# Optimal Control Algorithms with Adaptive Time-Mesh Refinement for Kite Power Systems

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Kite Power System Model

#### 2.1. 2D Kite Model

**Global G:**- A Cartesian coordinate system $(x,z)$ where x is aligned according to the wind direction ${\mathbf{v}}_{w}=({v}_{w},0)$− basis $({\overrightarrow{\mathbf{e}}}_{x},{\overrightarrow{\mathbf{e}}}_{z})$,
**Local L:**- A polar coordinate system $(r,\beta )$ centred at the kite position—basis $({\overrightarrow{\mathbf{e}}}_{r},{\overrightarrow{\mathbf{e}}}_{\beta})$.

#### 2.2. Acting Forces

#### 2.3. 3D Model

**Global G:**- An inertial Cartesian coordinate system $(x,y,z)$ where the origin is on the ground at the point of attachment of the tether and x is aligned according to the wind direction ${\mathbf{v}}_{w}=({v}_{w},0,0)$, on the basis of $({\overrightarrow{\mathbf{e}}}_{x},{\overrightarrow{\mathbf{e}}}_{y},{\overrightarrow{\mathbf{e}}}_{z})$.We consider that the kite is positioned in a point $\mathbf{p}$ with coordinates $(x,y,z)$.
**Local L:**- A non-inertial spherical coordinate system $(r,\varphi ,\beta )$ on the basis of $({\overrightarrow{\mathbf{e}}}_{r},{\overrightarrow{\mathbf{e}}}_{\varphi},{\overrightarrow{\mathbf{e}}}_{\beta})$ (Figure 3a).
**Body B:**- A non-inertial Cartesian coordinate system attached to the kite body on the basis of $({\overrightarrow{\mathbf{e}}}_{1},{\overrightarrow{\mathbf{e}}}_{2},{\overrightarrow{\mathbf{e}}}_{3})$. ${\overrightarrow{\mathbf{e}}}_{1}$ coincides with the kite’s longitudinal axis pointing forward, ${\overrightarrow{\mathbf{e}}}_{2}$ in the kite transversal axis points to the left wing tip, and ${\overrightarrow{\mathbf{e}}}_{3}$ in the kite vertical axis points upwards (Figure 3b).

## 3. Optimal Control Problem

- (${P}_{1}$):
- The production cycle, when the terminal state is free;
- (${P}_{2}$):
- The reel-out and reel-in cycle, by imposing that the terminal state should be near the initial one: $\left|\right|\mathbf{x}\left({t}_{f}\right)-\mathbf{x}\left({t}_{0}\right)\left|\right|<\epsilon $.

## 4. A Multi-Level Adaptive Mesh Refinement Algorithm

## 5. Numerical Results

#### 5.1. 2D Problem Results

- ${\pi}_{\mathrm{ML}}$:
- The mesh generated by the adaptive refinement strategy
- ${\pi}_{\mathrm{F}}$:
- The equidistant spacing mesh considering the lowest $\Delta t$ of ${\pi}_{\mathrm{ML}}$

