A study on the platform-pitching vibration of floating offshore wind turbines based on classical control theory

Abstract

One of the issues of floating offshore wind turbines is the platform-pitching vibration generated by the blade pitch angle motion of the variable speed control. This study investigated the platform-pitching vibration based on the classical control theory using a transfer function between the generator speed and the nacelle pitch angle. This study also investigated the impact of the floating platform vibration control, which can suppress the vibration by adjusting the blade pitch angle according to the nacelle pitch angle, by using a transfer function to which floating platform vibration control is added. The stabilities of these transfer functions were determined using the Nyquist stability criterion, and the impact of the floating platform vibration control parameters was investigated using Bode diagrams.

Keywords

Offshore wind turbine floating wind turbine control transfer function classical control theory

Introduction

Wind power generation is becoming increasingly popular globally, and it is predicted that the installation of wind power will proceed continuously. Most wind turbines have been installed onshore; however, offshore wind turbines are also becoming popular. The WindEurope (2017) reported that the cumulative capacity of offshore wind turbines had reached 18.8 GW by the end of 2017. Almost all of these offshore wind turbines are of the bottom-fixed type. However, floating offshore wind turbines (FOWTs) are expected to be more widely used because they can be installed far from the coast and can utilize stronger and stabler wind power compared with turbines installed near the coast.

FOWTs can utilize wind power effectively; however, a well-known issue is that floating platform-pitching vibration occurs at rated operation conditions. Larsen and Hanson (2007) reported a low-frequency vibration when a FOWT operated within the rated conditions. This vibration is caused by variable speed control (VSC). As the platform-pitching vibration increases fatigue loads, the vibration in FOWTs needs to be suppressed.

Several studies have been conducted to suppress the platform-pitching vibrations. Skaare et al. (2007) proposed a blade pitch angle (BPA) control to suppress the platform-pitching vibrations and showed that it could stabilize the platform posture by generating a damping property for the platform-pitching motion. Larsen and Hanson (2007) proposed some countermeasures against the platform-pitching vibrations and stated that a VSC gain-tuning method applied to bottom-fixed wind turbines reduces the output power fluctuation, but cannot suppress the platform-pitching vibrations, and that a method based on stall control can stabilize both output power and pitching motion. Jonkman (2008) proposed three control methods to reduce platform-pitching vibrations: BPA control based on nacelle acceleration, stall control, and VSC with low control gains. The simulation results showed that the third control method was better but had limited effectiveness. Christiansen et al. (2011) focused on FOWTs with spar platforms and proposed a linear quadratic regulator (LQR) control based on an observer that predicts rotor angle and tower deflection. The proposed method controls the BPA and generator torque, and is able to not only reduce damage-equivalent fatigue loads but also stabilize output power. Guo et al. (2012) proposed an expert proportional–integral–derivative (PID) control that changes its gain according to the difference between the measured and target signals. The expert PID also has a function of adjusting the BPAs independently based on the direct-quadrature transform of the rotor angle. They show that it can stabilize both the output power and the platform-pitching motion. Betti et al. (2012) devised a two-dimensional FOWT model with a spar platform and proposed an H∞ control to stabilize the platform-pitching motion. Their proposed method suppressed the platform-pitching vibrations. Christiansen et al. (2012) proposed an LQR control that minimizes the rotor thrust force by reducing the rated speed more than the rated wind speed. Although it suppresses the platform-pitching vibrations, it also increases the roll angle of the floating platform. Bagherieh et al. (2014) proposed two types of control, an input/output feedback control and a sliding control, to reduce fluctuations of both the rotor speed and the platform-pitching motion, and showed that the latter could satisfy both fluctuations. Kakuya et al. (2018) reported a BPA control for suppressing platform-pitching vibrations (floating platform vibration control [FVC]) that has a mode-change function based on the output power. We also showed experimental results by using a full-scale 2-MW-demonstration FOWT with a hybrid spar-floating platform, and showed that FVC can suppress the platform-pitching vibrations not only at, but also below, rated operating conditions.

These studies are effective in suppressing the platform-pitching vibration. However, to our knowledge, no report has analyzed the causes of the platform-pitching vibration itself or clarified the difference in the stability of the floating platform posture with and without a control to suppress the platform-pitching vibration. In this work, we analyze floating platform-pitching vibration behaviors using the transfer functions of classical control theory. This report is organized as follows. Section “Target wind turbines” describes two target wind turbines: bottom-fixed and FOWTs. In section “Transfer function for floating platform-pitching vibration of FOWT,” we propose a transfer function that describes the floating platform-pitching vibration behavior, whose input and output are the generator speed and the floating platform-pitching angle (the nacelle pitch angle), respectively. The proposed transfer function consists of the FOWT system and a controller. In section “Stability criterion of the proposed transfer function,” we investigate the stabilities of the proposed transfer function by applying the Nyquist stability criterion. In section “Stabilization by FVC,” we investigate the impacts of adding FVC to the proposed transfer function, and then investigate how the FVC parameters change the stability of the proposed transfer function.

