A Simplified Modeling Approach of Floating Offshore Wind Turbines for Dynamic Simulations

  Citation: López-Queija, J.; Robles, E.; Llorente, J.I.; Touzon, I.; López-Mendia, J. A Simplified Modeling Approach of Floating Offshore Wind Turbines for Dynamic Simulations. Energies 2022,15, 2228. https://doi.org/10.3390/en15062228 Academic Editors: Davide Astolfi and Uwe Ritschel Received: 21 February 2022 Accepted: 15 March 2022 Published: 18 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article A Simplified Modeling Approach of Floating Offshore Wind Turbines for Dynamic Simulations Javier López-Queija 1,2,* , Eider Robles 1,3, Jose Ignacio Llorente 2, Imanol Touzon 1 and Joseba López-Mendia 1 1TECNALIA, Basque Research and Technology Alliance (BRTA), Parque Tecnológico de Bizkaia Astondo Bidea, Edificio 700 E, 48160 Derio, Spain; eider[email protected] (E.R.); [email protected] (I.T.); [email protected] (J.L.-M.) 2Mechanical Engineering Department, University of the Basque Country UPV/EHU, 48013 Bilbao, Spain; [email protected] 3Automatics and System Engineering Department, University of the Basque Country UPV/EHU, 48013 Bilbao, Spain *Correspondence: javier[email protected] Abstract: Currently, floating offshore wind is experiencing rapid development towards a commercial scale. However, the research to design new control strategies requires numerical models of low computational cost accounting for the most relevant dynamics. In this paper, a reduced linear time-domain model is presented and validated. The model represents the main floating offshore wind turbine dynamics with four planar degrees of freedom: surge, heave, pitch, first tower fore-aft deflection, and rotor speed to account for rotor dynamics. The model relies on multibody and modal theories to develop the equation of motion. Aerodynamic loads are calculated using the wind turbine power performance curves obtained in a preprocessing step. Hydrodynamic loads are precomputed using a panel code solver and the mooring forces are obtained using a look-up table for different system displacements. Without any adjustment, the model accurately predicts the system motions for coupled stochastic wind–wave conditions when it is compared against OpenFAST, with errors below 10% for all the considered load cases. The largest errors occur due to the transient effects during the simulation runtime. The model aims to be used in the early design stages as a dynamic simulation tool in time and frequency domains to validate preliminary designs. Moreover, it could also be used as a control design model due to its simplicity and low modeling order. Keywords: floating offshore wind turbine; simplified model; FOWT dynamics; aerodynamics; hydrodynamics; structural dynamics 1. Introduction In recent decades, wind energy production growth has pushed renewable energies to directly compete with fuel-based energies [ 1 ]. In order to continue rising, the wind energy market leverages the advantages of offshore locations where the wind resource is preferable, with higher and steadier winds. In 2019, offshore wind energy capacity installation reached a peak [ 2 ], which is expected to be surpassed shortly, supported by floating wind energy development. Different commercial projects have been deployed in this direction such as Hywind Atlantic and Kincardine projects [ 3 ]. Despite the increasing amount of investment in floating solutions, the capital costs of installing a floating offshore wind turbine (FOWT) make the production costs per unit generated still high [ 1 ]. However, a viable levelized cost of energy (LCOE) reduction path could be through the implementation and testing of appropriate control strategies. A FOWT is a highly complex nonlinear system whose representation requires considering hydrodynamics, aerodynamics, structural dynamics, mooring dynamics, controller dynamics, and their couplings. The FOWT system modeling requirements are conditioned Energies 2022,15, 2228. https://doi.org/10.3390/en15062228 https://www.mdpi.com/journal/energies Energies 2022,15, 2228 2 of 16 by the design stage. The detail and accuracy needed for the model will differ from the conceptual design stage to the pre-industrial stage. Detailed mathematical models of FOWT, including complex aerodynamics, hydrodynamics, mooring dynamics, and system structural dynamics, have been developed, allowing complex aero-hydrodynamic load representation and its influence on the system through high-fidelity time-domain simulations [ 4 , 5 ]. There are different software packages available to represent in detail these complex systems [ 6 ] that serve as analysis tools for system dynamics, turbine loads, fatigue damage, and cost assessment in the final design stages. In [ 