Study of a Particular Recovery Regime of the Synchronous Machine and its Applications

Aurel CAMPEANU, Ion VLAD, Sorin ENACHE, Augustin IONESCU

University of Craiova, Romania, e-mail: acampeanu@em.ucv.ro

Abstract – In the paper the steady state operation of two mechanically coupled and parallel electrical connected synchronous machines to the mains are theoretically studied and experimentally tested. This issue is of practical interest and there are no detailed references in the literature. The dynamic behavior of the synchronous machine is analyzed in [7] and it is proved that in all practical circumstances the parallel work of mechanical coupled synchronous machine to strong or weak mains is possible. The recoverable character of the synchronous machine work is fundamental and is used in this paper for the study of the rated loaded machines. The resulting test method is very suitable in comparison with other methods [8], [9] especially for high power machines.

I. INTRODUCTION Let 1, 2 be the two synchronous machines mechanically

coupled and parallel connected to the main. The parallel operation is defined by the following compact system of equations: =1I I1a(U,UeE1,θ)+I1r(U,UeE1,θ);

I2=I2a(U,UeE2,θ)+I2r(U,UeE2,θ); Ir=Ira(U,UeE1,θ)+I1rr(U,UeE1,UeE2,θ); (1)

PM1=PM1(U,UeE1,θ); PM2=PM2(U,UeE2,θ); PM2(U,UeE2,θ)= PM1(U,UeE1,θ)+Σ Pmec;

where I1, I2, Ir are the currents of the machines 1, 2 and the line input current respectively while Iia, Iir (i=1,2,r) are the active and reactive components of this currents. The phase shift between the e.m.f. 1eEU , 2eEU induced by the excitation currents of the coupled machines depends on the lowest electrical angle α between the axes of the two poles having the same polarity. θ is the internal angle of the machine 1 and obviously, (α-θ)is the internal angle of the machine 2; PM1 and PM2 are the electromagnetic power and ΣPmec is the total of the mechanical and ventilation losses. From the last equation (1) it results that for every pair of values (UeE1, UeE2)<0, or (UeE1, UeE2)>0 another values for the internal angle θ, positive or negative and also other values for the electromagnetic power and the active and reactive components of the currents correspond. The following conclusion results: a great change of the two machines excitation currents leads to large scale modifications of the active and reactive load; the selection of the appropriate values is possible so that for one of the two machines a determined ratio between its active and reactive load is established. In a particular case this load can be the rated load. The electrical angle α is essential for the second machine load establishing.

II. MAIN RESULTS

A. Particularities of the considered regime A.1. The synchronous machines 1-2 are considered as sunken poles constructions.

When considering all losses, the current of the machine 1 assumed generator has the form:

)(

111

1111 βγθγ −+−= j

s

eEj

se

ZU

jeZUjI UU = (2)

The following relation is valid for the electromagnetic power 1MP :

21111

1

11 )sin( eE

s

eEM UA

ZmUUP ++−= βγθ (3)

21

21

211

211

1

11

1

11

)2sin(sin

sm

mm

sm ZZZRZR

ZZA +

=−

−=γββ (4)

In (2) and (3) there have been noted [6]: 1

1111βj

mmmm ejZjXRZ −=+=

1112//

1 2 mm LKmR ω= 111/

1 2 mm LKmX ω=

where 1mR , 1mX 1mZ - the equivalent resistance and magnetization reactance, corresponding to the iron losses, the magnetization impedance, respectively; 111mL - one phase proper main impedance. The following known notations have also been used:

111 jXRZ += - the leakage impedance; 1

1111γj

sms ejZZZZ −=+= - the synchronous impedance. The value )( ϑα −− instead of ϑ will be considered in (2) and (3) for the machine 2 assumed as being motor. A.2. Static stability analysis The last equation from (1) is taken into account; it can be brought at the following form by computation:

CmA =−− )sin( βθα (5) The equation (5) plotted in the figure 1 leads to the conclusion that the group of the two machines behaves to

Fig. 1 The equivalent characteristic PM(α-θ)

mains as a single synchronous machine having the internal angle )( ϑα − , but its mechanical characteristic

)( ϑα −= fPM does not pass through the axes system origin,

being shifted with the angle β and the ratio s

eE

ZU

is given

by the relation: 2

2

20cos

2

2

1

122

1

1

+

+

=

sZeEU

sZeEU

sZeEU

sZeEU

sZeEU

α (6)

seE ZU , the equivalent polar e.m.f. and synchronous impedance, respectively.

