Concept and performance of a novel wave energy converter: Variable Aperture Point-Absorber (VAPA)

: Ocean waves are a huge and largely untapped resource of green energy. In order to 1 extract energy from waves, a novel wave energy converter (WEC) consisting of a floating, 2 hollow cylinder capped by a roof with a variable aperture is presented in this paper. The 3 power take-off (PTO) system is composed of a linear generator attached to the seabed, driven 4 by the heave motion of the floating cylinder through a tether line. The air pressure within the 5 cylinder can be modified by adjusting the roof aperture. The hydrodynamic characteristics of 6 this WEC are investigated through an analytical model based on potential flow theory, in 7 which the wave diffraction/radiation problems are coupled with the air pressure fluctuation 8 and PTO system. Analytical expressions are derived for the maximum power absorbed by the 9 WEC under different optimization principles, revolving around the PTO damping, roof 10 aperture damping and non-negative mooring stiffness. We find that the best power absorption 11 is obtained when the aperture is either completely open or entirely closed, depending on the 12 wave conditions. Intermediate values of the aperture are useful to minimize the heave motion 13 and thus ensure survivability under extreme sea states.

Point-absorbers are a particular type of WEC: floating devices smaller than the typical wave length and capturing wave power mainly through a translating motion relative to a reference point.Although not the most efficient WEC type, point-absorbers are advantageous considering total performance and energy costs (Sjolte et al., 2013a), their compact dimensions and simple construction (Chen et al., 2017).
Most of the point-absorbers developed so far are based on truncated cylinders, e.g., the Uppsala University heaving buoy (Figure 1), connected to a translator in a linear generator installed on the seabed (Hai et al., 2016).The translator has a limited stroke, and is equipped with springs to dampen endstop shocks.A peak force still occurs on the mooring line when the upper endstop spring is hit (Sjökvist and Göteman, 2017).The Ocean Power Technology PowerBuoy (Figure 2a) uses a damping plate for reference (Mekhiche and Edwards, 2014).Wavebob (Figure 2b) adopts a submerged float rather than a plate or the seabed for reference (Falcã o, 2010).The submerged float allows the tuning to the incident wave frequency.Other point-absorbers (BOLT, CETO and Wavestar) are described in Ding et al. (2016), Ransley et al. (2017), and Ulvin et al. (2012).
The present work is motivated by three main objectives: to enhance the survivability of the system under extreme conditions, to reduce its cost, and to improve its wave power absorption in terms of the peak value of the frequency response.To this aim, a novel WEC, VAPA (Variable Aperture Point-Absorber), which combines the advantages of traditional point-absorbers (truncated cylinders) and torus buoys, is proposed and investigated.VAPA is a hollow cylinder with an inner chamber covered by a roof that can be opened totally or partially, and open at its bottom, below the waterline (Fig. 4).As the cylinder oscillates under wave action, so does the water column in the chamber, causing the air pressure in the chamber to fluctuate.Unlike floating oscillating water columns, there is no turbine installed on the roof.Instead, power extraction is achieved by a linear generator on the seabed, connected to the cylinder through a tether.The air pressure effect can be adjusted by changing the roof aperture.When the aperture is totally open, VAPA performs as a torus buoy, which is beneficial for survivability under extreme wave conditions; by contrast, when the aperture is totally closed, VAPA works like a traditional solid point-absorber with the water enclosed performing as ballast, which is beneficial in terms of cost (less weight of steel required) and wave power extraction (in particular, vis-à-vis the peak value of the frequency response).Thus, VAPA can switch between two configurations, the traditional point-absorber and the torus buoy, by changing the roof aperture, by means of an intelligent control system, the details of which are beyond the scope of the present article.To determine the effect of the roof aperture on power extraction, we develop, validate and apply an analytical model.Fig. 4. VAPA schematic

