Large-eddy simulation of airborne wind energy farms

The future utility-scale deployment of airborne wind energy technologies requires the development of large-scale multi-megawatt systems. This study aims at quantifying the interaction between the atmospheric boundary layer (ABL) and large-scale airborne wind energy systems operating in a farm. To that end, we present a virtual flight simulator combining largeeddy simulations to simulate turbulent flow conditions and optimal control techniques for flight-path generation and tracking. The two-way coupling between flow and system dynamics is achieved by implementing an actuator sector method that we pair 5 to a model predictive controller. In this study, we consider ground-based power generation pumping-mode AWE systems (liftmode AWES) and on-board power generation AWE systems (drag-mode AWES). For the lift-mode AWES, we additionally investigate different reel-out strategies to reduce the interaction between the tethered wing and its own wake. Further, we investigate AWE parks consisting of 25 systems organized in 5 rows of 5 systems. For both liftand drag-mode archetypes, we consider a moderate park layout with a power density of 10MWkm−2 achieved at a rated wind speed of 12 ms−1. For the 10 drag-mode AWES, an additional park with denser layout and power density of 28MWkm−2 is also considered. The model predictive controller achieves very satisfactory flight-path tracking despite the AWE systems operating in fully waked, turbulent flow conditions. Furthermore, we observe significant wake effects for the utility-scale AWE systems considered in the study. Wake-induced performance losses increase gradually through the downstream rows of systems and reach in the last row of the parks up to 17% for the lift-mode AWE park and up to 25% and 45% for the moderate and dense drag-mode AWE parks, 15 respectively. For an operation period of 60 minutes at a below-rated reference wind speed of 10 ms−1, the lift-mode AWE park generates about 84.4MW of power, corresponding to 82.5 % of the power yield expected when AWE systems operate ideally and interaction with the ABL is negligible. For the drag-mode AWE parks, the moderate and dense layouts generate about 86.0MW and 72.9MW of power, respectively, corresponding to 89.2 % and 75.6 % of the ideal power yield.

. For airborne wind energy systems, wake effects are generally considered small (Kruijff and Ruiterkamp, 2018) or are often ignored (Echeverri et al., 2020) due to the large swept area and relatively small wing dimensions. Hence, recent studies investigating performance losses  or layout optimization (Roque et al., 2020) in airborne wind energy farms did not consider wake effects.
For individual systems, recent investigations of flow induction and wake effects were performed, mainly considering axis-30 symmetric AWES configurations in uniform inflows using either analyses based on momentum theory (Leuthold et al., 2017;De Lellis et al., 2018), vortex theory (Leuthold et al., 2019;Gaunaa et al., 2020) or the entrainment hypothesis (Kaufman-Martin et al., 2021), or high-fidelity CFD simulations (Haas and Meyers, 2017;Kheiri et al., 2018). Further numerical investigations of large-scale airborne wind energy systems in the atmospheric boundary layer (ABL) (Haas et al., 2019b) have shown that the wake development downstream of the system is considerable, hence suggesting that wake effects of utility-scale 35 airborne wind energy systems can not be neglected when operating in parks.
To quantify the wake-induced performance losses in a park of airborne wind energy systems, we have developed a computational framework combining large-eddy simulations (LES) and optimal control techniques. This framework couples the dynamics of the atmospheric boundary layer with the dynamics of several airborne wind energy systems operating in a park.
The last two decades have seen the establishment of large-eddy simulations as an emerging tool for the modelling of wind 40 turbines and wind farms in the atmospheric boundary layer (Jimenez et al., 2007;Calaf et al., 2010), hence we apply this technique in the context of airborne wind energy parks. The inherent unsteadiness of the wind, the effects of wake flows and ABL turbulence make it challenging to foresee the behaviour and performance of individual AWE systems in the park. When modelling AWE systems, the wind environment is often approximated as a height-dependent power law or logarithmic distribution (Archer, 2013;Zanon et al., 2013b;Horn et al., 2013;Bauer et al., 2018). A limited number of studies have used more realistic In the current study, we neglect Coriolis and thermal effects, such that we only need to consider the turbulent surface layer of the ABL. In addition, the atmospheric flow through a wind farm is associated to a high Reynolds number such that we can omit the resolved effects of viscosity. Hence, the ABL can than be simplified to a pressure-driven boundary layer (PDBL) (Calaf et al., 2010). The PDBL is adequately modelled by the filtered incompressible Navier-Stokes equations for neutral flows, expressed by means of the continuity and momentum equations which are shown to read The three-dimensional velocity field can be decomposed into its resolved componentṽ, for which we solve Eq. (1), and its residual component v . The filtered velocity field is parametrized by its streamwiseṽ x , spanwiseṽ y and verticalṽ z flow is incorporated into the filtered modified pressurep * =p/ρ − p ∞ /ρ + tr(τ )/3, while the deviatoric part of the residual-stress tensor τ sgs is modelled using a closure model. Here, we opt for an eddy-viscosity Smagorinsky model with an height-dependent Smagorinsky length scale (Mason and Thomson, 1992) which reinforces the log-law behaviour of the ABL near the surface.
Furthermore, an impermeable wall stress boundary condition is imposed at the bottom surface (Calaf et al., 2010). Finally, the body force term f i emulates the effects of each AWE systems on the flow by means of actuator methods and its derivation is 110 elaborated in Sect. 2.1.2.
Large-eddy simulation of AWE parks are performed with our in-house solver SP-Wind developed at KU Leuven. The governing equations are discretised using a Fourier pseudo-spectral method in the horizontal directions (x, y) and a fourthorder energy-conserving finite-difference scheme in the vertical direction z. These discretisation methods require periodic boundary conditions in the horizontal directions: we circumvent periodicity in the streamwise direction x by applying a fringe-115 region technique (Munters et al., 2016). This technique also allows one to provide unperturbed turbulent inflow conditions to the AWE park simulation by performing a concurrent-precursor periodic PDBL simulation in parallel to the main AWE park simulation. Time integration is performed using a classical four-stage fourth-order Runge-Kutta scheme.