Target wind turbines

The simulation model and the specifications of the target FOWT are shown in Figure 1 and Table 1, respectively. This FOWT was constructed originally for the GOTO Floating Offshore Wind Turbine Demonstration Project, funded by the Ministry of Environment (MOE), Japan. The spar-type floating platform has a hybrid station-keeping structure: the upper side is made of steel and the lower side is made of prestressed concrete. The simulation model and the specifications of the bottom-fixed wind turbine, used for comparison with the FOWT, are shown in Figure 2 and Table 2, respectively. Basically, both rotor nacelle assemblies are the same and the main difference is the type of the station-keeping structure. The Bladed^® program was used to build these simulation models.

Figure 1.

Floating offshore wind turbine originally constructed in the Ministry of Environment demonstration project (Utsunomiya et al., 2015).

Table 1.

Specifications of the floating offshore wind turbine (Utsunomiya et al., 2015).

Rated power	2000 kW
Rotor position	Downwind
Rotor diameter	80 m
Floating platform	Hybrid spar type
Mooring	Catenary, 3 chains
Hub height	56 m
Rated wind speed	12 m/s

Figure 2.

Bottom-fixed wind turbine.

Table 2.

Specifications of the bottom-fixed wind turbine.

Rated power	2000 kW
Rotor position	Downwind
Rotor diameter	80 m
Hub height	65.4 m
Rated wind speed	12 m/s

Transfer function for floating platform-pitching vibration of FOWT

In this section, we propose a transfer function that describes the relation between the generator speed and the floating platform-pitching angle. We define the nacelle pitch angle as being the same as the floating platform-pitching angle below.

Whole system

The BPA acts to keep the rotor (generator) speed rated, and this causes the floating platform-pitching vibration. The VSC generates the BPA motion and determines the BPA demand based on the measured generator speed. The BPA demand by the VSC changes the thrust force applied to the rotor by the wind force; therefore, the VSC causes the nacelle pitch angle to vibrate.

We define the transform function below based on these characteristics. The input and output of this function are generator speed and nacelle pitch angle, respectively. The proposed transfer function is derived based on the following assumptions.

(a) Assume a situation in which the variance in the nacelle pitch angle changes the generator speed under the operation condition to keep the generator speed constant.

(b) Assume a wind speed condition in which floating platform-pitching vibration occurs. In this work, the wind speed is 14 m/s. This defines a representative point of generator speed and BPA for defining the transfer function.

(c) The transfer function describes the characteristic that the change in nacelle pitch angle is generated by the change in thrust force; the latter change is generated by the variance in generator speed caused by the change in nacelle pitch angle.

(d) Each component of the proposed transfer function is not time-dependent (is constant).

(e) The position of the rotation center of the nacelle pitch angle does not change (is constant).

(f) The characteristics of each component contained in the proposed transfer function are determined by the simulation results.

The proposed transfer function describes the relation between generator speed and the nacelle pitch angle (Figure 3). The transfer function consists of transfer functions that express the following relations and characteristics.

The relation between thrust force and the nacelle pitch angle $G_{W T 1} (s)$

The relation between generator speed and thrust force $G_{W T 2} (s)$

The relation between BPA and thrust force $G_{W T 3} (s)$

The relation between the nacelle pitch angle and generator speed $G_{W T 4} (s)$

Characteristic of VSC $G_{C 1} (s)$

Figure 3.

Block diagram of the generator speed and nacelle pitch angle.

$G_{W T 2} (s)$ determines one of the thrust forces depending on the difference in generator speed, whereas $G_{W T 3} (s)$ determines the other thrust force depending on the output of $G_{C 1} (s)$ , which is generated depending on the difference in generator speed. The final thrust force is determined after the two thrust forces are added. Then, $G_{W T 1} (s)$ determines the nacelle pitch angle depending on the final thrust force. After differentiating the nacelle pitch angle, $G_{W T 4} (s)$ determines the variance of the generator speed caused by the nacelle pitch angular speed. Finally, the proposed transfer function feeds the variance of the generator speed back to the input signal (generator speed).

The abovementioned relations are determined by setting the BPA to the value of the representative point; the BPA value is derived from results of steady-state simulations based on the representative wind speed.

Components of the proposed transfer function

In this section, we describe the characteristics of components of the proposed transfer function.

Relation between thrust force and nacelle pitch angle

We assume that the pitching vibration of a representative point attached to the nacelle around the rotation center is a first modal vibration, and that this vibration has a second-order response characteristic. The representative point of the nacelle coincides with the intersection of the tower and rotor axes

F_{T} L = M \overset{\cdot\cdot}{θ} + D \dot{θ} + K θ

(1)

G_{W T 1} (s) = \frac{θ (s)}{F_{T} (s)} = \frac{L}{M s^{2} + D s + K}

(2)

where $F_{T}$ is the thrust force applied to the rotor, $L$ is the length between the rotation center of the nacelle pitch angle and the nacelle representative point, $θ$ is the nacelle pitch angle, $M$ is the modal inertia of the first modal vibration of the floating platform pitch angle, $D$ is the modal damping coefficient of the first modal vibration of the floating platform pitch angle, $K$ is the modal stiffness coefficient of the first modal vibration of the floating platform pitch angle, and $s$ is the Laplace operator.

