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

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372 11.1 The alternator - single-phase AC machine

to the power = voltage times current for DC), the active power, equation (11.15),

is the product

= (1/2) E

î cos

= E

eff

cos

M

 (11.16)

The factor 1/2 disappears. Only the term cos

M

reminds us that there is for AC

apart from the active power also the reactive power with a sin

M

term.

For the mean torque of a single-phase alternator it follows from equation

(11.11), with

= P

, that

)()(

)2(

LRR

rlBM

:



: , (11.17)

if small mechanical losses due to friction etc. are neglected.

The expression in parenthesis (2 B l r) is the machine constant for one winding

that has been analyzed so far. With increasing number N of windings the constant

is correspondingly larger. A closer look at the mean active power in equation

(11.11) reveals that it consists of useful output power at the load resistor R

, and a

power loss which is proportional to the internal resistance R

of the machine. The

following expressions can be used:

= P

use

+ P

loss

use

loss

)()(

)2(

)()(

)2(

LRR

rlBP

LRR

rlBP

:

use



. (11.18)

The power curve P

(

) and the torque curve M

(

) for different speeds are of

particular interest for the operation of a generator with a wind turbine. They result

from equations (11.11) and (11.18). It becomes clear that the alternator has a very

pronounced maximum torque, and that the power tends towards a finite maximum

value for high rotational speeds, Fig. 11-9.

11 Wind turbines for electricity generation - basics

373

For the electrical design of systems the behaviour of current and voltage are im-

portant. According to Fig. 11-9, the output voltage of the machine, and the current

tend towards a final value for high frequencies, similar to the power. The source

voltage, however, increases linearly with the rotational speed.

Changing the load resistance R

or the excitation (magnetic field B) shifts the

load curve, as illustrated in Fig. 11-10. This provides a means to match, within

certain limits, the load curve to a given rotor and generator, cf. also Figs. 11-1a

and 11-1b.

relative angular frequency

RR 



BlrM



max

BlrP





max

1 RR

Blr





Lmax

BlrRU



max

relative angular frequency

RR 



BlrM



max

BlrP





max

1 RR

Blr





Lmax

BlrRU



max

Fig. 11-9 Dimensionless power, torque, current and source voltage versus the relative angular

frequency for an alternator with permanent magnets and a load R

; reference values given at the

right margin of the figure

R increases

1 2

Relative angular

frequency

1 2

B increases

Relative angular

frequency

R increases

1 2

Relative angular

frequency

1 2

B increases

Relative angular

frequency

Fig. 11-10 Shifting of the torque curve by changing the load resistance or by changing the excit-

ing magnetic field

374 11.1 The alternator - single-phase AC machine

11.1.2 Types of excitation, internal and external pole machine

In the alternator described so far, the “permanent-excited dynamo”, the required

magnetic field is created by a permanent magnet. The result is that the output volt-

age of the machine can only be manipulated by the rotational speed. If the perma-

nent magnet is replaced by an electromagnet, the flux density B can be influenced

through the excitation current. The source voltage can now be manipulated for a

fixed rotational speed. This is called a separately-excited machine, Fig. 11-11b.

In a ferromagnetic circuit, there remains residual magnetism after switching off

the current. Therefore, the machine can produce a small voltage without any exci-

tation current, the residual voltage (U

res

). If it is fed back into the excitation wind-

ing, the machine can build up its excitation without an external voltage source.

The only requirement is a rectifier which converts the AC current provided by the

generator into the DC current required for the excitation. Self-excitation only

starts above a threshold of the rotational speed since the voltage drop of the recti-

fier (approx. 1.4 V) has to be exceeded first, Fig. 11-11c.

