Power capture performance of hybrid wave farms combining different wave energy conversion technologies: The H-factor

4 In this paper we consider hybrid wave farms, in which different types of WEC are combined, 5 through a case study involving oscillating water columns (OWCs) and point-absorbers (PAs). 6 A new parameter, called “ H-factor ”, is introduced to compare hybrid (multi-type) and 7 conventional (single-type) wave farms in terms of wave power capture. We develop an ad hoc 8 semi-analytical model to calculate the H-factor in a computationally efficient manner


Introduction
For wave energy to contribute significantly to decarbonizing the energy mix and meeting the carbon reduction targets, wave energy converters (WECs) will have to be deployed in arrays, or wave farms.Power extraction by wave farms has been the focus of intensive research (Babarit, 2010(Babarit, , 2013;;Borgarino et al., 2012;Nader, 2013;Sarkar et al., 2014;Astariz andIglesias, 2016a, 2016b;Veigas et al., 2015;Veigas and Iglesias, 2014;Konispoliatis and Mavrakos, 2016;Penalba et al., 2017).The variability of the resource at the site (Carballo et al. 2015a(Carballo et al. , 2015b)), the power output of the WECs (Carballo and Iglesias, 2012;Malara and Arena, 2013;Lopez et al., 2016;Carballo et al., 2019) and, on this basis, the power output of the wave farm as a whole were investigated.Another relevant line of research concerns the optimization of WECs and WEC arrays (Veigas et al., 2014), often tackled by means of CFD (Lopez et al., 2014;Elhanafi et al., 2018;Guo et al., 2018).Finally, in the assessment of any wave energy project, the Levelised Cost of Energy plays a fundamental role.The economic viability of wave farms has also been investigated, e.g., Astariz andIglesias (2015, 2016c) and Contestabile et al. (2016).
The benefits that can be obtained by combining wave and offshore wind energy systems have been investigated by, e.g., Astariz andIglesias (2016d, 2016e) and Perez-Collazo et al. (2018a, 2018b), and a method was suggested to assess the feasibility of co-locating wave and offshore wind energy systems, based on the Co-location Feasibility Index (Astariz and Iglesias, 2017).These benefits range from a smoothed power output (Astariz and Iglesias, 2016f) to economies in O&M tasks (Astariz et al., 2018).
Research on wave farms has focused on arrays with one single wave energy conversion technology, i.e., one type of WEC.The possibility of combining different types of WECs in the same wave farm, forming a hybrid wave farm, has not been sufficiently investigated so far.
Recently, a wave farm with two types of WECs was investigated from the point of view of wave-structure interaction (Zheng et al., 2018), without assessing the all-important energy production performance -one of the fundamental elements in assessing the economic viability of a wave project.This is the motivation for the present work, in which a hybrid farm consisting of two oscillating water columns (OWCs) and two point-absorbers (PAs) is considered and compared with non-hybrid or conventional wave farms (with only one type of technology).
The WECs in a wave farm are, by definition, in proximity; therefore, the wave interaction between individual WECs may be expected to play a role in the response and power performance of the system (e.g., Zhong and Yeung, 2019;Zheng et al., 2019a).Consequently, the total power captured by a wave farm consisting of N0 WECs will in general be different from N0 times the power of a single WEC working in isolation.This is known as the array or park effect, which can be assessed through the q-factor.Also known as gain factor, it is defined as the ratio of the maximum time-averaged power absorbed by the interacting WECs to the maximum time-averaged power absorbed by the WECs in isolation (McIver et al., 1995;Mavrakos and Kalofonos, 1997;Wolgamot, et al., 2012).Thus, the q-factor quantifies the influence of the hydrodynamic interactions within a given WEC array.Following Evans (1980) and Falnes (1980), the time-averaged power absorbed by a wave farm is the difference between the excitation power and the radiated power.Therefore, the maximum time-averaged power absorbed by the farm can be evaluated from the wave excitation forces and the radiation damping matrix directly, without detailed consideration of the power take-off system (PTO).Mavrakos and Kalofonos (1997) employed this approach to compute the maximum power extraction of a wave farm and revealed that the q-factor was sensitive to variations in the configuration of the farm.This method was also used by Fitzgerald and Thomas (2007), who proved that for arrays of heaving hemispheres, the interaction of the q-factor over the entire range of incident wave directions, [0, 2π], was equal to 2π.Later, Wolgamot et al. (2012) found that the directional behavior of the q-factor was valid under more general conditions, i.e., regardless of body dimensions and motion modes.However, in the above studies the wave farms were assumed to be composed of fully controlled WECs with both phase and amplitude optimized for the full range of wave conditions; in practice, however, such optimal control is not realistic.
