Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled modelling approaches

Renewable and Sustainable Energy Reviews ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled modelling approaches Michael Borg n, Maurizio Collu Whittle Building, Cranfield University, Cranfield MK43 0AL, United Kingdom

art ic l e i nf o

Keywords: Floating offshore wind turbines Vertical axis wind turbine Coupled dynamics Hydrodynamics Coupled modelling

a b s t r a c t The need to further exploit offshore wind resources has pushed offshore wind farms into deeper waters, requiring the use of floating support structures to be economically sustainable. The use of conventional wind turbines may not continue to be the optimal design for floating applications. Therefore it is important to assess other alternative configurations in this context. Vertical axis wind turbines (VAWTs) are one promising configuration, and it is important to first understand the coupled and relatively complex dynamics of floating VAWTs to assess the technical feasibility. As part of this task, a series of articles have been developed to present a comprehensive literature review covering the various areas of engineering expertise required to understand the coupled dynamics involved in floating VAWTs. This third article focuses on approaches to develop an efficient coupled model of dynamics (considering aerodynamics, hydrodynamics, structural and mooring line dynamics, and control dynamics) for floating VAWTs, as well as suitable ‘semi-analytical’ hydrodynamic models for this type of coupled dynamics models. Emphasis is also placed on utilising computationally efficient models and programming strategies. A comparison of the various forces acting on a floating VAWT with the three main floating support structure (spar, semi-submersible and tension-leg-platform) is also presented to highlight the relative dominant forces and hence importance of model accuracy representing these forces. Lastly a concise summary covering this series of articles is presented to give the reader an overview of this interdisciplinary research area. This article has been written both for researchers new to this research area, outlining underlying theory whilst providing a comprehensive review of the latest work, and for experts in this area, providing a comprehensive list of the relevant references where the details of modelling approaches may be found. & 2014 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Coupled modelling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1. Previous research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Hydrodynamic modelling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1. Frequency-domain modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2. Time-domain modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2.1. The Morison equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2.2. The Cummins equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2.3. Wave excitation forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2.4. Radiation Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2.5. Hydrostatic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.6. Viscous damping forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.7. Sea current forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3. Other modelling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

n

Corresponding author. E-mail addresses: m.borg@cranfield.ac.uk (M. Borg), maurizio.collu@cranfield.ac.uk (M. Collu). http://dx.doi.org/10.1016/j.rser.2014.10.100 1364-0321/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Borg M, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled.... Renewable and Sustainable Energy Reviews (2014), http://dx.doi.org/10.1016/j.rser.2014.10.100i

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3.4. Verification and validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Comparison of floating VAWT forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.1. Spar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2. Semi-submersible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3. TLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5. Current implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.1. Current challenges with coupled dynamics design codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6. Computational strategies and issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.1. Programming methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2. Limiting model complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.3. Utilising current computing technologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 7. Engineering models suitable for the preliminary design of floating VAWTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.

1. Introduction The need to increase the share of renewable energy in global energy production has pushed wind farms further offshore into deeper waters to exploit larger energy resource. In water depths greater than 30–50 m, bottom-mounted (i.e. fixed) support may not be a technical and economical viable option [1]. A transition from fixed to floating support structures is essential for deep offshore wind turbines to become economically viable in the near future. The onshore wind industry has reached a relatively mature level, and the large majority of large scale wind turbines share the same configuration: horizontal axis of rotation, three blades, upwind, variable-speed, and variable blade pitch (with feathering capability). This has been the result of several decades of research and development, and originally several configurations had been considered, including horizontal axis wind turbines (HAWTs) with a different number of blades, but also vertical axis wind turbine (VAWT) configurations. The conventional three-bladed HAWT design emerged as the optimum techno-economic trade-off for the onshore large scale wind market. The same “evolutionary process” did not take place for the offshore wind market, substituted by a “marinisation” of the trusted configurations used for the onshore market. It has been implicitly assumed that, despite the very different environmental conditions of an offshore environment, the optimum configuration for the wind turbine is the same: the conventional three bladed, upwind, horizontal axis wind turbine. This has been implicitly assumed not only for the bottom-mounted offshore wind turbine configurations, but also for the proposed floating systems. It is therefore important to assess the technical and economic feasibilities of alternatives in order to ensure that the most suitable configurations are employed. The first step is to understand the coupled and relatively complex dynamics of such a floating system subjected to the harsh offshore environment. As part of this task, a series of articles have been developed to present a comprehensive literature review covering the various areas of engineering expertise required to understand the coupled dynamics involved in floating VAWTs. In part one of this series, an in depth review of different aerodynamic engineering models for VAWTs and their suitability for floating applications is presented [2]. Part two focuses on structural and mooring line dynamics modelling of floating offshore wind turbines (FOWTs), with emphasis on computational strategies and characteristics of VAWTs [3]. This third article focuses on approaches to develop an efficient coupled model of dynamics (considering aerodynamics, hydrodynamics, structural and mooring line dynamics, and control

dynamics) for floating VAWTs, integrating engineering models presented in [2] and [3], as well as suitable ‘semi-analytical’ hydrodynamic models for this type of coupled dynamics models. This article has been written with researchers new to this research area in mind, outlining underlying theory whilst providing a comprehensive review of the latest work. It is important to note that in this article not all possible modelling methods are described; only those that are relevant and suitable for developing an efficient coupled model of dynamics that is capable of being run on a typical desktop PC within a reasonable amount of time. The main use of these types of models is in the preliminary design stage of FOWTs when engineers need to carry out thousands of simulations to assess concepts according to design guidelines and standards. Jonkman [4] sums this concept by stating that coupled dynamics models for floating wind turbines should be based on advanced engineering models derived from fundamental physical laws but with appropriate simplifications and assumptions, supplemented with experimental data and computational solutions where appropriate. The paper is organised as follows:


Section 2 discusses approaches to the development of coupled


dynamics models for FOWTs and research carried out so far for floating VAWTs. Section 3 discusses in detail hydrodynamic models suitable for coupled dynamics modelling and the hydrodynamic forces floating FOWTs may experience.

Fig. 1. Loads acting on a conceptual two-bladed H-rotor VAWT mounted on a semisubmersible support structure in the offshore environment.

Please cite this article as: Borg M, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled.... Renewable and Sustainable Energy Reviews (2014), http://dx.doi.org/10.1016/j.rser.2014.10.100i

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Section 4 presents a brief comparison of all forces floating





VAWTs are subjected to. Section 5 outlines some of the current implementations by researchers and challenges faced in the development of efficient coupled dynamics design tools. Section 6 outlines computational issues and strategies that may be adopted in the development of such design tools. Section 7 provides a summary of engineering models for floating VAWTs discussed throughout this series of articles. Section 8 gives some conclusions.

