Collaborative Optimization of Aerodynamics and Wind Turbine Blades

Abstract

This paper explores the application of multidisciplinary design optimization to the blades in horizontal-axis wind turbines. The aerodynamics and structural performance of blades are considered in the optimization framework. In the aerodynamic discipline, class function/shape function transformation-based parameterized modeling is used to express the airfoil. The Wilson method is employed to obtain the aerodynamic shape of the blade. Computational fluid dynamics numerical simulation is performed to analyze the aerodynamics of the blade. In the structural discipline, the materials and ply lay-up design are studied. Finite element method-based modal analysis and static structural analysis are conducted to verify the structural design of the blade. A collaborative optimization framework is set up on the Isight platform, employing a genetic algorithm to find the optimal solution for the blade’s aerodynamics and structural properties. In the optimization framework, the design variables refer to the length of the blade chord, twist angle, and lay-up thickness. Additionally, Kriging surrogate models are constructed to reduce the numerical simulation time required during optimization. An optimal Latin hypercube sampling method-based experimental design is employed to determine the samples used in the surrogate models. The optimized blade exhibits improved performance in both the aerodynamic and the structural disciplines.

1. Introduction

Wind energy has become one of the most important renewable energy sources owing to its unique advantages including a clean, free, renewable resource in the form of wind, the economic value it brings, and the low costs of maintenance. In addition to windmills, wind turbines are the main wind devices used to convert the kinetic energy of wind into electricity. In general, two primary types of wind turbines are in use: horizontal-axis wind turbines (HAWTs) and vertical-axis wind turbines (VAWTs). Comparatively, HAWTs are the most commonly used type since this type of wind turbine offers greater efficiency in generating electricity.

A typical HAWT possesses two or three blades that often resemble aircraft propellers in appearance. As a major part of HAWTs, the blade plays an important role in the overall operation of the wind turbine, being crucial to the turbine’s efficiency and performance. A well-designed wind turbine blade can significantly increase energy production while lowering maintenance and operating expenses. From the viewpoint of system engineering, designing wind turbine blades is a complicated procedure. Starting from specific needs such as power output and operating circumstances, designing wind turbine blades involves multiple domains such as aerodynamics, materials, structure, noise, and cost. Aerodynamics and structure are the main focus. In the optimization of aerodynamic performance, the airfoil and the aerodynamic shape of the blade are the main focus. Representative airfoils include the NACA 4-digital and 6-digital family [1,2], the NREL-S family [3], the Ris ϕ family [4], the DU family [5], and the FFA-W family [6]. Combined with design optimization, these airfoils are modified to achieve better aerodynamic performance. For example, Wang and Xi [7] obtained optimized airfoils by using a Kriging approximation model, in which the lift-to-drag ratio is increased by 81% compared with the original airfoil. Combined with numerical simulation, the approximation model-based optimization of wind turbine airfoils is confirmed [8]. Taking the maximum power coefficient as the objective and combining this with an airfoil noise prediction mode, Chen et al. [9] obtained a new airfoil with a higher power efficiency and better aerodynamic performance. Qiao et al. [10] developed the NPU-WA airfoil family characterized by high lift under high Reynolds numbers. Combining a Hicks–Henne-type function and a multi-island genetic algorithm, Hu [11] increased the lift-to-drag ratio of an airfoil by 15%. Li [12] developed the CQU-L airfoil family using Theodorsen theory and a B-spine curve-based integrated design method. Ma et al. [13] studied the influence of different modification methods of the airfoil trailing edge on aerodynamic performance and proposed a trailing edge symmetrical thickening method. Zhou et al. [14] presented the optimization design of an airfoil based on a Bezier curve and bend function. Results show that under certain attack angles, the lift-to-drag ratio is increased by 29.8%. Akbari et al. [15] applied the differential evolution optimization technique to obtain optimal airfoils at various wind potentials. In addition to the airfoil, the aerodynamic shape of the blade exerts an important influence on wind turbine efficiency. Classic design methods of the aerodynamic shape of blades include the Glauert method [16] and the Wilson method [17]. Novel blade shapes can be obtained by either the Glauert or Wilson method. Afterward, the designs can be improved by means of optimization design. For example, based on the Glauert method and lifting line theory, Badreddinne et al. [18] developed a simplified model that reveals a potential for predicting improved and higher rotor performances. Based on Glauert’s blade element momentum theory, Rajakumara and Ravindranb [19] presented an iterative approach to optimizing the coefficient of power, the coefficient of lift, and the wind drag of a wind turbine rotor. To optimize the aerodynamic shape of blades, Zahle et al. [20] focused on the wind turbine blade tip and proposed a surrogate-based optimization scheme to maximize energy production. Combining the lifting surface method with a free-wake model, Shen et al. [21] applied a microgenetic algorithm to optimize the blade chord length and twist angle. Combined with a blade element momentum method, Yirtici and Tuncer [22] presented a gradient-based optimization framework for blade profiles to minimize power production losses due to icing. Targeting the maximization of the axial moment, Farhikhteh et al. [23] applied the continuous adjoint method to the aerodynamic shape optimization of wind turbine rotor blades. Aiming at diminishing the curse of dimensionality in the design of blade shapes, Li et al. [24] proposed a deep learning-based optimization of the aerodynamic shape of wind turbine blades.

