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 ab- sorber 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 eld of civil engineering as a structural control device which can absorb energy from vi- brating 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 mod- els are assessed through numerical simulation studies using WAMIT software and it is shown that the current-controlled WEC with the proposed mech- anism achieves an 88% increase compared to the conventional one for the JONSWAP spectrum with 6 s peak period and 1 m signi cant wave height. Keywords: wave energy converter, two-body point absorber, tuned inerter, renewable energy, energy harvesting 1. Introduction 1 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, 4 Preprint submitted to Renewable Energy February 23, 2021 ocean energy conversion technology has not yet shown a strong presence in 5 the renewable energy market. Since the concept of wave energy converter 6 (WEC) was introduced by a former Japanese naval commander, Yoshio Ma- 7 suda (1925-2009) [1], considerable eﬀort has been devoted to develop a variety 8 of WECs [2, 3, 4, 5] to exploit wave power in the ocean eﬀectively. And vari- 9 ous types of WECs proposed so far includes oscillating water columns [6, 7], 10 oscillating bodies [8, 9, 10], and overtopping devices [11, 12, 13]. Among these 11 devices, point absorbers consisting oating bodies are categorized as the os- 12 cillating body WEC and more expectations have been placed on this type of 13 WEC because of its availability in deep oﬀshore regions and its extensibility 14 by arraying many buoys. 15 To improve the power generation performance of a conventional single- 16 body point absorber WEC, which has one oating buoy, the authors em- 17 ployed a tuned inerter mechanism [14, 15]. Originally, the tuned inerter 18 mechanism was proposed by [16] as a structural control device to absorb 19 vibration energy eﬀectively from vibrating civil structures such as buildings 20 induced by seismic disturbances and to mitigate damage. This mechanism 21 consists of a tuning spring, a linear damping element, and an inerter [17]. 22 The inerter is an element to produce a force proportional to the diﬀerence 23 between the accelerations of both ends and realized by devices such as a ball 24 screw and a rack and pinion. In the tuned inerter mechanism, the damping 25 element is installed in parallel with the inerter and these two elements are 26 connected to the spring in series. Thus, once the spring stiﬀness is tuned so 27 that the inerter resonates with the dominant frequency of the input vibra- 28 tion to the device, the deformation of the damping part is increased and the 29 vibration energy is dissipated more eﬀectively. 30 The authors employed a generator or a motor as the linear damping 31 element in the tuned inerter mechanism and showed that the mechanism 32 enhances the ability to extract energy from vibrating structures at a low 33 frequency of less than 10 Hz [18, 19]. In addition, the authors proposed a 34 single-body point absorber WEC with a tuned inerter and the eﬃcacy of 35 the present device was shown through numerical simulation studies [14] and 36 wave ume testing using a small-scale prototype model [15]. 37 However, generally, the single-body point absorber WEC needs to be 38 connected directly to the ocean oor. While, a two-body point absorber 39 WEC [20, 21, 22, 23] consisting of a oating buoy and a submerged body is 40 just moored, not xed to the sea bottom. Additionally, the two-body type 41 has the potential to improve the power generation capability more than the 42 2 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- 45 sorber WEC with a tuned inerter. A detailed analytical model including 46 the coupled force between the two bodies for the proposed WEC and the 47 drag force