An Open-Source Comprehensive Numerical Model

An Open-Source Comprehensive Numerical Model for Dynamic Response and Loads Analysis of Floating Offshore Wind Turbines M. Baroonia,1,∗, N. Ale Alia,2 , T. Ashurib,2 a

Department of Ocean Engineering, Khorramshahr Marine Science and Technology University, Khoramshahr, Iran b Department of Mechanical Engineering, Arkansas Tech University, Russellville, AR 72801, USA

Abstract This paper presents the development of a comprehensive open-source numerical model to study the dynamic response and load analysis of floating offshore wind turbines. The model accounts for the wind inflow, rotor aerodynamics, multibody structural model of the system, wave and current kinematics, hydrodynamics, and mooring-line dynamics. This coupled simulation tool can be used for analysis, optimization and preliminary design to determine the technical and economic feasibility. Several verification and validation cases are performed to show the correctness of the numerical simulations. The results show that the proposed approach provides an accurate estimate of the wind turbine dynamics and loads. The simulation tool is then applied in the analysis of a 5 MW wind turbine aimed to characterize the dynamic response and to identify potential loads and instabilities resulting from the dynamic couplings between the turbine and the external conditions. This opensource fully coupled aero-hydro-elastic model provides a modular framework to enable investigating a variety of wind turbine configurations, support systems, and mooring lines. Therefore, it is expected that researchers and design engineers worldwide use the model to study, investigate and analyze different aspects of floating offshore wind turbine design, which results is the promotion and advancement of science and technology for floating offshore wind turbines. Keywords: Floating offshore wind turbine, Dynamic response analysis, Load analysis, Wind turbine design, Numerical model.

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1. Introduction Global warming and climate change are among the grand challenges of the new millennium [1–3]. Renewable energies have the potential to address these challenges by providing a clean resource of sustainable energy [4, 5]. Among all sources of renewable energy, off∗

Corresponding author Email address: [email protected] (M. Barooni) 1 Graduate Student 2 Assistant Professor

Preprint submitted to Energy

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shore wind is promising, since it allows bulk generation of electricity [6]. However, the cost of offshore wind generated electricity is still on average higher than that of fossil fuels. This encourages the development of science and technology to reduce the costs by developing larger offshore wind turbines [7–10], novel support structures [11–13], advanced control algorithms [14, 15], and new design techniques [16–18]. A floating offshore wind turbine (FOWT) is among the concepts that has the potential to be effective and economic in extracting energy from the vast offshore wind resources in deep waters [19, 20]. A FOWT is a turbine mounted on a floating structure that allows the generation of electricity in deep water depths where fixed bottom-mounted towers are not economically feasible. However, FOWTs are one of the most challenging ocean structures to design, since they require a multidisciplinary approach to analyze different coupled disciplines such as aerodynamics, hydrodynamics, structural dynamics and controls [21, 22]. Therefore, FOWTs are still in their infancy stage of technology readiness, and continuous research and development is needed to better understand the system’s dynamics and propose cost effective designs [23, 24]. In the recent years, several researchers investigated different aspects of modeling FOWTs to better understand their complex dynamic behavior. Jonkman [25] presented a computational code to model the aerodynamics of the rotor, the structural motion, the hydrodynamics of the wave, and a blade-pitch and generator controller for time-domain simulation. Sclavounos et al. [26] developed two low-weight, motion resistant stiff floating wind turbine concepts for deployment in water depths ranging from 30 to several hundred meters. They obtained linear and nonlinear wave loads on the floater using uncoupled computational methods developed for the design of oil and gas offshore platforms. Karimirad and Moan [27] developed a stochastic dynamic model of a tension leg spartype wind turbine subjected to wind and wave action. They implemented the model using the HAWC2 aero-hydro-elastic code and analyzed the dynamic motion of the structure, power production and tension leg responses. Skaare et al. [28] performed an analysis between full-scale measurements from the floating wind turbine Hywind demo, and corresponding numerical simulation. Wang et al. [29] presented a stochastic dynamic response analysis of a 5 MW Floating vertical-axis wind turbine (FVAWT) based on fully coupled nonlinear time domain simulations. The turbine has a Darrieus rotor, and a semisubmersible floater subjected to various wind and wave loads. Vaal et al. [30] investigated the effect of a periodic surge motion on the integrated loads and induced velocity on a wind turbine rotor. Through the analysis of the integrated rotor loads, induced velocities and aerodynamic damping, it is concluded that typical surge motions are sufficiently slow to affect the wake dynamics predicted by engineering models. Vorpahl et al. [31] presented a benchmark study on aero-servo-hydro-elastic codes for offshore wind turbine dynamic simulation. The verified codes account for the coupled dynamic systems including the wind inflow, aerodynamics, elasticity and controls of the turbine, along with the incident waves, sea current, hydrodynamics and foundation dynamics of the support structure. As the literature shows, limited efforts have been made to develop a comprehensive open-source numerical model to be shared publicly for investigating different aspects of the design and analysis of FOWTs. Therefore, we still lack numerical models, and their 2

