synchronous machine modeling by parameter estimation
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Louisiana State University Louisiana State University
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LSU Historical Dissertations and Theses Graduate School
1976
Synchronous Machine Modeling by Parameter Estimation. Synchronous Machine Modeling by Parameter Estimation.
Chi Choon Lee Louisiana State University and Agricultural & Mechanical College
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Recommended Citation Recommended Citation Lee, Chi Choon, "Synchronous Machine Modeling by Parameter Estimation." (1976). LSU Historical Dissertations and Theses. 2972. https://digitalcommons.lsu.edu/gradschool_disstheses/2972
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X erox University Microfilms300 North Zeeb RoadA nn A rbor, M ichigan 48106
7 6 - 2 8 , 8 1 3
LEE, Chi Choon, 1941-SYNCHRONOUS MACHINE MODELING BY PARAMETER ESTIMATION.The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1976Engineering, electronics and electrical
Xerox University Microfilms, Ann Arbor, Michigan 48106
SYNCHRONOUS MACHINE MODELING
BY PARAMETER ESTIMATION
A Dissertation
Submitted to th e Graduate F acu lty o f th e Louisiana S ta te U n iv e rs ity and
A g ric u ltu ra l and Mechanical College in p a r t i a l fu lf i l lm e n t o f th e requirem ents fo r th e degree o f
Doctor o f Philosophy
in
The Department o f E le c tr ic a l Engineering
byChi Choon Lee
B .S ., Seoul N ational U n iv e rs ity , 1967 M .S., Louisiana S ta te U n iv e rs ity , 1970
August 1976
ACKNOWLEDGMENT
The author i s g ra te fu l fo r the f in a n c ia l support provided during
the course o f th is work by th e E le c tr ic a l Engineering Department in
the form of teach ing and research a s s is ta n tsh ip . The support of the
Air Force Propulsion Laboratory, Wright P atterson , OH, through
Georgia I n s t i tu te of Technology is a lso g rea tly acknowledged.
I t has been the a u th o r 's p leasure to study under the d ire c tio n of
Dr. Owen T. Tan. In add ition to h is v a s t knowledge and deep understand
ing in the a rea o f th is study , the energy, ded ication to purpose, and
frien d sh ip o f Professor Tan have been an invaluable source o f encour
agement and stim ulus to p rogress.
The author would lik e to express h is deep app recia tion to
Dr. Johnny R. Johnson, co-major p ro fesso r, Dr. Leonard C. Adams,
Dr. David E. Johnson, Dr. Paul M. Ju lic h of the E le c tr ic a l Engineering
Department and Dr. James R. Dorroh of th e Department of Mathematics
fo r serving as members o f the examining committee.
F in a lly , the au th o r 's sincere apprecia tion goes to h is w ife,
Grace, fo r her constant support and encouragement during a sometimes
d i f f i c u l t and try in g period .
TABLE OF CONTENTS
Page
ACKNOWLEDGMENT................................................................................................... i i
LIST OF TABLES.......................................................................................... v
LIST OF FIGURES..................................................... v i
ABSTRACT.................................................................................................................. v i i i
CHAPTER
1. INTRODUCTION ............................................................................................. 1
2. STATE SPACE MODEL OF A SYNCHRONOUS MACHINE................................. 6
2 .1 . S ta te Equation o f a S a lien t-P o le Machine.......................... 7
2 .2 . S ta te Equation of a Round-Rotor M ach ine .......................... 16
2 .3 . System Equations w ith Balanced Three-Phase Load . . . 18
2 .3 .1 . Generator w ith R esis tiv e Load .............................. 18
2 .3 .2 . Generator w ith RLC Load ........................................... 20
2 .3 .3 . Generator with RC L o a d ........................................... 22
2 .3 .4 . Generator w ith RL Load ........................................... 23
3. OUTPUT SENSITIVITY ANALYSIS OF A SYNCHRONOUS GENERATOR . . 24
3 .1 . Mathematical Formulation ..................................................... 24
3 .2 . A pplication to Generator-Load System ............................ 26
3 .3 . Numerical R e s u l t s ................................................................ 30
3 .4 . P ossib le Output E rro r Due to Parameter Inaccuracy . . 40
4. SYSTEM IDENTIFICATION............................................................................ 42
4 .1 . Problem F o rm u la tio n ............................................................ 42
4 .2 . W eighted-Least-Squares E stim ator ..................................... 45
4 .3 . Maximum Likelihood E s t i m a t o r ....................................... 49
CHAPTER Page
4 .3 .1 . Maximum Likelihood E stim ator:
Without Input Noise .................................................. 52
4 .3 .2 . Maximum Likelihood E stim ator:
With Input N o i s e ................................................ 53
5. SYNCHRONOUS MACHINE PARAMETER ESTIMATION........................... 56
5 .1 . Model a t Sudden Three-Phase S h o r t - C i r c u i t .............. 56
5.2 . Sim ulation R e s u l t s ........................................................... 60
5 .2 .1 . Estim ation by W eighted-Least-Squares Method . 60
5 .2 .2 . Estim ation by Maximum Likelihood Method . . . 81
5 .3 . E stim ation with Step-Function F ield-V oltage Input . . 91
5 .4 . E ffe c t of R esis tiv e Load on E s t i m a t i o n .................. 98
6. CONCLUSIONS............................................................................................101
REFERENCES........................................................................................................ 104
APPENDIX 1 ........................................................................................................ I l l
APPENDIX 2 ........................................................................................................ 117
APPENDIX 3 .........................................................................................................125
V I T A ................................................................................................................ 137
iv
T ab les
3 .1 .
3 .2 .
3 .3 .
3 .4 .
5.2.1.
5 .2 .2 .
5 .2 .3 .
5 .2 .4 .
5 .2 .5 .
5 .3 .1 .
5 .4 .1 .
LIST OF TABLES
Page
Synchronous Machine Data in P e r - U n i t ........................................... 31
Load Param eter Data in P er-U n it fo r V arious
Load P o w er-F ac to rs ................................................................................ 31
Peak Value o f S e n s itiv ity Functions fo r V arious
Load P o w er-F ac to rs ................................................................................. 33a
P ossib le Maximum V aria tion o f i . and P . . . . ....................... 41abcWLSE Without N o i s e ................................................................................ 65
WLSE With Output N o is e ......................................................................... 72
WLSE With Input and Output N o i s e s ............................................... 79
MLE Without Inpu t N o i s e .................................................................... 83
MLE With Input N o is e ............................................................................. 89
WLSE With Step-Function F i e ld - V o l t a g e ....................................... 94
V ariab le R e s is tiv e Load ....................................................................... 100
v
LIST OF FIGURES
Figure Page
2 .1 .1 . R epresentation o f a S a lien t-P o le Synchronous Machine
by Equivalent C ircu its ......................................................................... 8
2 .2 .1 . Representation o f a Round-Rotor Synchronous Machine
by Equivalent C irc u its ......................................................................... 16
2 .3 .1 . A Balanced Three-Phase RLC L o a d .......................................... 20
2 .3 .2 . A Balanced Three-Phase RC Load .......................................... 22
3 .3 .1 . Output S e n s it iv i ty Function w .r . t . X j a t Jtpf = 0 .1 leading 36
3 .3 .2 . Output S e n s it iv i ty Function w .r . t . X" a t Apf = 0 .1 Leading 37
3 .3 .3 . Output S e n s it iv i ty Function w .r . t . XJj a t Jlpf = 0.2 Lagging 38
3 .3 .4 . Output S e n s it iv i ty Function w .r . t . X" a t Jlpf « o .2 Lagging 39
4 .1 .1 . System Id e n tif ic a tio n by Parameter Estim ation .................... 44
4 .2 .1 . W eighted-Least-Squares Parameter Estim ation Algorithm . . 49
5 .2 .1 . Block Diagram fo r Machine Parameter Estim ation .................... 63
5 .2 .2 . Oscillogram o f D eterm in istic Output Currents a t
Sudden Three-Phase S ho rt-C ircu it .................................................. 64
5 .2 .3 . Ratio o f Estim ate to True Value fo r X” w ithout Noise (WLSE) 69
5 .2 .4 . Ratio o f Estim ate to True Value fo r X^ w ithout no ise (WLSE) 69
5 .2 .5 . Ratio o f Estim ate to True Value fo r
Random I n i t i a l Estim ates without Noise (WLSE)................. 70
5 .2 .6 . Oscillogram o f Noise-Corrupted Output Currents a t
Sudden Three-Phase S h o rt-C ircu it .................................................. 71
5 .2 .7 . Ratio o f Estim ate to Time Value fo r X\' withdOutput Noise (WLSE) ........................................................................... 76
LIST OF FIGURES (CONTINUED)
Figure Page
5 .2 .8 . Ratio o f Estim ate to True Value fo r X,. withail
Output Noise (WLSE)............................................................................... 76
5 .2 .9 . Ratio o f Estim ate to True Value fo r
Random I n i t i a l Estim ates w ith Output Noise (WLSE)................. 77
5 2 10. Ratio of Estim ate to True Value fo r
Random I n i t i a l Estim ates w ith Input and Output Noises (WLSE) 80
5.2.11. Ratio of Estim ate to True Value fo r X',' w ithaOutput Noise (MLE) ............................................................................... 87
5.2.12. Ratio o f Estim ate to True Value fo r X ^ w ith
Output Noise (MLE) ............................................................................... 87
5.2.13. Ratio of Estim ate to True Value fo r
Random I n i t i a l Estim ates w ith Output Noise (MLE) ................. 88
5.2.14. Ratio of Estim ate to True Value fo r
Random I n i t i a l Estim ates w ith Input and Output Noises (MLE) 90
5 .3 .1 . Oscillogram o f D eterm inistic Output C urrents with
Step-Function F ield-V oltage .......................................................... 93
5 .3 .2 . Ratio o f Estim ate to True Value fo r X with
Step-Function F ield-V oltage (WLSE) .............................................. 97
5 .3 .3 . Ratio of Estim ate to True Value fo r X,, w ithaftStep-Function Field-V oltage (WLSE) ............................. . . . . 97
v i i
ABSTRACT
Synchronous machine modeling i s considered in d e ta i l from th e
s tru c tu ra l and data id e n tif ic a t io n viewpoint w ith p a r t ic u la r emphasis
on s ta te -sp a ce model s tru c tu re s and param eter e stim ato rs . A s ta te -sp ace
model fo r a sa l ie n t-p o le synchronous machine w ith damper winding as
w ell as a so lid round ro to r synchronous machine i s derived from the
d ir e c t and quadra tu re-ax is equ ivalent c i r c u i t model and expressed in
terms o f IEEE standard machine param eters. By perform ing a s e n s i t iv i ty
an a ly sis on a genera to r-load system, th e e ffe c ts o f param eter v a r ia
t io n and load power fa c to r on the output v a ria b le s a re in v es tig a te d ,
from which th e need fo r a method to determ ine th e machine param eters
accu ra te ly i s shown. The w eigh ted -least-squares and maximum lik e lih o o d
estim ato rs a re discussed along with th e d e riv a tio n o f th e i r computaional
algorithm s. These estim ato rs a re then app lied to a sim ulated s a l ie n t -
po le synchronous machine to accu ra te ly determ ine the standard machine
param eters fo r a f i f th ro rd e r model. D ig ita l computation shows the
performance o f th e estim ato rs in terms o f r a t io o f estim ate to tru e
value as a function o f the number o f i te r a t io n s . The e ffe c ts o f in p u t,
load , and system noises on th e estim ation a re a lso in v es tig a te d . The
r e s u l ts of t h i s study show th a t the w eigh ted -least-squares and maximum
lik e lih o o d estim ato rs a re accu ra te , e f fe c tiv e and promising to determine
th e machine d a ta accu ra te ly .
v i i i
CHAPTER 1
INTRODUCTION
Ever since the synchronous machine has been introduced as an
energy converter, i t has been m odified, improved and developed in
various s iz e s to become one o f the most im portant and complicated com
ponents in power systems. At th e same tim e, models o f the machine have
been developed, m odified, and used to rep resen t the machine in a closed
form analysis o r computer sim ulation fo r the follow ing power system
s tu d ies .
Cl) T ransien t s t a b i l i ty study [1 -7]: This generally involves
the time domain sim ulation o f a number o f generators and th e i r co n tro ls
to determine the systems response to a major d isturbance where imme
d ia te lo ss o f synchronism i s o f g rea t concern.
(2) S tead y -s ta te s t a b i l i ty an a ly sis [8-12]: The s t a b i l i ty of
the system a t s te a d y -s ta te without a major d isturbance o r upset i s re
fe rred to as s te a d y -s ta te s t a b i l i ty . Therefore, the natu re o f the
s te a d y -s ta te s t a b i l i ty i s sm all-signal dynamic s t a b i l i ty and i f unstab le ,
the in s ta b i l i ty i s u su a lly o sc illa to ry about the equilib rium p o in t.
(3) E le c tr ic a l t ra n s ie n t ana ly sis [13-17]: The o v e ra ll e l e c t r i
ca l t ra n s ie n t behavior is made o f a su b tran sien t and tra n s ie n t p a r t .
Accurate determ ination o f the su b tran sien t and tra n s ie n t behavior fo r
d if fe re n t types o f loading condition i s p a r t ic u la r ly im portant in cases
where so lid s ta te devices a re used.
1
2
(4) Voltage and frequency-drop c h a ra c te r is t ic s [18-19]: In the
ap p lica tio n o f synchronous genera to rs, p a r t ic u la r ly in the sm aller
s izes l ik e in an iso la te d in d u s tr ia l p la n t , the determ ination o f the
la rg e s t load th a t can be applied involves generato r te rm ina l-vo ltage
d ips, frequency drops, and recovery times to the normal s ta te as these
fac to rs may a f fe c t the performance o f o th e r equipment.
Besides the aforementioned s tu d ies th e re a re many o ther cases
where a synchronous machine model i s requ ired , fo r in s tan ce , e x c ita tio n
system fo rcing e f fe c t on s t a b i l i t y [20-22], optimal load-frequency con
t r o l [23-27], optim al t ra n s ie n t con tro l [28,29], s t a b i l i t y improvement
[30-34], system s ta te estim ation [35-39], system id e n t i f ic a t io n [40],
e tc .
Thus, many im portant s tu d ies o f power systems a re based on models
rep resen ting the synchronous machine in various s i tu a t io n s . This im
p lie s th a t the v a lid i ty o f the conclusions made in such in v es tig a tio n s
depends on th a t o f the model used. In th is re sp e c t, the s ig n ifican ce
o f a c o rre c t model can hard ly be over-emphasized.
For a successfu l modeling [41], i t i s genera lly necessary to
s e le c t a proper model s tru c tu re and to determine a complete and co rrec t
s e t o f da ta w ell f i t t e d to th e s tru c tu re . The s tru c tu re o f a model
could be in the form o f a d i f f e r e n t ia l equation , equ iva len t c ir c u i t or
s ta te model and the degree o f complexity may vary depending on the pur
pose o f study and the requ ired accuracy. The d a ta needed would then be
the s e t o f c o e ff ic ie n ts o f the d if f e r e n t ia l equation , the equivalen t
c i r c u i t param eters, or th e c o e f f ic ie n t m atrices o f th e s ta te equation.
In some cases, the param eters d ire c tly or in d ire c t ly re la te d to theV. I
<
the m atrix elements o f a s ta te equation could be chosen as a s e t o f
proper data [42].
For synchronous machine modeling, th e model s tru c tu re s are almost
invariab ly based on the d ,q ,o -re fe ren ce frame obtained by P a rk 's tra n s
formation from the a ,b ,c -re fe re n c e frame [43-49] and generally expressed
in terms o f p e r-u n it q u a n tit ie s [50-52]. As the s ig n ifican ce o f com
p u te r sim ulation and the demand fo r more accura te engineering p red ic
tio n s increase rap id ly , a g rea t amount o f a tte n tio n has been given to
synchronous machine model s tru c tu re s . As a consequence, many equivalent
c ir c u its as w ell as s ta te -sp ace models were introduced. E specia lly , the
Jackson-W inchester model [53] provides a considerably d e ta ile d d ire c t
and quadrature-ax is equivalent c i r c u i t rep re sen ta tio n fo r s o lid - ro to r
tu rb ine genera to rs. However, th e complexity o f th is model i s judged
too great to allow i t s p ra c tic a l use in multimachine sim ulation s tud ies
o f large power system s.
To a l le v ia te the complexity o f the Jackson-W inchester model and
to make i t p r a c t ic a l ly usable, th i s model has been s im p lified [54] to
one having seven ro to r equivalen t c i r c u i t s . The s im p lified model has
been used in studying the e ffe c ts o f synchronous machine modeling on
la rg e -sca le system s t a b i l i ty [6] where the dynamic behaviors o f several
d if fe re n t models were compared w ith each o th e r. The r e s u l ts of th is
study show th a t the accuracy o f a model i s not only dependent on i t s
s tru c tu re but a lso on the choice and accuracy o f the d a ta s e t .
P a ra lle l with the development o f equ ivalen t c i r c u i t models, a
s ta te -sp ace model expressed in terms o f hybrid-param eters o f a synchro
nous machine was introduced [55]. This p a r t ic u la r model i s obtained by
expanding the c la s s ic a l d ,q -ax is model o f a synchronous machine and
ch arac te riz in g i t by p e rtin e n t o p en -c ircu it and s h o r t - c i r c u i t hybrid-
par^m eters. Another s ta te -sp a ce model, whose elements o f the m atrices(S’
are expressed in terms o f standard machine param eters, was presented
[56]. This model i s o f advantage since power engineers are most fami
l i a r w ith the natu re o f the data used.
As noted, the s tru c tu re s o f synchronous machine models have been
developed q u ite s a t i s f a c to r i ly , however, the method to accu ra te ly d e te r
mine the data fo r a given model s tru c tu re has not been s tud ied exten
s iv e ly . In the l i t e r a tu r e , th e IEEE standard t e s t code [59] p resen ts a
se rie s o f e s tab lish ed t e s t procedures to determine the standard machine
param eters. In a recen t study [60] these t e s t procedures have been
thoroughly examined and i t was po in ted out th a t the accuracy o f the
data is questionab le . Moreover, i t was a lso n o ticed th a t the data re
quired to construct some of the models are frequen tly not re a d ily a v a il
able from the design data nor are they obtained by standard t e s t s [48,
61].
Recently, using a frequency-response technique a new t e s t method
[62-64] has been developed to determine the machine parem ters accu ra te ly .
However, as is w ell known in co n tro l systems theo ry , th is method requ ires
judgment in loca ting the break p o in ts . Furthermore, i t s p ra c t ic a l imple
mentation needs a v a ria b le frequency source capable o f supplying r e la
tiv e ly la rge c u rre n ts .
Although some o ther methods are av a ilab le to determine machine
param eters, the need fo r a method to obtain more accurate da ta fo r a
given model s tru c tu re i s generally recognized [48,61]. This m otivates
an exp lo ra tion in th e area of system id e n tif ic a t io n [65-70] which leads
to the suggestion to employ s t a t i s t i c a l param eter estim ation theory in
synchronous machine modeling. In th is th e s is , th e re fo re , a method i s£
proposed to determine accurate da ta fo r a synchronous machine model by
form ulating the problem in the frame o f con tro l systems theory and
applying e x is tin g id e n tif ic a tio n techniques.
CHAPTER 2
STATE SPACE MODEL OF A SYNCHRONOUS MACHINE
A synchronous machine [18] c o n s is ts e le c t r i c a l ly o f two e s s e n t ia l
components; one i s a f i e ld winding to produce a magnetic f ie ld and th e
o th e r i s an arm ature winding to generate o r rece iv e e le c t r i c energy.
The f ie ld winding i s norm ally wound in the ro to r and produces a magnetic
f i e ld by e x c itin g i t w ith d ire c t c u rre n t. The arm ature winding i s then
wound on th e in s id e o f a s t a to r which supports and provides a magnetic
f lu x path fo r th e arm ature winding. For a th ree-phase synchronous
machine, th e arm ature winding i s composed o f th ree groups o f c o i l s —
phase windings—which are disposed around th e s t a to r a t angle in te rv a ls
o f 120 e le c t r i c a l degrees such th a t w ith a uniform speed o f the ro to r a
s e t o f vo ltages d isp laced 120 degrees in phase w il l be induced in the
arm ature phase w indings.
In g en era l, synchronous machines are co nstruc ted w ith s a l ie n t
poles o f lam inated po le faces except tu rb in e* g en era to rs . The s a l ie n t -
pole machines u su a lly have a damper o r am ortisseu r winding c o n s is tin g
o f a s e t o f copper o r b rass bars embedded in pole face s lo ts and sh o rt-
c irc u ite d a t the machine ends. The am ortisseur perm its [75] s e l f -
s t a r t in g o f th e machine, gives a damping e f f e c t on ro to r o s c i l la t io n s
in dynamic t r a n s ie n ts , reduces over-vo ltages under c e r ta in sh o rt-
c i r c u i t co n d itio n s , and a ids in synchron iza tion p rocesses . For tu rb in e
g en era to rs , th e r o to r i s made o f s o l id s te e l which a lso functions as a
damper winding.
In t h i s chap ter, a s ta te -sp a ce model o f a synchronous machine
w ill be derived by using the d ire c t and quadrature (d,q) axis theory .
The e ffe c ts o f damper winding w ill be included. This model is then
param eterized in terms o f IEEE standard param eters whose values w ill be
determined by estim ators in the l a t e r chapters.
2 .1 . S ta te Equation o f a S a lien t-P o le Machine
A sa lie n t-p o le synchronous machine can be rep resen ted by a s e t
o f m agnetically coupled equivalent c ir c u i ts as shewn in Figure 2 .1 .1 .
The th ree armature phase windings are depicted by the c ir c u i ts on the
a ,b ,c axes, and the c i r c u i ts fo r i ^ and i^ rep resen t resp ec tiv e ly the
d- and q -ax is equ ivalent c irc u its o f the damper winding. The f ie ld
winding producing a magnetic f ie ld in the d ire c t axis i s rep resen ted by
the c i r c u i t w ith cu rren t i^ . The ro to r i s ro ta tin g w ith an a rb i tra ry
speed co while the instan taneous angle between the axes o f ro to r and
phase winding a i s denoted by 9,
8
a -ax is
d -ax is6
q -ax is
b -ax is
Figure 2 .1 .1 R epresentation o f a S a lien t-P o le Synchronous Machine by Equivalent C ircu its
Assuming the re s is ta n c e r o f each armature phase winding to beSt
the same, the follow ing s e t o f voltage equations i s obtained fo r the
c ir c u i ts shown in Figure 2 .1 .1 :
r i + X a a a (2.1.1)*
Vb + \ (2 .1 .2)*
r i + X a c c (2 .1 .3)»
r f i f + Xf (2 .1 .4 )
r kd1kd + *kd (2 .1 .5 )
0 = r, i . +kq kq (2 . 1 . 6)
wheTe i = a ,b ,c , f ,k d ,k q axe th e f lu x linkages o f th e corresponding
c i r c u i t s . The flu x lin k ag es and c u rre n ts a re r e la te d by
X = Li (2 .1 .7 )
whereX S *e »W V
1 = [ i a \ d V T
L — i , j = a ,b ,c ,f ,k d ,k q
w ith b e in g th e s e l f o r mutual inductance o f c i r c u i t s i and j .
S u b s t itu t in g (2 .1 .7 ) in (2 .1 .1 ) through (2 .1 .6 ) y ie ld s a s e t o f
f i r s t o rd e r d i f f e r e n t i a l equations r e la t in g e i th e r v o lta g es and c u rre n ts
o r vo ltag es and flu x lin k a g e s . In e i th e r case , th e c o e f f ic ie n ts o f th e
r e s u l t in g d i f f e r e n t i a l equations a re tim e-vary ing s in c e th e inductance
m atrix L i s dependent on th e ro to r p o s i t io n . The tim e-vary ing d i f f e r e n
t i a l eq u a tio n s , however, can be converted in to t im e -in v a r ia n t equations
by app ly ing P a rk 's tran sfo rm a tio n [43,44] which maps th e a ,b ,c v a r ia b le s
in to d ,q ,o v a r ia b le s :
dqo abc
where £ may rep re se n t a v o lta g e , c u rre n t o r f lu x linkage and P i s given
by
c o s (0) co s(0-120°) c o s (6-240°)
- s in (0 ) -sin(0-12O °) -sin(0-24O °)
/ r-/?i
(2 . 1 . 8)
10
I t can a lso be re a d ily shown th a t P i s o rthogonal, i . e . ,
E = P^£ abc dqo
Applying tran sfo rm ation (2 .1 .8 ) to th e a ,b ,c -v a r ia b le s o f (2 .1 .1 )
through (2 .1 .3 ) and (2 .1 .7 ) y ie ld s
H erein ,
XD “ V d + WXQ + VD
XQ ~ RQXQ " W XD + VQ
XD = LDDl D
\ ~ LQQi Q
A = -L i o o o
(2.1.9)
(2 . 1 . 10)
(2 . 1 . 11)
(2. 1 . 12)
(2.1.13)
(2.1.14)
whereas
X£]T
1D = xkd i f f T
iia> 0 vf ]T
Art =
In =
v„ =
VT[i.[Tq o 1—
1
” r 0 0 “ I r 0 « 0a a
0 - r k d °
11pP W = 0 0
0 0 - r f 0 - r .kq 0 0
' Ldd Ldkd Ld f r-L l . “ ]qq qkq
ldd = -Lkdd Lkdkd Lkdf
ii
j*
_"Lfd Lfkd Lf f _ _ ^kqq Lkqkq
XI
I t i s noted th a t ^ , i , j = d ,q ,kd ,kq , and the zero-sequence
inductance Lq are a l l tim e -in v a rian t q u a n ti t ie s . Combining (2 .1 .9)
through (2.1.11) w ith (2 .1 .12) through (2 .1 .14) y ie ld s
— ““A„D
A„ _Q
*Ao
_ —
RDLDD
-W
0
W
rqlqq
0
0
-r /L a o
(2.1.15)
which i s a s ta te equation o f a synchronous machine in terms o f the
c ir c u it parameters L ,^ , i , j = d ,q ,o ,k d ,k q .
A fter sim plify ing the expressions fo r the elements o f the ma
t r ic e s and L~* by using the p e r-u n it system [50,51,52] such th a t
Ldkdc' !U) = W * " 13 = Lf d (p,°
Lq CP<0 = Lkq&u3
and describ ing the p e r-u n it c i r c u i t param eters h i , j = d ,q ,o ,f ,k d ,
kq, in terms of coupling fa c to rs k^.., i . e . , y h T L 7 , equation
(2.1.15) becomes
'D
■W
0
W
CQ °
0 -1/T
D
(2.1 .16)
where the subm atrices CD and are given by
In the equations, = Li / r x ’ * = d ,q ,o ,k d ,k q ,f , a re the time constan ts
o f the ind iv idual c i r c u i t s , and the c-param eters are functions o f only
the coupling fac to rs k „ , i j = d ,q ,k d ,k q ,f , as l i s te d below [76].
c^ = (1 - k ^ ) / A _ ^ d f ^ k d d ^ d f " kfd3d£ = kk d d A
cf " (1 " kkdd)/A2 kkddCkkddkkdf ~ fdj
ckd = C1 ' kf d ^ A fd kkdf AC2.1.19)
_ kkdf^kfdkkdf _ kkdd^ _ kfd^kfdkkdd “ kkdPcdkd ” k - , A ckdf = k, . . Aid kdd
kfd Ckfdkkd£ " kkdd) _ kkdd^kfd kkdd ” kkdf')
kdd Cfkd ' kfd d
with
A ~ kkdf + kkdd + kfd * 2kkdfkkddkfd 1 ;
The u ltim ate o b jec tiv e i s to express the model equation (2 .1 .16 )
in terms o f standard machine param eters as defined in the IEEE code,
i . e . ,
= d -ax is su b tra n s ie n t reac tan ce ,
= d -ax is t r a n s ie n t reac tan ce ,
X - d -ax is synchronous reac tance ,
X" = q -ax is su b tra n s ie n t rea c ta n ce ,q .
