INVESTIGATTON OF CQMPIJTATTON TRCHNTQIJES FOR STWJJJATION OF
LARGE SCALE POWER SYSTEM DYNAMICS
Sreerama Kumar R
Indian
InstituteIndia.
o f Science
Bangalore,
Tnternational Development and
Engineering Associates
Aangalore,India.
- T h i s paper d e s c r i b e s initial
investigatj ons towards arri vi ng a t a most.
suitable algorithm for real time simulation of
power s y s t e m dynamic.;. Rven though t h e
modelling and simulation aspects of real time
are based on ideas that have
simulation
already been laid down for conventional
t-ransieitt .;tab<1 ity anal ysi s , the choi re of
the best algorithm for real time simiilation
require.; dif ferent giii de 1 ine.;.
_
~~~~~~~~ 1
_
1
_
-
Khincha H P
Senior Member
Ramaniljam R
I
_
In transient stability analysis, the
s t r e s s i s o n accuracy. Tn real t-ime
simulation, computation time and niimeri cal
stability are more important. J n thi.; paper,
various algorithms, obtained through different
combinations of the choices of synchronous
machine m o d e l , s o l u t i o n a p p r o a c h . a n g l e
corrector formula, and
convergence criteria
for the network solution, have been compared
with respect to computation speed and soliiti on
accuracy.
Prjnci pa 1 Symbol s :
: general symbol for internal voltages of
machines (phasor)
Efd : exciter output voltage
Pe : generator p o w ~ roiitpiit
mechanj ral p o w ~ r
Em
position of
q-axis with
re.;pect to
network reference f o r phasor solution
[YI : bus admittance matrix
[ V I : bus voltage vector
[I1 : current in jectjon vector
8
: generator terminal voltage angle
w : angular frequency
W o : synchronous angular frequency
M
: inertia constant
h
: step size of numerical integration
E
: general symbol f o r convergence index
E
1 . INTRODUCTION
Transient stability a n a l y s j s and r R a 1
time simulation of power systems both require
the solution of the differentia-algebraic
system of equations of a power system.
In
transient stability analysis, the stres? is on
increaqe in
a c c u r a c y , and a marginal
computational costs can he tolerated, in order
to obtain the desired accuracy. In r e a l time
simulation, on the other h a n d , rompiitation
time and unconditional numerical stability are
important. implicit
integration methods are
used t o c o p e with the s t i f f n e q s of the
associated d i fferential eql~ationsand thric: to
S ~ C * C ? U P the solution c)P r h e p e equations [ I - 7 1 .
In t h i s paper, the performance of varjous
algorithms for the simul ation of power systerr
340
Jenkins L
Member
Tndian Institute of Science
Bangalore, Indi a.
dynamics have been investigated, and comparec
with respert to computation .;peed and soluti or
accuracy. T h e s e a l g o r i t h m s w e r e ohtainec
through di fferent combination.; o f the c h o i c e <
of the following features:
(i)
selection of a synchronoiic: machine model
( i i ) selection of an implicit integratior
method
for
the
soliltion
of
ths
differential equations,
(iiilinterfacing o f t h e .;oliitionc: o f t h f
algebraic and the differential equations,
(iv) better angl P corrprtor form111a , and
(v)
better
convergence criteria for network
solution.
This p a p e r will be iisefiil i n evaluating
the trade o f f s between these various
techniques, and also will be providing a basis
for t h e d e v e l o p m e n t of a most suitable
a1 gori t h m for real time s i mu1 at ion involving
individual g e n e r a t o r dynamics. Such an
a l g o r i t hm f i n d s appli r a t j on j n o p e r a t o r
training simulators.
2 . SYNCHRONOX MACHTNF MODEL
-
Common
representations for qynchronous
machines
which have been
propoc:ed f o r
transient stability simiilation are
model-0: Constant
internal voltage behind
transient reactancp
model -1 : Variahl P internal vol tageq hehi nd
transi ent reactances
m o d e l - 7 : V a r i a b l e internal vol tages behind
subtransient reactances
Model-;! is the m o q t nrciirat= of the t.hreP
models. A l s o , a s thP subtransi~nt saliency i s
much smaller than t h P
trxnci pnt sal i P n c y ,
the number of iterations dup to saliency would
be less
w i t h the m a r h i n r m o d ~ l - 2 , a s
compared with the model-1 . The model-;! i s t.hus
preferred for hot-h transi ent stabi 1 i ty and
real time simulation.
