Control of a Variable-Speed
Pitch-Regulated Wind Turbine
Torbjörn Thiringer and Andreas Petersson
Division of Electric Power Engineering
Department of Energy and Environment
CHALMERS UNIVERSITY OF TECHNOLOGY
Göteborg, Sweden 2005
Abstract
In this report, the speed control of a modern pitch-regulated wind turbine is investigated.
A controller structure is derived and the significance of various parameters in the controller
structure is investigated.
In particular the influence of the speed control bandwidth on the speed variations, torque
stresses and energy production is analyzed.
The control structure operates without any knowledge of the wind speed or shaft torque,
instead these quantities are estimated from the rotor speed and electric power.
The results are compared with measurements made on two commercial turbines and the
agreement between measurements and simulations is satisfactory.
Acknowledgements
The financial support given by the Swedish Energy Agency is gratefully acknowledged.
iii
iv
Table of Contents
Abstract
iii
Acknowledgement
iii
Table of Contents
v
1 Introduction
1
2 Wind Turbine Operation
2.1 Aerodynamic Conversion . . . . . . . . . . . . . . . . . . . .
2.1.1 Cp (λ, β)-relation . . . . . . . . . . . . . . . . . . . .
2.1.2 Linearized Aerodynamic Conversion Model . . . . . .
2.2 Steady-State Operation . . . . . . . . . . . . . . . . . . . . .
2.3 Drive Train Representation . . . . . . . . . . . . . . . . . . .
2.4 Speed and Torque Observer . . . . . . . . . . . . . . . . . . .
2.5 Finding Optimal Tip-Speed Ratio . . . . . . . . . . . . . . . .
2.6 Wind Speed Estimation for Pitch Controller . . . . . . . . . .
2.7 Wind Generation using Spatial and Rotational Sampling Filters
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5 Analysis
5.1 Bandwidth of Speed Control Loops . . . . . . . . . . . . . . . . . . . . .
5.1.1 Perfect Wind Speed and Shaft Torque Knowledge . . . . . . . . . .
36
36
36
3 Field Measurements
3.1 Jung Wind Turbine . . . . . . . . . . . . . . . . . .
3.1.1 Time Series . . . . . . . . . . . . . . . . . .
3.1.2 Statistical Data . . . . . . . . . . . . . . . .
3.2 Bast Wind Turbine . . . . . . . . . . . . . . . . . .
3.2.1 Time Series . . . . . . . . . . . . . . . . . .
3.2.2 Statistical Data . . . . . . . . . . . . . . . .
3.3 Comparison between the Bast and Jung wind turbine
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4 Control
4.1 Internal Model Control (IMC) . . . . . . . . . . . . . . .
4.2 Rotor-Speed Control with Generator . . . . . . . . . . . .
4.3 Rotor-Speed Control with Pitch . . . . . . . . . . . . . . .
4.4 Overall Control . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Implementation Example . . . . . . . . . . . . . .
4.5 Example of Low-Wind Speed operation . . . . . . . . . .
4.5.1 Decreased and increased speed control bandwidth
4.6 Example of Medium- and High-Wind Speed operation . .
4.7 Comparison Between Measurements and Simulations . . .
4.7.1 Low Wind Speed . . . . . . . . . . . . . . . . . .
4.7.2 Medium Wind Speed . . . . . . . . . . . . . . . .
4.7.3 High-Wind Speed . . . . . . . . . . . . . . . . . .
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5.1.2
5.2
5.3
5.4
6
Perfect Wind Speed and Shaft Torque Knowledge with Rotational
Sampling Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Operation based on Estimated Wind Speed and Shaft Torque . . . .
Influence of “Pre-pitching” . . . . . . . . . . . . . . . . . . . . . . . . . .
Influence of Transition Length . . . . . . . . . . . . . . . . . . . . . . . .
Turbulence Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion
39
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48
References
49
vi
1 Introduction
Wind turbines can operate with either fixed speed (actually within a speed range about 1 %)
or variable speed. For fixed-speed wind turbines, the generator (induction generator) is directly connected to the grid. Since the speed is almost fixed to the grid frequency, and most
certainly not controllable, it is not possible to store the pulsating energy capture due to the
turbulence of the wind in form of rotational energy variations. Therefore, for a fixed-speed
system the turbulence of the wind will result in power variations, and thus affect the power
quality of the grid [9] in a negative way. For a variable-speed wind turbine, the generator
is controlled by power electronic equipment, which makes it possible to control the rotor
speed. In this way the power fluctuations caused by wind variations can be more or less
absorbed by changing the rotor speed [11] and thus power variations originating from the
wind conversion and the drive train can be reduced. Hence, the power quality impact caused
by the wind turbine can be improved compared to a fixed-speed turbine [7]. One common
way of quantifying the power fluctuations is with the so called flicker coefficient. There are
also several other reasons for using variable-speed operation of wind turbines; among those
are possibilities to reduce stresses of the mechanical structure, acoustic noise and the possibility to control the reactive power [2]. Most of the major wind turbine manufactures are
developing new larger wind turbines in the 3-to-5-MW range [1]. These large wind turbines
are all based on variable-speed operation with pitch control.
Although there are a number of publications where the speed control of a pitch-regulated
turbine is presented, for instance [4, 8], no publications comparing the obtained results with
measurements have been found by the authors.
The main purpose of this report is to investigate how “hard” the rotor speed should follow
the “ideal” rotor speed in order to have a good balance between rotor speed variations and
flicker impact/torque stresses. Moreover, an aim is to study the influence of various aspects
on the performance of the control of the turbine. Finally a goal is to compare the simulated
results with data from field measurements of variable-speed wind turbines.
This report is organized as follows: Firstly, the wind turbine, some useful background
information and used observers are presented. Secondly, field measurements are presented.
Thirdly, the control laws used in the simulations are presented. Finally, analysis of the control
parameters is performed.
1
2 Wind Turbine Operation
2.1 Aerodynamic Conversion
In this section mathematical models of the system will be presented.
2.1.1
Cp (λ, β)-relation
Some of the available power in the wind is converted by the rotor blades to mechanical power
acting on the rotor shaft of the wind turbine. For steady-state calculations of the mechanical
power from a wind turbine, the so called Cp (λ, β)-curve can be used. The mechanical power,
PT , can be determined by [6]
1
PT = − ρAr Cp (λ, β)ws3 = −kCp (λ, β)ws3
2
ΩT rr
λ=
ws
(1)
(2)
where Cp is the power coefficient, β is the pitch angle, λ is the tip speed ratio, ws is the wind
speed, ΩT is the rotor speed of the turbine (on the low-speed side of the gearbox), rr is the
rotor-plane radius, ρ is the air density and Ar is the area swept by the rotor. The produced
torque, TT , of the turbine can be found from
PT
.
ΩT
TT =
(3)
The Cp (λ, β)-curve for the wind turbine investigated here can be seen in Fig. 1.
5
Cp
0
−5
−10
−15
40
20
0
β
−20
0
5
15
10
20
λ
Fig. 1: Power coefficient as a function of the tip-speed ratio and pitch angle.