X =* q -ax is synchronous reac tan ce ,
T'^o = d -ax is o p e n -c irc u it su b tra n s ie n t time co n stan t,
T^o = d -ax is o p e n -c irc u it t r a n s ie n t time c o n s ta n t,
■Pjo = q -ax is o p e n -c irc u it su b tra n s ie n t time c o n s tan t,
r = arm ature re s is ta n c e , a *= d -ax is arm ature leakage reac tan ce ,
r^ = f i e ld re s is ta n c e .
I t is noted th a t in th e s e t o f standard param eters an a d d itio n a l arma
tu re leakage reactance X ^ has been added to the conventional s e t o f
param eters, s in ce the conventional s e t alone does no t completely de
sc rib e the machine behavior. However, i f th e leakage inductance o f
damper winding in the d -ax is i s n e g lig ib le w ith re sp e c t to i t s s e l f
inductance, i t can be re a d ily shown th a t ■ xd-
Using th e re la tio n sh ip s
14
in (2 .1 .17) and (2.1.18) re s u l ts in
and
where
Cd/Td Cdkd/T d V d
CD = Ckdd/T d<, Ckd/Tdo CL i ' Tko (2 .1 .20)
Cfd /Tdo Cfkd/Tdo Cf /Tdo
Vq q
C , / Tqkq q
C . /Tn C /T" qkq' qo q qo
^ d d " C1 " kkdf5ckdd
Cl
Cl
'kd k^ )c kdfJ kd
'kdf kkdf?ckd£
(2 . 1 . 21)
(2 . 1 . 22)
By solving the coupling fac to rs k. i , j = d ,q ,k d ,k q ,f , in terms o f
the IEEE standard reactances X1, X^, X^, X ^ , XJj, and X^, and su b s ti
tu tin g them in (2 .1 .19) and (2 .1 .2 2 ), the c and c 1-param eters can be
expressed as follow s:
xd fxa - xd / < xd - XP (xd - XP (xi - xP xd^ Xd CX’ - X )^ X"d tXd d£J d
'dkd
'd f
- X’cPXd
<Xd - V Xd
^ Xd - Xd&KXd - Xd)Xd
CXd - Xd£3CXd “ Xd£>Xd
CXd ' XdA> Xd
cd = - Xd/Xd c = -X /X"q q q
C, k , = / ' X, /X? 2 - X, /Kqq q q q
(2 .1 .23)
15
cfd
c
T herefore, (2 .1 .16 ) to g e th e r w ith (2 .1 .2 0 ) , (2 .1 .21) and (2 .1 .23) rep re
sen ts th e s ta te -sp a c e model o f a s a l ie n t-p o le machine in terms of s ta n
dard machine param eters.
Comparing (2 .1 .16) w ith (2 .1 .9 ) through (2 .1 .1 1 ) , i t i s readily-
seen th a t
1D " *4 CD AD (2.1 .24)
and(2.1 .25)
from which i ,, i , and i_ a re expressed as
i d = Xd [Cd Cdkd Cdf^AD (2.1 .26)
(2 .1 .27)
(2 .1 .28)
The cu rren t expressions (2 .1 .26 ) through (2 .1 .28) a re independent o f
machine term ina l connections and can be considered as the output
eq u atio n s.
16
2.2. S ta te Equation o f a Round-Rotor Machine
For a round-ro tor synchronous machine, the s o l id - s te e l ro to r
functions as a damper winding. The equ ivalen t c i r c u i ts in the d and
q-axes o f th e machine are usua lly rep resen ted as in d ica ted in
Figure 2 .2 .1
d-ax is
q-axis
Figure 2 .2 .1 R epresentation o f a Round-Rotor Synchronous Machine by Equivalent C irc u its .
where the c i r c u i t s w ith i _ and i are to account fo r the t ra n s ie n t be-f g
hav io r and th e c ir c u i ts fo r i, , and i. are to include the su b tran sien tkd kqe ffe c ts .
Noting the s im ila r ity between the mutual re la tio n sh ip s o f the
d-axis equ ivalen t c ir c u its and those o f th e q -axis equ ivalen t c i r c u i t s ,
i t i s e a s ily observed th a t th e m atrices Cq and W become
V
C /Tq q
C' /T" kqq qo
C /T ?gq qo
°qkq/Tq
C' /T" kq qo
Cgkq//Tqo
C /Tqg g
C' /T" kqg qo
C / V g qo
(2 .2 .1 )
and
00 0 0
w = 0 0 0 (2 . 2 . 2)
0 0 0
where the c and c*-param eters a re given by
c = ■X /X"q q
c =(X - X')(X ' - X")XLs q ,.q....
(X' - X J 2X"q q
(X1c = —aqkq
X")Xq .q_
fX* - X I X " 'gkq
(X» - X")(X - X J X .q g q qft q&
fVI V "(2vH
18
T herefore , l e t t in g
kq
0
equation (2 .1 .16) w ith th e subm atrices and given by (2 .1 .20) and
(2 .2 .1 ) , and th e c and c ’ -param eters l i s t e d in (2 ,1 .23) and (2 .2 .3 )
rep resen ts th e s ta te -sp a c e model o f a ro u n d -ro to r synchronous machine
in terms o f standard machine param eters.
2 .3 . System Equations w ith Balanced Three-Phase Load
This sec tio n i l l u s t r a t e s th e use o f the synchronous machine
model (2 .1 .16) to o b ta in the equation o f a genera to r fo r se v e ra l load
co n d itio n s. The re s u l t in g system equations fo r c e r ta in load s tru c tu re s
w il l be employed fo r s e n s i t iv i ty a n a ly s is and param eter estim ation in
the follow ing chap ters .
2 .3 .1 . G enerator w ith R esis tiv e Load
Consider a balanced th ree-phase r e s i s t iv e load connected to the
term inals o f a s a l ie n t-p o le synchronous g en era to r. The g en era to r v o l t
ages v . a re then
vabc r L 1abc
and the corresponding d,q,o-com ponents are
(2 .3 .1 )
v = r . l q L q (2 .3 .2 )
( 2 . 3 . 3 )
1
19
S u b s titu tin g (2 .1 .2 6 ) and (2 .1 .27 ) in (2 .3 .1 ) and (2 .3 .2 ) and
re p la c in g vd and in equation (2 .1 .16 ) by (2 .3 .1 ) and (2 .3 .2 ) r e s u l t
in
c w c d tra+V Cdkd < V V Cdf (1)Ad xd Xd Xd
A ,,kdCkdd’ do
Ckdr do
Ckdf’ do
0 0
V - Cfddo
CfkdT1do
Cf 0 0
\q
-Ld 0 0 CW CE
xq
( r +r,)C i. v a V qkqXq
Kkq 0 0 0 cqkg'pi
q°J k
qo
11
0
*kd 0
Xf +Vf
^q0
*kq0
(2 .3 .4 )
Choosing th e c u rre n ts i „ i_ , and i as output v a r ia b le s , the outputci t q
equation i s given by
*d
xf-
iq
_ mmm
'fdr f Tdo
'dkd
fkdTdo
dfXJ
y . '[**£ do
C_S.Xq
(2 .3 .5 )
Equations (2 .3 .4 ) and (2 .3 .5 ) form a complete s ta te model o f a synchro
nous g en era to r connected to a balanced th ree -p h ase r e s i s t i v e load.
20
2 .3 .2 . G enerator w ith RLC Load
C onsider a balanced th ree -p h ase load which c o n s is ts o f a cap ac i
tance C in p a r a l le l w ith a s e r ie s connection o f r e s is ta n c e r and induc
tance L as shown in F igure 2 .3 .1 .
ca
cb
cc
Figure 2 .3 .1 . A Balanced Three-Phase RLC Load
The v o ltag es and c u rre n ts o f th e RLC load a re r e la te d by
^abc r *£,,abc + ^£ ,abc
^&,abc ^ *&,abc
i C,abc "" 1abc ” i AJabc C ^abc
o r , in terras o f d ,q ,o -v a r ia b le s ,
'q “ LC "£d ' C "d1 i A 1 .
V ^ = 0) V _ - T T T A n J + - 1
(2 .3 .6 )
(2 .3 .7 )
(2 .3 .8 )
(2 .3 .9 )
(2 .3 .10 )
XSLd " Vd " L X£d + wA£q
A0„ = v - wA0 , - - A, xq q Kd L xq
21
(2 .3 .11)
(2 .3 .12)
where A ^ and A axe th e d,q-components o£ th e load f lu x - lin k a g e .
Replacing and i in (2 .3 .9 ) and (2 .3 .1 0 ) by (2 .1 .2 6 ) and
(2 .1 .27 ) and combining (2 .3 .9 ) through (2 .3 .1 2 ) w ith (2 .1 .16) r e s u l t in
r
kd
kq
£d
*q
m r o a„ w r o a- (o r o arX^dkd X ^ d f
T ^ k d d f ^ k d T ^ k d f
T ^ f d T ^ fk d T ^ f
-01
U
id r o a_
CXT'd CX^dkd CX/'dfd
0
d
0
io r o a-Xq q V qk<1
rjT'qkq TjjTqqo qo n
1 0 0 0
0 0 0 0
0 0 0 0
0 1 0 0
0 0 0 0
0 “ -LC
CX~q cF q k qq qA .LC-io 0 0
1 0 - r
0 1 -to ~
1 r r iXd 0
Xkd 0
Xf vf
\
+
0
Akq 0
Vd 0
vq 0
X£d 0
h q 0_ J
(2 .3 .13 )
Thus equation (2 .3 .13 ) p rov ides a s t a t e model in term s o f s ta n d a rd
machine param eters fo r th e g en e ra to r w ith a load c o n fig u ra tio n consid
ered in F igure 2 .3 .1 . An ou tpu t equation can a lso be ob tained by
choosing a p ro p er s e t o f ou tpu t v a r ia b le s as in s e c tio n 2 .3 .1 .
22
2 .3 .3 . G enerator w ith RC Load
Consider a balanced th ree -p h ase p a r a l le l RC lo ad connected to th e
te rm in a ls o f a s a l ie n t-p o le synchronous g en era to r as shown in
F igure 2 .3 .2 .
Figure 2 .3 .2 A Balanced Three-Phase RC Load
In t h i s c a se , th e v o ltag e and c u rre n t re la t io n s h ip s a re given by
1= I * _____Vabc C xabc ~ rC Vabc (2 .3 .1 4 )
o r , in term s o f d ,q ,o -v a r ia b le s ,
v . = - — v , + a) v + 4 i j d rC d q C d
v = — v ■ (D v . + ^ i q rC q d C q
(2 .3 .1 5 )
(2 .3 .1 6 )
23
Replacing i , and i in (2 .3 .15) and (2.3.16) by (2 .1 .26) and (2 .1 .2 7 ) ,H
and combining the r e s u l t in g equations w ith (2.1.16) give an equation
s im ila r to (2 .3 .13).
2 .3 .4 . Generator w ith RL Load
Following a s im ila r procedure as shown in the previous se c tio n s ,
the d and q-voltage equations fo r a s e r ie s RL load are
vd = r i d - w L i +■ L i d (2 .3 .17)
v = r i + (o L i , + L i . (2 .3 .18)q q d q '
Using (2 .1 .26) and (2 .1 .27) fo r i , and i and combining the re s u ltin gu qequations with (2.1.16) y ie ld a s ta te -sp a c e equation fo r the generato r
with a balanced three-phase se r ie s RL load.
CHAPTER 3
OUTPUT SENSITIVITY ANALYSIS OF A SYNCHRONOUS GENERATOR
The req u ire d degree o f accuracy in determ ining th e param eter
va lues fo r a given model s t ru c tu re depends on how much th e v a r ia b le s o f
i n t e r e s t forming th e output are a ffe c te d by a param eter d e v ia tio n from
i t s t r u e value. T his can be conven ien tly in v e s tig a te d by an ou tpu t
s e n s i t iv i t y fu n c tio n analy sis [72 ], where th e ou tpu t s e n s i t iv i t y w ith
re sp e c t to a p aram eter v a r ia tio n i s determ ined.
The ou tpu t s e n s i t iv i ty [73, 74] i s u s u a lly defined as th e f i r s t
o rd e r v a r ia tio n <5y(t) o f an ou tpu t y ( t ) due to a v a r ia t io n Sa o f parame
t e r a . The norm on <5y(t) o r 9 y ( t ) /3 a se rv ing as a measure o f s e n s i
t i v i t y i s q u a l i ta t iv e ly , and o f te n q u a n t i ta t iv e ly , in d ic a t iv e o f the
o u tpu t d isp e rs io n from i t s nominal t r a je c to ry due to a param eter
v a r ia t io n .
In th is c h a p te r , th e system c o n s is tin g o f a synchronous genera
to r and RLC load as d iscussed in Chapter 2 i s considered . The p o ss ib le
maximum output e r r o r due to machine param eter inaccuracy as determ ined
by IEEE t e s t p rocedures w il l be ob ta in ed by perform ing an ou tpu t s e n s i
t i v i t y a n a ly s is .
3 .1 . M athematical Form ulation
Let a system be g enera lly described by
x = f (x , ot, u , t ) ; x ( tQ) = xQ , (3 .1 .1 )
y = g(x , a , t ) (3 .1 .2 )
24
25
where x i s an n - s t a te v e c to r , a i s a k -param eter v e c to r , u i s an
m -input v e c to r , and y i s a p -o u tp u t v e c to r.
For th e system d escribed by (3 .1 .1 ) and (3 .1 .2 ) , th e p a r t i a l
d e r iv a tiv e o f th e ou tpu t y w ith re sp e c t to param eter a a t a nominal
value a0 o f a can be ob ta ined in m atrix form as
where
3a a= 3 g (x ,a ,t ) 3x o 3x 3a
$L 4 3yi3a 3a.
_ J.
3x A 3x.i■3a = 3a.
_ 3.
H 4 N15a ~ 3a. _ 3.
3^A ~9*i3x L3xjJ
a3 g (x ,a ,t )
3a a1-
= 1 ,2 , . . . , p , j = 1 ,2 , k
x — 1 ,2 , . . . , n , u ***1,2, . . . , k
i 1 ,2 , . . . , p , j 1 ,2 , . . . , k
~ 1 ,2 , . . . , p , j = 1 ,2 , n .
(3 .1 .3 )
Taking th e p a r t i a l d e r iv a tiv e s o f (3 .1 .1 ) w ith re sp e c t to a and
e v a lu a tin g them a t th e nominal va lue a ° y ie ld th e s t a te s e n s i t iv i ty
equation
d 3x = 3f 3x _3f 3u + 3 fd t 3a ao 3x 3a o + 3u 3a(X a ° 3 a
3x,= 0. (3 .1 .4 )
a v
From (3 .1 .1 ) and (3 .1 .4 ) , x ( t) a o and I f a o can be so lv ed , a f t e r which
a*.3a a'o i s found from (3 .1 .3 ) . To o b ta in a convenient p h y sica l in te r p r e
ta t io n o f th e num erical r e s u l t s , th e ou tpu t s e n s i t iv i t y fu n c tio n in
m atrix form i s d e fined as
26
s ( t ) = [ s ^ O t) ] =3y.
a? 1j 3a. J a?
J
••*> P » j =l j2 , •. • , 1c
(3 .1 .5 )
where each element s ^ ( t ) can be in te rp re te d as th e increm ent o f ou t
pu t y\ in percen tage o f th e base value o f y . , i f a? i s in c re a sed by one
p e rc e n t.
3 .2 . A p p lica tio n to G enerator-Load System
C onsider th e system d escrib ed by equation (2 .3 .13 ) in s e c t io n
2 ,3 .2 where th e load s t ru c tu re i s f a i r l y g e n e ra l. Let th e ou tpu t
v a r ia b le s o f i n t e r e s t be
vabc - / « • : * - 3(3 .2 .1 )
c-1abc 7 ’ K - ‘ 3
(3 .2 .2 )
P = V ji , + v i d d q q (3 .2 .3 )
Q - v i , - v , i q d d q (3 .2 .4 )
where v and i a^ c are re s p e c t iv e ly measures o f th e phase v o lta g e and
c u rre n t m agnitudes, and P a n d Q a re th e in s tan tan eo u s power o u tp u t and
approxim ately a measure o f th e re a c t iv e power o u tp u t, r e s p e c t iv e ly .
Furtherm ore, assume th a t
(1) The f i e l d vo ltage v^ has been ap p lied a t t = -« w ith th e arma
tu re c i r c u i t s open,
(2) The value o f v^ i s such th a t a d e s ire d g e n e ra to r te rm in a l
v o lta g e , say one p e r - u n i t , i s ob ta ined a t t = 0" ,
[3) At t = 0, a balanced three-phase RLC load as shown in Figure
2 .3 .1 is connected to the armature term ina ls ,
(4) The generato r speed i s constan t.
Then, from (2 .3 .13) i t i s seen th a t a t t = 0~,
Here, E is the rms value o f the no-load phase voltage and the super
s c r ip t o re fe rs to th e s te a d y -s ta te value.
For t > 0, the system i s described by (2.3.13) where the f ie ld
vo ltage i s unchanged. T herefore, su b trac tin g (3 .2 .5 ) from (2.3.13)
y ie ld s
where AX rep resen ts the increm ental s ta te -v a r ia b le s and Au is given by
0 = A(ot)A° + u° (3 .2 .5 )
where A(a) i s given in (2.3.13)
0 v ! /3E 0 0 0 0f >
a
AA = A(a)AA + Au; AA(0) = 0 , (3 .2 .6 )
Au = [0 0 0 -> 3E 0 0 0 0 0]T .
Since v , , v , i , , i are a l l zero a t t = 0 d q* d qt > 0 become
* vabc* ^abc’ P and Q fo r
28
P = Av.Ai , + Av Ai d a q q
Q = Av A i, + Av ,Ai q d d q
(3 .2 .9 )
(3 .2 .10)
where A i, and Ai a re r e la te d to th e f lu x -lin k a g e s by (2 .1 .26 ) and4(2 .1 .27) w ith Ap and A^ re s p e c tiv e ly rep laced by AA and AAq as d e te r
mined by (3 .2 .6 ) .
T herefo re , th e ou tpu t s e n s i t iv i t y a n a ly s is can be c a r r ie d out by
applying th e form ulas in se c tio n 3 .1 to equations (3 .2 .6 ) and (3 .2 .7 )
through (3 .2 .1 0 ) . For convenience o f n o ta t io n , denote AA and Au by x
and u , re s p e c tiv e ly . Then (3 .2 .6 ) and (3 .2 .7 ) through (3 .2 .1 0 ) a re
rep resen ted su c c in c tly by
x = A(a)x + u ; x(0) = 0 ,
y = g(x , a , t )
(3 .2 .11)
(3 .2 .12)
w ith
g = [v,abc abc Q]
Taking th e p a r t i a l d e r iv a tiv e s o f (3 .2 .1 1 ) and (3 .2 .12 ) w ith re sp e c t to
a y ie ld s
d_d t
3x3a
&Ba
a= A (a ) |^ o ' 3a a 3a a
a= ig .
o 3x 3a a 3a a(3 .2 .14 )
where —g - ^ -x , and evaluated a t a0 are g iven by3x 3a
29
9A (a) =8a x
3£x 3£x 3fx 8f^ ■gsg ^ 0 0
^ 2 f f 2 ^ 2 ^ 2 ^ Q8a. 8a» 8a, 3a, 8a-
8aj 8a2 8a3 8a^
0 0
3f6 » Je ^ 6 ^ 6Sa1 8a2 8a3 8a4
0 0 0 0
0 0 0 0
0
53a,.
0 0
0 0
0
0
3fl"5a
0 0
3f4 9f4
10
0 0
0 0
8a„ 8a,8
8a7 3ag 8ag
58a10
0
3f? 3f?
0 0 0
0
0
with
9 8a. . (a)E 8cl X"t J *** ®* k ~ l>2, . . . 11
j = l K J
3£.i3“k
30
0 0 0 0 0 ! f i3 x ,o 3x^
3g2 3g2 ®g2 3*23xl ^ 2 3x3 Sx4 3*5
0 0
8g3 8g3 3g3 3g3 8g3 3g3 3g33x^ 3x2 3x3 3x.4 3xS 9x6 3Xy
**4 3g4 9g4 3g4 9g4 3g49x^ 3x2 3x3 3x4 8xs 3x6 3Xy
0 0 0 0 0 0 0 0
3g2 3g2 9g2 3g20 8g2
■55^ 0 3a7
% 3g3 8g3 3g3 n n Sg33a^ 3o2 8a3 3a4 U 3a^ 3as
3g4 3g4 3g4 3*4 n n 3g4
8“ l 3a2 3a3 3a4 U U 3a? 8a8
The d e ta i le d exp ression f o r each elem ent o f th e m atrices i s p resen ted
in Appendix 1.
3 .3 . Numerical R esults
A num erical example i s given where th e com putations were based on
a 120 kVA, 208 V, 400 Hz a i r c r a f t g e n e ra to r w ith the measured param eter
va lues as p rov ided by th e m anufacturer l i s t e d in Table 3 .1 (time
base = = 1/2513 s e c ) , except fo r whose value was unknown and
31
assumed h ere t o be equal to 0 .0 4 pu. The s e t o f load param eter va lu es
i s g iv en in T able 3 .2 fo r d i f f e r e n t load p o w er -fa c to r v a lu e s , b u t th e
same ra ted c o n d it io n .
Table 3.1
Synchronous Machine Data in Per-U nit
= 0.1863 Xd = 2.10 X = 0.786q
T»do = 521.5 R = a 0.0189
X = 0.2160 X" = 0.107q
T" = 18.24 doftt
qo = 11S.1 Xd* = 0.04
Table 3 .2
Load Param eter Data in P er-U n it fo r V arious Load Pow er-Factors
Load Pow er-Factor R L C
0.1 lead ing 6.4000 4.8000 1.0700
0 .4 lead ing 1.6000 1.2000 1.2165
0 .7 lead ing 0.9143 0.6857 1.2391
1.0 0.7111 0.0533 0.7500
0 .8 lagging 0.6125 0.6249 0.2162
0 .5 lagging 0.3200 0.7332 0.2796
0 .2 lagging 0.0500 0.4975 0.1010
A d i g i t a l computer program has been developed fo r th e s e n s i t iv i ty
fu n c tio n com putation as l i s t e d in Appendix 2 (program was w r i t te n fo r a
time base o f 1/ wq s e c ) . The ou tpu t s e n s i t iv i t y functions were computed
using (3 .2 .6 ) , (3 .2 .1 3 ) , (3 .2 .1 4 ) and (3 .1 .5 ) fo r a tim e d u ra tio n of
620 p e r -u n i t seconds fo r each load ing c o n d itio n . The num erical r e s u l ts
32
a re p a r t i a l ly d isp layed in graph form as shown in Figures 3 .3 .1 -3 .3 .4 .
For conqmrative purposes, the peak values o f th e s e n s i t iv i ty
functions fo r th e considered time range are l i s t e d in Table 3.3 in th e
column corresponding to th e load pow er-fac to r values shown on the top
row. In a d d itio n , an a s te r is k i s provided a t th a t peak value which i s
th e la rg e s t in th e considered range o f th e load pow er-fac to r va lues.
The num erical r e s u l ts show th a t
(1) The output v a ria b le s are extrem ely s e n s i t iv e to param eter
v a r ia tio n s a t 0.1 load pow er-fac to r lead ing .
(2) In gen era l, th e output v a ria b le s a re more s e n s i t iv e a t
lead ing pow er-factors than a t lagging p o w er-fac to rs .
(3) At lead ing pow er-fac to rs , = X1, = X^, = T^q ,
a 7 = XJj and ag = X are the param eters which a f fe c t th e output v a ria b le s
most. At lagging pow er-fac to rs , i t i s seen th a t th e output i s most
s e n s i t iv e to a , = X'l and a_ = X".1 d 7 q(4) The re a l and re a c tiv e powers a re more s e n s i t iv e with r e
sp ec t to param eter v a ria tio n s than the vo ltage and c u rre n t magnitudes.
(5) The s e n s i t iv i ty functions o f the output v a ria b le s w ith re
sp ec t to = r^ are a l l n e g lig ib le .
Table 3 .3
Peak Value o f S e n s i t iv i t y Functions fo r Various Load Power-Factors
* Maximum peak-value fo r considered range o f load pow er-factor
S e n s itiv ity leading lagging
Function lpf=0.1 lpf=0.4 lpf=0.7 lpf=1 .0 lpf=0.8 lpf=0'.5 lpf=0,2
S11 10.86* 3.000 1.613 -.4530 -2.281 -3.130 -3.167
S21 10.87* -6.350 -4.322 -3.487 2.619 -4.567 -3.911
S31 40.01* 23.11 13.37 -6.158 7.374 15.20 -22.88
S41 -385.4* -52.69 12.02 -4.628 -6.613 -8.454 -11.82
S12 -2.280* -0.993 -0.119 -0.071 0.071 0.068 -0.076
S22 -2.280* -0.993 -0.119 -0.773 0.071 0.677 -0.141
S32 -8.420* -7.650 -0.616 -0.777 -0.128 -0.184 -0.191
S42 80.83* 17.43 0.626 -0.514 0.091 -0.175 -0.391
S13 26.95* 9.185 0.655 -0.185 -0.430 -0.400 -0.358
S23 26.97* 9.196 0,655 0.655 -2.000 -0.400 -0.681
S33 99.53* 70.76 3.377 -1.199 -0.734 -0.404 -0,132
CO -956.4* -161.2 -3.436 -0.798 -0.551 -0.701 -1.103
S14 -0.610* -0.181 -0.014 0.006 0.009 0.008 0,008
S24 -0.610* -0.182 -0.014 0.065 0.009 0.008 0.016
S34 -2.240* -1.396 -0.067 0.053 0.018 0.010 0.009
S44 21.56* 3.184 0.068 0.035 0.013 0.018 0.034
Table 3 .3 (continued)
Peak Value o f S e n s it iv i t y Functions fo r V arious Load Power-Factors
* Maximum peak-value fo r considered range o f load power fa c to r s
S e n s itiv ity leading lagging
Function lpf=0.1 lpf=0.4 lp fs0 .7 lpf=1.0 lpf=0,8 lpf=0.5 lpf=0.2
S15 -1.623* -0.462 -0.034 0.023 0.021 0.020 0.024
S25 -1.623* -0.462 -0.034 0.247 0.021 0.020 0.045
S35 -5.972* -3.558 -0.161 0.240 0.047 0.055 0.058
S45 57.46* 8.111 0.164 0.159 0.035 0.053 0.120
S16 -17.43* -5.436 -0.421 0.149 0.264 0.251 0.254
S26 -17.44* -5.441 -0.421 1.617 0.264 0.251 0.482
S36 -64.27* -41.86 -2.111 1.025 0.520 0.294 0.118
S46 618.5* 95.40 2.147 0,681 0.390 0.509 0.874
S17 -4.032 -2.511 -1.700 -0.595 4.876 5.998 -6.437*
S27 -9.350* -6.771 -5.317 -2.833 -5.472 -6.678 3.847
S37 47.36* 31.02 19.67 -6.798 16.13 21.35 -12.81
S47 -21.01 -13.90 -9.079 -4.600 -10.43 -15.50 -28.21*
S18 -0.387 -1.919* -1.598 -0.027 -0,042 -0.065 0.043
S28 -0.392 -1.935* -1.606 -0.194 -0.053 -0.072 -0.037
S38 -1.459 -14.86* -7.815 -0.372 -0.190 -0.273 0.157
S48 13.81 33.79* 7.967 -0.210 -0.124 -0.168 -0.216
35
Table 3 .3 [continued)
Peak Value o f S e n s it iv i t y Functions fo r Various Load Power-Factors
* Maximum peak-value fo r considered range o f load pow er-factors
S e n s itiv ity lead ing lagging
Function lpf=0.1 lp f=0 .4 lpf=0.7 lpf=1.0 lp fs0 .8 lp f= 0 .5 lpf=0.2
S19 0.198 1.016* 0.689 0.019 0.042 0.056 -0.037
s29 0.210 1.018* 0.689 0.166 0.052 0.062 0.031
S39 0.786 7.848* 3.369 0.319 0.163 0.235 -0.135
CO £>
i L.