2. Coupled modelling approaches The offshore environment subjects a floating wind turbine system to loads from differing origins. These include aerodynamics, hydrostatics and hydrodynamics, mooring line dynamics, structural dynamics and control dynamics as illustrated in Fig. 1. An in-depth understanding of each of these aspects is needed to develop an appropriate, efficiently coupled model of dynamics. 2.1. Previous research The first major choice in modelling is to perform the analysis in the frequency or time domain. The benefits of using frequencydomain analysis are that it has been used extensively in the offshore oil and gas industry; it is also computationally very efficient and is very useful to determine the natural frequencies of the system, that are a major design driver as they should be outside the range of frequencies where significant wave energy is located to reduce platform motion response and loads. Frequencydomain methods have also been used for the preliminary design of a number of offshore floating wind turbine support structures (Fig. 2): the tri-floater concept (Bulder [5]), tension leg platforms (TLPs) (Lee [6], Wayman et al. [7]), barges (Wayman et al. [7]), and semi-submersibles (Collu et al. [8], Lefebvre and Collu [9]). Whilst frequency-domain analysis may be an important tool in the preliminary stages of design, it has some important disadvantages that limit its use in detailed design. The linearization required for frequency-domain analysis does not allow for any nonlinear dynamics to be easily accommodated. It also cannot capture transient events (such as start-up and shut-down procedures), which may be critical in the design of a floating wind turbine. Matha [10] found that certain couplings between the tower and horizontal axis blade assembly and platform modes

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were not captured in the frequency-domain analysis. The main cause of this was the use of rigid blades and tower in the frequency domain, with flexible components used in the timedomain. This results in different natural frequencies and system motions. Whilst it is possible to include a flexible structural model in the frequency-domain, the approach assumes small displacements, which is not valid in the case of floating wind turbines [10]. Note that displacements, as defined by Cordle [11], are small for translational displacements relative to the incoming wave lengths or characteristic length of the platform, and for rotational displacements relative to the wave steepness. Therefore a time-domain analysis, where the inclusion of a complete flexible model is possible, is preferential to investigate the transient and nonlinear dynamics of floating wind turbines. This is evident in all current floating wind turbine design codes [12]. A major contribution to time-domain integrated dynamics design codes was made by Jonkman [13]. Jonkman developed a comprehensive simulation tool for the coupled dynamic response of floating HAWTs, and then performed integrated dynamic analysis on an HAWT mounted on a barge-type platform according to the IEC 61400-3 design standard [14]. This tool has become integrated into FAST, one of the most-widely used offshore HAWT design codes. Most studies on the coupled dynamic response of floating HAWTs have used FAST, or a modified version of it (e.g. [15–19]). There are a number of other codes available and Cordle and Jonkman [12] performed a comprehensive review of all the current simulation codes available for floating horizontal axis wind turbines. The main drawbacks of time-domain simulations are that they are more computationally intensive than frequency domain simulations, and in most cases still require input data from frequency-domain simulations to construct the time-domain model (as described in Section 3.2.2). Relatively little research has been done on investigating floating VAWTs. Vita [20] analysed a Darrieus-type rotor mounted on a spar buoy rotating platform (Fig. 3(a)), both at a technical and economic level. The concept proposed was envisaged to be simple to construct and transport, thereby reducing costs. Some shortcomings were that since it has an extremely large draft, it may be used in water depths above 150 m and had power losses through friction between the rotating platform and water. Collu et al. [21] presented the preliminary conceptual design and optimisation of a floating support structure for the NOVA rotor (Fig. 3(b)). The concept of this novel vertical axis rotor is to reduce the overturning moment acting on the support structure whilst maintaining sufficient power output. The analysis was conducted

Fig. 2. Different floating wind turbine support structures [26].

Please cite this article as: Borg M, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled.... Renewable and Sustainable Energy Reviews (2014), http://dx.doi.org/10.1016/j.rser.2014.10.100i

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Fig. 3. Floating VAWT concepts (from top left, clockwise): (a) DeepWind concept [27]; (b) NOVA concept [28]; (c) floating axis wind turbine concept [22]; and (d) VertiWind concept [25].

on a semi-submersible similar to the Dutch tri-floater design (Fig. 3(d)). This preliminary analysis investigated the long-term aerodynamic and frequency-domain hydrodynamic performances separately. The above studies were not based on fully coupled analyses, which may lead to certain characteristics of the system being excluded, e.g., the augmentation of aerodynamic forces due to increased relative velocities arising from platform motion. There is a need to develop a general coupled model of dynamics for floating VAWTs such that their dynamic behaviour may be investigated in detail. To be able to develop such a model, one must first identify the most suitable approaches to model the various environmental loads experienced by and dynamics of the floating wind turbine system. Fig. 4. A typical six degree-of-freedom co-ordinate system used in the analysis of floating structures.

in the frequency domain, with aerodynamic forces from a fixedbase VAWT applied as an external load. Another concept proposed by Akimoto et al. [22,23] and Nakamura et al. [24] was the floating axis wind turbine (Fig. 3(c)). This concept differs from that proposed by Vita because the generator is located outside the floating platform, with roller bearings transferring torque from the rotating tower to generators around the tower. The main idea was to eliminate the need to have large bearings supporting most of the loads from the rotor and to allow for much easier access for maintenance. Another concept called VertiWind was proposed by Cahay et al. [25] for a 3-bladed H-type Darrieus rotor mounted

3. Hydrodynamic modelling approaches Before describing the different hydrodynamic modelling approaches, it is necessary to define the degrees of freedom (DOFs) of the floating structure. The six DOFs of a rigid floating structure are as follows: three translations DOFs (surge, sway and heave) and three rotational DOFs (roll, pitch and yaw), as is illustrated in Fig. 4. Note that throughout this paper, bold terms in equations represent 6 DOFs to allow for a clearer discussion. 3.1. Frequency-domain modelling The frequency-domain approach is based upon Eq. (1) and has been an attractive approach due to its relative ease of

Please cite this article as: Borg M, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled.... Renewable and Sustainable Energy Reviews (2014), http://dx.doi.org/10.1016/j.rser.2014.10.100i

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implementation and fast computational speed as a large number of simulations are usually required in order to satisfy the requirements specified by the standards.


  ω2 ½M RB þ AðωÞ  jωBðωÞ þ C xðjωÞ ¼ τ exc ðjωÞ

ð1Þ

Eq. (1) assumes steady-state conditions, where xðjωÞ is the complex response of the body, AðωÞ and BðωÞ are the frequencydependent added mass and damping matrices, M RB is the rigid-body inertia matrix, C is the hydrostatic restoring stiffness matrix, and τ exc ðjωÞ is the wave excitation force. As a consequence, the downfall of this approach is that it requires linearization of the system of dynamics, which results in the inability to capture nonlinearities and transient events, as illustrated by Wayman [7] and Matha [10] with regard to floating wind turbines. Philippe et al. [29] also performed a comparative study between frequency and time domain simulations of a floating wind turbine and highlighted the importance of including nonlinear effects. Whilst frequency-domain simulations are all but absent from floating wind turbine design codes, they may still play an essential role in the preliminary design of the floating structure, as they are very useful to establish the natural frequencies of the system (see e.g., Wayman [7]; Collu et al. [21]). With this knowledge and that of the wave energy spectrum, the preliminary design's natural frequencies can be shifted away from the high-energy wave frequencies.

