FIRST BENCHMARK MODEL FOR COMPUTER SIMULATION OF SUBSyNcHRoNoUS RFSONANm

IEEE Subsynchronous Resonance Task Force of the Dynamic System Performance Working Group

Power System Engineering Camnittee

ABSTRACT

A benchmark model for the s tudy of subsynchronous resonance i s p resen ted a long w i th two t e s t problems for use in canputer program comparison and develop- ment. Models were deve loped w i th t he minimum sophis- t > s a t i o n n e e d e d t o o b t a i n u s e f u l r e s u l t s a n d w i t h d u e r e g a r d t o t h e k i n d s o f m a c h i n e c h a r a c t e r i s t i c s g e n e r - a l l y o b t a i n a b l e . The machine and c i rcu i t parameters are real values taken f rom the Navajo Project .

INTFCODUCTION

Present ly , severa l canputer p rograms and ana ly t i - cal t o o l s are a v a i l a b l e f o r t h e s t u d y o f SSR caused by the i n t e rac t ion o f mu l t imass t u rb ine -gene ra to r s and series canpensa ted t ransmiss ion sys tems. Others are u n d e r d e v e l o p e n t . T h e r e i s a c u r r e n t n e e d i n t h e e l e c t r i c p o w e r i n d u s t r y t o compare study results, de- t e rmine t he r easons fo r d i f f e rences , and r ev i se mode l s and techniques as deemed necessary. To h e l p meet t h i s need, the IEEE Subsynchronous Resonance Task Force has prepared s tandard test cases t o f a c i l i t a t e the compar- i s o n o f c a l c u l a t i o n s a n d t h e d e b u g g i n g o f c a u p u t e r programs. Using t h e Navajo Project 892.4 MVA genera- t o r s a n d 5 0 0 kV t r a n s m i s s i o n s y s t e m a s a guide, a s t a n d a r d n e t w o r k , t w o t u r b i n e g e n e r a t o r m o d e l s a n d d a t a f o r two tes t ca ses have been p rov ided . F l ex ib i l - i t y is p rov ided fo r add i t ion o f new test cases and t he modeling is s u f f i c i e n t l y d e t a i l e d f o r s i m u l a t i o n o f mos t a spec t s o f t he SSR problem.

ELECTRICAL NE"W

Extensive SSR s t u d i e s o f the Navajo Project re- v e a l e d t h a t a s i m p l e r a d i a l RLC c i r c u i t , p r o p e r l y t u n e d , c a n p r o d u c e b o t h t r a n s i e n t and s e l f - e x c i t a t i o n problems as severe as any observed i n t h e a n a l y s i s o f t h e a c t u a l s y s t e m . The s i n g l e l i n e d i a g r a m shown i n F igure 1 rep resen t s such a s i m p l e c i r c u i t . T h e c i r c u i t

VOLTAGE UNITY

Fig. I Network for Subsynchronous Rescmonce Studies

€ 77 102-7. A paper reccanrended and mprvved by the IEEE Parer Sys+e.. Ergkneering Cannittee of the I E E Power Engineering Society for presentation at the IEEE PES Winter Meeting, New Y&, N.Y., January 30- February 4, 1977. Manuscript submitted Septdxr 7, 1976; nnde available for printing N o v e n h r 4, 1976.

Parameters expressed in per u n i t on t h e g e n e r a t o r MVA r a t i n g a t 60 h e r t z c o r r e s p o n d t o t h e Navajo-McCullough l i n e . R e a c t a n c e s a r e p r o p o r t i o n a l t o f r e q u e n c y ; r e - s i s t a n c e s a r e c o n s t a n t . The i n f i n i t e b u s is a three- phase 60 h e r t z v o l t a g e s o u r c e w i t h z e r o impedance a t a l l f r e q u e n c i e s . Two f a u l t l o c a t i o n s ( A and B) a r e des igna ted , and there is p r o v i s i o n f o r i n c l u s i o n o f a f i l t e r . % capac i to r spa rk gaps are provided: a low v o l t a g e g a p t o b y p a s s t h e c a p a c i t o r d u r i n g t h e f a u l t , t h e r e b y l i m i t i n g t h e f a u l t c u r r e n t a n d n e t w o r k s t o r e d energy and a h i g h v o l t a g e g a p t o p r o t e c t t h e c a p a c i t o r d u r i n g r e i n s e r t i o n . The low vo l t age gap i s i n s e r v i c e a t a l l times except dur ing the per iod beginning wi th the f au l t c l ea rance and con t inu ing fo r a s h o r t time a f t e r c a p a c i t o r r e i n s e r t i o n t o a v o i d r e s t r i k e . When used, the proper 60 h e r t z (ms) p r u n i t v o l t a g e s e t t i n g s on a l i n e t o n e u t r a l base are 3 . 3 3 X, f o r t h e bypass gap and 5.31 X, f o r t h e r e i n s e r t i o n g a p . T h i s network i s u s e d f o r b o t h t r a n s i e n t a n d s e l f - e x c i t a t i o n s t u d i e s .

