E L S E V I E R Electric Power Systems Research 41 (1997) 191-20l

ELECTRIO POW|n SWSTIITI$ RI8|nMH

Stand alone induction generators

Dan Levy *

Department of Electronic & Computer Engineering, University of Limerick, Limerick, Ireland

Received 4 June 1996

Abstract

This paper discusses three types of induction generators used in stand alone systems and compares their characteristics and suitability of application for variable speed, variable load, constant output voltage and frequency. The generators in question are the Squirrel Cage, the Doubly Fed and the Cascade Machines. © 1997 Elsevier Science S.A.

Keywords: Induction generators; Cascade machine; Double stator machine; Isosynchronous machine; Squirrel cage machine; Doubly fed machine; Stand alone power generation; Stand alone turbines and prime movers; Hydro and wind power generation

I. Introduction

From the theory of the induction generator [1], it can be seen, that the only requirement for ob- taining an output is a source of magnetising vats and a suitable load below the power limit. The relationship between power and vars for a typical induction generator can be calculated [2]. The re- quired vars can be supplied by shunt capacitors or by a synchronous machine. Thus an output can be obtained from this type of generator when it is connected to a system consisting only of shunt capacitors and load.

The frequency of operation of an induction generator with self-excitation is very close to the frequency corresponding to the generator speed, multiplied by the number of pole pairs, since the pull out slip is normally less than 10% and the machine will not operate at a load above the power limit.

For constant output frequency and constant output voltage at variable speed, some form of regulation needs to be introduced. This regulation can be carried out with the aid of a controller.

A practical controller can be a solid state con- verter and a power source which can supply reac- tive power only.

* Tel.: + 353 61 333644; fax: + 353 61 330316.

0378-7796/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 3 7 8 - 7 7 9 6 ( 9 6 ) 0 1 1 8 3 - 2

2. The stand alone Squirrel Cage induction generator system (Fig. 1).

This type of system consists of a Squirrel Cage machine connected to a load Z~ and a solid state converter with voltage/current transfer function e i~/2.

The converter consists of a constant voltage and fre- quency, D.C. to A.C. inverter. The inverter supply is from a fully charged battery which, in normal operating conditions, accepts and delivers no active power to the inverter.

The relationship between the variables is given by:

Ez +z, _zar,-a=F] Z, Z2 JLIrJ

E

(2.1)

(2.2)

and

r /= Re(E. /*) = Re(~) = 0 (2.3)

(2.1)-(2.3) are four equations with four unknowns, /~, It, 11 and the slip 'S' , therefore 'S ' or the machine speed will be a dependent variable of the load Z 1. In other words the generator cannot run at a variable speed- - constant load or at a constant speed--variable load. For light load, Z 3 can be assumed to be in parallel with the load Zj, therefore a simplified analytical solution can be obtained for the relationship between the load and the slip 'S'.

192 D. Levy/Elec tr ic Power Systems Research 41 (1997) 191-201

i o e

Reactive Power Supply (inverter)

E I Z I Z 2

I I r

Fig. 1. The stand-alone squirrel cage induction generator system.

8 2

+ ( R)2 + 2 -, R1 ~ + c')-(L, + L 2 -- 2M) 2

(2.4)

(2.4) suggests the slip should be negative or - S > R2/

R~, this in order that the power generation constraint by (2.3) will take place.

The reactive current supplied by the converter can be shown to be equal to

. 1 1 R, + ] (2.5)

-£ J

This current depends on the load which in turn depends on the speed (2.4). The machine torque per phase taking (2.4) into account.

T = - MRe(jlsI*r) = - - Re (2.6) o)

This torque is proportional to the load which in turn is proport ional to the speed.

o) Machine efficiency = ~ (f2 > co) (2.7)

Due to the sensitivity to the load, this machine is not practically suited to stand alone power generation. However, a stand alone system was developed by the University of Toronto [3] where the excess power gener- ated due to load or speed variation was dampened using variable resistance heating elements.

2. I. Computer simulation (Fig. 2)

Computer simulation for (2.1) and (2.2) was carried out for the case:

Re (E ' I* ) - r /= 0 (2.1.1)

where ~/is equal to the real input power to the converter. The simulation was carried out in order to obtain a more accurate study of the system at different speeds and loads. A six pole, three phase Squirrel Cage Generator with the following data was used for simulation:

Cony. PWR

, ' 1.2 1,14 i G4 O,D

1.09

2DO

W

,)

-40D

600

-800

~o ;o ?o ?0 ;o

Fig. 2. Converter power as a function of the load, machine speed as a parameter, squirrel cage machine.