These modal parameters are calculated by simulation results of the shutdown (SD) in operation conditions as follows. After an SD simulation in steady wind condition is conducted, the thrust force and nacelle pitch angle just before SD give the modal stiffness coefficient by the following equation

F_{T S D} L = K θ_{S D}

(3)

The thrust force and nacelle pitch angle are calculated by averages of data for a predetermined duration

F_{T S D} = \int_{T_{S D} - T_{a v e}}^{T_{S D}} F_{t} (t) d t - \int_{T_{e n d} - T_{a v e}}^{T_{e n d}} F_{t} (t) d t

(4)

θ_{S D} = \int_{T_{S D} - T_{a v e}}^{T_{S D}} θ (t) d t - \int_{T_{e n d} - T_{a v e}}^{T_{e n d}} θ (t) d t

(5)

where $F_{T S D}$ is the thrust force just before SD, $θ_{S D}$ is the nacelle pitch angle just before SD, $F_{t} (t)$ is time history data of the thrust force of the SD simulation, $θ (t)$ is the time history data of the nacelle pitch angle of the SD simulation, $t$ is the time of the SD simulation, $T_{S D}$ is the SD start time of the SD simulation, $T_{a v e}$ is the average calculation time, and $T_{e n d}$ is the end time of the SD simulation.

The modal inertia is calculated by the following characteristic of the second-order response

M = \frac{K}{2 π f_{P}}

(6)

where $f_{P}$ is the natural frequency of the nacelle pitch angle and is given by the frequency characteristic of the SD simulation result.

The modal damping coefficient is calculated according to the second-order response characteristic

D = 2 ζ M \times 2 π f_{P}

(7)

where $ζ$ is the convergence coefficient of the second-order response characteristic. This is given by the following equation

ζ = \sqrt{\frac{{(\log O_{s})}^{2}}{π^{2} + {(\log O_{s})}^{2}}}

(8)

where $O_{s}$ is the overshoot after SD.

The parameter $L$ is determined by simulation results in operation conditions. After simulations under variable wind speeds were conducted, the plot positions of representative points (nodes) attached to the station-keeping structure along the fore-aft and vertical axes of the absolute coordinate system were shown in Figure 4. Based on these results, we can determine points A and B; point A is the intersection of lines Front 1 and Front 2, which are located at the position when the FOWT is at the most forward position, and point B is the intersection of the lines Back 1 and Back 2, which are located at the position when the FOWT is at the most backward position. We define $L$ as the distance from the average position of points A and B to the average position of the representative nacelle points.

Figure 4.

Position change of representative points of the tower and floating platform.

Relation between generator speed and thrust force

We determined this relation based on the steady-state simulation results. The relation between generator speed and thrust force plotted by the steady-state simulation results is shown in Figure 5. From results like those shown in Figure 5, a quadratic curve is approximated by using data within the range of the lowest and highest generator speeds

F_{T} = a_{g} ω^{2} + b_{g} ω + c_{g}

(9)

where $a_{g}$ , $b_{g}$ , and $c_{g}$ are coefficients of the approximated curve and $ω$ is the generator speed.

Figure 5.

Relation of generator speed and rotor thrust force based on steady-state simulation results: (a) floating offshore wind turbine and (b) bottom-fixed wind turbine.

We define this relation based on the following equation differentiating the abovementioned approximated curve

G_{W T 2} (s) = (2 a_{g} ω_{r a t} + b_{g}) L

(10)

where $ω_{r a t}$ is the rated generator speed.

Relation between BPA and thrust force

Similar to the previous subsection, the relation between the BPA and thrust force was plotted based on the steady-state simulation results (Figure 6). A quadratic curve is approximated by using the data shown in Figure 6

F_{T} = a_{b} β^{2} + b_{b} β + c_{b}

(11)

where $a_{b}$ , $b_{b}$ , and $c_{b}$ are the coefficients of the approximated curve and $β$ is the BPA.

Figure 6.

Relation of the blade pitch angle and rotor thrust force based on steady-state simulation results: (a) floating offshore wind turbine and (b) bottom-fixed wind turbine.

We also define this relation based on the following equation differentiating the abovementioned approximated curve

G_{W T 3} (s) = (2 a_{b} β_{r e p} + b_{b}) L

(12)

where $β_{r e p}$ is the BPA at the representative wind speed.

