Pre-print version 0.5

[1]MatteoCapaldo [2]PaulMella

1]TotalEnergies OneTech, Palaiseau, France 1,2]Ecole Polytechnique, Institut
Polytechnique de Paris, Palaiseau, France

Matteo Capaldo<EMAIL_ADDRESS>
# Damping analysis of Floating Offshore Wind Turbine (FOWT): a new control
strategy reducing the platform vibrations

###### Abstract

In this paper, the coupled dynamics of the floating platform and the WTG rotor
is analysed. In particular, the damping is explicitly derived from the coupled
equations of rotor and floating platform. The analysis of the damping leads to
the study of the instability phenomena and it derives the explicit conditions
that lead to the Non Minimum Phase Zero (NMPZ). Two NMPZs, one related to the
rotor dynamics and the other one to the platform pitch dynamics, are analysed.
The latter is a novelty and it is analysed in this work, providing the
community of an explicit condition for its verification. The domain of the
instability of the platform is explicitly derived from the coupled system of
equations. In the second part of the paper, from the analysis of the damping
of the floating platform, a new strategy for the control of FOWTs is proposed.
This strategy allows one to impose to the controller an explicit level of
damping in the platform pitch motion without changing the period of platform
pitching. Finally the new strategy is compared to the one without compensation
by performing aero-hydro-servo-elastic numerical simulations of the UMaine
IEA15MW FOWT. Generated power, movements, blade pitch and tower base fatigue
are compared showing that the new control strategy can reduce fatigue in the
structure without affecting the power production.

Wind energy is an important source of renewable energy and it has a very high
potential in both onshore and offshore. In terms of installed capacity,
onshore wind is still the largest contribution. However, in the next years,
the new annual offshore installed capacity is estimated to exceed 30 GW by
2030, in order to stay on-track for a netzero/$1.5^{\circ}\ C$ pathway LEE et
al. (2022). In offshore, there is a growing interest in floating foundations.
In fact, FOWTs would allow access to good wind resource locations that are not
suitable for fixed-bottom foundations.

In that context, the levelized cost of energy (LCOE) of offshore wind farms
needs to be decreased to be competitive with respect to onshore wind. This is
especially true for the FOWT. One effective way to achieve this objective is
to investigate different strategies for the control of the FOWTs in order to
improve it with respect to the LCOE. As explained in BIANCHI et al. (2007),
the minimisation of the LCOE involves a series of partial objectives, energy
capture, mechanical loads and power quality. These objectives are actually
closely related and sometimes conflicting and they should not be pursued
separately. Hence, the question is to find a well balanced compromise among
them. Considering FOWT, this optimization problem increases in complexity
since the movements of the floating platform interact with the feedback
control loop. Moreover, the coupling between the platform movements, the rotor
dynamics and the blade pitch control can lead to oscillating (or not damped)
stability or even to unstable conditions LARSEN et al. (2007).

The nature and the set of control parameters leading to the instability can
variate from one platform design to another one, e.g., barge, spare, semi-sub
or tension-leg platform. However, for each of the platforms, there are sets of
design and control parameters leading to oscillating stability or instability
FLEMING et al. (2014). The issue come from the fact that in above rated wind
speed, the blade pitch regulates the speed by increasing the angle of attack
to feather. For a FOWT, when the platform has a forward motion, the rotor
experiences an increasing wind speed. Consequently, the blade pitch control
increases the angle of attack to feather and, hence, reduces the aerodynamic
torque and regulates the rotor speed. However, it also reduce the rotor thrust
that induces a further forward motion. So the blade pitch control amplifies
the original forward motion of the platform because the floating platform
surge and pitch natural frequencies are in the bandwidth of the blade pitch
control.

Solutions exist to avoid this phenomenon. The first one and the most common in
literature is to reduce the blade pitch control proportional and integral
gains in order to reduce its bandwidth and exclude the platform pitch and
surge natural frequencies JONKMAN et al. (2008); LARSEN et al. (2007).
However, this solution do not completely solve the problem and, moreover, the
price to pay is to have a less reactive blade pitch control that allows
important over speed of the rotor. Alternative methods use additional sensing,
such as nacelle fore-aft acceleration measurements or platform gyroscopes, to
improve the performance of the pitch controller. In LACKNER et al. (2009) and
LENFEST et al. (2020), the platform pitch velocity is used to adjust the rated
speed set point to reduce platform motions. However, the platform pitch
damping analysis is not investigated and the link with the compensation
parameter is not given. Higher order controllers, such as a linear quadratic
regulator (LQR) are applied and evaluated in MA et al. (2018). Disturbance
accommodating controller (DAC) is evaluated here MENEZES et al. (2018) and it
is coupled with individual pitch control (IPC) in this paper, NAMIK et al.
(2011). A nonlinear pitch and torque controller using wind preview is designed
in SARKAR et al. (2020) and SCHLIPF et al. (2013), giving promising results.

The control strategy proposed in this work shares some similarities to the one
introduced in LACKNER et al. (2009) and LENFEST et al. (2020). As introduced
in this paper, in fact, the rated generator speed is no longer a constant
value but a function of the platform pitch velocity and the blade pitch is
used to damp the floating platform pitch. However, differently from LACKNER et
al. (2009), the ratio between proportional and integral gains of the
correction can be considered different for the platform pitch motion and the
rotor speed. This choice, in fact, can be not optimal for some cases.
Differently from LENFEST et al. (2020), a mathematical frame is given to
define the compensation to the platform pitch and the explicit form of it is
analytically given. Moreover, a second compensation is considered in the
control strategy proposed in this work. It is to solve the Non Minimum Phase
Zero (NMPZ) effects caused by the blade pitch on the rotor rotational
dynamics. It is already introduced in STOCKHOUSE et al. (2021) and it is, in
this work, analytically developed. The study of the NMPZs brings to a new NMPZ
phenomenon, described for the first time in this work. This is the NMPZ caused
by the blade pitch on the platform dynamics. This new phenomenon is
analytically developed leading to the explicit condition to verify it.
However, the compensations proposed by this model can’t correct it.

The novelty of this work is related to the FOWT damping analysis, i.e., the
damping obtained by coupling the rotor and the platform pitch dynamics. This
damping is explicitly derived from the coupled equations of rotor and floating
platform. This analysis leads to the study of the instability phenomena
underlining the conditions leading to the NMPZ. One new NMPZ, never discussed
in literature, is discovered and analysed in this work. The domain of the
instability of the platform is explicitly derived from the coupled system of
equations. The control strategy proposed relies on this analysis and it allows
to impose an explicit level of damping in the platform pitch motion to the
controller without changing the period of platform pitching. This explicit
form of the damping in the platform pitch dynamics is a novelty of this work.

The chosen strategy is, then, compared to the one which does not consider a
platform pitch compensation. The latter neglects the following phenomenon: the
blade feather reduces the aerodynamic thrust and, thus, a part of the opposing
force on the platform is neglected.

The document is organized as follows. In Section 1, the equations of the FOWT
system are described with the considered degrees of freedom and their coupling
terms. Section 1.1 presents the control model considered in this work. The
closed loop feedback system is then analysed in Section 1.3, leading to the
definition of the boundaries of the instability domain. For this controller, a
new control strategy dedicated to FOWTs, named $\zeta_{plt}$-fixed is
presented and analytically derived in Section 1.4 and 1.5. Some numerical
tests are presented and commented in Section 2.

##  1 Floating Offshore Wind Turbine and its controller

The floating offshore wind turbine (FOWT) is represented by a system of two
degrees of freedom, namely the rotor speed $\Omega$ and the platform pitch
angle, $\Phi$. Two control parameters, $B$ (blade pitch) and $\tau_{g}$
(generator torque), and two external disturbances, $V$ (wind speed) and $W$
(waves speed) are considered. For all the values that form a given operating
point (namely $\Omega,\Phi,B,T_{g},V,W$), the notation $X=\bar{x}+x$ is
adopted, with $x$ being the small perturbation of a steady-state operating
point $\bar{x}$.

