Volume 2 wind energy 2 11 – wind turbine control systems and power electronics

2.11

Wind Turbine Control Systems and Power Electronics

A Pouliezos, Technical University of Crete, Hania, Greece
© 2012 Elsevier Ltd. All rights reserved.

2.11.1
Control Objectives
2.11.2
Wind Turbine Modeling
2.11.2.1
Mechanical Part
2.11.2.2
Electrical Part – Generators and Converters
2.11.2.2.1
Permanent magnet synchronous generators
2.11.2.2.2
Asynchronous (induction) generators
2.11.2.2.3
Doubly fed induction generator
2.11.2.2.4
Squirrel cage generator
2.11.2.2.5
Power converter
2.11.2.3
Full Model
2.11.3
Control
2.11.3.1
Overall Control Strategy

2.11.3.2
Pitch Control
2.11.3.2.1
Collective pitch control
2.11.3.2.2
Individual pitch control
2.11.3.3
Generator Control
2.11.3.3.1
Wound rotor doubly fed induction generator control
2.11.3.3.2
Asynchronous squirrel cage generator control
2.11.3.3.3
Permanent magnet synchronous generator control
2.11.3.4
Coupled Pitch–Generator Control
2.11.3.5
Grid Control
2.11.3.6
Yaw Control
2.11.3.7
Grid Issues
2.11.4
Fault Accommodation
2.11.5
Hardware
2.11.5.1
Sensors
2.11.5.2
Actuators

References
Further Reading
Relevant Websites

Glossary
Direct drive (DD) technology A design that eliminates
the need for gearboxes. With fewer moving parts, DD
technology can reduce maintenance costs and provide
higher wind turbine availability.
Double fed induction generator (DF) Sometimes referred
to as DFIG – has been widely used technology in wind
turbines for ten years in thousands of commissioned wind
turbines. It is based on an induction generator with a
multiphase wound rotor and a multiphase slip ring
assembly with brushes for access to the rotor. Recently,
designs without the brushes have been introduced.
Fault ride-through (FRT) A requirement of network
operators, such that the wind turbine remains connected
during severe disturbances on the electricity system, and

Nomenclature
A(γ(t)) turbine rotor swept area (time-varying due to yaw
error) (m2)
cP power coefficient

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returns to normal operation very quickly after the
disturbance ends.

Grid-connected A wind turbine is grid-connected when its
output is channelled directly into a national grid.
Power curve The relationship between net electric output
of a wind turbine and the wind speed measured at hub
height.
Rated wind speed The lowest steady wind speed at which
a wind turbine can produce its rated output power.
Reactive power An imaginary component of the apparent
power. It is usually expressed in kilo-vars (kVAr) or mega­
vars (MVAr). Reactive power is the portion of electricity
that establishes and sustains the electric and magnetic
fields of alternating-current equipment.

cT torque coefficient
Cd drive-train torsional damping constant
iSd(t), iSq(t), iRd(t), iRq(t) stator/rotor (d, q) current
components (A)

doi:10.1016/B978-0-08-087872-0.00212-2

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Wind Turbine Control Systems and Power Electronics

IT lumped rotational inertia of the turbine (rotor,

generator, etc.) (kg m2)

K lumped stiffness coefficient of the turbine

Kd drive-train torsional spring constant

Lm stator/rotor mutual inductance (H)

LS, LR stator/rotor inductances (H)

p number of generator pole pairs

qr(t) generator rotor azimuth angle (rad)

qt(t) turbine rotor azimuth angle (rad)

R turbine rotor radius (m)

RS, RR stator/rotor resistances (Ω)

Te(t) generator electromagnetic torque (electrical) (N m)

Tm(t) rotor aerodynamic torque (mechanical) (N m)

VSd(t), VSq(t), VRd(t), VRq(t) stator/rotor (d, q) voltage

components (V)

w(t) hub-height uniform wind speed across the rotor disk

(m s−1)

βi(t) blade i pitch angle (rad)

γ(t) rotor yaw angle (rad)

θS(t) stator flux position (Hz)

λ tip speed ratio

ρ air density (kg m−3)

ψRd(t) = LRiRd(t) stator/rotor (d, q) flux components

(weber (m2 kg s−2 A−1))

ψRq(t) = LRiRq(t) stator/rotor (d, q) flux components

(weber (m2 kg s−2 A−1))

ψSd(t) = LSiSd(t) stator/rotor (d, q) flux components

(weber (m2 kg s−2 A−1))


ψSq(t) = LSiSq(t) stator/rotor (d, q) flux components

(weber (m2 kg s−2 A−1))

ω(t) rotor angular velocity (mechanical) (rad s−1)

ωe(t) = pωm(t) electrical speed (electrical rad s−1)

ωS(t)¼dθS ðtÞ=dt stator field frequency (rad s−1)

Note: Tip speed ratio (TSR) is nondimensional. However,
since λ = ωR/v, the units appear to be (rad s−1)(m)
(m−1 s) = rad. But the SI unit of frequency is given as
hertz (Hz), implying the unit cycles per second; the SI unit
of angular velocity is given as radian per second.
Although it would be formally correct to write these units
as the reciprocal second, the use of different names
emphasizes the different nature of the quantities
concerned. The use of the unit radian per second for
angular velocity, and hertz for frequency, also emphasizes
that the numerical value of the angular velocity in
radian per second is 2π times the numerical value of the
corresponding frequency in hertz. The following table
summarizes these facts.
Derived
quantity
Plane angle
Angular velocity

Name

Symbol

Expressed in terms
of SI base units

Radian
Radian per
second

rad
rad s−1

m m−1
m m−1 s−1 = s−1

Hence, λ has units (s−1)(m)(m−1 s): non-dimensional.

2.11.1 Control Objectives
The conversion of wind energy into electrical power is not as straightforward as it might seem at first glance.
Wind speed is highly unpredictable and volatile. Furthermore, wind potential is not evenly distributed across the globe
(Figure 1).
Wind generators cannot work optimally in every wind speed, thus they are designed for maximum production in a certain wind
speed margin. Categorization of wind turbines follows IEC’s 61400 classification as shown in Table 1 [1].
In stronger than rated winds, the generator is in danger of being damaged, while at weaker winds the generator produces less
than expected.
To increase power production in these ‘nonrated’ wind speeds, control and supervision systems are employed. In short, the most
important objectives of a wind turbine control and supervision system are
•
•
•
•

to maximize efficiency at every operating point;
to minimize the structural load on the wind turbine;
to meet strict power quality standards (power factor, harmonics, flicker, etc.); and

to transfer the electrical power to the grid at an imposed level for a wide range of wind speeds.

To meet the above objectives, the control and supervisory system of large, variable-speed machines should consist of three main
subsystems (Figure 2).
Smaller wind turbines which frequently have no blade pitch control have no active speed and power control. Instead, passive
aerodynamic power control is achieved by exploiting blades stall effect while speed remains almost constant as it is fixed to the
system frequency. But even in this simpler version, a supervisory control system is necessary for operation monitoring and
controlling the operating sequence.
Control requirements depend mainly on the two ends of the wind power conversion process: the turbine rotor side and the grid
side.
On the turbine generator end, we distinguish between fixed- and variable-speed operation. Although these modes require
different control strategies, it is common in large megawatt turbines to adopt a discrete strategy, depending on the wind regime that
the turbine is actually operating (Figure 3). Region 1 describes start-up when wind speeds are below cut-in speed. Region 2 is


Wind Turbine Control Systems and Power Electronics

Annual 50 m Wind Speed

July 1983–June 1993

90
60
30
0
–30
–60
–90
–180
0.0

–120
1.3

2.7

3.5

–60
4.5

5.0

Region average = 6.808 4

0
5.5

6.0 6.5
(m/s)

60
7.0

7.5

120
8.0

8.5

180

9.0

>12.0

NASA /SSE 13 Sep 2004

Figure 1 Global wind potential. Reproduced from NASA (www.nasa.gov).

Categorization of wind turbines (IEC’s 61400 classification)

Table 1

Wind turbine class

I

II

III

IV

vave (average wind speed at hub height (m s−1))
v50 (extreme 50-year gust (m s−1))

10.0
70

8.5
59.5

7.5
52.5

6.0
42.0

Aerodynamics

Electromagnetic
subsystem

Grid connection
subsystem

Pitch
Yaw

Variable-speed
control

Output power
conditioning

Figure 2 Overall control system. NREL (www.nrel.gov/wind).

