# Modelling a Heaving Point-Absorber with a Closed-Loop Control System Using the DualSPHysics Code

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## Abstract

**:**

## 1. Introduction

## 2. Smoothed Particle Hydrodynamics

#### 2.1. Governing Equations

**r**is the position,

**v**is the velocity, p is the pressure, ρ is the density, m is the mass, c is the numerical speed of sound, and

**g**is the gravitational acceleration. The artificial viscosity (Π

_{ab}) proposed in [39] will be used and the density diffusion term proposed by [41] is applied in the present simulations using δ = 0.1.

**r**,

**v**, p and ρ. The system can be closed by solving a Poisson-like equation, where the fluid is considered fully incompressible (ISPH), or by adding an equation of state, where the fluid is treated as weakly compressible (WSPH). In the case of DualSPHysics, the fluid pressure at a certain particle is obtained as a function of its density, according to Tait’s equation of state:

_{0}= 1000 kg∙m

^{−3}the reference density of the fluid. The speed of sound, c, is artificially lowered to keep a reasonable time step value (calculated according to the Courant–Friedrichs–Lewy condition) throughout the simulation, while also guaranteeing that density variations are lower than 1%.

#### 2.2. Boundary Conditions and Fluid-Driven Objects

**f**, of each floating particle q is computed as the summation of the contributions of all surrounding fluid particles b:

**V**the linear velocity, I the moment of inertia,

**Ω**the rotational velocity and

**R**the center of mass. By integrating Equations (6) and (7) in time, the values of

**V**and

**Ω**at the beginning of the next time step can be predicted. The velocity of each particle of the floating object is given by:

#### 2.3. Coupling with Project Chrono

#### 2.4. High-Performance Computing

## 3. Experimental Campaign

## 4. Radiation Test

_{i}, while the output obtained from the radiation test is the heaving velocity, v, which can be calculated as

#### 4.1. Monochromatic Force Signal

_{1}, is employed. According to [60], it can be defined in a general way as:

_{1}is bounded between zero and one, where one represents perfect agreement between the two signals.

_{1}confirm that results are highly accurate for the three values of dp and only a slight improvement is achieved when the two finest resolutions are used.

#### 4.2. Polychromatic Force Signal

_{i}(ω), can be modelled as:

_{f}, the radiation force is defined by the radiation damping, b

_{r}, and the added mass, m

_{a}, the hydrostatic force depends on the hydrostatic coefficient, k

_{hs}, and the inertia force simply depends on the mass of the device, M. It is important to note that Equations (11) and (12) allow attainment of the resonance frequency of the device, i.e., the frequency at which resonance is achieved. As stated in [62], resonance occurs when the heave velocity is in phase with the excitation force. Thus, the resonance frequency is the one that guarantees that Im(Z

_{i}) = 0 Ns/m (which directly implies ∠Z

_{i}= 0 rad).

_{i}), Figure 8 demonstrates a high level of agreement between the SPH, experimental and WAMIT results. The resonance frequency, obtained as the frequency at which the imaginary part (or, equivalently, the phase) of the intrinsic impedance crosses the x-axis, is around 0.65 Hz for all cases. Regarding the plot of the real part of Z

_{i}shown in Figure 8, the difference between the experimental and WAMIT curves is roughly constant with frequency, revealing an offset caused by the fact that WAMIT is neglecting the viscous effects, which have a significant impact on Re(Z

_{i}), as shown in Equation (12). This mismatch was investigated in depth in [61]. The real part of the intrinsic impedance obtained with SPH nicely matches the experimental data at high frequencies; however, when low frequencies are considered, Re(Z

_{i}) is highly underestimated. This means that when the frequency is high enough, the SPH simulation is able to reproduce the physics of the problem correctly, but when the frequency is low, there is a phenomenon, most likely related with viscosity, that is not being adequately captured.