3 . SOLUTTON
MFTHODS
Transient stability simiil ati on
three distinct t a s k s , namely
j nvolves
(i)
netwe.2 s o l i ~ +t r n
(ii) nun - i c a l
iptegration o f djfferentjal
equations,and
(iii) interfacjng
the solutions o f
the
algebraic and the differential equation9
3 . 1 . Network S o l u t i o n
v.
Corrept
the
m a c h i n e rot-or angles f r o m
t h e s o l u t i o n 0.E t h e swing e q u a t j o n s .
The n e t w o r k
soliltion
i s o b t a i n p d by
solving i t e r a t i v e l y the matrix
equation
vi.
C o m p a r e t h e c o m p u t e d r o t o r a n g l ~ sw i t h
t h e i r p a r l i p r v a l u e s . I f t h e y a r e iir,f
c l o s e e n o u g h , i - e t u r n to s t e p ( i i i 1 .
(3.l)
f o r biis v o l t a g e s [V], g i v e n t h e c i i r r e n t v e r t o r
[ I ] a t p a c h t i m e s t P p . The e l e v p n t c , of [TI a r e
non- zero
o n l y f o r g p n e r a t o r n o d e s . The
c o r n p i i t a t i o n o f [TI d e p e n d s on t h e N o r t o n
r e p r e s e n t a t i o n of t h e g e n e r a t o r s .
O p t i m a l o r d e r i n g , Spar.:P m a t r i x 181 a n d
s p a r s e v e c t o r method.:
/ 9 ] h a v e h e m employed
t o q p e e d lip t h e
so7iition of eqiiation ( - 4 . l i ,
and t o a c h i e v e s a v i n g i n s t o r a g e r e q i i i r e m ~ n t s .
Eumeri ca1. m q r a t i o n
3.2.
The g e n e r a l
form of
the
ordinary
d i f f e r e n t i a l e q u a t i o n for a v a r i a b l e ' y ' w i t h
time c o n s t a n t ' T ' i s
p y = ( 1 IT! ( A X
-
y)
(3.21
w h e r e t h e i n p u t v a r i a b l e ' x ' may a l s o b e a
s t a t e v a r j a b l e . Disrreti 7 a t j on o f eqiia t i on
( 3 . 2 ) by a n i m p l i c i t i n t e g r a t i o n method yield.;
a d i f f e r e n c e e q u a t i o n of t h e form
y ( t + h ) = F(t)
+
Rrlt+h)
(-4.3)
vii.
this
study,
algorithms
which
a l g ~ b r a i 7 e t h e d i f f e r e n t i a l e q i i n t i o n c . on t h e
b a s i s o f t h e Backward E u l p r f o r m i l l a and
T r a p ~ 7 o di a l r u l P h q v e hPPn c o n s i d e r e d ,
the
Tnterfacinq
OF
thP A l o e h r a i c
p i f f e r e n t i a 1 ~ g u a T i o nS n l i i t i o n s
ix.
Update t h e
past
s t a t e v a r i a b l el$.
x.
Advance
the
s t ~ g( i i 1 .
4.
f
.
ii. Predict
rotor
anglee: f o r a l l g e n e r a t o r s
u s i n g % h e formiJ7a [ I 1
i h Z/M)
S(t-+h)
=
2S(t-h\-s(t-3R)+
(P,,-Pp(+-h)
)
(l+hD/2M)
the
r i g h t hand s i d e
of
equa+ion
( 3 . 1 ) u s i n g t h e compiited r o t o r a n g l e ? ,
and t h e t e r m i n a l v o l t a g e v e r t o r a t t h e
previoiis i t e r a t i o n .
and
thP
to
go
for
m a c h i n e a n g l e c o r r i a c t i o n . T h r e q i i a t i on.:
r o t o r a c c e l e r a t i o n and s w i n g a r e
= P,-P,
-*
(4.1)
W-Wc
(4.2)
A n g l e c o r r e r t o r fo.rmir1ae
o h t a i n e d thrniiglv a r i o u s t e c h n i q u e s have been i n v e s t i g a t n d sn
a s t o s t i i d y t h f i r e f f e r c t on t h e
niimher o f
i t e r a t i o n s reqi.iired f o r t h r network s o l i ) t i , - ) n .
