energies Article
Multi-Objective Aerodynamic and Structural Optimization of Horizontal-Axis Wind Turbine Blades Jie Zhu 1, *, Xin Cai 2 and Rongrong Gu 3 1 2 3
*
College of Civil Engineering and Architecture, Jiaxing University, Jiaxing 314001, China College of Mechanics and Materials, Hohai University, Nanjing 210098, China;
[email protected] Architecture Engineering Institute, Jinling Institute of Technology, Nanjing 211169, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-573-8364-6050
Academic Editor: Frede Blaabjerg Received: 19 September 2016; Accepted: 4 January 2017; Published: 15 January 2017
Abstract: A procedure based on MATLAB combined with ANSYS is presented and utilized for the multi-objective aerodynamic and structural optimization of horizontal-axis wind turbine (HAWT) blades. In order to minimize the cost of energy (COE) and improve the overall performance of the blades, materials of carbon fiber reinforced plastic (CFRP) combined with glass fiber reinforced plastic (GFRP) are applied. The maximum annual energy production (AEP), the minimum blade mass and the minimum blade cost are taken as three objectives. Main aerodynamic and structural characteristics of the blades are employed as design variables. Various design requirements including strain, deflection, vibration and buckling limits are taken into account as constraints. To evaluate the aerodynamic performances and the structural behaviors, the blade element momentum (BEM) theory and the finite element method (FEM) are applied in the procedure. Moreover, the non-dominated sorting genetic algorithm (NSGA) II, which constitutes the core of the procedure, is adapted for the multi-objective optimization of the blades. To prove the efficiency and reliability of the procedure, a commercial 1.5 MW HAWT blade is used as a case study, and a set of trade-off solutions is obtained. Compared with the original scheme, the optimization results show great improvements for the overall performance of the blade. Keywords: multi-objective optimization; horizontal axis wind turbine; wind energy; blade mass; finite element method
1. Introduction With the world’s ever-increasing energy demands in the presence of continuous fossil fuel shortage and environmental pollution, various renewable energy sources are currently being investigated [1–4]. Wind energy has been widely accepted because of its great advantages, such as inexhaustibility and environmental friendliness [5,6]. Vigorous development and utilization of wind energy is of great significance to improving energy structure and achieving sustainable development. A typical arrangement to extract wind energy is the use of wind turbines, especially the horizontal-axis wind turbine (HAWT), which is the most valuable and widely used form at present [7]. In order to make wind energy more competitive with other energy sources, the wind turbines are designed aiming at minimizing the cost of energy (COE), namely, increasing the annual energy production (AEP) and bringing down the total cost. Blade is the key component of wind turbine to capture wind energy, it is of course not excluded from the overall optimization goals. The blade design process mainly includes two stages: the aerodynamic design and the structural design. The two stages are separated by the conventional methods to simplify the process, and the aerodynamic design is paid more attention to. However, with Energies 2017, 10, 101; doi:10.3390/en10010101
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of 18 with the growth in turbine size and blade length, multi-megawatt (MW) sized blades are 2now requiring designers to consider the structure of the blade earlier in the design process [8]. Modernin blades of multi-megawatt glass fiber reinforced plasticblades (GFRP) its low the growth turbineare sizeprimarily and blademade length, (MW) sized are due nowto requiring density and superior mechanical properties. As the blade length increases, there is a rapid growth designers to consider the structure of the blade earlier in the design process [8]. in the blade blades mass (approximately as aofcubic of length,plastic while(GFRP) the trend Modern are primarily made glasspower fiber reinforced dueof to many its lowmodern density blades has maintained 2.3 power [8]) as well as the loads. Even GFRP cannot satisfy the structural and superior mechanical properties. As the blade length increases, there is a rapid growth in the requirements, which leads toasthe employment lighterwhile and stronger materials as carbon blade mass (approximately a cubic power ofoflength, the trend of manysuch modern bladesfiber has reinforced plastic (CFRP) [9]. Since CFRP is much more expensive than GFRP, the obvious maintained 2.3 power [8]) as well as the loads. Even GFRP cannot satisfy the structural requirements, disadvantage cost,ofbut