Offshore and onshore ground-generation airborne wind energy power curve characterization

Abstract. Airborne wind energy systems (AWESs) aim to operate at altitudes well above conventional wind turbines (WTs) and harvest energy from stronger winds aloft. While multiple AWES concepts compete for entry into the market, this study focuses on ground-generation AWES. Various companies and researchers proposed power curve characterizations for AWES, but no consensus for an industry-wide standard has been reached. An universal description of a ground-generation AWES power 5 curve is difficult to define because of complex tether and drag losses as well as alternating flight paths over changing wind conditions with altitude, as compared to conventional WT with winds at fixed hub height and rotor area normalization. Therefore, this study determines AWES power and annual energy prediction (AEP) based on the awebox optimal control model for two AWES sizes, driven by representative 10-minute onshore and offshore mesoscale WRF wind data. The wind resource is analyzed with respect to atmospheric stability as well as annual and diurnal variation. The wind data is categorized using 10 k-means clustering, to reduce the computational cost. The impact of changing wind conditions on AWES trajectory and power cycle is investigated. Optimal operating heights are below 400 m onshore and below 200 m offshore. Efforts are made to derive AWES power coefficients similar to conventional WT to enable a simple power and AEP estimation for a given site and system. This AWES power coefficient decreases up to rated power due to the increasing tether length with wind speed and the accompanying tether losses. A comparison between different AEP estimation methods shows that a low number of clusters 15 with three representative wind profiles within the clusters yields the highest AEP, as other wind models average out high wind speeds which are responsible for a high percentage of the overall AEP.

of WRF simulation between September 2015 and September 2016. The area surrounding the airport mostly consists of flat agricultural land with the town of Pritzwalk to the south and is therefore a fitting location for wind energy generation (See (Sommerfeld et al., 2019a) and (Sommerfeld et al., 2019b) for details). The FINO3 research platform in the North Sea (lat: 55 • 11, 7 N, lon: 7 • 9, 5 E) was chosen as a representative "offshore" location due to the proximity to several offshore wind farms and the amount of comprehensive reference measurements (Peña et al., 2015). The offshore simulation covers the time 70 frame between September 2013 and September 2014.

Mesoscale model
The mesoscale simulations in this study were carried out using the weather research and forecasting (WRF) model from (Skamarock et al., 2008). The onshore simulation was performed with version 3.6.1 before the 2018 release of WRF version 4.0.2 1 in which the offshore simulations were computed. The setup of the model has been adapted and constantly optimized 75 for wind energy applications by the authors in the framework of various projects and applications in recent years (Dörenkämper et al., 2015(Dörenkämper et al., , 2017Dörenkämper et al., 2020;Hahmann et al., 2020;Sommerfeld et al., 2019c).
The focus of this study is not on the detailed comparison between mesoscale models, but on AWES performance subject to representative onshore and offshore wind conditions determined based on clustered wind profiles (described on section 3). To that end, both WRF models provide adequate wind data for our purposes. data Both simulations consist of three nested domains 80 centered around either the FINO3 met mast (see Figure 1) or the Pritzwalk Sommersberg airport. Atmospheric boundary conditions are defined by ERA-Interim (Dee et al., 2011) for the onshore location and by ERA5 (Hersbach and Dick, 2016) reanalysis data for the offshore location, while sea surface parameters for the offshore location are based on OSTIA (Donlon et al., 2012).
These data sets have proven to provide good results for wind energy relevant heights and sites (Olauson, 2018;Hahmann et al., 2020). Both simulations use the MYNN 2.5 level scheme for the planetary boundary layer (PBL) physics (Nakanishi 85 and Niino, 2009). While the onshore simulation was performed in one 12 month simulations (01.09.2015 -31.08.2016), the offshore simulation period consisted of 410 days (30.08.2013 -14.10.2014) that were split into 41 simulations of 10 days each with an additional 24 h of spin-up time per run. The data from the mesoscale models' sigma levels (terrain-following) were transformed to the geometric heights using the post-processing methodology described in (Dörenkämper et al., 2020).  High-Performance Computing clusters at the University of Oldenburg. Directional variability decreases and wind speed increases with height, following the expected trends in the northern hemi-95 sphere (Arya and Holton, 2001;Stull, 2012). Average onshore wind direction rotates about 14 • between 100 and 500 m, whereas average offshore wind direction only changes approximately 5 • . Offshore conditions veer about 10 • degree above 2 EDDY: HPC cluster at the Carl von Ossietzky Universität Oldenburg, see: https://www.uni-oldenburg.de/fk5/wr/hochleistungsrechnen/hpc-facilities/eddy/ 500 m, resulting in the same westerly wind direction at high altitudes. Due to prevailing unstable conditions offshore, a strong mixing with height is found resulting in less veer across the heights investigated in this study The relative wind speed increase of the offshore location is lower compared to the onshore location due to lower surface roughness and the already high wind 100 speeds at lower heights. Figure 3 shows the annual horizontal wind speed probability distribution for both locations. These statistics give an insight into the overall wind conditions, but the actual profile shapes, which are important for AWES power and trajectory optimization, are lost in this evaluation. The chosen nonlinear color range allows for the representation of the entire relative probability range.