They a r e :
Appl i c a t i o n o f t h e T r a p e 7 o i d a l
e q u a t i o n r , ( 4 . 1 1 .and ( 4 3 1 l e a d ?
solution
Linear Tnterpolation
of
to
rill F
t o
t h e
Pe
I n t h i s m e t h o d , i t i s assiirnrd t h a t t h P
v a r i a t i o n o f t h e e l e c t r i c a l p o w ~ ro u t p u t o f
e a c h m a c h i n e i s 1 i r t r a r o v p r on^ t i m e s t e p .
T h e n t h e e q u a t i o n s ( 4 . 1 ) a n d ( 4 . 2 1 a r e qo1vi.d
analytically. T h e .;olut;on obtil;i?F-;l h y t h i s
m e t h o d [I01 i s
(3.4)
i i i . Form
step,
timp
of
I n the algorithms developed,
the
s o l u t i o n of t h e s w i n g e g i i a t i o n is i i s e d f o r t h e
4.2.
Form b u s a d m i t t a n c c m a t r i x [ Y ]
terms
ANCT9R CORRKCTOR FORMIJT*A
pk
I n t h e s i m u i t a n e o u s sol i i t i o n method, t h e
algebraized d i f f e r e n t i a l e q u a t i o n s and thP
n e t w o r k e q u a t i o n . : a r e c o m b i n e d to f o r m a
composite model. The a l g o r i t h m f o r thP
S i m u l t a n e o u s I m p l i c i t (SI) s o l u t i o n a p p r o a c h
c o n s i s t s of t h e f o l l o w i n g s t e n ? :
hiqtory
I n t h e algoi.ithm
which
1 1 9 ~ 4 t h e
methnd
Partitioned Tmplirit (PT) s o l u t i o n
[5,6],
linear
p r e d i c t i o n and conseqiient
correction of the internal voltages
arc
incorporated,
i n o r d e r to a v o i d t h e p r o b l e m
of i n t e r f a c e e r r o r s b e t w e e n t h e s o l i i t i o n q c . f
t h e a l g e b r a i c e q u a t i o n s and t h a t o f t h p
d i f f e r e n t i a l e q u a t i o n s . Tn t h i . :
schemti, a f t e r
g e t t i n g t h e network s o l i i t i o n , ( i . P . , a f t e r q t p p
( v i i i ) ) , t h e d i f f e r e n t i a l e q u a t i o n s a r e t o hFi
solved e x p l i r i t l y Ibefore
t h e p a s t hi.:tnry
terms a r e u p d a t e d .
igd
T h e r e a r e b a s i c a l l y t w o a p p r o a c h ~ e :t o
t h e s o l u t i o n of t h e a l g e b r a i c snd d i f f e r e n t i a l
e q u a t i o n s : 44 m:11 t a n e o ? i s . ; u l i i t i o n
method, i n
which t h e machine
Ffpt-pntial e q u a t i o n s and
t h e s y s t e m a l g e b r a : c e q u a t i one: a r e s o 1 v e d
s i m u l t a n e o u s l y , and p a r t i t i oned s o l i l t i n n
method, i n
which t h e s e e q u a t i o n s a r e s o l v e d
alternately.
*r
viii.Check
for
exciter
o u t p u t vol rag"
convergence. T F convergence h a s n o t '@em
o b t a i n e d , go t o s t e p ( i i i )
MpW
3.3.
TT
~
.
where F r e p r e s e n t t h e p a s t h i s t o r y t e r m of t h e
v a r i a b l e y , a n d B is a c o n s t a n t . R o t h F a n d R
a r e d e p e n d e n t on t h e method o f i n t e g r a t i o n .
In
Check
for
saljency
converqPnr*
c o n v e r g e n c e h a s n o t b e e n o b t a i n e d , gc;
step (iii)
6 ( t + h ) = (-h2/6M)Peit+h)
+
m(t)
(4.5)
where
a ( t ) = 6 (t1 + i W ( t ) -W,,)h+ (h2/7M1 ( P m - ( 2 / 3 ) P, C t 3
iv.
