Investigation into the Potential Use of Damping Plates in a Spar-Type Floating Offshore Wind Turbine

Abstract

Spar is one of the promising floating platforms to support offshore wind turbines. Wind heeling moment is large in the case of floating offshore wind turbines and, therefore, it is important to reduce the pitch motion of the floating platform. To address this issue, a spar platform with damping plates is proposed and investigated in this study. (i) Type-A, (ii) Type-B, and (iii) Type-C models of 1/120 scale were fabricated with similar stability parameters. Type-A is a classic spar, Type-B and Type-C are spar with damping plates by replacing the ballast water part with horizontal plates and vertical plates, respectively. The rotor model consists of (i) no disk and (ii) with disk conditions. A series of model scale experiments were carried out in the water tank in regular waves, and motion response was measured. A calculation method based on classic frequency-domain was developed to incorporate damping plates and validated with the experiment results in no disk and with disk conditions. When pitch response of Type-B and Type-C were compared with respect to Type-A, it was found that the spar platform with damping plates reduced the pitch response in most wave frequencies.

1. Introduction

Offshore wind energy has seen tremendous progress over the past decade [1,2]. One of the main challenges for commercialization of offshore wind energy is to secure the wind turbines with a support structure in the ocean. The support structure in the ocean can be broadly divided into two types, the bottom-fixed type such as monopile or jacket foundations, and floating type such as spars, semi-submersibles, or TLPs [3]. The floating type support structure helps access the deep waters of the ocean where the wind is stronger with greater consistency. Further, larger wind turbines can be installed as a farm to produce more energy per unit, reducing costs [4]. Therefore, floating offshore wind turbines are getting attention worldwide and floating platforms are emerging as one of the key technologies to install wind turbines in deep waters [5].

The primary objective of a floating platform is to provide buoyancy to support the weight of the wind turbine, and to provide hydrostatic stability for constant generation of electricity under combined wind and wave loads. In the case of floating offshore wind turbines, the wind thrust at the top of the tower leads to a large wind heeling moment, and further increases the pitch motion of the floating platform [6,7,8]. It is, therefore, essential to tackle and stabilize the pitch motion of the floating platform. It becomes imperative to reduce the platform motions, especially the pitch motion to avoid misalignment to the wind direction during energy production and to avoid large loads during extreme events as observed in [9,10,11,12].

Spar, which consists of a long cylindrical body is one of the promising floating platforms owing to its simplistic design and limited platform response in waves. The world’s first commercial floating wind farm, Hywind Scotland (5 × 6 MW turbines), commissioned in 2017, uses a spar platform [13,14,15]. In Japan, a hybrid spar (2 MW turbine), made of steel and concrete, was installed in 2013, and is still operating off the coast of Goto Islands [16,17,18]. The basic design of a classic spar platform revolves around achieving stability through ballast water tank installed at the lower part of the cylindrical body. The ballast water is adjusted such that the center of gravity lies at a low position. However, it is possible to replace the ballast water part with another non-watertight structure because this part does not contribute to hydrostatic performance. The replacement of the ballast part can also result in a lighter spar platform, leading to reduced materials and costs.

Generally, the spar platform performs well in a heave response under waves, provided the diameter at the waterplane is kept to a minimum value. Further, several studies have been conducted on spar platforms that use heave plates to reduce the heave response by increasing added mass and heave damping [19,20,21]. The hydrodynamic effects of heave plates which are usually attached at the bottom of the platform to reduce heave motion have been researched in various studies [22,23,24,25]. However, there are limited studies to counter the large wind heeling moment, and to reduce the pitching motion of the spar platform. In this study, an attempt was made to address this issue by introducing damping plates. The idea is inspired by truss spars used in the oil and gas industry, which have been found to be cost-effective and reduce drag loads by current more than the classic spar [26,27]. Previously, one of the authors conducted a preliminary experimental study on the damping plates [28], but the hydrodynamic mechanism has not been confirmed.

The novelty of this study is to understand the hydrodynamic mechanism of the damping plates in order to counter the large heeling moment and reduce the pitch motion response of the spar platform. The basic mechanism of damping plates is to tap into the complex fluid–structure interaction between the incoming wave and damping plates. The added mass and the wave radiation damping is a function of incoming wave frequency, and, therefore, by altering the body form of the classic spar, it is possible to alter the hydrodynamic characteristics and overall platform response.