Wind regime
Onshore (left) wind speeds have a fairly narrow range below 300 m, due to dominant surface effects. Above this height the 105 distribution broadens, but a high probability of low wind speeds remains up to high altitudes. This leads to the development of bimodal characteristics caused by different atmospheric stratification. Low wind speeds are commonly associated with unstable and high wind speeds with neutral or stable atmospheric conditions (see sub-section 3.2).
Such multimodal distributions at higher altitudes are better described by the sum of two or more probability distributions, as standard Weibull or Rayleigh distributions can not capture this phenomenon (Sommerfeld et al., 2019a). Offshore (right) 110 wind speeds on the other hand have a wider distribution at all heights as they are less inhibited by surface effects. Similar to onshore, the offshore frequency distribution also shows a high probability of lower wind speeds (between 5-10 ms −1 ) at all heights. As mentioned above, the relative wind speed increase with height is less pronounced offshore than onshore. Higher wind speeds at lower altitudes benefits conventional WT and weakens the argument for offshore AWES as one of their benefits would be to harness energy from the stronger winds at higher altitudes. However, offshore AWES will also benefit from higher 115 5 https://doi.org/10.5194/wes-2020-120 Preprint.  Figure 2. Annual onshore and offshore wind direction and speed statistics for 100 and 500 m presented as wind roses. On average wind direction onshore rotates about 14 • while offshore winds rotate about 5 • between 100 and 500 m. Onshore shows a higher wind shear due to higher surface roughness and relatively high wind speeds offshore. offshore winds and move offshore for other reasons such as safety or land use regulations. Another benefit of offshore AWES in comparison to conventional WT is the smaller and cheaper support structure.
Atmospheric stability of the boundary layer, which highly affects the wind speed profile shape, is commonly categorized using the Obukhov length L (Obukhov, 1971;Sempreviva and Gryning, 1996). Here the application is extended to midaltitudes. L is defined by the simulated friction velocity u * , virtual potential temperature θ v , potential temperature θ, kinematic 120 virtual sensible surface heat flux Q S , kinematic virtual latent heat flux Q L , the von Kármán constant k and gravitational acceleration g:  Figure 3. Comparison of WRF-simulated annual wind speed probability distribution between onshore (left) and offshore (right) up to 1000 m. A nonlinear color scheme was chosen to represent the high probability of low altitude onshore winds while still differentiating the lower, wide spread frequencies at higher altitudes.
We chose the same classification as in (Sommerfeld et al., 2019b) for consistency. Neutral stratification occurs approximately 20% of the year at both locations. The lower heat capacity of the land surface leads to a faster heat transfer and a quicker surface cool-off which favors the development of stable stratification (≈17% onshore vs ≈6% offshore). The offshore location has a higher probability of unstable conditions which is likely caused by a warmer ocean surface compared to the air above (Archer et al., 2016).

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Both unstable and stable conditions can lead to non-logarithmic and non-monotonic wind speed profiles. Unstable conditions are often accompanied by almost uniform wind speed profiles due to increased mixing, whereas low level jets (LLJs) can develop during the nocturnal stable onshore boundary layer (Banta, 2008). Both locations have a high chance of unassigned "other" conditions which are mostly associated with low wind speeds (see figure: 8).

