AckermannTh. (ed) Wind Power in Power Systems

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Finally, we will describe the modelling of the turbulence. The turbulence component of

the wind speed is characterised by a power spectral density. Here, the following power

spectral density is used:

ðf Þ¼l



1

1 þ 1:5



5/3

; ð25:4Þ

in which P

is the power de nsity of the turbulence for a certain frequency (W/Hz); f is

the frequency (Hz); h is the height at which the wind speed signal is of interest (m), which

normally equals the height of the wind turbine shaft; v

is the mean wind speed (m/s); l

is the turbulence length scale (m), which equals 20 h if h is less than 30 m, and equals 600

if h is more than 30 m; and z

is the roughness length (m) . Through the parameter z

,we

take into account the dependence of the turbulence intensity on the landscape where the

wind turbine is located. The roughness length depends on the structure of the landscape

surrounding the wind turbine. Table 25.1 gives the values of z

for various landscape

types (Panofsky and Dutton, 1984; Simiu and Scanlan, 1986).

The fact that PSDSs are carried out in the time domain, whereas the turbulence of the

wind is described by a power spectral density given in the frequency domain, raises a

specific problem: the translation of a power spectral density into a time sequence of

values with the given power spectral density. To solve this problem, we use a method

that is described in Shinozuka and Jan (1972). This method works as follows. A power

spectral density can be used to derive information abou t the amplitude of a signal’s

component with a given frequency. Then, a large number of sines with a random initial

phase angle and an amplitude calculated from the power spectral density are added for

each time step. Thus we can generate a time domain signal with a power spectral density

that is a sampled equivalent of the original power spectral density. The smal ler the

frequency difference between the components, the better the power spectral density of

the artificially generated signal resembles the original power spectral density. The

required computation time will increase, though. For more information on the method

that we used for generating time domain signals with a given power spectral density, the

reader is referred to Shinozuka and Jan (1972). Figure 25.2 shows an example of a wind

speed sequence generated using this approach.

Table 25.1 Value of roughness length, z

, for various landscape types

Landscape type Range of z

(m)

Open sea or sand 0.0001–0.001

Snow surface 0.001–0.005

Mown grass or steppe 0.001–0.01

Long grass or rocky ground 0.04–0.1

Forests, cities or hilly areas 1–5

Sources: Panofsky and Dutton, 1984; Simiu and Scanlan, 1986.

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25.5.3 Rotor model

The following well-known algebraic equation gives the relation between wind speed and

mechanical power extracted from the wind (Heier, 1998; Patel, 2000):



ð; Þ

; ð25:5Þ

where P

is the power extracted from the wind in watts;  is the air density (kg/m

); c

the performance coefficient or power coefficient;  is the tip speed ratio v

, the ratio

between blade tip speed, v

(m/s), and wind speed at hub height upstream of the rotor, v

(m/s);  is the pitch angle (in degrees); and A

is the area covered by the wind turbine

rotor (m

). Most constant-speed wind turbines are stall controlled. In that case,  is left

out and c

is a function of  only.

Manufacturer documentation shows that the power curves of individual wind tur-

bines are very similar. We therefore do not consider it necessary to use different

approximations for the c

() curve for different constant-speed wind turbines in PSDSs.

Instead, a general approximation can be used. We would like to stress that this does not

necessarily apply to other types of calculations, such as energy yield calculations for

financing purposes. Here, we use the following general equation to describe the rotor of

constant-speed and variable-speed wind turbines:

ð; Þ¼c



 c

  c



 c



exp

c





; ð25:6Þ

where

 þ c









þ 1



1

: ð25:7Þ

0102030405060

Time (s)

Wind speed (m/s)