In (5) s

eEm Z

UUA =

meceEeE pUAUAC Σ++= 222

221

and

22

0cos1

1

2

2

0sin1

1

βγ

α

α

β +−

+

=

sZeEU

sZeEU

sZeEU

arctg (7)

where )( 12120 ββγγαα −−−+= .

The losses occur both when the abscissa β where the sine curve passes through zero is determined, and in the value of C representing the synchronous machine mechanical load. Since C is a power of the same magnitude order as the losses, the equivalent synchronous machine is practically discharged from mechanical power. The operation is possible if an intersection point a can be established (fig. 1), so if the pairs of values ( 21, eEeE UU )fulfill the condition:

( )2222

221

2

2

20

2

2

1

12

1

12 cos2

meceEeE

s

eE

s

eE

s

eE

s

eE

pUAUA

ZU

ZU

ZU

ZU

U

Σ++

≥

+

+

α

It is noticed from (7) that the excitations influence the value of β and therefore the value of )( ϑα − . It results that the adjustment of the two mechanically coupled machines excitations leads to large variations of the active loads. The analysis of the mechanical phase shift and of the excitations influence on stability, leads to the conclusions that for given excitations, the lower the value of 0α is the greatest this influence is; for a given 0α , mA is modified with the excitations after some hyperbola type curves (fig.2, [2]) which show that the static stability reserve increases with excitations when 2/0 πα < and may increase or decrease when 2/0 πα > .

Fig.2. Characteristic curves for mA .

B. Consideration of the machines1, 2 having salient poles The phasors diagram corresponding to the coupled machines (1) is depicted in the figure 3.

Fig.3. Phasors diagram when the losses are considered. B.1. Static stability analysis In order to simplify the conclusions, the ideal case when the losses are neglected is considered further on. As in the case of the sunken poles machine an equivalent mechanical characteristic is also established, ///

MMM PPP += ,in which:

[ ]ϕϑαβϑα −−=−−= )(2sin);sin( /// rUPAP MmM (8) It can be considered that there are two equivalent machines on the same shaft: one of them corresponding to the excited sunken poles machines ( /

MP ) and one of them corresponding

to the unexcited machines with salient poles ( //MP ); by

overlapping the effects the equivalent mechanical characteristics is obtained. In (8) β,mA are given by (7) where the losses are neglected, r and ϕ are established as functions of the two axes reactances and the mechanical decalage. The figure 4 presents the curves MP for 01 =epU and

02 <epU (demagnetizing excitation current) so that

IIIeIIeIe III 222 << .

Fig. 4. The equivalent characteristics PM(α-θ)

In order to establish analytically the internal angle for a pair of given values 21, epep UU , there are computed ϕβ ,,, rAm and

the equation 0=mP is solved; from the stability condition for the equivalent machine, as for every synchronous machine, only the solutions fulfilling the condition

0)( )(

>

− − i

ddPM

θαϑαare kept from the solutions ( t)ϑα − .

The equation resolution is difficult. A fourth order equation in x is obtained by writing x=− )sin( ϑα ; if all the solutions

i)( ϑα − are real, as it is also shown from the curve 1 from the figure 4 meeting four times the abscissa during a period, two of them are adequate from the stability point of view, if two solutions are imaginary, a single possible solution remains (the one also fulfilling the condition that the synchronized power in the given operation point to be positive). The possible value or values i)( ϑα − are more easily established in graphic way by depicting in the same axes system the fundamental sine curve /

MP and of frequency //MP

and by plotting the resulting curve MP . Its intersections with the abscissa lead to the searched solutions (for example the solutions i)( ϑα − in the case of the curve 1from the figure 4

are the ones corresponding to the points // 4,4,1,1 ; only the values given by the abscissa /1,1 for which it is shown that