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Wind energy in general and AWES in particular are mainly affected by wind velocity and its evolution in time and variation with height. Many temporal and spatial averages, correlations and approximations are used to describe the constantly varying wind conditions and their affect on the device. Instead, here representative wind velocity profiles are chosen to avoid excessive averaging and compare AWES performance as realistically as possible. The onshore (Pritzwalk) and offshore (FINO3) data are classified to determine representative profiles. Classifying the wind regime using atmospheric stability is an accepted method-140 ology to describe the near-surface atmosphere. A common proxy for atmospheric stability is the Obukhov length (Obukhov, 1971;Sempreviva and Gryning, 1996), a metric that exclusively uses surface data (see section 2.2 and equation 1). Previous studies (Sommerfeld et al., 2019b) showed that Obukhov-length-classified wind speed profiles diverge with height, especially during neutral and stable conditions. This indicates vertically heterogeneous atmospheric stability and suggests that surfacebased stability categorization is insufficient for higher altitudes. Clustering the wind speed or velocity profiles purely based on 145 data similarity on the other hand results in more cohesive profile groups (see figure A1 and A2 in the appendix) (Schelbergen et al., 2020). In contrast to classifying the wind regime by atmospheric stability, which requires temperature and heat flux data, mathematical clustering only uses wind velocity or speed data at multiple heights. Therefore, clustering can also be applied to wind-only measurements such as LiDAR.
The k-means clustering algorithm (Pedregosa et al., 2011) used in this study was chosen for its ease of use and scalability, due 150 to the high dimensionality of the data set. Many other algorithms produce similar results, but a comparison between clustering algorithms is beyond the scope of this research.
Before clustering the two horizontal wind velocity components u and v, whose vertical variation define the wind velocity profile, are rotated such that the main wind component (average wind direction up to 500 m) u main points in positive x direction and the deviation u deviation is perpendicular to it, pointing in positive y direction. This removes the directional dependency 155 of the wind velocity profiles, allows for more homogeneous clusters and simplifies the comparison of awebox results. It is analogous to assuming omnidirectional operation while the flying wing still needs to adjust to wind condition which are changing with height. The algorithm assigns each wind velocity profile up to 1000 m, comprised of approximately 30 heights and 2 directions, to one of k clusters defined by their respective cluster mean also referred to as centroid. These centroids are calculated such that they minimize the sum of the Euclidean distances, i.e. the cost function of the algorithm, also referred to 160 as "inertia" or "within-cluster sum-of-squares", to every data point within each cluster. As such, the centroids are usually not 8 https://doi.org/10.5194/wes-2020-120 Preprint. Discussion started: 19 November 2020 c Author(s) 2020. CC BY 4.0 License.
actual data points, but rather the average of that cluster. The resulting cluster labels are random results of initialization and are therefore insignificant. Later evaluation uses clusters sorted by average wind speed up to 500 m.
The variable k refers to the fixed, predefined number of clusters. The choice of k significantly affects the accuracy of the resulting power and AEP predictions (see section 5.5) as well as the computational cost associated with clustering (pre- velocity profiles up to 1000 m. Bottom left: Silhouette score (average of the silhouette coefficients on the right) over number number of cluster k for both locations. Right: Silhouette coefficients (top onshore, bottom offshore) which express the distance to neighbouring clusters, for a representative k=10. Negative values indicate that the sample is closer to neighbouring clusters than to the one they are assigned to. The red dashed line represents the silhouette score. k is often chosen at a point where the inertia reduction becomes marginally small with increasing number of clusters, often represented by a sharp bend or elbow in the inertia trend. Absolute values of inertia are somewhat meaningless as it is not a normalized metric and therefore scales with size of the considered data set. A majority of the difference between on-and 170 9 https://doi.org/10.5194/wes-2020-120 Preprint. Discussion started: 19 November 2020 c Author(s) 2020. CC BY 4.0 License. offshore is likely due to different number of vertical grid cells which the algorithm interprets as dimensions (see table 1). The silhouette coefficients on the other hand are normalized between -1 (worst) and 1 (best) and indicate the membership of a data point to its cluster in comparison to other clusters. A negative value suggests that a data point is assigned to the wrong cluster.
The silhouette score is the average of all silhouette coefficients for a fixed number of clusters k. Its trend is shown in the bottom left of figure 4 . The top right depicts the onshore and the bottom right the offshore silhouette coefficients for a representative k of 10. Note that the clusters are unsorted as a result of the random initialization process. Therefore, their labels (1 to 10) are omitted. Silhouette coefficients and the resulting silhouette score illustrate that the offshore clusters are more coherent than the onshore clusters. Onshore clusters also have more negative silhouette coefficients which could indicate too many or too few clusters. Another possible explanation could be that the continuous nature of wind which results in a high cluster proximity as well as the high variability of profile shapes onshore led to a worse score. The following sub-section shows that non-monotonic 180 wind velocity profiles (e.g. profiles with low level jets (LLJs), which are more common onshore, intersect with other clusters and therefore reduce the overall silhouette score. As expected offshore (right) low altitude wind speeds are higher and wind shear is lower than onshore (left). Overall, offshore centroids are wider spread and distinct in comparison to the onshore profiles which explains the higher silhouette score (see figure 4). The associated annual centroid frequency of occurrence for k=10 is shown below in figure 5. Wind speeds of the first 190 and sixth offshore centroid decrease at higher altitude. This could be caused by directional differences which are not depicted in a 2D wind speed plot, different large-scale weather phenomenon, or indicate the usage of too many clusters as both clusters have a very low probability. The first three onshore and offshore clusters exhibit very low wind shear with almost constant wind speed above 200 m. Onshore cluster 5, which seems to comprise of non-monotonic profiles as its centroid has a distinct LLJ nose at about 200 m, occurs about 5% of the time. Onshore centroids 7 and 8 also show a slight wind speed inversion at 195 higher altitudes.

Analysis of clustered profiles
Evidently, the wind speed magnitude plays a determining role in clustering as the resulting centroids are nearly stacked in terms of speed, especially offshore. This can lead to profiles whose shape significantly differs from the one of the centroid to be assigned to a cluster due to similar average wind speed. A clearer wind profile shape distinction could have been achieved . Onshore (left) and offshore (right) average annual wind speed profiles (or centroids) resulting from the k-means clustering process for k = 10 over height (top). Comprising WRF simulated wind velocity profiles depicted in grey. Centroids are sorted, labeled and colored in ascending order of average wind speed up to 500 m. The corresponding cluster frequency f for each cluster C is shown below.
considered. AWES therefore need to be able to either operate under such low speed conditions or be able to safely land and take-off. Both locations follow a distinct annual pattern (see figure 6) during which profiles associated with high wind speeds increase during the winter months and profiles with low wind speeds are predominantly found in summer. The two onshore and offshore clusters associated with the highest wind speed are almost exclusively present during November to February. Offshore data shows almost no diurnal variability (see figure 7) with only a slight increase of clusters associated with lower wind speeds during daytime. Onshore clusters on the other hand are more dependent on the diurnal cycle with a higher 215 likelihood of low speed clusters after sunrise. The frequency of onshore cluster 5, which comprises a LLJ nose (see figure 5), drops to almost zero during daytime and increases during nighttime, substantiating the assumption that this cluster is associated with nocturnal LLJs. The clustered wind velocity profiles and their associated speed and shape correlate with atmospheric stability as expected (see figure 8). Low wind speed clusters (categorized as "other" according to atmospheric stability summarized in table 2) have 220 a low impact on wind power assessment, but highly influence AEP because they make up about 20% to 30% of the annual wind resource. Unstable (U) and near unstable (NU) conditions are associated slightly higher wind speeds than "other" at both locations. The highest wind speeds are develop during neutral (N) and near stable (NS) conditions. LLJ profiles associated with onshore cluster 5 are most likely to develop during stable (S) and very stable (VS) conditions.  In conclusion, k-means clustering is able to capture and reveal temporal variations in the wind regime as well as location spe-225 cific wind profile shapes up to high altitudes. Wind speed magnitude seems to determine the resulting clusters more than profile shape. However, less common non-monotonic profiles with LLJs were identified. Normalizing the profiles before clustering should give more insight into the different vertical profile shapes. The corresponding cluster frequency follows the expected temporal trend and atmospheric stability association.

