ZHENG Yuqiao，LIU Zheyan，MA Huidong，ZHU Kai

School of Mechanical and Electronical Engineering，Lanzhou University of Technology，Lanzhou 730050，P.R.China

Abstract: This study develops a method for the full-size structural design of blade，involving the optimal layer thickness configuration of the blade to maximize its bending stiffness using a genetic algorithm. Numerical differentiation is employed to solve the sensitivity of blade modal frequency to the layer thickness of each part of blade.The natural frequencies of first-order flapwise and edgewise modes are selected as the optimal objectives. Based on the modal sensitivity analysis of all design variables，the effect of discretized layer thickness on bending stiffness of the blade is explored，and 14 significant design variables are filtered to drive the structural optimization. The best solution predicts an increase in natural frequencies of first-order flapwise and edgewise blade modes by up to 12% and 10.4%，respectively. The results show that the structural optimization method based on modal sensitivity is more effective to improve the structural performance.

Key words：composite blade；discrete layer thickness；modal sensitivity；optimal thickness distribution；bending stiffness

0 Introduction

The large horizontal-axis wind turbine blade，one of the most critical components of the wind pow?er system，is characterized by slender shape，vary?ing composite materials configurations，including sandwich construction［1-2］. Besides，blades are sub?jected to aerodynamic loading while working，which further increase the challenge of maintaining a safe distance between blades and tower［3］. Either glass-fi?ber or car-bon-fiber-reinforced polymers，or even hybrids of both fibers is utilized as the primary loadbearing materials［4-5］. Its long span，limited capacity to control blades tip deflection for ensuring a safe distance between blades tip and tower，along with the trend of individual wind turbine capacity increas?ing year by year，all suggest that it requires the high?er bending stiffness than other small and medium blades［6］. For these horizontal-axis wind turbines（HAWTs）， studies have found that the blade weight grows with a rotor radius at R2.3while the ro?tor power grows with R2.1［7］. To ensure a safe dis?tance between the blade and the tower，the blade stiffness must increase at least in proportion to its weight as the rotor radius grows. In addition to im?provements in the blade bending stiffness in the de?sign and manufacture process，structural optimiza?tion can be a practical approach to increase it［8-9］.

Most of the published literature have described some specific issues of the structural optimization of the composite blades. Barr and Jaworski［10］explored the concept of passive aero elastic tailoring to maxi?mize the power extraction of National Renewable Energy Laboratory（NREL）5 MW wind turbine blade and presented a variable-angle tow composite materials model along blade span to couple the bendtwist deformations under aerodynamic load. The re?sulting computational formulation predicted an in?crease of 14% in turbine power extraction when the optimization is performed around the cut-in wind speed，and by 7% when the blade is optimized near the rated wind speed. In Ref.［11］，a high-fidelity multi-disciplinary optimization capability is em?ployed for the structural optimization of wind tur?bine blades. The optimal fiber angles distribution through-out the internal structure of the blade were sought to minimize a stress parameter for each of several load cases. The result showed that the driv?ing stress for fatigue has a reduction of 18%—60%after optimization. Almeida et al.［12］presented a methodology to perform cross-section evolutionary optimization for a topologically-optimized structure using a genetic algorithm. The structure with both topology and cross-section optimization accom?plished a specific stiffness 330% higher than the structure of the quasi-isotropic stacking sequence.Buckney et al.［13］utilized the topology optimization to find optimal structural configurations for a 3 MW wind turbine blade and saved weight by up to 13.8% compared to a conventional design. There are two different approaches to achieve structural op?timization of the blade. The first approach is the op?timization in spanwise material distribution，the se?lection of materials，size of parts such as spar flange and shear webs through the knowledge of typical blade build-up and constraint，e.g. Refs.［10?11］.The other is topology optimization，which seeks the optimal material distribution，e. g. Refs.［12?13］.Here，the first approach will be applied to the struc?tural optimization of the blade.