-7.021 -17.82* -3.425 0.281 0.107 0.145 0.186
Sl,1 0 -0.378 -0.410* -0.223 -0.057 -0.177 -0.244 -0.156
S2,10 -0.913* -0.764 -0.596 -0.614 -0,205 -0.272 -0.283
S3,10 -3.572* -3.163 -1.873 -1.165 -0.537 -0.993 -0.702
S4,10 11.72* 7.204 1.178 -0.456 -0.401 -0.692 -0.847
36
1150'
V-
- s o -
S2150^
S3150
0
-SOs
50
-50
-ISO-
- 2 0 0 J
T320p e r-u n it tim e
620
F igu re 3 . 3 . 1 Output S e n s i t i v i t y Function w . r . t . XJJ a t Apf = 0 .1 lea d in g
37
1725
-25 '>2? 25 1
- -j ) jf+
-25
^ / V i y \ A y \ A H / v r \ / v a ■»>— *-
37SO -
25 -
-25 '
4725-
-25-
I----
—I--10
- r — ii--- r -20 26 320 620p er-u n it tim e
Figure 3 . 3 . 2 Output S e n s i t iv i t y Function w . r . t . XJJ a t £p f = 0.1 lea d in g
11
25
-25
S 21
25
iA>V^A/\y\^AAAA/W vV r^ —jV
■vtf/iA^vvWVWWVWVWW IV*
38
-25 ■
S31
25 1
iS-
-25
S41 25 4
-25 -
v W rH h
1“0
— I—
10 20 f t 1 r*26 320 620
p e r-u n it time
Figure 3 . 3 . 3 Output S e n s i t iv i t y Function w . r . t . X” a t Jlpf = 0 .2 lagg in g
0
-25 H
S2725
-25 '
S37 25 -
“A ' ■ ■**»■ | |.
■25 -
S47
25
0
-25
"1 1 1 %\ 1 "'■ " ) ~ ' - 1--0 10 20 26 320 620
per-u n it tim e
Figure 3 . 3 . 4 Output S e n s i t iv i t y Function w . r . t . X" at Apf = 0.2 la g g in g
3 .4 . P o ss ib le Output E rro r Due to Param eter Inaccuracy
40
Consider th e g en e ra to r param eter va lues l i s t e d in Table 3.1 as
given by i t s m anufacturer. Among th e se v a lu e s , only th e d i r e c t- a x is
param eters X j, X^, and w ill be taken to i l l u s t r a t e th e p o ss ib le
e r ro r due to param eter inaccuracy . E v id en tly , th e values o f th e parame
t e r s X1 , X^, the s h o r t - c i r c u i t t r a n s ie n t tim e-co n stan t and su b tran
s ie n t tim e-co n stan t T1 were taken as th e average o f th e th re e values
determ ined from th e s h o r t - c i r c u i t t r a n s ie n ts in th e th re e phases as can
be observed from th e va lues in th e Sth and 6th columsn o f Table 3 .4 .
The param eters T^q and TJQ can be computed from X^, X' , T^ and TjJ as
shown in th e ta b le .
Thus, u sing th e nominal va lues o f th e g e n e ra to r param eters as
given by th e m anufacturer fo r th e s im u la tio n , th e tru e param eter values
may d ev ia te from th e assumed values by th e percen tages given in th e 7th
column o f th e ta b le . From th e s e n s i t iv i t y a n a ly s is r e s u l t s in Table 3 .3 ,A
th e p o ss ib le maximum changes o f th e p h a se -c u rren t magnitude i a^ c and
ou tpu t power P due to a change o f 1% in th e param eter a re g iven resp ec
t iv e ly in th e 8th and 9 th columns. This seems to in d ic a te t h a t due to
a p o s s ib le param eter v a r ia t io n o f th e amount in th e 7th column, th e p e r
centage v a r ia t io n o f and P could be as la rg e as those va lues given
in th e 10th and 11th columns.
The f ig u re s in th e l a s t two columns show th a t th e param eter values
determ ined by th e conventional t e s t p rocedures a re no t s a t i s f a c to r y i f a
reasonably accu ra te a n a ly s is o f the system is d e s ire d . T h ere fo re , the
need f o r a method by which th e machine param eter va lues can be ob ta ined
much more a c c u ra te ly i s e v id e n t. To meet t h i s demand, param eter estim a
t io n methods a re proposed and d iscussed in the fo llow ing c h ap te rs .
Table 3 .4P ossib le Maximum V aria tion o f i . and Pabc
Machine
Parameter
Obtained from Phase Average of
3 Phases
Nominal
Value
P ossib le Dev ia tio n from Nom.Val. (%)
Max.Sens .Fen.of Possib le Max.Var.of
1 2 3 PC%) P(%)X'd 0.2225 0.2105 0.217 0.216 0.216 3 2.28 8.42 6.84 25.26
xa 0.204 0.1869 0.168 0.1863 0.1863 9.8 10.87 40.01 106.53 392.1
. Td 0.02776 0.02925 0.02975 0.02892 0.02892•pitXd 0.00526 0.00775 0.00575 0.00625 0.00625Tdo=
W Xd 0.262 0.292 0.288 0.281 0.207 41 17.44 64.24 715.04 2635.1
Tdo=X'T'VX’i d a d 0.00574 0.00873 0.00743 0.00725 0.00725 21 1.62 5.97 34.01 125.37
CHAPTER 4
SYSTEM IDENTIFICATION
A system id e n tif ic a t io n problem a ris e s when the mathematical
model fo r a system of in te r e s t cannot be completely sp e c if ie d without
being id e n tif ie d from an ac tu a l observation o f the system inpu t and
ou tpu t. This problem has been the su b jec t o f l iv e ly d iscussion by many
in v e s tig a to rs in recent years and a no tab le number o f repo rts have been
generated [65]. Apparently the in te r e s t in th is to p ic has been rooted
in the s ig n ific an c e o f a c o rre c t model, a v a i la b i l i ty o f estim ation
techniques and computing f a c i l i t i e s , and the d e f in ite demand fo r a
b e t te r knowledge o f the phenomena not only in engineering but a lso in
many o th er areas such as b io logy , economics, and s t a t i s t i c s , e tc .
I t i s no t the in te n tio n o f th is chap ter to p resen t a broad d is
cussion on the id e n t i f ic a t io n problem. However, a tte n tio n w ill be
focused on the na tu re of the problem and two param eter estim ation tech
niques which w il l be used to determine th e param eters o f a synchronous
machine in the following chap ter. A d e ta ile d computational algorithm
w ill a lso be developed fo r th e d ig ita l computer sim u la tions.
4 .1 . Problem Formulation
Suppose an unknown re a l system i s given and i t i s desired to
obtain a mathematical model s a tis fy in g a c e rta in c r i te r io n by observing
the system response to a p roperly chosen in p u t. This i s known as the
system id e n tif ic a t io n problem, and the usual approach to the so lu tio n
42
43
i s as fo llow s.
Let s r be an unknown re a l system to be id e n t i f i e d , s = {s^} be a
c la s s o f models o f the r e a l system s^ , yr be th e output o f th e re a l
system s r to a known in p u t tim e fu n c tio n u f t ) , y be the ou tpu t o f th e
model s^ to th e same in p u t fu n c tio n u ( t ) , [0 , T] be an o b se rv a tio n tim e
range, and J ty r » y , u , T) be a lo ss fu n c tio n a l imposed on yr ( t ) , y ( t )
and u [ t ) fo r te [0 , T ]. The system id e n t i f i c a t io n problem can then be
s ta te d as an o p tim iza tio n problem , i . e . , f in d a model s Q in s such th a t
th e lo ss fu n c tio n a l J i s m inimized.
With t h i s fo rm u la tion , th e f i r s t su b je c t to th in k o f i s how to
choose the c la s s o f m odels. In g e n e ra l, the models can be c h a ra c te r iz e d
by th e i r s t ru c tu re and a s e t o f param eters which may appear in th e model
equation l in e a r ly o r n o n lin e a r ly . Thus the s e le c tio n o f a c la s s o f
models reduces to f in d in g a model s t ru c tu r e and a corresponding s e t o f
param eters . U n fo rtu n a te ly , th e genera l method to determ ine them co r
r e c t ly i s l i t t l e known. In p r a c t ic e , however, th e model s t ru c tu r e and
i t s s e t o f param eters can be hypo thesized on th e b a s is o f experience
and knowledge o f th e p h y sica l laws governing th e system.
The choice o f th e lo ss fu n c tio n a l can be v a rie d depending on th e
id e n t i f i c a t io n method used , e . g . , th e lo ss fu n c tio n a l i s a p e n a lty
fu n c tio n on th e output e r r o r f o r th e w e ig h te d -le a s t-sq u a re s e s tim a tio n
method, a lik e lih o o d fu n c tio n fo r th e maximum lik e lih o o d e s tim a tio n
method o r an a p r io r i d e n s ity fu n c tio n fo r th e Bayesian e s tim a tio n
techn ique .
Once th e c la s s o f models and lo ss fu n c tio n a l have been p ro p erly
s e le c te d , th e model can be id e n t i f i e d by determ ining the optim al numeri
c a l va lues o f th e unknown param eters op tim izing th e lo ss fu n c tio n a l.
44
T herefo re , th e system id e n t i f i c a t io n problem reduces to a param eter
e s tim atio n problem. T his s i tu a t io n i s i l l u s t r a t e d in F igure 4 .1 .1 ,i-N
where th e sampled in p u t-o u tp u t d a ta (u C i) , yr (i)}^_^ o f th e r e a l system
s^, and an a p r i o r i knowledge o f the system ; v iz , th e s t a t i s t i c s o f th e
no ises a n d /o r p r o b a b i l is t ic in fo rm ation o f th e unknown param eter a a re
used to o b ta in a param eter e s tim a te .
System Noise Measurement Noise w v
u ( i ) Param eterE stim ate
a p r io r i knowledge
Sampling
Real System
Param eter
E stim ato r
F igure 4 .1 .1 System Id e n t i f ic a t io n by Param eter E stim ation
The th re e most w ell-developed techn iques fo r param eter e s tim a tio n
a re the w e ig h te d -le a s t-sq u a re s method, maximum lik e lih o o d method, and
Bayesian method. The w e ig h te d -le a s t-sq u a re s method i s th e o re t ic a l ly
reasonably sim ple and com putationally l e a s t involved , thus i t i s very
45
a t t r a c t iv e fo r the p ra c t ic a l increm entation. The maximum lik e lih o o d
method has many d esirab le c h a ra c te r is t ic s , but i s com putationally q u ite
involved. The Bayesian method i s th e o re tic a l ly most a t t r a c t iv e . How
ever, i t i s com putationally very much involved and im p rac tica l.
The question o f how to determine the b e s t inpu t sig n a l has been
discussed by severa l in v e s tig a to rs [85-88]. Some authors have formu
la te d the input s igna l design problem as an optim al con tro l problem
using the trac e o f the F isher inform ation m atrix as the cost fu n c tio n a l.
However, many questions r e la te d to th e input sig n a l design problem have
not y e t been fu l ly answered.
In summary, many aspects o f the id e n tif ic a t io n problem are not
completely known y e t. N evertheless, w ith in the realm o f p ra c t ic a l a p p li
ca tions i t has been shown th a t param eter estim ation techniques can be
su ccessfu lly implemented in many cases w ithout major d i f f ic u l t i e s . In
the following se c tio n s , the w eigh ted-least-squares and maximum l i k e l i
hood estim ators w ill be fu r th e r discussed along w ith the development o f
a d e ta ile d estim ation algorithm which w ill be implemented by sim ulation
in the next chapter.
4 .2 . W eighted-Least-Squares E stim ator
Suppose no a -p r io r i inform ation o f the system noises i s a v a ila b le
a t a l l and the model o f a r e a l physical system is described by
x = A(a)x + B(a)u; x (0) = xq (4 .2 .1 )
y = C(a)x, (4 .2 .2 )
where x and y are the s t a te and observable output v ec to rs , re sp e c tiv e ly ,
and u rep resen ts the inpu t v ec to r assumed to be a given time function .
46
The c o e ff ic ie n t m atrices A (a), B(a) and C(a) axe dependent on a parame
t e r vecto r a whose value i s to be id e n tif ie d . Note th a t the elements of
the c o e ff ic ie n t m atrices may be l in e a r o r non linear functions o f the
unknown param eter vecto r a.
Then the w eighted-least-squares estim ato r (WkSE) i s defined as
the param eter estim ato r which determines the values o f the unknown
param eter a minimizing the w eighted-squares loss functional
N „j = I EyTCi) - y ( i) ] W[y Ci) - y ( i ) ] (4 .2 .3)
i= l r r
where y^ and y are re sp e c tiv e ly the ou tpu ts o f the re a l system and model,
and W i s a p o s itiv e sem idefin ite symmetric weighting m atrix chosen on
the b asis o f engineering judgment.
Any e s tab lish ed op tim ization technique can be used to minimize
(4 .2 .3 ) . For s im p lic ity o f computer programming, the m odified Newton-
Raphson method [67] i s employed here to ob tain a recu rsive formula. For
easy re fe ren ce , the m odified Newton-Raphson method i s summarized in the
follow ing s te p s .
(1) Make an educated guess o f ao , the i n i t i a l value fo r parame-
v ec to r a .
(2) L inearize the model output y about the param eter value aQ
and s u b s titu te the r e s u l t in the cost functional [4 .2 .3 ) .
(3) Determine a new param eter value minimizing the cost func
t io n a l obtained in step 2.
(4) Perform the computations in s tep s 2 and 3 repeated ly u n t i l
no s ig n if ic a n t changes occur in two successive i te r a t io n s .
Following the steps shown above, a f te r an i n i t i a l value a o fo r
47
param eter a has been chosen, the output y i s expanded in a Taylor s e r ie s
about i t s t ra je c to ry a t a = aQ to ob tain
y = y * I 2-a 3a oa (a - aQ) + h igher o rder term s, (4 .2 .4)
o
where
3y9a
a _ a.+ 3CCa} * 3a x (4 .2 .5)
9xThe time-dependent v a riab le s a are found from the p a r t i a l deriva
t iv e o f (4 .2 .1) with respec t to a and evaluated a t a Q.
_ddt
*
3x■
+ 3ACoi) + 3B(a)3a a a 8“ a 3“
.o 0 0
ua
3x3a 0 ,
(4 .2 .6)
and equation (4 .2 .1 ) i t s e l f evaluated a t aQ. S u b s titu tio n o f (4 .2 .4).
in (4 .2 .3) y ie ld s
NJ= I
i=lyr ( i ) - yCD 3y(j)
a 3a o a C“ ' “o5 oWyr (i) - yCi) a .
_ syci)3a a
(4 .2 .7)
where the h igher-o rder terms o f the expansion a re ignored. With
(a - aQ) se lec te d such th a t equation (4 .2 .7 ) is minimized,
3J3(a-ao)
= 0 , (4 .2 .8)
from which follows a s e t o f l in e a r a lg e b ra ic equations fo r (a - aQ) :
48
NI
_i=l( V i ) ] I 3a
TWfayC1)}
9aS J
(a - a ) =ao L i i i t 30 J
TW(yr ( i) - y ( i) )
(4 .2 .9 )
where a denotes the a th a t minimizes (4 .2 .7 ) .
From equations (4 .2 .1 ) , (4 .2 .2 ) , (4 .2 .5 ) , (4 .2 .6 ) and (4 .2 .9 ) ,
an i te r a t iv e so lu tio n fo r a can be obtained . Upon w ritin g (a - aQ) = Aa,
the recursion formula fo r computing successive estim ates o f a is
ak = ak_1 + Aak (4.2.10)
where k (= 1,2 . . . ) re fe rs to the i te r a t io n count and a 0 is some
i n i t i a l guess made by engineering judgment. Since th e d e riv a tio n
assumes a s u f f ic ie n t ly small Aa due to tru n ca tin g the Taylor s e r ie s , i t
i s proposed to modify (4 .2 ,10) in
a k = a k" 1 + G(k)Aak (4.2.11)
where G(k) i s a gain m atrix , whose elements are se le c te d on the b asis
o f computational judgment. G(k) con tro ls the ex ten t o f co rrec tio n in
each i te r a t io n s te p . An excessively large value o f G(k) may cause a
to overshoot and go in to divergence or o s c i l la t io n about the optim al
param eter estim ate a*, whereas an overly sm all value o f G(k) causes
very slow convergence o f ak to a*. One o f the many p o ss ib le v a ria tio n s
i s to take
G(k) = diag [g^O O ] , (4.2.12)
where the s c a la r g ^ i ^ are c^osen *n some proper manner. A block d ia
gram fo r th is w eigh ted-least-squares param eter estim ation algorithm is
shown in Figure 4 .2 .1 .
49
u(t)R E A L S Y S T E M
yr (t)
Equation(4 .2 .1 )
x (t)
a
Equation(4.2.2)
y ( t)
u(t)l
Equation(4,2 .6)
Sx/3aEquation(4 .2 .5 )
3y/3a
Equation(4.2.11)
Equation
(4 .2 .9)
Figure 4 .2 .1 W eighted-Least-Squares Parameter Estim ation Algorithm
4 .3 . Maximum Likelihood Estim ator
Suppose the s t a t i s t i c s o f the system noises a re av a ilab le as
a -p r io r i knowledge such th a t a s to c h a s tic model
x = A(a)x + B(a)u + Dw;
y = C(a)x + v
E{x(0)} - E{x0} = 0 , (4 .3 .1 )
(4 .3 .2 )
can be formed where the inpu t noise w, output no ise v and i n i t i a l value
xq o f the s t a te va riab les a re assumed to be m utually independent and
charac te rized as follow s: w i s a q-zero-mean gaussian white noise with
E{w(t)wT(T)} = Q(t)6 ( t - t)
where Q is a p o s itiv e sem idefin ite m atrix , v is a p-zero-mean gaussian
w hite noise w ith
E {v(t)v (x )} = R ( t)6 ( t - t)
50
where R is a p o s i t iv e d e f in i te m a tr ix , and xq i s an n-zero-mean
gaussian random v e c to r w ith
E{x xT) - P o o o
H ere, E and 6 s ta n d fo r th e e x p e c ta tio n o p e ra to r and th e D irac d e l ta
fu n c tio n , r e s p e c tiv e ly .
Furtherm ore, assume th a t th e re a re no s t r u c tu r a l e r ro rs f o r the
proposed model, i . e . , th e re a c tu a l ly e x is ts some unknown value o f the
param eter a such th a t i f t h i s param eter value i s used in the model, i t s
ou tpu t y corresponds e x a c tly to th e r e a l system o u tp u t y^ fo r any in p u t.
The o b je c tiv e i s to determ ine th e most p robab le value o f a by
observ ing the system output y , i . e . , to choose th e value o f a a t which
p (aiy)> the a - p r io r i c o n d itio n a l d e n s ity fu n c tio n o f a fo r a given
value o f ou tpu t y , a t ta in s a maximum. U n fo rtu n a te ly , th e knowledge o f
th e a -p r io r i d is t r ib u t io n o f ct i s n o t e a s i ly o b ta in ed . However, i t can
be derived [70, 83] th a t
max p (a |y ) = max p (y |a ] . (4 .3 .3 )
H ere, the exp ress ion p (y |a ) regarded as a fu n c tio n o f y fo r a g iven
v a lu e o f a i s a c o n d itio n a l d e n s ity fu n c tio n o f y given a . I f t h i s
same q u a n tity i s regarded as a fu n c tio n o f a fo r a given value o f y , i t
i s no longer a d e n s ity fu n c tio n b u t r e f e r r e d to as a lik e lih o o d fu n c tio n
and i s denoted by p (y :a ) . T h ere fo re , by determ ining the value o f a
which maximizes th e lik e lih o o d fu n c tio n , the d e s ire d param eter va lue
can be o b ta in ed . T his i s r e f e r r e d to as a maximum lik e lih o o d e s tim a to r (MLE).
In g e n e ra l, i t i s n o t p o s s ib le to o b ta in th e lik e lih o o d fu n c tio n
f o r a continuous system model given by (4 .3 .1 ) and (4 .3 .2 ) s in c e th e
51
continuous observation gives r i s e to an in f in i te dimensional density
function . However, since the unknown param eters are u sua lly estim ated
by d ig i ta l computation using sampled observation data a t d isc re te time
in s ta n ts , the like lihood function needed can be obtained from the co r
responding d isc re te s to c h a s tic model
x ( i + 1) = F ( i+ l , i ;a ) x ( i ) + G ( i+ l , i ;a )u ( i) + H ( i+ l ,i )w ( i) ; E{xq } “ 0 ,
(4 .3 .4 )
y ( i + 1) = C(a)x(i+1) + v (i+ l) (4 .3 .5 )
These equations are obtained from (4 .3 .1 ) and (4 .3 .2 ) assuming th a t fo r
u ( t) - u (t^ ) = u ( i) = constantand
w(t) = w(t^) = w(i] - constan t.00
Here, {w (i), v(i)}^_^ a re gaussian white no ise sequences with s t a t i s t i
cal inform ations s im ila r to those o f the continuous case , v i z . ,
E{w(i)} = 0, E{w(i)wT( j ) } = Q ( i)« ( i- j ) ,
where Q(i) i s a p o s itiv e sem idefin ite m atrix ,
E{v(i)} = 0, E{v(i)vT(j)} = R ( i)6 ( i- j ) ,
where R(i) i s a p o s it iv e d e f in ite m atrix ,
E{x } = 0, E{x xT} = P ,L o o o o
where P0 i s a p o s itiv e sem idefin ite m atrix , and w (i) , v ( i ) , xq a re a l l
assumed to be m utually independent.
Using the d isc re te system equations (4 .3 .4 ) and (4 .3 .5 ) , the
e x p lic i t expressions fo r the like lihood function w ithout and with input
noise w ill be considered in the follow ing two se c tio n s .
52
4 .3 .1 . Maximum Likelihood Estim ator: Without Input Noise
Assume w(i) - 0, fo r a l l i and l e t y ( i ) , i = 1,2 . . . N, be the
observed output da ta . Since v i s a gaussian white n o ise , using the
product ru le the lik e lih o o d function i s expressed as
NpCyN:oO = n p (y (i)|cO (4 .3 .6 )
i= l
where p (y ( i) | a) and y^ a re given by
p(y(i)ict) =Exp{-±[y(i) - C (c0 x (i)]T R_1( i ) [ y ( i ) - C (a)x (i)]}
y (27T)P| R-(i) [ (4 .3 .7 ) and
yN = [yTCD yT(2) . . . yT(N)]T . (4 .3 .8)
From (4 .3 .6 ) and (4 .3 .7 ) , i t i s seen th a t maximization o f p(y^:a) is
equ ivalen t to minim ization o f -Jin p (y ^ :a ) , thus an equivalent co st
func tiona l i s obtained by tak ing the n a tu ra l logarithm o f (4 .3 .6 ) , i . e . ,
I NNp£n27r + I {*n[R (i)| + [y (i) - C (a )x ( i) ]T R“X(i) EyCi)
i= l. (4 .3 .9 )
- C (a)x (i)]}
Since R(i) i s independent o f param eter v ec to r a , i t i s c le a r th a t mini
m ization o f (4 .3 .9 ) with respec t to ct is equ ivalen t to m inim ization of
NI [yC i ) - C (a )x ( i) ]T R- 1 ( i ) [y ( i ) - C (a )x (i)] . (4 .3 .10)
* i= l
Note th a t y ( i ) , i = 1,2 . . . N, are the observed output da ta ,
th e re fo re , the cost functional (4 .3 .10) i s the same as given by (4 .2 .3)
53
w ith (4 .2 .2 ) , except th a t th e weighting m atrix W i s now rep laced by the
maximum lik e lih o o d estim ato r in th is case degenerates to the weighted-
least-sq u ares e s tim ato r where the weighting m atrix i s determined by the
s t a t i s t i c a l a -p r io r i knowledge. Thus, h igher accuracy in the estim a
tio n re s u lts can be expected.