## 5. Static and Dynamic Response Test

#### 5.1. Static Response Test

_{1}, defined in Equation (10) is used here as a quantitive measure of the total error in each case. Its values, shown in Table 3, certify the good matching observed in Figure 11, as well as the fact that the accuracy slightly improves when decreasing dp. Table 3 also includes the runtimes of the execution of 26 s of physical time using the same GPU card indicated before.

#### 5.2. Dynamic Response Test

_{a}. This actuator force may be either predefined externally (open-loop system) or defined during the test by means of the feedback provided by the PTO (closed-loop system). Both possibilities, depicted in Figure 12 via block diagrams, are considered in this section based on Case149 of the MASK3 experimental campaign [57]. During the experiment, the actuator force applied to the WEC is the reaction of the PTO force, i.e., F

_{a}= − F

_{PTO}; F

_{PTO}being modelled as a proportional–integral (PI) controller that depends on the device’s heave position, z, and velocity, v [63]:

_{PTO}and b

_{PTO}are, respectively, the stiffness and damping coefficient of the PTO system or, equivalently, the integral and proportional gains of the PI controller.

_{PTO}= 3527 N/m and b

_{PTO}= −1754 Ns/m. The wave conditions remain constant throughout the entire experiment: regular waves of height H = 0.11 m and period T = 1.58 s, which qualifies as a deep-water wave with a wavelength of L = 3.90 m. Figure 13 shows that, as expected, the frequency of the actuator force is coincident with the wave frequency and that, after the transient observed during the first period, its amplitude is approximately constant around 160 N. This initial transient is avoided in the simulations by considering the time series from t = 273 s.

#### 5.2.1. Open-Loop System

_{a}can be effectively used instead. Three different phases are simulated simply by delaying the time at which the external actuator force is applied in each simulation (cases A, B and C). The actuator force and heave position time series for the three numerical cases are shown, along with the experimental ones, in Figure 14. Regarding the numerical results, the actuator force signal is shifting in phase while the heave position is not, causing a phase difference between them of about π rad in case A, 3π/4 rad in case B, and π/2 rad in case C. The experimental phase between F

_{a}and z, observed in the bottom plot of Figure 14, is very similar to the one obtained with case C, i.e., approximately π/2 rad.

_{a}and z approaches the experimental one (Figure 14), so do the amplitudes of the heave position and velocity. However, the phase difference is not known before running the simulation, thus it becomes a trial-and-error process. This approach, besides being inefficient, is also unrealistic because the actuator force is very rarely given by a predefined signal, but rather by its PTO force.

#### 5.2.2. Closed-Loop System

_{PTO}and b

_{PTO}, as described in Equation (13). Project Chrono allows numerical implementation of this feedback loop simply by setting k

_{PTO}= 3527 N/m and b

_{PTO}= −1754 Ns/m. Therefore, thanks to the coupling of DualSPHysics with Project Chrono it is possible to simulate the dynamic response test with its corresponding closed-loop system using the numerical tank defined in Figure 9 and the wave conditions previously described (H = 0.11 m and T = 1.58 s).

_{a}and z now matches the experimental one. This is automatically accomplished because the actuator force is being calculated at each time step as a function of the instantaneous heave position and velocity, instead of being pre-imposed externally.

_{1}= 0.92 is obtained for both z and v when using a closed-loop system, which is significantly higher than the one obtained with the most accurate case when using an open-loop system (d

_{1}= 0.74). By comparing results in Figure 17 with the ones in Figure 15, it can be also seen that the amplitude overestimation obtained now with a closed-loop system is lower than in any of the three cases with an open-loop system. Note as well that the amplitude of the heave movement is of approximately 0.02 m, which is the value of the dp used, meaning that small motions can be correctly simulated even with relatively coarse resolutions. Therefore, this numerical simulation satisfactorily reproduces the response of the WEC equipped with a proportional–integral controller.