The researchers considered the size of blade parts as continuous design variables in the optimiza?tion problems. Chen et al.［14］developed a procedure to optimize composite structures of the wind turbine blades，which not only allows thickness variation but also considers it as a continuous variable. The results showed that the optimal mass of the blade is reduced significantly compared to the initial blade.Bottasso et al.［15］presented a similar parameteriza?tion method but with the addition of leading edge re?inforcement and skin core，and this method resulted in 53 continuous thickness design variables. Two loops achieved the optimization. The inner loop used a 2-D cross-section finite element model by which the gradient-based optimization was finished.The criteria utilized in the inner loop was calibrated in turn by a finer 3-D finite element model used in the outer loop.

A discrete sizing of plies is achieved by formu?lating the optimization problems. Sjolund and Lund［16］minimized the mass of wind turbine blade while considering manufacturing constraints，tip dis?placement，buckling，and static strength criteria.Researches divided the blade into patches and select?ed the layer thicknesses as design variables，which were assumed discrete in the optimization. This gra?dient-based sizing optimization resulted in a reduc?tion in mass and many practical constraints across multiple load directions. Fagan et al.［17］performed experimental testing on a 13 m long composite wind turbine blade and a design optimization to minimize the material used in blade construction by consider?ing the discrete thickness distribution of the compos?ite materials. While ample investigation is available to demonstrate that discrete layer size can be deter?mined as design variables to drive structural optimi?zation，the effect of discretized layer size on dynam?ic blade characteristics from a full-size perspective is still a challenging task.

To address this knowledge gap，first of all，the five main components of the blade，including the leading edge，leading edge panel，spar flange，trail?ing edge panel，and trailing edge，are equally divid?ed into 27 segments along the blade span. The layup thickness of each patch，coupled with blade root and its transition，are considered as design variables，and natural frequencies of the first-order flapwise and edgewise modes are determined as the optimal objectives of this work. Final 14 design variables are selected by the modal sensitivity analysis of all de?sign variables，and this optimization problem is solved using a genetic algorithm.

1 Methods and Models

1.1 Blade geometry

A 40.5 m blade design from a 1.5 MW turbine has been selected to conduct flapwise and edgewise bend stiffness optimum，and its tip displacement varies linearly with its bend stiffness. For this rea?son，the bending stiffness of a long blade must main?tain a safe distance between the blade tip and the tower under complex aerodynamic loads. The corre?sponding design parameters of the blade are listed in Table 1［18］.The aerodynamic airfoil surfaces of the blade are shown in Fig.1.

Table 1 Design parameter description[18]

Fig.1 Blade geometry surface model

1.2 Blade structure

The structural model of the blade，including the construction of the composite structure，was cre?ated in ANSYS，a commercial finite element analy?sis code［19］. For a finite element analysis process，one of the most essential steps in executing the finite element analysis code is the creation of a structural finite element grid，and this blade structural grid has a total of 27 549 elements.

The design of the blade layup is to determine the angle and number of each layup angle according to the performance requirements and load character?istics of different parts of the blade. This process is decisive for the blade weight，performance and life.So it needs repeated iteration and continuous im?provement. Glass fiber reinforced composite materi?al is selected as the primary laminated material of the blade；other auxiliary materials include Balsa wood，polyvinyl chloride（PVC）foam，and rein?forcement material［20］. The mechanical properties of these materials applied to the optimization of this work are listed in Table 2.

