Article Wave Energy Harnessing in Shallow Water through Oscillating Bodies
Marco Negri * and Stefano Malavasi
Department of Civil and Environmental Engineering, Politecnico di Milano, 20133 Milan, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-02-2399-6260
Received: 25 July 2018; Accepted: 10 October 2018; Published: 12 October 2018
Abstract: This paper deals with wave energy conversion in shallow water, analyzing the performance of two different oscillating-body systems. The first one is a heaving float, which is a system known in the literature. The second one is obtained by coupling the heaving float with a surging paddle. In order to check the different behaviors of the multibody system and the single-body heaving float, physical models of the two systems have been tested in a wave flume, by placing them at various water depths along a sloping bottom. The systems have been tested with monochromatic waves. For each water depth, several tests have been performed varying the geometrical and mechanical parameters of the two systems, in order to find their best configurations. It has been found that the multibody system is more energetic when the float and the paddle are close to each other. Capture width ratio has been found to significantly vary with water depth for both systems: in particular, capture width ratio of the heaving float (also within the multibody system) increases as water depth increases, while capture width ratio of the paddle (within the multibody system) increases as water depth decreases. At the end, the capture width ratio of the multibody system is almost always higher than that of the heaving float, and it increases as water depth increases on average; however, the multibody advantage over single body is significant for water depth less than the characteristic dimension of the system, and decreases as water depth increases.
Keywords: wave power; wave energy converter; heave; surge; shallow water; capture width ratio
1. Introduction Renewable energies are increasingly becoming a main topic of interest among scientists due to the need for sustainable development, the increasing energy world request and the expected depletion of fossil fuels. Among the third generation renewable energies, wave energy is one of the most promising. Since the second half of the last century many attempts to harvest wave energy have been made, but no device has yet to reach full commercial production, while several have achieved the prototype stage, e.g., the Wavestar (Wave Star Energy A/S, Brøndby, Denmark www.wavestarenergy.com), the Seabased AB (Seabased industry AB, Lysekil, Sweden www.seabased.com), the Ceto (Carnegie Wave Energy, Belmont, Australia https://www.carnegiece.com/wave/), and many other important ones like the Oyster (Aquamarine Power, Edinburgh, UK) and the Pelamis (Pelamis Wave Power, Edinburgh, UK) whose further development ceased. Wave energy converters (WECs) are very various and can be classified according different criteria: a common subdivision of WECs is according to their working principle [1], which are oscillating water column (OWC), oscillating body, overtopping. Other important classifications are according to location (offshore, nearshore, shoreline) and directional characteristic with respect to the oncoming wave (point absorber, attenuator, terminator) [2]. Reviews [1,2] are extensive works including the characterization
Energies 2018, 11, 2730; doi:10.3390/en11102730 www.mdpi.com/journal/energies Energies 2018, 11, 2730 2 of 17 of wave energy resource and the compendium of WEC technologies. A more recent review of wave energy technology can be found in reference [3]. Wave energy exploitation still has high investment costs, therefore, in the last ten years, hybrid solutions with other renewable sources like wind and solar have been proposed; this allows limiting installation costs and achieve a more regular energy production (see for example [4]). On the other hand, wave energy exploitation is economically advantageous for islands and remote coastal areas, where the electric grid is absent and fuel supply is difficult; reviews of wave energy and hybrid (wave, tidal, wind, solar) technologies suitable for these areas can be found respectively in references [5,6]. One of the principal issues with WECs is the ratio between energy production and costs. Offshore system (water depth >50 m) seems to be preferable due to the greater energy amount of the open sea waves, while shoreline systems usually have a lower energy availability, due to energy dissipation caused by the travelling of the waves in shallow waters. Exceptions to this generalization are those particular places (promontories or isles), called hot spots, where, for refraction and diffraction effects, a concentration of wave energy can be found [7]. On the other hand, offshore WEC have higher costs of installation, maintenance and connection to the electrical grid compared to shoreline system, due to obvious reasons of shore proximity. The reduction of wave energy from offshore to nearshore is better explained in reference [7]: If it is true that the gross wave energy significantly reduces, on the contrary the exploitable wave energy can be quite similar. The exploitable wave energy is directional-resolved and does not include the sea storms. Folley and Whittaker [7] found, analyzing two sea sites in North Atlantic coast of Scotland, that the exploitable wave energy is reduced only by 7% and 22% from offshore to nearshore. In the same work, it is also shown that the bottom slope influences wave energy dissipation, and that a steeper bottom causes less energy dissipation by bottom friction, as waves travel a shorter distance in shallow depths to reach the nearshore zone. In reference [8], numerical simulations of wave propagation along a sloping beach are performed, considering various wave heights and periods, and beach slopes. It is shown that, for example, a wave with significant wave height Hm0 = 4 m loses only about 20% of its power when travelling from deep water to a depth of 10 m, on a 1:100 sloping beach. Thanks to these considerations we can understand why nearshore sites have been revalued for wave energy harvesting, and several studies are being made to estimate the energy availability in many nearshore places, e.g., references [9–11]. Indeed, wave energy dissipation in the nearshore zone depends on so many factors, like wave properties, angle of propagation, bottom slope and bottom roughness [12] so that wave energy availability may significantly vary along the coast, and each site should be individually characterized. Besides water depth, also local irregularities of the seabed affect the WEC behavior, due to their influence on the wave propagation (see for example [13]). Therefore, localizing a suitable position for a WEC is a matter of fundamental importance; in reference [14], the numerical simulation of various well-known WECs in several points of the Portuguese nearshore also showed that the local energy availability is not the only factor affecting the WEC energy production. It is known that water particle trajectories in shallow water waves are different than the ones in deep water; the elliptic shape of the trajectories causes nearshore waves to have a higher surge force than offshore waves; in reference [15] it is shown that the capture width ratio (CWR) of a surging oscillating flap in 10 m water depth can be more than double than in deep water. For the Oyster, a bottom-hinged pitching-surging flap of 12 m of width installed in 12 m water depth (the flap was surface-piercing), CWR up to 70% was evaluated [16]. Nearshore sites are still good places for heave oscillating bodies, as evidenced by the Danish Wavestar prototype located at Hanstholm, which is placed in a water depth of about 6 m. The Wavestar prototype is composed of two hemispherical pitching-heaving floats of 5 m diameter, each of which is supported by an arm that is pivoted to a fixed structure above water [17]. A peculiarity of Wavestar is that the floats are aligned along the wave propagation direction, in order to supply a continuous force to the power take-off (PTO) system as wave travels through the floats. Energies 2018, 11, 2730 3 of 17
Energies 2018, 11, x FOR PEER REVIEW 3 of 17 In this work, we have compared the performances of two similar WECs in shallow water placed on a slopingIn this bottom,work, we in have water compared depths the of theperformances order of the of two WEC similar diameter. WECs One in shallow is a pitching-heaving water placed float,on a similar sloping to bottom, the Wavestar in water system depths [ 17of ],the except order for of the WEC fact that diameter. the arm One that is a supports pitching-heaving the float is alignedfloat, similar along waveto the propagationWavestar system direction [17], except (hereinafter for the we fact will that refer the arm to this that system supports as “float-only”).the float is Thealigned heaving along float wave is thepropagation most basic direction oscillating-body (hereinafter WECwe will (see refer reference to this system [1] for as some “float-only”). examples). Generally,The heaving the heaving float is floatthe most is studied basic inoscillating-b deep waterody (>40 WEC m) and,(see toreference the author [1] knowledge,for some examples). there is not a specificGenerally, study the onheaving the influence float is studied of water in depth; deep wate in thisr (>40 work, m) weand, investigate to the author the knowledge, behavior of there a heaving is floatnot in a waterspecific depth study ranging on the influence between 0.6of water and 2 depth; float diameters. in this work, we investigate the behavior of a heavingThe other float in WEC water tested depth in ranging this work between is the 0.6 Energy and 2 float Double diameters. System (EDS), a multibody system that consistsThe