FIRST BENCHMARK MODEL FOR COMPUTER SIMULATION OF SUBSyNcHRoNoUS RFSONANm
IEEE Subsynchronous Resonance Task Force of the
Dynamic SystemPerformanceWorkingGroup
Power System Engineering
Camnittee
ABSTRACT
A benchmarkmodel
forthestudyofsubsynchronous
r e s o n a n c e i s p r e s e n t e da l o n gw i t h
two t e s t problems
f o ru s ei nc a n p u t e rp r o g r a mc o m p a r i s o na n dd e v e l o p ment. Models were d e v e l o p e dw i t ht h e
minimum s o p h i s t>sationneededtoobtainusefulresultsandwithdue
r e g a r dt ot h ek i n d so fm a c h i n ec h a r a c t e r i s t i c sg e n e r a l l yo b t a i n a b l e .
The m a c h i n ea n dc i r c u ipt a r a m e t e r s
are real v a l u e s t a k e n f r o m
the N a v a j o P r o j e c t .
INTFCODUCTION
P r e s e n t l y ,s e v e r a lc a n p u t e rp r o g r a m sa n da n a l y 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
t h ei n t e r a c t i o no fm u l t i m a s st u r b i n e- g e n e r a t o r sa n d
series c a n p e n s a t e dt r a n s m i s s i o ns y s t e m sO
. thers
are
u n d e dr e v e l o p e n tT. h e r e
i s a currenn
t e e di nt h e
electricpowerindustryto
comparestudy results, determinethereasonsfordifferences,andrevisemodels
a n dt e c h n i q u e s as deemed n e c e s s a r y . To h e l p meet t h i s
n e e d , t h e IEEE SubsynchronousResonanceTaskForcehas
p r e p a r e d s t a n d a r d t e s t cases t o f a c i l i t a t e the compari s o no fc a l c u l a t i o n sa n dt h ed e b u g g i n go fc a u p u t e r
programs.Using
t h e N a v a j oP r o j e c t 892.4 MVA genera5 0 0 kV t r a n s m i s s i o ns y s t e ma s
a guide, a
t o r sa n d
s t a n d a r dn e t w o r k ,t w ot u r b i n eg e n e r a t o rm o d e l sa n d
t o t e s tc a s e sh a v eb e e np r o v i d e d .F l e x i b i l d a t af o r w
i t y is provided for addition of
new t e s t c a s e s a n d t h e
modeling i s s u f f i c i e n t l yd e t a i l e df o rs i m u l a t i o no f
most aspects of the
SSR problem.
P a r a m e t e r se x p r e s s e di n
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
lineR
. e a c t a n c e sa r ep r o p o r t i o n a tl of r e q u e n c y r; e s i s t a n c e sa r ec o n s t a n t .
The i n f i n i t eb u s i s a threephase 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
Two f a u l tl o c a t i o n s
( A and B) a r e
a l lf r e q u e n c i e s .
d e s i g n a t e d ,a n dt h e r e
i s p r o v i s i o nf o ri n c l u s i o no fa
are provided: a low
f i l t e r . % c a p a c i t o rs p a r kg a p s
v o l t a g eg a pt ob y p a s st h ec a p a c i t o rd u r i n gt h ef a u l t ,
t h e r e b yl i m i t i n gt h ef a u l tc u r r e n ta n dn e t w o r ks 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 gr e i n s e r t i o n .
The low v o l t a g eg a p i s i n s e r v i c e
a t a l l times e x c e p td u r i n gt h ep e r i o db e g i n n i n gw i t h
t h ef a u l ct l e a r a n c ea n dc o n t i n u i n gf o r
a s h o r t time
a f t e rc a p a c i t o rr e i n s e r t i o nt oa v o i dr e s t r i k e .
When
60 h e r t z ( m s ) p r u n ivto l t a g e
u s e dt h,per o p e r
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
bypassgapand
5.31 X, f o rt h er e i n s e r t i o ng 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
studies.
ROTOR MODELS FOR TRANSIENTSTUDIES
mass
F i g u r e 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
model p r o v i d e df o rt r a n s i e n ts t u d i e s .
The e l e c t r i c a l
a n dm e c h a n i c a lp o r t i o n s
are s i m p l er e p r e s e n t a t i o n s of
thenavajorotors,developedwiththe
minimum l e v e l o f
s o p h i s t i c a t i o nr e q u i r e df o rt h ec a u p u t a t i o n
of transientshaft torques. Constant field voltage
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.