Analytical model
For a preliminary performance assessment, the roof aperture effect is modelled as a linear damping, and nonlinear, viscous effects are neglected.Under an incident wave train of small amplitude, A, and angular frequency, ω, the free surface displacement in the chamber may be written as , with Q the complex amplitude, t time, and i the imaginary unit.Air pressure in the chamber may be written as , with p the complex amplitude.Assuming the mass flux across the roof aperture to be proportional to the pressure, and considering the effect of air compressibility, which results in a phase lag between Q and p, following Falcão and Sarmento (1980) and Sarmento and Falcã o (1985) we have: where cr is a damping coefficient representing the damping effect induced by the aperture on the roof.More specifically, it is related to the volume flux across the roof due to unit air pressure in the internal chamber.When cr=0, no volume flux will be excited regardless of the value of the internal air pressure, i.e., the roof aperture is totally closed; by contrast, when cr=∞, volume flux can be very easily excited with a small value of internal air pressure, i.e., the roof aperture is totally open.V0 is the air chamber volume, ca denotes the sound velocity in air, and ρ0 represents the static air density.
For small-amplitude regular waves, Q results from scattered (incident and diffracted) and radiated waves, which are induced both by cylinder oscillation and pressure oscillation.This also applies to the hydrodynamic forces acting on the float.

Governing equations and boundary conditions of wave diffraction and radiation problems
Let a vertical truncated circular cylinder of radius R with a moonpool of radius Ri float in water of finite depth h, with draught d.A Cartesian coordinate system is adopted, with the xyplane at the mean water surface, the Ox-axis in the incident wave direction, and the Oz-axis along the cylinder axis, pointing upwards (Fig. 5).The cylinder has three DoFs: surge, heave and pitch.A local cylindrical coordinate system (Orθ) is defined with r measuring radially from the z-axis and θ from the positive Ox-axis.The rotation center (r=0, z=z0) may serve as the reference point to calculate the pitch wave excitation moment and hydrodynamic coefficients in relation with the oscillation in pitch mode.Assuming the fluid to be isotropic, incompressible and inviscid, and the wave amplitude to be small, linear potential flow theory may be adopted to describe the hydrodynamic problem.The total spatial velocity potential  may be decomposed into the incident, I  , diffracted, D  , and radiated wave spatial potential, ( ) ( ) where j  is the complex velocity amplitude of the chamber oscillating in j-th mode (with j=1,2,3 denoting surge, heave, and pitch, respectively);  is the spatial velocity potential due to a unit amplitude velocity oscillation in j-th mode; and  is the spatial velocity potential due to a unit air pressure oscillation.
The spatial velocity potential for the undisturbed incident regular waves propagating along the positive Ox axis may be written as where Eq.(3a) employs the Cartesian coordinate system (Oxyz) and Eq.(3b) the local cylindrical coordinate systems (Orθz); k0 is the wave number, which satisfies the dispersion relation, ω 2 =gk0tanh(k0h); and g is the gravitational acceleration.
The free-surface and body boundary conditions to be satisfied by D  and ( ) R j  can be found in Mavrakos and Konispoliatis (2012), Zheng et al. (2018).

Spatial potentials in subdomains
The spatial potentials D  and ( ) R j  (j=0,1,2,3) in fluid subdomain Region n can be written in a unified format as n   , in which χ='D' and '(j)' represent the wave diffracted potential and the radiated potential due to air pressure oscillations inside the chamber (j=0) and cylinder motions in j-th mode (j=1,2,3), respectively.Applying the method of separation of variables in different regions, the general spatial potentials may be expressed by complex Fourier series as follows: 1) In Region 1 Im is the modified Bessel function of first kind and order m; Km is the modified Bessel function of second kind and order m; , ml A  and , ml C  are unknown coefficients; l  is the l-th eigenvalue:  is a particular solution; for χ='D', 1,p   =-I  ; for χ='(j)' (j=0,1), 1,p   =0; and for χ='(j)' , , e 00 where , ml D  is the coefficient to be solved; Jm is the Bessel function of order m; l k is the eigenvalue (Falnes, 2002).

3) Region 3
The spatial potential in Region 3 represents the wave travelling outwards from the cylinder, and can be written as an eigen-function expansion, , , e 00 where Hm is the Hankel function of first kind of order m, and , ml B  are unknown coefficients to be determined.

