# BEM Turbine Model and PID Control System of a Floating Hybrid Wind and Current Turbines Integrated Generator System

^{*}

## Abstract

**:**

## 1. Introduction

_{L}) and the drag coefficient (C

_{D}) of each airfoil. The fluid modeling was performed with ANSYS Fluent [29], and they compared the results with the experimental data obtained in a wind tunnel.

^{®}, and they called the resulting code FHYGSYS. In [36], the kinematic, dynamic, and inertial models were presented and the way in which the added mass of the floating system was computed. On the other hand, in [37], the method to calculate the hydrodynamics and the rest of the forces acting on the floating hybrid system was described in detail including the modeling of the turbines using One-Dimensional theory.

^{®}. From the point of view of computing time, the simulation of the three turbines of the floating hybrid system using the BEM theory was longer than using One-Dimensional theory. However, from the point of view of the quality of the modeling of the turbines, the use of BEM theory offers significant improvements.

## 2. Materials and Methods

#### 2.1. Introduction to Turbine Modeling Using Blade Element Momentum Theory

#### 2.2. Application of Blade Element Momentum Theory

_{Tower−top}.

#### 2.2.1. BEM Algorithm Application

_{Hub−loss}(t)). The total turbine losses (${F}_{loss}\left(t\right)$) are obtained by multiplying these two factors [43], as shown in Equation (23).

#### 2.3. Obtaining Values of the Wind Turbine Magnitudes

#### 2.3.1. Moment about the Shaft

#### 2.3.2. Blade Azimuth Position

#### 2.3.3. Vector of Wind Turbine Forces

#### 2.3.4. Additional Magnitudes

#### 2.4. Wind Turbine Control System

#### 2.4.1. Generator Torque Controller

_{GEN}(t)) can take.

#### 2.4.2. Collective Blade Pitch Angle Controller

_{Pitch}(t)) can take.

#### 2.5. Application of BEM Modeling to Marine Current Turbines

#### 2.6. Marine Current Turbines Control System

## 3. Results

#### 3.1. Test 1. Wind Speed Sweep from 3 to 25 m/s

#### 3.2. Test 2. Sub-Surface Current Speed Sweep from 0.5 to 3 m/s

#### 3.3. Comparison of Mooring Line Tensions between Tests 1 and 2

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Acronym list | |

ANSYS | Swanson Analysis Systems |

BEM | Blade Element Momentum theory |

CFD | Computational Fluid Dynamics |

DOF | Degrees Of Freedom |

FAST | Fatigue, Aerodynamics, Structures, and Turbulence |

FHYGSYS | Floating Hybrid Generator SYstems Simulator |

HAWC2 | Horizontal Axis Wind turbine simulation Code 2nd generation |

HAWT | Horizontal Axis Wind Turbine |

MCT | Marine Current Turbines |

MIMO | Multi-Input Multi-Output |

NREL | National Renewable Energy Lab |

OC3 | Offshore Code Comparison Collaboration |

OLAF | cOnvecting LAgrangian Filaments |

OWT | Offshore Wind Turbines |

PID | Proportional–Integral–Derivative |

PLC | Programmable Logic Controller |

RTHS | Real-Time Hybrid Simulation |

SHYFEM | System of HYdrodynamic Finite Element Modules |

SOWFA | Simulator for Wind Farm Applications |

WRP | WAMIT Reference Point |

Symbol List | |

$anif\left(t\right)$ | angular induction factor of the wind turbine (or marine current turbine) |

$ani{f}_{\left(AV\right)}\left(t\right)$ | mean value of angular induction factor |

${A}_{Turbine}$ | area of the wind turbine (or marine current turbine) |

$axif\left(t\right)$ | axial induction factor of the wind turbine (or marine current turbine) |

$axi{f}_{\left(AV\right)}\left(t\right)$ | mean value of axial induction factor |

$axi{f}_{max}$ | value of Glauert correction whose value is 1/3 |

${\alpha}_{rotor}\left(t\right)$ | low-speed shaft angular acceleration of the turbine rotor |

$\alpha $ | power law exponent |

${C}_{D}$ | aerodynamic (or hydrodynamic) drag coefficient |

${C}_{L}$ | aerodynamic (or hydrodynamic) lift coefficient |

${C}_{N\theta}\left(t\right)$ | aerodynamic (or hydrodynamic) normal coefficient |

center of mass of the wind turbine | |

$Co{M}_{WTurbine\left(BODY\right)}\left(t\right)$ | center of mass of the wind turbine expressed in the mobile coordinate system |

${C}_{P}\left(t\right)$ | power coefficient |

${C}_{T}\left(t\right)$ | thrust coefficient |

${C}_{T\theta}\left(t\right)$ | aerodynamic (or hydrodynamic) tangential coefficient |

$D\left(h\right)$ | tower diameter to calculate the vector ${\overrightarrow{v}}_{WINDtower-shadow\left(BODY\right)}\left(h\right)$ |

$\Delta {z}_{pni\left(MCT\right)}$ | value $\mathrm{that}\mathrm{corrects}\mathrm{the}\mathrm{Z}\mathrm{component}\mathrm{of}\mathrm{the}{p}_{ni\left(MCT\right)}\left(t\right)$ To calculate the shadow effect in a marine current turbine |

$d{\overrightarrow{F}}_{bladeij\left(BODY\right)}\left(t\right)$ | resultant of the differential forces of blade element $i$ of blade $j$ expressed in the mobile coordinate system |

differential of thrust force | |

differential of thrust force of blade element $i$ of blade | |

$d{\overrightarrow{F}}_{thrustij\left(BODY\right)}\left(t\right)$ | differential of thrust force vector of blade element $i$ of blade $j$ expressed in the mobile coordinate system |

$d{F}_{torque}$ | differential of torque force |

$d{F}_{torqueij}\left(t\right)$ | differential of torque force of blade element of blade $j$ |

$d{\overrightarrow{F}}_{torqueij\left(BODY\right)}\left(t\right)$ | differential of torque force vector of blade element $i$ of blade expressed in the mobile coordinate system |

$d{\overrightarrow{M}}_{bladeij\left(BODY\right)}\left(t\right)$ | resultant of the differential moments of blade element $i$ of blade $j$ expressed in the mobile coordinate system |

$d{M}_{pitching}$ | differential of pitching moment |

$d{p}_{ni}$ | blade elements or blade differentials |

$\delta $ | direction of the wind velocity vector |

$e\left(t\right)$ | error value of a PID controller |

$\epsilon $ | direction of the sub-surface current velocity vector |

${\epsilon}_{a}$ | $\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{percent}\mathrm{relative}\mathrm{error}\mathrm{to}\mathrm{calculate}\mathrm{the}\mathrm{values}\mathrm{of}axif\left(t\right)$ and $anif\left(t\right)$ |

${\epsilon}_{a-min}$ | minimum value of the percent relative error whose value is $5\xb7{10}^{-9}$ |

${\overrightarrow{F}}_{bladej\left(BODY\right)}\left(t\right)$ | resultant of the forces of blade $j$ expressed in the mobile coordinate system |

