A control-oriented model for floating wind turbine stability and performance analysis

A well-designed floating turbine controller should lead to small power fluctuations while keeping the system stable. Floating wind turbines can experience instability when operating above rated wind speed, due to the blade pitch controller. This phenomenon, known as negative aerodynamic damping, is the reason why a controller needs to be re-designed for floating applications. One solution is to de-tune the blade pitch controller and make it slower than the floater pitch motion. This solution, although simple, reduces the controller’s ability to react to changes in the inflow and results in large power fluctuations. A more sophisticated approach involves an additional loop feeding back the fore-aft nacelle velocity to the blade pitch or generator torque. The choice of gains, however, is not trivial. This work proposes a control-oriented model derived from first principles to investigate different controller configurations in terms of both stability and turbine performance. The linear model with seven degrees of freedom (six rigid-body motions and drivetrain) is formulated analytically and coupled with the controller. The linear formulation allows us to investigate stability via eigenvalue analysis and performance via frequency-domain response, thus bypassing the need for time-domain simulations. Parametric studies with thousands of different controller configurations can be completed in a matter of minutes.


Introduction
Offshore winds are typically stronger and steadier and to fully exploit the good wind resources, there is a growing interest in placing wind turbines on a floating platform in deep waters.Locating the turbine on a floating platform complicates the turbine design.One well-known challenge is the negative damping problem.The turbine blade pitch control regulates the rotor speed in above-rated wind conditions by pitching the blades.It does not only reduce the rotor aerodynamic torque but also the aerodynamic thrust, which creates an interaction between the turbine blade pitch loop and the fore-aft motion of the turbine rotor.Compared to a bottomfixed turbine, the fore-aft dynamics of a floating turbine is typically lightly damped with a lower natural frequency.Such coupling would inevitably decrease the damping of the closed-loop system due to the fast pitch regulation compared to the low natural frequency of the platform motion.That would cause a large fluctuation in power, rotor speed and turbine response.Therefore, this motivates the development of floating turbine control.The negative damping problem has been widely studied over the past decade.The simplest approach is to de-tune the onshore controller or reduce the control bandwidth, preventing the blade pitch from responding fast enough to excite the platform motion [1,2].In other words, the coupling between the blade pitch loop and platform pitch motion is weakened by reducing the controller gains.This method has been proven successful in many experimental studies [3,4,5].A study by [6] developed a gain-scheduled de-tuning method to overcome the system nonlinearity and later, a study by [7] extended the method with a robust formulation.Despite its simplicity, the efficacy of this method depends largely on the type of floater and the performance criteria in terms of power quality and turbine structural loads.
Alternatively, the use of an additional sensor has been proposed.A 2008 patent [8] introduced an additional loop by adding the feedback of the nacelle velocity to the blade pitch.The capabilities of the method was further demonstrated by [2,9,10].In contrast, a study by [11] took a different approach by feeding the nacelle velocity to the generator torque, which allows the blade pitch control bandwidth to increase further but at the expense of the generator torque.Besides using simple gain and filters, higher-order controllers were also studied.For example, [12] and [13] proposed linear quadratic regulator (LQR) and disturbance accommodating control (DAC) blade pitch controllers, respectively.A study by [14] developed a wave forecast algorithm combined with feed-forward LQR.A nonlinear model predictive controller proposed by [15] exploits both blade pitch and generator torque.
Instead of using feedback, [16] introduced an additional LiDAR-assisted feed-forward blade pitch signal overcoming the negative damping problem.In addition, [17] used the nacelle velocity signal to adjust the rated rotor speed set-point, aiming to reduce the platform motion.The method was revisited by [18], who demonstrated that the platform pitch motion could be damped without changing the platform pitching frequency.An overview of the recent development in floating turbine control can be found in [19,20].This paper proposes a control-oriented model derived from first principles with two main advantages.First, the analytical and simple formulation of the model offers an understanding of how the controller affects the floating turbine in terms of contributions to the mass, damping and stiffness matrices.Second, the linear formulation enables efficient stability assessment via eigenvalue analysis, and fast response computation via a transfer function.Both insights render controller tuning for floating turbines more intuitive and computationally efficient.