Actuator methods
Actuator methods have proven their abilities to accurately reproduce wake characteristics of conventional wind turbines, such 120 as the actuator line method (Sorensen and Shen, 2002;Troldborg et al., 2010;Martinez-Tossas et al., 2018) and the actuator sector method (ASM) (Storey et al., 2015). The dynamics of AWE systems and ABL flow span over a large range of spatial and temporal scales: while the AWES wing experiences local wind speed fluctuations within a few meters along the wingspan, the ABL domain covers several kilometres. In addition, the flight dynamics of AWE systems are extremely fast and require a temporal resolution down to 10ms whereas the large-scale flow structures evolve one to two orders of magnitude slower. 125 Therefore, we opt for an actuator sector method that can both capture the local variations of aerodynamic quantities along the wingspans of individual AWE systems and accurately operate across a larger range of temporal scales.
The time step of the LES is denoted by ∆t and its value is set to ∆t = 0.25 s throughout the simulations for the chosen grid resolution of the ABL, with cell size ∆x = 10 m in the axial direction. The dynamics of the AWE systems are however much faster and require a smaller time step δt. A stable simulation of the system dynamics is achieved for δt = 10 ms. Hence, for the 130 duration of an entire LES time step, ie. between the instants t n and t n+1 = t n + ∆t, the kinematics of the AWE system (see Sect. 2.2.3) are solved for each t l = t n + lδt ∈ t n , t n+1 while assuming a frozen flow fieldṽ n ≡ṽ(t = t n ).
At every sub-step t l , we compute the local aerodynamic forces per unit segment length f l q (s k ) from the procedure outlined in Sect. 2.2.2. Subsequently, the local aerodynamic forces of each segment are smoothed out over the LES grid cells in the vicinity of the wing using a Gaussian convolution filter G(x) = 6/(π∆ 2 ) 3/2 exp −6||x|| 2 /∆ 2 , where the width of filter 135 kernel is set to ∆ = 2∆ x (Troldborg et al., 2010). The instantaneous, spatially filtered forces, integrated over the complete normalised wingspan s ∈ [−1/2, +1/2], read When flying crosswind manoeuvres at high speed, the AWE system may fly through several LES cells within one simulation time step ∆t. For conventional wind turbines, blade tips sweeping several mesh cells in one time step result in a discontin-140 uous flow solution in the near wake (Storey et al., 2015). Hence, the contributions of the spatially distributed forces f l are subsequently weighted in time using an exponential filter (Vitsas and Meyers, 2016) The filter parameter γ is defined as γ = δt/(τ f +δt) with the filter constant τ f = 2∆t LES . Accordingly, the complete force distribution f accurately captures the fast and local dynamics of each individual AWE system and when added to the momentum 145 equation, Eq. (1b), emulates their collective effects onto the boundary layer flow.

Virtual plant: AWE system dynamics
The simulation strategy applied to AWE systems is shown in Fig. 1, part B. To describe and simulate the dynamics of AWE systems, there exist numerous models of varying complexity. These models range from fast quasi-steady models (Schelbergen and Schmehl, 2020), to simple 3DOF point-mass models (Zanon et al., 2013a), to complex 6DOF rigid-body models (Malz 150 et al., 2019). The point-mass model allows the modelling of large-scale AWE systems while it does not require an extensive knowledge of the aircraft dimension and inertia, nor aerodynamic parameter identification, as is the case for the rigid-body model (Licitra et al., 2017). The point-mass model is therefore highly scalable. The model is also highly versatile and allows one to easily adapt generic wing designs to both ground-based and on-board generation AWE systems. In addition, the point-   and outlined in Sect. 2.2.3.

Point-mass model
The architecture of AWE systems can be divided into three main key components: the ground station, the tether and the airborne device. In this work, we assume that the ground station has no influence on the atmospheric flow and can therefore be neglected.
In addition, we neglect tether dynamics such as tether sag and represent the tether as a rigid rod of varying length (Malz et al.,170 2019), which is solely subject to aerodynamic drag. At last, we limit our investigation to rigid-wing aircraft. Given the large level of approximation of the model, we simply assume that the wing planform is elliptical and without twist, which further allows us to simplify the aerodynamic properties of the finite wing (Anderson, 2010).
Hence, the tethered wing is modelled as a 3DOF point-mass based on a representation of the system in non-minimal coordinates as proposed by Zanon et al. (2013). The system state vector x = q,q, C L , Ψ, κ, l,l,l is composed of position q and 175 velocityq of the wing, the instantaneous aerodynamic quantities such as lift coefficient of the finite wing C L , roll angle Ψ and on-board turbine thrust coefficient κ as well as the tether variables l,l andl, representing the length, speed and acceleration of the tether respectively. In absence of explicit control surfaces such as ailerons, elevators and rudders, we assume that the aerodynamic variables and the tether reel-out speed are directly and instantaneously controlled, hence we introduce the control vector u = Ċ L ,Ψ,κ, ... l which consists of the time derivatives of lift coefficient, roll angle and thrust coefficient, and the 180 tether jerk.
The 3DOF point-mass model only considers the translational motion of the AWE system, while the rotational motion is neglected. Hence, the aircraft attitude, commonly defined by the basis frame of the rigid body, is not known. Instead, we define the reference frame in which the external aerodynamic forces F q acting on the system are expressed. First, we introduce the unit vector e q = q/||q|| and the apparent wind speed v a measured at the point mass. The apparent wind speed is defined as where v w is the three-dimensional wind speed vector, computed from the neighbouring LES cells surrounding the point-mass position q using trilinear interpolation. Subsequently we define the basis vector e v = v a /||v a ||, pointing along the apparent wind speed. Next, we introduce the perpendicular axis e ⊥ = e q × e v and the upward-pointing axis which is orthogonal to the plane spanned by {e v , e ⊥ } and reads e u = e v × e ⊥ . We apply a roll rotation of the perpendicular and normal axes about the 190 axis e v ≡ e D by an angle corresponding to the state variable Ψ, which results in the transversal and lift axes e T = sin(Ψ)e u + cos(Ψ)e ⊥ (5a)

