I

Excitation of isolated three-phase induction generator by a single capacitor

Y.H.A. Rahim

Indexing terms: Generators, Induction motors

~

Abstract: This study attempts to determine the steady-state performance of an isolated, self- excited, induction generator when a single capa- citor is connected across one phase or between two lines. The work is concentrated on the case when the machine is supplying one or two loads. With the aid of the equivalent circuit, an analyti- cal technique has been used to obtain nonlinear algebraic equations for operation frequency and magnetising reactance. A numerical solution method is applied to solve the nonlinear equa- tions. Performances for different load config- urations, and different load conditions have been considered. Where possible, computed results have been confirmed with experimental results.

List of symbols

V = voltage (V) I = current (A) X = reactance (ohm) R = resistance (ohm) F = per-unit frequency v = per unit rotor speed k = effective turns ratio

Subscripts 1 = stator winding 2 = rotor referred f = forward sequence components b = backward sequence components c = excitation capacitance components m = load side winding n = capacitor side winding g = air gap

1 Introduction

The phenomenon of self-excitation in isolated induction machines has been known since the 1930s 111. The util- isation of such an idea in the generation of electric power was realised after the recently energy crisis, and the growing interest in the use of other energy sources. This has been motivated by concern to reduce pollution by the use of renewable energy resources such as wind, solar, tidal and small hydropotential.

Preference is given to self-excited induction generators, in conjunction with the conversion of wind energy,

Paper 9069B (PI), first received 24th January and in revised form 30th June 1992 The author is with the Electrical Engineering Department, King Saud Univerwy, PO Box 800, Riyadh 11421, Saudi Arahia

44

because of their ability to convert mechanical power to electrical power over a wide range of speeds. These machines are also characterised by their simple construc- tion, the absence of current collection gear, their robust- ness, and their low unit price and maintenance costs. This type of conversion has been found particularly conven- ient for isolated and remote loads.

Considerable work has been done on the analysis of capacitor-excited, balanced, three-phase induction gener- ators 12-81, However, the unbalanced operation of such machines has been given no attention. This mode of operation may sometimes be of great interest for various small-scale applications where balanced conditions are not necessary, such as single-phase emergency supplies, portable sources for remote construction sites and iso- lated line repeaters. In case of the failure of one or two capacitors in a machine with balanced excitation, the drop in power output will not be very great if the remain- ing capacitor is used for excitatlon - a matter which does not affect unbalanced loads.

In this study, attempts have been made to determine the steady-state performance of an isolated induction generator when a single excitation capacitor is connected across one phase (phase excitation) or between two lines (line excitation), Because there are many modes of oper- ation for unbalanced operation, this work concentrates on the case when one or two loads are connected to the free lines or phases. Symmetrical components have been used to obtain the equivalent circuit. With the aid of this circuit, an analytical technique has been applied to obtain nonlinear algebraic equations for circuit param- eters, load impedance, operation frequency and magnet- ising reactance. A numerical method was applied to solve the nonlinear equations, and hence the machine per- formance. Where possible, computed results have been confirmed with experimental results.

2 Mathematical model

Some of the possible connections of the stator of a three- phase induction generator, whereby the machine can be excited by a single capacitor, are shown in Fig. 1. A con- nection which is analogous to that of Fig. la, is when the loads and the capacitor are connected across phases, and the one that is analogous to that of Fig. Ib, is when the load and capacitor exchange places. The case where the load is connected across the capacitor has the disadvant- age that the excitation current is sometimes high enough to make the winding to which they are connected operate near to its heating limit. The little residual flux available on the rotor is enough to produce small voltages in the stator winding upon the movement of the rotor. With sufficient capacitance connected across one of the stator windings, this voltage will be enough to begin the process

/ E t PROCEEDINGS-8, Vol. 140, No. I , J A N U A R Y IYY3

I

of build-up of the excitation current. In such a case, the stator flux and the residual magnetism lie on the same axis, thus aiding each other to produce more voltage and more excitation current. As the excitation current builds up, the resulting air-gap flux drives the machine into

0

I

<

b Fig. 1 erutor when excrted by one cupacitor and supplying U single loud

Some possible stator connections for three-phuse induction yen-

saturation. Thus the magnetising reactance is gradually decreased until a stable operating condition, determined by the saturation characteristic, excitation capacitance, rotor speed and terminal load, is reached. Relationships between these variables are obtained by considering the equivalent circuit of the machine.

