# Ship Towed by Kite: Investigation of the Dynamic Coupling

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}research program launched by Yves Parlier and managed in partnership with French Ministry of Defence with the support of French Environment and Energy Management Agency (ADEME). The project attempts to develop a tethered kite system as an auxiliary device for the propulsion of merchant ships. The existing knowledge on ships towed by kite has demonstrated great prospects for this technology in terms of CO

_{2}emissions and fuel savings. However, studies on the limitations of this concept to guarantee the safety and the integrity of the ship are very limited.

## 2. Ship Towed by Kite Modeling

#### 2.1. Reference Frame and Parametrization

#### 2.2. Ship Modeling

#### 2.2.1. Ship Motions

#### 2.2.2. Ship Modeling Validation

^{−1}and with frequency head waves from 1 rad.s

^{−1}to 7.5 rad.s

^{−1}. ${O}_{s}$ was defined at the ship center of gravity, hence ${l}_{x}=LCG$ and ${l}_{z}=VCG$. Figure 4 and Figure 5 plot the heave and pitch transfer function obtained with the experimental data, with the STF strip theory and with the present modeling. The experimental data are obtained for different wave steepness ${s}_{w}=\left\{0.025,\phantom{\rule{4pt}{0ex}}0.05,\phantom{\rule{4pt}{0ex}}0.075\right\}$. The amplitude of the transfer function for heave motion (a) is directly the ratio of the heave amplitude motion to the wave amplitude. The amplitude of the transfer function for the pitch motion (c) is given by the ratio of the motion amplitude (in radians) to the wave amplitude multiplied by the wave number k. The phase angle of the present modeling is obtained by cross correlation between the free surface elevation and the ship motion time series.

^{−1}in order to obtain the same roll amplitude at the natural frequency. As shown in Figure 6, for frequencies greater than 1 rad.s

^{−1}, the roll motion amplitude is well predicted despite a slight difference in phase. Nevertheless predictions appear to be less accurate for the lower frequencies, except for the natural roll frequency, ${\omega}_{roll}=0.56$ rad.s

^{−1}, where, by identification, both modelings match.

#### 2.3. Kite Modeling

#### 2.3.1. Short Literature Survey

^{2}kite with 360 m tether. Wellicome extended the zero-mass modeling defining a figure-8 trajectory for the dynamic mode [20,28]. Argatov et al. [29] proposed more recently a simpler trajectory definition. In addition, Argatov definition allows a much faster calculation and its formulation avoids discontinuities in kite acceleration. The size of the trajectory can easily be modified thanks only to azimuth and elevation amplitude parameters. In 2006, Naaijen et al. [2] computes the fuel savings on a full range of true wind angles for the British Bombardier, a 50000 DWT tanker. In 2011, Dadd et al. [21] improved the calculations with more realistic trajectories using the Wellicome formalism. In 2014, Leloup et al. [1,30] proposed a fully analytical solution for the zero-mass modeling that allows faster and more reliable computations avoiding optimization process failures. Leloup takes into account adequately the wind gradient effect and takes advantage of the Argatov trajectory definition for even faster computations.

#### 2.3.2. Zero-Mass Kite Model

#### 2.3.3. Kite Control According to a Trajectory

#### 2.3.4. Theoretical Lemniscate Trajectory

#### 2.4. Two Coupling Methods

#### 2.4.1. A Monolithic Approach

#### 2.4.2. A Segregated Approach

## 3. Results

#### 3.1. Validation and Comments on the Kite Aerodynamic Characteristics

^{®}Switchblade2016 of 5 m

^{2}designed for kite-surfing. The tether length was 80 m long. During the run, the kite performed eight shaped trajectory controlled by an autopilot based on the algorithm proposed in [34,35]. The experimental kite position is determined with a 3D load cell assuming that the tethers are straight. The evolution of the wind velocity with the altitude was identified thanks to a SOnic Detection And Ranging (SODAR). The experimental results presented here correspond to a phase averaging post-processing of a 5-minute kite flight run. For the presented case, the wind velocity was interpolated from the SODAR data with the following linear function:

^{2}Cabrinha

^{®}Switchblade is $2.23\sqrt{{A}_{k}}\approx 5.0$ m, Which is a little less than twice the projected wingspan. Given that the lift coefficient must have a strictly positive value in order to fly, we are here outside the validity domain of the proposed modeling. However, the order of magnitude is confirmed by Fagiano who evaluated the minimum turning radius at 2.5 times the wingspan [38,39]. The coefficients ${\u03f5}_{0}$, ${\kappa}_{\u03f5}$, ${C}_{l0}$ and ${\kappa}_{l}$ are dimensionless quantities. Consequently, the presented modifications in Equations (27) and (28) are retained as formulated for the rest of the paper.