Lifting-line model
The point-mass model provides the state and control variables that describe the flight path of the AWE system. The AWE 200 system however operates in a turbulent boundary layer characterised by local fluctuations of the wind speed magnitude and direction, which given the large-scale of the system's wing, considerably influence the force distribution along the wingspan.
Therefore, we supplement the original 3DOF point-mass model with the lifting-line model, which allows us to consider local fluctuations of (aero)dynamic quantities along the wingspan without adding complexity to the dynamics modelling and control of the aircraft. Hence, the wing is modelled as an aligned collection of discrete wing sections of width δs, for which we 205 compute the aerodynamic forces based on the local wind condition.
The orientation of the wing is derived from a series of state-based assumptions building on the point-mass basis B pm introduced above and resulting in a new orientation basis B = {e c , e t , e n }. First, we introduce the model-equivalent angle of attackα which we derive from the aerodynamic state C L and can be computed asα = C L /a + α L=0 for elliptical wings (Anderson, 2010). This model-equivalent angle of attack defines the a priori unknown orientation of the aircraft chord line 210 e c = cos (α) e D − sin (α) e L . The transverse axis of the body-fixed frame e t is aligned with the transverse axis of the pointmass frame e T and the normal axis of the body-fixed frame is given as the cross product e n = e c × e t .
The rotation matrix R associated to the body-fixed basis B can be formulated in terms of Euler angles (Zanon, 2015) and can further be expressed as an extrinsic sequence of three consecutive rotations about the axes of the inertial frame with the Euler angles Θ = (ψ, θ, φ), respectively representing yaw, pitch and roll angles, and the associated rotations The Euler angles are extracted from the entries r ij of the orientation matrix R (Licitra et al., 2019): 220 ψ = arctan (r 12 , r 11 ) , ψ ∈ (−π, π).
The angular velocity ω of the aircraft can be derived from the Euler angle dynamicsΘ . Accordingly, we first approximateΘ by a discrete backward differencė and finally compute the angular velocity as where, for the given sequence of rotations (Alemi Ardakani and Bridges, 2010), the mapping function inverse reads The approximations of the angular velocity ω and the body-fixed basis frame R can then further be used to define the position and speed of discretized wing elements of the AWE system.

230
From the state-based approximations of attitude and angular velocity of the aircraft, we can now evaluate the position and speed of each wing section k, defined at the section center, expressed in the inertial frame of reference as where s k is the distance from the point mass to the segment center given in the body-fixed frame. In this study, each AWES 235 wing is discretized using 61 actuator segments. The spatial discretization of the wing in the lifting-line model allows to account for the local fluctuations of the unsteady, three-dimensional wind speed. The local wind speed vector v w,k is hence interpolated at the locations q k from the wind speedṽ measured at the surrounding LES cells. We define the local apparent wind speed v a,k = v w,k −q k and compute the effective angle of attack α k seen by the airfoil of the wing section as

240
The local lift and drag coefficient c l (α k ) and c d (α k ) of each wing section are determined from aerodynamic polars obtained from 2D airfoil analysis mentioned in Sect. 3.1.
The air density at each wing section is computed from the standard atmosphere density model (Archer, 2013) and reads where p 0 = 1013.25 hPa and T 0 = 288.15 K are the reference pressure and temperature measured at sea level, g = 9.81 m s −2

245
is the gravity constant, Γ = 6.5·10 −3 K/m is the average lapse rate in the atmosphere and R = 287.058 J/(kg K) is the specific gas constant of dry air.
The chord length of each wing section,c k is computed from the elliptical planform distribution of the wing according tõ where b is the wingspan and AR the aspect ratio of the AWES wing. The values of b and AR are further discussed in Sect.
Similarly, for drag-mode AWE systems, the additional drag of on-board turbines is distributed over n g actuator segments, 255 covering the central third of the wingspan. We assume that the individual on-board turbines all operate at the same value of the generator thrust coefficient κ, such that the local generator drag reads for specific segments k ∈ (1, n g ) or is zero otherwise. The instantaneous, local section force per unit segment length f l q (q k ) at the time instant t l contains the sum of local lift forces, drag forces and eventual on-board turbine forces. This total force reads and can further be broadcast to Eq.
(2) to be later added to the LES flow equations, Eq. (1b), or integrated over the complete wingspan to define the instantaneous, total aerodynamic forces acting on the wing F l

AWE system dynamics
In the current procedure, the system configuration is described by its set of generalized coordinates q and its motion is re-265 stricted by the constraint c(q) = 0, to which we associate the Lagrange multiplier λ. For the tethered single-wing AWE system configuration considered in this study, the point-mass is constrained to the manifold described by In order to obtain an index-1 DAE that can be integrated with Newton-based numerical methods, an index reduction is performed by differentiating the constraint (22) twice . Baumgarte stabilization combined with periodicity 270 constraints in the optimal control problem then ensures that the consistency conditions c(q) = 0 andċ(q,q) = 0 are satisfied during the entire power cycle (Gros and Zanon, 2017). The motion of the AWE systems is then defined by the set of equations c + 2pċ + p 2 c = 0.
In Eq. (23a), m = m W + 1 3 m T and m = m W + 1 2 m T define the effective inertial mass and the effective gravitational 275 mass (Houska, 2007) with m W and m T representing the mass of wing and tether, and F q the aerodynamic forces acting on the system. In Eq. (23b), p = 10 is a tuning parameter of the Baumgarte stabilization scheme.
The total aerodynamic force acting on the system F q consists of the sum of wing forces F W from Eq. (21) and tether drag forces F T . The total tether drag (Argatov and Silvennoinen, 2013) is given by 280 with the tether diameter d T and the drag coefficient of a cylinder C cyl = 1.0 at high Reynolds numbers.
The kinematics of the systems (25) are further integrated in time using an internal four-stage fourth-order Runge-Kutta scheme 285 with a sub-step size δt as previously mentioned in Sect. 2.1.2.
Unfortunately, directly feeding the lifting line forces F W (21) into the AWE system dynamics (23) in combination with the NMPC controller, presented later in Sect. 2.4, is not possible: The mismatch between the aerodynamic forces generated by the lifting line and the aerodynamic model of the point-mass assumed by the controller does not lead to a stable system in our implementation. Therefore, in order to reduce the model mismatch between plant and controller, we compute the total wing 290 force at the point-mass position using the aerodynamic states as F W = F L + F D + F G , where the individual contributions of wing lift, wing drag and on-board turbine drag are given by

295
This approach, while it still takes into account the unsteadiness of the LES wind field when computing the apparent wind speed v a at the point-mass location from Eq. (4), results in a satisfactory tracking performance of the controller.

Controller: park supervision
The procedure defining the supervision level of the AWE park is shown in Fig. 1, part C. The supervision level contains two modules: the wind speed estimator, presented in Sect. 2.3.1, and optimal trajectory generator, presented in Sect. 2.3.2. For each 300 AWE system, the wind speed estimator samples the local wind conditions monitored at the wing and derives an approximation of the associated vertical wind profile. These wind profiles are further communicated to the trajectory generator and used to generate periodic power-optimal reference trajectories flown by each system individually.

