Reliability-based design optimization of offshore wind turbine support structures using RBF surrogate model

Abstract

An efficient reliability-based design optimization method for the support structures of monopile offshore wind turbines is proposed herein. First, parametric finite element analysis (FEA) models of the support structure are established by considering stochastic variables. Subsequently, a surrogate model is constructed using a radial basis function (RBF) neural network to replace the time-consuming FEA. The uncertainties of loads, material properties, key sizes of structural components, and soil properties are considered. The uncertainty of soil properties is characterized by the variabilities of the unit weight, friction angle, and elastic modulus of soil. Structure reliability is determined via Monte Carlo simulation, and five limit states are considered, i.e., structural stresses, tower top displacements, mudline rotation, buckling, and natural frequency. Based on the RBF surrogate model and particle swarm optimization algorithm, an optimal design is established to minimize the volume. Results show that the proposed method can yield an optimal design that satisfies the target reliability and that the constructed RBF surrogate model significantly improves the optimization efficiency. Furthermore, the uncertainty of soil parameters significantly affects the optimization results, and increasing the monopile diameter is a cost-effective approach to cope with the uncertainty of soil parameters.

Graphical abstract

Keywords

reliability-based design optimization / offshore wind turbine / parametric finite element analysis / RBF surrogate model / uncertain soil parameter

1 Introduction

Offshore wind turbines (OWTs), which benefits from the rapid development of wind energy to the have garnered significant attention from the renewable energy industry. The annual offshore wind installed capacity is estimated to increase from 21.1 GW in 2021 to 54.9 GW in 2031 [1]. By the end of 2019, more than 80% of the installed OWTs worldwide are supported by monopiles [2]. The size of monopile OWTs has increased with the rated power recently, and the cost incurred by the support structures constitutes more than 20% of the total cost [3]. To balance between safety and the economy, researchers have focused on the design optimization of support structures.

The method of applying a partial safety factor to loads and material properties has been widely used in the lightweight design of OWTs [4–7]. However, the safety of OWTs is related to uncertainties in environmental loads, material properties, manufacturing errors, etc.; as such, the abovementioned method, which cannot quantify the reliability of structures, is no longer applicable. In practical engineering, to consider the uncertainty of variables when field measurement data are insufficient, the variables are typically assumed to be random variables with selected distribution types, such as normal [8], Gumbel [9], and lognormal [10]. Lee et al. [11] incorporated reliability analysis into the optimization of a transition piece for monopile OWTs to solve the problem wherein deterministic design optimization (DDO) results cannot satisfy the fatigue reliability requirements. Yang et al. [12] discovered that compared with reliability-based design optimization (RBDO), deterministic optimization achieves a lower structural stress reliability than the expected level in the lightweight design of a substructure for tripod OWTs. Kamel et al. [13] used different RBDO and DDO methods to optimize the tower of a 2 MW OWT and discovered that the reliability level of the DDO results could not be controlled. These studies indicate that DDO typically obtains the optimal solutions near the constraint boundaries and leaves less room for uncertainties than RBDO.

The uncertainties of soil parameters are primarily due to spatial variability, geologic uncertainty, and transformation uncertainty [14], which occur widely in geotechnical engineering. Huang et al. [15] and Gong et al. [16] analyzed soil variability in the optimal design of pile-anchor structures for slopes and support piles in deep foundation pits, separately. To the best of our knowledge, OWT optimization considering the stochastic nature of seabed soil has not been conducted. Haldar et al. [17] investigated the effect of uncertainty in soil parameters on the dynamic response of OWTs in clay and discovered that an increase in the coefficient of variation (COV) of undrained shear strength adversely affected the turbine fundamental frequency and mudline rotation. Carswell et al. [18] discovered that as the COV of the soil friction angle increased, the reliability of OWTs decreased nonlinearly. Oh and Nam [19] proposed a new Monte Carlo simulation (MCS) method for the rapid prediction of the overall failure probability of OWTs caused by stochastic soil variability and discovered that soil stochasticity significantly affected the natural frequency of OWTs. The uncertainty of soil properties should be considered in the RBDO of the integrated support structure of monopile OWTs.