it will mitigated somewhat bycarbon a reduction in blade plastic mass. which leads to is theincreased employment lighter andbe stronger materials such as fiber reinforced Therefore, is anCFRP important issue to arrange thethan GFRP and the CFRP in a sufficient way to reach the (CFRP) [9].itSince is much more expensive GFRP, obvious disadvantage is increased optimal utilization of the material. Many researches [9–13] have recently been carried out to deal cost, but it will be mitigated somewhat by a reduction in blade mass. Therefore, it is an important with such problems using numerical methods or optimization techniques, the results showed issue to arrange the GFRP and CFRP in a sufficient way to reach the optimal utilization of the material. improvements in [9–13] structural of the blades, the aerodynamic performances were not Many researches havebehaviors recently been carried outbut to deal with such problems using numerical considered. methods or optimization techniques, the results showed improvements in structural behaviors of the A successful blade design must integrate blades, but the aerodynamic performances wereboth not aerodynamic considered. and structural concerns to get the overall optimal solutions. However, only a limited number of papers [2,14–18] concerns are focused on the the A successful blade design must integrate both aerodynamic and structural to get optimization for this purpose. Most of the works among these papers either taken a single objective overall optimal solutions. However, only a limited number of papers [2,14–18] are focused on the function in the or used beam models calculate structural the objective material optimization forproblem this purpose. Most of the worksto among thesethe papers either behaviors, taken a single layup was not discussed. Our recent research [19] described a multi-objective optimization method function in the problem or used beam models to calculate the structural behaviors, the material layup for the aerodynamic and structural integrated design of blades to maximize the AEP and minimize was not discussed. Our recent research [19] described a multi-objective optimization method for the the blade mass, element method (FEM) was applied so thatthethe layup variation the could be aerodynamic andfinite structural integrated design of blades to maximize AEP and minimize blade described, but the cost was not evaluated due to no CFRP usage. mass, finite element method (FEM) was applied so that the layup variation could be described, but the Thisnot paper presents procedure for the multi-objective aerodynamic and structural cost was evaluated due toa no CFRP usage. optimization ofpresents HAWT ablades based one in [19].aerodynamic The blade element momentum (BEM) This paper procedure foron thethe multi-objective and structural optimization theory is used to evaluate the aerodynamic performances, the FEM method is applied to calculate of HAWT blades based on the one in [19]. The blade element momentum (BEM) theory is used theevaluate structural Moreover, the non-dominated sorting genetictoalgorithm (NSGA) II is to thebehaviors. aerodynamic performances, the FEM method is applied calculate the structural adapted for the multi-objective optimization of the blades. Materials of CFRP combined with GFRP behaviors. Moreover, the non-dominated sorting genetic algorithm (NSGA) II is adapted for the are used in the spar cap, and aofmaterial costMaterials function is the procedure. The goal is to find multi-objective optimization the blades. ofadded CFRP in combined with GFRP are used in thea balance the aerodynamic and design minimize the COE ofbetween HAWT spar cap,between and a material cost functiondesign is added in structural the procedure. Thetogoal is to find a balance blades. the aerodynamic design and structural design to minimize the COE of HAWT blades.
2. Modeling 2. Modeling of of the the Blade Blade 2.1. Geometry Shape and Aerodynamic Loads The geometry blade is shown in Figure 1, which generally includes three geometryshape shapeofofa atypical typicalHAWT HAWT blade is shown in Figure 1, which generally includes regions: the root normally with circular cross-section, the aerodynamic region with thinner three regions: theregion root region normally with circular cross-section, the aerodynamic region with airfoils capture wind energy theand region from the rootthe to the at 15% thinner to airfoils to capture wind and energy the transition region transition from rootairfoil to thesection airfoil section radius a high to chord ofratio up toofabout The three of the airfoil at 15% with radius with thickness a high thickness toratio chord up to50%. about 50%. Theparts threeconsist parts consist of the series their the chord twist airfoil and series andlocations, their locations, the and chord anddistributions. twist distributions.