ELECTRICAL NE"W
E x t e n s i v e SSR s t u d i e so f
t h e N a v a j oP r o j e c t rev e a l e dt h a t
a s i m p l er a d i a l
RLC c i r c u i t ,p r o p e r l y
and s e l f - e x c i t a t i o n
t u n e d ,c a np r o d u c eb o t ht r a n s i e n t
intheanalysisof
problems as s e v e r e as anyobserved
t h ea c t u a ls y s t e m .
The s i n g l el i n ed i a g r a m
shown i n
F i g u r e 1 r e p r e s e n t ss u c h a s i m p l e c i r c u i t . T h e c i r c u i t
UNITY
VOLTAGE
Fig. I
Network for Subsynchronous Rescmonce Studies
€ 77 102-7.
A paper reccanrended and mprvved by
the IEEE P a r e r 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 30February 4, 1977. Manuscript submitted Septdxr 7,
1976; nnde available for printing N o v e n h r 4, 1976.
Fig. 2
Rotor Model for Transient Studies
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TABLE I
TABLE I1
Rotor Circuit Parameters
For Transient SSR Studies
Parameter
0.326 0.062
0.13
3 7 7 4 5.3
Xf
377% 3.1
k'
a'
XL
Generator Impedances and
Time Constants
X = 1.79
d
Q-axis
D-axis
0.53
I
pu
Tdo
= 4.3
Xd = 0.169 pu
Tio
= 0.032 s
Xi = 0.135 pu
T'
qo
T"
qo
= 0.85
s
= 0.05
s
5
= 0.0
I
1.54
0.0055
1.66
0.095
1.58
0.13
XL = 0.13
pu
= 1.71
pu
X
Table I shows theimpedances for the rotor circuit
expressed in per uniton the machineMVA base. Current,
voltage, torque and rotor speed are also expressed in
per unit. For transient representation, divide reactance in ohms by 377
to obtain the inductance in
henries.
q
X' = 0.228 pu
9
X" = 0.200 pu
q
TABLE I11
The generator standard impedances and
time conRotor Spring Mass Parameters
stants from which the circuit parameters
of Table I
were derived are shown
in Table 11. The
term XL is
armatureleakagereactance.Circuitparametersare
Inertia
Spring
Constant
obtained from the standard impedances and time constants by an iterative process outlinea in the appenMass
Shaft H(seconds)
K_o
dix. Where the treatment
of these standard machine
impedances and time constants
is not mathematically
HP
0.092897
reducible to the rotor network definedin Figure 2 and
HP-IP
7,277
Table I, the results
will differ.
IP
0.155589
13,168
IP-LPA
Table I11 shows the inertias and spring constants LPA
0.858670
for the spring mass model. Inertia is expressed
in 19,618
LPA-LPB
terms of the inertia constant
H based on rated kVA.
LPB
0.884215
The base torque is that required at synchronous speed 70
26,713
LPB-GEN
to deliver mechanical power in kilowatts equal to the
GEN
0.868495
rated (base) kVA value. Base angle is 377 radians, the
1,064
GEN-EXC
angle
of shaft rotation in one second (the.base time).
EXC
0.0342165
The spring constantK 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:
*D
T(pu) = 2 H5
+
D6
+
s
pu Torque/rad
K8
52.038
.E58
2.822
+
0
id
The steady state mechanical torque is apportioned
among the turbine sections
HP, IP, LPA andLPB, respectively as follows: 30%, 26%, 22% and 22%. The exciter
steady state torqueis assumed to be zero.
MODELSFOR SELF-EXCITATION
34.929
I
The spring constant is also given in per unit torque
For transientstudies,themechanical
perradian.
damping is assumed to be zero.
ROTOR
19.303
ed
Q-axis
S'WDIES
A simpler version of the rotor circuitry is provided (see Figure 3) for self-excitation studies. The
rotor circuit provided for transient studiesis inadebe
quate for self-excitation studies because it cannot
fitted satisfactorily
to the rotor impedance versus
frequencycharacteristicsfurnished
by themachine
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 butm r e complex model to cover the entire range
of torsional frequencies.
CircuitparametersforFigure3areshown
in
Table rV. Note that the elements XL, Xad, and Xaq are
model.
