Abstract
In this article we investigate the problem of generating electricity through an underwater kite power system (UKPS). For this problem, we develop the dynamical model for the UKPS and we formulate an optimal control problem to devise the trajectories and controls of the kite that maximize the total energy produced in a given time interval. This is a highly nonlinear problem for which the optimization is challenging. We also develop a numerical solution scheme for the optimal control problem based on direct methods and on adaptive time-mesh refinement. We report results that show that the problem can be quickly solved with a high level of accuracy when using our adaptive mesh refinement strategy. The results provide a set of output power values for different design choices and confirm that electrical energy that can be produced with such device.
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Abbreviations
- A :
-
Wing reference area of kite \(\left( \mathrm{m}^2\right) \)
- :
-
Wing aspect ratio
- \(a_t\) :
-
Tether reel-out acceleration \(\left( {\mathrm{m\,s^{-2}}}\right) \)
- \(c_{\mathrm{D}}\) :
-
Aerodynamic drag coefficient
- \(c_{\mathrm{L}}\) :
-
Aerodynamic lift coefficient
- E :
-
Energy produced \(\left( {\mathrm{Ws}}\right) \)
- \({\mathbf {F}^\mathrm{aer}}\) :
-
Aerodynamic forces \(\left( \mathrm{N}\right) \)
- \({\mathbf {F}^\mathrm{drag}}\) :
-
Aerodynamic drag force \(\left( \mathrm{N}\right) \)
- \({\mathbf {F}^\mathrm{cent}}\) :
-
Centrifugal force \(\left( \mathrm{N}\right) \)
- \({\mathbf {F}^\mathrm{cor}}\) :
-
Coriolis force \(\left( \mathrm{N}\right) \)
- \({\mathbf {F}^\mathrm{lift}}\) :
-
Aerodynamic lift force \(\left( \mathrm{N}\right) \)
- \({\mathbf {F}^\mathrm{inert}}\) :
-
Inertial forces \(\left( \mathrm{N}\right) \)
- \({\mathbf {F}^\mathrm{th}}\) :
-
Tether force \(\left( \mathrm{N}\right) \)
- m :
-
Mass \(\left( {\mathrm{kg}}\right) \)
- P :
-
Power produced \(\left( \mathrm{W}\right) \)
- \(\mathrm{R}_{\mathrm{G}\mathrm{L}}\) :
-
Rotation matrix from G to L
- r :
-
Tether length \(\left( \mathrm{m}\right) \)
- \(\rho \) :
-
Fluid density \(\left( {\mathrm{kg\,m}}^{-3}\right) \)
- s :
-
Wing span \(\left( \mathrm{m}\right) \)
- T :
-
Tether tension \(\left( \mathrm{N}\right) \)
- \(\mathbf {v}_a\) :
-
Apparent fluid velocity \(\left( {\mathrm{m\,s}}^{-1}\right) \)
- \(\mathbf {v}_w\) :
-
Fluid velocity \(\left( {\mathrm{m\,s}}^{-1}\right) \)
- \(v_t\) :
-
Tether reel-out velocity \(\left( {\mathrm{m\,s}}^{-1}\right) \)
- \(\alpha \) :
-
Angle of attack \(\left( \mathrm{rad}\right) \)
- \(\phi \) :
-
Azimuthal angle \(\left( \mathrm{rad}\right) \)
- \(\phi _{{\mathrm{ref}}}\) :
-
Reference angle \(\left( \mathrm{rad}\right) \)
- \(\omega \) :
-
Weight coefficient
- \(\mathbf {u}\) :
-
Control vector
- \(\mathbf {x}\) :
-
State vector
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Acknowledgements
We acknowledge the support of FEDER/COMPETE2020/NORTE2020/POCI/PIDDAC/MCTES/FCT funds through grants SFRH/BPD/126683/2016, UID/EEA/00147/2013\(\vert \)UID/IEEA/00147/006933–SYSTEC, PTDC-EEI-AUT-2933-2014\(\vert \)16858–TOCCATA, and 02/SAICT/2017-31447\(\vert \)POCI-01-0145-FEDER-031447\(\vert \)FCT–UPWIND.
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Paiva, L.T., Fontes, F.A.C.C. Optimal electric power generation with underwater kite systems. Computing 100, 1137–1153 (2018). https://doi.org/10.1007/s00607-018-0643-4
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DOI: https://doi.org/10.1007/s00607-018-0643-4
Keywords
- Nonlinear systems
- Optimal control
- Real-time optimization
- Continuous-time systems
- Adaptive algorithms
- Time-mesh refinement
- Kite power systems
- Underwater kites
- Airborne wind energy