Relation between nacelle pitch angle and generator speed

We define the relation between the nacelle pitch angle and generator speed as follows. We assume that all wind power generated by the motion of the nacelle pitch angle is fully converted to rotor power, and that the rotor power is fully converted to electrical power (generator torque and generator speed). The rotor power is calculated using the following equation

P = q ω = \frac{1}{2} ρ A C_{p} v^{3}

(13)

where $P$ is the rotor power, $ρ$ is the air density, $A$ is the rotor plane area, $C_{p}$ is the power coefficient of the rotor, $v$ is wind speed, and $q$ is generator torque. We assume that the infinitesimal difference of the nacelle pitch angle causes the infinitesimal differences of the generator speed and wind speed. Differentiating equation (13) gives the following

\frac{d q}{d t} ω + q \frac{d ω}{d t} = \frac{1}{2} ρ A \frac{d C_{p}}{d t} v^{3} + \frac{3}{2} ρ A C_{p} v^{2} \frac{d v}{d t}

(14)

The following equation is given by equation (14) based on the assumption that $C_{p}$ and $q$ are not time-dependent, based on the abovementioned assumption (d)

\frac{d ω}{d t} = \frac{3 ρ A C_{p} v^{2}}{2 q} \frac{d v}{d t}

(15)

The wind speed generated by the nacelle pitch angle motion is calculated based on the nacelle pitch angle velocity

v = - L \frac{d θ}{d t}

(16)

Differentiating equation (16) gives us the following

\frac{d v}{d t} = - L \frac{d^{2} θ}{d t^{2}}

(17)

By substituting equation (17) into equation (15), we get

\frac{d ω}{d t} = - \frac{3 ρ A C_{p} v^{2} L}{2 q} \frac{d^{2} θ}{d t^{2}}

(18)

The Laplace transform of equation (18) gives the following

ω (s) = - \frac{3 ρ A C_{p} v^{2} L}{2 q} s θ (s)

(19)

Extracting only the gain of equation (19) gives us the relation between the nacelle pitch angle and generator speed

G_{W T 4} (s) = \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}}

(20)

where $v_{r e p}$ is the representative wind speed, $C_{p r e p}$ is the power coefficient at the representative wind speed, and $q_{r e p}$ is the generator torque at the representative wind speed.

Characteristic of VSC

VSC is based on proportional–integral control

G_{C 1} (s) = K_{P} + K_{I} \frac{1}{s}

(21)

where $K_{P}$ is the proportional gain and $K_{I}$ is the integral gain.

Stability criterion of the proposed transfer function

In this section, we determine the stability of the nacelle pitch angle using the proposed transfer function. To determine the stability, we used the Nyquist stability criterion and Bode diagram to see the characteristics of the whole system. The open-loop transfer function of the proposed transfer function shown in Figure 3 is as follows

G_{O L 1} (s) = \frac{[{(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} s + (2 a_{b} β_{r e p} + b_{b}) L K_{I}] \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}}}{M s^{2} + D s + K}

(22)

The closed-loop transfer function of the proposed transfer function shown in Figure 3 is as follows

G_{A L L 1} (s) = \frac{{(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} s + (2 a_{b} β_{r e p} + b_{b}) L K_{I}}{D_{A L L 1} (s)}

(23)

\begin{array}{l} D_{A L L 1} (s) = M s^{3} + [D + {(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}}] s^{2} \\ + {K + (2 a_{b} β_{r e p} + b_{b}) K_{I} \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L^{2}}{2 q_{r e p}}} s \end{array}

(24)

The constants for the calculation of these transfer functions are shown in Table 3. Nyquist and Bode diagrams of the proposed transfer function are shown in Figure 7. The small arrows in Figure 7(a) indicate the directions of the vector locus of the bottom-fixed turbine and FOWT; both are clockwise (CW). The Nyquist stability criterion defines that a system is stable if the poles of an open-loop transfer function exist on the left side of a real-image plane and its vector locus sees $(- 1, j 0)$ on the left side. Therefore, when the direction of the vector locus is CW and makes a circle like in Figure 7, the system is stable if $(- 1, j 0)$ exists outside of the circle vector locus. We confirmed that all the poles of the open-loop transfer functions of both types exist on the left side of a real-image plane.

Table 3.

Constants for the block diagram described in Figure 3.

