*2.1. Model of Mooring Supported Turbine*

The mooring lines are assumed to continuously be in tension during operation. Therefore, this system can be modelled as an inverted flail pendulum in order to calculate its dynamics; Figure 3 provides the model for the three elements in flail pendulum. Equations of motion of the pendulum system can be derived using the following Lagrange's equation:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \theta\_i}\right) - \frac{\partial L}{\partial \theta\_i} = Q\_{i\nu} \tag{1}$$

where L = T − V is defined as the Lagrangian of the system, T is the kinetic energy and V the potential energy of system.

**Figure 3.** Multi-pendulum system with a finite number of rods and masses.

When an external force function *Q*<sup>i</sup> is not considered, then the point of anchor is chosen, which is suspension of the first pendulum as the origin, and angles are measured from the vertical line; as shown in Figure 3, the Lagrangian of the system can be written as:

$$\begin{aligned} L &= \frac{1}{2}(m\_1 + m\_2 + m\_3)l\_1^2 \dot{\theta}\_1^2 + \frac{1}{2}m\_2l\_2^2 \dot{\theta}\_2^2 + \frac{1}{2}m\_3l\_3^2 \dot{\theta}\_3^2 \\ &+ m\_2l\_1l\_2\theta\_1\theta\_2\cos(\theta\_1 - \theta\_2) + m\_3l\_1l\_3\theta\_1\theta\_3\cos(\theta\_1 - \theta\_3) \\ &+ (m\_1 + m\_2 + m\_3)gl\_1\cos\theta\_1 + m\_2gl\_2\cos\theta\_2 + m\_3gl\_3\cos\theta\_3. \end{aligned} \tag{2}$$

where *m*<sup>1</sup> is the lumped mass of three mooring lines at the connection node, *m*<sup>2</sup> represents the mass of buoy, *m*<sup>3</sup> represents the mass of the turbine. *l*1, *l*<sup>2</sup> and *l*<sup>3</sup> are the length of each segment. *θ*1, *θ*<sup>2</sup> and *θ*<sup>3</sup> are generalised coordinates as shown in Figure 3.

It is assumed that the turbine and the buoy are neutrally buoyant, so the potential energy terms in Equation (2) can be eliminated. The new Lagrangian of the system becomes:

$$L = \frac{1}{2}(m\_1 + m\_2 + m\_3)l\_1^2 \dot{\theta}\_1^2 + \frac{1}{2}m\_2l\_2^2 \dot{\theta}\_2^2$$

$$+ m\_2l\_1l\_2\theta\_1\theta\_2\cos(\theta\_1 - \theta\_2)$$

$$+ \frac{1}{2}m\_3l\_3^2\dot{\theta}\_3^2 + m\_3l\_1l\_3\dot{\theta}\_1\dot{\theta}\_3\cos(\theta\_1 - \theta\_3). \tag{3}$$

Substituting Equation (3) into Equation (1) yields the Euler–Lagrange differential equations of the system:

$$\begin{aligned}(m\_1 + m\_2 + m\_3)l\_1^2 \ddot{\theta}\_1 + m\_2 l\_1 l\_2 \ddot{\theta}\_2 \cos(\theta\_1 - \theta\_2) \\ -m\_2 l\_1 l\_2 \dot{\theta}\_2^2 \sin(\theta\_1 - \theta\_2) + m\_3 l\_1 l\_3 \ddot{\theta}\_3 \cos(\theta\_1 - \theta\_3) \\ -m\_3 l\_1 l\_3 \dot{\theta}\_3^2 \sin(\theta\_1 - \theta\_3) = Q\_1 \end{aligned} \tag{4}$$

$$\begin{aligned} m\_2 l\_2^2 \ddot{\theta}\_2 + m\_2 l\_1 l\_2 \ddot{\theta}\_1 \cos(\theta\_1 - \theta\_2) \\ -m\_2 l\_1 l\_2 \dot{\theta}\_1^2 \sin(\theta\_1 - \theta\_2) = Q\_2 \end{aligned} \tag{5}$$

$$\begin{aligned} m\_3 l\_3^2 \theta\_3 &+ m\_3 l\_1 l\_3 \theta\_1 \cos(\theta\_1 - \theta\_3) \\ -m\_3 l\_1 l\_3 \dot{\theta}\_1^2 \sin(\theta\_1 - \theta\_3) &= Q\_3 \end{aligned} \tag{6}$$

