Starting transients of induction motors connected to unbalanced networks

Electric Power Systems Research, 1 7 (1989) 189 - 197

189

Starting Transients of Induction Motors Connected to Unbalanced Networks RONALD G. HARLEY, ELHAM B. MAKRAM and EDWIN G. D U RA N

Electrical and Computer Engineering Department, Clemson University, Clemson, SC 29634-0915 (U.S.A.) (Received May 30, 1989)

ABSTRACT

The effects o f an unbalanced supply on induction m o t o r overheating has been k n o w n for some time. However, little or no attention has been paid to the dynamic response o f such a system. This paper evaluates the effects o f an unbalanced distribution system and unbalanced single-phase loads on the dynamic behavior o f a three-phase induction motor. Results are presented to illustrate that a significant error can be made under certain conditions by ignoring the imbalance or even by simulating such a system by an equivalent balanced one.

INTRODUCTION

Induction motors are being used more than ever before because t h e y are versatile, dependable and economical. Individual machines of up to 10 MW in size are no longer a rarity. In m a n y cases they represent 25% or more of certain industrial loads [1]. The use of these motors creates a burden on the subtransmission and distribution network and some of the k n o w n problems are voltage dips, high inrush current and a low lagging power factor. On the other hand, the distribution system affects the steady-state and the dynamic behavior of the induction motor. Feeder impedance, even if balanced, give rise to a reduced m o t o r terminal voltage, increased startup time, larger steady-state slip and therefore a reduced m o t o r efficiency [2]. Relatively little attention has been paid to m o t o r behavior in the presence of an unbalanced distribution system. Imbalance refers to the case of single-phase loads on the threephase system, the use o f an unbalanced distribution system network or energization of the three-phase bus with an unbalanced set of 0378-7796/89/$3.50

three-phase voltages. In addition, an unbalanced distribution configuration occurs in the case of single-phase taps off the three-phase substation bus, cases of a single conductor outage in a three-phase network, untransposed feeder circuits, or similar configurations that result in a dissimilar voltampere characteristic among the three phases. Unbalanced currents can cause problems. Negative-sequence currents may cause overheating and reduced torque of electrical machines [3, 4]. Ground currents (zero-sequence currents) can give rise to improper operation of protective relaying. Zero-sequence currents greatly increase the effect of inductive coupling between parallel transmission lines or feeders. This paper reports on an investigation into the effects of an unbalanced distribution system on the dynamic behavior of an induction motor. Attention is paid to the system in Fig. 1 and to the effects of a single-phase shunt load at the m o t o r terminals, two such single-phase loads, three such single-phase loads, and unbalanced feeders when the m o t o r and its load are started up. The results show that an error of as much as 7% in predicted start-up time can occur by replacing an unbalanced shunt load by an averaged balanced load and by assuming an unbalanced feeder to be an equivalent balanced feeder. Of these two, the error due to the imbalance in the feeder is by far the more 2

l I'

Vebcl

,I I°b'~ ®

z,bo ~ labc

'1 IabcL ~--LOAD I

Vabc2 Fig. 1. Distribution system containing induction m o t o r and shunt load. © Elsevier Sequoia/Printed in the Netherlands

190

important one. The error due to the imbalance in the shunt load is probably within the same range as the error caused by uncertainty in system parameters. The error also depends upon the m o t o r parameters.

I2n = (Vtn/Zn)[Zm,/(Zmn + Z2n)]

(4)

Ten = II2n!2R2n/(2--s)

(5)

The equation for motion of the rotor is given by pc0 m = (Tep - - Ten - - TL)/(2H/coz)

THEORY

Machine equations The induction m o t o r is simulated by its positive- and negative-sequence AC equivalent circuits shown in Fig. 2.

,°

i

tp

;Rm I2° x2° s

Xrnp

(a) In

Rm

2n

where H (sec) is the inertia constant, ~z the synchronous speed in rad s -1 , TL the load torque in p.u., and COm the shaft speed in rad s- 1 For the purpose of this paPer the m o t o r load is assumed to be a p u m p or a fan with a typical characteristic of T L = A + BCOm2

2Rp

zp

(6)

X2n

2__.2.n R 2-S O

(b) Fig. 2. I n d u c t i o n m o t o r AC equivalent circuit: (a) positive-sequence circuit; (b) negative-sequence circuit.

