Enhanced power absorption of a point absorber wave energy converter using a tuned inertial mass

A novel point absorber wave energy converter with a tuned inertial mass (TIM), which is capable of signiﬁcantly increasing the energy absorption and broadening the e ﬀ ective bandwidth, is proposed in this paper. The mechanism of the TIM has originally been introduced in the ﬁeld of civil engineering as a passive energy absorber for structures subjected to external loadings such as earthquakes. It relies on attaching an additional tuning spring and a rotational inertial mass to the primary system, to improve the energy absorption performance by amplifying the displacement of the damper. Thus, considering typical point absorbers modeled as a mass-spring-dashpot system similar way to civil structures, the application of the TIM to wave energy converters can be expected to have a signiﬁcant effect. In this paper, numerical investigation on the power generation performance of a point absorber with the TIM is conducted under random sea waves. The amplitude response and power generation performance are compared with the conventional point absorber, considering both non-resonant and resonant buoy cases. It is shown that by properly designing the tuning spring sti ﬀ ness and generator damping, the rotation of the generator can be ampliﬁed compared to the buoy, increasing the power absorption drastically.


Introduction
First stimulated by the oil crisis 40 years ago, harvesting energy from ocean waves is now considered as an alternative approach to meet renewable energy targets [1,2].The enormous wave power potential has attracted engineers worldwide with more than 1,000 patents in Japan, North America, and Europe [3], and among them are oscillating water column, oscillating body, and overtopping devices [4,5,6,7,8,9,10].Out of these, the point absorber classified as the oscillating body, is considered one of the most promising concepts which make up a great proportion of existing full-scale prototypes.It consists of a floater with dimensions much smaller than the incident wave length, which converts the motion of ocean waves to produce power.Compared to other concepts, point absorbers can exploit more powerful wave regimes available in deep offshore regions, and absorbs energy from all directions.Moreover, the buoys can be combined in arrays to produce sufficient energy for commercial exploitation.
Until now, a great amount of effort has been devoted to improving the efficiency of point absorbers based on mathematical models [11].It is well known that a point absorber in resonance with the incoming wave has significantly high power absorption due to its enhanced amplitude [12].However, in practice, this frequency matching meets with serious difficulty because real waves are not single frequency, and the natural frequency of the buoy tends to be much higher than typical ocean wave frequencies for a reasonable size buoy [4,13].Therefore, methods to force the system into resonance via active control have been considered.Latching the point absorber at fixed positions to achieve phase control was first proposed by Budal and Falnes [14] and the phase control strategies including latching control and reactive control continue to be an active area of research [15,16,17,18,19,20].This approach requires the prediction of the incoming wave some time in the future, along with solving the complicated control problem to determine the latched time intervals.On the other hand, methods to simply engineer the frequency response of the point absorber have also been considered.Shadmen et al. [21] presented a geometrical optimization method based on tuning the system to oscillate in the range of predominant sea states.Engstrom et al. [22] presented phase control by using a supplementary submerged body to shift the natural frequency of the buoy to coincide with typical sea states.Both of these studies discussed that enhanced amplitude response of the buoy leads to a narrow resonance bandwidth which is in conflict with the desire to correspond to the wide spectrum of ocean waves.Thus, the compromise between maximum power absorption and broad resonance bandwidth has always been an issue associated with the optimization of point absorbers.
In this study, a point absorber with a tuned inertial mass (TIM) is proposed to increase the power absorption and broaden the effective wave frequency range.The TIM has a different configuration from the traditional tuned mass damper (TMD) [23] and the mechanism has originally been introduced in the field of civil engineering as a passive vibration control device for structures subjected to exter-nal loadings such as earthquakes [24,25,26].It relies on attaching an additional tuning spring and a rotational inertial mass to the primary system, to amplify the displacement of the viscous damper by taking advantage of the resonance effect.In the theory of point absorbers, it is typical to model the power take-off (PTO) system as a linear damper [27], resulting in a mass-spring-dashpot system.Hence, the application of the TIM to point absorbers can be expected to have a significant effect in terms of energy absorption.In contrast to previous studies which couple the PTO with the buoy's oscillation, this study inserts the tuning spring which decouples the PTO from the buoy's oscillation and couples with the oscillation of the rotational inertial mass.Therefore, by appropriately designing the tuning spring, rotational inertial mass, and damping coefficient, the rotation number of the rotational inertial mass can be amplified compared to the buoy, which increases the displacement of the PTO to achieve substantial power absorption.
The objective of this paper is to verify the effectiveness of applying the TIM to a point absorber.First, in Section 2, the theoretical models of the conventional system and proposed system with the TIM are developed, following the modeling of stochastic sea states using the JONSWAP spectrum.Then, in Section 3, the state-space representation is given to evaluate the amplitude response along with the power generation.Finally, in Section 4, numerical studies are carried out focussing on the optimization of design parameters, amplitude response, and power generation of the proposed system compared with the conventional system.

