Design Optimization of Floating Offshore Wind Turbine Substructure Using Frequency Domain Modelling and Genetic Algorithm

Floating offshore wind turbines are becoming increasingly popular as a promising technology for producing cleaner energy. However, in order to be competitive with fixed offshore wind projects or even onshore wind projects, the costs associated with floating offshore wind turbines must be significantly reduced. To tackle this challenge, this paper presents a comprehensive framework for optimizing the design of floating offshore wind turbine substructures and their components. The innovative open-source frequency-domain dynamic model RAFT is used to take into account the aerodynamic, hydrodynamic and mooring forces that impact the stability and dynamics of the floating system. The numerical model is coupled with an efficient genetic algorithm to minimize the structural mass of the floating platform while maintaining its stability and dynamic performance. The geometrical parametrization of the substructure, the implementation of the numerical model, and the overall optimization process are all thoroughly detailed in this paper, along with preliminary results that demonstrate the potential cost reductions that can be achieved through this framework.


Introduction
Offshore wind turbines (OWT) have gained popularity worldwide, especially in Europe, due to the decreasing levelized cost of energy (LCoE) associated with this technology over the last few decades [1].However, conventional bottom-fixed turbines cannot harness a significant portion of the world's offshore wind resources due to technical and economic limitations.Floating foundations for offshore wind farms have the potential to unlock these untapped wind resources, particularly in water depths over 50 metres, but remain a costly alternative.In order to compete with fixed offshore wind technologies and even onshore wind projects, floating offshore wind turbines (FOWT) must significantly reduce their LCoE.
The design of FOWT substructures plays a crucial role in the overall performance and economic viability of floating wind farms.A wide range of concepts have been proposed and studied, each with its own design, fabrication process, installation technique, or stabilizing principle.In recent years, there has been a growing trend towards optimizing substructure designs for cost-effectiveness.This has resulted in the development of more streamlined and modular floating platforms, made up of standardized components that can be prefabricated and assembled efficiently in docks, enabling mass production and reducing the costs of fabrication and installation for each unit.One notable example of a cost-effective FOWT substructure concept is the tension leg platform (TLP) developed by the GICON group and its partners since 2009.The latest design of the GICON ® -TLP consists of four cylindrical buoyancy bodies arranged in a rectangular shape, connected by horizontal bracing tubes and vertical and diagonal tubes to the wind turbine transition piece.This TLP design has undergone several key design modifications over the years, and its development has been documented in [2] and is depicted in Figure 1.Additionally, the universal buoyancy body (UBB) concept, proposed in [3], offers a straightforward solution as a modular component for cylindrical buoyancy bodies.The design of the UBB, inspired by monopiles and wind turbine towers, aims to provide a cost-efficient solution for FOWT substructures.Optimizing the design of FOWT is essential to enhance their performance and efficiency, and to reduce the overall costs, not only for individual turbines but also for entire wind farms.optimization efforts can be directed towards various components of the complex system, such as rotor blades, towers, foundations or mooring systems, to name a few.The substructure is one of the most significant components of FOWT in terms of cost [5] and is a key area for optimization [6].A well-optimized design of the floating platform can significantly impact the overall performance and cost-effectiveness of the system.Numerical modelling of FOWT is vital for understanding and assessing the behavior of the floating turbine, such as its static properties or dynamic response, and is a critical aspect of the design process.The harsh environment in which FOWT operates results in different types of loads with non-linear effects, and the complex interactions between the different components of the system further complicate the analysis.Finding a proper balance between accuracy and computational efficiency is essential for any optimization work.Complex analysis with few assumptions may lead to a slow optimization process but with potentially more accurate results, while quick analysis using simple models is easier but may omit some important aspects of the modelling.
The main objective of this study is to develop a design optimization framework for the GICON ® -TLP, with the aim of reducing its structural mass and associated material costs.The framework utilizes an open-source frequency-domain code to model the FOWT and includes the parameterization of the substructure, along with the implementation of an efficient genetic algorithm (GA) for the optimization process.The following sections outline the methodology used and provide a detailed description of the optimization process.The paper concludes with preliminary results from a case study using the GICON ® -TLP, which demonstrate the potential for cost reduction and identify areas for future improvement.

