Control method for a floating wind turbine

Abstract

A controller for a floating wind turbine is adapted to cause the wind turbine to extract energy from wave-induced motion of the turbine. The controller controls the rotor speed of the turbine by controlling the torque of the load presented to the rotor such that the rotor speed varies in response to wave-induced motion.

Claims

1. A controller for a floating wind turbine, the controller being adapted to cause the wind turbine to extract energy from wave-induced motion of the turbine, the turbine having a rotor with a variable speed and a plurality of blades, wherein the controller controls the rotor speed of the turbine by controlling a torque of a load presented to the rotor such that the rotor speed varies in response to wave-induced motion.

2. A controller as claimed in claim 1 arranged to enable the blade pitch of the turbine to remain substantially at its minimum setting whilst the turbine is controlled during wave induced oscillations.

3. A controller as claimed in claim 1 arranged such that the rotor speed of the turbine may substantially maintain its optimum tip speed ratio as the turbine structure moves in wave-induced oscillations.

4. A controller as claimed in claim 1, wherein the torque of the load presented to the rotor is controlled based upon an input indicative of the motion of the turbine.

5. A controller as claimed in claim 4, wherein the input signal is based upon the tower top velocity of the structure.

6. A controller as claimed in claim 5, wherein the input signal based on the tower velocity is used to determine a wave component of the desired rotor velocity due to wave-induced motion of the tower based upon the desired tip speed ratio.

7. A controller as claimed in claim 6, wherein a desired reference rotor velocity is defined as the sum of the wave-induced component and a steady-state wind component, the reference velocity being used to calculate the desired torque of the load presented to the rotor.

8. A controller as claimed in claim 7, wherein the controller calculates two components of the desired torque of the load presented to the rotor, one based on the low pass filtered rotor velocity wind component and one based on the wave-induced rotor velocity component, the two components being summed to produce the desired torque value.

9. A controller as claimed in claim 1, comprising: a control loop having first and second branches, the first branch having as its input the rotor speed of the turbine, wherein a low pass filter is applied thereto and the wind component of the desired torque is obtained using the filtered value; the second branch having as its input the difference between the rotor speed and a reference rotor speed and being arranged to calculate the wave component of the desired torque using a proportional or proportional derivative controller; the outputs of the two branches being summed to determine the desired value of torque.

10. A controller as claimed in claim 1, the controller having inputs for signals representing measured tower velocity and an estimate of mean wind speed.

11. A controller as claimed in claim 10 comprising a single control loop wherein a value of the actual rotor velocity is compared to a reference rotor velocity, the reference rotor velocity being based on the tower velocity and the mean wind speed, the difference being used to determine the desired torque value using a proportional or proportional derivative controller.

12. A wind turbine comprising a controller as claimed in claim 1.

13. A floating wind turbine structure comprising a buoyant tower having a wind turbine mounted thereto and a controller, the wind turbine being arranged to drive a load which presents a torque thereto and being adapted to extract energy from wave-induced motion of the turbine, the turbine having a variable rotor speed, wherein the controller controls the rotor speed of the turbine by controlling the torque of the load such that the rotor speed varies in response to wave-induced motion.

14. A method of controlling a floating wind turbine such that the wind turbine extracts energy from wave-induced motion of the turbine, the turbine having a rotor with a variable rotor speed and a plurality of blades, wherein the rotor speed of the turbine is controlled by controlling a torque of a load presented to the rotor such that the rotor speed varies in response to wave-induced motion.

15. A method as claimed in claim 14, wherein the blade pitch of the turbine remains substantially at its minimum setting whilst the turbine is controlled during wave induced oscillations.

16. A method as claimed in claim 14, wherein the rotor speed of the turbine may substantially maintain its optimum tip speed ratio as the turbine structure moves in wave-induced oscillations.

17. A method as claimed in claim 14, wherein torque of the load presented to the rotor is controlled based upon an input indicative of the motion of the turbine.