The analysis of the machine is based upon the follow- ing assumptions:

(i) Of all machine parameters, only the magnetising reactance is affected by magnetic saturation. The magnet- ising reactance is assumed to be proportional to fre- quency at fixed flux levels.

(ii) Core losses are neglected. (iii) Space harmonics on the air-gap flux are ignored.

Saturation may have some effect on the values of the leakage reactance, but this is expected to have little effect on the results. For low frequencies and low levels of sat- uration, core losses are small and in most cases little error is introduced by neglecting them. Ignoring air-gap harmonics means neglecting the effect of the 5th and 7th components, which are usually dominant in induction machines. Good winding designs may reduce or totally eliminate the effect of these components.

To simplify the analysis, the case where the machine is excited by a capacitor, connected across phase or line, and supplying a load connected to one of the other lines or phases, is considered first. This may then be extended to cover the operation of the machine when supplying two loads. As only two windings are involved, the two- phase symmetrical component transformation method [9] can be used to develop the equivalent circuit shown in Fig. 2. In this circuit, the effect of the winding to which

/ E PROCEEDINGS-B, Vol. 140, No. I , J A N U A R Y I Y Y 3

the capacitor is connected, and that of the rotor, has been transferred to the winding to which the load is connected, by the use of referred parameters. Because isolated induc- tion generators are characterised by their operation at variable frequencies, all parameters are referred to the operating per-unit frequency F. Also in the equivalent circuit, the per-unit rotor speed 1' has been used in place of the slip. Core losses can be taken into account by placing a resistor in shunt with the magnetising react- ance, but the derivations will be lengthy.

The relation between the two-phase symmetrical com- ponent voltages and currents, shown in the equivalent circuit, and the actual voltages and currents of the wind- ings are given by the following:

where the subscripts and , refer to the stator windings across which the capacitor and load, respectively, are connected and I and , refer to the symmetrical forward and backward components, respectively, and k is the effective ratio of the turns of winding to those of winding m . The two loop equations for the equivalent circuit are

V F 2 = Z , , I , - Z , , I ,

where

Zl, Z , " z,, =-+++ZzI 2 2k

z , , = - z,, + 2 Z," + z, 2 2k

R l , Z , , = - + j X , , F

RI" Z, , = - + j X , , + Z , F

(3)

(4)

Winding , is terminated by the excitation capacitor, whereas winding is terminated by the load impedance Z = R + j F X . Hence the terminal voltages are given by

V, = 21, = (II + 1,)Z (6) V, = 2, I , = j ( I , - I,)Z,/k (7)

45

I

If these voltages are substituted in eqn. 1, the sequence voltage will be given by

V, = (I,(Z - Zc/k2) + Ib(Z + Z,/k2))/2 V, = (I , (Z + Z , / k z ) + I,(Z - Z,/k2))/2

(8)

(9)

tJX R,,/F JXI, Fig. 2 Equivalent cirruit of a three-phase induction generator when an excitation capacitor i s connected across one winding and a load is con- nected across another

If the sequence voltages are then substituted in eqns. 3 and 4, with a little manipulation it can be seen that

( Z - 2 F Z , , - Z , / k 2 ) ( 2 - 2FZ2,) - ( Z + 2 F Z , , + Z/k2)' = 0 (10)

Substituting for Z , ,, Z , , , Z , , , Z , , Z, and Z, from eqn. 5 into eqn. 10, and equating the real and imaginary parts to zero, leads to the following two nonlinear simultan- eous equations in terms of the per-unit frequency F and the magnetising reactance X , of the winding to which the load is connected.

f ( F , X , ) = ( C , + C , X , + C, X i ) F 5 + ( C , + c, x, + c, X 3 F 3 + ( C , + C 8 X , + c , X ; ) F = 0 (11)

g(F , X , ) = (Dl + D , X , + D , X i ) F 4 + (D4 + D , X , + D, X i ) F 2 + D, + D , X , + D y X i = 0 (12)

where the coefficients C , to C y and D, to D, are given in the Appendix. For capacitive loads, similar forms of equation can be derived. In case of no load, all coeffi- cients tend to go to their limit when the load resistance R goes to infinity.