The generator
the same as for the transient
rotor impedances at the torsional frequencies to which
elements of Figure 3 were fitted are shown
in Table V.
W
Fig. 3
Rotor Model for Self-ExcitationStudies
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Where t h et r e a t m e n to ft h e s er o t o ri m p e d a n c e s
i s not
m a t h e m a t i c a l l yr e d u c i b l et ot h er o t o rn e t w o r kd e f i n e d
IV, the results w i l l differ.
i nF i g u r e3a n dT a b l e
The r o t o sr p r i n g
mass
model
used
f o tr r a n s i e n t
studies is alsousedforself-excitationstudiesexcept
f otrhien 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
14
Mode
32.28
25.55
20.21
Frequency
15.71
(hz)
2
3
XL
0.13
Rrd PU
0.00587
0.13
0.00686
0.00764
0.13
0.00825
X r d p'
0.04786
0.04401
0.04080
0.03823
xad PU
1.66
1.66
1.66
1.66
0.13
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
turbinedampingsandshaftdampings
are n o t o b t a i n a b l e .
as o b t a i n e d by test are p r o v i d However,modaldampings
un. S i n c et h e s e
ed i n terms of t h ed e c r e m e n ft a c t o r
v a l u e s are load dependent they
are f u r n i s h e d a l o n g w i t h
c a n b ea d j u s t e df o r
t h e case d e s c r i p t i o n wherethey
load.
No l o am
d e c h a n i c adle c r e m e nf ta c t o rfsot rh e
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 uesrangeuptotwentytimeslarger.
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 gt oc h e c kt h e i rc a l c u l a t i o n s
byhand,
a modal
mechanicalmodelhasbeenprovidedfor
the f i r s t f o u r
modes (see F i g u r e4 ) .
The mass Hn h a sb e e na d j u s t e dt o
s t o r et h e
same mode energy as t h e sixlnass model when
its v e l o c i t yd e v i a t i o nc o r r e s p o n d st ot h a t
of t h eg e n e r a t o r mass. With n e g l i g i b l ee r r o r ,e l e c t r i c a lt o r q u e s
when a p p l i e d i n p h a s e w i t h t h e a n g u l a r d i s p l a c e m e n t a n d
a n g u l a rv e l o c i t yo f
mass Hn w i l l c h a n g et h e
mode f r e quencyand mode dampingby
t h e same amount a s i f t h e s e
mass i n t h e sixt o r q u e sw e r ea p p l i e dt ot h eg e n e r a t o r
mass model.
The per u n i ts y s t e m f o rt h er o t o r
electrical and
mechanicalmodels
i s i d e n t i c a lt ot h a td e s c r i b e df o r
is assumed.
t r a n s i e n ts t u d i e s .C o n s t a n tf i e l dv o l t a g e
are assumed
Electrical torque variations on the exciter
to be zero.
TRANSIENT CASE DESCRIPTION
T a b l e V I shows t h e minimum a d d i t i o n a li n f o r m a t i o n
r e q u i r e dt os p e c i f y
a t r a n s i e n t case.
S t u d yr e s u l t s
should show a tl e a s t h ef o l l o w i n g
as a f u n c t i o no f
time :
TABLE V
P e r Unit Rotor
Impedance a t SubsynchronousFrequency
M u l t i p l i e d by ( 6 0 / f n )
Rotor Frequency
fn
D-axis
paxis
+
+
+
+
+
+
0.02119
15.71
20.21
0.01932
0.01708
25.55
0.01465
32.28
+
+
j0.0468
0.03182
j0.0431
0.02621
j0.0400
0.02214
jO.0375
0.01900
j0.0467
j0.0456
j0.0446
j0.0437
a ) G e n e r a t o rp h a s ec u r r e n t s
b)Phasevoltagesonbus
A (see F i g u r e 1)
cC) a p a c i t ovr o l t a g e s
d )G e n e r a t o rr o t o rs p e e dd e v i a t i o n
e) E l e c t r i c atlo r q u e
f) S h a f tt o r q u ef o re a c hs h a f ts e c t i o n
For
case
1-T, X,
i s t u n gtdaop p r o x i m a t e l y
40
h e r t zb o t hd u r i n ga n da f t e rt h ef a u l t oe x c i t et h e
s e c o n dt o r s i o n a l mode.The
f a u l t impedance XF hasbeen
a c a p a c i t ot r a n s i e nv to l t a g e
a d j u s t ept odr o d u c e
a p p r o a c h i n gt h el o w e rg a ps e t t i n gd u r i n gt h ef a u l t .