${F}_{center-shadow}$ | factor that represents the shadow effect of the central tube of the marine current support |

${F}_{down-shadow}$ | factor that represents the shadow effect of the lower tube of the marine current support |

${F}_{Hub-loss}\left(t\right)$ | hub-loss factor |

${\overrightarrow{{F}_{k}}}_{\left(BODY\right)}^{WINDTURBINE}\left(t\right)$ | vector of wind turbine forces and moments expressed in the mobile coordinate system |

${F}_{loss}\left(t\right)$ | total turbine losses |

${F}_{thrust}$ | thrust force |

${\overrightarrow{F}}_{Thrust\left(BODY\right)}\left(t\right)$ | total thrust vector produced by the wind turbine expressed in the mobile coordinate system |

${F}_{Tip-loss}\left(t\right)$ | Prandtl’s tip-loss factor |

${F}_{torque}$ | torque force |

${F}_{Trigg}$ | pitch controller trigger offset factor |

${F}_{Trigg(\%)}$ | trigger offset factor |

${F}_{up-shadow}$ | factor that represents the shadow effect of the upper tube of the marine current support |

$i$ | symbol to refer to a specific blade element |

${I}_{rotor}$ | total inertia of the wind turbine rotor |

${I}_{WTurbine}$ | wind turbine moment of inertia about this x axis (${I}_{WTxx}$) |

${I}_{MCTurbine}$ | marine current turbine moment of inertia about this x axis (${I}_{MCTxx}$) |

${I}_{MCTurbine\left(SUM\right)}$ | marine current turbine moment of inertia about this x axis (${I}_{MCTxx\left(SUM\right)}$) to calculate added mass |

${I}_{WTxx},{I}_{WTyy},{I}_{WTzz}$ | wind turbine moments of inertia |

${I}_{MCTxx},{I}_{MCTyy},{I}_{MCTzz}$ | marine current turbine moments of inertia |

${I}_{MCTxx\left(SUM\right)},{I}_{MCTyy\left(SUM\right)}$ ${I}_{MCTzz\left(SUM\right)}$ | marine current turbine moments of inertia to calculate added mass |

$j$ | symbol to refer to a specific blade |

${K}_{anif}\left(t\right)$ | term of $anif\left(t\right)$ equation |

${K}_{axif}\left(t\right)$ | term of $axif\left(t\right)$ equation |

${K}_{p}$ | proportional gain of a PID controller |

${K}_{{p}_{GEN}}$ | proportional gain of the torque PID controller |

${K}_{{p}_{Pitch}}$ | proportional gain of the collective blade pitch angle PID controller |

${L}_{Chord}$ | chord length of a blade element |

${M}_{BODY}^{INERTIAL}$ | transformation matrix that allows changing between the inertial coordinate system and the mobile coordinate system |

${\overrightarrow{M}}_{aero\left(BODY\right)}\left(t\right)$ | total moment caused by the aerodynamics on the turbine |

${M}_{aero-shaft}\left(t\right)$ | low-speed shaft aerodynamic torque or moment (torque) about the shaft of the turbine |

${\overrightarrow{M}}_{blade\left(BODY\right)}\left(t\right)$ | moment vector caused by the blades on a turbine |

${\overrightarrow{M}}_{bladej\left(BODY\right)}\left(t\right)$ | resultant of the moments of blade $j$ expressed in the mobile coordinate system |

${M}_{GEN}\left(t\right)$ | high-speed shaft generator torque |

${M}_{GEN-shaft}\left(t\right)$ | low-speed shaft generator torque |

${\overrightarrow{M}}_{hub\left(BODY\right)}\left(t\right)$ | total moment vector originated by the force vector from the bases of the blades to the center of mass of the turbine |

${\overrightarrow{M}}_{hubj\left(BODY\right)}\left(t\right)$ | moment vector originated by the force vector ${\overrightarrow{F}}_{bladej\left(BODY\right)}\left(t\right)$ from the bases of the blades to the center of mass of the turbine |

${M}_{HT}{}_{BODY}^{INERTIAL}$ | homogeneous transformation matrix that allows for changing between the inertial coordinate system and the mobile coordinate system |

${M}_{IT\left(WTurbine\right)}$ | wind turbine rotor inertia tensor |

${M}_{IT\left(MCTurbine\right)}$ | marine current turbine rotor inertia tensor |

${M}_{IT\left(MCTurbine\left(SUM\right)\right)}$ | marine current turbine rotor inertia tensor to calculate added mass |

${M}_{pitch}$ | resultant of pitching moment on a turbine |

pitching moment | |

${\overrightarrow{M}}_{pitching\left(BODY\right)}\left(t\right)$ | total pitching moment vector |

${M}_{pitchingj}\left(t\right)$ | pitching moment originated in blade $j$ |

${\overrightarrow{M}}_{pitchingj\left(BODY\right)}\left(t\right)$ | pitching moment vector originated in blade $j$ expressed in the mobile coordinate system |

${M}_{rotor}\left(t\right)$ | low-speed shaft torque |

${\overrightarrow{M}}_{rotor\left(BODY\right)}\left(t\right)$ | low-speed shaft torque vector expressed in the mobile coordinate system |

${\overrightarrow{M}}_{Thrust\left(BODY\right)}\left(t\right)$ | total moment vector produced by the wind turbine at the origin of the mobile coordinate system expressed in the same coordinate system |

${\overrightarrow{M}}_{Thrustj\left(BODY\right)}\left(t\right)$ | moment produced by the resultant of the forces of blade $j$ at the origin of the mobile coordinate system expressed in the same coordinate system |

$\mathrm{difference}\mathrm{between}\mathrm{the}\mathrm{angular}\mathrm{speed}\mathrm{to}\mathrm{be}\mathrm{reached}($$)\mathrm{and}\mathrm{the}\mathrm{high}-\mathrm{speed}\mathrm{shaft}\mathrm{angular}\mathrm{speed}($) | |

${\omega}_{GEN}\left(t\right)$ | high-speed shaft angular speed of the generator |

${\omega}_{rotor}\left(t\right)$ | low-speed shaft angular speed of the turbine rotor |

${\mathsf{\Omega}}_{rotor}\left(t\right)$ | low-speed shaft angular speed of the turbine rotor expressed in rpm |

${\mathsf{\Omega}}_{SP}\left(t\right)$ | set point of low-speed shaft angular speed expressed in rpm |

${\omega}_{SP}$ | set point of high-speed shaft angular speed |

${P}_{AERO}\left(t\right)$ | rotor aerodynamic power |

${P}_{ELE}\left(t\right)$ | electrical generator power |

${P}_{MEC}\left(t\right)$ | mechanical power or low-speed shaft power |

${p}_{ni}$ | center of each of the blade elements |

${p}_{ni\left(BLADE\right)}$ | center of each of the blade elements expressed in a blade coordinate system |

${p}_{ni\left(BODY\right)}$ | center of each of the blade elements expressed in the mobile coordinate system |

${p}_{ni\left(INERTIAL\right)}$ | center of each of the blade elements expressed in the inertial coordinate system |