4 .3 .2 . Maximum Likelihood Estim ator: With Input Noise
Suppose the system in p u t no ise w(i) i s no t zero , and l e t
y ( i ) , i = 1,2 . . . N, be the observed output da ta . Using the product
ru le , the lik e lih o o d function can be expressed by
inverse covariance m atrix R~*(i) o f the output no ise v. T herefore, the
(4.3.11)
where
p (y d )!y o ;°0 = p (y d ) !<*) ■
Since y ( i ) , i = 1,2 . . . N, are jo in t ly gaussian , upon w ritin g
y ( i |i - l jo t ) = E{y(i) |y i _1;a}
= C(cx) E{x(i) |y i _1 ;a}
:= C(a) x ( i | i - l ; a ) (4.3.12)and
Py Ci | i - 1 = Cov{y(i) I y^_^
= C(a) C ov{x(i)iy ._1 ;a}CT(a) + R(i)
:= C(a) P ( i |i - l ;c O CT(a) + R ( i) , (4 .3 .13)
th e l ik e lih o o d fu n ctio n p(y^:a) becomes
54
N Exp{-%[y(i) - y ( i | i - l ; c O J T P ^ C i l i - l j a ) [y (i) - y ( i | i - l ; a ) ] }P(yN:a) = n ------------------------------------- y ‘
1=1 / C 2> r ) f | P y C i | i - l ; o ) l ( 4 . 3 . 1 4 )
Here, since independent gaussian inpu t and output no ises are assumed,
x ( i | i - l ; a ) and P ( i | i - l ; a ) are re sp ec tiv e ly th e optimal sin g le stage
p red ic tio n o f x (i) and the p red ic tio n e rro r covariance m atrix fo r a
given observation y^_^ and param eter a . Moreover, i t i s shown [82]
th a t they s a t i s f y , dropping a fo r n o ta tio n a l convenience,
x ( i | i - 1 ) = F ( i , i - l ) x ( i - l | i - 1 ) + G ( i , i - l ) u ( i - l ) (4 .3 .15)
P ( i j i - l ) = F ( i , i - l ) P ( i - l | i - 1 ) F T( i , i - l ) + H ( i ,i - l)Q ( i- l )H T( i , i - l ) .(4 .3 .16)
Here, x ( i - l | i - l ) and P ( i - 1 |i - 1 ) are re sp ec tiv e ly the optimal f i l t e r in g
estim ate o f x ( i - l ) and the f i l t e r in g e rro r covariance m atrix fo r given
y^_j and a , which s a t is f y the Kalman f i l t e r equation
x ( i+ l |i+ l) = x ( i+ l |i ) + K(i+1) (y (i+ l) - C (a )x ( i+ l |i ) ) (4 .3 .17)
with
K(i+1) = P ( i+ l |i )C T(a) (c (a )P ( i+ l |i )C T(a) + R (i+ l) )_1 (4 .3 .18)
P (i+ l|i+ 1 ) = (I - K (i+l)C(a)) P ( i+ l| i ) . (4 .3 .19)
Equations (4.3.15) through (4 .3 .19) c o n s titu te a recu rsive
formula to compute x ( i j i - l ;o t ) and P ( i | i - l ; a ) which are requ ired in
(4 .3 .12) and (4.3.13) to obtain y ( i | i - l ; a ) and P y ( i | i - l ; a ) used in the
lik e lih o o d function p(y^:a) in (4 .3 .1 4 ).
55
Taking -S-n p (yN:a) and d isc a rd in g th e term Np Jtn2ir r e s u l t in
NJ = h I {£n|P ( i | i - l ; c O [ + [y ( i) - y ( i | i - l ; a ) ]
i= l y
• P~1 C iI i— 1 [yCi) - y ( i | i - l ; a ) ] } . (4 .3 .20 )
M inim ization o f J in (4 .3 .2 0 ) r e la t iv e to a w ith c o n s tra in t equations
(4 .3 .12 ) through (4 .3 .13 ) and (4 .3 .15 ) through (4 .3 .1 9 ) w il l p rov ide
th e maximum lik e lih o o d e s tim a to r .
CHAPTER 5
SYNCHRONOUS MACHINE PARAMETER ESTIMATION
In t h i s c h ap te r , bo th th e w e ig h te d -le a s t-sq u a re s e s tim a to r and
maximum lik e lih o o d e s tim a to r a re used to determ ine th e IEEE s tan d ard
machine param eter d a ta fo r a s a l ie n t-p o le synchronous machine w ith
damper winding. The s ta n d a rd machine param eters a re e stim ated by ob
se rv in g the arm ature and f i e l d c u rre n ts during a sudden th ree -p h ase
s h o r t - c i r c u i t . This t e s t procedure has se v e ra l fav o rab le f e a tu re s ,
v i z . , wide i n i t i a l e stim ate range , f a s t e r convergence, r e l a t iv e ly sh o r t
o b serv a tio n p e r io d , and redundancy o f in p u t design.
In th e fo llow ing s e c t io n s , a sudden th ree -p h ase s h o r t - c i r c u i t
model o f a s a l ie n t-p o le synchronous machine w ith damper w inding w i l l
f i r s t be o b ta in ed . The w e ig h te d -le a s t-sq u a re s and maximum lik e lih o o d
param eter e s tim a to rs w il l th en be implemented by computer s im u la tio n
f o r various c o n d itio n s . F in a l ly , the e f f e c ts o f in p u t and load on th e
e s tim a tio n w il l be in v e s tig a te d .
5 .1 . Model a t Sudden Three-Phase S h o r t-C irc u it
Consider a s a l ie n t- p o le synchronous machine w ith damper w inding.
As shown in C hapter 2 , the d e te rm in is t ic machine equations w r i t te n in
s ta te - s p a c e form as req u ire d by equation (4 .2 .1 ) aTe
56
57
V " c / rd W Td Cd f /Td £l) 0 Vd
\ d Ck d /Tdo Ckd/Tdo W ’ So 0 0 0
= Cfd /Tio Cfkd/Tdo V Tio 0 0 + Vf»Xq
-0) 0 0 C /Tq q
C , /T qkq q vq
Xkq 0 0 0 C . /T" qkq' qo C /T" q qo 0
(S .1.1)
Herein, v^ and are the d,q components of the armature vo ltages and
represen ts the f i e ld vo ltage. The time constan ts Tj and T , and
the c and c '-c o n s ta n ts a re e x p lic i t ly expressed in standard param eters
as l is te d in (2 .1 .1 7 ), (2.1.18) and (2.1.23) o f Chapter 2.
A sudden s h o r t-c ir c u it i s taken from a s te a d y -s ta te no-load
cond ition , where the f i e ld curren t i ^ i s ad justed to obtain some desired
value o f the no-load armature vo ltage E. Thus, b e fo re the sh o rt
c i r c u i t ,
v , = 0, v = /3 E, V.J = v^ (5 .1 .2 )a q r x
and the constan t s ta te X° i s given by
A° = [Lmd i f = Lmd * f * ^ E Lf ^ 0 °JT * C5.1.3)
In (5 .1 .2 ) , v° is the f i e ld voltage corresponding to i^ , whereas L£ and
L ^ in (5 .1 .3 ) a re , re sp ec tiv e ly , th e s e l f inductance o f the f i e ld
winding and the mutual inductance between the th re e c irc u its in the
d -ax is . During s h o r t - c i r c u i t , the term inal cond itions are
58
Upon w riting
X* = X - X° , (5 .1 .5 )
the s ta te equation fo r the flux -linkage change X* during s h o r t-c ir c u it
can be w ritten as
11
cLdTd
CdkdTd
Cd fTd
U) 0
1CL
*t
■ 0
*kdCkddTdo
fkdTil
doCk fTir
do0 0 *kd 0
= Cfddo
CfkdT'!do
CfT'l do
0 0 + 0
•>*Aq -w 0 0
c_ aTq
^aia.T
q \ - 1/3 E
■
kq 0 0 0 Cqkqnnn
qo
cq
TTIqo
0
(5 .1 .6)
with = 0 * (5 .1 .7 )
I f th e three-phase s h o r t - c i r c u it cu rren t i a^ c > and thus i t s d ,q compo
nents i^ and i by applying P a rk 's transfo rm ation , and the f ie ld cur
re n t change i | = i ^ - i^ are considered as the observable ou tpu t, then
the output equations are described by
59
!■H
L
W0Cdxd
“oCdkdXd
“ oCd fXd
0 0
i*l f- cfd "Cfkd *Cf 0
r f Tdo r T1 f do r f Tdo
1o*
•HI
0 0 0u C
° qXq
0) c . o qkq
kd
1*Af
X*q
kq
(5 .1 .8)
where i s the field -w inding re s is ta n c e .
Equations (5 .1 .6 ) and (5 .1 .8 ) provide the de te rm in istic sudden
three-phase s h o r t - c i r c u i t model o f a s a l ie n t-p o le synchronous machine.
The s to c h a s tic model i s rea d ily obtained by adding input and output
n o ises;
1• K
I
CdTd
CdkdTd
CdfTd
0) 0
1<-<
1
0 W1
X*kdCkddT tr
do
c'kdMo
Cid fMo
0 0 Xkd 0 w2
X*A f - CfdT»Mo
CfkdTdo
cfT1Mo
0 0 xf + 0 +D W3
-03 0 0c_SLTq
C tqkqT
q-/3E W4
»
kq 0 0 0 cqfcq»rtf
qoT ff
qoXkq 0 w5
(5 .1 .9 )
60
*d“oCd
xdW0Cdkd
Xd“ oCd£
Xd
i* "cfd ~Cfkd "CfXf r T* r f *do r f Tdo r f Tdo
iq
0 0 003 C_°_3.
0
03 C , o qkq
*2
1*Akd
*f
i*kq
(5.1.10)
H erein, w^, i = 1,2 . . . 5, and v^, i = 1 ,2 ,3 , a re assumed to be zero-
mean gaussian white noises w ith known covariance m atrices Q and R,
re sp ec tiv e ly . The m atrix D can be determined from the inform ation as
to how the inpu t noises a f f e c t the system s ta te s .
5 .2 . Simulation Results
5 .2 .1 . Estim ation by W eighted-Least-Squares Method
The param eters to be id e n tif ie d are X£, X^, X^, X ^ , T£0 , T^q,
X J, X^, T1 , r &, r^ . Note th a t r^ in equations (5 .1 ,8 ) and (5.1.10)
i s a lso considered not known, th e re fo re , i t has to be estim ated to
gether with the standard param eters. Equations (5 .1 .6 ) , (5 .1 .7 ) and
(5 .1 .8 ) are now considered to correspond to the general model equations
(4 .2 .1 ) and (4 .2 .2 ) . In equation (5 .1 .6 ) , - /3 E i s regarded as an
inpu t with the abso lu te p e r-u n it value o f E equal to the p e r-u n it no-
load voltage o f th e generator before s h o r t - c i r c u i t , thus known from
the s h o r t - c i r c u it t e s t . The remaining equations needed to construct
the param eter estim ato r are obtained by applying equations (4 .2 .5 ) ,
61
(4 .2 .6 ) , (4 .2 .9) and (4.2.11) to equations (S .1.6) and (5 .1 .8 ) , where
3A(a)/3a, 3B(a)/3a and 3C(a)/3ct have to be evaluated.
A block diagram fo r the estim ation algorithm i s given in
Figure S .2 .1 , where the re a l output cu rren ts i ^ jCCA) and i^(A) a re
expressed in amperes. T herefore, these cu rren ts a re divided by th e i r
base values ig and i^g re sp ec tiv e ly , befo re they a re compared w ith the
corresponding p e r-u n it cu rren ts o f the model re fe rence . The base cur
ren t ig i s e a s ily computed from the machine ra tin g . Depending on the
choice o f p e r-u n it system fo r the ro to r q u a n ti t ie s , a base c u rren t i^g
can be obtained, e .g . , th a t value o f th e f ie ld cu rren t which produces
ra te d no-load vo ltage . I t i s noted, however, th a t i^g only a f fe c ts the
p e r-u n it value of r£ but none o f the o th e r param eters to be id e n tif ie d .
The param eter e stim ato r was app lied to a 120 kVA 208 V, 400 Hz
a i r c r a f t generator whose output s e n s i t iv i ty has been analyzed in Chap
te r 3. The s h o r t - c i r c u i t takes place a t un ity no-load vo ltage , E = 1,
which corresponds w ith a f ie ld voltage v^ o f 0.003637 and cu rren t i^
o f 0.088335 fo r the sim ulated generato r. The s h o r t - c i r c u i t c u rren tX X
^"abc ant* c u rren t i f i n p e r-u n it are displayed in Figure 5 .2 .2 .
The w eigh ted-least-squares param eter estim ation algorithm was
performed d ig i ta l ly with a sampling time o f 0.2 pu, whereas the t o ta l
number o f samples i s 600 corresponding to an observation time leng th of
120 pu (the Fortran computer program used i s l i s te d in Appendix 3 ).
The weighting m atrix was se lec te d equal to the id e n ti ty m atrix. Fur
thermore, a diagonal gain m atrix G(k) was chosen w ith the element
g^^(k) sp e c if ie d by
62
g i i Ck) = 1 i f |A a J / a*"1) < 0.25
gu (k) = 0.25 a^- 1 ^ |Aa^| i f l A a J / a * - 1 ! > 0 .2 5 .CS.2.1)
Several se ts o f i n i t i a l param eter estim ates were subsequently
t r ie d with values equal to 40%, 60%, 80%, 120%, 160% and 200% o f the
tru e values, and f in a l ly w ith values chosen a t random between 40% and
200% o f the tru e values. The re s u lts a re summarized in Table 5 .2 .1 .
The behavior o f successive estim ates fo r X'J as a function of the numberao f i te r a t io n s is displayed in Figure 5 .2 .3 , which i s ty p ic a l fo r the
e n tire se t o f param eters, except xdr whose behavior i s v isu a lized in
Figure 5 .2 .4 . Figure 5 .2 .5 shows the behavior fo r a l l param eter e s t i
mates with th e i r i n i t i a l values chosen a t random.
The re s u lts in d ic a te th a t the estim ates fo r a l l param eters,
except X ^ , converge to the tru e values in a few i te r a t io n s with le s s
than 1%, while i t takes some more i te r a t io n s fo r X ^ to converge. I t
i s a lso observed th a t the convergence i s f a s te r fo r i n i t i a l estim ates
w ith sm aller dev ia tions from the tru e va lues.
To in v es tig a te the e f fe c t o f system noises on the w eigh ted-least-
squares estim ato r, a zero-mean gaussian w hite noise w ith constant s tan -
dard dev ia tion o f 5% o f the s te a d y -s ta te value of i a^ c was added to the
output o f the sim ulated generator. Keeping a l l o th er conditions the
same as befo re , the standard machine param eters were estim ated and the
numerical r e s u lts a re summarized in Table 5 .2 .2 . The no ise-corrup ted
output and some ty p ic a l convergence behaviors o f param eter estim ates
a re shown in Figures 5 .2 .6 through 5 .2 .9 .
W) o
REALSYNCHRONOUS
MACHINE
ADJUSTER
-J3 E L J MODEL“ REFERENCE
Eqns 5 .1 .6 - 8
1. r1abc
•bdqo
TRANSFORMATIONdq
i f (A)
i f CA)
lfB
x d, x ds x ^ , x i t x q, x ;V T 1 T u T n a t 'do» 'do* 'qo
MACHINEPARAMETERESTIMATOR
E q n s4 .2 .8 - 9 4.2.12-14
Figure 5 .2 .1 Block Diagram For Machine Param eter Estim ation
64
/ V / V A A A A ^
i \ w w w v
1000
p e r-u n it timeFigure 5 .2 .2 Oscillogram o f D eterm inistic Output Current
a t Sudden Three-Phase S ho rt-C ircu it
0123456789101112131401234567891011121314
Table 5 . 2 , 1 WLSE Without N oise
Parameter E stim ates as F rac tio n o f True ValueA a A A A A A A A Ax3 X'
Ad xd xd*Til
do do X"q
Xq
Tilqo ra
2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 01 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 01 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 • 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1250 . 9 8 5 0 . 9 7 7 0 . 9 3 1 0 . 9 5 7 0 . 9 2 5 0 . 9 2 0 0 . 9 8 5 0 . 9 6 3 0 . 9 5 2 0 . 9 6 41 . 0 0 0 0 . 9 9 9 0 . 9 9 4 0 . 9 7 5 0 . 9 9 4 0 . 9 9 4 1 . 0 0 0 0 . 9 9 8 0 . 9 9 8 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 .0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 .0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 • 0 0 0 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 0
1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 01 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 00 . 9 6 1 0 . 9 3 6 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 6 2 0 . 9 0 1 0 . 9 0 0 0 . 9 5 90 . 9 9 8 0 . 9 9 6 0 . 9 8 7 0 . 8 7 8 1 . 0 0 9 0 . 9 8 8 0 . 9 9 8 0 . 9 9 0 0 . 9 8 9 0 . 9 9 81 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 3 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 • 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
Table 5 . 2 . 1 (Continued]Number of
I te r a t io n
Parameter E stim ates as F raction o f True ValueA
xaA
XA >< >
P* h o
A
TAoA
X"QA
Xq
A'Pll*qo
A
r aA
r f0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 01 0 . 9 6 1 0 . 9 3 6 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 6 2 0 . 9 0 1 0 . 9 0 0 0 . 9 S 9 0 . 9 8 22 0 . 9 9 8 0 . 9 9 6 0 . 9 8 8 0 . 8 5 0 t . 0 1 2 0 . 9 9 0 0 . 9 9 8 0 . 9 9 0 0 . 9 8 9 0 . 9 9 8 1 . 0 0 33 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 6 0 . 9 9 9 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 04 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 05 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 06 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 07 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 08 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 09 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
10 1 . 0 0 0 l . O G O 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 011 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
12 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 013 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
14 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 6 0 0 0 . 8 0 0 0 . 8 0 0
1 0 . 9 6 0 0 . 9 4 9 0 . 8 9 6 0 . 6 0 0 0 . 8 9 8 0 . 8 8 7 0 . 9 6 0 0 . 9 2 2 0 . 8 9 6 0 . 9 6 0 0 . 9 7 3
2 0 . 9 9 8 0 . 9 9 4 0 . 9 8 4 0 .7 5 0 0 . 9 8 4 0 . 9 8 6 0 . 9 9 8 0 . 9 9 0 0 . 9 8 6 0 . 9 9 8 1 . 0 0 1
3 1 . 0 0 0 1 .000 1 .0 0 0 0 . 9 3 7 0 . 9 9 9 1 . 0 0 1 1 . 0 0 0 1 .0 0 0 1 . 0 0 0 1 .0 0 0 1 . 0 0 0
4 1 .0 0 0 1 .000 1 .0 0 0 0 . 9 9 9 1 . 0 0 0 1 .0 0 0 I .0 0 0 1 . 0 0 0 1 . 0 0 0 1 .0 0 0 1 . 0 0 0
5 1 .0 0 0 1 .0 0 0 1 .0 0 0 1 . 0 0 0 1 .0 0 0 1 .0 0 0 1 .0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
6 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
7 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
8 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 ic O O O 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
9 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
10 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
11 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
12 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
13 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
14 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
Table 5 . 2 . 1 (Continued)Number o f
I t e r a t i o n
P a r a m e t e r E s t i m a t e s a s F r a c t i o n o f T r u e V a l u e
X"Ax d
Ax d*
AT"
do
ATd o
AX"
qAX
q T"qo
ra
ATf
0 0 * 6 0 0 0 * 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 01 0 . 7 5 0 0 . 7 5 0 0 . 6 9 3 0 . 4 5 0 0 . 7 4 5 0 . 6 7 2 0 . 7 5 0 0 . 7 2 5 0 . 6 3 3 0 . 7 5 0 0 . 7 5 02 0 . 9 3 7 0 . 9 1 5 0 . 8 2 0 0 . 3 3 7 0 . 8 5 6 0 . 8 1 4 0 . 9 3 7 0 . 8 3 1 0 . 7 4 2 0 . 9 3 7 0 . 9 3 73 0 . 9 9 6 0 . 9 8 6 0 . 9 5 9 0 . 4 2 2 0 . 9 8 0 0 . 9 6 6 0 . 9 9 6 0 . 9 5 6 0 . 9 2 8 0 . 9 9 6 1 . 0 0 24 1 . 0 0 0 0 . 9 9 9 0 . 9 9 8 0 . 5 2 7 0 . 9 9 8 1 . 0 0 1 1 . 0 0 0 0 . 9 9 7 0 . 9 9 6 1 . 0 0 0 0 . 9 9 8
S 1 . 0 0 0 1 . 0 0 0 1 . 0 0 1 0 . 6 5 9 1 . 0 0 0 1 . 0 0 3 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
6 1 . 0 0 0 1 . 0 0 0 1 . 0 0 1 0 . 8 2 4 1 . 0 0 1 1 . 0 0 2 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 07 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 6 0 . 9 9 9 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 08 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 • 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
9 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 010 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 011 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 012 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 013 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 014 I . 0 0 0 1 . 0 0 0 1 • 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
0 ' 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 ’ 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0
1 0 . 5 0 0 0 . 5 0 0 0 . 4 7 5 0 . 3 0 0 0 . 5 0 0 0 . 4 3 0 0 . 5 0 0 0 . 4 7 1 0 . 3 1 3 0 . 5 0 0 0 . 5 0 0
2 0 . 6 2 5 0 . 6 2 5 0 . 5 4 1 0 . 2 2 5 0 . 6 2 5 0 . 5 0 6 0 . 6 2 5 0 . 4 5 7 0 . 2 3 7 0 . 6 2 5 0 . 6 2 5
3 0 . 7 8 1 0 . 7 8 1 0 . 6 6 2 0 . 1 6 9 0 . 7 8 1 0 . 6 3 3 0 . 7 8 1 0 . 5 1 2 0 . 2 7 6 0 . 7 8 1 0 . 7 8 1
4 0 . 9 5 4 0 . 9 2 7 0 . 8 1 7 0 . 1 2 7 0 . 8 8 1 0 . 7 9 1 0 . 9 5 6 0 . 6 4 0 0 . 3 4 6 0 . 9 5 0 0 . 9 7 7
5 1 . 0 0 0 0 . 9 9 0 0 . 9 6 1 0 . 1 5 8 1 . 0 2 9 0 . 9 6 8 1 . 0 0 1 0 . 7 7 0 0 . 4 3 2 0 . 9 9 5 1 . 0 0 8
6 1 . 0 0 2 1 . 0 0 0 1 . 0 1 0 0 . 1 9 8 1 • 0 3 5 1 . 0 1 9 1 . 0 0 1 0 . 8 3 9 0 . 5 4 0 0 . 9 9 8 0 . 9 9 9
7 1 . 0 0 1 0 . 9 9 9 1 . 0 0 8 0 . 2 4 7 1 . 0 1 6 1 . 0 1 5 1 . 0 0 0 0 . 8 9 6 0 . 6 7 5 0 . 9 9 9 0 . 9 9 9
8 1 . 0 0 0 0 . 9 9 9 1 . 0 0 4 0 . 3 0 9 1 . 0 0 4 1 . 0 0 9 1 . 0 0 0 0 . 9 5 0 0 . 8 4 4 0 . 9 9 9 0 . 9 9 9
9 1 . 0 0 0 0 . 9 9 9 1 . 0 0 1 0 . 3 8 6 0 . 9 9 9 1 . 0 0 4 1 . 0 0 0 0 . 9 8 9 0 . 9 7 8 1 . 0 0 0 0 . 9 9 9
10 1 . 0 0 0 0 . 9 9 9 1 . 0 0 0 0 . 4 8 3 0 . 9 9 9 1 . 0 0 2 1 . 0 0 0 1 . 0 0 0 0 . 9 9 9 1 . 0 0 0 0 . 9 9 9
11 1 . 0 0 0 0 . 9 9 9 1 . 0 0 1 0 . 6 0 3 1 . 0 0 0 1 . 0 0 3 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
12 1 . 0 0 0 1 . 0 0 0 1 . 0 0 2 0 . 7 5 4 1 . 0 0 2 1 . 0 0 4 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
13 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 0 . 9 4 3 1 . 0 0 0 1 . 0 0 1 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
14 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 0 . 9 9 9 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
Table 5 . 2 . 1 (Continued)Number o f
I te ra t io n
Parameter Estim ates as F raction of True ValueA
Xd X*(1A
XdA
xd* *doA
doA
Xq Xq
<T>Hqo
Ar a
Ar a
0 1 * 6 0 0 1 . 4 0 0 1 . 2 0 0 1 . 8 0 0 0 . 5 0 0 0 . 7 0 0 0 . 6 0 0 0 . 8 0 0 0 . 4 0 0 2 . 0 0 0 0 * 9 0 01 1* 2 0 0 1 . 0 5 0 0 . 9 0 0 2 . 2 5 0 0 . 3 7 5 0 . 8 2 8 0 . 7 5 0 0 . 9 6 9 0 . 5 0 0 1 . 5 0 0 0 . 6 7 52 0 * 0 4 8 0 . 9 3 9 0 . 9 3 1 2 . 8 1 2 0 . 4 6 9 0 . 9 0 8 0 . 9 3 7 0 . 8 1 7 0 . 6 0 9 1 . 1 2 5 0 . 8 0 03 1 . 0 0 4 0 . 9 8 6 0 . 9 4 1 2 . 4 3 3 0 . 5 8 6 0 . 8 9 0 1 * 0 0 1 0 . 9 1 8 0 . 7 6 1 0 . 9 9 5 0 . 9 5 04 I . 0 0 1 0 . 9 8 6 0 . 9 6 6 1 • 8 2 5 0 . 7 3 2 0 . 9 4 6 1 . 0 0 0 0 . 9 7 5 0 . 9 5 0 1 . 0 0 0 0 . 9 9 3S 1 . 0 0 0 0 . 9 9 4 0 . 9 8 9 1 . 3 6 8 0 . 9 1 6 0 . 9 8 2 1 . 0 0 0 0 . 9 9 9 0 . 9 9 8 1 . 0 0 0 0 . 9 9 86 1 * 0 0 0 1 . 0 0 0 0 . 9 9 8 1 . 0 5 1 0 . 9 9 2 0 . 9 9 6 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 * 0 0 07 1 * 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 2 I . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 08 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 00 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 09 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 l.COO 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 010 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . ooo 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 I . 0 0 011 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . o o o 1 . 0 0 0 1 . 0 0 012 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 013 1 . 0 0 0 l . O C O 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 014 1 . 0 0 0 1 . 0 0 0 I . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0
69
• r t
1 . 6
0.8
0.40 2 4 6 8 10 12 14
Number o f I te r a t io n s
F igure 5 .2 .3 R atio o f E stim ate to True Value fo r XJj w ithou t Noise (WLSE)
2 . 0 q -o
0)+JcdE■H•Pto
Ui
0.4
0.00 2 4 6 108 12 14
Number o f I te r a t io n s
F ig u re 5 .2 .4 R atio o f E stim ate to True Value fo r w ithout Noise (WLSE)
Estim
ate
Rat
io
70
3 .0
2 . 0
1.8
1 . 6
1.4X"
1 .2X’
1.0
0 .8 dodoT",qo
0 . 6
0 .4
0 . 2Number o f I te r a t io n s
F igu re 5 . 2 . 5 R atio o f E stim ate t o True Value fo rRandom I n i t i a l E stim a te w ith ou t N o ise (WLSE)
71
^ y / V v W V \ /
-10 -
. r
5
0
-5. ri c10
5
0
-5
. r
0.5-
10001200p e r -u n i t time
F igure 5 .2 .6 O scillogtam o f N oise-C orrupted Output Currenta t Sudden Three-Phase S h o r t-C irc u it
4) 1 \ / \ / \ / \ / \ / \ / \
012345678910111213
i !012345678910111213
Table S . 2 . 2 WLSE With Output N oiseParam eter Estim ates as F raction o f True Value~ ~ 1 a — ^ ■ ■ ■ ,.