The sequence c o m p o n e n t voltages Vtp and Vtn are obtained from the following transformation from the phase voltages Vabe2 at the m o t o r terminals

(7)

where A sets the initial zero-speed load torque and B sets the final equilibrium speed where m o t o r torque is equal to load torque. Equations (1) - (7) represent a nonlinear system and they can only be solved by a numerical step-by-step integration process on a digital computer.

Network equations The abc voltage vector at node 2 in Fig. 1 is given by

ira211.1 zabzaclila} Ybx/-

Ve2J

Ve 1J

<8>

k Zea Zeb ZeeA Ie

or

(1) LVtnJ

a2

a

or Vt0pn = A-1Vabe2

where a is the complex operator a = 1/120 °= --0.5 + j0.866. Values of VabeZ are given b y eqn. (14). The current I2p in Fig. 2(a) is given by (2)

where Zp is the input impedance; Zmp and Z2p are the positive-sequence magnetizing and rotor impedances. This gives the positivesequence torque

Tep = 112pi2R2p/S

-

-

Zabelabe

(9)

where Zab e represents the feeder impedance matrix,/abe the feeder currents, and Vabel the infinite bus voltages. /abe in turn is related to the m o t o r and shunt load currents by

LVe2A

I2p = (Vtp/Zp)[Zmp/(Zmp + Z2p)]

Vabe2 = Vabe,

/abe = labem + labeL

Combination of eqns. (9) and (10) yields [/abe2 = Vabcl - - Z a b c l [ l a b c m + labcL]

(11)

Now labem and label Can in turn be defined in terms of the m o t o r and the shunt load admittance matrices as follows:

(12)

(3)

In a similar way, the following expressions are found for the negative-sequence circuit:

(10)

-Yearn Yeb~ Yee~l LV~2_I or

191

labcm = YabcmVabc2

Yaam = Ybb~ = Yecm = (Yp + Y n ) / 3

(20a)

and

Yabm = Ybem = Ycam = ( a Y p + a 2 Y n ) / 3

(20b)

Yaem = YD,~ = Ycbm = (a2yp + a Y n ) / 3

(20c)

[/aL1

o

:

I-I~LJ

0

Y~oL

Vb: (13)

or / a b e l = YabcL Vabc2

Substituting eqns. (12) and (13) into (11) and rearranging, we have

Vabc2 = { U + Zabc[Yabcm + YabcL]}-lVabcl

(14) where U is the identity matrix. All the variables on the right-hand side of eqn. (14) are k n o w n at the instant when the infinite bus voltages Vabc 1 are switched onto a stationary motor. This allows Vab¢2 to be c o m p u t e d and used in eqn. (1) to find Vt0pn which in turn is used by eqns. (2) - (6) to yield the acceleration pedro. A small time step At of integration then finds the new value of the speed C~m and slip s which are used to calculate the new matrix Yab¢m which in turn in eqn. (14) yields a new value for Vabc2. This process continues until the m o t o r has run up to a steady-state condition. A post start-up drop in supply voltage can be simulated by redefining Vabcl and substituting this into eqn. (14) and repeating the integration process. N o w eqn. (14) uses the matrix Yabcm which is found as follows: labcm = Yabcm Vabc2

(15)

which can be rewritten as Al0p n = YabcmAVtopn

Imbalance in the shunt load can be represented by substituting suitable values for YaaL, YbbL, and YccL in eqn. (13). For a balanced load these three admittances would be equal and for no shunt load whatsoever they are all three put equal to zero. Feeder impedance equation

The impedance matrix of the three-phase configuration can be obtained in general for any configuration as shown in Fig. 3 as an example. L

Obc-6ft

~l~

o a

b

O bz

C

@ E Fig. 3. Feeder configuration.

