# Floating Solar Systems with Application to Nearshore Sites in the Greek Sea Region

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Hydrodynamic Model for Simple Floating Structures Supporting PV Modules

**x**= (x

_{1}, x

_{2}, x

_{3}) was used, where x

_{1}, x

_{2}and x

_{3}are the longitudinal, transverse and vertical axes in the local coordinate system of the hull, respectively. The origin was located at the structure’s center of flotation, with the x

_{3}-axis pointing upwards. Following linear water wave theory, the velocity field is represented by the gradient of the potential function Φ:

^{2}/g is the frequency parameter, with ω being the angular frequency and g being the acceleration due to gravity.

_{3}= 0), as follows:

_{4}) of the structure, which dynamically alters the tilt angle of the solar panels on deck, while the linear oscillatory motions are considered not to impact the solar irradiance received. Using standard floating body hydrodynamic theory [14,15], the complex potential can be decomposed as follows:

_{d}is the diffracted field, ${\phi}_{4}$ is the radiation field induced by the angular roll oscillation of the structure about the longitudinal axis x

_{1}, ξ

_{4}is the complex amplitude of the structure’s response in roll motion and ξ

_{0}= ξ

_{d}= A = 1. The incident wave field is considered to be known and equal to

#### 2.1. BEM Hydrodynamic Model

_{3}= −h and ${\sigma}_{n}(x),x\in \partial D,n=d,4$ denotes the source–sink strength distribution defined on $\partial D$, corresponding to the diffraction field (n = d) and the roll radiation field (n = 4), respectively.

_{2}direction, an absorbing layer technique was adopted, consisting of an absorbing layer which was used to attenuate the outgoing waves in an optimal way, preventing reflections from the outer boundary; see, e.g., [17]. The thickness of the layer was of the order of the local wavelength λ and the implementation of the absorbing layer was achieved by making the frequency parameter complex inside the absorbing layer, as

_{PML}is the distance from the body where the PML is activated, and the optimized PML parameters c and q and effective length are defined depending on the angular wave frequency ω. Details concerning the values used can be found in Table 1 of Ref. [18].

_{ij}of the influence matrix

**A**was calculated by the induced potential and velocity from the j–source–sink panel j to the i-collocation point and corresponds to the discretized form of the left-hand side of Equation (9), while the component b

_{j}of the right-hand side contains the values of ${N}_{n},n=d,4,$ given by Equation (12) and evaluated at the i-collocation point. The piecewise constant values of the source/sink strength distribution defined on the boundary $\partial D$ were then used to evaluate the potential and the velocities in the domain, as follows:

_{4}) was evaluated by means of the equation of motion, as follows:

_{44}= MR

_{44}, where M = ρV is the mass of the structure and V = LBT is the submerged volume. The parameter C

_{44}, modeling the hydrostatic restoring roll moment, equals C

_{44}= gM·GM, where GM denotes the metacentric height, evaluated as GM = KB + BM − KG, where K is the reference point at the keel of the structure, G is the center of gravity, B denotes the center of buoyancy located at x

_{3}= −T/2, and the metacentric radius BM is evaluated as BM = I/ρ∇, where I is the second moment of area of the waterplane along the longitudinal axis x

_{1}, which in the case of the floating pontoon is given by Ι = LB

^{3}/12.

#### 2.2. Numerical Results and Hydrodynamic Model Verification

_{44}= 0.4B. Numerical and experimental results regarding the above configuration have been presented by Pinkster and van Oortmerssen [21]. In the latter work, model tests were carried out in the shallow water laboratory of the Netherlands Ship Model Basin, which measures 210 m in length and 15.75 m in breadth, and the water depth is equal to 1 m. The tests were carried out using a model at a scale of 1:50. Regular waves were generated at one end of the basin via a flap-type wave maker, while a perforated sloping beach at the other end of the basin served as a wave damper to minimize reflections. Concerning the present discrete BEM model, a minimum of 20 boundary elements per wavelength was applied to the free-surface boundary, while the number of equally distributed panels on the wetted surface of the pontoon cross section was 300, which was found to be sufficient for numerical convergence.

## 3. Offshore-to-Nearshore Transformation of Wave Conditions

^{3}and the draft of the structure is T = 3 m, and therefore the total mass of the FPV is M = 2.075 × 10

^{6}kg and the roll moment of inertia is I

_{44}= 7.47 × 10

^{7}kgm

^{2}. Moreover, the center of gravity is assumed to be located at a vertical distance of 3 m above the keel.

_{s}, mean energy period T

_{e}and mean wave direction θ

_{m}) at the two considered locations, including standard deviation and minimum (min) and maximum (max) values, are comparatively presented in Table 1.

## 4. Responses of the FPV Structure

^{2}is the acceleration due to gravity. Also, $a$ is a normalization parameter suitably defined such that ${H}_{s}^{}=\hspace{0.17em}4\sqrt{{m}_{0}^{}}$, where ${m}_{0}^{}$ is the zeroth-order spectral moment. Finally, the parameter $\gamma $ and the function $\delta \left(f\right)$ are defined as follows:

_{s}= 0.79 m and the corresponding value of the mean energy period T

_{e}= 3.41 s. In this case, the frequency spectrum using the JONSWAP model shown in Figure 9a is combined with the roll response (RAO) of the FPV plotted in Figure 9b, and we derived the roll motion spectra as shown in Figure 9c for two incident angles (β = 90° for beam waves, and β = 135° for quartering seas).

_{H}= 0° and thus β = θ.

