AckermannTh. (ed) Wind Power in Power Systems

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modulation (Heier, 1998). The last condition is not fulfil led during grid faults. However,

when a fault occurs, variable-speed wind turbines are promptly disconnected to protect

the power electronic converter. Further, there are very-high-frequency phenomena when

the power elect ronic converter responds to a voltage drop. PSDSs are unsuitable for

investigating this topic. Therefore, the model incorporates a low-frequency representa-

tion of the behaviour of the converter during faults, similar to high-voltage direct-

current (HVDC) converters (SPTI, 1997).

The converters are represented as current sources, and therefore the current set

points equal the rotor currents. The current set points are derived from the set points

for active and reactive power. The active power set point is generated by the rotor

speed controller, based on the actual rotor speed value. The reactive power set point

is generated by the terminal voltage or power factor controller, based on the actual

value of the terminal voltage or the power factor. Both controllers will be discussed

below.

When the stator resistance is neglected and it is assumed that the d-axis coincides

with the maximum of the stator flux, the electrical torque is dependent on the quad-

rature component of the rotor current (Heier, 1998). If u

equals u

, from the

assumption that the d-axis coincides with the maximum of the stator flux, then it

can be derived from Equations (25.13) and (25.16) that the following relation between

and T

holds.

ðL

s

þ L

: ð25:18Þ

where u

is the terminal voltage.

The reactive power exchanged with the grid at the stator terminals is dependent on the

direct component of the rotor current. From Equations (25.16) and (25.17), neglecting

the stator resistance and assuming that the d-axis coincides with the maximum of the

stator flux, it can be shown that

¼

s

þ L



ðL

sþL

: ð25:19Þ

However, the total reactive power exchanged with the grid depends not only on the

control of the generator but also on the control of the grid side of the converter feeding

the rotor winding. The following equations apply to the con verter:

¼ u

þ u

;

¼ u

 u

;

)

ð25:20Þ

in which the subscript c stands for converter. In this equation, P

is equal to the rotor

power of the doubly fed induction generator P

given in Equation (25.19). P

may be

multiplied with the converter efficiency if the converter losses are to be included. The

reactive power exchanged with the grid equals the sum of Q

from Equation (25.19) and

from Equations (25.20). Q

depends on the control strategy and the converter rating

but often equals zero, which means that the grid side of the converter operates at unity

power factor (see Chapter 19).

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25.6.5 Protection system model

The goal of the protection system is to protect the wind turbine from damage caused by

the high currents that can occur when the terminal voltage drops as a result of a short

circuit in the grid. It also has the task of preventing islanding. Islanding is a situation in

which a part of the system continues to be energised by distributed generators, such as

wind turbines, after the system is disconnected from the main system. This situation

should be prevented because it can lead to large deviations of voltage and frequency

from their nominal values, resulting in damage to grid components and loads. It can

also pose a serious threat to maintenance staff, who incorrectly assumes that the system

is de-energised after the disconnection from the transmission system.

The thermal time constants of semiconductor components are very short. Therefore,

the converter that feeds the rotor winding is easily damaged by fault currents, and

overcurrent protection is essential for the converter. The generator itself is more capable

of withstanding fault currents and is therefore less critical. The working principle of

overcurrent protection is as follows. When a voltage drop occurs, the rotor current

quickly increases. This is ‘noticed’ by the controller of the rotor side of the power

electronics converter. Then the rotor windings are shorted using a ‘crow bar’ (basically

turning the generator into a squirr el cage induction generator). A circuit braker between

the stator and the grid is operated and the wind turbine is completely disconnected. The

grid side of the converter can also ‘notice’ the volta ge drop and the corresponding

current increase, depending on the converter controls. After the voltage is restored, the

wind turbine is reconnected to the system.

The anti-islanding protection of the wind turbine acts in response to voltage and/or

frequency deviations or phase angle jumps. The grid side of the converter measures the

grid voltage with a high sampling frequency. There are criteria implemented in the

protection system for determining whether an island exists. If these criteria are met, the

wind turbine is disconnected. The criteria that are applied are a trade-off between the

risk of letting an island go undetected, on the one hand, and of incorrectly detecting an

island when there is none, leading to ‘nuisance tripping’, on the other.

