The technical progress in the development and industrialization of floating offshore wind turbines (FOWTs) over the past decade has been significant. Yet, the higher levelized cost of energy (LCOE) of FOWTs compared to onshore wind turbines is still limiting the market share. One of the reasons for this is the larger motions and loads caused by the rough environmental excitations. Many prototype projects tend to employ more conservative substructure designs to meet the requirements for motion dynamics and structural safety. Another challenge lies in the multidisciplinary nature of a FOWT system, which consists of several strongly coupled subsystems. If these subsystems cannot work in synergy, the overall system performance may not be optimized. Previous research has shown that a well-designed blade pitch controller is able to reduce the motions and structural loads of FOWTs. Nevertheless, due to the negative aerodynamic damping effect, improvement in the performance by tuning the controller is limited. One of the solutions is adding tuned liquid multi-column dampers (TLMCDs), meaning that there is a structural solution to mitigate this limiting factor for the controller performance. It has been found that the additional damping, provided by TLMCDs, is able to improve the platform pitch stability, which allows a larger blade pitch controller bandwidth and thus a better dynamic response. However, if a TLMCD is not designed with the whole FOWT system dynamics taken into account, it may even deteriorate the overall performance. Essentially, an integrated optimization of these subsystems is needed. For this paper, we develop a control co-design optimization framework for FOWTs installed with TLMCDs. Using the multi-objective optimizer non-dominated sorting genetic algorithm II (NSGA-II), the objective is to optimize the platform, the blade pitch controller, and the TLMCD simultaneously. Five free variables characterizing these subsystems are selected, and the objective function includes the FOWT's volume of displaced water (displacement) and several motion and load indicators. Instead of searching for a unique optimal design, an optimal Pareto surface of the defined objectives is determined. It has been found that the optimization is able to improve the dynamic performance of the FOWT, which is quantified by motions and loads, when the displacement remains similar. On the other hand, if motions and loads are constant, the displacement of the FOWT can be reduced, which is an important indication of lower manufacturing, transportation, and installation costs. In conclusion, this work demonstrates the potential of advanced technologies such as TLMCDs to advance FOWTs for commercial competitiveness.

Structural control techniques play an important role in mitigating undesired motions or loads across various disciplines. For floating offshore wind turbines (FOWTs), the implementation of tuned mass dampers (TMDs) or tuned liquid column dampers (TLCDs) has been extensively investigated in recent years. Passive and active TMDs installed in the nacelle are investigated and compared by

Nevertheless, the application of TLMCDs alone may not fully address the unique challenges posed by FOWTs. While studies on nacelle-based TMDs have already addressed the importance of the control strategies for the damping effectiveness

Given these challenges, integrated optimization techniques are the most suitable solutions to this problem. By exploring the design space of the whole FOWT system, design parameters that provide the best synergy between the coupled subsystems can be found. General best practices for design optimization of FOWTs have been extensively addressed in previous research. The aim of such optimization work is to reduce the levelized cost of energy (LCOE) of FOWTs. For the research community, this is usually represented by a reduction in size and weight, motion, and loads. A lot of work has been done on design optimization.

Building on this prior research and its findings, this work explores the limits of the TLMCD's contribution to mitigating motions and loads of the FOWT system. This is done through a multi-objective optimization approach that incorporates control co-design (CCD) to simultaneously optimize the design of the substructure, the TLMCD, and the blade pitch controller. By coordinating the functions of these subsystems, the framework is expected to achieve the best possible synergy to maximize the potential of TLMCDs in improving the performance of FOWTs at a systems engineering level.

As emphasized in the previous introduction, the dynamic responses of FOWTs are significantly affected by the blade pitch controller and the TLMCD. In this section, the impacts of these influential subsystems are visually demonstrated, which provides a comprehensive understanding of these effects.

The original NAUTILUS-DTU 10MW FOWT

Design and installation of the TLMCDs for the NAUTILUS-DTU 10MW FOWT.