L~ = 0.01087 H; L 2 = 0.1135 H; M = 0.105 H; RI = 0.33 f~; R 2 = 0.83 f2; E = 220 V effective; f = 50 Hz; Nominal power = 25 kW.

The machine was designed in particular to work as a generator rather than a motor with a higher rotor resistance.

Fig. 2 shows r/ as a function of the load where the speed is used as a parameter. As expected, the charac- teristics are not horizontal lines and, therefore, the speed needs to be increased with the load if 'q ' is held constant (or equal to zero). For the case q = 0, the generator operation can be regarded as being in parallel with the grid where it dampens the excess power as a generator or absorbs power as a motor.

3. Stand alone doubly fed induction generator system (Fig. 3)

A frequency converter connected between the stator and the rotor can supply reactive power to the genera- tor [4,5]. Such an ideal converter will consume no real power. The operation of such a system is similar to the 'static scherbius' method used for feeding power to the utility grid. However, for stand alone operation, a battery is added to the frequency converter in order to supply a reactive power while starting the generator. Voltage stabilisation is necessary in order to keep the output voltage constant, independent of the load and

D. Levy ,,/'Electric Power Systems Research 41 (1997) 191 201 193

speed variation within the working range of the system. The torque/speed characteristic for this type of opera- tion is almost constant power or more precisely:

r . a ; e Re +I~R1 +I~R2 (3.1)

or in terms of the rotor current Ir:

Rlo92M 2 [- E 2 2E imag(/~)-lj 4 R-~- ~ -~ ~Lo----~ + ~ -~ (3.2)

For most prime movers, a constant power load sys- tem is unstable. In other words, the generator torque/ speed characteristic slope should be greater than the load/speed characteristic slope. In particular for wind power generation, the torque/speed slope should be positive. From (3.2), in order to achieve a positive torque/speed slope, the rotor current should be in- creased with speed in such a way that the generator loss in (3.2) will be a dominant factor on the machine characteristic. Furthermore, the rotor power is indepen- dent of the type of control, is given by:

R~(V,/*)

I( SR'°92M2\-T*

( + S R~ + (O2Ll ~ ~- E2R~ (3.3)

Since the practical operating speed is close to S = 0, this power is positive and therefore the direction of the power flow will be from the stator to the rotor through the converter. It is worth mentioning that the power flow reversal will happen at much higher speed than the practical maximum operating supersynchronous speed because the rotor input power is

v . . E i o 9 ) e J "

!., z , z~ i i ~. f.

[ ~ Z t ]~

Z,= R,+ p(L,- M)

R Z~ = ~ + p(L~ - M)

Z~ = pM//RL, .

Fig. 4. The stand-alone cascade induction generator system.

- - g r Re( Vr/r* ) (but not-~- • 1")

Therefore, due to the complexity of the converter which requires its output frequency to vary as the slip fre- quency and the complexity of the controller necessary to keep the generator efficiency within an acceptable level, this type of application is not practical. It is desired therefore, to find a machine which possesses the following characteristics: 1. A torque/speed characteristic with a positive slope

and independent of the type of control used. 2. Both rotor and stator supply power to the load at

the practical operating speed range. Furthermore, the converter should operate at a fixed frequency which is the load frequency and at a fixed output voltage which is the load voltage.

3. Excess power to be dissipated on a fixed load such as the machine itself.

The above requirement will simplify the converter and its control elements for stand alone power generation.

It can be shown that a machine with rotor voltage equal to ~'r/S (rather than l~r) will meet the above requirements.

E ]-, Z I

~equelx:y voltoOo

Co~werle¢

Z2 "i', T~ V~

7-,= R,+ p(L,- M)

Z~ = RS-~ + IKL,- M)

Fig, 3. The stand-alone doubly-fed induction generator system.

4. The stand alone cascade (double stator) induction machine system (Fig. 4)

The machine consists of two wire wound rotor induc- tion machines [6] with their rotors connected in cascade and rotating in opposite directions. The frequency of both stators will therefore be fixed and independent of the speed of rotation, in contrast to the doubly fed induction machine. Furthermore, the rotor voltage sup- plied by the second stator can be regarded to be equal to Vr/S (Fig. 4). This is in contrast to the double fed machine which is equal to I~r . This difference doesn't allow the cascade machine to generate power at subsyn- chronous speed.