Numerical model formulation
We consider the floating wind turbine shown in fig. 1.The origin is located at the flotation point, with positive x pointing downwind and positive z upwards.The rigid-body motion of the structure is caused by hydrodynamic loads F wave and by the aerodynamic thrust T , which attacks at a height z hub from the waterline.The rotor speed Ω and blade pitch θ affect the aerodynamic torque Q, which counteracts the generator torque Q g .
The model is formulated with all variables referring to the deviation from the steady-state operation point, e.g. for the thrust ∆T (t) = T (t) − T op .The steady states depend on the mean wind speed V op , and their calculation is not part of the present model.For simplicity, we make the following assumptions: • All structural elements are rigid bodies (i.e.no tower or blade deformations).
• The environmental loads are applied with the floater in the upright position.
• The aerodynamic thrust acts as a horizontal force applied at the hub.• The turbine includes a direct-drive generator (i.e.no gearbox).
• The generator torque Q g does not excite the roll degree of freedom.
• The turbine always operates in the full-load region.

The floater motion
The motion of the floating wind turbine is governed by the 6×6 system of equations where M is the structural mass matrix, A is the added mass matrix, B represent hydrodynamic damping and C is a stiffness matrix including hydrostatic and mooring effects.Further, the displacement from the steady state in surge, sway, heave, roll, pitch and yaw is given by ∆ξ while the vector of environmental loads is

The drivetrain motion and controllers for blade pitch and generator torque
The drivetrain motion is governed by the equation where I dt is the drivetrain inertia, η is the generator efficiency, and the aerodynamic torque For simplicity, we adopt the following notation for the rotor speed: Since we pursue a linear model, we linearize the aerodynamic torque using a first-order Taylor expansion: We note that all the gradients are evaluated at the steady-state operation point.The generator torque, considering both constant torque and constant power strategies, can be linearized in a similar way, A constant torque strategy, which gives ∂Qg ∂Ω =0, is commonly used for floating wind turbines.For a classical PI (proportional-integral) blade pitch controller, the deviation in blade pitch ∆θ includes a proportional term acting on the rotor speed error, and an integral term acting on the integrated error, defined as follows:

Introducing additional feedback loops
To overcome the negative damping problem, the feedback of the nacelle velocity is sometimes used for controlling the blade pitch and/or generator torque.From fig. 1 one can see that the fore-aft nacelle velocity ẋnac is a linear combination of the floater velocities in surge and pitch, ẋnac = ξ1 + z hub ξ5 .
Besides the classical PI blade pitch controller, in floating configurations one can add a feedback loop from the nacelle velocity to the blade pitch (sometimes called tower-top velocity loop, nacelle feedback loop or pitch damper ), Similarly, an additional feedback loop from the nacelle fore-aft velocity to the generator torque (sometimes referred to as torque compensator ) is defined as follows, where the negative sign has been introduced to avoid working with negative values of the gain k q .

Coupling the floater, drive-train and controllers
First, the perturbation in wind speed at the hub ∆V is a result of turbulence and floater motion, thus After combining eqs.( 10) and ( 12) with eq. ( 6), the aerodynamic torque becomes Subsequently, combining eqs.( 11) and ( 13) with eq. ( 4) and knowing that by definition Q op = 1 η Q g,op yields: Next, the aerodynamic thrust can be linearized using a similar first-order Taylor expansion on T = T (V, Ω, θ): Making use of eqs.( 10) and ( 12) again, we can write The next step is to extend the 6×6 system of eq. ( 1) to 7×7 by adding an extra row and column of zeros to M , A, B and C, as well as an extra row with zero to ∆ξ and ∆F .This 7 th degree of freedom is the integral of the rotor speed error, ϕ.The 7×7 system thus becomes Now we insert eqs.( 14) and ( 16) into ∆F at the right-hand side, and all motion-dependent terms are moved to the left-hand side as contributions to the matrices M , B and C.These aero-servo-elastic contributions to the different matrices are detailed in Appendix A. The final equations of motion for the 7×7 system are We note that the matrices M , B and C in eq. ( 18) are those in eq. ( 17) plus the aero-servoelastic contributions shown in Appendix A. In eq. ( 18) all elements on the right-hand side depend on external excitation (wind turbulence and wave loads).The system matrices contain all the information necessary to investigate the dynamic stability of the coupled system for different mean wind speeds V op and different combinations of the gains k p , k i , k b and k q .We further note that by setting k b =0 and k q =0 the nacelle fore-aft velocity feedback control loops are disabled and the blade pitch controller is reduced to a standard onshore turbine controller.