#### 3.2. Study Case

^{−1}corresponding to the high range of a fresh breeze from the Beaufort scale was considered. The ship speed was set to ${U}_{h}=$7.5 m.s

^{−1}since it corresponds to a common sailing speed condition of the world merchant ship fleet. A kite with an area of ${A}_{k}=$500 m

^{2}and with the aerodynamic characteristics determined in Equation (28) was used. The tether attachment point was in the center plane, 7.9 m above the water line and 25 m in front of the center of gravity.

^{−1}. The 0.56 rad.s

^{−1}wave frequency corresponded to the natural roll ship frequency. For all following results, the simulation time was 1640 s with a time step of 0.3 s.

#### 3.3. Calm Water Case

#### 3.3.1. Kite Excitation Spectrum

#### 3.3.2. Comparison of the Segregated Approach with the Monolithic Approach

#### 3.4. Regular Beam Wave Case

^{−1}, 0.56 rad.s

^{−1}and 0.8 rad.s

^{−1}. As for the calm water case, the frequency domain of kite excitation is scanned with different tether lengths from ${L}_{t}=360$ m to ${L}_{t}=990$ m with a tether length step of 10 m.

#### 3.4.1. Ship Vertical Motion Influence on Kite Trajectory

#### 3.4.2. Interactions with Regular Beam Waves

^{−1}is 60.7°. For the case with the frequency wave ${\omega}_{w}=0.8$ rad.s

^{−1}, the interaction between the kite and the ship is less significant since the wave frequency is far from the most powerful kite harmonic frequencies. Moreover, in contrary to the calm water case, the segregated approach is not necessarily conservative with respect to the monolithic approach as shown by Figure 16a.

^{−1}and a tether length ${L}_{t}=840$ m, a drop of the ship roll amplitude can be noticed. Figure 17 shows that no principal kite harmonic frequency corresponds to the wave frequency. However, due to coupling between kite and ship motions, secondary harmonics appear. The secondary harmonic frequencies are denoted ${\omega}_{ki}^{{}^{\prime}}$. This phenomenon occurs when the frequency gap between the closest principal harmonic frequency and the wave frequency is a submultiple of the first kite harmonic frequency. For the case presented in Figure 17, ${\omega}_{k1}=3{\omega}_{k1}^{{}^{\prime}}$. Furthermore, it should be noted that these drops in roll amplitude are due to the interactions between the kite and the ship. With the monolithic approach, these drops are independent from initial conditions. The most important drops in roll amplitude occur when ${\omega}_{k1}={\omega}_{k1}^{{}^{\prime}}$ or ${\omega}_{k1}=2{\omega}_{k1}^{{}^{\prime}}$. The importance of the phenomenon occurring at ${\omega}_{k1}=n{\omega}_{k1}^{{}^{\prime}}$ decreases with the increasing value of the integer n. Moreover, the importance of the phenomenon decreases when the interaction with the wave concerns high harmonic orders.

#### 3.4.3. Kite Lock-in Phenomenon

^{−1}and ${L}_{t}=470$ m, because the interaction between the kite and the ship is win-win. Indeed, for this case the mean kite towing force predicted by the monolithic approach is 8.0% more important than the mean kite towing force predicted with the segregated approach. In addition, the ship roll amplitude is 1.4% weaker than without kite.

