1   Introduction

The importance of climate change and environmental issues is growing as global energy use rises, and thus, the employing of renewable and clean energy sources, such as coastal wind energy, is imperative. Many studies have investigated this field, and coastal wind energy has attracted considerable interest worldwide. Offshore wind turbines are highly considered due to the concentration of populations near coastal areas in most countries. Considerable progress in offshore wind turbine design has been accomplished through the adoption and transfer of offshore wind sources. Offshore wind energy is expected to play an essential role in future energy systems as it does not cause any damaging losses or air pollution. According to a global wind report (Lee and Zhao, 2021), the global wind power volume in 2021 will reach 733 GW. The report considers countries with wind power plants' capacity of over 10 GW, such as the United States, Germany, India, Spain, and the United Kingdom. Offshore wind power technology has made substantial progress, according to current research results. In Europe, wind turbine manufacturers and wind power plant installations have achieved 90% and 75% volumes, respectively. In shallow waters, offshore wind turbines are stabilized by jacket or single-column structures, which is a common foundation, but the use of these structures is limited by geological conditions and water depth. In deep waters, fixing the piers to match a wind turbine design is difficult because of the low natural frequencies, which are closer to the frequencies of overcoming waves. For water depths greater than 50 m, a floating foundation is recommended. Floating Wind Turbines (FWTs) are costeffective options for such depth because of their low overall cost. These systems are installed far from the shore and are thus not restricted by noise and other regulations. Additionally, FWTs can be used in waters with a depth of up to 700 m because they do not require tall towers and are made of unique construction materials (Hansen, 2006). Henderson et al. (2009) and Wang et al. (2010) discussed technical challenges associated with different types of FWTs. The concept of a Floating Offshore Wind Turbine (FOWT) was introduced by Heronemus (1972). Several other FOWT ideas were investigated in the 1990s, including barge or semisubmersible and spar platforms. A spar wind turbine is a vertical cylinder that has a rotor nacelle assembly on top of a tall tower, like an oil and gas platform. A floating foundation ensures that an FWT stands upright in the sea and during towing, offering a high bending arm and high inertial resistance against twisting movements. The efficiency of a wind-float semisubmersible FOWT hull and OC4 was compared under similar control conditions and environmental factors with a simulation program that paired a turbine-floater-mooring system and utilized a 5 MW wind turbine and catenary mooring system (Kim et al., 2017b). Additionally, a semisubmersible floating base with multiple wind turbines was proposed and analyzed using a dynamic analysis tool (Kim et al., 2017a; Jang et al., 2019; Bae and Kim, 2014). Kampitsis et al. (2022) conducted a unified simulation of offshore wind turbines with vibration control systems using Finite Element Analysis-Computational Fluid Dynamics (FEA-CFD). They showed that the extended K-Damper can be used to reduce the vibration of a monopile offshore wind turbine under the combined action of wind and sea wave excitation.

In the offshore oil and gas industry, TLPs are commonly used because they have lower torsion than fixed land wind turbines (Oguz et al., 2018). Barge-type platforms use shallow and wide barges as floating bases but are prone to wave and rotational movements. Although reducing the weight of steel coms in these platforms can greatly lower construction costs, large waves cause movements that can affect the platforms' stability. To minimize the lifter's movement and maintain the stability of a roller, the infrastructure of a floating foundation should be greater than or at least equal to the height of the center above sea level. Tables 1 and 2 present the advantages and disadvantages of fixed and floating foundations, respectively. Growing interest in offshore wind energy has prompted research on optimization methods and overall offshore wind turbine design based on time-domain, static, and frequency-domain analyses. Figure 1 shows the number of publications on optimizing offshore wind turbines. Some optimization algorithms used in optimizing offshore wind turbine design are discussed in this article.

Figure  1  Number of publications about offshore wind turbine optimization

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2   Design and optimization of offshore wind turbines

This section explains the design process of offshore wind turbines, covering various fields, such as frequency domain, static, and time domain. Offshore wind turbine structures are classified into three levels: macro, meso, and micro. The macro level deals with geometric dimensions similar to those of buildings or universal duty in structural treatment (Figure 2). The meso level has a specific function in a structural system, and the microlevel has a small geometric size and specialized structural task. The design process involves various steps, such as location selection, wind turbine dimension selection, subsurface consideration, geohazard evaluation, base and support structure selection, extension of design load conditions, and geotechnical and constructive analysis. Figure 3 illustrates the typical process for designing offshore wind turbines.