AWES trajectory optimization 230
Generating dynamically feasible and power-optimal AWES flight trajectories for given wind profiles is a nontrivial task given the nonlinear and unstable system dynamics and the presence of nonlinear flight envelope constraints. Optimal control methods are a natural candidate to tackle this problem, given their inherent ability to deal with nonlinear, constrained multiple-inputmultiple-output systems. In periodic optimal control, an optimization problem is solved to compute periodic system state and control trajectories that optimize a system performance index (here average AWES power output P ) while satisfying the system 235 dynamic equations. The initial and final state of the trajectory are freely chosen by the optimizer but must be equal to ensure 13 https://doi.org/10.5194/wes-2020-120 Preprint. Discussion started: 19 November 2020 c Author(s) 2020. CC BY 4.0 License. periodic operation. We here apply this methodology to generate realistic single-wing, ground-generation AWES power curves and AEP estimation based on simulated wind velocity profiles using the awebox. Take-off and landing are not considered in this paper. Instead only the trajectory during the production cycle is optimized.
4.1 Optimization model overview 240 We consider a 6 degree of freedom (DOF) rigid-wing aircraft model. It uses pre-computed quadratic approximations of the aerodynamic coefficients which are controlled via aileron, elevator and rudder deflection rates (Malz et al., 2019). The tether is controlled by the tether jerk ( ... l tether ) from which tether acceleration (l tether ), speed (l tether = v tether ) and length (l tether ) are derived. The tether is modeled as a single solid rod which can not be subjected to compressive forces (De Schutter et al., 2019).
The rod is divided into n aero = 10 elements and tether drag is calculated individually for each element relative to apparent 245 wind speed (Bronnenmeyer, 2018), with a tether drag coefficient of c tether D = 1. Wind profiles are implemented as 2D wind components rotated such that the main wind direction is in positive x direction and the deviation from it in y direction. This is equivalent to assuming omnidirectional AWES operation with the wing still needing to adjust to changing wind conditions with height. Furthermore, we include a simplified atmospheric model based on international standard atmosphere to account for air density variation. Lift is assumed to behave linearly in between the angle of attack constraints, visualized by black, vertical, dashed lines.

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Changes in the drag coefficient on the other hand are implemented by a quadratic approximation. This study compares two aircraft sizes, one with a wing area of A = 20 m 2 and another one with A = 50 m 2 . Aircraft geometry such as aspect ratio is kept constant (AR = 10). The aircraft mass and inertia were scaled relative to wing span b (see equation 2), based on the Galileo's square-cube law. However, we chose a rather optimistic κ of 2 (pure geometric scaling would assume κ = 3), assuming design and material improvements with scale. The wing loading of approximately 12.25 kgm −2 is consistent with 260 the AP2 reference data. This results in an overestimation of output power and lower cut-in speed in comparison to a heavier aircraft. The focus of this paper is on the derivation and investigation of the AWES power curve and not on realistic system design which will be subject of a future paper on scaling study of AWES. According to Loyd (Loyd, 1980) the ratio c 3 L /c 2 D determines the maximum power of any crosswind AWES. Figure 9 (right) therefore gives an estimate of optimal reel-out phase alpha. From this it is possible to estimate optimal power production for 265 this specific wing. (2)

Ground station model
The ground station constraints play a decisive role in the overall power of ground-generation AWES. The optimal reel-out (Loyd, 1980) and thereby limited by the prevailing wind conditions which hardly exceed 20 ms −1 .

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The reel-out to reel-in ratio is limited to 2 3 , e.g vout vin = 10 ms −1 15 ms −1 , to comply with ground station design limitations. A maximum tether acceleration ofl = 20 ms −2 is imposed to comply with generator torque limits. Tether diameter and maximum tether force are calculated from a pre-optimization, due to the complexity of the system which makes an a priori estimation difficult.
This calculation optimizes the AWES trajectory and tether diameter to maximize average cycle power subject to the same tether speed and acceleration constraint. A simple logarithmic wind speed profile was used as wind inflow (reference speed 275 of U sizing (z = 10 m) = 8 ms −1 ). Constraints on the tether force enforce it to be positive whilst not exceeding the maximum tether stress, to which a safety factor of 3 is applied. This results in a tripling of the tether area. These ground station and tether constraints do not represent an optimized AWES, but rather a representative system. Table 3. Aircraft design parameters for the two different sizes (Awing = 20, 50 m 2 ) analyzed in this study and for the reference AP2 aircraft.
Values in square brackets represent flight envelope bounds, which are implemented as inequality constraints of the optimization.