The model is, then, based on the two fundamental equations:

$\dot{\Omega}_{g}=\frac{N_{g}}{J_{r}}(T_{a}-N_{g}T_{g})$ (1)
$J_{t}\ddot{\Phi}+D_{t}\dot{\Phi}+K_{t}\Phi=h_{t}F_{a}+\tau_{w}$ (2)

where $\Omega_{g}$ is the generator speed, hereafter noted $\Omega$, $T_{a}$
and $T_{g}$ are the aerodynamic and electric torque, $N_{g}$ is the gearbox
ratio and $J_{r}$ is the rotor inertia. $J_{t}$ is the total system moment of
inertia about the pitch rotation, $D_{t}$ is the natural damping coefficient
(assumed constant), $K_{t}$ is a spring-like restoring coefficient (mainly
given by the mooring lines of the floating platform), $h_{t}$ is the height of
the rotor (approximately the tower length), $F_{a}$ is the aerodynamic force
flowing from the rotor to the system and $\tau_{w}$ is the overturning moment
given by the waves.
Once a steady-state operating point $\bar{x}$ is reached, the same two
equations can be applied to any small variations $x$ around this operating
point. Equations 1 and 2 applied on $X$ can be written:

$\dot{\omega}=\frac{N_{g}}{J_{r}}(\tau_{a}-N_{g}\tau_{g})$
$J_{t}\ddot{\phi}+D_{t}\dot{\phi}+K_{t}\phi=h_{t}dF_{a}+d\tau_{w}$

The infinitesimal thrust and torques satisfy $T_{g}=\bar{\tau}_{g}+\tau_{g}$,
$T_{a}=\bar{\tau}_{a}+\tau_{a}$, $F_{a}=\bar{F_{a}}+dF_{a}$ and
$\tau_{w}=\bar{\tau}_{w}+d\tau_{w}$. By considering small perturbations of a
steady-state operating point (given by $\Omega,\Phi,B,T_{g},V,W$), it allows
one to use the following linear forms:

$\tau_{a}=\frac{\partial\tau_{a}}{\partial\omega}\omega+\frac{\partial\tau_{a}}{\partial
v}v_{r}+\frac{\partial\tau_{a}}{\partial\beta}\beta$ $dF_{a}=\frac{\partial
F_{a}}{\partial\omega}\omega+\frac{\partial F_{a}}{\partial
v}v_{r}+\frac{\partial F_{a}}{\partial\beta}\beta$
$d\tau_{w}=\frac{\partial\tau_{w}}{\partial w}w$

Note that $F_{a}$ and $\tau_{w}$ are separated for clarity, which explains why
$dF_{a}$ does not depend on a $w$ perturbation. Moreover, the hypothesis of
$\phi$ being small allows one to remove the terms
$\frac{\partial\tau_{a}}{\partial\phi}\phi$, $\frac{\partial
F_{a}}{\partial\phi}\phi$ and $\frac{\partial\tau_{w}}{\partial\phi}\phi$.

Relative wind $v_{r}$ is the wind velocity in rotor reference frame, it is
computed from $v$ by:

$v_{r}=v-h_{t}\dot{\phi}$

Under the assumption of $\phi$ being small, $h_{t}\dot{\phi}$ represents the
rotor fore-aft velocity in a fixed global reference frame.

Equations 1 and 2 applied to small perturbations of a steady-state point can
therefore be expressed in the linear form:

$\dot{\omega}=\frac{N_{g}}{J_{r}}\left(\frac{\partial\tau_{a}}{\partial\omega}\omega+\frac{\partial\tau_{a}}{\partial
v}(v-h_{t}\dot{\phi})+\frac{\partial\tau_{a}}{\partial\beta}\beta-
N_{g}\tau_{g}\right)$ (1’)
$J_{t}\ddot{\phi}+D_{t}\dot{\phi}+K_{t}\phi=h_{t}\left(\frac{\partial
F_{a}}{\partial\omega}\omega+\frac{\partial F_{a}}{\partial
v}(v-h_{t}\dot{\phi})+\frac{\partial
F_{a}}{\partial\beta}\beta\right)+\frac{\partial\tau_{w}}{\partial w}w$ (2’)

Those coupled second order equations yield the following four dimensional
state-space model:

$\dot{x}=A_{0}x+B_{c}u_{c}+B_{d}u_{d}$ (3)

Where $x=(\theta,\dot{\theta},\phi,\dot{\phi})^{T}$, $\theta=\int\omega$ (i.e.
$\dot{\theta}=\omega$), $u_{c}=(\beta,\tau_{g})^{T}$ and $u_{d}=(v,w)^{T}$,
and :

$A_{0}=\begin{bmatrix}0&1&0&0\\\
0&\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\omega}&0&-h_{t}\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial
v}\\\ 0&0&0&1\\\ 0&\frac{h_{t}}{J_{t}}\frac{\partial
F_{a}}{\partial\omega}&-\frac{K_{t}}{J_{t}}&-\frac{1}{J_{t}}(D_{t}+h_{t}^{2}\frac{\partial
F_{a}}{\partial v})\\\ \end{bmatrix}\qquad B_{c}=\begin{bmatrix}0&0\\\
\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}&-\frac{N_{g}^{2}}{J_{r}}\\\
0&0\\\ \frac{h_{t}}{J_{t}}\frac{\partial F_{a}}{\partial\beta}&0\\\
\end{bmatrix}B_{d}=\begin{bmatrix}&0&0\\\
&\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial v}&0\\\ &0&0\\\
&\frac{h_{t}}{J_{t}}\frac{\partial F_{a}}{\partial
v}&\frac{1}{J_{t}}\frac{\partial\tau_{w}}{\partial w}\\\ \end{bmatrix}$

###  1.1 Control model description

In this section the pitch controller model is described.

The present control model considers $\omega_{r}$ as the reference for the
$\omega$ and $0$ as the reference for $\dot{\phi}$. It is based on several
SISO (single-input-single-output) feedback loops. It can be seen as a multi-
SISO:

  * •

Proportional : $\beta_{P}=k_{P}(\Omega-\Omega_{r})$

  * •

Integral : $\beta_{I}=k_{I}\int(\Omega-\Omega_{r})=k_{I}(\theta-t\Omega_{r})$

  * •

Beta platform pitch compensation :
$\beta_{comp}=k_{\beta}(\dot{\Phi}-\dot{\Phi}_{r})$

  * •

Generator torque platform pitch compensation :
$\tau_{g,comp}=k_{\tau_{g}}(\dot{\Phi}-\dot{\Phi}_{r})$

Controllers described by the literature considering the same compensations
ABBAS et al. (2021); STOCKHOUSE et al. (2021) aim at maintaining $\omega$
steady near its rated value by acting on the blade pitch $\beta$ to variate
the aerodynamic torque $\tau_{a}$ with the opposite sign with respect to the
rotor relative speed $\omega$. However, this strategy neglects the following
phenomenon: the blade feather modifies the aerodynamic thrust $F_{a}$. Thus, a
part of the opposing force on the platform is neglected. The strategy
developed in this paper aims at minimizing $\phi$ variations with the
constraint of maintaining a constant $\omega$. Such a control strategy should
reduce the loads on the structures (nacelle, tower and floater). Section 2
considers a full aero-hydro-servo-elastic model to verify this assumption. It
is analysed the performance of the control strategy in a FOWT realistic
environment reproduced by a numerical twin.

###  1.2 Global State-Space description

For a FOWT, the objective ot the pitch control is to remain in the equilibrium
operating point. It translates in: $\bar{\omega}=\Omega_{r}$ and
$\bar{\dot{\phi}}=\dot{\Phi}_{r}=0$. This objective allows one to justify the
linear form for of the global equations 1’ and 2’. For constant inputs
$\bar{v}$ and $\bar{w}$, this operating point is reached by the appropriated
constants $\bar{\beta}$ and $\bar{\tau_{g}}$, the fine pitch and fine electric
torque, respectively.

The control model is, here, introduced into the wind turbine state space
description. For small perturbations of this steady-state operating point, the
PI controller described previously becomes :

  * •

Proportional:

$\beta_{P}=k_{P}\omega$ (4)

  * •

Integral:

$\beta_{I}=k_{I}\int\omega=k_{I}\theta$ (5)

  * •

Beta platform pitch compensation:

$\beta_{comp}=k_{Beta}\dot{\phi}$ (6)

  * •

Generator torque platform pitch compensation:

$\tau_{g,comp}=k_{\tau_{g}}\dot{\phi}$ (7)

Figure 1 shows the entire picture of the control model. This control strategy
acts on the two dynamic systems, platform and rotor. Hence, one can appreciate
how the bottom-fixed scheme acting on the rotor ($\omega$) with a proportional
integral scheme is then corrected by the $\beta_{comp}$ and the $\tau_{g}$
that depends on the platform pitch dynamics error.

