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Identification of Wind Turbine Dynamics
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IDENTIFICATION OF WIND TURBINE DYNAMICS S. E. Mattsson /) I'/)(Jrlllll'lll or iI /llo/lwl it COl/lml. 1./11/(1 I lI.I l illlll' 0( T I'I-/II/olog.'. 1'. 0 . H ox 11 8 .
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Abstract. The energy conversion from wind energy to electrical energy of a large horizontal axis wind turbine can be controlled by changing the pitch angle. Thus , it is of interes t to identify the dynamics from wind and pitch angle to electrical power . This paper presents results from identification of wind turbine dynamics . A large data set has been collected from a prototype 3 MW system called the WTS-3 located in southern Sweden. Models of different structure have been estimated and compared with physical models . Keywords . Wind power; wind turbine dynamics ; identification; parameter estimation.
(2)
1. INTRODUCTION The energy conversion from wind energy to electrical energy of a large horizontal axis wind turbine can be controlled by changing the pitch angle of the blades. This paper presents results from identification of wind turbine dynamics of interest for pitch angle control.
For the turbine speed ~ we have tJ.~ '" tJ.l . Let ~O denote the synchronous turbine speed. then generated power IS
A large data set has been collected from a prototype 3 MW system called the WTS-3 located in southern Sweden. It is built by Karlskronavarvet AB . Sweden and Hamilton Standa rd a division of United Technologies Inc. USA . A 4 MW plant (the WTS- 4) of the same design is built in Medicine Bow. Wyoming. USA . The WTS-3 is designed to supply power in parallel with other generators to a large power utility grid and to operate in wind forces of 5-26 m/s o The rated power is reached at 14 m/so The WTS-3 has a horizontal axis wind turbine with two blades . The power is controlled by changing the pitch angle. The blade actuators are hydraulic positioning systems. The blades have a length of 39 m. and a weight of 14000 kg each. The tower is 78 m high . The turbine rotates at 25 rpm and a multistage planetary gearbox steps up the rotation to 1500 rpm . The generator is a three-phase . 50 Hz. 1500 rpm synchronous machine .
The driving aerodynamical torque T is a nonlinear function of the pitch angle (3 , the mean wind sp~ed U. over the rotor disc of the oncoming wind and >1>. The Iinearized expres s ion for tJ. T will be written as
(3)
(4)
This section presents physical models of interest for pitch angle control. A thorough discussion can be found in Mattsson (1984) .
The relations (I) and (2) are important and fundamental for pitch angle control. They show that the variations in the aerodynamical torque above the natural frequency "' =./f\(J of the torsional mode are attenuated when s~owing up in the shaft torque. The designers of the WTS-3 have taken advantage of the low pass dynamics and deliberately designed the drive train to be torsionally soft. This is achieved by a soft mounting of \pe plilfetary gearbox. The nominal values J. 5.1·10 kgm and K • 7 .7 MNm/rad give "'I • 1.2 rad/s . The aerodynamical damping of the turbine is low. because the blades are designed to give minimum losses . A synchronous generator provides hardly any damping for the torsional mode. The coupling between a synchronous generator and the grid is much stiffer than the drive train. The damping prov ided by the drive train is also low . Thus the controller mus t provide damping. The natural frequency", influences the bandwidth needed of the 1 pitch angle c 0ntrol . since it is no sens e to choose the cros sover frequency much higher than "'I ' For the WTS- 3 a cro s sover frequency of 2-3 rad/s is needed (Matts son. 1984) .
Turbine Dynamics
Blade Servo
In the frequency range of interest for pitch angle control the dynamics of the synchronous generator can be neglected. The basic dynamics is given by the oscillation of the turbine against the electrical system. A model for this oscillation around a stationa ry operating point is
The blade servo is designed to be a position servo with a time constant of 0 .4 s . However. the servo speed is limited to 4· Is . For the blade servo we thus have the first order model
The paper is organized as follows . Physical models presented in Section 2. In Section 3 measured data analysed. identification of models is discussed and resulting models are compared with physical models . results are summarized in Section 4.
are are the The
2 . PHYSICAL MODELS
Jd~ + Ddl + Kd~
= dT
(5)
(I )
where T is the aerodynamical torque . ~ is the torsion of the drive train , J is the turbine inertia and K and Dare the torsional spring and damping coefficients of the drive train . This model gives the electrical torque as
Wind Characteris tics For control design the wind speed can be viewed as consisting of one constant part and one stochastic part .
1890 The constant part represents the mean wind speed over 10 minutes and the stochastic part the turbulence . The wind speed varies over the turbine area . The major influence of the wind can be described by the mean wind speed U. over the turbine area. A simple model for the stochastic part ~U. of U. is (Holley and others. 1981)
(6) where w is white noise with zero mean and noise intensity rand (1 is the standard deviation of ~U • . W The time co"hstant T is of the order of 5 - 30 seconds . w
Transformation to Discrete Time Input/Output Models The measured data are sampled and the identification procedure used gives discrete time models on the form A(q
-1
)y(t)
2
B(q
-1
)u(t) + ~C(q
-1
(7)
)e(t)
where y is the output. u is the input of the system. e is white noise with zero mean_i'nd variance one. A. Band C is the delay operator; a!:f polynomials and q q x(t). x(t-h) with h being the time between two sampling instances. The model may have several inputs. Since the physical models presented above are continuous time models. they must be transformed to discrete time input/output models to compare with the experimental data. The physical models can be written as X
Y - Hx + Du
(9)
with F. G. Hand D being constant matrices and v being multivariable . continuous time . white noise with zero mean and the noise intensity R l ' A discrete time model with sampling interval h is then given by x(t+h) - ~x(t) +
h
J o
e
F(h-s)
~ _ e Fh
(12)
where w is multivariable discrete time white noise with zero mean and variance _
h
J eF(h-s)R
0
u(t+s) - u(t) + Ol·s/h · (u(t+h)-u(t))
(eF(h-S))TdS
( 13)
1
To calculate the transfer function ~C/A from e to y a spectral factorization must be performed. If we like. we can introduce white measurement noise < with variance Rand uncorrelated with w ; < y(t) = Hx(t) + Du(t) + < (t) (14) Let P be a symmetric. non negative definite solution to the discrete time. algebraic Riccati equation (15) Introduce K defined as (16) and let I be an non unity matrix. where dimensionnof the matrix F. then
varies ( 19)
For Ol • 0 u is constant between the sampling instances. For Ol • 1 u is continuous and varies linearly between the sampling ins tances. Under this assumption ~x(t)+ru(t)+Olrl(u(t+h)-u(t))+w(t)
x(t+h)
(20)
h
J o
r .