Figure 1. 1. The the blade. blade. Figure The geometry geometry shape shape of of the
In order to define the blade geometry shape, seven control points (CPs) with fixed locations are In order to define the blade geometry shape, seven control points (CPs) with fixed locations applied for the chord and twist distributions, as shown in Figure 2. CP1 is at root, CP3 is at the are applied for the chord and twist distributions, as shown in Figure 2. CP1 is at root, CP3 is at the maximum chord station and CP7 is at tip. The chord of CP2 is equal to CP1, and the twists of CP1–3 maximum chord station and CP7 is at tip. The chord of CP2 is equal to CP1, and the twists of CP1–3 are the same. The airfoil locations are reflected by the percent thickness distribution, as illustrated in are the same. The airfoil locations are reflected by the percent thickness distribution, as illustrated Figure 3. Six more CPs, which represent the airfoil locations and percent thicknesses, are used for the
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3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1 2 1.5 1.5 1 2 1.0 1.0 0.5 0.5 0.0 0.00.0 0.0
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Control points Control points
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7 7 0.8 0.8
Control points Control points
4 4
5 5
6 6
0.4 0.6 0.4 length (r/R) 0.6 Blade Blade length (r/R)
13 13 1 2 1 2 9 9
Twist Twist (°)(°)
Chord Chord (m)(m)
of 18 CPs, which represent the airfoil locations and percent thicknesses, are 3used for the percent thickness distribution. the chord, twist percent thicknessare aredefined defined with percent thickness distribution. Then,Then, the chord, twist andand percent thickness percent thickness distribution. Then, the chord, twist and percent thickness are defined with B-Spline curves. B-Spline curves.
5 5
1 1
1.0 1.0
-3 -30.0 0.0
0.2 0.2
0.4 0.6 0.4Blade length0.6 Blade length
6 6
7 7 0.8 0.8
1.0 1.0
Percent Percent thickness thickness (%)(%)
(a) (b) (a) (b) Figure 2. (a) Chord and (b) twist distribution of the blade. Figure 2. 2. (a) (a) Chord Chord and and (b) (b) twist twist distribution distribution of of the the blade. blade. Figure 100 100 8 8 80 80 60 60 40 40
Control points Control points
9 9
20 20 0 00.0 0.0
0.2 0.2
10 10
11 11
0.4 0.6 0.4 0.6 Blade length (r/R)
12 12 0.8 0.8
13 13 1.0 1.0
Blade length (r/R)
Figure 3. Percent thickness distribution of the blade. Figure Figure 3. 3. Percent Percent thickness thickness distribution distribution of of the the blade. blade.
Once the geometry shape of the blade is fixed, the aerodynamic loads can be calculated using Once the geometry shape of the blade is fixed, the aerodynamic loads can be calculated using BEMOnce theory by dividing several loads the[20,21] geometry shape ofthe theblade blade into is fixed, the independent aerodynamic elements. loads can Aerodynamic be calculated using BEM theory [20,21] by dividing the blade into several independent elements. Aerodynamic loads including operational and ultimate case forseveral the optimization design in thisAerodynamic paper are the loads same BEM theory [20,21] bycase dividing the blade into independent elements. including operational case and ultimate case for the optimization design in this paper are the same as the loads in [19]. case The and operational case for takes into account design the maximum rootare flap including operational ultimate case the optimization in this paper thebending same as as the loads in [19]. The operational case takes into account the maximum root flap bending moment under operational state, computed follow: the loadsMinflap[19]. The operational casewhich takes is into accountas the maximum root flap