other of theWEC pitching-heaving tested in this work float is the plus Energy a pitching-surging Double System (EDS), paddle a (Figuremultibody1). systemBy removing that consists of the pitching-heaving float plus a pitching-surging paddle (Figure 1). By removing the the paddle from EDS, the float-only system is obtained. In the perspective of an array arrangement, paddle from EDS, the float-only system is obtained. In the perspective of an array arrangement, the the EDS modules would be aligned parallel to wave front, unlike Wavestar system whose modules are EDS modules would be aligned parallel to wave front, unlike Wavestar system whose modules are orthogonal to the wave front. In the EDS system, the paddle is supported by an arm hinged on the main orthogonal to the wave front. In the EDS system, the paddle is supported by an arm hinged on the arm. The original idea behind this WEC is to harness both heave and surge wave forces, in order to ensure main arm. The original idea behind this WEC is to harness both heave and surge wave forces, in order a largeto ensure exploitation a large exploitation of the wave resourceof the wave and resource to obtain and a good to obtain flexibility a good toward flexibility different toward sea different conditions. seaUsually, conditions. multibody WECs are composed by two bodies that exploit the same wave force (typically heave force,Usually, see referencemultibody [ 1WECs]); the advantageare composed of these by WECstwo bodies is that that the PTOexploit can the use same the relative wave force motion between(typically the heave two bodiesforce, see instead reference of reacting [1]); the againstadvantage a fixed of these point WECs on theis that seabed the PTO or on can the use ground. the Inrelative this way, motion there between is the additional the two bodies benefit instead of avoiding of reacting the practical against a issues fixed ofpoint foundation on the seabed construction. or on Examplesthe ground. of heaving In this way, two-bodies there is WECsthe additional are [18, 19benefit]. of avoiding the practical issues of foundation construction.The peculiarity Examples of EDS of heaving is thatthe two-bodies two bodies WECs composing are [18,19]. it exploit different wave forces, heave and surge.The Inpeculiarity this work, of weEDS analyze is that the the two advantages bodies co ofmposing coupling it exploit a heaving different float andwave a surgingforces, heave paddle inand a sole surge. WEC, In andthis work, we verify we analyze whether the it advantages is energetically of coupling advantageous a heaving compared float and to a surging the heaving paddle float, givenin a thatsole WEC, the paddle and we would verify be whether a relatively it is energe simpletically addition advantageous to the latter. compared to the heaving float, givenEDS that has the some paddle important would advantages,be a relatively such simple the relatively addition to low the costs latter. due to its nearshore positioning, the possibilityEDS has to besome attached important to existing advantages, structures, such and the the relatively modularity. low These costs characteristicsdue to its nearshore make EDS particularlypositioning, suitable the possibility for ports, whereto be piersattached and breakwatersto existing structures, would provide and athe base modularity. to attach the These system orcharacteristics lay the foundation make ofEDS EDS. particularly The energy suitable produced for ports, could where be directly piers and utilized breakwaters for port would operations provide and a base to attach the system or lay the foundation of EDS. The energy produced could be directly for providing electrical power to the ships while stationary (cold ironing), increasing the environmental utilized for port operations and for providing electrical power to the ships while stationary (cold sustainability of the port. The natural variation during the year of the available wave energy fits with ironing), increasing the environmental sustainability of the port. The natural variation during the the activity of the boats inside the port: most of the yearly wave energy comes from autumn to spring, year of the available wave energy fits with the activity of the boats inside the port: most of the yearly and this is also when more energy is required because of the dry berthing and maintenance of the wave energy comes from autumn to spring, and this is also when more energy is required because of boats.the dry Insummer, berthing whenand maintenance the available of wavethe boats. energy In summer, and the energywhen the demand available for wave boat energy management and the are smaller,energy EDS demand could for even boat bemanagement stored out are of water,smaller, especially EDS could if even it impedes be stored the out boat of trafficwater, throughespecially the port,if it which impedes is greaterthe boat in traffic the hot through season. the port, which is greater in the hot season.
FigureFigure 1.1. EDSEDS concept.concept. ( aa)) side side view; view; (b (b) )top top view. view.
EDSEDS is is a a point point absorber/terminatorabsorber/terminator WEC: WEC: it is it de issigned designed for harnessing for harnessing waves waves shortly shortly before they before theybreak, break, when when their their energy energy content content is still is relevant still relevant.. The ideal The depth ideal is depth 3–10 m, is depending 3–10 m, depending on the sea on
Energies 2018, 11, 2730 4 of 17 the sea spectrum and site characteristics. Both paddle and float are “light systems”, as they are not resonant with waves. Paddle predominately works in the direction of wave propagation, pushed by the drag force originated by the mass transport of waves just before they break. A paddle of this type is potentially more energetic in shallow water than in deep water. Hazlett et al. [20] tested a similar paddle in deep water: it was a surface-piercing or half-immersed paddle that had the possibility to translate along the wave propagation direction, pushed by the waves. They measured a maximum CWR of 15% when the paddle was artificially driven so that it stayed on the wave crest; otherwise, CWR did not exceed 5%. A first scale model of EDS was studied in reference [21], providing early indications about system potentiality. In this work the EDS model has been significantly improved to make it more realistic and the influence of several parameters on system performance has been deeply investigated. The first part of this work deals with the description of the experimental model. Then, the results of the experimental tests on the float-only system and on EDS are presented. Finally, the performances of the two systems are discussed in more details, and they are compared with the ones of the the Wavestar lab model [22].
2. Experimental Setup and Method This part is subdivided in three sections: the first one describes the laboratory model of the EDS, the second one describes the data processing calculation of the power absorbed by the EDS model, and the third one describes the laboratory waves used in the experiments.
2.1. EDS Scale Model The laboratory model of EDS is shown in Figure2. It has been realized at 1:25 scale of the project prototype, and it is composed of a main arm OA, which supports the float and which is pivoted at O above the water surface. Another arm CE supports the paddle and it is pivoted in point C on the main arm, between the pivot O and the float. The two degrees of freedom of the system are θ1 and θ2, angles of rotation of the float around O pivot, and of the paddle around C pivot, respectively. θ1 and θ2 are both absolute coordinates: they express rotations around the equilibrium position of the main arm and paddle arm respectively. The float is composed by a hemispherical part at the bottom with radius 0.107 m and height 0.075 m, and a cylindrical upper part with diameter 0.204 m and height 0.062 m. Total height of the float is 0.137 m. The paddle has a cylindrical shape with radius 0.104 m and a chord of 0.12 m. The paddle width is 0.204 m. Main characteristics of the model and their uncertainties are reported in Table1. In order to support the float PTO, the main arm extends on the left hand side of O pivot up to point B. Paddle PTO is placed between the paddle arm (point D) and the main arm (point F). Both PTOs are heaving discs in not-pressurized oil (density 878 kg/mc, dynamic viscosity 0.12 Pa·s at 20◦), connected through leverages to the main arm or to the paddle arm. The float-only system is obtained by removing the paddle apparatus, which consists of the paddle, its arm (CE) and its PTO with the leverages (DF); in this way the system turns into a one-degree-of-freedom system and it becomes similar to the Wavestar model studied in reference [22]. With respect to the EDS model of the previous work [21], the main improvements made in this work are:
(1) Float geometry: the shape of the float has been improved, by further rounding the basis of the float, making it more hydrodynamic. Moreover, the new float is shorter, the cylindrical part is shorter and the spherical part is taller. This make the new float quite similar to the one used in [22]. (2) Inertia: also related to the float geometry, the inertia of the new model is lower. This make the new model more realistic. (3) Paddle PTO: the new model has a dissipative system for measuring energy absorbed by the paddle, while in the previous model the absorbed energy was estimated by the elongation of a spring connected to the paddle. That was a conservative system, hence the absorbed energy could not be accurately measured. Energies 2018, 11, 2730 5 of 17 Energies 2018, 11, x FOR PEER REVIEW 5 of 17
FigureFigure 2. 2.(a ()a EDS) EDS scheme schememodel; model; ((bb)) EDSEDS lab model; model; ( (cc)) EDS EDS lab lab model, model, particular. particular.