Dynamic model
In this study, the Response amplitudes of floating turbines (RAFT) open-source code [7] is used to model a multi-megawatt wind turbine mounted on the GICON ® -TLP.This frequency-domain dynamic model has the capability of evaluating the static properties and dynamic response of FOWT.The code has been validated against higher-fidelity tools such as OpenFAST and has shown good agreement while also being computationally efficient.This makes it an ideal choice for quickly evaluating various substructure designs under different environmental conditions.The complete description and theory behind RAFT can be found in [8], as here only some of the aspects of the modelling are highlighted to provide context for the design optimization framework presented.
The rotor-nacelle assembly (RNA) is represented as a lumped mass with three identical rotor blades with specific dimensions and distributed aerodynamic properties.The rotor aerodynamics at a particular mean wind speed are modelled using a steady-state blade-element-momentum theory solver that computes the mean system loads and the aerodynamic damping contribution to the system's dynamics.The wind turbine tower is represented as a single cylindrical member with any number of sections, tapers, and wall thickness variations.The floating substructure is also represented as a combination of cylindrical members with their geometric properties and hydrodynamic coefficients.The strip-theory hydrodynamics approach is based on the relative form of the Morison equation for transverse flow across a cylinder strip, accounting for wave and body velocity.The mooring system is simulated through a quasi-static mooring system model that includes floating bodies with linear hydrostatic properties.The FOWT is modelled as a rigid-body with six degrees of freedom, namely surge, sway, heave, roll, pitch, and yaw.The system responds linearly to each excitation frequency, allowing for the superposition of harmonic responses.The complex amplitude of the response X at a frequency ω can be obtained by solving the generic frequency-dependent equation of motion of the floating system, as described in [8]: where M is the FOWT's mass and inertia matrix, M a the hydrodynamic added mass matrix, B the damping matrix including the linearized quadratic drag effects from the Morison equation and damping coefficients from rotor aerodynamics, K h the total hydrostatic stiffness matrix of the floating turbine accounting for water-plane area effects of the substructure and moment arm effects due to the total structure weight, K m the mooring system stiffness matrix, and Fext the complex excitation vector from inertial and drag contributions from the Morison equation.In Equation 1, the frequency-domain dynamics are assumed to operate around a specific point, which is defined as the system's mean state.The mean position X is obtained by solving the static equilibrium equation, as in [8]: where F ext is the mean wind and wave load, and F m the non-linear reaction force of the mooring system accounting for the effective mooring stiffness.The maximum platform response X max is then approximated by the mean state value plus three times the standard deviation of the complex response spectra computed.Similarly, the maximum tower base fore-aft bending moment T max , can be estimated from the mean bending moment T , including the effects of the turbine's weight and thrust acting on it, and the standard deviation of the dynamic moment response spectra T (ω) which includes effects from inertial and aerodynamic reactions.

optimization process
In the literature, GA are often used when dealing with FOWT design optimization due to the complex behavior of the system and formulation of the problem.GA are a type of evolutionary algorithm that draw inspiration from natural selection process in biological evolution.Initially, a population of potential solutions is created and evolves through genetic operations like selection, crossover, and mutation to generate new candidate solutions.Each individual in the population is represented by a set of design variables and its fitness which is evaluated based on the objective and constraints of the optimization problem considered.Over the generations, the population evolves towards better solutions with the aim of converging to the optimal solution.The aim of the design optimization framework outlined in this study is to reduce the mass of the GICON ® -TLP.In addition, constraints are formulated to take into account the dynamic performance of the FOWT under different environmental conditions, as well as geometric ranges limited by manufacturing capabilities.Consequently, the constrained single-objective optimization problem can be expressed as follows: where m is the overall mass of the substructure and v the selected design variables.The subscripts L and U denote the lower and upper bounds of the design variables, which define the design space.Additionally, X lim and T lim are the maximum platform response and tower base fore-aft bending moment limits, respectively, that must be respected for all the defined environmental conditions.To consider these constraints, the most common approach when using GA is to apply penalty functions.Even though penalty functions are frequently used with evolutionary algorithms, they require a clear definition and relevant tuning of their parameters, often resulting in problem-dependent penalty functions.Here, the penalty function p i associated with constraint i is defined proportional to the normalized amount of the constraint's violation, as follows: where γ i is a tuning parameter to prevent any bias on the constraint, and g i lim the limit of the constraint.Eventually, considering that P is the sum of all the penalty functions, the fitness f of one individual (which here is to be maximized by the GA) is computed as: The implemented GA is inspired by the process used in the cumulative multi-niching genetic algorithm (CMNGA), presented and detailed in [9].Unlike classical GA, the population extends over the generations and the genetic operations.To generate the initial population of n min individuals, the Latin hypercube sampling (LHS) method is used.This method ensures that the entire design space is sampled and that there is no overlap between the selected points, thus reducing the risk of obtaining redundant or unrepresentative design variable samples.After the initial population is generated, offsprings are added to the existing population using a variation of the classical selection, crossover, and mutation operations until the total population reaches a fixed maximum number of individuals n max or the maximum number of generations n gen is exceeded.The selection process involves choosing two parents from the existing population.The first parent is selected using the roulette wheel selection (RWS) method, a common fitness proportionate selection method, so that fitter individuals in the population have a higher chance of being selected as parents.The second parent is chosen to be close to the first one, using a distance proportionate RWS method to select a pool of individuals.The fittest individual in the pool is then selected as the second parent for the crossover operation.This approach forces the selection process to occur in the fitter regions of the design space.The simulated binary crossover (SBC) method is used for the crossover operation, which is particularly adapted for real-valued optimization problems and continuous design spaces [10].The design variables of the two new offsprings, v o 1 and v o 2 , are computed as: where v p 1 and v p 2 are the design variables of the selected parents and β is a tuning coefficient that determines how close or far the generated offsprings are from their parents.The mutation operation is performed using random mutation, generating mutants with random design variables chosen uniformly within the entire design space.To prevent new individuals (offsprings and mutants) from being too similar to the existing population, an adaptive fitness-related proximity constraint is implemented before evaluating their fitness.This ensures that the population converges around fitter regions of the design space, resulting in a sparse population density in less fit regions.This approach prevents an excessive number of objective function evaluations, which can often be a problem with other conventional evolutionary algorithms.

