Design optimization of a TetraSpar-type floater and tower for the IEA Wind 15 MW reference wind turbine

We present an optimization study for the conceptual design of wind turbine floaters of the TetraSpar type. The optimization variables include all geometric dimensions of the floater, keel, mooring lines and tower design. A gradient based optimization method is applied to a mass proportional objective cost function. The objective function accounts for the different weight components of the floater, including secondary steel, the wind turbine tower, and the mooring system. A frequency domain response method is utilized, so that each design evaluation also takes into account the dynamic response for 12 wind speeds with associated wave conditions. Nineteen constraints are applied for static and dynamic response, natural frequencies, and fatigue at the bottom of the tower. Two reference designs are presented, namely one with a soft–stiff tower and one with a stiff–stiff tower. Due to the anti-phase coupling of the floater pitch and tower vibration, the soft–stiff tower needs a stronger floater stiffness in pitch. This design thus has a larger water plane area moment than the more compact stiff–stiff floater, which is found to be the least economical. A constraint analysis is next presented based on Lagrange multipliers and a relative cost index. We find that the strongest cost influence is exerted by the 3P tower frequency constraint for the stiff-stiff and soft-stiff designs. Finally, a third design variant with a free optimizable tower frequency is introduced. This design is found to be 11% cheaper than the soft–stiff design and highlights the potential cost savings of tower designs within the 3P region.


Introduction
Today, most offshore wind turbines are installed in depths below 50 m, on monopile, jacket, or gravity-based substructures [1].Space-frame support structures (e.g.jackets) can push the installation of offshore wind turbines to water depths up to approximately 60 m.However, almost four fifths of the global wind resource potential is located in water depths beyond 60 m, where floating support platforms are the most economical solution.Ireland, Norway, Italy, the US Pacific Coast, Morocco and Philippines are examples of potential markets for floating wind turbines [2].Floating wind turbines are complex, coupled multidisciplinary systems, and their design is a challenging and time-consuming task.Although the floating wind industry is still in the early stages of development, it is expected that 15% of offshore wind energy will be generated by floating wind turbines by 2050 [3].Several challenges need to be solved along the way, many of them leading to the reduction of the associated costs.For this reason, in this work we discuss the optimization-based design of a floating support structure for the IEA Wind 15 MW reference wind turbine [4], where the mooring system and the wind turbine tower are also simultaneously optimized.
Optimization of floating support structures for offshore wind turbines has already attracted significant attention from the research community.This is reflected in the increasing amount of literature on this subject.Detailed state-of-the-art reviews on the analysis and optimization of floating support structures for offshore wind turbines can be found in [5,6].Fylling and Berthelsen [7] presented an early study on the optimization of a floating wind turbine support structure of the spar buoy type, including the mooring system and the power take-off cable.The objective function minimized was the spar buoy cost, and the mooring line and cable costs.Hall et al. [8] discussed an optimization framework that focused on a parametrization of the design space capable of describing existing and new design concepts.Their intention was to provide a framework that could be applied to a given siting scenario to produce a list of the most promising floating support structure configurations.These configurations could then serve as starting points for more detailed design processes.The framework proposed by Hall et al. adopted a genetic algorithm to optimize the size of semi-submersible platforms made of cylindrical components, while considering linearized hydrodynamic forces, mooring forces, and turbine effects.Not long after, Hall et al. [9] proposed a hydrodynamics-based approach to optimize new designs.A collection of unique reference platform designs defined by their hydrodynamic performance coefficients was used.In the optimization process, the linear combination of these reference designs was optimized in order to approximate the characteristics of the resulting designed platform.The design optimization of a spar-buoy floater for a 5 MW offshore wind turbine with a genetic algorithm has been discussed recently by Leimeister et al. [10].Seven design variables were used to define the height and diameter of different segments of the floater, and a mass-proportional objective function was minimized.
The inclusion of the wind turbine control system has traditionally been ignored in studies on the optimization of floating wind turbines.Hegseth et al. [11] demonstrated that this does not always have to be the case and that the wind turbine controller can be included in the design optimization process with interesting results.They adopted a linearized aero-hydro-servo-elastic floating wind turbine model to optimize the design of the floating platform, tower, mooring system, and blade-pitch controller.The goal of the optimization was to minimize a weighted combination of system costs and rotor speed variation, which was used as a measure of power quality.An integrated design optimization approach for the design of the blade-pitch controller and the support structures was discussed by Hegseth et al. [12].A 10 MW floating wind turbine mounted on a spar buoy and four different control strategies were considered, in order to evaluate the effect of the controller on the structural design and the associated costs.The results showed that the controller had mostly an effect on the structural response of the tower in terms of fatigue damage, since the storm environmental condition with a parked wind turbine was found to govern the extreme responses of the system considered.
Frequency domain models have been utilized in order to provide rapid response predictions within the optimization process.Karimi et al. [13] discussed the multi-objective optimization of floating wind turbine support structures for a design space spanning three stability classes: tension-leg, spar buoy, and semi-submersible floating platforms for offshore wind turbines.The approach they adopted for analyzing any of the particular platform configurations used a simplified frequency domain dynamic model based on a linear description of the dynamics of the floating platform, mooring system, and of a reference 5 MW wind turbine.In their work, Karimi et al. considered aerodynamic and structural models, including the mooring system, that were obtained with FAST, a time-domain aeroelastic simulation tool [14].For calculating the hydrodynamic properties, they used WAMIT, a radiationdiffraction solver.More recently, Dou et al. [15] discussed the optimization-based conceptual design of a spar buoy for a 10 MW floating wind turbine.By adopting the QuLAF frequency response model of Pegalajar-Jurado et al. [16], the stochastic response to wind and wave forcing was obtained rapidly, through the solution of linear response equations.The wind turbine was subjected to pre-computed aerodynamic loads and linear hydrodynamic loads, and the optimization analysis was performed with a gradientbased algorithm with constraints on the system static and dynamic responses of interest.The work presented by Dou et al. [15] was extended by Pollini et al. [17] to optimize a spar buoy floater for the IEA Wind 15 MW reference wind turbine, Gaertner et al. [4].An optimization procedure based on a genetic algorithm has been described by Ferri et al. [18].A 10 MW floating wind turbine was considered with the purpose of finding the optimized sizes of a semi-submersible supporting platform.Response amplitude operators, namely the linearized transfer functions of the system, were adopted as objective functions, while considering the response of the system in resonant conditions.In very recent work, Hall et al. [19] presented a model called RAFT (Response Amplitudes of Floating Turbines) which is an open-source frequency-domain framework for floating wind turbine design optimization.It incorporates quasistatic mooring reactions, strip-theory and potential-flow hydrodynamics, blade element momentum aerodynamics, and linear wind turbine control.The purpose of the framework is to provide an integrated, open-source, and automated tool for quick evaluation of floating wind turbine designs.
The results presented in this paper represent a novel contribution to the design optimization of floating wind turbines, which significantly extends the initial results presented by Dou et al. [15] and Pollini et al. [17].We present the first optimization approach for the simultaneous design of a steel TetraSpar floater, the mooring system, and the tower for the IEA Wind 15 MW reference wind turbine, Gaertner et al. [4].The TetraSpar Demonstrator, installed in 2021 in Norway with a 3.6 MW turbine, is extensively described by Borg et al. [20].An earlier version of the TetraSpar floater designed for the DTU 10 MW turbine [21] was tested at DHI Denmark in 2017 [22,23] and the measurement data has been used for Operational Modal Analysis [24] and phase separation studies [25].
In this work, a gradient-based optimization framework is utilized, which ensures that modern numerical optimization methods relying on first-order information can be used to solve the design problem at hand.The wave loads on the floating wind turbine system is computed based on the Morison equation.The aerodynamic loads and damping are pre-computed in FAST [14], which, for a given wind turbine, are extracted only once for each mean wind speed from time-domain aero-elastic rotor computations.These aerodynamic properties of forcing and linear damping are thus assumed to be independent of the design variables of the problem at hand.In the optimization analysis, a mass-proportional cost function is minimized.The mass accounts for the volume of steel of the floater (including the keel and keel cables), the tower, and the mooring lines.Constraints are imposed on the static and dynamic response of the system to a range of environmental conditions, defined by wind speeds and associated sea states.We discuss the numerical results obtained for two cases, associated to soft-stiff and stiff-stiff tower designs.The different designs obtained are compared, with particular insights on the effect of design-driving constraints.The numerical results show that the proposed framework effectively identifies innovative designs that are not intuitive, while requiring modest computational resources on standard desktop computers.The main contributions of this paper are: • The first optimization study for the conceptual design of a TetraSpar-type floater, the mooring system, and the tower for a 15 MW wind turbine; • The inclusion of several practical design requirements in the design optimization analysis, both on the static and dynamic response of the floating wind turbine system; • The analysis of two different design scenarios, associated to soft-stiff and stiff-stiff tower designs; • The study of design-driving constraints and their effect on the final floater, mooring system, and tower designs.
The proposed framework should be seen as a feasible numerical toolchain capable of assisting offshore structural engineers in the conceptual design phases of the support structure, the tower, and the mooring system of offshore floating wind turbines.In this context, the optimized design proposed by this framework should be further analyzed with time domain simulations considering a full Design Load Basis (DLB).
The remainder of the article is organized as follows: In Section 2 we present the modeling and analysis approach.Important details are provided on: the design variables; the governing equations; the hydrodynamics, wind turbine tower, and mooring system modeling; and the fatigue and extreme dynamic responses considered.Section 3 presents the formulation of the optimization problem with particular attention to the formulation of the objective function to be minimized, and to the linear and nonlinear constraints.The numerical results for design optimization cases associated with stiff-stiff and soft-stiff tower designs are presented and discussed in Section 4. In Section 5 we analyze the design-driving constraints, and final conclusions are drawn in Section 6.