Wind speed estimator
The controller does not have knowledge of the unsteady, three-dimensional wind field from the LES and hence assumes that 305 the wind field is given by a one-dimensional, height-dependent profile v w = (U (q z ), 0, 0), where the streamwise wind speed The logarithmic distribution is parametrized by a reference speed U ref given When extracting power from the wind, conventional and airborne wind energy systems alike exert a thrust force on the incoming flow field (Jenkins et al., 2001). This process results not only in a velocity decrease downstream of the system, the wake, but also in a velocity decrease upstream of the system, the induction region. Upstream flow induction reduces the 315 available power that wind energy systems can harvest. From one-dimensional momentum theory, it was shown for conventional wind turbines modelled as actuator disks that the velocity decrease across the actuator can be defined as where the unperturbed inflow wind speed is denoted as U ∞ , while U D and U W respectively represent the wind speed at the 320 actuator disk and in the wake. The axial induction factor a quantifies the strength of the induction phenomenon. The values of axial induction factors reported in AWE literature (Leuthold et al., 2017;Haas and Meyers, 2017;Kheiri et al., 2018) are lower than the known Betz limit a = 1/3 of conventional wind turbines (Jenkins et al., 2001), suggesting that axial induction is less significant for airborne wind energy systems although it cannot be fully neglected. In the current approach, the LES-based wind velocity sampled by the wind speed estimator already contains the effects of axial induction. Hence, optimal trajectories the mean wind speed measured at the point-mass for the duration of the sampling period T s is given by The mean wind speed, integrated from the logarithmic profile over the same interval h, is a linear function of the actuator speed and is given by Hence we can derive the reference value of the logarithmic wind profile that best approximates the wind conditions monitored by each AWE system during the sample period T s as We further use this value to generate the associated reference trajectories as we show in Sect. 2.3.2.

Generation of reference trajectories
In this study we generate a set of reference power-optimal flight trajectories by using optimal control techniques. The Lagrangian based modelling procedure applied to a tethered point-mass proposed Sect. 2.2.1 results in an implicit index-1 differ-345 ential algebraic equation (DAE) given by (23) and (25). This representation can be summarized by In addition to the state and control vectors x and u, and the algebraic variables z = (λ) previously defined, we introduce an additional optimization parameter θ and a set of constant parameters p, which allow us to apply the optimization problem for a number of AWE designs and wind conditions. The optimization parameter θ = d T is the tether diameter and is optimized During operation, the AWE systems have to satisfy a set of constraints h(ẋ(t), x(t), u(t), z(t), θ, p) ≤ 0 that ensure a hardware-friendly operation and the manoeuvrability of the aircraft. Hence, for the dynamics (32) and constraints h, we can 355 formulate a periodic optimal control problem (POCP) with a free time period T p as: The cost function is defined as the average mechanical power output of the system generated over a full cycle of period T p .
The instantaneous power generated by lift-and drag-mode AWE systems respectively read For lift-mode AWE systems, power is generated by unwinding the tether from the winch at a speedl under the high tether 365 tension T T = λl generated by the aerodynamic forces acting on the wing. For drag-mode AWE systems, power is extracted from the wind by on-board generators. These generators are modelled as actuator disks such that the power is equivalent to the turbine thrust force F G = κ||v a || 2 multiplied by the wind speed measured at the disk η r ||v a ||. Here, η r = 0.8 is the rotor efficiency and characterizes the ability of the on-board turbines to extract power from the surrounding flow, ie. the axial induction associated to the actuator disk assumption. In line with (Echeverri et al., 2020), where η r is defined as "rotor efficiency 370 from thrust power to shaft power", Eq. (35) defines the mechanical power that drag-mode AWE systems can extract from the wind.
The trajectory and performance of AWE systems depend on the path constraints and variable bounds specified in the periodic optimal control problem 33c. Tables 1 and 2 summarize the chosen constraints and bounds specified for trajectory optimization. The path constraints and variable bounds used in this work are compiled from the available literature (Gros and Diehl,375 2013; Licitra et al., 2016;Zanon et al., 2013b;Kruijff and Ruiterkamp, 2018) and are adapted to large-scale AWE systems.
These choices are motivated hereafter: We set an operational upper limit to the tether length l max = 1000 m. We also limit the operation range of the winch by setting bounds to the tether speedl and accelerationl. In addition, the aerodynamic lift coefficient of the wing C L is restricted to the range [0.0, C L,opt ], where C L,opt = 1.142. The lower bound ensures that the tether tension remains positive at all time while the upper bound limits local stall along the wing. The roll angle Ψ is bounded 380 in order to ensure safe operation, ie. avoid collision between the tether and the airframe, and also take into account the reduced manoeuvrability of the large-scale aircraft. Next, the maximal tether tension and the maximal acceleration are limited to ensure the integrity of the aircraft. The former also ensures that maximum tether stress is not exceeded by considering a safety factor f s = 3 and a tether yield strength σ max = 3.09 GPa. Further, we constrain the operation range of the control variables u = Ċ L ,Ψ,κ, ... l in order to avoid aggressive pitch and yaw manoeuvres which are not suited to large-scale aircraft. The 385 spatial footprint of the trajectories is limited by means of spatial constraints limiting the axial, lateral and vertical extents of the trajectories in order to achieve system packing densities of 2/L 2 , as later discussed in Sect. 3.3.
In practice, the reference trajectories are optimized off-line for a specific range of wind speeds. We aim to operate the AWE systems in below-rated from regime, the so-called region II (Leuthold et al., 2018), where the harvested power increases with the wind speed, hence we generate a library of optimal trajectories (OTL) for a range of actuator-based wind speeds  Table 1. Bound constraints applied to state variables x used in optimal control problem (33). The maximal lateral extent of the flight path and the maximal tether length are respectively set to ymax = 250 m and lmax = 1000 m to limit the footprint of the trajectories. The optimal value of the lift coefficient is CL,opt = 1.142.