Solve equation
(3.1)
for
[V].
34 1
(4.61
Linear Interpolation ef & Tonrther with
the_Use
f _
Newton‘s
Method
_
- _o_
___-
4.3.
With model-2,the generator power output is
(4.7)
which is obtained by linear
interpolation on Pe can be rearranged to get
The equation ( 4 . 5 )
P,
=
(-6M/h2)6 + 9
(4.8)
Where
p = (6M/h2)(6rt)+h(W(t)-W0)) + 3Pm - 2 P e ( t )
(4.9)
From equations ( 4 . 7 ) and !4.8),and u-inn the
Newton’s method, i t can h r shown that the
angle correction i s
( -Eq“ I V
d6 =
and
I ) /Xd” ) Sin ( 6 - 0 ) - (6M/h2)6+8
(Rq” IV I ) /Xd”)C o s ( 6 - 0 )+ ( 6 M / h 2 )
6(t+h) = 6 ( t )
4.4 The Use
of
+
d6
(4.10)
(4.71)
Backward Guler Formula
Tntegration o f the swing equation by the
use of Backward ’uler formula y i e l d s
W(t+h) = W(t,)
+
6(t+h) = 6(t)
+ h(W(t)-Wo)
( h / M ) (Pm-Pe(t+h\I
(4.13)
(4.17)
5 . STMUIaATXON RRSULTS
S e v e r a l c o m b i n a t i o n s of s y n c h r o n o u s
m a c h i n e model.,
sollition m e t h o d , angl e
corrector formula and convergence criteria
have been examined.Thr distjnguishing features
of the corresponding algorithms, the results
of which
a r e reported in thjs paper, a r e
given in Table-I.
The performance evaluation
of
these
algorithms has been done on the W S f C nine h i i s ,
three generator standard teqt qystem [II]. The
distiirbance sjmiilated waq a three p h a c , e faiilt
on bus 7 at time t = 0 , with the fault being
cleared a t time t = 0.087 nprondc, ( 5 c y c l e s
of 60 Hz) hy opening the line 5-7. T e s t s h a v e
been carried n i i t with the thrpe grnrrators
r e p r e s e n t e d b y d i f f e r ~ n tcomhin.=tions of
models. Typi ral simiil atinn rec,iilts h a v e been
presented o f varioiis schemes obtained with the
generator 2 represented
by
model-:!
and
generators 1 and 3 by model-0.
5.1. Comparison
of
Alqorithms
Table IT gives the d e t a i l s of tha number
of iterations
(average taken o v e r 20 steps
following the initiation of the disturbance)
taken by these algorithms whrn h = 16.667mq.(3
cycle of 60 Hz) and when h = ZlOms. Comparing
t h e r e s u l t s of s r h e m e 1 and ~ c h e m e 3 ,
computational advantage o f s i m u 1 taneous
implicit approach over p a r t i t i oned imp1 ici t
approach can he seen.
The results of the stridieq indicate that
with Backward E u l e r formula chosen for angle
c o r r e c t i o n , both t h e a l t e r n a t j v e s f o r
discreti~ation o f the d i fferentj a1 eqilations
provide higher damping . H O W P V P ~ ,rompari ncJ
schemes 1 and 3 (scheme I has Trape7nidal riile
a p p l i e d to a 1 1 d i f f e r e n t i a l ~ q i i a t i o n s
including Yhe swing eqiiation, whpreas q r h e m e 7
uses Backward E u l e r formilla for the same), it,
is seen that the Rackward Eirler formiilir usage
t a k a s m o r e nirmhpr o f i t e r a t i n n s
This
indicates the choice o f T r w p e ~ o j d a l rule for
discretization o f the system differentiql
equations.
