AckermannTh. (ed) Wind Power in Power Systems

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We can introduce subscripts to refer S(t) to the two different frames as follows:

(t) is the space vector referred to the stator reference frame;

(t) is the space vector referred to the rotor reference frame.

In order to decrease the number of indices, the time-dependence (t) is included for all

quantities but is not indexed in subsequent formulae.

The coordinate transformation of

S from rotor to stator reference frame can be

expressed as follows:

¼ S

j

¼ S

þ j S



j

; ð26:2Þ

where 

is the angle between the two reference frames, which in our application is the

rotor angle of the machine. S

is the real part of S

, and S

is the imaginary part of S

The above transform ation is necessary if the connection of the machine is to be

described for the power system. The power system requires all quantities to be expressed

in the stator reference frame.

Identifying real and imaginary parts in Equation (26.2), we obtain:

¼ S

cos 

 S

sin 

; ð26:3Þ

¼ S

cos 

þ S

sin 

: ð26:4Þ

Furthermore, the inverse transformation is required in order to go from the stator to the

rotor reference frame. The equations for this transformation are as follows:

¼ S

j

; ð26:5Þ

identifying real and imaginary parts in Equation (26.5), we obtain:

¼ S

cos 

þ S

sin 

; ð26:6Þ

¼ S

cos 

 S

sin 

: ð26:7Þ

This coordinate transformation will be carried out for all stator quantities (i.e. for the

voltages, currents and fluxes) as all equations will be formulated and solved in the rotor-

fixed reference frame.

The projection of the space vectors over the phase axis has to coincide with the

corresponding phase quantities. This requirement results in the following relationship

between the phase and dq components for the rotor:

¼ Re½S

¼S

; ð26:8 Þ

¼ Re S

j2=3

¼

ﬃﬃﬃ

; ð26 :9Þ

¼ Re S

j2=3

¼



ﬃﬃﬃ

: ð26:10Þ

It should be noted that the real axis (i.e. the d axis) is collinear to the phase a winding

axis. From the equations above, it can be derived that the space vector can be expressed

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as a function of phase quantities. For the rotor, we arrive at the following relationship

between dq and phase components:

þ j S

þ

þ j

ﬃﬃﬃ

þ

 j

ﬃﬃﬃ

: ð26:11Þ

Identifying real and imaginary parts in Equation (26.11), we obtain:

ð2S

 S

Þ; ð26:12Þ

ﬃﬃﬃ

 S

ðÞ: ð26:13Þ

From Equations (26.12) and (26.13) it can be concluded that instantaneous three-phase

quantities [real-valued S

(t), S

(t)andS

(t)] can be transformed into a unique space

vector with real and imaginary parts S

(t) and S

(t), respectively.

Note that the complex magnitudes described above are different from the complex

magnitudes used in the so-called j! method. The j! method assumes the complex

magnitudes to rotate with constant speed, !, whereas here ! is allowed to vary freely.

26.4.2 Notation of quantities

In the following, we use the subscript italic s for quantities that are expressed in the stator

reference frame and the subscript italic r for quantities that are expressed in the rotor

reference frame. Superscript upright s represents the index of a stator quantity and super-

script upright r represents the index of a rotor quantity. Therefore,

represents the stator

voltage in the rotor reference frame, and

represents the stator voltage in the stator

reference frame. The notation

u represents a complex quantity, in this case a voltage.

26.4.3 Voltage equations of the machine

The stator voltage u

, referred to the stator frame is:

¼ i

; ð26:14Þ

where

is the complex stator current and

is the complex stator flux. r

is the stator

winding resistance.

The coordinate transformation of any quantity be tween the rotor reference frame and

the stator reference frame was derived in Equation (26.2). By applying this equation to

the three complex quantities in Equat ion (26.14), we trans form the equation from the

stator reference frame to the rotor reference frame. The result is:

¼ i

j

; ð26:15Þ

¼ u

j

; ð26:16Þ

j

: ð26:17Þ

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Substituting Equations (26.15) – (26.17) into Equation (26.14), we get

j

¼ i

j



: ð26:18Þ

The last term of Equation (26.18) is expanded as follows:

j



¼ e

j



j



¼ e

j



dj



j

¼ e

j



þ j!

j

;

ð26:19Þ

where !

is the speed of the machine. Inserting Equation (26.19) into Equation (26.18),

we obtain:

j

¼ i

j

þ e

j



þ j !

j

: ð26:20Þ

By multiplying both sides of Equation (26.20) by e

j

, we arrive at the final stator

voltage equation in the rotor-fixed coordinate frame:

¼ i



þ j !