1
(21)
eF(h-s)G ds
(22)
h
.! J s
r
eF(h-s)G ds
h 0
The transfer function B/A 'from u to y becomes B(q
-1
)/A(q
-1
) - D+H(ql
n
...
-~)
-1
(r + Olr (q-l)) l
(23)
3. IDENTIFICA nON
(11)
w
u
( 10)
Gu(t+s)ds + wIt)
y(t) : Hx(t) + Du(t)
R
Let t be a sa mpling instance. Assume that between the sampling instances as (0 :5 s < h)
(8)
Fx + Gu + v
m
sampling instances . This is typically the case when u is an output from a digital controller . However . in our case the inputs are analog . If the sampling frequency is sufficiently high it is reas onable to assume that u varies linearly between sampling instances. However . we must not choose too high a sampling frequency. since it gives numerically bad conditions for the e s tima~on. If p is a pole of the continuous time model then e P is a pole to the discrete time model. If h is small compared to the time constants of the s y s tem . the discrete time model . thus has its poles close to one and the calculation of the time cons tants of the continuous time model is ill-conditioned. It is possible to es timate the parameters directly from discrete time measurements . See Astr6m and Kallstr6m (1976). However. this approach requires a much greater effort. both in form of development of software and computing time .
n
is
the
( 17) ( 18) To calculate the transfer function B/ A from u to y we must evaluate the integral on the right side of (10) . This requires knowledge of the shape of u between the sampling instances . If we can choose the representation for u . it is convenient to let it be constant between the
Three measurement series from the WTS-3 recorded on the 11th October 1983 are used for identification: Series 1 at 11 :05. 240 seconds long. 24 000 data points Series 2 at 12:52. 240 seconds long . 24 000 data points Series 3 at 13:04. 210 seconds long. 21 000 data points The measurements were filtered with a sixth order Bessel filter with a cut- off frequency of 25 Hz and sampled with 100 Hz . One difficulty is that the system may be poorly identifiable. because it is controlled. To increase the identifiability the power reference of the controller was switched between 2 MW and 3 MW according to a PRBS-sequence . The interactive program package ldpac (Wieslander . 1980) was used to analyse the data . Fig. 1 shows a power spectrum for the electrical power PE calculated from measurement series 1. The series was cut into blocks containing 2048 data points each. Discrete Fourier transform with a Blackman-Harris time window with a sidelobe level of -74 dB (Harris . 1978) was used to calculate a spectrum for each of the first ten data blocks . Fig. 1 shows the mean of these spectra. The power spectra for the turbine speed ~ and the servo reference f3 shown in Fig. 2 and 3 were calculated in the same Jay . The power spectra from the three different measurement series are very similar. The power spectra for PE and ~ have several peaks. It is convenient to introdllce a normalized frequency P. such that lP corresponds to the synchronous turbine speed. which is 2.618 rad/s for the WTS-3 . There are large peaks at 2P. 4P . 6P and 8P. There are also peaks at odd integer multiples of lP. but they are much smaller . These variations are caused by the spatial variation of the wind speed over the turbine area . The variations at odd integer multiples of lP are smaller. because the turbine has two blades. There are three major sources of the variations at integer multiples of P . First. the mean wind
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I H9 1
10- 2 10- 3
10- 4 10- 5 10- 6 ;;-
10
100
rod/s
Fig . 1. Power s pec trum for the electrical power PE [MW) from meas urement series 1.
5
=
C 0 Cl> "0 0
0
iD
o
.... o
t
Cl>
Qi
"8 ~
100
120
11.0
s
Fig . 4. Meas urements from series 2 and the model error of the model (24) with T • 0 .65 s and a • 1. 10
100
rod/s
Fig. 2. Power spectrum for the turbine speed ~ [rad/s) from meas ure ment s er ies 1.
10- 5
If the generator dynamics is included in the physical models. they indicate that the system has its second torsional mode at 25 rad/s . The second torsional mode is thus a pos s ible explanation to why the peak at BP (-21 rad/s) is wider than those at 4P and 6P. Blade Servo Sampling of the servo model gives with a • e -hit
10-&
(l - aq -1 )Af3. [ (1-a)q - 1+a(1+(1-a)/ln a)(1-q -1 ) ] Af3 (24) r
10- 7 10- 8
10-9
10
100
rod / s
Fig. 3. Power s pectrum for the servo reference f3 [rad) r from mea s urement series 1.
speed varies with altitude. Second. the tower blocks the airflow for a rotating blade when it passes the tower . The blockage is of s hort duration. which means that at leas t 3 - 4 of the fir s t ev en harmonics have amplitudes of the s ame magnitude . Third . a rotating blade encounters tu r bulence whose characteristics are quite different from tu r bulence encountered in a fixed po int. The mid frequency r egion is depleted and the removed energy is distributed into the high frequency end with spikes at integer multiples of IP . An intuitive explanation is that a rotating blade chops across the eddies whereas a small fixed anemometer is totally immers ed in the turbulence eddies . The 2P variations in the aerodynamical torque are one good rea s on for having a s oft shaft. The 2P v ariations in the aerodynamical torque are attenuated more than eight time s when s howing up in the electrical torque. The 2P v a riations in the generated power has an amplitude of 1- 3% of rated power.
For each meas urement series the mean value of f3 was subtracted from f3 giving tl.f3 . The blade angle tl.~ was calculated in the s.fme way . Ttfe signals tl.f3 and tl.f3 were filtered with a second order Butterworth fnter having the cut-off frequen cy 10 rad/s and resampled with h • 0 . 1 s . Since both tl.f3 and tl.f3 were filtered with the same filter, the trans fer function between them remains unchanged. The filter ing does of course modify the importance of the behaviour at different frequencies when estimating the models. The ML-procedure and Idpac's evaluating procedures sugges t that the servo dynamics can be modelled by (5) with T · 0.65 :t 0.05 s. The model error, which is the difference between measured output and model output. s hows that the model is good . For a • 0 as well as for a = 1 the model error of (24) with T = 0 .65 s is for all three measurement series less than 0.20 and the s tandard deviation is 0 .060. The s hape of tl.f3 between the sampling is unimportant . because h is smafi compared to r. A typical res ult is illustrated in Fig. 4 . Turbine Speed and Electrical Power To get some insight into the validity of the relation (3) the coherence between turbine speed and electrical power w as determined . Fig . 5 s hows that the coherence is good in the frequency range 0 .5 - 5 rad/s . Low coherence is caus ed by nonlinearities or other input sources . The quantization in the turbine speed measurements is a significant nonlinearity . The quantization unit is 0 .0016 rad/s and the difference between minimum and maximum turbine speed for these three measurement series is less than 0 .05 rad/s . This implies that the
s.