2
2.1.2
Linearized Aerodynamic Conversion Model
Before linearizing the aerodynamic model we will study the partial derivatives of Cp (λ, β),
i.e.,
∂Cp (λ, β)
∂Cp (λ, β) ∂λ
∂Cp (λ, β) ∂λ
rr ΩT ∂Cp (λ, β)
=
=
=− 2
∂ws
∂ws ∂λ
∂λ
∂ws
ws
∂λ
∂Cp (λ, β) ∂λ
∂Cp (λ, β) ∂λ
rr ∂Cp (λ, β)
∂Cp (λ, β)
=
=
=
∂ΩT
∂ΩT ∂λ
∂λ
∂ΩT
ws
∂λ
(4)
(5)
The partial derivatives of TT can now be found as
∂TT
ws2
ws3 ∂Cp (λ, β)
ws2
∂Cp (λ, β)
= −3k Cp (λ, β)
−k
= −3k Cp (λ, β)
+ k w s rr
(6)
∂ws
ΩT
ΩT
∂ws
ΩT
∂λ
∂TT
w3
w3
w3 ∂Cp (λ, β)
w2 rr ∂Cp (λ, β)
= k 2s Cp (λ, β) − k s
= k 2s Cp (λ, β) − k s
(7)
∂ΩT
ΩT
ΩT ∂ΩT
ΩT
ΩT
∂λ
∂TT
w3 ∂Cp (λ, β)
= −k s
.
(8)
∂β
ΩT
∂β
T
For a given operating point (x0 = ws0 ΩT 0 β0 ) we have that
!
2
∂TT
ws0
∂Cp (λ, β0 ) (9)
= −k
3Cp (λ0 , β0 ) − λ0
∂ws
ΩT 0
∂λ
x=x0
!
3
∂TT
ws0
∂Cp (λ, β0 ) (10)
= −k 2
−Cp (λ0 , β0 ) + λ0
∂ΩT
ΩT 0
∂λ
x=x0
3
∂TT
∂Cp (λ0 , β) ws0
(11)
= −k
∂β
ΩT 0
∂β
x=x0
and the corresponding linearized torque becomes
∂TT ∂TT ∂TT ∆TT =
· ∆ws +
· ∆ΩT +
· ∆β.
∂ws x=x0
∂ΩT x=x0
∂β x=x0
(12)
In Fig. 2 the numerical partial derivatives of the Cp (λ, β)-curve with respect to λ and β
can be seen.
2.2 Steady-State Operation
In Fig. 3 the steady-state operating charcteristic of the investigated variable speed wind turbine can be seen. In the figure the two operational areas, i.e., the low wind speed area and
the high wind speed area, can be seen. In the low wind speed area (ωs < 8 m/s) the rotor
speed is controlled in order to keep the tip-speed ratio, λ, at its optimal value and the pitch
angle is zero. Then when the nominal power and maximal rotor speed is reached (ωs > 12
m/s), the rotor speed is kept around its nominal value. Moreover, when the nominal power is
reached, the blades of the turbine are pitch out of the wind in order to limit the turbine power
to its nominal value, this can also be seen in the figure. In between the high and low wind
speed region there is an operating region 8middel wind speed region) where the rotor speed
3
b)
2
100
1
0
∂Cp /∂β
∂Cp /∂λ
a)
0
−1
40
−100
−200
40
20
20
0
β
0
20
20
10
0
β
λ
10
0
λ
Fig. 2: Partial derivatives of the power coefficient Cp (λ, β) with respect to a) Tip speed ratio λ. b)
Pitch angle β.
a)
b)
100
3
2.5
ΩT [rad/s]
PT [%]
80
60
40
2
1.5
20
0
1
5
10
15
20
25
5
10
ws [m/s]
d)
c) 30
20
25
20
25
9
8
25
7
20
6
15
λ
β [deg]
15
ws [m/s]
5
10
4
5
3
0
2
5
10
15
20
25
5
ws [m/s]
10
15
ws [m/s]
Fig. 3: Steady-state operation of the WT. a) Power PT . b) Rotor speed ΩT . c) Pitch angle β. d) Tip
speed ratio λ.
4
is around its nominal while the produced power not yet has reached the rated one (8 m/s <
ωs < 12 m/s).
Fig. 4 shows the partial derivative of Cp with respect to the pitch angle for the operating
condition presented in Fig. 3. Note that in the figure the partial derivative is only shown for
pitch angles greater than zero. The average value of the partial derivative of Cp with respect
−0.6
−0.7
∂Cp /∂β
−0.8
−0.9
−1
−1.1
−1.2
−1.3
−1.4
5
10
15
20
25
ws [m/s]
Fig. 4: Partial derivative of Cp with respect to the pitch angle as a function of the wind speed. The
operating condition is given in Fig. 3.
to the pitch angle in Fig. 4 is approximately kCp = ∂Cp /∂β = −0.96. This value will be
used later on when designing control laws for the pitch angle.
2.3 Drive Train Representation
In this work, the drive train has been modeled by a stiff shaft. Accordingly the drive-train
dynamics are given by
(JT + JG )
dΩT
= Te − TT .
dt
(13)
In order to study the impact of a soft shaft, the following modifications can be made:
dΩT
= ξ(ΘG − ΘT ) + ζ(ΩG − ΩT ) − TT
dt
dΩG
JG
= Te − ξ(ΘG − ΘT ) − ζ(ΩG − ΩT )
dt
JT
(14)
(15)
If we define ∆Θ = ΘG − ΘT , the above drive train equations can be rewritten as
dΩT
= ξ∆Θ + ζ(ΩG − ΩT ) − TT
dt
dΩG
JG
= Te − ξ∆Θ − ζ(ΩG − ΩT )
dt
∆Θ
= ΩG − ΩT .
dt
JT
5
(16)
(17)
(18)
Writing the above equations in state-space form yields

 

 

−1/JT
0
ΩT
−ζ/JT ζ/JT
ξ/JT
ΩT
d 
TT







0
1/JG
ΩG +
ΩG = ζ/JG −ζ/JG −ξ/JG
.
Te
dt
0
0
∆Θ
−1
1
0
∆Θ
The characteristic polynomial of the system becomes
(JT + JG )
(JT + JG )
2
p p +
ζp +
ξ .
JT JG
JT JG
If the system is undamped, i.e., ζ = 0, the oscillation frequency can be found as
s
(JT + JG )
ξ.
ω=
JT JG
(19)
(20)
(21)
2.4 Speed and Torque Observer
In order to track the turbine torque we will use a slightly modified observer of the one proposed in [10], i.e.,
dT̂T
= γ1 (ΩG − Ω̂G )
dt
Te − T̂T
dΩ̂G
=
+ γ2 (ΩG − Ω̂G ).
dt
JG + JT
(22)
(23)
The system described by (22) and (23) have an equilibrium point at (T̂T , Ω̂G ) = (Te , ΩG )
and the following characteristic polynomial
γ1
p 2 + γ2 p +
.
(24)
JG + JT
If the parameters are set as γ1 = (JG + JT )ρ2 and γ2 = 2ρ, then ρ can be adjusted to the
desired bandwidth of the observer.