In the case of an array of two semi-submerged cylinders moving only in the heave mode, Babarit (2010) calculated the q-factor by setting the PTO stiffness to zero, and tuning the PTO damping in order to achieve the maximum energy absorption at the natural frequency of an isolated device.Additionally, a modified factor denoted as qmod was proposed as the ratio of the difference between the time-averaged power absorbed by a WEC in the farm and the timeaveraged power absorbed by the same WEC in isolation, to the maximum time-averaged power absorbed by the WEC in isolation.qmod was also adopted by Borgarino et al. (2012) and Renzi et al. (2014) to study the power absorption of a wave farm consisting of nine to twenty-five cylinder WECs and two to three flap-type WECs in the nearshore.Given that the value of qmod is specific to each WEC in the wave farm, in the case of farms composed by many WECs many values of qmod will need to be calculated, thus increasing the complexity of the performance evaluation.Penalba et al. (2017) investigated the influence of the slenderness and the number of WECs in a wave farm on the hydrodynamic performance in realistic wave climates with a numerical method.The PTO damping coefficient was assumed to be the same for all the WECs in the farm, as no significant improvement had been reported from individually optimizing the PTO coefficients of each WEC (Ricci et al., 2007).The PTO damping coefficient was optimized for each sea state, as in the case of a single device.It was found that for farms consisting of no more than ten WECs, there was an optimal inter-device distance for maximizing the power absorption of the farm.
Besides the q-factor, the wave power capture factor, also known as the capture-width ratio, is of interest here.It is defined as the ratio of the time-averaged power absorbed by the wave farm to the mean wave power flux across a "characteristic width" of wave front.For an array of submerged spherical WECs, Wu et al. (2016) chose the sum of their diameters as the characteristic width.The wave power capture factor was also used to investigate the performance of an array of OWCs (Nader, 2013;Konispoliatis and Mavrakos, 2016), with the characteristic width set as the sum of the inner diameters of the OWC chambers.Major coefficients calculated from Eqs. ( 19) and ( 20), are plotted against kh.For kh ranging from 0.1 to 6.0, ηOWC-kh presents a bimodal and a unimodal curves for Case I and Case II, respectively.
The first peak value of ηOWC for Case I is 0.25 at kh=3.1,only 32.5 percentages as large as that of the single peak for Case II, which is 0.78 occurring at kh=3.2.The second peak value of ηOWC for Case I is 0.52, occurring at kh=4.4.The peaks at kh=3.1 and 4.4, might be explained by the resonances induced by water column oscillation and the heave motion of OWC chamber, respectively.Note that for kh<2.5, there is nearly no power can be extracted by the free floating OWC (Case I), so is it for kh>5.5.Due to no power radiated away induced by oscillation of the chamber, the fixed OWC (Case II) shows an overall better power absorption ability for all the tested wave conditions, except at 4.2<kh<4.6,where the second peak of ηOWC for Case I happens.