*dX1x d xd dJt
TMdo Ti o X"
q xq•TH
qo r a2 . C 0 C 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 ' 2 . 0 0 01 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 01 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 50 . 9 8 5 0 . 9 6 8 0 . 9 4 6 0 . 9 3 3 0 . 8 7 6 0 . 9 4 5 0 . 9 8 4 0 . 9 4 3 0 . 9 2 8 0 . 9 8 51 . C 0 C 0 . 9 9 3 0 . 9 9 4 1 . 1 0 2 0 . 9 5 2 0 . 9 9 6 0 . 9 9 9 0 . 9 7 8 0 . 9 7 3 1 . 0 0 11 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 • 131 0 . 9 6 0 0 * 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . C 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . C 0 C 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . c o o 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . C 0 C C . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . C O O 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1; i . 6 o o 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 01 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 00 . 9 6 1 0 . 9 3 4 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 6 1 0 . 9 0 0 0 . 9 0 0 0 . 9 6 10 . 9 9 8 0 . 9 9 2 0 . 9 8 5 0 . 9 6 6 0 . 9 7 4 0 . 9 8 7 0 . 9 9 7 0 . 9 7 3 0 . 9 6 9 0 . 9 9 9l . C O C 0 . 9 9 6 0 . 9 9 7 1 • 1 3 3 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 11 . 0 0 0 0 * 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . C O O 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . C O O 0 . 9 9 6 0 . 9 9 7 1 • 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . C O O 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . C O O 0 . 9 9 6 0 . 9 9 7 1 • 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . C O O 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 11 .COO 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1I . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 • 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1l . C O O 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 « 0 0 |
0
Table 5 . 2 . 2 (Continued]Number o f
I t e r a t i o n
P a r a m e t e r E s t i m a t e s a s F r a c t i o n o f T r u e V a l u e
sro<
X
< X A
xdA
haA
T i ll d o do
A
X"q
A
x q
A
11qo
A
r a
A
r f
0 1 * 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 01 0 . 9 6 1 0 . 9 3 4 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 6 1 0 . 9 0 0 0 . 9 0 0 0 . 9 6 1 0 . 9 7 92 0 . 9 9 8 0 . 9 9 2 0 . 9 8 5 0 . 9 6 7 0 . 9 7 4 0 . 9 8 7 0 . 9 9 7 0 . 9 7 3 0 . 9 6 9 0 . 9 9 9 1 . 0 0 23 l . C O C 0 . 9 9 6 0 . 9 9 7 1 . 1 3 3 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 9 9 84 l . C O C 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 85 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 • 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 86 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 87 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 88 1 . C 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 89 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 81 0 l .C O O 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 81 1 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 81 2 1 . 0 0 0 C . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 813 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 814 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 * 9 7 6 1 . 0 0 1 0 . 9 9 8
0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 6 0 0 0 . 8 0 01 0 . 9 6 0 C. 9 4 9 0 . 9 2 2 0 . 6 0 0 0 . 9 2 7 0 . 9 3 1 0 . 9 5 9 0 . 9 1 3 0 . 8 8 6 0 . 9 6 0 0 . 9 7 62 0 . 9 9 8 0 . 9 9 2 0 . 9 9 1 0 . 7 5 0 0 . 9 5 8 0 . 9 9 5 0 . 9 9 7 0 . 9 7 3 0 . 9 6 6 0 . 9 9 9 0 . 9 9 93 l . C O C 0 . 9 9 5 0 . 9 9 7 0 . 9 3 7 0 . 9 5 9 0 . 9 9 8 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 9 9 84 1 .CCC C . 9 S 5 0 * 9 9 8 1 . 1 2 9 0 . 9 5 9 0 . 9 9 8 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 9 9 85 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 2 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . C 0 1 0 . 9 9 86 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 87 l .C O O 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 * 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 88 l . C O C 0 . 9 9 6 0 . 9 9 7 1 • 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 89 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 810 t . a o o 0 . 9 9 6 0 . 9 9 7 1 • 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 * 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 81 1 l . C O O 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 81 2 l . C O C 0 . 5 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 813 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 * 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 814 l . C O O 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 8
Table 5 . 2 . 2 (Continued)Number o f I te r a t io n
Parameter Estim ates as F rac tio n o f True Value
xd*X' Ad
4xd
I< X A
Tildo
A
T'do
AX"
q
A
xq
A
Jitqo
A
r aA
r f0 0 . 6 0 C 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 6 0 0 0 . 6 0 0 0 . 6 0 01 0 * 7 5 0 C . 7 5 0 0 . 7 4 9 0 . 4 5 0 0 . 7 5 0 0 . 7 5 0 0 . 7 5 0 0 . 7 2 8 0 6 3 7 0 . 7 5 0 0 . 7 5 02 0 . 9 3 7 0 . 9 2 0 0 . 8 8 7 0 . 3 3 7 0 . 9 1 5 0 . 9 0 6 0 . 9 3 7 0 . 8 3 2 0 7 4 5 0 . 9 3 7 0 . 9 3 73 0 . 9 9 6 0 . 9 6 7 0 . 9 8 3 0 . 4 2 2 0 . 9 6 4 0 . 9 9 3 0 . 9 9 5 0 . 9 4 6 0 9 19 0 . 9 9 7 0 . 9 9 84 l . C G C 0 . 9 9 5 0 . 9 9 7 0 . 5 2 7 0 . 9 5 7 0 . 9 9 9 0 . 9 9 9 0 . 9 7 8 0 9 7 3 1 . 0 0 1 0 . 9 9 75 1 . G 0 0 0 . 9 9 5 0 . 9 9 7 0 . 6 5 9 0 . 9 5 7 0 . 9 9 9 0 . 9 9 9 0 . 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 86 1 . 0 0 0 0 . 9 9 5 0 . 9 9 7 0 . 8 2 4 0 . 9 5 8 0 . 9 9 8 0 . 9 9 9 0 . 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 87 l . C O O 0 . 9 9 5 0 . 9 9 8 1 . 0 3 0 0 . 9 6 0 0 . 9 9 8 0 . 9 9 9 0 . 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 88 l . C O C 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 6 1 . 0 0 1 0 . 9 9 89 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 6 1 . 0 0 1 0 . 9 9 810 l . C O C 0 . 9 96 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 6 1 . 0 0 1 0 . 9 9 811 l . C O C C . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 6 1 . 0 0 1 0 . 9 9 812 l . C O C 0 . 9 9 6 0 . 9 9 7 1 . 1 3 1 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 6 1 . 0 0 1 0 . 9 9 813 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 6 1 . 0 0 1 0 . 9 9 814 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 . 131 0 . 9 6 0 0 * 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 6 1 . 0 0 1 0 . 9 9 8
0 !o. 4 oo 0 .4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 4 0 0 0 . 4 0 0 0 .4 0 01 0 . 5 0 0 0 . 5 C 0 0 . 5 0 0 0 . 3 0 0 0 . 5 0 0 0 . 5 0 0 0 .5 0 0 0 . 4 7 8 0 3 2 1 0 . 5 0 0 0 . 5 0 02 0 . 6 2 5 0 . 6 2 5 0 . 6 2 5 0 . 2 2 5 0 . 6 2 5 0 . 6 2 5 0 . 6 2 5 0 . 4 8 3 0 2 6 0 0 . 6 2 5 0 , 6 2 53 0 . 7 8 1 C . 7 8 1 0 . 7 7 6 0 . 1 6 9 0 . 7 8 1 0 . 7 8 1 0 . 7 8 1 0 . 5 3 6 0 3 0 1 0 . 7 8 1 0 . 7 8 14 0 . 9 5 5 0 . 9 3 7 0 . 9 1 2 0 . 1 2 7 0 . 9 3 4 0 . 9 2 9 0 . 9 5 5 0 . 6 6 6 0 3 7 6 0 . 9 5 1 0 . 9 7 75 1 . C 0 0 0 . 9 9 0 0 . 9 9 2 0 . 1 5 8 0 . 9 8 3 1 . 0 0 1 0 . 9 9 9 0 . 7 9 7 0 4 7 0 0 . 9 9 7 0 . 9 9 86 1 . 0 0 1 0 . 9 9 4 1 . 0 0 0 0 . 1 9 8 0 . 9 7 6 1 . 0 0 5 1 . 0 0 0 0 . 8 5 9 0 5 8 7 0 . 9 9 9 0 . 5 9 67 I . 0 0 1 0 . 9 9 4 0 . 9 9 9 0 . 2 4 7 0 . 9 6 4 1 . 0 0 4 0 . 9 9 9 0 . 9 1 1 0 7 3 4 1 .COC 0 , 9 9 78 1 . 0 0 0 0 . 9 9 4 0 . 9 9 8 0 . 3 0 9 0 . 9 5 7 1 . 0 0 2 0 . 9 9 8 0 . 9 5 5 0 9 1 7 1 . 0 0 1 0 . 9 9 79 l . C O O 0 . 9 9 4 0 . 9 9 7 0 . 3 8 6 0 . 9 5 6 1 . 0 0 1 0 . 9 9 9 0 . 9 7 9 0 9 7 3 1 . 0 0 1 0 . 9 9 710 l .C O O 0 . 9 9 4 0 . 9 9 7 0 . 4 8 3 0 . 9 5 6 1 . 0 0 0 0 . 9 9 9 0 . 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 711 1.CC0 0 . 9 9 5 0 . 9 9 7 0 . 6 0 3 0 . 9 5 7 0 . 9 9 9 0 . 9 9 9 0 . 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 712 1 . 0 0 0 0 . 9 9 5 0 . 9 9 7 0 . 7 5 4 0 . 9 5 7 0 . 9 9 9 0 . 9 9 9 0 . 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 813 1 . 0 0 0 0 . 9 9 5 0 . 9 9 8 0 . 9 4 3 0 . 9 5 9 0 . 9 9 8 0 . 9 9 9 0 * 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 814 l . C O O 0 . 9 9 5 0 . 9 9 8 1 . 1 3 0 0 . 9 5 9 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 9 7 5 1 . 0 0 1 0 . 9 9 8
Table 5 . 2 . 2 (Continued)Number of I te ra t io n
Parameter Estim ates as F raction o f True ValueX"Ad X'
xdA
xd
A A
T i ldo
A
doA
X"q
A
Xq
ynqo
A
r a r a0 1 . 6 0 0 1 . 4 00 1 . 2 0 0 1 8 0 0 o . s o o 0 . 7 0 0 0 . 6 0 0 0 . 8 0 0 0 . 4 0 0 2 . 0 0 0 0 . 9 0 01 I . 2 0 0 1 . 0 5 0 0 . 9 0 0 2 2 5 0 0 . 3 7 5 0 . 8 7 5 0 . 7 5 0 0 . 9 7 4 0 . 5 0 0 1 . 5 0 0 0 . 6 7 52 0 . 9 4 9 0 . 9 5 0 0 . 9 5 9 2 8 1 2 0 . 4 6 9 0 . 9 4 3 0 . 9 3 7 0 . 8 2 0 0 . 6 1 1 1 . 1 2 5 0 . 8 1 43 1 . 0 0 5 C . 9 8 6 0 . 9 7 2 2 3 6 0 0 . 5 8 6 0 . 9 3 6 1 . 0 0 0 0 . 9 1 2 0 . 7 6 3 0 . 9 9 6 0 . 9 5 44 1 . 0 0 1 0 . 9 8 4 0 . 9 8 3 1 7 7 0 0 . 7 3 2 0 . 9 7 3 0 . 9 9 9 0 . 9 6 3 0 . 9 3 9 1 . 0 0 1 0 . 9 9 2S l . C O C 0 . 9 9 1 0 . 9 9 3 1 3 2 7 0 . 9 0 7 0 . 9 8 9 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 9 9 76 l . C O C 0 . 9 9 5 0 . 9 9 7 1 1 4 7 0 . 9 5 7 0 . 9 9 6 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 87 1 .COO 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 88 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 89 l . C O O 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 810 l .C O O 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 811 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 812 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 813 l .C O O 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 814 1 . 0 0 0 0 . 9 9 6 0 . 9 9 7 1 131 0 . 9 6 0 0 . 9 9 7 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 8
in
76
o• HPcdas<upcd6
■ H
0. 8
0.4
0.020 4 6 8 10 12 14
Number o f I te ra tio n s
Figure 5 .2 .7 Ratio of Estim ate to True Value fo r X" w ith Output Noise (WLSE) d
0)pcdS• H PWuq
0.4
Number o f I te ra tio n s
Figure 5 . 2 . 8 Ratio o f Estim ate to True Value fo r X, w ith Output N oise(WLSE)
77
X"
o -
0.8do
qo0.6
0.4
Number of I te ra tio n s
Figure 5 . 2 . 9 R atio o f Estim ate to True Value fo rRandom I n i t ia l E stim ate w ith Output N oise (WLSE)
78
From the numerical r e s u l ts , the following i s observed:
(1) The convergence ra te i s about the same as in the determ in is
t i c case.
(2) The estim ates fo r param eters X' , X^, X^, T^q , X” , r a and r^.
converge to the tru e values in a few i te ra t io n s w ith le s s than 1% e rro r .
(3^ The estim ates fo r X ^ , T'^q , X and T”q converge to b iased
values re su ltin g in e rro rs between 2% and 13% with X ^ showing the
la rg e s t e rro r .
For fu r th e r in v e s tig a tio n o f the noise e f f e c t , an input noise as
w ell as output noise was added to the sim ulated generato r. The input
no ise was generated to be a zero-mean gaussian white process with s ta n
dard deviation o f 1% of th e s te a d y -s ta te value of th e corresponding
s ta te v a ria b le s . An id e n ti ty c o e f f ic ie n t m atrix fo r th e input noise
was used. Keeping a l l o ther conditions the same as in the determ inis
t i c case , the standard machine param eters were estim ated fo r the ran
domly chosen i n i t i a l e s tim ates . The sim ulation r e s u l ts a re given in
Table 5 .2 .3 and Figure 5 .2 .1 0 .
These r e s u l ts show th a t
(1) The convergence ra te i s about the same as in the determ in is
t i c case,
(2) The estim ates fo r param eters XJJ, X^, X|j, r^ and r^ converge
to the tru e values in a few i te ra t io n s w ith less than 1% e rro r ,
(3) The estim ation e rro rs of the parameters X^, X ^ , T”q, T^o,
X and T^q are between 2% and 50% w ith X ^ having the la rg e s t e rro r .
In conclusion, the numerical re s u l ts in th is sec tion show th a t the
w eigh ted-least-squares e s tim ato r is not adequate i f th e system is sub
je c t to s ig n if ic a n t environmental d istu rbances.
Table 5 . 2 . 3 WLSE With Input and Output N oisesNumber o f
Iteration
Parameter Estimates as Fraction o f True ValueAX"xd
AX 1xd
A
xd d£
AT M1
do
A
Tdo
AX"
q
AX
q
A'J’Tf
qo
Ara ra
0 1 .600 1.4C0 1 . 2 0 0 1 . 8 0 0 0 . 5 0 0 0 . 7 0 0 0 . 6 0 0 0 . 8 0 0 0 . 400 2.CG0 0 .9001 1 .200 1 .0 5 0 0 . 9 0 0 2 .2 S 0 0 . 3 7 5 0 . 8 7 5 0 . 7 5 0 0 . 9 7 7 0.5C0 1.500 0 . 6 7 52 0 .9 4 8 0 . 9 5 0 0 . 9 4 6 2 .8 1 2 0 . 4 6 9 0 . 9 3 2 0 . 9 3 7 0 . 8 2 0 0 . 6 1 0 1. 125 0.810-3 1.004 0 . 986 0 . 9 5 6 2 . 384 0 .5 8 6 0 . 9 a 4 1 .000 0 . 9 1 3 0 . 7 6 3 0 . 9 9 6 0 .9484 l.COC 0 . 983 0 .9 6 5 1 . 7 8 8 0 .7 3 2 0 . 9 5 6 0 . 9 9 8 0 .964 0 . 938 1 .C01 0 . 9 8 65 0 . 9 9 9 0 . 9 8 8 0 . 9 7 0 1 . 5 2 5 0 . 8 4 9 0 . 9 6 5 0 . 9 9 9 0 . 9 8 0 0 .974 1.001 0 .9 8 96 0 . 9 9 9 0 . 990 0 .971 1 .4 9 6 0 . 8 6 9 0 . 9 6 6 0 . 9 9 9 0 . 9 8 1 0 .974 1 .001 0 .9 8 97 0 . 9 9 9 0.991 0 .971 1 . 4 9 5 0 . 8 7 0 0 . 9 6 S 0 . 9 9 9 0 .9 0 1 0 .974 1 .001 0 . 9 8 9S 0 . 9 9 9 0.991 0 .971 1 . 4 9 6 0 . 8 7 0 0 . 9 6 5 0 . 9 9 9 0 .9 8 1 0 .974 1 .001 0 . 9 8 99 0 . 9 9 9 0 .991 0 .971 1 . 4 9 6 0 . 8 7 0 0 . 9 6 5 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1.001 0 . 9 8 910 0 . 9 9 9 C.991 0 .971 1 . 496 0 .8 7 0 0 . 9 6 5 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1.C01 0 . 9 8 911 0 . 9 9 9 C.991 0 . 971 1 .496 0 .8 7 0 0 . 9 6 5 0 .9 9 9 0 . 9 8 1 0 . 9 7 4 1 .00 1 0 . 9 8 912 0 . 9 9 9 C.991 0 .971 1 . 496 0 . 8 7 0 0 . 9 6 5 0 . 9 9 9 0 . 9 8 1 0 .974 1.001 0 . 98913 0 . 9 9 9 0 .991 0*971 1 .4 9 6 0 . 8 7 0 0 . 9 6 5 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1.001 0 .9 8 914 0 . 9 9 9 0 .991 0 . 9 7 1 1 . 4 9 6 0 . 8 7 0 0 . 9 6 5 0 . 9 9 9 0 .9 8 1 0 .974 1 .001 0 . 9 8 9
'-jto
Estim
ate
Rat
io
80
X”
-a
do,T"
0.4
0 2 4 6 8Number o f I te ra tio n s
Figure 5.2 .10 Ratio o f Estim ate to True Value fo r Random I n i t i a l Estim ate with Input and Output Noises (WLSE)
81
5 .2 .2 . E stim ation by Maximum L ikelihood Method
Consider th e s to c h a s t ic model equations (5 .1 .9 ) and (5 .1 .1 0 ) o f
a s a l ie n t-p o le synchronous machine w ith damper w inding. Let th e ou tpu t
n o ise v be the same as th e one considered in th e p rev ious s e c t io n , i . e . ,
a zero-mean independent gaussian random process w ith standard d e v ia tio n
o f 5% o f the magnitude o f the f i e l d and arm ature c u rre n ts a t s tead y -
s t a t e . The in v erse o f the covariance m atrix R w i l l then be
-1
0.02205
0 3.845 0
0 0.02205
x 10 (5 .2 .2 )
Furtherm ore, i t i s assumed th a t th e inpu t n o ise i s n e g lig ib le . This
assum ption i s q u i te r e a l i s t i c , s in c e during th e s h o r t - c i r c u i t t r a n s ie n t
p e rio d th e magnitude o f the d e te rm in is t ic component o f th e s t a te
v a r ia b le s in c re a se s to se v e ra l tim es i t s s te a d y - s ta te va lu e , thus th e
no ise to s ig n a l r a t i o becomes extrem ely sm a ll. Keeping a l l o th e r con
d i t io n s th e same as w ith the w e ig h te d -le a s t-sq u a re s e s tim a to r excep t
th a t th e w eighting m atrix W i s rep laced by R~*, th e machine param eters
were e stim ated f o r various i n i t i a l e s tim a te v a lu es as shown in
Table 5 .2 .4 . The ty p ic a l convergence c h a r a c te r i s t i c s o f su ccessiv e
param eter e stim ates a re g ra p h ic a lly dep ic ted in F igures 5 .2 .11 through
5 .2 .1 3 .
From th ese s im u la tio n r e s u l t s , i t i s seen th a t
(1) The maximum lik e lih o o d e s tim a to r p rov ides about th e same
convergence r a te as th e w e ig h te d -le a s t-sq u a re s e s tim a to r ,
82
(2] The estim ation accuracy i s s ig n if ic a n t ly improved compared
w ith the w eighted-least-squares e s tim a to r, v i z . , a l l param eter estim ates
converge to the t ru e value in a few i te r a t io n s w ith less than 1% e rro r ,
except T! , X„ and T” which have an e r ro r between 1.5% and 2.5%. r do q qoThese figu res are to be compared w ith 2% and 17% fo r the weighted-
le a s t - squares estim ato r.
To in v e s tig a te the e ffe c t o f inpu t noise on the estim ation re
s u l t s , the same inpu t no ise used fo r th e w eigh ted -least-squares e s t i
mator was employed. The standard machine param eters were estim ated
w ith randomly chosen i n i t i a l v a lu es . The numerical re s u l ts a re l is te d
in Table 5 .2 .5 and the p lo ts o f the convergence c h a ra c te r is t ic s o f suc
cessive estim ates a re given in Figure 5 .2 .14 .
The numerical r e s u l ts in d ic a te th a t
(1) The maximum like lihood e s tim a to r w ith input noise provides
about the same convergence ra te as th e w eigh ted-least-squares estim ato r,
(23 The estim ation accuracy, however, i s much improved, v iz . ,
the param eter estim ation e rro rs are le s s than 1% except fo r X ^ , T^o,
X and T" whose e rro rs are between 1.1% and 4%. Note th a t in the q qocase o f the w eigh ted-least-squares e s tim ato r, the estim ation e rro rs o f
*d* ^dJl* ^do’ ^q an<* ^qo ranSe f Tom 2% to 50%.
In conclusion, th e re s u lts ob tained in th is sec tio n support the
su p e rio rity o f the maximum like lihood estim ato r over the weighted-
lea s t-sq u a re s estim ato r i f system n o ises are o f s ig n if ic a n c e .
Table 5 .2 .4 MLENumber of
I te ra t io n
Parameter Estim ates as Fraction o f True Valuew
Yrfxd
X'Ad
i< X A
x d*
A
*doA
T1do X"q
A
Xq
A
*^11
qoA
r aA
r f0 | 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0 2 . 0 0 0
1 1 . 5 0 C 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0 i . 5 0 0 1 . 5 0 0 1 . 5 0 0 1 . 5 0 0
2 1 • 1 2 5 1 . 1 2 5 1 • 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 . 1 2 5 1 • 1 2 5 1 . 1 2 5
3 C . 9 8 4 0 . 9 7 0 0 . 9 6 2 0 . 8 4 4 0 . 9 0 3 0 . 9 7 1 0 . 9 8 4 0 . 9 4 2 0 . 9 2 5 0 . 9 8 6 0 . 9 9 3
4 1 . 0 0 0 0 . 9 9 6 1 . 0 0 1 0 . 9 9 9 0 . 9 8 0 1 . 0 0 5 0 . 9 9 8 0 . 9 7 8 0 . 9 7 3 1 . 0 0 1 1 . 0 0 0
5 l . C O C 0 . 9 9 8 1 . 0 0 2 1 . 0 1 0 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
6 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
7 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
3 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 S 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
9 i . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
10 1 . 0 C 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
11 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
12 1 . 0 0 0 0 . 9 9 8 I . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
13 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
14 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
0 |1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0 1 . 6 0 0
1 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0
2 0 . 9 6 0 0 . 9 2 6 0 . 9 0 0 0 . 9 C 0 0 . 9 0 0 0 . 9 1 4 0 . 9 6 0 0 . 9 0 0 0 . 9 0 0 0 . 9 6 2 0 . 9 8 5
3 0 . 9 9 6 C . 9 9 4 0 . 9 8 9 0 . 9 0 0 0 . 9 9 8 0 . 9 9 4 0 . 9 9 7 0 . 9 7 3 0 . 9 6 8 1 .COO 1 . 0 0 2
4 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 1 2 0 - 9 8 4 I . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 9 9 9
5 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
6 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
7 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
8 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
9 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 * 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
10 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
11 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
12 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
13 1 . c o o 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
14 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 • 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
Table 5 . 2 . 4 (Continued)Number o f
I te ra t io nParameter Estim ates as F raction o f True Value
*dA
Y »*d
A
xdA
dA {doA
TdoA
X"q
A
Xq
A
TnqoA
r aA
r f0 1 . 2 0 0 1 . 2 C 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 01 0 * 9 6 0 0 . 5 2 6 0 . 9 0 0 0 . 9 0 0 0 . 9 0 0 0 . 9 1 4 0 . 9 6 0 0 . 9 0 0 0 . 9 0 0 0 . 9 6 2 0 . 9 8 52 0 . 9 9 e 0 . 5 9 4 0 . 9 9 0 0 . 9 0 0 0 . 9 9 7 0 . 9 9 4 0 . 9 9 7 0 . 9 7 3 0 . 9 6 8 1 . 0 0 0 1 . 0 0 23 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 1 2 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 9 9 94 1 . 0 0 0 0 . 5 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9S 1 . 0 0 0 C . 5 9 3 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 5 96 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 97 l . C O O 0 . 5 5 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 98 1 . 0 0 0 0 . 9 5 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 99 1 . 0 0 0 0 . 9 5 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 * 0 0 1 0 . 9 9 910 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 * 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 911 1 . 0 0 0 0 . 5 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 912 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 913 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 . 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 914 1 . 0 0 0 0 . 5 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 01 0 . 9 5 8 0 . 9 3 4 0 . 9 2 0 0 . 6 0 0 0 . 7 7 9 0 . 9 3 3 0 . 9 5 8 0 . 9 1 2 0 . 8 8 0 0 . 9 6 1 0 . 9 7 4
2 0 . 9 9 8 0 . 9 8 9 0 . 5 9 5 0 . 7 5 0 0 . 9 5 5 1 . 0 0 7 0 . 9 9 7 0 . 9 7 2 0 . 9 6 4 1 .COO 1 . 0 0 03 l . C O C C . 9 9 7 1 . 0 0 2 0 . 9 3 7 0 . 9 8 4 1 . 0 0 6 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 5 9 9
4 1 . 0 0 0 0 . 5 9 8 1 . 0 0 2 1 . 0 1 0 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 95 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
6 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 - 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
7 l . C O C 0 . 5 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
8 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 99 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 I . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
10 l . C O O 0 . 5 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
111 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 5 9 9
12 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
13 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
14 l . C O Q 0 . 9 5 8 1 . 0 0 2 1 . 0 0 9 0 . 9 Q 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
Table 5 . 2 . 4 (Continued)Number o f
I te r a t io n
Parameter Estim ates as F raction o f True Valuea
xdA
X1 xd
A
xdA
dAA
Tl*do
A
TVdoA
Xq
X > A
J l l
qo r aA
r f0 .0* 6 0 C 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 01 0 . 7 5 0 0 . 7 5 0 0 . 7 3 8 0 . 4 5 0 0 . 4 5 0 0 . 7 5 0 0 . 7 5 0 0 . 7 2 1 0 . 6 1 6 0 . 7 5 0 0 . 7 5 02 0 . 9 3 6 C . 9 C 4 0 . 8 7 4 0 . 3 3 7 0 . 5 2 7 0 . 8 8 3 0 . 9 3 S 0 . 8 2 0 0 . 7 1 7 0 . 9 3 7 0 . 9 3 73 0 . 9 9 0 0 . 9 6 9 0 . 9 8 9 0 . 4 2 2 0 . 6 5 9 1 . 0 1 6 0 . 9 9 4 0 . 9 4 0 0 . 8 9 7 0 . 9 9 7 1 . 0 0 04 1 . 0 0 1 0 . 9 8 5 1 . 0 1 4 0 . 5 2 7 0 . 8 2 4 1 . 0 3 6 0 . 9 9 9 0 . 9 7 8 0 . 9 7 3 I . 0 0 1 0 . 9 9 85 l . C O O 0 . 5 9 4 1 . 0 0 6 0 . 6 5 9 0 . 9 6 4 1 . 0 1 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
6 l . C O C 0 . 9 9 7 1 . 0 0 2 0 . 8 2 4 0 . 9 8 4 1 . 0 0 6 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 97 1 . 0 0 0 0 . 9 9 7 1 . 0 0 2 1 . 0 1 3 0 . 9 8 4 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 98 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 99 l . Q O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 * 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 910 l . C O C 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 911 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 912 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
13 1 . 0 0 0 0 . 9 5 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 914 l . C O O C . 5 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
0 0 . 4 Q 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 Q 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0
1 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 0 . 3 0 0 0 . 3 0 0 0 , 5 0 0 0 . 5 0 0 0 . 4 8 4 0 . 3 1 4 0 . 5 0 0 0 . 5 0 0
2 0 . 6 2 5 C . 6 2 5 0 . 6 0 7 0 . 2 2 5 0 . 2 2 5 0 . 5 6 1 0 . 6 2 5 0 . 4 7 2 0 . 2 3 6 0 . 6 2 5 0 . 6 2 5
3 C . 7 8 1 0 . 7 8 1 0 . 6 8 9 0 . 1 6 9 0 . 1 6 9 0 . 6 3 4 0 . 7 8 1 0 . 5 0 1 0 . 2 4 3 0 . 7 8 1 0 . 7 8 1
4 0 . 9 5 3 0 . 8 9 3 0 . 8 2 7 0 . 1 2 7 0 . 2 0 2 0 . 7 9 3 0 . 9 5 3 0 . 6 0 6 0 . 3 0 4 0 . 9 5 4 0 . 9 7 7
5 1 . 0 0 9 0 . 9 3 7 0 . 9 8 8 0 . 1 5 8 0 . 2 5 3 0 . 9 9 1 0 . 9 9 9 0 . 7 4 0 0 . 3 8 0 0 . 9 9 7 1 . 0 1 0
6 1 . 0 1 2 0 . 9 4 8 1 . 0 6 2 0 . 1 9 8 0 . 3 1 6 1 . 1 2 7 1 . 0 0 2 0 . 8 0 4 0 . 4 7 5 0 . 9 9 8 0 . 9 9 1
7 1 . 0 0 9 0 . 9 5 6 1 • 0 6 5 0 . 2 4 7 0 . 3 9 5 1 . 1 2 9 1 . 0 0 1 0 . 8 6 7 0 . 5 9 4 0 . 9 9 9 0 . 9 9 0
8 1 . 0 C 6 0 . 9 6 5 1 • 0 4 8 0 . 3 0 9 0 . 4 9 4 1 . 1 0 0 1 . 0 0 0 0 . 9 1 7 0 . 7 4 2 1 . 0 0 0 0 . 9 9 4
9 1 . 0 0 3 0 . 9 7 4 1 . 0 3 2 0 . 3 8 6 0 . 6 1 7 1 . 0 7 1 0 . 9 9 9 0 . 9 5 9 0 . 9 2 8 1 . 0 0 0 0 . 9 9 6
10 1 . 0 0 1 C . 9 e 3 1 . 0 1 9 0 . 4 8 3 0 . 7 7 1 1 . 0 4 4 0 . 9 9 9 0 . 9 8 0 0 . 9 7 7 1 . 0 0 1 0 . 9 9 7
11 1 . 0 0 0 0 . 9 9 1 1 . 0 0 8 0 . 6 0 3 0 . 9 5 0 I . 0 2 1 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 8
12 l . C C O 0 . 9 9 7 1 . 0 0 2 0 . 7 5 4 0 . 9 8 3 1 . 0 0 6 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
13 1 - c o o 0 . 9 9 8 1 . 0 0 2 0 . 9 4 3 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 5 1 . 0 0 1 0 . 9 9 9
14 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 1 0 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 . 0 . 9 9 9
Table 5 . 2 . 4 (Continued)Number o f
I t e r a t i o n
P a r a m e t e r E s t i m a t e s a s F r a c t i o n o f T r u e V a l u e
x > A
xd
A
Xd
A
Xd * ^ 0 do
A
X"q
A
Xq
A'J’l!