Controller: flight path tracking
The procedure defining flight path tracking is shown in Fig. 1, part D. In the LES-generated virtual wind environment, the operation conditions of the AWE systems differ substantially from the model assumptions in Eq. (27). The complex dynamics make the motion of the AWE system highly sensitive to fluctuations. Therefore a control algorithm is required to lead the 400 system onto its reference trajectory. Accordingly we apply non-linear model predictive control (NMPC) .
For a moving time horizon, the NMPC repeatedly computes the optimal control inputs that reduce the error between the current states x and the states x r of the reference flight path taken from the OTL. Therefore, we can formulate for the prediction horizon  Table 1 and Table 2.
The cost function (36a) is formulated as a least-square objective, defining the tracking error between the current states x(t) 410 and the given reference x r (t), with a penalization on the deviation from both the reference states and controls. Just like for the generation of periodic optimal trajectories, the constraints (36b) and (36c) enforce the system dynamics and path constraints, while the initial condition (36d) ensures that the initial states match the current estimatex 0 . An additional terminal condition (36e) ensures the system finds back the reference at the end of the prediction horizon. Some of the path constraints in Eq.
(36c), in particular the aerodynamic states and the control variables, are relaxed compared to the POCP bounds in tables 1 and 415 2. This allows the controller to handle wind disturbances and plant-controller model mismatch with a wider range of steering strategies. Table 3 summarizes the NMPC constraints and bounds adapted for trajectory tracking. The maximal allowed lift coefficient C L,max ≈ 1.37 corresponds to the upper bound of the validity range of the linear lift slope. In addition, the maximal allowed tether tension and acceleration are relaxed by respectively 10% and 20%.
To solve the optimal control problems, we use the awebox toolbox (awebox, 2021). In the toolbox, both trajectory gener-420 ation and flight path tracking OCPs are discretised using the direct collocation approach based on a Radau scheme with order 4 polynomial. The resulting nonlinear program is formulated using the symbolic framework CasADi (Andersson et al., 2018) and solved with the interior-point solver IPOPT (Wächter, 2002) using the linear solver MA57 (HSL, 2011). More information about the implementation can be found in (De Schutter et al., 2019).

Simulation setup
The different AWES park configurations investigated in the current work are presented hereafter. First, we detail the design choices behind the large-scale lift-and drag-mode AWE systems in Sect. 3.1. Next we discuss the mode-specific operation of both the lift-and drag-mode AWE systems, in particular the effects of wind speed on the system performance, in Sect.
3.2. Finally, we specify the computational settings of the performed LES of the atmospheric boundary layer and present the 430 different AWES park layouts used in this study in Sect. 3.3.

Design of large-scale AWES
For given wind conditions and space constraints, wind farm developers design their parks such as to optimize several design metrics. Traditionally, these metrics include the annual energy production (AEP) and the levelized cost of energy (LCOE) (Dykes, 2020). In order to minimize the LCOE, wind farms of conventional wind turbines in the range 10-15 MW target power densities of approximately 5 MW km −2 (Bulder et al., 2018). For the large-scale deployment of their technologies, AWE manufacturers have engaged in the development of utility-scale multi-megawatt AWE systems for offshore operation generating up to 5 MW of power (Kruijff and Ruiterkamp, 2018;Harham, 2012). Also, the AWE manufacturer Ampyx Power aims at achieving farm level power densities in the range of 10-25 MW km −2 with their commercial systems (Kruijff and Ruiterkamp, 2018). Accordingly, when designing parks of AWE systems, one has to define first the type of AWE systems used 440 to harvest power and second the layout of the farm that will ensure a safe operation while maximizing the power density.
Hence, we propose a generic design for lift-and drag-mode AWE systems that is easily scalable and allows one to model the operation of AWE systems in various wind conditions. The wingspan b of the wing is chosen as the driving design parameter to describe the dimensions of the different systems while the wing aspect ratio is fixed to AR = 12, similar to the AWE systems developed by Ampyx Power. Design considerations such as manufacturability or structural analysis of the wing lie outside of 445 the scope of the study. Therefore, a wing with elliptical planform is chosen here for simplicity. Furthermore, elliptical wings exhibit a low induced drag and a constant downwash along the wingspan, and also provide an analytical formulation of the aerodynamic lift and drag coefficients of the wing (Anderson, 2010) where α g is the geometric angle of attack of the wing. In addition, α L=0 and C d,0 respectively represent the zero-lift angle of 450 attack and profile drag coefficient of the airfoil. The lift slope a of the finite wing is given by a = a 0 / (1 − a 0 /(πAR)) with a 0 = 2π the airfoil lift slope from inviscid flow theory.
In order to achieve a high lift-to-drag ratio, we opt for airfoil sections of type SD 7032, also used by the manufacturer TwingTec (Gohl and Luchsinger, 2013). The remaining necessary dimensions of the AWE systems are derived from a semi-empirical study based on available manufacturer data, where we derive the wing mass and tension limits from the wingspan b. We set an upper bound to the tether 470 tension and assume that it scales linearly with the surface area of the wing. Further we assume that the targeted AWE system with a 60 m wing with aspect ratio 12 can sustain up to approximately 850kN of pulling force from the tether. Further, the wing mass is estimated by fitting a power law onto data from the manufacturer Makani Power taken from (Harham, 2012) as shown in Fig. 6. Hence wingspan-based relations for the maximal tether tension and the wing mass read with the parameters γ 1 ≈ 2840.24, γ 2 ≈ 0.1478, γ 3 ≈ 2.662. Note that for lift-mode AWE systems, the wing mass is reduced by 25% in order to account for the absence of on-board turbines. Finally, the tether material is assumed to be Dyneema for both types of AWE systems. This material is commonly used by AWE manufacturers and has a density ρ T = 970.0 kg/m 3 . 12.0 m s −1 . We design and optimize one drag-mode AWE system and two different lift-mode AWE systems, of which we discuss the operation characteristics in Sect. 3.2.1. of reel-out speed on the instantaneous flow conditions experienced by the system. Therefore, we use two different reel-out strategies for the operation of lift-mode AWE systems, which we discuss in Sect. 3.2.2. The details of the single AWE system analysis are given in appendix A. The first strategy uses the hypothetical physical limitations of the winch, where we assume that the maximal reel-out speed isl max = 20.0 m s −1 as given in Table 1. However, given that the optimization procedure does not take the generation of the system wake into account, the wing can potentially interact locally with its own wake during 490 the reel-out phase as discussed in Sect. 3.2.2. Therefore, the second strategy limits the maximal reel-out speed to the expected advection speed of the wake U W as estimated in Eq. (28). While the value of the induction factor a is a priori unknown, we opt for the conservative guess a = 0.25, such that the maximal reel-out speed of the tether becomes wind speed-dependent and is shown to reaḋ

Operation at rated wind speed
At rated wind speed, the drag-mode AWE system and the two lift-mode AWE systems all generate 5 MW of power. However, their flight path and power generation profile differ substantially as shown in Fig. 7. A complete description of all system states, controls and additional metrics is given in appendix B, and their main features are summarized here.   With the second strategy, the reel-out speed of the tether is capped by the induction-based constraint such that the tether is reeled out at a nearly constant speed. Accordingly, the generated power remains nearly constant during the generation phase. Given the reel-out speed limitation, part of the kinetic energy of 520 the system is used to accelerate the system during the downward flight resulting in large fluctuations of the flight speed in the range 60 m s −1 to 90 m s −1 .

