Multi-Objective Optimization Design of a Mooring System Based on the Surrogate Model

Abstract

As the development of floating offshore wind turbines (FOWTs) progresses from offshore to deeper sea, the demands on mooring systems to ensure the safety of the structure have become increasingly stringent, leading to a concomitant rise in costs. A parameter optimization method for the mooring system of FOWTs is proposed, with the mooring line length and anchor radial spacing as the optimization variables, and the minimization of surge, yaw, and nacelle acceleration as the objectives. A series of mooring system configuration samples are generated by the fully analytical factorial design method, and the open source program OpenFAST is employed to simulate the global responses in the time domain. To enhance the efficiency of the optimization process, a multi-objective evolutionary algorithm, Non-dominated Sorting Genetic Algorithm II (NSGA-II), is utilized to find the Pareto-optimal solutions, alongside a Kriging model, which serves as a surrogate model for the FOWTs. This approach was applied to an IEC 15MW FOWT to demonstrate the optimization procedure. The results indicate that the integration of the genetic algorithm and the surrogate model achieved rapid convergence and high accuracy. Through this optimization process, the longitudinal motion response of FOWTs is reduced by a maximum of 6.46%, the yaw motion by 2.87%, and the nacelle acceleration by 11.55%.

1. Introduction

With the increasing emphasis on offshore wind power, the scale of offshore wind farms continues to expand, and the development of wind power is gradually transitioning from shallow to deep waters to harness more wind energy resources. As a critical subsystem of FOWTs, an unsuccessful design of mooring systems could result in the redundancy of materials, excessive drift motion of the floater, or even accidental collisions with adjacent offshore structures. Therefore, it is imperative to identify a solution that ensures the safe and reliable operation of the entire system, thereby promoting offshore wind power generation [1].

Mooring system design entails the consideration of various factors, including the composition of mooring lines, the type of platform, and environmental conditions. The system’s capacity to maintain the platform’s position is critical, as it significantly impacts the integrity of the risers and the overall stability of the floating platform [2]. There are several parameters that should be taken into consideration for the design of a mooring system. For example, anchor types, the mooring line strength, anchor positions, the seabed pattern, environmental conditions, and the platform life [3]. The selection of parameter values determines the capability of the mooring system. Meanwhile, the choice of methods for mooring system design and optimization dictates the effectiveness and efficiency of the related research.

Sclavounos et al. [4] conducted a pioneering parametric design study of the FOWT and its mooring system using a coupled frequency domain (FD) model. This model is a six-degrees-of-freedom (DoF) platform model, where the contributions of the mooring lines and the wind turbine are modeled as additional inertia, damping, and stiffness matrices. Philippe et al. [5] and Karimi et al. [6] also proposed similar models. Karimi et al. [7] conducted FD-based multi-objective optimization on a 5 MW FOWT, identifying the Pareto front that minimized platform cost while maximizing wind turbine performance. Brommundt et al. [8] used a FD model to optimize the mooring system by minimizing the line length and the cost. Bruschi et al. [9] and Barbanti et al. [10] investigated the optimal mooring weight mass and position with the objective of minimizing the platform response. The above researchers all utilized frequency-domain analysis methods to design efficient and reliable mooring systems, which improved the overall performance of floating wind turbines. However, FD models are limited in their ability to account for nonlinear effects, leading to optimization results that do not satisfy the requirements [11].

Additionally, some scholars have also conducted mooring optimization based on time-domain models. Mahdi et al. [12] developed a genetic algorithm for mooring system optimization using a time-domain analysis tool to capture the nonlinear characteristics. Bruno et al. [13] optimized mooring systems using a PSO algorithm based on a fully coupled nonlinear time-domain model of the FPSO–mooring–riser, providing valuable references for engineering practice. In this optimization process, fully nonlinear time-domain simulations are computationally expensive, significantly increasing CPU costs.

Surrogate models can effectively approximate numerical analysis results and address the challenge of balancing accuracy and efficiency. They capture the relationships between the objective or constraint functions and the design variables using simplified equations. Surrogate models typically require only a limited number of costly numerical analyses, which can greatly reduce the computational effort needed to find the optimal design. Consequently, this approximation method has been extensively utilized in engineering design and optimization to reduce the computational burden [14,15]. In recent years, some scholars have begun to use surrogate models to optimize the design of FOWTs. Zhang et al. [16] and Zhang et al. [17] employed a radial basis neural network algorithm to construct a surrogate model as an alternative to the fully coupled computational model, based on three-dimensional potential flow theory using AQWA and OpenFAST. Sun et al. [18] developed a Kriging surrogate model for the rapid geometric design of friction ring dampers (FRDs) with complex geometries. Thapa et al. [19] and Mao et al. [20] used a support vector machine with a grid-search-optimized mixed kernel method (SVM-GSM) to develop a surrogate model, which significantly reduces the computational time required for optimization analysis.

In addition, the accuracy and cost of multi-objective optimization for mooring systems also depend on the optimization algorithm used. Optimization techniques are generally divided into two main categories: gradient-based algorithms and metaheuristic algorithms. Gradient-based algorithms are classical approaches that effectively solve problems represented by differentiable objective functions. However, they tend to struggle with real-world scenarios that are non-differentiable, discontinuous, nonlinear, or multimodal. One way to apply gradient-based algorithms is by simplifying numerical models or analytical methods, though this can lead to discrepancies from actual outcomes [21,22]. Metaheuristic algorithms, in contrast, do not depend on the structural information of the objective function and are highly effective in solving complex engineering optimization problems. The core concept of metaheuristic algorithms is designing intelligent, iterative search processes that mimic behaviors observed in biology, nature, or society, which is why they are often referred to as intelligent optimization algorithms [23]. Examples of metaheuristic algorithms include genetic algorithms, differential evolution algorithms, and particle swarm optimization algorithms, all of which have been successfully applied to address engineering optimization challenges. For a detailed review of the specific applications of these optimization algorithms, please refer to references [24,25,26].