(21)

Z a a = r a + r d + j(o]g l n ( D e / D s ) ---- Z b'b = Z e 'c -- Z~'g

Zab = r d +

jcdk i n ( D e / D a b

) = Zba

(23)

Zca = r d + jcok l n ( D e / D c a ) = Z ' c

(24)

z ' ag = rd + jcok ln(De/Dag) =

(25)

' Zga

Zbg' = rd + jcok l n ( D e / D b g ) = Zgb' =

r d + jcdk In(De/Dog ) = Zg' c

10pn ---A-IYabcmAVt0pn

= Y0pnVtopn

(17)

Hence Yopn = A - I Y a b c m A

(18)

or

Yabcm = AYopn A-1

(19)

where Yo = 0 for an ungrounded stator neutral or a delta stator winding, Yp = 1 / Z p a n d Y. = 1/Zn.

Manipulating eqn. (19), we have

(26) (27)

where r a = rb = rc

or

(22)

Z~¢ = r d + joJk l n ( D ~ / D b c ) = gob

' Zcg

(16)

,-}

O~-2f~

(28)

Equations (21) - (28) are in ~ / u n i t length and Dsa = D s b = Dsc = D s is the conductor self geometric mean distance in ft. De is a function of both the earth resistivity p (taken to be 100 ~ m) and the frequency f, and is defined by the relation [5] De = 2 1 6 0 ( p / f ) i n = 2790 ft for f = 60 Hz

r d = 1.588 × 10-3f ~ / m i l e The constant cok is 0.121 34 when the unit of length is in miles and co = 2uf.

192

The result o f the above eqns. (21) - (28) leads to the primitive 4 × 4 im )edance matrix

fsi

gab Zac I gag t ! I t Zbb Zbc I Zbg

Z~bCG =

Z~b Z~¢ i ' . . . . . i Zcg

L

!

r

t

(29)

t

Zgb Zgc = Zgg

B a s e case: b a l a n c e d f e e d e r , n o s h u n t load

Reducing along th e indicated partitioning, we c o m p u t e the Z~B~ matrix for the feeder:

[Zaa Zab c = / Z b a

Zab Z a c ]

Zbc/

Zbb

LZ a

Two types o f feeder configuration were considered: firstly, a balanced triangular spacing of 5.33 ft between phases, and, secondly, with the dimensions shown in Fig. 3. This yielded a feeder Zab¢ matrix (see eqn. (30)) with the same diagonal but different offdiagonal values for the two types of feeders.

(30)

Zcc]

RESULTS

In order to evaluate t he effects of imbalance on the system in Fig. 1, t he m o t o r was started up when the feeder was unbalanced and with different a m ount s of unbalanced shunt load. The m o t o r parameters used were (all in per unit) Rip = R~n = 0.021

X W = X l n = 0.049

Rmp = R m n = 0 . 0

Xmp = X~n = 3.038

R2, = R2n = 0.057

X 2 v = X2n = 0.132

f = 60 Hz

H = 0.128 s

U n b a l a n c e d f e e d e r : n o s h u n t load

The m o t o r ' s mechanical load parameters (see eqn. (7)) were A = 0.1 p.u. and B = 0.6 p.u. which gives a final steady-state m o t o r t o r q u e o f a b o u t 0.64 p.u., as seen in Fig. 4. 1.2 1.0

The length of feeder was chosen such that its positive-sequence impedance was equal to (0.0556 + j 0 . 1 4 ) p.u. This corresponds to a long feeder and gave a m o t o r terminal voltage of only 56% at switch-on, but was nevertheless chosen as a worst-case condition to evaluate the effects o f later unbalanced feeder and load effects. Curve T E l in Fig. 4 shows a starting t orque of about 0.45, a pull-out t o r q u e of about 1.14 and a final steady-state value o f 0.64 p.u. It is well k n o w n that this t o r q u e curve is proportional to the square of the applied voltage, hence if started from rated voltage, all points on the curve would be multiplied by (1/ 0.56) 2 = 3.2 and the m o t o r would run up much sooner than the 1.3 s seen in Fig. 4. The magnitudes of the phase currents Ial, I b , , and I¢~ in Fig. 5 are identical for this base case, and so are the voltages V~, Vb~, and Vet in Fig. 6. The starting t o r q u e curve TE2 in Fig. 4 now consists of a positive-sequence component TEP and a negative-sequence c o m p o n e n t TEN. Of great interest is that the feeder im-

.'P-~j~" \~E1

TE2=TEP-TEN

~'0.8 o

0.6

(.... "'0.4 "'0.2

sEou.)

