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In this paper we formulate and solve optimal control problems for power generating kite systems. Here, the kite generates energy by periodically pulling a generator on the ground while flying fast in a crosswind direction. We are searching for an intrinsically open-loop stable trajectory such that the kite generates as much power as possible without needing feedback, while neither the kite nor the cable should touch the ground in the presence of wind turbulence. As the wind turbulences are unknown, robustness aspects need to be taken into account. The formulation of the associated optimal control problem makes use of periodic Lyapunov differential equations in order to guarantee local open-loop stability while robustness aspects are regarded in a linear approximation. The main result of this paper is that open-loop stable kite orbits exist and that open-loop stability only costs approximately 23% compared to the power-optimal unstable orbit.