qor
a r f
0 1 . 6 0 0 1 . 4 0 0 1 . 2 0 0 1 . 8 0 0 0 . 5 0 0 0 . 7 0 0 0 . 6 0 0 0 . 8 0 0 0 . 4 0 0 2 . 0 0 0 0 . 9 0 01 1 . 2 0 0 1 . 0 5 0 0 . 9 0 0 1 * 3 5 0 0 . 3 7 5 0 . 8 7 5 0 . 7 5 0 0 . 9 9 0 0 . 5 0 0 1 . 5 0 0 1 . 0 9 62 0 . 9 6 2 0 . 9 5 9 1 . 0 2 3 1 . 0 1 2 0 . 4 6 9 1 . 0 5 4 0 . 9 3 7 0 . 8 1 3 0 . 6 2 0 1 . 1 2 5 1 . 0 0 43 1 . 0 0 7 0 . 9 7 4 1 . 0 2 7 1 . 0 9 4 0 . 5 8 6 I . 0 5 9 1 . 0 0 0 0 . 9 1 7 0 . 7 7 4 0 . 9 9 6 0 . 9 9 94 1 . 0 0 1 0 . 9 8 3 1 . 0 1 6 1 . 0 2 8 0 . 7 3 2 1 . 0 3 6 0 . 9 9 8 0 . 9 6 5 0 . 9 4 2 1 . 0 0 1 0 . 9 9 95 1 .COO 0 . 9 9 1 1 • 0 0 7 1 . 0 1 5 0 . 9 1 6 1 . 0 1 8 0 . 9 9 9 0 . 9 7 9 0 . 9 7 5 1 . 0 0 1 0 . 9 9 96 1 . 0 0 0 0 . 9 9 7 1 . 0 0 2 1 . 0 1 0 0 . 9 8 1 1 . 0 0 6 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 97 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 I . 0 0 1 0 . 9 9 98 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 99 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 910 I . c o o 0 . 9 9 8 1 • 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 911 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 912 1 . 0 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 913 l . C O O 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 . 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 5 914 1 . C 0 0 0 . 9 9 8 1 . 0 0 2 1 . 0 0 9 0 . 9 8 5 1 * 0 0 5 0 . 9 9 9 0 . 9 8 0 0 . 9 7 6 1 . 0 0 1 0 . 9 9 9
87
o■ H+Jcd£Ka+jcde•H4Jwin
0 . 8
0.4
20 4 6 8 1210 14Number o f I te ra tio n s
Figure 5.2.11 Ratio of Estim ate to True Value fo r X*,' with Output Noise (MLE) d
C D4JCd6■ H
0.4
2 60 4 8 10 12 14Number o f I te ra t io n s
Figure 5 .2 .12 Ratio o f Estim ate to True Value fo r X, with Output Noise (MLE)
88
doX” doqo0.6
0 2 4 6 8Number o f I te r a t io n s
F igure 5 . 2 . 1 3 R atio o f E stim ate to True V alue fo rRandom I n i t i a l E stim a te w ith Output N o ise (MLE)
Table 5 . 2 . 5 MLE With Input N oise
Number of
I te ra t io n
Parameter Estim ates as F raction o f True Valueyn<1 *4 x
»
l do TJ.doAX"
q
AX
qT"
q°
Ar a
Ar f
0 i . e o c 1 * 4 0 0 1 * 2 0 0 1 . 8 0 0 0 * 5 0 0 0 . 7 0 0 0 . 6 0 0 0 . 8 0 0 0 * 4 0 0 2 . 0 0 0 0 . 9 C 01 1 . 2 C C 1 . 0 6 0 0 * 9 0 0 1 . 3 5 0 0 . 3 7 5 0 . 8 7 5 0 . 7 5 0 0 - 9 9 3 0 . 5 0 0 1 . 5 0 0 1 . 1 1 62 0 * 9 6 3 0 * 9 6 4 1 * 0 1 9 1 * 0 1 2 0 * 4 6 9 1 . 0 5 8 0 . 9 3 7 0 . 8 1 2 0 . 6 1 9 1 * 1 2 5 1 . 0 3 23 1 * 0 0 7 0 * 9 7 8 1 . 0 2 5 0 . 5 3 9 0 . 5 8 6 1 * 0 6 5 1 . 0 0 1 0 . 9 1 8 0 . 7 7 4 0 . 5 9 5 0 . 5 5 24 1 . C 0 1 C . 9 8 6 1 * 0 1 3 0 * 8 8 0 0 . 7 3 2 1 . 0 4 4 0 . 9 9 8 0 . 9 6 6 0 . 9 4 3 1 . 0 0 1 0 . 5 9 45 l . C C C 0 * 9 9 5 1 . 0 0 4 0 . 8 6 8 0 * 9 1 6 1 . 0 2 4 0 . 9 9 9 0 . 9 8 0 0 . 9 7 4 1 . 0 0 1 0 . 9 9 46 1 * 0 0 0 1 . C C 2 0 . 9 9 7 0 . 8 6 2 1 . 0 0 1 1 . 0 1 0 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 9 9 47 l . C C C 1 . C C 3 0 . 9 9 6 0 . 8 6 1 1 * 0 1 1 1 . 0 0 7 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 9 5 48 l . C C C 1 . 0 0 3 0 . 9 9 6 0 . 8 6 1 l . O U 1 . 0 0 7 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 5 5 49 l . C O C 1 . C C 3 0 * 9 9 6 0 . 8 6 1 1 . 0 1 1 1 . 0 0 7 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 9 9 410 l . C O C 1 . C C 3 0 . 9 9 6 0 . 8 6 1 1 . 0 1 1 1 . 0 0 7 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 »C01 0 . 9 9 411 l . C C C 1 . C C 3 0 . 9 9 6 0 . 8 6 1 1 . 0 1 1 1 . 0 0 7 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 5 9 412 l . C C C 1 . C 0 3 0 . 9 9 6 0 . 8 6 1 1 . 0 1 1 1 * 0 0 7 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 5 9 413 1 . 0 0 0 1 . C C 3 0 . 9 9 6 0 . 8 6 1 1 . 0 1 1 1 * 0 0 7 0 . 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 5 9 414 l . C O O 1 . C C 3 0 . 9 9 6 0 . 8 6 1 1 * 0 1 1 1 . 0 0 7 0 * 9 9 9 0 . 9 8 1 0 . 9 7 4 1 . 0 0 1 0 . 5 9 4
90
o■H+JcdCrf<U+JcdE•H+JWW
do0.6
0.4
0 2 4 6Number o f I te ra tio n s
Figure 5 .2 .14 Ratio o f Estim ate to True Value fo r Random I n i t i a l Estimate w ith Input and Output Noises (MLE)
91
5 .3 . Estim ation with Step-Function F ield-V oltage Input
In the previous two se c tio n s ,th e machine param eters were e s t i
mated by observing the armature and f ie ld cu rren ts during a sudden
three-phase s h o r t - c i r c u i t . The se le c tio n o f th e b e s t t e s t procedure
has t a c i t ly been based on a -p r io r i knowledge o f the e le c t r ic a l tra n s ie n t
behavior o f a synchronous machine during s h o r t - c i r c u i t . In f a c t , i f the
param eter estim ation problem is s t r i c t l y considered from a con tro l sys
tems view point, i t seems to be n a tu ra l to e x c ite th e system with a
s tep -fu n c tio n f ie ld -v o lta g e w ith the generator term inals s h o r t-c irc u ite d
and estim ate the machine param eters by observing the ou tpu t. This ap
proach would appear to be equivalent to the sudden s h o r t - c i r c u i t case.
However, since the su b tran sien t behavior of the armature cu rren ts are
p ra c t ic a l ly not no ticeab le compared with th a t o f a sudden three-phase
s h o r t - c i r c u i t , la rg e r estim ation e rro r , slower convergence r a te o r even
divergence can be expected. This m atter is now fu r th e r in v es tig a ted .
Let the s ta te and output equations o f a sa lie n t-p o le synchronous
machine be re sp ec tiv e ly given by equations (5 .1 .1 ) and (2 .3 .5 ) .
I t is desired to determine the standard machine param eters by exc iting
the machine with a s tep -fu n c tio n f ie ld -v o lta g e and observing the f ie ld
and armature cu rren ts w ith s h o r t-c irc u ite d te rm in a ls . Here, the f ie ld -
voltage magnitude was chosen such th a t the magnitude of the re su ltin g
s te a d y -s ta te arm ature-currents i s the same as in the sudden three-phase
s h o r t- c ir c u it case w ith no-load vo ltage E. This condition makes the
re s u l ts o f th is sec tion comparable with those o f the sudden sh o rt-
c i r c u i t case, and is met by se le c tin g the magnitude o f the f ie ld -
voltage as determined by
Using equations (2 .3 .S 3 , (5.1.13 and (5.3,13 w ith a voltage
E = 1, the standard param eters were, estim ated by the w e igh ted -least-
squares method in a way s im ila r to th a t in section 5 .2 .1 . Figure 5 .3 .1
shows the oscillogram of th e armature and f ie ld cu rren ts o f the simu
la te d generator. Note the s ig n if ic a n t d iffe ren ce o f th e behavior o f
th ese cu rren ts in the su b tra n s ien t and tra n s ie n t p e riod from th a t shown
in Figure 5 .2 .2 . The num erical re s u lts fo r various i n i t i a l param eter
estim ates are l i s t e d in Table 5 .3 .1 . The p lo ts o f the convergence be
h av io r of some ty p ica l param eters are shown in Figures 5 .3 .2 and 5 .3 .3 .
From th e numerical r e s u l t s , the follow ing is observed:
(13 While with a sudden s h o r t - c i r c u i t , a l l the q -ax is param eters
converge to th e tru e values f a s t and accu ra te ly fo r any choice o f i n i
t i a l estim ate values between 40% and 200% o f the tru e v a lu es, i t i s no t
so when a s tep -fu n c tio n f ie ld -v o lta g e was used in the estim ation .
(23 Many parameter estim ates show the tendency o f divergence a t
th e end of the fourteen th i te r a t io n . In f a c t , fo r th e s e t of i n i t i a l
values randomly chosen between 40% and 200% o f the tru e values, the
estim ates d iverge in less than ten i te r a t io n s .
In conclusion, i f th e r e s u lts in th is sec tion a re compared w ith
those in sec tio n 5 .2 .1 , i t can be s ta te d th a t for the same sampling and
observation tim e, the sudden s h o r t - c i r c u i t scheme i s su p e rio r to the
s tep -fu n c tio n f ie ld -e x c ita t io n method w ith respect to convergence r a te ,
i n i t i a l estim ate range, and accuracy.
M
U
ij cu
"-4 in
I-I —
. H
HH
93. ri a
>»* v % ^ ^ y ' t A A / v v \ A A / \ A A / — —\ y \ / V / \ / \ / V / V y
-5 -
* \ / \ / \ / \ / \ / \ / \ .
0.5-
“ i------- \\----------------------- 1----120 1000
p e r -u n i t tim e
Figure 5 .3 .1 O scillogram o f D e te rm in is tic Output C urrentw ith S tep-F unction F ie ld -V o ltage
Table 5 . 3 . 1 WISE With Step-F unction F ie ld -V o ltageNumber o f
I te ra t io n
Parameter Estim ates as F raction o f True Value
X"cl XdA
Xd£T”l do
A
Tdo
A
X"q
Xq
A
qo r aA
Tf0 2 . 000 2 .000 2 . 0 0 0 2 . 000 2 . 000 2 . 0 0 0 2 .0 0 0 2 .0 0 0 2 . 0 0 0 2 .000 2 . 0 0 01 2 . S 0 0 1 . 6 7 5 2 . 5 0 0 2 . 5 0 0 1 .500 2 . 5 0 0 1 . 5 0 0 1 . 500 1 . 500 1 . 500 1 .5002 1 .875 1 .256 3. 125 1 • 875 1 . 8 7 5 3 . 1 2 5 1. 1 25 1 . 1 2 5 1. 125 1 . 125 1 . 1253 2. 344 1 . 135 2 . 344 1 . 4 06 2 . 3 4 4 2 . 3 4 4 1 . 4 0 6 1 . 406 0 . 9 5 7 1 .406 0 . 9 0 74 1. 758 1.110 2 . 9 3 0 1 . 055 2 . 93 0 2 . 9 3 0 1 . 7 5 8 1 .758 0 . 9 9 2 1.758 1.0005 1 .318 1 . 0 2 7 2 . 197 1 . 143 3 . 6 6 2 2 . 1 9 7 1 .3 13 1 . 3 1 8 1 . 003 1 . 318 1 .0126 1 .648 1 . 058 2 . 7 4 7 0 . 857 4 . 5 7 8 2 . 7 4 7 1.548 1 .64 8 0 . 9 9 7 1 . 648 0..9997 . 1. 236 0 . 987 2 . 060 1 . 071 5. 722 2 . 060 1 . 2 3 6 1 .236 1 . 0 0 8 1 .236 1.0 118 1 . 5 4 5 1 . 044 2 . 5 7 5 0. 804 7 . 153 2 .451 1 . 545 1 . 5 4 5 0 . 9 9 8 1 . 5 4 5 0 . 9 9 89 i . 159 1.030 1 .931 0 . 906 8 .941 1 . 8 3 8 1 .93 1 1 .931 1 .005 1 .931 1 . 0 0310 1 .217 1 . 045 1 .851 0 . 6 7 9 6 . 7 0 6 1 . 6 7 9 1 .643 1 . 646 0 . 9 9 9 1 .647 1 .00911 1.441 1 .090 1 .92 8 0 . 8 4 9 8 . 382 1 .72 0 1 . 9 8 5 1 .985 0 . 9 9 7 1 .985 1 . 0 0 512 1. 590 1. 120 2 . 124 1. 062 10 .4 77 1 . 3 6 4 2 . 4 8 2 2 .481 1 . 0 0 0 2 . 4 8 1 1 .00613 1 .988 1 . 1 96 2 . 5 1 5 1 . 3 2 7 13 .097 2 .01 7 3 .1 02 3 . 102 1 .001 3 . 1 0 1 1 .00714 2 .4 8 S 0 .897 3. 144 0 . 9 9 5 9 . 8 2 3 2 . 5 2 2 3 . 8 7 8 3 . 8 7 7 1 . 074 3 . 8 7 7 0 . 7 5 6
0 1 . 6 0 0 1 . 600 1 .60 0 1 .600 1 . 6 0 0 1 *600 1 . 600 1 .600 1 . 6 0 0 1 .600 1 .6001 1 .200 1 . 200 2 . 0 0 0 1 . 200 2 . 0 0 0 2 . 000 2 . 0 0 0 2 . 0 0 0 1 .200 2 . 0 0 0 1 .2002 1 . 2 3 4 1 .0 13 1 .718 1 . 5 0 0 1 . 5 0 0 1 . 5 7 8 1 .5 0 0 1 .500 0 .90 1 1 .500 0 .9693 1 . 4 9 2 1 . 1 41 2 . 1 4 7 1 . 125 1 . 8 7 5 1 . 9 7 3 1 . 4 0 3 1 . 372 0 .98 I 1 .39 1 0 .9954 1 .119 1.121 2 . 5 2 6 1 . 4 0 6 2 . 3 4 4 2 . 2 2 5 1 . 0 5 3 1 .0 2 9 1 . 0 0 9 1 .0 4 3 1 .0055 1 . 3 9 9 1 .119 2 . 9 4 3 1 . 758 1 . 7 5 8 2 . 6 6 7 1 .3 1 6 I . 2 8 6 0 . 992 1 .304 0 .9906 1 .049 1 .089 3 . 172 2 . 197 2 . 197 2 . 852 0 .9 8 7 0 .9 6 5 1 . 0 0 2 0 . 9 7 8 1 .0077 1.141 I . C 66 3 . 2 5 3 1 . 972 1 .6 7 4 3 . 009 1 . 2 3 3 1 . 2 0 6 0 . 9 9 3 I . 223 1.0018 1. 134 1 . 0 63 3 .0 7 7 1 . 5 6 9 1 . 2 5 6 2 . 8 9 0 1.3 10 1 .308 0 . 9 9 9 1.311 1.0019 1 . 188 l.C>73 3 . 0 9 9 1 .961 0 . 9 4 2 2 . 9 0 1 1 .173 1 . 1 6 8 1 . 000 1 . 175 0 .998
10 1 . 0 8 5 1.0 56 2 . 9 6 2 1 .471 1 .061 2 . 7 9 9 1 . 0 8 9 1 . 0 8 9 1 . 000 1 .088 1 .00111 1 . 139 1 . 0 6 6 3 . 000 1 . 6 4 6 0 . 8 3 3 2 . 8 1 9 1 .130 1 . 130 I . 000 I . 128 0 . 9 9 9
12 1 . 0 6 0 1.0 58 2 . 9 6 5 I . 2 3 5 1 . 0 4 2 2 . 7 9 9 1 . 059 1 . 058 1 .000 1 .059 1 .000
13 1 . 0 8 9 1 . 062 2 . 9 8 1 1 . 2 4 5 0 . 9 4 8 2 . 8 0 7 1 . 0 7 6 1 .077 1 . 000 1 .076 1 .000
14 I . 0 7 8 1.061 2 . 980 1 . 167 0 . 9 7 4 2 . 8 0 7 1 . 0 7 6 1 .076 1 .000 1 .076 1 .000
Table 5 . 3 . 1 (Continued)Number o f
I te ra tio n
Parameter Estim ates as Fraction o f True ValueaX"
AX 1d
Axd
Ttt*do
AT i do
AX"
q
AX
q
A■Jit
qoAr a
ATf
0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0 1 . 2 0 0
1 1 . 5 0 0 1 . 2 13 1 . 5 0 0 1 . 5 0 0 0 . 9 0 0 1 . 3 5 3 1 . 5 0 0 1 . 5 0 0 0 . 9 0 0 1 . 5 0 0 0 . 9 5 1
2 1 . 1 2 5 1 . 0 2 4 1 . 6 3 0 1 • 1 2 5 1 . 1 2 5 1 . 6 2 0 1 . 1 2 5 1 . 1 2 5 1 . 0 2 4 1 . 1 2 5 0 . 9 8 7
3 0 . 8 4 4 0 . 9 8 6 1 . 6 7 9 0 . 8 4 4 1 . 4 0 6 1 . 7 1 0 0 . 8 4 4 0 . 8 4 4 1 . 0 0 1 0 . 8 4 4 I . 0 0 3
4 0 . 9 9 3 1 . 0 2 0 1 . 2 5 9 0 . 9 5 9 1 . 0 5 5 1 . 2 8 3 0 . 9 4 1 0 . 9 4 2 0 . 9 9 9 0 . 9 4 2 0 . 9 9 9
5 1 . 0 4 7 1 . 0 0 6 0 . 9 5 4 1 . 1 9 9 0 . 9 1 0 0 . 9 8 6 0 . 9 7 9 0 . 9 8 1 1 . 0 0 0 0 . 9 7 6 1 . 0 0 0
6 1 . 0 7 0 1 . 0 1 2 1 . 0 0 1 1 . 4 3 6 0 . 8 6 1 0 . 9 9 5 1 . 0 1 8 1 . 0 2 2 1 . 0 0 0 1 . 0 1 8 1 . 0 0 0 '
7 0 . 8 0 2 0 . 7 5 9 0 . 8 2 7 1 . 0 7 7 1 . 0 7 6 1 . 0 7 8 0 . 7 6 3 0 . 7 6 6 0 . 9 9 5 0 . 7 6 3 1 . 0 1 3
8 0 . 7 6 9 0 . 9 2 0 0 . 6 2 0 0 . 8 0 8 1 . 3 4 5 0 . 8 0 8 0 . 8 5 1 0 . 8 2 1 0 . 9 9 8 0 . 8 2 6 1 . 0 0 6
9 0 . 5 7 7 0 . 8 6 0 0 . 7 7 5 0 . 6 0 6 1 . 6 8 1 0 . 9 9 9 0 . 6 3 8 0 . 6 1 5 0 . 9 8 7 0 . 6 1 9 1 . 0 1 3
10 0 . 4 3 2 0 . 7 5 3 0 . 5 8 1 0 . 4 5 4 2 . 1 0 2 0 . 7 4 9 0 . 4 7 9 0 . 4 6 2 1 . 0 3 5 0 . 4 6 5 0 . 9 9 5
11 0 . 5 4 1 0 . 9 0 6 0 . 7 1 4 0 . 3 4 1 2 . 6 2 7 0 . 8 3 3 0 . 5 9 8 0 . 5 7 7 0 . 9 8 5 0 . 5 8 1 1 . 0 0 9
12 0 . 6 7 6 1 . 0 66 0 . 8 9 2 0 . 4 2 6 1 . 9 7 0 1 . 0 4 1 0 . 7 4 8 0 . 7 2 1 0 . 9 9 3 0 . 7 2 6 1 . 0 0 0
13 0 . 8 4 5 1 . 0 1 0 1 . 1 1 5 0 . 4 3 9 2 . 0 1 0 1 . 1 3 9 0 . 8 9 0 0 . 8 8 7 0 . 9 8 6 0 . 8 9 3 1 . 0 1 1
14 0 . 7 9 8 0 . 9 9 a 1 . 1 1 6 0 . 3 5 9 1 . 5 8 4 1 . 1 1 7 0 . 8 7 9 0 . 8 7 9 1 . 0 0 0 0 . 8 7 9 1 . 0 0 1
0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0 0 . 8 0 0
1 1 . 0 0 0 1 . 0 0 0 0 . 6 1 0 1 . 0 0 0 0 . 6 0 0 0 . 6 0 0 1 . 0 0 0 1 . 0 0 0 0 . 9 1 6 1 . 0 0 0 0 . 9 5 8
2 1 . 2 5 0 1 . 0 2 3 0 . 4 5 7 1 . 2 5 0 0 . 4 5 0 0 . 4 5 0 1 . 2 5 0 1 . 2 5 0 0 . 9 9 7 1 . 2 5 0 0 . 9 9 7
3 0 . 9 3 7 0 . 7 6 7 0 . 3 4 3 1 . 5 6 2 0 . 3 3 7 0 . 3 3 7 0 . 9 3 7 0 . 9 3 7 1 . 0 3 7 0 . 9 3 7 1 . 0 0 9
4 0 . 7 9 7 0 . 6 2 8 0 . 2 5 7 1 . 9 5 3 0 . 4 2 2 0 . 2 5 3 0 . 7 0 3 0 . 7 0 3 1 . 0 1 2 0 . 7 0 3 1 . 0 0 2
5 0 . 9 9 6 0 . 8 4 8 0 . 1 9 3 2 . 4 4 1 0 . 3 1 6 0 . 2 2 1 0 . 5 8 9 0 . 5 9 0 0 . 9 9 8 0 . 5 9 4 0 . 9 9 8
6 0 . 9 5 0 0 . 3 4 2 0 . 2 1 7 2 . 3 9 4 0 . 2 3 7 0 . 2 5 9 0 . 7 3 6 0 . 7 3 8 0 . 9 9 9 0 . 7 4 2 0 . 9 9 7
7 0 . 9 7 0 U „ 3 4 6 0 . 2 2 1 2 . 4 7 5 0 . 2 9 7 0 . 2 6 3 0 . 9 2 0 0 . 9 2 2 I . 0 0 0 0 . 9 2 8 0 . 9 9 7
8 0 . 9 6 7 0 . 8 4 6 0 . 2 2 1 2 . 4 8 1 0 . 3 7 1 0 . 2 6 3 1 . 0 7 8 1 . 0 8 0 1 . 0 0 0 1 . 0 8 1 0 . 9 9 8
9 0 . 9 6 7 0 . 8 4 7 0 . 2 2 3 2 . 4 8 7 0 . 4 6 3 0 . 2 6 5 1 . 1 9 2 1 . 1 9 4 1 . 0 0 0 1 . 1 9 4 0 . 9 9 9
10 0 . 9 6 7 0 . 8 5 0 0 . 2 2 7 2 . 4 8 8 0 . 5 7 9 0 . 2 6 8 I . 2 5 6 1 . 2 5 6 1 . 0 0 1 1 . 2 5 6 0 . 9 9 9
11 0 . 9 6 5 0 . 8 5 3 0 . 2 3 1 2 . 4 8 4 0 . 7 2 4 0 . 2 7 2 1 . 1 7 8 1 • 1 7 8 1 . 0 0 2 1 . 1 7 6 0 . 9 9 9
12 0 . 9 6 5 0 . 8 5 6 0 . 2 3 4 2 . 4 6 6 0 . 5 8 2 0 . 2 7 5 1 . 0 9 3 1 . 0 9 3 1 . 0 0 2 1 . 0 9 2 0 . 9 9 8
13 0 . 9 7 0 0 . 8 5 9 0 . 2 3 9 2 . 4 7 9 0 . 6 7 2 0 . 2 7 9 1 . 1 5 2 1 . 1 5 3 1 . 0 0 1 1 . 1 5 2 0 . 9 9 9
14 0 . 9 7 2 0 . 3 6 1 0 . 2 4 2 2 . 4 7 4 0 . 6 3 1 0 . 2 8 2 1 . 1 3 4 1 . 1 3 5 1 . 0 0 2 1 . 1 3 4 0 . 9 9 9
Table 5 . 3 . 1 (Continued)Number of
I te ra t io n
Parameter Estim ates as F raction o f True Value"
*d
AX' Xd
A
***ATit
do do X"q
Xq
'['11qo
A
r aA
Tf0 u.ouO 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 0 0 . 6 0 01 0 , 7 5 0 0 . 7 5 0 0 . 7 5 0 0 . 7 5 0 0 . 4 5 0 0 . 7 5 0 0 . 7 5 0 0 . 7 5 0 0 . 6 2 8 0 . 7 5 0 0 . 7 5 02 0 . 9 3 7 0 . 9 3 7 0 . 6 5 7 0 . 9 3 7 0 . 3 3 7 0 . 6 1 0 0 . 5 6 2 0 . 5 6 2 0 . 7 8 6 0 . 5 6 2 0 . 9 3 13 1 . 1 7 2 1 . t 7 3 0 . 8 2 2 1 • 1 7 2 0 . 4 1 6 0 . 7 6 3 0 . 5 1 7 0 . 4 6 5 0 . 9 4 9 0 . 4 9 1 0 . 9 8 74 0 . 8 7 9 0 . 8 0 5 0 . 8 6 1 0 . 8 7 9 0 . 5 2 1 0 - 9 5 4 0 . 6 4 6 0 . 5 8 2 0 . 9 9 4 0 .6 1 3 0 . 9 9 6S 0 . 7 3 3 0 . 8 7 2 0 . 6 4 6 0 . 6 5 9 0 . 6 5 1 0 . 7 1 5 0 . 8 0 8 0 . 7 2 7 1 . 0 0 2 0 . 7 6 7 1 . 0 0 36 0 . 7 0 9 0 . 9 S 3 0 . 7 5 2 0 . 4 9 4 0 . 8 1 3 0 . 7 8 4 0 . 7 2 8 0 . 7 2 3 0 . 9 9 5 0 . 7 3 6 0 . 9 9 97 0 . 7 8 5 0 . 9 6 5 0 . 7 6 0 0 . 4 1 8 1 . 0 0 8 0 . 7 9 1 0 . 7 4 4 0 . 7 4 4 1 . 0 0 0 0 . 7 4 4 0 . 9 9 3S 0 . 7 4 3 0 . 9 6 1 0 . 7 5 9 0 . 3 2 9 1 . 2 6 1 0 . 7 9 2 0 . 7 4 1 0 . 7 4 1 1 . 0 0 0 0 . 7 4 1 0 . 9 9 99 0 . 7 4 6 0 . 9 6 0 0 . 7 6 0 0 . 3 5 3 1 . 4 1 9 0 . 7 9 2 0 . 7 4 2 0 . 7 4 2 1 . 0 0 0 0 . 7 4 2 1 . 0 0 010 0 . 7 4 2 0 . 9 6 0 0 . 7 6 0 0 . 3 5 0 1 . 4 3 8 0 . 7 9 2 0 . 7 4 2 0 . 7 4 2 1 . 0 0 0 0 . 7 4 2 1 . 0 0 011 0 . 7 4 1 0 . 9 6 0 0 . 7 6 0 0 . 3 4 9 1 . 4 4 1 0 . 7 9 2 0 . 7 4 2 0 . 7 4 2 1 . 0 0 0 0 . 7 4 2 1 . 0 0 012 0 . 7 4 1 0 . 9 6 0 0 . 7 6 0 0 . 3 4 9 1 . 4 4 2 0 . 7 9 2 0 . 7 4 2 0 . 7 4 2 1 . 0 0 0 0 . 7 4 2 1 . 0 0 013 0 . 7 4 1 0 . 9 6 0 0 . 7 6 0 0 . 3 4 9 1 . 4 4 2 0 . 7 9 2 0 . 7 4 2 0 . 7 4 2 1 . 0 0 0 0 . 7 4 2 1 . 0 0 014 0 . 7 4 1 0 . 9 6 0 0 . 7 6 0 0 . 3 4 8 1 . 4 4 2 0 . 7 9 2 0 . 7 4 2 0 . 7 4 2 1 . 0 0 0 0 . 7 4 2 1 . 0 0 0
