Control of Variable Speed Wind Turbines
N. Caliao and A. Zahedi
Solar Energy Applications and Research Group
Department of Electrical and Computer Systems Engineering
Monash University
Clayton, Victoria 3168
Australia
E-mail: nolan.caliao@eng.monash.edu.au
ahmad.zahedi@eng.monash.edu.au
Abstract
Uncontrolled wind turbine configuration, such as stall-regulation captures, energy relative to the amount of
wind speed. This configuration requires constant turbine speed because the generator that is being directly
coupled is also connected to a fixed-frequency utility grid. In extremely strong wind conditions, only a fraction of
available energy is captured. Plants designed with such a configuration are economically unfeasible to run in
these circumstances. Thus, wind turbines operating at variable speed are better alternatives.
This paper focuses on a controller design methodology applied to a variable-speed, horizontal axis wind turbine.
A simple but rigid wind turbine model was used and linearised to some operating points to meet the desired
objectives. By using blade pitch control, the deviation of the actual rotor speed from a reference value is
minimised. The performances of PI and PID controllers were compared relative to a step wind disturbance.
Results show comparative responses between these two controllers. The paper also concludes that with the
present methodology, despite the erratic wind data, the wind turbine still manages to operate most of the time at
88% in the stable region.
1. INTRODUCTION
Recently, the operation of variable-speed wind turbines received better feedback over fixed speed operation.
Wilmshurst (1988) and Freris (1987) both claimed increases in energy capture from 3% to 20% by operating the
wind turbine at variable speed. The use of blade pitch allows a horizontal axis wind turbine (HAWT) to operate
over a wide range of wind speeds. This enables the HAWT to increase energy capture which a constant speed
operated wind turbine is unlikely to be capable of delivering. The operation of variable wind speed can be
divided into three different regions. The operation below the cut-in wind speed is called Region 1. In this region,
the wind turbine is operated from rest to the cut-in speed. Although most wind turbines are wind started, the
generator coupled to the wind turbine, in some cases, is used as a motor to drive the turbine until the rotor is
accelerated to the cut-in speed. At cut-in speed, the wind turbine reaches enough momentum to run
independently. Operating the wind turbine just above the cut-in wind speed and below a certain upper bound
(Region 2), the wind turbine starts to generate energy proportional to the cube of the wind speed experienced. In
Region 2, the power generated is practically controlled by the variation of pitch blade angle subject to the
instantaneous wind speed. In high or extremely high wind speed operation (Region 3), wind turbines experienced
severe fatigue which is detrimental to the blade's life. In this region, a better control system or strategy enables
the wind turbine to capture the energy that might have been lost had the wind turbine been operated at constant
speed. Furthermore, wind turbines are frequently subjected to sudden and huge step variations in wind speed,
causing large flapwise loads and bending moments at the roots of the blades, but these can be minimized by the
controller (Kendall et al, 1997).
This paper investigates the use of a simple, rigid and first order nonlinear model of wind turbines. This nonlinear
model was linearised relative to the PI and PID controls. Operating points were selected based on the
performance coefficient curves of the wind turbine under study. Responses of the controller were investigated
for the various regions mentioned.
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2. WIND TURBINE MODEL AND CONTROL METHODOLOGY
A simple, rigid, nonlinear turbine model for the purpose of controller design was used in this section. As
illustrated in Figure 1, the model that is based on the Sørensen et al. (1995) includes both the generator and
turbine inertias. The generator inertia can be neglected in controller design because of its relatively insignificant
magnitude as compared to the turbine inertia.