Constraints
The tether constraints such as tether length, speed and force are summarized in table 3 (see sub-section 4.3). Flight envelope 280 constraints include limits on acceleration, roll and pitch angle (to avoid collision with the tether) or angle of attack, in between which the lift is assumed to be linear. Furthermore, a minimal operating height of z min = 50 + Awing 2 m is imposed for safety reasons.

Wind boundary condition
AWES trajectories depend on prevailing wind conditions as they greatly benefit from dynamically adapting their operational 285 altitude, tether speed and path to maximize power production and minimize losses. The above described AWES were subjected to several different wind conditions to compare the impact on their trajectory, estimate the power curve and AEP. Logarithmic included in the power curve estimation, their contribution to AEP will be very low as their frequency of occurrence is close to zero. Results are compared to clustered, WRF simulated, onshore and offshore wind conditions which were interpolated by a differentiable function (here a Lagrange polynomial). Three actual wind profiles with a p-value of 5,50, 95, based on average wind speed up to 500 m within every cluster, were chosen to assess the AWES power curve. A representative k=20 is a reasonable choice according to the elbow method and silhouette score described in section 3. To estimate AEP, cluster centroids across the range of k = 5 − 100 were implemented. Wind conditions for the AEP estimation are based on the cluster centroids for k = 5 − 100 due to the high computational cost of running multiple profiles per cluster. These results are compared to the AEP calculated from power of k=20 p5, p50 and p95 wind profiles.

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AWES trajectory optimization is a highly nonlinear and non-convex problem which likely has multiple local optima. Therefore, the particular results generated by a numerical optimization solver can only guaranty local optimally, and usually depend on the chosen initialization. This can result in unwanted or unrealistic AWES trajectories, which implies that the quality of all solutions needs to be evaluated a posteriori.
A periodic optimal control problem is formulated to maximize the average cycle power (P ) of a single AWES subject to 305 equality and in-equality constraints described above (De Schutter et al., 2019;Leuthold et al., 2018). The trajectory optimization problem is discretized into 100 intervals using direct collocation.An initial guess is generated using a homotopy technique similar to (Gros et al., 2013) with an estimated circular trajectory based on a fixed number of loops (here n loop = 4) at a 30 • elevation angle and an estimated aircraft speed. The homotopy technique initially fully relaxes the dynamic constraints using fictitious forces and moments to reduce model nonlinearity and coupling, improving the convergence of Newton-type opti-310 mization techniques. The constraints are then gradually re-introduced until the relaxed problem matches the original problem.
The resulting nonlinear program (NLP) is formulated in the symbolic modeling framework CasADi for Python (Andersson et al., 2012) and solved using the linear solver MA57 (HSL) in IPOPT (Wächter and Biegler, 2006).

Results
In this section we compare representative onshore (Pritzwalk) and offshore (FINO3) trajectories and time series trends. Build-315 ing on that onshore and offshore operating height statistics and tether length trends are examined. AWES power curves are determined based on average cycle power and wind speeds at different reference heights. From these power curve trends we determine an AWES power coefficient c AWES p similar to conventional WT to allow for a quick estimate of AWES power based on wing area, path length and wind speed. Lastly, the annual energy production (AEP) and capacity factor (cf) estimates of different number of clusters are compared to Rayleigh distributed log-profiles as they are defined by IEC standards.  nents as experienced by the AWES in color. As expected based on theory (Stull, 2012) and from the wind rose in section 2.2, the onshore profiles veer more than offshore profiles. The two center plots show the optimized trajectory in side view (center top, x-y plane) and top view (center bottom, x-y plane).
When maximum tether force is reached the system starts to de-power while maintaining the same high tension (right, 1st from top in figures 10 and A3). Such trajectories often extend perpendicular to the main wind direction (y-direction). This often 335 results in odd and unrealistic or unexpected trajectories, even though these local minima are within the system constraints (roll rate etc.). De-powering by increasing the elevation angle is also possible and likely to happen, but harder to determine as it is not easily identifiable whether the elevation angle increased due to better wind conditions or to de-power the wing. Reducing the angle of attack (right, 3rd from top) while maintaining constant maximum tether force (right, 1st from top) can be observed in the highest onshore wind speed trajectory (green). The exact reason why the angle of attack at almost all wind speeds is 340 lower offshore than onshore is hard to determine. A possible reason could be that the wind conditions offshore are so beneficial that the system can operate at lower altitudes and therefore closer to optimal directly down wind conditions with optimal c 3 L /c 2 D at around α ≈ 8 • predicted in figure 9. Onshore conditions on the other hand do not seem to allow for angle of attack close to this optimal.
The algorithm seems to always maximize tether force and vary tether speed (right 2nd from top) close to optimal reel-out 345 speed (v out ≈ 1 3 v wind (Loyd, 1980)) to maximize average cycle power. At high wind speed the tether speed constraint is active during the reel-in phase, presumably to keep this phase as short as possible. In these cases the trajectory starts to differ from its predefined shape with distinct loops to de-power, visible in the power development during the production phase (green).
Trajectories for such high speed wind conditions without a tether force constraint, where the tether diameter is adjusted to the wind conditions, would be closer to the looping paths seen for lower wind speeds (blue, orange, red). The time history of 350 instantaneous power (P current ; right bottom) clearly distinguishes the production and consumption phase of pumping-mode (ground-generation) AWES. However, all optimized trajectories have a close to zero power usage during reel-in as they reduce the angle of attack to near zero lift conditions. One commonality between all time series is that they all almost have the same flight time independent of location, wind speed or aircraft size. The flight time is almost solely determined by the initial number of loops, here five, used in the initialization procedure. Based on previous analyses, mechanical AWES power output seems to 355 be insensitive to number of loops and flight time.