bending moment moment Mflap under operational state, which is computed as follow: Mflap under operational state, which is computed R as follow: M flap = R 4πρu22 a(1 - a)Fr 22 dr (1) 0 M flap Z= 4πρu a(1 - a)Fr dr (1) R0 2 2 M fwind 4πρu (1 −axial a) Frinduction dr (1) lap = speed, a isathe factor, F is the Prandtl’s tip where ρ is the air density, u is the 0 a is the axial induction factor, F is the Prandtl’s tip where ρ is the air density, u is the wind speed, and root correction factor, and R is the blade length. and root factor,uand R is the blade length. where ρ iscorrection the air density, the wind speed, a is the axial F is the Prandtl’s tip into and The maximum bendingis moment can be derived frominduction dividing factor, the range of wind speeds The maximum bending moment can be derived from dividing the range of wind speeds into root correction is the length. different valuesfactor, everyand 0.5 R m/s, andblade the corresponding wind speed u can be determined. Therefore, different values every 0.5 m/s,moment and thecan corresponding winddividing speed u the can range be determined. Therefore, The maximum bending be derived from of wind speeds into the lift force L and drag force D per unit length can be calculated as follow: the lift force L and drag force D per unit length can be calculated as follow: different values every 0.5 m/s, and the corresponding wind speed u can be determined. Therefore, 1 be2 calculated as follow: can the lift force L and drag force D per unit length L = 1 ρW 2 c (r )Cl L = 2 ρW c(r )Cl ( (2) 2 2 c2(r )C (2) L D ==1211ρW l W c r C ρ ( ) (2) D =12 ρW2 2 c (r )Cd D = 2 ρW c(r )Cdd 2 where W is a relative velocity related to u, c(r) is the local chord, Cl and Cd are the lift and drag where velocity related to u,to c(r)u,isc(r) the local theClift andthe drag where W Wisisa arelative relative velocity related is thechord, local Cchord, and d are liftcoefficients. and drag l and CC d lare coefficients. Then, the lift and drag forces are projected into normal to and tangential to the rotor plane coefficients. Then, the lift and drag forces are projected into normal to and tangential to the rotor plane directions obtain the forces dpN and in Equation (3):and tangential to the rotor plane Then,tothe lift and drag forces aredpprojected into normal to T , as show directions to obtain the forces dpN and dpT, as show in Equation (3): directions to obtain(the forces dpN and dpT, as show in Equation (3): dp N = dLcosφ + dDsinφ = 211ρW 22c(r )(Cl cosφ + Cd sinφ)dr (3) dpN = dLcosφ + dDsinφ = 11 ρW2 2 c (r )(Cl cosφ + Cd sinφ)dr dp T = −φdDcosφ dpdLsinφ + dDsinφ== 22ρW cosφ +−CC φ)dr)dr = dLcos ρW cc((rr)( )(Cll sinφ d cosφ d sin N (3) 2 1 (3) 2 = φ φ dp dLsin dDcos = 1 2 c (r )(C l sinφ − C d cosφ)dr T dp = dLsinφ - dDcosφ = 2 ρρW φ − φ W c r C sin C cos dr ( )( ) l d T 2 where dpN is the force normal to the rotor plane, dpT is the force tangential to the rotor plane, ϕ is the where dpN is the force normal to the rotor plane, dpT is the force tangential to the rotor plane, ϕ is the angle between the rotor plane and the relative velocity, as shown in Figure 4. angle between the rotor plane and the relative velocity, as shown in Figure 4.
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where dpN is the force normal to the rotor plane, dpT is the force tangential to the rotor plane, φ is the angleEnergies between rotor plane and the relative velocity, as shown in Figure 4. 2017, the 10, 101 4 of 18 Energies 2017, 10, 101
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Figure 4. The local loads on a blade cross section.
Figure 4. The local loads on a blade cross section. Figure 4. The local loads on a blade cross section.
