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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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71. M. M. Elmetwally and N. D. Rao, " S e n s itiv ity analysis in power sys­tem dynamic s ta b i l i ty s tu d ie s ," IEEE Trans. Power App. S y s t.,vo l. PAS-91, No. 4, pp. 1692-1699, July/August 1972.

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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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

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DO 2 9 6 J = 1 * 52 9 6 C R ( I . J ) = C ( I . J )

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

2 C 2 A ( I * J ) = 0 .DO 2 0 1 J = l . l l

SX( I . J 1=0.2 0 1 A P ( I * J ) = 0 .

DO 2 0 3 1 = 1 . 3DO 2 0 4 J = 1 , 5

2 0 4 C < I . J 1 = 0 .DO 2 0 3 J = l . l l C P ( I . J 1 = 0 .

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 .

2 0 6 S U M ( I . J 1 = 0 .DO 2 0 9 1 = 1 . 1 1

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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