2.5 Finding Optimal Tip-Speed Ratio
The “optimal” WT torque can with relatively good agreement be approximated as
TT = Kopt Ω2T
(25)
where the constant Kopt is determined from steady state values in Fig. 3. This means that
in order to operate the WT at optimal tip-speed ratio, the rotor speed reference should be set
according to
s
T̂T
Ωref
(26)
T =
Kopt
where T̂T is the turbine torque estimation obtained as described in Section 2.4. In Fig. 5 a
comparison of the rotor speed reference as a function of the torque for known wind speeds
(optimal tip-speed ratio) and using the approximation in (26). It is seen in the figure that for
the turbine investigated here the approximation works well.
6
1.1
Rotor speed [p.u.]
1
0.9
0.8
0.7
0.6
0.5
0.4
0
20
40
60
80
100
Turbine torque [%]
Fig. 5: Rotor speed speed reference as a function of the torque. Black line is with known wind speed
and red line is with the approximation in (26).
2.6 Wind Speed Estimation for Pitch Controller
The steady-state operation as given in Fig. 3 is used to estimate the wind speed. When
the generator is operated in speed-control mode, the wind speed will be estimated from the
power in Fig. 3a) with a second-order polynomial as
ŵs,low = a1 + a2 P + a3 P 2
(27)
where P is the wind turbine power. When the pitch angle is used to control the speed of
the turbine, the pitch angle is used for the estimation. In this case a first-order polynomial is
used
ŵs,high = b1 + b2 β.
(28)
Then, the estimated wind speed is low-pass filtered as
dŵs
= αws ((1 − ktr )ŵs,low + ktr ŵs,high − ŵs )
dt
= αws (1 − ktr ) a1 + a2 P + a3 P 2 + ktr (b1 + b2 β) − ŵs
(29)
where αws is the bandwidth of the estimator and ktr is a transition parameter between speed
control of the generator and pitch angle control. Fig. 6 shows how well the approximation
in (27) and (28) holds. The accuracy of the steady-state performance of the wind speed
estimator as a function of the real wind speed can be seen in Fig. 7. From Fig. 6 and Fig. 7 it
can be seen that the estimation works relatively well even though there are some deviations
especially around 12 m/s. However, as already mentioned the estimated wind speed is only
used for “gain scheduling” of the pitch controller, while the rotor speed reference for low
wind speeds is found from (26). The speed of the estimator is set by the bandwidth of the
low-pass filter αws .
7
b) 25
Wind speed [m/s]
Wind speed [m/s]
a) 12
10
8
6
4
20
15
10
0
50
100
0
10
20
30
Pitch angle [deg]
Power [%]
Fig. 6: Wind speed approximation. Black line is the “real” wind speed and red line is from the
approximation. a) Estimation at low wind speeds. b) Estimation at high wind speeds.
Estimated wind speed [m/s]
25
20
15
10
5
0
0
5
10
15
20
25
Wind speed [m/s]
Fig. 7: Estimated wind speed as a function of the “real” wind speed.
2.7 Wind Generation using Spatial and Rotational Sampling Filters
In [15] it is shown how a wind field arriving at the rotor disc can be modelled. However, in
a time-critical application it may be unfeasible to determine the driving shaft torque, as described above, for each time step. An alternative approach applicable to fixed-speed systems,
which is less time-consuming and still provides similar results, has been suggested in [12]
and makes use of the aerodynamic filter approach. This approach has been used in this work.
Wind at one point only (at the hub level) is generated and two filters are used to represent the
impact of the rotating blades. The first filter is the Spatial Filter (SF),
√
2 + bs
HSF (s) = √
√
( 2 + bs a)(1 +
8
√b s)
a
(30)
where:
a = 0.55
b = γ ∗ R/U
R
the turbine radius
U
the average wind speed at the hub height
γ
the decay factor over the disc (γ = 1.3)
The spatial filter damps higher frequency components that are present in the wind. In this
way the filtering property of the rotor blades is represented. The transfer function in (30) can
be simplified to a first order transfer function with a negligible effect on its characteristic,
HSF (s) =
1
=
sb + 1
1
s
2πfcut
+1
(31)
where fcut is the cut-off frequency of the filter.
The second filter represents the rotational sampling of the wind by the turbine rotor and
is therefore called rotational sampling filter (RSF), in this paper slightly modified to
HRSF (s) =
1
(s
ω2
+ ω)2
1
[s + (d + jω)][s + (d − jω)]
(ω 2 +d2 )
(32)
where
n
ω = 2πN 60
N
the number of blades
n
the rotor speed in [rpm]
d
damping factor
This filter amplifies the variations at a frequency region around the blade passing frequency.
In other regions this filter has a gain of nearly one. The expression in (32) can only be used
for fixed-speed wind turbines since ω has to be constant. However, if (32) is transformed to
the time domain it can be used for variable-speed wind turbines, i.e.,
0
1
0
ẋ(t) =
w
(t)
(33)
x(t) +
2
2
1 s,SF
−d − [ω(t)] −2d
and
2 2
d2
d2
d (d + [ω(t)]2 )
ws,RSF (t) = −
2 −1 −
+
+ ω(t) x(t)
[ω(t)]2
[ω(t)]2 ω(t)
d2
+ 1+
ws,SF (t)
[ω(t)]2
9
(34)
3 Field Measurements
The measurement data used in this report comes from two different wind turbines. The
first turbine is located in Jung and the second in Bast. Both locations are in the county of
Västergötland.
When presenting statistical data the standard deviation will be used. The standard deviation, σ(x), of a measured signal x is calculated with
n
1 X
(xi − x)2
σ(x) =
n − 1 i=1
(35)
where n is the number of samples in x and x is the average value of x, i.e.,
n
1X
x=
xi .
n i=1
(36)
3.1 Jung Wind Turbine
The Jung turbine is a VESTAS V-52 850 kW WT, located (≈ 100 km from the west coast).
The wind turbine is located in a flat surrounding and is connected to the 10-kV distribution
grid via a transformer, which transforms the voltage to the wind-turbine voltage of 690 V.
See Fig. 8 for a picture of the turbine and the data acquisition computer. The VESTAS V-52
Fig. 8: Jung wind turbine and the data acquisition computer.
is a variable-speed wind turbine with pitch control and with a doubly-fed induction generator. In Table 1 some data of VESTAS V-52 850 kW WT is given. The stator currents and
stator voltages are measured using transformers, which transform the current to 5 A and the
voltage to 110 V. In addition, both stator and rotor currents are also measured directly using
LEM modules. The rotor speed is determined from the rotor currents while the pitch angle
is measured directly. The bandwidth of the measured signals is in the kHz range. In addition
10
Table 1: Data of VESTAS V-52 850 kW WT [14].
Rated voltage (Y)
Rated power
Rotor diameter
Rotor speed
Cut-in wind speed
Nominal wind speed
Maximum wind speed
690 V
850 kW
52 m
14.0–31.0 rpm (26 rpm)
4 m/s
16 m/s
25 m/s
to these quantities, the wind speed signal is also available from the main wind turbine controller, but with a lower bandwidth.
3.1.1
Time Series
In Fig. 9 the grid power, wind speed, rotor speed and pitch angle are presented for the case
with low wind speed. The average power production in the measurement in Fig. 9 is 98 kW
and the average wind speed is 5.7 m/s.
