*4.1. Comparison with Other Methods and Verification*

The results obtained by the previously described numerical model are here compared to previous research for verification purposes. The results concern a twin-hull floating structure whose individual hulls are cylindrical, with a draft equal to the radius, which results in wetted surfaces whose cross-section shapes are semicircles. Numerical results regarding the above configuration were presented by Ohcusu [26] in 1969 and Rhee [27] in 1982 concerning the amplitude ratio of the radiated fields' wave height away from the body divided by the amplitude of the forced oscillation that excites the field itself in calm water.

Figure 2 illustrates the aforementioned ratio regarding the heave and sway motions in the case of unit amplitude of the twin-hull structure. The wetted surface of each hull is semicircle of radius *R* in the *x*2*x*<sup>3</sup> plane, while each of the two semicircles' centers are at a distance *P* from the origin, following the notation of Rhee [27]. Therefore, the two centers are at a distance *2P* apart and the configuration is defined so that 2*P*/*R* = 3. The results concern the radiation fields that propagate in deep water, which is achieved in the present numerical model by setting the depth as a constant and equal to half the wavelength, for each simulated frequency, as calculated from the dispersion relation for deep-water (*λ* = 2*πg*/*ω*2).

**Figure 2.** Amplitude ratios for heave and sway for a twin-hull floating structure with semicircle hull cross-sections (2*P*/*R* = 3, *h*/*R* = ∞).

The domain extends to three wavelengths away from the floating body in both directions and the free surface elevation is evaluated by the discrete BEM model at the last free surface boundary element away from the structure (adjacent to the first boundary element of the radiation boundary). The amplitude ratios of Figure 2 are presented as functions of the non-dimensional frequency parameter *ω*2*R*/*g*.

Indicative results are illustrated in Figure 3 concerning wave fields generated by unitamplitude forced oscillations of the twin hull in sway and heave, with the non-dimensional frequency parameter *ω*2*R*/*g* set to 1. The amplitude ratios are equal to 0.992 and 0.520 for sway and heave, respectively, as also shown in Figure 2.

**Figure 3.** Sway and heave radiation fields for 2*P*/*R* = 3, *h* = *λ*/2 and *ω*2*R*/*g* = 1. (**a**,**b**) Real and imaginary part of the normalized sway field *ϕ*2(**x**) and corresponding free-surface elevation. (**c**,**d**) Real and imaginary part of the normalized heave field *ϕ*3(**x**) and corresponding free-surface elevation.

An identical twin-hull structure was studied by Dabssi et al. [28] in 2008 regarding its hydrodynamic coefficients. Figure 3 illustrates the added mass and damping of the floating structure in heave (*ξ*3). The added mass (*A*33) has been divided by the structure's mass, while the damping coefficient for heave (*B*33) has been divided by the mass times the angular frequency *ω* so that all presented quantities are non-dimensional. It is noted that the displacement in this case does not need to be numerically calculated since it equals the sum of volumes of two half-cylinders of radius R that were considered to extend to unit length in the transverse direction (*x*1) and therefore is equal to *πR*2. The results of Figure 4 concern the heaving motion of the twin-hull in a finite water depth *h*, where *h*/*R* = 2. The calculated data sets are presented as functions of the non-dimensional wavenumber *kR*.

**Figure 4.** Added mass and damping coefficient of heave, for a twin-hull floating structure of semicircle hull cross sections (2*P*/*R* = 3, *h*/*R* = 2).

#### *4.2. Hydrodynamic Analysis of Floating Twin-Hull Structure in Waves*

The effect of sloping seabed environments on the hydrodynamic characteristics of a twin hull is here illustrated by considering the case of a structure of non-dimensional total breadth equal to *B*(*T*)/*h* = 2/3, with the non-dimensional breadth and draft of each hull set to *B*(*H*)/*h* = *T*/*h* = 1/10, where *h* denotes the mean water depth of the inhomogeneous domain *D*. The individual hulls that make up the twin-hull layout were modeled via the cross-section of a Wigley hull at *x*<sup>1</sup> = 0, which is given by the analytical relation:

$$\mathbf{x}\_2 = \mp \frac{B\_{(H)}}{2} \cdot \left[ 1 - \left( \frac{\mathbf{x}\_3}{T} \right)^2 \right]. \tag{15}$$

The configuration is considered to be located in an inhomogeneous region (see Figures 5 and 6). The center of gravity was selected to coincide with the center of flotation. The center of buoyancy (*B*), which is calculated as the center of area of the submerged volume's cross-section, is located at (*x*<sup>2</sup> = 0, *x*<sup>3</sup> = −0.375 · *T*) and, thus, the non-dimensional metacentric height of this layout is *GM*/*h* = 1.179.