Wind turbine type	Floating	Bottom-fixed
Rotor position	Downwind
Rotor diameter	80 m
Floating platform	Hybrid spar type	N/A
Mooring	Catenary, 3 chains	N/A
Hub height (m)	55.88 (from MSL)	65.38
Rated wind speed (m/s)	12	12
Representative wind speed for transfer functions of platform-pitching characteristic $v_{r e p}$ (m/s)	14	14
Length between the rotation center of the nacelle pitch angle and the nacelle representative point $L$ (m)	101.88	65.38
Natural frequency of nacelle pitch angle $f_{n}$ (Hz)	0.085 ${f_{n}}^{o n s h o r e}$	${f_{n}}^{o n s h o r e}$
Thrust force just before in SD simulation $F_{T S D}$ (kN)	1.37 ${F_{T S D}}^{o n s h o r e}$	${F_{T S D}}^{o n s h o r e}$
Nacelle pitch angle just before in SD simulation $θ_{S D}$ (rad)	13.598 ${θ_{S D}}^{o n s h o r e}$	${θ_{S D}}^{o n s h o r e}$
Overshoot of nacelle pitch angle in SD simulation $O_{s}$ (rad)	19.732 ${O_{s}}^{o n s h o r e}$	${O_{s}}^{o n s h o r e}$
Convergence coefficient. $ζ$ (−)	0.49 $ζ$	$ζ$
Modal inertia of first modal vibration of floating platform pitch angle $M$ (Nms²/rad)	216.008 $M^{o n s h o r e}$	$M^{o n s h o r e}$
Modal damping coefficient of first modal vibration of floating platform pitch angle $D$ (Nms/rad)	0.908 $D^{o n s h o r e}$	$D^{o n s h o r e}$
Modal stiffness coefficient of first modal vibration of floating platform pitch angle $K$ (Nm/rad)	0.157 $K^{o n s h o r e}$	$K^{o n s h o r e}$
Coefficient of $G_{W T 2} (s)$ $a_{g}$ (kNs²/rad²)	0.003	0.0028
Coefficient of $G_{W T 2} (s)$ $b_{g}$ (kNs/rad)	0.1844	0.032
Coefficient of $G_{W T 2} (s)$ $c_{g}$ (kN)	−69.656	−4.4566
Rated generator speed $ω_{r a t}$ (rad/s)	${ω_{r a t}}^{o n s h o r e}$
Coefficient of $G_{W T 3} (s)$ $a_{b}$ (kN/rad²)	−5514.1	−3329.3
Coefficient of $G_{W T 3} (s)$ $b_{b}$ (kN/rad)	1354.5	550.03
Coefficient of $G_{W T 3} (s)$ $c_{b}$ (kN)	102.55	186.88
Representative blade pitch angle for transfer functions of platform-pitching characteristic $β_{r e p}$ (rad)	${β_{r e p}}^{o n s h o r e}$
Air density $ρ$ (kg/m³)	1.225
Rotor plane area $A$ (m²)	$A^{o n s h o r e}$
Power coefficient at representative point $C_{p r e p}$ (−)	${C_{p r e p}}^{o n s h o r e}$
Generator torque at representative point $q_{r e p}$ (kNm)	${q_{r e p}}^{o n s h o r e}$
Proportional gain for PI control $K_{P}$ (s)	${K_{P}}^{o n s h o r e}$
Integral gain for PI control $K_{I}$ (−)	${K_{I}}^{o n s h o r e}$

SD: shutdown; PI: proportional–integral; MSL: mean sea level.

Figure 7.

Comparison of (a) Nyquist and (b) Bode diagrams of the transfer function of the generator speed and nacelle pitch angle for bottom-fixed and floating offshore wind turbines.

Figure 7(a) shows clearly that the bottom-fixed turbine is stable and that the FOWT is unstable. As shown in the gain characteristic of Figure 7(b), the nacelle pitch angle of an FOWT may vibrate at 0.035 Hz because there is a peak of gain near the frequency range. The frequency is the natural frequency of the nacelle pitch angle. By contrast, although the bottom-fixed type also has a peak around 0.4 Hz, nacelle-pitching vibration may converge because $(- 1, j 0)$ exists outside of its vector locus.

The reason for the FOWT instability is as follows. The comparisons of the modal parameters of the bottom-fixed type and the FOWT are shown in Table 4. The ratios of the modal damping and stiffness coefficients relative to the modal inertia of the FOWT are much smaller than those of the bottom-fixed type. We utilize the Routh stability criterion to clarify which coefficient causes the FOWT instability, and the characteristic equation of the system shown in Figure 3 is derived from the following equation

Table 4.

Comparisons of the modal characteristic parameters.

Wind turbine type	Floating	Bottom-fixed
Ratio of modal damping coefficient D relative to modal inertia M	0.042 ${R_{1}}^{o n s h o r e}$	${R_{1}}^{o n s h o r e}$
Ratio of modal stiffness coefficient K relative to modal inertia M	0.007 ${R_{2}}^{o n s h o r e}$	${R_{2}}^{o n s h o r e}$

D_{A L L 1} (s) = 0

(25)

The Routh arrays are given by the coefficients of equation (25), and the values of the Routh arrays are shown in Table 5. As shown in Table 5, one of the Routh arrays, $a_{1}$ , is negative and the signs of the Routh arrays change. Consequently, the system shown in Figure 3 is unstable. Because $a_{1}$ is equal to the second term on the right-hand side in equation (24), we can investigate the cause of FOWT instability based on the following equation

a_{1} = D + {(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}}

(26)

Table 5.

Values of the Routh arrays of the system shown in Figure 3.

