Skip to main content

Advertisement

SpringerLink
Log in

Optimal electric power generation with underwater kite systems

  • Published:
Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

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

References

  1. Ahrens U, Diehl M, Schmehl R (eds) (2013) Airborne wind energy. Green energy and technology. Springer, Berlin

    Google Scholar 

  2. Ampyx Power (2017) Airborne wind energy. http://www.ampyxpower.com

  3. Betts JT (2001) Practical methods for optimal control using nonlinear programming. SIAM, University City

    MATH  Google Scholar 

  4. Betts JT, Huffman WP (1998) Mesh refinement in direct transcription methods for optimal control. Opt Control Appl Methods 19(1):1–21. https://doi.org/10.1002/(SICI)1099-1514(199801/02)19:1%3c1::AID-OCA616%3e3.0.CO;2-Q

    Article  MathSciNet  Google Scholar 

  5. Canale M, Fagiano L, Milanese M (2010) High altitude wind energy generation using controlled power kites. IEEE Trans Control Syst Technol 18(2):279–293

    Article  Google Scholar 

  6. Diehl M (2001) Real-time optimization for large scale nonlinear processes. Ph.D., Univ. Heidelberg

  7. Diehl M (2013) Airborne wind energy: Basic concepts and physical foundations. In: Airborne Wind Energy, pp 3–22. Springer. https://doi.org/10.1007/978-3-642-39965-7-1

  8. Douglas MR, Gomes MW (2014) Dynamics of a river kite power production system with a kinetic energy storage device. In: Proceedings of the international conference of control, dynamic systems, and robotics (Ottawa, Canada, 2014), pp 1–6. http://avestia.com/CDSR2014_Proceedings/papers/55.pdf

  9. Fontes FA, Frankowska H (2015) Normality and nondegeneracy for optimal control problems with state constraints. J Optim Theory Appl 166:115–136

    Article  MathSciNet  Google Scholar 

  10. Ghasemi A, Olinger DJ, Tryggvason G (2015) Computational simulation of the tethered undersea kites for power generation. In: ASME 2015 International mechanical engineering congress and exposition, pp V06BT07A043–V06BT07A043. American Society of Mechanical Engineers. http://proceedings.asmedigitalcollection.asme.org/proceeding.aspx?articleid=2500876

  11. Glenn Research Center N (2017) Aerodynamics of kites. https://www.grc.nasa.gov/WWW/K-12/airplane/kiteaero.html. Accessed 1 Feb 2017

  12. Grinsted TW, Watchorn MJ (2005) Extracting power from moving water. http://www.google.com/patents/US6849963. U.S. Classification 290/42, 290/43, 290/53, 290/54; International Classification F03D5/06, F03B17/06; Cooperative Classification F05B2210/16, F03B17/06, Y02E10/28; European Classification F03B17/06

  13. KiteGen (2017) http://kitegen.com

  14. Li H, Olinger DJ, Demetriou MA (2015) Control of a tethered undersea kite energy system using a six degree of freedom model. In: 2015 54th IEEE conference on decision and control (CDC), pp 688–693. https://doi.org/10.1109/CDC.2015.7402309

  15. Loyd ML (1980) Crosswind kite power. J Energy 4(3):106–111. https://doi.org/10.2514/3.48021

    Article  Google Scholar 

  16. OECD: OECD green growth studies: energy Tech rep, OECD (2011). https://www.oecd.org/greengrowth/greening-energy/49157219.pdf

  17. Paiva LT (2013) Optimal control in constrained and hybrid nonlinear system: solvers and interfaces. Tech rep, Faculdade de Engenharia, Universidade do Porto

  18. Paiva LT, Fontes FACC (2015) Adaptive time-mesh refinement in optimal control problems with state constraints. Discret Contin Dyn Syst 35(9):4553–4572. https://doi.org/10.3934/dcds.2015.35.4553. http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=10999

    Article  MathSciNet  Google Scholar 

  19. Paiva LT, Fontes FACC (2017) Optimal control of underwater kite power systems. In: 2017 International conference in energy and sustainability in small developing economies (ES2DE), pp 1–6. https://doi.org/10.1109/ES2DE.2017.8015353

  20. Patterson MA, Hager WW, Rao AV (2014) A ph mesh refinement method for optimal control. Opt Control Appl Methods. https://doi.org/10.1002/oca.2114

    Article  Google Scholar 

  21. Penedo RJM, Pardal TCD, Silva PMMS, Fernandes NM, Fernandes TRC (2013) High altitude wind energy from a hybrid lighter-than-air platform using the magnus effect. In: Ahrens U, Diehl M, Schmehl R (eds) Airborne wind energy, green energy and technology. Springer, Berlin, pp 491–500

    Chapter  Google Scholar 

  22. Power) DVLM (2015) Developing a 600 kw airborne wind turbine. In Airborne wind energy conference 2015

  23. UPWIND: UPWIND Project. http://www.upwind.pt

  24. Vinter RB (2000) Optimal control. Springer, Berlin

    MATH  Google Scholar 

  25. Zhao Y, Tsiotras P (2011) Density functions for mesh refinement in numerical optimal control. J Guid Control Dyn 34(1):271–277. https://doi.org/10.2514/1.45852

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. SYSTEC–ISR, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal

    Luís Tiago Paiva & Fernando A. C. C. Fontes

Authors
  1. Luís Tiago Paiva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando A. C. C. Fontes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luís Tiago Paiva.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-018-0643-4

Keywords

Mathematics Subject Classification

Search

Navigation