The original alternator (dynamo) according to Fig. 11-4 is an external pole

machine since the exciting magnet is located on the outside in the stator. The dis-

advantage of this arrangement is that the power has to be transferred out through

slip rings.

a) Permanent magnetic field excitation

Advantage: no additional energy source;

high efficiency

Disadvantage: output voltage cannot be

easily manipulated by the excitation

current

b) Separate excitation

Advantage: output voltage can be

easily manipulated by the excitation

current

Disadvantage: an additional energy

source has to provide the required

excitation energy; complicated

c) Self-excitation

Advantage: no additional energy source

is required for the excitation of the

machine; output voltage is easily

manipulated

Disadvantage: complicated, compara-

tively low efficiency

Fig. 11-11 Types of excitation

11 Wind turbines for electricity generation - basics

375

Fig. 11-12 A.C. machine, excitation on the rotor, internal pole arrangement [7]

e(t)

i(t)

sin ȍ

-e(t)

sin ȍ t

e(t)

i(t)

sin ȍ

-e(t)

sin ȍ t

Fig. 11-13 a) Equivalent circuit diagram of a single-phase synchronous machine; (b) process of

synchronization

For a large power, this is expensive and susceptible to wear. Therefore, usually,

the magnet revolves, the so-called revolving field, and the coil is in the stator, cf.

Fig. 11-12. Separately excited internal pole generators still require slip rings for

the excitation current, but now the power to be transmitted is small in comparison

to the rated power (approx. 2 to 10%).

376 11.1 The alternator - single-phase AC machine

11.1.3 Alternator (single-phase AC machine) in grid-connected

operation

Bringing a generator on-line requires a great deal of care. At the very moment the

generator starts feeding current into the grid, the frequencies (i.e. speed), ampli-

tudes and phase angles of the voltages have to be equal. If these three conditions

are fulfilled, there will be no transient terms. The differential equation reads

)(sin

sGii

tetU

LiR :  . (11.19)

The right side of the differential equation (11.19) disappears completely if in the

very moment of connection, it is valid that U

= E

und

= 0 for the

source voltage e(t)

e(t) = E

sin (:t +



) . (11.20)

In that case, there will be no transients. Once the synchronous machine is con-

nected to the grid there is only one speed

. Therefore, the index S will be

dropped from the rotational speed.

By means of a stronger excitation in the revolving field, the source voltage ampli-

tude E can now be increased E > U

, or alternatively decreased. At this point a

general sinusoid will be assumed for the current (where I

is in phase with E

), cf.

equation (11.5):

i(t) = I

sin :t + I

cos :t (11.22a)

= : I

cos :t - : I

sin :t . (11.22b)

Applying a driving torque to the shaft, causes a leading load angle , Fig.

11-14. The result is that the maximum value of the source voltage e(t) precedes

the maximum of the grid voltage (by the angle ): both were in agreement at the

moment of synchronization, with (almost) no driving torque. For the source volt-

age, it is now valid that

e(t) = E sin(t + )

= E (sin cos t + cos  sin t)

= E

cos t + E

sin t . (11.21)

ϑ ϑ

11 Wind turbines for electricity generation - basics

377

This is inserted into the differential equation (11.19). We use the simplifying

assumption that the internal resistance is very small, R

| 0. This is valid for

medium-sized and large synchronous generators.

With R

<< : L

and equations (11.21) and (11.22), the differential equation

(11.19) becomes

:L

cos :t – I

sin :t) = (U

– E

) sin :t – E

cos :t (11.23)

Fig. 11-14 Left: Synchronous machine at the moment of synchronisation, vector diagram and

voltages e(t), u(t); right: rotor load angle

due to the driving torque, vector diagram and voltages

Usin tΩ

()

Esin t+

tΩ

378 11.1 The alternator - single-phase AC machine

Comparing the coefficients of the sine terms and cosine terms, respectively, we

obtain the current amplitudes:



sin

(active current), (11.24a)



cos

GSG

(reactive current). (11.24b)

The power results from P(t) = u(t) i(t), and is

P(t) = U

sin

:t + U

sin :t cos :t, (11.25)

Active power Reactive power

with the first term describing the active power and the second term the reactive

power, cf. Fig. 11-7. The negative sign in equation (11.24) is due to the fact that

the referencing in Fig. 11-13 treats the grid as a load [2]. This is the opposite of

the reference in Fig. 11-6. The power can now be stated in the form

)2cos1(

sin

)(

tP :

  (Active power), (11.26a)

EUU



 2sin

)cos(

(Reactive power). (11.26b)