Modeling and analysis approach
We present in brief the details of the three systems of equations adopted to describe the response of the floating wind turbine considered.These are: the equations of motion in the frequency domain; the equations for the system eigenvalues; and the equations for the static response.The analysis approach adopted herein to evaluate the response of floating wind turbines is based on QuLAF [16], as in [15,17].QuLAF is an efficient frequency-domain approach for quick load analysis of floating wind turbines.However, compared to our earlier work where the geometry of a simple spar-buoy floater was optimized, in the present study we introduce two new major elements: (i) we consider a more complex floater topology inspired by the TetraSpar Demonstrator geometry [20] by Stiesdal Offshore; (ii) we include the tower geometry in the optimization problem.QuLAF has been extensively compared to FAST in previous work [16,26], and generally a good agreement was found for the damage-equivalent bending moment at the tower base.The floater motion was also well estimated, with some under-prediction for strong sea states due to the absence of viscous hydrodynamic forcing.
In the model considered here, the system has seven degrees of freedom, namely the floater surge, sway, heave, roll, pitch, and yaw, and the first tower fore-aft bending mode.The system response is evaluated for simultaneous wind and wave loads acting in a 2D plane where the degrees of freedom in surge, heave, pitch, and the tower fore-aft bending mode are directly excited.The aerodynamic loads and damping are pre-computed, as in [17].For a given rotor and controller they are extracted only once for each mean wind speed from time-domain aero-elastic rotor computations.The aerodynamic properties are thus assumed to be independent of the design variables for the floater, mooring system and tower.We note that while the rotor geometry does not change with the floater design, the controller settings, which influence the aerodynamic damping, may change due to variations of the natural frequencies of the system.