1892
F. r>.lattSSOIl D2 '" O. Fig. 6 shows a typical result of the identification. Tile model error in the electrical power of this model is for all three meas urement series less than 0.08 MW and its standard deviation is 0 .025 MW . The high pass filtering was necessary in order to obtain a good result. The model error of the above identified model has low frequency «0.2 rad/s) components giving model errors of 0.5 MW . if the measured turbine speed and electrical power are used without being high pass filtered. Identification using signals without high pass filtering gives K • 2 .2 MNm/rad and the same D and D as above . The standard deviation of the model error is if.13 MW .
1.0
O.B
0.6 0.4
0.2 0 1.0
0.1
10.0 rod Is
Fig. 5. The coherence between ~ and PE for measurement series 1. turbine speed measurement is quantized into only 30 different values. The resolution in the power measurements is better. The quantization is 0 .0025 MW and the variation was 1 MW implying over 300 different values. Low resolution in the turbine speed measurements is a probable reason for the poor correlation at high frequencies . Furthermore . Fig. 2 shows that the spectrum of '" has a peak at 12-13 rad/s . but the spectrum of P has no such peak. No explanation to this has been foun!: Artefacts in the measurement device for the turbine speed may be an explanation. When deriving relation (3) it was as sumed that a~ = a~ and hence that a", = al. This as sumption is violated if the bus frequency varies. Low frequency variations in the bus frequency may cause a significant difference between al and a",. This is one possible explanation for the poor coherence between turbine speed and electrical power below 0.5 rad/s. If the bus frequency is assumed to be constant . the relation (3) indicates that the spectral density of a~ should be small for low frequencies. Low frequency measurement disturbances when measuring the turbine speed have then large impact on the spectral properties for low frequencies and these disturbances are thus another possible explanation for the poor coherence below 0 .5 rad/s . The drive train is equipped with hydraulic dampers . Measurements made by the manufacturers show that these hydraulic dampers have a pure quadratic characteristic with no linear term up to 0 .03 rad/s. Above 0 .03 rad/s the hydraulic dampers give a constant torque . For all three measurements series the deviation from mean turbine speed is less than 0.03 rad/s . This suggests an extension of the model (3) to
The identification indicates that there are other sources of damping than the hydraulic dampers that are important for sma ll os c illations. If the linear damping is elimin ated from the model by setting D to zero. identification gives the 2 same value for K and D2 = 460 ± 50 MNm(s/rad) . The givi'n value for the hydraulic dampers is 488 MNm(s/rad) . This model gives a poorer fit to the data than the linear model. The standard deviation of the model error in the electrical power is 0.034 MW. Le.t us return to the linear model (3) . We can view either a", or aPE as input and use ML-identification. [t turns out that it IS not possible to improve the system model by increasing the model order. It is only the noise model that is improved. Errors in the observations of the turbine speed and the e[ectrical power are a basic problem. When using ML-identification it is assumed that the inputs are the true ones. Nothing more can be done w ithout a priori knowledge about the measurement noises. [n summary. the relation (3) is at least good in the frequency range 0.5 - 5 rad/s . which is an important frequency range for pitch angle control.
0.01
-0.01
3
~
(1.'"
0 .2
~
Cb
~
0
a. (25)
0
If a~ . a~ t a~ I and a", are viewed as inputs and aPE as output . the parameters can be estimated by a least square fit. However. the values for a", are not available . but have to be calculated from a~. If it is assumed that a~ varies linearly between the sampling instances we get
ti
";:
2
UJ
3
~
g Cb
Kh{
~~(ih) + (6~(nh)-6~(0))/2}
0.1 -0.1
Cb
n-1 +
-0 .2
"0 0
(26)
i=O Guided by the coherence . the meas ured a~ and ap were processed in the following way: First they were pass filtered with a second order Butterworth filter with the cut-off frequency 5 rad/s and resampled with h • 0.1 s. The resampled signals were then high pass filtered with a second order Butterworth filter with the cut -off frequency 0.5 rad/s. A least square fit to the model (26) gave the following parameter estimates K = 9.4 :10.2 MNm/rad. D = 4.3 ±0.3 MNms/rad and
10£
~
50
60
70
BD
90
1005
Fig . 6 . Measurements from series 3 and the model error of the model (26) with K c 9 .4 MNm/rad. D • 4.3 MNms/rad and D2 • O. The measurements are low pass filtered witH a second order Butterworth filter with the cut -off frequency 5 rad/s and high pass filtered with a second order Butterworth filter with the cut-off frequency 0.5 rad/s .
Idelltificatioll of Wind Turbine Dynamics
189~
Wind Speed The unavailability of representative wind speed measurements causes many problems . [t is difficult to get such measurements, because the wind speed at a point may be poorly correlated with the mean wind speed U. sensed by the turbine. First, the wind speed at a point of the rotor disc is in s ome weather conditions poorly correlated with the mean value over this disc. Second, the plant may al s o itself induce local disturbances so that an anemometer gives bad measurements . There is an anemometer on the nacelle. A few hundred meters from the plant there is a ma s t with anemometers . The output UN of the anemometer on the nacelle is very strange. [t has dips down to 8 - 10 m/s with a duration of about one second. The output U of the anemometer on the M mast at an altitude of 75 m nas no such dips.
1.0
0.1
0.01 r - - r - - - r - - r - - " T - - r - - " T - - r - - 4 - i 0 .01 0.1 1.0 rod's
deg [n the following measurement series 3 was used, since it had the highes t mean wind speed. The mean values m(U ) s 16.3 m/s and m(U ) s 16.7 m/s and the standarJl de vi ations O'(U ) c 2 . ob/- m/s and O'(U ). 0.60 m/s o N From ClU (s ampling frequency 2 Hz) the ~L-procedure M gave a fir'sl order model
o
-50
(27) where TM = 5 ± 1 s and v is white noise with zero mean and noise intens ity r . Holley, Thresher and Un (1981) propos e a procedurev for calculating the parameters in model (6). It gives the time constant T • 7.9 s and the standard deviation 0' = 0.44 m/so w w
-100
0 .01
O. ,
1.0
rod's
From Blade Servo and Wind To Electrical Power The values in Table 1 represent our a priori knowledge of the system . The derivatives of the aerodynamical torques are calculated under the assumption that the mean wind speed O. is 16.7 m/s and that the generated power PE is 2 .5 MW. The uncertainties in the calc~lations are indicated in the following. An increase of U. with 1 m/s gives an increase of 10% in TR and an increase of 5% in TU ' An increase of PE wllh 0.5 MW gives a decrease of 5% in T i3 and an increase of 4% in TU ' If the sampling frequency was chosen to 10 Hz when identifying a model with the output ClP and the inputs E Cli3r. and ClU , the ML-procedure mainly attempted to N moael the nOIse. Models for the 2P, 8P, 4P and 6P variations were obtained with increasing model order . [dpac's ML- procedure minimizes the one-step prediction error, which means that the noise characteristics are more important than the system dynamics for short sampling periods . To eliminate the 4P, 6P and 8P variations the measured data were filtered with a second order Butterworth filter with the cut-off frequency 5 rad/s . The signals were resampled with 2 Hz . [f it is assumed that Cli3 varied linearly between the sampling instances the tim[ discrete model gets a direct term . This is unfortunately not identifiable , since the controller also has a direct term. But fortunately, this is not too serious. The dis crete time transfer functions calculated using a s 0 or a = 1 have the same poles. Below 2 rad/s the difference in amplitude is les s than 10% and in phase less than 25·.