For the structural optimization of wind turbine blades, the materials and lay-up design are of interest. Anderson et al. [25] investigated the composite fiber angles throughout the internal structure of the blade and used gradient-based optimization to minimize a scalar stress parameter that correlates with the accumulation of fatigue damage. Regarding the blade chord, twist, layer number, layer location of the spar cap, and the position of the shear web as variables, Chen and Chen [26] investigated the strength, stiffness, and stability of an optimized model of wind turbine blades using an evolutionary algorithm. Chen et al. [27] developed a procedure combining finite element analysis and the particle swarm algorithm to optimize composite structures of the wind turbine blades, which not only allows thickness variation but also permits variation in spar cap location over the structure. Bagherpoora and Li [28] proposed a generic structural optimization procedure for the 2MW composite wind turbine blade in which the layout thickness of the root is considered. Combined with the finite element method, Tarfaoui et al. [29] presented the optimal blade structural configuration in which the skin thickness, thickness, and width of the spar flange, and thickness, location, and length of the front and rear spar web are considered. Lee and Shin [30] addressed a genetic algorithm (GA)-based two-stage design optimization procedure in which the ply lay-up pattern of the spar cap and cross-sectional design optimization at several spanwise locations are investigated. Zhao and Zhang [31] proposed a GA-based neural network to optimize the ply angle, ply thickness ratio, and ply stacking sequence of the blade and increase the static strength and stiffness of the blade. Couto et al. [32] presented a multi-objective structural optimization framework for a composite wind turbine blade, in which the blade mass, the maximum blade tip displacement, the natural frequencies of vibration, and the critical load factor are considered.

Traditionally, the strategy of series design is employed in the process of design of wind turbine blades. At each stage of the design, a single design domain is considered while all other domains are ignored. For instance, the aerodynamics domain is only concerned with obtaining the optimal lines of the blade. Afterward, structure performance is analyzed based on the obtained lines. If the demands of structure performance are not satisfied, the blade lines need to be re-designed. Such a design methodology results in low efficiency and the loss of an overall optimal solution. In fact, in the optimization design of blades, there are coupling relationships between different domains, referring to the design variables, optimization objectives, and constraints. To achieve the best overall performance of a wind turbine blade, both aerodynamic and structural requirements are considered by researchers. For example, Wang et al. [33] presented an aerodynamic/aero-elastic model that includes the structural dynamics of the blades and blade element momentum (BEM) theory. Wang [34] developed a multi-objective optimization method for the aerodynamics and mechanics of a large-scale wind turbine blade using an evolutionary algorithm and extreme load calculation. Pourrajabian et al. [35] proposed an aero-structure multi-objective optimization framework for a small wind turbine blade in which the coupling between structure and aerodynamics is considered. Dal Monte et al. [36] used a multi-objective genetic algorithm to carry out a coupled aerodynamic–structural optimization and obtained an improved distribution of both chord values and twist angle. Yang et al. [37] proposed a discrete aero-structural multi-objective optimization framework for wind turbine blades in low wind speed areas. Wang et al. [38] presented a coupled aero-structural multi-objective optimization method to simultaneously optimize the outer shape and the interior structural layout of a wind turbine blade. Yao et al. [39] performed an aero-structural design and optimization exercise to reduce the blade mass/cost by obtaining the optimal blade chord and airfoil thickness.

It is noted that in most studies on the coupled aero-structural optimization of wind turbine blades, a multi-objective optimization (MOO) strategy is preferred. Although MOO can provide a Pareto optimal solution, the collaboration between different disciplines is not comprehensively explored, especially in terms of inconsistency or conflict between disciplines. Multidisciplinary design optimization (MDO) provides an alternative to the aero-structural optimization of wind turbine blades. By establishing a hieratical optimization framework, MDO can improve optimization efficiency since parallel optimization of disciplines can be performed. Moreover, interdisciplinary inconsistency can be minimized [40]. In the field of wind turbine design, MDO has been applied to some aspects such as platforms, rotor blades, and towers. For example, Naishadh and Gangadharan [41] defined an MDO process for a wind turbine blade to maximize aerodynamic efficiency and structural robustness. Pavese et al. [42] defined an MDO process that carries out a simultaneous optimization of the aerodynamic and structural characteristics of a swept wind turbine blade design. McWilliam et al. [43] presented a design of a 100 kW bend–twist coupling blade with a monolithic MDO framework. Batay et al. [44] offered an analysis of a wind turbine blade’s aero-structural optimization based on an open-source MDO framework. Meng et al. [45] proposed an MDO framework for an offshore wind turbine tower design. Scott [46] developed an MDO framework for the aero-servo-elastic tailoring of a 20 MW wind turbine platform. Mangano et al. [47] used an MDO framework to perform aero-structural optimization of a wind turbine rotor using high-fidelity numerical simulation.

In the study, a more comprehensive application of MDO in the aero-structural optimization of wind turbine blades is presented. A collaborative optimization (CO) framework is established to analyze the aerodynamic and structural disciplines simultaneously. The main contributions of the study include: (1) A CO framework is introduced to the design of wind turbine blades to improve optimization efficiency; (2) parameter sensitivity analysis is performed to determine the design variables in the aerodynamic and structural discipline analyses; (3) in the optimization loop, surrogate models are designed to replace the high-fidelity numerical simulation modules used for aerodynamics and structure analyses; (4) combined with a class function/shape function transformation (CST) method, parameterized modeling is carried out to obtain an airfoil design of the blade; (5) and the Isight integrated optimization platform is employed to achieve the automatic design and optimization of the wind turbine blades. Compared with existing methods of design optimization of wind turbine blades, a novel optimization framework that combines CST parametric modeling, collaborative optimization, and surrogate model is proposed for the design of wind turbine blades.