Design variables
We parametrise the floater and the mooring system with a total of  = 14 design variables, which are collected in the vector .These design variables are (see Figs. 1, 3(a), and 3(b)):    3).
•  12 depth of the mooring fairlead connection, •  13 mooring anchor radius (measured from the central axis of the central column), •  14 length of the mooring line.
Fig. 1 shows a schematic representation of the TetraSpar floater geometry considered.The red segments of the central column and the diagonal braces are dry.That is, they are assumed to be outside of the water.As a consequence, the water plane area is composed of a circular cross section due to the central column, surrounded by three ellipses due to the diagonal braces.Fig. 2 shows the resulting water plane area for the initial design guess of the floater considered in the optimization analysis, namely the design point  0 in Table 3.We assume that the diagonal braces and the central column meet 10 m above the mean sea level.In this way, the water plane area is made up of four disjointed areas that do not overlap (see Fig. 2).The keel cylinders are assumed to be fully filled with concrete.The keel diameter is not an independent design variable, but is calculated based on the vertical buoyancy equilibrium for a given design of the floater, the mooring line system, and the tower.The length of the keel cylinders is the same as that of the lateral braces.The structural components of the floater are considered rigid and their thickness is predefined and kept fixed during the optimization process.The thickness value of the floater components herein is set to 5 cm [27].Moreover, during the optimization process, the internal forces in the floater structural elements are neither computed nor constrained.Figs.3(a) and 3(b) show a schematic view of the models considered for the tower and the mooring lines.More details are given in Sections 2.3 and 2.5, respectively.

Modeling of the hydrodynamics
The floater and keel are represented by an ensemble of cylindrical members, for which the hydrostatic and hydrodynamic properties are computed internally in every design iteration by the code developed by the authors.Calculations of added mass and wave loads are efficiently formulated by analytical integration of the Morison equation [28] along a cylinder with an arbitrary orientation.This yields a closed-form complex valued transfer function from the Fourier amplitudes of the free surface elevation signal to the integrated hydrodynamic loads, which can be used for any random realization of the waves.Although we only consider strictly linear wave loads, viscous damping is included and estimated from the Morison equation and the dimensions of each cylinder.Consistent with slender-body theory, radiation damping is neglected, and the added mass is frequency independent.This efficient and general wave load transfer function formulation allows its application to any floater consisting of slender cylinders, thus bypassing the use of third-party radiation-diffraction solvers and speeding up the optimization analysis framework.

Modeling of the tower
We include in the model the fore-aft bending mode of the wind turbine tower.To this end, the tower is modeled with 2D Euler-Bernoulli beam elements, with two degrees per node (one rotation and one horizontal displacement).The tower is 129.495 m long [29], and we define 150 nodes equally spaced along its length.This results in a wind turbine tower made of 149 beam finite elements, each with constant mechanical properties.We parametrise the tower with the top and bottom values of diameter and thickness, as indicated in Fig. 3(a).These quantities are linearly interpolated along the height of the tower.The fore-aft bending mode is used to calculate the modal contributions of the tower to the matrices of the system mass  and stiffness .In the design optimization process for the floater and the tower, among the constraints we also include limits to the tower first fore-aft natural frequency.This allows for the design of stiff-stiff or soft-stiff towers, depending on the specific tower frequency constraints imposed.

Governing equations
The linear equations of motion in the frequency domain for a floating wind turbine system are written as follows: or in compact form In Eq. ( 1),  is the vector of the design variables which include the geometric properties of the floater, the mooring system and the tower; ξ = [ ξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 , ξ7 ]  is the vector of the Fourier motion amplitudes in the frequency domain associated to the th environmental condition; and Fℎ  (, ) and F  () are the hydrodynamic and aerodynamic loads in the frequency domain.The hydrodynamic loads are computed internally by the Morison equation for each design update and environmental condition.The aerodynamic loads are pre-computed beforehand in FAST [14] considering rigid-blade and fixed-nacelle rotor computations, hence they do not depend on the design updates.() and () are the structural mass and hydrodynamic added mass matrices, which are computed internally and updated during the optimization process; the damping matrix () accounts for contributions from: the structural damping on the tower, only affecting the tower degree of freedom; the pre-computed aerodynamic damping, affecting surge, pitch and tower degrees of freedom; and hydrodynamic viscous damping, affecting surge, heave and pitch, computed internally for each design update.For further details on the damping calculations we refer the reader to Pegalajar-Jurado et al. [16].() is the restoring matrix, which includes the hydrostatic and mooring contributions and it is recomputed after every design update during the optimization analysis.We note that in the formulation considered herein the added mass matrix () does not depend on frequency, since it is based on the Morison equation.The aerodynamic damping, on the other hand, is frequency-dependent.Although other authors have investigated frequency-dependent aerodynamic inertia [30], here we include only the aerodynamic damping.
The optimization problem formulation considers also limits on the undamped natural frequencies through the system eigenvalues,   .They depend on the design variables and are calculated from the following system: where   are the eigenmodes associated to the eigenvalues   .
The static surge and pitch ( ,1  ,5 ) are also constrained in the optimization problem formulation.They are calculated considering the rated thrust (  ) and its associated moment (  ) around the point of flotation shown in Fig. 1.The static forces are collected the static load vector   = [  , 0, 0, 0,   , 0, 0]  .Other mean-inducing loads, such as hydrodynamic drift loads or drag forces on the tower, were not included because the aerodynamic thrust is dominant when the wind turbine is in operation, as in all the cases considered here.The static pitch and surge are calculated by solving the following linear system: where  is the restoring matrix, and   = [ ,1 ,  ,2 ,  ,3 ,  ,4 ,  ,5 ,  ,6 ,  ,7 ]  .

Mooring system
We consider a mooring system made of three identical catenary lines uniformly distributed along the external circumference of the floater, defined by the radial braces.We consider only the stiffness matrix contribution of the mooring system in the initial configuration, which is added to the global stiffness matrix of the entire system  in Eqs. ( 1), (3), and (4).In particular, in this work we rely on the results presented by Al-Solihat and Nahon [31], where an analytical derivation of the stiffness matrix for such mooring systems is provided.

