Robustness and stability optimization of power generating kite systems in a periodic pumping mode

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.

[1]  Moritz Diehl,et al.  An approximation technique for robust nonlinear optimization , 2006, Math. Program..

[2]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[3]  Patrizio Colaneri,et al.  Stability Analysis of Linear Periodic Systems Via the Lyapunov Equation , 1984 .

[4]  R. Kálmán.  LYAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E IN AUTOMATIC CONTROL. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects , 2003, Comput. Chem. Eng..

[6]  M. Diehl,et al.  Optimal Control of Towing Kites , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[7]  M. Canale,et al.  KiteGen project: control as key technology for a quantum leap in wind energy generators , 2007, 2007 American Control Conference.

[8]  B. Lansdorp,et al.  The Laddermill: Innovative Wind Energy from High Altitudes in Holland and Australia , 2006 .

[9]  R E Kalman,et al.  CANONICAL STRUCTURE OF LINEAR DYNAMICAL SYSTEMS. , 1962, Proceedings of the National Academy of Sciences of the United States of America.

[10]  A. L. Schwab,et al.  MODELLING OF THE DYNAMIC BEHAVIOUR OF A LADDERMILL, A NOVEL CONCEPT TO EXPLOIT WIND ENERGY , 2003 .

[11]  Wim Michiels,et al.  The Smoothed Spectral Abscissa for Robust Stability Optimization , 2009, SIAM J. Optim..

[12]  M. L. Loyd,et al.  Crosswind kite power (for large-scale wind power production) , 1980 .

[13]  R. E. Kalman,et al.  Control System Analysis and Design Via the “Second Method” of Lyapunov: I—Continuous-Time Systems , 1960 .

[14]  Filip Logist,et al.  Approximate robust optimization of time-periodic stationary states with application to biochemical processes , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[15]  Paolo Bolzern,et al.  The periodic Lyapunov equation , 1988 .

[16]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization: Part II: Software aspects and applications , 2003, Comput. Chem. Eng..

[17]  B. Lansdorp,et al.  The Laddermill: work in progress , 2004 .

[18]  Stephen P. Boyd,et al.  Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems , 2006, Math. Program..

[19]  A. R. Podgaets,et al.  Problem of Pareto-Optimal Control for a High Altitude Energy System , 2006 .

[20]  Richard D. Braatz,et al.  Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis , 2004 .

[21]  Moritz Diehl,et al.  ACADO toolkit—An open‐source framework for automatic control and dynamic optimization , 2011 .

[22]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[23]  M. Canale,et al.  Control of tethered airfoils for a new class of wind energy generator , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[24]  Moritz Diehl,et al.  Stability Optimization of Hybrid Periodic Systems via a Smooth Criterion , 2009, IEEE Transactions on Automatic Control.

[25]  Moritz Diehl,et al.  Robust nonlinear optimal control of dynamic systems with affine uncertainties , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[26]  P. Williams,et al.  Optimal Cross-Wind Towing and Power Generation with Tethered Kites , 2007 .

[27]  Moritz Diehl,et al.  Optimal control for power generating kites , 2007, 2007 European Control Conference (ECC).

[28]  Wubbo Ockels,et al.  Ship propulsion by Kites combining energy production by Laddermill principle and direct kite propulsion , 2006 .

[29]  Moritz Diehl,et al.  Nonlinear MPC of kites under varying wind conditions for a new class of large‐scale wind power generators , 2007 .

[30]  Adrian S. Lewis,et al.  Optimization and Pseudospectra, with Applications to Robust Stability , 2003, SIAM J. Matrix Anal. Appl..

[31]  B. Lansdorp,et al.  Comparison of concepts for high-altitude wind energy generation with ground based generator , 2005 .

[32]  A. R. Podgaets,et al.  Flight Control of the High Altitude Wind Power System , 2007 .