For the stability margin, a Nyquist plot is used to visualize how system stability changes with control designs. Figure

Nyquist plot of the open-loop transfer function

In addition to the stability margin, the step responses are also an important measure to describe the dynamic behavior. How the step responses react to wind change with control gains and TLMCD setups is presented in Fig.

Step response of generator speed to unit step wind with different control gains at 16 m s

Combining the observations on the sensitivity margin in Fig.

The linear analysis clearly demonstrates the significant influence of both the TLMCD and the blade pitch controller on the system stability, as well as their role in shaping the closed-loop dynamic responses. However, it is still unclear how these subsystems interact and affect the coupled dynamic responses in real operating conditions. In order to gain a deeper insight into the coupled dynamics, a more comprehensive study using coupled aero-hydro-servo-elastic time simulations is conducted, and the statistics of the simulation results are presented in Fig.

Comparison of relative system statistical responses of the NAUTILUS-DTU 10MW FOWT with respect to the case with a reference controller and without a TLMCD at different operating wind speeds (sea states based on the design load cases at a site in the LIFES50

The first step is to understand the impact of the TLMCD. In both cases (i.e., with the reference controller and with the redesigned controller), the TLMCD can improve the pitch motion and the tower-base bending moment and do so more significantly at higher wind speeds. However, this improvement, especially the rotor speed performance, is independent of the blade pitch controller. When the TLMCD is active, redesigning the controller can significantly improve the rotor speed performance, which is evident when comparing the solid yellow and black bars. As the generator torque is constant in the simulation, the rotor speed also represents the power production quality.

In summary, adding structural damping to an actively controlled system does not automatically guarantee improved overall system performance. It is crucial to optimize the controller in conjunction with the damping system to achieve synergy and maximize benefits. This finding inspires the work described in the next section, where CCD optimization techniques are employed. By systematically considering the effects of the substructure design, the additional damping, and the active control system, the overall performance of the system is expected to be maximized.

This section outlines the setup of the optimization framework, which involves the design space, optimizer, constraints, and cost model. These elements are essential to ensure that the optimization process is well defined and that all subsystems can be optimized to maximize the desired performance while staying within the constraints. The entire framework is implemented in MATLAB^{®} using the linearized SLOW for controller design and the nonlinear SLOW for coupled simulations.

The optimization framework for the FOWT system includes several subsystems (i.e., the floating platform, the blade pitch controller, and the TLMCD). To ensure a manageable computational complexity, it is beneficial to limit the number of free variables for each subsystem. For the platform, the design space is derived from

Illustration of the design variables of the substructure used for the optimization

The fairleads of the mooring system are attached to the outer walls of the four vertical columns. As the column spacing is a free variable, the fairleads' positions in the body frame of the floater are affected by the column spacing,

Regarding the controller design, the automated design procedure in

As for the TLMCD, its horizontal arm must have a length of

Table

Free variables defined for the optimization process.

When evaluating the system performance of FOWTs, various factors come into play to define the cost model and assess the effectiveness of the FOWT design. These factors include structural loads, energy production, motions, and more. To consolidate these factors into a single objective function, weight coefficients are generally used to obtain the optimal solution. However, determining these coefficients poses challenges, especially in academic research settings where input from industry experts on realistic weighting factors may not be available. Furthermore, the choice of weight coefficients has a significant impact on the optimization process and final outcomes. Recognizing this, a multi-objective optimizer is chosen to address these complexities. By employing a multi-objective optimizer, designers can obtain a range of favorable designs across different scenarios, rather than a single unique optimum, which offers valuable flexibility and provides designers with a comprehensive understanding of trade-offs and design considerations.