is introduced and the equation of motion and the state-space rep- 48 resentation are developed. Then the energy harvesting performance for the 49 JONSWAP spectrum is assessed by comparing with a typical two-body point 50 absorber WEC without the proposed mechanism. Secondly, the eﬀectiveness 51 of the current control of the generator for the power take-oﬀ (PTO) system 52 of the proposed WEC is investigated. As control algorithms, two controllers 53 proposed in [24], i.e., static admittance (SA) control and performance guaran- 54 teed (PG) control, are applied to capture more energy. The obtained results 55 of the numerical studies using WAMIT [25] are shown and conclusions gained 56 from this research follow. 57 It should be noted that we make use of the short-hand G[AB CD] 58 to imply G(s) =C[sIA]1B+Dwhere A,B,C, and Dare the state, 59 input, output, and feedthrough matrices of a state-space representation, re- 60 spectively, and ^f(!) denotes the Fourier transform of a function f(t) in this 61 article. Also, note that jis the imaginary unit such that j2=1 and that 62 the expected value is denoted by Efg. 63 2. Modeling 64 To implement numerical studies, the analytical model of the proposed 65 WEC is developed here as well as the models of the hydrodynamic forces, 66 drag forces, and the JONSWAP spectrum. For simplicity, we consider only 67 the heave direction as in the literature [20, 21, 23] because this motion be- 68 comes dominant for the power extraction of wave energy. Additionally, the 69 generated power is de ned in this section. Note that the oating buoy and 70 the submerged body are indicated by 1st and 2nd bodies, respectively, in this 71 article. 72 2.1. Conventional two-body point absorber WEC 73 For comparison, a conventional two-body point absorber WEC consist- 74 ing a oating buoy (1st body) and a submerged body (2nd body) shown 75 schematically in Fig. 1 (a) is reviewed brie y rst. A PTO system including 76 3 kPTO (a) (b)msCPTO Lfw,1 m1 m2z1 z2fw,2hdD1 D2cs PTOkPTO hH1Fd,1 Fd,2Figure 1: Conventional two-body point absorber WEC: (a) Schematic illustration, (b) Model. a generator is placed between these two bodies. In this research, it is assumed 77 that the oating buoy has a circular cylinder shape with a diameter D1and 78 the submerged body is a sphere with a diameter D2. Also, the distance be- 79 tween these bodies is hdat the static equilibrium position and these bodies 80 are connected by a spring whose stiﬀness is kPTO. 81 The model of the conventional type is illustrated in Fig. 1 (b). Let 82 zk;(k= 1;2) be the displacement of the kth body and zsbe the rotational 83 displacement of the generator. Then we have 84 zs=z1z2 (1) Hence the equation of motion of the oating buoy would be 85 m1z1+ms(z1z2) + (cs+CPTO)( _z1_z2) +kPTO(z1z2) =Fd;1+fw;1(2) where m1is the mass of the oating buoy, msis the inertance caused by the 86 generator itself, csis the unwanted inherent mechanical damping coeﬃcient 87 caused in the PTO system, CPTOis the damping coeﬃcient of the generator. 88 Also, fw;1andFd;1are the hydrodynamic force and the drag force acting on 89 the oating buoy, respectively. 90 While, the equation of motion of the submerged body is developed as 91 m2z2ms(z1z2)(cs+CPTO)( _z1_z2)kPTO(z1z2) =Fd;2+fw;2(3) where m2is the mass of the submerged body and fw;2andFd;2are the 92 hydrodynamic force and the drag force on the submerged body, respectively, 93 similarly to the oating buoy. 94 4 kt zsms CPTO kPTO (a) (b)Lfw,1 m1 m2z1 z2fw,2hdD1 D2cs PTOTuning spring kt kPTO H1 hFd,1 Fd,2 InerterFigure 2: Two-body point absorber WEC with a tuned inerter: (a) Schematic illustration, (b) Model. 