3 Computation and results

A three-phase 1 hp, 380/220 V, Y/A, 60 Hz, four-pole induction machine was found suitable for the laboratory tests. The parameters of this machine were obtained using the standard DC, locked rotor and no-load tests, with the machine running as a single-phase motor, sup- plied across one phase or between two lines. To obtain more accurate results for the magnetisation character- istics, the machine was driven, at synchronous speed, by a synchronous motor, and the input impedance was meas- ured at different values of supply voltage. It has been assumed that the leakage reactance of the stator winding is equal to that of the rotor when referred to the stator. The parameters of the machine for the phase and line supplies are shown in Table 1. The saturation curves,

46

expressed in terms of air-gap voltage V,/F and magnet- ising reactance X , for one-line and one-phase supplies, are shown in Fig. 3. For computation purposes, the satu- ration curves are represented by appropriate curve fits.

Table 1 : M a c h i n e parameters for phase and l ine suppl ies

Measured parameters SUPPlY connected

across

phase line

Stator resistance (ohm) 21.2 10.6 Stator leakage reactance (ohm) 15 0 32.4 Rotor resistance referred 7 6 18.3

to stator (ohm)

250r

i $ 1 0 0 ~

150 175 0 z501f magnetising 1;s reactance,ll a

100)

xx

0 250 300 350 400 450 500 550 600

magnetising reactance.ll

b

5011 ' ' ' ' ' ' : ' ' Fig. 3 Magnetisation curves n Single phase h line supply

x x x exwrimental fitted curve ~-

The induction machine was coupled to a variable- speed DC motor. A bank of capacitors is connected to the terminals of the machine, as appropriate. Per- formance results were recorded for different speeds, excitation capacitance and load conditions.

The computation of results is based upon the numeri- cal solution of the two nonlinear simultaneous eqns. 11 and 12. In this study, the numerical method of Newton- Raphson [2,4] has been found suitable to determine the values of F and X , . An initial guess for the unknowns has to be provided, together with the rotor speed, excita- tion capacitance and load impedance. The saturation curve fits are then used to obtain the air-gap voltages

IEE PROCEEDINGS-B. Vol . 140, No. I , J A N U A R Y 1993

I

600-

500-

400 . m

300- 0

700-

100-

I$,/F and VJF. The symmetrical component voltages V, and V, in Fig. 1 may then be calculated and, hence, other performance indices.

/

~

35r phase excitation

1000 1500 2000 2500 3000

speed. r p rn Fig. 4 excitation

Minimum capacitance and speed requirement for phase and line

no-load ~- with fixed resistance

01 1 1500 1750 2000 2250

speed,r p m Fig. 5 Variations ofthe no-load line terminal uolfages ofa three-phase induction generator when a capacitor of 10 p F is connected across one line

