# Dynamic Modeling of an Offshore Floating Wind Turbine for Application in the Mediterranean Sea

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## Abstract

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## 1. Introduction

- Construction of a state of the art mathematical model for onshore wind turbines, in order to implement the aerodynamics and finally verify the results with FAST, in terms of control on the blade pitch, generated power and loads discharged at the tower base.
- Construction of a state of the art mathematical model for a platform immersed in water, in order to implement hydrodynamics and finally verify the results with FAST, in terms of hydrostatic and hydrodynamic loads, moorings and displacements of the hull.
- Combination of the aerodynamics and hydrodynamics macro-blocks: the loads at the base of the tower act on the hull, the movements of the platform influence the orientation of the turbine which is integral with it.

## 2. Materials and Methods

#### 2.1. Dynamic Modeling

- Waves: for each timestep it provides the loads in the 6-DOFs acting on the hull and derived from waves.
- Wind: for each timestep it provides the wind speed components on the three axis of the FRA (U-wind, V-wind, W-wind).
- Wind turbine: for each timestep it provides loads in the 6-DOFs acting at the turbine tower base on the hull. It takes as input wind speeds and 6-DOFs positions of the hull, from which it calculates loads on the turbine that generate power and reaction loads at the base.
- Moorings: for each timestep it provides loads in the 6-DOFs acting on the hull due to mooring lines for stability maintenance. Moorings block outputs depend on hull 6-DOFs position.
- Hull: for each timestep it takes loads by waves, moorings and wind turbine. Through hydrodynamics laws 6-DOFs position are detected.

#### 2.2. Hydrodynamics

#### 2.2.1. Irregular Waves

#### Standard Wave Spectra

#### 2.2.2. Hydrostatic ‘Restoring’ Load

- the $xz$ and the $yz$ planes are symmetry planes, so ${K}_{34}={K}_{35}={K}_{45}=0$. This approximation holds because the elements violating symmetry-the rotating blades and the nacelle-have negligible masses and their centres of mass are very close to the z axis
- in the small amplitude approximation, the centre of buoyancy and the centre of gravity stay vertically aligned, so ${K}_{46}={K}_{56}=0$.

#### 2.2.3. Froude-Krylov Load

#### 2.2.4. Diffraction Load

#### 2.2.5. Radiation Load

#### 2.2.6. Drag Load

#### 2.3. Moorings

- Free–Hanging Line
- Line Touching the Bottom

- Free–Hanging Line$$l=\frac{H}{\omega}+[ln(\frac{V}{H}+\sqrt{1+{\left(\frac{V}{H}\right)}^{2}})-ln(\frac{V-L\omega}{H}+\sqrt{1+{\left(\frac{V-L\omega}{H}\right)}^{2}})]+\frac{HL}{EA}$$$$h=\frac{H}{\omega}+[\sqrt{1+{\left(\frac{V}{H}\right)}^{2}}-\sqrt{1+{\left(\frac{V-L\omega}{H}\right)}^{2}}]+\frac{1}{EA}\left(VL-\frac{{L}^{2}\omega}{2}\right)$$$$\omega =gA\left({\rho}_{cable}-\rho \right)$$
- Line Touching the Bottom$$l={L}_{B}+\frac{H}{\omega}[ln(\frac{V}{H}+\sqrt{1+{\left(\frac{V}{H}\right)}^{2}})]+\frac{HL}{EA}+\frac{{C}_{B}\omega}{2EA}\left[{x}_{0}\lambda -{L}_{B}^{2}\right]$$$$h=\frac{H}{\omega}[\sqrt{1+{\left(\frac{V}{H}\right)}^{2}}-1]+\frac{{V}^{2}}{2EA\omega}$$$$\lambda =\left\{\begin{array}{cc}{L}_{B}-\frac{H}{{C}_{B}\omega}\hfill & {x}_{0}>0\hfill \\ 0\hfill & otherwise\hfill \end{array}\right.$$

- H: Horizontal fairlead force (Figure 5);
- V: Vertical fairlead force (Figure 5);
- A: Cable cross-section line;
- E: Young’s modulus;
- g: Gravity acceleration;
- ${\rho}_{cable}$: Cable density;
- $\rho $: Fluid density;
- L: Unstretched line length;
- ${L}_{B}$: Line length resting on the seabed (Figure 5B);
- ${C}_{B}$: Seabed friction coefficient;
- ${x}_{0}$: Horizontal force transition point (Figure 5B).