IT
ngear
K
T
T
Aero
dynamic
Low
speed
shaft
Gear
box
High
speed
shaft
Figure 1: Physical model of drive train, I = turbine and generator inertias, K = spring stiffness
If the drive train is neglected, the simple mathematical model of the first order is still a good representative of a
variable-speed wind turbine and can be stated as:
I T θ&T = T A − T E
(1)
where IT is the moment of inertia of turbine rotor; θT is the angular shaft speed; TE is the mechanical torque
necessary to turn the generator, which was neglected because of its negligible value. The aerodynamic torque, TA
is expressed in terms of the variable torque coefficient Cq(λ,β) which is dependent on the tip speed ratio (tsr or
λ). This term describes the ability of the wind turbine to convert kinetic energy of moving air to mechanical
torque (QA) (Novak, et al 1995). The torque coefficient is directly dependent on the performance coefficient
Cp(λ,β) or Cp(λ,β) = λ Cq(λ,β) (refer to Figure 3 for various Cq at various blade pitch angle). It is the maximum
value of Cq(λ,β) that a controller is trying to track during partial load operation. With the above assumptions,
Equation 1 becomes
I T θ&T = K1 ⋅ C q ⋅ (λ , β )(v w ) 2
(2)
where, K1 = 0.5 ρ ⋅ A ⋅ R , ρ is the air density, β is the blade pitch angle, A is the rotor swept area, R is the rotor
radius and vw is the wind speed.
2.1.
Controller Design Procedure
The most popular controller used among control engineers is probably one or a combination of, proportionalintegral-derivatives (PI, PD or PID) because of the easy implementation. However, the performances of these
controllers are sometimes dependent on the ability or experience of the control engineer in tuning their gains.
Hand & Balas (2000) have developed a systematic controller design methodology to fine tune PID gains by
generating contour plots to determine the optimum combination of the gains to achieve the appropriate damping
of the wind turbine. Finding the exact gain values is still based on good intuition. A simple yet effective method
used by Kendall et al. (1997) was to inject a step function into a system. This approach was implemented in this
study using Hand & Balas results as a reference.
In system control design, linearising a model is one of the powerful options available because it offers flexibility
in the analysis of the system. To linearise a model, the Taylor series expansion of the mathematical model was
obtained. The degree of accuracy of the model is relatively dependent on the order of the series. However, in
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practical cases higher-order terms are neglected, especially when these terms have negligible deviations from the
operating condition. A linear system model of the wind turbine is simply
IT θ&T = α1 ⋅ ∆θT + α 2 ⋅ ∆vw + α 3 ⋅ ∆β .
(3)
where the linearisation coefficients are given by
α1 = I T ⋅
∂C q
∂θ&T
= K1 ⋅ R v w op
∂θ T
∂λ
(4)
op

∂C q 

= K1 ⋅ v w op 2 C q
− λ op
op


∂λ
op
op 

∂θ&
2 ∂C q
δ 2 = IT ⋅ T
= K1 ⋅ v w op
∂β
∂β
α 2 = IT ⋅
∂θ&T
∂θ T
op
(
)
(5)
(6)
op
The symbol ∆ denotes deviations from the operating point (op) and the partial derivatives of the Cq represents
slopes of the Cq - λ or Cq - β curves. Since the torque coefficient is a function of both tsr and the blade pitch
angle, it is partially differentiated with respect to one variable while taking the other variable as constant (Leigh,
1992).
2.2.
Operating Point
Choosing the operating point of the linearised wind turbine dynamics requires critical judgment to preserve the
aerodynamic stability of the system (Kendall et al., 1997). Theoretically, there are two regions where a wind
turbine operates. The two regions are separated at the critical tip speed ratio (λcrit) which corresponds to the
maximum Cq. As illustrated in Figure 2, values to the right of λcrit with negative slopes correspond to the
unstalled (stable) region while values to the left side of λcrit denote the stalled region (arrows pointing to the right
and left respectively). The tip speed ratio corresponding to the maximum Cp is called the operating tip speed
ratio (λop). This point is obviously the objective of the controller, that is to set the blade pitch (βop) at a certain
operating value so as to attain and maintain maximum Cp as much as possible during operation.
Figure 2: Cq under the normal and stall regions. Regions belonging to below λcrit and above λcrit
represent the normal and the stall regions respectively.
