Enhanced Dynamic Game Method for Offshore Wind Turbine Airfoil Optimization Design

Abstract

The novel enhanced dynamic game method (EDGM) is proposed to advance game-based design approaches, with a focus on enhancing solution distribution, precision, and the ability to reveal the dynamic influence sensitivity of design variables on objective functions. An integrated mathematical model is developed by combining EDGM with PARSEC and CST parameterization methods, forming a systematic framework for offshore wind turbine airfoil optimization. Targeting airfoils with approximately 30% and 35% thickness, the study aims to improve annual energy production (AEP) and optimize the polar moment of inertia. Redesigned airfoils using the EDGM-integrated model exhibit significant enhancements in aerodynamic performance and anti-flutter capability compared to baseline airfoils DU97W300 and DU99W350. The methodology’s superiority is validated through analyses of pressure distributions, lift-to-drag ratios, and streamline patterns, as well as comparative evaluations using HV and Spacing metrics, demonstrating EDGM’s potential for broader engineering applications in complex multi-objective optimization scenarios.

1. Introduction

As fossil fuel reserves deplete and the environment deteriorates, countries worldwide are increasingly prioritizing clean energy to promote a low-carbon economy [1,2]. Among various renewable sources, wind energy stands out for its environmental benefits and high conversion efficiency, making it one of the fastest-growing sectors. The aerodynamic performance, structural design, and operational stability of wind turbine blades are all based on the aerodynamic characteristics, geometric parameters, and environmental adaptability of airfoils. They form a deep correlation through the logic of “local airfoil characteristics spanwise integrated to get blade performance” with airfoil performance being the core determinant of blade performance. Ali et al. [3] analyzed the impact of different airfoil shapes on stress distribution and fatigue resistance of wind turbines through numerical simulations, and the results showed that the NACA 0030 airfoil performed best, reducing strain by approximately 30% and fatigue damage by 25% compared to traditional structures. Lin et al. [4] calculated the lift–drag coefficients of multiple airfoils and found that airfoil design for offshore wind turbine blades was the basis for optimizing blade chord length, twist angle, and matching thrust performance. Su et al. [5] found that the performance of the airfoils was a key factor determining the overall effectiveness of the blade, and the performance of the airfoils directly affected the blade’s flutter boundary and torsional vibration. Chuang et al. [6] studied the impact of icing on the aerodynamic characteristics of airfoils, and the results indicated that changes in airfoil aerodynamic characteristics lead to significant variations in the power curve, directly related to the output power of wind turbines.

In offshore applications, where wind resources are more abundant and stable, turbines are designed with increasingly longer blades to maximize power output. However, the resulting increase in structural flexibility, combined with the harsh marine environment, significantly elevates the risk of blade flutter—a dynamic aeroelastic instability that can induce severe vibrations and compromise structural safety [7]. This study focuses on airfoil optimization to enhance not only power generation efficiency but also resistance to flutter. Airfoil shape construction is a prerequisite for optimization. Since an airfoil’s geometry is controlled by hundreds of discrete points, directly designing these coordinates would increase design variables and reduce optimization convergence speed. Therefore, airfoil parameterization methods are essential, including function-based approaches like the Hicks-Henne [8], PARSEC [9], and CST [10] methods, as well as spline-based methods like B-splines [11], Bezier curves [12], and NURBS [5].

Methods such as the particle swarm optimization algorithm [13], genetic algorithm [14], and evolutionary algorithm [5] have been applied to the optimization design of wind turbine blades or airfoils. However, the aforementioned methods have the following shortcomings: (1) They cannot reveal the dynamic influence relationships between design variables and objective functions. Different design variables have varying degrees of influence on the objective functions, and such influence relationships change with optimization iterations. Nevertheless, the above-mentioned methods fail to reveal such relationships. Yet, the influence relationships of design variables on objective functions are of great significance to designers, as they can provide valuable references. (2) When dealing with complex multi-dimensional optimization design problems (characterized by a large number of design variables, complex constraint conditions, and strong coupling of objective functions), these methods face issues such as high computational difficulty and may become trapped in local optimal solutions. Game theory, as a theory for resolving conflicts, seeks optimal decision-making solutions through mathematical modeling in the context of multiple participants’ confrontation and conflict. It is very suitable for solving such complex optimization problems. Compared to these multi-objective optimization methods [8,13,14], optimization methods based on game theory can convert a multi-objective problem into strategic interactions among multiple players. At the same time, high-dimensional and intricate issues are simplified into several lower-dimensional problems, making them easier to solve. Additionally, these methods help uncover how design variables affect sub-objectives by assigning them to specific strategy spaces for each player, which supports engineers in achieving an optimal design [15,16]. Game theory has also found widespread application in multi-objective optimization problems across a variety of engineering fields. Hati and Panda [17] evaluated the best strategies for improving the residual life of aging structures based on Nash non-cooperative and super standard cooperative games. Multi-objective robust design based on game theory was carried out by Saeedi et al. [18] and Ahmadi et al. [19], where cooperative and non-cooperative game models were applied. Tao and Andrés [20] proposed the potential game approach to search for the Nash equilibrium of the layout of the wind farms. Mo et al. [21] proposed a multi-objective collaborative game optimization method to solve the conflict of interest among multiple energy suppliers. In addition, game theory has also been applied to parallel manipulators design [22], wind turbine blade optimization [15], intelligent transportation [23], parameter optimization of thin-walled pressure vessel and twin hull ship [24], scheduling of parallel machine production and transportation [25], and aerodynamic shape design of a hypersonic air-breathing vehicle [26]. Although game theory-based optimization methods offer advantages, most of them only provide a single solution within a fixed game model. The Pareto optimal solution set is vital for multi-objective optimization. Meng et al. [27] proposed a full information cooperative game that generates solutions near the Pareto frontier. However, the static strategy space limits solution distribution and precision. To overcome this, this paper proposes an enhanced dynamic game method (EDGM), where the strategy space evolves dynamically, yielding better-distributed and more accurate solutions.

To verify the feasibility of EDGM, it is combined with the PARSEC and CST parameterization methods to optimize offshore wind turbine airfoil design. The chapter structure is as follows: Section 2 introduces the theory behind generating airfoil geometry using PARSEC and CST methods. Section 3 details the airfoil optimization model, including objective functions, design variables, and constraints. Section 4 elaborates on the theoretical foundations and solution process of EDGM. Section 5 applies EDGM with CST and PARSEC methods to regenerate airfoils with thicknesses of approximately 30% and 35%, comparing them with DU97W300 and DU99W350. Section 6 presents the conclusions.

2. Airfoil Parameterization Model

2.1. PARSEC Parametric Method

The PARSEC parameterization method primarily employs 12 parameters to describe the original airfoil [9], as illustrated in Figure 1. The meaning of the parameters highlighted in Figure 1 is reported in

Table 1. Physical meanings of PARSEC parameter method.