ROTOR MODELS FOR TRANSIENT STUDIES

Figure 2 shows t h e r o t o r c i r c u i t a n d a s p r i n g mass model p r o v i d e d f o r t r a n s i e n t s t u d i e s . The e l e c t r i c a l and mechanical por t ions are s imple r ep resen ta t ions of the nava jo ro tors , deve loped wi th the minimum l e v e l o f s o p h i s t i c a t i o n r e q u i r e d f o r t h e c a u p u t a t i o n of tran- s i e n t s h a f t t o r q u e s . C o n s t a n t f i e l d v o l t a g e i s assumed. E l e c t r i c a l t o r q u e v a r i a t i o n s on t h e e x c i t e r are assumed to be zero.

Fig. 2 Rotor Model for Transient Studies

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TABLE I

Rotor Circuit Parameters For Transient SSR Studies

Parameter D - a x i s Q-axis

3 7 7 4 0.53 5.3

377% 1.54 3.1 Xf 0.062 0.326

'k 0.0055 0.095 'a 1.66 XL 0.13

1.58 0.13

Table I shows the impedances for the rotor circuit expressed in per unit on the machine MVA base. Current, voltage, torque and rotor speed are also expressed in per unit. For transient representation, divide reac- tance in ohms by 377 to obtain the inductance in henries.

The generator standard impedances and time con- stants from which the circuit parameters of Table I were derived are shown in Table 11. The term XL is armature leakage reactance. Circuit parameters are obtained from the standard impedances and time con- stants by an iterative process outlinea in the appen- dix. Where the treatment of these standard machine impedances and time constants is not mathematically reducible to the rotor network defined in Figure 2 and Table I, the results will differ.

Table I11 shows the inertias and spring constants for the spring mass model. Inertia is expressed in terms of the inertia constant H based on rated kVA. The base torque is that required at synchronous speed to deliver mechanical power in kilowatts equal to the rated (base) kVA value. Base angle is 377 radians, the angle of shaft rotation in one second (the.base time). The spring constant K is given in per unit where base spring constant is defined as base torque divided by base angle. The simple second order torque equation in this system of units is:

T(pu) = 2 H 5 + D6 + K8

The spring constant is also given in per unit torque per radian. For transient studies, the mechanical damping is assumed to be zero.

The steady state mechanical torque is apportioned among the turbine sections HP, IP, LPA and LPB, respec- tively as follows: 30%, 26%, 22% and 22%. The exciter steady state torque is assumed to be zero.

ROTOR MODELS FOR SELF-EXCITATION S'WDIES

A simpler version of the rotor circuitry is pro- vided (see Figure 3) for self-excitation studies. The rotor circuit provided for transient studies is inade- quate for self-excitation studies because it cannot be fitted satisfactorily to the rotor impedance versus frequency characteristics furnished by the machine manufacturer. The model provided follows the trend in the industry to improve the representation by using simple but separate models for each mode rather than a single but m r e complex model to cover the entire range of torsional frequencies.

Circuit parameters for Figure 3 are shown in Table rV. Note that the elements XL, Xad, and Xaq are the same as for the transient model. The generator rotor impedances at the torsional frequencies to which elements of Figure 3 were fitted are shown in Table V.

Mass

HP

IP

-

LPA

LPB

GEN

EXC

TABLE I1

Generator Impedances and Time Constants

X = 1.79 pu Tdo = 4.3 s

Xd = 0.169 pu Tio = 0.032 s

Xi = 0.135 pu T' = 0.85 s

XL = 0.13 pu T" = 0.05 s

X = 1.71 pu 5 = 0.0

X' = 0.228 pu

I

d I

qo

qo

q

9

q X" = 0.200 pu

TABLE I11

Rotor Spring Mass Parameters

Inertia Spring Constant

- Shaft H(seconds) K_o pu Torque/rad

0.092897

0.155589

0.858670

HP-IP 7,277 19.303

IP-LPA 13,168 34.929

LPA-LPB 19,618 52.038 0.884215

LPB-GEN 26,713 70 .E58 0.868495

0.0342165 GEN-EXC 1,064 2.822

D* + 0 I

id

ed

Q-axis

W

Fig. 3 Rotor Model for Self-Excitation Studies

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Where the t rea tment o f these ro tor impedances i s not ma themat i ca l ly r educ ib l e t o t he ro to r ne twork de f ined in F igure 3 and Table I V , t h e r e s u l t s w i l l d i f f e r .