Table TT: Comparison of Algnri thms
[e6=0.0001rd, ep=O .001pu., r V = o .Ol p u . 1
Table ‘I: Sali ent P e a t i i r ~ sof the Algorithms
Scheme
Featiirp
I
I.Sol rition Method:
I
I
I
I
I
Trapezoidal rul e
8
3
6
j
4-15
j
In order to choose the angle cori-ector
formula, results o f schemes 1 4 , 5 and 5 are
compared. It is observed that for c m a l l step
siics (of the order of I c y c l e ) schpmes 1 and
5 take less number of iterations a s compared
to s c h e m e s 4 a n d 6. A t h e
step s i z e
increases, scheme 6 shows better results a s
compared to schemes 1 , 4 a n d 5 From Table T T ,
it can be seen that when the step s i z e i s one
cycle, schemes 3 and 5 take t h e same n1Jmhe.r of
terations for network s o l u t i o n However, the
342
i n c r e a s e i n t h e number of i t e r a t i o n s w i t h
s c h e m e 5 , a s t h e c;tPp s j 7 e i n c r ; . a s e c ,
i s less
than t h a t w i t h s c h e m e 1 . Thlls s c h e n e 5 t 1 1 r n s
o u t t o be b e t t e r t h a n s c h e m e 7
whpn t h e %:t%p
size is m o r e t h a n 1 c y c l e .
T t
is s p e n
From Fig.2
f-hii?
r ) 7 ~
i t e r a t i o n s d u e t r i e a r i t a t ; o n syctelr c
a1
be b y p a s s e d
w i t h o u t cht. a c c i i r i i c v k ~ :
a f f e c t e d . Table V showq t h e r e d i i c t ~ o ~i?
n u m b e r o f i t e r a t i o n s t h a t can hp a c h i p v ? . d
making t h e a l g o r i t h m q n o n i t ~ r a t i v P
Tn s c h ~ m e s7 a n d 8 , q P n e r a t o r power o ~ l t g ~ l t s a l i e n c y and e x c i t a t i c,n s p s t c w .
(P,) i s taken a s the= r o n v s r g e n c p c r i t e r i a for
n e t w o r k s o l t i t i o n . When t h e r p q i l l t s o f t h p s e
T a b l e I V : G f f e c : of S n r - i
s c h e m e s a r e coroparerl w i t h t h e r e s i i l t ~o f
the
Sg.;tem ( G e n e r a
r e m a i n i n g srhemen, t h a t a r e h a s c d o n ~ n g l e
c o n v e r g e n c e c r i t e r i a , i t c a n he i n f e r r e d t h a t
wheii t h e s t e p s i z e i s s m a l l
t h P qchcmes hiised
on power c o n v e r g e n c e r - r i t c - r i a a r p l e s s
a t t r a c t i v e . W o w e v e r , whpn s t e p s i 7 e i s
increased
scheme
8
shows
a
definite
coniprit a t io n a 1 a d v a n t a g e .
T h u s i t c a n be c o r i r 1 1 1 d e d t h a t w h i l e
s r h e m p 5 g i v e s h e f t p r r e s x l t q whpn t h e % t e p
s i z e i s s m a l l (of t h e 0 r d r . r o f o n e o r t w o
cycles),
srhemr
8
tiirnsoi~t t o
he
c o m p u t a t i o n a l l y t h e b e s t a t l a r g e step s i r e s .
H o w e v e r , when t h a s t e p s i 7 e ic; l a r g e , ac; the
a p p l j ed s y s t e m d i s t u r b a n c e c a n h e seen b y t h e
p r o g r a m o n l y a f t e r somi= d e l a y , a h i g h e r
o v . e r s h o o t m a y be e x p ~ c t e d i n t h e r e l a t i v e
swing.; of g e n e r a t o r s , a n d t h e s o l i i t i o n a c c t i r a c y
w o u l d h e af f e c t e d , T h u s re1 a I-ivt.1 y s m a l l e r s t e p
size
of t h e o r d e r o f o n e o r t w o c y r l ~ + .i s
p r e f e r r e d f o r t r a n s i e n t s t a b i l i t y s t ~ i d i ~a ns d
h e n r e t h e c h o i c e of scheme 5 f o r siich
s i m u l a i - i o n i s o b v i o u s . On t h e o t h e r - h a n d , as
t h e s c h e m e 8 p r o v i d ~ s b e t t e r qcopt. f o r
i n c r P a 9 i n g t h e s t e p r ; i 7 ~ ,t h i q s c h e m e i s
p r e f ~ r r p d f o r r p a l t i m ~s < r n i i l a t i o n . T h 1 1 s
f l i r t h e r a t t e n t i o n i s confine8 t o o n l y srherneq
5 and 8.
