Linear coupled model for floating wind turbine control

Abstract

This work deals with an analytical linear coupled model describing the integrated aero-hydrodynamics of floating offshore wind turbines. Three degrees of freedom (platform surge, platform pitch and rotor azimuth) were considered with the goal of building a reduced-order model suitable for being integrated in control design algorithms as well as to be used for a straightforward evaluation and comprehension of the global system dynamics.

Keywords

Floating offshore wind turbines reduced-order modeling pitch control scale model testing

Introduction

Within last years, offshore wind power has been recognized as a high potential source of renewable energy, with different research projects currently being developed to make possible and cost-effective the deployment of multi-megawatt wind turbines in deep waters. However, several technical and scientific challenges arise when designing a floating wind turbine. Platform motion causes the rotor to operate under unsteady flow regimes (Bayati et al., 2017b) which result in variable aerodynamic loads. Moreover, the interaction between the platform rigid-body motion-modes and the wind turbine controller makes the traditional control strategies, commonly used to regulate power in high winds, not directly applicable.

Recently, dedicated numerical tools were developed to support the design of floating wind turbines. These codes are usually capable of replicating the complex aero-hydrodynamic loads experienced by the floating structure and are used for high-fidelity time-domain simulations. Nevertheless, reduced-order models are needed: by describing only the most significant wind turbine degrees-of-freedom (DOFs) in a control design perspective, it is possible, even in the early design stages, to relate more clearly the effect of the wind turbine controller action onto the floating offshore wind turbines (FOWT) dynamics and optimize the overall system design. Finally, reduced-order models are helpful for scale model testing activities, where they can be used to clarify the effects of a scaled environment on the global closed-loop response of the floating wind turbine.

Linear coupled model

A reduced-order linear coupled model was primarily designed by the authors for solving control design issues related with the implementation of a power control algorithm on a 1/75 scale model (Bayati et al., 2016, 2017, 2017) of the DTU 10 MW reference wind turbine (RWT; Bak et al., 2013) to be used for hybdrid hardware-in-the-loop (HIL) wind tunnel tests (Bayati et al., 2017). The model describes the floating wind turbine dynamics focusing on the interaction between rotor and the along-wind platform rigid-body motion-modes, surge $x$ , and pitch $θ$ , commonly recognized, for example, by Lemmer et al. (2016), as the most critical for control design, with the pitch mode setting a hard limit to the implementation of traditional control strategies (Larsen and Hanson, 2007). Tower and blade deformable modes are neglected as well as out-of-plane dynamics (i.e. heave, sway, roll, and yaw) since the model was originally thought to be applied to the rigid version of the PoliMi wind turbine model installed on a 2-DOFs HIL system (Bayati et al., 2017). According to the above-mentioned assumptions, the wind turbine is considered to be composed of three rigid bodies, platform, tower, and rotor-nacelle assembly (RNA) as shown in Figure 1. When it comes to tuning the wind turbine controller to achieve a desired performance, it is possible to consider the free response of the system and the floating wind turbine as operating in still water. The dynamic equilibrium of the floating structure is then expressed by the nonlinear equation of motion (equation (1))

\begin{array}{l} {[M]}_{f} {\underline{\overset{\cdot\cdot}{q}}}_{p} = {\underline{F}}_{h s t} + {\underline{F}}_{g r a v} + {\underline{F}}_{m o o r} + {\underline{F}}_{r a d} + \\ + {\underline{F}}_{M o r} + {\underline{F}}_{a d d} + {\underline{F}}_{a e r o} \end{array}

(1)

where vector ${\underline{q}}_{p} = [x θ]^{T}$ collects the two platform DOFs. The inertia forces, given by the two-by-two floater mass matrix ${[M]}_{f}$ and the acceleration of the platform DOFs ${\underline{\overset{\cdot\cdot}{q}}}_{p}$ , are balanced by hydrostatic restoring forces ${\underline{F}}_{h s t}$ , gravitational loads ${\underline{F}}_{g r a v}$ , the forces ${\underline{F}}_{m o o r}$ exerted on the platform by the mooring system, radiation forces ${\underline{F}}_{r a d}$ , Morison’s forces ${\underline{F}}_{M o r}$ , added damping loads ${\underline{F}}_{a d d}$ , and the aerodynamic forces ${\underline{F}}_{a e r o}$ developed by the turbine rotor. The rotor dynamics are described by the single-DOF equation of motion (equation (2))