Power

Wind power

Aerodynamic power

Rated power

wcut-in

wrated

Region 1 Region 2
Figure 3 Wind turbine operational regions.

wcutout

Region 3 Region 4

Wind speed

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Wind Turbine Control Systems and Power Electronics

between cut-in and rated wind speeds, just before the turbine generates rated power. The main objective of a controller in this region
is to capture the maximum amount of energy from the wind. This is achieved by keeping blade pitch approximately constant and
using generator torque to vary the rotor speed. With small pitch changes about the optimal angle, a controller can also reduce

dynamic loads in the structure. In Region 3, between rated and cutout wind speeds, wind power must be shed by the rotor to limit
output power to the rated value. This is usually accomplished by keeping generator torque constant and commanding blade pitch
angles. Structural fatigue loads can also be reduced in Region 3 via individual pitch commands. The overall goal of the control
system is to meet different performance objectives in each operating region and make the transition between Regions 2 and 3
proceed smoothly to avoid load spikes. Finally in Region 4 the controller should stall the machine.
This strategy, as shown in the generic block diagram of Figure 4, is supervised by the wind turbine’s supervisor system.
On the grid side, modern megawatt turbines employ full- or partial-load frequency converters to convert variable-frequency,
variable-voltage current into constant-frequency, constant-voltage current (Figure 5). This enables decoupled regulation of active
and reactive power, wherever the type of generator allows it. Hence, it is acceptable to consider generator control separately from
grid control. Having said that however, it must be pointed out that the two subsystems may be coupled in the case of systems
capable of tolerating grid faults.
It has been difficult to gather information on the design and field performance of industrial controllers employed in real wind
turbines. Furthermore, although there is an abundance of research papers on wind turbine controllers, ranging from the simpler
proportional-integral-derivative (PID) to more exotic fuzzy or neural versions, the performance of most of these is judged from
computer simulations of mathematical models of the wind turbine system. Even though these models are sometimes quite
complicated and therefore quite accurate, it is impossible for them to capture every detail of the real world. Another paradox is
the well-known fact that despite an abundance of theoretical work on almost every type of controller available, industry, to the best
of my knowledge, still uses the solid PID controller, with various modifications, on all machines.
Some of the material that follows is taken from the US’s Department of Energy National Renewable Energy Laboratory (NREL)
research. This is tested both on simulated environments and on their Controls Advanced Research Turbine (CART) (Figure 6). A
short description of CART follows.
The CART is actually a Westinghouse WTG-600 two-bladed, teetered, upwind, active-yaw wind turbine. It is of variable speed,
and each blade can be independently pitched with its own electromechanical servo. The pitch system can pitch the blades up to

vcut-in
Power
Production

Failures

No

v>vcutout
v
Figure 4 Wind turbine generic structure for supervisor control.

Figure 5 Power electronics converter.

Idle

Yes


Wind Turbine Control Systems and Power Electronics

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Figure 6 NREL’s CART (Controls Advanced Research Turbine). Reproduced from US’s Department of Energy National Renewable Energy Laboratory NREL
(www.nrel.gov/wind).

18° s−1 with pitch accelerations up to 150° s−2. The squirrel cage induction generator with full-power electronics can control torque
from minus rating (motoring) to plus rating (generating) at any speed. The torque control loop has a high-rated bandwidth of
500 rad s−1. Rated electrical power is 600 kW at a low-speed shaft (LSS) speed of 41.7 rpm. Power electronics is used to command
constant torque from the generator and full-span blade pitch controls the rotor speed.
The machine is equipped with a full complement of instruments that gather meteorological data at four heights. Blade root flap
and edge strain gauges, tower-bending gauges, and low- and high-speed shaft (HSS) torque transducers gather load data.
Accelerometers in the nacelle measure the tower’s fore–aft and side–side motion. Absolute position encoders gather data on
pitch, yaw, teeter, and LSS and HSS positions. These data are sampled at 100 Hz.

In order to design any controller, a model of the controlled process is usually needed. Non-model-based methods, collectively
known as ‘intelligent control’, are also possible (fuzzy, neural, etc.). Models can be either linear or nonlinear, with linear models
usually being linearizations of the corresponding nonlinear ones about an operating point.
As already stated, most controllers actually installed on wind turbines are of the standard PID type. Their tuning is necessarily
based on linearized models of the processes involved. This fact imposes the first of the two main problems that designers are facing,
namely, that of degraded performance in other operating points. This problem is usually faced by ‘gain scheduling’ methods, even
though more advanced techniques, including nonlinear multi-input multi-output (MIMO) approaches, have been proposed. It is a
general feeling, however, that these highly complex techniques do not greatly outperform a well-tuned PID controller.
The second major problem is that of the wind disturbance, the wind being highly stochastic and unpredictable in nature. A
number of techniques have been proposed for overcoming this problem also, the most prominent of which is wind information
ahead of the turbine. This information is then fedforward, thus making the whole system respond better to sudden wind
fluctuations such as gusts or lulls.
Other problems include quality of sensor data (rotation speed, angle, etc.), a major source of performance degradation in every
feedback control system. Solutions come in the form of suitable filters or better sensors.
Finally, additional control objectives such as load mitigation or energy maximization have to be cast in a multivariable
framework and solved by vector optimal control or other performance-driven methods.

2.11.2 Wind Turbine Modeling
For control design purposes, it is imperative to establish a relation connecting the wind turbine system’s inputs, outputs, and
disturbances, possibly utilizing internal ‘state variables’.


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Wind Turbine Control Systems and Power Electronics

The output of the turbine is its actual rotating speed, which in turn affects the captured aerodynamic power. Control inputs are
(wherever applicable) pitch angle and generator currents. Yaw angle control could also be included. Lastly, wind power is the
disturbance term.

2.11.2.1

Mechanical Part

A process model should be as simple as possible for the specific control problem. Therefore, for simple, collective pitch control, a
simple, scalar, spring-mass description is adequate, while in individual pitch control (IPC), possibly coupled with vibration
attenuation, a more complex vector procedure is called for.
Let us therefore start with a spring-mass system describing in simple terms the wind turbine dynamics given by
IT

d2 qðtÞ
þ Kqðt Þ ¼ Tm ðtÞ − Te ðtÞ
dt 2

½1Š

If K is neglected, a simpler form results:
dωðtÞ
¼ Tm ðtÞ − Te ðt Þ
½2Š
dt
What eqn [1] actually says is that rotor rotation is a balance between the aerodynamic torque applied by the wind and the electrical
torque applied by the generator. Furthermore,
IT

Tm ðtÞ ¼

1
ρAðtÞRcT ðωðtÞ; wðtÞ; βðtÞÞw2 ðtÞ
2

½3Š

To complete the model, an expression for Te is needed, which depends on the specific type of generator employed. Usually the
expression for Te involves (d, q) axis stator and rotor current components. Such a formula for the doubly fed induction generator
(DFIG) is
Te ðtÞ ¼

À
Á
3
pLm iSq ðtÞiRd ðtÞ − iSd ðtÞiRq ðtÞ
2

½4Š

Plugging eqns [3] and [4] into eqn [1] results in a highly nonlinear system in both the states and the control. Furthermore, the
system is coupled since, for example, generator electromagnetic torque Te(t) affects rotor speed ω(t).
It is common in wind turbine control systems to use pitch angle β(t) and generator current i(t) in different operating regions.
Therefore, let us look at these different situations, starting with the case in which generator electromagnetic torque is kept constant,
Te ðtÞ ¼ Te .
The only control input to this model will be rotor collective pitch. This means that the pitch angle β(t) of each blade is identical.
Since Tm is a continuous function of w, ω, and β, it can be expanded in a Taylor series about an equilibrium point as
Tm ðw0 þ δw; ω0 þ δω; β0 þ δβÞ ¼ Tm ðw0 ; ω0 ; β0 Þ þ

∂Tm
∂Tm
∂Tm
δw þ
δω þ

δβ þ higher order terms
∂w
∂ω
∂β

⇒ Tm ðw; ω; βÞ ≅ Tm ðw0 ; ω0 ; β0 Þ þ αδw þ γδω þ ηδβ

½5Š
½6Š

where subscript 0 denotes values at some equilibrium point, usually at rated configuration.
At equilibrium, acceleration is zero (d2q(t)/dt2 = 0); therefore, aerodynamic and generator torque must cancel each other. Hence,
using eqn [6], eqn [2] becomes
IT δ€q ¼ α δw þ γ δω þ η δβ þ ðTm ðw0 ; ω0 ; β0 Þ − Te Þ
⇒ IT δ€q ¼ α δw þ γ δω þ η δβ

½7Š

Finally, since by definition q_ ¼ ω, eqn [7] becomes
δω_ ¼

γ
η
α
δω þ δβ þ δw
IT
IT
IT

½8Š

Equation [8] constitutes the linearized wind turbine model in terms of the deviation variables. We can rewrite eqn [8] in other
forms, suitable for single-input single-output (SISO) or MIMO control. By Laplace-transforming eqn [8], for example, we get the
transfer function
~ ðsÞ ¼
ω

a
η
~ ðsÞ
β~ ðsÞ þ
w
ðsIT − γÞ
ðsIT − γÞ

½9Š

where the tilded variables denote deviation variables.
Alternatively, eqn [8] can be written in (trivial) state space form as
x_ ðtÞ ¼ AxðtÞ þ BuðtÞ þ GdðtÞ
yðtÞ ¼ CxðtÞ