- N: Elements i at node j
- ${M}_{j}$: Point mass applied to the node jth. Used to evaluate the clump weight;
- ${B}_{j}$: Displaced volume applied to node jth;
- ${F}^{ext}$: external force applied to node jth;
- ${\alpha}_{i}$: Angle between ith angle and global X direction.

#### 2.4. Aerodynamics

#### 2.4.1. Wind-Speed Profile

#### Turbulence

“There are basically two methods for determining wind turbulence by theoretical means. One is via the energy spectrum of the turbulence, the other is by means of an actual wind speed time history. Independently of the chosen method, one phenomenon affecting the reaction of the wind rotor to turbulence must not be overlooked. In the open atmosphere, wind speed and turbulence are always unevenly distributed in space over the rotor-swept area. Many gusts strike the rotor not as a whole, but only on one side or only partially. This fact is significant for the response of the structure as regards the rotating rotor. The rotor blades "beat" into the gusts, i.e., the local wind speed changes, at their tangential speed. An observer travelling with the rotor blade experiences these speed changes considerably more strongly than he would in the steady-state system. Moreover, depending on the duration of the gust and the speed of the rotor, the rotor blade can encounter the same gust several times”

#### 2.4.2. Shaft Torque Balance

#### 2.4.3. Aerodynamic Loads

- blades are divided into small elements represented by 2D airfoils which are only subject to local physical events (blade element model); this means that all blade sections are independent and any spanwise evolution is neglected. The rotor disk area is divided into annuli of thickness dr; in each annulus there are Z blade elements of length dr, where Z is the number of wind turbine blades (Figure 7). The forces contribution from all annuli are summed along the span of the blade to calculate the total loads on the rotor.
- the rotor acts as an actuator disk (momentum theory) removing kinetic energy from the wind and thus gradually slowing down the stream, making the streamlines diverge; the disk is considered frictionless and the flow stationary, incompressible and frictionless; the momentum loss in the rotor plane is used to calculate the axial tangential velocities, that affect the forces calculated from the blade element theory.

#### Corrections to the BEM Model

- Prandtl’s tip-loss factor: considers that the airflow is not parallel near the tip of the blade.
- Glauert correction: in reality the thrust force on the turbine will not decrease when $a>0.4$, but its value will increase beyond 1. It avoids overestimating the tangential load with turbulent wind.
- Skewed wake correction: considers the deflection of the inflow wind that increases the induction factor; in fact, the wind direction is not perpendicular to the plane of rotation of the blades.

