*2.3. Mooring Line System*

A uniformly distributed four-line spread mooring system is applied for the integrated system. The fairleads are located at the outside heave plates, at a depth of 35 m below the sea water level and at a radius of 62 m from the cage center. The anchors are located at a water depth of 200 m and at a radius of 891.6 m from the cage center. The mooring lines are made from a studless grade 4 chain, with a chain nominal diameter of 0.153 m [19]. The unstretched length of each mooring line is 880 m, a submerged unit mass of 447 kg/m, an axial stiffness of 2.1 × 106 kN and a catalogue breaking strength of 2 × <sup>10</sup><sup>4</sup> kN [19]. Note that the length of 880 m is determined after some study on the mooring system of the DeepCWind semisubmersible platform at the same water depth [20]. All hydrodynamic coefficients are determined according to the chain's nominal diameter. The drag and the added mass coefficients are assumed based on DNVOS-E301 [21]. Tables 3 and 4 list the configurations and properties of the mooring line system, respectively. Figure 5 shows the mooring line system configuration.

**Table 3.** Configurations of mooring line system.


**Table 4.** Properties of mooring line system.


#### *2.4. Wind Turbine System*

In this work, the upper wind turbine of the integrated FOWTs is selected as the NREL offshore 5 MW baseline wind turbine [22]. For the convenience of understanding, the properties of the wind turbine will be described briefly in this section. More details can be found in Ref. [22].

To support the preliminary study aimed at assessing the integrated system composed of a floating offshore wind turbine and a fishing cage, the NREL offshore 5 MW baseline wind turbine, a representative utility-scale multi-megawatt turbine, is applied in this work. This wind turbine is a conventional three-bladed upwind variable-speed variable-pitch turbine. Table 5 shows the main properties of the NREL offshore 5 MW baseline wind turbine. More details can be referred to Refs. [22,23].

**Figure 5.** Top view of the mooring line system configuration.

**Table 5.** Main properties of the NREL offshore 5 MW wind turbine.


The tower base is coincident with the top of the central column of the fishing cage and is located at an elevation of 10 m above the still water level. The tower top is coincident with the yaw bearing and is located at an elevation of 87.6 m above the still water level. This tower-top elevation (90 m above the still water level) is consistent with the land-based version of the NREL 5 MW baseline wind turbine, as described in Ref. [23]. These properties are all relative to the undisplaced position of the fishing cage.

The diameter at the tower base for the NREL 5-MW offshore baseline wind turbine is 6.5 m, which matches the diameter of the central column of the fishing cage. The tower–base thickness, top diameter and thickness are 0.027, 3.87 and 0.019 m, respectively. The effective mechanical steel properties of the tower are determined according to the DOWEC study [24]. Young's modulus and shear modulus are set to 210 GPa and 80.8 GPa, respectively. The effective density of the steel is set as 8500 kg/m3, which is meant to be an increase above steel's typical value of 7850 kg/m3, to consider paint, bolts, welds and flanges that are not included in the tower thickness data [15]. The tower radius and thickness are assumed to be linearly tapered from the tower base to the tower top.

The overall tower mass is 249,718 kg and is centered at 43.4 m along the tower centerline above the still water level. This is obtained from the overall tower length of 77.6 m. A structural damping ratio of 1% critical is specified for all modes of the isolated tower, which corresponds to the values used in the DOWEC study [24]. Table 6 gives the undistributed tower properties.

**Table 6.** Undistributed tower properties.


#### **3. Dynamics Modeling**

In the present work, the fully coupled aero-hydro-elastic-servo-mooring model of the integrated FOWT is established through a coupling framework based on FAST [25] and ANSYS/AQWA [26]. The upper wind turbine is modeled on FAST. The turbine dynamic responses are represented by external forces and added mass within each invocation, then are passed to the AQWA solver to be combined with the hydrodynamic loads and mooring line forces to calculate the dynamic responses of the aquaculture cage. For the convenience of understanding, the subsequent sections will briefly describe the basic theories applied in FAST and AQWA for structural modeling. Ref. [27] gives more implementation details.

Figure 6 shows the modeling overview of the integrated system through FAST and AQWA in this work.