For the isolated PA, as kh increases from 0.1 to 6.0, ηPA increases first, and then stabilizes at 0.4 after kh reaching 4.0.Compared with the fixed OWC, PA performances better in capturing power for short waves.Whereas the fixed OWC is prefered for 2.2<kh<3.6,where for some specified wave conditions, the maximum power capture factor of OWC can be two times as large as that of PA.Variation of and with kh shows rather different trends (Fig. 3b).
decreases monotonically with the increasing of kh.While, -kh turns to be a unimodal curves curve, regardless of Case I and Case II, whereas the the peaks of for Case I and Case II occur at different kh, i.e. 4.4 and 3.2, which correspond to the resonant frequencies of water column oscillation and OWC chamber heaving motion, respectively.In addition, the peak value of for Case I is much larger than that for Case II.In the following parts, the optimal PTO damping coefficients for isolated OWC and PA (Fig. 3b) are adopted in the wave farm of OWCs and/or PAs.Two array configurations of a hybrid wave farm consisting of two OWCs and two PAs, denoted as H1 and H2, respectively, are examined in this section (Fig. 4).The OWCs and PAs are placed at the tops of a square.The distance between two OWCs/PAs in the same row/column is L.  the PAs of P2 plays a constructive role for 4.6<kh<6.0,where the maximum power capture factor of P2 is 0.45, going up by 12.5% compared with that of the isolated PA.Whereas for the rest wave conditions, especially for 3.0<kh<4.5,a destructive effect of the interaction in P2 is found.Compared with isolated free floating OWC, constructive effect for P1 of Case I appears at 3.6<kh<4.0and 4.6<kh<6.0.Note that a rather destructive influence also occurs at 3.0<kh<3.5.For the hybrid wave farm H1 of Case I, its power capture factor (ηH1) for kh<2.5 is approximately 50% of that of P2 (ηP2).This is mainly due to the much less power absorption by the free floating OWCs for these wave frequencies.When comparing with P1 of Case I, H1 of Case I holds an obviously larger power capture factor for almost all the studied wave conditions.If OWCs and PAs are arranged at diagonal corners of the square, i.e., H2 (see Fig. 4b), rather than in parallel with wave front line ( H1, see Fig. 4a), power absorption of the hybrid farm can be significantly improved for kh ranging from 2.8 to 5.5, except around kh=4.4 (strictly, 4.3<kh<4.5)where the sharp peak occurs.
Comparison of Figs.6a and 6b shows that power extraction of the farm consisting of fixed OWCs (Case II) is much larger than that of the farm with OWCs free floating.This is clearly a consequence of the performance difference between the isolated OWC of Cases I and II as discussed above about the results illustrated in Fig. 3a.The power capture factors of these six farms/single WEC for Case II are quite close to each other for any specified wave frequency in the range of kh<2.0 (Fig. 6b).Although the "wave capture factor-kh" curves representing isolated fixed OWC and the array of fixed OWCs ( P1 of Case II) are both unimodal, the latter one holds a lower peak, nevertheless a wider bandwidth.Specifically, for 2.5<kh<3.9,ηP1<ηOWC, and the peak value of ηP1 is 0.55 at kh=3.1, much smaller than that of ηOWC, which is 0.78 occurring at kh=3.2.Whereas for 1.3<kh<2.5 and 3.9<kh<5.8,ηP1>ηOWC.For the two deployments of hybrid wave farm both consisting of two fixed OWCs and two PAs ( H1 and H2 of Case II), H1 holds a better performance in power absorption at 2.4<kh<4.1,where for H1 and H2 of Case I the comparing result is mostly opposite as discussed above.For short waves, the hybrid wave farm H2 absorbs more power from ocean waves than H1.
It is worth noting that, for 3.3<kh<3.7,ηH1 is obviously larger than both ηP1 and ηP2.This implies that, for a certain range of wave conditions, mixing OWCs and PAs in the wave farm are rather beneficial to the power absorption of wave farm.Power absorption capacity of the wave farm (Fig. 6) is given in terms of wave capture factor, which does not separate the effect induced by mixing different type of WECs in the wave farm, i.e., hybrid effect, from the effect of hydrodynamic interaction among WECs array, i.e., array effect.
