A Dynamic Parametric Wind Farm Model for Simulating Time-varying Wind Conditions and Floating Platform Motion
aa r X i v : . [ phy s i c s . f l u - dyn ] O c t A Dynamic Parametric Wind Farm Model for Simulating Time-varyingWind Conditions and Floating Platform Motion
Ali C. Kheirabadi and Ryozo Nagamune The University of British Columbia, Vancouver Campus, 2054-6250 Applied Science Lane,Vancouver, BC Canada V6T 1Z4October 21, 2020
Abstract
This paper introduces a dynamic parametric wind farm model that is capable of simulating floating windturbine platform motion coupled with wake transport under time-varying wind conditions. The simulator is namedFOWFSim-Dyn as it is a dynamic extension of the previously developed steady-state
Floating Offshore Wind FarmSimulator (FOWFSim). One-dimensional momentum conservation is used to model dynamic propagation of wakecenterline locations and average velocities, while momentum recovery is approximated with the assumption ofa constant temporal wake expansion rate. Platform dynamics are captured by treating a floating offshore windfarm as a distribution of particles that are subject to aerodynamic, hydrodynamic, and mooring line forces. Thefinite difference method is used to discretize the momentum conservation equations to yield a nonlinear state-space model. Simulated data are validated against steady-state experimental wind tunnel results obtained fromthe literature. Predictions of wake centerlines differed from experimental results by at most 8 .
19 % of the rotordiameter. Simulated wake velocity profiles in the far-wake region differed from experimental measurements byless than 3 .
87 % of the free stream wind speed. FOWFSim-Dyn thus possesses a satisfactory level of fidelityfor engineering applications. Finally, dynamic simulations are conducted to ensure that time-varying predictionsmatch physical expectations and intuition.
Since the introduction of parametric wake models by Jensen [1] and Kati´c et al. [2], such wind farm simulatorshave served as essential tools for enhancing wind farm performance. This enhancement has been achieved via twodistinct fields of study. The older of the two is layout optimization , wherein the optimal installation locations of windturbines are computed with the objective of maximizing annual revenue [3]. Since such optimization problems aresolved offline prior to wind farm construction, steady-state wake models have sufficed for estimating annual energyproduction. The field of study that has more recently experienced a surge in interest is wind farm control , whichinvolves real-time wind turbine actuation for the purpose of manipulating the wind field to achieve some wind farm-level objective [4]. This ultimate goal may be efficiency maximization, or power output tracking with turbine loadalleviation [5]. In either case, since actuators are adjusted in real-time, dynamic wake phenomena such as turbulence,transport delay, time-varying mean wind speed and direction, and floating platform motion (for deep-water offshorewind farms) are pertinent when evaluating controller performance.Steady parametric wake models have been used successfully to raise wind farm efficiency in large eddy simulations(LES) [6] and field tests [7]. Further, in one instance, Gebraad et al. [8] reported no significant performance gainswhen using a dynamic wake model for wind farm control in contrast to using a steady wake model. Nonetheless,there are benefits associated with using dynamic wake models. First, traditional state and parameter estimationtechniques may be used to adapt such models to time-varying wind conditions [8]. Second, low-fidelity dynamicwake models may be used to test controller robustness against time-varying wind conditions prior to dedicatingtime and resources to conducting high-fidelity simulations and field tests (as performed by Johnson and Fritsch [9]and Gebraad and van Wingerden [10]). Further research comparing wind farm control based on dynamic versussteady parametric models may reveal additional benefits. Such progress will only be possible, however, provided theavailability of various dynamic wake models.Reviews of wake modeling may be found in the works of Boersma et al. [11], G¨o¸cmen et al. [12], Vermeer et al. [13],and in our previous review article [4]. We will focus our current discussion on parametric dynamic wake models. The1arliest of such models found application in power de-rating wind farm control research conducted by Gebraad andvan Windgerden [10], Johnson and Fritsch [9], and Ahmad et al. [14]. Power de-rating involves reducing the thrustforce exerted onto the wind by upstream wind turbines as a means of increasing the fluid momentum available todownstream machines [15]. Since this application involves neither wake deflection nor wind turbine relocation, wakedynamics in these studies were modeled using time-delays in computed steady-state incident wind speeds. Thesetime-delays represented the duration required for changes in the wind field at some upstream turbine to propagateto downstream machines. This approach is valid as long as the wind direction remains constant, and wake centerlinedeflection and turbine relocation are not pertinent.In order to account for transport delay of steered wakes, Gebraad and van Wingerden [16] developed the FlowRedirection and Induction Dynamics (FLORIDyn) model. Their approach involved representing flow within a wakeusing translating points that were initialized at each turbine and then transported downstream. Each point containedinformation regarding its corresponding turbine’s operating parameters at the instant in time at which the pointwas initialized. Using this information, wake properties at the downstream location of the translating point wereobtained using the Flow Redirection and Induction in Steady-state (FLORIS) wake model [6], which utilizes integralforms of mass and momentum conservation to compute downstream wake properties. More simply put, FLORIDyntransports steady-state wake characteristics computed with FLORIS in the free stream wind direction. Time varyingwind direction and floating platform motion are not considered in FLORIDyn however.In an alternative approach, Shapiro et al. [17] used the differential forms of mass and momentum conservationto simulate dynamic wake behavior. Local and convective wake accelerations in the free stream wind direction weredescribed by material derivatives, and these accelerations were equated to force terms representing turbulent mixingand rotor thrust. The advantage of this modeling approach was that wake transport would be inherently capturedby convective acceleration terms, thus eliminating the need for the translating points employed by the FLORIDynmodel. Instead, all wake characteristics were functions of a fixed grid in the downstream direction. Shapiro et al. [18]later extended their model to capture wake redirection resulting from rotor yaw misalignment. Prandtl’s liftingline theory was used to compute transverse wake velocities, shed circulation, and vortex properties immediatelydownstream of yawed rotors. Wake centerline deflection in the free stream wind direction was then computed byequating the material derivative of the centerline position to the transverse component of the wake velocity. Theseworks do not capture floating platform motion or time-varying wind speed and direction.Finally, Boersma et al. [19] developed the
Wind Farm Simulator (WFSim), which is a control-oriented dynamicwake model based on the two-dimensional form of the unsteady turbulent Navier-Stokes equations. The major benefitof WFSim is that individual wake expansion and the interaction of multiple wakes are inherently captured by themixing length turbulence model employed. In the previously discussed models, the rate of linear wake expansion waseither estimated or assumed. Further, the previous models simulated flow behavior in regions with overlapping wakesby assuming that the effective kinetic energy deficit in the wind field is equal to the sum of deficits corresponding toall pertinent wakes. Despite its higher-fidelity, WFSim requires approximately 1000 sec of computation for 1000 sec ofsimulation in comparison to previously discussed models (several seconds of computation for 1000 sec of simulation).Moreover, WFSim does not model floating platform motion or time-varying wind speed and direction.In the current paper, we loosely follow the approach of Shapiro et al. [17], whereby partial differential equationsare used to capture wake transport, and we develop a dynamic parametric wake model capable of simulating time-varying wind speed and direction, along with platform motion for floating offshore wind farms. The novelty ofthis paper therefore includes the following: (i) additional terms in the wake momentum conservation equations tocapture time-varying free stream wind velocity effects; and (ii) a coupled dynamic model that captures planar floatingwind turbine motion in the presence of aerodynamic interaction. Our approach is physics-based with the rate ofwake expansion as the only parametric assumption. This model serves as a dynamic extension of our previouslydeveloped steady-state tool [20], which was named the
Floating Offshore Wind Farm Simulator (FOWFSim), andwill henceforth be referred to as FOWFSim-Dyn. Fixed-foundation wind farms may also be modeled by simplydeactivating turbine platform motion.The remainder of this paper is organized as follows: Section 2 provides a detailed mathematical description ofFOWFSim-Dyn along with a discussion of its limitations. In Section 3, we perform a mesh convergence study andvalidate FOWFSim-Dyn using steady-state experimental results reported by Bastankhah and Port´e Agel [21]. Wealso present dynamic simulation results that demonstrate the various capabilities of FOWFSim-Dyn. We finallyconclude the paper in Section 4 by listing potential research directions for enhancing FOWFSim-Dyn.2igure 1: Schematic of a general floating offshore wind farm with semi-submersible platforms used as a basis forFOWFSim-Dyn’s mathematical model.