Point absorber with tuned inertial mass
We begin by reviewing the conventional point absorber illustrated in Fig. 1(a).A semi-submerged cylinder with diameter D and draft L is considered as the wave interacting part of a point absorber.The generator is anchored rigidly to the ocean floor, and the supporting spring with stiffness k s is used to prevent the wire from sagging.For simplicity, we consider the heave motion only as this becomes dominant for the power extraction of wave energy [28].
Let z be the vertical displacement of the buoy from its equilibrium state.Since the displacement of the buoy is coupled with the rotation of the generator, the conventional point absorber can be modeled as a single-degree-of freedom (SDOF) system with the following equation of motion:  where M is the buoy mass, m s is the inertial mass of the generator and pulley, c s is the mechanical damping including friction, u is the electromagnetic damping force which acts as a control force given by and f w is the hydrodynamic force acting on the cylinder.The corresponding model is shown in Fig. 1(b).
Based on linear potential wave theory, the hydrodynamic force f w is described by where f a is the excitation force, f b is the hydrostatic force due to buoyancy, and f c is the radiation force.The relation between the excitation force f a and the amplitude of the incident wave a is given in the frequency domain using a transfer function F a (ω) as fa (ω) = F a (ω)â(ω) Note that f (ω) denotes the Fourier transform of a function f (t).The hydrostatic force becomes a linear function of z given as where K w is given with the gravity acceleration g and sea water density ρ by and the radiation force f c is given as where m a (ω) is the added mass and c r (ω) is the radiation damping.Next, to increase the power absorption and widen the effective bandwidth, a point absorber with TIM is proposed.As illustrated in Fig. 2(a), a tuning spring is installed between the buoy and generator, and a relatively small physical mass which rotates with the generator is attached.When the vertical motion is transformed into rotary motion, the small physical mass can produce an amplified equivalent mass effect on the order of a thousandfold due to the rotary inertia [24].As shown in Fig. 2(b), the proposed system is modeled as a two-degreeof-freedom (2DOF) system, noting that the rotational displacement of the inertial mass is coupled with the generator.
Defining z, z s , and z t as the displacement of the buoy, rotational inertial mass, and the elongation of the tuning spring, respectively, the relationship among these variables are given by z = z s + z t (8) The equations of motion for the proposed TIM system are derived as where f t is the force from the tuning spring and the the electromagnetic damping force is defined by u = C PT O żs (11) in this case.Also, denoting the stiffness of the tuning spring as k t , this force becomes proportional to the relative displacement given as

Power take-off system
In this study, the generator is assumed to be a three-phase permanent magnet synchronous machine, interfaced with a centralized DC power bus.For this assumption, it is reasonable to assume that the quadrature field is explicitly controlled to regulate power conversion while the direct stator field is maintained at zero.Thus, the three phase voltage and current vectors are transformed to "quadrature components", i.e., effective scalar quantities are used for the back-EMF v and corresponding current i.More details can be found in [29,30].Assuming an ideal generator with linear behavior and minimal core loss, results in linearity between the back-EMF and the velocity coupled with the generator.Therefore, the equation is defined separately for the conventional system and TIM system, given as respectively, where K e is a constant associated with the back-EMF of the generator.By reciprocity, the electromagnetic force and generator current has the following linear relationship In the study of wave energy converters (WECs), it is common to assume viscous dampers in place of the PTO system.This constitutes the imposition of a feedback law where Y is the admittance of the generator.Applying this to Eq. ( 14) with Eq. ( 13) yields for the individual systems.Thus the generator damping C PT O for both cases can be given as expressing how the generator damping, i.e., the electromagnetic damping force u is controlled by the admittance Y.