(J_{R} + τ^{2} J_{G}) {\overset{\cdot\cdot}{θ}}_{R} = Q_{R} - τ Q_{G}

(2)

where $θ_{R}$ is the rotor azimuth angle. The overall drivetrain inertia, given by the rotor inertia $J_{R}$ and the generator inertia $J_{G}$ , reflected at rotor by the gearbox ratio (high-speed to low-speed) $τ$ , is balanced by the aerodynamic torque at rotor $Q_{R}$ and the generator torque $Q_{G}$ .

Figure 1.

Reduced-order model scheme and conventions.

Linearized platform loads

It is possible to get a linearized representation of the floating wind turbine from Taylor expansion of equations (1) and (2) in the neighborhood of any steady-state operating conditions defined for the nonlinear system. The first three terms on the right-hand side of equation (1) represent the restoring loads due to buoyancy, gravity, and the mooring system of the floating platform. The corresponding forces can be expressed as

{\underline{F}}_{h s t} ({\underline{q}}_{p}) = {\underline{F}}_{h s t} ({\underline{q}}_{p, 0}) - {[C]}_{h s t} {\bar{\underline{q}}}_{p}

(3)

{\underline{F}}_{g r a v} ({\underline{q}}_{p}) = {\underline{F}}_{g r a v} ({\underline{q}}_{p, 0}) - {[K]}_{g r a v} {\bar{\underline{q}}}_{p}

(4)

{\underline{F}}_{m o o r} ({\underline{q}}_{p}) = {\underline{F}}_{m o o r} ({\underline{q}}_{p, 0}) - {[K]}_{m o o r} {\bar{\underline{q}}}_{p}

(5)

The first term on the right-hand side of equations (3) to (5) represents, respectively, the hydrostatic, gravitational and mooring line loads that define the static equilibrium position ${\underline{q}}_{p,0}$ . The variation of recall forces around any static equilibrium position is given by the linear hydrostatic stiffness matrix ${[C]}_{h s t}$ , commonly obtained from hydrodynamic panel codes, the gravitational stiffness matrix ${[K]}_{g r a v}$ , and the equivalent stiffness matrix ${[K]}_{m o o r}$ computed from the linearization of mooring line loads in the undisplaced platform position.

The time-varying pressure field associated with the platform oscillation in the absence of incident waves defines the radiation loads ${\underline{F}}_{r a d}$ , function of platform hull shape, oscillation frequency, and history of motion. The so-called free-surface memory effect is accounted, in the time domain, by the convolution integral of the retardation matrix (Cummins, 1962) or, in the frequency domain, by the product between the Fourier transform of $[K]$ and the platform velocities

{\underline{F}}_{r a d} (ω) = (ω^{2} {[A]}_{\infty} - j ω [K (ω)]) {\underline{Q}}_{p} (ω)

(6)

where ${\underline{Q}}_{p} (ω) = F ({\underline{q}}_{p} (t))$ . The Fourier transform of the retardation function in equation (6) can be decomposed into a real and imaginary parts as

[K (ω)] = [B (ω)] + j ω ([A (ω)] - {[A]}_{\infty})

(7)

where the $[B (ω)]$ and $[A (ω)]$ are the frequency dependent added damping and added mass matrices, respectively. Concerning control engineering, in Perez and Fossen (2009) confirmed, there is a description of a method to get a state-space representation of the radiation matrices based on the addition of radiation memory states. However, to keep the model complexity low and clearly investigating how the platform hydrodynamics affects the global FOWT response, it was preferred to approximate $[A (ω)]$ and $[B (ω)]$ as the amount of added mass and damping at a single-oscillation frequency. In Jonkman (2007), the dynamics of a controlled floating wind turbine is studied by writing a single equation of motion for the platform pitch rotation and simplifying the radiation problem considering the added mass and damping corresponding to the platform pitch natural frequency only. Nevertheless, in this work, two platform DOFs were considered, thus it was necessary to select a single-oscillation frequency among the rigid-body motion-modes taken into account by the reduced-order model. The latter was primarily developed as a tool to address the FOWT control issue where the overall system damping plays a crucial role, and for this reason, it was decided to introduce in the model the amount of added damping and added mass resulting from the radiation matrices $[B (ω)]$ and $[A (ω)]$ evaluated at the natural frequency of the less damped mode.