0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0 0 . 4 0 0
1 0 . 5 0 0 0 . 5 0 0 0 .3 0 0 0 . 5 0 0 0 . 3 0 0 0 . 3 0 0 0 . 5 0 0 0 . 5 0 0 0 . 3 0 0 0 . 5 0 0 0 . 5 0 0
2 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 6 2 5
3 0 . 2 8 1 0 . 2 8 1 0 . 2 8 1 0 . 2 8 1 0 . 4 6 9 0 . 2 8 1 0 . 2 8 1 0 . 2 8 1 0 . 4 6 9 0 . 2 8 1 0 . 7 8 1
4 0 . 3 5 2 0 . 3 5 2 0 . 3 5 2 0 . 3 5 2 0 . 3 5 2 0 . 2 1 1 0 . 3 5 2 0 . 3 5 2 0 . 3 5 2 0 . 3 5 2 0 . 9 7 7
5 0 . 2 6 4 0 . 2 6 4 0 . 2 6 4 0 . 2 6 4 0 . 4 3 9 0 . 1 5 8 0 . 2 6 4 0 . 2 6 4 0 . 3 3 8 0 . 2 6 4 1 . 2 1 3
6 0 . 3 3 0 0 . 3 3 0 0 . 3 3 0 0 . 3 3 0 0 . 3 3 0 0 . 1 9 8 0 . 3 3 0 0 . 3 3 0 0 . 2 5 3 ' 0 . 3 3 0 1 . 3 0 3
7 0 . 4 1 2 0.<* 12 0 . 4 1 2 0 . 4 1 2 0 . 2 4 7 0 . 2 4 7 0 . 4 1 2 0 . 4 1 2 0 . 1 9 0 0 . 4 1 2 0 . 9 7 7
8 0 . 5 1 5 0 . 5 1 5 0 . 5 1 S 0 . 5 1 5 0 . 1 8 5 0 . 3 0 9 0 . 5 1 5 0 . 5 1 5 0 . 1 4 9 0 . 5 1 5 1 . 0 4 3
9 0 . 3 8 6 0 . 6 4 4 0 . 6 4 4 0 . 6 4 4 0 . 1 3 9 0 . 3 8 6 0 . 3 8 6 0 . 3 8 6 0 . 1 8 6 0 . 3 8 6 1 . 0 4 1
10 0 . 4 8 3 0 . 8 0 5 0 . 6 6 2 0 . 8 0 5 0 . 1 0 4 0 . 2 9 0 0 . 4 8 3 0 . 4 8 3 0 . 2 3 2 0 . 4 8 3 1 . 0 5 1
11 0 . 6 0 3 1 . J 0 6 0 . 8 2 7 1 . 0 0 6 0 . 0 7 8 0 . 2 4 6 0 . 6 0 3 0 . 6 0 3 0 . 2 9 0 0 . 6 0 3 1 . 0 0 3
12 i 0 . 4 5 3 0 . 7 5 4 0 . 6 2 0 0 . 7 5 4 0 . 0 9 8 0 . 3 0 3 0 . 4 5 3 0 . 4 5 3 0 . 3 6 3 0 . 4 5 3 0 . 9 6 5
13 0 . 3 3 9 0 . 6 0 3 0 . 4 6 5 0 . 5 6 6 0 . 1 2 2 0 . 2 8 6 0 . 3 3 9 0 . 3 3 9 0 . 4 5 4 0 . 3 3 9 1 . 0 0 0
14 0 . 2 5 5 0 . 4 5 2 0 . 3 4 9 0 . 4 2 4 0 . 1 5 3 0 . 2 1 5 0 . 2 5 5 0 . 2 5 5 0 . 5 3 2 0 . 2 5 5 1 . 0 4 3
Estim
ate
Ratio
Es
timat
e R
atio
97
0.8
0 2 4 6 8 1210 14Number of I te ra t io n s
Figure 5 .3 .2 Ratio o f Estim ate to True Value fo r X'J with Step-Function F ield-V oltage (WLSE)
0.4
4 60 2 8 1210 14Number of I te ra t io n s
Figure 5 . 3 . 3 Ratio o f E stim ate to True Value fo r X, withStep-Function F ie ld -V o lta g e (WLSE)
98
5 .4 . E ffec t o f R esistive Load on Estim ation
As a continuing e f fo r t to in v e s tig a te the e ffe c t o f various con
d itio n s on the estim ation accuracy, i t i s n a tu ra l to ask what kind of
load provides the h ighest accuracy o f the param eter e stim ates . This
question leads d ire c tly to the load design problem.
The load design problem involves the determ ination o f load s tru c
tu re as w ell as i t s numerical d a ta which i s as d i f f ic u l t as the system
id e n tif ic a t io n problem i t s e l f . Furthermore, even i f the load s tru c tu re
i s given, d ire c t determ ination o f the load data is not p o ss ib le since
the system param eters a re unknown. A sequen tia l design could be a t
tempted; the computation, however, i s very much involved.
Although the load design problem cannot be s a t is f a c to r i ly solved
a t th is p o in t, some in v es tig a tio n was made on the e ffe c t o f a r e s is t iv e
load on the estim ation r e s u l ts . With a balanced th ree-phase re s is t iv e
load, the w eigh ted-least-squares estim ato r was t r ie d fo r several per-
u n it values o f lo ad -re s is tan c e , v iz . ,
Rl = 0 .0 , 0.25, 0.50, 0 .75, 1.00.
Using the same gain m atrix G(k), sanqpling time and observation time
period as in sec tio n 5 .2 .1 , the machine param eters were estim ated w ith
i n i t i a l estim ates o f 200% of th e i r tru e values and with i n i t i a l values
chosen a t random. The sim ulation re s u lts are l is te d in Table 5 .4 .1 .
Table 5 .4 .1 shows th a t the param eter estim ation accuracy i s a f
fec ted by the load re s is tan c e value. The h ighest accuracy occurs a t
RL = 0 .0 . I f necessary , however, a load re s is tan c e le s s than
R = 0.5 pu can be added w ithout s a c r if ic in g the accuracy but at the
expense o f a few more i te r a t io n s . This i s very desirab le since in re a l
w orld im plem entation the s h o r t - c i r c u i t connections w i l l have some
f i n i t e re s is ta n c e . I t i s s tro n g ly no ted th a t the r e s u l t s in th is sec
t io n to g e th e r w ith those in s e c tio n 5 .3 support th e adequacy o f
choosing the sudden th ree -p h ase s h o r t - c i r c u i t t e s t fo r th e param eter
e s tim a tio n .
Table 5 . 4 . 1 V ariab le R e s is t iv e Load
rl
Number of
I te ra t io n
Parameter E stim ates as F raction o f True Value ( I n i t i a l E stim ates:2.03
*dAX •xd *d Xd&
ATH
do T1*doA
X"q
AXq
piqo r a r f
0.00 5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000.25 5 1.001 0.999 0.989 1.318 0.988 0.991 1.000 1.000 1.000 0.999 0.998
8 1.000 1.000 1.000 1.004 1.000 1.000 1.000 1.000 1.000 1.000 1.0000.50 5 1.000 1.001 0.992 1.101 1.000 0.987 1.000 1.000 1.000 1.001 0.999
8 1.000 1.000 1.000 1.001 1.000 1.000 1.000 1.000 1.000 1.000 1.0000.75 5 0.998 0.996 0.890 1.318 1.318 0.800 1.003 1.002 1.004 1.012 1.000
9 1.000 1.000 0.999 1.001 1.001 0.999 1.000 1.000 1.000 1.000 1.0001.00 5 0.791 1.318 2.197 1.318 2.197 2.197 1.113 1.044 1.067 1.388 1.084
10 0.986 2.898 0.521 0.861 6.706 0.521 1.047 1.033 1.066 1.074 1.818
Number o f Parameter Estim ates as F raction of True Value ( I n i t i a l E stim ates:Random)
RI. I te ra t io nAX"
A Ax.
Ax.„
AT i i Ti X" X
A ,*pi r
AT j -d d d d& do do q q qo a f
0.00 8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000.25 8 1.004 0.980 1.009 1.390 0.644 1.015 1.000 1.000 1.000 1.000 1.000
13 1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.0000.50 8 1.000 0.992 1.015 0.926 0.928 1.026 1.000 1.000 1.000 1.000 1.000
11 l.QOO 1.000 1.000 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.0000.75 diverge1.00 diverge
CHAPTER 6
CONCLUSIONS
The th e s is has endeavored to in v e s tig a te the a p p lic a b i li ty o f
the w eigh ted-least-squares and maximum lik e lih o o d estim ators to syn
chronous machine modeling fo r determ ining the machine data accu ra te ly
and e ffe c tiv e ly . A d e ta ile d computational algorithm has been developed
and applied to the f i f th o rder model of a sa lie n t-p o le synchronous
machine to determine the standard machine param eters. The computa
tio n a l algorithm was a lso te s te d with system input and output noises
which were modeled by zero-mean gaussian white processes. F in a lly , the
e ffe c ts of input and load on param eter estim ation have been s tud ied by
employing a s tep -fu n c tio n f ie ld -v o lta g e as input and using d if fe re n t
load re s is ta n c e s .
Based on the computational re s u lts obtained in th is study, the
follow ing conclusions can be drawn:
(1) In the implementation o f the computational algorithm , no con
vergence problems were encountered fo r i n i t i a l estim ates w ith in
the in te rv a l from 40% to 200% of th e tru e values. With the same
weighting and gain m atrices, s im ila r re s u l ts have been obtained
fo r d if fe re n t s e ts o f param eter values ty p ic a l fo r large
machines. For i n i t i a l guesses ou tside the above-mentioned
in te rv a l , the algorithm converged slower and might even diverge,
so th a t a d if fe re n t gain m atrix fo r the recursion formula may
be needed. This i s not su rp ris in g since the estim ato r equations
101
102
are highly non linear in the standard param eters, p a r t ic u la r ly
X ^ . In the p ra c tic a l im plem entation, however, i n i t i a l e s t i
mates can be taken equal to the values obtained from a reference
tab le [2], design data and/or conventional t e s t s , so th a t con
vergence problems should not occur.
(2) The sim ulation re s u l ts show th a t the estim ators a re very
accura te , v iz . , i f th e machine under te s t is not too se n s itiv e
to environmental d isturbances and accurate measuring devices
are a v a ila b le , the estim ation e r ro r fo r a l l param eters i s ex
pected to be le ss than 0.1% o f the tru e value, which cannot be
obtained by conventional IEEE t e s t procedures.
(3) Even i f the system is exposed to a no ticeab le amount o f no ise ,
reasonably accurate param eter estim ation can be a n tic ip a te d by
employing the maximum like lihood estim ato r. The maximum l i k e l i
hood estim ato r provides a means o f u t i l iz in g th e av a ilab le
a -p r io r i knowledge o f the measurement noise to improve the
estim ation accuracy. No advantage o f th is a -p r io r i knowledge
can be taken in the standard t e s t procedures.
(4) The estim ato rs are very e ffe c tiv e considering th a t a l l parame
te rs a re sim ultaneously obtained from the data o f a sin g le t e s t ,
where i t i s not even necessary to reach the s te a d y -s ta te condi
tio n . Note a lso th a t the q -axis param eters a re extremely d i f f i
c u lt to determine accu ra te ly by conventional t e s t procedures
[61].
103
(5) In th is s tu d y , the estim ation a lgorithm has been app lied to the
f i f th - o r d e r model s tru c tu re and th e IEEE s ta n d a rd machine pa
ram eters were ob ta ined . However, the a lgo rithm can be ap p lied
to any o th e r type o f model s t r u c tu r e and s e t o f param eter d a ta .
For example, a th ird -o rd e r model s t ru c tu re ig n o rin g the e f f e c ts
o f th e damper winding could a ls o be used. Then, minimizing the
d ev ia tio n o f i t s ou tpu t t r a j e c to r y from th e r e a l one would r e
s u l t in a s e t o f m odified s ta n d a rd param eter va lues fo r the
th ird-O Tder m odel.
(6) The study on th e e f f e c t o f r e s i s t i v e load on param eter estim a
tio n in d ic a te s th a t the b e s t r e s u l t s a re ob ta ined w ith a zero
load r e s is ta n c e . Furtherm ore, i t i s shown th a t th e sudden
th ree -phase s h o r t - c i r c u i t t e s t p rov ides su p e r io r output d a ta i f
compared w ith th e s te p -fu n c tio n f ie ld -v o lta g e t e s t w ith sh o r t-
c ir c u ite d arm ature winding. T h ere fo re , th e sudden s h o T t-c irc u it
t e s t i s an ex trem ely proper cho ice fo r param eter e s tim a tio n .
In summary, th e param eter e s tim a to rs considered in th i s th e s is
promise a p o te n t ia l method to determ ine th e machine param eters accu
ra te ly and e f f e c t iv e ly , which i s u rg e n tly needed in power system
s tu d ie s .
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72. E. K reindler, "On the d e fin itio n and a p p lica tio n o f the s e n s i t iv i ty function ," J . Frank. I n s t . , 285, pp. 26-36, 1968.
73. M. L. Bykhovskiy, " S e n s it iv i ty and dynamic accuracy of con tro l system s," Eng. Cyb., pp. 121-134, 1964.
74. A. P. Sage, Optimum Systems C ontro l, P ren tice -H all, In c ., Englewood C lif f s , N .J . , 1968.
75. S. B. Crary and W. E. Dungan, "Am ortisseur windings fo r hydrog enera to rs ," E le c tr ic a l World, vo l. 115, pp. 2204-2206, June 28, 1941.
76. F. Shokooh, "Synchronous machine analysis using s ta te equations with re lev an t c o e ff ic ie n ts ," M.S. T hesis, Louisiana S ta te U niversity , 1975.
77. V. E. Mablekos and L. L. Grisby, "S ta te equations o f dynamic systems with tim e-varying c o e f f ic ie n ts ," IEEE Trans. Power App. S y s t.,v o l. PAS-90, pp. 2589-2598, November/December 1971.
78. V. E. Mablekos and A. H. El-Abiad, "An algorithm fo r the u n if ie d so lu tio n o f the s ta te equations o f e le c t r ic a l ly excited systems with tim e-varying c o e f f ic ie n ts ," IEEE Trans. Power App. S y s t., v o l. PAS-90, pp. 2726-2733, November/December 1971.
79. N. D. Rao and M. M. Elmetwally, "Design o f low s e n s i t iv i ty optim al reg u la to rs fo r synchronous machines," 1972 IEEE Summer Power M eet., Paper No. C72 489-3.
110
80. P. Subramaniam and 0. P. M alik, "C losed loop o p tim iza tio n o f power systems w ith tw o-axis e x c i ta t io n c o n tro l ," IEEE W inter Power M eet., 1972, Paper No. T72 237-1.
81. W. D. Humpage, J . R. Smith and G. J . Rogers, "A pp lica tion o f dynamic o p tim iza tio n to synchronous-genera to r e x c i ta t io n c o n tro l le r s ," Proc. IEE, v o l. 120, No. 1, pp. 87-93, January 1973.
82. R. Kalman and R. Bucy, "New r e s u l t s in f i l t e r i n g and p re d ic tio n th e o ry ," ASME Jo u rn a l o f Basic E ng ., v o l. 83, pp. 95-108,March 1961.
83. D. J . S ak rison , Communication Theory: T ransm ission o f Waveformsand D ig ita l In fo rm ation . John W iley and Sons, New York, 1968.
84. J . S. M editch, S to c h a s tic Optimal L inear E stim ation and C ontro l. McGraw-Hill, New York, 1969.
85. V. C. Levadi, "Design o f in p u t s ig n a ls fo r param eter e s tim a tio n ," IEEE Trans. Auto. C o n tro l, vo l. AC-11, No. 2, pp. 205-211,A pril 1966.
86. M. Aoki and R. M. S ta le y , "On in p u t s ig n a l sy n th e s is in param eter id e n t i f i c a t io n ," Autom atica, v o l. 6 , pp. 431-440, 1970.
87. R. K. Mehra, "Optimal in p u t s ig n a ls fo r param eter e s tim a tio n in dynamic system s—survey and new r e s u l t s , " IEEE T rans, Auto.C on tro l, v o l. AC-19, No. 6 , pp. 753-768, 1974.
88. N. E. Nahi and D. E. W allis , J r . , "Optimal in p u ts fo r param eter e s tim a tio n in dynamic system s w ith w hite o b se rv a tio n n o is e ," P re p r in ts , JACC, Boulder, C o lo ., August 1969.
APPENDIX 1
ELEMENTS OF THE MATRICES and | j
Let the elements a^ , i = 1 ,2 ,3 . . . 11, o f the parameter vecto r
and x^, i = 1 ,2 , 9, o f the s t a t e vec to r x be re sp ec t iv e ly defined by
a .
“ 2 = Xd
“ 3 = Xd
a 4 ” XdP.
a c = T"5 do
a* =6 do a_ = X"
7 q
a s = X n8 q
CJa = T"9 qoa.„= r10 a
a i r r f
X, = \
X2 = Akd
x4 = Xq
x5 ■ %
itX- = V7 q
Tf ~ >*8 I d
X3 " Xf X6 Vd X9 = *Iq
Then the expression fo r each element o f the m atrix —4^- —■ x in (3.2.13]dxi s as follows:
3 fl Moa 103a.l a ,
9C, 3C ,.jd dkd3a. X1 + 9a.i i
9Cdf 2 ~5a7 x3i
3f CD-5—“ = — (C ,x, + Cj, J X_ + C JJ. x_)
10 a3
* J i - ±3a. a_ i 5
3Ckdd , 9ckd . 3Ckdf 3a. X1 3a. x2 3a. x3
9f2 -1■557 ' 2 CCkdd X1 + Ckd X2 + Ckdf V 5 a„
, i = 1 ,2 ,4
i = 1 ,2 ,3 ,4
I f s . i3a. ~ cl, i 6
3Cfd x. + 3Cfkd 3Cfx„ + “— x.3a. 1 3a. 2 3a. 3i l l, i = 1 ,2 ,3 ,4
111
where
113
9Cdf a3Ca3- a 2HCa1-a 4)Ca2-a4J + ( c y a ^ C o i j - a ^ }3a,
3Ckdd
axCa2-a 4) 2 ta 3-a4) 2
V a 23a,
kdd9a,3
= 0
3 Cl Akd a 23a^
“l3Ckd
3a30
3C'kdf (a2 - a 3 )o.4
3a1 (a3-a4^al3cfd fV “4 )0l4
80,C«2 - “4 3«i
3Cfd (0 , - 0 4 )8 a 3 0 ,(0 2 -0 4 )
3Cfd o4(o2-o4) ♦ a 2 Co4 - a 3)3a,4 0 , ( 0 2 - 0 4 ) 2
3cfkd (a4_a3 a2a 4So,
(O2 -O4 ) 0 ,
3C£kd o4 (o3 - a 4) ( 2a , - a 2 -a4)3a2 o,(o2 -o4 ) 3
fkd (a2-ai ) (a2(a3-a4) + '
kdd3a„
1_a.
3C3akdd = ^
a,
f k3a„
H i3a„
z La i
= o
kdf3a„
a.vvv3Cfd = CW t y o » , ]
H (a2 - “ 4 ) 2 o,
3Cfkd _ C V V “4"5a Ca2-a 4) a 1
3Ckdf C W a43a3 a 1(a3-a 4) 2
3a,
3cf"Sa^
“ i C V VX __c3a,
3Cdkd3a, 1 - v w
a3 C«2- V
3Ckdf Ca3“a 2) a 33a, ,24 a ^ - t x , )
114
53a,
3Cc3a,
3Cdkdact. 1 - V y V
‘ j t V V+ c
a4 t y ydkd , ,2
W y
sc,
3a,
3C __c3a,
acdkd3ot, 1 - W a45
“ 3 t W
a*Jdkd , . 2
Ca 2“a4 a 3
act.ac.__c3a,
3Cdkd3a, 1 - y y y
a3 (a2-a 4) 'dkd(a3-a 4)c.2 + (a4- a 2)a 4
“ 3(°2-“4)2
The expression for each element o f the matrices in (3.2.14)
i s given in the following:
3gl P F ~ x6 3gl H T x7/ 2 2 7 / 2 2
/ X6 + X7 V x6 + X7
3g„ 3 i ,a ^ = I D a3T , i = 1,2,31 X
ago 3i3J7= IQ 5 ^ , 5 =4 , 5
ag3 3 id3x7 = xe axT ' i = 1,2,3
3g3 3iqaTT = x7 a3^ ' J = 4,5
3 3
• !? S = •8Xg ” 1d 3x^ 1q
ag4 a id3xi ' x7 3x. ’ 1 “ 1,2,3
3g4 A ,3 7 7 = "X6 H 7 , J = 4,5
! ! i = „■ ! f i = *3Xg ~1q 3x^ Xd
115
3g5 “c fd 9gS "Cfkd 3g5 ' Cf3xl a6a l l 3x2 a6a l l * 3x3 a6a l l
The expression fo r each element o f the matrix in (3.2.14) isdOtl i s t e d below.