7 ], a recent review of the FOWT dynamics and modeling approaches is presented. However, the most extended simulation tool for wind turbine design, both onshore and offshore, is the software named OpenFAST [8]. Although reliable FOWT modeling tools are available, which are generally based on very detailed and complex mathematical descriptions, it is of interest to develop simpler models that accurately represent the main FOWT dynamics. These simplified models can be used to: provide a clear understanding of the main system dynamics, easily modify the model parameters to check different system configurations, design controllers, and quickly test both system designs and controllers in early design stages. In the same way, these models can be helpful for scale prototype test activities. In [ 9 , 10 ], reduced-order models are presented for time-domain simulations. A similar model is described in [ 11 ] for dynamic performance evaluation, but simulated in the frequency domain. This low-order modeling approach is also being developed for scale prototype activities such as in [ 12 ], where the model is used for a hybrid hardware-in-the-loop test. A similar model derivation is proposed in [13], where the authors validate the model against a scaled prototype. In general, the FOWT control research field trends towards designing controllers able to lead with more than one objective at the same time, which is basic for these systems where platform stabilization and power production are competing for wind turbine operation above rated wind [ 14 ]. A review of the current control strategies applied to different FOWT concepts can be found in [ 15 ]. The model typology for multivariable control design requires representing the system dynamics with the minimum number of variables. The control design is usually performed in a final stage through a sequential design approach [ 16 ]. However, the complexity level needed to assess control influence on global loads and motions is low, allowing modeling of the system with the dominant system dynamics considering only a few degrees of freedom [ 17 ], enabling the use of simplified models. In [ 18 ], a simplified dynamic model for control development is presented and then utilized for its purpose in, for example, [ 19 ]. A similar control-oriented model is proposed in [ 20 ] which is used to design a robust controller after being validated with OpenFAST. In this paper, a simplified mathematical model is proposed. The developed model is simple and represents the system dynamics well enough to be used for advanced controller design. The model’s architecture is thought to allow changing between FOWT concepts by changing the inputs to the system. The mathematical expression of the model is obtained considering the FOWT system as two rigid solids: rotor–nacelle assembly (RNA) and substructure linked by a flexible beam representing the tower. Using a 5 MW wind turbine atop a spar substructure, aerodynamic and hydrodynamic loads acting on the system are explained. A comparison of the presented model against OpenFAST is performed to validate the system dynamic response both in time and frequency domains. The paper is structured as follows. In Section 2, the FOWT case study definition is given, defining those properties needed to build the model. Through Section 3, the modeling approach is introduced, explaining the aerodynamic, hydrodynamic, and structural dynamic representation used in the model. Section 4summarizes the load cases and the comparison of the proposed model performance against the state-of-the-art model OpenFAST. Finally, in Section 5, some conclusions are drawn considering the results from the previous section, and possible future research directions are also mentioned in this last section. Energies 2022,15, 2228 3 of 16 2. Case Study The presented model is validated using the OC3 Hywind Spar buoy shown in Figure 1 as a case study. The floating system’s main properties including the platform, the wind turbine, and the mooring system are summarized based on the description provided in [ 21 ]. Energies 2022, 15, x FOR PEER REVIEW 3 of 17 2. Case Study The presented model is validated using the OC3 Hywind Spar buoy shown in Figure 1 as a case study. The floating system’s main properties including the platform, the wind turbine, and the mooring system are summarized based on the description provided in [21]. Figure 1. OC3 Hywind Spar buoy illustration [21]. The platform hull reaches 10 m above sea water level (SWL) and has a draft of 120 m. The stability of this floating platform concept is achieved by a restoring moment generated by the difference in height between the mass and buoyancy centers. In Table 1, the main properties used for the development