**Figure 6.** Modeling overview through FAST and AQWA.

#### *3.1. Aerodynamics and Structural Dynamics*

FAST is an aero-hydro-elastic-servo wind turbine simulation tool developed by the National Renewable Energy Laboratory (NREL) [25]. The aerodynamic loads on the blades are calculated by the blade element momentum (BEM) theory within the AeroDyn module [28,29]. The rotor thrust (*T*) and torque (*M*) are calculated as:

$$\begin{cases} \text{d}T = \frac{1}{2}\rho\_{\text{air}}\mathcal{W}^2 c(\mathcal{C}\_l \cos \varphi + \mathcal{C}\_d \sin \varphi)\text{d}r\\ \text{d}M = \frac{1}{2}\rho\_{\text{air}}\mathcal{W}^2 c(\mathcal{C}\_l \sin \varphi - \mathcal{C}\_d \cos \varphi)r\text{d}r \end{cases} \tag{2}$$

where *ρ*air is the air density, *Cl* and *Cd* are, respectively, the lift and drag coefficients of the airfoil, *W* is the absolute wind speed, *c* is the chord length of a blade element, *ϕ* is the inflow angle and *r* is the local radius of a blade element.

More detailed information on the aerodynamic calculation can be found in Refs. [28,29]. When the aerodynamic loads are calculated, they are then imported into the electrodynamic

module of FAST to resolve the motion equations of the wind turbine. Kane's kinetic approach is used, which is defined as

$$\mathbf{F}\_r^\* + \mathbf{F}\_r = \mathbf{0} \tag{3}$$

where **F**∗ *<sup>r</sup>* and **F***<sup>r</sup>* are the generalized inertia force vector and the generalized active force vector, respectively.

The generalized inertia forces are composed of the inertia forces of the nacelle, tower, hub and blades. The generalized active forces consist of aerodynamic loads **F**aero, elastic restoring forces **F**elas, gravity **F**grav and damping forces:

$$\mathbf{F}\_r = \mathbf{F}\_{\text{zero}} + \mathbf{F}\_{\text{elas}} + \mathbf{F}\_{\text{grav}} + \mathbf{F}\_{\text{darmp}} \tag{4}$$

The wind turbine is modelled as a multi-body system including rigid bodies and flexible bodies. The hub and nacelle are modelled as rigid bodies. The tower and blades are treated as flexible bodies. The generalized inertia force of a rigid body is represented by the same formula. Further information on the motion equations can be referred to in Refs. [30,31].

#### *3.2. Mooring Line Dynamics*

The finite element approach within AQWA is applied to consider the dynamic effects of mooring lines. Each line is discretized into several finite elements, and the mass of each element is concentrated into a corresponding node, as shown in Figure 7. In the figure, *Sj* is the length of an unstretched line between the anchor and the *j*th node, and *De* represents the local segment diameter of one line. Each line is treated as a chain of Morison elements subjected to various external forces. According to Ref. [32], the motion equation of each line element is defined as:

$$\begin{cases} \frac{\partial \mathbf{T}}{\partial S\_v} + \frac{\partial \mathbf{V}}{\partial S\_v} + \mathbf{w} + \mathbf{F}\_{\mathbf{h}} = m\_e \frac{\partial^2 \mathbf{R}}{\partial t^2} \\ \frac{\partial \mathbf{M}}{\partial S\_v} + \frac{\partial \mathbf{R}}{\partial S\_v} \times \mathbf{V} = -\mathbf{q} \end{cases} \tag{5}$$

where **T** and **V** are the tension force and shear force vectors of the first element node, respectively; **R** is the position vector of the first element node; *S*<sup>e</sup> is the length of an unstretched element; **w** and **F**<sup>h</sup> represent the weight and hydrodynamic load vectors per unit element length, respectively; *m*e is the mass per unit length; **M** is the bending moment vector of the first element node; and **q** is the distributed moment load per unit length.

The bending moment and tension force vectors are calculated by:

$$\begin{cases} \mathbf{M} = EI \cdot \frac{\partial \mathbf{R}}{\partial \mathbf{S}\_{\varepsilon}} \times \frac{\partial^2 \mathbf{R}}{\partial \mathbf{S}\_{\varepsilon}^2} \\ \mathbf{T} = EA \cdot \varepsilon \end{cases} \tag{6}$$

where *EI* and *EA* are, respectively, the bending stiffness and axial stiffness of the line, and *ε* is the strain of the line.