2.2. Two-body point absorber WEC with a tuned inerter 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 ktis added between the oating 98 buoy and the PTO system. Also, an additional rotational mass such as a 99 ywheel producing suﬃciently large inertance msis 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 satis ed 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 oating buoy and the submerged body are given 106 by 107 m1z1+kPTO(z1z2) +kt(z1z2zs) =Fd;1+fw;1 (4) 108 m2z2kPTO(z1z2)kt(z1z2zs) =Fd;2+fw;2 (5) 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 111 mszs+ (cs+CPTO) _zs=kt(z1z2zs) (6) 5 2.3. Hydrodynamic force 112 The hydrodynamic force fw;kacting on the kth body is described based 113 on the linear potential wave theory by 114 fw;k=fa;k+fb;k+fc;k (7) where fa;kis the excitation force, fb;kis the hydrodynamic forces due to 115 buoyancy, and fc;kis the radiation force. 116 The relationship between the excitation force fa;kand the amplitude of 117 the incident wave a(t) is given in the frequency domain using a transfer 118 function Fa;k(!) as 119 ^fa;k(!) =Fa;k(!)^a(!) (8) The hydrostatic force fb;1on the cylindrical oating buoy becomes a linear 120 function of z1given as 121 fb;1=Kwz1; K w=g(D1 2)2 (9) where gis gravitational acceleration and is the sea water density. While 122 the hydrostatic force of the submerged body is constant, thus fb;2can be set 123 as 124 fb;2= 0 (10) at the equilibrium position in the equation of motion. 125 Next, de ne l= 1;2;(l̸=k), then the radiation force fc;kon the kth body 126 including the coupled force aﬀected by the lth body is given by 127 ^fc;k(!) =(j!A kk(!) +Bkk(!))^_zk(j!A kl(!) +Bkl(!))^_zl (11) where AkkandBkkare the added mass and the radiation damping of the kth 128 body, and AklandBklrepresent the coupled added mass and the coupled 129 radiation damping from the lth body to the kth body. 130 2.4. Drag force 131 The drag force Fd;kacting on the kth body is modeled according to the 132 nonlinear Morison equation given by [26] 133 Fd;k=1 2SkCd;kj_zkj_zk (12) 6 where Skis the characteristic area and Cd;kis the dimensionless drag coeﬃ- 134 cient. As stated before, in this research, the shapes of the oating buoy and 135 the submerged body are assumed to be a cylinder and a sphere, respectively, 136 thus we have 137 S1=D2 1 4; S 2=D2 2 4(13) However, it would be cumbersome to deal with nonlinear equations in 138 the frequency domain, Eq. (12) is linearized under the condition of irregular 139 wave as [27, 26] 140 Fd;k=1 2SkCd;k√ 8 _zk_zk (14) where _zkis the standard deviation of _ zk. While, in general, the drag force 141 of the oating buoy is negligible compared to the hydrodynamic force [28], 142 thus we assume that 143 Fd;1= 0 (15) From, Eq. (14), the linearized drag force on the submerged body is given 144 with the viscous damping coeﬃcient cv;2by 145 Fd;2=cv;2_z2; c v;2=1 2S2Cd;2√ 8 _z2 (16) Hereafter, in this article, _zis used to represent the standard deviation of _ z2 146 for simplicity. 147 2.5. Stochastic sea state model 148 In simple theoretical models of WECs, it is typical to assume the incident 149 waves to be regular. For a more realistic model, irregular waves are used with 150 time-domain analysis which requires much more computing time [4]. An 151 alternative method with less computation for modeling true sea states is the 152 stochastic modeling. We assume the wave amplitude a(t) to be a stationary 153 stochastic process with spectral density Sa(!) which is characterized by the 154 JONSWAP spectrum [29] with its mean wave period T1, signi cant wave 155 height Hs, and peak enhancement factor expressed as 156 Sa(!) = 310 H2 s T4 1!5exp[944 T4 1!4] W(17) 7 where 157 W= exp[ (0:191!T11p 2)2] ;  ={ 0:07 : !T15:24 0:09 : !T1>5:24(18) The JONSWAP spectrum can also be represented by the peak period Tp158 using the well-known relationship T1= 0:834Tp. 159 2.6. Power take-oﬀ system 160 In this study, the generator is assumed to be a three-phase permanent 161 magnet synchronous machine (PMSM). However, the three phase voltage 162 and current vectors can be transformed to "quadrature components". More 163 details on this transformation can be found in [24, 30]. Then, assuming an 164 ideal generator with linear behavior and minimal core loss results in linearity 165 between the back-EMF eand the velocity coupled with the generator _ zs. 166 Therefore, the equation relating to eand _zsis given as 167 e=Ke_zs (19) where Keis a constant associated with the back-EMF of the generator. By 168 reciprocity, the electromagnetic force and generator current ihas the follow- 169 ing linear relationship 170 CPTO_zs=Kei (20) In the case of single-directional converter used in [24], the input current 171 to the generator ican be expressed as 172 i=Y e (21) where Yis the admittance of the generator restricted in ideal conditions by 173 [24] 174 Y2[0;1=R] (22) andRis the internal or coil resistance of the generator. Applying Eq. (21) 175 to Eq. (20) with Eq. (19) yields 176 CPTO=Y K2 e (23) which expresses how the generator damping CPTOis controlled by the ad- 177 mittance Y. 178 The total power generation is de ned 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., Ri2, then we have the power generation as 181 Pg=eiRi2(24) 8 3. State-space representation 182 In this section, to assess the power generation by the current controllers 183 under stochastic sea states, a state-space form [31] of the proposed WEC 184 augmented with the JONSWAP spectrum is developed. Also, the controllers 185 for the current of the generator for power generation are reviewed brie y. 186 3.1. Two-body point absorber WEC with a tuned inerter 187 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 192 { !2(m1+A11(!)) +j!(cv1+B11(!)) + ( Kw+kPTO+kt)} ^z1 +{ !2A12(!) +j!B 12(!)(kPTO+kt)} ^z2 =Fa;1(!)^a+kt^zs(25) and 193 { !2A21(!) +j!B 21(!)(kPTO+kt)} ^z1 +{ !2(m2+A22(!)) +j!(cv2+B22(!)) + ( kP T O+kt)} ^z2 =Fa;2(!)^akt^zs(26) respectively. Hence, Eqs. (25) and (26) can be combined and written in 194 matrix form as 195 [Z11Z12 Z21Z22][^z1 ^z2] =[Fa;1 Fa;2] ^a+[kt kt] ^zs (27) where 196 Z11(!) =!2(m1+A11(!)) +j!(B11(!)) + ( Kw+kPTO+kt) Z12(!) =!2A12(!) +j!B 12(!)(kPTO+kt) Z21(!) =!2A21(!) +j!B 21(!)(kPTO+kt) Z22(!) =!2(m2+A22(!)) +j!(cv;2+B22(!)) + ( kPTO+kt)(28) Solving Eq. (27) for[ ^z1^z2]Tyields the expressions of form 197 ^z1=Ga;1(!)^a+Gz;1(!)^zs (29) ^z2=Ga;2(!)^a+Gz;2(!)^zs (30) 9 When Ga;1,Gz;1,Ga;2, and Gz;2are approximated by nite-dimensional sys- 198 tems, we have representations as 199 Ga;1[Aa;1Ba;1 Ca;10] ; G z;1[Az;1Bz;1 Cz;10] (31) 200 Ga;2[Aa;2Ba;2 Ca;20] ; G z;2[Az;2Bz;2 Cz;20] (32) It should be noted that the function Fa;kin Eq. (8) is non-causal which will 201 be problematic when approximating Ga;1andGa;2by a nite-dimensional 202 state-space. Therefore the technique of spatial delay proposed by Falnes [32] 203 is used, de ning a(t) as the wave amplitude at a distance of din front of the 204 buoy. 205 Once Eqs. (31) and (32) are obtained by system identi cation techniques, 206 the identi ed systems Eqs. (29) and (30) are represented in the time domain 207 as 208 _x1(t) = A1x1(t) +G1a(t) +E1zs(t) (33) z1(t) = C1x1(t) (34) where A1,G1,E1, and C1are expressed by Aa;1,Ba;1,Ca;1,Az;1,Bz;1, and 209 Cz;1. and 210 _x2(t) = A2x2(t) +G2a(t) +E2zs(t) (35) z2(t) = C2x2(t) (36) where A2,G2,E2, and C2are expressed by Aa;2,Ba;2,Ca;2,Az;2,Bz;2, and 211 Cz;2as well. 212 Also, considering z1z2as an input, we have a state-space representation 213 about the tuned inerter part where the output is the velocity of the generator 214 as 215 _xs(t) = Asxs(t) +Bsi(t) +Es(z1(t)z2(t)) (37) _zs(t) = Csxs(t) (38) where the state vector is de ned as xs=[ zs_zs]Tand 216 As=[0 1 kt mscs ms] ;Bs=[0 ce ms] ;Es=[0 kt ms] ;Cs=[ 0 1] (39) 10 De ne the state vector as xh=[ xT 1xT 2xT s]T. Then we have a state- 217 space