**Figure 5.** Outline of the modeled configuration and basic dimensions.

**Figure 6.** (**a**) Floating body and domains of transmission (*B*(*T*)/*h* = 2/3, *B*(*H*)/*h* = *T*/*h* = 1/10). (**b**) Hydrodynamic forces *F*1 *<sup>k</sup>*, *k* = 2, 3. (**c**,**d**) RAOs in sway and heave motions, respectively. (**e**,**f**) Hydrodynamic coefficients *A*22, *B*<sup>22</sup> and *A*33, *B*33, respectively. All quantities were plotted vs. the non-dimensional wavelength *λ*/*h*, where *h* denotes the average water depth.

Numerical results are presented in Figures 6 and 7 concerning the hydrodynamic behavior of this floating structure in constant depth and over two linear shoals characterized by (constant) bottom slopes of 10 and 20%, respectively (see Figure 6a). The shoaling environments were achieved using a linear depth reduction of 2*h*/3 and 4*h*/3, respectively, over a depth variation distance of 10*B*(*T*), with the mean water depth of all three domains of transmission being equal to *h*. The results concerning the homogeneous domain were plotted using solid lines, while the results concerning the inhomogeneous transmission domains with bottom slopes of 10 and 20% were plotted using dashed lines and dotted lines, respectively.

**Figure 7.** (**a**) RAO in roll motion. (**b**) Hydrodynamic moment *F*1 4. (**c**,**d**) Hydrodynamic coefficients *A*44, *B*<sup>44</sup> and *A*24, *B*24, respectively. All quantities are plotted vs. the non-dimensional wavelength *λ*/*h*, where *h* denotes the average water depth.

In particular, Figure 6b illustrates the normalized hydrodynamic forces as functions of the non-dimensional wavelength *λ*/*h* for all three considered domains of transmission, where *λ* = 2*π*/*k*<sup>0</sup> is the wavelength corresponding to the mean water depth *h*, as obtained through application of the dispersion relation: *<sup>ω</sup>*<sup>2</sup> = *<sup>k</sup>*0*<sup>g</sup>* · tanh (*k*0*h*). The normalization used for the hydrodynamic forces is *F*1 *<sup>k</sup>* = *Fk*/*ρghA*, *k* = 2, 3, where *A* is the incident wave amplitude.

Figure 6c,d depict the twin hull's response amplitude operators (RAOs) associated with its two linear motions, i.e., sway (*ξ*2) and heave (*ξ*3). The body's linear responses were normalized as *RAOk* = *ξ* 1 *<sup>k</sup>* = |*ξk*|/*A*, *k* = 2, 3. Finally, in Figure 6e,f corresponding results concerning the hydrodynamic coefficients are presented. The matrix **A**(3×3) of added inertial coefficients and the matrix **B**(3×3) of hydrodynamic damping coefficients were normalized as:

$$
\tilde{\mathbf{A}} = \frac{\mathbf{A}}{\rho} \begin{pmatrix} h^{-2} & h^{-2} & h^{-3} \\ h^{-2} & h^{-2} & h^{-3} \\ h^{-3} & h^{-3} & h^{-4} \end{pmatrix}, \\
\tilde{\mathbf{B}} = \sqrt{\frac{h}{\mathcal{S}}} \cdot \frac{\mathbf{B}}{\rho} \begin{pmatrix} h^{-2} & h^{-2} & h^{-3} \\ h^{-2} & h^{-2} & h^{-3} \\ h^{-3} & h^{-3} & h^{-4} \end{pmatrix}. \tag{16}
$$