The purpose of this study is to firstly design a spar platform with damping plates. To this end, in this research, three models of spar platform were designed, (i) Type-A, (ii) Type-B, and (iii) Type-C. Type-A was the classic spar without damping plates. Type-B and Type-C were spar platforms replacing the ballast water with horizontal damping plates, and vertical damping plates, respectively. To be able to compare the motion performance of the three types, the stability parameters (Weight × GM) were kept similar. The working mechanism of the damping plates needs to be verified and since there are no established methods, a simplistic calculation method was proposed and developed in this study. The objective of the proposed calculation method is to understand the hydrodynamic effects of damping plates in the early stage of development and obtain quick results.

The paper is organized as follows. Section 2 describes the experiment details of the spar models with damping plates. It also describes the principal particulars of the spar models, rotor model, along with the experiment setup and conditions. The basis for numerical calculation is confirmed from experiment results. Section 3 exemplifies the proposed calculation methodology and the equation of motion. Section 4 shows the calculation results and validates the implemented methodology by comparing it with model scale experiments. It also investigates the potential use of damping plates to reduce the pitch motion of the spar platform. The findings of the study are concluded in Section 5.

2. Experiment

This section describes the details of the model scale experiment models, experiment setup, and experiment conditions. The experiment results of the spar models in regular waves are also presented.

2.1. Spar Models

As defined in the introduction, the objective of this study was to understand the potential use of damping plates in the spar platform by conducting model scale experiments. The model scale of the experiment was chosen as 1/120 to suit the two-dimensional wave tank facility at the Institute of Ocean Energy, Saga University, Japan. All tests were conducted in this wave tank which measured 18.1 m in length, 0.8 m in width, and 1.0 m in depth. Correspondingly, the three spar models, (i) Type-A, (ii) Type-B, and (iii) Type-C were designed as shown Figure 1. Type-A was a classic spar, consisting of a long cylindrical structure with water ballast at the lower part of the spar platform. The spar models with damping plates, i.e., Type-B and Type-C basically consisted of a long cylindrical structure and damping plates introduced at the lower part of the spar platform.

Type-B was the spar model replacing the ballast water part of the Type-A model with two horizontal plates. The outer diameter of the acrylic horizontal plate was 110 mm and the inner diameter was 60 mm. The thickness of the two outer layers was 10 mm, and the thickness of the two inner layers was 5 mm. All the plates were placed in parallel, and the top and bottom plates were connected using metal fittings.

Type-C was the spar model replacing the ballast water part of the Type-A model with three vertical plates. It had three rectangular acrylic vertical plates sandwiched between the top and bottom horizontal plates. Each vertical plate had a length of 40 mm, a thickness of 5 mm, and a height of 210 mm. Further, each vertical plate consisted of 20 holes.

Table 1. Principal particulars of Type-A, Type-B, and Type-C spar models.

Spar Model Specification (Scale: 1/120)		Type-A	Type-B	Type-C
Diameter of the main part (m)		0.11
Diameter of the water plane (m)		0.07
Diameter of the tower (m)		0.04
Draft (m)		0.845
Mass (kg)		7.49	5.71	5.84
Pitch moment of inertia (kgm2)		1.27	1.23	1.25
Metacentric height, GM (m)		0.084	0.114	0.112
Stability parameter, weight × GM (Nm)		6.17	6.39	6.42
Keel to center of gravity, KG (m)		0.308	0.345	0.337
Heave natural period (s)	No disk	2.79	2.62	2.56
	With disk
Pitch natural period (s)	No disk	3.16	2.97	3.05
	With disk	3.37	3.77	3.23

shows the principal particulars of the crafted spar models. It can be seen in Figure 1 that the diameter of the spar cylindrical body was reduced at the water plane which helps minimize the spar platform’s response, especially heave. A smaller diameter at the water plane reduces wave exciting force and, in turn, reduces the restoring force of heaving, which leads to longer natural period than incoming ocean waves. The diameter of the main part was 0.11 m and was reduced to 0.07 m at the water plane. The draft of the platform was 0.845 m. In this study, to be able to compare Type-A, Type-B, and Type-C models, the stability parameter (Weight x GM) was kept almost same for all the models.