Symbols of the Routh array	Values
Symbols of the Routh array	Floating	Bottom-fixed
a ₀	7.652 ${a_{0}}^{o n s h o r e}$	${a_{0}}^{o n s h o r e}$
a ₁	−0.868 ${a_{0}}^{o n s h o r e}$	0.159 ${a_{0}}^{o n s h o r e}$
b ₁	0.344 ${a_{0}}^{o n s h o r e}$	2.289 ${a_{0}}^{o n s h o r e}$

Based on Table 3 and equation (26), the FOWT instability is because $a_{b}$ , one of the characteristics of wind turbines, is negative. Because it is multiplied by $K_{p}$ , the system in Figure 3 becomes stable if it has no VSC (i.e. $K_{p} = 0$ ). In addition, $a_{1}$ is given only by the modal damping coefficient $D$ , not the modal stiffness coefficient $K$ . To change the sign of $a_{1}$ to be positive, $D$ must be larger than 1.746 $D^{o n s h o r e}$ , and this value is 1.922 times the value in Table 3. Therefore, the system instability is because the modal damping coefficient is small.

As shown in Table 3, $a_{b}$ of the bottom-fixed type is also negative. However, the signs of the Routh arrays of the bottom-fixed type in Table 5 do not change. This is because the values of $D$ of the bottom-fixed type are larger than that of the FOWT and the values of $L$ of the bottom-fixed type in Table 3 are smaller than that of the FOWT.

Stabilization by FVC

In this section, we determine the stability of the transfer function-added FVC to the proposed transfer function shown in Figure 3.

Whole system of transfer function with FVC

Figure 8 shows a block diagram–added FVC to the block diagram shown in Figure 3. Compared with Figure 3, the FVC characteristic $G_{C 2} (s)$ was added. $G_{C 2} (s)$ consists of a band-pass filter (BPF) and a proportional–derivative (PD) control as follows

G_{C 2} (s) = \frac{2 ζ_{c} ω_{c}}{s^{2} + 2 ζ_{C} ω_{c} s + {ω_{c}}^{2}} (B_{P} + B_{D} s)

(27)

B_{D} = B_{P} t_{D}

(28)

where $ζ_{C}$ is damping coefficient of the BPF, $ω_{c}$ is frequency of the BPF, $B_{P}$ is the proportional gain of the PD control, $B_{D}$ is the derivative gain of the PD control, and $t_{D}$ is the derivative time of the PD control.

Figure 8.

Block diagram of generator speed and nacelle pitch angle; floating platform vibration control is added.

As shown in Figure 8, the final BPA is determined by adding the BPAs generated by VSC and by FVC because FVC generates a BPA according to the nacelle pitch angle. That is, the controller of the transfer function shown in Figure 8 consists of VSC and FVC.

Change in stability by adding FVC

In this subsection, the stability of the transfer function shown in Figure 8 is determined as in the previous section. The open- and closed-loop transfer functions of Figure 8 are as follows

G_{O L 2} (s) = \frac{N_{O L 2} (s)}{D_{O L 2} (s)}

(29)

\begin{array}{l} N_{O L 2} (s) = {(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}} s^{3} \\ + [{(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} + (2 a_{b} β_{r e p} + b_{b}) L K_{I}] \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}} s^{2} \\ + [{(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} + 2 (2 a_{b} β_{r e p} + b_{b}) L K_{I} ζ_{c} ω_{c}] \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}} s \\ + (2 a_{b} β_{r e p} + b_{b}) L K_{I} {ω_{c}}^{2} \frac{3 ρ A C_{p r e p} {v_{r e p}}^{2} L}{2 q_{r e p}} \end{array}

(30)

\begin{array}{l} D_{O L 2} (s) = M s^{4} + (2 M ζ_{c} ω_{c} + D) s^{3} \\ + {M {ω_{c}}^{2} + 2 D ζ_{c} ω_{c} + K + 2 (2 a_{b} β_{r e p} + b_{b}) L B_{D} ζ_{c} ω_{c}} s^{2} \\ + {D {ω_{c}}^{2} + 2 D K ω_{c} + 2 (2 a_{b} β_{r e p} + b_{b}) L B_{P} ζ_{c} ω_{c}} s + K {ω_{c}}^{2} \end{array}

(31)

G_{A L L 2} (s) = \frac{N_{A L L 2} (s)}{N_{O L 2} (s) + D_{O L 2} (s)}

(32)

\begin{array}{l} N_{A L L 2} (s) = {(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} s^{3} \\ + [2 {(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} ζ_{c} ω_{c} + (2 a_{b} β_{r e p} + b_{b}) L K_{I}] s^{2} \\ + [{(2 a_{g} ω_{r a t} + b_{g}) L + (2 a_{b} β_{r e p} + b_{b}) L K_{P}} {ω_{c}}^{2} + 2 (2 a_{b} β_{r e p} + b_{b}) L K_{I} ζ_{c} ω_{c}] s \\ + (2 a_{b} β_{r e p} + b_{b}) L K_{I} {ω_{c}}^{2} \end{array}

(33)

where $ω_{P} = 2 π f_{P}$ is the natural frequency of the nacelle pitch angle.