Non-dominated Sorting Genetic Algorithm II (NSGA-II), a major subset of metaheuristic algorithms, is one of the most widely used intelligent optimization algorithms worldwide. It is based on genetic algorithms and achieves multi-objective optimization through non-dominated sorting, crowding distance calculation, and an elitism mechanism. NSGA-II is highly effective in handling complex optimization problems, particularly in situations where there are conflicting objective functions. It is commonly applied in fields such as engineering design, resource allocation, and other areas where balancing multiple optimization objectives is critical.

Currently, research on the mooring systems of FOWTs lacks a simple and effective optimization method. Considering the accuracy and cost of optimization, this paper proposes a method that combines a Kriging-based surrogate model with a genetic algorithm for parameter optimization of mooring systems. In this study, we conducted optimization of the mooring system for the IEA 15MW semi-submersible FOWT. Multi-objective optimization was conducted with the mooring line length and anchor radial spacing as the optimization variables and the minimization of surge, yaw, and nacelle acceleration of FOWTs as the optimization objectives. Global responses calculations were performed using OpenFAST 11.0 software, and a surrogate model was constructed using the Kriging algorithm. The genetic algorithm was then used to optimize the mooring line parameters, yielding optimal values for each parameter.

The present paper is structured as follows. In Section 2, the relevant theories are introduced. Section 3 introduces the parameters of FOWTs and the optimization functions. Section 4 describes the construction of the surrogate model and the optimization results. Section 5 offers concluding remarks.

2. Surrogate Models for the Optimization of FOWTs

In the fully coupled time-domain model, the dynamic response of FOWTs needs to be solved iteratively by the power equation. In contrast, a surrogate model bypasses this iterative process by capturing the nonlinear mapping relationship between the structural parameters of FOWTs and their power response. By leveraging surrogate models, continuous optimization of FOWTs within the design space becomes both feasible and efficient.

2.1. Kriging Surrogate Modeling

2.1.1. Kriging Algorithm

The surrogate model uses the approximation function $\widehat{y}=\widehat{f} \left(x,\omega \right)$ to replace the complex mapping relationship $f\left(x,y\right)$ between the variable $x$ and the outcome $y$, where $\mathbf{\omega }$ is a parameter that determines the exact state of the model. Using N sample points to build the approximation function can be written as follows:

$$\widehat{y}=\widehat{f} \left(x,\mathbf{\omega }\right)={\mathbf{\omega }}^{\mathit{T}}\mathbf{\Psi }={\displaystyle \sum }_{i=1}^{\mathrm{N}}{w}_i\psi \left(x-x_i\right)$$

(1)

where $\psi \left(x-x_i\right)$ is the basis function describing the correlation between any point x in the design space and the first $i$ sample point $x_i$.

The Kriging model is employed to build the surrogate model [27]. The Kriging neural network algorithm uses the basis function $\psi =e^{-{\displaystyle {\displaystyle \sum }_{j=1}^n}{\theta }_j*{\left|x_j^{\left(i\right)}-x_j\right|}^{P_j}}$. The correlation between any two of the N sample points, $x^{\left(p\right)}$ and $x^{\left(q\right)}$, is expressed using Equation (2):

$$cor\left[Y\left(x^{\left(p\right)}\right),Y\left(x^{\left(q\right)}\right)\right]=e^{-{\displaystyle {\displaystyle \sum }_{j=1}^n}{\theta }_j*{\left|x_j^{\left(p\right)}-x_j^{\left(q\right)}\right|}^{P_j}}$$

(2)

where $n$ is the dimension of the random variable, and $\theta$ and $P$ are parameters that adjust the smoothness of the basis functions [28]. $P\in \left\{1,2\right\}$. The value of $\theta$ can be obtained by searching for the maximum value of the likelihood function of $y$ when the dimensionality of the variable is low.

$$L={\displaystyle \frac{e^{\left[-\frac{{\left(y-1\mu \right)}^T{\mathbf{\Psi }}^{-1}\left(y-1\mu \right)}{2{\sigma }^2}\right]}}{{\left(2\pi {\sigma }^2\right)}^{\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left|\mathbf{\Psi }\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$$

(3)

$$\mathbf{\Psi }=\left(\begin{array}{c}cor\left[Y\left(x^{\left(1\right)}\right),Y\left(x^{\left(1\right)}\right)\right]\dots cor\left[Y\left(x^{\left(1\right)}\right),Y\left(x^{\left(N\right)}\right)\right]\\ ⋮\\ cor\left[Y\left(x^{\left(N\right)}\right),Y\left(x^{\left(1\right)}\right)\right]\dots cor\left[Y\left(x^{\left(N\right)}\right),Y\left(x^{\left(N\right)}\right)\right]\end{array}\right)$$

(4)

where $\mu$ and $\sigma$ are the variances and means of the sample points. $\mathbf{\Psi }$ is the correlation matrix between the $N$ sample points at $\omega ={\mathbf{\Psi }}^{-1}y$. The covariance matrix of the sample point response values can be written as follows:

$$cov\left(\mathbf{Y},\mathbf{Y}\right)={\sigma }_Y^2\mathbf{\Psi }$$

(5)

where ${\sigma }_Y^2$ is the variance of the response value at the sample point. Based on the correlation matrix, the response value $\widehat{y}$ at unknown point $x$ can be obtained using the Kriging surrogate model with known values of $\theta$ and $P$, and the correlation vector between the predicted and sample points is the following:

$$\mathbf{\psi }=\left(\begin{array}{c}cor\left[Y\left(x^{\left(1\right)}\right),Y\left(x\right)\right]\\ ⋮\\ cor\left[Y\left(x^{\left(n\right)}\right),Y\left(x\right)\right]\end{array}\right)=\left(\begin{array}{c}{\mathit{\psi }}^{\left(1\right)}\\ ⋮\\ {\mathit{\psi }}^{\left(n\right)}\end{array}\right)$$