Numerical model implementation
The model described in section 2 has been implemented numerically for the IEA Wind 15MW reference wind turbine [21] mounted on UPC's WindCrete spar-buoy [22].With basis on eq. ( 1), and with the goal of making the model as simple as possible, the added mass matrix has been taken as the zero-frequency limit calculated in WAMIT [23].A constant damping matrix that represents radiation and viscous effects has been included as well.The stiffness matrix contains WAMIT-based hydrostatic contributions and a mooring matrix obtained for each wind speed, as in [24].On the right-hand side, the wave loads are obtained using WAMIT's linear transfer function together with the Fourier amplitudes of free-surface elevation for a given sea state.
The natural periods of the rigid-body motion, obtained by eigenvalue analysis of the 6×6 system, are shown in table 1.When compared to the natural periods reported in [22], the values estimated by the model are within 9% error for the horizontal degrees of freedom and within 6% error for the vertical ones.This indicates that the dominant source of error is the mooring system linearization.We note that, although the sway and roll natural periods are not included in [22], they are expected to be very close to the surge and pitch periods, respectively.On the rotor side, the aerodynamic gradient values are taken from [25] (see fig. 2).These were calculated by central finite differences, using HAWC2 [26] to evaluate the thrust and torque on the rotor for each operation point needed.Figure 2. Aerodynamic gradients calculated in HAWC2 as given in [25].

Results
The results presented in this section correspond to a generic environmental condition with wind and waves.The sea state is described by a Pierson-Moskowitz spectrum with a significant wave height of 6 m and a peak period of 10 s.The mean wind speed is 12 m/s with a turbulence intensity of 0.14.All dynamic results are based on a 10-minute time series of response calculated by the numerical model.
When referring to a certain controller, we report the natural frequency f c and damping ratio ζ corresponding to the one-degree-of-freedom drivetrain system (without coupling to the floater motion), which are directly related to the proportional and integral controller gains k p and k i .
In addition, we report the gains for the two nacelle-velocity feedback loops, namely k b and k q .We call de-tuned controllers those with a natural frequency f c below the floater pitch frequency at 0.0231 Hz, and fast controllers those with a higher natural frequency, that would be unstable without the nacelle-velocity feedback loops.

Dynamic response
First, the dynamic response for a de-tuned controller (f c =0.02 Hz, ζ=0.7, k b =0, k q =0) is shown in fig. 3.Here we show time series and PSD (Power Spectral Density) plots of free-surface elevation, floater surge, heave and pitch on the left half, as well as wind speed at the hub, rotor speed, blade pitch and power on the right half.For validation purposes, we show in blue the response obtained using a 4 th -order Runge-Kutta time-marching algorithm.In red we show the response obtained in the frequency domain and transformed to the time domain via inverse fast Fourier transform.The legend shows the standard deviation of the given signal from the time-and frequency-domain calculation, respectively.To avoid the transient present in the timedomain solution, the first 10 min have been left out.As expected, we observe that after the transient the two solutions agree completely.In the PSD plots we see that all responses except heave are dominated by the pitch natural frequency at 0.0231 Hz.This is because although this de-tuned controller does not trigger the floater pitch instability, its natural frequency is not too far from the stability limit, hence the damping ratio for the floater pitch mode is small.A similar analysis has been carried out for a faster controller with the nacelle-velocity-togenerator-torque loop active (f c =0.05 Hz, ζ=0.7, k b =0, k q = 2.5e6 Nm/(m/s)), see fig. 4. When comparing to fig. 3, a few differences can be observed.First, the responses are not anymore entirely dominated by the pitch natural frequency, thus the footprint of the wind and wave spectra can be seen, for example in the PSD of surge or power.Second, all standard deviation values are smaller for the faster controller than for the de-tuned controller (especially for rotor speed, where a factor 4 reduction is achieved).

