# State-Space Approximation of Convolution Term in Time Domain Analysis of a Raft-Type Wave Energy Converter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Device

_{0}between them, as shown in Figure 1b. The length, diameter, and density are denoted as L, D, and ρ

_{0}, respectively. The motion characteristic of the device is formulated in a Cartesian coordinate (x, y and z) system with its origin O coincident with the center of the joint, where the z-axis is in the vertical direction, while the x- and y-axes are taken along the length and the diameter direction of the rafts in still water, respectively, as shown in Figure 1a. For the fore raft, the displacements are labeled as x

_{1}-surge, z

_{1}-heave, and θ

_{1}-pitch. For the aft raft, the displacements are labeled as x

_{2}-surge, z

_{2}-heave, and θ

_{2}-pitch. The surge and heave displacements of the joint are denoted as x

_{0}and z

_{0}, as shown in Figure 1c.

_{w,j}(t) (here, subscript j = 1, 3, and 5 indicate the surge, heave, and pitch modes of the fore raft, respectively; j = 1′, 3′, and 5′ indicate the surge, heave, and pitch modes of the aft raft, respectively). As the wave passes along the length of the rafts, the rafts output a relative pitch motion around the joint, which drives the hydraulic PTO unit to convert wave energy.

## 3. Frequency Domain Analysis

**x**(t) = [x

_{0}z

_{0}θ

_{1}θ

_{2}]

^{T}to describe the motion characteristic of the device and given a four-DOF frequency domain model for waves in the positive x-direction based on the Lagrange’s equations, which can be written as:

**M**is the generalized mass matrix;

**A**

_{add}(ω) is the generalized added mass matrix;

**B**(ω) is the generalized radiation damping matrix;

**K**is the generalized hydrostatic restoring stiffness matrix;

**C**

_{pto}and

**K**

_{pto}are the damping matrix and stiffness matrix of the PTO unit, respectively;

**X**(ω) is the complex amplitude of the generalized displacement vector

**x**(t); and

**F**

_{e}(ω) is the complex amplitude of the generalized wave excitation force vector. Their expressions are given in Appendix A.

#### 3.1. Frequency Response Function

**G**(ω), can be described by [27]:

_{w}is the wave amplitude and

**Γ**

_{e}(ω) is the complex generalized excitation force coefficient vector (force vector per unit incident wave amplitude). The expression of

**Γ**

_{e}(ω) is given in Appendix A.

_{v_rp}, is:

**ζ**= [0 0 1 −1].

#### 3.2. Power Capture Ability

## 4. Time Domain Analysis

**A**

_{add}(∞) is the limiting value of the generalized added mass matrix

**A**

_{add}(ω) for ω = ∞;

**f**

_{e}(t) is the generalized wave excitation force vector;

**f**

_{pto}(t) is the generalized force vector applied by the PTO unit; and

**h**(t) is the retardation function matrix. The time and frequency domain representations of the retardation function matrix are [28]:

_{n}, ε

_{n}, and a

_{w}(ω

_{n}) are the wave frequency, random phase angle, and wave amplitude of the n-th wave component, respectively.

#### 4.1. State-Space Model of Convolution Term

**μ**(t):

**μ**(t), it also has a form as:

_{p,q}(t) in the convolution term

**μ**(t) can be replaced by a state-space model [29,32].

^{(q)}(t) is the q-th element of the generalized displacement vector

**x**(t); the sizes of matrices

**A**

_{s}

^{(p,q)},

**B**

_{s}

^{(p,q)}, and

**C**

_{s}

^{(p,q)}are (n

_{p,q}× n

_{p,q}), (n

_{p,q}× 1), and (1 × n

_{p,q}), respectively; and n

_{p,q}is the number of states corresponding to the state vector

**x**

_{s}

^{(p,q)}.

