The optimal control problem for flight trajectories of Fly-Gen airborne wind energy systems (AWESs) is a crucial research topic for the field, as suboptimal paths can lead to a drastic reduction in power production. One of the novelties of the present work is the expression of the optimal control problem in the frequency domain through a harmonic balance formulation. This allows the potential reduction of the problem size by solving only for the main harmonics and allows the implicit imposition of periodicity of the solution. The trajectory is described by the Fourier coefficients of the dynamics (elevation and azimuth angles) and of the control inputs (onboard wind turbine thrust and AWES roll angle). To isolate the effects of each physical phenomenon, optimal trajectories are presented with an increasing level of physical representation from the most idealized case: (i) if the mean thrust power (mechanical power linked to the dynamics) is considered as the objective function, optimal trajectories are characterized by a constant AWES velocity over the loop and a circular shape. This is done by converting all the gravitational potential energy into electrical energy. At low wind speed, onboard wind turbines are then used as propellers in the ascendant part of the loop; (ii) if the mean shaft power (mechanical power after momentum losses) is the objective function, a part of the potential energy is converted into kinetic and the rest into electrical energy. Therefore, the AWES velocity fluctuates over the loop; (iii) if the mean electrical power is considered as the objective function, the onboard wind turbines are never used as propellers because of the power conversion efficiency. Optimal trajectories for case (ii) and (iii) have a circular shape squashed along the vertical direction. The optimal control inputs can be generally modeled with one harmonic for the onboard wind turbine thrust and two for AWES roll angle without a significant loss of power, demonstrating that the absence of high-frequency control is not detrimental to the power generated by Fly-Gen AWESs.

Airborne wind energy (AWE) is the branch of wind energy which aims at harvesting energy from the wind using airborne systems. Airborne wind energy systems can be classified according to the flight operations, which are linked to the power generation technique. The flight operations can be divided into crosswind, tether-aligned and rotational, as discussed by

The first analytical power equation of crosswind AWESs was derived by

Higher-fidelity, but still computationally efficient, dynamic models are developed by

The dynamic models just introduced are particularly suitable to be used within optimal control studies for their computational inexpensiveness and for the reduced number of nonlinearities compared to even higher-fidelity codes, such as kiteFAST

As the aim of this work is to interpret optimal trajectories in a physical way, a low-fidelity dynamic model, similar to the one proposed by

Even though the frequency-domain formulation can be used for any periodic flight trajectories (i.e., circular and figure of eight), only circular trajectories are analyzed here to limit the paper scope and length. Figure-of-eight trajectories are intended to be analyzed and intensively compared with circular trajectories in a future work.

The paper is organized as follows: in Sect.

Two coordinate systems (Fig.

The wind velocity is in the positive

To describe the AWES attitude, a non-sideslip velocity constraint is included in the modeling. Indeed, the wing operates at the highest performance under this condition. To impose this constraint implicitly, the unit vector

The spanwise unit vector

Note that

With this formulation,

The aerodynamic lift

The dynamic equations of motion in compact form read

As the objectives of the optimal control problems are linked to the power production, three different power quantities are defined. The thrust power

The shaft power

Finally, the electrical power exchanged with the grid

When power is generated (

Frequency-domain formulations may present clear advantages when solving for periodic solutions of dynamic and control problems. They have the capability of solving for both stable and unstable branches of periodic solutions in an efficient way. Moreover, they potentially use fewer variables to describe the same problems. Since the problem of optimal trajectories for AWESs has a periodic nature, the flight dynamic model just introduced is expressed in the frequency domain. The harmonic balance methodology is then used to transform the differential equations of motion into a set of nonlinear algebraic equations

The first and second time derivatives of the state vector can be found analytically:

Similarly, the control inputs, assumed to be periodic, can also be expressed as a Fourier series of order

The Fourier coefficients of the equations of motion are found numerically by applying the Fourier coefficient definition to the time series, which should have a minimum size of

In this work, the frequency-domain formulation is included within an optimal control problem (OCP). A generic optimization problem can be written as

Graphical representation of the optimal control problem setup.