The ultimate case is derived from the parked 50 year extreme wind condition, which can be
The ultimate case is by derived fromformula the parked parked wind condition, which can be The ultimate case is derived from the 50 year yearextreme extreme wind condition, which can be calculated approximately empirical [22]: 50 calculated approximately by empirical formula [22]: calculated approximately by empirical formula [22]: 2
hhub + 2 3 R 2 1 p(r ) = 1 ρC f Vb2 kt2 + 3 c (r ) (4) ln hhub + 2 3 R 2 2 2) 2 z2/3R p(1r ) = ρC V k ln + 3 c ( r 0 (4) h + f b t hub z0 p(r ) = ρC 2f Vb2 k2t ln (4) + 3 c(r ) 2 z0 where C f is a force coefficient, Vb is a basis wind speed, kt is a terrain factor, hhub is the hub where C f is a force coefficient, Vb is a basis wind speed, kt is a terrain factor, hhub is the hub where C f isand a force Vblength. is a basis wind speed, kof terrain is the hub height t isa awind z0 iscoefficient, a roughness If the surrounding farmfactor, has nohhub nearby obstacles height z is a roughness length. If the surrounding of a wind farm has no nearby obstacles and and zheight is a roughness length. If the surrounding of a wind farm has no nearby obstacles and very low 0 0 and very low vegetation, kt = 0.17 and z0 =0.01 m . k = 0.17 and z =0.01 m . and very low vegetation, vegetation, k t = 0.17 and z0 =t 0.01 m. 0
2.2. Structural form and Finite Element Method Model of the Blade
2.2. Structural formform andand Finite Element of the theBlade Blade 2.2. Structural Finite ElementMethod Method Model Model of
Figure 5 shows a typical structural form of the blade cross section, which can be divided into
5 shows a typical structural form ofand the blade section, which be divided into Figure 5 shows a typical structural form of the bladecross cross section, which can be divided four Figure parts: leading edge, spar cap, shear webs trailing edge. The spar capcan mainly consists of into four parts: leading edge, spar cap, shear webs and trailing edge. The spar cap mainly consists of suchcap, as GFRP CFRP, while the leading shearcap webs and trailing four unidirectional parts: leadinglaminates edge, spar shearorwebs and trailing edge. edge, The spar mainly consists of unidirectional such as GFRP or CFRP, while the leading edge, shearwebs webslaminates andtrailing trailing edge consist oflaminates sandwich structure materials. Material properties ofedge, the unidirectional are edge unidirectional laminates such as GFRP or CFRP, while the leading shear and edge consist of sandwich Material properties the unidirectional laminates are shown in Table 1, the coststructure of CFRPmaterials. is Material set as 10 times than that ofofunidirectional GFRP [12]. Usually the spar consist of sandwich structure materials. properties of the laminates arecap shown shown in the Table 1, the cost of CFRP is set as 10 times thanblade, that of GFRP [12]. in Usually the spar cap provides stiffness and strength requirements of the its thickness the middle of the in Table 1, the the coststiffness of CFRPand is set as 10 times than that of GFRP [12]. Usually the sparmiddle cap provides the provides strength requirements of theparts. blade,Consequently, its thickness the in the of thea blade is typically large in comparison to those of other spar cap makes stiffness and strength requirements of the blade, its thickness in the middle of the blade is typically blade typically large in blade comparison to those of other parts. Consequently, theattention spar captomakes a major is contribution to the mass and cost [23], which should be paid more during largemajor in comparison to to those of other parts. Consequently, the sparbecap makes aattention major contribution to contribution the blade mass and cost [23], which should paid more to during the optimization process. the blade mass and cost [23], which should be paid more attention to during the optimization process. the optimization process.
Figure 5. A typical structural form of the blade cross section. Figure 5. A typical structural form of the blade cross section.
Figure 5. A typical structural form of the blade cross section.