where *Q*1, *Q*<sup>2</sup> and *Q*<sup>3</sup> are now generalised moments with respect to *θ*1, *θ*<sup>2</sup> and *θ*3. In this case, *Q*<sup>3</sup> equates to the momentum thrust develop by a turbine loading, where this is operating under combined wave and currents conditions. *Q*<sup>2</sup> relates to the buoyant forces occurring on the floater, which are considered in the form of buoyancy and wave excitation forces. *Q*<sup>1</sup> will be obtained from the relationship between *Q*<sup>2</sup> and *Q*3. According to Anli and Ohlhoff [14,15], the generalised force can be obtained as:

$$Q\_k = \sum\_{i=1}^n F\_i \frac{\partial r\_i}{\partial q\_k},\tag{7}$$

where *Qk* is the Generalised force associated with the *kth* Euler-Lagrange differential equation, *F*<sup>i</sup> is the external force, *r*<sup>i</sup> is the position of the point of application and *qk* is the generalised coordinate.

Thus, substituting Equation (7) to the generalised coordinates with respect to *θ*1, *θ*<sup>2</sup> and *θ*3, the generalised moments for this system are given as:

$$Q\_1 = F\_3 l\_1 \cos \theta\_1 - F\_{2b} l\_1 \sin \theta\_1 + F\_2 l\_1 \cos \theta\_1 \tag{8}$$

$$Q\_2 = F\_2 l\_2 \cos \theta\_2 - F\_{2b} l\_2 \sin \theta\_2 \tag{9}$$

$$Q\_3 = F\_3 l\_3 \cos \theta\_3. \tag{10}$$

When the generalized moments are obtained, the Euler–Lagrange differential equations of the system can be solved with given initial conditions. Substituting Equations (8)–(10) into Equations (4)–(6) then dividing by *l*1, *l*<sup>2</sup> and *l*<sup>3</sup> yields:

$$\begin{aligned} (m\_1 + m\_2 + m\_3)l\_1 \ddot{\theta}\_1 + m\_2 l\_2 \ddot{\theta}\_2 \cos(\theta\_1 - \theta\_2) \\ -m\_2 l\_2 \dot{\theta}\_2^2 \sin(\theta\_1 - \theta\_2) + m\_3 l\_3 \ddot{\theta}\_3 \cos(\theta\_1 - \theta\_3) \\ -m\_3 l\_3 \dot{\theta}\_3^2 \sin(\theta\_1 - \theta\_3) \\ = F\_3 \cos \theta\_1 - F\_{2b} \sin \theta\_1 + F\_2 \cos \theta\_1 \end{aligned} \tag{11}$$

$$\begin{aligned} &m\_2l\_2\ddot{\theta}\_2 + m\_2l\_1\ddot{\theta}\_1\cos(\theta\_1 - \theta\_2) \\ &-m\_2l\_1\dot{\theta}\_1^2\sin(\theta\_1 - \theta\_2) \\ &= F\_2\cos\theta\_2 - F\_{2b}\sin\theta\_2 \end{aligned} \tag{12}$$

$$\begin{aligned} -m\_3 l\_3 \ddot{\theta}\_3 + m\_3 l\_1 \ddot{\theta}\_1 \cos(\theta\_1 - \theta\_3) \\ -m\_3 l\_1 \dot{\theta}\_1^2 \sin(\theta\_1 - \theta\_3) \\ = F\_3 \cos \theta\_3. \end{aligned} \tag{13}$$

The external forces on the buoy and turbine are depicted in Figure 4. In this paper, the wave excitation forces on the turbine are assumed to be ignored.

**Figure 4.** Forces on the system.

Wave excitation is considered to be a factor of the system. The buoy will be excited by the wave at a different magnitude based on its shape. The wave excitation in two directions can be defined as the exciting force and drift force, based Wu [16]. The thrust on the turbine can be obtained using an ESRU in-house BEMT code [11]. Modifications have been made in the original code owing to the relative velocity between the turbine and inflows:

$$\mathbf{U}\_{\cdot} = \begin{array}{c} \mathbf{u} - \mathbf{U}\_{\mathsf{T}\prime} \end{array} \tag{14}$$

where **u** is the inflow velocity, which is calculated using the wave–current interaction model; **U**<sup>T</sup> is the inertia velocity of the turbine, which is the velocity generated by the motions of turbine in waves. It can be calculated in vertical and horizontal directions as:

$$\mathbf{U}\_{\rm T} = \frac{\partial}{\partial \Delta t} (\mathbf{X}\_{\rm turbine}^{\rm it} - \mathbf{X}\_{\rm turbine}^{\rm it-1})\_{\prime} \tag{15}$$