Stochastic sea state model
In simple theoretical models of WECs, it is typical to assume the incident waves to be regular.For a more realistic model, irregular waves are used with time-domain analysis which requires much more computing time [4].An alternative method with less computation for modeling true sea states is the stochastic modeling.We assume the wave amplitude a(t) to be a stationary stochastic process with spectral density S a (ω) and variance given by Characterizing S a (ω) by the JONSWAP spectrum [31] with peak wave period T p , significant wave height H s , and peak enhancement factor γ derives the following where and The peak enhancement factor γ is constrained by 1 ≤ γ ≤ 3.3, with γ = 1 describing a fully developed sea.
To model the wave amplitude, we find a finite-dimensional noise filter such that its power spectrum is close to the JONSWAP spectrum, i.e., S a (ω) ≈ |F w (ω)| 2 , for a unit intensity white noise input.It should be noted that we make According to the simplified procedure advocated by Spanos [32], F w can be approximated by a fourth-order controllable canonical form of where the filter parameters a 1 , a 2 , a 3 , a 4 , and c 3 are chosen to minimize the meansquare error 2 dω, while constraining a 1 through a 4 so that the system poles are in the open left half plane.Fig. 3 shows an example of a JONSWAP spectrum for T p = 6 s, H s = 1 m, γ = 1 and its fourth-order finitedimensional approximate system.We can confirm in the figure that the fourthorder F w estimates the JONSWAP spectrum very well.

State-space representation
In this section, to evaluate the amplitude response and assess the power generation for stochastic sea states, a state-space form of the proposed device with TIM is developed.The derivation of the state-space form for the conventional point absorber can be found in [33], which is not delved into in this paper.

Amplitude response of buoy and rotational inertial mass
In the following, the equations of motion for the buoy and rotational inertial mass are expressed in state-space form separately, and then augmented to assess the amplitude response relating the wave elevation.
Taking into account the hydrodynamics acting on the buoy, we take the Fourier transform of Eq. ( 9) with Eqs. ( 3), ( 4), (5), and (7) inserted, which gives the transfer function relating a and f t to z as where The infinite-dimensional systems G a and G f are approximated into finite-dimensional systems, i.e., We must note, the function F a (ω) in Eq. ( 25) is noncausal which will be problematic when approximating G a (ω) by a finite-dimensional state-space.Therefore the technique of spatial delay proposed by Falnes [34] is used, defining a as the wave amplitude at a distance of d in front of the buoy.Once Eq. ( 27) is obtained, the identified systems are augmented as where the augmented matrices A m , B m , E m , and C m are and the equivalent state-space representation is On the other hand, the equation of motion for the rotational inertial mass in Eq. ( 10) can be transformed into state-space form directly.Defining the state vector as where Now that the state-space form for the buoy and rotational inertial mass are determined, Eqs. ( 30) and ( 32) can be augmented by defining where The generator current and the force from the supporting spring can be expressed using the state variable x h as where Thus, when the design parameters Y and k t are determined, the amplitude response of the buoy and rotational inertial mass to the wave elevation can be assessed by defining a transfer function for each.Substituting Eqs. ( 36) and (37) into Eq.( 34) yields the closed loop system where Therefore, the transfer function from the wave elevation to the buoy displacement is given as Similarly, the transfer function from the wave elevation to the rotational displacement of the inertial mass is given as