**μ**(t), sixteen state-space models are presented. Then, for reducing the model complexity and computational load, four state-space models can be assembled by these sixteen state-space models to replace each convolution term μ

_{p}(t), and finally, a total state-space model can be assembled by the four state-space models. Figure 2 shows the general idea of assembling the state-space model. Thus, the convolution term μ

_{p}(t) can be rewritten as:

**A**

_{s}

^{(p)},

**B**

_{s}

^{(p)}, and

**C**

_{s}

^{(p)}are assembled by matrices

**A**

_{s}

^{(p,q)},

**B**

_{s}

^{(p,q)}, and

**C**

_{s}

^{(p,q)}, respectively; the state vector

**x**

_{s}

^{(p)}is assembled by state vectors

**x**

_{s}

^{(p,q)}; the sizes of matrices

**A**

_{s}

^{(p)},

**B**

_{s}

^{(p)},

**C**

_{s}

^{(p)}, and

**x**

_{s}

^{(p)}are ($\sum}_{q=1}^{4}{n}_{p,q}}\times {\displaystyle {\sum}_{q=1}^{4}{n}_{p,q$), (${\sum}_{q=1}^{4}{n}_{p,q}}\times 4$), ($1\times {\displaystyle {\sum}_{q=1}^{4}{n}_{p,q}}$), and (${\sum}_{q=1}^{4}{n}_{p,q}}\times 1$), respectively. The expressions of

**A**

_{s}

^{(p)},

**B**

_{s}

^{(p)},

**C**

_{s}

^{(p)}, and

**x**

_{s}

^{(p)}are shown in Appendix A.

**A**

_{s},

**B**

_{s}, and

**C**

_{s}are assembled by matrices

**A**

_{s}

^{(p)},

**B**

_{s}

^{(p)}, and

**C**

_{s}

^{(p)}, respectively; the state vector

**x**

_{s}is assembled by state vectors

**x**

_{s}

^{(p)}; and the sizes of matrices

**A**

_{s},

**B**

_{s},

**C**

_{s}, and

**x**

_{s}are ($\sum}_{p=1}^{4}{\displaystyle {\sum}_{q=1}^{4}{n}_{p,q}}}\times {\displaystyle {\sum}_{p=1}^{4}{\displaystyle {\sum}_{q=1}^{4}{n}_{p,q}$), (${{\displaystyle \sum}}_{p=1}^{4}{{\displaystyle \sum}}_{q=1}^{4}{n}_{p,q}\times 4$), ($4\times {{\displaystyle \sum}}_{p=1}^{4}{{\displaystyle \sum}}_{q=1}^{4}{n}_{p,q}$), and (${{\displaystyle \sum}}_{p=1}^{4}{{\displaystyle \sum}}_{q=1}^{4}{n}_{p,q}\times 1$), respectively. The expressions of

**A**

_{s},

**B**

_{s},

**C**

_{s}, and

**x**

_{s}are shown in Appendix A.

**A**

_{s}

^{(p,q)},

**B**

_{s}

^{(p,q)}, and

**C**

_{s}

^{(p,q)}. It can be seen from Equation (16) that the Laplace transform of the retardation function h

_{p,q}(t) is the transfer function of the system with scalar input ${\dot{x}}^{\left(q\right)}\left(t\right)$ and scalar output μ

_{p,q}(t). Thus, the relationship between the state-space model and the transfer function is:

**μ**(t) (see Equation (18)) can be expressed by:

**A**

_{s}

^{(p,q)},

**B**

_{s}

^{(p,q)}, and

**C**

_{s}

^{(p,q)}by identification using the generalized added mass matrix

**A**

_{add}(ω) and radiation damping matrix

**B**(ω). Moreover, for the two rafts with the same geometric parameter, it should be mentioned that some elements of the symmetrical transfer function matrix

**H**(s) would satisfy the following relationships: H

_{12}(s) = H

_{21}(s) = 0, H

_{13}(s) = H

_{14}(s), H

_{23}(s) = −H

_{24}(s), and H

_{33}(s) = H

_{44}(s). Therefore, the identification only need to be carried out for H

_{11}(s), H

_{13}(s), H

_{22}(s), H

_{23}(s), H

_{33}(s), and H

_{34}(s), i.e., to determine the transfer function matrix

**H**(s) of the total system, the identification procedure only needs to be carried out six times.

#### 4.2. Transfer Function Estimation Using Regression in Frequency Domain

**ϕ**is a vector containing the estimated parameters, and defined as:

**ϕ**that gives the best non-linear least square (NL-LS) fitting to the frequency response.

_{n}is the weighting coefficient and “arg min” represents the minimizing argument.