Figure 25.2 Example of a simulated wind speed sequence, with the following input values:

average wind speed, v

, 11.5 m/s; start time of wind speed ramp, T

, 5 s; end time of wind speed

ramp, T

, 35 s; amplitude of wind speed ramp,

, 4 m/s; start time of wind speed gust, T

,5s;

end time of wind speed gust, T

, 15 s; amplitude of wind speed gust,

¼3 m/s; and roughness

length, z

, 0.01 m

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The structure of this equation originates from Heier (1998). However, the values of the

constants c

to c

have been changed slightly in order to match the manufacturer data

better. To minimise the error between the curve in the manufacturer documentation and

the curve we obtained by using Equations (25.6) and (25.7), we applied multidimensional

optimisation. Table 25.2 includes both the original parameters and the parameters used

here. Figure 25.3 depicts the power curves of two commercial constant-speed wind

turbines, together with the generic numerical approximation from Table 25.2.

High-frequency wind speed variations are very local and therefore even out over the

rotor surface, particularly when wind turbines become larger. To approximate this

effect, a low-pass filter is included in the rotor model. Figure 25.4 illustrates the low-pass

Table 25.2 Approximation of power curves

Heier (1998) 0.5 116 0.4 0 — 5 21 0.08 0.035

Constant-speed wind turbine 0.44 125 0 0 0 6.94 16.5 0 0.002

Variable-speed wind turbine 0.73 151 0.58 0.002 2.14 13.2 18.4 0.02 0.003

0 5 10 15 20 25

0.2

0.4

0.6

0.8

1.2

1.4

Output Power (p.u.)

Wind speed (m/s)

Figure 25.3 Comparison of the numerical approximation of the power curve of a stall-controlled

wind turbine (solid curve) with the power curves of two commercial stall-controlled wind turbines

(dotted lines)

Wind

speed

(m/s)

Filtered

wind

speed

(m/s)

τs

Figure 25.4 Low-pass filter for representing the evening out of high-frequency wind speed

components over the rotor surface

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filter. The value of the time constant  depends on the rotor diameter as well as on the

turbulence intensity of the wind and the average wind speed (Petru and Thiringer, 2000).

For the wind turbine analysed here,  was set to 4.0 s.

Finally, we would like to include the tower shadow in the rotor model. This can be

done by adding a periodic pulsation to the mechanical power that is the output of the

rotor model, as calculated with Equation (25.5). The frequency of this pulsation depends

on the number of blades (normally three) and the rotational speed of the wind turbine

rotor. The amplitude of the pulsation is in the order of a few percent. The tower shadow

is particularly important in investigations concerning power quality an d the mutual

interaction between wind turbines that are situated close to each other, electrically.

25.5.4 Shaft model

It has been repeat edly argued in the literature that the incorporation of a shaft repre-

sentation in models of constant-speed wind turbines (Type A) is very important for a

correct representation of their behaviour during and after voltage drops and short

circuits (e.g . see Akhmatov, Knudsen and Nielsen, 2000). This is because of the fact

that the low-speed shaft of wind turbin es is relatively soft (Hinrichsen and Nolan, 1982).

The models presented here are to be used for PSDSs. These are, among others, used for

analysing a power system’s response to the mentioned disturbances. It is therefore

essential to incorporate a shaft representation into the constant-speed wind turbine

model. However, only the low-speed shaft is included. The gearbox and the high-speed

shaft are assumed to be infinitely stiff. The resonance frequencies associated with

gearboxes and high-speed shafts usually lie outside the frequency bandwidth that we

deal with in PSDSs (Akhmatov, Knudsen and Nielsen, 2000; Papathanassiou and

Papadopoulos, 1999). Therefore, we use a two-mass representation of the drive train.

The two-mass representation is described by the following equations:

 K



;

 T

;

d

¼ 2fð!

 !

Þ;

;

ð25:8Þ

in which f is the nominal grid frequency; T is the torque;  is the angular displacement

between the two ends of the shaft; ! is frequency; H is the inertia constant; and K

is the

shaft stiffness. The subscripts wr, m and e stand for wind turbine rotor, generator

mechanical and generator electrical, respectively. All values are in per unit, apart from

,  and f, which are in p.u./el. rad., degrees, and hertz, respectively.