Tether length and altitude
This sub-section compares tether lengths and operating altitudes for onshore and offshore wind conditions for a wing size of A wing = 20 m 2 . Results for the A wing = 50 m 2 design can be found in the appendix in figure A6. The data is based on the p5,p50, p95-th wind profiles of k=20 onshore and offshore clusters (see sub-section 4.5). Onshore tethers are generally longer as operating altitudes tend to be higher due to higher wind shear and typically higher winds 365 offshore. Where a tether length of approximately 600 m suffices for the entire offshore wind regime, onshore tethers need to be at least 1000 m long, except for a few outliers which would benefit from an even longer tether. The gradual increase of tether length with wind speed offshore is probably due to lower wind shear and more homogeneous wind regime (see sub-section 2.2). Onshore tethers on the other hand already exceed a length of 400 m from U (z ref = 200 m) > 5 ms −1 .
Operating altitudes over the entire wind regime, both off-and onshore, are almost never higher than 500 m above ground, 370 confirming findings in (Sommerfeld et al., 2019a,b). Low altitude offshore winds seem to be so favorable that AWES operate approximately 75% annually below 250 m. This also has implications for tower-based, conventional wind turbines as these results suggest that the benefit of going towards higher altitudes might not outweigh the costs. This is seen in large WTs, such as the IEA 15 MW reference turbine (Gaertner et al., 2020), with hub height smaller than one rotor diameter. Multi-wing AWES could benefit from higher operating altitudes due to their higher lift to tether drag and weight ratio. However, more detailed 375 analysis are required. The A wing = 50 m 2 aircraft both on-and offshore seems to benefit from higher operating altitudes and longer tethers (see figure A6) which could be due to the higher lift to tether drag ratio. However, optimal operating altitudes exceed heights above 600 m at either location only 5% of the time. A future analysis of even larger systems will investigate whether this trend continues. 20 https://doi.org/10.5194/wes-2020-120 Preprint. Discussion started: 19 November 2020 c Author(s) 2020. CC BY 4.0 License.

Power curve
This sub-section compares AWES power curve representations based on various wind profile inputs over different reference heights. Clustered WRF profiles are compared to logarithmic wind speed profiles, as defined in the IEC standards (International Electrotechnical Commission, 2010). Due to many conceptually different AWES designs and the novelty of the technology, there is no unanimously accepted AWES power curve definition. Therefore, no standard reference wind speed, equivalent to wind speed at hub height for conventional WT, has been agreed upon. Similarly, no standard wind speed probability distribution 385 such as the Rayleigh or Weibull distribution for conventional wind has been defined. Determining these parameters is more complex than for conventional wind turbines as AWES power is highly dependents on the wind speed variation with height and the resulting flight trajectories. height z WT is assumed to be 100 m for both onshore and offshore WT. The swept area of the turbine A WT is chosen such that its rated power is equivalent to the AWES using: Cut-in and cut-out wind speeds were not used for either the AWES or WT to not limit specific designs. Therefore, energy offshore power curves must be caused by different wind profile shapes as both systems are otherwise exactly the same. This highlights that for AWES power predictions is not just influenced by design, but also by flight trajectory in a given boundary layer. The AWES power curves align well with a c WT p = 0.3 (see figure A9 in the appendix). A better AWES design with 415 higher c 3 L /c 2 D should increase the equivalent WT power coefficient. This however is subject to further investigation and will be included in a future study.
The annual energy production distribution is derived from the integral multiplication of the mean power curve (top) and the wind speed probability distribution at reference height (center). Its total accumulates to the annual energy production (AEP) 22 https://doi.org/10.5194/wes-2020-120 Preprint. Discussion started: 19 November 2020 c Author(s) 2020. CC BY 4.0 License. further described in sub-section 5.5. AWES energy production distribution shifts towards higher wind speeds due to higher 420 operating heights and their higher wind speeds. Similarly, the maximum onshore wind speed at 100 m is lower than offshore, while wind speeds at other reference heights are similar to offshore. Table 4 compiles the AEP of both system sizes and both locations. The table also includes the estimated WT AEP for reference. Overall energy estimates for one system size and location are fairly consistent with each other. However, energy estimates of the larger wing (A wing = 50 m 2 ) onshore shows more variability due to the wider range of wind conditions and 425 operating heights. This indicates that this effect scales with system size which will be investigated further in a future study.
The smaller AWES with a wing area of A wing = 20 m 2 outperforms the WT with the same rated power onshore, whereas the larger wing does not. This indicates that onshore wind conditions favor higher operating altitude due to higher wind shear. Furthermore, the relative reduction in AWES energy with size could be related to additional losses associated with a longer tether and heavier aircraft. Offshore, the WTs outperform the AWESs for both sizes as the lower wind shear favors 430 lower operating altitudes. The offshore AEP is about 25 % larger than onshore for both AWES sizes, while WT performance increases about 50% offshore in comparison to onshore due to better wind resource. This main difference between WT and AWES can be explained by the high c WT p = 0.45 while the wind turbine equivalent of AWES power is closer to c WT p = 0.3.
We assume that the best reference wind speed would be the wind speed along the actual AWES trajectory. Since this is hard to estimate before site selection, a better reference wind speed would be calculated from the average between 100 and 600 m 435 since this is the height at which most onshore and offshore AWES operate (see figure 11). Choosing one fixed reference height might be an inadequate choice as larger AWES sweep a larger altitude range. Table 4. Annual energy predictions (AEP) and capacity factor (cf) results for Awing = 20, 50 m 2 subject to 3 wind velocity profiles within each of the k=20 onshore and offshore clusters. AEP calculated from power curve and wind speed probability distributions at various reference heights (see figure 12 and A8). AEP results for logarithmic wind speed profiles with Rayleigh wind speed probability distribution and WTs (size in rotor diameter dWT) with same rated power as AWESs and wind speed probability distribution at zW T = 100m for reference.