Parameters	Physical Meaning	Parameters	Physical Meaning
$x_{up}$	Position of upper crest	$z_{up}$	Upper crest point
$x_{lo}$	Position of lower crest	$z_{lo}$	Lower crest point
$r_{le,up}$	Upper leading edge radius	$z_{xx,up}$	Upper crest curvature
$r_{le,lo}$	Lower leading edge radius	$z_{xx,lo}$	Lower crest curvature
$z_{te}$	Trailing edge offset	${\alpha }_{te}$	Trailing edge direction angle
$∆z_{te}$	Trailing edge thickness	${\beta }_{te}$	Trailing edge wedge angle

.

The PARSEC parametric method divides the airfoil into upper and lower airfoil surfaces, which are fitted to the upper and lower surfaces of the airfoil, respectively, with the following expressions:

$$z_u\left(x\right)={\displaystyle \sum _{n=1}^6{a}_n^{(u)}{x}^{n-\frac{1}{2}}}$$

(1)

$$z_l\left(x\right)={\displaystyle \sum _{n=1}^6{a}_n^{(l)}{x}^{n-\frac{1}{2}}}$$

(2)

$z_u$ is the required $z$ coordinate for the upper surface, $z_l$ is the required $z$ coordinate for the lower surface, and $a_n^{(u)}$, $a_n^{(l)}$ are the coefficients to be solved from the twelve control variables.

The coefficients $a_n^{(u)}$ and $a_n^{(l)}$ are determined based on the following formulas derived from the twelve control variables:

$$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ x_{te}^{{\displaystyle \frac{1}{2}}}& x_{te}^{{\displaystyle \frac{3}{2}}}& x_{te}^{{\displaystyle \frac{5}{2}}}& x_{te}^{{\displaystyle \frac{7}{2}}}& x_{te}^{{\displaystyle \frac{9}{2}}}& x_{te}^{{\displaystyle \frac{11}{2}}}\\ x_{up}^{{\displaystyle \frac{1}{2}}}& x_{up}^{{\displaystyle \frac{3}{2}}}& x_{up}^{{\displaystyle \frac{5}{2}}}& x_{up}^{{\displaystyle \frac{7}{2}}}& x_{up}^{{\displaystyle \frac{9}{2}}}& x_{up}^{{\displaystyle \frac{11}{2}}}\\ {\displaystyle \frac{1}{2}}{x}_{te}^{-{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{3}{2}}{x}_{te}^{{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{5}{2}}{x}_{te}^{{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{7}{2}}{x}_{te}^{{\displaystyle \frac{5}{2}}}& {\displaystyle \frac{9}{2}}{x}_{te}^{{\displaystyle \frac{7}{2}}}& {\displaystyle \frac{11}{2}}{x}_{te}^{{\displaystyle \frac{9}{2}}}\\ {\displaystyle \frac{1}{2}}{x}_{up}^{-{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{3}{2}}{x}_{up}^{{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{5}{2}}{x}_{up}^{{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{7}{2}}{x}_{up}^{{\displaystyle \frac{5}{2}}}& {\displaystyle \frac{9}{2}}{x}_{up}^{{\displaystyle \frac{7}{2}}}& {\displaystyle \frac{11}{2}}{x}_{up}^{{\displaystyle \frac{9}{2}}}\\ -{\displaystyle \frac{1}{4}}{x}_{up}^{-{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{3}{4}}{x}_{up}^{-{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{15}{4}}{x}_{up}^{{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{35}{4}}{x}_{up}^{{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{53}{4}}{x}_{up}^{{\displaystyle \frac{5}{2}}}& {\displaystyle \frac{99}{4}}{x}_{up}^{{\displaystyle \frac{7}{2}}}\end{array}\right]\left[\begin{array}{c}{a}_1^{(u)}\\ a_2^{(u)}\\ a_3^{(u)}\\ a_4^{(u)}\\ a_5^{(u)}\\ a_6^{(u)}\end{array}\right]=\left[\begin{array}{c}\sqrt{2r_{le,up}}\\ z_{te}+{\displaystyle \frac{∆z_{te}}{2}}\\ z_{up}\\ \begin{array}{c}\tan ({\alpha }_{te}-{\beta }_{te})\\ 0\\ z_{xx,up}\end{array}\end{array}\right]$$

(3)

$$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ x_{te}^{{\displaystyle \frac{1}{2}}}& x_{te}^{{\displaystyle \frac{3}{2}}}& x_{te}^{{\displaystyle \frac{5}{2}}}& x_{te}^{{\displaystyle \frac{7}{2}}}& x_{te}^{{\displaystyle \frac{9}{2}}}& x_{te}^{{\displaystyle \frac{11}{2}}}\\ x_{lo}^{{\displaystyle \frac{1}{2}}}& x_{lo}^{{\displaystyle \frac{3}{2}}}& x_{lo}^{{\displaystyle \frac{5}{2}}}& x_{lo}^{{\displaystyle \frac{7}{2}}}& x_{lo}^{{\displaystyle \frac{9}{2}}}& x_{lo}^{{\displaystyle \frac{11}{2}}}\\ {\displaystyle \frac{1}{2}}{x}_{te}^{-{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{3}{2}}{x}_{te}^{{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{5}{2}}{x}_{te}^{{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{7}{2}}{x}_{te}^{{\displaystyle \frac{5}{2}}}& {\displaystyle \frac{9}{2}}{x}_{te}^{{\displaystyle \frac{7}{2}}}& {\displaystyle \frac{11}{2}}{x}_{te}^{{\displaystyle \frac{9}{2}}}\\ {\displaystyle \frac{1}{2}}{x}_{lo}^{-{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{3}{2}}{x}_{lo}^{{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{5}{2}}{x}_{lo}^{{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{7}{2}}{x}_{lo}^{{\displaystyle \frac{5}{2}}}& {\displaystyle \frac{9}{2}}{x}_{lo}^{{\displaystyle \frac{7}{2}}}& {\displaystyle \frac{11}{2}}{x}_{lo}^{{\displaystyle \frac{9}{2}}}\\ -{\displaystyle \frac{1}{4}}{x}_{lo}^{-{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{3}{4}}{x}_{lo}^{-{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{15}{4}}{x}_{lo}^{{\displaystyle \frac{1}{2}}}& {\displaystyle \frac{35}{4}}{x}_{lo}^{{\displaystyle \frac{3}{2}}}& {\displaystyle \frac{53}{4}}{x}_{lo}^{{\displaystyle \frac{5}{2}}}& {\displaystyle \frac{99}{4}}{x}_{lo}^{{\displaystyle \frac{7}{2}}}\end{array}\right]\left[\begin{array}{c}{a}_1^{(l)}\\ a_2^{(l)}\\ a_3^{(l)}\\ a_4^{(l)}\\ a_5^{(l)}\\ a_6^{(l)}\end{array}\right]=\left[\begin{array}{c}\sqrt{2r_{le,up}}\\ z_{te}+{\displaystyle \frac{∆z_{te}}{2}}\\ z_{lo}\\ \begin{array}{c}\tan ({\alpha }_{te}-{\beta }_{te})\\ 0\\ z_{xx,lo}\end{array}\end{array}\right]$$

(4)

where $x_{te}$ represents the horizontal coordinate at the trailing edge of the airfoil, $a_1^{(u)}~ a_6^{(u)}$ refers to the coefficients in the equations for the upper surface, and similarly, $a_1^{(l)}~ a_6^{(l)}$ denotes the coefficients in the equations for the lower surface.