_{n,g}= 1, and after a few iterations Q

_{pq}(−iω,

**ϕ**

_{g}) ≈ Q

_{pq}(−iω,

**ϕ**). Then, the optimal parameter of vector

_{g}_{−1}**ϕ**can be determined as

**ϕ***=

**ϕ**

_{g}. Thus, the parametric model ${\widehat{H}}_{pq}\left(s\right)$ can be obtained. Then, the constant matrices

**A**

_{s}

^{(p,q)},

**B**

_{s}

^{(p,q)}, and

**C**

_{s}

^{(p,q)}of the state-space model can be determined by using the Matlab function tf2ss. Thereafter, the matrices

**A**

_{s},

**B**

_{s}, and

**C**

_{s}can be assembled by the formulations shown in Appendix A. Finally, the identified state-space model can be applied to replace the convolution term in the time domain model.

#### 4.3. Power Capture Ability

_{0}is a moment when the device has come into a steady state of motion and T is the wave period.

## 5. Numerical Results and Discussion

#### 5.1. Identification of State-Space Model

_{uj}(ɷ), the radiation damping b

_{uj}(ɷ), and the complex excitation force coefficient Γ

_{j}(ɷ)) in ANSYS AQWA [34], commercial hydrodynamic software based on three dimensional radiation/diffraction theory.

_{11}(−iω), H

_{13}(−iω), H

_{22}(−iω), H

_{23}(−iω), H

_{33}(−iω), and H

_{34}(−iω) for the device with structure parameters L = 10 m, D = 1 m, d

_{0}= 1 m, and ρ

_{0}= 512.5 kg/m

^{3}. The sizes of constant matrices

**A**

_{s},

**B**

_{s}, and

**C**

_{s}are 87 × 87, 87 × 4, and 4 × 87, respectively, which are not shown here for their large sizes. The original data of retardation functions H

_{11}(−iω), H

_{13}(−iω), H

_{22}(−iω), H

_{23}(−iω), H

_{33}(−iω), and H

_{34}(−iω) calculated by Equation (9) are also presented in Figure 3. It can be seen that there is a good agreement between the identification results and those obtained by Equation (9).

#### 5.2. Validation of Time Domain Analysis

#### 5.3. Numerical Results in Regular Waves

#### 5.3.1. Influence of Wave Frequency

#### 5.3.2. Influence of Mounting Position r_{0}

_{0}of hydraulic cylinder is a key parameter influencing the capture width ratio of the device. It is assumed that the mounting position r

_{0}is not confined to the diameter of the rafts. The power capture ability of the device is examined by using the time domain analysis over a wide range of mounting positions r

_{0}. Figure 6a shows the variation of the capture width ratio of the device with a type-four parameter with the mounting position normalized by the diameter of the raft at four different wave periods. It can be seen that the capture width ratio η

_{cap}increases with an increasing normalized mounting position r

_{0}/D, and then decreases after reaching a peak value. This is not surprising. As shown in Equation (A.6), there is a quadratic function relationship between the rotational damper 2r

_{0}

^{2}c

_{pto}and the mounting poison r

_{0}. For any specified wave period T and any specified damping coefficient c

_{pto}, a too large mounting position r

_{0}(i.e., too large rotational damping 2r

_{0}

^{2}c

_{pto}) would lead to a small relative pitch motion, and consequently, the device outputs a little power; whereas a too small mounting position r

_{0}(i.e., too small rotational damping 2r

_{0}

^{2}c

_{pto}) would induce a small PTO force, which also results in a little captured power. Therefore, for any specified wave period and any specified damping coefficient c

_{pto}, there exists an optimal normalized mounting position r

_{0}*/D, which corresponds to a peak capture width ratio η

_{cap}*.

