Gasch R., Twele J. (Eds.) Wind Power Plants: Fundamentals, Design, Construction and Operation

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382 11.2 Three-phase machines

Adding graphically only the sum of the mean values remains

3 · (1/2) E

= 3 E

eff

cos

M

The troublesome oscillations are eliminated. The supply of power and torque is

constant. Synchronous machines are mainly used as generators today.

Instead of using a three-phase system with six wires to carry the current

between generator and motor, it is possible to use only four or three wires, due to

the symmetry of the voltages. The result is the star connection of the windings

shown in Fig. 11-17. It is mainly used for generators which often operate in an

asymmetric load regime, i.e. the phases are not loaded evenly. For larger three-

phase loads, e.g. motors, three-phase heaters, etc., the delta connection is used. It

does not require a star point, Fig. 11-18.

connection

terminal box

Fig. 11-18 Winding arrangement in the stator of a three-phase machine; possibilities of connect-

ing AC windings; labelling of terminals in the terminal box [8]

11 Wind turbines for electricity generation - basics

383

This is possible, as the load is usually symmetric in these cases, therefore the

phase currents add up to zero.

Using the delta connection, the power of generators and motors is increased by the

factor

333  because the voltage at the coil endings, e.g. between L1 and L2,

is larger by the factor

3 than for the star connection between L1 and N, Fig. 11-

18. the currents are increased by that factor as well. So it follows that

delta

= 3 P

star

The stator construction of the three-phase synchronous machine which we just

discussed, is similar to that of the three-phase induction machine (i.e. asynchro-

nous machine). In contrast to that, the rotor construction of these two classical

types of the three-phase machines differs significantly, Fig. 11-19.

Fig. 11-19 AC machines and their rotor types [2, 4]

384 11.2 Three-phase machines

The rotor of the three-phase synchronous machine carries a pole system that is

either a salient pole rotor or a smooth-core rotor. The excitation is caused by per-

manent magnets or electromagnets where the excitation current is supplied via slip

rings. Some machines are completely without slip rings. In that case, an auxiliary

machine is used. It is mounted on the same shaft and supplies the necessary

energy (brushless excitation).

For motors, mainly three-phase induction machines (asynchronous machines)

are used today. Their rotor is equipped with a three-phase winding, too. If the

windings are short-circuited directly in the rotor, it is called a squirrel-cage rotor.

This arrangement allows for a very robust and low cost construction of the

machine. If a slip ring rotor is used, the rotor circuit can be manipulated. This type

of machine carries a three-phase winding on the rotor. The ends of the windings

are accessible through slip rings.

If a rotating field (star or delta connection) is generated in the stator of an in-

duction machine, stator and rotor windings function similar to the primary and

secondary sides of a transformer. The rotating stator field induces a voltage in the

rotor windings, which results in large currents. These currents create their own

field.

This field lags behind the rotating field of the stator and causes the rotor of the

machine to rotate. With increasing rotor speed, the frequency of the secondary

side of the imaginary transformer decreases, and, therefore, the value of the

induced voltage, too. When the rotor and the rotating field generated by the stator

run synchronously, the values of frequency and induced voltage in the rotor are

zero. When used as a motor the rotational speed n of the machine is slightly below

synchronous speed n

. It runs asynchronously with a slip of a few percent.

The induction machine (asynchronous machine) is commonly used as genera-

tor

for medium-sized wind turbines up to 1000 kW. It has two advantages over the

synchronous machine. Due to its simple design the purchase price is quite low.

And, moreover, the synchronization with the grid is easy. It is connected to the

grid in the motor mode and accelerates. The driving torque of the wind turbine

rotor then pushes it into the generator mode “automatically”. A disadvantage is its

reactive power consumption

as we will see later. In this respect, the synchronous

machine is easier to manipulate.

11.2.2 The three-phase induction machine

Stator construction

As mentioned above, the stator construction of the three-phase induction machine

(i.e. asynchronous machine) is similar to the one of the synchronous machine,

11 Wind turbines for electricity generation - basics

385

Fig. 11-17 and 11-18. Three windings (p = 1) produce a rotating field with the

grid frequency f

, resp.

= 2

If the stator has a suitably connected system of 6 windings (p = 2), then the

synchronous angular frequency of the rotating field is halfed to

= 2

/ p. It

rotates with 25 Hz (1500 rpm) instead of 50 Hz (3000 rpm). In pole switchable

machines this is used to vary the synchronous frequency of the rotating field and

consequently the idling speed of the induction machine. In wind turbines with

power limitation by the stall effect pole switching of the induction machine is

often used: most often it is switched from p = 2 to p = 3, i.e. from a synchronous

speed of 1500 rpm ( in a 50 Hz grid) to 1000 rpm.