To ensure a unique solution to Equation (5), pinned connection boundary constraints (Equation (7)) are imposed on the top and bottom ends:

$$\begin{cases} \mathbf{R}(0) = \mathbf{P}\_{\text{bot}} \mathbf{R}(L) = \mathbf{P}\_{\text{top}}\\ \frac{\partial^2 \mathbf{R}(0)}{\partial S\_v^2} = 0, \frac{\partial^2 \mathbf{R}(L)}{\partial S\_v^2} = 0 \end{cases} \tag{7}$$

where **P**bot and **P**top are the position vectors of attachment points, and *L* is the length of the unstretched line.

**Figure 7.** Schematic diagram of the dynamic model of mooring lines.

#### *3.3. Hydrodynamics of Aquaculture Cage*

The hydrodynamic loads on the aquaculture cage are obtained based on the frequencydependent hydrodynamic coefficients, including the added mass, radiation damping and mooring line restoring forces. These coefficients can be obtained through a frequency domain analysis in AQWA.

#### 3.3.1. Hydrodynamics of Cage Support Structures

The hydrostatic and hydrodynamic analyses are performed by the panel method to solve the radiation and diffraction problems for the interaction of surface waves with main structures in the frequency domain. Based on boundary conditions, the velocity potentials can be solved on the mean body position, and the pressure can be obtained from the linear Bernoulli's equation [2]. The governing equation for the motion of a floating body in six degrees of freedom (DOFs) is expressed by [2]:

$$\left(-\omega^2(M+A(\omega)) + i\omega B(\omega) + \mathbb{C}\right)\mathbb{f}(\omega) = F(\omega) \tag{8}$$

where *M* is the mass matrix, *ω* is the angular frequency, *A*(*ω*) is the added mass matrix, *B*(*ω*) is the damping matrix, *C* is the hydrostatic stiffness matrix, ζ(*ω*) is the dynamic response vector and *F*(*ω*) is the dynamic load vector.

When using the potential flow theory in AQWA, the diffraction panel elements model does not include the effect of the viscous drag force. Instead, the Morison elements are usually applied for slender structures. Therefore, the cross-bracings that act as a support role are modeled through a series of Morison elements. According to the setting in Ref. [15], the drag coefficient for the cross-bracings in this work is set to 0.63. Figure 8 shows the panel model of the aquaculture cage modeled in AQWA.

#### 3.3.2. Net Hydrodynamics

In addition to the cage support structures, the nets of the fishing cage also need to be properly modeled. The nets are flexible and undergo various degrees of deformation depending on the wave and current conditions. The hydrodynamic force on nets and the coupling effect between nets and cage support structures have strong nonlinearity. Thus, a nonlinear model is usually needed to model the nets, and examples of flexible nets modeling can be found in Refs. [33–35].

Compared with the traditional flexible collar fishing cage, this semisubmersible rigid fishing cage has rigidity and larger supporting structures, so it is expected that the nets have less influence on the fishing cage. Therefore, in this preliminary study, the equivalent nets are modeled by Morison elements, which are considered to be stiff and rigidly connected to the cage to work as a whole rigid body [14,18]. Thus, the relative motions between the nets and the cage are neglected. It should be noted that this simplification may overestimate the hydrodynamic forces and viscous effects of the nets because deformation is not considered [18]. Nevertheless, it is still a good start to study the global dynamic responses of the integrated system with the inclusion of the nets modeled by the simplified rigid model.

**Figure 8.** Panel model of the aquaculture cage in AQWA.

Hydrodynamic forces on the nets may be divided into three components: inertia force, drag force and lift force. The inertia force and drag force for the nets can be calculated by using the Morison equation. The lift force can be calculated by the same drag force term in the Morison equation, but the lift coefficient, instead of the drag coefficient, must be introduced [36]. Considering that the mass of the nets accounts for a small proportion of the whole cage mass, the inertia force of the nets may be ignored [5,37,38]. Moreover, when the nets are attached to the cage sides and bottom, the wave and flow directions are almost 0◦ or 90◦ to the normal direction of the net plane where the lift force is almost zero [15,39]. Therefore, in this paper, only the drag force on the nets is considered.