The response of a doubly fed induction generator to a terminal voltage drop is a high-

frequency phenomenon. It cannot be modelled adequately with PSDS software. This

becomes clear when we look at the assumptions on which these programs are based (see

Section 25.2 and Chapter 24). Hence, if the protection system that is incorporated into a

doubly fed induction generator model is used in PSDS it will react to terminal voltage and

not to rotor current. It is thus a simplified representation of the actual protection system.

There are simulations in the literature that show the different responses of reduced-

order and complete models (Akhmatov, 2002). However, presently there are no quanti-

tative investigations that analyse the importance of the differences between the reduced-

order model and the complete mod el for the interaction with the system. The latter is,

however, the main point of interest in PSDS, so this topic requires further investigation.

Further, it must be noted that the protection system of a model of a doubly fed

induction generator that will be used in PSDSs can incorporate criteria only with respect

to the amplitude and frequency of the terminal voltage. As only the effective value of

fundamental harmonic components is studied, it will not be possible to detect phase

jumps.

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In addition to criteria for detecting whether the wi nd turbine should be disconnected,

the protection system also includes criteria for determining whether the wind turbine

can be reconnected, as well as a reconnection stra tegy. If a protection system of a model

of a doubly fed induction generator is used in PSDS, it consists of the following

elements:

a set of criteria for deciding when to disconnect the wind turbine;

a set of criteria regarding terminal voltage and frequency for determining when to

reconnect the wind turbine;

a reconnection strategy (e.g. a ramp rate at which the power is restored to the value

determined by the actual wind speed).

25.6.6 Rotor speed controller model

The speed controller of a variable-speed wind turbine operates as follows:

The actual rotor speed is measured with a sample frequency f

(Hz). The sample

frequency is in the order of 20 Hz.

From this value, a set point for the generated power is derived, using the characteristic

for the relationship between rotor speed and power.

Taking into account the actual generator speed, a torque set point is derived from the

power set point.

A current set point is derived from the torque set point, using Equation (25.18).

In a reduced-order model, this current set point is reached immediately as a result of the

modelling approach. In practice, the current set point will be used as input to the current

control loops and it will take a certain time to reach the desired value of the current, as

will be discussed in Chapter 26. However, the time necessary to reach the new current is

significantly below the investigated bandwidth of 10 Hz.

We will use the characteristic for the relationship between rotor speed and generator

power to arrive at a set point for generated real power. In most cases, the rotor speed is

controlled to achieve optimal energy capture, although sometimes other goals may be

pursued, pa rticularly noise minimisation. The solid line in Figure 25.8 depicts the

relation between rotor speed and power for optimal energy capture. At low wind speeds,

the rotor speed is kept at its minimum by adjusting the generator torque. At medium

wind speeds, the rotor speed varies proportionally to the wind speed in order to keep the

tip speed ratio, , at its optimum value. When the rotor speed reaches its nominal value,

the generator power is kept at its nominal value as well.

Controlling the power according to this speed – power characteristic, however, causes

some problems:

The desired power is not uniquely defined at nominal and minimal rotor speed.

If the rotor speed decreases from slightly above nominal speed to slightly below

nominal speed, or from slightly above minimal speed to slightly below minimal speed,

the change in generated power is very large.

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To solve these problems, we use a control characteristic that is similar to those that lead

to optimal energy capture. The dotted line in Figure 25.8 depicts this control characteristic.

Sometimes, this problem is solved by applying more advanced con troller types, such as

integral co ntrollers or hysteresis loops (Bossanyi, 2000). The location of the points at

which the implement ed control characteristic deviates from the control characteristic

leading to optimal energy capture is a design choice. If these points lie near the minimal

and nominal rotor speed, the maximum amount of energy is extracted from the wind

over a wide range of wind speeds, but rotor speed changes near the minimum and

nominal rotor speed result in large power fluctuations. If these points lie further from

the minimal and nominal rotor speed, the wind speed range in which energy capture is

maximal is narrowed, but the power fluctuations near minimal and nominal rotor speed

are smaller.