To reduce computational and experimental costs in reliability analysis, metamodels such as artificial neural networks [20], response surface methods [21], and kriging models [22], and support vector machines [23] are typically used instead of performing numerical calculations or experiments. Radial basis function (RBF) neural networks offer the advantages of simple structure, stable operation, and accurate modeling of arbitrary functions; thus, they have been used to optimize guardrails [24], turbine impellers [25], and jet pumps [26]. OWT structures are typically optimized using one of two algorithms, i.e., gradient optimization and metaheuristic algorithms [27]. In the optimization problem of marine structures, which is non-convex, the metaheuristic algorithm is preferred as it can obtain near-optimal global solutions more easily [4]. Meanwhile, the particle swarm optimization (PSO) algorithm has been applied to optimize OWTs as it requires fewer parameters to be adjusted, consumes less computer memory, and offers a high convergence speed [28,29].

Herein, a method that couples an RBF neural network and a PSO algorithm is proposed to perform RBDO on a support structure. This method computes the failure probability via the RBF surrogate model combined with the MCS method and considers the effect of soil parameter uncertainty on the optimization results. A 5 MW monopile OWT was used to optimize the volume of the support structures.

The remainder of this article is organized as follows: Section 2 presents a parametric finite element analysis (FEA) model for a typical 5 MW monopile OWT. Section 3 describes an RBF-based RBDO method. For optimization design, an RBF surrogate model is established and validated; subsequently, it is employed for the optimization design of support structures, and the effect of the COV of soil properties on the design RBDO results is discussed in Section 4. Concluding remarks are provided in Section 5.

2 Monopile offshore wind turbine and parametric finite element analysis model

2.1 Model description

The monopile OWT was developed by the National Renewable Energy Laboratory (NREL); it is rated at 5 MW [30] and has been widely utilized as a reference wind turbine model [31,32]. The rotor-nacelle assembly (RNA) of the turbine is reduced to mass points and the moment of inertia in the x-, y-, and z-direction.

As shown in Fig.1, the support structure comprises three components: a tower, transition piece, and monopile. The tower length is 77.6 m; the top and bottom diameters are 3.87 m and 6 m respectively; and the tower thickness increases linearly from 0.019 m at the top to 0.027 m at the bottom. The transition piece is used to connect the tower with the monopile and transmit the wind load. Its diameter is 6 m; the top and bottom thicknesses are 0.027 and 0.06 m respectively; and the thickness exhibits a linear relationship with the height. The monopile is a thin-walled cylinder with a thickness of 0.06 m and a diameter of 6 m; and its total length is 56 m, where 36 m is driven into the seabed.

2.2 Loading analysis

As depicted in Fig.2, the support structure is typically subjected to five types of loads. Design load cases (DLCs) of the start-up, power production, parked situation, etc., are typically considered [33], and the parked condition (DLC 6.1) has been proven to be one of the most critical DLC and thus has been applied in structural optimization design [34]. Therefore, DLC 6.1 was applied to a case under extreme environmental conditions with a 50-year return period. The site considered in this study is located in NL-1 in the North Sea of Netherlands. The details of the site are provided in Tab.1 [4].

The aerodynamic loads were calculated using the fatigue, aerodynamics, structures, and turbulence (FAST) code [35] and the Turbulence Simulator (TurbSim) [36]. The three-dimensional wind field information generated by TurbSim is input into the FAST code to perform a time-domain simulation for 630 s, with the first 30 s removed to exclude the effect of turbine start-up. Additionally, to ensure the statistical reliability of the maximum load effect estimated, six random seeds were simulated for each operating condition [37]. Yaw angles from −8° to + 8° were segmented into a discrete series to account for wind direction changes to which the turbine has not yet responded [33].