TableTable 1. 1.Characteristics Characteristics ofof the EDS laboratory model. model.
CharacteristicCharacteristic Symbol Symbol Value Value Float arm length l1 [m] 0.4 ± 0.0005 Float arm length l [m] 0.4 ± 0.0005 Main arm inclination α [°]1 25 ± 1 Main arm inclination α [◦] 25 ± 1 Paddle height sP [m] 0.12 ± 0.0005 Paddle height sP [m] 0.12 ± 0.0005 Paddle draft dP [m] 0.09 ± 0.001 Paddle draft dP [m] 0.09 ± 0.001 Float draft dF [m] 0.06 ± 0.001 Float draft dF [m] 0.06 ± 0.001 DiameterDiameter of of the the float, float, paddle paddle width width D [m]D [m] 0.204 0.204 ± ±0.00050.0005 2 MomentMoment of ofinertia inertia I [kgI [kgm ] m2] 0.450.45 ± ±0.020.02 HydrostaticHydrostatic stiffness stiffness of offloat float k [Nm/rad]k [Nm/rad] 36 36 ± ±1 1 Natural period of the float-only system TN [s] 0.84 ± 0.01 Natural period of the float-only system TN [s] 0.84 ± 0.01
Energies 2018, 11, 2730 6 of 17 Energies 2018, 11, x FOR PEER REVIEW 6 of 17
FigureFigure3 is 3 ais schematic a schematic of of the the PTO PTO systems systems of of float float and and paddle. paddle. TheThe float float damper damper is is a a single single heavingheaving disc, disc, supported supported by by a a stem stem that that can can slideslide throughthrough bearings.bearings. The The bearings bearings are are fixed fixed at atthe the cylinder,cylinder, which which is fixedis fixed to to the the ground. ground. The The amount amount ofof dampingdamping was changed changed by by varying varying the the diameter diameter of theof the heaving heaving disc. disc. The The diameters diameters of of the the discs discs usedused werewere 93, 96, 99, 100 100 mm; mm; the the internal internal diameter diameter of of thethe cylinder cylinder was was 104.8 104.8 mm. mm.
FigureFigure 3. 3.PTO PTO of offloat float ((aa)) andand paddlepaddle ( (b).).
TheThe paddle paddle damper damper is similar, is similar, except except for two for characteristics.two characteristics. The firstThe one first is one that is the that stem the supports stem twosupports heaving two discs, heaving a porous discs, one a porous fixed on one the fixed stem, on and the anotherstem, and one another that can one slide that can along slide the along stem the for a shortstem distance. for a short This distance. system allowsThis sy forstem a differentallows for amount a different of damping amount inof thedamping two ways in the of two motion ways of of the stem.motion Depending of the stem. on theDepending way of motion,on the way the of discs motion, are attachedthe discs are together attached or detachedtogether or and detached the holes and of thethe fixed holes disc of arethe closedfixed disc or are freed. closed The or diameter freed. The of di theameter fixed of disc the fixed is 23 disc mm, is with 23 mm, eight with holes eight of holes 3 mm diameter;of 3 mm the diameter; diameter the of diameter the cylinder of the iscylinder 24 mm. isThe 24 mm. other The difference other difference between between the float the and float paddle and PTOpaddle is that PTO the latteris that reacts the latter against reacts the against main armthe main in point arm F in and point not F against and not the against ground. the ground. Point F isPoint at the leftF hand is at the side left of hand point side O (see of point Figure O3 (see) and Figure the length 3) and of the segment length of OF segment (see Figure OF (see2a) isFigure0.130 2a)± is0.005 0.130 m. A sensor± 0.005 distancem. A sensor and distance a load celland werea load placed cell were on placed each PTO on each system, PTO insystem, order in to order measure to measure the relative the relative motion 𝑥(𝑡) between the disc and the cylinder and the force exerted on the disc. motion x(t) between the disc and the cylinder and the force exerted on the disc.
2.2.2.2. Measurement Measurement Method Method
The distance sensors measure the relative position x1 and x2 between the stem and the cylinder The distance sensors measure the relative position x1 and x2 between the stem and the cylinder (x1 for the float PTO, x2 for the paddle PTO). Note that x2 measures the lengthening and shortening of (x for the float PTO, x for the paddle PTO). Note that x measures the lengthening and shortening 1 FD , which is embedded2 between the main arm and the paddle2 arm, so it is a relative coordinate. The of FD, which is embedded between the main arm and the paddle arm, so it is a relative coordinate. coordinates 𝜃 and 𝜃 are calculated from the coordinates x1 and x2. The load cells measure the total The coordinates θ1 and θ2 are calculated from the coordinates x1 and x2. The load cells measure the forces F1 and F2. F F total forcesSince 1theand motion2. is periodic, all measured signals are phase-averaged taking the position of the Since the motion is periodic, all measured signals are phase-averaged taking the position of the disc x1(t) as phase reference; in this way the mean oscillation cycle (whose duration is equal to the discwavex1(t )period as phase 𝑇) of reference; the measured in this quantities way the are mean derived. oscillation cycle (whose duration is equal to the wave periodThe totalT) of dynamic the measured force 𝐹 quantities acting on arethe derived.disc can be calculated by subtracting the static forces 0 (weightThe total and dynamicconstant part force ofF buoyancyacting on of the the discstem can and be the calculated disc) from bythe subtracting total measured the force static F forces (weightThen, and constant the damping part of force buoyancy on the of discs the stem Fd can and be the obtained disc) from by thesubtracting total measured the time-varying force F. buoyancyThen, the −𝜌𝑔𝐴 damping 𝑥 and force the oninertia thediscs (𝑚+𝑚Fd can )𝑥 be from obtained the total by dynamic subtracting force the 𝐹 time-varying: buoyancy .. 0 −ρgAsx and the inertia (m + mA)x from the total dynamic force F : 𝐹 =𝐹 +𝜌𝑔𝐴 𝑥−(𝑚+𝑚 )𝑥 (1) 0 .. Fd = F + ρgAsx − (m + mA)x (1)
Energies 2018, 11, 2730 7 of 17
where ρ is the oil density, As is the wet section of the stem, m is the total mass of disc and stem, mA is the added mass of the disc. If x(t) is periodic and differentiable, the energy E12 dissipated by the disc between time instants t1 and t2 where disc velocity vanishes is independent on inertia and stiffness forces, and it is:
Z t2 Z t2 Z t2 . 0 . E12 = Pdt ≡ Fdxdt = F xdt (2) t1 t1 t1 where P is the instantaneous power dissipated by the disc. E12 represents the mechanical energy extracted by the float or by the paddle from the waves during the time interval included between t1 and t2, which can be the instants that subdivide a whole cycle or the ascent and descent phases of the 1 3 cycle (nominally half-cycle). The disk added mass is equal to mA = CAmA0, where mA0 = 3 ρD is the theoretical added mass for a heaving disc in unbounded flow [23]. The coefficient CA takes into account the confinements effects; it was calculated using the Fourier analysis, according to [23]:
Z T 1 0 CA = F (t) sin(ωt)dt (3) ωXπmA0 0 where X is the motion amplitude and ω = 2π/T is the motion (and wave) pulsation. This optimization gave CA ≈ 2 for all the discs used. It must be said that the value of CA does not affect the estimation of the energy dissipated by the disc during the oscillation cycle and during the ascent and descent phases, but only little affects the estimation of instantaneous damping force and dissipated power. As an example, Figure4 shows the phase-averaged temporal story of the heaving disc of float PTO. Position x1, velocity, forces and dissipated power are reported. In order to calculate the equivalent linear damping of the PTOs, we calculated the harmonic approximations θe1(t) and θe2(t) of the phase-averaged rotations of the main arm θ1ph(t) and of the paddle arm θ2ph(t) for each test.