2. Collaborative Optimization

The optimization procedure is the core component of MDO. Representative procedures are the multidisciplinary feasible method (MDF), individual discipline feasible (IDF), simultaneous analysis and design (SAND), collaborative optimization (CO), concurrent subspace optimization (CSSO), analytical target cascading (ATC), and bi-level integrated system synthesis (BLISS). In the study, CO is employed since it has the advantages of easy integration of software and parallel analyzing of different disciplines [48]. Unlike direct optimization, CO is characterized by a bi-level structure. A system-level problem coordinates the interdisciplinary coupling and consistency between the discipline-level subsystems is emphasized to improve the system-level performance objective.

A typical CO framework is shown in Figure 1. It divides a coupled system into a two-level optimization structure consisting of a system level and a discipline level. At the system level, the coupling relationship between disciplines is processed and the globally optimal solution is solved. At the discipline level, each discipline independently undergoes optimization without considering the coupling relationship between disciplines. The information of design variables circulates between the system level and the discipline level until the optimization goal is achieved. At the discipline level, optimization is performed according to the desired design variables given by the system level. The obtained results then return to the system level to allow analysis of the constraint and optimization goal at the system level. After the two-level optimization is completed, the optimal design variables can be obtained and resultant disciplinary performance can be evaluated.

The mathematical model at the system level optimization can be described as:

$$\left\{\begin{array}{l}\underset{\mathrm{d}.\mathrm{v}.}{\mathrm{Min}}\hspace{1em}f\left(z\right)\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{J}_i^{\ast }\left(z\right)={\displaystyle \sum _{j=1}^N{\left(x_{ij}^{*}-z_j\right)}^2=0}\\ \mathrm{d}.\mathrm{v}.\hspace{1em}z=\left[z_j\right]\end{array}\right.,$$

(1)

where $f\left(z\right)$ is the optimization objective; $J^{*}\left(z\right)$ is the constraint; $x_{ij}^{*}$ is the optimal design variable; and $z_j$ is the design variable.

At the discipline level, the optimization formulation for the i-discipline can be described as

$$\left\{\begin{array}{l}\underset{\mathrm{d}.\mathrm{v}.}{\mathrm{Min}}\hspace{1em}{J}_i\left(x_i\right)={\displaystyle \sum _{j=1}^N{\left(x_{ij}-z_j^{*}\right)}^2}\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{g}_i\left(x_i\right)\le 0\\ \mathrm{d}.\mathrm{v}.\hspace{1em}{x}_i=\left[x_{ij}\right]\end{array}\right.,$$

(2)

where $J_i\left(x_i\right)$ is the optimization objective at the discipline level; $x_{ij}$ is the design variable at the discipline level; $z_j^{*}$ is the desired design variable obtained from the system level; and $g_i\left(x_i\right)$ is the constraint.

As can be seen from (1), the consistency constraint at the system level is an equality constraint, which leads to poor robustness and slow convergence in optimization. Moreover, because the Karush–Kuhn–Tucker necessary condition is difficult to satisfy, a locally optimal solution might be obtained. To this end, relaxation of the constraint is required. In the study, a slack factor is introduced to relax the constraint. Combined with a slack factor, the constraint in (1) is rewritten as

$$J_i^{\ast }\left(z\right)={\displaystyle \sum _{j=1}^N{\left(x_{ij}^{*}-z_j\right)}^2\le \xi }$$

(3)

where $\xi$ is the slack factor, selected as a small positive constant.

3. Discipline Analysis

In the study, the aerodynamic and structural disciplines are studied in the design of wind turbine blades. To evaluate the aerodynamic and structural performance, high-fidelity numerical simulation is performed. However, in the optimization loop, surrogate models are designed to replace the time-consuming numerical simulation modules. In this section, the theoretical and numerical simulation methods applied to the aerodynamic and structural analysis of the wind turbine blade are discussed, while in the next section, the surrogate models will be illustrated. The main steps in numerical simulation are modeling, meshing, defining inputs and boundary conditions, and running the simulation based on numerical methods.

3.1. Aerodynamic Discipline

The aerodynamic characteristics of a wind turbine blade depend on its geometry, which mainly refers to the airfoil and aerodynamic shape.

3.1.1. Airfoil Design and Optimization

The design of the airfoil is fundamental to blade design and crucial to the overall performance of the wind turbine. Due to the complex geometry of airfoils, parameterized modeling is preferred in most studies. In general, the perturbation-based shape function method, orthogonal basis function method, parametric section method, and CST method are available. Comparatively, CST has the advantages of better fitting efficiency, numerical stability, and universality. In the study, the CST method is employed.

According to the CST method [49], the coordinates of an airfoil $\left(\psi ,\xi \right)$ can be described as

$$\left\{\begin{array}{l}{\xi }_U\left(\psi \right)=C\left(\psi \right)·S_U\left(\psi \right)+\psi ·\Delta {\xi }_U\\ {\xi }_L\left(\psi \right)=C\left(\psi \right)·S_L\left(\psi \right)+\psi ·\Delta {\xi }_L\end{array}\right.$$

(4)

where $C\left(\psi \right)$ is the class function while $S_i\left(\psi \right)$ is the shape function; $\Delta {\xi }_i$ is the trailing edge thickness; and the subscripts U and L denote the upper and lower surfaces of the airfoil, respectively.

