Numerical Simulation of a Floating Offshore Wind Turbine Incorporating an Electromagnetic Inerter-Based Device for Vibration Suppression and Wave Energy Conversion

Asai, Takehiko; Tsukamoto, Shota; Nemoto, Yudai; Yoshimizu, Kenji; Watanabe, Urara; Taniyama, Yoshihiro

doi:https://doi.org/10.1155/2023/5513733

Structural Control and Health Monitoring

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Abstract Introduction Numerical Simulation Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5513733 | https://doi.org/10.1155/2023/5513733

Numerical Simulation of a Floating Offshore Wind Turbine Incorporating an Electromagnetic Inerter-Based Device for Vibration Suppression and Wave Energy Conversion

Takehiko Asai,¹Shota Tsukamoto,²Yudai Nemoto,³Kenji Yoshimizu,³Urara Watanabe,³and Yoshihiro Taniyama³

Academic Editor: Suparno Mukhopadhyay

Received25 Jan 2023

Revised19 Jul 2023

Accepted03 Aug 2023

Published14 Aug 2023

Abstract

Offshore wind turbines (OWTs) are considered vital to the promotion of the development of renewable energy. Especially, floating OWTs can be deployed over a larger area than bottom-fixed OWTs. The floating OWTs, however, are vulnerable to vibration induced by disturbances and require a backup power supply in the case of power outage. On the one hand, various kinds of inerter-based devices have been proposed especially for vibration suppression of civil structures subjected to earthquake loadings. Recently, combined with electromagnetic devices, the inerter technologies have also been applied in the field of vibration energy harvesting such as point absorber wave energy converters. Thus, this paper proposes a novel floating OWT consisting of two bodies combined with inerter-based power take-off (PTO) devices which accomplishes vibration suppression and wave energy conversion at the same time. To investigate the vibration suppression and energy conversion capabilities of the proposed floating OWT with a variety of inerter-based PTO devices for ocean waves, numerical simulation studies employing WEC-Sim are conducted, and the performance of each system is compared for regular and irregular waves. Results show that the proposed floating OWT with the appropriately designed inerter-based PTO devices for the incident wave period has great potential for both vibration suppression and wave energy conversion in a specific frequency range.

1. Introduction

In recent decades, social demand for renewable energy has been increasing worldwide due to concerns about global warming and depletion of fossil fuels. Wind power is expected to be one of the most promising renewable energy sources; however, the land area on which wind turbines can be built is limited. Therefore, offshore wind turbines (OWTs), which can harness the vast offshore wind power, have been attracting attention [1–4], and the OWTs can be divided into two categories: bottom-fixed and floating OWTs. Although the development of the bottom-fixed OWTs has progressed considerably and its practical application has already been achieved, the bottom-fixed ones can only be installed in waters up to about 50 meters deep. On the other hand, while the floating OWTs can be constructed in deeper waters, this type is still in the development stage, and there are many issues to be resolved for practical use.

One of the challenges of the floating OWTs to be addressed is the platform oscillation due to disturbances such as winds and waves. The same challenge can be seen in other civil structures subjected to earthquake loadings, and one of the strategies that have received the most attention in the field of structural vibration control recently is inerter [5, 6]. Originally, the inerter has been proposed by Smith [7] for vibration suppression in automobiles, and it produces a force proportional to the relative acceleration across its two terminals. The proportional constant between the force and the relative acceleration is called inertance, which has the same physical unit as mass. It should be also noted that a large inertance can be realized easily through a ball screw or a rack and pinion from relatively small actual masses. Some inerter-based devices have already been applied to structures on the ground, and their applications to floating structures such as floating OWTs for vibration suppression are currently being studied. For example, Hu et al. [8] investigated the structural control performance of various kinds of inerter-based devices located at the nacelle part by using the FAST-SC code [9]. Zhang and Fitzgerald [10] installed a tuned mass damper-inerter (TMDI) at a location close to the tip of each blade for the mitigation of edgewise blade vibrations, and the TMDI and a tuned mass damper fluid-inerter (TMDFI) are also applied by Sarkar and Fitzgerald [11, 12]. Moreover, Zhang and Høeg [13] applied a rotational inertia double-tuned mass damper (RIDTMD) to a spar-type floating OWT. Ma et al. [14, 15] also examined the heave motion mitigation of a semisubmersible platform with the inerter technologies.