Pg [%]
30
20
10
0
ws [m/s]
7
0
30
60
90
120
0
30
60
90
120
0
30
60
90
120
0
30
60
90
120
6
ωr [p.u.]
5
1
0.9
0.8
0.7
0.6
β [deg]
0
−1
−2
t [s]
Fig. 9: A measured two minute time serie of the Jung wind turbine at low wind speeds.
The wind speed signal is acquired from the main controller and it can be observed that
11
the updating interval is 10 s. Moreover it can be observed that the pitch angle slightly varies
also at low wind speed, which has not been accounted for in the simulations presented in this
report.
In Fig. 10 the corresponding time series at medium wind speed can be seen. The average
power production in this case is 485 kW and the average wind speed is 10 m/s.
Pg [%]
100
50
0
ws [m/s]
15
0
30
60
90
120
0
30
60
90
120
0
30
60
90
120
0
30
60
90
120
10
5
ωr [p.u.]
1.1
1
0.9
0.8
β [deg]
5
0
−5
t [s]
Fig. 10: A measured two minute time serie of the Jung wind turbine at medium wind speeds.
Finally in Fig. 11 the time series at high wind speeds are presented. Now the average
power production is 840 kW and the average wind speed is 20 m/s.
At high wind speed it can be noted that the pitch angle is constantly changing, while the
output power of the turbine is constant. The rotor speed varies with about 10% around the
nominal value.
12
Pg [%]
100
50
0
ws [m/s]
30
0
30
60
90
120
0
30
60
90
120
0
30
60
90
120
0
30
60
90
120
20
10
ωr [p.u.]
1.1
1
0.9
0.8
β [deg]
40
20
0
t [s]
Fig. 11: A measured two minute time serie of the Jung wind turbine at high wind speeds.
13
3.1.2
Statistical Data
Fig. 12 shows the measured steady-state characteristics of the Jung wind turbine.
100
b) 1.1
80
1
ΩT [p.u.]
PT [%]
a)
60
40
0.9
0.8
0.7
20
0
0.6
5
10
15
20
25
5
10
ws [m/s]
15
20
25
20
25
ws [m/s]
d) 20
c) 50
40
15
20
λ
β [deg]
30
10
10
5
0
−10
0
5
10
15
20
25
ws [m/s]
5
10
15
ws [m/s]
Fig. 12: Steady-state operation of the WT. a) Power PT . b) Rotor speed ΩT . c) Pitch angle β. d) Tip
speed ratio λ.
Fig. 13 shows the average value and standard deviation of the pitch angle as function of
the average grid power.
Fig. 14 shows the average value and standard deviation of the rotor speed as function of
the average grid power.
Fig. 15 shows the standard deviation of the grid power as function of the average grid
power.
The standard deviation of the estimated electromechanical torque (Pg /ΩG ) can be seen
in Fig. 16. Before calculating the standard deviation the electromechanical torque has been
high-pass filtered with a cut-off frequency of 0.1 Hz and low-pass filtered with a cut-off
frequency of 20 Hz.
It can be noted that the measured steady-state characteristics for the Jung turbine is very
similar to the previously presented theoretical one.
14
b) 40
a) 50
40
σ(β) [deg]
β [deg]
30
30
20
10
20
10
0
−10
0
0
25
50
75
100
0
25
Pg [%]
50
75
100
Pg [%]
Fig. 13: Statistical data of the pitch angle as a function of the average grid power. a) Average value
of the pitch angle. b) Standard deviation of the pitch angle.
ΩG and σ(ΩG ) [p.u.]
120
100
80
ΩG
60
σ(ΩG )
40
20
0
0
20
40
60
80
100
Pg [%]
Fig. 14: Average value and standard deviation of the rotor speed as a function of the average grid
power.
3.2 Bast Wind Turbine
The description of the measurement set-up at Bast can be found in [13]. The Bast turbine is
a 600 kW Enercon turbine, which is equipped with a full-power converter system. The grid
voltages and grid currents are measured using high-bandwidth equipment while mean values
of the rotor speed and wind speed is obtained from the main wind turbine controller.
The Bast turbine is located about 130 km from the west coast, also in a flat landscape.
3.2.1
Time Series
Fig. 17 shows a time series of power delivered to the grid from the Bast wind turbine. The
average wind speed is approximately 16 m/s and the rotor speed is about 33 rpm. In Fig. 18
the frequency spectra of the grid power can be seen.
15
200
σ(Pg ) [%]
150
100
50
0
0
20
40
60
80
100
Pg [%]
Fig. 15: Standard deviation of the grid power as a function of the average grid power.
35
30
σ(T̂e ) [%]
25
20
15
10
5
0
0
20
40
60
80
100
Pg [%]
Pg [%]
Fig. 16: Standard deviation of the estimated electromechanical torque as a function of the average
grid power.
100
50
0
30
60
90
t [s]
Fig. 17: Output power from the Bast WT at 16 m/s.
16
120
0
10
−2
∆Pg [%]
10
−4
10
−6
10
−8
10
−2
10
−1
0
10
1
10
2
10
3
10
10
Frequency [Hz]
Fig. 18: Frequency spectra of the grid power of the Bast wind turbine.
3.2.2
Statistical Data
In Fig. 19 the steady-state behavior of the Bast wind turbine is shown. The data used in Fig.
b) 1
ΩT [p.u.]
PT [%]
100
50
0
c) 15
0.8
10
0.6
λ
a)
5
0.4
0.2
5
10
15
ws [m/s]
20
0
5
10
15
ws [m/s]
20
5
10
15
20
ws [m/s]
Fig. 19: Steady-state operation of the Bast WT. a) Power PT . b) Rotor speed ΩT . c) Tip speed ratio
λ.
19 comes from the main wind turbine controller of the wind turbine.
An observation that can be done is that the Bast turbine reaches the maximum speed at
a higher wind speed. This leads to that the turbine will be more silent in the middle wind
speed region. This is a setting that also could have been done on the Jung turbine.
3.3 Comparison between the Bast and Jung wind turbine
Fig. 20 shows the grid power spectra for the two turbines at low wind speed operation.
Although it is obvious that the two turbines have different magnitudes in different frequency regions, it can be stated that the performance is fairly similar.
Fig. 21 shows the grid power spectrum at about half rated operation and in Fig. 22 the
grid power spectrum at close to rated operation is presented.
17
2
10
0
∆Pg [%]
10
−2
10
−4
10
−6
10
−2
10
−1
0
10
10
1
10
2
10
Frequency [Hz]
Fig. 20: Grid power spectrum of the Bast (black) and Jung (red) wind turbine. The turbines are
operated at about 10% of rated operation
2
10
0
∆Pg [%]
10
−2
10
−4
10
−6
10
−2
10
−1
0
10
10
1
10
2
10
Frequency [Hz]
Fig. 21: Grid power spectrum of the Bast (black) and Jung (red) wind turbine. The turbines are
operated close to half of rated operation
For the two latter cases it can be concluded that the output power spectra are almost identical and accordingly the Jung turbine measurements that are more detailed (and in particular
is equipped with a high-performing pitch angle and rotor speed measurement system) can be
used for the further evaluation.