Estimation of lifetime fatigue damage
In the optimization analysis we estimate the lifetime fatigue damage and we constrain it.In particular, we transform the response of the wind turbine system obtained in the frequency domain with QuLAF to the time domain using the inverse fast Fourier transform (IFFT).We note that an IFFT is a very efficient way to get back to the time domain, where post-processing steps such as rainflow counting and fatigue estimation are more naturally carried out.Subsequently, for each th environmental condition we use the obtained system responses in time to compute discrete time series of the bending moment at the tower base,   (  ) for   = 0, … ,   .We expect the tower base to be the most critical location from a structural point of view, since both wind loads and turbine-related inertial loads create a larger moment here due to the longer arm.Nevertheless, the structural integrity both for fatigue and extreme loads of other tower locations should be considered in the subsequent time-domain analysis.Assuming that the influence of the axial loads is negligible, we convert the bending moment to stress using the classic equation where  is the outer radius of the tower-base circular cross section, and  is its second moment of area.A standard rainflow counting method is applied on the stress time history, from which a table of load range versus number of cycles is obtained.The mean stress is not considered in the calculations.According to the DNV recommended practice report for fatigue design of offshore structures [32], the fatigue damage for a given simulation of environmental condition EC  is where   is the number of entries in the rainflow table,   is the th stress range, and   is the number of cycles with range   .For non-welded hollow steel cross sections and a number of cycles below 10 7 , the values  = 4 and log 10 ā = 15.117 from [32] are used.
Once the above analysis has been done for different environmental conditions, we upscale the damage of each EC  according to its lifetime probability of occurrence   , the expected service life of the wind turbine    (taken here as 25 years), and the simulation time   (one hour) as We consider a non uniform probability of occurrence of the environmental conditions.The probabilities of occurrence considered are shown in Fig. 4.They have been defined by interpolating the probabilities of occurrence provided by Vigara et al. [33] for the COREWIND project, Site B -Gran Canaria Island.In our optimization framework, the fatigue lifetime damage at the base of the tower is constrained to    ≤ 1.

Estimation of extreme responses
We consider constraints limiting the dynamic response of the system.To this end, we evaluate the peak surge, pitch, and tower top acceleration in time, and constrain them to predefined allowed limits.In particular, given a response in time (  ), with   = 1, … ,   we calculate the maximum response as  2.
where  is a high even number, e.g. 10 6 .Eq. ( 8) approximates the maximum absolute value of the vector   , with increasing precision for increasing values of .In Appendix we provide additional details on the approximation of the max function given in Eq. ( 8).Thus, for each th environmental condition considered, the maximum surge    1, , pitch   5, , and tower top acceleration ü 7, in time are evaluated with Eq. ( 8) and constrained to be less or equal than predefined maximum allowed limits.More details are given in Section 3.2.

Optimization problem formulation
The optimization problem considered in the following is a nonlinear and non-convex one.It consists in minimizing a massproportional objective cost function, which accounts for the costs of the floater, the tower, and the mooring system.Constraints are imposed on the eigenvalues, the dynamic and the static responses of the floating wind turbine system considered, as well as on the fatigue damage at the tower base.Formally, the optimization problem is stated as follows: minimize In Eq. ( 9)  is an index that runs over the constraint functions considered, and  is an index that runs over the design variables considered in the optimization problem.  and v are the lower and upper bounds for the th design variable   .The matrices   and  contain the coefficients of the linear equality and inequality constraints, respectively.The right-hand-side terms   and  in our case are vectors with all entries equal to zero.Lastly, () is the objective function minimized and ()  for  = 1, … ,  are nonlinear inequality constraints.More details about the objective function, and the linear and nonlinear constraints of problem (9) are given in Sections 3.1 and 3.2, respectively.
It should be noted that the optimization problem at hand is highly nonlinear, non-convex, and thus potentially characterized by several local minima.The solutions obtained should thus be considered as local minima of the problem that improve an initial design guess while fulfilling the design constraints.The codes for the evaluation of the wind turbine system response, and for the optimization analysis used for the solution of (9) were implemented in MATLAB.For optimization, we used the SQP algorithm implemented in the fmincon function, which is part of the MATLAB Optimization Toolbox.For more information on SQP, the interested reader is referred to Nocedal and Wright [34].The gradients of the objective function and the constraints are calculated by the fmincon function with finite differences.During initial numerical experiments, we observed that forward finite difference approximations of the gradients, with fmincon default step-size, led to a stable convergence towards final optimized solutions.fmincon also automatically ensures that the step size of the design variables does not violate the variables' bounds.Thus in the final numerical experiments, we used forward finite difference approximations with fmincon default step size, which is √  ≈ 1.5 × 10 −8 .In the following, we provide the details of the objective function and constraints considered.

Objective cost function
The objective cost function minimized in the optimization problem ( 9) is: where:   is the steel mass of the floater and keel calculated for each design iteration, and it is increased by 15% to include the additional mass associated to the secondary structures [27];   is the mass of the tower;   is the mass of the mooring lines; and   is the mass of the cables connecting the keel to the floater.The parameters   ,   ,   , and   are the different unit costs, here considered non-dimensional and whose purpose is mainly to adjust the relative weight between the single objective function components.More details on the numerical values of the parameters involved in the characterization of the objective function defined in Eq. ( 10) are given in Section 4.