5.7.
Effect
zf
L o 1 1 ~ c e 2 cf_ Network
-I
1
( 3 ) 11.70(3.65) 17
( 6 ) 15 4 0 ( 4 . % 5 )
j
'-l-------l__l-*
(Numbers i n b r a c k e t s i n d i c a t e t h e 9 h q ~ r v a 7 5 o n e
w i t h o u t cxri t a t i o n system)
8
14
T a b l e V: E f f e c t of B y p a s s i n g T t e r a t i o n s 611e t o
B o t h S a l i iancy a n d F x c i t a t - i o n Sysi-ern
1
h = 16.667 mn.
h = 1 1 0 ms.
i
I
(Numbers i n b r e c k s t
sali.em
iterations includec
wjth the
Soliltion
T I- e r a t ion s
Num (3 Iic a I Qn:-gr a t i on
__-______-I
R e c a i l ~ c . t h ~m a r h i n r q d o n o t e x h i h i t
p r o n o u p c e c l s a l iP n c y i n t h P ~ i i h t r a n s c-nt
i
state,
a s a v 5 n T i n r o m p l i t a t i o n * i m p r a n be a r h i e v e d
by
making
t h ~ q a l ie n r y
c o r r e c : t io n
n o n i t r r % i :Vi P
T h @ (1f T i 4 r S
of
bypass7 nrj
i t e r a t i o n s diie t o q a l i e n c y ;'i g i v e n i n T a b l e
T T T . From F i g . 1
i f C ' A ~k , ~ q+@n t h ? t
t-hr
a c c u r a c y 7's n r J t 4 F f e f : P r t w j .
T a h l e T r T : E f f e r t of' R y p n s c , i r l i j T t e r w t i o n r :
d u e + P RaTiednry
t h e l e a s t w i t h sclhmie 8 . t h i s s c h e m e i s r ; j a e 2 - s )
to f i n d t h e e f f e c t of f u r t h e r increase i n tbp
step size o f n u m e r i r a l i n t e g r 3 t - i 011. 'ra
s h o w s t h e d e t a i l s o f iterations o r thi.: b r
with a
s t e p s i z p o f 3h0 ms. I t I $ ohc;+=
t h a t even a t t h i o l l a r n e q t e p R ~ ? B ,
bha
b y p a s s i n g o f t h P i t r r a t i n n s (3uc t o hot-.;?
saliency a n d pxci t a t i n n syat-em doer: n o t s P C - r t
t h e acriirary of
r e s i 1 1t s
RpprPci ; I b ? y , F , g '1
s h o w s t h p n w i n g c i i ~ v c s of thjTi r i l s i p , w i t h t ' f
s s l i e n c y a n d t h p p : y r i 6 a t i o n s y q t e r n -i t + v a t i n r i c :
disabled.
T a b l P VT:
Srherne 8 nl-, h =. 1 6 0 m s
I' n o t Mnde'1,
\Numbers i n brackets i n d i c a t p t h e o h q e r v a t i n n s
p~btafn;.rl
w i t h
thp s q l i ~ n c yiteration
included)
115
o r d ~ yt *
qturly
th?
~ f f + - r to f
t h e
i n c l u s i o n o f t h e excitdt-ion qystern on t h e
number of i t e r a t i o n s f o r n e t w o r k q n l l ~ t i r t n ,
T R E E t y D P - 1 m o d e l is u s e d [ 1 2 1 , w i t h t h p
exciter s a t ~ i r a t , oi n a p p r o x i m a t e d by a two-ql ope
p i e c e - w i s e l i n e a r i 7 t . d c u r v e [ I ? . From T a h l p
J V , i t c a n be observed t h a t t h e nilmher o f
i t e r a t j o n s for n e e w o r k s 0 3 1 1 t i o n i n r r + n . ; P c , W J t b
t h e incliision of e x c i t a t i o n system.
N u m e r i c a l i n s 1 a b i l j t y i q o b s e r v e d b W-i n
t h e s t a ~ s) j z p i s i n ( - r a a s e d t o 1 7 0 ilici.