The negative value of the mean thrust power

Inequality constraints ^{®} environment and solved with the interior-point algorithm implemented in

To compare the results of the optimal control problem with idealized analytical expressions, the main results from a refined version of the Loyd power equation

Finally, the power generated and sent to the grid takes into account the efficiencies of the electrical conversion:

For high

Normalized power

Maximum normalized power

Reference values for the examples (values from the Makani MX2 description;

The tether force can be evaluated as

In this idealized case, the turning radius is

In addition to the non-dimensional mass parameter, the Froude number, which weights the fluid inertial forces to gravity forces, is used in this work:

In the following sections, the results will be generalized as a function of the non-dimensional parameters just introduced. Input parameters from the Makani

To make sure the frequency-domain formulation is well implemented and finds solutions which respect the equations of motion, they are compared with the solution coming from a time integration scheme. The model described in Sect. ^{®}

As the analysis is limited to circular trajectories, a cylindrical reference frame

Cylindrical reference system

To increase complexity incrementally, the optimal control problems (OCPs) are modified from the most idealized case to a realistic one. For the idealized cases analyzed in this section, uniform incoming wind speed (

For the most idealized case, the gravity is null

Optimal trajectory.

For the solution to be optimal, it is found that the AWES span is perpendicular to the wind speed, or, in analytical terms, that

Optimal opening angles

The values of

Settings of the two optimal control problems maximizing the mean thrust power considering gravity.

Gravity is now included in the modeling, and the objective function is taken as the mean thrust power

By solving the OCPs, it is found that the optimal solutions have a negative mean elevation of

Optimal trajectory for OCP A (

Optimal

Figure

Figure

Optimal

Figure

Norm of the optimal AWES velocity

Figure

Tether force

To compare the two OCPs and draw some conclusions, the power output, shown in Fig.

Optimal thrust power production and consumption

As the two analyzed OCPs are almost equivalent, the optimal trajectories are characterized by the perpendicularity of the AWES span with respect to the wind (

Following these considerations, the power trend, as shown in Fig.

The onboard wind turbine thrust can be approximated with

Figure

Amplitude of the first Fourier coefficient of

Figure

First Fourier coefficient of elevation

In this section, the onboard wind turbine induction is included in the power evaluation, and the mean shaft power

Settings of the two optimal control problems maximizing the mean shaft power considering gravity.

Figure

Optimal trajectory (–) and a circle with radius

Figure

Optimal

Optimal

Figure

In Fig.

Norm of the optimal AWES velocity

To conclude the analysis of the example, Fig.

Optimal shaft power production and consumption

In Fig.

Optimal shaft power production normalized with the analytical expression of thrust power as a function of

Optimal values of

Optimal values of

In Sect.

Finally, Fig.

In this section, the electrical efficiency is included into the optimal control problem, and the mean electrical power is considered as objective function. Two OCPs, whose characteristics are given in Table

Settings of the two optimal control problems maximizing the mean electrical power considering gravity.

Figure

Optimal trajectory for OCP A (

Figure

Optimal

Optimal

Norm of the optimal AWES velocity

Figure

Figure

The electrical power as a function of the angular position is shown in Fig.

Optimal electrical power production as a function of the angular position.

One could try to investigate how the optimal values evolve for an increasing wind speed. Figure

Optimal values of

In Fig.

Normalized electrical power for a case with

To conclude, Fig.

Optimal values of

In this section, the wind shear is included in the problem. The reference altitude is

Two OCPs are solved and they are summarized in Table

Settings of the two optimal control problems maximizing the mean power considering gravity and wind shear.

Figure

Optimal trajectory for case A (

The roll angle

Optimal

Figure

Optimal

Figure

Wind velocity as a function of the angular position.