A
f a u l t d u r a t i o n of.075seconds(fourandone-halfcycles
a t 60 h e r t z ) was chosen t o p r o d u c e a n o t c h i n t h e g e n of t h e
e r a t o r power e n v e l o p el a s t i n g three h a l f c y c l e s
r o t o rs e c o n dt o r s i o n a lf r e q u e n c y .T h i sn o t c h i si n t e n d 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 D e s c r i p t i o n
Case 1-T
27.8
6.92
Mode
-
Hn
fn (hZ)
%(no b d )
-
I
2
3
2.7
4
3.92
15.71
20.21
25.55
32.28
Fig. 4
0.05
0.11
0.028
0.028
Modal MIckonical Spring Mau Modd
Generator power o u t p u t Po
Generator power f a c t o r PF
F a u l tr e a c t a n c e XF, (L-G)
0.9
0.9
0.04
Fault location
Type o f f a u l t
Prefault phase voltage
B ( F i g u r e 1)
Simultaneous 3L-G
Clear 1st p h a s e , i = 0
C l e a r 2nd phase
Clear 3rd phase
. 0 7a5f ftsae ur l t
n e x t current z e r o
nextcurrentzero
Xc
C a praecaicttoarn c e
C a p abcyviupots(oalentrsdaosg
)t e
C a p a c i troeri n s e r t i vo on l t a(gneuoste d )
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va
pu
pu ( l a g g i n g )
PU
= 0
pu0.371
ator mass and the
cal model. Their
SELF-EXCITATION CASE DESCRIPTION
Table VI1 shows the infomation required to specify a self-excitation case. Note
that each case pertains to the investigation of a single torsional mode.
Study results should include one or all of the following :
1.
Six-Mass
nodel
1.
The torsional mode decrement factor Uric
(reciprocal time constant), forthe entire
coupled electrical-mechanical system.
The additional mechanical damping required
to just obtain sustained torsional oscillation, expressed in terms of incremental
decrement factor ADn or incremental
damping factor 4. For consistency,
apply all dash pot damping on the generator mass of the six-mass model.
3.
The additional armature resistance AR1
required to just obtain sustained torsional oscillation.
4.
Reduction in network series compensation
AXc to just obtain sustained torsional
oscillation.
Self-Excitation Case Description
Case 1-S
Capacitor reactance
0.287
Xc
Filter
an
2.722
15.3
0.465
0.461
Reduction in series compensation
for sustained oscillations
AXctper unit)
TABLE VI1
Mode
Rotor decrement factor
Mechanical damping 41
2.743
75.9
Additional armature resistance for
sustained oscillations
ARl(per unit)
4.
.-1.503
Additional mechanical damping for
sustained oscillations
A u (per
~
second)
Data shown f o r case 1-S was calculated to produce
maximum negative damping of the third torsional mode.
For case 1-S, use circuitparameters shown in Table IV
under mode 3. For study of other modes use the appropriate modal data shown in Table IV and Figure 4.
Generator power output Po
Generator power factor PF
-1.525
b3(per unit)
3.
Modal
Model
Coupled electrical-mechanical
system decrement factor
A u (per
~ ~second)
2.
2.
other member used the modal mechanistudy results are sunmarized below:
0.0465
0.0470
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 computational results of the various programs now being applied 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 subsynchronous resonance problem.
0
-
The task force plans to develop more complex system models for future application.
3
0.028
0.77504
CHAIRMAN’S NOTE
The chairman wishes to acknowledge the contribution of Eli Katz who provided the major portion of the
effort in the-preparation of the benchmark model and
comparison of case studies.
pu
none
RESULTS OF TRANSIENT CASE 1-T
APPENDIX
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 selected for presentation and these are shown in Figure5.
The solid line is a
composite of the response curves
submitted by thesethree task force members.