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3.2. Time-domain modelling 3.2.1. The Morison equation The Morison equation [30] is one of the most widely-used modelling tools in wind turbine simulation tools, shown in the following equation:   _ t Þ ¼ ρC m V u_ ðt Þ þ ð1=2ÞρC d Au_ ðt Þu_ ðt Þ F ðu; ð2Þ Where ρ is the fluid density, C m is the added mass coefficient, V is the volume displaced by the body, uðt Þ is the flow velocity, u_ ðt Þ is the flow acceleration, A is the cross-sectional area of the body and C d is the drag coefficient. Both the added mass and drag coefficients are found through experiments. It is an empirical relation describing the viscous and inertial loads on a vertical slender cylinder subjected to small amplitude, high frequency oscillations. This implies the slender body does not modify the incident wave characteristics and hence diffraction loads are not considered. Whilst it has been used successfully for fixed offshore wind turbine structures and for spar-type floating support structures (e.g. [20,31–33]), the slender-body assumption does not hold for most floating wind turbine support structure concepts as many designs involve the use of large-volume structures (cf. Figs. 2 and 3). A large-volume structure modifies the characteristics of the incident waves and is also subjected to large-amplitude motion. Chakrabarti [34], and subsequent recommended practices such as DNV-RP-C205 [35] classify wave forces regimes based on wave height, wavelength and platform characteristic length, as is shown in Fig. 5 [35]. The horizontal axis represents the characteristic length to wavelength ratio and the vertical axis represents the wave height to characteristic length ratio. This graph allows design engineers to easily assess the wave force regime acting on a platform, and used appropriate methods to obtain first estimates of wave loading. 3.2.2. The Cummins equation The first ab initio approach to the time-domain hydrodynamics modelling of floating bodies was done by Cummins [36] (see Eq. (3)), with later expansion of its practical implementation by Ogilvie [37]. Z t   € þ _ tÞ M RB þA xðtÞ K ðt  τÞx_ ðτÞdτ þCxðtÞ ¼ τ exc ðx; x; ð3Þ 1

Fig. 5. Wave forcing regimes according to characteristic length, D, wavelength, λ, and wave height, H [34].

M RB is the rigid-body inertia matrix, A is the infinite-frequency added mass matrix, xðtÞ is the body's acceleration, K is the retardation function matrix, x_ is the body's velocity, C is the hydrostatic restoring stiffness matrix, xðtÞ is the body's displacement, and τ exc is the wave excitation force and any other external loads. In this approach the loads acting on the floating body are separated into three problems that are solved separately, relying on the linear superposition of the effects. They consist of radiation, diffraction and hydrostatic loads. The use of this approach allows the inclusion of nonlinear effects such as viscous damping in the equations of motion represented as external loads (including the possible use of the Morison equation is to integrate the viscous

Table 1 Hydrodynamic models comparison.

Complexity Ease of implementation Computational effort Geometric constraints Viscous model

Morison equation

Cummin's equation

Cummin's equation w/state-space model

Low Easy Low Limited to slender bodies Implicit

Medium Medium High Not constrained Explicit

Medium–high Hard Low–medium Not constrained Explicit

Please cite this article as: Borg M, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled.... Renewable and Sustainable Energy Reviews (2014), http://dx.doi.org/10.1016/j.rser.2014.10.100i

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term into the time-domain model to account for loads on slender structural members, such as braces and secondary struts). This method has been implemented by a number of leading floating wind turbine design codes (see e.g., [16,38–40]). Table 1 gives a summary on the different semi-analytical hydrodynamic modelling approaches available, outlining their relative advantages and disadvantages. The next sections give a brief overview of the various hydrodynamic forces that should be considered. 3.2.3. Wave excitation forces Wave excitation forces originate from incident waves interacting with the support structure. In the majority of floating wind turbine literature first order wave forces has been implemented, and only recently has the importance (and inclusion) of second order wave forces been illustrated in literature, as will be discussed below. In the following two subsections, first and second order wave forces are briefly highlighted, and the reader is directed to Faltinsen [41] and Journée and Massie [42] for more in-depth discussions of the underlying theory. First order wave excitation forces are calculated assuming the structure is fixed, that is, it is not allowed to oscillate freely in the presence of incident waves. The wave excitation force under this condition can be split into two parts: (a) a force due to the undisturbed pressure field surrounding the submerged structure, also known as the Froude–Krylov force and (b) a force from the modified pressure field due to the presence of the submerged structure, also known as the diffraction force. These forces are calculated under the assumption that the incident waves are sinusoidal and regular. In the context of the Cummins equation, these forces are calculated by frequencydomain fluid-structure interaction software (such as WAMIT [43] and AQWA [44]) for single incident wave frequencies in the form of response amplitude operators (RAOs)1. The superposition principle is then used to establish irregular sea state conditions. Thus the first order wave excitation force, τ exc ð1Þ , for a unidirectional sea state is given by the following equation:  N

 τ exc ð1Þ ðt Þ ¼ ∑ ρg F wave ð1Þ ωk ; β Ak cos ωk t þ ϕk þ ϵk ð4Þ k¼1

where g is the acceleration due to gravity; F wave ð1Þ is the first order wave force RAO as a function of the kth incident wave frequency ωk and wave direction β (constant for unidirectional case); Ak is the kth wave frequency amplitude; ϕk is the kth wave frequency phase; and ϵk is the kth wave frequency random phase angle. The evaluation of these forces using the above mentioned software usually involves discretising the submerged surface of the floating structure into a number of panels. Particular attention should be paid to mesh density at surface discontinuities and ensuring that the panel length should be 1/8th of the shortest wavelength considered to avoid numerical errors. Second order wave excitation forces follow the assumption that the support structure is oscillating with first order harmonic motions induced by first order wave forces. The second order force has five components, as described by Pinkster [45] and Matos et al. [46]:


Rotation of the first order fluid force relative to the body axes
Second order potential If one were to observe the equation of the second order wave force, it can be seen that it is composed of both second order terms as well as products of first order terms (for a practical derivation and discussion, refer to Chapter 9 in Journée and Massie [42]). This aspect is important from a computational standpoint since some of these terms may hence be found from the first order potential alone. Calculation of the second order potential is very cumbersome and computationally intensive since it is forced by the first order solution and requires numerical discretisation of the free surface around the floating body. It is apparent in literature (e.g. [47,48]) that sum-frequency and difference-frequency forces (which related to the first order relative wave elevation) are of great importance in the analysis of floating wind turbines. Difference frequency forces fall in the same range of low frequencies where one would usually find the certain mode natural frequencies for catenary-moored structures. Sum-frequency forces are found at higher frequencies than first order wave forces and could coincide with natural frequencies of tension-moored structures, turbine tower and blade structural modes, as well as interact with VAWT aerodynamic forces. The sum- and difference-frequency forces based on quadratic transfer functions (QTFs) are given by the following equations: ( ) N

i¼1k¼1

( ðtÞ ¼ F ð2Þ dif f

Re

N

∑ ∑ Ai Ak X sum ðωi ; ωk Þe jðωi þ ωk Þt

2Þ F ðsum ðtÞ ¼ Re

N

N

∑ ∑ Ai Ak X dif f ðωi ; ωk Þe

i¼1k¼1

ð5Þ

) jðωi  ωk Þt

ð6Þ

An example difference-frequency QTF for a typical FOWT semisubmersible support structure is illustrated in Fig. 6. Note that as can be seen in Eqs. (5) and (6), it is a function of two wave frequencies, rather than one wave frequency as seen in first order wave excitation forces. 3.2.4. Radiation Forces As the floating support structure is oscillating it produces, or radiates, waves. This radiation of waves imparts forces on the support structure known as radiation forces are represented by the convolution integral within the Cummins equation, presented in the following equation: Z t F radiation ðx_ ; t Þ ¼ K ðt  τÞx_ ðτÞdτ ð7Þ 1

Note that radiated waves represent a loss of energy to the support structure and thus radiation forces effectively dampen the motions


First order relative wave elevation
Products of the gradient of first order pressure and first order motions


Quadratic components of first order velocities 1 RAOs essentially are transfer functions that relate the response to a unit amplitude of excitation (in this case incident wave amplitude) as a function of incident wave frequency. Force RAOs give the force imparted on the body in a specific DOF per unit wave amplitude. Motion RAOs give the motion of the body in a DOF per unit wave amplitude.

Fig. 6. example difference-frequency surge quadratic transfer function for the OC4 semi-submersible support structure (adapted from Coulling et al. [47]).

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of the support structure. As can be observed from the convolution integral limits, waves radiated in the ‘past’ still affect the support structure motion in the ‘present’. The downfall of the convolution integral in the Cummins equation is that it is computationally expensive since it has to be evaluated numerically. This can be remedied by truncating the integral to a set length of time beyond which there is no noticeable contribution, as done by Jonkman [38] where the convolution integral was truncated to 60 s of memory. Another approach is to substitute the convolution integral with an approximation. One approximation method is to use the standard quadrature rule as applied by Wacher and Neilsen [49] to a wave energy converter, given in the following equation: Z

t 1

N

K ðt  τÞx_ ðτÞdτ ¼ ∑




n¼1

 1 K ðndt Þx_ ðt  ndt Þdt þ K ð0Þx_ ðt Þ 2

ð8Þ

Note that N is usually limited to around 40 as including additional terms would not yield better approximations [49]. Another approach that is more elegant is to approximate the convolution integrals with linear state space models as done by [50–53]. The substitution of the convolution integral with the state space model is shown in the following equation: 

 M RB þ A x€ ðtÞ þ τ rad ðx_ ; tÞ þ CxðtÞ ¼ τ exc ðx_ ; x; tÞ _ t Þ ¼ A' zðt Þ þ B' x_ ðt Þ z_ ðz; x; _ t Þ ¼ C' zðt Þ þ D' x_ ðt Þ τ rad ðz; x;

ð9Þ

Here z and z_ are the internal state vectors and the A' , B' , C' and D' are the constant matrices. One of the major advantages of state space modelling is that it is inherently computationally very efficient, with Taghipour et al. [53] finding that the state space approach runs about 80 times faster than by calculating the actual convolution integrals. It was noted that this difference increased with smaller time steps and longer simulated times. Hence this method is very beneficial for the detailed analysis of floating structures, and matches very well with the requirement of having very fast computational models. Linear state space models are readily applicable in model linearisation and hence are attractive when designing and implementing control systems. As the state space approach is an approximation to the convolution integral, it is important to identify the errors involved and if they are small enough to be acceptable. Fig. 7 shows an example impulse response of the direct convolution integral and its state-space approximation, with very good agreement even at relatively low state-space model orders. During the construction of the state space representation, frequency-dependent hydrodynamic coefficients are required. These are usually obtained from potential-flow software packages such as SESAM Wadam [54] and WAMIT [43]. With these in hand, there are two approaches to obtaining the state space models: direct time-domain system identification and frequency-domain system identification. Taghipour et al. [53,55] describe in detail the advantages and disadvantages of the different state space model identification methods, with the frequency-domain variant being more accurate. Briefly, the frequency-domain identification is carried out by first establishing the frequency response functions of the convolution integrals (which beforehand were constructed from the frequency-dependent hydrodynamic coefficients). Rational transfer functions are then fitted to the frequency response functions through regression and these are then converted into a state-space model that can easily be used in the time domain.

Fig. 7. Comparison of the impulse response of the radiation-force convolution integral and the state-space approximation with different orders (adapted from [52]).

3.2.5. Hydrostatic forces Hydrostatic forces originate from buoyancy forces due to the displacement of water by the submerged support structure. As can be seen from the third term in Eq. (3), hydrostatic forces vary as a function of the platform's displacement. Since the buoyancy force always acts vertically upwards, hydrostatic forces are nonzero only in the heave, roll and pitch DOFs. In the context of Eq. (3), there are two methods to calculating the hydrostatic forces: the linear and nonlinear approach. When applying the linear approach, the hydrostatic stiffness of the support structure is calculated assuming the support structure is displaced from the equilibrium position and the waterline level is constant, and this value is kept constant throughout the numerical simulation. In the nonlinear approach, the instantaneous wetted surface of the displaced support structure, and subsequently the hydrostatic stiffness, is calculated at each instant in time. If the support structure displacements are small relative to its characteristic length, the linearised approach is can adequately represent the hydrostatic forces as the waterplane area will not vary significantly over time. This approach has been used in the majority of floating wind turbine literature. When a floating wind turbine undergoes large-amplitude motions or is operating in extreme sea states there may be significant differences in hydrostatic stiffnesses over time (see e.g., Philippe et al. [29,40]), highlighting the need to include nonlinear hydrostatic forces in higher-fidelity hydrodynamic models. Appropriate nonlinear hydrostatic models have been by proposed by Senjanović et al. [56] and Sclavounos [57].

3.2.6. Viscous damping forces An important assumption in the Cummins approach is that software packages used to evaluate all the relevant hydrodynamic characteristics adopt a potential flow approach. Hence the fluid interacting with the support structure is assumed to be inviscid and no viscous effects are incorporated. Therefore it is imperative to incorporate, at least, relatively simplistic viscous models within a coupled dynamics model to better represent the hydrodynamic forces acting on the floating VAWT. Whilst there is some theoretical background to attempt to derive the viscous damping characteristics, experience and experimental data are usually employed to fine tune these models (see e.g., Coulling et al. [58]).