½10Š


Wind Turbine Control Systems and Power Electronics

335

where

~ ðtÞ; uðt Þ ¼ βðt Þ; dðtÞ ¼ wðtÞ
xðtÞ ¼ ω
!
!
!
γ
η
α
A¼
; B¼
; G¼
; C ¼ ½1Š
IT
IT
IT

½11Š

and measurement of rotor speed is assumed.
If IPC is possible, a more complicated multivariable model is needed. Furthermore, since individual pitching provides more
degrees of freedom, other control goals besides constant speed may be fulfilled, such as blade load mitigation, especially in large
megawatt turbines. Actually, it may not be an exaggeration to state that in current and future design control, interest has shifted from
maximization of power capture to minimization of blade and tower load. At best, an optimal compromise between the two may be
sought after.
Although many different versions of suitable models exist in the literature, we will outline the model used in NREL’s laboratories [2],
since, as explained, it is one of the few that has actually been field tested. This model uses 15 state variables in its more general form. In
the sequel, a model aiming at ensuring stability and adequate damping of the first tower fore–aft mode will be presented. Other control
objectives may require a different model to be employed, but the general control design procedure shown is still applicable.
The appropriate model uses nine state variables and its state space form is
x_ ðtÞ ¼ AxðtÞ þ BuðtÞ þ Bd ud ðtÞ

yðtÞ ¼ CxðtÞ

½12Š

where the state x, control u, and disturbance ud are defined as follows:
h
i
xðtÞ ¼ ~q 1ðtÞ ~q_ 1 ðtÞ ~q 2 ðtÞ ~q_ 2 ðt Þ ~q_ 4 ðtÞ Kd ð~q 4 ðtÞ − ~q15 ðtÞÞ ~q_ 15 ðtÞ ~q 7 ðtÞ ~q_ 7 ðtÞ T
!
!
~ 1 ðtÞ
w
β~ ðtÞ
uðtÞ ¼ ~ 1
; ud ðt Þ ¼
~ 2 ðtÞ
w
β 2 ðtÞ

½13Š

The meaning of the various quantities is as follows (see also Figure 7): x1 ð¼ ~q1 ðtÞÞ is the blade 1 perturbed flap tip
displacement; x2 ð¼ ~q_ 1 ðtÞÞ the blade 1 perturbed flap tip velocity; x3 ð¼ ~q 2 ðtÞÞ the blade 2 perturbed flap tip displacement;
x4 ð¼ ~q_ 2 ðtÞÞ the blade 2 perturbed flap tip velocity; x5 ð¼ ~q_ 4 ðtÞÞ the perturbed rotor rotational speed (Ωrot); x6 ð¼ Kd ð~q 4 ðtÞ − ~q 15 ðtÞÞÞ
the perturbed drive-train torsional spring force (ψ , ψ ); x7 ð¼ ~q_ ðtÞÞ the perturbed generator rotational speed (Ω ); x8 ð¼ ~q ðtÞÞ
rot

gen

gen

15

7

the perturbed tower-top first fore–aft mode deflection; x9 ð¼ ~q_ 7 ðtÞÞ the perturbed tower-top first fore–aft mode velocity, ui ð¼ β~ i ðtÞÞ
^ i ðtÞ the wind speed impinging on blade i.
the blade i pitch perturbation angle; and w
Finally, matrices A, B, and Bd are obtained through linearization about the chosen operating point and C depends on the
available measurements. The linearization is performed using any suitable software such as SymDyn [3].
Blade 1 flap
δq1

Kd (δψrot − δψgen)
δΩgen

δΩrot

δq1

δq2
Blade 2 flap

Figure 7 CART (Controls Advanced Research Turbine) state variables. Reproduced from Wright AD (2004) Modern control design for flexible wind
turbines. NREL Report No. TP-500-35816. Golden, CO: National Renewable Energy Laboratory [2].


336

2.11.2.2

Wind Turbine Control Systems and Power Electronics
Electrical Part – Generators and Converters

The generator consists essentially of two elements: the field and the armature. The field winding carries direct current and produces a
magnetic field which induces alternating voltages in the armature windings. The normal practice is to have the armature on the stator.
Slowly rotating generators, such as those employed in wind turbines, have a rotor structure with a large number of salient (projecting)
poles and concentrated windings. Figure 8 shows the cross section of a three-phase synchronous generator with one pair of poles.
The magnitude of the stator magnetomotive force (MMF) wave and its relative angular position with respect to the rotor MMF
→
wave depend on the load (MMF is defined as the line integral of the magnetic field intensity H in the machine’s air gap). The
electromagnetic torque on the rotor acts in a direction so as to align the magnetic fields. In a wind turbine generator, the rotor field
leads the armature field due to the forward aerodynamic torque.
Voltage differential equations that describe the performance of AC machines are usually time-varying, except when the rotor is
stalled. A change of variables is often used to reduce the complexity of these equations. This general transformation refers machine
variables to a frame of reference that rotates at an arbitrary angular velocity.
This idea was contributed by Park [5], who in the 1920s formulated this concept, which in effect replaces all variables (voltages,
currents, and flux linkages) associated with the stator windings of a synchronous machine with variables associated with fictitious
windings rotating with the rotor. This, in effect, eliminates all time-varying inductances from the voltage equations. It was later
discovered that this transformation admits a general structure applicable in all AC machines, be it synchronous or asynchronous. In
this approach, all stator and rotor variables are referred to a frame of reference that may rotate or remain stationary. It is worth
noting that in the case of synchronous machines this idea works only if the reference frame is fixed to the rotor.
The general equations of this transformation are [6]
2
3



2π
2π

cos θðtÞ cos θðtÞ −
cos θðt Þ þ
6
3
2
2
3 6
3
3 7



7
f qs ðt Þ
6
7 f as ðt Þ
2π
2π 76 f ðtÞ 7 ¼ Kf ðtÞ
4 f ds ðtÞ 5 ¼ 6
½14Š
6
74 bs 5
abcs
sin θðtÞ þ
6 sin θðtÞ sin θðt Þ − 3
3 7
f 0s ðt Þ
6
7 f cs ðt Þ
4 1

5
1
1
2
2
2
where
dθðtÞ
½15Š
dt
In eqn [14], f can represent voltage, current, flux linkage, or electric charge. Subscript s denotes stationary circuits. The frame of
reference may rotate at any constant or varying angular velocity ω or remain stationary.
The inverse transformation is obtained by inverting K:
2
3
cos θ 
sin θ  1
6
7
2π
6 cos θ − 2π
sin θ −
17
6
7
3
3
K −1 ¼ 6
½16Š
7





6
7
4
5
2π
2π
cos θ þ
sin θ þ
1
3
3
ωðt Þ ¼

q-axis

Axis of phase b

Field winding

Armature winding
a
c�
N

S
b

b�

d-axis

ωr Rotor
θ

Air gap
c
a�

Axis of phase a

Stator

Axis of phase c
Figure 8 Three-phase synchronous generator. Reproduced from Kundur P (1994) Power System Stability and Control. New York, NY: McGraw-Hill, Inc. [4].


Wind Turbine Control Systems and Power Electronics

337

ω

fb s
fq s

θ

fa s

fc s

fd s

Figure 9 Trigonometric sketch of d–q transformation (eqn [14]). Reproduced from Krause PC, Wasynczuk O, and Sudhoff SD (2002) Analysis of Electric
Machinery and Drive Systems. New York, NY: IEEE Press [6].

It is often convenient to visualize the transformation equations as shown in Figure 9. As seen, the transformation may be thought of
as if the fds and fqs variables are directed along paths orthogonal to each other and rotating at an angular velocity ω, whereas fis may
be considered as variables directed along stationary paths displaced by 120°.
Electrical generators are systems that are generally controlled by power electronics. Two different types of self-commutated
converters can be implemented: the voltage source inverter or converter (VSI or VSC) and the current source inverter (CSI). The first
has a capacitor in the DC circuit and works with a relatively constant DC voltage; the second has an inductance in the DC circuit and
works with a relatively constant DC current. The power production of the turbine is controlled by controlling the generator current.
A chopper can be used to change the DC voltage and to keep it constant; it also provides an additional way to control generator
torque. VSI converters are preferable in modern applications due to several advantages they present regarding control performance
and power quality issues. These converters, as their name implies, use voltage as the control input.
The building block of VSC is the insulated gate bipolar transistor (IGBT). It is capable of handling large phase currents and can be
used in converters rated up to 1700 V.
The generator-side converter or rectifier and the grid-side converter or inverter usually exploit pulse width modulation (PWM)
techniques. In this case, the power converter comprises six valves with turn-off capability and six antiparallel diodes (Figure 10). The
valves are typically realized by IGBTs because they allow for higher switching frequencies than classical gate turn-off thyristors
(GTOs). The converter is fed from a DC source, usually a rectified three-phase AC source.