Additional damping terms are introduced in the model by ${\underline{F}}_{a d d}$ and ${\underline{F}}_{M o r}$ , that are responsible, respectively, of linear and quadratic damping. Depending ${\underline{F}}_{M o r}$ on the square of the relative fluid velocity, it represents a higher order term when the still water assumption is introduced in the linear model, so it can be neglected when studying the system stability.

Equivalent mass, damping, and stiffness terms resulting from the linearization of platform loads are summed to the floater and drivetrain structural properties giving the equivalent second-order system of equation (8)

{[M]}^{*} \underline{\bar{\overset{\cdot\cdot}{q}}} + {[R]}^{* h} \underline{\bar{\dot{q}}} + {[K]}^{*} \underline{\bar{q}} = [\begin{matrix} {\bar{Q}}_{R} - τ {\bar{Q}}_{G} \\ \bar{T} \\ h_{n} \bar{T} \end{matrix}]

(8)

where

\underline{\bar{q}} = \underline{q} - {\underline{q}}_{0}

(9)

defines an infinitesimal variation of the system DOFs ( $\underline{q} = [θ_{R} x θ]^{T}$ ) in the neighborhood of the linearization (i.e. steady-state) condition ${\underline{q}}_{0}$ . The equivalent mass, damping and stiffness matrices ${[M]}^{*}$ , ${[R]}^{* h}$ , and ${[K]}^{*}$ are constant and are set only by the wind turbine drivetrain structure, the platform geometric and hydrodynamic properties, the mooring system characteristics. On the right-hand side of equation (8), the variation from steady-state value of aerodynamic torque ${\bar{Q}}_{R}$ , thrust $\bar{T}$ , and generator torque $\bar{Q_{G}}$ are collected.

Linearized aerodynamics

Rotor torque and rotor thrust can be written as a function of non-dimensional coefficients as

Q_{R} = \frac{1}{2} ρ π R^{3} C_{Q} U^{2}

(10)

T = \frac{1}{2} ρ π R^{2} C_{T} U^{2}

(11)

where $R$ is the wind turbine rotor radius, $ρ$ is the air density, $U$ is the hub-height wind speed, $C_{Q}$ is the torque coefficient, and $C_{T}$ is the thrust coefficient. The previously mentioned coefficients are function of the aerodynamic configuration of the turbine rotor that can be expressed, at any operating point, by the combination of two independent coordinates, the full-span rotor-collective blade-pitch angle $β$ and the tip–speed ratio (TSR; $λ = ω R / U$ ). Linearized aerodynamic loads are obtained starting from Taylor expansion of equations (10) and (11). For rotor torque, the following expression holds

\begin{array}{l} Q_{R} ≃ Q_{R} (β_{0}, ω_{R,0}, U_{0}) + \frac{\partial Q_{R}}{\partial β} |_{0} (β - β_{0}) + \\ + \frac{\partial Q_{R}}{\partial ω_{R}} |_{0} (ω_{R} - ω_{R,0}) + \\ + \frac{\partial Q_{R}}{\partial U} |_{0} (U - U_{0}) \end{array}

(12)

where $β_{0}$ , $ω_{R,0}$ , and $U_{0}$ are the steady-state values of blade pitch, rotor speed, and wind speed. In a similar fashion, for rotor thrust, it is possible to write

\begin{array}{l} T ≃ (β_{0}, ω_{R,0}, U_{0}) + \frac{\partial T}{\partial β} |_{0} (β - β_{0}) + \\ + \frac{\partial T}{\partial ω_{R}} |_{0} (ω_{R} - ω_{R,0}) + \\ + \frac{\partial T}{\partial U} |_{0} (U - U_{0}) \end{array}

(13)

Equations (12) and (13) may be written in a more compact form as

Q_{R} ≃ Q_{R,0} + K_{β Q} \bar{β} + K_{ω Q} \bar{ω_{R}} + K_{U Q} \bar{U}

(14)