of the reduced model are summarized. Table 1. OC3 Hywind platform properties [21]. Parameter Value Units Depth to platform base below SWL (Draft) 120 m Elevation to platform top above SWL 10 m Taper top depth below SWL 4 m Taper bottom depth below SWL 12 m Platform diameter above the taper 6.5 m Platform diameter below the taper 9.4 m Platform mass 7,466,330.0 kg Platform center mass (CM) below SWL 89.9155 m Platform pitch inertia about CM 4,229,230,000 kg m2 Additional linear damping (𝐵11 𝑙𝑖𝑛𝑒𝑎𝑟) 100,000 N s m−1 Additional linear damping (𝐵33 𝑙𝑖𝑛𝑒𝑎𝑟) 130,000 N s m−1 The wind turbine modelled is the NREL5MW [22], which is a variable-speed variable-pitch (VSVP) controlled turbine often used as a reference turbine for research purposes. The hub height of the onshore turbine is 90 m above ground, so the tower height of the model is shortened to match the same hub height of the onshore turbine due to the platform hull height above water level. Tower top and bottom diameters are unchanged. The wind turbine properties are shown in Table 2. Figure 1. OC3 Hywind Spar buoy illustration [21]. The platform hull reaches 10 m above sea water level (SWL) and has a draft of 120 m. The stability of this floating platform concept is achieved by a restoring moment generated by the difference in height between the mass and buoyancy centers. In Table 1, the main properties used for the development of the reduced model are summarized. Table 1. OC3 Hywind platform properties [21]. Parameter Value Units Depth to platform base below SWL (Draft) 120 m Elevation to platform top above SWL 10 m Taper top depth below SWL 4 m Taper bottom depth below SWL 12 m Platform diameter above the taper 6.5 m Platform diameter below the taper 9.4 m Platform mass 7,466,330.0 kg Platform center mass (CM) below SWL 89.9155 m Platform pitch inertia about CM 4,229,230,000 kg m2 Additional linear damping (Blinear 11 )100,000 Nsm−1 Additional linear damping (Blinear 33 )130,000 Nsm−1 The wind turbine modelled is the NREL5MW [ 22 ], which is a variable-speed variablepitch (VSVP) controlled turbine often used as a reference turbine for research purposes. The hub height of the onshore turbine is 90 m above ground, so the tower height of the model is shortened to match the same hub height of the onshore turbine due to the platform hull height above water level. Tower top and bottom diameters are unchanged. The wind turbine properties are shown in Table 2. Energies 2022,15, 2228 4 of 16 Table 2. OC3 Hywind wind turbine properties [22]. Parameter Value Units Rated power 5 MW Rotor diameter 126 m Hub height 90 m Rotor mass 110,000 kg Rotor inertia 35,444,067 kg m2 Generator inertia 534.116 kg m2 Generator friction 16.5489 N s/m Gearbox ratio (high-speed to low-speed) 97 - Nacelle mass 240,000 kg Nacelle CM above tower top 1.96 m Tower mass 347,460 kg The mooring system properties are described in [ 21 ] and summarized in Table 3. It consists of three catenary lines attached to the platform via delta connection. In the model, this configuration is simplified to reduce the system complexity as was done in the OC3 project. The fairleads are located at 70 m below SWL and symmetrically spread at a 5.2 m radius from the platform centerline, mounting a mooring system configuration where lines are 120◦separated. Anchors are fixed at a radius of 853.87 m from the platform centerline and at a depth of 320 m. Table 3. OC3 Hywind mooring system properties [21]. Parameter Value Units Unstretched mooring line length 902.2 m Mooring line diameter 0.09 m Equivalent mooring line mass density 77.7066 kg/m Equivalent mooring line weight in water 698.094 N/m Equivalent mooring line extensional stiffness (EA) 384,243,000 N 3. Modeling Approach As it is explained in [ 23 ], a control-oriented wind turbine model is usually derived using the Multibody System approach (MBS). With this technique, reduced low-order models can be obtained allowing the consideration of only those degrees of freedom that are directly coupled to the controller actions. In a variable-speed variable-pitch (VSVP) wind turbine operation, the speed control interacts with the modes in the rotation frame such as drivetrain torsion mode and blade edgewise bending modes. However, these modes’ natural frequencies fall beyond the controller bandwidth frequency [ 23 ], allowing the simplification. The pitch control not only affects the power production through the aerodynamic torque but also changes the thrust excitation force. In consequence, tower bending mode and floating system rigid solid modes, affected by the thrust excitation force, should also be considered. In the present work, a planar MBS is derived, presenting the FOWT as two lumped masses, platform, and rotor nacelle assembly (RNA), connected by a flexible tower. The model describes the FOWT dynamics attending to the interaction between the rotor and the along-wind modes. Surge, heave, and pitch rigid-body modes are considered in the model since they are identified as the most critical modes for FOWT control design [ 24 ], with the pitch mode a limitation for traditional control strategies [ 25 , 26 ]. The first fore–aft tower modal deflection is also included because it is significantly excited by aerodynamic forces due to its low natural frequency. These loads can be reduced using the blade pitch control for wind speeds above the rated value [ 27 ] and, consequently, a lighter tower and foundation could be designed. Finally, the rotor dynamics are also represented by a single degree of freedom equation of motion. Energies 2022,15, 2228 5 of 16 The time-domain equation of motion is based on Newton’s second law: [M].. x(t)+[C]. x(t)+[K]{x(t)}={FExt(t)}(1) where the motion vector {x(t)} comprises the motions in each of the considered degrees of freedom (DOF). [M] , [C] , and [K] are, respectively, the mass, damping, and stiffness matrices of the system. All the previously introduced matrices are 5 × 5 according to the represented system motions. The system total mass sums the contribution of the structural mass and the hydrodynamic added mass from the inertial component of the radiation force. The damping matrix is mounted from the contribution of three damping terms. First, the linear hydrodynamic damping is presented before in the case study section. The second is the structural damping of the system. Thirdly, the damping contribution from the radiation force is added to the model through direct integration of the convolution of the product between radiation impulse response and the state velocity. Finally, the contribution from the hydrostatic restoring and structural stiffness are included in the system stiffness matrix. The last term, {FExt(t)} , represents the external forces acting on each of the system DOFs. In the present model, this load vector encompasses wind {Fa} , wave {Fh} , excitation forces, and mooring loads {Fmoor} . In Figure 2, a block diagram of the proposed FOWT model is presented: Energies 2022, 15, x FOR PEER REVIEW 5 of 17 foundation could be designed. Finally, the rotor dynamics are also represented by a single degree of freedom equation of motion. The time-domain equation of motion is based on Newton’s second law: [𝑀]{𝑥󰇘(𝑡)}+[𝐶]{𝑥󰇗(𝑡)}+[𝐾]{𝑥(𝑡)}={𝐹𝐸𝑥𝑡(𝑡)} (1) where the motion vector {𝑥(𝑡)} comprises the motions in each of the considered degrees of freedom (DOF). [𝑀], [𝐶], and [𝐾] are, respectively, the mass, damping, and stiffness matrices of the system. All the previously introduced matrices are 5 × 5 according to the represented system motions. The system total mass sums the contribution of the structural mass and the hydrodynamic added mass from the inertial component of the radiation force. The damping matrix is mounted from the contribution of three damping terms. First, the linear hydrodynamic damping is presented before in the case study section. The second is the structural damping of the system. Thirdly, the damping contribution from the radiation force is added to the model through direct integration of the convolution of the product between radiation impulse response and the state velocity. Finally, the contribution from the hydrostatic restoring and structural stiffness are included in the system stiffness matrix. The last term, {𝐹𝐸𝑥𝑡(𝑡)}, represents the external forces acting on each of the system DOFs. In the present model, this load vector encompasses wind {𝐹𝑎}, wave {𝐹ℎ}, excitation forces, and mooring loads {𝐹𝑚𝑜𝑜𝑟}. In Figure 2, a block diagram of the proposed FOWT model is presented: Figure 2. Block diagram of the proposed model considering a controller action. In the following subsections, the different terms of the previously presented dynamic equation are obtained for the case study model. 3.1. Aerodynamics The interaction between the wind and the turbine is defined by the aerodynamic model. Commonly, the Blade Element Momentum (BEM) theory is applied to define the wind turbine aerodynamic loads [28]. This theory is a complex computational method that requires iterations to obtain the axial and tangential induction factors needed to calculate the lift and drag forces in each of the blade sections. The use of this theory is avoided during simulation runtime due to the numerical effort required for induction coefficient determination. However, it is applied in a pre-processing stage where the wind turbine aerodynamic properties are obtained using Aerodyn [29] for different rotor speeds and pitch angles assuming nacelle motions are small so the aerodynamic properties remain Figure 2. Block diagram of the proposed model considering a controller action. In the following subsections, the different terms of the previously presented dynamic equation are obtained for the case study model. 