This study investigates the performance of pumping-mode ground-generation airborne wind energy systems (AWESs) by determining cyclical, feasible, power-optimal flight trajectories based on realistic vertical wind velocity profiles. These 10 min profiles, derived from mesoscale weather simulations at an offshore and an onshore site in Europe, are incorporated into an optimal control model that maximizes average cycle power by optimizing the trajectory. To reduce the computational cost, representative wind conditions are determined based on

Airborne wind energy systems (AWESs) aspire to harvest stronger and less turbulent winds at mid-altitude, here defined as heights between 100 and 1000 m, presumably beyond what is achievable with conventional wind turbines (WTs). The prospects of higher energy yield combined with reduced capital cost motivate the development of this new class of renewable energy technology

Recent consensus among the scientific community defined a power curve as the maximum average cycle power, which is the combination of consecutive reel-out and reel-in phases, as a function of wind speed at pattern height, which is the time-averaged height during the reel-out power-producing phase

In their study,

This paper's main contribution is the examination of how realistic onshore and offshore wind profiles, compared to a standard log profile, affect the power-optimal performance of AWESs, as well as how the choice of reference height impacts the power curve, particularly given the wide range of wind speed profile shapes. This study is a continuation of previous analyses of lidar measurements

The

The structure of this research is as follows. Section

Section

This study compares the performance of airborne wind energy systems (AWESs) at two locations in Europe, one onshore and one offshore (Fig.

Map of northern Germany with the representative onshore (Pritzwalk) and offshore (FINO3) locations highlighted by black dots.

The mesoscale simulations used the Weather Research and Forecasting (WRF) model

Key setup parameters of the onshore and offshore mesoscale WRF simulations to generate the wind data used in this study.

Each simulation consists of three nested domains around their respective location (black dot in Fig.

Figure

Wind roses of annual wind direction and speed statistics at Pritzwalk (onshore) and FINO3 (offshore) for heights of 100 and 500 m during the simulated years.

Directional variability decreases, and the wind speed

Figure

Comparison of WRF-simulated annual wind speed

Such multimodal distributions at higher altitudes are better described by the sum of two or more probability distributions, as standard Weibull or Rayleigh distributions cannot capture this phenomenon

The Obukhov length

Stability classes based on Obukhov length

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 (

Representative wind velocity profile

Both unstable and stable conditions can lead to non-logarithmic and non-monotonic

An accepted method to describe the near-surface atmosphere is atmospheric stability, commonly quantified by the Obukhov length

The

The

Onshore

The wind velocity data up to 1000 m are comprised of data points at 30 height levels and in two directions. The clustering algorithm assigns each data point to one of the

Absolute values of inertia (Fig.

Vertical onshore

For visualization purposes, the following subsections describe the wind conditions at both locations using only

Figure

As expected, offshore

Vertical offshore wind speed profiles categorized using the

Clusters

Monthly frequency of

This subsection examines the relationship between the clusters and monthly, diurnal, and atmospheric stability. These analyses reveal patterns that give insight into the wind regime and the resulting changes in AWES performance.
Subsequent sections examine wind data from

Both locations exhibit a clear annual pattern as shown in Fig.

Offshore data indicate minimal diurnal variation as shown in Fig.

Diurnal frequency of

Atmospheric stability (U: unstable; NU: nearly unstable; N: neutral; NS: nearly stable; S: stable; VS: very stable) distribution of

The wind velocity clusters (Fig.