2.11.2.2.1

Permanent magnet synchronous generators

The main type of synchronous generator used in large wind turbines is the multipole permanent magnet synchronous generator
(PMSG). In this concept, the PMSG is commonly directly driven, that is, the wind turbine is gearless, and connected to the grid
through full-load frequency converters. The complicated electromagnetic construction of the permanent magnets presents a draw­
back of the whole design. Several types of PMSG exist, but the most widely used is the radial flux permanent magnet (RFPM)
generator.
iDC

+

SA+

SB+

SC+
DB+

DA+

C
VDC

A

iA

B

SA−

iB

C

SB−
DA−

DC+
iC

Z

SC−
DB−

DC−
n

−

N

Figure 10 Three-phase inverter circuit.

Z

Z


338

Wind Turbine Control Systems and Power Electronics

S

ve, fe

Power
converter

v, f

Grid

R

Figure 11 Permanent magnet synchronous generation grid connection.

Usually, the permanent magnets are placed in the generator rotor. In wind turbine applications, the stator is connected to the AC
grid via full-scale frequency converters (Figure 11). In variable-speed machines, the use of frequency converters is essential, since the
generator output frequency is time varying and equal to
f e ðtÞ ¼ f m ðtÞp

½17Š

where fm is the turbine rotor frequency (mechanical) and p is the number of pole pairs. From eqn [17], the rated (=maximum)
electrical frequency of the machine may be derived as
f er ¼ f mr p

½18Š

where superscript r denotes ‘rated’.
In order to obtain a dynamic model of the PMSG, we consider a symmetrical, three-phase generator where the developed stator
flux is sinusoidal. This in turn implies that the electromotive forces (EMFs) are also of the same type.
In the case of PMSG, the angular velocity of the stator space vectors, ωe, is equal to the mechanical angular velocity ω. This means
that the coordinate system rotates field synchronously and fixed to the rotor. Therefore, if the coordinate system is chosen to match
the direct d-axis or field flux, the coordinate system represents the desired field flux orientation. Therefore, for studying synchronous
machine characteristics, the transformation axes are defined as (Figure 12) [4]
• the direct (d) axis, centered magnetically in the center of the north pole; and
• the quadrature (q) axis, 90 electrical degrees ahead of the d-axis.
Then the equations describing its operation are given by [7]
ψ_ ds ðtÞ ¼ − vd ðtÞ − Rs ids ðtÞ − ωe ðtÞψ qs ðtÞ
ψ_ qs ðtÞ ¼ − vq ðtÞ − Rs iqs ðtÞ − ωe ðtÞψ ds ðtÞ

½19Š

and
ψ ds ðtÞ ¼ Lds ids ðtÞ þ ψ m
ψ qs ðtÞ ¼ Lqs iqs ðtÞ
Τ

½20Š

Τ

Τ

where VG = [vd vq] is the terminal generator voltage vector, ψG = [ψds ψqs] is the stator flux vector, iG = [ids iqs] is the machine
current vector, and ψm denotes the magnitude of the permanent magnet flux linked to the stator windings.
Furthermore, the electric torque of the PMSG is



Te ðtÞ ¼ p ψ ds ðtÞiqs ðtÞ − ψ qs ðtÞids ðtÞ
Á
À
½21Š
¼ p ψ m ðtÞiqs ðtÞ þ ðLds − Lqs Þiqs ðtÞids ðtÞ
Rotation

ωr

elec. rad s −1

d-axis
efd

q-axis

ifd

ib
θ

ikq
Axis of phase a
ikd

eb

ψb

ψc

b

ψa
ea

ia

a

ec

Rotor

Stator

ic

c

Figure 12 Stator and rotor circuits of synchronous machines. Reproduced from Kundur P (1994) Power System Stability and Control. New York, NY:
McGraw-Hill, Inc. [4].


Wind Turbine Control Systems and Power Electronics

339

Finally, the active, Ps(t), and reactive, Qs(t), power delivered by the stator are given by

Ps ðtÞ ¼ vd ðtÞids ðtÞ þ vq ðtÞiqs ðtÞ
Qs ðtÞ ¼ vd ðtÞiqs ðtÞ − vq ðtÞids ðtÞ

2.11.2.2.2

½22Š

Asynchronous (induction) generators

An induction generator carries AC currents in both the stator and rotor windings. In a three-phase generator, the stator windings
are connected to a balanced three-phase supply. The rotor windings are connected through slip rings to a passive external circuit.
The distinctive feature of the induction generator is that the rotor currents are ‘induced’ by electromagnetic induction from
the stator.
When current of frequency fs Hz is applied to the stator, its windings produce a field rotating at synchronous speed given by
ns ¼

120fs
ðr min − 1 Þ
p

½23Š

where p is the number of poles (two per three-phase winding set).
Whenever there is relative motion between the stator field and the rotor, voltages are induced in the rotor windings. The rotor
current reacting with the stator field produces a torque, which accelerates the rotor in the direction of the stator field rotation. The
slip speed of the rotor is defined as
s¼

ns − nr
ðper unit of synchronous speedÞ

ns

½24Š

where nr is rotor speed and ns stator field speed. To act as a generator, the rotor speed must be greater than ns, that is, with negative
slip. The resulting torque is in this case opposite to that of rotation.
An induction machine differs from the synchronous one in the following points:
• The rotor has a symmetrical structure, making the (d, q) axes circuits equivalent.
• The rotor speed varies with load.
The above characteristics make a synchronously rotating reference frame an appropriate choice for transforming induction generator
relationships.

2.11.2.2.3

Doubly fed induction generator

In the DFIG, the wound rotor has conventional three-phase windings brought out through three slip rings on the shaft so that they
can be connected to a frequency converter.
As stated, the induction generator model can be expressed in (d, q) axes, rotating at synchronous speed ωs. The q-axis is assumed
to be 90° ahead of the d-axis in the rotation direction. If the d-axis is chosen so that it coincides with the phase axis a at t = 0, its
displacement from axis a at any time is ωst.
To express the dynamic equations describing the operation of the DFIG, expressed in the (d, q) frame, in state space form, let us
choose the following state and control vectors:
2
3
2
3
Vds ðtÞ
ids ðtÞ
6 iqs ðtÞ 7

6 Vqs ðtÞ 7
7
6
7
xðtÞ ¼ 6
4 idr ðtÞ 5; uðtÞ ¼ 4 Vdr ðtÞ 5
iqr ðtÞ
Vqr ðtÞ

½25Š

With this choice of variables, the DFIG can be expressed in standard control notation as [8]
x_ ðtÞ ¼ AðωÞxðtÞ þ BuðtÞ
yðtÞ ¼

½26Š

3
pLm ðx2 ðtÞx3 ðtÞ − x1 ðtÞx4 ðtÞÞ ≡ Te ðtÞ
2

½27Š

where
2

pωL2m
RS
ωS þ
6

σL
σL
S
S LR
6 

6
2
pωL
R
6
S
m
−
6 − ωS þ
6
σLS LR
σ LS
AðωÞ ¼ 6
6
pωLm
Lm RS
6
−
6
σLS LR
σLR
6
4
Lm RS

pωLm
σLS LR
σLR
−

Lm RR
σLS LR
pωLm
−
σLS
RR
−
σLR
pω
−ωS þ
σ

pωLm
σ LS
Lm RR
σLS LR
pω
ωS −
σ
RR
−
σLR

3
7

7
7
7
7
7
7
7
7
7
7
5

½28Š


340

Wind Turbine Control Systems and Power Electronics
2

1
6 σLS
6
6
0
6
6
B¼6 L
6− m
6

6 σLS LR
4
0

0
1
σLS

−

0
Lm
−
σ LS LR

0
1
σLR

Lm
σLS LR

σ ¼ 1−

3

Lm
σLS LR

0

and

2.11.2.2.4

−

0
1
σLR

0

7
7
7
7
7
7
7
7
7
5

½29Š

L2m
LS LR

Squirrel cage generator

The squirrel cage induction generator rotor consists of a number of uninsulated bars in slots, short-circuited by end rings at both
ends. Thus, the squirrel cage model can be obtained from the corresponding DFIG by setting Vdr(t) = Vqr(t) = 0.
Let
!
Vds ðtÞ
uðtÞ ¼
½30Š
Vqs ðtÞ
Then the relevant space state equations governing the electrical operation are
x_ ðtÞ ¼ AðωÞxðtÞ þ B1 uðtÞ
where

2

3

1
σLS

6
6
6
0
6
6
B1 ¼ 6 L
6− m
6
6 σLS LR

4
0

½31Š

0
−

Lm
σLS LR
0

−

Lm
σLS LR

7
7
7
7
7
7
7
7
7
5

½32Š

and A(ω) is given by eqn [28].
Equation [31] is complemented by the electromagnetic torque equation
yðtÞ ≡ Te ðtÞ ¼

2.11.2.2.5

3
pLm ðx2 ðtÞx3 ðtÞ − x1 ðtÞx4 ðtÞÞ
2

½33Š

Power converter

To model the generator-side converter, it is common to neglect switching dynamics, ripple currents, and other fast dynamics in
the electrical system and model the generator-side converter as a simple time delay. Thus, voltages on the generator clamps are
given by
Vds ðtÞ ¼ Vc; d ðt − td ÞVdc ðtÞ
Vqs ðtÞ ¼ Vc; q ðt − td ÞVdc ðtÞ