For parallel grid operation, the stators of the two machines are constructed so that, one stator is fixed but the other is mechanically adjustable. In other words,

194 D. Lew / Electric Power Systems Research 41 (1997) 191 201

one stator is made to be positioned at angle 0 relative to the other stator. This arrangement permits torque or power regulation of the generator and matches its characteristic to the prime mover.

The system can be designed as a single phase or polyphase generator with any number of poles as usual.

The two stators of the machine are normally con- nected together to the power lines when it is used to feed power in parallel with the national grid system.

The two machines together can be designed as a single unit with double stator windings [7-13] of differ- ent numbers of poles and a single special shape squirrel cage rotor. By this means, the machine has no slip rings or commutators and it is reliable as the standard squir- rel cage machine. However, an external phase shifter (normally an electronic converter) is used for adjust- ment of the machine characteristic. The two stator windings are effectively decoupled from each other due to the difference in the number of poles. The load is normally connected to one stator winding as shown in Fig. 4. The converter shown can be any type, voltage source or current source. This converter supplies reac- tive power to the machine at constant frequency. Prac- tically, it will be a constant voltage/constant frequency converter which has the same voltage level in its input and output, but, with a phase difference 0. This phase is therefore automatically adjusted to keep the load voltage constant independent of the speed and load variation. An ideal converter of this type consumes no power.

Since the load is normally connected to one stator winding, the power sharing of the two stators is equal. It could happen, depending on the type of converter used, that one stator supplies power to the load but the other absorbs it.

'The linear relationship between voltages and currents of this machine and the difference between its two sections are given in Appendices A and B.

4. I. Analysis

IrI'

(4.3) indicates this type of system can be operated at supersynchronous speed only or in other words 0 > S > -- ( R 2 / R J),

Therefore, the rotor current here is a dependent variable in contrast to the doubly fed system in which it is an independent variable. The torque is therefore:

T = R e ( l ~ E2 R2 (4.4) \Z IJ (o R 2 + SR 1

(4.4) indicates the torque increasing with speed and it is independent of the type of control used, and therefore stability of the operating speed is assured for most turbines, in particular wind turbines.

It is important to note, the stator resistance R 1 has an important role in dictating the slope of the charac- teristic as in the doubly fed machine.

Any excess power generated, will therefore be dissi- pated on the rotor and stator resistance which should be taken into account in any design.

For a lossless converter, the power output of each stator is calculated: for the load stator power

PI E2RI coME ( J[*, R~ q- (o2L 2 + Re R 1 @,)Li - ,] (4.5)

and the controlled stator power

Since the rotor current phase in (4.5) is an independent variable, then P~ and P2 depend on the type of control used. For a very light load, one stator supplies power but the other absorbs power.

4.2. Cascade machine operation under voltage controlled converter (Appendix C)

In contrast to the doubly fed induction machine with complicated torque/speed characteristic given in (3.2), the characteristic of the cascade machine can be shown to be given by:

, /.Z2\ T = - 2MIrlrRe~j Zk ) (4.1)

(4.1) is an exact relationship without any approxima- tion. For Z , &j(oM, (4.1) will be reduced to

R2 T = - 2IrI* (4.2)

(.o - f2

For a lossless converter of any type, it can be shown the rotor current should be constrained by the relation- ship

A common practice of operating a cascade machine is by a voltage source converter with equal voltage in its input and output, but phase shifted by 0. The transfer function of the converter:

[2: e a° (4.2.1) V,1

where 1?< = E = constant. The converter power loss r/is given by

Re(El* - Ee°I*2) = q (4.2.2)

Adding (4.2.1) and (4.2.2) to the cascade machine linear equations, it can be shown that such control is feasible for the r /= 0, as long as a solution for 0 can be obtained from:

D. Levy .'Electric Power Systems Research 41 (1997) 191 201 195

COs0=R.(~+Zkkj\I)(r2+

where

R2 r A R m + - -

S

L = L l + L2 -- 2 M

L 2 ) + 1 (4.2.3)

(4.2.4)

(4.2.5)

4.2. I. C o m p u t e r s imu la t ion

A precise compu te r s imula t ion was carr ied out for two s imilar wire w o u n d ro to r induct ion m o t o r s wi th

da t a as follows: L~ = 0.0656 H; L2 = 0.06557 H; M = 0.06325 H; R~ = 0.78 f~:

R 2 = 0.62 f~; Z 1 = 10 f~ (or 5 kW/phase ) and Z I = 20 f~ (or 2.5 kW/phase ) ; E = 220 V effective;

f = 50 Hz; I ron loss equivalent resistance = 250 f~ per phase; N o m i n a l power = 7.5 kW; N o m i n a l efficiency = 50% at full load as a motor . Fig. 5 shows the conver te r losses r/ as a funct ion o f

the speed, for 0 = constant . In o rde r to keep r / = 0 , over the speed range, 0

should be ad jus ted as per Fig. 6. The conver te r input power which is equal to its

ou tpu t power (for the case q = 0) or the con t ro l sta- tor power, is shown in Fig. 7. The power is negative,

DELAY THETA

-40

i -80

E -]O0

-160

2O

-180 ~ L 1 I 15 :. 11, ~13 ,.~ i 116 NORMALISED SP{~3)

Fig. 6. 0 as a function of speed for the case ~? = 0, voltage controlled converter, cascade machine, load as a parameter = 10 and 20 fh

tha t is the con t ro l s t a to r is supply ing power to the load at all speeds. Fig. 8 shows the power ou tpu t o f the load s tator . A t low speeds, up to 15% o f the synchronous speed, this s ta tor is also supply ing power to the load. However , a t h igher speed, it abso rbs power in o rde r to diss ipate the excess power gener-

a ted on the mach ine itself.

+i0 25C

2,]C

iG0

100

5O

0

-5C

CONV. PWR

- 8 4

2(}

11 I ~ A4 !15 16 i. 1.2 1.3 i. 1,

NOP, NAI.ISED SPEF~

Fig. 5. Voltage controlled converter losses q as a function of speed. 0 as a parameter, cascade machine.

,~c' CONV INP PWR 0

-20

-40

W -6C

26

-100

-120

-]40

-]60

Fig. 7. Voltage controlled converter, input power as a function of speed, q = 0 (equal to stator No 2 input power), cascade machine, load as a parameter = 10 and 20 fL

CONV. PWR

*I0 1000

-52

-180

0oo

w

~oo

4oo

2oo

o

-200

i l l 112 113 i14 l,'fi 116 /O0~IALIS]~B SP~)

Fig. 8. Stator No 1 input power as a function of speed t/= 0, voltage controlled converter, cascade machine, load as a parameter = 10 and 20 f2.

S1 PWR

4.3. Cascade machine operation under current controlled

conl)erter

Under current control led converter, an example o f a transfer function can be assumed in the form (Appendix C):

~2 = eio (4.3.1) Ii

The converter power consumpt ion q will be given by:

R e ( E " / * ) - Re(ffs2' e - J ° ' [ *) = r/ (4.3.2)

Adding (4.3.1) and (4.3.2) to the cascade machine linear equation, it can be shown that such control is feasible for the case r /= 0, if a solution for 0 can be obtained f rom

cos 0

A2+82+ + )

toM (4.3.3)

where

A g R ~ +

r = R l + S

L ~ L 1 + L 2 - - 2 M

DELAY THETA

1°000

]50

500 T

200

0

-2f0

196 D. Levy/Electric Power Systems Research 41 (1997) 191-201

1~1 ~12 1~3 21, 110 11,

Fig. 9. Current controlled converter, losses ~/ as a function of the speed, 0 as a parameter, cascade machine.

4.3.1. Computer simulation A precise compute r simulation was carried out for a

practical machine with the measured data shown in Section 4.2.1.

Fig. 9 shows the converter losses ~/as a funct ion o f the speed for 0 = constant . In order to keep q = 0, over the speed range, 0 should be adjusted as per Fig. 10.

The converter input power which is equal to its output power (for the case 1/= 0) or the control s tator power is shown in Fig. 11. This power is negative at low speed, that is this stator is supplying power to the load. At higher speed, this stator absorbs power in order to d u m p excess power on the machine itself.

-20

-30

-40

-50

-60

-70

-80

-90

111 112 113 ,i, 115 116

Fig. t0. 0 as a function of speed for the case r/= 0, current controlled converter, cascade machine, load as a parameter = 10 and 20 ~.