T ≃ T_{0} + K_{β T} \bar{β} + K_{ω T} \bar{ω_{R}} + K_{U T} \bar{U}

(15)

where $T_{0}$ and $Q_{R,0}$ are the steady-state thrust force and aerodynamic torque, respectively, balanced at any operating point by platform recall forces and generator torque $Q_{G,0}$ . The unsteady part of aerodynamic loads is expressed by $K_{β Q}$ , $K_{ω Q},$ and $K_{U Q}$ , the aerodynamic torque sensitivities to rotor-collective blade-pitch angle, rotor speed, and wind speed, and $K_{β T}$ , $K_{ω T}$ , and $K_{U T}$ , the rotor thrust sensitivities to rotor-collective blade-pitch angle, rotor speed, and wind speed. The hub-height wind speed seen by the rotor when operating in a laminar wind field is given by the combination of a mean component and unsteady component associated with the nacelle speed induced by the platform motion

U = U_{0} - (\dot{\bar{x}} + \dot{\bar{θ}} h_{n})

(16)

By substituting equation (16) into equations (14) and (15)

Q_{R} ≃ Q_{R,0} + K_{β Q} \bar{β} + K_{ω Q} \bar{ω_{R}} - K_{U Q} (\dot{\bar{x}} + \dot{\bar{θ}} h_{n})

(17)

T ≃ T_{0} + K_{β T} \bar{β} + K_{ω T} \bar{ω_{R}} - K_{U T} (\dot{\bar{x}} + \dot{\bar{θ}} h_{n})

(18)

From equations (17) and (18), it is evident how the unsteady part of the aerodynamic forces depends on the rotor-collective pitch angle, the main control input in the above-rated region, and the system states taken into account by the reduced-order model. In particular, any term depending on the system states acts on the floating wind turbine as an equivalent damping and may be moved on the left-hand side of equation (8) that becomes

{[M]}^{*} \underline{\overset{\cdot\cdot}{\bar{q}}} + ({[R]}^{* h} + {[R]}^{* a}) \underline{\dot{\bar{q}}} + {[K]}^{*} \underline{\bar{q}} = [H] \underline{\bar{F}}

(19)

The overall FOWT damping is given by the superposition of the constant equivalent hydrodynamic damping matrix ${[R]}^{* h}$ and aerodynamic damping matrix ${[R]}^{* a}$

{[R]}^{* a} = [\begin{matrix} - K_{ω Q} & K_{U Q} & K_{U Q} h_{n} \\ - K_{ω T} & K_{U T} & K_{U T} h_{n} \\ - K_{ω T} h_{n} & K_{U T} h_{n} & K_{U T} h_{n}^{2} \end{matrix}]

(20)

By looking at the equivalent aerodynamic damping matrix, reported in equation (20), it becomes evident how the aerodynamic forcefield is responsible of the coupling between the rotor DOF and the platform DOFs. The aerodynamic coupling is function of the mean wind speed and so of the wind turbine operating point. The input matrix, function of the considered operating point, is

[H] = [\begin{matrix} K_{β Q} & - τ \\ K_{β T} & 0 \\ K_{β T} h_{n} & 0 \end{matrix}]

(21)

and defines how a variation in the control inputs from their steady-state value $\underline{\bar{F}} = [\bar{β} {\bar{Q}}_{G}]^{T}$ affects the system response at different wind speeds.

Controlled FOWT response

Literature has shown how the interaction between low-frequency platform modes and a pitch control system designed for onshore operations may lead to a low-damped or unstable motion of the floating system. Numerical simulations were performed on the FAST v8 model of the DTU 10 MW (Bak et al., 2013), coupled with the Stuttgart Wind Energy (SWE) TripleSpar floating platform developed by Lemmer et al. (2016a), in order to assess the nonlinear controlled system response and to verify the influence of the power controller on overall stability.

Decay simulations in still water and under a laminar wind field were performed enabling the action of the power controller designed for the onshore version of the RWT. In Figure 2, the FOWT response in terms of rotor speed, platform surge, platform pitch, and rotor-collective pitch angle is shown for two above-rated wind conditions, where the rotor speed is regulated by the pitch controller, while the generator torque is held constant (see Hansen and Henriksen, 2013).

Figure 2.

Response of the DTU 10 MW on the SWE TripleSpar platform for above-rated conditions and original pitch controller gains.