0)(/ >−ϑαddPM are kept. The analysis leads to the conclusions: -unlike the sunken poles machines for which there is a single accepted solution for )( ϑα − in an interval of π2 , owing to the salient poles it is possible to determine in general two positions of stable synchronous operation in the same interval, the probability being greater when the excitations are lower. In determined conditions, for internal angles around the limit values (in the fig. 4 for the case of point 2) the excitations modification may cause the saltus displacement of the rotors from one position to another stable position; -for mechanically coupled salient poles machines it is always possible to establish at least one position of stable operation, irrespective of the excitations magnitude and sense because the curve MP in a period must pass through zero at

least two times. The synchronism cannot be kept in a single

case: for 2πα = and unexcited identical machines when

0,0 == rAm and therefore the resulting curve MP merges with abscissa; -the angles i)( ϑα − established graphically or analytically are different in a certain extent from the real ones because the real resistant torque to the equivalent machine shaft is different from zero. The smaller the equivalent synchronizing power is the greater the deviations are. However owing to the small values of the losses torque 1/( 0 <<nMM and decreases with the increase of the machines power), its influence becomes important only around the values

lim)( ϑα − ;-the excitations modification leads to the equivalent

mechanical characteristic displacement and therefore to the modification of the active and reactive loads as in the case of the sunken poles machines.

III.CAPITALIZATION OF RECOVERY CHARACTER OF CONSIDERED REGIME

For full load test of high power synchronous machines, the

literature [8], [9], indicates recoverable methods involving besides the studied machine a great number of supplementary machines rated at comparable powers. This fact limits the possibility of utilization for powers of 1000 ÷ 2000 kW, otherwise the test stand would have to be fitted out with too great machines, which does not occur in practice. In the case when the powers exceed the powers available in stands, artificial tests are made because they need a reduced test power but they do not provide conclusive results. On the basis of the presented method there is shown the possibility of direct, recoverable loading, at rated load for synchronous machines rated at powers greater than the installed power of the test stands, without being necessary special installations and measures.

Fig. 5. Phase diagram for P2=const.,α=var.

In accordance with the method, the tested machine is coupled mechanically rigid with an identical machine or a machine rated at a power which is at least equal; the group machines is supplied by a common mains. The required active and reactive loads, particularly the rated regime, can be established for one machine by adjusting the excitations. Convenient values for the mechanical decalage of the rotors α must be taken in order to result admissible values for loadings corresponding to the auxiliary machine and to the mains. In the simplified diagram (fig. 5) in which the losses are neglected, it is considered that 2 is the machine operating as over-excited motor at given values 22 cos, ϕI date. Further on it is considered that 1 is the auxiliary machine. It is noticed that this loading can be achieved for different values of α .

There is established 2/1 II −= and 0=rI for the mechanical

decalage /α , therefore an alternating current mains is not necessary anymore. An auxiliary direct current or alternating current motor is necessary for drive, its power being small, equal to the losses from the machines system. It is shown that the active and reactive loadings cannot be modified separately, the given values 22 cos, ϕI can be established

only for /αα = , therefore it is necessary to couple the rotors with a well established mechanical decalaj. For all other values of 0, ≠rIα an external mains is necessary. If IVI1 and

///1I are admissible values for the machine 1 currents, the

mechanical decalage may get any value fulfilling the condition IIIIV ααα << , if the mains allows to establish currents varying between IV

rI and ///rI ; when the mains is if

low power, the range of the admissible values for α is restricted to such limits so that admissible values correspond to the current rI .

The ones presented in the figure 5 correspond to the ideal case when the losses are neglected. In reality, the active powers exchanged by the machines 1, 2 with the mains are different ( 12 PP > ) and the current rI has also an active component of variable magnitude dependent on the losses occurring at different loadings. The advantages of the connection on external mains over the first discussed case ( /αα = ) result: -the rotor coupling under a precise mechanical decalage is not necessary anymore, the admissible values of α may oscillate in a certain range; -the drive motor is not necessary anymore; -there is the possibility for improving the mains ϕcos (for

example in the case ///αα = ) during the test. In the real case when the losses are considered, for the rated load on one of the coupled machines, the optimum value of

1αα = is established on graphic and analytical way. In order to simplify the things the iron losses are enclosed in the no-

load losses and when choosing the mains current which is to establish, only its reactive component is taken into account, the only one which can be modified as we want, the active component dependent on losses modifying in restricted limits. The stages for establishing 1α are the following ones: First of all the current of the auxiliary machine 1 is established as magnitude and phase as follows: -the efficiency 2η of the machine 2 corresponding to the given rated load ( nnI 22 cos, ϕ ) is determined; -the current of the machine 1 is determined approximately by making the geometric sum between the current 2I and the mains current which is to establish and the efficiency 1η is pre-determined on this basis; -the group efficiency is established