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The torque coefficient curve (Figure 2) was used for choosing the rotational speed operating point. It is desirable
to have a constant rotational speed of 6.5 rad/sec (62 rpm - rated). In Figure 3, at 13 m/s rated wind speed, the
3°-pitch angle would provide the maximum Cq or maximum power which has an equivalent tsr of 7.5. However,
slight variation from this point towards the left side of the maximum point means that the wind turbine is
operating in a stalled condition, which would decrease the power capture. The wind turbine that was used in this
study allows a maximum of 8 rad/sec (74 rpm) rotational speed. If the wind turbine experienced 25 m/sec wind
speed (cut-out wind speed of the turbine) at 8 rad/sec rotor speed, the equivalent tsr is 5. This corresponds to a
deviation of 2.5 tsr from the 3°-pitch of 7.5. It should be noted that somewhere within the 2.5-deviation, the wind
turbine might operate in a stalled condition. Choosing an 8-degree with 7.5 tip speed ratio operating point will
somehow decrease the torque coefficient, however, the deviation of the tip speed ratio could still be tolerated.
With all the assumptions that have been made, the 3°-pitch angle is considered the reference, and the 8°-pitch
angle operating point and the 12°-pitch angle form as an alternative linear point for comparison purposes (See
Table 1 for details).
Figure 3: Reference and linearised operating points.
Table 1: Reference and Linearised Operating Points.
2.3.
Blade pitch angle
Tip speed ratio
Wind speed
Reference Point
3°
7.5
13 mps
Linear A
8°
7.5
13 mps
Linear B
12°
6.5
13 mps
Linear C
15°
5.5
13 mps
Simulation Model
There are two wind speed data used in the simulation to obtain the performance of the controller (Figure 4).
Statistics show that the low wind traces with mean speed of 9.5 m/s have lower fluctuation (4.2 m/s deviation)
while the high wind traces mean speed with mean speed of 15.7 m/s have higher fluctuation (6.9 m/s deviation).
Also, there are quite a number of instances where the 25 m/s wind speed (cut-out wind speed of the wind turbine
used in the study) is exceeded. These data sets would be a good representation of a severe step disturbance in the
wind turbines. The performance of a wind turbine in step disturbance is easily modeled by using a step-input
from the Matlab/Simulink block.
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Figure 4: Wind speed data (high wind, mean = 15.7 m/s; standard deviation = 6.9 m/s and low wind, mean
= 9.5 m/s; standard deviation = 4.2 m/s).
The block diagram in Figure 5 depicts the simulation model. The reference rotor speed is subtracted from the fed
back rotor speed output of the linearised wind turbine model resulting in a new rotor speed ∆ω (error). This rotor
speed error is the input of the controller, which commands a change in blade pitch angle based on the error. It
should be noted that one of the fundamental objectives of the controller is to minimise the error, that is, make it
zero as much as possible. In particular, the root-mean square error (RMS) of the rotational speed is to be
minimised. The reference pitch angle is added to the output angle from the controller at a particular instant
resulting in a new pitch angle, ∆β. This new angle is limited to being within 3° to 60°. The actuator's life is
prolonged by limiting its pitch rate within a certain range (between -10 and +10 deg/sec). The actuator's life or
the actuator duty cycle (ADC) is simply the total number of degrees pitched over the duration of simulation.
Such a measure ensures the temperature of the actuator (which is a motor) is to be within the tolerable level. Too
much actuation can result in a thermal overload that can reduce the motor's life or damage it. The dead zone is a
measure that prevents noise in the simulation. Here, ± 0.1 deg/sec was assumed reasonable.
Wind
Speed
Reference
Pitch (βref)
Ref. Rotor
Speed (ωref)
+
Controller
(PI/PID)
∆ω
+
+
Pitch Angle
Limit
Actuator
Wind
Turbine
Rotor
speed (ω)
Figure 5: Block diagram of the simulation model.
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3. RESULTS AND DISCUSSION
The performances of the PI and PID controllers on a step wind disturbance are illustrated in Figure 6. This figure
shows a critically damped rotor speed response of 1-m/s wind speed step, that is from 13 m/s to 14 m/s wind
speed rise. The rotational speed of the wind turbine never exceeds 0.6 rad/sec from the rated rotor speed of 6.5
rad/sec. The overshoot of 0.6 rad/sec. corresponds to just below 10% of the desired value, which could be
assumed permissible for a highly unstable system. Approximately, the settling time of the rotor speed response is
about 10 seconds from the simulation. The PID controller has lesser overshoot values for all the models as
compared to the PI controller. As expected the linearised model performs better than the referenced model.