(6)

Based on this, the correlation matrix of the augmentation can be written as follows:

$$\tilde{\mathbf{\Psi }}=\left(\begin{array}{cc}\mathbf{\Psi }& \mathbf{\psi }\\ {\mathbf{\psi }}^{\mathit{T}}& 1\end{array}\right)$$

(7)

When replacing the original correlation matrix with the augmented correlation matrix, the likelihood function of the augmented data can be obtained through a series of calculations:

$$l{n}^{\left(L\right)}={\displaystyle \frac{1}{2{\widehat{\sigma }}^2\left(1-{\mathbf{\psi }}^{\mathit{T}}{\mathbf{\Psi }}^{-1}\mathbf{\psi }\right)}}{\left(\widehat{y}-\widehat{\mu }\right)}^2+\left({\displaystyle \frac{{\mathbf{\psi }}^{\mathit{T}}{\mathbf{\Psi }}^{-1}\left(y-1\widehat{\mu }\right)}{\widehat{\sigma }\left(1-{\mathbf{\psi }}^{\mathit{T}}{\mathbf{\Psi }}^{-1}\mathbf{\psi }\right)}}\right)\left(\widehat{y}-\widehat{\mu }\right)$$

(8)

This is a quadratic function on $\widehat{y}$ whose maximum value is easily obtained. The maximum likelihood estimate (MLE) of $\widehat{y}$ is as follows:

$$\widehat{y}=\widehat{\mu }+{\mathbf{\psi }}^{\mathit{T}}{\mathbf{\Psi }}^{-1}\left(y-I\widehat{\mu }\right)$$

(9)

where $\widehat{\mu }$ and ${\widehat{\sigma }}^2$ are maximum likelihood estimates of the mean and variance $\mu$ and ${\sigma }^2$ of the sample points.

$$\widehat{\mu }={\displaystyle \frac{{\mathbf{l}}^{\mathit{T}}{\mathbf{\Psi }}^{-1}y}{1^{\mathit{T}}{\mathbf{\Psi }}^{-1}1}}$$

(10)

$${\widehat{\sigma }}^2={\displaystyle \frac{{\left(y-1\mu \right)}^T{\mathbf{\Psi }}^{-1}\left(y-1\mu \right)}{n}}$$

(11)

2.1.2. Surrogate Modeling

(1) After determining the design variables in the optimization problem, full analytic factor sampling is used in the experimental design. The objective is to generate a certain number of sample points distributed in the design space to obtain a comprehensive characterization of the model response.

(2) Basis for Numerical Modeling

The numerical model is then constructed based on the mooring configurations obtained from the design variables at each sampling point. A series of time-domain simulations of the FOWT are performed to calculate target values, such as platform offsets.

(3) Kriging Surrogate Model

A Kriging surrogate model is developed using the sample points from step (1) as inputs and the corresponding target values from step (2) as outputs. This model establishes the functional relationship between the input and output data, thereby facilitating a more efficient search for the optimal solution in subsequent steps.

2.2. Global Response Calculation of FOWTs

A floating wind turbine system is a complex multi-body system including blades, a nacelle, tower, floating platform, and mooring system. The fully coupled motion equation for the FOWT in the time domain is as follows [29]:

$$M_{kj} \stackrel{..}{xj}=F_j(xj,\stackrel{.}{xj},t)$$

(12)

$$F_j(x_j,\stackrel{.}{x_j},t)=F_j^{\mathrm{rw}}+F_j^{\mathrm{tw}}+F_j^{\mathrm{w}}+F_j^{\mathrm{s}}+F_j^{\mathrm{c}}$$

(13)

where $xj$, $\stackrel{.}{xj}$, and $\stackrel{..}{xj}$ represent the displacement, velocity, and acceleration; $M_{kj}$ is the mass matrix; and $F_j$ is the force matrix, which includes the aerodynamic force on wind turbine $F_j^{\mathrm{rw}}$, aerodynamic force on tower $F_j^{\mathrm{tw}}$, wave force on the platform $F_j^{\mathrm{w}}$, hydrostatic restoring force $F_j^{\mathrm{s}}$, and current force $F_j^{\mathrm{c}}$.

(1)

Aerodynamic Load

The aerodynamic loads on wind turbine $F_j^{\mathrm{w}}$ are calculated using the Blade Element Momentum (BEM) theory. This involves defining the axial induction factor $a$ and the tangential induction factor $a^{\prime }$, and iteratively computing the aerodynamic loads at each radial position until the values of $a$ and $a^{\prime }$ converge. From this, the overall thrust $T$ and torque of the wind turbine $P$ are determined.

$$T={\displaystyle {\int }_0^{R}d}T(r)=4\pi {\rho }_{\mathrm{air}}B{u}_0^2{\displaystyle {\int }_0^{R}a}(1-a)rdr$$

(14)

$$P={\displaystyle {\int }_0^{R}d}M(r)·\omega =4\pi {\rho }_{\mathrm{air}}B{u}_0{\omega }^2{\displaystyle {\int }_0^R{a}^{\prime }}·(1-a)r^{3}dr$$

(15)

where $B$ represents the number of blades, and ${\rho }_{\mathrm{air}}$ represents the air density.

(2)

Fluid Loads

The load of fluid on the floating wind turbine platform can be divided into $F_j^{\mathrm{w}}$, $F_j^{\mathrm{s}}$, and $F_j^{\mathrm{c}}$.