Figure 7a illustrates the twin hull's RAO associated with the angular motion, i.e., roll (*ξ*4). The body angular response is normalized as *RAO*<sup>4</sup> = *ξ* 1 <sup>4</sup> = |*ξ*4|/*kA*, with *k* being the wavenumber corresponding to the mean water depth *h*. The corresponding normalized hydrodynamic moment *<sup>F</sup>*1<sup>4</sup> <sup>=</sup> *<sup>F</sup>*4/*ρgh*2*<sup>A</sup>* is shown in Figure 7b. Figure 7c depicts the diagonal elements (*A*44, *B*44) of the added inertia and damping matrices. Moreover, the (non-diagonal) elements of the symmetric added inertia and hydrodynamic damping matrices are shown in Figure 7d. All results are plotted as functions of the non-dimensional wavelength *λ*/*h*.

Finally, indicative results regarding the total induced wavefields are depicted in Figure 8 for non-dimensional wavelength equal to *λ*/*h* = 2.4. In particular, Figure 8a illustrates the real part of the total potential *ϕ*(**x**); see Equation (4) for the three considered cases of 0, 10 and 20% bottom slope (see Figure 6a) in the general vicinity of the floating twin-hull structure. Figure 8b depicts the imaginary parts of the corresponding potential functions. The configuration was made dimensional by setting *h* = 30 m and an incident field of amplitude *A* = 1.5 m has been considered. The alterations to the wavefields due to the non-uniform topography profiles are clearly seen, as the equipotential lines intersect each seabed boundary section perpendicularly, which implies the satisfaction of the impermeability boundary condition.

**Figure 8.** (**a**) Real and (**b**) Imaginary Part of the total complex wave potential and corresponding free surface elevation for a twin-hull floating structure of breadth *B* = 20 m and three considered cases of bottom slope 0, 10 and 20% in an environment with a mean depth of *h* = 30 m in the case of an incident wave of wavelength *λ*/*h* = 2.4 and amplitude *A* = 1.5 m.

#### **5. Effects of Floating Structure Response in Waves on Floating PV Performance**

The energy efficiency of a floating photovoltaic (FPV) unit is based on several parameters, many of which are the result of the surrounding marine environment. Some of the factors that affect the energy efficiency of FPV are common with corresponding land-based units, with similar power output levels, while others are absent in land installations.

In open seas, there is generally a higher level of humidity than inland, as well as lower ambient temperatures. The decreased temperatures are a result of various factors, which among others, include [29] the water's transparency, which results in the incoming solar radiation being transmitted to inner layers of the medium rather than just the surface layer, as well as the fraction of incident irradiation that is naturally used for evaporation. Furthermore, the wind speed is usually higher due to long fetch distances compared to land. The above parameters can help to maintain a low operating temperature of the solar cells, which, in turn, leads to close-to-optimal performance of the solar panel. The latter's efficiency decreases with increasing temperatures. More importantly, PV efficiency is strongly dependent on the angle of incidence (AOI) of solar irradiation, which, in offshore FPV installations, is directly affected by the dynamic wave-induced motions. In particular, the power output of photovoltaic cells is strongly affected by the angle of incidence (AOI) of solar irradiation and the plane of array (POA) irradiance, which is given by the following equation:

$$POA = DNI \cos(AOI) + DHI + RI,\tag{17}$$

where *DNI, DHI* and *RI* are the direct, diffuse and reflected irradiance components on a tilted surface, respectively. To provide indicative results regarding the effect of waveinduced motions of the floating structure on the power output, we considered an offshore

installation in the geographical sea area of the southern Aegean Sea. For the latter area, the optimized values for tilt and azimuth angles of the photovoltaic installations, respectively, are *θ<sup>T</sup>* = 30*<sup>o</sup>* and *θ<sup>A</sup>* = 135*<sup>o</sup>* using data extracted from the Sandia Module Database, which is provided by the PV\_LIB toolbox (https://pvpmc.sandia.gov/applications/pv\_ lib-toolbox/ (accessed on 12 August 2021)).

In this work, a preliminary assessment of a floating photovoltaic system's energy efficiency is made for twin-hull structures, taking into account data regarding the dynamic motions of the floating unit carrying the panels, as derived by the hydrodynamic model presented earlier, while the interesting effects of temperature and humidity will be studied in future work. The linear motions, i.e., sway (*k* = 2) and heave (*k* = 3), are considered to have no important effect on the tilt angle of the panels and, therefore, the angle of incidence. Hence the effect of the unit's mobility is limited to the angular oscillation i.e., the roll motion (*k* = 4) under excitation from the beam incident waves.