#### BEM Model Algorithm

- Discretize the blade into ${n}_{nodes}$ nodes, each i-th node of length ${l}_{BE,i}$ representing the blade element in that section
- Find the loads for each node, through the following steps (BEM theory):
- (a)
- (b)
- Iteration cycle for convergence of the induction factors, which provides:
- Calculate the angle of attack by mean of Equation (57).$$\alpha ={\varphi}_{0}-(\beta +{\theta}_{p})$$
- Read the lift and drag coefficients from airfoil tables for $\alpha $$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {C}_{l}={C}_{l}\left(\alpha \right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {C}_{d}={C}_{d}\left(\alpha \right)\hfill \end{array}$$
- Calculate normal and tangential forces coefficients by mean of Equations (61) and (62).$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {C}_{n}={C}_{l}\left(\alpha \right)cos{\varphi}_{0}+{C}_{d}\left(\alpha \right)sin{\varphi}_{0}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {C}_{t}={C}_{l}\left(\alpha \right)sin{\varphi}_{0}-{C}_{d}\left(\alpha \right)cos{\varphi}_{0}\hfill \end{array}$$
- Calculate Prantdtl’s tip- and hub-loss factor$${F}_{Pr}\left(r\right)={F}_{Pr,tip}\left(r\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{F}_{Pr,hub}\left(r\right)$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {F}_{Pr,tip}\left(r\right)=\frac{2}{\pi}{cos}^{-1}\left({e}^{-{\displaystyle \frac{B}{2}}{\displaystyle \frac{R-r}{rsin{\varphi}_{0}}}}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {F}_{Pr,hub}\left(r\right)=\frac{2}{\pi}{cos}^{-1}\left({e}^{-{\displaystyle \frac{B}{2}}{\displaystyle \frac{r-{R}_{h}}{rsin{\varphi}_{0}}}}\right)\hfill \end{array}$$
- Calculate the axial and tangential induction factor using Glauert correction:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& a=\left\{\begin{array}{cc}{\displaystyle \frac{\kappa}{\kappa +1}}\hfill & \hfill \phantom{\rule{1.em}{0ex}}{\varphi}_{0}>0,\kappa \le \frac{2}{3}\\ {\displaystyle \frac{{\gamma}_{1}-\sqrt{{\gamma}_{2}}}{{\gamma}_{3}}}\hfill & \hfill \phantom{\rule{1.em}{0ex}}{\varphi}_{0}>0,\kappa >\frac{2}{3},{\gamma}_{3}\ne 0\\ 1-{\displaystyle \frac{1}{2\sqrt{{\gamma}_{2}}}}\hfill & \hfill \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\varphi}_{0}>0,\kappa >\frac{2}{3},{\gamma}_{3}=0\\ {\displaystyle \frac{\kappa}{\kappa -1}}\hfill & \hfill \phantom{\rule{1.em}{0ex}}{\varphi}_{0}<0,\kappa >1\\ 0\hfill & \hfill \phantom{\rule{1.em}{0ex}}{\varphi}_{0}<0,\kappa \le 1\end{array}\right.\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {a}^{\prime}={\displaystyle \frac{{\kappa}^{\prime}}{1-{\kappa}^{\prime}}}\hfill \end{array}$$$$\kappa =\frac{\sigma {C}_{n}}{4\phantom{\rule{0.166667em}{0ex}}{F}_{Pr}{sin}^{2}{\varphi}_{0}};\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\kappa}^{\prime}=\frac{\sigma {C}_{t}}{4\phantom{\rule{0.166667em}{0ex}}{F}_{Pr}sin{\varphi}_{0}cos{\varphi}_{0}}$$$$\left\{\begin{array}{c}{\gamma}_{1}=2\phantom{\rule{0.166667em}{0ex}}{F}_{Pr}\kappa -\left(\frac{10}{9}-{F}_{Pr}\right)\hfill \\ {\gamma}_{2}=2\phantom{\rule{0.166667em}{0ex}}{F}_{Pr}\kappa -{F}_{Pr}\left(\frac{4}{3}-{F}_{Pr}\right)\hfill \\ {\gamma}_{3}=2\phantom{\rule{0.166667em}{0ex}}{F}_{Pr}\kappa -\left(\frac{25}{9}-2\phantom{\rule{0.166667em}{0ex}}{F}_{Pr}\right)\hfill \end{array}\right.$$
- Calculate the flow angle by mean of Equation (58).
- Check for convergence on the flow angle ($\epsilon $ is an arbitrary tolerance)$$|\varphi -{\varphi}_{0}|<\epsilon $$
- If the cycle has not converged yet, use the values a, ${a}^{\prime}$ and ${\varphi}_{0}=\varphi $ as new attempt values for the iteration, otherwise, continue. The relative tolerance for convergence is 0.01${}^{\circ}$ and a maximum number of iterations is 1000.

- (c)
- Once the values of a and ${a}^{\prime}$ are obtained, skew correction is applied on a:$${a}_{skew}=a\left(1+\frac{15\pi}{64}\frac{r}{R}tan\frac{\chi}{2}sin\psi \right)$$$$\chi =(0.6a+1){\chi}_{0}$$$${\chi}_{0}=\frac{{V}_{Avg,x}}{\sqrt{{V}_{Avg,x}^{2}+{V}_{Avg,y}^{2}+{V}_{Avg,z}^{2}}}$$
- (d)
- Calculate the relative wind speed by Equation (51).
- (e)
- Calculate the normal and tangential loads per unit length$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {p}_{n}=\frac{1}{2}\rho {C}_{n}{V}_{rel}^{2}c\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {p}_{t}=\frac{1}{2}\rho {C}_{t}{V}_{rel}^{2}c\hfill \end{array}$$