iωt θe1(t) ≡ Re Θˆ 1e iωt (4) θe2(t) ≡ Re Θˆ 2e
The complex Fourier coefficients of the first harmonic Θˆ 1 and Θˆ 2 were calculated as (for simplicity in the continuous form): T Θˆ = 1 R θ (t)e−iωtdt 1 2T 0 1ph (5) ˆ 1 R T −iωt Θ2 = 2T 0 θ2ph(t)e dt
Then, the equivalent linear torque damping b1eq and b2eq applied to the float and to the paddle are:
E E ≡ 1 = 1 b1eq . 2 2 (6) R T ωπΘ1 0 θe1dt
E2 E2 b2eq ≡ = (7) . . 2 2 2 R T ωπ Θ1 + Θ2 − 2Θ1Θ2 cos(φ1 − φ2) 0 θe2 − θe1 dt where Θ1, Θ2 are the real amplitudes and φ1, φ2 are the phases, calculated from the complex amplitudes Θˆ 1 and Θˆ 2. Equation (7) takes into account the rotation of the main arm θ1 as well, because θ2 is an absolute coordinate, while paddle damping is embedded between the main arm and the paddle arm. In order to give a more realistic estimation of paddle damping, which was intentionally different in the forward and backward phases, the subsequent expressions were used: Energies 2018, 11, 2730 8 of 17
E2, f orth b = (8) 2eq, f orth . . 2 1 R T 2 0 θe2 − θe1 dt Energies 2018, 11, x FOR PEER REVIEW 8 of 17
E2,back b = 𝐸 , (9) 𝑏 , 2eq,back= . . 2 1 (9) 1 𝜃R T −𝜃 𝑑𝑡 2 2 0 θe2 − θe1 dt where 𝐸 , and 𝐸 , are the energy dissipated in the forward and backward phases. where E2, f orth and E2,back are the energy dissipated in the forward and backward phases.
Figure 4. Phase averaged oscillation of the heaving disc of the float. hF/D = 0.7, wave A (see Table2),
discFigure diameter 4. Phase 96 mm. averaged (a) Position oscillation and of dissipatedthe heaving power;disc of the (b )float. Forces; hF/D ( =c )0.7, Velocity wave A and (see damping Table 2), force. Thedisc total diameter dynamic 96 forcemm. ( ina) Position (b) is the and total dissipated measured power; force (b) minus Forces; the (c) staticVelocity contributions and damping (weight force. and constantThe total part dynamic of buoyancy force in of (b the) is discthe total and measured stem). force minus the static contributions (weight and constant part of buoyancy of the disc and stem). For the float PTO, four discs of different diameter were used, in order to generate a different amount ofFor damping: the float PTO, depending four discs on theof different disc diameter, diameter equivalent were used, linear in order torque to generate damping a different ranged from 2 to 5amount Nms/rad. of damping: For the paddledepending PTO, on the disc same diameter, disc configuration equivalent linear was torque used indamping all tests. ranged The equivalentfrom 2 to 5 Nms/rad. For the paddle PTO, the same disc configuration was used in all tests. The equivalent linear torque damping generated on the paddle was b2eq,forth = 0.5 Nms/rad in the forward phase and linear torque damping generated on the paddle was b2eq,forth = 0.5 Nms/rad in the forward phase and b2eq,back = 0.2 Nms/rad in the backward phase. b2eq,back = 0.2 Nms/rad in the backward phase. TestsTests made made on on the the float-only float-only configurationconfiguration were were al alsoso useful useful to quantify to quantify the benefits the benefits brought brought by by the paddlethe paddle to theto the system; system; to to make make aa reasonablereasonable comparison, comparison, we we decided decided to set to the set same the same mass massin the in the twotwo configurations. configurations. System System mass mass is oneis one of of the the parameters parameters that that most most affects affects response response motionmotion andand CWR; moreover,CWR; moreover, it is linked it is to linked the structureto the structure costs. costs. Hence, Hence, the the inertia inertia moment moment aroundaround O O was was the the same same for the float-onlyfor the float-only system system and forand the for EDS,the EDS, the the latter latte beingr being calculated calculated considering the the second second degree degree of of freedom blocked (𝜃 =𝜃 , and paddle arm in vertical position at rest). Moment of inertia I is freedom blocked (θ2 = θ1 , and paddle arm in vertical position at rest). Moment of inertia I is reported reported in Table 1. Natural period TN of the float-only configuration (Table 1) was measured through in Table1. Natural period TN of the float-only configuration (Table1) was measured through free oscillationsfree oscillations in calm in water, calm water, by giving by giving an initial an initial displacement displacement to to the the float float and and then then releasingreleasing it. it.
2.3.2.3. Laboratory Laboratory Waves Waves The EDS model was tested inside the wave flume of the Hydraulics Laboratory of Politecnico di The EDS model was tested inside the wave flume of the Hydraulics Laboratory of Politecnico di Milano. The scheme of the wave flume is shown in Figure 5. The flume was 30 m long, 1 m wide, 0.7 Milano.m high The and scheme it was ofequipped the wave with flume a piston is shown wavemaker. in Figure A smooth5. The beach flume with was a 30 slope m long, of 7° was 1 m placed wide, 0.7 m ◦ highat and the itend was of equippedthe channel. with The awater piston depth wavemaker. in the channel A smooth was the beach same for with alla the slope tests, of h7 = 0.4was m. placed at theWave end properties of the channel. were measured The water by capacitive depth in thewave channel gauges, was without the samethe wave for allenergy the tests,converterh = 0.4 m. Wavebeing properties in the channel. were measured by capacitive wave gauges, without the wave energy converter being in the channel.
EnergiesEnergies2018 2018, 11, 11, 2730, x FOR PEER REVIEW 99 ofof 17 17
FigureFigure 5. 5.Scheme Scheme of of the the wave wave flume. flume.
TheThe methodmethod ofof Goda & Suzuki Suzuki [24] [24 ]was was used used to to separate separate the the incident incident and and reflected reflected waves. waves. The Thecoefficient coefficient of reflection of reflection was was similar similar for for the the three three waves waves considered. considered. Waves Waves properties properties are are reported reported in inTable Table 2.2 .