Therefore, the NSGA-II (non-dominated sorting genetic algorithm II) is chosen as the optimization method in this study. A detailed description of the algorithm can be found in

The cost model is an essential part of the optimization framework that can significantly influence the optimization results. While minimal LCOE is generally accepted as a good objective function in the wind industry, it is derived from a wide range of factors, some of which are not relevant to the subsystems under investigation, such as policy, market, or supply-chain-related issues. Moreover, certain components of the LCOE may vary in different markets or change with suppliers, making it less informative and potentially unable to reveal the influence of the design parameters. As a result, indicators that not only reflect the LCOE, but also have physical meanings and strong correlations with the design variables are used for optimization purposes.

Three indicators are selected for the optimization: a motion indicator, a load indicator, and a cost indicator. The motion indicator will be measured using sensors for the platform pitch and nacelle fore–aft acceleration, while the load indicator will be determined by measuring the tower-base bending moment and mooring fairlead tension. Instead of calculating the actual cost for materials, manufacturing, transportation, and so on, the total displaced tonnage (i.e., the weight of water displaced by the FOWT in normal operation) is used as the cost indicator. Although the term tonnage can have different meanings in the shipping industry depending on the loading condition of the vessel, the displaced tonnage is an important measure that can provide a qualitative indication of manufacturing, operation, and maintenance costs. Unlike direct cost calculation, which has many uncertainties and can vary over time and markets, displaced tonnage is a physical value that can be accurately calculated from the structural model. Therefore, it is the only measure used here to indirectly represent all costs associated with material, manufacture, transportation, and installation.

To account for the different units and magnitudes of the selected indicators, they are normalized by comparison to the original NAUTILUS-10 design. Denoting the displacement, the damage equivalent load (DEL) of the tower-base bending moment and the mooring fairlead tension, and the standard deviation (SD) of the platform pitch motion and the tower-top acceleration as

As can be seen, both

In order to accelerate the optimization process, a set of design constraints has been defined to eliminate unfeasible designs. These constraints can be classified into two types: static and dynamic. The static constraints are applied at the beginning of the optimization process and immediately reject any designs that fail to meet the requirements, thus reducing the number of designs to be simulated. The dynamic constraints are applied during the simulation and take into account the behavior of the designs under various load conditions.

The static constraints are checked before a computationally intensive time simulation is conducted. If an individual design fails to satisfy the constraint criteria, it will be excluded from further evaluation. The algorithm will continue to search for potential candidates in order to maintain the size of the design candidates which are to be evaluated. The constraints on the natural frequencies of the floater are primarily intended to avoid the wave frequency range. These static constraints are defined as follows:

The displaced tonnage should not be more than twice as large as that of the NAUTILUS-10 design (i.e.,

The maximum static pitch angle should be smaller than 10°.

The heave natural period should be greater than 15 s.

The pitch natural period should be greater than 18 s.

To further refine the optimization process, dynamic constraints are defined based on the statistical analysis of dynamic simulations. If a design exceeds these constraints, the algorithm sets high values to the cost model

The generator overshoot should be lower than 30 %.

The maximum dynamic pitch should not exceed 12°.

The nacelle acceleration should be smaller than

Workflow of evaluating the offspring of the

It is worth noting that ultimate limit state (ULS) and fatigue limit state (FLS) are not considered. The selection of constraints is based more on experience and established rules of thumb derived from previous research projects. This approach not only guarantees timely convergence, but also allows for a wider range of potential solutions to be explored within the design space. However, it is important to emphasize that the values chosen may be conservative and may not necessarily align with commercial standards.

The most important and computationally intensive step is the performance evaluation based on the objective functions, which includes the design and modeling of the subsystems, the execution of coupled simulations in the time domain, and the post-processing of the simulation data to compute the objective functions. Figure

Design load cases used for optimization.

The substructure design module is the first step of the optimization process. It takes the design variables of each offspring created by the generic algorithm as input and calculates the inertial properties of the FOWT based on its geometrical variables. At the same time, the module generates a mesh for the wet surface of the substructure and the associated panel coordinates. The data produced here are then passed on to the hydrodynamic module. The main function of this module is to generate the hydrodynamic coefficients using the panel code Ansys Aqwa. In addition, the module calculates the response amplitude operator (RAO) and natural frequency of the platform pitch motion, which serve as input data for the TLMCD design module. These design modules are developed in

Mutual domination rate of the optimization process, showing the convergence of the optimization process.

