Numerical evaluation of a two-body point absorber wave energy converter with a tuned inerter

To increase the amount of energy captured from a vibrating buoy in the ocean with a simple mechanism, this paper proposes a two-body point absorber wave energy converter (WEC) with a tuned inerter. The tuned inerter mechanism consists of a spring, a linear damping element, and a component called inerter. This mechanism was originally proposed in the (cid:12)eld of civil engineering as a structural control device which can absorb energy from vibrating structures eﬀectively by taking advantage of the resonance eﬀect of the inerter part. In addition to this mechanism where a generator is used as the linear damping element, the current of the generator for the power take-oﬀ system is controlled based on the algorithms proposed in literature to achieve further improvement of the power generation capability. In this research, a detailed analytical model of the proposed WEC is introduced and developed. Then the power generation performances of full-scale WEC models are assessed through numerical simulation studies using WAMIT software and it is shown that the current-controlled WEC with the proposed mechanism achieves an 88% increase compared to the conventional one for the JONSWAP spectrum with 6 s peak period and 1 m signi(cid:12)cant wave height.


Introduction
Ocean has been expected to be a promising renewable energy source since 2 more than 70% of the Earth's surface is covered with oceans. However, com-3 pared with other renewable energy sources such as wind and solar energy, single-body type by designing the two bodies to achieve a greater relative 43 velocity between the two bodies. 44 The primary objective of this paper is to propose a two-body point ab-  95 Next, the proposed two-body point absorber WEC with a tuned inerter 96 shown in Fig. 2 (a) is considered. As can be seen, unlike the conventional two-97 body type, a tuning spring whose stiffness is k t is added between the floating 98 buoy and the PTO system. Also, an additional rotational mass such as a 99 flywheel producing sufficiently large inertance m s is mounted intentionally 100 on the generator shaft.

101
The model of the present device is shown in Fig. 2 (b) and the equations 102 of motion of the device are derived as follows. In contrast to the conventional 103 type, Eq. (1) is not satisfied because the proposed system becomes a three-104 degree-of-freedom system due to the tuning spring. Then, for this model, the 105 equations of motion of the floating buoy and the submerged body are given respectively. And considering the fact that the force of the tuning spring 109 equals the force of the PTO system, the equation of motion of the inerter is 110 derived as

Hydrodynamic force
The hydrodynamic force f w,k acting on the kth body is described based 113 on the linear potential wave theory by where f a,k is the excitation force, f b,k is the hydrodynamic forces due to 115 buoyancy, and f c,k is the radiation force.

116
The relationship between the excitation force f a,k and the amplitude of 117 the incident wave a(t) is given in the frequency domain using a transfer The hydrostatic force f b,1 on the cylindrical floating buoy becomes a linear 120 function of z 1 given as where g is gravitational acceleration and ρ is the sea water density. While Next, define l = 1, 2, (l ̸ = k), then the radiation force f c,k on the kth body 126 including the coupled force affected by the lth body is given by where The JONSWAP spectrum can also be represented by the peak period T p 158 using the well-known relationship T 1 = 0.834T p . 159 2.6. Power take-off system 160 In this study, the generator is assumed to be a three-phase permanent Therefore, the equation relating to e andż s is given as where K e is a constant associated with the back-EMF of the generator. By to the generator i can be expressed as where Y is the admittance of the generator restricted in ideal conditions by and R is the internal or coil resistance of the generator. Applying Eq. (21) which expresses how the generator damping C PTO is controlled by the ad- The total power generation is defined as the extracted power minus the 179 electrical loss [10]. In this paper, we assume that the current-dependent loss 180 is resistive, i.e., Ri 2 , then we have the power generation as

State-space representation
In this section, to assess the power generation by the current controllers Next, state-space representation for the proposed device is developed here.

188
As state-space representation for the conventional two-body WEC can be 189 developed in a similar way, its derivation is omitted.

190
Substitute the equations for the hydrodynamic and drag forces into Eqs.

191
(4) and (5), then taking Fourier transform gives and respectively. Hence, Eqs. (25) and (26) can be combined and written in 194 matrix form as ] T yields the expressions of form When G a,1 , G z,1 , G a,2 , and G z,2 are approximated by finite-dimensional systems, we have representations as It should be noted that the function F a,k in Eq. (8) where A 1 , G 1 , E 1 , and C 1 are expressed by A a,1 , B a,1 , C a,1 , A z,1 , B z,1 , and 209 C z,1 . and where A 2 , G 2 , E 2 , and C 2 are expressed by A a,2 , B a,2 , C a,2 , A z,2 , B z,2 , and 211 C z,2 as well.
where the state vector is defined as Define the state vector as such that its power spectrum is close to the JONSWAP spectrum, i.e.,

226
S a (ω) = |F w (ω)| 2 , for a unit intensity white noise input w(t). Then we According to the simplified procedure advocated by Spanos [33], F w can be 229 approximated by a forth-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 231 mean-square error disturbance input is white noise w(t) expressed as where

244
To improve the power generation performance and to examine the effec-  24) can be written as using Eq. (47).
In this algorithm, the admittance is controlled by where Note that Y c is the constant value for the SA control. In this case, the current 276 i is expressed by And the generated average energy would be which guarantees the inequality given by Eq. (54).  The parameter values for the two-body point absorber used here is deter-287 mined based on the study conducted in [20], which are summarized in Table   288 1.

289
The added mass and the radiation damping of the floating buoy and  and G z,1 with 2 zeros and 4 poles, and G z,2 with 4 zeros and 5 poles. These    where Sż = S Ṫ z > 0 is the solution to the Lyapunov equation   Fig. 9 (b).

338
This is referred to the flowchart in [26] as well as Fig. 9 (a). For the process, 339 σż is first set to σż 0 = 0.1 m/s as well as before and c v,2 is investigated in 340 the range of the peak wave period T p from 2 s and 12 s, i.e., we set T p0 = 2 341 s and T pe = 12 s.

342
The result for the case of the two-body point absorber with a tuned inerter 343 of m s = 6, 900 kg is shown in Fig. 11. We can find that c v,2 is much larger

347
Finally, the average power generations for the JONSWP spectrum with