In this paper, the coupled dynamics of the floating platform and the wind turbine rotor are analyzed. In particular, the damping is explicitly derived from the coupled equations of the rotor and floating platform. The analysis of the damping leads to the study of instability phenomena, thus obtaining the explicit conditions that lead to the non-minimum phase zero (NMPZ). Two NMPZs are analyzed, one related to the rotor dynamics and the other one to the platform pitch dynamics. The latter introduces a novelty, and an explicit condition is provided in this work for its verification. In the second part of the paper, from the analysis of the damping of the floating platform, a new strategy for the control of floating offshore wind turbines (FOWTs) is proposed. This strategy allows one to impose on the controller an explicit level of damping in the platform pitch motion that adapts with wind speed and operating conditions without changing the period of platform pitching. Finally the new strategy is compared to one without compensation and one with a non-adapting compensation by performing aero-hydro-servo-elastic numerical simulations of a reference FOWT. Generated power, motions, 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 very high potential both onshore and offshore. In terms of installed capacity, onshore wind is still the largest contributor. However, the new annual offshore installed capacity is estimated to exceed 30 GW by 2030 in order to stay on track for a net zero/1.5

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 FOWTs.
One effective way to achieve this objective is to investigate different strategies for the control of FOWTs.
As explained in

These phenomena can be classified into two families: one is related to undesired motions of the platform, even if the system is still stable. These are the non-minimum phase zeros (NMPZs). They are associated with the zeros of the transfer function describing the system. The other family is associated with the damping of the system, which is related to the poles of the transfer function, and can affect the system stability.

The nature and the set of control parameters leading to these phenomena can vary from one platform design to another, e.g., a barge, spar, semi-sub or tension-leg platform.
However, for each of the platforms, there exist sets of design and control parameters leading to undesired behaviors

Bottom-fixed control strategies normally consider a squared law for the electrical torque control (below-rated wind speeds) and
a set of integral and proportional coefficients (the pitch scheduling) to control the rotational speed by the blade pitch and operate the wind turbine in the desired steady-state conditions (above-rated wind speeds)

To adapt this control strategy to FOWTs, a first compensation is considered in this work.
It aims at solving the NMPZ effects caused by the blade pitch on the rotor rotational dynamics.
This solution has already been introduced in

A second compensation considered in this work aims at solving the issue of the coupling between the platform motions and the rotor dynamics, leading to non-damped oscillations of the systems. This phenomenon can even be amplified when the bottom-fixed pitch control is considered for a floating wind turbine. The issue comes from the fact that in above-rated wind speed, the blade pitch regulates the speed by increasing the angle of attack to feather. For an FOWT, when the platform has a forward motion, the rotor experiences an increasing wind speed; this means an increasing aerodynamic torque, which tends to accelerate the rotor. 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 reduces the rotor thrust, which 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 controller.

Solutions exist to avoid this phenomenon.
The first one and the most common in the literature is to reduce the blade pitch control proportional and integral gains in order to reduce its bandwidth and to exclude the platform pitch and surge natural frequencies

The control strategy proposed in this work shares some similarities to the ones introduced in

Higher-order controllers, such as a linear quadratic regulator (LQR), are applied and evaluated in

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 the 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 before in the literature, is discovered and analyzed 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 us to impose an explicit level of damping in the platform pitch motion on the controller without changing the period of platform pitching. The chosen strategy is then compared to one without platform pitch compensation (with detuning) and one that considers a single value of the compensation parameter for every wind speed and operating condition.

The paper is organized as follows. In Sect.

The floating offshore wind turbine (FOWT) is represented by a system of 2 degrees of freedom, namely the rotor speed

A scheme of the considered system with 2 degrees of freedom,

The surge degree of freedom is not considered in this model.
In fact, the surge speed of the FOWT can be neglected with respect to the speed at nacelle generated by the pitch motion of the platform. This has already been mentioned in

Two control parameters,

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

Once a steady-state operating point

The relative wind

Equations (

The pitch controller model is described in this section.

The present control model considers

proportional,

integral,

blade pitch (

generator torque (

Controllers described by the literature considering the same compensations

For an FOWT, the objective of the pitch control is to remain in the equilibrium operating point.
It translates to

The controller 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,

integral,

blade pitch (

generator torque (

Figure

Block diagram of the controller model.

The linear expression of

The time domain system can be rewritten in the Laplace complex domain. Using the following notation,

By defining

Here,

This section analyzes the problem of negative damping by addressing the positions of the zeros of each component of

This translates the fact that

Physically

For the rest of the section, an open-loop control on

The gain equation in the Laplace domain for

The condition for the NMPZ on

Intuitively, this corresponds to an operating point where

In the absence of NMPZ, i.e., Eq. (

When Eq. (

Figure

The set of parameters to show the NMPZ of Eq. (

Platform pitch

The gain equation in the Laplace domain for

Hence the condition for NMPZ on

This corresponds with an operating point where

In the absence of NMPZ, i.e., Eq. (

The set of parameters chosen to show the NMPZ of Eq. (

Rotor speed (

The comparison between Figs.

NMPZ

In order to complete the analysis of NMPZ phenomena related to FOWT systems,
a hypothetical situation where both Eq. (

A hypothetical situation where both Eq. (

The set of parameters chosen to show the instability given by NMPZs of Eqs. (

Pole–zero analysis of the systems described in Tables 1, 2 and 3:

A pole–zero plot is a commonly used synthesis of both NMPZ and stability issues. The above case-by-case analysis highlighted the drawback of allowing a zero of the transfer function in the right half of the complex plane. Similarly, the stability of a system can be well synthesized by the position of the poles of the transfer function. Poles of the transfer function situated in the right half of the plane result in a global instability, such as the one observed in Fig.