0.0 !

0.0

|

0.2

I

0.4

!

!

0.8

1.0

I

0.6

1

I

1.2

1.4

TIME, SEC Fig. 4. I n d u c t i o n m o t o r t o r q u e d u r i n g s t a r t - u p for b a l a n c e d a n d u n b a l a n c e d feeders, b u t n o s h u n t load: T E l , base-case b a l a n c e d feeder; TE2, u n b a l a n c e d feeder.

193

3.2

~-? 2.4

Ib2

I

Z

~2.0

•< 1.2

~ , ~

0.8 I

I

0.0

0.2

1

I

""

0.4

0.6

|

I

$

I

0.8

1.0

1.2

1.4

TIME, SEC Fig. 5. Armature phase currents during start-up for balanced (Ial, Ibl and Icl) and unbalanced (la2, Ib2 and Ic2) feeders, but no shunt load.

1.0

~0.9 ~E

~0.8 ..mJ

.fY

Zo.7 ~-0.6

0

___I_:

___2__:~-"-

.....................

0

............

0.5 I

I

0.0

0.2

~

I

0.4

t

I

[

I

I

I

0.6

O.B

1,0

1,2

1.4

TIME, SEC Fig. 6. Terminal voltage during start-up for balanced (Val, Vbl and Vcl ) and unbalanced (Va2, Vb2 and Vc2 ) feeders, but no shunt load.

balance increases the positive-sequence torque b y a larger a m o u n t than it produces negativesequence torque. This arises because the imbalance n o t only creates a negative-sequence voltage, b u t also increases the positivesequence voltage of the balanced feeder. Owing to the nature of the induction m o t o r any positive-sequence voltage produces much more torque than the same amount of negative-sequence voltage. The unbalanced feeder therefore enables the m o t o r to have a faster dynamic response. The three unbalanced currents Ia2, Ib2, and It2 appear in Fig. 5 and the voltages Va2, Vb2, and Vc2 in Fig. 6. Unbalanced feeder: unbalanced shunt load Figures 7 and 8 summarize the torque and speed start-up results for the base case and the

unbalanced feeder with no shunt load considered so far, as well as the unbalanced feeder with a 21% shunt load first on phase A and then on phase C; these are compared with a balanced feeder and a balanced 7% shunt load on each phase. Consider a practical case (curve TE3) of an unbalanced feeder and a 21% load on phase A. Moving the shunt load to phase C {curve TE4) makes little difference to the dynamic response. However, a much greater error is incurred b y assuming the feeder to be balanced ( T E l ) and w i t h o u t any shunt load at all; the worst case occurs by assuming that a 21% load on phase A o f an unbalanced feeder (TE3) can be approximated by a balanced 7% load per phase on a balanced feeder (TE5). Curves TE3 and TE5 (and W3 and W5 in Fig. 8) thus illustrate the inaccuracy of repre-

194

1.2 i Q.

-1.0 T E 4 ~ /

' ~ 3

IX

~0.8 ,~O.S =_1

0.4 0.0

[

I

I

I

0.2

0.4.

0.6

0.8

"

I

[

I

1.0

1.2

1.4

TIWE, SEC Fig. 7 Starting torque for balanced and unbalanced feeders, with unbalanced shunt load: TEl, balanced feeder, no shunt load; TE2, unbalanced feeder, no shunt load; TE3, unbalanced feeder, 21% shunt load on phase A; TE4, unbalanced feeder, 21% shunt load on phase C; TE5, balanced feeder, balanced 7% shunt load. 0.95

............