## 4. Discussion

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

dof | Degree(s) of freedom |

EFD | Experimental Fluid Dynamic |

Representations | |

$\mathbf{n}$ | Earth fixed reference frame (inertial) $\left({O}_{n},{\underline{x}}_{n},{\underline{y}}_{n},{\underline{z}}_{n}\right)$ |

$\mathbf{s}$ | Ship fixed frame $\left({O}_{s},{\underline{x}}_{s},{\underline{y}}_{s},{\underline{z}}_{s}\right)$ |

$\mathbf{h}$ | Hydrodynamic reference frame $\left({O}_{h},{\underline{x}}_{h},{\underline{y}}_{h},{\underline{z}}_{h}\right)$ |

$\mathbf{rw}$ | Relative wind frame $\left({\underline{x}}_{rw},{\underline{y}}_{rw},{\underline{z}}_{rw}\right)$ |

$\mathbf{k}$ | Kite reference frame $\left(K,{\underline{x}}_{k},{\underline{y}}_{k},{\underline{z}}_{k}\right)$ |

Representations | |

${l}_{z}$ | Vertical position of ${O}_{s}$ with respect to ship the baseline [m] |

${l}_{x}$ | Longitudinal position of ${O}_{s}$ with respect to the oft perpendicular of the ship [m] |

Variables | |

${\underline{V}}_{s}$ | Generalized velocity vector of the ship at ${O}_{s}$ with respect to the $\mathbf{n}$ frame expressed in the $\mathbf{s}$ frame [m.s^{−1}, rad.s^{−1}] |

$\underline{\xi}$ | Generalized position vector of the ship with respect to the $\mathbf{h}$ frame expressed in the $\mathbf{h}$ frame[m, rad] |

$\underline{S}$ | Generalized position vector of the ship with respect to the $\mathbf{n}$ frame expressed in the $\mathbf{n}$ frame[m, rad] |

${\underline{U}}_{h}$ | Mean ship forward speed [m.s^{−1}] |

Parameters | |

K | Kite position (${k}_{x},{k}_{y},{k}_{z}$) |

A | Tether attachment point |

${O}_{s}$ | Origin of the Ship reference frame |

${O}_{h}$ | Origin of the hydrodynamic reference |

${O}_{n}$ | Origin of the Earth reference frame |

${O}_{k}$ | Origin of the kite reference frame |

${\underline{U}}_{rw}$ | Relative wind velocity [m.s^{−1}] |

${\underline{U}}_{tw}$ | True wind velocity [m.s^{−1}] |

${\underline{U}}_{10}$ | Reference wind speed at 10m altitude [m.s^{−1}] |

${\underline{U}}_{A}$ | Kite attachment point velocity [m.s^{−1}] |

${\underline{U}}_{aw}$ | Apparent wind velocity to the kite [m.s^{−1}] |

${\underline{U}}_{k}$ | Kite velocity with respect to the $\mathbf{rw}$ frame [m.s^{−1}] |

Parameters | |

$\omega $ | Angular frequency of the motion [rad.s^{−1}] |

${\omega}_{w}$ | Angular frequency of the wave [rad.s^{−1}] |

${\omega}_{e}$ | Angular frequency of encounter [rad.s^{−1}] |

${\omega}_{e}$ | Angular frequency of encounter [rad.s^{−1}] |

${\omega}_{ki}$ | Angular frequency of the ith principal kite roll moment harmonic [rad.s ^{−1}] |

${\omega}_{ki}^{{}^{\prime}}$ | Angular frequency of the ith secondary kite roll moment harmonic [rad.s ^{−1}] |

k | Wave number [m^{−1}] |

${L}_{t}$ | Tether length [m] |

${\underline{T}}_{k}$ | Tether tension [N] |

${\u03f5}_{k}$ | Glide angle of the kite [rad] |

${C}_{lk}$ | Kite lift coefficient |

${A}_{k}$ | Kite surface area [m^{2}] |

${z}_{ref}$ | Measurement altitude of the wind [m] |

SODAR | Sonic Detection And Ranging |

FFT | Fast Fourier Transform |

g | Gravitational constant (9.81) [m.s^{−2}] |

${S}_{w}$ | Wave spectrum |

${\psi}_{w}$ | Wave direction with respect to the $\mathbf{c}$ frame [rad] |

Parameters | |

$\underset{=}{\mathcal{K}}$ | Impulse response function of the retardation matrix |

$\underset{=}{K}$ | Laplace transform of the retardation matrix |

$\mathcal{C}$ | Kite trajectory |

${\underset{=}{R}}_{\mathbf{s}}^{\mathbf{n}}$ | Transformation matrix of the time derivatives of the Euler’s angle $\mathbf{s}$ with respect to $\mathbf{n}$ to the turning rates in $\mathbf{s}$ |