As the yaw error increased, the aerodynamic thrust force increased gradually (see Fig.3). Therefore, the mean value of the aerodynamic load was the largest with a yaw angle of −8°, and the wind seed whose maximum value was the most similar to and greater than the mean value was regarded as the maximum extreme event. In this study, the loads on the hub center were regarded as a random field, and the average values of these loads are shown in Tab.2.

The wind pressure P_{\mathrm{w}} exerting on the unit area of the tower can be calculated as follows:

(1)P_{\mathrm{w}}=\frac{1}{2}{C}_{\mathrm{p}}{\rho }_{\mathrm{a}}{u}^2\left(\mathit{\text{z}}\right),

where C_{\mathrm{p}} is the pressure coefficient, which is equal to 1.0 for horizontal and vertical surfaces [38]; {\rho }_{\mathrm{a}} is the density of air, i.e., 1.225 kg/m³; and u(z) is the wind velocity at height z. The wind shear profile is generally expressed via a power law as follows:

(2)u\left(\mathit{\text{z}}\right)=u_{\mathrm{h}\mathrm{u}\mathrm{b}}{\left(\frac{\mathit{\text{z}}}{{\mathit{\text{z}}}_{\mathrm{h}\mathrm{u}\mathrm{b}}}\right)}^{\alpha },

where u_{\mathrm{h}\mathrm{u}\mathrm{b}} and {\mathit{\text{z}}}_{\mathrm{h}\mathrm{u}\mathrm{b}} are the wind velocity and height of the hub center, respectively; z is the distance of the calculated section from the mean sea level (MSL), which is positive above the MSL; \alpha is the power law exponent for offshore wind, which is 0.115 [39].

For slender members, when the wavelength is greater than five times the diameter in water, the Morison equation can be used to calculate the wave load, and the wavelength is calculated via a simplified formula as follows [38]:

(3)\lambda =1.56T^2,

where T is the wave period. Based on Eq. (3), the wavelength \lambda is 92 m, which satisfies the requirement. The combined wave and current-induced load F_{\mathrm{w}} exerting on the unit area of the monopile can be calculated using the Morison equation shown in Eq. (4) [40].

(4)\begin{array}{rl}{F}_{\mathrm{w}}=& \frac{\text{π}}{4}{\rho }_{\mathrm{w}}{C}_{\mathrm{M}}{D}^2{u}_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^{\prime }\left(\mathit{\text{z}},t\right)+\frac{1}{2}{\rho }_{\mathrm{w}}{C}_{\mathrm{D}}D(u_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}\left(\mathit{\text{z}},t\right)\\ & +u_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}\left(\mathit{\text{z}},t\right))\left|u_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}\left(\mathit{\text{z}},t\right)+u_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}\left(\mathit{\text{z}},t\right)\right|,\end{array}

where {\rho }_{\mathrm{w}} is the density of seawater, which is 1025 kg/m³; D is the outer diameter of the monopile; C_{\mathrm{M}} and C_{\mathrm{D}} are the mass and drag coefficients, respectively, which are equal to 2.0 and 1.2 for a smooth tubular section, respectively [41]; u_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}\left(\mathit{\text{z}},t\right) and u_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^{\prime }\left(\mathit{\text{z}},t\right) are the horizontal components of the velocity and acceleration of the wave-induced particle, respectively, where the wave elevation profile is described by second-order Stokes wave theory [42]; and u_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}\left(\mathit{\text{z}},t\right) is the horizontal velocity of the wave-induced particle.

The hydrostatic pressure at the z-section of the monopile foundation P_{\mathrm{h}}\left(\mathit{\text{z}}\right) is calculated as follows:

(5)P_{\mathrm{h}}\left(\mathit{\text{z}}\right)={\rho }_{\mathrm{w}}g\left|\mathit{\text{z}}\right|,

where g is the acceleration of gravity, which is 9.8 m/s² .