For this purpose, response data was simulated by assuming specific sea conditions. The latter are characterized by a frequency spectrum used to describe the incident waves. We considered the floating twin-hull structure of total breadth *B*(*T*) = 20 m examined in the previous section in constant water depth *h* = 30 m. The sea state is described by a Brettschneider spectrum model (see [30], Chapter 2.3), as follows:

$$S\left(\omega; H\_{s}, T\_{p}\right) = \frac{5}{16} H\_{s}^{2} \frac{\omega\_{p}^{4}}{\omega^{5}} \exp\left[-\frac{5}{4} \left(\frac{\omega}{\omega\_{p}}\right)^{-4}\right] \tag{18}$$

where *Hs* is the significant wave height, *ω<sup>p</sup>* = 2*π*/*Tp* is the peak frequency and *Tp* the corresponding peak period.

The roll responses calculated by the present model, as discussed in the previous section, were used to evaluate the fluctuations of the AOI and the effect on the power output performance of a PV system consisting of panels, with the aforementioned values of tilt (relative to the horizontal deck of the structure) and azimuth angles. Specifically, the roll spectrum was calculated using the RAO of the roll motion (see Figure 7a) of the twin-hull structure using:

$$S\_4(\omega) = R A O^2(\omega) k^2 S(\omega) \tag{19}$$

where the wavenumber *k* is given by the dispersion relation of water waves for the water depth considered. Based on the calculated roll spectrum, time series of the roll motion *ξ*4 *t* ; *Hs*, *Tp* of the above floating twin-hull structure were simulated, for the considered configuration (structure and coastal environment) and incident waves, characterized by the parameters *Hs*, *Tp* using the random-phase model [30], Chapter 8.2, (see also [31]).

The results were normalized using the value corresponding to calm water (flat horizontal deck of the structure) in the same sea environment, which results in the following definition of the performance index:

$$PI(t) = \frac{a \cdot \cos(a\_m + \xi\_4(t)) + b}{a \cdot \cos(a\_m) + b} \tag{20}$$

where *a* = *DNI* and *b* = *DHI* + *RI* for the geographical area and sea environment considered, respectively, and *α<sup>m</sup>* is a representative value for the angle of incidence.

As an example, the numerical results concerning the calculated roll response of a floating twin-hull structure of breadth *B*(*T*) = 20 m at depth *h* = 30 m with an incident wave spectrum (dashed line) and roll angle spectrum (solid line) of the structure in the case of incident waves of significant wave height *HS* = 0.5 m and peak period *TP* = (2*π*/*ωp*) = 4 s are presented in Figure 9. Based on the calculated roll spectrum, the simulated time series of the roll motion of the above floating twin-hull structure for the considered coastal environment and incident waves are shown in Figure 10 for a time interval of 1 h. Furthermore, in the same figure, a representative small interval of 3 min was obtained using the randomphase model [30,31] for a sea state that was characterized by significant wave height *HS* =

0.5 m and peak period *TP* = 4 s, as generated by winds corresponding to the Beaufort scale levels *BF* = 1 − 2. In this case, indicative results concerning the effect of waves and roll responses of the structure on the performance index are shown in Figure 11, as calculated by Equation (20) using a representative value of the mean angle of incidence *α<sup>m</sup>* = 5◦ and omitting, as a first approximation, the effect of diffuse and reflected irradiance components (*b* ≈ 0). The value of the performance index in calm water was *PICALM* = 0.9962. In the considered case of incident waves, which were characterized by a very low energy content, the *RMS* value of the estimated performance index dropped to *PIRMS* = 0.9947. The latter's mean value, as well as the corresponding calm-water value, are shown in Figure 11 using cyan and red lines, respectively.

**Figure 9.** (**a**) Roll response of the floating twin-hull structure of breadth *B* = 20 m at a mean depth *h* = 30 m. (**b**) Incident wave spectrum (dashed line) and roll angle spectrum (solid line) of the structure in the case of incident waves with a significant wave height *HS* = 0.5 m and a peak period *TP* = 4 s.