3.1. Aerodynamics The interaction between the wind and the turbine is defined by the aerodynamic model. Commonly, the Blade Element Momentum (BEM) theory is applied to define the wind turbine aerodynamic loads [ 28 ]. This theory is a complex computational method that requires iterations to obtain the axial and tangential induction factors needed to calculate the lift and drag forces in each of the blade sections. The use of this theory is avoided during simulation runtime due to the numerical effort required for induction coefficient determination. However, it is applied in a pre-processing stage where the wind turbine aerodynamic properties are obtained using Aerodyn [ 29 ] for different rotor speeds and pitch angles assuming nacelle motions are small so the aerodynamic properties remain constant. These properties, power coefficient, CP(λ,β) , and thrust coefficient, CT(λ,β) , Energies 2022,15, 2228 6 of 16 shown in Figure 3, are computed as functions of the blade pitch angle and the tip speed ratio (TSR or λ ), which is the relation between the blade tip linear speed and the incident wind: λ=ΩR vw(2) Energies 2022, 15, x FOR PEER REVIEW 6 of 17 constant. These properties, power coefficient, 𝐶𝑃(𝜆,𝛽), and thrust coefficient, 𝐶𝑇(𝜆,𝛽), shown in Figure 3, are computed as functions of the blade pitch angle and the tip speed ratio (TSR or 𝜆), which is the relation between the blade tip linear speed and the incident wind: 𝜆=Ω𝑅 𝑣𝑤 (2) Figure 3. Power coefficient (left) and Thrust coefficient (right). In the present model, the wind turbine aerodynamics are written based on the nondimensional power and thrust coefficients represented as: 𝐹𝑎=12· 𝜌𝑎𝑖𝑟·𝐴𝑅𝑜𝑡𝑜𝑟·𝐶𝑇(𝜆,𝛽)·𝑣𝑅𝑒𝑙 2 (3) 𝑀𝑎=12· 𝜌𝑎𝑖𝑟·𝐴𝑅𝑜𝑡𝑜𝑟·𝐶𝑃(𝜆,𝛽) Ω·𝑣𝑅𝑒𝑙 3 (4) where 𝜌𝑎𝑖𝑟 is the air density, 𝐴𝑅𝑜𝑡𝑜𝑟 is the rotor area, Ω is the rotor speed, and 𝑣𝑅𝑒𝑙 is the relative wind speed at hub height, calculated as the wind speed (𝑣𝑤) reduced by the hub velocity (𝑣ℎ𝑢𝑏): 𝑣𝑅𝑒𝑙=𝑣𝑤−𝑣ℎ𝑢𝑏 (5) 3.2. Structural Dynamics Floating wind turbine motions and structural responses are consequence of rigidbody motions rather than elastic deformations [30]. Hence, rigid-body theory is accurate enough to represent the FOWT dynamics for control design. However, the proposed model is augmented to a multi-body elastic deformation model to consider the tower as a flexible element due to the influence of the wind excitation force into the tower dynamics and controller design. The tower is a continuous structure that can be discretized in various ways. If shear deformation and lateral inertia effects are neglected, the tower deformation can be modelled with a generalized displacement in combination with the principle of virtual displacements, as is proposed in [31] and later used to model a FOWT in [11,32]. In the proposed model, this theory is followed to describe the tower response. The mode is described by a shape function; thus, the accuracy of the modelled response depends on how well the deformation is captured by the shape function. These shape functions are commonly chosen to be the most relevant eigenmodes of the system. In this case, it is the tower’s first fore-aft bending mode. In the proposed model, the tower bending mode shape function is obtained using BModes [33]. Figure 3. Power coefficient (left) and Thrust coefficient (right). In the present model, the wind turbine aerodynamics are written based on the nondimensional power and thrust coefficients represented as: Fa=1 2·ρair·ARotor·CT(λ,β)·v2 Rel (3) Ma=1 2·ρair·ARotor·CP(λ,β) Ω·v3 Rel (4) where ρair is the air density, ARotor is the rotor area, Ω is the rotor speed, and vRel is the relative wind speed at hub height, calculated as the wind speed ( vw ) reduced by the hub velocity (vhub): vRel =vw−vhub (5) 3.2. Structural Dynamics Floating wind turbine motions and structural responses are consequence of rigid-body motions rather than elastic deformations [ 30 ]. Hence, rigid-body theory is accurate enough to represent the FOWT dynamics for control design. However, the proposed model is augmented to a multi-body elastic deformation model to consider the tower as a flexible element due to the influence of the wind excitation force into the tower dynamics and controller design. The tower is a continuous structure that can be discretized in various ways. If shear deformation and lateral inertia effects are neglected, the tower deformation can be modelled with a generalized displacement in combination with the principle of virtual displacements, as is proposed in [ 31 ] and later used