Method of computation for unknown coefficients
The expressions of the diffracted spatial potential and radiated potentials, Eqs.(4)~( 12) in Section 2.2, should satisfy the conditions of continuity for pressure and normal velocity on the interfaces of the two adjacent subdomains, i.e., at r=R and r=Ri, as follows.: 1) Pressure at the boundary r=R: 2) Pressure at the boundary r=Ri: 3) Normal velocity at the boundary r=R: For 0 in which δ is the Kronecker delta function.

Wave excitation volume flux/forces
The rate of free surface displacement inside the chamber due to the contributions of the undisturbed incident wave and the diffracted wave can be written as

Re
e t F  −   , where, with utilization of Eq. (3) and Eq. ( 8), The wave excitation forces due to the incident wave acting on structures which are stationary can be computed from the incident wave potential and the diffracted potential.The generalized wave excitation force on the WEC chamber in j-th mode (j=1,2,3) is , where is the unit normal vector directed into the fluid domain at the wetted surface of the cylinder.

Hydrodynamic coefficients
An upward flux at the water surface inside the chamber (radiation volume flux) and forces on the floats (radiation forces) can be induced when the air pressure inside the chamber or the cylinder oscillate in the absence of an incident wave.
The complex amplitudes of the radiation volume flux due to a unit amplitude velocity oscillation of the WEC chamber oscillating in j-th mode (j=1,2,3) and a unit air pressure oscillation inside the WEC (j=0) can be written, respectively, as: where 0, j a and 0, j c are called the hydrodynamic coefficients.
Similarly, the complex amplitudes of radiation force exerted on the WEC chamber in j'th mode (j'=1,2,3) due to unit amplitude velocity oscillation of the chamber oscillating in j-th mode and unit air pressure oscillation inside the WEC (j=0) can be respectively written in terms of the hydrodynamic coefficients , jj a  and , jj c  as: The method for calculating the hydrodynamic coefficients as given in Eqs. ( 19)-( 20) is straightforward based on the definitions of radiation volume flux and radiation forces.Hence it is referred henceforth as the "direct method (DM)".In fact, there is a Haskind relation (HR) between wave diffraction and radiation problems (Falnes, 2002), and a number of hydrodynamic coefficients can be written in terms of the wave excitation volume flux and wave excitation forces as: where '*' denotes the complex-conjugate, and vg is the wave group velocity expressed as ( ) 2.6 Response and power absorption of the VAPA WEC For the novel WEC under regular waves of small amplitude, after coupling the chamber oscillation with the air pressure fluctuation and PTO system, the matrix equation of motion in the frequency domain may be written as where X is the motion/pressure response vector written as , in which the motion response of the floats are given in terms of velocities, 'T' denotes the transpose; Fe represents the wave excitation volume flux/force acting on the device, and it is a 4×1 vector, written as . Ma and Cd are two 4×4 square matrices of addedmass and radiation damping coefficients due to wave radiation, which can be calculated, together with Fe, from Sections 2.4 and 2.5.MPTO is a diagonal matrix of mass coefficients of Power Take-Off system (PTO) in the device, the diagonal elements of which can be written as d is adopted with ca=340 m/s and ρ/ρ0=1000, following Martins-rivas and Mei ( 2009).The non-vanishing elements involved in MPTO are used to consider the effect of compressibility of air in the chamber.CPTO represents a diagonal matrix of the damping coefficients of the PTO written as diag(CPTO)=[0,0,cPTO,0] T , in which cPTO represents the PTO damping induced by the linear generator connected to the WEC; Cr is a matrix used to consider the damping effect induced by the aperture size of the roof, the volume flux created by the heaving motion of the WEC chamber, and the force on the horizontal roof of the WEC due to its inner pressure.These effects are reflected by the non-vanishing elements, cr, 2 i πR and 2 i πR − , located at the first row and the first column, the first row and the third column, and the third row and the first column of Cr, respectively.M and Ks are the mass matrix and hydrostatic stiffness matrix of the device.For the effect of hydrostatic stiffness on the air pressure enclosed by the chamber has already been included in radiation coefficients (Falnes, 2002), different from those for traditional floats, no separate term is required in Ks for the air pressure.We assume the WEC is half submerged at equilibrium with the mass uniformly distributed all over its chamber body.Km is the restoring stiffness matrix induced by the mooring lines.Here we consider mainly the spring effect on the heave motions, which is the most prominent influence of the mooring system, and disregard other effects, such as damping or inertia.Thus, there is only one non-vanishing element, located on the diagonal line as: diag(Km)=[0,0,km,0] T , where km is the moorings restoring force coefficient in heave mode of the WEC.Actually, the stiffness in the PTO system can also be treated as a part of km.
In regular waves, the time-averaged absorbed power of the novel WEC can be expressed as: The capture factor, also called the relative capture width, can be defined by in 2 where Pin represents the incoming wave power per unit width of the wave front (Zheng and Zhang, 2018).