where it is time step count number, **X**turbine is the turbine position and Δ*t* is the time difference. The buoy is assumed to be spherical. The drag force is also calculated from the BEMT code. The relative velocity between the buoy and inflows must be considered:

$$\mathbf{V}\_{\perp} = \mathbf{u} - \mathbf{U}\_{\mathbf{B}}.\tag{16}$$

**u** can be calculated using the wave–current model based on the coordinates of the buoy. **UB** is the inertia velocity of the buoy:

$$\mathbf{U}\_{\rm B} \quad = \begin{array}{c} \boldsymbol{\Theta} \\ \frac{\boldsymbol{\Theta}}{\boldsymbol{\Theta} \boldsymbol{\Delta} t} (\mathbf{X}\_{\rm buoy}^{\rm it} - \mathbf{X}\_{\rm buoy}^{\rm it-1}) . \end{array} \tag{17}$$

#### *2.2. Flow Diagram*

The original in-house BEMT code for wave–current environments is based on a rigid supported turbine, where the position of the turbine does not change with time. However, the coordinates of the mooring supported turbine are variable with time and the relative velocity must be calculated using the relative motion between the turbine and wave–current inflow. Figure 5 depicts the main process of the simulation. The process nodes with the dark background are works based on this study, which are different from the original BEMT.

**Figure 5.** Solution process.

There are some variable input parameters that can be defined by users for different mooring turbine devices. The state of motion for the system in the first time step is defined by the initial conditions. The loads on each mooring element and buoy are obtained as the external forces, whereas the load on the turbine is calculated using the BEMT code while modifying the relative velocity. After the external forces are determined, the governing equation for the "flail" system with a finite number of segments can be solved by an ordinary differential equations solver to obtain the new state of motion of the system. The new values are used as the initial conditions for the next time step until the simulation culminates.

For the wave–current model module, calculations of the horizontal and vertical particle velocities are related to the horizontal and vertical coordinates of the turbine in different wave theories. The coordinates of the turbine and buoy hinge nodes are *x*turbine, *y*turbine, *x*buoy and *y*buoy. Because the vertical particle velocities are varied along blades, the blade element coordinates are:

$$\mathbf{x}\_{\text{element}} = \mathbf{x}\_{\text{turbine}} + h\_{\text{element}} \mathbf{c} \mathbf{s} \boldsymbol{\phi} \tag{18}$$

$$y\_{\text{element}} = \\_\\_y\_{\text{turbine}} + h\_{\text{element}} \text{sir} \phi\_\prime \tag{19}$$

where *h*element is the element position on the blade and *φ* is the pitch angle of the turbine. Assume that the turbine is nearly horizontal during the operation and the pitch angle is 0 degree. The element coordinates become:

$$
\begin{array}{rcl}
\boldsymbol{\lambda}\boldsymbol{\pi}\_{\text{element}} & = & \boldsymbol{\lambda}\_{\text{turbine}} \\
& & & \\
\end{array}
\tag{20}
$$

$$\text{у}\_{\text{element}} = \, \|\, \text{y}\_{\text{turbine}} + h\_{\text{element}} \,\text{t} \,\tag{21}$$

Next, substitute these coordinates into wave models. Then, this module is modified to work for the mooring supported turbine.

The relative velocity modification of the inflow velocity must be considered in not only the BEMT equation but also the dynamic wake model and Morison equation.

Based on the methodology adopted, the dynamic inflow affects the BEMT model [17–19]. On a blade element bounded by radii *R*<sup>1</sup> and *R*<sup>2</sup> as shown in Figure 6, the momentum thrust equation depends on the time derivative of axial induction factor *a*˙ as:

$$\mathbf{d}F\_{\mathbf{A}} = \mathbf{2}\mathbf{u}a\mathbf{it} + \mathbf{u}m\_{\mathbf{A}}\mathbf{d}\_{\prime} \tag{22}$$

where *m*˙ is the mass flow through the intersecting fluid annulus, *a* is the axial induction factor, and *m*<sup>A</sup> is the apparent mass of the blade section.

**Figure 6.** Blade element bounded by radii *R*<sup>1</sup> and *R*2.

The mass flow through the annular element can be calculated as:

$$
\dot{m} \quad = \quad \rho \mathbf{u} (1 - a) \mathbf{d} A\_\prime \tag{23}
$$

where *ρ* is the density of water and d*A* = *π*(*R*<sup>2</sup> <sup>2</sup> − *<sup>R</sup>*<sup>2</sup> 1).