The class function $C\left(\psi \right)$ is calculated as

$$C\left(\psi \right)={\psi }^{N_1}·{\left(1-\psi \right)}^{N_2}$$

(5)

where N₁ and N₂ are constants indicating the class of geometry shape of airfoil. For a representative airfoil shape with a rounded front and pointed rear, one can choose $N_1=0.5,N_2=1.0$.

Based on the Bernstein polynomial, the shape function $S_i\left(\psi \right)$ in (4) can be expressed as

$$S_i\left(\psi \right)={\displaystyle \sum _{j=0}^{N_i}{A}_i}(j){\displaystyle \frac{N_i!}{j!\left(N_i-j\right)!}}·{\psi }^j·{\left(1-\psi \right)}^{N_i-j},\left(i=U,L\right)$$

(6)

where $A_i(j)$ are the coefficients of shape function and $N_i$ is the order of polynomial.

To verify the CST method, the NACA4412 airfoil is investigated. The order of the Bernstein polynomial is selected as 7. Given the original discrete points on the surfaces of the airfoil, the coefficients $A_i(j)$ can be calculated. The shape function $S_i\left(\psi \right)$ can be further determined. The approximation results on the basis of (4) are shown in Figure 2. As calculated by using 18 points, the standard deviation $\sigma$ is 8.68 × 10⁻⁵, which verifies the CST method.

After the parameterized model of the airfoil is obtained, optimization can be performed to obtain an optimal airfoil that relates to the optimal aerodynamic performance. In the study, a high-fidelity numerical simulation, namely computational fluid dynamics (CFD), is conducted to analyze the aerodynamics of the airfoil. Computer simulation is very useful in the design of wind turbine blades since it helps improve the blade design before manufacture. By computer simulation, one can predict the blade’s performance in various environmental conditions and modify the designs to achieve optimal efficiency.

An automatic optimization framework is established on the Isight platform to obtain the optimal airfoil. The framework consists of four modules: the modeling module, meshing module, CFD module, and optimization algorithm module. Several different software packages are employed in the four modules. In detail, CAESES is used in the parameterized modeling of the airfoil; ICEM is used for meshing, based on the model obtained by CAESES; and FLUENT is used in the CFD calculation, based on the meshing results by ICEM. Based on the CFD calculation results, optimization is conducted in combination with an optimization algorithm. Note that CAESES, ICEM, and FLUENT can be integrated as a Workbench module. A brief optimization framework of the airfoil in Isight is shown in Figure 3, in which the maximization of the lift-to-drag ratio (Cl/Cd) is set as the optimization objective.

In the CFD calculation, the calculation domain is set as shown in Figure 4, where c denotes the chord length of the airfoil. The inlet is set as a velocity inlet while the outlet is set as a pressure outlet. Bound 1 and bound 2 are set as walls when the angle of attack of the airfoil is zero and as velocity inlets when the angle of attack is nonzero. The boundary of the airfoil is set as a wall.

In meshing, a structured C-block grid is employed. The meshing results are shown in Figure 5. To analyze the effects of the computational mesh on the calculation results, a mesh independence check and quality check are performed. The results of the independence check are listed in

Table 1. Mesh independence check.

	Scheme 1	Scheme 2	Scheme 3	Scheme 4	Scheme 5
Number of grids	184,863	306,083	378,695	415,955	557,703
Lif coefficient	0.4037	0.39404	0.3933	0.3900	0.38917
Error	3.7%	1.2%	1.1%	0.2%	N/A

, in which five schemes on grid number are tested. The relative errors with respect to the finest mesh scheme (i.e., Scheme 5) are calculated. As can be seen, the fourth scheme is reasonable since the accuracy approximates that of the fifth scheme while the calculation burden can be reduced. As a result, the number of grids is selected as 415,955. Figure 6 presents the results of the mesh quality check, based on the grid number adopted from Scheme 4. As can be seen, the mesh quality is over 0.85, which equates to a high quality of meshing.

If we assume an inlet velocity v = 8 m/s and select the SST k − ω turbulence model, the calculation results of the lift-to-drag ratio under the angle of attack α ∈ [0, 24°] are shown in Figure 7. As can be seen, the optimal angle of attack is 5.5° since it relates to the maximal lift-to-drag ratio.

Figure 8 shows the pressure and velocity distribution around the airfoil. As can be seen, the upper surface of the airfoil is at negative pressure and high velocity while the lower surface of the airfoil is at positive pressure and low velocity. This observation is consistent with both the Bernoulli equation and reality.

Based on the optimal angle of attack, optimization is performed to achieve the optimal airfoil profile. As presented in Figure 2, a 7th-order polynomial can accurately describe the airfoil. Therefore, the coefficients of the shape function $A_i(j)(i=U,L;j=0,1,\dots 7)$ in (6) can be set as design variables. The optimization objectives are maximization of lift and lift-to-drag ratio:

$$\left\{\begin{array}{l}\max C_l, \max {\displaystyle \frac{C_l}{C_d}},\\ \mathrm{s}.\mathrm{t}. C_l>C_{l0},C_d\le C_{d0}\end{array}\right.$$

(7)

where $C_{l0},C_{d0}$ are the lift coefficient and drag coefficient, respectively, with respect to the baseline airfoil.

An improved non-dominated sorting genetic algorithm, NSGA-II, is selected as the optimization algorithm. This algorithm is a very popular evolutionary algorithm for dealing with multi-objective optimization problems. Incorporating this algorithm into the “optimization” module shown in Figure 3, one can run an optimization loop. The optimization results are listed in

Table 2. Optimization results of lift, drag, and lift-to-drag ratio.