Maximization of power absorption
Although the system has four degrees of freedom, Eq. ( 25), the surge motion (and also the pitch motion) are decoupled from the heave motion and the internal air pressure; therefore, the advantage of the hollow cylinder in terms of survivability thanks to its weaker surge motion still applies to the VAPA WEC.The heave motion used to capture wave power is only coupled with the air pressure enclosed by the WEC chamber.Therefore, a two DOF motion matrix equation as given below can be used also to evaluate the heave motion of VAPA: where in which s0 denotes the cross-sectional area of the device.The expression of the heave velocity can be derived as: ) The power absorbed in the PTO damping is: ) There are three variables involved in the expression of P, i.e., PTO c , km and cr.The more variables that are optimized at the same time, the more complicated the design/control system that is required, with the consequent difficulties for practical applications.Therefore, in addition to the optimization of two or three variables at the same time, the optimization of individual variables is considered in the following.

1) Optimization of the PTO damping coefficient
We note that P=0 for PTO c =0 and for PTO c =∞, and that P> 0 for 0 < PTO c <∞. Thus there is a maximum of absorbed power when ∂P/∂ PTO c = 0, which occurs if: where κ1 and κ2 are two real parameters introduced from ( ) in which κ1 is found and can also be proved positive regardless of the WEC scales (see Eq. (A1) in Appendix A).Note that both κ1 and κ2 are dependent of cr.Therefore, referring to Eq.
(32), the optimal PTO c for maximizing power absorption, i.e. ( ) , is influenced by both cr and km.
The corresponding maximum of absorbed power is 2) Optimization of the mooring stiffness With reference to Eq. ( 31), if only km is variable, the maximum power absorption occurs when km/ω+κ2=0, i.e., ( ) which is only affected by cr, regardless of PTO c .
The corresponding maximum of absorbed power is In practice, km should be non-negative, hence Eqs. ( 35) and ( 36) are rewritten as: 3) Optimization of the roof damping coefficient The analysis the effect of cr on the power absorption is obviously more complicated than those for the optimization of PTO c and km.After making some rearrangement, the power absorbed by the novel WEC as expressed in Eq. ( 31) can be rewritten as: There can be two different solutions of cr satisfying ∂P/∂cr=0.It is found through analytical experiments that only one of the two roots is positive, which is written as: ) where ξ1, ξ2, ξ3 and ξ4 are four real parameters introduced from According to Haskind relation, it can be known from Eqs.( 21)~( 23) that  40) minimizes power absorption rather than maximizes it.Hence the cr obtained from Eq. ( 40) can be denoted as cr,min.This is reasonable for the roof aperture exerts a linear damping, implying power dissipation, which results in diminished power absorption by the WEC.Therefore, cr,min may be seen as the optimal option for reducing the heave oscillation of the WEC, i.e., for survivability under extreme wave conditions.The corresponding minimum absorbed power ( ) r min P may be easily evaluated by substituting Eq. ( 40) into Eq.( 39).
The maximum power absorption can be evaluated after making a comparison between the results with cr=0 and ∞; its analytical expression is: for which the corresponding optimal cr is ( ) implying that to improve power capture width of the novel WEC, the roof should either be entirely open, or be completely closed.