#### 5.2. 3D Problem Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | wing reference area of kite [m^{2}] |

$\mathit{A}\phantom{\rule{-3.33328pt}{0ex}}\mathrm{R}$ | wing aspect ratio |

${a}_{t}$ | tether reel–out acceleration [m s^{−2}] |

${c}_{\mathrm{D}}$ | aerodynamic drag coefficient |

${c}_{\mathrm{L}}$ | aerodynamic lift coefficient |

E | energy produced [J] |

${\overrightarrow{\mathrm{F}}}^{\mathrm{aer}}$ | aerodynamic force [N] |

${\overrightarrow{\mathrm{F}}}^{\mathrm{drag}}$ | drag force [N] |

${\overrightarrow{\mathrm{F}}}^{\mathrm{cent}}$ | centrifugal force [N] |

${\overrightarrow{\mathrm{F}}}^{\mathrm{cor}}$ | Coriolis force [N] |

${\overrightarrow{\mathrm{F}}}^{\mathrm{lift}}$ | aerodynamic lift force [N] |

${\overrightarrow{\mathrm{F}}}^{\mathrm{inert}}$ | inertial forces [N] |

${\overrightarrow{\mathrm{F}}}^{\mathrm{th}}$ | tether force [N] |

g | gravitational acceleration [m s^{−2}] |

m | mass [kg] |

P | power produced [W] |

R_{90} | 90° anticlockwise rotation matrix |

R_{GL} | rotation matrix from G to L |

R_{LG} | rotation matrix from L to G |

r | tether length [m] |

$\rho $ | air density [kg m^{−3}] |

s | wing span [m] |

T | tether tension [N] |

t | time [s] |

${\mathbf{v}}_{a}$ | apparent wind velocity [m s^{−1}] |

${\mathbf{v}}_{w}$ | wind velocity [m s^{−1}] |

${v}_{t}$ | tether reel–out velocity [m s^{−1}] |

$\mathbf{u}$ | control vector |

$\mathbf{x}$ | state vector |

$\alpha $ | angle of attack [rad] |

$\varphi $ | azimuthal angle [rad] |

$\beta $ | elevation angle [rad] |

$\theta $ | polar angle [rad] |

$\psi $ | roll angle [rad] |

## References

- Global Wind Report 2016 | GWEC. Available online: http://gwec.net/publications/global-wind-report-2/global-wind-report-2016/ (accessed on 1 October 2017).
- Ampyx Power: Airborne Wind Energy. Available online: http://www.ampyxpower.com (accessed on 1 October 2017).
- KiteGen. Available online: http://kitegen.com (accessed on 1 October 2017).
- Lind, D.V.M. Developing a 600 kW Airborne Wind Turbine. In Proceedings of the 2015 Airborne Wind Energy Conference, Delft, The Netherlands, 15–16 June 2015. [Google Scholar]
- Penedo, R.J.M.; Pardal, T.C.D.; Silva, P.M.M.S.; Fernandes, N.M.; Fernandes, T.R.C. High Altitude Wind Energy from a Hybrid Lighter-than-Air Platform Using the Magnus Effect. In Airborne Wind Energy; Ahrens, U., Diehl, M., Schmehl, R., Eds.; Green Energy and Technology; Springer: Berlin/Heidelberg, Germany, 2013; pp. 491–500. [Google Scholar]
- Ahrens, U.; Diehl, M.; Schmehl, R. (Eds.) Airborne Wind Energy; Green Energy and Technology; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Fagiano, L.; Milanese, M. Airborne Wind Energy: An overview. In Proceedings of the 2012 American Control Conference (ACC), Montreal, QC, Canada, 27–29 June 2012; pp. 3132–3143. [Google Scholar]
- Cherubini, A.; Papini, A.; Vertechy, R.; Fontana, M. Airborne Wind Energy Systems: A review of the technologies. Renew. Sustain. Energy Rev.
**2015**, 51, 1461–1476. [Google Scholar] [CrossRef] [Green Version] - Loyd, M.L. Crosswind kite power. J. Energy
**1980**, 4, 106–111. [Google Scholar] [CrossRef] - Paiva, L.T.; Fontes, F.A.C.C. Mesh-Refinement Strategies for Fast Optimal Control and Model Predictive Control of Kite Power Systems. In Book of Abstracts of the 2015 International Airborne Wind Energy Conference; Delft University of Technology: Delft, The Netherlands, 2015; p. 101. [Google Scholar]
- Paiva, L.T.; Fontes, F.A.C.C. Optimal control of kite power systems: Mesh–refinement strategies. Energy Procedia
**2017**, 136, 302–307. [Google Scholar] [CrossRef] - Findeisen, R.; Allgöwer, F. An Introduction to Nonlinear Model Predictive. In Proceedings of the 21st Benelux Meeting on Systems and Control, Veldhoven, The Netherlands, 19–21 March 2002; pp. 1–23. [Google Scholar]
- Diehl, M.; Bock, H.; Schlöder, J.P.; Findeisen, R.; Nagy, Z.; Allgöwer, F. Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Process Control
**2002**, 12, 577–585. [Google Scholar] [CrossRef] - Fontes, F.A.C.C. A general framework to design stabilizing nonlinear model predictive controllers. Syst. Control Lett.
**2001**, 42, 127–143. [Google Scholar] [CrossRef] - Fontes, F.A.C.C.; Magni, L.; Gyurkovics, É. Sampled-data model predictive control for nonlinear time-varying systems: Stability and robustness. In Assessment and Future Directions of Nonlinear Model Predictive Control; Springer: Berlin/Heidelberg, Germany, 2007; pp. 115–129. [Google Scholar]
- Paiva, L.T.; Fontes, F.A.C.C. Adaptive time-mesh refinement in optimal control problems with state constraints. Discret. Contin. Dyn. Syst.
**2015**, 35, 4553–4572. [Google Scholar] [CrossRef] - Paiva, L.T. Numerical Methods in Optimal Control and Model Predictive Control. Ph.D. Thesis, Universidade do Porto, Porto, Portugal, 2014. [Google Scholar]
- Canale, M.; Fagiano, L.; Milanese, M. High Altitude Wind Energy Generation Using Controlled Power Kites. IEEE Trans. Control Syst. Technol.
**2010**, 18, 279–293. [Google Scholar] [CrossRef] - Diehl, M. Real-Time Optimization for Large Scale Nonlinear Processes. Ph.D. Thesis, University Heidelberg, Heidelberg, Germany, 2001. [Google Scholar]
- Vinter, R.B. Optimal Control; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
- Betts, J.T. Practical Methods for Optimal Control Using Nonlinear Programming; SIAM: Philadelphia, PA, USA, 2001. [Google Scholar]
- Gerdts, M. Optimal Control of ODEs and DAEs; De Gruyter: Berlin, Germany; Boston, MA, USA, 2011. [Google Scholar]
- Gavriel, C.; Lopes, S.; Vinter, R. Regularity of minimizers for higher order variational problems in one independent variable. Ann. Rev. Control
**2011**, 35, 172–177. [Google Scholar] [CrossRef] - Paiva, L.T. Optimal Control in Constrained and Hybrid Nonlinear System: Solvers and Interfaces; Technical Report; Faculdade de Engenharia, Universidade do Porto: Porto, Portugal, 2013. [Google Scholar]
- Betts, J.T.; Huffman, W.P. Mesh refinement in direct transcription methods for optimal control. Optim. Control Appl. Methods
**1998**, 19, 1–21. [Google Scholar] [CrossRef] - Patterson, M.A.; Hager, W.W.; Rao, A.V. A ph mesh refinement method for optimal control. Optim. Control Appl. Methods
**2014**, 36, 398–421. [Google Scholar] [CrossRef] - Zhao, Y.; Tsiotras, P. Density Functions for Mesh Refinement in Numerical Optimal Control. J. Guid. Control Dyn.
**2011**, 34, 271–277. [Google Scholar] [CrossRef] - Fontes, F.A.C.C.; Frankowska, H. Normality and Nondegeneracy for Optimal Control Problems with State Constraints. J. Optim. Theory Appl.
**2015**, 166, 115–136. [Google Scholar] [CrossRef] - Falugi, P.; Kerrigan, E.; Van Wyk, E. Imperial College London Optimal Control Software. User Guide (ICLOCS); Department of Electrical Engineering, Imperial College London: London, UK, 2010. [Google Scholar]
- Fontes, F.A.C.C.; Paiva, L.T. Guaranteed Collision Avoidance in Multi–Kite Power Systems. In Book of Abstracts of the 2017 International Airborne Wind Energy Conference; Albert-Ludwigs University Freiburg: Freiburg, Germany, 2017; p. 103. [Google Scholar]

