⚡ electrical engineering

Optimal Path Planning of Airborne Wind Energy Systems with a Flexible Tether

This paper establishes an optimal control framework for airborne wind energy systems with flexible tethers by formulating the problem as an index-1 differential-algebraic system using a minimal coordinate representation and homotopy strategy, demonstrating through simulations that accounting for tether flexibility is essential for accurate trajectory optimization compared to rigid tether models.

Original authors: Omid Heydarnia, Jolan Wauters, Tom Lefebvre, Guillaume Crevecoeur

Published 2026-07-02

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CC BY 4.0

Original authors: Omid Heydarnia, Jolan Wauters, Tom Lefebvre, Guillaume Crevecoeur

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to generate electricity by flying a giant kite high into the sky, where the wind is stronger and more consistent. This is the goal of Airborne Wind Energy Systems (AWES). Instead of a massive, stationary wind turbine with a tower, you have an aircraft tethered to a winch on the ground.

Here is how the system works, according to the paper:

The Power Phase: The kite flies in fast, looping patterns (like a figure-eight or a circle). This pulls the tether out, spinning a generator on the ground to make electricity.
The Reset Phase: Once the tether is fully out, the kite needs to come back. The generator switches to "motor" mode, using some of the stored energy to reel the kite back in. During this time, the kite flies in a way that creates very little lift to make reeling it in easier.

The Problem: The "Saggy" String

Most computer models used to plan the best flight paths for these kites treat the tether (the string) as a perfectly rigid rod. They assume the string is always straight and tight, like a steel pole.

The authors of this paper argue that this is a bad idea for the "Reset Phase." When the kite is being pulled back in, the string isn't under high tension; it can go slack and sag due to gravity and wind resistance, just like a heavy rope hanging between two points. If you ignore this sag, your computer model thinks the kite is in a different place than it actually is, and it might plan a flight path that is inefficient or even dangerous.

The Solution: A "Smart" Flexible String Model

The researchers created a new way to model this system that accounts for the string's flexibility without making the computer calculations take forever.

The Analogy: Imagine the tether isn't one solid stick, but a chain of heavy beads connected by stretchy rubber bands. The computer calculates the position of each bead, accounting for gravity pulling them down and the wind pushing against them.
The Trick: To keep the math from becoming too heavy, they used a "quasi-static" approach. This means they assume the beads settle into their shape instantly based on the forces acting on them, rather than calculating every tiny vibration. This balances accuracy (it looks like a real sagging rope) with speed (the computer can still solve the problem in a reasonable time).

The Experiment: Rigid vs. Flexible

The team used their new model to find the "optimal path" (the most energy-efficient flight pattern) for a large, megawatt-scale kite. They compared two scenarios:

The Old Way: Treating the string as a rigid, straight rod.
The New Way: Treating the string as a flexible, sagging rope.

What They Found

Even though the flight paths looked very similar on a map (both kites flew in circles or figure-eights), the physics were quite different:

The "Pull" is Different: When the kite was being pulled back in (the retraction phase), the flexible model showed the string pulling at a different angle than the rigid model. Because the string was sagging, the force wasn't pointing straight at the winch.
The Power Mistake: The rigid model (the straight string) overestimated how much power could be generated. In some cases, it thought the system would produce 33% more power than it actually would. This is because the rigid model didn't account for the energy lost to the string sagging and the extra drag it creates.
Control Changes: To fly the same path, the kite with the flexible string needed different control inputs (like turning its wings differently) to compensate for the weird pull of the sagging rope.

The Takeaway

The paper concludes that if you want to design a real, working airborne wind energy system, you cannot treat the tether as a stiff stick. You must account for the fact that the rope sags. Ignoring this sag leads to overly optimistic predictions about how much electricity the system can generate.

By using their new "flexible string" math, engineers can plan safer, more accurate flight paths that know exactly how the rope will behave, ensuring the system generates the maximum amount of power possible without crashing or running out of energy.

Technical Summary: Optimal Path Planning of Airborne Wind Energy Systems with a Flexible Tether

Problem Statement
Airborne Wind Energy Systems (AWES) aim to harvest energy from high-altitude winds using aircraft and tethers. A critical challenge in AWES operation is establishing optimal flight paths that maximize power generation while ensuring safety during traction (power generation), retraction (aircraft recovery), and phase transitions. While the dynamics of the aircraft are well-studied, modeling the tether presents significant difficulties. Most existing optimization frameworks assume the tether is a rigid rod, an assumption valid during the high-tension traction phase but inaccurate during retraction and transitions where tether sag and elastic deformation occur. Previous attempts to model flexible tethers often rely on complex lumped-mass or finite-element approaches that drastically increase system dimensionality and computational cost, making them unsuitable for real-time or efficient optimal control. Furthermore, many high-fidelity models utilize index-3 differential-algebraic equations (DAEs), which introduce consistency condition problems and numerical drift.