This section details the mathematical formulation behind FOWFSim-Dyn. First, the problem setup, solver blockdiagram, and resulting equations of motion are presented in Sections 2.1–2.7. Finally, important assumptions andlimitations pertaining to FOWFSim-Dyn are discussed in Section 2.8.
Figure 1 shows a top view schematic of the general floating offshore wind farm that we model in the current work.Floating wind turbines are treated as a system of particles that are distributed along the two-dimensional oceansurface. Throughout this paper, we consider only three-cylinder semi-submersible floating platforms as per thebaseline design presented by Robertson et al. [22]. Each floating structure is therefore connected to three anchorsvia mooring lines for the purpose of station-keeping.We define the set F = { , , · · · , N } to denote the indices of the N floating wind turbines within the wind farm,and we refer to each individual turbine using the identifier i . We then number the wind turbines in ascending orderbased on their downstream location. That is to say, the most upstream turbine is numbered by i = 1, while the mostdownstream machine is identified by i = N .The fixed global frame of reference is identified by the ˆ x and ˆ y axes. Each wind turbine also possesses a localnon-inertial translating (though not rotating) reference frame that is attached to its center of gravity. We identifythe reference frame that is fixed to turbine i as frame i . Further, the axes of frame i are referred to as ˆ x i and ˆ y i .We assume that a predominant wind direction exists, and that it is aligned with the positive ˆ x axis. The freestream wind velocity is then denoted by the vector V ∞ ( t ), which contains ˆ x and ˆ y components U ∞ ( t ) and V ∞ ( t ) asfollows: V ∞ ( t ) := (cid:2) U ∞ ( t ) V ∞ ( t ) (cid:3) T . (1) U ∞ ( t ) therefore represents the free stream wind speed in the predominant wind direction, while V ∞ ( t ) accounts forfluctuations in the transverse free stream wind speed. Ultimately, FOWFSim-Dyn takes the following nonlinear state-space form:˙ x farm ( t ) := f ( x farm ( t ) , u farm ( t ) , V ∞ ( t )) , (2)where the wind farm state vector x farm ( t ) combines the floating wind turbine state vector x ( t ) with the wake statevector x w ( t ) as follows: x farm ( t ) := (cid:2) x T ( t ) x Tw ( t ) (cid:3) T . (3)3igure 2: Block diagram showing the computation modules of FOWFSim-Dyn along with information transfer routes.The wind turbine state vector x ( t ) comprises the position and velocity vectors of all floating wind turbines withinthe wind farm as follows: x ( t ) := (cid:2) r T1 ( t ) r T2 ( t ) · · · r T N ( t ) v T1 ( t ) v T2 ( t ) · · · v T N ( t ) (cid:3) T , (4)where r i ( t ) and v i ( t ) are vectors containing ˆ x and ˆ y components of the position and velocity of turbine i as follows: r i ( t ) := (cid:2) x i ( t ) y i ( t ) (cid:3) T , (5) v i ( t ) := (cid:2) v x,i ( t ) v y,i ( t ) (cid:3) T . (6)The wake state vector x w ( t ) contains the states of the wakes generated by the N floating wind turbines as follows: x w ( t ) := (cid:2) x Tw , ( t ) x Tw , ( t ) · · · x Tw ,N ( t ) (cid:3) T . (7)Assuming that the states of wake i are defined at N p ,i discrete points along the downstream direction, x w ,i ( t )comprises the states of wake i at each of these discrete points as follows: x w ,i ( t ) := (cid:2) x Tw ,i, ( t ) x Tw ,i, ( t ) · · · x Tw ,i,N p ,i ( t ) (cid:3) T . (8)The state vector x w ,i,p ( t ) at each point p along wake i then consists of the wake centerline location y w ,i,p ( t ), wakevelocity components u w ,i,p ( t ) and v w ,i,p ( t ), which correspond to the ˆ x i and ˆ y i directions, and the wake diameter D w ,i,p ( t ) as follows: x w ,i,p ( t ) := (cid:2) y w ,i,p ( t ) u w ,i,p ( t ) v w ,i,p ( t ) D w ,i,p ( t ) (cid:3) T . (9)These wake characteristics are portrayed in Fig. 6 and discussed in Section 2.6.The wind farm input vector u farm ( t ) contains the input vectors for the N floating wind turbines as follows: u farm ( t ) := (cid:2) u T1 ( t ) u T2 ( t ) · · · u T N ( t ) (cid:3) T , (10)where u i ( t ) consists of the axial induction factor a i ( t ) and yaw angle γ i ( t ) of turbine i as follows: u i ( t ) := (cid:2) a i ( t ) γ i ( t ) (cid:3) T , (11)with all yaw angles defined as positive counter-clockwise from the ˆ x axis. The block diagram for FOWFSim-Dyn is shown in Fig. 2. The simulator consists of two main modules. The aerodynamics module requires the states x ( t ) and inputs u ( t ) of all turbines, along with the free stream wind velocityand acceleration vectors V ∞ ( t ) and ˙ V ∞ ( t ) at time t . Its function is to compute the effective wind velocity vector V i ( t ) that is incident on the rotor of turbine i for all i ∈ F .The Floating turbine dynamics module uses these incident wind velocity vectors, along with turbine states andinputs, to compute the rates of change of turbine states ˙ x i ( t ) at time t . Using a standard ordinary differentialequation solver, state derivatives are integrated to compute state trajectories over time. This module also computesthe power outputs of individual wind turbines as well as that of the entire wind farm P farm ( t ).4igure 3: Schematic of floating platform velocity vector v i ( t ), incident wind velocity vector V i ( t ), and the relativeincident velocity vector V rel ,i ( t ) at the location of turbine i . The total power output of the wind farm P farm ( t ) is computed as the sum of electricity production from all windturbines as follows: P farm ( t ) = X i ∈ F P i ( t ) , (12)where P i ( t ) is the power output of turbine i , and is estimated assuming steady-state performance as follows [23]: P i ( t ) = 18 C p ,i ( t ) ρ a πD i k V rel ,i ( t ) k . (13) D i is rotor diameter of turbine i , ρ a is the density of air, and V rel ,i ( t ) is the wind velocity that is incident uponthe rotor of turbine i from the perspective of an observer who is fixed to turbine i . Referring to Fig. 3, V rel ,i ( t ) isdefined as follows: V rel ,i ( t ) = V i ( t ) − v i ( t ) , (14)where v i ( t ) is the velocity vector of turbine i , and V i ( t ) is the wind velocity vector (in the global frame) that isincident upon the rotor of turbine i with the following ˆ x and ˆ y components: V i ( t ) := (cid:2) U i ( t ) V i ( t ) (cid:3) T . (15) V i ( t ) is calculated using the wake interaction model discussed in Section 2.7.The power coefficient C p ,i ( t ) of turbine i is computed