4) Optimization of the PTO damping coefficient and the roof damping coefficient
The expressions of these optimal values of PTO c , km and cr as derived above are obtained when each of them is regarded as the only variable parameter.Furthermore, when both PTO c and cr can be arbitrarily specified, the maximum power could be: where in which ζ1 and ζ2 are two real parameters satisfying The corresponding optimal values of PTO c and cr are written as: Note: 1  can be treated as an special root of κ1 with cr =0, thus we have 1  >0 as well.

5) Optimization of the PTO damping coefficient and the mooring stiffness
It can be seen by inspection of Eqs. ( 32)~(34) that if PTO c and km can be chosen such that then the maximum absorbed power is Considering km to be non-negative, the maximum absorbed power and the corresponding optimized PTO c and km can be rewritten as 6) Optimization of the roof damping coefficient and the mooring stiffness Similar to the optimization of PTO c and cr, when both cr and km can be arbitrarily specified the maximum absorbed power is The corresponding optimal cr and km are written as ,, , 0, , With consideration of the non-negative property of km, Eqs. ( 54) and ( 56) can be rewritten as 57) where ( ) and 7) Optimization of the PTO damping coefficient, the roof damping coefficient and the mooring stiffness Furthermore, referring to Eq. ( 54), if PTO c is also included as a variable parameter in the optimization, i.e., if PTO c , cr and km are optimized concurrently, the maximum absorbed power can be written as: where in which P1 represents the maximum absorbed power by the WEC with its roof completely open; whereas P2 denotes the one when the roof is entirely closed.In fact, it can be proved that P2≡P1, as given in Eq. (A3) in Appendix A. Hence Eq.( 60) simplifies to: The corresponding optimal PTO c , cr and km have two solutions, wrtten as The capture factors corresponding to ( ) max P can be denoted as ( ) max  , in which from Eqs.
(21), ( 27) and ( 62), we have (64) Actually, with heave motion as the only mode of oscillation for any single axisymmetric body, it has been first derived independently by Budal and Falnes (1975); Evans (1976); Newman (1976) that the maximum absorption width (defined as the ratio between P and Pin) is equal to 1/k0.The results of Eq.( 64) reveals that when PTO c , cr and km can be optimized at the same time, the theoretical maximum absorbed power and wave capture factor of the novel WEC are all the same to those of a solid cylinder with the same radius, regardless of compressibility of the air in the chamber.
If we consider the mooring stiffness non-negative, we have:

Validation of the analytical model
The dimensionless quantities of the non-vanishing wave excitation volume flux/forces and hydrodynamic coefficients are defined by: where i=0 for j =1,2; whereas i=1 for j=3.

Wave diffraction and radiation
Nader (2013) applied a three-dimensional FEM (Finite Element Method) model to a heaving cylindrical OWC with the following dimensionless parameters: R/h=0.25,Ri/h=0.2,d/h=0.2.The FEM model is based on linear potential flow theory and the discretisation of the entire computational water domain into a finite number of elements, where the quantity of interest is approximated.Neither the wave excitation forces nor the hydrodynamic coefficients related to the surge or pitch modes were considered in this FEM model.The corresponding coefficients can be evaluated with commercial codes based on the conventional BEM (Boundary Element Method), such as WAMIT and ANSYS-AQWA.In this section, the present analytical model is applied to study wave diffraction and radiation from the VAPA WEC with the same basic dimensionless parameters used by Nader (2013).For validation the analytical results are compared with numerical results from both FEM (Nader, 2013) and ANSYS-AQWA (ANSYS AQWA, 2011) codes.
Figure 6 presents the results of wave excitation forces and volume flux using different methods.It is apparent that the analytical results agree well with those from other methods.Note that the case studied in this paper is a circular truncated cylinder with a circular moonpool; therefore, in addition to the plane y=0, x=0 is also a plane of symmetry.It follows that n1 and n3 in Eq. ( 20) are odd functions of x, whereas are satisfied for (j=1,2,3; j'=1,2,3) and (j=1,2,3; j'=0 or j=0; j'=1,2,3 ), respectively, for the wave radiation problem of the present case (Falnes, 2002).Therefore, the only nonvanishing off-diagonal elements of the radiation hydrodynamic matrix are    with the present analytical model and by trial and error.The trial and error ("brute force") method may be described as an exhaustive search approach characterized by repeated, varied attempts until success without any intelligent algorithms employed.In Figures 9 and 10 these two methods, the present model and trial and error, are employed to evaluate the maximum and minimum power absorption of the device when only cr can be varied.Figure 11 presents the results when both cr and km can be optimized.As shown in Figs.8~11, there is excellent agreement between the analytical and numerical optimization results, corroborating the correctness of the expressions derived in Section 2.7.

