Nonlinear Dynamics

, Volume 92, Issue 2, pp 181–202 | Cite as

Control of an oscillating water column wave energy converter based on dielectric elastomer generator

  • Gastone Pietro Rosati Papini
  • Giacomo Moretti
  • Rocco Vertechy
  • Marco Fontana
Original Paper


This paper introduces a model-based control strategy for a wave energy converter (WEC) based on dielectric elastomer generators (DEGs), i.e. a device that can convert the energy of ocean waves into electricity by employing deformable elastomeric transducers with variable capacitance. The analysed system combines the concept of oscillating water column WEC with an inflated circular diaphragm DEG (ICD-DEG). The device features strongly nonlinear dynamics due to the ICD-DEG electro-hyperelastic response and the compressibility of the air volume comprised between the water column and the ICD-DEG, while the hydrodynamic loads can be approximated as linear. The optimal control solution that maximises the power extraction of the device is numerically investigated in the case of monochromatic waves over the typical frequency and amplitude ranges of sea waves. The more realistic case of panchromatic waves is also analysed through the implementation, in simulation environment, of a real-time controller. This regulator is based on a simple sub-optimal control logic that is deduced from the monochromatic case. The performance of the proposed control strategy is illustrated in comparison with unoptimised algorithms.


Smart materials Energy harvesting Wave energy Dielectric elastomers Hydrodynamics EPAM DEG 

List of symbols

Physical constants


Acceleration of gravity (\(\hbox {m}/\hbox {s}^{2}\))


Atmospheric pressure (Pa)

\(\gamma \)

Air’s heat capacity ratio

\(\rho _\mathrm{w}\)

Sea water density (\(\hbox {kg}/\hbox {m}^{3}\))

Hydrodynamic model (continuous time)

\(a,\ b,\ c,\ d\)

OWC collector dimensions (m)


jth amplitude of irregular waves harmonic components (m)

\(\varvec{A_r},\ \varvec{B_r},\ \varvec{C_r}\)

Radiation state-space model matrices

\(\varvec{A_c},\ \varvec{B_c},\ \varvec{C_c}\)

Hydrodynamic state-space model matrices

\(\mathcal {E}_\mathrm{a}\)

Energy absorbed by the PTO machinery (J)


PTO force (N)


Excitation force (N)

\(f_\mathrm {MAX}\), \(f_\mathrm {MIN}\)

Upper and lower envelopes of the PTO force profiles (N)


Regular wave height (m)


Significant irregular wave height (m)


Radiation force kernel (N/m)

\(m_\infty \)

Added mass at infinite frequency (kg)


Dimension of \(\varvec{x_c}\)


Air chamber relative pressure (Pa)


Pressure derivative threshold (Pa/s)


Dimension of \(\varvec{x_r}\)


Water column cross section (\(\hbox {m}^{2}\))

\(S_{\omega }(\omega )\)

Wave spectrum (\(\hbox {m}^2\hbox {s}\))


Regular wave period (s)


Irregular wave energy period (s)


State-space hydrodynamic model input (\(\hbox {m}/\hbox {s}^{2}\))


Hydrodynamic model state vector


Radiation state vector

\(\hat{\varGamma }(\omega )\)

Wave excitation coefficient (N/m)

\(\varDelta \omega \)

Step between consecutive values of \(\omega _{j}\) (rad/s)

\(\delta \)

Sign of the air pressure time derivative

\(\eta \)

Water column displacement (m)

\(\tau \)

Time (s)

\(\phi _j\)

jth phase of irregular waves harmonic components (rad)

\(\omega \)

Angular frequency (rad/s)

\(\omega _{j}\)

jth frequency of irregular waves harmonic components (rad/s)

Hydrodynamic model (discrete time)

\(\varvec{A}_{\varvec{d}},\ \varvec{B}_{\varvec{d}},\ \varvec{C}_{\varvec{d}}\)

Discretisation of \(\varvec{A_c},\ \varvec{B_c},\ \varvec{C_c}\)


Number of samples in a discretised horizon

\(\varvec{P},\ \varvec{V}\)

Matrices to extract velocity and water column displacement from \(\varvec{x_d}\)


Discretisation time step (s)


Discretisation of \(u_c\) (\(\hbox {m}/\hbox {s}^2\))

\(\varvec{X},\ \varvec{Z}\)

Sparse matrices defined in Eq. (36)


Discretisation of \(\varvec{x_c}\)

\(\varvec{\varPhi }\)

Prediction system matrix

\(\varvec{\varGamma }\)