based on the vortex cylinder model of a yawed actuatordisc as follows [24]: C p ,i ( t ) = 4 a i ( t ) (cos γ rel ,i ( t ) − a i ( t )) cos γ rel ,i ( t ) + tan χ i ( t )2 sin γ rel ,i ( t ) − a i ( t ) sec χ i ( t )2 ! , (16)where a i ( t ) is the axial induction factor of turbine i and, as per Fig. 3, γ rel ,i ( t ) is the yaw misalignment of turbine i relative to V rel ,i ( t ) as follows: γ rel ,i ( t ) = γ i ( t ) − θ i ( t ) . (17)In the above expression, γ i ( t ) is the yaw angle of turbine i and θ i ( t ) is the angle of V rel ,i ( t ) relative to the positiveˆ x axis as follows: θ i ( t ) = tan − V i ( t ) − v y,i ( t ) U i ( t ) − v x,i ( t ) . (18)Finally, χ i ( t ) is the wake skew angle immediately past the rotor of turbine i and is approximated as follows [24]: χ i ( t ) = (0 . a i ( t ) + 1) γ rel ,i ( t ) . (19)5igure 4: Schematic of aerodynamic thrust force F a ,i ( t ), hydrodynamic drag force F h ,i ( t ), and mooring line forces F m ,i,k ( t ) acting on wind turbine i with a semi-submersible floating platform. The rates of change of the position and velocity of turbine i are expressed as follows:˙ r i ( t ) = v i ( t ) , (20)˙ v i ( t ) = F i ( t ) m i + m a ,i , (21)where m i is the mass of floating wind turbine i . The added mass m a ,i associated with turbine i will be discussedalong with the hydrodynamic drag force.As shown in Fig. 4, the total force F i ( t ) acting on turbine i is the sum of its respective aerodynamic, hydrodynamic,and mooring line forces as follows: F i ( t ) = F a ,i ( t ) + F h ,i ( t ) + F m ,i ( t ) . (22)The aerodynamic thrust force F a ,i ( t ) acting on the rotor of turbine i is expressed as follows: F a ,i ( t ) = 18 C t ,i ( t ) ρ a πD i k V rel ,i ( t ) k n i ( t ) , (23)where the thrust coefficient C t ,i ( t ) is computed based on the vortex cylinder model of a yawed actuator disc asfollows [24]: C t ,i ( t ) = 4 a i ( t ) (cid:18) cos γ rel ,i ( t ) + tan χ i ( t )2 sin γ rel ,i ( t ) − a i ( t ) sec χ i ( t )2 (cid:19) , (24)and n i ( t ) is a unit vector normal to the rotor of turbine i as follows: n i ( t ) = (cid:2) cos γ i ( t ) sin γ i ( t ) (cid:3) T . (25)Based on elementary fluid mechanics principles concerning immersed bodies, F h ,i ( t ) is approximated by summingthe drag force contributions of all submerged components of turbine i as follows: F h ,i ( t ) = 12 X j ∈ D i C d ,i,j A d ,i,j ρ w k w ( t ) − v i ( t ) k ( w ( t ) − v i ( t )) , (26)where ρ w is the density of ocean water, and w ( t ) is the ocean current velocity vector (which we assume to be w ( t ) = 0 m / s in this work). Let the set D i = { , , · · · , N h ,i } denote the indices of all submerged componentsthat contribute to the hydrodynamic drag force acting on turbine i , with N h ,i being equal to the total number ofsubmerged components of turbine i . C d ,i,j and A d ,i,j are thereby the drag coefficient and reference area of the j th submerged component of turbine i . Added mass accounts for hydrodynamic loads that act upon an object that is accelerating with respect to the surrounding fluid. Itcompounds with hydrodynamic drag forces, which are typically modeled as functions of instantaneous velocity only. k of turbine i .In a similar manner, the total added mass m a ,i associated with turbine i is estimated by summing the addedmass contributions of all submerged components of turbine i as follows: m a ,i = ρ w X j ∈ D i C a ,i,j A a ,i,j , (27)where C a ,i,j is the added mass coefficient of the j th submerged component of turbine i , and A a ,i,j is the added massreference area of the same component.Let the set M i = { , , · · · , N m ,i } denote the indices of all mooring lines connected to turbine i , with N m ,i beingequal to the total number of mooring lines attached to turbine i . F m ,i ( t ) may then be expressed as the sum of allmooring force contributions acting on turbine i as follows: F m ,i ( t ) = X k ∈ M i F m ,i,k ( t ) , (28)where F m ,i,k ( t ) is the restoring force exerted on turbine i by its k th mooring line. This force is calculated by firstfinding the magnitude of the horizontal component of tension within mooring line k of turbine i , and then projectingthis tension in the appropriate direction as follows: F m ,i,k ( t ) = − H F ,i,k ( t ) r F / A ,i,k ( t ) (cid:13)(cid:13) r F / A ,i,k ( t ) (cid:13)(cid:13) . (29)The function H F ,i,k ( t ) outputs the horizontal component of tension along the k th mooring line of turbine i . Thisfunction is generated by solving the static differential equations describing a suspended cable which is either partiallycontacting or fully lifted above the seabed. The relevant formulae are provided in Appendix B and are also availablein out previous publication [20].As shown in Fig. 5, the term r F / A ,i,k ( t ) describes the position vector from the anchor of the k th mooring line ofturbine i to the corresponding fairlead, and is expressed as follows: r F / A ,i,k ( t ) = r i ( t ) + r F / G ,i,k − r A ,i,k , (30)where r F / G ,i,k is a constant position vector from the center-of-gravity of turbine i to the fairlead that connects tothe k th mooring line of the same turbine, and r A ,i,k is a constant position vector representing the location of theanchor of the same mooring line. In Eq. (29), dividing r F / A ,i,k ( t ) by its Euclidean norm therefore produces a unitvector that points from the anchor of the k th mooring line of turbine i to the corresponding fairlead. The restoringforce associated with this mooring line pulls the turbine in the opposite direction. Fig. 6 shows the characteristics of interest when modeling wake i , which is the wake generated by the rotor ofturbine i . These characteristics include the wake’s centerline position y w ,i (ˆ x i , t ) relative to the ˆ x i axis, its averagevelocity vector v w ,i (ˆ x i , t ) measured in frame i , and its diameter D w ,i (ˆ x i , t ).Two key assumptions are necessary for justifying the mathematical formulation presented in this section. First,if fluctuations in the wind direction relative to the ˆ x i axes are presumed to be small, then all wake characteristicsmay be defined as smooth functions of only ˆ x i and t . Furthermore, wake cross-sections may be assumed to alwaysremain normal to the predominant flow direction, which corresponds to the positive ˆ x and ˆ x i axes in our work.7igure 6: Schematic of characteristics necessary for modeling the wake generated by turbine i . The wake centerlineposition y w ,i (ˆ x i , t ), average wake velocity v w ,i (ˆ x i , t ), and wake diameter D w ,i (ˆ x i , t ) are defined within the referenceframe that is fixed to turbine i .Second, if the free stream wind speed is presumed to be significantly larger than the velocities of floating platforms,then the equations of motion describing any wake may be defined relative to a reference frame that is fixed to thewake-generating turbine. The frame of reference shown in Fig. 6 is therefore non-inertial and translates with turbine i ,while y w ,i (ˆ x i , t ), v w ,i (ˆ x i , t ), and D w ,i (ˆ x i , t ) are defined in this translating frame. This approach eliminates the needto model wake behavior upstream of turbine i , while removing time-dependency from the wake centerline boundarycondition ( i.e. y w ,i (ˆ x i , t ) is always equal to zero at ˆ x i = 0 m).Granting these preliminaries, the equations of motion describing wake i may now be derived. Specifically, weshall present partial differential equations that model wake average velocities, wake centerline locations, and wakediameters over space and time. Let the vector L i (ˆ x i , t ) describe the linear momentum deficit of wake i per unitlength along the ˆ x i axis as follows: L i (ˆ x i , t ) = ρ a π D ,i (ˆ x i , t ) [ V ∞ ( t ) − ( v i ( t ) + v w ,i (ˆ x i , t ))] . (31)As v w ,i (ˆ x i , t ) is measured in frame i , the term v i ( t ) + v w ,i (ˆ x i , t ) redefines the velocity of wake i in the global frame.Since no external forces impact wake i , the time-derivative of L i (ˆ x i , t ) must equate to zero, which results in thefollowing momentum conservation equation: ∂ v w ,i (ˆ x i , t ) ∂t + ( U ∞ ( t ) − v x,i ( t )) ∂ v w ,i (ˆ x i , t ) ∂ ˆ x i =˙ V ∞ ( t ) − ˙ v i ( t ) + 2 D w ,i (ˆ x i , t ) dD w ,i (ˆ x i , t ) dt ( V ∞ ( t ) − v i ( t ) − v w ,i (ˆ x i , t )) . (32)The time-derivative of y w ,i (ˆ x i , t ) must equate to the ˆ y i component of v w ,i (ˆ x i , t ), which results in the followingexpression describing the wake centerline location: ∂y w ,i (ˆ x i , t ) ∂t + ( U ∞ ( t ) − v x,i ( t )) ∂y w ,i (ˆ x i , t ) ∂ ˆ x i = v w ,i (ˆ x i , t ) . (33)In Eqs. (32) and (33), u w ,i (ˆ x i , t ) and v w ,i (ˆ x i , t ) are the ˆ x i and ˆ y i components of v w ,i (ˆ x i , t ), v x,i ( t ) is the velocity ofturbine i in the ˆ x direction, U ∞ ( t ) is the free stream wind speed in the ˆ x direction, and the term U ∞ ( t ) − v x,i ( t )serves as the transport speed in the ˆ x i direction. When modeling fluids using the three-dimensional Navier-Stokesequations, the transport and fluid velocities at any given point are equal. Following this logic, the transport speed inEqs. (32) and (33) should simply be u w ,i (ˆ x i , t ). However, when neglecting three-dimensional effects, it is debatableexactly how the transport velocity should be defined. Our simulations indicate that defining the transport speedas the free stream wind speed (defined in frame i ) yields predictions closer to experimental observations than doessetting the transport speed to u w ,i (ˆ x i , t ).In steady-state parametric wake models, the wake diameter is typically assumed to grow at a constant spatialexpansion rate k x along the downstream direction. When modeling wakes dynamically, however, we assume thatwake diameters grow at a constant temporal expansion rate k t . In other words, the time-derivative of D w ,i (ˆ x i , t )must equate to k t as follows: ∂D w ,i (ˆ x i , t ) ∂t + ( U ∞ ( t ) − v x,i ( t )) ∂D w ,i (ˆ x i , t ) ∂ ˆ x i = k t . (34)8f the spatial expansion rate k x under steady-state conditions is known for some reference free stream wind speed U ∞ , ref , the temporal expansion rate at U ∞ , ref must be k t = k x U ∞ , ref . Assuming that the free stream wind speed k V ∞ ( t ) k does not vary significantly from U ∞ , ref , then k t may be assumed to remain constant.In order to obtain the wake states employed in Eq. (9), the spatial gradients in Eqs. (32), (33), and (34) mustbe discretized over some fixed downstream distance using the finite difference method, which would yield a systemof nonlinear ordinary differential equations that would be rearranged to state-space form. We will not present thediscretized forms of these equations as the finite difference method is an elementary numerical technique.When implementing the above solution, we recommend the following initial conditions: y w ,i (ˆ x i ,
0) = V ∞ (0) U ∞ (0) ˆ x i , (35) v w ,i (ˆ x i ,
0) = V ∞ (0) − v i (0) , (36) D w ,i (ˆ x i ,
0) = D i + k x ˆ x i , (37)which ensure, respectively, that all wake centerlines are initially aligned with the free stream wind, wake velocitiesare initially equal to the free stream wind velocity, and that wake diameters initially grow at a predefined spatialrate k x . Note that D i is the diameter of turbine i . With regards to boundary conditions, the following are necessarybased on assumptions inherent to FOWFSim-Dyn: y w ,i (0 , t ) = 0 , (38) v w ,i (0 , t ) = v w , init ,i ( t ) , (39) D w ,i (0 , t ) = D i , (40)Equation (38) states that the centerline of wake i at ˆ x i = 0 m must always correspond to the location of turbine i ,which is in fact the origin of frame i . Equation (39) states that the velocity of wake i at ˆ x i = 0 m must always beequal to the wake velocity v w , init ,i ( t ) immediately downstream of the rotor of turbine i . Finally, Eq. (40) requiresthat the diameter of wake i at the location of turbine i is always equal to the rotor diameter of this turbine.We calculate the velocity vector v w , init ,i ( t ) based on simplifications made to Glauert’s momentum theory [24] byBastankhah and Port´e Agel [21] as follows: v w , init ,i ( t ) = k V rel ,i ( t ) k q − C t ,i ( t ) (cid:20) cos ( ξ w , init ,i ( t ) + θ i ( t ))sin ( ξ w , init ,i ( t ) + θ i ( t )) (cid:21) , (41)where ξ w , init ,i ( t ) is the initial wake skew angle, which is expressed as follows based on a momentum conservationderivation reported by Jim´enez et al. [25]: ξ w , init ,i ( t ) = − C t ,i ( t )2 cos γ rel ,i ( t ) sin γ rel ,i ( t ) . (42)The derivation by Bastankhah and Port´e Agel [21] assumes that the free stream wind velocity is aligned with the ˆ x axis. As a result, the addition of θ i ( t ) to ξ w , init ,i ( t ) in Eq. (41) accounts for the misalignment of V ∞ ( t ) relative tothe ˆ x axis. When a wind turbine rotor is influenced by wakes that are generated from multiple upstream turbines, a wakeinteraction model is necessary for approximating the resultant