TABLE 1: Numerical values used in calculations with the physical models . 6
K
7 . 7.10
J
6 5. 1 . 10 kgm
<1
w
Ti3 T·
0»
0 .44 9 .8 . 10
- 6 . 0 . 10
Nm/rad 2
m/s 6
5
Nm/rad Nms/rad
D
6 3.0 . 10 Nms/rad
T
0.4
s
T w
7 .9
s
TU
2.1 . 10
Fig. 7. Bode plots of the transfer function from the white noise e to ClP [MW) . The bold lines are for the estimated sixttf order model with the input Cli3 . The thin lines are for the models (1)-(4) and (~) with the parameter values given in Tab[e 1.
1.0
0 .1
0 .01 0.01
0.1
1.0
rod's
0.1
1.0
rod's
deg
o
- 50
-100
5
Ns
0.01
Fig. 8 . Bode plots of the transfer functions from the white nois e e to ClP E [MW) for the estimated sixth order model and Four models picked with the random distribution given by the estimated parameters and their cQvariance matrix .
1894
S. E. Matlssoll
10.0
1.0
0.1 0.01
0.1
1.0
rod/s
o deg -100
-200
-300
0.01
The ML-procedure and Idpac's test procedures show that ~UN has low explanatory significance for variations in the electrical power. The variations are picked up in the noise model. A fifth order model is obtained if both ~i3 and ~UN are used as input. A sixth order model i~ obtained if only ~i3 is used as input. The fifth order model is just on tfie borderline for acceptance . Fig . 7 shows the Bode plot of the transfer function from the white noise e to ~P for the estimated model. It is hard to analyse the uncer'lainties in the transfer function . The ML-procedure gives a covariance matrix for the uncertainties of the parameters. Idpac's RANPA-command provides a facility which can give us an idea of the uncertainties . It generates a new model by picking parameters with the random distribution given by the estimated parameters and their covariance matrix. Fig. 8 shows the transfer functions for four such models. If ~U is also used as an input , the Bode plot of the ideMified model does not differ from those given in Fig. 8. The physical noise model lies within the uncertainties of the estimated noise model. Note that the 2P variations is not included in the physical noise model. The low explanatory power of ~UN is unfortunate, since it implies a larger statistical uncertainty. It also introduces numerical problems. The A-polynomial needs a zero close to one to model the turbulence. To eliminate that pole in the transfer function from ~i3 the B-polynomial must have a zero close to one. This if hard to obtain numerically. The location of this pole _is rather uncertain. It can eliminated by using (l-q )~~ as input and (l-q- )~PE as output when mI"king ML-identification of the deterministic part. A fourth order model is then obtained. See Fig.O and 10. It is hard to estimate the static gain, because, as the model (6) indicates, the wind has much energy at low frequencies.
¥.
0.1
1.0
rod/s
Fig. O. Bode plots of the transfer function from ~~ [rad] to ~PE [MW] . The bold lines are for th~ estimated fourth order model. The thin lines are for the models (1)-(5) with the parameter values in Table 1 and Cl< O. D
10.0
The estimated model has one complex pair for the 2P variations and one complex pole pair at 0 .62 ± 0 .52i. which in continuous time corresponds to -0.4 ± 1.4i. Interpretation of this pole as the po~~ of the soft shaft· m'2de _~ives (D-T;l/J. 0.85 (s rad) and K/J z 2. 13 (s rad) '6 The ~tim':!:ted values for K and D then give J • (5±0.5)·10 kgm. A possible explanation to the absence of the servo pole is that it is hard to detect because the continuous time pole is close to the zero at -D/K. 4. CONCLUSIONS
1.0
0.01
0.1
1.0
rod/s
de~'-------------~---==---------I
In this paper identification of wind turbine dynamics of i"terest for pitch angle control has been considered. The results show a good agreement with simple physical models. The unavailability of representative wind measurements forced us to model the wind variations by a noise model in the identification. This gave uncertainties in the estimated parameters of 10-20%. No dynamics were found below 5 rad/s that could not be explained by the physical models or the statistical uncertainty. ACKNOWLEDGEMENTS This work was supported by Sydkraft AB . MalmO, Sweden. I want to thank Mr . Eskil UI~n for his assistance at the recording of the mea surements and Professor Karl Johan Astr6m for many valuable dis cussions.
-200
REFERENCES
-400
-600;---r---~-'r-~---'r--r--~--~
0.01
0.1
1.0
rod/s
Fig. 10. Bode plots of the transfer functions from ~i3 [rad] to ~PE [MW] for the estimated fourth ordeF model and Tour models picked with the random distribution given by the estimated parameters and their covariance matrix .
Astr6m, K.J . and KallstrOm , C.G. (1976). Identification of Ship Steering Dynamics. Automatica . 12. 0-22. Holley . W.E.. Thresher. R.W . and Lin. S-R. (1981). Wind Turbulence Inputs for Horizontal Axis Wind Turbines . In Wind Turbine Dynamics. NASA-CP-2185 . DOE CONF- 810226. pp. 101-112. Mattsson . S .E. (1984). Modelling and Control of Large Horizontal Axis Wind Power Plants . TFRT-l026. Department of Automatic Control. Lund Institute of Tec hnology. Lund . Sweden. Wieslander . J. (1980). Idpac Commands - User's Guide. TFRT -3157. Department of Automatic Control. Lund Institute of Technology. Lund . Sweden.