The hydrodynamic drag force of the nets is closely related to the net solidity ratio. In Refs. [36,40], a drag coefficient of nets, *C*D, was estimated using the following equation:

$$\mathcal{L}\_{\rm D} = 0.04 + \left(-0.04 + 0.33S\_{\rm n} + 6.54S\_{\rm n}^2 - 4.88S\_{\rm n}^3\right)\cos\theta\tag{9}$$

where *S*<sup>n</sup> is the net solidity ratio, *θ* is the angle between the inflow direction and the net normal.

In this work, it is assumed that the angle of inflow to the net normal is 0◦ for both wave inflow to the side nets and vertical motion inflow to the bottom nets. The drag coefficient *C*<sup>D</sup> = 0.2 is determined according to the selected solidity ratio, which is applied for the equivalent nets at both the side and bottom. Note that the possible influence of biofouling on the drag coefficient is neglected in this work, which requires more accurate biofouling hydrodynamic characteristics. This issue needs to be solved by performing experiments in the future.

#### **4. Control System**

As studied by Jonkman [22], the operation regions of a wind turbine are divided into five parts: 1, 1.5, 2, 2.5, and 3, as illustrated in Figure 9. In Region 1, the wind speed is lower than the cut-in wind speed, thus, no electrical power is output. In this region, the rotor is accelerating for a start-up. Region 1.5 is a linear transition between Region 1 and Region 2. In Region 2, the controller adjusts the generator torque according to the generator speed while keeping the blade pitch angles at the optimal value. In Region 3, the blade pitch angles are tuned by a collective variable-pitch strategy to maintain the rated generator speed. The demanded blade pitch angles are provided through a gain-scheduled proportional-integral (PI) controller, depending on the speed error between the filtered and the rated generator speeds. Moreover, in order to resist negative damping in the rotor speed response, the generator-torque control law in Region 3 is set to a constant generator-torque control. The constant generator torque is set to the rated value, 43,093.55 Nm [23]. Region 2.5 is a smooth transition region between Regions 2 and 3. This region is also applied to limit tip speed and noise emissions.

**Figure 9.** Relationship between generator torque and generator speed for the baseline control system.

The blade pitch control system can be represented by the following equation of motion [41]:

$$\underbrace{\left[I\_{\text{driver}}+\frac{1}{\Omega\_{0}}\left(-\frac{\partial P}{\partial \theta}\right)N\_{\text{gez}}K\_{\text{D}}\right]}\_{M\_{\text{\theta}}}\dot{\eta} + \underbrace{\left[\frac{1}{\Omega\_{0}}(-\frac{\partial P}{\partial \theta})N\_{\text{gez}}K\_{\text{P}} - \frac{P\_{0}}{\Omega\_{0}^{2}}\right]}\_{C\_{\text{\theta}}}\dot{\eta} + \underbrace{\left[\frac{1}{\Omega\_{0}}(-\frac{\partial P}{\partial \theta})N\_{\text{gez}}K\_{\text{I}}\right]}\_{K\_{\text{\theta}}}\dot{\eta} = 0 \tag{10}$$

where *I*drivetrain is the drivetrain inertia cast to the low-speed shaft; *N*gear is the gearbox ratio; Ω<sup>0</sup> is the rated rotor rotational speed; *P*<sup>0</sup> is the rated mechanical power; *∂*P/*∂θ* is the sensitivity of aerodynamic power to the rotor collective blade pitch angle; *K*P, *K*<sup>I</sup> and *<sup>K</sup>*<sup>D</sup> are the blade pitch controller proportional, integral and derivative gains, respectively; . *ϕ* = ΔΩ is the rotor speed error.

The rotor speed error responds as a 1-DOF dynamic system with natural frequency *ωϕ*<sup>n</sup> and damping ratio *ζϕ*: <sup>⎧</sup>

$$\begin{cases} \omega\_{\mathfrak{q}\mathfrak{n}} = \sqrt{\frac{K\_{\mathfrak{p}}}{M\_{\mathfrak{q}}}}\\ \mathcal{L}\_{\mathfrak{q}} = \frac{\mathcal{C}\_{\mathfrak{q}}}{2M\_{\mathfrak{q}}\omega\_{\mathfrak{q}\mathfrak{n}}} \end{cases} \tag{11}$$

when designing a blade pitch controller, the PI gains can be calculated by ignoring the derivative gain and negative damping term [42]:

$$\begin{cases} \begin{cases} \ K\_{\rm P} = \frac{2I\_{\rm divivetrain} \, \Omega\_{\rm Q} \mathbb{I}\_{\rm gv} \omega\_{\rm gen}}{N\_{\rm gv} \left( -\frac{\partial P}{\partial P} \right)}\\\ K\_{\rm I} = \frac{I\_{\rm divivetrain} \, \Omega\_{\rm Q} \omega\_{\rm gen}^2}{N\_{\rm gvar} \left( -\frac{\partial P}{\partial P} \right)} \end{cases} \tag{12}$$