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corresponding mathematical formulation [32–34]. To address this shortcoming, this paper presents the development of the governing equations of a fully coupled nonlinear FOWT, and it presents a public open-source computational code for analysis, optimization and preliminary design. This new simulation code is named BA-Simula, and it is available to download from https://github.com/tashuri. In BA-Simula, the dynamic response due to external and inertial loads is obtained using a fully coupled numerical model implemented in MATLAB® . Blade element momentum (BEM) theory is used to determine aerodynamic loads on the rotor [35]. Panel method and Morison’s equation are used to calculate the hydrodynamic loads considering the instantaneous position of the wind turbine system [36]. Mooring loads are calculated by a quasi-static equilibrium at different time steps [37]. The fully coupled equations of motion are solved using Runge-Kutta method [38]. The accuracy of the code is tested using several model-to-model comparisons, and validation with experimental data as found in the literature. This gives the confidence to perform more thorough analyses, and it provides the users the necessary insight needed to analyze the dynamic behavior and design of FOWTs. This contributes to advancement of science and technology for FOWTs, which in turn reduces the costs. The remainder of the paper is organized as follows. First, the methodology and the mathematical formulation to model the fully coupled nonlinear FOWT is presented. Second, the verification and validation of the code is illustrated. Third, numerical simulations are performed to investigate the influence of several input conditions on a spar type FOWT dynamic response and loads. Finally, conclusions and future works are drawn and presented.

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2. Methodology

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This section presents the development of a comprehensive numerical model to study the dynamics and load analysis of FOWTs. As figure 1 shows, there are several support structure configurations used for FOWTs. A catenary moored spar type is the commonly used configuration that consists of a single floating cylindrical spar-buoy moored by catenary cables. This research uses the 5 MW National Renewable Energy Laboratory (NREL) wind turbine that has a catenary moored spar type support structure. As figure 2 shows the turbine has a draft of 120 m at a water depth of 320 m [39]. Table 1 presents the main properties of the 5 MW NREL FOWT. To describe aerodynamic, hydrodynamic and structural loads, four different coordinate systems are used as presented in figure 2. The first coordinate system is global, and it is fixed at sea bed. The three other coordinate systems are local. One is located at the center of gravity (CG), the other at hub height, and another in the blade root.

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2.1. Aerodynamic loads

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Blade aerodynamic loads are calculated using BEM theory [35]. BEM combines momentum theory, and blade element theory to compute the loads on the blade iterativly. First, the momentum balance on a rotating annular stream tube passing through a turbine is computed. Second, lift and drag forces on spanwise cross sections of the blade are computed. These two separate theories run iterativly until the desired convergence on

3

Figure 1: Different FOWT support structure used as the floater in large water depths [40].

Table 1: Overall system properties of the 5 MW NREL wind turbine [39]

Property Value Rotor configuration (-) 3 blades, upwind Rotor and hub diameter (m) 126, 3 Hub height (m) 90 Cut-in, rated, and cut-out wind speed (m⁄s) 3, 11.4, 25 Cut-in, and rated rotor speed (rpm) 6.9, 12.1 Draft (m) 120 Diameter above taper (m) 6.5 Diameter below taper (m) 9.4 Total mass (kg) 8.06E8 Overall center of gravity (along the centerline of the 78.0 platform) (m) Pitch inertia about the center of gravity (kg m2 ) 6.80E10 2 Yaw inertia about the centerline (kg m ) 1.92E8

4

(a) Global, Center of gravity, and hub height coordinate system and their locations with respect to wind turbine.

(b) Blade coordinate system mounted at the blade root. Figure 2: Configuration of the 5 MW NREL floating wind turbine, and the used coordinate systems.