In summary,

This section introduces the dynamic AWES model used in the

The rigid-body model considers a 6-degrees-of-freedom (DOF) fixed-wing aircraft which is connected to the ground via a straight tether. The introduction of the tether reduces the DOFs to 5

Ampyx Power AP2 reference kite aerodynamic lift

The longitudinal dynamics of the tether is controlled via the tether jerk

The presented model utilizes the Ampyx Power AP2 aerodynamic coefficients from

Published actual (circle) and anticipated (square) aircraft mass scaling provided by Makani (red color scheme)

The lift coefficient

The aircraft dynamics are described by a single rigid body of mass

AWESs need to dynamically adapt to changing wind conditions to maximize power generation and ensure safe operation. Section

Maximizing the average cycle power can be formulated as a trajectory optimization problem, which takes into account the interaction between the tether, kite, and ground station. This study analyzes the mechanical power produced by a single aircraft tethered with a straight line throughout one production cycle, including reeling in and out, while disregarding take-off and landing. Power production is intrinsically linked to the aircraft's flight dynamics, as the AWES never reaches a steady state over the course of a power cycle. Generating dynamically feasible and power-optimal flight trajectories is nontrivial, given the nonlinear and unstable system dynamics and the presence of various flight envelope constraints. Optimal control methods are a natural candidate to tackle such problems, given their inherent ability to deal with nonlinear, constrained multiple-input–multiple-output systems

Several important constraints define the operational envelope. The most important constraints such as tether length, tether reeling speed, and tether force are summarized in Table

Selected AWES design parameters for the original AP2 reference system

The trajectory optimization process is highly nonlinear and non-convex, resulting in multiple local optima. These solutions depend on the chosen initial conditions. Some of the locally optimal solutions may be feasible and within the constraints but may have undesirable characteristics such as looping maneuvers during reel-in or excessively high operating altitudes.
As a result, it is necessary to evaluate the quality of all solutions. To solve the complex optimization problem, initial guesses are generated using a homotopy technique similar to

Representative WRF-simulated vertical onshore wind speed profiles

To reduce the computational cost while maintaining an adequate representation, we only implement three wind velocity profiles from each cluster into the trajectory optimization toolbox. More profiles could be chosen for an in-depth analysis. The power for a total number of 60 wind profiles, three profiles for each of the

The vertical wind velocity profiles

The

For comparison, logarithmic wind speed profiles,

The reference wind speed

This section introduces a simplified quasi-steady-state AWES reference model (QSM) (Sect.

The QSM estimates the mechanical power of ground-generation AWES based on the assumption that the trajectory of the tethered aircraft can be approximated by a progression through steady equilibrium states where tether tension and total aerodynamic force are aligned. We simplify the QSM by approximating the reel-out and reel-in trajectory with a single state and neglecting the effects of gravity. The QSM, based on

The average cycle power

During the reel-in and reel-out phases, we assume that the tether force

Tether tension is a function of the apparent wind speed

This section introduces a simplified steady-state reference WT model that calculates power as

This section analyses the optimization results and compares them to the reference models. Section

Representative WRF-simulated vertical offshore wind speed profiles

Figure

Figure

Time series of instantaneous tether tension

Time series of instantaneous tether tension

The optimizer maximizes tether tension by adjusting the reel-out speed and angle of attack (Fig.

Figure

Tether length range

Onshore

None of the optimizations reach the maximum tether length constraint of

The power curve of wind energy converters quantifies the power that can be harvested at a given reference wind speed. For conventional WTs the wind speed at hub-height is commonly used as the reference wind speed. Whether this is appropriate for ever-growing towers and longer rotor blades is debatable

Each data point corresponds to one of the sampled WRF-simulated wind velocity profiles

Average cycle power

The more homogeneous offshore wind conditions result in less power variation.
The three different reference heights have almost no impact on the offshore power curve up to the rated wind speed. Above

Figure

The QSM and WT reference model use the same sampled WRF-simulated wind data.
No cut-out wind speed is defined. The cut-in wind speed of

The logarithmic wind speed profiles (Eq.