Principle of operation

The field which rotates at grid frequency in the stator of the induction motor

induces a voltage in the short-circuited rotor winding. The frequency of this volt-

age depends on the rotor speed. If the rotor does not move, this frequency is the

same as the grid frequency (synchronous frequency)

= 2

. The system

works like a transformer with a short-circuited secondary side. Rotor current and

air-gap field, cause a tangential force on the rotor which moves the rotor in the

direction of the rotating field.

If a load-free rotor is starting up (M = 0, load-free idling), Fig. 11-20, it will

accelerate up to the synchronous speed

. At synchronous speed, the rotating

magnetic field of the stator does not intersect the windings (slip ring rotor) or the

conductor bars (squirrel-cage rotor) any longer, since the rotor moves at the same

speed as the stator field. Therefore, no rotor current is induced, and there is no

torque, either.

If a load is applied to the rotor (M

> 0), its speed decreases slightly relative to

the synchronous speed, n < n

. The angular frequency

of the rotor voltage is

then equal to the difference between the rotating field speed n

, and the mechanical

speed n multiplied by 2

:

= (n

- n) 2

= s :

. (11.30)

Here s is the slip, i.e. the difference between synchronous speed n

and real speed

n divided by the synchronous speed



. (11.31)

The currents now induced in the rotor cause torque. During normal operation, the

slip is not very large, s < 10 %, i.e. the induction motor normally runs almost at

synchronous speed.

386 11.2 Three-phase machines

If instead of a load a driving torque (M < 0) is forced upon the rotor, it accelerates

to a speed that is larger than the synchronous speed, n > n

: now the machine acts

as a generator, Fig. 11-20.

n/n

Fig. 11-20 Torque-speed characteristic of an induction machine; pull-out torque M

, slip at pull-

out torque s



)

Fig. 11-21 Equivalent circuit diagram of a transformer

1ı 2ı

Fig. 11-22 Equivalent circuit diagram of an induction machine, single phase model

In the generator mode the driving torque must never be allowed to exceed the pull-

out torque M

which is 2 to 3 times the rated torque M

, otherwise the rotor would

reach excessively high speeds almost immediately. In the motor mode, the rotor

would stop abruptly and overheat if the load torque was higher than the pull-out

torque.

The similarities between a transformer and an induction machine were men-

tioned before. The transformer circuit diagram in Fig. 11-21 can be used to derive

the voltage equations [10, 13].

11 Wind turbines for electricity generation - basics

387

They are as follows:

LiRtu

1111

)( 

(11.32a)

LiRtu

2222

)( 

Ҡ (11.32b)

with the main inductances L

= /

and L

= /

in the circuit diagram on

the right and the left. M

is the mutual inductance M

= /

which couples the

two circuits via the windings w

and w

is the magnetic permeance of the main

circuit. R

and R

are the ohmic resistances of the coils, and L

and L

are the

leakage inductances. They represent magnetic fluxes that are not contained in the

iron core, and, therefore, do not link both coils.

For the induction motor with its short-circuited rotor winding it is valid that

= 0. The frequency in circuit two is determined by the slip n

= n

s, i.e.

s. This differs from a transformer.

The equivalent circuit diagram of Fig. 11-22 is a fairly good description of an

induction machine. For more details, see

[3, 10, 13, 15]. The apostrophe (‘) indi-

cates values which have been transferred to the primary side using the turns ratio.

The current

i is calculated from the current of the secondary circuit i

, accord-

ing to Fig. 11-22 by multiplying with the reciprocal value of the turns ratio

. (11.33)

For the other quantities, the following is valid:

(11.33a)

(11.33b)

. (11.33c)

388 11.2 Three-phase machines

The fact that circuit two has the angular frequency

is apparent from the equiva-

lent resistance of the rotor, which depends on the slip. A plausibility check of this

equivalent circuit diagram considers s = 0, i.e. the rotor moves at synchronous

speed. As initially discussed, no voltage or current is induced in the rotor circuit,

for s = 0. In the circuit diagram, this is due to the resistance

sR /

which becomes

infinite then.

For large induction machines the stator resistance is small compared to the

main reactance

, therefore the equivalent circuit diagram of Fig. 1-22 can be

simplified further, Fig. 11-23 top.

The calculation of the currents in an induction machine is beyond the scope of

this book. Instead, the Heyland diagram, or Ossanna’s circle, will be used. It is

based on the short-circuit and idling tests. This diagram shows the amplitude and

the phase angle

of the stator current I

relative to the grid voltage U

, depending

on the slip, Fig. 11-23 bottom.