Using the three-dimensional potential flow theory, the wave force $F_j^{\mathrm{w}}$ can be separated into the wave excitation force $F_j^{\mathrm{exc}}$ that arises from the incident potential ${\mathrm{\Phi }}^I$ and diffraction potential ${\mathrm{\Phi }}^D$, and the radiation force $F_j^{\mathrm{R}}$ that is generated by the radiation potential ${\mathrm{\Phi }}^R$ in the frequency domain.

$$F_j^{\mathrm{w}}(\omega )=-{\rho }_{\mathrm{water}}{\displaystyle \int {\displaystyle \int \frac{\partial ({\mathrm{\Phi }}^I+{\mathrm{\Phi }}^D+{\mathrm{\Phi }}^R)}{\partial t}}}{n}_{j}dS$$

(16)

In the time domain, the wave excitation force is decomposed into the first-order wave force $F_j^{\mathrm{exc}\left(1\right)}$ and second-order wave force $F_j^{\mathrm{exc}\left(2\right)}$. These forces are obtained by convolving the impulse response function $h$ with the wave elevation $\zeta$.

$$F_j^{\mathrm{exc}\left(1\right)}={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{h_j}^{(1)}}(t-{\tau }_1)\zeta ({\tau }_1)d{\tau }_1$$

(17)

$$F_j^{\mathrm{exc}\left(2\right)}(t)={\displaystyle \frac{1}{4{\pi }^2}}{\displaystyle \int {\displaystyle {\int }_{-\infty }^{+\infty }{h_j}^{(2)}}(t-{\tau }_1,t-{\tau }_2)\zeta ({\tau }_1)\zeta ({\tau }_2)d{\tau }_{1}d{\tau }_2}$$

(18)

The hydrostatic restoring force $F_j^{\mathrm{s}}$ can be calculated from the hydrostatic stiffness.

(3)

Mooring Line Tension

The lumped mass method is generally used for modeling the mooring line [30,31]. The motion equation of each mooring line node is as follows:

$$M_i{a}_i=F_{Ti}-F_{Ti-1}+F_{Di}+F_{Ai}-W_i$$

(19)

$$F_{Di}={\displaystyle \frac{1}{2}}{\rho }_{\mathrm{water}}l{C}_{D}D{u}_N|u_N|$$

(20)

$$F_{Ai}={\rho }_{\mathrm{water}}{C}_{M}V(-a_i)$$

(21)

where $a_i$ is the node acceleration vector, $M_i$ is the mass of the node, $W_i$ represents the gravity acting on the node, $F_{Ti}$ and $F_{Ti-1}$ are the tensions in the mooring line elements $i$ and $i-1$, respectively, $F_{Ai}$ and $F_{Di}$ correspond to the fluid inertial force and drag force at the node, $D$ is diameter of the mooring line, $C_D$ is the drag coefficient of the mooring line, and $C_M$ is added mass coefficient of the mooring line.

2.3. NSGA-II Algorithm

The NSGA-I algorithm is a computational model established by simulating the mechanism of evolution theory and genetics, with good convergence and diverse operation results, solving multi-objective optimization problems in many fields. It has also been used for algorithmic improvement and comparative experiments many times, and it has now become one of the most classical algorithms of MOEAs.

This study used the NSGA-II algorithm developed by Deb [32]. The method uses a genetic algorithm to perform the search, using mutual dominance relationships and spacing between design alternatives to drive the genetic process. The entire population is collated and sorted by dominance, and a penalty function is used to give the search a wider range. Building an optimization model based on the NSGA-II algorithm consists of the following steps:

(1) Define the objective function and decision variables: first specify the objective function that the optimization problem requires to be maximized or minimized, as well as the range and constraints of the decision variables.

(2) Initialization population: randomly generate a certain number of initial solutions as the initial population.

(3) Genetic operations: populations are iteratively updated through genetic operations such as selection, crossover, and mutation; selection operations are performed based on non-dominated ordering and crowding distances to ensure that the selected parents are of good quality and diversity.

(4) Non-dominated sorting and crowding distance calculation: in each generation iteration, non-dominated sorting and crowding distance calculation are performed on the population to select the best individuals for the next generation.

(5) Termination condition: repeat the above iterative process until the termination condition is satisfied, e.g., a preset number of iterations is reached or the quality of the solution meets the requirements.

The multi-objective optimization problem consists of design variables, an objective function, and constraints, the expression of which is shown in Equation (22):

$$\left\{\begin{array}{l}\min f\left(x\right)=\left[f_1,f_2,f_3,\dots ,f_m\right]\\ x=\left[x_1,x_2,x_3,\dots ,x_n\right]\\ \text{s.t.}\left\{\begin{array}{l}{g}_i\left(x\right)\ge 0 i=1,\dots ,k\\ h_i\left(x\right)=0 i=1,\dots ,h\end{array}\right.\end{array}\right.$$

(22)

where the function $f\left(x\right)$ is an $m$ dimensional multi-objective function. $x$ denotes a design variable with an $n$ dimensional array. $g_i$ and $h_j$ denote the $i$th inequality constraint and the $j$th equality constraint, respectively.

2.4. Optimization Process Based on the Surrogate Model

As shown in Figure 1, the surrogate model optimization process is as follows.

Figure 1. Flowchart of the optimization methodology.

(a)

Sampling

After determining the design variables in the optimization problem, full analytic factor sampling is used in the experimental design. The objective is to generate a certain number of sample points distributed in the design space, to obtain a comprehensive characterization of the model response.

(b)

Numerical Modeling

The numerical model performs a series of numerical modeling steps of the moored floating platform in the time domain to compute the target values based on the mooring configurations obtained from the design variables at the sample points.