Design parameterization
The presented framework aims to optimize the design of the GICON ® -TLP, and thus requires a parametrization of the substructure.The latest design of this TLP consists of four cylindrical UBB arranged in a rectangle shape, connected by horizontal bracing tubes and vertical and diagonal tubes to the wind turbine transition piece.To simplify the parametrization, the focus is on the geometric properties of the cylindrical UBB used in the GICON ® -TLP design.The shape of the four buoyancy bodies is defined by a set of chosen design variables as follows: where l is the length of the cylindrical buoyancy bodies, d their diameter, and z the vertical position of their centre from the mean sea level (MSL).Figure 2 shows the design parameterization and member-based modelling of the substructure using RAFT and the chosen design variables for this study.The remaining design parameters of the GICON ® -TLP are kept constant for this study, including the lengths and diameters of all other pipes in the substructure's design.Additionally, the thickness of all the members involved in the numerical model remains the same throughout the entire optimization process.The height of the upper node above the MSL, the draft, and all parameters related to the mooring system or the wind turbine remain unchanged.It's worth noting that a safety margin has been included to ensure that the top of the buoyancy body doesn't come too close to the mean sea level, and the bottom of the UBB doesn't collide with the horizontal pipe.

Design optimization framework
The design optimization framework described in this article is implemented using Python's object-oriented programming language, which ensures short and readable code.As RAFT is also written in Python and is open-source, building the optimization framework on top of it is relatively straightforward.The framework includes a main optimization module that implements the previously described optimization process and interacts with RAFT to evaluate the fitness of individuals in the population as the algorithm progresses through generations.As GA can be easily parallelized, the optimization framework applies multi-threading to evaluate multiple individuals simultaneously using a pool of n core threads during the fitness function computation for initialization and addition operations.At the beginning of the optimization process, an input file is read to specify all the GA parameters, the design space, the constraint limits, as well as the wind turbine and mooring system to be used for the substructure.At the end of the optimization process, the fittest individual in the final population represents the solution to the defined optimization problem, and the associated RAFT model can be obtained.Figure 3 provides an overview of the framework's structure, including the files and modules involved, and a representation of the initial and final populations during the optimization process.

Study case
In this section, we consider the design optimization of the GICON ® -TLP in combination with the IEA 15-MW reference wind turbine [11].The turbine model used was adapted from the VolturnUS-S semisubmersible [12] RAFT model available at [7].Adjustments were made to the tower to accommodate the GICON ® -TLP's higher transition piece and to adjust the tower's mass distribution.The platform is equipped with four vertical mooring lines arranged in a square pattern, each connected to a suction anchor installed on the seabed at a water depth of 200 m.The model is evaluated in terms of dynamic responses under a stochastic sea state and steady wind, represented by a mean wind speed W s and a Pierson-Moskovitz wave spectrum of significant wave height H s and peak period T p .While the environmental conditions used in this study are not specific to any site, they were chosen from previous optimization and simulation work on FOWT to represent a typical design load case, as in [13], from which three representative environmental conditions were chosen.Figure 4 highlights the different components of the RAFT model of the GICON ® -TLP and the environmental conditions it is exposed to for this analysis.
It is worth noticing that there is no wind-wave misalignment, and that the efforts are normal to the rotor plane.
To determine the mass of the GICON ® -TLP, a steel density of 7850 kg.m 3 is considered.The dynamic performance of the system is restricted as in Equation4, with limit values for the maximum platform response in surge, heave, and pitch and for the maximum tower base fore-aft bending moment.The maximum constraint value among all the various environmental states considered for the analysis is compared to the constraint limit, whose generic value was selected based on other optimization and simulation work, as in [13].The steel pipes used to manufacture the UBB are derived from wind turbine tower and monopile designs and typically have diameters ranging from 5 m to 15 m.To explore different lengths, a slightly wider range of 5 m to 20 m is considered, and the vertical position range is limited by the platform's draft and the safety margin before the MSL.Table 1 summarizes the inputs used for this design optimization case study, including the GA parameters used for the optimization results presented next.