18
2
10
0
∆Pg [%]
10
−2
10
−4
10
−6
10
−2
10
−1
0
10
10
1
10
2
10
Frequency [Hz]
Fig. 22: Grid power spectrum of the Bast (black) and Jung (red) wind turbine. The turbines are
operated close to rated operation
19
4 Control
4.1 Internal Model Control (IMC)
Due to the simplicity of IMC for designing controllers, this method is used throughout this
report. The idea behind IMC is to augment the error between the process, G(p), where
p = d/dt, and a process model, Ĝ(p), by a transfer function C(p), see Fig. 23. Controller
iref
+ P
v
C(p)
i
G(p)
−
Ĝ(p)
+
− P
F (p)
Fig. 23: Principle of IMC.
design is then just a matter of choosing the “right” transfer function C(p). One common way
is [3]
α n
C(p) =
Ĝ−1 (p).
(37)
p+α
where n is chosen so that C(p) become implementable, i.e., the order of the denominator is
equal to or greater than that of the numerator. The closed-loop system will be
−1
C(p)
(38)
Gcl (p) = G(p) 1 + C(p)[G(p) − Ĝ(p)]
which can be simplified to
α n
Gcl (p) = G(p)C(p) =
p+α
(39)
when G(p) = Ĝ(p). The parameter α is a design parameter, which for n = 1, is set to the
desired bandwidth of the closed-loop system. The relationship between the bandwidth and
the rise time (10%–90%) is α = ln 9/trise . The controller, F (p), (inside the dashed area in
Fig. 23) becomes
−1
C(p).
(40)
F (p) = 1 − C(p)Ĝ(p)
For a first-order system, n = 1 is sufficient. The controller then typically becomes an ordinary PI controller:
F (p) =
ki
α −1
Ĝ (p) = kp + .
p
p
20
(41)
4.2 Rotor-Speed Control with Generator
Here a rotor-speed controller for the generator will be derived. Moreover, when designing
the speed controller it will be assumed that the generator itself is controlled very fast. This
means that we will assume that the electrodynamic torque equals its reference value. For a
stiff shaft the drive train dynamics are given by (13). Now, by choosing the electrodynamic
torque as
Te = Te′ − Ba ΩT
(42)
it is possible to rewrite the drive-train dynamics (13) in as
dΩT
= Te′ − Ba ΩT − TT .
(43)
dt
In the above expressions Ba is used for damping of disturbances. How to chose Ba will
be discussed later on in this section. The transfer function from Te′ (p) to ΩT can then be
expressed as
(JT + JG )
G(p) =
1
ΩT (p)
=
′
Te (p)
(JT + JG ) p + Ba
(44)
with IMC the following controller is obtained
F (p) =
αB
ki
αs −1
s a
Ĝ (p) = kp + = αs JˆT + JˆG +
.
p
p
p
(45)
Moreover, the transfer function from a “disturbance” TT to ΩT becomes with ideal parameters
G
p
p
GT Ω (p) =
(46)
=
=
2
1 + FG
(JT + JG ) p + (Ba + kp ) p + ki
(JT + JG ) (p + αs )2
where the latter equality holds if Ba = αs (JT + JG ). This means that the parameters of the
controller can be summarized as
(47)
ki = αs2 JˆT + JˆG
kp = Ba = αs JˆT + JˆG
4.3 Rotor-Speed Control with Pitch
In this section a rotor-speed controller acting on the pitch angle will be derived. This means
that the rotor speed is controlled by the turbine torque TT . The pitch controller will only
be in action during the higher wind speeds. The drive-train dynamics in (13) can now be
expressed as
dΩT
= Te − TT = Te − (TT 0 + ∆TT )
(48)
dt
where the turbine torque has been linearized, i.e. TT = TT 0 + ∆TT . Then by using (12) we
can further rewrite the drive-train dynamics as
∂TT ∂TT ∂TT dΩT
· ∆ws −
· ∆ΩT −
· ∆β.
= Te − TT 0 −
(JT + JG )
dt
∂ws x=x0
∂ΩT x=x0
∂β x=x0
(49)
(JT + JG )
21
TT
Ωref
G + P
ki
kp +
p
−
+P
−
−
+P
1
ΩG
JT p
Ra
Fig. 24: Illustration of the speed-control loop of the generator.
If we let
∂T
∂T
T
T
Ted = Te − TT 0 −
· ∆ws −
· ∆ΩT
∂ws x=x0
∂ΩT x=x0
(50)
then (49) becomes
3
∂C
(λ
,
β)
∂T
w
dΩT
p
0
T
s0
= Ted −
· ∆β = Ted + k
∆β.
(JT + JG )
dt
∂β x=x0
ΩT 0
∂β
x=x0
(51)
Under the assumption that kCp = ∂Cp /∂β is constant (or at least close to constant) in the
operating point the drive-train dynamics can finally be approximated as
(JT + JG )
By choosing ∆β as
dΩT
w3
≈ Ted + k s0 kCp ∆β.
dt
ΩT 0
∆β = (∆β ′ − Baβ ΩT )
Ωref
T
3
kws0
kCp
(52)
(53)
then if ΩT 0 ≈ Ωref
T it is possible to approximate
(JT + JG )
dΩT
≈ Td + ∆β ′ − Baβ ΩT .
dt
If Td is treated as a disturbance the controller gains can, as previous, be found as
kiβ = αβ2 JˆT + JˆG
kpβ = Baβ = αβ JˆT + JˆG
(54)
(55)
where αβ is the desired bandwidth. This means that the controller becomes
∆β =
kpβ e + kiβ
Z
e dt − Baβ ΩT
where e is the control error.
22
Ωref
T
3
kws0
kCp
(56)
wind turbine
Ωref
G
Ωref
G + P
kiβ + P
kpβ +
p
−
−
ŵs
Ωref
G
k ŵs3 kCp
Tenom
Aero.
+
−P
1
ΩG
JT p
Baβ
Fig. 25: Illustration of the speed-control loop with the pitch angle.
4.4 Overall Control
The overall control law can be summarized as
Z
ref
Te = Tnom ktr + kp e + ki e dt − Ba ΩT (1 − ktr )
Z
Ωref
T
ref
kpβ e + kiβ e dt − Baβ ΩT ktr
β = β0 +
3
kws0
kCp
(57)
(58)
where ktr is a transition parameter. This means that when ktr = 0 the generator is operated
with speed control and the pitch angle is set to β0 , while for ktr = 1 the generator torque is
set to its rated and the speed is controlled by the pitch angle. The angle β0 can be used to
pre-pitch the blades in order to simplify the control in the upper medium wind speed region.
The effect of this pre-pitching will we demonstrated in the Analysis chapter.
In Fig. 4.4 shows an illustration of the overall control. Even though the generator is
Pitch contr.
Ωref
G + P
−
β ref
Aerodyn.
TT
−
P
ktr
Gen. contr.
Teref
+
Gen.
1
ΩG
JT p
Te
Fig. 26: Illustration of the overall control algorithm.
included in the figure it is modeled as ideal, meaning that Te = Teref .