**Figure 1.**Kite power system components. ACU: airborne control unit; CKM: controlled kite module; GGM: ground generator module.

Parameter | Value |
---|---|

$\rho $ | 1.2 kg m^{−3} |

${v}_{w}$ | 10 m s^{−1} |

g | 9.8 m s^{−2} |

m | 1 kg |

A | 10 m^{2} |

${\mathit{\pi}}_{\mathit{j}}$ | ${\mathit{N}}_{\mathit{j}}$ | $\Delta {\mathit{t}}_{\mathit{j}}$ | ${\mathit{I}}_{\mathit{j}}$ | Objective | ${\left|\left|{\mathit{\epsilon}}_{\mathit{x}}^{\left(\mathit{j}\right)}\right|\right|}_{\infty}$ | CPU Time (s) | |
---|---|---|---|---|---|---|---|

Solve | ${\mathit{\epsilon}}_{\mathit{x}}$ | ||||||

${\pi}_{0}$ | 51 | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$50$}\right.$ | 253 | 53.216968 | 1.795 × 10^{−2} | 4.754 | 0.107 |

${\pi}_{1}$ | 474 | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$500$}\right.$ | 122 | 54.156063 | 2.328 × 10^{−3} | 4.722 | 0.856 |

${\pi}_{2}$ | 2877 | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5000$}\right.$ | 26 | 54.266839 | 3.168 × 10^{−4} | 6.107 | 3.383 |

${\pi}_{\mathrm{ML}}$ | 2877 | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5000$}\right.$ | 401 | 54.266839 | 3.168 × 10^{−4} | 15.583 | 4.346 |

${\pi}_{\mathrm{F}}$ | 5001 | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5000$}\right.$ | 1342 | 54.266873 | 2.201 × 10^{−4} | 431.234 | 10.913 |

© 2018 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**

Paiva, L.T.; Fontes, F.A.C.C.
Optimal Control Algorithms with Adaptive Time-Mesh Refinement for Kite Power Systems. *Energies* **2018**, *11*, 475.
https://doi.org/10.3390/en11030475

**AMA Style**

Paiva LT, Fontes FACC.
Optimal Control Algorithms with Adaptive Time-Mesh Refinement for Kite Power Systems. *Energies*. 2018; 11(3):475.
https://doi.org/10.3390/en11030475

**Chicago/Turabian Style**

Paiva, Luís Tiago, and Fernando A. C. C. Fontes.
2018. "Optimal Control Algorithms with Adaptive Time-Mesh Refinement for Kite Power Systems" *Energies* 11, no. 3: 475.
https://doi.org/10.3390/en11030475