18. A method as claimed in claim 17, wherein the input signal is based upon the tower top velocity of the structure.

19. A method as claimed in claim 18, wherein the input signal based on the tower velocity is used to determine a wave component of the desired rotor velocity due to wave-induced motion of the tower based upon the desired tip speed ratio.

20. A method as claimed in claim 19, wherein a desired reference rotor velocity is defined as the sum of the wave-induced component and a steady-state wind component, the reference velocity being used to calculate the desired torque of the load presented to the rotor.

21. A method as claimed in claim 20, wherein the method comprises calculating two components of the desired torque of the load presented to the rotor, one based on the low pass filtered rotor velocity wind component and one based on the wave-induced rotor velocity component, the two components being summed to produce the desired torque value.

22. A method as claimed in claim 14, comprising: providing a control loop having first and second branches, inputting into the first branch the rotor speed of the turbine, applying a low pass filter thereto and obtaining the wind component of the desired torque using the filtered value; inputting into the second branch the difference between the rotor speed and a reference rotor speed and calculating the wave component of the desired torque using a proportional or proportional derivative controller; summing the outputs of the two branches to determine the desired value of torque.

23. A method as claimed in claim 14, comprising inputting signals representing measured tower velocity and an estimate of mean wind speed.

24. A method as claimed in claim 23, comprising providing single control loop in which a value of the actual rotor velocity is compared to a reference rotor velocity, the reference rotor velocity being based on the tower velocity and the mean wind speed, and using the difference to determine the desired torque value using a proportional or proportional derivative controller.

25. A software product comprising instructions which when executed by a processor cause the processor to control a floating wind turbine structure such that the wind turbine extracts energy from wave-induced motion of the turbine, the turbine having a variable rotor speed and a plurality of blades, wherein the rotor speed of the turbine is controlled by controlling a torque of a load presented to the rotor such that the rotor speed varies in response to wave-induced motion.

26. A product as claimed in claim 25, wherein the software product is a physical data carrier.

27. A method of manufacturing a software product which is in the form of a physical carrier, comprising storing on the data carrier instructions which when executed by a processor cause the processor to control a floating wind turbine structure such that the wind turbine extracts energy from wave-induced motion of the turbine, the turbine having a variable rotor speed and a plurality of blades, wherein the rotor speed of the turbine is controlled by controlling a torque of a load presented to the rotor such that the rotor speed varies in response to wave-induced motion.

28. A controller as claimed in claim 1, wherein the load comprises a generator load.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Certain embodiments of the invention will now be described, by way of example only, and with reference to the accompanying drawings:

(2) FIG. 1 is a graph showing non-dimensional values of the wind speed (100), aerodynamic torque (101) and rotor speed (102) for a typical wind turbine as a function of time;

(3) FIG. 2 is a graph showing rotor speed as function of wind speed for a wind turbine that operates with an optimal tip speed ratio (103) and an actual curve for a typical wind turbine as implemented in HAWC2 (104);

(4) FIG. 3 is a graph showing aerodynamic power reference as function of wind speed for a wind turbine that operates with an optimal tip speed ratio (103) and an actual curve for a typical wind turbine as implemented in HAWC2 (104);

(5) FIG. 4 is a graph showing aerodynamic power as function of rotor speed for a wind turbine that operates with an optimal tip speed ratio (103) and an actual curve for a typical wind turbine as implemented in HAWC2 (104);

(6) FIG. 5 is a graph showing the power coefficient as function of tip speed ratio for zero blade pitch angle for a typical wind turbine;

(7) FIG. 6 is a graph showing the aerodynamic power as function of time during simulation with a constant wind speed of 6 m/s and a sinusoidal tower motion with a velocity amplitude of 1.18 m/s and a period of 9 seconds;

(8) FIG. 7 is a graph showing the aerodynamic power as function of relative wind speed during simulation with a constant wind speed of 6 m/s and a sinusoidal tower motion with a velocity amplitude of 1.18 m/s and a period of 9 seconds;