The heave and pitch natural periods measured in the free-decay tests are also summarized. The heave natural period of Type-A was 2.79 s, which corresponds to about 30.6 s in a real sea state. The design idea was to have the natural period of the spar platform far away from the sea state and avoid resonance. It can be seen from Table 1 that the heave natural period followed the trend, Type-A > Type-B > Type-C. However, the pitch natural period (no disk) followed the trend, Type-A > Type-C > Type-B. Further, the pitch natural period with a disk was longer than the no disk condition due to additional damping from the disk.

2.2. Rotor Model

The spar models were designed to support NREL 5 MW reference wind turbine [29], with the scale of 1/120. The maximum wind thrust load of the NREL turbine was about 830 kN at the rated wind speed, which translated to about 0.485 N in model scale. In this study, two sets of experiment models, (i) no disk and (ii) with disk, were adopted as shown in Figure 2. The intention of the model scale experiment with disk was to simulate the wind load by assuming a non-rotating flat disk at the top of the tower. For weight reduction, the disk was made of a styrene board. Considering the fluctuating aerodynamic force when the spar was pitching in waves, the diameter of the disk was decided to be 0.7 m, which was 70% of the scaled NREL turbine.

2.3. Experiment Setup and Conditions

Figure 3 shows the experiment setup. The spar models were moored horizontally at around mid-draft position by wires and springs of 0.0102 N/mm. Wave elevation was measured at 8 positions and the reflected wave components were eliminated in the analysis. The experiments were conducted in regular waves with the wave frequencies of 2.7 to 7 rad/s which correspond to wave periods of 25.5 s to 9.8 s in a real scale. A wave height of 8 cm was adopted in the experiment which corresponds to 9.6 m in real scale. The motion capture system based on LED markers (see Figure 4) tracked and measured the rigid body motion of the spar body in waves, and the three degrees of freedom motion in surge, heave, and pitch was obtained. The measured motion response data were analyzed to obtain the response amplitude operator (RAOs) in surge, heave, and pitch. Surge and heave were divided by the wave amplitude, and pitch was divided by the wave amplitude and wave number for normalization.

2.4. Spar Motion Characteristics in Waves

Figure 5 shows the experiment result of spar motion in one wave cycle for wave frequencies of 2.7, 4.5, and 7.0 rad/s as an example. Each black arrow line represents the motion of the spar platform at every 0.1 s. The blue concentric circles denote the motion of water particles, and z = 0 denotes the still water surface. Spar motion in waves is unique because of a long shape and low center of gravity position. The center of rotation is far below the bottom (keel point) of the spar platform and, therefore, the motion of the below part is smaller than the motion at the upper part of the spar body. Since the motion of the below part where the damping plates are attached is small, it can be said that the relative velocity with the incoming wave is also small and flow separation is limited. It is, therefore, possible to use conventional frequency-domain linear equations based on linear potential theory to study the effect of damping plates in a spar platform.

In the model scale experiments, three trials were conducted for each wave frequency and the motion response was measured. The average of the motion amplitudes was obtained from the measured data by employing Fourier analysis based on wave frequency. Figure 6 shows the experiment result of spar motion response in surge and pitch with respect to Type-A for the no disk condition. The experiment results revealed the potential of damping plates in reducing the surge and pitch response of the spar platforms in most wave frequencies. However, it is difficult to understand how the replacement of ballast water with damping plates reduces the motion response. Therefore, to understand the hydrodynamic mechanisms of the spar with damping plates, and to clarify the experiment results, numerical calculations were conducted. The basis of the numerical calculations, methodology and validation results are presented in Section 3.

3. Calculation Methodology

For evaluating the behavior of offshore wind turbines, great efforts are being made worldwide. To solve the complicated aero–hydro–servo–elastic coupling problem, many codes have been developed and validated [30]. These non-linear, time-consuming, time-domain calculations are indispensable for safe, economical design and certification. However, classic frequency-domain, linear calculations, which save time and cost, are still useful in the early stages of development, especially to understand the essentials of hydrodynamics of the new floater which supports the wind turbines. Further, it has been confirmed in Section 2.4 that the flow separation is limited and, therefore, a time and cost-saving calculation methodology was implemented. This section explains the calculation methodology based on the classic frequency domain, and equation of motion used to incorporate the proposed damping plates.