Figure  2  Structural decomposition of an offshore wind turbine. Reproduced with permission from (Bontempi et al., 2008)

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Figure  3  Design of a typical offshore wind turbine (Malhortra, 2011)

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2.1   Design method based on static analysis

Optimizing wind turbines is a crucial task that requires accurate finite element models, and static techniques are often used to minimize the weight and maximize the stiffness of offshore structures. These tasks are accomplished by altering a structure's geometry, particularly its thickness and diameter, and preventing system buckling. Uys et al. (2007) optimized a 1 MW turbine by adjusting the number of ring stiffeners and wall thickness variables and prevented buckling; these techniques were used for onshore wind turbines. Chantharasenawong et al. (2011) investigated a 1.0 MW turbine and were able to reduce tower weight by over 20% by increasing the diameter and decreasing the thickness of the section; these methods decreased the bulk factor for buckling failure within a permissible range. In a similar study, Gencturk et al. (2012) designed a 100 kW wind turbine and revealed that centralizing the parameters of a grid tower's transfer line decreased the turbine's weight by approximately 20%. Long and Moe (2012) used static analysis and buckling checks to determine the optimal lower base spacing for a 5 MW offshore wind turbine. Damiani et al. (2013) utilized optimization techniques to investigate a measurement tool for defining the main dimensions and topology of a jacket. van der Giessen (2021) investigated the optimization of the mooring system for FWTs in deep water through static analysis. He examined whether the mooring line specifications of a semi-taut mooring system with drag-embedded anchors for a 12 MW wind turbine supported via a semi-submersible can be optimized in deep water; they performed static analysis on the ultimate limit state scenario.

2.2   Review of domain analysis as a design method

Time domain techniques are recommended for thorough design assessment based on structural code and design standards. Yoshida (2006) investigated the optimization of the dimensions of a 2 MW onshore steel turbine by using a time domain and Genetic Algorithm (GA). Gutierrez et al. (2017) described the use of the FAST method in optimizing offshore wind turbine design. Ashuri (2012) examined forecasting offshore wind turbine design using optimization methods. Haghi et al. (2014) optimized the support structure of a 3.6 MW offshore wind turbine's monopile, reducing its weight by 12% relative to that of an initial design. Zwick et al. (2012) designed a jacket support structure by using time-domain simulation analysis. Chew et al. (2014, 2015, 2016) used iterative algorithms to compare three- and four-legged structures, and their results showed that the former is superior to the latter as an economic model. Schafhirt et al. (2014) used GA and time-domain analysis and optimized the OC4 jacket support structure; they demonstrated that the GA was slower than the time-domain analysis. Schafhirt et al. (2016) optimized an offshore wind turbine design on the basis of fatigue criteria with a gradient approach, but they did not consider the results of changes in single pipe dimensions. Oest et al. (2017) used a gradient method to optimize the total mass of offshore wind turbines with OC4 casings. Metaheuristic optimization techniques, such as GA, were reviewed by AlHamaydeh et al. (2015, 2017).

Bárcena Pasamontes et al. (2014) used a GA to minimize the weight of the offshore wind turbine and investigate the behavior of OC4 jackets. Chen et al. (2016) obtained an optimal hybrid offshore wind turbine infrastructure by using a Particle Swarm Algorithm (PSO) and time-domain analysis considering fatigue criteria. Cheng and Wang (2008) proposed a technique that utilizes time-domain analysis to identify damage to offshore platform structures.

2.3   Design method based on domain frequency analysis

One of the methods for structural investigation based on time is frequency-domain analysis, which has a low computational cost. Gentils et al. (2017) performed GA and finite element analysis to optimize the weight of a 5 MW offshore wind turbine; they found that stress, vibration, buckling, deformation, and fatigue restricted optimization. Arany et al. (2016) proposed a promising technique for designing monopile foundations on the basis of frequency-domain analysis; it can be used in designing single columns according to essential data, such as site, turbine, and terrain specifications.Thiry et al. (2011) developed a method for the optimization of single-pile steel structures with GA. Van Der Tempel (2006) and Ziegler et al. (2015) expressed a technique to calculate the exhaustion age of offshore wind turbines by utilizing frequency-domain analysis and the method of Dirlik (1985). Long and Moe (2012)investigated the fatigue loads of jacket substructures. Brommundt et al. (2012) utilized a spectral technique to optimize the self-control system of a floating structure.