According to the study by Larsen [43], the smallest natural frequency of the blade pitch controller must be less than the smallest critical natural frequency of the support structure to ensure that the support structure motions of a floating offshore wind turbine with active pitch-to-feather control remain positively damped. Therefore, the blade pitch controller's natural frequency of 0.032 Hz (which is below the cage-pitch natural frequency of about 0.06 Hz) and a damping ratio of 0.7 [23] is used in this paper. The resulting proportional gain and integral gain are 0.006275604 s and 0.0008965149, respectively.

#### **5. Results and Discussions**

### *5.1. Free Decay Test*

Free decay tests in six degrees of freedom (DOFs) are performed to determine the natural frequencies of the integrated system. The test results are shown in Figure 10 and the natural frequencies are listed in Table 7.

**Figure 10.** Free decay motions of the integrated system in six DOFs.



It should be noted that because of the coupling of different DOFs, the integrated system will always generate some motions in other DOFs when performing free decay tests, especially the surge-pitch DOFs and sway-roll DOFs. For the free decay in the surge DOF (the initial surge displacement is set to 7 m), the largest pitch amplitude is almost 0.2◦, while for the free decay in the pitch DOF (the initial pitch displacement is set to 5◦), the largest surge amplitude is almost 3 m. Similar results are observed for the free decay tests in the sway and roll DOFs. These values are not so large when compared to those in the DOFs that are being tested. Thus, the results obtained from the free decay tests are considered reasonable.

#### *5.2. Uniform Wind with Regular and Irregular Waves Test*

The uniform wind with regular waves is firstly tested. For the uniform wind, a constant wind speed of 18 m/s, without considering wind shear, is applied. The regular waves are generated using the Airy wave theory. The significant wave height and peak period are set as 4.1 m and 10.5 s, respectively [19]. The simulation time is 700 s, and the results corresponding to the first 100 s are omitted. Figure 11 shows the time series of cage motions in six DOFs under the uniform wind with regular waves condition. It is found that the surge and have motions are more significant than the sway motion. For the rotational motions, the roll and yaw motions are close to zero. With the main action of aerodynamic loads, a mean pitch rotation of 0.48◦ is induced.

**Figure 11.** Time histories of cage motions under the uniform wind with regular waves. (**a**) Translational motion. (**b**) Rotational motion.

For the irregular waves, the sea state is generated by the JONSWAP spectrum. Figure 12 shows the time series of cage motions in six DOFs under the uniform wind with irregular waves condition. For the translational motions, the surge motion is more significant than the heave and sway motions. A mean surge motion of approximately 2.7 m is mainly caused by the aerodynamic loads. For the rotational motions, the roll and yaw motions are close to zero. With the main action of aerodynamic loads, a mean pitch rotation of 0.18◦ is induced. Additionally, it is expected that no cage-pitch resonance is found because the blade pitch controller's natural frequency is below the cage-pitch natural frequency. Overall, it is observed that, compared with the regular wave condition, the irregular condition causes more significant cage motions.

(**b**)

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**Figure 12.** Time histories of cage motions under the uniform wind with irregular waves. (**a**) Translational motion. (**b**) Rotational motion.

Figure 13 illustrates the time series of blade pitch angles and wind turbine power output under the uniform wind with irregular waves condition. Due to the blade pitch angle variation tuned by the blade pitch controller, a mean generator power of 5,000 kW is output, with a small stand deviation of approximately 57 kW. The mean value of the blade

pitch angle is approximately 15◦ and there is only slight oscillation. This is attributed to the stability of the cage pitch motion, as shown in Figure 12.

**Figure 13.** Time histories of pitch angles and power output of the wind turbine under the uniform wind with irregular wave.