The Nyquist and Bode diagrams of the system with FVC added are shown in Figure 9, and the FVC parameters used to get the results of Figure 9 are shown in Table 6. The FVC parameters were set based on the impacts described in the next subsection.

Figure 9.

(a) Nyquist and (b) Bode diagrams of the transfer function of the generator speed and nacelle pitch angle of the bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Table 6.

Parameters of floating platform vibration control described in Figure 8.

Names	Units	Values
Proportional gain of PD control $B_{P}$	–	12.0
Derivative time of PD control $t_{D}$	s	8.0
Damping coefficient of the BPF $ζ_{C}$	–	3.5
Frequency of the BPF $ω_{c}$	rad/s	2.5 $ω_{P}$

PD: proportional–derivative; BPF: band-pass filter.

As can be seen in “Floating with FVC” in Figure 9(a), the circle is smaller than the others and $(- 1, j 0)$ exists outside of it. Therefore, we can recognize that the system described in Figure 8 is stable. It is clear that FVC can stabilize the system significantly because its vector locus is also far from $(- 1, j 0)$ . Figure 9(b) indicates that adding FVC can suppress the platform-pitching vibration because the peak around the natural frequency of the nacelle pitch angle decreases.

Impact of FVC parameters

In this subsection, we clarify how the FVC parameters change the stability of the transfer function shown in Figure 8. Table 7 shows the parameter sets used in this subsection. The shaded cells indicate parameters changed from those at baseline. Figure 10 shows the Nyquist and Bode diagrams of the baseline parameters. Comparing “Floating” with “Floating with FVC,” the gain characteristics of both Bode diagrams are similar; however, “Floating with FVC” is stabler than “Floating” (see Figure 10(a)).

Table 7.

Parameter conditions of floating platform vibration control.

Case No.	Proportional gain of PD control $B_{P}$	Derivative time of PD control $t_{D}$	Damping coefficient of BPF $ζ_{C}$	Frequency of BPF $ω_{c}$	Figure No.
Case No.	–	s	–	rad/s	Figure No.
Baseline	5.0	1.0	0.5	2.0 $ω_{P}$	10
B-1	1.0	1.0	0.5	2.0 $ω_{P}$	11
B-2	20.0	1.0	0.5	2.0 $ω_{P}$	12
B-3	50.0	1.0	0.5	2.0 $ω_{P}$	13
T-1	5.0	0.0	0.5	2.0 $ω_{P}$	14
T-2	5.0	4.0	0.5	2.0 $ω_{P}$	15
T-3	5.0	16.0	0.5	2.0 $ω_{P}$	16
Z-1	5.0	1.0	0.1	2.0 $ω_{P}$	17
Z-2	5.0	1.0	1.0	2.0 $ω_{P}$	18
Z-3	5.0	1.0	5.0	2.0 $ω_{P}$	19
O-1	5.0	1.0	0.5	1.0 $ω_{P}$	20
O-2	5.0	1.0	0.5	3.0 $ω_{P}$	21
O-3	5.0	1.0	0.5	6.0 $ω_{P}$	22

PD: proportional–derivative; BPF: band-pass filter.

Figure 10.

(a) Nyquist and (b) Bode diagrams of the baseline parameters: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Proportional gain of PD control

Figures 11 to 13 show Nyquist and Bode diagrams in the case that the proportional gains of PD control are changed. As shown in Figure 11(b), if the proportional gain is smaller than that at baseline, the gain characteristic of the Bode diagrams of “Floating with FVC” is similar to that of “Floating”; the system is still unstable since $(- 1, j 0)$ exists inside of its vector locus circle. Figure 12(a) indicates that increasing the proportional gain can stabilize the system because $(- 1, j 0)$ is inside of the vector locus. However, Figure 13(a) shows that increasing the proportional gain too much can make the system unstable. Therefore, we should set an appropriate value for the system stabilization. It should be noted that increasing the proportional gain allows the peak of the gain characteristic of the Bode diagram to shift to a higher frequency. This means that a high proportional gain can generate higher frequency than the natural frequency of the nacelle pitch angle. We have to consider frequency shifts such as this because they can increase the motion of BPA around the frequency.

Figure 11.

(a) Nyquist and (b) Bode diagrams of B-1: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 12.

(a) Nyquist and (b) Bode diagrams of B-2: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 13.

(a) Nyquist and (b) Bode diagrams of B-3: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Derivative time of PD control

Figures 14 to 16 show the Nyquist and Bode diagrams when the derivative times of the PD control are changed. The three Nyquist diagrams state that the derivative times do not make the systems unstable because $(- 1, j 0)$ exists outside of the vector locus. This means that the derivative time does not affect the system stability. As shown in the three Bode diagrams, it is clear that with increasing derivative time, the peak of the gain characteristic in the Bode diagram becomes lower; however, Figure 16(b) shows that a high derivative time can generate low and high frequencies near the natural frequency of the nacelle pitch angle.