©
I F .. \( : Idt"1I 1ificl l io n and S ~' S ll ' llI Pa r;tlllt"1t:r Estim atioll I ~ ' ~'G. Yo rk . l ' K . I !I~F)
IDENTIFICATION OF WIND TURBINE DYNAMICS S. E. Mattsson /) I'/)(Jrlllll'lll or iI /llo/lwl it COl/lml. 1./11/(1 I lI.I l illlll' 0( T I'I-/II/olog.'. 1'. 0 . H ox 11 8 .
.\'-221
()(J
/./11/(1.
S 1I'1'I1I' 11
Abstract. The energy conversion from wind energy to electrical energy of a large horizontal axis wind turbine can be controlled by changing the pitch angle. Thus , it is of interes t to identify the dynamics from wind and pitch angle to electrical power . This paper presents results from identification of wind turbine dynamics . A large data set has been collected from a prototype 3 MW system called the WTS-3 located in southern Sweden. Models of different structure have been estimated and compared with physical models . Keywords . Wind power; wind turbine dynamics ; identification; parameter estimation.
(2)
1. INTRODUCTION The energy conversion from wind energy to electrical energy of a large horizontal axis wind turbine can be controlled by changing the pitch angle of the blades. This paper presents results from identification of wind turbine dynamics of interest for pitch angle control.
For the turbine speed ~ we have tJ.~ '" tJ.l . Let ~O denote the synchronous turbine speed. then generated power IS
A large data set has been collected from a prototype 3 MW system called the WTS-3 located in southern Sweden. It is built by Karlskronavarvet AB . Sweden and Hamilton Standa rd a division of United Technologies Inc. USA . A 4 MW plant (the WTS- 4) of the same design is built in Medicine Bow. Wyoming. USA . The WTS-3 is designed to supply power in parallel with other generators to a large power utility grid and to operate in wind forces of 5-26 m/s o The rated power is reached at 14 m/so The WTS-3 has a horizontal axis wind turbine with two blades . The power is controlled by changing the pitch angle. The blade actuators are hydraulic positioning systems. The blades have a length of 39 m. and a weight of 14000 kg each. The tower is 78 m high . The turbine rotates at 25 rpm and a multistage planetary gearbox steps up the rotation to 1500 rpm . The generator is a three-phase . 50 Hz. 1500 rpm synchronous machine .
The driving aerodynamical torque T is a nonlinear function of the pitch angle (3 , the mean wind sp~ed U. over the rotor disc of the oncoming wind and >1>. The Iinearized expres s ion for tJ. T will be written as
(3)
(4)
This section presents physical models of interest for pitch angle control. A thorough discussion can be found in Mattsson (1984) .
The relations (I) and (2) are important and fundamental for pitch angle control. They show that the variations in the aerodynamical torque above the natural frequency "' =./f\(J of the torsional mode are attenuated when s~owing up in the shaft torque. The designers of the WTS-3 have taken advantage of the low pass dynamics and deliberately designed the drive train to be torsionally soft. This is achieved by a soft mounting of \pe plilfetary gearbox. The nominal values J. 5.1·10 kgm and K • 7 .7 MNm/rad give "'I • 1.2 rad/s . The aerodynamical damping of the turbine is low. because the blades are designed to give minimum losses . A synchronous generator provides hardly any damping for the torsional mode. The coupling between a synchronous generator and the grid is much stiffer than the drive train. The damping prov ided by the drive train is also low . Thus the controller mus t provide damping. The natural frequency", influences the bandwidth needed of the 1 pitch angle c 0ntrol . since it is no sens e to choose the cros sover frequency much higher than "'I ' For the WTS- 3 a cro s sover frequency of 2-3 rad/s is needed (Matts son. 1984) .
Turbine Dynamics
Blade Servo
In the frequency range of interest for pitch angle control the dynamics of the synchronous generator can be neglected. The basic dynamics is given by the oscillation of the turbine against the electrical system. A model for this oscillation around a stationa ry operating point is
The blade servo is designed to be a position servo with a time constant of 0 .4 s . However. the servo speed is limited to 4· Is . For the blade servo we thus have the first order model
The paper is organized as follows . Physical models presented in Section 2. In Section 3 measured data analysed. identification of models is discussed and resulting models are compared with physical models . results are summarized in Section 4.
are are the The
2 . PHYSICAL MODELS
Jd~ + Ddl + Kd~
= dT
(5)
(I )
where T is the aerodynamical torque . ~ is the torsion of the drive train , J is the turbine inertia and K and Dare the torsional spring and damping coefficients of the drive train . This model gives the electrical torque as
Wind Characteris tics For control design the wind speed can be viewed as consisting of one constant part and one stochastic part .
1890 The constant part represents the mean wind speed over 10 minutes and the stochastic part the turbulence . The wind speed varies over the turbine area . The major influence of the wind can be described by the mean wind speed U. over the turbine area. A simple model for the stochastic part ~U. of U. is (Holley and others. 1981)
(6) where w is white noise with zero mean and noise intensity rand (1 is the standard deviation of ~U • . W The time co"hstant T is of the order of 5 - 30 seconds . w
Transformation to Discrete Time Input/Output Models The measured data are sampled and the identification procedure used gives discrete time models on the form A(q
-1
)y(t)
2
B(q
-1
)u(t) + ~C(q
-1
(7)
)e(t)
where y is the output. u is the input of the system. e is white noise with zero mean_i'nd variance one. A. Band C is the delay operator; a!:f polynomials and q q x(t). x(t-h) with h being the time between two sampling instances. The model may have several inputs. Since the physical models presented above are continuous time models. they must be transformed to discrete time input/output models to compare with the experimental data. The physical models can be written as X
Y - Hx + Du
(9)
with F. G. Hand D being constant matrices and v being multivariable . continuous time . white noise with zero mean and the noise intensity R l ' A discrete time model with sampling interval h is then given by x(t+h) - ~x(t) +
h
J o
e
F(h-s)
~ _ e Fh
(12)
where w is multivariable discrete time white noise with zero mean and variance _
h
J eF(h-s)R
0
u(t+s) - u(t) + Ol·s/h · (u(t+h)-u(t))
(eF(h-S))TdS
( 13)
1
To calculate the transfer function ~C/A from e to y a spectral factorization must be performed. If we like. we can introduce white measurement noise < with variance Rand uncorrelated with w ; < y(t) = Hx(t) + Du(t) + < (t) (14) Let P be a symmetric. non negative definite solution to the discrete time. algebraic Riccati equation (15) Introduce K defined as (16) and let I be an non unity matrix. where dimensionnof the matrix F. then
varies ( 19)
For Ol • 0 u is constant between the sampling instances. For Ol • 1 u is continuous and varies linearly between the sampling ins tances. Under this assumption ~x(t)+ru(t)+Olrl(u(t+h)-u(t))+w(t)
x(t+h)
(20)
h
J o
r .