~ ~~ voltage across capacitor x x , 00. + + experimental results

~~~ voltage across open circuited lines

When a second load is connected to the machine, the method of superposition can be used, after applying the above procedure to each load separately. The use of this method in a nonlinear system may produce some errors, but the analysis would otherwise be cumbersome.

3.1 Onset of self-excitation The process of self-excitation in induction generators is in some way similar to that in DC self-excited generators. For each terminal capacitance, the machine continues to operate successfully until the speed or excitation capa- citance goes below specific critical values. Because the line and phase saturation characteristics are different, dif- ferent critical speed curves for line excitation and phase excitation are also expected. Fig. 4 shows the variations in minimum capacitance required for machine excitation, with speed, for phase and line excitations. Critical speeds for line excitation are lower, because of the relatively higher line magnetising reactance. For some remote applications, a load of fixed impedance may be per-

I E E PROCEEDINGS-B, Vol. 140, N o . I , J A N U A R Y 1993

manently connected across the machine terminals. For this case, a fixed resistance, that would draw rated current at rated voltage, is connected across the load ter-

10 0 , 1000 1500 2000

0

0

I

100

1500 2000 2500 speed,r p m

b

Fig. 6 Variations with speed of open-circuit voltage and voltage across capacitor U line excltallan h phase excitation ~ ~~~ voltage across capacitor

00 + + experimental results voltage across phase or between lines

minals. The computed results of critical speed and capa- citance are included in Fig. 4. It is apparent that, for a certain speed, more capacitance is required to start the excitation, in the presence of a load.

3.2 Open circuit characteristics The open-circuit performance of the machine with line and phase excitation is considered. It is interesting to show, first, the voltage across the three lines, for line excitation. The results for a capacitance of 10pF are shown in Fig. 5 . The agreement between the measured and the calculated values is good, except for some dis- crepancy at higher speeds. This may be due to a failure to obtain accurate measurements for highly saturated values of the magnetising reactance, because of the danger of overheating the winding. It can be seen that the line volt- ages are not equal. The net air-gap flux produces equal EMF in the three lines. The excitation current in the line containing the capacitor leads the induced E M F in that line by a right-angle, thus causing a higher terminal voltage. With the aid of a simple phasor diagram, it can be shown that the voltage drop due to this current in the phases common with open circuited lines, affects the

47

I

magnitude and phase of the respective terminal voltages, but to different extents. When the excitation capacitor is connected across one phase, the terminal voltages of the two other phases are almost equal.

4OOr

300 -

> - a

0 0 - -

01 a

r

0 0 5 10 15 2 0 25 load current, A

b Fig. 7 Variations of load characteristic and excitation and common phase currents with load current, when excitation capacitor and load are connected across different lines U load characteristic h excitation and common phase currents __ excitation current ~~-~~ common phase current 00 + + measured values

Fig. 6 shows the variations with rotor speed, of the no-load voltage across the free line and phase, for line and phase excitation and a range of excitation capa- citances. The figure also includes variations of voltage across the excitation capacitor. The agreement between the measured and the calculated results is reasonable. It can be seen that, for phase excitation, and at the begin- ning of onset, the voltage between the terminals across which the capacitor is connected, and that of the open circuited line, are nearly equal. This is also true for the voltage between the terminals of the line across which the capacitor is connected, and that of the open circuit phase. When excitation starts, the rate of increase of the open circuited line voltage is greater than the corresponding phase voltage. The results in Fig. 6 also show that phase and line excitation complement one another, in the sense that, for the same range of capacitance, phase excitation

48

operates in the range above rated speed, whereas line excitation operates in the range below rated speed.

3.3 Load characteristics According to the available range of capacitance and speed, loads may be connected across lines or phases. Fig. 7 shows the load characteristics of the generator

I \

D P

/ CzlOpF C=ZOuF N=2220 r p m

P

100 1 ?': CzlOpF

N ~ Z O r p m

'0 0'2 0'4 06 O b l b 1'2 1'4 1'6 1; load c u r r e n t , A

Fig. 8 Variations ofload coltage with load current _ _ capacitor a c m b line and load across phase. predicted

00 + + predicted measured .~ ~ capacltor across phase and load across line

when line excitation is used and a pure resistive load is connected to one of the free lines. The results include variations in the load voltage and excitation current, in addition to the current in the phase that is common between the load and the capacitor circuit, for two values of excitation capacitance, namely 10 and 2 0 p F . The correlation between the measured and the calculated results is good. In the normal operation regions of the machine, the maximum error in voltage is about lo%, and that in the excitation current is about 7% of the measured values. The load voltage gradually decreases with increased load, until i t doubles back on itself in a way similar to that in DC shunt generators. The two voltage curves are almost parallel to one another. The excitation current takes a shape similar to the shape of the load voltage. In such cases, one should be concerned that the current in the common phase does not exceed the winding heat limit, until rated power is supplied. The results show that this current decreases sharply with increased load, leaving no doubt about any possibility of going above the no-load value. This may not be true for other types of load, or when a second load is present.

Phase and line excitation are also compared by con- necting the capacitor across a line, and a load across a free phase, and vice versa. The load characteristics for the two cases are compared in Fig. 8, for the same rotor speed. Although a higher no-load voltage is used in the case of phase excitation, a relatively lower range of oper- ation and less output power is obtained. Comparison of the results with those in Fig. 7 shows the superiority of line excitation and line supply.

3.4 General performance Having established the validity of the computer model, the technique is used to study the performance of the machine under different operating conditions. Because of

IEE PROCEEDINGS-B, Vol . 140, No. I , J A N U A R Y I Y Y 3

I

its superiority over phase excitation, the following dis- cussion will be limited to the case of line excitation, when equal load impedances are connected to the free lines. To generalise their applicability, the results in this section will be presented as per-unit values for the test machine.

Fig. 9 shows the variations in the supply frequency and terminal voltage of the two loads, with load currents

01 1 0 0 5 1 0 15 2 0 2 5

load current, pu Fig. 9 speed = 1800 r.p.m., C = 15 p F

~ load voltage (unity pi) . . . . . . . load voltage (0 8 pi lag)