Figure 6: Step disturbance response (13-14 m/s wind speed step change).
Figure 7 shows the wind turbine response under high wind series using the PID controller. The upper part of the
figure categorically depicts the rapid response of the blade pitch to a variation in wind speed. Even for sharp
wind variation, the pitch was able to be kept tracked making the rotor speed error minimum. On the average, the
rotor speed error falls within the acceptable levels except for a wind speed of more than 25 m/s in which the
rotor speed error rose significantly.
In Table 2, the RMS and ADC values of the linearised models were notably better than for the reference model.
With respect to the use of the controller, the PID categorically was the preferred approach. When comparing the
power production on these controllers, they have insignificant discrepancy (Figure 8). The graph below Figure 8
provides the tip speed ratio of the wind turbine during the whole simulation. This would provide a clearer picture
of whether the turbine is operating in a stable or unstable region. The wind turbine which is being studied allows
a maximum rotor speed of about 8 rad/sec (≈ 74 rpm) or an equivalent critical tsr of 4.6. The wind turbine is
running at an average tsr of 7.1 (0.4 below the objective tsr of 7.5). For a deviation of 2.7, the wind turbine
would be running between 4.4 and 9.8. Therefore, the wind turbine is running 4% of the simulation time in the
unstable region. This value is considered a minimum considering a highly erratic wind speed. On the contrary,
the wind turbine is running about 88% of the time in the stable region.
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Figure 7: Wind turbine response at high wind speed (Linear A - PID controller).
Figure 8: Available wind turbine power and tip speed ratio during high wind (Linear A).
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Table 2: RMS and ADC values for each model.
Model
Reference Point
Linear A
Linear B
Low wind
High wind
Performance
check
PID
PI
PID
PI
RMS
0.0187
0.1331
0.0254
0.1377
ADC
0.0898
0.0480
0.2661
0.1901
RMS
0.0180
0.1311
0.0247
0.1362
ADC
0.0738
0.0500
0.2618
0.1938
RMS
0.0156
0.1282
0.025
0.1344
ADC
0.0728
0.0447
0.2578
0.1886
4. CONCLUSION
The paper described a simple and rigid wind turbine model for controller design. Although, PID has better
performance over the PI controller for a wind step disturbance, it does not provide a significant difference in the
power. Comparatively, PID and PI controllers have almost the same performance in terms of the regions where
the wind turbine operate (stable or unstable regions). With the present methodology, despite the erratic wind
data, the wind turbine still manages to operate in the stable region at an average of 88%. Although, the above
results are reasonably impressive, it is vital that these values be compared with the actual wind turbine
performance and measurements.
REFERENCES
Freris, L.L. (1987), Generator Option for Variable and Fixed Rotational Speed Operation, Electrical Generation
Aspect of Wind Turbine Operation, Workshop BWEA 1987, Leicester UK, 9-23.
Hand, M.M. (2000), Systematic Controller Design Methodology for Variable-speed Wind Turbines, Wind
Engineering, 24(3), 169-187.
Kendall, L, Balas M.J., Lee, Y.J. and Fingersh, L.J. (1997), Application of Proportional Integral and Disturbance
Accommodating Control to Variable Speed Pitch Horizontal Axis Wind Turbines, Wind Engineering, 21(1), 21-38.
Novak, P., Ekelund T., Jovik, I. & Schmidtbauer, B. (1995), Modelling and Control of Variable-Speed Wind
Drive-System Dynamics, IEEE Control System Magazine, 15(4), 28-38.
Thiringer, T. and Jan Linders, J. (1993), Control by Variable Speed of a Fixed-Pitch Wind Turbine Operating in a
Wide Speed Range, IEEE Transactions on Energy Conversion, 8(3), September 1993, 520-526.
Wilkie, J., Leithead W.E. and Anderson, D. (1990), Modelling of Wind Turbines by Simple Models, Wind
Engineering, 14(4), 247-274.
Wilmshurst, S.M.B. (1988), Control Strategies for Wind Turbines, Wind Engineering, 12(4), 236-249.
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