Table 2 Blade material properties

In Table 2，Exand Eyare the Young’s modu?lus in two principle directions，respectively，Gxyis the in-plane shear modulus，υ denotes Poisson’ra?tio，and ρ represents the material density. The wind turbine blade is a typical composite material struc?ture. Its manufacture can be divided into five stag?es：materials，fabrics，stackups，sub laminates，blade. Materials：This stage is mainly to prepare the basic composition materials of the blade in Ta?ble 2. Fabrics：This stage aims to produce a single layer of fabric corresponding to each material in the first stage. The mechanical properties of each singlelayer fabric are shown in Fig.2. Stackups：The lami?nation of single-layer fabrics forms the basic struc?ture of blade manufacturing，such as uniaxial glass，biaxial glass，etc. Sub laminates：This stage aims to complete the layup of each sub-region of the blade according to the blade layup scheme. Blade：The overall composite structure of the blade is ob?tained by combining the sub-laminates of the fourth stage. It can be seen that the mechanical properties of the blade depend on the mechanical properties of the materials in Table 2 and the layup scheme of the blade. Therefore，these material properties play role through the layup scheme of the blade.

According to the layup materials in Table 2，uniaxial fabrics，biaxial fabrics，triaxial fabrics，bal?sa fabrics，PVC fabrics，and reinforcing material fabrics were manufactured. Corresponding mechani?cal properties of these fabrics were calculated and il?lustrated in Fig.2.

Fig.2 Mechanical properties of single-layer fabrics

Fig.2 indicates elastic modulus is the largest among all single-layer fabrics，which is 33.2 GPa，and its shear modulus has a maximum value of 8.3 GPa in the ± 45° directions. The biaxial fabric has the largest elastic modulus in the directions of± 45°，reaching 17.1 GPa，while the maximum val?ue of the shear modulus occurs at the main direc?tions of 0° and 90°. It is evident that the anisotropy of the biaxial fabric is not as significant as the uniaxi?al fabric. For Balsa single-layer fabric，its elastic modulus is much lower than that of uniaxial and bi?axial fabrics，and there is only a second principal elastic modulus with a maximum value of 2.07 GPa.Its shear modulus has a maximum value of 1.04 GPa in the ± 45° directions. PVC single-layer fabric is almost an isotropic material，its elastic mod?ulus and shear modulus are the lowest among all sin?gle-layer fabrics，and the corresponding maximum values are 58.5 MPa and 21.9 MPa，respectively.Reinforcement and Gelcoat single-layer fabrics are two isotropic materials in all single-layer fabrics and have similar mechanical properties.

The blade root consists of the 0.6 mm-thick tri?axial fabric， and its total thickness reaches 96.6 mm. The spar flange is laid by the 0.97 mmthick uniaxial fabric with two layers of 0.6 mm-thick Gelcoat fabrics on the outside of the blade. The large stack-ups of uniaxial fabric on the spar flange to increase the bending stiffness so that the tip dis?placements of the designed blade have a similar am?plitude to those operated in real conditions. The shear webs are laminated from a 30 mm-thick core material（PVC）with two layers of ±45° glass fiber（0.57 mm-thick biaxial fabric）on one side and two layers of ±45° glass fiber on the other side. The lamination on the shear webs greatly enhances the shear resistance of the blade，and the effect of bendtwist coupling. The trailing edge panel has a similar construction as the shear webs，but the core materi?al PVC is replaced with the 20 mm-thick Balsa wood. The entire inner surface of the blade is laid with two layers of 1.2 mm-thick reinforcement mate?rial，and the blade skin is made of two layers of 0.6 mm-thick Gelcoat，which generates a smooth aerodynamic surface of the blade. Because the layup of the blade is carried out in sections and regions，there will be many areas of abrupt thickness drop be?tween any two sections. For these areas，the epoxy resin is applied to fill for reducing local stress concentration. The direction from the root to the tip of the blade as a 0° of ply orientation and fiber angles are defined as unfavorable if they are biased towards the leading edge for 0° direction of ply orientation.The leading and trailing edges of the blade mainly consists of uniaxial fabric and reinforcing material.

1.3 Modal analysis and natural frequencies

From the perspective of vibration analysis and control of a structure，any vibration system can be uniformly characterized by the source-path-receiver model，as shown in Fig.3.