Table 1. Material properties of the unidirectional laminates. GFRP: glass fiber reinforced plastic; and Table Material properties ofplastic. the unidirectional laminates. GFRP: glass fiber reinforced plastic; and CFRP: 1. carbon fiber reinforced TableCFRP: 1. Material of the unidirectional laminates. GFRP: glass fiber reinforced plastic; and carbon properties fiber reinforced plastic. ρ (kg/m3) Cost (m−3) Material E1 (GPa) E2 (GPa) G12 (GPa) v12 CFRP: carbon fiber reinforced plastic. (kg/m3) Cost (m−3) ρ Material E 1 (GPa) E 2 (GPa) G 12 (GPa) v 12 GFRP 42.19 12.53 3.52 0.24 1910 1 42.19 12.53 CFRP 130.00 10.30 Material GFRP E1 (GPa) E2 (GPa) CFRP
GFRP CFRP
130.00
42.19 130.00
10.30
12.53 10.30
7.17 G12 3.52 (GPa) 7.17
0.24 0.28v 12 0.28
3.52 7.17
0.24 0.28
1910 1 1540ρ (kg/m3 )10 1540 10
1910 1540
Cost (m−3 ) 1 10
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The FEM modeling process of the blade is complicated and time-consuming, in order to simplify the creation andblade analysis the models,and including parametric studies to The and FEMexpedite modeling process of the is complicated time-consuming, indesign order to simplify create variousthe models, the ANSYS parametric design language (APDL) in ANSYS software and expedite creation and analysis the models, including parametric design studies to create(Version various 14.0, ANSYS Inc., Canonsburg, PA, USA) is applied. After inputting the geometry shape, the models, the ANSYS parametric design language (APDL) in ANSYS software (Version 14.0, ANSYS structure of the blade is modeled with shell elements (SHELL99 element type for the spar cap and Inc., Canonsburg, PA, USA) is applied. After inputting the geometry shape, the structure of the blade SHELL91 for the other three parts). Thethe two of SHELL91 elements element are capable of is modeledelement with shelltype elements (SHELL99 element type for spartypes cap and type for representing layer characteristics by defining the real constants, namely, the material properties, the other three parts). The two types of elements are capable of representing layer characteristics by orientations and constants, thicknesses. Appropriate number of real constants should be determined in order defining the real namely, the material properties, orientations and thicknesses. Appropriate to find a balance between accuracy and computation cost. Moreover, a mapping meshing method is number of real constants should be determined in order to find a balance between accuracy and applied to prevent erroneous results. computation cost. Moreover, a mapping meshing method is applied to prevent erroneous results. 3. Optimization Optimization Model 3.1. Design Design Variables Variables 3.1. Once the the airfoil airfoil series series and and the Once the blade blade length length are are confirmed, confirmed, the the geometrical geometrical shape shape contributes contributes directly to to the Therefore, the the chords, chords, twists twists and and airfoil airfoil locations locations of of the the directly the aerodynamic aerodynamic performances. performances. Therefore, CPs in Figures 2 and 3 are selected as aerodynamic variables. The root diameter is set not to change, CPs in Figures 2 and 3 are selected as aerodynamic variables. The root diameter is set not to change, which means the chords are used used which means the chords of of CP1 CP1 and and CP2 CP2 are are fixed, fixed, so so five five aerodynamic aerodynamic variables variables (x (x11 to to xx5)) are for the the chords chords of of CP3–7. CP3–7. As As the the twist twist remains remains constant constant inboard inboard of of the the maximum maximum chord, chord, another another five five for aerodynamic variables (x to x ) are employed for the twist of CP3–7. The location of CP8 is at root aerodynamic variables (x66 to x1010) are employed for the twist of CP3–7. The location of CP8 is at root four more aerodynamic variables (x11(xto11 xto needed for the 14 )xare and the location location of ofCP13 CP13isisatattip, tip,thus thus four more aerodynamic variables 14) are needed for locations of CP9–12 (airfoils with percent thickness of 40%, 30%, 25% and 18%). Furthermore, the rated the locations of CP9–12 (airfoils with percent thickness of 40%, 30%, 25% and 18%). Furthermore, rotational speed of the rotor has aalso great influence on the aerodynamic performances, thus it is the rated rotational speed of also the rotor has a great influence on the aerodynamic performances, selected the lastasaerodynamic variable (x 15 ). thus it is as selected the last aerodynamic variable (x15). A typical material layup in the spar cap of commercial