laha asoke kumar 1972 - university of saskatchewan
TRANSCRIPT
:' -.
.
, ..'
'
.. �:.
.' .'.
•'
•••••• * ••••',' :.:.
. ':. :
"'.'
.'. . .'. '.
.'. ",
..'
: ".'
.
:'
. ':,
:.
. TEdiNtt)UES FOR DESIGNING CONTROLLERS".
.. OF A. MULTI-MACHINE .�YSTEM ...
: .
.
.. '. '.
.
,
A �heS1s'.
S"bmitted· ·to the Faculty o':Sraduate· ..Stud1e.·S.. '. '. .
.
in Partial· Fulfilment of the ftequ1r.eme"t$·: ..
". ftr the ()egr.e. Qf. .:.
.
.
'
.".ster '.'of':Scitrtte
.
," .:
.
.
. '.' :. . '.
"-:
-,': .....
'.
••< ':. :"."
..•..... bY···.'.
.
.'..
.
.
:... :.�... ..'. :
......... ; . ................'
.... '
..
.
"
.'
.
'. �:.. ,
••;'••••••• I' . ::, '
...... .: ':".
• saskatoon."Saskatch,ewa� •.. '
•
JU1y.:,:i�72'
..
. ::.!'
.. :", ..
. ". .... .' � .
"
'
..... .' ;':: .
. . ".".
"
.'
.
The' auth«r claims 'copyright� Use shalf' not be· made of the fftlterfal' .'.:.
.
contain. ·hert!1n without' proper acknowledge"""t••• ,:"clh:ated on....
.
the fol1�nq paqe.··." .
.
".
""'..'
• ••
d
. "'.'
.
,. ' .'
�,•
#•
"::. '.',
.''.
'
....
..; '.
''
•'
... '. '.1 .
�..
. .'
,; ." .
'. ,.... .
.............
,'
••i
,
T�e author has agreed that :·the : ·Ub.�.�. �1ft"1 ty of ,s..skatchewan .·
shall' make this' thesis' freel, av.ilable for inspection. Moreover. theauthor has agreed that parmisst. for ext�stve copyfn, of'thh thesis .•for scholarly purposes may bf! grantecfbyl,tJte pr�essor' or. professors
·
Who supervised the. thesis work .·��o�ed· heNi,,·.. or� in the·i'r:.ab$ence. by' ..'.'
the Head of the Department' or the nean of tht Col'"e ,fn Which the......
thesis work was ··date. It is 'understOOd that due reCOgnition will tie·.
'" given to. the. autho_" of thft. thesis' aft(f: to··'the U,ftfYers1;�y of ,Sask.tch.wan·.1n any use· of material in this thei.f$.· Copyfnt or.: ..pu6·l1catton:. G!' any .
.
other use of the thesis 'for ffnan�i.l· O,f", w1� ItPp�".l·�y :the .:.
· Unh'ersity of Saskatchewan. and ttl'- .ut�or·"·s ··Wit.. '""'ils1on is'..
.
prdafbfted.·..
';', _.: ".,. :
..i:' ':' :. :.
.
. Req",sts for'periiissfon to·t�py:.. or:. to :,,�.·:'Other·us" .C)f·'ma�r1·al";·in this thesis frt �ole 0; "n' p�,.t sho�i4··.be ��"'�� toi:: '::'
'. '... .' '.'.
. '. .
�. : ". '.' ." , '.' .•
' .. � .'.... . . .',
. '. 7' c,. ';" I ' ....I '. '�. '. ' .
..
.....:..': : ...'
..
..
'."
", '.' .:..
: .. '. Head �:.th�·.Depar_flt· rt Electricai.f!n91ne.pf"�.·:. �"
,,:.,:..'
�fv'"-f;ty of saska�ch""ari. .
.
. .' ': : .•.••....•.•..• '
Sask'toa.. Canada. .•.. .".
'. , .. ,
. .' ,.' ':" .
.'.
.
.':. '.,�..
..... '. ".
; ..... ," ". '" .
: ....
'\;- . "
...'
'"'.
, :. ...
'
.� .. '
.'.'
.. '
.'..
.
. fi·'··..
: ..:...,:,
.
" ".
.
....
,,:.
", ..
.' .' .
.
'""
,
. .. .: " . ,
"
"
.
",
,,' .
. '.; .'
'..... ".. .. : .....
, " ACKNOWLEDGEMENTS" "'
'
The a�thor wfshes to express his 1ndebtednes$' to' Pt'Ofess�K�E. Bol11nqer far sugqesting th1,s project arid ,for:'his �u1dance.
'
encouragement and 'assistance dQr1n�J the cCllf.se' c1. the i-esea.-eh and,.
. . .' .
.
, ,
preparation, of this thesis.,'. ' '\. .
.
The 'author also wishes to acknowledge the saskat�hewan POwer.
Corporation for providing the data used in this thesis. '
, ',Acldu.ledgellient of fineneil'l �UPf)or� by the ,N'at1dri., _.searchCouncil of Ca�.da, under:Gra"t No�' :A5124 fs' 'made. ·
'
• .' �'. M •
. . :'
"
'.'.'
.'
0,'.' • ;.. t' .�' "
'.' � ,r: \0
, '
..�
...
...
. ....
: .' .' � "".' .-'
'..
'
'··.0.
.
.�.: " ,
'�
.
0: .
:'. :..
'
.. ' .:.:
... ,., .
",: itl, .....
"..
.'. :
. .
.
:'. .
..
':.::
. .'.:' .'..
'.". .
.
..
UNIVERSIty '* SAsKATCH�Aif.
.
Electrical Eno1neerinq Abstract,72A147' ".
• '..""'. . .�. .'
.
.. .•...
.'': ":'. i·· ....
", '.'
. .'
.
" ,.'
.• -TECHlugUES· 'Oli D�S!GI!I"G.
ctIlTROLlERS OF A HULTI-MCHINE SYSTEM" ..
. ; �
Student: AsOke KlIMr Lalla·...
... .. .. S�.GI': It.�. 601Hnlll'r
..
-. M�Sc. Thesis pr&$ented to the C�Uege of G�ad.uate· Studies·, du·ly .,.'1912 ••.'. .'
. ." .
.
. .
.' . . . ..
.'
.
..
.
A�STRACT .•
'. .
... In recent years consider,ble' _phasis hi. bt�n. placed orrtt)e ..'
.
• enhance.ment � dynl�ic stability'Of ,�r S.��t_�;"�hfS;fnvolves the •..'
.. ,' ".:' :.
addition'of external coriboilers or .�""$at�S. to the eXi$t1t'tQ'" ..
. '
. ..'. . .. . .. . .
.
.... ';',� ..... ". '.. '.
.
.
power p lan·t..
.. v .
•.
•
'. ;' .', i.'
'. ·.ThiS '. thesis'. describes' the ,·a·PPneat.1,,", o'.�.ffe"n�<tech",gUes·.. "
•.,
'...
• •
":'"
.: ," .> .'•
1nclud1nq �timal 'cmt�ol theory in ilnPro.v.irn.tth•. dY.n.Jmi� stabn+ty· •.•.......
.. .. .'.' :' .... .' .':. .'
�
: . .'. :' . .'
of the multi.machine system. the c'�l�)t: �.:r� 'sY$tetYi 1s con�fdf!red.
as a dual .. leve' forin 0' systeftl M,prese,ntati()n, rathe� ·than: a s1n('lle.'
mac:hine�1nfin1te 'bus 'SY$�; 'The' pel'ti�tm�' s".11�,qn�1 :equatfms.
'. . . .
,
.
.: ...' .
'" • t' .�
. : ,"-, '. . :"': �:.. . ". .
.
.'. .
are derived and· 'the. state .Plce "',�etJs,:deve,lt)�. Different tech- .....
n1ques' ,including optimal control the'cry lllld. its modff.1eatiofts' and',
conventf�naJ 'eigenvalue searCh' .thods: 'have been' .pplf,d in design-.
1ng' dif'e�nt types of c�trol1ers 1" D'der to �1ve improved'system .•.per'forNnce, in the dynam�c· qleratfng' r'�ge of' a· power syst.... 'Aprllct1ca' appl1cati:Qn, fnvolv1nq' the �e�i'" of f.,;cen�r,011�r fqi', the ....
.
'.
Squaw Ral'fd$ Gener.ating Stat.1on,of s�;�atchew.n' P_�'·CO""4)�8t10n-'1:S. '
..'.
'.' .
..
. :.;'"..
". ".
':'. :', ':.'. ,
,TABLE OF CONTENTSPaae
COPYRIGHT H
ACKNOWLEDGEMENTS'
i {i
ABSTRACT iv,
, TABLE OF CONTENTS '
'
v
LIST OF ILLUSTRATIONS AND FIGURES vHi
LIST OF TABLES xii
"1. INTRODUCTION t
,1.1 (;eneral 1.. ..
1.2 Review of the previous'work 2
1.3 'lAtHne of research reported in this thesis 7.
.
. .
2. THE APPLICATION, OF OPTI";AL CONTROL THEMY TO PCMEP. SYSTEt-4 10
2.1 State space modellin� 10.
.
.
.
.
.
. .�. .
2.2 Pertinent optimal control theory' ,13
2.3 Power system representati,on 19
3. STATE SPACE MODELLINn 'OF A MULTI-MACHINE SYSTEM 22
3.1 Introducti on 22
3.2 t.1athe�tical model of tie line equati ms '23'
3.3 Mathematical model of qenerator equations, 32
'3.4 Mathematical mode' of the plant controller, 39
3.4.1'
Governor and pr1.me mover 39, '
3.4.2 Automatic voltaqe regulator'
41
3.5 Transf�tion into state space form 44
3.5.1 Stat� variable, fO","ulat1on for th� • study' system '
'���'' �
,
3.5�2 State variable formulation, for the 'external'system 'generators 50
.3�5.3 State variable formulation for the tfovernor-. prime mover of .the 'study'system and 'external'
.
system plants'.
51.
3.5.4 State variable fonnulati·on for the' automatic. voltage regulators of the 'study' system plants 51
3.6 Reduced state space model neqlecting transfomer action 53
4. DESIGN OF THE OPTIMAL STATE REGULATOR 58
4.1 Introduction 58
4.2 Solv1nq the matrix-Riccat1 eguati on 60
4.3 Basic system specifications 6S
4.4 Desi� of state regulator and trlfncated state regulator 68
4.5 Time domain results 83
5. DESIGN OF A SINr,LE TRANSFER. FUNCTI� DERIVED FROM TUE fPTI"1ALSTATE REGULATOR 92
5.1 In.troc:fucti (J1 92
5.2 Modification of Pearsm's Theory.
. :94
.5.3 Des.ign of the opti�l transfer function for a low order 10.0system
5.4 .Design of the optimal transfer function for the system: '.' under �study ..
·
.
107
6. ADDITIONAL STUDIES . 114
5.3. Feedback of twO. control variables.
114
6.2 Design of d(llble feedback optimal state regulator '. 117
7. LEAD-LAG AUXILIARV CONTROLLER-DESIGN AND COMPARISON 128
7.1 Introduction 128
7.2 General form of a lead",lag feedback.. 129
7.3 Selection of the optimum parameters of .the c�trollersusing eige.nvalue search 133
7.4 Comparison of results . 139
8. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARC� 142
9 •. LlST OF REFERENCES 146
.
vi .
10 •. APPENDICES 153
'10.1 Appendix. A 153..
10.2 Ap�ndix B 157
10.3 ·Appendix C 174
10.4 ,Appendix 0 179
10.5 Appendix E·· 182
vii
<
••
••
LIST' OF 'ILtUsrRATlcIIS MD FlUES
.�� �1.1 AsS_d structure of overall feedback control sYs" 3
2.1 The block df.gr.. of • system represented by equations(2.1) and (2.2) 12
.'.: .
.
'...'
3.'1 Gene...,.conffgu....tton of ttle subsysa. 24
3.2 $Ct._tfc representatfon shawlng the cOnnection of the.
jth node to other nodes'..'
25
3.3 Df.,.._ sbowing the positions of Q axis'� dfffeNnt.
•chines with 'respect to the.NfeNnce·· f... 27
3.4.
Gene... '· block'dt.g....of the govemOl'-PMme ."e..
..., of ·study" syste".. and ·extem.'· systelt plants 40.3.5 Voltage NgUla�·canponents· for a· ..otary ty.,e exciter .42
3.6 Block diag.... for the autoMatic voltage regulato.. ·of . the.
·study··.systetn generating plants ." 43
O1agnll showtn, the ,art •.tion 01 K( or) with " :. . .
SchelBtie representation of • portion of the Saskatchewanp.,... COl'poratfon Systell used as en illustrative ex..."l.otlgmt showing the loeet10ns of doIrInant syst. roots f.n
cOllJPlet.e S-planeSquaw Raptds Sp"d error ,ersus time for the uneOMPBns.tedsYStc. (tncl-..df1'g transfcr.-r action) .
.....
67
74
85
4.4b Queen Eliza�th speed error versus tt. fo.. the. unc..,en-sated system (tneludfng transfonaer action) 85
4.6a Squaw .Rapids speed error .."US ti. for the unc.-n,.tedsystem (excluding' ·transf�r aetton) 86
4.Sb Queen EUza.,.th speed e...-or versus time for the ....e..nsateet
S1$_ (exclud1n, tr.nsfo.....ctton) 86
viii
Page
'4.6a sq-.w Rapids speed :em»r "rsU$ ti••n the systan is
coupled with ',the' state' regulator 88
4.6b Queen Elizabeth spaid err"" versus ti. when the systemis '�oupled with the state regulator ,
88
4.7a $qu. Rapi4s,speed emr versUs ti. when the systeM is'coupled with the t.runcated state regulator' ,'" 89
4.7b Queen Elizabeth speed error versus ti. when the systemis coupled with the truncated 'state regulator,
'
89
4.81 Squaw Rapids speed error versus tile'<a> UftCOlllf)ensated system
,
(b) apprac11t1te state regulator(c) ,stlte regulator 91
,'
4.81;) Queen Elizabeth speed error versus ti.
(a). UncOlPlftsated systeM ,:
(b) approxi•• state regulator.(c)' state regulator 91
,,
5.1 Gefter,' block diagram or a plant coupled with the optf.''
'
transfer function '98
5.2 T1. dcafn resul ts. ffir hypothetical s15_ for all the
three cases 106
5.3 Block ,�fagrllt fI �e. ·autOMatic voltage regul.tor. of the'
Squaw Rapids plant cdlpled with the' opt1a., transfer',function. 108
'S.4a '$qua" Rapids speed ,ferror ,ersus ti.' when the sy.stem is
coupled 'wt�b the opti., transfer'fuoct1on5.Se Squaw Rapids speed error verS.us ti. for t... ",c__nsated
sY$_ 113
5.5b· Que.. Elfabeth speed error ve"�s' time .10r the unCClq)en�
sated system
112
113
1x
. P.ge·. . "
.. . .
6.1 Sch_tic.
di....·
01 the· excttatfon.
and' gcwemor-,n·t.lIWer '" with two utemal' feedback variables' 116
6.2a squaw. Rapfds speed elTOr versus tf. �en the system fs
. coupled wf·th opti., state regulater with two feedbacks 124
6.2b.
Queen Elizabeth speed elTOr versus tf. when the systeMis coupled with op.tf.' state regulator with two
feedbacks .".
. 124'.
6.31 Squaw •. Rapids speed error Versus' t1. for: the un�en';'-.
sated system .
6.3b Queen Elizabeth spaid error 'WI'SUS tf. for the.'.
anc(llllpen$ated system .
. 6. 4a Squaw Rapi'dS' speed e,,"r versus' ti. when the sYsteit 1 s .
.:
. coupled 'with opti.,' state regulator with one feedback
6.4b Queen Elizabeth 'speed'errCl' versus ti. when the systelltfs coupled with Optf., state regulator with one
feedback:
125
125
126
'.126
"..
7.1 . Block df.agl'lJl of aux111•.., control' 129
7.2a Squaw Riptds speed err.. versus ti. for the sys. coupled .
with lead-lag feedback auxn1ary contro11e.. , 136
.7.2b Queen Elizabeth speed errOf' versus ti. for the·s1st.coupled with ,.d.llg feedback aux111ary controlle.r 136
7.3 Ge�eral auxfliary controller conlllCted to the Squaw Rapfdsplant 131
7." Squ.. Rapfds speed error versus ti.Ca.) unc;�.ted· sys_'(b) �pproxf..te s�� regul.tor(cl 's�te regulator
'
..
'
(cI) le,d.lag feedback 140'
'. : "
'.'
..�'.
.
,; :
.' .: .'
....:.
. :'. .....
'. '.
7.4b QuMn Elfzabeth iPeed error versUs tt.,"
(a), uncGMpefts.�d systell"
(b) approxt.te state regulator'(c)' state regul.to, ',",(d) lead.1.g f�ck
'
.
,
...
" .... -'
"
."
.'. :':"':".
.' .' '. .
.: xi '
... :....
';'
" ."
.
'.'
.... , :
Page
,140'
.
LIST' OF .
TABLES .
Table
4.1 The.char.Ct8rtstfc ..trtx· A of the unc....sated systeM'.
(with transf"'r .ctton excluded)· .
.
.. .
.
4.2A· Eigenv.'ues of the utlCaMpens.ted systeM. :W1th tr.nsfo,.r
.ct1an exCluded'.' .
.
4.28 DaMinant roots 'of. the above ....t1oned 51stan.
.
. ·4.3A Eigenv.1Ues of. t... .....,.....tecI systeM with tranifonner.' actton 1"cluded '. . '. .'
.
.
'. .'
.
4.38: DGllfnant roots of the. above _tfoned system4.4
.
"11 _t,.tx.. ".4.5 . "21' ·.trtx ....
.'
4.6 K* .trtx·
.
4.7 The char.cteristic· .trfx A'll of· tilt· cCJllpenSated .,s.couPled with the stete. regUlator'
'..
. .
..4.8A· �fgeftv.'ues of the' c""sated syS_ coupled.with stlte
regUl.tor·.
4.88. OoMflJlftt roots of the abewe .ntfoned sYsteM4.9A Eigenvalues. Of the caapensatecl. systeM caupled.:with .
'. truncated state regul.tor.
.
4.98 DaIit1nant ,.ts Of the above .ntfoned s,stell. .
.. .'
.5.1 Etgtnval. or the lQlCClllpensated sys_.
5.2 Eigenvalues of the c..ns.ted s1$- with the optf.'state re,,'ator .... . .
.
5.3 E'genvllues of the e�nsated s'"teIl coupled with tht
cptt., transfer '...etton..
5'.4 ConIpar1son of d"'nan� $"_ roots'
P.ge
69
71
71
.,'
72.
72
76
77
78
80
81
81
84
84
102
103
104 .
110
.
6.1 . K* .trtx 110 .
6.2 The character1·stfc .trt)( A. ·of:the COIIIP'ltSated sY$.t- 121
."..t••
xii
Page
6.3 Eigenvalues of the e......sated systeM eq�1pped with two
feedback GPt1.' control variables 122. 6.4 CaMp_rison of doIIfnant systelll roots
. 123
7.1 SysteM roots for varying. auxiliary c.troller paraMeters 138
7.2 CoIIparisan of daaiMnt s,Stell··roots 141
xiii
1
1.1. a.eNl.
Power SysteM voltage and frequency are controlled near accepted
values through the use of autGllttc voltage and' frequency. regulation. .
devices. DeViations' fran s.tepoint values of voltage and frequency. artIe
as a. result of cansUlller load', delJllnd fluctuatfons or system dfsturbances.
. . .
due to faults or s�tch1ng. For many years fnvestigations ,of power. .,
.
system stabflfty was ltmfted to two specfal facets of the prtblem -
'
tNnsfent stabf lit, and steady state stabilfty. To detentfne whether•
the p_' systeM fs stable after a disturbance ft is necessary to plot.:: ... .
... .::
.
and fnspect the swing curves. '!f the system fs stable fn steady state. .
and also on the ffrst .�wfng. it is a well established fact that it will
not be usvall,Y' unstable fn "ttMt subsequent perf.od,· for there � 11.
be
enough d�fng to reduce the ..1ft� of the mng� The.
latest 'Uter-. ... .. . .
ature1,2 'has shoWl'sOlle interesting developments in the :i"" of stability."
It has been reported by S�leff !t,.!!..1 that stabi1fty curves begin to
.
shelf divergence �th machine groups punf"g out of step after several
osctllaticras rather than the first swing after a .disturbante. In short
'"sufffcieflt damping often beca. evident. Hanson !!!l.2, whileinvestigating frequency oscillatfons bebeen the fnterconnected utility
systems of S�katch_n. Manitoba and �tarfo (_st) had drawn sftrilar
" cORcl"sfcns. The reason is that although there Ire sever., SOll�s of,
.
positive daMpfng fn p�r systems, there are also sources of ne�t1ve
,41....,1"g, nC)tably voltage ,,"ul.tors and sl*d governor$. The s.,1
synchronizing power' coefficfents' due te lirge electric distances between',
,
majGr groups 'of gene,.atfon and low time constants 'Gf mode"" high-speed,
,'
'VGltage·�gulators aggravate the situatfon. Furthel"lllOre although' theinherent pOSitive damping predcmfnatesunder most situation�,'frt so.
, ,
ciJ-CUlllStan,ces· the net damping fs' negative. With negative net d.mpin�. anqular
swings Of' the machine. instead of dyfng out. increase _til either'equilibrium IIIIPUtude (If.it CYcle) is reiched or syncbrOnism i5,lGSt•.. : . .:.. ..
. .:". '.'.' '.
If the inherent damping fS' negative ane c_'re is to, add arti1icf.""pMitive damping. Fran the'dffferent stud'es and field tests it has
been found that f"'Proved system dutpfng can be achieved by adding
supplementary. tit' stabilizing. ,control to the voltage regulatGrs at
selected plants. III tIM! contrGl theory terms this leads one fn visual ..
1z1ng that one "st construct a ��YSfCal d�v1ce which shall be talled the.
.
.
.
'canpensator.' or 'c:uatro}ler'" and which when driven by the, sensor ._asu-
,rement signals' will gtnerate the c_nded inputs �o the phySical'.' .'
'... ..'. .
.
'. .'
process such that the ti. evolut.ion of the physical plant state ",
.
.
. . '. .
variables is' satisfactory for the applfcatfor, at hand. The' feedback
structure is fllustr�ted 1n Figure 1..1.
1.2. Revf.., Gf the Previous Work
Power system dynlmfc stabi1fty has �een th, $ubject of e.xtens1ve
investigations by man� authors, to ascertafn a control system design"
,'",'atf" to, the p,rfonnance of fndfvid"", ""chines. A rlc'"t paper3 has.
'.
.
.
,
s...rized the backgrQUnd of eire_tenees and' hi$torf�l develOJ)111ents
of the stability probl_ fn the w.stern $tates and �he work 'that has been
... 3
.,.
-COMMANDED· .INPUTS TO ..
..
, .... PHYSICAL . P.ROCE·SS...
.
..
.
..
. .
..
. ·PLANT::
,.
. .',.':.
.
_____
L. COMPENSATOR �
".
___..
.....__-----II (to be des.ig"rled.) ��---
.. SENSOR·MEAsUR�MENT.
·SIGNALS
Figure 1.1 -. AssUllled structure of ove..an ·f_edback control· syst_ •
. ....
',:'" .:
' ..
4 ..
. ... �
done on the ·prob lem.· ...
The. methods that have been used· so. far ina, be. classified as .
f�l1ows:.
.. ·a) Rotith-Hurw1tz criterion: In .this method the sign .of the.·1'eal
parts. of the roots· of the .characterfstfc equatfon is calculated on
the basis of· the coefficients of the equatfons. Thfs approach has·
been effectively used by Concordia. vi. and Gthers5,6.7,8. Its _jor.. disadvantage is that. rio fnformation regarding the degree of·stabilfty
.
is avaflable..
.
. ". .
.
.
. ... ." .". .
...b) Nyquist Criterion: This classfca 1 c�trol technique has been
used by Mess��le and ·Bru�k9. Jacovides and Adkfns10• Aldred and·
Shackshaft11 and··Ewart;and De�ell012�··Even though this .thod is ..
capab·le of showing the degree· of stability and indicating the possibleproc�dures for fts fmprovemt!nt•. it suffers from the disadvantage of .
extensfve canPutation time..
..
...
... .. .. .
c) Analog Computation: This method ·has been used by Messerle13...
and Heffron and Ph1llfPs14.Whf1e ft pennfts the detenninatfon of
stabflity.by dfrect cbservation of the resulting perlormance� it is
lengthy· and tfme consunrfng.. ....
d) Root· LocuS; Using standard root locus techniques, Stfpleton.1Sinvestigated the effect or varfous voltage ""ulator paruteten .0" .
stability and dyna�c response.
e)· D..Part1on or �in separation: TMs method has �en lISed by ...
.
. 16.
·7 12 18 19... ...
....
Stlrana·· and Harfharan and others '. • .' .•. Since 1� renders possible
5
,
"
the graphical ,representatf'(;n of stab111�y Hm1ts in the plane of one
or �wo parameters of interest. ft appears to have considerable app1i�"
'
cation to problems involving the selection of regular parameter
,'values for optfmum results.
f) Matrix Techniques: These techniques whfch are based: 'on "'odem'
cOntl'O1 theory, have been suggested by Vln Ness30, LlughtOn20 and
Undrn121• Using 'the cOn,capt of state variables, 'the dffferent1�'i:�'.
...
equations deSCribing the small�signal perfonnance of the synchronous,� .. .. .
..
machines' and' assocfated control' apparatus are expressed in,:,'vector -'
matrix fonn. '�ce such a fOrmulation is achieved, a variety of met�odsthat have been'dev�loped for'thfS fo� of representation can'be're�dnyapplied.
'
Prev'ioUs studies clearly advocated the', necessity,0' a pOwer, systemstabilizer,tIIe functten o{ which ,is to ,sense shaft'spfted or" 'requency
.
.... .. .. .
'.�fltfon. conditiOn the Signal thu� obtained. and then'; feed it into '"
. ."
the voltage regulator as a supplementary signal. The subsequent 'studies
had concentrat4td mainly on ,the methods of '(les1gr) of sueh cattronerl �,."'
the pioneer being F.R. Schlief 1.22-25. His philosophy of design was to
calculate the frequency response" of the system from field tests and. .
..
the" adjust the gains and time constants of the stabilizer to canpensate
for the phase lag of the syStem. 'n his recent paper25 Schl'fefcalculated the damping coefficient Owh1ch is given by O-'M Sfn: e
where M is the' ga,1n an4. e is the phase lag obta1n8d frOm the f.req�encyresponse curves. His analY$.1s showed ,that for all frequencies �f inter.est tht �a�1ng coeff1c1ent is n�at1ve for the �"compens��d system but
.
. ..
" .
6
pOliti ••; "for the'cattpeftsated system. Similar 'approaches had 'been:
reported,1n O.ther literature :26.27.28'. Bollinger and Flem1ng29 used. .. .....
f�uency response technique to' dei1gn the auxl1iary controller with. ".
.
'.'
,
the help of a� analOg contpUter. other researchers att8lllpted to s1mulat.
a ponion O.f the poWer sys. on the analog computer22 as we11' as' on'
'the digital cc.puter2•31 and then adjust the para_tars of the control
ler to fncrease the dlMPing O.f the overall sYstem.'laughton32 and Kasturf
et al.33 designed the controller using sensitivity 'analysis. Grainger�-". . .' '.
'"
and' Abinerf34 and Undri1135 utilized Lyapunov's method to increase the
d..�ng of the pCMr system. Krause and Tow1.36 investigated tl1e effects
of two 'field windings instead' of a conventional one to' achieve the', sameobjective.
Recent, cleve1�nts of optimal control, theory ,prtlllpted many
researchers to' widen thefr methods of attack. The prime' philoSophy O.f'optimal control technique is ,to' select, Control over, an admfssible
'control· set such that a cost (or� performance) functional is minimized
(or max1m1zecJ ). 'u !!!l.37 'designee. the opt1.' signals fed 'to the:
vol�age regulator and the governor Qf the power plants on the basfs of '
Pontry,g1n's .xillUl pri"ciple. ,In another paper38 the authors conceived
the voltage re.gu'�tor .s the plant and then designed the optimizing ,
,,
'
stab1lf�1ng s1gnal feedback to the voltage regulator. The stabilizer
thus 'designed .s tried on an actual nem-linear model of ,the powr system.Davison and Rau39 used the theQr,y O.f para_tel' oPt1m1zatf�ndeveloped by
Athens and Levfns40 to desfgn the QPtillla1 controller b�sed on the output
7
..'
, "
feedback. All the above pape" used the quadratic cost functional.Yet another gm..rp of. researchers41,42 have thought of the problem as
•
....
.
, ,
a' time ·.opt1.' one. Their controller seeks to drive the powr syStem to.
a stable target state in mtn1l11UM time. This fonnulat1on leads to a
classical bang-bang control through the solution of a non lfnear �
point bouiu:fary value, prOblem.: nits theory has been ,used in the past in'
transient stabflity studf" of, the power sYstem by Jones43 and Sm1thlM.A review of all the _bove papers reveals that the solution of the
, .
probl. is far fl'an complete, although a large number of papers have. .:.
been' published so far. 'The appl1catfon (jf optt., contr()l, theory is"handicapped by the abSence of a well..oestabl1shed theory of outp�t feed
back. :Moreove� the mathematical model of the plant used by mOst of the
authors was the simpliffed single machi,ne-infinite, bus system�.
"
. .,' ..',: ..
..
.
"
1.3� 'Outline of Research ..,.rted in This Thesis. ','.
In'this thesis � power system comprtzing i number of generating, '
,
stations has �en modelled in a dual-level form of representatiOn. The
pertf·nent algorithm has . been developed to accOIIIodate the above fom
of representation. Optimal cOJ'ltrol theory and other techniques are then
applfed to this mathemati�l model in o�r to,invest1gate the effects
of different forms of controllers on' system damping.
In the ffrst etlJpter the pertinent background, meterial has been
revf_d briefly to l1lustrate the methods which are now used to design
au�111ary controllers or compensators. These techniques are quite arbi-'
'. '.
trar,y and require a cons1derable measure of subjective judgement and,
....
�.. ...
..
'.8..
e�per1eftce 11 they are to b.e used stiecesS.full.v•.·The ·probl... of t.rOv1n,·•...,
.s)is_d.pt"g for'
•.�'t1",chfne sys_ :ust'n� .' 'dua,,,:'e.'·.· 10l'Il of .
...
'. . .. ... .'
syStem'rep,.sentatton fs··taken up fn the :subs..uent chlpt..,. of thfsthes1·s•.•.
'
.. In ·the second' chapter the var1OU�' iIIth_tf�l COfttepb requfY'ed ....
to tmpl.ent the .•thod are discussed br1ef1Y�''rile par.t1�lar:"th�·::..
' .tics·Nqufred is dnwn '"'" the areas : of.,state: spece theory"and'.
optfmal control theory... .
' .
. The applicatfon of any contrOl theor,y' necessitates the fafthful.... .' .. .
math_ttta'" description ·of. the' :e,xfstfng pl.nt. "... thfr.d .t:h.pte;rf$; .''
..
.
'devoted 'to the'mathe.'ti·cal·
...nu,. ':Of� the ,"'ti..thf�'· ."S-'"
.
including the plant :contrct1ters•. Iftftf:ally, th. �11e<i"fcm.,"1 5YS_.
.: .
..' .
..�
,.
'. .Odel1fng fs described and later tltt recI�C.ed· 1.0"' is' outli·"cl. ..
.:'
..
The' f�rth ':chapter is' de�oted ',to tilt:",.'01 '�';Optf_l' and"'"'.
... . .. .
. .; .
•ans of a 'SP"fff'C .n "feat e...,l•• ·'. ,: ",' ..
.' :
. '. I,n the fifth Chapter the"single capt,., trans'er f,unetten based on·
.
the' output· feedback is dfsigned� The pertinent' thttotY 'fs developed 'and.;, .'.,
...*...
. ... .
then' applfed to. the design of. ,. eontroller..ff.rst 'for. ttYpothetictl'system. ind'la,ter. for. pra�tfcal Powe�:�'y�,.· .•dt up of $Iveral·.chfnes•.". .:The'siXth ch."ter '1nVest1gites; the posst·bit-fty ..o' t.rt_t 'fn>,
system dalllpfng by feeding bact. coftt�' Yar1abl�;'1nto,:the'90verno" and"
automatf.c :voltage reg�1ator� The optf.' state regqlato" is: designed for'.
'. this case. and the results 'N,e�red W:1·'th the ,'..s..,� of.,t.�l':"'" .
. .' : .. .. .) �.. "
.
.'.: ': .... '. '.,' .,: .: ': .: .
..
.
'......'
"
.'
"
: .. '
,
"
9, '
'.: .. :
'
..".
"
In' the" seventh chapter '.' c.p.nsator consfstf,ng of • thrft.stage, '" '
.
'. ... .' '. . .
..
. ... : .
.
transfer' functiOn is ass...d as a' gen....� f�....of the control1.� and'
then with, the help of 'eigenvalue' se.rc�· the par_ters of the cOntrol-"lers are' determtned� The tf. d0llli1n results are calculated and c"red
,"
,
,with the rf!sults of other caapensa'ors. ,',In, the eighth chapter the conclusiOns draWn based on 'the overall
"
�S.rch project are present�. s_" rec....d.tions', ,for futu" , work,
arei also pr'Oposed fn thfs chIP"". '
t,
:. : '.':
.... ,
", ',., : '.,
.. ;..,.
. ; ......... -:
'
... :.: ..'
..'. : .':.' ;: :.
".:.:'
v ", ",� ;,r. ·
.....
.
.'
.
. ''
.
10
2. THE APPLICATIm' OF· OPTIMAL CONTROl THEORY
TO ELECTRIC PClfER'SYSTEM
2.1. State'Space Modelling
The state space fonn of system representation 1·s a tnnsfoPlltltion
of the original sets of differential equations of all orders to a set of
first'order differential equations. The transformed set of differential
equations with provisions for additional control can be written in the
fonn•
X • AX + BU + GF
y. ex
. (2.1)
(2.2)Where,
X ... nx 1 vector describing the .st�tes of the. system :
. ..
'. X - n·x·1 vector containing the f1rst:der1vatives of •
the state variables.
A - n x n coefficfent matrix
8 - n x m control matrix ..
U - m x 1 vector containing the designated �tput variablesfrom the contra11er
F ,.. E x 1 vector containing controllable input variables
G -. n x E coupl1"g matrix.