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3.2.6.1. Linear viscous damping. In this approach the viscous damping force, F visc , is represented as a linear damping coefficient, Blin visc , such that the damping force is a function of the platform velocity: _ F visc ðx_ r ; t Þ ¼ Blin visc :xr ðt Þ

ð10Þ

Blin visc

is usually applied as a fraction of the critical damping in pheave, roll and pitch (Critical damping is defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n ðm þma Þ Uc where m is the structure inertia, ma is the added mass and c is the restoring stiffness coefficient [42]). From vibrational analysis of a mass-spring-damper system (as is the case in these DOFs), Blin visc can be found if a damping ratio, ζ, is set (usually between 0.1 and 0.2, see Fossen [59]):   Bpot þ Blin ð11Þ visc ¼ 2ζωn where Bpot is the potential damping at the natural frequency ωn (Potential damping is the frequency-dependent damping derived from potential flow theory and represents radiation-induced forces). Following on from Fossen [59], Blin visc for surge, sway and yaw can be found based on the time constant 2, T, in these DOFs: Blin visc ¼

I þ A0 T

ð12Þ

where I is the platform DOF inertia and A0 is the zero-frequency added mass coefficient. 3.2.6.2. Quadratic viscous damping. In this approach the viscous damping force, F visc , is represented as a quadratic damping coefficient, Bquad visc , such that the damping force is a function of the platform velocity squared:   _  _ F visc ðx_ r ; t Þ ¼ Bquad ð13Þ visc :xr ðt Þ: xr ðt Þ Whilst models have been developed to calculate Bquad visc for ships, there have not been many (if any) developed for the novel support structures being used for floating wind turbines (cf. Fig. 1). Hence deducing accurate values for Bquad visc is not trivial, and may not be possible without detailed computational fluid dynamics (CFD) simulations or experimentation. In fact in a recent study applying quadratic drag, Coulling et al. [58] obtained Bquad visc through trial and error with experimental data. 3.2.6.3. Morison viscous drag. For large-volume structures that have slender members such as braces and struts (see e.g. Fig. 3(d)), the viscous drag term of the Morison equation (Eq. (2)) can be used to more accurately represent hydrodynamic forces from these parts of the structure. Philippe et al. [40] found that the addition of this drag produced noticeable differences in the floating wind turbine motion response in storm conditions, so it seems that it should be considered in a coupled model of dynamics. 3.2.7. Sea current forces A moored floating structure can experience significant loading due to the relative flow with the water induced by sea currents. As the current speed magnitude and direction can vary temporally and spatially depending on site conditions, appropriate current velocity profiles should be considered during coupled model simulations, as specified by design standards such as DNV-RPC205 [35]. The inclusion of this environmental load follows a similar approach as the Morison equation. The selection of drag coefficients is heavily dependent on the geometry of the floating 2 The time constant for an uncontrolled floating structure is given by T ¼ m=btotal , where m is the structure inertia and btotal is the total linear damping coefficient (see Fossen [59]).

Fig. 8. The three main classes of floating support structures currently being adopted [70].

structure, with cylindrical spar type structures utilising typical Morsion drag coefficients, and large-volume structures may utilised approaches developed for ships if similar cross-sectional geometries are apparent. Suitable methods include that proposed by Hoerner [60] and by the International Tank Towing Conference Recommended Procedures [61]. 3.3. Other modelling methods There are a number of other, more detailed approaches to modelling the hydrodynamic behaviour of FOWT support structures. These usually require fine discretisation of the simulated environment, requiring extensive computing resources to carry out simulations in reasonable time scales. The use of computational fluid dynamics (CFD) and higher order boundary element method (HOBEM) have been somewhat limited in FOWT mainly due to the difficulty in interfacing multiple domains (solid and fluid) and computational effort required. However recently there has been an increased number of studies using these higher fidelity numerical models (see e.g. [62–64]), as research interests advance to investigate specific operational events (such as steep and extreme waves, and vortex induced motion). Smooth particle hydrodynamics (SPH) is also another novel approach to modelling floating wind turbines (see e.g. [65,66]). 3.4. Verification and validation An essential part of numerical model development and simulations is verification and validation of the numerical tool implemented. Verification of numerical tools against other design codes and/or higher-order models provides confidence that the mathematical models have been correctly implemented numerically, whilst validation again model- or full-scale experimental data establishes the physical relevance of the design tool. Verification for floating HAWT design tools has been carried out through codeto-code comparisons, providing insights into model limitations and improvements as discussed by Robertson et al. [67]. More recently this has also been done for floating VAWT design tools by Borg et al. [68], accelerating the development of these tools. Whilst it is difficult to obtain full-scale experimental data from floating HAWT prototypes such as Hywind [32] and WindFloat [19] due to commercial sensitivity and currently there are no floating VAWT prototypes in operation, scale-model experimental campaigns in controlled ocean basin environments provide significant validation data that provide insights into the dynamics of floating wind turbines that may have not been foreseen with

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2E+16 2E+15 2E+14 2E+13

Pitch PSD

2E+12 2E+11 2E+10 2E+09 2E+08 2E+07 2E+06 2E+05 2E+04 0.05

0.25

0.45

0.65

0.85

1.05

Wave Frequency (rad/s) Aerodynamic Forces

Wave Excitation Forces

Hydrostatic Forces

Gyroscopic Forces

Radiation Forces

Mooring Forces

Viscous Damping Forces

Fig. 9. Spar pitch moments power spectral densities.

numerical tools. This was experienced in the DeepCwind project that involved systematic testing of three different floating HAWTs, where Robertson et al. [69] found that current state-of-the-art coupled dynamics design tools had to be improved to more accurately capture second-order hydrodynamic forces and viscous damping in severe sea states.

4. Comparison of floating VAWT forces In this section, a comparison of first order forces acting on a floating VAWT is presented to provide the reader with an initial understanding of the dominant forces on the major types of FOWT support structures. The 5 MW VAWT developed in the EU-FP7 DeepWind project (see Vita [20]) was mounted on three different platforms: spar, semi-submersible and TLP (originally developed for floating HAWTs in the DeepCwind project, see Goupee et al. [70]). These support structures are depicted in Fig. 8 and further modelling details may be found in Borg and Collu [71]. A typical North Sea short-crested sea state was simulated with a wind speed equivalent to the VAWT rated wind speed, using the FloVAWT design tool [72]. The mean wave direction and wind direction were both aligned along the x-axis, which is perpendicular to both the vertical (yaw) axis and pitch axis (cf. Fig. 4). For brevity loads in pitch shall be presented here through power spectral densities (PSDs), as pitch is one of the critical DOFs of a FOWT. Note that for comparative reasons the PSDs are shown in semi-logarithmic graphs, as different force PSDs can vary by several orders of magnitude for the range of frequencies that captures hydrodynamic and aerodynamic forces. Furthermore, since the time-domain simulations were carried out for a finite period of time (three hours), noise is inadvertently introduced during frequency analysis of data. Whilst using techniques such as detrending, cosine tapering and windowing does reduce noise somewhat, a moving average filter was still applied to the PSDs generated. The forces that will be presented are as follows: first order wave excitation forces; radiation forces; hydrostatic forces;