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the aerodynamic loads is achieved [41]. In BEM theory, the perpendicular force (Fp ), and tangential force (Ft ) are expressed as a function of the drag, D, and lift, L, forces [42]. That is: Fp = Nb (L cos φ + D sin φ) = 1/2ρair W 2 Nb c(Cl cos φ + Cd sin φ)∆r

(1)

Ft = Nb (L sin φ − D cos φ) = 1/2ρair W 2 Nb c(Cl sin φ − Cd cos φ)∆r

(2)

Here, ρair is air density, W is resultant wind velocity, Nb is number of blades, c is chord length, Cl is lift coefficient, Cd is drag coefficient, φ is inflow angle, and ∆r is the length of blade element. As shown in figure 3, the resultant wind velocity W is the vector sum of the axial (Vax ), and tangential (Vt ) wind velocities on the blade element, and it can be expressed as:

Chord line L D Plane of rotation

W

Figure 3: Velocity and force components at the rotor plane.

q 2 +V2 W = Vax t 99 100 101 102 103

(3)

In this figure, α is the angle of attack, and γ1 and γ2 are the blade twist and pitch angle, respectively. Due to the presence of the rotor, the axial and tangential wind velocities are not identical to free stream and rotational speed. Therefore, axial (a), and tangential (a0 ) induction factors are used to represent the fractional change in wind velocity due to rotor presence as: Vax = V∞ (1 − a)

(4)

Vt = Ωr(1 + a0 )

(5)

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Where V∞ is free stream velocity, Ω is angular velocity of the rotor, and r is blade element radius. Knowing the induced wind velocity at each blade element in time, and taking into account velocity of blade elements induced by blade motion, the loads on each blade element can be calculated as:
L = 1/2ρair c Cl (Vax − x˙ p )2 + (Vt − x˙ t )2 ∆r

(6)


D = 1/2ρair c Cd (Vax − x˙ p )2 + (Vt − x˙ t )2 ∆r

(7)

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Where x˙ p and x˙ t are velocities of blade element in perpendicular and tangential directions to rotor plane, respectively. In the above formulation, the free stream velocity is computed at the hub-height of the wind turbine, and it can be either steady or turbulent [43, 44]. For the simulation of the turbulent wind, the von Karman spectrum is used [45].

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2.2. Hydrostatic and restoring loads

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Hydrostatic loads are found using Archimedes’ principle [46]. Using this principle, the total weight of the wind turbine, platform, and mooring cables in water is equal to the weight of the displaced water volume. Buoyancy force (Fb ) is calculated at the center of buoyancy (CB). The general form of buoyancy force equation can be expressed as: Fb = ρsea g (V0 − Asurf uZ + Asurf ηsurf )

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Here, ρsea is water density, g is the gravitational acceleration, V0 is displaced water volume, Asurf is spar area at water surface, uZ is translation in Z-direction in global coordinate system, and ηsurf is wave elevation. The restoring moment arm of the buoyancy force is given by: |CB − CG| =

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(8)

(V1 r1 + V2 r2 + V3 r3 ) (V1 + V2 + V3 )

(9)

Where, V1 , V2 and V3 are displaced water volumes by spar upper, middle and lower sections, respectively, and r1 , r2 and r3 are the corresponding vertical distances between each volumetric section CG and the entire system CG. These are shown in figure 4.

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V1

6.5

MWL

V2

108

V3

Uz

r3 r2 r1

CG

9.4

Figure 4: Spar platform diagram to compute the buoyancy force.

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2.3. Hydrodynamic loads Hydrodynamic loads are calculated by combining Morison equation and strip panel method. Morison equation is widely used in the analysis of fixed bottom offshore structures such as spars [36]. In strip panel method, the structure is split into a number of elements [47], and Morison equation is used to compute linear wave, and nonlinear viscous drag loads on each element [39]. In this paper, Morison equation is modified to take into account the moving element due to the motion of the entire structure [36]. That is: 1 (10) fhydro = ρsea Cd ds (u + c − q˙s ) |u + c − q˙s |dz + ρsea As (cm η¨ − ca q¨s ) dz 2 Here, ds is the diameter of spar, u and c are wave particle and current velocities, q˙s and q¨s are the body velocity and acceleration, As is cross-sectional area of spar, η¨ is the local acceleration of wave particles, and dz is the unit length of spar section. Cd , Cm and Ca are viscous-drag, inertia and added mass coefficients that are found empirically [48]. These parameters are functions of Reynolds number, Kaulegan-Carpenter number, relative current number, and surface roughness ratio [49]. Wave particle kinematics is obtained for each time step based on the Airy linear wave theory [50], and used for calculating the hydrodynamic forces for each section considering the instantaneous position of each strip. Due to deep draft of the spar, an alternative approach for calculating hydrodynamic forces on the bottom section of the spar is used, which consist of Morison inertia term and a fluid damping term [51] as: Z Fhydro = −Bm z¨s − Cvd |z˙s |z˙s

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(11)

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Where, Bm is the vertical added mass that is assumed to be equal to half of a sphere of water below the spar [36], and z˙s and z¨s are vertical velocity and acceleration components of spar, respectively, and Cvd is fluid damping.