The r o t o r s p r i n g mass model used f o r t r a n s i e n t s t u d i e s i s a l s o u s e d f o r s e l f - e x c i t a t i o n s t u d i e s e x c e p t f o r t h e i n c l u s i o n of mechanical damping. Individual

TABLE IV

Rotor C i r c u i t Per Unit Impedance a t 60 Hz f o r S e l f - E x c i t a t i o n S t u d i e s

Mode 1 2 3 4 Frequency (hz) 15.71 20.21 25.55 32.28

XL 0.13 0.13 0.13 0.13

Rrd PU 0.00587 0.00686 0.00764 0.00825

Xrd p' 0.04786 0.04401 0.04080 0.03823

xad PU 1.66 1.66 1.66 1.66

Rrq PU 0.00884 0.00936 0.00998 0.01081

Xrq Pu 0.04742 0.04648 0.04556 0.04469

Xaq PU 1.58 1.58 1.58 1.58

TABLE V

Pe r Unit Rotor Impedance a t Subsynchronous Frequency

M u l t i p l i e d by (60/fn)

Rotor Frequency fn (hz ) D-axis paxis

15.71 0.02119 + j0.0468 0.03182 + j0.0467

25.55 0.01708 + j0.0400 0.02214 + j0.0446 32.28 0.01465 + jO.0375 0.01900 + j0.0437

20.21 0.01932 + j0.0431 0.02621 + j0.0456

- Mode - Hn fn (hZ) %(no b d )

I 2.7 15.71 0.05 2 27.8 20.21 0.11 3 6.92 25.55 0.028 4 3.92 32.28 0.028

- -

Fig. 4 Modal MIckonical Spring Mau Modd

turbine dampings and shaf t dampings are not ob ta inable . However, modal dampings as obta ined by test are provid- ed i n terms of the dec remen t f ac to r un. S i n c e t h e s e v a l u e s are load dependent they are furnished a long with t h e case d e s c r i p t i o n where they c a n b e a d j u s t e d f o r load. No load mechanica l decrement fac tors for the f i r s t f o u r modes a r e shown i n F i g u r e 4. F u l l l o a d v a l - ues range up to twenty t imes l a rger .

As a n a l t e r n a t i v e a n d f o r t h e c o n v e n i e n c e o f t h o s e w i s h i n g t o c h e c k t h e i r c a l c u l a t i o n s by hand, a modal mechanical model has been provided for the f i r s t f o u r modes (see F igure 4 ) . The mass Hn has been ad jus t ed t o s t o r e t h e same mode energy as t h e sixlnass model when its v e l o c i t y d e v i a t i o n c o r r e s p o n d s t o t h a t of the gen- e r a t o r mass. With n e g l i g i b l e e r r o r , e l e c t r i c a l t o r q u e s when app l i ed i n phase w i th t he angu la r d i sp l acemen t and angu la r ve loc i ty o f mass Hn w i l l change the mode f r e - quency and mode damping by t h e same amount a s i f t h e s e torques were appl ied to the genera tor mass i n t h e six- mass model.

The per u n i t s y s t e m f o r t h e r o t o r electrical and mechanical models i s i d e n t i c a l t o t h a t d e s c r i b e d f o r t r a n s i e n t s t u d i e s . C o n s t a n t f i e l d v o l t a g e is assumed. E l e c t r i c a l t o r q u e v a r i a t i o n s o n t h e e x c i t e r are assumed t o b e z e r o .