0.90 0.85

/S "/"

d 0.80

t.,,,i

0.75 0.70 0.65 0.90

I

[

I

I

I

I

]"

0.95

1.00

1.05

1.10

1.I5

1.20

1.25

•

1.30

TIUE, SEC Fig. 8. Speed run-up for balanced and unbalanced feeders, with unbalanced shunt load: Wl, balanced feeder, no shunt load; W2, unbalanced feeder, no shunt load; W3, unbalanced feeder, 21% shunt load on phase A; W4, unbalanced feeder, 21% shunt load on phase C; W5, balanced feeder, balanced 7% shunt load.

senting an u n b a l a n c e d f e e d e r and u n b a l a n c e d load b y a b a l a n c e d f e e d e r a n d balanced load, as assumed b y all stability p r o g r a m s c u r r e n t l y available. T h e r e is a b o u t a 5% d i f f e r e n c e in t i m e t a k e n b y W3 and W5 t o r e a c h a speed o f 0.9 p.u. H o w e v e r , it is i m p o r t a n t t o realize t h a t t h e a p p r o x i m a t i o n 'fails to s a f e t y ' in t h a t it predicts a s l o w e r d y n a m i c r e s p o n s e t h a n w o u l d a c t u a l l y be t h e case. Three d i f f e r e n t m o t o r s In o r d e r t o evaluate the effects o f m o t o r p a r a m e t e r s o n t h e t r a n s i e n t response, the start-up characteristics o f the first m o t o r s h o w n in Figs. 7 and 8 are r e p e a t e d f o r a s e c o n d a n d t h i r d m o t o r a n d the results f o r all t h r e e m o t o r s a p p e a r in Figs. 9 a n d 10. These clearly illustrate t h a t t h e spread o f results due t o i m b a l a n c e is greatest f o r m o t o r No. 2 and

the least f o r m o t o r No. 1. T h e m e a n i n g o f each curve in each set in Figs. 9 and 10 is the same as f o r the sets in Figs. 7 and 8, respectively. In the case o f m o t o r No. 2 t h e r e is a b o u t a 7% d i f f e r e n c e in t i m e t a k e n b y curves W3 and W5 t o reach a speed o f 0.9 p.u. Second motor: R ] p = Rln = 0 . 0 2 4

Xlp = Xln = 0 . 0 7 7

Rmp = R u n = 0.0

X m , -----Xmn = 2.2

R2p = 0 . 0 2 9

R2n = 0 . 0 7 8

X2p = 0.177

X2n = 0 . 0 7 4

f = 60 Hz

H= 0.128 s

Third motor: Rlu = Rln = 0.024

Xlp = Xln = 0.077

Rmp = R u n = 0.0

Xmp = Xmn = 2.2

195

THIRD

1.3 1.2 1.1 ~.o 0.9

FIRST"

/

~0.8

I~TOR

/ .,Yi\t

No.7 ~0.6 -~0.5 ~0,4 ~,0.3 ~ 0.2 0.1 0,0 I"

I

0.0

~

0.5

I

1.0

I

I

1.5

2.0

'

I

I

2.5

[

3.0

3.5

T I I~E, SEC Fig. 9. Starting torque of three different motors with unbalanced feeders and unbalanced shunt load.

1.0 0.9

~~

o.B

}~// R

~IRSI:

o,

/

'" ' .~ ~

D

/

0.6 "0,5 0.4 0.3 0.2 0.1 0.0

.

i

'

0.0

I

0.5

'

I

'

'

1.0

I

I

1.5

2.0

f I ME,

1

2.5

|

3.0

I

3.5

SEC

Fig. 10. Speed run-up of three different motors with unbalanced feeders and unbalanced shunt load.

R2p = 0.029

R2n 0.032

X2v = 0.065

X2n = 0.064

f = 60 Hz

H = 0.128 s

Incorrect definition o f voltage imbalance The most precise and meaningful definition of voltage imbalance at a point in a netw o r k is the ratio o f the negative- to the positive-sequence voltage components at that point. However, in practice, it is difficult to measure these sequence c o m p o n e n t s w i t h o u t special instrumentation. Certain utilities [6] in the U.S.A. instead use an approximate equivalent measure o f imbalance equal to max. deviation from average voltage average voltage