Constraints of the design optimization problem
The optimization problem formulation includes several constraint functions that impose limits on the floating wind turbine system response and on the design variables.In the following, we provide the details of the constraints considered.
The matrix   has dimensions [2 × ], i.e. there are two linear equality constraints.In the first row,   (1, 1) = 1 and   (1,12) = −1, which impose that  1 =  12 .In words, this means that the depth of the fairlead has to coincide with the depth of the bottom end of the central column of the floater ( 1 =  12 ).In the second row,   (2, 4) = 1 and   (2, 6) = −1, which means that the diameter of the radial brace is equal to the diameter of the lateral brace ( 4 =  6 ).The matrix  has dimensions [1 × ], i.e. there is one linear inequality constraint.In particular, (1, 1) = 1 and   (1,11) = −1, which means that the keel must be below the central column of the floater ( 1 ≤  11 ).As mentioned above, both   and  are vectors with zero entries.
We consider two constraints on the static surge and pitch of the TetraSpar floater, which are calculated with Eq. ( 4).The two constraints read as follows: where   ,1 = 12.5%   [27], with   the water depth.We consider five additional constraints on the natural frequencies of the system, calculated by solving the eigenvalue problem stated in Eq. ( 3).In particular, we impose an upper bound on the surge frequency   1 , and upper and lower bounds on the heave and pitch frequencies, i.e.   3 and   5 respectively.These constraints are defined as follows: We constrain also the natural frequency of the tower.We consider two design cases, and associated sets of constraints.In one case, we pursue a stiff-stiff tower design, with the tower natural frequency placed above the upper-limit of the 3 range: In the second case, we choose a soft-stiff tower design, with the tower natural frequency placed between the 1 and 3 ranges: Here we considered safety frequency margins of   = 15% with respect to the allowed tower frequencies  1 and  3 .
To keep certain resemblance with the TetraSpar concept and avoid that the optimizer finds waterplane-stabilized solutions, we impose also a constraint that ensures that the center of mass   of the floater is below its center of buoyancy   : The mooring line length  14 has the following lower and upper bounds: where:   is the length for which the line is fully stretched between the anchor point and the equilibrium position of the fairlead;   is the length for which the line is completely slack (this length is equal to the anchor radius plus the vertical distance from the sea bed to the fairlead point);   is a safety coefficient set to 5%.Both   and   depend on the radius of the anchor  13 , the diameter of the central column  2 , and the length of the radial braces  3 .We also constrain the vertical component of the tension force in the mooring lines at the fairlead.The purpose is to ensure that the mooring lines are never fully suspended, thus avoiding vertical loads at the anchors.This constraint considers the static tension in the mooring line calculated by solving the catenary equation for a mooring line resting on a seabed.More details on the catenary equation can be found in [31].
This constraint is thus defined as follows: where   is the vertical component of the tension force  in the mooring line at the fairlead.Further,   is the weight of the mooring line in water, and is defined as   =    14   , where   is the weight of the mooring line per meter,  14 is the variable length of the mooring line, and   is the number of mooring lines, set to three in this work.
During initial numerical experiments, we observed that the optimizer tended to converge towards final floater designs with a small footprint (short radial braces compared to the length of the central column) and a very deep keel.Essentially, these designs resembled a spar-buoy design configuration.To ensure that the obtained designs were characterized by reasonable ratios between the length of the radial braces and the central column from an engineering point of view, we added two constraints.One on the inclination of the diagonal braces, and one on the vertical hydrostatic equilibrium during the towing phase.Thus, herein we constrain the inclination of the diagonal brace   as follows: where   is the maximum allowed angle of inclination.To define its value we referred to Borg et al. [20], where the diagonal brace inclination is 40 deg with respect to an horizontal axis.
For the full-scale TetraSpar Demonstrator, once the floater is built and the wind turbine tower and the rotor-nacelle assembly are mounted on the floater, the wind turbine is towed from the harbor to its assigned site.During this phase, the keel is not deployed and is considered neutrally buoyant.Only when the towing boat reaches sufficient water depth, the keel can be lowered.Thus, we consider a constraint that ensures vertical hydrostatic equilibrium in the initial phases of towing when the keel has not yet been lowered.The vertical equilibrium of the wind turbine system is provided by the buoyancy of the radial and lateral braces.Thus: In Eq. ( 19),   is the weight of the wind turbine tower, the rotor-nacelle assembly, and the floater without the keel.  is the buoyancy force generated by the radial and lateral braces.Here we consider a scenario where the cylindrical elements of the lower triangle at the floater base are half-submerged.The parameter  is set to a small value (e.g. 10 −2 ), and it should be seen as a tolerance.In fact,  15 is a relaxation of the equality constraint   =   , and it is convenient from a numerical point of view for a smoother convergence of the optimization analysis.We consider a set of environmental conditions (EC) and we evaluate the time series of dynamic response of the wind turbine system for each EC with Eq. ( 1).In particular, we constrain the maximum pitch rotation, surge displacement, 2 and tower-top acceleration to predefined allowed values.This results in the following additional dynamic constraints for each th EC considered, with  = 1, … ,   : In Eq. ( 20), û1 is the maximum surge displacement allowed and it is here set to 25% of the water depth   [27]; û5 is the maximum pitch rotation allowed; and û7 is the maximum tower-top acceleration allowed.The extreme responses in time for each th EC (  1, ,   5, , ü 7, ) are calculated as described in Section 2.7.Lastly, we impose a constraint on the lifetime fatigue damage evaluated at the tower bottom.This constraint is defined in Eq. ( 7): The details of the parameters involved in the definitions of the optimization constraints described in this section are provided in Section 4 and Table 1.