3g= 0 , i = 1,2, . . . 11
3g2 3‘*'d3 ^ = 10 a 5 T ’ i = 1 >2’3>4i i
3g2 3 iq3^7 = IQ 3 ^ ’ J - 7>83 33g3 3xd3o7 = x6 3cT ’ 1 = 1>2>3>4l i
3g3 3iq3a. ~ x7 3a. ’ 3 = 7,83 39g4 3 id3a. “ x7 3a. 5 1 = 1 >2 >3 >4l i
3g4 3 iq3a. = "x6 3a. * 3 7,83 33gq 3i_3 ^ = ^ , i - 1 ,2 ,3 ,4 ,6 ,1 1
where
1d = a j CCdXl + CdkdX2 + S i f V
wi = — (C x. + C , xc) q ag q 4 qkq
xf " ™ a 1 CCfdxl + Cfkdx2 + Cfx3:)11 D
116
S i d “ o c d 9 i , o> C „ , d _ o dkda„
3 id . “oCdf9x~ a„
9i u C 3. = _ U3x4 b,
9i (u C , a _ O qkq9xr a 8
9 ld 1 3fl9a. a1{) Bcl ' 1 " 1*2»3 ' 4
3io i 9f^ a _ __i____49a. a iri 9a. *
i 10 ii = 7,8
5 e9a.i
1 M San1 9a.11 i
, i = 1 ,2 ,3 ,4 ,6
9a11 “n l f
ID- f l {
. 2 .2i , + ld qIQ - / v
/. 2 . 2i , + ld q
117
APPENDIX 2
7 0 1
10
104
SENSITIVITY ANALYSIS PROGRAM
DIMENSION TAXIS{ 102) ,SAXISI 102) .BUFFER 18000)DIMENSION X 1 9 ) . S X ! 9 , 1 1 ) , A P 1 9 . 1 1 > , S Y 1 5 , 1 1 ) , G X ( 5 . 9 ) . A ( 9 , 9 ) DIMENSION GP ( 5 . I 1 )‘»P ( 1 1 ) .ST (S» 1 1 *100) *U(9)*XS(9) .Y{5) DIMENSION US( 9)DIMENSION RR t 7 ) . ELEL(7) .CC(7)COMMON A.U.US PLDEL T=0.025 TAXIS!10 1)=0 «TAXIS!102)=1.0 SAXIS(101)=0.SAXIS!1 0 2 ) - 1 . 0 TAXIS!1 ) = 0 .
DO 701 1=2 ,100TAXIS!I> =TAXIS(I- 1 ) +PLDELT NRELC=7 NV = 9 NSTEP=2 KNSTEP=99 NRUNGE=2 DELT=0.1 OMEGA=1•RL = 0.READ!5 . 1 0 ) P t , P 2 , P 3 , P 4 , P 5 , P 6 , P 7 , P 8 . P 9 . P 1 0 , P 11 FORMAT!8F1 0 . 5 / 3 F 1 0 . 5 )WRITE 16* 104) P 1 . P 2 . P 3 . P 4 •PS, P 6 , P 7 . P8. P 9 . P 1 0 . P I 1FORMAT!1 HI . I OX.»TRUE PARAMETER VALUES• / / / ! 1 OX. 1 F 1 0 . 5 ) )P l l ) = P lP ( 2 )=P2P!3)=P3P ( 4 )= P 4P!5)=P5PC 6)=P6P ( 7 ) =P 7PIS )=P8PC 9)=P9PI 10 ) =P 1 0P! 11 )=P11CD=-P3/P1CDKD=P3*(P2-P1) / ( P 1 4 1 P 2 - P 4 ) )C D F= P 3 * I P 1 - P 4 ) * !P 3 -P 2 J / I P 1 * ( P 3 - P 4 ) * ! P 2 - P 4 ))CF=CD+CDKD*1I . -P4*!P3-P4) / (P3*(P2-P4) ) )CFD=!P1-P4)*(P3-P4 > / ! P l * ( P 2 - P 4 ))CFKD=( P2-P 1)* ( P3-P 4)+P 4 / ( P1* (P 2-P4)**2 )CKDD=!P2-P4) / P I
118
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CALL P L O T ( 1 ■ • 1 6 * * —3 )J = 1Y S C A L E = 5 0 .DO 7 2 5 N P L O T = l , 2
W R I T E ( 6 , 5 0 0 ) ( ( S T ( I , J , K ) , I = 1 , S ) . K = 1 , 1 0 0 > FORMAT i 1 H 1 » * S E N S I T I V I T Y * . / / ( 5 F 2 0 . 5 ) >
OO 7 1 5 1 = 1 , 5CALL P L O T ( 5 . , 0 * , - 3 )CALL P L O T ( 3 * * 0 , * 2 )CALL P L O T C O . , 4 , , 3 )CALL P L O T t 0 . , - 4 , . 2 )
DO 7 2 0 K = 1 , 1 0 0 S A X I S ( K > = S T ( I , J , K ) / Y S C A L E CALL L l N E t T A X I S , S A X I S , 1 0 0 , 1 , 0 , 0 )CONTINUE J = 7CONTINUE M = 0K N S T E P = 1 0 0NRUNGE=3CD E L T = 0 , 2CONTI NUEK N S T E P = 9 9NRUNGE=2O E L T = 0 . 1CONTINUESTOPEND
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F U NC T I ON FCN< T . X . K . N D E T )D I M E N S I O N F ( I O ) . X M O )D I M E N S I O N A ( 9 , 9 ) . U ( 9 ) , U S < 9 )COMMON A . U . U SF ( K > = 0 .DO 1 0 0 J = 1 * 9 F { K ) = F ( K } + A ( K , J ) * X ( J >I F ( N D E T • E Q « 3 ) GO TO 3 0 0 F ( K ) = F ( K ) + U ( K )GO TO 4 0 0F ( K J = F ( K ) + U S ( K )F C N = F ( K )RETURNEND
SUBROUTI NE RUNGE( T . D L T . X . NV* N DE T )D I M E N S I O N X N ( I O ) . K U O ) * T A Y ( 4 . L 0 )DO I M = l , N V
TAY( I . M) = F C N ( T . X * M . N D E T ) * DLT DO 2 0 L = 2 . 3 DO 2 M= 1 i N VX N ( M ) = X < M ) + T A Y I L - 1 * M ) / 2 .DO 2 0 N = 1 , N VT A Y ( L , N ) = F C N ( T + D L T / 2 . . X N . N . N D E T ) *DLTDO 3 0 N = 1 . N VX N ( N ) = X ( N ) + T A Y ( 3 . N )DO 3 M - l , N VT A Y I 4 . M ) = F C N ( T + D L T . XN » M » N D E T ) *DLT DO 4 M= 1 , N VXN(M ) =X( M) + ( T A Y U , M) + 2 . * T A Y ( 2 * M ) + 2 . *T A YI 3 • M ) +TAY C 4 , M ) ) / 6 , X ( M ) = XN < M )RETURNEND
124
125
APPENDIX 3
WLSE PROGRAMCc * * * * * PFACTR I S PARAMETER ERROR FACTOR OF I N I T I A L GUESSC * * * * * OELFAC I S FOR THE INCREMENT OF P F ACTRC * * * * * NI NP AR THE NUMBER OF REPEAT OF PF ACTR+ DELF AC C * * * * * N R E P I T I S NUMBER OF I T E R A T I O N OF PARAM E S T I M A T I O Nc * * * * * n s a m p l i s f o r t h e n u m b e r o f s a m p l e s
c * * * * * NIRUNGE I S THE NUMBER OF RUNGE RO U T I N E REPEAT P F A C T R = 2 «0 D E L F A C = - 0 . 4 N I N P A R = 7 N R E P I T = I A N I R N G E ^ l P L I M I T = C . 2 5
R L = 0 •O M E G A = I .
D £ L T = 0 * 2N V - 5D I M E N S I O N L L M T X I 5 ) , M M M T X ( 5 )
D I M E N S I O N S S A 1 ( 5 )D I M E N S I O N XR( 5 ) . A R I 5 . 5 ) , U ( S ) , Y R ( 3 ) » C R < 3 . 5 )D I M E N S I O N X( 5 ) * A ( 5 * 5 ) • Y ( 3 ) «C {3 * 5 )D I M E N S I O N S X ( 5 * 1 1 } , A P < 5 , 1 1 ) , S Y ( 3 , 1 1 ) , C P ( 3 , 1 I }D I M E N S I O N X S ( 5 ) , U S ( 5 > , W I N V ( 3 . 3 ) , S Y T < 1 1 , 3 )D I M E N S I U N SYT W( 1 1 , 3 ) , S U M { 1 1 , 1 1 ) , T S U M ( 1 1 , 1 1 )D I M E N S I O N Y D ( 3 ) • YW(1 1 ) , TYW( 1 1 ) , T S U M l N t 1 1 , 1 1 )D I M E N S I O N D E L P t l 1 ) , P N { 1 1 )D I M E N S I O N P T R U E ( 1 1 ) » P R A T I Q { 1 1 )D I M E N S I O N LMTX ( 1 1 ) » M M T X ( 1 1 )D I M E N S I O N E S T M A T t I S , I 1 )COMMON A R , A * U , US
U ( I ) = 0 .U ( 2 ) = 0 .
U ( 3 > = S O R T ( 3 . ) * 0 . 0 0 2 1 U ( 4 ) = 0 .
U ( 5 ) = 0 .READ( 5 , 1 0 ) P I , P 2 , P 3 , P 4 , P 5 , P 6 . P 7 , P 8 , P 9 , P 1 0 , P 11
1 0 F O R M A T ! 8 F 1 0 . 5 / 3 F 1 0 . 5 )W R I T E ( 6 , 1 0 4 ) P l , P 2 . P 3 , P 4 , P 5 , P 6 , P 7 . P 8 , P 9 , P 1 0 , P l 1
1 0 4 FORMATI 1 H 1 , 1 0 X , * TRUE PARAMETER V AL UE S * ✓ / / ( 1 OX, 1 F 1 0 . 5 ) ) P T R U E I 1 ) = P l P T R U E ( 2 ) = P 2 P T R U E ( 3 ) = P 3 P T R U E I 4 ) = P 4 P T R U E ( 5 ) = P 5 P T R U E 16 > = P 6 P T R U E ( 7 ) = P 7 P T R U E ( 8 ) = P 8
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WR I T E ( 6 , 1 0 1 ) ( CAR ( I • J ) * J = 1 * S ) . 1 = 1 . 5 )1 0 1 F O R H A T < I H 0 . 1 0 X , ' M T X A R • / / / ( 1 OX* 5 F 2 0 • 7 > 1
W R I T E ( 6 * 1 0 2 > { ( CR C I » J ) » J = 1 » 5 ) *1 = 1 • 3 )1 0 2 FOHMATC 1 H 0 < 1 0 X , ' M T X C R • / / / { 1 O X . 5 F 2 0 . 7 ) J
CC C * * * * * + * * * * * * * * * I N P U T N O I S E I S A D D E D * * * * * * *STARTC
CALL M I N V ( A . 5 . D D . L L M T X . M M M T X )CALL G M P R D C A . U . S S A I , 5 . 5 . 1 )
DO 2 8 3 1 = 1 * 5 2 8 3 S S A I I I ) = - S S A I I I )
W R I T E ( 6 * 2 8 2 ) I S S A I { I » . 1 = 1 . 5 )2 8 2 F O R M A T ! 1 H 0 . 5 F 2 0 . 5 )
D I N P U T=0 * 01 CGMAU1= S S A I C 1 ) * D I N P U T C G M A U 2 = S S A I C 2 ) * D I N P U T C G M A U 3 = S S A I ( 3 ) * D I N P U T C G M A U 4 = S S A I ( 4 ) * D I N P U T CGMAU5 = S S AI C 5 ) * D I N P U T CGMAU1=ABS( CGMAU1 )CGMAU2=ABS( CGMAU2)CGMAU3=ABSCCGMAU3)CGMAU4=ABS( CGMAU4)CGMAU5=ABS( CGMAU5 )
DO 2 0 5 1 = 1 . 3 DO 2 0 5 J = 1 * 3
2 0 5 WINVI I . J ) = 0 .WINV( 1* 1 1 = 0 . 0 2 2 0 4 7 WINVC 2 * 2 ) = 3 . 8 4 5 WI NVC3 * 3 ) = 0 * 0 2 2 0 4 7 WINVC 1 * 1 ) = 1 * 0 W IN VC 2* 2 ) = 1 .WINVC 3* 3 ) = 1 • 0
DO 8 8 8 N P A R A M = l , N I N P A R PNC 1 ) = P T R U E < 1 ) * P F A C T R PNC 2 ) = P T R U E C 2 ) * P F A C T R PNC 3 ) = P T R U E ( 3 ) * P F A C T R PN C 4 > = P TRUE C 4 ) + P F AC T R P N C 5 ) = P T R U E C 5 ) * P F A C T R PN C 6 1 = P TRUE C 6 ) * P F A C TR PNC 7 ) =PTRUE C 7 ) * P F A C T R PNC 8 ) = P T R U E C 8 ) * P F A C T R PNC 9 ) = P T R U E C 9 ) * P F A C T R
127
P N ( 1 0 ) = P T R U E ( 1 0 ) * P F A C T R P N ( 11 } = P T R U E ( 1 1 } * P F A C T R I F ( N P A R A M . L T . 7 ) GO TO I t P N ( I 1 = P T R U E ( 1 1 * 1 . 6 P N ( 2 > = P T R U E ( 2 ) * 1 . 4 P N ( 3 1 = P T R U E ( 3 ) * 1 . 2 P N ( 4 l = P T R U E ( 4 ) * l , 8 P N ( 5 ) - P T R U E ( 5 1 * 0 . 5 P N ( 6 ) =P T RUE( 6 ) * 0 • 7 P N l 7 > = P T R U E t 7 ) * 0 . 6 P N ( 8 ) =P T R U E ( 8 1 * 0 . 8 P N ( 9 1 = P T R U E ( 9 1 * 0 . 4 P N { 1 0 ) = P T R U E ( 1 0 1 * 2 . 0 P N ( 11 ) = P T R U E ( 1 1 1 * 0 . 9
11 0 0 3 1 = 1 «1 IPRAT 1 0 ( I > = P N ( I 1 / P T R U E ( I 1
3 CONTINUEDO 4 1 = 1 , 1 1
4 E S T M A T I 1 , 1 > = P R A T I 0 ( I 1DO 9 9 9 N I T = 1 . N R E P I T
1 X 1 = 5 1 X 2 = 5 5 1 X 3 = 5 5 5 1 X 4 = 5 5 5 5 1 X 5 = 5 5 5 5 5 I D N = 9 9 1 I F N = 3 3 3 I Q N = 7 7 7 0 0 2 0 1 1 = 1 . 5X R ( I 1 = 0 .X< I 1 = 0 .D 0 2 0 2 J = 1 . 5
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2 0 3 S Y l I . 3 1 = 0 .DO 2 0 6 1 = 1 . 1 1
DO 2 0 6 J = 1 . 1 1 TSUM( I » J ) = 0 .
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6 0 CONTINUEC P ( 1 1 ) = AP I 1 , 1 ) / C P 1 0 + R L )C P t 1 2 ) - A P U . 2 ) / I P 1 0 + R L )CP I 1 3 ) =AP I 1 . 3 ) / I P 1 0 + R L )C P I 1 4 ) =AP I 1 , 4 ) / I P l 0 + R L )CP ( 2 1 ) =—A P I 3 , 1 ) ✓P I 1CP< 2 2 ) =—AP I 3 , 2 ) / P I 1C P I 2 3 ) = - A P ( 3 , 3 ) / P 11C P t 2 A ) = - A P ( 3 . 4 ) / P I 1C P I 2 6 ) = —A P I 3 . 6 ) / P I 1C P I 2 11 ) =CP I 2 . 6 ) * C P 6 / P I 1C P t 3 7 ) =AP I 4 , 7 ) / I P 1 0 + R L )C P t 3 8) =AP I 4 , 8 ) / ( P 1 0 + R L )CALL G M P R D l C . S X , S Y , 3 , 5 , l l )DO 6 5 1 - 1 , 3 DO 6 5 J = l , 11
6 5 S Y ( I * J > = S Y t I . J ) + C P ( I , J )CALL G M P R D l C R » X R , Y R , 3 , 5 . 1 1 CALL G A U S S ( I O N , 0 . 0 3 3 6 8 , 0 . , V I )CALL G A US S ( I F N , 0 . 0 0 2 6 , 0 . , V 2 )CALL G A U S S ! I Q N . O . 0 3 3 6 8 , 0 . , V 3 )Y R ( 1 ) = Y R { 1 l + V I
YR1 2 ) = Y R ( 2 ) +V2 Y R ( 3 ) = Y P ( 3 ) + V 3 CALL G M P R D ( C . X . Y . 3 . 5 , I )DO 8 0 1 = 1 , 3
8 0 Y D ( I ) = YR( I 1 — Y CI )
FOR 5TH ORDER MODEL VtlTH 11 MACHINE PARAMETERS
DO 7 0 1 = 1 , 3 DO 7 0 J = l , l l
7 0 S Y T I J , I ) = S Y t I , J )CALL G M P R D I S Y T , W I N V , S Y T W , 1 1 , 3 , 3 )CALL G M P R D ( S Y T W , S Y , S U M , 1 I , 3 , 1 1 )DO 7 5 1 = 1 . 1 1 DO 7 5 J = l , l l
7 5 TSl iMt 1 , J > = TSUM( I • J ) + S U M < I , J )CALL GMPRD( SYTW,YD « Y W , 1 1 , 3 , 1 )DO 8 5 1 = 1 , 1 1
8 5 TYWl I ) = T Y W t I ) + Y W I I )9 0 1 CONTI NUE
CALL MI N V C TSVJM , 1 1 , D , L M T X ,MMTX )CALL G M P R D ( T S U M , T Y W , D E L P , 1 1 , 1 1 , 1 )DO 4 0 1 1 = 1 , 1 1P P G A I N = A B S C D E L P I I ) / P N ( I ) 1 - P L I M I T I F I P P G A I N . G T . O * ) D E L P C I ) = P N ( I ) * ( D E L P I I ) /
132
nn
n
♦ A B S t D E L . P t I ) ) ) * P L I MI T P N ( I ) =PN< I ) + D E L P ( I )
4 0 1 CONTI NUE
FR 5TH ORDER MODEL WITH 11 MACHINE PARAMETERS
DO 7 1 = 1 * 1 1 PRAT I O( I ) = P M( I ) / P T R U E ( 1 )
7 CONTINUEDO 5 1 = 1 * 1 1
5 E S T M A T t 1 + N 1 T . I ) = P R A T l Q ( t )9 9 9 CONTI NUE
N R E A P = N R E P I T + 1 WR I T E ( 6 * 9 )
9 FORMATt 1 H 1 . 1 O X * * R A T I Q » / / / )W R I T E f 6 * 6 ) ( ( E S T M A T t I . J ) * J = 1 . 1 1 ) . 1 = 1 *NREAP)
6 FORMA T ( 2 Q X * 1 1 F 7 . 3 )I F t NPARAM* G T * 3 ) D E L F A C = - 0 . 2 P F A C T R = P F A C T R + D E L F A C
e a a c o n t i n u eSTOP
END
SUBROUTI NE G M P R D ( A * 8 * R * N * M * L ) D I M E N S I O N At 1 ) , B ( 1 ) , R ( 1 )IR = 0 IK =-M
DO 10 K = 1 , L I K = I K + M DO 1 0 J = 1 . N I R = I R + 1
J I = J - N I B = I K R t I R ) = 0 .DO 1 0 I = 1 * M J 1 = J I +N I B = 10 +1
1 0 R t I R ) = R ( I R ) + A t J I > * B I I B )RETURNEND
SUBROUTINE MI N V <A , N • D , L * M) D I M E N S I O N AC 1 ) , L C I ) , M < 1}D=1 . 0 NK=- NDO 8 0 K = 1 , N NK=NK+N L ( K ) = K M ( K ) = K KK=NK+K 8 IGA=A CKK)DO 2 0 J = K ■N I Z = N * ( J - 1 >DO 2 0 I = K , Ni j = i z + i
1 0 I F { ABSC B I G A ) —A8 S ( A ( I J > ) ) 1 5 , 2 0 . 2 0 1 5 B I G A = A ( I J )
L ( K ) = I M ( K ) = J
2 0 CONTINUE J= L (K >I F C J - K ) 3 5 , 3 5 , 2 5
2 5 K I = K - NDO 3 0 1 = 1 , N K I = K t + N H O L D = - A ( K I >J I = K I ~ K + J AI K I ) =A C J I )
3 0 A ( J I ) =HOLD 3 5 I =MCK)
I F C I - K ) 4 5 , 4 5 . 3 8 3 8 J P = N * C I - 1 )
DO 4 0 J = 1 , N J K = N K + J
J I = J P + J H O L D = - A C J K )A C J K ) = A C J I 1
4 0 AC J I ) =HOLD4 5 I F ( B I G A ) 4 8 , 4 6 . 4 84 6 D = 0 . 0
RETURN4 8 DO 5 5 I - l . N
I F C I - K ) 5 0 . 5 5 , 5 0 SO I K = N K+ I
AC I K ) =AC I K l / C - B I G A )5 5 CONTI NUE
DO 6 5 1 = 1 , N IK=NK +1
134
H£JLD=A( I K )I J - I —N DO 6 5 J = 1 , N I J = I J + NI F t I —K ) 6 0 , 6 5 , 6 0
6 0 I F ( J - K ) 6 2 , 6 5 * 6 2 6 2 K J = I J - 1 + K
A ( I J ) =HGLD*A t K J ) + A 1 1 J ) 6 5 CONTINUE
K J = K - N DO 7 5 J = 1 , N K J =K J 4-NI F ( J —K) 7 0 * 7 5 , 7 0
7 0 A ( K J J = A ( K J ) / B I G A 7 5 CONTINUE
D=D* BI GA A ( K K ) = 1 , 0 / B I G A
8 0 CONTINUE K- N
1 0 0 K = ( K - 1 )I F ( K ) 1 5 0 , 1 5 0 , 1 0 5
1 0 5 I = L ( K )I F ( I - K ) 1 2 0 . 1 2 0 , 1 0 8
1 0 8 J Q = N * ( K - 1 )J R = N * { 1 - 1 )DO 1 1 0 J = l , N
J K = J Q + J H O L D = A ( J K )J I = J R + J
A I J K ) = - A ( J I )1 1 0 At J I ) =HOLD 1 2 0 J = M ( K )
I F t J - K ) 1 0 0 , 1 0 0 , 1 2 5 1 2 5 K I = K - N
DO 1 3 C 1 = 1 , N K I = K I + N H O L D = A ( K I )
J I =K I - K + J A t K I J = - A ( J I i
1 3 0 A ( J I ) = H n t D GO TO ICC
1 5 0 RETURN END
Wtn
1,SUBROUTI NE G A U S S C I X . S . A M . V ) A = 0 , 0DO 5 0 1 = 1 . 1 2CALL R A N D U C I X , I Y , Y >I X - I Y
5 0 A=A+YV = ( A— 6 ■ 0 I * S+AMRETURNEND
SUBROUTI NE RANDU( I X « I Y • Y F L )I Y = I X * 6 5 5 3 9I F ( I Y ) 5 , 6 . 6
5 I Y = I Y + 2 1 4 7 4 8 3 6 4 7 + 16 Y F L = I Y
Y F L = Y F L * . 4 6 5 6 6 1 3 £ - 9RETURN
END
FUNCT I ON F C N ( T . X . K . N D E T )D I M E N S I O N F U O ) . X ( I O )D I M E N S I O N ARC 5 , 5 ) . A ( 5 . 5 ) . U ( 5 ) . U S C 5 ) COMMON A R . A . U . U S F ( K ) = 0 .I F C N D E T . E Q . 1 ) GO TO H O DO 1 0 0 J = 1 , 5
1 0 0 F ( K ) = F C K ) + A ( K , J ) * X ( J )GO TO 1 5 0
H O DO 1 2 0 1 —1 * 5 1 2 0 F ( K ) = F C K ) + A R ( K * I ) * X ( I )I S O I F C N D E T . E Q , 3 ) GO TO 3 0 0 2 0 0 F ( K ) = F ( K )4-U ( K )
GO TO 4 0 0 3 0 0 F C K ) = F ( K ) + U S ( K )4 0 0 FCN=FCK>
RETURNEND
136
VITA
Chi Choon Lee was bom on January 6, 1941, in Seoul, Korea as the
son of Hyung Mann Lee and Su Boon Kim. A re s id e n t o f Seoul a l l o f h i s
p reco llege l i f e , he a ttended p u b lic school in Seoul. A fte r graduation
from Kyotong High School in March 1960, he en te red Seoul National
U n ive rs i ty where he earned h i s B.S. degree in e l e c t r i c a l engineering
in February 1976, a f t e r se rv ing in the R.O.K. Army fo r th re e years .
With h is B.S. degree, th e au thor worked fo r Han Young E le c t r i c a l manu
fac tu r in g Company, Seoul, in 1967. In March 1968, he jo ined United Power
Systems Control Company, S e a t t l e , Washington before he en ro lled a t th e
graduate school of Louisiana S ta te U n iv e rs i ty , from which he received
h is M.S. degree in e l e c t r i c a l engineering in August 1970. A fte r working
fo r Federal P a c if ic E le c t r i c Company, M ilp ita s , C a l i fo rn ia , from 1972
through 1974, he re tu rn e d to Louisiana S ta te U n iv e rs i ty and received
h is Ph.D. degree in August 1976.
E ffe c t iv e August 17, 1976, Dr. Lee jo ined th e f a c u l ty o f th e
E le c t r i c a l Engineering Department, U n ivers ity o f Colorado a t Denver,
Denver, Colorado, where he i s c u r re n t ly teaching and conducting resea rch
in the a re a o f power and c o n tro l systems, which c o n s t i tu te h is major
i n t e r e s t . He has co n tr ib u te d a number o f a r t i c l e s t o the c o n tro l and
power systems l i t e r a t u r e and p resen ted r e s u l t s from h is resea rch a t
reg ional and n a t io n a l symposia.
Dr. Lee i s a member o f th e honor o rgan iza tion o f E ta Kappa Nu and
Phi Kappa Phi as w ell as th e p ro fe s s io n a l o rgan iza tion of IEEE. He i s
p re sen tly a member o f th e Power Engineering and Control Systems so c i
e t i e s of IEEE.
137
EXAMINATION AND THESIS REPORT
Candidate: Chi Choon Lee
Major Field: Electrical Engineering
Title of Thesis: Synchronous Machine Modeling by Parameter Estimation
Approved:
Major Professor and Chairman
Dean of the Gradcafe School
EXAMINING COMMITTEE:
'i L
C ,
Date of Examination:
June 25/ 1976