To have a better understanding of the effect caused by the hybrid WECs in the hybrid wave farm, power capture ability of H1 and H2 for Case I and II, L/h=0.75, β=0 in terms of the new proposed factor, i.e., "H-factor", are plotted in Fig. 7.For kh<2.0 of Case I (Fig. 7a), both HH1 and HH2 equal to 1.0 approximately, implying a negligible hybrid impact for these wave conditions.This also applies for H2 of Case II for kh <3.0 as shown in Fig. 7b.As kh increases towards 6.0, the values of HH1 and HH2 for Case I and II oscillate around 1.0, which means that, for L/h=0.75, β=0, constructive and destructive hybrid effect appear alternately with the variation of wave conditions.In Case I (Fig. 7a), for 2.0<kh<2.7,HH1>1 and HH2<1; while for 3.0<kh<3.6,HH1<1 and HH2>1.That is to say, hybrid effect plays opposite roles for different deployment of the hybrid wave farm consisting of free floating OWCs (Case I) for those wave conditions.Such opposite hybrid effect can also be found for the hybrid wave farm with fixed OWCs (Case II, Fig. 7b), for wave conditions: 3.3<kh<4.0and 4.3<kh<5.7.Note for 3.3<kh<3.6,opposite hybrid effect happens not only for different deployments of hybrid wave farm, it also occurs for the same deployment by using different cases of OWCs.Take H1 as an example: if free floating OWCs are adopted in such hybrid wave farm (Case I), we have The exact values of the H-factor together with the corresponding q * -factor, q-factor (qOWC and qPA) for some selected wave numbers are presented in Table 1.For kh=3.5, although the qfactor values for conventional OWCs (Case II) and PAs wave farms are only 0.721 and 0.936, respectively, demonstrating the destructive array-effect, the q * -factor of the hybrid wave farm H1 (Case II) reaches 1.074.This proves that this hybrid wave farm greatly benefits from the dramatic constructive hybrid-effect (HH1=1.345).In the case of kh=6.0,although the q * -factor of the hybrid wave farm H1 (Case I) is q * H1=0.997, which looks acceptable, the corresponding H-factor is only 0.901, indicating a destructive hybrid-effect and revealing the weakness of the hybrid wave farm H1.It is apparent in these examples that the H-factor is useful in revealing the advantage or disadvantage, as the case may be, of the hybrid wave farm relative to a conventional wave farm, and that this information cannot be obtained from the q *factor.

Effect of incident wave direction
In this section, the effect of the incident wave direction (β) on the power absorption of the hybrid wave farm is investigated.First, the frequency response of wave capture factor of the farm with conventional OWCs/PAs (#P1 and #P2) under regular waves propagating in different directions (β=0, π/8 and π/4) is evaluated (Fig. 8).The effect induced by increasing β from 0 to π/4 can be generally divided into three levels: negligible for long waves, enhanced effect for medium waves and inhibited effect for short waves (Table 2).For the wave farm of PAs (i.e., #P2), the inhibited effect strength of β at 3.9<kh<6.0 is much stronger than in the case of the enhanced effect, occurring at 2.5<kh<3.7.Due to the symmetries of the deployment of #P1 and #P2, the power absorbed by the conventional wave farm with β=0 is the same as with β=π/2 and π.Similarly, β=π/8 and π/4 may also be used to represent situations with β=-π/8 and -π/4, respectively.
