# System Identification of a Heaving Point Absorber: Design of Experiment and Device Modeling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Identification: Overview

#### 2.1. Model Categories

#### 2.2. Experiments for System Identification

**Improved signal-to-noise ratio**—For the same frequency resolution and RMS value, the signal-to-noise ratio is $\sqrt{2}$ smaller; or for the same signal-to-noise ratio and RMS value, the measurement time is half as long.**Increased range of physical regimes**—Experiments where the system is tested using one input at the time (dual SISO) do not mimic the operational conditions, which may be a problem if the system behaves nonlinearly (i.e., single input tests may not reach the relevant physical regimes, therefore the test fails to observe important system dynamics).

## 3. Description of Experimental Setup

## 4. WEC Modeling in the Classical Framework: Radiation and Excitation

#### 4.1. Intrinsic Impedance and Radiation Impedance

#### 4.1.1. Nonparametric Models

#### Radiation Force Modeling

#### 4.1.2. Parametric Models

#### Black Box Modeling

`tfest`(included in the System Identification toolbox), which computes an estimation of continuous-time transfer functions. The data used for the SID is the same that has been used for the derivation of the nonparametric models described in Section 4.1.1. Bode plots of the parametric models of ${Y}_{i}$ are shown in Figure 18; the same figure also shows nonparametric FRFs of ${Y}_{i}$. Each curve corresponds to a different experiment, which makes it very clear that the system shows a nonlinear behavior. This can be further confirmed from the pole-zero map shown in Figure 19.

#### Grey Box Modeling

`tfest`. The algorithm used by the function

`tfest`considers the system to be described by a parametric model in the “Output-Error” form, and it calculates the coefficients of the transfer function by minimizing the prediction error [5].

#### Comparison of Grey Box and Black Box Models, and Cross-Validation

#### 4.2. Excitation Force Modeling

#### 4.2.1. Estimation of the Excitation FRF from Diffraction Tests

#### 4.2.2. Estimation of the Excitation FRF without Locking the Buoy

- Execute forced oscillation experiments in calm water to obtain a model of the intrinsic impedance as described in Section 4.1.1 and obtain either a parametric or nonparametric model for ${Z}_{i}$.
- Execute the forced oscillation experiment in presence of waves. In this case, the available measurements are the actuator force (${F}_{a}$), the buoy velocity (v) and the surface elevation ($\eta $). By using the frequency-domain equation of motion$${\widehat{F}}_{e}\left(\omega \right)+{\widehat{F}}_{a}\left(\omega \right)={Z}_{i}\left(\omega \right)\phantom{\rule{0.166667em}{0ex}}\widehat{V}\left(\omega \right),\phantom{\rule{2.em}{0ex}}\mathrm{with}\phantom{\rule{2.em}{0ex}}{\widehat{F}}_{e}\left(\omega \right)=H\left(\omega \right)\widehat{\eta}\left(\omega \right),$$$$H\left(\omega \right)=\frac{{Z}_{i}\left(\omega \right)\phantom{\rule{0.166667em}{0ex}}\widehat{V}\left(\omega \right)-{\widehat{F}}_{a}\left(\omega \right)}{\widehat{\eta}\left(\omega \right)}.$$

#### 4.3. Validation of Combined Model

`ss`included in the Control System toolbox, which returns a linear model in state space form, when the input is a transfer function. Cross validation results are shown in Figure 29, for both the Black box and Grey box models, where data from experiment 115 has been used for validation. In this case, the fit for the Black box model, according to the metric in (25), is 85%, whereas the fit for the Grey box model is 83%.

#### 4.4. WEC Model as Multiple-Input Single-Output System

## 5. WEC Modeling Using Pressure

`n4sid`, included in the System Identification toolbox; this function implements the subspace identification algorithm [5], which is a time-domain method that provides a parametric model of the system in state space form. The identification is carried out using data from experiment 105, in which both input signals are non-periodic. The force signal is a band-limited white noise and the waves have a Bretschneider spectrum with repeating period of 2 hours; however, the duration of the experiment is 30 minutes. Data from experiment 115 is used for validation; in this case, both inputs are pink multisines. Figure 36 shows a 50 s interval comparing the measured buoy’s velocity against the simulated velocity. In this case the fit, measured using the metric in (25), is 87%. We have found that the fit is generally good when using MISO models with pressure as an input. This shows a good potential for state estimation and model based control design.