$\underset{=}{A}$ | Generalized added mass matrix with respect to $\mathbf{s}$ for a given frequency of motion and $\underset{=}{\tilde{A}}$ at infinite frequency; ${\underset{=}{A}}^{*}$ with respect to $\mathbf{h}$ |

$\underset{=}{B}$ | Generalized damping matrix with respect to $\mathbf{s}$ for a given frequency of motion and $\underset{=}{\tilde{B}}$ at infinite frequency; ${\underset{=}{B}}^{*}$ with respect to $\mathbf{h}$ |

$\underset{=}{C}$ | Generalized restoring matrix with respect to $\mathbf{s}$; ${\underset{=}{C}}^{*}$ with respect to $\mathbf{h}$ |

${\varphi}_{8}$ | Azimuth of the centre of the kite trajectory [rad] |

${\theta}_{8}$ | Elevation of the centre of the trajectory [rad] |

${\chi}_{8}$ | Rotation angle of the trajectory around the axis defined by its centre and the tether attachment point A |

${\varphi}_{s}$ | Heeling angle of the ship [rad] |

${\varphi}_{w}$ | Phase angle of the Froude-Krilov force with respect to the free surface elevation [rad] |

${\underline{U}}_{ref}$ | True wind velocity at altitude of measurement [m.s^{−1}] |

${R}_{\mathcal{C}}$ | Trajectory radius of curvature [m] |

${\Delta}_{{K}_{k}}$ | Amplitude of the kite roll moment [N.m] |

${\Delta}_{{\varphi}_{s}}$ | Amplitude of the ship heeling angle [rad] |

$\mathcal{A}$ | Amplitude of the Fourier transform |

$\Psi $ | Phase of the Fourier transform |

## Appendix A. Ship Modeling

#### Appendix A.1. Time Domain Equations of Motion

#### Appendix A.2. Hydrodynamic Data and Linear Convolution Term

^{®}assuming an infinite depth. The frequency range of the data depends on the ship size, but for a commercial ship, the low frequency limit is generally about 0.1 rad.s

^{−1}and the high frequency limit does not generally exceed 3 rad.s

^{−1}. To improve the quality of the identification method an extrapolation of the hydrodynamic data toward the asymptotic value is necessary. As shown by Newman [5], assuming a potential flow, at zero and infinite frequency, the sectional damping is zero. Hence, each sectional damping are then extrapolated at high frequency with the function in Equation (A17), proposed by Greenhow [50]:

^{®}function “imp2ss” to control the order. This method is efficient but the identified transfer function has the following form:

^{®}system identification toolbox. This function use local optimization under constraints algorithm based on gradient methods. The structure of the transfer function, as proposed in Equation (A14), can be imposed to the frequency domain optimization algorithm. These two steps are repeated for several transfer function orders, for instance from 2 to 10. Then, the best transfer function order is selected according to a criterion based on the normalized quadratic error ${e}_{tot}$ from:

#### Appendix A.3. Wave Forces

^{−1}and the angle of the waves with respect to the ship heading. ${\beta}_{w}$ is given by ${\beta}_{w}={\psi}_{s}-{\psi}_{w}$, where ${\psi}_{w}$ denotes the wave angle with respect to ${\underline{x}}_{n}$. With $i\in \u27e61;6\u27e7$, each component ${f}_{wi}$ of the Froude–Krilov and diffraction forces generated by a single unit wave can be expressed by the following expression:

## References

- Leloup, R.; Roncin, K.; Behrel, M.; Bles, G.; Leroux, J.B.; Jochum, C.; Parlier, Y. A continuous and analytical modeling for kites as auxiliary propulsion devoted to merchant ships, including fuel saving estimation. Renew. Energy
**2016**, 86, 483–496. [Google Scholar] [CrossRef] - Naaijen, P.; Koster, V.; Dallinga, R. On the power savings by an auxiliary kite propulsion system. Int. Shipbuild. Prog.
**2006**, 53, 255–279. [Google Scholar] - Ran, H.; Janson, C.E.; Allenström, B. Auxiliary kite propulsion contribution to ship thrust. In Proceedings of the 32nd International Conference on Ocean, Offshore and Arctic Engineering, Nantes, France, 9–14 June 2013; pp. 1–9. [Google Scholar]
- Bigi, N.; Behrel, M.; Roncin, K.; Leroux, J.B.; Nême, A.; Jochum, C.; Parlier, Y. Course Keeping of Ship Towed by Kite. In Proceedings of the 15th Journees de l’Hydrodynamique, Brest, France, 22–24 November 2016; pp. 1–14. [Google Scholar]
- Newman, J.N. Marine Hydrodynamics; MIT Press: Cambridge, MA, USA, 1977. [Google Scholar]
- Molin, B. Hydrodynamique des Structures Offshore; Editions Technip: Paris, France, 2002. [Google Scholar]
- Faltinsen, O.M. Hydrodynamics of High-Speed Marine Vehicles; Cambridge University Press: Cambridge, UK, 2005; pp. 1–476. Available online: http://xxx.lanl.gov/abs/arXiv:1011.1669v3 (accessed on 5 June 2020).
- Bertram, V. Practical Ship Hydrodynamics; Elsevier: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Salvesen, N.; Tuck, E.; Faltinsen, O. Ship motions and sea loads. Trans. SNAME
**1970**, 78, 250–287. [Google Scholar] - Lee, C.H. WAMIT Theory Manual; Massachusetts Institute of Technology, Department of Ocean Engineering: Cambridge, MA, USA, 1995. [Google Scholar]
- Skejic, R. Ships Maneuvering Simulations in a Seaway—How close are we to reality? In Proceedings of the International Workshop on Next Generation Nautical Traffic Models, Delft, The Netherlands, 6 November 2013; pp. 91–101. [Google Scholar]
- Böttcher, H. Simulation of Ship Motions in a Seaway; Technical Report; Institut für Schiffbau der Universität Hamburg: Hamburg, Germany, 1989. [Google Scholar]
- Sutulo, S.; Guedes Soares, C. An implementation of the Method of Auxiliary State Variables for solving seakeeping problems. Int. Shipbuild. Prog.
**2005**, 52, 357–384. [Google Scholar] - Sutulo, S.; Soares, C. A generalized strip theory for curvilinear motion in waves. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering—OMAE, Estoril, Portugal, 15–20 June 2008; Volume 6, pp. 359–368. [Google Scholar] [CrossRef]
- Cummins, W. The Impulse Response Function and Ship Motions; Technical Report; David Taylor Model Basin: Washington, DC, USA, 1962. [Google Scholar]
- Kristiansen, E.; Hjulstad, A.; Egeland, O. State-space representation of radiation forces in time-domain vessel models. Ocean Eng.
**2005**, 32, 2195–2216. [Google Scholar] [CrossRef] - Fossen, T.I.; Smogeli, Ø.N. Nonlinear time-domain strip theory formulation for low-speed manoeuvring and station-keeping. Model. Identif. Control
**2004**, 25, 201–221. [Google Scholar] [CrossRef] - Pérez, T.; Fossen, T.I. Time-vs. frequency-domain Identification of parametric radiation force models for marine structures at zero speed. Model. Identif. Control
**2008**, 29, 1–19. [Google Scholar] [CrossRef][Green Version] - Behrel, M.; Roncin, K.; Leroux, J.B.; Montel, F.; Hascoet, R.; Neme, A.; Jochum, C.; Parlier, Y. Application of phase averaging method for measuring kites performance: Onshore results. J. Sail. Technol.
**2018**. [Google Scholar] - Wellicome, J.F.J.F.; Wilkinson, S. Ship Propulsive Kites: An Initial Study; Technical Report; Department of Ship Science, University of Southampton: Southampton, UK, 1984. [Google Scholar]
- Dadd, G.M.; Hudson, D.A.; Shenoi, R.A. Determination of kite forces using three-dimensional flight trajectories for ship propulsion. Renew. Energy
**2011**, 36, 2667–2678. [Google Scholar] [CrossRef] - Fossen, T. A Nonlinear Unified State-Space Model for Ship Maneuvering and Control in a Seaway. Int. J. Bifurc. Chaos
**2005**, 15, 2717–2746. [Google Scholar] [CrossRef] - The University of Iowa. EFD Data, The University of Iowa, Iowa City, USA. 2013. Available online: https://www.iihr.uiowa.edu/shiphydro/efd-data/ (accessed on 30 June 2020).
- Irvine, M.; Longo, J.; Stern, F. Pitch and Heave Tests and Uncertainty Assessment for a Surface Combatant in Regular Head Waves. J. Ship Res.
**2008**, 52, 146–163. [Google Scholar] - Ikeda, Y.; Himeno, Y.; Tanaka, N. A Prediction Method for Ship Roll Damping; Technical Report; Department of Naval Architecture, University of Osaka Prefecture: Osaka, Japan, 1978. [Google Scholar]
- Loyd, M.L. Crosswind kite power (for large-scale wind power production). J. Energy
**1980**, 4, 106–111. [Google Scholar] [CrossRef] - Paulig, X.; Bungart, M.; Specht, B. Conceptual Design of Textile Kites Considering Overall System Performance. In Airborne Wind Energy; Springer: Berlin/Heidelberg, Germany, 2013; pp. 547–562. [Google Scholar]
- Wellicome, J. Some comments on the relative merits of various wind propulsion devices. J. Wind Eng. Ind. Aerodyn.