3.1. Calculation Technique

Figure 7a shows the flow of the calculation methodology to model the damping plates in the spar models. The replacement of damping plates at the lower part of the spar platform leads to a change in the added mass and wave radiation damping, leading to different motion characteristics. Therefore, to capture this phenomenon, the damping plate configurations were modeled in Rhino 7 [31], a CAD software. The surface mesh was imported into NEMOH [32,33,34,35], an open-source boundary element method (BEM) hydrodynamic solver based on the linear potential flow theory. The added mass, wave radiation damping coefficients, and wave excitation force were obtained at various wave frequencies tested in the same range as the model scale experiments. The equation of motion was then set up and the motion responses were obtained in MATLAB R2023b.

3.2. Equation of Motion

The equation of motion setup is described here. The six degrees of freedom linear equation of floating oscillating body in waves can be written as:

$$\left[M+A\left(\omega \right)\right]\ddot{\mathit{x}}+B\left(\omega \right)\dot{\mathit{x}}+C\mathit{x}={\mathit{F}}_{\mathit{e}}$$

(1)

where the matrix $M$ is mass or mass inertia; $A\left(\omega \right)$ is the added mass or added moment of inertia, which is a function of wave frequency; $B\left(\omega \right)$ is the wave radiation damping, which is also a function of wave frequency; $C$ is the restoring stiffness or stability parameter; and ${\mathit{F}}_{\mathit{e}}$ is the wave excitation force or moment.

Considering the oscillation in regular waves, the wave excitation force is expressed as:

$${\mathit{F}}_{\mathit{e}}={\tilde{\mathit{F}}}_{\mathit{e}}\left(\omega \right)e^{i\omega t}$$

(2)

Then, $\mathit{x}$, $\dot{\mathit{x}}$, and $\ddot{\mathit{x}}$, which are the displacement, velocity, and acceleration of the oscillating body, are complex numbers.

In Equation (1), the $A\left(\omega \right)$, $B\left(\omega \right),$ and ${\mathit{F}}_{\mathit{e}}\left(\omega \right)$ are obtained from NEMOH calculations. The objective is to solve for the displacement amplitudes.

To do this, displacement is assumed to be:

$$\mathit{x}={\tilde{\mathit{x}}e}^{i\omega t}$$

(3)

where $\tilde{\mathit{x}}$ is the complex motion amplitudes to be solved. Based on this assumption in Equation (2), the following can be written

$$\dot{\mathit{x}}=i\omega \tilde{\mathit{x}}{e}^{i\omega t} ; \ddot{\mathit{x}}=-{\omega }^2 \tilde{\mathit{x}}{e}^{i\omega t}$$

(4)

Equation (1) can be rearranged as,

$$\left[-{\omega }^2 \left(M+A\left(\omega \right)\right)+i\omega B\left(\omega \right)+C\right]\tilde{\mathit{x}}={\tilde{\mathit{F}}}_{\mathit{e}}\left(\omega \right)$$

(5)

The above equation of motion was set up in MATLAB and solved for motion response in six degrees of freedom as shown in Equation (6):

$$\left|\tilde{\mathit{x}}\right|=\left[\begin{array}{c}\mathrm{S}\mathrm{u}\mathrm{r}\mathrm{g}\mathrm{e} \mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\\ \mathrm{S}\mathrm{w}\mathrm{a}\mathrm{y} \mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\\ \mathrm{H}\mathrm{e}\mathrm{a}\mathrm{v}\mathrm{e} \mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\\ \mathrm{R}\mathrm{o}\mathrm{l}\mathrm{l} \mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\\ \mathrm{P}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h} \mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\\ \mathrm{Y}\mathrm{a}\mathrm{w} \mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\end{array}\right]$$

(6)

where $\left|\tilde{\mathit{x}}\right|$ gives the absolute value of the complex motion amplitudes. These calculation results are then non-dimensionalized to obtain RAOs and compared with the experiment results.

3.3. Modeling of Wind Drag Force Due to Disk

The damping due to the disk oscillation in air under the action of regular waves was considered into the calculation. In this study, the effect of air drag force on the disk was evaluated by a simplified formula. The drag coefficient was assumed constant and, in order to include it in the equation of frequency domain, the force was linearized. As a result, Equation (1) can be rewritten as:

$$\left[M+A\left(\omega \right)\right]\ddot{\mathit{x}}+\left[B\left(\omega \right)+B_{disk}\right]\dot{\mathit{x}}+C\mathit{x}={\mathit{F}}_{\mathit{e}}$$

(7)

where $B_{disk}$ is the air damping coefficient proportional to the horizontal velocity $U$ at the nacelle position; in which the surge and pitch velocities are combined:

$$B_{disk}=\frac{8}{3\pi }\frac{1}{2}{\rho }_{a}A{C}_d{\overline{x}}_{nac}\omega U$$

(8)

where ${\rho }_a$ is air density, $A$ is the area of the circular disk, $C_d$ is the constant drag coefficient for an oscillating disk, ${\overline{x}}_{nac}$ is the nacelle horizontal displacement, and $\omega$ is the wave frequency. With this methodology, the wind drag force acting on the spar models due to the non-rotating disk can be modelled.