The rotor network parameters are obtained from the
generator standard inpedances and time constants by
solution of the following simultaneous equations which
are based onthe material contained in reference [l] :
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-EXCITATIONCASE 1-S
Two members of the task force have provided results for the self-excitation case. One
member used
the six-mass model with a single dash pot on the gener-
o =
+- 1
(5)
o =
+- 1
(6)
+- 1
(7)
o =
xad
w
1
Xa-377T;i0qd
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xad
0
( a ) CapacitorVoltage,
A Phase
in
5 0
92
50
3
a
Seconds.
0
( b ) Generator
Current,
0
( c )Torque
Electrical
Generator
0
( dSl h a f Tt o r q u eL, P A L- P B
Phase
A
w
,
4
t
W
Seconds
0
GEN-EXC
Torque,
( e ) Shaft
Seconds
O_I
9
----Fig. 5
Response
Curves
For
Transient
Case
1569
Rotor circuit os defined in Fig. 2 and Table I
Different rotor circuit modeling
1-T
o =
q
X
1
1
-377T" R
fgqfoq
1
'
xkq-377T;0%q
1
o = Xfd-377ThRfd
'
1
X,d-377T!&d
1
o = xkq-377T'S(
Xfq-377T'R
+
q
O =
fq
+Xa q
1
q
(9)
+ 2 + I_
(10)
'aq
1
Xfd-377TxRfd
t
1
Xkd-377T'&
1
o = x
+ -'adI + -L'1
1 + 1
+-
'ad
1
fq-377TiRfq
xL
+ 1+
X kq-377T"\q
'aq
(11)
L
'
(12)
L
'
REFERENCES
General Theoryof
Electrical
Machines.
London:
Chapan
& Hall
Ltd.,1957,
pp.101-124,pp.145-151.
[l] BernardAdkins,The
IFSE SUBSYNCHRONOUS RESONANCE TASK FORCE
R. G. Farmer, Chairman
ArizonaPublicService
Company
Equations (5) through (8) d e f i n et h e open c i r c u i t
time c o n s t a n t s w h i c h a r e d e r i v e d
by s e t t i n g t h e o p e r a t i o n a la d m i t t a n c eo ft h e
three p a r a l l e lb r a n c h e sf o r
e a c hr o t o rn e t w o r kt oz e r o .E q u a t i o n s( 9 )t h r o u g h
(12)
d e f i n et h es h o r tc i r c u i t
time c o n s t a n t s which are der i v e d i n t h e same way e x c e p t t h a t t h e l e a k a g e r e a c t a n c e
i s added i pn a r a l l ewl i t thhoe t h ebr r a n c h e s .
For
cases where the
m a t u r e resistance R1 is notzero,the
XL i n e q u a t i o n s ( 9 ) through
armatureleakageimpedance
(12) is replacedby:
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
C. H a l l
Southern California Edison Capany
M.
Shawky Hammam
Clarkson College
X~-377TR1
where T is t h e a p p r o p r i a t e t i m e c o n s t a n t .
R. A. Hedin
Allis-Chalmers Corporation
An i t e r a t i v e me.thod i s r e q u i r e df o rs o l u t i o no f
To start t h ep r o c e s s ,t h e f o l l o w t h e a b o v ee q u a t i o n s .
i n g approximations may be used:
John S. Joyce
Allis-Chalmers PowerSystems,Inc.
Eli
Kat2
Los AngelesDepartmentof
Water and Power
Lee K i l g o r e
W e s t i n g h o u s eE l e c t r i cC o r p o r a t i o n
D. G. Ramey
WestinghouseElectricCorporation
Caleb H. D i d r i k s e n , Jr.
C h a r l e s T. Main I n c .E n g i n e e r s
Robert Quay
GeneralElectric
Company
Harlow Peterson
Salt River Project
K. R. Shah
Commonwealth Associates, I n c .
Rfd
R
fq
=
Xfd + xad
377Tho
John DOrney
Public Service
Company o f New Mexico
.
xaq
377T'
qo
Xfq +
(22)
Edgar R. T a y l o r , Jr
Westinghouse E l e c t r i c C o r p o r a t i o n
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
\q
%q =
Th
xkq'fq
'aq
'aq
(377T" (X
+ X
90f q a q
+
+
Company
Audrey J . Smith
B e c h t e l Power Company
'fq
EdwardKimbark
B o n n e v i l l e Power A d m i n i s t r a t i o n
= aT h ,
d
1570
Discussion
Yao-nanYu and M. D. Wvong (University of British Columbia, Vancouvei, Canada): The authors are to be congratulated for the excellent
work they have done in developing this first benchmark model for computer simulation of SSR. It is not only useful for comparing computer
programs, but also convenient for stabilizer design. We would appreciate
comments upon thefollowing 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
computerprogramcomparison.andthatthesupplementalexcitation
control, if included, could create a very large electric torque on the
exciter shaft.
