Robustness and Stability Optimization of Open-Loop Controlled Power Generating Kites

Author

• B. Houska

Reference

• Diploma thesis, University of Heidelberg, 2007.

Abstract

In the first part of this thesis we present optimization studies for kites that produce wind energy by periodically pulling a generator on the ground while flying fast in a crosswind direction. We derive a model for a single kite and formulate an optimal control problem with periodic boundary conditions and free cycle duration. The objective function is the average power at the generator. We solve this nonlinear and unstable optimal control problem numerically with Bock’s direct multiple shooting method. Here, the main result is that we can attain about 5 MW with a 500 m^2-kite at 10 m/s wind speed. In addition, we consider a system of two ”Dancing” kites and show that such systems can further increase the efficiency.

In the second part of this thesis we introduce novel techniques to increase the robustness and stability of periodic dynamical systems with respect to small disturbances on an infinite time horizon in the past. Therefore, we start with linear time-periodic systems and consider the corresponding periodic Lyapunov differential equations. This allows us on the one hand to compute confidence ellipsoids of the states with respect to a white noise disturbance, and on the other hand, we link the existence of periodic and positive definite solutions of the periodic Lyapunov differential equation to the asymptotic stability of the underlying linear system. We extend these considerations to linear time-periodic systems with inequality constraints. In addition we consider random disturbances with bounded autocorrelation functions. We transfer the results to nonlinear systems formulating nonlinear optimal control problems that allow us to take robustness and stability aspects into account. Finally, we apply our robustness and stability optimization techniques to the power generating kite system. We discuss how to find trajectories and corresponding controls for the kite such that the following requirements are satisfied:

• The kite should produce as much power as possible.

• The kite should fly on an open-stable orbit (without any feedback).

• The kite and the cable should not touch the ground even if wind turbulences are present.

Surprisingly, such robust and open-loop stable trajectories exist and were found by the presented stability optimization techniques. We present long time simulation of open-loop stable kite systems testing the robustness with respect to random wind turbulences.

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• Diploma Thesis

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