Figure

The issue related to the

The complete prevention of the problem can be obtained by several sets of parameters that involve the WTG,
the floating platform and the control setup.
In fact, for this NMPZ, Eq. (

In the left plot of Fig.

Pole–zero analysis of the systems described in Tables 1 and 3, with the parameter

The right plot of Fig.

In Sect.

Considering the whole system, with both degrees of freedom

The term in square parentheses represents the coupling term between the dynamics of the platform (

In this coupled form, it is complicated to explicitly determine the damping of the system.
In the next paragraph, under a hypothesis, the coupled system can be separated into two second-order systems, one related to the rotor dynamics

Defining a damping coefficient (or a damping ratio) requires us to reduce the global system to a second-order oscillatory system.
Equations (

Under such assumptions, the linear form of Eq. (

Thus, when all interactions with platform pitch are neglected, the rotor behaves like a second-order oscillatory system. The corresponding filter

In the case of

The above formulas enable one to obtain explicitly

They are well known: several controllers, such as in

Similarly to what is done in the previous paragraph, here the global system of Eqs. (

We consider

Looking at the

Thus, when all interactions with rotor dynamics are neglected, the platform behaves like a second-order oscillatory system.
The corresponding filter

By knowing the features of the FOWT, one can impose a given level of damping and obtain an explicit expression for

The strategy is such that

In

Figure

Bode diagram of the second-order low-pass filter

It can be observed that

In this part the first-order effect of the

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

The first inequality comes from Eq. (

In this section, it is analyzed how the new control strategy described in the previous section affects platform and rotor dynamics and, especially, tower loads and rotational speed.

The simulation tool used is OpenFAST v3.4.0 (

The controller strategies are implemented in the ROSCO environment

For the still wind and monochromatic wave condition, the

For the readers more used to quality factors, the corresponding values are

Environmental conditions for the numerical test cases.

This test considers fixed operating points; thus,

Compensation gain (

Figures

Platform pitch

Platform pitch

Figure

Blade pitch (deg) evolution over time for a monochromatic wave of a period of 28.75 s (test case 1).

Tower base moment (MN m) evolution over time for test case 1

Tower base moment (MN m) evolution over time for test case 2

The

Rotor speed (rpm) evolution over time for test cases 1

Rotor speed (rpm) evolution over time for test cases 2

To conclude this part of the tests, the

For angular frequencies

For angular frequencies outside the previous set, the

The tests presented hereafter are more representative of what is typically done during the design or verification of offshore wind structures. They are inspired by DLC1.2, for normal power production in normal turbulence and a normal sea state, as described in the IEC standards. This kind of load case aims 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 ranging from 4 to 24 m s

Environmental conditions for DLC1.2.

As with the previous test cases, the

The level of damping imposed on the platform is

Table

The controller strategies considered for the DLC1.2 tests.

Figure

The evolution of

From the time series of the tower bottom moment, a rainflow algorithm is used to count the cycles following the American Society for Testing and Materials (ASTM) normative

Comparison results for the DLC1.2 for the UMaine floater with the IEA 15 MW WTG. Outputs show statistics for platform pitch (average and std), blade pitch, tower bending moment (max and damage equivalent load), rotor speed (average and max) and generator power (average and max).

Around rated wind speed, the

An extract of the time series concerning the case with a mean wind speed at 10 m s

Figure

Statistics for results concerning the case with a mean wind speed at

The fatigue cumulative damage at the tower bottom by using rainflow counting and Miner's linear rule. The damage is obtained considering the tower base design proposed by UMaine and Wöhler's bi-linear curve with

A specific fatigue analysis of the pitch bearing is realized by following

The bearing life is inversely proportional to the cube of the bearing loading.
From the overturning moment acting on the bearing, the equivalent loading at

Damage equivalent loading at

The first part of this paper presents the analysis of the NMPZ related to the system of equations describing the dynamics of a floating offshore wind turbine (FOWT).
The equation of the rotor dynamics and the one of the platform dynamics are analyzed in the complex domain to explicitly derive the conditions leading to their respective NMPZs.
One of those NMPZs, i.e., the instability given by the blade pitch on the rotor dynamics, is already known in the literature, and a compensation already exists to avoid it.
The other one, i.e., the instability given by the blade pitch to the platform dynamics, is a novelty in the community. The effects of the NMPZs are analyzed on two analytical examples: at the beginning both

In the second part of the paper, the damping analysis is further investigated while proposing a new strategy control for FOWT, named

This is different with respect to the existing platform pitch compensation strategies, which aim to decouple rotor and platform dynamics.
The difference is underlined by the value of

This work highlights the importance of defining proper controller strategies for FOWTs in order to reduce loads on the structure or to improve the performance. Accordingly, it is useful for helping the industry to achieve its objective in terms of LCOE reduction.

The original ROSCO controller code is available at this address:

MC contributed to the original idea of the new control strategy, and the development of the numerical twin and the numerical tests, and he is the main contributor to the paper in terms of editing. PM developed the mathematical framework and contributed to the original idea of the new control strategy and to the editing of the paper.

The contact author has declared that neither of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful for the support of the R&D Wind program of TotalEnergies OneTech, which made this research possible.

This paper was edited by Erin Bachynski-Polić and reviewed by three anonymous referees.