_{0}*/D with wave period T is illustrated in Figure 6b. It is found that the optimal normalized mounting position r

_{0}*/D presents an approximately linear relationship with wave period, and a smaller damping coefficient c

_{pto}gives a larger gradient of the approximate linear relationship.

#### 5.3.3. Influence of Damping Coefficient c_{pto} and Stiffness k_{pto}

_{pto}is assumed to be zero, while in fact, the stiffness plays a significant role in power extraction. In order to see how the stiffness k

_{pto}and the damping coefficient c

_{pto}affect the power capture ability of the device, the capture width ratio is examined by using the time domain analysis over a wide range of damping coefficients c

_{pto}and stiffnesses k

_{pto}. Figure 7 shows the influence of the damping coefficient c

_{pto}and stiffness k

_{pto}on the capture width ratio of the device with a type-four parameter.

_{pto}*, which only depends on the wave period T. For a specified wave period T and a specified stiffness k

_{pto}, there exists an optimal damping coefficient c

_{pto}*. The optimal damping coefficient c

_{pto}* decreases with increasing stiffness k

_{pto}, and then increases after reaching a minimum value, and is symmetric to the optimal stiffness k

_{pto}*, as shown in Figure 7b. When the optimal damping coefficient c

_{pto}* is obtained in the condition of optimal stiffness k

_{pto}*, an optimal combination of the optimal damping coefficient and the optimal stiffness is achieved. This optimal combination could improve the power extraction ability significantly. As is shown in Figure 7a,b, the capture width ratio obtains a value of 0.0769 at k

_{pto}= 0 kN/m and the corresponding optimal damping coefficient, whereas it can reach 0.2868 at the optimal combination, i.e., an increase of more than 270%. Therefore, if the stiffness k

_{pto}of the PTO unit can be adjusted to be an optimal stiffness k

_{pto}* in the varying wave states, which makes the resonant frequency close to the considered wave frequency, the capture width ratio of the device can be improved dramatically. However, as it is revealed in [35], the nonlinear viscous damping would play an important role in the dynamic response of a WEC, especially in the vicinity of the resonant frequency. Hence, the present numerical model based on inviscid flow theory may overestimate the captured power, and the capture width ratio may thus be overpredicted.

_{pto}* and the optimal stiffness k

_{pto}* in the optimal combination with wave period is presented in Figure 7c. It can be learned from Figure 7c that the optimal stiffness k

_{pto}* in the optimal combination is normally negative, which means that the PTO unit may output power to the rafts in some period of its cycle. To be scientific, it is a kind of reactive control [28], but it is difficult to implement practically without a complicated PTO unit. It can also be seen from Figure 7c that the optimal damping coefficient c

_{pto}* in the optimal combination increases with the increase of wave period, then decreases after reaching a peak value, whereas, generally, the optimal stiffness k

_{pto}* in the optimal combination decreases monotonously.

#### 5.3.4. Influence of Surge and Heave Motions of Joint

_{v_rp}for the device with a type-one parameter. Figure 8b shows the corresponding capture width ratio η

_{cap}, obtained by the model with and without a consideration of surge and heave motions of the joint, whereas the variation of capture width ratio η

_{cap}with the damping coefficient c

_{pto}is presented in Figure 8c.

#### 5.3.5. Influence of Quadratic Damping PTO

_{x}

_{0}(t), f

_{z}

_{0}(t), f

_{θ}

_{1}(t), and f

_{θ}

_{2}(t) are the PTO forces acting on the generalized modes x

_{0}, z

_{0}, θ

_{1}, and θ

_{2}, respectively; and β is the quadratic damping coefficient.

_{cap}increases with an increasing quadratic damping coefficient β, and then decreases after reaching a peak value. For the wave states with the same period, a larger wave amplitude gives a higher capture width ratio at small quadratic damping coefficients β, while the peak capture width ratio η

_{cap}* has the same value at different wave amplitudes.