The use of state variables offers considerable benefits concep';'
tually, notat1�fll1y lind analytically. The cOfl(:ept1onal convenience 1$
derived from the .representation of the instantaneous:cond1tion of the
system by the notion of the system 'state' which usually relates in a
one ·to 'One manner to the prob1etn. ,variable. The notational and ·analyti.cal
.''
.
.
.. ':';.' 11
convenfences come thrOugh the use: of vecto.....trfx repreSentatfon which".
'
..allows the .s,ystellt equations �ncI' the···fO",,' of the soluttons to bt written...
' ..
'. '.. .
..
cOlllplctly. The state space approach petWft$ ....dy · •.,.,·ltcatfon of .tty.
"
_thematical.thods (e.g. oPt1.1 cantro; theory) whfch' have been ...
devel�d foi-· the soll1tfon. or the :qu.litatfft d1·scU.:sfon ef··these.
.'
. . .: ..
. .... ..
.
stlte space ,equat1ons.1he system tig,.ll flGl fo"" of, eqUation (2.1) .nd.
.
"'(2.2) can be portrayed by Melnsof the.
block df'agranfshCMI' 1ft Ffgure '2.1.Figure 2.1 is a con-ventent way of represent1ng the linear. plant
and the general extema, contrOl l00p�. Open·c,f.cuftfI\9 thefeedbeck'
path gives the schematfc of ·ihe s"'"ll"d�·unc.ns... ·
f)l'''t� :11", :,'. .
..
. . .
, . .
cOntrolle��y be a l1:near.cGnibfnatt. of·the .tI�s or- of'*.l.c�;.····".
states as'described 1n :the S_'bsequeftt·.teMI1.'·Thedott.r1;.n6S ',.p;e-''.' sent �the·c.se:wh.re:·not all ttlestate. a",':fed b.�: •• ·.:tht.'f,,"t t'o
. tbe controller.' Al�mat1vely the:�ontr.ol1er lIllY eofItafn '.d�1.U _t-: .•s as':"desctibed: l ..ter�· 'Fo.r thl's' "synthesis to ;.&e····feas1ble:·:"ttae stite
.
..
.
.' .
.
'...' .'. '.'
'.'.
space model.of 'a .'ti'...chfne.syst...'fs �u1red ,...4h" to :.triceS··'· ..
'
A. B. G:andC Of:equatfans (2�1) a�d (2.�). This state··S�'C.�fonR i� the
.: major cmtrlbut1cn of this thes1s •..A:.seccnd co\t"1.but�fon fs.:t�"'COMpar-··. tsan· of .the different forms C;f'cmtr.oll_.n·, "rGii optflllll tC)·�the' ...
fntuft1ve·.1e.ad-lag·whfCh 1s';tained by �Ply1·n.gdfffe.Hn.t$.vnthes1s.,'I .
.
.' .' .
tecbn1�" to this s·tate $pace "1. Before .scl'ibiht the ;.pM... :sYstem.
·1IIOde111ftg pr.ocedure· a brief descrfptf.". ,
or.the.: pe"�in."� .'.opt".:'. ,.
... .
..'.
.
"
.
c�t"ol theory':'" 1-1 be 91ven/.
.' ....
.
.....'
.
.
: .,:.
.....
,
.
.
'.: :.: . :'�" ..<
.'•
'
.. :"'.
'.' ..,
... '..
• "<' '.',' .",
':."
:. " ,
,'
i.
'...
'.,'
. .''.
" .. '
.
B····
.".
,'.. :'
A·' .
,
, .. ';
. :.'
.'.
. .
:u ".
�--� CO·NT;R,QtJ�- �"'.' .
. ER
'.": .
12
y.
C:
i .:II
'·1.
.. .. :,. I..
. J ..
'
1.1I
. ]�._ ....... ....__ -J ,
. ", ':'. .'.
'. '.
.
.....•,
'
Of'; ::�.;' .
'
-, :' '.' (�r .... ' '
.... '
.,
,
" ",...
'.
F1qure 2.1. The blOCk d1a!lNIIt of • 5ystat representee..- by equations (2.1)..
..' . '. .
and (2.2). '.
".\ 13
2.2. Pertinent ·�ti.' Control Theory. .
.
.
With the help .of opt1.1control theory one fs .abl. to minimize
system disturbances by applying these methods to the state space form.
.
.'. .
.
to generate an optfma' controller. Other Methods described in ··the.
cont1'01 lfterature28•29 are essentially proce'sses' of para.ter. opt1.·.fzat1en. or tn o�r wotels. the structure of the cont1'OUer fs fixed':f1 rst and then an attenlpt is .de to detemne the OptilUl par..ters
of the assumed controller•
. In 'determ1q1ng the optfmal cOntrOl strategy, ane dGeS, not mike any.
.: .' �r·. .'
pMOr assUMptfons or c0llR1t..ents -that would fix the coftt1'011er structu.....
,tt is assumed th••t' a iltathentatfeallftadtl of the P,.t·, ts '1(_. The aim. .'
.. : . .
'..
.
.
.
. is to ff"d .a 'unction Which when applied· to the plant 'inPut wi·'" Opt••
tze the syst..'s perfcnance with .respect �"s_ 'cost:crtte�fon•..
Paraphrasing. t he 10...' stateMent of �tht optf-.fzai1On "...,.. is as.
, "
"
"" ""
" " ""
10110W$:". """j
",
Gtven that the plant is described by the gen.ral vector equation.
"�
..
.
'x • f (X,u.t) ,(2.3) .
"
"""
"
.
"
"
. ffnd u(t)., tE [�o.T] such tha.t the cost functional :op pertor-. mince 'index J, g1v�n by
J. +(X(T» + jL(X,U;.t) dt ..
1i;.
is Mfnf"'zed. ,Since in dYftllR1c' stabflity studies the ,pl.nt 15 MOdelled""
" "" """
in small signal 10"", one can �'te that:,
+(X(T» • O. .' {2.5) .
. 14
Thus·equat1on (2.4) can be modified to
J.
.;. ,T ·L(X.U,t)dt
toThe b8S�C mathematical descriptions leading to the equation linking
the elements of an optimal controller to the existing plant parameters
(2.6)
. ..
.
is Bell.-n's functional equation. It represents a necessary condition.
for opt1malfty and is given by
_ &J(X,t) .•. III1n·· t L(X.U.t) + < aJ(X;t) f(X,Utt»] (2.1).
&t . U(t) ax
. :If there is no constraint on. the magnitude .01 U and. if L a.nd f possess. . . ..
first partial derivatives with· respect to U. then U*(t). the optimal·
eontrol.ler output varfable or vector, can be found by diffe....t1at1ng
the left hand side of equation (2.7) and setting th� result to zero.
This. yields the condition'
(2.8)
where .
pet) • aJ(X,t)aX
and is known as the adjoint or co-state ve�tOr. If a quadratic pertormance·'ndex is selected. (the reason fer this chofce is explained. in
the last part of ·th1s. section) one can write
t . • .� [IlXI12Q + IIUl12R]where. q .. a pO$it1� �em1-dE!fin1te sytWtric matrix·
.. R·;" a positive definite .$ynanetrfc .trix.
(2.9)
15
Thus
,.!l:. • RU,
aU(2.10)
,
Equating equation (2.3) wtththe standard fom of state spac�,'.'
equation given by X • AX .... BU, yields•
X • f(X.U.t) • AX .... SU (2.11)
Thus
'.!t.S"aU
,Substituting equations (2.10) and (2.12) into (2.8) leads to
, (2.12)
U* • _R-1STpAccording to Pontryagin's in1nfnun principle,
.', aHp • - --, (2.14),
aX
(2.13)
where H is defined as the Hamiltonian expression and is given, ,by"
H • min[l + fT (X.U.t)P)' (2.15)
U'
It follows then that
• ' ,
aHp. -��X
,(2.16)
For conveniellce. let
, p .. KX (2.17)
, 16.'
Thus'.
.: .. .
p.• ·KX.+ KX..
' (2.18):.'
.
'.
and" equation' (2.16l reduces to the.�ne"l .xpressfon.
:Ki'. Ki • -QX .. ATp.
..
Subst1�utfng equation (2.11) into (2.19) gives• .,....
. .'. .
..
···T.
KX + KAX .+ KBU • -QX;. A KX .:
.
(2.19)
(2.20)
or .
.
'"
. ..
.
KX + KAX· ..·
KBR-1BTKX .• '
-QX' _ ATKX .'.
(2.21)
. or.. �K". KA· .. KBR-IBTK. ATK + 'Q'
.
(2.22) .'
this 1ast.equattcm is the well-known .tl'i:x Rf�tf d1ffet'en�1al··equation subject to the' boundary �ond1tie»n at the' terminal tflle T
.•
K(T) • 0
�trfx K* is. g1.en· by,'K* .. 11m K(T) where l' • T .: t
t-toQ ."(2.23)
•
As 1-+0 , K • 0 .
thus equation (2.22) reduces to
o • K*A .. K*8R-1bTk* + ATK* + Q
. The a�ove �uat1on is known as the 'algebraic: .irfx "R1cc:atf .qu,�1qn�•.Matrix K* can be foun.fby solVing the above non-linear :equatton (2.24).
.
The optfnt1zed 10"' of equat1e»n (2.1)-15 now given b;..'
� i • AX - BR-1BTK*X + GF.
(2.25)
or
.
(2.26)
," ...
.
11"
•
.X • A*X + GF
where, .
'A* :' A.- BR-1BTK*.The applicatiOn' of opti.'·· cantrol: theory. has thus··::tMnsfoiWd .....
the Orfginal'c�f�1efent _trfx " frOl:� _rginall�' stable condition ..
to 'the well _ed fol'lll given by eqUation (2.27).
'
... (2.27)
..
:. ...'
.' .
.
.
·2.2.1. Selection of a quadNt�c pe�,..nce index
The choice of a syst" peMtorlillnce index wh1chfs a reason
ab.l. _asure of transient responSe fl"Clll time :te to T is o� �at. 1l11POrtan� to a···.s1gner when applying OPtf.l cantrOl thf!Ory.· The;'
.'
quadratic cO$t functional or perfo....ce index (QPI) is :'prefltftd over
" the others Ixtcause of the 1011ow1 ng reasons:· .... ..... ..
.
. ....
.) ',The Ql'I .f� the c_fn.�f� of fntegril Iq"''''� (' I X112Ql·.
and.�ate of ene..gy expenditure (llul r2R) of .the pl.t an4 hence .it· ..reflects a 'desired perlonnance constraint' of the powe.. systelll unde.. .
study'.'.
. ." .....
.' .
.
.
b) The choice of perfo,.nce· index also reduces the effect of;.
......
:... "
....' ,',
..,' ..
neglecting second order te.. ··in the -..th_t,cal mode.1 0" the system.
Th1s:can be illustrated by con�idering the lfn.�,zfng ·PNCectU".'The ..dfffeNntfal. equations' r.ep....."tfng. the·dynUrles.of·the ;'_"
. s1.S_ are essentfally. non-Un... fn 'nature.' For dyn..'e 'stability"studie�.the·�$.11· signal' 'lfnear'equ�tf�$ are develOp� Pout a .
.
'
qufescent Ope"tfng point (X�(tl.tJo(t». :The Tayl0� se"._:_'x""S10n48
.' ".'
leads.tO...... ,
18,
f(X(t).U(t» • f(XoCt).UO(t)) +!i Ix.xo 6X(t)
+.!! I' aU{t) ,
+ ' F(aX(t).aU(t»,
aU U·U "
,
,
, 0 '
" "
,
(2�28)
where F denotes the higher order tenns. F can be represented exactly
by the following expression'
F(ax( t) .aU{ t» .;.l2
.
'1'
,
,
' '2f', '
"
+ auT(t), .!!:.!t 6U(t) ,
'
, ,',', aU2fu-n ,"
,
f " ,,'
, + 24xT(t) 82.1' I,' 8U(t)}ax(t)aU(t} u.u '
1\'.x·X·
where ai• the natural bastc vectors 1n the vector subspace R,.
'"'
X • X + a.ax I"
'1\' , ,0 < 8 < 1
U • U + a.au','
In order to reduce the error of the small-signal model. one must
ensure that the higher order �nns F are indeed small for all t C[�o.T].Since from equatiOll (2.a9) ().Re can readily see that they are quadratic
in X(t) and U(t}. the obvious choice is to se,ect the performance,
index as described by equation (2.6) �o be quadratic.
19
2.3. POWI' System Representation'Two .1n app'rOlches can in· general be taken when' considering '
..•>. ". .:
-,
• .' '.
t.chntques for. stooying the stability of complex power syst_. One,
.
.>.", " . ',,'. .
..
can rely··on· the concept' of.
input-oUtput stabili ty and study the non�
11near system 1n the tfllle dOMain OJ' alternatively one can utilfze the.
... .
.
.'
.concept 'of dynamic system stab11ity on. the l1nea'rized small si�ar
. .
· model. in the sense of Lyapuntlf.. Although the fdea of fnput-output. .
. .
stability appears to be lION suited for certain applications in.
partfcular' for systeMS driv-m by constantly actfng disturbanCeS. it'. .' ". .'
.
..
. '..
is. not.ll sutted fol' deal1ng with the. dynamic stability problem of
POWI' systems. The' pOwer system i.s safd to be dynllltfcally ·stable at·
an equf1fbrf .. pofnt if the state of the power 'systeM ...1ns.elOse to·
. .... "..' .
this point··for small'fnft1er deviations of :the system var1abl" (PoWer
input. load ·angle. ,.ehine speed, ·etc).· from thefr, equ1.1fbrhlll ·vllues•.In' other words the system does not· ·lose· synchroni'sm for·' $l1li11 variations
..
.
.
..'.
. .
of the variables. ·Thfs property is eq'uivalent to loCal stabf:'ity fn the.
sense of.. Lyapunov and can be �nalyzecl by considerfng 'the l1ne�r1zedpower system.·· .
.
. .'. .
.
.'
.
. The general power system ..urtc control' prObl.. consists basieally. ". .'
of two parts - voltage control and speed ·control. These twc;:control .
, '
processes .interact to a certain extctnt a.,d should,both be,fnclu.d.ff
llleanfngfu1. �sults, are to be obtained f.-ollt studies whfch ,tncorporate ·
th, plant _del. With the size of interconnected areas be�o..fng 1ncreas-· 1;'91y large .. and .nON cCIIIPlex the concept of treating tile power Syst., '
,
•• ,. of
., ....
as • ,1ragle ..chint-1nfinite bus -'ctel' is being discarded fn fayorof the lIult1-machine systetn.
'
...
.. ;
'The dynamics of the phys1calplant _de up of IIIny C�1natfonsof IIIch1nes can be described �th..t1cal1y by extending the equations
of the single .Chine syst•• The end resul.t is a' set of state space,·
equations of the fonn given by equation (2�1) and (2.2) ·de.scl'fbed. sch_t1cal1y in Ffgure 2.1.
. The' power system equations can' Hst be descHbed by treating': the'·.· system description .as groups of'differentfal equations deScribing"
.
.
. ,"
major portions of the system 'as fonows: •..a). The differential and algebra1c..:equattons des.crtbing the power
. flow in the tntertonn�tec" tfet1nes. ".'. . . '..
....
::·:r: ..... , ..
. .
". generators ... '..
......
: .. : .
.
.. 'c) ···The differential equations :deserfbfng .
the 'dynaMiCS 'of "the'.speed· gOYe",ing devices •. e.g. governon and: prime movers •.
:
.
4) .The· di ffet'entia1 equat10fts desc:rtbi ng.
the (Jjna....es . 01; the
excftatfon:systems. i.e�. the exc1·�". Irtd voltage ·,regUlato\-S.::: ..
.
.
The abo� equations are essentfal'y non-11near and· since one fs.'
inteNsted 1�: .·i.ocll stabilitY'. �e linearized fo"" of the equat16ns" '.'.
.. ..
,..... . .,.' ..
· 1.s required.: These are obtained bY'expand,ng the non-litle.r �uatton$..
about .1' quiescent . operating point•.IS 'per equation" '(2.28). and' n8g1ec�ting the .hfgher order �rms F. The resulting 11near equations .are· ';'
referred. te ·ls····'.. ,1 signal' equations'. D�J4t4.ff· the �a· nO_,t1on .
'
..
�: .'
t., '
: "<, "
:' ,'..
.. :;.
.
.. ,'
21.
which sign,1,.s the small signal variables in eq�at1on (2�8), the
resulting equations can be WTitten in the state spice form
• •
X • AX + BU + GF .
·U. KX.
(2.30)
(2.31)\ .
.
.
rhese two equations describe. the dynam1� of a power system .
.
.
.
.
.
.
'. '. ..
.
consisting of the process .nd the auxiliary controller, [K), whic� is
to be designed.
Results obtained from stability investigations on the small signal
representation are valid only in a small neighbourhood near the
quiescent operating point. No conclusions can be drawn �th regard to
the stability of the system for large excursions of the variables on·. .
.
the basis of the results obtained in a study using 'small signal' fonDs.
The linearized model is a valid representation of the actual system.
when the generator is subjected to small cyclic load variations, a
.. condition which predominates in the norwqal operation of a power system•.
22
3. STATE SPACE MODELLING OF A MULTI.MACHINE'SYSTEM
3.1. Introduct1�
As outlined in the preceeding chapters ,the dynamic stability.
..
.
.'. . .
pr.Oblem �st be approached from an overall system basis. In a mult1-.. . .' .
.',,'
machine system when there are a 'number of generitlJr'S involved, it is
frequently 1mprac'tica 1 or uneconomical to simulate on a df�iital
computer th,e cOliplete sets of differential and al�bra1c "uationsdesf;r1b1ng the whole interconnected system in detail. In' order to
adequately study the resPonse of interconnected machines or groups of..' .
machines, 'ft is conVenient to represent those synChrtftous machines in, ,
detail where dampinq forces are felt to be significant. In contrast to
this it is expedient it') areas remote frem, the machines being ,1nYest1"'!'
'gated and where the amplitudes and phase changes of the system voltages
are s,man, to s1mp11fy the system representat'm. It is therefore '
beneficial to divide the complex power systel'l, havin,g " machines into':the foll�n9 sUb-system:-
a) Study System - This system contains m plants made un of those'
, ,
machines which are closely electricilly coupled to the plant which is
of fn�erest. Each simulated plant model in this block includes,
g�vemor. p,rime mover and voltage regulat'(Jt" representat10n and the
�nerators are modelled us11')9 Park's equations or. slight mGtif1cations
thereof.
b) Externa,',
Syst.em "" This part canteins p qe"_nt1"� pl�nts which
are of second�ry importance.' The madels include 1nel't1al and ,�peed
23
damping effe.cts only and· are represented by I cOnsqnt. voltage behind
the equiValent ·.transient reactance.'.c) Inf�.nfte 8us - The balance 'of:maehines .(n-m-p) are represented
'.
by an equ1va'lent 'Infinite'Bus' ",odel�' This is includ� if 'the m + p. .
. .
Rlachine's have a total capability wttich 1$ s.11 relative. to a coneen-.
.
"
trated energy source ',n the syStem.· In s;stem (a) and. (b) it is also
assumed that when two or mOre' �im11ar machines are connected to the.
same node, they can be representeef by. an"equivalent parallel machine.'. .
.
..'
.
.
The above subsystems are illustrated by F1qure 3.1.
The· "st· Of this chapter 15 devoted to dev.1Op1n� the·: small signal
equations of: the tfel1ne.: qene�ator's, prime mtYIet's and !tovernors and
automatic v�ttage.
regulators. ·A.list of constants and Ya��ab.1es used
in the fol'lG-iing equations is given in Appendf'x A� •"
. '.... .
.
.
·'.3.2. Mathematical Mode' of Tie-line Equations': , .
..
:. : ". ,
. \,,'
.'The· genera1 configuration ,,1 an interconnect,d .,_r' sYst. : tI .
'. .'. ". .
. .
) .
.
n 'nodes:ts shom in Figure 3.2 ••...
The c�t t.f in· 1:h� jth node of Ftgore 3.2 can be Wl'ftte.. as
1j• t1 ·/(v:.-vk)'dt·.".·.·.·· (a.n
.
.
.k-l L1k.··.· ,J '.
.
k,rj··
where th� tie.line resistances and admittances are negleeted�'.
",' . . ..
".. .
',' ... ,"
Equations descr1b1n� the behavior of the entire s�stent a� '
..
.
developed �. the basis 'of a Nferenee' frame' rotating at- sYnchronous·.
. .
angular ;v,.lOc1t;v. Because of the different al'lgular position's 'of the .:
.
quadrature "axis: Of. eath machine r�or' ..e'latfv� to the. ref'eren,ce p�asor,
.
..j '".. NODE OF THE SruDY SYSTEM ( Is: �$·III)
r---- �-
� -----�----.----�------�.--,-..,.---��..... - ...
I .
1,
II11II
.
,.II·I·I.
IIIII1I·I.I•,II . 1L
.� ..."'!'"'------�------:...----------- ..----.... - ........
.24
Ui'.MITilUI
• r--,TIl- LI" ---: :
I I
...IiM.ATIOIt I:I
L__J
I
L_� �.;.. .. .:.- ....--.,.-....--�--..;------...----J ...
Figure 3.1 General configUration. of the S.._ys".
'. '··25
VrJ
v·I
) i·J
,',. .:.'
..
':..;..
Figure 3.2.' ScheNt1e representation showing the 'ccnnection 0' the
jth node to the other nodes.
26
.
the reference frames far each machine is different and durinq distur
bances they oscillate With respect to the synchronously rotating·.
connon reference frame. Machines are connected to the.network at
..
. �'.. .
·
their specific nodes. The DQO voltaqes and currents of these ftOdes are.
.
.
. ..
· ·related:·to each other by an axis transfonnat16n. The relative positions
of the Q-ax1s or the rotor posf.t1ons of different machines with respect
to the COJllllOn rotating reference frame are shown in Figure 3.3.
The relation of the phase ·voltaqes and currents with respect to· the DQO components of the jth machine is given by ec:tuations (3.2)' and
. .' ..
(3.3).
(3.2)
.
dj·.
qj."Yk • vk
.
Cos 8j vk Sin 8j ; 1 � k � n (3.3)
where 8f is the.intantaneous angle betwen jth rotor axis and fixed
st. tor frame..
.
Equation (3.1) can be .rewritten as
d n 1.
- (1 j) • Vj'; --dt. k-l L,ikktfj
(3.4)
or
. d. (1 ). 1 '1-.(1t j ._-j
Lpj . (3.5) \
1 1 ·1·+ - +- + - .. - +- ·(3.6)
Lj1 Ljk .. Ljn
27
.. Substituting'· (3.2) ··into (3.5) leads to
dj C· 6 qj S·1 e. Yk
.
OS.vf- �k. n
j (3.7).Ljk .
er .
(3.8)
RfFERf;NC:E FRAIU ..
����--�--------
F1gu�· 3.3; Diagram shorlng thtt position$ 'of 0 ax1$ of cUff�rentmachines with re$J')ect to the refer..ce frame•.
28
..Equating the coefficients of Cos' !.1 and Sin c!.1' one can write
S f dj _ " qj's e .. V..J_dj _ � Vkd.1
. j .i j. .
k�l- (3.9)
.
lp'� k,.j Ljk
.
.'.
v q' n 'v- q' .
- 1jdj S 8j - s f ..qJ .._� + a .:.
k.. '
(.3.10).
.
. ,J kal-
.
Lpj k,.j· Ljk
where S a Laplacian ooerator
Equatfon (3.9) in small signal form fs given by
S 1 dj_ (1 qj + f qj)(W + s � ) - I qjw a
j j. ,1 j j. j . jdj' n y.d,1L� r k (3.11)kal-
l j k,..f ljkP.
or
s ., dj _ I qj S 6 -'f Q,f W' a Vjdj _; 'kdj '.. (3'.• 12)'j j j j. j k·l-
Lpj. k,.j· .l.fk..
.
. Perunitizing all the varfables in equation (3.12) using the Xad base
per unit system yields,
S i dj. dj' n y' dj
3 a' I qj .:..:t + i qj + i - t ..:.L..Wb···
. j Wb. ,1
lpj' kal lk,rj' jk
(3.13)
Similarly equation (3.10).1n small si�al and per unit form is (fiven by,
S 1 qj' "IdjS6 .
V qj n vkqj3 a -1 dj_ ·3 ·1 + :L. - t (3.14)-
w. j , .
w� Lpj kal Ljkbk,r.1
.
29
..
In'order to maniPulate' the tie-line equations given by (3.13).
.
. ." .
. '. .''
.
.
and (3.14) into a f"nn that can be directly used in state space
modelling, the. small signal fonn of the vOltages vkd.1. vkqj are.
. '. .
.'.
.
replaced by voltage components referred to their own 'reference axis.. .
The relationship between the' D and (\ axiS 'components of voltages of the jth';
and kth machine contained in equations (3.13) and (3.14) :i5 (Jiven by '.
.
(3.15)
(3.16)
The sman signal fonn of the above two equations (3.15) and (3.16)
yields,.
dj''. dk'" .
.
.' dk .
. .. '.Yk ....Yk. Sfn (�kjH. cSk �j) + vk Cos (cSkj)·
.. - ykqk COs (6kj)( 6k • 6j> - 'kqksfn hkj) (3.17).
. ......
.... .
'..
:. .
. .
ykqj • ykdk Cos (cSkj)(cSk ... cSj>, +V kdk Sin (cSkj)
(3.18).
. . where 1 < k' < 1ft- -
. .
For t.he 'extenta l' system plants where each generator is repre
sented by a constant voltage.behind the transient reactance, equations
(3.17) and (J.lS) are m�1f1'ed as follows;-
.. '.
., die. 0Ie
v. qle. 0.k .
.
V .dle. 0.... ". :.Ic •...
'
.. '.' ",
.
.
Vieqk ,,·.'Vk:
where In < k � (In ... p)
. 30 .
.
(3.19).
(3.20) .
i
.Thus the small: s1gn'a1 forn; of tile ·v.,lta(te equat1'ons' for the .
.
'external' "systent IIlchines 'can be obta1n&d by substituting (3� 19).. into (3.17) and (3.18). fhe' resulting equati'ons' are given by (3.20)
and (3.21) •.'
.. dj .
(" )"( ).
"k .• -Vk Cos &kj . cSfc ... 6,1 .'
. (3.20).
. ,....
�'kqj -: .�Vk S1n (cSfcj)(&1e -a:;l: ....'
.
(3.21)
SiMilarly fen' the infinite bus systelll, The follllWfng 1IIOdiffca. tims' are ..de, to' (3�17) and· (3.18).;'
vdfc �, 0
.. k ...
,V' qk it' y' "
.
. k. .
. Ie
':y' dk. • 0·".Ie .
V.fcqlc ... 'VIc
where'. (III ..+.p): < .. � !. n "
.
. ..:
.
..
,
.
..
:
'(3.22)
) ..
t3.�3).
The small sign'al :fonn of the ··�r.t1nent equations for fnfin1� ·bus .
. ..' .'
system are ·obtained by' substitutf"9 (3.2t.l1nto (3.17)'.an'd ('3�18) �.
The'" ',' .. .
',.; ..
31
."
.resulting equ.t1ons are given by.
.
'kdj - -Vk Cos (6kj)(6k :- 6jl - Vk Sfn (6kj) .
,�qj - .Yk Sin (c5kj)(6k .. 6j> + Vk Cos (4kj)
(3.24)
(3.25)
substituting equations (3.17) it . (3.18) •. (3.20). (3.21). (3.24)and (3.25) into (3.13) and (3.14) the generalized direct axis node
equation for the .1th machine of the m machines represented in detiii.·is'given by.
I Qj Y. dj mI. dk.:L ..
s �j +' 1:r ..+ .J...... (CQ
.
- t - "'k os 6k1W L 1(-1 l . ,
b..pj. .It".i jk.
n
+.1:k-m+p+l
v Sin �. m' y" dk S1··n· �.. +'y qk Co's' .',. k . Qkj � [to .
k. Qkf· It· . Qkj
ljk ·k-l ljkk"j
n+ t
k..i (3.2�)
. arid the quadrature axis eQuition.. by•.
S qj Ijdj S 6'
.
y.-qj m.. j .
1 dj t.+ .:.L. 1: ·1- •
j- -
k-l._
,Wb W Lpj lkj. b k"j
'.." .. 32
.
. In·'+ [( t
kal. k"j
(3�27)
··Equations (3.26) and (3.27)' describe' the· tie.Hne 1nte..eonnEt(tions for',
.'. . .
'..
t�e n machine systehl. The equivalent, 'tie-Une Plr.�ters can be Nldi.ly.
obtaf.d from "etWork' reduction �C:"hn1que.,� All steady state, yoltaqes·. :'" .:' .�.: ..
a,ut angles can. be obtained directly. from a load. flow' fOl' .
a given' set ··Of.
operating conditions. .
.
.
.
'� ..
"
..
\.
\
3.3.' Mathematical Model of Generator EquationsThe dynamics of the' synchronous:gene.-.tors are descrfbefJ by· Park IS
eqUation's .tn a reference frame f1�ed to:1t$ field winding and'rotating'with it -.The general equations desc�bing >the characteristics. �f" ,'"
. . '.. .
synchronous generator can be written in per. uniti%�d form using Xad ..:
base(�.:4) '5 follows:�' ... ; ..' :_ ..... : ...
:.._...
"..' .• . ··l"S.. ... .;:
..
_: >:,�·.·iisS1·f : La;';kd .... .:� .
'd • -(R� +- ) .1d + hid,:-.+ .. 'il '.+ .
W� lad ....kq. (3.28)
: Wb .. b ..
·.
b
'".
.
,."
� -,
"., .:: .... :
...�:. ':::� �:.. :'. ':. :' :'i
;
33
(3.29)
(-3.30)
(3 •.32)
t-
-' t .... 2HS CI) + 0,.,. m e. '; .' .'
te = .:�. (Ladiflq + (L,,-Ldlidlq .. L.dlkdVlaafknfdJ(3.33)
(3.33a)
.. .
An. initial asstmption in the • study· . systPm mcdels is' that t\-/O or more sim-.
l1er eenerators which are . connected to a COOlmOO node can be represented. . .
. ..
" . .
by a $1nrJle-mach1ne ecluivalEmt. The equivalent reststances and reactance
parameters are. obtained by treating them as if the corresnondtne resis
tances or reactances of the individual 'machines �re coonected in·parallel.
The eouivalent inertia constant is the sum of the inertia constants of
the individual machines. The other assumptiC1l is that the damper
circuits and currents ikd and iko are also neglected. The electrical
effects of the ·damner windin(Js on the stator and rotor coils is assuMed'
to be neq1iait"lle for the linearized model and the me:chanical da"",inQ.
.
". ..
contributed by these w1ndfnos is assumed to be included in the da�er :. .'
con�tant n a lana. �rlth the �rime mover and external system da�"ina. other-
56.
.
".studies done indicated that the overa.ll effect of·damoer windinns is
'34
to generate two large rea,l eigenvalues which have ,little effect on the"
dynam1� stability 01 the system.'
Park's �uat1on (3.28) - (3.33a) can be r.ewrftten with the above
ass�t1ons' "�cluded', for the jth Nchine of the' stud;' system generators
as follows;.'
L S 1 " L 'S1 djffj , fj. R' 1 ,+ adj j +'y,
-
1j fj W fj"Wb b
" . LS1dj LSi, 'V
dj • _ R 1 d,f' + dj j + L .. 1 qj + adj fjj " a.1 j "
Wb ,',qj j j W,b
,
'
" ',"
qjqj
,
dj qj , LgjS1 ..,
Vj • - Ld�1"jij "
-, Rajij -
W". '+ Ladj�jffj'
, b
,
(3.34),
,
(3.35)
(3.37)
The small signal gen.rator equatiOns can be derived from the above.
.: ..."...
. . .
eq�ations by expanding them about an operating point according to
eQlAat10n (�.28)' and neglecting 'the, h1qher order temS F. The resultinq
equations listed �low are in a fo",,', which allows them to be tran's-,
"
,"
fo�d di�ctly 'into ttl, state �pace notation,"
djlff! S if! Ladj S 1
jat a& " - Rfj 1 fj+ , + Vfj
Wb Wb,(3.38)
. !
···.35. !••
1
.. '.: .....
.
'
. '.
.
elj..
.''.
.l·. I' qj.. dj .. ' 'dj ' ..Ldj S 1j'�'" ..
j j jV '. :;.,. R j1j.. +. - - + L i q
: + 9 . Saj. , ,a.. ,. .
'Wb. .
. qj j Wb' j
..'
(3�39)�.'. '. .
(3.40)
2M s1. a. I)' SIS
. if '. '.', '.".. j J .' - \nj .,. tdj ". 3 . t+ ::stULqj <II .ldjlwb " '. wb:· wb
.'
.'
qj dj' .'.
"
qj -.
'
.' dj'Ij 1j + Ladj Itj 1j . + (Lqj.,. Ldj)lj ,
. qj '.
q:t ..,�., ..,
ij + Ladj Ir� 1fj}
.... ',.'.:"
. '.
(3.41)/",. >
The generat.or, mOdels f� the plants extemalto '''studytt systemare repre'sented by a constant voltage behind trinsient reactar)ce. The,
.
. ',:. ; ":, ..
."
'..'. ..'. ",
�
.
',.:..
equations .-elating the mechanical torque, electr1c�l oUtput and,inertiascan be written for the Kth generator in per,' unit forM as:�
.
.
.''.. .
,
. 2H '526 '.
'.
D', S, "',..
.... '
.
'.. .' Ie Ie ·k k (. tmk + �dk • tek +.
,
+.. 3�42)
. .
Wh, Wb
1It:< K i;�(III+p).:
.
. ..
',',,", .. r,' ;-, ',.
,,'
36 ..
.
.',..
. ". ..' .: .." ".
.
.' '. .
The elect,.1cal torque tek can be' Written as.
-.
:
Wmk..
. '.' • ·'.qk.'."
.
'. dk
'
qk.'.
tele·'Vb
[Ladle ff(( fie': + (Ldk • Lqk) fie fie ] (3.4S.)
Since the plants belonging to t"':·extemal· System generators a ....,'. '. . .' ,', '. '. ,: .
.', .
represented by ·a constant voltage behind transfent Nactance. the'.
-...' '.
.. .
..
equat10n (3.43) reduces to .
.
W.
t... mk L
.
f 1 qle'.
ele·- adle fk IeWb
(3.44)
I.
or'.' W
.
' .....
t'. ink' f qk ..
'
. ek· - �. Ie ...'. ,Vb
.
(3.45)
", ':.
The exp��sfo" for fieqk: �.n be obtained from the tfe�11ne equations
as de�fved. f�' the: 'study' system generators. This expressfon for 'f'kqk :"sgiven by equatfon (3.26) and repeated· here for c(llvenfence.