Fig.3 Source-path-receiver model

Fig.3 displays that the modal analysis mainly explores the second part of the model，namely the structural vibration characteristics，to obtain the nat?ural vibration characteristics such as the modal pa?rameters of the structure. The modal parameters of the blade mainly consist of natural frequencies，mode shape，and damping coefficient，among which the natural frequency and damping coefficient are global characteristics，and the mode shape presents local characteristics. Because the investigation main?ly focuses on improving the bending stiffness of the blade by varying the layer thickness of the blade，the characteristics of the natural blade frequency are highlighted.

According to the calculation formula of the nat?ural frequency of the structureis the structural stiffness matrix and M the structural mass matrix），when the structural weight remains unchanged，its natural frequency grows as the struc?tural stiffness increases. In order to improve the flap?wise and edgewise stiffness of the blade，the first two natural frequencies of the blade will be selected as the optimal objective. Using both Block Lanczos（BL）and test method to determine the modal pa?rameters of the blade，and the first two orders’natu?ral frequencies of the blade are extracted，as shown in Table 3. This table contains the calculated and tested modal parameters of the blade. The calculat?ed modal parameters are solved by commercial finite element software ANSYS.The tested modal param?eters are provided by companies that have a coopera?tive relationship with authors.

Table 3 First two orders natural frequencies of the blade

1.4 Modal sensitivity

Modal frequency sensitivity is defined as the rate of change of the structure’s natural vibration frequency with the structure parameters. The differ?ential equation of undamped free vibration of a struc?ture can be expressed as

where λirepresents the ith eigenvalue，and fithe ith natural frequency of the structure.

Assuming the jth design variable is xj，the dif?ferential of Eq.（1）to xjcan be stated by Eq.（2）

Multi ply fiTon the left side of Eq（.2）and com?bine Eq（.1）. The result can be formulated by Eq（.3）

Therefore，the eigenvalue sensitivity of equa?tion can be calculated as

In the present work，the differential approach with the advantage of relatively high calculation ac?curacy and low solution cost was employed to calcu?late the sensitivity of stiffness and mass matrices along the blade. The frequency sensitivity of the structural vibration equation can be obtained by

2 Parameter Optimization

2.1 Design variables

The natural frequencies of the blade are propor?tional to the stiffness of the blade when the blade’s weight is constant. Here，the aim of bending stiff?ness optimization of the blade can be achieved by maximizing the natural frequencies of the blade，af?fected by the composite layup thickness in the spar flanges，leading and trailing edge，and leading and trailing panel. These parts are equally divided into 27 segments along the blade span，and the layup thickness of each point was considered a design vari?able for the bending stiffness optimization. A cubic spline interpolation was carried out between the points where the layup thickness is explicitly defined to obtain the layup thickness at any area along the blade span. The blade layup thickness is symmetri?cal about the spar flange to reduce computational costs and the complexity of blade layup operations.In other words，the design variables at the leading and trailing edges of the blade are equal in magni?tude，and so are the design variables at the leading and trailing panels，as shown in Fig.4.

Fig.4 illustrates the symbolic representation of the design variables. Blade root and its two transi?tion positions are denoted by capital letters M，N and P. Leading edge，leading panel，spar flange，trailing panel，and trailing edge are represented by capital letters A，B，C，D，and E，respectively.This simple representation，which is exclusive for each design variable，is a combination of letters and numbers. For example，the three design variables highlighted in yellow in Fig.4 can be expressed as A4，B5 and C6.

2.2 Optimization model

The bending stiffness of the blade mainly in?cludes the flapwise bending stiff-ness along the axial of the blade rotor and the edgewise bending stiffness in the rotation direction of the blade rotor，and the corresponding natural modes of these two bending stiffnesses are also significantly different. To create a positive correlation between natural frequencies and bending stiffness of the blade，the two optimiza?tion strategies are proposed. The first strategy deter?mines the blade weight as a constraint to ensure the blade weight does not exceed the existing blade weight and the other strategy regards the blade weight as an optimization goal. The optimization problems are formally stated as

where f1is the first-order flapwise natural frequen?cy，f2the first-order edgewise natural frequency，m the blade weight，m0the existing blade weight，and xithe design variables. xiland xiuare the lower and upper limits of the design variables in the optimiza?tion problems.