½34Š

where Vc,d and Vc,q are the controller’s outputs.
The dynamic equation for the DC-link voltage Vd is given by
dVd ðtÞ
1
ð−Pc ðtÞ þ Ps ðtÞÞ
¼
dt
CVd ðt Þ

½35Š

neglecting the losses from the converters, where Pc and Ps are the active power from the grid-side converter and the stator-side
converter, respectively, and C is the DC-link capacitance.
The grid is usually modeled as an infinite bus with voltage Vgrid and frequency ωgrid. The grid-side converter voltage is given as
Vc ðtÞ ¼ mVd ðtÞeja

½36Š

where m and a (phase angle between Vc and Vex) are controlled by the amplitude and phase controllers on the grid side.
The currents through the grid-side converter are given as a function of the grid voltage,
Vdc ðt Þ − Vdex ðt Þ
Xt
Vqc ðtÞ − Vqex ðtÞ
idc ðtÞ ¼ −
Xt
iqc ðt Þ ¼

½37Š


Wind Turbine Control Systems and Power Electronics

341

where Xt is the grid-side inductance and subscript c indicates values flowing through the grid-side converter while ex indicates values
at the grid. Hence, the reactive power flowing into the grid, Qgrid, is given by
Qgrid ðtÞ ¼ Vdex ðtÞiqc ðtÞ − Vqex ðtÞidc ðtÞ

½38Š

while the active power flowing into the grid, Pc, is
Pc ðtÞ ¼ Vqc ðtÞiqc ðtÞ þ Vdc ðtÞidc ðtÞ

2.11.2.3

½39Š

Full Model

For transient stability analysis, a dynamic mathematical model comprising every wind turbine subsystem is essential. The level of
detail of this complete system varies, and is dependent on the particular application. Some, or all, of the modules shown in
Figure 13 can be included. However, most models are linearized versions of the nonlinear equations, around some suitable
operating point, usually the nominal.
If driveshaft, rotor torque, tower, thrust, and wind speed are taken into account, the following state space model of the
mechanical system is derived [9]:
2 D þγ
Ds
Ks
s
r
−
−
0
6
Ir
Ir Ν g
Ir
6

2
3 6
Ds
Ds
Ks
ω_ r ðtÞ
6
−
0
2
6 ω_ g ðtÞ 7 6
I
Ν
Νg
I
Ν
I
g
g
g
6
g
g
6
7
6 _ ðt Þ 7 ¼ 6
6
1
6 Δ 7 6
1

−
0
0
4 x_ t ðt Þ 5 6
Ν
g
6
€x t ðtÞ
6
0
0
0
0
6
4
γt
Kt
−
0
0
Mt
Mt
or
!
à βðt Þ
Â
x_ 0 ðtÞ ¼ A0 x0 ðtÞ þ Bβ0 BΤ0
þ G0 wðt Þ
Te ðtÞ

αr
Ir

3

2 η
r
7
3
72
I
6
r
ω
ð
t
Þ
7
r
6
76
6
0
76 ω g ðt Þ 7
7 6 0
76
7
6
76
6

76  Δ ðt Þ 7
7þ6 0
7
0
7 6
76
74 x t ðt Þ 5 6
6 0
7
4 η
1
7 x_ t ðt Þ
5
− t
Dt α t
Mt
− −
Mt Mt
−

0
1
Ig
0
0
0

−

3

2
7
6
7
7
6
7
! 6
7 βðt Þ
6
7
6
7 T ðt Þ þ 6
7 e
6
7
6
7
4
5

αr
Ir
0
0
0
αt
Mt

3
7
7
7
7
7
7wðt Þ
7
7
5

½40Š

where subscripts r and g denote corresponding turbine rotor and turbine generator quantities and the rest of the variables are
depicted in Figures 14 and 15.
Equation [40] can be augmented with actuator dynamics, for example,
3 2
 β
Ã
3 2
3
3 2
2
2 3
!
x 0 ðt Þ
x_ 0 ðtÞ
0 0
G0
Β0T

A0
B0 0
βref ðt Þ
6
7 4
4 x_ β ðt Þ 5 ¼ 4
5
5
4
þ 4 0 5wðt Þ
½41Š
0
Aβ
0 5 þ xβ ðtÞ þ Bβ 0
Tref ðt Þ
0 BT
0
x_ T ðt Þ
xT ðt Þ
Αg
0
0

Rotor dynamics and
fore–aft tower
movement

Structural
motion

Structural
forces

Reaction torque

Shaft speed
Wind

Drive-train

dynamics


Aerodynamics

Converter
electronics

Power

Hub torque

Aerodynamic
pitching moment

Pitch actuation

Actuator
dynamics

Generator
rotor
speed

Torque demand

Control
dynamics

Figure 13 Full system interconnection. Reproduced from Henriksen LC (2007) Model Predictive Control of a Wind Turbine. MSc Thesis, Technical
University of Denmark [9].


342

Wind Turbine Control Systems and Power Electronics

xt
Dt

Mt

Kt

Qt

Figure 14 Wind tower variables. Reproduced from Henriksen LC (2007) Model Predictive Control of a Wind Turbine. MSc Thesis, Technical University of
Denmark [9].

φg Ωg

,
Ng Ng
φr, Ωr

NgQg
Ds
Ks

Ng2Ig

Qr
Ir

Figure 15 Drive variables. Reproduced from Henriksen LC (2007) Model Predictive Control of a Wind Turbine. MSc Thesis, Technical University of
Denmark [9].

Equation [40] can be further extended if generator dynamics are added. In this case, however, it may not be possible to get linear
expressions, since in some cases generator torque is bilinearly related to the control variables (see, e.g., Reference [4]).
On the other hand, if simpler models are desired, one can attempt to use black box identification through step response data. In
Reference [10], such a procedure is described whereby a PID pitch angle controller for a fixed-speed active-stall wind turbine is
designed using the root locus method. The purpose of this controller is to enable an active-stall wind turbine to perform power
system stabilization. Considering the open-loop system of Figure 16, the step response of Figure 17 was obtained.

PSet

PDiff

PErr

θ

ferror
Droop

P

Wind
turbine

Pel

Pem
Generator

Figure 16 Open-loop block diagram for wind turbine identification. Reproduced from Jauch C, Islam SM, Sorensen P, and Jensen BB (2007) Design of a
wind turbine pitch angle controller for power system stabilisation. Renewable Energy 32: 2334–2349 [10].


Turbine power (kw)

2000

0.12
0.1
0.08
0.06
0.04
0.02
0
−0.02

1500
Wind turbine response

1000

Frequency error step change

500
0
−500
−1000
3

4

5

6

7

8


9

10

11

12

343

Frequency error (Hz)

Wind Turbine Control Systems and Power Electronics

13

Time (s)

Figure 17 Wind turbine power open-loop step response. Reproduced from Jauch C, Islam SM, Sorensen P, and Jensen BB (2007) Design of a wind
turbine pitch angle controller for power system stabilisation. Renewable Energy 32: 2334–2349 [10].

As seen, the response resembles a second-order transfer function response,
Kω2n
ferr ðsÞ
≡ GðsÞ ¼ 2
s þ 2ζ ωn þ ω2n
Pe ðsÞ

½42Š

Identification of parameters K, ζ, and ωn is straightforward from the characteristics of the step response, such as overshoot and
settling time. As is often the case, a set of such transfer functions must be obtained corresponding to different operating points due
to the nonlinearity of the process.

2.11.3 Control

2.11.3.1

Overall Control Strategy

As outlined previously, current wind turbines aim at maximizing power output while maintaining grid quality standards. However,
in isolated areas, such as islands, power limitation may be imposed by the grid operator.
The basic relation concerning this goal is the well-known equation connecting mechanical power Pm and power coefficient Cp:
Pm ðtÞ ¼ 0:5ρCp ðλ; βÞAðγÞv3 ðtÞ
λðtÞ ¼

½43Š

ωðtÞR
; tip speed ratio ðTSRÞ
vðtÞ

½44Š

From eqn [43] it can be clearly seen that power maximization depends mainly on Cp and secondarily on A(γ), with all other
parameters being external disturbances. Since the yaw angle γ is regulated independently in its optimum position, Pm depends on λ
(or equivalently ω) and β.
The graph of Cp(λ, β) is a three-dimensional surface, obtained for each turbine by modeling or experiments. Figure 18 shows
such a typical curve, drawn using [11]
Cp ðλ; βÞ ¼

4 X
4
X

αij βi λj

½45Š

i¼0 j¼0

If β is fixed, Pm depends only on λ. Figure 19 shows a graph relating Pm and ω for various v.