TRANSIENT CASE DESCRIPTION

Table V I shows t h e minimum add i t iona l i n fo rma t ion r e q u i r e d t o s p e c i f y a t r a n s i e n t case. S t u d y r e s u l t s should show a t l e a s t t h e f o l l o w i n g as a func t ion of time :

a) Genera tor phase cur ren ts b) Phase voltages on bus A (see Figure 1) c ) Capac i to r vo l t ages d ) Gene ra to r ro to r speed dev ia t ion e) E l e c t r i c a l t o r q u e f) Shaf t t o rque fo r each sha f t s ec t ion

For case 1-T, X, i s tungd to approximately 40 h e r t z b o t h d u r i n g a n d a f t e r t h e f a u l t t o e x c i t e t h e second to rs iona l mode. The f a u l t impedance XF has been ad jus ted to p roduce a c a p a c i t o r t r a n s i e n t v o l t a g e approach ing t he l ower gap s e t t i ng du r ing t he f au l t . A f a u l t d u r a t i o n of .075seconds ( four and one-half cycles a t 60 h e r t z ) was chosen t o produce a no tch in the gen- e r a t o r power enve lope l a s t ing three h a l f c y c l e s of t h e ro to r s econd t o r s iona l f r equency . Th i s no tch i s i n t end- ed t o i n c r e a s e t h e s h a f t t o r s i o n a l r e s p o n s e .

TABLE V I

T r a n s i e n t Case Descr ip t ion

Case 1-T

Generator power o u t p u t Po 0.9 pu Generator power f a c t o r PF 0.9 pu ( lagging) Fau l t r eac t ance XF, (L-G) 0.04 PU

F a u l t l o c a t i o n B (F igure 1) Type o f f a u l t Simultaneous 3L-G Pre fau l t phase vo l t age va = 0

Clear 1st phase, i = 0 . 0 7 5 s a f t e r f a u l t Clear 2nd phase Clear 3rd phase

next current zero nex t cu r ren t ze ro

Capaci tor reactance Xc 0.371 pu Capac i tor bypass vo l tage (no t used) C a p a c i t o r r e i n s e r t i o n v o l t a g e ( n o t u s e d )

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SELF-EXCITATION CASE DESCRIPTION

Table VI1 shows the infomation required to speci- fy a self-excitation case. Note that each case per- tains to the investigation of a single torsional mode. Study results should include one or all of the follow-

ator mass and the other member used the modal mechani- cal model. Their study results are sunmarized below:

ing :

1.

2.

3.

4.

The torsional mode decrement factor Uric (reciprocal time constant), for the entire coupled electrical-mechanical system.

The additional mechanical damping required to just obtain sustained torsional oscil- lation, expressed in terms of incremental decrement factor ADn or incremental damping factor 4. For consistency, apply all dash pot damping on the genera- tor mass of the six-mass model.

The additional armature resistance AR1 required to just obtain sustained torsion- al oscillation.

Reduction in network series compensation AXc to just obtain sustained torsional oscillation.

Data shown for case 1-S was calculated to produce maximum negative damping of the third torsional mode. For case 1-S, use circuit parameters shown in Table IV under mode 3. For study of other modes use the appro- priate modal data shown in Table IV and Figure 4.

TABLE VI1

Self-Excitation Case Description

Case 1-S

Generator power output Po Generator power factor PF

0 - Mode 3 Rotor decrement factor an 0.028 Mechanical damping 41 0.77504

Capacitor reactance Xc 0.287 pu Filter none

RESULTS OF TRANSIENT CASE 1-T

Transient response curves for case 1-T based on the rotor model defined in Figure 2 and Table I were provided by three task force members using different computer programs and problem formulations. The results were overlayed and found to be in close correspondence. From among the sixteen curves specified, five were se- lected for presentation and these are shown in Figure5. The solid line is a composite of the response curves submitted by these three task force members.

A fourth task force member provided response curves based on a treatment of the rotor circuits which was not mathematically equivalent to that used by the other three members. These response curves, shown dashed in Figure 5, indicate the variation in response that can occur with a different modeling technique.

RESULTS OF SELF-EXCITATION CASE 1-S

Two members of the task force have provided re- sults for the self-excitation case. One member used the six-mass model with a single dash pot on the gener-

1.

2.

3.

4.

Six-Mass Modal nodel Model

Coupled electrical-mechanical system decrement factor

A u ~ ~ (per second) -1.525

Additional mechanical damping for sustained oscillations

.-1.503

A u ~ (per second) 2.743 2.722

b3(per unit) 75.9 15.3

Additional armature resistance for sustained oscillations

ARl(per unit) 0.461 0.465

Reduction in series compensation for sustained oscillations

AXctper unit) 0.0470 0.0465

CONCLUSIONS

Simple models and test cases have been presented for the study of subsynchronous resonance. The models and test cases provide a basis for comparison of compu- tational results of the various programs now being ap- plied in the industry. In addition, the models and test cases are useful in computer program development, in the investigation of more sophisticated modeling, and in the discussion of other aspects of.the subsyn- chronous resonance problem.