Michailides and Angelides (2012) and Hall et al. (2013) described the multi-objective method for optimizing floating structures with GA. Dou et al. (2020) proposed a technique for optimizing the design of FWT support structures through frequency-domain analysis and by using analytical gradients; they utilized a four-degree-of-freedom frequency-domain model to understand the dynamic reaction of an FWT under wave and wind loads; their approach that optimized an integrated design including the geometrical features of a floater and mooring system and the coverage of long realizations of many load cases. Oh et al. (2013) performed hydrodynamic load analysis in the frequency domain and assessed the dimensions of a single pile foundation.

3   Issues in the hydrodynamics and aerodynamics of offshore wind turbines

The issue of yield and security in offshore wind turbines is a vital topic that has attracted the attention of researchers in various fields, including hydrodynamics, aerodynamics, structural dynamics, and elasticity. This section focuses on investigating the problems related to hydrodynamics and aerodynamics in the optimization of offshore wind turbines.

Micallef and Rezaeiha (2021) identified the current gaps in knowledge and challenges in the aerodynamics of FOWT rotors by leveraging previous research on fixed rotor aerodynamics. They presented the relationship between FOWT aerodynamics and other related fields in Figure 4. Gao et al. (2022) investigated essential hydrodynamic issues related to wind turbines, such as resource evaluation, wave loads on fixed turbines, rotors for single units and arrays, and floating turbines. For instance, they examined the hydrodynamic behavior of a generic bottom-fixed offshore wind turbine with a large monopile. They found that the installation, maintenance, and decommissioning steps involved specific vessels and components for the tower, which can lead to significant nonlinear hydrodynamics interplay. The density of turbines has increased, which in turn amplifies the aerodynamic interplays between the turbines. The interplays are a result of the farm layout, which is affected by macro/micro siting optimization.

Figure  4  Studies on the circumambient FOWT aerodynamics. Reproduced with permission from (Micallef and Rezaeiha, 2021, Elsevier) © (2021)

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He et al. (2022) introduced the modern optimization method that merges fluid mechanics techniques and mathematical models to investigate the optimal array layout of large offshore wind farms; they suggested a united interposition model explaining the aerodynamic interaction between any two adjacent turbines to determine the maximum power generation. Then, they investigated the variable connection of the layout of wind turbines and the power generation of the system and three various scenarios with equaled and staggered layouts subject to variant wind situations. Their results showed that the maximum power yield of a large-measure offshore wind farm is nearly related to dominant wind orientations. In addition, they concluded that the maximum power production of a wind farm was obtained when the largest number of wind turbines was used; however, the maximum power production of the system was not obtained, and thus, the power production yield of each wind turbine in the system decreased. In Figure 5, the process of calculating the optimal layout of their research is shown.

Figure  5  Calculation flowchart of the optimal layout of the wind farms. Reproduced with permission from (He et al., 2022, Elsevier) © (2022)

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Chen et al. (2020) proposed four parametric models to describe the geometry of model blades that were combined individually into the optimization model; they utilized high-order redesign techniques for wind turbine blade optimization based on aerodynamic likeness to offset the effects of scale efficacy on the result of model tests for FOWTs; this approach considered aerodynamic similarity in generating an optimized lightweight blade; they concluded that the deflection between the empirical and numerical model blades subject under specific conditions was extremely small (only 3.16%). The model blade with aerodynamic similarity can amend the rotor thrust by 106.33% compared with the Froude-scaled model blade. According to Figures 6 and 7, the chord length of the model blades with aerodynamic similarity increased compared with the scaled model blade, and the major regions are concentrated near the tip of the blade compared with the scaled model blade.

Figure  6  Comparison between the optimized and scaled model blades using polynomial curve fitting. Reproduced with permission from (Chen et al., 2020, Elsevier) © (2020)

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Figure  7  Comparison between the optimized and scaled model blades using Bessel curve fitting. Reproduced with permission from (Chen et al., 2020, Elsevier) © (2020)

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Table 3 lists the main articles about the optimization of FOWT aerodynamics, providing a consolidated survey of research endeavors in the past years.

