The effect of machine saturation on small disturbance stability

Electric Power Systems Research, 22 (1991) 173 - 180

173

The E f f e c t of M a c h i n e S a t u r a t i o n on S m a l l D i s t u r b a n c e Stability S. B. P A N D E Y

New Jersey Institute of Technology, Newark, NJ 07102 (U.S.A.) ( R e c e i v e d J u n e 19, 1991)

Turbogenerator

ABSTRACT

V

Vr I

The effect of machine saturation on small disturbance stability of a power system whose machine excitation is controlled by a fast-acting three-term automatic voltage regulator (A VR) has been investigated. It is seen that the system's stability limits were practically unaffected but the optimum settings of the A VR gain control parameters were affected significantly at the higher operating conditions. 1. I N T R O D U C T I O N

Small disturbance stability analysis neglecting s atu r a t i on may lead to unduly pessimistic estimates regarding the ability of a machine whose excitation is controlled by a continuously acting rheostatic regulator to remain in synchronism in a power system [1, 2]. Techniques for estimating the proper values of saturated reactances involve not only the machine s at ur a t i on characteristics but also the system's operating conditions. The effect of saturation, when considered, is usually taken into account by suitably modifying the synchronous r eact ance [3-7]. The purpose of this paper is to analyze the effect of machine s at ur a t i on in a power system whose excitation is controlled by a three-term (proportional, differential and double differential channels) automatic voltage regulator (AVR). The investigations are aimed at finding the effect of machine saturation on the system's stability limits as well as on the optimum settings of the voltage regulator gain control parameters at different operating conditions. 2. SYSTEM INVESTIGATED A steam t u r b o g e n e r a t o r connected to an infinite bus th r ough a long lossless transmis-

Tr.

~f Operating

line i.b. J condition 9

Fig. 1. S y s t e m i n v e s t i g a t e d .

sion line has been investigated. The main excitation of the machine is assumed to be controlled by a three-term (proportional, differential and double differential channels) angle-actuated AVR [8], as shown in Fig. 1. The angle between the generator bus and the infinite bus is defined as the operating condition. Algorithms have been developed to obtain the active and reactive powers at the generator bus at any operating condition. To achieve this objective the expressions for the overall A B C D constants of the power network under study were obtained first: Ao = cos 2 - Xt sin 2

(1)

Bo = j(sin 2 + Xt cos 2)

(2)

Co = j sin 2

(3)

Do = cos 2

(4)

where 2 = 360 ° x fl/3 x 105 and, at f = 60 Hz, 2 = 7.2 x 10-2l

deg

(5)

Algorithms for the active and reactive powers at the generator bus were then obtained: Ps = V s ( C o A ~ ° ) s i n O

= Usz Do vs / Co

(6)

AoDo'~

)

cos 0

(7)

where V~ is the generator bus voltage (p.u.) and 0 is the operating condition (deg).

174

2.1. Estimation of machine saturated reactance An accurate determination of the saturated reactance is r at her complicated and very involved as it depends on the flux distribution inside the machine. Following Kimbark [9], the saturation synchronous reactance can be estimated by (8)

where X~,t is the saturated d-axis reactance (p.u.), Xp the Potier reactance (p.u.), X~. . . . t the u n s atu r ated d-axis reactance (p.u.), K the saturation factor, and a and b are constants. The factors K and a/b are determined for a flux loading corresponding to the voltage E , behind the Potier reactance and the field excitation voltage E~, as described in the literature [9]. 2.2. Modeling for stability analysis The following assumptions were made in the analysis: (1) the saturation is a function of the resultant airgap flux only; (2) saturation of parameters X~ and X~ is omitted as they have less influence than the parameter Xa [1]; (3) the effect of governor action is ignored; (4) the damper windings are not considered. The dynamic behavior of a power system following small perturbations can be described by a set of linearized models as follows. Gen er ato r rotor dynamics: (9)

Transient flux changes in field winding: (10)

Field excitation control system:

K~A~ (1 + sTy)(1 + sTy)