effective wind speed that is incident on the downstreamrotor. The most commonly used wake interaction technique is based on the assumption that the effective kineticenergy deficit at the location of the downstream rotor must be equal to the sum of kinetic energy deficits of allpertinent wakes [2]. As a result, the effective wind speed at the downstream rotor is a function of the root-sum-square of relevant wake velocity deficits. Further enhancement may be obtained by approximating wake velocityprofiles using Gaussian distributions [21]. We continue to make use of this wake interaction methodology.Let the set U i = { , , · · · , i − } denote the indices of all turbines that are located upstream of turbine i . Theeffective wind velocity vector that is incident on the rotor of turbine i may therefore be expressed as follows: V i ( t ) = k V ∞ ( t ) k − s X q ∈ U i ( k V ∞ ( t ) k − v w ,q → i ( t ) · n ∞ ( t )) n ∞ ( t ) . (43)9here n ∞ ( t ) is a unit vector aligned with V ∞ ( t ) as follows: n ∞ ( t ) = V ∞ ( t ) k V ∞ ( t ) k , (44)and v w ,q → i ( t ) is the effective velocity of wake q that is incident upon the rotor of wake i . Equation (43) projects v w ,q → i ( t ) along the free stream wind direction (hence the dot product operation with n ∞ ( t )), and then computesthe velocity deficit in this direction. Average wake velocities perpendicular to the free stream wind direction areassumed to be negligibly small far enough downstream; their effects are therefore neglected.We now describe our procedure for computing v w ,q → i ( t ). Let v w ,q → i ( t ) denote the average velocity of wake q atthe location of wake i as follows: v w ,q → i ( t ) = v q ( t ) + v w ,q ( x i ( t ) − x q ( t ) , t ) . (45)Since the average velocity vector of wake q is defined in frame q , the substitution ˆ x q = x i ( t ) − x q ( t ) into v w ,q (ˆ x q , t ) isnecessary for identifying the location of turbine i in frame q . The addition of the turbine velocity vector v q ( t ) thentransforms v w ,q ( x i ( t ) − x q ( t ) , t ) to the global frame.The next step is to generate a Gaussian profile ˘ v w ,q → i ( r, t ), where r is the radial distance from the centerlineof wake q , to approximate the continuous velocity distribution of wake q at the location of wake i . Imposing arequirement that the total momentum deficit of V ∞ ( t ) − ˘ v w ,q → i ( r, t ) per unit length must equate that of a top-hatdistribution with amplitude V ∞ ( t ) − v w ,q → i ( t ) as follows: Z ∞ ρ a πr ( V ∞ ( t ) − ˘ v w ,q → i ( r, t )) dr = ρ a π D w ,q → i ( t ) ( V ∞ ( t ) − v w ,q → i ( t )) , (46)the following Gaussian profile is then obtained: V ∞ ( t ) − ˘ v w ,q → i ( r, t ) = 18 (cid:18) D w ,q → i ( t ) σ (cid:19) ( V ∞ ( t ) − v w ,q → i ( t )) exp − r σ , (47)where D w ,q → i ( t ) is the diameter of wake q at the location of turbine i as follows: D w ,q → i = D w ,q ( x i ( t ) − x q ( t ) , t ) . (48)The standard deviation σ in Eq. (47) may be estimated based on experimental or high-fidelity numerical data.Finally, the effective velocity v w ,q → i ( t ) is obtained by averaging ˘ v w ,q → i ( r, t ) along the rotor area A i of turbine i .This task is achieved by numerically computing the following integral at each time-step: v w ,q → i ( t ) = 4 πD i Z A i ˘ v w ,q → i ( r, t ) dA. (49) Several assumptions have been made when developing FOWFSim-Dyn which impose limitations on its fidelity andapplicability. The current subsection summarizes these limitations.
The first and most crucial of these assumptions is that floating platform motion may be adequately captured using atwo-dimensional planar model. That is to say, we neglect floating platform heave, yaw, pitch, and roll. In consequence,FOWFSim-Dyn fails to capture dynamic effects induced by ocean waves and oscillatory wind conditions on platformrotation. FOWFSim-Dyn remains appropriate for wind farm controller design and testing, since this application isprimarily concerned with average rotor positions over extended periods of time. However, any attempt to controlor evaluate individual wind turbine dynamics requires the use of three-dimensional multi-body nonlinear modelingtools.
Although we present a dynamic model, mooring line tensions are found based on the solution to a static suspendedcable problem. It has been reported by Hall et al. [26] that such static models accurately predict mooring line loadsand floating wind turbine motion; thus rendering them appropriate for wind farm control. However, Hall et al. [26]also mentioned that use of such models may lead to large inaccuracies in turbine load predictions. Therefore, analysisand control of individual turbine motion must consider higher-fidelity modeling techniques such as a lumped-massdynamic mooring line model [27]. 10 .8.3 Steady-state turbine aerodynamics
Turbine power outputs and thrust forces (Eqs. (13) and (23)), along with their respective coefficients (Eqs. (16) and(24)), are calculated based on steady-state actuator disc theory. This approach assumes ideal rotors and fails tocapture unsteady aerodynamic effects and asymmetric rotor loadings. These phenomena significantly influence bladeloads when yaw misalignment occurs; however, for the purpose of wind farm control, our focus lies on the overallinfluence of rotor operation on fully-developed wake regions. Nonetheless, any turbine-level analysis requires moredetailed fluid-structure interaction modeling.The computation of v w , init ,i ( t ) in Eq. (41), which is the average wake velocity immediately downstream of tur-bine i , relies on a steady-state momentum balance on a control volume spanning across the rotor of turbine i . As aresult, momentum fluxes into and out of this control volume are considered, while the rate-of-change of momentumwithin the control volume is neglected. Given the low density of air, these inertial effects may be neglected, althoughtheir significance should be investigated. FOWFSim-Dyn does not capture wake centerline deflection caused by rotor rotation. This phenomenon was firstobserved in high-fidelity simulations conducted by Gebraad et al. [6]; however, more recent work by Fleming etal. [28] showed that the scale of this phenomenon is insignificant. Instead, Fleming et al. [28] observed that vorticesgenerated by turbine rotors induce wake deflection past downstream machines, even if their rotors are not operatedwith yaw offset. Additional terms may be added to Eqs. (32) and (33) to account for such phenomena.