Optimized designs for stiff-stiff and soft-stiff tower
In the following, we present several numerical results.We solved problem (9) for different design cases, defined by different specific settings of the constraints discussed in Section 3.2.We consider the following settings for the numerical examples:  13)  3 = 0.378 Hz and in Eq. ( 14)  1 = 0.126 Hz and  3 = 0.250 Hz; • minimum and maximum lengths of the mooring lines • for the mooring lines we consider a mass in water   of 300 kg/m; •   is set to 40 deg; • in Eq. ( 19)  is set to 10 −2 ; • the maximum dynamic surge û1 , pitch û5 , and tower top acceleration û7 are set to 80 m, 11 deg, and 2 m∕s 2 , respectively.
Table 1 provides an overview of the constraints considered in the optimization analysis (Section 3.2), with details on the upper or lower bounds considered.For the evaluation of the dynamic response constraints of Eq. ( 20) the environmental conditions considered are listed in Table 2.
The unit costs   ,   ,   , and   of the objective function defined in Eq. ( 10) are set to one.During initial numerical experiments, we also investigated different values of these cost parameters.Nevertheless, we observed a tendency of the optimization algorithm to converge towards similar final optimized design solutions, independently of the specific values assigned to the cost parameters.To calculate the mass of the floater structural components we consider: steel density equal to 7850 kg∕m 3 ; dry mass per unit length of the mooring lines equal to 300 kg/m; and mass per unit length of the cables connecting the keel to the floater equal to 15 kg/m as in [20].
The first numerical examples that we present consist of optimizing the floater, the mooring system, and the tower for stiff-stiff and soft-stiff tower design configurations.By stiff-stiff tower design, we mean a tower design such that its coupled natural frequency lies above the 3 frequency range.For this, we consider the constraint of Eq. (13).By soft-stiff tower design, we mean a tower design such that its natural frequency lies between the 1 and 3 frequency ranges.This is achieved considering the constraints of Eq. ( 14).Table 3 lists the lower and upper bounds of the design variables (i.e.,   ,   ), as well as the initial and final optimized designs obtained in the two design cases, i.e.  0 ,     , and     .The stiff-stiff optimization analysis case converged in 26 iterations to a final design with an associated objective function value of 10.65 10 6 kg.The soft-stiff optimization analysis case, instead, converged in 86 iterations to a final design with an associated objective function value of 10.19 10 6 kg.For comparison, we performed an optimization analysis in which we did not consider the 1P and 3P constraints for the tower frequency ( 8 and  9 from Eqs. ( 13) and ( 14)).We refer to this design as the free tower design,     .In this case, the final objective function value was 9.06 10 6 kg, and the optimization converged in 47 iterations.We note that, although no constraint was applied to the tower frequency in this case, the constraint on the lifetime damage  19 =    − 1 ≤ 0 was satisfied.
The optimized layouts of the two floaters for the stiff-stiff and soft-stiff design cases are shown in Figs.5(a) and 5(b).It can be observed in Fig. 5 that the two designs of the floater are quite different.The most striking difference is that the floater obtained

Table 3
Lower and upper bounds (  ,   ) of the design variables, initial design ( 0 ), final optimized design solution for the stiff-stiff (    ), soft-stiff (  for the stiff-stiff tower design case has shorter radial braces ( 3 = 39.1 m), compared to the floater obtained for the soft-stiff tower design case ( 3 = 52.0m).In the stiff-stiff tower design, the stiffness of the tower is sufficient to meet the structural deformation demand of the wind and wave loads.In the soft-stiff design case, instead, the tower is more compliant, and the optimizer identifies a wider floater to reduce the floater motion.Because the floater and tower bend in anti-phase in the global tower vibration mode, the global eigen frequency is shifted upward relative to that of a clamped tower.Hence, the larger pitch stiffness of the wider floater helps to keep the global tower frequency below the 3P region.The evolution of the values attained by the objective functions and the envelope of the constraints during the optimization analyses for the stiff-stiff and soft-stiff design cases are shown in Fig. 6(a) and in Fig. 6(b).In both design cases, a decrease of more than 60% of the objective function value with respect to the initial design guess is observed.Moreover, we can see from the two figures that for the final optimized designs all the constraints' values are less than or equal to zero, within given acceptable tolerance levels.This confirms that for the optimized designs all the constraints are satisfied.Fig. 7 shows the frequency content of the wind and wave loads for EC 4 (see Table 2), as well as the 1 and 3 frequency ranges considered in the optimization.In the figure, the first fore-aft frequencies of the optimized stiff-stiff, soft-stiff, and free tower designs are also shown with black lines.Thus, it is possible to identify the available ranges in which the optimizer can place the tower frequency for the soft-stiff and stiff-stiff design cases.We can observe that in the soft-stiff case, the tower frequency falls  2), 1P and 3P frequency ranges, tower fore-aft frequency for stiff-stiff, soft-stiff and free tower design cases.
where the wave loads have a high energy content.Hence, the wave loads tend to excite the tower fore-aft frequency through the floater motion, and as a consequence the optimizer proposes a larger floating platform with the goal of reducing the wave-induced floater motion and thus the loads on the tower.This is also supported by the constraints analysis in Section 5, where the fatigue and acceleration constraints are active for this design (see also Table 4).
In Figs. 8 and 9 we compare the response of the stiff-stiff and soft-stiff optimized designs for EC 4 (associated to rated wind speed) and EC 12 (associated to cut-out wind speed), respectively.The response associated to the stiff-stiff design case is plotted in blue, whereas the data associated to the soft-stiff design case is plotted in red.In each figure we present time series of the wind speed for a selected time window, the free-surface elevation, the surge displacement, the heave displacement, the pitch rotation, and the nacelle acceleration.The figures also include power spectral density (PSD) plots of the time series, and the probability of exceedance of the peak values in the time series.These plots are shown to compare the different responses of the two designs, and to understand which loads affect the most those responses.For rated wind speed (Fig. 8), we observe that the two designs have similar surge and heave displacements, as well as pitch rotations.In the soft-stiff design case (red curves), by looking at the frequency content of the nacelle acceleration, we observe a high energy content in the frequency range of the wave loads, where the tower natural frequency is located.Moreover, we observe that the heave response signal in Fig. 8 has a mean displacement of approximately −0.1 m.This offset is the result of the vertical component of the rated mean thrust force.For the cut-out wind speed (Fig. 9) we can make observations similar to those for the rated wind speed case.The surge, heave, and pitch time histories are still similar, even though the heave response has a higher energy content in the frequency range close to that of the wave loads for the stiff-stiff design case.Also in this case, the nacelle acceleration of the soft-stiff design case shows a higher energy content in the frequency range of the wave loads.
Table 4 lists the values of the Lagrange multipliers of the nonlinear inequality constraints (see Section 3.2) in correspondence of the final optimized stiff-stiff and soft-stiff design solutions.In order to keep the amount of data listed in the table concise, only the Lagrange multipliers of the dynamic constraints  16 ,  17 ,  18 of Eq. ( 20) associated to EC 4 and 11 are reported (see Table 2 for a full list of the EC considered).The values of the Lagrange multipliers are non-negative, i.e.   ≥ 0 for  = 1, … ,   where   is number of nonlinear inequality constraints.In correspondence of an optimal design point  * (even in a local sense) the Karush-Kuhn-Tucker (KKT) conditions hold, Nocedal and Wright [34].In such a case,  * is said to be a KKT point of the problem at hand.Among the KKT conditions, there are the complementary conditions which are stated as follows: (23) Eq. ( 23) implies that for an optimal point  * , if the th constraint is active then   = 0 and  > 0, otherwise   < 0 and  = 0 if it is not active.When the constraint is active, i.e.   = 0, then the value of the Lagrange multiplier indicates how quickly the value of the objective function would change if we were to vary the constraint.Thus, the higher the value of a Lagrange multiplier, the higher is the influence of the associated constraint on the optimized design solution.At the end of the optimization analysis for the stiff-stiff tower design case, six of the constraints are active.That is, six of the constraints presented in Section 3.2 have a value close to zero at the end of the optimization analysis.These constraints are  1 ,  2 ,  8 ,  12 ,  14 , and  15 .At the end of the optimization analysis for the soft-stiff tower design case, six of the constraints are active.These constraints are  1 ,  8 ,  12 ,  15 ,  18,11 , and  19 .Lastly, for the free tower design case, seven nonlinear constraints are active.These are  1 ,  2 ,  12 ,  14 ,  15 ,  18,11 , and  19 .