0.5
0.4

Cp


0.3

0.2
0.1
0
20

15
10
5

Tip speed ratio

0

5

10

15

20

Pitch angle (deg)

Figure 18 Cp(λ, β) surface. Reproduced from Li S, Haskew TA, and Xu L (2010) Conventional and novel control designs for direct driven PMSG wind
turbines. Electric Power Systems Research 80: 328–338 [11].


344

Wind Turbine Control Systems and Power Electronics

1200

1000

P (kW)

800
13 m/s
600

11 m/s

400
9 m/s

200
7 m/s
0

0

10

20

5 m/s
30
40

50

60

70

80

90

100

Rotational speed (rpm)
Figure 19 Pm vs. ω for the NTK 500-41 wind turbine.

Now, from eqn [44] it follows that
ωà ðtÞ ¼

λà vðtÞ
R

½46Š

is a suitable wind-dependent optimal speed trajectory (set point) for less than rated wind speeds (red line in Figure 19).
As explained in the introduction, the control process of the wind turbine consists mainly of two phases: (1) a constant pitch,
variable torque and (2) a variable-pitch, constant torque. Other, less important from a control viewpoint, processes include start-up
and shutdown and transition from phase (1) to phase (2). These phases are traditionally termed regions of operation in wind
turbine design, as shown in Figure 3. Figure 20 shows a generic control block diagram of the whole process, depicting the most
important control loops.
In variable-speed machines, the generator is connected to the grid by a power electronics system. For synchronous generators and
for induction generators without slip rings, this system is connected between the stator of the generator and the grid, where the total
Wind
speed
disturbance
v

vrated
Pitch
controller

ωr = ωrated

β

–

Turbine
+
Generator

ω

Turbine

vcut-in
vv>vrated

ωr = λ*/vR
βr = −1°

ωr = 0

Control supervisor
Figure 20 Overall wind turbine control block diagram.

–

Generator vd
controller

Tm

Te
Generator
–

ω


Wind Turbine Control Systems and Power Electronics

345

power production must be fed through it. For induction generators with slip rings, the stator of the generator is connected to the grid
directly and the rotor of the generator is connected to the grid by an electronic inverter. This has the advantage that only a part of the
power production is fed through the inverter; therefore, the nominal power of the inverter system can be less than the nominal
power of the wind turbine.
There are two different types of inverter systems: grid commutated and self-commutated. The grid commutated inverters are
mainly thyristor inverters, for example, 6 or 12 pulse. This type of inverter produces integer harmonics like the 5th, 7th, and so on
(frequencies of 250 Hz, 350 Hz, and so on), which in general must be reduced by harmonic filters. On the other hand, thyristor
inverters are not able to control the reactive power. Self-commutated inverter systems are mainly pulse width modulated, where
IGBTs are used. This type of inverter has the advantage that in addition to the control of active power, reactive power is also
controllable. One disadvantage is the generation of interharmonics, in the range of some kHz, necessitating filters to reduce their
effect. In modern wind turbines, transistor-based inverter systems are usually used.
Figure 21 [12] shows a schematic of the wind turbine control processes where the main elements (inputs, outputs, and controls)
are clarified (modified from Reference [13]).
Figure 21 depicts a doubly fed generator configuration whose rotor windings are connected to the grid by a partial power
converter, but it is nevertheless quite representative of all configurations. As seen, there are two main control blocks, a mechanical
(wind turbine) and an electrical (generator and grid). As explained these blocks are quite decoupled. Let us look at the signals
involved in some detail.
Reference signals. The reference signals influencing the control system operation are the requirements of the grid, at some point M, as
Ã

.
expressed by the quantities Pgà (active power set point) and QÃg (reactive power set point) and the converter’s DC voltage VDC
The active power set point may influence both the mechanical and electrical converter, but is usually set at the rated power of the
turbine. Special circumstances may dictate a value less than the rated, for example, in cases where maximum power cannot be
absorbed by the grid.
Reactive power usually influences only the grid-side portion of the converter, since rotor and grid control are also decoupled by
suitable transformations. Its value is usually zero unless the turbine operates in a weak grid or a grid fault situation demands grid
voltage support from the wind park.
The converter’s reference DC voltage depends on the size of the converter, the stator/rotor voltage ratio, and the modulation factor of
the power converter.
Control signals. The control mechanism generates two control signals: pitch angle β and converter modulation signals pulse width
modulated based on the desired generator power Peà . These control signals affect essentially the rotational speed in two distinct
ways depending on the region where the turbine operates.
Disturbance signals. The main disturbance signal is wind speed v. However, in this case, wind speed is a ‘measurable’ disturbance, a
fact that is utilized in the control algorithms in order to generate the control signals. Even though most current wind sensors do
not provide accurate measurements, the employment of new, laser-based equipment will surely correct this deficiency.
Measurement signals. In order to keep controlled variables in their reference limits, a number of measurements must be made (apart from
wind speed). Although specific measurements depend on machine configuration and control strategy, it can safely be stated that
generated active and reactive power, PG and QG, DC-link voltage, VDC, generator speed, ωm, and rotor current, iAC, are usually measured.

2.11.3.2
2.11.3.2.1

Pitch Control
Collective pitch control

Pitch control is used in both fixed- and variable-speed machines, serving similar purposes. In a fixed-speed pitch-regulated turbine,
an induction generator is connected directly to the AC network and rotates at a nearly constant speed. In a variable-speed turbine,
pitch control comes into play in operating Region 3, where again the electrical power is at rated levels.
As the wind speed varies, the power produced will vary roughly as the cube of the wind speed. At the rated wind speed, the

electrical power generated becomes equal to the rating of the turbine. As the wind speed rises above the rated wind speed, the blades
are pitched in order to reduce the aerodynamic efficiency of the rotor and limit the power to the rated value. The usual strategy is to
pitch the blades in response to a set-point error, such as the power error or rotational speed error, defined as the difference between
the rated value and the actual value achieved, as measured by a transducer.
Most pitch controllers use the simple linearized model in eqn [9], which is reproduced here:
~ ðsÞ ¼
ω

η
a
~ ðsÞ
β~ ðsÞ þ
w
ðsIT − γÞ
ðsIT − γ Þ

½9Š

The control block diagram for this problem is shown in Figure 22.
An example of synthesizing such a proportional integral (PI) pitch controller is detailed in Reference [14]. The turbine data refer
to NREL’s CART design. The goal of this controller is to regulate rotational speed to 41.7 rpm (rated speed for the CART).
Since this is a typical disturbance attenuation problem, the disturbance to output transfer function of Figure 22 is found by
plugging in the controller transfer function into eqn [9],


M
Reference values
Measurement signals
Control signals
Disturbance signals

N

AC

DC
DC
T

PWMr

β

AC
PWMg

VDC
iAC
Generator control level

v
Rotor side
controller

Grid side
controller

ir
Pe*

Qg*

Pg
Qg

*
VDC

ωm
Wind turbine control level

Power limitation
controller

Power
maximization
controller

Pg

Grid
operator
Pg*

Figure 21 Schematic of wind turbine control. Reproduced from Hansen A, Jauch C, Sørensen P, et al. (2003) Dynamic wind turbine models in power system simulation tool DIgSILENT. Report Risø-R-1400(EN).
Roskilde, Denmark: Risø National Laboratory [12].


Wind Turbine Control Systems and Power Electronics

347

Wind
disturbance
w

α

Kp +

Ki

s

~
β (rad)

η

1
sIT − γ

~
ω (rps)

Figure 22 Pitch control block diagram.

~ ðsÞ
ω
T ðsÞ ≡
¼

~ ðsÞ
w

α
α
s
αs
sIT − γ
I
T

 ¼ 2
¼
ðγ þ ηKP Þ
η
η
KP s þ KI
s IT − ðγ þ ηKP Þs − ζ KI
s2 −
s − KI
1−
IT
IT
sIT − γ
s

½47Š

To proceed let us substitute for the constants γ/IT = −0.194, η/IT = −2.65, and α/IT = 0.069 (for linearization about w0 = 18 m s−1,
ω0 = 41.7 rpm, θ0 = 11°). With these values, the denominator of eqn [47] becomes

s2 þ ð0:194 þ 2:65KP Þs þ 2:65KI ¼ 0
A simple way to choose the controller parameters is to relate them to the standard second-order damping ratio ζ and the natural
frequency coefficient ωn. This means
ω2 ΙΤ
2ζ ωn Ι Τ − γ
KI ¼ − n ; KP ¼ −
½48Š
η
η
−1
Usually the damping ratio is in the range 0.6–0.8 for underdamped response and ωn ≈ 0.6 rad s for acceptable attenuation
time. In Figure 23 are shown responses for ωn = 0.6 rad s−1 and ζ = 0.3 (red), 1 (blue), and 2 (green) for a wind unit step disturbance
(from 17 to 18 m s−1). Calculated controller gains are KI = 0.136 and KP = 0.38.
Although simple, this controller suffers from poor performance in different operating points, that is, different from the ones that
it is designed for. This is illustrated in Figure 24 for ζ = 1, where a much greater overshoot is observed (red line) compared to the one
previously attained (blue line).
This problem can be overcome by an approach termed ‘gain scheduling’. In this method, the overall controller gain is not
constant but varies depending on the actual operating point [13, 15]. The relationship for the multiplying factor is
1

gK ðβi Þ ¼ 
β
1þ i
βK
where βK is a pitch angle chosen as follows.
42.6
Rotor speed (rpm)

42.4
42.2

42.0
41.8
41.6
41.4
41.2
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
Time (s)
Figure 23 Pitch control unit disturbance step responses. Reproduced from Wright AD and Fingersh LJ (2008) Advanced control design for wind
turbines part I: Control design, implementation, and initial tests. Technical Report NREL/TP-500-42437. Golden, CO: National Renewable Energy
Laboratory [14].