Power generation
The stochastic sea state model in Eq. ( 22) combined with the WEC dynamics in Eq. ( 34) yields the following augmented system, with state vector where and w(t), the input to the noise filter, is white noise with unit intensity.Eqs. ( 36) and (37) can be rewritten as where In this paper, the total power generation is defined as the extracted power minus the electrical loss [29], i.e., P g = −iv − P d (50) where P d is the electrical loss dominated by conductive dissipation.While the expression for P d depends on many parameters of the electronic hardware, we make the simplifying assumption that the current-dependent loss is resistive, i.e., where R includes the stator coil resistance of the generator and an approximate transmission resistance for the drive.From v = Cx and Eq.(51), Eq. ( 50) would be To maximize the averaged value of Eq. (52) denoted by Pg , the admittance Y and the tuning spring stiffness k t are treated as control gains subjected to a white noise input as in [35].Substituting Eqs. ( 47) and (48) into Eq.(45) yields the closed loop system ẋ = (A − YBC + k t FT)x + Gw (53) In this study, the static admittance control strategy introduced in [33] is utilized for the current i.Thus both Y and k t are treated as constant values.Therefore in stationary stochastic response, the average power generation Pg is given by [36] where the covariance matrix S = ε[xx T ] is computed as the solution to the Lyapunov equation For an ideal system, the admittance value to control the input current to the generated is restricted by to assure the definiteness of S. Thus, the objective becomes the optimization of Y and k t to maximize Pg .

Numerical simulations
To validate the power generation performance of the point absorber with TIM proposed in Fig. 2, numerical studies are carried out and compared with the conventional SDOF system illustrated in Fig. 1.A reasonable size cylindrical buoy with diameter 5 m, draft 1 m, in water depth 30 m is considered.The hydrodynamic coefficients are calculated with the software WAMIT [37] which uses boundary element method based on linear potential theory.The resulting added mass m a (ω), radiation damping c r (ω), and the magnitude and phase of the transfer function F a (ω) defined by Eq. ( 4) are shown in Fig. 4. The parameters for the generator and buoy used in this study are summarized in Table 1.
Fig. 5 shows the frequency response data G a and G f given by Eq. ( 25) and ( 26), respectively, and the finite dimensional approximations expressed by Eq. (27).For accuracy, G a is approximated with 6 zeros and 7 poles, and G f is approximated with 3 zeros and 4 poles.The wave amplitude a is taken to be the wave amplitude d = 10 m ahead of the buoy in the propagation direction.It can be seen that the approximated model matches the frequency response data very well over the dominant wave frequencies.Previous studies which consider irregular waves typically select some number of sea states as representatives of a deployment site.While, in this study, a more comprehensive analysis is carried out, focussing on the variation of peak wave periods since the system's amplitude response strongly depends on this characteristic parameter.Therefore, stochastic sea state models are determined for each sea state with peak wave period ranging from 2 s to 12 s and H s = 1 m, γ = 1 for the JONSWAP spectrum defined by Eq. ( 19).

TIM system 4.1.1. Design
For the proposed TIM system, the design parameters Y and k t must be decided for the rotational inertial mass employed.In this study, a rotational inertial mass of m s = 8264 kg is examined which corresponds to 30% of the effective oscillating mass M + m ∞ where m ∞ denotes the infinite frequency added mass having the value of 23,547 kg.This large inertial mass can be realized by a relatively small physical mass due to the rotational amplifying effect [24].
It is well understood that the damping coefficient C PT O strongly affects the magnitude of the generated power.This can be adjusted through the admittance Y of the electronics, thus the damping can be optimized for each sea state.On the other hand, adjusting the stiffness of the tuning spring after deployment is regarded as unpractical.Therefore, the value for k t is decided so that the power generation is maximized for the predominant sea state assumed as JONSWAP spectrum with T p = 6 s.The optimized value of k t is obtained through the contour plot shown in Fig. 6, which demonstrates how the admittance and tuning spring stiffness relate to the power generation.The upper limit of k t is set to 1.0×10 5 N/m which is large enough to consider the spring to be rigid, while the admittance is constrained by Eq. ( 56).The contour plot shows a clear peak at k t = 1.72 × 10 4 N/m and Y = 0.0044 Ω −1 giving maximum power of Pg = 1795 W. Hence, the optimum tuning spring stiffness is set to k * t = 1.72 × 10 4 N/m and held fixed for the varying sea states.
In the following simulation, the effect of adjusting the rotational inertial mass after deployment is also explored.Therefore, two more values, m s = 5509 kg  and 11019 kg corresponding to 20% and 40% of the effective oscillating mass are considered.Note, the value of m s can be adjusted by either changing the attached mass or the radius in which it rotates.