AWES power coefficient
To simplify the AWES power estimation, we derive the power coefficient for AWES c AWES p similar to conventional WT from equation 5. The reference area is calculated from the swept area A swept along the traveled path length l path and wing span b 440 (see table 3). The equation uses the wind speed U along the path and the average cycle power P . Air density ρ air is calculated from the linear approximation described in sub-section 5.3.
(5) Figure 13 shows the previously described power curve (top) over average wind speed along the path U (z operating ). Data points are based on awebox optimizations for wing areas of 20 and 50 m 2 with three wind velocity profiles (p-value of 5,50, 445 95 within each cluster) within each of the k=20 clusters. Onshore and offshore power curves of the same system size are close to each other, however onshore power is slightly lower due to unfavorable wind conditions.
Similarly, the onshore path length (second from top) is generally smaller than offshore. Before reaching rated power, the onshore path length for A wing = 20 m 2 is approximately 18% smaller than the length of A wing = 50 m 2 ; offshore this ratio is only 12%. third sub-figure from the top shows the power coefficient c AWES p , which is calculated from equation 5, for two 450 different AWES sizes at both locations. c AWES p is location and wind velocity profile independent as the reference wind speed on the abscissa is the average speed along the trajectory. The difference between sizes amounts to the chord c wing , which scaled with wing size since the aspect ratio AR = 10 of the wing is kept constant. A possible explanation for this difference is that the mechanical power of a ground-generation AWES is the product of tether force and tether speed. Tether force scales with wing area and tether speed increases because the tether length increases while the total cycle time remains almost constant (see is in good agreement with the data.

AEP
This sub-section contrasts annual energy predictions (AEPs) and capacity factor (cf) based on the various power estimates and wind statistics. Figure 14 compares results for an increasing number of clusters (k= 2,5,10, 20, 30, 40, 50, 100) to results using p5, p50, p95 wind velocity profiles for k=20 to assess the necessary number of clusters and therefore optimization runs needed are solely based on each cluster's centroid which is equivalent to the average wind velocity profiles of all data points within the respective cluster. Here we assume that the power calculated from each centroid is constant within and representative of the entire cluster. Therefore, AEP is the sum of the product of average power P i and cluster probability f i over all clusters k multiplied by the number of hours in a year.
Conventional WT energy (dashed line) is estimated from a simple static power approximations (described in sub-section 5.3, equation 4) using cluster centroid wind speed at 100 m and the same cluster frequency as the AWES.
Both onshore and offshore AEP vary with number of clusters, however above k=10 the variation is negligible and the possible improvement in energy prediction does not justified the increased computational cost. Similarly, WT AEP does not 475 vary significantly for more than 10 clusters. However, AEP and cf are consistently higher than those of AWES. Compared to these results, AEP calculations based on an estimated power curve from three representative wind profiles per cluster k=20 ( see sub-section 5.3 ; color refers to location, onshore: blue, offshore: orange) yield a higher energy estimate. Estimates using just the centroid have lower AEP because of averaging effects within each cluster. High wind speed profiles, which are responsible for a considerable percentage of the cluster energy due to the nonlinear power to wind speed relationship, are 480 averaged out. We therefore believe that a power curve estimation together with wind speed probability distribution for a lower number of total clusters and multiple profiles within a cluster yield better AEP estimates than just using the cluster centroids.
Reference AWES AEP and cf are depicted as dotted lines These data are based on power curves for logarithmic wind speed profiles (with z 0 = 0.1 onshore and z 0 = 0.001 offshore) and Rayleigh wind speed probability distributions (U onshore ave = 10 ms −1 and U offshore ave = 12 ms −1 ) (International Electrotechnical Commission, 2010) . Offshore AEP estimates based on 485 logarithmic wind profiles are closer to power curve estimates based on WRF data than similar onshore results. This implies that offshore wind conditions (wind profile shape and probability) are better represented by logarithmic wind speed profiles than onshore conditions.