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2.4. Mooring loads

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The mooring lines are modeled using a quasi-static method that finds the tension of the cables by assuming its static equilibrium in each time step. The model accounts for the apparent weight in fluid, elastic stretching and seabed friction of each line, but it neglects the individual line bending stiffness, and the inertia and damping of the mooring system. However, the nonlinear geometric restoration of the whole mooring system is considered [25]. Knowing the fairlead position at any instant of time, the following two nonlinear equations are solved using Newton-Raphson technique [52]. As figure 5 presents, the horizontal and vertical components of the effective tension in the mooring line denoted by HF and VF are computed as: 

s



HF  VF VF + ln + 1+ ω ω HF     CB ω VF 2 VF HF + −(L − ) + L− − max L − 2EA ω ω CB w xF (HF , VF ) = L −

s

zF (HF , VF ) =

HF  1+ ω



VF HF

s

2 −

1+ 1 + EA

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2



 + HF L EA  VF HF − ,0 ω CB ω VF HF

VF − wL HF

2

W L2 VF L − 2

(12)

   (13)

Where, L is the total unstretched length, ω is the apparent weight in fluid per unit length, EA is extensional stiffness, CB is coefficient of seabed static-friction drag, and xF and zF represent the fairlead location relative to the anchor. As illustrated in figure 6, the mooring reaction forces in global coordinate system are found by: X Fmooring = HF 1 cos (βline1 ) + HF 2 cos (βline2 ) + HF 3 cos (βline3 )

(14)

Y Fmooring = HF 1 sin (βline1 ) + HF 2 sin (βline2 ) + HF 3 sin (βline3 )

(15)

Z = VF 1 + VF 2 + VF 3 Fmooring

(16)

Here, βline is the mooring line angle as shown in figure 6.

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L, ,EA,

Anchor

Figure 5: Mooring line in a local coordinate system.

Displaced position of spar Y

X

Anchor Initial position of spar

Figure 6: Mooring line positions and net forces.

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2.5. Coordinate systems As mentioned earlier, four different coordinate systems are used to describe the equations of motion. One global coordinate system that is fixed at the sea bed and defined by ˆ J, ˆK ˆ unit vectors, and three local coordinate systems. The first local coordinate system I, ˆ The second local coordinate system is located at the system’s CG with unit vectors ˆi, ˆj, k. ˆ is positioned at the hub center with pˆ, qˆ and l unit vectors. Unit vector pˆ is aligned with the main shaft of the turbine, and qˆ and ˆl form a perpendicular plane to it. The fourth coordinate system is defined at the blade root with unit vectors b1 , b2 and b3 . Transformation between different coordinate systems is accomplished with a transformation matrix. As an example, the transformation between the CG coordinate, and the global coordinate system is given by:  ˆ    ˆi I  Jˆ  = R  ˆj  (17) ˆ ˆ k K With: ˆ  ˆ ˆi) cos(I, ˆ ˆj) cos(I, ˆ k) cos(I, ˆ  ˆ ˆi) cos(J, ˆ ˆj) cos(J, ˆ k) R =  cos(J, ˆ ˆ ˆi) cos(K, ˆ ˆj) cos(K, ˆ k) cos(K, 

(18)

179

A similar matrix is used to make the transformation across different coordinates. As an example, a chain of coordinate system transformation is used to first compute the aerodynamic loads in the blade coordinate system, and then transform it to the hub, and then CG coordinate system.

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2.6. Kinematics of FOWT

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The rotational velocity, ω, of the FOWT can be determined with respect to CG as: ω = [θ˙ cos φ cos ψ + φ˙ sin ψ]ˆi − [θ˙ cos φ sin ψ + φ˙ cos ψ]ˆj + [ψ˙ + θ˙ sin φ]kˆ

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Here, φ, θ and ψ are relative angles between the unit vectors of the global coordinate system and CG, and the dot overhead shows the derivative with respect to time. To account for relative velocities experienced by the rotor due to the motion of the structure, the system’s rotational velocity computed at CG is transferred to the blade as: ω = ωx b1 + (ωy cos Ωt + ωz sin Ωt) b2 + (−ωy sin Ωt + ωz cos Ωt) b3

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(20)

Here, Ω is the rotational velocity of the rotor, and t is time. Thus, the blade’s rotational velocity vector can be expressed as: ω ~ b = Ωb1 + ω ~

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(19)

(21)

Therefore, the components of ω ~ b are: ωb1 = ωx + Ω 11

(22)