**1985**, 20, 111–142. [Google Scholar] [CrossRef] - Argatov, I.; Rautakorpi, P.; Silvennoinen, R. Estimation of the mechanical energy output of the kite wind generator. Renew. Energy
**2009**, 34, 1525–1532. [Google Scholar] [CrossRef] - Leloup, R.; Roncin, K.; Bles, G.; Leroux, J.B.; Jochum, C.; Parlier, Y. Kite and classical rig sailing performance comparison on a one design keel boat. Ocean Eng.
**2014**, 90, 39–48. [Google Scholar] [CrossRef] - Group, I.Q.S. ITTC Symbols and Terminology List Version 2014. Technical Report. In Proceedings of the International Towing Tank Conference, Copenhagen, Denmark, 31 August–5 September 2014. [Google Scholar]
- Leloup, R. Modelling Approach and Numerical Tool Developments for Kite Performance Assessment and Mechanical Design; Application to Vessels Auxiliary Propulsion. Ph.D. Thesis, Université de Bretagne Occidentale, Brest, France, 2014. [Google Scholar]
- Behrel, M.; Roncin, K.; Leroux, J.B.; Neme, A.; Jochum, C.; Parlier, Y. Experimental set up for measuring onshore and onboard performances of leading edge inflatable kites—Presentation of onshore results. In Proceedings of the 4th Innov’Sail Conference, Lorient, France, 28–30 June 2017. [Google Scholar]
- Fagiano, L.; Zgraggen, A.U.; Khammash, M.; Morari, M. Automatic control of tethered wings for airborne wind energy: Design and experimental results. In Proceedings of the European Control Conference (ECC), Zurich, Switzerland, 17–19 July 2013; IEEE: Piscataway, NJ, USA, 2013; pp. 992–997. [Google Scholar]
- Fagiano, L.; Zgraggen, A.U.; Morari, M. On Modeling, Filtering and Automatic Control of Flexible Tethered Wings for Airborne Wind Energy. In Green Energy and Technology; Springer: Berlin/Heidelberg, Germany, 2013; pp. 167–180. [Google Scholar]
- Leloup, R.; Roncin, K.; Blès, G.; Leroux, J.B.; Jochum, C.; Parlier, Y. Estimation of the effect of rotation on the drag angle by using the lifting line method. In Proceedings of the 13th Journees de L’Hydrodynamique, Chatou, France, 21–23 November 2012. [Google Scholar]
- Duport, C.; Leroux, J.B.; Roncin, K.; Jochum, C.; Parlier, Y. Comparison of 3D non-linear lifting line method calculations with 3D RANSE simulations and application to the prediction of the global loading on a cornering kite. In Proceedings of the 15th Journees de L’Hydrodynamique, Brest, France, 22–24 November 2015. [Google Scholar]
- Fagiano, L. Control of Tethered Airfoils for High–Altitude Wind Energy Generation. Ph.D. Thesis, Politecnico di Torino, Torino, Italy, 2009. [Google Scholar]
- Fagiano, L.; Milanese, M.; Piga, D. Optimization of airborne wind energy generators. Int. J. Robust Nonlinear Control
**2012**, 22, 2055–2083. [Google Scholar] [CrossRef] - Longo, J.; Stern, F. Resistance, sinkage and trim, wave profile, and nominal wake tests and uncertainty assessment for DTMB model 5512. In Proceedings of the 25th American Towing Tank Conference, Iowa City, IA, USA, 1998; Available online: https://pdfs.semanticscholar.org/3799/7310985bd531109d91e38c08e6508353f4c9.pdf (accessed on 30 June 2020).
- Bearman, P.W. Vortex shedding from oscillating bluff bodies. Annu. Rev. Fluid Mech.
**1984**, 16, 195–222. [Google Scholar] [CrossRef] - Sarpkaya, T. A critical review of the intrinsic nature of vortex-induced vibrations. J. Fluids Struct.
**2004**, 19, 389–447. [Google Scholar] [CrossRef] - Behrel, M.; Roncin, K.; Iachkine, P.; Hascoet, R.; Leroux, J.B.; Montel, F.; Parlier, Y. Boat towed by kite: Methodolgy for sea trials. In Proceedings of the 16th Journées de l’Hydrodynamique, Marseille, France, 27–29 November 2018. [Google Scholar]
- Martin, H.R.; Kimball, R.W.; Viselli, A.M.; Goupee, A.J. Methodology for Wind/Wave Basin Testing of Floating Offshore Wind Turbines. J. Offshore Mech. Arct. Eng.
**2014**, 136, 021902. [Google Scholar] [CrossRef] - Giberti, H.; Ferrari, D. A novel hardware-in-the-loop device for floating offshore wind turbines and sailing boats. Mech. Mach. Theory
**2015**, 85, 82–105. [Google Scholar] [CrossRef] - Ogilvie, F.T. Recent Progress Towards the Understanding and Prediction of Ship. In Proceedings of the 6th Symposium on Naval Hydrodynamics, Bergen, Norway, 10–12 September 1964; pp. 3–79. [Google Scholar]
- Riemann, B. Ueber die Darstellbarkeit Einer Function Durch eine Trigonometrische Reihe; Dieterichschen Buchhandlung: Göttingen, Germany, 1867. [Google Scholar]
- Routh, E.J. A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion; Macmillan and Company: London, UK, 1877. [Google Scholar]
- Hurwitz, A. Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann.
**1895**, 46, 273–284. [Google Scholar] [CrossRef] - Greenhow, M. High- and low-frequency asymptotic consequences of the Kramers-Kronig relations. J. Eng. Math.
**1986**, 20, 293–306. [Google Scholar] [CrossRef] - Kung, S.Y. A New Identification and Model Reduction Algorithm via Singular Value Decompositions. In Proceedings of the 12th Asilomar Conference on Circuits, Systems and Computers, Pacific Grove, CA, USA, 6–8 November 1978; pp. 705–714. [Google Scholar]