21ηηη = (9) -the current 1I is determined. There are the following relations between the active and reactive components of the currents 1I and 2I

rrrraa IIIII =+−= 21211η

(10)

rrI is considered known so the new more exact value of 1Ican be found from (10), therefore the efficiency 1η can be established in the second approximation and the precision level for the current 1I determination can be increased on this basis. The results given by (10) are usually fully satsfactory at their first utilization. If the machine 2 must operate as over-excited motor at given values 22 cos, ϕI then the auxiliary machine current in magnitude and phase is given by the relations

( )1

221

2221

sin1sin

cos1

ϕϕ

ϕϕ

ηϕ

rr

rr

III

II

tgtg

±=

±=

(11)

In (11) the sign (+) is considered if rI2 has the same sense with rrI . The internal angles of the machines 1, 2 are determined in accordance with the following The internal angles of the machines 1, 2 are determined in accordance with the following relations by knowing the active and reactive components of the currents 21, II :

aqr

rqa

aqr

rqa

IXIRIXIRU

ctg

IXIRIXIRU

ctg

1111

11111

2222

22221)(

+

−+=

+

+−=−

ϑ

ϑα

(12)

and therefore the searched decalage 1α .The relations (12) correspond to the operation of the

machine 2 as over-excited motor and of the machine 1 as under-excited generator.

In [2] there is a graphic way for establishing 1α , as well as the way for determining further on on the basis of the ones presented before the excitation currents when the saturation is considered. The possible values of the mechanical decalage are analyzed because in the real case of the machines coupled by bolts, one is interested in obtaining values α as close as possible by the ones obtained by computation. It is shown that this one can be modified by relative rotation of the couplings, by terminals permutation and by the rotation sense reversal (when the machines do not have determined rotation sense). If

0gα is the geometric decalage between the rotor axes and the

coupling has n bolts, the possible values of iα measured in electrical degrees are given by the relation:

3

220

ππαα ±

+=

nKp igi (13)

For the real mechanical decalages iα given by (13) which deviate with α∆± over the pre-determined 1α , one is interested especially in the modification way of the main necessary power. If the losses are neglected, the mains current (and therefore the power) in values related to the rated current of the main machine 2 is given by the relation

)(1)( 121

2 αϑ ∆±+−= ctgIX

IPI aq

rrrr (14)

If the pre-determination of iα is made so that ,0=rrI then

1ϑ from (14) is given by

aq

rq

IXIX

arcctg21

211

1−=ϑ (15)

From the analysis of the relation (14) it results that it is sufficient from practical point of view that 010≅∆α for which nr PP 225,0≅ . It is shown that this value can be obtained without special measures by taking into account all means for obtaining a α∆ as small as possible. It is possible to emphasize the important conclusion that the test of the synchronous machines rated at powers approximately four times the mains power can be made without difficulties, over this ratio of the powers a series of measures must be taken, but these meaasures are not difficult from practical point of view (for example by considering some cuple with great number of buloane).

IV.EXPERIMENTAL VERIFICATIONS A.Experimental study of the steady state synchronous regime operation for mechanically coupled machines The variation curves of the currents and powers with the excitations are plotted for two identical machines 1,2 rated at 150 kVA (fig.6).

Fig. 6. Curves P1(IeE2), P2(IeE2) for IeE1=const.

Its form confirms the theoretical conclusions presented before. The possibility for modifying in large limits the active and reactive loadings and the fact that the scheme has a good static stability are verified experimenatally. B. Experimental verification of the proposed method for rated loading The test has been performed on identical machines rated at 630 kVA . Identical machines rated at 150 kVA have been used as source. The determinations have been performed both on analytical way and on graphic way when the losses are considered and neglected. The table 1 systematizes the results obtained at the test of the machine rated at 630 kVA operating as motor.

