  In this work, we establish an optimal control framework for airborne wind
energy systems (AWESs) with flexible tethers. The AWES configuration,
consisting of a six-degree-of-freedom aircraft, a flexible tether, and a winch,
is formulated as an index-1 differential-algebraic system of equations (DAE).
We achieve this by adopting a minimal coordinate representation that uses Euler
angles to characterize the aircraft's attitude and employing a quasi-static
approach for the tether. The presented method contrasts with other recent
optimization studies that use an index-3 DAE approach. By doing so, our
approach avoids related inconsistency condition problems. We use a homotopy
strategy to solve the optimal control problem that ultimately generates optimal
trajectories of the AWES with a flexible tether. We furthermore compare with a
rigid tether model by investigating the resulting mechanical powers and tether
forces. Simulation results demonstrate the efficacy of the presented
methodology and the necessity to incorporate the flexibility of the tether when
solving the optimal control problem.