## 6. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BEM | Boundary element method |

FRF | Frequency response function |

FRM | Frequency response matrix |

FFT | Fast Fourier transform |

LTI | Linear time invariant |

IRF | Impulse response function |

MASK | Maneuvering And Sea Keeping |

MISO | Multiple input single-output |

NRMSE | Normalized root mean square error |

PTO | Power take-off |

SID | System identification |

SISO | Single input single output |

WEC | Wave energy converter |

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**Figure 2.**Block diagram of the WEC base on radiation/diffraction model. The surface elevation is denoted by $\eta $, the actuator force is ${F}_{a}$ and the velocity is v.

**Figure 6.**Basin layout for testing with locations of wave probes (“WP1”, “WP2”, and “WP3”) and test device (“Float”). Wave propagation was at ${70}^{\circ}$ with respect to the x-axis.

**Figure 8.**Block diagram of the system when including effects of radiated and scattered waves on the surface elevation measurements. The measured surface elevation ${\eta}^{tot}$ includes terms due to the scattered waves ${\eta}^{s}$ and radiated waves ${\eta}^{r}$, in addition to the incoming wave train $\eta $.

**Figure 9.**The diagram in Figure 8 can be rearranged as a two input system where the total surface elevation is considered as an input.

**Figure 12.**FRF of the intrinsic impedance ${Z}_{i}$ calculated from one experiment. The figure also shows the FRF of ${Z}_{i}$ calculated numerically using WAMIT and the physical properties of the buoy.

**Figure 13.**Intrinsic impedance estimated from all the experiments. Frequency smoothing has been performed over a window of 12 frequencies.

**Figure 14.**Equivalent linear damping is linearly proportional to the inverse of the 2-norm of the buoy velocity (${R}^{2}=0.993$).

**Figure 15.**By using the linearized equivalent damping in (12), the offset in the real part of the intrinsic impedance has been significantly reduced.

**Figure 16.**FRFs of the velocity $\widehat{V}$, of the estimated noise ${\widehat{n}}_{V}$ and of their ratio $\widehat{V}/{\widehat{n}}_{V}$.

**Figure 17.**Comparison of radiation damping and added mass provided by WAMIT and calculated from all the forced oscillation experiments 081-091.

**Figure 18.**FRFs and Bode plots of parametric models of the intrinsic admittance ${Y}_{i}$ for different experiments, showing nonlinear behavior as the response depends on the experiment.

**Figure 19.**Pole-zero maps of the intrinsic admittance ${Y}_{i}$ from different experiments (experiments: 081, 082, 086, 087 and 089).

**Figure 22.**Comparison of Bode plots and FRFs of ${\tilde{Z}}_{r}$ obtained from different experiments (experiments: 081, 082, 086, 087 and 089). The 95% confidence interval is the shaded area around the magnitude and phase plots of the parametric models.

**Figure 23.**Pole-zero maps of the parametric models for ${\widehat{Z}}_{r}$ obtained from different experiments (experiments: 081, 082, 086, 087 and 089). The 95% confidence interval is shown as a circle around the poles and a line for the zeros.

**Figure 26.**Time-domain responses of the both the black box model and the grey box model, compared to the experimental response. The identification has been carried out using experiment 091 and validation has been carried using data from experiment 085, where the linear equivalent damping has been adjusted according to (12).

**Figure 28.**Excitation FRFs obtained from experiment with locked and unlocked device. The surface elevation is measured using the capacitive probe named “WP1”.

**Figure 29.**Validation for Grey box and Black box models using experiment 115, where waves have a pink multisine spectrum and also the actuator’s force has a pink multisine spectrum.

**Figure 30.**FRFs of the 2-input, 1-output black-box model for the WEC. Here, the surface elevation is measured in inches and the force in kN.