Figure 9 illustrates the variation in power capture factor of the hybrid wave farm #H1 (ηH1 for Cases I and II) with kh for different β ranging from 0 to π with step π/4.The main effect of β for Case I occurs at kh>2.5 (Fig. 9a).For 2.9<kh<4.0,ηH1 stays at a rather low level for incident waves coming from the OWCs side, i.e., β=0 and β=π/4, generally below 0.17.By contrast, when waves impinge from the side of the two PAs, i.e., β=3π/4 and π, the response of ηH1 peaks at 0.43 and 0.47 for β=3π/4 and π, respectively, both appearing at kh=3.2.For 4.6<kh<6.0, the hybrid wave farm #H1 has an obviously better performance in extracting power from waves incoming at β=π/2 and π compared with β=0 and, especially, π/4 and 3π/4.This rule also applies to #H1 with all OWCs fixed for 4.2<kh<6.0(Case II, Fig. 9b).Such results do not appear as a surprise, for both arrays of conventional OWCs/PAs are better at absorbing power for short waves with a smaller β (Fig. 8).However, if we review the wave capture factor response of the conventional wave farms #P1 and #P2 (see Fig. 8) at 2.8<kh<3.5,where both ηP1 (Cases I and II) and ηP2 reach the smallest at β=0 (π/2 and π), and have a comparison with the response of ηH1 (Cases I and II, see Fig. 9), a completely opposite effect of β on power absorption of the hybrid wave farms is observed, for ηH1 achieves the largest at β=π (equivalent to β=0 for #P1 and #P2) in some certain subrange among 2.8<kh<3.5.This interesting phenomenon can be clearly explained using the H-factor (Fig. 10).Although both #P1 and #P2 capture the least power (i.e., worst array effect) at β=π, for 2.8<kh<3.4 and 3.2<kh<3.6for Cases I and II, respectively (Fig. 8), the corresponding H-factor, HH1, on the contrary, is the largest (i.e., best hybrid effect, Fig. 10), and the latter overcomes the former, leading to a better power absorption for β=π than β with any other values (Fig. 9).This implies that the hybrid effect, which can be quantified through the H-factor proposed in this work, may in fact overcome the destructive array effect.It should also be noted that the hybrid effect does not control the power absorption for all wave conditions.For example, HH1 for β=3π/4 is larger than that for β=0 in 4.5<kh<5.4and 4.2<kh<4.9for Cases I and II, respectively (Fig. 10), while due to the contrary β-based array effect happened in these wave conditions as demonstrated in Fig. 8, less power can be captured for β=3π/4 compared with β=0 (Fig. 9).As discussed above, the better performance of the hybrid wave farm #H1 at β=π with all OWCs free floating or fixed for 4.6<kh<6.0and 4.2<kh<6.0,respectively, is likely due to a better array effect.From Fig. 10 we infer that HH1>1.0 is satisfied for all these wave conditions, and can be even as large as 1.43 and 1.56, occurring at kh=5.2 and 5.0, respectively.This indicates that the better performance of the hybrid wave farm #H1 at β=π results not only from the constructive array effect but also the constructive hybrid effect.This constructive hybrid effect of #H1 at β=π for short waves can be explained from the point of wave reflection and transmission.The scale of PA in terms of radius and draft is smaller than that of the OWC, hence when the waves propagate from the PAs side, after being partially absorbed by the PAs, they can pass through the PAs toward the OWCs to be further absorbed by the OWC.
Furthermore, thanks to the large wet area of the OWC, some waves are also reflected from the OWC towards the PAs, which are thus excited.However, if waves propagate from the OWCs side a significant part of wave power will be reflected by the OWCs, resulting in less power being transmitted to the PAs.For #H1, Case II, there is yet one more advantage: HH1>1.0 applies to all wave conditions (Fig. 10b).
We now turn to the hybrid wave farm #H2, in which the two OWCs and two PAs are arranged at the two diagonal corners (Fig. 4b).The frequency response of ηH2 and HH2 for the farm of Cases I and II under waves from different incident directions is presented in Figs.11 and 12, respectively.For #H2 with all OWCs free floating (Case I), Figs.11a and 12a, ηH2 and HH2 depend strongly on β when kh ranges between 3.0 and 4.0.Unlike the performance of #H1 in 3.0<kh<4.0,#H2 presents a better power absorption in terms of both ηH2 and HH2 at β=π/4, rather than β=π for #H1.ηH2 and HH2 for β=π/4 reach their peak values 0.62 and 2.61, respectively, both at kh=3.3, where the corresponding values of ηH2 and HH2 for β=-π/4 are only 0.20 and 0.83, less than one third of those for β=π/4, demonstrating the strong dependence on the incidence direction.Generally, for 3.0<kh<4.0,the more β is away from π/4, the less HH2 and also the less power can be captured by #H2 (Case I).