4   Investigation of optimization algorithms in coastal wind turbine design studies

Research on offshore wind turbines is expected to grow because of the development of new optimization algorithms. Artificial intelligence is used in the computing performance–based development of novel algorithms that can incorporate an optimization problem into an optimal solution, greatly improving offshore wind turbine designs. However, various numerical simulation techniques have some shortcomings, especially in terms of dynamic effects on turbines, mooring systems, and hull-containing nonlinear factors subject to wave–hull interactions. Thus, efficient optimization technologies that can reduce computational costs are crucial. One such technology is the accelerated boundary element method, which effectively simulates offshore wind turbines under different field conditions and in environments and reduces computational costs (Willis et al., 2007; Li et al., 2019). Another important topic is the huge computational costs due to the slow convergence of optimization algorithms. To address this issue, an improved GA (Vairavamoorthy and Ali, 2005) can be utilized for the optimization of wind turbine designs. This algorithm has a fast convergence speed and accurate and is thus an ideal solution for optimizing wind turbines. Another way to extend design optimization methods, accelerate optimization, and increase accuracy is to replace a complex numerical model and use an approximate model. By training an example through machine learnin, Choe et al. (2021) with an artificial neural network or support vector machine, an approximate model can be obtained. The collection of data used for training is critical for this approximation rather than the use of direct scalar tools. However, access to the data set still limits the use of this technique. The size of the data set is often limited, and the state of the data cannot be ascertained. Therefore, this issue can be addressed without receiving new data through data augmentation, which can increase the diversity and facilitate data measurement. This specification can be helpful to the optimization of offshore wind turbine designs, especially for people lacking experience in this field. Research on deep-water applications should be focused on optimizing the design of mooring systems, and a critical factor is the simultaneous optimization of the restrain line system design and infrastructure according to the kind of moors in the restrain line system, geometry, and the specifications of materials.

Shourangiz-Haghighi et al. (2020) reviewed advances in wind turbine efficiency optimization utilizing CFDs; they reviewed the objective functions to distinguish the efficiency of wind turbines, CFD techniques utilized in the simulation of wind turbines, and optimization algorithms for wind turbine yields. Kale and Varma (2014) conducted an aerodynamic design and optimization of a small horizontal axis wind turbine blade according to the NACA 4412 airfoil specification, and the chord and twist angles were determined from the initially designed blade; optimization was performed to improve power and reduce startup time, and the results showed a 24% reduction in the optimized blade chord and 44% reduction in thickness; the power factor of the optimized vane increased up to 30% relative to that of the normal blade. Karthikeyan et al. (2015) investigated the aerodynamic optimization of the shapes of the blades of a small horizontal-axis wind turbine. Therefore, they optimized the profile of the blades and the geometry of the airfoils and achieved a high-power factor in Re of less than 5×10⁵ in the wind turbine; they concluded that changes in the trailing edge and thickness considerable affected noise performance, and the conditions under which the airfoil started working were characterized. Some unique metaheuristic algorithms for offshore wind turbine design have been discussed. In general, optimization algorithms are divided into three categories (Figure 8). Figure 9 shows how some algorithms can address problems in offshore wind turbines.

Figure  8  Classification of optimization algorithms

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Figure  9  Percentage of using some algorithms in offshore wind turbine optimization according to the number of publications

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In the optimization of offshore wind turbines, the selection of a suitable objective function is crucial. Evaluating the easy objective functions is extremely fast but may be imprecise or result in undesirable outcomes; in addition, involved objective functions can be extremely time-consuming. For the optimization of objectives, suitable candidates are as follows:

• Maximization of energy production

• Maximization of profit

• Minimization of the cost of energy

Reducing capital cost is an important duty for designing wind turbines because the utilization and maintenance costs often represent only a small fraction of the capital cost. To reach an aerodynamically effective rotor in wind turbines, they should be well designed with a low unit energy production cost. Designing the rotor has an essential role in the total design process of a wind turbine. To optimize the rotor of wind turbines, the objective function is determined as:

where the cost of energy of a wind turbine rotor is Z; the total cost for producing, transporting, and erecting a wind turbine rotor is C_rotor, and annual energy production is AEP. Thus, the whole cost of a rotor, C_rotor, is a self-relative amount determined as follows:

To calculate AEP, one should combine the possible wind density and power curve. The function of possible density is determined as:

The scale parameter, shape coefficient, and wind speed are denoted as A[-], K[-], and V (m/s), respectively. According to Rayleigh distribution, the shape coefficient K is 2. For example, if a wind turbine operates for 8 760 h per year, its AEP can be assessed as follows:

where P(V_i) is the power at the wind velocity of V_i. The power P can be calculated as follows:

The mass flow rate, wind velocity, air density, and wind space are denoted as ṁ, V₀, ρ, and A₀, respectively. The geometry of the blade should be displayed for the reliable optimization of a wind turbine blade, and many design variables are needed. For the control of the rotor form, airfoil specifications, rotational velocity, and pitch angle, the design variables are selected. The constraints of the design variables are as follows:

where X_{i, min} is the lower limit, and X_{i, max} is the upper limit for the chord, twist angle, and relative thickness of the blade, respectively (Eke and Onyewudiala, 2010).

4.1   Genetic algorithm

GAs are used to optimize the designs of offshore wind turbines, particularly for electrical connections and location selection. Häusler and Owman (2002) utilized a GA to optimize the electrical connection design of offshore wind farms. Zhao et al. (2004, 2006, 2009) used a GA to explore a similar issue for wind farm shape optimization. Chen et al. (2008) investigated the application of a GA in electrical system optimization for offshore wind farms. The reliability of collection systems has been the focus of some studies (Yang et al., 2009). For a large offshore wind farm, Huang et al. (2009) optimized an electrical connection design by utilizing a GA. Lee et al. (2010) utilized a GA to maximize the energy density that meets the maximum distance from the shoreline and the criteria of maximum water depth of offshore wind turbines.

Gonzalez-Longatt et al. (2011) optimized the electrical grid design of large offshore wind farms by using an improved GA. Hall et al. (2013) optimized FOWT infrastructure by using the frequency-domain model efficiency of the FOWT to movements in six degrees of freedom and used a GA for local optimization. In 2013, Polat and Tuncer (2013) optimized the aerodynamic shapes of the blades of a 1 MW Nordtank horizontal axis wind turbine to maximize power generation with a GA; they based their studies on the movement range of a blade element and considered a certain speed for the wind and the speed and diameter of the rotor, using a specific number of blades for the rotor; the optimal distribution of the chord and twist angle of the airfoils that consisted the blades was due to the optimal geometry of the turbine blades, and the annual power production increased by 10%. To optimize the design of OC4-type support jacket structures, Bárcena Pasamontes et al. (2014) utilized a GA; each project for the load in the time domain with a complete wind turbine model was analyzed.

Pillai et al. (2016) used a GA to optimize an offshore wind farm and determine positions where turbines may be placed. Gentils et al. (2017) performed coupled parametric finite element analysis and used GA to optimize the elements of support structures, such as grouts, monopiles, transmission parts, and towers. Based on MATLAB programming, a method using GA for the optimal design of horizontal axis wind turbine blade structure was proposed by Bagherpoor and Xuemin (2017). Karimi et al. (2017) optimized FWTs with three fixed classes of spars and FWT infrastructure with a GA. Ahmadpour (2021) investigated wind efficiency with a minor horizontal axis by using the NSGII algorithm. The design of cylindrical floating hulls for the floating substructures of offshore wind turbines was optimized by Benifla and Adam (2022). Kirchner-Bossi and Porté-Agel (2024) conducted a study on the optimization of the power density (PD) in wind farms and analyzed its sensitivity to the available area size. They introduced a novel GA that optimizes PD and turbine layout. The algorithm is self-adaptive to the PD and the diversity of solutions. Table 4 lists some studies on the use of GA in wind turbine optimization between 2005 and 2021.

4.2   Sequential quadratic programming (SQP)

Optimizing the design of offshore wind turbines requires several nonlinear variables and equations. The objective function of nonlinear programming problems has conditions or nonlinear functions. Compared with linear programming problems, linear programming problems are much more complicated. The SQP algorithm, which is the most efficient technique for solving constrained nonlinear optimization problems, optimizes the designs of wind turbine blades. Kenway and Martins (2008) applied SQP to optimize blade shapes. The combined algorithm of GA and gradient-based method has been used in the optimization of thick wind turbine airfoils, 3D aerodynamic shapes, complex design in CFD (Foster and Dulikravich, 1997), and airfoil foil design (Vicini and Quagliarella, 1999). Bizzarrini et al. (2011) used a hybrid algorithm based on gradient algorithms, such as SQP and the GA, to optimize wind turbine airfoils. The SQP convergence method for wind turbine performance optimization was investigated by Ning and Petch (2016). Schröder et al. (2016) investigated damage localization in wind turbine support structures by updating the finite element model and using SQP.