25.6.7 Pitch angle controller model

The pitch angle controller is active only in high wind speeds. In such circumstances, the

rotor speed can no longer be controlled by increasing the generated power, as this would

lead to overloading the generator and/or the converter. Therefore the blade pitch angle

is changed in order to limit the aerodynamic efficiency of the rotor. This prevents the

rotor speed from becoming too high, which would result in mechanical damage. The

optimal pitch angle is approximately zero below the nominal wind speed. From the

nominal wind speed onwards, the optimal angle increases steadily with increasing wind

speed, as can be seen in Figure 25.7. Equations (25.6) and (25.7) are used to calculate the

impact of the pitch angle, , on the performance coefficient. The resul ting value can be

inserted in Equation (25.5) in order to calculate the mechanical power extracted from

the wind.

It should be taken into accou nt that the pitch angle cannot change immediately, but

only at a finite rate, which may be quite low because of the size of the rotor blades of

modern wind turbines. Blade drives are usually as small as possible in order to save

0.8

1.0

0.6

0.4

0.2

0.4 0.6 0.7 0.8 0.90.5

Power setpoint (p.u.)

1.0 1.21.1

Rotor speed (p.u.)

Figure 25.8 Optimal (solid curve) and practical (dotted curve) rotor speed-power characteristic

of a typical variable-speed wind turbine

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money. The maximum rate of change of the pitch angle is in the order of 3–10 degrees

per second, depending on the size of the wind turbine. As the blade pitch angle can

change only slowly, the pitch angle controller works with a sample frequency f

, which

is in the order of 1–3 Hz.

Figure 25.9 depicts the pitch angle controller. This controller is a proportional (P)

controller. Using this controller type implies that the rotor speed is allowed to exceed its

nominal value by an amount that depends on the value chosen for the constant K

Nevertheless, we use a proportional controller because:

a slight overspeeding of the rotor above its nominal value can be tolerated and does

not pose any problems to the wind turbine construction;

the system is never in steady state because of the varying wind speed. The advantage

of an integral controller, which can achieve zero steady state error, would therefo re be

hardly noticeable.

25.6.8 Terminal voltage controller model

A variable-speed wind turbine with a doubly fed induction generator is theoretically

able to participate in terminal voltage control, as discussed in Chapter 19. Equation

(25.19) shows that the reactive power exchanged with the grid can be controlled,

provided that the current rating of the power electronic converter is sufficiently high

to circulate reactive current, even at nominal active current.

The first term on the right-hand side of Equation (25.19) determines the net reactive

power exchange with the grid, which can be controlled by changing the direct compon-

ent of the rotor current, i

. The second term represents the magnetisation of the stator.

Equation (25.19) can be rewritten in the following way

¼

ði

dr; magn

þ i

dr; gen

s

þ L



ðL

s

þ L

; ð25:21Þ

in which i

has been split in a part magnetizing the generation (i

;

magn

) and a part

generating reactive power (i

;

gen

It can be seen that i

dr, magn

, the rotor current required to magnetise the generator

itself, is given by:

Rotor

speed

(p.u.)

Maximum

rotor speed

(p.u.)

Pitch

angle

set point

(deg) (deg)

Pitch

angle

Rate

limiter

(deg/s)

–

Figure 25.9 Pitch angle controller model. Note: K

is a constant; f

is the sample frequency of

the pitch angle controller

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dr; magn

¼

: ð25:22Þ

The net reactive power exchange between the stator and the grid is then equal to

¼

dr; gen

s

þ L

: ð25:23Þ

Figure 25.10 depicts a terminal voltage controller for a doubly fed induction generator.

The value of K

determines the steady state error and the speed of response. This value

should not be set too high, because this leads to stability problems with the controller.