Analysis of the design-driving constraints
From Table 4 it is possible to understand which of the constraints affects the most the stiff-stiff and soft-stiff final designs.Thus, we now relax some of the design-driving constraints and rerun the optimization analyses, to get a better understanding of their effect on the final optimized design obtained.We start by reducing the safety frequency gap of the constraints  8 and  9 , which was initially set to   = 15% (Eqs.( 13) and ( 14)).To this end, we now set it to   = 0%, and rerun the optimization analyses.Table 5 Fig. 9. Comparison of the dynamic responses at cut-out wind speed of the optimized stiff-stiff (in blue) and soft-stiff (in red) designs.The mean wind speed is 25.00 m∕s, significant wave height is 5.11 m, and the wave peak period is 9.00 s.

Table 5
Final values of the objective functions of the stiff-stiff and soft-stiff design cases for different values of the parameter   in Eqs.(13)   Next, we consider the constraints on the mooring line length, namely  11 and  12 .In particular, we study the effect of a relaxation of the safety coefficient   in Eq. ( 16), originally set to 5%.We now set it to   = 2.5%.As it can be observed in Table 6, the reduction of   from 5% to 2.5% causes a decrease of 1.4% and 0.79% in terms of the final objective function value in the stiff-stiff and soft-stiff design cases, respectively.Similarly to the previous case, these results translate directly into lower values of structural mass, because the objective function cost components are all set to one.
In Table 4 it is possible to observe that the static surge and pitch constraints,  1 an  2 (Eq.( 11)), are active for the stiff-stiff design case, whereas for the soft-stiff design case only the constraint on the static surge is active.It seems therefore interesting to explore also their effect on the obtained final designs.We first relax the limit on the static surge displacement allowed.To this end, we modify   ,1 by increasing its initial value of 40 m by 10%, leading to   ,1 = 44 m.From the results listed in Table 7 we learn that in the stiff-stiff design case a 10% relaxation of the static surge constraint causes a decrease of approximately 1.89% in the objective cost value.For the soft-stiff design case, such relaxation results in a 1.23% decrease of the final objective function value.Next, we relax the static pitch constraint.We modify   ,5 by increasing its initial value of 5 deg by 10%, leading to   ,5 = 5.5 deg.Table 8 shows that in the stiff-stiff design case, a 10% relaxation of the static pitch constraint causes a decrease of 0.26% in the objective cost value.For the soft-stiff design case, such relaxation does not lead to a change in the final objective function value.
Lastly, the results listed in Table 4 show that for the soft-stiff design case the constraint on the tower-top peak acceleration is active for EC 11.Thus, we relax the constraint  18 to study its effect on the final value of the objective function.In particular, we increase its initial value of 2 m∕s 2 by 50%, leading to û7 = 3 m∕s 2 in Eq. ( 20), which is regarded as an allowable value.Table 9 shows that in the soft-stiff design case, the relaxation of the dynamic tower top acceleration constraint causes a decrease of 12.03% in the objective cost value, which is a significant saving.For the stiff-stiff design case, such relaxation does not change the final objective function value.As a final measure of the effect of the constraints' limits on the final optimized designs and their associated objective function values, we define the following parameter: In Eq. ( 24),  =   −  0 where  0 is the final objective function value associated to the     or     solutions listed in Table 3, and   is the final objective function value obtained when the relaxed constraints are considered.Similarly,  ḡ = ḡ − ḡ0 and ḡ0 is the initial constraint limit, whereas ḡ is the relaxed constraint limit.Essentially,  − measures the normalized change of the final objective function value with respect to a change in a constraint limit.For each of the constraints considered in this section, we report in Table 10 the associated value of the parameter  − .It can be observed that for the stiff-stiff design case, we achieve the highest relative change in the final objective function value when the limit of the constraint  8 on the tower fore-aft frequency is relaxed (Eq.( 14)).However, looking at Table Table 10, it can be observed that a relaxation of the constraint  1 (Eq.( 16)) for the maximum static surge also seems to have a significant effect on the final design.For the soft-stiff design case, the highest change in objective function is also achieved when the constraint  8 on the tower fore-aft frequency is relaxed (Eq.nEq.( 14)).