Fig.5 shows the steps that these four schemes are performed in the optimization procedure.

Fig.5 Flow chart of performing optimization

Scheme 1：Optimize blade flapwise and edge?wise bending stiffness according to the optimal mathematical model shown in Eq.（6）.

Scheme 2：Optimize blade flapwise and edge?wise bending stiffness according to the optimal mathematical model shown in Eq.（7）.

Scheme 3：Based on the results of scheme 1，scheme 1 is performed one more time.

Scheme 4：Based on the results of Scheme 2，Scheme 1 is performed one more time.

The precondition gradient-based solver in AN?SYS is select as the optimizer to solve this optimiza?tion problem. To guarantee that the optimal solution converge on the global maximum in the design space，the genetic algorithm with a robust search ca?pability is selected to determine it.

3 Results and Discussion

3.1 Effect of layup thickness of blade on its nat?ural frequencies

The modal sensitivity of all design variables from the six parts of the blade，blade root and its transition， leading edge， leading panel， spar flange，trailing panel and trailing edge is calculated and plotted in Fig.6（a）—Fig.6（e）. A positive mod?al sensitivity indicates a positive correlation between the layer thickness and the natural frequency，while a negative modal sensitivity means a negative corre?lation. The larger the absolute value of the modal sensitivity，the greater the influence of the parame?ters on the natural frequency of the blade. In the op?timization of blade stiffness，parameters with rela?tively high sensitivity should be selected，while pa?rameters with relatively low sensitivity should be discarded.According to the sensitivity calculation re?sults in Fig.6，remove the design variables with the absolute value of modal sensitivity below 0.14 and select the 14 significant design variables.

Fig.6 Modal sensitivity of design variables at each part of the blade

Fig.6（a）depicts the modal sensitivity of design variables at the leading edge，blade root，and transi?tion. The design variables of these parts have signifi?cant effects on the first-order flapwise natural fre?quency of the blade are N，P，A1，A3，A14，and A21，the maximum sensitivity to the design vari?able N is 0.122 4 Hz/mm. These design variables influence the first-order edgewise natural frequency of the blade include N，P，A1 and A2. The maxi?mum sensitivity to the design variable A1 of the first-order edgewise modes is 0.49 Hz/mm.

The modal sensitivity of layup thickness at the leading panel is marked in Fig.6（b）. The design variables with relatively high first-order flapwise modal sensitivity in this position are B4，B8—B16，and B19，among which，the design variable B4 has the maximum sensitivity value of 0.092 Hz/mm.The effect of the design variables B8，B22 and B25 on the first-order edgewise natural frequency is rela?tively significant and B22 with the modal sensitivity value of 0.055 Hz/mm has the most conspicuous in?fluence. Further analysis shows that all design vari?ables at this part have a little effect on the natural frequency of the first-order flapwise and edgewise modes，and their modal sensitivity values are all less than 0.1 Hz/mm. This phenomenon occurs be?cause the leading-edge panel is mainly subjected to shear loads and contributes trivially to flapwise and edgewise bending stiffness.

The modal sensitivity of layup thickness at the spar flange is shown in Fig.6（c）. Since the spar flange mainly bears the flapwise load，the design variables at this position have a significant effect on the first-order flapwise natural frequency of the blade，but these design variables correspond to low first-order edgewise modal sensitivity. The maxi?mum absolute values of modal sensitivity for the first-order flapwise and edgewise mods are 0.4 Hz/mm and 0.13 Hz/mm，respectively.

Fig.6（d）illustrates the modal sensitivity of de?sign variables at the trailing panel. The modal sensi?tivity of the design variables at the trailing panel has the same characteristic as the design variables at the leading panel，and both modal sensitivity values for the first-order flapwise and edgewise modes are all less than 0.1 Hz/mm. Therefore，the design vari?ables of these two parts will be neglected in the sub?sequent structural optimization of the blade.