In addition, the inerter-based devices are applied to energy harvesting from vibrations of structures. So far, various kinds of inerter-based devices utilizing electromagnetic machines, such as electromagnetic inertial mass damper (EIMD) [16, 17], tuned inertial mass electromagnetic transducer (TIMET) [18, 19], and electromagnetic tuned inerter damper (E-TID) [20], have been developed. The EIMD consists of an inerter and a motor which are arranged in parallel. In the TIMET, a spring is connected to the EIMD to increase the energy absorption capability by taking advantage of the resonance effect of the inerter. The E-TID also consists of an inerter, a spring, and a motor as well as the TIMET, and the configuration of these elements are different from that of the TIMET: the spring and the motor are parallelized and the inerter is added in series with them. The energy harvesting potential of these devices were already shown, and particularly, a point absorber wave energy converter (WEC) employing the TIMET was introduced by the first author [21–24], showing that the mechanism of the TIMET was able to increase the wave energy absorption capability compared to the conventional WEC.

On the other hand, hybrid systems combining floating OWTs with various kinds of WECs [25, 26] such as torus [27–30], point floater [31], flap-type [32], oscillating water column [33], and wavestar-type [34, 35] have been proposed to harness both wind and wave energy. The common concept of these studies is to submerge WEC devices in water, and these devices are designed to mitigate the vibration of the floating OWT by absorbing the wave energy.

The aim of this paper is to present a novel spar-type floating OWT incorporating electromagnetic inerter-based devices as power take-off (PTO) systems to achieve further improvements in the performance of power generation and structural control for wave disturbances and to assess and compare these performances through numerical simulation studies using WEC-Sim (Wave Energy Converter SIMulator) [36]. Unlike the other WEC devices introduced in the literature, in the proposed system, the spar-type floating OWT is separated into two parts: the submerged body and the superstructure, and is modeled as a two-degree-of-freedom (2DOF) in the heave direction. Then, these two bodies are connected by an elastic body, which can be modeled by a linear spring, and a PTO system combined with the inerter technologies. The spring stiffness is adjusted intentionally so that the second mode of the floating buoy itself amplifies the vibration, and then the PTO system is designed and installed so that the vibration energy is absorbed effectively and the amplified vibration being suppressed due to the second mode.

This article is organized as follows: first of all, analytical models for the proposed systems are developed and the response amplitude operators (RAOs) are calculated to examine the displacement response of the heave direction. Subsequently, numerical simulation studies for regular and irregular waves are carried out on a spar-type floating OWT model created based on the OC3 [37, 38] model, which is developed by the National Renewable Energy Laboratory (NREL). Finally, conclusions obtained from this research follow.

2. Problem Formulation

This section introduces the mechanism of the proposed floating OWT. Moreover, four types of PTO systems, three of which are coupled with electromagnetic inerter-based devices, for wave energy conversion and vibration suppression are reviewed briefly. Then, analytical models considering only the heave motion are derived, and the power generation of the PTO system is defined.

2.1. Floating OWT

The schematic diagram of the present floating OWT is illustrated in Figure 1. A typical spar-type floating OWT is shown in 1(a) where the whole structure consisting of a nacelle, a tower, and a platform can be modeled as a single rigid body. Also, any additional damping devices are not installed on the model shown in Figure 1(a). Generally, the natural period of a spar-type OWT in the heave direction is much longer than the wave period to avoid the resonance with incident waves. While as can be seen in Figure 1(b), the floating OWT proposed in this research is developed by separating the conventional floating OWT into two parts, submerged body and superstructure. The submerged body is a hollow platform and the superstructure is composed of the nacelle, tower, and platform. Part of the tower of the superstructure reaches into the interior of the hollow platform, and these two parts are connected by an elastic body and the PTO system, allowing each part behaves separately in the heave direction. The inherent damping also exists between the two parts. The concept of this floating OWT is developing a system consisting of two separate structures, which can be modeled as a 2DOF system in the heave direction. The proposed system is designed so that its second mode matches with the period of the incident waves. Then, the vibration energy excited resonantly in the second mode is absorbed by the PTO systems to harvest sufficient electrical energy from wave-induced vibration and to suppress the response of the floating OWT simultaneously without amplification due to the second mode.

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The analytical model considering only the heave motion is shown in Figure 1(c). The displacements of the submerged body and the superstructure are defined as and , respectively. Let the force produced by the PTO system be , then the equations of motion of the submerged body and the superstructure are given as follows:andrespectively, where is the mass of the submerged body, is the mass of the superstructure, is the stiffness of the elastic body inserted between the submerged body and the superstructure, is the unavoidable inherent damping coefficient caused between them, and is the additional linear damping coefficient considered in [38]. Furthermore, is the hydrodynamic force acting of the submerged body, which is described based on the linear potential wave theory [39, 40] bywhere is the excitation force, is the hydrodynamic force due to buoyancy, and is the radiation force.