The resonance frequency of the shaft’s torsional mode was experimentally determined

as 1.7 Hz (Pedersen et al ., 2000). When resonance frequency and the inertia constants of

the generator and the turbine rotor are known, K

can be calcul ated by using the general

equation describing a two-mass system (Anderson, Agrawal and Van Ness, 1990). The

shaft is depicted schematically in Figure 25.5, which includes some of the quantities

from Equation (25.8).

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25.5.5 Generator model

The voltage equations of a squirrel cage induction generator in the d – q (direct –

quadrative) reference frame, using the generator convention, can be found in the

literature (Kundur, 1994) and are as follows

¼R

 !

;

¼R

þ !

;

¼ 0 ¼R

 s!

;

¼ 0 ¼R

þ s!

;

ð25:9Þ

in which s is the slip, u is the voltage, i is the current, R is the resistance, and is the flux.

All quantities are in per unit. The subscripts d and q stand for direct and quadrature

component, respectively, and the subscripts r and s for rotor and stator, respectively.

The generator convention is used in this equation, which means that a current leaving

the machine is positive, whereas a current entering the machine is negative. The opposite

of the generator convention is the motor convention, where a current entering the

machine is positive whereas a current leaving the machine is negative.

The slip is defined as follows:

s ¼ 1 

; ð25:10Þ

in which p is the number of poles. With respect to per unit (p.u.) quantities, it should

be noted at this point that the goal of using them is to make impedances independent

Shaft Generator

rotor

Wind

turbine

rotor

Figure 25.5 Schematic representation of the shaft, including some of the quantities from Equ-

ation (25.8): H

and H

¼inertia constant for the wind turbine rotor and generator rotor,

respectively; !

and !

¼ angular frequency of wind turbinerotor and generator rotor,

respectively;  ¼ angular displacement between shaft ends

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of voltage level and generat or rating by expressing them as a percentage of a

common base value. For an elaborate treatment of p.u. calculation and the correl-

ation between physical and p.u. values, see general textbooks on power systems, such

as Grainger and Stevenson (1994) . The flux linkages in Equation (25.9) can be

calculated by using the following equations, in which, again, the generator conven-

tion is used:

¼ðL

s

þ L

Þi

 L

;

¼ðL

s

þ L

Þi

 L

;

¼ðL

r

þ L

Þi

 L

;

¼ðL

r

þ L

Þi

 L

;

ð25:11Þ

In these equati ons, is flux linkage and L is the inductance. The indices m, r and  stand

for mutual, rotor and leakage, respectively. By inserting Equations (25.11) in Equations

(25.9), while neglecting the stator transients, in agreement with the assumptions

discussed above, the voltage current relationships become:

¼R

þ !

½ðL

s

þ L

Þi

þ L

;

¼R

 !

½ðL

s

þ L

Þi

þ L

;

¼ 0 ¼R

þ s!

½ðL

r

þ L

Þi

þ L

þ

;

¼ 0 ¼R

 s!

½ðL

r

þ L

Þi

þ L

þ

;

ð25:12Þ

The electrical torque, T

, is given by:



; ð25:13Þ

and the equation of the motion of the generator is:

ðT

 T

Þ: ð25:14Þ

The equations for active power generated, P, and the reactive power consumed, Q , are:

¼ u

þ u

;

¼ u

 u

)

ð25:15Þ

Because only the stator winding is connected to the grid, generator and grid can

exchange active and reactive power only through the stator terminals. Therefore, the

rotor does not need to be taken into account. The values of the various parameters are

dependent on the generator rating and can be derived from tables and graphs (Heier,

1998). Table 25.3 includes the generator parameters used in this Chapter.