~~~~ load voltage (0 9 pf. lag) ~ frequency

for different values of power factor. The frequency results for the considered cases are almost identical. Within the operation range, the results show very little change in fre- quency. As expected, the voltage regulation increases with the decrease in power factor. The regulation for the line with the higher open circuit voltage is greater than that of the other.

Power calculations have shown that the total power supplied to the two loads at rated conditions is about 85% of that supplied by the machine when balanced excitation is used. If, in addition, a small load is con- nected across the capacitor, the load supplied to the two other loads will decrease a little, but the overall load will be increased to about 91 %.

Another family of load characteristics for different values of excitation capacitance is shown in Fig. 10. The

Effect of power .factor on load voltage and supply frequency.

C=lOpF 0

1 0 2 0 3 0 40 5 0 load current, pu

Fig. 10 ~ load voltage allrnc I

load voltage of line 2

Effect ofexcitation capacitance on the load characteristics

~

I E E PROCEEDINGS-B, Vol. 340, N o . I, J A N U A R Y I993

curves, which are almost parallel, indicate a regular increase in the terminal voltage with excitation capa- citance. However, the rate of this increase goes down with increased capacitance, a matter that indicates an operation in the deep saturation region.

The inherent characteristic of wind energy implies changes in the prime mover speed over a wide range. To provide suitable excitation and regulators, the effect of speed on the load voltage should be determined. Fig. 11

"0 10 2 0 3 0 40 5 0 6 0 7 0 load current, pu

Fig. 11 ~ load voltage of h e I

~ ~ load voltage of line 2

Effect of roror speed on the load characteristics

shows the family of load characteristics for a fixed capa- citance of 15 p F and a wide range of rotor speeds. The terminal voltages have a regular increase with speed. Also the voltage regulation shows greater reduction with speed.

4 Conclusion

In this study the steady-state performance of an isolated, self-excited, induction generator, excited by a single capa- citor connected across one phase, or between two lines, has been described. The work is concentrated on the case when the machine is supplying one or two loads. Sym- metrical components have been used to obtain the equiv- alent circuit, which is used to derive nonlinear algebraic equations for operation frequency and magnetising react- ance. The numerical method of Newton-Raphson has been found suitable to solve the nonlinear equations, and hence the machine performance.

The computed and the measured results, which show reasonable agreement, confirm that the induction gener- ator can be successfully excited by a single capacitor. In general excitation with a single capacitor, the result causes some differences in the line and phase voltages of the generator. Results have shown that phase and line excitation complement one another in the sense that, for the same range of capacitance, phase excitation operates in the range above rated speed, whereas line excitation operates in the range below rates speed. However, line excitation gives a wider range of operation and a greater output power. The machine thus provides a good poten- tial for wind generation, in almost balanced and unbal- anced modes of operation. The results have shown the ability of the machine to convert power comparably to when balanced excitation is used, but the supply would then be suitable for unbalanced loads.

49

I

5 References

I WAGNER, C.F.: 'Self-excitation of induction motor with senes capa-

2 RAHIM, Y.H.A., MOHAMADIEN, A.L., and AL KHALAF, A S citors', Trans. AIEE, 1969, IO, pp. 1241-1247

'Comparison between the steady-state performance of self-excited reluctance and induction generator', IEEE Trans., 1990, EC-5, (3) pp. 519-525

3 ARRILAGA, J., and WATSON, D.B.: 'Static power conversion from self-excited induction generators', IEE Proc., 1978, 125, (81, pp 743- 746

4 MURTHY, S.S., MALIK. O.P., and TANDON. A.K.: 'Analysis of self-excited induction generators', 1 E E Proc. C.. 1982, 129, (6) . pp. 260-265