AP~:;~ = P2 AE~, + $2 A5

(13)

where P~,P,~, S~ and $2 are defined as the power derivatives, their expressions being: -

~P~:~

(?Ed $1 = i~PEd

e~

-

Vr sin 5 XdZ

=--EdVr cos 6

P2 = ~,E~ - XaZ' sin 5

(16)

~5

E',~Yr XdY.--XdZ' -- Z~----~cos 5 -- V# ~ - ~ - - X ~ cos(25)

(17)

It is to be noted that A P E = APEa = APE;,. Also, the numerical values of each of eqns. (14) -(17) are affected by the machine saturation. An overall system's characteristic equation can be derived from eqns. (9) -(17): aos 7 + a~s ++ a2s ~ + a3s 4 + (a4 + K2a4~)s 3 + (a~ + K2as, + Kla52 + Koa53)s 2 + (a6 + K,a~2 + Ko%3)s + (aT + KoaTa) = 0

(18) where ao = MT'd TeT1T2 T3

+ T 1 T 2 T3(T e + T~)] a2= M[T'd(TI T.2 + T, T3 + T2T3)

+ TSTo(T, + T~+ T3) + TeT~(T, + T~)

+ To(T~ + T2 + T3) + TAT1 + T~)]

Going through eqns. (1)-(3), we find five unknown variables, namely, AS, APE, AE~+, AE~, and AE~. In order to solve these equations simultaneously, we require two more linearized models which can be derived easily from a phasor diagram following the literature [10]. These new linearized models can be

(15)

$2 - ~ P E ' d

a3 = M[T'd(T, + T2 + T3 + Te)

(11)

(14)

xaz

+ T, r~(r~ + T3)] + S2T'~ToTIT2T~

g~s ~

~E~o= K0+ 1+sT + 1 + ~ ; / x

(12)

a, = M[T'dT+(T, T2 + T1Ta + T2T3)

Ms 2 h(~ -k- AP E = 0

g~s

APEa = P~ AE~ + S~ A(t

P1

x~ . . . . t - x ~ X~,t = Xv + K(1 + a/b) ~/2

AEa+ = AE~ + T~os AE'~

expressed in the form

+ S, Te T, T~ T~ + S~[T'~ To(T~ T~ + TIT~+ T2T3) + T'~TIT~T~] a 4 = M ( T ' d + T~ + T2+ T3+ 7"+) + S, [T~ T3(T, + 7"2) + T, T2(T~ + T3)]

+ S~[T'~Te(T, + T~ + T3) + T'~(T,T~ + T~ T~+ T~T~)]

175

a~, = P~ T1

a~ = M + S, [TAT, + T2 + T~) + T~(T~ + T~) + T, T2] + Sz[T'~(Te+ T, + T2+ T3) a~, = P2,

a52 = P2 T2,

a53 = P2 T, T2

a6 = S~(T~ + T1 + T2 + T3) + S2T'd a6~ = P2, a~ = S~,

a63 = P2(T~ + T~)

w h e r e = is the degree of stability. Initially, the v a l u e of ~ can be set to zero and a stability region c o n s t r u c t e d as co varies in the plane (K~, K2). N e x t we assume a low v a l u e of ~ and r e p e a t the process of o b t a i n i n g a new stability c o n t o u r and c o n t i n u e until one of very small size is obtained. This can be used for the o p t i m u m settings of the AVR gain control parameters.

av~ = P~

Ko, K~, and Ke are the u n k n o w n gain c o n t r o l p a r a m e t e r s of the AVR, and

T'~ = T~o(P~/Pe) Since the last term of c h a r a c t e r i s t i c equation (18) c o n t a i n s only the p a r a m e t e r Ko, its critical v a l u e can be d e t e r m i n e d from the expression Ko >1S,/P~. In the p r e s e n t study K o was fixed at 1.80.