linear viscous drag forces; VAWT aerodynamic forces; VAWT gyroscopic forces; and linearised mooring line forces. 4.1. Spar Fig. 9 presents the PSDs for the various pitch moments acting on the spar platform. As the spar is a slender structure with respect to the incident linear sea waves, viscous forces tend to be dominant over the wave frequency range. As can be seen in Fig. 9, there is a clear impact of the aerodynamic forces at 1.1 rad/s (twice the turbine rotational speed3) on all other forces bar the wave excitation force, that in the case of linearity (i.e. small amplitude displacements) is independent of platform pitch motion. Around this frequency, the affected forces PSDs see an increase of between two to three orders of magnitude. Between 0.3 and 0.9 rad/s, the range where the majority of the wave energy spectrum is found, first order wave excitation and viscous forces PSDs are four to six orders of magnitude higher than other forces PSDs. This indicates the importance of adequately modelling viscous forces for the spar platform (cf. §3.2.1 and Fig. 5). 4.2. Semi-submersible Fig. 10 presents the PSDs for the pitch moments outlined above, acting on the semi-submersible platform. In contrast to the spar platform, the semi-submersible is a large-volume structure with respect to the incident linear sea waves, and hence wave excitation forces tend to be dominant. Also since the semi-submersible achieves stability mainly through hydrostatic restoring, the hydrostatic moment also relatively dominant over the range of wave frequencies shown. Similar to the spar platform, the augmentation of all forces except wave excitation is clearly seen around 1.1 rad/s. 3 Due to the nature of VAWT aerodynamics, where a single blade produces sinusoidal forces as the turbine rotates, the frequency at which most energy is found for aerodynamic forces is equal to the number of blades multiplied by the rotational speed of the turbine.

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8E+16

8E+15

8E+14

Pitch PSD

8E+13

8E+12

8E+11

8E+10

8E+09

8E+08

8E+07 0.05

0.25

0.45

0.65

0.85

1.05

Wave Frequency (rad/s) Aerodynamic Forces

Wave Excitation Forces

Hydrostatic Forces

Gyroscopic Forces

Radiation Forces

Mooring Forces

Viscous Damping Forces

Fig. 10. Semi-submersible pitch moments power spectral densities.

Compared to the spar, radiation forces seem to play a larger role in platform motion of the semi-submersible. This is due to the geometry of the semi-submersible, due to the larger waterplane area, transferring more energy in to radiated waves as the platform moves. The peaks seen in most force PSDs at 0.21 rad/s are present since this is the natural frequency of the platform in pitch. The increased platform velocities close to this frequency induce larger forces (except for wave excitation forces, for the same reason stated in the previous section). 4.3. TLP Fig. 11 presents the PSDs of the pitch moments of forces interacting with the TLP. The characteristics of TLP pitch forces differ somewhat to the other two platforms, as the TLP maintains stability through excess buoyancy and pre-tensioned mooring cables. In fact, the mooring restoring force is the dominant force through the frequency range, responding to the highest excitation force at any particular frequency (e.g., as aerodynamic force PSD becomes greater than wave excitation force PSD at 1.05 rad/s, the mooring restoring force PSD then follows the aerodynamic force PSD). The importance of the mooring system for TLPs implies that appropriate models should be used during preliminary design stages, and that the linearised stiffness matrix might not capture all the relevant characteristics of the tendons, particularly in extreme met-ocean conditions when snap loading can occur (see companion paper Borg et al. [3]). Due to the very small water plane area of the TLP, the hydrostatic restoring force PSD is on average three orders of magnitude lower than that of the semisubmersible. This same rationale also applies for the radiation force PSD, with the same order of magnitude difference as the hydrostatic force PSD when compared to the semi-submersible. When comparing both hydrostatic force and radiation force PSDs of the TLP to those of the spar, they are of similar magnitudes as the waterplane areas of both platforms are similar (see Goupee et al. [70] for platform geometry)

5. Current implementations Until recently offshore wind turbine codes (adopting time analysis approach) have almost exclusively been based on the Morison equation [73] although there now is a clear trend to implement the linear Cummins approach [74], more recently also including second order wave forces (see e.g., Bae and Kim [48] and Coulling et al. [47]) and quadratic damping (see e.g., Philippe et al. [40]). These studies were all investigating floating HAWTs. Recently the radiation-force state-space approximation has also been implemented in the popular floating HAWT design code FAST by Duarte et al. [75]. In his study of a floating VAWT, Vita [20] implemented the Morison equation since the floating structure was a slender spar with aeroelastic code HAWC2. Only recently have there been efforts to develop a dedicated coupled model of dynamics for floating VAWTs:


Researchers at Cranfield University are currently developing a







generic coupled model of dynamics for floating VAWTs, the progress of which is outlined by Collu et al. [72,76]. This coupled model employs an in-house aerodynamic momentum model (see companion paper [2]) and the hydrodynamic model is based on the Cummins equation with radiation-force statespace approximation (see Borg et al. [76–78]) based on the MATLAB/Simulink Marine Systems Simulator (MSS) toolbox developed by Fossen and Perez [79]. Researchers at NTNU and DTU Wind have also developed a coupled model of dynamics as outlined by Wang et al. [80]. In this model an aerodynamic momentum model is also used, and is coupled to the hydrodynamic/structural Simo/Riflex software. A generator control module is also incorporated. Researchers at Texas A&M University and Sandia National Laboratories (SNL) have developed an open-source finite-element toolkit to accurately model the structural dynamics of floating VAWTs, as outlined by Owens et al. [81]. Whilst it is

Please cite this article as: Borg M, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part III: Hydrodynamics and coupled.... Renewable and Sustainable Energy Reviews (2014), http://dx.doi.org/10.1016/j.rser.2014.10.100i

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2E+16 2E+15 2E+14 2E+13

Pitch PSD

2E+12 2E+11 2E+10 2E+09 2E+08 2E+07 2E+06 2E+05 2E+04 0.05

0.25

0.45

0.65

0.85

1.05

Wave Frequency (rad/s) Aerodynamic Forces

Wave Excitation Forces

Hydrostatic Forces

Gyroscopic Forces

Radiation Forces

Mooring Forces

Viscous Damping Forces

Fig. 11. TLP pitch moments power spectral densities.




solely a structural model, it has been partially coupled to the CACTUS aerodynamic software developed by SNL [82]. It may also be coupled with the new FAST framework in the near future. Le Cunff et al.[83] have developed a coupled dynamics model based on the DeepLines software that incorporates aerodynamics, hydrodynamics, structural and control dynamics. Although this model has been developed mainly for floating HAWTs it also has the capability to model floating VAWTs using a simplified aerodynamic momentum model.