- Calculate the aerodynamic loads at the nacelle as the integral of the normal and tangential loads over the length of each of the 3 blades, defined r the node position between hub radius ${r}_{h}$ and the blade length ${r}_{b}$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& F{x}_{aer}={\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,1}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,2}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,3}dr\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& F{y}_{aer}={\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,1}sin{\psi}_{1}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,2}sin{\psi}_{2}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,3}sin{\psi}_{3}dr\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& F{z}_{aer}={\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,1}cos{\psi}_{1}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,2}cos{\psi}_{2}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,3}cos{\psi}_{3}dr\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& M{x}_{aer}={\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,1}rdr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,2}rdr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{t,3}rdr\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& M{y}_{aer}={\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,1}rsin{\psi}_{1}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,2}rsin{\psi}_{2}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,3}rsin{\psi}_{3}dr\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& M{z}_{aer}={\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,1}rcos{\psi}_{1}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,2}rcos{\psi}_{2}dr+{\int}_{{r}_{h}}^{{r}_{b}}{p}_{n,3}rcos{\psi}_{3}dr\hfill \end{array}$$

#### 2.4.4. Control System

- A generator-torque controller is designed to maximize the power extraction below the nominal point
- A full-span rotor-collective blade-pitch controller is designed to regulate generator speed above the nominal point

#### Generator-Torque Controller

#### Blade-Pitch Controller

## 3. Results

#### 3.1. Choice of the Best Location

- The distance from shore, that impacts on the installation costs (longer route for laying vessels) and visual pollution (the greater the distance from the coast, the less the wind farm will be visible).
- The sea depth, which should be considered for the mooring costs (shorter bathymetry means shorter length of the mooring lines).
- Migration of birds: considered not to disturb the migration [63].
- Geo-political reasons. All other conditions being equal, areas not close to foreign borders are preferred.

#### 3.2. Comparison with FAST v8.16

#### 3.3. Choice of the Best Floating Platform

- ${I}_{0}$: Capital expenditure (CAPEX) in €
- ${A}_{t}$: Annual operating costs (OPEX) in year t
- ${M}_{el}$: Produced electricity in the corresponding year in kWh
- i: Weighted average cost of capital (WACC) in %
- n: Operational lifetime in years
- t: Individual year of lifetime (1, 2, .., n)

#### 3.4. Offshore vs. Onshore Investment

## 4. Conclusions

- Computational burden: based on the performed comparison, the in-house model requires about twice the computational time to carry out a simulation compared to FAST. However, it is noticeable that this time is still less than the simulation time, thus not limiting the possibility of carrying out real-time simulations.
- The platform, tower, nacelle and blades are currently modeled as rigid bodies.
- Concerning aerodynamics, the implementation of the BEM theory is currently characterized by a tolerance value higher than that of FAST, in order to enhance the resolution algorithm convergence and shorten the calculation time.
- The used mooring theory, MAP++, is quasi-static, while mooring dynamics can also be affected by considerable inertial loads.
- The model is very limited in the representation of the overall power output, as the electrical conversion is currently simply represented by a generator efficiency value.

- The elastic theory could be applied to the modeling of platform, tower, nacelle and blades, in order to further increase the model accuracy.
- The MAP++ quasi-static theory could be replaced with a dynamic theory, leading to higher fidelity in representing mooring effects on the floating platform.
- The integration of accurate electrical machines and converters models could provide important information for the study of arrays, power connections and, more in general, on the impact of large floating wind farms in power networks.
- The insertion of the in-house model in a genetic algorithm in order to obtain new and innovative substructure concepts.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LCOE | Levelized Cost of Energy |

CF | Capacity Factor |

PV | Photovoltaic |

CFD | Computational Fluid Dynamics |

FRA | Fixed Reference Axis |

LSA | Local Structure Axis |

DOF | Degree of Freedom |

LTI SSM | Linear-time-invariant state-space model |

COG | Center of Gravity |

SWL | Still Water Level |

NREL | National Renewable Energy Laboratory |

MSQS | Multisegmented quasi-static |

BEM | Blade Element Momentum |

PI | Proportional-integral (control) |

RMSE | Root Mean Square Error |

CS | Commercial stage |

CAPEX | Capital expenditure |

OPEX | Annual operating costs |

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**Figure 3.**Superposition of different regular waves, from [32].

**Figure 4.**Spatial arrangement of the line [41].

**Figure 6.**Effect of an uneven wind-speed distribution over the swept rotor area on the upwind velocity of the rotating rotor blade [47].

**Figure 7.**Blade elements of a three-bladed turbine [53].

**Figure 8.**Loads on a blade airfoil [53].

**Figure 9.**Triangles of velocities on a blade airfoil [53].

**Figure 11.**Specific producibility of onshore wind at 100 m (above ground level) and offshore wind at 100 m (above sea level) [57].

**Figure 14.**Positions and performances comparison with H

_{s}= 0.75 m, T

_{p}= 5.5 s, V

_{0}= 8.5 m/s.