4. Results and Discussion

In this section, the spar motion characteristics in the experiment is first discussed. The calculation results are then shown and compared between the three spar models. The stability parameter of the spar models was kept similar and, therefore, the added mass, wave radiation damping and wave excitation force were compared directly to understand and address the effect of damping plates. It is noted that these are the input parameters to obtain the motion response of the spar models.

4.1. Added Mass and Wave Radiation Damping

Figure 8a,b show the added mass obtained from the calculations in surge and heave directions, and Figure 8c shows the added moment of inertia in the pitch direction for Type-A, Type-B, and Type-C spar models. In the equation of motion, they correspond to $A_{11}$, $A_{33}$, and $A_{55}$ terms. As in the model scale experiments, the added mass and added moment of inertia were plotted against the varying incoming wave frequencies of 2.7 to 7 rad/s. The difference in surge added mass between the models can be explained based on the decreased lateral projected area for Type-B and Type-C models. It can be observed that the added mass in the surge direction was largest for the Type-A spar model, and least for the Type-B spar model due to the replacement of the lower part of the model by horizontal damping plates. In the case of the Type-C spar model, the added mass in the surge direction is less than the Type-A model, but slightly larger than the Type-B model due to the influence of vertical plates.

However, in the heave direction, the added mass of Type-A spar model was the least due to no damping plates, and the added mass of Type-B was the largest due to horizontal plates. The added moment of inertia in pitch followed a similar trend as surge. The Type-C spar model with vertical plates increased the added moment of inertia than the Type-B spar model with horizontal plates in the pitch direction. Further, it can be observed that the added moment of inertia slightly increased as the wave frequency increased.

Figure 8d–f show the variation of wave radiation damping with respect to the incoming wave frequencies for Type-A, Type-B, and Type-C spar models. In the equation of motion, they correspond to $B_{11}$, $B_{33}$, and $B_{55}$. It can be seen that the wave radiation damping increases with the increasing wave frequencies and Type-A was found to have a larger value than Type-B and Type-C. It is hypothesized that, in the case of Type-B and Type-C, reduced volume of the spar model is less exposed to the waves and, therefore, these values are largest for the Type-A model. Now, comparing Type-B and Type-C models, the wave radiation damping of Type-B and Type-C spar models was almost same in the surge direction; however, the wave radiation damping of Type-C spar model was found to be slightly larger than the Type-B spar model in the pitch direction especially for larger wave frequencies. It was also found that the variation trend in surge and pitch directions was similar which is interesting as it means the pitch–surge coupling effects were dominant for the spar models. Section 4.2 further investigates this coupling effect.

4.2. Pitch–Surge Coupling Effects

This section describes effects of pitch–surge coupling for spar models. Figure 9a shows the additional moment in pitch direction due to the surge motion. In the equation of motion, this corresponds to the term $A_{51}$. It was also found that the pitch–surge-coupled additional moment of Type-B was slightly larger than Type-A and Type-C. The order of magnitude follows the trend, Type-B > Type-C > Type-A. This can be explained by considering the geometry of the spar models. When spar moves sideways, for Type-A the force works symmetrically from bottom to top and, therefore, the moment is small. For Type-B, the force at the horizontal plates is smaller than other parts and, therefore, the moment is slightly larger. This is an important finding as it can be said that the damping plates will affect the coupled motion response of the spar models differently. It is, therefore, important to evaluate the pitch–surge coupling effects when considering spar models with damping plates.

The wave radiation damping due to the pitch–surge coupling effect is shown in Figure 9b related to the term $B_{51}$ in the equation of motion. The wave radiation damping increased with the increasing wave frequencies due to pitch–surge coupling and Type-A had a larger value because of more volume exposed to incoming waves.

4.3. Wave Excitation Force

This section discusses the amplitude and phase of wave excitation forces and moment in surge, heave, and pitch directions. It is noted here that these results were obtained for unit wave amplitude.