T1b e p o s s i b l e t o a l l ~ v i a t et h i s p r o b l a r p
n u m e r i c a l " b l o w u p " b y i n $ - r o d u c i n g a sr"
damping on machtne r a t 4 n g in the s
e q u a t i o n . T h i s d e m s i t d s f i i r t h ~ ri n v e q t i g 8 t
343
lG0
6 . CONCLUSTONS
h = 16.67 rns
Exciter not modelled
a-wlthout saliency Itcration
Time in seconds
- Rclative swing between Generator 2
and Generator 1
a ) Scheme 5
I
160(
Exclter
not
modelled
This paper reports t h e r P s i i J t C i o f
investigations towards arriving a t a most
suitable algorithm for real t i m e qimulation of
power system dynamics. The selection o f a
synchronous machine model , the - 0 1 i r t i on
approach, and a convergence criteria for the
network s o l \,tion have heen deal t w f th. E f f e c t
o f various angle corrector €ormuIae o n
computation time has
also b e e n examined.
Further work
is
needed to
maki. the
algorithms fully noniterative and to increase
the step size of numerical integration, while
keeping the solution numerical 1 y qtabl P .
80
REFERENCES
o-wlth sallency iteratlon
mwithout sa 1 iency lteratlor,/
Ob
io
210
610
Id0
8'0
Tlme I n secords
b + Scheme 8
Relative swlng between Generator 2
and Generator 1
Figure 5: Sensitlvlty o f saliency iteration.
-
160
-
T i m e in seconds
a) Scheme 5 Relative swing between Generator 2
and Generator 1
h.= 110 ms
Generator 2 with exciter
120
with saliwry ard excik ltwation
CwItkut sdliency 8 exciter iteratict
0.
,
0
'I
[2]
R B I Johnson, M J Short, R J Cory,
"Improved Simulation Tachni quec for
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[3] G Zross, A R Bergen, " A Class o f New
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computation
of
Power
System
Dynamic
Response", TEEF Trans. on
PAS, Vol .PAS-96,
No.1, pp.293-301, Jan./Feb. 1977.
h = 16.67 ms
Generator 2 with exciter
160
[ l J H W Dommel, N Sato,
F a s t Transient
Stability Solutions",TEFF T r a n s . o n P A S ,
Vol.PAS-91, pp. 1643-1650, J~lly/Ail~.
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[4]
B Stott, "Power System Dynamic ReSpon-3e
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[5] K Saikawa, M G o t o , Y Tmamura, M T a k a t o , T
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[ti]
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[ 7 ] Sreerama Kumar R , Ramanujam R , Khincha H
Jenkins L ,
"
Synchronoiis
Mac
Interfacing Techniques for Transient Stab
and Real Time Sinulation",Report No.EE/78/Nov.
1989, Department of Electrical Engj neering,
Indian Institute o f Science, Rangalore, Tndia.
P,
-
h
I t 0 rns
Exclter m t modelled
60
30
0
i
t
6.0
8-0
seconds
a1 With saliency tteration avoided
2.0
L.0
100
T h e In
Generator 2 with exclter
181 W F Tinney, J W Walker, "Direct Solutions
of Sparse
Network
Equations
by Optimally
Ordered Triangular Factorization", Proc. IEEE,
V01.55, pp. 1801-1809, Nov.1967.
[9] W F Tinney, V Brandwajn, M S Chan. "Sparse
Vector Methods", IEEE Trans. O R P A S , V o l
PAS-104, pp. 295-301, Feb. 1985.
.
[lo] H W Dommel," Exact Solution of Simplified
Machine with Damping) by Linear Tnterpolatioii
of P". , Personal Communication.
[111 P M Anderson , A A Fouad, "Power system
Control and Stability ".T h e Towa S t a t e
University eress,Ames,Towa,7977.
J
1
L.0
6-0
8.0
10.0
Time in seconds
b ) !Jlti- saliency aid exciter Iterations
0
2.0
avotcI?d
I;igure 3:Scheme 8
- Relative
swing betweerl
C c c w s t o r 2 and Generator 1
[I21
T E R G Cornmittpa Report , " C o m p u t ~ r
Representation of Excitation systems", TFFF
Trans.or! P A S , Vol .PAS- 87,p p -1460-1 464,Jiina 1968