2.3. Construction of a stability region Since the n u m e r i c a l v a l u e of p a r a m e t e r K 0 is fixed, the stability r e g i o n can be c o n s t r u c t e d in the plane of the r e m a i n i n g two AVR gain c o n t r o l p a r a m e t e r s K, and K2 by following the two-dimensional f r e q u e n c y domain t e c h n i q u e [3, 10] w h i c h is briefly described as follows. We r e a r r a n g e the c h a r a c t e r i s t i c e q u a t i o n (18) as K1A(s) + K2B(s) + C(s) = 0

(19)

w h e r e A(s) are those terms of the characteristic e q u a t i o n s which are associated only with p a r a m e t e r KI, B(s) are the terms associated only with p a r a m e t e r K2, and C(s) are the terms i n d e p e n d e n t of b o t h K 1 and K2. Next, we replace s by ( - ~ +jco) in eqn. (19) and t h e n s e p a r a t e it out into real and imagin a r y parts. Finally, we get

K1A~ (~, co) + K 2 B I ( ~ , co) + CI(Zt, co) = 0

(20)

K~A2(a, co) + K2Be(~,co) + C2(a, co) =O

(21)

3. RESULTS AND DISCUSSION

F o r the system u n d e r investigation, active and r e a c t i v e power flows at the g e n e r a t o r bus were c a l c u l a t e d as shown in Fig. 2. Following the two-dimensional d o m a i n technique, the stability regions were c o n s t r u c t e d at different o p e r a t i n g conditions r a n g i n g from light to full loads. The stability results are s h o w n in Figs 3 - 11 by the full curves. This analysis does not include the m a c h i n e s a t u r a t i o n . The o p t i m u m settings of the AVR gain control p a r a m e t e r s are also indicated. The system's stability limits were found at 0 = 79 °.

1.5

1.0

0.5

Solving for K, and K2, we get A~(a, co) A2(a, co) K, = A l ( a ' co)

- C 1 ( ~ , co) - C 2 ( = , co) Bl(a, co)

A2(~, co)

B2(~, co)

(22) $

40 °

Operating

-Ci(~, co) B,(~, co) g~=

- C2 (~, co)

B2 (~, co)

AI(~, co) B,(~,~) A2(~, co) B2(~, co)

60 °

80 ° Condition

i00 °

120 °

j

-0.3

(23) Fig. 2. Active and reactive power flows at the generator bus.

176

+2 K1

3

4

xx ii t

fo~ K I - K.

-2

<'I

z

6

?

9

Unsaturated - Saturated ......

/

i i ~ ~ /

5

K2

•

~

n'5-

-4

O'~-

-4

/'

-5

-8 Fig. 3. Stability region at the operating condition 0 = 20 °.

f

2

K1

"="

.

Fig. 4. Stability region at the operating condition Et= 30 °.

I

case:

2 Case:

K1 ~-~. "~

4 3 z / Pl'~-

1 " ~ •~

unsaturated _ Saturated .... 3 4 5 K2 ----~

Unsaturated _ _ Saturated

6 Region f o r / / .~ ~

~.~//~

setting ~'2~.~-opt imu~Kl_K2.~ .(~-

for

{~:

- -- --~-~-~-- - -- -~-c--~-- - k~--

Fig. 5. Stability region at the operating condition 19= 40 °.

K2 - - ~

x ~x%%%~

°'"

" " - "-3- . . . .

-5

"xx~

5"-~- . --~/- ~

"~

-4 Fig. 6. Stability region at the operating condition O = 50

177

3.1. Computation of machine saturated reactance T h e c o m p u t a t i o n c o m p r i s e d the following steps. Step 1. F o r a c h o s e n o p e r a t i n g c o n d i t i o n find the a c t i v e a n d r e a c t i v e p o w e r s f r o m Fig. 2. Step 2. E s t i m a t e the m a c h i n e ' s i n t e r n a l v o l t a g e s b e h i n d the P o t i e r a n d s y n c h r o n o u s d-axis r e a c t a n c e s :

3 Case: Unsaturated

KI saturate(]

2

Region for 1 optmm settlng for~ K1 - }{2

/ _ ~ _ . _ . ~

""

~

~ c.'~-

-t

.....

~¢=o ..........

~ :

[(

.,.-to-

Ep=

.........