Conclusions and outlook
We characterized ground-generation AWES power, annual energy production and capacity factor based on representative, 490 mesoscale onshore wind data at Pritzwalk in northern Germany and offshore wind data at the FINO3 research platform in the North Sea. The analysis is deduced from path optimization using awebox toolbox, with the objective to maximize average cycle power. Representative wind velocity profiles based on k-means clustering were chosen to reduce computational cost.
As long-term high resolution high altitude measurements with sufficient data availability are scarce, wind data are based on mesoscale WRF simulations. These simulations span an entire year with a temporal resolution of 10 minutes, thereby including 495 seasonal, synoptic and diurnal variations at a higher resolution than re-analysis data sets. The annual wind roses for heights of 100 m and 500 m confirm the expected wind speed acceleration and clockwise rotation at both locations, with generally lower offshore wind shear and veer than onshore. Annual wind speed statistics reveal that while average wind speeds increase with height, low wind speeds still occur at a fairly high probability up to 1000 m. To further dissect wind conditions essential to the design and operation of AWES, representative wind velocity profiles were 500 chosen based on k-means clustered data. This algorithm groups similar profiles together into a fixed, predetermined number of k clusters represented by the mean of each cluster. For a representative k of 10 a more extensive analysis and comparison between onshore and offshore wind conditions revealed that average wind speed, rather than profile shape, plays a decisive role in the assignment of profiles to a certain cluster. However, the algorithm was able to identify and define a cluster for onshore LLJs as well as various non-logarithmic wind profiles at both locations. Further analysis revealed seasonal and diurnal wind 505 speed and atmospheric stability dependent cluster correlation, which generally agrees with literature predictions. We therefore believe that k-means wind velocity clustering yields coherent data that provides good insight into the wind regime, especially for higher altitudes. The derived groups represent the annual variation better than traditional logarithmic or exponential wind speed profiles.
The 5th, 50th and 95th percentile wind velocity profiles within each cluster for k=20 as well as logarithmic reference wind 510 speed profiles were implemented into the airborne wind energy trajectory optimization toolbox awebox to estimate average cycle power of ground-generation AWES. Two scaled Ampyx AP2 aircraft sizes (A wing = 20, 50 m 2 ) are compared in terms of trajectory, operating altitude, instantaneous tether force and length as well as power. AWESs at both location rarely operate above 400 m, with offshore systems mostly flying below 200 m, due to fast wind speeds at low heights and low wind shear.
These results weaken the claim of increased power harvest above 500 m for AWES, but also obviate airspace restriction 515 challenges for AWES. A wing with the given size, aerodynamic and mass properties achieves a similar power curve as a similarly rated wind turbine with a power coefficient of approximately 0.3. As expected, offshore AWES generally outperform onshore AWES in terms of AEP and capacity factor. Furthermore, social acceptance of such systems will likely be higher offshore.
From this analysis we derived an AWES power coefficient c AWES p based on average AWES power curves, path lengths and 520 wing spans. The decrease in c AWES p with wind speed can be attributed to the increase in tether length and the accompanying weight and drag losses. Scaling these power coefficients by the inverse wing chord (c AWES p /c wing ) lead to a collapse of both location and both size trends to a single quadratic, decreasing progression.
We compared AEP and cf estimates for each system based on different power and wind speed probability description and conclude that the highest, and probably most realistic AEP prediction, is based on an average power curve which is derived 525 from multiple wind profiles within a cluster (p5, p50, p95) for a small number of clusters (k=10). The wind speed probability distribution is ideally derived from the wind speed along the flight path. As this is difficult to predetermined before operation, we recommend to use average wind speeds between 100 and 400 m. Offshore this choice seems to be less significant as winds are less sheared and are more monotonic than onshore. Therefore, AEP estimates based on logarithmic wind profiles and Rayleigh distribution give similar results as the clustered profiles.

530
In summary, k-means clustering provides adequate categorization and provides realistic, representative wind velocity profiles for AWES trajectory optimization. This increases the power prediction accuracy in comparison to logarithmic wind speed profiles. Furthermore, clustering reduces the computational cost of AEP estimates as only a few number of clusters suffice.
Best AEP results and power curve description can be achieved by using multiple representative profiles within each cluster instead of using the cluster centroid. A nonlinear AWES power coefficient to approximate AWES power up to rated power 535 gives reasonable results. We expect further work, field tests and other research studies with different AWES sizes, aerodynamic coefficients and flight paths to confirm our findings.
Based on these results, we will describe the design space and weight budget of ground-generation AWES in a future sizing study using the here described clustered wind data. To that end, we will compare the performance of a high lift airfoil to the here used AP2 aerodynamic reference model and determine the maximum weight for different aircraft sizes. Furthermore, we 540 will investigate the impact of a nonlinear lift coefficient. These results should inform researchers and industry on the scaling potential of AWES. An interesting research question is the seasonality of AWES performance in comparison to WT. wind speed probability distribution (center) based on WRF simulation and Rayleigh distribution (red) with Uave = 10 ms −1 (onshore) and 12 ms −1 (offshore) for reference. Energy production distribution (bottom) shows the distribution of annual produced energy over wind speed which is the product of power and wind speed probability distribution. Integrating this product results in the AEP. . Annual wind speed probability distribution (center) based on WRF simulation and Rayleigh distribution (red) with Uave = 10 ms −1 (onshore) and 12 ms −1 (offshore) for reference. Energy production distribution (bottom) shows the distribution of annual produced energy over wind speed which is the product of power and wind speed probability distribution. Integrating this product results in the AEP.