(9) FIG. 8 is a diagram of a conventional torque controller;

(10) FIG. 9a is a diagram of a torque controller according to an embodiment of the invention;

(11) FIG. 9b is a diagram of a torque controller according to an alternative embodiment of the invention;

(12) FIG. 9c is a diagram showing an optional additional feature for the controllers of FIGS. 9a and 9b;

(13) FIG. 10 is a Bode diagram for the closed loop rotor speed P-controller that may be used in the embodiments;

(14) FIG. 11 is a snapshot graph of tower top velocity during simulations of floating wind turbines with conventional control (blue) and optimal rotor speed control (red);

(15) FIG. 12 is a snapshot graph of rotor speed during simulations of floating wind turbines with conventional control (105) and optimal rotor speed control (106) together with the optimal rotor speed reference signal (107);

(16) FIG. 13 is a snapshot graph of aerodynamic power during simulations of floating wind turbines with conventional control (105) and optimal rotor speed control (106);

(17) FIG. 14 is a snapshot graph of generator power during simulations with floating wind turbines with conventional control (105) and optimal rotor speed control (106); and

(18) FIG. 15 is a wind turbine incorporating a controller according to an embodiment of the invention.

DETAILED DESCRIPTION

(19) Turning first to FIG. 15, there is illustrated a floating wind turbine assembly 1. It comprises a turbine rotor 2 mounted to a nacelle 3. The nacelle is in turn mounted to the top of a structure which comprises a tower 4 secured to the top of a floating body 5 in the general form of a spar buoy. The floating body is secured to the sea bed by one or more anchor lines 7 (only one is shown). The nacelle contains an electrical generator which is connected to the turbine rotor by means of a reduction gearbox in the conventional manner (these items are not shown). Alternatively, the assembly could comprise a direct driven generator. The nacelle also contains a control unit.

(20) When the rotor is turned by the wind it causes the generator to produce electricity in the well known manner. The rotor comprises variable pitch blades whose pitch angle may be adjusted by the control unit. Its minimum pitch setting is defined as the zero degrees position. Other settings are represented by a positive angle. The optimal setting of pitch angle with respect to the power coefficient is zero with small variations around this value.

(21) The controller also acts to vary the torque which the generator provides as a load to the turbine rotor 2. Thus, for a given rotor speed, the energy obtained from the wind, and hence the output power from the generator, can be varied.

(22) Variation of the blade pitch and/or the torque is used to ensure that the turbine operates within its rotor speed and output power limitations. The lowest wind speed at which maximum power can be obtained is termed the rated wind speed for the turbine.

(23) Since the turbine assembly 1 is floating in the sea (or other large body of water), it is subject to wave-induced motion. (The waves 9 on the water's surface are shown schematically.) As the turbine assembly 1 moves back and forth relative to the wind due to the wave excitation, it is possible to extract wave energy under operation below the rated wind speed for the turbine (it would not be useful to do so above the rated wind speed). Under the assumption of steady wind and harmonic motion for a floating wind turbine, the relative velocity between the wind and the turbine can be written as
U.sub.r=U.sub.wU.sub.A cos(t+)(1)
where U.sub.r is the relative wind speed, U.sub.w is the incoming wind speed, U.sub.A is the velocity amplitude of the harmonic motion of the floating wind turbine, is the frequency of the harmonic motion, t is time and is a phase angle. By assuming constant power coefficient C.sub.p, the aerodynamic power delivered by the turbine can be written as

(24) $\begin{array}{cc}P=\frac{1}{2}{}_{a}A{C}_p{U}_r^3,& \left(2\right)\end{array}$
where P is the aerodynamic power from the turbine, .sub.a is the density of air and A is the area swept by the rotor. By substituting (1) into (2), the energy production over one cycle of oscillation is found to be

(25) $\begin{array}{cc}E={}_0^{T}P\left(t\right)t=\frac{1}{2}T{C}_p{}_a{\mathrm{AU}}_w^3\left(1+\frac{3}{2}\frac{U_A^2}{U_w^2}\right),& \left(3\right)\end{array}$
where