When a 12 m/s wind step is applied to the floating wind turbine, this is interested by an unstable response. Platform pitch oscillations are amplified by the pitch controller action and affect the response of the other DOFs, platform surge and rotor azimuth. The same behavior is seen for wind steps up to 17 m/s. When the FOWT undergoes a wind step of 18 m/s or higher, the system response is stable, showing positively damped oscillations.

Linear closed-loop dynamics

The linear model presented in the previous section was used to study the closed-loop dynamics of the DTU 10 MW on top of the SWE TripleSpar platform, so to clarify how the control system action affects the system response at different operating conditions. A Laplace domain representation of the floating wind turbine can be directly obtained from equation (19) as

({[M]}^{*} s^{2} + {[R]}^{*} s + {[K]}^{*}) \underline{Q} (s) = [H] \underline{F} (s)

(22)

where

{[R]}^{*} = [R]^{* h} + {[R]}^{* a}

(23)

is the equivalent damping matrix due to the hydrodynamic and aerodynamic forcefields at the considered operating point. The analytical transfer function matrix from control inputs $\underline{F} (s)$ to the system DOFs is evaluated from equation (22) as

[G (s)] = ([M]^{*} s^{2} + {[R]}^{*} s + {[K]}^{*})^{- 1} [H]

(24)

and the three-by-two matrix is

[G (s)] = [\begin{matrix} G_{β θ_{R}} & G_{Q_{G} θ_{R}} \\ G_{β x} & G_{Q_{G} x} \\ G_{β θ} & G_{Q_{G} θ} \end{matrix}]

(25)

From the analysis of the FOWT transfer function matrix $[G (s)]$ , it is possible to predict the system response at different wind speeds (i.e. steady-state operating points) and understand the reasons behind the previously mentioned instability problem.

The transfer function $G_{β Ω} (s) = s G_{β θ_{R}}$ describes how rotor speed (controlled variable) responds to a collective pitch angle (control input) variation and defines the pitch controller plant. The bode diagram of $G_{β Ω} (s)$ computed at four different operating points in the above-rated region is shown in Figure 3 and the corresponding pole-zero ( $\times$ for poles, $○$ for zeros) map is shown in Figure 4.

Figure 3.

Bode diagram on the FOWT transfer function from rotor-collective pitch angle to rotor speed $G_{β Ω} (s)$ .

Figure 4.

Pole-zero map of the FOWT transfer function from rotor-collective pitch angle to rotor speed $G_{β Ω} (s)$ .

The transfer function shows four complex conjugate poles, corresponding to the platform rigid-body motion-modes, a real pole, associated with the drivetrain mode, and four complex conjugate zeros at frequencies close to those of the platform modes. The surge poles and the corresponding zeros are always close to the imaginary axis at a frequency around 0.005 Hz (see Table 1) and are almost constant for different operating points. The same is not true for pitch poles (the SWE TripleSpar pitch mode natural frequency is 0.038 Hz in still air, see Table 1) and zeros that move significantly in the complex plane considering different mean wind speeds. In particular, above 11.4 m/s (the DTU 10 MW rated wind speed), the pitch zeros are in the right half-plane (RHPZ) and move toward the imaginary axis, crossing it at 17 m/s. For higher wind speeds, the platform pitch zeros have a negative real part. The single drivetrain pole is always in the left half-plane and, for increasing wind speeds, moves far from the RHP.

Table 1.

Main parameters of the SWE TripleSpar platform.

Parameter
Water depth	180 m
Draft	56 m
Tower base above SWL	25 m
Overall platform mass	2.857 × 107 kg
Water displacement	29,498 m³
Number of lines	3
Surge natural frequency	0.005 Hz
Pitch natural frequency	0.038 Hz

In order to study the FOWT closed-loop response and tune the wind turbine pitch controller gains, the plant transfer function $G_{β Ω} (s)$ obtained from the reduced model was combined with the transfer function of the proportional and integral (PI) pitch controller, as shown in the block diagram of Figure 5.

Figure 5.

Block diagram of the pitch-controlled offshore DTU 10 MW.