_{cap}* obtained by using linear damping and quadratic damping. It can be seen that the peak capture width ratio η

_{cap}* obtained by using quadratic damping is slightly larger than that obtained by using linear damping, especially in the vicinity of wave period T = 3.3 s. For 2 s < T< 2.6 s or 2.6 s < T< 5 s, the difference in peak capture width ratios obtained by using linear damping and quadratic damping increases with increasing wave period, and then decreases after reaching a local maximum value. The maximum peak capture width ratio η

_{cap}* obtained by using quadratic damping is 0.1086, whereas that obtained by using linear damping is 0.1031, and they are all obtained at wave period T = 3.3 s.

#### 5.4. Numerical Results in Irregular Waves

_{s}is the significant wave height and T

_{p}is the peak wave period.

#### 5.4.1. Influence of Mounting Position R_{0}

_{0}*/D, and the optimal normalized mounting position r

_{0}*/D presents an approximately linear relationship with wave period. In this section, we intend to investigate how the capture width ratio varies with mounting position r

_{0}in irregular waves. A wide range of mounting positions r

_{0}are examined. Figure 11a shows the variation of capture width ratio η

_{cap}with normalized mounting position r

_{0}/D for the device with a type-four parameter, whereas the variation of the optimal normalized mounting position r

_{0}*/D with peak wave period is presented in Figure 11b.

_{p}and a specified damping coefficient c

_{pto}, there exists an optimal normalized mounting position r

_{0}*/D. However, unlike the results in regular waves, the relationship between the optimal normalized mounting position r

_{0}*/D and the peak wave period T

_{p}presents obvious nonlinear characteristics in irregular waves. This is perhaps due to the random nature of the irregular waves.

#### 5.4.2. Influence of Damping Coefficient c_{pto} and Stiffness k_{pto}

_{pto}and the stiffness k

_{pto}play a significant role in the capture width ratio, we wonder whether the damping coefficient c

_{pto}and the stiffness k

_{pto}play a similar role in the capture width ratio in irregular waves. Figure 12a,b show the influence of the damping coefficient c

_{pto}and stiffness k

_{pto}on the capture width ratio of the device with a type-four parameter in irregular waves. Unlike the results in regular waves, it can be seen from Figure 12b that the optimal stiffness k

_{pto}* not only depends on the peak wave period but also increases slightly with an increasing damping coefficient c

_{pto}, and the optimal damping coefficient c

_{pto}* is not symmetric to the optimal stiffness k

_{pto}*. The variation of the optimal damping coefficient c

_{pto}* and the optimal stiffness k

_{pto}* in the optimal combination with peak wave period is presented in Figure 12c. Similar to the results in regular waves, the optimal stiffness k

_{pto}* in the optimal combination is negative and decreases with increasing peak wave period, whereas the optimal damping coefficient c

_{pto}* in the optimal combination increases with increasing peak wave period, and then decreases after reaching a peak value.

#### 5.4.3. Influence of Surge and Heave Motions of Joint

_{cap}with peak wave period, whereas the variation of capture width ratio η

_{cap}with damping coefficient c

_{pto}is presented in Figure 13b.

#### 5.4.4. Influence of Quadratic Damping PTO

_{cap}* obtained by using quadratic damping is slightly larger than that obtained by using linear damping in regular waves, it is necessary to investigate how quadratic damping β influences the capture width ratio η

_{cap}in irregular waves. Figure 14a shows the variation of capture width ratio η

_{cap}with quadratic damping coefficient β, whereas the variation of peak capture width ratio η

_{cap}* with peak wave period is presented in Figure 14b. As shown in Figure 14a, there exists an optimal quadratic damping coefficient β*, which corresponds to a peak capture width ratio η

_{cap}*. However, unlike the results in regular waves, it can be seen from Figure 14b that the peak capture width ratio η

_{cap}* obtained by using quadratic damping is almost the same as that obtained by using linear damping.