Characteristics of lift-mode operation
In order to assess the different reel-out strategies, we investigate in details the operation of individual lift-mode AWE systems.
For this analysis, the systems operate in a high-resolution logarithmic inflow with reference wind speed U D = 10.0 m/s without 525 ambient turbulence and their trajectories are optimized for the same wind conditions. We further assume that the wind conditions remain constant over time and the systems are "perfectly steered", suggesting that the systems can perfectly follow their reference flight path with the reference controls, hence making the controller part of the framework obsolete for this analysis.
The detailed analysis is given in appendix A, which also includes a grid convergence study and an evaluation of the different actuator methods introduced in Sect. 2.1.2. Figure 9. Instantaneous values of streamwise velocity vx measured at several locations along the wingspan for (a) lift-mode AWE system with original bounds and (b) lift-mode AWE system with induction-based limitation of reel-out speed. The prefixes "p" and "s" stand for the port and starboard sides of the wing, while the prefix "c" stands for the center, ie. the point-mass. Furthermore, the numbers stand for the segment positions on each side of the wing, from 0 at the center to 30 at the wing tip. Figure 8 shows the structure of the wake flow for both lift-mode AWE systems at the end of the reel-out phase. During the power-generation phase, the wing is subject to high lift forces such that tip vortices emanate from the wing tips and are transported downstream by the background flow. These wing tip vortices generated during the individual loops start to interact with each other in the wake and eventually merge into a more significant single structure, which eventually breaks down further downstream due to the effect of mixing. In particular, the figure shows the effect of the reel-out strategy: while with the first 535 reel-out strategy the individual tips vortices of each loop cannot be precisely identified, the second, induction-based strategy limits the interaction between each loop, resulting in more distinct structures. The interaction can however not be completely prevented, such that the individual structure eventually merge later downstream. The effects of the reel-out strategy can also be seen on the instantaneous flow conditions monitored at seven equidistant locations along the wingspan as shown in Fig. 9.
For the first strategy, we observe large fluctuations of the instantaneous wind speed. Sharp drops of velocity are measured 540 during the lower part of each power-generation loop suggesting that the wing suddenly encounters a region of low wind speed that we can identify as the wake. Furthermore, the intensity of the velocity drop increases as the system reels out further downstream, indicating that the individual wakes of each loop are combined into a single wake. With the second strategy, the patterns are less distinct and the magnitude of the fluctuations less significant, hence indicating that the interaction between wing and wake is limited. We still observe that the outer section of the wing, referred as p020 and p030, experiences very sharp 545 fluctuations at the end of reel-out phase, suggesting that the outer wing tip flies through some wake region. However, these fluctuations are very local and temporary, while for the first reel-out strategy most of the wing is interacting with the wake.
Given the limited interaction between the wing and the wake when using the second reel-out strategy, ie. with a inductionbased upper limit of the reel-out speed, we will perform the lift-mode AWES farm simulation with the second design of lift-mode AWE system.

Operation of large-scale AWES in below-rated regime
When operating in a park, AWE systems are subject to large-scale wind speed variations and wakes of upstream systems. As a result, the operation of each individual system evolves as it experiences different wind conditions. Hence, we discuss here the effect of varying wind speed on the trajectories of drag-and lift-mode AWE systems. The power curves and selected metrics for the expected range of wind speeds 5.0 m s −1 ≤ U D ≤ 12.0 m s −1 are shown in Fig. 10, while the the wind-speed dependency 555 of the flight path is shown in Fig. 3. A complete description of the effects of varying wind conditions on all system states, controls and additional metrics are given in appendix B For the expected range of wind speeds, the generated power ranges from approximately 1 MW to 5 MW for both operation modes. For lift-mode AWE systems, decreasing wind speeds mainly affect the reel-out patterns. At low wind speeds, the system only reels-out, and hence generates power, during the downward part of the loops. In comparison, the upward flight 560 is performed at constant tether length under fluctuating tether tension. Consequently, the increase in tether length during the reel-out phase is much more limited and hence the flight path more compact.
For the drag-mode AWE system, the shape of the single-loop flight path barely changes for varying wind speeds. However, the most noticeable difference is the contribution of the on-board turbines at low wind speeds. During the upward flight, the on-board turbines switch to propeller-mode in order to overcome gravity and keep a constant flight speed and tether tension, 565 and hence consume some power.

LES setup and AWES park layout
The turbulent ABL is modelled according to the procedure introduced in Sect. 2.1. In order to capture all the relevant spatiotemporal scales of the turbulent ABL in the precursor simulations and the wake effects in the AWE park simulations, the flow domain spans over several kilometres and is resolved with high spatial resolution. The dimensions of the simulations domain are 570 L x ×L y ×L z = 10.0×4.0×1.0 km 3 . The domain is discretized into cells of size ∆ x ×∆ y ×∆ z = 10.0×10.0×5.0 m 3 , resulting in a grid resolution N x × N y × N z = 1000 × 400 × 200 of up to 80 millions cells for both main and precursor domains. Note that temporally resolved subsets of the generated precursor simulations are available for download from the Zenodo platform (Haas and Meyers, 2019).
For the utility-scale deployment of their technology, Ampyx Power envisions park power densities in the range of 10 − 575 −25 MW km −2 while targeting a system packing of about 2/L 2 , where L is the maximal tether length (Kruijff and Ruiterkamp, 2018). Accordingly, we investigate three different AWE parks configurations with the same packing density, where each park is composed of 25 systems ordered in 5 rows and 5 columns with an equidistant spacing between each system, ie. L in the streamwise directions and L/2 in the spanwise direction. The top and side views of the three different AWE park layouts are shown in Fig. 11. The first AWE park operates with lift-mode AWE systems and the reference layout length is set to L = 1000 m. This park layout is the densest possible layout to be operated with the designed lift-mode AWE system, such that the allocated areas do not overlap, and will further be referenced as L1 (Lift-mode park 1). The second AWE park operates with drag-mode AWE systems and the same layout as L1 and is referenced as D1 (Drag-mode park 1). These two AWE parks can achieve farm power densities up to 10.0 MW km −2 when all systems operate at rated wind speed. The third AWE park also operates with 585 drag-mode AWE systems but with a reference layout length L = 600 m. This AWE park, referenced as D2 (Drag-mode park 2), corresponds to the densest possible layout for the designed drag-mode AWE system and can achieve a farm power density up to 27.8 MW km −2 in rated wind conditions. Note that the flight path constraints in Table 1 ensure a safe operation in accordance with the area allocated to each system in all park configurations.