V. STATOR2 CONV INP PWR

%°

qco 20

w

Fig. 11. Current controlled converter input power as a function of the speed, r/= 0 (equal to the stator No 2 input power), cascade machine, load as a parameter = 10 and 20 ~.

300

250

200

15C

i00

20

20

• I I I , 14 . . i 12 i . ] . . 1 ,15 1 1 6

Fig. 13. Stator No 2 voltage as a function of speed, r/= 0, current controlled converter, cascade machine, load as a parameter = 10 and 20 [2.

The load s ta tor p o wer as in Fig. 12 is a lways

negat ive , tha t is, this s ta tor is the oppos i t e to the vo l tage con t ro l l ed conver te r , in a lways s u p p l y i n g power to the load.

The con t ro l l ed s ta to r vol tage, is therefore var i - ab le a n d is s h o w n in Fig. 13. Th i s vo l tage is very low at h igh speed.

5. Torque and efficiency

As ca lcu la t ed earl ier , the t o r q u e a n d efficiency o f the m a c h i n e are n o t affected by the type o f

con t ro l . These two p a r a m e t e r s are s h o w n in Figs. 14

a n d 15 respect ively a n d they m a t c h the ana ly t i ca l ca lcu la t ion .

1-02~ ~ 0 A

-30~

W -~CO

-50C

STI PWR

D. Levv ,"Electric Power Systems" Research 41 (1997) 191 201 197

il ] 112 l , I ] 11~ t.51 !16

Fig. 12. Stator No 1 input power as a function of speed, r/=0, current controlled converter, cascade machine, load as a parameter = 10 and 20 Q.

35

3O

N

~ 25

2O

10

1 ~1 1.12 | .I] 114 1 ,t5 1 .i6

Fig. 14. Torque/speed characteristic of the stand alone cascade induc- tion generator, load as a parameter = 10 and 20 f~.

198 D. Levy /'Electric Power Systems Research 41 (1997) 191 201

0,9

0.8

O.?

0.6

0.5

0.4

0.3

EFF

Io

2o

1.1 1.2 1.3 1.4 1.5 1.6

Fig. 15. Efficiency/speed characteristic of the stand alone cascade induction generator, load as a parameter = 10 and 20 ~.

pendent of the speed, therefore the power converter and the control mechanism of the system are much sim- plified.

Appendix A. Cascade (double stator) induction machine a n a l y s i s - - F i g . 4

The relationship between voltages and currents of the cascade induction machine are given by the following linear relationship

Z1 + Zk -- Zk is2

L z~ - z k 2 ( z2 + zo L L (1)

where

6. Conclusions Z 1 = R 1 + p ( L 1 -- M )

Ideal stand alone generators are almost constant power machines with decreasing torque/speed charac- teristic for constant output voltage and load. Induction generators are not an exception. However, practical machines, due to their losses, can offer an increasing torque/speed characteristic which is stable for electric power generation from most prime movers in particu- lar, wind turbines.

The squirrel cage machine, in practice is not suitable for stand alone power generation due to its sensitivity to the load. Excess power generated cannot be dumped on the machine itself or on a fixed load but on an auxiliary controlled load. The latter increases the con- verter complexity.

In the doubly fed induction generator, it is not practically feasible for the reactive power necessary for stand alone power generation, to be fed to the rotor, due to the large amount of losses involved which reduces much of the efficiency of the machine to an unacceptable level. Furthermore, for any type of con- trol, the rotor absorbs power at the practical operating speed range of the machine. Rotor power reversal happens only at a very high speed above its maximum capability.

The cascade (double stator) machine is different. The slope of its torque speed characteristic is positive inde- pendent of the type of control. Both stators supply power to the load at the practical operating speed range and any excess power is dissipated on the machine itself without an extra auxiliary controlled load. Due to its construction, both stators accept fixed frequency inde-

R 2 Z2 = --~ + p ( L 2 - M )

Z k = j o M or (jc0M in parallel with the equivalent iron losses resistance). For stand alone generator, l?sl is an independent variable Vsl ~ E = Constant. When the machine is connected to a power converter as in Fig. 4, Vs2 will be a dependent variable which satisfies:

Re (E '~ Vs2 * - "I~2) = r/ (2)

the converter input current ~ is also a dependent variable for given converter losses ~/(q = 0 for an ideal converter). [1 satisfy the following:

E /, = - ~ 1 - - - ( 3 )

Zl

/1 may be eliminated from (2) to give:

-- • T-* R e E l s l + Z --t- Vs2 I s 2 = - - Y/ ( 4 )

(1) and (4) are four equations with 17s2, /~2, Isl, Ir as unknown.