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25.6 Model of a Wind Turbine with a Doubly fed Induction Generator

25.6.1 Model structure

Figure 25.6 depicts the general structure of a model of variable-speed wind turbines with

doubly fed induction generators (Type C). It shows that a wind turbine with a doubly

fed induction generator is much more complex than a con stant-speed wind turbine (see

Figure 25.1). Compared with a constant-speed wind turbine, a variable-speed wind

turbine has additional controllers, such as the rotor speed controller and the pitch angle

controller. Additionally, if it is equipped with terminal voltage control, it also has a

terminal voltage controller. We will now discuss each block in Figure 25.6, with the

Table 25.3 Simulated induction generator parameters

Generator characteristic Value

Number of poles, p 4

Generator speed (constant-speed; rpm) 1517

Generator speed (doubly fed; rpm) 1000–1900

Mutual inductance, L

(p.u.) 3.0

Stator leakage inductance, L

s

(p.u.) 0.10

Rotor leakage inductance, L

r

(p.u.) 0.08

Stator resistance, R

(p.u.) 0.01

Rotor resistance, R

(p.u.) 0.01

Compensating capacitor (constant-speed; p.u.) 0.5

Moment of inertia (s) 0.5

Wind

speed

model or

measured

sequence

Wind

speed

Mechanical

power

Rotor

model

Model of

doubly

fed

induction

generator

Rotor

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

Fundamental

frequency

grid

model

Converter

and

protection

system

Figure 25.6 General structure of a model of a variable speed wind turbine with doubly fed

induction generator (type C)

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exception of the grid model and the wind speed model. The wind speed model is

identical to that in the constant-speed wind turbine model, described in Section 25.5.2.

The grid model is not discussed for the same reasons mentioned in Section 25.5.1.

Figure 25.6 does not include a shaft model, in contrast to Figure 25.2. The reason

is that in variable-speed wind turbines the mechanical and elect rical part, to a large

extent, are decoupled by the power electronics. Therefore, the control approach of

the power electronics converter determines how the properties and behaviour of the

shaft are reflected in the terminal quantities of the generator. The mutual interde-

pendencies between shaft, control of the power electronic converter and output power

pattern are a very advanced topic which will not be treated here. If it is nevertheless

desired to incorporate a shaft representation in a model of a variable-speed wind

turbine with a doubly fed induction generator, the approach described in Section

25.5.4 can be used.

25.6.2 Rotor model

Again, Equation (25.5) and a numerical approximation of the c

(, ) curve based on

Equations (25.6) and (25.7) are used to represent the rotor (see Section 25.5.3). How-

ever, it is assumed here that the variable-speed wind turbine is pitch controlled. The

performance coefficient is thus dependent not only on the tip speed ratio, , but also on

the pitch angle, . Therefore, a new numerical approximation for the c

(, ) curve has

been developed, using manufacturer documentation of variable-speed wind turbines and

multidimensional optimisation. Table 25.2 includes the values for the parameters in

Equations (25.6) and (25.7), which are used to represent the rotor of a variable-speed

wind turbine. Figure 25.7(a) depicts the resulting power curve, together with the power

curves of two commercial variable-speed wind turbines. Figure 25.7(b) shows the pitch

angle deviation that is necessary to limit the power to the nominal value. In this case,

documentation from only one manufacturer was available.

Again, the rotor model of the simulations includes the low-pass filter depicted in

Figure 25.4 in order to represent the smoothing of high-frequency wind speed compo-

nents over the rotor surface. The simulations are carried out to validate the models and

are described in detail in Section 25.8. This issue is less critical for variable-speed wind

turbines, though, as rapid variations in wind speed are not translated into output power

variations because the rotor functions as an energy buffer. For the same reason,

representation of tower shadow was not included either, because in variable-speed wind

turbines the tower shadow hardly affects the output power because of the decoupling of

electrical and mechanical behaviour by the power electronics (Kru

ger and Andresen,

2001; Petru and Thiringer, 2000).

25.6.3 Generator model

The equations that describe a doubly fed induction generator are identical to those of

the squirrel cage induction generator, [i.e. Equations (25.9) – (25.14); see Section 25.5.5].