**Figure 32.**FRFs of the radiation and excitation models using pressure (from experiments 089 and 010 respectively).

**Figure 34.**FRFs of the MISO model and of the dual SISO model (shown in Figure 33).

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

Rigid-body mass (float & slider), M (kg) | 858 |

Displaced volume, ∀ (m^{3}) | 0.858 |

Float radius, r (m) | 0.88 |

Float draft, T (m) | 0.53 |

Water density, $\rho $ (kg/m^{3}) | 1000 |

Name | Type | x-Location (m) | y-Location (m) |
---|---|---|---|

Float | NA | 37.9 | 78.5 |

WP1 | Capacitive | 19.7 | 28.9 |

WP2 | Sonic | 27.2 | 20.1 |

WP3 | Sonic | 21.0 | 77.4 |

**Table 3.**Experimental datasets utilized in this study. For experiment 105, both the wave train and actuator input are non-periodic: in particular, the actuator input is a band-limited white noise (BLWN) and the wave spectra is Bretschneider (BS) (* indicates reseeded phasing).

Test ID | Actuator Input | Actuator Freq. (Hz) | Actuator Gain | Wave Input | Wave Freq. (Hz) | Wave Gain |
---|---|---|---|---|---|---|

010 | None | – | – | Pink | $0.25<f<1.0$ | 1.00 |

081 | White | $0.25<f<1.0$ | 1.00 | None | – | – |

082 | White | $0.25<f<1.0$ | 1.50 | None | – | – |

083 | White | $0.25<f<1.0$ | 0.50 | None | – | – |

084 | White | $0.25<f<1.0$ | 1.25 | None | – | – |

085 | White | $0.25<f<1.0$ | 0.75 | None | – | – |

086 | Pink | $0.25<f<1.0$ | 1.00 | None | – | – |

087 | Pink | $0.25<f<1.0$ | 1.50 | None | – | – |

088 | Pink | $0.25<f<1.0$ | 0.50 | None | – | – |

089 | Pink | $0.25<f<1.0$ | 2.00 | None | – | – |

090 | Pink | $0.25<f<1.0$ | 0.75 | None | – | – |

091 | Pink | $0.25<f<1.0$ | 1.25 | None | – | – |

105 | BLWN | $0.25<f<1.0$ | 1.00 | BS | ${T}_{p}=3.08$ s | ${H}_{s}=0.121$ m |

109 | Pink | $0.25<f<1.0$ | 1.00 | Pink | $0.25<f<1.0$ | 1.00 |

110 | Pink | $0.25<f<1.0$ | 0.50 | Pink | $0.25<f<1.0$ | 1.00 |

111 | Pink | $0.25<f<1.0$ | 2.00 | Pink | $0.25<f<1.0$ | 1.00 |

112 | Pink | $0.25<f<1.0$ | 1.00 | Pink | $0.25<f<1.0$ | 2.00 |

113 | Pink | $0.25<f<1.0$ | 0.50 | Pink | $0.25<f<1.0$ | 2.00 |

114 | Pink | $0.25<f<1.0$ | 2.00 | Pink | $0.25<f<1.0$ | 2.00 |

115 | Pink * | $0.25<f<1.0$ | 2.00 | Pink | $0.25<f<1.0$ | 1.00 |

116 | Pink * | $0.25<f<1.0$ | 0.50 | Pink | $0.25<f<1.0$ | 1.00 |

117 | Pink * | $0.25<f<1.0$ | 1.00 | Pink | $0.25<f<1.0$ | 1.00 |

© 2017 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/).

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**MDPI and ACS Style**

Bacelli, G.; Coe, R.G.; Patterson, D.; Wilson, D.
System Identification of a Heaving Point Absorber: Design of Experiment and Device Modeling. *Energies* **2017**, *10*, 472.
https://doi.org/10.3390/en10040472

**AMA Style**

Bacelli G, Coe RG, Patterson D, Wilson D.
System Identification of a Heaving Point Absorber: Design of Experiment and Device Modeling. *Energies*. 2017; 10(4):472.
https://doi.org/10.3390/en10040472

**Chicago/Turabian Style**

Bacelli, Giorgio, Ryan G. Coe, David Patterson, and David Wilson.
2017. "System Identification of a Heaving Point Absorber: Design of Experiment and Device Modeling" *Energies* 10, no. 4: 472.
https://doi.org/10.3390/en10040472