2.2. CST Parametric Method

The CST method involves category and shape functions. The category function defines the base airfoil type, while the shape function refines it for precise geometric adjustments. As outlined by Akram and Kim [28], the CST parametric method is expressed as follows:

$${\displaystyle \frac{y_f}{c}}=C(\tau )S(\tau )+\tau {\displaystyle \frac{∆y_{te}}{c}}$$

(5)

$$\tau ={\displaystyle \frac{x_f}{c}}$$

(6)

$$C(\tau )={\tau }^{N_1}{(1-\tau )}^{N_2}$$

(7)

$$S(\tau )={\displaystyle \sum _{i=0}^n{a}_i}{B}_n^i(\tau )={\displaystyle \sum _{i=0}^n{a}_i}\left[{\displaystyle \frac{n!}{i!(n-i)!}}\tau {(1-\tau )}^{n-i}\right]$$

(8)

where let $x_f$ and $y_f$ represent the horizontal and vertical coordinates of the airfoil, respectively. c denotes the chord length of the airfoil, while $S(\tau )$ refers to the shape function. $B_n^i$ represents the Bernstein polynomial, and n stands for the order of the Bernstein polynomial. $a_i$ serves as the parameter for controlling the shape, and $∆y_{te}$ represents the thickness of the airfoil’s trailing edge. $C(\tau )$ is defined as the category function. N₁ and N₂ are the geometric shape-control coefficients of the category function. In this paper, N₁ = 0.5, N₂ = 1, n = 3.

3. Multi-Objective Optimization Design Model

3.1. Design Variables

When using the PARSEC parameterization method to construct the airfoil, the twelve parameters can be treated as design variables for airfoil optimization and can be expressed as: $x=(x_1,x_2,\dots ,x_{12})=(r_{le,up},r_{le,lo},x_{up},z_{up},z_{xx,up},x_{lo},z_{lo},z_{xx,lo},∆z_{te},z_{te},{\alpha }_{te},{\beta }_{te})$, the specific meanings of each design variable are detailed in Table 1. The upper and lower limits of design variables of PARSEC parameterization method are shown in

Table 2. Upper and lower limits of design variables of PARSEC parameterization method.

Design Variables	Lower Limit	Upper Limit	Design Variables	Lower Limit	Upper Limit
$x_{up}$	−0.15	0.15	$z_{up}$	−0.15	0.15
$x_{lo}$	−0.15	0.15	$z_{lo}$	−0.15	0.15
$r_{le,up}$	−0.15	0.15	$z_{xx,up}$	−0.15	0.15
$r_{le,lo}$	−0.15	0.15	$z_{xx,lo}$	−0.15	0.15
$z_{te}$	−0.15	0.15	${\alpha }_{te}$	−0.15	0.15
$∆z_{te}$	−0.15	0.15	${\beta }_{te}$	−0.15	0.15

.

When using the CST parametric method to fit the geometric shape of the airfoil, the design variables of the airfoil can be expressed as: $\mathit{x}=(x_1,x_2,\dots ,x_9)=({\mathit{a}}^{(u)},{\mathit{a}}^{(l)},∆{\mathit{y}}_{te})=\left(a_0^{(u)},a_1^{(u)},\dots ,a_3^{(u)},a_0^{(l)},a_1^{(l)},\dots ,a_3^{(l)},∆y_{te}\right)$, where, ${\mathit{a}}^{(u)}$ and ${\mathit{a}}^{(l)}$ denote the design variables for the upper and lower surfaces of the airfoil, respectively. The range of values for the design variables in the CST parametric method is presented in

Table 3. Upper and lower limit of design variables of CST parameterization method.

Design Variables	Lower Limit	Upper Limit	Design Variables	Lower Limit	Upper Limit
$a_0^{\left(u\right)}$	−0.15	0.15	$a_0^{\left(l\right)}$	−0.15	0.15
$a_1^{\left(u\right)}$	−0.15	0.15	$a_1^{\left(l\right)}$	−0.15	0.15
$a_2^{\left(u\right)}$	−0.15	0.15	$a_2^{\left(l\right)}$	−0.15	0.15
$a_3^{\left(u\right)}$	−0.15	0.15	$a_3^{\left(l\right)}$	−0.15	0.15
$∆y_{te}$	−0.15	0.15	-	-	-

.

3.2. Objective Functions

3.2.1. Anti-Flutter Performance

In the study conducted by Gao et al. [29], key factors influencing the anti-flutter performance of offshore wind turbine airfoils were thoroughly investigated. These factors include torsional stiffness, the position of the center of gravity, air density, and the location of the elastic axis. Among these, torsional stiffness, which is directly related to the polar moment of inertia I, plays a critical role in enhancing anti-flutter performance. Therefore, the optimization objective is defined as $f_1(x)=1/I$. Figure 2 presents the schematic of the airfoil and its associated parameters.

The calculation method for the polar moment of inertia I follows the approaches outlined by Gao et al. [29] and Gustavsson et al. [30].

The process of calculating the polar moment of inertia starts with the determination of the area of airfoil, denoted as A, which is given by:

$$A={\displaystyle {\iint }_{D}dxdy={\displaystyle {\int }_0^{c}dx{\displaystyle {\int }_{{\phi }_1(x)}^{{\phi }_2(x)}dy={\displaystyle {\int }_0^c\left[{\phi }_2(x)-{\phi }_1(x)\right]}}}}dx$$

(9)

where x and y represent the transverse and longitudinal coordinates of the airfoil, respectively. The chord length is denoted by c, ${\phi }_1(x)$ and ${\phi }_2(x)$ represent the functions of the lower and upper airfoil curves, respectively.