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ωb2 = ωy cos Ωt + ωz sin Ωt

(23)

ωb3 = −ωy sin Ωt + ωz cos Ωt

(24)

Similarly, the acceleration vector of the FOWT with respect to the CG coordinate system is: α = αxˆi + αy ˆj + αz kˆ In this formulation, the components of α are given by:

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αx = θ¨ cos φ cos ψ + φ¨ sin ψ − θ˙ψ˙ sin φ cos ψ − θ˙ψ˙ cos φ sin ψ + φ˙ ψ˙ cos ψ

(26)

αy = −θ¨ cos φ sin ψ + φ¨ cos ψ + θ˙ψ˙ sin φ sin ψ − θ˙ψ˙ cos φ cos ψ − φ˙ ψ˙ sin ψ

(27)

αz = ψ¨ + θ¨ sin φ + θ˙φ˙ cos φ

(28)

This allows the computation of the relative rotational acceleration vector of the blade

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as: αb = ω˙ b1 b1 + ω˙ b2 b2 + ω˙ b3 b3 + ω ~ × (Ωb1 )

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The equations of motion for the blade are defined as: M1 = Me − I1 αb1

(30)

M2 = I2 αb2 − (I2 − I1 ) ωb1 ωb3

(31)

M3 = I3 αb3 − (I3 − I1 ) ωb1 ωb2

(32)

Where, M1 , M2 and M3 are moments around each blade axis, I1 , I2 and I3 are the mass moment of inertia around each blade axis, and Me is the blade torque defined as:

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(33)

Here, Ftaero is the tangential aerodynamic force on the blade, and r is the moment arm. The modified Euler equation is used for deriving the dynamic equations of motion for translational modes by [53]: ~ + F~ aero + F~ hydro + F~ mooring M~u¨ = W

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(29)

2.7. Equations of Motion

Me = Ftaero r 198

(25)

(34)

Here, M is the mass matrix of the entire system, ~u¨ is translational acceleration vector and ~ is weight vector. This equation is solved in time to obtain the dynamics of the FOWT W 12

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using the 4th -order Runge-Kutta method. Dynamic equations of motion for rotational modes are given as: Ixx αx + (Izz − Iyy ) ωy ωz = Mx + M1

(35)

Iyy αy − (Izz − Ixx ) ωx ωz = My + M2 cos Ωt − M3 sin Ωt

(36)

Izz αz − (Ixx − Iyy ) ωx ωy = Mz + M2 sin Ωt + M3 cos Ωt

(37)

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Where Ixx , Iyy and Izz are system’s mass moments of inertia with respect to CG, and Mx , My and Mz are the total excitation moments.

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2.8. Block diagram of the code

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As Figure 7 shows, BA-Simula consists of five function blocks. Each block calculates the loads acting on the FOWT at different time steps. The equations of motion function block solves iteratievly the translational and rotational modes for the user defined initial conditions, and inputs. 𝑉∞

Controller

Ω

Mooring 𝐹⃗ 𝑚𝑜𝑜𝑟𝑖𝑛𝑔 ⃗⃗⃗𝑚𝑜𝑜𝑟𝑖𝑛𝑔 𝑀

Ω, 𝛾2 Aerodynamic

⃗⃗⃗𝑎𝑒𝑟𝑜 𝐹⃗ 𝑎𝑒𝑟𝑜 , 𝑀 S

S

Dynamic equation solver (ode45)

S

S 𝐹⃗ℎ𝑦𝑑𝑟𝑜 ⃗⃗⃗ℎ𝑦𝑑𝑟𝑜 𝑀

𝐻𝑤𝑎𝑣𝑒 , 𝑇𝑤𝑎𝑣𝑒

𝑢, 𝜂̈

Hydrodynamic

Wave

S

S, t

𝐹⃗𝑏𝑜𝑢𝑦𝑎𝑛𝑐𝑦 ⃗⃗⃗𝑏𝑜𝑢𝑦𝑎𝑛𝑐𝑦 𝑀 Hydrostatic & 𝜂𝑠𝑢𝑟𝑓

Restoring

Figure 7: BA-Simula code structure. In this figure, Hwave and Twave are wave high and period, respectively, and S is the output list of all translational and rotational modes of the wind turbine.