The only exception is that the rotor wind ing is not short-circuited. Therefore, in the

expressions for rotor voltages u

and u

in Equations (25.9) and (25.12), these voltages

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are not equal to zero. The flux equations are identical to those of the squirrel cage

induction generator, given in Equations (25.11).

If we want to obtain the voltage–current relationship using the voltage and flux

linkage equations, first the stator transients must again be neglected. This way, the

generator model corresponds to the assumptions used in PSDSs. This time, we also

neglect the rotor transients (Fujimitsu et al., 2000). To take into account the rotor

transients would require detailed modelling of the converter, including the semiconduc-

tor switches and the current control loops. The result would be time constants that are

significantly lower than 100 ms, the typical minimum time constant studied in PSDSs.

Further, the resul ting model would be much more complex and therefore difficult to use,

and it would require many more parameters, which are often difficult to obtain, in

practice. To avoid this, we assume that the VSCs with current control loops can be

modelled as current sources, as already observed in Section 25.4. Chapter 26 presents a

complete model of the doubly fed induction generator including the stator and rotor

transients.

1.2

1.0

0.8

0.6

0.4

0.2

51520

Output power (p.u)

0 5 10 15 20 25

Wind speed (m/s)

Pitch angle (deg)

(a)

(b)

Figure 25.7 (a) Comparison of the numerical approximation of the power curve of a pitch-

controlled wind turbine (solid curve) with the power curves of two existing pitch-controlled wind

turbines (dotted curves). (b) Pitch angle deviation above nominal wind speed, based on a

numerical approximation (solid curve) and manufacturer documentation (dotted curve)

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The following voltage – current relationships result in per unit quantities:

¼R

þ !

½ðL

s

þ L

Þi

þ L

;

¼R

 !

½ðL

s

þ L

Þi

þ L

;

¼R

þ s!

½ðL

r

þ L

Þi

þ L

;

¼R

 s!

½ðL

r

þ L

Þi

þ L

:

;

ð25:16Þ

Note the differences between Equations (25.16) and Equations (25.12): the rotor volt-

ages do not equal zero and the d /dt terms in the rotor equations have been neglected.

The generator parameters are given in Table 25.3.

The equation giving the electrical torque and the equation of motion of a doubly fed

induction generator are again equal to those of a squirrel cage induction generator,

given in Equations (25.13) and (25.14), respectively. The equations for active and

reactive power are, however, different because the rotor winding of the generator can

be accessed. Thi s leads to the incorporation of rotor quantities into these equations:

P ¼ P

þ P

¼ u

þ u

;

Q ¼ Q

þ Q

¼ u

 u

þ u

 u

)

ð25:17Þ

It should be emphasised that the reactive power, Q, in Equation (25.17) is not necessar-

ily equal to the reactive power fed into the grid, which is the quantity that must be used

for the load-flow solution. This depends on the control strategy for the grid side of the

power electronic converter that feeds the rotor winding. This does not apply to the

active power. Even though the converter can generate or consume reactive power, it

cannot generate, consume or store active power – at least not long enough to be of any

interest here. The expression for P in Equations (25.17) gives, therefore, the total active

power generated by the doubly fed induction generator, apart from the converter

efficiency, which can be incorporated by multiply ing the last two term s of this expres-

sion (i.e. u

and u

) with the assumed converter efficiency. That means that all

active power fed into or drawn from the rotor winding will be drawn from or fed into the

grid, respectively.

25.6.4 Converter model

The converter is modelled as a fundamental frequency current source. This assumption

is, however, valid only if the following conditions are fulfilled:

the machine parameters are known;

the controllers operate in their linear region;

vector modulation is used;

the terminal voltage approximately equals the nominal value.

The wind turbine manufacturer is responsible for the first two conditions and we assume

that these conditions are fulfilled. The thir d requirement is met, because the control of

the co nverter used in variable-speed wind turbines is ne arly always based on vector

570 Reduced-order Modelling of Wind Turbines