The coordinates of the airfoil’s centroid are calculated as follows:

$$\left\{\begin{array}{l}{x}_g={\displaystyle \frac{1}{A}}{\displaystyle {\iint }_{D}x}dxdy\\ y_g={\displaystyle \frac{1}{A}}{\displaystyle {\iint }_{D}y}dxdy\end{array}\right.$$

(10)

For an airfoil of homogeneous material with uniform density, the above equations simplify to:

$$\left\{\begin{array}{l}{x}_g={\displaystyle \frac{1}{A}}{\displaystyle {\int }_0^{c}x}\left[{\phi }_2(x)-{\phi }_1(x)\right]dx\\ y_g={\displaystyle \frac{1}{A}}{\displaystyle {\int }_0^c\left[\left({\phi }_2^2(x)-{\phi }_1^2(x)\right)/2\right]} dx\end{array}\right.$$

(11)

The moments of inertia, $I_x$ about the $x$-axis and $I_y$ about the $y$-axis, are calculated as follows:

$$\left\{\begin{array}{c}{I}_x={\displaystyle \underset{D}{\iint }{y}^2}dxdy={\displaystyle {\int }_0^c[({\phi }_2^3(x)-{\phi }_1^3(x))/3]}dx\\ I_y={\displaystyle \underset{D}{\iint }{x}^2}dxdy={\displaystyle {\int }_0^c{x}^2[{\phi }_2(x)-{\phi }_1(x)]}dx\end{array}\right.$$

(12)

When shifting the axis to the centroid, the adjusted moments of inertia are:

$$\{\begin{array}{l}{I}_{x1}=I_x-A{y}_g^2\\ I_{y1}=I_y-A{x}_g^2\end{array}$$

(13)

Ultimately, the polar moment of inertia is computed as follows:

$$I=I_{x1}+I_{y1}$$

(14)

3.2.2. Annual Energy Production

The output power of the airfoil was calculated using the improved fixed-point iteration method, which is an improvement on the blade element momentum (BEM) and addresses convergence failures in traditional numerical algorithms [31]. It is described in the form of pseudocode in Algorithm 1.

Assuming that the annual operation time of the wind turbine is 8700 h, which is a commonly used simplification and adjustments for real-world hours are feasible according to scheduled maintenance. The annual energy production (AEP) per unit length at the spanwise $r$ is calculated by the output power of the airfoil. The objective function is defined as $f_2(x)=-\mathrm{AEP}$. The AEP can be expressed as:

$$\mathrm{AEP}={\displaystyle \sum _{i=1}^{N-1}\frac{1}{2}}[dP(V_i)+dP(V_{i+1})]f(V_i<V_0<V_{i+1})\times 8700$$

(15)

where $dP(V_i)$ is the power per unit span of the airfoil when the wind speed is $V_i$. $f(V_i<V_0<V_{i+1})$ denotes the probability that the wind speed is greater than $V_i$ and less than $V_{i+1}$, obtained from the Weibull distribution: $f(V_i<V<V_{i+1})=\exp (-({\displaystyle \frac{V_i}{A}}{)}^k)-\exp (-({\displaystyle \frac{V_{i+1}}{A}}{)}^k)$, where A and k represent the scale factor and shape factor, respectively. In this study, A = 6.8 and k = 1.9. It is notable that A and k can be adjusted by airfoil designers according to local actual wind speed conditions, and wind speed statistics beyond the Weibull Distribution can refer to [32].

Algorithm 1: Improved fixed-point iteration method

Input: Air density $\rho$, number of blades B, wind speed V, blade rotation angular velocity $\omega$, tip speed ratio $\lambda$, rotor radius R, chord length c, twist angle $\theta$$, \mathrm{airfoil} \mathrm{coordinates} \mathit{C}\mathit{o}\mathit{o}\mathit{r}\mathit{d}$.

Output: Output power of airfoil dP(V)

1:   Step 1: Construct the fixed-point error function        2:   Define the function $g_{\varphi }=f_{\varphi }({\varphi }_i)-{\varphi }_i$, where ${\varphi }_0=f_{\varphi }({\varphi }_i)$ is the fixed-point iteration function.        3:   $f_{\varphi }({\varphi }_i)$ maps $\varphi$ to a and b through the BEM equation, then maps a and b to $\varphi$ through $\varphi =\arctan {\displaystyle \frac{(1-a)V}{(1+b)\omega r}}$. Here, a and b refer to the axial and circumferential induction factor. $\varphi$ and r represent inflow angle and radial position.        4:   Step 2: Interval search prejudgment        5:   Compute the output $\left\{{\varphi }_i(1), {\varphi }_i(2), \dots , {\varphi }_i(N)\right\}$ for the input $\left\{{\varphi }_o(1),{\varphi }_o(2),\dots ,{\varphi }_o(N)\right\}$, where ${\varphi }_i(n)=90°{\displaystyle \frac{n-1}{N-1}}$, ${\varphi }_o(n)=g_{\varphi }({\varphi }_i(n))$.        6:   Compute the product of consecutive elements in the array $\{{\varphi }_o(1){\varphi }_o(2), {\varphi }_o(2){\varphi }_o(3), \dots , {\varphi }_o(N-1){\varphi }_o(N)\}$.        7:   if ${\varphi }_o(n){\varphi }_o(n+1)\le 0$ then        8:   A zero point of $g_{\varphi }$ exists in the interval $[{\varphi }_i(n),{\varphi }_i(n+1)]$.        9:   end if       10:   Step 3: Compute a more precise interval       11:   Use the bisection method to calculate a more precise interval $[{\varphi }_i^{(L)},{\varphi }_i^{(R)}]$, where ${\varphi }_e={10}^{-5}$ represents the tolerance of the interval.       12:   Step 4: Existence check of the solution       13:   Compute ${\varphi }_o^{(L)}=g_{\varphi }({\varphi }_i^{(L)})$ and ${\varphi }_o^{(R)}=g_{\varphi }({\varphi }_i^{(R)})$.       14:   if $\left|{\varphi }_o^{(L)}-{\varphi }_o^{(R)}\right|<k_e{\varphi }_e$ then       15:   The solution to the BEM equation corresponds to the inflow angle $\varphi \hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{{\varphi }_i^{(L)}+{\varphi }_i^{(R)}}{2}}$, and compute a and b using BEM.       16:   else       17:   A jump discontinuity of $g_{\varphi }$ exists in the interval $[{\varphi }_i^{(L)},{\varphi }_i^{(R)}]$, and the BEM equation has no physical solution.       18:   Set a = b = 0.       19:   end if       20:   Step 5: Calculate normal and tangential force coefficients       21:   Calculate angle of attack α by $\alpha$ = $\varphi$ − $\theta$, and then $C_L$ and $C_D$ are computed using Xfoil (an open source software), where $C_L$ and $C_D$ refer to lift and drag coefficient.       22:   Extend the aerodynamic parameters for the angle of attack from −180° to 180° using the Montgomerie method [33] for extrapolation.       23:   Calculate the normal coefficient $C_N$ and tangential coefficient $C_T$ using blade element momentum theory.       24:    $\left\{\begin{array}{l}{C}_N=C_L(\alpha )\cos \varphi +C_D(\alpha )\sin \varphi \\ C_T=C_L(\alpha )\sin \varphi -C_D(\alpha )\cos \varphi \end{array}\right.$       25:   Step 6: Compute output power of airfoil       26:   The output power per unit span of the airfoil $dP(V)={\displaystyle \frac{1}{2}}\omega B\rho c{V}_{rel}^2{C}_{T}r$, where $V_{rel}$ is relative wind speed.