In addition to the environmental loads, the gravitational loads generated by the RNA mass m_{\mathrm{R}\mathrm{N}\mathrm{A}} and support structure mass were considered. The gravitational load G_{\mathrm{R}\mathrm{N}\mathrm{A}} due to the RNA mass can be calculated as follows:

(6)G_{\mathrm{R}\mathrm{N}\mathrm{A}}=m_{\mathrm{R}\mathrm{N}\mathrm{A}}g.

2.3 Parametric finite element analysis model of 5 MW offshore wind turbine on monopile

A geometric model was established based on the OWT specifications presented in Subsection 2.1, as shown in Fig.4. The geometric model comprised four components: a tower, transition piece, monopile, and soil layer. The RNA was simplified to a mass point and moments of inertia. In the pile soil analysis (PISA) project, the FEA model was proven to be more accurate than the classical p–y curve method in describing the mechanical behavior between large-diameter piles and soil [9]. Therefore, solid elements were used to model soil in this study. The soil model diameter and the distance between the pile bottom and soil bottom boundary were 120 and 60 m, respectively, which were sufficiently large to exclude the effects of artificial boundary conditions on the calculation results.

The material of the support structure was set as S355 steel, whose material properties are listed in Tab.3. The effective density of the tower was set as 8500 kg/m³, which was increased from 7850 kg/m³ of S355 steel to consider the effects of paint coating, bolts, flanges, etc., which were not specified for computation [43].

The plasticity of the soil was defined using the Mohr–Coulomb constitutive model. The soil was categorized into three layers based on the density of sand. The specific parameters, as presented in Tab.4, were obtained from Ref. [44]. According to Kawa et al. [45], the uncertainties of soil internal friction angle, effective unit weight, and elastic modulus significantly affect FEA simulation results; therefore, they were used in the stochastic modeling of soil.

The soil model was meshed using eight-node hexahedral elements, and the mesh sizes on the radial edge from 1 m at the center were decreased gradually to 6 m at the boundary. Because the support structure of the OWT was a thin-walled structure, using the four-node shell elements is more suitable than using solid elements. Mesh convergence tests were performed to determine the most suitable mesh size. For control the mesh more effectively, the following four parameters were defined: the number of divisions along the circumference edges of the support structure N_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}, the longitudinal element sizes of the support structure E{S}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}, the number of divisions along the circumference edges of the soil N_{\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{l}}, and the longitudinal element sizes of the soil E{S}_{\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{l}}. Three meshing cases were performed to validate mesh convergence, and the calculated tower top displacements are shown in Tab.5. The relative difference between the calculated tower top displacement for Cases 2 and 3 was only 1.18%, whereas the calculation time of Case 2 was 86.25% lower than that of Case 3. Therefore, Case 2 was chosen as the meshing method in this study. Fig.5 shows a sectional view of the created mesh.

The bottom fixed and side radial constraints were applied to the soil model. In addition, the aerodynamic loads were reduced to three forces and three moments on the RNA mass point, and the hydrodynamic loads were simplified to a horizontal thrust and a bending moment on the top of the monopile. Meanwhile, wind pressure and hydrostatic loads were applied as surface pressure exerting on the tower and monopile, respectively.

2.4 Finite element analysis model verification

To avoid the resonance in the OWT structure, the natural frequencies of the OWT must be controlled to a certain range. Based on Tab.6, the first two frequencies calculated by the current parametric FEA model agreed well with the values presented in Ref. [44]. The difference in the second-order fore-aft mode was the greatest; however, it satisfied the engineering requirement of 5% accuracy.

To further verify the FEA model, a static analysis was conducted. In this case, the static analysis was performed under two loads, i.e., a concentrated force of 3.5 MN generated by the RNA mass and a concentrated force of 2 MN generated by the wind turbine thrust, where both loads were applied at the RNA’s center of mass. The accuracy of the results was verified by comparing the displacement at the RNA mass point and tower base center.