Table 2. Laboratory waves. Table 2. Laboratory waves. Incident Reflected Theoretical Experimental Incident Wave Incident Reflected Wave Theoretical Incident Reflection Wave Wave Wave Wave Wave ExperimentalWave Wave Reflection Type Wave Wave Period Wave Wave Coefficient Type Amplitude Amplitude Period Length WaveLength Length Power Coefficient Amplitude Amplitude Length Power aI [mm] aR [mm] T [s] Lth [m] Lexp [m] PI[W/m] r [%] aI [mm] aR [mm] T [s] Lth [m] Lexp [m] PI[W/m] r [%] WaveWave A 24.4 1.6 1.02 1.52 1.50 2.70 6.4 Wave B24.4 25.4 1.6 1.6 1.02 1.36 1.52 2.29 2.30 1.50 3.972.70 6.26.4 A Wave C 31.7 2.1 1.20 1.92 1.91 5.45 6.8 Wave 25.4 1.6 1.36 2.29 2.30 3.97 6.2 B WaveWave power per unit crest width was calculated from linear theory (see for example [25]): 31.7 2.1 1.20 1.92 1.91 5.45 6.8 C 1 2 PI = ρg(2aI ) Cg (10) Wave power per unit crest width was calculated8 from linear theory (see for example [25]): 1 where Cg is the group celerity, calculated𝑃 according= 𝜌𝑔(2𝑎 to) the𝐶 water depth in the channel and using the 8 (10) dispersion relation (see for example [25]). whereThe C waveg is the length group was celerity, also experimentallycalculated according measured, to the through water depth cross in correlation the channel of theand signals using the of thedispersion two wave relation gauges. (see In for Table example2 the theoretical [25]). and experimental values of wave length are reported. Finally,The the wave capture length width was ratio also CWR experimentally of EDS was measur calculateded, through as: cross correlation of the signals of the two wave gauges. In Table 2 the theoretical and experimental values of wave length are reported. E1+E2 Finally, the capture width ratio CWR of EDS wasPa calculatedT as: CWR = 𝐸= +𝐸 (11) PI D PI D 𝑃 𝑇 CWR = = (11) where Pa is the average power absorbed by the𝑃 𝐷 system𝑃 𝐷 and PI D is the wave power of a wave front a 𝑃 𝐷 whosewhere width P is the isequal average to thepower width absorbed of the wave by the energy system converter. and E is1 andthe waveE2 are power the energy of a wave absorbed front whose width is equal to the width of the wave energy converter. E1 and E2 are the energy absorbed by float and paddle during one oscillation cycle, T is the wave period, D is the float diameter, PI the waveby float power and for paddle unit crestduring width one calculatedoscillation on cycle, the basisT is the of thewave incident period, wave D is height.the float For diameter, the float-only 𝑃 the wave power for unit crest width calculated on the basis of the incident wave height. For the float- system E2 = 0. only system E2 = 0. 3. Results 3. Results Float-only system was tested first. The system was not in resonance with any of the waves tested, becauseFloat-only its natural system period was was tested smaller first. thanThe system any wave was periods.not in resonance The water with level any in of thethe channelwaves tested, was thebecause same its for natural all tests, periodh = 0.4 was m. smaller The system than wasany testedwave periods. at different The distances water level from in the the channel shoreline was (and the hencesame differentfor all tests, depths h = 0.4hF m.), andThe withsystem different was tested values at different of damping distancesb1. The from position the shoreline of the system (and hence was varieddifferent by rigidlydepths moving,hF), and with along different the channel, values the of structure damping that b1. The supports position the of WEC the modelsystem(the was points varied fixedby rigidly on the moving, structure along are the the pivot channel, O and the the structur cylindere that of the supports float damping the WEC system). model In(the the points condition fixed ofon the the maximum structure are proximity the pivot of O the and system the cylinder to the shoreof thethat float was damping tested, system). water depth In the at condition the float of was the hmaximumF = 0.12 m (proximityhF/D = 0.6); of thisthe system was the to limit the beyondshore that which was the tested, float touchedwater depth thebottom at the float when was oscillating hF = 0.12 underm (hF/ theD = wave 0.6); this tested. was the limit beyond which the float touched the bottom when oscillating under the wave tested.
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Energies 2018, 11, x FOR PEER REVIEW 10 of 17 Energies 2018, 11, x FOR PEER REVIEW 10 of 17 FiguresFigures6– 86–8 show show the the CWR CWR of of the the float-only float-only system,system, for the the three three waves waves considered considered (waves (waves A, A, B, B, C respectively).Figures 6–8 The show dimensionless the CWR of waterthe float-only depth h system/D is, reportedfor the three inthe waves abscissa. considered The diameter (waves A, of B, the C respectively). The dimensionless water depth hFF/D is reported in the abscissa. The diameter of the C respectively). The dimensionless water depth hF/D is reported in the abscissa. The diameter of the circlescircles is proportionalis proportional to to the the equivalent equivalent linearlinear torquetorque damping applied applied to to th thee float, float, whose whose amount amount is is circles is proportional to the equivalent linear torque damping applied to the float, whose amount is reportedreported in in the the legend. legend. For For each eachh Fh/F/D, different values values of of damping damping fo forcesrces were were tested, tested, but but optimal optimal reported in the legend. For each hF/D, different values of damping forces were tested, but optimal dampingdamping was was not not reached reached in in any any cases. cases. ForFor aa givengiven hhFF/DD, ,it it can can be be said said that that optimal optimal damping damping was was damping was not reached in any cases. For a given hF/D, it can be said that optimal damping was approximatelyapproximately reached reached if if the the uppermost uppermost circlecircle stands over over both both a a smaller smaller and and a abigger bigger circle, circle, which which approximately reached if the uppermost circle stands over both a smaller and a bigger circle, which meansmeans that that the the damping-CWR damping-CWR trend trend at that at thathF/D hshowsF/D shows a relative a relative maximum; maximum; thus thethus distance the distance between means that the damping-CWR trend at that hF/D shows a relative maximum; thus the distance thebetween circles atthe a givencircles hatF /Da givenprovides hF/D anprovides indication an indication about how about damping how damping affectsCWR. affects CWR. between the circles at a given hF/D provides an indication about how damping affects CWR. ItIt can can be be seen seen that that maximum maximum CWRCWR meanlymeanly incr increaseseases as as water water depth depth increases increases and and it ithas has an an It can be seen that maximum CWR meanly increases as water depth increases and it has an oscillating trend. This is reasonable due to the fact that wave reflection from the beach causes the oscillatingoscillating trend. trend. This This is reasonableis reasonable due due to theto the fact fa thatct that wave wave reflection reflection from from the the beach beach causes causes the the wave wave height to oscillate along the channel (i.e., [26]). Figure 6 shows that optimal damping for wave heightwave to height oscillate to oscillate along the along channel the channel (i.e., [26 (i.e.,]). Figure [26]). Figure6 shows 6 shows that optimal that optimal damping damping for wave for wave A does A does not vary much with water depth, corresponding often to the small circles, 2–3 Nms/rad. For notA varydoes muchnot vary with much water with depth, water corresponding depth, correspondi oftenng to often the small to the circles, small circles, 2–3 Nms/rad. 2–3 Nms/rad. For wave For B wave B and C (Figures 7 and 8), the dimension of the uppermost circles (that indicates the optimal andwave C (Figures B and C7 and(Figures8), the 7 and dimension 8), the dimension of the uppermost of the