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In the following section, we present the main outcomes of the simulations of AWE parks in the turbulent boundary layer.  Figure 13 shows time-averaged contours of normalized streamwise velocity deficits and turbulent kinetic energy averaged over 600 one hour of AWE park operation. The wake field develops through the parks as individual wakes add up with the wakes induced by consecutive AWE systems. We observe in the three cases that downstream system rows operate in fully waked conditions. At the last row of the park, the mean velocity deficit, relative to the ABL precursor, reaches about 10% for the lift-mode park (L1), while for the drag-mode parks the velocity deficit increases to about 20% (D1) and up to 30% (D2). In the trail of the annuli swept by the wings, these velocity deficits are further accentuated and can reach up to 20% for the lift-mode AWE 605 systems (L1) and respectively 40% and 50% for the drag-mode AWE systems (D1 and D2). For the drag-mode AWES park with dense layout (D2), the blockage effect of the drag-mode AWE systems also results in an acceleration of the mean flow up to 10% around the park, which cannot be observed for the two AWE parks with moderate layout (L1 and D1).
The mean turbulence intensity T I of the current ABL simulation ranges between 2% and 8% and is about 3% at operation altitude. Hence the power extraction and the wake mixing in the AWE park lead to large increases of TKE levels. The observed  The higher packing density of the second drag-mode AWE park (D2) exhibits the same phenomena as park D1 but with much higher intensity, given the reduced space available for wake recovery.

Flow characteristics inside the parks
We can further deepen the analysis by looking at vertical and horizontal profiles of velocity deficit and turbulent kinetic energy 620 as shown in Fig. 14. The profiles are shown one diameter upstream of the trajectory center for AWE systems located in the central column of the parks. For the lift-mode system, the diameter of the trajectory is approximately 240 m and its center is located 645 m downstream of the ground station at an altitude of 220 m. Equivalently, for the drag-mode system, the diameter

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The profiles provide an overview of the local flow distribution experienced by each system in downstream locations of the AWE column. The profiles confirm the prior observations relative to the downstream development of the wakes through the park: for the lift-mode AWE park, the wake is much more widespread and therefore induces a weaker velocity deficit. In contrast, for drag-mode AWE parks, the wake exhibits more localized and stronger velocity deficits, highlighting the annular shape of the wake. Each park configuration shows higher velocity deficits at the bottom half of the loops, induced by higher 630 loadings on the wing. In addition, wake recovery in the upper part of the loop is eased by higher turbulence levels.
The wake impacts are much stronger for the drag-mode AWE park (D2) with a higher packing density. The horizontal profiles show that the wake merges with the wakes of neighboring AWES columns as it radius extends laterally the further downstream it progresses. Hence, wakes do not only aggregate as they advect downstream but also merge in the spanwise direction, therefore resulting in the stronger velocity deficits and higher TKE levels observed for the higher farm power density. with some delay through the downstream rows of the park. However, the large decreases of tracked speed for the downstream systems are due to the effects of upstream wakes. As discussed above, drag-mode AWE systems experience much stronger wakes and hence track lower wind speeds than lift-mode AWE systems. In particular, a larger decrease is observed from the front row to the second row. 650 We can further look at the instantaneous streamwise wind speed measured in-flight at the wing as shown in Fig. 16 for two systems located in the front and back rows. Despite the ambient turbulence of the ABL flow, the wind speed monitored by  This indicates that the wing may still interact with its own wake in some sections of the flight path. In order to ensure a reliable operation, the reel-out speed limitation was lifted off for flight path tracking to allow a better reactivity of the systems to the unsteady wind conditions. Hence, the systems can achieve higher reel-out speeds than the reference.
The strong velocity deficits and the higher levels of TKE observed in the back rows of the parks greatly impact the instanta-660 neous wind speed monitored. We observe a much larger range of fluctuations for AWE systems located in the last row than in the first row. The effects described for the front row lift-mode system are accentuated for the back row system, suggesting the presence of compact wake regions from upstream systems advecting through the park.
The unsteady wind conditions also impact the tracking accuracy of the NMPC controller. Figure 17 shows

Power performance at system and park level
The unsteady wind conditions not only affect the tracking behaviour of each AWE system but also impact their performance. tension that requires to be compensated by steering manoeuvres. These tether tension variations negatively impact the amount of generated mechanical power such that we observe power losses of about 10% to 15% relative to the reference power of the tracked trajectories. For the two drag-mode park configurations, the wind fluctuations barely impact the operation of the systems and we observe a good tracking of the power generation profiles. In the back row however, the tracked flight path and the low wind speed available require the on-board turbines to operate as propellers in order to overcome gravity while flying 680 upward. Hence, some drag-mode systems end up also consuming some of the harvested power. Note that the tether tension maxima monitored at each system match the relaxed upper bound used for flight path tracking as discussed in Sect. 2.4.
We further address the power extraction process from the perspective of the complete AWE park, and in particular from the system location in the park. Figure 20 shows the mean power generation of each row relative to the reference power of the tracked trajectories for the three different park configurations. In this way we can quantify the power losses due to local wind   drag-mode AWE parks, the wake losses are much more important and reach approximately 25% in the last row of AWE park D1 and up to 45% for the AWE park with denser layout D2. While for the lift-mode AWE park the power decrease across the subsequent rows is progressive, for drag-mode AWE parks, the power losses increase abruptly between the first and second row. This drop account for about 62% and 52% of the row losses for the park configurations D1 and D2, respectively.