: ð26:21Þ

The voltage equation for the rotor circuit in the rotor reference frame can be directly

expressed as

¼ i



; ð26:22Þ

where

is the complex rotor voltage; i

is the complex rotor current;

is the complex

rotor flux; and r

is the rotor winding resistance.

26.4.3.1 Per unit system of the machine

So far, all equations have been written and deduced using real values of the variables and

parameters involved. In general, it is very useful to work in a per unit (p.u.) system, which

will be developed in this section. In order to arrive at a per unit system, we have to define a

base system first (see also Section 24.5). There is a number of base systems from which we

can choose. We will, however, use peak phase voltage and peak phase current as the base

values. The other base values can be calculated from these two base values, as shown in

Table 26.1. It has to be stressed that the base power, S

base

, is the three-phase power of the

induction generator, S

, (where a subscript N indicates the nominal value).

Equations (26.21) and (26.22) in p.u. of the machine base are:

¼ i



þ j

; ð26:23Þ

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¼ i



: ð26:24Þ

Note that the speed of the machine !

in Equation (26.23) is in [radians per second].

In our equations, machine speed, !

, as well as turbine speed, !

, will be measured in

physical units. The angular speed, !

, in Equations (26.23) and (26.24) is equal to 2f

where f

is the power frequency (e.g. 50 Hz).

26.4.4 Flux equations of the machine

In order to solve the system of equations for the induction generator it is necessary in

addition to formulate equations representing the relationship between currents and

fluxes. In the following stator flux equation, the stator current and the stator flux are

referred to the rotor reference frame:

¼ i

þ i

; ð26:25Þ

where x

is the stator self-reactance, equal to

¼ x

þ x

; ð26:26Þ

and where x

is the leakage reactance of the stator and x

is the mutual reactance

between the stator and rotor windings. Equation (26.25) is given in p.u.

It can be shown that the values of the reactances and inductances coincide if they are

given in p.u. Therefore, the reactances in Equation (26.25) can be replaced with the

inductances given on the same p.u. base.

Table 26.1 Per unit bases

Quantity Expression

base

(kV) U

base

¼ U

ﬃﬃ

base

(kA) I

base

ﬃﬃ

base

(MVA) S

base

¼ S

base

() Z

base

(rad/s) !

base

¼ !

¼ 2f

base

(MWs) T

base

(Wb)

base

ﬃﬃ

Note: U ¼voltage; I ¼current; S ¼apparent power; Z ¼impedance;

! ¼ rotational speed; T ¼torque; ¼ flux; subscript ‘base’ indicates

the base value for the per unit system; subscript N indicates the

nominal value; U

is the nominal root mean square (RMS) value of

the phase-to-phase voltage of the machine.

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In the following rotor flux equation, the stator current is referred to the rotor frame:

¼ i

þ i

; ð26:27Þ

where x

is the mutual reactance between the stator and rotor windings and x

is the

rotor self-reactance, which is equal to

¼ x

þ x

; ð26:28 Þ

and where x

is the leakage reactance of the rotor.

By combining Equation (26.25) with Equation (26.27), the stator and rotor currents

can be expressed as functions of stator and rotor fluxes:



.

1 



; ð26:29Þ



.

1 



; ð26:30Þ

and, by introducing the total leakage factor, , given by

 ¼ 1 

; ð26:31Þ

we can rewrite Equations (26.29) and (26.30) as









; ð26:32Þ









: ð26:33Þ

Equations (26.32) and (26.33) describe the currents of the machine as functions of the

stator and rotor fluxes. All quantities are in the rotor reference frame.

26.4.5 Mechanical equations of the ma chine

There are two different mechanical bodies in the system we want to represent – the

induction generator and the wind turbine. This section contains the mechanical equa-

tions of the induction generator.

The mechanical equations of the induction generator can be represented as follows :

ðT

þ T

SHAFT

Þ; ð26:34Þ

d

¼ !

; ð26:35Þ

where T

is the elect rical torque produced by the induction generator; T

SHAFT

is the

incoming torque from the shaft connecting the induction generator with the wind

turbine; and H

is the inertia constant of the inducti on generator. The angle 

is the

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machine angle, and !

is the speed of the induction generator. All quantities, except for

the machine speed, !

, are given in p.u.

The expression for the electrical torque in physical units is



: ð26:36Þ

The derivation of Equation (26.36) is not included here; for further details, see Kova

cs,

1984. The asterisk indicates the complex conjugate of the flux, . Using Equation

(26.32) for the stator current in the rotor reference frame,

, the equation for the

electrical torque can be rewritten as











Þ;

ð26:37Þ

where real and imaginary parts of the stator and rotor fluxes are shown in parentheses in

the final expression. Their relation to the complex stator and rotor fluxes is as follows:

þ j

; ð26:38Þ

þ j

: ð26:39Þ

Expressed in the p.u. system, the equation for the electrical torque, Equation (26.37), is





Þ; ð26:40Þ

where the reactances, total leakage factor, fluxes and torque are given in p.u. of the

machine ratings.