uppermost circles (that circles indicates (that indicates the optimal the damping)optimal damping) seems to fairly vary with depth, but it is misleading: due to the fact that the same range of seemsdamping) to fairly seems vary to fairly with vary depth, with but depth, it is misleading:but it is misleading: due to due the to fact the that fact thethat samethe same range range of floatof float damping was utilized for the three waves, CWR variability with respect to damping is smaller dampingfloat damping was utilized was utilized for the for three the waves, three waves, CWR variabilityCWR variability with respect with respect to damping to damping is smaller is smaller as T/T N as T/TN increases, where T is the wave period. The mean optimal damping for the three waves, increases,as T/TN whereincreases,T is where the wave T is period.the wave The period. mean The optimal mean damping optimal fordamping the three for waves,the three obtained waves, by obtained by averaging the optimal damping at each hF/D, is b1eq,opt = 2.51, 3.41, 3.16 Nms/rad, averagingobtained theby optimalaveraging damping the optimal at each dampingh /D, isatb each =hF 2.51,/D, is 3.41, b1eq,opt 3.16 = 2.51, Nms/rad, 3.41, 3.16 respectively Nms/rad, for respectively for waves A, B, C; this agreesF with the1eq,opt periods of the waves, since a higher optimal wavesrespectively A, B, C; for this waves agrees A, with B, C; the this periods agrees of with the waves, the periods since aof higher the waves, optimal since damping a higher corresponds optimal damping corresponds to a higher wave period, as long as T/TN > 1. Indeed, optimal damping should todamping a higher wavecorresponds period, to as a longhigher as waveT/TN period,> 1. Indeed, as long optimal as T/TNdamping > 1. Indeed, should optimal have damping a minimum should and have a minimum and CWR should have a maximum for T/TN = 1; according to this fact, CWR of the have a minimum and CWR should have a maximum for T/TN = 1; according to this fact, CWR of the CWRsystem should with have wave a maximumA is on average for T higher/TN = than 1; according CWR with to thisthe other fact, CWRtwo waves, of the on system the contrary with wave there A is system with wave A is on average higher than CWR with the other two waves, on the contrary there onis average not much higher difference than CWRbetween with CWR the with other wave two waves,B and C. on the contrary there is not much difference betweenis not much CWR difference with wave between B and C.CWR with wave B and C.
Figure 6. CWR of the float-only system as a function of float damping b1eq, water depth at the float FigureFigure 6. 6.CWR CWR of of the the float-only float-onlysystem system asas aa functionfunction of float float damping damping bb1eq1eq, water, water depth depth at atthe the float float hF/D. Wave A. hF/DhF/D. Wave. Wave A. A.
FigureFigure 7. 7.CWR CWR of of the the float-only float-only system system asas aa functionfunction of floatfloat damping bb1eq1eq, water, water depth depth at atthe the float float Figure 7. CWR of the float-only system as a function of float damping b1eq, water depth at the float hF/DhF/D. Wave. Wave B. B. hF/D. Wave B.
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Energies 2018, 11, 2730 11 of 17 Energies 2018, 11, x FOR PEER REVIEW 11 of 17
Figure 8. CWR of the float-only system as a function of float damping b1eq, water depth at the float
hF/D. Wave C. 1eq FigureFigure 8. CWR8. CWR of of the the float-only float-only system system as asa afunction function ofof floatfloat damping bb1eq, ,water water depth depth at atthe the float float hF/D. Wave C. hF/D. Wave C. Figure 6 shows that the highest CWR almost always occurs with the minimum damping applied. However,FigureFigure it6 was shows 6 shows not that possible that the the highest tohighest reach CWR CWR the optimal almost almost always alwadampingys occurs for wave with the theA duemi minimumnimum to the dampingphysical damping applied.limitation applied. ofHowever, PTOHowever, amplitude it wasit was not notand possible possible the necessity to to reach reach to the the avoid optimal optimal the dampingfloatdamping touching for wave wave the Achannel A due due to to thebottom the physical physical when limitation limitationhF/D was of PTO amplitude and the necessity to avoid the float touching the channel bottom when hF/D was low.of PTO Again amplitude in Figure and 6, the for necessity low depths to avoid hF/D the= 0.6 float and touching hF/D = 0.7, the channelthe applied bottom damping when hdoesF/D wasnot low. Again in Figure 6, for low depths hF/D = 0.6 and hF/D = 0.7, the applied damping does not significantlylow. Again inaffect Figure the6 ,CWR for low of the depths float. hThisF/D fact= 0.6 is andless evidenthF/D = for 0.7, waves the applied B and dampingC in Figures does 7 and not 8.significantly significantly affect affect the the CWR CWR of of the the float. float. This This fact fact is is less less evident evident forfor waveswaves BB andand CC in Figures 77 andand8 . 8. Then, the complete EDS system was tested.tested. Different distances between paddle and floatfloat ddP-FP-F,, Then, the complete EDS system was tested. Different distances between paddle and float dP-F, water depth at the floatfloat hhFF and float float external damping bb1eq1eq werewere varied varied in the tests. The The configuration configuration water depth at the float hF and float external damping b1eq were varied in the tests. The configuration of paddle PTO was the same for all the tests: two discs inside the cylinder were used in order to of paddle PTO was the same for all the tests: two discs inside the cylinder were used in order to generate asymmetric damping. Due Due to to the the great great number of parameters that characterizes the EDS generate asymmetric damping. Due to the great number of parameters that characterizes the EDS system, only some of them have been optimized in this work, while other ones, like paddle damping, system,system, only only some some of of them them have have been been optimized optimized inin this work, while while other other ones, ones, like like paddle paddle damping, damping, will be investigated in future works.works. The paddle immersionimmersion atat restrest waswas ddP = 0.09 m in all the tests; this will be investigated in future works. The paddle immersion at rest was dP = 0.09 m in all the tests; this valuevalue was was found found to to be be optimal optimal in in the the previous previous workwork [[21].[21].21]. FiguresFigures9 9–11–11 9–11 show show show CWRCWR CWR ofof of thethe the EDSEDS EDS systemsystem system for forfor the thethe threethree waveswaveswaves A,A, B, B,B, C. C.C. Not NotNot all allall the thethe combinations combinationscombinations of theof the the three three three parameters parameters parameters (h ((hFh/DFF/D, b, 1eqbb1eq1eq, d, ,P-FdP-Fd)P-F )are are) arereported: reported: reported: wewe have we have neglectedneglected neglected the the configurations configurations the configurations that that were were that wereclearlyclearly clearly less less efficient. less efficient. efficient. For For each Foreach wave, each wave, wave, float float floatoptimaloptimal optimal dampingdamping damping of thethe of EDSEDS the is EDSis similar similar is similar to to the the toone one the of oneof the the of float-onlythefloat-only float-only system: system: system: averaging averaging averaging the the best the best best configurations configurations configurations forfor each for each hFF/D/Dh,, F wewe/D , obtainobtain we obtain b 1eq,optb1eq,opt b=1eq,opt =2.17, 2.17, =3.77, 2.17,3.77, 3.15 3.77,3.15 Nms/rad3.15Nms/rad Nms/rad for for waves forwaves waves A, A, B, B, A,C. C. B,This This C. Thisresult result result suggests suggests suggests thatthat float thatfloat floatoptimal optimal dampingdamping damping is is independent independent is independent from from the from the presencethepresence presence of ofthe of the thepaddle. paddle. paddle.