After all inputs for the dynamic plant are set up, steady states for various operating wind speeds can be simulated and calculated. These steady states are then used to linearize the model. The linearization of the FOWT is introduced in

For the automated design of the blade pitch control for a specific FOWT, a loop-shaping-based methodology is used. A comprehensive description of the theoretical basis and evaluation based on coupled numerical simulation can be found in

Pareto fronts resulting from the two-variable substructure optimization, showing the trade-off between the relative displaced water volume and the relative costs defined in Eq. (

Once the dynamic plant and controller are established, the final step is to perform coupled time-domain simulations using a subset of design load cases recommended in the LIFE50

This section discusses the results of two rounds of optimization performed to determine optimal designs. The first round only considers the geometric optimization of the substructure using two design variables: column spacing

For the first round of optimization, where only the substructure is optimized, a population size of 20 is used. For the second round, which involves the TLMCD and blade pitch controller in the optimization loop, the population size is increased to 50. Checking for convergence is essential to demonstrate the validity of the optimal solutions generated by the optimizer when using genetic algorithms. In this respect, Fig.

Comparison of Pareto optimal surfaces between the two-variable geometric optimization (in black) and the five-variable TLMCD-aided CCD optimization (in yellow).

As mentioned earlier, the optimization process involves two rounds with different objectives. The following section discusses how the definition of the objectives can influence the final optimal solution.

The results of a two-objective optimization are presented in Fig.

Relative reduction in displacement contributed by the TLMCD-aided CCD optimization.

Looking at the black dots on the plots where only two variables related to the substructure geometry are optimized, two main observations can be made. Firstly, there is a strong inverse correlation between the DEL cost/SD cost and the relative displacement (displaced water volume). This correlation is nearly linear and is evident in the range where the relative displacement is approximately 20 % above and below zero. While decreasing the displacement can lead to a reduction in the total material, construction, transportation, and installation costs, this always comes at the expense of higher DEL and SD. It is worth noting that the DEL cost and SD cost are only slightly reduced when the relative displacement reaches 0.5, indicating a 50 % increase in displacement compared to the NAUTILUS-10. The second observation is that the point

Geometric decision space on the Pareto surface.

The impact of the TLMCD and the blade pitch controller on the substructure geometry optimization can be seen by comparing the yellow and black dots. Obviously, the Pareto fronts are shifted to the left side, indicating an improved overall dynamic response. Despite this, the shape of the Pareto front is very similar to the one obtained from pure substructure geometry optimization; therefore, the previous observations still apply. However, a notable improvement in the platform pitch motion can be found as the SD cost is reduced by 5 % to 10 % for the same displacement. In terms of the DEL cost, the contributions of the TLMCD and the blade pitch controller are limited when the displacement is relatively small. These designs are typically lighter and have higher natural frequencies, which are closer to the wave frequency range. This makes them more susceptible to wave-induced excitation, and the dynamic response cannot be significantly improved even with additional damping. Of course, the mass of the TLMCD also plays an important role. As smaller substructures have a TLMCD with less fluid mass, their ability to compensate for motion induced by aerodynamics is limited. This explains why the designs with larger displacement in the optimization process can achieve a bigger improvement by including the TLMCD.

The Pareto front obtained from a two-objective optimization only shows the optimal solutions for each optimization case, with only two objectives being optimized at a time. However, it is important to note that the decision space may be different for each of the cases shown in Fig.

Optimal head loss

Figure

Controller decision space on the Pareto surface.

Properties and costs of the two selected designs on the optimal Pareto surface.

To gain a better understanding of the displacement reduction contributed by the TLMCD, the contour lines of the displacement reduction plotted against the SD cost and DEL cost are presented in Fig.

Comparison of statistical responses between the original NAUTILUS-10 design and two selected designs with similar SD cost and DEL cost on the optimal Pareto surface.