4.4.1
Implementation Example
Below an example of how the control algorithm can be implemented, is presented. The
implementation presented here is given in the C language.
23
WGref = sqrt(abs(TTh)/Kopt); if (WGref > WGmax)
WGref = WGmax;
e = WGref - WG;
// Control error
// Generator control
Te_ref2 = Tnom*ktr+(kp*e+ki*I-Ba*WG)*(1-ktr);
fun_limit(Te_ref2,Tnom,-Tnom,&Te_ref);
I += Ts*(e + (Te_ref-Te_ref2)/kps);
if (ktr >= 1)
I = (Ba*WG+Te_ref)/ki;
// Pitch control
beta0_2 += Ts*wc_ktr*((wsh-ws_p0)*beta0_max/(ws_p1-ws_p0)-beta0);
fun_limit(beta0_2,beta0_max,beta0_min,&beta0);
beta_ref2 = beta0+WGref*(kpb*e+kib*Ib-Bab*WG)/(k*wshˆ3*kCp)*ktr;
fun_limit(beta_ref2,beta_max,beta_min,&beta_ref);
Ib += Ts*(e+(beta_ref-beta_ref2)/kpb);
if (ktr <= 0)
Ib = (Bap*WG)/kib;
// Transition between generator and pitch control
if ((fabs(TeLP) > Tnom) && (e < 0))
ktr_direction = 1;
if ((pitchLP < beta_min) && (e > 0))
ktr_direction = -1;
ktr = ktr + ktr_direction*Ts/transition_length; if (ktr > 1) {
ktr = 1;
ktr_direction = 0;
} if (ktr < 0) {
ktr = 0;
ktr_direction = 0;
}
TeLP
+= Ts*wc_ktr*(Te_ref2-TeLP);
pitchLP += Ts*wc_ktr*(beta_ref2-pitchLP);
// Wind speed estimation
wsh += Ts*aws*((a1+a2*PG+a3*PGˆ2)*(1-ktr) + (b1+b2*beta_ref)*ktr);
// Update states of observers
WGh += Ts*((Te_ref-TLh)/(JGh+JTh)+2*rho_obs*(WG-WGh));
TTh += Ts*(JGh+JTh)*(-1)*rho_obsˆ2*(WG-WGh);
24
4.5 Example of Low-Wind Speed operation
In Fig. 27 an example of the operation at low wind speed can be seen. The average wind
speed in the simulation is 6 m/s. The parameters are set as follows:
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition length
a)
value
1 rad/s
0.1 rad/s
0.1 rad/s
0.5 s
ws [m/s]
8
6
4
b)
PT [%]
30
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
20
10
0
c)
ωr [p.u.]
1
0.5
d)
PG [%]
30
20
10
0
t [s]
Fig. 27: Low-wind speed operation. a) Wind speed, ws . b) Turbine power, PT . c) Rotor speed, ΩT
(black actual, red optimal). d) Generator power, PG .
4.5.1
Decreased and increased speed control bandwidth
In order to demonstrate the impact of the bandwidth of the speed controller, this is reduced
by a factor of 6. The result can be studied in Fig. 28. The parameters are set as follows:
25
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition length
a)
value
2π · 0.02 = 0.13 rad/s
0.1 rad/s
0.1 rad/s
0.5 s
ws [m/s]
8
6
4
PT [%]
b)
30
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
20
10
0
1
ωr [p.u.]
c)
0.5
PG [%]
d)
30
20
10
0
t [s]
Fig. 28: Low-wind speed operation, low rotor speed controller bandwidth. a) Wind speed, ws . b)
Turbine power, PT . c) Rotor speed, ΩT (black actual, red optimal). d) Generator power, PG .
Now, the bandwidth of the speed control loop is set to twice the original value and the
result can be found in Fig. 29. The parameters are set as follows:
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition length
value
2π · 0.3 = 1.9 rad/s
0.1 rad/s
0.1 rad/s
0.5 s
From the curves presented in this section it is quite clear how the speed controller bandwidth affects the operation. A lower speed controller bandwidth gives a smoother operation
while a higher bandwidth gives rise to more output power pulsations. It is not possible to
reduce the bandwidth of the speed controller too much, because this will lead to that the
26
a)
ws [m/s]
8
6
4
PT [%]
b)
30
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
20
10
0
1
ωr [p.u.]
c)
0.5
PG [%]
d)
30
20
10
0
t [s]
Fig. 29: Low-wind speed operation, high rotor speed controller bandwidth. a) Wind speed, ws . b)
Turbine power, PT . c) Rotor speed, ΩT (black actual, red optimal). d) Generator power, PG .
turbine in this case is operating at rotor speeds that deviate from the ideal tip-speed ratio too
much, with a resulting energy loss.
4.6 Example of Medium- and High-Wind Speed operation
In Fig. 30 an example of the operation at low wind speed can be seen. The average wind
speed in the simulation is 11 m/s. The parameters are set as follows:
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition length
value
1 rad/s
0.1 rad/s
0.1 rad/s
0.5 s
The trace of the pitch angle and the transition constant have been added to the figure,
since the turbine now from time to time hit the upper power limit. As can be observed from
the figure, the control algorithm manages well to keep the output generator power not to
exceed the rated one. It can be noted that the pitch angle is already changed before the wind
speed exceeds the rated one. This is due to the pre-pitching strategy used in this work, in
which the pitch angle already is increased at 10-12 m/s before it is really necessary. The
27
a)
ws [m/s]
14
12
10
8
PT [%]
b) 150 0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
100
50
ωr [p.u.]
c)
1
PG [%]
d)
100
50
0
β [deg]
e)
40
20
0
ktr [-]
f)
1
0.5
0
t [s]
Fig. 30: Medium-wind speed operation. a) Wind speed, ws . b) Turbine power, PT . c) Rotor speed,
ΩT . d) Generator power, PG . e) Pitch angle, β. f) Transition parameter, ktr .
purpose is to avoid too large pitch actions in case of a sudden wind guest. For the used blade
profile, the problem was that in order to have a substantial effect of the pitch angle at the
rated wind speed, the pitch angle first had to change from 0 to 6 degrees before the power
significantly dropped.
28
In the lowest time trace, the transition constant (between generator torque control and
pitch angle control) can be observed, which nicely changes between 0 and 1 depending on
the wind speed condition. The parameters are set as follows:
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition length
value
1 rad/s
0.1 rad/s
0.1 rad/s
0.5 s
In Fig. 31 an example of the operation at high wind speed can be seen. The average wind
speed in the simulation is 15 m/s.
Here it can be noted that the incoming power that has been converted from the wind is
varying strongly. If this had been a fixed-speed turbine, these variations would have been
injected into the grid. However, due to the possibility of changing the rotor speed, the output
power from the wind turbine can be kept fairly constant.
29
a)
ws [m/s]
20
15
10
PT [%]
b) 150 0
40
80
120
160
200
240
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
100
50
ωr [p.u.]
c) 1.05 0
1
0.95
PG [%]
d)
100
50
0
β [deg]
e)
40
20
0
ktr [-]
f)
1
0.5
0
t [s]
Fig. 31: High-wind speed operation. a) Wind speed, ws . b) Turbine power, PT . c) Rotor speed, ΩT .
d) Generator power, PG . e) Pitch angle, β. f) Transition parameter, ktr .