## 6. Conclusions

- (1)
- State-space approximation of the convolution term through regression in the frequency domain has sufficient accuracy. The time domain analysis with the convolution term approximated by a state-space model could be used to investigate the performance of the device.
- (2)
- To obtain a relatively large capture width ratio, the resonant frequency of the designed device should be as close to the considered wave frequency as possible. When there is no control included in the PTO unit, the arrival of resonance is usually at the cost of a relatively large raft size.
- (3)
- For a certain wave period (or peak wave period), there exists an optimal mounting position r
_{0}*, corresponding to a peak capture width ratio η_{cap}*. In regular waves, the relationship between the optimal normalized mounting position r_{0}*/D and the wave period is approximately linear, and a smaller damping coefficient c_{pto}gives a larger gradient of the approximately linear relationship; however, this relationship presents nonlinear characteristics in irregular waves. - (4)
- In regular waves, the optimal stiffness k
_{pto}* only depends on wave period, and the optimal damping coefficient c_{pto}* relies on wave period and stiffness k_{pto}, and is symmetric to the optimal stiffness k_{pto}*. However, in irregular waves, the optimal stiffness k_{pto}* depends on not only the wave period, but also the damping coefficient c_{pto}, and the optimal damping coefficient c_{pto}* is not symmetric to the optimal stiffness k_{pto}*. The optimal damping coefficient c_{pto}* in the optimal combination increases with increasing wave period (or peak wave period), and then decreases after reaching a peak value, whereas the optimal stiffness k_{pto}* in the optimal combination is usually negative and decreases monotonously. - (5)
- The surge motion of the joint could be neglected. The motion equation of the device can be reduced to a three-DOF model only with the consideration of heave motion of the joint and two pitch motions of the two rafts.
- (6)
- In regular waves, the peak capture width ratio η
_{cap}* obtained by using quadratic damping is slightly larger than that obtained by using linear damping; however, this advantage vanishes in irregular waves.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

_{c}is the rotary inertia about the centre of mass; a

_{uj}and b

_{uj}are the frequency-dependent added mass and radiation damping at mode u due to the motion of mode j (u = 1, 3, 5, 1′, 3′, 5′; j = 1, 3, 5, 1′, 3′, 5′), respectively; k

_{33}and k

_{3′3′}are the hydrostatic restoring stiffnesses of the fore and aft rafts caused by the their heave motions, respectively; k

_{55}and k

_{5′5′}are the hydrostatic restoring stiffnesses of the fore and aft rafts caused by the their pitch motions, respectively; Γ

_{j}(ω) is the complex excitation force coefficient at mode j (force per unit incident wave amplitude); the modulus and argument of Γ

_{j}(ω) are

**|**Γ

_{j}(ω)

**|**and ∠Γ

_{j}(ω), respectively; r

_{0}is half the distance of the mounting position between the top and bottom hydraulic cylinders, as shown in Figure 1b; and c

_{pto}and k

_{pto}are the equivalent damping coefficient and stiffness of the hydraulic cylinder, respectively.

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**Figure 1.**Schematic of a raft-type WEC consisting of two rafts and a PTO unit: (

**a**) Front view (

**b**) Initial position and (

**c**) Position after motion.

**Figure 3.**Identification results of retardation functions: (

**a**) H

_{11}(−iω); (

**b**) H

_{13}(−iω); (

**c**) H

_{22}(−iω); (

**d**) H

_{23}(−iω); (

**e**) H

_{33}(−iω); (

**f**) H

_{34}(−iω).

**Figure 4.**Performance of the device in regular waves (FD = Frequency domain TD = Time domain): (

**a**) Amplitude of G

_{v_rp}; (

**b**) P

_{ave_cap}; and (

**c**) η

_{cap}.

**Figure 5.**Influence of wave frequency in regular waves (FD = Frequency domain TD = Time domain): (

**a**) P

_{ave_cap}; and (

**b**) η

_{cap}.

**Figure 6.**Influence of mounting position in regular waves: (

**a**) Variation of η

_{cap}with r

_{0}/D; and (

**b**) Variation of r

_{0}*/D with T.

**Figure 7.**Influence of damping coefficient and stiffness in regular waves: (

**a**) 3D η

_{cap}in T = 6 s; (

**b**) η

_{cap}contour in T = 6 s; and (

**c**) Optimal damping coefficient and optimal stiffness in optimal combination.

**Figure 8.**Influence of surge and heave motions of joint in regular waves: (

**a**) Variation of Amplitude of G

_{v_rp}with kL; (

**b**) Variation of η

_{cap}with kL; and (

**c**) Variation of η

_{cap}with c

_{pto}.