The task force plans to develop more complex sys- tem models for future application.

CHAIRMAN’S NOTE

The chairman wishes to acknowledge the contribu- tion of Eli Katz who provided the major portion of the effort in the-preparation of the benchmark model and comparison of case studies.

APPENDIX

The rotor network parameters are obtained from the generator standard inpedances and time constants by solution of the following simultaneous equations which are based on the material contained in reference [l] :

o =

o =

o = 1 Xa-377T;i0qd

1 + - (5) xad

+ - 1 (6) w 1 (7) + -

xad

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0 ( a ) Capacitor Voltage, A Phase

in 5 0 9 2 3

50 a

0 Seconds.

( b ) Generator Current, A Phase

0 ( c ) Generator Electrical Torque

w

0

t 4, ( d l S h a f t T o r q u e , L P A - L P B

W

Seconds

0 ( e ) Shaft Torque, GEN-EXC

O _ I 9 Seconds

Rotor circuit os defined in Fig. 2 and Table I ----- Different rotor circuit modeling

Fig. 5 Response Curves For Transient Case 1-T

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o = 1 1 1 X -377T" R

+ - f q g o f q ' xkq-377T;0%q X aq

o = 1 1 + - I + - 1 Xfd-377ThRfd ' X,d-377T!&d 'ad 'L (9)

o = 1 + 1 Xfq-377T'R xkq-377T'S( + 2 + I_ (10)

q fq q q 'aq xL

O = 1 t Xfd-377TxRfd Xkd-377T'& 1 1 1 + - + - (11)

'ad 'L

o = x 1 1

fq-377TiRfq X kq -377T"\ q q 'aq 'L + 1 + (12)

Equations (5) through (8) d e f i n e t h e open c i r c u i t time cons tan ts which a re der ived by s e t t i n g t h e o p e r a - t i ona l admi t t ance o f t he three p a r a l l e l b r a n c h e s f o r each ro tor ne twork to zero . Equat ions (9) th rough (12) d e f i n e t h e s h o r t c i r c u i t time c o n s t a n t s which are de- r i v e d i n t h e same way excep t t ha t t he l eakage r eac t ance is added i n p a r a l l e l w i t h t h e o t h e r b r a n c h e s . For cases where the m a t u r e r e s i s t a n c e R 1 is no t ze ro , t he armature leakage impedance XL i n e q u a t i o n s (9) through (12) is replaced by:

X~-377TR1

where T is t he app ropr i a t e t ime cons t an t .

t h e ing

An i t e r a t i v e me.thod i s r e q u i r e d f o r s o l u t i o n o f above equat ions. To start t h e p r o c e s s , t h e f o l l o w - approximations may be used:

Xfd + xad Rfd - 377Tho

R = X f q + xaq

f q 377T' qo

\q 'aq + xkq 'fq + 'aq 'fq %q = (377T" (X + X

90 a q f q

(22)

REFERENCES

[l] Bernard Adkins, The General Theory of Electrical Machines. London: Chapan & Hall Ltd. , 1957, pp. 101-124, pp. 145-151.

IFSE SUBSYNCHRONOUS RESONANCE TASK FORCE

R. G. Farmer, Chairman Arizona Publ ic Service Company

C. E. J. Bowler G e n e r a l E l e c t r i c Company

C. V. C h i l d e r s , S e c r e t a r y Idaho Power Company

M. C. Hal l Southern Cal i forn ia Edison Capany

Shawky Hammam Clarkson College

R. A. Hedin Allis-Chalmers Corporation

John S. Joyce Allis-Chalmers Power Systems, Inc.

E l i Kat2 Los Angeles Department of Water and Power

Lee Kilgore West inghouse Electr ic Corporat ion

D. G. Ramey West inghouse Electr ic Corporat ion

Caleb H. Didr iksen, Jr. Char les T. Main Inc. Engineers

Robert Quay G e n e r a l E l e c t r i c Company

Harlow Peterson S a l t R i v e r P r o j e c t

K. R. Shah Commonwealth Associates, Inc.

John DOrney Publ ic Serv ice Company of New Mexico

Edgar R. Taylor , Jr . Westinghouse Electric Corporat ion

John M. U n d r i l l Power Technologies, Inc.