**Figure 4.**Heave and pitch transfer function at function of the frequency of encounter ${\omega}_{e}$. Amplitudes are plotted for heave (

**a**) and pitch (

**c**). Phases are plotted for heave (

**b**) and pitch (

**d**). The results are obtained with the frequency domain and time domain approaches, experimental data for different wave steepness ${s}_{w}$ and with the STF strip theory.

**Figure 5.**Heave and pitch transfer function at $U=1.53$ m.s

^{−1}according to the frequency of encounter ${\omega}_{e}$. Amplitudes are plotted for heave (

**a**) and pitch (

**c**). Phases are plotted for heave (

**b**) and pitch (

**d**). The results are obtained with the frequency domain and time domain approaches, experimental data for different wave steepness ${s}_{w}$ and with the STF strip theory.

**Figure 6.**Roll response amplitude operator and phase at $U=7.716$ m.s

^{−1}as function of the frequency of encounter ${w}_{e}$ with DTMB 5512 at full scale. The results are obtained with the present model and the STF strip theory with the roll damping modeled with the method proposed by Ikeda et al. [25].

**Figure 7.**Schema describing the target point to control the kite velocity direction ${\underline{x}}_{vk}$.

**Figure 9.**Evolution of the kite velocity (

**a**) and of the tether tension at A, along the average trajectory (

**b**); comparison between the zero mass model with constant aerodynamics and the model with a linear modification of the kite aerodynamic characteristics using phase averaging of experimental data from Behrel et al. [33].

