  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.