representation in which the inputs are the current iand the wave height 218 aand the output is the voltage efrom Eq. (19) and Eqs. (33) through (39) 219 as follows: 220 _xh(t) = Ahxh(t) +Bhi(t) +Gha(t) (40) e(t) = Chxh(t) (41) where Ah,Bh,Gh, and Chcan be composed of A1,A2,B1,B2,G1,G2, 221 C1,C2,As,Bs,Es, and Cs. 222 3.2. JONSWAP spectrum 223 First, a state-space model of the wave amplitude is derived. We nd a 224 nite-dimensional noise lter 225 Fw[AwBw Cw0] (42) such that its power spectrum is close to the JONSWAP spectrum, i.e., 226 Sa(!) =jFw(!)j2, for a unit intensity white noise input w(t). Then we 227 have 228 _xw(t) = Awxw(t) +Bww(t) (43) a(t) = Cwxw(t) (44) According to the simpli ed procedure advocated by Spanos [33], Fwcan be 229 approximated by a forth-order controllable canonical form of 230 Aw=2 6640 1 0 0 0 0 1 0 0 0 0 1 a1a2a3a43 775;Bw=2 6640 0 0 13 775;Cw=[ 0 0 c30] (45) where the lter parameters a1,a2,a3,a4, and c3are chosen to minimize the 231 mean-square error∫1 1(Sa(!) jFw(!)j2)2d!, while constraining a1through 232 a4so that the system poles are in the open left half plane. 233 For example, Fig. 3 shows a JONSWAP spectrum for Tp= 6 s, Hs= 1 234 m, = 3:3 and its fourth-order nite-dimensional approximate system. We 235 can con rm in the gure that the fourth-order Fwestimates the JONSWAP 236 spectrum very well. It should be noted that Hs= 1 m, = 3:3 are xed in 237 the numerical simulation studies in this paper. 238 11 0 0.5 1 1.5 2 2.5 ω (rad/s) 00.511.5 S ω (m 2s) JONSWAP Model Figure 3: JONSWAP spectrum with Tp= 6 s, Hs= 1 m, = 3.3 3.3. Augmented system 239 Finally, combining Eqs. (40) and (41) with the stochastic sea state model 240 given by Eqs. (43) and (44) gives the augmented system where the external 241 disturbance input is white noise w(t) expressed as 242 _x(t) = Ax(t) +Bi(t) +Gw(t) (46) e(t) = Cx(t) (47) where 243 x=[xh xw] ;A=[AhGhCw 0 A w] ;B=[Bh 0] ;G=[0 Bw] ;C=[ Ch0] (48) 3.4. Controller 244 To improve the power generation performance and to examine the eﬀec- 245 tiveness of the current control on the proposed device, the current of the 246 generator iis controlled by the two control laws introduced in [24] which are 247 reviewed brie y here. Before we go any further, it should be noted that the 248 average of the generated power de ned as Eq. (24) can be written as 249 Pg="{[x i]T[01 2CT 1 2C R][x i]} (49) using Eq. (47). 250 12 3.4.1. Static admittance control 251 For the SA control, a constant feedback gain Ycrestricted by Eq. (22) is 252 adopted so that the value de ned by Eq. (49) is maximized. The method to 253 search for such a value is reviewed here. 254 From Eqs. (21) and (47), the input current to the generator is expressed 255 as a function of the state variable x, i.e., 256 i(t) =YcCx(t) (50) Substituting Eq. (50) into Eq. (46) yields the closed-loop dynamics having 257 the form 258 _x(t) = (AYcBC)x(t) +Gw(t) (51) Let the average power by the SA control be PSA g. Then for any time-invariant 259 Ycsatisfying Eq. (22), it is a standard result that the power generation 260 objective can be written as [34] 261 PSA g=tr[GTSG] (52) where S=ST<0 is the solution to the Lyapunov equation 262 (AYcBC)TS+S(AYcBC) +CT(Yc+Y2 cR)C=0 (53) Ycmust be less than or equal to 1 =R, so the last term on the left-hand side 263 of Eq. (53) is negative-semide nite for all Yc. Thus, since AYcBC is 264 asymptotically stable, the de niteness of Sis assured by Lyapunov's second 265 theorem [35]. Then the optimal value for Ycis chosen so that Eq. (52) is 266 maximized. 