The relationship between the excitation force in the heave direction and the amplitude of the incident wave is given in the frequency domain using a transfer function aswhere denotes the Fourier transform of a function .

Assume that the submerged body is a tapered cylinder as shown in Figure 1 and its diameter at the water level is . Thus, the hydrodynamic force becomes a linear function of aswhereand is gravitational acceleration and is the sea water density.

Finally, the radiation force is given bywhere and are the added mass and the radiation damping in the heave direction of the submerged body.

2.2. PTO Systems

To absorb the vibration energy and convert it to electrical energy, PTO systems are installed between the submerged body and the superstructure of the floating OWT introduced above. This paper investigates the three kinds of inerter-based electromagnetic devices, i.e., EIMD, TIMET, and E-TID. For comparison, an ordinary PTO system without an inerter mechanism is examined as well. The models of these four PTO systems are depicted in Figure 2.

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2.2.1. Electromagnetic Device without Inerter (2DOF)

When the PTO system without an inerter as shown in Figure 2(a) is installed between the submerged body and the superstructure, the device force becomeswhere is the damping coefficient of the motor for the PTO system. Thus, equations (1) and (2) can be written asrespectively.

Substituting, equations (4), (5), and (7) and taking the Fourier transform of these equations giveswhere each element of is given as

This case is referred to 2DOF model in this article.

2.2.2. Electromagnetic Inertial Mass Damper (EIMD)

Then, the EIMD [16, 17] shown in Figure 2(b) is applied. In the same way as the 2DOF model, the device force is given by

The system still can be expressed as a 2DOF system, and the equations of motion are given byrespectively.

The Fourier transform of these equations has the same form as equation (10); however, each element of is given as

2.2.3. Tuned Inertial Mass Electromagnetic Transducer (TIMET)

A model for the floating OWT with the TIMET [18, 19] shown in Figure 2(c) is developed. Let the deformation of the inerter be and consider the equilibrium of force in the TIMET. Then, the force produced by the TIMET can be expressed as

Substituting (15) into equations (1) and (2) yieldsandand rewriting equation (15) gives

Thus, the degree of freedom for this system becomes three.

Taking the Fourier transform of equations (16)–(18) giveswhere each element of is given as

2.2.4. Electromagnetic Tuned Inerter Damper (E-TID)

Finally, the E-TID [20] shown in Figure 2(d) is installed on the floating OWT. As in the previous TIMET case, the system becomes a three-degree-of-freedom model, and the device force for the E-TID is expressed aswhere is the displacement of the inerter as in the TIMET case. Then, the equations of motion are as follows:and

The Fourier transform of equations (22)–(24) has the same form as equation (19), and each element of is given as

2.3. Power Generation Objective

To assess the power generation capability, the power generated by the PTO system is defined here. Let the back-EMF constant of the motor of the PTO system be , and assume the relationship between the current and the voltage of the motor to bewhere is the electrical resistance of the circuit including the internal resistance or coil resistance in the motor. Thus, the damping coefficient of the motor, , is given by [41]

This equation shows that is variable by controlling the electrical resistance.

The absorbed mechanical energy by the PTO system is defined as

Strictly speaking, to calculate the generation power, the power loss caused by the internal resistance or coil resistance in the motor should be considered. However, with a motor of this size, the power loss is negligibly relative to the absorbed power. Therefore, for the sake of brevity, the power generation is defined as the absorbed energy, i.e.,

Moreover, the average power generation from time 0 to is defined as

3. Numerical Simulation

In this section, numerical simulation studies are performed using WEC-Sim to evaluate the structural control performance and the wave energy conversion efficacy of the proposed system to various wave inputs. WEC-Sim is an open-source code developed in MATLAB/Simulink [42] for simulating the behaviors of WECs.

3.1. Parameters

The parameters for the spar-type floating OWT are decided based on the OC3 whose total length is 210 m developed by the NREL [37, 38] and are summarized in Table 1. Since the detailed shape of the OC3 platform is not given in the literature, this research assumes that the part that submerged in the water is a perfect cylinder except for the tapered portion of the platform and that the mass distribution of this part is uniform. These assumptions lead to an error in buoyancy; thus, the parameters in Table 1 are slightly different from the values in the literature to adjust the draft depth in static equilibrium state.

The transfer function vector for the wave excitation force , added mass matrix , and radiation damping matrix for the WEC-Sim simulation are produced by the WAMIT software [40].