The torque per phase:

T = M R¢(j/~, .I*) - M R~(jls2" I*) = M R~j(~I -- I~2)I*

(5)

The mechanical power per phase:

P m = T" f~ (6)

The electrical power per phase:

D. Levy/Electric Power Systems Research 41 (1997) 191-201 199

The efficiency

Pm

(7)

(8)

~1 (t) ( (± T" ~ (

_o,(,) ( Y < -i 1 (t) -i 3 (t) -e3 (t)

Fig. 16. Voltage source phase shifter.

Appendix B. Comparison of the two sections of the cascade machine

Machine 1 Machine 2

Speed f~ - f2 Slip (co-f~)/co ( - c o + ~ ) / - c o

= S = S Rotor frequency c o - f~ - co + Stator frequency co co Torque T t T 2 Mechanical power Tlf~ - T2f~ Rotor current /~ - - I r

Rotor current @ 0 0 ~Tsl ~ V s 2

Torque @ l?sl = l?s2 0 0

Appendix C. Phase shift realisation

The phase shift between the two stators of two wire wound rotors induction machines with grid con- nection is normally carried out mechanically. One sta- tor is fixed, the other is mounted on bearings and geared to a small servo motor. Another possibility is the use of an additional wire wound rotor induction machine as a phase shifter. The adjustment of its ro- tor angle position relative to its stator will provide the necessary phase shift for controlling the torque/ speed characteristic. This arrangement is ideal for wind turbines in parallel grid connection which pro- vides adjustable torque at almost constant speed and therefore adjustable power.

With a stand alone system, a passive phase shifter is not relevant since it shifts both voltages and cur- rents by the same amount. Therefore, an electronic phase shifter is necessary. This can be done with the aid of converters in a closed loop. Voltage source and current source converters are possible.

C.I. The voltage source converter (Fig. 16)

The converter consists of a constant voltage and frequency D.C. to A.C. converter and A.C. and D.C.

converter. The connection of the two converters is made through a voltage source (energy sink) arrange- ment and a D.C. voltage source E which ideally sup- plies no current to the converter except at the starting period.

The two converters are chopper type and are inde- pendent, with the following defined and dependent variables, voltages and currents.

C.I.I. D.C. to A.C. Chopper

Defined input voltage E 2 & E [V] D.C. Defined output current ij(t) & 11 cos(cot - q)) Dependent output voltage el(t) & (4/rc)E cos cot Dependent input current 12 = I~ cos q) [A] D.C.

C.I.2. A.C. to D.C. Chopper (controlled rectified)

Defined input current i3 & ~ cos cot Defined output voltage E2 & E [V] D.C. Dependent input voltage e3(t) = (4E/zr) cos(cot - 6) Dependent output current I 2 = I 3 c o s J [A] D.C. where 13 cos 5 = 11 cos ~0.

The converter transfer function will be:

J~3 _j~

El

C.2. The current source converter (Fig. 1 7)

The converter consists of a constant voltage and frequency D.C. to A.C. converter and an A.C. to D.C. converter. The connection of the two converters is made through a D.C. current sink arrangement and a D.C. current source (consists of a battery and a high resistance in series). Ideally, the battery supplies no current to the converter except at the starting period.

12 E 2 12

-e I ( t ) ) T ( T T > % e)

-i I (t) -i 3 (t)

Fig. 17. Current source phase shifter.

200 D. Levy~Electric Power Systems Research 41 (1997) 191-201

The two converters are chopper type and are inde- pendent, with the following defined and dependent variables, voltages and current.