Final considerations
In this article, we presented a new framework for the simultaneous optimization of a TetraSpar-type floater, the mooring system, and the tower for the IEA Wind 15 MW reference wind turbine.The optimization framework is based on fast frequency domain analysis to evaluate the dynamic response of floating wind turbine systems.Several design variables are considered during the optimization analysis process.These include the dimensions (length and diameter) of the TetraSpar floater cylinders, the length, radius, and fairlead position of the mooring lines, and the tower thickness and diameter at the top and bottom ends.In the optimization problem formulation, a mass-proportional cost function is minimized.Constraints are imposed on static and dynamic responses, as well as on the frequencies of the floating wind turbine systems.Additional constraints are imposed from design requirements for the floater and mooring line systems.The optimization problem is solved with a gradient-based algorithm.Two main design cases are considered.Namely, the cases of stiff-stiff and soft-stiff tower designs, for given 1P and 3P frequency ranges.In the numerical examples, the effect of several design-driving constraints is studied and reported in detail.Through these experiments, it has been possible to gain a better understanding of which of the design constraints considered are the most influential on the final design obtained and to what extent a relaxation of the constraint limits leads to different final optimized solutions.Based on the numerical results obtained, the following observations can be made: • The stiff-stiff tower design criterion results in a floater with shorter radial braces compared to the soft-stiff design case.This can be explained by the need to reduce the floater motion and associated tower loads in the soft-stiff design case; • The numerical results suggest that it is possible to obtain a feasible conceptual design solution also in the case in which the 1P and 3P tower frequency constraints are not considered.In this article, we refer to this design as free tower design.In our case, this led to the cheapest floater and tower designs that met all of the design constraints considered.This result thus indicates that the most cost-effective tower designs may be associated with natural frequencies within the 3P region; • Among the design-driving constraints there are the static surge displacement constraint, the static pitch rotation constraint, the constraint on the minimum length of the mooring line, the natural frequency constraints of the tower, the maximum acceleration constraint of the nacelle, and the fatigue constraint at the tower bottom.Relaxation of these constraints can thus be used to reduce the costs; • The stiff-stiff and soft-stiff tower design cases lead to very different designs of the associated TetraSpar floaters.This indicates that there is a tight link between the floater and tower design, and thus the design of these components should ideally be carried out simultaneously.
The design optimization approach discussed in this paper could serve offshore structural engineers in the conceptual design phases for the floating structures, the tower, and the mooring system of next generation offshore floating wind turbines.In addition, the settings adopted and the results obtained in this work could serve as a reference for future studies on this topic.Lastly, it should be mentioned that, in practical design scenarios, the designs obtained with the proposed approach should be further analyzed with time domain simulations considering a full Design Load Basis (DLB).

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.Schematic view of the floater layout with the associated optimization variables and point of flotation.

Fig. 2 .
Fig. 2. View of the water plane area for the initial design  0 (see Table3).

Fig. 3 .
Fig. 3. Schematic view of the tower layout and of one mooring line with the associated optimization variables.

Fig. 4 .
Fig.4.Probability of occurrence  for each wind speed WS associated to the environmental conditions EC defined in Table2.

Fig. 5 .Fig. 6 .
Fig. 5. Optimized designs for the results discussed in Section 4. The red segments of the central column and diagonal braces are assumed to be outside of the water.

Fig. 7 .
Fig. 7. Shown in the figure: frequency content of wind and wave loads for EC 4 (Table2), 1P and 3P frequency ranges, tower fore-aft frequency for stiff-stiff, soft-stiff and free tower design cases.

Fig. A. 10 .
Fig. A.10. Function considered for the calculation of the maximum absolute value with the p-norm approximation (Fig. A.10(a)), and error of the p-norm max function approximation with respect to the ground truth for varying values of  (Fig. A.10(b)).
central column, •  3 length of radial brace, •  4 diameter of radial brace, •  5 diameter of diagonal brace, •  6 diameter of lateral brace, •  7 tower diameter at bottom, •  8 tower diameter at top, •  9 tower thickness at bottom, •  10 tower thickness at top, •  11 depth of the keel from mean sea level,

Table 1
(9)linear design constraints definitions (Section 3.2) and associated limits considered in the optimization problem(9).n/a stands for not applicable.When the upper or lower bound is design dependent its value is not fixed, and we indicate this with n/a.
a stiff-stiff tower design case.b soft-stiff tower design case.cthenvironmental condition EC, see Table2.

Table 2
[33]ronmental conditions EC with wind speed   in m/s, significant wave height   in m, and peak period   in s.EC 4 is associated to the rated wind speed of the wind turbine.The ECs have been defined by interpolating the values provided in[33]for the COREWIND project, Site B -Gran Canaria Island.
lists the optimization results obtained for   = 0% and those initially obtained for   = 15% for comparison.As it can be observed, the reduction of   from 15% to 0% causes a reduction of 5.65% and 11.05% in terms of final objective value for the stiff-stiff and soft-stiff design cases, respectively.As the objective function cost components are all set to one, these results translate directly into reductions of structural mass.

Table 6
Final values of the objective functions of the stiff-stiff and soft-stiff design cases for different values of the parameter   in Eq. (16).

Table 8
Final values of the objective functions of the stiff-stiff and soft-stiff design cases for different values of the parameter   ,5 in Eq. (11).

Table 9
(20)l values of the objective functions of the stiff-stiff and soft-stiff design cases for different values of the parameter û7 in Eq.(20).

Table 10
Values of the parameter  − defined in Eq. (24) for the constraints analyzed in Section 5.