4.3   Particle swarm algorithm

Liao et al. (2009) optimized wind turbine blades with an improved PSO algorithm; according to the results obtained between optimal and non-optimal blades, they concluded that this method is suitable for offshore wind turbine systems. To decrease wind turbine costs, they also investigated the minimum blade mass by utilizing an improved PSO algorithm with the FAST program (Liao et al., 2012). Wan et al. (2010) introduced PSO to solve the wind turbine location problem and ensure that this algorithm has a set of particles in the solution area, each of which shows the order of the turbine; during the progress of the group, each particle randomly moved in optimal coordinates. In another study, to reclaim the performance of PSO, they described a Gaussian particle swarm with a local differential search technique (Wan et al., 2012).

Chowdhury et al. (2013) concluded that using turbines with optimized hub height, the PSO algorithm can increase the normalized output power. Si et al. (2013) optimized offshore wind turbine blades by PSO. The pitch angle distribution along the blade span and chord length were optimized in specific wind conditions to recover the aerodynamic performance. The location of wind turbines in a large-scale wind farm by utilizing PSO was optimized by Hou et al. (2015). Chehouri et al. (2015) focused on maximizing energy production in a certain period (of the order of one year) and the spectrum of wind speed (not just a specific speed) and provided an overview of optimization techniques and strategies; they compared different algorithms and compared multi-objective optimization results (PSO and GA) and the results of the CFD solution. In this research, energy cost was minimized, electricity production was maximized, blade mass was reduced, and a suitable model for optimizing wind turbine performance was presented. To design the jacket infrastructure of a 5 MW turbine, Häafele and Rolfes (2016) proposed a general PSO-based method with some modifications; the method decreased the operational costs of offshore wind turbines.

Hou et al. (2017) proposed an integrated optimization method that simultaneously optimized the locations of FWTs and offshore substations and cable coupling configuration; they utilized a mixed integer particle swarm optimization algorithm to minimize the levelized production cost of the wind farm, and the results showed that the proposed approach can decrease the levelized production cost by 5.00%; they determined that the approach was better than the usual technique, which only achieved a 1.45% levelized production cost decreation but increased energy efficiency by 3.95%. By utilizing PSO and FAST programs, a time domain calculation model and hybrid optimization design method for wind turbine blades were developed by Ma et al. (2019). Shin et al. (2020) investigated the solution to the turbine arrangement problem in offshore wind farms by using a metamodel and EA/PSO. A foundation's effectiveness affects the energy performance of wind farms as the sizes of the wind farms increase. The behavior of wind turbines affects costs and contributes to losses due to reduced energy production. Song et al. (2023) developed a particle swarm optimization–based method capable of globally optimizing a farm layout. This method considered yaw angles for various wind speeds and directions in large-scale nonconvex joint optimization.

4.4   Other algorithms

Similar to PSO and GA, ant colony optimization and coral reef optimization use bio-inspired optimization methods (Salcedo-Sanz et al., 2014). Pouladi et al. (2013) selected an optimal location for a wind turbine farm by utilizing an improved ant colony algorithm; the performance of the proposed method contributed to the optimal localization of turbine farms. By using a developed RS algorithm, Feng and Shen (2017) examined the design optimization of offshore wind farms with different kinds of wind turbines.

Kaveh and Sabeti (2018) used the Colliding Body Algorithm (CBO) to investigate the supporting frame of offshore wind turbines. By utilizing the optimization method, the weight of the construction was decreased by about 50%. By utilizing the quantum particle swarm optimization algorithm, Furlanetto et al. (2020) minimized the weight of tubular steel towers of wind turbines. They used continuous design variables. Using three algorithms (NSGA-Ⅱ, PESA-Ⅱ, and NSGA-Ⅲ), Mellal and Pecht (2020) proposed the identification and optimization of acceptable solutions with multi-objective wind turbine design; they aimed to minimize energy expenditure and maximize nominal power according to the limitations of the structure design, considering the radius of the rotor and the height of the hub. Settoul et al. (2021) used a salp swarm algorithm and addressed problems in the optimal placement of wind turbines on the basis of decreasing active power loss in the power broadcasting network of Constantine City.