.
S fledk '. Ikq.. Ie. sak'" .
.
. �vdie ... ' .. . '. .
• + f.kqle + -l-.... .
-.t ...!.. ( ,'I' dj Cos.a· •.
. j.·I· j.... kj.Wb ' Vb Lple' Ljlc .
. .'
. "
f·�"·"dj'.
'qj... >: .
-L'( -V'j '. S1n6kj. + Vj .. COS4k.j)"j
jk .
.
.
".
.' .
n ,
... t
.'
j.....l.j� .
.< .
"
. : . 37
..
'
..'
, For, the generators,belongi�q to the "extemal' 'system, the
derivative tems of the :current can be �eglec:ted.:Moreover, there will '
.'
be·.·�o direct, aXi.S component of' voltage vkdk. These cons�r.aints can be'.',
.':' :' '. ". '.' '.'
expressed mathematically as,
Vkdk '. 0
S1kdk--'-0'(3.48)
Vb
By substitUting .(3.48) into (3.47)ikqk can be obtained as a'fu�ct1on of,,'.
.
'.. .
,
the currents and voltages of' all other' .odes. Tbe result is given byequation (3.49).·.
ikqk • _Ikqk
S4k + ;. II ('tjdj Cos4kj -+ ¥j
qj Sin4kj)•...
,Vb ,', j-l jk ''
, ,
m. + [ 1:,
j-1
...; ..... :
n'
V! COS6k.1 "
'
E � -)5 (3.49)
J--.&.1'
t'"
,k'-nIT' jk
',
�,.
.
'.'
...... ...
' .
. ... : .... :.:.:. .
Substituting (3.49) into '(3.46) and th� resultant i,nto·(3.42) yields,
38
o ._ Wilt\' . I qk.
k It''' k k .
b '.
). S "k
\4mk m Vie. dj qj'+ - t - (Yj Cos�k� + v ... S1.n�k")W j.l L .J J.. J
b. jk
Wnt m. Vie qj
.
dj- - E - ( V
j COS&kj - Yj . Sf:n�kj)cSjWb j·l Lik. ."
(3.50)
Although the generalized equations in summation fo� as developed
in the pre.ceeding pages appear to be carlplex. the state space 10m
describing ·the dynamics of the generators belonging to· the 'study'
system and 'external' system can be easily derived from them, The
details. of 'd1fferel1t'matrices" associated with the states describ1nq the
dynamics of the. generators are «jiven in Appendix B. The above equations·
are sufficient to calculate the dynamic stability if the effects of the
plant controllers are neglected, or in other words, if the input power to
39
the generators and f;eld voltage are assumed to be' constant. J10wever.
. �.
.
.
in this thesis the effects of the plant controllers are included and
their dynamics are described in the next section.
3.4. Mathematical Model of the Plant Controllers
3.4.1. Governor and Prime MOver
A generalized block diagram �f the govern'or-prime moyer system
is Shown in Figure 3.4. This form of controller representation is
identical to the one suggested by General Electric to represent
hydraulic turbine in their transient stability proqrain and is flexible.
.
.
..
enough to cover most other types of:governor-pr.1me mover combinations
encountered in various' power 'plants.
In deriving the transfer function shown in Figure 3.4 the, non
linear differential equations descr1bin� the operation of the prime
mover �re expanded about an operating point in the, same manner as
the generator equations. The, differential equations leading to the
state space form of the governor and prime 'moyer are obtained from,� ,
Figure 3.4. They are,
+T43 T
3JtrJ ..
'
T4.1T3J
"
Tsj
Tc.1 T5.1 T
cj
(3.51)
.....
1 hj· 1+ ST3·.·____ .....��..... JI + S TeJ I + S Tlj
CJ· . I + S r.j1+ S T5j
.. tmJ
S Ii
.•. i,
Figure 3.4. General block diagram of the qowerno....pr1"" �0'Ie" MOdel
.
I
Rj
.
...
of ·It.lfdyll system and -Ext.mi,- systilt plants.
41
T TT � Sgj • hj· :(1 -�) - g� - �S,I T ,I Tcj . cj
• tj_.:.:.at - hjr ..
WbRj
(3�52)
(3.53)
3.4.2. Autanatic Voltage Regulator.
The stabili:ty of. a power system is usually very sensitive to
the parameters of the voltage regulator loop. and hence in ord�r to ...
.. .
obtain nean1ngful predictive ·results these elements·must be modelled.
.
fn detail. Excitation systems for synchronous generators have been dealt
with in general. by Kimbark45• A schematic· of a basic automatic voltage
regul:ator .ts shCM1.in Figure 3.5.. . .
.
The fonn of voltage regulator (selected· for the� study) was· comp�:rfsed of· a magnetic amplfffer-amplfdyne. combination. as shOt'" in
Figure 3.6. In this block diagram the transfer function Kaf (1 + TanJ S)
1 + Tadj S
is equivalent to the combination of the m�9netic ampl1ffer..ampl1dyne
equivalent and the amplifier feedback path. The ·last block � is
Vfbj.the conversion factor calculated on the basis of th� Xed base. This.constant nonnalfzes the actual exciter voltage with the base field.voltage. ·This type of excitation system representation closely rese�bles
Type 1 of the four ex.citation systems to be used in computer represen
tation as recommended by IE�E Workin9 Coomfttee46, Various other types
42. i
;
"
vr SENSING Si GN "l P'OW£R MAl N v-
10,- ..
CIRCUIT AMPLIFIER AMPLIFIER EXCIT�� ..
" ,
AMPLIFIE,fSTABILI �ER
,
..
. : " ' '
, ,
:" :: '2t, '
' ,
MAIN EXCITER " '
,'
�
STA81LIJER-
"
" .<;'
":
TERMINAL VOLTA-....;,. , Vt
GE STA81LliER,
,
. .' .
, Ftgure 3.5. Volta�' t"eCJUl�tO,. e�ts for a rotary t"" exciter.,
,
43
K.j Vtoj. KajU+Tqni.S) V.xj.
I.. T. J S.
1+ Tadj S .
I
Vtbj
i a f b j K a fb j S.......----i--_--'
.1.·.TofbjS·
Figure 3.6 Block· diagraM for the automatic voltage regulatOr01 the ·study" system generating plants.
,. 44·""
of regulators are in use and these forms: can be readily. incorporated .
.
into .the modelling procecure,
The differential equations deScribing the' voltage retlutator
dynam.1cs can be obtained directly from Figure 3.6 and are given by...
. .
. ... .. .". ". .
.. .... .
.
S (Tadj Vexj � KaJ Tanj
Vtaj) .� Ka,1 Vta.i
- Vexj.
'5 Tej Vtaj • - Vtaj .+ KeJ vrj + Ke.1 'Uj - Kej'1afb':
.
(3.54) ..
. .
.
...... .;. Kej iYfbj - Ke�iVj'
.
S (Tafbj 1 afbj-'Kafbj Vexj') .•
'
.;. ·i·afbj·
(3�55) '.
.
'(3.56)· .' : . '. .
'..
S. ,(TVfbj.·.fVfbj ... Kvfbj· Vj) .••1·vfbj ..
·
.... ," .(3.57)
· .Although the models .of· the governor' prime" mov&rs aod vol�age .'
regulators as shO\�n 1nFigu�es 3•• 4 and 3.6 app�ar tobe restr1cted� 'the'computer pr'ogram:used for this .study 1s.:.flexible enough.. to.atcomodate··
other types' of nlant controllers. In the *pecific exa�ple described. in.
the next chapter,involving'the three-machine reduced' model of a p�rt of. .
�.
Saskatchewan Power' Corporation. the plant controller models of Squaw. .
Rapids and ,Queen Elizabeth plants are similar.to the one$ described ....
.
..
.. ..
.
in' Figures 3.,4 and 3.6.' ...
·..:. '. ,",:'
. 3;5. Transformation· Into State-Space Fonn: '..
.
.
.. As 'ex.Plained' in·Chapi;e.r 2.1. the state spa.c.e f.orm .of the ;set$·..of ..
'. d1ffe.r.e,rti�1 equatf.ons .. des·cr1b1ng. the dynamic�' of. the Pow.e.r' 's'yst� can
.
'
.. :
.
, .
..', . . ',
.• ,'.,0'
.'.
. .
··.·1 . ","I
45
be written as:
•
X = AX + BU + GF (2.1)
(2_2)y == ex
where the vector X describes the states of the system.
The above two equations can be derived as follows:-·. . .
By sequentially defining a vector X for. the variables on the left
hand side of the equations developed in sections 3.2. 3.3 and 3.4. it
is possible to set up a matrix equation of the form
•
OX .• QX.+ EZ + RU + SF•
Z • JX + PX
(3.58)
(3.59)
where:·
X is the .vector describing the states of the system•....
' .
..' .".
.
.
-...'. .
.. :" .' ..
"
.
X 'is the vector containing the firSt derivatives of the' state
'. vari ab les •.
. ..
. .. .
z is the vector describing the linearly .. dependent variables.
of the'system•
.
U is the vector contain1nc:t the deSignated output variables from
the controller.
F is the vector containing the' input variables. and.
. .
D. Q. E, R. S, P·and J are the 'coupling matrices.
Substituttnq (3.59) into (3�58) leads to,.
"
..
.
(0 .. EP)X • (Q+ EJ)X + RU + SF
..
(3.60)or
i.' (0 - EP)-l (Q + EJ) X + (0 - EPr-l RU + (0 - Errl SF (3.6i)
·.·46
let
(0 _ EP)-l (Q + EJ) .; P-
(0 - EP)-l R -.B
(0 � EP);'1 S . • G.
.
(3 .•62)
. .
Thus equat10n (3.61) reduces to•
X • AX + BU + GF
which is the same as equation (2 ..J).. .
.
In order to derive the coupling matrices O� 0, ·E, R, S,· J� P of" .
. .
the. �uat1on$ (3.58) and (3.59), the state space equati·ons of each. ... .
.
section of the p·ower system is first derived, and . then they are inter-
linked ·to each other by algebraic manipulatiOn •.. ". ....
• .'I
"
Le·t .
... :..... .
. ..
.. . .
.
.
Xl � the � vector representing the sta.tes· which describe· the· .
.'. .
·dynamfes of the Synchr010us generators 01 the. "study".
'. '. .
.
system (Ref. equations 3.26, 3�27, 3.37, 3�38·, 3.3�. 3.41)•.
X2 .,- ,the vectpr. representi n� the states whi ch deseribe the·
dynami cs .
of :the 'external' sy$terll genera:tors (ref. equation.
j�50).·'"
" .
.. ..
'. '" .
.
.. Xi, ·x4 �··the vec�ors rep'resenting �he.·states wh"chde$·c�be the·. .' . .... .
-: dynamics of governor-prime· movers of the 'study'. system atid.
'external' sys·tem respectivelY(ref. equations· 3.51, 3.5·2•.. . '. �'.
and ·3.53)
. X5 - the: vector· reJ)1'.'esentfng t� sta·tes which ·.de.scr1be the dynamics·. . .
. .
.
of .the·autOlMt1c ·voltaqe.
regulato" of the: ·stud.\f·;··System .
generators (ref. equations 3.54 ..... 3.57)�'. '.' \1"..
•
. :.',
" ,'"'" ....
.
'The vector X can therefore be·written·as,.
.;. �'. .
. .
':.
.
.. ,.'..
X •
X ... 1X.2X3
X4X5
" '. .' . . .
EqUations (3�58) and (3.59) can therefore be rewritten in
47
(3�63).
partitioned fom as follows:";•
Oil °12 °13 D14 015.
X···. 1•
021 ...... °22 °23 D24 °25 X2•
°31 D32 °33· ... D34 0 X3 •
·35.• .
°41 °42 D43· ··°44.
°45 X..
4•
°51 .052 • °53 °54 °55 X5
. �
Qll· •.. Q12 Q13·· . Q14 Q15 Xl .E1
Q21 °22 Q23 Q24····· Q25 X E22
031 °32..
Q. Q34 (f X3 + £3 Z +...
.
33 35
Q41 '142 °43··· °44 Q45 X4 E. 4
051 Q52 .°53 Q54 Q55 Xs ...• ES
..
;...
48
.. ,
..-;
" RI, SU' S12 S13'F.j "
,R"
'S21 S22' , ..
,
S23,2, '
'R,' , U, + S31 " S32 533, F' (3.64) "3
" 2
R4 S41 S42 S43RS ,SSt SS2 SS3, F3
,Z Ii [JIIJ2IJ3IJ4I'Js] ,tXtlX21X3'IX41XS1T,'
.
'. • . e. 4 .. ,� • T+[PIIP2IP3IP4IPSl (Xl'X�IX3IX�IX5]
"
·
The rest of this section involves the definitiOn Of all the
,.
'
, (3.65)
".
.
.
'. '. '.
,
vari ab les 'and" techn" ques to ea leul ate the subritatr1 ees � "
,
"
,'3.5.1. State Variable Formulation for the, 'Studyi 'SYS�,,''. Genera tors.EQuations (3.26), �(3.27) and' (3.41) indicate' that there'
are five first order differential equati "'s requirtd to describe the.'.. .... , .
',: :,.'
dynamics of each • study' system �enerator a"d hence five 'state vart'ables'
are required for them. let' fjdJ, Jjqj, lijt c5j and S�jbe selected as z
the state variables for the ,1.th machiri�. ,,"'
. :',
(3.66)
the • study' system qenerators. ...
�"
.' �" . �.,,'
: ','.
.� .
..
"
.;'
or Z.·
y dl1
.
.
,. ql. !1
. d2Y2Y q22
·dmYmY qm:m
. 49
Z€R_· .
. ·"Zm (3.67)
. For this particular configuration the vector U CGnsists of ml
controllab�e outputs for m1 'study'· sys_ generators. It 15 .sslIned.
that the external cOQtrollers are to be connected to ml' generators.
U1
U"'! -1
U�It follows· that. m1 �m
The vector F contains two subsets of vectors - ·F1 and F2asspc1ated wfth the 'study' system generators..
. .. .
.
The d1s�urbance vector Fl comprises 4. and "t>- tile angle and
�oltage variation of the equivalent 1n.f1n1te bus.
or U·
U·2
.
(3.68)
(3.69).
(3.70)
50
. ... .
... .
·
.' .... . .•... ..•. ..•...•. .
th':
.
.
.'·
. The vector F2 comprises tdj' td arid 'Yrj for th'e j ma�h1ne 01
'the .' study'. system' generators • :
"
. �.
.
. or.
T"--------
.' tdm tnn vnnl.·'.'
.
(3.71)�
....."
"
.
.
Thus. USing state, space techniques on equations (3.36). (3.37).'. .
. (3.28). (3.39) � and (3.'41) the coefficient matrices associated with.
vector' Xl can bedeveloped. These I'1atrices are given tn deta111n
Append_1-�. ·13.
3.5.2. State Var1'able Fo""ulat10n for the 'External' SystemGene rato
.rs •
.
Equation (3�50) indicates that two state':variables:are.
necessary to describe .thedynamics for each generator of the 'external'
system. Let 6k andS�\' descr."be the states of the: kth lex.terna.l' systemge.nerator,
.
.. .'
.
Thus. one '.can wrf te
.
. x2• [�l S151 c$2: S152 --...�..�-- 6p S8plTX2£ R2D (3.72)
Vector F3 contatns the disturbance input variables tdk and trk .
for the kth machine' of th� 'externale. system. .
'"
or. .
.' ...: ". .
.
:F3 •..[t�11 trl td2 tr2.With the abOVe def'fnitiQns the state '�paee fonn .of the :coeff1cfent: :'
· '., . . " .,":..
matrices. associated wf,th vector X2can·be·wrftten. The� are g·fven....
··
,
51 ,"
'
in Appendix'B.,
',
3.5.3. State Variable Formul�tion 'or 'the, Governor,,"riwne movers,
,
of the ·Study· System and "Extemal' System Plants.. .
. . .
Since the mathematical model' chosen "or the cH"ernor-prime.
.'.
mover is essentially a 3rd 'order system as can b� seen from Fiqure 3.4,: '.
.
. .' '.' .... . ,',',..
.'
',and equations (3.51),. (a.52), and, (3.53). :the variables tid' ,9j' h,1 can,
, be designated as the state variables 'or the jth plant. This leads to,
:,
(3.74)
.end:
.
.. .
X4 • [tm1 91 hl tm2 92 h2,
The coe'ficient matric.es ass()cfated: with the vectorcs X3 andX4 are"
developed f�, the equations,..(3.�1). (3.52.).. and (3.53). ,'They are '9iven
in AppendiX B.
, It, should be poi,nted out that Jf ,a,ny other prf� 'rrtOvet:'-govemor.
.. . '
.. ,
.
model is' used. it can be tran,sfom.ed into the saine f()m as sho.wn inFf<ture 3.4 ot" the ,coupling matrices· aSSOC1'ated, with the v�ctors X3 �n� ..
X4 or both can' be' changed.''
3.5.4. State Varfable'Fo.nnulation 'or the Autanatic Voltaqe,
Re�lators of the ·Study" System Plants.
The transfer, function. rep�senting the; automatic v:oltage'
regul'ator as shown fn Figure 3.6 has 'our characteristtc roots a�d'�herefore 'our ',state variables are re�u1red' to describe its dynamics •
."
. .
.. 52.
.
The v�riables (Taaj' Vexj - Kaj' T�nj Vtaj)t Vtajt (Tafbjiafbj'.. Kafbj "eXj>. and (Tvfb3 1vfbj .. Kvfbj' Y.1' : have been selected as' the
.
four states for the jth plant� The reason for selecting these variables.
'as >states is to. make 0ssd1agonal and hence O· alin�t diagonal. sothat.
.
it will take much less cOmputer time to·calculate the inverse matrix•...
..."....
Thus'the .vector 'Xs can be defined as..
. X ..S
Tadl Ve)(l" Kat Tanl Vta1'. Vta1
'. '. Tafbl iafbl ... ·Kafb1· Vex!.
'.T'fbl·i,ffbl .. Kvfbi VI' .
Tad2 ·.VeX2· ... Ka2 Tan2 Vta2 .....
Vta2Tafb2 ia�'2
.. Kaf.b2. Vel_{2 .
.
.. .. .
:..
Tvfb2 1 vfb2..
'
'Kvfb2 y2
....,
"
.
..... Tadm. Vexm ".Kam ta�m.Vtam •
Vtam.
.
.'. ..' ""
.
· Tafbm 1 afbm.. Kafbln Vexm
. .
..
�
.. Tvfbm 1vfbm·"· KYfbm Ym
.:.4'"
x.s. €. R4m: (3.76)'
,
53
T.he coefffcientmatrices assoc1atedw1th the vector'X5 are
developed fram equations' (3.54) - (3�57) and are given In Appendix O.,
"
....
"; ...'.
-.
'Thus with reference to the'submatr1ces given in Appendix B.,,
,
equations' (3.64) and (3.65) are definedln a fonn that is readily.
. . ... . .
,
adaptable t'o' the dig1,tal c�uter.' Using the transfomation defined,
', ,
'
.,' , ."
' "
by equation (3.62) the final state space fonn X • AX + BU + 'GF of.equation (2.1)' can � readily obtained. The total' number of state
,
variables required to describe the models cMsidered in this, study'
is given by,
q .,12m + 5p' (3.77)
This can be reduced to (10m +",5'-) if the tr4ris,fo�r action in.'". '. '"
.,'.
the statorrw1ndings and the tie lin, is 'neglected as described in' the, '
, "
nel(tsect1on.",",'
.
,',:",
.,:.'.' '.:' .. ,': ., .
',.. -, .. "
..'
,
3�6.' Redticed State Space'Model'Negl�ct,ing Trans10nner Action.,
"
In ·the' foregoing' anal.Y$f.s: the "tranSf()hne�' voltages,: hi ,the direct'
.: . .
'" '. . : . .
.
, ".
. . . .' .
:. ..' '. .'.
. .... .
consfde�d� Further Simplification is possible by ,neglectfn!J' these,'
. '. ...
voltage terms. This assumption 'is, justified as the predorrrinant roOts:,aN little ,affected :by this assumption. The adva�tage lies ,in 'the·:fact
" that since' �1s asslJlnpt10n :,mak�S tlte der1vative,!�enns ,of: ij��' and ijqj,
equ�l to ze�'C). '1jdj �nd 1jqf can be conceived of as dependent varfables,
instead of states and hence the number of state var1:ables w111 be
reduced by 2' per• study' system \gener�tor or, from (12m + 5p,) to';
,
"
(1Om'+ 5P).,' in, total.' This "'in' turn' redue.e$:�e, colnputer,t1me,:�ons1derably ,. . .
: : '.'.
.
. '. '. . \'.' �.
.
','. .
,', .:
#: . "
.. : '",.
.,',"'_ , :. !', ','"
.
. �..
.' :.�.' ,,':: '"
'."':'.
.
. � .
.. :; ..•'. ... ....
'. :-
54,
with a minimal reduction in accuracy.
The state space equations of the reduced system model can be
. deri ved by e1 ther of the fon ow1 ng two methods:-
Method 1
, S'1nce the state space model of the power system with transformer
,action included are already developed as given by equations (3.58) and
(3.59), the state space fo� of the reduced system can be obtained
indirectly by perf'onning algebraic manipulations on these equations.,
Let the new state variables' be defined as
I1
X2
l' • X' Ie Ref'3
X4
Xs
(3.78)
where ' q.. 10m + ,5p,
The vector Il can be defined as per equation (3.80)
TII • [al �1 S61 a2 62 S�2 am 6", S6m}
(.3.79)
", (3.80)
La,d 1 1 <ii,'where "
81,. 1 f1
-
iLffi'
(3.81)
55
.'...
'')he, new dipe�ent v''''lb le'
vector 1" ,. defined. IS , '
, (3.�)
(3.83).
.
..' .
.
Let' the, new, state SPIce" 8quation be wi t_" IS,
...
.,.' .
"
(3.84),
, (3.85)tt·,'J"T+RU:..8F,,'
, ,', .i:; , : 1;� "1,
,, ,
The p�lem therefore is to de"ive �he t".nslonned .trices 11. tr. r..
.
.
"..
'. '. ',. .....
I. I.�. J'. Iti an.d 81 f"om equations (3.68): and' (3.69). The teChnique, '
used ,for 'this 1s'desc"ibed in Appendfx C.'
MethOd g'The 'above method wis, used in the cQIIPuter pr09ram fo" ,the' analY$fs
'
"of the II1Ult1-lIIch1ne systeM.' It allowed fo" the inclusion 0" exclusion '
of tNnsfo..r act.ion. On the othe" hand if one prefers 'to �velop a
comPuter PY'09". fo" the reduced model direct,ly fram' ,the system' equa-, ,
.
tfons,., the following' procedure can be adopted. '
,The equations ,g�m1ng the dynamics of the •study' system",
'macbines can be obtafn�d f� (3.26). (3.27>' ('3'.37). (3�38). �nd (3.39)
by neglecting the i"ansfonlllar a��fon. 'They Ire given by,'
- 1 jflj • ttjs�;t.y�� • �•.. J.. '''tell< Cos4kj - "kqk Sinakj)Wb ' lp4,., k 1 l"k' ,",,
' :' " """ k,.j
,
,oJ
"
. : .. : '.
..•• m.·+ t·
.
k·l.
k,ej
.
.
".:.'."
..,'. :.' .','
'Lt.. (Vkdk S1n.s·k"j' + vkq.k Cos.a'k'··j)�k +..'.� .: Vk COS6k:f'\k-m+l. L·
.
jk . .
. jk
n .'
+ t·•.: kam+p+l
.
.
V· dk S1 ·v· qk c: .
. '.'.
'. y .5in4 .'. m .' k . n6k3 +k. . OS6kj.·:...k . kj -I t
.. L' . k-l Ljk .'. k,rj" jk
. {3.86)
I d.1 S4..
V qj1 dj
.
j J +.:L_. ._�. i· (.y. dk. S1'n'� . y' qk Cos�. )j.• ..
. W .• k.l L- >k .'. Ukj .... k .'. '.' Ukjb
. lpj k�j jk
m. 1 (" dk.
qk .
).. n.· � Vk. S1n4k3, .
- t ..-. . Yk. COS6·kj.� Vk ..... S11'1�kj 6k + =.....1·· 'L ... ,
.' .4k.
-. �;� Ljk . ..
.
'.'. '.. .
.''.
• Jk.'
.
.', ,'. .'
.... � . vk 51n6k3 )]6 ..•..k L'
. j...1 '
.. jk '
(3.87)
.\ :
........
'
::.
.
'. ': : .
..
..
.'.' '. qj
'V dj • R' 1 dj + L 1 qj.' Lgj I". ·s � .
j .
-
aj j . qj j .:.
. ...; ., .
."j.
. . �b. ..,
,. (3.89)
........... �.'
57
The other equations describing the dynamics of the system w111 be'
unchanged from the ones used when' transfonner action is ·included •.
'The new state vectol"S.l' and l't will .be identical to those de.fined
. by equ�tions (3.78) and (3.80). Thus with the definitf()n .oftl as per
.. (3.80) and with 'the help of:the equat1·ons.(3.86). (3�87),.(3.88). (3�89).
(3.90) and (3.41l1t is. possible to Write the reduCed s'tate spaCe fonn.' :
.
.
'..'
.' . .' '.
.
for the '.study' system generators. The state space models associated ...
with other vecu,rs, X2' X3• )(4 and Xs 'remain unchan�•.'
The state space model of the: multi-machine Sy.st�ni has been derived
using a 'dual level i form of $ystt'l!1 representat10n�" In the 'next ch4pterthf·s· model is used to design an' oPt'iina 1 state regulator fn order to .'
.
. .'
.''.,
. .'. '.' '. ".
enhance the damping' of this sys·tem.· TM s . CompeAsated perforNnce is...,
.'.
. � ..
..' .'
·then· cOlripared with the' dynamics" of the sistem compensated by- �t�er. ,:""': .
controllers.' ..
. '.'
...•.
\,' .. :
.
. ..�"
..
..
.
-;'.: ". .
.;
... ',.
.... ;;.,
. ,
::', .. :'./
. ':'....
,."
58"
..'
.
' .. '. ..,. :.., ..
4. "DESIGN OF THE OPTIMAL STATE REGULATOR
4.1 Introduction
As explained" in Chapter 1. to enhance"
the damping of the power
"system." which 15 essential fr. a dynamic stability. pOint of view,
.y require additional feedback elements in addition to the voltage
regulators and governors. As outlined in Chapter 2 optimal control" "
"
theory provides the designer with a mechanism for finding a general.
. .'
"stabflizing signal by adjusting all of th, control parameters s1mul ..
taneously. thus mfninrtzing the syStem response and control signal
excursions. This can be" effected by .pplyin� opt1.' control theory
outl1ned in Chapter 2 and utilizing the math_tical models of the
power system whfch. have �een developed in Chapter. 3." "
The -problem" can be restated as follows:-
Given that the plant equation is given by•
X • AX + BU
and "the cost"" functional J 1$ given by
(4.1)
"
(4.2)"
""
the Optimal control variable U as derived 1n Chapter 2. can be written
as
(4.3)
where K*fs the solutton of the IJIItrix..R1ccat1 algebraic equation given
by"
"
-1 T T"
"
"
o 1!11 K*A.. K*B � .8" K*' + A K* + " Q
.59
..The first step in applying q>timal control theory 'to the nwlti�
machine controller q>timizat1on problem is to select the weightinQ.
..
...
_trices Q and R. There .s no mathematical guideline: in this respect
but the following rules can.
.,e· accepted as a starting pOint:.·
1) The larger elements in Q lead to a larger' gain matrix (R-laTK)and a faster time response.
.
.'..'
....
2) The larger elements in R lead to a smaller gain matrix and a.
.
. ..
slower.t1me response.
In, the subsequent study described in this �hesis Q is selected as
a diagonal matrix because of computational convenience. This �ll not
yield any error because. all the .states are already coul)led to each
.
other by the coeffici'ent' matrix A� Because of the fact that t"e' basic'
requirements of the controller is to minimize the 'angle and frequency.. . . .
..
.
deviation o'f all machines in the system. heavier "enalties have been.
.'
imposed on' the corre·sponding states. After a trial and error method
us1ng'different Q matrices the .we1qht1ng ma.trix was selected as the.
unit matrix except. for a wei�hting of 10000 for the state variables
repre�entinq ,the speed error and angle change 'at the 'study' system
and 'external' system' pJants. In light of rule (2) as stated above,
. the R matrix was chosen to be'�nfty.
The 'important hurdle is to find the solution of the non-11neer .
matrix-Ric;catf equatton �e$crf�ed by equation (4.4). �he next section'
describes the pertinent al qorithm used to solv.e .th1s 'equation.
4.2.
Solving the.Matr1x-R1ccati Equation
The matr1x-R1ccat1 differential equation can be written as
'. -K(t');. K(t)A + ATK(t) -K(t) aR-1aTK(t) + Q
Putting .-r • T - t , where T + CI
equation (4.5 is changed to .
.
� T .
.'
1 tK(-r) • K(-r) A.+A K(T) - K(-r) aR- a K(T) + Q
60
. (4.5)
(4.6).
The variation of K(T) with time can be graphically "presented as shown
in Figure 4.1.
K(t') .
,
.'
.
'. .�.
Figure 4.1. Diagram showing the variation of K(T) with .T.
,.'.
.
61:
..,. ..
.
.
'Since it 'I' ... u , K('l')'. 0, the steady state solution K* can be obtained
as'.'
.,. ,',
LimK*· . ,
.
Kh). '1"'" u··
(4.7). .
.
�e Obvious way of f1ndfng K* fs to fntegrite n....feeny' 4Mtuatfan .
.
'. .... . '. .'. ... ,,: '.
(4.6) until the steady s�te is Nached, 1.e. until K('t') becomes ze1"O•.
AlthOugh quite adtquate for. the solutfon of a low ,order' sy'stem, this'system suffers 'from 1 eng, computation time, as the step size must,' fn...ost eases, 'be quite small to insure· numerfcal accuracy. A computer
run Using the high speed CON of the IBM '360/50, waS unable to obtain'.
_. steady state solutton for' a l$th order system encountered in this.
.'.
. .
.
.. .
..
.
.
project wf�fn 60: nrlnutes. So other methods described in the literature
were ..tnvest1gated•.
AlthGUgh there are several 'other methods available fn the literature
for solving equation (4.6), most of them were found to',be n..r1cally
unstable or inefficient when IPplf,ed to the higher oreIe ... system en.coun';'
tered in �h1S application ·where the' coeffiCient, matrfx A of equation
(4.1) is sparse. An alternate method suggested by Fath53 was used to
detennfne the matrix J<*'. the pertfnent algorithm fs described belOW.
Equat10ns (4.1) and (4.3) yield
X(t) • AX(t) .... Br�lBrp(t) (4.8,)
where p(t). KX(t) 1s the co-state vector.
Equation (2.16) can �e rewritten as
"p(t) --Qx(t) - 'ATp(t) (2.16).' .'
.
.
The .•bow two equat1(1flS can be written f'n .trfx fom 1$ .
62
•
. _BR ...1B! . (4.9).X(t) A X(t)
'. ..• , Tp.(t) . .Q .-A . p(t)'
Def1n1�g l' •.T-t equatfGn (4.9) changes to
•
·BR-1BT. X(or)x(T) -A.. (4.10)
•
AT . .
p( 'f)p(1:) Q
or...•.
X(t) .
[M]Xed
• (4.11)i
p(1:) . p( 1:)
'. (4.i2) .
(4.13)
The solutf. Of equatfon (4�11) can therefore be obtained as·
.. .. ..
. [. �(t)] =.. [Atl(') ..
. PCd �IC1:) .
'..
'..
· �::; ] .• I :::LJ i (4.14).
I·..
. .
:.
Since p(1:) Ii 0 froM ffgure (4.1). equatfan (4.14) reduces to -.
[::::J = [��::: 11 X(.) ., (4.15)
01" ph)· �lh) sa 11-1(1:) X (d
-1.
Thus Khl • "21( 1:) 011 (d.
(4.16)
(4.17) .
63
.The eigenvalues of M are synnetric with respect to the real and
.
. .', .
imaginary axfs and are ass..d to be distinct. For the real efgenvalue .
Ai' define • Ix 1 matrix Cf as
(4.18).
.
For a set of complex ei�envalues Arf :t Alf .: deffne a.2 x 2 matrix Cf �s.
C1 . [AM·
�11 .]' .
.
.'
Ali Ari
Let C be defined as the q x q block diagonal matrix
C· DIA (Cil all i: such that the diagonal. elements.
. are posftive]. . '" .. ,
..' .
(4.20)
'(4.19)
and 'C" as
'C" .: D�A [ttl all 1 s'uch that the. diagonal .,..ts .
are negative] ,
'.
Let 'w be defined as' the tr�nsformatfon matrix' such that· .
W-1MW .•. OIA [c.'C"]
(4.2�)
(4.22)
The fundamental matrix o(�) can therefore be written with the
(4.23)
V12]..
Vu •.. (4.24)
.', .. 64
Equation (4.24) can be. reduced to
wllec-r in • W12,T:'t·1l21.n( 't). • (4.25)
C't '" u' 'f't 'IT.
\II e yr. +. "22e . "'2221 '. 12
substituting the values of 0l1(tl ando 'll( t) f�an e"uatfon (4.25)
into (4.17) yields
K('t) • (w21eCt "11' +'. W'l.2e'f't V21) (WlleC't '11.
+
.
w12e"f't V21) -1 ('4.26) .
as. 't + (I , eT!'t + 0 .nd hence the steady state solut1.on K* is obtained
fran equatfon(4.'l6)as
(4.27)
. ..
.
..' ..,
.
1l •...
'K* • W21WU- (4.28) .
:
.
As can be noted from equation (4.28) only' twO·blocks Of the tt'ans
formation matrtx W are needed to calculate the CGnstant coefficientfeedback matrix K�.
To solve for W it is re(lu1red to generate the eigenvectors. �f; M.
These e·igenvectors fann the colUlll1s of a 2q x '2q auxiliary matrix. W*� .'.
�.
The first q' columns are the eigenvectirs corresponding to the'efge�valuesof M having positive real parts. Thus w* is the transformation matrix by
wbich matrix M can be cClnverted to the canm1ea1 fo,,"_ 1 �e•.
'.
(4.29)
A,be1ng the eigenvalues (real ·or canplex) of the augmented matrix M..
.'.. . ...
.. .' .
Since matrix W transforms M into the block dfagen.l fonn as .'
. defined in equation (4.21), W is related to W* by the equation (4.30).