${p}_{n0j\left(BODY\right)}\left(t\right)$ | center point at the base of blade $j$ |

${p}_{nij\left(BODY\right)}\left(t\right)$ | center of blade element $i$ of blade $j$ expressed in the mobile coordinate system |

${\phi}_{Attack}\left(t\right)$ | $\mathrm{angle}\mathrm{of}\mathrm{attack}\mathrm{between}\mathrm{velocity}{V}_{rel}\left(t\right)$ and the chord line of each blade element |

${\phi}_{Attack\left(AV\right)}\left(t\right)$ | mean value of angle of attack |

${\phi}_{Pitch}\left(t\right)$ | collective blade pitch angle |

${\phi}_{Twist}$ | twist angle of a blade element |

${\psi}_{rotor}\left(t\right)$ | angular position of the turbine rotor |

${Q}_{aero}\left(t\right)$ | magnitude of vector |

${r}_{Turbine}$ | turbine radius |

$RtAeroF$ | total rotor aerodynamic forces (thrust) |

$RtAeroM$ | total rotor aerodynamic moments (torque) |

${\rho}_{AIR}$ | density of air whose value is $1.225\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\rho}_{SEAWATER}$ | density of seawater whose value is |

${\sigma}_{i}$ | local solidity |

${T}_{d}$ | derivative time of a PID controller |

integration time of a PID controller | |

${T}_{{i}_{GEN}}$ | integration time of the torque PID controller |

${T}_{{i}_{Pitch}}$ | integration time of the collective blade pitch angle PID controller |

${T}_{Thrust}\left(t\right)$ | magnitude of vector ${\overrightarrow{F}}_{Thrust\left(BODY\right)}\left(t\right)$ |

$TSR\left(t\right)$ | tip-speed ratio |

$\theta \left(t\right)$ | angle between ${V}_{rel}\left(t\right)$ velocity and a blade vertical plane |

$u\left(t\right)$ | control action of a PID controller |

${u}_{{M}_{GEN}}\left(t\right)$ | control action of the torque PID controller |

${u}_{Pitch}\left(t\right)$ | control action of the collective blade pitch angle PID controller |

${\overrightarrow{u}}_{Bjthrust}$ | $\mathrm{thrust}$ |

${\overrightarrow{u}}_{Bjthrust\left(BODY\right)}$ | $\mathrm{thrust}\mathrm{unit}\mathrm{vector}\mathrm{or}\mathrm{vector}\mathrm{perpendicular}\mathrm{to}\mathrm{blade}$ expressed in the mobile coordinate system |

${\overrightarrow{u}}_{Bjtorque\left(BODY\right)}$ | torque unit vector expressed in the mobile coordinate system |

$\mathrm{pitching}$ | |

${\overrightarrow{u}}_{Mjpitching\left(BODY\right)}$ | $\mathrm{pitching}$ expressed in the mobile coordinate system |

${\overrightarrow{u}}_{shaft}$ | shaft unit vector |

${\overrightarrow{u}}_{shaft\left(BODY\right)}$ | shaft unit vector expressed in the mobile coordinate system |

${V}_{0}\left(t\right)$ or ${V}_{0ij}\left(t\right)$ | magnitude of vector ${\overrightarrow{v}}_{niEFF-Bjthrust\left(BODY\right)}\left(t\right)$ |

${V}_{0\left(AV\right)}\left(t\right)$ | mean of the magnitude of the effective wind velocity vector |

${V}_{1-2}\left(t\right)$ | normal velocity to the vertical plane of each blade |

${V}_{rel}\left(t\right)$ | magnitude of the wind velocity reaching each blade element |

${V}_{rot}\left(t\right)$ | tangential velocity to the vertical plane of each blade |

${V}_{W-REF}$ | magnitude—or mean wind speed—of the wind velocity vector at the reference height |

${V}_{WIND}\left(h\right)$ | $\mathrm{wind}\mathrm{speed}\mathrm{at}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{center}\mathrm{point}\mathrm{of}\mathrm{each}\mathrm{blade}\mathrm{element}({p}_{ni}$) |

${\overrightarrow{v}}_{niEFF-Bjthrust\left(BODY\right)}\left(t\right)$ | effective wind velocity vector at node $i$ $\mathrm{of}$ expressed in the mobile coordinate system |

${\overrightarrow{v}}_{niEFF-WIND\left(BODY\right)}\left(h,t\right)$ | effective wind velocity vector at node $i$ expressed in the mobile coordinate system |

${\overrightarrow{v}}_{pni\left(BODY\right)}\left(t\right)$ | $\mathrm{velocity}\mathrm{vector}\mathrm{of}\mathrm{the}\mathrm{center}\mathrm{point}\mathrm{of}\mathrm{each}\mathrm{blade}\mathrm{element}({p}_{ni}$) expressed in the mobile coordinate system |

${\overrightarrow{v}}_{pni\left(INERTIAL\right)}\left(t\right)$ | $\mathrm{velocity}\mathrm{vector}\mathrm{of}\mathrm{the}\mathrm{center}\mathrm{point}\mathrm{of}\mathrm{each}\mathrm{blade}\mathrm{element}({p}_{ni}$) expressed in the inertial coordinate system |

${\overrightarrow{v}}_{SSCUR\left(BODY\right)}\left(h\right)$ | sub-surface current velocity vector of the vector field rotated by the corresponding angle $\epsilon $ expressed in the mobile coordinate system |

${\overrightarrow{v}}_{SSCURsupport-shadow\left(BODY\right)}\left(h\right)$ | sub-surface current velocity vector considering the support shadow effect |

${\overrightarrow{v}}_{WIND\left(BODY\right)}\left(h\right)$ | wind velocity vector of the vector field rotated by the corresponding angle $\delta $ expressed in the mobile coordinate system |

${\overrightarrow{v}}_{WIND\left(INERTIAL\right)}\left(h\right)$ | wind velocity vector of the vector field rotated by the corresponding angle $\delta $ expressed in the inertial coordinate system |

${\overrightarrow{v}}_{WINDtower-shadow\left(BODY\right)}\left(h\right)$ | wind velocity vector considering the tower shadow effect |

${x}_{pni},{y}_{pni},{z}_{pni}$ | x, y and z components of ${p}_{ni\left(BODY\right)}\left(t\right)$ point |

${x}_{pni\left(MCT\right)},{y}_{pni\left(MCT\right)},{z}_{pni\left(MCT\right)}$ | x, y and z components of ${p}_{ni\left(MCT\right)}\left(t\right)$ point |

${z}_{REF}$ | reference height of the wind velocity vector |

${z}_{WIND}\left(t\right)$ | height to calculate the magnitude of the wind velocity vector |

## Appendix A

Node | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{L}\mathit{A}\mathit{D}\mathit{E}\right)}\left(\mathbf{m}\right)$ | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{O}\mathit{D}\mathit{Y}\right)}\left(\mathbf{m}\right)$ (Blade 1) | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{O}\mathit{D}\mathit{Y}\right)}\left(\mathbf{m}\right)$ (Blade 2) | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{O}\mathit{D}\mathit{Y}\right)}\left(\mathbf{m}\right)$ (Blade 3) |
---|---|---|---|---|