## 6. Conclusions

_{i}at low frequencies, that is significantly underestimated. On the other hand, WAMIT also provides good results except for Re(Z

_{i}), which is underestimated at all frequencies. This offset, which is roughly constant with frequency, is due to the fact that the potential-flow solver neglects viscosity. Thus, the most efficient way to numerically obtain the radiation model is to use WAMIT (or another potential-flow solver) but adding a correction that takes into account the viscous effects. The proven ability of the SPH method to reproduce the viscous effects at high frequencies allows for an estimation of the viscous correction needed for the potential-flow solver. Therefore, a fast and reliable procedure for system identification can be set up by combining the results obtained with a CFD simulation and a potential solver.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Numerical and experimental time series of the heave position (

**a**) and velocity (

**b**) for three values of dp.

**Figure 8.**Intrinsic impedance calculated from the physical tests, using WAMIT and using DualSPHysics.

**Figure 12.**Block diagrams for the dynamic response of the WEC with an open-loop system (

**a**) and a closed-loop system (

**b**).

**Figure 14.**Numerical and experimental time series of the actuator force (left axis) and heave position (right axis) for three different phases (π, 3π/4, and π/2 rad) with an open-loop system.

**Figure 15.**Numerical and experimental time series of the heave position (

**a**) and velocity (

**b**) for different phases with an open-loop system.

**Figure 16.**Numerical and experimental time series of the actuator force (left axis) and heave position (right axis) with a closed-loop system.

**Figure 17.**Numerical and experimental time series of the heave position (

**a**) and velocity (

**b**) with a closed-loop system.

Parameter | Symbol (Unit) | Value |
---|---|---|

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

Displaced volume | V (m^{3}) | 0.858 |

Float radius | R (m) | 0.88 |

Float draft | D (m) | 0.53 |

**Table 2.**Number of particles, number of total time steps, runtime and index of agreement for three values of dp.

dp (cm) | Particles | Time Steps | Runtime (h) | d_{1} | |
---|---|---|---|---|---|

z | v | ||||

1.5 | 24.1 × 10^{6} | 3.4 × 10^{5} | 132.6 | 0.97 | 0.96 |

2.0 | 10.3 × 10^{6} | 2.6 × 10^{5} | 53.6 | 0.97 | 0.96 |

3.0 | 3.1 × 10^{6} | 1.7 × 10^{5} | 10.6 | 0.96 | 0.96 |

**Table 3.**Number of particles, number of total time steps, runtime and index of agreement for three values of dp.

dp (cm) | Particles | Time Steps | Runtime (h) | d_{1} |
---|---|---|---|---|

1.5 | 28.1 × 10^{6} | 3.7 × 10^{5} | 157.8 | 0.90 |

2.0 | 12.1 × 10^{6} | 2.8 × 10^{5} | 63.5 | 0.89 |

3.0 | 3.7 × 10^{6} | 1.9 × 10^{5} | 10.4 | 0.89 |

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Ropero-Giralda, P.; Crespo, A.J.C.; Coe, R.G.; Tagliafierro, B.; Domínguez, J.M.; Bacelli, G.; Gómez-Gesteira, M.
Modelling a Heaving Point-Absorber with a Closed-Loop Control System Using the DualSPHysics Code. *Energies* **2021**, *14*, 760.
https://doi.org/10.3390/en14030760

**AMA Style**

Ropero-Giralda P, Crespo AJC, Coe RG, Tagliafierro B, Domínguez JM, Bacelli G, Gómez-Gesteira M.
Modelling a Heaving Point-Absorber with a Closed-Loop Control System Using the DualSPHysics Code. *Energies*. 2021; 14(3):760.
https://doi.org/10.3390/en14030760

**Chicago/Turabian Style**

Ropero-Giralda, Pablo, Alejandro J. C. Crespo, Ryan G. Coe, Bonaventura Tagliafierro, José M. Domínguez, Giorgio Bacelli, and Moncho Gómez-Gesteira.
2021. "Modelling a Heaving Point-Absorber with a Closed-Loop Control System Using the DualSPHysics Code" *Energies* 14, no. 3: 760.
https://doi.org/10.3390/en14030760