267 3.4.2. Performance guaranteed control 268 For comparison, the eﬃcacy of time-varying gain Ybased on the PG 269 control algorithm proposed in the literature is examined. This algorithm is 270 operated with a single-directional converter and the admittance Ybecomes 271 a function of time varying within the range of Eq. (22) so that the generated 272 average power PPG gmust be larger than PSA g, i.e., 273 PPG gPSA g (54) In this algorithm, the admittance is controlled by 274 Y(t) = [0;1=R]sat{Kx e} (55) 13 where 275 K=1 R( BTS+1 2C) (56) Note that Ycis the constant value for the SA control. In this case, the current 276 iis expressed by 277 i(t) =8 < :iu(t) : iue+i2 uR0 0 :iue+i2 uR > 0 and iue >0 e(t)=R : otherwise(57) where 278 iu=Kx (58) And the generated average energy would be 279 PPG g=PSA g+RE{ (iu+Yce)2(iui)2} (59) which guarantees the inequality given by Eq. (54). 280 4. Numerical simulation 281 To verify the eﬃcacy of the proposed two-body point absorber WEC, nu- 282 merical simulation studies are carried out in this section. First, the parame- 283 ter values of the model used here are developed, then the power generation 284 performance is assessed. 285 4.1. Model development 286 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 oating buoy and 290 the submerged body calculated using WAMIT software [25] are shown in 291 Figs. 4 and 5, respectively. The magnitude and the phase of the transfer 292 functions given by Eq. (8) are shown in the gures as well. Moreover, the 293 coupled added mass and radiation damping acting on the oating buoy from 294 the submerged body, i.e., A12andB12and vice versa, i.e., A21andB21are 295 shown in Fig. 6. 296 Then, from Eqs. (27) and (28), the frequency response data for Ga;1,Gz;1, 297 Ga;2, and Gz;2in Eqs. (29) and (30) are calculated as depicted in Figs. 7 and 298 14 Table 1: Parameter values for numerical simulation Parameter Value Parameter Value m1 58,075 kg m2 34,515 kg D1 6.0 m D2 4.0 m H1 2.5 m L 2.0 m h 400 m hd 20 m kt 10,000 N/m kPTO 100,000 N/m cs 50 Ns/m Cd;2 0.1 R 25 Ω  1,027 kg/m3 8 by solid lines. Then, to express these in state-space form as expressed by 299 Eqs. (31) and (32), Ga;1andGa;2are approximated with 5 zeros and 6 poles, 300 andGz;1with 2 zeros and 4 poles, and Gz;2with 4 zeros and 5 poles. These 301 numbers are chosen by trial and error so that the nite-dimensional models 302 shown by dashed lines approximate the frequency response data very well. 303 This derivation is carried out using MATLAB [36] in this research. Note that 304 the wave amplitude a(t) is taken to be the wave amplitude d= 10 m ahead 305 of the buoy in the propagation direction. 306 4.2. Inerter design 307 Next, the inertance msfor the proposed WEC needs to be designed. In 308 this study, this value is chosen so that the average power generation for the 309 JONSWAP spectrum with Tp= 6 s is maximized when the SA controller is 310 applied. While the damping coeﬃcient of the drag force on the submerged 311 body given by (16) depends on the standard deviation of the velocity _ z2. 312 Thus, the value msis designed according to the owchart as shown in Fig. 313 9 (a) here, in which  _zand_zare standard deviations at the previous and 314 current iterations, respectively. This owchart is made based on the method 315 to determine _zintroduced in [26]. For this iteration process, the initial value 316 for_zis set to _z0= 0:1 m/s and the range for msis set between ms0= 5;000 317 kg and mse= 10;000 kg and the best value is sought with respect to each 318 100 kg iteratively. 319 During the iteration, the standard deviation of _ z2is calculated simply 320 from 321 2 _z=GTS_zG (60) 15 05A11 (kg) ×10 4 0 0.5 1 1.5 2 2.5 ω (rad/s)05B11 (Ns/m)×10 4024|Fa, 1| (N/m)×10 5 0 0.5 1 1.5 2 2.5 ω (rad/s) -2 02 Fa, 1(rad) (a) (b)(c) (d)Figure 4: Hydrodynamic parameters for the heave mode of the oating buoy : (a) Added massA11, (b) Radiation damping B11, (c) Magnitude of Fa;1(!), (d) Phase of Fa;1(!). 