**Figure 10.** Simulated time series of the floating twin-hull structure's roll motion. Total breadth *B*(*T*) = 20 *m* at a depth *h* = 30 *m* in the case of incident waves with a significant wave height equal to *HS* = 0.5 *m* and a peak period *TP* = 4 s. (**a**)A1h long time series and (**b**) indicative roll motion over a 3 min long interval.

**Figure 11.** Performance Index of FPV on the floating twin hull structure of breadth *B*(*T*) = 20 m at a depth *h* = 30 m in the case of incident waves with a significant wave height *HS* = 0.5 m and a peak period *TP* = 4 s. Roll motion time series (**a**) in a 1 h long time interval and (**b**) in an indicative 3 min long time interval.

#### **6. Discussion**

Following previous works [32,33], concerning the investigation and modeling of marine renewable energy systems, the present method focused on the estimation of the effect of wave-induced responses on the performance index of a twin-hull FPV structure in various sea conditions, as defined by the wave climatology of the offshore–coastal site where the system was deployed. As an example, the results concerning the performance index that is associated with the wave effects (Equation (20)) of the floating twin-hull structure of breadth *B*(*T*) = 20 m at a water depth *h* = 30 m that are presented and discussed above are given in Table 1 for wind waves corresponding to the Beaufort scale from *BF* = 1 (relatively calm sea) to *BF* = 5 − 6 conditions.


**Table 1.** Performance indexes for different sea conditions.

We observe that the effect of roll responses results in fluctuations of the AOI that could cause a significant drop in the performance index as the sea condition changed from calm to moderate and higher severities. A more complete picture of the sea state's effect on the FPV module's power output, as estimated using the present method, is shown in Figure 12, indicating its usefulness for supporting the systematic analysis and design of the system, including the offshore structure, as well as the electric production and storage subsystems. Although inevitable fluctuations of the AOI due to waves in offshore PV units reduce the power output, this negative effect could be balanced or even reversed by the cooling effect and other factors resulting from the marine environment, which is a subject that is left to be considered in future work.

**Figure 12.** Contour map of the normalized performance index as a function of the prevailing sea state (significant wave height *HS* and peak period *TP*).

#### **7. Conclusions**

A BEM model was developed and applied to the hydrodynamic analysis of twin-hull structures in variable bathymetry regions and was used to predict their hydrodynamic responses and their effects concerning the power output of offshore FPV systems. The analysis was restricted to two spatial dimensions for simplicity. After verification of the method with comparisons against data from the literature, the method was systematically applied and the derived numerical results are presented for floating bodies of simple geometry, lying over uniform and sloping seabeds. With the aid of systematic comparisons, the effects of bottom slope on the hydrodynamic characteristics (hydrodynamic coefficients and responses) of the floating bodies were illustrated and discussed. Finally, response data that was simulated for specific sea conditions, characterized by frequency spectra, were considered to describe the incident waves interacting with a floating twin-hull structure, in order to evaluate the effect of wave-induced fluctuations on the power output performance of the floating PV system. The effects of waves on the floating PV performance are presented, indicating significant variations of the performance index ranging from 0 to 15% depending on the sea state. Future work will be directed toward (a) the detailed analysis of wave and wind environmental factors and their effects on the resulting system's performance, (b) the extension of the model to 3D including 6-DOF wave motion analysis of the floating structures over general bathymetry and evaluation of their performance and (c) a systematic application of the present method to realistic cases that support the optimized design of floating PV modules in specific marine environments.

**Author Contributions:** Conceptualization, K.B.; methodology, K.B. and A.M; software, A.M. and K.B.; validation, A.M.; writing—original draft preparation, A.M.; writing—review and editing, K.B. and E.R.; visualization, E.R.; supervision, K.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the research project DREAM (Dynamics of the Resources and Technological Advance in Harvesting Marine Renewable Energy), funded by the Romanian Executive Agency for Higher Education, Research, Development and Innovation Funding–UEFISCDI, grant number PNIII-P4-ID-PCE-2020-0008.

**Acknowledgments:** The doctoral studies of A. Magkouris are currently supported by the Special Account for Research Funding (E.L.K.E.) of the National Technical University of Athens (N.T.U.A.) operating in accordance with Law 4485/17 and National and European Community Law.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