to model a FOWT in [ 11 , 32 ]. In the proposed model, this theory is followed to describe the tower response. The mode is described by a shape function; thus, the accuracy of the modelled response depends on how well the deformation is captured by the shape function. These shape functions are commonly chosen to be the most relevant eigenmodes of the system. In this case, it is the tower’s first fore-aft bending mode. In the proposed model, the tower bending mode shape function is obtained using BModes [33]. Three rigid-body modes of motion and one flexible mode are considered in the model, namely, surge, heave, pitch, and the first tower fore-aft bending mode, respectively. Motions are referenced to the center of gravity of the whole system. The tower mode introduces off-diagonal terms in the system matrixes to couple the bending mode with surge and pitch modes. The actual bending mode shape needed to represent the tower fore-aft mode is not Energies 2022,15, 2228 7 of 16 known and is, therefore, obtained from an eigenvalue solution. Consequently, it is assumed that the shape of the fore-aft mode is kept in the coupled model. An additional variable is added to represent the rotor dynamics. Assuming a rigid drivetrain, a first-order dynamic equation is applied to consider the rotational speed of the rotor in the proposed model: IR−rT·Ig·. Ω=Ma−rT·Mg(6) where the overall drivetrain inertia is computed by the rotor inertia ( IR ) and the generator inertia ( Ig ) expressed in the low-speed shaft by the gearbox relation ( rT ). The inertia is balanced by the aerodynamic torque ( Ma ) and the generator torque ( Mg ), which is kept constant to obtain the required rotor speed (Ω). 3.3. Hydrodynamics In a spar-type foundation, one of the most widely used methods is related to the Morison equation and strip theory [ 34 ]. The theory is applied for a certain environmental condition where waves are large, and the floating structure is considered slender. Since the presented model aims to be used for control strategy design, which is assessed within the wind turbine operational range, the environmental sea conditions are smoother, enabling solution of the radiation–diffraction problem with the linear potential flow theory [ 35 , 36 ]. The frequency-dependent added mass and radiation damping are precomputed in panel code software, such as the programs AQWA [ 37 ] or WAMIT [ 38 ], for the specific OC3 platform shape. The hydrostatic restoring matrix including the contribution from the buoyancy center (CB) and waterplane area together with the wave excitation force are obtained from the panel code solver based on the previously mentioned linear potential flow theory. To compute the time-domain hydrodynamic radiation force, the sum of added mass and radiation damping, the so-called free-surface memory effect, is considered by the convolution integral of the retardation function. In [ 39 , 40 ], the approach based on the potential theory results is developed and validated. The time-domain values for the added mass and radiation damping are computed as follows: A∞=a(ω)+Z∞ 0B(τ)·sin(ωτ)dt (7) B(τ)=2 π·Z∞ 0b(ω)·cos(ωτ)dω(8) where a(ω) is the added mass and b(ω) is the radiation damping, both function of frequency. B(τ) is the retardation function obtained from the cosine transformation of the impulse radiation function. Additionally, the hydrodynamic damping model is augmented with additional linear damping, which is added to match the free-decay test, as is detailed in [ 21 ] and summarized in Table 1. Second-order effects would be of significant influence in surge motions and need to be accounted for to conclude the benefits of specific control strategies. However, this aspect falls beyond the scope of this paper and has not been introduced. Slowly varying second-order drift forces will be considered in subsequent versions of the presented model, enabling its applicability to assess wind turbine control strategies. 3.4. Mooring Dynamics The floating system is anchored to the seabed by three catenary mooring lines, with a delta connection to increase the stiffness in yaw. However, the simplifications assumed in [ 21 ] are also considered. Each of the mooring lines is supposed as a continuous cable with homogeneous properties. If the forces from inertia, viscous drag, internal damping, and bending and torsion modes are neglected, a quasi-static analysis approach can be applied to obtain the non-linear catenary stiffness as a function of the platform movement, as is justified in [ 41 ]. In a preprocessing step, the mooring stiffness forces are obtained for Energies 2022,15, 2228 8 of 16 different platform displacements that are then used as a look-up table to compute the forces in each simulation time