3.2. PTO System Design

To examine the effect of the inertance and the damping coefficient of the inerter-based PTO systems shown in Figure 2 on the vibration response and power generation efficacy, three different values for and , respectively, are adopted, and then nine cases (Case A through Case I) summarized in Table 2 are applied for each PTO system. The stiffness of the spring used in the TIMET and E-TID is fixed to a constant value and given in Table 2 as well. These values are determined by trial and error so that the response amplitude operator (RAO) [43] introduced later is suppressed to waves with a period of around 10.0 s successfully. Note that the 2DOF model have neither a spring nor an inerter; thus, for the 2DOF model, cases A, B, and C are identical, and similarly cases D, E, and F and cases G, H, and I are identical, respectively. Also note that the EIMD device does not have a spring.

3.3. Response Amplitude Operator (RAO)

Prior to performing the analysis in WEC-Sim, the RAO [43] for the heave direction is derived here. The RAO is a transfer function showing the relationship between the incident wave amplitude and the response displacement amplitude in the frequency domain. Let the inverse of derived in the previous section be , then the RAO for the superstructure when the abovementioned four devices are installed on the floating OWT can be expressed aswhere is the element of the second row and first column of .

The RAOs obtained from equation (31) are shown in Figure 3 as well as the single-degree-of-freedom (SDOF) model for comparison, in which the submerged body and superstructure are rigidly connected without any PTO systems, i.e., the original spar-type floating OWT, as shown in Figure 1(a). The mass of the SDOF model is assumed to be . Note that the RAOs for the SDOF model in all cases in Figure 3 are identical. As stated previously, for the 2DOF model, the RAOs for cases A, B, and C, cases D, E, and F, and cases G, H, and I are identical, respectively, because the 2DOF model does not have an inerter.

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It is observed in Figure 3 that the 2DOF model increases the RAO compared to the SDOF model due to the excitation in the second mode as intended. Then, it can be confirmed that the floating OWT with the inerter-based PTO systems such as EIMD, TIMET, and E-TID have the potential of suppressing the RAO and that the effective period range depends on the inertance value. Especially, the TIMET and the E-TID work well and show better performance than the SDOF model in the specific period range due to the additional spring, while these PTO systems have a risk of increasing the RAO outside the intended period range. From these results, a mechanism that controls the inertance for the EIMD, TIMET, and E-TID according to the change of the incident wave period is considered very promising.

3.4. Time History Analysis in WEC-Sim

Time history response analysis is conducted employing WEC-Sim here. The wave excitation force, added mass, and radiation damping obtained by WAMIT are used. Regular and irregular waves with a duration of 1000 s are input as incident waves. Although the proposed systems aim to suppress vibration of the floating OWT in the heave direction, the influences in other directions are examined. The analytical model developed for the WEC-Sim simulation studies is assumed to be symmetric; thus, without loss of generality, the analysis in WEC-Sim can be constrained to three directions: heave, pitch, and surge. For the simulation, the coordinates (, , ) of the center of gravity of the submerged body and the superstructure are set to (0, 0, −89.9 m) and (0, 0, 64.0 m), respectively, and the pitching moments of inertia are kg·m² and kg·m², respectively. Note that the coordinate system in WEC-Sim is defined as shown in Figure 4, in which the origin is placed on the water line and the incident wave direction defined in the same direction as the -axis is also depicted.

3.4.1. Regular Wave

To investigate the performance for different wave periods, regular waves with a period of 9.0 s, 10.0 s, and 11.0 s are used as incident waves. The wave height of these waves is set to 6.0 m. These conditions are determined with reference to various literature [37, 38, 43]. The RMS (root mean square) response values of the heave, pitch, and surge of the superstructure calculated by WEC-Sim for 1000 s are summarized as well as the average power generation calculated by equation (30) in Tables 3–5. In particular, the heave response ratio to the SDOF model and the power generation are compared in Figure 5. Moreover, the time histories for 100 s of the heave response and power generation for Case E to each input are compared in Figures 6–8, respectively.