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In this study we have investigated the wake interaction and power extraction for large parks of utility-scale AWE systems.
Flight path tracking by means of model predictive control (MPC) has proven to be a robust control strategy for both lift-and drag-mode systems when operating in turbulent wind conditions: the path tracking results in position offsets generally up to 12 m, while satisfying the set variable bounds and tracking quite accurately the power profiles, in particular in drag mode.
Lift-mode AWE systems encounter however important fluctuations in local wind speeds during operation, which can lead to 705 temporary large decreases of tether tension and significant reductions of instantaneous power. First, we have observed that lift-mode systems can interact with their own wake: during reel-out, the wing can fly (partially) through or in the direct vicinity of the wake induced in the previous power-generating loops. Hence, the interaction with its own wake needs to be considered already during the design phase: when generating optimal reference trajectories, it is crucial to define reeling strategies that avoid these situations and incorporate induction phenomena in the modelling procedure. This is an active topic in current AWE 710 research. Second, these large wind speed fluctuations can be associated to the characteristic wakes induced by lift-mode AWE systems: while the flow forcing is negligible during the retraction phase, the wakes induced during consecutive loops of the power-generation phase tend to merge into a single wake structure. This interaction leads to compact, low-speed flow regions which get advected downstream with the mean flow and negatively impact the operation of downstream systems.
Moreover we showed that for the three configurations investigated with our simulation framework, wake losses of utility-715 scale AWE parks are significant. We observed wake-induced performance losses of up to approximately 15% in the last row for lift-mode AWE park. For the two drag-mode AWE parks with moderate and dense farm power densities, the wake-induced power losses increase up to 25% and 45% respectively in the last row of the parks. Hence, the layout of the farm also plays a considerable role in the performance of the park. However, note that wake losses and tracking losses/gains can not completely be isolated from each other: improving the tracking behaviour of lift-mode systems by limiting the large tension fluctuations 720 in the tether would result in a stronger axial flow induction, hence leading to larger wake losses and vice versa.
Furthermore, the induced wakes display very characteristic annular shapes: the strength and the width of the wakes highly depends on the operation mode and the park layout. The wake profiles show quite large deviations from the logarithmic mean flow distribution assumed by the high-level controller. Therefore in the future, we can implement supervision strategies that take the wake shape into consideration and make use of the inherent flexibility and adaptability of AWE systems: we can 725 further use the wake profiles to generate new optimal flight trajectories in which individual systems adapt the elevation and azimuthal angles of operation to eventually avoid the upstream wakes. Furthermore, we can improve both the modelling of the atmospheric boundary layer and the AWE systems: first, the description of the ABL flow can be enhanced by including thermal effects and geostrophic wind into the LES framework (Allaerts and Meyers, 2015). Second, the complex AWE system dynamics can be better characterized by more detailed models such as the 6DOF rigid-wing model (Malz et al., 2019). The   Ahead of the investigation of utility-scale AWE parks, we have conducted a series of single-AWES analysis in order to assess the computational setup used in this study. In this appendix, we address in particular the tuning of the actuator parameters, the 745 detailed analysis of wake structure by means of high-resolution simulation, the grid-dependency of wake characteristics and the convergence of the supervision level of the controller. Figure A1 shows the computational setup used in this study. The domain dimensions are L x × L y × L z = 2800 × 1000 × 1000 [m 3 ]. We use three different resolutions to discretize the flow domain as specified in Table A1. The AWE systems operate in a sheared inflow without any ambient turbulence which is parametrized by a logarithmic profile given in Eq. (27) for comparison: Over the period of one full periodic cycle, the model mismatch between the forces computed with actuator methods and the reference OCP forces is less than 4%.

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Given the high flight speed of the system, the wing navigates through several grid cells during one time step for the simulations with ∆t = 0.1 s and ∆t = 0.05 s. Therefore, the ALM requires the smallest time step, ∆t = 0.025, to ensure a smooth transition as the wing sweeps across the flow domain. This results however in an fourfold increase of the computational effort.
The ALM simulation with ∆t = 0.025 hence serves as reference simulation. The ASM technique can be applied to simulations with ∆t = 0.1 and allows to approximately reproduce the unsteady forcing of the reference ALM with an error less than 2%.

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Increasing the value of the time filter constant τ f tends to reduce the error but also increases the computational cost. Hence, in order to keep a reasonable computational cost and achieve a sufficient accuracy, we perform the AWE farm simulations with the ASM techniques using τ f = 2∆t. Figure A3 shows the structure of the wake flow for each AWE system simulated with ASM on the fine grid by visualizing positive levels of Q-criterion (Jeong and Hussain, 1995). For the drag-mode AWE system, tip vortices emanate continuously 770 from the wing tips and are transported downstream in a coherent manner by the background flow. For the lift-mode AWE systems, the wing tip vortices generated during the individual loops start to interact with each other in the wake and eventually merge into a more significant single structure, which eventually breaks down further downstream due to the effect of mixing.
The pumping behaviour of the lift-mode systems is also clearly visible: during the retraction phase, the aircraft takes a different path while being subject to a much smaller wing loading, hence interrupting the wake generation for a short amount of time.

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The grid resolution and the actuator settings enable to resolve individual flow structures, and provide interesting insights in the complex characteristics of the wake flow of single AWE systems. Further, we can use the flow statistics gathered with the fine grid resolution as a reference when assessing the grid dependency of the simulation framework. Figure A4 shows time-averaged profiles of axial flow velocity and turbulent kinetic energy at different locations in the wake of the drag-mode AWE system for three different grid resolution. The velocity profiles show 780 that all resolutions can capture the general annular shape of the wake and its strength up to 1200 m behind the system. This good agreement allows us to use the coarse grid resolution for the AWE park simulations, where the downstream spacing between two rows of AWE systems does not exceed 1000 m, and which would be too computationally expensive to perform on the finer resolutions. In terms of resolved turbulent kinetic energy, the medium and coarse grid cannot capture the same levels of turbulence as the fine grid. This is however less crucial given that in the fully turbulent ABL simulations, ambient turbulence 785 triggers wake breakdown and turbulent mixing earlier than in the turbulence-free, sheared inflow investigated here.
Finally we can also verify the convergence of the supervision level of the controller and assess the different reel-out strategies employed for the lift-mode systems. Figure A5 shows the tracking profiles of the three AWE systems by visualizing the timeaveraged wind speed values sampled by the controller and the associated reference velocity of the tracked trajectories. All systems show good convergence towards an asymptotic value between 9 m/s and 10 m/s. As discussed in Sect. 3.2.2, lift-mode  the induction-based reel-out strategy used by lift-mode system (1) allows to track higher wind speed references, showing that the wing experiences less interaction with its wake. Therefore, this strategy also allows to operate the system along trajectories generating more power, and hence increases the performance of the AWE park. -drag_mode_awes_opt_trajectories.pckl: Drag-mode AWES.
The states, controls and other quantities can be visualized over a periodic cycle, as shown for lift-mode AWE systems in Figures   B1 and B2 and for drag-mode AWE systems in Figures B3 and B4, using the Python script: -visualize_reference_trajectories.py The instantaneous in-flight values of states, controls and additional metrics are monitored and can be compared to their -drag_mode_farm_layout1_awes_XXX.pckl: Drag-mode AWES park D1.
The record of tracked references is also organized as Python dictionary and is stored as pickle object for the three park configurations of the study. The size of the tracking record files is about 1.4 MB and the files are stored as:
Each monitored quantity can be compared to its reference for a given time horizon, as shown for lift-mode AWE systems in Figures B5 and B6 and for drag-mode AWE systems in Figures B7 and B8, using the Python script: