind speed and time-varying fluctuating wind speed, respectively. By referring to Griffin (2001), the average wind speed for wind turbine at hub height zhub is defined by Equations (11) and (12): v¯ (z) = 5.8
z
zhub = 1.3D
hub
10
0.143 (11) (12)
The calculation of fluctuating wind speed is beyond the scope of this article. Appendix A in Law, Bu, and Zhu (2005) gives a detailed explanation of the calculation. The simulation of a timevarying wind is realized by MATLAB code, and Figure 6(a) shows a realization of stochastic wind speed in 600 seconds. The relationship between wind speed and pressure is defined by Equation (9). Figure 6(b) shows the maximum stress ratio distribution during 600 seconds by dynamic nonlinear implicit analysis in SAMCEF. 3.2. Rainflow counting algorithm The rainflow counting algorithm is used in the analysis of fatigue data in order to reduce a spectrum of varying stress into a set of simple reversals. Its importance is that it allows application of Miner’s rule to assess the fatigue life of a structure subjected to complex loading. Details of the rainflow counting algorithm are given in the ASTM standard (ASTM 2005). To illustrate the rainflow counting algorithm, a simple load signal is depicted in Figure 7(a). An example of the resulting
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Engineering Optimization
Figure 6.
(a) A realization of stochastic wind speed; (b) maximum stress ratio distribution.
three-dimensional histogram showing the number of cycles presented in the maximum stress ratio distribution (Figure 6b) is depicted in Figure 7(b). Downloaded from MATLAB Central online, the rainflow counting algorithm code (Nieslony 2009) has been prepared according to the ASTM standard (ASTM 2005) and optimized considering the calculation time.
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Figure 7. (a) Rainflow counting algorithm illustrated by a simple load signal; (b) resulting threedimensional histogram showing the number of cycles presented in the maximum stress ratio distribution.
3.3. Fatigue life prediction The fatigue life prediction for the blade is based on Miner’s rule (Sutherland 1999), which can be applied in the case of a machine part operating under alternative stress with variable amplitudes. This theory assumes that every operating cycle consumes a percentage of the part life. If stresses with ratio level R1 , R2 , . . . , Rk are applied to a part for a total number of cycles n1 , n2 , . . . , nk ,
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respectively, and the lives (the allowable number of cycles) corresponding to these stresses are N1 , N2 , . . . , Nk , then the fatigue damage Dfat in test time Tfat can be estimated by adding the percentage damage consumed by each cycle. Each element in the blade finite element model has a different stress ratio value, and the maximum value among stress ratio values of all the elements is selected as the ratio level Ri for the fatigue life prediction because the maximum value is the most critical one for the fatigue life prediction. The total damage is stated mathematically as Equation (13). The fatigue strength is presented by an S-N curve, which is a linear line in a log-linear basis (Equation 14). Thus the predicted fatigue life Ttot is equal to test time Tfat divided by fatigue damage Dfat as formulated by Equation (15): Dfat =
k nk i=1
Ni
σi = 1 − β log(Ni ) σ0 Tfat = k
nk
Ri = Ttot
i=1
4.
(13) (14) (15)
10(1−Ri )/β
Design optimization procedures
Automatic design optimization procedures dramatically reduce the calculation time and increase the efficiency, especially when the design problem is complex, e.g. there are large numbers of design variables and a single finite element analysis cost is expensive. In this part, the definitions of the design variables, multi-objective and constraints are explained in Sections 4.1, 4.2 and 4.3, respectively. Considering the layer symmetry of composite materials, the design variables are reduced by the design variable linking (DVL) method. For a conservative design, partial safety factors are adopted in constraints. Then, the design optimization problem of this study is formulated in Section 4.4. Section 4.5, explains why the evolutionary algorithm (EA) was chosen as an optimizer for solving the design optimization problem formulated in the previous section 4.1.
Design variables
The blade layers are mainly made of CFRP and GFRP. The thickness and fibre angle of each layer play important roles in determining material properties. Thus, three kinds of variables for each layer are important: material type (CFRP or GFRP), thickness and fibre angle. There are altogether 50 layers in the blade: 20 layers (L1–L20) for the skin, 10 layers (L21–L30) for the shear web and 20 layers (L31–L50) for the root. If the material type, thickness and fibre angle of every layer are selected as design variables, the number of design variables will be 150. Considering the material selection and the symmetry of the composite material structure preferred in real manufacturing, however, DVL (Vanderplaats 1999) is applied as shown in Table 2. The materials for the skin and root are specified as GFRP and CFRP, respectively, while only the material types of shear web layers are selected as design variables LM 1 –LM 5 All 20 layers of the skin and the root are assumed to have identical thicknesses denoted as LT 1 and LT 7 , respectively, and LT 2 –LT 6 denote layer thicknesses of the shear web layers. LA1 –LA5 , LA6 –LA10 and LA11 –LA15 are design variables representing the fibre angles of composite layers used in the skin, the shear web and the root, respectively. Consequently, there are 27 design variables in total.
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Table 2.
Design variable linking.
Material type
Layer ID
Thickness
Layer ID
Fibre angle
LM 1 LM 2 LM 3 LM 4 LM 5
L21, L30 L22, L29 L23, L28 L24, L27 L25, L26
LT 1 LT 2 LT 3 LT 4 LT 5 LT 6 LT 7
L1–L20 L21, L30 L22, L29 L23, L28 L24, L27 L25,L26 L31–L50
LA1 LA2 LA3 LA4 LA5 LA6 LA7 LA8 LA9 LA10 LA11 LA12 LA13 LA14 LA15
Layer ID L1, L6, L15, L20 L2, L7, L14, L19 L3, L8, L13, L18 L4, L9, L12, L17 L5,L10,L11,L16 L21, L30 L22, L29 L23, L28 L24, L27 L25, L26 L31, L36, L45, L50 L32, L37, L44, L49 L33, L38, L43, L48 L34, L39, L42, L47 L35, L40, L41, L46
4.2. Design objectives The multi-objective consists of mass and cost. The mass is obtained from the SAMCEF batch file directly, while the cost is defined by Equation (16). Here the Cost is a nominal cost in order to increase the usage range of this approach: Cost = 20A1 f (1)LT1 + 2A2
5
f (LMq )LTq+1 + 20A3 f (2)LT7
(16)
q=1
where A1 , A2 and A3 are the areas of skin, shear web and root, respectively; f (LMq ) is the unit volume cost of GFRP and CFRP defined by Equation (17): 1 if LMq = 1 f (LMq ) = , q = 1–5 (17) 10 if LMq = 2 The final multi-objective function is defined by Equation (18), where λ1 and λ2 are the weight coefficients for the normalized cost and mass, respectively. Cost and Mass are the cost and mass of the initial design, respectively: F(LMq , LTp , LAs ) = λ1
Cost Mass + λ2 Cost0 Mass0
(18)
An appropriate weight value for each objective function depends on the relative importance of the cost and mass in the multi-objective function. Given a specific blade optimization problem, the values of λ1 and λ2 can be set accordingly. In the current optimization problem, the reductions of cost and mass are assumed to be equally important. Thus, values of λ1 = 0.5 and λ2 = 0.5 are set. 4.3.
Design constraints
The constraints in this study consist of six inequality and three equality constraints. Among the six inequality constraints, a flapwise bending strength constraint value and a tip deflection constraint value are evaluated using the BEWiL analysis described in Section 2.2, while a fatigue life constraint value is evaluated using the fatigue analysis described in Section 3.
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The flapwise bending strength constraint is defined by Equation (19) considering the partial safety factors (IEC 2005): σ0 γn γf σmax ≤ (19) γm
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where the value of γf for this load case is 1.35, that of γm for analysis of tensile or compression strength is 1.3, and that of γn for the wind turbine class I is 0.9. σmax is the maximum applied stress and σ0 is the maximum allowable strength. By using the Tsai–Wu failure criterion (Vinson and Sierakowski 2008) code in SAMCEF, the strength constraint could be represented by the maximal strength ratio Rmax (the ratio between maximum stress and ultimate strength) defined by Equation (20): Rmax (LMq , LTp , LAs ) =
σmax 1 ≤ σ0 γm γn γf
(20)
The constraint derived from critical deflection indicates that the value of tip deflection W must be smaller than 5% of blade length L (Todoroki and Kawakami 2007) as shown in Equation (21). The blade in this study has a 2 MW rated power with 40 m length. γn W (LMq , LTp , LAs ) ≤ 0.05L
(21)
The blade should work for at least 20 years under normal wind load. Combining Equation (15), the fatigue life constraint is defined by Equation (22): Ttot (LMq , LTp , LAs ) ≥ 20 years
(22)
From the standpoints of structural robustness and manufacturing, it is favourable to keep composite materials balanced, for instance, keep the same proportion of opposite-angled layers in each part. Thus, the numbers of layers with opposite fibre angles in each part are enforced to be equal. These constraints are expressed as Equations (27)–(29) in the following section. In addition, the numbers of 0◦ layers are limited for an effective composite material design. These constraints are expressed as Equations (30)–(32) in the following section. 4.4.