In the current paper, we have assumed that the free stream wind velocity is uniform throughout the wind farm,which is why the variable V ∞ ( t ) is solely a function of time. This variable may readily be expressed as V ∞ (ˆ x, t ) ifspatial variations of the free stream wind velocity are known. Furthermore, in order to represent wake characteristicspurely as a function of the downstream distance along the ˆ x i axes, while ignoring changes in the cross-sectionalareas of wakes, we assumed that variations in the free stream wind direction are small relative to the ˆ x axis. The ˆ y component of the free stream wind velocity must therefore remain small in comparison to its ˆ x component. In this section, we first perform a mesh sensitivity analysis to ascertain the dependency of model predictions upon thesize of finite difference elements in Section 3.1. We then validate FOWFSim-Dyn against steady-state experimentalresults reported by Bastankhah and Port´e Agel [21] in Section 3.2. Finally, we present dynamic simulation results forvarious scenarios to ensure that model predictions are in line with physical expectations and intuition in Section 3.3.
For a mesh sensitivity study, we simulate the experimental setup employed by Bastankhah and Port´e Agel [21].Namely, the wake of a single fixed-foundation turbine with diameter D = 15 cm is simulated with a steady freestream wind speed of U ∞ = 4 .
88 m / s. The turbine’s axial induction factor is set to the optimal value of a = 1 / γ = 20 deg is implemented to observe mesh effects on wake deflection. To approximate steady-stateresults, all simulations are run for a duration of 5 sec and data is extracted from the final time-step. The Gaussianprofile standard deviation is set to σ = 0 . x + 0 .
396 m based on experimental data reported by Bastankhah andPort´e Agel [21]. The (diametrical) spatial wake expansion constant is set to k x = 0 .
08 as per the recommendationby Shakoor et al. [3].Simulated wake centerlines and normalized velocity deficit profiles at a downstream distance of 7 D are plottedin Fig. 7 for different finite difference element sizes. Qualitatively, it is apparent that the evolution of the wakecenterline is insignificantly influenced by the mesh size. At ˆ x/D = 16, the centerline deflection obtained using anelement size of 8 D only differs by 5 % relative to the value corresponding to an element size of 0 . D . As a result,we solely utilize the maximum normalized velocity deficit as a convergence criterion.Fig. 1 lists the computation times corresponding to different element sizes from Fig. 7 as well as predictedmaximum normalized velocity deficits. Dynamic simulations were performed using the MATLAB fourth-order Runge-Kutta solver implemented on a laptop computer with a 2 .
80 GHz Intel Core i7-7700HQ processor. Figure 1 also lists Velocity profiles corresponding to a yaw angle of γ = 0 deg from Fig. 21 in the paper by Bastankhah and Port´e Agel [21] weredigitized and Gaussian function curve fitting was used to compute the standard deviation. b)a) Figure 7: Effects of various finite difference mesh element sizes on a) the steady-state wake velocity profile at adownstream distance of 7 D , and b) the steady-state wake centerline evolution. Simulation parameters: D = 15 cm, U ∞ = 8 m / s, a = 1 / γ = 20 deg, k x = 0 . σ = 0 . x + 0 .
396 m.Table 1: Computation times and maximum normalized velocity deficits corresponding to different simulated meshelement sizes from Fig. 7. The final column lists the convergence of the maximum normalized velocity deficit. Inother words, it contains the relative difference in the maximum normalized velocity deficit that would be obtainedif each element size was halved. For instance, if the element size were to be reduced from 8 D to 4 D , the predictedvelocity deficit would change by 19 .
46 %. The computation times correspond to 5 sec long simulations.
Elm. size (D) Comp. time (sec) Max. velocity deficit (-) Rel. diff. (%)8 0.397 0.261 19.464 0.703 0.219 7.192 1.496 0.204 2.001 3.237 0.200 0.690.5 8.948 0.199 0.170.25 24.035 0.198 - the convergence of the maximum normalized velocity deficit as the element size is decreased. We observe that meshsensitivity is sufficiently reduced at an element size of 1 D since further reduction to 0 . D only results in a 0 .
69 %change in the predicted maximum normalized velocity deficit. An element size of 1 D is also appropriate from thestandpoint of time-efficiency as it requires 3 . FOWFSim-Dyn predictions of steady-state wake centerlines and normalized velocity profiles are compared againstexperimental results reported by Bastankhah and Port´e Agel [21] in Fig. 8. Wake centerline evolutions are well-predicted for all simulated yaw angles and downstream locations. For yaw angles of γ = 0, 10, and 20 deg, maximumdiscrepancies between predicted wake centerlines and experimental measurements are 6 .
87, 7 .
60, and 8 .
19 % of therotor diameter, respectively.Simulated normalized velocity profiles deviate significantly from experimental measurements at downstream lo-cations closer than 7 D . For instance, at a yaw angle of γ = 0 deg, the root-mean-square error (RMSE) betweenexperimental and predicted velocity profiles ranges from 12 . x = 4 D to 4 . x = 7 D . Such inaccuracies at close downstream distances are expected since FOWFSim-Dyn does not consider theinviscid nature of flow within the near-wake region. Beyond ˆ x = 7 D , velocity profiles are well-predicted with RMSEvalues that remain below 3 .
87 % of the free stream wind speed. Validating predictions of dynamic wake behaviour is not possible at this time due to the absence of high-fidelity simulation toolscapable of modeling floating offshore wind farms; we thus defer this process to future work. Sim. wake centerlineSim. wake boundariesExp. wake centerlineSim. deficit profilesExp. deficit profiles a)b)c)
Figure 8: Comparison between FOWFSim-Dyn predictions and experimental results reported by Bastankhah andPort´e Agel [21]. Each figure shows steady-state wake centerlines and normalized velocity profiles corresponding toyaw angles of a) γ = 0 deg, b) γ = 10 deg, and c) γ = 20 deg. Normalized velocity profiles range from zero to oneusing the same scaling as the ˆ x/D axis, but have been shifted to the downstream location where they are measured.Simulation parameters: D = 15 cm, U ∞ = 8 m / s, a = 1 / k x = 0 . σ = 0 . x + 0 .
396 m.
Our final tasks are to demonstrate the capability of FOWFSim-Dyn to capture the intended dynamic phenomenaand to ensure that predicted turbine and wake behaviors respect physical intuition. The wind farm configurationthat is used for dynamic simulations is shown in Fig. 9. This plant contains a single row of three floating offshorewind turbines that are aligned with the predominant free stream wind direction. The neutral positions of thefloating turbines are spaced 7 D apart. All wind turbines are based on the National Renewable Energy Laboratory’s(NREL’s) 5 MW baseline design presented by Jonkman et al. [29], and all floating platforms and mooring subsystemsare modeled after the design described by Robertson et al. [22]. Details corresponding to these designs are listed inAppendix A. In all simulations, we increase the lengths of mooring lines from their baseline values ( i.e. L = 835 m) to L = 900 m to render floating platform motion more notable. In all the following cases, less than 10 sec of computationtime was required to complete simulations on a laptop computer with a 2 .