While the focus so far has been on achieving optimal objectives, it is also essential for system designers to examine the design choices that can ensure good performance. The following section discusses the decision space, which represents the optimal subsystem designs chosen by the optimizer. This will provide a better insight into how design choices affect the overall performance of the system.

Figure

The design of the TLMCD involves only one free variable, namely the head loss

The last two design variables are associated with the blade pitch controller – specifically, the rise time,

Comparison of frequency responses between the original NAUTILUS-10 design and two selected designs with similar SD cost and DEL cost on the optimal Pareto surface at wind speed 13.9 m s

The results of the TLMCD-aided CCD optimization show that the optimizer can potentially reduce the displaced tonnage of a FOWT by up to 20 %. While the optimizer only considers the defined design objectives, it is still important to examine the dynamic responses of the optimal designs. Hence, two designs with similar DEL costs and SD costs to the original NAUTILUS-10 design are selected. The corresponding design space and decision space are listed in Table

Design 1 and Design 2 are two optimal designs on the Pareto surface that are selected for further analysis. Design 1 has a column spacing that is similar to the original NAUTILUS-10, but its draft is 4 m shorter, resulting in a reduction in the displaced tonnage of the FOWT by 11.6 %. In contrast, Design 2 has a larger column spacing and a further reduced draft, resulting in a reduction in the displaced tonnage of 20 %. Both designs have very similar DEL costs to the original NAUTILUS-10 design (less than 1 % difference). Additionally, Design 1 has a lower SD cost, while Design 2 has a 7.1 % increase in SD cost, which is a cost for the significantly reduced displacement.

In Fig.

Figure

Integrating a tuned liquid damping system into a FOWT presents a substantial system optimization challenge. The potential benefits of such a damping system to the overall system depend largely on its interaction with the turbine control system, the design of which is also linked to the substructure geometry. Due to the physical coupling between aerodynamics and hydrodynamics, the design of each subsystem has a significant impact on the overall system dynamics. The optimization of these individual subsystems, as well as their efficient cooperation, essentially influences the final system performance of the FOWT. To address this challenge, a multi-objective CCD optimization framework has been developed to optimize the substructure geometry, the TLMCD, and the blade pitch controller, systematically incorporating a tuned liquid damping system into a FOWT. The framework explores the design space of all three subsystems simultaneously, searching for the optimal synergy between them to achieve a good balance between production cost and response performance.

A case study based on the LIFES50

While the initial concept-level results show promise, further analysis tailored to industrial application is imperative. In particular, integrating considerations for the platform's structural integrity into the optimization framework as design constraints is essential. The current optimization findings suggest that larger column spacing between platforms tends to improve the dynamic responses; yet, this may conflict with the structural integrity of the heave plate. Further investigation of this component would provide a more realistic design assessment. Due to computational limitations, the platform geometry optimization currently focuses on only two free variables. Exploring enhancements by optimizing the column diameter could yield considerable benefits. Additionally, introducing more industry-specific objective functions that accurately reflect the LCOE will contribute to the derivation of more realistic and optimal designs.

The work presented uses SWE's in-house code for reduced-order wind turbine modeling and simulation, and simulation management, which have not yet been published. However, SWE is open to scientific collaborations in which the simulation models can be made available.

The data for the figures can be accessed via

WY: conceptualization, methodology, implementation of the framework, analysis, and writing (original draft preparation). STZ: parameterization of the NAUTILUS-10 semi-submersible substructure and writing (reviewing and editing). FL: supervision and writing (reviewing and editing). PWC: supervision and writing (reviewing and editing).

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research was partly supported by the CROWN project. The CROWN project has received funding from the Eurostars-2 joint program, with co-funding from the European Union Horizon 2020 research and innovation program.

This research has been supported by the Eurostars-2 program (grant no. E!113443). This open-access publication was funded by the University of Stuttgart.

This paper was edited by Amy Robertson and reviewed by two anonymous referees.