Mass | 4605 t |

Displacement | 5250 m${}^{3}$ |

Draft | 12 m |

Center of gravity from the FRA | (0, 0, −55) m |

Roll moment of inertia | 34,803,381,981 kg m${}^{2}$ |

Pitch moment of inertia | 1,579,004,565 kg m${}^{2}$ |

Yaw moment of inertia | 34,818,228,666 kg m${}^{2}$ |

number of mooring lines | 6 |

angle between adjacent lines | 60${}^{\circ}$ |

seabed depth | 120 m |

mooring leg composition | chain |

unstretched lengths | 473.3 m |

type | chainup |

diameter | 80.9 mm |

mass per unit length | 138.7 kg/m |

min. breaking load | 3380 kN |

axial stiffness | 1071 MN |

cable/seabed friction coefficient | 1 |

rated wind speed | 11.4 | m/s |

rated rotor speed | 12.1 | rpm |

rated generator speed | 1173.7 | rpm |

rated generator torque | 43,094 | Nm |

rated mechanical power | 5.30 | MW |

rated electric power | 5.00 | MW |

Site A | Site B | Site C | |
---|---|---|---|

Sea depth | 120 m | 420 m | 115 m |

Distance from shore | 21.5 km | 8.3 km | 26.5 km |

Productivity | 4235 MWh/MW | 4320 MWh/MW | 4205 MWh/MW |

C.F. | 48.3% | 49.3% | 48.0% |

Coordinates | LAT 37.01${}^{\circ}$ N, LONG 12.02${}^{\circ}$ E | LAT 36.54${}^{\circ}$ N, LONG 11.54${}^{\circ}$ E | LAT 36.37${}^{\circ}$ N, LONG 11.48${}^{\circ}$ E |

Spar-Buoy | Saipem’s Hexafloat | ||
---|---|---|---|

Platform diameter | 14.75 m | Central column diameter | 8.38 m |

Platform height | 88 m | Central column height | 35.25 m |

Concrete height | 15.64 m | Side length | 30 m |

Sea water height | 17.85 m | Ballast ${z}_{CoG}$ | −99.45 |

Ballast diameter | 10 m | ||

Mass (steel) | 23,230 t | Ballast mass | 3304 t |

Density (S355 steel) | 8500 kg/m^{3} | Density magnetite | 5200 kg/m^{3} |

Spar-Buoy | Saipem’s Hexafloat | |
---|---|---|

Capacity factor | 45.2% | 45.5% |

LCOE Parameter | Spar-Buoy | Saipem’s Hexafloat |
---|---|---|

CAPEX | 121.74 M€ | 106.63 M€ |

OPEX | 2.14 M€ | 2.14 M€ |

WACC (i rate) | 8% | 8% |

Useful life | 25 years | 25 years |

Capacity factor | 45.22% | 45.53% |

Productivity net | 98.99 GWh | 99.64 GWh |

LCOE | 136.82 €/MWh | 121.70 €/MWh |

LCOE Parameter | Value |
---|---|

CAPEX | 28.61 M€ |

OPEX | 0.92 M€ |

WACC (i rate) | 8% |

Useful life | 25 years |

Capacity factor | 38.74% |

Productivity net | 84,832 MWh |

LCOE | 42.45 €/MWh |

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## Share and Cite

**MDPI and ACS Style**

Cottura, L.; Caradonna, R.; Ghigo, A.; Novo, R.; Bracco, G.; Mattiazzo, G. Dynamic Modeling of an Offshore Floating Wind Turbine for Application in the Mediterranean Sea. *Energies* **2021**, *14*, 248.
https://doi.org/10.3390/en14010248

**AMA Style**

Cottura L, Caradonna R, Ghigo A, Novo R, Bracco G, Mattiazzo G. Dynamic Modeling of an Offshore Floating Wind Turbine for Application in the Mediterranean Sea. *Energies*. 2021; 14(1):248.
https://doi.org/10.3390/en14010248

**Chicago/Turabian Style**

Cottura, Lorenzo, Riccardo Caradonna, Alberto Ghigo, Riccardo Novo, Giovanni Bracco, and Giuliana Mattiazzo. 2021. "Dynamic Modeling of an Offshore Floating Wind Turbine for Application in the Mediterranean Sea" *Energies* 14, no. 1: 248.
https://doi.org/10.3390/en14010248