(b) The parameter equations
based upon the simple parallel circuits of Figs. 2 and 3 of the paper are clearly and concisely expressed in
equations (1) through (24).
We would like to point out that theleakage reactance XL must be
properly defined. Some details may be found in reference [ A ] .
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, theresulting transient waveforms are as shown below.
0
.e
P
(a) Capacitor Voltage,
A Phase
9
0 0
’9c
(bJGenerator Current,
A Phase
02
03
0.4
(c) Generator Electrical
Torque
(d) Shaft Torque,
LPA LPB
-
REFERENCE
c.2
[A] Yao-nanYu and H. A. M. Moussa, “Experimental Determination
of Exact Equivalent CircuitParameters of SynchronousMachines”,
IEEETrans.
on PowerApparatusandSystems,Vol.
PAS-90,
pp. 2555-2560, Nov./Dec. 197 1.
(e) Shaft Torque,
Gen Ex
-
Fig. 1. Response for IEEE Test Case I-T
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 t o
facilitate the comparisonofcalculationsand
the debugging ofcomputer programs for studying subsynchronous resonance problems. Two
test cases are provided including solutions that can be duplicated precisely 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 sensitivity 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 involving generator torsional interactions canbe performed using the concept of steady-state electrical impedances. The major
assumptions are
that the mechanicaltorsional resonantfrequency will not besignificantly 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 torquecomponent in phase with velocity and the oscillating current component in
phase with voltage. Additionally, the generator is modeledby the induction machineequivalentcircuit
forthesynchronous plusandminus
the torsional resonant frequency. When the electrical interaction is expressed as a mechanical damping coefficient, the total damping becomes
In this equation the impedances R+ and Xt are those of the series electric circuit formed by the induction machine circuit of the generator,
the unit transformer, and the external
systemimpedance at the synchronousfrequency(fs)plusthe
torsional frequency(fn).The
impedanc‘esR- and X- are the impedancesof the same circuit at frequency 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 corresponding total systemresistanceandreactances
are R- = 0.00874,
X- = 0.0, R+ = 0.04796, and X+ = 1.0268. When these values are substituted in the above formula,they resultin a value of D equal to
-76.29. Thus, AD3 (per unit) = 76.29 compared with a value of 75.9
using an eigenvalue program.
Similar checks can be madeoftheamount
of series resistance
which must be added and the amountof reduction in series capacitance
necessary to make AD3 = 0. The values are A R 1 = .464 p.u. and
AXc = ,0507. The largest deviation from the values given is less than
10 percent. This small error shows that the major SSR interaction can
be expressed by use of a single algebraic equation.
The dominant factor in the transient torque SSR problem is the
magnitude and frequency of the stator current that is induced by 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
These waveforms are very similar to those in Fig. 5 of the paper. There
isless than 10 percentdifferencein
the magnitudeofthetransient
shaft torque peaks, so the same engineering decisions would result.
The primarydifferencebetween
the simulationtechnique
described aboveand that suggested by thepaper is thatthe torsional
oscillation df thegenerator rotor wouldinfluence the magnitudeof
subsynchronous voltages, currents, and torques in the more complete
modeldescribed in thispaper. With the generatorrepresented as a
60 Hz voltage source, this interaction is neglected. The close match in
transient magnitudes and waveforms between
the two methods shows
that the error in this simplification is of secondary importance in transient torque studies.
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
the 1977 IEEE Power Engineering Society Winter Power Meeting,
January 30-February 4, 1977.
T. J. Hammons (Glasgow University, Glasgow G12 8Q0, U.K.): The
authors are to be commended for their effort
in developing a benchmark model to provide a basis for comparison of computational results
of the various programs now under development.
Peak shaft torques fluctuate cyclically in a complex manner following fault clearance or faulty synchronization,and the highest torques
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 stipulated fault conditions in a given supply network.
In general, torque values are greatest at the generator/turbineinterface, and decrease progressively along theturbineshaft.