Structural design for wind turbines can be categorized into two fields: robust optimization design and probabilistic design. Robust optimization design refers to the optimization of wind turbine structures under certain constraints in the structure's performance. By contrast, probabilistic design deals with structural design issues related to variables and parameters that may cause possible problems. While robust optimization methods are commonly used in offshore wind turbine design, there are few studies on probabilistic design for offshore wind turbine support structures. Yang et al. (2015) proposed a practical method for design optimization based on reliability for offshore wind turbine substructures subject to dynamic response. Jin et al. (2018) utilized an artificial fish swarming algorithm to optimize the mass damper parameters of FWTs. They investigated the performance of the mass damper location on loads of key components of FWT, vibration control, and the displacements of FWT. Stieng and Muskulus (2020) described a general method that integrates the use of analytical gradients as reliability-based design optimization methods. This article separated the nondeterministic response of the offshore wind turbine into two parts: probabilistic and deterministic. Furthermore, this method separates the reliability analysis from the design optimization function as a computational method to decrease the high computational cost of this factorization. Hajinezhad Dehkharghani et al. (2021) combined a nonprobabilistic method with artificial neural networks to detect damage to an FWT.

In 2015, Wang et al. (2015) proposed a method for postoptimality sensitivity analysis in multi-objective robust optimization problems. They introduced two robustness indexes, named IRS and IRL. The design parameters were divided into two types: DVs and DEPs. DVs are controllable parameters, in contrast to DEPs. The IRS determines the robustness of multi-objective optimization problems against small differences in DVs and DEPs. The IRL determines the robustness of multi-objective optimization problems against large differences in DEPs. He et al. (2022) proposed a progressive optimization technique that utilizes mathematical models and fluid mechanics approaches to analyze the optimal arrangement of offshore wind farms; they suggested using an integrated interference model to maximize power generation and considered the aerodynamic interaction between any two near turbines; they then studied the effect of the layout of wind turbines and the power generation of the system. Furthermore, they conducted a sensitivity analysis to evaluate the impact of multiple dominant wind directions on the power yield of wind farms.

Lumbreras et al. (2015) introduced a new approach that combines ordinal optimization and mixed-integer programming to efficiently solve large-scale problems with a statistical optimality guarantee; they applied this algorithm to a case study of the Barrow offshore wind farm in the East Irish Sea.

Pérez-Rúa et al. (2022) introduced an optimization framework for the global optimization of the cable layout topology for offshore wind farms; the framework was designed to compare closed-loop and radial layouts for the collection system of OWFs. They have used a two-stage stochastic optimization program based on a mixed-integer linear programming model; for the radial layout, they used a hop-indexed full binary model. The objective function allows the simultaneous optimization of three factors: (ⅰ) initial investment (network topology and cable sizing), (ⅱ) total electrical power loss costs, and (ⅲ) operation costs resulting from energy curtailment due to cable failures; the results indicate that (ⅰ) the profitability of each topology type depends heavily on the project size and wind turbine rating. Closed-loop systems may offer a competitive solution for large-scale projects subjected to considerable energy curtailment. (ⅱ) The stochastic model presents difficulties when dealing with large-scale instances, requiring increased computing time and memory resources. (ⅲ) Strategies must be implemented to simplify the mathematical models used in stochastic optimization for modern offshore wind farms, either analytically or numerically.

Göçmen et al. (2019) used the PossPOW algorithm, which uses only 1 Hz turbine data as inputs and provides an estimate of possible power outputs; it was trained and validated in Thanet and Horns Rev-Ⅰ offshore wind farms under nominal operations, where the turbines followed the optimum power curve; the findings indicated that the exceptional performance of the algorithm; in the Horns Rev-Ⅰ wind farm, the more than 70% of the strict power system requirements were met for a 24 h data set on which the algorithm was evaluated. Geng et al. (2021) proposed a graph neural network for offshore wind speed prediction and named it the spatiotemporal correlation graph neural network.

5   Conclusions

The objective of optimization algorithms based on natural and biological processes is to efficiently solve problems and find appropriate solutions. This paper elaborates on the various metaheuristic algorithms used in optimizing the designs of offshore wind turbines. The use of metaheuristic algorithms for structural optimization is a rapidly growing field and has considerably improved the designs of offshore wind turbines. The results suggest that metaheuristic algorithms are effective in optimizing the design and development process of offshore wind turbines.