30
4.7 Comparison Between Measurements and Simulations
In order to validate the control law with respect to the measured data from the Jung wind
turbine the frequency spectra will be studied, i.e., the simulated frequency spectra of ∆ΩG =
ΩG − ΩG and ∆β = β − β will be compared to measured one.
4.7.1
Low Wind Speed
In Figs. 32 and 33 the frequency spectra of the rotor speed and grid power at low wind speed
can be seen.
0
10
−2
∆ΩG [p.u.]
10
−4
10
−6
10
−8
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 32: Frequency spectra of rotor speed, low wind speed. Red, blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
It can be noted that the measured speed variations are slightly higher than the simulated
ones, the reason for this is unknown. The output power pulsations on the other hand agrees
well with the measurements.
As mentioned earlier, the pitch angle has been considered to be constant in the simulations at low wind speeds. Measurements on the other hand, see Fig. 34 show that the pitch
angle also varies during low wind speed operation.
4.7.2
Medium Wind Speed
In Figs. 35 and 36 the frequency spectra for the rotor speed and grid power at medium-wind
speed can be seen.
It can be observed that the simulated result agree well with the measured ones.
In Fig. 37, the simulated spectrum of the pitch angle is compared with some measured
ones. Again the agreement is good.
31
2
10
0
∆Pgrid [%]
10
−2
10
−4
10
−6
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 33: Frequency spectra of grid power, low wind speed. Red, blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
0
10
−2
∆β [deg]
10
−4
10
−6
10
−8
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 34: Frequency spectra of pitch angle, low wind speed. Red, blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
4.7.3
High-Wind Speed
In Figs. 38 and 39 the frequency spectra of the rotor speed and pitch angle at high-wind
speed can be seen.
Here it can be noted that in the simulations the rotor speed variations for frequencies
above 1.5 Hz are more damped than in the measured cases. A possible reason is that the
implemented aerodynamic model is too crude. In the ideal world of simulations, moreover,
the simulations predicts a constant output power which, of course, does not exist in reality.
32
0
10
−2
∆ΩG [p.u.]
10
−4
10
−6
10
−8
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 35: Frequency spectra of rotor speed, medium wind speed. Blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
2
10
∆Pgrid [%]
0
10
−2
10
−4
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 36: Frequency spectra of grid power, medium wind speed. Blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
Accordingly, the simulated grid power is not visible in the spectra of the output powers, se
Fig. 40.
33
2
10
0
∆β [deg]
10
−2
10
−4
10
−6
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 37: Frequency spectra of pitch angle, medium wind speed. Blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
0
10
−2
∆ΩG [p.u.]
10
−4
10
−6
10
−8
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 38: Frequency Spectra of rotor speed, high wind speed. Red, blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
34
2
10
0
∆β [deg]
10
−2
10
−4
10
−6
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 39: Frequency Spectra of pitch angle, high wind speed. Red, blue and black spektrum are from
measurements on the Jung wind turbine and green spektrum is from a simulation using the above
mentioned control law.
2
10
0
∆Pgrid [%]
10
−2
10
−4
10
−6
10
−2
10
−1
0
10
10
1
10
Frequency [Hz]
Fig. 40: Frequency Spectra Pgrid. Red, blue and black spektrum are from measurements on the Jung
wind turbine and green spektrum is from a simulation using the above mentioned control law.
35
5 Analysis
In this section, the significance of various aspects on the performance of the derived control
law will be analyzed.
5.1 Bandwidth of Speed Control Loops
5.1.1
Perfect Wind Speed and Shaft Torque Knowledge
The figures (Fig. 41 – Fig. 44) presented in this section is without rotational sampling filter,
torque and wind speed estimation. In the table below the parameters are given.
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition time
value
α rad/s
0.02 Hz
0.1×α rad/s
0.5 s
The turbine torque TT has been low-pass filtered with ρ = 2π · 0.02 rad/s before calculating the reference speed. See [5] for detailed definition of flicker coefficient. Shortly it can
be mentioned that the highest value in the wind speed range up to the rated one is the one
that determines the total flicker coefficient for the wind turbine. Here a purely resistive grid
has been used, so the flicker impact of the reactive power pulsations have been ignored. In
order to refer to measurements it can be stated that a flicker coefficient of 4 is very good for
a wind turbine while 8 is less good.
In Fig. 41 the flicker impact is presented, in Fig. 42 the standard deviation of the rotor
speed is presented, in Fig. 43 the energy capture is presented and finally in Fig. 44 the
standard deviation of the torque pulsations is presented. Fig. 44 shows the standard deviation
of the electromechanical torque. Before taking the standard deviation, the electromechanical
torque has been high-pass filtered with a cut-off frequency of 0.1 Hz in order to remove the
influence of the very low-frequency power fluctuations.
All curves are presented as a function of the bandwidth of the rotor speed controller.
The results show that the most sensitive point of operation is just before we reach the
rated power, where we have the highest flicker emission and highest torque stresses. Moreover the results confirm the suspected results that the torque stresses, flicker emission and
energy capture increases with a quicker rotor speed controller while the rotor speed variations
decrease with a quicker rotor speed controller.
36
3.5
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
Flicker coefficient [-]
3
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 41: Flicker coefficient as a function of the bandwidth of the speed control loop for different
average wind speeds.
12
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
σ(ΩG ) [%]
10
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 42: Standard deviation of the rotor speed in percentage of the average speed as a function of the
bandwidth of the speed control loop.
37
100
99.5
Energy [%]
99
98.5
98
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
97.5
97
96.5
96
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 43: Energy as a function of the bandwidth of the speed control loop.
6
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
σ(Te ) [%]
5
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 44: Standard deviation of the electromechanical torque in percentage of the average torque as a
function of the bandwidth of the speed control loop.
38
5.1.2
Perfect Wind Speed and Shaft Torque Knowledge with Rotational Sampling Filter
The figures (Fig. 45 – Fig. 48) presented in this section is without torque and wind speed
estimation. The parameters are set as follows:
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition time
value
α rad/s
0.02 Hz
0.1×α rad/s
0.5 s
This means that only the rotational sampling filter has been added in comparison to the
previous section. The turbine torque TT has been low-pass filtered with ρ = 2π · 0.02 rad/s
before calculating the reference speed.
Flicker coefficient [-]
5
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 45: Flicker coefficient as a function of the bandwidth of the speed control loop for different
average wind speeds.
By comparing the curves with the ones presented in the previous section, it can be noted
that the addition of the rotational sampling filter only slightly adjusts the results. In particular
the flicker emission increases. The speed deviations are hardly affected due to the fact that
the RSF filter adds ( from a rotor point of view) very quick perturbations which are well
damped by the rotor inertia.
39
12
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
σ(ΩG ) [%]
10
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 46: Standard deviation of the rotor speed in percentage of the average speed as a function of the
bandwidth of the speed control loop.
100
99.5
Energy [%]
99
98.5
98
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
97.5
97
96.5
96
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 47: Energy as a function of the bandwidth of the speed control loop.