Prediction input matrix

\(\varvec{\varUpsilon }\)

Matrix used to formulate the terminal state constraint

\(\varvec{\varOmega _\mathrm{P}},\ \varvec{\varOmega _\mathrm{V}}\)

Matrices that correlate the vectors of velocity and position to the input vector

ICD-DEG model


Capacitance (F)


Electric field at the tip element (V/m)


Breakdown electric field (V/m)

\(\mathcal {E}_\mathrm{e}\)

Electrical energy supplied by the DEG (J)


Flat DEG pre-stretched radius (m)


DEG unstretched radius (m)


Tip displacement (m)

\(h_\mathrm{MAX},\ h_\mathrm{MIN}\)

Maximum and minimum admissible tip displacements (m)


Current (A)

\(I_\mathrm {MAX}\)

Maximum admissible current (A)


Gent invariant parameter


Gent stiffness parameter (Pa)


Number of in-parallel layers


Charge (C)


ICD-DEG curvature (m)


Flat DEG pre-stretched thickness (m)


DEG unstretched thickness (m)


Voltage (V)

\(\epsilon \)

Dielectric constant (F/m)

\(\zeta \)

Ratio of generated energy (per cycle) to maximum cyclic convertible energy

\(\lambda \)

Stretch at the tip element

\(\lambda _1,\ \lambda _2\)

Meridian and circumferential stretches

\(\lambda _\mathrm{p}\)


\(\lambda _u\)

Rupture stretch

\(\sigma \)

Stress at the tip element (Pa)

\(\varOmega \)

Current air chamber volume (\(\hbox {m}^3\))

\(\varOmega _{d}\)

Volume subtended by the ICD-DEG (\(\hbox {m}^3\))

Operators and notation

\(\varvec{0}_{\varvec{a}\times \varvec{b}}\)

Matrix of zeros \(\in \mathbb {R}^{a\times b}\)

\(\varvec{1}_{\varvec{a}\times \varvec{b}}\)

Matrix of ones \(\in \mathbb {R}^{a\times b}\)


Identity matrix of size a

\(\dot{\xi }\)

Time derivative of the generic variable \(\xi \).

\(\varvec{\xi }^\mathrm{T}\)

The transpose of generic vector \(\varvec{\xi }\)

\(\xi _{200}\)

Value/profile of generic variable \(\xi \) in the presence of maximum current constraint \(I_\mathrm {MAX}=200\) A

\(\underset{i \rightarrow j}{\varvec{\xi }}\)

Short form for \(\begin{pmatrix}\varvec{\xi }^\mathrm{T}[i], \varvec{\xi }^\mathrm{T}[i+1], \ldots , \varvec{\xi }^\mathrm{T}[j] \end{pmatrix}^\mathrm{T}\)

\(\underline{\varvec{\xi }}\)

Short form for \(\underset{0 \rightarrow N-1}{\varvec{\xi }}\)



The research leading to these results has received funding from the European Union Seventh Framework Programme [FP7/2007-2013], project PolyWEC, under Grant Agreement No. 309139, and from the European Union Horizon 2020 Program, Project WETFEET, under Grant No. 646436.