80 GHz Intel Core i7-7700HQ processor.Figure 9: Schematic of the 1 × D used for dynamic simulations. All windturbines are based on the NREL 5 MW baseline design presented by Jonkman et al. [29], and all floating platformsand mooring subsystems are modeled after the design described by Robertson et al. [22].13 -2-1012-2-1012-2-1012 Figure 10: Velocity contours at various time-steps of simulation scenario 1 ( i.e. fixed wind condition and turbineoperating parameters, while platform motion is permitted). The white + symbols represent the neutral positionsof the floating platforms. All floating platforms are held fixed at their respective neutral positions for the first1000 sec of simulation. Simulation parameters: U ∞ ( t ) = 8 m / s, V ∞ ( t ) = 0 m / s, a ( t ) = a ( t ) = a ( t ) = 1 / γ ( t ) = γ ( t ) = −
20 deg and γ ( t ) = +20 deg, k x = 0 . σ = 0 . x + 0 .
396 m.
The first of three simulated scenarios maintains constant wind speed and direction with U ∞ ( t ) = 8 m / s and V ∞ ( t ) =0 m / s, while rotor yaw angles are fixed at γ ( t ) = γ ( t ) = −
20 deg and γ ( t ) = +20 deg. All axial induction factorsare maintained at a ( t ) = a ( t ) = a ( t ) = 1 /
3. All floating platforms are locked at their neutral positions for thefirst 1000 sec of simulation, after which they are permitted to relocate. The aim of this scenario is to assess floatingplatform motion. Snapshots of velocity contours for simulation scenario 1 are shown in Fig. 10. As expected, thealternating assignment of yaw angles causes adjacent floating platforms to shift in opposite directions over time.Further, the leading turbine displays the greatest amount of relocation from its neutral position ( i.e. the left-mostwhite + symbol) since its incident wind speed is the largest ( i.e. its incident wind speed is the free stream windspeed uninhibited by upstream rotors). The trailing turbine undergoes the smallest amount of relocation over timesince its incident wind speed is diminished by the velocity deficits of wakes 1 and 2.14 -2-1012-2-1012-2-1012
Figure 11: Velocity contours at various time-steps of simulation scenario 2 ( i.e. fixed wind condition and sinusoidallyvarying yaw angles, while platform motion is permitted). The white + symbols represent the neutral positions ofthe floating platforms. All floating platforms are held fixed at their respective neutral positions for the first 1000 secof simulation. Simulation parameters: U ∞ ( t ) = 8 m / s, V ∞ ( t ) = 0 m / s, a ( t ) = a ( t ) = a ( t ) = 1 / γ ( t ) and γ ( t )defined in Eq. (50) and γ ( t ) defined in Eq. (51), k x = 0 . σ = 0 . x + 0 .
396 m.
The second simulation sinusoidally varies the yaw angles of the three turbines between ±
20 deg with a period of400 sec. Specifically, the following yaw angle expressions are used for t ≥ γ ( t ) = γ ( t ) = ( −
20 deg) sin (cid:20) π
400 ( t − (cid:21) , (50) γ ( t ) = (+20 deg) sin (cid:20) π
400 ( t − (cid:21) . (51)Velocity contours for this case are plotted in Fig. 11. The sinusoidal yaw angle fluctuations cause oscillations of floatingplatforms in the ˆ y direction with the expected 400 sec excitation period. In terms of wake behaviour, the transporteffect is clearly observed. As floating turbines shift in the ˆ y direction, the corresponding effects on their respectivewakes are transported downstream at approximately 8 m / s. For instance, at t = 1400 sec, the leading turbine islocated at a peak value past its neutral position in the +ˆ y direction. Given that U ∞ = 8 m / s , then 200 sec later,the centerline of the leading turbine’s wake must peak in the +ˆ y direction at ˆ x = 8 m / s ×
200 sec = 1600 m = 12 . D .Observing the velocity contours 200 sec later at t = 1600 sec, such a peak is observed at just under ˆ x = 12 D .15 -2-1012-2-1012-2-1012 Figure 12: Velocity contours at various time-steps of simulation scenario 3 ( i.e. fixed turbine operating conditionsand fluctuating wind speed in the ˆ y direction, while platform motion is prohibited). The white + symbols representthe neutral positions of the floating platforms. The white arrows denote the free stream wind direction. All floatingplatforms are held fixed throughout the simulation. Simulation parameters: U ∞ ( t ) = 8 m / s, V ∞ ( t ) defined inEq. (52), a ( t ) = a ( t ) = a ( t ) = 1 / γ ( t ) = γ ( t ) = γ ( t ) = 0 deg, k x = 0 . σ = 0 . x + 0 .
396 m.
The third scenario assesses the impacts of time-varying wind direction, which is modeled by maintaining U ∞ ( t ) =8 m / s and fluctuating V ∞ ( t ) sinusoidally between ± / s with a period of 200 sec. Specifically, V ∞ ( t ) is expressedas follows for t ≥ V ∞ ( t ) = (2 m / s) sin (cid:20) π
200 ( t − (cid:21) . (52)All yaw angles in this scenario are maintained at γ ( t ) = γ ( t ) = γ ( t ) = 0 deg. Velocity contours for simulationcase 3 are shown in Fig. 12. The notable expectation here is that, as the wind direction changes, wake centerlinesare transported in tandem with the free stream wind in both ˆ x and ˆ y directions. For instance, at t = 1000 sec, thecenterline of wake 1 is aligned with the ˆ x axis since γ ( t ) = 0 deg and V ∞ ( t ) had been equal to zero at all previoustimes. By t = 1050 sec, the effects of turbine 1 on the wind field should only be transported downstream by a distanceof 8 m / s ×
50 sec = 400 m = 3 . D . Therefore, for ˆ x < . D , we expect variations in the curvature of the centerlineof wake 1 due to the presence of turbine 1, while for ˆ x > . D , this curvature should remain unchanged. Instead,for ˆ x > . D , the centerline of wake 1 should be shifted in the +ˆ y direction as a result of V ∞ ( t ) having held positivevalues for the past 50 sec. Observing velocity contours at t = 1050 sec, it is evident that the centerline curvature ofwake 1 remains flat at all downstream distances past approximately ˆ x = 3 D , while having been shifted in the +ˆ y direction. 16 Conclusions and recommendations for future research
This paper extended FOWFSim [20], which is a steady-state modeling tool that may be used for simulating and opti-mizing floating offshore wind farms, by adding capabilities that captured time-varying free stream wind velocities andfloating platform motion. In addition to presenting a mathematical formulation, we performed a mesh convergencestudy and validated steady-state predictions on wake behaviour against experimental data obtained from existingliterature. It was demonstrated that the limited number of tunable parameters produced wake centerline deflectionand velocity deficit results that matched experimental observations with reasonable similarity for engineering anal-ysis. We then conducted simulations under various wind and turbine operating conditions to assess the dynamicbehavior of FOWFSim-Dyn. It was observed that FOWFSim-Dyn captures dynamic floating wind farm phenomenasuch as wake transport, time-varying wind speed and direction effects, and floating platform motion in line withphysical reasoning and intuition.For the purposes of further developing and enhancing the current framework, several recommendations on poten-tial research directions are made. First, to this date, no LES-based wind farm simulators are capable of capturingfloating platform motion. Developing wind farm CFD tools that consider such dynamics would therefore permit morecomprehensive validation of FOWFSim-Dyn predictions pertaining to both platform motion and wake behaviour.Complementing this point, scaled wind tunnel experiments of floating wind turbines would also enable validation ofdynamic FOWFSim-Dyn predictions.Second, we did not model turbulence in the current framework. This feature may be incorporated by addingmeasurement noise to model outputs, or by including temporally and spatially distributed turbulence accelerationterms in the equations of motion. Finally, additional force gradients may be included in the equations of motion tocapture complex wake phenomena such as secondary steering [28] and wake deflection due to rotor rotation [6].