Figure 25.10 is, howeve r, only one example of a voltage controller for a wind turbine

with a doubly fed inducti on generator. There are also other possible controller topo-

logies. However, any voltage controller will correspond to the basic principle that the

terminal voltage is controlled by influencing the reactive power exchange with the grid.

If the value of K

is changed to zero, it results in a controller that keeps the power

factor equal to one. Currently, this is the dominant mode of operation of wind turbines

with doubly fed induction generators, because it requires only small converters and

reduces the risk of islanding. Islanding requires a balance between both active and

reactive power consumed. If the wind turbines are prevented from generating any

reactive power this situation will hardly occur.

25.7 Model of a Direct drive Wind Turbine

Figure 25.11 depicts the general structure of a model of a variable-speed wind turbine

with a direct-drive synchronous generator (Type D). Only the generator model and the

voltage controller model are discussed below. The wind speed model is identical to that

in the case of a constant-speed wind turbine model, described in Section 25.5.2. The

rotor model as well as the rotor speed and pitch angle controllers are identical to those

used in the doubly fed induction generator and were described in Sections 25.6.6 and

25.6.7, respectively. The converter and the protection system of a wind turbine with a

Terminal

voltage

(p.u.)

Voltage

reference

(p.u.)

Set point

for i

dr,gen

Set point

for i

dr,magn

(p.u.)

K v

–u t

–

Figure 25.10 Voltage controller model for a wind turbine with a doubly fed induction generator

(Type C). Note: K

is the voltage controller constant; u

is the terminal voltage; !

is the angular

frequency of the stator; L

is the mutual inductance; i

is the direct component of the rotor current;

dr,magn

and i

dr,gen

are the currents to magnetise the generator and in the generator respectively

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doubly fed induction generator are different from that of a direct-driv e synchronous

generator. However, because of the simplifying assumptions used in models for dynamic

simulations, the analyses in Sections 25.6.4 and 25.6.5 also hold for the model of the

converter and the protection system of a direct-drive wind turbine. For the reasons

mentioned in Section 25.5.1, we will not discuss the grid model.

25.7.1 Generator model

According to Kundur (1994), the voltage equations of a wound rotor synchron ous

generator in the dq reference frame, taking into account the assumptions in Section

25.4, are:

¼R

 !

;

¼R

þ !

;

¼ R

;

ð25:24Þ

The subscript fd indicates field quantities. Note that for the stator equations the

generator convention is used (i.e. positive currents are outputs). The flux equations are:

¼ðL

þ L

s

Þi

þ L

;

¼ðL

þ L

s

Þi

;

¼ L

;

ð25:25Þ

All quantities in Equations (25.24) and (25.25) are in per unit values.

In the case of a permanent magnet rotor, the expressions for u

and

Equations (25.24) and (25.25) disappear because they refer to field quantities, and

expression for

in Equations (25.25) becomes

Wind

speed

model or

measured

sequence

Wind

speed

Mechanical

power

Rotor

model

Model of

direct

drive

snchro-

nous

generator

Active

and

reactive

power

Stator

currents

Active

power

set point

Reactive

power

set point

Active

and

reactive

power

Voltage

and

frequency

Rotor

speed

Pitch

angle

Pitch

angle

controller

Rotor

speed

controller

Terminal

voltage

controller

Funda-

mental

frequency

grid

model

Converter

and

protection

system

Figure 25.11 General structure of a model of a variable-speed wind turbine with a direct-drive

synchronous generator (Type D)

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¼ðL

þ L

s

Þi

; ð25:26Þ

in which

is the amount of flux of the permanent magnets mounted on the rotor that

is coupled to the stator winding. When neglecting the d /dt terms in the stator voltage

equations, the voltage flux relationships become:

¼R

þ !

ðL

s

þ L

Þi

;

¼R

 !

ðL

s

þ L

Þi

;

¼ R

;

ð25:27Þ

The d /dt terms in the expressions for u

and u

are neglected because the associated

time constants are small, and taking them into account would result in the need to

develop a detailed representation of the power electronic converter. That would include

phenomena that we are not interested in at this point. For a complete model of this wind

turbine type, including the d /dt terms in the stator equations and a full converter

model, see for instance, Chen and Spooner (2001).

