Optimal control of crosswind kite systems with an engineering wake model based on vortex loops and dipoles

Abstract. Modern crosswind kite systems provide a technological means to exert large aerodynamic forces in wind fields above the reach of conventional wind turbines, with applications such as airborne wind energy (AWE) generation and ship towing. A central challenge in kite system optimization is to both accurately and efficiently model self-induction effects on system design, performance, and operation. Vortex-based models are a natural candidate for this task as they provide a wake resolution that allows us to consistently capture kite-specific operating conditions. Existing vortex-based approaches have been developed under the assumption of static, axisymmetric flight, which is typically violated in practice. Therefore, we propose a vortex-based continuous-time wake model for simulation and optimal control of crosswind kites that is capable of capturing the unsteady, non-axisymmetric flight conditions induced by skewed inflow and gravity. The model represents the shed vorticity as a hybrid distribution of infinitesimal vortex loops and dipole elements and shows good agreement with simulation results obtained with the free-vortex solver DUST, with the remaining discrepancies largely due to convection-velocity selection. As a second contribution, we introduce a transcription strategy to efficiently incorporate the new model into periodic optimal control problems (OCPs). In a numerical case study based on a dual-kite system, we examine the sensitivity of the OCP optimal cost (average tether force) and design to the key transcription parameters. Based on this sensitivity analysis, we find an efficient transcription that solves the problem at three times the computational cost of the OCP without any wake model while still retaining accuracy (in terms of combined error of optimal cost and design) within 5 % compared to a highly resolved transcription of the wake-aware problem. In contrast, the solution of the no-wake problem deviates by 145 % compared to this highly resolved reference solution. Overall, the framework enables the efficient and wake-aware optimal control of crosswind kite systems and can be readily applied to industry-relevant applications such as single-kite airborne wind energy systems.