5.1. Current challenges with coupled dynamics design codes A current issue with performing coupled dynamics simulations is the interfacing of different software packages to provide a fully integrated numerical simulation. Problems with the communication of data between different programs lead to instabilities and longer simulation times. This was seen in Cermelli et al. [15], where a number of codes were coupled together to analyse a floating HAWT. Whilst the interfaced codes provided the required results, the authors noted that there were issues with the interfaces. Another issue following on from the above is the need to have faster and more efficient design codes. This originates from design guidelines and standards requiring a vast number of simulation test cases be carried out that encompass most conceivable environmental and operating conditions. Organisations such as Det Norske Veritas [84] and the American Bureau of Shipping [85,86] have recently developed new guidelines specifically for floating wind turbines outlining the simulation requirements.

by Hall et al. [87] that required a large number of simulations (in the order of 1000) to identify an optimal support structure in a design space that had a number of local optima. Muskulus [73] also highlighted the fact that most engineers involved in FOWT research and design usually only have access to desktop computers that have limited computational power, hence requiring the use of efficient design codes for carrying out simulations. 6.1. Programming methodologies There is a need to establish stable and efficient real-time interfaces between the different modules of the coupled model of dynamics code. Attempts by researchers to couple different commercial and/or open-source design software has led to a large reduction in simulation performance [88]. This is due to the limited communication in the interface between different codes as well as the restricted use of less efficient numerical solvers [88]. Furthermore Brommundt et al. [31] found that different numerical solvers may have a significant effect on simulation performance. One approach may be to develop the code in a single programming language as well as ensure uniform interfacing between all modules of the code. Researchers at the National Renewable Energy Laboratory, USA have recently made extensive efforts to apply this programming methodology to the FAST design code for floating HAWTs [4]. In fact, such is the extent of the modularity of this tool that it may be possible to couple it with an aeroelastic model for VAWTs. The importance of maintaining numerical stability and accuracy when coupling the multi-physics modules in FAST has been presented by Gasmi et al. [89]. The use of ‘loose’ or ‘tight’ coupling between modules found to provide different results and it is imperative to consider the coupling approach and its effect on the accuracy of simulations.

6. Computational strategies and issues 6.2. Limiting model complexity With current design codes achieving a ratio of 1:1 between simulated and simulation times, there is a great need to have faster codes to allow for accelerated design cycles and optimisation. A prime example is the use of a genetic-algorithm optimiser

Some initial research to increase the efficiency of a coupled design code was done by Karimirad and Moan [33]. The authors simplified the analysis such that it still gave acceptable results. The

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Table 2 Summary of suitable engineering models currently available for floating VAWTs. Complexity

Aerodynamics BEM Cascade Vortex Panel Hydrodynamics Morison Cummin's Cummin's w/state-space model

Ease of implementation

Computational effort Rotor-wake/multiple rotor interaction

Unsteady conditions

Restricted to known aerofoils

Limited Limited Yes Yes

Yes Yes Yes No Viscous model

Low–medium Low–medium Medium–high High

Easy–medium Easy–medium Medium Hard

Low Low Medium–high Medium–high

No/no No/no Yes/limited Yes/yes Geometric constraints

Low Medium Medium–high

Easy Medium Hard

Low High Low–medium

Limited to slender bodies Not constrained Not constrained

Implicit Explicit Explicit

Accommodate nonlinearities Mooring line dynamics Force–disp. relation Quasi-static Multibody Finite Element Structural dynamics Rigid-body Modal Multibody Finite element

Low Low Medium High

Low Low Medium High

Easy Easy–medium Easy–medium Hard

Easy Easy–medium Easy–medium Medium–hard

Low Low Low-medium Medium–high

Low Low Low-medium High

authors claim to have achieved simulation-to-simulated time ratios of 1:4, and whilst this is an improvement, the reduction in accuracy of the simulation might not warrant the reduction in computation time. Further work by Karimirad [90] investigating the inclusion of second order hydrodynamics (see e.g., Faltinsen [41]) found that this extra level of detail increased computational times by 10–15 times without appreciable differences to the global motions of floating spar-type HAWT modelled. Whilst this also highlights the trade-off between computational effort and model accuracy, the results should be treated with caution as these second-order forces may be highly dependent on the nature of the submerged structure (such as TLPs). There is also the possibility of interaction with the high-frequency oscillatory aerodynamic forces produced by a VAWT4. As mentioned before, the validity of certain models could be questionable when applied to the floating offshore environment. This is evident with the use of aerodynamic momentum models, as the assumption of momentum balance may not necessarily apply in the unsteady conditions found offshore, which has been discussed in detail by Sebastian [91].

Very limited Limited Yes Yes Elastic analysis

Detailed stress analysis

No Yes Yes Yes

No No Limited Yes

FOWTs. As the underlying fundamental engineering models are inherently parallelisable, the use of GPU programming would drastically increase the computational efficiency of design codes (with a speed-up factor of 10 that can be easily achieved [88]). One drawback of this approach is the limited programming language available for use in codes utilising GPUs, as well as the laborious task of rewriting any established software. Another approach that bypasses the use of GPUs (albeit at the cost of computational speed) would be to utilise the number of central processing units (CPUs) available in modern desktop computers, which usually have 2–4 CPUs. The approach would be to run individual modules of the coupled dynamics code on separate CPUs. Currently running a conventional design code as a single executable process on a multi-CPU computer only makes use of one CPU. Hence it would be worthwhile to utilise all CPUs by, for example, running the aerodynamic, hydrodynamic and structural dynamics modules concurrently on separate CPUs.

7. Engineering models suitable for the preliminary design of floating VAWTs

6.3. Utilising current computing technologies As mentioned earlier, the typical FOWT research and design engineer only has access to a desktop computer for carrying out coupled dynamics simulations, and usually do not have a large budget to upgrade and expand computational resources. As discussed by Muskulus [73], the advent of low-cost graphical processing units (GPUs) available as ‘add-ons’ to desktop computers paves the way for a new generation of coupled design codes for 4 Aerodynamic forces produced by a VAWT are roughly sinusoidal in shape with a frequency equal to the rotational speed (see e.g. Shires [92]). With multimegawatt VAWTs, this frequency approaches the higher end of typical wave spectra, possibly inducing interactions.

Building upon the review carried out in this article and companion papers [2,3], a summary of suitable aerodynamic, hydrodynamic, structural and mooring line dynamics engineering models for use in coupled dynamics analysis of floating wind turbines during the preliminary design stages can be presented. Table 2 was constructed to concisely illustrate this summary. In aerodynamics modelling, the blade element momentum model is the ideal starting point to obtain a first insight into the behaviour and power performance of a floating VAWT rotor. Vortex and panel models are then more suitable once the preliminary design has been set and investigations into specific issues such as rotor-wake interactions and the influence of closely spaced rotors on one another.