Result
The magnitudes of the transfer functions H b and H r defined by Eqs. ( 43) and (44) with the obtained optimum value k * t = 1.72 × 10 4 N/m are compared in Fig. 7.The damping coefficient is set to the optimum value for T p = 6 s in each plot.For the SDOF system, an inertial mass of m s = 50 kg caused by the generator and pulley is assumed.Fig. 7 shows that the amplitude response of the rotational inertial mass is significantly higher than the buoy response, indicating that the tuning spring amplifies the oscillation of the rotational inertial mass compared to the buoy.This leads to increased amplitude and velocity of the generator movement due to its rigid connection with the rotational inertial mass.Moreover, the frequency for optimum amplitude response of the TIM system decreases with increasing m s .This points out that even when the buoy's natural frequency is apart from the dominant wave frequency, enhanced amplitude response can be gained, and shifting of the frequency is possible through the value of the rotational inertial mass.
Fig. 8 shows the power generation for various sea states, with the optimum damping coefficient given below.It can be clearly seen that all three TIM systems perform better than the SDOF system, due to the increased amplitude and velocity of the rotational inertial mass and generator.This results in lowering the optimum damping coefficient, opening up the possibility to have small generator units.Furthermore, when the value of m s is changed, the peak of the power curve is shifted, which indicates that adjusting the rotational inertial mass allows the TIM system to correspond to the variability of T p .

Phase controlled TIM system 4.2.1. Design
Considering the significant effect of the TIM system observed from the amplitude response, a superior effect can be expected for when the oscillation of the buoy itself is large, i.e., when the buoy resonates with the wave.In the following example, the effectiveness of the TIM system is validated for when the buoy's natural frequency coincides with the dominant wave period.To shift the heave oscillating period, we assume phase control is applied by adding a fully submerged body [22,38].The supplementary inertia due to the submerged body mass and added mass allows the buoy to resonate with the dominating sea state.By placing the submerged body at sufficient depth, the radiation damping and excitation force remains the same with the case of the single cylinder, which is explored before.The performance of the TIM system is compared with the SDOF system, having the buoy for both systems to be in resonance with the wave motion.Thus, the supplementary inertia due to the submerged body mass and added mass is set to 200 × 10 3 kg to make the buoy's oscillation tuned to the sea state of T p = 6 s.Fig. 9 shows the finite dimensional approximation of the frequency response data G a and G f with the same orders, displaying a distinct peak at 0.9 rad/s due to the resonance of the buoy.For the rotational inertial mass, the same three values (i.e., m s = 5509 kg, m s = 8264 kg, and m s = 11019 kg) are considered.The optimization of the design parameters are carried out in a similar process, following the amplitude response and power generation.
The stiffness of the tuning spring k t is optimized through the contour plot given in Fig. 10.A sharper peak can be observed at k t = 6.06 × 10 3 N/m and Y = 0.0036 Ω −1 giving maximum power of Pg = 17.1 kW.Hence, the optimum tuning spring stiffness is set to k * t = 6.06 × 10 3 N/m.