**Figure 9.**Performance of the device with quadratic damping PTO in regular waves: (

**a**) Variation of ${\dot{\theta}}_{1}-{\dot{\theta}}_{2}$ and f

_{θ}

_{1}(t) with time; and (

**b**) Variation of η

_{cap}with β.

**Figure 10.**Variation of η

_{cap}* with wave period for the device with linear damping or quadratic damping.

**Figure 11.**Influence of mounting position in irregular waves: (

**a**) Variation of η

_{cap}with r

_{0}/D; and (

**b**) Variation of r

_{0}*/D with T

_{p}.

**Figure 12.**Influence of damping coefficient and stiffness in irregular waves: (

**a**) 3D η

_{cap}in T

_{p}= 6 s; (

**b**) η

_{cap}contour in T

_{p}= 6 s; and (

**c**) Optimal damping coefficient and optimal stiffness in optimal combination.

**Figure 13.**Influence of surge and heave motions of joint in irregular waves: (

**a**) Variation of η

_{cap}with T

_{p}; and (

**b**) Variation of η

_{cap}with c

_{pto}.

**Figure 14.**Influence of quadratic damping PTO in irregular waves: (

**a**) Variation of η

_{cap}with β; and (

**b**) Variation of η

_{cap}* with T

_{p}.

Parameter | Type-One | Type-Two | Type-Three | Type-Four |
---|---|---|---|---|

Raft length L (m) | 10 | 20 | 20 | 30 |

Raft diameter D (m) | 1 | 1 | 2 | 2 |

Raft space d_{0} (m) | 1 | 1 | 1 | 1 |

Raft density ρ_{0} (kg/m^{3}) | 512.5 | 512.5 | 512.5 | 512.5 |

Raft volume V/m^{3} | 7.8540 | 15.7080 | 62.8319 | 94.2478 |

Damping coefficient c_{pto} (kN/m/s) | 500 | 500 | 500 | 500 |

Mounting position r_{0} (m) | 0.5 | 0.5 | 1 | 1 |

Optimal ratio kL | 3.1163 | 3.6184 | 3.4987 | 3.6715 |

Optimal ratio of raft length to wavelength | 0.4960 | 0.5759 | 0.5568 | 0.5843 |

Resonant frequency of relative pitch velocity ω_{r}_{p} (rad/s) | 1.7485 | 1.3322 | 1.3100 | 1.0957 |

Resonant frequency of heave velocity ω_{h} (rad/s) | 1.4078 | 1.2354 | 1.2100 | 1.0179 |

Resonant frequency of surge velocity ω_{s} (rad/s) | 1.0144 | 0.7370 | 0.7322 | 0.6021 |

Amplitude of FRF of wave amplitude to relative pitch velocity at ω_{r}_{p} (rad/s/m) | 0.4281 | 0.3181 | 0.3054 | 0.1799 |

Amplitude of FRF of wave amplitude to heave velocity at ω_{h} (m/s/m) | 1.2623 | 1.5099 | 1.4735 | 1.2589 |

Amplitude of FRF of wave amplitude to surge velocity at ω_{s} (m/s/m) | 0.7881 | 0.5624 | 0.5655 | 0.4674 |

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## Share and Cite

**MDPI and ACS Style**

Liu, C.; Yang, Q.; Bao, G.
State-Space Approximation of Convolution Term in Time Domain Analysis of a Raft-Type Wave Energy Converter. *Energies* **2018**, *11*, 169.
https://doi.org/10.3390/en11010169

**AMA Style**

Liu C, Yang Q, Bao G.
State-Space Approximation of Convolution Term in Time Domain Analysis of a Raft-Type Wave Energy Converter. *Energies*. 2018; 11(1):169.
https://doi.org/10.3390/en11010169

**Chicago/Turabian Style**

Liu, Changhai, Qingjun Yang, and Gang Bao.
2018. "State-Space Approximation of Convolution Term in Time Domain Analysis of a Raft-Type Wave Energy Converter" *Energies* 11, no. 1: 169.
https://doi.org/10.3390/en11010169