**Figure 10.**Kite flight trajectory parameter versus tether length. (

**a**) Trajectory angle ${\chi}_{8}$; (

**b**) Azimuth of the center of the trajectory; (

**c**) elevation of the center of the trajectory.

**Figure 11.**With a tether length ${L}_{t}=500$ m; (

**a**) spectrum of the kite excitation moment around the longitudinal ship axis ${\underline{x}}_{s}$; (

**b**) time history of the kite excitation moment around the longitudinal ship axis ${x}_{s}$ over the last loop.

**Figure 12.**(

**a**) Amplitude of the ship roll motion, (

**b**) first kite harmonic frequency and (

**c**) amplitude of the kite moment of excitation for different tether lengths from 360 m to 990 m by step length of 10 m in calm water.

**Figure 13.**Kite and ship path with respect to $\mathbf{n}$ for ${L}_{t}=390$ m with a wave of 2.5 m high at the frequency ${\omega}_{w}=0.8$ rad.s

^{−1}.

**Figure 14.**(

**a**) Amplitude of the ship roll motion, (

**b**) first kite harmonic frequency and (

**c**) amplitude of the kite moment of excitation for different tether lengths from 360 m to 990 m by step length of 10 m with a beam regular wave of 2.5 m high at a frequency of ${\omega}_{w}=0.4$ rad.s

^{−1}.

**Figure 15.**(

**a**) Amplitude of the ship roll motion, (

**b**) first kite harmonic frequency and (

**c**) amplitude of the kite moment of excitation for different tether lengths from 360 m to 990 m by step length of 10 m with a beam regular wave of 2.5 m high at a frequency of ${\omega}_{w}=0.56$ rad.s

^{−1}.

**Figure 16.**(

**a**) Amplitude of the ship roll motion, (

**b**) first kite harmonic frequency and (

**c**) amplitude of the kite moment of excitation for different tether lengths from 360 m to 990 m by step length of 10 m with a beam regular wave of 2.5 m high at a frequency of ${\omega}_{w}=0.8$ rad.s

^{−1}.

**Figure 17.**With a tether length ${L}_{t}=840$ m at the wave frequency ${\omega}_{w}=0.4$ rad.s

^{−1}; (

**a**) Spectrum of the roll motion of the ship; (

**b**) Spectrum of the kite excitation moment around the longitudinal ship axis.

**Figure 18.**With a tether length ${L}_{t}=470$ m at the wave frequency ${\omega}_{w}=0.56$ rad.s

^{−1}; (

**a**) Spectrum of the roll motion of the ship; (

**b**) Spectrum of the kite excitation moment around the longitudinal ship axis ${\underline{x}}_{s}$.

Parameter | Units | 5512 | Full Scale |
---|---|---|---|

Scale ratio | - | 46.6 | 1 |

Length, ${L}_{pp}$ | m | 3.048 | 142.04 |

Beam, B | m | 0.405 | 18.87 |

Draft, T | m | 0.132 | 6.15 |

Weight | Kg - t | 86.6 | 8763.5 |

LCG | m | 1.536 | 71.58 |

VCG | m | 0.162 | 7.55 |

Pitch radius of gyration, ${k}_{5}$ | m | 0.764 | 35.6 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bigi, N.; Roncin, K.; Leroux, J.-B.; Parlier, Y. Ship Towed by Kite: Investigation of the Dynamic Coupling. *J. Mar. Sci. Eng.* **2020**, *8*, 486.
https://doi.org/10.3390/jmse8070486

**AMA Style**

Bigi N, Roncin K, Leroux J-B, Parlier Y. Ship Towed by Kite: Investigation of the Dynamic Coupling. *Journal of Marine Science and Engineering*. 2020; 8(7):486.
https://doi.org/10.3390/jmse8070486

**Chicago/Turabian Style**

Bigi, Nedeleg, Kostia Roncin, Jean-Baptiste Leroux, and Yves Parlier. 2020. "Ship Towed by Kite: Investigation of the Dynamic Coupling" *Journal of Marine Science and Engineering* 8, no. 7: 486.
https://doi.org/10.3390/jmse8070486