The role of β is also significant in the case of #H2 with all OWCs fixed (Case II, Figs.11b    and 12b), and even for a wider range of kh.For β=π/4, HH2 >1.0 is satisfied for 1.5<kh<4.0,and HH2 is also larger than those for any other value of β studied (Fig. 12b).After taking consideration of the influence of β on the array effect, i.e., not enhanced for all 1.5<kh<4.0but even inhibited for large frequencies (see Fig. 8 and Table 2), the final power absorption of #H2 (Case II) for β=π/4 turns to have a larger value for only 1.5<kh<3.8.Note for kh>4.1, on the contrary, HH2 for β=π/4 is the smallest one among different β (see Fig. 12b), which, with account of the inhibited influence of β on array effect for both #P1 (Case II) and #P2 as listed in Table 2, results in the worst power absorption (Fig. 11b).

Effect of distance between two OWCs/PAs in the same row/column
We have learnt from Section 3.2 that, β=π and π/4 are good choices for the hybrid wave farms #H1 and #H2, respectively, for some specified wave conditions and L/h=0.75.Hence, in this section, we discuss separately the effect of L/h on the performance of wave farms with β=π and π/4 employed for #H1 and #H2.Before that, however, we consider the power capture factor of the conventional wave farm versus kh for different L/h is plotted in Fig. 13.Both the amplitude of the first peak of ηP1 and ηP2 and their corresponding kh are found to be affected by L/h.Detailed results are listed in Table 3.  with any specified L/h can absorb more power than the other farms with different L/h for a certain range of wave conditions (Fig. 13), but it may also capture the least for other wave conditions.A discussion on the effect of the spacing between WECs on the power absorption of a conventional (single type of device) wave farm consisting of OWCs/PAs can be found in (Nader, 2013;Konispoliatis and Mavrakos, 2016;André s et al., 2014;Borgarino et al., 2012;Penalba et al., 2017).
Figures 14 and 15 present how power capture factor and H-factor of hybrid wave farm #H1 (Cases I and II) are affected by L/h.For the hybrid wave farm with all OWCs free floating (Fig. 14a), the maximum values of the first peak of ηH1 for L/h=1.25 and 1.5 are achieved at kh=3.2 and 3.1, respectively.The first peak value of the farm with L/h =0.75 is a little smaller compared to L/h =1.25 and 1.5, and the corresponding conventional farms with L/h =0.75 are not attractive in extracting power either (Fig. 13), while thanks to the wider and stronger constructive hybrid effect occurring at 2.3<kh<3.6 and 4.3< kh <6.0 (Fig. 15a), #H1 with L/h =0.75 holds a wider bandwidth of peak at 2.5< kh <4.0 and a larger capture factor at 4.7< kh <5.6.It should also be noted that HH1 for L/h =1.0 is larger than 1.0 for nearly all the kh ranging from 0.1 to 6.0, and the power capture factor around kh =4.0 is much better than the farms with the other values of L/h.What is better, different from those with other L/h, #H1 with L/h =1.0 is almost never the worst one in capturing power for any wave conditions.
For #H1 of Case II, as shown in Fig. 14b, L/h =1.25 performs better than the others for wave conditions around kh =3.2 from the view of peak value of ηH1 and its bandwidth.This is contributed to by both the array effect (Fig. 13) and the constructive hybrid effect (Fig. 15b).