(c)

Kriging Surrogate Modeling

Taking the sample points in step (a) as inputs and the target values in step (B) as outputs, the Kriging surrogate model is established, and the smoothing parameter of the modified model basis function in the Kriging surrogate model is adjusted to determine the functional relationship between the input and output data, thereby accelerating the subsequent search for the optimal solution.

(d)

Mooring System Optimization

A Pareto solution for multi-objective optimization of mooring systems is obtained by combining the Kriging surrogate model with the NSGA-II algorithm.

3. Case Study

3.1. The Parameters of the FOWT Model

3.1.1. Semi-Submersible Platform

As shown in Figure 2, the semi-submersible floating platform comprises a central column and three peripheral columns, with the wind turbine mounted on the central column. The specific structural parameters are listed in Table 1.

Figure 2. Schematic of 15 MW semi-submersible platform.

Table 1. Principal particulars of platform.

Parameter	Unit	Value
Water depth	m	200
Displacement	m3	20,206
Draft	m	25.474
Center of mass	m	−18.162
Center of buoyancy	m	−16.757
Platform mass	kg
Moment of inertia about the x-axis	kg·m2	5.873 × 1010
Moment of inertia about the y-axis	kg·m2	5.873 × 1010
Moment of inertia about the z-axis	kg·m2	2.367 × 1010

The heave plates are installed at the bottom of the outer columns to reduce the motion of the FOWT, as shown in Figure 3. The location and structural parameters of the heave plates are provided in Table 2.

Figure 3. Schematic of the heave plate.

Table 2. Principal particulars of heave plate.

Parameter	Unit		#1	#2		#3
x	m		−45.141	0		−30.161
y	m		22.571	39.092		−30.161
z	m		26.306	−45.561		−30.161
Diameter		m			18.900
Thickness		m			1.890
A33		kg			2.679× 107
B33		N/(m/s)			1.02 × 106
Cd		-			6.203
Cm		-			6.437

3.1.2. Wind Turbine and Tower

This paper focuses on the IEA Wind 15 MW wind turbine as the research subject [33]. The wind turbine has a rotor diameter of 240 m, a cut-in wind speed of 3 m/s, a cut-out wind speed of 25 m/s, and a rated wind speed of 10.59 m/s. The specific parameters of the IEA Wind 15 MW wind turbine are detailed in Table 3.

Table 3. Principal particulars of the IEA 15 MW wind turbine.

Parameter	Unit	Value
Number of blades	-	3
Cut-in wind speed	m/s	3
Rated wind speed	m/s	10.59
Cut-out wind speed	m/s	25
Rotor diameter	m	240
Hub height	m	150
Minimum rotor speed	rpm	5
Maximum rotor speed	rpm	7.56

The diameter of the tower varies linearly from bottom to top, with specific parameters listed in Table 4.

Table 4. Principal particulars of tower.

Parameter	Unit	Value
Tower mass	kg	1.26 × 106
Tower length	m	129.49
Base diameter of the tower	m	10.00
Top diameter of the tower	M	6.50

3.1.3. Initial Mooring System

A catenary mooring system with nine mooring lines is designed, arranged as shown in Figure 4. The mooring system consists of three 900 m long chain mooring lines. Each mooring line connects at the fairlead to one of the three side columns of a platform located 15 m above SWL, which are distributed at 120-degree intervals. The radial distance from the centerline of the tower to the anchor points is 837.60 m (Table 5).

Figure 4. Schematic of the mooring line layout.

Table 5. Mooring line configuration.

Parameter	Unit	Value
Number	-	3
Radius of anchor points from centerline	m	837.6
Radius of fairlead holes from centerline	m	58
Diameter	mm	185
Dry weight	kg/m	685
Stiffness	MN	3270
Length	m	900

3.2. Design Load Condition

Rated environmental conditions are selected in the optimization model. These environmental conditions are listed in Table 6, where the JONSWAP wave spectrum and Kaimal wind spectrum are used. The center of the wind field is at the hub.

Table 6. Environmental conditions.

Condition	Tp (s)	Hs (m)	Peak Enhancement Factor	Wind Speed (m/s)	Turbulence Intensity (%)
-	12.51	5.5	1.0	10.59	5.5

3.3. Design Variables

For the overall performance design of the floating wind turbine, we selected optimization design variables, as shown in Table 7.

Table 7. Optimization design variables.

Design Variables	Variable Name	Initial Value	Lower Bound	Upper Bound
$x_1$	anchor point location Xanch	900	500	1500
$x_2$	mooring line length	−837.6	−500	−1500

3.4. Objective Function and Constraints

During the optimization process, the surge motion response of the FOWT and the geometric configuration of the mooring lines need to be constrained. For the platform surge motion response, a permissible offset-to-water-depth ratio of 0.15 is used as a constraint [33]. For the mooring line geometric configuration, it is necessary to ensure that the triangle side-length criteria and the required length of the catenary section of the mooring lines are met. The specific details of the constraints are provided in Table 8.

Table 8. Constraints in the optimization process.

No.	Parameter	Constraint
1	surge motion	${\displaystyle \frac{MP{M}_{\mathrm{surge}}}{z_{depth}}}\le 0.15$
2	mooring line length	$L>\sqrt{{\left(x_{anch}-x_{Fair}\right)}^2+{\left(z_{anch}-z_{Fair}\right)}^2}$
3	catenary section length	$L_B=L-{\displaystyle \frac{2\left|V_{Fair}\right|}{{\omega }_{moor}}}\ge {\displaystyle \frac{L}{10}}$

Here, $MP{M}_{\mathrm{surge}}$ represents the statistical maximum value of the surge motion response, $x_{anch}$ and $x_{Fair}$ are the horizontal coordinates of the anchor point and fairlead of mooring line #1, and $z_{anch}$ and $z_{Fair}$ are their respective vertical coordinates. $V_{Fair}$ denotes the maximum tension at the fairlead, and ${\omega }_{moor}$ is the wet weight per unit length of the mooring line. The optimization function is given by Equation (23).