Coefficient	Baseline Airfoil	Optimized Airfoil	Variation (%)
$C_l$	0.92564	0.93292	0.7
$C_d$	0.02163	0.0209	−3.375
$C_l/C_d$	42.794	44.634	4.299

. As can be seen, the lift and lift-to-drag ratio are both improved after optimization while drag is decreased, compared with the baseline airfoil.

A comparison between the baseline airfoil profile and the optimized airfoil profile is shown in Figure 9. As can be recognized, the bending curvature of the airfoil decreases after optimization, which accounts for the slight increase in lift, as shown in Table 2. However, the maximal airfoil thickness decreases and its corresponding location moves forward, which accounts for the obvious decrease in drag and the resulting increase in lift-to-drag ratio.

3.1.2. Aerodynamic Shape Design

Based on the optimized airfoil, the aerodynamic shape of the wind turbine blade is further investigated. A wind turbine blade is typically a complicated, curved structure that generates lift and drag when the wind blows across the blade surface. In the study, the Wilson method is employed to obtain the aerodynamic shape of the blade. Based on BEM theory, the power coefficient with a blade element with a height of dr can be calculated as

$$\mathrm{d}{C}_p={\displaystyle \frac{8}{{\lambda }_0^2}}b\left(1-a\right)F_P{\lambda }^3\mathrm{d}\lambda$$

(8)

where ${\lambda }_0$ and $\lambda$ are the tip speed ratio and local speed ratio, respectively; a is the axial induction factor; b is the tangential induction factor; and F_P is the Prandtl tip loss factor [17], defined as

$$F_P={\displaystyle \frac{2}{\mathsf{\pi }}}\arccos \left(e^{-f}\right)$$

(9)

$$f={\displaystyle \frac{N_b}{2}}{\displaystyle \frac{R-r}{R\sin \phi }}$$

(10)

where N_b is the number of blades; R is the wind rotor radius; r is the blade local radius; φ is the incoming flow direction angle, as shown in Figure 10, where dF is the force impacted on the blade element; dF_n, dF_t, dF_L, dF_D are projection components of dF in different reference frames; α is the angle of attack while θ is the twist angle; and V is the wind velocity far upstream while V₀ is the relative wind velocity. Obviously, the relationship between velocity ${\overrightarrow{V}}_0=\left(\left(1+b\right)\omega r,\left(1-a\right)V\right)$ and incoming flow direction angle φ is

$$\phi =\arctan {\displaystyle \frac{\left(1-a\right)}{\left(1+b\right)}}{\displaystyle \frac{V}{\omega r}}$$

(11)

and the relationship between twist angle θ, direction angle φ, and angle of attack α is

$$\theta =\phi -\alpha$$

(12)

As can be seen from Figure 9, the force dF is closely related to the relative velocity V₀. Furthermore, the V₀ components projected in the axial and tangential directions depend on the induction factors a and b. According to BEM theory, the variations in a and b can be obtained as

$$a\left(1-a{F}_P\right)=b\left(1+b\right){\lambda }^2$$

(13)

and the relationship between chord and axial induction factor is

$$c={\displaystyle \frac{r}{N_b{C}_l}}{\displaystyle \frac{\left(1-a{F}_P\right)a{F}_P}{{\left(1-a\right)}^2}}{\displaystyle \frac{8\mathsf{\pi }{\sin }^2\phi }{\cos \phi }}$$

(14)

where c is the chord length and C_l is the lift coefficient, which is dependent on Reynolds number and angle of attack [19].

Based on the above calculations, the process of aerodynamic shape design of the blade is as shown in Figure 11. The main steps are as follows.

(1)

Global parameters including the number of blades, tip speed ratio, rated power and wind velocity, rotor angular velocity, and blade length are determined according to the specific needs.

(2)

One divides the blade into n parts alongside the length of the blade. Thus n + 1 sections can be obtained and each section can be viewed as an airfoil. The local radius r_i and speed ratio λ_i can also be obtained.

(3)

After initialization of induction factors a and b, one can take the maximization of the power coefficient (8) as the objective function and expression (13) as the constraint and then perform an optimization calculation to obtain updated induction factors (a_i, b_i) and tip loss factor F_Pi.

(4)

Based on the optimized {a_i, b_i, F_Pi}, one can obtain the direction angle φ_i according to (11) and chord length ci according to (14). The twist angle θ_i is then determined according to (12).

In the study, the blade is divided into 18 parts alongside the length of the blade. Based on the above calculation, the results for chord length and twist are as shown in

Table 3. Chord length and twist angle.

Station No.	Radius Ratio r/R	Chord Length c (m)	Twist Angle $\mathit{\theta }$ (Degree)
1	0.15	0.65	22.2401
2	0.2	0.72	18.1109
3	0.25	0.6707	14.6253
4	0.3	0.5927	11.7227
5	0.35	0.5266	9.3421
6	0.4	0.4711	7.4228
7	0.45	0.4248	5.9038
8	0.5	0.3866	4.7244
9	0.55	0.3551	3.8237
10	0.6	0.329	3.1409
11	0.65	0.3071	2.6151
12	0.7	0.2881	2.1856
13	0.75	0.2706	1.7914
14	0.8	0.2534	1.3717
15	0.85	0.2353	0.8657
16	0.9	0.2149	0.2126
17	0.95	0.1909	−0.6485
18	1	0.1621	−1.7784

.