3.3. Design Constraints

To ensure the optimized airfoil satisfies geometric integrity, aerodynamic performance, and structural reliability, the design process need to satisfy the following constraints:

$$\left\{\begin{array}{l}{x}_{\min }\le x\le x_{\max },\hfill \\ {\mathrm{AEP}}_{\mathrm{opt}}\ge {\mathrm{AEP}}_{\mathrm{ori}},\hfill \\ t_1\le t\le t_2.\hfill \end{array}\right.$$

(16)

where $x_{\min }\le x\le x_{\max }$ represents the limits of design variables to ensure geometric integrity and prevent unreasonable shape, ${\mathrm{AEP}}_{\mathrm{opt}}\ge {\mathrm{AEP}}_{\mathrm{ori}}$ ensures that the optimized airfoil achieves higher AEP compared to the original airfoil, $t_1\le t\le t_2$ represents the airfoil thickness constraints, and thickness deviation range is ±0.6%. The specific meanings and ranges of the parameters are provided in Section 2 and Section 3.1.

4. Enhanced Dynamic Game Method

The formulation of the multi-objective optimization problem, which involves m objectives and n design variables, is given by:

$$\left\{\begin{array}{c}\underset{x}{\mathrm{m}\mathrm{i}\mathrm{n}} \mathit{F}(\mathit{x})=[f_1(\mathit{x}),f_2(\mathit{x}),\dots ,f_m(\mathit{x})]\\ \mathrm{s}.\mathrm{t}. \mathit{x}\in \{(x_1,x_2,\dots ,x_n){\left|x\right.}_{\min }\le x_i\le x_{\max },i=1,2,\dots ,n\}\end{array}\right.$$

(17)

The enhanced dynamic game method (EDGM) applies game theory to transform multi-objective optimization problems into game model. m optimization objectives are treated as m game players, and each objective function $f_i$ is mapped to the cost function $u_i$ according to the cooperative mode with different cooperation coefficient. The design variables $X=(x_1,x_2,\dots ,x_n)$ are assigned to each game player to form their respective strategy spaces $(S_1,S_2,\dots ,S_m)$ according to partition method of strategy space, where, $S_1=(x_i,\dots ,x_j),\dots ,S_m=(x_k,\dots ,x_l)$ and $S_1\cup S_2\dots \cup S_m=X$; $S_a\cap S_b=\Phi$$(a,b=1,\dots ,m\dots ;a\ne b)$. Notably, the strategy space is dynamically updated in each game round, with the multi-objective optimization constraints aligning with those of the game model. Strategies for each player are determined through iterative repetition and convergence evaluation under different weight cooperation coefficients. The final solution, closely approximating the Pareto front, is derived from the strategy combinations across varying weight coefficients and game rounds.

4.1. Strategy Space Partition

In this paper, based on the concepts of spatial distance and moment [34], as well as the partitioning of strategy space [15], the design variables are allocated to each player’s strategy space accordingly. The algorithm is as follows:

(1)

The influence $\Theta (j,i)$ of design variables on the objective functions is represented as follows:

$$\Theta (j,i)={\displaystyle \frac{1}{2∆x_j}}{\displaystyle \underset{x_{ji}-∆x_j}{\overset{x_{ji}+∆x_j}{\int }}\left|\frac{\partial f_i}{\partial x_j}\right|}dt$$

(18)

where $∆x_j$ is the calculating step size. The influence $\Theta (j,i)$ is normalized and the impact index of $x_j$ on the objective $f_i$ is obtained, its expression is as follows:

$$∆(j,i)={\displaystyle \frac{\Theta (j,i)}{{\displaystyle \sum _{l=1}^n\Theta (l,i)}}}(j=1,2,\dots ,n;i=1,2,\dots ,m)$$

(19)

(2)

The spatial distance $d(j,i)$ between the design variable and the objective function indicates the degree of influence, with a negative correlation between the distance and the variable’s impact. The formula is as follows:

$$d(j,i)={\displaystyle \frac{1/∆(j,i)}{{\displaystyle \sum _{h=1}^{m}1/∆(j,h)}}}(j=1,2,\dots ,n;i=1,2,\dots ,m)$$

(20)

(3)

Considering the overall effect of the design variables on all objective functions, the moment $Mo(j)$ of the design variable $x_j$ is calculated as:

$$Mo(j)={\displaystyle \frac{1}{{\displaystyle \sum _{h=1}^{m}1/∆(j,h)}}}(j=1,2,\dots ,n)$$

(21)

The moment threshold $\lambda$ is defined as:

$$\lambda ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^{n}Mo(j)}$$

(22)

(4)

In the actual allocation, design variables are ranked according to spatial distance $d(j,i)$ and $∆(j,i)$, and those with smaller distances and higher $∆(j,i)$ are prioritized for allocation. All design variables are assigned to the corresponding game players according to the corresponding instructions [15].

4.2. Construction of the Game Cost Function

The enhanced dynamic game establishes a cooperative model by combining the individual players’ cost functions with those of other participants [27]. The final cost function $u_i$ of the i-th game player is expressed as follows:

$$\left(\begin{array}{c}{u}_1\\ u_2\\ \dots \\ u_m\end{array}\right)=\left(\begin{array}{cccc}{\overleftrightarrow{u}}_{11}& {\overleftrightarrow{u}}_{12}& \dots & {\overleftrightarrow{u}}_{1m}\\ {\overleftrightarrow{u}}_{21}& {\overleftrightarrow{u}}_{22}& \dots & {\overleftrightarrow{u}}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {\overleftrightarrow{u}}_{m1}& {\overleftrightarrow{u}}_{m2}& \dots & {\overleftrightarrow{u}}_{mm}\end{array}\right)\mathbf{*}\left(\begin{array}{cccc}{w}_{11}& w_{12}& \dots & w_{1m}\\ w_{21}& w_{22}& \dots & w_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ w_{m1}& w_{m2}& \dots & w_{mm}\end{array}\right)\left(\begin{array}{c}1\\ 1\\ \dots \\ 1\end{array}\right)\to \min$$

(23)

where ${\overleftrightarrow{u}}_{ij}$ is the relative cost of the i-th player when adopting a specific strategy, calculated as ${\overleftrightarrow{u}}_{ij}={\displaystyle \frac{f_j(S_i,{\overline{S}}_i^{(0)})}{\left|f_j(S_i^{(0)},{\overline{S}}_i^{(0)})\right|}}$, where $f_j$ is the objective function, $S_i^{(0)}$ denotes the initial strategy set of the i-th player, and ${\overline{S}}_i^{(0)}$ represents the complementary strategy set associated with the initial strategy of the i-th game player. $w_{ij}$ denotes the cooperation coefficient matrix, which indicates the level of collaboration between players, with higher values indicating stronger cooperation. The symbol “*” represents the Hadamard product, “min” indicates that a smaller value of $u_i$ corresponds to a more optimal solution.