Result
The amplitude response of the buoy and the rotational inertial mass are shown in Fig. 11.The solid line in Fig. 11 the SDOF buoy is greatly enhanced due to phase control, while the bandwidth has become narrow.This has always been an issue associated with the design of point absorbers since a compromise must be made between maximum amplitude response and wide resonance bandwidth over the dominant wave frequencies.On the other hand, the significant effect of the TIM system is that enhanced amplitude response and widened bandwidth is gained at the same time as shown in Fig. 11(b).Especially for the TIM system with m s = 8264 kg in which the tuning spring was optimized, the benefit of 2DOF is gained, considerably widening the bandwidth.We must emphasize that for the proposed TIM system, increased amplitude response of the rotational inertial mass is desired since the generator is coupled to its displacement.While the buoy's amplitude response is reduced for the TIM system, the amplitude response of the rotational inertial mass is significantly increased.This indicates that a large amount of energy from the wave to the buoy is successfully transferred to the rotational inertial mass for the TIM system.The power generation of the two systems with phase control is compared in Fig. 12.A similar feature with the amplitude response can be observed, with the TIM system having increased power generation and widened bandwidth.Particularly for the sea state in which resonance is gained, the power generation for the TIM system is increased to more than two times the power of the SDOF system.Contrary to the non-resonant case, the effect of adjusting the rotational inertial mass can not be observed, which indicates that designing k t for a particular m s value is sufficient to gain the benefit of the TIM system.As with previous results, a decrease in optimum damping coefficient can be observed for the TIM system, while the optimum damping is less sensitive to the variation of the sea states.

Conclusions
A theoretical model for a point absorber with tuned inertial mass has been proposed.Numerical simulations in stochastic sea states using the JONSWAP spectrum were carried out to compare the power generation of the proposed device with the conventional SDOF point absorber.For the TIM system with the buoy's natural frequency apart from the dominant wave frequency, increased power gen- eration was gained along with the shifting of the power curve depending on the rotational inertial mass value.On the other hand, for the TIM system with a resonant buoy, a significantly high amplitude response of the rotational inertial mass was gained, while the response of the buoy was reduced.This indicated that substantial wave energy was transferred to the rotational inertial mass, leading to increase the power generation to more than two times the power of the conventional system at the resonance frequency.
For future work, methods to determine the optimum rotational inertial mass should be established, along with its sensitivity to the wave spectrum.In addition, algorithms to control the stiffness and damping of the PTO system with regard to the change of the sea state are desired.Furthermore, experimental investigation [39] to assess the feasibility of the device under real sea condition is required.This includes the consideration of locking the PTO system in survival mode and understanding the buoy response to highly non-linear extreme waves.

2 Figure 3 :
Figure3: Example of Jonswap spectrum with T p = 6 s, H s = 1 m, γ = 1 (solid) and the spectrum of the fourth order finite dimensional approximate system (dashed).

ωFigure 4 :
Figure 4: Hydrodynamic parameters for the heave mode of a circular cylinder with radius 2.5 m, draft 1 m, in water depth 30 m: (a) Added mass, (b) Radiation damping, (c) Magnitude of F a (ω), (d) Phase of F a (ω).

Figure 6 :
Figure 6: Contour plot of power generation showing the effects of the tuning spring stiffness k t and admittance Y.The plots are for m s = 8264 kg with sea state T p = 6 s, H s = 1 m, γ = 1.

Figure 7 :
Figure 7: The amplitude of the transfer functions when the tuning spring stiffness is k * t = 1.72×10 4 N/m and the damping coefficient C PT O is set to the optimum value for T p = 6 s, (a) H b (ω), (b) H r (ω).

Figure 8 :
Figure 8: Power generation with optimum damping for each sea state.The tuning spring for the TIM system is set to k * t = 1.72 × 10 4 N/m.

Figure 10 :
Figure 10: Contour plot of power generation showing the effects of the tuning spring stiffness k t and admittance Y.The plots are for TIM system applied to a resonant buoy with m s = 8264 kg and supplementary inertia of 200 t.Sea state is T p = 6 s, H s = 1 m, γ = 1.

Figure 11 :
Figure 11: The amplitude of the transfer functions when the tuning spring stiffness is k * t = 6.06 × 10 3 N/m and the damping coefficient C PT O is set to the optimum value for T p = 6 s: (a) H b (ω), (b) H r (ω).

Figure 12 :
Figure 12: Power generation with phase control and optimum damping.For the TIM system, the tuning spring is set to k * t = 6.06 × 10 3 N/m.

Table 1 :
Parameter values used for numerical simulation studies