0 (blade base) | (0, 0, 0) | (−5.126, 0, 91.52) | (−5.322, −1.299, 89.28) | (−5.322, 1.299, 89.28) |

1 | (0, 0, 1.3667) | (−5.067, 0, 92.88) | (−5.441, −2.482, 88.60) | (−5.441, 2.482, 88.60) |

2 | (0, 0, 4.1) | (−4.948, 0, 95.61) | (−5.679, −4.846, 87.25) | (−5.679, 4.846, 87.25) |

3 | (0, 0, 6.8333) | (−4.828, 0, 98.34) | (−5.917, −7.211, 85.90) | (−5.917, 7.211, 85.90) |

4 | (0, 0, 10.25) | (−4.679, 0, 101.8) | (−6.214, −10.17, 84.21) | (−6.214, 10.17, 84.21) |

5 | (0, 0, 14.35) | (−4.500, 0, 105.9) | (−6.571, −13.72, 82.19) | (−6.571, 13.72, 82.19) |

6 | (0, 0, 18.45) | (−4.322, 0, 110.0) | (−6.927, −17.26, 80.16) | (−6.927, 17.26, 80.16) |

7 | (0, 0, 22.55) | (−4.143, 0, 114.1) | (−7.284, −20.81, 78.14) | (−7.284, 20.81, 78.14) |

8 | (0, 0, 26.65) | (−3.964, 0, 118.1) | (−7.641, −24.36, 76.12) | (−7.641, 24.36, 76.12) |

9 | (0, 0, 30.75) | (−3.785, 0, 122.2) | (−7.997, −27.90, 74.09) | (−7.997, 27.90, 74.09) |

10 | (0, 0, 34.85) | (−3.606, 0, 126.3) | (−8.354, −31.45, 72.07) | (−8.354, 31.45, 72.07) |

11 | (0, 0, 38.95) | (−3.427, 0, 130.4) | (−8.711, −35.00, 70.04) | (−8.711, 35.00, 70.04) |

12 | (0, 0, 43.05) | (−3.249, 0, 134.5) | (−9.067, −38.55, 68.01) | (−9.067, 38.55, 68.01) |

13 | (0, 0, 47.15) | (−3.070, 0, 138.6) | (−9.424, −42.09, 65.99) | (−9.424, 42.09, 65.99) |

14 | (0, 0, 51.25) | (−2.891, 0, 142.7) | (−9.781, −45.64, 63.97) | (−9.781, 45.64, 63.97) |

15 | (0, 0, 54.6667) | (−2.742, 0, 146.1) | (−10.08, −48.60, 62.28) | (−10.08, 48.60, 62.28) |

16 | (0, 0, 57.4) | (−2.623, 0, 148.9) | (−10.32, −50.96, 60.93) | (−10.32, 50.96, 60.93) |

17 | (0, 0, 60.1333) | (−2.503, 0, 151.6) | (−10.55, −53.33, 59.58) | (−10.55, 53.33, 59.58) |

18 (blade tip) | (0, 0, 61.5) | (−2.444, 0, 153.0) | (−10.67, −54.51, 58.91) | (−10.67, 54.51, 58.91) |

19 (Surge positive unit) | (1, 0, 0) | (−4.127, 0, 91.47) | (−4.329, −1.337, 89.17) | (−4.329, 1.337, 89.17) |

20 (Sway negative unit) | (0, −1, 0) | (−5.126, −1, 91.52) | (−5.398, −0.7990, 88.41) | (−5.247, 1.799, 90.14) |

21 (Heave positive unit) | (0, 0, 1) | (−5.082, 0, 92.52) | (−5.409, −2.164, 88.78) | (−5.409, 2.164, 88.78) |

Node | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{L}\mathit{A}\mathit{D}\mathit{E}\right)}\left(\mathbf{m}\right)$ | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{O}\mathit{D}\mathit{Y}\right)}\left(\mathbf{m}\right)$ (Blade 1) | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{O}\mathit{D}\mathit{Y}\right)}\left(\mathbf{m}\right)$ (Blade 2) |
---|---|---|---|

0 (blade base) | (−1.1, 0, 0) | (−0.9, 17.1, −19) | (−0.9, 17.1, −21) |

1 | (−1.1, 1.075, 0) | (−0.9, 17.1, −18.93) | (−0.9, 17.1, −21.08) |

2 | (−1.1, 1.3607, 0) | (−0.9, 17.1, −18.639) | (−0.9, 17.1, −21.36) |

3 | (−1.1, 1.7821, 0) | (−0.9, 17.1, −18.22) | (−0.9, 17.1, −21.78) |

4 | (−1.1, 2.2036, 0) | (−0.9, 17.1, −17.80) | (−0.9, 17.1, −22.20) |

5 | (−1.1, 2.625, 0) | (−0.9, 17.1, −17.38) | (−0.9, 17.1, −22.63) |

6 | (−1.1, 3.1342, 0) | (−0.9, 17.1, −16.87) | (−0.9, 17.1, −23.13) |

7 | (−1.1, 3.7313, 0) | (−0.9, 17.1, −16.27) | (−0.9, 17.1, −23.73) |

8 | (−1.1, 4.3283, 0) | (−0.9, 17.1, −15.67) | (−0.9, 17.1, −24.32) |

9 | (−1.1, 4.9253, 0) | (−0.9, 17.1, −15.08) | (−0.9, 17.1, −24.93) |

10 | (−1.1, 5.5223, 0) | (−0.9, 17.1, −14.48) | (−0.9, 17.1, −25.52) |

11 | (−1.1, 6.1193, 0) | (−0.9, 17.1, −13.88) | (−0.9, 17.1, −26.12) |

12 | (−1.1, 6.7164, 0) | (−0.9, 17.1, −13.28) | (−0.9, 17.1, −26.72) |

13 | (−1.1, 7.3134, 0) | (−0.9, 17.1, −12.69) | (−0.9, 17.1, −27.31) |

14 | (−1.1, 7.9104, 0) | (−0.9, 17.1, −12.09) | (−0.9, 17.1, −27.91) |

15 | (−1.1, 8.5074, 0) | (−0.9, 17.1, −11.49) | (−0.9, 17.1, −28.51) |

16 | (−1.1, 9.1045, 0) | (−0.9, 17.1, −10.89) | (−0.9, 17.1, −29.11) |

17 | (−1.1, 9.7015, 0) | (−0.9, 17.1, −10.30) | (−0.9, 17.1, −29.70) |

18 (blade tip) | (−1.1, 0, 10) | (−0.9, 17.1, −10) | (−0.9, 17.1, −30) |

19 (Surge positive unit) | (−0.1, 0, 0) | (0.1, 17.1, −19) | (0.1, 17.1, −21) |

20 (Sway negative unit) | (−1.1, −1, 0) | (−0.9, 16.1, −19) | (−0.9, 18.1, −21) |

21 (Heave positive unit) | (−1.1, 0, 1) | (−0.9, 17.1, −18) | (−0.9, 17.1, −22) |