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From Tables 3–5 and Figure 5, the same trend as in Figure 3 can be confirmed for the heave direction, i.e., the inerter-based PTO systems show great potential for heave response reduction in the specific period range when the PTO parameters are appropriately designed. For example, for the waves of period 10.0 s, cases C, F, and I of the EIMD, cases B and C of the TIMET, and cases of B, C, E, F, H, and I of the E-TID are superior to the SDOF model in the heave response. Especially, the PTO system utilizing the E-TID works better than the other systems in reducing the heave response when the parameters of the device are well matched to the incident wave period, while the E-TID has a risk of amplifying the heave response drastically to the unexpected incident wave period and is inferior to other schemes in power generation efficiency. Thus, it can be found out that the E-TID is very sensitive to the time period of the incident waves. This corresponds to the results reported in [44]. Hence, the inertance value of the E-TID should be designed carefully. Moreover, it can be found that the proposed systems utilizing an inerter have little effect on the responses in the pitch and surge directions. These results also show that a more effective system for vibration suppression over a wider period range can be developed if the inertance is variable and controllable based on the incident wave period. Also, it is observed that the power generation performance of the systems with the inerter-based PTOs is not as good as that of the 2DOF system, even when the response is reduced successfully. This is because the response reduction depends not only on energy absorption but also on the input energy to the floating OWTs from the waves. As explained previously, the inerter-based PTOs are designed so that the response is reduced, leading to less input energy and less power generation. However, the proposed systems realize that wave energy conversion is not possible with the conventional SDOF model in addition to the response reduction.

The time histories of the heave response and the power generation for 100 s to each regular wave input are compared in Figures 6–8, respectively. Generally, the wave height is defined as the vertical distance between the crest and the trough of a wave, so it is equivalent to twice the amplitude of the sinusoidal wave. Taking this into account, the heave responses in time histories agree very well with the RAO, as shown in Figure 3.

3.4.2. JONSWAP Spectrum Wave

Then, irregular waves with the JONSWAP spectrum [43] are employed to evaluate the proposed system. The JONSWAP spectrum in WEC-Sim is defined as [36]where is the significant wave height, is the peak wave frequency (), is the peak wave period, is the wave frequency, and the peak-shape parameter exponent, , is defined as

The other peak-shape parameter, , is determined based on the following equation:

In this study, two irregular waves produced by the abovementioned equations are used as JONSWAP spectrum waves. The first one has a significant wave height of 7.0 m and a peak wave period of 9.0 s, and the second one has a significant wave height of 7.0 m and a peak wave period of 10.0 s. The peak-shape parameter, , of each case becomes 5 and 4.07, respectively, from equation (34). The spectra of these two JONSWAP waves are compared in Figure 9, which shows that the JONSWAP spectrum contains broadband waves.

In the same way as with the regular waves, the RMS responses and average power are summarized in Tables 6 and 7, the heave response ratio and power generation are compared in Figure 10, and the time histories of the heave response and power generation are shown in Figures 11 and 12. As can be seen, the proposed systems utilizing the inerter-based PTO systems deteriorate the heave response in most cases compared to the SDOF model. This is because the irregular waves contain the waves in the period range where the PTO system does not suppress the heave response, as shown in Figure 3. Particularly, the E-TID did not work at all to the waves with a peak wave period of 10.0 s due to the long-period or low-frequency wave components included in the JONSWAP spectrum. However, considering the additional benefit of the WEC and the fact that the heave response does not deteriorate significantly, the EIMD and the TIMET still have sufficient advantages over the conventional floating OWT even to waves containing broadband components.

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4. Conclusions

This article introduced the novel floating OWT coupled with inerter-based PTO systems. To investigate the effectiveness, the three well-known configurations of the electromagnetic inerter-based devices, i.e., EIMD, TIMET, and E-TID, were applied to the PTO system for the spar-type floating OWT separated into two parts. Numerical simulation studies implemented in WEC-Sim showed that the inerter-based PTO systems worked well on the present floating OWT to improve its structural vibration suppression in the heave direction and wave energy harvesting capabilities when the parameters of the PTO systems were appropriately designed for the particular incident wave period. In other words, the performance was highly dependent on the incident wave period and the inertance value; thus, it was observed that the proposed system had a risk of deteriorating the structural control performance for waves with unexpected periods.

To address this issue, not only optimal design methods for the present OWT but also variable inerter mechanisms, which can change the inertance value, should be developed and applied to the inerter-based PTO systems for future work. This improvement is especially necessary for the E-TID, which is sensitive to the inertance value. In addition, to control both inertance value and damping coefficient of the PTO system for better performance, algorithms based on feedback control strategies using the measured wave data on-site to adapt the change of the dominant period of the incident waves should be developed. Moreover, other types of floating OWTs such as semisubmersible type should be used for the proposed inerter-based PTO systems, and their structural control performance in the pitch and surge directions when not only waves but also wind loadings are applied should be evaluated as well. Finally, experimental verification is also necessary to show the efficacy of the proposed systems and find out practical difficulties.

Data Availability

The simulation data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was partially supported by Japan Science and Technology Agency (JST) (grant no. JPMJFR215I). This support is gratefully acknowledged.

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Copyright © 2023 Takehiko Asai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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