_{s}= 0.79 m, peak period T

_{e}= 3.41 s and mean direction θ

_{m}= 24.37° (corresponding to the climatological mean values at the FPV coastal location of the SE Evia Island region), is presented in Figure 10. In the sequel, the short-time roll responses ${\xi}_{4}\left(t;{H}_{s},{T}_{e},{\theta}_{m}\right)$ for each wave condition were used to calculate the angle of incidence (AOI) at the FPV and the resulting effect on the power output performance of a PV system, in conjunction with other data concerning the tilt (with respect to the deck of the structure) and their orientation (azimuth angle), as described in the following section.

## 5. Effects of Waves on FPV Module Power Performance

_{PM}= 33.9 V and I

_{PM}= 33.9 V, were considered, with module efficiency of 15% at T

_{STC}= 25 °C. Thus, the total panel area of panels on the FPV structure was A = 605.44 m

^{2}and k

_{p}≈ 0.4%/°C was used as an approximate value for the silicon panel technology. The total radiation G received by the panels consisted of the direct (beam) radiation B and the diffuse horizontal irradiation D, and it also included the reflected irradiance component R. The latter components were calculated as follows (see, e.g., [26]):

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Elongated floating pontoon structure supporting photovoltaic panels and parameters of the hydrodynamic model.

**Figure 3.**Normalized non dimensional roll moments (

**a**,

**c**) and roll motion responses (

**b**,

**d**), as functions of the non dimensional frequency, as calculated via the present method and as measured using model tests [21] for incident wave fields propagating at (

**a**,

**b**) β = 90° and (

**c**,

**d**) β = 135°. Froude–Krylov, diffraction and total roll moments, as calculated by the present BEM scheme, are plotted using thin, dashed and thick lines, respectively.

**Figure 4.**Nearshore areas considered for FPV module deployment. (

**a**) Pagasistikos Gulf area, and (

**b**) SE coastal region of Evia Island.

**Figure 5.**(

**a**) Nearshore area considered for the FPV deployment in western Pagasitikos Gulf region. (

**b**) Calculated waves using SWAN model for the following offshore data: H

_{s}= 0.42 m, T

_{e}= 4.44 s, mean wave direction = 141°, wind speed = 6.33 m/s, wind direction = 180°. The position of the considered FPV structure is shown by using yellow rectangle.

**Figure 6.**Nearshore area considered for the FPV deployment in SE region of Evia Island. The position of the considered FPV structure is shown using yellow rectangle.

**Figure 7.**Annual distribution of wave parameters at the nearshore point TP in the Pagasitikos Gulf region.

**Figure 8.**Annual distribution of wave parameters at the nearshore point TP5 of SE region of Evia Island.

**Figure 9.**(

**a**) Wave frequency spectra corresponding to the mean values of significant wave height and mean period of the nearshore data in the SE coastal area of Evia Island in the case of incident waves of significant wave height ${H}_{s}=0.79\mathrm{m}$ and energy period ${T}_{e}=3.41\mathrm{s}$. (

**b**) Roll response operators (RAOs) and (

**c**) response frequency spectra of the floating pontoon-type FPV structure of length L = 45 m, breadth B = 15 m, draft 3 m, located at depth h = 15 m for beam β = 90° and quartering β = 135° seas.

**Figure 10.**Simulated time series of the pontoon-type FPV structure’s roll motion. Length L = 45 m, breadth B = 15 m, draft 3 m, depth h = 15 m. Incident waves of significant wave height equal to ${H}_{s}=0.79\hspace{0.17em}\mathrm{m}$ and energy period ${T}_{e}=3.41\hspace{0.17em}\mathrm{s}$. (

**a**) Simulated 1 h-long time series data and (

**b**) indicative roll motion in a 2 min long time interval.

**Figure 11.**Simulated time series of temperature and power performance for land-based and FPV 100 kWp configuration considered at the geographical location of Pagasitikos Gulf region. Ambient and cell temperature of (

**a**) land-based unit and (

**b**) FPV, and (

**c**) comparative daily power production in a TMY.

**Figure 12.**Simulated time series of temperature and power performance for land-based and FPV 100 kWp configuration considered at the geographical location of the SE Evia Island nearshore region. Ambient and cell temperature of (

**a**) land-based unit and (

**b**) FPV, and (

**c**) comparative daily power production in a TMY.

Nearshore Point | H_{s},_{mean} | T_{e},_{mean} | H_{s},_{std} | T_{e},_{std} | H_{s}min/max | T_{e}min/max | R (H _{s},T_{e}) | θ,_{mean}(deg) | θ,_{std}(deg) |
---|---|---|---|---|---|---|---|---|---|

Pagasitikos Gulf | 0.25 m | 3.57 s | 0.20 m | 1.12 s | 0.03/1.50 m | 1.63/7.16 s | 0.796 | 54.04 | 41.07 |

SE Evia Island | 0.79 m | 3.41 s | 0.73 m | 1.45 s | 0.01/6.98 m | 1.31/9.40 s | 0.877 | 24.37 | 38.73 |

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**MDPI and ACS Style**

Magkouris, A.; Rusu, E.; Rusu, L.; Belibassakis, K.
Floating Solar Systems with Application to Nearshore Sites in the Greek Sea Region. *J. Mar. Sci. Eng.* **2023**, *11*, 722.
https://doi.org/10.3390/jmse11040722

**AMA Style**

Magkouris A, Rusu E, Rusu L, Belibassakis K.
Floating Solar Systems with Application to Nearshore Sites in the Greek Sea Region. *Journal of Marine Science and Engineering*. 2023; 11(4):722.
https://doi.org/10.3390/jmse11040722

**Chicago/Turabian Style**

Magkouris, Alex, Eugen Rusu, Liliana Rusu, and Kostas Belibassakis.
2023. "Floating Solar Systems with Application to Nearshore Sites in the Greek Sea Region" *Journal of Marine Science and Engineering* 11, no. 4: 722.
https://doi.org/10.3390/jmse11040722