Node | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{L}\mathit{A}\mathit{D}\mathit{E}\right)}\left(\mathbf{m}\right)$ | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{O}\mathit{D}\mathit{Y}\right)}\left(\mathbf{m}\right)$ (Blade 1) | ${\mathit{p}}_{\mathit{n}\mathit{i}\left(\mathit{B}\mathit{O}\mathit{D}\mathit{Y}\right)}\left(\mathbf{m}\right)$ (Blade 2) |
---|---|---|---|

0 (blade base) | (−1.1, 0, 0) | (−0.9, −17.1, −19) | (−0.9, −17.1, −21) |

1 | (−1.1, 1.075, 0) | (−0.9, −17.1, −18.93) | (−0.9, −17.1, −21.08) |

2 | (−1.1, 1.3607, 0) | (−0.9, −17.1, −18.639) | (−0.9, −17.1, −21.36) |

3 | (−1.1, 1.7821, 0) | (−0.9, −17.1, −18.22) | (−0.9, −17.1, −21.78) |

4 | (−1.1, 2.2036, 0) | (−0.9, −17.1, −17.80) | (−0.9, −17.1, −22.20) |

5 | (−1.1, 2.625, 0) | (−0.9, −17.1, −17.38) | (−0.9, −17.1, −22.63) |

6 | (−1.1, 3.1342, 0) | (−0.9, −17.1, −16.87) | (−0.9, −17.1, −23.13) |

7 | (−1.1, 3.7313, 0) | (−0.9, −17.1, −16.27) | (−0.9, −17.1, −23.73) |

8 | (−1.1, 4.3283, 0) | (−0.9, −17.1, −15.67) | (−0.9, −17.1, −24.32) |

9 | (−1.1, 4.9253, 0) | (−0.9, −17.1, −15.08) | (−0.9, −17.1, −24.93) |

10 | (−1.1, 5.5223, 0) | (−0.9, −17.1, −14.48) | (−0.9, −17.1, −25.52) |

11 | (−1.1, 6.1193, 0) | (−0.9, −17.1, −13.88) | (−0.9, −17.1, −26.12) |

12 | (−1.1, 6.7164, 0) | (−0.9, −17.1, −13.28) | (−0.9, −17.1, −26.72) |

13 | (−1.1, 7.3134, 0) | (−0.9, −17.1, −12.69) | (−0.9, −17.1, −27.31) |

14 | (−1.1, 7.9104, 0) | (−0.9, −17.1, −12.09) | (−0.9, −17.1, −27.91) |

15 | (−1.1, 8.5074, 0) | (−0.9, −17.1, −11.49) | (−0.9, −17.1, −28.51) |

16 | (−1.1, 9.1045, 0) | (−0.9, −17.1, −10.89) | (−0.9, −17.1, −29.11) |

17 | (−1.1, 9.7015, 0) | (−0.9, −17.1, −10.30) | (−0.9, −17.1, −29.70) |

18 (blade tip) | (−1.1, 0, 10) | (−0.9, −17.1, −10) | (−0.9, −17.1, −30) |

19 (Surge positive unit) | (−0.1, 0, 0) | (0.1, −17.1, −19) | (0.1, −17.1, −21) |

20 (Sway negative unit) | (−1.1, 1, 0) | (−0.9, −16.1, −19) | (−0.9, −18.1, −21) |

21 (Heave positive unit) | (−1.1, 0, −1) | (−0.9, −17.1, −20) | (−0.9, −17.1, −20) |

## Appendix B

#### Appendix B.1. Test B1. Wind Speed 11 m/s and Direction 70 Degrees

Section | Parameter | Original Value | Modified Value |
---|---|---|---|

PropagationDir | 0 | −70 | |

Parameters for steady wind conditions | HWindSpeed | 0 | 11 |

Section | Parameter | Original Value | Modified Value |
---|---|---|---|

Initial conditions | RotSpeed | 11.89 | 0 |

NacYaw | 0 | 70 |

**Figure A3.**Linear and angular degrees of freedom of Test B1: (

**a**) linear degrees of freedom; (

**b**) angular degrees of freedom.

**Figure A4.**Torque and power of Test B1 yielded by the wind turbine modeling: (

**a**) rotor aerodynamic torque; (

**b**) rotor aerodynamic power; (

**c**) low-speed shaft torque; (

**d**) low-speed shaft power; (

**e**) electrical generator torque; (

**f**) electrical generator power.

**Figure A5.**Thrust and rotor speed of Test B1 yielded by the wind turbine modeling: (

**a**) rotor thrust; (

**b**) rotor speed.

**Figure A6.**Collective blade pitch angle and outstanding coefficients of Test B1 yielded by the wind turbine modeling: (

**a**) blade pitch angle; (

**b**) power coefficient; (

**c**) tip-speed ratio; (

**d**) thrust coefficient.

**Figure A7.**Total integrated hydrodynamic loads from both the potential flow and strip theory at the WRP point and mooring system fairlead forces of Test B1: (

**a**) surge and sway forces; (

**b**) roll, pitch, and yaw moments; (

**c**) heave force; (

**d**) force on the fairleads of the floating platform of each mooring line. T[1], T[2], and T[3] are the nomenclature used by FASTv8 for the fairlead tensions in mooring lines 1, 2, and 3.

#### Appendix B.2. Test B2. Wind Speed 15 m/s and Direction 0 Degrees

Section | Parameter | Original Value | Modified Value |
---|---|---|---|

PropagationDir | 0 | 0 | |

Parameters for steady wind conditions | HWindSpeed | 0 | 15 |

Section | Parameter | Original Value | Modified Value |
---|---|---|---|

Initial conditions | RotSpeed | 12.1 | 0 |

NacYaw | 0 | 0 |

**Figure A9.**Linear and angular degrees of freedom of Test B2: (

**a**) linear degrees of freedom; (

**b**) angular degrees of freedom.

**Figure A10.**Torque and power of Test B2 yielded by the wind turbine modeling: (

**a**) rotor aerodynamic torque; (

**b**) rotor aerodynamic power; (

**c**) low-speed shaft torque; (

**d**) low-speed shaft power; (

**e**) electrical generator torque; (

**f**) electrical generator power.

**Figure A11.**Thrust and rotor speed of Test B2 yielded by the wind turbine modeling: (

**a**) rotor thrust; (

**b**) rotor speed.

**Figure A12.**Collective blade pitch angle and outstanding coefficients of Test B2 yielded by the wind turbine modeling: (

**a**) blade pitch angle; (

**b**) power coefficient; (

**c**) tip-speed ratio; (

**d**) thrust coefficient.

**Figure A13.**Total integrated hydrodynamic loads from both the potential flow and strip theory at the WRP point and mooring system fairlead forces of Test B2: (

**a**) surge and sway forces; (

**b**) roll, pitch, and yaw moments; (

**c**) heave force; (

**d**) force on the fairleads of the floating platform of each mooring line. T[1], T[2], and T[3] are the nomenclature used by FASTv8 for the fairlead tensions in mooring lines 1, 2, and 3.