4.3. Solution Process of Enhanced Dynamic Game Method

Consider that there are Q groups, each characterized by distinct weight coefficients, and let the total number of game rounds be represented by M. Each player provides N strategies per round. By transforming the multi-objective optimization problem into a game, the enhanced dynamic game method is applied to find a solution. The weight settings of q are as follows [27]:

$$\left(\begin{array}{cccc}{w}_{11}& w_{12}& \dots & w_{1m}\\ w_{21}& w_{22}& \dots & w_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ w_{m1}& w_{m2}& \dots & w_{mm}\end{array}\right)=\left(\begin{array}{cccc}{\displaystyle \frac{q}{Q+1}}& {\displaystyle \frac{Q-q+1}{(m-1)(Q+1)}}& \dots & {\displaystyle \frac{Q-q+1}{(m-1)(Q+1)}}\\ {\displaystyle \frac{Q-q+1}{(m-1)(Q+1)}}& {\displaystyle \frac{q}{Q+1}}& \dots & {\displaystyle \frac{Q-q+1}{(m-1)(Q+1)}}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{Q-q+1}{(m-1)(Q+1)}}& {\displaystyle \frac{Q-q+1}{(m-1)(Q+1)}}& \dots & {\displaystyle \frac{q}{Q+1}}\end{array}\right)$$

(24)

The flowchart of the enhanced dynamic game method is shown in Figure 3.

5. Airfoil Optimization Design Using Enhanced Dynamic Game Method

5.1. Airfoil Optimization Design Setup

Based on the 5MW wind turbine blade model [35], two typical airfoils, DU97W300 and DU99W350, are chosen as references, as detailed in

Table 4. Airfoil parameters at selected spanwise positions.

Airfoil	Span r (m)	Chord Length c (m)	Twist Angle θ (deg)
DU97W300	22.55	4.249	9.011
DU99W350	14.35	4.652	11.48

.

Based on the settings of Reynolds number and Mach number for different spanwise positions of a 5 MW wind turbine blade described in the book by Chen and Wang [36], the Reynolds and Mach numbers for the DU97W300 airfoil are set to $Re=5\times {10}^6$ and $Ma=0.15$, while those for the DU99W350 airfoil are set to $Re=2\times {10}^6$ and $Ma=0.15$.

Section 3 provides a detailed introduction to the optimization design variables, objective functions, and constraints. The optimization is carried out using the EDGM presented in Section 4, where the strategy space for each player is divided according to the method outlined in Section 4.1.

Set the values of Q, M, and N to 10, 10, and 100, respectively. The strategy sets for each player are randomly generated within the design space, serving as the initial optimization starting point. After each round of the game, the strategy space for each player is dynamically updated according to the method in Section 4.1, and the players submit their new strategies to start the next round of the game. The process of airfoil optimization is given by Figure 4.

Three constituent modules of offshore wind turbine airfoil optimization design based on enhanced dynamic game method are shown in Figure 4.

5.2. Computation Results

Figure 5 illustrates the solution sets optimized using the EDGM methodology in conjunction with two parameterization techniques, alongside the -AEP and 1/I result derived from fitting the coordinates of the DU97W300 and DU99W350 airfoils. F_P300 and F_P350 represent the objective functions for airfoils with approximately 30% and 35% thicknesses, respectively, using EDGM and PARSEC, while F_C300 and F_C350 correspond to the objective functions for airfoils derived from EDGM and CST. The results show that the objective function values for the EDGM and PARSEC have a broader distribution compared to those from EDGM and CST.

Figure 6a shows new airfoils derived from EDGM and PARSEC, with thicknesses ranging from 29.41% to 30.24%, while Figure 6b displays airfoils derived from EDGM and CST, with thicknesses from 29.47% to 30.38%. In Figure 6c, airfoils from EDGM and PARSEC are shown, with thicknesses from 34.61% to 35.45%, while Figure 6d presents airfoils from EDGM and CST, with thicknesses from 34.51% to 35.47%. These variations are constrained by strict thickness requirements, with a deviation range of ±0.6%.

Figure 6. Comparison of the new and original airfoils. (a) DU97W300 and airfoil series of F_P300; (b) DU97W300 and airfoil series of Fc₃₀₀; (c) DU99W350 and airfoil series of F_P350; (d) DU99W350 and airfoil series of F_C350. As described in Section 4.1, the strategy space of each player is dynamically partitioned in each game round. Table 5 illustrates the dynamic partition of strategy space, S₁ and S₂, over a ten-round game process, where X_P300, X_C300, X_P350, and X_C350 refer to the design variables of F_P300, F_C300, F_P350, and F_C350, respectively. The results clearly demonstrate the dynamic evolution of each player’s strategy space across the game rounds. For instance, for airfoils with a thickness of approximately 30% generated using the EDGM and PARSEC, the design variables x₂ to x₇ predominantly fall within the strategy space S₁ in most game rounds, significantly influencing the anti-flutter performance. In contrast, x₉ to x₁₁ have a greater impact on AEP. Similarly, for airfoils with a thickness of approximately 30% generated using the EDGM and CST, the design variables x₁ to x₄ and x₆ to x₈ are primarily located in strategy space S₁ in most game rounds, playing a critical role in anti-flutter performance. Meanwhile, x₉ mainly belongs to the strategy space S₂, contributing more significantly to AEP.

Figure 6. Comparison of the new and original airfoils. (a) DU97W300 and airfoil series of F_P300; (b) DU97W300 and airfoil series of Fc₃₀₀; (c) DU99W350 and airfoil series of F_P350; (d) DU99W350 and airfoil series of F_C350. As described in Section 4.1, the strategy space of each player is dynamically partitioned in each game round. Table 5 illustrates the dynamic partition of strategy space, S₁ and S₂, over a ten-round game process, where X_P300, X_C300, X_P350, and X_C350 refer to the design variables of F_P300, F_C300, F_P350, and F_C350, respectively. The results clearly demonstrate the dynamic evolution of each player’s strategy space across the game rounds. For instance, for airfoils with a thickness of approximately 30% generated using the EDGM and PARSEC, the design variables x₂ to x₇ predominantly fall within the strategy space S₁ in most game rounds, significantly influencing the anti-flutter performance. In contrast, x₉ to x₁₁ have a greater impact on AEP. Similarly, for airfoils with a thickness of approximately 30% generated using the EDGM and CST, the design variables x₁ to x₄ and x₆ to x₈ are primarily located in strategy space S₁ in most game rounds, playing a critical role in anti-flutter performance. Meanwhile, x₉ mainly belongs to the strategy space S₂, contributing more significantly to AEP.