65'
(4.30)
(4.31). �. .
whereas matrix'S ·'DIA [Z1]I
and
1 for real eigenvalues, and''
It '[I-j 1+"J' for pair 01' �amplex eigenvalu,es."
'
l+j I-j ',,',
,
.
'
'
Thus the computat1an of K* is divided into the following four'
steps:-. .'
'.
.
Step 1. Fonn the auqmented matrix M as defined by equation (4.12).
Step 2. Calculate the eigenvectors, of M and fonn the transformation "
" matrix W*..' ....
Step �. Fo"" the matrix S accordinq to (4.31) and calculate W with
�he help of ,equatf� (4.30).
Step 4. Calculate K*·, fran equation (4�28).� . . . .
The eigenvectors of a ncn ..s�tr1c matrix can be calculated, using
a, computer program developed by Yan Ness. It was written: on: the bash
of the algarithm knClfn as the • Inverse Iteration MtthOdLas desCl'fbed
in reference 54, and was obta,fned fran th, NOrthwestern' University,. "
. .
Il11nois. This program was used in conjunction with the algorithm',. . .
.
, derived above to obtain the eigenvectors and subsequent optimal con-
troller K*. The problem encountered'using other techniques 'for solvinq
the _trix-Rtccatf equati()ft were eliminated by this approach.
,4.3 Basfe System Specifications,
A practical application involving the use of the 'dual' model
involVed a study into the effects 'of addt" an auxf'1iary e�trol1er at
66,
, the Squ.w Rapids plant of the SiskatChewan Power Corporation SyStem.,'
The reduCed' system,is :shown in� Figure 4.2.
In this case the system consisted of:
1) The six-machine Squaw Rapids plant. expressed as' a single-'
machine equivilent. including excitltion, governor. prime mover and
syn,chranous, generator details.,
2) ,The two-mach1ne Queen EHzabethplant expressed as I single
macht,ne equivalent. but'including inertial and st)eed damping effeets, "
only and assuming a constant voltaqe behind the equivalent transient'
relctance. "
. .'.
�.
, 3) The, power network ext?ressed in tems of a simple delta-connec-'.
. .
. .: ."' .'
ted system :with nodes at the Squaw Rapi�s 'and Queen EHzabeth 14.,4 KY
buses "and at, the Boundlry Dam 230: KV bus. the "atter being cCI"sidered
IS an infinite hus.,
'
The Squlw Rapids plant consists of:-. '. '.
.'
.
Ca) hydr�uHc turbine with teqJor.ary",droop type governor'
,
(b) rotary automatic voltage regu"ator.
The Queen Elizabeth plant consists of:;;o
(al non-rehelt steam turbine.
The 'system speetfica,ttons ind all the constants used 'or this study
are tabulated in Appendt)C D. All the values are in p.u.' unless otherwise
stated.
67
"STUDy"SYSTEMSQUAW RAPIDS PLANT
INFINITE BUS.. EXTERNAL' SYSTEM
.
SOUTH EAST GENERATION Q. E. PLANT
GOV. w.AND .--
. PM MVR
2·3.0 KV .
. BUS
tml S.R .
. 6-MACH·PAAK
r-&--_. EQlJlV 1---j�I---I-""----"""';'�'lm�-"'!""��_"-",\BUS
: �
.' .'..
LI2 . ····.,1- INERTIA.MODEL WITHGOVERNOR
• CONSTANTVOLTAGE ..
BEHIND "d
AV R· ........------t·14· 4 t( V
.
BUS
� ',: .''-:-
,', .. ', .
.
",".
.
. Ffgure 4.2•.Schanatfc representatfon of a portiM of the Sask.tchewan·Power Corporatian system used as an fllust...tfve ,Xllllf)le.: .:;:
•• 7 .:':
68
.'. .
.
4.4 Design of State Requlator and iruncated State Regulator
. The state N!IU latol' . WlIS destqIled on the basts of the al!101'f tlmt
derived in Chapter 3 and usinq the syst.. parameters of Appendix 0.
Using the approximated fo� of the generator and tie line equations
the ve�tor rel�tion between the state variables arid the physical
va�1ables is given by:
(4.32)
where'. ·tad1a -1f1 - -
.
Lffli dl1
c -. Vta1. d - Tafbl 1 ifbl
- Kafbl Vext .
� - Tvfbl ivfb1 - Kvfbl VIThe state equations are qiven by
..
,':
•
X - AX + au
where
(4.33)
.. '. TX - [Xl Xz ----------.---------- X1S) .
and U is the scaler control variable to be applied' to the input of the
automatic voltige regulator of the Squa" Rapids plant•.
The numerical values of the elements of the 15th'order coefficient
matrix A obtained from the digital computer is given in Table 4.1.
TABLE 4.1. THE CHARACTERISTIC "1ATRIX A OF THE UNCOMPENSATED SYSTEM
..
(With Transfonner Action' Excluded)
..o�1731 0.0 0.0 . 0.0. . O�O 0.0 0.0 .". 0.0 ..
'
.'
0.0 '·0.0 .0.0 . 0.0258 1.29' 0.0 . 0.0 .
0.0 0.0 1.0 0.0 0.0 0.0 ·0.0. 0.0 0.0· . 0.0 0.0'
.
0.0 0.0..
0.0.
0.0'
-19.7 '-18.0 ...0.149 17.8 0.0 . 18.1''.
0.0 0.0' . 0.0 0.0' 0.0 .0.0.
0.0 O�O 0.0
0.0 0.0' 0.0· 0.0 1�0 .
. 0.0 0.0 0.0 0.0' 0.0 0.0 0.0 0.0 0.0' 0.0
3.03 19.8 .0.0136-22.2· .0.102 0.0 0.0' 0.0 18.2 0.0'
-,'0.0 .0.0 0.0 0.0' 0.0
0.0 0.0 0.0537 . O�O .. 0.0 .0.909 0.915 . 0.352. 0.0' 0.0 0.0 0.0 0.0 0.0 0 ..0
0.0 0.0 -0.0268' 0.0 .: 0.0 0.0 -0.0029 -0.176 0.0 . 0.0 0.0 .0.0 . O�O O.G 0.0.
. .
00 0 ..0 -0.937 0.0·' 0.0 .0.0 . 0.0. .
.
-6.25 . 0.0 0.0 0.0..
0.0 0.0 .. 0.0 . 0.0....
·0.0 . 0.0 0.0 . 0.0 . 0.0 O�O . 0.0·· .. 0.0 -20.0 20.0 0.0 0.0.
.
O�()· 0.0.
0.0
0.0.
0.0 0.0 O�O.
0.0 00",
0.0 .:
'
0.0 .. 0.0 .. -6.67 .. 6.67 .0.0 . 0.0 0.0 . 0.0'..
0.0 0.0 0.0·.
0.0" ....1.38 0.0; ,,' 0.0 0.0·.' 0.0
'.
0.0 -12.5 '·.0.0' 0.0 0.0· . 0.0
0.0 0.0. 0.0. 0.0 . 0.0 . 0.0 0.0 0.0.'. 0.0 0.0 0.0 -30.3-1015.1 0.0 '. 0.0
�3.55.. 0.141 -0.0159 -0.142 0.0 O�O 0.0' . 0.0 .. 0.0 0.0
'.
0.0..
-2.93 .149.0 �19.3. '-8.51
0.0..
. 0.0.
. 0.0 .' 0.0 .'. 0.00.0 . 0.0 0.0 .0.0' 0.0· ..... 0.0 -0.689 -34.4 ·-4.54 .0.0:
.0.313· 0.•0129-0.00141-0.0125 0.0. 0.0 0.0·
... 0.0 0.0 0.0 0.0. 0.0 0.0 0.0 ;..2.0.'
..
0.co
" ,70
,
The co..,pl1ng matrix B is given by
B. [0" 0 0 0 0 0' 0 0 0 0 0 0 4.26 0 O}T (4.34)
In order to justify the necessity of the ad�ft1onal ccntroller '
, '
it is required to f1rid the root location of the uncompensated syst...
The eigenvalues can be readily obtained frain the s.tate space model of
the system mce the prot; lem 'equations have been tl"lnsfo�d into, this.
. .
..
convenient fOnD. The system is stable if the real l)a1"ts of all ei(len-'
values are negative. In the absence of ,
any control variable U. the. ..
.. .
coefficient matrix of the ,system is unchanged from ma,trix' A. The', ,
characteristic roots of A we:re calculated using the IBM scientific'
, subroutines HSBG and ATEtG. These roots are tabulated'in Table 4�2A.
The predominant roots of the system are shoWn in Table 4.28. .,
:."
It is worthwhile to cCJIII)art'the ,eigenvalues 'in' Table:'4.2A wfth
'those"1f the transfonner action is not neglected. In this case "idl'and "
,i1Q1 can 'no,'on�r �'considered'is,dependent variibletbecauseof the
presence of 'their first derivative �rms in the state equatiOnsas shown
in Chapter 3. The order of the system is thus increased to 11. The
coefficient matrix A was aqain canputed and the eigenvalues and the
pred(ftinant roots are shown in Tables 4;.3A and 4.3B. :'
,A ccnparative study of Tables 4.2 and'4.3 clearly demonstrates
that there is relatively little change in the magnitude of the pre
dominant eigenvalues :by neglecting the ,transfontle', action in the stator
windings and th� tie'line. Thus little mathematical accuracy is sacri
ficed by considering 'Ildl and "1Qlasdependent Variables', instead of
11
,', TABLE·4.2.
A. EIGENVALUES OF THE UNCOMPENSATED. SYSTEM .
WITH TRANSFORMER ACTION EXCLUDED
No. Eigenvalues
1 ..113.9 + j 0.0.2 .. 17.42 + j 1.82·
3 ... 17.42 .. j 1.824 .. 9.46 + j·O.O5 .. 6.38 + j 0.06. '. 0.044 + j 6.687 .. 0.044 .. j 6.68'8 .. 3�26 .+ j 0.0·9· .. 0.63 + ,1 1.3410 .. 0.63 ..
. j'I.3411 .- 1.98 + j 0.0'
0.81•
"12' - + J 0.0 .
13 - 0.20" + j 0.2914 .. 0.20 .. J 0.29
15 .. 0.0029 + j 0.0, .
B. DOMINANT. ROOTS OF THE ABOVE .MENTIONED SYSTEM
No. E1ge"v.lues
1 ... 0.044 t j 6.68'2 .. 0,044 - j 6.683 .. 0.63 .+ j 1.344 .. 0.63,; 't .. j 1.34
•* ',.
,.
'5 _. 0.20 , + j 0.296 "
, .. 0,20 .. j 0.29,
,
".,,� . .'
72
,
, TABLE 44!3'
,,
A� EIGENVALUES OF THE UNCOMPENSATED SYSTEr.1 '
, ,'
WITH TRANSFORMER ACTION INCLUDED
No. Eigenvalues
1 _
,
0.67 + j ,376.972 _ 0.07 '_ j' 376.973 -,173.85 + j 0.0 ,
4 - 17.42 + j 1.825 - 17.42 - j 1.826 _ 9.46 + j 0.07 .. 6.38 + j 0.08 ..
" 0.046 + j 6.719 - ,'0.046 .. j 6.7110 '
.. 3.23 .. j' 0.0"11 .. 0.66 + j' 1.3712 .. 0.66 .. j 1�37,
13 .. 1.98 + j 0.0"14 • 0.,81 + j 0.015 - 0'.22 + j 0.31,16 .. 0�22 '- j 0�3117 .. 0.0017 + j 0
B. DOMINANT ROOTS OF THE �BOVE MENTICWEQ SYSTEM
No. E1q.nvilues
1 .. 0.046 + j 6.71'2 - 0.046 - j 6.71'3 .. 0.66 '+ j ,1.374 .. 0.66 - j 1.37,5 - 0�22 + j 0.316 .. 0.22 - j 0.31
"
73',3
.
-,
states. Fran a computational point of view this asstanpt10n is hiqhly. .. ....,'. '.' ,..,' '. .
welcome as it .would save considerable computer tiM 'in system analysis·. . '" .
because of the' fact that the system order would be �duced by �ce the
nu""er of study systC!flt machines. All analyses in this thesis were
.
carried out using the. mathematical model which neqlects the transformer.
voltage in the line and stator cons of: the generators.
Table 4.2A.conta1ns the combination of real and complex eiqen-'.' values of the uncompensated sYstem ... A real negative root corresponds
to an exponentially decayinq non-oscillatory nl()de. A pair of c.lex
conj·ugate eigenvalues yields a sinusoidal mode whose frequency is given
by the imaginary �rt of the e1getwalue and whose decay rate is fixed
t»y the negative real part� The eigenvalues as shown .in Table 4.28 ....
clearly ind1·.ca�· the necessity of. an additional. ext.",a' controller to
enhance the system damping. The predominant roots of Table 4.28 are'.
. . '.'. �. .
fairly close·to the imaginary aXis. as shown in Figure 4.3 and hence
.
they would produce sustained mechanical osCillations of tha : rotors. .
.
follow1ng a system disturbance. The purpose of the external contreller·is
to move these roots away from the ..imaginary axis.
In order to design the optimal state regulator the penalty matrices. .
..
Q and R were selected from a number of trials. The following' values.provided the·best system characteristics.
Q • DIA (1 10000 10000 10000 10000 .1 1.1 1 1 '1 L
1'.1 I] (4.35)
.... (4.36).... R. 1.0
. ".
•
••
.
.. 5 .:
)C•
•
·74.
.+ - UNCOMPENSATEDSYS'TEM
.
j5 .: .)( � 'STATE REGULATOR
a-APPROX' STATE'REGULATOR
. j3
....j I
. . .. .
•.
'. ,. I • .,
3.·,' ,·5 .
.' .'
.' .' .
Figure 4.3 Dflgra. shOwing the locattans·.of dGllfnant systeM. NOts 1" cGllplex S-pltne. .
.+
+
-j3
75
.... .... '
...'
.
. .... :.
'.' ....
' ..The mitrix-R1ccit1 equation is fonnulated from equation (4.4) by
..
... . . .
.
.
substituting A. O. Q' and R matrices as defined above. The steady state.
··solution K* .was found using the iE1�envector' approach described in.
...
the preceding section. The transformation-,matriX W was formulated from
...
.:.. .
.
. equation (4.30) and the submatrfces WI1 and W21 which were requfred to
'calculate K* were solved for;, the.se are given by Tables 4.4 and 4.5.
'The solution matrix .K* was next found from equation (4.28). This'
is given by Table 4.6.
Sylvester's criterf'on52 was ·."plied to check '1f K*'was positive.
. ',' .
definite or not. This' gives an indication if the solution to the'matrix-.
Rfcca,ti equation is correct.
The optiNl control variable u*,'was. obtained from equatfon • (4.3).
It was found ·to be .
..
tJ*. - (654.0 182.8' -142.6 . -261.0.
1.11 .";592�6 . -81.2
-16.4 2.6 ;.(i1.O .54.4 -0.12 8.1 -8.7 �4.11 (X} . (4.37).
These results �re obtained from a di�ital comPuter p�ram ,Which�. .
gave the optimal controller parameters for an arbitrary combination of
generatorS in the ·study· and 'ext�mal' ,system generators., .To generate the compensited characte,ristic matrix"requires s�bsti-'
tuting equa.tion (4.3) into (4.1).' This yields·.
(4.38) .
. or
•
X. A*X (4.39)
Where.
A* .[) - BR-1 OT K*J.
(4.40)
TABLE 4.4
Wu MATRIX.
(All ttle elements of the I"tatr1x are e�pressed in the fo"" mEn which is eaual tf' m x IOn)
-0.1199£-02 '().7asu..(16 '()�9312£..(16 ,.0.4011£-93 0.8215E'()5 -0.1610£-04 0.8991£-05 .0.30_-05 0.4445£004 0.275SE004 ,.0.2353£..04 '().SS57£-G5 ,.0.2794£-04 ,.0.1917£,.03 0.1808[-04O.7859E-06 0.4383£-05 0.558tlE-OS .O.15Z2E..04 -O.4221E.()S -O.7OS7E.()5 0.4742£-04 0.3821£.()4 0.1676£-05 1J.9347£-05. 0.334lE-04 . 0.2548£..04 0.1003£-03 c).1031£-03 .().U86E"()S
-O.13a4£'()3 ..().8S98E.:.o. -CI.89SlE-04 -O.S126E"()3. ·1).5225E-Cl4 -0.1983£-04 �.W2E"()3 -8 • .1062E-03 0.1238£-03 ..o.4678E-04 -O.1301E-83 .o.5091E.()4 -O.9899E-04 ;.0.9445£-04 O;U3SE"()7-0.1202£;'06 0.9518£.04· .O�a2S9E-04 ··0.1639£.-05 0.9010£-OS O.2823E'()S -0.1183£-05 0.2722£-04 0.111SE-04 0.8033E-04 0.3327£..05 0.2642£-04 .0.9281£.ci4 0.7072£;.04 0.1411£-05··0.2085£-04 -0.1_£,.02 -1).1265£..02 ..o.2387E-04 -O.295a£-04 0.5700£..0. ·0.7542£-05 .::0.1111£-03 0.6044E� -8.3097£-03 0.7555E..05 -0.5184£-04 .;0.91_.04 -G.6481£-04 -o.tI3OEo01
O�44OaEo07 O.ll!l7E:.os· 0.2S95£..06 0:3620£-05·0.2229(-06 oO.3S7lE-06 O.I62OE-03.o.321i£-06 0.1914£..os -0.2741£-04 0._4£-04 .0.1351£-05 ..().l690E-04 oO.1713r-03 1).1112£-04
oO�2181£-07 '().I663E..(16 ..0.2_..(16 ..().1646£-O5 -O.4169Eo07 0.i719E..06 -O.647aE-04 -0.5030£-07 "().8731£..06 -O.�S9E-04 0.4UII6E-04 0.....(16 0.5714£-116 0.7331£.006 0.1l0aE-04-O.7636E..06 ;.o�5832E.05 -0._..(16 oO.5782E..04 -O.I�-05 0�59!1OE-OS oO.2Z96£..o2 .:o.12�-05 ..()••3E..04 0.2352£.04 0.4M3E..04 O.I123E-04 O.IWE.004 O.i6s8E-04 -O.1928Eo08
0.9319E-09 0.1956E..02 O�ll1SE..()2 O.4988E-04 -O.SOl6E.os "().1524E.os -O.5GSn.:ll4 -O.335iE-os -0.2381£-04 0�1474£-O3 -0.3556£-114 0.1104E-04 0.1357£..04. 0.9325£;.05 -0• .3600£-09
-0.7152£-08 0.15iO£.;.o3. 0:_-03 o.I35SE-04 �.4195E-05 ..0.3070£-05 -o.3a62£-04 0.1019£-05 -b"�2123E-04 O.1l93E-03 -0.2599£.0. O�9!l32E..os O.moE-04 0._-05 ;.o.2905E-o!I.
O�i189E..(16 ·-0.3314£;.03 ..o�4781E..o3 .0.1667£-04 -o.3742E-06 .o.6144E"()5 -0.1701£-05 0.ilSx-04 -0.1240£-04 0.4t53E,.04 O.3516E-05 0�,""-05 O.1099E.04 f).7732E-05 -0.3898£-09
o.ioooE+oi 0.1115£.02 �.1_..()2 0.1000E '01 -O.6133E-02 .Ml46E...()2 -0.1207£-01 0.17�-Ol· o.4!109E� 0.2025£.01 -O.2925E'()l -O.lOlME:-Ol -O.1974£-G2 -O.IZ32E"()l -0.2404£-03
0.1411E00 -O.1858E-04 -O�-04 -0.1551£-01 0.1683E-03 0.7330E-04 '().2�-O3 -0.'.571£-03 '().1912E.()3 ...0.5352£-03 O.7546E"()' 0.2799£..04 0.57OOE-04 0.3M9E-03 ,..7163£-05
G.3284£-01 0.3766E..o.. o.3,2.ttE-G4 O.l543E-oi :-Q.301SE.()3' O.IMD6E-04 -1).812&£-03 O.12�;.02 .0.4398£-03 o�i!IUE-m -O.979OE-02 -0.1071£-03 .0.1697£-03 .,0.1047£-02 ..o.1694E�-o.2187E-Os 0.I788E-G7. O.�\I84E-07 -O.1009E..(14 0.4OO8E-06 0.4685£-06 0.4�2E-Q6 .O�4_-05 0.4405E� .o.2556E� 0.2642£-114 -O.li_-Ol .0._.-05 ·0.5583£;.04 -0.3931£-05
. . .. . . '.�' . .., .
. ..
;;c ..
TABLE 4.5
\�21 MATRIX
(All the elements of the "'atrfx are exeressed in the fo"" mEn which is eoual to � x IOn)
0.1229£..02 ..0.5540£..01 ..0.1425£..01 0.4922£..00 ..0.4341£ 00 ..o.4597E·00 ,o.I000E 01 0.1000E 01 0.1000E 01 O.I000E..ol O.I000E �1 O.I000E..ol.
..o.I376E 00 ..0.8691£..00 ..0.1923£..01
0.9842£..03 ..q.8337E 00 ..o.824-1E 00 0.3711£ 00 ..0.9190[ 00 ..0.9291£ 00 0�7904E 00 0.1843£ 00 0••4£..01 ..0.7103£ 00 0.5712£ 00 0.2401£ 00 O.iOOOE 01 0.1000E 01 0.5617£..03
..0.7846[..02 ..0.8611£ 01 ..0•.1076£ 00 ·.,0.3241£ 00 0.1506£ 00 ..0.8748£· 01 ..o�3457£ 00�.3605[ 00 0.2681[..01 ..0.2816£ 00 ..0.2081£ 00 ..0.1302£ 00 .0.923O£..oz 0.545....01. 1).3094E..o3
..0.9501£..03 O.iOOOE 01 �.999E 00 ..o.3329[ ,')0 O.lOOOE 01 0.1000[ (1-1 ..0.6693£ 00 0.14-10£..01 o.i541E 00 o.ll94E �l -o.3669E 00 0.17_ 00 0.9361E 00 0.7625£ 00 . O.IMOE..oZ
O.l195E..ot -0.8922£ 00 -0.7512£ 00 ..o.�E-01 ..o.Z303E 00 0.1788E 00 -O.849Z£-OI-O.316OE 00. ..0.1267£-02 -0.3922£ 00 -0.8335£-01 ..0.8116£ 00 O.8628E-Ol 0.5422£ 00 0.6162£..02
...0.8157£..03 -0.7389£041.00.1138£ 00 -0._ 00· 0.3325£ 00 0.3038£ 00 -0.8605£ oO:.o.I�I£ 01 ..0.8495£ 00,-0.1000£ 01 ..o.8569E 00 -O.8117E 0«) O.8828E-Ol O.5422E 00 0.6162£..oz.
-O.43OOE..os ..o.moE..oz �.6312£ 02 .oO.2388E-01 .0.3238£-01 !).5l25E..ol ..0.1235£ 00 0.5492£-02 -0.2880[.00 -o.2i86£ 00 -o.U03£ 00 ;.0.3707£ 00 0.8164£-01 t\.5409E 00 0.1000£ 01.
..o.�l599E,.()$ -0.9441£..03. ..0.1720£-02 ..o.�..o2 0'.2584£-OZ 0.1549£-01 ..0.2248£-01-0.2698£-01 -0.3819.£..01 .;.0.3035£-01 -o.269sE-01 -O.2675E-Ol 0.2313£..02 0.13.-01 -0.2783£-01.
0.1127£-03 -0.4158£ 00 -0.3862£ 00 -0.2034£-01 -0 ..2243£ 00 0�a305£-01 -O.5870E-Ol..o •.Z49OE 00 ..o.3196E..oi . .o.� 00 -0.6794£-01..0.8930£-01 .0.6598£..oz- 0;3839£..01' 0.7918£-03'.
0.1250£...04 -o.3i92E 00 -0.3447£ 00' -O.1917E..oi -0.4354£ 00 -0.1593£ 00 -O.9OO4E.oJ -0."78£ 00 ..0.2432£ 00 -o.S613£ 00 -0.14. 00 ..0.2061£ 00 0.1724£-01 0.1013£ 00 0.2406[-OZ
.0.4491£4 -0.6623£-01 ..o.8088E-01 -O.4725£..oz ..0.1-406£ 00 ..0.1463£ 00 ..o.318iE..ol-o.16S4E .00 -0.1438£ 00 -0.2272£ 00 -0.6_-01 ..0.9475£-01 0.852«-02 O.S03lE-01 0.1283£-02.0.5103£..oz 0.4557£-04. 0.6127£..04 .0.,3684£-01 ..0.315.8£-03 -O.3402E-03 .f).7909E.03 O.1189£..oz 0.7594£-03 0.1478£-02 -0.1714£..02 0.8506£"," -0.1787£..03 .o�1124E4 -0.2402£-04
-O.1592E..oi ..0.8463£.03 ..o.6199E-03 :.0.2866£ 00 -0.1424£..03 �.Z543£-02 O.5204E.o2-O.2OO9E..oz· 0.5857£-02 ..0••931£..02 0.41_-81 0.2462£-01 0.1683£..03 0._.-03 .o.1418E�
11.1914£,.02 O.6!170E�3 0.60!i0E-03. 0.2911E 00 .!).4064E..q! 0.3851£-02 .,.8.9292£,.02 0.1066£-01 -O.I086E-01 0.1526£-01 -O.!l574£-O1 �.7273E-Ol "o.619lE-03 ..o.36oaE-02 . 0.5591E-04
0.7721£..03 0.3425£.03 0.30_-03 0.1474£ 00 -O.Z566£-02 0.1405£-02 ..o.5289E�Z 0.7543£..oz -O.53OSE-02. O�t045E-01 -0.5943£..01 .:0.5522£..01' .0.4766£-03 -o.2797£..oz . '0.5682£-04.
....
...,
TABLE 4.6
K*.�TRIX
(All the elements of· the matrix are expressed in. the form mEn which '1s etlual to m x 10n)
0.5387£ 05 0.1555£ 05 -O.l1fiOE 05 .0.2199£ 05 -0.7025£ 01 .fl;4871£ os -0.6497£ �. -O.1356E 04. O.lUSE 03 -0.4846£ 04 -0.4323£ 04 0.2479£ 02 0.33OOE 03 -0.2545£ 03 ..0.7930£ 02
0.1554£ 05 0.8164E OS 0.5537£ 04 .0.6941E 05 O.S66IiF 04 -0.5275£ n4 0.4048E 03 .0.3372£ 03 0.8415£ 04 0.3054E \l$ 0.1486£ os 0.8331£ 01 .o.!J187E 02 .,.'}.5057£ 02 -0.1380£ 0.2 '
-o.U43E 05 0.5330£ 04 0.5630£ 04 �.2U7E 04 .0.Ziltj.' 113 1).1214E 05 (i.493fi� 03' o.2l64E 03 0.24� 03 0.�84E 04 0.2931E'" �.3785E 01 -O.711J9E 02 0.2657£ 02 0.4299£ 01
-o ..21s2E 05 -0.6954£ 0. -0.2364£ 0... 0.8149[ 05 .(1).3197£ 04 0.1204(05 -O.��,.t" 03 .
0.4653£ 03 .0.6_ 04 -0.2755£ 1)5 .o.l383E os ..o�I051E 02 -0.1265£ 03 0.6096£ 02 '0.1460£ 02. 0.1129£.03 0.5524£ 04 -0.2781£ 03. -O.30S8E 04 1).4334E .04 1).6$15£ 03--O.� 02 O.n84E 02 .D�3484E 04 0.4784£ 04 0.1041E 04 -O.6G63E 00 O�1496£. 01 0.1214£ 02 0.3914£ 1)1
..o.4896E os -0.5117£ D4 0.1231£·lIS 0.1195£ 05 0.7597£ 03 0.4697£ 05 0.3813[ 04 0.132ot 04 0.9is6£..o3· 0.889OE 04 0.6227£·04 -8.2162E 02 -0.3004£ 03 0.2126£ 03 i).5969£ 02
",0.3255£ 04 -0.3799£ 04 0.1341E 1)4 O.4361E 04· 0.1191E 04 .o.1361E 04 0.6293£ 05 -O.2� 04 0.142GE 04 0.9683£'4 0.7172£ 04 -0.4790£ 01 -O.4514E 02 0�22!i9r 03 0.3389£ 02-i).15m 04 -li.1S02E 03.0.2204£ 03 .0.26i8E 03 1).41iSE � 0.1523£ 04 -O.11illOE 'l2 O.I'l9OE 03 0.2_ OZ �.187SE 02 -O.2293E 02 ..o.w9E 110. -ci._E 01 0.3167£;;1 0.2106E.OlO.3088E 03 0.8244£ 04. 0.1986£ 03 .o.6�3E 04 9.3498E.44 0.8379£ 03 .o;1109f � .0.6333£ 02 0.3187[·04. 0.5722£ 04 0.1793£ 04 -O�3201E 00 0.2168£ oi 0.1009£ 02 0.322OE 01
�._ 04 9.2941.£ 05 9.4023£ 04 -0.2642£ 05 0.49200.04 0.7744E 04' 0.5255£ 03 0.2074£ 03 0.5781£ 44 0.1739£ os 0.7942£ 1)4 .0.15_ 01 .o.2438£.Q2 0.3141£ 02 0.8371£ 01
.0.3533£ 04. 0.1412E 05 0.2759£ 04 -O.130SE 05 0.1137£'04 0.5468£ D4. 0.4036£ �3 0.IU2E 03 '0.1836£ 04 0.7947£. 04. 0.4198£,04 -0.1236£ 01 ..0.2302£ 02 0.1788£ 02 0.4366£ 01
0.245� � .0.•8673E 01 .-O.3nSE 01 -O.IC)88E 02 .0.6747.£ 00 -O.21.teE 02 ..Q.3953E 0•. ..0.6658€ 00 �.3788E 00 -O.lOO2E 01 -O.l525E -01 M_..ol -O.W4E-Dl -O.2374E�1 -O.1026E 01
O.3273E.o3 0.9144£ � -O�7133E 0.2 .o.I305E 03 0.557SE 00 ..0.2963€ 03 ,J).4061E ot -8.81nE .01 . o ..129l£· 01 -O.3049E 02 -0.2721£ 02 -0.6382£.01 0.4068E 01 -0.4368£ 01 -8.2049£ 01
-o.Z34sE 03 -0.7287£ 02. 0.2108[.02 0.8433£ 02 0.1407£ 02 1J.l923E 03 c).SOIlE 02 0.6735£ 01 0'-1I!52E 02 .0.2874£ 02 0.1646E 02 -O.161gr-Ol-O.4257E 01 0.1036£ 02 0.�261E 01
-O.7n2E 02 .,.0.1574£ 02. 0.4512E 01 O.I66QE 02 1).4398£ 01 0.5739£.�2. 0.3371£ 02 tM843E 01 0.3744£ 01 0.1204E 02 0.7051£ 01 ..o.1043E 01 .0.2049£ ')1 ,0.6322£ 01 G.47ft£ 01
.. ,1,- •
.
•... CCI
, 79
The new �haracteristic matrix A*, because of the presence of the.' .
.. .
.
'.'
opti,.' controller, is a modified fo"" of the orf�fnal A'matrix, as
described in Table 4.7.,
''
It is of fnterest to compare the eiqenvalues of A* with those of..'
.. .
..
,
A. The. eigenvalues of A* were' calculated usfng the' IBM 'SSP subroutines
HSBQ and ATEtG and are tabulated in Tables 4.8A arid 4.88. The pre
dominant roots are p,otted fn Figure 4.3.
: A comparison between the" relative, locations' of the predominant'
'ro()ts in the complex S-plane fs shown "in Figure 4.3. these· results.
show that' an improvement'1,n system dampfng by :a 'factor of' 25' has' been
, '
achieved with the', addition of the optimal controlleJ"$I! The most dOmin-
ant,roots hive been changed from - .044 :I: j '6.68 to,- 1.1 1: j 6.69.. .
.
. .'.'
.
The time: domain' results as shown in the next seetion' 'give'.' pictorial,indication of the"improvement in $ystemdamping�' ".
Oesi9" ,of a Truncated' State,Regulator'.
The basic difficulty with the state regulator fs that it 15 often
impractical to implement althoUgh mat.,.tfcally feasible� Typically, "
the feedback portion of the optimal control system is a function of all. .
,
.
'. ,,'.'
the states of the system. This woUld be, satisfactory provided all the',states are accessible for measurement, however t�is generally is inot
..
�.
the case. From the'se consider.ations it 1$ desirous to develop a ·sub
optimal· control so ,that the performance of the suboPt1Nl sYstem is.
."
.
.
.
:.:, ',.." .
I near' to that of the optimal system. One approach is to follow the
"Minimum no",," criterion as suggested by Kosut47 as described below••
TABLE 4.7.THE CHARACTERISTIC MATRIX A* OF THE COMPENSATED ·.SVSTEf.1 COOPtED WITH THE STATE. REGULATOR'
'-0.1730.0 0.0 0.0 .
0.0 0.0 0.0 0.,0 0.0 0.0 0.0 0.0258 1.29 ". 0.0 0.00.00.0 1.0 0.0 0.0 0.0 0.0 . 0.0 0.0 0.0 0.0 00 . 0.0 0.0 0.0 .
.
.. .
-19.7�18.0 -0.149 17.8 0.0 18.1 0.0 .
'
0.0 0.0 . 0.0 0.0 0.0 0.0 0.0 0.00.0.
0.0 0.0.
0.0 .. 1.0, 0.0 0.0 0.0 0.0.
0.0 O�O .0.0 0.0 0.0 . 0.03.03'19.8 0.0136-22.2 -0.'102 . 0.0 0.0
.
' 0.0 .18.2.
0.0 0.0 0.0 0.0 0.0 00. .
.0.00.0 0.0537 0.0 0.0 -0.909.0.915. .0.352 0.0 0.0 0.0 0.'0 0.0 0.0 0.0
0.00.0.-0.026.8' 0.0' . 0.0 0.0 -0.0029 -0.176 '. O�O 0.0' .: 0.0 0.0 O�O 0.0 0.0
0.0.'0.0 '. -0.937 0.0 '
0.0 0.0 0.0'
-6.25 '·0.0 . 0.0' . 0.0 0.0 0.0 0.0 . 0.00.00.0 '. 0.0 .: ··00.·.· 00" 00 0.0 .. O�O .
-20.0 20.0 . 0.0.
0.0 . 0.0 .
·
.• ··0.0 0.0 ..
.• e... e.
,.0.00.0 . 0.0 .... 0.0"
0.0 0.0 0.0 "
.: 0.0 0.0 . -6.67 6.67 0.0 0.0 0.0' . 0.0 .
0.00.0' 0.0 0.0 -1.38 .' 0.0 0.0 . 0.0 0.0 0.0 -12.5 . 0.0.