Figure 1: Block diagram of the controller model

The linear expression of $u_{c}=(\beta,\tau_{g})^{T}$ as a function of
$x=(\theta,\dot{\theta},\phi,\dot{\phi})^{T}$ is $u_{c}=K_{0}x+u_{c,ol}$
where:

$K_{0}=\begin{bmatrix}k_{I}&k_{P}&0&k_{\beta}\\\ 0&0&0&k_{\tau_{g}}\\\
\end{bmatrix}$

is the matrix of the control gains and $u_{c,ol}$ is an optional additional
control (open loop) that can be considered. This is useful to analyse the NMPZ
in the next section. By replacing it in equation 3, it leads to:

$\dot{x}=(A_{0}+B_{c}K_{0})x+B_{c}u_{c,ol}+B_{d}u_{d}$ (8)

Which leads to define the global matrix of the system of equations:

$A=(A_{0}+B_{c}K_{0})=\begin{bmatrix}0&1&0&0\\\
k_{I}\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}&\frac{N_{g}}{J_{r}}(\frac{\partial\tau_{a}}{\partial\omega}+k_{P}\frac{\partial\tau_{a}}{\partial\beta})&0&\frac{N_{g}}{J_{r}}(-h_{t}\frac{\partial\tau_{a}}{\partial
v}+k_{\beta}\frac{\partial\tau_{a}}{\partial\beta}-k_{\tau_{g}}N_{g})\\\
0&0&0&1\\\ k_{I}\frac{h_{t}}{J_{t}}\frac{\partial
F_{a}}{\partial\beta}&\frac{h_{t}}{J_{t}}(\frac{\partial
F_{a}}{\partial\omega}+k_{P}\frac{\partial
F_{a}}{\partial\beta})&-\frac{K_{t}}{J_{t}}&\frac{1}{J_{t}}(-D_{t}-h_{t}^{2}\frac{\partial
F_{a}}{\partial v}+k_{\beta}h_{t}\frac{\partial F_{a}}{\partial\beta})\\\
\end{bmatrix}$ (9)

The time domain system can be rewritten in the complex domain. Using the
following notation, $\mathscr{L}\\{x(t)\\}=\underline{X}$, equation 8
translates into:

$\underline{X}=(sI-A)^{-1}(B_{c}\underline{U}_{c,ol}+B_{d}\underline{U}_{d})$
(10)

This leads us to define

$B=\begin{bmatrix}B_{c}&|&B_{d}\end{bmatrix},\;u=\begin{pmatrix}\beta_{ol}\\\
\tau_{g,ol}\\\ v\\\ w\\\ \end{pmatrix}\;$

and

$G(s)=(sI-A)^{-1}B=\frac{1}{det(sI-A)}Com(sI-A)^{T}B=\frac{1}{\chi_{A}(s)}N(s)$
(11)

in order that :

$\chi_{A}(s)\underline{X}(s)=N(s)\underline{U}(s)$ (12)

G(s) is a $4\times 4$ matrix of which every component can be written as the
quotient of a polynomial in s and $\chi_{a}(s)$.

###  1.3 Non minimum phase zeros analysis and resolution (negative damping on
the control)

In this section, we address the problem of negative damping by addressing the
positions of the zeros of each component of G in the complex plane, i.e. the
points where equation 12 is well defined but becomes:

$\chi_{A}\ X_{0}^{T}\cdot\underline{X}=0.$ (13)

This translates the fact that $s$ is a NMPZ if it exists a specific
$X_{0}^{T}$, such as, for any value of $\underline{U}(s)$, the linear
combination $X_{0}^{T}N(s)\underline{U}(s)$ gives $X_{0}^{T}=0$.

Physically a $X_{0}^{T}$ is equivalent to an infinitesimal shifting along a
specific direction of a steady-state point that can not be obtained with any
infinitesimal shifting of the input. This phenomenon is better illustrated
case by case.
For the rest of the section, an open loop control on $\beta$, is considered in
order to highlight the NPMZ. Hence, $\beta_{ol}$ is added to the multiple SISO
as already described in 8. Since the feedback control on $\beta$ can’t erase
the NMPZ condition, to lighten the formulas, the notation $\beta$ will be used
instead of $\beta_{ol}$ in this section.

####  $\phi$-NMPZ: negative damping on $\phi$ control by $\beta$

Gain equation in the Laplace domain for $\beta\rightarrow\phi$ control is
obtained by projecting 12 on the $x=(0,0,\phi,0)$ axis and considering only a
$\beta$ perturbation, ie. an input $u=(\beta_{ol},0,0,0)$. The resulting
equation is:

$\chi_{A}(s)\underline{\phi}(s)=N_{3,1}(s)\underline{\beta}(s),$ (14)

$N_{3,1}(s)=\frac{J_{r}}{N_{g}}\frac{\partial
F_{a}}{\partial\beta}s^{2}+\left(\frac{\partial\tau_{a}}{\partial\beta}\frac{\partial
F_{a}}{\partial\omega}-\frac{\partial
F_{a}}{\partial\beta}\frac{\partial\tau_{a}}{\partial\omega}\right)s$ (15)

The condition for the NMPZ on $\beta\rightarrow\phi$ control is that $N_{3,1}$
has a root with a real part strictly positive. Assuming that
$\beta=\beta_{f}$, with $\beta_{f}$ the fine pitch, the previous derivatives
are all negative. Hence, the root research of $N_{3,1}$ leads to:

$\frac{\partial\tau_{a}}{\partial\omega}\ /\
\frac{\partial\tau_{a}}{\partial\beta}\quad<\quad\frac{\partial
F_{a}}{\partial\omega}\ /\ \frac{\partial F_{a}}{\partial\beta}$ (16)

Intuitively, this corresponds to an operating point where $\tau_{a}$ is rather
influenced by $\beta$ and $F_{a}$ is rather influenced by $\omega$. This NMPZ
does not depend on parameters of the platform, it is only related to WTG
performances. However, the amplitude of the phenomenon is related to the
platform properties. It is to be noted that this NMPZ has never been
highlighted in literature and the control model (with compensations of the
platform motions) introduced in Section 1.1 in the feedback control loop does
not prevent it. It results only from the characteristic of the FOWT system.
Further works should focus on this phenomenon and introduce corrections to
prevent it for any FOWT system.

In absence of NMPZ, ie. eq. 16 being false, increasing $\beta$ from a steady-
state operating point (ie. setting $d\ddot{\beta}>0$, $d\dot{\beta}>0$ and
$d\beta>0$) will always imply a reduction of $\phi$. In presence of NMPZ, ie.
if eq. 16 is true, the reduction or the increase of $\phi$ depends on the
ratio between $\ddot{\beta}$ and $\dot{\beta}$.

The latter only happens when eq. 16 is verified: increasing blade pitch
reduces $\tau_{a}$ more than it increases $F_{a}$ (because $\tau_{a}$ is
rather influenced by $\beta$), thus $\omega$ increases and causes $F_{a}$ to
decrease (because $F_{a}$ is rather influenced by $\omega$) and then $\phi$ to
increase. If eq. 16 is not verified, increasing blade pitch $\beta$ from a
steady-state operating point always reduces platform pitch $\phi$. In
practice, this effect can become an issue for a control algorithm mainly
focused on $\omega$ stabilization since it generates unexpected platform
dynamics. Figure 2 reproduces in the time domain $\phi$ and $\omega$ responses
to a $\beta$-step input (at $t=10s$): values (resumed in Table 1) are chosen
so that eq. 16 is false: $\phi$ decreases. On the right, values are chosen so
that the eq. 16 is true: $\phi$ increases even though $\beta$ has step up.

