The aim of this work is to propose engineering models to include the external forces acting on the tether into the power estimation of an Airborne Wind Energy System flying crosswind. The influence of aerodynamic drag, centrifugal force and gravitational force on power production is investigated. These three types of external loading acting on the tether are lumped with the kite and the tether is considered straight. In this way, a set of equations describing the tether shape and its influence on the power production is reduced to just one power equation. The formulation with straight tether and lumped quantities is suitable for preliminary designs and analyses and points out how the tether influence the system. On top of the results concerning the tether modelling, a new analytical model which includes the centrifugal forces into the power estimation for circular paths is presented.
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.
Abstract. 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.
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