Methodology
The authors propose an optimal control framework that integrates a quasi-static flexible tether model into a computationally efficient optimization problem. The methodology is structured as follows:

Modeling: The system model builds upon the MegAWES toolbox. It utilizes a minimal coordinate representation for the aircraft's attitude (Euler angles) rather than rotation matrices to avoid singularities and reduce complexity. The core innovation is the inclusion of a quasi-static flexible tether model. This model discretizes the tether into $N+1$ lumped masses connected by massless elastic segments. It accounts for gravitational sag and elastic deformation (via Hooke's Law) and drag forces, while neglecting dynamic elastic vibrations to maintain computational tractability. The model is formulated as an index-1 DAE system, where the tether force at the winch is an input, and the position of the aircraft is determined via a recursive calculation and a root-finding problem to satisfy geometric constraints.
Optimization Formulation: The problem is defined as a periodic continuous-time Optimal Control Problem (OCP) aimed at maximizing average power output over a cycle. The cost function includes negative power output and penalties on aggressive maneuvers (angular acceleration and side-slip). Path constraints include tether force limits, flight boundaries, and collision avoidance between the tether and aircraft.
Solution Strategy: To address the non-linearity and under-actuation of the system, the authors employ a Penalty-based Interior-Point Homotopy (PIPH) approach. The optimization is solved in three stages:
1. Initialization: Fictitious forces and torques are applied to the aircraft to generate a feasible initial trajectory.
2. Transition: The fictitious forces are gradually removed, and a propulsion/braking force is used to maintain feasibility while restoring real aerodynamics.
3. Optimization: The cost function shifts focus to maximizing power generation.
  The problem is transcribed using direct multiple shooting and solved using the IPOPT solver with CasADi.

Key Contributions

Flexible Tether in Optimization: The paper integrates a physically accurate flexible tether model (accounting for sag and elasticity) into an optimal path planning problem without incurring the prohibitive computational cost associated with full dynamic lumped-mass models.
Index-1 DAE Formulation: The authors introduce a semi-explicit index-1 DAE formulation with minimal coordinates. This eliminates the consistency condition problems and numerical drift often associated with index-reduction techniques used in higher-index DAE models (e.g., index-3).
Comparative Analysis: The study provides a direct comparison between rigid and flexible tether models within the same optimization framework, isolating the effects of tether flexibility on control inputs and power estimation.

Results
Simulations were conducted for a megawatt-scale AWES with a 42.5m wingspan, testing both circular and lemniscate flight paths under a logarithmic wind shear profile (15 m/s).

Trajectory Similarity: The optimal flight paths generated for rigid and flexible tether models appear visually similar.
Control and Power Discrepancies: Despite trajectory similarities, significant differences were observed in control inputs and harvested power. The rigid tether model consistently overestimated the harvested power compared to the flexible model. For the tested scenarios, the rigid model overestimated power by up to 33%.
Retraction Phase Dynamics: The most pronounced differences occurred during the retraction phase and the transition between phases. The flexible model captured tether sag, which altered the direction of the tether force acting on the aircraft's body frame. This change in force direction necessitated different control inputs to maintain stability and efficiency.
Computational Cost: The inclusion of the flexible tether (with 10 segments) increased computational time compared to the rigid model, particularly for the more complex lemniscate trajectories. However, the quasi-static approach kept the computation time manageable on standard hardware.

Significance and Claims
The paper claims that while rigid tether assumptions may suffice for rough estimations during high-tension phases, they fail to accurately predict system performance during retraction and transition phases. By neglecting tether sag, rigid models lead to an overestimation of harvested power and potentially unsafe or suboptimal control strategies. The proposed index-1 DAE formulation with a quasi-static flexible tether offers a balanced approach, providing higher physical fidelity than rigid models while remaining computationally efficient enough for optimal control applications. The work demonstrates that accurate modeling of tether flexibility is essential for the precise path planning and performance evaluation of large-scale AWES.

The Problem: The "Saggy" String

The Solution: A "Smart" Flexible String Model

The Experiment: Rigid vs. Flexible

What They Found

The Takeaway

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