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**Figure 1.**Degrees of freedom of the floating hybrid system, inertial and mobile coordinate systems, fluid velocity vector fields, and blade elements.

**Figure 3.**Blade elements ($d{p}_{ni}$) whose centers are the ${p}_{ni}$ points resulting from the discretization of a wind turbine blade.

**Figure 5.**Blade thrust (${\overrightarrow{u}}_{Bjthrust}$), turbine shaft moment (${\overrightarrow{u}}_{shaft}$), and pitching moment (${\overrightarrow{u}}_{Mjpitching}$) unit vectors of the wind turbine.

**Figure 9.**Block diagram of the control system interacting with a floating hybrid system turbine modeled with BEM theory.

**Figure 10.**Comparison of the wind turbine speed (${\mathsf{\Omega}}_{rotor}\left(t\right)$) with and without the torque control system in operation, using the version of OC3-Hywind implemented in FHYGSYS: (

**a**) test conditions; (

**b**) results.

**Figure 11.**Comparison of the wind turbine speed (${\mathsf{\Omega}}_{rotor}\left(t\right)$) in normal conditions, only operating the torque control and without any control system in operation, using the version of OC3-Hywind implemented in FHYGSYS: (

**a**) test conditions; (

**b**) results.

**Figure 13.**Blade thrust (${\overrightarrow{u}}_{Bjthrust}$), turbine shaft moment (${\overrightarrow{u}}_{shaft}$), and pitching moment (${\overrightarrow{u}}_{Mjpitching}$) unit vectors of marine current turbines.

**Figure 15.**Wind speed and electrical power generated by the wind turbine: (

**a**) wind speed sweep; (

**b**) electrical generator power, Equation (65).

**Figure 16.**Linear and angular degrees of freedom in the wind turbine test: (

**a**) linear degrees of freedom; (

**b**) angular degrees of freedom.

**Figure 17.**Aerodynamic and electrical torque, rotor speed, and collective blade pitch angle in the wind turbine test: (

**a**) rotor aerodynamic torque, Equation (47); (

**b**) rotor speed, Equation (58); (

**c**) electrical generator torque, Equation (81); (

**d**) collective blade pitch angle, Equation (87).

**Figure 18.**Aerodynamic thrust and power and their coefficients in the wind turbine test: (

**a**) rotor thrust, Equation (62); (

**b**) rotor aerodynamic power, Equation (67); (

**c**) thrust coefficient, Equation (71); (

**d**) power coefficient, Equation (70).

**Figure 19.**Tip-speed ratio and $27{C}_{P}/16-TSR$ curve in the wind turbine test: (

**a**) tip-speed ratio, Equation (72); (

**b**) $27{C}_{P}/16vsTSR$ graphic representation.

**Figure 20.**Control and integral control actions of the wind turbine PID controllers: (

**a**) generator torque control action, Equation (79); (

**b**) collective blade pitch angle control action, Equation (85); (

**c**) generator torque integral control action; (

**d**) collective blade pitch angle integral control action.

**Figure 21.**Test 2 conditions. Sweep of the sub-surface current speeds from 0.5 to 3 m/s and direction of 0 degrees.

**Figure 22.**Wind speed and electrical power generated by the marine current turbines: (

**a**) wind speed sweep; (

**b**) electrical generator power, Equation (65).

**Figure 23.**Linear and angular degrees of freedom in the marine current turbines test: (

**a**) linear degrees of freedom; (

**b**) angular degrees of freedom.

**Figure 24.**Aerodynamic and electrical torque, rotor speed, and collective blade pitch angle in the marine current turbines test: (

**a**) rotor aerodynamic torque, Equation (47); (

**b**) rotor speed, Equation (58); (

**c**) electrical generator torque, Equation (81); (

**d**) collective blade pitch angle, Equation (87).

**Figure 25.**Aerodynamic thrust and power and their coefficients in the marine current turbines test: (

**a**) rotor thrust, Equation (62); (

**b**) rotor aerodynamic power, Equation (67); (

**c**) thrust coefficient, Equation (71); (

**d**) power coefficient, Equation (70).

**Figure 26.**Tip-speed ratio and $27{C}_{P}/16-TSR$ curve in the marine current turbines test: (

**a**) tip-speed ratio, Equation (72); (

**b**) $27{C}_{P}/16vs.TSR$ graphic representation.

**Figure 27.**Control and integral control actions of the marine current turbine PID controllers: (

**a**) generator torque control action, Equation (79); (

**b**) collective blade pitch angle control action, Equation (85); (

**c**) generator torque integral control action; (

**d**) collective blade pitch angle integral control action.

**Figure 28.**Tension on the fairleads of the mooring lines in Tests 1 and 2: (

**a**) mooring line tensions in Test 1; (

**b**) mooring line tensions in Test 2. T[1], T[2], and T[3] are the nomenclature used by FASTv8 for the fairlead tensions in mooring lines 1, 2, and 3.

Property ^{1} | Value | Symbol |
---|---|---|

Elevation to tower top above still water level | 87.6 m | ${h}_{Tower-top}$ |

Elevation to tower base above still water level | 10 m | ${h}_{Tower-base}$ |

Tower top diameter | 3.87 m | ${D}_{Tower-top}$ |

Tower base diameter | 6.5 m | ${D}_{Tower-base}$ |

^{1}The data are from [20].

Property ^{1} | Value | Symbol |
---|---|---|

Rated electrical power | 5 MW | ${P}_{ELE-rated}$ |

Number of blades | 3 | $B$ |

Hub height | 90 m | ${h}_{Hub}$ |

Hub radius | 1.5 m | ${r}_{Hub}$ |

Blade length | 61.5 m | ${L}_{Blade}$ |

Precone | −2.5 deg | ${\phi}_{Precone}$ |

Shaft tilt | 5 deg | ${\phi}_{ShaftTilt}$ |

Rated rotor speed | 12.1 rpm | ${\mathsf{\Omega}}_{rated}$ |

Gearbox ratio | 97:1 | $gearR$ |

Electrical generator efficiency | 0.944 | $genE$ |

Generator inertia about high-speed shaft | 534.116 kg·m^{2} | ${I}_{GEN}$ |

Minimum generator speed to connect the torque controller | 670 rpm | ${\omega}_{GEN-min}$ |

Rated generator torque | 43,093.55 N·m | ${Q}_{GEN-rated}$ |

Maximum generator torque rate | 15,000 N·m/s | $\Delta {Q}_{GEN-rate}$ |

Minimum blade-pitch setting | 0 deg | ${\phi}_{Pitch-min}$ |

Maximum blade-pitch setting | 90 deg | ${\phi}_{Pitch-max}$ |

Maximum absolute blade pitch rate | 8 deg/s | $\Delta {\phi}_{Pitch-rate}$ |

^{1}The data are from [14].