HH1>1 applies to all wave conditions for L/h =1.0, especially at kh=3.3 and 5.0, where two peaks occur (Fig. 15b).Similarly, ηH1-kh for L/h =1.0 (Fig. 14b) also present a bimodal curve and it could be welcome for the sea site with bimodal type wave conditions.which have two peaks at least, the one for #P1 with L/h =1.5 has only one single peak (Fig. 16a).For the #P1 (Case II), Fig. 16b, the ηP1 (Case II)-kh curves have one single peak for L/h =0.75, 1.0, and 1.75; whereas for L/h=1.25 and 1.5, there is a second peak, at kh=4.9 and 3.9, respectively, where ηP2 (#P2, Fig. 16c) are also peaked.The improvement for such wave conditions is caused by the diffracted and radiated waves between the WECs in the farm.The larger the L/h, the smaller the kh value for which the improvement induced by the array effect occurs.The single peak of ηP1 for L/h =1.75 occurs at kh=3.4 (Fig. 16b), which is obviously larger than the kh where the peak happens for the farm with different L/h ratios.Given that ηP2 peaks at kh=3.45 (Fig. 16c), this phase difference may well be caused by hydrodynamic interaction among the WECs.

Discussion
In practice, hybrid wave farms may be advantageous relative to conventional (single-type) wave farms in many situations, as proven in the case study.However, it is important to clarify that the theoretical maximum absorbed power over all incidence angles cannot be improved by the hybrid concept.If we consider a hybrid wave farm consisting of M PAs and N fixed OWCs as an example, the expression of the theoretical maximum absorbed power, PMAX, over all incidence angles is ( ) ( ) the derivation of which is given in Appendix A.
This theoretical maximum absorbed power corresponds to an ideal situation in which the configuration of the wave farm has been optimized and each WEC is controlled optimally, i.e., both damping and stiffness/mass are adopted and ideally optimized.In that case, resonance is achieved by all the WECs in the wave farm.In practice, however, it will often be impossible to attain such an ideal situation.Involving both damping and stiffness/mass in the PTO system and optimizing them simultaneously may well prove too complicated and expensive in the real world.For this reason, most researchers so far have studied the performance of a wave farm assuming that the PTO system of its WECs consists of PTO damping only (Babarit, 2010;Penalba et al., 2017).Therefore, in the present study of the hybrid effect, and of its quantification by means of an ad hoc H-factor, the effect of the PTO system for each WEC is merely represented by a damping coefficient (the effect of air compressibility is also considered as a kind of "added mass" for the OWCs) and the theoretical maximum absorbed power is not considered.
In other words, in the ideal conditions of optimum damping and stiffness/mass for all the WECs, a hybrid wave farm cannot theoretically absorb more energy than a conventional wave farm over all incidence angles.The interest of such conditions is limited, however, given that they are all but impossible to attain in practice.Importantly, in the practical, more easily achievable conditions (i.e., with optimal PTO damping coefficient adopted), a hybrid wave farm can indeed enhance power absorption in certain cases.For example, when the layout of a wave farm is fixed, e.g., due to marine space considerations, or environmental or economic constraints, a conventional wave farm may not be the most efficient option as a result of a destructive array effect; in that case, a hybrid farm may well be more efficient if a constructive hybrid effect occurs.The hybrid effect, which can be quantified through the H-factor proposed in this work, may in fact overcome the destructive array effect, as was proven in the case study.

Conclusions
In this paper the power performance and the motion and pressure response of a hybrid wave farm were investigated by means of an ad hoc semi-analytical model.A novel parameter, the H-factor, was proposed to estimate the effect of the hybrid nature of the farm, i.e., of combining different types of WECs, on its power capture performance.More specifically, two configurations consisting of two OWCs and two PAs were considered: parallel (#H1) and diagonal (#H2).The roles of the incident wave direction and the spacing between the WECs were investigated and discussed.The following conclusions may be drawn.