design optimization results
Figure 5 shows the evolution of the GICON ® -TLP' mass and its corresponding dynamic response and tower base fore-aft bending moment throughout the generations of the optimization process.The plot displays the fittest individual within the population for each generation as the population size increases.These results demonstrate the successful implementation of the optimization framework, resulting in an optimized design of the GICON ® -TLP that satisfies the required dynamic constraints while minimizing the mass and, hence, material cost.It is worth noting that the optimization process initially identified a design with lower mass.However, this potential design exceeded the defined constraint limits, resulting in a low fitness value at a later generation and leading the optimization to favour other designs with higher mass but with better dynamic behavior.Later during the optimization process, once a design seems to satisfy the constraints defined, the algorithm identifies new potential design with lower mass.To account for the inherent randomness of GA, multiple optimization runs were carried out to obtain a reliable characterization of the performance of the design optimization framework.The results presented here are a general representation of all the runs performed.Table 2 summarizes the optimized design variables and provides the maximum platform offsets of the optimized system, as estimated by the RAFT model, along with the tower-base bending moment for the most demanding environmental condition.The optimal design found features a long UBB design with a larger length than the diameter with a relatively low centre.This design has the advantage of minimizing the substructure mass due to its relatively small geometry but ensures sufficient buoyancy to the system so that its dynamics are positively impacted.It is important to notice that the results obtained are quite sensitive to the inputs of the problem (design space and constraint limits), therefore a certain control of these parameters is necessary in order to have confidence in the design optimization results obtained.Figure 5 shows the RAFT model of the optimized GICON ® -TLP in its unloaded (no wind and still water) and displaced positions under the most demanding environmental conditions.Additionally, the distribution of the design variables (normalized to the design space boundaries) of the final population is plotted.This highlights the fact that the algorithm did not add new individuals in uninteresting areas of the design space.As generations progress, the population converges towards fitter regions of the design space, resulting in a sparse population density in less fit regions.This prevents unnecessary design evaluations of potentially low fitness solutions, leading to a lower number of fitness function evaluations to reach convergence compared to other GA. Figure 6.Optimized GICON ® -TLP in its unloaded (black) and displaced (blue) positions under the most demanding environmental conditions.Design variables distribution at the end of the optimization process.

Conclusion and Outlook
In conclusion, floating offshore wind is a rapidly expanding industry that requires costeffective solutions to unlock the vast potential of offshore wind resources.Innovative concepts such as the GICON ® -TLP and the UBB provide such solutions for FOWT substructures, enabling mass production and reducing fabrication and installation costs.
This paper presents a design optimization framework specifically developed for the GICON ® -TLP.By implementing an efficient GA and the validated frequency-domain dynamic model RAFT, a cost-optimized platform design was obtained that met the defined dynamic constraints for varying environmental conditions.The framework has several advantages, such as being fast due to the frequency approach and the low number of fitness function evaluations, which prevents unnecessary design evaluation.Future improvements are necessary to enhance the design optimization framework presented in this study.The dynamic results obtained using RAFT accounted for strip-theory hydrodynamics and linearized quasi-static mooring dynamics, whereas other models can better capture mooring dynamic or hydrodynamic effects.More complex models, such as potential flow solutions for the hydrodynamic aspect, will be analyzed and implemented in the framework to improve the numerical model used.Fatigue loads derived from time domain simulation are also used to design FOWT, and time domain numerical models will be implemented using OpenFAST and RAFT as an input pre-processor.Additionally, other design variables such as thickness, which highly impacts the total mass, and the geometry of other pipes, will be considered to span a wider range of the design space.

Figure 2 .
Figure 2. Design parametrization and member-based modelling of the GICON ® -TLP using RAFT with the selected design variables.

Figure 3 .
Figure 3. Structure and modules of the design optimization framework coupling RAFT and the GA described, with representation of the population evolution.

Figure 4 .
Figure 4. RAFT model of the GICON ® -TLP and environmental conditions for the design optimization study case.

Table 1 .
Input parameters and GA settings for the design optimization study case.

Table 2 .
Optimized design variables of the GICON ® -TLP and associated maximum platform offsets and tower base fore-aft bending moment.