Node | r_{ni}(m) | dp_{ni}(m) | φ_{Twist}(deg) | L_{Chord}(m) | Airfoil |
---|---|---|---|---|---|

1 | 2.8667 | 2.7333 | 0 ^{2} | 3.542 | Cylinder 1 |

2 | 5.6 | 2.7333 | 0 ^{2} | 3.854 | Cylinder 1 |

3 | 8.3333 | 2.7333 | 0 ^{2} | 4.167 | Cylinder 2 |

4 | 11.75 | 4.1 | 13.308 | 4.557 | DU 40 |

5 | 15.85 | 4.1 | 11.48 | 4.652 | DU 35 |

6 | 19.95 | 4.1 | 10.162 | 4.458 | DU 35 |

7 | 24.05 | 4.1 | 9.011 | 4.249 | DU 30 |

8 | 28.15 | 4.1 | 7.795 | 4.007 | DU 25 |

9 | 32.25 | 4.1 | 6.544 | 3.748 | DU 25 |

10 | 36.35 | 4.1 | 5.361 | 3.502 | DU 21 |

11 | 40.45 | 4.1 | 4.188 | 3.256 | DU 21 |

12 | 44.55 | 4.1 | 3.125 | 3.01 | NACA 64–618 |

13 | 48.65 | 4.1 | 2.319 | 2.764 | NACA 64–618 |

14 | 52.75 | 4.1 | 1.526 | 2.518 | NACA 64–618 |

15 | 56.1667 | 2.7333 | 0.863 | 2.313 | NACA 64–618 |

16 | 58.9 | 2.7333 | 0.37 | 2.086 | NACA 64–618 |

17 | 61.6333 | 2.7333 | 0.106 | 1.419 | NACA 64–618 |

^{1}The data are from [14].

^{2}Value used in FHYGSYS.

Wind Speed ^{1}${\mathit{V}}_{\mathit{W}-\mathit{R}\mathit{E}\mathit{F}}(\mathbf{m}/\mathbf{s})$ | Wind Turbine Speed ^{2}${\mathbf{\Omega}}_{\mathit{S}\mathit{P}}\left(\mathit{t}\right)\left(\mathbf{rpm}\right)$ | Proportional Gain ${\mathit{K}}_{{\mathit{p}}_{\mathit{G}\mathit{E}\mathit{N}}}(-)$ | Integration Time ${\mathit{T}}_{{\mathit{i}}_{\mathit{G}\mathit{E}\mathit{N}}}\left(\mathbf{s}\right)$ |
---|---|---|---|

3 | 6.97 | 0.09 | 15 |

4 | 7.18 | 0.08 | 15 |

5 | 7.51 | 0.07 | 15 |

6 | 7.94 | 0.06 | 15 |

7 | 8.47 | 0.05 | 20 |

8 | 9.16 | 0.04 | 30 |

9 | 10.3 | 0.03 | 40 |

10 | 11.43 | 0.02 | 50 |

11 | 11.89 | 0.018 | 100 |

11. 4 | 12.1 | 0.019 | 100 |

12 | 12.1 | 0.018 | 100 |

13 to 25 | 12.1 | 0.01 | 200 |

^{1}Wind speeds at a 90-m reference height.

^{2}The data are from [49].

${\mathit{u}}_{{\mathit{M}}_{\mathit{G}\mathit{E}\mathit{N}}}\left(\mathit{t}\right)$ | ${\mathit{M}}_{\mathit{G}\mathit{E}\mathit{N}}\left(\mathit{t}\right)$ |
---|---|

1 | 0 |

0 | ${Q}_{GEN-rated}$ |

Wind Speed ^{1}${\mathit{V}}_{\mathit{W}-\mathit{R}\mathit{E}\mathit{F}}(\mathbf{m}/\mathbf{s})$ | Wind Turbine Speed ^{2}${\mathbf{\Omega}}_{\mathit{S}\mathit{P}}\left(\mathit{t}\right)\left(\mathbf{rpm}\right)$ | Proportional Gain ${\mathit{K}}_{{\mathit{p}}_{\mathit{P}\mathit{i}\mathit{t}\mathit{c}\mathit{h}}}(-)$ | Integration Time ${\mathit{T}}_{{\mathit{i}}_{\mathit{P}\mathit{i}\mathit{t}\mathit{c}\mathit{h}}}\left(\mathbf{s}\right)$ | Trigger Offset Factor F_{Trigg(%)} (%) |
---|---|---|---|---|

3 to 11 | 6.97 to 11.89 | 0 | 1 | 0 |

11.4 | 12.1 | 0.0023 | 7 | 11 |

12 | 12.1 | 0.0025 | 7 | 11 |

13 | 12.1 | 0.0016 | 7 | 7 |

14 | 12.1 | 0.0013 | 7 | 4.5 |

15 | 12.1 | 0.0012 | 7 | 3 |

16 | 12.1 | 0.0011 | 7 | 3 |

17 | 12.1 | 0.0011 | 7 | 3 |

18 | 12.1 | 0.0011 | 7 | 3 |

19 | 12.1 | 0.0010 | 7 | 3 |

20 | 12.1 | 0.0010 | 7 | 3 |

21 | 12.1 | 0.0010 | 7 | 3 |

22 | 12.1 | 0.0010 | 7 | 3 |

23 | 12.1 | 0.0010 | 7 | 3 |

24 | 12.1 | 0.0010 | 7 | 3 |

25 | 12.1 | 0.0010 | 7 | 3 |

^{1}Wind speeds at a 90-m reference height.

^{2}The data are from [49].

${\mathit{u}}_{\mathit{P}\mathit{i}\mathit{t}\mathit{c}\mathit{h}}\left(\mathit{t}\right)$ | ${\mathit{\phi}}_{\mathit{P}\mathit{i}\mathit{t}\mathit{c}\mathit{h}}\left(\mathit{t}\right)$ |
---|---|

−1 | ${\phi}_{Pitch-max}$ |

${F}_{Trigg}$ | 0 |

Property | Value | Symbol |
---|---|---|

Length from the hub center to tube junction | 4.427 m | ${l}_{hub-junction}$ |

Length from the hub center to floating platform | 12.09 m | ${l}_{hub-platform}$ |

Maximum Length between the center of the main tube and the others | 2 m | ${l}_{tube-max}$ |

Turbine support tube diameter | 0.6 m | ${D}_{tube}$ |

**Table 9.**Origin of the marine current turbine coordinate systems. The points are in their initial position for a sub-surface current direction ($\epsilon $) equal to zero degrees.