Figure 9. CWR of EDS as a function of flat damping b1eq, distance between paddle and float dP-F/D,
Figurewater 9. depth CWR at of the EDSEDS float asas h aFa/ functionDfunction. Red circles, ofofflat flat dP-F damping damping/D = 0.8; blueb 1eqb1eq, ,circles, distancedistance dP-F betweenbetween/D = 1.1; green paddlepaddle circles andand d float P-Ffloat/D d =dP-F 1.4.P-F//D D,, waterWave depth depth A. at at the the float float hF/hDF./ RedD. Redcircles, circles, dP-F/Dd =P-F 0.8;/D blue= 0.8; circles, blue d circles,P-F/D = 1.1;dP-F green/D = circles 1.1; green dP-F/D circles = 1.4. WavedP-F/D A.= 1.4. Wave A.
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Figure 10. CWR of EDS as a function of flat damping b1eq, distance between paddle and float dP-F/D, Figure 10. CWRCWR of of EDS EDS as asa function a function of flat of flatdamping damping b1eq, distanceb1eq, distance between between paddle paddle and float and dP-F float/D, water depth at the float hF/D. Red circles, dP-F/D = 0.8; blue circles, dP-F/D = 1.1; green circles dP-F/D = 1.4. dwater/D depth, water at the depth float at h theF/D. floatRed circles,h /D. d RedP-F/D circles, = 0.8; blued circles,/D = 0.8; dP-F blue/D = 1.1 circles,; greend circles/D = d 1.1P-F/D; green = 1.4. WaveP-F B. F P-F P-F circlesWave B.dP-F /D = 1.4. Wave B.
FigureFigure 11. 11. CWRCWR of ofEDS EDS as asa function a function of flat of flatdamping damping b1eq, distanceb1eq, distance between between paddle paddle and float and dP-F float/D, Figure 11. CWR of EDS as a function of flat damping b1eq, distance between paddle and float dP-F/D, waterdP-F/D depth, water at the depth float at hF the/D. floatRed circles,hF/D. d RedP-F/D circles, = 0.8; bluedP-F circles,/D = 0.8; dP-F/D blue = 1.1; circles, greend P-Fcircles/D = dP-F1.1;/D green= 1.4. water depth at the float hF/D. Red circles, dP-F/D = 0.8; blue circles, dP-F/D = 1.1; green circles dP-F/D = 1.4. Wavecircles C.dP-F /D = 1.4. Wave C. Wave C.
TheThe best configuration configuration of EDS is almost always the one with dP-F/D/D == 0.8, 0.8, for for all all the the three three waves. waves. The best configuration of EDS is almost always the one with dP-F/D = 0.8, for all the three waves. There is is no no appreciable appreciable difference difference between between the the configurations configurations with with dP-Fd/DP-F =/D 1.1= and 1.1 anddP-F/DdP-F = 1.4./D =Like 1.4. There is no appreciable difference between the configurations with dP-F/D = 1.1 and dP-F/D = 1.4. Like theLike float-only the float-only configuration, configuration, the EDS the CWR EDS meanly CWR meanlyincreases increases as water asdepth water increases depth and increases it oscillates. and it the float-only configuration, the EDS CWR meanly increases as water depth increases and it oscillates. Again,oscillates. for Again,water depth for water hF/D depth < 0.8, hCWRF/D
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Energies 2018, 11, x FOR PEER REVIEW 13 of 17 Energies 2018, 11, x FOR PEER REVIEW 13 of 17 a slightly smaller T/TN than the one of [22]: this could bring a higher CWR for our float. Though it is difficultdifficult toto make make a a precise precise comparison, comparison, the the present presen resultst results of CWR of CWR are comparable are comparable with thewith ones the found ones byfound [22]. by [22]. InIn FigureFigure 1313 optimaloptimal dampingdamping ofof thethe float-onlyfloat-only systemsystem is compared with the one found by [22]. [22]. Again,Again, our results are are presented presented in in terms terms of of mean mean optimal optimal damping damping ± standard± standard deviation deviation along along the thebeach beach for fora agiven given wave wave frequency. The dimensionlessdimensionless optimaloptimal damping damping was was calculated calculated as as 𝑏 ( /(𝜌𝑉 𝜔𝑙) ), where 𝑉 is the immersed volume of the float at rest and l1 is the float arm (OA in basext 𝑏/ ρ/(𝜌𝑉V0ω l1𝜔𝑙, where), whereV0 𝑉is is the the immersed immersed volume volume of of thethe floatfloat atat rest and and ll11 isis the the float float arm arm (OA ( OA in inour our system). system). Present Present results results agree agree with with the the ones ones of of [22] [22 ]except except for for wave wave A. A. This This disagreement couldcould bebe causedcaused byby thethe factfact thatthat wewe couldcould notnot furtherfurther decreasedecrease floatfloat dampingdamping andand reachreach thethe optimaloptimal valuevalue duedue toto thethe physicalphysical constraintsconstraints ofof thethe model.model.
Figure 12. CWR as a function of wave dimensionless frequency. Wave A: H/L = 0.03; wave B: H/L = FigureFigure 12. CWRCWR as as a afunction function ofof wave wave dimensionless dimensionless frequency. frequency. Wave Wave A: H A:/L H= /0.03;L = wave 0.03; waveB: H/L B: = 0.02; wave C: H/L = 0.03. H0.02;/L = wave 0.02; C: wave H/L C:= 0.03.H/L = 0.03.
FigureFigure 13. Dimensionless optimal damping for the float-only configuration. Wave A: H/L H= 0.03;L wave Figure 13. DimensionlessDimensionless optimal optimal damping damping for for the the float-only float-only configuration. configuration. Wave Wave A: H A:/L = 0.03;/ = wave 0.03; waveB: H/L B: = H0.02;/L = wave 0.02; C: wave H/L C:= 0.03.H/L = 0.03. B: H/L = 0.02; wave C: H/L = 0.03. For each wave, the best configuration of EDS was compared with the best configuration reached For each wave, the best configuration of EDS was compared with the best configuration reached with the float-only system (Figure 14). On average, CWR of EDS is 5% (in absolute) higher than the one with the float-only system (Figure 14). On average, CWR of EDS is 5% (in absolute) higher than the of the float-only system; relatively speaking, the CWR of the system can increase, with the addition of one of the float-only system; relatively speaking, the CWR of the system can increase, with the theaddition paddle, of bythe 10paddle,÷50% by for 10÷50%hF/D ≤ for0.8. hF This/D ≤ advantage0.8. This advantage tends to decreasetends to decrease as hF/D increases.as hF/D increases. This is addition of the paddle, by 10÷50% for hF/D ≤ 0.8. This advantage tends to decrease as hF/D increases. due to the fact that, increasing depth, the horizontal wave force decreases and the energy captured by This is due to the fact that, increasing depth, the horizontal wave force decreases and the energy thecaptured paddle by decreases. the paddle For decreases.hF/D < 0.8, For CWR hF/D of < EDS0.8, CWR is up toof 1.5EDS times is up CWR to 1.5 of times the float-only CWR of system.the float- captured by the paddle decreases. For hF/D < 0.8, CWR of EDS is up to 1.5 times CWR of the float- Figures 15–17 show how CWR of EDS is distributed among float and paddle, for the best only system. configuration of EDS and float-only system, for low hF/D; furthermore, energy is split into the upward and downward phases of the float and forward and backward phases for the paddle.
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Figure 14. ComparisonComparison betweenbetween CWRCWR ofof EDSEDS andand floafloatt -only-only systemsystem atat bestbest configuration.configuration.