Parametric study
The linear response format of the numerical model allows the efficient evaluation of a large number of parameter combinations.In fig. 5 we show parametric results for 10,000 different de-tuned controllers, a calculation that took approximately 3 min of CPU time in a standard laptop.We have kept f c =0.02 Hz and ζ=0.7, while varying k b ∈[0,0.1]rad/(m/s) and k q ∈[0,5e6] Nm/(m/s).These ranges of k b and k q are rationally chosen to be within physical limits of the given wind turbine (i.e.maximum changes of 5.73 deg in blade pitch and 25% of the rated generator torque for a nacelle velocity of 1 m/s).For each single controller, the damping ratios of the surge, pitch and drivetrain modes are obtained by eigenvalue analysis.These are shown as contour plots on the top row of fig. 5, where the damping ratio grows from red to blue.In addition, the dynamic response to wind and waves is obtained in the frequency domain and the standard deviation of rotor speed, blade pitch and power is calculated.These are shown in the contour plots on the bottom row, where the standard deviation grows from blue to red.This choice of color scales makes the interpretation of the contours easier, since blue always corresponds to desirable scenarios with high damping ratios for the floater motion and small standard deviations for the turbine operation.For example, in fig. 5 the most damped floater pitch motion is achieved for low values of k b and high values of k q , whereas k b =0 and k q =0 (a pure de-tuned controller) yields the most stable power output.
A similar parametric study was carried out for 10,000 fast controllers, as shown in fig.6.Here we have fixed f c =0.05 Hz and ζ=0.7, and varied k b ∈[0,0.1]rad/(m/s) and k q ∈[0,5e6] Nm/(m/s).The unstable combinations with a negative damping ratio are shown as white regions in the contour plots.As expected, disabling the nacelle-velocity feedback loops leads to instability (bottom-left corner in the contour plots where k b =0 and k q =0).From that corner, stability can  Parametric study for 10,000 fast controllers, where blue represents the highest damping ratios (top plots) or the lowest standard deviations (bottom plots).be reached by increasing k q alone or together with k b , but not by increasing k b alone.As already observed for the de-tuned controllers, the floater pitch motion is most damped for k b =0 and the largest k q , whereas the smallest power fluctuations happen for k b =0 and k q ≈1.3e6 Nm/(m/s).

Conclusions
In this work, a control-oriented model derived from first principles was presented.The proposed model was formulated based on six degrees of freedom of the floater motion and one degree of freedom of the rotor drive-train.Subsequently, the linear response model was coupled with the controller, forming a closed-loop system in a mass-spring-damper representation.Such a representation offers insights into how negative damping is introduced by the control feedback.In addition, the simplicity of the model enables fast computation of stability and response of the closed-loop system, which is particularly relevant for controller tuning.
The simplified model was implemented numerically and demonstrated for both de-tuned and fast controllers.Taking advantage of the model's efficiency, parametric studies were utilized to evaluate the stability and performance of thousands of different combinations of controller parameters in a few minutes.Future work will include validation of the control-oriented model against a time-domain aero-hydro-servo-elastic code, and inclusion of the excitation of the outof-plane degrees of freedom through the generator torque.

Figure 1 .
Figure 1.A floating wind turbine in wind and waves.

Figure 3 .
Figure 3. Dynamic response for a de-tuned controller obtained with time-domain (blue) and frequency-domain (red) methods.

Figure 4 .
Figure 4.Dynamic response for a fast controller obtained with time-domain (blue) and frequency-domain (red) methods.

Figure 5 .
Figure 5. Parametric study for 10,000 de-tuned controllers, where blue represents the highest damping ratios (top plots) or the lowest standard deviations (bottom plots).

Figure 6 .
Figure 6.Parametric study for 10,000 fast controllers, where blue represents the highest damping ratios (top plots) or the lowest standard deviations (bottom plots).

Table 1 .
[22]ral periods calculated in the numerical model compared to those reported in[22].