step. 3.5. Modeling of Wind and Wave Resource Wind and wave modeling is simulated following the guidelines provided in [ 42 ]. The Kaimal spectrum model is selected to include the turbulent component of the wind and the JONSWAP spectrum is used to represent different sea states. The wind speed Kaimal spectrum is defined by the following expression: Sk(f)=σ2 U 6.868 LU U10 1+10.32 f LU U10 5/3 (9) In this expression, f denotes the frequency, σU is the wind standard deviation, U10 is the 10 min mean wind speed at 10 m height above the still water level, and LU is the integral length scale of the wind speed process. This last parameter can be obtained as: LU=300 z 3000.46+0.074 ln z0(10) where zis the height above sea water level and z0is the terrain roughness calculated by: z0=Ac g  kaU(z) lnz z0  2 (11) where ka is the von Karman’s constant ( ka= 0.4), g is the gravity acceleration, and Ac is the Charnock’s constant, which is dependent on the wave velocity and the available water fetch. The definition of the implemented method for Ac is given in [ 43 ]. The simulations carried out in Section 4are computed using the turbulent wind speed time series developed with the proposed model; see Figure 4. Energies 2022, 15, x FOR PEER REVIEW 9 of 17 The wave elevation used in Section 4 simulations is obtained using this spectrum. In Figure 4, the irregular wave elevation time series can be seen. Figure 4. Wind and wave time series. 4. Model Validation The model is implemented in Python and its accuracy is tested against OpenFAST [8]. OpenFAST is a multi-physics, multi-fidelity tool for simulating the coupled dynamic response of wind turbines, both fixed and floating. The OC3 project [44] collects results from different modeling tools including OpenFAST. To validate the proposed low-order model, some of the load cases simulated in that project are used. In Table 4, a summary of the different load cases is shown: Table 4. Load cases for model comparison. Load Case Models Wind Waves Analysis 1 OpenFAST Reduced Model None None Eigenanalysis Decay Tests 2 OpenFAST Reduced Model None Regular RAOs 3 OpenFAST Reduced Model None Irregular JONSWAP spectrum 𝐻𝑠 = 6 𝑚 𝑇𝑝=10 𝑠 Time series PSDs 4 OpenFAST Reduced Model Turbulent: Kaimal spectrum 𝑉𝑤 = 8 𝑚/𝑠 None Time series Statistics 5 OpenFAST Reduced Model Turbulent: Kaimal spectrum 𝑉𝑤 = 18 𝑚/𝑠 Irregular: JONSWAP spectrum 𝐻𝑠 = 6 𝑚 𝑇𝑝=10 𝑠 Time series PSDs Statistics In the following, the comparison of the results between the proposed model and OpenFAST is presented. All the simulations have a simulation length time of 1800 s, and to avoid initial transient effects the first 800 s are neglected. The controller dynamics are simulated for a specific operation point, which means a constant rotor speed and a constant blade pitch angle for the complete simulation. The values for the wind turbine steady Figure 4. Wind and wave time series. The waves are modelled following the Airy wave theory, which considers that the fluid layer has a uniform depth and its flow is inviscid, incompressible, and irrotational [ 39 ]. The JONSWAP wave spectrum is utilized due to its validity to represent a sea state in a fetch limited situation, and it is formulated by: Sj(ω)=AγSPM(ω)γexp (−0.5(ω−ωp σ ωp)) (12) Energies 2022,15, 2228 9 of 16 where the Pierson–Moskowitz spectrum (SPM) is defined as: SPM(ω)=5 16 H2 Sω4 pω−5exp −5 4ω ωp−4!(13) The parameter HS is the significant wave height, ωp is the angular spectral peak frequency obtained from the peak period value ( Tp ), and ω is the angular frequency. Aγ is a normalizing factor function of the non-dimensional peak shape parameter, γ , and σ is a spectral width parameter. The values used for the parameters are those proposed in [ 42 ]. The wave elevation used in Section 4simulations is obtained using this spectrum. In Figure 4, the irregular wave elevation time series can be seen. 4. Model Validation The model is implemented in Python and its accuracy is tested against OpenFAST [ 8 ]. OpenFAST is a multi-physics, multi-fidelity tool for simulating the coupled dynamic response of wind turbines, both fixed and floating. The OC3 project [ 44 ] collects results from different modeling tools including OpenFAST. To validate the proposed low-order model, some of the load cases simulated in that project are used. In Table 4, a summary of the different load cases is shown: Table 4. Load cases for model comparison. Load Case Models Wind Waves Analysis 1OpenFAST Reduced Model None None Eigenanalysis Decay Tests 2OpenFAST Reduced Model None Regular RAOs 3OpenFAST Reduced Model None Irregular JONSWAP spectrum Hs=6 m Tp=10 s Time series PSDs 4OpenFAST Reduced Model Turbulent: Kaimal spectrum Vw=8 m/s None Time series Statistics 5OpenFAST Reduced Model Turbulent: Kaimal spectrum Vw=18 m/s Irregular: JONSWAP spectrum Hs=6 m Tp=10 s Time series PSDs Statistics In the following, the comparison of the results between the proposed model and OpenFAST is presented. All the simulations have a simulation length time of 1800 s, and to avoid initial transient effects the first 800 s are neglected. The controller dynamics are simulated for a specific operation point, which means a constant rotor speed and a constant blade pitch angle for the complete simulation. The values for the wind turbine steady operation are obtained from [ 22 ]. This is done to reduce the number of variables acting on the system’s dynamic response. 