Results and discussion
In this section the validated analytical model is employed to investigate the power extraction by a VAPA WEC with the following dimensionless parameters: R/h=0.15,Ri/h=0.1, d/h=0.1.After solving the wave diffraction and radiation problems, the excitation forces and volume flux, and the hydrodynamic coefficients with respect to the oscillating water column and the heave motion of the chamber are presented in Figure 12.As shown in Fig. 12a, ( ) F reach their peak values of 6.44 and 0.97, respectively, at kh = 6.2 and 6.1.We have ( ) 2 e F = 0 at kh = 7.1.In Fig. 12b, 0,0 a = 0 occurs at kh = 6.2, which corresponds to the resonant wave frequency of the device as a fixed OWC.In Fig. 12c, apart from 2,2 a and 2,2 c , a combination parameter ( ) ( ) versus kh is also plotted into a black dot The power absorption of the novel WEC can be evaluated by combining the solutions of the diffraction/radiation problems with power take-off systems by means of Eq. ( 25). Figure 13 presents variation of η with kh for different r c , i.e., different aperture size of the roof, and m k =0, PTO c =5.As indicated, changing the size of the roof aperture leads to obvious changes in the frequency response of η.With the aperture entirely closed the device captures more power than with the aperture completely open for most wave conditions, except in the range 5.5 < kh < 6.1.As r c increases from 0 towards ∞, the kh corresponding to the peak of η-kh curve increases, whereas the peak value of η first decreases and then increases after reaching a minimum value.Among the six cases with different values of r c as plotted in Fig. 13, the minimum peak value of η is 0.46 occurring at kh=5.4 with r c =10, while the maximum peak value of η is 0.57 occurring at kh=5.7 with r c =∞, which is 1.24 times as large as the minimum one.In addition to the curves for the novel WEC, the power absorption of a conventional (solid cylinder) point-absorber with the same scales of R and d is plotted as well.It is found that the novel WEC with the roof entirely closed works almost all the same in absorbing power with traditional point-absorber using solid cylinder.Since the surge motion is decoupled from the heave motion, surge is not affected by the size of the roof aperture, i.e., the surge motion of the novel WEC is independent of the aperture size, and is the same as that of a hollow cylinder without a roof.This means that a hollow cylinder with the roof aperture completely closed performs similarly to a solid cylinder in capturing wave power; however, since the displacement of the hollow cylinder is much smaller than that of the solid cylinder, from a cost point of view, the novel WEC could be more attractive than traditional pointabsorber.Additionally, compared with the solid cylinder, the hollow cylinder might be advantageous in terms of survivability as well because it presents less motion in surge mode, as reported by Engström et al. (2017); Gravråkmo (2014) and Göteman (2017).