For the mean value of the active power and the mean torque we now have



sin

(11.27)

sin



. (11.28)

Active current I

, equation 11.24a, and active power, equation (11.27), are caused

by the load angle . If > 0 (leading), the machine acts as a generator, power is

fed into the grid (P

< 0). The sign is negative, but that was neglected in the equa-

tions (11.27) and (11.28). If a torque is required at the machine shaft (motoring

mode), the load angle becomes negative, < 0. Therefore, the sign of sin

changes, i.e. power is consumed from the grid (P

> 0). For rated power, the load

angle is usually 20° to 30°. If due to overloading, the load angle is forced to

ϑ ϑ

11 Wind turbines for electricity generation - basics

379

become larger than 90q, the machine slips out of step. During motoring, the speed

will then drop to a speed below synchronous speed, depending on machine

parameters and load moment. In the generating mode, the speed increases steeply.

Both would lead to the machine’s destruction due to the speed oscillations and the

currents. Therefore this state has to be avoided.

The sign of reactive current, equation (11.24b), and reactive power can be

manipulated by the value of E

= E cos , i.e. by the excitation current of the

revolving field which causes the source voltage amplitude E.

If = 0° and for the case of

overexcitation, E

> U

, this equation shows that reactive power is fed

into the grid (capacitive mode).

In the case of

underexcitation, E

< U

, reactive power is taken from the grid

(inductive mode).

Often, grid-connected synchronous generators operate with overexcitation in order

to compensate for the reactive current consumption of the numerous induction

motors, reactance coils and transformers, etc. which are always present in the grid.

Fig. 10.15 shows the rotating vectors of voltages E and U

, and current I which

can be projected to obtain the time curves, cf. Figs. 11-8 und 11-14.

Comparing the active current component I

= I cos

in Fig. 11-15 to the compo-

nent calculated according to equation (11.24a), we obtain the relationship between

the current phase angle

, and the load angle

I cos

= -

sin

. (11.29)

Iȍ L

Generator mode

Overexcited

Underexcited

Iȍ L

Generator mode

Overexcited

Underexcited

Fig. 11-15 Vector diagram of voltages U

, E, and current I during electricity generation; E

> U

overexcitation, E

< U

underexcitation

380 11.2 Three-phase machines

90



E/U 1.5

Fig. 11-16 Grid-connected synchronous machine: a) Heyland’s vector diagram at E = 1,5 U

;

b) torque-speed diagram

This is also the instruction for scaling the length of the current vector in Fig. 11-15

and 11-16. The latter figure gives an overview of the total behaviour of the grid-

connected synchronous machine.

11.2 Three-phase machines

11.2.1 The three-phase synchronous machine

The single-phase alternator - which has been discussed so far - has the disadvan-

tage that the electrical and the mechanical power oscillates between zero and a

maximum value, cf. Fig. 11-7. To prevent this, the single-phase machine was

extended into a three-phase machine with the three windings arranged in the stator

spatially offset by 120q, Fig. 11-17 and 11-18. Machines with such a stator are

called three-phase machines.

The three-phase synchronous machine works just like the single-phase alternator.

The single-phase equivalent circuit diagram is therefore applicable to the three-

ϑ ϑ

11 Wind turbines for electricity generation - basics

381

phase machine. The calculations necessary for determining its operating behaviour

can be derived from those for the

alternator (cf. section 11.1). The voltage curves

correspond to Fig. 11-17. They are those of the

alternator supplemented by a

phase offset at 120q and one at 240q.

The same applies to phase currents and phase powers. The mean total power of

the machine is tripled. The individual phase powers complement one another in

such a way that the power output is constant without any oscillation. This becomes

clear if the active power curve of Fig. 11-7, bottom,

sin

:t = (1/2) (1 – cos 2:t) E

is drawn three times with the respective 120° phase shifts.

Star point

A.C. Generator A.C. Motor

ĳ resp. t

Star point

A.C. Generator A.C. Motor

ĳ resp. t

Fig. 11-17 Three-phase (A.C.) generator and motor: a) voltages in the three phases, b) applica-

tion with a star connection