4.1. Load Case 1 The system natural frequencies are calculated with the proposed model by solving the eigenvalue problem mathematically described as: −nω2 no[M]+[K]{ˆ x(ω)}={0}(14) To obtain the natural frequencies in OpenFAST, a PSD of the decay test revealed the system’s damped natural frequency. A comparison of natural frequencies is given in Energies 2022,15, 2228 16 of 16 21. Jonkman, J. Definition of the Floating System for Phase IV of OC3; National Renewable Energy Laboratory (NREL): Golden, CO, USA, 2010. 22. Jonkman, J.; Butterfield, S.; Musial, W.; Scott, G. Definition of a 5-MW Reference Wind Turbine for Offshore System Development; National Renewable Energy Laboratory (NREL): Golden, CO, USA, 2009. 23. Bianchi, F.; De Battista, H.; Mantz, R. Wind Turbine Control Systems: Principles, Modelling and Gain Scheduling Design; Springer: London, UK, 2007. 24. Lemmer, F.; Schlipf, D.; Cheng, P.W. Control Design Methods for Floating Wind Turbines for Optimal Disturbance Rejection. J. Phys. Conf. Ser. 2016,753, 092006. [CrossRef] 25. Larsen, T.J.; Hanson, T.D. A Method to Avoid Negative Damped Low Frequent Tower Vibrations for a Floating, Pitch Controlled Wind Turbine. J. Phys. Conf. Ser. 2007,75, 012073. [CrossRef] 26. Jonkman, J. Influence of Control on the Pitch Damping of a Floating Wind Turbine. In Proceedings of the 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 7 January 2008. 27. Liu, J.; Thomas, E.; Goyal, A.; Manuel, L. Design Loads for a Large Wind Turbine Supported by a Semi-Submersible Floating Platform. Renew. Energy 2019,138, 923–936. [CrossRef] 28. Burton, T.; Sharpe, D.; Jenkins, N.; Bossanyi, E. Wind Energy Handbook; John Wiley & Sons: Chichester, UK, 2001. 29. Jonkman, J.M.; Hayman, G.J.; Jonkman, B.J.; Damiani, R.R.; Murray, R.E. AeroDyn V15 User’s Guide and Theory Manual; NREL Draft Report; NREL: Golden, CO, USA, 2015; p. 46. 30. Karimirad, M.; Moan, T. Extreme Dynamic Structural Response Analysis of Catenary Moored Spar Wind Turbine in Harsh Environmental Conditions. J. Offshore Mech. Arct. Eng. 2011,133, 3393. [CrossRef] 31. Naess, A.; Moan, T. Stochastic Dynamics of Marine Structures; Cambridge University Press: Cambridge, UK, 2012; ISBN 978-0-52188155-5. 32. Hegseth, J.M.; Bachynski, E.E. A Semi-Analytical Frequency Domain Model for Efficient Design Evaluation of Spar Floating Wind Turbines. Mar. Struct. 2019,64, 186–210. [CrossRef] 33. BModes. Available online: https://www.nrel.gov/wind/nwtc/bmodes.html (accessed on 2 December 2021). 34. Karimirad, M. Offshore Energy Structures; Springer International Publishing: Cham, Switzerland, 2014; ISBN 978-3-319-12174-1. 35. Det Norske Veritas. DNV-OS-J101: Design of Offshore Wind Structures; Det Norske Veritas: Høvik, Norway, 2014; p. 238. 36. Fossen, T.I. Handbook of Marine Craft Hydrodynamics and Motion Control; John Wiley & Sons: Hoboken, NJ, USA, 2011; p. 582. 37. Ansys|Engineering Simulation Software. Available online: https://www.ansys.com/ (accessed on 15 February 2022). 38. Wamit, Inc. The State of the Art in Wave Interaction Analysis. Available online: https://www.wamit.com/ (accessed on 15 February 2022). 39. Journeé, J.M.J.; Massie, W.W. Offshore Hydromechanics; Delft University of Technology: Delft, The Netherlands, 2001. 40. Touzon, I.; Nava, V.; Gao, Z.; Mendikoa, I.; Petuya, V. Small Scale Experimental Validation of a Numerical Model of the HarshLab2.0 Floating Platform Coupled with a Non-Linear Lumped Mass Catenary Mooring System. Ocean Eng. 2020 ,200, 107036. [CrossRef] 41. Touzon, I.; Nava, V.; de Miguel, B.; Petuya, V. A Comparison of Numerical Approaches for the Design of Mooring Systems for Wave Energy Converters. JMSE 2020,8, 523. [CrossRef] 42. DNV GL. DNVGL-RP-C205: Environmental Conditions and Environmental Loads; DNV GL: Høvik, Norway, 2017; p. 259. 43. Astrup, P.; Larsen, S.E. WAsP Engineering. Flow Model for Wind over Land and Sea. ETDEWEB 1999,27, 700466. 44. Jonkman, J.; Musial, W. Offshore Code Comparison Collaboration (OC3) for IEA Task 23 Offshore Wind Technology and Deployment. Renew. Energy 2010,74, 112.