Based on the obtained chord length and twist angles, CAESES is employed to run the parameterized modeling. The results are shown in Figure 12, in which Figure 12a shows the 3D model of the blade while Figure 12b depicts the variation in blade elements along the length of the blade.

Based on the model, a CFD calculation is carried out to analyze the aerodynamics of the wind turbine blade. Considering the rotation of the rotor blade, a multiple reference frame (MRF) technique is employed. The calculation domain is divided into two areas, a rotational in-fluid area and a stationary out-fluid area. The number of blades is selected as three. Therefore, both the in-fluid area and out-fluid area are set as one-third cylinders to save calculation time. The distance from the inlet to the blade is set as 3D while the distance from the blade to the outlet is 5D (where D is the length of the wind turbine blade). The radius of the out-fluid one-third cylinder is set as 2D while that of the in-fluid is set as 1.05D. The length of the in-fluid one-third cylinder is set as 6.6 times the maximal chord length, i.e., 6.6c_max. The calculation domain relating to the wind turbine blade is shown in Figure 13. Corresponding boundary conditions are set as follows. The inlet is set as a velocity inlet while the outlet is set as a pressure outlet. The boundary of the rotational in-fluid area is set as frame motion while that of the out-fluid is set as stationary. Correspondingly, the cylindrical wall of the out-fluid area is set as a stationary wall. The boundary of the interface relating to the in-fluid and out-fluid areas is set as interior. Since only a one-third cylinder is considered in the CFD calculation, the plane walls of the out-fluid are set as a periodic boundary. The boundary of the blade is set as a moving wall.

Considering the complex curves of the blade, an unstructured grid is employed in the meshing. In generating surface grids, shared topology is used to deal with interfaces. In generating volume grids, the Poly-Hexcore mosaic meshing technique is employed. Refinement is conducted for the rotational in-fluid area. After an independence check, the number of grids is determined as 1,170,775, out of which 677,134 grids make up the in-fluid area and 493,641 grids the out-fluid area. The meshing results are shown in Figure 14.

Combined with the SST $k-\omega$ turbulence model, the FLUENT calculation results are shown in Figure 15. The blade is assumed to be rotating counterclockwise. As can be seen, the windward surface is a pressure side since positive high pressure exists on this side, while the leeward surface is a suction side since most pressure on this side is negative. Moreover, the high pressure concentrates on the front edge of the blade. In general, the pressure distribution obtained from the CFD calculation is consistent with reality.

3.2. Structure Discipline

The analysis of structure mainly concerns the material and ply lay-up design to satisfy the required structural performance, which can be evaluated in terms of frequency, strain, stress, and deflection. The materials are stacked and molded to create the blade shape, which determines the aerodynamic and structural performances of the blade.

3.2.1. Material Selection

Currently, wind turbine blades are usually made from composite materials to achieve stiff and lightweight blades with long life and resilience against fatigue and corrosion. Glass fiber, carbon fiber, and resin composite are commonly used. A typical blade structure consists of several parts: the leading edge, the trailing edge, and the spar. In this study, these three parts are designed as sandwich sections. For the skin material, epoxy E-glass UD composite is selected. For the sandwich core, PVC foam is selected for the leading edge and the trailing edge while spar cap mixture is selected for the spar cap. The reason for selecting a different core material is that the spar resists the main load impacted on the wind turbine blade, i.e., the bending load. The material properties of the selected materials are listed in

Table 4. Material properties of blade.

Property	Epoxy E-Glass UD	PVC	Spar Cap Mixture
Ex (MPa)	45,000	70	25,000
Ey (MPa)	10,000	70	9230
Ez (MPa)	10,000	70	9230
PRxy	0.3	0.3	0.35
PRyz	0.4	0.3	0.35
PRxz	0.3	0.3	0.35
Gxy (MPa)	5000	27	5000
Gxy (MPa)	3846	27	5000
Gxy (MPa)	5000	27	5000
ρ (kg/m3)	2000	60	1750

, where Ei represents the elastic modulus, PR_ij represents Poisson’s ratio, G_ij represents the shear modulus, and ρ represents the density.

3.2.2. Lay-Up Design

Lay-up design plays an important role in the structural performance of wind turbine blades due to the different mechanical behavior of different parts. As stated above, the blade is formed of three parts, the leading edge, spar, and trailing edge. Comparatively, the material thickness of the spar is larger than those of the contiguous two edges since the bending moment is the main internal force that the wind turbine blade has to resist. Around the area of the blade root, the thickness of the three parts should be much larger than in other areas since flapwise bending should also be considered.

For the leading edge and trailing edge, the internal forces include the bending moment and torsion. Based on the material selection of E-glass UD composite for the skin and the PVC foam for the sandwich core, the fiber orientation is ±45°. The same fiber orientation is used for the spar, which resists most of the bending load imposed on the wind turbine blade. E-glass UD composite is selected for the skin of the spar and spar cap mixture is selected for the spar cap.

The Ansys Composite PrepPost (ACP) component in Workbench is employed to conduct the lay-up design. First, the blade is partitioned alongside the length of the blade and the length of the chord. Afterward, finite element analysis (FEA) meshing is performed. A typical shell181 element is adopted and 46024 elements are obtained. A detailed lay-up design is then carried out in the Setup module of the ACP component. The lay-up results of a section are shown in Figure 16. The thickness distribution alongside the length of the blade is shown in Figure 17. As can be seen, the thickness decreases from the root to the tip and from the spar to the edges.