changes towards a bimodal curve, in which the second peak appears at kh=8.3 where resonance occurs.Meanwhile, a vanishing power absorption point is also obtained at kh=7.1.This is due to no wave excitation force acting on the chamber (see Fig. 12a) and very limited interacting air/hydrodynamic force exerted on the roof/chamber bottom because of the negligible air pressure.Although the peaks of ( ) .The comparison between Fig. 15a and Fig. 13 shows that the power absorption of the device can be significantly improved in short waves by properly increasing mooring stiffness.When the novel WEC is subjected to extreme waves, it may be required to restrict its heave motion for the sake of survivability.For the VAPA WEC, the air pressure within the cylinder can be modified by adjusting the roof aperture.This may be used to minimize the heave motion, which naturally reduces power capture.The contrary of Fig. 16, Fig. 17 presents the results of ( ) r min  and ( ) r min c when the power absorption of the device is minimized with a proper value of cr.Comparing the two figures it is apparent that ( ) r min  is much smaller than ( ) r max  .For example, the values of ( ) r max  at kh = 5.5 are 0.46, 0.58, 0.59, 0.55 and 0.44, respectively, for PTO c =1, 2, 3, 5 and 10; whereas the values of ( ) r min  are merely 0.34, 0.45, 0.47, 0.44 and 0.32, leading to a reduction in heaving amplitude of 13.9%, 11.7%, 10.6%, 10.8% and 14.0%, respectively.Under longer waves, e.g., kh=4.0, for PTO c =1, 2, 3, 5 and 10, a proper selection of the aperture size of the roof might result in the maximum reduction in heaving amplitude of 6.2%, 7.7%, 9.3%, 12.4% and 17.6%, respectively.
Another important aspect with reference to extreme waves is that viscous effects become relevant; under such conditions the linear model may overpredict the motion and power absorption of the WEC. is no more than 1/(2kR), which is the ratio of analytical maximum power capture width by a vertical asymmetrical heaving buoy relative to 2R.For large mooring stiffness, e.g., m k =1.0, 1.5 and 2.0, a bulge occurs at 5.5<kh<6.2,where the corresponding ( ) PTO,r opt,r c =∞.The sharp peak of the bulge occurs at kh=6.0 exactly, where the peak of 2,2 a happens as shown in Fig. 12c.c =∞ is 2.8 times as large as that for r c =0, implying that to achieve the same power absorption a much larger heave motion is required for the device with the roof completely open compared to that with the roof entirely closed.Given that the optimal mooring stiffness is independent of PTO c , as derived in Eqs. ( 35) and ( 50) , the values of ( ) PTO,m opt k (Fig. 19c) are found to be similar to those of ( ) m opt k (Fig. 15b).When the PTO damping coefficient is large enough, e.g., PTO c ≥ 5, the device without any roof covering has a better performance in power extraction for certain wave conditions, e.g., 5.4 < kh < 6.2 (Fig. 20b).Notwithstanding, for generic (unconstrained) wave conditions the device with the roof entirely closed is preferable.Thanks to the positive mooring stiffness for kh > 6.0 (Fig. 20c), the maximum power capture factor of the device can be increased significantly, which is apparent when comparing Figs.20a and 16a.For kh < 4.0, the device with a larger value of PTO c can capture more power from waves; however, for wave conditions with large frequencies such that kh > 6.5, a large value of

