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# Optimizing the design of a kite-based generator using a simple mechanical model

## Antonin Bavoil
## Université Côte d'Azur, CNRS, LJAD, Inria
antonin.bavoil@univ-cotedazur.fr

**Keywords.** airborne, kite, optimization design

Using kites to collect wind power and generate energy has been intensively studied in the last decade, see*e.g.* the survey by M. Diehl et al. in [1]. In the framework of the KEEP (Kite Electrical Energy Power) funded by CNRS and gathering researchers from ENSTA Bretagne (well acquainted with the topic after previous studies on kites [2], most notably for boats [3]) and Université Côte d'Azur, we are interested in the analysis of a simple device composed of a kite attached to an arm; having the kite running along a well chosen curve will move the arm and generate electricity.

[Figure : device en 3D]

Our main objective with this work is to determine the maximum average power generated by our device. Our model should be simple to focus only on the choice of design parameters, and not worry about the trajectory of the kite nor its control. The model established by Kelvin Desenclos [2] achives this by using a simple point-mass mechanical model where the kite motion is prescribed to a eight-shaped conical surface, hence eliminating the need for control. With some refinement we were able to implement it as a dimension $5$ ordinary differential equation (ODE) ($2$ dimensions for the state, $2$ for its rate of change, $1$ after integrating the generated power over time). We were also able to use this model in order to compute the average power for a given set of design parameters by running a simulation over an extended period of time. The next step is then to optimize these parameters. A crucial fact is that, as time evolves, the kite converges towards a limit cycle. We conducted some numerical analysis to confirm the result and found that any starting configuration converges rapidly towards one of two limit cycles, depending on the initial conditions. We are now able to compute the exact average power by 1. finding the limit cycle and 2. integrating the instantaneous power over the cycle (about three seconds of simulated time), instead of integrating it over a very large time span (six hundred seconds for a reasonable yet approximate average power). Using this knowledge, we devise an optimization problem whose solution gives us the parameters, initial condition and period such that after one period, the system goes back to its initial condition, and the average power is maximized. This problem can be implemented using three different levels of implicitness:

- given a set of parameters, the period and the initial conditions are obtained by a root finding algorithm (initial state - final state = $0$) that solves the ODE, the dynamics is integrated over one limit cycle to obtain the average power;
- given a set of parameters, an initial condition, and a period, an ODE solver is used to compute the state after one period, and we add the constraint that the final state must equal the initial state;
- given a set of parameters, a period, a number of integration steps $N$, and all the states at each integration step, we add $N$ constraints that ensure the state at each integration step respects a given numerical method for integrating the ODE (*e.g.* Forward Euler or 4th order Runge-Kutta).

In the more implicit formulation, we delegate solving parts of the problem to other function so that the optimizer focuses only on finding the optimal parameters; while in the more explicit one, we hope to leverage the capabilities of powerfull optimizers like Ipopt. Once an optimum is found, we conduct a sensibility analysis around it to determine the most impactful parameters and guide the design process of a potential prototype.

**References**

[1] Ahrens, U.; Diehl, M.; Schmehl, R. (eds.) Airborne Wind Energy, Springer, 2013

[2] Desenclos, K.; Nême, A.; Leroux, J.-B.; Jochum, C. A novel composite modelling approach for woven fabric structures applied to leading edge inflatable kites. *Mech. Composite Materials* **58** (2023), no. 6, 867—882.

[3] Podeur, V.; Merdrignac, D.; Behrel, M.; Roncin, K.; Fonti, C.; Jochum, C.; Parlier, Y.; Renaud, P. Fuel economy assessment tool for auxiliary kite propulsion of merchant ship. **Houille Blanche** *1* (2018), 5—7.