For a turbine of radius *R*, Tuckerman [20] suggests that the apparent mass acting on the rotor can be approximated by an enclosing fluid ellipsoid, which can be expressed, through the use of potential flow theory, as:

$$m\_{\rm A\_{\rm A}} = -8/3\rho R^3. \tag{24}$$

Substituting Equations (23) and (24) into Equation (22) and dividing each term by *π*, *ρ*, **u**, and d*A* and multiplying by 2, then replacing the in flow velocity with the relative velocity *Ux* which is the horizontal component of relative velocity **U**. The final form of the unsteady thrust coefficient for an annulus can be obtained as:

$$\mathcal{C}\_{\rm FA} = -4a(1-a) + \frac{16}{3\pi Ul\_{\rm x}} \frac{(R\_2^3 - R\_1^3)}{(R\_2^2 - R\_1^2)}\hbar. \tag{25}$$

Substituting thrust coefficient *C*<sup>T</sup> = 4*a*(1 − *a*) into Equation (25) gives:

$$\mathcal{C}\_{\rm FA} = \,^\*\mathcal{C}\_{\rm T} + \frac{16}{3\pi dL\_{\rm x}} \frac{(R\_2^3 - R\_1^3)}{(R\_2^2 - R\_1^2)}\hbar. \tag{26}$$

The last term on the right-hand side of Equation (26) can be used to calculate the additional force from the dynamic wake effects.

The inertial force caused by fluid acceleration, which is the added mass around a rotating blade section can be expressed as Morison equation, as presented by Buckland [21] and Chapman [22]. The inertial force per unit length, d*l* in the wave propagation direction on a submerged body can be calculated as:

$$\mathrm{d}F\_{\mathrm{in}} = -\rho \mathrm{C\_{m}} A \frac{\partial \mathbf{u}}{\partial t} \mathrm{d}l\_{\prime} \tag{27}$$

where *A* is the cross horizontal sectional area parallel to the flow and *C*<sup>m</sup> is the inertia coefficient, which is expressed as:

$$\mathbb{C}\_{\text{M}} = \begin{array}{c} \text{1} + \text{C}\_{\text{A}} \end{array} = \text{1} + \frac{M\_{\text{A}}}{\rho A \text{d} \text{l}^{\prime}} \tag{28}$$

where *M*<sup>A</sup> is the added mass for a blade element.

For blade elements, the added mass in axial and tangential directions can be approximated with that of a fixed pitched plate as per Theodorsen's theory [23]:

$$M\_{\rm A,axial} = -\rho \pi (\frac{c \sin \beta}{2})^2 \text{d}l \tag{29}$$

$$M\_{\rm A,tan} = -\rho \pi (\frac{\alpha \cos \beta}{2})^2 \text{d}l,\tag{30}$$

which are the masses of the enclosing fluid cylinders with radii *r* of half the vertical and horizontal chord components *c* of the respective blade sections with section angle *β* [24].

Substituting Equations (28)–(30) into Equation (27) and plugging the components of relative velocity **U** into the equation gives the equations for the inertia forces in the axial and tangential directions for a blade element as:

$$\mathrm{d}F\_{\mathrm{in,axial}} = \rho (1 + \frac{\pi((\mathrm{csin}\beta)/2)^2}{A\_a}) A\_a \frac{\partial \mathrm{d}I\_x}{\partial t} \mathrm{d}r \tag{31}$$

$$\mathrm{d}F\_{\mathrm{in,tan}} = \rho (1 + \frac{\pi((\mathrm{ccos}\beta)/2)^2}{A\_{\mathrm{d}}}) A\_{\mathrm{d}} \frac{\partial L\_{\mathrm{y}}}{\partial t} \mathrm{d}r,\tag{32}$$

where *Aα* is the cross-sectional area of the airfoil at the blade section.

When the external forces based on the initial conditions for the first time step are calculated using the BEMT equations with the wave–current model, dynamic wake model, and Morison equations, the system Lagrange equation solver begins to solve the differential equations of motion for the mooring supported turbine. Next, the new values and angular

velocities ˙ *θ<sup>i</sup>* for each segment are obtained to serve as the new initial conditions for the next time step. This loop continues until the time step reaches the end time of the simulation.

#### **3. Initial Conditions and System Parameters**

This study focuses primarily on external forces, which are buoyancy and wave–currentcoupled forces. Sea states with regular and irregular waves were investigated. The wave data were collected by the UK Offshore Operators Association and were provided by British Oceanographic Data Center [25]. In this study, parameters of a 1 MW turbine and mooring system were applied to deep water. Parameters given below were fixed to control the number of variables. The material for the mooring line was chosen to be Dyneema, whose density is close to that of water. There were three sets of system parameters in each model, two of them were Dyneema mooring and one was steel mooring.