The rotor-collective pitch angle set-point $β$ is the output of a PI controller with feedback on the rotor speed error $e_{ω} (s) = Ω_{R} (s) - Ω_{R,0} (s)$

G_{c} (s) = \frac{β (s)}{e_{ω} (s)} = k_{P} (1 + \frac{1}{T_{I} s})

(26)

where $T_{I}$ is the integral time constant and $k_{P}$ is the proportional gain resulting from the combination of the aerodynamic gain scheduling function $η_{A}$ and the proportional gain at rated conditions $k_{P,0}$

k_{P} = η_{A} \cdot k_{P,0}

(27)

The aerodynamic gain scheduling $η_{A}$ is a quadratic function of pitch angle $β$ (see Hansen and Henriksen, 2013), and within region 3, it is generally different from 1. In this work, it was assumed that at any considered operating point, $η_{A}$ is the function of the steady-state pitch angle $β_{0}$ and does not vary significantly with a perturbation of the pitch angle $\bar{β}$

η_{A} = \frac{1}{1 + \frac{β_{0}}{K K_{1}} + \frac{β_{0}^{2}}{K K_{2}}}

(28)

where $K K_{1}$ and $K K_{2}$ are constants obtained from curve fitting of the pitch-to-torque sensitivity $K_{β Q}$ performed by Hansen and Henriksen (2013).

The open-loop transfer function $G (s)$ of the controlled drivetrain is given by the product of $G_{β Ω} (s)$ and the PI pitch controller transfer functions of equation (26)

G (s) = \frac{Ω_{R} (s)}{e_{ω} (s)} = G_{c} (s) \cdot G_{β Ω} (s)

(29)

The Bode diagram of $G (s)$ computed at four different wind speeds with the gains of the onshore DTU 10 MW (see Table 2) is visible in Figure 6. From 12 to 17 m/s, the phase margin is negative and the closed-loop system unstable.

Table 2.

Main parameters of the DTU 10 MW pitch controller.

Parameter
Rated rotor speed $ω_{R,0}$	9.6 r/min
Integral time constant $T_{I}$	3.088 s
Proportional gain $k_{P,0}$	0.436 rad/s
Linear gain scheduling constant $K K_{1}$	198.329°
Quadratic gain scheduling constant $K K_{2}$	693.222°

Figure 6.

Bode diagram of the open-loop pitch-controlled drivetrain $G (s)$ .

The root locus of the pitch-controlled drivetrain is shown in Figure 7 for wind speeds considered in the rest of the analysis. At any operating point, distinguished by a different line color, if the proportional gain $k_{P}$ is increased, the closed-loop poles move far from the open-loop poles (colored ×). Closed-loop poles obtained for the onshore gains of Table 2 are marked in the diagram by black ×, and pitch poles are in the RHP for wind speed between rated and 17 m/s. This is in agreement with numerical simulations on the nonlinear FAST model of the floating system (see Figure 2) where the platform pitch DOF was interested by instability starting from rated up to 17 m/s.

Figure 7.

Root locus of the pitch-controlled drivetrain for onshore gains of Table 2. Colored $\times$ and $○$ for $G (s)$ poles and zeros and black $\times$ for $L (s)$ poles.

Onshore controller detuning

The typical approach adopted in literature to solve the pitch controller instability problem is to reduce the controller gains (see, for example, Larsen and Hanson, 2007) limiting the bandwidth to a fraction of the frequency of the platform mode recognized as the most critical for the FOWT response. A reduction of the pitch controller gains results in a lower coupling between the drivetrain and platform dynamics, causing the FOWT to respond more as an open-loop system (Jonkman, 2007). Accordingly, it was decided to tune the DTU 10 WM pitch controller gains, studying the characteristics of the poles of the closed-loop transfer function $L (s)$ computed from equation (30)

L (s) = \frac{G (s)}{1 + G (s)}

(30)