Optimization problem formulation
Now, the multi-objective discrete design optimization problem of this study can be formulated as: Find LMq , LTp and LAs to minimize F(LMq , LTp , LAs ) = λ1
Table 3.
Cost Mass + λ2 Cost0 Mass0
Evolutionary algorithm parameter setting.
Parameter
Value
Population size Maximum number of generations Violated constraint limit Mutation probability Selection probability
10 250 0.003 0.01 0.15
(23)
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subject to Rmax (LMq , LTp , LAs ) ≤
1 γm γn γf
(24)
γn (LMq , LTp , LAs ) ≤ 0.05L
(25)
Ttot (LMq , LTp , LAs ) ≥ 20 years
(26)
No. of unbalanced layers in skin = 0
(27)
No. of unbalanced layers in shear web = 0
(28)
No. of unbalanced layers in root = 0
(29)
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◦
No. of 0 layers in skin ≤ 8
(30)
No. of 0 ◦ layers in shear web ≤ 4
(31)
◦
No. of 0 layers in root ≤ 8
(32)
where q = 1–5, p = 1–7, s = 1–15 LMq ∈ [1, 2] LAs ∈ [−45, 0, 45, 90] Table 4.
Comparison of design variables at the initial and optimal designs.
Material type LM 1 LM 2 LM 3 LM 4 LM 5
Initial
Optimal
Thickness
Initial
Optimal
Fibre angle
Initial
Optimal
2 1 2 1 2
1 1 1 1 1
LT 1 LT 2 LT 3 LT 4 LT 5 LT 6 LT 7
2.6 1.5 1.5 2.0 2.5 2.0 2.6
2.5 1.0 1.0 1.6 2.0 1.5 2.0
LA1 LA2 LA3 LA4 LA5 LA6 LA7 LA8 LA9 LA10 LA11 LA12 LA13 LA14 LA15
−45 45 0 0 90 45 −45 −45 90 0 −45 45 −45 0 45
45 90 0 0 −45 0 −45 90 90 45 −45 90 0 45 90
Table 5.
Comparison of performances at the initial and optimal designs.
Performance Rmax γn W Ttotal No. of unbalanced layers in skin No. of unbalanced layers in shear web No. of unbalanced layers in root No. of 0◦ layers in skin No. of 0◦ layers in shear web No. of 0◦ layers in root Cost Mass F (weighted sum of objectives)
Initial
Optimal
0.4183 1724.39 24.58 0 2 0 8 2 4 6.30 6466.05 1.0000
0.5117 1938.53 24.22 0 0 0 8 2 4 4.94 6155.61 0.8683
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Figure 8. Convergence histories of objective functions and design constraints related to structural performances: (a) weight; (b) cost; (c) F (weighted sum of objectives); (d) maximal strength ratio Rmax ; (e) tip deflection, γn W ; (f) fatigue life, Ttotal .
LTp ∈ [2.0, 2.1, . . . , 2.9, 3.0] for p = 1, 5, 7 LTp ∈ [1.0, 1.1, . . . , 1.9, 2.0] for p = 2, 3 LTp ∈ [1.5, 1.6, . . . , 2.4, 2.5] for p = 4, 6 No. of unbalanced layers in skin = |No. of 45◦ layers in skin − No. of − 45◦ layers in skin| No. of unbalanced layers in shear web = |No. of 45◦ layers in shear web − No. of − 45◦ layers in shear web|
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Figure 9. Convergence histories of design constraints related to the number of unbalanced layers: (a) skin; (b) shear web; (c) root.
