Floating Offshore Wind Turbine: Active Wave Damping

Floating Offshore Wind Turbines (FOWTs) represent the frontier of renewable energy in deep coastal waters. Unlike fixed-bottom turbines anchored directly to the seabed in shallow waters (< 50 m), floating foundations such as the NREL OC3-Hywind spar-buoy operate at depths of 100 to 300+ meters.

While deep waters offer stronger and steadier wind resources, they introduce severe structural dynamics challenges. A floating spar-buoy behaves in pitch (rotation $\theta$ ) as a massive submerged pendulum. Because radiation damping from surrounding seawater is exceptionally small ( $\zeta \approx 0.03$ ), ocean wave excitation at or near resonant frequencies builds compounding rotational oscillations that threaten platform stability and structural life.

This case study investigates active wave damping using a reduced-order 1-DOF spar-buoy pitch model, demonstrating why naive feedback fails and how a second-order notch filter solves the fundamental frequency conflict.

The Physical Setup

The platform is based on the benchmark NREL OC3-Hywind spar-buoy concept. In pitch rotation (mode 5 in naval architecture), the dynamics are governed by Newton-Euler rotational equilibrium:

I_{\text{tot}} \ddot{\theta} + B_{55} \dot{\theta} + C_{55} \theta = M_{\text{wave}} - M_{\text{ctrl}}

Parameter	Meaning	Benchmark Value
$I_{55}$	Structural pitch moment of inertia	$5.00 \times 10^{10}\ \text{kg}\cdot\text{m}^2$
$A_{55}$	Hydrodynamic added mass (entrained water)	$1.70 \times 10^{10}\ \text{kg}\cdot\text{m}^2$
$I_{\text{tot}}$	Total virtual pitch inertia ( $I_{55} + A_{55}$ )	$6.70 \times 10^{10}\ \text{kg}\cdot\text{m}^2$
$B_{55}$	Hydrodynamic radiation damping	$8.65 \times 10^8\ \text{N}\cdot\text{m}\cdot\text{s/rad}$
$C_{55}$	Hydrostatic restoring stiffness	$3.111 \times 10^9\ \text{N}\cdot\text{m/rad}$
$f_n$	Natural pitch resonance frequency	$0.0343\ \text{Hz}\ (\omega_n = 0.2155\ \text{rad/s},\ T_n \approx 29.2\ \text{s})$
$\zeta$	Open-loop damping ratio	$\approx 0.03$ ( $Q \approx 16.7$ , $+24.5\ \text{dB}$ resonance peak)

The Degrees of Freedom ("55" Convention)

In naval architecture and marine hydrodynamics, any floating vessel has 6 standard degrees of freedom:

Surge (longitudinal translation)
Sway (lateral translation)
Heave (vertical translation)
Roll (rotation about longitudinal axis)
Pitch (rotation about transverse axis)
Yaw (rotation about vertical axis)

The index $55$ denotes pitch reaction caused by pitch motion (uncoupled single-degree-of-freedom pitch dynamics).

Transfer Function & Spectral Divide

The open-loop transfer function from total input torque to spar-buoy pitch angle $\theta$ is:

G(s) = \frac{\Theta(s)}{M(s)} = \frac{1}{I_{\text{tot}} s^2 + B_{55} s + C_{55}}

In the frequency domain:

The spar-buoy structural pitch resonance occurs at $\omega_n = 0.2155\ \text{rad/s}$ ( $f_n = 0.0343\ \text{Hz}$ ).
Dominant ocean wave excitation occurs around $f_p \approx 0.10\ \text{Hz}$ ( $\omega_p = 0.628\ \text{rad/s}$ , period $T_p \approx 10\ \text{s}$ ).

Because the wave frequency is nearly three times the structural pitch resonance ( $\omega_p / \omega_n \approx 2.91$ ), there exists a distinct frequency divide:

Structural resonance is low-frequency ( $0.034\ \text{Hz}$ ).
Wave swell excitation is higher-frequency ( $0.10\ \text{Hz}$ ).

Why Naive PD Control Fails

A conventional PD controller employs derivative action ( $K_d s$ ). Derivative gain increases at $+20\ \text{dB/decade}$ with frequency:

|K_d \cdot j\omega| = K_d \cdot \omega

At the wave frequency ( $0.10\ \text{Hz}$ ) the derivative term answers every radian of wave-induced tilt with $\sqrt{K_p^2 + (K_d\omega)^2} \approx 561\ \text{MN}\cdot\text{m}/\text{rad}$ of counter-torque, although the waves are not what it needs to damp. With the waves of this lesson ( $20\ \text{MN}\cdot\text{m}$ ) the peak torque is only about $1.25\ \text{MN}\cdot\text{m}$ — $2.5\%$ of the $\pm 50\ \text{MN}\cdot\text{m}$ limit, which would be reached only for a wave-frequency tilt of about $5.1^\circ$ . Nearly all of that effort is spent at the wave frequency, where it does nothing for the resonance; the PD does raise the resonance damping from $\zeta = 0.030$ to about $0.054$ .

Active Wave Damping with a Notch Filter

The control solution places a second-order notch filter $N(s)$ centered at the peak wave frequency ( $f_0 = 0.10\ \text{Hz}$ ) directly into the feedback path:

N(s) = \frac{s^2 + \omega_0^2}{s^2 + 2\zeta_n \omega_0 s + \omega_0^2} \quad \text{with } \omega_0 = 2\pi f_0

At the wave excitation frequency ( $s = j\omega_0$ ), the numerator becomes zero:

|N(j\omega_0)| = 0 \quad (-\infty\ \text{dB rejection})

The filter makes the controller blind to the first-order wave motion, allowing the vessel to ride the wave crests compliantly: in this lesson the peak torque and the dissipated energy drop by roughly 70%. It is not free: at the structural resonance ( $0.034\ \text{Hz}$ ) the notch has $|N| \approx 0.93$ but lags the feedback by $21^\circ$ , so the damping the PD adds shrinks by about 70% (closed-loop $\zeta \approx 0.037$ with the notch, $0.054$ without, $0.030$ open loop).

Reduced-Order Scope & Visual Context

This lesson models the platform's primary uncoupled pitch dynamics (1-DOF reduced-order spar-buoy). The turbine rotor, mooring catenary lines, and wave heave visual elements serve as illustrative context to aid physical intuition without burdening the fundamental control concepts with multi-megabyte hydroelastic simulation overhead.