3.2.3. Modal Analysis

Based on the lay-up design and FEA mesh, modal analysis is performed to determine the natural frequency and modal shapes, which assists in resonance avoidance. Treating the blade as a cantilever beam, one can use the Lanczos method to calculate the modals. The results are shown in

Table 5. Frequencies of modal shapes.

Mode	Frequency [Hz]
1	3.2396
2	7.8134
3	12.087
4	20.028
5	30.848
6	36.809

and Figure 18. As revealed, the first mode is flap while the second mode is flapwise. The third, fourth, and fifth modes are coupled flap and flapwise. The sixth mode is torsion. The first natural frequency is 3.2396 Hz. As calculated, the revolution of the rotor is 107 rpm, resulting in a frequency of 1.7833 Hz, less than the first natural frequency of the blade. Therefore, the lay-up design satisfies the vibration requirement.

3.2.4. Static Structural Analysis

Static strength and stiffness analysis is further performed to guarantee the safety of the wind turbine. As with the modal analysis, the blade is regarded as a cantilever beam. The forces imposed on the blade include gravity, centrifugal force, and pressure induced by wind. The stress distribution and deflection distribution are shown in Figure 19 and Figure 20, respectively. As can be seen, the blade experiences the highest stress in the area approaching the root, reaching a maximum of 40.435 MPa. The maximal deflection occurs at the tip of the blade, with a value of 48.279 mm. When the angle of incline of the blade relative to the wind turbine tower is 6°, the minimum distance between the blade tip and the tower is calculated to be 0.52 m. Therefore, the maximal deflection is much less than this distance.

4. MDO of the Wind Turbine Blade

On the Isight platform, a CO framework is established in which two disciplines of wind turbine blade design, aerodynamics and structure, are analyzed in parallel. In the framework, to improve optimization efficiency, surrogate models are constructed to replace the numerical simulations, i.e., the CFD simulation and the FEA simulation. For the optimization algorithm, at the system level, the artificial intelligence-based NSGA-II algorithm is employed to obtain the globally optimal solution, while at the discipline level, a gradient-based sequential quadratic subroutine, the NLPQLP, is used to guarantee optimization efficiency.

4.1. Optimization Objective

In the framework of the collaborative optimization of the aerodynamics and structure of the blades of a wind turbine, the design variables circulate between the system level and the discipline level. As discussed in the discipline analysis section above, the chord length and twist angle are the main factors governing aerodynamic performance, while the chord length, twist angle, and ply thickness are the main factors governing structural performance. As a result, chord length, twist angle, and ply thickness are selected as the design variables. The scatter relating to chord length and twist angle in Table 2 can be regressed by a third-order polynomial. Therefore, regarding the aerodynamic discipline, four design variables can be selected for the chord length, denoted as a_i (i = 1, 2, 3, 4). In the same way, the number of design variables relating to twist angle is selected as four, denoted as b_i (i = 1, 2, 3, 4). For the ply thickness, since the area around the blade root contributes a lot in withstanding external loads on the blade, limited sections around the blade root are optimized. Considering the dominant role of the spar in resisting external loads, in this study, four spar ply thicknesses are selected as the design variable relating to the structure discipline, denoted as d_i (i = 1, 2, 3, 4).

In a CO framework, the optimization objective at the system level is a combination of performance indices relating to both disciplines. In the study, the performance index from the aerodynamic discipline is selected as torque since it is closely positively related to the power coefficient of the wind turbine. The performance index from the structural discipline is selected as the mass of the blade, while the stress and deflection of the blade are taken as the constraints. It is understandable that in the case of guaranteed strength and stiffness, a blade with a lower mass can lower the manufacturing cost. Thus, the optimization objective at the system level is designed as

$$J=\max \left\{F=k_{1}D-k_{2}M\right\}$$

(15)

where D denotes the torque of the blade; M is the mass of the blade; and k_i is the weight of discipline i in the objective.

Unlike multi-objective optimization, CO performs a hierarchical optimization. At the discipline level, the optimization objective is not related to the discipline performance but discrepancies associated with design variables. Since different design variables exert different influences on the discipline performance, the effects of design variables are analyzed. To this end, parameter sensitivity analysis is performed to determine the effects of the design variables in the aerodynamic and structural discipline analyses. The principles adopted in the sensitivity analysis are described in [50]. Four chord length a_i, four twist angles b_i, and four spar ply thicknesses d_i (i = 1, 2, 3, 4) are evaluated to determine their importance. For the aerodynamic discipline, design variable effects on the torque of the blade determined by sensitivity analysis are presented in Figure 21a. For structural discipline, the effects of design variables on the mass of the blade determined by sensitivity analysis are presented in Figure 21b.

Based on the results of the sensitivity analysis, one can select important variables as the disciplinary design variables. The two following optimization objectives at the discipline level can be obtained as

$$f_1={\left(b_4^{*}-b_4\right)}^2+{\left(b_3^{*}-b_3\right)}^2+{\left(b_2^{*}-b_2\right)}^2+{\left(a_4^{*}-a_4\right)}^2$$

(16)

for the aerodynamics discipline, and

$$f_2={\left(d_1^{*}-d_1\right)}^2+{\left(d_2^{*}-d_2\right)}^2+{\left(a_4^{*}-a_4\right)}^2+{\left(d_3^{*}-d_3\right)}^2+{\left(d_4^{*}-d_4\right)}^2$$

(17)

for the structure discipline. In objectives (16) and (17), the variables with a star superscript denote the design variables from the system level, i.e., the desired design variables.