Table 1 1α )(1 AI rI iα α∆

simpl. /04076 60,8 0 069 /0407Analytical method

exact. /01073 54,9 6,5 069 /0104simpl. 077 60,8 0 069 08Graphic

method exact. /03073 55,0 7,0 069 /03040

Experimental results

02

2

69630

,8,60,6000

==

==

ikVAP

AIVU

α

)(1 AI )(1 AIe )(2 AIe )(AIr rP(kVA)

55,6 - - 15 156 51,5 122,5 224 10,25 106,5 56,0 - - 15,5 161 51,5 124,1 224,5 10,5 109 53,1 130,2 229,4 9,7 101

kWPra 66=

The angle o771 =α has been determined analytically and

graphically; the angle oi 69=α has been the closest available

one in accordance with (13). It is noticed that: -the two ways for the graphic and analytical resolution provide practically the same results or identical conditions and therefore they may be used in equal extent; -it results that it is necessary to consider the losses when 1αis established because there occur important deviations of the power rP determined by simplified methods over the measured one. -the possibility for testing machines rated at powers exceeding the power of the test stand is confirmed. In the presented case the power ratio has been 23,6/2 =rn PP .

V. CONCLUSIONS

The study concerned by the behaviour of two mechanically coupled synchronous machines, connected to mains, prsented in this paper, is materialized in the folowing things: -when operating at synchronism the variation of the coupled machines excitations modifies in large limits their active and reactive loadings; -the active and reactive incarcari can be modified as we want, their variation limits for each machine are determinable, by considering different parameters influence on these limits; -the group of the coupled machines can be considered from the static stability point of view as a single synchronous machine at no-load operation and threfore it is characterized by a good static stability irrespective of the electrical angle αbetween the rotors; -from the detailed theoretical and experimental analysis of the transient regime, it results that the two machines parallel coupling to the mains can be performed irrespective of weak mains [5] or strong mains [7] and of rotors decalage angle α ;

-the analytical expressions deduced for the dynamical mechanical characteristics and the active powers corresponding to the machines and to the mains in transient regime being verified experimentally, have practical value. From the analysis of the presented method for the rated load incarcare there result:

-it is recoverable and therefore economic; it is proved theoretically and verified experimentally that until powers reaching approximately four times the mains power, the metgod can be applied without special measures; -the method is also applicable on a test stand whih is less fitted out by eliminating the machines chain; -the scheme operation is stable, there is no danger for the starting of oscillations or for the synchronism loss; -the active power required from mains is less than in the case of the recoverable methods with machines chain indicated in literature, because the losses of the machines from chain do not occur anymore; -the differences occurring between the pre-determination theoretical results of the loadings on the coupled machines when the losses are considered and the experimental ones show that this consideration is correct and reflects the reality in a sufficient practical extent. On the whole, the theoretical study has analyzed aspects concerned by the behaviour to the mains of the mechanically coupled synchronous machines and has emphasized the possibility for establishing on coupled machines some regimes which are interesting in practice. The original proposed method has been used for over five years by an industrial factory and this utilization has allowed both the verification of the precision and of the problem correct resolution and its utility for the industrial practice.

VI. REFERENCES

[1] M. Buhler, Einfuhrung in die theorie geregelter drehstromantriebe, Birkhause, Basel, 1997.

[2] A. Campeanu, C. Nica, Synchronous electrical machines, special regimes (in Romanian), Sitech Publ. House, Craiova, 2001.

[3] A. Campeanu, “Uber eine besondere betriebsweise des ruckarbeitsbetriebs von synchronmacschinen”, A.f.E., Vol. 61, 1979.

[4] A. Campeanu, “Static stability of the synchronous machine in recovery regime” (in Romanian), Electrotehnica, No. 11, Bucharest,1971.

[5] A. Campeanu, M. Rădulescu, M. Bădică, R. Prejbeanu: “Connection to weak electrical supply network of two synchronous machines operating in recovery regime”. ICEM 2006, Creta, Grecia.

[6] T. Dordea, “On a.c. machines equations”, St. şi Cerc. Energ. şiElectrot, Vol. 16, 1966

[7] Al. Fransua, A. Campeanu, “Dynamic stability of two synchronous machines in recuperation regim”, St. şi Cerc. Energ. şi Electrot., Vol. 16, 1966.

[8] W. Nurenberg: Die prufung elektrischen maschinen, Springer Verlag, Berlin, 1959.

[9] R. Richter: Electrische Maschinen, IIB, Synchronmaschinen und einankerumformer, Verlag Birkhauser, Basel/Stuttgart, 1954.