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As discussed in detail earlier the most suitable hydrodynamics model is based on the Cummins equation with a number of additions, namely the radiation-force approximation and additional viscous drag. Whilst more detailed models (such as HOBEM, CFD and SPH) could be implemented, their computational effort restricts their use in preliminary FOWT design for the foreseeable future. The structural behaviour of the floating wind turbine system can be represented by a number of models, ranging from the simplified rigid body model to finite element method. The most popular approach using the multi-body formulation (MBF) as it provides the best trade-off between computational speed and physical accuracy. Lastly, modelling of the mooring lines can be performed through the use of a linearised force–displacement relation or quasi-static catenary model (except for TLP support structures). From the literature review it is evident that there is a gradual shift to implementing MBFs to account for inertial and hydrodynamic drag effects. This is convenient for structural dynamics modelling as the same underlying principles are used and hence common computational routines can be used. Also the use of the MBF allows for the prediction of significantly more accurate mooring line tensions that the quasi-static catenary model, making it suitable for later stages of design.

8. Conclusions As highlighted in Section 2, developing an efficient ‘aerohydro-servo-elastic’ coupled model of dynamics for floating VAWTs is of great importance in assessing the technical and economic feasibility of these systems. Computational efficiency is major challenge and one of the most important factors as these models need to run in a short span of time on desktop PCs by engineers assessing a large number of configurations and operating conditions. Whilst the trend has been to develop time-domain models that are able to incorporate the myriad of nonlinearities present, frequency-domain analyses are still crucial as they allow engineers a first glimpse at the performance of a given floating VAWT. The most suitable hydrodynamic model in current FOWT research is based on the Cummins equation with a number of modifications and additions. The substitution of the radiationforce convolution integral with approximations leads to greater computational efficiency, following up on the above-mentioned challenge. The inclusion of second order wave excitation forces and higher-fidelity viscous damping models allow for more accurate prediction of the device performance in adverse environmental conditions, even if it may be considered negligible for the majority of the operational load scenarios. Nonetheless, and in particular for VAWTs, the interaction between sum-frequency second order wave forces and aerodynamic forces may be very important as these forces exist within the same range of frequencies for the multi-megawatt VAWT rotors envisaged to be deployed far offshore. As has been highlighted in recent literature, more accurate viscous damping models are required, and the use experimental data to tune these models is a promising approach (see Coulling et al. [58]). Readers are also directed to the companion papers of this article for further information on modelling VAWT rotor aerodynamics, as well as structural and mooring dynamics. Table 2 has been constructed to summarise the review of suitable engineering models to model a floating VAWT. As is the case with most engineering areas, different stages of design require models of varying complexity and there is no ‘onesize-fits-all’ model available to be used throughout the design

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process. A major challenge in this area of research is establishing an efficient coupled model of dynamics, with a code that provides stable interfaces between modules and is flexible enough to accommodate future concepts. As can be seen throughout, there is also a need to take an integrated approach to modelling floating wind turbines, considering the interactions between different aspects of the system.

Acknowledgements The research leading to these results has been performed in the frame of the H2OCEAN project (www.h2ocean-project.eu) and has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under Grant agreement no. 288145. It reflects only the views of the author(s) and the European Union is not liable for any use that may be made of the information contained herein. References [1] Jonkman JM, Matha D. Dynamics of offshore floating wind turbines-analysis of three concepts. Wind Energy 2011;14(4):557–69. [2] Borg M, Shires A, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part I: aerodynamics. Renew Sustain Energy Rev 2014;39:1214–25. [3] Borg M, Collu M. Offshore floating vertical axis wind turbines, dynamics modelling state of the art. Part II: Struct Mooring Line Dyn Renew Sustain Energy Rev 2014;39:1226–34. [4] Jonkman JM. The New Modularization Framework for the FAST Wind Turbine CAE Tool. In: proceedings of the 51st AIAA aerospace sciences meeting, January 7–10, Dallas, Texas, AIAA; 2013. [5] Bulder. Study to Feasibility of and Boundary Conditions for Floating Offshore Wind Turbines. 2002-CMC-R43, Drijfwind; 2002. [6] Lee KH. Responses of floating wind turbines to wind and wave excitation. Cambridge, MA: Massachusetts Institute of Technology; 2005. [7] Wayman E. Coupled dynamics and economic analysis of floating wind turbine systems [M.Sc. thesis]. Massachusetts Institute of Technology; 2006. [8] Collu M, Kolios AJ, Chahardehi A, Brennan F. A comparison between the preliminary design studies of a fixed and a floating support structure for a 5 MW offshore wind turbine in the north sea. RINA, Royal Institution of Naval Architects – Marine Renewable and Offshore Wind Energy – Papers, vol. 63; 2010. [9] Lefebvre S, Collu M. Preliminary design of a floating support structure for a 5 MW offshore wind turbine. Ocean Eng 2012;40:15–26. [10] Matha D. Model development and loads analysis of an offshore wind turbine on a tension leg platform, with a comparison to other floating turbine concepts, NREL/SR-500-45891, NREL, Colorado, USA; 2010. [11] Cordle, A. State-of-the-art in design tools for floating offshore wind turbines, UpWind D4.3.5, UpWind; 2010. [12] Cordle A, Jonkman J. State of the art in floating wind turbine design tools. Proc Int Offshore Polar Eng Conf 2011;1:367. [13] Jonkman J. Dynamics modeling and loads analysis of an offshore wind turbine. NREL/TP-500-41958, NREL, Colorado, USA;2007. [14] IEC 61400-3. Wind Turbines – Part 3: Design Requirements for Offshore Wind Turbines. International Electrotechnical Commission; 2007. [15] Cermelli C, Aubault A, Roddier D, McCoy T. Qualification of a semi-submersible floating foundation for multi-megawatt wind turbines. Proc Annu Offshore Technol Conf 2010;2:1592. [16] Bae YH, Kim MH, Im SW, Chang IH. Aero-elastic-control-floater-mooring coupled dynamic analysis of floating offshore wind turbines. Proc Int Offshore Polar Eng Conf 2011;1:429. [17] Robertson AN, Jonkman JM. Loads analysis of several offshore floating wind turbine concepts. Proc Int Offshore Polar Eng Conf 2011;1:443. [18] Sebastian T,Lackner M. Unsteady near-wake of offshore floating wind turbines. In: proceedings of the 50th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, 09–12 January 2012, Nashville, Tennessee, AIAA; 2012. [19] Roddier D, Cermelli C, Aubault A, Weinstein A. WindFloat: a floating foundation for offshore wind turbines. J Renew Sustain Energy 2010;2(3):033104. [20] Vita L, Paulsen US, Madsen HA, Nielsen PH, Berthelsen PA,Carstensen S (), Design and aero-elastic simulation of a 5 MW floating vertical axis wind turbine. In: ASME 2012 31st international conference on ocean, offshore and arctic engineering, American Society of Mechanical Engineers; 2012. p. 383. [21] Collu M, Brennan FP, Patel MH. Conceptual design of a floating support structure for an offshore vertical axis wind turbine: the lessons learnt. Ships Offshore Struct 2012;1:1–19. [22] Akimoto H, Tanaka K, Uzawa K. Floating axis wind turbines for offshore power generation – a conceptual study. Environ Res Lett 2011;6(1):044017.

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