Richard H. Webster P a c i f i c Gas a n d E l e c t r i c Company

Audrey J. Smith Bechtel Power Company

Edward Kimbark Bonnevi l le Power Adminis t ra t ion

Th =a Th, d

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Discussion

Yao-nan Yu and M. D. Wvong (University of British Columbia, Van- couvei, Canada): The authors are to be congratulated for the excellent work they have done in developing this first benchmark model for com- puter simulation of SSR. It is not only useful for comparing computer programs, but also convenient for stabilizer design. We would appreciate comments upon the following points:

(a) It appears that the exciter electric torque must be specified if the exciter mass is to be included in the linear multi-mass model for computer program comparison. and that the supplemental excitation control, if included, could create a very large electric torque on the exciter shaft.

(b) The parameter equations based upon the simple parallel cir-

equations (1) through (24). cuits of Figs. 2 and 3 of the paper are clearly and concisely expressed in

We would like to point out that the leakage reactance XL must be properly defined. Some details may be found in reference [A].

REFERENCE

[A] Yao-nan Yu and H. A. M. Moussa, “Experimental Determination of Exact Equivalent Circuit Parameters of Synchronous Machines”, IEEE Trans. on Power Apparatus and Systems, Vol. PAS-90, pp. 2555-2560, Nov./Dec. 197 1.

Manuscript received February 2 2 , 1977

D. G. Ramey (Westinghouse Electric Corporation, East Pittsburgh, PA): The stated objective of this paper is to provide reference test cases to facilitate the comparison of calculations and the debugging of com- puter programs for studying subsynchronous resonance problems. Two test cases are provided including solutions that can be duplicated pre- cisely if the machine and system models outlined in the paper are used.

It is of considerable interest t o most researchers t o know the sensi- tivity of the solution to variations in the models or in the data used in the models. We have found that very much simpler models than those implied in this paper can give quite accurate results and at the same time impart more understanding of the physical processes involved in the SSR phenomenon [ 11. The study of self-excited oscillations in- volving generator torsional interactions can be performed using the con- cept of steady-state electrical impedances. The major assumptions are that the mechanical torsional resonant frequency will not be signifi- cantly altered when it is coupled to the electrical system, and that the only components of the interaction between electrical and mechanical quantities that must be considered are the oscillating torque compo- nent in phase with velocity and the oscillating current component in phase with voltage. Additionally, the generator is modeled by the induc- tion machine equivalent circuit for the synchronous plus and minus the torsional resonant frequency. When the electrical interaction is ex- pressed as a mechanical damping coefficient, the total damping becomes

In this equation the impedances R+ and Xt are those of the series elec- tric circuit formed by the induction machine circuit of the generator, the unit transformer, and the external system impedance at the syn- chronous frequency (fs) plus the torsional frequency (fn). The im- pedanc‘es R- and X- are the impedances of the same circuit at fre- quency fs - fn.

This formula can be easily compared with the results given in the paper for case I-S. For this comparison the average of the D and Q axis generator rotor resistances for the third torsional mode is used in the induction machine model. This gives an induction machine resistance (Rr/s) of -0.01 126 at 34.45 Hz and +0.02796 at 85.55 Hz. The cor-

X- = 0.0, R+ = 0.04796, and X+ = 1.0268. When these values are sub- responding total system resistance and reactances are R- = 0.00874,

-76.29. Thus, AD3 (per unit) = 76.29 compared with a value of 75.9 stituted in the above formula, they result in a value of D equal to

using an eigenvalue program.

which must be added and the amount of reduction in series capacitance Similar checks can be made of the amount of series resistance

necessary to make AD3 = 0. The values are AR1 = .464 p.u. and AXc = ,0507. The largest deviation from the values given is less than

be expressed by use of a single algebraic equation. 10 percent. This small error shows that the major SSR interaction can

magnitude and frequency of the stator current that is induced by the The dominant factor in the transient torque SSR problem is the

application or removal of the short circuit i.n the transmission system. This can be seen by representing the generator as a 60 Hz voltage source behind subtransient reactance for the transient case I-T. Instantaneous

Manuscript received February 28. 1977

electrical power, which in per-unit would be generator torque, is the sum of the products of instantaneous phase current and voltage. If this torque is applied to a spring-mass model of the rotor, the resulting tran- sient waveforms are as shown below.