An illustration of the effect of the design vari?ables at the trailing edge on the modal sensitivity of first-order flapwise and edgewise mode is shown in Fig.6（e）. The trailing edge is a weak part of the blade，which basically does not bear the load.Therefore，the design variables at this part have a general trivial effect on the bending stiffness of the blade，but a few of these variables have a significant effect on the bending stiffness and cannot be ig?nored. For example，the design variables that have a non-negligible effect on the first-order flapwise nat?ural frequency are E1，E16 and E21，and the de?sign variables that have a significant impact on the first-order edgewise natural frequency are E1，E2，E3，E8，E22，E23 and E25.

3.2 Optimization results

3.2.1 Optimal results of the natural frequen?cies

The computed natural frequencies of first-order flapwise and edgewise modes of the blade and its weight from the four optimization schemes are dis?played in Table 4.

It is evident from Table 4 that four optimiza?tion schemes all meet the weight constraints，andthe natural frequencies from the four blades are in?creased by 7.5%—12% compared to the first blade.Further analysis showed that the optimization Scheme 3 increased the natural frequencies of the first-order flapwise and edgewise modes by up to 12% and 10.4% and obtained the best optimization results.

Table 4 Optimization results

3.2.2 Optimal results of the layup thickness

The resulting curve of optimal layup thickness versus blade span for optimization Scheme 3 is plot?ted in Fig.7.

Fig.7 Comparison of layup thicknesses for initial and opti?mal blades.

Fig.7 shows that the initial parameters and opti?mal solutions are driving towards a similar trend，and the optimal solutions of optimization Scheme 3 vary greatly in maximum layup thickness for each part of the blade. For example，the optimal solution shows that the maximum layup thickness of the spar flange（4.6 mm reduction）compared to the initial blade，the leading and trailing panels（2.8 mm re?duction），and the leading and trailing edges have a reduction of 4 mm. Another apparent trend is that the layer thickness of the areas near the blade root transition and the tip is increased to some extent af?ter optimization，especially the layer thickness on the spar flange near the root transition with an incre?ment of 5.5 mm. In general，the optimal solution shows that the thickness transition for any two adja?cent areas is smoother than the initial’s along the blade span.

The strength of the blade optimized in Scheme 3 is checked under extreme conditions，and the re?sults show that its strength met the requirements.Therefore，the sensitivity analysis can provide the modification direction for the dynamic modification of the blade structure，avoiding the blindness of the structural modification. Although manual modifica?tion can also improve the blade performance to a cer?tain extent，only by using the optimized design based on the sensitivity analysis to realize the auto?matic optimization of the structure can we ensure the best blade performance，which ensures that the blade dynamic performance is optimized without in?creasing its weight.

4 Conclusions

This work develops a new selection approach on design variables from six parts of the blade in the structural design optimization of the wind turbine blades considering the flapwise and edgewise bend?ing stiffness of the blade，and the selection of signifi?cant design variables based on the modal sensitivity analysis of layer parameters of the blade.

The analysis results show that the design vari?ables that significantly affect the natural frequency of the first-order flapwise mode of the blade are basi?cally distributed on the spar flange of the blade，and the modal sensitivity values of the design variables at this position are mostly higher than 0.1 Hz/mm.The design variables that have a significant effect on the natural frequency of the first-order edgewise mode are mainly located at the leading and trailing edges，especially the design variable A1 with a sen?sitivity value of 0.49 Hz/mm. Based on the modal sensitivity analysis of the layup thickness，a total of 14 design variables were selected to drive the struc?tural optimization of the blade.

Four different strategies were explored to find an optimal distribution of composite layup thickness that improves the natural frequencies of a blade in both flapwise and edgewise directions. The third case increased the natural frequencies of the first-or?der flapwise and edgewise by 12% and 10.4%，re?spectively，achieving the desirable optimization re?sults.