C.2.1. D.C. to A.C. chopper

Defined input current I2 ~ I [A] D.C. Defined output voltage el(t) & E cos cot Dependent input voltage E 2 = (2/rc)E cos 0 I Dependent output current i j ( t )= (4/~r)I cos(cot - 01)

C.2.2. A.C. to D.C. chopper (controlled rectifier)

Defined input voltage e3(t ) & E 3 cos cot Defined output current 12 & I Dependent input current i3(t)= (4/zc)I cos(cot - 02) Dependent output voltage E2 = (2/zr)E3 cos02 where E cos 01 = E3 COS 0 2

The converter transfer function will be:

/ • _3 eJ(01 -- 02)

Ii

C.3. Cyclo converter chopper

A bidirectional chopper operated at twice the load frequency with a chopping phase 0 relative to the input voltage will offer an ideal phase delay between its input and output as follows:

Defined input voltage & E cos cot Defined output current & I cos(cot - (p) Dependent output voltage = (2E/zr) cos(cot - 0) Dependent input current = (2I/rr) cos(cot - 0 + q)) The voltage transfer function = (2/zr)e j0 The current transfer function = (rc/2)e-J~2~° 07 The chopping phase 0 is to be adjusted in such a way

that the load voltage will be constant and equal to E. The load frequency is therefore dictated by the chopper frequency.

Appendix D. List of symbols

or ~ or A.C. to D.C. converter input current i3(t)

/r rotor current ImagO ima~nary part j x / - 1 Ll stator self inductance L2 rotor self inductance M mutual inductance, rotor/stator P1 input power to stator No. 1 P2 input power to stator No. 2 Pe electrical output power Pm mechanical input Power p j.co Rk iron loss equivalent resistance R1 stator resistance R 2 rotor resistance Re0 real part S slip T total torque T1 machine No. 1 torque /'2 machine No. 2 torque Vr rotor voltage V_-s, stator No. 1 voltage Vs2 stator No. 2 voltage Z1 RI +p(L1 - M)

Z2 R2 + p ( L 2 _ M)

Z 3 pSM or pM/'/Rk Zk p M or p M / / R k Z 1 load independence

Greek symbols c5 phase angle between voltage and cur-

rent 0 delay angle 01 phase angle between voltage and cur-

rent 02 phase angle between voltage and cur-

rent (p phase angle between voltage and cur-

rent ~/ converter Power Losses co load angular frequency f~ rotor angular speed

E o r /~ or el(t)

E2 E 3 or /~3 or

e3(t)

f

5, Is2 I~ or [l or

il(t) Iz or I

D.C. to A.C. converter output voltage or load voltage intermediate D.C. converter voltage A.C. to D.C. converter input voltage

frequency stator current stator No. 1 current stator No. 2 current D.C. to A.C. converter output current

intermediate D.C. Converter current

Speed Normalised S p e e d - Synchronous Speed

References

[1] J.E. Barkie and R.W. Ferguson, Induction generator theory and application, A1EE Trans., 73 (1954) 12 19.

[2] Gary L. Johnson, Wind Energy Systems, Prentice Hall, N.J., 1985, pp. 252-263.

[3] R. Bonert and G. Hoops, Stand alone induction generator with terminal impedance controller and no turbine controls, IEEE Trans. Energy Conversion, 3 (1) (1990) 28 31.

D. Levy~Electric Power Systems Research 41 (1997) 191 201 201

[4] B.T, Ooi and R.A. David, Induction generator/synchronous condenser system for wind-turbine power, Proc. IEE,, 126 (1) (1979) 69-74.

[5] D.B. Watson and J, Arrilage, Controllable D.C. power supply from wind-driven self excited induction machines, Proc. lEE, 126 (12) (1979) 1245-1248.

[6] D. Levy, Analysis of double stator induction machine used for VSCF small scale hydro/wind electric power generator, Electr. Power Syst. Res., 11 (3) (1987) 205-223.

[7] L.J. Hant, The cascade induction motor, lEE J., 52 (1914) 406-434.

[8] F. Creedy, Some developments in multi-speed cascade induction motors, lEE J., 59 (1921) 511-537.

[9] A.R.W. Broadway and L. Burbridge, Self cascade machine: A low-speed motor or high frequency brushless alternator, Proc. lEE, 117 (7) (1970) 1277-1290.

[10] B.H. Smith, Synchronous behaviour of doubly fed twin stator induction machine, IEEE Trans. Power Appar. Syst., PAS-86 (10) (1967) 1227-1236.

[11] B.H. Smith, Theory and Performance of a Twin Stator Induction Machine, Vol. PAS-85, No. 2, 1966, pp. 123-131.

[12] H. Owen Whiney, Induction Machine, US Patent 4,228,391, October 14, 1980.

[13] Rugi Li, Ren6 Sprc, Alan K. Wallace and G.C. Alexandra, Synchronous drive performance of brushless doubly-fed motors, 1992 IEEE Industry Applications Soc. Annual Meet., Houston, 1992, pp. 631-637.