4.2. Surrogate Model

Numerical simulation has been widely applied in many engineering areas since it can provide accurate calculation results. However, the calculation might be lengthy, especially for complicated systems. Directly incorporating a numerical simulation module into an optimization farmwork degrades the efficiency. The use of surrogate models provides an effective way to perform discipline analysis in an MDO application since the use of an accurate surrogate model can significantly improve optimization efficiency. In this study, two surrogate models are constructed to replace the CFD calculation in the aerodynamic analysis and the FEA calculation in the structural analysis.

A typical process of surrogate model construction is shown in Figure 22. Starting from data sampling, one might select a type of surrogate model and then generate the model by using the samples. Afterward, by performing an error analysis, one can confirm if the generated surrogate model satisfies accuracy requirements and can be applied to the optimization. In the step of data sampling, a design of experiment (DOE) method, optimal Latin hypercube design (Opt LHD) is employed. In the step of model selection, the classical Kriging model is selected. To verify the Kriging surrogate model, determination coefficients are calculated.

As stated above, torque is selected as the performance index for the aerodynamic discipline while mass is selected as the performance under the stress and deflection constraints. The torque can be calculated by the CFD technique. The mass, stress, and deflection can be determined by FEA. In the study, Kriging surrogate models are constructed with respect to these indices/constraints.

Table 6. Determination coefficients.

	D	M	${\mathit{w}}_{\mathbf{max}}$	${\mathit{\sigma }}_{\mathbf{max}}$
R2	0.93107	0.97377	0.9183	0.96663

lists the determination coefficients for the surrogate models. In the table, R² denotes the determination coefficient, denotes the maximal deflection of the blade, and denotes the maximal stress. As can be seen from the table, the surrogate models have high accuracy, which implies that they can replace the numerical simulation modules in an optimization project.

The CO framework on the Isight platform combined with the surrogate models is shown in Figure 23.

By running the two-level optimizers shown in Figure 23, one can obtain the optimization results, as shown in

Table 7. Optimization results of the wind turbine blade.

Variable	Initial Values	Range	Optimized Results
a1	−1.7135	[−1.74, −1.7]	−1.7242
a2	3.9152	[3.9, 3.94]	3.9108
a3	−3.3233	[−3.34, −3.3]	−3.3081
a4	1.2836	[1.26, 1.3]	1.2907
b1	−81.1153	[−82, −80]	−80.4060
b2	177.4159	[176, 178]	177.7467
b3	−137.1777	[−138, −136]	−136.2567
b4	44.5987	[43, 45]	44.4364
d1 (mm)	20	[15, 25]	17.7579
d2 (mm)	20	[15, 25]	16.0984
d3 (mm)	18	[15, 25]	15.3651
d4 (mm)	16	[15, 25]	15.1506
D (N·m)	295.22	N/A	325.04 (+10.1%)
M (t)	0.40475	N/A	0.3845 (−5%)

, in which the design variables and the optimization objectives, i.e., the torque and the mass of the blade, are listed. As can be seen, the torque is increased by 10.1%, which implies that the power coefficient of the wind turbine can be improved. Similarly, the mass is decreased by 5%, which implies that the cost of the wind turbine can be reduced.

The optimized chord and twist angle of the wind turbine blade are shown in Figure 24. As can be seen, the chord length of the blade increases to some extent. The twist angle becomes smaller around the root and larger around the tip, which contributes to the improvement in the torque. In terms of contributions to the reduction in mass, although the chord length and the twist angle do not change sharply, the thickness of the blade noticeably decreases, as can be seen in Table 7. It is noted that in this study, the optimized design is not a completely different design but an improvement on the original NACA 4412 airfoil.

5. Conclusions

Designing a wind turbine blade is a complicated system engineering procedure, in which many disciplines should be considered to achieve a blade that can maximize wind energy extraction while reducing the manufacturing cost and the effect of internal and external disturbances. In this study, the aerodynamic and structural performance of a horizontal-axis wind turbine blade are studied in a collaborative optimization framework. Through design optimization, the torque provided by the blade improves while the mass of the blade decreases under the stress and deflection constraints. Based on the results, the following conclusions can be made.

(1)

Collaborative optimization provides an effective optimization strategy for the design of wind turbine blades. In the strategy, multiple disciplines can be optimized simultaneously. Unlike direct optimization, the consistency between disciplines is considered, which helps in obtaining an overall globally optimal solution.

(2)

In MDO and multiobjective optimization, design variables might be different for different disciplines. Parameter sensitivity analysis emphasizes the important design variables in a specific discipline. As can be seen from the results of the parameter analysis, the chord length and twist angle exert an important influence on the torque while spar ply thickness greatly affects the mass of the blade.

(3)

Surrogate models in an MDO framework can effectively replace the time-consuming use of numerical simulation modules, with high accuracy and high efficiency. In the study, an optimal Latin hypercube design-based Kriging surrogate model is established.

(4)

Parameterized modeling can effectively improve the modeling accuracy of an airfoil and help the subsequent optimization of the wind turbine blade. In the study, class function/shape function transformation is shown to be feasible for representing the airfoil of the wind turbine blade.

In this study, only the aerodynamic and structural performance criteria of the wind turbine blade are considered. Other important performance criteria such as noise and aeroelastics are not considered. Similarly, fatigue and instability are not considered in the structural discipline analysis. In future work, these aspects will be studied. Moreover, the fluid–structure interaction (FSI) issue will be addressed in the next work. In the paper, the optimization work is based on numerical simulation; in future work, experiments will be performed to validate the numerical simulation.