The incoming torque from the shaft to the induction generator, T

SHAFT

, in Equation

(26.34), consists of the two terms T

TORSION

and T

DAMPING

SHAFT

¼ T

TORSION

þ T

DAMPING

: ð26:41Þ

Equation (26.41) represents a drive-train system, where T

TORSION

represents the elasti-

city of the shaft and T

DAMPING

represents the damping torque of the shaft (see Novak,

Jovile and Schmidtbauer, 1994 and Akhmatov, 2002).

TORSION

is expressed as a function of the angle of the wind turbine, 

, and the angle

of the machine, 

,as

TORSION

¼ Kð

 

Þ; ð26:42Þ

where K is the shaft torsion constant (p.u. torque/rad); that is, it is the effective shaft

stiffness.

DAMPING

is related to the speed of the wind turbine, !

, and the speed of the

machine, !

, as follows:

DAMPING

¼ Dð!

 !

Þ; ð26:43Þ

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where D is the shaft damping constant [p.u. torque/(rad/sec)] and represents the damp-

ing torque in both the wind turbine and the induction generator.

It is important to note that the speed of the machine, !

, and the speed of the wind

turbine, !

, are measured in radians per second and that the angle of the machine, 

and the angle of the wind turbine, 

, are measured in radians.

26.4.6 Mechanical equations of the wind turbine

The mechanical equations of the wind turbine are represented by the following two

differential equations:

ðT

MECH

 T

SHAFT

Þ; ð26:44Þ

d

¼ !

; ð26:45Þ

where T

MECH

is the mechanical torque produced by the wind turbine; T

SHAFT

is the

torque from the shaft co nnecting the induction generato r with the wind turbine [see

Equation (26.41)]; and H

is the inertia constant of the turbine. The angle 

represents

the position of the wind turbine, and !

is the speed of the wind turbine. All quantities

except !

and 

are given in p.u.

In this chapter, we assume that the mechanical torque, T

MECH

, produced by the wind

turbine is constant. However, the constant T

MECH

can be changed to a varying T

MECH

as described in Section 24.3.1

The equations of the machine given in this section (Section 26.4) can also be found in

Centeno, 2000.

26.5 Voltage Source Converter

It is assumed that the converters in Figure 26.1 are arranged in a VSC. In this chapter,

we will not implement the detailed modulation scheme but, rather, assume that the

switching frequency is infinite. If the detailed modulation scheme were implemented it

would slow down the simulations significantly, as each switching operation of the

converter would be han dled by the simulator as a separate event. The switching opera-

tions of the converters can be neglected without significantly affecting the result.

In this chapter, the VSC and its control is condensed and lumped together in a unique

module. Therefore, the voltage that the voltage source control impresses onto the rotor of

the machine is represented as a direct output of the voltage source converter control, (

Figure 26.3). In spite of this simplification, we take into account the limitations in the voltage

generation that the converter imposes. These limitations will be implemented as limitations

to the voltage and torque controllers (U-reg and T-reg, respectively, in Figure 26.3).

The strategy used for the converter control is field-oriented control. The input signals

to the VSC are:

the stator voltage in the stator reference frame of the machine, u

;

the speed of the machine, !

;

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the angle of the machine, 

;

the electrical torque of the machine, T

Figure 26.3 shows the input parameters to the converter control: the stator winding

resistance, r

; the stator self-reactance, x

; the rotor winding resistance r

; the rotor self-

reactance, x

; and the mutual reactance between the stator and rotor windings x

; !

is equal to 2f

,wheref

is the power frequency (e.g. 50 Hz). The input parameters x

and x

were defined in Equations (26.26) and (26.28), respectively. The constant 

was defined in Equation (26.31). The following two reference values are also fed into

the VSC: !

gref

and u

ref

, each set by the user to a constant value throughout the simulation.



RFC

in Figure 26.3 is the stator flux in the rotor flux coordinate system RFC. V

RFC

is a

complex quantity which real part is the output signal from the voltage controller V

RFC

and its imaginary part is the output signal from the torque controller V

RFC

Figure 26.3 is the rotor flux in the RFC system, a real quantity,  is the rotor flux angle in

the rotor coordinate system, T

Erefwlim

is a torque reference which is the output signal from

the speed control !-reg. In Figure 26.3 s represents the Laplace operator.

The constants 

and 

in Figure 26.3 are defined as follows:



; ð26:46Þ



: ð26:47Þ

Erefwlim

gref

–s

(1–σ)r

σx

(1–σ)r

σx

–r

–s

RFC

– RFC

– r

RFC

–

RFC

ref

–s

–

1+σ

σx

–

(θ

+ ρ)

(θ

+ ρ)

–

jρ

U-reg

ω -reg

-reg

–s

–

–RFC

Figure 26.3 Control system configuration for the voltage source converter; for definitions of

variables, see section 26.5

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The speed, voltage and torque controllers – !-reg, U-reg, and T-reg, respectively – are

modelled as PI controllers (proportional–integral controllers).

26.6 Sequencer

The sequencer is a module that handles the different modes of operation of the system

that consists of the DFIG, the wind turbine, the VSC and its control. The need for such

a sequencer arises from the fact that whenever there is a fault in the interconnected

power system, the system of the DFIG must be able to handle the resulting high currents

without causing any damage to the equipment. Another reason for having different

modes is that the system has to recover from a fault as soon as possible in order to

remain in phase. The sequencer basically controls the rotor current level and the stator

voltage level and sets the current mode of operati on.

The aim of the sequencer is to protect the VSC from high currents as well as to optimise the

behaviour of the system. The changes of mode are required during transient conditions. There

are three different modes, with mode 0 corresponding to the mode for normal operation.

If the rotor current exceeds 2 p.u., the VSC has to be disconnected because it cannot

resist such a high current. Therefore, the rotor is short-circuited during such an event. If

the machine works as a short-circuited machine, the system operates in mode 1. If the

machine works in mode 1 and the stator voltage magnitude has been greater than 0.3 p.u.

for 100 ms, the system switches to mode 2. During mode 2, the machine is still short-

circuited, but additional resistances are connected into the rotor circuit. In that case, the

equivalent rotor resistance is 0.05 p.u., which forces the rotor current to decrease.

Finally, if the system works in mode 2 and the stator voltage level has exceeded

0.85 p.u. for 100 ms and the rotor current has been lower than 2 p.u. for 100 ms, the

system returns to normal operation (i.e. mode 0). Once the system has returned to

normal operation the machine first tries to get magnetised and then starts producing

torque. That is done by setting the torque reference to 0 during 400 ms, once the

disturbance has been cleared.

The left-hand side of Figure 26.6 (page 601) depicts the modes of operation during a

simulation.

26.7 Simulation of the Doubly-fed Induction Generator

In this section we will test the setup of the DFIG. The machine is incorporated into a

small power system where it is connected to an infinite bus via a line impedance,

Z,of0:06 þ j0:60 (). In the power system, there will be a three-phase fault at t ¼36 s at

30% of the full distance from the infinite bus (see Figure 26.4). The three-phase fault has a fault

resistance, R

fault

,of0:30 (). The power system is modelled in instantaneous value mode.

Prior to the fault, the DFIG operates at nominal speed. In Figure 26.5, the terminal

voltage is shown on the left-hand side and the stator fluxes

and

are shown on the

right-hand side.

In Figure 26.6, the rotor current and the mode of operation is shown on the left-hand

side and the speed of the machine, !

, on the right-hand side. From Figure 26.6 we can

see that the rotor is short-circuited (mode of operation equal to 1) approximately

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between 36.0 < t < 36.1 (s). When the mode of operation is equal to 2, between

36.1 < t < 36.2 (s), the rotor is still short-circuited but the rotor current is slowly

decreasing as additional resistances are switched into the rotor circuit.

After a few seconds, the DFIG model returns to its initial state. However, this is not

shown in Figures 26.5 and 26.6.

26.8 Reducing the Order of the Doubly-fed Induction Generator

Different models are available to simulate the behaviour of the DFIG under different

conditions. The machine described in Section 26.4 is a 6th-order model, as the model

contains the following six state variables:

, !

and 

and

are

the real and imaginary parts of the stator flux

, and

and

are the real and

imaginary parts of the rotor flux

, as shown in Equations (26.38) and (26. 39).

The 6th-order model developed in this chapter is an extension of the 5th-order model

described in Thiringer and Luomi (2001). The extension consists of the machine angle,

Infinite bus Fault

DFIG bus

Figure 26.4 Test system for the doubly-fed induction generator (DFIG)

36 36.2 36.4 36.6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Time (s)

Terminal voltage (p.u.)

36 36.2 36.4 36.6

–1.5

–1.0

–0.5

0.5

1.5

Time (s)

and ψ

(p.u.)

Figure 26.5 Terminal voltage and stator fluxes

and

of the doubly-fed induction generator

(DFIG) during a three-phase fault in the power system

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