0.0 ·0.0 0.0"
0.00.00.0 0.0 0.0.
" 0.0 0.0 . 0.0 0.0 .O�O 0.0 . 0.0 -30.3 -1015.1 0.0 .
.•�1311.55-3($5.4 285.2. 521.86 -2.22 1185.2.' 162�4 .32.8 -5.2 122.0. 108.8 -2.69 -165.2' -1.9 -0.31
O�O0.0 0.0 ·O�O· " 0.0 . 0.0 0.0 . 0.0 . 0.0 0.0 . 0.0 �0.689·-34.4 .4.54 '·'0.0-0.313'0.0129-0.00141-0.0125 0.0 0.0 0.0 '. 0.0 0.0 O�o. . 0.0 0.0 '. 0.0 0.0 -2.0
!.
,.
TARLE· 4.8
. A. EIGENVALUES OF THE COMPENSATED SYSTEM
COUPLED WITH THE. STATE REGULATOR
No." . Eigenvalues..
1 .. 173.5 + j 0.0
·2 .. 17.42 + j 1.82 '
.
3 ... 17.42 .. j 1.82
4 ,. 14 •.55. + j 0.05 .. 1.1g + j 6.69
..
6 .' 1.19 j 6.69... ..
7. .
6.37·.. + j 0.08 .. "263 + j 3.35. ."
9 .. '2.63 .. 'J .3.3510 ... 3.88 + j
.
0.1011 .. 388 '•
. j '0.10..
.
12 �. 20.
+ j 0.0. .
13 .... .. 0.98 + j.
0.014 .' 0.915 + j 0.015 .. 0.0026. + j. 0.0 ,
B. DOMINANT ROOTS OF THE ABOVE tENTtmED SYSTEM.
No. Eigenvalues
1 .. l�19 .. j 6.69.
2 .. .
.
1.19 . .. j 6.693 .. 2.63 :+ j 3�354 ..
. 2.63 • j 3.355 - 3.88 + j, 0.106 .. 3�88 .. j 0.10
81'
.82 ...
If U*. _R-1 8T K* X for optimal state requlator. or, .
u* • .S*X
and if the suboptimal control vector UO is defined as
.
UO • -SoX·
.. then the proJ_)lem is to find SO such that •.Min ...
.
S 1'1 .. SO - S* IIo
.
. (4.41)
(4.42)
(4.43).
. .' ':'. ". ','.'.
let Z denote the feedback variables to be used in suboptimal control.
or
Z· MX
From equations(4.43) and (4.44) ·it· can be derfved47 that
SO • S MT (M MT).IM
(4.44).
(4.45)
From equation ·(4.45) it can be concluded that the mini.·· n�"" sub..
�Ptimal control vector is nothing but the optimal control vector where.
.
the ·t�nns· involving the undesirable �tates are deleted. Equation (4.39)
fndicates the· eXistence of �ik couDl1ng between u* and some of thestates. Th.1s implies· that: there are some states in the syStem which
·
offer little contribution in enha�C1"9· system damping. It is of int�rest
.
to decouple the fnsfqnfffcant states by setting the corresponding ele-.
.
ments in this vector to zero. The result is·· given by equati·on (4.46) •..
U*.. ;"(654.0 182.8. -142.6.
-261.0· O· -592.6.·
o· 0 0 0 0 0 0 0 OJ [Xl (4.46)" '.. "
.'. t"
. ..
..
It was of 'importance to find the stability of the closed· loop.
."
.
system with the truncated regulator included. The new characteristic, .
83
matrix:c'A-BSO) was computed and its eigenvalues :were calculated. These
are qiven in Tables 4.9A and 4.98. It can be seen from Tables 4.88 and'.
. .
.
4.98 that the predominant roots do not differ significantly frem'the
optimal 'case by truncating the ,state regulator. The suboptimal system.
..
. . . ..
performance was expected to be very sim'liar to that of the optimal
system. The time domain results :of the. 'subopti'mal controller are
included with those of the optimal res'ults in the next section'. The
time d_in transients for the uncOlltpensated system are also included
for comparf son purposes.'
4.5 Time Domain Results
The 'eigenvalues which are essenti,al1y the po'es of the closed 'lOop
system' as giVen in Tables 4.2,' 4 ..3, 4.8 and 4�9do not' give'.'1:he,..
"
complet. picture of, the 5YSt., dynant1csbeca�se of th� lack Of:,,'knoWtedgeof the relatiVe importance 'of the different predominant \roots.: Calcu-,'
. ......
.
..
'lat1nq the zeros of the transfer function would give a direct indicae'
tion of their relative effects. The alternate WlY to qet the most
tangible evidence of the effects of thedifferent forms of 'the contro'
lers is to display the transients of the compensated and the uncompen
sated system in the time d�ain, using, the iterative solutions on the
state space equations. The res"lts a� shQWn fn.Figure 4.4� 4.5. 4.6 ,""
and 4.7..
.' .',... ': '.
The state space equations were solved numerically usinq ,the''
standard fourth order Runga-Kutta method. In the Case of the deta11ed
mathematical m,odel whiCh included transfOrmer action. the, ti. interval,.
,".
TABLE 4.9,
,
A.' EIGENVALUES OF THE COMPENSATED SYSTEM
COUPLED WITH THE 'TRUNCATED STATE REGULATOR
No�,
Eigenva"ues
,I ·.164.1 + ,j 0.0
2 - 17.42 ' + j 1�82'3 ' . 17.2' '
• j 1.824 • 9.18 + j 0.05 .', 0.91, + j 5.23
6 .. 0.91 - ' j 5.23
7 - 6.42 + j 0.0
8 • 2.99" + f 3.51'; 9' • 2.99 '. j 3.51
,
10 - 3.02 + j 0.65,11'
' '
3 02 j 0.65- ' -,.
12: ..
:'._ ,2.00' + j 0.0,,' ....
,,
"
13 -, 1.00 + j, 00.,
14 "- , 9.832 � j O�O
15 • 0.0024 + j 0.0
B. DOMINANT ROOTS oF THE :A80YE MENTIONED SYSTEM,
No. Eigenvalues,
1 - 0.91, + j 5.23 '
,
2 • 0.91 '
. j 5.233 • 2.99 + j 3.51-
'4 _ 2.99 - j 3.515 '_ 3.02 + j 0.65
'
6 -, 3.02 - j ,0.65
84'
o.o
Time in seconds
Figure 4.4' Ca) .•.. Squaw' Rapids speed error vers�s tt_ fO"th,'uneGlllpensated system (including trans'.,.."Icti era)..
Figure 4.4 Cb) .Queen Elizabeth speed errcr versus tiMe fo" t.he .
uncompensated syst.. (includfng traftsfo".,,·..
. .
action).
'85
', I
. ..
Figure 4.5 (a) ·Sq",w Rapids speed error verSus. time for the ..
uncCinpensated system (excluding transformer· .
.'.
.
action).8' .'
w
. � !lH�-·.o.�.. �!�:f! .
.I ....."1,'"1
ililillh . 1m!v:
. nHP:II!·· TinilY.: I,., •• l l l , ••
"1".'.: ·:'lilit:Qn,TI'IIi.!1!!�: 'I .. .Ip" . .: ....
1'.e: ! i!fI"j'dIlHIII!I!j.n,� g-j ihl'I' Inillllll:.!in�lll:l.o' .,'" I ru "litH "
� ... : i HiLiUt'ltt1I',:ni rll.
,.: .
:lpl:u:n hi ::qq-l.. � I' dIP!!I' !! j qill'lq�.! ...... •
"'I'll' 11 .. 1
'''' 1101� ",. ·I't· i" "'1' I" .,
i! 1:lli Ihhli!lid ilhl!!!t: 1111 umn Ijlll,lllljjl:11 1. ••
c: \,ndih:q:I'I'!IL11i!!iII'I'!i:·I· .1!h. 'lilt- : -.1:,n:lII:"I' 'Id '1::::":::' .n::: .T_
-111'1'... :
Plillt·j,l,iH·ln:ln,i'ILHPIPI'!lq·I'•• !inl::·1 ·IIPI- TIllh:o : T·,'.I II: n l'n ::P! i: : nUln • d' : 'I'll!i !i",:H!'1 I!I I ill:illillliiilfjl'IIIi!:jll!!!lflili ih. iiililli illll!I)I'\o:{ :'j '1Iiq":I:n:qlll·1d'I:·'I· (11:111:1:- T':'II �110I1 l' t'II'111!'.'0. o::!:I,II: dd!! unn .. J!:;:.!!: '11,1,1'1'" :,', II, TI!! -I !: !I' .. II! '1:111 II qt·t:"II I't .... 11, •• t '11'1 "t" 1 .: .... I ··· ..
'1·'.. .1 • .1 •••• -I' I , 1'111'• w·· . , .. 1·'·'I,·f 1 •• 1.1. " I •••••• 1 ••• "., I. Ii::, .e ········1····
. ,. I.
t',
# o:!l 'I II til:npl PIHIII!lt!!! liIUU1P'!I!Plj!1 Illlll Ii tll!'!I! !iil!l!I!'lqjl ,II'!!!'eII,,1 II,
IlI'l 1111' II II"·' 1-111111 "II' "PI''''t, !'II II
.... I' I",
1'1�ltli.lll!.ll .tt .l.ll.llt,.UU ... lil.l lttl.t J.ldl.lllillit. tll.l.l.l1[lll.ltd lHltlll i .llllUlld.. l.lllUll., 0-0 I'S 30 4·5 Hrolll. in "co.d.
.S·
F1�ure 4.5 (b) Queen El1zabeth speed error versus time for the.
uncGnpensated sys_ (excludfng transfomer
action) •
,
used in the numerical 1nteqrat1on was e.hosen as 0.00375 sec. This,,
'
.. .
. .
, low ,value was required to �et rid of the numerical instability"'. .... .
.' ..
because of the presence of high frequency transients. In the case of.
, ..
the approximated Mod�fwhere transfomer action WI'S deleted. higher'.
..' .
, ,,
frequency terms were no longer present. Hence in the latter case a
larger time fnte"al of 0.01 sec. was used whiCh,' in tum. saved,
a considerable amount of camputer time. The time dONin results are
ShCMn in Figures 4.4 and 4.5 for the uncompensated sYSttm us1n., these
two, different mathematical models. The variables plotted are the
Squaw Rapids Sf)eed Error and t)ueen Elizabeth, Speed Ertor respectively.,
, ,
The disturbance to the sYstem to produee :the transients was a si"u-.
.'. '. .
.
.
.
..
l.ted 10% step chan� in the' electrical power d_Jid at Squaw Ra.,fds�The s1m1larftiesbetweerl Figure<4.4 and' Figure:4.S' justify the decision
. ..
. ..
. .
, to use, the· approximated'maiel'·instead· of. 'the "detailed' (JI'Ie•• "conclusion'reached earlier b.Y cOlQftaring the systeln"eigenvalUes. '
'
'
The' �ha"acterist1cs ,of the 'transients sho"" in Figure 4'.5 can be
directly related, to, the uncompe.,sated $�tem' ri)()ts q1v�n in T'able,4.2A. "
" ,
The predominant root w,th a damped' frequency of 6�:7 radians/second has,
a period of appreximately 1 second. th� transient,s art! fairly' ose11';'
latory for the uncompensated system. In addftf on ',the 23 second time.'.
. .
constant of ,'the envelope of the damped sinusoid.: as' pr.edictecf by:the, '
. '. . ..
system root locations. is certainly a�p'arent in the persisten,t 05cl1-
latims:of these transients. The transients for the optima" and sub- '
.. 88
·0
:0 .
-,or
:51 it iii"!, '. .
,;0; iII"!,' II � .
�: lllh 1 i l l i .
;; j °I'!lI'!I"I'!,i!lji'J! i !I III l'ill'll'.
-. i l1111111i! 11!,i il,1 il '..
0, 'III" ! 11' I It��i IfI,lI'l' IIII!II I''Il IIII I! II '.
..
�O! II I·
"11"""......'. .
:! I uruuu ill". . :
!! I I n '11".
..
.
l i I
I I I' i. .
.
I !I
II
I i 1 Ii"" .... ". .'.
0.
I I II I II it '111"lmllW "
.
g1...l... . .. \ ..... \ ..... .t. ..... ; ... \ ...�., .. ll .. l, .. \.IHIlUlmmnminmth\mmmm��.t..0-_ 1·6. 3 . .0· ... . ....
.. T .... I...,....... '. . .
Figure 4.6 (a) Squaw �pids speed errGi'" versus. time when. the.:
system is coupled with the state regillator.
8 .
....
...
�� ��..
�o� 1 � ng i iii 11;; : .11: II
: 1. ! i j! 111. i i lillollls8!.· iil,i II'w: .. i 0 I
�� i i lilt .
.
Ii 111 III'· .11"1"1'11"11 .
! I!! '11'11II I" "II" Ih .
.
! 1"1 I I' I"I 111" . np .
. .
.
ti.. ,11111 . .I.t ..ll.t .....J I ..l.l.lt \IIIII IUIlhltlllltl'll'\!Itn;m It'l!'"''''""""'"... ·w0..fl...., Itt ...�......
:.
.'..
'. '..
..
.
Figure 4.6 (b) Queen Elizabeth sPeed error versus time when the .'system is coupled with the state. regulator.:
89 '
'0o '
tt: n'�! nu .
��I ill' Iilil! L jll 11111! : i Ii I I'
.. !s!lr··· I..
;�:!::i I II I 1111 " " '" I " II" "
·0:'I :..... :
.
CIt !
I
Jl0.:.' •
00
'F1gure 4.7' fa) 'Squaw RaJ)ids ,speed e"to," versus time when the systeM,
: is coupled with the truncated state JreCJul.tor. "
'
��1 ...1, •
..... 0: ,It
f�i ' III15 i I· '
.
. � :
� i.. i .; .' .
Gill' ,
5g: 1 It."
;,j' ".I'j , .. ,
.....••....jt
i I I, .'
III
t I) Pit·111 Iii .
" ,'.', ,.' ••,:, ",', ',',
.
I lill 1111111,11 ijrllllllil1�11j!ill!' ' ....•.'. .
,., '..
" �)t 11 h. W.W..lll..l.\i.ll.!.w.tthUlW.uHltj ttUmlmt\t,,,,"''1'1'1\'''I. �.'00 I s o 4& 60
'
Tillie in '.'Qlld., ,
Figure 4�1 (.b) Queen' Elizabeth speed e.,.or ""US tiN \rfhen the systesnis coupled with the truncated state ",,'et.,...
.
"," .... ., 90
optiinal 5yste", are shoWn fn 'F1'oures 4.6 and '4.7. As expected the.
.
.
tranSients using the truncated state' regulator compare 'qufte favor.ably
with those us1nq the c�lete state reoulator. The effect of adding.
. .
.....
either of these controllers to the system is' quite evident. A compar-
ison of F1�ures 4.5 and 4.6' clearly dem01strate that the introduction'.
of the.�tiN 1 cmtroller increases the, systel't damping to a qreat '
. .. .
extent. The close simUaritY'between Fi«'fures 4.6 and 4.7 $uqqests that. -
· by a proper. choice' of state variables it fs·. possible to build an
.
..
. .
.. ;.
�tiNl centroller based. m the measurable .states only. A better.
. '. .
"
.
, canparison can ,be' obtained by d'rawing Figures, 4�5t. ',4.6, and 4.71n :. .
. .
one plane. This is ·dme in F1qure,4.8.
,
"
·
system transients. have ,been ,ca1culated �sin� an IBM, 360150 df9itll.
·
computer and: the results are:. compa�d. 'The improveMent in systewn .
damp1nq'due to the addition of the state r.equlator' has :been ,·verified.
both' by cmparing root . locations as we,n. as fr� time ,d_in results.
In the next chapter an al�..,r1thM is developedby Which the optimal
state requlator ca� be. trans.fOTmed into a simple transfer funetfM ....form and nu.nerfcalexamples·are used' to illustrate the techn.fque.
.. 01..
�
_j�...
gt> O!l
�101
u'..
�
� 03..
......a.<II
(>I
10 2-0. &0
·01
.
·()3
Ffg�re 4.8 Ca) Squaw Ra.pfds speed er..... venus tiM
'Ca) UncClnpensated systeM. Cb) Approxfllllte state .-.gul.tOt'
. (e) State "gulato".
� 07
,..�o
,;?
";;05
"
·Ev
�
t 03
..
.,
..
.."ov
01
\ r-
\ V �"\\�� \
ter :
�
10
Figure 4.8 (b).·Queen Elizabeth speed .....01' ""US tf.
Ca) Uncampeftsated s,stent .
(b) App..acf.te state regul.tcr(c) State .-.gulato...
91.
·92
· 5. DESIGN OF· A S.INGLE TRANSFER· FUNCTI (If . DERIVED FROM.
THE . OPTIMAL STATE REGULATOR.
5.1 Introduction·
· The design· of an optimal state regulator requhoes access to all
the.states for measurement •. These are·seldom available in a.practical.
application.· Even if sn the states are measurable in man.v weakly·
.
.'
coup'ed;la�e scale sys.teros. this requires a very·laroe.number of
feedback loops. Considerable control system· simplifications ·w1thout
serious deqradation of the overall system performance can be often
. cbtained by having each control·variable dependent only on .some but.
. .. "
not all �ystem state variables. This giv" r1se to what ·1s· known as
the design of a 'sub-optimal' cmtroller. �e such way of desiqn1na ..
a �ub-�timal eantroller has been discussed in Chapter. 4. DavfsC)n
and· Rau39 have discussed alternative means of deS1Q�i·ng a Suh-of)timal. .
controller for a pOWer. system based on' Qltput feedback.·.
A great number of authors48.51.60 had dealt:with the problem of·feedbaCk control of linear dynamic systems with limited (lItput measure
ments. and feedback dynamics. Some autnors49,50 have approached this.
class. of preble",s fran a .stochastfc viewPoint where the s,vstem 15
assumed to have input noise and a random initial state whose statistics.
are known and the feedback law is assumed to·cons.ist of fixed, or time-
. varyinQ gains, on the system output. Several oth�r authorsS1,50 have.
: ..
tried to construct a Lueflberqer mserve·r for the 'unmeasurable states
and then operate upon· these state esti..tes with feedback qains so as to
93
compute .the cO'ltrol yariables� �e difficulty with a 'luenber�er (bserver
is that one can mathematicallv push:the.e1�nvalue of the observer,. . .
.
.
which are to be assumed, tQ>lartis' mi'nus irtfinity··yield1nq extremely'. '." .
rapid convergenc�. This tends however to make the"observer act like a.
.
differentiater and thereby become h1qhlY's'ensitive to noise. A few
papers55•59 have been published describfn� techniaues for desiqnin� a.
· suboptimal controller based'O'I the Kalman�8ucy' filter. 'e,"(1158 had. . .
investigated the problem .of desfanin� a linear compensator in such a
manner that the quadratic performance index 1$ minimized in the same. . '.
. .
·
manner as' whe" all states are available. Althou�h this 'methoci is nuite'
'attractive it was difficult to ·imr>lem.eflt for this system considered '1n
thh study, because of the raundfnq'errors ,nd n..wer1cal instabilities
arising ·when 'tackHn� Ii hi�her order syStem•. All of:these results ha�e.
�lat1ve advantaqes' and disadvantaoes in ee.-tafn .eeses, un'ortunate1�no organized attempt has been reported so far: to ;haye� a c�ar.tivestUdY'· of a 11 the�e technfttues, 'their properties, 'their ':uselulness 'for
· a li� order system, as well as their imel1cation from'a pract.fcal. .
. .' " '. . '.
pOint of vieW. '
.
.
." . ..... •.
.'. . ..'
The alqorithm which is developed in the follCl1in� oa�es is in'.
essence a modification of Pearson.',$ ,",�hod.; The technfque is modified,
so as to (Jet ·rid of the numerical di·fffc:u·lties encountered in his
method. ()) ihe, basis'. 01' this alq()r.�thin :the.:sinql. t.r.nsfer functionoptimal compensator was' desi.qned fer two dffferen·t·, cases, .:
.
. a). a. hypotheti ca15th' order "system .
b) the 15th order pewer system under study (the .-educed ;3 mach1ne
94
system of Saskatchewan Power Corporation).
5.2 Modification of Pearson's Theory
The optimal cmtrol variable u* is the linear cOmbination of all
the states defining the dynamics of the system (let the number of.
states be e), Because ·of the non-availability of all the states, one
is interested in relating u* with the output variable Y·. It is obvious
that U* cannot be a linear cm:aination of.Y alone� In order to ensure
that th� behavior of U* should be. same, the effect of the other (q-l)
states can be �placed by couplinq· u* with,. throuah 8. transfer funct":
ion whos� order will be (q - 1) and this will qenerate additional
(q - l)state variables.· The follow1n� al�orithm �alculates the coeff
icients· .of this transfer function.
let the dynami cs of the plant be �1 ven by•
·X • ·AX + BU AE. Rq x q
C E.Rl x q
(5.1)
(5.2).
Y • ex
where,
X is the vector descrfbina the states of the· system. .
.
.
.: .
Y is the output variable
U is· the sca ler control vari ab le •.
From optimal control theory (refer to equation (4.3».
U .';'R-1 sT K* X
or
U.· III HX (5.4)
95·
where.
H • _R-1 aT K*
Substituting (5.5) 1nto (5.1) y1elds·
(5.5)
..
X • (A + BH)X (5.6)
or
•
X • A*X (5.7)
Where.
A* Ii A + B H (5.8)
A* is the new coeffic1ent matrix of· the System..
. . �.
Now dffferent1atinq (5.4) y1elds .
•
Su • HX ·(5.9)
where,
S 1s the Laplacian operator
Substftuting equat10n (5.7) into (5'.9) I)fte:can write
SU • HA",X
Differentiating (5.10) aq.1n, one·canwrite
S2U •. HA*X
(5.10)
(S.l1) .
or
.
.
:.
.
". .
By successive differentiation, the last te"" een be written as
sq-1u • HA*q�lx (5.13)
Fran the above equations the following I!1iltrix can be derived
.or.
u. ..
H
SU HA*
S2U HA.2S3U' •
.
HA*3 X (5.14)
96
11 • PX (5.15)
where.
U • ,[U SU S2U S3U _. _ _ _ Sq-1U)T
and.
(5.16)
(5.17).
P is a q x q non-sfnqular square matrix and therefore its inverse
exists.
From equation (5.15) one can write
X • p-l U
HA*
P • HA.2HA*3
.
Substituting (5.18) into (5.2) :v1eJds·
Y. Cp:-l nor
or
"
:� .. ',
.
','
(5.18)
.
(5.19)
(5.20)'
(5.21)
97"
>
• .'. ..:',',.". •
'. •
.''. •
,
Thus the t�ansfer funetion 'RCS) - U/' can be obtained 1rOl'1, eQuation,
,,(5.21). It is given b!l equation ,(5.22) •
."
.
"
,', R(S) :: __1__
q-1' 1t Q1s'1-0
(5.22)
The addition of the optimal transfer function to the'uncomoensated
power system increases the ,order of the system from q to, (2q..1).'
Equation, (5,.22) has been derived by al�ebraic manipulation without any
assumptions or approximations to equati()fts (5� 1). (5.2). ,and (5.3) which'" . . . .
are",the basic ,equations describinq 'the dynamics of, the' p1,ant wi th an"
,optimal state regulator. 'It can therefore. be concluded that the behav
ior of, the :plan,� with optimal transfer furtction as des1Qned above'should
,be identical' to' that of the ',plant with th� 'oPtimal state regulator.,
The block diaQram of the plant with the 'optimal transfer function'.'
.
.
, is shown in Fi�ure 5.1.
In o��r to ascertain the perfOrmance of the plant with the op't1mal '
transfer function: i'nclu�ed. it is necessary 'to develop the sta,te'space
model of the total composite system. In addition to the exfst1nc.t state'"variables' of, the' uncompensated plant, the state space"",Odel of the:"
optimal tran'sfer function is required. '
'
.'
. . . ..' :'..
"
. .'
Let "the new state variables of the optima,l transfer' fUnction be,,
,
def,ned as follows:
,
"
. ,:,:,.....
; _-".
. '�
FG
-
·'·X.
.. .... �
.
f
-.
98.
x
·s·· A.
y
R(S)
. .:,", ..
__---.OPTIMAL TRANSFER' FUNCTION ..
Figure 5.1. General block diaf!rallt of a plant cou.,led with the _till'al
transfer function.
u
I .
I!.
99
(5.23)
. ." .....
.
The stlte space' model 'of equation (5.21) Cln therefore be Wr1'tten.
...... :
as follows: .
•
. Xq+l· '.
Xq+2..
'.
.Xq+3 .
0 1 0 0 0 Xq+l 0
0 0 1 :0 �, 0 Xq+2 0
0 o· 0'· 1: 0 Xq+3 0
..
+. (5.24)
',
0 0 0 0 1 X2q�2 0
_Qo - Q1 ··_Q2 .;..Q3 ., Qg-2- - - - X2q_1Qq-1 Qq-l. Qq-l
..
Qq-l Clq-l
--
•
X2q..2.:
X2q.ioi.
•
Xq+1· .......
-
Xq+!•
o
o .
'0
o
(5.25)
o
·100...
where C1 is the i th element of the rOtl.matrix C.
Equation (5.25) can be .written in the campaet form.
...
'.' .' .
(t] .. or) [Xl (5.26)'
I f the dts turbanee vector F is ..nc 1 uded equati on (5.26) expands· to. .
. .
. .
..'
.
.
.
[I] � ['K] (X] +. ['ft"] [F) (5�27)
Thus .the new 'coefficientmatrix hasb�en chin<l.edio � 1m,!' A 'Of .
.
equation (5.1).
The'rest of 'this chapter is devoted to the ap"l1catfon of this
alqorithm'to l'IO<I1fyino. the state' requlator ee the o"timal transfer.
form. F1 rst t a hypothetical 5th order system is cOns1 dered and then.
the t�chniQue is applied to the plant considered previously Whieh was'
. the reduced 3 machine system ofSaskatehewan Powe� Corporation. 'In ·both.�ases if'l1)rovement in systeI'Y dampin(J was found to be sim1:1at to 'that
obtained w1:th the 'optimal' state reGulator•... .' ,,'
.
5.3" oesf�n Of an �timal Transfer Function· for a Low Order System
A sfmple low order system,.was used as an example to 1l1�strate
the application of the al(Jor1thm derived in the preceding $��.t1on. The'.plant' considered for this 'case was a. 5th order' system37 t the dltnamicsof which is �1ven by the equation ..
•
X • AX + BU + GF
y .. • ex
.
','"
(5.28)
.: ':,,' ,',.�: "
. �"'. .
: .....
101
, The matrices A. B. r, and r. were chosen as follows:-
,-24 0 ,', -130 10325 ' 10325
0, -24 130 0 0
A • 56'82 ..5682' 0' 0 0 (5.29)
-0.18 0 0 0 0
0 0 0 0 -loon
B • [0 , 0 0 0 III (5.30)
100 0
0 100
r, • 0 0 (5.31)
0 ()
0 0
C • (0 0 I, 0, 0) , (5.32).
:. .' .
The variables X, U. F and r, were as defined'befof"e.
To beqfn, with, the eigenvalues of the matrix, A which are the
roots of the uncompensated system, were'computed usin(J,subrouttnes
HSBG' and'ATElr;. The roots are given in TaMe 5.1. It can be seen from,
"
,Table 5.1 that the predominant'troublesome roots are ..11.99 :I: .1 27.8S
'
with' a damp1nrr l"atto «() eaual to '0.396" The matn purp()Se of addinq'
an external controller would be to move these roots away from, the"
,
ima(Jinary axh thereby, increasing ,the damp1n!t r�tio.'
, (
.
"
....
.' .,;. i� .
:
� ,
102
TABLE 5.1
Ei�envalues of the Uncompensated Syste�
.
No.Poles of th
Real
1 .- 1000.0 0
.2 - 11.99 - 27.88
. 3 . 11.99 +'27.88
4 - 11.99 -1215.0
5 - 11.99 +1215.0
.
The optimal state requ'lator was designed next us:fnQ the alqorithm
. described in Chapter 4. The quadratic cost functional was assumed and
the syrmtatr1c Q and R matrices were arbitrarfl.v selected. The.v are oiven
by the equations(5.J3).and (5.34).
Q'. OIA [20 20 10·20 20]
R • 1.0
(5.33)
(5.34)
The alqebraic matrix Riccati equation was then solved usinq the eitJen
vector method des'cribed in section 4.2. The optimal state variable U·
of equation (5.6) was found to be
U • [-25.7 . 19.0 4.1 -238.0 -238.0] [XJ .
. (5.35)
The new coefficient matrix A* defined by equation (5.8) was next. .
... .. .
comO�ted ared the eiqenvalues of A* were calculated it. order to find
the ffl1Proveynent in system dam�fn9. they arE! qiven .1n Table 5.2.
103
TABLE 5.2.
Eigenvaluesof �he Compensated SystemWith the Optimal State Regulator
No. Poles of the SYstem. Real Imaainary
1 -1012.4 O' .
2 - 27.9.
-10.49
3 .. 27.9 +10.49
4 � 108.7 -1225.4
5 .,; 108.7 +1225.4
A comparis(J'1' of the results aiven in Tables 5.1 and 5.2 'indicate.
..
that·the p�dom1tiantpair of troublesome r'Oots have been. shifted to .
-27.9 ±.:1 10.49 which corresponds to a dampinf} ratio of' 0�934.
The optimal transfer function was next desf oned, 'The matrix H of
• equation (5.4) was obtained from equation (5.35). It was found to be
. H • [ ..25.7 19.0 4.1 -238.0 -238.0] (5.36)
With the help of the dio.ital cOMputer proqra� written on the basis of'
the alQorithm described previo�slYt the couplinQ matrix Q was calcu- .
. lated. This is ('fiven. by equation (5.37).
Q ... [3.29 x 10-2 1.41 x 10-3 2.88 x 10-6 1.34 x 10-9 1.69 x 10-12] (5.37)
.. The new.coefffcfent matrix X of equation (5.26) 'was then calcu-
lated with the help of equation (5.25)� It was found to be .
.
·104
�24 . 0 -130 10325 ··10325 0 0 0 0
0 -24 130 0 0 0 0 0 0
5682 -5682 0 0 0 0 01 0 ·0
- 0.18 0 0 0 O· 0 0 0 0
'A. 0 0 0 0 -1000 . 1.0 0 0 0 (5.38)
0 0 0 0 0 0 1.0 .f) 0
0 0 0 0 0 0 0 1.0 . 0
0 0 0 0 0 0 0 ··0 .• 1.0..
_5.9X1021 0 0.
10 . 8.
60 0 -1.9xlO �8.3xl0 .1.7xlO -7.9x10
..
..
The next 10qi·cal step was to calculate the characteristic roots of..
matrix X in o"der to compare them with the roots of the system coup··led
with the .0Dtimal state .requlatnr. The .ei�enval.ues were llftain c"""uted
and they are shown in Table 5�3.
TABLE 5.3.
..
Eiqenvalue of the COIIlPensated System CoupledWith the Ootimal Transfer Function
Poles �, the· SYl!O;-�".
No. Real ImaQi naY'Y .
1 •1012.4 0
2 - 27.9 ...10�49 ...
J - 27.9 +10.494 .
- 108.7 1225.05 - 108.7 ; 1225.06 - 34.8 1224.0·7 - 34.8 1224.08 ... 456.8 0
9 . - 28.·42 0
105
.
.' .
.
The results of Table 5.3 are reassuriOft. By e.,arinq �he results of
Tables 5.2, and 5.3 it can be noted' that the first five roots are
relatively unchanqed. The increase in the order of the system due to
"
the optimal transfer 1unction has created four additional roots. Two
'of these addition'al roots (-34.8 ± j 1224.0) would ,prOduce exponentially'. ....
..
decayinq oscillations at 196 HZ while the other two have only expon-
,
,
entially decayin� effects. Therefore these four roots have a negligible
effect on the overall system dynamics':which was governed mainly by the.
. '. .
predominant pair �27.9 ± j 10.49. Thus the optimal transfer function
enhanced, the system damping in the' same manner, as ,the optimal state,,
,
requlator by shifting the predominant pair Of 'roots 'rom ..11.99:!: r
27.88 to -27.9 ± j 10.49.
The fourth order Run�e Kut:ta method wa$ used 'to �lcullte the.'
.
'
'. .'
time d(Jn,ain system response f�r this, hYPc'thetieal,system. the disturb..
, ance to the 5y'stem was selected as 8,1tli step incw-eAse of varlabl&, F2• ,
, State Xl was arbitrarily selected f�r comparison purposes." The time"
domain transients of the state Xr were plotted for all the three,
cases, viz�:,
a) uncompensated case,
b) coinpensated system coupled with optimal state reqUla�to",, ;
,
c) compensated system coupled with optimal transfer function.
They are shfMl in Fiqure 5.2. A marked improvement in system dampfnq.
.
. .'
'due, to the addf'tion of'either 'form of the optimal reGulator to the,
, plant can bf! easily seen by comparing th�se results.'
,
1111.111111i 11111111111.1 II! IIIII II II! 1 i ii Iii II I I I!
'111.1,1.11 II', '"
'II!I ! I"
j lliii llii 1,1 I Iii,Llll!!:ii!PI !!id!I'! ,·"ilmmi!!tn ..
1:11!1111111111111111!1I i!! " ..... ,! !ll111111111111!1:lllillllllll00 CH 02 03
.�.
i
i?...
1IIi
... 0
al�,:>
...
..
� q.'"......
. ....N
c?
ali:,.,
.�0
o.o
i;
00. ". 01,
�o
I;;
Q iM.'.' 1w :... '
. : �. ir I
!
�...
i
�
g l!lllllllill!ill!ill!liii0'0' 01
SYSTEM
lilillilll[ill III! 11111l! i:i04 .0$
TIMlIseeoNoSl
COMPENSATED SYSTEM ICOIIPL.EO WITH STAll _.....TO'U
t
p,
!
101
5.4 Design of the Optimal Transfer Function fo';' the Power System �odel
,
The techniQues for deriving the single transfer function fom of
the optima 1 requlator can be readf1y apT'11ed to, the power system ",ode 1
consisting", of. the three machine 'reduced model of a portion of the ';ask ..
atchewan Pm'ler Corporation considered previously. The control, variable. .
.
'. . .
.'
u, in this case is ,fed into the voltage reQulator of the !iqua., �apfds', . '
plant from the oPtimal transfer function with input siflnal S6. The
schematic is shown in Fiqure 5.3.
The rrjatrices A. C of ettuatinn (5.1) have ',been previously ce lcu- "
.. '.
." .