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Inputs of the code are: wind data, wave and sea data, rotor aerodynamic chord and twist data, rotor polar data, turbine operational data, mass and geometry of the turbine, blade-pitch and generator controller data, mooring and spar data, as well as the initial condition and simulation data. Outputs of the code are: time-series of the loading on the structure at locations such a blade root, tower, and mooring lines, displacements, velocities and accelerations, as well as the plot of important information as presented in the results section of the paper. 13

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This section presents the verification and validation of different function blocks of BASimula. The 5 MW NREL wind turbine is used for this purpose. First, the aerodynamic module of BA-Simula is verified with QBlade [54]. QBlade is an open-source wind turbine code developed by Hermann F¨ottinger Institute of TU Berlin, and distributed under General Public License. Figures 8 to 11 illustrate important parameters in aerodynamic analysis module such as axial and tangential induction factors, and axial and tangential forces on individual blades. These results are for a constant wind velocity of V = 11.4 m/s. As the figures show, a good agreement exists between a model-to-model comparison in BA-Simula and QBlade. Minor differences can be explained by the way QBlade models the airfoil polar data. In QBlade, airfoil polar data are modeled using the XFLR5 code [55]. XFLR5 is integrated into QBlade to generate airfoil lift and drag coefficients for turbine aerodynamic simulations. In BA-Simula, the user has the freedom to use any airfoil polar data as long as they follow the format used in BA-Simula. Therefore, small changes in the airfoil polar data are responsible for the minor differences in these figures. Axial induction factor (-)

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3. Verification and validation

8000

0.6 Axial blade force (N)

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QBlade BA-Simula

0.4

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0

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6000 4000 2000 0

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QBlade BA-Simula

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800 600 400 200 0 -200

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Figure 10: Tangential induction factor. 235

40

Figure 9: Axial force.

Tangential blade force (N)

Tangential induction factor (-)

Figure 8: Axial induction factor.

0

30

Radial position (m)

Figure 11: Tangential force.

Figures 12 and 13 show the rotor thrust and tension in the mooring line. A code-tocode comparison is made with FAST [56]. FAST is an aero-hydro-elastic code developed 14

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Rotor thrust (kN)

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by NREL. The shown distance in figure 13 is the horizontal distance between fairlead and anchor. This good agreement indicates the correct implementation of the aerodynamic and mooring line modules. 1000

15000

800

12500 Tension (kN)

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600 400 200 0

10000

BA-Simula NREL FAST

0

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BA-Simula NREL FAST

7500 5000 2500 0 700

1000

750

Time (s)

Figure 12: Rotor thrust comparison.

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Figure 13: Mooring line tension comparison.

0.03

15 BA-Simula NREL FAST

10 5 0 -5

0.01 0 -0.01

-10 -15

BA-Simula NREL FAST

0.02 Sway (m)

241

850

Figures 14 to 16 compare the platform surge, sway and heave free decay time series in BA-Simula and FAST. For this comparison, all external forces are eliminated, and an identical initial displacement in the surge motion is considered. As the results show, BASimula and FAST exhibit comparable free decay motion.

Surge (m)

240

800

Displacement (m)

0

200

400

600

800

-0.02

1000

Time (s)

0

200

400

600

800

1000

Time (s)

Figure 14: Free decay surge motion time history.

Figure 15: Free decay sway motion time history.

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0.1 BA-Simula NREL FAST

Heave (m)

0 -0.1 -0.2 -0.3

0

200

400

600

800

1000

Time (s)

Figure 16: Free decay heave motion time history.

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Surge (m)

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To investigate the forced dynamic response of BA-Simula model, two load cases are considered and compared with FAST. First, a regular wave with a wave height of H = 1.4 m, and period of T = 10 s is considered. Figures 17 and 18 show the surge and heave motions. The results show in general a good agreement. Small variations in these results relate to the amount of damping used in each code. In BA-Simula, we added a constant damping for all modes. To confirm this, we varied the damping value in BA-Simula to see the sensitivity of the results, and if we can obtain closer results to FAST predictions. We observed that the amount of damping can result in a better match of the dynamic response outputs. However, it is not clear which code uses a more realistic value for damping and further investigation is needed to fully address this issue. Therefore, no conclusion can be made about the accuracy of each code. This shows the importance and the need for conducing experiments to advance this field of science and technology. 20 15 10 5 0 -5 -10 -15 -20

0.4 BA-Simula NREL FAST

BA-Simula NREL FAST

0.2 Heave (m)

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Time (s)

0

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Time (s)

Figure 17: Surge motion time history (only wave). Figure 18: Heave motion time history (only wave). 256 257 258 259 260 261 262

Second, a regular wave with a wave height of H = 1.4 m and period of T = 10 s is considered, while the turbine is in operation at a wind speed of V = 11.4 m/s. The results are shown in figures 19 and 20 for surge and heave modes, respectively. Again, a good agreement is found between BA-Simula and FAST. Here, damping terms are added to BA-Simula to make up for diffraction and radiation that is not represented in Morison equation. These added damping terms can cause instability in some load cases, since they are sensitive to initial conditions. 16

26 BA-Simula NREL FAST Heave (m)

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24 22 20 18 16 14

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Figure 19: Surge motion time history (wind & wave).Figure 20: Heave motion time history (wind & wave).