Table 1: Set of parameters to show the NMPZ of eq. 16 | eq. 16 false | eq. 16 true
---|---|---
$\frac{\partial\tau_{a}}{\partial v}$ | $2980.9\leavevmode\nobreak\ kN.s$ | $3079\leavevmode\nobreak\ kN.s$
$\frac{\partial F_{a}}{\partial v}$ | $354.8\leavevmode\nobreak\ kN.s.m^{-1}$ | $355.6\leavevmode\nobreak\ kN.s.m^{-1}$
$\frac{\partial\tau_{a}}{\partial\omega}$ | $-58597.1\leavevmode\nobreak\ kN.m.s.rad^{-1}$ | $-55499.5\leavevmode\nobreak\ kN.m.s.rad^{-1}$
$\frac{\partial F_{a}}{\partial\omega}$ | $-5658.0\leavevmode\nobreak\ kN.s.rad^{-1}$ | $-5820.4\leavevmode\nobreak\ kN.s.rad^{-1}$
$\frac{\partial\tau_{a}}{\partial\beta}$ | $-152347.8\leavevmode\nobreak\ kN.m.rad$ | $-160140.5\leavevmode\nobreak\ kN.m.rad$
$\frac{\partial F_{a}}{\partial\beta}$ | $-16052.2\leavevmode\nobreak\ kN.rad$ | $-15260\leavevmode\nobreak\ kN.rad$

Figure 2: $\phi$ and $\omega$ responses to a $\beta$-step input (at $t=10s$):
on the left, values (Table 1) are chosen so that eq. 16 is false: $\phi$
decreases. On the right, values (Table 1) are chosen so that the eq. 16 is
true: $\phi$ increases even though $\beta$ has step up.

####  $\omega$-NMPZ: negative damping on $\omega$ control by $\beta$

Gain equation in the Laplace domain for $\beta\rightarrow\omega$ control is
given by:

$\chi_{A}(s)\ \underline{\omega}(s)=N_{2,1}(s)\ \underline{\beta}(s)$ (17)
$N_{2,1}(s)=\frac{J_{t}}{h_{t}}\frac{\partial\tau_{a}}{\partial\beta}s^{3}+\left[\frac{D_{t}}{h_{t}}\frac{\partial\tau_{a}}{\partial\beta}+h_{t}\left(\frac{\partial\tau_{a}}{\partial\beta}\frac{\partial
F_{a}}{\partial v}-\frac{\partial
F_{a}}{\partial\beta}\frac{\partial\tau_{a}}{\partial
v}\right)-k_{\tau_{g}}N_{g}\frac{\partial
F_{a}}{\partial\beta}\right]s^{2}+\frac{K_{t}}{h_{t}}\frac{\partial\tau_{a}}{\partial\beta}s$
(18)

Hence the condition for NMPZ on $\beta\rightarrow\omega$ control is:

$h_{t}^{2}\left(\frac{\partial F_{a}}{\partial
v}-\left(\frac{\partial\tau_{a}}{\partial
v}+k_{\tau_{g}}\frac{N_{g}}{h_{t}}\right)\frac{\frac{\partial
F_{a}}{\partial\beta}}{\frac{\partial\tau_{a}}{\partial\beta}}\right)<-D_{t}$
(19)

This corresponds with an operating point where $\tau_{a}$ is rather influenced
by $v$ and $F_{a}$ is rather influenced by $\beta$. In presence of NMPZ, i.e.
if eq. 19 is true, the sign of $d\omega$ depends on the choice of
$d\ddot{\beta}$, $d\dot{\beta}$ and $d\beta$. Intuitively, the latter only
happens when eq. 19 is verified: increasing blade pitch will reduce $F_{a}$
more than it increases $\tau_{a}$ (because $F_{a}$ is rather influenced by
$\beta$), thus $\dot{\phi}$ will decrease and cause relative wind
$v_{r}=v-h_{t}\dot{\phi}$ to increase. As $\tau_{a}$ is rather influenced by
$v$, this will reduce $\omega$ in the end. In practice, this effect can become
an issue if a $\omega$ control algorithm obtains the opposite result than what
was expected.

Because of this issue, literature has concerned the NMPZ phenomenon on
$\beta\rightarrow\omega$ control and suggested several control corrections.
Due to the nature of this phenomenon, any correction concerning $\beta$
control, introduced in eq. 6, can’t prevent this NMPZ. However, detuning the
PI controller (by lowering $k_{P}$ and $k_{I}$ gains) or using the $\beta$
platform pitch compensation $d\beta_{comp}=k_{\beta}\dot{\phi}$ as suggested
in ABBAS et al. (2021), can mitigate the effect of NMPZ when eq. 19 is true.

However, the complete prevention of the problem can be obtained by several set
of parameters that involves both the WTG, the floating platform and the
control set-up. In fact, for this NMPZ, equation 7 of the control model
described in Section 1.1 allows one to avoid NMPZ by choosing a well-suited
value of $k_{\tau_{g}}$. This compensation has been already introduced by
STOCKHOUSE et al. (2021) with the formula:

$k_{\tau_{g}}=-m_{\tau_{g}}\frac{h_{t}}{N_{g}}\frac{\partial\tau_{a}}{\partial
v},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
m_{\tau_{g}}\in[0,1]$ (20)

The authors notice however it usually needs to be saturated because of turbine
generator design constraints. In absence of NMPZ, i.e. eq. 19 being false,
increasing $\beta$ from a steady-state operating point will always imply
reducing $\omega$.

In order to visualize this NMPZ, Figure 3 shows $\omega$ responses to a
$\beta$-step input (at $t=10s$). On the left, parameters (Table 2) are chosen
so that the condition eq. 19 is false: $\omega$ decreases. On the right,
parameters are chosen so that condition eq. 19 is true: at first $\omega$
increases even though $\beta$ has step up.

Table 2: Set of parameters to show the NMPZ of eq. 19 | eq. 19 false | eq. 19 true
---|---|---
$\frac{\partial\tau_{a}}{\partial v}$ | $2980.9\leavevmode\nobreak\ kN.s$ | $2838\leavevmode\nobreak\ kN.s$
$\frac{\partial F_{a}}{\partial v}$ | $354.8\leavevmode\nobreak\ kN.s.m^{-1}$ | $303.0\leavevmode\nobreak\ kN.s.m^{-1}$
$\frac{\partial\tau_{a}}{\partial\omega}$ | $-58597.1\leavevmode\nobreak\ kN.m.s.rad^{-1}$ | $-59428.7\leavevmode\nobreak\ kN.m.s.rad^{-1}$
$\frac{\partial F_{a}}{\partial\omega}$ | $-5658.0\leavevmode\nobreak\ kN.s.rad^{-1}$ | $-6282.9\leavevmode\nobreak\ kN.s.rad^{-1}$
$\frac{\partial\tau_{a}}{\partial\beta}$ | $-152347.8\leavevmode\nobreak\ kN.m.rad$ | $-133058.7\leavevmode\nobreak\ kN.m.rad$
$\frac{\partial F_{a}}{\partial\beta}$ | $-16052.2\leavevmode\nobreak\ kN.rad$ | $-18247.0\leavevmode\nobreak\ kN.rad$

Figure 3: $\omega$ responses to a $\beta$-step input (at $t=10s$). On the
left, parameters (Table 2) are chosen so that the condition eq. 19 is false:
$\omega$ decreases. On the right, parameters are chosen so that condition eq.
19 is true: at first $\omega$ increases even though $\beta$ has step up.

#### Conclusion of NMPZ analysis:

Comparison between Figures LABEL:fig:NPMZ and 3 enlightens what really happens
after a step input, with and without NMPZ: at the beginning both $\omega$ and
$\dot{\phi}$ always decrease just after the step. However, when both NMPZ
conditions eq. 16 and eq. 19 are false, those tendencies don’t change.
Conversely, when eq. 16 is true, we observe that $|\dot{\omega}|$ is so big
that $\dot{\phi}$ jumps into positive values. Similarly, when eq. 19 is true,
we observe that $|\dot{\phi}|$ is so big that $\omega$ jumps (only for a short
time) into positive values. NMPZ, as we have seen in the examples, can cause
important shifts and unexpected behaviors for both $\omega$ and $\phi$.

The NMPZ $\beta\rightarrow\phi$ doesn’t depend on above defined parameters:
the model in Section 1.1 does not give solution to always prevent it, but
condition eq. 16 forecasts which operating points it affects. On the other
hand, a wise choice of $\tau_{g}$ avoids $\beta\rightarrow\omega$ NMPZ, which
is the main reason why this compensation has already been introduced by
STOCKHOUSE et al. (2021) and in the control model in Section 1.1.