Tab.7 shows that the results calculated using the proposed model agreed well with the results presented in [46]. In particular, the maximum difference was only 1.14%, which confirms the accuracy of the current FEA model.

3 Radial basis function-based reliability-based design optimization method

3.1 Reliability-based design optimization and deterministic design optimization methods

The minimum volume of the OWT support structure was specified as the optimization goal, and the optimal solution for the structure was identified based on its diameter and thickness ranges via reliability theory. For comparison, a deterministic optimization design was performed, and the reliability of the system was considered based on a material safety factor and a load safety factor [37]. The mathematical models of the RBDO and DDO are shown below.

RBDO:

(7)\begin{array}{c}\text{Find}{D}_1,D_2,D_3,T_1,T_2,T_3\\ \text{Min}V\left(D_1,D_2,D_3,T_1,T_2,T_3\right)\\ \text{s}\text{.t}\text{.}\{\begin{array}{c}P\left(g_i⩽0\right)⩽1-\mathrm{\Phi }\left(\beta \right),i=1,2,...,6,\\ D_1⩽D_2⩽D_3,T_1⩽T_2⩽T_3,\\ D_g^{\mathrm{L}}⩽D_g⩽D_g^{\mathrm{U}},T_g^{\mathrm{L}}⩽T_g⩽T_g^{\mathrm{U}},g=\text{1,2,3},\\ \frac{D_3}{T_3}⩽120,\end{array}\end{array}

DDO:

(8)\begin{array}{c}\text{Find}{D}_1,D_2,D_3,T_1,T_2,T_3\\ \text{Min}V\left(D_1,D_2,D_3,T_1,T_2,T_3\right)\\ \text{s}\text{.t}\text{.}\{\begin{array}{c}\frac{{\sigma }_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}}{{\gamma }_{\mathrm{m}}}-{\sigma }_{max}⩾0,\\ L_{\mathrm{m}}-L_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{n}}⩾0,\\ 1.25\frac{H_{\mathrm{t}}}{100}-U_{\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}⩾0,\\ {\theta }_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}⩾0,\\ \frac{f_{1\mathrm{P}}}{{\gamma }_{1\mathrm{P}}}⩽f_{1\mathrm{s}\mathrm{t}}⩽\frac{f_{3\mathrm{P}}}{{\gamma }_{3\mathrm{P}}},\\ D_1⩽D_2⩽D_3,T_1⩽T_2⩽T_3,\\ D_g^{\mathrm{L}}⩽D_g⩽D_g^{\mathrm{U}},T_g^{\mathrm{L}}⩽T_g⩽T_g^{\mathrm{U}},g\text{= 1,2,3},\\ \frac{D_3}{T_3}⩽120,\end{array}\end{array}

where \beta is the target reliability index, which is generally set as 3.71, based on det norske veritas (DNV) standard [37]; and g_i represents the limit state function of the ith constraint, where the constraints are specified in Subsections 3.4 and 3.5. To reserve a tolerance of 5%, {\gamma }_{1\mathrm{P}} = 0.95 and {\gamma }_{3\mathrm{P}} = 1.05 were specified as the frequency safety factors of f_{1\mathrm{P}} and f_{3\mathrm{P}}, respectively [13]. Additionally, the load safety factors for the ultimate and serviceability limit states were 1.35 and 1.0 during the DDO, respectively [33].

The reliability was evaluated using the MCS method during the RBDO. Unlike DDO, RBDO can quantify the reliability and ensure that the index \beta satisfies the requirements. The MCS method requires a certain number of sample points during each iteration; additionally, using the RBF surrogate model will significantly improve the computational efficiency (see Fig.6).