(26) $T=\frac{2}{}$
is the period of one cycle of oscillation. This equation provides the maximum obtainable value of E, i.e. where the power coefficient C.sub.p is kept at a constant, maximum value. C.sub.p is a function of both blade pitch and tip speed ratio (the rotor tip speed divided by the wind speed, i.e. =r/U.sub.R where r is the rotor radius) so this in turn requires that the rotor speed varies with the harmonic motion such that the tip speed ratio is kept at a constant, optimal value. In other words, to achieve the maximum value of E, the rotor speed must vary with the variation of the relative wind speed during each cycle of oscillation.

(27) Leaving aside for the time being the matter of wave-induced changes in relative wind speed, the wind turbine is arranged to keep the tip speed ratio at the optimal value (or at least as close to it as possible) in response to changes in wind speed when the turbine operates below the rated wind speed. In order to do this the generator torque for a variable-speed wind turbine such as this can be controlled in a known manner. (This operation regime is often referred to as the maximum power regime.)

(28) The ideal the operating point of the turbine is seen to be on the top of the parabola in the power coefficient curve shown in FIG. 5, with minimum blade pitch (i.e. =0) and this is the setting used in the embodiments described herein. However, in alternative embodiments, small changes to this setting may be made whilst operating in the maximum power regime to further optimise operation.

(29) An optimal generator torque curve as function of the rotor speed can be found in the following manner:

(30) Since power P is the product of angular velocity and torque, it follows that T.sub.EL=P/.sub.t and so the desired, or optimal, generator torque transformed to the low speed side of the gear as a function of rotor speed is given as

(31) $\begin{array}{cc}\begin{array}{c}{T}_{\mathrm{EL}}\left(w_t\right)=\frac{1}{2}\frac{1}{w_t}{r}_a{C}_p\left(l_{\mathrm{opt}},0\right){\mathrm{pr}}^2{U}_r^3\\ =\frac{1}{2}{\mathrm{rC}}_p\left(l_{\mathrm{opt}},0\right)p\frac{U_r^3}{w_t^3{r}^3}{r}^5{w}_t^2\\ =\frac{1}{2}{\mathrm{rC}}_p\left(l_{\mathrm{opt}},0\right)p\frac{1}{l_{\mathrm{opt}}^3}{r}^5{w}_t^2\\ =C_{\mathrm{EL}}{w}_t^2,\end{array}& \left(4\right)\end{array}$
where T.sub.EL is the generator torque transformed to the low speed side of the gear assuming an ideal gear without losses, .sub.t is the rotor speed, .sub.opt is the optimal tip speed ratio, r is the rotor radius, and the constant C.sub.EL is given as

(32) $C_{\mathrm{EL}}=\frac{1}{2}{\mathrm{rC}}_p\left(l_{\mathrm{opt}},0\right)p\frac{1}{l_{\mathrm{opt}}^3}{r}^5.$
Equivalently, since for gear ratio n:1 the generator torque is related to the torque seen by the rotor by T.sub.G=T.sub.EL/n and .sub.t=.sub.g/n the optimal generator torque as function of the generator speed on the high speed side of the gear can be written as

(33) $\begin{array}{cc}\begin{array}{c}{T}_G\left(w_g\right)=\frac{1}{n}{T}_{\mathrm{EL}}\left(\frac{1}{n}{w}_g\right)\\ =\frac{1}{n^3}{C}_{\mathrm{EL}}{w}_g^2\\ =C_G{w}_g^2,\end{array}& \left(5\right)\end{array}$
where .sub.g is the generator speed, n is the gear ratio, T.sub.G is the generator torque at the high speed side of the gear and the constant C.sub.G is given as