The goal of the case study is to minimize the maximum platform displacement, nacelle acceleration, and yaw under environmental loads as much as possible. The objective function can be expressed as follows:

$$\left\{\begin{array}{l}\min \left\{\begin{array}{l}{f}_1=MP{M}_{surge}\\ f_2=ACC{E}_{nacelle}\\ f_3=MP{M}_{yaw}\end{array}\right.\\ x=\left[x_1,x_2\right]\\ \text{s.t.}\left\{\begin{array}{l}{g}_1\left(x\right)={\displaystyle \frac{\Delta x_{offset}}{Z_{depth}}}-0.15<0\\ g_2\left(x\right)=\sqrt{{\left(x_{anch}-x_{Fair}\right)}^2+{\left(z_{anch}-z_{Fair}\right)}^2}-L<0\\ g_3\left(x\right)={\displaystyle \frac{L}{10}}-\left(L-{\displaystyle \frac{\left|V_{Fair}\right|}{{\mathbf{\omega }}_{moor}}}\right)<0\end{array}\right.\end{array}\right.$$

(23)

4. Results and Discussion

4.1. Surrogate Modeling and Validation

4.1.1. Sample Point Selection

To build the surrogate model, it is essential to base it on specific discrete sample points that represent the entire function space as accurately as possible. In this study, the full-factorial design method was used in conjunction with the constraints to determine the sample set of design variables, comprising 58 design variable values. The response values for these sample points were obtained through time-domain calculation models. The design variable values are shown in Table 9.

Table 9. Summary of training set.

No.	Length [m]	Coordinates [m]
1	550	−500
2	600	−550
3	600	−500
4	650	−600
5	650	−550
6	650	−500
7	700	−650
8	700	−600
9	700	−550
10	750	−700
11	750	−650
12	750	−600
13	800	−750
14	800	−700
15	800	−650
16	850	−800
17	850	−750
18	850	−700
19	900	−850
20	900	−837.5
21	900	−800
22	900	−750
23	950	−900
24	950	−850
25	950	−800
26	1000	−950
27	1000	−900
28	1000	−850
29	1050	−1000
30	1050	−950
31	1050	−900
32	1100	−1050
33	1100	−1000
34	1100	−950
35	1150	−1100
36	1150	−1050
37	1150	−1000
38	1200	−1150
39	1200	−1100
40	1200	−1050
41	1250	−1200
42	1250	−1150
43	1250	−1100
44	1300	−1250
45	1300	−1200
46	1300	−1150
47	1350	−1300
48	1350	−1250
49	1350	−1200
50	1400	−1350
51	1400	−1300
52	1400	−1250
53	1450	−1400
54	1450	−1350
55	1450	−1300
56	1500	−1450
57	1500	−1400
58	1500	−1350

4.1.2. Global Responses of the Sample Set

Table 9 includes each sample point of the design variable combinations considered candidate solutions for the optimization problem. Therefore, a total of 58 time-domain simulations were required to build the sample database. The design load conditions were applied to the numerical model to obtain the responses. The statistical values of these responses are shown in Table 10.

Table 10. Global responses of the training set.

No.	Surge [m]	Heave [m]	Pitch [°]	Yaw [°]	ACCE [m/s2]	VFair [N]
1	3.720	2.060	0.109	0.095	0.857	4.752 × 106
2	3.640	2.011	0.111	0.098	0.864	4.757 × 106
3	4.060	2.073	0.131	0.099	0.880	4.405 × 106
4	3.620	2.004	0.110	0.093	0.845	4.771 × 106
5	4.030	2.092	0.124	0.094	0.960	4.438 × 106
6	4.100	2.105	0.138	0.097	0.918	4.362 × 106
7	3.640	2.030	0.111	0.098	0.851	4.787 × 106
8	4.000	2.141	0.126	0.096	0.932	4.429 × 106
9	3.900	2.153	0.138	0.097	0.956	4.349 × 106
10	3.690	2.038	0.110	0.092	0.825	4.795 × 106
11	4.090	2.131	0.128	0.095	0.894	4.442 × 106
12	3.600	2.133	0.135	0.089	0.927	4.383 × 106
13	3.740	2.039	0.109	0.091	0.801	4.804 × 106
14	3.770	1.948	0.126	0.089	0.931	4.429 × 106
15	3.900	2.092	0.133	0.091	0.874	4.415 × 106
16	3.700	2.039	0.109	0.090	0.801	4.809 × 106
17	3.690	2.057	0.123	0.091	0.868	4.490 × 106
18	4.000	2.185	0.131	0.089	0.904	4.365 × 106
19	3.780	2.025	0.108	0.091	0.795	4.805 × 106
20	3.810	1.921	0.120	0.089	0.863	4.533 × 106
21	3.550	2.013	0.119	0.090	0.849	4.450 × 106
22	4.000	2.149	0.132	0.095	0.889	4.357 × 106
23	3.850	1.990	0.109	0.094	0.817	4.812 × 106
24	3.850	2.164	0.123	0.090	0.884	4.534 × 106
25	4.000	2.082	0.134	0.090	0.945	4.410 × 106
26	3.810	2.027	0.108	0.094	0.777	4.810 × 106
27	3.910	2.046	0.118	0.093	0.789	4.480 × 106
28	3.700	1.946	0.129	0.093	0.940	4.388 × 106
29	3.810	1.945	0.106	0.092	0.850	4.825 × 106
30	3.750	2.087	0.121	0.089	0.880	4.538 × 106
31	4.000	2.068	0.127	0.091	0.875	4.396 × 106
32	3.860	1.988	0.108	0.087	0.809	4.816 × 106
33	3.880	2.086	0.118	0.087	0.779	4.481 × 106
34	3.800	1.979	0.128	0.086	0.931	4.394 × 106
35	3.740	1.873	0.108	0.087	0.843	4.825 × 106
36	3.680	2.011	0.122	0.087	0.878	4.537 × 106
37	4.000	2.159	0.131	0.087	0.890	4.423 × 106
38	3.820	2.026	0.108	0.087	0.765	4.812 × 106
39	3.720	2.036	0.121	0.087	0.881	4.538 × 106
40	4.000	2.211	0.128	0.087	0.875	4.402 × 106
41	3.830	2.077	0.107	0.088	0.766	4.833 × 106
42	3.640	2.032	0.118	0.092	0.842	4.526 × 106
43	3.600	1.862	0.125	0.091	0.812	4.447 × 106
44	3.850	1.967	0.107	0.087	0.796	4.825 × 106
45	3.760	2.006	0.120	0.087	0.836	4.526 × 106
46	4.100	2.121	0.132	0.091	0.867	4.428 × 106
47	3.840	1.949	0.107	0.090	0.804	4.837 × 106
48	3.550	1.934	0.120	0.090	0.828	4.554 × 106
49	4.000	2.119	0.129	0.089	0.841	4.454 × 106
50	3.700	2.102	0.109	0.0871	0.753	4.894 × 106
51	3.710	2.088	0.119	0.0886	0.818	4.554 × 106
52	3.800	2.141	0.129	0.084	0.865	4.439 × 106
53	3.830	1.938	0.107	0.084	0.837	4.822 × 106
54	3.980	1.969	0.114	0.084	0.794	4.570 × 106
55	3.600	1.954	0.121	0.084	0.851	4.442 × 106
56	3.840	1.989	0.108	0.085	0.776	4.835 × 106
57	3.870	2.177	0.117	0.084	0.787	4.541 × 106
58	3.800	2.050	0.127	0.084	0.873	4.444 × 106