3.2 Radial basis function surrogate model

The RBF neural network offers the advantages of simple structure, stable operation, and accurate modeling of arbitrary functions. In the present study, we used the RBF to develop a surrogate model for the optimal design of OWT support structures. The RBF surrogate model was established by interpolating discrete sample points and exhibited high nonlinear fitting capability, which allowed it to predict output the responses efficiently. The Gauss function is typically applied in the hidden layer as the RBF to perform a nonlinear transformation on the input vector. The Gauss function of the jth neuron in the hidden layer is as follows:

(9)G_j\left(x_i,c_j\right)=\exp \left(-\frac{{\Vert x_i-c_j\Vert }^2}{2{\sigma }_j^2}\right),i=1,2,...,m;j=1,2,...,n,

where x_i is the vector of variables at the ith sample point; m is the number of sample points; n is the number of neurons in the hidden layer; c_j and {\sigma }_j are the center and width of the jth neuron in the hidden layer, respectively; and \Vert x_i-c_j\Vert is the Euclidean distance from x_i to c_j. The output layer is the linear weighted sum of the hidden layer output, which can be expressed as

(10)y_k=\sum _{j=1}^n{\omega }_{jk}\exp \left(-\frac{{\Vert x_i-c_j\Vert }^2}{2{\sigma }_j^2}\right),j=1,2,...,n;,k=1,2,...,l,

where y_k is the output of the kth neuron in the hidden layer; {\omega }_{jk} is the connection weight between the jth hidden layer neuron and kth output layer neuron; and l is the number of output layer neurons.

The coefficient of determination (R²) is used to verify the accuracy of the model, and it is expressed as

(11)R^{\text{2}}=1-\frac{\sum _{i=1}^{n_t}{\left(y_i-{\hat{y}}_i\right)}^2}{\sum _{i=1}^{n_t}{\left(y_i-\overline{y}\right)}^2},

where y_i and {\hat{y}}_i are the ith accurate and predicted values of the response obtained using the FEA and RBF surrogate model, respectively; and \overline{y} is the average response values of the test set.

3.3 Variable definition

The variables comprise design and stochastic variables. After defining the variables, the parametric modeling method in Section 2 can be employed to perform random finite element simulations. The model was defined using the diameter and thickness of the tower, transition piece, and monopile, which resulted in six design variables. To ensure structural continuity, the diameter and thickness of the support structure were fixed from the top of the tower to the bottom of the monopile during optimization. The locations of the variables are shown in Fig.2 and Fig.7.

The uncertainties of the geometry dimensions (design variables) and properties of steel were described in the form of a normal distribution with a COV of 0.01 [11], and the loads were assumed to reflect a Gumbel distribution with a COV of 0.1 [9]. In addition, the RNA mass was described in the form of a normal distribution with a COV of 0.02 [47]. Meanwhile, the soil properties were assumed to show normal distributions with a COV of 0.03 [45]. The stochastic variables and the COV of the parameters are listed in Tab.8.

3.4 Optimization design constraints

Sizing optimization is performed on the support structure to minimize its volume such that the safety requirements of the industry standards are satisfied. The safety requirements generally pertain to the strength, stiffness, stability, natural frequency, and dimensions of a structure. Fatigue life significantly affects the optimal design of the support structure. However, it is not considered in this study but will be considered in future studies.

The maximum von Mises stress under the ultimate limit state is typically used to determine the ability of a wind turbine structure in resisting plastic yielding. The limit function for stress is as follows:

(12)g_1={\sigma }_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}-{\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}},

where {\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}} and {\sigma }_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}} are the maximum and allowable von Mises stress of the support structure, respectively; specifically, {\sigma }_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}} is 355 MPa for S355 steel.

Buckling analysis must be performed on the wind turbine structure, which is a thin-walled and slender structure. To avoid stability failure to the support structure, the minimum buckling load multiplier L_{\mathrm{m}\mathrm{i}\mathrm{n}} should be greater than the allowable load multiplier L_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}} of 1.45 [48]. The performance function is as follows:

(13)g_2=L_{\mathrm{m}\mathrm{i}\mathrm{n}}-L_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}.

