The optimal control problem for flight trajectories of Fly-Gen airborne wind energy systems (AWESs) is a crucial research topic for the field, as suboptimal paths can lead to a drastic reduction in power production. One of the novelties of the present work is the expression of the optimal control problem in the frequency domain through a harmonic balance formulation. This allows the potential reduction of the problem size by solving only for the main harmonics and allows the implicit imposition of periodicity of the solution. The trajectory is described by the Fourier coefficients of the dynamics (elevation and azimuth angles) and of the control inputs (onboard wind turbine thrust and AWES roll angle). To isolate the effects of each physical phenomenon, optimal trajectories are presented with an increasing level of physical representation from the most idealized case: (i) if the mean thrust power (mechanical power linked to the dynamics) is considered as the objective function, optimal trajectories are characterized by a constant AWES velocity over the loop and a circular shape. This is done by converting all the gravitational potential energy into electrical energy. At low wind speed, onboard wind turbines are then used as propellers in the ascendant part of the loop; (ii) if the mean shaft power (mechanical power after momentum losses) is the objective function, a part of the potential energy is converted into kinetic and the rest into electrical energy. Therefore, the AWES velocity fluctuates over the loop; (iii) if the mean electrical power is considered as the objective function, the onboard wind turbines are never used as propellers because of the power conversion efficiency. Optimal trajectories for case (ii) and (iii) have a circular shape squashed along the vertical direction. The optimal control inputs can be generally modeled with one harmonic for the onboard wind turbine thrust and two for AWES roll angle without a significant loss of power, demonstrating that the absence of high-frequency control is not detrimental to the power generated by Fly-Gen AWESs.

The flight mechanics of rigid wing Airborne Wind Energy Systems (AWESs) is fundamentally different from the one of conventional aircrafts. The presence of the tether largely impacts the system dynamics, making the flying craft to experience forces which can be an order of magnitude larger than those experienced by conventional aircrafts. Moreover, an AWES needs to deal with a sustained yet unpredictable wind, and the ensuing requirements for flight maneuvers in order to achieve prescribed control and power production goals. A way to maximize energy capture while facing disturbances without requiring an excessive contribution from active control is that of suitably designing the AWES craft to feature good flight dynamics characteristics. In this study, a baseline circular flight path is considered, and a steady state condition is defined by modeling all fluctuating dynamic terms over the flight loop as disturbances. In-flight stability is studied by linearizing the equations of motion on this baseline trajectory. In populating a linearized dynamic model, analytical derivatives of external forces are computed by applying well-known aerodynamic theories, allowing for a fast formulation of the linearized problem and for a quantitative understanding of how design parameters influence stability. A complete eigenanalysis of an example tethered system is carried out, showing that a stable-by-design AWES can be obtained and how. With the help of the example, it is shown how conventional aircraft eigenmodes are modified for an AWES and new eigenmodes, typical of AWESs, are introduced and explained. The modeling approach presented in the paper sets the basis for a holistic design of AWES that will follow this work.

The power equations of crosswind Ground-Gen and Fly-Gen airborne wind energy systems (AWESs) flying in circular trajectories are refined to include the contribution from the aerodynamic wake, modeled with vortex methods. This reveals the effect of changing the turning radius, the wing geometry and the aerodynamic coefficients on aerodynamic performances and power production. A novel power coefficient is defined by normalizing the aerodynamic power with the wind power passing through a disk with a radius equal to the AWES wingspan, enabling the comparison of different designs for a given wingspan. The aspect ratio which maximizes this power coefficient is finite, and its analytical expression for an infinite turning radius is derived. By considering the optimal wing aspect ratio, the maximum power coefficient is found, and its analytical expression for an infinite turning radius is derived. Ground-Gen and Fly-Gen AWESs, with the same idealized characteristics, are compared in terms of power production, and later three AWESs from the literature are analyzed. With this modeling framework, Ground-Gen systems are found to have a lower power potential than Fly-Gen AWESs with the same geometry because the reel-out velocity makes Ground-Gen AWESs fly closer to their own wake.

Understanding and modeling the aerodynamic wake of airborne wind energy systems (AWESs) is crucial for estimating the performance and defining the design of such systems, as tight trajectories increase induced velocities and thus decrease the available power, while unnecessarily large trajectories increase power losses due to the gravitational potential energy exchange. The aerodynamic wake of crosswind AWESs flying circular trajectories is studied here with vortex methods. The velocities induced at the AWES from a generic helicoidal vortex filament, trailed by a position on the AWES wing, are modeled with an expression for the near vortex filament and one for the far vortex filament. The near vortex filament is modeled as the first half rotation of the helicoidal filament, with its axial component being neglected. The induced drag due to the near wake, built up from near vortex filaments, is found to be similar to the induced drag the AWES would have in forward flight. The far wake is modeled as two semi-infinite vortex ring cascades with opposite intensity. An approximate solution for the axial induced velocity at the AWES is given as a function of the radial (known) and axial (unknown) position of the vortex rings. An explicit and an implicit closure model are introduced to link the axial position of the vortex rings with the other quantities of the model. The aerodynamic model, using the implicit closure model for the far wake, is validated with the lifting-line free-vortex wake method implemented in QBlade. The model is suitable to be used in time-marching aero-servo-elastic simulations and in design and optimization studies.

The aerodynamic design of the fly-gen airborne wind energy system aircraft, referred to as a windplane here, is a largely unexplored yet crucial problem for improving power production. To this end, an engineering model for the aerodynamics of the onboard turbines, the aerodynamics of the wing, and their interactional aerodynamics is developed and coupled to a steady-state windplane model and a far-wake model. This novel comprehensive model is then used to design the windplane aerodynamics for a given wingspan. Initially, a design space exploration study reveals that placing the turbines at the wing tips and rotating them inboard down increases the power production compared to other locations and rotation directions. This improvement arises because the turbines' wake swirl reduces the wing's induced drag, which increases flight speed and, consequently, the generated power. Moreover, airfoils with a high lift-to-drag ratio are found to be optimal for windplanes. As a consequence, NACA4421 airfoils are used for the design of the wing and the tip-mounted turbines. The trapezoidal wing with constant twist which maximizes power production has an aspect ratio of 5.1 and a taper ratio of 0.60. By design, the onboard turbines operate at a low tip speed ratio of 1.9 to increase the wake swirl. The results from the vortex models of the wing, the turbines, and their interaction show very good agreement with the lifting line, the vortex lattice method, and the vortex particle method implemented in the well-established code DUST. Finally, the windplane is studied with DUST at different wing angles of attack and at different turbine tip speed ratios to characterize its behavior away from the design point. The maximum in power production is observed near to the design point, well separated from the stall regions of both the wing and the turbines.