blade isblade shownisinshown Figure in 6. As mentioned typical material layup in the spar cap of commercial Figure 6. As earlier, the earlier, middle the region (shown in green) sparof cap most to themost blade and mentioned middle region (shownofinthe green) thecontributes spar cap contributes to mass the blade cost. Hence, the Hence, materialthe layup in thislayup regioninand width and are selected as structural variables. Eight mass and cost. material thisitsregion its width are selected as structural discrete CPs withdiscrete different number and location of layers used of to simulate layup, as shownthe in variables. Eight CPs with different number and are location layers arethe used to simulate Figure as 7. CP16–19 has7.three variables thevariables number of GFRP layers, the number CFRP layup, shown ineach Figure CP16–19 eachthat has are three that are the number of GFRPoflayers, layers and the location of layers, while the other four points each has two variables that are the number the number of CFRP layers and the location of layers, while the other four points each has two of GFRP layers andthe thenumber numberofofGFRP CFRPlayers layers.and In addition, the of number layersInofaddition, CP17–18 the are variables that are the number CFRP of layers. defined the same. of Therefore, structural variables are used for thestructural material layup, x16 to for number of layers CP17–18eighteen are defined the same. Therefore, eighteen variables arex22 used the number of GFRP layers, x to x for the number of CFRP layers and x to x for the location of for the material layup, x16 to 23 x22 for29the number of GFRP layers, x23 to x2930for the 33 number of CFRP layers. and Another x34 location is used for width of the spar cap. xIn the blade mass andspar cost layers x30 tovariable x33 for the of the layers. Another variable 34 addition, is used for the width of the can further decrease the shear webs positioned appropriately, of the shear webs is cap. In addition, theif blade mass andarecost can further decrease so if the position shear webs are positioned selected as the so lastthe structural (x35 ).webs is selected as the last structural variable (x35). appropriately, positionvariable of the shear
Figure Figure 6. 6. Material Material layup layup of of the the spar spar cap. cap.
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30 0.0
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0.1
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Figure 7. Material layup simulation of the spar cap. Figure 7. Material layup simulation of the spar cap.
3.2. Objective Functions 3.2. Objective Functions The purpose of the present work is to minimize the COE of the blade, namely, increasing the The purpose of the present work is to minimize the COE of the blade, namely, increasing the AEP and reducing the cost. In addition, the blade mass is another important goal for blade design. AEP and reducing the cost. In addition, the blade mass is another important goal for blade design. The reduction in blade mass could not only reduce the cost but also the loads, which has a The reduction in blade mass could not only reduce the cost but also the loads, which has a multiplier multiplier effect throughout the system including the foundation [24]. Therefore, an AEP function, a effect throughout the system including the foundation [24]. Therefore, an AEP function, a mass function mass function and a material cost function are taken into account as the three objectives of the and a material cost function are taken into account as the three objectives of the problem. In order problem. In order to enable visual comparisons between the optimization results and original to enable visual comparisons between the optimization results and original values, dimensionless values, dimensionless method is applied to the functions. method is applied to the functions. The first objective function is computed by normalizing the current AEP to the value of the The first objective function is computed by normalizing the current AEP to the value of the original blade AEP0 , which is defined as: original blade AEP0 , which is defined as:
f (1) = max(AEP / AEP0 )
f (1) = max ( AEP/AEP0 ) N 1 AEP N = [P (ui ) + P (ui+1 ) ] × f (ui < u0 < ui +1 ) × 8760 1 i =1 2 AEP = ∑ [ P(ui ) + P(ui+1 )] × f (ui < u0 < ui+1 ) × 8760 2 i =1 R P(ui )Z= 4πr 3ρui ω2 a'(1 - a)Fdr R0
2 0
3
(5) (5) (6) (6) (7) (7)
4πr ρui ω a (1 − a) Fdr k ui +1 k − ui − exp − f (ui < u0 < ui +1 ) = exp (8) k uA Aui+1 k i (8) f (ui < u0 < ui+1 ) = exp − − exp − A A where thepower powerofofthe thewind windturbine, turbine,f (uf(u< u0 < < uui +1 ) )isis the i