Figure

Norm of the optimal AWES velocity

Figure

Optimal power production

As carried out in the previous section, trends are studied as a function of the Froude number for the optimal control problem B. Figure

Optimal values of

Optimal values of

To conclude, Fig.

In this work, a novel methodology to study optimal trajectories for Fly-Gen AWESs is introduced. The chosen low-fidelity dynamic model is characterized by 2 degrees of freedom (the AWES is modeled as a point mass with constant tether length) and two control inputs. The degrees of freedom are the elevation and the azimuth angle. The control inputs are the roll angle, defined as the rotation around the relative velocity direction, and the onboard wind turbine thrust coefficient. An optimal control problem is formulated in the frequency domain through a harmonic balance method. Working with the Fourier coefficients of the time series, instead of the time series themselves, allows the potential reduction of the problem size, the implicit imposition of periodicity and the acquisition of an intuitive understanding of the results by analyzing the harmonic contributions. Moreover, the analytical gradient of the objective function and the constraints with respect to the optimization variables can be provided to the solver, allowing for a deep and fast convergence of the optimal solutions.

The MX2 design from

If the mean thrust power (mechanical power neglecting onboard wind turbine induction) is the objective function, the optimal trajectories are circular, have a constant AWES velocity and the wing span is perpendicular to the incoming wind. To obtain this condition, all the potential energy is converted into electrical energy by the onboard wind turbines. At low wind speed, onboard wind turbines are then used as propellers in the ascendant part of the loop. The optimal power, the trajectory shape and the production strategy can be accurately approximated with analytical expressions.

If the mean shaft power (mechanical power considering onboard wind turbine induction) is the objective function, the potential energy, in the descending leg, is partially converted into electrical energy and partially into kinetic energy. This is because the power conversion penalizes solutions with high onboard wind turbine induction. Therefore, the velocity fluctuates over the loop, and the trajectories are squashed along the vertical direction to decrease the potential energy exchange.

If the mean power electrical provided to the grid is the objective function (i.e., the electrical efficiency is included), the onboard wind turbines never operate as propellers. If operated as propellers, power would be converted from mechanical into electrical while descending and from electrical into mechanical while ascending, leading to large power losses due to the electrical efficiency. This effect is found only at low wind speed, when propelling the AWES in the climbing leg maximizes the mean shaft power. Past a given wind speed, using the onboard wind turbines as propellers does not maximize the mean shaft power, and the influence of the electrical efficiency on the production strategy vanishes.

For all the analyzed cases, additional analytical approximations characterizing the solution are introduced. These approximations are found by modeling the control inputs with the lowest number of harmonics. The onboard wind turbine thrust can be modeled with just one harmonic and the roll with two harmonics without loss of generality of the results.

The results of this work align with the discussions in

Azimuth and elevation of the trajectory found with the harmonic balance method and the time integration scheme for a circular-shaped trajectory.

Time series of the control inputs provided to the harmonic balance method and the time integration scheme for a circular-shaped trajectory.

Azimuth and elevation of the trajectory found with the harmonic balance method and the time integration scheme for a figure-of-eight-shaped trajectory.

Time series of the control inputs provided to the harmonic balance method and the time integration scheme for a figure-of-eight-shaped trajectory.

The code was developed for this publication. For inquiries about it, please reach out to the corresponding author.

No data sets were used in this article.

FT, ICF and GP conceptualized the study and the research methods. FT developed the research methods. FT and ICF developed the code. FT produced the results and wrote the draft version of the paper. CEDR and AC supervised the research. ICF, GP, CEDR and AC reviewed the draft version.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work by PoliMI had no external funding and was therefore self-funded by the research team. The work by ICF was carried out under the framework of the GreenKite-2 project (PID2019-110146RB-I00) funded by MCIN/AEI/10.13039/501100011033.

This paper was edited by Roland Schmehl and reviewed by Nikolaus Vertovec and two anonymous referees.