[(

"a . . . .

x

I

"

Fig. 7. Stability region at the operating condition 0 = 60+.

3 Ca~: Unsaturated

K1 2

Reqian for

saturated

.....

KI - I(2

£ . . . . . r . . . . -4--%- - -

t

s

6

Fig. 8. Stability region at the operating condition 0 = 65+.

Qs apt2

V++

Vs / + \

(Ps/p/211/2

V+ / J

(24) (25)

Step 3. U s i n g the r e s u l t s of step 2 a n d the s a t u r a t i o n g r a p h g i v e n in the l i t e r a t u r e [9], r e p r o d u c e d h e r e in Fig. 12, g e n e r a t e a new g r a p h b e t w e e n Ep a n d KEp. This g r a p h is s h o w n in Fig. 13 w i t h the o p e r a t i n g c o n d i t i o n m a r k e d by the open circles. Step 4. D r a w a t a n g e n t at the c h o s e n opera t i n g c o n d i t i o n a n d o b t a i n the v a l u e s of cons t a n t s a a n d b, as depicted in Fig. 13. Step 5. Finally, i n s e r t the v a l u e s of p a r a m e t e r s K, a, b Xp a n d Xd . . . . t in e x p r e s s i o n (8) a n d e s t i m a t e the v a l u e of the s a t u r a t e d reactance Xdsat. F i g u r e 14 p r o v i d e s the m a c h i n e ' s s a t u r a t e d d-axis r e a c t a n c e at different o p e r a t i n g conditions. T h e r a n g e of the s a t u r a t e d r e a c t a n c e was f o u n d to be 0 . 5 7 - 0 . 9 2 p . u . It is also a c o m m o n p r a c t i c e to choose X d s a t : 0 . 8 X d . . . . t for light loads a n d 0.6Xd . . . . t for full loads [1]. 3.2. Stability results with saturated reactance

Case: UnSaturated - Saturated

3

I,

.....

opt llvedmsetting KI - K2

2

0

1

2

3

4

5

6

K2 Fig. 9. S t a b i l i t y

region

at the operating

condition

0 = 7 0 °.

By r e p l a c i n g u n s a t u r a t e d d-axis r e a c t a n c e by s a t u r a t e d d-axis r e a c t a n c e in the s y s t e m ' s characteristic equation where appropriate and applying a two-dimensional frequency d o m a i n m e t h o d , the s t a b i l i t y r e g i o n s were c o n s t r u c t e d at different o p e r a t i n g conditions, as s h o w n by b r o k e n lines in Figs. 3 - 11. T h e o p t i m u m s e t t i n g s for the AVR g a i n control p a r a m e t e r s are also s h o w n in t h e s e Figures. T h e s y s t e m ' s s t a b i l i t y limits, w h e n the s a t u r a t i o n was considered, w e r e f o u n d at 0 = 80 °. It is seen t h a t the s y s t e m ' s s t a b i l i t y limits r e m a i n p r a c t i c a l l y the s a m e c o m p a r e d w i t h t h o s e of a s y s t e m in w h i c h s a t u r a t i o n was n o t a c c o u n t e d for in the a n a l y s i s ( r e f e r to Fig. 11).

178 Case: Unsaturated Saturated

..... o.~

K1 Region for optimum setting K1 - ~

1

"'6

~ ' ~ ~ °

2

3

4

6

5

K2

Fig. 10. Stability r e g i o n at t h e o p e r a t i n g c o n d i t i o n 0 = 75 °.

15

L i m i t i n g case

@ = 79 °

I-t~I_~IL~_

1.~

.~ff/L=

1.7

/~1// I111£ AIIil IIIIZ4 I/WI /1111I IliJl

CaSe : K1

1.6

Unsaturated _ _ Saturated

.....

L/lnlting case @ = 800

:,'~,., . . . . . .

"i_

l~_

k

1

L4

I

MII

/111 ~¥1 I I

1.1

1

2

3

4

LO0

5

0.2

~4

0.6

O.8

].0

L2

IA

K 2 --~

Fig. 11. S y s t e m ' s s t a b i l i t y limits.