In order to complete the analysis of NMPZ phenomena related to FOWT system, a
hypothetical situation where both conditions eq. 16 and eq. 19 are true has
been simulated and reported in Figure 4. At first, the dynamics are always the
same: both $\dot{\phi}$ and $\omega$ decrease, but soon they both diverge
because of the negative damping. It is to be noted that, $k_{P}$/$k_{I}$
corrections (without compensations $\tau_{g}$ and $k_{\beta}$) can delay this
divergence but can not avoid it.

Table 3: Set of parameters to show the instability given by NMPZs of eq. 16 and eq. 19. | eq. 16 and eq. 19 true
---|---
$\frac{\partial\tau_{a}}{\partial v}$ | $3105.0\leavevmode\nobreak\ kN.s$
$\frac{\partial F_{a}}{\partial v}$ | $293.0\leavevmode\nobreak\ kN.s.m^{-1}$
$\frac{\partial\tau_{a}}{\partial\omega}$ | $-51356.5\leavevmode\nobreak\ kN.m.s.rad^{-1}$
$\frac{\partial F_{a}}{\partial\omega}$ | $-7150.0\leavevmode\nobreak\ kN.s.rad^{-1}$
$\frac{\partial\tau_{a}}{\partial\beta}$ | $-148063.0\leavevmode\nobreak\ kN.m.rad$
$\frac{\partial F_{a}}{\partial\beta}$ | $-16543.6\leavevmode\nobreak\ kN.rad$
Figure 4: an hypothetical situation where both conditions eq. 16 and eq. 19
are true. At first, both $\dot{\phi}$ and $\omega$ decrease, but soon they
both diverge because of the negative damping ($k_{P}$/$k_{I}$ corrections can
delay this divergence but can not avoid it).

###  1.4 Damping analysis

In section 1.3 the issue of NMPZ, ie. the issue of negative damping in the
control/input side of the equation, has been analysed.

On top of setting boundaries on gains $k_{P}$, $k_{I}$, $k_{\beta}$ and
$k_{\tau_{g}}$ (cf. Section 1.3), this section analyses how they affect the
explicit value of the damping. The goal is to optimize (or tune) the stability
of $\omega$ and $\phi$ responses to an external ($v$ and $w$) disturbance. In
other words, the goal is to obtain an explicit expression of the damping of
the FOWT system with respect to the control parameters, $k_{P}$, $k_{I}$,
$k_{\beta}$ and $k_{\tau_{g}}$, such that, for an imposed level of damping,
one can obtain a value of the control parameters. This is a powerful result
for the floating wind community and a novelty of this work with respect to the
existing literature.

Considering the whole system, with both degrees of freedom $\omega$ and $\phi$
and their coupling, $\chi_{A}(s)$ of equation 8 in the complex domain leads to
the equation 12. The study of the damping is related to
$\chi_{A}(s)=det(sI-A)$. The explicit form of $\chi_{A}$ is:

$\chi_{A}(s)=\chi_{rot}(s)\chi_{plt}(s)+\frac{N_{g}h_{t}}{J_{r}J_{t}}s\left[(k_{p}s+k_{I})\frac{\partial
F_{a}}{\partial\beta}h_{t}\frac{\partial\tau_{a}}{\partial
v}+\left(\frac{J_{r}}{N_{g}}\frac{\partial
F_{a}}{\partial\beta}(k_{P}s+k_{I})+\frac{\partial
F_{a}}{\partial\omega}\right)\left(k_{\beta}\frac{\partial\tau_{a}}{\partial\beta}+k_{\tau_{g}}N_{g}\right)\right]$
(21)

where:

$\begin{split}\chi_{rot}(s)&=s^{2}-\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\omega}s-\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}(k_{P}s+k_{I})\\\
\chi_{plt}(s)&=s^{2}+\frac{1}{J_{t}}(D_{t}+h_{t}^{2}\frac{\partial
F_{a}}{\partial v}-k_{\beta}h_{t}\frac{\partial
F_{a}}{\partial\beta})s+\frac{K_{t}}{J_{t}}\\\ \end{split}$ (22)

The term in square parenthesis represents the coupling term between the
dynamics of the platform ($\phi$) and the dynamics of the rotor ($\omega$).

In this coupled form, it is complicated to explicit the damping of the system.
In the next paragraph, under some hypothesis, the coupled system can be
separated in two second order systems, one related to the rotor dynamics
$\omega$ and the other one related to the floating dynamics $\phi$. In
particular, for the latter, it is possible to define a damping for the
floating platform and obtain an explicit form for the compensation term
$k_{\beta}$ related to the imposed damping.

#### Simplified analysis of rotor dynamics:

Defining a damping coefficient (or a damping ratio) requires to reduce the
global system to a second order oscillatory system. Equations 8 couples rotor
and platform pitch dynamics, they hence involve a $4^{th}$ order polynomial
expression. In order to deal with rotor dynamics independently of the
platform, it is supposed:

$h_{t}\dot{\phi}\ \ll\ v$ (23)

For large FOWT systems, this hypothesis is, generally, respected. It implies:

$\begin{split}N_{g}k_{\tau_{g}}\dot{\phi}&\ll\frac{\partial\tau_{a}}{\partial
v}v\\\
\frac{\partial\tau_{a}}{\partial\beta}k_{\beta}\dot{\phi}&\ll\frac{\partial\tau_{a}}{\partial
v}v\\\ \end{split}$ (24)

Under such assumption, the linear form of equation 1 becomes:

$\dot{\omega}=\frac{N_{g}}{J_{r}}\left(\frac{\partial\tau_{a}}{\partial\omega}\omega+\frac{\partial\tau_{a}}{\partial
v}v+\frac{\partial\tau_{a}}{\partial\beta}\beta-\tau_{g}\right)$ (1”)

and the control is described by the PI controller:
$\dot{\beta}=k_{P}\dot{\omega}+k_{I}\omega$, such that the resulting Laplace
transform equation is

$\underline{\omega}(s)=G_{rot}(s)\underline{v}(s)$ (25)

where, considering a $k_{I}>0$,

$\begin{split}G_{rot}(s&)=\frac{\frac{\partial\tau_{a}}{\partial
v}s}{s^{2}-\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\omega}s-\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}(k_{P}s+k_{I})},\\\
i.e.\quad
G_{rot}(j\nu&)=\frac{1}{1+\frac{j}{2\zeta_{rot}}\left(\frac{\nu}{\nu_{rot}}-\frac{\nu_{rot}}{\nu}\right)}\frac{-\frac{\partial\tau_{a}}{\partial
v}}{\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\omega}+\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}k_{P}}\\\
\end{split}$ (26)

with:

$\nu_{rot}=\sqrt{-\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}k_{I}}\;,\quad\zeta_{rot}=-\frac{\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\omega}+\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}k_{P}}{2\sqrt{-\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}k_{I}}}$
(27)

Thus, when all interactions with platform pitch are neglected, the rotor
behaves like a second order oscillatory system. The corresponding filter
$G_{rot}$ is a second order band-pass filter with cutoff angular frequency
$\nu_{rot}$. In case $k_{I}\leq 0$, $\nu_{rot}$ and $\zeta_{rot}$ would be
imaginary according to the formulas above. $G_{rot}$ is no longer a band-pass
filter.

The above formulas enable one to obtain explicitly $k_{I}$ and $k_{P}$.

They are well known: several controllers, such as ROSCO ABBAS et al. (2021),
suggest to define:

$\left|k_{I}\right|=\left|\frac{\nu_{rot}^{2}}{\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}}\right|\quad\text{and}\quad\left|k_{P}\right|=\left|\frac{\left(\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\omega}+2\zeta_{rot}\nu_{rot}\right)}{\frac{N_{g}}{J_{r}}\frac{\partial\tau_{a}}{\partial\beta}}\right|$
(28)

#### Simplified analysis of platform dynamics:

Similarly as what is done in the previous paragraph, here the global system 1
is reduced to a second order oscillatory system that allows us to have a
better understanding of platform dynamics.

Considering $k_{P}=k_{I}=0$ and assuming:

$\frac{\partial F_{a}}{\partial\omega}\omega<<\frac{\partial F_{a}}{\partial
v}v_{r}+\frac{\partial F_{a}}{\partial\beta}\beta$ (29)

The latter is the condition to decouple the global system. It enables to
consider $\phi$ response independently of $\omega$, and as a second order
oscillatory system’s degree of freedom. The resulting Laplace transform
equation is:

$\begin{pmatrix}\underline{\phi}\\\ \underline{\dot{\phi}}\\\
\end{pmatrix}(s)=G_{plt}(s)\underline{u}_{d}(s)$ (30)

$u_{d}=\begin{pmatrix}v\\\ w\\\ \end{pmatrix}$: the input array is reduced
because only the damping in the output side is analysed and it is not
necessary for this to consider any additional control input.