R´

/ s

= L

+ L´

i´

(t) =

sin

R´

/ s

= L

+ L´

i´

(t) =

sin

R´

/ s

= L

+ L´

i´

(t) =

sin

R´

/ s

= L

+ L´

i´

(t) =

sin

, s = 0,

No load

Motor

Generator

s = 1, short-circuit at start-up

s = 

1.0

0.5

-0.5

Braking

Motor

s = 1

s = 

s = 0

Generator

Braking

, s = 0,

No load

Motor

Generator

s = 1, short-circuit at start-up

s = 

1.0

0.5

-0.5

Braking

Motor

s = 1

s = 

s = 0

Generator

Braking

Motor

s = 1

s = 

s = 0

Generator

Braking

Fig. 11-23 Top: Simplified single phase equivalent circuit diagram of an induction machine;

bottom: Heyland vector diagram; stator current vector dependent on slip s resp. phase angle ĳ;

active current I

= I

cos ĳ, reactive current I

= I

sin ĳ and slip scale s

11 Wind turbines for electricity generation - basics

389

In the idling test (s = 0) the rotor resistance approaches infinity because of the slip

s in its denominator. The idling current I

is purely inductive, has the phase angle

= 90° and is found on the horizontal axis. In the short-circuit test – very first

moment of starting – the slip is s = 1. The inductances of stator and rotor together

with the resistance

R determine the high starting current I

and its phase angle.

This is shown in the circle diagram.

The case of total braking, s = , would lead again to a current vector which is

horizontal at

= 90° because the current path through the rotor does not contain a

resistance any more. However, the practical realisation of this test is difficult.

Nonetheless, the circle diagram can be determined using only the idling test and

the short-circuit test [13]. An auxiliary line is drawn from the vector tip of the

idling current I

to the vector tip of the starting current I

. From the centre of

the auxiliary line a perpendicular line is constructed towards of the horizontal axis.

The intersection of the perpendicular line with the horizontal axis gives the

centre of the circle on which the current vector moves for motor, generator or

braking mode:

Motor mode: 0 < s < 1

Braking mode: 1 < s < 

Generator mode: s < 0

The slip is scaled linearly and its scale is obtained by a similar construction [13].

In Fig. 11-23 the slip scale shows for slip at pull-out torque a value of approx.

0.25.

The electrical power can be determined using the Heyland vector diagram:

Reactive power:: Q = 3 U

sin

 (11.34a)

Active power: P = 3 U

cos

M

 (11.34b)

The factor 3 is due to the three phases of the stator coils.

If we ignore, somewhat generously, the copper and iron losses, the stator con-

verts the active power into a torque which revolves with the synchronous fre-

quency

P = 3 U

cos

= M :

(11.34c)

The rotor experiences this torque at its circumference but follows it, due to the

slip, only with the angular frequency of

(1- s). Therefore, the motoric me-

chanical power is slightly smaller than the electrical power:

)1()1(

cos3

mech

sPs

 :

(11.35)

390 11.2 Three-phase machines

In the generator mode (s < 0) it is larger than the electrical power. Slip represents

loss. Precisely speaking: for both cases 3% slip equals 3% loss! Certainly, there

are additional copper and iron losses. Nonetheless, large induction generators

reach an efficiency of more than 95%.

Fig. 11-24 shows the torque-speed (resp. slip) characteristic of a large induction

machine including current curve I

and efficiency curve. These curves may be

measured directly or derived from the vector diagram (as approximation).

The starting and short-circuit currents are quite high, they reach multiples of

the rated current. Therefore, the star connection is used for starting and then it is

switched to the delta connection, Fig. 11-18.

The torque-speed curve which is important for wind turbine design can be

described quite simply by the Kloß’ formula referring to pull-out torque and pull-

out slip



. (11.36)

Generator

Motor

cosĳ

-1

cosĳ

Generator

Motor

cosĳ

-1

cosĳ

Fig. 11-24 Torque versus slip of an induction machine; stator current I

, rotor current I

efficiency Ș and cos

11 Wind turbines for electricity generation - basics

391

During start-up (s >> s

), the torque approximately follows a hyperbola,

M/M

= 2 s

/s. Near rated power, it is a straight line through the synchronous

point. In the design phase, pull-out torque and pull-out slip can be calculated. For

an existing induction machine, it will best be experimentally determined [10 to

15]. The big advantage of the induction machine, compared to the synchronous

machine, is the self-starting behaviour and the easy transition from motor mode to

generator mode. Synchronisation is not required. This and the ability for rotor

speed switching (Fig. 11-26) made the induction machine very popular for the

stall-controlled wind turbines of the Danish concept. The reactive power

consumption I

sin

is disadvantageous and occurs in both, motor and generator

mode, as can be seen in Fig. 11-23.

Manipulation of the torque-speed characteristics

The position of the pull-out torque in the characteristics, Fig. 11-20 resp. 11-24,

can be influenced by the resistance R´

as the equation for the pull-out slip shows:

with

LLL



(11.37)

It depends on the leakage inductances L

of stator and rotor as well as the rotor

resistance R´

[10, 14]. The pull-out slip also determines the slope of the torque

characteristic close to the synchronous point (s = 0). Contrary to that, the magni-

tude of the pull-out torque M

is not influenced by a change of R´

, Fig. 11-25,

since it is valid that [10, 14]

. (11.38)

Fig. 11-25 Torque characteristics: influence of the rotor resistance and of the grid voltage U