Description | Value (m) | Symbol |
---|---|---|

Origin of clockwise turbine coordinate system | (0, 17.1, −20) | p_{oC−MCT (BODY)} |

Origin of counterclockwise turbine coordinate system | (0, −17.1, −20) | p_{oCC−MCT (BODY)} |

Property | Value | Symbol |
---|---|---|

Rated electrical power ^{1} | 550 kW | ${P}_{ELE-rated}$ |

Number of blades ^{1} | 2 | $B$ |

Hub depth ^{1} | −20 m | ${h}_{Hub}$ |

Hub radius ^{1} | 1 m | ${r}_{Hub}$ |

Blade length ^{1} | 9 m | ${L}_{Blade}$ |

Precone ^{1} | 0 deg | ${\phi}_{Precone}$ |

Shaft tilt ^{1} | 0 deg | ${\phi}_{ShaftTilt}$ |

Rated rotor speed ^{1} | 11.5 rpm | ${\mathsf{\Omega}}_{rated}$ |

Gearbox ratio ^{2} | 97:1 | $gearR$ |

Electrical generator efficiency ^{2} | 0.944 | $genE$ |

Generator inertia about high-speed shaft ^{2} | 534.116 kg·m^{2} | ${I}_{GEN}$ |

Minimum generator speed to connect the torque controller ^{3} | 325 rpm | ${\omega}_{GEN-min}$ |

Rated generator torque ^{3} | 4971.531 N·m | ${Q}_{GEN-rated}$ |

Maximum generator torque rate ^{3} | 1000 N·m/s | $\Delta {Q}_{GEN-rate}$ |

Minimum blade-pitch setting ^{2} | 0 deg | ${\phi}_{Pitch-min}$ |

Maximum blade-pitch setting ^{2} | 90 deg | ${\phi}_{Pitch-max}$ |

Maximum absolute blade pitch rate ^{3} | 4 deg/s | $\Delta {\phi}_{Pitch-rate}$ |

Node^{1} | p_{ni}(m) | dp_{ni}(m) | ϕ_{Twist}(deg) | L_{Chord}(m) | Hydrofoil |
---|---|---|---|---|---|

1 | 1.075 | 0.15 | 0 ^{2} | 0.8 | Cylinder 3 |

2 | 1.3607 | 0.42143 | 12.86 | 0.87812 | NACA 63–424 |

3 | 1.7821 | 0.42143 | 12.86 | 1.1802 | NACA 63–424 |

4 | 2.2036 | 0.42143 | 12.86 | 1.5287 | NACA 63–424 |

5 | 2.625 | 0.42143 | 12.805 | 1.7023 | NACA 63–424 |

6 | 3.1342 | 0.59702 | 10.727 | 1.6304 | NACA 63–424 |

7 | 3.7313 | 0.59702 | 8.8252 | 1.5404 | NACA 63–424 |

8 | 4.3283 | 0.59702 | 7.5219 | 1.4565 | NACA 63–424 |

9 | 4.9253 | 0.59702 | 6.5262 | 1.372 | NACA 63–424 |

10 | 5.5223 | 0.59702 | 5.7432 | 1.2867 | NACA 63–424 |

11 | 6.1193 | 0.59702 | 5.1057 | 1.2002 | NACA 63–424 |

12 | 6.7164 | 0.59702 | 4.5608 | 1.1116 | NACA 63–424 |

13 | 7.3134 | 0.59702 | 4.0772 | 1.0214 | NACA 63–424 |

14 | 7.9104 | 0.59702 | 3.6195 | 0.9299 | NACA 63–424 |

15 | 8.5074 | 0.59702 | 3.1795 | 0.83481 | NACA 63–424 |

16 | 9.1045 | 0.59702 | 2.7241 | 0.73776 | NACA 63–424 |

17 | 9.7015 | 0.59702 | 2.2413 | 0.63825 | NACA 63–424 |

^{1}Data deduced from the blade properties described in [34].

^{2}Value used in FHYGSYS.

Current Speed ^{1}${\mathit{V}}_{\mathit{S}\mathit{W}\mathit{L}}(\mathbf{m}/\mathbf{s})$ | MCT Speed ^{2}${\mathbf{\Omega}}_{\mathit{S}\mathit{P}}\left(\mathit{t}\right)\left(\mathbf{rpm}\right)$ | Proportional Gain ${\mathit{K}}_{{\mathit{p}}_{\mathit{G}\mathit{E}\mathit{N}}}(-)$ | Integration Time ${\mathit{T}}_{{\mathit{i}}_{\mathit{G}\mathit{E}\mathit{N}}}\left(\mathbf{s}\right)$ |
---|---|---|---|

0.5 | 3.37 | 0.300 | 30 |

0.65 | 4.363 | 0.150 | 30 |

0.8 | 5.356 | 0.100 | 30 |

0.95 | 6.349 | 0.090 | 30 |

1.1 | 7.35 | 0.080 | 30 |

1.25 | 8.355 | 0.070 | 40 |

1.4 | 9.36 | 0.060 | 50 |

1.55 | 10.21 | 0.050 | 60 |

1.7 | 10.77 | 0.045 | 70 |

1.85 | 11.32 | 0.045 | 80 |

2.0 | 11.47 | 0.040 | 80 |

2.03 | 11.5 | 0.040 | 150 |

2.1 to 3.0 | 11.5 | 0.030 | 150 |

Current Speed ^{1}${\mathit{V}}_{\mathit{S}\mathit{W}\mathit{L}}(\mathbf{m}/\mathbf{s})$ | MCT Speed ^{2}${\mathbf{\Omega}}_{\mathit{S}\mathit{P}}\left(\mathit{t}\right)\left(\mathbf{rpm}\right)$ | Proportional Gain ${\mathit{K}}_{{\mathit{p}}_{\mathit{P}\mathit{i}\mathit{t}\mathit{c}\mathit{h}}}(-)$ | Integration Time ${\mathit{T}}_{{\mathit{i}}_{\mathit{P}\mathit{i}\mathit{t}\mathit{c}\mathit{h}}}\left(\mathbf{s}\right)$ | Trigger Offset Factor F_{Trigg(%)} (%) |
---|---|---|---|---|

0.5 to 2.0 | 3.37 to 11.47 | 0 | 1 | 0 |

2.03 | 11.5 | 0.0015 | 9 | 12 |

2.1 | 11.5 | 0.0012 | 10 | 10 |

2.25 | 11.5 | 0.0012 | 10 | 10 |

2.4 | 11.5 | 0.0012 | 10 | 10 |

2.55 | 11.5 | 0.0012 | 10 | 10 |

2.7 | 11.5 | 0.0012 | 10 | 10 |

2.85 | 11.5 | 0.0012 | 10 | 10 |

3 | 11.5 | 0.0012 | 10 | 10 |

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## Share and Cite

**MDPI and ACS Style**

Tamarit, F.; García, E.; Quiles, E.; Correcher, A.
BEM Turbine Model and PID Control System of a Floating Hybrid Wind and Current Turbines Integrated Generator System. *J. Mar. Sci. Eng.* **2023**, *11*, 1634.
https://doi.org/10.3390/jmse11081634

**AMA Style**

Tamarit F, García E, Quiles E, Correcher A.
BEM Turbine Model and PID Control System of a Floating Hybrid Wind and Current Turbines Integrated Generator System. *Journal of Marine Science and Engineering*. 2023; 11(8):1634.
https://doi.org/10.3390/jmse11081634

**Chicago/Turabian Style**

Tamarit, Fernando, Emilio García, Eduardo Quiles, and Antonio Correcher.
2023. "BEM Turbine Model and PID Control System of a Floating Hybrid Wind and Current Turbines Integrated Generator System" *Journal of Marine Science and Engineering* 11, no. 8: 1634.
https://doi.org/10.3390/jmse11081634