Conclusions
In this paper a novel WEC, Variable Aperture Point-Absorber (VAPA), was proposed; it consists of a hollow cylinder capped by a roof with an aperture of variable size.To extract wave power the cylinder is connected by a tether to a linear generator on the seabed.The characteristics of power absorption of the WEC can be modified by adjusting the aperture on the roof.To study the performance of VAPA, the wave diffraction and radiation problems are solved with an analytical model.The influence of the PTO system and the roof aperture is represented by linear damping coefficients.
The power absorption of the novel WEC was found to be strongly dependent on three parameters: the PTO damping coefficient, the roof aperture damping coefficient and the nonnegative mooring stiffness.A systematic analytical derivation of the maximum absorbed power was carried out under different optimization principles revolving around these three parameters.The following conclusions may be drawn.
First, changing the roof aperture modifies the frequency response of the wave capture factor.
Second, for unspecified wave conditions, the device generally captures more wave power with the roof aperture completely closed than with it completely open.Furthermore, with the roof aperture completely closed, the novel WEC performs similarly to a conventional (solid cylinder) point-absorber in terms of power capture.The VAPA WEC has, however, two significant advantages, a lower cost and enhanced survivability, thanks to its smaller displacement and lower surge motions.
Third, opening the roof aperture leads to a narrower bandwidth and a larger peak value of power capture relative to the configuration with the roof aperture closed.This may be advantageous when the wave conditions match the peak of the response of the device.
Fourth, if the configuration of the PTO is such that its damping can be tuned to the wave conditions, then increasing the size of the roof aperture leads gradually to a bimodal response, with the second peak (at kh = 8.3) corresponding to resonant conditions.This configuration would be ideal for bimodal sea states, when a swell and a wind sea coexist.
Fifth, the optimal mooring stiffness for the novel WEC was found to be independent of the PTO damping coefficient.Finally, the Variable Aperture Point-Absorber, VAPA, presents the best power absorption when the roof aperture is completely open or entirely closed for any specified wave conditions.Intermediate values of the roof aperture are preferable, however, in storm conditions, for the adequate aperture was found to minimize power extraction and heave motions -an advantage for survivability.
In sum, a novel WEC concept, Variable Aperture Point-Absorber (VAPA), was presented and investigated by means of an ad hoc analytical model.A thorough analysis was carried out to determine its performance and optimize the values of PTO damping, roof aperture damping and mooring stiffness for power capture.Unlike conventional point-absorbers, VAPA is capable of minimizing heave motions, hence forces on the mooring lines, under extreme wave conditions.This is a significant advantage in that it can be the difference between surviving a storm or not.
The wave power absorption of the VAPA WEC proposed in this work might be further enhanced to some extent by capturing the surge or pitch motion for power generation.However, this must be balanced with the greater cost and, possibly, smaller robustness under extreme sea states of the more complicated PTO system that would be required -which will be considered in future work.FF in terms of c0,0 and c2,2.


>0 is also satisfied regardless of PTO c and km which can be proved in Eq.(A2), as given in Appendix A. Since 3 >0, the value of cr calculated from Eq. ( induced by the linear generator ( PTO c ), the damping induced by the size of the aperture on the roof (cr), and the moorings restoring force coefficient (km) are normalized as: of the optimal PTO damping and the moorings restoring force coefficient corresponding to ( ) max P are denoted as( ) Fig. 6.Real and imaginary parts of the dimensionless wave excitation volume flux and forces against kh, (a) wave excitation volume flux; (b) surge wave excitation force; (c) heave wave excitation force; (d) pitch wave excitation moment.Note that the case studied in this paper is a circular truncated cylinder with a circular moonpool; therefore, in addition to the plane y=0, x=0 is also a plane of symmetry.It follows frequency domain as a function of kh are plotted in Fig.7.

Figure
Figure 8 presents the variation of ( ) PTO max 

Fig
Fig. 8. Variation of ( ) PTO max  Fig. 12. Dimensionless wave excitation force/volume flux and hydrodynamic coefficients regarding oscillating water column and heave motion of the chamber against kh for R/h=0.15,Ri/h=0.1, d/h=0.1:(a)

Fig. 13 .
Fig. 13.Variation of  with kh for different r c and m k =0, PTO c =5.The results as shown in Fig.13are those without optimization of any parameters.The maximum power extraction of the device with different optimization principles as derived in Section 2.7 is presented and discussed in the following sections.

Figc
Fig. 15.Variation of ( ) m max  and ( ) m opt k with kh for different values of r c ( PTO c =5).

Fig. 17 .
Fig. 17.Variation of ( ) r min  and ( ) r min c with kh for different PTO c and m k =0.

Fig
Fig. 19.Variation of ( Fig. 20.Variation of ( ) r,m max  , ( ) r,m opt c and ( ) r,m opt k with kh for different

Table 1
Comparison of the hydrodynamic coefficients using DM and the Haskind relationIn this section, we consider the case of a VAPA with dimensionless parameters: R/h=0.15,Ri/h=0.1, and d/h=0.1, as an example to validate the power absorption optimization by means of the analytical model.