263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294

BA-Simula is also compared with a code-to-code comparison study present in the literature [31]. The comparison is performed using a regular wave with a wave height of 6 m, a wave period of 10 s, and a constant wind speed of 8 m/s. Figures 21a to 21d show the results for surge, heave and yaw motion, as well as fairlead loads. Variation in the results is related to different damping values, different equation solvers and time steps, and different assumptions and simplifications used in each code. Figure 21a shows the platform surge displacement, and all codes except HAWC2 agree on the amplitude of oscillation. For the heave motion depicted in figure 21b, BA-Simula shows a heave overestimation as the result of having less hydrodynamic restoring forces and damping in heave motion. This high amplitude motion is related to the use of Morison equation for the calculation of hydrodynamic excitations in vertical direction that is not well damped. The yaw oscillation shown in figure 21c agrees reasonably well among all codes and the low amplitude in BA-Simula is the result of replacing mooring delta line with an equivalent spring stiffness that can be adjusted to agree with other codes. However, this parameter is not changed intentionally here to show the impact of making different modeling assumptions. Figure 21d shows the variation in the fairlead tension, and it can be concluded that BA-Simula shows comparable results to other codes. The main reason for having slightly different dynamic response among these codes is the hydrodynamics model. Two different approaches are used to model offshore sparbuoy floating wind turbines. In the first approach, hydrodynamics loads are modeled using Morisons equation that is augmented with hydrostatics and wave excitation heave forces. In the second approach, hydrodynamic loads are modeled using potential-flow theory that is augmented with the nonlinear viscous drag term from Morisons equation. Both hydrodynamic models are valid to predict equivalent hydrodynamic loads on the spar, since in most of the conditions the radiation damping is negligible. For codes such as BA-Simula that employ the first approach, the heave force can be calculated based on the variation in buoyancy force, and using a direct integration of the time-dependent hydrostatic pressure and wave elevation. There are some additional linear hydrodynamic damping based on measurements to be included in the model. Codes that neglect this additional damping exhibit less overall damping in their responses, as is the case for BA-Simula as well [31]. 17

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Another difference among these models is the physical representation of the mooring system. The two main approaches for modelling the mooring system dynamics are the dynamic formulation and the quasi-static formulation. In the dynamic formulation, the mooring system is modeled using time-accurate Finite Element Methods. The quasi-static formulation is time-dependent but is slow enough to ignore the inertial effects. All of the compared codes use the quasi-static formulation of the complete mooring system, and are therefore only valid for small time-increments and small displacements. Depending on the amount of damping used in each code, some load cases may exhibit numerical instability if large time-steps are selected [31]. In general, it is difficult to conclude which one of these codes predict more accurate results. This is due to the fact that the limited existing numerical codes are at the stage of infancy when it comes to modeling harsh environmental conditions. Therefore, they require more revision and future work for validating their underlying model assumptions with experimental data as it is evident from the comparisons presented in this work.

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Figure 21: Time series with regular wave with a wave height of 6 m, a wave period of 10 s, and a constant wind speed of 8 m/s.Platform surge, heave and yaw displacements; and downstream fairlead tension.

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BA-Simula is also validated against the limited publicly available experimental data in the literature. The experimental validation cases are based on a scaled model of the 5 MW NREL spar-type floating wind turbine [57]. In the first validation case, the heave motion time history of the wind turbine is compared with the experimental data as presented in figure 22a. The comparison is performed using an irregular wave with a wave height of 4 m, a wave period of 13 s, and a constant wind speed of 17 m/s. In the second validation case, the pitch motion time history is compared with experimental data as depicted in figure 22b. In this case, an irregular wave with a wave height of 15.3 m, a wave period of 15.5 s, and a constant wind speed of 30 m/s is used. 2

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Figure 22: Validation of BA-Simula against experimental data using a scaled model of the 5 MW NREL wind turbine.

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As the figures show, the results of BA-Simula match well with the experimental data. It should be noted that the stochastic wave kinematics used in BA-Simula are imported from ANSYS AQWA® for this study.