(34) $\begin{array}{c}{C}_G=\frac{1}{n^3}{C}_{\mathrm{EL}}\\ =\frac{1}{n^3}{C}_{\mathrm{EL}}\\ =\frac{1}{2n^3}{\mathrm{rC}}_p\left(l_{\mathrm{opt}},0\right)p\frac{1}{l_{\mathrm{opt}}^3}{r}^5.\end{array}$

(35) If a generator torque controller is based upon (4) and (5) alone, it is important to note that the optimal tip speed ratio is reached only in the steady state. There is a certain time constant from a change in the wind speed to a corresponding change in the rotor speed. The optimal tip speed ratio is therefore only achieved around a mean value of the wind speed. First, there is a time constant from a change in the wind speed to a change in the aerodynamic torque. Secondly, there is a time constant from a change in the aerodynamic torque to a change in the rotor speed due to the large moment of inertia of the rotor.

(36) This is illustrated in FIG. 1 where non-dimensional, transient values of the wind speed, aerodynamic torque and rotor speed are plotted as a function of time for a step in wind speed from 6 to 7 m/s. All variables have been transformed to take values between 0 and 1 in order to simplify the comparison of the time constants. The aerodynamic torque reaches its steady state value in 1.5 seconds while the rotor speed reaches its steady state value after 85 seconds, which is far greater than the typical period of wave-induced motion.

(37) The turbine used in the embodiment of the invention is a typical 2.3 MW turbine. FIGS. 2-5 illustrate some relationships between power coefficient, tip speed ratio, aerodynamic power, rotor speed and wind speed for such a turbine and the corresponding optimal curve.

(38) The rotor speed is shown as function of the wind speed for a turbine that operates with an optimal tip speed ratio and the actual curve for the turbine in FIG. 2. The reason for the large differences for wind speed above about 8 m/s is due to restrictions in the maximum allowable rotor speed for the turbine, which is equal to 1.78 rad/s for this specific turbine.

(39) The effect of not being able to operate the turbine optimal due to the rotor speed restrictions shown in FIG. 2 is shown in the corresponding power curves of FIG. 3. It is seen that the actual power curve is below the optimal power curve from a wind speed of about 8 m/s and up to the rated wind speed for the turbine.

(40) The aerodynamic power curve as function of rotor speed is shown in FIG. 4, and this curve corresponds to the relationship between rotor speed and aerodynamic torque as given in equation (4). The power coefficient as function of the tip speed ratio is shown in FIG. 5, where the optimal tip speed ratio is seen to be around 9.

(41) In contrast to the conventional controller, in order to achieve increased wave energy extraction below rated wind speed, the control unit of the first embodiment employs a generator torque controller with a novel rotor speed controller part as well as the conventional torque controller as described in equations (4) and (5) above. This additional part has an additional input based upon the tower velocity and will be described further below.

(42) The reference signals to be used in the generator torque controller of the embodiment are:

(43) $\begin{array}{cc}{}_{\mathrm{ref}}=-\frac{{}_{\mathrm{opt}}}{R}{\stackrel{.}{x}}_{\mathrm{top}}+{}_{\mathrm{lp}}& \left(6\right)\\ {\stackrel{.}{}}_{\mathrm{ref}}=-\frac{{}_{\mathrm{opt}}}{R}{\stackrel{\mathrm{.Math.}}{x}}_{\mathrm{top}},& \left(7\right)\end{array}$
where .sub.lp is the low pass filtered signal derived from the measured rotor speed, . It is assumed that the positive direction of the horizontal motion of the nacelle corresponds to the positive wind direction, such that it is optimal to reduce the rotor speed when the turbine is moving in the same direction as the wind.

(44) It will be seen that equation (6) is a sum of two velocities. The first is the contribution to the rotor velocity by the wave-induced motion, assuming optimum tip speed ratio is achieved. The second is the no-wave wind-induced part. Equation (7) assumes that .sub.lp is sufficiently constant that its derivative approximates to zero, i.e. that the rate of changes of the rotor speed due to wave-induced motion is much larger than those to due variations in the wind speed.