  1. 1.
    Babarit, A., Hals, J., Muliawan, M., Kurniawan, A., Moan, T., Krokstad, J.: Numerical benchmarking study of a selection of wave energy converters. Renew. Energy 41, 44–63 (2012)CrossRefGoogle Scholar
  2. 2.
    Camacho, E.F., Alba, C.B.: Model Predictive Control. Springer, Berlin (2013)Google Scholar
  3. 3.
    Carter, D.: Estimation of wave spectra from wave height and period (1982)Google Scholar
  4. 4.
    Chiba, S., Waki, M., Wada, T., Hirakawa, Y., Masuda, K., Ikoma, T.: Consistent ocean wave energy harvesting using electroactive polymer (dielectric elastomer) artificial muscle generators. Appl. Energy 104, 497–502 (2013)CrossRefGoogle Scholar
  5. 5.
    Cretel, J.A.M., Lightbody, G., Thomas, G.P., Lewis, A.W.: Maximisation of energy capture by a wave-energy point absorber using model predictive control. IFAC Proc. Vol. 44(1), 3714–3721 (2011)Google Scholar
  6. 6.
    Cruz, J.: Ocean Wave Energy: Current Status and Future Prespectives. Springer, Berlin (2007)Google Scholar
  7. 7.
    Evans, D.: Maximum wave-power absorption under motion constraints. Appl. Ocean Res. 3(4), 200–203 (1981)CrossRefGoogle Scholar
  8. 8.
    Falcão, A.: Wave energy utilization: a review of the technologies. Renew. Sustain. Energy Rev. 14(3), 899–918 (2010)CrossRefGoogle Scholar
  9. 9.
    Falcão, A.d.O., De, O.: The shoreline OWC wave power plant at the Azores. In: Fourth European Wave Energy Conference, Aalborg, Denmark, pp. 4–6 (2000)Google Scholar
  10. 10.
    Falnes, J., et al.: Optimum control of oscillation of wave-energy converters. Int. J. Offshore Polar Eng. 12(02) (2002)Google Scholar
  11. 11.
    Fletcher, R., Leyffer, S., Toint, P.L.: On the global convergence of a filter-sqp algorithm. SIAM J. Optim. 13(1), 44–59 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hals, J., Falnes, J., Moan, T.: Constrained optimal control of a heaving buoy wave-energy converter. J. Offshore Mech. Arct. Eng. 133(1), 011401 (2011)CrossRefGoogle Scholar
  13. 13.
    Halvorsen, E., Le, C., Mitcheson, P., Yeatman, E.: Architecture-independent power bound for vibration energy harvesters. In: Journal of Physics: Conference Series, vol. 476, pp. 012026. IOP Publishing (2013)Google Scholar
  14. 14.
    Hamdi, A., Abdelaziz, M.N., Hocine, N.A., Heuillet, P., Benseddiq, N.: A fracture criterion of rubber-like materials under plane stress conditions. Polym. Test. 25(8), 994–1005 (2006)CrossRefGoogle Scholar
  15. 15.
    Henderson, R., Hunter, R.: Control requirements for wave energy converters. Landscaping study. Tech. Rep., Wave Energy Scotland (2016)Google Scholar
  16. 16.
    Henriques, J., Falcao, A., Gomes, R., Gato, L.: Latching control of an oscillating water column spar-buoy wave energy converter in regular waves. J. Offshore Mech. Arct. Eng. 135(2), 021902 (2013)CrossRefGoogle Scholar
  17. 17.
    Horn, R.A.: The Hadamard product. In: Proceedings of Symposia in Applied Mathematics, vol. 40, pp. 87–169 (1990)Google Scholar
  18. 18.
    Hosseinloo, A.H., Turitsyn, K.: Fundamental limits to nonlinear energy harvesting. Phys. Rev. Appl. 4(6), 064009 (2015)CrossRefGoogle Scholar
  19. 19.
    Hudson, J., Phillips, D., Wilkins, N.: Materials aspects of wave energy converters. J. Mater. Sci. 15(6), 1337–1363 (1980)CrossRefGoogle Scholar
  20. 20.
    IPOPT.: (2016). Accessed 04 Oct 2016
  21. 21.
    Jean, P., Wattez, A., Ardoise, G., Melis, C., Van Kessel, R., Fourmon, A., Barrabino, E., Heemskerk, J., Queau, J.: Standing wave tube electro active polymer wave energy converter. In: SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring. International Society for Optics and Photonics (2012)Google Scholar
  22. 22.
    Kaltseis, R., Keplinger, C., Baumgartner, R., Kaltenbrunner, M., Li, T., Mächler, P., Schwödiauer, R., Suo, Z., Bauer, S.: Method for measuring energy generation and efficiency of dielectric elastomer generators. Appl. Phys. Lett. 99(16), 162904 (2011)CrossRefGoogle Scholar
  23. 23.
    Kaltseis, R., Keplinger, C., Koh, S.J.A., Baumgartner, R., Goh, Y.F., Ng, W.H., Kogler, A., Tröls, A., Foo, C.C., Suo, Z., et al.: Natural rubber for sustainable high-power electrical energy generation. RSC Adv. 4(53), 27905–27913 (2014)CrossRefGoogle Scholar
  24. 24.
    Koh, S.J.A., Keplinger, C., Li, T., Bauer, S., Suo, Z.: Dielectric elastomer generators: how much energy can be converted? IEEE ASME Trans. Mech. 16(1), 33–41 (2011)CrossRefGoogle Scholar
  25. 25.