Fig. 12. S a t u r a t i o n c h a r a c t e r i s t i c s .

On comparing the results of the optimum settings for the gain AVR control parameters without and with saturation, we found that they were greatly affected at the higher operating conditions, especially in the latter case (refer to Figs. 7- 10).

4. C O N C L U S I O N S

Machine saturation is an important factor in small disturbance stability studies. A power system's small disturbance stability limits remain practically unaffected by

179

1.0

T

0

0.5

1.0

2.0

1.5 KEp

Fig. 13. Relationship between Ep and

KEp.

Xp = 0.11 p.u. K e = 1.0 T1 = 0.03 s T3 = 0.04 s Vr = 1.0 p.u.

1.0 0.9 0.8 0.7 0.6

Tao = 8.0 s T o = 1.0s T2 = 0.05 s l = 400 km

0.4

NOMENCLATURE

0.4 0.3

Ea, E'~,Ep

0.2

Eae

0.i

go

0 O

i0°

20~

30a

40©

50°

Operating

60°

70°

80°

90

Condition

1 M

P+,Q+

Fig. 14. S a t u r a t e d r e a c t a n c e characteristic. S

m a c h i n e s a t u r a t i o n , especially w h e n m a c h i n e e x c i t a t i o n is c o n t r o l l e d by a three-term AVR, b u t m a c h i n e s a t u r a t i o n does affect the optim u m settings of the AVR gain c o n t r o l parameters at the h i g h e r o p e r a t i n g conditions.

Tao T~

Tt T~

APPENDIX

System specifications M = 0.0395 p.u. X d = 1.11 p.u.

V~ = 1.0 p.u. Xh = 0.35 p.u.

T~ X~Z X~lZ X~

m a c h i n e i n t e r n a l voltages, p.u. exciter o u t p u t voltage, p.u. exciter gain c o n s t a n t t r a n s m i s s i o n line length, km m a c h i n e inertia c o n s t a n t , p.u. active a n d r e a c t i v e powers at gene r a t o r bus =d/dt, differential o p e r a t o r d-axis a r m a t u r e w i n d i n g open-circuit time c o n s t a n t exciter field w i n d i n g time constant AVR differential c h a n n e l time constant AVR double differential c h a n n e l time c o n s t a n t rectifier unit time c o n s t a n t

=xd+x~ = x'a + x+ t r a n s f e r r e a c t a n c e b e t w e e n genera t o r and infinite buses, p.u.

180

xt

transformer equivalent reactance, p.u. 4

A 5 0

deviation from steady-state value rotor angle with respect to intlnite bus operating condition

5

6 REFERENCES 7 1 M. S. Sarma, Synchronous M a c h i n e s - - T h e i r Theory, Stability and Excitation, Gordon and Breach, New York, 1979. 2 S. B. Pandey and V. V. Chalam, Dynamic stability investigations--effect of damping, saturation, governor action and stabilizing link, Electr. Power Syst. Res., 2 (1979) 155- 164. 3 S. M. Peeran, Consideration of saturation in the

8 9 10

region of dynamic stability of power systems, Correspondence, Proc. Inst. Electr. Eng., 116 (1969)610-612. G. R. Slemon, Analytical models for saturated synchronous machines, IEEE Trans., PAS-71 (1971) 409417. J. Lemay and T. M. Barter, Small perturbation linearization of the saturated synchronous machine, I E E E Trans., PAS-91 (1972) 1115- 1130. G. Shackshaft, Model of generator saturation for use in power system studies, Proc. Inst. Electr. Eng., 126 (1979) 759- 763. D. W. Auckland, S. M. L. Kabir and R. Shuttleworth, Generator model for power system studies, Proc. Inst. Electr. Eng., Part C, 137 (1990) 383-390. A. Barzam, Automation in Electrical Power Systems, Mir, Moscow, 1977. E. W. Kimbark, Power System Stability: Synchronous Machines, Dover, New York, 1965. V. A. Venikov, Transient Processes in Electrical Power Systems, Mir, Moscow, 1977.