, ,
lated in �hapter 4. Si:nce S5' was selected as, the ,inpu't to the trans'fer
function. C is written as.
C = :[0: 0 1 0 0 0 0 0 0 0 0 0 0 0 0],
(5.39)
The optimal f�edback matrix Hof equation (5.4) is obtaine<f f'r�
Chapter 4 where the ortinal state re�ulat()r ,,;as dp.si�ned for this'
system., It is 'Jiven by equation (5,,40).
H =-[654.0 182�8 ;'142.6 '.261.0 1.11 ,-592.6' -Rl.2 �16�4'
2.6' -61.0 -54.4 -0.12 8.1 -8.7 -4.1] (5.40)
The next step was to compute the !'latrix P (fiven b.v enuatfon (5.14) and
calculate the couol1nq matrix Q given by (5.19). The evaluation of
matrix Q renutred the inversion of matrix P. Because of the tYDical,
.
nature of'
the matr1� P \-/hose nth row 1$ HA*n-l"( the martn1tude of each
eleP.lP.nt of a column increases as it is moving �ownwards. This created,
large roundin� �rrors and n�r1cal 1nstabfli�y when tryino to invert
the M(ltrix,p us1nq direct method. In order to, get rid of the nUf"lerical
..1+1815
i·vfbl
. /. / .
R(S)
108
OPTIMAL . TRANSFER FUNCTION·
. VIal . KaI(r+T��1 S).
vexa1+ Tadl 5
Kafbl5I +TatblS
Kvfbl S
I +Tvfbl S
1
Vfbl·
the SQuaw Rop1ds plant co�pled with the optfNl
.. tra"sfer function.
Figure 5.3. Block diaqram of the aut�t1c volta�e requlator at. .
109
error,. Q of equation (5�20) was redefined as follows:
Q .; Cp-l '(5.41)
or-1 .
Q • (C p-lpT )pT. T
.
\'1here P is the transpose of p. .'
let "T" .• ps
(5.42)
(5.43)
. Thus (5.42) yields
Q • CP5.-1pT (5.44)'
or.
Q:.•: C PINV' .: .
where PiNV.'ps... lpT
.
..' (5.45)
. (5·•.46)
.The·necessity of generatinq Ps stemmed from the fact that apart
from its ,symMetricity, all of its elements were more or less of the
same order as compared to matrix P and hence were less orone to
rounding.e�rors.·
PINy'was thenmult1plied ,",yP to check the accuracy of the
inversion process • .The produ�t indicated a s.1inht rounding error and
a further standard iteration method6! .wa$ used to br1t1q the results
within acceptable limits.
A brief summary of the steps re�u1red.to solve for the matrix �
is in order.'
a) Scale the matrix P
6) ·Multiply P by its transpose "T to generate Psc) F1r'1d the inverse of p
sand later PINV.
,.
·110
" .... ;
d) CCllllpute Q with the help of equatf'On (5.20)�
The Q .trix· for thts Clse was calculated· to' be:
.. Q • -, 11.47 x 10.2, 1.44 x 10-1 .
1�18:x 10-1 '
5�59 x 10-1 1.12 x 10..1 .
.
3.66 )( 10.2 5.4� x·lO·3 5.55 x 10.4.
3.35 x 10.5 . 1.16 'x 10.1 .
.1�43 x -19-' . 9.•01 x 10.9 . 1��. x 10..10 4.44 x 10-13 ·.2.12 x 10·15J (5.41)
. . .' .
. . "'.
..
.' The new charactert.stlc .trix X' g1ven by equations . (5'�25l and
(5.26) was calculated aitd ·1ts efgenvalues were' then cOMPuted fOr. . .
..
.
conpaMson purpose$� The predamtnant roots alOftiJ. with the roots of ·the·.'
uncGllpensated syStem are shown .1n Table 6.4•.. ".
Table 5.4'
...
. ". . .
.
.'
: '. ..
. CGMpar1son :01 DOMinint'S1S,. �Oots ..-, .. ", ..
:,' .
. _
.....
DOIII1nant·· .Roots.. .
..
ContrOl. System..
.1"1«71 t,.
«72:;* j-z «73 t .1-3'.
-.
. �.
. ,'.
,
IMccmpensated._'
System ';'0.044 t . .1 6.68' ·.,..0.:63'.* .1 1.34' " ·-0.20 t j 0.29.. .
' .
....-.
CampensatedSystem· coupled' ;
.. ., ,"J: ...� .
.with· Optimal··.· -1.28 tj 6.99' 2"73 t 3.42 .3�89 t." 0.10......
Tranifer'..
':.
Function "
". ..... ." '.
It can be eaS11y $,.n that Wftt.·.:titt·'·addftton. of .n:·OptiNl.
transfer f:unet1an' cartsfderabl.· fMprOve.nt 1n system ."'."9 was obtained
over the uncanpensated system and 1n additfon the Nsv;t. are 'sftllflai' .- .
. :..
Wtth those 'obtained for the' optlIn11'regul.tor. This ... observation ''.'
,
can.' be deduced by plottf.ng the t1. �.. in results. :The '11�late� di$t�· _:;'.·urbance to � system was a 10� 'step.' f.ncrease "n loact<Cientan.d at Squaw
....
... :..
...'.
111
R�pfdS plant. The Squaw. Rapid. speed error and the Queen El1zabet�
speed error are shown in Figures '5.4 (a) and' (b). The tranSient·'.responses for these � variables for the uncompensated systm are
....
..
plotted again in Ffgures 5.5 (a) and (b). for c.-rison. It can be
noted that the fairly 'oscillatory nature of' these two variables for.
.
'the uncompensated s"tem·.re notiCeably suppressed'whenthe, optimal..
.
.
. transfer function fs. added.
An �lgorfthm for the design of an optimil transfer function for
the single input case has been developed in this chapter. It was
then applied to two different plants and the results obtained were .'
found to be canparable with those obtained using the' optimal .state
regulator..
.. .
. ..: . .... .
.
So far the phl1osoptly behind improyf·ng system damping. of multi-.' .
• .chine power systeM is to feedback the speed deviation through i
controller. 'In the next chapter to further uti1fze the 'duellevel'
power system model the effect of :'cons1dering the gove",�r input as
an additional feedback on sys� d�1ng is fnvestfgated.
112
''0...,e,VI'
i
1lII1lllllili � . .
[llttl ll l1.llllll(lll t l \ t \ \ � \ \ \ � \"" \ .... "·r'.
.
4'� 6 O'
.
Time .in ,,,orid,
3'.0
Figure 5.4 (a) Squaw Rapids speed error versus time when the syst_is coupled with the optimal transfer function.
8'i.lon'0'.
�� I
I I'3 0;'': 0 i� � �
. U tn:
.s � i6
I:::...
; 11:llltl1il,l �! ,lillill i llllll! llllllllllllllll.l tilItu\\iiIII!111111U l \\, ...\u1HHi \H\th.....�,,,\\\\\It"'1'
00 I� 30. 4'�.
60
Tim. III IIC:OIId·,
Figure 5.4. (b) Queen Elizabeth speed eM'Or venus time When the systemis coupled with
..the·.opt1mal tr.nsfer function.
·0o
..
Figure 5.5. (a) Squaw Rapids speed errOr versus time for
. the uncCJIIpeftsated system"
o.0
III
f.::,.:
. I:::.
6�. �:!!n
:,'!,!, i, ii,:,'l.· iliijl!!: !ilh
:d!:P!i, 1ddlfv· ; P:qd!i:f .'I!!HP:: .'!!!·'I"I1t:l nl'li� .
: Iln:II'III:I'1 ".1"tC• 1'1"11, I 1'1 1'1111,• I II I. I I
11'1'·0: u '11,1:'1' III': lid....-: '
.. I.ln '1111 1'1'11::.GO: rlll uuuum :.111111-.. w: :l:nid,nlpn :IIi III... : uuuuru ,: nl uu
-: 2; llillll!'lll! Hlllll!i!PIIJi'; 0: : ill': ! u I! P! ll u I'j P!.. '. .1 I" l'II'j' I I',I.
=: n. 111'1'11 : III '11'Plii·-: 1111 1,1 ,II , ,I 'II .
•
� tlflilffllffnnfff:fl:flff:i. ,h. tTl,c:. '0' "1'11'1'111111"1'11"11' ."'1' • .1 .. ,
. 111, .. I I "f'l II I 't I' I"" 1"'1'.� . • "'1'1"I I'll' 1111 U'I' ·"11' tt-· ttl ,
: Inn 'I ::.11::1 : tn. '11"1,:t :'1'11':.1 'I'j t:lj'lj'l� : �lil'llljjliPli:li'lli:l1l11 :'lil!l!h!II:iIiI' .•. .1:1 i� .. I11II Ii.
:: l���llill iil'lllil'illlIllll:lliiiiilliil i.
ill llill'l i,l ,il h: giP!ill'II'!I'lliIHl.il'I!lllllliillllli!lI!II'lIiliilll'h �llI!lii 1!11I1,ll1li �'III!PII','II!i,i�'11• ::111 I III 1"'1111 1111 'I'Pl"I"'1 II· .:::t·· �II':I:: li·�1 "II' 111'1 ,
i �I.lll.ll.lIJ.llllll.lllllll.l tllllli.J l t l tl.l.l.t I. tl.l.l.lll t1....tlll.ll.lllllhdlll.llll.llllllllll t lUlllltltm0·0 1-11 10 4·5 '"'
TIIII. In ..coll4a
Figure 5.5. (b) Queen Elizabeth speed error versus time
for the uncompensated system.
·.113
114
6. ADDITIONAL STUDIES •
6.1 Feedback of Two Control Variables
Any researcher wOrkin� with the dynamic stability aspects of power
,systems would be confronted with the question - is it desirable to
control both the exciter and qovernor inputs or exciter only� As Out-.
.
. '.
.
'.' '..
H,ned in Chapter I, most of the papers published so far had accep,ted '
the philosophy of adding a stabilizing signal to the exciter as a means
of ,'enhancinq system dam"inq. Much of the literature'describes the
theoretical studies for designing an auxiliary controller for this
purpose. But'some of the earlier papers2•35 had reported the effects, "
of the governor settines on the' dynami c s tabl1i ty. rwo recent papers
37,39 ap'plyinq optimal control theory to the power system had also, ,
considered governor input as a, control variable in addition to the
excitatiOn, ,1nDtJt. Yet no attempt had been reported so far re'larding,
the relative importance and hence justification of consideration of
the governor input as a, control variable on the dynamic stability"problem. T�e $ubse'luent material stems mainly from the above consider-'
etion. The optimal state regulator is designed for a suhportion of the
SPC syste," consider1nq its exciter and gOvernor inputs as two control
variables. The 1mnrovetnent in system damPing is then compared with the
res�lt$ of'the ,optimal state'requlator having the exciter input as the
only control variable.·
The system,considered fQr this study was the sa� three-mach1n�
sys�e� c�s1stinq of a reduced model of a·part of Saskatch�an.Power
115
. ..
.
Corporation, as described in s�et1on 4.3 •. ·The dynamics. of the system is..
governed by the following state space equation•
X·.' AX + 8U + GF «.»
where.
vector X defines the states of the system. The approximate model. ..
.
.
of the. system excludinq transforme.r act10ri was developed in Chapter 3, .
subsequently used in Chapters 4 and 5, and is used here. The nUltlber of.
. .. .
."
state 'variables describing the system given by equation (6.1) leads to.
• 15th order characterist1ctnatrix A and a 15 x 7 coupl1nQ matrix G.
The contw:-01 vector U in this case consists of two scalar variables
Ul and U2' i.e.. (6 ..2) ..
The va�iables U1 and U2 are fed back to the exciter and govern�r Of •
the Squaw' Rapids ':plant of th-e" • study' . system. The schematic'diagram :,
comprising these' cont.rol variabl,.s ts shown in Figure 6.t.·.' ".' ..
. Because'of the presence of two control variables in the coupling.
.. .
.'.'
..
.
_trix B of equatfan (6.1) "s different fron the one used in the �re-'.
'. '..
. ..
ceding chapterS. The B matrix for this case is give" by equiticn (6.3).
0 00 00 o·0 00 00 ()
B •• 0 0 (6.l)1 .
0 -.
Tel0 00 00 00 0
Kel 0-
Tel ' .
0 00 0
116 .
K••. '
KaI(1 +Tan,8). Vtal'.
'1... , . I vfl..
.----
I. T•• 8 1 :.- lode 8. V,.,I
iVfbl
;Kofbl S .. ·
....-------
1 .. Tofl)18.:
Kyfbl S......---.....-----\
1 �Tvfl)l S .
( ..
... �.
I + TCI $
.
I ... T31 S··1 +,TSI S
9t '.
1 + T41 $ .'
..'+ T5,I S..
'.
I .
"'R ....
I .
.
1....---.-.��----t. '-
Wb
Figure 6.1. Schematic diagraM' or the excft.tian and' gp"mor;': ...pri. move.. ftiodel With':. e)C�.ma.l. feedback. var1abl.s. · i
,'" .
,.
117
Substituting the .numerical valuesof the parameters of the Squaw Rapids.
plant as .(Jiven in section 4.3. the B matrix can be derived as aiven
by equation (6.4) •
. 0ooo·oo
B· 0ooO·oo4.2563o'0
oo.oo·0oo6.2o'0ooo·0
.
0.
(6.4)
The characteristic matrix A of (6.1) is unchanged from Chapter 4 where
it· is defined by Tabl� 4.1.
The rest of this chapter is devoted to the desi�n of the optimal.
. .
.. .
state regulator and the qualitative analysis of the results.
6.2 �sign of Double Feedback �timal State Regulator
The design of the double feedbaek optimal.state requlitor involved
the selection of a cost functional or perfonnance index which was'
assumed to be quadratic in nature and the' solution.of al�ebraic matrix
Riccat1 equation. 'This was outlined ·1n· Chapter 4 •.
The. cost functional J is !liven by
(6.5)
118
, ,'
as stated previously in equation (4.2). The matrix Q given by equation,
'(6.6) for ·the double feedback' system was selected to impose heavy
penalties on the an9le and speed deviations of the Squaw Rapids and
Quee� El1tabeth plants •.
Q .• DIA [l.0 10000.0' 10000.0 10000.0 10000.0, 1 .1 1. 1,
1 :,1 1 1 1 1]. (6.6).
.
The R matrix was assumed.,to be un1'ty; or in .other words..
. ..'
.
[1.0 OJ '
R •.
"
0'1.0"
The optimal control vector u* .ts related to the state vector X by
means of. ,the 11nea .. feedblck "'lations�fp, ..
(.6.8).'
where K* is. the solution.of .the .1qebra1c ",atr1x-Riccati equation'.
0 • K*A+ ATK*�*8R-IBTK*.+ Q.
.' (6.9)
The .tr1x-R1cca�1equati4)n is fOl"nlUlated fran equation (6.9) by
substituting "for the A. B. Q and R matrices a� defined aboVe. The.
. ..
equatfOll was then solved using the alqorithm described in section 4.2•.. .
The solution matrix K* was calculated and .·1s gfven in Table 6.1.
Sylve$ter's c:r1ter1qn52 waS agai,. applied to check 11 the Riccat1
matrix,,,,,,s positive definite or not. The detenninant of, K* and the
principal minors ,of the determinant were calculated and were found.
to be .,os1 t1ve indicatfng the correct solut1on.
The control inputs U1* and u2* obtained from equation (6.8). are
!liven by eqUlt10n (6.10>-
TABLE 6.1
. K* HJ1T�rX
(All the e lerents of the rratrix are expressed in th� �om mEn Ml1ch is equa 1 t� m x IOn)
0.86OOE 04 0.2754£ 04 -O.I64SE 04 -6.3619£ 04 0.6078£ 03 -O.535� 04 -0.5671£ 04 -0.3555£ 02 C.576lE 03 0.7003£ 03 0.5561£ 02 0.6460£ 01 0.2982E 02 -0.1'708E 112 _O.I!347f 1)1
O.l769£ 04 0.5914£ 05 0.5261£ 04 -0.4565£ 05 0.5681£ 04 0.4587£ 04 0.4342£ 03 0.3370£ 02 0.7187£ 04 0.2266£ 05 0.1054£!l5 0.3807£ 01 0.2627£ 01 -O.933!1£ 01 -0.2511£ 01.
-0.1644£ 04 0.526lE 04 0.1984£ 04 -0.3879£ 04 0.2877£ 03 0.2257£ 04 0.2577£ "3 0.2195£ 02. 0.5194£ 03 .0.2346£ 04 0.1240£.04 -0.5466£ 00 -0.�73£ I'll 0.4126£ 00 0.1788£ 1)0
-0.3627£ 04 -0.4SESE .05 -0.3880£ 04 0.5511£ 05 -0.3259£ 04 -0.3304£ f\4 �.5!1i;2f. !l� -0.2034£.02 -0.5383£ (,4 -0.2034£ 05 -0.1003£ 05 -0.4148£ 01 -0.6751£.01
O.6083E 03. 0.5682£ 04 0.287� 03 -0.3260£ 04 . 0,3387£ 04 0.1224£ 113 -0.3391£ 112 0.5838£ 00 0.2699£ 04 0.3853£ 04 0.9516£ 03 0.2C152[ 00 0.2807£ 01
-0.5353£ 04 O.es94£ 04. ·0.2257£ 1)4 -O.3308E 04 0.1227£ 03 0.4655£ 04 0,2999£ 04 .0·.4093E 02 O.3872E 03 0�2399£ 04 0.1388£ 04 -0.3469£ 01 -0.2093£ 02
-0.5669£ 04 0.4493£ 03 0.2585t 03 -O.604OE 03 -0.3365£ 02 0.2999f 04 0.5973£ 1)4 -O.2286f; 00 1).2297£ 02 0.4221£· 03 ".29311£ 03 _0.4832£ 01 -0.1741£ 02
-0.3556£ 02 0.3370£ 02 ·0.2195£ 1'2 -0.2034£ 02 0.5859£· 00 0.4094£ 02 -0.2213£ no 0.7233£ 00 0.2739£ 01 0.2007£ 02 0.1206£ 02 -0.1818£-01 -0.1588£ no 0.2221£-01 0.3!l35F.;'02
0.577* 03 n.719OE 04 0.5196£ 03 .(1.5385£ 04 0.2699E 04 0.3865£ 03 1).2153£ 02 0.2737£ 01· 0.2451£ 04. 0.4392£ 04 0.1398£ 04 0.4120£ 00. ·0.2129£ 01 -0.2679£ 00 0.1371£ 0')
0�7038E 03 0.2267£ 05 0.2347£ 04 -0.2035£ 05 .0.3852£ 04 0.2399£ 04 ·0.41117£ 03 0.2006£ 02 0.4391£ 04 0�U99E 05 0.5139£ 04 O.l138E 01 ..0.5946.£-01 -0.2007£ 01 -0.3620£-01
O,5385E 02 0.1054£ es 0.12.t(JE 05 -0.1003£ ()5 1).9511£ 03 ".131111E"4 1'1.24151£ 113 0.1205£ fJ2 O.I396E 04 0.5136£ 04 6.2512£ 04 .0.4552E 00 -".1541lf 01 -0.11175£ 01 -O.665Rf-Ol
0.6459£ 01 0.379A[ 01 -O.5471E 00 -0.4140£ 01 0.2946E 00 -O.3471E 01 -0.4834£ 01 -O.1817E-01 0,4102£ 00 0.1134£ 01 0.4567E 00 0.2841£-01 -0.118U: 00 1).27651:-01 1).•5943£-02.
0.2982E 1)2 0.2588£ 01 .,0.8476£ !II ';'.6732E 01 1).2l1l)I;£ IIi -0.201)3£ n2 -0.1741£ 02 "O.I588E 00 0.21241' 01 -0.6829E-Ol -0.1534£ (11 .(I.I18lE 00 0.1181E 01 -0.8711E 00 _O.4293r 00
.-O�1708£ 02 -0.929U 01 ,0.4143£ 00 0.9874£ 01 0.1878£ 00 0.8576£ (11 0.1545E 02 6.2219£-01 -0.2620£ 00 -0.1994£ 01 -0.1079£ 61 0.2765£-01 -o.s7il£ 00 0.2303£ III l),l366£!)l
-0.8355£ 01 -0.2486£ 01 0.1809£:lO 0.2625£ 01 0,2732£ 00 0.4277£ 01 .t).8401E 01 :I.393!1£-02 0.1383£ 00 -0.2945E-Ol -0.6747£-01 0.5936E-02 ..0.4293£ 00 0.1366£ 01 0,1242£ 01
0.990.1E 01 0,2649£ 01
0.1861F 00 0.275OF. Of)
0.8574£ 01 0.4270£·010.1545£ 02 "',8391£ 01
....
...
...'"
120
x-
_-
126.9 11.01 -36.06 -28.'64 11.94 -89.09 -74.10
-0.67 9.03 -0.29 -6.53 .0.50 5.02 -3.7 -1.82
"U2* -222.2 210.6 137.2 -127.12 3.66 255.87 '�1.38
4.5 17.11 125.4 75.35 -.113 -0.99 0.139 0.024
(6.10),
The state ,space equation of the system equipped �th an optimal
state regulator is given by.
'
x'. A*X + GF (6,.11)
where
'A* • [A - BR-1aTK*] (6.12)
S�bstituting the values of B, R, A and K* fram eouat'f(ms (6.6), (6.7)
and Tables (4.1) and (6.1) i�to (6.12) yields the tra"sf'orm.ed character-.
.
. .
.
-
1st1c matrix A*. This is given in Table 6.2. '
,
Equation (6.11) with A* defined by Table 6.2 was used 'or stab ..,
.
il1ty analysis in the complex S-plane and for the time domain,solilt1,on. '
, ,
'F�r stability analysis in the S-plane the eiqenvalues of the character-
istic equation I A* - Ail • 0 were 'calculated. These are given' by'Table 6.34!
,,
The relative improvements in system damp1nq is apparent' by eompir.;.
fng the dominant eigenvalues of the system with the d�l. feedback,
optimal state regulator with those obtained 'when the' 5ys-,1s coupled·with a ,single feedback optimal state regulator, and with those Obtained
from the uncompensated system ccnsidered previously. These comparative
results are shown in Table 6.4.
TABLE 6.2
THE CHARACTERISTIC MATRIX A* OF THE' COMPENSATED SYSTEM· .
. ,
-0.17310.0. O�O 0.0 0.0 0.0 . 0.0. '.0.0 0.0 O�O 0.0.
'. 0.0258' 1.29' 0.0 0.00.00.0 1.0 0 ..0 0.0 0.0 0.0 . 0.0 .0.0 O�O 0.0 0.0 '0.0 0.0 0.0-19.7-18•.0 '.0.1-49 ·.17.8 0.0 18.1 0.0 : :0.0
.
'0.0. . 0.0 0.0 0.0 0 •.0 '0.0 .'.0.00.00.0
.
0.0 ·0.0 .: 1.0 0.0. 0.0.
0.0 0.0 .0.0· 0.0 0.0 0.0 0.0 0.03.0319.8 0.0136-22.2 -0.102.0.0 O�O 0.0 18.2 0.0" 0.0 0.0 0.0
.
0.0 0.00.00.0 0.0537 0.0 0.0 -0.909 0.91.5 .' 0.352'. 0.0 0.0 , 0.0 O.Q 0.0 0.0 0.00.00.0: -0.0268 0.0 0.0 0.0 ',-0.0029 ·-0.176 0.0 0.0
·
0.0 0.0· . 0.0 0.0 00..'
..
"!'1388.7'1316.2 856 •.6 -794.5' 22.87 1'599,.1 .' . -8.62 21.87 106.9 . 783.7 '470.9 -0.70 -6.i9 0.87 0.150.0O�O 0.0 0.0 .·0.0 . 0.0 . 0.0
.
. .. 0.0 -20.0 20.0 .
·
0.0 . 0.0 0.0 0.0 00....·0.00.0 0.0 0.0 0.0 0.0
.
0.0 0.0 0.0 . -6.67 · 6.67.
0.0.
0.0 0.0 0.0'·0.00.0 0.0 . 0.0 -1.38 . 0.0 .. 0.0 0.0 0.0 0.0 .-12.5' 0.0 0.0 0.0 ·0.0.0.0.0.0 0.0 0.0 0.0' 0.0 0.0 0.0 0.0 0.0 0.0' .30.3 -1015.1 0.0 0.0
.
..536.747.01-.153.49 -122.04 SO.82"!'379.2 ';'315.4.'· -2.85.'
38.43 -1.23 -27.79 -2.13 21.36-15.·75· ",,7,.75'·'0.0..0.0. 0.0 0.0 0.0 0.0 0.0 ., 0.0 ·.0.0 0.0 0.0 -0.689 -34.4 -4.54 00..-0.3130.0129-0.00141-0.0125 0.0· 0.0
.
0.0 .. <>, O�O .:.. 0.0 ·0.0 0.0 0.0 0.0 0.0' -2.0
...N...
"
Tlble6.3. ...
.
.
EIGENVALUE· OF nt£ COMPENSATED SYSTEM EQUIPPED WITH.
. . : TWO FEEDBACK OPTIMAL COOROL VARIABLES. , ..
.
Mo. EigenvllUes.'
1 -l72.10 + j 0.0'2 - 26.42 + j 0.03 .- 0.21 .+ j 0.04 - 0.46 +. j 0.0'S - 1.0 + j 0.0
"
6 - 2.0 + j 0.0
1" .
1.81 j 5.85• -
8 :.- . 1.81 + j 5.85
9 • .4.33 - j.
0.03310 - 4.33 + j 0.033
.: 11 • 17 44 .- j 1 81... ..
12 .. 17.44 + j 1.81
13� '!' 16.45· + j 0.014. '.
_: 8.06 - j 13.31
15 .,- 8.06 + j .13.31..
. " ,"..
122
, ,,:
123
Table 6.4,
' .,
. C_aRI$(IC. OF :�NANT . SYSTEM. RQ9TS .
. . .' . . .
. ,*,nant . Roots
'Cant"' .$,.",., . 411 :t.�� .. C12::t.j� ... «13 :t jfal3
.
..
..
Clse' .1: ..unc......ted-
�.044.
sys•• ·, :t j 6.68 -0.63 .:1: j 1.34 .•0.20, :I: j 0.29
Case 2t o,t1.1. sQte. , ,...,.tor with
..
on. 'eedback '. ...1.1' :t j 6.69 �2.63 t j. 3�35 -3.est j 0.10"
"
'.
Case 3: opt'..l state.regulator wfth
�1.81 5.85 . -11.44 t,j 1.81. ·two�Cks. :t j -4.33 t .10.033,
',
':', ',' " ,
'
,
'" ' , , '
.
It· is "_fltelY obvious frem Table 6.4 that a' greater ''''Pro_ntin ·syst.ee: _tn�tby.: ratt0 of 41 WIS, obtained' over the' uftCOIIPtnsated
, ,',,
'
"
, ,
'
, ,,
'. systeM by cons.fdeHng:·.th... ·exe1ter .nd goyem.or-prl_ lIIOver fnp-..ts as
the cont..ol""'ibles. �t Cln be further "otteed for. Cise 3 that the
control· of bOth .C1tet- and 9oYtmo;.pri.IltCMtr inputs had bitter
effect•. 1",oft, as the enhanC8lWlent in system ."'ng was Coftcemed as
CCJlIWPINd to: c.•• 2 ",,..... 'y the control of .cite" input was consider
ed. T,hf. can be further -.onstNted by the time domafn' results. The"
"
' '
,
",
'ou"" Order Rung. Kut-e. .thOd was used to ca leulate the t1. respon-, ',. '
ses fOr the double '"db_cit ,yst_. Th.• dtsterbance to the s,Y$teIn was
a ·st.lated 101' step fncNase 1n the load delltand at the Squaw Rapids
plant•.:'.
'The transients 'of �,i Rfpfds speed error and the Q.,.en ElfZa-.
.
beth speed effC)r are shown 1n F1gUre 6.2. The transients' for the
124
81&1
&L.... : ..
'
- � :
.: i::�:::; IiI... i.::..
;.::.. �.!.o,· .1
" lHh1 11l! Hi!!!,ti.....
0'0 j·5
'0:.
"••
� 1111 l!lllmllUill!h!!"'<tbH"" .. , ,
.3(1 .
.
45 6·0
Time jll IfCOftCls
Figure 6.2 (a) Squaw Rapfds speed e�or Yersus tiMe when the.
systelt is coupled wtth optfl.l state regula_with two ·feedbacks
oo
i r r.
::::.II � � :!!::: :
:tIJii!
...
i;;'=:.:.,'
lll.llU.lltlll (\lll� � \ lHi \ll. \� ;.lhlm·I�\;���'�:.o.,," i .
•0 .5' 60
Time.III ... "oneil
FigUre 6.2 (b) Queen Elizabeth speed error versus ti_ Mten the
sys. is coupled wt� opt1.1 state N9ulatoPwi th two feedbacks.
Figure 6.3 (I) Squaw Rapids speed error versus tt_ for the·. uncClllPenslted systeM
I
..,
r: lie
..,:. �:::co·.
• I lIe
6•.: ::n:
nun.1'11'11
. !H!i1! �i!� !i!nnn !iPl
;,: llllllllll t:::::t.: PP':PIJ, .-n:p::.. :
.111I1"l!m:lq!:1!1l lillI', '111ii!11I:!!!"ll• I" I 'III "I"'" t.! 0: '1'11'1 n .p: nn.:::,. 0: .L: :,npi.l::I'II:::n.� w! 1",l:tq.",'II': '11\lnq.... : 'l'I!f!',i:qi.! il! 1"IIf,"I .,: : I' IIt: I " tit .. , : "
Co) � : , !'!iiPHiHI!!!t::nt- .... : .,11 •• 1"'1" ... , 1'11'1":.. V
'1,""" ,.ltl' 1II1 "I;: fP uuu 1':1'1 II":: qf 1 IIi "Iililili ill illlll!h: i1lfl!!L!!P!',!fI!:!fl fqln .'. -,c: :: .. , .... " lli''''I'I'' p. .I:. tl ••. - : .: Ilq !!I!!!!!:'!!!! :.iP,11 -lllll .r. _nil.: ::::,1:1:: l'I::::III::::lI::! itt nl,h·
.::, .::. :.. : _,'U, 'I:::' ,1'1'1' 111111"'111'•• ""'11 I.,.
"111,15 �·,'l1l!linlq'lliillliillllllllll;,'ll.lllljlllHlilii iii TlnLil'TI ill I,',liT.
ru II ,
11"1"1"1'11':""1' ,'":' .1111:'1 I • '1, '.It.
"11 I,"1 8iP!,,!!I','If!!tlll"I'1 i,I,,!f!',I,l iI,fiIHiHii!,'rnl ·"fll'II'I't 1111Ii!.it .1111 if '1:111.".1.,', 1'1 "I" I 1"'" un ',',It "' I... " ... 1, dl'· ·1' 'I" "'II"i iiili! iilil'liii'I'III,l'liiiiii'llllilllllililliilj il!l!"I'jj TIi!i!! ilWillllllllllt'iidlll'�t.olt�lli.l.i .lli.t�t[li d�llll.i 1!�l�li�!I.l.l.ll!U!l!ik. illtl.l.ll Jlhiit tWIt!. I. .l[l[uHlt!ll lHlll ..,-. 1'5 3() 4·11 .
. &0TI",. ill ..cond.
Figure 6.3 (b) Queen El1zabeth speed error versUS· tf_ fflrthe unCOIIIP8ftSlted system
125
o
�. ;.:.... .0:
:�� n .i!li
tl illl:.:dlllllI . .
.
� j Ii! lil!iHljiip: 1 tl, IIIII!I !,'II,-' rlll"1'111' 'jj
•
<>! ,i' i Ij. 1 'il. 10: II! ,I I II I :}'II�l 11111'111111 ill
I
11 '
:1111 1,111 II, ',lli]"- I" ,'III! 111111111111111111 iit.: II II ,:
II
Ijlll I:'
I'
..
r I IIIII 1IIII11WWrl i" , ,
gl l .. ; l.l.u ll .. � . .llll.ll !.l .. dll.lthlllHn:lmlmlthmlmmm�mm•.{I.O. HI 3·0 .. &Tutt. .. ·,ec...d,
Figure 6.4 (a) Squaw Rapids speed error versus time when the
system fs coupled with optf.l state regulator.
wi.th one feedback
Figure 6.4 (b) queen Elfzabeth speed errol' versus t1. whenthe systell is coupled with o,t1l.1 state
.
regulator with one feedback.
126
127 '
same state ,var'ables ftll" Case 1 and. 2 wre calculated in Cha1)ter 4.
They are, reproduced in Figures 6.3 and 6.4 for comparison purposes.'.
. ...
.
The ,improvement in system damping weI' the 'uncompensated system
.can be observed by c,omparing Figure 6.2 with Figure 6.3. The'
oscillations present in the uncompensated system were virtually'. :
absent in this study.' A c.,anson of Figure 6.2 with 6.4 reveals
that a better improvement in system damping was obtained in Case
3 or Case 2. This agrees with earlier conclusions drawn on the basis
of comparing the root placements as shown in Table 6.4. This impro
vement would not appear to be in proportion to the c.'exfty:,
involved 1,n' designing the, controller for Case 3, as the number:,of '
.''.
feedback paths would be double for Case 3 as coq>ared to Case :.2.,
,,'
A system analyst might be satisfied with a ftasonable atnOUnt of.'
improvement in damping by controlling enly,the exciter input, alone'.' .
.
,,'as the' additional control of governo....pr". IIIOYer,inl)ut does'not'
enhance ,the system damping to a considerable extent in this Cise.
This cOlllpletes the appl1catian of the ....lti4chine litod,l1fng,
procedure to optfmel regulator design using various forms of optimal
regulators. In the next section the conventionallea�lag speed
stabilizer presently being u�ed,'n,' marty articles i� used as the aux
fliary controller ,nd its parameters 'are, being optimfzed by the so
called 'eigenvalue search'. A caaPlrison ,is then made be�en,the
,
, previous results and those obtained with this campensator in' the system� ,
then compared with those obtained usinq optimal state requlators.,
7.2 'General Form of a lead-La� Feedback.
,'129 "
Three cascaded lead-lag transfer functions were considered as
the eeneral fonn of the auxiliary controller to enhance the plantdynaJlti cs, The block di at'fram of the compensator is shown in F1 (lure 7.1.
Figure 7.1. Block Diagram of the Auxiliary Control'
K2j+T12j S Vyj, Kaj (1+Tllj S) Vzj,1'+ T,3jS 1+ 114jS
"
AVR
1 + TI6jS
I,
,
I-
130
.
.
.
.. .
.
.'. .
.
.