4.1.3. Reliability Analysis of Surrogate Model

The establishment of the Kriging surrogate model requires constant correction of the model basis function smoothing parameters to ensure the accuracy of the mapping between input and output parameters, and the values of the corrected model basis function smoothing parameters are shown in Table 11.

Table 11. Smoothing parameters in Kriging surrogate model.

	${\mathit{\theta }}_{\mathit{d}1}$	${\mathit{\theta }}_{\mathit{d}2}$
K-f1	23.784	50.000
K-f2	23.784	47.880
K-f3	6.484	5.221

To verify the accuracy of the surrogate model and prevent overfitting, the model needs to be tested. The testing involves selecting design parameters not included in the sample points to form a test set. The results from the time-domain simulations are compared with those from the surrogate model to validate its accuracy. In this case, a total of 10 test set sample points were selected, and their design parameters are shown in Table 12.

Table 12. Summary of test set.

No.	Length [m]	Coordinates [m]
1	702.4	−624.3
2	758.5	−625.5
3	882.0	−796.0
4	951.3	−837.6
5	1037.8	−963.5
6	1106.9	−994.0
7	1204.3	−1100.8
8	1257.9	−1123.8
9	1356.8	−1220.3
10	1450.6	−1358.7

Table 13 and Table 14 present the results of time-domain simulations and surrogate model predictions for the test sample. Figure 5 illustrates the error for each group of surrogate model results. The errors in surge, heave, pitch, yaw, nacelle acceleration, and mooring tension are all below 5%, indicating the high accuracy of the surrogate model.

Table 13. Global responses of test set based on time-domain model.

No.	Surge [m]	Heave [m]	Pitch [°]	Yaw [°]	ACCE [m/s2]	VFair /[N]
1	3.720	2.060	0.109	0.098	0.857	4.75 × 106
2	3.640	2.011	0.111	0.096	0.864	4.76 × 106
3	4.060	2.073	0.131	0.097	0.880	4.41 × 106
4	3.620	2.004	0.110	0.092	0.845	4.77 × 106
5	4.030	2.092	0.124	0.095	0.96	4.44 × 106
6	4.100	2.105	0.138	0.089	0.918	4.36 × 106
7	3.640	2.030	0.111	0.091	0.851	4.79 × 106
8	4.001	2.141	0.126	0.089	0.932	4.43 × 106
9	3.901	2.153	0.138	0.091	0.956	4.35 × 106
10	3.690	2.038	0.110	0.090	0.825	4.80 × 106

Table 14. Global responses of test set based on surrogate model.

No.	Surge [m]	Heave [m]	Pitch [°]	Yaw [°]	ACCE [m/s2]	VFair /[N]
1	3.831	2.082	0.111	0.099	0.876	4.85 × 106
2	3.803	2.029	0.114	0.097	0.901	4.90 × 106
3	4.210	2.118	0.132	0.098	0.895	4.54 × 106
4	3.677	2.012	0.111	0.094	0.857	4.82 × 106
5	4.195	2.108	0.128	0.099	0.998	4.56 × 106
6	4.186	2.128	0.139	0.090	0.928	4.43 × 106
7	3.760	2.034	0.114	0.092	0.882	4.88 × 106
8	4.088	2.174	0.126	0.09	0.939	4.51 × 106
9	4.040	2.198	0.140	0.093	0.972	4.48 × 106
10	3.734	2.103	0.110	0.092	0.835	4.83 × 106

Figure 5. The error of the surrogate model when compared with the time-domain model.