Table 5. Dynamic partition of strategy space over a ten-round game process of EDGM.

Strategy Space (S1 and S2)
Game Rounds	By XP300	By XC300	By XP350	By XC350
1	{1,2,3,4,5,6,7,8,12} {9,10,11}	{1,2,3,4,7,8,9} {5,6}	{2,3,4,5,7,8,9,12} {1,6,10,11}	{1,2,3,4,5,6,7,8} {9}
2	{2,3,4,6,7,8} {1,5,9,10,11,12}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,5,6,7,8,9} {10,11,12}	{1,3,5,6,8,9} {2,4,7}
3	{2,3,4,5,6,7,8,9,12} {1,10,11}	{5,8,9} {1,2,3,4,6,7}	{1,2,4,6,7,8,9,10,12} {3,5,11}	{3,4,6,7,8,9} {1,2,5}
4	{1,2,3,4,6,7,8} {5,9,10,11,12}	{1,2,3,4,6,7,9} {5,8}	{1,2,3,4,5,6,7,8,9,12} {10,11}	{1,2,3,4,5,6,7,8} {9}
5	{2,4,5,6,7,8} {1,3,9,10,11,12}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,7,9} {5,6,8,10,11,12}	{2,4,5,7} {1,3,6,8,9}
6	{1,2,3,4,5,6,7,8,11,12} {9,10}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,6,7,8} {5,9,10,11,12}	{1,2,3,4,5,6,7,8} {9}
7	{1,2,3,4,6,7,8,9} {5,10,11,12}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,5,6,7,9} {8,10,11,12}	{1,3,4,6,7,8} {2,5,9}
8	{1,2,3,4,5,6,7,8,12} {9,10,11}	{2,3,4,6,7,9} {1,5,8}	{1,2,3,4,5,6,7,8,9,11,12} {10}	{2,4,5,6,7,8} {1,3,9}
9	{1,2,3,4,5,6,7,8,9} {10,11,12}	{1,2,3,4,6,7,8} {5,9}	{1,2,3,4,5,6,7,8,9} {10,11,12}	{1,3,5,7} {2,4,6,8,9}
10	{1,2,4,5,6,7,12} {3,8,9,10,11}	{1,2,3,4,5,6,7,8} {9}	{3,4,7,8,10} {1,2,5,6,9,11,12}	{1,2,3,4,5,6,7,8} {9}

Table 5. Dynamic partition of strategy space over a ten-round game process of EDGM.

Strategy Space (S1 and S2)
Game Rounds	By XP300	By XC300	By XP350	By XC350
1	{1,2,3,4,5,6,7,8,12} {9,10,11}	{1,2,3,4,7,8,9} {5,6}	{2,3,4,5,7,8,9,12} {1,6,10,11}	{1,2,3,4,5,6,7,8} {9}
2	{2,3,4,6,7,8} {1,5,9,10,11,12}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,5,6,7,8,9} {10,11,12}	{1,3,5,6,8,9} {2,4,7}
3	{2,3,4,5,6,7,8,9,12} {1,10,11}	{5,8,9} {1,2,3,4,6,7}	{1,2,4,6,7,8,9,10,12} {3,5,11}	{3,4,6,7,8,9} {1,2,5}
4	{1,2,3,4,6,7,8} {5,9,10,11,12}	{1,2,3,4,6,7,9} {5,8}	{1,2,3,4,5,6,7,8,9,12} {10,11}	{1,2,3,4,5,6,7,8} {9}
5	{2,4,5,6,7,8} {1,3,9,10,11,12}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,7,9} {5,6,8,10,11,12}	{2,4,5,7} {1,3,6,8,9}
6	{1,2,3,4,5,6,7,8,11,12} {9,10}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,6,7,8} {5,9,10,11,12}	{1,2,3,4,5,6,7,8} {9}
7	{1,2,3,4,6,7,8,9} {5,10,11,12}	{1,2,3,4,5,6,7,8} {9}	{1,2,3,4,5,6,7,9} {8,10,11,12}	{1,3,4,6,7,8} {2,5,9}
8	{1,2,3,4,5,6,7,8,12} {9,10,11}	{2,3,4,6,7,9} {1,5,8}	{1,2,3,4,5,6,7,8,9,11,12} {10}	{2,4,5,6,7,8} {1,3,9}
9	{1,2,3,4,5,6,7,8,9} {10,11,12}	{1,2,3,4,6,7,8} {5,9}	{1,2,3,4,5,6,7,8,9} {10,11,12}	{1,3,5,7} {2,4,6,8,9}
10	{1,2,4,5,6,7,12} {3,8,9,10,11}	{1,2,3,4,5,6,7,8} {9}	{3,4,7,8,10} {1,2,5,6,9,11,12}	{1,2,3,4,5,6,7,8} {9}

5.3. Analysis and Discussion

To further validate the effectiveness of the proposed method, the full information cooperative game (FICG) [27] is used for comparison. The solution sets are compared using two metrics: hypervolume (HV) [37] and Spacing [38]. The HV metric measures the coverage of the objective space by the non-dominated solution set, with a larger HV value indicating better convergence and diversity of the algorithm. The Spacing metric measures the uniformity of the distribution of solutions in the set. It calculates the standard deviation of the minimum distances from each solution in the set to the other solutions. A smaller Spacing value indicates a more uniform distribution of the solution set. The specific values of these metrics are shown in

Table 6. Comparison of performance indicators between EDGM and FICG.

Result	Objective Function	Method	HV	Spacing
Figure 7a	FP300	EDGM	0.1486	1.2438
		FICG	0.1632	0.9712
Figure 7b	FC300	EDGM	0.1154	0.0989
		FICG	0.1128	0.1297
Figure 7c	FP350	EDGM	0.1623	0.2727
		FICG	0.1492	0.9825
Figure 7d	FC350	EDGM	0.1148	0.1131
		FICG	0.1125	0.2302

, and the comparison of solution set distributions is displayed in Figure 7. EDGM outperforms FICG in three of the Spacing metrics, indicating that the solution sets generated by EDGM have a more even distribution. Furthermore, EDGM surpasses FICG in three of the HV metrics, highlighting the superior convergence and diversity of the EDGM solution sets. Based on these two metrics, the overall performance of the solution sets obtained by EDGM is superior to that of FICG. This advantage stems from the new method’s capability to dynamically regulate the strategy space of each player in every round of the game. This dynamic adjustment not only enables more in-depth game computation but also enhances the diversity and convergence of game solutions.