40
7
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
6
σ(Te ) [%]
5
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 48: Standard deviation of the electromechanical torque in percentage of the average torque as a
function of the bandwidth of the speed control loop.
41
5.1.3
Operation based on Estimated Wind Speed and Shaft Torque
The figures (Fig. 49 – Fig. 52) presented in this section is with rotational sampling filter,
torque and wind speed estimation. The parameters are set as follows:
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition time
value
α rad/s
0.02 Hz
0.1×α rad/s
0.5 s
Flicker coefficient [-]
5
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 49: Flicker coefficient as a function of the bandwidth of the speed control loop for different
average wind speeds.
12
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
σ(ΩG ) [%]
10
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 50: Standard deviation of the rotor speed in percentage of the average speed as a function of the
bandwidth of the speed control loop.
42
100
99.5
Energy [%]
99
98.5
98
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
97.5
97
96.5
96
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 51: Energy as a function of the bandwidth of the speed control loop.
7
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
6
σ(Te ) [%]
5
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Bandwidth [Hz]
Fig. 52: Standard deviation of the electromechanical torque in percentage of the average torque as a
function of the bandwidth of the speed control loop.
The results show that the estimation works very well and the curves are very similar to
the ones presented in the previous section.
43
5.2 Influence of “Pre-pitching”
Fig. 53 shows two simulations, one with (black) and one without pre-pitching (red). When
the system is pre-pitch the pitch angle is linearly pitch between 10–12 m/s. This means that
the pitch angle is 0 deg at 10 m/s and 4 deg at 12 m/s. The parameters are set as follows:
Parameter
Bandwidth speed control loop
Bandwidth torque observer
Bandwidth wind estimator
Transition time
value
1 rad/s
0.1 rad/s
0.1 rad/s
0.5 s
In the simulations the produced energy when using pre-pitching is 0.5% lower, i.e., by
using pre-pitching we have lost 0.5% of the energy (in the above simulation). On the other
hand the the standard deviation of the pitch angle has been lowered by 17% giving less
fatigue. If the total energy capture for a whole year is studied the energy loss due to the
pre-pitching is found to be about 0.1 %.
44
a)
ws [m/s]
14
12
10
8
PT [%]
b) 150 0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
0
40
80
120
160
200
240
100
50
ωr [p.u.]
c)
1
PG [%]
d)
100
50
0
β [deg]
e)
40
20
0
ktr [-]
f)
1
0.5
0
t [s]
Fig. 53: High-wind speed operation. a) Wind speed, ws . b) Turbine power, PT . c) Rotor speed, ΩT .
d) Generator power, PG . e) Pitch angle, β. f) Transition parameter, ktr .
45
5.3 Influence of Transition Length
The duration of the transition length between generator torque control and pitch control
influences the performance of the control. The figures (Fig. 54 – Fig. 57) presented in this
section is with rotational sampling filter, torque and wind speed estimation. Fig. 57 shows
4.5
Flicker coefficient [-]
4
3.5
3
2.5
9 m/s
10 m/s
11 m/s
12 m/s
15 m/s
2
1.5
1
0.5
0
2
4
6
8
10
Transition length [s]
Fig. 54: Flicker coefficient as a function of the bandwidth of the speed control loop for different
average wind speeds.
0.8
9 m/s
10 m/s
11 m/s
12 m/s
15 m/s
σ(ΩG ) [%]
0.7
0.6
0.5
0.4
0.3
0
2
4
6
8
10
Transition length [s]
Fig. 55: Standard deviation of the rotor speed in percentage of the average speed as a function of the
bandwidth of the speed control loop.
the standard deviation of the electromechanical torque. Before taking the standard deviation,
the electromechanical torque has been high-pass filtered with a cut-off frequency of 0.1 Hz,
as in the previous cases.
The impact of the transition length is as strongest for 11 m/s since it is for this wind speed
that the transitions take place most frequently.
46
100
99.5
Energy [%]
99
98.5
98
9 m/s
10 m/s
11 m/s
12 m/s
15 m/s
97.5
97
96.5
96
0
2
4
6
8
10
Transition length [s]
Fig. 56: Energy as a function of the bandwidth of the speed control loop.
7
6
σ(Te ) [%]
5
4
3
9 m/s
10 m/s
11 m/s
12 m/s
15 m/s
2
1
0
0
2
4
6
8
10
Transition length [s]
Fig. 57: Standard deviation of the electromechanical torque in percentage of the average torque as a
function of the bandwidth of the speed control loop.
For the quickest transition lengths, the flicker emission is increased. While the energy
capture increases as a function of transition length for 12 and 15 m/s it decreases for higher
transition lengths for 11 m/s.
Based on the results a suitable transition length is around 0.5–4 s.
47
5.4 Turbulence Intensity
The control was tested for turbulence intensities up to 20 %. The developed control structure
operated well also with this higher turbulence intensity that can occur in a rough landscape.
The results was that the flicker emission, rotor speed and torque variations increased.
The increase in flicker emission with a turbulence intensity of 20 % instead of 10 % was
about 50 % for the highest values at 12 m/s. The rotor speed and shaft torque variations
also increased by about 50 %. When the turbulence in the wind increases, there is also more
energy in the wind but it is also more difficult to extract the full potential. The available,
“convertible” energy in the wind in fact decreases at higher and medium wind speeds. Fig.
58 shows the relative energy production with a turbulence intensity of 20% compared to a
turbulence intensity of 10% as a function of the bandwidth of the torque observer
104
6 m/s
8 m/s
10 m/s
12 m/s
15 m/s
20 m/s
Energy [%]
102
100
98
96
94
0
0.01
0.02
0.03
0.04
0.05
Bandwidth [Hz]
Fig. 58: Energy production with a turbulence intensity of 20% relative 10% as a function of the
bandwidth of the torque observer.
From the figure the following observations can be made: For the highest wind speed, 20
m/s the turbulence level did not influence the energy capture so much. A higher turbulence
led to more occasions when the wind speed was below the rated one, this effect is however, much stronger for 12 m/s as can be seen in the figure, where the energy capture has
been substantially lowered. At very low wind speeds the fact that the power in the wind is
proportional to the cube of the wind speed is clearly obvious (se 6 m/s).
6 Conclusion
In this work, the speed control of a modern pitch-regulated wind turbine is investigated. A
controller structure is derived and the significance of various parameters in the controller
structure is investigated.
Various speed control bandwidths are investigated from 0.01 rad/s up to 0.3 rad/s. It
seems like the lower bandwidths are the ones to prefer. When comparing with measurements,
the lower bandwidths match the measured results at low wind speeds best. For higher wind
speeds the bandwidth of 0.1 rad/s has been used which match the measurements well.
48
Initially, the wind speed and shaft torque is assumed to be measured, but the main evaluation in the report is made using a wind speed and shaft torque observer.
At low wind speeds the rotor speed is controlled using the electrodynamic torque of the
generator and at high wind speeds the rotor speed is controlled using the pitch angle. A
transition constant is used and the speed of this is evaluated in order to assure a smooth
transition between these two modes. A transition constant of about 0.5–4 s is found to be
appropriate.
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50