0 - P 0 0 .e 9 (a) Capacitor Voltage, ’9 A Phase c

0 2 03 0.4

c.2

0 2 03 0.4

c.2

(bJ Generator Current, A Phase

(c) Generator Electrical Torque

(d) Shaft Torque, LPA - LPB

(e) Shaft Torque, Gen - Ex

Fig. 1. Response for IEEE Test Case I-T

These waveforms are very similar to those in Fig. 5 of the paper. There is less than 10 percent difference in the magnitude of the transient shaft torque peaks, so the same engineering decisions would result.

scribed above and that suggested by the paper is that the torsional The primary difference between the simulation technique de-

oscillation df the generator rotor would influence the magnitude of

model described in this paper. With the generator represented as a subsynchronous voltages, currents, and torques in the more complete

60 Hz voltage source, this interaction is neglected. The close match in transient magnitudes and waveforms between the two methods shows

sient torque studies. that the error in this simplification is of secondary importance in tran-

REFERENCE

[ 11 L. A. Kilgore, D. G. Ramey, M. C. Hall, “Simplified Transmission and Generation System Analysis Procedures for Subsynchronous Resonance Problems”, IEEE Paper No. F 77 066-4, presented at

January 30-February 4, 1977. the 1977 IEEE Power Engineering Society Winter Power Meeting,

T. J . Hammons (Glasgow University, Glasgow G12 8Q0, U.K.): The authors are to be commended for their effort in developing a bench- mark model to provide a basis for comparison of computational results of the various programs now under development.

lowing fault clearance or faulty synchronization, and the highest torques Peak shaft torques fluctuate cyclically in a complex manner fol-

do not necessarily occur in the first few cycles after fault clearance. They must be calculated over a period long enough to ensure that the highest peaks have been found. These peaks usually occur in the shaft assembly at different times.

Shafts and couplings must be designed to withstand these torques without immediately failing, but as shaft torque oscillates, the possible effects of cumulative fatigue should be carefully assessed. Since shaft torques depend very much of the mechanical parameters of the shaft assembly, each generator must be investigated in detail, even for stipu- lated fault conditions in a given supply network.

face, and decrease progressively along the turbine shaft. Since shaft In general, torque values are greatest at the generator/turbine inter-

diameters tend to decrease in a similar manner, stresses may neverthe- less be high. The oscillatory torques in the exciter may become exces- sively large.

Faulty synchronizing at a displacement angle of approximately

not exceed those resulting when a 3-phase fault on the HV busbar is 120’ also gives rise to high shaft torques. These torques may or may

cleared. Peak torques increase as network and fault impedances are made lower, and can become excessively large if subsynchronous resc- nance occurs.

One method of limiting the torques on shafts and couplings is by means of generator coupling bolts that shear at a torque below that which might occur under the worst conditions, or when damage might otherwise result in the shaft. Use of such couplings have been shown to produce a significant reduction in torques along the shaft. Proper means must be employed, however, to avoid excessive axial float-of the gen- erator rotor after the bolts fail.

Manuscript received March 1, 1977.

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Peak acceleration of the last LP turbine section following shearing

case, which has been examined, [A] the peak acceleration was found to of the generator/turbine coupling bolts can be excessively large. In one

be 1600 rad@, the corresponding acceleration under non-shearing con- ditions being 750 rad@. The frequency of the vibration set up in the last LP stage by coupling failure was 103 Hz. The hgher acceleration which results on coupling failure may be acceptable provided the struc- tural limitations of the LP blading are not exceeded. It may therefore be necessary to consider for these higher accelerations pulsating at high frequency the inertia stresses induced in the blades by movement of their roots. Similar phenomena mght be ’induced by subsynchronous resonance which could lead to displaced resonant frequencies and addi- tional material damping which would effect system response.

An up to date assessment of this phenomena in relation to sub- synchronous resonance and related transient peak shaft stresses would be timely in this case.

REFERENCE

[AI Hammons, T. J. , “Effect of Three-phase Faults and Faulty Syn-

erators”, Rev. Gen. Elect., Vol. 86, (749, July - August 1977, chronization on the Mechanical Stressing of Large TurbineGen-

pp 558-580.

Eli Katz: Messrs. Yu and Wvong have suggested correctly that exciter torques are important when dealing with the effects of the power sys- tem stabilizer or when designing the exciter controls for damping rotor oscillations. The important role played by the exciter in the self-excita- tion problem is illustrated by the need to remove the power system stabilizers from service at the Navajo Plant, because of their destabilizing tendency, until they could be equipped with filters to desensitize them to torsional frequencies.

with accurate rotor impedance at the modal frequencies. When the As noted in the paper, self-excitation studies require a rotor model

exciter is permitted to play a role, the model must a1.w accurately represent the variation in h d caused by variations in Efd at each tor- sional frequency. These machine characteristics are not generally avail- able, but they can be obtained by test or calculated by the machine de- signer. When these characteristics are available, I believe that a simple three parallel branch model might be satisfactorily fitted for the study of one mode at a time.