Under the serviceability limit state, the maximum displacement of the tower top and the maximum rotation of the monopile at the seabed typically satisfy the requirements. The limit state functions are as follows:

(14)g_3=U_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}-U_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.25\frac{H_{\mathrm{t}}}{100}-U_{\mathrm{m}\mathrm{a}\mathrm{x}},

(15)g_4={\theta }_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}},

where U_{\mathrm{m}\mathrm{a}\mathrm{x}} and {\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}} are the maximum tower deformation and mudline rotation, respectively; U_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}} is the allowable tower deformation; and H_{\mathrm{t}} is the distance from the seabed to the tower top. According to the DNV standard, the total rotation for a pile at a seabed is 0.5°, with consideration of the installation tolerance; therefore, the allowable rotation {\theta }_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}} is 0.25° [37].

For the 5 MW monopile OWT investigated, the potential danger of resonance caused by rotor rotation must be considered. The first natural frequency f_{1\mathrm{s}\mathrm{t}} between the rotational frequency of the rotor f_{1\mathrm{P}} and the blade passing tower frequency f_{3\mathrm{P}} is considered be the most important parameter for an economical design. The limit state functions of the first natural frequency can be written as follows:

(16)g_5=f_{1\mathrm{s}\mathrm{t}}-f_{1\mathrm{P}},

(17)g_6=f_{3\mathrm{P}}-f_{1\mathrm{s}\mathrm{t}}.

The cut-in and rated rotor speeds were 6.9 and 12.1 r/min, respectively; thus, f_{1\mathrm{P}} and f_{3\mathrm{P}} were 0.202 and 0.345 Hz, respectively.

3.5 Variable constraints

For the OWT support structure, the resultant loads were transmitted from the tower top to the monopile therefore, the diameter and thickness of the support structure should increase from top to bottom. Thus, the constraint is defined as

(18)D_1⩽D_2⩽D_3,T_1⩽T_2⩽T_3.

Restrictions must be imposed on the diameter-to-thickness ratio of the tubular support structures subjected to external pressure [49].

(19)\frac{D_g}{T_g}⩽120,g=1,2,3.

To ensure that the size optimization of the support structure is feasible, as shown in Tab.9, the upper and lower limits of the wind turbine dimensions (design variables) were set based on actual values in practical engineering. Additionally, the following constraint was imposed:

(20)D_g^{\mathrm{L}}⩽D_g⩽D_g^{\mathrm{U}},T_g^{\mathrm{L}}⩽T_g⩽T_g^{\mathrm{U}},

where D_g^{\mathrm{U}} and T_g^{\mathrm{U}} are the upper bounds of the gth diameter D_g and thickness T_g, respectively; and D_g^{\mathrm{L}} and T_g^{\mathrm{L}} are the respective lower bounds, respectively.

Thus, the PSO algorithm is combined with the RBF neural network for the RBDO of the OWT support structure. The optimization process is shown in Fig.8.

4 Optimization results and discussion

4.1 Establishment and validation of radial basis function surrogate model

The surrogate model was assumed to exhibit good predictive accuracy with an R² greater than 0.9. Therefore, when constructing the surrogate model, 360 design sample points were obtained using the Latin hypercube sampling method. Subsequently, responses were obtained via FEA simulation, as shown in Fig.9. The RBF model was trained through 300 sample points containing variables and their responses. Additionally, cross-validation was performed to evaluate the accuracy of the surrogate model by testing a set comprising 60 sample points.

Based on Fig.9, the tower top displacement, mudline rotation, and first natural frequency of the OWT are more likely to exceed the constraint limit. Fig.10 shows a comparison of the FEA results with the results predicted by the RBF surrogate model. After processing, the R² of the tower top displacement, mudline rotation, and first natural frequency were 0.976, 0.969, and 0.978, respectively, which indicates the high precision of the RBF surrogate model.