The

This research compares the optimal performance of a single-aircraft, ground-generation AWES using mesoscale wind data simulated by the WRF model to the performance calculated using standard logarithmic wind profiles. It also describes trajectories, instantaneous performance, and trends in tether length and operating height. These analyses use 1 year of onshore wind data at Pritzwalk in northern Germany and 1 year of offshore wind data at the FINO3 research platform in the North Sea to drive the

The optimization model is able to determine power-optimal trajectories for complex, non-monotonic wind velocity profiles. The optimized results are only marginally lower than those obtained using the simplified QSM, which neglects the effects of gravity and only simulates a single reel-in and reel-out state instead of the entire trajectory. The predicted onshore AWES power exceeds the WT reference model. This is because AWES can adapt their operating altitude to benefit from higher wind shear or LLJs. Offshore wind velocity profiles are generally more monotonic and exhibit higher wind speeds with less turbulence and wind shear. As a result, offshore winds produce average power that is similar to their logarithmic approximation. Due to the initialization of the

An investigation of the time series data shows that the optimizer first maximizes tether tension by adjusting reel-out speed and angle of attack. As the wind speed increases, the tether reel-out speed approaches its maximum limit and becomes more constant, whereas at lower wind speeds, the reel-out speed varies more. Up to rated wind speed, when average tether tension and tether reeling speed are maximized, the optimizer increases the deployed tether length and reduces the elevation angle to operate at optimal height. At higher wind speeds, the elevation angle increases to de-power the system and stay within design constraints. As a result, approximately 75 % of the optimal onshore and offshore operating heights are below 400 m. The offshore power curve appears to be independent of the reference height due to the lower number of non-monotonic wind speed profiles and lower wind shear. In contrast, the choice of reference height is more important for the onshore power curve.

The mesoscale wind simulations, which include a year’s worth of wind data with a temporal resolution of 10 min, are analyzed and categorized for both onshore and offshore locations. The annual wind roses for heights of 100 and 500 m confirm the expected wind speed increase and clockwise rotation at both locations. Offshore shows a lower wind shear and veer than onshore.
Annual wind speed statistics reveal that low wind speeds still occur at a fairly high probability up to 1000 m at both locations. The

As a continuation of this study, the power curves and realistic wind conditions described here could be utilized to calculate AEP estimations.
Further research is required into AWES power curves and their reference wind speed, which could be accomplished by deriving shape-specific power curves from normalized wind speed profiles or by considering the correlation between wind speeds at different reference heights. Future work should include a variable number of loop maneuvers as a variable in the optimization objective function. Using the same data and model, it is possible to investigate the annual and diurnal AWES power variation in comparison to WT performance.
A parallel sizing study

Time series of instantaneous tether tension

High-wind-speed WRF-simulated vertical offshore wind speed profiles

The simulated WRF data set can be accessed
via

MS evaluated the data and wrote the manuscript in consultation with and under the supervision of CC. MD set up the numerical offshore simulation and contributed to the meteorological evaluation of the data and reviewed the manuscript. JDS co-developed the optimization model and helped to write and review the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank the BMWi for the funding of the “OnKites I” and “OnKites II” projects (grant numbers 0325394 and 0325394A) on the basis of a decision by the German Bundestag and project management Projektträger Jülich. We thank the PICS, NSERC, and the DAAD for their funding.

We thank the Carl von Ossietzky University of Oldenburg and the Energy Meteorology research group for providing access to their high-performance computing cluster

We further acknowledge Rachel Leuthold (University of Freiburg, SYSCOP) and Thilo Bronnenmeyer (Kiteswarms Ltd.) for their help in writing this article, great technical support, and continued work on the

This research has been supported by several sources. These include student scholarships from the Pacific Institute for Climate Solutions (PICS), the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Deutscher Akademischer Austauschdienst (DAAD).

Additionally, projects and researchers whose contributions were essential to the writing of this article were funded by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie (grant agreement no. 642682), as well as the Bundesministerium für Wirtschaft und Energie (BMWi) (grant no 0324005, OnKites I grant no. 0325394, and OnKites II grant no. 0325394A).

This paper was edited by Roland Schmehl and reviewed by Mark Schelbergen, Maximilian Ranneberg, and one anonymous referee.