(45) Considering firstly the conventional controller shown in FIG. 8, the input to the (physical) system is the aerodynamic torque, T.sub.A, and the output from the system is the rotational speed of the rotor , in other words the aerodynamic torque that the wind acting on the rotor creates causes the rotor to run at speed . The measurement in the system is . The generator torque seen on the low speed side of the shaft is T.sub.G (Which corresponds to T.sub.EL in the previous discussion) and can be represented/calculated as a function T.sub.G() of the measured rotor speed. T.sub.G is the output from the controller.

(46) In the dynamic process, the rotor acceleration is given as {dot over ()}=1/J*(T.sub.AT.sub.G), where J is the moment of inertia, by assuming a stiff shaft and using Newton's second law.

(47) The controller of the first embodiment can be regarded as a modified version of the above conventional controller and is shown in FIG. 9a. As may be seen, the rotor speed input to the conventional torque reference is the low pass filtered measurement of the rotor speed, while the additional rotor speed control loop contains a PD (proportional differential) controller and a rotor speed reference trajectory that is based on the measured tower top motion x in order to obtain a desired tip-speed ratio.

(48) More specifically, as before the input to the physical system is the aerodynamic torque, T.sub.A. The rotor speed reference, .sub.ref is given by equation (6) above. The output from the physical system is the rotational speed of the rotor . The measurement in the system shown is the rotor speed. In addition to this, the nacelle velocity is measured and used in the calculation of .sub.ref.

(49) The generator torque seen on the low speed side of the shaft is T.sub.G, which is the output from the controller. It consists of two components that are added. The first is a rotor speed tracking controller that gives an additional contribution to the conventional controller such that the rotor's speed varies with the wave induced motions of the nacelle in an optimal way. The rotor speed tracking controller takes the difference between a reference speed .sub.ref and the measured rotor speed as input. The output is a generator torque signal. The second is a torque controller that behaves like the conventional torque controller of FIG. 8. This controller consists of a low pass filter (or band pass filter) that removes the wave frequencies, and the original torque controller function T.sub.G(). The output is a generator torque signal with zero mean. The rotor acceleration {dot over ()} is as given above.

(50) An alternative embodiment is shown in FIG. 9b. In this embodiment, input to the system is (again) the aerodynamic torque, T.sub.A, and the rotor speed reference, .sub.ref is given by:

(51) $\begin{array}{cc}{}_{\mathrm{ref}}=-\frac{{}_{\mathrm{opt}}}{R}{\stackrel{.}{x}}_{\mathrm{top}}+\frac{{}_{\mathrm{opt}}}{R}{\hat{u}}_{\mathrm{mean}},& \left(6a\right)\end{array}$

(52) The output from the system is the rotational speed of the rotor, . The measurement in the system is again the rotor speed. In addition to this, the nacelle velocity is measured and the mean wind speed is estimated and used to calculate .sub.ref according to equation 6a above. The generator torque seen on the low speed side of the shaft is T.sub.G, which is the output from the controller. The rotor speed tracking controller takes the difference between a reference speed and the measured rotor speed as input. The output is the generator torque signal.

(53) In the embodiment described above it is assumed that the generator dynamics are fast compared to the rotor dynamics, such that the generator torque is the same as the commanded generator torque that is actually the output from the torque controller (this is a common assumption). However, in a modified embodiment, the generator dynamics are taken into account: to do this the features shown in the block diagram of FIG. 9c are connected between the controller output and the generator torque in FIGS. 9a and 9b.

(54) The performance of the controllers described above, and in particular the FIG. 9a version will now be considered. It is assumed that the conventional torque controller part in FIGS. 9a and 9b will not affect the speed tracking controller since the two controllers operate in different frequency domains. The contribution from the conventional torque controller is assumed to be slowly varying since the controller is based on low pass filtered rotor speed with filter frequency below the wave frequency area, while the rotor speed tracking controller will be designed to operate in the wave frequency area in order to obtain a desired tip speed ratio. For this reason, the conventional torque controller part of the controller in FIG. 9a is neglected in the following discussion when considering the speed tracking controller around the low pass filtered rotor speed.