The following equation gives the electromechanical torque:



: ð25:28Þ

Based on this equation, the set point for the stator currents can be calculated from a

torque set point generated by the rotor speed controller. The equation of motion is given

by Equation (24.14) (page 545). The active and reactive power of a synchronous

generator are given by:

¼ u

þ u

;

¼ u

 u

)

ð25:29Þ

It has to be emphasised that the generator is fully decoupled from the grid by the power

electronic converter. Therefore, the power factor of the generator does not affect the

reactive power factor at the grid connection. The latter is determined by the grid side of

the converter and not by the operating point of the generator. The expression for Q

Equations (25.29) is therefore of limited interest when studying the grid interaction but

is important when dimensioning the converter. The generator parameters are given in

Table 25.4.

25.7.2 Voltage controller model

The voltage controller applied in a direct-drive wind turbine is different from that in a

wind turbine with a doubly fed induction generator because the generator is fully

decoupled from the grid. It is therefore not the generator that generates active power

and controls the terminal voltage but the power electronic converter. The generated

reactive power is given by the expression for Q

in Equations (25.20). The model of the

voltage controller is depicted in Figure 25.12. It assumes that the term inal voltage u

equal to u

in Equations (25.20). Similar to the terminal voltage controller of a doubly

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fed induction generator, there may be alternative controller topologies here, too. How-

ever, they are all based on the principle that the reactive power exchange with the grid

has to be controlled in order to influence the terminal voltage. To simulate wind

turbines operating at unity power factor, i

must be kept at zero, and the voltage

controller can be removed.

25.8 Model Validation

25.8.1 Measured and simulated model response

In this section, the model’s responses to a measured wind speed sequence will be

analysed and then compared with measurements. The measurements were obtained

from wind turbine manufacturers under a confidentiality agreement. Therefore, all

values except wind speed and pitch angle are in per unit and their base values are not

given. We used Matlab

to obtain the simulation results.

Figure 25.13(a) depicts a measured wind speed sequence. In Figures 25.13(b)–

25.13(d), respectively, the simulated rotor speed, pitch angle if applicable and output

power are depicted for each of the wind turbine types. Figure 25.14(a) shows three

measured wind speed sequences. In Figures 25.14(b) and 25.14(c), respectively, the

measured rotor speed and pitch angle of a variable-speed wind turbine with a doubly

Table 25.4 Simulated synchronous generator parameters

Generator characteristic Value

Number of poles, p 80

Generator speed (rpm) 9–19

Mutual inductance in d-axis, L

(p.u.) 1.21

Mutual inductance in q-axis, L

(p.u.) 0.606

Stator leakage inductance, L

s

(p.u.) 0.121

Stator resistance, R

(p.u.) 0.06

Field inductance, L

(p.u.) 1.33

Field resistance, R

(p.u.) 0.0086

Inertia constant, H

(s) 1.0

Terminal

voltage

(p.u.)

Voltage

reference

(p.u.)

Set point

for i

(p.u.)

–

Figure 25.12 Voltage controller model for wind turbine with direct drive synchronous generator.

Note: K

is the voltage controller constant; i

is the direct component of the converter current;.

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0102030405060

02030405060

0102030405060

Time (s)

Time

(

)

Wind speed (m/s)Rotor speed (p.u)Pitch angle (deg)Output power (p.u.)

0.6

0.8

1.0

1.2

–1

0.92

0.94

0.96

0.98

1.00

1.02

1.04

Constant speed

Direct drive

Doubly fed

Doubly

fed

Doubly fed

Direct

drive

Direct

drive

Direct

drive

Constant speed

(a)

(b)

(c)

(d)

Figure 25.13 Simulation results: (a) measured wind speed sequence; (b) simulated rotor speed;

speed ¼constant-speed wind turbine; Direct drive ¼variable-speed wind turbine with a direct-drive

synchronous generator; Doubly fed ¼variable-speed wind turbine with a doubly fed induction generator

580 Reduced-order Modelling of Wind Turbines