Acknowledgment
The authors are grateful for the financial support provided by the Natural Sciences and Engineering Research Councilof Canada (NSERC). 17
Wind farm properties
Table 2: List of floating wind farm properties used during simulations that are discussed in Section 3. All windturbines are based on the NREL 5 MW baseline design presented by Jonkman et al. [29], and all floating platformsand mooring subsystems are modeled after the design described by Robertson et al. [22].
External properties ρ a (cid:0) kg / m (cid:1) ρ w (cid:0) kg / m (cid:1) m i (kg) 1 . × Mass D i (m) 126 Rotor diameter A i (cid:0) m (cid:1) π D i Rotor area η p .
786 Electrical power conversion efficiency [6] p p .
88 Power coefficient tuning parameter [6]Floating platform hydrodynamic properties C d ,i, → .
61 Drag coefficients of three top cylinder portions C d ,i, → .
68 Drag coefficients of three bottom cylinder portions C d ,i, .
56 Drag coefficient of middle cylinder D d ,i, → (m) 12 Diameters of three top cylinder portions D d ,i, → (m) 24 Diameters of three bottom cylinder portions D d ,i, (m) 6 . L d ,i, → (m) 14 Submerged lengths of three top cylinder portions L d ,i, → (m) 6 Submerged lengths of three bottom cylinder portions L d ,i, (m) 20 Submerged length of middle cylinder A d ,i,j (cid:0) m (cid:1) L d ,i,j D d ,i,j Drag reference area of any cylinder C a ,i,j .
63 Added mass coefficients of any cylinder A a ,i,j (cid:0) m (cid:1) π L d ,i,j D ,i,j Added mass reference area of any cylinderMooring system properties r TF / G ,i, (m) (cid:2) . . (cid:3) Position vector from turbine center to first fairlead r TF / G ,i, (m) (cid:2) − . (cid:3) Position vector from turbine center to second fairlead r TF / G ,i, (m) (cid:2) . − . (cid:3) Position vector from turbine center to third fairlead r TA ,i, (m) r Tneutral ,i + (cid:2) .
80 725 . (cid:3) Location of first anchor of any turbine r TA ,i, (m) r Tneutral ,i + (cid:2) − . (cid:3) Location of second anchor of any turbine r TA ,i, (m) r Tneutral ,i + (cid:2) . − . (cid:3) Location of third anchor of any turbine z F (m) 186 Fairlead distance above seabed L (m) 835 Cable length w (N / m) 1065 . A m E (N) 753 . × Cable tension per unit strain µ s B Formulae for computing mooring line tension
This appendix section briefly details the formulae used to calculate the horizontal component of tension within anymooring line cable. Derivations of the following formulae may be found in our previous work [20]. For readability,we drop functional time-dependency indicators and subscripts ( i.e. H F ,i,k ( t ) simply becomes H F ) since the discussedsolution is static and all formulae remain the same for any individual mooring line cable.We begin by defining three zones of mooring line operation. The first zone is in effect when the fairlead is closeenough to its respective anchor that the cable is vertical at the fairlead location; the resulting horizontal componentof tension is zero in this case. The second zone of operation occurs when the cable is partially contacting the seabed,while the third zone is relevant when the cable is fully lifted off of the seabed. Based on these definitions, we define H F as follows: H F = x F ≤ x F , → ,f if x F , → < x F ≤ x F , → ,f if x F ≥ x F , → , (53)where x F is the horizontal distance from the fairlead to its respective anchor as follows: x F = (cid:13)(cid:13) r F / A ,i,k (cid:13)(cid:13) , (54) Anchors are located at angles of 60, 180, and 300 deg along a circle of radius 837 . r neutral ,i oftheir respective turbines. Simulations corresponding to Figs. 10 to 12 use longer cable lengths of L = 900 m. x F , → = L − z F , (55) x F , → = H → w (cid:20) wLA m E + sinh − wLH → (cid:21) . (56)The parameters L , w , A m , and E represent the length, specific weight in water, cross-sectional area, and elasticmodulus of the cable, z F is the vertical distance between the fairlead and its respective anchor, and H → is thehorizontal tension within the cable at the transition between operating zones 2 and 3, which has been derived to givethe following expression: H → = wL " − (cid:18) z F L − wL A m E (cid:19) z F L − wL A m E (cid:19) − . (57)The function f from Eq. (53) solves the following system of nonlinear equations for the horizontal and verticalcomponents of cable tension H F and V F : x F − L s = H F w (cid:18) V F A m E + sinh − V F H F (cid:19) , (58) z F = 1 w V A m E − H F − s (cid:18) V F H F (cid:19) . (59)These equations correspond to a catenary profile that is partially contacting the seabed. The term L s is the lengthof the cable portion that is contacting the seabed, which we derive to yield the following expression: L s = L − V F w + (cid:16) H F A m E (cid:17) − (cid:20)(cid:16) H F A m E (cid:17) − µ s wA m E x s (cid:21) µ s wA m E − x s . (60)The term x s represents the location along the seabed-contacting portion at which the total static friction force equatesthe cable tension. Our derivation for x s is expressed as follows: x s = min (cid:20) L − V F w , H F µ s w (cid:18) H F A m E (cid:19)(cid:21) . (61)Similarly, the function f from Eq. (53) solves the following system of nonlinear equations for the horizontal andvertical components of cable tension H F and V F : x F = H F w (cid:18) wLA m E + sinh − V F H F − sinh − V F − wLH F (cid:19) , (62) z F = LA m E (cid:18) V F − wL (cid:19) + H F w s (cid:18) V F H F (cid:19) − s (cid:18) V F − wLH F (cid:19) . (63)These equations correspond to a catenary profile that is fully-lifted off of the seabed. References [1] N. O. Jensen. A Note on Wind Generator Interaction. Technical report, Risø National Laboratory. Reportnumber: Risø-M-2411, 1983.[2] I. Kati´c, J. Højstrup, and N.O. Jensen. A simple model for cluster efficiency.
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