No. of unbalanced layers in root = |No. of 45◦ layers in root − No. of − 45◦ layers in root| 4.5.
Optimization algorithm
In the manufacturing process, the layer thickness and reinforcement angle are discrete. The values of layer thicknesses are selected from the discrete sets shown above. The reinforcement angles are often limited to a discrete set such as 0◦ , ±45◦ and 90◦ to obtain a costeffective design (Camanho et al. 2008). Thus, all 27 design variables including the material types of shear web layers are discrete. Also, six constraints expressed in Equations (27)–(32) should have discrete values. To effectively solve this discrete design optimization problem, the authors chose to employ an EA with the parameter values listed in Table 3 as an optimization technique because the EA is a wellknown optimization techniques for discrete optimization and particularly suitable for solving discrete combinatorial optimization problems. It is well known that the EA generally requires a large number of function evaluations to obtain the global optimal solution. However, a large number of function evaluations cannot be afforded because one analysis run evaluating the multi-objective values and all constraint values required a time-consuming structural simulation (about 15 minutes). To obtain a good design solution in a reasonable amount of time, the population size and maximum number of generations were specified as 10 and 250, respectively. These small parameter values were
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Engineering Optimization
Figure 10. (c) root.
17
Convergence histories of design constraints related to the number of 0◦ layers: (a) skin; (b) shear web;
chosen because the combinatorial design space to be explored was not very large (the discrete values allowed for the material type, fibre angle and layer thickness were just 2, 4 and 11, respectively). Note also that three equality constraints of Equations (27)–(29) and three inequality constraints of Equations (30)–(32) among nine constraints were simple functions of fibre angles only.
5.
Results
Table 4 summarizes the design variables of the initial and the optimal designs and Table 5 summarizes the performances of the initial and the optimal designs. The materials in the shear web all converge to GFRP, which indicates that the higher cost of CFRP makes the optimizer choose GFRP, although CFRP is lighter and stronger than GFRP. Compared to the initial layer thickness design variables, the optimal ones are significantly reduced, which also contributes to reducing the total cost. The convergence history curves of objective functions and design constraints are shown in Figures 8–10. In total, 2500 function evaluations (10 chromosomes × 250 generations) were consumed to complete the design optimization. At the optimum solution, the nominal Cost is reduced by 21.6%, from 6.30 to 4.94, and Mass is reduced by 4.8%, from 6466.05 kg to 6155.61 kg. All nine constraints including three equality constraints in the optimal design are satisfied, while the margins between optimal values of inequality constraints and limit lines are reduced compared to those of the initial design. Among six inequality constraints, two inequality constraints (the numbers of 0◦ layers in skin and shear web) are found to reach their upper limits, i.e. active,
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W. Hu et al.
while the other four (the maximum stress ratio, tip deflection, fatigue life) are inactive at the optimal design.
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6.
Conclusions
Based on the integrated structural analysis and optimization procedure, as well as the results obtained, the following summary is provided and conclusions are drawn. The article describes a robust automated design-optimization procedure for a HAWT composite blade based on an integrated structural analysis. The design problem is mathematically formulated and optimized. A similar design-optimization procedure could be utilized in other industrial fields, e.g. aerospace and automotive industries. Extreme operating gust analysis and fatigue life prediction under stochastic normal wind loads have been investigated. A new structural analysis named the BEWiL analysis is proposed. The BEWiL model can reflect the wind load dependency on the location of each blade element, and also involve the blade rotation effect on the wind load. By applying the BEWiL model, more severe tip deflections and stress ratios have been obtained. Consequently, the optimum design obtained using the BEWiL model should be safer than the design using single EOG loaded model that underestimates the real tip deflection and stress ratio. The structural analysis results indicate that the stress-critical spots are located in the interface between the skin and root. A time-varying fluctuating wind has been simulated and applied to the dynamic analysis for fatigue life prediction. The procedure couples the characteristics of the composite materials and the wind turbine blade. The optimal results provide a guide to blade manufacturing in terms of composite material type, layer thickness and reinforcement orientation. Acknowledgements This work was supported by the Second Brain Korea 21 Project in 2010 and the Korean NRF grant funded by the Korea government (MEST) (no. 2012-0005530). The authors also express gratitude to PIDOTECH Inc., which provided PIAnO software as a PIDO tool for this study.
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