The variation in field current caused by armature current varia- tions need not be precisely modeled so long as thtre is a damper wind- ing. Exciter torques produced by this field current are comparatively inconsequential in both self-excitation and transient studies. For transient studies, the exciter torque caused by exciter voltage variation cannot be ignored unless the exciter is desensitized to those torsionai

play a dominant role in shaft torque buildup. frequencies with large exciter motion. If this is not done, the exciter can

D. G. Ramey presents some very useful approximations which im- part considerable understanding of the SSR phenomena. His interaction equation gives the added generator mass dampbg required for the inter- action to just sustain oscillations. Although the equation is derived for a machine with a symmetrical rotor, it is also quite accurate for ma-

Table VI11 Mode 3 Rotor Circuit Per Unit Impedances

at 60 hz for Self-Excitation Study Pole Face Damper Removed

Nonsymmetric Rotor Symmetric Rotor

XL .13 .13 Rrd BO7841 ,0325455

xrd .050268 .135274

xad 1.66 1.62

Rrq .057250 .0325455

xrq

xaq 1.58 1.62

.220280 .135274

Manuscript received April 18,1977.

chines such as the Navajo units which, because of their pole face damper windings, are nearly symmetrical. Less accuracy can be expected for machines not so equipped.

To illustrate how the symmetrical and nonsymmetrical interaction compare, we have remn the self-excitation case using the modal model

data furnished for the same machine before the pole face damper was but this time with rotor circuit parameters based on manufacturer’s

added. Table VI11 shows the parameters for the new rotor circuit and the parameters for the symmetrical comparison case which was o b tained by averaging the direct and quadrature axis elements. The cir- cuit series capacitor was increased from .287 per unit to .3 1 per unit to compensate for the added circuit inductance and thereby maintain maximum third mode interaction.

Table IX shows the change in third mode decrement factors for these two cases as the line resistance is varied from .01 to .6 per unit. Using Mr. Ramey’s equation, the line resistance for zero damping was calculated to be S224 which corresponds closely to that shown for the

resistance is .4429. symmetrical rotor. For the nonsymmetrical rotor, the comparable line

Table IX Third Torsional Mode Decrement Factors

Comparison of Symmetric and Nonsymmetric Machines X,= .31 pu

Line Resistance Per Unit

.o 1

.02

.03

.030355

.030356

.036985

.036990

.04

.05

.1

.2

.4

.442905.

.5 324378 .6

Decrement Factor Nonsymmetric Rotor Symmetric Rotor

- .83243 - .76104 - 1.21123 - 1.00821 - 1.93901 - 1.45696 - 1.97398 + 1.83941

- 2.020 12 + 1.8896 1

+ 1.13938 + 1.63349 + .76016 + 1.08284 + .23850 + .33285 + .07502 + .lo796 + ,00655 + .01997

0 - ,00699 + .00312

0 - .01603 - .00080

fortunately they were not available in time to be included in Figure 5 Mr. Ramey’s transient response curves speak for themselves. Un-

along with those submitted by other members of the Task Force. Mr. T. J . Hammons has provided an interesting review of the mech-

anical problems related to faulty synchronizing, an allied subject which the Task Force has elected to exclude from its scope of activity. In his discussion, Mr. Hammons raises the hope of using shear bolts to limit shaft torque caused by subsynchronous resonance, an idea that also occurred to some of us in 1972 when we were up to our ears in this problem. However, engineers who are familiar with turbine-generator design tell us that it is a very bad idea for the following reasons:

1. Shear bolt failure would subject the turbine buckets to large bending stress because of the sudden release of shaft-stored energy.

2. Sudden loss of generator load and inertia would result in ex- cessive turbine overspeed subjecting the longer blades to excessive radial stress and possible contact with stationary parts.

3. It would be nearly impossible to design the shear bolts so they would have sufficient strength to withstand normal full load torque and at the same time provide reliable protection against the relatively modest cyclical torques which are capable of causing shaft failure. Because of the serious consequences of Items 1 and 2 above, we could not accept the risk of an occasional unnecessary failure of shear bolts.

On behalf of all the task force members, I wish to thank the dis- cussers for their thoughtful and informative comments.

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