Figure 10a shows the wave excitation force in the surge direction; the left figure is the amplitude and the right figure is the phase relative to the incident wave. It can be inferred that the amplitude is the largest for Type-A, and Type-C has a slightly larger value than Type-B. This tendency is similar to the added mass in the surge direction (Figure 8a). The trend of the amplitude, Type-A > Type-C > Type-B, can be explained based on the lateral projected area of the lower parts of the models. When wave frequency increases, the amplitudes almost increase, but the difference between Type-A and Types-B, -C becomes small. This is because the pressure of lower parts becomes smaller as the wave frequency becomes larger. As for phase angle, all types had the value of about 90 degrees, regardless of wave frequency. This is characteristic of spar and means that the wave excitation force becomes maximum when the model is located at the middle of front slope of wave, where the horizontal acceleration of water is maximum.

Figure 10b shows the wave excitation force in the heave direction. It can be inferred that the amplitude decreases to a low value around the wave frequency range of 3 to 3.5 rad/s for all the spar models. It can be explained based on the balance of upward pressure acting at the bottom of the spar and downward pressure on the joint part of circular cylinders. Further, in the case of Types-B and Type-C models, the balance of the upward pressure at the bottom surface of the spar models and the vertical downward pressures act on other horizontal surfaces. As the wave frequency increases beyond 3 rad/s, the amplitude of wave excitation force increases with the increasing wave frequency. Further, looking into at the phase angle, it can be inferred that phase angle changes to 180 degrees, i.e., reversal of phase to incoming waves around the wave frequency range of 3 to 3.5 rad/s for all the spar models. Therefore, it is necessary to ensure water-tightness of the upper structure as the relative water motion becomes large. From this, it can be said that it is important to understand not only the amplitude, but the phase angle of the wave excitation forces especially for the heave motion of the spar models.

Figure 10c shows the wave excitation moment in the pitch direction. It can be inferred that the amplitude increases with the increasing wave frequency. The amplitude is almost the same for all the spar models in the lower wave frequency range of 2.7 to 4.5 rad/s. However, as the wave frequency increases, the Type-A spar model was found to have a slightly larger moment compared to the other types. As far as phase angle is concerned, it was found that the phase angle was about 90 degrees, similar to surge direction.

The motion response of all spar models in waves was calculated using these parameters of added mass, wave radiation damping, and wave excitation force. Further, they are compared with the experiment results in the next section.

4.4. Motion Response of Spar Models—Validation

This section compares the experiment and calculation results for all the spar models and validates the developed calculation method. The motion response calculations were conducted after setting up the input parameters in the equation of motion as explained in Section 3.2. It is noted here that the surge amplitudes were obtained at the water surface level rather than at the center of gravity for comparison between the spar models. The heave and pitch amplitudes were obtained at the center of gravity. Figure 11, Figure 12 and Figure 13 show the validation results for Type-A, Type-B, and Type-C models, respectively.

Figure 11 shows the response amplitude operator (RAO) of surge, heave, and pitch motions plotted against wave frequency for the Type-A classic spar model. The experimental findings were compared with the calculations in no disk and with disk conditions (Figure 11a and Figure 11b, respectively). The effect of rotor disk was included by linearizing the drag force due to the disk. The surge, heave, and pitch amplitudes peaked close to the natural period of the floating body which was about 2 rad/s. The pitch–surge coupling effect was included in the calculation and, therefore, it can be seen that there was only small difference between the experiment and calculation results. Further, the heave amplitude reduced to a low value around 3 rad/s, as observed in the wave excitation force. The pitch RAO followed a similar trend of surge response due to pitch–surge coupling being dominant for the spar floating body. From these figures, it can be said there is a good correlation between experiment and calculation results for surge, heave, and pitch motions.

When the RAOs of no disk and with disk conditions are compared, it can be said that the disk induces some effect of damping and, therefore, the surge and pitch amplitudes with disk is less than the no disk condition. Even with disk condition, the experiment and calculation results agreed well for surge, heave, and pitch motions.

Figure 12 and Figure 13 show the response amplitude operator (RAO) of surge, heave, and pitch motions plotted against wave frequency for Type-B and Type-C spar models, respectively. The pitch RAOs of Type-B and Type-C spar model show that the calculation overpredicts the experiment results, especially at lower wave frequencies (2.7 to 3.5 rad/s). However, the difference was small and it can be said that there is a good agreement between the experiment and calculation results for both Type-B and Type-C spar models. As seen with the Type-A spar model, the surge and pitch amplitudes with disk were less than the no disk condition. It can be concluded that the calculation methodology can incorporate damping plates through these validations for all the spar models. After validating the calculation methodology, the effect of damping plates on pitch motion response was studied in the next section.