Four solutions with relatively balanced AEP and anti-flutter performance are selected from the non-dominated solutions by EDGM. They are located near the middle of non-dominated solutions, with relatively balanced improvement trends in the two objective performances. The core selection criterion is “balancing AEP and anti-flutter performance”. The specific locations of four selected solutions are shown in Figure 8.

Compared to the DU97W300, solution I achieves improvements of 6.31% in AEP and 8.69% in anti-flutter performance, while solution II results in improvements of 8.22% in AEP and 3.47% in anti-flutter performance. Compared to the DU99W350, solution III brings about improvements of 4.45% in AEP and 1.14% in anti-flutter performance, whereas solution IV gains improvements of 5.38% in AEP and 0.63% in anti-flutter performance. The strict constraint on thickness deviation (±0.6%) and the well-inherent aeroelastic stability of DU99W350 lead to a relatively low improvement in anti-flutter performance. The corresponding airfoils generated by these solutions are shown in Figure 9.

Figure 10a illustrates that the newly designed airfoils exhibit a superior lift-to-drag ratio compared to the DU97W300 within the angle of attack range of [−5°, 9°]. The lift-to-drag ratio only begins to show a slight decline, falling below that of DU97W300, when the angle of attack exceeds 10°, but this occurs beyond the optimal operational range. Similarly, Figure 10b reveals that the newly generated airfoils outperform the DU99W350 in terms of lift-to-drag ratio across the angle of attack range of [−5°, 7°]. The lift-to-drag ratio drops below that of DU99W350 only when the angle exceeds 8° that the lift-to-drag ratio falls below that of DU99W350, again beyond the optimal range. The optimized airfoils, when evaluated at the optimal power coefficient, demonstrate higher annual energy generation compared to the original airfoils. Therefore, the aerodynamic performance of the newly generated airfoils surpasses that of the original airfoils.

When analyzing the pressure coefficient distribution at angles of attack of 7° and 10°, corresponding to conditions where the lift-to-drag ratio is higher than the original airfoil and slightly lower, respectively, Figure 11 presents the pressure distributions for six airfoils.

The results indicate that at an angle of attack of 7°, a notable increase in pressure difference across the upper and lower surfaces is observed for the four newly designed airfoils compared to the DU97W300 and DU99W350, which leads to a significant enhancement in lift. A similar trend is observed at an angle of attack of 10°, where the pressure difference also increases for the new airfoils.

The wind turbine blade functions under the influence of a multitude of factors, with the local angle of attack at each airfoil section varying along the span. Leveraging the parameters of the airfoils at different positions along the blade, along with key factors such as tip speed ratio, Mach number, and Reynolds number discussed in Section 5.1, the local angles of attack for both the optimized and original airfoils can be calculated, employing the methodologies detailed in the work of Schaffarczyk and Alois Peter [39].

Table 7. Lift–drag ratio of the airfoil at the local angle of attack.

Airfoil	Local AoA°	Lift–Drag Ratio
DU97W300	5.06°	99.635
Solutions I	4.32°	106.95
Solutions II	4.09°	114.332
DU99W350	8.20°	84.318
Solutions III	6.96°	96.314
Solutions IV	6.88°	92.208

presents the local angles of attack and the corresponding lift-to-drag ratios for each airfoil along the blade. The results demonstrate that the newly designed airfoils exhibit superior lift-to-drag ratios under local angles of attack when compared to the original airfoils. The streamline diagrams for each airfoil at their respective local angles of attack are presented in Figure 12. The study reveals that the streamline patterns for both the reference and newly generated airfoils remain relatively smooth, with no notable vortices or flow separations detected across all six airfoils. When comparing the DU97W300 airfoil with the newly generated designs, the vortices in the latter are notably smaller. Similarly, when comparing the DU99W350 airfoil to the new designs, the airfoil generated using the EDGM and PARSEC methods exhibits slightly larger vortices, whereas the airfoil derived using the CST method shows smaller vortices. The new airfoils exhibit no flow separation and outperform the original ones in terms of lift-to-drag ratio, thereby indicating an enhancement in aerodynamic efficiency. Furthermore, they demonstrate improvements in the uniformity of pressure distribution, the stability of lift–drag ratio, and the smoothness of flow field, which collectively confer potential adaptability to environmental disturbances. For instance, the more stable surface pressure distribution could mitigate turbulence-induced load fluctuations, while the optimized aerodynamic configuration might reduce sensitivity to aerodynamic efficiency degradation caused by sea spray.

6. Conclusions

To address the shortcomings of current multi-objective game optimization approaches, a novel enhanced dynamic game method (EDGM) is proposed. This method is then utilized to redesign airfoils to enhance both AEP and anti-flutter performance.

(1)

The results of comparison between four optimized airfoils and two reference airfoils show that the EDGM and PARSEC-based airfoil (approximately 30% thickness) improves AEP by 6.31% and anti-flutter performance by 8.69%, while the EDGM and CST-based airfoil achieves 8.22% and 3.47% improvements, respectively. For the approximately 35% thickness airfoil, the EDGM and PARSEC-based airfoil improves AEP by 4.45% and anti-flutter performance by 1.14%, whereas the EDGM and CST-based airfoil shows gains of 5.38% in AEP and steady anti-flutter improvements. Enhanced pressure distributions and lift-to-drag ratios further confirmed the methodology’s effectiveness.

(2)

By comparing the HV and Spacing metrics with the full information cooperative game method, it is demonstrated that a series of solutions generated by EDGM outperforms the full information cooperative game in terms of distribution, diversity, and convergence. This advantage is due to the ability of the new method to dynamically update the strategy space of each player in every round of the game, which facilitates deeper computation and improves the multiplicity of game solutions.

(3)

The method demonstrates good generality and scalability. Although the optimization objectives for the airfoils listed in this paper are annual energy production and polar moment of inertia, designers can set different objective functions based on their specific design needs and apply this method to obtain airfoils with improved performance. Also, the method can be applied to a broader range of engineering optimization problems.

(4)

EDGM exhibits good performance in airfoil optimization (2D parameterized models), but in scenarios involving high-dimensional objectives (e.g., simultaneously considering aerodynamic performance, structural strength, fatigue life, etc.) or multi-player game scenarios (with more than 3 objective functions), the dynamically updated strategy space may lead to increased computational complexity. It is therefore necessary to further integrate dimensionality reduction techniques to improve efficiency.

(5)

Future research plans will extend EDGM to full-machine multi-objective optimization incorporating environmental parameters, which encompasses blade twist distribution, pitch control systems, and marine environmental load coupling, to realize progressive full-chain performance enhancement from components to systems. Further refinement of quantitative analysis on the mapping between airfoil optimization and overall performance will be conducted to ensure the research retains methodological innovation while demonstrating practical value in offshore wind engineering, thereby enhancing real-world relevance and balancing the integration of methodology and applications.