    Koh, S.J.A., Zhao, X., Suo, Z.: Maximal energy that can be converted by a dielectric elastomer generator. Appl. Phys. Lett. 94(26), 262902 (2009)CrossRefGoogle Scholar
  26. 26.
    Lee, C.H., Newman, J.N.: Wamit User Manual. WAMIT, Inc (2006)Google Scholar
  27. 27.
    Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Moretti, G., Fontana, M., Vertechy, R.: Model-based design and optimization of a dielectric elastomer power take-off for oscillating wave surge energy converters. Meccanica 50(11), 2797–2813 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Moretti, G., Rosati Papini, G.P., Alves, M., Grases, M., Vertechy, R., Fontana, M.: Analysis and design of an oscillating water column wave energy converter with dielectric elastomer power take-off. In: ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers (2015)Google Scholar
  30. 30.
    Moretti, G., Rosati Papini, G.P., Fontana, M., Vertechy, R.: Hardware in the loop simulation of a dielectric elastomer generator for oscillating water column wave energy converters. In: OCEANS 2015-Genova, pp. 1–7. IEEE (2015)Google Scholar
  31. 31.
    Mørk, G., Barstow, S., Kabuth, A., Pontes, M.T.: Assessing the global wave energy potential. In: ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering, pp. 447–454. American Society of Mechanical Engineers (2010)Google Scholar
  32. 32.
    Opti toolbox.: (2016). Accessed 03 Oct 2016
  33. 33.
    Pelrine, R., Kornbluh, R.D., Eckerle, J., Jeuck, P., Oh, S., Pei, Q., Stanford, S.: Dielectric elastomers: generator mode fundamentals and applications. In: SPIE’s 8th Annual International Symposium on Smart Structures and Materials, pp. 148–156. International Society for Optics and Photonics (2001)Google Scholar
  34. 34.
    Rossiter, J.A.: Model-based predictive control: a practical approach. CRC Press, London (2003)Google Scholar
  35. 35.
    Scherber, B., Grauer, M., Kllnberger, A.: Electroactive Polymers for Gaining Sea Power (2013).
  36. 36.
    Teillant, M., Rosati Papini, G.P., Moretti, G., Vertechy, R., Fontana, M., Monkand, K., Alves, M.: Techno-economic comparison between air turbines and dielectric elastomer generators as power take off for oscillating water column wave energy converters. In: 11th European Wave and Tidal Energy Conference. Nantes, France (2015)Google Scholar
  37. 37.
    Tóth, R., Felici, F., Heuberger, P., Van den Hof, P.: Crucial aspects of zero-order hold lpv state-space system discretization. In: IFAC Proceedings, vol. 41, No. 2, pp. 4952–4957 (2008)Google Scholar
  38. 38.
    Tröls, A., Kogler, A., Baumgartner, R., Kaltseis, R., Keplinger, C., Schwödiauer, R., Graz, I., Bauer, S.: Stretch dependence of the electrical breakdown strength and dielectric constant of dielectric elastomers. Smart Mater. Struct. 22(10), 104012 (2013)Google Scholar
  39. 39.
    Vertechy, R., Fontana, M., Rosati Papini, G.P., Bergamasco, M.: Oscillating-water-column wave-energy-converter based on dielectric elastomer generator. In: SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, pp. 86870I–86870I. International Society for Optics and Photonics (2013)Google Scholar
  40. 40.
    Vertechy, R., Fontana, M., Rosati Papini, G.P., Forehand, D.: In-tank tests of a dielectric elastomer generator for wave energy harvesting. In: SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, pp. 90561G–90561G. International Society for Optics and Photonics (2014)Google Scholar
  41. 41.
    Vertechy, R., Fontana, M., Stiubianu, G., Cazacu, M.: Open-access dielectric elastomer material database. In: SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring. International Society for Optics and Photonics (2014)Google Scholar
  42. 42.
    Vertechy, R., Frisoli, A., Bergamasco, M., Carpi, F., Frediani, G., De Rossi, D.: Modeling and experimental validation of buckling dielectric elastomer actuators. Smart Mater. Struct. 21(9), 094005 (2012)CrossRefGoogle Scholar
  43. 43.
    Vertechy, R., Rosati Papini, G.P., Fontana, M.: Reduced model and application of inflating circular diaphragm dielectric elastomer generators for wave energy harvesting. J. Vib. Acoust. 137(1), 011004 (2015)CrossRefGoogle Scholar
  44. 44.
    Yu, Z., Falnes, J.: State-space modelling of a vertical cylinder in heave. Appl. Ocean Res. 17(5), 265–275 (1995)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Scuola Superiore Sant’AnnaPisaItaly
  2. 2.University of BolognaBolognaItaly
  3. 3.University of TrentoTrentoItaly

Personalised recommendations