1
(21)
eF(h-s)G ds
(22)
h
.! J s
r
eF(h-s)G ds
h 0
The transfer function B/A 'from u to y becomes B(q
-1
)/A(q
-1
) - D+H(ql
n
...
-~)
-1
(r + Olr (q-l)) l
(23)
3. IDENTIFICA nON
(11)
w
u
( 10)
Gu(t+s)ds + wIt)
y(t) : Hx(t) + Du(t)
R
Let t be a sa mpling instance. Assume that between the sampling instances as (0 :5 s < h)
(8)
Fx + Gu + v
m
sampling instances . This is typically the case when u is an output from a digital controller . However . in our case the inputs are analog . If the sampling frequency is sufficiently high it is reas onable to assume that u varies linearly between sampling instances. However . we must not choose too high a sampling frequency. since it gives numerically bad conditions for the e s tima~on. If p is a pole of the continuous time model then e P is a pole to the discrete time model. If h is small compared to the time constants of the s y s tem . the discrete time model . thus has its poles close to one and the calculation of the time cons tants of the continuous time model is ill-conditioned. It is possible to es timate the parameters directly from discrete time measurements . See Astr6m and Kallstr6m (1976). However. this approach requires a much greater effort. both in form of development of software and computing time .
n
is
the
( 17) ( 18) To calculate the transfer function B/ A from u to y we must evaluate the integral on the right side of (10) . This requires knowledge of the shape of u between the sampling instances . If we can choose the representation for u . it is convenient to let it be constant between the
Three measurement series from the WTS-3 recorded on the 11th October 1983 are used for identification: Series 1 at 11 :05. 240 seconds long. 24 000 data points Series 2 at 12:52. 240 seconds long . 24 000 data points Series 3 at 13:04. 210 seconds long. 21 000 data points The measurements were filtered with a sixth order Bessel filter with a cut- off frequency of 25 Hz and sampled with 100 Hz . One difficulty is that the system may be poorly identifiable. because it is controlled. To increase the identifiability the power reference of the controller was switched between 2 MW and 3 MW according to a PRBS-sequence . The interactive program package ldpac (Wieslander . 1980) was used to analyse the data . Fig. 1 shows a power spectrum for the electrical power PE calculated from measurement series 1. The series was cut into blocks containing 2048 data points each. Discrete Fourier transform with a Blackman-Harris time window with a sidelobe level of -74 dB (Harris . 1978) was used to calculate a spectrum for each of the first ten data blocks . Fig. 1 shows the mean of these spectra. The power spectra for the turbine speed ~ and the servo reference f3 shown in Fig. 2 and 3 were calculated in the same Jay . The power spectra from the three different measurement series are very similar. The power spectra for PE and ~ have several peaks. It is convenient to introdllce a normalized frequency P. such that lP corresponds to the synchronous turbine speed. which is 2.618 rad/s for the WTS-3 . There are large peaks at 2P. 4P . 6P and 8P. There are also peaks at odd integer multiples of lP. but they are much smaller . These variations are caused by the spatial variation of the wind speed over the turbine area . The variations at odd integer multiples of lP are smaller. because the turbine has two blades. There are three major sources of the variations at integer multiples of P . First. the mean wind
Idl'll1ifi( atioll o f Wind TlIrbille Ihnatlli rs
I H9 1
10- 2 10- 3
10- 4 10- 5 10- 6 ;;-
10
100
rod/s
Fig . 1. Power s pec trum for the electrical power PE [MW) from meas urement series 1.
5
=
C 0 Cl> "0 0
0
iD
o
.... o
t
Cl>
Qi
"8 ~
100
120
11.0
s
Fig . 4. Meas urements from series 2 and the model error of the model (24) with T • 0 .65 s and a • 1. 10
100
rod/s
Fig. 2. Power spectrum for the turbine speed ~ [rad/s) from meas ure ment s er ies 1.
10- 5
If the generator dynamics is included in the physical models. they indicate that the system has its second torsional mode at 25 rad/s . The second torsional mode is thus a pos s ible explanation to why the peak at BP (-21 rad/s) is wider than those at 4P and 6P. Blade Servo Sampling of the servo model gives with a • e -hit
10-&
(l - aq -1 )Af3. [ (1-a)q - 1+a(1+(1-a)/ln a)(1-q -1 ) ] Af3 (24) r
10- 7 10- 8
10-9
10
100
rod / s
Fig. 3. Power s pectrum for the servo reference f3 [rad) r from mea s urement series 1.
speed varies with altitude. Second. the tower blocks the airflow for a rotating blade when it passes the tower . The blockage is of s hort duration. which means that at leas t 3 - 4 of the fir s t ev en harmonics have amplitudes of the s ame magnitude . Third . a rotating blade encounters tu r bulence whose characteristics are quite different from tu r bulence encountered in a fixed po int. The mid frequency r egion is depleted and the removed energy is distributed into the high frequency end with spikes at integer multiples of IP . An intuitive explanation is that a rotating blade chops across the eddies whereas a small fixed anemometer is totally immers ed in the turbulence eddies . The 2P variations in the aerodynamical torque are one good rea s on for having a s oft shaft. The 2P v ariations in the aerodynamical torque are attenuated more than eight time s when s howing up in the electrical torque. The 2P v a riations in the generated power has an amplitude of 1- 3% of rated power.
For each meas urement series the mean value of f3 was subtracted from f3 giving tl.f3 . The blade angle tl.~ was calculated in the s.fme way . Ttfe signals tl.f3 and tl.f3 were filtered with a second order Butterworth fnter having the cut-off frequen cy 10 rad/s and resampled with h • 0 . 1 s . Since both tl.f3 and tl.f3 were filtered with the same filter, the trans fer function between them remains unchanged. The filter ing does of course modify the importance of the behaviour at different frequencies when estimating the models. The ML-procedure and Idpac's evaluating procedures sugges t that the servo dynamics can be modelled by (5) with T · 0.65 :t 0.05 s. The model error, which is the difference between measured output and model output. s hows that the model is good . For a • 0 as well as for a = 1 the model error of (24) with T = 0 .65 s is for all three measurement series less than 0.20 and the s tandard deviation is 0 .060. The s hape of tl.f3 between the sampling is unimportant . because h is smafi compared to r. A typical res ult is illustrated in Fig. 4 . Turbine Speed and Electrical Power To get some insight into the validity of the relation (3) the coherence between turbine speed and electrical power w as determined . Fig . 5 s hows that the coherence is good in the frequency range 0 .5 - 5 rad/s . Low coherence is caus ed by nonlinearities or other input sources . The quantization in the turbine speed measurements is a significant nonlinearity . The quantization unit is 0 .0016 rad/s and the difference between minimum and maximum turbine speed for these three measurement series is less than 0 .05 rad/s . This implies that the
s.
