A quaternion-based model for optimal control of the SkySails airborne wind energy system

TL;DR

This paper introduces a quaternion-based model for tethered kite dynamics and formulates an optimal control problem to maximize energy generation in airborne wind energy systems, demonstrating efficient computation and doubled power output.

Contribution

The paper presents a singularity-free quaternion model for kite dynamics and an optimal control framework for pumping cycles, enabling efficient optimization of energy extraction.

Findings

Optimal control finds the best orbit in minutes.

Power output doubles compared to initial guesses.

Model is derived from first principles and is singularity-free.

Abstract

Airborne wind energy systems are capable of extracting energy from higher wind speeds at higher altitudes. The configuration considered in this paper is based on a tethered kite flown in a pumping orbit. This pumping cycle generates energy by winching out at high tether forces and driving a generator while flying figures-of-eight, or lemniscates, as crosswind pattern. Then, the tether is reeled in while keeping the kite at a neutral position, thus leaving a net amount of generated energy. In order to achieve an economic operation, optimization of pumping cycles is of great interest. In this paper, first the principles of airborne wind energy will be briefly revisited. The first contribution is a singularity-free model for the tethered kite dynamics in quaternion representation, where the model is derived from first principles. The second contribution is an optimal control formulation and numerical results for complete pumping cycles. Based on the developed model, the setup of the optimal control problem (OCP) is described in detail along with its numerical solution based on the direct multiple shooting method in the CasADi optimization environment. Optimization results for a pumping cycle consisting of six lemniscates show that the approach is capable to find an optimal orbit in a few minutes of computation time. For this optimal orbit, the power output is increased by a factor of two compared to a sophisticated initial guess for the considered test scenario.