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4. Numerical simulation results

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To show some of the capabilities of BA-Simular, and the type of analysis the user can perform, this section presents few case studies using the 5 MW NREL wind turbine. In the first case study, the turbine is analyzed using a constant wind speed of 11.4 m/s, a single sinusoidal wave height of 1.4 m and a wave period of 6.5 s. The motion trajectory graph illustrated in figure 23 indicates the path of CG in 3D space for 5000 sec simulation time for this case study. As it can be seen, the system oscillates around an equilibrium region, and the closed elliptical orbits indicate that the 3D motion is stable.

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Figure 24 shows the Poincare map. This map shows that the system is chaotic, and the 3D motion of the system is n-periodic. When a system is exhibiting a chaotic response, it means that the behavior of the dynamical system is highly sensitive to initial conditions. In such a situation, each point in a chaotic system can arbitrarily be approximated by other points, with significantly different future paths, or trajectories. Therefore, an arbitrarily small change, or perturbation of the current trajectory may lead to significantly different future behavior [58]. Fourier fast transformation (FFT) method is used to derive the power spectrum of motion modes in frequency domain. In figures 25 and 26, the excitation frequency spectrum, and spectrum of surge motion are represented as an example.

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Figure 24: Poincare map.

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Figure 23: 3D motion trajectory.

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Figure 26: Surge excitation frequency spectrum.

As figure 25 shows, the surge motion response has a maximum amplitude at zero frequency that indicates non-vibrating motion. The surge motion response also is centralized in low frequency zone, which coincides with its natural frequency. Eliminating external excitation forces, and considering initial conditions of 5, 2 and 0.5 m for surge, sway and heave, respectively, and 1% damping for all frequencies, table 2 presents the natural damped frequencies and periods.

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Table 2: Natural damped frequencies and periods.

Motion Surge Sway Heave Pitch Roll Yaw 347 348 349

Frequency (Hz) 0.0069 0.0069 0.0314 0.0224 0.0224 0.1140

Period (s) 144.1871 144.1871 31.8362 44.5702 44.5702 8.7737

In the second case study, we assumed a steady wind of 11.3 m/s, and three wave profiles as presented in in table 3. This is to study the dynamic response of the system for varying wave condition. Table 3: Wave load cases

Load cases C1 C2 C3 350 351 352 353 354

Wave properties H (m) T (s) 1.4 6.5 2.44 8.1 3.66 9.7

Wind velocity (m/s) 11.4 11.4 11.4

Frequency spectrum of roll motion for these load cases is illustrated in figure 27 to investigate in details the effect of hydrodynamic loads on the system response. As it can be seen, different wave load cases do not have a significant impact on system response. For these load cases, the dominant response frequency does not change, and only a slight variation in response amplitude occurs. 12 C1 C2 C3

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Figure 28 shows the effect of rotor gyroscopic moment on turbine power generation. The simulation is performed using a regular wave with a wave height of 1.4 m, a wave 21

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period of 6.5 s, and a constant wind speed of 11.4 m/s. As the figure shows, the gyroscopic moment increases the turbine power out, but this is a negligible amount of 0.1%. This slight increase in power generation of the turbine is due to the small increase in rotor acceleration, and it is not statistically of any significant importance. Therefore, it can be concluded that the gyroscopic motion does not alter the power output, and it is only important to consider for the loading and the motion of the structure.

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Figure 28: Turbine generated power with and without gyroscopic moment.

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5. Conclusion This paper presented the mathematical formulation used to develop an open-source numerical model for the analysis of FOWTs. The model is verified and validated using several benchmark cases with the existing studies in the literature. The results of the fully coupled nonlinear model showed agreement with results in the literature, and it can be concluded that the model gives reasonable results. Several simulation studies were also performed to understand the dynamics of the integrated FOWT. It was observed that aerodynamic loads govern the system response compared to hydrodynamic and mooring loads. This indicates that for the design of FOWT, the rotor design plays a significant role to achieve the desired system’s response. The developed open-source coupled aero-hydro-elastic numerical model of this paper allows researchers worldwide to use the code for the analysis and design of FOWTs. It also enables them to modify and further develop the source-code to advance the field. Since the methodology is characterized by a modular integration of a several function blocks, with little modifications, it is extendable for studying other FOWT support structures as well. However, several challenges still remain when it comes to the analysis and design of FOWTs. For future work, the development of a fully-nonlinear mooring dynamic model is recommended. In addition, improvement of BA-Simula’s hydrodynamic module is required to consider the effect of nonlinear stochastic waves. To investigate different operational conditions of the wind turbine, developing a robust dynamic controller should also be considered. If these challenges are resolved using continuous research and development, FOWT technology can become an economical way of extracting energy from the vast offshore

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resources in deep waters. This would enable generating most of the world’s energy consumption with a clean and sustainable source of energy.

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