1892
F. r>.lattSSOIl D2 '" O. Fig. 6 shows a typical result of the identification. Tile model error in the electrical power of this model is for all three meas urement series less than 0.08 MW and its standard deviation is 0 .025 MW . The high pass filtering was necessary in order to obtain a good result. The model error of the above identified model has low frequency «0.2 rad/s) components giving model errors of 0.5 MW . if the measured turbine speed and electrical power are used without being high pass filtered. Identification using signals without high pass filtering gives K • 2 .2 MNm/rad and the same D and D as above . The standard deviation of the model error is if.13 MW .
1.0
O.B
0.6 0.4
0.2 0 1.0
0.1
10.0 rod Is
Fig. 5. The coherence between ~ and PE for measurement series 1. turbine speed measurement is quantized into only 30 different values. The resolution in the power measurements is better. The quantization is 0 .0025 MW and the variation was 1 MW implying over 300 different values. Low resolution in the turbine speed measurements is a probable reason for the poor correlation at high frequencies . Furthermore . Fig. 2 shows that the spectrum of '" has a peak at 12-13 rad/s . but the spectrum of P has no such peak. No explanation to this has been foun!: Artefacts in the measurement device for the turbine speed may be an explanation. When deriving relation (3) it was as sumed that a~ = a~ and hence that a", = al. This as sumption is violated if the bus frequency varies. Low frequency variations in the bus frequency may cause a significant difference between al and a",. This is one possible explanation for the poor coherence between turbine speed and electrical power below 0.5 rad/s. If the bus frequency is assumed to be constant . the relation (3) indicates that the spectral density of a~ should be small for low frequencies. Low frequency measurement disturbances when measuring the turbine speed have then large impact on the spectral properties for low frequencies and these disturbances are thus another possible explanation for the poor coherence below 0 .5 rad/s . The drive train is equipped with hydraulic dampers . Measurements made by the manufacturers show that these hydraulic dampers have a pure quadratic characteristic with no linear term up to 0 .03 rad/s. Above 0 .03 rad/s the hydraulic dampers give a constant torque . For all three measurements series the deviation from mean turbine speed is less than 0.03 rad/s . This suggests an extension of the model (3) to
The identification indicates that there are other sources of damping than the hydraulic dampers that are important for sma ll os c illations. If the linear damping is elimin ated from the model by setting D to zero. identification gives the 2 same value for K and D2 = 460 ± 50 MNm(s/rad) . The givi'n value for the hydraulic dampers is 488 MNm(s/rad) . This model gives a poorer fit to the data than the linear model. The standard deviation of the model error in the electrical power is 0.034 MW. Le.t us return to the linear model (3) . We can view either a", or aPE as input and use ML-identification. [t turns out that it IS not possible to improve the system model by increasing the model order. It is only the noise model that is improved. Errors in the observations of the turbine speed and the e[ectrical power are a basic problem. When using ML-identification it is assumed that the inputs are the true ones. Nothing more can be done w ithout a priori knowledge about the measurement noises. [n summary. the relation (3) is at least good in the frequency range 0.5 - 5 rad/s . which is an important frequency range for pitch angle control.
0.01
-0.01
3
~
(1.'"
0 .2
~
Cb
~
0
a. (25)
0
If a~ . a~ t a~ I and a", are viewed as inputs and aPE as output . the parameters can be estimated by a least square fit. However. the values for a", are not available . but have to be calculated from a~. If it is assumed that a~ varies linearly between the sampling instances we get
ti
";:
2
UJ
3
~
g Cb
Kh{
~~(ih) + (6~(nh)-6~(0))/2}
0.1 -0.1
Cb
n-1 +
-0 .2
"0 0
(26)
i=O Guided by the coherence . the meas ured a~ and ap were processed in the following way: First they were pass filtered with a second order Butterworth filter with the cut-off frequency 5 rad/s and resampled with h • 0.1 s. The resampled signals were then high pass filtered with a second order Butterworth filter with the cut -off frequency 0.5 rad/s. A least square fit to the model (26) gave the following parameter estimates K = 9.4 :10.2 MNm/rad. D = 4.3 ±0.3 MNms/rad and
10£
~
50
60
70
BD
90
1005
Fig . 6 . Measurements from series 3 and the model error of the model (26) with K c 9 .4 MNm/rad. D • 4.3 MNms/rad and D2 • O. The measurements are low pass filtered witH a second order Butterworth filter with the cut -off frequency 5 rad/s and high pass filtered with a second order Butterworth filter with the cut-off frequency 0.5 rad/s .
Idelltificatioll of Wind Turbine Dynamics
189~
Wind Speed The unavailability of representative wind speed measurements causes many problems . [t is difficult to get such measurements, because the wind speed at a point may be poorly correlated with the mean wind speed U. sensed by the turbine. First, the wind speed at a point of the rotor disc is in s ome weather conditions poorly correlated with the mean value over this disc. Second, the plant may al s o itself induce local disturbances so that an anemometer gives bad measurements . There is an anemometer on the nacelle. A few hundred meters from the plant there is a ma s t with anemometers . The output UN of the anemometer on the nacelle is very strange. [t has dips down to 8 - 10 m/s with a duration of about one second. The output U of the anemometer on the M mast at an altitude of 75 m nas no such dips.