In particular, the target of the analysis was to find a combination of gains which maximize the platform damping given and a drivetrain damping around 70%. The damping ratio $h$ of the drivetrain and platform pitch poles was evaluated according to equation (31) (see Ogata, 2010) for different values of $T_{I}$ and $k_{P}$ at a wind speed slightly above rated (12 m/s), where the real part of the closed-loop poles is more positive (i.e. stability is more critical)

h = - \frac{R e (λ_{L})}{\sqrt{R e {(λ_{L})}^{2} + I m {(λ_{L})}^{2}}}

(31)

where $λ_{L}$ is the generic complex pole of the closed-loop transfer function $L (s)$ . The surge mode was not included in the analysis since, as shown in Figure 7, it is less sensitive than the other DOFs to a variation of the controller gains. In Figure 8, the damping ratio (fraction of critical one) of closed-loop platform pitch poles $h_{θ}$ and drivetrain poles $h_{R}$ is shown for different combinations of the integral time constant $T_{I}$ and proportional gain $k_{P}$ . For any given integral time constant $T_{I}$ (e.g. black line in Figure 8), if the proportional gain $k_{P}$ is increased, the drivetrain damping rapidly increases up to 1, with the closed-loop drivetrain poles being real. At the same time, the platform pitch poles damping decreases to 5%. When $k_{P}$ is fixed (e.g. red line in Figure 8), if the integral time constant $T_{I}$ is increased, the drivetrain poles become rapidly real, while the platform pitch poles damping remains almost constant. Due to these considerations, the best compromise, here considered optimal, was found for a total integral time constant $T_{I} = 12$ and a total proportional gain equal to the 45% of the original gain (intersection of black and red lines in Figure 8).

Figure 8.

Damping ratio of platform pitch mode $h_{θ}$ (top) and drivetrain mode $h_{R}$ (bottom) for different pitch controller gains $k_{P}$ and $T_{I}$ combinations.

Detuned system dynamics

The closed-loop FOWT response with detuned pitch controller gains is assessed from the root locus of the open-loop pitch-controlled drivetrain transfer function $G (s)$ shown in Figure 9. For wind speeds between rated and 17 m/s poles of the closed-loop transfer function $L (s)$ (black ×) are always in the left-hand plane and the system response is consequently stable.

Figure 9.

Root locus of the pitch-controlled drivetrain for detuned gains. Colored $\times$ and $○$ for $G (s)$ poles and zeros and black $\times$ for $L (s)$ poles.

The dynamic characteristics of closed-loop poles at a wind speed of 12 m/s with original and detuned gains are compared in Table 3.

Table 3.

Closed-loop poles at 12 m/s with original and detuned gains.

Mode	Original		Detuned
Mode	$h$ (–)	$f$ (Hz)	$h$ (–)	$f$ (Hz)
Drivetrain	0.564	0.045	0.744	0.013
Surge	0.006	0.006	0.014	0.006
Pitch	−0.098	0.029	0.190	0.036

Numerical simulations were performed on the fully nonlinear FAST v8 model of the floating DTU 10 MW in order to assess the effectiveness of the procedure used to design the turbine pitch controller. In Figure 10, the detuned system response resulting from a 12 m/s wind step is compared with the outputs of the corresponding simulation performed with the original pitch controller tuning (see Figure 2). As visible, the new system response is stable, with pitch oscillations damped after few cycles. Almost no rotor speed oscillations are seen, in agreement with the results of linear analysis, being closed-loop drivetrain poles very close to real axis for wind speed above 12 m/s (see root locus of Figure 9 and damping ratios of Table 3).

Figure 10.

Response of the DTU 10 MW on the SWE TripleSpar platform for a wind speed of 12 m/s (above rated) and detuned pitch controller gains.

Conclusion and future developments

This work presented the analytical formulation of a simple reduced-order linear model useful to describe the dynamics of a generic FOWT. From the model’s coupled matrices, it was possible to directly correlate the platform hydrodynamics to the rotor aerodynamic properties and to extract analytical transfer functions to be used for control design tasks. A systematic procedure for tuning pitch controller gains based on linear analysis was introduced and applied to a floating system composed of the DTU 10 WM RWT and SWE TripleSpar platform. The performance of the detuned pitch controller and the effectiveness of the design procedure were finally assessed through numerical simulations on the fully nonlinear system.

Specific tests were recently performed to characterize the PoliMi wind turbine model aerodynamic performance and the response of the generator and pitch actuators. By introducing experimental data in the linear coupled model presented in this work, it will be possible to clarify the effects of a scaled environment on the FOWT closed-loop response simulated during hybrid HIL tests. A specific tuning procedure will be introduced to work around scaling issues and match the controlled full-scale system response.

Footnotes

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

This project has partially received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 640741.

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