Since shaft
diameters tend to decrease in a similar manner, stresses may nevertheless be high. The oscillatory torques in the exciter may become excessively large.
Faulty synchronizing at a displacement angle of approximately
120’ also givesrise to high shaft torques. These torques may or may
not exceed those resulting when a 3-phase fault on the HV busbar is
cleared. Peak torques increase as networkandfaultimpedancesare
made lower, and can become excessively large if subsynchronous rescnance 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 torquesalong the shaft. Proper means
must be employed, however, to avoid excessive axial float-of the generator rotor after the boltsfail.
Manuscript received February 28. 1977
Manuscript received March 1, 1977.
1571
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Peak acceleration of the last LP turbine section following shearing
of the generator/turbine coupling bolts can be excessively large. In one
case, which has been examined, [A] the peak acceleration was found to
be 1600 rad@, the corresponding acceleration under non-shearing conditions 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 structural 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 m g h t be ’induced by subsynchronous
resonance which could lead to displaced resonant frequencies and additional material damping which would effect system response.
An up to date assessment of this phenomena in relation to subsynchronous 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 Synchronization on the Mechanical Stressing of Large TurbineGenerators”, Rev. Gen. Elect., Vol. 86, (749, July - August 1977,
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 system stabilizer or when designing the exciter controls for damping rotor
oscillations. The important role played by the exciter in the self-excitationproblem 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.
As noted in the paper, self-excitation studies require arotor model
withaccurate rotor impedance atthemodal
frequencies. When the
exciter is permitted to play arole, the modelmust a1.w accurately
represent the variation in h d caused by variations in Efd at each torsional frequency. These machine characteristics are not generally available, but they can be obtained by test or calculated by the machine designer. When these characteristics are available, I believe that a simple
three parallel branch model might be satisfactorily fitted for the study
of one modeat a time.
The variation in field current caused by armature current variations need not be precisely modeled so long as thtre is a damper winding. 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
frequencies withlarge exciter motion. If this is not done, theexciter can
play a dominant role in shaft torque buildup.
D. G . Ramey presents some very useful approximations which impart considerable understanding of the SSR phenomena. His interaction
equation gives the added generator mass dampbg required for the interaction to just sustain oscillations. Although the equation is derived for
a machine with a symmetrical
rotor, it is also quite accurate for maTable VI11
Mode 3 Rotor Circuit Per Unit Impedances
at 60hz for Self-Excitation Study
Pole Face Damper Removed
Nonsymmetric Rotor
Symmetric Rotor
XL
Rrd
xrd
xad
Rrq
.135274
xrq
xaq
.13
BO7841
.050268
1.66
.057250
.220280
1.58
Manuscript received April 18,1977.
.13
,0325455
.135274
1.62
.0325455
1.62
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.
T o illustrate how the symmetrical and nonsymmetrical interaction
compare, we have remn the self-excitation case using the modal model
but this timewith rotor circuitparameters based on manufacturer’s
data furnished for the same machine before the pole face damper was
added. Table VI11 shows the parameters for the new rotor circuit and
the parameters forthe symmetricalcomparison case which was o b
tained by averaging the direct and quadrature axis elements. The circuit series capacitor was increased from .287 per unit to .3 1 per unit to
compensate forthe addedcircuitinductance
andthereby 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 S 2 2 4 which corresponds closely to that shown for the
symmetrical rotor. For the nonsymmetrical rotor, the comparable line
resistance is .4429.
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
- 1.21123
- 1.93901
- 1.97398
+ 1.83941
+ 1.13938
+
-
.76016
.23850
.07502
,00655
0
,00699
-
.01603
+
+
+
- .76104
- 1.00821
- 1.45696
- 2.020 12
+ 1.8896 1
+ 1.63349
+ 1.08284
+ .33285
+
+
.lo796
.01997
+
.00312
0
-
.00080
Mr. Ramey’s transient response curves speak for themselves. Unfortunately they were not available in time to be included in Figure 5
along with those submitted by other members of the Task Force.
Mr. T. J . Hammons hasprovided an interesting review of the mechanical 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 subsynchronousresonance,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 excessive turbine overspeed subjecting the longer blades to excessive radial
stress and possible contact with stationaryparts.
3. It would be nearly impossible to design the shear bolts so they
would have sufficient strength to withstand normal full load torque and
atthe
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 discussers for their thoughtful and informative comments.
1572
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