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
  • 200 Downloads

Abstract

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.

Keywords

Smart materials Energy harvesting Wave energy Dielectric elastomers Hydrodynamics EPAM DEG 

List of symbols

Physical constants

g

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

\(P_\mathrm{atm}\)

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)

\(A_{j}\)

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)

\(f\)

PTO force (N)

\(f_\mathrm{e}\)

Excitation force (N)

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

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

H

Regular wave height (m)

\(H_\mathrm{s}\)

Significant irregular wave height (m)

\(k_\mathrm{r}\)

Radiation force kernel (N/m)

\(m_\infty \)

Added mass at infinite frequency (kg)

n

Dimension of \(\varvec{x_c}\)

\(p\)

Air chamber relative pressure (Pa)

\(\dot{p}_\mathrm{th}\)

Pressure derivative threshold (Pa/s)

r

Dimension of \(\varvec{x_r}\)

S

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

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

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

T

Regular wave period (s)

\(T_\mathrm{e}\)

Irregular wave energy period (s)

\(u_c\)

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

\(\varvec{x_c}\)

Hydrodynamic model state vector

\(\varvec{x_r}\)

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

N

Number of samples in a discretised horizon

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

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

\(T_\mathrm{s}\)

Discretisation time step (s)

\(u_d\)

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

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

Sparse matrices defined in Eq. (36)

\(\varvec{x_d}\)

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

\(C\)

Capacitance (F)

\(E\)

Electric field at the tip element (V/m)

\(E_\mathrm{BD}\)

Breakdown electric field (V/m)

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

Electrical energy supplied by the DEG (J)

\(e\)

Flat DEG pre-stretched radius (m)

\(e_0\)

DEG unstretched radius (m)

h

Tip displacement (m)

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

Maximum and minimum admissible tip displacements (m)

\(I\)

Current (A)

\(I_\mathrm {MAX}\)

Maximum admissible current (A)

\(I_1\)

Gent invariant parameter

\(k_1\)

Gent stiffness parameter (Pa)

\(n_l\)

Number of in-parallel layers

\(Q\)

Charge (C)

R

ICD-DEG curvature (m)

\(t\)

Flat DEG pre-stretched thickness (m)

\(t_0\)

DEG unstretched thickness (m)

\(V\)

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

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

Pre-stretch

\(\lambda _u\)

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

\(\varvec{I_{a}}\)

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

Notes

Acknowledgements

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.

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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

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