05A22 (kg) ×10 4 0 0.5 1 1.5 2 2.5 ω (rad/s)050 100B22 (Ns/m)0510 |Fa,2| (N/m) 0 0.5 1 1.5 2 2.5 ω (rad/s) -2 02 Fa,2(rad) ×10 3(a) (b)(c) (d) Figure 5: Hydrodynamic parameters for the heave mode of the submerged body : (a) Added mass A22, (b) Radiation damping B22, (c) Magnitude of Fa;2(!), (d) Phase of Fa;2(!). where S_z=ST _z>0 is the solution to the Lyapunov equation 322 (AYcBC)TS_z+S_z(AYcBC) +CT _zC_z=0 (61) and 323 _z2=C_zx (62) Then, if the diﬀerence between the obtained _zand the previous value  _z324 is larger than 0.001, the damping coeﬃcient for the linearized drag force is 325 16 -5000500A12 (kg) -5000500A21 (kg) 0 0.5 1 1.5 2 2.5 ω (rad/s)-5000B12 (Ns/m) 0 0.5 1 1.5 2 2.5 ω (rad/s) -5000B21 (Ns/m)(a) (b)(c) (d)Figure 6: Coupled hydrodynamic parameters for the heave mode of the two bodies : (a) Added mass, A12(b) Radiation damping B12, (c) Added mass Added mass, A21, (d) Radiation damping B21. recalculated using the newly obtained _zfrom Eq. (16). This procedure 326 continues until the value of the standard deviation converges enough. And 327 the average power for the SA control is calculated using the converged _z328 from Eq. (52) 329 The average power obtained in the process of the owchart is plotted in 330 Fig. 10. As can be seen, the generation performance peaks at ms= 6;900 331 kg, which is used for the numerical simulation studies. 332 4.3. Drag force model 333 The damping coeﬃcient cv;2for the linearized drag force acting on the 334 submerged body as expressed by Eq. (16) still needs to be obtained for 335 each peak wave period Tp. The inertance value is set to ms= 6;900 kg as 336 determined before. Then we seek for the damping coeﬃcient value for each 337 Tpof the JONSWAP spectrum based on the owchart shown in Fig. 9 (b). 338 This is referred to the owchart in [26] as well as Fig. 9 (a). For the process, 339 _zis rst set to _z0= 0:1 m/s as well as before and cv;2is investigated in 340 the range of the peak wave period Tpfrom 2 s and 12 s, i.e., we set Tp0= 2 341 s and Tpe= 12 s. 342 The result for the case of the two-body point absorber with a tuned inerter 343 ofms= 6;900 kg is shown in Fig. 11. We can nd that cv;2is much larger 344 than the radiation damping on the submerged body denoted by B22and 345 con rm that the drag force on the submerged body should not be ignored. 346 17 010 20 0 0.5 1 1.5 2 2.5 (rad/s)-10-5 0012 0 0.5 1 1.5 2 2.5 (rad/s) -5 05| G z, 1| G z, 1 (rad) | G a, 1| G a, 1 (rad) ω ω(a) (b)Figure 7: Frequency domain data (solid) and nite-dimensional approximation (dashed) : (a)Ga;1(!) , (b) Gz;1(!). 010 20 | G a, 2| 0 0.5 1 1.5 2 2.5 ω (rad/s)-20-100 G a, 2 (rad) 012| G z, 2| 0 0.5 1 1.5 2 2.5 ω (rad/s) -5 05 G z, 2 (rad) (a) (b) Figure 8: Frequency domain data (solid) and nite-dimensional approximation (dashed) : (a)Ga;2(!) , (b) Gz;2(!). 4.4. Results 347 Finally, the average power generations for the JONSWP spectrum with 348 the peak wave period from 2 s to 12 s are calculated from the parameter values 349 determined above.The results obtained from the SA and the PG controllers 350 are compared in Fig. 12 (a), in which the proposed point absorber WEC with 351 a tuned inerter and the conventional WEC cases are denoted by 2BwTI and 352 2B, respectively. The average power for the SA control cases can be given by 353 the closed form expressed by Eq. (52), however, for consistency with the PG 354 control cases, the average values shown in Fig. 12 (a) are calculated from Eq. 355 18 |σz-σz-| > 0.001 ms < mse Start EndSet σz = σz0 Set ms = ms0 Determine admittance Yc Determine cv, 2 using Eq. (16) Calulate standard deviation σz using Eq. (63) Caluculate average power PgSA with Eq. (55) ms = ms + 100 kg No No YesYes|σz-σz-| > 0.001 Tp < Tpe Start EndSet σz = σz0 Set Tp = Tp0 Determine admittance Yc Determine cv, 2 with Eq. (16) Calulate standard deviation σz with Eq. (63) Tp = Tp + 0.1 s No No YesYes (a) (b)Figure 9: Flowchart: (a) Inerter design, (b) Drag force model. 