1.0
0.1
0.01 r - - r - - - r - - r - - " T - - r - - " T - - r - - 4 - i 0 .01 0.1 1.0 rod's
deg [n the following measurement series 3 was used, since it had the highes t mean wind speed. The mean values m(U ) s 16.3 m/s and m(U ) s 16.7 m/s and the standarJl de vi ations O'(U ) c 2 . ob/- m/s and O'(U ). 0.60 m/s o N From ClU (s ampling frequency 2 Hz) the ~L-procedure M gave a fir'sl order model
o
-50
(27) where TM = 5 ± 1 s and v is white noise with zero mean and noise intens ity r . Holley, Thresher and Un (1981) propos e a procedurev for calculating the parameters in model (6). It gives the time constant T • 7.9 s and the standard deviation 0' = 0.44 m/so w w
-100
0 .01
O. ,
1.0
rod's
From Blade Servo and Wind To Electrical Power The values in Table 1 represent our a priori knowledge of the system . The derivatives of the aerodynamical torques are calculated under the assumption that the mean wind speed O. is 16.7 m/s and that the generated power PE is 2 .5 MW. The uncertainties in the calc~lations are indicated in the following. An increase of U. with 1 m/s gives an increase of 10% in TR and an increase of 5% in TU ' An increase of PE wllh 0.5 MW gives a decrease of 5% in T i3 and an increase of 4% in TU ' If the sampling frequency was chosen to 10 Hz when identifying a model with the output ClP and the inputs E Cli3r. and ClU , the ML-procedure mainly attempted to N moael the nOIse. Models for the 2P, 8P, 4P and 6P variations were obtained with increasing model order . [dpac's ML- procedure minimizes the one-step prediction error, which means that the noise characteristics are more important than the system dynamics for short sampling periods . To eliminate the 4P, 6P and 8P variations the measured data were filtered with a second order Butterworth filter with the cut-off frequency 5 rad/s . The signals were resampled with 2 Hz . [f it is assumed that Cli3 varied linearly between the sampling instances the tim[ discrete model gets a direct term . This is unfortunately not identifiable , since the controller also has a direct term. But fortunately, this is not too serious. The dis crete time transfer functions calculated using a s 0 or a = 1 have the same poles. Below 2 rad/s the difference in amplitude is les s than 10% and in phase less than 25·.
TABLE 1: Numerical values used in calculations with the physical models . 6
K
7 . 7.10
J
6 5. 1 . 10 kgm
<1
w
Ti3 T·
0»
0 .44 9 .8 . 10
- 6 . 0 . 10
Nm/rad 2
m/s 6
5
Nm/rad Nms/rad
D
6 3.0 . 10 Nms/rad
T
0.4
s
T w
7 .9
s
TU
2.1 . 10
Fig. 7. Bode plots of the transfer function from the white noise e to ClP [MW) . The bold lines are for the estimated sixttf order model with the input Cli3 . The thin lines are for the models (1)-(4) and (~) with the parameter values given in Tab[e 1.
1.0
0 .1
0 .01 0.01
0.1
1.0
rod's
0.1
1.0
rod's
deg
o
- 50
-100
5
Ns
0.01
Fig. 8 . Bode plots of the transfer functions from the white nois e e to ClP E [MW) for the estimated sixth order model and Four models picked with the random distribution given by the estimated parameters and their cQvariance matrix .
1894
S. E. Matlssoll
10.0
1.0
0.1 0.01
0.1
1.0
rod/s
o deg -100
-200
-300
0.01
The ML-procedure and Idpac's test procedures show that ~UN has low explanatory significance for variations in the electrical power. The variations are picked up in the noise model. A fifth order model is obtained if both ~i3 and ~UN are used as input. A sixth order model i~ obtained if only ~i3 is used as input. The fifth order model is just on tfie borderline for acceptance . Fig . 7 shows the Bode plot of the transfer function from the white noise e to ~P for the estimated model. It is hard to analyse the uncer'lainties in the transfer function . The ML-procedure gives a covariance matrix for the uncertainties of the parameters. Idpac's RANPA-command provides a facility which can give us an idea of the uncertainties . It generates a new model by picking parameters with the random distribution given by the estimated parameters and their covariance matrix. Fig. 8 shows the transfer functions for four such models. If ~U is also used as an input , the Bode plot of the ideMified model does not differ from those given in Fig. 8. The physical noise model lies within the uncertainties of the estimated noise model. Note that the 2P variations is not included in the physical noise model. The low explanatory power of ~UN is unfortunate, since it implies a larger statistical uncertainty. It also introduces numerical problems. The A-polynomial needs a zero close to one to model the turbulence. To eliminate that pole in the transfer function from ~i3 the B-polynomial must have a zero close to one. This if hard to obtain numerically. The location of this pole _is rather uncertain. It can eliminated by using (l-q )~~ as input and (l-q- )~PE as output when mI"king ML-identification of the deterministic part. A fourth order model is then obtained. See Fig.O and 10. It is hard to estimate the static gain, because, as the model (6) indicates, the wind has much energy at low frequencies.
¥.
0.1
1.0
rod/s
Fig. O. Bode plots of the transfer function from ~~ [rad] to ~PE [MW] . The bold lines are for th~ estimated fourth order model. The thin lines are for the models (1)-(5) with the parameter values in Table 1 and Cl< O. D
10.0
The estimated model has one complex pair for the 2P variations and one complex pole pair at 0 .62 ± 0 .52i. which in continuous time corresponds to -0.4 ± 1.4i. Interpretation of this pole as the po~~ of the soft shaft· m'2de _~ives (D-T;l/J. 0.85 (s rad) and K/J z 2. 13 (s rad) '6 The ~tim':!:ted values for K and D then give J • (5±0.5)·10 kgm. A possible explanation to the absence of the servo pole is that it is hard to detect because the continuous time pole is close to the zero at -D/K. 4. CONCLUSIONS
1.0
0.01
0.1
1.0
rod/s
de~'-------------~---==---------I
In this paper identification of wind turbine dynamics of i"terest for pitch angle control has been considered. The results show a good agreement with simple physical models. The unavailability of representative wind measurements forced us to model the wind variations by a noise model in the identification. This gave uncertainties in the estimated parameters of 10-20%. No dynamics were found below 5 rad/s that could not be explained by the physical models or the statistical uncertainty. ACKNOWLEDGEMENTS This work was supported by Sydkraft AB . MalmO, Sweden. I want to thank Mr . Eskil UI~n for his assistance at the recording of the mea surements and Professor Karl Johan Astr6m for many valuable dis cussions.
-200
REFERENCES
-400
-600;---r---~-'r-~---'r--r--~--~
0.01
0.1
1.0
rod/s
Fig. 10. Bode plots of the transfer functions from ~i3 [rad] to ~PE [MW] for the estimated fourth ordeF model and Tour models picked with the random distribution given by the estimated parameters and their covariance matrix .
Astr6m, K.J . and KallstrOm , C.G. (1976). Identification of Ship Steering Dynamics. Automatica . 12. 0-22. Holley . W.E.. Thresher. R.W . and Lin. S-R. (1981). Wind Turbulence Inputs for Horizontal Axis Wind Turbines . In Wind Turbine Dynamics. NASA-CP-2185 . DOE CONF- 810226. pp. 101-112. Mattsson . S .E. (1984). Modelling and Control of Large Horizontal Axis Wind Power Plants . TFRT-l026. Department of Automatic Control. Lund Institute of Tec hnology. Lund . Sweden. Wieslander . J. (1980). Idpac Commands - User's Guide. TFRT -3157. Department of Automatic Control. Lund Institute of Technology. Lund . Sweden.