4.2 Reliability-based design optimization and deterministic design optimization results

Both the RBDO and DDO used the PSO algorithm, where the number of particles was set to 40 and the iterative optimization was performed 100 times. An MCS was performed to count each subpoint 50000 times during the optimization process to calculate the reliability index. Fig.11 shows the convergence history of the volume change of the support structure, and the details of the optimization results are shown in Tab.10.

As shown in Fig.11, the DDO and the RBDO calculations converged rapidly and achieved convergence at approximately the 35th iteration, which required approximately 1400 FEA simulations. However, the construction of the RBF model required only 360 calculations, which translates to a reduction in the computational cost by 74.3%.

Based on Tab.10, the volume reduction yielded by the DDO was 22.62%, whereas that yielded by the RBDO was only 10.95%. This is because improving the structural reliability requires a larger diameter and thickness, which slightly increases the structural volume. In addition, the results obtained via the RBDO satisfied all constraints, and the reliability level satisfied the required level of 3.71. Meanwhile, the solution yielded by the DDO was located in a safe region that satisfied the design constraints and variable constraints; however, the reliability level was significantly lower than the required level.

Fig.12 shows a comparison between the initial values of the design variables with the results yielded by the DDO and RBDO. The thickness of the three components of the support structure obtained via the two optimization methods reduced, indicating that thickness contributes significantly to the volume reduction of the structure. Additionally, the most sensitive design variable affecting the total volume was the tower top thickness T_1, which decreased by 37% via both optimization methods, followed by T_2, which decreased by 18% and 26% via the DDO and RBDO, respectively.

4.3 Effects of coefficient of variation of soil parameters on design reliability-based design optimization results

As presented in Subsection 4.1, the design results are dominated by the tower top displacement, mudline rotation, and first-order natural frequency. In this section, the reliability index calculated using the three limit state functions above is considered. Soil parameters significantly affect FEA results; therefore, the effect of soil parameter uncertainty must be considered when evaluating the structural response of wind turbines. To the best of four knowledge, the optimization of OWTs that accounts for the uncertainty of soil parameters has not been investigated. Hence, the effect of the different COV of soil parameters on the optimization results are considered in this section. The discussion herein is based on three sets of COV (i.e., 0, 0.03, and 0.05) and the assumption that the soil parameters are independent of each other. Using the effective unit weight of soil as an example, the cumulative probability distributions are as shown in Fig.13.

Based on Fig.13 and Fig.14, when the COV of the soil parameters increased, the soil parameters that show the normal distribution were more discrete, and the overall diameter of the structure after optimization increased, particularly for the monopile diameter. This is because the increase in the COV of the soil parameters significantly affects the uncertainty of soil stiffness. When the soil stiffness decreases, the monopile foundation must provide greater stiffness to ensure that the overall stiffness remains unchanged to achieve the safety requirements. Additionally, changing the diameter is more effective than changing the thickness when the COV of the soil parameters changes.

5 Conclusions

In this study, an RBDO method was proposed for the integrated support structure of OWTs to solve the problem wherein DDO cannot achieve the required reliability index. This method considers five limit states, i.e., strength, tower top displacement, mudline rotation, buckling, and natural frequency, in optimizing the outer diameter and thickness of the OWT support structure to minimize the structural volume. A 5 MW monopile OWT developed by the NREL was used as an example, a parametric FEA model was established while considering stochastic variables, and stochastic FEA simulation results were input into the RBF neural network to construct a surrogate model. Additionally, we compared the RBDO results obtained using different COV of soil parameters to account for the pile–soil interaction. The conclusions can be summarized as follows.

1) Compared with DDO, which fails to satisfy the target reliability index, RBDO not only ensures that the reliability index exceeds 3.71, as required by standards, but also effectively reduces the support structure volume by 10.95%.

2) The proposed RBF surrogate model, which was used to replace the time-consuming FEA, reduced the computational cost by 74.3% as well as the product development time.

3) The stochastic characteristics of soil must be considered when performing RBDO. The results showed that increasing the monopile diameter effectively mitigated the uncertainty of soil parameters.

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