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 Inertance (kg)400060008000100001200014000Average power (W) Figure 10: Average power vs inerter. (24) using the numerical simulation results for 100,000 s. The admittances for 356 the SA controller are compared in Fig. 12 (b) which shows the diﬀerence of 357 19 10 11 12 T p (s)050010001500cv,2 (Ns/m) 2 3 4 5 6 7 8 9Figure 11: Damping coeﬃcient cv;2for the drag force on the submerged body. the optimized admittance between the proposed and the conventional WECs. 358 For the assessment of the conventional WEC, the same parameters as the 359 proposed WEC are employed except that ms= 50 kg. As you can see in Fig. 360 12 (a), the proposed WEC shows the better power generation performance 361 than the conventional WEC in the wide range of the peak wave period Tpeven 362 though the eﬀectiveness deteriorates in a certain range of the lower period. 363 Especially, the superiority of the proposed WEC is notable around Tp= 6 s 364 to which the inerter of the proposed WEC is designed to be tuned. Also, the 365 PG controller works well to improve the power generation especially for the 366 proposed WEC. Speci cally, the calculated average powers of 2B-SA, 2B-PG, 367 2BwTI-SA, and 2BwTI-PG cases when Tp= 6 s are 7,180 W, 7706 W, 12,030 368 W, and 13530 W, respectively. Thus, the present WEC controlled based on 369 the PG algorithm achieves an 88% increase compared to the conventional 370 WEC. 371 4.5. Discussion 372 It is shown through the numerical simulation studies that the tuned in- 373 erter mechanism works well on the two-body point absorber WEC. However, 374 to enhance the credibility of the device, more elaborate studies employing a 375 more accurate model considering non-linearity and behaviors other than the 376 heave direction are necessary. Also, the optimum design for the shapes of the 377 oating buoy and the submerged body should be examined more thoroughly 378 and it is worth trying other simulation schemes, for example, a CFD (com- 379 putational uid dynamics) technique and WEC-SIM [37]. Moreover, various 380 20 050001000015000Average power (W)2BwTI-SA 2BwTI-PG 2B-SA 2B-PG 10 11 12 T p (s)00.010.020.030.04Admitance Y c ( Ω-1 ) 2BwTI 2B 2 3 4 5 6 7 8 9(a) (b)Figure 12: Results. (a) Power generation, (b) Admittance. control algorithms such as reactive controllers should be applied to the gen- 381 erator current of the proposed WEC or more eﬀective controllers need to be 382 developed. For a practical perspective, experimental veri cation is still desir- 383 able and the capture width ratio and the levelized cost of electricity (LCOE) 384 of the present device should be calculated and compared to other devices. 385 5. Conclusions 386 This article introduced the two-body point absorber WEC with a tuned 387 inerter and developed the detailed analytical model including the coupled 388 force between the two bodies. Then, its eﬀectiveness was veri ed by compar- 389 ing the conventional two-body point absorber WEC through the numerical 390 simulation studies using the values obtained from WAMIT software. The re- 391 sults for the JONSWAP spectrum showed that the tuned inerter mechanism 392 not only increased the power generation performance but also broadened the 393 eﬀective range of the peak period of the JOWNSWP spectrum. Moreover, 394 this research showed that the performance of the proposed WEC was im- 395 21 proved further by controlling the current of the generator. Considering the 396 results obtained in this research, we conclude that the tuned inerter mech- 397 anism has great potential to improve the power generation performance of 398 the two-body point absorber WEC. 399 Acknowledgments 400 The authors gratefully acknowledge support for this research through 401 JSPS KAKENHI Grant number 18K189808 and TEPCO (Tokyo Electric 402 Power Company) Memorial Foundation. 403