A rigidly-convected, lifting-line vortex model in the optimal control of airborne wind energy systems

Abstract. Solving optimal control problems (OCPs) to estimate the performance of airborne wind energy (AWE) systems can raise the question: "How should one include a vortex-filament model into an OCP on an irregular flight path, specifically where there are also a substantial number of inequality constraints?" Since a rigidly-convected, lifting-line vortex (RLL) model might balance computational expense against sensitivity to geometric assumptions, we construct a solvable problem by judiciously lifting certain algebraic variables into an AWE OCP’s decision variables. This RLL model makes predictions that are consistent considering modeling assumptions with independently-published Large Eddy Simulations, during rotationally-steady as well as dynamic situations. Further, demonstrating the importance of including wake-modeling in airborne wind energy applications, solving an AWE OCP including this RLL model finds a substantial difference in power compared to solving an AWE OCP assuming negligible induced velocity; this paper’s demonstration problem, with a two-kite (Ampyx-AP2) system and a single- winding trajectory, predicts a 20 %–30 % loss in harvestable power (depending on initialization and objective). Unfortunately, solving such an RLL AWE OCP is also quite expensive: in the problem instances and solver configurations tested within this work, the required computational memory scales with the total number of vortex elements instantiated in the entire problem, while the processor time scales with more than that number squared. In conclusion, it is possible to include a RLL model into a solvable AWE OCP with an irregular trajectory and many inequality constraints, though the solution is quite expensive and is heavily influenced by the homotopy strategy used for initialization.