Figures 15–17 show how CWR of EDS is distributed among float and paddle, for the best configuration of EDS and float-only system, for low hF/D; furthermore, energy is split into the upward configuration of EDS and float-only system, for low hF/D; furthermore, energy is split into the upward and downward phases of the float and forward and backward phases for the paddle. FigureFigure 14. 14.Comparison Comparison between between CWR CWR of of EDS EDS and and float floa -onlyt -only system system at at best best configuration. configuration.
Figures 15–17 show how CWR of EDS is distributed among float and paddle, for the best configuration of EDS and float-only system, for low hF/D; furthermore, energy is split into the upward and downward phases of the float and forward and backward phases for the paddle.
FigureFigure 15. 15.CWR CWRCWR distribution distributiondistribution among amongamongfloat floatfloat and andand paddle,paddle,paddle, and motion phases phases of of each each body. body. Wave Wave A. A.
Figure 15. CWR distribution among float and paddle, and motion phases of each body. Wave A.
EnergiesFigure 2018, 16.11, xCWR FOR PEER distribution REVIEW among float and paddle, and motion phases of each body. Wave B.15 of 17 Figure 16. CWRCWR distributiondistribution amongamong floatfloat andand paddle,paddle, and motion phases of each body. Wave B.
Figure 16. CWR distribution among float and paddle, and motion phases of each body. Wave B.
FigureFigure 17. 17.CWR CWR distribution distribution among among float float andand paddle,paddle, and motion phases phases of of each each body. body. Wave Wave C. C.
The charts show that the energy absorbed by the paddle in the forward phase is much higher than in the backward phase, because of the asymmetrical external damping and of the mass transport of the waves in shallow water. For hF/D ≤ 0.9, the CWR of the paddle is 10–15%, with about 10% in the forward phase, which is higher than the 5% found in reference [20] for a surging paddle pushed by the waves in deep water; this is likely due to the fact that the surge wave force is higher in shallow water than in deep water. On the contrary, the surging flap tested in reference [15] showed a CWR significantly higher (>20%) than the EDS paddle, probably due to the fact that, in reference [15], the damping applied to the flap was optimized. CWR of the float in the upward and downward phase tends to be equal as water depth increases, due the reduction of wave nonlinearity. As seen before, the presence of the paddle increases the total CWR of EDS, though generally decreases the float CWR, with respect to the case of the float-only system.
5. Conclusions The behavior of two oscillating-body WECs on a sloping bottom has been investigated: a pitching-heaving float and the EDS, an innovative multibody WEC composed by the pitching- heaving float and a pitching-surging paddle. The position of the WEC was found to be very influential on its performance. This is imputable to the properties of the waves, which change as water depth changes, and to the wave reflection from the sloping bottom: a high variation of CWR with the position of the WEC is observed. On average, CWR of the WEC increases as water depth increases, but presents significant oscillation due to the wave height oscillation along the wave propagation direction. This is very important in order to localize optimal positions for a nearshore WEC. The main parameters that characterize EDS have been investigated, and indications about its optimal configuration have been found. EDS performances were compared with the ones of the float- only system with the same inertia, and the results obtained show that EDS almost always has a higher CWR. This is even more interesting, considering that the implementation of a paddle on a system with a heaving float does not require significant changes in the supporting structure. Optimal damping of the float is generally independent from water depth, except for water depth hF/D < 0.8, where a wider range of damping brings a similar power production. This is valid both for the float- only system and for the EDS: for a given system mass, optimal float damping for the float-only system and the EDS is similar. For the water depths tested hF/D = 0.6–1.4, EDS is more efficient than the float-only system, more significantly in water depth hF/D = 0.6, where the CWR of EDS is about 1.5 times the CWR of the float- only system. EDS advantage on the float-only system decreases as water depth increases, although it is still present, at least up to hF/D = 1.4. The EDS advantage on the float-only system is due to the fact that the energy contribution of the paddle exceeds the energy decrease of the float when inside the EDS system.
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The charts show that the energy absorbed by the paddle in the forward phase is much higher than in the backward phase, because of the asymmetrical external damping and of the mass transport of the waves in shallow water. For hF/D ≤ 0.9, the CWR of the paddle is 10–15%, with about 10% in the forward phase, which is higher than the 5% found in reference [20] for a surging paddle pushed by the waves in deep water; this is likely due to the fact that the surge wave force is higher in shallow water than in deep water. On the contrary, the surging flap tested in reference [15] showed a CWR significantly higher (>20%) than the EDS paddle, probably due to the fact that, in reference [15], the damping applied to the flap was optimized. CWR of the float in the upward and downward phase tends to be equal as water depth increases, due the reduction of wave nonlinearity. As seen before, the presence of the paddle increases the total CWR of EDS, though generally decreases the float CWR, with respect to the case of the float-only system.
5. Conclusions The behavior of two oscillating-body WECs on a sloping bottom has been investigated: a pitching-heaving float and the EDS, an innovative multibody WEC composed by the pitching-heaving float and a pitching-surging paddle. The position of the WEC was found to be very influential on its performance. This is imputable to the properties of the waves, which change as water depth changes, and to the wave reflection from the sloping bottom: a high variation of CWR with the position of the WEC is observed. On average, CWR of the WEC increases as water depth increases, but presents significant oscillation due to the wave height oscillation along the wave propagation direction. This is very important in order to localize optimal positions for a nearshore WEC. The main parameters that characterize EDS have been investigated, and indications about its optimal configuration have been found. EDS performances were compared with the ones of the float-only system with the same inertia, and the results obtained show that EDS almost always has a higher CWR. This is even more interesting, considering that the implementation of a paddle on a system with a heaving float does not require significant changes in the supporting structure. Optimal damping of the float is generally independent from water depth, except for water depth hF/D < 0.8, where a wider range of damping brings a similar power production. This is valid both for the float-only system and for the EDS: for a given system mass, optimal float damping for the float-only system and the EDS is similar. For the water depths tested hF/D = 0.6–1.4, EDS is more efficient than the float-only system, more significantly in water depth hF/D = 0.6, where the CWR of EDS is about 1.5 times the CWR of the float-only system. EDS advantage on the float-only system decreases as water depth increases, although it is still present, at least up to hF/D = 1.4. The EDS advantage on the float-only system is due to the fact that the energy contribution of the paddle exceeds the energy decrease of the float when inside the EDS system. These results confirm that EDS is particularly interesting for shallow water, and, thanks to its structure, it is particularly suited to be docked on the external side of breakwaters of ports. The distance between paddle and float also affects the CWR of EDS: best results were obtained with the paddle close to the float. The results obtained on EDS are encouraging, also because the system still has room for improvement. In our opinion, the optimization of paddle damping in particular may further increase the CWR of EDS.
Author Contributions: M.N. contributed in designing and developing the experimental setup, designing and realizing the experiments, analyzing the results and writing the paper. S.M. supervised the research and contributed in designing the experimental setup and the experiments and writing the paper. Funding: This research was partially funded by Regione Lombardia (POR FESR 2007–2013, Azione B–Bando 2009 Efficienza Energetica). Energies 2018, 11, 2730 16 of 17
Acknowledgments: The authors would like to remember and thank Arturo Lama, the inventor of the EDS system, for sharing his contagious enthusiasm and supporting the project with tireless passion. Moreover, the authors would like to thank Tecmomac s.r.l. (www.tecnomac.it) for all the support provided. Conflicts of Interest: The authors declare no conflict of interest.
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