5 ELDER. J.M.. BOYS, J.J. , and WOODWORD. J.: 'Self-excited induction machine as a small low-cost generator'. IEE Proc. C, 1984, 131, pp. 33-41

6 OUAZENE, L., and MrPHERSON, G.: 'Analysis of the isolated induction generator', IEEE Trans.. 1983, PAS-102, ( X ) , pp. 2793-2797

7 MURTHY, S . S . NAGARA, H.S., and KURIVAN, A.: 'Design-based computational procedure for performance prediction and analysis of self-excited induction generator using motor design packages'. IEE Proc. E. , 1988, 135, ( l ) , pp. 8-16

8 MURTHY, S.S.. and SINGH, B.P.: 'Studies on the use of conven- tional induction motors as self-excited lnduction generator', IEEE Trans., 1988, EC-3, (4). pp. 842-848

9 HANCOCK, N.N.: 'Matrix analysis of electrical machinery'. 2nd ed. (Pergamon Press, 1974)

6 Appendix

6.1 The voltage equations for the two involved windings are:

Derivation of the equivalent circuit

Substituting for V , I and V, in eqns. 13 and 14 in terms of the symmetrical components of eqns. 1 and 2 gives

VJ. + 5 = Z,,(l, + I b ) + Z,l, + Z,l , F F

(1 5 )

When these two equations are solved simultaneously, the results are given by

The simplified form of these two expressions is given by eqns. 3 and 4. Since the machine is terminated by the load impedance Z across winding m, then

Substituting for the values of V, and V, in eqns. 17 and 18 gives two loop equations which can be expressed by the circuit in Fig. 2.

50

6 . 2 Coefficients of eqns. 1 1 and 12 c, = x,x,,x: C , = 2 X , X , , X 2 + k 2 X l X : + X , , X : C , = k 2 X , X 2 + k 2 X : + X , X , , + X , , X , C,= - R , R , , X ; - 2RlR ,X , ,X2 - v ~ X , X , , X :

- X l X , X : - R : X , X l n - 2 R 1 , R , X 1 X 2

C, = -2 (kZR ,R , X , + R , R , X I , + R , R l n X Z ) - V2kZXlX; - vzx,,x; - 2 v ~ x , x , , x , - k Z R : X , - 2 X i X , X , - X , X : - R : X , ,

- 2 R , , R , X , - 2R, ,R ,X2

C , = - k Z R , R , - R,R, , - v Z k 2 X , X , - vZk2X:

- kZR: - R,,R, - v z X , X , ,

- x ,x , - x,x, - v 2 x , , x , C , = R, (R , ,R ; + 2 R 2 X , X , + v2R,,X:)

+ v 2 X , X , X : + R ; X , X , C 8 = 2R, (R ,X , + vzR, ,X,)

+ v z ( 2 X , + X , ) X , X , + R : X , c, = V ~ R , R , , + V ~ X , X , + v 2 x , x , D l = - R , X , , X : - 2 R 2 X , X , , X 2 - R , , X , X :

D, = - k Z R , X : - 2 R 1 X , , X 2 - 2 k Z R 2 X , X 2

- 2 R z X i X 1 , - 2 R , X , , X ,

- 2 R , , X , X , - R, ,X:

D, = - k 2 R , X 2 - R I X , , - k 2 R , X I - 2k2R2X2

- RZX," - Rl"X1 - RI"X2

D, = 2R,R , ,R ,X2 + 2 R , X l X , X 2 + R, ,R:X i + R i X , X : + v2X: (R i ,X1 + R,X,,) + R l R : X , ,

D, = R , , R: + 2R2 X , X , + 2R2 X , X I + 2R1RinR2 + 2R1X, X2 + v 2 ( 2 R l X l n + 2R, ,X i + R,,X,)X, + kZR,R: + v2k2R,X:

D, = R , X , + R , X , + Y ~ R , X , , + v2R,,(Xl + X,) + v2kZR,X2

D , = -R,(R: + v 'X;)X, D, = - 2 v Z R i X , X ,

D , = - v Z R i X ,

R I = R + R I , x , = x + x,,

The resistance of the capacltor may be included in R, , . I E E PROCEEDINGS-B, Vol 140, No. I , JANUARY 1993