(55) The PD controller in FIG. 9a can be represented with the transfer function
h.sub.PD(s)=K.sub.P+K.sub.Ds,(7)
where K.sub.p and K.sub.D are the proportional and derivative gains, respectively. By neglecting slowly varying effects, the dynamics of the closed system in FIG. 9a can be developed:

(56) 0$\begin{array}{cc}=\frac{1}{\left(J+K_D\right)s+K_P}{T}_{A\_\mathrm{dyn}}+\frac{K_{D}s+K_P}{\left(J+K_D\right)s+K_P}{}_{\mathrm{ref}},& \left(8\right)\end{array}$
where J is the moment of inertia for the rotor and the generator and T.sub.A.sub._.sub.dyn is the dynamic part of the aerodynamic torque. Further, the loop transfer function for the dynamic part of the system is given as

(57) $\begin{array}{cc}\begin{array}{c}{h}_0\left(s\right)=\frac{K_P+K_{D}s}{\mathrm{Js}}\\ =\frac{K_P}{J}\frac{1+\frac{K_{D}J}{K_P}s}{s}\end{array}& \left(9\right)\end{array}$

(58) The transfer function representing the control system's ability to follow a reference signal becomes

(59) $\begin{array}{cc}M\left(s\right)=\frac{K_{D}s+K_P}{\left(J+K_D\right)s+K_P},& \left(10\right)\end{array}$
while the transfer function representing the error between a desired reference signal and the measurement becomes

(60) $\begin{array}{cc}N\left(s\right)=\frac{1}{\left(J+K_D\right)s+K_P}& \left(11\right)\end{array}$

(61) The embodiment as described above employs a PD controller. However, a purely proportional gain controller can be introduced by setting the parameter K.sub.D in equations (9) and (10) equal to zero, resulting in the transfer functions

(62) $\begin{array}{cc}=\frac{1}{\mathrm{Js}+K_P}{T}_{A\_\mathrm{dyn}}+\frac{K_P}{\mathrm{Js}+K_P}{}_{\mathrm{ref}}& \left(12\right)\\ h_0\left(s\right)=\frac{K_P}{\mathrm{Js}}& \left(13\right)\\ M\left(s\right)=\frac{K_P}{\mathrm{Js}+K_P}& \left(14\right)\\ N\left(s\right)=\frac{1}{\mathrm{Js}+K_P}& \left(15\right)\end{array}$

(63) The bandwidth of the system with the P-controller is

(64) ${}_{0\mathrm{dB}}=\frac{K_P}{J},$
and the Bode plot for the system is shown with K.sub.P/J=.sub.0 dB=8.49 in FIG. 10.

(65) The benefits of the invention can be better understood by considering some simplified theoretical calculations of wave energy extraction. These are based on the actual power curve in FIG. 2, together with the power coefficient curve in FIG. 4. The following three cases are considered: Fixed: Power extracted from a fixed foundation wind turbine during operation at a constant wind speed of 6 m/s. Actual: Power extracted from a floating wind turbine during operation at a constant wind speed of 6 m/s and with sinusoidal tower top velocities with an amplitude of 1.18 m/s and a period of 9 seconds (this corresponds to the case with waves with 2 m amplitude and a period of 9 seconds in Simo-Riflex-Hawc2 simulations) under the assumption that the turbine operates with constant rotor speed corresponding to the optimal rotor speed at 6 m/s. Optimal: Power extracted from a floating wind turbine during operation at a constant wind speed of 6 m/s and with sinusoidal tower top velocities with an amplitude 1.18 m/s and a period of 9 seconds under the assumption that the turbine operates with the desired tip speed ratio for the turbine, corresponding to the actual power curve in FIG. 2.