½49Š


348

Wind Turbine Control Systems and Power Electronics

42.6
Rotor speed (rpm)

42.4
42.2
42.0
41.8
41.6
41.4
40

42

44

46

48

50
52
Time (s)

54

56

58

60

Figure 24 Pitch control at different operating points. Reproduced from Wright AD and Fingersh LJ (2008) Advanced control design for wind turbines
part I: Control design, implementation, and initial tests. Technical Report NREL/TP-500-42437. Golden, CO: National Renewable Energy Laboratory [14].

An operating point in the beginning of Region 3 is chosen. For the turbine examined, this may be w0 = 13.7 m s−1, ω0 = 41.7 rpm,
and β0 = 0.53°. At this operating point, the controller gains are calculated as KI = 0.78 and KP = 2.35. Next an operating point is found
at which η is doubled. Simulation studies have shown that around w0 = 14.1 m s−1, ω0 = 41.7 rpm, and β0 = 2.62°, η is doubled. Thus,
1

gK ðβi Þ ¼ 
β
1þ i

2:62

½50Š

Figure 25 shows the revised controller in actual implementation. Figure 26 shows the improved performance of this controller at
different operating points. An alternative method for computing the parameters for use in the above expression is described in
Reference [15] based on a best-fit least-squares estimate of the pitch sensitivity for various blade pitch angles.
This controller is in operation on all wind regimes but takes effect only when winds approach regime 3. When the controller
starts operation in this way, a phenomenon known as ‘integrator windup’ is observed. This results in a momentary large overshoot
in the rotor speed and a corresponding long settling time. This problem can be treated by the addition of an additional gain term in
the integrator whose value can be found through trial and error. The modified diagram is shown in Figure 27. The results are
depicted in Figure 28, where the benefits of the anti-windup mechanism are clearly observed.

2.11.3.2.2

Individual pitch control

For IPC, eqn [12] is used. The capability of individually regulating blade pitch allows us to add to the standard control objective of
speed regulation the following:
•
•
•
•

first drive-train torsional mode stabilization;
enhanced damping of the tower first fore–aft mode;
stabilization of rotor first symmetric and asymmetric flap mode; and
attenuation of blade flap wind shear response.

Control action is usually calculated as a function of states or measurements, that is,

uðtÞ ¼ KðtÞxðtÞ or uðtÞ ¼ KðtÞyðtÞ

½51Š

Wind
disturbance
~
β

1
1+

β

2.348 +

0.78
s

2.62

β0

Figure 25 Gain scheduling pitch controller.

~
ω (rps)

(rad)
Turbine

+


Wind Turbine Control Systems and Power Electronics

349

Rotor speed (rpm)

42.4
Gust2
Gust1

42.2
42.0
41.8
41.6
41.4
40

42

44

46

48

50

52

54

56

58

60

Time (s)
Figure 26 Gain scheduler performance. Reproduced from Wright AD and Fingersh LJ (2008) Advanced control design for wind turbines part I: Control
design, implementation, and initial tests. Technical Report NREL/TP-500-42437. Golden, CO: National Renewable Energy Laboratory [14].

Wind
disturbance
2.348

1

~
β

β
1+
2.62

(rad)

w

Pitch saturator

~
ω (rps)
Turbine

–

0.78
s

β0

1
1+

β
2.62

Kaw

Figure 27 Anti-windup, gain scheduling pitch controller.

Rotor speed no anti-windup
Rotor speed with anti-windup
Blade pitch no anti-windup
Blade pitch with anti-windup

80

38
33

60
50

28
23

40

18

30

13

20

8

10

3

0

Blade pitch (deg)

Rotor speed (rpm)

70

–2
30

40

50
Time (s)

60

70

Figure 28 Anti-windup results. Reproduced from Wright AD and Fingersh LJ (2008) Advanced control design for wind turbines part I: Control design,
implementation, and initial tests. Technical Report NREL/TP-500-42437. Golden, CO: National Renewable Energy Laboratory [14].

There are a variety of methods for calculating the time-varying feedback matrix. A rather simple, yet quite powerful technique is
the well-known linear quadratic regulator (LQR) method. In this method, K(t) is calculated as the (constant) matrix that
minimizes the infinite horizon criterion,
∞È
É
J ¼ ∫ xT ðtÞQxðtÞ þ uT ðtÞRuðtÞ dt
0

½52Š

350

Wind Turbine Control Systems and Power Electronics

subject to the plant dynamics given by eqn [12]. The matrices Q and R are design matrices used for weighting the importance of
control effort and state departure from equilibrium. What actually eqn [52] means is that its minimization will result in
minimization of the relevant ‘deflection’ angles and consequently of the loads experienced by the blades and tower.
Before proceeding with the details of solving eqn [52], let us address two other problems that are inherent in our formulation:
the problem of the wind disturbance and the problem of state estimation.
• Wind disturbance
Wind can be thought of as a highly stochastic disturbance input. Disturbance rejection or attenuation is the primary goal of any
control system. How one achieves this goal depends on whether the disturbance can be measured, forecasted, or is completely
unpredictable. In the following, one such method will be outlined, namely, the disturbance accommodation control (DAC) [16]. In
this method, the disturbance is modeled as a dynamic system of known structure, such as
z_ d ðtÞ ¼ Fzd ðtÞ; zd ð0Þ ¼ z0d ðunknownÞ

½53Š

This model is then ‘augmented’ with the original system to form the plant-disturbance model given by
!
!
!
!
_
A Bd
xðtÞ
B
xðtÞ
¼
þ

uðtÞ
0 F
zd ðtÞ
0
z_ d ðtÞ
!
xðtÞ
yðtÞ ¼ ½ C 0 Š
zd ðtÞ

½54Š

Furthermore, if the feedback control law is calculated as
uðtÞ ¼ GxðtÞ þ Gd zd ðtÞ

½55Š

and eqn [55] is substituted in eqn [54], we get
!
!
!
!
_
A Bd
xðtÞ
B
A þ BG
xðtÞ
þ
¼

fGxðtÞ þ Gd zd ðtÞg¼
z_ d ðtÞ
0 F
0
0
zd ðtÞ

Bd þ BGd
F

!

xðtÞ
zd ðtÞ

!
½56Š

• State estimation
A usual problem associated with eqn [55] is that the full state x and disturbance zd are not measured; therefore, they must be
retrieved (estimated) from the available measurement vector y. The standard procedure is the use of observers, either in their ‘full’ or
‘reduced’ version. The full version is given by
^x_ ðtÞ ¼ A^x ðtÞ þ BuðtÞ þ KðyðtÞ − ^y ðtÞÞ; ^x ð0Þ ¼ 0
^y
ðtÞ ¼ C^x ðtÞ

½57Š

As seen, eqn [57] uses the same inputs and outputs as the actual system to generate the state estimates. The gain matrix K determines
how fast the estimate will converge to the true value, since it can be proved that the estimation error’s equation is

e_
ðtÞ ¼ ðA − KCÞeðtÞ ⇒ eðtÞ ¼ e ðA − KC Þ t eð0Þ
Thus K can be chosen by either specifying its eigenvalue set or solving an LQR problem.
Now let us use eqn [57] to design a full-state estimator for the augmented system (eqn [56]). We get
!
!
!
!
^x ðtÞ
B
A Bd
^x_ ðt Þ
uðt Þ þ K ðyðt Þ − ^y ðt ÞÞ
þ
¼
^z d ðt Þ
0 F
0
^z_ d ðtÞ
!
^x ðtÞ
^y ðtÞ ¼ ½ C 0 Š
^z d ðt Þ

½58Š

½59Š

and eqn [55] becomes
^ þ Gd^z d ðtÞ

uðtÞ ¼ GxðtÞ

½60Š

Gain matrices G and K exist if certain conditions on original matrices A, B, and C are fulfilled [17]. Coefficient matrix Gd can be
chosen to minimize.
‖BGd þBd ‖

½61Š
−1

Now, for the problem at hand, the linearization is performed around w0 = 18 m s , θ0 = 12°, and ω0 = 42 rpm, which are rated
values for the CART turbine. Measurement matrix C is
2