$G_{plt}(s)=(sI-A_{plt})^{-1}B_{d}$ (31) $A_{plt}=\begin{bmatrix}0&1\\\
-\frac{K_{t}}{J_{t}}&\frac{1}{J_{t}}(-D_{t}-h_{t}^{2}\frac{\partial
F_{a}}{\partial v}+k_{\beta}h_{t}\frac{\partial F_{a}}{\partial\beta})\\\
\end{bmatrix}$ (32)

$A_{plt}$ is the right part of $A$ defined in 1.2 as $B_{d}$.

Looking at the $\phi$ degree of freedom, eq. 30 gives:

$\underline{\phi}(s)=G_{plt,1,1}(s)\underline{v}(s)+G_{plt,1,2}(s)\underline{w}(s),$
(33)

with:

$\left(G_{plt,1,1},G_{plt,1,2}\right)(s)=\begin{pmatrix}\frac{\frac{h_{t}}{J_{t}}\frac{\partial
F_{a}}{\partial v}}{s^{2}+\frac{1}{J_{t}}(D_{t}+h_{t}^{2}\frac{\partial
F_{a}}{\partial v}-k_{\beta}h_{t}\frac{\partial
F_{a}}{\partial\beta})s+\frac{K_{t}}{J_{t}}},\frac{\frac{1}{J_{t}}\frac{\partial\tau_{w}}{\partial
w}}{s^{2}+\frac{1}{J_{t}}(D_{t}+h_{t}^{2}\frac{\partial F_{a}}{\partial
v}-k_{\beta}h_{t}\frac{\partial
F_{a}}{\partial\beta})s+\frac{K_{t}}{J_{t}}}\end{pmatrix},$ (34)

i.e.:

$\left(G_{plt,1,1},G_{plt,1,2}\right)(j\nu)=\frac{1}{1-\left(\frac{\nu}{\nu_{plt}}\right)^{2}+2j\zeta_{plt}\frac{\nu}{\nu_{plt}}}\begin{pmatrix}\frac{h_{t}}{K_{t}}\frac{\partial
F_{a}}{\partial v},\frac{1}{K_{t}}\frac{\partial\tau_{w}}{\partial w}\\\
\end{pmatrix},$ (35)

$\nu_{plt}=\sqrt{\frac{K_{t}}{J_{t}}}\;,\qquad\zeta_{plt}=\frac{1}{2\sqrt{K_{t}J_{t}}}\left(D_{t}+h_{t}^{2}\frac{\partial
F_{a}}{\partial v}-k_{\beta}h_{t}\frac{\partial F_{a}}{\partial\beta}\right)$
(36)

Thus, when all interactions with rotor dynamics are neglected, the platform
behaves like a second order oscillatory system. The corresponding filter
$G_{plt}$ is a second order low-pass filter with cutoff angular frequency
defined by $\nu_{plt}$ and damping ratio defined by $\zeta_{plt}$.

###  1.5 Artificial damping of the platform : $\zeta_{plt}$-fixed strategy

By knowing the features of the FOWT, one can impose a given level of damping
and obtain an explicit expression for the $k_{\beta}$:

$k_{\beta}=\frac{1}{h_{t}\frac{\partial
F_{a}}{\partial\beta}}\left(D_{t}+h_{t}^{2}\frac{\partial F_{a}}{\partial
v}-2\sqrt{K_{t}J_{t}}\zeta_{plt}\right)$ (37)

The strategy is such that $k_{\beta}$ is a negative number instead of what is
proposed in ABBAS et al. (2021); STOCKHOUSE et al. (2021). In those articles,
$\beta_{comp}=k_{\beta}\dot{\phi}$, a positive value, is introduced in order
to erase at first order the coupling between platform and rotor dynamics, and
therefore is defined by :

$k_{\beta}=h_{t}\frac{\partial\tau_{a}}{\partial
v}/\frac{\partial\tau_{a}}{\partial\beta}$ (38)

#### Expected effect on platform dynamics

Figure 5 shows the second order low-pass filter $G_{plt}$. In other words, it
is how $\zeta_{plt}$ value can affect the damping of platform oscillations.

It can be observed that $\zeta_{plt}$ has a significant effect on the damping
of platform oscillations only for angular frequencies
$\nu\approx\nu_{plt,natural}$. Yellow vertical band in figure 5 shows the
interval of angular frequencies $I_{damped}$, arbitrarily defined by:

$I_{damped}=\left[\frac{\nu_{plt,natural}}{\sqrt{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \sqrt{2}\nu_{plt,natural}\right]$ (39)

that are directly damped when $\zeta_{plt}$ increases. Therefore, it is to be
expected that $\zeta_{plt}$-fixed strategy will be well fit to reduce platform
motion and tower loads when their variations happen at an angular frequency
$\nu\in I_{damped}$.

Figure 5: Bode diagram of second order low-pass filter $G_{plt}$

#### Expected effect on rotor dynamics

In this part the first order effect of $\zeta_{plt}$-fixed strategy on the
rotor dynamics is analysed. The state space representation of the FOWT
dynamics is given by equation 8. By considering $0$ input (ie.
$u_{c,ol}=u_{d}=0$), this leads to the following linear equation, truncated at
first order:

$\dot{\omega}=\ddot{\theta}=A_{2,1}\theta+A_{2,2}\dot{\theta}+A_{2,4}\dot{\phi},$
(40)

where

$\begin{split}A_{2,4}&=\frac{N_{g}}{J_{r}}\left(-h_{t}\frac{\partial\tau_{a}}{\partial
v}+k_{\beta}\frac{\partial\tau_{a}}{\partial\beta}-k_{\tau_{g}}N_{g}\right)\\\
&=\frac{N_{g}}{J_{r}}\left(-h_{t}\frac{\partial\tau_{a}}{\partial
v}+\frac{\frac{\partial\tau_{a}}{\partial\beta}}{h_{t}\frac{\partial
F_{a}}{\partial\beta}}\left(D_{t}+h_{t}^{2}\frac{\partial F_{a}}{\partial
v}-2\sqrt{K_{t}J_{t}}\zeta_{plt}\right)-k_{\tau_{g}}N_{g}\right).\\\
\end{split}$ (41)

Moreover, the following inequalities are verified for an above-rated operating
point:

$\begin{split}h_{t}\frac{\partial\tau_{a}}{\partial v}+k_{\tau_{g}}N_{g}&\geq
0\\\ \frac{\frac{\partial\tau_{a}}{\partial\beta}}{\frac{\partial
F_{a}}{\partial\beta}}&>0\\\
\frac{1}{2\sqrt{K_{t}J_{t}}}\left(D_{t}+h_{t}^{2}\frac{\partial
F_{a}}{\partial v}\right)&\leq\zeta_{plt}\\\ \end{split}$ (42)

First inequality comes from 20 (notice that in section 2, it is considered
$\tau_{g}=0$). Third inequality is a consequence of the assumption that
$\zeta_{plt}$-fixed strategy aims at increasing the damping of the platform.
This implies that:

$\frac{\partial}{\partial_{\zeta_{plt}}}\left|A_{2,4}\right|>0$ (43)

meaning that the first order coupling between platform dynamics and rotor
dynamics will increase when $\zeta_{plt}$ increases if $\zeta_{plt}$-fixed
strategy is applied. Thus, if the characteristic time of platform dynamics is
small enough, the equation truncated at first order is valid and it is to be
expected that, at least for some tunings of the PI controler,
$\zeta_{plt}$-fixed strategy would increase rotor speed variations.

##  2 Numerical tests with time domain simulations

In this section, it is analysed how the new control strategy described in the
previous section affects platform and rotor dynamics, and especially the
impact on tower loads and rotational speed. The reference is the control
strategy one without compensation, with $k_{\beta}=0$.