The· eM'or s1gnal1rcm.the frequency de�iit1on·transduc:et (Sc!j). was· use� .
as the ·'n""t to the stabilizer. The stabilizer feeding to the exe1ter
pt'CJl1ded a 'phase lead to overcome the phase ·'a� that·· is inherent in
the .automat1c voltage �·'ator. ·1he lag time constants were necessary.
.. '..
.. .
..' .
.
fran f}r.ct1al holse considerations •.Each lead cOlftpensatfon element
provides ga1n which fnereases at the rate· of 20 db· per decade of· frequ.
e.ncy fnCre.st·.• 'Noise in a control· system at 'requenc1es above th�e
..
us� for con:t�l can be amplified greatly by lead elements and 'may
overload thi ampliffers. laq· el�nts are: used to limft the qa1n of
these freq�nef�s'� the corner freauency of the lag circuits 1� in
gene",� 1.n the ·order of 10": times the corner frequency of the ,'ead. .
..
.
..
.
cfrcu1:t. 'The sfgnal reset tltadul. 1ncor'porates an additional long time
constant which 1s',n�rmal'1y set at 10 sec:mds or ',MOre2�.tt prevents the.
.... ",
stabU�zfng· sfgrtal, frem biasing tem1na1 voltage for' (I'Iger frequency
.. excurstons.· eeCluse Of its .1on�rtfme cd)stant, f,t ha$ little effect·.
.'' .
. on the ·phase ,.. ga1n Of· the stabilizing s1gnal oyer the frequency range
.of interest.
The pertinent equations describing the dynamics of the auxiliary. .
.
.. cOnt�'1e... connected to the jth node of the system are:-.. ..
,
.
t,
J16iH(t • T153s
:3 _ 'zj·b
.. : T14j S Vyj • KtjVzj + S Ktj T11., Vzj .. Vyj
fli.t .�v;,d .�.�tj v.yj'+""i"t'�' V.yj � V�,i
(7.1)
..•. (7.2)
1.31.
Let the pertinent state varfables be defined as follow$:-
(7.4)
(7.5)
(7.6)
Equations (7�4) and (7.6) yield
(7.7)
.KlJ. Tll�+ Vz.1 ..T14.1
or
(7.8)
v =.�'+ T12Jx.i. T·· T Vyj
'13.1 ···13j
or
(7.10) .
: ., ..
·132.
The state space model of t� auxiliary controller can now be obtained .'
by substituting equations (7.7). (7.8) and (7.9) into the ,equations'o.n, '(7.2) and (7.3). The final form is given by equation (7.U).
where
(7.11 )
.
.
T_Zj iI [Z1.1 Z2j Z3.1] .
Wj ill S6.1
where
. (7.12)
(7.13)
.(7.14)
.
The coeffictent rnatri x Tjis 9i ven by
1--
Tj • T16j
KlJ. _ �WbTlij T14j
o· .. 0
1-- o
T14j- 1---.
T13j
.
The cropl1nQ m�1:rix Sj is �,.'ve" by equat1pn (7.16).
_ T15J.·T16.1 ..
KIJ �)Tl5,1 { ...
HpTIf\j. . T14j
s •. j .:
(7.tS)
(7.16)
133
. ..
.. .
The state space model of the aux111 arv cmtroller can �ow· be.
.... -;
coupled with that of the plant. The pertinent alqcr.1thm is der�ved .
in Af'lp.endix E,with the pertinent equations brouqht forward.
1;he dynamics of·the cOl'tDensated·plant as described in· equation. ..
. ..,. � . .
(E.21) are given by
(7.17)
where· r - . the vector de$cribinq the state variables of the compen-.
sated plant ....
'
r .. the vector descr1binq the first der1vati¥lS of t� state.
.
·variahles' ofthe'cnn1pensateit·plant·F ... the di sturbance vector·· of the· cnnpensated ,,1ant .
1; .. the (Ci' + 3m1) x (0 + 3"'1)· 'coeffici�nt matrix.,
..� .:. «f + 3t1ii) x«2n .. in) coonl1na.:·matrix..
The parameters of the controller are calculated in the next sect1C1'_l.
.
.
'. ..
7.3·. Selec;tion of the:'Optimum Parameters of the Cnntrol1ers usin�
'Ei�envalue' Search.
As an l1lustrati·ve example the mathefl1atical model of 'the reduced(
.
.
Saskatchewan Power Cnrporati m . system was used to '"nvestiQate the
effects of the parameter,adJ�stn'teht$ of the 'lead-lag' compensator•.
The compensatinrr network used Was of the fom .shown in Ff�ure 7.1 .with
.
a tr_nsfer function containing three poles and ·three zeroes. Start1.nq
from eauation (3.1). the state s�ace form of·the uncompensated.system,
equation. (7.17) can be easily derived with the help of alqorithm
134
p.xplained in AJ'Dendix E. The J matrix was, ftu(1mented to include the
stab1lizin� scheme. The elements of the J matrix were com�uted and
its eiqenvalues were calculated on the digital computer using the
IB'·1 Scientific Subroutine HSBG and ATEIA. A measure of system stah.,
1lfty can be assessed by observinq the relative magnitudes of the
real pa'ns of the system roots. In order, to establ1sh the 'best
values' of the constants of the transfer function, ft,WlS necessary
to make several runs on the diqital computer varyinq each of the
controller parall1eters in turn. Fran Table 4.2B which is reDroduced
here, it can be observed that there are three pair 01 dominant com-, ' ,
, ,
, ,
plex roots which dIctate the fnnn of the system transients. The,,
,
TABLE 4.28 '
Ominant R('Iots of the Uncomoensated'System,
No. E1qenvalues
1 -0.044, + .1 6.682 -0.044 - j 6.68,3 ...0.63 + .1 1.344 -0.63 - .11.345 ..0.20 + J 0.29
, ,
6 .0.20, - j O�29
significant point to note about the data in Table 4.28 is that the
system dynamics are detenn1ned prf�rl1y by the roots located at:
,'
,
-.044 ± j 6.�8 for the uncompensated, system. Any improvement in the
system perfomance would only cane about if this troublesome complex
, i ..
...... ,..'.'
, .
....'.
.... 135
.
.
roOt· cOuld,:b.i moved to 'the le1t whl1e' 'st111 retafning a 1a1"ly' stable'catd1t1cr. fof the prtdan1nant �_.-. frequency root. listed 1n the
·
fibl. � .:A' ,tr1illnd I'rot' process· was uSld 'to detamine the most.
effect1� �.in�ti.on Of ccmtro11er sItt1ngs that would prov1'de the'gtea.,t tnhlnCetnet,t Of I1$tllll d,,'ng.· The S1gnff1�nt roots 10r
.
'.. ..... .
.
different· cOrt6f"at1_s of the parameters are shown· in·Tab'e 7.1..
.'..
.
t�1:tfan.Y the derivative ·ci,.cuit WIS. excluded (i .1. switch 51,was' closed) wM'ch �ant that thl compensating sche.. consisted of
·.
. .' .
anlY .tw1i'1 ::"ad.'lg· networks. The 'best' values of the time constants
·
Till' T t�i.' Tl�l' T141 were f�nd for this' ease. Us1n�, these tfme
· constants th.·,swi.tch $1 was opened and the. 'best value' 01 the other'· per.ters. _": found.•. 1he • optimUM' comb1natton in each set of runs
is n(')ted,by _j.arr.<.ws in Table '.1. As expected the· additfon of the.
..
.
..
.
the p....dom1.�ant roots as can be seen from this table. An improvement.
.. .....
in system lta�ntty by a factor of 22 was obtafned over the uncompen-
sated .ys:tenr� .the proper chOice. of the parameters for the auxiliary.
(:ont.-o11.r•. 'The : time danain results were then. calculated for SQuaw
Rapids spe�d.· E�.or M.td Oueen El1zabeth Speed Error. They are shown in
Fiqur.es· 7.2(.).. :."_U'.) 1he'marked.
improvement in system damping over.. : '.:.. . . . '.
the ·unc�.nsated system'can be 'readily observed from. these time
domain resw1ts. AS. �red1cted.1n the best case the one cycle per second
oscill.t1� ha. v,rt.. lly d1sappeared in approximately 1 s.econd •
...
, .: .....
.
.., .:.�.::.. :._.:
136
oo
.
III: �h
�I JllllnllIIlh..•
e· : :Hi:::::in:::n.!:.id •.
.
.: i HllllllI1illll!flHi!!: . . .
.
� I! Ill! llllil ! !! Ii III !l! III:.. .. Tlrli ijHiijPlii1',:PIillllillHp,lliijj �1.I.I.I.I.!.l t 1.1.1.l.l.ll.1.[1.1. ttll!.!.!.tI11.11llJ.lnh:.�::.:.l:.l1.:;,ll.ll.lllll.lll.ll.l.ll.lltl.l.tll[ttl.ll.1.tll..ll.u .ttl.. l.I.!.\ ...
0·0 '·5 3-0 4·5 .0TI... ill ••co"d.
Figure 7.2. (a) Squaw Rapids speed error �ersus time for the systemcoupled with lead.lag feedback auxi1,ary controller.
Figure 7.2. (b) Queen Elizabeth speed error versus time for the systemcoupled with lead-lag feedback auxiliary controller.
Figure 7.3 General auxflta1'Y cantroller connected .to the
Squaw Rapids plant�
AVR
137
"--
,
"
138
I Dcitfftlftt Root, (-0. :t J"'). I
Tm T121,
1131 T141 ' 1151 1161 �1 �l "I � °2 -z "3 "3 toMettts
.. .'
. - - .. ,- - .044 6�68 .628 1.34 .205 .287 IIIIcoNptns. ted 115 tllll
'1.5 0.0 : .002 '�03 .. - 10 i.e .048 6.69 .. .625 1.38 .2U .2" swttcll 511S, closed1.5 0.0
'
.001 .03 .. - 100 1.0 .089 6.75 .624 '1.65 ' .261 .1211.5 0.0 .001 .03 .. - 1000 1.0 .728 7.28 .953 3.09 .. -
1.0 0.0 .001 .03 - - 10 1.0 .046 6.69 ' '.619 1.36 .218 .?a2 V.l'y1nCl '1,.TI31 'lild Tl4l '
1�0 0.0 ,.001 �"3 ,- - 100 1.0 .011 6.73 ' .5n 1.5$ ,.304'
.09A1.0
'
0.0 .001 .03 .. ., 1000 1.0'
�4t4 7.12 .722 2.73 ' .855 ' .109 ,
1.0 0.0 .0001'
.06 - .. , 10 1�0 .043 6.69 �613 1.38 .219 .-1.0 0.0 .0001 .06 - .. 100 1.0 .1)59 6.73 : �51 1.56 .29IIi .0321.0 0.0 .0001 .06 - .. 1000 1.1) .284 7.01 .867 3.02 .8. .0951.0 0.0 .0001 .06 , .. - 1200 1.0 .423 7.3i .647 2.97 .866, .0981.0 0.0 .0001 .06 ' - ..
, 1400 1.0 .364 ' 7.32 .912 3.16 .906 .1291.0 0.0 .0001 .06 - - 1600 1.0 .559 7.54 .763 3.25 .884 .1061.0 0.0 .0001 .06 .. .. l80Q .1.0, .735 7.56 ' .792, 3�58 .894, .1021.0 0.0 .0001 .06 ,- .. 2000 1.0 �61Z,' 8.14 .491 3·Ot ' �902 '.092
-,
1.0 0.0 .0001 .01 '�.
'' .. 1000 1.0 .555 i.,i7 .689 2,.71' '.852 .085 , '
:
1.0 ' 0.0 .0001' .01 - - 1500 1.0 .822 7.18 .936 3.07 .1IB4 .U91.0 1.0 .06 .06 .. - 1000 1.0 , .809 5.49 2.96 1.15' .758 '".Q47...Mttt 1_OYtIIIIn,t -over IlfiCfll'l.1.0 1.0 .06 .0& - .. 1500 1.0 .653, 5.35 3.44 1.02 .788 .076 ___ted ease (f.ctor af m1.0 0.0 .0001 ,.06 - .. 2500
'
1.0 1.58, 9.59 ,'.451,3.47' .. .', .
1.0 0.0 .0001 ,.06 - 3000 1.0 1.5&,' 8.41"
.727 4.48, .9,15 It.079..
1.0 20.0 20�O .06 .; - 1000 0.0 .317 7.16' .681 2.75
'
.861 .1071.0 20.0 20.0 .06 - .. 1500 0.0 .524 7.45 ' .758 3.18 .- .0981.0 29.0 20.0 .06 .. - 2000 0.0 .764 7.78 .794 3.53 .�: .087
'."
1.0 ' 1.0 .06 �06 10 10 1000 1.0 .196 5.50 3�05 1�15 , .018 .-1.0 1'.0 .06 .� 10 10 1500 1.0 .644, 5.35 3•• 1.04 .019 .062 , SWt teh Sl 0IItII, ,
1.0:
1.0 1.0 .06 .06 10 ' 10 1800 .587 5.31 3.61 0.951 .012 .045, Va"..noCU ..TI51 ,and TIU1.0 1.0 .06, .06 10 10 2000 ,1.0 .556 5.30' 3.68 0.909 .012 .0451.0 1�0 .06 .06 20 ' 20 1000 1.0 ' .8()4 5.�9 3.00 1.15 .02i .0411.0 1.0 .06 .06' 20 20 1500
" 1.0 .649 ,5.� 3.41 1.04 .014, .0'34 '
1.0 1.0 .06 .06 20 20' 1800 1.0 .S88 !!.31 ' 3.&1 .:973 .0'12 �0301.0 1.0 .06 .06 20 20 2000 ' i�o, .557. S�29 3�67 •.917 .011 .0291.0 1.0
'
.06 .06 30 ,30 100 1,.0 .3' 6.68 ,.90 ' 1.51 .25 .12 ,,'1.0 1.0 .06 06' :30 30 ZOO 'l�O: .60 6.58 1.11 1.66,. ,
..
.. -
1.0 1�0 .06 .0'6 30 30 " 400 1.0 .92 ' 6.20 1.67 1.59 .04 ' .031.0 1.0' .06 .06 30 30 500 1.0 •• 6�01 1.92 J.53 �O3. ••031'-'t l�t over COlllDen. ,
,
"
,.ted S1l"", (i.ct.. of 2�)1.0 1.0 .06 .06 30 30 '600 1.0' .116
'
5.... 2.17 1.43 ' .030 ,.031'
"
, ' '
1.0 1.0 .04 ,�06 30 30 700 1.0 .93'
,5.71 2.40 1.32 .025 ' .0301.0 1.0 .06 .06 30, '30 800 1.0 .89 ' 5.62 ,2.63 1,23 , .022 .0301.0 1.0 �06 .06 ,30 30 1000 1.0 •• 5.49 ',3.00 1.15 .017 .0291.0, 1.0 .06 ' .06 " 30 30 1500 1.0 .650 5 ..35 3.47 1.03 ,OU .02'41.0
'
1.0 .06'
,
�06 30 30 1800 1.0 ' .Sto 5,31 3.60 ._1 .010, " ,.0231.0 1.0' .06 .06 30 30 2000 1.0 .55' 5.2t 3,66 .- .010 .022
roo... ,
,'
TAB,Lt,7.t
Syst.. Roots, 'Of' ,.rytnQ Awentl" c.ntroll ... P.......,.
,'
,
.
.
.
..
"
.
..
..'
..
.
.. The geMr.l root 10C1tfG,� is desfQII.ted b'y ""'1 + �. The lIIICIIIitude of 01"
is iild1cat1ve of the _...t
of Ita..,fn!, assoct.teet with elClt dec:a.v1l1fJ s1!'11lsoid of. ,..dian f�,".
139
.. 7.4 ComparisM of Results.
..
. .
Since the optimal state retlulators as described in Chapter,4 and..
.
.
...
'..
.the lead-lag stabilizer described in Chapter) are two alternate means.
for synthesizing it controller to pr(Wide an· auxiliary signal to the
voltage regulator, it is of interest to compare their relative con- .
tribtition to system :dampinn and stabil1ty. This can be. done by comp
aring the. root locations of all the compensated system with the un-.·compensated as well as noting the correspMdin� time domain perfor
mances of the systems for each case for an identical disturbance. The
dominant system rcots for III cases are listed in Table 7.2 for
comparison purposes..'
. .
.
The time danain ·transients for the Squaw Rapids Speed ErrOr and·
the Queen EliJ.!lbeth Speed Error ··for all these cases are compared in .
..
.
Figures 7.4(1) & (bh tn.all these eases the simulated disturbance was
a step ··change of 10% load demand at Squaw Rlpids�;·.
. It is ·obvious from Table 7.2 and fiqures··7.4(.r and (b), that the. .
controller designed on the basis f'f imp\i'oved rtlOt'locatl'on ClaVe ",'. . .....
comparable results to the opti",al regulator· cases and�the system
analyst miqht he justified in this case in selecting: the 1 eaci.lag,. . ." '. : .
controller as the system performance is comparable ·to the 'optimal'
regulator case.
In this section· a lead-laq �etwork having a transfer fUnction.
.
.
.�:
".
.
.
containing three zeroes and three poles has been ','cMsidered as the.·. .
..
..'
general fonn of the auxiliary. controller. Thf$ was coupled to the
140:
....- ....
r
o 6·0
.-0'1Time (seconds)
Figure 7�4. (a) Squaw Rapids speed etror v&rsus time
(a) Unc�ensated system" (b) Approximate state regulator. (c) State regulator Cd) Lead�lag feedback.
I)
..
:=
-e..
�
/. II
1'0 '\ 40. ,I
.' .\5·0':.\ : \._ ..
I
'. I .,
\ .I Time '(seconds)\_/
0:
F1gure 7••·• (b) Queen Elizabeth speed error. versus time
(a) Uncompensated syst.. (b) Appt_1mate state regulator.�
(c) State regulator (d) Leed-lag feedback.
141.
.
TABLE' 7.2 .'
Comparison' of Dominant System Roots
Control Dominant.RootsSystel'1
'a1 :t .1� �2 :t j"2 a3 :t..l'3
a -0.044 ± j.6.68 ",,0.63 :t j 1.34 ...0.20 :tj 0.29
b �0.91 .+ j 5.23.
-3.0' :t j 3.51 -3.02 :t j 0.65
.c -1.1 .:t :f 6�69 .2.63 :t. j 3.35 . ';'3.8 :t j 0.10
d' -0.96 .:t .1 5.01 ;'1.92 :t j 1.53 �0.03 ± .1 0.03
(a) UncOfl'IPensated System.
. (b) Truncated State Regulator·
(c) Optimal State Re�ulator
(d) Lead-lag Feedb.acic
.
mathematical model of the multi-ltl8chine system and the gatn and time
co.nstants were optil'tized usinq an 'eigenvalue search' method. The time
domain results were then c'anpared .with those obtained using a state.'
regulator as explained in Chapter 4. It was found that a similar
system improvement in dan1nant .root locations was achieved for these·
two. different types of controllers.
142
.. .
.
'. .
• 8,· CmClUSIONS AND RECM1ENDATIONS FOR FUTURE· RESEARCH
8.1 Conclusions
In this thesis the dyriam1e· stabl1i·ty c�"iter10n of a power: system
.'" ".
.
.'
...
has been consid,red' on an overall system basis. rather 'than ,from the'.
.,... '.
conventional mathematical lI!odel of a sinq,Je machine. - 'infinite. bus'.s,Vs tem used "i� - inos.� 0;· the ·"i te�ature pub11 shed Sf) �f�; � '. The .
di ffe,re.�ti a 1
- "equations describing the dYf\am1cs"oi-the power systeRJ ha�e' been der1vel '
.
'. . .�. .... '.
.
.
,here 'in a f.orii Wh�ch �an be' e�sfly 'incorporat�� into the ·state 's�ace"
...
. .
,,'...
..
moclel •.
In this thesis'the mathematical model ·of a power system has been .... .
.
. .
,
.
deve loped 'in two degrees of detaf 1:
a) 1nclud1ng ·transformer action. and. .
b) .c1udtn, transfonner ict10n in the stator wfnd1·ngs and the
tie line.
The M!sults confinned the.
intuitive observation that the. tran-sfo""er
voltages have 'no significant· influen.ce on the dynamic stability
criterion. other than generating rapidly decaying high frequency oscil
·lations in the system. As a matter of fnterest. it has· been observed
that these.hi gh frequency roots are important when constdering th.e time
fnterval for,numer1cal integration in order tf) avoid numerical 1nstab-�.'
flit.y. A lll"ger time interval een be u$ed wtaen transformer. aetion i� .
neglected.
Later studies (not "ported 1n this thesis) have thmm addf,tionallight on the effects of neglecting transformer actio",. The technique
143
derived in . this thesis was applied to ft larqer five machine system.
where two machines belaiq to 'study' system. two Jft8chines belono to
'external' system and the other one is considered as an eauivalent
infinite. bus • .Inclusion of transformer action both in the stator
windfnas and· tieline qave expected results. A"t.ifieial instability
occurred when transformer action in the stator co11s was nealected. .
and tie-line inductance effects were retained for convenience sake •
. These unstable conditions resultinq from the modelling procedure can'.' .
be avoided by 'cmsistently includinq or excluding all transfomer tems
in the'O and"Q axis of stator circuits •. Further research would be"
required in order to draw any further conclusions on ·this mservatfon.
The optimal cUitrol the:�ry and its ,"odifications were used in.
.
.
.
thiS thesis' '1n designing optimal' requlatorS. "The transient behavior.
.' .
.
.
.
of the system wi th the addi ti on of these controllers was found to be
sfnrilar. The comparable results obtained by using the truncated state.
.
.
.
.
.
.
regulator shows the presence of some states which have little influence
on· system damptnq.·· .
.
The effect of ccnsiderfnq the qoYernor-prime mewer input as a.
controL variable in addition to excitation cmtrol was 1nvest1q�ted.The optimal state regulator for this case was designed and the time
domain results and eiqenvalues were fCllnd· to be similar to those when
only exc1.tation inputs ,were controlled. It was therefore felt'that
considerinq the 'relatively little impr'Ovement in system. damping. the
canplexity in the design of the ccntrol1er due to the feedback of two
144
control variables is not warranted.
The conventimal 'eiqenva lue search' was used to select the best
values of the parameters of a transfer'function considered as a general,
fo� of controllers. It was found that the transients obtained from
this case were similar to those obtained from the optimal state regu
lator. The 10�1cal cCllclusion is that, it is possible to desiqn a. .' .
controlle,r" CJl the bash of trial and error method, that gives results
c�arable to the Opt'ima1 state regulator based on the mathematical
logic.
The favorable results obtained fran the study suggested several'. ..... .
areas for ,future research as outlined in the next section'.
8.2 Reconwnendations For Future,'Research .
'
Several 1nwnediate area,s of research a� sug�sted frG't theresults of this endeavOr.
a) 'Because of the nCII-Hnear 'structure of the power system.'
the
operat1n� point has considerable importance on the deqree of stibility.It would be a worthwhile project to investiqate the effects of operatinq
points CJl
I)' the controller parameters. and
2) the degree of stability ,if the controller desiqned on the
basis of one operating point is kept unchanqed�
b) It 'WOUld be interesting to compare the performal)ces of the',
controllers de,veloped in this thesis with those published so far. There
145
are quite a few methods reported in the control literature to design ,
a controller on the basis of Gpti., cantrol theory andlOl' other
techniques. It will be an important project'to cOlllPare all these methods, , ,
in light of its sUitabi1t,ty and :l1nrltatf. to enhance the sySteII dlllPfng.,
c) the method described in Chapter 5 describes the sfmple input-.
..
.
simple ouq,ut opti., ...,ulatOl'. Further .,"" can be done to extend it
fOl' .'tiple input-output systeMs. Also it would be worthwhile to
de�emne if the _thed applies in all cases and to, issess the effeet
of 'the additional poles on the systeM,stability for the general case.,
d) All of the existing literature and also the work reported 1ft
thi,s thesis,ha.' neglected the effects, of htuNtf.. It . .,.r,d be, ,
interesting to tnclude the effects of saturation of the machine tn' the
state space modelling' of the IllUltt ....chine system. The'saturation cum '
: ". ..... '.
can be cons'tdered as a combtnation of stNfght lines of different""
, ,
slopes., The effect of saturation on the eigenvalues and systeat transients" '
,
should be investigated.
e) The relative effect of a non-l1ne�r controller'on the daMpingof the nultf-machfne system using the ._..tical ntod,l deVe10ped in
this thesis can be a good topic of research.
146
9 •. 1IST OF REFEREHCES..
.
I. F.R. Schleif, H.D. Hunldns, G.E. Martin··and E.E. Hatton, "Exci·tation
control to ·improve power system stability-. IEEE Traps. on PAS, "01.
98, pp :1426-34, llune, 1968.
2. P.N.• Hanson. C.,l. �oodwin and P ..L, Datiden�, l'Influence of excitationa�d speed' cmtrol paraMete�s tn st�bniZinq tntersvstem osci 11 ati"ons II
•..
IEEE Tra�s •. on '!'AS, Vol. 97,. pp 1306";14. ��ay, lQ63.', ••
t
J. A •. kfopfenste1n, ItExnerience with system stabi1izinfl excitation'control� M 1he aeneration of the Southern Cal1foni1a E�ison COl'lpanv".
.
.' .
IEEE Trans, on PP$., Vol. 90, pp 698-706 ••�arch/l\pMl,.1971.
4� K�E.� Rollinqer. "P��lt!r systP-m tiamoinit usfnQ sDeed·�ianais", ·"·.Sc•...
Thesb ·in Eiectrfcal F.nti1neerinrr, University of Saskatchew�n, it'l6S.·..
5. C. Concordia; "Ste�d.v s�te stahi1it.y of synchronttls r'lachinp.s as
aff�cted .h.y voltane reQulat(\r charectertstfcs," .I'.l!:E ·E1ec .. F.nflfl.Trans. Vol. 63, pp 215-220. �a.v, 1944. . .
6. C� Concord1a. "Effects·of buck-boost voltaqe reQulator on steadystate stability liMit," AlEE El�.· Enqq. Trans •. Vol. 69, pp.38fl-385. 1�50.
7. Y.H. Yu and K. von�$uriya, "Steady state stab111t.v limits of (\
regulatei synchr(JloUs r'l8ch1ne connected to an ipfinite s,Yste",",IEEE Trans. on PAS •• Vol. 85-, Df' 759 ...767, July. i�66 •.
.
B. q.,�. r,ove, ""e()l!1etric .copstructfnn:of the stability li",its of
synchronous· machines,'1 Proc, lEE.Vol. 112. pp, 977-985, Ma.v, lQ65.
147
. ....
9. H.,K., Messerle an,d R.W. Bl"Uck, "Steady' state stability of synchronousgenerators as aff�cted ,by regulators and 9ovemors." Proc. tEE.,Part C, vol. 112, pp 24-34:, 1955.
'
10. L.J. Jacovides and B., Adkins, "Effect of eXc1 tat1 on regullt1� on.: ..
.
synchronous machine·stllb111ty." Proe. lEE" Vol. 113, P1). 1021';1034"June, 1966.
. ..
.
•
,'. • ':,' • •. �. • ',,:.• •• '.
•
•• " •••• ,
,'4, ,,"" .:., 11. A.S. Aldred �nd G. Shackshaff, "A frequency response met�od for the
predeteminat1on, of synchronous maeh1:ne stabi,l1ty," Proc. tEE"
Vol. ,107C. 'pp. 2-10, 1960.'", .: ,
'
12. 'D.N. Ewart arid F.P.' De�l1�', "A digital 'computer prooram for the'"
,
automatic dete�nat1on of dynamiC stability limits," IEEE
Trans. On PAS•• Vor.' 86�, pp.' 867-875, July, '1967�
13. H.K. Messerle, 'Relative 'dynamic stabl1ity of la�� synchronous" '
generators," Proc. tEE, Vol. t03C, pP. 234-242,,1956.
14. W.G. Heffron and R.A. Ph" lips , ,"Effects of a modern ,a"",l1dynevoltage regulator on 'under ,excited operation of large"turbi"e ,
"
generators,"' AI,EE Trans�.Vol. 71, pp 692"!'697, AU�h 1952.,
'
,
,,
15. C.A. Stapleton� "Root locus study of synchronous,'machine regulation,".
. ." .
.
Proc. lEE" Vol. Itt, pp 761.768.,Aprfl, 1964.'
,
,
16'. S. Surana and M.V. Har1haran. "Transie.nt respon$e and' ,t,,,ns1entr' "
"
,
stabi11ty of a p�r Syste". Pmc. lEE, Vo1.-:115, pp. 114-120,'. . .
. .... .
January, 1968,
17., V.A. Yeo1kov and V.A. Stroev. "P�r system stabHity' as affected,
by aut_tic con,trq,l"oi generatorS ',-,5- method' of analy.s1s andsynth�$is." Paper No� 71-TP�21-PWR, presented at IEEE W1�ter 'Power',�_t1ng. 197,�.
148
18. J. Nanda', "OntiJTIization of vo'ltane re(1ulator !lain by the D�decOlT1f)osition technique for steady state stability 11�1t," Paner No. 71-
TP;'105.P\'IR, presented at IEEE Winter Power Meetin�, 'January, ,1971.
19••J. r{anda, "Analysis of steady state stability of two machine systemby D-decomposit1on technique," IEEE Trans. at PAS., "01. 90, nn
1848-55, July /Auq•• 1971.
20. �1 .. A. Lauohton, "t�at�ix analysis of dynamic stability'1n synchronousmulti-machine , system," Prec, tEE, Vol. 113, pr) 325-336, Feb., 1966.
21. J.M. lfndri1l, "Dynamic stability .calculations for an Ilrb1trary number
of interconnected s.ynchrm.(lfs machines," IEEE Trans� on ·f)AS., V01.
87, PD 835-844; �arch, 1968.,
22. F.R. Schleif and J.H. l�hite, "Dampinq for the northwest-southwest
t1el1ne oscillati00 - an analo!, study," IEEE····Trans. on PAS., "01. 85,pp 1239-43, 1966.
23. F.R. Schleif, t,.E. Martin and R.R. Anqell, ,"[)aminQ of systel'l oscil
lations ·with a hydro-Qeneratinq unit," IEEE Trans. on' P'AS�, :Vol. 86,pp , 438-442, April, 1967.
,
24. F.R. Schleif, 11.0. Hunk1n, E.R. Hutton and H.B� G1sh, "Control Of'
. .. ..
rotatin!-T exciters for poWer' system damping -"pilot application and
exper1ence."IEEE Trans. on PAS., "01. 88, pp·1259-66. AUQust, 1969.
25. E.J. Uarchol, F.R. Schleif, \�.D. t;ish and J�R. Church,' "Al1llflmentand mode111 nq of Hanford exci ta t1 on .contra 1 for system dampine, "
IEEE Trans. on PAS •• Vol. 90'; po 714�724. March/April, 1971.
26. F.P. DeMello and C. Concordia, "Concept of �ynchronrus machine
stabil1ty as affected,by excitation control," IEEE Trans. on PAS,Vol. 88. po 316-329·, April, 1969.
149
.27. Gerhart, Hil1esland Jr., J.F. Luini, Rockfield, Jr., "POWer systemstabilizer - field test and di9ital simulation," Paper rfo. 71-TP-
77-PWR, presented at IEEE Winter Power Meetinq, January, 1971.
28. F.W. Keay and \�.H. South, "Design of a power s.ystE!fl'l stabilizer
senstne freouency deviation, II IEEE· Trans. on PA!\ •• Vol. 90,· pp 705-
713, f�arch/Apr1l, 1971.
29. K.E. Bollinqer and. P..J. Flel'l11nQ,. "Oesiqn of speed stabl1itint'J traM';'·
fer function for a synchronous. qenerator, II IEEE Winter P�r �1eetinq
January, 1969.
30. J.E. Van Ness and William F. Goodard, "Formation of the· c(M!fflcient
matrix of a larqe dynamic system," IEEE Trans. a't PI\S. t Vol� 87,pp 80-83, January, 1968.
31 -. R.H. Shier and A.L. Blyth·, '�Field tests. of dynamic stability usfn"
a stabilizing sianal and computer Drooram verifications,lt IEEE
Trans. un PAS., Vol. 87, pp 315-322, February, 1968.
32. M.A. Lauqhton; "Sensitivity in dynamic system analysis," Journal·
of Electronics and Control. Vol. 17, D,r 577-591. Nnvel'l1ber, 1965.
33. R. Kasturi and P. Oorara.1u, "Sensitivity of power syst�," ·IEEE
Trans. on PAS., Vol. 88. en 1521-1529, Octobe". 19.69.:
34. J.J. Grainqer and R. Abmeri." The effect of non-dynamic parametersof excitation system on stability perfomance," Paper No. 71-CP-
.
185-PWR. presented at IEEE Winter �ower �etfnt'J. 1971.
35. J.M. lIndri 11. "PCMer systE!fl'l stability studies by the method of
Liapunov I and II,". IEEE Trans. on PAS •• Vol. 86, July. 1967.
150
,
36. p.e; Kr�use and J.N� Towle, "Synchronous machine dampinn byexcitation control wi�h direct and nuadrature a�is field' '�indinq. ti
'
rEEE Trans. on PAS •• \fol. 88, PJ> 1266-1274, AUQust, 1969.
'37. V.N.,Yu"K. Vonqsur1ya andl. Hedmanff, "Applfcation of ,an,optimalcOfltrol theory to a pm;,er svsten," IE£E Trans. on DAS •• Vol. gg,pp 55-62, January, 1970.
38. Y.N. Yu and C. S1Qgers, "Stabil1zation 'and' ,optimal' control stenals'
for a power system," IEEE Trans. on PAS.,'Vol. 90, pp 1469-1481�,.Jul.v/Auc;1ust, '1971.'
39� 'E.J� Davison ,and M.S.' Rau. "The optimal output feedback control' of'.
. ".
.
a synchronous machine," IEEE Tr,ans. on P,4S."Vol. 90, nn 2123-
2134, Sept/Oct. 1971.
40. H.S. levine and'M. Athens, "On the' detem1nat10ri of �he ootif'lal,
consta,nt output, feedback gains for linear mult1variahle system:IEEE Trans. on'Autornatic Control, Vol. 15, DO 44-48, Febru�ry. 1970."
,
.
'" .' .'"
,
.
.
.' ..'.
.
.
.
.
. '.
41: n�H. Kelly' and A.,A. Rahim, �Closed loop OPti�l exc1tatfon c'ontrolfor power system stability, I� IEEE TraOs. on PAS •• Vol. 90. nn 2135-
2141, Sept/"ct. 1971.
4�. �.�1. Yousif, "Ovnam'fc of)timizatiofl of, pm�r s,yste!'l," IEEE Winter.'