0:5
C¼4 0
0

0 −0:5
0
0
0
0

3
0 0 0 0 0 0
0 0 0 1 0 05
0 0 0 0 0 0

½62Š


Wind Turbine Control Systems and Power Electronics

351

where it is assumed that perturbed generator speed, tower-top fore–aft deflection, and the rotor first asymmetric mode deflection
are measured.
The rest of the system matrices are evaluated using standard linearization routines. Wind disturbance is modeled as
2

0
z_ d ¼ 4 −ω2
0

3
1 0
0 05
0 0

½63Š

With this formulation, the disturbance vector is
ud ¼

!
1 0 0
z
0 0 1 d

½64Š

Note that in this way the first two states of eqn [63] model the azimuth-dependent wind component, while the third state models a
uniform step wind disturbance.
Design feedback matrix G in eqn [60] was found through eqn [52], with
2

0
60
6
60
6
60
6
60
Q¼6
60
6
60
6
40
0

0
0
0
0
0
0

0
0
0

0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0

0
0
0
0
0
0

0
0
0
0:0001
1 Â 10 −12
0
0
0
0
0
0
0

0 0
0 0
0 0
0 0
0 0
0 0
1 0
0 10
0 0

3
0
0 7
7
0 7
7
7

0 7
1
0 7
7; R ¼ 0
0 7
7
0 7
7
0 5
0:1

0
1

!
½65Š

It has to be remembered that these weight matrices balance orders of magnitudes of variables involved as well as weighting their
relative importance. Design estimator matrix K in eqn [59] is similarly calculated through a suitable LQR formulation with
0
B
B
B
B
B
B
B
B
B
Qe ¼ B

B
B
B
B
B
B
B
B
@

1

500

C
C
C
C
C
C
C
0
C
1
C
C; Re ¼ @
C
C
C
C

C
C
C
C
A

50 000
500
50 000
120
800
220
20
2500
105
106

1
A

1

½66Š

1

105
Note that Qe is 12 Â 12 to accommodate the nine system states plus the three disturbance states.
The resulting system was simulated in the FAST and ADAMS simulation environments using a turbulent wind inflow. The results
are shown in Figures 29 and 30.

As seen, generator speed is kept well within the set point of 42 rpm, while blade pitch slightly exceeds its mechanical limits. One
way to alleviate this could be to use generator torque as an additional control input.
The described multivariable linear quadratic Gaussian (LQG) approach may seem quite complicated, while at the same time
requiring considerable computer power to work online. These problems have led to an alternative formulation, leading to an
‘almost decoupled’ system, on the lines of generator control.
This d–q axis representation transforms the three blade root load signals into a mean value and variations about two orthogonal
axes (the ‘direct’ and ‘quadrature’ axes). Recently, it has been shown that it is possible to treat the d- and q-axis as being almost
independent. This means that conventional classical design techniques can be applied to generate a SISO controller that can be
applied separately to the d- and q-axis. A conventional PI controller in series with a simple filter provides very satisfactory control
action. In practice, there is some interaction between the two axes, but this can be accounted for by introducing a simple azimuthal
phase shift into the d–q axis transformation, that is, adding a constant offset to the rotor azimuth angle used in the transformation.
The direct transformation is given by [18]

βd
βq

!

2
26
¼ 6
34

cos θ
sin θ



2π
cos θ þ

3


2π
sin θ þ
3

3

4π
2 3
β1
cos θ þ
3 7
74
 5 β2 5

4π
sin θ þ
β3
3

½67Š


352

Wind Turbine Control Systems and Power Electronics

Generator speed (rpm)

45

ADAMS
FAST

44
43
42
41
40
60

20

100

140

180

220

260

300

Time (s)

Blade 2 pitch rate (deg s–1)

Figure 29 Generator speed response. Reproduced from Wright AD (2004) Modern control design for flexible wind turbines. NREL Report
No. TP-500-35816. Golden, CO: National Renewable Energy Laboratory [2].

50

ADAMS

40

FAST

30
20
10
Pitch rate
limit

0
–10
–20
–30
–40
20

60

100

140

180
Time (s)

220

260

300

Figure 30 Control effort: pitch rate. Reproduced from Wright AD (2004) Modern control design for flexible wind turbines. NREL Report
No. TP-500-35816. Golden, CO: National Renewable Energy Laboratory [2].

while the inverse relations are
2

3
cos θ
sin θ




6
" #
2π 7
β1
6 cos θ þ 2π
7 βd
sin θ þ
6

4 β2 5 ¼ 6
3
3 7
7
6



7 β
β3
4
4π
4π 5 q
cos θ þ
sin θ þ
3
3

2

3

½68Š

where θ is the angle of blade 1 and the direct axis formation. The inverse transformation [68] is used to generate the individual pitch
demand increments for the three blades from the d–q axis pitch demands generated by the LQG algorithm.

2.11.3.3

Generator Control

Generator control is active in Region 2 so as to speed up the turbine rotor and bring it closer to its optimal value (Figure 20). It could
also be used in Region 3, whenever pitch angle saturates, in order to slow down the rotor speed.
Electrical generators are systems that are generally controlled by power electronics. Hence, controlled electrical generators are
systems whose inputs are stator and rotor voltages, having as state variables the stator and rotor currents or fluxes [19]. They are
composed of an electromagnetic subsystem, which outputs the electromagnetic torque decelerating the generator rotor coupled to
the turbine rotor.
As explained earlier, control of wind turbine AC generators falls into two main categories. The first comprises synchronous and
induction generators without slip rings, controlled by full power converters, whereas the second refers to induction generators with
slip rings that are controlled by partial power converters.


Wind Turbine Control Systems and Power Electronics

2.11.3.3.1

353

Wound rotor doubly fed induction generator control

The stator of the DFIG is directly connected to the grid while the rotor is connected through a power electronics converter
(Figure 31). The aim of the controller is to regulate stator active and reactive power through a variable-frequency and
variable-magnitude rotor voltage supplied to the slip rings.
The power converter consists, as usual, of two converters, that is, a machine-side converter and a grid-side converter. A DC-link
capacitor is placed between them, as energy storage, in order to keep the voltage variations (or ripple) in the DC-link voltage small.
With the machine-side converter it is possible to control the torque or the speed of the DFIG and also the power factor at the stator
terminals, while the main objective of the grid-side converter is to keep the DC-link voltage constant.
The DFIG is described by the differential equations [41]. In order to design efficient controllers, it is useful, as mentioned
previously, to transform these equations in a suitable decoupled form. This is achieved by using a (d, q) reference frame that is
synchronized with the stator flux, that is, the d-axis is aligned with the stator flux vector. Consequently,

ψ ds ¼ ψ s ;

ψ qs ¼ 0

while the electromagnetic torque is given by
Te ¼ −p

Lm
iqr ψ s
Ls

½69Š

The dynamics of stator flux is controlled only by the stator voltage. Stator voltage is imposed by the network; stator flux is
established very quickly. We can thus admit the following simplifying relation and consider that the flow of the stator evolves in
a static way. Thus, according to the reference frame carried by the stator vector voltage and neglecting the effect of stator resistance,
we have the following relationships:
ψ ds ¼ Vs ¼ Vqs ¼ V ðgrid voltageÞ
Vds ¼ψ qs ð¼0Þ
Furthermore, stator active and reactive power are now given by
Vs ðt ÞLm
iqr ðt Þ
Ls

½70Š

Vs2 ðt Þ Vs ðt ÞLm
idr
−
Ls

Ls

½71Š

Ps ðtÞ ¼ −
Qs ðtÞ ¼
while



L2
Vs ðt ÞLm
Lr − m idr ðt Þ þ
Ls
ωs Ls


L2m
ψ qr ðtÞ ¼ Lr −
iqr ðtÞ
Ls

ψ dr ðtÞ ¼

½72Š

Inspection of eqns [69]–[71] reveals that, as designed, stator reactive power depends on idr, while stator active power depends on
only iqr. Then eqn [26] reduces to
2
pω 3

!
0
t
i
ð
Þ
7
6
dr
σ
þ
5
4
Lm
RR
iqr ðt Þ
−
−
σLS LR
σLR

2
RR
!
−
_i dr ðtÞ
6
σL
R
¼4

pω
i_ qr ðt Þ
−ωS þ
σ

ωS −

1
σLR
0

32
3
Vs ðt Þ
74
5
1 5 Vdr ðt Þ
Vqr ðt Þ
σLR
0

½73Š

If eqn [73] is expanded,

RR
pω 
1
iqr ðtÞ þ
Vdr ðt Þ

idr ðtÞ þ ωS −
σ
σ LR
σ LR

1
Lm
pω 
RR
i_ qr ðtÞ ¼ − ωS þ
idr ðt Þ −
iqr ðt Þ þ
Vqr ðt Þ −
Vs ðt Þ
σLS LR
σ
σLR
σLR
_i dr ðtÞ ¼ −

V, f
S

R

Figure 31 Schematic of doubly fed induction generator wind turbine control.

Partial
power
converter

Grid

½74Š