The $\zeta_{plt}$-fixed strategy has been implemented in the ROSCO environment
ROSCO (2021), replacing the existing pitch control. The rest of the controller
remains basically the same. In the next, $\zeta_{plt}$-fixed strategy is
compared to the one without compensation, named reference. The only difference
between the two terms of comparison concerns the $k_{\beta}$. For the
$\zeta_{plt}$-fixed strategy, it is given by 36. For the reference, it is
$k_{\beta}=0$.

###  2.1 Test cases

The simulation tool used is OpenFast v2.4.0
(https://github.com/OpenFAST/openfast) and the FOWT model considered is the
IEA 15 MW wind turbine mounted over the UMaine VolturnUS-S semi-submersible
floater ALLEN et al. (2020). Initially simple constant wind and monochromatic
waves are testes in order to verify the analytical developments of the
previous section. Then a test case more representative of the industrial
design of FOWT is considered by testing a DLC1.1. For the latter, simulation
consider only $1$ seed of $3600$ seconds with aligned wind and irregular
waves. For this time simulation, this is statistically equivalent to $600$
seconds and $6$ seeds.

###  2.2 Tuning of the PI controller

Values of $k_{P}$, $k_{I}$, $k_{\beta}$ are continuously updated following the
explicit expression given in equations 27 and 36, then low-pass filtered. This
means that globally, for each of these test cases and each set of parameters,
$k_{P}$, $k_{I}$, $k_{\beta}$ have almost fixed values. After some tests, for
all the considered simulations, $\zeta_{rot}=0.6$ and $\nu_{rot}=0.01$ are
chosen for the PI controller’s tuning. This choice ensures that most of the
wave spectrum (which peaks at $T\approx 11\leavevmode\nobreak\ s$, ie.
$\nu\approx 0.57\leavevmode\nobreak\ rads^{-1}$) and platform dynamics natural
angular frequency ($\nu_{plt}\approx 0.22\leavevmode\nobreak\ rad.s^{-1}$)
fall outside of $G_{rot}$ pass-band. In the next, several values of platform
damping are analyzed for the $\zeta_{plt}$-fixed strategy and compared to the
reference strategy for different test case.

###  2.3 Still wind and monochromatic wave

For the still wind and monochromatic wave condition, two $\zeta_{plt}$ are
tested: $\zeta_{plt}=0.1$ and $\zeta_{plt}=0.25$. The corresponding quality
factors are: $Q=5$ and $Q=2$, respectively. Thus, the platform is expected to
behave like an under-damped second order oscillatory system. Table 6 states
external conditions for test cases with still wind and monochromatic wave.
Hereafter, results are plotted over time are drawn for a $100$ seconds time
interval in a simulation on a long period of time, so that the operating point
is reached. When necessary for a better understanding, results are reported
for a longer interval.

Table 4: Environmental conditions for the numerical test cases. case |  wind speed $V$ ($ms^{-1}$) |  wave period $Tp$ ($s$) |  wave height $H_{w}$ ($m$)
---|---|---|---
(1) |  11 |  11 |  1.5
(2) |  11 |  28.75 |  1.5
(3) |  22 |  11 |  1.5
(4) |  22 |  28.75 |  1.5

Table 5 gives the mean value of $k_{\beta}$ for test cases (1) to (4). Cases
(1), (2) and cases (3), (4) are gathered together as they use the same mean
value of $k_{\beta}$.

Table 5: compensation gain ($k_{\beta}$). case |  $\zeta_{plt}=0.10$ |  $\zeta_{plt}=0.25$ |  reference
---|---|---|---
(1) and (2) |  $k_{\beta}=-8.6$ |  $k_{\beta}=-42.7$ |  $k_{\beta}=0.0$
(3) and (4) |  $k_{\beta}=-7.4$ |  $k_{\beta}=-34.8$ |  $k_{\beta}=0.0$

Figure 6: Platform pitch $\Phi$ (deg) for a monochromatic wave of period 11s
(test case (1) on the left and (3) on the right

Figure 7: Platform pitch $\Phi$ (deg) for a monochromatic wave of period
28.75s (test case (2) on the left and (4) on the right

Figures 6 and 7 show the forced oscillations of the platform when it is
subjected to a monochromatic wave of period $11s$ (which corresponds to the
realistic fundamental period of a wave) and $28.75s$ (which is the natural
period of the platform). As shown by the analytical development, increasing
$\zeta_{plt}$ reduces platform oscillations, especially when the wave period
is close to the natural period of the platform (see diagram 5). This is shown
in Figures 8 and 9 where the density of occurrence of each value of tower
bottom reaction moment is reported. For the monochromatic wave with period
$11s$ the damping of the $\zeta_{plt}$-fixed strategy is less evident. More
floating wind test cases are reported in the next section.

Figure 8: Tower base moment (MN.m) densities for a monochromatic wave of
period 11s (test case (1) on the left and (3) on the right

Figure 9: Tower base moment (MN.m) densities for a monochromatic wave of
period 28.75s (test case (2) on the left and (4) on the right

Figure 10: Rotor speed (rpm) evolution over time for test cases (1) on the
left and (2) on the right

Figure 11: Rotor speed (rpm) evolution over time for test cases (3) on the
left and (4) on the right

$\zeta_{plt}$-fixed strategy’s effect on rotor dynamics is not easily
described by a second order linear equation: it involves the coupling between
platform and rotor dynamics, which was analysed at first order in 1.5. From
this analysis, $\zeta_{plt}$-fixed strategy was expected to increase the
coupling between platform and rotor dynamics for a short characteristic time.
In Figure 10 and 11 it can be observed that for a short characteristic time
($11s$) $\zeta_{plt}$-fixed strategy increases rotor speed variations, but not
for a longer characteristic time, such as $28.75s$ (for which it behaves
slightly better than $k_{\beta}=0$ strategy).

To conclude this part of the tests: $\beta$-compensation strategies perform
very differently depending on the oscillatory frequency of the platform:

  * •

For angular frequencies $\nu\in
I_{damped}=\left[\frac{\nu_{plt,natural}}{\sqrt{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \sqrt{2}\nu_{plt,natural}\right]$), $\zeta_{plt}$-fixed
strategy is very effective when it comes to the damping of platform
oscillations, as seen in paragraph 1.5 (cf. figure 5). The tests highlight
that $\zeta_{plt}$-fixed strategy is reducing both tower loads and rotor speed
variations in turbulent wind conditions.

  * •

For angular frequencies outside the previous set, $\zeta_{plt}$-fixed strategy
is less effective for damping platform oscillations. Tower loads reduction by
$\zeta_{plt}$-fixed strategy is therefore barely visible, whereas rotor speed
variations are actually amplified, especially when comparing this strategy to
reference strategy.

###  2.4 DLC1.2 tests

Tests presented hereafter are more representative of what is done during
design or verification of offshore wind structure. As already done for the
previous test cases, the $\zeta_{plt}$-fixed strategy is compared to the
reference strategy, $k_{\beta}=0$. They are inspired by the DLC 1.2, for
normal power production in normal turbulence and normal sea state, as
described in IEC standards. This kind of load case aim at assessing the
fatigue design criteria. Kaimal’s turbulence model is considered following IEC
61400 v.3 for a Wind Turbine of turbulence type B, for average wind speeds
from $12ms^{-1}$ to $24ms^{-1}$, as described in Table 6. The wind box is
generated by TurbSim tool developed by NREL. For the waves, JONSWAP
distributions are considered with significant wave height $Hs=1.5m$, wave
period $Tp=11.0s$ and $\gamma=2.0$. Wind and waves are considered aligned in
the same direction.

For those test cases, the level of damping imposed to the platform is
$\zeta_{plt}$-fixed = $0.1$. This value is found to be the interesting to be
tested for this floater and WTG configuration. Other tests with higher values
of imposed damping show less interesting results. The choice of the right
$\zeta_{plt}$ for each FOWT system is important and demands for some
iterations before concluding.

Table 6: Environmental conditions for DLC 1.2. Time sim [$s$] |  w.speed [$m.s^{-1}$] |  w. condition |  $Tp$ [$s$] |  $H_{s}$ [$m$] |  $\gamma$ |  waves dir.
---|---|---|---|---|---|---
$3600$ |  $12.0-24.0$ |  Normal turbulence B |  $11.0$ |  $1.5$ |  $2.0$ |  co-linear
Figure 12: evolution of $k_{\beta}$ for some of the simulations.