Po\"er r'1eetinQ, 1972.
43. ,G.A. Jones. "Irenstent s�bi11ty of. a svnehronous Q�n�ratt)r underconditi ons of ban", bano �xcitat10n schedul fnn, II IEEE Trans. on PI\S ..
: vet, a4, pp 114.121, February,1965.'
, ,
46. O.J,.�. Smith. "Optil!uiftran$ient removal in a pwer system," IEEE
Trans, on PI\S •• Vol. 84, PO 361,.374. M�y, 1965.
151
45. E.W. K1f1lbark, "Power systernstabil1ty, Vol. nr,· John Wiley and
Sons, New York.� 1967.
46. IEEE, Committee Report, "Computer representation of excitation system, n
,
IEEE Trans. on PAS., Vol. 87. PI' 1460-1463, June, 1968. '
47.' R.l. Kosut. "Suboptimal control of lineflr time-invariant syste",sub.ject to control structure constraints,· Proceedinos of JACC,pp 820-828, June, 1970.,
48. M. Athens, "The role and use of the stochastic linear-ouadratic- ,
gaussian Droblem in control system design,· IEEE Trans. on Automatic
Control, Vol. 16, pp 529-551; December, 1971.
49. P.,). Maclane, "linear optimal stochastic control usfn9 instantaneousoutput feedback," Int. J. Contr. Vol.' 13, Feb, 1967.
',' 50.,M. Athens, "The role anduse of .the stocha'stic-Hnear-nuadratic
gaussian problem in control system desf<Y,t't"" IEEE Trans. on AutOMatic
Control. Vol. 16, pp 529-552, December, 1971.
52. K. ·:Ogata, "State space analysis of control systeMS,· Prentice-Hall,Englewood Cliffs, N.J., 1967.
53. A.F. Fath, ·Computational aspects 0" the linear oDt'1mal 'ref}ulatorproblems," presented at the JACC, 1969.
54. J.E. 'Van Ne,ss, "Inverse 1terat1on method for ffndfn� e1'�envalue$, It ,
IEEE Trans. on Autanatfc Control, pp 63-69, February, 1969.
152
55. R.E. Kalman and ,R.S. Bucy, "He", results in linear fl1terin� and
prediction theory," Trans. ASHE., J. Basic Enq•• Vol. 83, pp 95-
107, Oec. 1961.
56. J. Lemay and T.H. Barton, "Small perturbation l1'1earization of the
saturated synchronous machine equatiens," Pa�er No. 71-TP-515 P�IR
presented at IEEE Summer Power Meet1n�. 1971.
57. J. Chand" "Stability analYSiS, of a H.V.O.C. tranSMission link.", 'M.Sc. Thesis .tn Electrical Enqineerin!,!. University 'Of Saskatchewan.NoveMber, 1971.
58. J.B. Pearson, "Compensator desiQf1 for dynamic optfmitation," Int�J'. Control, Vol., 9. pp 473-482. I'pri1 • .1969*
59. K.H .. Simon and A.R. Stuhberud. "Reduced order Kalman filter." Int. '
J .. Co.,trol, Vol. 10. pp 501-509. NoveJ11ber. 1969.''
60. H.M., NewMann. "OptiMal and sub.,optiMal cnntrol usinq an observer
when S9"'@ of the state variables are not measurable.1t Int. J.
C()ntrol" Vol. 9. pp 281-290, March, 1969.'
61. J.Ii. \4ilk1nson. "Poundinq errors in algebraic ereeesses," Prentice
Hall, Enqlewood Cliffs. N.J., 1964 •
.
.
,.
153
.
10. APPENDICES ..
10.1 Appendix A
L1st of SYf!!bo1 s
Tie Line:.
Lkj.
W.
.
b
y.j .
dj.
Y .
.j
y qj,1
1.1
I;/jI ojj
IfjW .
mj
angle between Q axis of the kth node and 0 axis of the .1thnode (steady-state) .
.
.
..
..
tie line inductance connect1ncr kth node and Jth node.
base electrical frequency (radians per second)
tennftial voltaae of jth node
direct. axis component of Vjquadrature axis component of Vi .
1...
equivalent stator current of jth l'1achine.
.. .
. '.. .
direct axis component of Ij .
quadrature axis· COffloonent 01.Ijfield current ·0' jth machitie
mechantal anqu)ar speed!: (rdI'ens '�r se�) 0" ,1th machine
Generator Constants:
Raj eouivalent stator resistance of Jth machine
Ld.i
Loj
ladj
Rfj
.
.. th···equivalent n-axts stator inductance of.i fltachine·
.
.
.
.'
..
equivalent O..axis stator inductance of .1 th machine
•................... �..
direct axis magnetiZing inductance of.1 Machine
. field resistance of jth machine
154
'-ffj "field self-inductance of jth machine
Governor Constants of the jth maChine' of the "study" and "external"SystemS:
, T5j,
T4j
T3j
TejRj
Hj
OJ
Tsj
water s,tartinq time/2.0, "
-2.0 X'T5jreset time constant
servo-motor time constant'
equivalent steady-state speed droop,
equivalent i'n.rt1a constant•• "
,
'
,
equ1v�lent damping constant'" Rj + 6
'
(R
) Tcj where 6 is the temporar.y speed drop
j
Constants of AVR ofjth machine:
K' ' ati'lpl1dyne gain
, aj
Kej main exciter gain
,Kafbj , matn excite", feedback qa1n "
Kvfbj, voltage feedback gain
, Vfbj Xad field voltage base
Tanj, ampHdyne lead t,1me constant.> •• >;<. ,
•
Tadj amplfdyne lag time constant
Tafbj main,exc1ter feedback time 'constant
Tvfbj· voltarye feedback time constant
Te.1 .
Main exciter time constant·
.
Constants of Auxiliary Controller of jth machine:
TUjT12'3
T13j
T14j
T15j. . .
.
T16j ...• feedback controller time constant
K1•1
K2.1
feedback controller time constant
feedback controller time constant
feedback controller time constant
feedback controller time constant
feedback controller ti,", constant
feedback controller qa1n
feedback controller gain
Variables:
i dj o-ax1s stator current of jth machinej
f qj n-exts stator current of Jth machine.1 .
1fj
6.1
tmj
9jh ..
j
.. f1 e 1 d current of ,1th
.
Machi ne
rotor an�le (radians) of jth machine
mechanical torque stmpl1es to .1th·mac:hine
wicket �ate pos1ti.on of jth machine'.
'.: ..
convenience variables for qovemors of jth machine
th .
field voltage of j machine
·155.
. Vtajiafbj
fVfbj. VXj
Vyj.VZj .•..Y· dj'v qjj • j
Tdj
trj
Vrj· '.
ab·vb
156
'. .
. .
..
. ..
ampl1d,yne termfnal voltage of jthmachine
amp11dyne feedback. output of· jth .
machine'.
main exciter feedback output of jth mach.ine
. output sfgnal fran auxiliary controller to AVR of jthmachine
.
'.
'.
.
intermediate signal of the luxf11ar,y controller
D-axfs and O-axis tenninal voltage of jth _chine respecfvely .
mechinfcal torque disturbance of jth ...chine
. sPeed reference to j th machine"'.'
terminal· voltage reference to jth .chine
. infinite bus angle and voltage-dfsturbance.,input.
General':
E. be1OfIgs to
aeRn .
.
a is a vector of·lenqth n
baRmxn b is a matrix of dimensian III x n
t&(to.t,J to �t �tf (closed set)'
tE(to.tfl.
to < t � tf (semi-open set) .
tfii[to.tf) to � t < tf (s�-open set)
t4i.(to.tf) to'< t< tf (.en set).:
< .
. >'" inner product
157
10.2 .. ApPEn,d1 x 8
Technique to Derive the Sub-matr1ce.s of Eguations (3.64) and
.(3.65)
The state space form of the mUlti-machine system is given by
equations (3.58) and' (3.59) and are reproduced here•...
.OX - QX'" EZ ... RU ... SF (3.58)
•
Z: - ·JX ... PX (3·.59)
The coefficient matrices O,Q,E,R,S,J and P can be written in.
partitioned form as per equations. (3.64) and ·(3.65). With the help of
equations describing thedynamfcs of the multi-machfne system as
derived in Chapter 3 the· different matrices are. derived as given tn
the folloWing· ·pages.Matrix D.·
o �tr1x ·is a dtagonal matrix, or in other words ..
0f"l .- Null matrixoJ. 1�j ..
The submatr1x 0Il is given by..
0Il - OIA (OU1 0112 .. - - 0Uj - - - DUm] 0uE::·:..xSmwhere 011j is defined by equation (8.3)
.
1 '1·· �ffj -!!10l1j - OIA [- - . 1· ] 011jERS x 5.
.
Wb Wb Wb . Wb
The subnlatr1x 022 is given by
-2Hm+1 ... 2Hm+2.
-2H· .
022 \II OIA (1.
1 ----�-... 1.'!!tB )022.€ R2px2p (a��)
Wb.
Wb· .
Wb
(8.1) .
(8.2) .
.(P.3)
158
.' :.I,
The submatrices 033' 044 and 055 associate� with state vectors X3•, X4 and Xs are also diagonal andare given by eq�.ti�s (8.5). (8.6) •
and (8.7).
033 • OIA [lS1 !Sl Tel TS2 TSZ Ta - � � �- T5111 T5m Tem)
(B.5),
"
044 ,. DIA[T5m+1 TSInfol TCnt+i T5m+2 TSm+2 TCm+Z '_ - '_ • - T511+p TSIt+p Tcin+p], °44€ R3px3p ,,(8.6)
055 • DIA (1 Tell 1 � Te2 1 1 - - - - 1 Tem III 055 €'R4nlX4nI (8.7),
Matrix g. .
....
:.
. ". ..
. .
,
The subtnatrix Qll can be written in ',anft1oned form IS, pe,. .•aifon'
,
.'
(8.7) .,',
, (8.7)
Wi ttl the. help of equltions (3.26). (3.21). (3.36,). (3.38) and·'
"
(3.41) the submatrices JQU (1 .j) are, written as follows:
QI','I(i ,j) • Ijqj,'"0, ',' 1 '0 '
ajj1� \
t dj,
-I, 0 0 bjj -�,WI)o
(B.8)
0 ,0, 0 0 1
cj d ej 0 �j"b
, ,
"
'. 159
. QU(1.j)· • 0 0 0 a1j 0
it/j.
o· 0 0. b1j o· (B.9).'
0 0 0 0 0
0 0 0 0 0
0 0 o .
0 0
(8.10)
(8.1.1)
(8-.-12)
. ·(8.14).
.
....
The subntatrix Q12 can be written in ,artftt.ned fo"" as .per eQuatfon
(B.15).
160·
Q12· - [Q .. ) .. Q12E RSmX2p (8.15). 12(1.,j). 1-1.m
j.l.p
where
... Q12<1.j) - Vm+� cosa�� 11 • 0
L(m+j).1 ••.
Vm+� S1nam+�11.
0 (8.16)L(m+j).10 0
0 0
0 0
In a similar manner Q13 is (liven by equation (8.17).
Q13·· - . (f)13(1.j)1 ·1-1.m:.
Q13E: Rsmx3m. j-l.",·
where
. Q13(1,j) :; 0 0 01-j
0 0 0
o· 0 0
0 0 0
.-1 0 0
and Q13(t.j)lt�j •. Null _trtxThe submatrfx Q14 - Null matrix
(8.1.7)
.. (8.18)
(8.19) .
. .
The submatr1x qls in pa�titfoned form is given by
•. QIS • [QIS(1 ,j)] i-I,m QlSCRSmx4m
.
..
.
j-I,m
where
• o
o
o
o
o o
I KajTanjVfbjTadj VfbjTadj
• >.'
Q15 (f ,j) • Nun matrix
'"j
With ·the help of equation (3.50) the submatrfces 021'Q22 etc"..
associated wfth state vector X2 can be calculated as follows �'
. let. K Ii m + ,.
The submat",x Q21·'s given by··
Q21 • rQ21 (1,j)J. 1 • l,p. L .. j • I,m
where
o
o
o
o .. o
o
o
-. 0
o
o
o
o
o
o
161
.. (8.22)
. Q21(i,j) • [0 0 0 0 :l. 0 0 0 'fj . (8.23)
and. W.Vk qj
.
dj kj)aU • - (.Vj COS�kj - Vj Sin4
ljk .
.
The submatr.1 x Q22 1 s 91 ven by
QZ2 • [Q22(1.j)] 1 • 1.p Q22 E: �px2pj • lIP
where
. Q22( 1 ,j)..
-
[0f#3 .
aU·.
+ � Vj tose5kj 1.. j .......l Ljkj�
. Wmk kDk - Vb Vk tkq
c1 -_-----Wb
There is no cOUpling betweer;- vector X2 and either of X3 or XS.Thus Q23 - Q2S • Null :matrixThe submatrix Q24 is given by
Q24 - (Q24(i j)1 Q24 E::.... �px3p,.. i-l,p. ..
.j.l.p·
162.
(Q.24).
.
.(8.25)
(B.26)
(B.27)
163
. ... ..
The submatMces Q31' Q32 etc.' associated wfth state vector X3 can
be found out 'reM equations (3.51). (3.52) and (3.53). Since there is.
no coupltng between X3 and e'ther of � or X4 or X5• f.t can be concluded
that
The subtnatMx Q31 is g1 ven by
Q31 - [Q31(i • .1)11�1.11 Q31, E: R3mkSmj-l.p
where
Q ( ) I- Null mit",1x
. . 31 1 • .1 1';.1and '.
.
Q31(f ,3) If''3• o _
14jT3jTSjTc.1"a,Rj
.
_ T33. TCjWbR.1 .
.
1 .
--
o o o
.0 o. o· o
o 0 0 o
(8.29)
(8.30)
(8.31)
..
The subtnatrix Q33 15 defined as ...
o -1 ... 1 _
T33
Tej ..
-11 o
16.4
(8.32)
(8.33"·
There is n� couplfng between X4 andefther of Xl or X3 or XS. Thisleads to
Q41- Q43 - 045 .� NuH lnatrfx ..
.
The $ubmatr1x Q42 is deffned as.
Q -t<) (. )142Sl 42 f .j f-l.. .p
.
j·l.p. where
.
Q • Null II'IItrfx. 42(1.j) f"j·
and· .
o ..
o
. 1stTckWbRt
T3k- _...,._
, TCk.Wb\.1-.-o
.
(8.35)
(8.36)
The submatrfx Q44 is given by equation (8.37) •
Q44(3). •
-1
-1.
-1
o o
165 .
(8.38)
The submatrices Q51' Q52' etc. assocf�ted with state vector Xs can ..
be obtained from. "uatfons (3.54). (3.55). (3.56) and (3.57). The
absence of couplfng between Xs and either of Xt. or X2. or X3 or X4leads to
Q55 is given by.
Q55 • OiA [Q55(.f)1 Q55 E. RsmxSrnIJ j-l,II.where
QSS(j) •1
aj 0 0-
. T.djb .
cj - Ke� • Ke�j Tafbj !vfbjd
.
ej1
.
0-.
j .
Tafbjo. 0 0 . 1
-
Tvfbj
(8-.39)
(8.40)
(8.41)
·166
(8.42)
MAtrix E.
The submatr1x·Ef is ·derived from equations (3.24)., (3.27). (3.38)..
a�d (3.41). It is: defined by equation (8.43).
El • [E1(1.j)l'�ltm El E Rsmx2mj.l.m
where
[l(f ,.1) If •.1• L:.1o
o
1-
Lpj..•
C 0 ..
o
o
O·
(8.43)
(B.44) .
� Sin�jf•. L,no·
o
o
The submatrix E2 is defined as.
.
_ COS�3i.. Lji
o
o·
o
E2 • [E2(1,j)] i-l,p £2 e: �px2mj·l,m
where
E2(1,j) •
and e
.k •
There is no CQupling between land efther of X3 and X4•Thf.s leads to
.
E3 • E4 .. Null matrfx
The submatrfx E5· is given by
E • [E] E E R·· .
5 ...
5(1,j) .
i.l,m 5 4mx2m.
j·l,m
.
167.
(8.46)
(8.47) .
·(8.48) .
where
E • Null matrix5(i .j) ifj·
E5(foJ)I'f.J s
__KVfbjTvfbj .:
0 0
a V dj ajvjoJj j .
0 0
dj qjbjVj . bjVj
168
(8.49)
Matrix R.
Since there is no coupling between vector U and either Of Xl' X2•
X3 and X4• one can write
(8.50). : .
The subtnatrix..Rs· can be derived front equations (3.54). (3.55). (3.56).
and (3�57). It is given by
Rs - [R5(f.j>] i-l.mj.l.m1
where RS(1.j)· Null matrix except
RS(f.j) i!!"l• 0
1-j Kejo
o
. (8.51).
. ·169
Mat...i" S.
The submatrices SU' 512• S13 associated with state Vector Xl can· be .
. ..... ..: ..
derived with the help of equations (3.26). (3.27), (3.38) and (3.41).
5ince there is no coupling �tween Xl and F3, one can write that.. ..
· 513 • Null matrix (8.52"
The submitrfx 511 is given by
.
511 •. [ 511(1 ,1)Ji•1,m S11 e: Rsmx2 (8.53)
where·
· .511(1,1>. •
.Vb Cos\i 5.1n\1. Lib lib
Vb Sin\f . Cos\. . 1...
/ ··.Lib Lib0 0
0 0
0 0
,/.
The subMltrtx 512 1$ given by
512 .• [512(1,J)] i.l,m· 512 e: R5mx3mj-1,'"
where·
(8.55) ..
· SI2(1,J)1 f�j• Null matrix
512(1,3)11-"- 0 ·0 0
0 .0 0
0 0 0
0 0 0
·1 0 .0
. 170.
'. 'The cOlq)lfng _trices S21- S22 and ,s23·:assocfated with vector' .� ..
. can be obtained from equation (3.50). Because of the fact that X2 is.
not coupled with F2"leads to ..
. S22 • Null matrix. .
.. .
The submatrix $21' can be'.wr1tten as·.
S21 •. [S21(1 ,1)} i.l,p 521 € R2px2
where
(B.56)
o
Wmk VkVbCOS4kb '.- -
. Wb' Lkb
·0
.
v: WmtVt .' S1n'kb.................. . ... (8.57)Wb .... Lfcb,;and.
. .
The sublnatrix �23' is defined by equation (B.58)�st3· [�3(I.j)11.1.p· 523 E: R2pxlp
'.
j·l • .,
(B.58)
Where: .
5 ( ) I • Null matrix23 f ,j 1"j.o
o
(B.59)
The 'subtnatriees S31 and S33 Unking X3 with Fl a�d "2 will be. iero.because of the absence of any coupling between th�.· This can be expressed
mathematically as,
S31 • 533 • Null enatrix (B.60) .
The other sublllatrix 532 is given by
532 '- DIA [ , 532(j)] "
532, � R3mx,ln" ,
j-l,m,
where
532(j) -
o
o
o
T4jT3jTSjT�j
'21TCj ,
1
o
o
o.
. ...
In a similar manner, the submatMx S43 is given by,
S • f' S ' ] 5 e:. R43 L 43(1.j) i-l,p 43" 3px2p
"
;
j-l,p,
where
543(1,j),
1t'f- Nun matrix'
S ()-
43 i,j ,
1-j ,
o T4kT3k"
T5kTck
Tlk-,
171
(B.61),
(B.62)
(B.63)
(B.6�)o
The other submatrices assOciated with X4 are zero, or in other words,"
S41 - S42 - Nun matrix'
(B.65)
As the vector Xs 1 s not coup led with 81 ther of F1 or F3 as can be
, seen fran eqUations (3.54) • (3�57), one can write
551 - S53 - Null matrix (B.66)
172
Following the same procedure as applied before, the submatt1x..
S52 can be written as
SS2 - [S52(1,j)] 1-1,111· S52 e R4Inxlmj-l,m
where.
S I- Null matrix52(1,j) 1"j.
S.52(i,j) I 1-j.- 0 0 0
0 0 Kej0 0 0
0 0 0. Matrices J and P ..
. :".
(8.67).
.(8.68) .
As can be seen from equations (3.39) .and (3�40) the ·dependent .. ., ..
vector Z is ·onlY related to sta.te Yettor Xr Thus one can write
J2 - J3 - "4 - "S - P2 - P3·-· Ps - Ps - Null matrix (8.69)
The subniatr1x "1 is given by
J - [J ..] ..
:1- . 1(1.�) i-I,m
. .
jwl,m
where
J I· - Null .t"X1(1.,j).• '"j
. and
. 0 .: r . 0
(8.71)
173
In a s1m11ar manne.. the submat..1x ,Pi is given by',
PI - [P1(f,j)] i-l.in PI E:_' R2mxSm',,' " ,
"
'j-l m,
' ' , .
(8.72)
:
where'
Pl(,·j}I'f'j "
- Null matrix
ind, '2
P I.' ldj + (.ladj)
,
0 ladJ,1(1"j) 1-j "t, lffjWb lffjWb
,�0 ,_ 0'W
, b
o o
(B�73)
o 0
TMs completes the derivation of all submatrfees of equations.. . ..
.
.'
(3.64) and (3.65)., 'Thus the equations (3.58) and (3.59) are' well-deffnedand usfnq the tl'lnsfonnat1on, defined by equati on (3.:62) the' final state
•
space fortli 'X - AX + BU + GF of equatfon (2.1) can be readily obtained. '
... �
174"
..
10.3 APpendix C. .
Algorithm to Derive: Reduced'Stitt' SHe. Model'f.' the Detal1ed
one.-.
..
.. . -,
As.outlined 'fn SectiCII 3.6. it fs'pC)Ssfble tod...ive the state .
space equatfon of the reduced system m�el dfreetly from the detailed
one. This method was used fn the digital computer program to allow for
the aption of excluding the transfonner action in the tie lfne and·
stator windings.
The'state space equation of the d�tafled system (including trans·
foner action) a.re given by equations (3.58) and (a.59) «:•
. DX • QX + EZ + RU + SF•
.Z • JX + PX
(3.$8).
(3.59).
.. .
The' state.
$.,a�e .
equatfon.
of.the reduced system (e�e1uding
.
tr.n��.
fOmer action)' are gfven by tquatfons (3.84) and (3.85)..
.
.
....
15'1·· 1ft+I!+tu +rF
'R' r • '1- r + RlU + G1F (3.86)
It is therefore reQu1.red to .dertve the coefffc1_nt ..trices of
equatfons (3.84) and (3.85) fran eqUations. (3.58) and (3.59). The
.
technique to calculate them are developed in the follqwfng p.ges,
Matrix O.
With the de'f"itfon of new state v.rfables r as gtven by equation
(3.78) tr win be $trfctly diagon.'.
Let for 1 <.. 3m,"
...
1· 3 (1(-1) + i1j .. (K.l).5 + "1." 2. k • 1,,,,
'0.1'1>3111 j.,+211
...•. 'If can be found. fram, the following equation •...o (1,1) • 0 o.n .U'E� _.
'. ." q x qwhere
q • 10m + 5p
'175
· (c.l).'
Matrices q and r.
Let f ,l, ,1,j1' 12,32, be defined as follows:-
It 1 � 3m f • (k.1).3 + '3 '3 • 1;2,3
.
1f 1 > 3m
'f 3 < 3m-
.
'1.' +2m'.J .• (�1-1).3 + 3331 • (kl-,l)�5 + 2 + 33 '.
.
'f j !I' 3m ' jl • 3 + 2m
3 Ii (kl-2) + 34,32 • (k1-1).5 ... j4'
.
, 1f j <. 2m..- ..
� IIItrix can therefore be written ·as '
,
'if (1,l) • Q (11'31) If E. Rq xi'le.t the matrices El and' £2 be defined a� follCM:- '
El (f,j). Q. (1l'j2)E2 (1,j5) • E (11,35) for all j5€[ '1,2m )
.
Thus r tin be written as
. r c: b· .
.' .... q x m
'. ., ...
j3 • 1.2,3
kl• I,m
34 • 1.2
kl • I,m
j'
(C.2)
·
(C.3)· (C.4)
(C.5)
,
Matrices l' and If ,
,Let i, and f I be defined such that,
'
"fo�' f'!.. 3In' t • (k-I).3 +:1'2 :'
"1 • (k-I).5 + 2 + "2, "
"2 • 1.,2.3
k • l�m
,and for.. • 3m
The ,matrices 'It and I' are given by equations' (C;.6) and (C�7) •
176
'It (1,j)'. R (1,j) far all j E [1,m] (C.6)
tr (1,j) • G(il'j) for all j E [1,. (3m .. 2p + 2») (C.7)
"
The dimensions of i and I are (q. x m) ·and q x (3m + 2p + 2)
respectively.
.
.
. ..
Let the ptrt1nent viriables be defined as follows:-
For 1 < 2m-,
'
k • l.m
"2 • 1,2"
For j > 2m.
,i • (k-l).2 + "2"1 · (k-l).5 + "2"1 • 1 .. 2m'-
j • (kl-I).2 + j2'
jl • (kl�1).5 + j2 ' ''
'
jl • j - 2m
j2. 1,2
:kl • I,m
For 1 >'2m
For j � 2",
Thus the matrix r can be written as follows:-'
For. 1,j � 2111"
,r (1,j) • -Q (11Jl),
For i � 2m and j .2m r (1,j) • -E,Ul'jl)For .. > 2m. and 3!. 2m K (1,3) .' -J ,,(fIji)For 1.j > '2m K (1,j)· Un1tymatrix I
(C.8)
Rl (i,k·2) • R(1l'�)Gl(i,k3) • G(fpk3)
R1(1,k2) • Null matrix
G1(1.k3) • Null matrixThe dimensions of r, Rl and G1 are 4m x 4m. 4m x m and 4m x
(3m .. 2p + 2) respectively.
For' 1 < 2m-
For 1 > 2m
Matrix J'
Let 1,j, 11.Jl be defined as follows:-
For 1 !. 2m 1 • (k;'l) 2 + "2 '.tt � (k-l).5 • "2,j • (kl-1).3 + j2jl • (k1-U.5 + 32 + 2
For j < 3m.
-
177
for all � E. (I,m].
for all K3 E U,3m+2p+1]
(e.9)
. "2 • 1,2
k • l.m
j2 • 1,2,3.
. kl .. l,m
The required matrix J" is derived as follows:-
For 1 !. 2m and. j !.. 3m,
J (1,j) it Q (1pj!).:
'
For 'I !. 2m and j > 3m J" (1,j) • Q (11'3+2m)For 1 > 2m and j � 3m J" (1,j) • . J(1-2m,jl)For 1 > 2m and j > 3m J" (i,j)· J(1-2m. j+2m).
Thus using the above algorithm the state spaCe equations (3.84)
and (3.85) describing the dynamics.of the reduced system can be
calculated. In order to transfonn these equations 'into the standard
(C.tO)
•
state space fonn 1'. X Y + 'f u + t F, the pertinent steps are
described below.
178
Equation (3.85l1eads to.
.
! = r1 '3' Y + r1 R1U + rl G1F·.
tc.m
Substituting (C. II) into (3.84) yields
wi'. (if + r r1 J) t + (l + rr1Rt) U+ Jr + rrl Gt'·F (C.12)or
•
.. y •.Xl' + 'If u + t F (C.13).
where·
x •• 1j1ro- + . r rl J')
'If • 11"'1 (If + £r1 R1) (C.t4)- 0-1.(1 + rrl G )G •
I .
:.:.
179
10.4 Appendix 0
System Specifications of Reduced Three-Machine'Model of'
Saskatchewan Power Corporation.
Subscripts 1,,2, 3 stand for Souaw Rapids nlant, nueen Elizabeth.
:.
..
.
olant and infinite bu� respectively. All values are in P.U. unless
otherwise stated.
(a) Study System
Plant: Squaw Raoids ( � 1 to i 6 machines)
�enerat�r Specifications:"
,
33.5 MW; , 37"5, MVA; 14.4 KV; 120 RPM; 60 HZ"
Base Values usino Xad base per Unit System:,
MVA KV - Amps ,Z (ohms) L (henrv)
Stator 100 11.7 5670 2.07 5.5 x 10-3Rotor 100 113 880 129 ' 3.42 x lO�l'
-Basic constants of the single machine eo..,1valent,
, G£NE�ATOR AUTOMATIC VOLTAGE REGULATOR
Ldl 0.25 Kal ',500.0
Lql 0.16 Ket 2.0 '
Ladl 0.70 Kafbl ' 0.005
Rfi 0.OOl8 Kvfb1 0.3
Lff1 3.92 Vfb1 113000
HI 10.4, Tanl 0.10
180
PRIME MOVE R AND GOVERNOR . AUTOMATIC VOLTAGE REGULATOR
TS1 1.1
T41 -2.2
TSI 349.0
. T31 10.0
Tel 0.16 .
R' 0.01771
Dl 5.0
(b) External System
Tadl 0.033
Tafbl .0.22
Tvfb1 0.50
Tel 0.47
.
Plant: Queen E11 zabeth (tf I and tf 2 machines)
66 MW; 82.5 MVA; 14.4 KV; 3600 RPM; .60 HZ·
Basic Constants of the Single Machine Equivalent
PRIME MOVER AND GOVERNOR
T52 . 0.5
T42 .0.0· .
.
T52 0.15
Tc2 0.08
R 0.0242
H2 10.34
. °2 2.12.
..
(c) Operating Point
VI 1.0 ..
. ql .0.96V2V2 1.0
. dl 0�36V3Y 1.0 ql .
o 93.3 V3 .
.
. i.
dl 0.27.dl
. ! .
0.87VI . I 1 ., .
V ql I ql:
0.96 1.691 . 1
V dl 0.28 Ifl 1.692. .
The Delta (4) matri" (values are in degrees)
(413 • 41 - 4jl· (steady.state)
0 '16.26 20.9. ,
..
-16.26 0 4.63
...20.9 - ·4.63 0
The tie line reactan�e matrix L:.
,
L1j • transfer -impedance betWeen 1 th and jth node.
LU • Lpi • Self impedance of i th node.
0.86 0.88.
.. 51.3.
7.40.88 .
. 0.79
7.4 ·6.4751.3
!
181
182
10.5 Appendix E
,
State Space Equations of the Multi-Machine System Coupled, "
Wit.., Lead-lag AuXiliarY Controller
The dynamics of the uncompens'ated system as described by
equation (2.1) are qiven by,•
X • 'AX + BU -+ GF (E.l)
where
x -' (i" x:l vector describing the state variables of the
uncompensated syste� ,
•
X - '0 x 1 vector describing the first derivatives of the
state variables
V -, m1'xl' vector descri'b1 nq the' �utPut of the. contro11ers
to ,be 'fed back to the pla"t''
F - (2n + IIJ) x,l dist�rbance vector
,
A '- (i" x a coefficient matrix ".,
',",
"I '
'B' - q x 'ml con�rol'matrix',
:.
. ". . � .
.
G - a x (2n + m) matrix coupling the states with the
disturbance vector.
,
The algorithm to calculate the I'tatrice$ A, a a"d, G are
descr-ibed in Olapter 3,.
Suppose that pu,t of the m �study system" generators'rol plantshave lead-lag auxiliary cpntrollers connected to ,them (m1 � m). For
convenience sake without any'mathematical :sacrifice the nodes can be
renumbered so that the first ""1 nodes ;are those se,lected generators
where the control1er� are connected.
183
.
Because of the addftion of these controllers the numbers of
.
states are f�creased. Let t be the new vector representinq the state .
variables of the compensated system�·Thus Y is defined as
.. (E.2)
(E.3)
and Zj as defined by eQuation (7.12) represents the sta,te var.iab.les of
the auxiliary controller connected to the jth node of.'the system •.
The· length of· the vector tis (a·.+ 3ml)where. q • 10m + 5"
From Figure 7.1, f.t can be written as..
Uj • VXj (F..4)
Substituting vJlue of Vxj from equation (7.9) .vields to
(E.5)
.
or:1 < f < "'1- -
(E.6)
where Kj is a row matrix of dimension 1 x (q + 3m)�From equation (7.21) .
184'
it can be concluded that
Kj '.' Null matrix except
aJ 112j T153T13j T14j
'(E.7),
,, Ij 112j
,
,Kj(1, q+ 3j,- 2) it
T13,1 T14jT'12j
,"
Kj (1, q + 3j - 1,) • T13.1 T 14.1
Kj(1, q ... 3j) • 1/113i. ". '
.' ,'.'.
" ..
.
The control vectorUfs theNfOl'e !Ii"" by
U • [KJ [1')
where Kj is the jth,row of th� matrix K. Kj can be calculated with,'the help of equation' (E.7).,
'
Substituting equation (E�8) into (E.l) yields,•
x,. [A 0] (t] ... [8] (K] [Ii + [G) [FJ'
Defining
(E.9) ,
A • [A 0 J + [8} [KJc
(E.I0),
equati9fl ',,(E.9) is chang� to,",
•
X'. A I + GF, c
(E.11) ,
.,
,
, 185 '
Equation (7.11) can be rewritten as
(7.11)
Fran equation (7.13). W.f. Sc5,1or
(E.12),
where OJ is the row matrix of length ('q + 3ml) and 15 defined as,
'
'
0,1 • Null matrix except,
'Dj(l. Sf) -1.0 J (E.13)
The vector ZJ is related to the state vector I by the followinflrelation, '
EiE:: R3 x (q + 3ml) (E.14)" '
The transformation matrix Ej can'be found oOt as
E.1 • Null matrix except
• 1.0,i • 1.2,3 ,
(E. 15)
.
. .". .
Substitut1riq equations (E.14) 'and (E.12) into (7.11) yields
[ij] • [T.,HEJ][I] + [5 j][D.1][t] " {E.16,
. : .,
£ljl • [PjH� ],
where
P.1 • [Tj1[Ej] + [5,1][O.1J,',,
(E.18)
�<186 "
From equations (E.17) and (E.3) it can be written as
.
".. .'. .' .
Z • p y ,
, p� R' �
,
� )' (E.i9), � x ,q + ""'1 ,
"
Where Pj as defined b,Y eauatim (E.18) is the .1hsubmatrix of the
,
matrix P.
CormininQ the equat,ions (£nl.),� (E.11) and (E.19) yields,• A
,+ [�] ,[F)r • [..£] ['X'],
, por
•
y • Xl + �r
(E.20
(E.21),
A 1: E. R(Q' + 3m1) x ((f + 3ml)where 7( • If],_ G
' (E.22),
G • ltf] , ,'If €. R(ci + 3m1) x (2n + m).
.
.'.
.'
Thus the addition of the controller chanCles the coefficient matrix
of the system fran A of equation (E.1) to 1: of equatien (£.21).
,'