First, the combination of different types of WECs in a hybrid farm may enhance or reduce power capture performance relative to a conventional (single-type) wave farm.This hybrid effect may be easily quantified through the H-factor, with H>1 or H<1 signifying enhanced or reduced performance, i.e., constructive or destructive hybrid effect, respectively.For some specified cases, constructive hybrid effect (i.e., H>1) is valid for the whole range of wave conditions studied (0.1<kh<6.0), demonstrating the advantages of hybrid wave farms relative to conventional wave farms in certain conditions.. Second, the power capture performance of the farm depends on the type of OWC adopted (free-floating or fixed), the wave conditions (frequency and direction) and the configuration of the farm (#H1 or #H2).For the hybrid wave farm with spacing distance L/h=0.75 working in the specified wave conditions, kh=3.3,#H1 is found to be a better choice in capturing power from waves with a broad range of incident directions.By contrast, for waves with a narrow range of incident directions, #H2 could be more beneficial to power absorption thanks in part to a constructive hybrid effect, as demonstrated by the H-factor taking on a value above unity.
Third, regarding the type of OWC, the hybrid wave farm generally absorbs more power motion/pressure response vector velocity of the i-th WEC oscillating in j-th mode ca the sound velocity in air Cd matrix of wave damping coefficients ch,n PTO damping coefficient of PA n cp,n PTO damping coefficient of OWC n CPTO matrix of damping coefficients of PTO system in a wave farm dn submerged depth of WEC n E constraint matrix Fc restraining forces/moments to fix the OWC chambers Fe wave excitation volume/force acting on the wave farm g Km mooring line restoring stiffness matrix km,n mooring line restoring stiffness of WEC n Kp a skew-symmetric matrix connecting the motions of OWC and its chamber Ks hydrostatic stiffness matrix of a wave farm L The distance between two OWCs/PAs in the same row/column M mass matrix of a wave farm M number of OWCs in a wave farm m0 mass of PA Ma matrix of added-mass coefficients MPTO matrix of mass coefficients of PTO system in a wave farm N number of PAs in a wave farm pi air pressure inside the chamber of OWC i Pin incoming wave power per unit width of the wave front pOWC power absorbed by an OWC in isolation POWC power absorbed by an OWC wave farm pPA power absorbed by a PA in isolation PPA power absorbed by a PA wave farm qOWC array factor of an OWC wave farm qPA array factor of a PA wave farm q * array factor of a hybrid wave farm Ri,n inner radius of OWC n Rn outer radius of WEC n vg wave group velocity Vn air chamber volume of OWC n Ẋ motion/pressure response vector η wave power capture factor of a hybrid wave farm ηOWC wave power capture factor of an OWC in isolation ηPA wave power capture factor of a PA in isolation ηP1 wave power capture factor of an OWC wave farm ηP2 wave power capture factor of a PA wave farm ηH1 wave power capture factor of the hybrid wave farm #H1ηH2 wave power capture factor of the hybrid wave farm #mass of OWC n in isolation due to its air pressure oscillation -mass of OWC n in isolation due to its air pressure oscillation -mass of WEC n in isolation due to its heave motion of OWC n in isolation due to its air pressure oscillation of OWC n in isolation due to its heave motion ( ) heave radiation damping of OWC n in isolation due to its air pressure oscillation heave radiation damping of WEC n in isolation due to its heave motion

Fig. 3 .
Fig. 3. Wave power capture factor of a floating/fixed OWC and a PA in isolation, and optimized PTO damping versus kh.

Figure 6
Figure 6 presents a power capture factor comparison between the hybrid wave farm H1, H2, the conventional wave farm P1 (all OWCs) and P2 (all PAs), and the isolated OWC and PA for both Cases I and II, L/h=0.75, β=0.The hydrodynamic interaction between

Fig. 13 .
Fig. 13.Power capture factor of the conventional wave farm versus kh for different L/h, β= π: (a) OWCs farm of Case I; (b) OWCs farm of Case II; (c) PAs farm.

Table 2 .
Frequency ranges and levels of effect of β.

Table 3 .
First peak value of wave capture factor of #P1 (Cases I and II), #P2 and the corresponding kh for different L/h, β=π.