4.2. Mooring System Optimization

4.2.1. Pareto Solution

In applying the Kriging surrogate model to replace the costly finite element simulations, the initial values of the design variables were based on the initial mooring design mentioned in Section 3.1.3. After optimizing the mooring system using the NSGA-II algorithm, the Pareto solutions listed in Table 15 were obtained. As shown in Figure 6, the Pareto solutions are concentrated in the regions of smaller surge, yaw, and nacelle acceleration in the spatial domain. Considering the various optimization objectives, Pareto optimal solution No. 8 was selected as the final design scheme.

Table 15. Pareto solutions optimized using surrogate model.

No.	Length [m]	Coordinates [m]	Surge [m]	Heave [m]	Pitch [°]	Yaw [°]	ACCE [m/s2]	VFair [N]
1	626.6	−604.6	3.510	1.998	0.105	0.0928	0.821	4.998 × 106
2	884.0	−803.2	3.523	2.008	0.114	0.0976	0.818	4.556 × 106
3	882.8	−809.6	3.537	2.003	0.112	0.0964	0.810	4.605 × 106
4	606.5	−583.6	3.552	2.006	0.104	0.0923	0.837	4.995 × 106
5	606.5	−583.6	3.553	2.006	0.104	0.0923	0.837	4.995 × 106
6	875.8	−806.8	3.554	2.012	0.111	0.0960	0.806	4.639 × 106
7	881.9	−815.3	3.562	2.002	0.110	0.0954	0.803	4.652 × 106
8	874.6	−811.2	3.572	2.012	0.110	0.0912	0.798	4.679 × 106
9	875.5	−816.0	3.588	2.009	0.109	0.0903	0.786	4.706 × 106
10	678.9	−676.4	3.604	2.018	0.099	0.0917	0.801	5.202 × 106
11	873.2	−826.0	3.652	2.013	0.107	0.0933	0.792	4.800 × 106
12	1381.2	−1351.5	3.656	2.093	0.106	0.0890	0.768	5.035 × 106
13	1377.3	−1356.3	3.658	2.086	0.105	0.0889	0.776	5.096 × 106
14	1389.9	−1359.1	3.673	2.099	0.107	0.0887	0.767	5.033 × 106
15	1394.3	−1350.8	3.678	2.104	0.108	0.0891	0.756	4.941 × 106
16	1400.8	−1347.5	3.702	2.102	0.109	0.0896	0.752	4.870 × 106
17	1401.8	−1357.9	3.716	2.094	0.108	0.0889	0.759	4.939 × 106
18	1219.7	−1183.3	3.839	2.064	0.104	0.0889	0.763	4.921 × 106
19	1222.1	−1190.1	3.860	2.069	0.104	0.0888	0.762	4.957 × 106
20	1225.8	−1198.1	3.879	2.072	0.103	0.0888	0.761	4.994 × 106

Figure 6. Pareto fronts based on the Kriging surrogate model.

4.2.2. Results for Optimized Configurations Based on Time-Domain Model

Based on the design variables in the optimized Pareto optimal solution No. 8, the time-domain model was built and simulated, and the time-domain curve is shown in Figure 7.

Figure 7. Time history of global responses for optimized configurations: (a) surge; (b) heave; (c) pitch; (d) yaw; (e) nacelle acceleration; (f) tension.

Table 16 shows that the error between the optimization results from the surrogate model and the time-domain optimization results does not exceed 5%, confirming the accuracy of the surrogate model.

Table 16. Global response statistics for optimized configurations based on time-domain model.

	Surge [m]	Heave [m]	Pitch [°]	Yaw [°]	ACCE [m/s2]	VFair [N]
	3.564	1.991	0.102	0.087	0.784	4.61 × 106
Error	2.05%	0.22%	1.03%	4.64%	1.78%	1.41%

Compared to the initial design, the optimized design achieved a 2.8% reduction in mooring line length and a 3.14% reduction in anchor radial spacing. While ensuring that the mooring line tension at the fairlead and the touchdown segment length meet regulatory requirements, the maximum reduction in the surge motion response of the FOWT is 6.46%, the pitch motion response is reduced by up to 2.87%, and the nacelle acceleration is reduced by a maximum of 11.55%.

5. Conclusions

In this paper, we have proposed an optimized design method for the mooring system of an FOWT. To minimize the surge, yaw motion response, and nacelle acceleration, the mooring line length and anchor radial spacing were defined as design variables. The optimization procedure integrates the Kriging surrogate model with the NSGA-II method to evaluate the candidate solution. The primary conclusions of this study are summarized as follows.

(1) The Kriging function was used to develop a surrogate model FOWT, which built a functional relationship between the design variables and objective. The results produced by the surrogate model were within a 5% error margin compared to time-domain model results, indicating the high accuracy of the surrogate model.

(2) The traditional, time-consuming time-domain finite element calculations are discarded in this optimization framework. During the optimization process, both time and computational costs are significantly reduced, with the majority of the time spent on building the sample database. Additionally, the optimization framework demonstrates a rapid convergence rate.

(3) Since the optimization is multi-objective, the results are not unique. The optimization results need to be selected based on their impact on the safety of the wind turbine system. Compared to the initial design, the optimized mooring line length is reduced by 2.8%, and the anchor radial spacing is reduced by 3.14%. Under the condition that the fairlead tension and catenary section length meet the regulatory requirements, the maximum reduction in surge motion response is 6.46%, the maximum reduction in roll motion response is 2.87%, and the maximum reduction in nacelle acceleration is 11.55%.

There were some limitations in the present study. This study adopted only one sea state for the calculations. Future studies will focus on more complex environmental conditions, with additional sea states and incidence directions. Further improvements will include adding more optimization objectives and design variables, such as considering the economic benefits of the floating wind power system (LCOE), and the layout of pontoons and sinkers, for more comprehensive and detailed optimization work.