4.5. Effect of Damping Plates on Pitch Response

Now to understand the effect of damping plates on pitch motion, the pitch response of Type-B and Type-C spar models were compared with respect to the Type-A classic spar model. Figure 14a,b show the experiment and calculation results of pitch response, respectively, for the no disk condition.

It is clear from the experiment result (see Figure 14a) that the pitch response of both the Type-B model with horizontal plates, and the Type-C model with vertical plates, has the potential to reduce the pitch motion of the spar platform except at low wave frequencies. The calculation result (see Figure 14b) also predicts the same for Type-B. However, for the Type-C spar model, the pitch motion reduction was predicted for wave frequencies greater than 4.5 rad/s in the calculation result, unlike 3.5 rad/s in the experiment result. This can be due to the limitation of calculation methodology which is based on linear potential theory and does not consider non-linear wave damping or viscous effects. Nevertheless, the trend between the experiment and calculation results was similar, i.e., the potential of pitch motion reduction using horizontal plates (Type-B) was higher than using vertical plates (Type-C) for most wave frequencies. Similarly, Figure 14c,d show the experiment and calculation results for the with disk condition. It can be concluded that both Type-B and Type-C spar models have the potential to reduce the pitch response of the platform at most wave frequencies.

The moment of inertia and added moment of inertia of the proposed spar models (Types-B and C) were smaller than the conventional model (Type-A) and the wave excitation moment was almost the same amplitude for all the spar models. These quantities should lead to a larger pitch motion of Type-B and Type-C models. However, this does not happen due to the pitch–surge coupling effects, wherein all the three models have similar coupling coefficients and the surge amplitudes of the proposed spar models were significantly smaller than the conventional spar. From this, it can be said that the proposed damping plates reduce the pitch response of the spar model through pitch–surge coupling.

The hydrodynamic mechanism of damping plates can, therefore, be confirmed based on the calculation methodology implemented. The current calculation methodology is simplistic and quickly helps to evaluate the potential use of damping plates for a spar platform. In the future, as the size of wind turbine increases, the platform also becomes larger relative to ocean waves. This means that the platform response in large wave frequencies region becomes important. Therefore, for practical purposes, the current methodology is useful to find new geometries of floating structures of wind turbines, especially the design and operating range of large wave frequencies.

5. Conclusions

In this study, a spar platform with damping plates which has the potential to reduce pitch motion was investigated with a series of model experiments and numerical calculations. Three models, (i) Type-A, which is a classic spar with a water ballast at the lower part of the platform; (ii) Type-B, which is a spar with the ballast part replaced by horizontal damping plates; and (iii) Type-C, which is a spar with the ballast part replaced by vertical plates, were designed with similar stability and 1/120 model scale experiments were conducted in regular waves. A frequency-domain calculation method by potential theory was adopted and validated with the experimental findings. Specific findings are summarized as follows.

• Characteristic motions of spar in waves are presented. The center of rotation was low in wide wave frequencies; mostly lower than the bottom of the floater because surge and pitch motions were in phase. This can be explained by the congruence of phases of wave exciting force/moment in surge and pitch directions.
• Among 6DOF motions, pitch is generally considered the most important for FOWT not only on power generation but bending moment at the tower root. Focusing on pitch motions in regular waves, RAOs of the proposed spar types (Types-B and C) were smaller than the conventional type (Type-A) in most wave frequencies.
• Comparing proposed spar types with the conventional type, the moment of inertia and the added moment of inertia were smaller and wave exciting moments were almost the same amplitude. These quantities lead to larger pitch motion. However, as for pitch–surge coupling, the coupling coefficients were almost the same among the three types and the surge amplitudes of the proposed types were significantly smaller than the conventional type. From this comparison, it was found that the reformed ballast parts work to reduce pitch motion through pitch–surge coupling.
• It is concluded from the validation results that the proposed calculation methodology by potential theory can be used to incorporate the effect of damping plates in spar platforms. It is considered that this approach is useful when flow separation at the plates is limited due to small relative water velocity. This kind of methodology and hydrodynamic considerations can help find new geometries of floating structures of wind turbines.