Figure 14.

(a) Nyquist and (b) Bode diagrams of T-1: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 15.

(a) Nyquist and (b) Bode diagrams of T-2: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 16.

(a) Nyquist and (b) Bode diagrams of T-3: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Damping coefficient of BPF

Figures 17 to 19 show the Nyquist and Bode diagrams when the damping coefficients of FVC are changed. Three Nyquist diagrams state that an appropriate value for system stability exists because the lower and higher damping coefficients have the areas of the vector locus becoming wider and $(- 1, j 0)$ is inside them. In addition, we should recognize that the peak of the gain characteristic of the Bode diagram can shift to a higher frequency than the natural frequency of the nacelle pitch angle.

Figure 17.

(a) Nyquist and (b) Bode diagrams of Z-1: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 18.

(a) Nyquist and (b) Bode diagrams of Z-2: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 19.

(a) Nyquist and (b) Bode diagrams of Z-3: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Frequency of BPF

Figures 20 to 22 show the Nyquist and Bode diagrams when the BPF frequencies are changed. As shown in the Nyquist diagrams in the three figures, it is clear that the BPF frequency should be set appropriately to stabilize the system. In addition to the Nyquist diagrams in Figures 9 and 10, the appropriate values of the BPF frequency are in the range of 2.0−3.0 $ω_{P}$ . The Bode diagrams in Figures 21 and 22 indicate that a BPF frequency higher than the abovementioned range tends to enhance the peak of the natural frequency of the nacelle pitch angle.

Figure 20.

(a) Nyquist and (b) Bode diagrams of O-1: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 21.

(a) Nyquist and (b) Bode diagrams of O-2: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Figure 22.

(a) Nyquist and (b) Bode diagrams of O-3: bottom-fixed, floating, and floating with floating platform vibration control (FVC) turbines.

Discussion

This work revealed two issues: one is that the small modal damping coefficient of the nacelle-pitching characteristic of the FOWT causes the floating platform-pitching vibration, and the other is the manner by which the FVC parameters can shift the frequency of the nacelle and BPAs. In particular, the achievements related to the FVC parameters are as follows:

Proportional gain of the PD control: An appropriate value should be set to suppress the platform-pitching vibration; however, a value that is too high can disturb the stabilization. A high value can generate a higher frequency of the nacelle and BPAs than the natural frequency of the nacelle pitch angle.

Derivative time of the PD control: A high value can suppress the platform-pitching vibration; however, a high value can generate a lower and higher frequency of the nacelle and BPAs than the natural frequency of the nacelle pitch angle.

Damping coefficient of the BPF: An appropriate value should be set to suppress the platform-pitching vibration. Values that are too low or too high cannot maintain the stabilization, and a high value can shift the peak of the gain characteristic of the Bode diagram to a higher frequency than the natural frequency of the nacelle pitch angle.

Frequency of the BPF: An appropriate value range exists for system stability; this range is a little higher than the natural frequency of the nacelle pitch angle, that is, from 2.0 to 3.0 $ω_{P}$ .

Of course, the suitable FVC values to stabilize the system can change depending on the characteristics of the FOWT. Although the final FVC values should be selected based on the actual responses of an actual FOWT, the findings of this work can help in deciding on some directions for FVC implements.

As mentioned above, we assumed that the components of the proposed transfer functions are not time-dependent and the position of the rotation center of the nacelle pitch angle is constant. Although variations of those parameters may affect the system stability to a certain extent, we suppose that its extent is a little because those parameters are based on average characteristics given by simulation results. In future work, we will investigate impacts of the parameter variations.

Conclusion

In this work, we proposed a transfer function between the generator speed and nacelle pitch angle (floating platform-pitching angle) using classical control theory. The proposed transfer function consists of controllers and system characteristics and can be used to investigate system stability using Nyquist and Bode diagrams. We clarified that the small modal damping coefficient of the nacelle-pitching characteristic of FOWTs can cause platform-pitching vibrations. Next, we added FVC to the proposed transfer function and showed that it could suppress the platform-pitching vibration. Moreover, we clarified how the FVC parameters can shift the frequency of the nacelle and BPAs. Although here we focused on the rated power operation condition, in future work, we plan to investigate not only impacts of the parameter variations of the transfer functions but also variable and rated speed conditions that are within the rated power condition using similar transfer functions.

Footnotes

Acknowledgements

The results of this study were obtained based on demonstration results using a commercial-scale demonstration FOWT constructed by the GOTO Floating Offshore Wind Turbine Demonstration Project, funded by the MOE, Japan. We thank MOE and the participants for allowing us to use it.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hiromu Kakuya

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