(66) The results of the power calculations for the three different cases are shown as functions of time and relative wind speed in FIG. 5 and FIG. 6, respectively, and some key values are listed in Table 1. Note that use of equation (2) gives an estimate of the wave extraction part of the aerodynamic energy of 5.80% for velocity amplitude of 1.18 m/s, which corresponds to the calculations with the floating wind turbine with optimal rotor speed control in Table 1 below.

(67) FIG. 6 shows the aerodynamic power for a fixed wind turbine (line 108), a floating wind turbine operating at optimal tip speed ratio (line 109) and a more realistic floating wind turbine operating with the rotor speed corresponding to the mean wind speed (line 110). The mean values for the aerodynamic power during operation with the optimal and the more typical tip speed ratios are shown in lines 111 and 112, respectively.

(68) FIG. 7 shows the aerodynamic power for a fixed wind turbine (line 108), a floating wind turbine operating at optimal tip speed ratio (line 109) and a more realistic floating wind turbine operating with the rotor speed corresponding to the mean wind speed (line 110). The mean values for the aerodynamic power during operation with the optimal and the more typical tip speed ratios are shown in lines 111 and 112, respectively.

(69) TABLE-US-00001 TABLE 1 Simple calculations of aerodynamic power INCREASED AERODYNAMIC MEAN S.D. POWER PRODUCTION POWER [kW] [kW] [%] Fixed Foundation 318 0 0 Wind Turbine Floating Wind 328 134 3.03 Turbine With Conventional Control Floating Wind 337 135 5.80 Turbine With Optimal Rotor Speed Control
Simulation Study

(70) The simulations in this section are carried out with the coupled analysis tool Simo-Riflex-Hawc2 with the concrete hull tower structure and a 2.3 MW turbine.

(71) The environmental conditions in the simulations are a constant wind speed of 6 m/s and regular waves with an amplitude of 2 m and a period of 9 seconds.

(72) The simulations in this section were carried out with a P-controller for rotor speed tracking control having the same parameters as used for plotting the Bode diagram of FIG. 10.

(73) A snapshot of the tower top velocity is plotted in FIG. 11 for a floating wind turbine with a conventional torque controller and a torque controller for optimal rotor speed control. It can be seen that the turbine motions are not affected significantly by the choice of controller.

(74) The corresponding rotor speeds are plotted in FIG. 12 together with the optimal rotor speed reference signal. It may be seen that using a conventional controller there are only small reactions to the wave motion and the rotor speed is also out of phase with the optimal rotor speed reference signal. This indicates that the assumptions behind the actual curve in the simple calculations herein are reasonable. It is clear that the rotor speed follows the optimal rotor speed signal when the optimal rotor speed tracking controller is used.

(75) A snapshot of the aerodynamic power with use of the two controllers is shown in FIG. 13, while some key data for the generator power, also with comparison to a fixed foundation wind turbine, are shown in Table 2 below.

(76) TABLE-US-00002 TABLE 2 Key data for the generator power in Simo-Riflex-Hawc2 INCREASED GENERATOR MEAN STD POWER PRODUCTION POWER [kW] [kW] [%] Fixed Foundation 284 0 0 Wind Turbine Floating Wind 291 13 2.46 Turbine with Conventional Control Floating Wind 302 1338 6.69 Turbine with Optimal Rotor Speed Control

(77) A simulation snapshot of the generator power is shown in FIG. 14, while the key comparative data are given in Table 2 (above). It may be seen that the optimal rotor speed tracking controller requires large torque contributions such that energy is also extracted from the grid in parts of the fluctuating cycle. Note that the simple calculations for the increased aerodynamic power given in Table 1 coincide well with the calculated increased generator power that is found for the numerical simulations with use of Simo-Riflex-Hawc2 in Table 2.

(78) It should be apparent that the foregoing relates only to the preferred embodiments of the present application and the resultant patent. Numerous changes and modification may be made herein by one of ordinary skill in the art without departing from the general spirit and scope of the invention as defined by the following claims and the equivalents thereof.