Figure 12 shows how $k_{\beta}$ evolves for some cases of the simulation pool.
The below-rated behaviour is more dynamic than the over-rated, where the
floating feedback is more stable. As remarked in 1.5, the $k\beta$ for the
$\zeta_{plt}$-fixed is a negative value. This is the novelty of this strategy
with respect to other floating feedback strategies, as the one proposed in
ABBAS et al. (2021).

From the time series of the tower bottom moment, a rainflow algorithm is used
to count the cycles following ASME normative for steel structures ASME (2011).
The DEL is obtained by using a Wohler’s curve with a single slope of exponent
$m=3.0$. Platform pitch, power, rotor speed, blade pitch, tower load and tower
DEL results of the simulations for the comparison are resumed in Figure 13.
The $\zeta_{plt}$-fixed strategy reduces platform pitch motion for all wind
speeds. As expected this is done by the coupling with the rotor and the use of
the blade pitch. In fact, blade pitch is more used by considering the
$\zeta_{plt}$-fixed strategy. This was expected by the analytical development.
As shown, the coupling between platform pitch and rotor dynamics is increased.
The increase in pitch utilisation explains the slight decrease in rotor speed
for the above rated wind speeds. The generator power is also affected,
slightly increased for lower wind speeds and slightly decreased for higher
wind speed (around $+/-1\%$).

Loads in the tower are reduced and fatigue Design Equivalent Load (DEL) also.
This is remarkable for around rated speed. For high wind speeds, the gain is
less evident. In average there is a gain around $15\%$ of the DEL. Table 7
shows, for the $10$ $ms^{-1}$ case, a deeper comparison by reporting the
statistics of the quantities of interest extracted from this simulation. The
difference in the minimum value of the moment at the tower base is remarkable:
passing from $80.97$ $MNm$ for the reference up to $165.40$ $MNm$ for the
$\zeta_{plt}$-fixed strategy. This is also clear in the difference of the
standard deviations. The amplitudes of the oscillations of this load are
reduced.

It is interesting to analyse Figure 14. This figure reports a deeper analysis
of the fatigue damage. In fact, the stress in the tower bottom section is
obtained by considering the design proposed by UMaine in ALLEN et al. (2020).
Then, an offshore Wohler’s curve is considered with two slopes in the log-log
domain: $m=3.0$ for loads with less than $1.0$ million cycles and $m=5.0$ for
loads with higher number of cycles. Those are typical values proposed by DNV
for offshore steel structures. This analysis leads to obtain an estimation of
the $25$ years damage at the tower bottom. The gain is much more evident than
the DEL. This is due to the second slope, $m=5.0$, which amplifies the changes
in the load amplitudes. Offshore WTG in production are mostly subjected to a
very high number of cycles of small amplitudes. This figure shows also the
effect of the turbulence on the fatigue. In fact, looking at the reference, up
to $12$ $ms^{-1}$, the shape of the damage distribution follows the one of the
thrust curve. However, since the turbulence is a percentage of the average
wind speed, from $16$ $ms^{-1}$, the damage starts increasing again.

Figure 13: Comparison results for the DLC1.2 for the UMaine floater with IEA15MW WTG. The imposed level of damping in the platform dynamics for the $\zeta_{plt}$-fixed strategy is $0.10$. Output show statistics for platform pitch; blade pitch; Tower bending moment, max and damage equivalent load; rotor speed and generator power.  Table 7: Statistics for results concerning the case with mean wind speed at $10$ $ms^{-1}$. For each quantity of interest, there is the comparison of the minimum, maximum, mean and standard deviation values produced by the $\zeta_{plt}$-fixed control strategy and the reference. |  $\zeta_{plt}=0.1$ |  $k_{\beta}=0$ |  $\zeta_{plt}=0.1$ |  $k_{\beta}=0$ |  $\zeta_{plt}=0.1$ |  $k_{\beta}=0$ |  $\zeta_{plt}=0.1$ |  $k_{\beta}=0$
---|---|---|---|---|---|---|---|---
|  min |  min |  mean |  mean |  max |  max |  st.d. |  st.d
PtfmPitch [deg] |  2.84 |  1.09 |  4.55 |  4.56 |  5.62 |  5.91 |  0.51 |  0.59
TwrBsM [MNm] |  165.40 |  80.97 |  366.66 |  367.79 |  493.80 |  505.10 |  43.20 |  46.82
GenPwr [MW] |  7.85 |  5.49 |  12.09 |  12.01 |  15.15 |  15.19 |  1.29 |  1.47
BldPitch [deg] |  0 |  0 |  0.218 |  0.176 |  6.651 |  8.33 |  0.394 |  0.729
RotSpeed [rpm] |  5.23 |  5.34 |  6.90 |  6.92 |  7.90 |  8.00 |  0.61 |  0.60
Figure 14: Fatigue cumulative damage at tower bottom by using rainflow
counting and linear Miner’s rule. The damage is obtained considering the tower
base design proposed by the UMaine, a Wohler’s curve bi-linear with $m=3.0$ up
to $10^{6}$ cycles and $m=5.0$ after, as proposed by DNV for Offshore steel.
The probability of occurrence of each wind is equal, without any weibull
distribution.

The first part of this paper presents the analysis of the NMPZ related to the
system of questions describing the dynamcs of a floating offshore wind turbine
(FOWT). The equation of the rotor dynamics and the one of the platform
dynamics are analysed in the complex domain to explicit the conditions leading
to the two respective NMPZs. Two analytical examples are considered, one per
each NMPZ condition, to compare the behaviour of a system affected by a NMPZ
with another one outside the NMPZ domain. One of those NMPZ, instability given
by the blade pitch on the rotor dynamics, is already known in literature and a
compensation already exists to avoid it. Another one, related to instability
given by the blade pitch to the platform dynamics, is a novelty in the
community. The effects of the NMPZs are analysed on two analytical test cases:
at the beginning both $\omega$ and $\dot{\phi}$ always converge to the right
solutions just after the first steps. When both NMPZ conditions are not
verified, those tendencies don’t change. However, when the $\phi$-NMPZ is
verified, $|\dot{\omega}|$ becomes so big that $\dot{\phi}$ jumps into
unexpected values without converging to the expected solution. Similarly, when
the $\omega$-NMPZ condition is verified, $|\dot{\phi}|$ becomes so big that
$\omega$ oscillates before converging to the expected solution. NMPZs can
cause important shifts and unexpected behaviors for both $\omega$ and $\phi$.

In the second part of the paper, the damping analysis is further investigated
proposing a new strategy control for FOWT, named $\zeta_{plt}$-fixed. This
strategy focuses on the coupling between the rotor dynamics and the floating
platform dynamics and it is based on a compensation parameter $k_{\beta}$
proportional to the platform pitch velocity. The idea behind the control
strategy developed in this work is to use the blade pitch to damp the platform
motions. An explicit expression linking $k_{\beta}$ to $\zeta_{plt}$ (damping
ratio imposed to the platform) is derived by defining a second order filter on
the equation of the platform dynamics. This is different with respect to
already existing strategies based on platform pitch compensation ABBAS et al.
(2021); STOCKHOUSE et al. (2021) which focus on decoupling rotor and platform
dynamics. This difference is underlined by the values of $k_{\beta}$, which is
negative for the new control strategy, while it is positive for the ones
existing in literature. The performances of the $\zeta_{plt}$-fixed strategy
are tested analytically first and, then, numerically by considering an
OpenFAST numerical twin of the Umaine IEA15MW FOWT. For a test representative
of the DLC1.2, the $\zeta_{plt}$-fixed strategy allows to reduce the loads at
the tower foundation interface for all the considered wind speeds, without
significant losses in terms of power production. The damage analysis shows a
remarkable gain in terms of fatigue lifetime. The blade pitch use slightly
increases remaining in the bounds of a standard controller limitations. For
each FOWT system, some iterations are necessary in order to find the optimum
value for $\zeta_{plt}$. This work highlights the importance of defining
proper controller strategies for FOWT in order to reduce loads on the
structure or improve the performances and, then, helps the industry to achieve
the objective in terms of LCOE reduction.

Matteo Capaldo has contributed for the original idea of the new control
strategy, the development of the numerical twin and the numerical tests and he
is the main contributor for the paper editing. Paul Mella developed the
mathematical framework, he has contributed for the original idea of the new
control strategy and he has contributed for the paper editing

The authors declare there are not competing interests in this work.

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