× 100%

where voltage is measured from phase to neutral and average voltage is defined to be the average of the magnitudes o f the threephase to neutral voltages. For the purpose of this paper, the precise definition will be referred to as 'sequence imbalance' and the approximate one as 'voltage imbalance'. Figure 11 shows h o w the percentage imbalance varies with time during start-up, calculated according to b o t h definitions and for different degrees and conditions of n e t w o r k imbalance. When comparing curves with the same number in Figs. l l ( a ) and (b), it is clear that a considerable error is incurred b y using the approximate definition of voltage imbalance. Taking values at zero time, the precise value of curve 1 (Fig. l l ( a ) ) is almost three times the value of the approximate curve 1 (Fig. l l ( b ) ) ; similarly, the precise curve 3 is

196

?I. . . . .

t

3

'

6

I

'

I

l

I

2

3

TIME,5EC

(a) 9.

? ..J O

i

2

'i

b

O

N

4. 3.

2.

1

(b)

2

3

TIME,SEC

Fig. 11. Percentage voltage imbalance during start-up of the first motor with unbalanced feeders and unbalanced shunt load: (a) precise definition; (b) approximate definition. Curve 1, unbalanced feeder, no shunt load; 2, unbalanced feeder, 21% shunt load on phase A; 3, unbalanced feeder, 21% shunt load on phase C. initially m o r e t h a n twice the value o f the a p p r o x i m a t e curve 3. In c o n t r a s t t o these results, t h e precise value o f curve 2 is initially slightly less t h a n t h e a p p r o x i m a t e value, b u t t h e s t e a d y - s t a t e a p p r o x i m a t e value is a b o u t five times greater t h a n t h e precise value. These results t h e r e f o r e illustrate t h a t t h e app r o x i m a t e d e f i n i t i o n gives significant errors o f b o t h a positive and a negative n a t u r e a n d can t h e r e f o r e n o t b e used as a conservative 'fail safe' alternative d e f i n i t i o n t o t h e precise definition. CONCLUSIONS This p a p e r has p r e s e n t e d t h e t h e o r y o f an u n b a l a n c e d d i s t r i b u t i o n s y s t e m s u p p l y i n g an

u n b a l a n c e d s h u n t load and an i n d u c t i o n m o t o r . Results o f several case studies have s h o w n a worst-case e r r o r in m o t o r start-up t i m e o f a b o u t 7% w h e n forcing an equivalent balanced m a t h e m a t i c a l m o d e l o n t o an unbalanced system. The amount of error depends o n t h e length and t y p e o f feeder, a m o u n t o f f e e d e r imbalance, a m o u n t o f s h u n t load and t y p e o f m o t o r . However, t h e single greatest cause o f e r r o r appears t o be t h e f e e d e r imbalance and n o t the load imbalance. T h e p a p e r has also s h o w n t h a t the direct use o f m e a s u r e d phase to n e u t r a l voltages t o calculate i m b a l a n c e can lead t o a significant e r r o r c o m p a r e d t o using the precise ratio o f positive- t o negative-sequence voltage, b u t w h i c h is difficult t o measure in practice.

197 REFERENCES

1 D. J. N. L i m e b e e r and R. G. Harley, Subsynchronous resonance of single-cage i n d u c t i o n motors, Proc. Inst. Electr. Eng., Part B, 128 ( 1 9 8 1 ) 3 3 -42. 2 A. A. M a h m o u d , T. H. Ortmeyer, R. G. Harley and C. Calabrese, Effects of reactive compensation on i n d u c t i o n m o t o r dynamic performance, IEEE Trans., PAS-99 (1980) 841 - 846. 3 R. G. Harley and M. A. Tshabalala, I n d u c t i o n

m o t o r behaviour in the presence of unbalanced voltages, Trans. SAIEE, 76 (2) (1985) 51 - 55, 4 J. E. Williams, Operation o f 3-phase i n d u c t i o n m o t o r s on unbalanced voltages, Trans. AIEE, P o w e r A p p . Syst., 73 (1954) 125 - 133. 5 P. Anderson, Analysis o f Faulted Power Systems, Iowa State Univ. Press, 1981. 6 R. W. Fischer, Voltage unbalance on three-phase distribution systems, Southeastern Electric Ex-

change 1977 Annu. Conf., Engineering and Operation Division, N e w Orleans, April 1977.